pneumatic and hydraulic control systems. seminar on pneumohydraulic automation (first session)

Seminar on Pneumohydraulic Automation (First Session) PNEUMATIC A N D

HYDRAULIC CONTROL SYSTEMS

In Two Volumes

VOLUME 1

U N D E R T H E E D I T O R S H I P O F

M.A. A I Z E R M A N Doctor of Technical Sciences

T R A N S L A T E D F R O M T H E R U S S I A N B Y

P . L I N N I K

T R A N S L A T I O N E D I T E D B Y

J . K . R O Y L E A N D F . P . S T A I N T H O R P

P E R G A M O N P R E S S

O X F O R D · L O N D O N · E D I N B U R G H · N E W Y O R K

T O R O N T O · S Y D N E Y · P A R I S · B R A U N S C H W E I G

Pergamon Press Ltd . , Headington Hi l l Hal l , Oxford

4 & 5 Fitzroy Square, London W . 1

Pergamon Press (Scotland) Ltd . , 2 & 3 Teviot Place, Edinburgh 1

Pergamon Press Inc., 44-01 21st Street, L o n g Island City, N e w Y o r k 11101

Pergamon of Canada, Ltd . , 6 Adelaide Street East, Toronto, Ontario

Pergamon Press (Aust . ) Pty. Ltd . , Rushcutters Bay, Sydney, N . S . W .

Pergamon Press S . A . R . L . , 24 rue des Écoles, Paris 5e

Vieweg & Sohn G m b H , Burgplatz 1, Braunschweig

Copyright © 1968

Pergamon Press Ltd.

First English edition 1968

This is a translation of the Russian book

B o n p O C H Π Η Θ Β Μ Ο - H Γ Η ^ ρ θ - a B T O M a T H K H

(Voprosiy pnevmo- i gidro-avtomatiki)

published by A k a d . N a u k SSSR

Library of Congress Catalog Card N o . 66-19864

1103/68

TRANSLATOR'S FOREWORD

I N A collection of this kind, it is inevitable that some overlapping occurs be-

tween works on closely related subjects, particularly where the same author

is represented by a paper read at a conference, as well as by a magazine

article, which did not appear in the original book. As a rule, in such cases too

severe pruning is undesirable, because it would lead to untidy structure, un-

warranted deviations from the author's line of reasoning, and overburdening

by cross-references. Only in a few instances the considerations of economy

were allowed to prevail. The sequence of articles was changed in order to

bring together the related works, and it has been found that some further

rearrangements, authorized by the Editor of the Russian original, were needed

to improve the "balance" of the English edition.

P. L.

ix

TO ENGLISH READERS FROM THE EDITOR OF THE RUSSIAN ORIGINAL

P E R G A M O N PRESS , in agreement with the International Book Society, has decided to publish in English the collection of Russian works on the subject of pneumatic and hydraulic automatic control. This collection includes the papers presented to the "Symposium on Pneumo- and Hydro-automation" in Moscow in 1957, and published by the Academy of Sciences U.S.S.R. in 1959, as well as works by several authors, printed in the magazine Avtomatika i Tekmekhanika (Automation and Remote Control), in the years from 1947 to 1959.

No t long ago, it seemed to the majority of automatic control specialists that remarkable successes of electronics have predetermined the main line of development in automation techniques, and the importance of pneumatic and hydraulic equipment is bound to decline from year to year. As often happens, a rapid growth of one or another branch of engineering led to ex-cessive enthusiasm and rash conclusions. N o w it is already certain that pneumatic and hydraulic means of automation have held their ground against the powerful onrush of new developments and new technological ideas. In recent years, one hears more and more often the opinion that pneu-matic and hydraulic automatic control has, and will continue to have, its own domain of applications, which is still expanding. As a result, there is a rapidly growing interest in scientific research and design investigations in the field of pneumatics and hydraulics, as applied to automation.

In the light of these facts, the decision of Pergamon Press is undoubtedly useful and timely, furthering the development of scientific contacts also in this branch of engineering. It must be kept in view that the present collection incorporates only a small part of Russian works on pneumatic and hydraulic automatic control, and can provide but an incomplete picture of the develop-ment of this branch of science and engineering in the U.S.S.R.

M . A . A I Z E R M A N

χ

PNEUMATIC EXTREMUM CONTROLLERS I AT A N U.S.S.R.

Y u . I . O S T R O V S K I I

T H E O B J E C T O F E X T R E M U M C O N T R O L L E R S

Extremum controllers are required to find and maintain to a required ac-curacy an extremum of a measured quantity, which varies with time in an unspecified way. The location of the extremum is accomplished by an auto-matic survey.

Extremum regulation is advantageous only when (a) optimum operation of a process in a controlled system depends on an extremum (maximum or minimum) of a certain characteristic parameter (for example, maximum ef-ficiency, minimum fuel consumption, or consumption of a similar agent at a given output of the plant, etc.) and (b) , when the magnitude of this charac-teristic parameter, which has an extremum value under certain conditions, is determined not by one but by several other parameters. These parameters can be divided into two groups : the first group would comprise parameters which cannot be influenced by a controller (e.g. the kind of fuel, the condition of the surrounding atmosphere, the degree of wear of a plant, the load, etc.), where the prediction of these parameters by calculations is either very dif-ficult or impossible. The second group comprises parameters which are af-fected by a controller. Their optimum values, which define the extremum, depend on parameters of the first group.

Let us consider a concrete example. Figure 1 shows the block diagram of a control system for a heating furnace. The temperature controller TC main-tains the desired temperature t0 of the object Ο by acting on the control valve ROx. This alters the fuel consumption Qf. The corresponding air consumption Qa is established by a ratio controller RC, which maintains a pre-set ratio QajQf = oc. The values of t0 and oc are determined by setting controls SCt and SC2.

The controlled furnace can be considered as a series connection of two elements—the flame chamber FC and the heated object O. The temperature ίφ of the flame in the chamber depends on the regulated quantities Qf and Qa, and also on unpredictable air leaks Q'a, the condition of fuel nozzles Φ (their erosion, dirtiness, etc.) and on other factors. The temperature t0 of the object at a steady state depends on the flame temperature, on the heat trans-

3

4 P N E U M A T I C A N D H Y D R A U L I C C O N T R O L

fer coefficients kt and k2 (from flame to object, and from object to its en-vironment), and on the load L (e.g. on the discharge rate of the heated agent).

Obviously, the optimal running of the process from the viewpoint of economy requires a desired temperature t0 with the minimum fuel consump-tion.

The minimum fuel consumption, which may be attained at the optimum value of oc = Qa/Qf (a parameter of the second group), depends on the load,

Air

i SC?

RC

R02 Qg

R01 Qf

Fuel

Qa Φ

TC

ki ko

FC

i

^

i

0 FC 0

in

F I G . 1. Block diagram of the furnace control without an extremum controller.

F I G . 2. Block diagram for the control of air-fuel ratio by an extremum controller EC

acting on the setting of the ratio controller RC. Minimum fuel consumption is main-

tained.

P N E U M A T I C E X T R E M U M C O N T R O L L E R S I A T A N U . S . S . R . 5

properties of the fuel, heat transfer coefficients k1 and k2, and the condition of the nozzles. All these factors cannot be calculated accurately, and vary with time. They can be considered as belonging to the second group of parameters.

The optimum value of oc depends also on the properties of the fuel, and on non-calculable, chance air leakage.

As can be seen, the two conditions given above which make the applica-tion of extremum control advantageous are fully present here.

Figure 2 shows one of the possible layouts for this particular case. The extremum controller EC acts continuously on the setting controls SC2 of the ratio controller RC. The temperature controller TC suitably alters the fuel fl°wC?/ »

t o maintain a given temperature 10. The extremum controller chooses

that setting of SC2 which ensures the minimum value of Qf. In fact, the setting of SC2 oscillates with a relatively small amplitude about the deter-mined optimum value.

If the object Ο has a substantial thermal capacity, the variation in SC2 set-ting (searching the extremum) must be carried out relatively slowly; other-wise the changes of fuel flow Qf would lag behind the changes of the SC2

setting, and the air flow would fluctuate with a large amplitude. As a result, the economy of the plant would suffer. A low rate of searching the extremum, however, is only possible if the parameters of the second group, which in-fluence the optimum ratio, change infrequently and slowly. But if some of them (e.g. load, or properties of fuel) are subject to frequent variations, the scheme given above may prove to be impracticable. In this case the layout shown in Fig. 3 may be used. Here the extremum controller acts on the setting control of the ratio controller, in order to determine the maximum flame temperature ί φ. With an increased flame temperature, the temperature of the object will rise and the temperature controller TC will accordingly reduce the fuel rate. If the response rate of the circuit RC—R02—FC—EC is much greater than that of the circuit TC—RO^—FC—0, then, as a result of simultaneous action of the controllers, the fuel consumption will take a minimum value and will not oscillate appreciably, while the air consumption will oscillate about the optimum value, and the temperature t0 will oscillate about the maximum for a given fuel consumption.

The layout shown in Fig. 4 differs in having the extremum controller acting not on the setting control of the ratio controller, but directly on the control valve R02, which controls the air flow. This scheme is suitable when a high searching rate can be used, in order to cope with drastic changes of the working conditions (for example, with sudden changes of load).

Figure 5 shows the relationship between the flame temperature of a metallurgical heating furnace and the air consumption at a constant rate of consumption of fuel oil. The curves were obtained by A.Butkovskii of the Steel Institute for one of the furnaces of the "Sickle and Hammer" Works. The curves were taken at various times. They indicate that the actual setting


F I G . 3. Blockdiagram for the control of air-fuel ratio by an extremum controller EC

acting on the setting of the ratio controller RC M a x i m u m flame temperature is

maintained.

of the air-fuel ratio controller has not been at the optimum. At the optimum setting, the flame temperature could have been increased, on the average, by 30°C, or the same flame temperature could have been obtained at a lower rate of fuel consumption by the simultaneous use of a temperature controller and an extremum controller.

Fuei ΤΓ ΤΓ

TsCi

F I G . 4. Block diagram for the control of air-fuel ratio by an extremum controller

acting directly on the control valve.


1*50

Air consumplion per cent of max.

F I G . 5. Relationship between flame temperature and air consumption at a constant

rate of consumption of fuel oil. 181—conditions at the beginning of the measure-

ments.

T H E P R I N C I P L E OF T H E P N E U M A T I C E X T R E M U M

C O N T R O L L E R

The extremum controller described is based on the principle of "memorizing of maximum". If the task is to find a minimum and not a maximum, then the sign of a measured quantity is inverted in the transducer.

The measured quantity y (in the example given above—temperature) is transformed into a proportional air pressure, which enters the input of the extremum controller. The air pressure Px at the output of the extremum controller determines the magnitude of the input of the system (on the schemes of Figs. 2 and 3 this input is a particular setting of the control SC2

at the ratio controller, and in Fig. 4 the position of control valve R02).

An example of a static characteristic y = f(x) of a system is shown in Fig. 6, a. Figure 6, c shows the change of pressure Px as a function of time, and Fig.6,b corresponding changes of pressure Py. A t the instants tel the pres-sure Py attains its maximum value Pymax and with further changes of Px be-gins to decrease. With an "inertial" system, the value of Pymax will be smaller than Pyinax corresponding to the static characteristic. The value of Pymax at-tained is registered by a special device, a "memory unit" (see Fig. 7).


F I G . 6. Operation of the pneumatic extremum controller.

The pressure Pm in the "memory chamber" of this unit follows the pressure Py when Py increases, but when Py begins to diminish, the magnitude of Pm

remains constant (the dotted line in Fig. 6).

CPD

F I G . 7. Layout of the pneumatic extremum controller. M—memory unit; CU—com-

parison unit; RS—step switch; CPD—constant pressure drop unit; Ο—controlled

object; RO— control valve; T— transducer; PR—pressure reduction valve; Th—

controllable throttles; FR—fixed restrictors; N— nozzles; Β—baffles; S—springs;

V— seated valve; R—sealing ring; RV— rotary valve; Ρ—pressures.


C O N S T R U C T I O N OF T H E P N E U M A T I C E X T R E M U M C O N T R O L L E R

The type of extremum controller shown in Fig. 7 permits the use of stan-dard parts for the pneumatic instruments of the "Aggregate Unified Sys-tem". The controller consists of four units : memory unit (Fig. 8) ; comparison unit; step switch (F ig .9) ; and the constant pressure drop unit.

In the "comparison unit" (Fig. 7), the current value of the pressure Py is compared with its maximum, retained by the "memory chamber". A t the instant tpl when the difference δ = Pm — Py reaches a certain pre-set value o n a x at the output of the "comparison unit", a pressure impulse Pt (Fig. 6,d) occurs which is fed to a step switch (RS in Fig. 7). Pressure PP at the out-put of the step switch can have one of the two discrete values: Pp = P0

and Pp = 0. Switch-over occurs at the instant of the impulse Pt. In the inter-val between two impulses the pressure Pp has the value established by the preceding impulse. Pressure Pp is fed to the input of the "constant pressure drop unit", CPD (Fig. 7). \ΪΡΡ = P0, then the pressure Px at the output from this unit increases at a constant rate. If Pp = 0, then Px decreases at the same rate. Pressure Px governs the input of the controlled object. In addition to reversing the input of the object, the Pt also performs the fol-lowing functions: the self-locking of the "comparison unit", the cancellation of the "memorizing process", and (due to the time lag) subsequent release of the self-locking of the "comparison unit".

The cancellation of the memorizing operation is effected at the instant of time tpl by connecting the "memory chamber" with the measured pressure line. This equalizes the pressures Pm and Py. Due to a time lag, thelocked-in pressure Pt in the "comparison chamber" is held for a certain time. This is necessary for the following reason: if the object is an "inertial" one, the output represented by y proportional to the pressure Py still decreases for a certain time, after the reversal of the input. The memory unit, which is again switched on at the instant tpl, would "memorize" the cor-responding value of Py. If now the output fell so much that β > ôinax

(see Fig.6,b), and if at this instant the impulse Pt were removed, then we would obtain a "false" reversal of direction in the change of Px. T o avoid this, the impulse Pt is removed with a certain time lag. At the time instant tci

the pressure impulse Pc (Fig. 6, e) occurs at the output of the delaying element. This impulse removes the self-locking of the comparison unit. Pressure Pt in the main pipeline drops to zero. From this instant, the memory unit is again able to fix the maximum of Py. The drop of pressure Pt also causes the drop of pressure Pc (with a certain time lag). A t the instant te2 the pressure Py

again reaches the maximum, and the memory unit fixes the corresponding value of Pyinax. A t the instant tp2 the sign of the change of Px is reversed again, and the searching cycle repeats itself.


The pressure PyJ[ proportional to the output of the controlled system, is

fed to the chamber 20 of the memory unit, and to the chamber 14 of the

comparison unit. When the valve V is open, the main pipeline Py is also

connected to the memory chamber 23. The valve V is closed by the spring

S5, and opened by the pressure in chamber 22. The absence of air leakage is

ensured by a rubber ring R{. The pressure Pm from the memory chamber is

fed to chambers 13 and 21, where it is compared with pressure Py.

If Py > P,„, then the nozzle N3 is open, and in chambers 8, 9, 10, 11, 12,

15, 16, 17, 18 and 19 the pressure is atmospheric. Under these conditions,

F I G . 8. Memory unit. F I G . 9. Step switch.

the baffle B5 closes the nozzle N5, which controls the opening of the valve V;

as a result, the valve V opens and connects the memory chamber with the main pipeline Py. In this way, as long as the output quantity of the system increases, the pressure Pm in the memory chamber follows the pressure Py;

but when it ceases to increase, the pressures Pm and Py are equalized, and this causes the baffle B5 to move from its nozzle N5, thus reducing the pressure in chamber 22, and closing the valve V. During the decrease of Py, the valve V remains closed. Therefore a pressure equal to the current maximum Py

establishes itself in the memory chamber. If the pressure Py diminishes to the extent that the difference Pm - Py becomes equal to o , n a x , then the baffle B3

will close the nozzle 7V3, with a suitable setting of the spring S2. The pressure Pu at the output of the comparison unit will increase. Due to the positive feedback connexion with chamber 11 of the comparison unit, the impulse

f In the following text, letters Ρ with various subscripts are used to denote both a

pressure and a corresponding pipeline or passage (Translator).


Pi increases in jumps. This impulse turns the rotary valve R V of the step

switch by an amount corresponding to one ratchet tooth. In the position

shown in the scheme, the rotary valve connects the pipeline Pp to atmosphere.

The next impulse Pt will connect the pipeline Pp with the pipeline P0. The pres-

sure Pi9 fed to the memory unit, closes the nozzle 7V4 by means of the baffle

2?4. Due to this, the pressure in chambers / 9 and 22 increases, and this causes

valve V to open and, consequently, interconnects the memory chamber and

the pipeline Py. The pressures in the chambers 13 and 14 of the comparison

unit are equalized, but the nozzle N3 remains closed, due to the pressure in

chamber 11.

After a certain time lag, determined by the resistances of the throttles Th2

and Th3, the pressure in the chamber 8 of the delaying element attains a level

sufficient to overcome the force of spring St ; the nozzle N2 will be closed by

the baffle B2. The appearing impulse Pc enters the chamber 72, the nozzle N3

opens and this causes the removal of impulse Pt. After that, the scheme is

restored to its initial position.

Let us now consider the operation of the constant pressure-drop unit.

When there is no pressure in the chamber 2, the pressure P0 in the chamber

3 is balanced by the difference of pressure Px and P5 in the chambers 4 and

5 (note that Px > P5). Air is discharged from the diaphragm unit of the control

valve RO into the atmosphere. The pressure drop across the throttle Thl

remains at a constant value. If, for example, this pressure drop were reduced,

the gap between the baffle Bx and its nozzle Nt would increase and this

would cause a reduction of pressure in chambers 6 and 5, and the restoration

of the set value of the pressure drop. In this way the speed of operation of

the control valve RO is kept constant.

If the pressure Pv is fed to the chamber 2, the diaphragm unit of the

control valve is charged at a constant rate, determined by the pressure drop

in the throttle Th1.

R E S U L T S O F L A B O R A T O R Y T E S T S

Figure 10 shows a block diagram of the test rig used to test an experi-mental rig of the controller.

For a system with a non-linear static characteristic

y = k(x — a)2 + b

the electronic simulator E M U - 5 has been used. Constants k, a and b could be altered in the course of the work by means of a variable resistance Rk and potentiometers Ra and Rb.

The model of the object consisted of an inertia-less non-linear element NE

with a parabolic characteristic, and two inertia links of the first order, Ul

and U2, situated at both sides of element NE. The air pressure Px at the out-put of the extremum controller £ C was transformed into an electric signal by a


potentiometer transducer E D M U - 1 . The output signal of the model was

transformed into an air pressure Py by an electro-pneumatic transducer EP,

developed at I A T A N U.S.S.R. Pressures Px, Py9 Pm and Pt were recorded

with the aid of tensometric (strain-gauge) transducers.

The oscilloscope record in Fig. 11 illustrates the behaviour of the extremum

controller with an inertia-less object (time constants of the links UY and U2

EDMLf Px EC Py

EP

J L

F I G . 10. Block diagram of the test rig for tests of the extremum controller prototype.

7sec

F I G . 11. Oscilloscope record for the operation of the extremum controller, with an

inertia-less object.


were equal to zero). The oscilloscope record in Fig. 11, a shows the steady

self-excited oscillations near the extremum, and a transitional process, which

arose because of a change in coefficient b of the static characteristic of the

object. In Fig. 11,b the oscillations are recorded, and also the transitional

effects due to the change of coefficient a. The rapid changes of pressure Py

at the instants of the impulses Pt are caused by the release of pressure from

the "memory chamber" into the line Py. The tests of an experimental proto-

type of the extremum controller lead to the following conclusions:

1. The scheme ensures a definite "memorizing" of the maximum. The

pressure fall in the memory chamber was 1 mm mercury/hr, with an initial

pressure Pm = 0-5 atm, and Py = 0 atm (1 atm abs.). I f necessary, the leak-

age from the memory chamber could be still further reduced.

2. A well-defined operation of the extremum controller has been obtained

at o m a x = 0-015 kg/cm2 (the working range of Py is 1 kg/cm

2) . It appears that

further "tuning" of the controller would result in a reduction of (5 m a x.

3. The extremum controller searches a maximum when it is displaced

slowly, and finds a new maximum value after a sudden shift ("horizontal" or

"vertical").

4. The time lag for the reversal of Px after attaining a desired value of

(5 m ax is approximately 0-3 sec.

PNEUMATIC RATIO CONTROLLERS (WITHOUT MECHANICAL DIVIDERS)

G . T . B E R E Z O V E T S

T H E CONTROL of a ratio between two parameters finds a wide field of applica-tion in the chemical industry, metallurgy, oil refineries, etc. Sometimes the task is only to maintain a certain ratio, but in other cases it is necessary to introduce an automatic correction of ratio between two parameters in ac-cordance with a third parameter. For example, sometimes it is necessary to vary the ratio between the flow of two chemical reagents, depending on the temperature in the plant, where the chemical reaction is taking place; the temperature itself may be subject to change because of a varied concentra-tion of the solutions.

Up to now our industry produced pneumatic ratio controllers of one type only—44-MS-700. This instrument is intended to maintain a ratio at a desired value, and cannot be used when it is required to perform an auto-matic correction in accordance with a third parameter. Nor can the ratio relay, made at present by the "Tizpribor" works as one of instruments of pneumatic "unit block system" ( A U S ) , be used for this purpose.

The Institute of Automatics and Telemechanics of the Academy of Sciences U.S.S.R. has developed two pneumatic ratio controllers RSI , de-signed to maintain a ratio at a pre-set constant value, and RS2, in which the set value of the ratio is cascaded from a third parameter. Both controllers are built on the basis of the integral action unit 5RB-9A | of the "unit block system", and have no mechanical dividers. The work of a divider is per-formed by a pneumatic "flow chamber" (Fig. 1). It has a fixed restrictor 2, which restricts the flow of entering air, and a variable restriction 3, restricting the discharge. Experimental investigations on the "flow chamber" have shown that by suitable choice of these restrictors it is possible to obtain a practically linear relationship between the pressures P2 (in the chamber) and Pl (before the fixed throttle). The curves of its static characteristics are shown in Fig. 1 : each of them represents an equation,

P2 = kP1.

In this equation the coefficient k, which determines the slope of each charac-

t Detailed description of this unit is given in Ref. 1.

14

P N E U M A T I C R A T I O C O N T R O L L E R S 15

teristic, depends on the amount of opening of the variable restrictor 3. When it is fully closed, pressures ΡΓ and P2 are equal, and therefore k = 1 ; as the throttle is opened, the pressure in the chamber drops, and consequently k

diminishes, approaching zero. Obviously, pressure P2 is \\k times less than pressure P X , and this makes it possible to use the flow chamber as a divider. The ratio controllers in question are based on this principle.

P2kg/cm2

Phkg/cm2

F I G . 1. Arrangement of the flow chamber, and its static characteristics.

The Ratio Controller RSI (Fig. 2) differs from the controller 5RB-9A [1] only in having its desired value chamber a and measuring chamber b made as flow chambers.

There are fixed restrictors 1 and 2 in their inlets, and they receive from transducers the pressures, proportional to parameters A and B, the ratio between which must be maintained constant. The adjustment of the set value of this ratio is effected by variable restrictors 3 and 4, located at the outlets from these chambers.

As already mentioned, the flow chambers work according to equations:

PIA = k A P 1 A and P 2 B = k B P 2 B,

where P 1 A and P 1 B are pressures before throttles 1 and 2, proportional to


measured parameters A and B\ P2A and P2B are pressures in chambers a and b\ and kA and kB are coefficients of proportionality, set by throttles 3 and 4.

As the controller is of the integral action type, its equilibrium is only pos-sible when the pressures in chambers a and b are equal, that is when

PIA — P IB\ or

kAP IA = k BP ι B.

From this it follows that

P\A\P\B = kBjkA = k.

This means that the controller RSI will maintain a given ratio/: between the pressures P1A and P2A, and consequently a ratio between parameters A and

F I G . 2. Arrangement of the ratio controller R S I .

B. The value of k may be varied between wide limits (from 5:1 to 1:5, and if necessary these limits can be widened further).

The ratio controller RS2 (Fig. 3) is intended to control the ratio between two parameters A and Β with automatic correction according to a third parameter C. Such a controller can be developed from the controller R S I , if one of the coefficients kA ovkB\s made to be dependent from the third para-meter. This has been achieved by replacing the adjustable throttle 3 of the controller RSI by a nozzle 5 and baffle <5, the position of the baffle being determined by the parameter C.


As the working stroke of the baffle 6 is very small (of the order 0-05 mm),

it is installed in a special set, consisting of chambers c and d9 analogous to

the system of chambers of an integral action element. Pressure from the

transducer measuring the third parameter enters the chamber c. Chamber d

is connected through the fixed throttle 7 to the supply pressure. Chambers c

and d are separated by the diaphragm 8, to which is fastened the baffle 6.

F I G . 3. Arrangement of the ratio controller RS2.

As the pressure in the chamber c varies, so does the pressure in chamber d,

with an exact follow-up of the pressure in chamber c. If these pressures at one instant are unequal, the diaphragm 8 moves out of its position of equilibrium, until the opening of nozzle 9 would be changed sufficiently to equalize pressures in chambers c and d. The exhaust nozzle 9, and nozzle 5 which replaces throttle 3, are made as a concentric pair of nozzles, covered by the same baffle 6. The stroke of the diaphragm and baffle is proportional to the third parameter C . When this parameter increases, the baffle 6 begins to close the nozzle 5, and the coefficient kA of the chamber a begins to increase, approaching to / . When the parameter C diminishes, the coefficient kA diminishes also.


T o ensure the possibility of altering the degree of influence of parameter C on the ratio between A and B, the chamber c is made as a flow chamber, i.e. the pressure from the transducer is fed through a fixed restrictor 10, and at the exit there is a variable restrictor 77. When 77 is closed the degree of influence of C is at its maximum and is reduced by opening the restrictor. This provides additional possibilities for the adjustment of the controller.

Figure 4 shows the change of coefficient kB of the flow chamber b[(kB)

= PIBIPIB] as the restrictor 4 is opened. The degree of opening is measured as the angle oc, by which the head of the throttle is turned (at oc = 0 the throttle

KB 1-0

0-8

0-6

0*

F I G . 4 . Relationship between coefficient kB and opening of throttle 4,

F I G . 5. Relationship between coefficient k A li pressure P 1 C, and opening of

throttle 11 (angle a ) .


is closed). The values of kB are calculated on the basis of the characteristics

shown in Fig. 1.

Figure 5 shows the dependence of coefficient kA = P2AIP\A on the pres-

sure P 1 C , proportional to the third parameter C and from the opening of

throttle 11.

The adjustments of integral action time and proportional band are the

same as for the control unit 5 R B - 9 A ; that is, the proportional band can be

altered from 10 per cent to 250 per cent, and the integral action time from

3 sec to 1 hr and more.

R E F E R E N C E S

1. G . T . B E R E Z O V E T S and I . F . K O Z L O V , Information on Scientific Research Works, Vo l . 29,

N o . 1-56-200, V I N I T I , 1956.

2. G . T . B E R E Z O V E T S , Avtomatika i Telemekhanika, V o l . X V I I , N o . 5, 1956.

PNEUMATIC CONTROLLERS WITH AUTOMATIC RE-ADJUSTMENT

ACCORDING TO THE LOAD

T . K . B E R E N D S and A . A . T A L '

O U T L I N E OF T H E P R O B L E M

Automatic control of industrial processes usually is effected by controllers

with proportional and integral action. These have two adjustments, which

are used to establish the proportional control factor (gain) kp and the integral

action time T{. These quantities, and consequently the settings of both terms,

are chosen after taking into account the dynamic characteristics of the con-

trolled plant so as to give to the control system certain dynamic properties.

This creates some difficulties, because the properties of the plant vary with

substantial changes of load (or, more generally, with changes of working

A it

F I G . 1. Block diagram of the control of a simple plant by an integral action controller.

conditions) while the adjustments remain constant. Consequently, the dyna-mic properties of a control system, with the terms adjusted for a certain range of loads, will be altered by the change to any other range of loads, and may become quite unsuitable.

T o prove this, let us consider a control system (Fig. 1) consisting of the simplest object

Τ α ^ + φ = Κμ (1) at

20

A U T O M A T I C R E - A D J U S T M E N T A C C O R D I N G T O T H E L O A D 21

Τ ^ RR

1 r = a dt

a = η + Ii + l 2

η = -Âr Py

Ii = -Μ

dl2 Γ, + | 2 = j "

(2)

Here 99 is the deviation from the controlled condition, and Μ the relative dis-

placement of the correcting element. The equation (2) of such a controller

can also be written in the following form:

r i . r f ^ + (7-i + r f ) ^ = - ^ ( , + r f ^ dt

2 dt \ dt

(3)

The plant (1) and the controller (3) form a system, described by a differen-

tial equation of the third order:

where

* - * + C LL - 2 - + C 2 * L + C I V- O , dt

3 dt

2 dt

ι 1 1 c, = — + — + —

Τα Γ, Tc

r = — f — + 1 + *«M C z τ.\τ, Tc J

(4)

c3 = kak p

Τ TT Λ Α

1 Ι

Λ C

( 5 )

With the introduction of non-dimensional time τ

r = C \ '3t =

the equation (4) takes the form

d3w „ d

2w „ dœ

— Ï - + Λ — 2 - + Β—ί- + φ = 0. dir

3 d-r

2 dr

(6)

(7)

and an integral action controller, with a linear servo-motor


This equation contains two basic parameters:

\τα τ, TJ\ kakp ,

B = c c-2/3 = _L (1_ + 1 + *ΛΛ ^ V / 3

τα \ Tt Tc J \ kakp

(8)

which determine the dynamic properties of the system. From the expressions (8) it follows directly that the change of properties of the object (ka and Ta) is determined by a change to other working conditions, leads to a change in the dynamic properties (A and B) of the system if the adjustments of the controller (kp and Γ ( ) remain constant.

Adjusting a P. and I . controller involves the establishment of such values of its gain kp and integral action time Tt, which would give to the system the required dynamic properties A and B. The time constant Tc of the servo-motor, and the characteristics of the object—its gain ka and time constant Ta—should be considered as given. Equations (8) would enable the required values of kp and Tt to be calculated :

According to (9) and (10), the definite dynamic properties A and B, at a given time constant Tc of the servo-motor, can remain constant (as ka and Ta vary) only if the controller is being re-adjusted to alter kp and Tt:

kP=faka,Ta)\

Τ>=ΜΤα). J ( H)

The properties of the object are determined by its load, and (in a steady state) a definite position .v of the control valve corresponds to a value of load. Therefore:

ka = ka(x) \

T.-TJM] ( 1 2)

Taking into consideration (12), the relationships (11) can be expressed in the following form: , ,

kp = k„(x), (13)

Tt = Tt(x). (14)


D E S I G N O F T H E P N E U M A T I C C O N T R O L L E R W I T H

A U T O M A T I C R E - A D J U S T M E N T A C C O R D I N G T O T H E L O A D

First Alternative

Figure 2 represents the design of such a regulator, based on the elements of Aggregate Unified System ( A U S ) . This controller consists of the same elements as a standard control unit,t with the difference that its adjustments for the proportional band [DD = 100/&p], and for integral action time Ti9

are remotely controlled, and also a special device is provided, linking the output of the controller with both adjustments, a functional transformer being inserted in each connecting line.

The remotely controlled restrictor, used as an adjuster, consists of two "nozzle-baffle" elements. One of them performs the functions of a working restrictor, being included in the feedback line of the controller, and the other one is a follow-up element, used for the purpose of equalizing pressures in the follow-up and command chambers of the adjuster. The baffles of these

t See, for example, Refs. 1, 2, and 3.

2 Aizerman I

Hence, it is seen that the relationships (11), which ensure the constancy of dynamic properties of the system, can be attained automatically by providing connexions between the positions of the regulating organ and the organs of adjusting.

In existing controllers, the levelling out of dynamic properties of a system at various loads is effected by suitable selection of the characteristics of the elements of the regulator (usually control valve). These means can only ensure the constancy of the plant's gain ka, and the dynamic properties of the system could remain unchanged only if the time constant Ta of the object remains constant. But if the time constant varies with the load, then it is impossible to level out the dynamic properties of the controller, even at constant ka; this follows from (11). Being of a limited usefulness in principle, the "profiling" method has also some practical defects. The control valves, regulating the flow of a controlled stream, require the use of special alloy steels which are in short supply. Design and making of such "profiled" valves is laborious and expensive.

The problem of levelling out the dynamic properties of a controlled system in a working range of loads can be solved radically by the use of a special device between the controller and the valve, in order to ensure the realization of relationships (13) and (14). Two devices of this kind were developed in the Laboratory of Pneumo- and Hydro-Automation I A T of the Academy of Sciences U.S.S.R. In conjunction with the controller unit of the pneumatic Aggregate Unified System, it provides a pneumatic controller with auto-matic re-adjustment according to the load.


two elements are attached to a double diaphragm, which divides the follow-up and command chambers, and therefore are rigidly tied together. This ar-rangement, which equalizes pressures in the follow-up and command cham-bers, ensures a definitive relationship between the position of the baffle of the working restriction and the command pressure.

Va

IF

m <

p3Se\ value

H ρ From t r ansduce r_ _

Supply

—ι—

I F

Ιϊα

P.

To control valve

F I G . 2. Arrangement of a pneumatic integral action controller with automatic re-

adjustment (first alternative). /—amplifier; / /—comparing element; ///—integral

action element; IV—device for automatic re-adjustment of the proportional band;

I Va—functional transformer (P7 = Fkp(P2))', V—device for automatic adjustment

of integral action time; Va—functional transformer (Ps = FTk(P2))-

Command pressures P7 and P8 of the remotely controlled restrictions IV and V are determined by two functional transformers I Va and Va, to the input of which is fed the output pressure of the controller P2, which de-termines the position χ of the controlling valve. Therefore,

Pi = Fkp(P2), (15)

Ρ s = FTi(P2). (16)

Functions (15) and (16), realized by functional transformers, must ensure the fulfilment of conditions (13) and (14), necessary for stabilizing the dyna-mic properties in relation to load. As the conditions (13) and (14) may differ


substantially from one object to another, and often cannot have an exact analytical expression, the functional transformers must be so designed that the functions which they generate can be altered in operation.

The first alternative, consisting of the pneumatic controller with automatic re-adjustment according to load (to the position of the control valve), can only be realized when functional transformers with the required properties have been fully developed. Work on such functional transformers is proceeding in I A T , Academy of Sciences U.S.S.R. (See the article by L. A.Zalmanzon and A . I . Semikova, "The application of jet-tube elements for non-linear trans-formations in pneumatic systems" in this book, p. 59).

A controller equipped with adjusters /Kand V, remotely controlled by the output pressure of the control unit and transformed in functional transfor-mers IVa and Va, will ensure the continuous rigid interdependence between the position χ of the control valve and the adjustable parameters of the controller (ka and Ta). By suitably setting the functional transformers (i.e. by establishing relationships (15) and (16)), the fulfilment of conditions (13) and (14) will be ensured, hence constant dynamics properties of the closed-loop control system will be maintained as the load varies.f

I IF W IF

F I G . 3. Arrangement of a pneumatic integral action controller with automatic re-

adjustment (second alternative). / , / / , / / / , IV— sections of automatic re-adjustment

unit; I—proportional band adjusters; 2—integral action time adjusters; la, 2a—

needle-valves; lb, 2b—stop valves; 3—command capacities; 4—command relays;

5—blocking valves; 6—fixed restrictors.

Second Alternative

Figure 3 shows the arrangement of the second alternative of a pneumatic controller with automatic re-adjustment. Like the first alternative, it is based on A U S components. It consists of a standard A U S control unit, with the adjustments separated to form a special attachment—the automatic ad-justment unit.

f Throughout this article it has been assumed that oscillations of the system are small.


This unit consists of a number of similar sections /, / / , / / / , . . . Each section comprises two manually adjusted needle-valves la and 2a, with built-in stop valves lb and 2b, one command capacity 3, one manually adjusted command relay 4, and one blocking valve 5.

The output pressure of the controller P2 is fed to the inlets of needle-valves la and 2a, and to a blind chamber of the command relay 4. The outlets of the valves are connected through the stop valves lb and 2b respectively with the positive feedback chamber, and with the blind chamber of the integral action of the control unit. With this arrangement, each section has its ad-justments (for proportional band, and for integral action time), ready to be switched into the operation of the controller. Capacity 3 of each section is connected at one side through a fixed throttle 6 to the feed line and, at the other side, to the atmosphere, through the valve of the command relay 4 of a given section and through blocking valves 5 of a given and all following sections.

The command relay 4 of each section is tuned by adjusting the valve spring to correspond to a certain pressure P2. When this pressure is attained, the relay trips and opens its valve. The value of this "trip pressure" P2 is in-creased from section to section. With this scheme, in the conditions when P2n < P2 < P2(n+1) only one section η has its command capacity con-nected to the atmosphere, because in all the following sections the valves of command relays 4 are closed and, in the section η — 1, as is the blocking valve 5. The throttles la and 2a (also closed) of section η are included in the feedback line of the controller, because the stop valves lb and 2b are open. The valves of section η will determine the adjustment of the controller, as long as ~ ~

Pin < P2 < Plin+D-

The automatic adjustment unit of this type, being attached to a controller, retains its adjustment within certain ranges of load, and automatically changes it during change-over from one range to another. The limits of these ranges as determined by the adjustment of command relays 4, and the values of kp and Tt within each range as determined by the adjustment of needle-valves la and 2a, should be chosen so as to approximate the conditions (13) and (14), which should equalize the dynamic properties of a control system at any plant loading.

R E F E R E N C E S

1. V . R . A N D E R S and N . F . P A N T A E V , Automatic Control of Oil Refining Processes (Avtoma-

ticheskoe Regulirovaniye Processov Pererabotki Nefti) . Gostoptekhisdat, 1954 .

2. G . T . BERESOVETS, Investigations of a pneumatic controller. Reports of the Second and

Third Conferences of Young Scientists IAT. Published by Academy of Sciences, U . S . S . R .

3. A . J . Y O U N G , Control of chemical processes. Instruments and Automation, Vo l . 27 ,No . 3,

1954 .

PNEUMATIC SWITCHING CIRCUITS

T . K . B E R E N D S and A . A . T A L '

I N T R O D U C T I O N

Modern systems of automatic process control contain, as their most im-portant components, various computing devices, and devices performing certain logical functions (switching schemes).

Pneumatic systems are widely used in industrial automation; this is due to the advantages of pneumatic servo-motors, as well as to the fact that pneumatic elements are simple, reliable, and essentially free from fire- and explosion-hazards. There are many plants in which all automatic control is effected by pneumatic means. I f these control systems incorporate com-puting elements or switching schemes, and if it is not essential to have a high operating speed of these elements, it is definitely advantageous to use their pneumatic alternatives.

When evaluating the possibilities of the application of pneumatic com-puters and switching schemes, it should be kept in mind that their speed of action is quite adequate for chemical industry, oil refining, heat and power engineering, metallurgy, etc. Also, pneumatic devices are inherently suitable for work at high temperatures.

The Laboratory of Pneumo- and Hydro-Automation I A T A N U.S.S.R. is working on pneumatic analogue computers and switching schemes. The present paper describes the possibilities of pneumo-automatic means for these purposes.!

The designs described below are based on the same contact-relay principle as used in electro-mechanical relay systems [1]. In the pneumatic alternatives, relay chains appear in a similar way, but the relays and other elements of a chain are pneumatic devices. The signal is also pneumatic action: in the designs under consideration, the application of a pressure of 1 atm will be referred to as the signal with symbol " 1 " , and 0 atm as the signal with symbol "0" . The designs also incorporate fixed, non-regulated resistances.

The pneumatic switch (Fig. l ,a) , like an electric switch or relay, has two main elements: an actuator and contacts. The contacts are represented by a

f Results of work in I A T A N U .S.S.R. were reported to the conference on the theory

and applications of discrete automatic systems, 26 September 1958. In the work also par-

ticipated L . I . R o z o n o e r , A . A.Tagayevskaya, N . V . Grishko, Τ. K . M o r o z o v a , P . A . L e b e -

dev, and M . M . Sharafetdinov.

27


nozzle-baffle pair. When this element is included into a pneumatic line, it

produces a resistance, which depends on the distance between the nozzle and

the baffle, and may vary from 0 (when the baffle is far enough from the

nozzle) to infinity (when it is tightly pressed to the nozzle). If the baffle can

have only these two positions, the element will carry out the functions of a

contact, that is, it will either completely interrupt the pneumatic circuit, or

connect it with practically no additional resistance. Henceforth, we shall say

that the pneumatic contact is on when the baffle is removed from the nozzle,

and off when it closes the nozzle completely. The actuator consists of three

flat diaphragms. Their centres are connected by a rigid rod (the stem) the

travel of which is limited by two stops. Their peripheries are clamped in the

instrument body. Command pressure signals are fed into the blind chambers,

(a)

(b)

(C)

F I G . 1. Components of pneumatic schemes : a—pneumatic switch ; b—fixed resistance

(throttle); c—adjustable resistance (throttle).

P N E U M A T I C S W I T C H I N G C I R C U I T S 29

separated by the diaphragms. They result in a certain force, moving the stem

from one stop to the other. One of the stops is the nozzle, and the end of the

stem corresponds to the baffle. The actuator and the pneumatic contact are

hence incorporated in one unit.

Fixed pneumatic resistances are incorporated in a pneumatic scheme either

at the points where the pneumatic circuit connects with the supply pressure

1 atm, or at the vent to atmosphere. Then, if the output signal is formed

in the section between the fixed resistance and the rest of the unit, the input

signal 1 (0) in the first case results in a 0(1) output signal, and in the second

case in a 1 (0) output signal.

When used in this way, the size of the pneumatic resistances affect the

speed of action, the air consumption, and the deviation of signals from their

nominal levels (0 and 1 atm). If we consider, however, that pneumatic sys-

tems can work effectively over a fairly wide range of the above quantities, it

will be appreciated that the magnitude of a fixed resistance is never critical,

and generally the element is simple and undemanding. Practically, these

resistances are constructed from lengths of capillary tubes (Fig. l ,b) .

Pneumatic switches and fixed resistances are the main units for construct-

ing a pneumatic switching scheme. Often it is necessary also to use adjust-

able resistances. A diagram of such an element is shown on Fig. l ,c.

Naturally, the fact that pneumatic switching devices are analogous to the

corresponding electrical devices means that they have the same functional

possibilities. It is shown below, how these elements should be used to obtain

systems which can be described by a general term "finite automats" [2]. First

we shall consider devices, performing elementary logical functions. They can

be used as a basis for designs which accomplish operations, determined by

logical functions of any degree of complexity—so-called primitive schemes.

Then we give the design of the most important link of finite automats—the

delay element. Further, it is shown how the elements of primitive schemes,

and the delay element, can be combined to produce so-called non-primitive

schemes.

The use of these elements for the construction of devices for the conver-

sion of continuous quantities into discrete ones is also described. In conclu-

sion, some examples of practically important applications are given. The

main subject matter is treated in the form of diagrams, and explanations are

kept to the minimum for brevity.

N O T A T I O N S A N D S Y M B O L S

Graphic symbols, shown in Fig. 2, are used for pneumatic switching dia-grams. There are also special symbols which indicate the manner of operating the actuator of a pneumatic switch. These symbols are similar to those used for electric switching schemes [1].


The main variable in pneumatic schemes is the air pressure. If this pressure can only take either of two values (0 and 1 atm), it will be denoted by Ρ with subscripts, but if it can have any intermediate value between 0 and 1 atm, it will be denoted by Ρ with subscripts. Another important variable is x, which defines the condition of the contact : χ = 1 when the contact is on (the baffle is away from the nozzle), and χ = 0 when the contact is off (the nozzle is

(a)

(b)>

id)-

IH> Ο F I G . 2. a—pneumatic pipeline; b—feed end of a pipeline; c—"earthed" (vented)

end; d—fixed throttle; e—adjustable throttle; f—pneumatic contact; g—actuator;

h—pneumo-capacity.

closed). The variable χ is the output of the switch, determined by the input— that is, by pressures P fl and P f t, fed to the actuator of the switch. Generally, this can be considered as a dependence χ = χ (Ρ α; Ρ*,). Its concrete form is determined by the peculiarities of pressure signals and the ways of actuating the switch. In the designs considered below, five modes of this dependence are encountered :

(a) χ = Sg(Pa — Pb) = (P fl — P ö ) . A pneumatic switch which works according to this equation will be called a receiving switch; it is operated as shown in diagram A, Fig. 3. The pressure signals P a and Pj, fed to its reacting organ can vary within the range 0-1 atm.

T A B L E 1 T A B L E 2

Pa 0 1 Pt 0 1

X = Pa 0 1 x = Ph ι ο

(b) λ* = Ρ a (Table 1). T o produce this mode, the switch is also connected according to the layout Λ , Fig .3 , but has only one input Pa which takes either the value 1 atm or 0. The pressure fed to the other con-nexion remains constant or "fixed", equal to about 0-5 atm.


(c) λ' = Pb (Table 2). In this case the switch is also connected according

to diagram^, and has one input Pb; the other connexion is then fixed

at 0-5 atm.

(d) χ = Pa-* Pb (Table 3). Here both inlets are "free", that is, both Pa

and Pb are variable, taking values 0 or 1. T o produce this mode, the actuator is provided with a spring as shown in diagram Bl9 Fig. 3.

(e) χ = Pb -* Pa (Table 3). This case differs from the preceding one in

the location of the spring (diagram B2, Fig. 3).

T a b l e 3

Pa 0 1 0 1

Pb 0 0 1 1

x = Pa Pb 0

1

1

1

0 0

χ = Pb-> Pa

0

1

1

1 0 1

The symbols and labels as shown in Fig. 3 refer to the above five modes.

The method of labelling provides complete information about the way in

which a certain switch has been operated.

Marking of contacts

PA" ?bJ

(b)-Pa

(O-

Actuator Scheme of switch connexions

Idi-Pa-Pb

- Ο ο— Pa

" φ

Pb Pb-JlThr

Τ 1

-±. -L B,

Fb-Pa -o o—

((Fig. 3))

Ί Pb Γ

1_ B2

F I G . 3. Relationships, realized in pneumatic switches, and corresponding symbolical

representation of actuators and contacts:

a—χ = Sg(Pa - Pb) = [ P FL - P B ] ; b—x = Pb;

c—x = Pb; ci .ν Pa Pb; e—x = Ph->Pa.

2 a Aizerman I


3. R E A L I Z A T I O N OF E L E M E N T A R Y L O G I C A L O P E R A T I O N S .

P R I M I T I V E S C H E M E S

Any primitive scheme, as follows from Ref. 3, can be constructed by using only one elementary logical operation, for example a "Scheffers stroke". But it is much more convenient, even from the theoretical viewpoint, to use for this purpose two elementary logical operations, for example negation and conjunction, or negation and disjunction. Still more convenient is the widely used system, based on three elementary logical operations—negation, con-junction and disjunction. Naturally, a still wider system of elementary logical operations, while not introducing any essentially new possibilities, may have some practical advantages.

It is shown below how the operations of negation, repetition, Scheffers stroke, conjunction, disjunction, implication, equivalence, and logical sum can be realized by pneumatic means. The above set of elementary logical operations is certainly sufficient for constructing a logical function of any degree of complexity.

Al l these operations can be divided into three groups:

(1) one-input group (Table 4)—negation (P = P{), and repetition (P

(2) rc-input group (Table 5)—Scheffers stroke (P = PY1P2

! · · • jPn\ con-

junction (Ρ = Pl Λ Ρ2 Λ · · · Λ Ρ„) and disjunction (Ρ = Ρι ν Ρ2

ν - ν Ρ„);

(3) two-input group (Table 6)—implication (Ρ = P1 - » P 2 ) , equivalence

(Ρ = ρί ~ p2) and logical sum (Ρ = P x V P 2 ) .

Diagrams of devices performing the operation of the first group are shown

in Fig. 4. All these schemes are based on the use of pneumatic switches of the

type χ = Pa or χ = Pb (Fig. 3), and a certain scheme corresponds to each

F I G . 4. Arrangements of devices, realizing elementary logical functions of one

variable ( T a b l e 4 ) ; a—negation "no", Ρ = P x ; b—repetition "yes", Ρ = Pl.

= P i ) ;

P N E U M A T I C S W I T C H I N G C I R C U I T S

T A B L E 4

33

Λ 0 1

P = P1 1 0

P = Pi 0 1

T A B L E 5

0 1 0 1 0 1 0 1

Pi 0 0 1 1 0 0 1 1

p 3 0 0 0 0 1 1 1 1

Pn 0 0 0 0 1 1 1 1

Ρ = ΡίΙΡ2/Ρ3Ι IPn 1 1 1 1 1 1 1 0

Ρ = Pi Λ P2 Λ i>3 Λ - Λ Ρ η 0 0 0 0 0 0 0 1

P = Pi V P2 V P* V - V Ρ η 0 1 1 1 1 1 1 1

T A B L E 6

Pi 0 1 0 1

Pi 0 0 1 1

Ρ = Ρι->Ρι 1 0 1 1

P = P i ~ P 2 1 0 0 1

Ρ = Λ Ν P2 0 1 1 0

type of the switch. Diagrams of the devices realizing operations of the second

group are shown in Fig. 5. They are based on use of the same types of swit-

ches. T w o alternative ways of constructing a scheme—by series or by parallel

connexion of contacts—are given for each operation.

Diagrams of devices realizing the operations of the third group are given

in Fig. 6. They are based on the use of switches of the χ = Pa -> Pb or

χ = Pb-* Pa type (Fig. 3). Here also each alternative corresponds to a certain

type of switch. Under each of these three illustrations (Figs. 4, 5 and 6) of a

device is given the symbol for the operation realized by that device.


β -p2-

(a)

Pi-p2-

Pn-

^Ρ-Ρ^ΛΛΡη

(b)

Pj-p2- ^yP-P^v-vPn

(C)

F I G . 5. Arrangements of devices, realizing elementary logical functions of //

variables (Table 5 ) . a—Scheffers stroke, Ρ = P1/P2I ··· IP ; b—conjunction, "and"

Ρ = Pi A P2 Λ • · · Λ Λ , ; c—disjunction, "or" Ρ - Ργ ν Ρ2 V · · · V Ρη.

© Ρι + Ρζ

>—χ—4—0 ο—|ι. > -

L-Q Ο—*

Py*P2 c

- Ο Ο < •!(- •—ο ο—ο ο-—|ι·

F I G . 6. Arrangements of devices, realizing elementary logical functions of two

variables: a—implication, Ρ=^Ρν-+Ρ2\ b—equivalence, P \ ~ P 2 \ c—logical

sum, Ρ = Pi V P2.

4. R E A L I Z A T I O N O F T H E " D E L A Y " O P E R A T I O N .

L I N E S O F D E L A Y " R I N G " C O M P U T I N G S C H E M E S

Figure 7 shows the arrangement of an element performing the operation of one-step delay (temporary retardation for a unit interval). This element has two independent inputs Pl and Pt and one output P. Input Pt has a specific duty: the appearance of "0" and " 1 " signals here determine the division of physical time into "steps" (unit intervals), in the way shown in Fig. 7. Hence-


forth, we shall distinguish at the time of every step change-over two instants:

one tbn corresponds to the end of «-th stop, and the other ta(n+1) to the be-

ginning of (n + l)-thstop. In an idealized scheme these two instants coincide,

but we shall consider that tan = tbin_n + 0. The operation of delay, which is

MM HL

WJ • T, relative units

n+1 relative units

F I G . 7. Element of one-step delay, Ρ = DiP1.

realized by the element in question, correlates its input and output by the

relationship Ρ = D1PU which in our notation has the following meaning:

and P(t) = Pt(t - 0) if t = tan (when t - 0 = f M l I_ 0 )

^(0 = const, if tan < t < tain+1).

The diagrams of Fig. 7 illustrate the work of the delay element, which actually is nothing else but "memorizing" the input quantity for a length of time, corresponding to one step. Each step can be divided into two parts; in one of them Pt = 0, and in the other Pt = 1. Accordingly, the element of delay consists of two sections; in one of them "memorizing" is affected for the time when Pt = 0, and in the other when Pt = 1. The series connexion of delay elements into one chain produces a line of delay (Fig. 8), which per-forms the operation:

P2 = DlP1, P3 = D

2P P n ,

If we connect the output Pn+1 of the last element of the delay line to the input P± of the first element, we shall have a "ring" computing scheme (Fig. 9). Figures 8 and 9 illustrate the operation of delay line and ring schemes.


ß-D'P, ^ß-D'Pj Pn-D"~'P,^PnH'DnP,

1'ot

P ' l

nnl-

JnnmL

3 4 η n+1

F I G . 8. Delay line.

n+2 X

1

0ML

Pnffi1

0W

9iL

P1

\

1 3 *

J i m lïïil

J L

n+1

F I G . 9. "Ring" computing scheme. P2 =• DlP, ΡΛ =- D

1P2

Pi — Ρ η + 1 ·

r

Λι+ 1

5. N O N - P R I M I T I V E S C H E M E S

The addition of the delay element to systems which perform elementary logical functions changes qualitatively their scope. Without the delay element, only primitive schemes can be realized, but with its introduction, it becomes possible to construct non-primitive schemes of any degree of complexity, i.e. any finite automats. This is confirmed by the following considerations: it is


known from existing theoretical premises that any finite automat can be con-

structed from universal elements. One of the universal elements is the model

of a nerve cell (neuron) proposed by McCulloch and Pitts [2].

The main property of the McCulloch-Pitts neuron (Fig. 10, a) is this: in the

instant t + 1 symbol 1 emerges at its output Ρ (the output is excited) then,

RAPo

(a) fa

sPd'PgVPbVPc

l-APrPd^A'-AP^p^

lb)

F i g . 10. McCulloch-Pitts neuron, a—theoretical scheme of a neuron, having η

exciting and m suppressing inputs, with the threshold of action //. b—realization of

a neuron, having 7 / ^ 3 and h — 2 at the elements "no", "and", "or", and "delay".

and only then, when in the instant t not less than A out of its exciting inputs are activated, and none of m its suppressing inputs is activated. Figure 10,b shows a neuron with η = 3 and A = 2; its logical formula has the form:

Ρ = £ 1 { [ ( Λ Λ P2) V (P2 Λ P3) V ( Ρ , Λ Ρ3)] Λ Ρ3 + ί Λ ... Λ P3 + m } .

This formula is based on the use of operations "not", "and", "or" and "delay"; a logical formula for any other neuron is also based on these ele-

M L

1TTÎTMTTÏ1

Ρ 7

M frm i l / 2 3 4 τ

F i g . 11. Impulse divider.


ments. Therefore, the possibility of realizing these four operations deter-

mines the possibility to construct any neuron, and consequently any finite

automat. We have shown already that the above four operations can be per-

formed by pneumatic means. From this it follows that it is possible to render

any neuron and any finite automat by these means.

As an example of realizing a non-primitive scheme, Fig. 11 gives the ar-

rangement of a pneumatic impulse divider, a pneumatic trigger. This very

simple but practically important scheme consists of a "no" element having

its output connected back to its input through the element "delay". The

logical formula of this scheme is: Ρ = DlP. The diagrams of Fig. 11 illus-

trate the operation of the scheme.

6. D E V I C E S F O R C O N V E R S I O N O F C O N T I N U O U S

Q U A N T I T I E S I N T O D I S C R E T E O N E S

( A N A L O G U E T O D I G I T A L C O N V E R T E R S )

The above paragraphs deal with finite automats, realized by pneumatic

means, but the systems of automation often contain also discrete devices

which do not belong to the class of finite automats—for example, converters

of continuous quantities into discrete ones.

F i g . 12. Interval indicator.

Ρti = SgÇPa2 - Ρ ) , Pb3 - Sg(Pa3 - Ρ ) , . . . Pbn = Sg(?ull - Ρ ) ,

Pl — Pb2i Ρ2 — Pb2 Pb3> Ρ3 =" ΡV>3 ~> Λ>4 - · · , Pn ~ Pbn

The main element of such devices is the receiver switch of the type

x = S g ( P e - P > ) ( F i g . 3 ) .

The receiver switch applied as an analogue to a digital converter, together

with logical elements described above, enables us to construct schemes of

various receiver devices. For example, Fig. 12 shows the arrangement of a

so-called "interval indicator" of a continuous quantity. In it pressures Pal

divide the whole range of input pressure Ρ into η intervals; to each pressure


P A I in the device corresponds a separate section with the output Pb (note that

P „ i = 0).

In this way an /-th interval of the input quantity Ρ corresponds to an out-

put Pai. I f during the work of the device the pressure Ρ takes the value of the

/-th interval, the value of Pt which corresponds to this interval will be 0, and

all other sections will have the output pressure = 1.

7. S O M E A P P L I C A T I O N S O F D I S C R E T E P N E U M A T I C D E V I C E S

Pneumatic Integral-action Controller with Automatic Re-adjustment accord-

ing to the Load (see also the preceding article in this book). A P . + I.-action controller of the "Aggregate Unified System" ( A U S ) generates proportional and integral control, and has two adjustments. One adjusts its gain, and the

F I G . 13. Integral action pneumatic controller with automatic re-adjustment : 1—con-

trol unit; 2—set of restrictorsΛ and β with valves; 3—interval indicator.

other the integral action time. These adjustments remain constant during the work of the instrument. In a number of cases, e.g. when the properties of the object are strongly influenced by the load, and the latter varies within a wide range, it is difficult to choose an adjustment which would be equally suitable for all conditions of work. The preceding article proves that in such cir-cumstances it is advantageous to supplement the controller by an attachment which can readjust it to suit the changed conditions. This attachment


(Fig. 13) consists of a set of restrictors with valves, and an interval indicator. Automatic re-adjustment is effected when the pressure Ρ (input of the inter-val indicator) changes over from one interval to another.

Pneumatic Semi-proportional Controllers. Three-term controllers are fre-quently used in industry; they realize one or another combination of pro-portional, integral and derivative action. Reference 4 describes the advan-tages of introducing also a "semi-proportional" (non-linear) way of control, proposed by V. Ferner. Figure 14 shows how a standard A U S controller can be converted into a semi-proportional controller by the addition of an at-tachment having two receiver switches, one logical element ("equivalence"), and two valves.

Automatic Optimum Controller with Several Regulating Organs. The existing optimum controller for general industrial applications (see the article in this book, p. 3) is designed for work with objects in which the quantity to be optimized depends from the position of only one regulating organ. It is possible, however, to adapt it for cases when the quantity to be optimized depends on the positions of several control valves (Fig. 15). In such cases it is proposed to attain the optimization by moving the valves in a sequence into positions which guarantee reaching a "conditional extremum" of the controlled quantity. It is assumed that each regulating organ can find such a position in three "search" moves. The scheme shown in Fig. 15 is constructed, having in view this method of optimization. It contains all the elements of the standard optimum controller with one regulator: an impulse measuring element, which detects the moves away from the extremum (memorizing and comparing units) ; an impulse divider (step-by-step, or ratchet switch) ; and an integrator (unit of the constant pressure drop). In addition to this, the scheme incorporates an impulse counter, which registers the emergence of

[ Petes "PmeasJ o-

rmeasured

F I G . 14. Semi-proportional controller.


4 C

i * C — i r ßl

Yi

O _csi Ο

Ϊ2

r

f | 4 ! ! ll

Yn

F I G . 15. Automatic optimum controller with several regulating organs. 1—object

with η regulating organs ; 2—memory unit and comparison unit ; 3—impulse divider;

4—integrator; 5—throttle set of the integrator, with valves; 6—set of amplifiers,

with valves; 7—impulse counter; 8—"ring" computing scheme.

R E F E R E N C E S

1. M . A . G A V R I L O V , Theory of Switching Schemes (Teoria releino-kontaktnykh skhem.)

Published by Academy of Sciences U . S . S . R . , 1 9 5 0 .

2 . Automata Studies, Collection of W o r k s , Ed . by C .E .Shannon and J. McCarthy . Prince-

ton University Press, 1 9 5 6 .

3. D . H I L B E R T and W . A C K E R M A N N , Grundzüge der theoretischen Logik. Berlin, 1 9 3 8 and

1949 . ( A l s o in English translation: D . H I L B E R T and W . A C K E R M A N N , Principles of

Mathematical Logic. Chelsea Publishing C o . , N e w Y o r k , 1 9 5 0 . )

4 . V . F E R N E R , D e r halbproportional wirkende Regler—ein neuer erfolgversprechender

Reglertyp. Die Technik, 11, Jahrg. 12 , 8 1 1 - 8 1 5 , 1 9 5 6 .

5. Y u . I . O S T R O V S K I I , Pneumatic extremum controllers I A T A N U S S R (this book, p. 3 ) .

every third impulse; a "ring" computing scheme with the number of sections

corresponding to the number of regulating organs of the object; a set of

throttles and repeaters with valves.

UNIVERSAL PNEUMATIC M U L T I P L Y I N G - D I V I D I N G DEVICE

A N D DEVICE FOR SQUARE ROOTING

Y u . T . I v L i C H E V and E . M . N A D Z H A F O V

I N A U T O M A T I C control engineering, it is often necessary to perform operations of multiplication or division with control signals, or to obtain square roots from them. Simplicity and reliability of pneumatic devices makes them parti-cularly suitable for this application.

The present work contains the description and analysis of the universal pneumatic device developed for this purpose in the Laboratory of Pneumo-and Hydro-Automation of the Institute of Automatics and Telemechanics, Academy of Sciences, U.S.S.R. The multiplying and dividing device permits the performance of the operation

P.-ψ. where P4 is the output pressure, and Ρλ, P2 and P3 are the input pressures. Al l pressures are expressed as gauge units (above atmospheric). By fixing, in turn, one or other of the three input quantities, it is possible to perform also the following simpler operations:

(1) Multiplication of two signals: PA = kP1P2;

(2) Squaring of a signal: P 4 = kP2;

(3) Division of one signal by another: P4 = kPl/P3;

(4) Division of the square of one signal by another signal: PA = P2\P3\

(5) Amplification of a signal, with a wide range of gain: P 4 = kPY\

(6) Obtaining a reciprocal: P4 = k\P3.

In all these formulae k is the scaling coefficient, which may be varied over wide limits. All pressures (input and output) are within the range 0-1 atm. Therefore, multiplication and squaring are applicable to any value within this range. Division, multiplication by a constant k, and obtaining a recipro-cal are limited by the requirement that the output signal shall not exceed 1 atm.g.

Equation (1) indicates that the device is built according to the principle of an equalizing bridge, which ensures the relationship P±\P2 = Λ/Λ>·

42

U N I V E R S A L P N E U M A T I C M U L T I P L Y I N G - D I V I D I N G D E V I C E 43

Operations of the multiplying-dividing device necessitate the condition P 4 < P2. In the case of the general operation (1) it is necessary to fulfil the condition P3 > Pt.

T o widen the range of application, any of the variables Pv, P2 and P3 may be divided by a predetermined constant k by means of a pneumatic di-vider, consisting of a flow chamber with inlet and outlet throttles. One of these throttles is adjustable, which permits the variation in the value of k.

If in equation (1) the output quantity P 4 is taken to be equal to the quan-tity P3, it is formally possible to perform the square rooting operation, ac-cording to the equation:

Ps = J(PiP2), (2)

where, in this particular case, P3 is the output quantity of the square rooting device. In practice, it is difficult to obtain square roots in this way, and an-other scheme, based on the multiplying-dividing device, is proposed.

By fixing one of the input quantities P1 or P2, it is also possible to perform the operation of square rooting of one signal, with an appropriate scale con-stant—that is, the operation P3 = k yjΡ where k = y/Pl, or k = ^JP2. The scale constant k may be altered within wide limits. For the operation (2) it is necessary to observe the condition P3 < P2.

1. D E S C R I P T I O N O F T H E M U L T I P L Y I N G - D I V I D I N G U N I T

If in a nozzle-baffle relay (Fig. 1) the input pressure Pt and the position of the baffle are altered, the pressure Px in the intermediate chamber will change with these two quantities. Generally, Px is a non-linear function of either one of the two variables mentioned above when the other variable is

F I G . 1.

fixed—even if the linearity of the flow-pressure relationship of both throttles is ensured. I f the opening of the nozzle is a function of another external pressure P2, then at certain defined conditions (termed here the conditions of compensation) it is possible to ensure the relationship

Px = kP,P2. (3)

This relationship is the basis of the universal multiplying-dividing unit which has been developed.


The line diagram is shown in Fig. 2. The device has two pneumatic relays,

with two nozzles closed by a common baffle. Input pressures P2 and P3 are

connected to the fixed resistances of these relays, and the input pressure Pt

to the blind chamber of the diaphragm set. The diaphragm set works as a

follow-up system. In the chamber Β there is always a pressure Px, equal to

the pressure P± in the chamber A. The change of pressure P1 alters the posi-

tion of the centre rod of the diaphragm set, which serves as the baffle for two

nozzles. The pressure Px in the chamber Β changes, until there is equilibrium

A

Ham =4

Β Px C P, T T " ' 1P<

F I G . 2 .

at some other valve of the nozzle opening. This is ensured by the negative feedback, transmitting the pressure Px into the chamber Β of the diaphragm set. In the case of a constant pressure P1, but with variations of pressure P3

before the fixed resistance of this relay, the equilibrium of pressures in chambers A and Β is attained by varied opening of the nozzle. These equi-librium conditions, determined by equality of pressures Pl and Px, are ob-tained with quite small movements of the diaphragm centre-piece (about 0-05 mm). In this way, pressure Pt or pressure P3 governs the opening of the nozzle, throttling the exit of air from the chamber B. As the displacement of the end of the centre-piece controls the opening of two symmetrically located nozzles, pressures i \ and P3 also control the opening of the nozzle exit from chamber C. Pressure in chamber C of the second relay depends on the extent of the nozzle opening (and therefore on pressures Ργ a n d P 3) as well as on pressure P2 at the entry into the fixed resistance. The output pres-sure of the multiplying-dividing device is the pressure P4. A n increase of pressures P1 and P2 causes increase of pressure P 4 , and an increase of pres-sure P3 causes its decrease. The attainment of the balance conditions for two pneumatic relays, governed by one common baffle, will lead to the relation-ship of the form (1) .

2. S T A T I C P R O P E R T I E S OF T H E M U L T I P L Y I N G - D I V I D I N G U N I T

T o evaluate the static properties of the unit, let us derive the equations of steady-state processes in the flow chambers of pneumatic relays, and the equation of force equilibrium at the centre-piece of the diaphragms.


W e assume a linear relationship between mass flow and pressure dif-ferences at every throttle. If this condition is fulfilled, steady-state processes in pneumatic relays can be described by equations:

(4) Pz - P

* _ P*

Re l

Pi - PA PA

P-2 Pc 2 ( 5 )

where R is the pneumatic resistance of a throttle, defined as R = APjG. Here zlP is the pressure difference, and G the mass flow. Neglecting small terms (such as the weight of moving parts, jet reactions, and elastic forces of dia-phragms), and assuming equal effective areas of diaphragms, we obtain from the condition of force equilibrium at the centre-piece:

Λ = Px. (6)

Assuming that the effective flow areas of variable throttles (nozzle-baffle sets) are proportional to the movement of the baffle, the resistance of these throttles can be determined by the following expressions, well supported by experimental results:

Rcl = R ' ^ , RC2 = R"!±, (7) h h

where h is the distance from the nozzle to the baffle ; hm the practical maximum value of this distance; and R' and R" the resistances of nozzles at hm.

The set of equations (4)-(7) describes the static properties of the unit. Solving these equations in relation to the output quantity P 4 produces:

P* = ^ 7 ν · <

8>

V RR2 )

T o obtain the relationship ( 1 ) requires also that the additional condition of compensation shall be satisfied:

^ = — , (9) R2 R"

that is, the requirement of proportionality of throttle resistances. In any particular case, this is realized if the respective resistances are equal:

R1 = R29 R' = R"


The realization of the condition (9) becomes difficult practically for the case of proportionality of resistance, because this would entail proportional, but not equal, gaps between nozzles and baffles. This involves substantial design difficulties.

Therefore the condition of compensation (9) will be taken for the particular case of equal resistances.

In this way, the device shown schematically in Fig. 2 realizes the operation P4 = PiP2jP3, provided a linear relationship between discharge rate and pressure drop is ensured under steady-state conditions for all fixed and variable throttles, and if the condition of compensation (9) is satisfied. The linearity of discharge rate and the condition of compensation are achieved by suitable choice of resistances and the use of nozzles with wide end surfaces.

Should the above conditions be fulfilled exactly, the operation PA = P1P2/P3

would be carried out without error. In practice, there are three factors which introduce errors:

(i) the deviation of the discharge characteristics of throttles from linearity;

(ii) inaccuracy in satisfying the condition of compensation (9 ) ;

(iii) the effect of smaller factors, neglected in formulating equation (8 ) : the weight of moving parts, the reaction of jets, elastic resistance of diaphragms, etc.

Let us consider the influence of these factors on the accuracy of the device.

(a) Evaluation of the Effect of Non-linearities in Discharge Rate Charac-teristics. When writing the equations (4) and (5), a linear relationship be-tween discharge rate and pressure drop at all openings has been assumed. Actually, this is not so, for two reasons: non-linearity of resistances, and compressibility of air. T o evaluate the effect of these factors, let us introduce non-linear terms into the equations of the steady-state processes in pneuma-tic relays:

3. S T A T I C E R R O R S OF T H E

M U L T I P L Y I N G - D I V I D I N G U N I T

R (10)

Ρ 2 - Ρ*

Ri + φ2(Ρ2, PA) =

PA + ψα(Ρ*)> ( Π )

where <ρΐ9φ29 ψ€ι > ψ ε ί are terms representing the deviation of discharge rates


ρ

R2

a 1 + — ι P2

1 + (a - 1) A + A { ψ ί _ V c l )

P3 P3

(12)

where a = R"RxjR'R2. T o determine the relative errors, the expression (12) may be written in the

form:

pA = -i-L(\+àl), (13)

where

( P \ R R 1 - — I + α — (<p2 - (fc2) -(φ, - q>cl)

u, = h i ^ h . (14)

l + ( f l- l ) _ L + _ L ( y 1 - . V c l)

Let us apply the condition of compensation (9), and assume, for simplicity, that R1 = R2 = R> W e obtain, as a result:

ô = Ρζ(ψ2 - Ψα) - Ρ2(ψι - ψ ci) ( l 5)

ρ {Ρ^ ι Ν

Pi I — + ψι - <Pci

It can be seen that a smaller relative error can be obtained by satisfying the conditions of non-linear compensation,

Ψ2 = (Pel, ψΐ = <Pcl, (16)

but this is difficult to achieve. T o evaluate the error, let us assume that the differences between deviations of discharge rate from linearity are equal for both relays, and have a maximum value

(ψι - <Pci) = (Ψ2 - <Pci) = Δφ. (17) Then

Introducing the notation P3/R = G, we obtain the formula for relative error:

Αφ

G

from linearity. Taking into account equations (6) and (7), we obtain

(ψι - ψ cl)


If. V/G is sufficiently small, this is given approximately by:

(19)

For example, if the capillaries are of 0-22 mm diameter and 40 mm length, and the nozzles have 0-5 mm hole diameter and 6 mm end-face diameter, the error does not exceed 0-2 per cent.

(b) Evaluation of Errors Due to Inaccuracies in Complying the Compensation Conditions (9). Let us assume that the conditions of compensation (9) are not completely satisfied, and that there is a primary error:

ε, = 1 . R'R2

Equation ( 7 ) then takes the form:

(1 + ει)ΡιΡ2

PA = P?, + ε, Λ

The relative error, caused by inaccurate compensation, is determined by the expression :

0, = — ^ . (20)

Pi

I f the primary error ε! is small, the maximum relative error is

δ 2 < ε ι . (21)

In practice, the error Ά2 can be made very small. (c) Evaluation of Errors Due to Neglecting Minor Terms. When such minor

terms as the weight of the diaphragm centre rod and reaction force of jets are taken into account, the equation of force equilibrium (6) takes the form:

PiFcr - PxFef + Q- N c l - Nc2 = 0, (22)

where F e f is the sum of the effective areas of diaphragms, Q the weight of moving parts, referred to the diaphragm rod; Nci and Nc2 the reaction forces of jets. In equation (22), the elastic forces of the diaphragms are disregarded, because the travel of the baffle is very small.

Denoting the effective areas of nozzles as Fcl and Fc2, the jet reaction for-ces can be evaluated as

Ncl < P3FC], Nc2 < P2Fc2

U N I V E R S A L P N E U M A T I C M U L T I P L Y I N G — D I V I D I N G D E V I C E 49

P3 + \Pl +

Q * P*F« - P*F" I (a - 1 )

where a has the same meaning as in equation (12). Further, the expression

ε2 = Q ~ P i F ci - P l F e2 (25)

will be called a "primary force error". ] f the compensation condition (9) is fulfilled a = 1 and equation (24) takes the form:

P. = ul±±Hl± . (26)

The relative error due to the minor terms is:

3 3 < . (27) Pi

When Ργ = 1, the weight of the moving parts is 50 g, total effective areas of the diaphragms is 20 cm

2 and the nozzles have the dimensions given above,

this error amounts to 0-15 per cent. The overall static relative error can be defined as the sum

δΣ = ô, + ô2 + 03 (28)

and in our example amounts to 0-35 per cent. For the case where one of the pressures serves to introduce a scaling factor, and should be kept constant, the variations of this pressure may introduce additional errors.

4. S O M E A S P E C T S O F T H E D Y N A M I C S O F T H E

M U L T I P L Y I N G — D I V I D I N G U N I T

T o evaluate the dynamic properties of the unit, we shall deduce its equa-tion, assuming that the static compensation condition (9) is fulfilled, that the mass of moving parts, referred to the centre rod, is negligible, and that the

Equation (22), written for the case of the highest possible values of jet reaction forces, will take the form:

PiFe( - PxFef + Q - P3Fcl - P2Fc2 = 0. (23)

Taking into consideration equations (4), (5) and (7), we have

P i + Q - P3Fcl - P2Fc2l pi

PA = — , (24)


movements of the centre rod lie within the working range of the nozzle.f With these assumptions, the processes in the flow chambers are described by the equations:

D D D A D

(29) ^ 3 - Px Px

Pel ât

P2 - PA PA r dP4

— C 2 Rc2

àt (30)

where C1 — V±jRT is the total capacity of the flow chamber and feedback cham-ber; C2 = V2jRT the total capacity of the output flow chamber and load; Vx,

V2 the volumes of the respective chambers ; R the gas constant ; and Τ the absolute temperature.

If we combine equations (6) and (7) with (29) and (30), we shall obtain the differential equation for the dynamical behaviour of the unit.

F lT 2 ^ ± + P J a ( p ^ T l ^ ) + Λ ( 1 - a) dt \ at

= PiP2, (3D

where Ti = R^C^ and T2 = R2C2 are the time constants of the flow cham-bers. When a = 1, the equation takes the form:

âP / άΡ \ PiT2 — ± + Pa [ P I - Ά —î- U PXP2. (32)

àt \ dt J

As can be seen from (31), relationship between input and output pressures even in the case of the simplest model leads to a complicated differential equation. Although the attainment of static compensation conditions sim-plifies this equation, it still remains rather complicated. Equation (32) be-comes a linear differential equation only for the particular case Pl = const and P2 = const.

5. D E S C R I P T I O N O F T H E G E N E R A L D E V I C E F O R

S Q U A R E R O O T I N G

As already mentioned, the operation of square rooting can be performed if, in equation (1), we take P 4 = P3. It is, however, impossible to obtain high accuracy because one relay loads another relay, if the feedback arrange-ment is of such a type. In this case, the conditions of compensation (9) would have to be fulfilled not by equalizing resistances, but by choosing different resistances so as to fulfil the conditions of compensation, and to

t The pneumatic relay has, when working, a gain of the order of 4000, and incorporates

a rigid negative feedback.


reduce the influence of load. Such a solution would be very complicated and difficult to realize practically. It is much simpler to perform square rooting by applying the follow-up chamber.

The line diagram of a device for square rooting, based on these principles, is shown in Fig. 3. It works in the same way as the multiplying unit, taking into consideration that P3 = P4. There is an additional negative feedback

RC1I\\\

RC2

F I G . 3.

connexion from the output of the device to the input of the first relay. Be-cause of the feedback, the output quantity variations are reduced.

I f the conditions of compensation (9) are satisfied, the device performs, with a limited accuracy, the operation (2).

6. S T A T I C E R R O R S OF T H E S Q U A R E R O O T I N G D E V I C E

If the conditions assumed in the analysis of the multiplying-dividing unit were fulfilled precisely, then in the absence of internal loading, the operation p3 = yj

/(P1P2) would be performed without error.

In the square rooting unit, the errors arise due to the following causes:

(a) influence of internal load ;

(b) deviation of discharge-rate characteristics from linearity;

(c) inaccuracies in satisfying the compensation conditions (9) ;

(d) influence of minor factors, disregarded when deriving the equa-tion ( 8 ) (e.g. weight of moving parts, jet reactions, elastic forces of diaphragms, etc.).

The effects of these factors is discussed below. (a) Errors Due to Internal Load. The equilibrium of the forces acting on a

diaphragm of a follow-up system occurs when the pressures above and below the diaphragm are equal :

P3 = Py. (33)


If there is only one controllable outlet throttle closed or opened by the dia-

phragm, such as to give a sufficiently low pressure in the chamber below the

diaphragm, the pressure in the throttle chamber follows the controlling pres-

sure perfectly, whatever the resistance of the inlet throttle. It is necessary,

however, to have the feed pressure not less than the controlling pressure.

However, if there is an additional fixed outlet resistance, the follow-up re-

lationship is not necessarily maintained.

When the controlling pressure is low, the additional air outflow rate from

the chamber does not introduce errors. Equality of pressures is maintained

by regulating the opening of the outlet nozzle so as to achieve a balance

of inflow and outflow rates, corresponding to the equilibrium of pres-

sures.

When, however, the controlling pressure is at its upper limit, and the out-

let nozzle is completely closed, the pressure in the chamber will be lower than

the controlling pressure. The lower the resistance of the second outlet

throttle, the greater will be the difference between these two pressures.

In the diagram of the follow-up system described, the effect of feedback

causes further changes, which result in a difference between the control pres-

sure and the pressure in the chamber, when the control pressure is less than

its maximum. The error, however, is always smaller than that for the case of

maximum control pressure.

T o evaluate these errors, let us consider the isolated follow-up system at

maximum load. The steady-state equation for the flow through the chambers

of the follow-up system is

where R0 is the resistance of feed throttle; Rc3 the resistance of the follow-up throttle; and P0 the feed pressure.

The precise value of the output pressure P£, which corresponds to the no-load condition, would be obtained from (34) if Rl = oo ; that is:

(34)

P* = Po (35)

1 +

When there is a load, it follows from (34) that

(36)


and the relative error will be determined by

zl , < ^ . (37)

1 + + A Rc3 Rc ι

If zl, is small, then from (37) it follows that

< Rc3

(38)

In order to choose the values of the resistances, the condition (38) must be supplemented by the condition outgoing flows of the chamber of the follow-up system, when the nozzle is fully open. T o simplify, we take the load to be negligible, and obtain:

Po - Ps = _Pj_ ( 3 9)

Ro R,n

where P3 is the magnitude of the pressure P3 in the chamber when the nozzle is fully open and R'" is the inherent resistance of the fully open nozzle.

Denoting the relative pressure P3l

!P0 by σ{, we obtain:

σ, = 1

. (40)

1 + R

°

If σ, is small, we can assume:

R'"

R"r

* i = · (41)

The greatest error occurs when Rc3 = R'". Substituting (41) into (38) we ob-tain:

A < 1/1,(1 + σ 1 ) | . (42) Ri

In this way, then from (41) and (42), at a given ratio of resistances R0/Ri we can find the ratio R'

,f/R0 (or vice versa) if we assume a permissible error Δ x .

In pneumatic computers, it is reasonable to choose the order of ογ as 0-01. For given geometrical dimensions of throttles, the errors due to the load

do not exceed 0-25 per cent. All other errors in the square rooting unit are about a half of those of the multiplying-dividing unit.

The total static error may be defined as the sum of all these errors, and amounts to 0-5 per cent.


7. D Y N A M I C A L E Q U A T I O N S OF T H E

S Q U A R E R O O T I N G D E V I C E

It is assumed that there are no static errors, and that the displacement of the diaphragm rod occurs within the working region. The dynamical equa-tions of the square rooting unit can be derived from those of the multiplying-dividing unit (32), with added equations describing the dynamic processes in the follow-up system.

In the absence of internal load, these processes are determined by the

equation

F o~

F 3 - - ^ = C 3 ^ - , (43)

R0 *c3 àt

where C 3 = V3jRT is the total capacity of the load and the follow-up chamber; V3 the volume of the respective chambers; andi? c3 the resistance of the outlet nozzle.

Rc3 = R'" —, (44) ζ

where ζ is the instantaneous travel of the follow-up chamber diaphragm, and zm its maximum travel.

The equilibrium equation of the follow-up chamber diaphragm is

{Py-P3)FU + Nc3 = 0. (45)

Here F'ef is the effective area of the diaphragm and Nc3 the jet reaction force of the nozzle Rc3 :

N c 3 = ^ ^ F c 3 P 3 , (46)

where Fc3 is the effective area of the follow-up nozzle. From the set of equations (32), (43)-(46), and taking into account that for

the square rooting device P 4 = Py and neglecting minor terms, it is easy to obtain the dynamical equation of the unit:

g{T2T3 p ά*Ρ^ + Γΐσ,Τ, p _ τ ρ _ σ,Τ,Τ, _ O j T l d P 3 Ί

k 1

at2 L k 3 2 1 k

k at \

dP x — i - - Ρ 3

2 = - Λ / > 2 . (47) at

The following notation is introduced here: T3 = RQC3 and k = F^jFc3. The equation is essentially non-linear.


8. R E S U L T S OF E X P E R I M E N T A L I N V E S T I G A T I O N S

OF T H E M U L T I P L Y I N G - D I V I D I N G D E V I C E

The device has been tested for hysteresis; for linearity (when working as a multiplier; for consistency of the square law when working as a squaring device); for performing definite mathematical operations; and, by changing the order of the factors, when multiplying.

P+yatm

10

Oß

0-6

04

02

0 0-2 04 0-6 0-8 fyatm 10

F I G . 4.

The results of hysteresis tests are shown in Fig. 4. The tests were con-ducted in the following manner: the dependence of P4 on Pl was measured at P2 = 1 and 0*5 atm, with P3 fixed as 1 atm. Measurements were made as Pt

was increased from 0 to 1 atm and reduced from 1 atm to 0, with the pressure held at the maximum point for 15 min, to allow permanent setting of the diaphragm. Experimental points on the diagram are shown by circles. The pressure was measured by master gauges of the 0-2 class of precision, with the range 0-1 atm. Experiments prove that there is practically no hysteresis. This can be explained by the use of a symmetric arrangement of diaphragms, in which the permanent deflections of diaphragms cancel each other (at static equilibrium). Flat diaphragms, made of diaphragm fabric MP, were used.

Results of linearity tests in multiplying regime are shown in Figs. 5 and 6. They were obtained by taking functions P4(Pi) at fixed values of P2, and ^ 4 ( ^ 2 ) at fixed values of Px. Two identical families of straight lines were

3 Aizerman I

T o improve the dynamic properties of the device, it is necessary to keep as small as possible the time constants Tx and T2, the resistance ratio σί9 and increase as far as possible the ratios of effective areas k.


obtained, with scatter not exceeding half of a division on the master gauge

dial. This represents an accuracy of 0-35 per cent (including gauge errors).

The results of test for accuracy in squaring are shown in Fig. 7. Deviations

of experimental points from a parabola do not exceed half of a gauge divi-

sion, and errors are of the same order as in multiplying. Other mathematical

fy, aim P^,atm

0 0-2 04 0-6 08Pvotm1-0 0 0-2 04 06 08&afmW

F I G . 5. F I G . 6.

operations, covered by equation (1), were also performed with errors of the

same order.

Tests have shown that the response time of the device is 0-5 sec. This time

may be shortened by reducing the volume of the chambers. Adding a capa-

city to the output line increased the time. I f a substantial capacity is added,

F I G . 7.


0 x 0 = 0 -000 , 0-3 χ 0-2 = 0 0 6 0 ,

0 χ 1 = 0 -000 , 0-2 χ 0-3 = 0 0 6 0 ,

1 x O = 0 -000 , Ο2

= 0 0 0 0 ,

1 χ 1 = 1-000, I2

= 1 0 0 0 ,

1 χ 0-5 = 0 -498 , 0 · 72

= 0 -488 ,

0-5 χ 1 = 0 - 5 0 1 , 0 - 52

= 0 -250 .

It can be seen that the operations are performed with three-place accuracy, and the order of factors can be changed.

9. R E S U L T S O F E X P E R I M E N T A L I N V E S T I G A T I O N S

OF T H E S Q U A R E R O O T I N G D E V I C E

The device was tested for hysteresis, for consistency of results, for definite square-rooting operations, and for rapidity of action. Experiments have shown the practical absence of hysteresis and a satisfactory consistency. The accuracy corresponded to class 0-5 (measurements were made by master gauges of class 0-2). The response time (without any attachments) was 0*5 sec.

Figure 8 shows the family of curves, obtained experimentally as Ργ is varied, with pressure P2 as parameter.

Below are given examples of operations, from which the accuracy of the device may be judged.

V O - 0 4 = 0-2 , VO-81 = 0 -905 ,

V O - 0 9 = 0 -295 , y/l-0 = 0 -996 ,

V0-16 = 0 -403 , 7(0-2 χ 0-3) = 0 -246 ,

V0-25 = 0-505 , V « M x O - 9 ) = 0 -303 ,

V0-36 = 0-606 , 7(0-5 χ 0-7) = 0 -498 ,

V O - 4 9 = 0 -705 , V ( 0 - 7 x 0-7) = 0 -697 ,

v/ 0 - 6 4 = 0-806 , V(0-7 χ 0-9) = 0 -795 .

it is necessary to provide a cascade amplifier, in order to obtain a satisfac-tory response rate.

Below are given examples of calculations performed by the device, which illustrate the degree of accuracy and the effect of changing the order of fac-tors Ργ and P2:


All input and output quantities are in atm (gauge). It can be seen that the

device works with three-place accuracy.

The developed multiplying-dividing and square rooting devices can find a

wide field of application in pneumo-automation and pneumatic computers.

P^kg/cn

10,

0-8

0-6

04

02

^y^X\P2=1-0 kg/cm

2

χ

(P2=0-81

À J x ^ x

Cß'0'25

' Ρ2=0 ^χ

β z/s

J x ^ x

Cß'0'25

Ρ2=0-36

-016

02 04 0-6

F I G . 8 .

08 ΡΊ kg/cm2 VO

These devices can be used also for hydro-automation, directly in combined

pneumo-hydraulic systems, and also in a fully hydraulic system, providing

means of obviating silting-up of small orifices are incorporated.

R E F E R E N C E S

1. G . T . B E R E Z O V E T S , V . N . D M I T R I Y E V and E . M . N A D Z H A F O V , Priborostroyeniye, 4, 1957.

2. V . N . D M I T R I Y E V and A . G . S H A S H K O V , Force of the jet action on the baffle in pneu-

matic and hydraulic control units (this book, p. 2 7 2 ) .

3. V . N . D M I T R I Y E V , Collection of Papers on Automation. Academy of Sciences U . S . S . R . ,

1956.

THE APPLICATION OF JET-TUBE ELEMENTS

FOR N O N - L I N E A R TRANSFORMATIONS IN PNEUMATIC SYSTEMS

L . A . Z A L M A N Z O N and A . I . S E M I K O V A

JET RELAYS, controlling the motion of pistons, are well known (Fig . l ,a ) . They

are, however, not used at present in pneumo-automation, because the

"nozzle-baffle" elements, which have a number of advantages, are used in-

stead. It may be questioned whether the complete elimination of jet-tube

elements is justified. In particular, the question arises as to whether they

could be used to solve certain problems of functional transformations in a

simpler way than is possible with nozzle-baffle elements. Such questions

arose in the course of preliminary studies of jet-tube elements. Results of

these studies lead to the conclusion that the elements, which consist of a jet

of air issuing from a nozzle and a Pitot tube moving within a jet (Fig. l ,b) ,

can be used for certain functional transformation.

The characteristics of jet-tube elements were investigated in the Pneumo-

and Hydro-Automation Laboratory of I A T A N U.S.S.R. The present work

is the result of the first stage of these investigations.!

Investigations of jet-tube elements are based on certain conclusions from free turbulent jet theory [2, 3, 4]. From the law of velocity distribution in a free turbulent jet, it is easy to obtain approximate equations for the cal-culation of velocity head at various points of a jet. In relative quantities (ratios of respective distances to nozzle diameter), the equations take the

t Certain schemes of jet devices used to perform functional operations are given in

Ref. 1. See also the article in the present book ( p . 87) by the same authors. + See Appendix I I .

C H A R A C T E R I S T I C S OF T H E J E T - T U B E E L E M E N T S

following form*

(1)

59

F I G . 1. Jet-tube element, and its characteristics.

P N E U M A T I C A N D H Y D R A U L I C C O N T R O L 60

J E T - T U B E E L E M E N T S F O R N O N - L I N E A R T R A N S F O R M A T I O N S 61

for its initial part {hjd < 5). Here Ρ is the velocity head at a given point of the jet ; Pf the pressure (above

ambient) at the entry into the nozzle; d the nozzle diameter; h the distance from the nozzle end to the plane of the tip of the tube; and S and b the dis-tances, as shown on Fig. l,c (cross-sections / and 77).

The following considerations were taken as a starting point; it is known that the geometrical relationships, which determine the profile of the jet, are practically independent of the pressure Pf. As follows from equations (1) and (2), the ratio PjPf also does not vary with changing Pf, and for a given jet cross-section depends only on the function of S/d, because then h = const, and also b = const, since the geometrical profile of the jet does not vary. A t a given d, the ratio P/Pf depends only on S. Characteristics PjPf = f(S), calculated from (1) and (2), have the shape shown on Fig. l,d. The curve 7, relating to the initial part of the jet, has a flat upper part, which corresponds to the core of constant velocity (outlined by broken lines on Fig. l ,c) .

The ratio PjPf is independent of Pf9 and for a given jet cross-section is only a function of coordinate S; in other words, the pressure Ρ at any given point is proportional to Pf. Figure l,e shows the family of curves Ρ = f(S). If we select parts of these curves, with limits determined by the left or right sloping part, and the adjoining horizontal "plateau", we obtain families of curves shown on Fig. l,f.

It is essential to note that the horizontal "plateau" may be extended to any length by using the slot nozzles, as illustrated by Fig. l,g. Left and right branches of the slot nozzle characteristics are shown on Fig. l,h and i.

The above considerations apply to free turbulent jets. When calculating the characteristics of these jets, it is assumed that the dimensions of the tube are small compared with the dimensions determining the profile of the jet. With the jet-tube elements intended for use in devices of pneumatic control this is not always possible, and actual characteristics of jet-tube elements may differ from those of free turbulent jets. The difference may be caused by flow disturbances introduced by the tube, and by pressure variations across the tube opening, which covers a relatively large part of the jet cross-section.

T o investigate the possibility of obtaining actual characteristics, similar to those shown on Fig. l,h and i, various nozzles were tested. Figure 2,a and b shows experimental characteristics Ρ = f(S), obtained at several values of supply pressure Pf for a round nozzle with d = 0*8 mm and h = 1*7 mm and for a slot nozzle with the height of the slot 0-8 mm and h = 0-3 mm. The tubes in both cases had an inner diameter of 0-27 mm and an outer of

(2)

for the main part (hjd > 5) of the axially-symmetrical jet, and


0-5 mm, and a length of 7 mm. Results of these tests lead to the conclusion that the actual characteristics Ρ = f(S) resemble those shown on Fig. l,h and i. The level of the plateau can be altered by regulating the adjusting pressure Pf. The distance on the S axis corresponding to the sloping parts of the curves is practically independent of the pressure Pf. For the curves shown on Fig.2,b this length (öS) can be taken as 0-4 mm.

Pi aim fjatm

F I G . 2. Experimental characteristics of jet-tube elements.

I N C R E A S E D P O S S I B I L I T I E S OF F U N C T I O N A L

T R A N S F O R M A T I O N S B Y T H E S I M U L T A N E O U S U S E

OF S E V E R A L J E T - T U B E E L E M E N T S

Let us take an arbitrary function Ρ = f(S), for example the curve 7 on Fig. 3,a (equally, any other function might be taken), and consider what must be done to reproduce this function by the use of jet-tube elements.

The curve can be replaced by the broken line approximation consisting of a number of straight lines, chosen so that for each of them the corresponding length on the S axis is OS (Fig.3,b). Then the elementary characteristics can be drawn, as on Fig.3,b, c and d, with the slopes equal to those of the initial broken line. Adding these elementary characteristics produces a broken line which differs from the initial one only in being offset in the direction of the P-'àxis—that is, instead of the desired broken line 2 we may obtain line 3 or 4 (Fig.3,e).


|AAAA)

Poutp-

F I G . 3. Approximation to arbitrary non-linear functions by several jet-tube elements.

Thus, the operations needed to obtain a desired function Ρ = f(S) are as follows:

(1) to obtain elementary characteristics Ρ = f(S), as shown on Fig.3,b,c, and d;

(2) to co-ordinate these elementary characteristics along the .S-axis, as shown on the same Fig. 3;

(3) to add up the elementary characteristics;

(4) to displace, if necessary, the resulting characteristics by a constant amount

in the direction of the P-axis.

The first operation has been discussed in the first part of the article. The second operation can be effected by the relative location of nozzles, as

3 a Aizerman I


illustrated by Fig. 3,f; according to this scheme, the nozzles remain station-ary, and the set of tubes moves as a whole along the coordinate .S.| The problems of the third operation—adding the pressures—are discussed on p. 65. The fourth operation—offsetting by a constant amount the whole characteristic—may be accomplished by the device shown schematically in Fig.3,g (see also p. 90, Fig. 4,a and b). Pressure Poutt>, which is estab-lished as the output from this device, always differs from the input pressure Pk by a constant determined by the spring setting.

F I G . 4. External view of test rig for jet-tube elements. J—tubes, 2—screw, moving

the set of tubes horizontally (along the coordinate 5 ) , 4—nozzles, 5—slide for

shifting a nozzle relative to other nozzles, 6—screws for the adjustment of relative

position of nozzles, 7—screw for setting the spacing between nozzles and tubes

(coordinate h).

An experimental rig has been built to test the possibility of combining the elementary characteristics of the jet-tube elements which work simultaneous-ly, and to investigate the problems of adding these characteristics. The ex-ternal view of this test rig is shown in Fig. 4. It incorporates four jet-tube elements; the location of the nozzles along the S-axis could be changed in the way illustrated by Fig.3,f. The tubes were mounted on a common

f The curve shown on Fig. 3,a is approximated by three elementary characteristics;

Fig. 3,f shows the set of four elements, and this number has been used in all further exam-

ples in this article. Depending on the shape of a given curve, and the required accuracy of

approximation, various numbers of elements may be needed—from one or two to ten

or more.


A D D I T I O N O F P R E S S U R E S

Addition of the pressures in the tubes of a set of jet-tube elements can be done by a mechanical summator (Fig. 5,a), the pressure from each tube being

ι ι ι !

F I G . 5. Diagrams illustrating possibilities for summators,

and their characteristics.

slide, the horizontal motion of which traversed the tubes across the jets

(along the coordinate 5 ) . In addition to this, provisions were made for

varying the distance from nozzles to tubes (coordinate A), adjustment of

tubes in the vertical direction, and angular position changes for one of the

nozzles. The characteristics of the set of jet-tube elements, obtained with the

aid of this test rig, are given in the next section.


connected to one of the sections; but this solution is considered unaccept-able, a mechanical summator being too cumbersome and requiring an auxiliary transformer to scale down the added pressures. A t the outset of the present work it had been found that the usual collector chamber (on Fig. 3,f—the chamber connected to the tubes) under certain conditions could work as a summator.f Let us consider a chamber into which air is fed through m inlet throttles, and flows out through η outlet throttles. Depending on the relationship between external pressures and pressure in the chamber, the values of m and η may change (as in some throttles the direction of flow may be reversed), but the sum total m + η remains constant. Let us accept also, that under steady-state conditions, for an z-th inlet throttle the mass flow of incoming air is:

Gt = kt(Pt - Pk)

and for ay-th outlet throttle, the mass flow of outgoing air:

Gj = kj(Pk - Pj),

where Gt and Gj are the weight discharges per second; Pt and Pj the external pressures; Pk the pressure in the chamber; and kt and k3 the slope co-efficients of the discharge characteristics for respective throttles.

As the flow rates of incoming and outgoing air flows may be equated we have:

m η

lki(Pi - Λ) = lkj(Pk-Pj), i=1 j =1

from which we obtain

«ι η

Σ *,Λ + Σ kjPj

A = - ^ ψ · (3)

1=1 j =1

The magnitude of the denominator in (3) is independent of m and η taken separately, and is determined only by the value of m + η = const, and by values of the coefficients k for each throttle. Therefore, even when coefficients ki and kj are unequal, the dependence of Pk on Pt and P3 remains linear. When all the coefficients k{ and kj are equal, the equation (3) is reduced to :

m η

Σ Λ· + Σ ρ j

Pk=l=l i^—. (4) m + η

f The idea of using a pneumatic chamber as a summator has been independently ar-

rived at, in the Pneumo- and Hydro-Automation Laboratory I A T A N U . S . S . R . , by the

authors and by Yu.I . Iv l ichev.


It follows from these equations that in order to use a collector chamber as a summator it is sufficient to have linear relationship between mass flow and pressure for all restrictors (in this particular case for all tubes), and equal co-efficients in these relationships. In this case it is immaterial that in operation the flow in some tubes may be reversed, from inlet to outlet, or vice versa; in all cases the pressure in the chamber is equal to the sum of pressures before all the inlet throttles and behind all the outlet throttles, divided by their total number.

It must be noted that for independent adjustment of separate jet-tube elements, in order to obtain a desired function Ρ = f(S), it is only necessary to ensure the condition of linearity, and while the condition of equality of coefficients is desirable, it is not essential. For example, let us consider a characteristic 7, shown on Fig.5,b, obtained by the action of several nozzles (arranged as on Fig.3,f) in the range S = S1 (one nozzle is not operated). Then if a constant (over the whole range S) pressure is now applied by this nozzle (lines 2 and 3 on Fig. 5,b) it only displaces the characteristics of other elements (4 and 5 on Fig. 5,b) by a constant amount, without altering their slope, if all elements have linear characteristics. I f the condition of linearity is not realized, the changes of "plateau" level of characteristic 2 or 5 would alter the angles of characteristic 4 or 5 (Fig. 5,c), which makes the adjust-ment very difficult.

It may be added that none of these conditions (neither linearity nor equal-ity of coefficients) would be needed if the desired characteristics are limited to those having as dP/dS=f(s) SL positive, non-diminishing function (Fig.5,d). Then the additional action of any subsequent nozzle does not influence the characteristic Ρ = f(S) obtained at a smaller value of S, and no special summator would be required.

Initially, the manufacture of suitable restrictors with linear characteristics presented certain difficulties. A method of evaluating the influence of non-linearity of throttle characteristics on the summator performance has been developed. Also, the factors causing deviation of discharge characteristics of capillary tubes from linearity were investigated. Brief information on these tests is given below; the conclusions from the analysis of capillary tube characteristics can be found in Appendix I .

The following method has been adopted to evaluate the performance of a pneumatic chamber as a summator. With restrictions situated as shown in Fig.5,e, the characteristic Pk = f(Pi) has been taken at P2 = P3 = 0. The same process has then been repeated at certain constant values of P2 and P3. A summator producing practically linear and parallel characteristics would be considered satisfactory. The tests have demonstrated that even with the patently non-linear characteristics of throttles, if P2 = P3 = 0, i.e. if the direction of flow through the throttles does not change, linear Pk = / ( P i ) characteristics were sometimes obtained, due to the mutual compensation of non-linearities. But this linearity disappeared, when constant pressures P2


and P3 were chosen so that the direction of flow through some restrictions

changed during the process of taking the characteristics.

Initial experiments with the summator, using the Pitot tubes of the jet-

tube elements as direct restrictions, gave unsatisfactory results because it

Ρ, aim χ

TO

0-8

06

04

1

02 0<t 06 0-8 S,mm

ι 8 h î 1

m

0 02 04 0-6 0-8 WPtfltm

b

0-5 1-5 δ) mm 20

Pklatm

04

0-3

02

0-1

lPi=0-5 1^-04

-—-c

-—-t

1-3

12

-0-1

0 0-2 04 0-6 0-8 P2]atm

c

0 0-2 04 0-6 08 10 P1tatm

9

F I G . 6. Experimental characteristics of a pneumatic summator.


was difficult to use tubes with an inner diameter d less than 0-27 mm and length / less than 7 mm ; and at these magnitudes the characteristics are already non-linear. Figure 6,a shows the characteristics Ρ = f(S), obtained with the test rig shown in Fig. 4, for tubes with d = 0-27 mm and 1=7 mm. Curve 1 has been taken with pressures fed to three nozzles and zero pressure to the fourth one. Curve 2 refers to the same pressures at three nozzles and a constant pressure to the fourth one, working within the hori-zontal part of the characteristic. Figure 6,b shows characteristics taken as described above (Fig. 5,e), with various values of P2, and with P3 = 0. Here the tubes have d = 0-27 mm and / = 15 mm. In Fig.6,c these charac-teristics are redrawn with coordinates Pk = f(P2), and every curve corre-sponds to a certain value of Pt. The essential non-linearity of tube charac-teristics was verified also by direct measurement of air discharge through each tube at various pressure differences.

The conclusions deduced from the analysis of factors which affect the linearity of discharge characteristics (Appendix I ) may serve as a basis for further investigations as to the possibilities of using Pitot tubes also as throttles of a summator. At the present stage, however, a practical solution has been found, namely the installation of special annular restrictors after the tubes at the entry into the summator. The advantage of an annular re-strictor (Fig.6,e) lies in retaining laminar flow at much higher pressure dif-ferences. Consider an annulus with the same length as a cylindrical capillary, the latter having a critical pressure difference APcritl = (P0 — Pi) corre-sponding to a flow Qcrit ι, at a critical Reynolds number. Then if the radial clearance of the clearance throttle is chosen so as to obtain the same flow at the same ΔΡ = APcrit,, the transition from laminar to turbulent flow will occur at the pressure difference APcrit2, which can be determined approxi-mately as

where D and d are the diameters of the restrictions, shown in Figs. 6,e and 6,d, and where the ratio can be made sufficiently large.

Experiments were carried out with annular restrictors 7 mm long, shaped as a conical frustrum with a semi-apex angle of 2° and least diameter of 1 mm. Throttles were adjusted for the mass flow G = 2 χ 10~

3 g/sec at Ρ

= 1 kg/cm2 ; this corresponds to the working setting of restrictions in con-

trollers of A U S systems. The pressure in the chambers between tubes (which were of the same dimensions as above) and throttles was practically in-dependent of the pressure in the summator chamber. The tests on the sum-mator according to the method described above produced characteristics shown on Fig. 6,g, which satisfied the requirements. Preliminary tests of the summator, connected to the set of jet-tube elements, also gave satisfactory results (see Fig. 6,f ; compare with 6,a). Further tests confirmed the possibility


of using the collector chamber as a summator for performing prescribed functional transformations. For the operation of adding characteristics Ρ = f(S), the shape of nozzles, their distance from tubes, etc. are immaterial. In this respect the characteristics shown on Fig. 7 are significant. They were

P^fm

0-6

k /1

2

Im ja » )

\ Λ if

\ lit ~

ft \ Iff

ψ in

Iff 3

Jk

• - I — Λ

> 0 —

I ^ ^ V

0-5

0+

03

0-2

0-1

Ό 05 10 15 20 25 30S,mm 35

F I G . 7. Resultant characteristics of a set of jet-tube elements and a summator.

obtained with two round and two elongated (slot type) nozzles, with hi = 0-8 mm, hjj = 0-04mm, h u l — 0,hIV = 0-3 mm. On Fig. 7 the line / is the broken straight line approximation to the desired characteristic Ρ = f(S); 2 is the characteristic, from the summator, connected to four jet-tube elements simultaneously ; 3, 4, 5, 6 are characteristics obtained by connecting to the summator one element at a time; 7 is the characteristic obtained by adding the ordinates of 5, 4, 5 and 6.


P O S S I B L E A P P L I C A T I O N S O F J E T - T U B E E L E M E N T S

I N P N E U M A T I C C O N T R O L

The conclusions from the previous sections lead to the possibility of making a functional transformer with the jet-tube elements, to transform a linear motion (or an input pressure proportional to it) into an output pressure according to any desired law.

The need for such devices arises with the development of self-adjusting and complex systems of pneumatic control.t

From these considerations, a layout of a functional transformer using jet-tube elements is shown in Fig. 8. Air under the input pressure 7^ is connected to the bellows 7, which is followed up by bellows and slide 2, which carries tubes 3 and clearance throttles 4. The feed pressure Pf supplying the nozzles is reduced by the needle valves 5. The nozzles can be moved by screws 6 (only one needle 5 and screw 6 are shown on the layout). Pressure from the sum-mator chamber 7 is ducted to a chamber at the right hand side of diaphragm

F I G . 8. Layout of a non-linear pneumatic transformer with jet-tube elements.

8. The output pressure differs from the pressure in this chamber by a constant value, determined by the setting of screw 9. The functional transformations discussed above depend on the transverse position (S) of the tube in a cross-section of the jet. It also appears to be possible to use for the same purpose the longitudinal position (h) of the tube in an axial (or parallel to axis) sec-tion of the jet. Figure 9,a shows the characteristic Ρ = f(h), for the nozzle with d = 0-8 mm, and tube with an inner diameter 0-27 mm and outer diameter 0-5 mm, the tube being moved along the axis of the jet, with the constant pressure Ρ = 1 atm at the entry to the nozzle. It is noteworthy that the working range of the characteristic Ρ = f(h) corresponds to a variation of h

-ο

outp.

f See, for example, the article by Τ. K.Berends and A . A . Tal ' in this book ( p . 20).


up to a few centimetres. There is a portion of the curve which approximates

to a straight line, and corresponds to several millimetres change of h. The

shape of such characteristics, and its extension along the h axis, may vary

substantially, depending on the nozzle diameter, on the relative dimensions

Ptatm

/ •Or

075

0-5

025

30 htmm 0

/ /IS*

4-*

1-2 S.mm

15 S,mm

0-8 S,mrn 12

F I G . 9. Characteristics of jet-tube elements.

of nozzle and tube, and on the transverse position if the tube is moved parallel to the jet axis. Figure 9,b shows the family of Ρ = f(S) curves, taken at various values of h.

On the basis of the characteristic shown in Fig. 9,a it is possible to devise follow-up elements with comparatively large linear displacements. For ex-ample, feedback in a functional transformer is simpler than feedback in systems using nozzle-baffle elements. A transformer with this kind of feed-


F I G . 10. Layout of a non-linear pneumatic transformer, using jet-tube elements in

the transforming unit and in the feedback system of the moving block.

example, it is often necessary to obtain the characteristic Ρ = f(S)9 consist-ing of one or two sections, each of nearly linear form. Such characteristics can be easily obtained, using one or two jet-tube sets. Also, other approaches may be used (see, for example, the characteristics shown on Fig.9,c and d, obtained by adding in a second jet to the steep portion of the Ρ = f(S) char-acteristic of a primary jet).

A P P E N D I X I

Notes on the effect of various factors on the non-linearity offlow characteristics in capillary

tubes

1. The main factor causing deviation of the flow-pressure characteristic from linearity

is the transition from laminar to turbulent flow (point at ΔΡ = APCTit on Fig. 11 , a ) .

back is shown in Fig. 10. The nozzle / is moved together with the main set

of tubes, and the pressure in the feedback chamber 2 is determined by the

distance h between the nozzle / and tube 3. By the choice of a spring 4, and

by the use of diaphragms with suitable effective areas, it is possible to work

within the virtually linear part of the Ρ = / ( A ) characteristics.

When designing devices for industrial pneumatic control, or controllers for

other branches of engineering, such as jet and rocket motors, it is often

necessary to perform comparatively simple functional transformations. For


Even with laminar flow, however, the characteristic G = f(AP) (where G is the mass

flow and ΔΡ the pressure difference) is not always linear. This is caused by the additional

resistance at the entry section, where the laminar flow develops, and by compressibility. It

has been found that these influences have opposite effects. The quantitative estimation of

these factors is given below.

0 10 20 30 0 10 20 30

{^/Re)x103

c

F I G . 11. Characteristics of restrictions.

2. The value of the critical pressure difference APcrit, at which laminar flow changes to

turbulent flow, depends fundamentally on the air density. A t a given temperature, this varies

inversely as the mean absolute pressure

ΔΡ


where P* is the absolute pressure at the outlet from the capillary. This follows from the

equation : / u

2

APctlt= 32ReCTit— · α

ά ρ

where ReCTit = 2300; μ is the viscosity; ρ the mean air density which, at a given tempera-

ture, is proportional to the mean pressure P*. This equation is obtained from the usual

formula for a laminar flow resistance. Figure l l , b shows the curves APCTit = f(d) for a

capillary tube 15 mm long, at P * = 1 and 2 atm abs.

3. The development of laminary flow in the entry length causes a deviation of the G

= f(AP) characteristic from linear which corresponds to established laminar flow. This is

shown in Fig. 11,a, where curve 3 takes this factor into account. HereAPH is the value of

A P which the development length extends the whole length of the capillary (at AP

< APH the development length is only a fraction of the total length). The quantitative

evaluation of this factor is given in Appendix I I , para. 2.

4. For isothermal laminar flow of air through capillaries, it is possible to use the Poi-

seuille formula, if the change of velocity along the capillary is slight, t The mean air density

ρ is then determined by the mean pressure Pm = (P0 + Pi)/2 (notation as in Fig. 6,d). The

relationship between G and Ρ can be expressed as G = CAP -f- C*AP2, where C and C*

are constants, for a given pressure Ρ at the outlet of the capillary, determined as follows:

c = nd+Px c+ = nd*

\2fylRT 9

256μ//?Γ'

For the proof, see Appendix I I , para. 3.

In the equation G = CAP + C* AP2 the member CAP corresponds to the relation-

ship between G and AP according to the Poiseuille formula, with the density referred to

the pressure at the outlet from the capillary. The term C * AP2 represents the deviation from

linearity as illustrated by curve 4 in Fig. 11,a. The ratio of C* AP2 to CAP, as a percent-

age, is 50 APIPt.

Experiments were carried out in order to determine the influence of density variations

on the deviation of the curve G = f(AP) from linear. T o minimize the effect of resistance

on the entry length, a very long capillary with a small inner diameter was used ( / = 722 mm,

d = 0*23 mm). The inlet pressure was varied over a wide range, from 0 to 3 atm. The char-

acteristic which was obtained, G = f A(P), is shown in Fig. l l , d .

5. T o obtain the closest approximation of the G = f(AP) characteristic to a straight line,

the dimensions of the capillary should be chosen so as to compensate, within the working

range of pressures, the influence of increased resistance in the entry length of the tube (the

length of which depends on AP) by the effects of air density variation. If the capillaries are

either too long or too short, the results may be unsatisfactory.

A P P E N D I X I I

1. Proof of formulae (1 ) and (2) . Theoretical methods can be applied to a uniform

velocity field at the outlet cross-section of the nozzle. Practically, however, the velocity

distribution in the initial section of the jet is not uniform, and the formulae given in Refs.2

and 3 use a certain coefficient a, which depends on the ratio of maximum velocity to the

mean velocity at the outlet section of the nozzle. In the "core" of constant velocity at the

t If the change of velocity is considerable, the curve G =f(AP) deviates from a straight

line, due to inertial effects, in the same general direction as the deviation caused by the

increased resistance in the entry length.


0-145 + ah Id

Assuming a = 0-067, which corresponds to a transition from the initial conditions of the

jet to the main part of the jet at h Id = 5, then for the main part of the jet:

ve 0-3 -t 0'\4h/d

and for the initial portion:

For any point, either in the main jet, or at the boundary of the initial part, the velocity ν

can be determined from the equation:

^ X 3 / 2 - , 2

where the meaning of b and S is defined in Fig. l,c.

Using these equations for v0/ve and vjV0, and assuming that P\Pf = (vjve)2, we obtain

formulae (1) and (2).

2. The analysis of the influence of resistance change in the entry length on the deviation of

the characteristic G = f(àP) from a straight line. The entry length in which the laminar

flow develops is taken as lH = 0O2$15dRe ^ 0-029dRe. For this initial length ΔΡ

= (A/Re)(lld)(v2Ql2) [5 and 6] where the coefficient A is determined by the relationship

A = f [(lid)/Re] obtained by Frenkel [5 and 6] as curve 1 on Fig. 1 l,c. Approximating this

curve by a hyperbola

Re A = 7 0 + 0-15 (curve 2 on Fig. 1 l ,c)

Ijd

we obtain the equation

ΔΡ = d G + C 2 C2,

where

/ μ C x = 45-5 q — , C 2 = 0-122 d gd

3g' g

2od

g is the acceleration due to gravity, and all other symbols are as given in the previous

formulae.

Numerical calculations prove that for the capillary tubes the second rrember C2G2 is

negligible, thus

Re d 2

At Δ Ρ = ΔΡΗ, when the development part extends to the whole length of the tube

70 / v2o

ΔΡ = - . Re d 2

But when Δ Ρ - > 0, and therefore ν - > 0 and Re - > 0, the length of this development length

(/„ = 0029dRe) also approaches zero. Therefore, at small values of ΔΡ, the Poiseuille for-

beginning of the jet, the velocity at any point is equal to the efflux velocity. In the main

part of the jet, the ratio of the velocity at the axis of the jet v0 to the efflux velocity ve is:

vn 0-48


mula is applicable practically to the whole length of the tube, that is:

A p 64 / v*q_

" Re' d' 2

These considerations lead to the stated conclusion as to the effects of laminar flow develop-

ment in a capillary on the deviation of G = f(AP) from linearity (curve 3 on Fig. 11 ,a) .

3. Analysis of the effects of air density on the characteristics of G = f(AP). F o r a length

dx of a capillary (Fig . 6,d) at the distance χ from the outlet, the air density ρ may be as-

sumed constant, and so, according to the Poiseuille formula:

Re 2d

Taking into account that

Ρ n vod „ nd2

Re = ; G = ν og, gRT μ 4

the Poiseuille formula takes the form:

\2SGRTu ,

PdP = T-t-dx. nd*

Assuming that the conditions are isothermal (T = const), the integration with limits 0-/,

for Ρ changing from PY to P0, gives:

Pi - P\ _ 128G/?7>//

2 nd*

A s

P è ~

Pl = (Po - Pi)

P° t

P l = APPm = Z I P + —

therefore • ( " + ί γ ) ·

G = nd*Pm Ap = nd\mg

\2$,ulRT 128/1/

(Pm and om are the mean pressure and mean air density); or

nd*AP

~ \2SfiiRT

which can be written as

G = AP + U d

" AP\ nSfilRT 256// /ΛΓ

The ratio of the second term of the right hand side of this equation to the first term is

(SOAP/Pi) per cent.

4. Derivation of equation (5 ) . For a cylindrical capillary (Fig. 6,d) the volumetric flow

rate of air is

nd*AP

128/// '

where AP = P0 — PY.

A s the critical Reynolds number = Qvd/μ = 2300, and QCTlt = (nd2f4)vcrit = 1800(μ/ρ) d,

the value of AP at which laminar flow in a cylindrical capillary becomes turbulent


δ =

Let us take for an annular restrictor

(«Ο

and consider that

* * c r l t = Μ Η = 1 1 00

\ ίι / c r i t

Ô c r i t 2 = v c t l t 2n D ö = nDô.

2ρό

Substituting this expression for Q c r it 2 and the value of δ obtained above into the expres-

sion

1 D _ 1 2 f t / g c r i t2 J

^c r i t 2

- nDô*

we find that

^ « 1 1 2 = 7-01 x l O4

^ - ^ 4 ' 0 dô d

Hence

APcrit , d d

The fact that in practice clearance throttles usually have a characteristic G = f(AP) which

is nearly linear at fairly high A Ρ is confirmed by experimental results quoted in Ref. 7.

R E F E R E N C E S

1. V . F E R N E R , Neue pneumatische bzw. hydraulische Elemente in der M e ß - und Re-

gelungstechnik. Die Technik, 6, 1954 .

2. G . N . A B R A M O V I C H , Turbulent Free Jets of Liquids and Gases (Turbulentnye svobodnye

strui shidkostei i gazov). Energoizdat, 1948 .

3. G . N . A B R A M O V I C H , Applied Gas Dynamics (Prikladnaia gazovaia dinamika). Gittl,

1953 .

4. G . S C H L I C H T I N G , Boundary Layer Theory (Teoriya pogranichnogo sloia). Izd-vo inostr.

lit., 1956 .

5. N . Z . F R E N K E L , Dissertation ( O nekotorykh elementakh teoreticheskogo rascheta kar-

byuratora) 1938 .

6. N . Z . F R E N K E L , Hydraulics (Gidravl ika) . Energoizdat, 1956 .

7. G . T . B E R E Z O V E T S , V . N . D M I T R I Y E V and E . M . N A D Z H A F O V , Priborostroyeniye, 4 , 1957.

T o distinguish between the symbols for a cylindrical capillary and for an annular re-

strictor,the former will be given the subscript 1, and the latter 2 (i.e. the ReCTit and APcrit

above will become Recxit χ and APCTit j while corresponding quantities for an annular re-

strictor will be denoted as ReCTit2 and APcrit2.)

F o r an annular restrictor, the volume flow Q = (πΌΙ12μΙ)δΑΡ (Fig. 6 ,e) .

If we take for an annular restrictor values of g and A Ρ corresponding to ReCTii for a

cylindrical capillary, and substitute into the above expression g = gcr i t 1 = 1 8 0 0 (μ/ρ) d,

and AP = APcrit ! = 7-37 χ 1 04 (μ

21ρ) ( / / < / ) ( 1 / r f

2) , then we find

NEW COMPACT PNEUMATIC INSTRUMENTS

FOR THE AUTOMATIC CONTROL

A N D REGULATION DEVELOPED IN

"NIITEPLOPRIBOR"!

I . F . K O Z L O V

T H E PRESENT stage of automation is characterized by the transition from the

solution of problems of regulating separate parameters to the complex

automation of the whole technological process, which needs must be ac-

companied by the centralization of the control and direction of the process.

If large-dimensioned instruments are used in these circumstances, the in-

strument panels grow to such dimensions that it is difficult, sometimes even

impossible, to observe how the process is running. In order to make the panels

easily observable, it is necessary to minimize areas occupied by instruments

on panels, and it is desirable to have a graphic scheme of the process shown

on the panel. According to foreign practice, smaller instruments made it

possible to reduce the lengths of panels to a half or a third of those used

before.

When designing such smaller instruments, the accuracy of measurements

and visibility of dials should not be sacrificed.

The tendency to increase the speed of technological processes puts up new

requirements for the means of automation:

(1) The instruments and control systems must have adequately rapid

response;

(2) It is necessary to aim at a "block" system, which enables the realization of complicated control schemes by simple means, and to replace promptly a "block" which has developed a fault.

"NIlTeplopribor" began in 1955, and completed towards the end of 1956 the development of a set of controllers and secondary instruments with small overall dimensions (referred to briefly as "compact instruments"). This work has been carried out, taking into account the above requirements as well as the experience gathered by the construction and more than two years' service of larger pneumatic control instruments, the suggestions of

t The Scientific Research Institute for Thermal Instruments.

79


users, and the experience of well-known instrument-making firms in U.S.A. and Great Britain.

When choosing the design of compact instruments, consideration has been given to the advantages of the aggregate principle, which enables the majority of control schemes to be built up using a limited range of standardized instruments, and also the advantages of basic designs. With this in view, a certain degree of flexibility has been provided for the development of control and automation schemes, based on compact instruments. This has been

F I G . 1. Controller unit 4 R B - 3 2 A .

shown, for example, in the combination of the recording instrument, desired-value indicator, indicator of the position of regulating mechanism, and re-mote control panel—all in one secondary unit; this resulted in a compact system, and simplified its installation and use.

In addition, the provisions are made for the installation of the control unit directly at the secondary unit. This reduces by half the number of pneumatic lines.

Al l these compact instruments are based on the principle of force com-pensation, which ensures a simple design, good sensitivity, and comparatively low inertia of instruments.

The Controller Unit. The controller unit 4RB-32A (Fig. 1) is the basic in-strument of this series. This is an integral action controller of the "block" type, with built-in "deflecting relay".

It has quick-release couplings, which can connect it either to the secon-dary recording or indicating instrument (Fig. 2), or to a special "nest"

I N S T R U M E N T S F O R A U T O M A T I C C O N T R O L A N D R E G U L A T I O N 81

F I G . 2. Controller unit 4 R B - 3 2 A attached to a secondary unit.

(Fig. 3), which serves also as a bracket for the installation of the controller directly at the source of measurements, or at the regulating mechanism. The proportional band may be adjusted from 10 to 250 per cent, and the integral action time from 3 sec to 100 min.

The Unit of Derivative Action ("anticipating unit"). T o obtain the control action in response to the speed of deviation of a regulated parameter from a desired value, the system is provided with a derivative action unit BP-28B

F I G . 3. Controller unit 4 R B - 3 2 A attached to a "nest".


F I G . 4. Derivative action unit B P - 2 8 B .

(Fig. 4). The derivative action time can be altered from 3 sec to 10 min by the use of an adjuster knob.

Secondary Instruments. The supervision of the process can be assisted by a secondary recording instrument 3RL-29B (Fig. 5), or a secondary scale instrument 2MP-30B (Fig. 6). A remote control panel is used to transmit desired-value setting and to effect the manual control of the process. These panels, built into secondary instruments, enable a smooth transition from

F I G . 5. Secondary recording instrument 3 R L - 2 9 B with a remote control panel and

a controller.


F I G . 6. Secondary scale instrument 2 M P - 3 0 B with a remote control panel.

automatic to manual control, and vice versa. Quick-release couplings of

pneumatic and electric lines facilitate the replacement of instruments with-

out serious interruptions of the control process.

The recording instrument traces a graph on a ribbon with usable width

100 mm. The graph is recorded in rectangular coordinates, which is con-

venient for the analysis and evaluation of the graph. The ribbon speed can

be 20, 40, 60 and 2000 mm/hr.

The scale instruments have rotating scales and stationary pointers. Re-

cording and scale instruments are made also without remote control panels

(Figs. 7 and 8).

F I G . 7. Secondary recording

instrument 1 R L - 2 9 A .

F I G . 8. Secondary scale

instrument 1 M P - 3 0 A .


F I G . 9. Secondary instrument 1SP-31A, indicating discharge or delivery, and

incorporating a summator.

T o determine discharge or delivery of a physical quantity during a certain time interval, there is a summator of the type 1 SP—31 A , with the counter and the scale showing instantaneous discharge or delivery at a given moment (Fig. 9).

Programme Setters. Regulation according to a programme is attained by the use of programme setting attachments to secondary recording or in-dicating "compact" instruments. They work as a part of the controller, for the purpose of automatic remote variation of a desired value of a quantity to be regulated.

For the setting of programmes as functions of time there has been de-veloped a setter PD-35A (Fig. 10), and for the programme setting to a para-meter a setter PD-36A (Fig. 11).

F I G . 10. Time programme

setter P D - 3 5 A .

F I G . 11. Parameter programme

setter P D - 3 6 A .


F I G . 13. Ratio relay R S - 3 3 A .

F I G . 12 (left). Adding relay (summator) B S - 3 4 A .

Computing Devices. The series of small pneumatic instruments includes also two simple computing devices: summator (adding relay) BS-34A (Fig. 12), which performs algebraic addition of up to three pneumatic signals (two with sign of + and one with — ) , and ratio relay RS-33A (Fig. 13), performing multiplication of pneumatic signals by a constant coefficient, which may be set from 4 to 02 .

Controller of the Two-step "On-ojf" Type. This is represented by the relay PS-37A (Fig. 14), which may be used as a signalling device.

F I G . 14. Controller of the two-step ("on-off") type.


The secondary instruments, controller, programme setter to a parameter,

ratio relay, and signal relay—all are designed for the same range of output

pressures, from 0-2 to 1 atm, and therefore may be combined with any primary

instrument (transducer) having output pressure within the same range. A t

the output of the controller and the derivative action unit, the pressure is

varied from 0 to 1 atm, which means that standard regulating (final) units are

suitable for systems using these instruments. Air supply pressure of all these

instruments is 1-4 atm. Variations of this pressure up to + 10 per cent do not

affect their work. Air consumption is moderate—from 1 to 2*5 free l./min per

one instrument.

Al l secondary instruments, with the exception of scale instrument 1MP-

30A, and all programme setters, have the unified front elevation dimensions

160 χ 190 mm. This leads to improved appearance of panels, as all openings

for instruments are of the same size. The secondary scale instrument 1 M P -

30A has the front elevation dimensions 80 χ 170 mm. The design of its casing

enables the assembling of clusters of these instruments for the control of any

number of parameters. The scales of instruments may be arranged horizon-

tally or vertically.

The developed compact pneumatic instruments are universal in their ap-

plications, performing the control and automatic regulation of any para-

meters of thermal technology, such as pressure, pressure drop, flow or dis-

charge, level, temperature, etc. In every case, only transducers and scales of

secondary instruments must be adapted to a particular parameter.

INVESTIGATIONS OF PNEUMATIC JET-TUBE ELEMENTS

L . A . Z A L M A N Z O N and A . I . S E M I K O V A

T H E ELEMENTS of pneumatic control devices at present are generally of the

"nozzle-baffle" type [1]. There are, however, many problems that might be

more effectively solved by the use of jet devices, referred to further as jet-tube

elements. Their action depends on the pressure distribution in a jet of air.

There are two types of jet-tube device. In the first one a Pitot tube is

moved across a jet issuing from another nozzle (Fig. l ,a). The second type

has no moving parts and the change of pressure in the Pitot tube is produced

by deflecting the jet by a second jet issuing from another nozzle (Fig. l ,b) .

lux (a) (b)

F I G . 1.

Compared with nozzle-baffle units, jet-tube elements have the following advantages: absence of precision-fitted details, no need for hermetically tight connexions, better reliability over widely varying temperatures. The dis-advantage is a comparatively heavy air consumption. This is unimportant when the controlled object itself contains a powerful source of air pressure (gas turbines, turbo-jet and ram-jet engines, air/gas devices in the petroleum in-dustry, etc.). Pneumatic instruments for general industrial purposes may achieve a reduction in air consumption by using lower pressures. The first type of jet-tube device can be used with advantage when the variation of pressure is combined with mechanical movement. The elements of the second class are purely aerodynamic in action, and are suitable for controlling one pressure as a function of other pressures. The present paper discusses ele-

4 Aizerman I

87


merits of the first type. The first section gives some examples of their pos-

sible use in pneumatic control systems. The following sections deal with the

theory of these elements. The nature of the pressure changes in the Pitot tube,

as it traverses a jet, are investigated. Specific problems arise here for nozzles

with small dimensions due to the possible influence of the tube size on the

pressure characteristics. Section 2 establishes the extent to which the results

obtained with free jets are applicable for calculations of the characteristics of

jet-tube elements. Section 3 gives the analysis of functional transformations,

as performed by these elements. Section 4 investigates possible discrepancies

between the characteristics of free jets and those of jet-tube elements. Here the

special case is considered in which the relative size of the tube is too large

(due to the small size of the nozzle orifice) to be neglected in comparison

with the jet cross-sectional area.

1. E X A M P L E S O F P O S S I B L E A P P L I C A T I O N S

OF J E T - T U B E E L E M E N T S I N P N E U M A T I C

C O N T R O L S Y S T E M S

Jet-tube elements can be used with success for converting mechanical dis-

placements into air pressure changes, and vice versa. For the first conversion,

a linear characteristic is sometimes required, but more often the device must

operate as a non-linear transducer.

s

F I G . 2.

The single jet-tube element is the simplest form of transducer. Movement of the tube at different cross-sections of the jet gives a variety of non-linear characteristics. For example, by varying the supply pressure and distance h, it is possible to obtain a family of curves Ρ = f(s), as shown in Fig.2,d. An element, producing such characteristics, serves as a model of the object to

I N V E S T I G A T I O N S O F P N E U M A T I C J E T - T U B E E L E M E N T S 89

be controlled by simulating the processes of control.! By using the nearly linear parts of the characteristics (e.g. lengths ac in Fig.2,a and b) , it is possible to reproduce practically linear functional conversions. A single jet-tube element can reproduce functions which otherwise might require com-plicated cam mechanisms, lever systems, etc.

c P. b

(a) a b'

The reproduction of non-linear functions of a more complex nature, e.g. Ρ = f(s) with two or more maxima (Fig. 2,e), can be attained by combinations of two or more jet-tube elements.

These elements can be used also for a universal non-linear converter, re-producing any desired function Ρ = f(s). Arbitrary functions can be ob-tained with a set of jet-tube elements, having nozzles with particular sections (Fig. 3,a). For each one, the function Ρ = f(s) has the shape shown in Fig. 3,b. The characteristics shown in Fig. 3,c and d, are obtained by limiting the tube displacement to sections ab'c' or b'c'd. Changes of supply pressure alter the magnitude of the horizontal "plateau", and correspondingly the slope of the side branches (Fig. 3,e and f ) .

The method of obtaining an arbitrary function Ρ = f(s) by a set of jet-tube elements has been described in detail in an earlier article (see p. 59).

Generally, a set of several nozzles and tubes can reproduce a curve closely approximating to the desired one, but it is often displaced in a positive or negative direction, corresponding to a constant pressure (see p. 63, Fig. 3). It can be adjusted to the desired level either by a nozzle-baffle device (Fig. 4,a)

t In a simulator, the shift of the maximum of the Ρ =f(s) curve along the j-axis may be

attained by moving the nozzle, and along the P-axis by altering the supply pressure. This

relates to static characteristics of the object; but a simulator can also be made to repro-

duce dynamic properties of an object by adding pneumatic capacitances (a chamber or a

system of chambers).


or by the method employing no moving parts, i.e. being based on purely

aerodynamic action (Fig.4,b). In the first case, the difference between in-

put and output pressures is adjusted by a screw bearing on a spring. In the

second case the adjustment is made by a pneumatic chamber, analogous to

the one used as a summator. When the pressure Pk varies, the pressure Λ > ι η Ρ

F I G . 4.

changes, being offset by a constant value determined by the setting of the

pressure ΡΛύ}.

The conversion of air pressure into a mechanical displacement is also sim-

plified, if a jet-tube element is used instead of a nozzle-baffle. For compari-

son, Fig.5,a shows a converter with a nozzle-flapper element; Fig.5, bone

with a jet-tube element. In the latter case, the jet and tube / form a feedback

(a) (b)

F I G . 5.

element. Due to the feedback, there is a definite relationship between the dis-tance from the nozzle to the tube and the pressure in chamber 3 (constant pressure is applied to chamber 2). By locating the tube in various cross-sections of the jet, it is possible to provide a wide range of mechanical dis-placements, even relatively large ones, with small nozzle dimensions. The displacements can be varied from tenths of a millimetre to several milli-metres, or even several centimetres.

The analysis of jet characteristics indicates that the potential applications of jet-tube elements is not limited to the applications given above. For example, it is possible to use them for the operations of multiplication, di-


vision, squaring, taking advantage of the fact that at a given position of the

tube the ratio of output to input pressure is independent of the latter. Here

gauge pressures are considered. Figure 6,a illustrates the principle of these

operations by using jet-tube elements.! Pressure Pt is fed to chamber 7, and

due to feedback from the jet-tube element 2 an equal pressure arises in

chamber 3. For this system P2 is the supply pressure. On the same moving

traverse is mounted the second nozzle 4. Nozzle diameters and tube dimen-

sions for 2 and 4 are equal. A t any given value of h for the element 2 there

is a relationship Ρ 2/ Λ = / ( A ) , and P 3 / P 4 = f(h). As h is the same for both

jet-tube sets, the values of f(h) must also be the same. Therefore, P2/Pi

= P3/P^, from which it follows tha tP 4 = PlP3/P2. If P2 = I, P4 = PlP3,

(a)

d7>d2>d3

F I G . 6.

or if P3 = 1, then P4 = Pl/P2. Likewise, if P2 = 1 and Px = P3, then i> 4

= P\ .* It is also possible to use for the same purposes the transverse motion of the tube across the jet (Fig.6,b). The scheme of Fig.6,a, with the intro-duction of an additional feedback, can be adapted for square rooting. For example, with the scheme shown in Fig.6,c, where P2 = 1 and P3 = Pl9 we obtain P 4 = P1P3/P2 = P\, or Ργ = > / P 4 , where PA = input pressure and Pl = output pressure. Models of the universal non-linear converter and the multiplying dividing device, described above, were built in the Pneumatic and Hydraulic Laboratory of I A T A N U.S.S.R. The tests confirmed the as-sumptions. The models had the moving unit suspended on flexible supports so that parallel motion of the nozzles was ensured. Diaphragms were fitted as shown on Fig. 6. The examples of the application of jet-tube elements dis-cussed above can be expanded to include hydraulic devices. The theory and calculations of jet-tube elements, as shown below, are based on the use of free turbulent jet characteristics, and these are common for air and liquids.

f A scheme for a calculating device, performing the same operations and using nozzle-

flapper elements, has been published (see p. 42). φ A n y single pressure can be considered as unity, and constants taken into account.

Experiments with multiplying and dividing devices along the lines suggested above were

carried out at P2 = 1 atm.

9 2 P N E U M A T I C A N D H Y D R A U L I C C O N T R O L

2. I N V E S T I G A T I O N S I N A P P L Y I N G F R E E

JET C H A R A C T E R I S T I C S T O T H E C A L C U L A T I O N

OF J E T - T U B E E L E M E N T S

The expansion of a free jet has been studied by aerodynamicists [4, 5]. The

experimental work in the main related to the flow of air from nozzles with

diameters of the order of several centimetres or decimetres. For instruments

of pneumatic control the nozzle diameters are usually in the region of tenths

of a millimetre, and only exceptionally a few millimetres. The question arises,

whether the experience accumulated during studies of air flow from relatively

large nozzles can be applied to the smaller units of pneumatic control devices.

(b)

F I G . 7.

First, let us consider some of the characteristics of free jets, with which the characteristics of jet-tube elements will be compared. The longitudinal sec-tion ("profile") of a free axially-symmetrical turbulent jet is shown on Fig. 7,a. The initial part of the length hn has a core of constant velocity (hatched on Fig. 7,a) in which the axial velocity is constant and equal to the nozzle-exit velocity. The remainder of the section at this length is the boundary region. In the main part of the jet at h > hn, the boundary effects influence the whole cross-section, and the velocity on the axis diminishes as h increases. The re-duction of velocity occurs with a constant momentum in the jet, and is due to the increase of mass, as more air is dragged in from the surrounding


atmosphere. The static pressure in the jet is assumed to be constant, equal to the pressure of the surrounding medium. When efflux occurs into the at-mosphere, the velocity head at the tube is equal to the total pressure.

The nature of the jet structure as described above is a simplification. In fact, the static pressure in the jet does not remain strictly constant. In the main part of the jet, the intensity of turbulence varies, and it would be more accurate to distinguish several zones of the main part. But the influence of these factors is relatively slight and, for free jets, it is possible to accept certain mean values, from which their characteristics can be calculated.

Experimental characteristics of jets, given in earlier works on free jet theory, have been analysed, in order to evaluate quantitatively the jet profiles. The results were compared with the results of experiments carried out in the course of the present investigation. The following considerations were taken into account.

1. From the point of view of control instrument applications, it is essential to know the distribution of pressures in the jet. Yet the references on free jets usually quote the data on velocity distribution which, however, were obtained in the first place by measuring pressures. As the total pressure at any point in the jet, measured as a gauge pressure, is also the velocity head, it is propor-tional to the square of velocity at that point. Therefore, the practical limiting contours of a jet as determined by velocity or pressure measurements at say 0-5 per cent of their maximum values are different for the two cases (com-pare magnitudes of b and c on Fig.7,b; Swis the value of S, at which Wis

0-5 per cent of the velocity on the axis Woc).

2. In experiments with free jets the tube is usually parallel to the jet axis. Therefore, although velocity vectors in the jet are not co-linear, it may be assumed that the axial velocity component is equal to the full velocity, as far as empirical formulae are concerned.

Let us examine, in the first place, the experimental characteristics using mean values as a basis of comparison of free jets and jet-tube elements. These main values were based on the theory of free turbulent jets, and on analysis of experimental data.t The following numerical values were taken to define the jet boundaries (see Fig. 7,a) : \& = 8° 40' on the velocity diagram, or \(% = I

e on the pressure diagram; hn = 5d, corresponding to \ß = 5°43'. There-

fore, the width of the boundary zone, defined as shown in Fig.7,a, is:

b = h tan 5C43' + h tan 8°40' = 0-252/* at — < 5,

d (1)

b =d

-2

(2)

t Experiments by Triipel, Fertmann, and Göttingen Aerodynamic Laboratory; for

sources, see [4].


For the calculation of pressures the following formulae were used (see p. 75, Appendix I I ) :

τ· - m

3 / 2 Ί4 h

for — < 5, d

3/2-12 \ 2

0-3 + 0-14 — d

for — > 5. d

(3)

(4)

In these formulae Pf is the pressure before the nozzle; Ρ the pressure at a given point in the jet; d the diameter of the jet, or its width (in the case of a slot); b the width of boundary zone; s the coordinate of a given point (see Fig.7,a); and h the distance from the nozzle to the tube.

W/Woc

0-8

0-6

04

0-2

\ »

o - /

• -2 v - J

\

VS.

k \ \

Nil

0-2

F I G . 8.

0-i C-ô

(b)

0-3 TO )

Also, Woc and Poc will denote the velocity and the pressure in the centre of the section, i.e. on the jet axis.

Substituting values of b from (1) and (2) into (3) and (4), it is easy to ob-tain P/Poc =f(s/b), and as WjWoc = J(P/P0C), also W\Woc =f{s\b).

Diagrams plotted from these formulae are shown in Figs. 8 and 9 by dot-ted lines. In Fig. 8,a is shown the profile of a jet. In these figures, the ex-perimental data for free jets and for jet-tube elements are given. In Fig. 8 the following notation is used: 7, points from data quoted by Trüpel for a


free jet, nozzle diameter 90 mm (Re = 550,000, M = 0-26); 2, points from the

data obtained at Göttingen Aerodynamic Laboratory for a free jet, nozzle

diameter 137 mm (Re = 380,000, M = 0-12); 5, points from the data by Fert-

mann for a slot nozzle with a width of 30 mm and a slot length of 650 mm,

for a free jet at Re = 70,000 and M = 0-09; 4, points obtained during the

P/Poc

to Ψ

0-8

0-6

04

0-2

ο

V I I

-h/d=10\p 1 . -h/d=6)

pr

1afm

Ά

0 0-2 04 0-6 0-8 Wç- 0

(a) P/Pf

TO

0-8

0-6

04

02

1 \

\ \ \ \

°v \ \ \ \

Pf=1atm

\ \

\ Λ \ \ \ \ \ ^

NN,

0 02 04 0-6 0-8 0 i (c)

F I G . 9.

12 16 20 i

id)

experimental investigation of a jet-tube element, nozzle diameter 0-8 mm, tube inner diameter 0-27 mm and outer diameter 0-5 mm, tube length 7 mm (Pf = 0-2-1 atm; M = 0-4-1; Re = 7500-31,000); 5, points from the ex-periments on a jet-tube element, with a slot nozzle 0-31 mm wide and 1-2 mm long (M « 1 ; Re = 10,000); the tube had the same dimensions as for the circular nozzle described above. In Fig.8,b, together with the calculated curve WjW0C = f(slb), the data of Göttingen Aerodynamic Laboratory are

4 a Aizerman I


P/Pf P.atm Pf'latm P,atm

F I G . 10.


3. C H A R A C T E R I S T I C S O F P R E S S U R E C H A N G E S O B T A I N E D

B Y P L A C I N G T H E T U B E

I N V A R I O U S C R O S S - S E C T I O N S OF J E T

Axial Movement of the Tube along the Jet. If the tube is co-axial with the circular nozzle, the characteristic Ρ = f(h) for the main part of the jet (hjd > 5) is given by the equation:

For the initial zone (hjd < 5),

Ρ 1· (6)

Equations (5) and (6) can be obtained from (3) and (4) by substituting S = 0. The characteristic P/Pf = f(h), according to (5) and (6), is shown in Fig. 10,a. From this characteristic, plotted in normalized coordinates, we can obtain various families of curves Ρ = f(h). For a given nozzle diameter d all ordinates of the characteristic Ρ = f(h) vary proportionally to supply pres-sure Pf (F ig . l0 ,b ) .*

A change of nozzle diameter d at a constant Pf is equivalent to a change

t For a section of a jet at a considerable distance from the nozzle, the function W\ Woc

= f{sjb) at low speeds is nearly linear from sjb = 0-3 to sjb = 1. φ Taking h as positive in the direction of the tube approaching the nozzle and suitably

choosing the origin of the h scale, we obtain the characteristics shown in Fig. 10,c.

shown for the nozzle 137 mm in diameter. Points marked on Fig.8,b by numbers 1, 2 and 3 are respectively for h = 0-6 m and hjd 6-7; h = 0*8 m and hjd 8-9; A = 1 m and h\d 1 l - l . f

In Fig. 9 is shown, in addition to a calculated characteristic, the graphs PjP0C = f(sjc) and (PjPr) = f(hjd), obtained experimentally for the jet-tube elements described above (Fig.9,a and b for the circular nozzle; Fig.9,c and d for the slot nozzle). Fig.9,c for hjd = 6 points are given for the tube in three different locations, as shown by the sketch.

On the basis of the results shown in Figs. 8 and 9 it may be said that ex-perimental results with jet-tube elements agree satisfactorily with those ob-tained experimentally and by calculation on free turbulent jets. It must be noted, however, that this conclusion is valid only for the condition that the ratio of nozzle sectional area to the jet sectional area is within the limits of the tests. The question of the influence of this ratio on the characteristics of pressure change in jets is investigated in section 4.


of scale of h\ this is shown in Fig. 10,d. Several other characteristics may be obtained by varying both Pf and d for specific changes of these variables. A peculiarity of the characteristics, shown in Fig. 10,b, is the comparatively large displacement, corresponding to a working range of pressure P. These displacements amount to several millimetres, or even centimetres. It is im-portant that a linear approximation to these characteristics can be made over distances of several millimetres.

Lateral Displacement of the Tube across a Jet. Substituting into (3) and (4) the values for b from (1) and (2), we obtain for the part of the jet (hjd > 5)

(Γι - 2-83 ( * ηγ Pf { 0-3 + 0-Uh/d >

and for initial region : . 3 / 2 -

— = Γΐ - 7-9 («LX (8)

For each value of hjd, equations (7) or (8) determine the function PjPf

= f(sjd). In Fig. 10,e is shown, as an example, the characteristics PjPf

= f(sjd), for hjd = 1, 3, 6 and 10. From any one of these it is possible to obtain a family of curves Ρ = f(s). For example, taking various values of the pressure Pf, for a nozzle with d = 0-8 mm, hjd = 6, i.e. h = 4-8 mm, we ob-tain a family of curves Ρ = f(s), shown in Fig. 10,f. Maintaining Pf = 1 atm, and varying d, we obtain for given ratios of hjd the family of curves shown in Fig. 10,g. For proportional changes of d and Pf9 the original curve PjPf

= f(sjd) shown in Fig. 10,e, for hjd = 6, is expanded into the family of curves Ρ = f(s) shown in Fig. 10,h, and so on.

Oblique Movement of the Tube, at any Angle φ to the Axis of the Jet. We will limit this case to tube displacement along the major axis of an ellipse, de-fined by the intersection of the main part of the jet and a plane, inclined at an angle φ to the axis (Fig. 10,i). Then:

1 _ 2-83' (sjd)sm<p

1 + 0-304(A0/rf ± (sjd) cos φ)j _, ,

Pf

v 0-3 + (M4(A 0/rf ± (sjd) cos φ)

The meaning of the symbols φ, 5 Φ and h0 is given in Fig. 10,i; otherwise the notation for (9) is the same as for the preceding equations. In Fig. 10, are also shown the limiting values of sv, that is 5 Φ Ι Γ Η ΑΧ and ^ 2 m a x. When 0 < ^ φ

< s V i m a x, in equation (9) there should be a + sign before the term (s(pjd) cos φ ; when 0 < 5 Φ < sV2max a — sign. Values of s V i m ax and sV2m.dX can be obtained


from the jet profile drawn in Fig.8,a or calculated by the formula:

0-152 ^5. + 0-5 0-152 - ^ - + 0-5 i f f l < n MÏ d , sfr mnv d

and d cos <p(tan φ — 0-152) d cos φ (tan φ + 0· 152)

(10)

The last equation is only valid if the intersecting plane lies beyond the initial region of the jet: this corresponds to the condition

tan<p > l'26l(h0/d- 5).

Otherwise, it is necessary to use the formula :

— - 5

cos φ (11)

The proof of formulae (9), (10), and (11) is given in paragraph 1 of the A p -pendix. Note that (7) is a particular case of (9) at φ = π/2; then h 0= h and 5 Φ = s. Also (5) is another particular case of (9) with φ = 0, h0 ± Ξφ = h.

In Fig. 10,1 are shown the characteristics P\Pf = f{sjd), calculated by (9) for h0/d = 6 at φ = 0°, 5°, 10°, 30° and 90°.

Further Possibilities. Families of characteristics Ρ = f(s), differing from those discussed above, can be obtained by moving the tube across a jet cross-section, not along a diameter, but along a chord. Then the function PjPf

= f{sjd) becomes :

V t W2 + W 2 ] V / 2 - ^ 2

(Γΐ - 2-83

L V 1 + 0 - 3 Ρ _ L V 1 + 0 - 3 0 4 ^ 7 _,

Pf { 0-3 + 0·14Α/έ/

where sx is the distance from the centre-line, measured along the chord, and a the distance of the chord from the axis. For sxmmjd in this case we have the formula:

(13)

The tube can also be moved along a straight line parallel to the jet axis, at a distance s from it. The family of characteristics is then defined by equation (7), in which it is necessary to take sjd as a constant, and consider h/d as a vari-able. The minimum value of the ratio h/d, at which the tube reaches the boundary of the jet, is:

± - 0 - 5


The proof for formulae (12)-( 14), and additional observations relating to the application of equation (7), are given in paragraph 2 of the Appendix. Further extension of the Ρ = f(s) characteristics can be obtained by moving the tube along chords of curves, defined by the interaction of the jet by an oblique plane.

All the relationships given above relate to the case where the axis of the tube is parallel to the axis of the jet (Fig. 10,j), and it has been assumed that the tube registers the full velocity head. The tube may also be placed at an angle θ to the jet axis (Fig. 10,k). A t large angles, however, the flow distur-bances caused by the tube are substantial, and it is necessary to correct the value of the velocity head, taking the velocity WQ = W cos Θ. Then, for example, (7) will be transformed into the equation

where k is the correction coefficient. On the basis of experiments with a tube having inner and outer diameters of 0-27 mm and 0-5 mm (nozzle diameter 0-8 mm), this coefficient may be determined as |

Thus, one more variable θ is introduced into the relationship between P/Pf

and s/d or h/d, originally determined by equation (7). For the particular case, θ = 90 — φ, in (15) cos θ is replaced by sin φ.

Expressions (5), (7), (9), and following equations relate to the main por-tion of the jet. Similar formulae can be derived for the initial part of the jet.

Calculations of the Parameters of a Jet-tube Element. Confining the argu-ment to the case of a tube moving along the axis of a jet (equation (5)), then only for this case the numerical calculations are reasonably simple; in other cases we have non-linear equations, which can only be solved graphically.

The following formulae determine the distance h0 from the nozzle to the tube, and the pressure before the nozzle Pf, at which the curve Ρ = f(h) passes through given points A(P0, h0) and Β(Ρί, h0 + Ahx); see Fig. 10,m.

(15)

(16)

h0 = - 2-14rf, (17)

(18)

t M o r e details in the following section.


The derivation of these formulae, as well as of (19), (20) and (21), is given in paragraph 3 of the Appendix.

The following numerical example illustrates the application of (17) and (18): P 0 = 0-65 atm; />, = 0-38 atm; Δ h y = 2 mm; d = 0-8 mm; from (17) and (18) we find h0 « 4-7 mm, and P f = 0-83 atm.

It may be required that the chara cteristic Ρ = /(A) should pass through the points A(P0, h 0 ) , B ( P i yh 0 +Δ h1),C(P2, h0 + Ah2), M(Pm, h0 + Ahm),

N(P„, h0 + Ah.,), with coordinates fulfilling the condition (see Fig. 10,n):

zlA, ΖΪΑΓ

^0 Λ

- 1

Pi

- 1

/ / P o

Po

(19)

I f it is more convenient to determine the characteristic Ρ = f { h ) by the derivative (àPjàh)0 at the point Ρ = P 0 , then the values of A0 and Pf can be calculated as follows :

2Po h0 =

Pf =

d P \

~dh)o

0-2%Pl

- 2-Ud, (20)

(21) d P \

dÄ "Jo

Notes on the Characteristics of Slot Nozzles. In section 2 it has been shown that the characteristics of slot nozzles (Fig. 3,a) obtained by moving the tube along the jet axis, or across the jet, parallel to axis ζ (Fig.9,c) can be ap-proximated by the above formulae. For practical applications it is important

P,atm

fOi

0-8

0-6

04

0-2

Λ j J t f J

β at m

f0\

0-81

0-6

04

0-2·

Pf-1-5 atm

0-6

02

02 04s,mmO6 ~08 -06 "04 -02 0 02 04 0-6s,mm

(a) (b)

F I G . 11.


to know the characteristics obtained by moving the tube along the major axis of the jet cross-section (parallel to x, in the plane xy\ see Fig.9,c). The uni-versal non-linear converter, described in section 1, depends on such charac-teristics. Experiments have shown that the pressure distribution on the mixing zone of a jet issuing from a slot nozzle differs little from that for a tube when a traverse is made parallel to axis ζ and to axis χ (as far as the sloping part of the curve is concerned). As an example, Fig.l l,a shows the corresponding parts of the characteristics Ρ = f(sz) and Ρ = f(sx), taken for an element with a slot 0-31 mm wide and 1-2 mm long, at h = 1-8 mm, and Pf = 1 atm. In Fig. 11,b are shown the experimental characteristics Ρ = f(Sx) for the same element at h = 0-3 mm and Pf = 1 atm. In Fig. l l , b are shown the experi-mental characteristics Ρ = f(Sx) for the same element at h = 0-3 mm and Pf

= 1,0-6 and 0*2 atm gauge. These characteristics are close to the theoretical ones shown in Fig. 3. The tube used for these experiments had inner dia-meter 0*27 mm, and outer diameter 0-5 mm.

4. I N V E S T I G A T I O N O F T H E D E V I A T I O N S

OF J E T - T U B E E L E M E N T C H A R A C T E R I S T I C S

F R O M T H O S E O F F R E E T U R B U L E N T JETS

The deviations are caused by the following factors : because of a relatively large size of the tube diameter (at a small nozzle diameter), the velocity head gradient across the jet may become appreciable. Then the pressure registered by the tube will be a certain mean value of the pressures in the plane of the

F I G . 12.


tube head. Also, a tube of relatively large dimensions may alter the pattern of the flow.

Investigation of Tube Influence. Let us assume that the velocity field before the tube is that of a free jet, i.e. that the tube does not cause any disturbance. The flow is assumed to be isentropic (later on we shall consider possible deviations from such idealized conditions). Under these assumptions, the stagnation pressure is determined by the magnitude of Pmf = J Pdf, where Pm is the mean pressure, as measured by the tube, and Ρ the pressure at a given point in the tube entry.

For the positions of the tube shown in Fig. 12,a, b, c, and d, the values of Pm can be calculated by the formulae given below (proof in paragraph 4 of the Appendix). In Fig. 12 the following notation is used: b the radius of jet cross-section; a the distance between the axes of tube and jet; s and φ the polar coordinates of a point in the jet in the plane of the tube mouth. Also, as in the preceding expressions, dis the nozzle diameter; h the distance from nozzle to tube; and Pf the supply pressure.

For the case shown in Fig. 12,a:

For the case of Fig. 12,b formula (22) is also valid, but the upper and lower

limits of integration are replaced respectively by bjd and ajd — rjd. For the

case of Fig. 12,c:

(23)

Formula (23) can also be applied to the case shown in Fig. 12,d if the upper limit in the first integral is taken as r\d, and the second integral is here equal to zero.

For the tube position shown in Fig. 12,a calculations by formula (22), with given values of hjd and r/d, are made as follows: taking into account the limits of aid (in the case under consideration r/d < a]d < (b — r)jd), several


values of ajd, within these limits, are chosen. For each one, the limits of integration are calculated and several values of s/d, within these limits, are chosen.

Taking these values, the function under the integral sign is calculated. The graph is plotted, and with the aid of a planimeter, the value of the integral in (22) is determined. The operations are also carried out for other chosen values of ajd. The second integral in equation (23) can be calculated in a similar way. The first integral of (23) does not include a/d, and calculation of this integral involves only the choice of several values of s/d within the limits 0 < s/d < (r/d — a/d), plotting a graph of the integrand and determining the area.

The above method has been applied to the calculation of the characteris-tics Ρ = f(h), obtained by moving the tube along the jet axis, for a nozzle diameter d = 0-8 mm, and two inner diameters of the tube, dT = 0-27 mm and 0-8 mm. These characteristics are compared with the corresponding characteristics calculated by formula (5), with the assumption dT = 0. Al l three characteristics are shown in Fig. 12,f.

Experimental Verification of the Method Given above. Effects of Flow Distur-bances Induced by the Tube. In Fig. 12,g is shown an experimental characteris-tic PjPf = f(hjd) (solid line), and one calculated by the method given above (broken line), both for dT = 0-27 mm. These characteristics are practically identical. Figure 12,h compares an experimental (solid line) and a calculated (broken line) characteristic PjPf =f(hld) for an element with dT = 0-8 mm (nozzle diameter d = 0*8 mm, as in the preceding example). This experi-mental characteristic lies below that for the tube with dT = 0-27 mm (Fig. 12,g). This confirms the trend of a decrease of Ρ with increase of dT, but the observed values of Ρ are higher than obtained by calculation. This may be explained by the effects of disturbances caused by the tube. Taking into account the losses occurring in retarding the flow, the pressure should be lower than that calculated for isentropic retardation. An increase of pressure, however, can occur due to the curvature of the streamlines (the energy of a larger portion of the jet being converted into velocity head compared with the ideal case). The pressure may also be increased due to entrainment from the surrounding region (Fig. 12,e).

In conjunction with this, it appears to be probable that the characteristics would be influenced not only by the inner diameter of the tube, but also by its outer diameter.!

The tube used for the experiments described above had an outer diameter of 0-5 mm and inner diameter of dT = 0-27 mm, and, for a second case, diameters of 1-26 and 0-8 mm respectively. The area corresponding to the outer diameter of the first tube was 2-5 times smaller than the area of the

t The flow about a tube can also be influenced by its length and the shape of the tip.

Al l experiments were carried out with tubes 7 mm long, with the plane of the tip square

to the tube axis.


- n-51 I I L _ F I G . 13 .

flow pattern, depending on the angle θ (Fig. 10,k). Figure 13 shows the ex-

perimental curve Ρ/Ρθ=ο = / ( θ ) (solid line) obtained when a tube is yawed

about a vertical axis through the tip (the centre being located on the axis of

the jet). The tube had inner and outer diameters of 0*27 mm and 0-5 mm, and

for the nozzle d = 0-8 mm.

The broken line on Fig. 13 represents the curve for k calculated from for-

mula (16).

A P P E N D I X

Derivation of Formulae for Calculations of Jet-tube Elements

1. Derivation of Equations ( 9 ) - ( l l ) . Let h -= h0 + Ah (Fig. 14,a) . Then from A ABC:

s = s<P sin φ, Ah = s<p cos φ, and h = h0 + % cos φ. F r o m Δ DEF: b — (d/2) = (h0-\-Ah)

x tan a /2 , and therefore b = (h0 + Ah) tan a /2 + d\2 = {h0 + s<p cos φ)0\52 + d\2. If h

= Iiq — Ah (Fig . 14 ,b) , then out of aA'B'C follows, as before: s = S(pS\r\(p, and Ah

= s φ cos φ. But in this case h = h0 — Sc, cos φ. From Δ D'E'F'v/e find b = (h0 — cos φ)

x 0-152 -\r\d. Using formula (7) to determine the pressure in points C or C", with the

values for s, h, and b as found above, we obtain ( 9 ) . Equations ( 1 0 ) are obtained in the

following way. From aAGH (Fig. 1 4 , C ) : bL = Ahimax tan φ. F r o m aGEF: c = (h0

4- Ahimax) tan Λ / 2 - (h0 + Zl /7 i m ax)0-152, and bY = 0 · 1 5 2 ( Λ 0 + ^ l m a x ) + d\2. Elim-

inating Ahimax from the expressions for b^, we find bx = [ ( 0 - 1 5 2 A 0 -F dj2) tan <p]/[tan φ

— 0-152] . From AAGH:SVimax = Αι/sin φ. Substituting the value of bl into the last

expression, we obtain the first formula ( 10) . In a similar way, we can obtain the formula for

Sç2maxld, but the latter is only valid as far as the limiting position of the intersecting plane,

shown in Fig. 14,d, where this plane intersects the boundaries of the jet at the point with

nozzle orifice, and for the second tube 2-5 times larger. T o determine the ef-

fects of a further increase of tube outer diameter, experiments were made

with a tube having an outer diameter of 3 mm (dT = 0-8 mm). Results are

shown in Fig. 12,i (curve / for the tube with 3 mm outer diameter; curve 2

with 1-26 mm). Virtually no difference is apparent.

All the experiments described were carried out with the head of the tube

in a plane at right angles to the jet axis. As already mentioned, if the tube is

positioned obliquely, it is necessary to take into account the change in the


abscissa 5d, corresponding to the transition from the initial part of the jet to the main jet.

For this position we find from AOKL (Fig. 14,d), / = 5*/tan α/2 - 0-16d. A l so NL

= l-26d, and from AÂLN it follows that tan φ = 1-26 (hold — 5). W h e n tan φ < 1-26

l(hold—5)9 as follows from Fig. 16,e, s<p2max = (h0 — 5i/)/cos φ and s<f2maxld = [ih0ld) — 5]

/cos φ; that is, the formula (11).

(h)

F I G . 14.

2. Brief Derivation of Formulae (12)-(14) , and the Application of Formula (7) for the

Case of a Tube Moved Parallel to the Jet Axis. Formula (12) is obtained from (7), substitut-

ing \j(s2

x + a2) for s (Fig. 14,f). Expression (13) follows from s x m ax = ^ (b

2 — a

2)

(Fig. 14,g). The application of formula (7) for the case of a tube moved along an axis,

parallel to the jet axis and at a distance from it, follows from Fig. 14,h. F r o m AEFD

we findCs — \d)jhm\rx = tana/2 = 0-152, which leads to (14).

3. Derivation of Formulae (17)-(21). Let us first consider the derivation of (17) and (18).

From (5) P0 =-· + 0 - 1 4 A 0W "2 and P1 = Ρ /{ 0 · 3 + 0·14[(Λ0 + Ah^/d]}'

2. There-

fore, P0IPi = ({0-3 + 0-14[A0 + AhA/d]I{0-3 + 0'14h0/d})2. Solving this equation for h0,

we arrive at (17). Substituting the value of h0 into P0 ^ Ρ / (0 ·3 + 0-\4hold)~2 we obtain

(18). Furthermore, considering Fig. 10,n : by analogy with (17), it is possible to deduce the

expression h0 = [Ah2/\(P0IP2) — I] — 2-\4d. This equation is compatible with (17) only


when AhilliXPo/Pi) — 1 ] = h2/[\(PolPi) — 1 ] · Analogous reasoning may be applied to

the equations relating to each length up to h0 + Ahn. A s a result, ( 1 9 ) is obtained. T o

obtain formula ( 2 0 ) we differentiate ( 5 ) ; from the expressions:

( d P / d A ) 0 = - [ 0 - 2 8 / y ( 0 - 3 + 0-\4h0ld)3d and P0 = P / 0 - 3 + 0-\4hold)~

2

we eliminate Pf, and, bearing in mind that (âPjdh0) < 0 , and therefore, | (dP/d/?) 0|

= — (dP/d/z)o, we arrive at ( 2 0 ) . Formula ( 2 1 ) is obtained by substituting the value of h0

from ( 2 0 ) into the expression P0 = P / 0 - 3 + 0'\4hQld)~2.

4. Derivation of Formulae ( 2 2 ) and ( 2 3 ) . W h e n the tube is located in the jet cross-sec-

tion, as shown in Fig. 12,a, the pressure at all points of the hatched element is constant

(it can be calculated from formula ( 7 ) ) , and the area of element is / = 2s(p as. Since cos φ

= [(a2 + s

2 — r

2)jlas], then / = 2s arc cos [(a

2 -f s

2 — r

2)j2as] ds. The element of

force acting on this element of area is:

Substituting for s, a, and r the relative quantities sjd, ajd and rjd, integrating with respect

to sjd between the limits a\d — r\d to a/d -\- rjd, and dividing the force obtained by the

area of the tube bore m2, we arrive at ( 2 2 ) . This reasoning covers the case shown in Fig. 12 ,b

as well, but the limits of integration are sjd = aid — rid, and sjd = b\d.

For the case shown in Fig. 12,c, we take the sum of two integrals, one corresponding to

the range 0 < s < r < a, and the other to the range r — a < s < r -\- a (the latter range

is represented by the hatched area on Fig. 12,c) . The derivation of formula ( 2 3 ) is similar

to that of ( 2 2 ) . The second integral in formula ( 2 3 ) is the same as in formula ( 2 2 ) ; it cor-

responds to the second range. F o r the first range, elementary areas are defined a s / = 2nsds

and this is used in determining the first integral in ( 2 3 ) , instead of the previous expression

/ = 2s arc cos [(a2 + s

2 — r

2)j2as]ds. A concentric position of the tube in the nozzle

(Fig. 12 ,d) may be regarded as a particular case of the position in Fig. 12,c. Then, only the

first integral remains in formula ( 2 3 ) , and the upper limit of the integration is equal to rjd,

as a = 0 .

R E F E R E N C E S

1. G . T . B E R E Z O V E T S , Avtomatika i Telemekhanika, Vo l . X V I I , N o . 1, 1956 .

2. V . F E R N E R , Die Technik, 6, 1954 .

3. Y U . I . I V L I C H E V and E. M . N A D Z H A F O V , Universal pneumatic multiplying-dividing de-

vice and device for square rooting (this book, p. 4 2 ) .

4. G . N . A B R A M O V I C H , Turbulent Free Jets of Liquids and Gases (Turbulentiye svobodnye

strui shidkostei i gazov). Gosenergoizdat, 1 9 4 8 .

5. G . S C H L I C H T I N G , Boundary Layer Theory (Teoriya pogranichnogo sloya). Izd-vo Inostr.

Literatury, 1956 .

PNEUMATIC AGGREGATE SYSTEM OF KB-TsMA

M . S . S H N E Y E R O V


Pneumatic techniques are still the main form of automatic control in

foreign practice, despite the development of various other techniques. Pneu-

matic installations are used in metallurgy, in gas-works and oil refineries, in

chemical plants, power stations, etc.

Some processes are automatically controlled entirely by pneumatic meth-

ods, and others use combinations of pneumatic and electronic devices.

There are several reasons for this widespread use of pneumatics in auto-

matic control. One of these is safety against explosions and fire. This con-

sideration, important though it was in the early days of pneumatic control,

cannot now be counted as a major issue. There are other important con-

siderations, such as:

(1) Simplicity of design and servicing, so that highly skilled personnel are

not essential.

(2) Low cost of manufacture.

(3) The ability to realize complex systems of control by simple methods.

(4) Wide range of adjustment : for the proportional term, from 0 to 1000 per

cent (and more); for the integral action time, from 1 sec to an hour or

more; the derivative action time may also be varied within wide limits.

(5) The ability to transmit signals for considerable distances (up to 300-

500 m).

(6) Good protection against corrosion, due to the flow of clean air through all essential parts.

The development of pneumatic-automation technique is connected with the three main stages of automation, each with its specific technology.

In the first stage, we have automation applied to separate technological processes. Here the main application was the automatic regulation of the separate parameters of a process, which involve large instruments, placed close to the production plant. Siting a few instruments on the working plant presented no difficulty. This period is characterized by the use of large in-

111


struments with easily visible dials, and justified the combination of measur-ing, desired-value setting, and regulation units in one instrument. Such is the origin of the basic designs of pneumatic controllers, typical of this first period.

The second stage was the complete automation of several separate pro-cesses in a manufacturing cycle. Here, the process is controlled by instru-ments, mounted in one panel. As the number of these instruments can be quite considerable, the panels would be several metres in length, if instru-ments of large overall dimensions were used. Such panels are inconvenient for observation. Therefore, already at this stage, it is necessary to reduce drastically the overall dimensions of the control equipment.

The third stage is the transition from automatic control of separate pro-cesses to complete automation of production units and workshops. The tendency to centralize all controls in one control room is typical of this period. T o facilitate supervision, panels with line diagrams of the control system are installed; the instruments are located at certain points of these diagrams, corresponding to the actual positions of the detecting and measuring elements on the plant.

T o build these panels, it is necessary to have small-sized, even miniature, instruments of automatic control and regulation. Auxiliary recording equip-ment, not requiring continuous observation, are installed outside these panels and may be of large overall dimensions.

During the transition to the second and particularly to the third stage of automation, basic designs of universal controllers were found inconvenient for complex types of control system. The need arose for pneumatic com-ponents arranged on the aggregate principle, with the main elements of the regulating device built as separate instruments. The input and output para-meters of these units are air pressures, which are varied over the same range. This design of control equipment permits interconnections in any com-binations and any number, and installation with a considerable distance from one to another. The elements of such aggregate installations are very amenable to standardization.

General purpose industrial equipment cannot always be used for certain processes in the non-ferrous metallurgical industry, particularly in hydraulic ore treatment and chemical metallurgy of rare metals and titanium, because these branches of industry have their own specific problems, due to the presence of corrosive gases. T o meet these conditions, the Design Bureau for Automation in the Non-ferrous Metals Industry ( K B - T s M A ) in 1955 started work on a system of pneumatic devices for the control of the technological processes involved in titanium manufacture. These devices, designed on the principle of force balance, form an aggregate system.

P N E U M A T I C A G G R E G A T E S Y S T E M O F K B - T S M A 113

P N E U M A T I C A G G R E G A T E S Y S T E M S

Figure 1 shows an example of the regulation of a single parameter using

the pneumatic aggregate system K B - T s M A . The pneumatic transducer 1

measures the parameter to be regulated and converts it to an air pressure.

This output pressure of the transducer 1 is directly proportional to the

measured parameter, and is connected to the secondary (indicating or re-

cording) instrument 2, and to the input of the controller unit 3. The parallel

F I G . 1. Pneumatic aggregate system K B - T s M A for the control of a single parameter:

/—primary instrument (pneumatic transducer); 2—secondary instrument (indicat-

ing or recording); 3—controller unit; 4—control panel and desired-value control ;

5—diaphragm regulating unit; 6—positioning device; 7—filter; 8—air pressure

reducing valve; 9—pressure gauge.

connexion of these two instruments enables them to be positioned in two separate locations, although the illustration shows a direct joint arrange-ment. The second chamber of the measuring element in the controller unit is connected to the desired-value setting device, built into the secondary in-strument. The output of the controller unit (the command pressure) is ducted to the pneumatic diaphragm of the regulating unit 5. T o compensate for dynamic errors in measurement, a derivative action element ("anticipating element") may be installed in series with the secondary instrument; this is not shown on the illustration. The secondary instrument has two alternative arrangements—with or without remote control elements.


The following considerations were taken into account during the design of

pneumatic equipment on the aggregate principle:

(1) The need for measuring the parameters of corrosive gases, when the

instruments are mounted in a harmful atmosphere.

(2) The requirement for systems with an accuracy not less than class 1-5.

(3) The least possible air consumption.

(4) The feasibility of control from a central panel.

(5) The feasibility of combining new devices with those already in quantity production.

The last condition determines the range of the input and output pressures

of the instruments; the range of pressure from OT to 1-0 atm has been chosen.

The upper limit of 1-0 atm was set because the secondary units were bellows

pressure gauges types MS and 04-MS with dials from 0 to 1Ό atm, and because

the batch-produced pneumatic valves were designed to adjust the pressure

over the 0-1-1-0 range. The lower limit of 0-1 atm was also chosen because

of the properties of the pressure gauges type MS, where the scales can be

extended by 10 per cent without other changes being necessary.

In instruments of the A U S system, and in the majority of foreign in-

struments, the initial pressure is set at 0-2 atm, in order to improve the static

characteristics of the transducer, which contains an open-nozzle amplifier

with a gain equal to 1. The initial pressure of 0-1 atm, chosen by K B - T s M A ,

satisfies the condition for linearity of the static characteristics of a transducer,

but does not ensure linearity of the static characteristics of the control unit,

as will be shown later. This consideration, however, also applies to the A U S

system. In order to make these characteristics more nearly linear, it would

be desirable to increase the initial pressure, but the optimum value has not

yet been determined.

P N E U M A T I C A M P L I F I E R S

The development of pneumatic instruments in K B - T s M A started from the basic unit of an aggregate system: an amplifier. Three types of amplifier, designed on the principle of force compensation, have been made: with an enclosed nozzle and a gain equal to 1, a diaphragm type (Fig.2,a) ; a bellows type (Fig.2,b); and a third type, with an open nozzle and a gain equal to 3 (Fig.2,c). These amplifiers have a natural frequency of oscillation which, however, is so high that no elements connected to an amplifier show any response.

These amplifiers have linear static characteristics (Fig. 3). The non-linear effects are within the limits of accuracy of measurement, and do not exceed ±0-5 per cent. The air consumption rate is from 0-8 to 1 -21. of free air per min. The time constant is about 1 sec. If the output of the amplifier is connected to a pipeline 5 mm dia., 150 m long, the time constant is 4-5 sec.


F I G . 2. Pneumatic amplifiers: a—diaphragm type, with an enclosed nozzle;

b—bellows type with an enclosed nozzle; c—with an open nozzle.

input; mm Hg

a)

—ft-

\ i 2

/

10 20 30 W 50 60

Stroke; microns

b)

F I G . 3. Static characteristics of amplifiers: a—with an enclosed nozzle : 1—output

pressure Pk as a function of the input pressure Pu \ 2—pressure drop Δ Ρ in the nozzle,

as a function of Pu; 3—air flow Q rate, b—with an open nozzle: 1—characteristics

of a primary relay; 2—characteristics of an amplifier.


P N E U M A T I C T R A N S D U C E R S

Following the work on amplifiers, K B - T s M A developed a range of pneu-

matic transducers which convert the magnitude of a controlled term into

a proportional air pressure.

A t present, K B - T s M A make the following transducers designed on the

principle of force balance :

(a) Pressure transducers, designed to measure pressures from 0-250, 0-

630, 0-1600, and 0-3000 mm of water, and containing a mechanical

range adjuster with a ratio 2-6; and a pressure transducer designed

to measure pressures from 0 to 2 atm (gauge), provided with a range

adjustment which varies the effective diaphragm area.

(b) Differential pressure transducers, designed for pressure differences

from 0 to 250 and from 0 to 630 mm of water, and from 0 to 100 mm

mercury.

(c) Vacuum pressure transducers, designed for a range from 0 to 250 mm

of water.

(d) Draught pressure transducers, designed to measure vacuum or pressure

over the range ± 125 mm of water.

Al l elements of the transducers (Fig. 4) are protected from the corrosive

atmosphere. The outer surfaces are painted with a special anti-corrosion

paint. A pliable diaphragm made of "ftoroplast-3"f is used as the sensing

F I G . 4. Pneumatic transducer: 1—sensing element; 2—zero-level adjuster; 3—cor-

recting device; 4—amplifier.

element. For this purpose, a technique has been evolved for making both flat

and corrugated diaphragms out of "ftoroplast-3". These diaphragms have

small hysteresis, high strength, are resistant to corrosive gases, and the

f A plastic material, based on polytetrafluoroethylene (Translator).


physical properties remain unchanged. There is a metallic separating dia-phragm made of stainless steel or of tantal at the point where an actuating lever is led out of the chamber containing the sensing element. This diaphragm ensures complete "sealing" of the sensitive element, low joint rigidity, and does not distort the linearity to any noticeable extent. This design also elim-inates the error due to the change of static pressure in pressure-drop trans-ducers.

In view of the low stiffness of the transducer elements, it is necessary to provide a zero-level adjuster, which ensures a stable zero setting, corre-sponding to a pressure of 0· 1 atm.

Three types of range adjuster have been developed. The first design had lever adjustments on the knife-edge pivots. Although the friction was re-duced, these were complicated and difficult to manufacture. As the movements involved are comparatively small, it has been found possible to replace the knife-edge and conical fulcrums by flexible metal bands in subsequent de-signs. This simplified the design considerably, and permitted an increase in the adjustment of the working range from 3 to 6.

The main disadvantage of transducers with enclosed nozzle amplifiers is the change in the displacements of the sensing diaphragm with variation in the setting of the range adjustment. As the main source of measurement error is diaphragm hysteresis, it is desirable to have transducers in which the dia-phragm displacement (stroke) remains constant for all working ranges. This condition is better satisfied by a transducer with an open nozzle amplifier. Such a transducer is shown in Fig. 5. It is designed to measure pressures from 0 to 2 atm. The output pressure, like that of other transducers, varies from 0· 1 to 1Ό atm. The range setting is performed by adjusting the gap between the nozzle and baffle—that is, by the displacement of the sensing diaphragm and feedback diaphragm relative to the plane in which they are clamped. Both diaphragms have a common push-rod. Because of this, the decrease of the effective area of one diaphragm is accompanied by the increase of effective area of the other. The working range adjustment in these transducers is 20-50 per cent of the measurement range. K B - T s M A make these transducers, designed for pressures 0-0-6, 0-1-0, and 0-2 atm.

The presence of large masses in the dynamic operation of a transducer, and the fact that an amplifier operates with self-excited oscillations, cause self-excited oscillations of the transducer. Indeed, absence of damping in a sensing element must produce self-excited oscillations of large amplitude. T o eliminate any possibility of oscillation, an investigation has begun on a damper with a small stroke. Particular conditions under which the trans-ducers operate, such as a corrosive atmosphere, do not permit the use of existing hydraulic dampers.

It should be noted that the successful application of these ideas is sharply dependent on the quality of the diaphragms, which must have the least pos-sible hysteresis.

1 1 8 P N E U M A T I C A N D H Y D R A U L I C C O N T R O L

F I G . 5. Pneumatic transducer with an open nozzle amplifier: 1—sensing element;

2—primary relay; 3—amplifier; 4—feedback chamber; 5—range regulating

device.

The small displacements in transducers, which amount to 0-02-0-05 mm, require high rigidity of the whole unit, and particularly of the plate on which the units are mounted. It should be mentioned that the diaphragms should undergo an ageing treatment in order to produce permanent calibration, and reduce the errors due to temperature effects. Other properties of transducers depend on the amplifier characteristics.

C O N T R O L L E R U N I T S

K B - T s M A have designed and are manufacturing two types of pneumatic controller: a two-term one with two adjustments (Fig. 6) and a three-term one with three adjustments (Fig. 7).

All transducers developed by K B - T s M A have stable linear static charac-

teristics. Variations of characteristics, as tested in the laboratory, are less

than ± 0 - 5 per cent, which puts them into class 1. Errors of measurement due

to temperature effects of these transducers amount to 0-15 per cent for 1 0 ° C .

P N E U M A T I C A G G R E G A T E S Y S T E M OF K B - T S M A 119

An important feature of these controller units, different from the A U S

type, is the single diaphragm between the chambers, so that they may be

installed in a corrosive gas.

A design feature of the three-term controller is a pneumatic device for

setting the desired value. Also, the element for producing the derivative ac-

tion term is installed at the input. This ensures that the derivative action

time is independent of the magnitude of the output signal.

The control units have quick-release pipe connexions, which facilitate their

installation or dismantling.

The range of adjustment of the two-term controller covers a proportional

range from 10 to 150 per cent, and an integral action time from 3 sec to 60 min.

The air consumption is about 4 l./min.

F I G . 6. Two-term controller unit. F I G . 7. Three-term controller unit.

The three-term controller allows adjustment of the proportional band from 3 to 3000 per cent, and of the integral action time and derivative action timefrom3secto80min. This controller has an air consumption of approxi-mately 8 l./min. It should be mentioned that with a three-term controller the precise adjustment of integral action time and derivative action time is dif-ficult if they exceed 5 min, due to the small volumes of the appropriate chambers.

5 Aizerman I


T H E D I A P H R A G M R E G U L A T O R

A N D P O S I T I O N I N G D E V I C E

The presence of corrosive gases makes it impossible to use valves and re-

gulating units incorporating packings, glands, bellows, etc. The best design in

this case is one without packing, containing a sealing diaphragm, and with

the inner surfaces of the valve lined by special anticorrosion materials.

K B - T s M A , together with TsKB Armaturostroyeniya (Central Design

Bureau for Tube Fittings and Valves), have developed a series of stop valves

and pneumatic regulating valves (Fig. 8 ) . For the diaphragm material, special

rubber and "ftoroplast-4" are used ; for the special lining, rubber, "faol i t -A",

F I G . 8. Regulating valve: 1—diaphragm regulator; 2—positioning device.

and "ftoroplast-3" are used. Stop valves and regulating valves for nominal diameters of 15, 25, 40 and 50 mm are in present production.

These valves have a practically linear relationship between flow rate and displacement. The static characteristics of a regulating valve with a dia-phragm of "ftoroplast-4" is influenced by the static pressure of the medium, and has considerable hysteresis, sometimes amounting to 30 per cent of the diaphragm movement. T o eliminate these defects, K B - T s M A have devel-


oped a positioning device. The output of this positioning device is de-

signed to position the regulating valve over a stroke range of 4-12 mm. The

accuracy is ±1-5 per cent of the magnitude of stroke. The air consumption

of the positioning device is 1-2 l./min.

Al l instruments of the aggregate system K B - T s M A have connexions for

plastic tubes. This permits the use of polyvinyl chloride tubes instead of

metal ones, for both the impulse and command pneumatic lines. This dras-

tically reduces the cost of installation and maintenance of pipelines, parti-

cularly in a corrosive atmosphere.

HYDRAULIC EQUIPMENT OF AUTOMATIC MACHINE-TOOL LINES

L . S . B R O N

H Y D R A U L I C and electro-hydraulic devices are widely used in automatic machine-tool lines, designed by SKB-1 (Machine-tool Design Bureau N o . 1 ) for quantity production. Their applications are as follows:

(a) Feed mechanisms of drilling, boring, milling and other similar

"powered heads", used in machining.

(b) Mechanisms for moving, transferring, locating, turning, tilting, and

generally for positioning parts.

(c) Mechanisms for holding and clamping parts and supporting jigs, as

well as clamping and releasing of parts, during loading and unloading

operations.

(d) Mechanisms for the removal of waste and swarf.

(e) Mechanisms for checking dimensions.

The control of the working sequence in these hydraulic and electro-hydraulic installations is effected either by the mechanical action of cams on hydraulic valves, or by the action of solenoids, operated by limit-switches and electro-hydraulic relays. Some systems make use of a pressure drop at a particular part of a hydraulic circuit.

These hydraulic systems are usually powered by twin-vane pumps—one with a large capacity and low pressure, for rapid movement with a small load, and a second, having a smaller capacity and working at a high pressure, for higher force-level operations, such as clamping, feeding, etc.

Several methods have been devised to bring into action the appropriate pump: (a) with automatic unloading of the larger-capacity pump, as soon as a certain pressure is reached; (b) using electrical signals, the larger-capacity pump may be unloaded at a desired value of the stroke.

H Y D R A U L I C F E E D M E C H A N I S M S

The most widely used feed mechanisms of "powered heads"—main cutting units of automatic lines—are hydraulic pressure compensated flow control valves, which ensure stability of speed, as determined by a throttle setting.

122

E Q U I P M E N T O F A U T O M A T I C M A C H I N E - T O O L L I N E S 123

Figure l,a represents the diagram of a hydraulic system with a flow regula-

tor valve / , and a throttle 2, installed at the entry to the volume of side 3

of the cylinder. The other side 4 of this cylinder is connected to the tank

through a back-pressure valve 5. Oil from the pump 6 passes through the

flow regulator valve 1 to the throttle 2 into the cylinder volume 3. A certain

To the tank

ι Pressure j

~J relay ι

ι I

a)

F I G . l ,a. Hydraulic layout for the feed control of self-contained powered heads.

Old design.

proportion of the oil from the pump is returned to the tank through the port 8 of the flow regulator valve. The amount of oil passing through the throttle 2, and consequently the speed of piston 7, is determined by the orifice area of the throttle 2, and the drop of pressure across it. For a given orifice area, the piston speed would remain constant if the pressure drop across the throttle 2 were constant. The flow regulator valve automatically holds a desired value of pressure drop at the throttle, as the small piston 15, which separates the chambers 10 and 13, responds to fluctuations in the pressure


drop, and controls the quantity of oil returned to the tank through the

port 8.

The desired value of the pressure drop, from 2 to 3-5 kg/cm2, is determined

by the setting of spring 9. The instantaneous value of the pressure drop, and

consequently the piston speed, remains approximately constant, despite the

F I G . l ,b . Hydraulic layout for feed control of self-contained powered heads. A n

improved design.

possible variations of the load (force R9 acting on the piston rod of the working cylinder). This is assisted by the changes of pressure in the cham-ber 77, connected by a small diameter hole to the end 12 of the valve spindle. Oil flowing from 13 to 8 through the slots of the valvef encounters the stop

t The cone end, as shown on Fig. l ,a , is merely a diagrammatic representation. A s can

be inferred from the text, the valve end has slots or grooves—probably " V notches"

(Translator).


5

3

2

1

0 1 2 3 i 5

Qj I/min

F I G . 2. Pressure-rise time in the cylinder, when the piston is stopped, plotted as a

function of flow through the throttle.

scheme Fig. 1 ,a. For simplicity, means of returning the piston 7 are not shown on either scheme. The curves of Fig. 2 show the time of pressure rise in the cylinder after its piston is stopped, plotted as a function of flow through the throttle (before the piston meets a stop). As can be seen from these curves, for equal flows and pressure drops AP = 35 atm, the time Tis smaller for the valve of the type shown in Fig. l,b (curve 7) than for the valve of Fig. l,a (curve 2).

For feed mechanisms requiring remote electric controls, hydraulic control-panels can be used which enable simultaneous and independent opera-tion of two and more feed mechanisms powered by one twin-delivery pump. Similar control panels are used for feeds with a sequence operation of two

14. The velocity of oil in this restricted space is considerable, and as it is reduced, an increase of pressure occurs at the spindle end 72. This pressure is transmitted to the chamber 77, produces a force tending to close the valve, because it acts at 72 on an area somewhat smaller than that of 77.

The maximum pressure in the system is limited by a relief valve 16. Figure l,b shows a similar scheme with an improved flow regulator valve.

Comparative tests have demonstrated a higher sensitivity, compared with

Tfsec 6


cylinders (powered heads for boring and internal facing, for milling, and similar operations) [2].

Figure 3 shows the layout of a hydraulic control system for a feed mech-anism with small loads. The volume on side 7 of the cylinder is connected to

I V P-

1 1 \

I " 1

[Pressure' ι relay , ι I

F I G . 3. Hydraulic layout for a feed mechanism for small loads.

the pump through a metering device incorporating a reducing valve 2 and throttle 3; the volume 4 (piston rod side) is connected directly to the pump. A peculiar characteristic of this arrangement is the reduction of the pressure difference across the piston as the load increases. This occurs be-cause the increasing load requires an increase of pressure in volume 7, which then approaches the pressure level in volume 4. Therefore with increase of load the leakage past the piston diminishes. This feature permits the use of pistons without seals, with the obvious advantage of reduction in the mag-


H Y D R A U L I C T R A N S F E R D R I V E S

Established designs of hydraulic-mechanical systems for the reciprocating

movements of automatic lines make use of standard hydraulic panels U 2423

[1], and speed-increase rack-and-pinion transmissions.

Figure 4 shows a diagram of a transfer drive, consisting of guides 7,

hydraulic cylinder 2, rack-and-pinion unit 5, and table 4, which is fastened

to the bracket connected to the drive. Slowing down of the table at the end-

stops is effected by a cam-operated deceleration valve for the forward stroke

(loaded with production parts), and by a dashpot 6 built into the cylinder head,

Reverse Forward

F I G . 4. D i a g r a m of a transfer drive.

forthereturn stroke(empty).The table position is controlled by a pressure switch 7, operated when the drive reaches the stop 8 (forward stroke), and by a limit-switch for the return stroke. The drive is powered by oil from a twin-vane pump ; this, together with a differential cylinder and a rack-and-pinion, allows the parts to move with a high speed, although the pump capacities are small.

5 a Aizerman I

nitude and range of variation of friction forces. This arrangement is parti-

cularly suitable when the feed force must be strictly limited to avoid tool-

breakage—for example, in small "powered heads" for drilling.

The high sensitivity of the arrangement shown on Fig. 3 makes it applicable

also for the automatic control of stepped feeds (for deep, small-diameter

drilling, etc.), where the signal for tool retraction is given by the increase of

axial force, or torque. There are also mechanisms for stepped feeds with

automatic retraction after a pre-determined length or time [3].


When designing such drives, it has been found necessary to solve the

problem of deceleration so that the production parts would not be thrown off.

The Institute of Machine Mechanics of the Academy of Science U.S.S.R.

(A.E.Tsukhanova, Candidate of Technological Sciences), in co-operation

with SKB-1 , have investigated the operation of transporter drives, and have

evaluated the relevant parameters for the retarding lands of the decelera-

9 1 V L

F I G . 5. Hydraulic layout for rotational movement.

tion valve, to obtain a deceleration of about 1 m/sec2, with the mean speed

of the translatory movement up to 12-15 m/min. The control of various rotating mechanisms in the automatic control

equipment is effected by special rotary valves. T o save time, rotation usually occurs simultaneously with the operations of clamping or release. The rotating mechanisms are actuated by hydraulic cylinders with dash-pots at both ends. These cylinders are controlled by rotary valves, which receive hydraulic signals from the clamping devices. The cylinders are usually fed by oil from the hydraulic supply to the transporters, with which they work in sequence. The end of rotation is controlled by electric limit-switches.

Figure 5 is a diagram of a typical rotating operation. It works in the fol-lowing way: when the parts are clamped, oil from the clamping system enters


H Y D R A U L I C C L A M P I N G D E V I C E S

Several types of hydraulic systems are used for location and clamping of

elements [1] :

(a) With the hydraulic panels U 2423 for direct clamping, when the pres-

sures available for clamping and release are equal.

(b) With the panels U 2424 for wedge clamping devices, with separate control of pressure for clamping and release; the latter is set to be higher.

(c) With the panels U 2425 for both location and clamping, where the

clamping operation is initiated by the increase of pressure when loca-

tion has been completed.

Usually a large number of cylinders (up to 50, or even more) are energized

in a clamping operation. The end of the clamping operation is controlled by

a pressure switch, which operates when the last piston has completed its

stroke.

In more complicated systems for location and clamping, with several operations in sequence, use is made of combined control systems with elec-trical, as well as hydraulic, signals.

Figure 6 shows the hydraulic system for locating and clamping for one of the automatic lines built by the Machine-tool Works S. Ordzhonikidze. In this system the operations of location and clamping are accomplished in four stages. Appropriate cylinders are brought into action by two solenoid spool valves 7 and 2, and two sets of auxiliary valves 3 and 4. The solenoid of the valve 2 is switched on by the action of pressure switch 9. Auxiliary valves re-spond to the pressure rise in cylinders 5 and 6, which occurs when their pis-tons reach the limits of stroke. The system is provided with relief valves for clamping (7 ) and release (6").

through pipeline 1 into the volume 2 of the rack-and-pinion actuator, mov-

ing its piston 3 to the left; this causes rotary valve 5 to turn through a ratchet

wheel 4, which connects the pressure line 6 with the volume 7 of the operat-

ing cylinder 8 ; the piston of this cylinder actuates the rotation of the part. When

the clamps are released, oil is directed to the pipeline 9, returning the piston 3 to

its initial position. The ratchet wheel, and consequently the rotary valve 5,

remain at standstill, being held by an index roller 10.

If it is also required to turn the parts during the release operation, the

ratchet-wheel mechanism is omitted.

130

PN

EU

MA

TI

C

AN

D H

YD

RA

UL

IC

C

ON

TR

OL

FIG. 6. Hydraulic layout for the locating and clamping devices for an automatic line.


H Y D R A U L I C M E A S U R I N G D E V I C E S

The checking of the main dimensions of some workpieces, machined on

the transfer line, and checking that holes which are to be threaded have been

drilled and are of the correct depth are accomplished by feeler devices. They are

moved by hydraulic cylinders. I f the depth is sufficient, and if the dimensions

ffl Pi Pi fi ft

10

9 I f 5

F I G . 7. Hydraulic layout for checking threaded holes.

are to the drawing, the plunger of the cylinder moves the feelers forward. They then operate limit-switches and so allow the subsequent machining operation to begin. The hydraulic feed of the feeling devices permits a limit to the forces which can be applied to feelers. This is particularly necessary when checking holes of small diameter.

Figure 7 shows the hydraulic layout of the feeler device for checking the depth of holes which are to be threaded. The device consists of a hydraulic control panel 1 and a plunger-type cylinder 2, or several cylinders connected in parallel. The system is usually supplied by oil from the hydraulic power


line to the clamping devices. In the clamping operation, oil is fed through the

pipelines 3 and 4 to the hydraulic control panel 7. The spool valve 5 directs

oil through the pipeline 6 to the cylinder 2, moving its plunger with feeler

pins 7. The speed of the plunger motion is limited by the throttle #, and the

pressure in the cylinder 2 by the pressure-reducing valve 9. The plunger is re-

tracted by the spring 10 simultaneously with the release of clamps. The line 4 is

then connected to the tank, and line 7 7 to the pressure. Oil from the cylinder 2

is drained into the tank through the passage 72.

H Y D R A U L I C V I B R A T O R S

Hydraulic vibrators shake out swarf and chips from machined cavities. Figure 8 is the diagram of the vibrator drive for an automatic transfer line. The vibrator is controlled by the panel 7, having a solenoid spool-valve 2 and

F I G . 8. Hydraulic layout of a vibrator.


throttle 3. The latter controls the amount of oil flowing to the vibrator drive

4, and consequently the frequency of oscillation of the piston 5. The opera-

tion of the piston is effected by the floating spool-valve 6 and reversing

spool-valve 7. The axial gap a and the stroke of the reversing valve 7 deter-

mine the amplitude of vibrations. The frequency is controlled by the throttle

5, and may be varied from 0*5 to 8 c/s.

G E N E R A L A R R A N G E M E N T O F H Y D R A U L I C U N I T S

The hydraulic equipment of automatic lines designed by SKB-1 is ar-

ranged in complete panels, in which the valves are located in the bores of a

single or several blocks, and inter-connected by drilled holes. This system

F I G . 9. Hydraulic tank. 1—pump unit; 2—feed control panels with solenoid con-

trol; 3—air and oil filters.

permits the setting-up and adjustment of hydraulic elements before the com-plete unit is installed in the line, facilitates repairs and replacements, and reduces the number of pipes, as well as the overall dimensions of a hydraulic unit [4].

Hydraulic panels are attached to totally enclosed tanks, insulated by filters from the atmosphere.

Figure 9 is a photograph of a hydraulic tank for one of the automatic lines, made by the Machine-tool Works S.Ordzhonikidze.


R E F E R E N C E S

1. L . S . B R O N , Stanki i Instrument}?, N o . 7, 1953.

2. L . S . B R O N , Stanki i Instrument)?, N o . 8, 1954.

3. L . S . B R O N , Stanki i Instrumenty, N o . 3, 1956.

4. L . S . B R O N , Stanki i Instrumenty, N o . 8, 1956.

ELEMENTS OF HYDRAULIC CONTROL SYSTEMS

B . F . S T U P A K

T H E ELEMENTS described in this article are used in hydraulic systems for the

control of variable delivery pumps (Fig. 1), in power systems with semi-

! I I

I I

To the pump

F I G . 1. Arrangement of a hydraulic amplifier unit.

135


rotary vane servo-motors (Fig. 2), and in reciprocating power systems with

piston or plunger type servo-motors. These control systems are often re-

ferred to as hydraulic amplifiers.

Most hydraulic amplifiers have an electrical input, i.e. are controlled by

an electrical input signal, with a mechanical output, at a force level of tens

c e b d

F I G . 2. Arrangement of a hydraulic amplifier for remote control.

to hundreds of kilograms, or a torque level of tens to hundreds of kilogram-metres.

Figures 1 and 2 show the layout of two hydraulic amplifiers. Each has a control unit a9 spool valve b, oil filter c, relief valve d, oil pump e, hydraulic motor / , and feedback device g.

In the first amplifier (Fig. 1) the hydraulic motor is not separated from the other elements, but is enclosed with them in a common housing, which forms the hydraulic tank. This is called an amplifier unit. The feedback linkage is mechanical.

The second amplifier (Fig. 2) has the hydraulic motors located remotely at a distance of tens of metres. This is termed a remote action amplifier. The feedback signal is obtained electrically.

C O N T R O L U N I T S

The electrical signal, fed to the control unit, is transformed into a mech-anical displacement in the form of a shaft rotation of the motor type A D P or NED-101P, or as a straight-line motion in the magnet units.

E L E M E N T S O F H Y D R A U L I C C O N T R O L S Y S T E M S 137

Hydraulic amplifiers, with a spool valve diameter usually from 5 to 20 mm

and supply pressure up to 25 kg/cm2, use as a control element an electric

motor type ADP-263A. This is an asynchronous two-phase capacitor-con-

trolled motor with a hollow non-magnetic rotor of small inertia, and with a

supply at 31 V nominal, frequency 500 c/s and torque 450 g/cm. The signal

from the controlling instrument to the field coil of the A D P is increased

through a magnetic amplifier.

Hydraulic systems containing a spool valve with hydraulic amplification

use an electric motor type ADP-123 at a nominal voltage of 110 V , fre-

quency 400-500 c/s, with a starting torque of 140 g/cm, or a motor type

ADP-135B, which differs from the previous type in having a starting torque

of 170 g/cm, or a converter of the induction type, described in Ref. 1.

Systems which do not contain a magnetic amplifier, which are used when

a lower degree of accuracy is acceptable, use for the motor the type NED-101

with a frequency of 427 c/s, for the control element the unit type ND-404P,

and for a feedback transmitter the unit type BS-404P.

There is also a design incorporating rotary transformers type S K W T , size

N o . l , for both input setting and as a feedback transmitter. N o magnetic am-

plifier is incorporated and the simple 8 mm diameter spool valve has no

hydraulic amplification, but is actuated by an electric motor type APD-363,

which has a starting torque of 500 g/cm.

In the control unit shown in Fig. 1, an angular displacement of ±330° of

the electric motor shaft displaces the spool valve by ±5-54 mm. This unit

consists of the motor 7, reduction gear 2, a safety spring device 3 of the

"scissors" type, centring spring 4, and an air dashpot 5. A control unit

having spool valves with hydraulic amplification is shown in Fig. 2. Here, the

shaft of the electric motor NED-101P has a rigidly connected T-shaped lever,

with one arm provided with a counterbalanced weight, a second arm with an

air dashpot, with the third connexion to a rod which controls a sleeve valve.

A third type of control unit is shown in Fig. 3,a, together with its spool valve.

This is an electromagnetic transducer, supplied by d.c. at a voltage varying

from 0 to 130 V ; the resistance of each coil is 6500 Ω and the windings are con-

nected differentially; the controlling power is 2-6 W , when the useful axial

load is 50 g, with a stroke 0*25 mm, and with a differential current in the

windings of 16-18 mA. In a dynamic converter the power is 10 W , with a

stroke of 0-6 mm, the control current is 37-50 mA, resistance of the control

coil, 4500 Ω, voltage from 0-200 V , stiffness (spring rate) of the diaphragm

1750 g/mm.

S P O O L V A L V E S

There are two types of spool valve used in hydraulic systems: those without hydraulic amplification, and those with it.

Spool valves without any hydraulic amplification have a diameter from

138

PN

EU

MA

TI

C

AN

D

HY

DR

AU

LI

C

CO

NT

RO

L

FIG. 3,a. Control unit: a -connected to the spool valve; b—with hydraulic amplification.


5 to 20 mm. The sleeve and spool are made of 12 KhN3A-type steel,| carbur-

ized and hardened to Rc = 58-62. Sliding surfaces are lapped. During the

course of manufacture, the bores in the valve bodies and sleeves are mea-

sured by pneumatic gauges type TF-17-12 of the "Kalibr" organization.

The fixed sleeve and the valve body have an interference fit:

Outer dia., mm Interference, mm

< 3 0 0-010-0-020

> 3 0 0-015-0-025

Between the sleeve and spool, or between a moving sleeve and a valve body,

we have the following clearances:

D i a . , mm clearance, mm

< 2 0 0-005-0-010

20-40 0-008-0-015

> 4 0 0-010-0-020

In certain cases the sleeve is fitted to the valve body to give a sliding fit of second class accuracy, and provided with O-rings of oil-resistant rubber.

In deciding the design of a valve with hydraulic amplification, three alter-natives have been tested: with a needle control; with a hollow pilot-spindle and piston; and with a hollow pilot-spindle and calibrated orifices. After comparative tests, the second alternative (Fig. 3) has been found preferable. It is more reliable than the first alternative and is free from the additional oil loss characteristic of the third alternative.

In spool valves of this type (Fig. 3), the hollow spindle 1 is displaced by the control unit a, and the main spool 2 follows the displacements of the pilot spindle, and so controls the oil flow rate. The follow-up action between the main spool and the pilot spindle is decribed in Ref. 2. Oil passes to the left-hand end of the main spool, through a calibrated orifice. Its diameter is 0-5 mm for a spool of 12 mm diameter, and 0-8 mm for a spool 45 mm dia. The diameters of the pilot spindle are 2-5 mm and 5 mm respectively. A specific feature of the valve shown in Fig. 3,b is the separate tank return from the pilot spindle,* which simplifies the design and does not require a relief valve and metering orifice at the entry to the pilot line. With this arrangement of the return oil, and with a pressure of 30 kg/cm

2 in the pilot line, the force needed to

t Containing chromium and nickel (Translator) . φ Figure 3,a is not suitable for an explanation of the valve's operation, because it is not

a diagram, but an assembly drawing. Figure 3,b (taken from Ref. 1, page 604) may be of

some assistance, though it does not necessarily accord with Fig. 3,a in everything; for

example, the separate return from the pilot line is not shown. References 1 and 2 give a

thorough mathematical analysis of the amplifier shown in the diagram (Translator).


move the pilot valve of 2-5 mm diameter does not exceed 4 g. The valve unit

shown in Fig. 3 is designed to be externally mounted at either the amplifying

system, or close to the output hydraulic motor; the latter arrangement elim-

inates the effects of elasticity in long pipelines on the performance of the

1 I ;

Return

Rassure

F I G . 3,b.

follow-up system. The sleeve of this valve is formed by a stack of separate

rings, having precise axial dimensions; the tolerance on ring thickness is

0Ό005 mm. This design of the sleeve (first used by the Machine-tool Works

S.Ordzhonikidze) improves the accuracy and eliminates the most laborious

F I G . 4. Spool valve without hydraulic amplification.


fitting operation—manual finishing of the sleeve ports. I f the valve is to be

attached directly to another unit, pipe connectors are omitted.

Figure 5 shows a spool valve with manual and solenoid control. The

electrical signal from a detecting unit is fed to the windings of the "pusher"

type solenoid / , which develop a force of 7 kg with a stroke of 5 mm and a

F I G . 5. Spool valve with manual and solenoid control.

current of 0-3 A . Each solenoid has two windings: a control winding, 197 Ω and a damping winding, 55 Ω. The quoted force is developed only when the control windings are energized. A double-piston adjustable oil dashpot 2 is provided to resist hydraulic shocks. The adjustment control for the dashpot is placed on the face of the instrument, above the manual control lever, and is protected by a special cover 3. When the solenoid operates, the manual control mechanism remains stationary, because the valve-spool extension is slotted to a suitable length. For manual control, turning the handle produces a displacement of a lever which takes up the backlash (half of the dif-


ference between the slot length and the thickness of the lever end), and dis-

places the valve spool. The return of the spool is brought about by the

spring 4, mounted on the spool extension, and the handle is returned by its

own spring, located on the axis and closed by the cover 5. The working

volume of the valve body is pressure-tested to 150 kg/cm2. An inner recess

ι ! ι ι ; ι

F I G . 6. Spool valve with manual, solenoid, and hydraulic control.

in the valve body houses the screw contacts for the solenoids and a signal bulb which glows when the solenoid control is operating. The electrical sup-ply cable is led in through a packing. The force at the operating handle does not exceed 6*5 kg. The current needed to move the spool with a pressure of 100 kg/cm

2 does not exceed 300 mA, and the release current is not less than

6 0 mA.

A spool valve with manual, electrical and hydraulic control (Fig. 6) has been developed for large oil flows, when spool valve control (manual and solenoid control particularly) becomes difficult^ Here the manual or elec-trical (by means of solenoid 7) system of control acts only on the pilot valve 2, and the main spool valve 5, which has a diameter 30 mm, is controlled

t A force of 2-5-3 kg is needed to move a 12 mm spool without springs or magnets, held

at rest for 5-8 min with a supply pressure of 100 kg /cm2.


O I L F I L T E R S

Gauze filters with a brass mesh N o . 0125 or N o . 0112 (Standard GOST 6613-53) are used for hydraulic amplifiers. The construction of these filters is shown in Fig. 7, and their characteristics in Table 1.

T A B L E 1

Flow capacity Dimensions, mm

Area of one N o . of

l./min Outer dia. Inner dia. Length element, c m2

elements

8 44 15 42 6-6 10

12 70 20 51 19-8 8

18 70 20 64 19-8 12

25 70 20 76 19-8 16

70 82 25 110 33-4 18

F I G . 7. Oil filter.

entirely hydraulically. The centring springs are here replaced by the cen-

tring pistons 4. Hydraulic resistances and special metering devices allow the

speed of the spool displacement to be controlled.

A t the limits of the main spool stroke, damping is assisted by notches

formed in the working edges of the spool lands. When the valve is controlled

manually, proportionality is ensured by a mechanical feedback link 5 be-

tween the pilot and main valves.


The calculated flow capacities are based on a flow of 0-12 l./min/cm2 for

the mesh No.0125, which is ten times the flow at 0-3 kg/cm2 pressure drop.

Filters for the protection of the duralumin bodies of the oil pumps are in-

stalled in the suction lines. The "hub" 1 is the base, on which all filters

elements are mounted. The cover 7 and five filter sections, each consisting of

two mesh elements 8 and two separating rings 5 and 6, are threaded onto the

"hub".

The whole assembly is clamped to the inner diameter by the nut 3, and is

closed at the outer diameter by the cover 4 and clamped by the nut 2. In

operation, the filter is completely submerged in oil. The oil flow direction is

shown in Fig. 7 by arrows. The hollow "hub" is screwed into the body of the

pump unit (Fig. 9).

O I L P U M P S

The filtered oil (the filter does not pass particles exceeding 0*045 mm) enters the gear pump (Fig. 8). The pump consists of two steel gears I and 2 with corrected teeth, duralumin body 3, and duralumin covers 4 and 5. T o eliminate axial forces, the shaft 6 is hollow, and the cover 5 has vent holes 7.

F I G . 8. Oil gear pump.


F I G . 9. Pump unit.

can be connected to a motor by a coupling, fastened with the key 11 and nut

72. The pump is fitted into a casing with the fit, class A / D , forming a pump-

ing unit together with the filter and the relief valve (Fig. 9). The inlet and

T A B L E 2

Capacity

l./min

Shaft

speed

rev/min

Working

pressure

k g / c m2

Dimensions, mm

Capacity

l./min

Shaft

speed

rev/min

Working

pressure

k g / c m2 Flange

dia. D

Overall

length /

Shaft and

distance

C from

flange face

Body dia.

d

3 1400 20 72 98 36 52

6 1400 20 90 102 35 68

4 1500 20 135 150 61 100

delivery ports of the pump are matched with the ports in the casing. Special tests have confirmed that with the fit of class A / D there is no reduction of pump capacity due to leakage between the pump body and casing. Table 2 gives the technical data of the pumps.

The driving shaft 8 has a double oil seal with garter springs. T o facilitate

replacements, oil seals are fitted into a bush 9, retained by a circlip. The

driving gear 1 is keyed to the shaft 8 by a key 10. The outer end of the shaft


R E L I E F V A L V E S

Relief valves, which return the fluid from the system to the tank, serve the

purpose of maintaining pressure in the system, and protecting the system

against excessive pressure. Two types of relief valve are used for hydraulic

amplifiers: the unbalanced, direct-acting type (Fig. 10), and the balanced

F I G . 10. Unbalanced direct-acting F I G . 11. Relief valve with a spool

relief valve. and ball-valve control.

type with a spool valve and a ball pilot-valve (Fig. 11). The latter is made for two alternatives: for internal fitting below the oil level, and for external fitting in pipelines. In the unbalanced relief valve (Fig. 10) oil is returned through the ports 4. The body 7 is made of steel St 35 and oxidized, and the valve itself 2 of steel 12KhN3A, hardened to 58-62 Rc. The valve is constructed with a free-running fit; a closer fit would result in the valve operating with a peculiar high-pitched noise.

Table 3 gives the technical details of unbalanced relief valves. The unbalanced relief valve has a comparatively large spring 3. Its ad-

vantage is the possibility of accurate and gradual adjustment. The range of


adjustment is usually smaller than that of a balanced relief valve. The un-

balanced valve is simple, small in overall dimensions, and light in weight. Its

main disadvantage is the violent vibration, accompanied by loud high-

pitched noises, which occurs when the nominal capacity is exceeded.

T A B L E 3

Capacity

l./min

Range of

pressures

kg / cm2

Outer dia.

of valve

Dimensions, mm

Capacity

l./min

Range of

pressures

kg / cm2

Outer dia.

of valve Overall

outer dia.

Length of

thread Thread

Overall

length

2 6-10 8 20 8 1M14 χ 1-5 56

3 0-25 8 22 8 1M14 χ 1-5 63

8 6-14 14 27 12 1M22 χ 1-5 80

12 0-25 14 35 12 1M22 χ 1-5 105

The balanced relief valve (Fig. 11) is a further development of the valve

G-52 designed by E N I M S and available in quantity production. The pres-

sure relief spool 2 is balanced axially, as long as the ball valve 3 is closed.

This is due to the pressure balance between the cavities 7 and 6 through the

passage holes 6", and between cavities 6 and 5 through the central hole in the

spool 2. When the pressure in these cavities increases sufficiently, the ball

valve 5, loaded by the spring 70, opens. Oil is released from the cavity 5

through passage 77, and as a result, the pressure in cavity 5 rapidly falls, as

the oil entering this cavity from 6 must pass through a calibrated restrictor

orifice 9, with a diameter 1-1 mm, formed in the plug 4. The spool 2 then

lifts and connects the cavity 6 with the return port 13.

T A B L E 4

Capacity

l./min

Range of Dimensions, mm

Capacity

l./min pressures

kg /cm2 Outer

dia. D

Hexagon

A . F .

Length

L

Length of

thread L

Thread

18 2-25 53-5 46 146 12 1M30 x 2 18 2-65 53-5 46 132 22 2M27 x 1-5

When the pressure falls, spring 72 returns the spool 2 to its lower position. This spring exerts a force of 1-7 kg. The body 7 of the internal type valve is made of duralumin, and that of the external (pipeline) type is made of the steel St35. Internal type valves are screwed into the casings of pump units.

Technical details of balanced relief valves are given in Table 4.


F I G . 12. Piston type hydraulic actuator (ram) with a double piston rod.

H Y D R A U L I C A C T U A T O R S

Hydraulic amplifiers use hydraulic actuators of the piston, plunger and

vane (abutment) type.


Piston actuators (rams or jacks) are made either with double piston rods

(Fig. 12) or with a differential piston (Fig. 13). The piston rod shown in Fig. 12

passes through end covers without any seals. This is permissible, because this

particular ram is located below oil level. As distinct from this type, the piston

rod shown in Fig. 13 is sealed by O-rings.

F I G . 13 . Piston type hydraulic actuator

( ram) with a differential piston.

F I G . 14. Plunger type hydraulic actuator

(ram) .

In both these hydraulic actuators the pistons have no seals; they are ac-curately machined and lapped, permissible clearances being from 0-03 to 0*04 mm. Plunger hydraulic rams are the cheapest and the most convenient for manufacture, because they require accurate machining only along the short length of cylinder bore taking the bush. The first type of plunger actuators


were provided with double U-seals (Fig. 14) and subsequent types with dou-

ble O-rings.

The vane-type hydraulic actuator (abutment engine) (Fig. 15,a and 15,b) is

very convenient for reciprocating angular motion. It consists of four basic

parts: the body / , two covers 3 and 4, and vane 5, rigidly connected to the

shaft 2. The cover 4 has a flange 6, which fixes it to the driven load. It has

been found in practice that the best way of mounting this vane-type actuator

is by using hardened steel stops or a ball joint 7, to restrain the body from

F I G . 15,a. Vane type hydraulic motor (details).

turning. This eliminates the possibility of warping causing misalignment be-tween the vane and the body. The vane is attached to the shaft by the serrated face 8. The shaft has corresponding serrations, and an internal thread for a tightening bolt. The main pipe connectors are located in the lower part of the body, and the upper part contains small air-bleed screws with non-return ball valves. The vane is sealed on its perimeter by a U-seal of oil-resistant rubber. To reduce wear of the rubber due to friction with the surface of the covers and body, these surfaces are polished and chrome-plated. For the bearings, the most suitable material proved to be textolite. [A fabric-reinforced phenol-formaldehyde material. Translator.] Calculations of textolite bearing dimen-sions were based on a unit load of 100 kg/cm2.

A peculiar feature of this hydraulic actuator is the hydraulic dashpots, which cushion the impact of the vane against the stop at the extreme of movement, with sharp reversal. When designing a drive with a vane type hydraulic actuator it is necessary to note that the inertial forces are greater than those with an electric motor and reducing gear drive where the moment of inertia of the parts on the final shaft is divided by the square of the transmission


620

F I G . 15,b. Vane type hydraulic actuator (general assembly).

T A B L E 5

Nominal

pressure

difference

at the

vane

kg /cm2

Nominal

torque

kgm

Effective

vane

area

c m2

Radius of

pressure

centre

cm

Unit

volume,

per 1°

of

rotation

c m3

Weight

kg

Leakages

past

vanes

cm3/min

M a x .

angle

of

turn

25 93 52 — — — ± 3 5 °

10 12 22-9 6-6 2-64 6 < 4 0 ± 1 5 °

20 88 52 10-5 9-25 17 < 1 0 0 ± 2 5 °

25 135 54 10-9 10-25 80 — ± 1 5 0 °

25 1000 235 20-25 83-00 160 - ± 2 2 °

25 93 52 — — — ± 1 0 °

6 Aizerman I


ratio of electric motor drive.f It is important to take this into account, parti-

cularly for drives with rapid reversing and high inertial loads.

In Table 5 are given the technical data of existing vane type hydraulic

actuators.

The main parts of hydraulic actuators—vane, body and covers—are made of

steel 40Cr with heat-treatment ensuring a yield point not less than 75 kg/mm2.

R E S T R I C T O R S ( T H R O T T L E S )

The speed of hydraulic amplifiers, supplied by oil from a common main

pressure line, is controlled by hydraulic resistances of the groove, screw or

needle types, which control the speed by varying the oil flow rate. Restrictors

d/am.2

*6 H

F I G . 16. Restrictor of the groove type.

of the groove type (Fig. 16) are most widely used. Their resistance is com-paratively less affected by changes in oil temperature. Restrictors of the groove type are suitable for the control of flows from 5 to 100cm

3/sec at a

t Logic of this statement appears to be doubtful. Since the mechanism under considera-

tion has stops at the output (slow-running) end, the effects of electric motor inertia would

be only detrimental, and the smallness of its counterparts in a hydraulic system an ad-

vantage (Translator) .

EL

EM

EN

TS

O

F

HY

DR

AU

LI

C

CO

NT

RO

L

SY

ST

EM

S

FIG. 17. Restrictor of the screw type.

153


pressure up to 100 kg/cm2. Control is effected by turning the plug from 0 to

210°.

With a stable temperature good results are obtained from a screw restric-tor (Fig. 17). It has a rectangular screw thread, the working length of which can be adjusted within ± 16 mm by turning the knob. The screw restrictor is much larger than the groove type, but has the advantage of a more gradual characteristic. It is designed for pressures up to 100 kg/cm

2, and flows from

1 to 10 cm3/sec.

R E F E R E N C E S

1. V . A . K O T E L N I K O V and V. A . K H O K H L O V , Avtomatika i Telemekhanika, Vo l . X V I I , N o . 7,

1956.

2. V . A . K H O K H L O V , Avtomatika i Telemekhanika, Vol . X V I I , N o . 10, 1956 .

THREE-TERM CONTROLLER SET KB-TsMA

M . L . P O D G O Y E T S K I I and E . M . B R A V E R M A N

T H E Design Bureau of Automation for Non-ferrous Metals Industry has de-

veloped a three-term pneumatic controller with adjustable proportional

band, integral action time, and derivative action time.

D E S I G N O F T H E C O N T R O L L E R

The controller (Fig. 1) consists of three units. The input to unit I is the

pressure Pt, proportional to the measured quantity. It is fed to chamber 2.

The output from the unit is the pressure P3 which is produced in chamber 3,

P10 = 0-5afm

Unit 1 Unit II Unit I

F I G . 1. Arrangement of the three-term controller K B - T s M A .

located between a fixed restriction a, adjustable restriction Tu capillary re-sistance R and the controller pilot-valve—a baffle-nozzle element Ca. The baffle of the element Ca moves together with a rigid stem connecting the dia-phragms of the unit. The stem balances the forces set up by pressures Pt (in the measuring chamber 2) , Pd (in the set value chamber 7), P 4 (in the negative feedback chamber 4) and P10 = 0-5 atm (gauge) (in chamber 70). A t equi-librium, the sum of all these forces acting on the stem is zero, and the pres-sures in chambers 3 and 4 are equalized. Then we have:

Λ - Λ ο = ^ ( Λ - Λ ) , 0) J 2

157


where / i is the effective area of the diaphragm between chambers 1 and 2 and f2 the effective areas of other diaphragms of the unit.

Figure 2,a shows a static characteristic of the unit 1 according to equa-tion (1).

The adjustable restriction Tl9 located between chambers 3 and 4, is used to adjust the derivative action time. Because of this, the whole unit is called the derivative action unit.

The input to unit / / is the pressure P3 and the output-pressure P6 in chamber 6, located between fixed restriction b, adjustable restriction 0, and baffle-nozzle element Cb. The baffle of Cb is in equilibrium when the pressure baffle in chamber 5, P5 = P10 = 0-5 atm. The pressure drop P3 - P5 across the capillary resistance R determines the flow of air through this line and, for a given setting of the adjustable restriction, the pressure drop P5 — P6

across this restriction. Taking the flow of air to be approximately propor-tional to the pressure differentials, we obtain:

kR(P* - P5) = kô(P5 - P6); (2)

as P5 = P10 = 0-5 atm, it follows that

Ρβ-Ριο= ~{Ρ*-Ριο)· (3) kô

Here kR and kb are the discharge coefficients of the capillary resistance R and throttle ô.

The static characteristics of unit II, as given by equation (3), are shown in Fig. 2,b. The slope of each graph is determined by the magnitude of k&, which can be varied from 0 to oo ; consequently kR\kb also can vary from 0 to oo (practically it is possible to realize 0-05 < kRjkô < 40). As the ratio kRjkô

ultimately determines the internal proportional control factor of the control-ler, and is adjusted by elements located in unit / / , the whole unit is called the proportional band adjustment unit.

The input to unit / / / is the pressure P6 in chamber 6. The output from this unit is the pressure Pouir>, which then goes to a regulating mechanism. The pressure P o u tp appears as the output of the power amplifier included in this unit and is controlled by pressure P9 in the chamber between the fixed restriction C and the nozzle-baffle element Cc. The baffle of Cc mo-ves with the stem, which balances the forces resulting from pressures P6

(in chamber 6), P10 (in chamber 10), P8 = Λ > υ ΐ Ρ (in the positive feedback chamber), and P7 (in the negative feedback chamber). I f the adjustable throttle T2 is completely closed, and pressure Ρη = P l 0 , then at the equi-librium:

Λ η π ρ - Λο = --r^-r ( p e - Λ ο ) . (4) J 2 ~ f l

T H R E E - T E R M C O N T R O L L E R SET K B - T S M A 159

Here fx and f2 have the same meaning as in (1). Figure 2,c is the static

characteristic of unit I I I , as given by equation (4), that is, for proportio-

nal action only. The adjustable restriction T2 between chambers 7 and 8 is

Pjjo/m.

1-0

08

0-6

^04

0-2

-06 -04 -0-2 0 02 04 06

AP = Pl-Pdafm.

a

Podμ*P8>Qtm

-10

04 0-6 0-t 05

P6;afm.

10 r

0-6

05

04

0-2

0,

m hi m hi

\\ ι 1

! 1 1

'01 0-2 0-4 0-6 0-8 10 05

P3, aim.

b

Poutp.'afm

-

10

0-8

0-6

04

0-2

-0-6-04 -02 0 0-2 04 0-6

AP=PL-Pd atm.

d

F I G . 2. Static characteristics of units of the three-term controller.

used for setting the integral action time. This unit is hence known as the integral action adjusting unit. From equations (1), (3) and (4) it follows that

Λ > α ί ρ - Λ ο = ^ ( Λ - Λ ) . (5)

Figure 2,d shows the static characteristics (as given by (5)) of the whole controller set when working in the proportional action mode.

S T A T I C C H A R A C T E R I S T I C S O F T H E C O N T R O L L E R

The static characteristics (P6 as a function of P3) of the controller, shown in Fig.2,b, are only a rough approximation, because they conform to equa-tion (2), which assumes a direct proportionality between air flow through the

6 a Aizerman I


restriction and pressure differential across it. For more accurate results, it is

necessary to replace equation (2) either by:

/ Γ / Ρ \2/"' / Ρ \ ( « + i ) / » n

" Λ Μ(Τ7) - { * ) ]

or by:

/ Γ / Ρ \2

lm / Ρ \ ( " '+ ΐ ) /»π

kR(P\ - Pi) = kô{P\ - Pi),

(6)

(7)

where SR and Sô are the orifice areas of restrictions R and ό, μ.Α and μδ their discharge coefficients, and m the adiabatic exponent. Equation (6) applies for turbulent flow through restrictions R and ό, when Re > 3000; then the dis-charge coefficients remain constant. Equation (7) applies for laminar flow when Re < 2300. Numerical calculations on the basis of equations (6) and (7) have shown that the static characteristics are closer to linearity under condi-tions of laminar flow through both restrictions. Therefore, the capillary re-sistance and the restriction should be chosen to ensure the laminar flow. Taking this for granted, the static characteristics P6 = F(P3) were calculated

P6,atm.

0-8

0-6

0+

02

\ , 7

0 02 04 06 0-8 10 12

atm.

F I G . 3. Static characteristics of the proportional band adjustment unit.

according to equation (7) ; these are shown in Fig. 3. An experimental static characteristic has been obtained for the case when kRjkô = 1, and the points plotted on Fig. 3. They confirm the suitability of equation (7).

The static characteristics of the controller are distorted by any inequality between the areas of the outer diaphragms, enclosing the chambers 7, 2 and

T H R E E - T E R M C O N T R O L L E R SET K B - T S M A 161

7, #, and also by inequality between the effective areas of the diaphragms of

unit / / . This inequality may be caused by an incorrect setting of the stem in

relation to the diaphragm seating.! Figure 4 illustrates the distortion of the

static characteristics in the case when the planes of the centre discs do not

02 04 06 08 f-0 1-2 M Pifz, atm.

atm.

. atm.

F i g . 4. The influence of diaphragm stem position on the static characteristics of a

relay: a—incorrect, and b—correct position of stem. Curve J—P3 = F(P2) at

Pi = 0; Curve 2—P3 = F(Px) at P2 = 1-2 atm.

coincide with the planes of the flange. In unit / such distortion of the static

characteristics leads to a difference between nominal and actual values of

parameters when the set value is altered ; in unit 77, to the displacement of the

"control point" when the proportional control factor is altered; and in unit

/ / / , to an offset, the amount of which is determined by the equation:

A = (fl ~ / ï } Üilh-, (8) fi ~ fi fif\ kô

where /2 and f'2 are the effective areas of the larger diaphragms of unit/ / /

when incorrectly set.

Static tests of the controller, and tests of a model, have confirmed the

necessity for an accurate adjustment of the diaphragm stem. For this pur-

pose, the baffles have been provided with wedges (Figs. 1 and 4) so that the

position of the stem can be altered by moving the flappers radially.

t See the article by V . V . Afanasyev, "Variations of the effective areas of diaphragms",

in the present book, p. 311.


C O N S T R U C T I O N A N D O P E R A T I O N A L E Q U A T I O N

OF T H E C O N T R O L L E R

Here we shall analyse the performance of the three-term controller (Fig. 1) as determined by its construction.

Three-term controllers used in industry work according to the following equation :

γ = κ(τίΡ+ 1 + (9)

where y is the output variable of the controller; χ the input variable; /Cthe gain (proportional action factor); Tt the derivative action time; 7} the inte-gral action time; and Ρ the differential operator.

K, Ti and Tj are the adjustment parameters of the controller. Equation (9), called the functional equation of a controller, can be realized by various combinations of dynamic and static elements—that is, by various construc-tions. Controllers so obtained would not have exactly identical properties, though they generate the same functional equation (9).

Further, we must know, at least approximately, the equation of a relay element when shunted by a linear feedback (Fig. 5) with the transfer func-tion Wt(p).

F I G . 5. Block diagram of a unit with a relay element: 1-

feedback link.

-relayelement; 2—linear

For very small deviations, the relay element may be considered as a static link with infinitely great gain. In this case, the overall transfer function of the whole link, shown in Fig. 5, is related to the transfer function of the linear feedback element as follows:

WQ(p) s - — . (10)

The approximate equity approaches the exact one as the amplitude of the input signal to the relay element approaches zero.

We assume that the system "baffle-nozzle" has infinite sensitivity (as far as the relationship between pressure and force applied to the baffle is concerned).

T H R E E - T E R M C O N T R O L L E R S E T K B - T S M A 163

If the pressure in the chamber preceding a nozzle is a controlling quantity for

a power amplifier, then the characteristics of the whole system (output pres-

sure of amplifier—force at the baffle) is also assumed to be infinitely sensitive.

Summator

PrPd

ML iL.

TiP+1

Poutp

F I G . 6. Block diagram of the three-term controller K B - T s M A .

The arrangement of the controller is shown in Fig. 6. Notations are the

same as for Fig. 1. The equation corresponding to this arrangement takes

the form:

k» \ T2pJ

where 7\ is the time constant of chamber 4 and T2 the time constant of chamber 7.

In this form, the equation will be called the operating equation, as distinct from (9) .

Comparing equations (9) and (11), we obtain

K=^L(I + 1L\, (12)

T,= Τ' , (13)

Tj = T2(l + l±). (14)

From (13) and (14), it follows that

T, T\ 2

and consequently if Tx = T2, we have

IL = (Ά) = I Tj \ Tj / ma x 4

(15)

(16)


Thus we arrive at the conclusion that this particular controller cannot be ad-

justed so that T, > Tj/4.

The relationships (12), (13) and (14) may also be expressed in the following

way:

Τ^ = Τ,Τ2, (17)

KTi = TJ-1-^—, (18) k6T2

— = - ^ - . (19) Tj kaT2

These equations indicate that the main parameters of the controller per-formance (Κ, Τι and Tj) remain unchanged, if adjustments of kô, T1 and T2

are carried out in such a way that TXT2 = const and kôT2 = const.

COMPACT HYDRAULIC CONTROLLERS IAT A N U.S.S.R.

B . M . D V O R E T S K I I

A T PRESENT the pneumatic devices of automatic control of the Aggregate

Unified System ( A U S ) , based on the force balance principle, are firmly es-

tablished in automatic control engineering. It is difficult, however, to build

a complete pneumatic control system, using these components, because of

the lack of good final regulators. The usual diaphragm regulators M I M are

not satisfactory if there is dry friction in the regulating element; and they also

necessitate a short travel of the regulating element. Existing hydraulic mech-

anisms, working on the follow-up principle, are free from these short-

comings, but it is difficult to build a complete hydraulic control system, be-

cause there are no satisfactory hydraulic controllers. The available jet-tube

type of hydraulic controller has a limited range of adjustment of integral

action time, and requires a prohibitive flow of working fluid. One is therefore

compelled to combine pneumatic controlling elements with hydraulic final

drives. This makes it necessary to have both a compressor plant for the

pneumatic controllers, and a pump installation for the hydraulics. If we are

to avoid this, it is necessary to have either reliable pneumatic regulators, or

a hydraulic controller able to compete with the pneumatic controllers of A U S .

Having in mind the second problem, the Institute of Automation and Tele-

mechanics of the Academy of Sciences U.S.S.R. carried out the design of a

hydraulic controller, based on the principle of force compensation, like the

pneumatic controllers A U S . The diagram of this controller is shown in

Fig. 1. The input signal is the difference between the measured pressure and

the pre-set pressure (Pt — Pd), ducted respectively to the measuring chamber

Kt and pre-setting chamber Kd.

The effective areas of the diaphragms Μ , , M2 and M3, which, together

with the rod Rl form the comparison element, are chosen so that when there

is a difference in pressures P{ and P2, the force acting on the rod is propor-

tional to this difference, and has the same sign. The movement of the rod Ri

alters the pressure Px in the chamber Cx, located between the fixed resis-

tance FRY and nozzle Nx. The pressure Px in the chamber Cx acts on the

diaphragm D9 of the secondary amplifying relay, which controls the output

pressure P2 of the unit by means of the moving nozzle N2 and a ball valve.

When the nozzle presses against the ball, chamber C2 is connected with the

165


pressure line, and this increases the pressure P2. When the nozzle moves

away from the ball, it connects the chamber C2 with the tank connexion, and

pressure P2 is reduced. Pressure P2, acting on the diaphragm D0, provides

the effect of negative feedback for the secondary relay.

In chamber C2, connected with chamber C 2 , the pressure is equal to the

output pressure. As it acts on the diaphragm it provides the negative feed-

F I G . 1. D i a g r a m of the hydraulic controller.

back effect. The chamber C 2 is also connected through a needle valve VR

(variable resistance) to the chamber C 3 , which in turn is connected through a fixed resistance FR2 to the chamber C 5 , where the pressure P5 is established equal to pressure P 4 in the integral action chamber C 4 .

In chamber C 3 the pressure P3 is dependent on the output pressure P2, on the ratio of the hydraulic resistances of the needle valve VR and resistance FR2, and also on the pressure P5.

C O M P A C T H Y D R A U L I C C O N T R O L L E R S Ι Α Τ A N U . S . S . R . 167

1-0

0-8

â Ο'δ

0-2

I 0 ' 02 0+ 06 0-8 W

• P[j atm.

F I G . 2. Static characteristics of the controller.

The pressure P3 acts on the diaphragm D3, and creates in the controller

an effect of positive feedback.

When the pressure P 4 is fixed (the valve V is closed) the unit becomes a

proportional controller. Figure 2 shows the static characteristics for this case.

Each curve corresponds to a particular setting of the needle valve VR. The

calibration characteristics of the needle valve VR is shown in Figure 3.

7 I I I I I I I I I

2

10

25

- 50

100

1 2 3 ï 5 6

— • Number of turns from

"closed "positions

7 8

F I G . 3. Calibration of needle valve DD

1-2


If the chamber C 4 is connected through the valve V to the chamber C 2 ,

the unit now has a certain integral action. The pressure P 4 depends on the

volume of fluid which has entered via the valve V into the chamber C 4 ,

formed by a spring-loaded bellows. The integral action time is determined

by the hydraulic resistance of valve V.

Μ ι 1 1 1 1 1— 1

T o obtain very large values of the integral action time, it is necessary to

ensure small flow rate of the working fluid through the valve V (of the order

of a few cubic centimetres per minute), at any pressure differences. If mineral

oil is used as the working fluid, this cannot be achieved simply by reducing

the flow area of the valve, because of silting. In the unit designed by I A T this

F I G . 5. External view of the controller.

F I G . 4. Calibration of valve V.

C O M P A C T H Y D R A U L I C C O N T R O L L E R S I A T A N U . S . S . R . 169

is avoided by using a lot of sufficiently large orifices in series. The adjustment of the integral action time is brought about by bringing into use the throt-tling orifices in the valve V. The relationship between the number of orifices and integral action time is shown in Fig. 4 (curves 7 and 2 correspond to different springs S2, that for curve 7 being twice as stiff as for 2). By using additional volumes connected to the chamber C 4 it is possible to obtain integral action times of the order of tens of minutes. The controller unit is made in the form of several discs, held together by threaded tie-rods (Fig. 5).

The amount of oil used by the controller does not exceed 250 cm3/min. It

can be used with mineral oils, various liquid fuels of low viscosity, and water; but in the latter case special provisions must be made against corrosion.

Specification of the Hydraulic Integral-action Controller

Supply pressure

Range of output pressures

Proportional band

Range of integral action time . .

D e a d zone in the working range

1-3-1-4 atm (gauge)

0-2-1-2 atm (gauge)

2-300 per cent

40-100 sec.

0-5 per cent of range

PROBLEMS IN THE DESIGN OF PRIMARY I N S T R U M E N T S -

DIFFERENTIAL PRESSURE TRANSDUCERS WITH FORCE BALANCE

S . M . Z A S E D A T E L E V and V . A . R U K H A D Z E

T H E PERFORMANCE of a pneumatic control system largely depends on the relia-

bility and accuracy of the primary instrument-transducer, which converts

the measured parameter into a proportional air pressure. Therefore, design

Supply

F I G . 1. Line diagram of a pneumatic pressure difference transducer.

organizations dealing with industrial pneumatic control systems spare no effort in the development of optimal types of transducers and rational design of elements in order to ensure the desired accuracy. During recent years, transducers based on the idea of force balance have increased in popularity.

In particular, this idea is used also for pressure-difference transducers. A typical scheme for such a transducer is shown in Fig. 1. Here the measured pressure drop P1 = Pa — Pb is taken up by the diaphragm 7. The resulting force is transmitted by tie-rod 2 to a lever, pivoting about the fulcrum 4. The

170

P R O B L E M S I N T H E D E S I G N O F P R I M A R Y I N S T R U M E N T S 171

lever passes from the chamber medium and is sealed by bellows 5. The sub-

unit of bellows 5 and the fulcrum 4 will be referred to as the "lead-out" of

the instrument. When the lever 3 moves, the distance between the baffle 69

attached to the lever 5, and the nozzle 7 is altered, and this affects the pressure

Pl in the nozzle line. The pressure Pt is transmitted by the amplifier 8 into

the output pressure Pk, which is transformed to some secondary instrument

(indicating, recording or regulating) and simultaneously acts on the feedback

element, producing a force on diaphragm 9, transmitted by an intermediate

lever 10 and tie-rod 77 to the main lever 3. A t equilibrium, the moment of

this force balances the moment due to the force of diaphragm 7.

Transducers of this type have a very small dead zone (0-01 per cent and

less), work very accurately (0-5 per cent and better), and react rapidly to

changes in the measured parameter.

There is little in the literature, particularly in Russian, on problems in-

volving the analysis and design of these transducers.

The present paper deals with some of the problems concerned with the

choice of design scheme and with the main parameters of transducers, based

on the design experience and investigations of such instruments at " N I I T e -

plopribor".t It appears that the subject matter may also be useful for the

development of other instruments, such as manometers, thermometers, in-

struments for measuring density, and level indicators, as well as controllers

and amplifiers, etc.

N O T E S O N M E T H O D S O N R A N G E A D J U S T M E N T

For general industrial automation it is necessary to have a series of dif-ferential pressure transducers, covering a range of pressure difference from a few millimetres of water to several atmospheres. In order to cut down on the number of variations in such a series, it is necessary to develop instru-ments with a wide range of adjustment. A t the same time, these instruments must be simple. Both these requirements greatly influence the choice of mechanical linkage of a transducer.

The simple single-lever mechanism (Fig.2,a) gives a variable range adjust-ment for pressure differences with a factor of three or four only; therefore it is not often used. The majority of instruments use a two-lever linkage (Fig.2,b). By moving the main sliding link 7, it is easy to obtain an adjustment range with a factor of six to eight. Here the limits of range adjustment are deter-mined by the fact that if the sliding link is too close to the fulcrums, the adjustment is too sensitive, small movements of the sliding link producing abrupt changes in calibration. A further increase in range can be obtained by the addition of further adjustable elements, such as the compensating element 2 (feedback element) with provision to alter its effective area. In some

t The Scientific Research Institute of Thermal Instruments.


cases, different sensing elements may be used with various effective areas. Al l

these methods allow us to increase the range of adjustment (for instruments

of the type shown in Fig. 2b) to 25 or more. This range of adjustment was tried

at "NIITeplopribor" with differential pressure transducers for small and

medium pressure differences. The coarse adjustment was brought about by

S S S S X

4 M

a

S S S S Pu

sss β,

F I G . 2. Lever mechanisms of transducers.

moving the sliding link 1 into one of several fixed positions, and the fine ad-justment by varying the compensating element 2. This method of dual ad-justment (coarse setting with the highly sensitive element and fine setting on the element with low sensitivity) is recommended for all instruments with a wide range of adjustment.

It may be noted that the demand for a very wide range of adjustment, ex-ceeding 20-25, makes it difficult to obtain a precision instrument, due to the fact that the forces acting on the levers vary over a wide range. When these forces are small the threshold level of the flexing elements is too pronounced, resulting in an increased dead zone, and when the forces are large the lack of stiffness of the levers is apparent.

The choice of the main dimensions for the lever systems of sensing and compensating elements is governed by consideration of the high accuracy required with an instrument of small overall size. In any case, it is advan-tageous to have the largest possible moment developed by the measured pressure difference at the main lever. The larger this moment, the smaller is the threshold effect due to rigidity of the flexing elements, which causes non-linearity and increases the error due to temperature variation; also, the smaller the influence of friction, and consequently the dead zone, the less is the relative effect due to the moment of the static pressure, which distorts the output signal of the instrument.

Lack of sensitivity, non-linearity, temperature errors, and errors due to static pressure level are the main defects of the majority of instruments based on the force balance idea. The origins of these defects, and methods for reducing or eliminating them, will be discussed below.


E R R O R S D U E T O T H E C H A N G E O F S T A T I C P R E S S U R E

Pneumatic instruments usually have a "lead-out" in the form of a bellows

or diaphragm. Bellows allow a lower rigidity, and are therefore used more

often, although the design of a bellows "lead-out" is more complicated. Let

us consider the errors caused by bellows "lead-out".

A change in static pressure leads to errors because of the following effects.

1. Curvature of the bellows axis (Fig.3,a). The internal and external pres-

sures tend to either straighten the axis, or to increase the curvature. The

resulting moment on the lever is proportional to the pressure difference.

2. Offset of the resultant force about the fulcrum (Fig.3,b). These two

effects can be eliminated or reduced by accuracy in manufacture and by

careful assembly and adjustment.

-3 τ

F I G . 3. Illustrations of the bellows "lead-out".

3. Displacement of the fulcrum under the static pressure load, with the consequent change of the distance e between the fulcrum and the line of action of the resultant force. This is caused by elastic deflection of the fulcrum, by inaccuracy of its location and also by deflections of the instrument frame.


As a result, the moment on the lever due to the static pressure force varies

in a non-linear manner with changes in this pressure, and the elimination of

errors becomes difficult.

4. Deflection of the lever under the load due to the measured pressure dif-

ference (Fig. 3,c). Bending of the lever causes a change in the distance be-

Pief

a

F I G . 4. Methods of adjustment of the bellows "lead-out".

tween the line of action of the resultant of the static pressure force and the fulcrum. Thus the errors due to change of static pressure are also influenced by the magnitude of the measured parameter. As distinct from the previous remarks, this source of error cannot be eliminated by adjustment.

5. Alterations in rigidity of the "lead-out" with changes of static pressure. This error manifests itself as the bellows is flexed by a rotation of the lever about its fulcrum. It cannot be eliminated by adjustment, but can be mini-mized by suitably choosing the dimensions of the "lead-out".

In order to eliminate or reduce the errors caused by the first two effects, the design of the "lead-out" must allow all the following adjustments:

(1) A reduction of the initial eccentricity e0 by displacement of the fulcrum by amount A (Fig.4,a).

(2) Provision of an additional moment due to static pressure, which would compensate for the moment due to eccentricity, and bellows curvature. This can be achieved by turning the lever through the angle θ (Fig.4,b).


-n-s

F I G . 5. Relation between the coefficients w g , mô and η and the parameters =- ajl.

fulcrum so as to obtain the minimum stiffness of the "lead-out". T o deter-mine the stiffness of the bellows "lead-out" in [5] the following formula has been obtained

K= —n + IPFJme, (2)

where Β is the stiffness in bending of the bellows, and η = 4(1 — 3o< — 3oc2),

(3) Provision of the additional moment by the parallel displacement of

lever and fulcrum by the amount δ (Fig.4,c).

The additional moment in the last two cases can be calculated by the usual

relations found in strength of materials, to give the following formulae (for

proof see [5]).

Μθ = PFtJ0mO9 Μδ = -pFe{ômô; (1)

where

me = - 9oc + 9 *2) , mb = 75(11 - 12a),

and

oc = —j (see Fig. 4),

Ρ is the static pressure and Fe{ the effective area of bellows. From the curves for the coefficients mQ and mb (Fig. 5) it is seen that the

efficacy of the adjustments in question depends on the lengthwise location of the fulcrum in the bellows. Using this fact, it is possible to design the "lead-out" to give easy adjustment, or to obtain dual adjustment, coarse and fine.

The problem of the optimum location of the fulcrum along the bellows is also connected with the question of errors due to the changes in bellows stiffness with change of static pressure. It is also desirable to position the


a factor depending on the position of the fulcrum (Fig. 5). The pressure Ρ is

considered positive if it is applied to the bellows externally.

Experimental investigations of bellows at "NIITeplopribor" confirmed

this formula to be satisfactory. The curve for the coefficient me (Fig. 5) shows

that the stiffness of the "lead-out" is not influenced by the static pressure at

(x'o = 0-127 and OCQ = 0-873. But in instruments of this type the angle of

rotation of the lever is usually very small, and the error due to change of

stiffness of the "lead-out" is therefore insignificant. In certain cases, there-

fore, it is better to locate the fulcrum not at the positions mentioned above,

but closer to the centre of the bellows, so that the stiffness of the "lead-out"

is a minimum (see the curve for the factor η in Fig. 5).

E R R O R S D U E T O N O N - L I N E A R I T Y O F T H E R E L A T I O N S H I P

B E T W E E N O U T P U T P R E S S U R E

A N D T H E M E A S U R E D P R E S S U R E D I F F E R E N C E

Non-linearity of the relationship between the output pressure Pk (Fig. 1)

and the measured pressure difference Pt = Pa — Pb is the result of the non-

linearity of pressure Px before the nozzle, with baffle 6 displacement. The latter

F I G . 6. Diagram of forces in the transducer linkage.

type of non-linearity would not cause the complete instrument to be non-linear, if its operation were entirely in accordance with the principle of force balance—that is, if the force due to the measured pressure difference were completely balanced by the force due to output pressure. Actually, however, only a part of the measured pressure force is balanced by the output pressure


force; the rest is taken up by the forces required to overcome the resistance

due to the rigidity of the moving parts of the instrument.

From the conditions of equilibrium (Fig. 6) and neglecting the force exerted

on the baffle by the jet of air and other small quantities, we obtain :

Λ-F - k(z + δ) = PkFkik + kk(zk - ôk) ik + kBz

+ kn(zn - ôn) in - P0in, (3)

where Pt and Pk are the measured pressure difference and output pressure; Fand Fk the effective diaphragm area of the sensing and feedback elements; k, kk, kB and kn are the stiffness of the measuring element, feedback ele-ment, "lead-out", and zero-setting spring; z, zk and zn are the displace-ment of the sensing element, feedback element, and zero-setting spring, cor-responding to movement of baffle z 3 , levers and other elements of the linkage assumed to be perfectly rigid; δ, ök and δη the movement of the sensing ele-ment, feedback element and zero-setting spring, due to elastic deflection of the levers and other elements in the linkage; i k, in and / 3 the transmission ratios ("leverage ratios") of the feedback element, spring and baffle displace-ments for a displacement of the sensing element; a n d i ^ is the initial force of the zero-setting spring.

Note that ζ = z 3 / 3 ; zk = ( i 3 / / k ) z 3 ; zn = (i3lHn) z 3 .

For a given "nozzle-baffle" pair it is possible to find experimentally or by calculation the relationship between the output pressure and the baffle dis-placement :

z 3 = z 3 ( P k ) .

Substituting this relationship into the equation of equilibrium (3), we can calculate the errors due to non-linearity.

In a more general form, the equation of equilibrium can be written :

Λ = P5 + Pr,

where Ps is the difference between the instantaneous value of the output pressure and its initial value, which is counterbalanced by the zero-setting spring, referred to the sensing element (i.e. multiplied by the ratio of the dia-phragm areas, and leverage ratio), and Pr is the sum of the pressures needed to overcome the threshold rigidity of the moving elements, referred to the sensing element. These pressures are shown in Fig.7,a as functions of the baffle movement z. The relationship Ps = / i ( z 3 ) is linear, but Ps = / Ί ( ζ 3 ) is non-linear because of non-linearity in the static characteristics of a "nozzle-baffle" system.

Figure 7,b shows the relationship Pr = F(P)t plotted from the values of Fig.7,a. It shows, in marked fashion, the errors due to non-linearity. These errors may be reduced either by making the characteristics of the nozzle-


baffle system more nearly linear or by reducing the stiffness of the sensing

element, "lead-out" and feedback element.

Comparison of various nozzle-baffle elements shows that the pneumatic

relay with an enclosed nozzle and constant pressure drop across the fixed

F I G . 8. Pneumatic amplifier with a method for maintaining constant pressure dif-

ference at the fixed restrictor. a—external view, b—line diagram.

F I G . 7. Force characteristics.


and variable throttles has an almost linear characteristic and the smallest

working movement. Nevertheless, relays with nozzles open to atmosphere

and constant pressure drop across the fixed restrictor can also give good

P^kg/cm2

F I G . 9. Ejector nozzle, a—line diagram, b—static characteristic.

results. The important advantage of the latter type is the great simplification

in the transducer design.t

A very nearly linear characteristic is obtained from the so-called "ejector

nozzle" (Fig. 9) proposed by Ferner [6]. A completely linear characteristic

can be obtained by the use of a moving nozzle (Fig. 10). Here it is possible to

0 -p,

F I G . 10. Pneumatic transducer with a moving nozzle.

select the pressure-operated element which moves the nozzle so that the sensing element is balanced at a constant position for any pressure (within certain limits).

t This method has been used with success in the amplifier developed in "NIITeplopr i -

bor" by Eng. V . V . Kerbunov. The line diagram and photograph are shown in Fig. 8. It is

small, simple and reliable in operation.


The reduction of stiffness can be attained by using diaphragms containing

over a certain range of their characteristics a negative component. With some

pre-tensioning of diaphragms, on assembly in accordance with the stiffness

of other elements, it is possible to obtain a very small stiffness over the work-

ing range (Fig. 11).

„ Working range

F I G . 11. Force characteristics: 1—of a diaphragm; 2—of another elastic element;

3—of both together.

S O U R C E S O F I N S E N S I T I V I T Y

Friction in the fulcrums can increase considerably the dead zone of an instrument. Cruciform flexible strips can be used as supports with advantage in order to reduce friction. Their use is justified only because the angular movements are small. Knife-edge and ball supports are simpler, but may cause considerable friction, if made or installed inaccurately.

High frictional forces, particularly detrimental in instruments with small forces operating on the sensing elements, occur at the junction of the "range-adjusting" sliding link with the lever. Thin, flexible strips in tension are therefore to be preferred here, and are used in the most recent foreign in-struments. This, however, leads to complications both in manufacture and in adjustment.

Proper attention must be paid to the design of the connexions between the diaphragms (or bellows) and levers. From the viewpoint of minimum fric-tion, the best solution would be a rigid fastening; but this usually leads to excessive rigidity. Recently a wider use has been made, particularly for sensing elements, of connexions in the form of a flexible strip (usually in tension).

A thorough examination of all the possible sources of friction enables us to obtain a transducer with a dead zone which is practically undetectable, and this is particularly important when the instrument is used in an auto-matic control system.


T E M P E R A T U R E E R R O R S

Errors due to changes of temperature are, probably, the most annoying and difficult to deal with. They may be considered as consisting of two compo-nents, one causing a change of the proportional control factor, and the other a displacement of the zero setting.

The first component results mainly from the change in the modulus of elasticity of materials. In instruments designed on the force balance principle this effect is usually insignificant, because the movement of the elastic elements is small; but it becomes noticeable if the elastic forces are of the same order as the feedback pressure forces. Special measures are then needed to combat it.

The displacement of the zero setting is caused by alterations in the dimen-sions of elements and in the pre-tension of the zero-setting spring and other elastic elements. These errors can be reduced by a suitable choice of ma-terials. In particular, the zero-setting spring must be made of an alloy with a modulus of elasticity insensitive to changes of temperature—for example, wire EI-702.

It is practically impossible to eliminate all temperature errors in mass-produced instruments, because the thermal expansion coefficients of the same grade of steel or non-ferrous alloy vary from batch to batch. Good results are obtained by using adjustable thermal compensation—for example, a baffle can be attached by an adjustable stud made of a material with a coef-ficient of thermal expansion differing from that of the levers and frame. Simi-larly, bimetal elements may be used.

Large temperature errors occur with instruments containing liquid as a means of protecting the sensing element (see following section). In such cases, temperature compensation is indispensable.

It sometimes happens that in operation one part of an instrument is war-mer than another—for example, due to heating along one side by radiation from nearby sources of heat. T o safeguard against this, it is advisable to provide the casing with lagging.

It may be noted that with a correct choice of materials the temperature error of a force-compensated instrument may be reduced to a value less than 0-01 per cent for 1°C.

P R O B L E M S O F R E L I A B I L I T Y

It is very important to ensure that a pressure-sensitive element is not dam-aged by excessive pressure. Safety devices used for this purpose may be divided into three groups:

1. Devices which eliminate the possibility of overload—for example, safety valves and shut-off valves (Fig. 12,a, b, c) .


2. Rigid stops which limit the travel of the sensitive element (Fig. 12,d).

Metal guards, with corrugations corresponding to those of the diaphragm

(with allowance for its deflection), are used for high-pressure diaphragms

(Fig.l2,d, e,f, g ) .

3. Safety devices with a liquid filling (Fig. 12,h, i, j ) .

One of the simplest means of protection is the rupture diaphragm which

is destroyed by overload, such that both the chambers of a differential in-

strument become interconnected (Fig. 12,a). This device, however, has earned

+ b

F I G . 12. Methods of overload protection of pressure-sensitive elements.


F I G . 13. Overload protection unit.

overload exceeds a factor of 1 , both the differential chambers are inter-connected, and the overloaded chamber is disconnected from the line.

Safety devices of the second group are used in a number of foreign in-struments. Devices of the third group can be considered to be the most re-liable. They are comparatively complicated in manufacture but do not suffer from dirt in the working chambers, and can protect the pressure-sensitive element from damage with very severe overloads.

There are a number of other problems, additional to the method of over-load protection, which must be considered in the development of these in-struments. For example, it is often necessary to consider systems involving corrosive media. The achievements of Russian metallurgy and chemical in-dustry have made it possible in recent years to make such instruments. Good results were obtained with the plastic material "ftoroplast 3", developed by the Leningrad Institute of Plastic Materials. A t present, a method exists for making corrugated diaphragms of this material. The diaphragms have a very low stiffness (of the order of a few grams per millimetre of deflection) and retain well the profile of the corrugation which is fixed during the poly-merization process. An ever-increasing field of applications exists for dis-persion-hardenable stainless steel El-702 with good elastic properties.

7 Aizerman I

a poor reputation in practice. Still less successful is the shut-off valve, intended

to disconnect the overloaded chamber from the supply line. A rapid rise of

pressure can damage the pressure-sensitive element before the chamber is

disconnected. Nevertheless, some instruments with safety devices of the first

group work quite reliably.

Figure 12,b shows the line diagram, and Fig. 13 a photograph of the over-

load protective unit, developed in "NIITeplopribor". This compact unit can

be installed in the input line of any industrial pressure transducer. When the


"NIITeplopribor"have developed methods for making welded bellows of this

steel, which have been used for experimental instruments and found to be

successful. It is also essential that the internal surfaces of chambers and tubes,

and the complete mechanism of the instrument, should be protected from

corrosion. New lacquers of high durability are very useful for this purpose.

In particular, good results have been obtained with epoxy resins, high tem-

perature lacquers, and similar protecting finishes.

The degree of reliability of an instrument is largely determined by its

convenience for installation and servicing. For example, the life of an in-

strument is increased and servicing is facilitated by the provision for self-

acting draining of the differential pressure chambers, and by the fitting of drain

plugs and air release plugs in the high parts of the measuring unit. The lack

of a pressure gauge in the feed line is a serious omission, which makes the

fitting and servicing a good deal more difficult.

For normal operation of an instrument, it is necessary to provide some

means of damping out pulsations of the measured parameter and output

pressure. The best results are obtained with totally enclosed hydraulic or

pneumatic dampers, but tests in "NIITeplopribor" have shown that open

hydraulic dampers are sufficiently effective in most cases. These are either of

the piston type, with a damping adjustment, or of the vane type.

Pressure transducers must also be able to withstand vibrations. T o elimi-

nate errors due to vibrations of the instrument frame under external ex-

citation, it is essential to balance carefully all the moving parts of the in-

strument. If this is done, the transducer may be mounted in any position.

When designing a transducer it is useful to make provisions for compensation

of residual unbalance.

B R I E F D E S C R I P T I O N OF I N S T R U M E N T S D E V E L O P E D

BY " N I I T E P L O P R I B O R "

The considerations given above on the construction of pressure trans-ducers designed on the principle of force balance were used in " N I I -Teplopribor" during the course of development of new instruments.

In view of the difficulties with the "lead-out" sub-unit, the first instruments of the force-balance manometers type M P K - 1 (Fig. 14) were built as in-struments for measuring pressure or vacuum on one side only. The pressure-sensitive element here is a bellows of 100 mm diameter, with six corrugations. It is made of stainless steel 1 Khl8N9T. The working range may be varied from 0 - 2 5 to 0 - 1 0 0 0 mm of water. The errors in measurement amount to ± 0 - 5 per cent of the upper limit of measurement.

Later, transducers for pressure difference measurement (differential mano-meters) were developed of type D M P K - 1 0 (sensitive element—a rubberized fabric diaphragm), and D S P K - 1 0 (sensitive element—a bellows 100 mm

P R O B L E M S I N T H E D E S I G N OF P R I M A R Y I N S T R U M E N T S 185

F I G . 14. Manometer M P K - 1 .

diameter, made of stainless steel). Experimental units are shown in Figs. 15

and 16, in the working position, with covers removed. Pneumatic transfor-

mer designs of these instruments differ only in the design of the dampers.

These instruments are designed for pressure differences from 0-25 to

F I G . 15. Differential manometer (pressure difference transducer) D M P K - 1 0 .


0-400 mm of water, at a static pressure up to 10 kg/cm2. Extensive laboratory

trials have shown that their accuracy is better than that required for instru-

ments of class 0-5.

Transducers of the DMPK-100 type for higher pressure differences and

for static pressures up to 100 kg/cm2 were also designed and made. They

belong to the accuracy classes 0-5-1-0.

F I G . 16. Differential manometer (pressure difference transducer) D S P K - 1 0 .

R E F E R E N C E S

1. G . T . B E R E Z O V E T S , Avtomatika i Telemekhanika, Vo l . X V I I , N o . 1, 1956.

2. G . T . B E R E Z O V E T S , V . N . D M I T R I Y E V and E . N . N A D Z H A F O V , Pribovostroenie, N o . 4, 1957.

3. J . R . D A V I D S O N , Some problems in the design of a differential pressure transmitter.

PaperN53-F-12 , A S M E .

4. F . K R E T Z S C H M E R , Regelungstechnik, N o . 3, 1956.

5. S . M . Z A S E D A T E L E V and V . A . R U K H A D Z E , Nauch. Dokl. Vyssh. Shkoly, N o . 1, 1958.

6. V . F E R N E R , Die Technik, N o . 6, 1954.

7. "NIITeplopr ibor" , Scientific Technical Report N o . 08034150.

ELECTRO-PNEUMATIC TRANSDUCERS

I AT A N U.S.S.R.t

Y U . V . K R E M E N T U L O

PROCESS control often requires the simultaneous use of electronic and pneu-

matic instruments. For example, in many cases it is advantageous to effect the

measurement electrically but to control pneumatically. Sometimes this is the

only possible way, for example when measuring very small voltages. For this

purpose, it is necessary to correlate the outputs of electric devices with the

inputs of pneumatic ones. This is performed by electro-pneumatic trans-

ducers. Also, the use of electro-pneumatic instruments in certain cases per-

mits the replacement of expensive pneumatic pipelines by cheaper electric

wiring. This not only reduces the cost of installations, but also increases the

speed of control action.

This paper discusses two types ( E P P - 1 and E P P - 2 ) of electro-pneumatic

transducers, developed in I A T A N U.S.S.R. In the transducer E P P - 1 the

electrical part is the same as in self-balancing Wheatstone bridges and poten-

tiometers. Both types were originally intended to receive input signals from

the Potentiometrie pick-off of the electronic instruments of A U S (Aggregate

Unified System).

The output signals of the first type of transducer ( E P P - 1 ) do not depend

on the supply voltage to the rheostatic pick-off. But when the second type of

transducer ( E P P - 2 ) is used with rheostatic pick-off there may be an error

from this cause. This, however, is not always essential. In particular, if the

transducer is incorporated into an automatic control scheme as on Fig. 8,

then the error caused by changes of supply voltages is equal to*

γ = Δ -oc,

where γ is the error of the transducer due to supply voltage changes; Δ the difference between instantaneous value of controlled quantity and its steady state value at the same adjustment; and oc the difference between actual and nominal supply voltages.

t Institute of Automatics and Telemechanics of the Academy of Sciences U . S . S . R . + In the case when the resistance of the pick-off is much smaller than the input resis-

tance of the transducer.

187


Therefore, as the system approaches steady state condition, when Δ -* 0, the error γ also approaches zero.

It has been found subsequently that the field of application of electro-pneumatic transducers is wider than originally anticipated. Some of these applications are discussed in the section "Applications of electro-pneumatic transducers" (see p. 193 ) .

E L E C T R O - P N E U M A T I C T R A N S D U C E R E P P - 1

The transducer of this type is intended for work with the rheostatic pick-off of the electronic branch of A U S (Aggregate Unified System) developed by one of the laboratories I A T A N U.S.S.R. It can also work with rheostatic pick-offs of such mass-produced instruments as E P P - 0 9 , E P P - 2 2 7 , E P P - 1 2 0 .

etc.

Design and Operation

The arrangement of the instrument is shown in Fig. 1. The rheostatic pick-off Rpl of the electronic measuring device is connected to the transducer, forming a bridge with its slide-wire Rp2. When the wipers 1 and 2 are out of

F I G . 1. Arrangement of electro-pneumatic transducer E P P - 1 . Rpl—slide-curve of

secondary electronic instrument, J—its wiper; Rp2—feedback slide-wire, 2—its

wiper; EU—electronic amplifier; F—feed unit; DA-1—balancing electric motor;

K4—cam; 3—baffle; C—nozzle; PDt, PD2—fixed restrictions; MKl9 MK2—

chambers between restrictions; POC—feedback adjustment; BP—secondary

pneumatic amplifying relay; Poc—feedback pressure; PK—reduction gear; S—

stem; Ml9 M2—diaphragms; Κ—flat valve; 7 W , Tr2—transformers.

E L E C T R O - P N E U M A T I C T R A N S D U C E R S I A T A N U . S . S . R . 189

t This amplifier, slide-wire Rp2, and motor DA-1 were developed in one of the labora-

tories I A T A N U . S . S . R . , and are used for the transducer as "ready-made" units.

balance, a signal arises in the diagonal of the bridge. This signal is fed to the input of electronic amplifier ££/,t consisting of input transformer Tri and phase-sensitive cascade with the valve 6P1P. In the anode circuit of the valve is included the control winding of the two-phase reversible motor DA-1.

The motor shaft is connected by the reduction gear with the wiper 2 of the rheochord Rp2.

The motor is so connected that the movement of wiper 2 is towards re-duction of the unbalance signal. Consequently, the wiper 2 always follows up after the wiper 7, and its position is always proportional to the measured parameter. On the shaft of wiper 2 is also mounted a cam , with the pro-file of an Archimedean spiral. Rotation of this cam moves the baffle 3 of the primary relay nozzle C. Air to the nozzle C is fed through the fixed restric-tion PZ>!. The motion of the baffle alters the pressure Py in the intermediate chamber MK2. This pressure is connected to the chamber of the secondary power-amplifying relay BP, which is of the usual non-bleed design.

When the pressure Py is increased the diaphragm Μγ of the secondary relay is flexed downwards. This causes the hollow stem S to press the flat valve K,

and air from the feed line Supply enters the output line. The pressure Pouip

will grow until the forces acting on the diaphragm Mx (due to pressure Py)

and the diaphragm M2 (due to pressure Poutp) attain equilibrium.

When equilibrium is achieved the valve Κ shuts against the air supply, and the stem S remains pressed to the valve, keeping the air vent closed. De-crease of pressure Py flexes the diaphragm Mx upwards. This causes the stem S to lift from the valve Κ and air from the chamber connected with the out-put line is exhausted to the atmosphere. The output pressure will drop until equilibrium between the forces acting on diaphragms M1 and M2 is re-established.

In order to obtain a linear characteristic P o u tp = /(<%), (where oc is the angle of rotation of wiper 7), and also to diminish the influence of changes in sup-ply pressure negative feedback is introduced. The feedback unit (UOS) con-sists of two bellows. The output from the relay is connected to the space be-tween these bellows. The nozzle C is rigidly connected with the bellows end-plate, and moves together with them, as the pressure varies. The feedback arrangement results in the nozzle C following up the baffle motions with an accuracy equal to the working gap, which is approximately 0-015 mm.

Linearity of the feedback unit characteristics, and a large feedback gain, ensure proportionality between output pressure and baffle position. The latter is proportional to the measured parameter; therefore the output pres-sure is also proportional to the measured parameter.


Results of Tests

Figure 2 shows the graph of the static characteristic of the transducer

P o u tp = / ( Λ ) , with the angle of rotation of wiper 1 measured in per cent of its

full angle. The linearity of this characteristic is obvious. The resolution of the

0 20 W 60 80 100 ^ <*,%

F I G . 2. Static characteristic of the transducer E P P - l .

transducer is ±0-7 per cent, dead zone is 0-25 per cent. Change in supply pressure by ± 15 per cent causes an additional error of about 0-5 per cent.

Technical Data on the Transducer EPP-l

1. Must be connected to a slide-wire pick-off with resistance 120-200 Ω.

2. The pick-off is supplied, through the feed unit F, with alternating cur-

rent at 24 V and 50 c/s.

3. Changes of supply voltage up to ± 1 5 per cent cause insignificant errors.

4. Power supply to the transducer is by compressed air at 1-3 atm (gauge), and alternating current at 220 V and 50 c/s.

5. Overall dimensions: 110 χ 140 χ 270 mm.

6. Weight, 5-5 kg.


E L E C T R O - P N E U M A T I C T R A N S D U C E R E P P - 2

The input signals to this transducer are transmitted by direct current, and the output is air pressure.

Design and Operation

The design of the transducer is shown in Fig. 3. The direct current iy is connected to the coil Wy, which is suspended by two centring flat springs Ρ

in the air gap of the permanent magnet M.

Poutp.

F I G . 3. Arrangement of electro-pneumatic transducer E P P - 2 : PD—fixed restriction ;

C—nozzle; 3—baffle; M—permanent magnet; BP—secondary relay; Wy—coil;

Ρ—flat spring; Py—pressure at the output of primary relay.

The pull of the coil is proportional to the current iy. Each value of current corresponds to a definite position of the coil in the air gap, determined by the pull force and the stiffness of the flat springs.

The pneumatic part of this transducer is similar to that of EPP-1 . In order to improve the sensitivity of the transducer (ratio of output pres-

sure changes to the changes of current iy), it is necessary to reduce the work-ing movement of the bellows of the feedback unit.| This, however, would in-

t Arbitrarily taken as the movement of bellows with the change of pressure from 0-2

to 1 atm.

7 a Aizerman I


crease two undesirable effects: (1) the error due to changes of supply pres-sure; and (2) the error due to non-linearity and instability of the primary relay. A reduction of the working gapt may be obtained by increasing the pressure gain APoutJAPy, where APy is the change of pressure Py fed to the secondary relay BP. This gain is approximately 2, and the corresponding working gap is 0Ό15 mm. The working movement of the bellows is 0-6 mm. In these conditions the deviation of the function Poutp = f(x3) (where x3 is the baffle movement) from linearity does not exceed 0T6 per cent.

There is one more consideration in favour of increasing the pressure gain of the secondary relay. Figure 4 shows the characteristic of the secondary relay Poutp = F(Py). When the pressure Py changes from 0-28 to 0-68 atm (points a and b on the graph), the output pressure changes from 0-2 to 1 atm.

10

09

0-8

07

0-6

£ i \ '0-5

04

0-3

0-2

0-1

0

a 1

0-1 0-2 03 04 0-5 06 0 0-28

Py kg /cm2

0-68

F I G . 4. Static characteristic of secondary relay.

Since the maximum pressure in the chamber between the restrictions is equal to 0-68 atm, a perfect closure of the nozzle by the baffle is not essential. Also, the primary relay works in a narrow zone of its characteristic, approaching closely to linearity.

Results of Investigations

The static characteristic of the transducer is shown in Fig. 5. A change of current from —3 to + 3 mA causes a change of output pressure from 0-2 to 1 atm. Inaccuracy, as found by numerous tests, is ±0-7 per cent. Deviation

t The corresponding change of gap between nozzle and baffle is called the "working

gap".


of characteristics from linearity does not exceed 1 per cent. Dead zone does not exceed 0-2 per cent. Changes in feed pressure by ± 1 0 per cent cause + 1 per cent error of output pressure within the range 0-2-0-85 atm, and ± 1-5 per cent, within the range 0-85-1 atm.

1-0 CVl

^1

f

M CVl

^1

f

1 08

CVl

^1

f 0-7

0ύ Ao-75+15t225+3

-3-2-25-1- 5-076 A jrn.t;

l0

04

0-3

0-2

04

0-3

0-2

04

0-3

0-2

F I G . 5. Static characteristic of transducer E P P - 2 .

The ambient temperature during the tests was varied between 2° and 50 °C, but the characteristics P o u tp = f(iy) remained unchanged. I f a constant vol-tage is maintained at the coil, an additional error arises because of the change of coil resistance. This may be countered by the usual methods of thermal compensation, as used for electric measuring instruments.

Technical Data on the Transducer EPP-2

1. Input, direct current from —3 to + 3 mA. 2. Output, air pressure from 0-2 to 1 atm gauge. 3. Supply air pressure 1-2 atm gauge. 4. Fundamental error ±0-7 per cent. 5. Dead zone ±0-2 per cent. 6. Resistance of controlling coil, 650 Ω. 7. Power required by coil (max.) about 6 mW. 8. Overall dimensions, 70 χ 110 χ 190 mm. 9. Weight % 1-4 kg.

A P P L I C A T I O N S OF E L E C T R O - P N E U M A T I C T R A N S D U C E R S

Some possible schemes of automatic control, incorporating electro-pneu-matic transducers, are shown in Figs. 6, 7 and 8.

The first two illustrate the uses of transducer E P P - 1 . Figure 6 shows a


F I G . 6. D iagram of control system with electronic measuring instrument (EIP),

electro-pneumatic transducer E P P - 1 , and pneumatic controller (RB). A—object;

D—pick-off; RD—slide-wire pick-off; 2—wiper of measurement; Pm—pressure,

proportional to measured quantity; Psci—pressure, proportional to set value;

MIM—diaphragm control valve.

F I G . 7. D iagram of a control system with electronic measuring instrument (EIP),

electronic control device (ERU) and transducer E P P - 1 . A—object; D—pick-off;

EU—electronic amplifier; DA-1—balancing electric motor; PK^, PK2—reducing

gears; PM—pneumo-mechanical part of transducer; MIM—diaphragm control

valve.


control system with the electronic measuring instrument EIP, electro-pneumatic transducer EPP-1 , and pneumatic control unit RB. The pressure Pm is proportional to the measured quantity at the output of the transducer. This pressure is connected to the measuring chamber of the controller RB,

and the pressure P s e t, proportional to the set value of the controlled quantity, is connected to its set value chamber. The output from RB is connected to the diaphragm control valve MIM.

RD

F I G . 8. D i a g r a m of a control system with electronic measuring instrument {EIP),

transducer Ε Ρ Ρ - 2 , and pneumatic controller (RB). A—object; D—pick-off;

RD—slide-wire pick-off; 1—set value; 2—wiper of measurement; Δ—deviation

signal; P6—pressure, proportional to deviation; P s e t—pressure set value; MIM—

diaphragm control valve.

The arrangement shown in Fig. 7 relates to the case when an electronic controller is employed. In this system the electro-pneumatic transducer EPP-1 is included after the controller. The electronic measuring instrument, connected to the pick-off, produces signals proportional to the deviation. These signals are the input to the electronic controller ERU, which generates an integral action control law. The integral action time is set by altering the ratio of reduction gear PKl, and the gain by the resistance i ? 4. The trans-ducer EPP-1 is connected to the electronic control device, and the transducer output to the control valve MIM.

A system employing electro-pneumatic transducer EPP-2 is shown in Fig. 8. The signal (proportional to deviation) from the slide-wire pick-off is fed to the input of the transducer. A t the output of the transducer the pres-sure Pô = —k(U3 — Ut) + P0 (if the resistance of the pick-off is much smaller than the input resistance of the transducer).


Here U3 is the voltage, proportional to the set value of the controlled

quantity; £/, the voltage, proportional to its current value; P0 the initial pres-

sure at the output of the transducer, at zero current in the coil ; and k the

proportionality constant.

In the summer of the pneumatic controller a signal is generated which is

equal to :

Δ = P 3 - Ρδ = P 3 - P0 + k(U3- Ut).

If Δ is to be proportional to U3 — Ui9 it is necessary to apply to the set value

chamber the pressure P3 = P0, then Δ = k(U3 — £/,).

There are also certain other possible applications of electro-pneumatic

transducers. Electronic controllers are often combined with relay action

regulations; in such cases the shaft of the electric motor also operates the

feedback rheostat, which is connected to the electronic controller. Here it is

possible to change over for pneumatic actuation, by using the electro-

pneumatic transducer EPP-1, its output operating a diaphragm control valve

MIM.

The transducer EPP-2 can be used with success for remote programming,

remote control, as a junction between "real" pneumatic apparatus and elec-

tronic simulators, and as a summator (in this case it is necessary to have a

coil with several independent windings). Transducers of this type may also be

used with automatic potentiometers for measuring the e.m.f. of a thermo-

couple, which have as their output not an angle of rotation, but a current,

proportional to the e.m.f. of the thermocouple.

PNEUMATIC RELAYS WITH CONSTANT PRESSURE DIFFERENTIAL RESTRICTIONS

V . N . D M I T R I Y E V

C O N T E M P O R A R Y instruments of pneumatic control widely use the pneumatic

relay of the "nozzle-baffle" type (Fig. 1). The usual static characteristic of

such a relay—that is the relationship between pressure Px in the chamber

h F I G . 1. Pneumatic relay.

after the fixed restrictor and the distance h between the nozzle and baffle— is shown in Fig. 2 by the curve 7. This shape of the static characteristic imposes severe limitations on the possibilities of improving the accuracy and

Pi

F I G . 2. Static characteristics of relays: 1—simple relay (P0 = const, P2 = const);

2—ideal characteristic; 3—relay with constant differential pressure over both

restrictors.

197


sensitivity of instruments incorporating the pneumatic relay. This would not be so with the ideal characteristic, shown on Fig. 2 as the curve 2. Therefore pneumatic relays are provided with special devices which enable a charac-teristic of the type represented by curve 3, sufficiently approaching to an ideal one, to be obtained. These devices maintain constant pressure differential across the fixed and variable restrictors. Figure 3 shows the arrangement of a pneumatic relay provided with such devices. The constant pressure dif-ference APf at the fixed restriction D is maintained by the action of the bel-lows, operating the ball valve Ky if the pressure Px changes, so causing the pressure P* to differ from the feed pressure P0. As the movement of the bellows is small, the pressure difference at the nozzle-baffle element is kept constant by means of the diaphragm-controlled ball valve K2. Since the de-flections of the diaphragm are small, and therefore the force exerted by the spring is nearly constant, the difference APV = Px — P* also tends to re-main constant. Detailed analytical and experimental investigations of the static properties of this pneumatic relay are given in the next article (by the same author; see p. 202). Below are quoted a few of the results of this investigation which show the influence of constant pressure drop on the shape of the static characteristic of the relay.

Po

F I G . 3. Pneumatic relay with devices for maintaining constant differential pressure

across both restrictors. C—nozzle; 3—baffle of the variable restriction ; P%—pres-

sure after the variable restriction ; D—fixed restriction ; P0—pressure in the feed line ;

P2—atmospheric pressure; Py—pilot (controlling) pressure; P x—output pressure

(in the chamber between restrictions); P$—pressure before the fixed restriction;

Κι and M x— v a l v e and bellows of the pressure difference regulator at the fixed re-

striction; K2 and M2—valve and diaphragm of pressure difference regulator at the

variable restriction.

Static Characteristic of the Relay with Constant Pressure Drop across the Fixed Restrictor (Fig. 4, curve 7). I f the constant pressure drop is artificially maintained at the fixed restrictor, that is, if P* — Pl = APf = const (P* is the pressure between the valve Kt and the fixed restrictor; see Fig.3), then

R E L A Y S W I T H P R E S S U R E D I F F E R E N T I A L R E S T R I C T I O N S 199

the static characteristic of such relay will have a break, dividing it into two sections. The first section (h0 > h > 0) coincides with the corresponding part of the characteristic of a simple relay, shown in Fig. 4 by the curve 2. The second section (h > h0) is the working one. This division into two sec-tions can be explained in the following way : when h = 0, the pressure in the

F I G . 4. Static characteristic of a relay with constant pressure differential at the fixed

restriction (APf = const).

chamber between the restrictions is equal to the supply pressure P0. T o ob-tain a desired pressure drop aPf?\ the fixed restrictor, which pressure drop will then be maintained constant, it is necessary to reduce the pressure to the point when the difference P* — Pi will be equal to the desired drop APf. T o obtain this, the baffle must move through a distance h0. Over this region P0 = const, and P2 = const; therefore the relay works as a simple one ( F i g . l ) .

The next article gives the formula for this distance:

h Jj_ /Γ ΔΡΤ{Ρ0 - APf) 1

4d2 \Il(Po-APf-P2)P2 j

(dY and d2 are the diameters of the fixed restrictor and nozzle). Further increase of h takes place with a constant differential pressure

across the restrictor, 7, so that as Pt is reduced so also is P * · In the following article it is shown that the mean slope of the working part

of the static characteristic is always greater for a relay with constant pressure drop at the fixed restrictor when compared with simple relay. The mean slope is measured as the tangent of the angles γχ and y2. These angles are formed between the abscissa and the lines joining P1 at h = h0 and Pt at h = Hl

and h = H2 i.e. the linear approximation over the working range. In the following article (p. 202) it is shown that with increase of d1 the value

h0 increases, and diminishes with increase of d2 (Fig. 1). The mean slope of the working range of the characteristic diminishes with increase of dx, and in-


creases with increase of d2. The increase of P0 results in a decrease of h0 and an increase of the mean slope. The increase of APf has the inverse effect: h0

increases, and the mean slope diminishes. Static Characteristic of the Relay with Constant Pressure Drop at the

Nozzle-baffle (Fig. 5, curve 7). Investigations of the static characteristics of this relay, carried out in the following article, lead to the following conclu-sions:

(1) As the back-pressure downstream of the nozzle-baffle increases, the pressure Pi in the chamber between the restrictors increases, and therefore the static characteristic of the relay with back-pressure P% runs higher than that of the simple relay, the static characteristic of which is shown in Fig. 5, curve 2.

(2) The mean slope tan y 3 of the working part of the characteristic for a relay with constant pressure drop across the nozzle-baffle may differ in either direction from that of a simple relay (tan y , ) . This depends upon the points compared.

(3) The provision of constant pressure drop only across the variable restrictor does not improve noticeably the static characteristic of the relay, in the sense of bringing it closer to the ideal one (line 2 on Fig. 2).

Static Characteristic of the Relay with Constant Pressure Drop across both

Restrictors (Fig. 6, curve 7). If there are constant pressure drops across both restrictors, the fixed one, and the nozzle-baffle unit, the static characteristic is also divided into two sections. The first region, where h0l > h > 0, pre-cedes the working section, where h > h 0 l. The first region coincides with the corresponding part of the characteristic of a relay with constant pressure

F I G . 5. Static characteristic of a relay with constant pressure differential at the

variable restrictor (APV = const).

constant drop over the nozzle-baffle (Fig. 6, curve 2). This is so because the constant drop across the variable restrictor may occur immediately after h = 0 while a pre-set constant drop at the fixed restrictor is established only at h = h 0 l. Thus, in the first region the relay works as one with constant drop at the variable restrictor, and in the second (working) region with constant

R E L A Y S W I T H P R E S S U R E D I F F E R E N T I A L R E S T R I C T I O N S 201

H Ht ^

F I G . 6. Static characteristic of a relay with constant pressure differential at both

restrictors (APf = const; APV = const).

ample, with P0 = 1 atm, P3 = 0-2 atm; P2 = Oatm; APf = 005 kg/cm2,

and ΔΡυ = 0-1 kg/cm2 (the latter being the constant pressure drop over

the flapper baffle); the ratio tan y 4/tan y2 = 10-3.

(3) Increase of P0 causes an increase of tan y 4 , and increase of APf and APV its decrease.

(4) Increase of P0 and APV decreases h 0 l, and increase of APf increases it. (5) The distance h0l in a relay of this type is larger than the corresponding

distance h0 in a relay with constant drop at the fixed restrictor only. For example,with P0 = 1 atm,P 2 = 0,APf = 005kg/cm

2 andAPv = 01 kg/cm

2,

the ratio hoxjh0 = 2-26.

The results of this analysis lead to the conclusion that the maintenance of constant pressure drops across both fixed and variable restrictions brings the static characteristic closer to an ideal one.

These results are confirmed by experiments. Some static characteristics, obtained experimentally, are given in the following article.

drop at both restrictors. Detailed investigations lead to the following con-clusions:

(1) The working section of the characteristic is close to a straight line, and has a steep slope (tan y 4 ) .

(2) The mean slope of the relay in question (tan y 4 ) is always steeper than that of a relay with constant drop at the fixed restrictor (tan γ2). For ex-

IMPROVEMENT OF STATIC CHARACTERISTICS OF PNEUMATIC RELAYS

BY USING CONSTANT PRESSURE DIFFERENTIAL RESTRICTIONS

T H E DESIRABILITY of having constant pressure differentials across the fixed and variable restrictions of a pneumatic relay is discussed in the previous article, which gives (Fig. 3) an arrangement for such a relay. The present work gives the theoretical foundation for the design of relays, having a constant dif-ferential pressure either over the fixed restrictor, or over the variable restrictor, or over both. The conclusions of the theory are confirmed by the results of experiments which are quoted briefly in the preceding article, and given here in more detail.

The equations of the static characteristics of pneumatic relays are deduced from the flow equations, under assumption of constant density γ of air. This, of course, gives only an approximate solution.!

For the sub-critical flow, the equation of discharge is :

Here G is the discharge, in weight units (weight flow) ; g the gravity accelera-tion; P i the pressure after the orifice; P0 the pressure before the orifice; T1

the absolute temperature of air (after the orifice); R the gas constant;/! the effective area;/j = μίΡι where μΛ is the discharge coefficient and F± is the orifice cross-sectional area.

t Investigations [6] prove that the use of this assumption, instead of exact Saint-Venant

and Wanzel formulae for adiabatic processes, gives the maximum error of 3-4 per cent in

the whole range of pressures (both sub-critical and trans-critical). Investigations were car-

ried out for air.

V . N . D M I T R I Y E V

1. B A S I C E Q U A T I O N S

C = / i / Γ — Λ ( Λ > - Λ ) 1 ( — >0'5) ( 1 )

and for trans-critical

( l a )

202

S T A T I C C H A R A C T E R I S T I C S O F P N E U M A T I C R E L A Y S 203

The static characteristic equation for the simple relay f can be found from

the condition of equal mass flow through the fixed and variable restrictions.

W e assume that the process is isothermal, that is, T0 = Tt = T2 = T. (For

the justification of this assumption see, for example, Ref. 7.) Having in view

mainly the qualitative evaluation of the influence of relay parameters, we also

assume that μι!μ2 = 1. Then, for the combination of flow conditions "s-s",*

Here h is the distance from nozzle to baffle, άγ and d2 the bore of fixed and variable restrictions respectively. Equations for the static characteristics of a simple relay for other combinations of flow conditions are given in Appen-dix 1, Table 2.

A more detailed analysis of the influence of relay parameters on its static characteristic will be carried out for the case ".y-.s", which is the most usual one in the industrial pneumatic control. Equations for other combinations of flow conditions are given in the appropriate tables of Appendix 1.

If a constant pressure difference P* — Pi = APf = const is maintained at the fixed restriction (see Fig. 3 of the preceding article), the static charac-teristic will consist of two distinct sections (Fig. l ,a) . The first section (h0

> h > 0) coincides with the static characteristic of a simple relay (curve / ) . The second section h > h0 is the working range of the relay (curve 2).

, p

i P

1 Pi

t A relay without any provisions for constant differential pressure across its restriction

(i.e. working at P0 = const, P2 = const) shall be called a "simple relay". φ

" 5 - 5 " denotes sub-critical flow through both restrictions; similarly, will mean

trans-critical flow through the fixed restriction, and sub-critical through the variable one;

" M " is trans-critical in both, etc. F o r the determinants of flow conditions see Appendix 1,

Table 1.

we have

(2)

2. R E L A Y W I T H T H E C O N S T A N T D I F F E R E N T I A L

P R E S S U R E O V E R T H E F I X E D R E S T R I C T I O N

F I G . 1.


This division of the characteristic into two sections is necessary because a certain travel, h0, of the baffle is required to lower the pressure by a certain amount P0 — P± = Pf, which then can be maintained at a constant level. Since for the first section P0 = const and P2 = const the relay works as a simple one. The distance h0 is determined by the formula:

h o = _d}_ J ΔΡΑΡο-ΔΡ,)

4d2 V (P0 - APf - P2) P2

Further increase of /*, from h0 on, occurs at a constant pressure drop, that is, at diminishing pressure P*. Therefore, the pressure in the chamber de-creases, not only because of increased h, but also because of decreased P* · As a result a much steeper slope of the curve is obtained. The first section of the characteristic is described by equation (2), and the second (the working range h > h0) by the equation :

Η-Λ. Ι Ρ*ΔΡ< . ( 4 ) 4d2 V (/>, - P2) P2

It can be proved that the mean slope of the working range of the characteristic (curve 2) is always greater than the mean slope of the initial section (curve 1).

The mean slope will be defined as the tangent of the angle formed by the straight line which connects the point on the curve which has the abscissa h = h0 with the end of working range,f in this case having abscissae A = Hi

and h = H2- Denoting the pressure at the end of working range by P3, and using equations (2)-(4) , we shall have:

tan j ' , = P0 - APf - P3

Ht - h0

P0 - AP, - P3

d\_ Γ IP3(PQ - P3) _ / (P0-APf)APf

<W2 L V ( P i - Ρ2) Pi V (Po - APf - Ρ2) Pi _

(5)

tan γ2 =

4d

P0 - ΔΡ, - P.

H2 - h0

P0-AP, - P3

d\ Γ / P3AP, _ I (P0-APf)APf

4d2bl (P3- P2)P2 V (P0- APf- P2)P2_

(6)

Comparing these expressions for tan γ2 and tan y , , one will see that the only difference is in the numerators of the first fractions under square roots. Since P0 — P3 is necessarily greater than APf (at P0 — P3 = APf the relay would

t The end of working range corresponds to the maximum possible travel of the baffle,

within the limits where it exerts a throttling influence on the nozzle—that is, 0 < A < d2\A.


be transformed into a simple relay), we obtain obviously tan y2 > t any x.

For example, if P0 = 2 atm abs., P3 = 1-2 atm abs., P2 = 1 atm abs., and

APf = 0-05 atm (gauge), then tan y 2/tan γγ = 8-25.

Curve 2 represents a monotonie decreasing function, which has a po-

sitive second derivative d2/

>

1/ d / 22 > 0. Therefore, the maximum slope of

the curve is in the point with the abscissa h = h0. This maximum slope can

be determined by finding the limit:

lim tan γ 2

which gives :

d/7

ah Sd2(P0 - APf - Ρ2Ϋ

12 lPo-AP

APfP2

Analysis of expressions (3) and (6) enables us to evaluate the influence of relay

parameters on the distance h09 and on the mean slope of the working range

characteristic. For example, increase of dx increases /?0, and increase of d2

reduces it. The mean slope decreases with increase of d{, and with decrease

of d2. Increase of P0 reduces h0 and increases the mean slope, and increase

of APf acts in the reverse direction. The proof of these statements is given in

Appendix 2.

3. R E L A Y W I T H C O N S T A N T D I F F E R E N T I A L P R E S S U R E

A C R O S S T H E V A R I A B L E R E S T R I C T I O N

The static characteristic of this relay (Fig. l,b, curve 3) is described by the

equation :

d\ I (P0 - Λ ) Λ h = 4d, APv(Pl -APV

(7)

Since there is a back-pressure at the variable restriction the pressure in the intermediate chamber increases, consequently the static characteristic runs above the one for a simple relay (Fig. l,b, curve 7). This can be shown by comparing the distance h at the same Ργ for the relay with constant pressure differential over the variable restriction and the simple relay (see equations (2) and (7)) .

(Λ - P2) Pi (Λ - Pi) Pi

ΑΡι(Ρί -ΔΡν) V (P > 1. (8)

It follows from this comparison that h for a relay with constant pressure dif-ference over the variable restrictor is always greater than h for a simple relay, because P2 > P2, P\*\PX > 0-5, and P2lPl > 0-5 (see Fig.2).


tan r 3 =

H3 - Ao i

- Δ Ρ

> - * (10) d\ Γ / P3(P0 -Ρ*) _ I (P0-APf)APf

4d2 LV APV(P3 - APV) V APV(P0 - APf - APV) _

The mean slope in this case may be greater or smaller than that of a simple

relay. This depends upon the chosen magnitudes of APf and P3 or, in other

words, upon the choice of the parts of characteristics, which would be com-

pared.

2

F I G . 2.

For example, if we calculate the ratio tany 3/tan y u using equations (10)

and (5), for a simple relay, at P0 = 2 atm abs., P3 = l-2atmabs., P2

= 1 atm abs. and for a relay with constant differential pressure over the

variable restriction (with the same parameters and APV = 0-1 atm), then, for

the cases APf = 0-05 atm and 0-3 atm, we obtain respectively tan y 3/tan γγ

= 0-8 and 1-14.

Since the whole characteristic is described by a single equation, and con-

sequently there is no sharp division between initial and working sections, we

shall take arbitrarily that the working range begins, as in the previous case,

at Pi = P0 — APf (here APf means merely the difference between feed

pressure and pressure in the chamber at h0i—see Fig. l,b—and not the

pressure difference, which must be kept constant). The distance h01 (Fig. l ,b)

is:

(9)

and the mean slope of the working range :


4. R E L A Y W I T H C O N S T A N T D I F F E R E N T I A L P R E S S U R E

O V E R B O T H R E S T R I C T I O N S

If a constant pressure difference is maintained at both the fixed and the variable restrictions, the characteristic of the relay will consist of two sections. The initial one (h0l > h > 0) is represented by curve J, Fig. l,c, and the second one, corresponding to the working range, by curve 4.

The initial section coincides with the static characteristic of a relay with constant differential over the variable restriction. This can be explained by the fact that a constant differential across the variable restriction is main-tained even at h = 0, while at a fixed restriction it cannot exist until h = h 0 l.

Because of this, the relay works first with a constant differential only at its variable restriction, and only after h = h01 is reached can the pressure dif-ferential over the fixed restrictor be maintained constant.

The distance h0l is determined by equation (9). Equation (7) describes the initial section of the characteristic (h0l > h > 0), and for the second section we have:

h = APfPx A .

Ad\ V ΔΡν(Ρχ - APr) (11)

The working section of the curve has a steep slope, and comes close to a straight line; this can be seen from the analysis of formula (11).

From (11) it follows that h is but slightly influenced by Px.

In order to find the mean slope of the working section of the characteristic, we assume that the end of this section corresponds to Pt = P3. Then, from (11) and (9), we obtain (see Fig. l , c ) :

tan } ' 4 = P0 - APf - P3

Ηι — h0l

P 0 - A P f - P3

1 " d2 L v

APf(P0-APf)

»v(P0-APf-APv)_

(12)

4d2 IV APV(P3 -ΔΡυ) V APr

The maximum slope corresponds to h = h 0 l, and can be found as the limit

UP, lim tan y 4 =

P3-P0-dPf dh

dP,

dh

(/>„ -APF- ΔΡΧΙ2ΔΡ,

d\ P0 - 2(P0 - Δ Ρ , - ΔΡ,^ΔΡ,,

Comparing (6) and (12), we can see that the mean slope tan y 4 (in the case of constant differentials over both restrictions) is greater than the mean slope


tan γ2 of the characteristic of relay with constant differential over the fixed restriction only. The proof of this is given in Appendix 3.

This can be illustrated by an example: the ratio t any 4/ t any 2 is 10-3 for relays with the following parameters: P0 = 2 atm abs., P3 = 1-2 atm abs., Ρ2 = 1 atm abs., APf = 0-05 atm and APV = 0-1 atm.

Considering the expression (12) for tan y 4 , we find that increase of P0 increases tan y 4 , while increase of APf and APV reduces it (see Appen-dix 3).

The influence of the various parameters of the relay on the distance h0l can be evaluated by analysing the expression (9). For example, in-crease of P0 and APV reduces A 0 1, and increase of APf increases it (see Appendix 4).

The distance h0l for a relay with constant pressure differential over both restrictions exceeds that for a relay with constant pressure drop at the fixed restrictor only. This is proved in a general way by (8). As an example, the ratio h0l/h0 for the relay with the same parameters as in the above example is 2*26.

5. E X P E R I M E N T A L I N V E S T I G A T I O N S

T o verify the results of the above analysis, a series of tests was carried out on the test rig shown in Fig. 3. The constant pressure drop over the fixed restriction was obtained for each measurement in the following manner.

F I G . 3.


0 001 0-02 h,mm

F I G . 4. 1—APf 0-033 atm; 2—APf = 0-1 atm; 3—APf = 0-2 atm;

4—APf = 0-3 atm.

touching the nozzle) was detected by a millivoltmeter 7. The pressure P0 was kept constant throughout the tests. When investigating the relay with a con-stant differential over the fixed restriction, and the simple relay, the chamber above the nozzle was connected to the atmosphere, and the baffle moved by the means of adjusting nut 8, since feedback was then absent.

Figure 4 shows the family of static characteristics of the relay with con-stant differential over the fixed restrictor. The relay had the following para-meters: d2 = 0-50 mm; di =0-18 mm; / = 18 mm (a capillary). The curves were obtained for several values of APf9 at P0 = 1 atm. The static charac-teristic of the simple relay is also shown. From Fig. 4, it is seen that the introduction of constant pressure difference substantially increases the slope of the characteristic, bringing it nearer to the "on-off " type. Increase of APf

displaces the characteristic to the right, increasing h0. The slope of the work-ing section is reduced by the increase of APf, and the total range of pressures for the working section is also reduced.

The static characteristics of a relay having d2 = 3-08 mm, with the con-stant pressure drop over fixed restriction, are shown in Fig. 5. They were taken for various diameters dx of the fixed restriction. It is seen that they coincide with the characteristic of a simple relay over the range 0 < h < h0. Increase

Yv XX \

With the aid of reduction valve 7, a given feed pressure P* was set before

the fixed restriction. Then the pressure Pl was set by adjusting the baffle, by

pressurizing chamber 3 or 4 through a reduction valve 5. A t the variable

throttle, the constant pressure difference was maintained automatically. The

gap between nozzle and baffle was measured by a micrometer 6, with an

enlarged circular scale, divided into steps of 1 μ. The zero point (baffle

Pi, atm. P0*1atm.

1

0-8

0-6

0+

0-2


04

0-2

0 0-04- 0-08 012 0-16 Q-20 0-21 h,mm

F I G . 5. l—di = 1-04 mm; 2—ci = 1-63 mm; 3—d = 2-00 mm.

of dY increases h0, and slightly reduces the slope of the working section

of the curve.

Figure 6 shows the static characteristics of a relay with constant pressure

drop at the variable restriction, having dx = 0*18 mm, d2 = 0-50 mm and

/ = 18 mm. Pressure P0 was maintained constant, equal to 1 atm. Curves were

obtained for a number of values of APV.

P7,o/m P0

= 1afm.

1

0-8

0-6

04

~~0·01 002 0-03 h,mm

F I G . 6. 1— APV ^ 011 atm; 2 — Δ Ρ ν & 017 atm; 3—ΔΡν = variable; P2 = 0 atm.

As can be seen from Fig. 6, the static characteristics of this type of relay run above those of a simple relay. With reduction of APV the curves are dis-placed towards higher pressures Ργ.

Pi,atm. APf =0033atm. P0 = 1atm.

1

0-8

0-6


Figure 7 shows the static characteristics of the relay (with dt = 0-18 mm,

d2 = 0-50 mm and / = 18 mm), taken at various values of APV, and at the

same APf for all curves. They demonstrate that the reduction of APV dis-

P7, atm. Δ Pf =-0033atm,P0=1atm.

U

002 h,mm

F I G . 7. 1—APV 0-14 atm; 2—APV & 0-35 atm; 3—APV = variable;

P2 = 0 atm.

places the working section of the characteristic to the right, and its slope be-

comes steeper.

Figure 8 shows the static characteristics of the same relay, taken at various

APf and constant APV. On the same diagram are also shown the static

F I G . 8. 1—APf = 0-03 atm; 2—APf = 01 atm; 3—APf = 0-2 atm;

4—P2 - 0 atm; 5—APf = 0-3 atm.


characteristics of a simple relay (curve 4), and a relay with constant APV, but

continuously varied APf (upper enveloping curve). Comparing this diagram

with Fig. 4, taken at a number of constant values of APf but without main-

taining a constant APf, it may be noted that the constant differential over

the variable restriction raises the initial section of characteristics towards

higher values of Pu and displaces the working sections to the right. The in-

crease of APf for a relay with constant differential over both restrictions in-

creases the distance h 0 i, as can be seen from Fig. 8. Figures 7 and 8 also in-

dicate that the addition of the constant pressure difference at a variable re-

striction increases the range of pressures within the working sections of the

characteristics.

C O N C L U S I O N S

1. Maintenance of a constant pressure difference over the fixed restriction

(A Pf) substantially increases the slope of the working section of the charac-

teristic. The increased value of a constant APf displaces the working section

to the right and reduces the slope.

2. Maintenance of a constant pressure difference over the variable restric-

tion (APV) displaces the characteristic towards higher pressures in the cham-

ber. Maintaining APV = const in itself does not improve substantially the

performance of a relay.

3. The constancy of both APf and APV substantially increases the slope of

the working section of the characteristic, its linearity, and moves the initial

section (preceding the working section) towards higher pressures in the cham-

ber. Increased value of a constant APf has the same effect as mentioned in

conclusion 1. Increased value of a constant APV increases the slope, and dis-

places the curve to the right.

In consequence it is now possible to select the required characteristic by

choosing various values of APf and APV.

A P P E N D I X 1

T A B L E 1. L I M I T S OF F L O W REGIMES

Conditions Exists when :

s-s Λ > APf; Ργ > 2APV

i s APf > Λ > 2APV

s-t 2APV > P , > APf

t-t P, < APf \ P j < 2APV


Conditions Formula

s-s d\ / ( Ρ ο - Λ ) Λ

1 Ad2 V ( Λ - P2) Pi

t-s h dl P

° y/lP2(Pi-P2)]

s-t

t-t , d\ P0

4d2 Λ

T A B L E 3. E Q U A T I O N S FOR / / 0. R E L A Y S W I T H APf = const

Conditions

t-s

t-t

Formula

//o d\ I APf(P0-APf) I APf

Ν (Po -

Po dj_

8^2 y/[P2(Po-APf-P2)]

dl I APf

ho

ho

2d2 Ν Pq— APF

d\ Po

4d2 P0 - APF

T A B L E 4. E Q U A T I O N S F O R W O R K I N G R A N G E S OF S T A T I C C H A R A C T E R I S T I C S

FOR R E L A Y S W I T H APf = COnSt

Conditions Formula ' Conditions Formula

SS , d\ I APfPx

4d2 V (P,~P2)P2

S h

d* lAPf

2d2 V Pl

t-s d\ APf + Px

8 i /2 y/[(Pi -p2)P2]

d\ (AP, \ = 4 ^ 7 \~pT /

T A B L E 2. E Q U A T I O N S FOR S T A T I C C H A R A C T E R I S T I C S OF A SIMPLE R E L A Y


T A B L E 5. M A X I M U M SLOPE OF S T A T I C C H A R A C T E R I S T I C S ;

R E L A Y S W I T H APf = const

Conditions Formula

s-s dPi Sd2 A , , 2 / P 0 - A P r

dA m a x~ d\ i Po Δ Ρ

* P 2)

V APfP2

t-s d/>, \6d2 (P0 - APf - Ρ2Ϋ

/\Ι P2

t-s , d// m ax " d\ (2P2 π 2APf - P0)

s-t άΡχ 4d2 (P0-APfY

/2

s-t

d/? Imax d\ yJ{APf)

d/>, ! Ad2 (P0 - APf)2

t-t d/7 : M A X" d\ APf

T A B L E 6. E Q U A T I O N S FOR Λ 0 1. R E L A Y S W I T H B O T H APf = const

A N D APV = const

Conditions Formula

s-s d\ I A Pf(P0 — A Pf)

'0 1 " 4d2 V APr(P0-APf-APv)

t-s d\ Po

h U Mi ^[APv(P0-APf-APt)]

s-t Same as h0 (Table 3)

t-t Same as h0 (Table 3)

T A B L E 7. E Q U A T I O N S OF S T A T I C C H A R A C T E R I S T I C S FOR R E L A Y S W I T H

APV = const

Conditions Formula

s-s d\ liPo-ΡΟΡχ

1 Ad2 V APtXP{ -APV)

t-s h dl P

° 8 i /

2 yj[APviPl -APV)]

s-t Same as h for simple relay (Table 2)

t-t Same as // for simple relay (Table 2)

S T A T I C C H A R A C T E R I S T I C S O F P N E U M A T I C R E L A Y S 2 1 5

Formula

" 4d2 *1\ΔΡΌ ( Λ -APv)j

= d\_ APf+ Px

' ~ %d2 ^[ΔΡν{Ργ — APV)]

Same as h for relay

with APf = const (Table 4)

Same as h for relay

with APf = const (Table 4)

T A B L E 9. M A X I M U M SLOPE O F S T A T I C C H A R A C T E R I S T I C S .

R E L A Y S W I T H B O T H APf = const and APV = const

Formula

\àPj I _ \6d2 (P0-APf-APvY/2APv

' d/î i m ax ~ d\ p0 _ 2(P0 - APf - APV) yJ(APt)

j dP1 j Sd2 (P0 - APf - ΑΡνΫ/2 y/(P0 - APf)

s-t Same as d/z

j for relay with ZIP/

max

= const (Table 5)

t-t Same as d/7

i for relay with APf

; max

= const (Table 5)

A P P E N D I X 2

The influence of changing parameters P 0 and APf on the distance h0 can be demon-

strated by analysing the derivatives dh0jdP0 and dh0\d APf. The first of these derivatives

is negative, the second one positive. W h e n considering the derivative dh0ldAPf, it is

necessary to take into account the inequality P0 > 2APfi which determines the sub-critical

flow through the fixed throttle.

T o demonstrate the influence of changing parameters APf and P0 on the slope tan γ2,

we alter the equation (6) thus:

tan y2 = 4

^2 {(P0 - APf - P2) ^[Ρ3(Ρ3 - P2)]

d\^{P2APf)

+ >/[<Λ> - APf) (Po - APf - P 2 ) ) ( P 3 - P i ) ) ,

from which it follows that increase of P0 increases tan γ2 and increase of APf reduces it.

Conditions

s-s

t-s

s-t

t-t

Conditions

s-s

8 Aizerman!

T A B L E 8. E Q U A T I O N S OF S T A T I C C H A R A C T E R I S T I C S F O R R E L A Y S W I T H B O T H

APf = const and APV = const


A P P E N D I X 3

The expression ( 1 2 ) differs from ( 6 ) in having P2 replaced by APV. Substituting in the

modified formula for tan γ2 (Appendix 2 ) P2 by APV, we obtain the expression for tan y 4.

From the resulting formula it can be seen that t a n y 4 decreases with increasing APV.

Similarly t a n y 2 decreases with increasing P2. Since P2 > APV, it follows that t a n y 4

> tan γ2.

A P P E N D I X 4

The positive sign of the derivative dh0ildAPf, and the negative sign of the derivative

dhoildP0, indicate that h 0l increases with increasing APf, and diminishes with increasing

P0. T o prove that h01 diminishes with increase of APV, we note that the denominator of the

term under the square root of equation ( 9 ) may be differentiated with respect to APV to

give:

d Γ 1 "I = - P 0 + APf + 2APv

d APV [APV(P0 - APf - APV)\ AP2(P0 - APf - APV)

2 '

T o show that this derivative is negative, we prove the inequality P0 > APf -f 2APV. A s

APf = P0 — Pl9 consequently P 0 > P0 — ( Λ — 2APV). N o w it is known that PY

> 2APV for the conditions of sub-critical flow and therefore the inequality is proven.

R E F E R E N C E S

1. V . N . D M I T R I Y E V , Avtomatika i Tekmekhanika, Vol . X V I I , N o . 9, 1956 .

2. A . G . S H A S H K O V , Theory of control devices of the "nozzle-baffle" type, working with

oil (this book, p. 2 8 5 ) .

3. E . S A M A L , Regelungstechnik, N o . 3, 1954.

4 . G . T . B E R E Z O V E T S , Avtomatika i Tekmekhanika, Vol . X V I I , N o . 1, 1956.

5. S . A . B E R G E N , Instrument Practice, N o . 4 , 1954 (also in Regelungstechnik, N o . 1, 1 9 5 4 ) .

6. G . T. B E R E Z O V E T S , V . N . D M I T R I Y E V and E . M . H A D Z A F O V , Priborostroyeniye, N o . 4 , 1 9 5 7 .

7. L . A . Z A L M A N Z O N , Diagrams for parameters of steady-state air flow through systems

of orifices in pneumatic controllers (this book, p. 3 5 5 ) .

A HYDRAULIC FOLLOW-UP POWER CONTROL UNIT FOR GENERAL

INDUSTRIAL USE

V . P . T E M N Y I

T H E Institute of Automation and Telemechanics of the Academy of Sciences U.S.S.R., together with the "Teploavtomat" factory in Kharkov, have de-veloped a hydraulic follow-up power control systems [1], which can exert a large force over a sufficiently long stroke, and at high velocity. This device (GSP-1) is intended for use in automatic control systems, with either a hydraulic or pneumatic controller of the compensation type, or with electric controllers. In the latter case a special attachment—an electro-hydraulic transducer—is required.

The developed unit finds applications in automatic control of power plants, in metallurgy, the chemical industry, gas works, oil refineries, furnaces, etc.

Figure 1 shows a line diagram of the hydraulic follow-up device and the connected electro-hydraulic transducer. If the device works with a pneumatic or hydraulic controller, the control signal (air or oil pressure) is transmitted into the outer chamber of control element 1 (in relation to bellows).

Simultaneously, the feedback valve pressure is transmitted to the inner chamber of the same bellows. This pressure is proportional to the piston stroke of the hydraulic motor. I f the forces produced at the control element by the input pressure signal and the feedback pressure are unequal, the out-of-balance force displaces the rigid centre cap of the bellows and the con-nected control rod of the hydraulic amplifier. The amplifier is used in order to improve the sensitivity of the system.

Hydraulic amplifier 2 is a unit in which a spool valve accurately follows the displacement of the control rod. When the spool is displaced from its neutral position, oil under pressure enters into one side of the hydraulic cylinder and moves the piston. Oil is displaced from the other side into the return line.

The displacement of the piston is transformed by a lever mechanism into an angular displacement of a shaft, which is connected by a linkage to a re-gulating unit.

A cam is fastened to the output shaft of the hydraulic motor. This cam moves the push rod of the feedback valve 4. This device ensures proportional-ity between the angular deflection of the shaft and the pressure in the feedback

219


F I G . 1. Line diagram o f the hydraulic follow-up unit with an electro-hydraulic

transducer. 1—control element; 2—hydraulic amplifier; 3—hydraulic motor;

4—feedback valve ; 5—electro-hydraulic transducer.

When the hydraulic follow-up unit is used in conjunction with an elec-tronic controller, a potentiometer at the output of the latter is included in a bridge circuit. An electrical signal, due to unbalance of the bridge, is

line. The piston of the hydraulic motor continues to move until the force in

the bellows of the control element due to the feedback pressure balances the

force due to the control signal, at which condition the spool settles in the

neutral position and the piston movement ceases.

H Y D R A U L I C F O L L O W - U P P O W E R C O N T R O L U N I T

F I G . 2. External view of the hydraulic follow-up unit.

r so /

/ A

/ >

A /

/ j V

/ J

0-2 0-v 0-6 0-8 1-0 1-2

pinp) of m

F I G . 3. Static characteristics of the hydraulic follow-up unit.

A 221


amplified, and operates the electro-hydraulic transducer 5. The resulting

pressure at the output of the transducer is used as a control signal, and is fed

to the chamber of the control element 7. The hydraulic motor, and the

electro-hydraulic transducer, are supplied by oil at 16 atm pressure through

the pressure line of a pumping set. The returned oil is fed to the tank of the

same pumping unit. In industrial hydraulic automatic control systems the

most widely used working fluid is transformer oil. A cheaper spindle oil,

V - 2 , with a greater viscosity may also be used. These oils have a high re-

sistance to oxidation, and are free from water and impurities. The external

view of the unit is shown in Fig. 2. The piston of the hydraulic motor has a

diameter of 80 mm, and develops 800 kg thrust with 16 atm supply pressure.

The maximum torque at the lever arm is 100 kg/m. The maximum speed of

the lever rotation is 45° per sec, with a pump delivery rate of 40 l./min. The

maximum angle of the lever arm rotation is 90°. The range of input signal

pressures is 0-1-1-1 atm., the sensitivity of the system 0-4 per cent. The

whole unit is a linear device, as can be seen from results given in Fig. 3, where

the abscissa is input pressure, and the ordinate angular deflection of the out-

put shaft.

R E F E R E N C E

1. V . P . T E M N Y I and V . A . K H O K H L O V , Peredovoi nauchnoteklmicheskii i proizvodstevennvi

opyt, 42, Ρ - 5 7 - 6 0 / 1 2 , U N I I T I , 1957.

EQUATIONS OF A STABILIZING SYSTEM, CONSISTING OF A H Y D R A U L I C

LINEAR MOTOR (RAM) CONNECTED TO THE CONTROL ELEMENT

BY PIPELINES

S. A . B A B U S H K I N

W E CONSIDER a system (Fig. 1) for the automatic stabilization of a certain

mechanical load 7. The motion of the load 7 is effected by a hydraulic servo-

cylinder 5 and the piston 6, which is controlled by a throttling element,

consisting of a rocker 2 and two pin valves 3 and 49 controlling the pressure

in the main hydraulic lines 10 and / / . Oil is pumped into each of these lines

F I G . 1. Line diagram of the automatic stabilization system.

8 a Aizerman I

223


by the gear pumps 9, and is by-passed through orifices, covered by the pins 3

and 4. The position of the pins, as determined by the angle β of the rocker's angular deflection, is dependent on the position of ζ and velocity azjat of the load 7. Signals proportional to ζ and azjat act through an electronic am-plifier 6" on the torque motor 7, and consequently on the rocker 2, rigidly fastened to the armature of the torque motor. Pipelines 10 and 11 incor-porate flexible rubber hoses and hydraulic accumulators 12 and 13. The purpose of the hydraulic accumulators is to obtain a stable and accurate control system.

The present paper deals with the derivation of the equations of this system, without neglecting the pipe elasticity of the connexions 10 and 11, and the compressibility of the working fluid. The analysis also takes into account the inertial forces due to the unsteady motion of the fluid.

B A S I C E Q U A T I O N S F O R T H E E L E M E N T S

OF T H E S Y S T E M

(1) The Equation of Motion of a Hydraulic Ram carrying a Load. Neglecting dry friction, we obtain:

M =

- Pkt)S- c/ r, (1) at

where ζ is the displacement of the load; Phl and Ph2 the pressures across the piston of the driving cylinder, connected to the lines 10 and / / ; Ut the ex-ternal load, acting on the output; M the total mass of the load, piston and piston rod; and S the effective piston area.

(2) Equation of Motion of the Rocker carrying the Pins. Again neglecting dry friction, we have:

+ r, (2) at

2 \ at J

whereβ is the angular displacement of the rocker ; Fx and F2 are the forces of the pins on the rocker; J the moment of inertia of the rocker and pins; r the dis-tance from the axis of the rocker to the pins; and Rv and R2 the coefficients giving the effects of displacement and velocity of the piston.

(3) Static Characteristics of the Pin Valves. The characteristics of the "ori-fice-pin

,, pair (Fig. 2) is given by the relationships

Qu = β.(*,/υ, (3) F=F(x,Pu), (4)

whereQu is the mass flow rate through the throttling element; PMthe pressure

E Q U A T I O N S O F A S T A B I L I Z I N G S Y S T E M 225

before the orifice; xj = b — (—l)Jrß the lift of the needle; and b is the

needle lift at β = 0. The relationship (3) can be calculated from the equation:

Qu = ßQfny/[2Q(Pu- Pr)h (5)

where μ0 is the discharge coefficient ; / „ the area of the orifice; Pr the external (return) pressure; ρ the fluid density; and / % ndcx sin oc the minimum clear area when the pin is moved by the amount χ (the area of the conical frustrum

F I G . 2. Sketch of the "orifice-pin" element.

with the generator AB) (Fig. 2). The relationship (4) may be found from the momentum equation, applied to the mass of fluid in the volume ABEFDC (Fig.2) . In the direction of the axis, the equation takes the form:

QU(W2 cos* - Wl) = (Pu- Pr) ndl

- F, (6)

where Wx = Qujofe is the mean velocity in section CD, with the area fe

= nd2c/4; and W2 = QjQfun is the mean velocity through the area of the

conical frustrum with the generator AB. From equation (6) we obtain

F=(PU- Pr) nd

2

c 4 tan c

4 ndcq tan a \x dc

Equation (7), together with (5), determines the relationship (4).

(7)


1 + —(/>' - λ>) Ε

( H )

where Q0,fo are the density and cross-sectional area at a pressure P' = P0; Kfi is the bulk modulus of the fluid; Ε the modulus of elasticity of the pipe material; and e a coefficient depending on the type of cross-section and the wall thickness of the pipe; for thin-walled pipes e = dm/ô, where δ is the wall thickness.

Equations (10) and (11) are applied to the entry cross-section of the length of pipe, to simplify further operations. Taking into account that

QV = Qfl = Qofol + P

~K

P°) > W

where Κ = 1 / (1 / /^ + e/E) is the apparent modulus of compression, and omitting indices of ρ and / , equation (9) can be written :

1 AP' Q'-Q" = —Qfl — . (13)

Κ dt

(4) Equations for the Hydraulic Supply Lines. Consider a hydraulic line con-sisting of several lengths, each of constant circular cross-section, connected through local resistances. In addition, we will include the possibility of blind branches and hydraulic accumulators, connected at the junctions.

(a) Equations of Motion of a Fluid in a Length of Pipeline of Constant Circular

Cross-section. The pressure and flow rate at the entry to this length are P'

and g ' , and at the exit P" and Q" respectively. These terms are related by the momentum equation, that is:

(8)

and the equation of mass continuity:

where / is the length of the line ; d the diameter of the pipe ; / = nd2/4 the cross-

sectional area; λ the coefficient of resistance; ρ the fluid density ; and F the volume of fluid in the length of the line.

The fluid is considered to be compressible, the pipe-walls to be elastic, and their thickness constant. Assuming that the fluid and the material of the pipes follow Hooke's law, we obtain:

(9)

(10)


In automatic control systems containing hydraulic components, certain connecting lines are sometimes formed as holes in the casing of a hydraulic mechanism, and these can be considered to be perfectly rigid, that is Κ = Kfl.

If the connecting line is a thin-walled tube, the elasticity of which cannot be neglected, then the apparent modulus Κ must be used.

Equations (8) and ( 13) | are valid for every z-th part of the connecting hydrau-lic system with a constant cross-section. Denoting the flow and pressure at the entrance into the /-th section as Qt and Pt, and the increments of these quantities along the part asZlg, and ZIP,, on the basis of equations (8) and (13) we obtain :

1 άΡ' AQt= -—qM—L, (14)

Ki at

( 1 5) fi àt 2 d, off

(b) Equation for the Pressure Drop at a Local Resistance. The lengths of the

line are joined through local resistances. The pressure drop at a resistance

between the i-th and / + 1-th sections is written as:

Qfhx ΔΡ,,,+ Ι = -ζ,,,+ ιζιμ., (16)

where £ * , ί + ι . . . the coefficient of local resistance. Equations (14), (15), and (16) enable us to find the flow rate and pressure

at one end of a hydraulic system, providing these terms are known at the other end. Denoting the pressure and flow rate at the beginning of the first section by P[ and Q\ and, at the end of the &-th section, by P'l and Ql we obtain :

Κ = P[ + Σ(ΔΡΧr +ζΐΛ-ι.ι) i= 1

k

* 1 άΡ'

Ql = ß i - Σ — 9/ώ -τ1- • (18)

i = 1 Κι Gt

(c) Equation of Flow in a Plugged Pipe. I f there is a branch consisting of a plugged pipe or channel (Fig. 3) between the sections (i — 1) and /, then

1 άΡ' Q'i = Ql-. + ^0, = β ? - 1 - — QlJp —L, (19)

K„ àt

t Equations (8 ) and (13) can also be obtained from the hydrodynamical equations for

the one-dimensional flow of a fluid, written in partial derivative form.


where AQP is the instantaneous flow rate out of the plugged pipe; lp and fp

are the length and cross-sectional area of the plugged pipe; and Kp is the apparent bulk modulus.

{A) Equation of Flow for a Hydraulic Accumulator. I f there is a branch lead-ing to an accumulator (Fig. 4) between sections (/ — 1) and /, then

Qt = QU + AQa = QU - 4- (QVa), (20) at

where AQa is the flow rate out of the accumulator and Va the volume of fluid in the accumulator at a given instant.

F I G . 3. Sketch of a plugged pipe. F I G . 4. Sketch of a hydraulic accumulator.

Assuming the accumulator to be cylindrical, the pressure under the piston Pa = Pl and with the notation fa = piston area; la = initial piston position; / = instantaneous piston position; P0 = initial pressure in the accumulator; c = spring stiffness; and neglecting the mass of the piston we obtain

1 dP'

Q\ = QU - —QiJa-r-, (2D Ka at

where \\Ka = (\/Kfl) + (fa/cla). (e) The Complete Equation for the Hydraulic Line. When calculating Pk and

Qk from equations (17) and (18) we shall assume that the pressure and flow at the /-th section are approximately equal to those at the beginning (or end) of


the hydraulic system under consideration, i.e. Q\ = Q\, P\ = P[ (or Q\

= Qk \ P[ = Pi). Then equations (17) and (18) may be written by incor-

porating (19) and (21):

Pl = p[ - N ^ - - λ ^ - 9 (22) at ρ

Q: = Q \ - - ^ - v e q ^ , (23) Kfl at

where Veq = + ' ι Λ ( * Α ) + Lfa{KfljKa\ the equivalent vol-/ = 1

ume of the hydraulic system; if there are no accumulators and

the walls of the pipes are rigid, it is equal to the actual volume of

fluid (tf, = Kp = Kfl);

k

Ν = ^ ( / / / / / ) an equivalent ratio of the length of the hydraulic system / = 1

to its cross-sectional area;

k (λ- /· 1 1 \

λ = X ( — — — H j Ci — ι, ζ ) , an equivalent resistance coefficient i = 1

\ 2 di fi fi J of the hydraulic system.

The quantities Veq, Ν and λ will be called the equivalent parameters of a hydraulic system.

( f ) Equations for the Main Hydraulic Lines of the Hydraulic Control System.

Each main line can be considered as consisting of two sections: from the gear pump to the pin valve, and from the gear pump to the driving cylinder. Gear pumps have a constant delivery rate Q0j. The pressure at the delivery ports of the pumps is P0J. According to equations (22) and (23) for the first of the two sections:

P u j = P ' u j - N u j ^ - X u j ^ L , (24) at ρ

U= 1,2)

Quj = Quj - ( VeXj ^ , (25) Kfl at

where PUJ, Quj are the pressure and flow rate at the beginning of the section ; Puj, Quj the pressure and flow rate at the end of the section (at the pin valve); and Nuj, Xuj, V{eq)Uj the equivalent parameters of the section.

For the second section, and remembering that the direction of flow is here determined by the direction of the piston motion relative to the cylinder, we


C O M P L E T E B A S I C E Q U A T I O N S

F O R A C L O S E D - L O O P C O N T R O L S Y S T E M

+ I Xuj - ( - \)j sign *L λ J QS

2 (*f)\ (31)

dt J \dt J

Quj = Qoj - ( - l )J

Q S ^ - - - 2 - [(Veq)UJ + (Veq)hJ] . (32) dt Kfl dt

In equation (31) it is assumed that QhJ = Q'hJ, and in equation (32) that

Phj = Puj — Ρuj-Substituting the value F3 from (7) into equation (2), and taking into ac-

count (5), and assumingμ οι — μ 0 2, we obtain:

Ô t V d t) (33)

have

p hj = p ' hj - ( - \ y s i gn f ! i (NhJ + 'hj 2Ά 9 ( 2 6)

df \ d / ρ /

( 7 = U2)

= - ( - Ο" sïgn ^ - i - (Κ,Λ,· ^ , (27) d/ Ä>, d/

where , QHJ are the pressure and flow rate at the beginning of the section ; Phj, Qhj the pressure and flow rate at the end of the section (at the giving cylinder); and NhJ, Xhj, (Veq)hJ the equivalent parameters of the section.

The equations for the flow rate and pressures at the junctions of the pump lines and the main (pin valve—cylinder) lines are:

ô ; = 0 o j - ( - l ) ^ i g n ^ Ô i j , (28) dt

U= 1,2)

P0J = P'UJ = P'hJ. (29)

(5) The Equation Relating Piston Velocity and Flow Rate. This equation is:

Qkj = QS^-. (30) dt

From equations (24)-(30) we obtain


nd2

A = r 4

" ι Λ 2 * · ~ Λ Ab λ" l — 4u,j — sin 2a 1 — 4 — tan α

ν </ )_ 5 = 8πμ&

3 s i n

2Λ ,

C = πμ\γ2ά sin 2 Λ ^ 1 — 8 — tan Λ \ .

d

Substituting the values of Quj from (5) into (32) we obtain:

d\\- ( - d ' - L / T I v î / v b

J*s

d z ρ d / ,u

df Jf,. v

àt Pr) = Qoj - (-Was — - - t - ( ^Λ ·

V I

(34)

where D = μ0π àb sin Λ V ( 2 ß ) ; ·/ = 1 ; 2^ = (^«V/ + (Κβ)ν is t ne

equivalent volume of the j-th cavity.

Finally substituting the values of PhJ from (31) into (1) we obtain

. d2z rr dz / d z \

2 · dz

( M + Afy,) — + / / , , — + ( ; . „ + λ„2) gS3[ — \ sign —

d /2 àt \dt / dt

- q S3

& 2 - λ. ,) f - ^ Y - - ( ^ 2 0 0 2 - KlQlù = S(Pu2 - pu1) - u„ \

d t J Q (35)

where MFL = (NUL + NU2 + NHL + NH2) qS2 is the mass of fluid referred to

the piston and Hfl = 2(ÀulQul + Xu2Qu2) S2 the damping coefficient of the

drive, including the hydraulic resistance of the pipelines.

L I N E A R I Z E D E Q U A T I O N S O F A C L O S E D - L O O P C O N T R O L

S Y S T E M

Equations (33), (34), and (35) are greatly simplified if both branches (pin

valve-pump piston) of the hydraulic system are symmetrical. Assuming this,

and confining the analysis to the object of stabilizing the system at a constant

load component U, it is possible to find the steady-state values of the term z,

Pul* Pu2> ß from the following static equations:

kxz + (A + Bß2) (Pu2 - Pui) + C(Pu2 + Λι - 2Pr)ß = 0, (36)

D^l-(-l)Jjß^y/(Pul-Pr) = Q09 (y = 1 , 2 ) (37)

S(Pu2 - Pui) = Ü. (38)

where


dt2 \ dt

dvi . dy T2—- + vx = α2μ- b 2 ^ , (40)

dr dt

dis dy T3 — + v2 = -α3μ + 6 3 — , (41)

dr dr

{ T, 2 + r, 2 } ^ + d / = ^ _ ^ _ ^ ( 4 2)

d/2 df

Here we use the non-dimensional terms:

Aß . zl /> ul .

β max ßm'in max

APu2 Az

u m m

•<2 = — , X = (43) Ρ —Ρ Ζ — ? • 1 m max * h m i n ^ m a x " m i n

and the following notation is used:

2Bß(Pu2 -PHl) + C(Pu2 + P u l - 2Pr)

2?Ve<lJ(Pu2 - Pr) r2 =

b

K„D[l + Lß b

= M 1

Hfl

Denoting the perturbation of the respective terms from their steady-state values by Zlz, ΔΡυί, APu29 Δβ and linearizing equations (33), (34) and (35) by the usual methods, the linear performance of the system is as follows:

T\ + μ = -ax ( τ * £ + χ \ - bxv2 + c,v,, (39)


__ ^ l (Zm a x ~

Zm i n ) .

a i ~ [2Bß(Pu2 - >„,) + C{Pu2 + - 2P,)](ßmax - ßmJ

b = (A + Βψ + Cß) (Paam - Pumin) .

' [2Bß(Pu2 - Pui) + C(Pu2 + Pal - 2Pr)](ßmax - ßmiS

= (A + Bß2 - Cß) (Pumax - PumJ .

C l [2B~ß(Pu2 - />„,) + C{Pu2 + Pul - 2Pr)](ß,nax - ßmiS

2 - (hi - Λ) & ßmax ßm'm

α2 =

b2 =

# 3

r ~ Ρ — λ ~ Ί β

p . r ~ Ρ — λ ~ Ί β

2@S y/(Pa2 - Pr) z m a x-— ^min

- p . 1 u m i n

2 (Pal - Pr) ßm'in

r ~ Ρ —Ρ Λ , ο « max

χ u m i n

l + Jß

b _ 2oSjÇPul-Pr) z„

3 / r ~\ P - P

η / 1 ι λ 1 umax -* umm

Dy + -b

ß

" 4 —

»

^ / l (Z

m a x — Z

m i n )

E/f - L /

ΗJl(zmax ~

Zm i n )

The suffices "max" and "min" are used to indicate the magnitude of the various terms when maximum or minimum constant load is applied to the output member of the control system.

Consideration of these equations leads to the following conclusions: 1. The inertia of fluid, included by defining the equivalent ratio of pipe

length to cross-sectional area, may influence the transient performance of an automatic control system. This effect will be particularly marked with a quick acting output motor (actuator), with a driving system with a low iner-tia, with a considerable distance between the control device and the output ram or motor, and when there is a rapidly changing acceleration demand.


2. The nature of the pressure changes in a hydraulic control system is

determined by its equivalent total volume.

3. The sum of the hydraulic resistances, as expressed by the effective equi-

valent resistance coefficients, exerts a damping effect on the action of an

automatic control system.

The "equivalent" parameters of hydraulic circuits may be varied to some

extent when designing new control systems in order to obtain the desired

results.

This paper gives the results of an investigation carried out by the author

under the direction of Prof. J. P. Ginsburg (University of Leningrad).

R E F E R E N C E S

1. J. A . C H A R N Y I , Unsteady Motion of Real Fluids in Tubes. Gostekhizdat, 1951.

2. J. P. G I N S B U R G and A . A . G R I B , VestnikL.G.U. (Bulletin of Leningrad University) N o . 8 ,

1954.

AUTOMATIC COMPRESSED AIR PLANT

V . S . P R U S E N K O

T H E EFFICIENCY of industrial automation systems which use compressed air

energy for measuring, controlling and regulating elements depends on the

uninterrupted supply of compressed air and on its quality.

Below is described an automatic compressed air plant which ensures, by

the simplicity of its design, a reliable supply of clean and dry compressed air.

a

F I G . 1. Centrifugal compressor with "hydraulic pistons".

The plant consists of a working compressor and a reserve; a dehydrator; a starting and controlling automatic system; receivers; and pipelines.

Figure 1 shows a centrifugal! compressor, sucking in and compressing air by means of "hydraulic pistons". In a cast housing 7, machined out to an elliptic shape, rotates in a clockwise direction rotor 2, with curved blades. The housing is partially filled with water. The turning of the rotor

t The term "centrifugal" is used here in a wider meaning. Strictly, this well-known

compressor type is an approximation to a positive displacement compressor, differing

fundamentally from what is generally known as a purely centrifugal (i.e. "dynamic")

machine (Translator).

237


causes the water to be driven by blades. Due to centrifugal force, water will follow the contour of inner walls of the housing, forming an elliptic ring. This water ring, as can be seen from Fig. 1, has its greatest distances from the rotor axis in points a, and in points b its shortest. Between each pair of blades the water ring forms a closed chamber, which expands as it moves towards points a, and contracts moving towards points b. During each re-volution of the rotor, each chamber has two expanding and two compressing strokes. The water ring acts here as a "hydraulic piston". The chambers are connected in turn with suction ports 4, and delivery ports 5, leading respec-tively to inlet 3 and outlet 6. Fresh air is sucked in by expanding chambers through ports 4 ; with the further turning of the rotor the volume of chambers is reduced, air is compressed and delivered through ports 5 into the outlet 6.

The capacity of the compressor is equal to twice the swept volume of the "hydraulic pistons" multiplied by the number of revolutions per minute of the rotor. It can be from 20 to 600 free m

3/hr, and the compression ratio (which

is the ratio of full volume of working chambers to the difference between this full volume and the swept volume of hydraulic pistons) may vary from 2 to 8.

T o avoid overheating and evaporation of water, the housing is cooled by water. The absence of friction parts eliminates the necessity for lubrication, and therefore the air is not contaminated by oil. The water ring ensures elimination of dust from the air. The compressor has practically only one moving part—the rotor. There are no friction parts, such as pistons, sealing rings, valves, gears, etc., which are usual for other compressors. Therefore the wear and running expenses are insignificant. The compressor is very re-liable, and requires practically no maintenance. It can be considered, by vir-tue of its above-mentioned features, to be a dependable source of com-pressed air for automation.

The air delivered by compressors is free from dust and oil, but it is still unsuitable for pneumatic apparatus, because of its high temperature and humidity. The specific moisture content (by weight) at a constant temperature can be considered with sufficient accuracy to be inversely proportional to absolute pressure. The relative humidity at a constant temperature (until a dew-point is reached) is directly proportional to the absolute pressure. Con-sequently, if usual atmospheric air, after being compressed, is cooled to the inlet temperature, a part of water will be condensed, and air itself would have 100 per cent relative humidity. Such air cannot be supplied for pneumatic instruments, because this would result in water condensation in units working at a lower temperature, and in icing up in winter. Therefore the compressed air must be dried and cooled. The drying must be carried out to such a de-gree that the dew-point would be lower than the lowest temperature likely to be encountered in any elements of an automatic system (regulators, pipe-lines, receivers, etc.).

In the described air plant, the cooling and drying of air from compressors is effected by a two-stage dehydrator. Its scheme is shown on Fig. 2. The

A U T O M A T I C C O M P R E S S E D A I R P L A N T 239

dehydration of air is carried out in two steps : first step in a water condenser / ,

and second in silica-gel absorbers 2 and 3.

In the condenser, which works like a usual heat-exchanger of the "tube

within a tube" type, compressed air is cooled to the temperature 15-20°C.

During this cooling, condensed water separates itself from air and, flowing

F I G . 2. Scheme of a two-stage dehydrator.

along the walls of the condenser, is collected in traps / ' , from which it is drained through solenoid valves 5CB and 6CB.

In usual circumstances, the condenser removes much more moisture than the absorbers do. In view of this, it would be possible in certain cases to limit the installation to a condenser alone. This is permissible, for example, when the automatic plant works in locations where the temperature through-out the year does not fall below 0°C.

In the described universal plant, however, there is a second stage of air-drying by means of silica-gel absorbers. Silica-gel is remarkable for its great capillarity. The volume of capillaries takes up from 40 to 50 per cent of


32

28

2V

20

18

12

8

¥

-20 0 20 ¥0 60 80 100 120 WO Tß°C

F I G . 3. Equilibrium characteristics of silica-gel. W—equilibrium moisture content

of silica-gel in per cent of its dry weight; TB—air temperature before the dryer;

TM—air temperature before the dryer, according to a wet thermometer.

equilibrium, shown on Fig. 3. Its actual moisture capacity when working in a flow of air is, however, only two-thirds or a half of that given by Fig. 3.

In addition to the condenser 7 and absorbers 2 and 3, the dehydrator (Fig. 2) has also a jet circulation pump 4, flow-meters 5, electric air heater r5, auto-transformer 7, solenoid valves 1CB to 7CB, one-way valves, and in-struments of automatic control, the work of which will be described below.

The dehydrator works as follows: compressed air from compressors enters the condenser 7, where it is cooled, and a substantial part of the humidity is removed. From the condenser air enters one of absorbers 2 or 3, where it is dried to a residual relative humidity of 2-3 per cent, and then to receivers. T o obtain uninterrupted work, there are two absorbers. While one of them is working, the other is being subjected to regeneration. Switch-over from

the volume of silica-gel proper, and the surface of capillaries of 1 kg of silica-gel reaches up to 400,000 m

2. Because of this, silica-gel is an absorbent of

great capacity, enabling to dehydrate air to a residual relative humidity of 2-3 per cent at temperatures of 15-20 °C. This corresponds to a dew-point at — 30 to — 40 °C. Silica-gel which has absorbed moisture can be easily re-generated by heating to 150-200 °C. The moisture capacity of silica-gel for various conditions of its work can be determined from the curves of its static

W, percent


moisture-absorbing to regeneration, and vice versa, is performed automatic-ally.

An important feature of the described dehydrator is the running of the regeneration cycle without releasing the pressure in the absorber, which is being regenerated. Because of this, damage to the silica-gel is avoided, its service life increased, and consequently maintenance expenses of the plant are reduced. Moreover, the regeneration of silica-gel is done without loss of air; all the air from the compressors is dried and delivered for use. When, for example, absorber 2 is working and absorber 3 is being regenerated, then the solenoid valves 1CB and 4CB are opened and solenoid valves 2CB and 3CB closed. Compressed air goes through the venturi 4, condenser 7, absorber 2, and left-hand one-way valve to the receivers. A proportion of air, after absorber 2, is by-passed through electric heater 6, heated to 200-250 °C, and passed through absorber 3, where it effects the regeneration of silica-gel. Farther on, this air with water vapour passes from the upper outlet of the absorber into the narrow section of venturi 4, and into the condenser, mixing with "raw" air from the compressors.

When the solenoid valves are switched over, absorber 3 changes over to working cycle, and absorber 2 to regeneration cycle.

The work of the dehydrator is fully automatic, and co-ordinated with the work of the compressor by automatic control instruments. The block diagram of the automatic control for the compressed air plant is shown on Fig. 4. It may be divided into two parts: (1) control of compressors and (2) control of the dehydrator.

Control of Compressors. As already mentioned, of the two compressors one is working, and the other is a stand-by. Accordingly, in the scheme of controls for the motors of compressors, there are two control keys 1Y and 2Y. Through them, either of the compressors can be put into the condition of automatic start from contacts 2P-3 or 2P-4 of the intermediate relay 2P, switched on by the minimum contacts of the pressure switch ("electro-con-tact manometer") EKM-M. Relay 2P will remain "on" until the action of relay IP, actuated by the maximum contact of the pressure switch EKM-B. Simultaneously with the start of the compressor through the contact 2P-2, the solenoid valve 7CB opens the line of cooling water for the condenser. The control scheme provides also a manual push-button control of the compressor motors by push-buttons Κ for start, and 0 for stop, and signal lamps 5L and 6L. A breakdown or emergency pressure fall below a permissible level, which may occur because of the line breakage or failure of a compressor to start, and also a fall of pressure of water for condenser cooling, are signalled by bellows-operated warning devices 2C and 7C, actuating a signal lamp 1L and a buzzer G. Check of the emergency signalling scheme is effected by the push-button KG, and cancellation of the buzzer by the push-button KGG.

Controls of Dehydrator. These perform the automatic change-over be-tween drying and regeneration cycles, and also automatic drainage of con-


densed water from the condensers. The change-over of absorbers may be ef-fected either directly in response to a measurement of residual air humidity, or to a measurement of silica-gel saturation, these two quantities being inter-dependent. The degree of silica-gel saturation in the described set-up is determined by the time of the compressor's work, and this time governs the change-over.

For this purpose, an electric timing device KEP-12Y, which operates solenoid valves in a time sequence, is included into the current supply line

^ IB M

m m I

/ y - / 2P-3 ©•

1M

1V-Z OK

tM-J

Compressor 1 Compressor 2

F I G . 4. Scheme of automatic control of the compressed air plant.

through the blocking contacts of the magnetic starting devices 1M-5 and 2M-5. This ensures that the timing device KEP-12Y runs only during the work of a compressor, and is off when both compressors are at standstill.

The drying cycle of an absorber is signalled by a lamp 2L or Silica-gel is regenerated by heating it to 150-200 °C; the regeneration cycle

follows immediately after the drying cycle, and is accompanied by switching on electric heater EP through an autotransformer LATP-1.

The flow of heated air into an absorber goes on until the temperature in the absorber reaches 150 °C. Then the maximum contact of the thermo-switch ("electro-contact thermometer") EKT operates, switching on an inter-mediate relay 5Ρ (or 6P), which switches off the electric heater. This com-pletes the cycle of regeneration. As the air continues to pass through the heater also after it has been switched off, the absorber enters the cooling


cycle, being cooled to a temperature of 15-20 °C. The sum of regeneration time and cooling time is equal to drying time. The proportioning of time between regeneration and cooling may be set at will by adjusting a manual control valve with the aid of flow-meters at the venturi, and adjusting the air tempera-ture by means of auto-transformer LATP-2 and a mercury thermometer, provided at the heater.

The scheme incorporates also solenoid valves 5CB and 6CB for periodical draining of condensed water.

The instruments for the automatic control of a compressed air plant are mounted on a compact control panel, in a special relay box, on the panels of the electric motors driving the compressors, and to some extent on thecondenser and absorbers. The instrumentation of automatic control enables the plant to be run without service personnel. For setting-up and adjustments after as-sembly and installation, the plant is provided by a sufficient number of mercury thermometers and pressure gauges, fitted directly where tempera-ture or pressure has to be measured.

The described compressed air plants are compact, fully automatic and highly productive. It may be expected that they will be widely used, not only as a source of compressed air for automation of industrial processes, but also for other purposes—for example, supply of dry compressed air for high voltage circuit breakers. It would be appropriate to organize in one of the instrument-making factories the production of a range of such plants, with capacities of 12-5, 25, 50, 100 and 200 free m

3/hr.

Pneumatic and hydraulic instruments and control devices often incor-

porate controllable throttles of the "nozzle-baffle" type (Fig. 1). The purpose

is to throttle the flow of the working fluid, by moving the baffle to alter the

• x

F I G . 1. D iagram of the "nozzle-baffle" element. 1—nozzle; 2—baffle; 3—radial

gap; rc—nozzle radius, m; r T—radius of nozzle end surface, m; h—distance be-

tween nozzle and baffle, m; P0—pressure before the throttle, kg /cm2 ; P1—pressure

after the throttle, k g / c m2; Q—volume flow, m

3/ sec; F—force due to the action of

flow on the baffle.

resistance. Quantitative characteristics of such devices are given by the re-lation between flow rate β and the pressure difference P0 — Pl9 and the dis-tance h between the nozzle and the baffle. The main object of this investiga-tion is to develop the method for calculating these characteristics. For the cases in which the external forces acting on the baffle are of the same order as the force of the jet of fluid, it is important to be able to calculate the force characteristic of the device—that is, the relationship between the force F due

9 Aizerman I

247


Y E . A . A N D R E Y E V A

CALCULATION OF STATIC CHARACTERISTICS OF NOZZLE-BAFFLE

ELEMENTS


to the action of the jet on the baffle, and the distance h between the nozzle and the baffle. As the nozzle-baffle element is used for hydraulic as well as pneumatic devices, it is necessary to carry out investigations for both a viscous incompressible liquid, and for a viscous ideal gas. The present work deals with the flow of a viscous incompressible fluid. The problem of cal-culating the discharge characteristic Q = Q(P0 — P^, h), and the force char-acteristic F = F(P0 — Ρχ, h) can be accomplished by solving the hydro-dynamic problem of a steady-state flow of a viscous incompressible fluid in the radial gap between the end surface of a nozzle and a baffle.

F U N D A M E N T A L C O N S I D E R A T I O N S F O R T H E

S T E A D Y - S T A T E F L O W O F A V I S C O U S I N C O M P R E S S I B L E

L I Q U I D I N A R A D I A L G A P

Taking into account the axial symmetry of the flow, and using a cylindrical system of co-ordinates (r, z) the problem can be reduced to determining a pressure field ρ = p(r, z). The basic equations, consisting of the equation of continuity :

dvr vr dvz

— + — + — = 0 (1) vr r dz

and the Navier-Stokes equations:

dvr dvr 1 dP (d2vr d

2vr 1 dvr vr \

vr — + vz — = + ν I + + , (2) dr dz ρ dr \ dr

2 dz

2 > r dr r

2 J

dvz dv, 1 dP ( d2vz d

2vz 1 dvz\

vr + vz — = + ν I + + ) (3) dz ρ dz \ dr

2 dz

2 r dz J

dr

are fundamental to the problem. In these equations, vr and vz are the radial and the axial components of the velocity vector in m/sec, ρ the fluid density in kg sec

2/m

4, and ν the kinematic viscosity in m

2/sec.

Further, we shall consider only those cases when h is small, and it can be assumed that:

P(r, z) = P(r) + P(r, z), where

Ρ < Ρ

and, consequently

dP _ d(P + P) _ d^

dr dr dr

dP _ dP

dz dz

and

C A L C U L A T I O N O F S T A T I C C H A R A C T E R I S T I C S 249

dvr dvr 1 dP ( d2vr d

2vr 1 dvr vr .

vr + vz = + ν j H 1 — ) . (5) dr dz ρ dr \ dr

2 dz

2 r dr

-2

Taking into account the following boundary conditions:

vr(r, 0) = 0, v2(r, 0) = 0,

vr(r,h) = 0, vz(r,h) = 0, (6)

ί vrdz = ^ , (7) ο 2nr

? ( γ Γ ) = Λ , (8)

the solution of equations (4)-(5) will give the distribution of the mean pres-

sure Ρ = P(r), and also the velocity fields vr = vr(R, z) and vz = v2(r, z).

If we required to determine the actual pressures ( P ) , and not the mean pres-

sures (P)9 then it would also be necessary to solve the following equation,

derived from (3) :

dvz dvz 1 dP (d2vz d

2vz 1 dvz\

vr— + v z — = + ν I + + ) . (9) dr dz ρ dz \ dr

2 dz

2 r dz J

Here the functions vr = vr(r9 z) and vz = vz(r, z) should be taken as known; they are found by solving the pair of equations (4)-(5) .

Introducing the stream function ψ = ^(r , z ) , which by definition converts the continuity equation (4) into an identity, it is convenient to represent the system of equations (4) and (5) and the corresponding boundary conditions (6), (7) and (8) as an equation, which determines the stream function :

) [dz2 dr r dz

2) \dr)\dz

V)

dip

Tz

k ä z4 dz

2 dr

2 r dz

2 dr

the relationships for determining the velocity field from the stream function

( 1 0 )

Under these conditions, equations (1) and (2) determine the problem:

^ l + i l + ^ l = 0) (4) dr r dz

( Π )


F I N D I N G T H E S T R E A M F U N C T I O N ,

A N D T H E V E L O C I T Y F I E L D

Using the approach of Ref. 1, we assume that a stream function can be

represented by a series:

^ , : ) = f ^ . (13) k = i r

Substituting (13) into equation (10) and with the conditions of (11), we obtain after certain substitutions:

r2 i=i r U = i / i = i r

X J Σ (' " k + ' ) (a'ka"-k + 2 - ^ i - f c + 2 ) |

/ ( / + 2) v, fl',v , ^

v ~ a)l3

Vr = Σ —> V

z = L(k - Ο Π Γ Γ Γ ·

( 1 5)

If we now equate in (14) coefficients of equal powers of r, we obtain the fol-lowing system of ordinary differential equations for the functions:

^ = 0 , ^v = 0,

IV A

t η IV Z

/

^ 3 = ai + 3 = \ a j a i - j + 2. (16)

V Vj=\ By integrating these equations, we obtain functions ak = ak(z) depending

on the values of the arbitrary constants. The substitution of these functions into (15) enables us, after finding the arbitrary constants from conditions (6) and (7), to determine the velocity distribution.

and an equation, giving the mean pressure from the velocity field:

L*L = -Vr**-0t*± + J ^ + ^ + ±^-!£\. ( i 2 ) ρ dr dr dz \dr

2 dz

2 r dr r

2 J

When solving equation (10) with the substitutions of (11), it is necessary to take into account the boundary conditions (6) and (7) and when integrating the equation (12) the condition (8) .


v = h " f i l . v = \ vs il

2 6i70C(l - 0 Re \ h ) ζ9 ' [>οί(1 - ζ)

35 r_f

Re h

ζ= ; Re = ; ν0 = - — - , (17) η ν 2nrh

and considering that ak = 0 for even numerical values of k, the equations for

the velocity distribution can be stated as follows :

V2 = 0; Vr = 0. (18)

As a second approximation (k = 3)

V2 = (\ - 0 ( 1 -2C)(C 2-C- D ,

F r = 7 ( Î 4 -2 Î 3 + i £

2 + i Î - ! ) . (19)

As a third approximation (k = 5)

vz = (l - C)(i - 2C)(C2 - C - D + ( 7 ) 7 3 ^ x

/ 4 _ 6 8 M 12 .3 4 „ 256 . 16\

χ —C7 - 2ζ6 + - ζ

5 ζ

3 + —ζ

2 + ζ

\ 9 3 5 3 15 5 /

h Re χ

r I - ζ

12 „ , 16 „ - 19 „ 9 48 „ 17 ... 6 „ 26 . 53 , χ - C

9 ί

8 + — ί

7 ζ

6 ζ* + — ζ*—ζ

3 ζ + — ,

\55 5 3 2 35 5 5 231 770/

K , _ 7 | i . - 2 r + ± c . + ±ç - r> + i±V 1

2 2 7 / W l - ί

,7 - 4C

6 + — £

5 - 3£

3 + — £

2 + — ζ x

3 3 5 5 / r 1 -ζ

x [ ^ c 9 - 8 C 8 + - C 7 - 9 C 6 - - C 5 + - f 4 - - C 3 - — ί + 5 3

5 7 5 10 2 154 1540y

(20)

T H E P R E S S U R E F I E L D

Substituting now expressions (18), (19) and (20) into equation (12) , | and integrating, having taken into account condition (8), we obtain the pattern

t It must be noted that here the functions substituted in (12), v- = vz(rf z) and

vr = vr(r, z ) , are represented by a finite number of terms of a series; this leads to a

contradiction — dP/dr appears to be dependent on z. T o eliminate this contradiction, mean

values a long ζ should be taken.

Introducing the notation :


of the pressure distribution, which takes the following form after suitable

transformations:

T o a first approximation:

Β = 6A \noc - \2-5A2(oc

2 - 1 ) . ( 2 1 )

T o a second approximation :

Β = 6A I n* - 1 6 · 7 8 Λ20 χ

2 - 1 ) . ( 2 2 )

T o a third approximation :

Β = 6A In oc - \&7SA2(oc

2 - 1 ) - \4-2Α

2β

2(** - 1 ) - 3 · 6 7 Λ

3( Λ

4 - 1 ) ,

( 2 3 )

where

A = -2-ß>; Ä = ^ A * f ; « = - ^ ; ß = — . (24) πνη ρν

2 r 10rT

The solution of the problem, neglecting the effects of viscosity, is now

Β = - 1 2 · 5 Λ2( α

2 - 1 ) , ( 2 5 )

and the more general solution, which takes into account the viscosity, but

neglects the inertial terms of the equation, is

B = 6A\noc. ( 2 6 )

The formula, but without the same transformation, is quoted in Ref. 2 . In the paper by McGinn [ 3 ] is given the formula

Β = 6A \noc - \9-3A2(oc

2 - 1 ) , ( 2 7 )

which is obtained by combining linearly the approximate solutions ( 2 5 ) and ( 2 6 ) , which ensures a parabolic relation for the radial component of fluid velocity in the axial direction.

In addition, it may be noted that the solution, obtained by N . P . Shumskii, published in the present book (p. 2 7 2 ) , is identical with ( 2 1 ) , and so gives the first approximation to an exact solution.

In Fig. 2 all the quoted solutions are compared, together with experimental results taken from Ref. 3. An analysis of these curves leads to the following conclusions :

1. Solutions ( 2 5 ) and ( 2 6 ) do not give a correct picture of the pressure dis-tribution ; corresponding curves 1 and 2 on Fig. 2 are far removed from the experimental results.

2 . McGinn's formula ( 2 7 ) gives a pressure distribution, in good agree-ment with the experimental results for a diverging flow (solid line 5 ) , and somewhat less so for a converging flow (broken line 3). As the formula is, however, partly empirical, it lacks a rigorous basis.


Β

1 r FIG . 2 . Pressure distribution along the radius of baffle. — = dimensionless

p — p . ( h \2 oc rT

radius; B= —h2 I 1 —dimensionless pressure: /v = 3-84 m m :

Qv2 \ ί ο / τ ;

rT = 52-4 mm. (1) Q = 36-4 cm3/sec, h - 0 - 3 5 7 mm, t = 1 2 ° C , Ο = experimental

points, calculated pressure distribution.

(2) Q = —9-61 cm3/sec, h = 0 -357 mm, / = 1 4 ° C , #—experimental points,

calculated pressure distribution. 1—distribution, calculated by f o r m u l a ( 2 5 ) ;

2—by ( 2 6 ) ; 5—by ( 2 7 ) ; 4—by ( 2 1 ) ; 5—by ( 2 2 ) ; 6—by ( 2 3 ) .


F I G . 3. Pressure distribution along the radius of the baffle at various flow rates of the

-dimensionless

-dimensionless flow rate. pressure ;

fluic


3. Formulae (21), (22) and (23), represented by curves 4, 5 and 6 in Fig. 2, prove that the greater the number of terms of the series that are used, the better is the agreement with the experimental results. For a converging flow, these formulae give a closer fit with the experimental points than McGinn's formula (27).

Formulae (21), (22) and (23), which are obtained from the solution of a hydrodynamic problem, and are in satisfactory agreement with ex-perimental evidence, can be used for the calculation of pressure distribution (in Fig. 3 are shown curves for Β = B(oc, A), calculated by (23) for the case when β = 0) as well as for determining the flow and static force charac-teristics.

T H E D I S C H A R G E C H A R A C T E R I S T I C S

If the fluid flowing through the nozzle could be considered inviscid, it would be possible to express the pressure Pc at the radius of the nozzle in terms of the pressure P0 before the throttle and the flow rate Q. In this case we have

Pc = Po - γ ΐ £ ,

where

vr dz =

Accordingly, we obtain :

Pc = Po

or, taking into account (24):

2nhrc

qQ2

Zn2h

2r

2

where

Bc = B0 - 12-5Λ2*

2,

Λ) - Pi h

2ß

2; occ

(28)

(29)

(30)

(31)

(32)

Applying now formulae (21), (22) and (23) to the point r = rC9 that is taking oc = occ and Β = Bc, and eliminating Bc by applying formula (31), we obtain the following discharge characteristics of a "nozzle-baffle" element:

T o a first approximation :

B0 = 6A \nac + 12-5Λ2.


B0 = 6A \nac + (16-78 - 4·28*2) A

2.

(33)

(34)

9 a Aizerman I


R E F E R E N C E S

1. Ν . A .SLEZKIN, Maternât, sbornik, Vo l . 42, N o . 1, 1935.

2. Τ . M.BASHTA, Aircraft Hydraulic Drives and Installations (Samoletnye Gidravlicheskiye

Privody i Agregaty) . Oborongiz, 1951.

3. I . H . M C G I N N , Observation on the radial flow of water between fixed parallel plates.

AppL Sei. Res., A 5, N o . 4, 1955.

2\2

(39)

(40)


B0 = \nac + (16-78 - 4*2U2

C)A2 - \4ΊΑ

2β

2{<χ* - 1) - 3·67Λ

3(α* - 1).

(35)

T H E F O R C E C H A R A C T E R I S T I C S

Formulae (21), (22) and (23) establish the pressure distribution for rc < r.

Also, the pressure at the centre of the baffle where r = 0 is known, that is Ρ = P0. Let us assume that between the points r = 0 and r = rc the pressure change is linear, that is:

P = Po- F

° ~ Pc

r. (36)

N o w we have a definite pattern for the pressure distribution over the whole surface of the baffle.

After integration, we obtain the following force characteristics of the nozzle-baffle element:

T o a first approximation:

D = 3A(oc2

c - 1) + \2-5A2(x

2

c{\ - 2 l n a c ) . (37)


D = 3A(oc2 - 1) + 16·78Λ

2α

2(0·248 - 2 1 n * c) . (38)


D = 3A(a2

c - 1) + 16·78Λ2<χ

2(0·248 - 2\nac) - \4-2Α

2β

2{\

- 3·675Λ3(1 - α

2)

2,

where

D= _ ^ _ ( 1 0 ^2

)2

. πρν

THE RESULTS OF EXPERIMENTAL A N D THEORETICAL INVESTIGATIONS

OF CONTROL DEVICES OF THE NOZZLE-BAFFLE TYPE

N . P . S H U M S K I I


A flat valve of the "nozzle-baffle" type (Fig. 1) is often used in automation.

This is justified by the fact that it is free from the shortcomings inherent in

sliding valves. The "nozzle-baffle" valve can be applied successfully to a

control system which contains no sliding surfaces in its construction. The properties of nozzle-baffle elements, until recently, were investigated almost exclusively by experimental methods, and cannot be considered sufficiently well known. The experimental results published [1] are limited in the range of pressure drop (ΔΡ < 1-4 kg/cm

2) as well as in the type of working fluid

(oil and air),f and characteristics quoted in this work require improvements in accuracy. With this in view, the author of the present work has carried out further experimental and theoretical investigations of nozzle-baffle elements. The results of these investigations are briefly outlined below.

t It is, however, important to have information as to the performance of "nozzle-

baffle" valves with kerosene.

F IG. 1. D i a g r a m of a valve of the "nozzle-baffle" type.

257


The following notation will be used here:

r0—radius of nozzle;

rj—outer radius of the nozzle end surface;

r—radius variable;

h—distance between the nozzle and baffle;

/—length of the nozzle;

U = 2h/r0—relative lift of the baffle;

Κ = rjr0—relative radius;

Kx = rl/r0—relative outer radius;

ξ = //2r0—relative length of the nozzle;

P0—pressure before the nozzle;

Ρ ι—pressure at the exit from nozzle (at the cross-section nrl)

P2—pressure at the entry into the cylindrical gap (with the sectional area 2nr0h);

Ρ Β—external pressure;

Δ Ρ = P0 — P g—pressure drop in the valve;

Q—flow rate through the valve;

R—force exerted by the jet impinging on the baffle;

Ho—velocity in the nozzle;

vv r 0—velocity at entry to the cylindrical gap (with sectional area 2nr0h)\

\vr—velocity variable;

MVi—velocity at exit from the valve (with the sectional area

2nrxh)

μ—discharge coefficient, based on the full pressure drop (P0 — PB), and the section 2nr1h at exit from the valve;

//,—discharge coefficient, based on the pressure drop (P0 — Pi) and sectional areajrr

2. at exit from the nozzle;

μ2—discharge coefficient of the transitional part of the flow from the nozzle exit to the entry into the cylindrical gap— corresponding to the pressure drop (P^ — P2) and sectio-nal area 2nr0h;

E X P E R I M E N T A L A N D T H E O R E T I C A L I N V E S T I G A T I O N S 259

2h w0 Re !

r0 W r 0 Re

Investigations and calculations on automatic control systems, incorporat-ing hydraulic control devices of the "nozzle-baffle" type, are feasible if we know their discharge characteristics and force characteristics:

Q = Q(AP,h), R = R{AP,h).

These relationships are the object of the experimental and theoretical in-vestigations in the present work. The flow rate Q and force R can be ex-pressed in the following forms:

ρ = μ2πΓιη l [2 { Po

" P b )

) , (1)

R = mlgwl + nrliP^ - PB) + 2π\ (P - PB)rar. ( 2 ) π ! \ ρ - PB) J r0

This indicates that it is necessary to know the coefficient of discharge μ, and the pressures P(r) and P{, in order to calculate characteristics of Q and R.

The pressure distribution Ρ = P(r) can be determined by solving the problem of the radial flow in the clearance to the baffle.

T H E F L O W OF A V I S C O U S I N C O M P R E S S I B L E F L U I D

I N A R A D I A L G A P

T o determine the function Ρ = let us consider the flow of a viscous incompressible fluid in the clearance formed by the baffle and the end of the nozzle. We shall use a system of cylindrical coordinates (see Fig. 1), and as-sume that the flow is axially symmetrical, the velocity component in the direc-tion of the oz axis is zero (w = 0), and that the fluid entering the diffuse has a uniform velocity wr0 at all points where r = r0. In this case, the Navier-

μ3—discharge coefficient of the diffuser, corresponding to a

pressure drop (P2 — PB), and sectional area 2πΑ*1Α, at exit

from the valve;

ρ—fluid density;

ν—kinematic viscosity;

Re ! = r0w0lv—Reynolds number, based on the flow in the nozzle;

Re = rwrjv—Reynolds number, based on the flow in the radial gap.

Obviously,

U


and hence v d

2v

ρ or ro r dz

2

where v{z) = rwr.

If we now put d2v!dz

2 = Cl9 we obtain, after integrating,

Ρ = _ J ^ I _ + C1vq\nr + C 2 ( z ) 2

1 dP d2wr

— + ν — - + ρ dr V dr2

dwr

dr r

1 ÔP υ2

(5)

r

It can be easily proved that the resulting set of functions of Ρ and Wr

satisfies the conditions of equations (3) and the condition dPjdz = 0 is

satisfied by the mean values of the functions wr and C2, with reference to the

thickness h of the radial clearance, as follows:

l ΓΛ /2

- 1 p/ 2

wr = — H ' r d z , C2 = — C 2 ( z ) d z .

h J - A / 2 ^ J - Λ / 2

(6)

The quantities C 3 , C 4 , Cx and C2 are determined in the following way:

(a) C 3 and C 4—from the boundary conditions due to the adhesion of fluid

to the solid boundaries:

νιγ(/ \ζ) = 0, ζ = ± — , 2

from which it follows that

and, consequently

(b) Ci—from the flow equation:

-2π Λ/ι/2

ΠΛ/2

ο

αφ dz, —Cl = nh

3

Stokes equation, and the equation of continuity, which determine the flow,

have the form

wr —- = + v[ + + — • dr ο dr \ dr

2 dz

2 r dr

(3)


and, consequently

or

- v = 4 ^ - ^ 1 1 - 1 ^ ) I; ( 8 ) 2 2m0n

(c) C2(z)—from the boundary conditions for the relations governing the distribution of pressure and velocity in the inlet (where r = r0) and outlet (where r = rx) sections of the diffuser. The magnitude of C 2 , generally speaking, differs for different streamlines, but we shall take its mean value C 2 . Considering the boundary conditions at the entry to the diffuser, we obtain:

n2h

2 Vo r

2 ) nh

3 r0

or

, = J p 2 + i L ( _ e _ ) 2 A - J _ - * ! ^ y (,o) 2 \2nr0h J \ K

2 ReU

2 J

If, however, we determine the magnitude of C 2 from the boundary conditions at the exit of the diffuser, then:

0 - 1 2 5 i ^ - f -L - ± U ^ £ m υ- , ( i d n

2h

2 \r

2 r\) nh

3 r

or

96 In 1 _ Κ

K\ U2Re

(12)

From equations (7) and (9) we may make the following conclusions:

1. Neglecting viscosity (with ν = 0), the solution obtained is that for po-tential flow.

2. The diverging flow in the diffuser described generally occurs against a pressure gradient, i.e. in the same way as in conventional diffusers with divergent sections. The distribution of velocity across the flow follows the "square law" parabola, i.e. the same as for a two-dimensional flow.

3. The difference between solution (10) for the pressure distribution and that quoted in Ref. 2 is in the presence of a term corresponding to potential flow, and therefore (10) is more general.

P N E U M A T I C A N D H Y D R A U L I C C O N T R O L

T H E F L O W C H A R A C T E R I S T I C S

T o determine the flow characteristic (1), it is necessary to know the relation between the discharge coefficient /i and the parameters of the valve and fluid. T o obtain this relationship, let us enumerate the important criteria. The flow of fluid in the valve is determined by the following six quantities:

ß ( — Y r ° ( m ) ' r i ( m ) > h(m>>> ν ( — ) · \ sec / \ sec /

According to the π theorem of dimensional analysis (model scaling laws) only four independent non-dimensional groups can be set up:

(a) Reynolds number Re = Qjlnrv

(b) relative baffle-gap U = 2h/r0

(c) relative outer radius of the nozzle Kx = rxjr0

(d) relative length of the nozzle ξ = l/2r0

Other non-dimensional parameters of the valve, including the discharge coefficient μ, must be definite functions of these four groups which determine the system, that is μ = μ-iRe, U, Κγ, ξ). The problem is to determine this relationship quantitatively.

It must be remembered that the motion of the fluid produces irreversible losses in the following elements of the valve: in the nozzle, at the change in direction of the flow, and under the baffle.

T o these losses correspond the discharge coefficients:

which shall be called "proper" coefficients of discharge, as they correspond to true pressure drops, established at a given section in accordance with the law of conservation of energy; they are referred to the exit sections of the elements.

The overall coefficient of discharge μ for the whole device according to (1) can be expressed as:

262

(13)

(14)


From (13) and (14) we obtain

,2 _ μ κ\υ2

K2(\ - υ2) ι - κ\

+ — - — - — · +

(15)

2 μ2

μϊ

Obviously, the form of the resultant equation for // remains valid for any finite number of resistances, connected in series.

λ 0-16

0-12

0Ί0

OOS

006

om

0-OZ

0 lei lei

2 J ¥ S

F IG. 2. The relation between resistance coefficient λ of a nozzle and the Reynolds

number, from the Karman-Prandt l formula.

From the relationship (15) it follows that, in order to find μ, it is necessary to establish the relationship of the proper coefficients μΛ, μ2 and μ3 from the criteria which determine the system.

The Discharge Coefficient μΛ . Evaluating the losses by the Weissbach equa-tion, we have, for the nozzle:

Λ) - Λ = 1 + — λ

2 l Λ Q

wo

2/<ο / 2

from which we obtain, taking into consideration (13),

μ\ = —[— · (16)

1 + ξλ

The coefficient λ can be determined from the Karman-Prandtl equation:

-L = 2 1 g ( R e l x A ) - 0 - 8 . (17)

In Fig. 2 the relationship (17) is shown as the graph λ = A(log R e x ) , and is very convenient for practical calculations.


The values of μ ΐ 5 calculated from equation (16), hardly differ from ex-perimental results, obtained with a nozzle having r0 = 0-75 mm at 5 < Δ Ρ < 80 kg/cm

2, where μΛ = 0-8 + 0-85.

The Discharge Coefficient μ2 is connected with the losses due to the change in direction of the flow and to entry into the radial gap. Figure 3 shows the experimental relation between this coefficient and the Reynolds number, and the relative baffle lift U.

/2 /6 20 2¥ ,28 32 36 ¥0 ¥9 ¥8 02 06 60 Re !0~

3

F IG. 3. The relation between the discharge coefficient μ2 and the Reynolds num-

ber and the relative baffle lift U. The relative radius Kx is here 1-4.

Ui = 0 0 1 6 0

U 2 =0-0213

U3 = 00267

U4 = 00533

U5 = 00800

U6 = 0 1 0 6 7

Un =0-1333

U8 =0-1867

U9 =0-2133

U l 0 =0-2667

UX1 =0 -4000

U l 2 =0-5333

Ul3 = 0-8000

For values of Re above 45 χ 103, the discharge coefficient μ2 ceases to

depend on Re ; the relationship μ2 = μ2ϋ within this region is shown in Fig. 4.

The Discharge Coefficient μ z . From equation (10), which gives the pressure distribution for radially diverging flow, we have

Ρ - Ρ - ρ( Q Χ (96]nK> + 2- - Λ 2 Β

2 \2nr0h) \ U2Rc Κ\ ) '

E X P E R I M E N T A L A N D T H E O R E T I C A L I N V E S T I G A T I O N S

96K2

U2Rc

In Kx - K\ + 1

265

(18)

The relationships (16), (18), and the experimental characteristics for μ2

(Figs. 3 and 4), enable us to find the required coefficient μ. For this purpose, equation (15) must be used. It may also be written in the following form:

μ = μι

κ\υ2 + κ](\ - υ2) ( — J + 0 - * î ) ( — J

(19)

In Figs. 5 and 6 are shown the relationships μ = / i (Re , U) and μ = μ(ϋ)9

obtained experimentally with a valve having the relative radius Kt = 1*4, and referred to the section 2nr0h. From Fig. 5 we can see that the discharge

1-0

08

06

Ο*

ν \ \

0-2 0¥ 06 0-8 10 U

F I G . 4. Relation between the discharge coefficient μ2 and the relative baffle lift U. Here

R e > 45 χ 103.

coefficient becomes independent of Re, when the latter exceeds 45 χ 103.

The region for Re > 45 χ 103 is given in Fig. 6.

When the flow parameters are such that the inertial terms of the Navier-Stokes equations can be neglected (this is so for very small Reynolds num-bers), equation (4) will have the form:

Hence

]_ dP _ d2v

ρ dr r dz2

6ovQ\nKi , Δ Ρ = — - - (see [2] ) .

π/*3

from which, taking into account (13), we obtain:


16 20 2¥ 28 32 36 10 ¥¥ ¥8 52 J6 60

Re-W3

F IG. 5. The relation between the discharge coefficient// of aflat valve and the Reynolds

number, and the relative baffle lift U. Here KY = 1-4.

Ux = 0 0 1 6 0

U2 = 00213

U3 = 0-0267

t / 4 = 0-0533

U5 = 00800

U6 = 0 1 0 6 7

V η = 0-1333

U« =0 -1867

U9 =0 -2133

Uio = 0-2667

Un = 0 - 4 0 0 0

U12 = 0-5333

U l 3 = 0-8000

C / 14 = 1-0667

Therefore

or

where

ο . "V 1 Ι/2ΔΡ\ 1(2ΔΡ\

12 v\nK, VV Q / V V Q )

Ο = uinr.h /

v

K\ In ^ 9 6 ν ν

With the limitations given above, assuming μι = 1 (which is quite permis-sible, as in this case the resistance of the nozzle is negligible), the discharge


μ Ki In Kt 96

or, referring the pressure drop to the area,

U2 Re

μ = Κ, In Kx 48

This result is also in agreement with the known theoretical solutions for the

flow of fluid through orifices and nozzles, for which // = Re/48 at Re < 25.

12

10

0-8

06

04

02

ο

IK X N X I A

χ 1 X ^ • v ^ x

- • — -• —

0 02 04 06 08 10U F IG. 6. The relation between the discharge coefficient μ of a flat valve and the relative

baffle lift U. Here R e > 45 χ 103.

T H E F O R C E C H A R A C T E R I S T I C S

T o determine the force characteristics (2), it is necessary to know the magnitude of pressure Px at the exit from the nozzle and the distribution of pressure under the baffle Ρ = P(r), which is established according to equa-tion (11).

From (13) we have

oQ2 1

2(nr2

0)2 ~μϊχ"

Pi = Po (20)

where μ ! is determined by the relationship (16). Substituting into (2) the value of Ρλ from (20), and that of Ρ from (11), we obtain:

2μ\) nr

2(P0 - P s) + Rlt

(21)

coefficient of a flat valve can be expressed theoretically as :

U2 Re


F I G . 7. The flow rate Q = Q(AP, h) and force R = R(AP, h) characteristics of a flat

valve, having r0 = 0-75 mm, Κχ = 1*4. W o r k i n g fluid—kerosene γ = 0-81 g /cm3,

/ = 10-20°C.


F I G . 8. F l o w rate Q = Q(h, AP) and force R = R(h, AP)characteristics, plotted from

the same data as Fig. 7.


F I G . 9. F low rate Q = Q(h,AP) and force R = R(h,AP) characteristics, plotted

from the same data as Fig. 7.


FORCE R ON THE BAFFLE DUE TO FLOW ACTION

For h = 001 mm For h = 0 0 2 mm For h = 004 mm For // 0-10 mm

AP Calcu- Experi- Calcu- Experi- Calcu- Experi- Calcu- Experi-

lated ment lated ment lated ment lated ment

15 0-365 0-280 0-264 0-250 0-230 0-210 0-200 0-150

20 0-443 0-400 0-349 0-360 0-303 0-310 0-280 0-240

30 0-675 0-630 0-500 0-585 0-452 0-520 0-456 0-430

40 0-928 0-880 0-705 0-820 0-635 0-740 0-633 0-620

50 1-140 1-130 0-955 1-060 0-771 0-965 0-800 0-790

T o conclude, Figs. 7, 8 and 9 show experimental curves of flow rate Q = Q(AP, A), and force R = R(AP, h).

R E F E R E N C E S

1. V . I . DMITRIYEV and A . G . SHASHKOV, Avtomatika i Telemekhattika, Vo l . X V I I , N o . 6,

1956.

2. T . M . B A S H T A , Aircraft Hydraulic Systems (Samoletnye Gidravlicheskiye Ustroystva).

Oborongiz, 1946.

After integration, we have:

J,, . -0-25 i f (l„ i - ajjL) + U - rl - Κ .„ IL) . π/*

2 V r 0 2r2 / A

3 \ Ό /

(22)

The values of the force JR, calculated by (21) and (22), in the range of the

working pressure of a controller, agree satisfactorily with the experimental

data quoted in the table below.

FORCE OF THE JET ACTION ON THE BAFFLE IN PNEUMATIC

A N D HYDRAULIC CONTROL UNITS

V . N . D M I T R I Y E V and A . G . S H A S H K O V


Despite the prevalence of baffle-nozzle elements in pneumatic and hydrau-lic control engineering, little is known about the forces acting on the baffle; yet it is far from being unimportant.

As a rule, a baffle is controlled by a sensing element of very small power.

The force acting on the baffle may be commensurable with the force developed

by the sensing element.

The force of the fluidf jet action on the flapper, being a non-linear function

of the gap between flapper and nozzle, may noticeably influence the form of

the static characteristic of a controller. The magnitude of this force often

limits the choice of sensing elements.

Previously, this force has been disregarded when making a performance

analysis of controllers. Yet experience proves that sometimes this leads to

serious errors.

The present work gives the results of theoretical and experimental inves-

tigations of these forces.

In the literature there are only isolated examples of quantitative studies on the forces exerted by a jet of fluid [1]. The problem is discussed in some text-books on mechanics when explaining the momentum theorem (see, for example, Ref. 2, p. 78) but without considering the specific features of nozzle-flapper elements.

1. T H E O R E T I C A L A N A L Y S I S O F F O R C E S

I N A B A F F L E - N O Z Z L E P A I R

When the gap between nozzle and baffle is such that the latter causes a throttling effect (Fig. 1), the force acting on the baffle is a sum:

Fx = FXl + FX2. (1)

t Here and in the following text the term "fluid" includes both liquids and gases.

272

F O R C E O F T H E JET A C T I O N O N T H E B A F F L E 273

Here Fxl is the force caused by the change of momentum of fluid. It would

exist also if the nozzle and baffle were made as one solid piece. This reaction

force, for the case when the stream of fluid alters its direction by 90°, can be

determined from the momentum theorem; it is equal to the mass flow of

fluid, multiplied by its velocity in the nozzle:

G_

g F = — V (2)

Here G is the weight flow (discharge per second, by weight); g the gravity

acceleration ; Vc = G\ycnr^ ; yc the density of the fluid in the nozzle.

Hence: ^ 2

(3) ™cgyc

The other force, Fx2, is the result of static pressure acting on the baffle. As

the baffle actually is not fastened to the nozzle, the static pressure acting on

F IG. 1.

the flat surface of the nozzle is taken up by the parts which hold the nozzle, and all the static pressure on the baffle is transmitted to the sensing instru-ment.

This latter force can be determined as :

FX2 = 2π J pr(x;p; R; r; rc;p0)râr - nR2p0,

where ρ is the feed pressure; p2 the pressure at the distance r from the nozzle axis (see Fig. 1), p0 the pressure at the surrounding medium. The above inte-

274 P N E U M A T I C A N D H Y D R A U LIC C O N T R O L

+ 2n Pr2(x;p; R; r; rc; p0) r dr - nR2p0.

If the nozzle is sharp-edged (R = rc) the second integral becomes 0. When

the gap χ is small, it may be assumed that within the circle of radius rc the

pressure prl = const = p. Then

G2

Fx = —; + πΓ

*(Ρ ~ P o)

nrcgyc

+ | ^ 2 π | p,2(x; p; R; r; rc; p0) r dr - np0(R2 - r c

2)J . (4)

At .v = 0, the flow G = 0, and the expression in square brackets becomes 0, then

Fx = nr2

c(p -po).

As can be seen from (4), the problem is essentially that of finding the func-tion pr2. This function can be found from the Bernoulli equation.

T o obtain this equation for cylindrical sections rc and R, we find the re-lationship between velocities in these sections from the equation of conti-nuity :

Here VR is the velocity at section R\ Vx the velocity at section r c; yx and yR

the density of fluid in sections rc and R.\ The Bernoulli equation for air, on the assumption of an adiabatic process, is:

k ρ + n = k pR { r\y\ V2

X | ζ r2

cy2

x V2

X ^

k - 1 γχ 2g k - \ γΛ R2y

2

R 2g R2y

2

R 2g

Here k = 1-4 is the ratio of specific heats; ζ the resistance coefficient. If we denote the pressure and density in a section r by pr and y r , we shall have

Here Ci is the flow resistance coefficient. When only the gap χ is variable, tqe

f In the case of incompressible liquid, yR = γ χ.

gral can be represented as a sum of two integrals:

FX2 = 2nj pri(x;p; R;r;rc;p0) rdr


quantities yr9 Vx, ζ can take such values that the pressure prl becomes

smaller than p0. Therefore Fx can be negative when

\nR2p0\ > 2n p,2(x; / ? , R, r,rc,pG)ràr + πΐ'

2ρ +

J r c ™cgyc

2. E X P E R I M E N T A L I N V E S T I G A T I O N S

(a) Description of Test Equipment and Methods

A test rig, the scheme of which is shown on Fig. 2, has been developed for measuring forces Fx. It consists of a balanced lever 7, with a pivot pin 2, which rests on ruby supports. One end of the lever carries the baffle, and the

F IG . 2.

other a small movable weight 4, used to balance the lever initially. The main movable weight 7\ balances the force FX9 produced by a jet of fluid flowing from the nozzle 5. Upstream of the nozzle there is a fixed restriction 6. The supply pressure pl and the pressure behind the nozzle ρ are measured by pressure gauges. The gap χ is measured by micrometer 7. Its additional large dial carries 0-001 mm divisions.


The following procedure is used when measuring the force Fx. After en-suring that the lever is horizontal (by adjusting the height of nozzle), the accuracy of fitting the baffle to the nozzle is checked. This is necessary to ensure the squareness of the baffle to the nozzle axis at χ = 0, and to avoid errors at small gaps. The accuracy was checked by placing a drop of petrol between the nozzle and the baffle when closed and applying air pressure. Absence of bubbles indicated a good fit.

The force Fx in grams is determined from the equation of moments

The gap χ was altered by moving the weight Τλ, and determined by two methods:

1. By hydraulic method of measurement [3], which consists in taking (in a separate series of tests) the calibration curves ρ = p(x) at pL = const, and ρ = ρ(ρι) at χ = const. Subsequently a gap χ can be determined by measuring the pressure p.

2. By electro-contact method. For this purpose, a battery 9 and voltmeter # are connected to the lever 7, which is insulated from the body of the test rig by its ruby supports. This enables the micrometer 7 to be set with a high accuracy, as contact between the nozzle and the baffle is regis-tered by the voltmeter.

In this case, the gap χ could be measured only by the hydraulic method, because oil affected the electric contact. The characteristics ρ = p(x), at px = const, were taken with ργ = 1, 1-5 and 2 atm.

(b) Results of Tests with Transformer Oil

p,atm

0-006 0-012 0-018 xtcm

FXlgW 20 0

20

F I G . 3.

Fx 19

: D = 0-294cm; : D = 0-160 cm.

(Fig. 2) :


The gap χ was changed by increments of 0-002 cm of the ranges 0-002-

0Ό04 to 0016-002 cm. The dimensions of the nozzles tested are given in

Table 1.

The curves of force Fx as a function of gap χ and pressurep, at pv = 1-5 atm,

dr = 0T45 cm and dA = 0Ό8 cm, and at various diameters D of the nozzle

TABLE 1

dc, cm D , cm D\dc

0-145 0-160 1100

0-145 0-232 1-600

0-145 0-294 2-030

0-181 0-218 1-200

0-181 0-332 1-840

face, are shown on Fig. 3. On the same diagram are shown the static characteristics ρ = p(x), which were used as calibration curves for deter-mining the gap χ by the hydraulic method.

fog

I ι ι ι ι ι ι ι 0 0-2 04 0-6 08 10 12 H p,atm.

F IG . 4. I—χ = 0003 cm; 2—χ = 0-004cm; 3—χ = 0-005 cm; 4—χ 0-006cm;

5—x = 0 0 0 9 c m ; 6—χ = 0-006cm; 7—χ = 0-010cm; 8—χ = 0 0 1 4 c m .

: air (d 0-223 cm, D = 0-336 c m ) ; : oil (d = 0-145 cm,

D = 0-160 cm).

From Fig. 3 it is seen that Fx = Fx(x), and Fx = Fx(p) are non-linear functions. As χ is reduced the pressure ρ and the force Fx increase. The dia-gram also shows that with increase of D the force Fx increases.

Figure 4 shows the relationship between Fx and the pressure ρ at χ = const. It shows that Fx = Fx(p) is practically linear, if χ < 0-01 cm.


Figures 5 and 6 show the curves Fx = Fx(x) at ρ = const, andFx = FX(D).

From Fig. 5 (curves 7 and 2) it is seen that the force Fx increased sharply when the gap χ is reduced to a small value.

μ , — xHj — 1 F I G . 5. 1—D = 0-294 c m ; 2 — D = 0-160 cm; 5—D = 0-445 cm;4—D = 0-210cm;

5—D = 0-140cm. : air (p = 0-5atm, de = 0-1 c m ) ; : oil (pe = 1-Oatm,

de = 0-45 cm); : conjectured extension of (air) .

I I ! I I I I 01 02 0-3 04 D,cm

F I G . 6 . : oil (d = 0-145 cm, * = 0-006cm,/? = 0-5 atm);

air (d = 0· 100 cm, χ = 0-004 c m , / ? . = 0-5 atm).

Figure 6 proves that increase of D substantially increases Fx.

During tests with nozzles of the dimensions given in Table 1, the negative force Fx did not appear.

(c) Results of Experiments with Air

The gap was varied from 0-002 to 0-014 cm. The dimensions of the nozzles are given in Table 2.

TABLE 2

dc, cm Z), cm D\dc

0100 0-140 1-400

0-100 0-210 2-100

0100 0-445 4-450

0-224 0-336 1-500


40

30

20

70

0

-10

F IG. 7. : ρ = 0-5 atm; : ρ = 1-0 atm; : ρ = 1-5 atm.

Curves of Fx = Fx(p) are shown in Fig. 4 for a number of values of x.

When these are small (x = 0-003 and 0-004 cm), the curves are nearly linear. Figure 5 shows the curves of force Fx as a function of gap x, at a constant pressure ρ and several values of D.

For the nozzle with D = 0-445 cm, a broken line shows the conjectured extension of the experimental curve. With the increase of χ (from the point b

on) the force decreases, and at χ = 0-007 cm (point c) becomes 0. In the interval 0-0464 > χ > 0-007 cm the force Fx is negative and at the point d it reaches its minimum. At χ > 0-0464cm, Fx increases, and then decreases to 0 at χ = oo. A t χ > 0-0265 cm = xk0 no tests were carried out because the smallest increase of χ (up to xkl = 0-053 cm) upset the balance of the lever 1.

10 Aizerman I

Curves of Fx as a function of x, for pressures ργ = 0-5, Ι Ό and 1-5 atm,

and for dc = 0-100 cm, D = 0-445 cm, and fixed restrictor diameter dt

= 0Ό32 cm, are shown on Fig. 7. On the same diagram are also shown the

static characteristics of the nozzle-baffle element calibration curves ρ = p(x).

Figure 7 shows that a steep slope of the static characteristic ρ = p(x) is as-

sociated with a steep slope of the Fx = Fx(x) curve. It can also be seen that at

certain values of χ the force Fx becomes negative.

?x,9

50


Only the point e has been determined experimentally. It should be noted that there are two points of equilibrium—stable at xy and unstable at xh.

The balanced lever with the baffle, if put into any position χ < xH, settles in the position xy.

-10L

F IG. 8. : ρ = 0-25 atm; : ρ = 0-50 atm; : ρ = 0-75 atm.

From Fig. 5 it is seen that the negative force was observed only with nozzles having D = 0-210 and 0-445 cm. The negative force is practically constant over the major part of its range.

Figure 8 shows how the force Fx depends on the gap χ at various pressures ρ for nozzles with D = 0-445 and d = 0· 100 cm. The force—whether positive or negative—increases with increasing pressure p. Figure 6 shows the rela-tionship between Fx and the outer diameter Z>, at ρ = 0-5 atm, χ = 0-004 cm,

Xy,cm x^/Cm

0-007 -0-05

0006 -0W

I ι I I I ι ι 1

02 Ot 0-6 08 10 1-2 H p,atm.

F I G . 9.


d = 0*100 cm. The values of the gap xy9 at which the baffle is in stable equi-

librium, and xH, when the equilibrium is unstable, are shown on Fig. 9, as a

function of pressure p. It follows from Fig. 9 that both xy and xH diminish

with increasing pressure p, tending towards a constant value.

Pr. gauge No.3

Pr. gauge No. 2

Pr. gauge No.1

F I G . 10. The end surface of the nozzle with holes for measuring pressure distribution

(dimensions in mm).

The maximum relative error in the measurement of force can be calculated by the formula:

y ÔF = àT1 + ôl,

where δΤγ and ôl are the maximum relative errors in determining the weight

and the length of the moment arm. The maximum F occurs when both 7\ and

/ have their smallest values, that is Τλ = 3g, I = 20 mm. As Τ and / were


F IG. 11.

3. C O M P A R I S O N B E T W E E N T H E O R E T I C A L

A N D E X P E R I M E N T A L D A T A

If the gap χ = 0, the theoretical force acting on the baffle is:

Fx = nr2

cp.

It has been found that this formula gives a good agreement with experi-

measured with absolute errors ΔΤγ = ±0-1 g, and ΔΙ = ±0-5 mm—the

max. relative error oFmax = 5-8 per cent.

(d) Experimental Determination of the Pressure Distribution

Over the Nozzle Face (establishing the function p^l for the nozzle

working with air).

For this purpose a special nozzle was made, with pressure tappings, as shown in Fig. 10.

The experimental curves pr2 = pri{r) for various gaps x, and pressure ρ

= 0-5 atm, are shown in Fig. 11. Broken lines represent the conjectured ex-trapolation for rc < r < 0-98 mm.


mental results only when the annulus at the face of the nozzle is small, i.e.

Djdc « 1. When this is not the case, the experimental value of Fx is larger

than nr2p9 and the difference is greater at the higher ratios D\dc. This can be

explained by recognizing that the fluid which flows between the nozzle and

the baffle is under a certain pressure. In view of this, it follows that:

pnr2 < Fx < pnR

2.

T o find the force Fx, it is necessary to determine the value of the integral:

This can be done, using the curves (Fig. 11), converted into curves pr2r = f(r),

shown on Fig. 12. The area under the curve represents the above integral. Measuring the area by a planimeter, and multiplying by 2π, one obtains the force on the area of the nozzle tip. The force on the area nr

2

c is nr2

cp.

F IG. 12.

The force due to change of flow direction has been calculated by formula (3) . The results of calculations for χ = 0-003, 0-004, 0-005 and 0-006 cm are shown on Fig. 13, together with experimental results obtained for the nozzle shown on Fig. 10 at 0-5 atm. Figure 13 demonstrates a good agreement be-tween calculated and experimental results.


F I G . 13. Experimental points— χ . de = 0-0953 cm; D = 0-5 cm; ρ = 0-5 atm.


1. The article provides a set of experimental data pertaining to fluid jet action on the baffle.

2. A formula is deduced, from which the force Fx may be calculated as a function of pressure, gap, and nozzle diameters dc and D.

3. It is established that at certain values of dc and D the force can be either positive (pushing the baffle away from the nozzle), or negative, acting in the opposite direction. There are stable and unstable equilibrium positions. With increase of outer diameter D, and of pressure p, the absolute value of a positive or negative force increases.

Therefore it is advisable to reduce the diameter D and the feed pressure ρ

in the cases when it is necessary to have the least possible force on the baffle.

R E F E R E N C E S

1. E . S A M A L , Pneumatische Meßwertumformer für Regelzwecke. Regelungstechnik, No. 3,

1954.

2. L .G.LOITSIANSKI and A . I . L U R Y E , Technical Mechanics, p. 2 (Tekhnicheskaya Mekha-

nika). Tekhteoretizdat, 1948.

3. V .A.TRAPEZNIKOV, I.E.GORODETSKI, B . N . P E T R O V and A.A.FELDBAUM, Automatic

Control of Dimensions (Avtomaticheski kontrol razmerov). Oborongiz , 1947.

THEORY OF CONTROL DEVICES OF THE "NOZZLE-BAFFLE" TYPE,

W O R K I N G WITH OIL

A . G . S H A S H K O V

T H E NOZZLE-BAFFLE control unit (Fig. 1) is widely used in various units of

hydraulic control, but its theory has not been fully developed. The con-

ventional approach usually assumes a certain law of discharge through the

fixed restriction and through the nozzle, which remains constant for the

whole working range [1, 2]. In fact, however, a linear law is applied for small

gaps between nozzle and baffle, and a square law for larger gaps.

Certain empirical relationships between discharge coefficients and Reynolds number have been obtained (see Appendix) which enable us to investigate analytically the static and dynamic characteristics of the nozzle-baffle con-trol elements, without recourse to the assumptions stated above.

If the time constant for pressure changes in the intermediate chamber is small compared with the period of the input signals (when the baffle is moving slowly), the control device may be considered instantaneous for changes in the input. In this case, the control device can be regarded as an ideal transducer, that is, one that can be fully characterized by its static properties only. But if the time constant for pressure changes in the inter-mediate chamber is of the same order as the period of the input signal, then these time constants must be taken into account.

The present work deals with both the static and dynamic properties of the nozzle-baffle control unit.

1 2 0 -x

1 M ^m^M

F IG. 1. 1—fixed restrictor; 2—nozzle; 3—baffle.

285


t This covers pressure differences from 0*2 to 2 kg /cm2, with the usual dimensions.

S T A T I C C H A R A C T E R I S T I C S

O F T H E N O Z Z L E - B A F F L E E L E M E N T

The relation between the pressure Px in the intermediate chamber and the distance χ between the nozzle and the baffle, at a constant supply pressure P{,

is called the static characteristic of the control element. The main problems in the static performance of the control device are: (1) T o deduce the equations for the static characterictics :

Px = Px(x), Px = const.

(2) T o analyse the effects of the parameters of the control device, which influence its static characteristic. (Main parameters are the diameter of the fixed throttle du nozzle diameter dc, and supply pressure

(3) T o choose these parameters so as to obtain the required static char-acteristic.

Let us consider the control unit (relay) to consist of a series connexion of a fixed resistance (restrictor), an intermediate chamber, and the nozzle-baffle element (variable resistance) (Fig. 1). In order to predict the static char-acteristics of such a device, it is necessary to know the fluid flow rate through the fixed and variable resistances, which depends on their discharge co-efficients.

Experimental investigations have shown that for a range of Reynolds numbers from 100 to 500 f the discharge coefficient of the fixed resistance hardly varies (see Appendix). When calculating the flow rate through the fixed throttle, the discharge coefficient may be taken as approximately con-stant, within the range 0-6-0-7.

Generally, the discharge coefficient is determined by the geometry of the boundaries, and the Reynolds number. The investigations, however, show that the discharge coefficient of a nozzle-baffle element is completely deter-mined by the Reynolds number alone:

1-095 m

"* =

/ Λ 305-5 X ' ()

where

Re = - ^ . ( l a ) ndcv

Qx is the flow rate of oil through the nozzle, ν the kinematic viscosity of oil.

T H E O R Y O F C O N T R O L D E V I C E S O F " N O Z Z L E - B A F F L E " T Y P E 287

1. Analysis of the Static Characteristics of the Control Unit, Taking into Account the Variation in Discharge Coefficient

When the oil flows from the nozzle into a constant pressure region down-stream (e.g. atmosphere), the pressure drop across the nozzle-baffle element is the pressure Px in the intermediate chamber. Therefore, the flow rate through the nozzle is given by the equation :

Qx = μχ/χ

2g fx<

ndl (2)

where fx = ndcx is the area of the annular gap between the nozzle and the flapper, g the acceleration due to gravity, and γ the specific weight of oil.

Substituting (1) into equation (2), we obtain

Introducing the notation :

1095

1 + 305-5

R e '2 5

fx i ( ^ p ,

2g 305-5(ndcvy

25 = fl, — (ndc)

2 = Z>, μ1/ι *

y We have finally:

Ql + at/(Ql) = V2bx2P: (3)

The oil flow rate through the fixed restriction is given by the equation:

Qi = J[y (Pi - Px)j = k V ( / > , - Px), (3a)

where ft = nd\jA is the cross-sectional area of the fixed restriction.

Ν Because Qx = Qi (for steady-state flow), we can substitute in equation (3) Qi for Qx, and solve the resulting equation for x:

— | 7 i -b LV Pi

kt/k\J\ Λ

1

pJ

1-2

This equation can be converted into a non-dimensional form :

(4)

C = μι

Pi + k\Jk 1 -

1

77 1-2 A 4dc

(4a)

10a Aizerman I


The physical meaning of C is the distance between the nozzle and the baffle, at which the cross-sectional areas of the fixed resistance and the nozzle-baffle element are equal. In general form, the static characteristic of the con-trol unit is described by the following equation:

Λ 1 1

1 + μχ/χ 1 +

I6d2

c

(5)

As \6d2idt = 1/C

2, we can write it in the form:

A =

1

Λ 1 + l c

(5a)

The term C can be considered as a constant, which fully determines the geometry of the whole unit.

Control units can have equal values of x/C, and can consist of geometric-ally similar elements (Fig. 2). For dynamical similarity, it is necessary to

c,cm

0018

0-016

OOH

0-012

0010

0-008\

owe

O-OOi

0-002\

:

C= ^~

3

Γ "7—1 1

0 0-10 012 OU 0-16 018 0-20 0-22 0-24- 026 028dCl.cm

F I G . 2 . l—dl — 0 - 0 4 c m ; 2—d1 = 0 - 0 6 c m ; 3—dx = 0 -08cm.

have equal Reynolds numbers (referred to the nozzle diameter) and dis-charge coefficients μΛ of the fixed resistances. I f these conditions are ful-filled, the non-dimensional static characteristic can be represented by a function

^ = φ ( - ; R e ) . Λ \C


a = a(dc), ν = var,

c = c(dc), dx = var,

k/μι = k(d,), γ = var.

α

2&0

260

2W

220

200

WO

160

HO

120

100

80

6-0 0Ό9 010 0-12 OH 016 018 020 dc,cm

FIG . 3a. 1—v = 0-22 cm2/sec; 2—ν = 0-21 cm

2/sec; 3—ν = 0-205 cm

2/sec;

4—v = 0-2 cm2/sec; 5—ν = 0-18 cm

2/sec ; 6—ν = 0-19 cm

2/sec;

7—v = 0-185 cm2/sec; 8—ν = 0-18 cm

2/sec.

In particular, for the control device in question, this relationship is given

by equation (4 a).

2. Construction of Static Characteristic Curves

In Figs. 2, 3a, 3b the coefficients c, a and k are shown graphically:


The relation between the discharge coefficient of the fixed resistance and the pressure difference is shown in Fig. 4.

The construction of the static characteristic curve is illustrated by the following example:

dc = 0-180 cm, P1 = 1500 g/cm2,

d1 = 0-082 cm, ν = 0-2 cm2/sec.

(a) From Fig. 4 with άγ = 0-082 cm and a pressure drop across the fixed

resistance ΔΡ' = Pl - Px = £Λ = 1500/2 = 750 g/cm2, we find/ij = 0-665.

029

027

0-25

0-23

021

0-19

017

015

0-13

011

009

007

005

1J

I 2

003 Mi 005 006 007 008d7cm

F IG. 3b. 1—γ = 0-875g/cm3; 2—γ = 1-0g/cm

3.

(b) From Fig. 2 for dl = 0-082 cm and </c = 0-180cm, w e find

C = 0-0089 cm.

μΥ€ = 0-665 χ 00089 = 00059 cm.


0-7

0-6

0-5

0Ί

03

0-2

01

·· · ********

— • J ,

^ w o ί

' t 35

j = l

1 ^ w o ί ί

1 1 1

0 * 0 * 0-tf 00

ΔΡ-Ρ,

TO 12 ΐδ 18

5 _η Μ /

F I G . 4.

Marking of

points di, cm i/o, cm /, cm

Ο

•

0 0 6

0-082

0-2

0-2

3-33

2-0

0-15

0-29

2-5

2-9

W e now have all the coefficients of equation (4 a). Results of further calculations are shown graphically in Fig. 5. On the same diagram are plotted an experimental curve, and two theoretical curves, one assuming a square law for both resistances and μχ/μι = 1, and the other a square law for the fixed resistance and a linear relation for flow rate and pressure drop

(c) From Fig. 3b, for ί/χ = 0Ό82 cm and γ = 0-875 g/cm3, we find

* K = 0-24. Hence k = 0-24 χ 0-665 = 0 159.

k\/k = 0-159^0-159 « 0 1 .

From Fig. 3a with ν = 0-2 cm2/sec and dc = 0-180 cm, we obtain the

coefficient a — 20-05.

P-1 0-8


of the flow between the end-face of the nozzle and the baffle.

_ ngx3Px

~~ ( D \ ' βγν In [ — l

where D = outer diameter of the nozzle end face.

From Fig. 5 it is seen that curve 2, calculated by this method, with

experimental values of the changes in discharge coefficient, agrees with the

experimental curve 3 better than the two other curves, calculated entirely the-

ft β

01,

6 10 12 n 16x10~êm

F I G . 5. 7: Ρχ

1 +

(τ)

3: experimental curve; 4: Pi

1 + (τ)'


oretically. It should be noted that a still better agreement with experimental

results could be obtained, if necessary, by taking into account the changes

in discharge coefficient for the fixed resistance.

3. Analysis of Static Characteristics

The sensitivity of a control device depends on the slope of the character-istic ôPxjdx. Assuming that μx is a function of x, and μx is independent of x, we obtain the following expression for the slope of the static characteristic:

The slope is increased by reduction of C. The magnitude of C,in turn, will

Px % ?Rtt

W%,Cr}

1'6

Η

1-2

TO

08

0-6

04

02

0

-0-2

-(Η

-0-6

A e _ J L _ . c ( A V Λ , * i \ Pi fx

Η

I V -0-75

I Ρχ

l | \/ •0-577

o- •8 1-2 V 6 2 0 *

F I G . 6.


be smaller if the ratio between the diameters of the nozzle and of the fixed throttle is increased.

If the control unit is a part of a control system, it is desirable to choose a value of the distance χ which would ensure the best sensitivity, and hence the best static accuracy, of the controller at the required value of the con-trolled quantity and average load. This position of the baffle corresponds to the point where the slope of the characteristic is a maximum, that is dPJdx = max. or d

2Pxl'dx

2 = 0.

A general solution of this problem is impossible, because there is no generalized relationship between μχ and x. Al l three relationships (function PxjPi =f(x/c), and its first and second derivatives with respect to x) are shown in Fig. 6.

Solving the equation d2Px/dx

2 = 0 with the assumption that μΧιΙμι = 1,

we can find the value of χ at which dPx/dx is a maximum: χ = 0-577C. This value of χ on the static characteristic corresponds to the point

Px = 0-75 Λ.

4. Choice of Control Unit Parameters to Suit a Given Static Characteristic Curve

Let us consider the following problems: (1) Obtain the flow characteristics which are required by the fixed and

variable resistances in order to obtain a given static characteristic for the control unit; and

(2) Decide the geometry of the fixed and variable restrictors to produce flow characteristics Qi = Qi(PL — Px) and Qx = QX(PX), x being variable as required.

For a solution of these problems, it is necessary to assume a flow character-istic for one of the resistances (fixed or variable one), a supply pressure Pl

and the diameter dx of the fixed resistance (or dc of the nozzle). It is desirable to have the nozzle diameter dc about 2-2-5 times the fixed restrictor dia-meter dl9 as such dimensions gives a sufficiently steep slope to the static characteristic.

With a flow characteristic of the nozzle-baffle element, Qx = QX(PX) (x = variable), and the static characteristic required, it is possible to con-struct graphically the required flow characteristic of the fixed throttle. We then determine the diameter of the fixed resistance, so as to have a flow characteristic approximating to that required. These operations can be con-veniently performed in the following order (see Fig. 7 ) :

1. The desired static characteristic Px = Px(x)9 Pi = const, is plotted in coordinates Px and x.

2. The flow characteristics Qx = QX(PX), x = variable, are plotted in co-ordinates Qx and Px (for different values of x); the ordinate Px is common to both graphs.

T H E O R Y O F C O N T R O L D E V I C E S O F ' ' N O Z Z L E S A F F L E " T Y P E 295

3. From the points of the static characteristic curves at these values of x,

horizontal lines are drawn to intersect the flow characteristic curves for the

appropriate x.

4. The points of intersection lie on a curve (solid line on Fig. 7) and give

the required flow characteristic of the fixed resistance.

5. Comparing the curve obtained with a series of flow characteristics

β ι = δ ι ( Λ - ^ 1 variable, we can choose the most suitable diameter

for the fixed resistance.

In passing, it may be mentioned that the static characteristics of the con-

trol unit are determined not by the curves of the absolute values of flow rate

through the fixed and variable resistances, but by the intersecting values.

This results in the fact that the static characteristics, obtained from the plotted

flow characteristics, are all contained in a comparatively narrow range.

*o *

Q W Λ

F IG. 7. F IG. 8.

T R A N S I E N T E F F E C T S I N C O N T R O L U N I T S

O F T H E N O Z Z L E - B A F F L E T Y P E

As a rule, the intermediate chamber is connected (or, as it is called, "loaded") with a variable capacity (Fig. 8). In this case the output quantity is the travel of a diaphragm, bellows, or piston, which is dynamically dependent on pressure Px.

The main aims of the further investigation are as follows: 1. Formulation of the theory for establishing a certain pressure in the

chamber.


2. Analysis of the effects of control unit parameters on its dynamic per-formance.

3. Determining these parameters for the required dynamic performance.

For unsteady flow through the control unit, the flow rate Qx through the fixed resistance is equal to the sum of the flow rates Qk into the intermediate chamber (due to changes of its volume) and Qx through the nozzle :

ö i = Qio + àQx, Qk = Qk0 + AQk, Qx = Qx0 + AQX. (7)

Here Qi0 is the flow rate through the fixed resistance under static conditions, Qx0 the flow rate through the nozzle under static conditions; and Qk0 = 0 (under static conditions, there is no change in the volume of the chamber). Henceforth the suffix "0" relates to static conditions.

Substituting (7) into (6), we have:

We shall also assume that:

(a) the supply pressure Px is constant (assuming a relief valve, and a sufficient supply);

(b) the ambient pressure around the baffle is constant, and equal to atmospheric pressure;

(c) the fixed resistance is in the form of a cylindrical tube. The flow rate AQk into the chamber may be caused either by changes of

its volume due to the "load" (diaphragm, bellows, or piston), or due to fluid compressibility, mainly due to air entrained in the oil.

In these cases, the flow rate AQk depends on the pressure perturbations APX in the chamber, and can be taken as AQk = W(p) APX where ρ = ajdt;

and W is a function, as yet unknown, of p.

1. Dynamical Equation of the Control Unit

Qi = Qk + Qx, (6)

AQl = AQk + AQX. (7a)

J 0 i = AQk + AQX, AQX = Q

JQk= W{p)APX9 AQX = Q

0ΛΔΡχ,μι), Qx(x,APx, μχ).

(8)

The linearized form of equations (8) are:

àQ, = AQk + AQX,

(9)

T H E O R Y O F C O N T R O L D E V I C E S O F " N O Z Z L E S A F F L E " T Y P E 297

Αμγ = 0.

Including equations (2), (3a) and (9), we obtain:

APX AQX = -

2R,

'AP

AQX = Kxx0 I - + 2 x0

Λ - Pxo

f^xO

P, -

Qio

μχ0πά, l(—Px0)

\ V y J

(10)

( Π )

(12)

The quantity Rx characterizes the hydraulic resistance of the fixed restrictor. Both Ri and Kx depend on the static condition of equilibrium, about which the initial equations have been linearized. From equations (7a), (10) and (11)

Mxo

Δχ Ρχο *0

Κ.χ

W(p)\

2Ri ΔΡΧ

F I G . 9.

we can construct a block diagram (Fig. 9). The diagram shows two indepen-dent inputs Ax and Αμχ, but actually the input of the control unit is the baffle position, determined by Ax, and all other quantities depend on it. Therefore, we must establish the relationship between the discharge coefficient μχ and the flow rate Qx.

It has been shown above that μχ alters only slightly with changes in

pressure drop, and over a wide range of the latter may be considered con-

stant. Therefore


As the duration of a control action is usually short compared with the

time needed to warm up (or to cool) the oil, the changes of viscosity due to

temperature effects can be neglected.

Then the coefficient μχ will be dependent only on the flow rate Qx:

In a linear approximation

Δμχ

μχ = μχ(ζ>χ)·

ομχ AQX.

Cx /Qx = Qxo*0

After substitution of (1 a) into (1) and differentiation, we have

Δμχ = AXAQX.

0-685

(13)

Here

A, = QxO VQx

Finally, the flow processes in the control unit are described by the following

set of equations:

AQ, = AQk + Qx.

APr AQ, = 2Rl

AQX = Kxx0

Δ Ρ γ ΡχΟ Λ ΡχΟ Λ

L· + _ H Δχ + Δμχ

2 χο μχο

(14)

Δμχ = AXAQX,

AQk = W{p)APx.

In Fig. 10 this set of equations (14) is shown as a block diagram.

r i 3 & 4M* Αν Κ

Δχ β(0 *0

ΔΟ,χ

Wip)\

AQi 2Ri

F I G . 10.

T H E O R Y O F C O N T R O L D E V I C E S O F ' ' N O Z Z L E S A F F L E " T Y P E 299

Let us determine, on the basis of (14) and Fig. 10, the transfer function

APx\Ax of the unit:

Qxo 2RXRC

ΔΡΧ x0 Rx + RCB

A X 2 R^ BW(P) + l

(15 )

Rx + RCB

Β = \ - AxKxx0 — = \ - Ax , J ? c = J _ = ^ L . ( 16 ) FTXO f*>XO Κχ

Χ0 QXO

The quantity Β represents the influence of changes in the discharge co-

efficient μΧ9 on the dynamical behaviour of the unit. The assumption of

constant μχ is equivalent to substituting Β = 1 into (15). Rc represents the

hydraulic resistance of the nozzle-baffle element, its magnitude depending

on the position of the baffle and the pressure before the nozzle.

2. Influence of the Control Unit Parameters on the Gain

and Time Constant

If the volume of the intermediate chamber remains constant during opera-

tion and fluid compressibility can be neglected, then AQk = 0, and equa-

tion (15) takes the form:

ΔΡΧ _ Qxo2RxRc _ K

Ax x0(Rx + RCB)

This is the linearized equation of a static characteristic, expressed as perturba-

tions. The gain ^characterizes as a linear approximation the static sensitivity

of the control unit. As lim A PxjAx = dPx/dx, the maximum value of Κ can Jv^O

be determined from the condition d2Px/dx

2 = 0.

If the change of μχ can be disregarded, then in the case given above,

χ = 0-577C, a n d P x = 0-75P x.

After substituting Rc into (17), the gain can be expressed as:

K = L ^ 2 ^ . (18) x0 Rx + RCB

From (18) it can be seen that Κ increases with a reduction of Rc and B, and diminishes with a reduction of Rx. As Β < 1, the minimum gain will be when Β = 1 (all other conditions being equal); that is, at a constant value of coefficient μχ. Then Κ is reduced for a smaller value of μχ.

The resistance of a fixed restrictor is inversely proportional to the square of its diameter dx, and the resistance of a nozzle-baffle element inversely pro-portional to the nozzle diameter dc. Therefore, to increase the gain, the fixed


throttle diameter dl must be reduced, and the nozzle diameter dc made larger. Let us now consider the influence of the control unit parameters on the

time constant. If the intermediate chamber is loaded by a cylinder with a piston, having negligible mass and viscous drag, the flow rate into the inter-mediate chamber is determined by the equation :

AQk = KmpAPx. (19)

Here Km = S2jß, where S is the piston area, and β is the stiffness ("rate")

of the spring. In this case W(p) = AQk/APx = Kmp.

Consequently, the transfer function of the control unit, for the case where the flow rate into the intermediate chamber is determined by equation (19), will have the form:

Qxo 2RXRC

= - V o * I + RCB

Introducing the notation Rx + RCB β

1 c Β = Γ, (21)

Rx + RCB β

and substituting the gain Kïvom (17), we can write equation (20) as follows:

APX Κ

Ax Tp + 1 (22)

Therefore, if we neglect the mass of the piston and its frictional resistance, the control unit can be represented as an aperiodical link of the first order. From (21) it is seen that the time constant Τ can be reduced by reducing resistances R, of the fixed restrictor, and Rc of the nozzle. The reduction of Rx, however, will also cause a reduction of Κγ, which may be undesirable. For example, a substantial part of the characteristic would fall into the region of very small values of x, prone to silting and coarser mechanical obstruction of the gap between the nozzle face and the baffle. Also, it might happen that the force of the jet on the baffle will be greater than is permissible.

The most effective way of reducing the time constant lies in the reduction of S

2/ß or, in other words, reducing the changes in the volume of the chamber.

Obviously, at β = o o , the time constant Τ = 0. When designing the control device, the stiffness of the spring (bellows or diaphragm) should be chosen from the condition:

+ G = PxnaxS. (23)

Here j m a x is the maximum piston travel (with respect to the deflection of the bellows or diaphragm), P x m ax is the pressure in the chamber, corre-


sponding to j m a x, and G the external load (constant in magnitude and direc-

tion) acting on the piston.

Hence,

S2 _ S S G_

Ρ ' . r m a x -* x i n a x Ρ

Because S2 Iß is included as a factor into the expression for T, the time con-

stant increases with S, ymax, and G:

T _ 2RXRCB /Symax + _ S _ G_\ ( 2 4)

Rx + RCB \ P x m ax ^ m a x β /

From (24) it is seen that the hydraulic resistance of the control unit depends neither on the magnitude of the variable capacity of the chamber, nor on its design peculiarities.!

The factor 2RlRcBl(Rl + RCB) characterizes the hydraulic resistance of the fixed restriction and the nozzle, and the quantity (5 /P x max)ymax + (S/Pxmax) (Gjß)

depends on the variable capacity with which the device is "loaded". Con-sequently, the time constant Tis determined not only by the variable capacity, which "loads" the chamber, but also by the hydraulic resistance of the fixed restriction and the nozzle-baffle element. For a quantitative evaluation of the dynamic properties of a control unit we shall calculate the time constant Τ

and gain AT of a unit with the following parameters:

dc = 0-180cm, P x = 1500 g / c m2, S = 28 cm

2, ^ = 0-082 cm.

^ c m a x = H 0 0 g / c m2, ymax = 0-1 cm.

The stiffness of the spring is found from (23):

ß = L ^ S = 1 4 00 X 2 8

- 392-000 g/cm J W 0-1

and, consequently,

S2 28

2

Λ Λ ΛΑ c i

= = 0-002 cm5/g .

β 392-000

The results of calculating Tand A^from equations (17) and (21) for various distances between the nozzle and baffle are represented graphically in Fig. 11.*

Broken-line curves are calculated without taking into account the changes of μχ. From Fig. 11 it is seen that the effect of changes in μχ is substantial, both for the gain, as well as for the time constant. The relative difference ÔK attains 69-5 per cent, and ÔT = 59-3 per cent. The influence is "towards im-

t This is valid, if the hydraulic resistance of the intermediate chamber can be neglected.

Usually this is permissible in view of the smallness of this resistance. φ Quantities needed to calculate Τ and Κ were taken from the experimental data of a

control unit.

F I G . 12. 1—χ = 0.006 cm ; 2— Λ: = 0.008 cm; 3— χ = 0.010cm; 4— χ = 0.012cm;

5—χ = 0.014 cm; (5 χ = 0.016 cm.

F I G . 11.

P N E U M A T I C A N D H Y D R A U L I C C O N T R O L 302


provement"; that is, if the variability of μχ is taken into account, the values

of Κ are then greater and those of Τ are smaller. The time constant at first

increases up to a certain maximum, and then diminishes as χ is increased.

This shape of the curve Τ = T(x) can be explained by the increase of R1 and

the reduction of Rc with increase in x. Consequently, the quantity

u1D3

I ^ 7

3

I

w=0â Of / W J

05^

f o i l Ί 05\

5^

U10 180 160 110 120 0 80 60 10 20 0

F IG . 13. / χ = 0-006cm; II—χ = 0-008cm; III—χ = 0-010cm;

IV—χ = 0-012cm; V—χ = 0-014cm; VI—χ = 0-015cm.

2RlRcBj(Rl + RCB) will have a maximum determined by the magnitude and rate of change of and RCB, with respect to x.

W e can now construct the step response, and amplitude-phase character-istics (frequency response), for various initial conditions about which the basic equations were linearized. The step response curves ΔΡΧ = ΚΔχ{\ — e " *

/ r) are

shown in Fig. 12. In all cases, Δχ = 0-002 cm has been taken as the unit step.

The frequency response characteristics W(ico) = —Κ/(Τιω + 1) are shown

in Fig. 13, from which it can be seen that with increase in χ both the phase change and the amplitude change are increased. In particular, for ω = 1 :

arc tan 1 3 ° ; ^( / Ω )

» - ° - ^/ Ω )

» - Μ 0 Ο « 2 · 7 % .

^ ( ' · ω ) ο , = ο

3. Choice of Parameters of the Nozzle-baffle Control Unit

Let us illustrate the choice of the parameters^, dC9 D, S, β, Pl9 P x m a x, ymax of a control unit, to comply with the following specifications:

1. The static characteristic must be linear within the interval from xt

to x2.

2. The slope of the static characteristic Px = Px(x), Ρ γ = const, must be equal to or greater than the required one, within the same interval:

dPx

dx


dr = 2 πΡ1

T o construct the flow characteristics of the nozzle-baffle element, we can apply the method of determining the discharge coefficient μχ for a given χ

and Px [5], and then, using the equation Qx = μχπά£χ ^J(2glyPx), plot the flow characteristics of the nozzle-baffle element:

Qx = Qx(Px)\ x = variable.

From the flow characteristics of the nozzle-baffle element, and the required static characteristic of the whole control unit, it is possible to determine graphically the required flow characteristics of the fixed restrictor (see Fig. 7), and to choose the diameter of the fixed restrictor so that its flow character-istic would be close to the required one.

The time constant Jean be determined from the relation:

r<r r e q. (25) t The magnitude of F is limited by the available power of the sensing system, which

moves the baffle. This question is discussed in Ref. 3.

3. The force Fx exerted by the oil jet on the baffle must not exceed a cer-

tain value F : |

FX<F.

4. The gain K, within the same interval, must be less than or equal to a

required value:

K ^ ^ > # r e q. CX

5. The time constant T, within the same interval, must not exceed a certain

value:

Τ < Τ Λ 1 req ·

Since the number of parameters which characterize the control unit is greater than the number of equations mutually connecting these parameters, it is necessary to fix some of these initially, for example the supply pressure Pl,

and then fixed the nozzle diameter from the condition Fx < F.

It has been established experimentally [4] that Fx -> F m ax when χ-> 0. Then the following relationship is valid:

\πά2Ργ < Fxmax < \πϋ

2Ργ,

from which the nozzle diameter can be found as:

\nd]Px < (Fxmax < F) < ίπ02Ρ{ ,

F

T H E O R Y O F C O N T R O L D E V I C E S O F " N O Z Z L E B A F F L E " T Y P E 305

The stiffness of the spring β referring to the piston (with respect to the

diaphragm or bellows) should be chosen from the condition:

ßymax + G = ^ m a x ^ from which it follows:

ß = P^S G ( 2 6)

Here the force G is known, and Pxmax is determined by the working move-ment of the baffle. The maximum travel y,aax of the piston (with respect to the diaphragm or bellows) should be considered as a given quantity, because it is determined by the specifications of the control unit. The relation (25), taking into consideration (26), will give the form :

t> < i r e q -R, + RCB SPxmax - G

From this we can determine the piston area S:

S =

pxm>* ± . / ( p x m n - 4 n

2 R l R; D B ^ G

Rl + RCB Treq

If the force G is negligible, this expression can be simplified:

Τ Ρ S =

1 r e q

x jcmax

2RXRCB '

Rl + RCB

Thus the control device can be designed to come close to the required static and dynamic performance. In the case where a specification cannot be met by a choice of parameters, it would be necessary to alter the design of the elements, or to design these for appropriate special working conditions—for example, to provide a constant pressure drop at the fixed resistance and at the nozzle-baffle element. The experiments of V. L. Lossiyevski [2] demonstrate that it is possible to improve the static sensitivity of the control device by altering the design of the nozzle.

The provision of a constant pressure drop at the fixed restrictor and at the nozzle is the most effective means, which is used in nearly every pneumatic instrument of the "Unified Block System".



1. The discharge coefficient μχ of the nozzle-baffle element is determined

by the Reynolds number, as given by equation (1).

2. The static characteristic of the control unit is determined by equation

(4 a), which gives the influence of the main parameters of the unit on the

static characteristic.

The present work includes the dynamical equation of the unit, block dia-

grams showing the interaction of its elements, and determines the transfer

function (20), taking into account the variation in discharge coefficient.

3. It is established that the time constant Γ and the gain calculated for the

assumption of constant discharge coefficient of the nozzle-baffle element,

differ substantially from those calculated with allowance for variations in

this coefficient. The relative differences amount to 60 per cent for Τ and

60-80 per cent for K.

A method of choosing the parameters of a unit, in order to satisfy a given

specification, is discussed.

4. Experimental results and theoretical investigations of the nozzle-baffle

control unit enable us to indicate the fields of application.

A P P E N D I X

Elements of nozzle-baffle control units were investigated experimentally in Ι Α Τ A N

(Academy of Sciences) U .S .S .R. on the test rig shown in Fig. 14.

FIG . 14. 1—oil tank; 2—gear pump; 3—accumulator; 4—oil filter; 5—air filter;

6—reduction valve for air; 7—release cock; 8—relief valve; 9—nozzle-baffle element.

TH

EO

RY

O

F

CO

NT

RO

L

DE

VI

CE

S O

F

"N

OZ

ZL

E-

BA

FF

LE

" T

YP

E

307

FIG. 15.


Oil from the tank is passed by a gear pump into an accumulator, to smooth out pressure

pulsations. A i r is pumped under pressure into the accumulator and can be released through

a cock 7, if required. Oil from the accumulator passes through a filter (an aircraft oil filter

with 1 0 , 0 0 0 apertures per 1 c m2 was used) to a structure holding the element under in-

vestigation. In this particular case, a nozzle-baffle element, with the baffle formed by the

pad of a micrometer, is installed. The pressure was adjusted by a relief valve.

During these tests, quantities involved in criteria of similarity (directly or indirectly)

were measured, at least three times, under steady-state conditions.

I f any variations of flow rate through the nozzle-baffle element were detected, the num-

ber of repeated tests was increased. A l l tests were conducted with transformer oil. The re-

sults of experiments with fixed resistances (plain cylindrical holes) are given in Fig.4f

and for nozzle-baffle elements in Fig. 1 5 (points and solid line). The broken line shows

the results of calculations of equations ( l a ) and ( 2 ) .

R E F E R E N C E S

1. G . WÜNSCH, Controllers of Quantities and Pressures (Regulatory kolichestva i davleniya).

Gosenergoizdat, 1932 .

2 . V.L.LOSSIYEVSKI, Automatic Controllers (Avtomaticheskiye regulatory). Oborongiz,

1944 .

3. V . N . D M I T R I Y E V and A.G.SHASHKOV , Article 2 3 in the present book.

4. A . G . SHASHKOV, A Dissertation, Moscow, 1956 .

5. N.N.SHUMILOVSKI , A Dissertation, M o s c o w - L v o v , 1947 .

f Figure 4 gives only certain curves μ = μ {ΔΡ) which are used in the present article.

Other data for dY = 0 - 2 - 1 - 2 mm are given in Ref. 4 and those of dx > 1-2 mm in Ref. 5.

VARIATIONS OF THE EFFECTIVE AREAS OF DIAPHRAGMS

V . V . A F A N A S Y E V

A N I M P O R T A N T parameter of fabric diaphragms as used in pneumatic appa-

ratus is the effective area; that is, the area whose product with the differential

pressure gives the force developed by the rigid centre of the diaphragm. The

effective areas of diaphragms used for instruments based on the force-

balance principle must not vary with the travel of their rigid centres, other-

wise the input signal would be distorted. The study of the relationship be-

tween the effective area of a diaphragm ( F i g . l ) and the size of its rigid centre

F I G . 1. A corrugated diaphragm.

has been carried out in K B T s M A f in order to determine the optimum dimen-

sions and shape, and the optimum operating conditions.

The effective area Fe of a diaphragm usually [1] is calculated by the for-

mula:

Fe = - — (D2 + Dd+ d

2), (1)

3 4

where D is the diameter of the outer diaphragm rim ; d the diameter of the rigid centre. This formula gives the effective area for the case of a neutral position of a diaphragm when its clamping plane in the outer rim coincides with that in the rigid centre. Calculations by formula (1) do not take into account the changes of effective area, due to the travel of the rigid centre.

Consideration of the equilibrium between the forces acting on the centre-piece of the diaphragm leads to the conclusion that the effective area can be

t Design Bureau for Automation in Non-ferrous Metals Industry.

11 Aizerman I 311

312 p n e u m a t i c a n d h y d r a u l i c c o n t r o l

(2)

where D e is the effective diameter, equal to the diameter of the circle at the crest of the corrugation.

ΤΎΤΠ ! M M κ ^ 7

F I G . 2 . The part of a corrugated diaphragm determining its effective area.

The assumption that the fabric can resist only tensile stresses leads to the conclusion that there are forces Τ parallel to the plane of the diaphragm ; then the resolution of the forces in the direction of the axis of the diaphragm

12 (*

— P + PcosxdF = R, 4 ) F

where Ρ is the difference of pressure; dFthe element of diaphragm surface; oc the angle between the direction of elementary force Ρ dF and diaphragm axis, R = FeP.

Since : Ρ cos oc dF = Ρπ

Dl - d2

4

we obtain formula (2). Let us assume now that at any position of the dia-

\C

(o

r

ΔΛ9

L Deo

F I G . 3. Diaphragm used for the deduction of formulae for the effective diameter.

determined by the following formula:

Fe = — ,

V A R I A T I O N S O F T H E E F F E C T I V E A R E A S O F D I A P H R A G M S 313

where

Λτ ^ D + d D - d ADe = De — De0; De0 = ; a = . (7)

2 2 The values of ADeja corresponding to the given parameters hja and a\I

may be found from the nomogram, Fig. 4, which has been constructed from equations (5) and (6).

It may be mentioned that formula (2) gives results for the neutral position of the diaphragm, differing but little from those of formula (1). In these conditions De = (D + d)/2, and equation (2) gives:

Fe2 = - - ( D2 + 2Dd+d

2),

4 4

while by equation (1)

Fel = - - ( D2 + Dd + d

2).

3 4

t This assumption, fundamental for the present work, requires a special proof, based

on the investigation of the actual form of corrugations. Such proof, however, is outside of

the scope of the present work.

phragm the cross-section of the fabric part is an arc of a circle.f This assump-

tion enables us to calculate the relationship De = De(h) where h is the travel

of the rigid centre from its neutral position. From the geometry of Fig. 3 it

follows:

A? = £>eo - 2y sin<%,

but 2y sin a = h cot β, therefore :

De = De0 - h cot β. (3)

Also,

ß = — ; sinß = — V(/*2 + *

2) ,

2r 2r

from which we obtain the following equation for

i ^ = i-V(*2 + « 2 ) . ( 4 ) ß

1

Formulae (3) and (4), suitable for calculations of effective diameter, can be conveniently represented in the final form:


Comparing these expressions, we obtain

For example, if d = 0-8A then Fel - Fe2 « 0003Z>2.

F I G . 4. N o m o g r a m for calculating the effective diameter De. ADe = De — De0;

De0 = (D + d)\2\a = (D — d)\2\d—rigid centre diameter; D—outer rim diameter;

/—length of corrugation's generator; h—height of rigid centre above the plane of

the clamping in the outer rim.


F3,cm< 1 8-6

7 f / J

R f /

///

/

DJ

&**^>

^ ^ ^ ^

9S

h) mm 06 o-v 0-2 -0-2 -0'¥ -0-6 -Ο'δ

F I G . 5. Function Fe = f(h) for a diaphragm with D = 30 m m ; </ = 24 mm.

7—calculated ; 2—experimental.

It has been established that characteristics Fe = f(h), obtained by cal-

culation according to formulae (2), (5) and (6), agree well with experimental

data (Fig. 5).

Some of the results of the investigation into the relationship Fe = f(h) and

certain related design studies carried out in K B T s M A are given below.

From (5) and (6) it follows that the function De = De(h) could be expressed

in the form: A

ADe , [h I ~Γ = Φ ~' a (a a

Such relationships, plotted on the basis of the nomogram Fig. 4, are shown

in Fig. 6.

ADe fa 10

0-9

08

0 7

0-6

05

OV

0-3

0-2

Of -0-3-02-01

/ / 1

/ * J / % %

-0-3

0-1 02 0-3 0¥ OS 06 0-7 08 0-9 fO 4-

F I G . 6. Function ADJa = 0(h/a, lid).

,2


The graphs of Fig. 6 indicate that a deeper corrugation gives a reduced

slope of the curve. In this connexion it is interesting to note that some

American firms use diaphragms with very deep corrugations. Such dia-

phragms are supported between the wall of the chamber and the high rim

of the rigid centre (Fig. 7). The shape of corrugation remains unchanged,

Ci

F I G . 7. Design of a corrugation having Ija > π/2.

even at rather large movements, and its crest remains on the same vertical

cylindrical surface; the effective area of the diaphragm does not change.

It is obvious also that variations of the effective area are at their minimum

when the rigid centre is in its neutral position. Then stretching the dia-

phragm does not cause the horizontal displacement of the corrugation crest,

and does not alter the effective area (Fig. 8,a). When the rigid centre is moved

ο b

F I G . 8. Changes in De due to stretching of fabric.

away from its neutral position (Fig. 8,b), stretching of the fabric alters the effective area. Similar considerations apply also to diaphragms which are sub-ject to a change in the direction of the applied differential pressure (Fig. 9).

The results of the present investigation were taken into consideration during the design of the integral action pneumatic controller K B T s M A . In its three-diaphragm set, adjustment has been provided for the rigid centre posi-


F I G . 9. Changes in De with the change in the direction of pressure action. Planes of

clamping in the rim and in the rigid centre are a—coincident; b—not coincident.

tion relatively to the housing, as shown on Fig. 10. In order to put all three diaphragms in the position corresponding to the least variability of effective area, there is a special wedge adjuster. By moving the wedge, it is possible to raise or lower the complete diaphragm unit.

In practice the adjustment of the diaphragm stack is carried out as follows: throttles 5 and 6 are fully closed, and atmospheric pressure is trapped in the chamber K. The initial magnitude of the output pressure Poutp is then deter-mined by the loading of spring 3. This pressure is recorded. Then the cham-

F I G . 10. Design of the integral action controller.


bers F and G are put under equal pressures, exceeding the recorded Poutp.

This produces a deflection in diaphragm 2, and one in the opposite direction in diaphragm 1. I f the diaphragm stack had been assembled incorrectly, a change of effective areas would result, and consequently the output pressure P o u tp would be altered.

The wedge-baffle 4 should be moved, until the output pressure returns to its initial value. Then the pressure in chambers F and G is released ; if this

causes a change of output pressure, the process of adjustment is repeated, until a position is found in which pressurizing chambers F and G, and sub-sequent release of pressure, does not alter the output pressure.

The alteration in the effective areas of fabric diaphragms when they are dis-

F I G . 11. Design of the pressure transducer.


11 a Aizerman I

placed is utilized in the design of a pressure transducer developed in

K B T s M A .

The design of this tranducer is shown in Fig. 11. The measured pressure is

fed to chamber / . This produces a force on diaphragm 1 which moves

the rod 2, and the attached screw 3, downwards. The screw presses on the

baffle 4, which closes the nozzle 5, connected with a pneumatic relay. Closure

of nozzle 5 causes an increase of output pressure, which is also connected to

the chamber K, where it produces a counteracting force acting on the feed-

back diaphragm 6. When the forces from the diaphragms 1 and 6 are

equal, an equilibrium state corresponding to a given output pressure is at-

tained. The adjustable spring 7 is used for zero-setting of the output pressure.

Range adjustment is effected by the screw 3, which raises or lowers the rod 2

with the rigid centres of diaphragms 1 and 6. This causes a reduction of the

effective area of one diaphragm, and an increase of that of the other, thus al-

tering the range of measurements.

A transducer of this type, developed by K B T s M A , originally was

intended to measure pressures from Oto 2 kg/cm2, but the use of the above

effect enabled the upper limit to be extended to 2-5 kg/cm2 or lowered to

1-6 kg/cm2.

R E F E R E N C E

1. V . L . L O S S I Y E V S K I I , Automatic Controllers (Avtomaticheskiye regulatory). Oborongiz ,

1944 .

CHARACTERISTICS OF DIAPHRAGMS USED IN SENSING ELEMENTS

OF CONTROLLERS

Y U . L . M A C H and G . P . S T E P A N O V

D I A P H R A G M S are often used in sensing elements of controllers, both for general industrial purposes and for special applications (such as the control of jet engines). The working medium which exerts the pressure on the dia-phragm is a gas or a liquid, sometimes a corrosive one. For jet engine con-trol, the working medium is either air or exhaust gases.

The main difference between the working conditions of industrial and jet engine controllers is a greatly extended temperature range in the latter case. Low temperatures correspond to winter conditions and flight at great heights, while high temperatures occur in the immediate surroundings of a controller, at high speeds being approximately equal to the stagnation temperature. The general principles are, however, common for both industrial and special pur-pose controllers, and the study of diaphragms, being applicable to both, can be of general interest.

There are many references on the subject, but in the main they deal with calculations on metallic diaphragms treated as flexible shells, and do not elucidate problems, which sometimes are decisive for the efficiency of a con-troller.

Experience with diaphragms proves that the problems of diaphragm mounting, permanency of their characteristics, hysteresis, etc. are very im-portant, and the accuracy of control is largely dependent on the solutions of such problems.

The present work includes:

(1) Investigation of the constancy of diaphragm characteristics, both re-peated tests of the same diaphragm and tests of several diaphragms made from the same material. Examination of the influence of the means of at-tachment of diaphragms to rigid bodies.

(2) Comparison of various materials used for the general class of rub-berized-fabric diaphragms.

(3) Investigation of the relationship between hysteresis and working am-plitude of rubberized-fabric diaphragms.

(4) Investigation of the influence of ambient temperature on the charac-teristics of rubberized-fabric diaphragms.

320

C H A R A C T E R I S T I C S O F D I A P H R A G M S 321

Methods of Test. One of the main characteristics of a diaphragm is the relationship between the centre travel and the pressure difference. T o study these characteristics, it is necessary to have a reliable technique for measure-ment of the centre travel. In certain previous investigations these measure-ments were performed with the aid of a dial gauge, while others used optical means.

The analysis of both these techniques shows that the use of a dial gauge introduces substantial errors. In characteristics taken by the gauge method

δ, m m

2-

0.

1 > - , JY' ^ *

y s

4 /

/ / /

/ 41

/ • ff

/ Ί 1 (2

/ / t

/ 1 k

l i V

Γ

1 / f /

/ / // à/

/

« s i

/

250 300 350 WO

Δ P, mm of water

F I G . 1. Characteristic of a diaphragm S = S(AP). Travel S measured: 1—by an

indicator; 2—by optical methods.

hysteresis appears to be greater, and the whole curve is displaced along the pressure drop Δ Ρ axis when compared with curves obtained for the same dia-phragm by optical means (Fig. 1). This can be explained by additional friction introduced by the gauge.

Therefore in the present work the centre travel was always measured by optical means, using a microscope with 20:1 magnification. The ocular of the microscope was moved by a micrometer screw with divisions of 0-01mm. The pressure difference was measured by a water manometer.

Tests were carried out with diaphragms made of various rubberized materials (silk, kapron,| glass fibre), and for metal diaphragms. Al l these diaphragms had nominally identical dimensions, as shown in Fig. 2.

Investigation of Consistency of Rubberized-fabric Diaphragm Characteris-tics. Figure 3 shows two characteristics of the same diaphragm fitted to the

t Synthetic fibre, similar to nylon.


body in two different ways. The solid lines refer to fitting with pre-tensioning

by air pressure, and with tightening of the clamping nuts by a calibrated

torque spanner, giving a 45 kgcm torque. The broken lines refer to fitting

without pre-tensioning, the tightening of the nuts being approximately uniform,

F I G . 2. Design of a rubberized-fabric diaphragm (dimensions in mm).

F I G . 3. Characteristic of a diaphragm fitted :

with pre-tensioning; — without pre-tensioning.


but the torque spanner was not used. As can be seen, the method of fitting

has a marked influence on the diaphragm characteristics, and therefore in all

following experiments the diaphragm was pre-tensioned, and clamped by

means of a torque spanner, at the air pressure differential 150 mm of water.

Under these conditions, repeated tests of the same diaphragm have shown

a satisfactory consistency of characteristics (Fig. 4).

A t the same time, it has been found that two diaphragms made of the

same material may have different characteristics, even if the above rules for

their fitting were complied with. Fig. 5,a, b and c, shows the characteristics of diaphragms made respectively of silk, kapron and glass-fibre cloth. Each graph shows the characteristics of two diaphragms made of the same ma-terial.

Possible sources of non-uniformity for diaphragms made of the same material are: a small shift of holes for clamping studs; non-uniformity of material in various points; some deformation of fibres during the fitting of a rigid centre-piece.

In view of the above circumstances, it is necessary to test each diaphragm individually, and maintain a uniform technique of fitting in such cases where the diaphragm must fulfil important requirements. In particular, it is ne-cessary to use pre-tensioning, and to tighten the clamping nuts with a torque spanner. It would be also of advantage if diaphragms with a definite location of the rigid centre-piece relative to the body were developed [1].

Comparison between Diaphragms of Various Rubberized Materials. From numerous tests of diaphragms made of various makes of silk, kapron and glass-fibre cloth, a choice could be made of those with the most satisfactory

F I G . 4. Characteristics of a diaphragm after repeated tests.

P N E U M A T I C A N D H Y D R A U L I C C O N T R O L

's X

/ γ i

Λ s 1 / */ Λ /

J i ι I

1 • / /

ft f 1 J L 250 300 350

Δ P,mm of water

s—

y /

/ f

ί r X

/ /

V L Ί ? î

«

Ι L Ί ?

/ I •

/ 1 f ;

/ / /2 /

250 300 350 WO

Δ P, mm of water

/ / / f // J 1 / /

/

r / / 200 250 300 350

Δ Ρ, mm of water

F I G . 5. Characteristics of diaphragms made of: a—silk; b— glass-fibre ; c—kapron.

324

( a )

( b )

(c)


Sjirim

1 1

H <

/ / /

11 / •

ml II

X -f-1

/

/ /

/ κ

-I X -f-1

/

/ /

t r

•V' i /

Ie c

If 7 / /

/ J h > / r

i A / / /

><

J f i 250 300 350 WO

Δ Ρ, mm of wa te r

F I G . 6. Characteristics of diaphragms, made of: a—silk; b—kapron; c—glass-

fibre cloth E S T B .

characteristics. The results shown in Fig. 6 were taken at the temperature 20 °C. The best characteristics are possessed by kapron diaphragms, which have the least hysteresis.

The Dependence of Hysteresis on the Working Travel of Rubberized-fabric Diaphragms. It has been found that the hysteresis loop diminishes substan-tially with reduction of diaphragm travel. Figure 7 shows the characteristics

Δ Ρ, mm of wafer Δ Ρ, mm of water

F I G . 7. Changes of hysteresis loop at varied ranges of pressure difference.


obtained at varied pressure differentials, and consequently different amounts of centre travel.

T o find out the origins of hysteresis, additional experiments were conduct-ed with a diaphragm which originally had a large hysteresis loop. First, a characteristic was taken at a travel 2-4 mm (Fig. 8, curve 7) and then the travel was limited by stops, so that it could not exceed ±0-2 mm from the neutral position. The pressure difference remained the same. As can be seen from Fig. 8, curve 2, hysteresis practically disappeared. On the basis of such experiments, we conclude that the amount of hysteresis can be reduced by

S,mm y>*^\

i

/ V 1

{

1

7 / / /

j Ι 1 ΐΦ— ι i—c^TIL

250 300 Δ Ρ, mm of water

F I G . 8. Changes of hysteresis loop, depending on the range of diaphragm travel.

allowing only small movement of the centre-piece. In this respect, transducers working on the principle of force balance can ensure a higher precision of control.

The Influence of Ambient Temperature. Diaphragms made of rubberized silk, kapron, and glass-fibre were tested at temperatures from —40° to 150

CC.

It was established that the characteristics of these diaphragms were strongly influenced both at low and high temperatures (see Fig. 9,a, b and c) , decreasing their slope in both cases; that is, the rigidity of the diaphragm material in-creased in both cases. At low temperatures this was caused by hardening of the rubber, while at higher temperatures it occurred partly because of changes in the physico-mechanical properties of the fabric, and partly because of tightening caused by thermal expansion of the housing. Diaphragms using silk and glass-fibre base gave moderate change of characteristics at increased temperatures (up to 120°). It has been noticed that after initial tests at ele-vated temperatures the return to 20 °C results in a characteristic differing

F I G . 9. Influence of temperature on the characteristics of diaphragms made of:

a—silk; b—kapron; c—glass-fibre cloth.



S,mm

2

250

• * / h i // 2 j

ψ * à

i l i

t h f ß

l JÄ? <f00

Δ P, mm of water

F I G . 10. Settling of a diaphragm after repeated tests at increased temperature.

Characteristics taken: 1—at 2 0 ° C ; 2—at 20°C after being loaded once at 120°C;

5—at 20 ° C , after being loaded many times at 120°C.

S,mm

0-5r

(Η

0-3

0-2

0-1

-0-1

-0·2\

-0-3

/ /

j / / /

/ /

/

ί

/ /

/ Λ /

f

7 2 3 4 / 5 6 7 8

/

/ •

/

k Δ Ρ, mm of wafer

F I G . 11. Characteristics of a beryllium-copper diaphragm, clamped in a brass

housing.


from the original one at 20 °C. After repeated high temperature test, how-

ever, the characteristics at 20° settle to a reasonable uniformity (Fig. 10).

Tests of Metallic Diaphragms. Beryllium-copper diaphragms with a work-

ing diameter of 120 mm and thickness from 0-05 to 0-06 mm were tested. Such

diaphragms have a high sensitivity, approaching that of rubberized-fabric

i>, mm

0-9

0-8

0-7

0-6

0-5

0i

0-3

02

0-1

0 1 2 3 * 5 ô 7 6 9 10

Δ Ρ, mm of waler

F I G . 12. Characteristics of a beryllium-copper diaphragm, clamped in a steel

housing.

diaphragms (sensitivity here can be defined as the maximum value of àSjaP). The sensitivity of rubberized-fabric diaphragms is 004-0 16 mm/mm of water, and of beryllium-copper diaphragms 0Ό5 mm/mm of water. Metal diaphragms differ advantageously from rubberized-fabric ones in having good linearity of characteristics and no hysteresis (Fig. 11).

When the coefficients of thermal expansion of diaphragm material and of housing material were very similar, an increase of ambient temperature up to 250 °C did not alter the linearity of the characteristics (see Fig. 11), but its slope slightly changed, and it was somewhat displaced (due to the movement of the rigid centre-piece at zero pressure difference). When the same dia-


R E F E R E N C E

1. Mesures et Contrôle Industriel, N o . 229, Juin, 1956.

phragm was tested in a steel housing, the distortion of its characteristics was

much more pronounced (Fig. 12). In view of this, it is necessary to introduce temperature compensation, if

great precision is required in the performance of control devices which use

metallic diaphragms and work over widely varying temperatures.

HYDRAULIC LOSS COEFFICIENTS A N D DISCHARGE COEFFICIENT

FOR THE PORTS OF SPOOL VALVES USED IN HYDRAULIC CONTROL SYSTEMS

V . A . K H O K H L O V

S P O O L valves are widely used for a number of purposes. It is very important

to know the pressure loss in these valves, since the major part of the pump

power is dissipated in overcoming the resistance of valves and pipes. The

pipe frictional loss is often small, sometimes negligible. For cases where the

pipe connexions are long and contain a great number of bends, the pressure

losses in them can be calculated, for example, by the methods given in Ref. 1.

Hydraulic losses in spool valves, however, have not been sufficiently in-

vestigated, and are treated incompletely in the literature. Publications on the

subject [2, 3] do not reflect the true character of these losses, as the pecu-

liarities of flow through narrow slots, and leakage through the clearance

spaces, are not taken into consideration.

The object of the present article is to construct and analyse diagrams for

the changes in the hydraulic loss coefficients, and of discharge coefficient, for

the ports of spool valves, taking into account the factors mentioned above.

1. H Y D R A U L I C L O S S E S I N S P O O L V A L V E S

The line diagram of the spool valve in question is shown in Fig. 1 in which κ denotes the opening or the distance between the controlling edges of the valve pair, projected on the valve axis (for open ports κ > 0, for closed ones κ < 0), and δ is the radial clearance between the spool and the bore.

The total hydraulic loss in the valve is defined as the sum of the local losses in the open ports and the losses along the annular gap (between the bore and the neck of the spool). Since the annular flow lengths are short, and have a large effective cross-sectional area, compared with the ports, the main part of the hydraulic losses occurs at the ports.

It is known that the local pressure losses can be evaluated by the local loss coefficient ζ. This is determined by the shape of the duct, and by the nature of the flow. With laminar flow, the coefficient ζ is a function of Reynolds

333


number. With turbulent flow, the effects of Reynolds number are small, and

it is accepted in practice that ζ depends only on the shape of the flow duct.

Due to the complexity of the flow as the fluid passes through the port of a

spool valve, it is difficult to predict analytically the transition from laminar

to turbulent flow and, consequently, to find a local loss coefficient. Moreover,

turbulent flow is not uniform. A laminar layer may exist very close to the

wall. Numerous investigations prove that the thickness of the laminar layer

is a function of Reynolds number, and can vary within wide limits [4].

F I G . 1. Line diagram of the spool valve.

T o determine the loss coefficients, and also the type of flow in the ports of spool valves, we shall use the results obtained by V. A.Leshchenko [5]. It contains the experimental relationship between the flow rate of a mineral oil and the port opening at fixed pressure drops. During these experiments, silting of small gapsf was eliminated by an oscillatory motion of the valve elements. It should be mentioned that the frequency of these oscillations / = 2-10 c/s and amplitude £ m ax = 0-01 mm, are similar to those often used in practice. Therefore, we may say that the results of these investigations reflect with suf-ficient accuracy the actual processes of flow through the ports of spool valves.

In all further analysis we shall evaluate the flow conditions with reference to the mean velocity ν in the geometrical cross-section of a flow passage:

yf

where Qm is the mass flow rate (mass discharge per second); / the passage area; γ the specific weight of fluid. In this particular case, mean velocities have been determined by taking the value of Qm from Ref. 5, and the flow areas from the dimensions of the experimental valve. Assuming that the

f On silting phenomena, see the articles in the present book, pp. 382 and 393.

H Y D R A U L I C L O S S C O E F F I C I E N T S 335 sectional flow area of a valve port is a narrow rectangular slot, the formula

for the mean velocity is:

ν =

yV-D^(a2 + y

2)

(V(<32 + κ2) = δ at κ < θ ) , (1)

where D3 is the nominal diameter of the valve; ψ the total effective angle of

the port (usually 2π).

Figure 2 shows the diagram of mean velocity as a function of the port

opening.

From these curves, it may be noted that the mean velocity at a given pres-

sure drop Λ Ρ is fully determined by the gap κ.

υ, cm I sec. 5000

L Λ Y ΔΡ * 13ati 77

IM/*

ΔΡ *Waïm

IM/*

ΔΡ = 7 atm Αη_ ΔΡ- ο aw

-0-Q6 -m -002 0 0Ό2 OM 000 OOS 010 042 Olk 0J6 0/0 0-20

s, mm F I G . 2. Curves of mean oil velocity in the spool valve ports, as functions of the

opening κ.

Let us determine the local loss coefficient ζ. Relating the local losses to the

velocity head,t we have

4 «. γνΖ

Δρ = ζ — , 2g

and

C = γν

2

(2)

(3)

Using equation (3) and the curves given above for the mean velocity (Fig. 2)

we find, for various pressure drops, the local loss coefficient as a function of

the opening κ (Fig. 3).

t Having in mind the mean velocity in the passage area.


The following conclusions can be drawn from Fig. 3.

1. For a given pressure drop, the loss coefficient is determined by the port

opening.

2. With an increase in port opening, the loss coefficient approaches a con-

stant value, which, for "Turbine Oil L " , often used in hydraulic systems,

is equal to 3-1. The value of the port opening which corresponds to this

constant coefficient depends on the pressure drop.

\ \ς 17/7

\ V P-3c um

ΔΡ^ 5 atm

\ -\j0 *ΔΡ- 13 at η

1U

-0-06 -004 -002 0 0ΰ2 m 006 000 Ο'/Ο 0/2 0/4 0-/6 Ο/β Χ, mm

F I G . 3. Curves of the local loss coefficient ζ, as functions of the opening κ.

3. An increase in pressure drop beyond 7-8 atm has virtually no effect on the loss coefficient (when the port is open).

4. A t small port openings, the loss coefficient is a minimum, which depends on the pressure difference.

It may be conjectured that the reduction of loss coefficient and the increase of fluid velocity at small gaps can be explained by a reduction of viscosity, due to heating of the oil in the immediate vicinity of the edges.

In order to determine the type of flow, it is necessary to investigate the effects of loss coefficient on the Reynolds number.

The Reynolds number for the fluid flow through long narrow slots is usually calculated by the formula:

ν

where A is the geometric width of the slot, and ν the kinematic viscosity. From equation (4) the Reynolds number can be found as a function of the port opening.

H Y D R A U L I C L O S S C O E F F I C I E N T S 337

Thus, for a given pressure difference, we have a set of equations, in para-

metric form:

C = C(«),

Re = R e t » ,

where κ is the parameter. Eliminating κ, we obtain

ζ = C(Re).

The relation between ζ and Re, at fixed pressure drops, is shown in Fig. 4. In order to analyse the curves of Fig. 4, they may be divided into five sec-

tions. 1. The first section (from the right) covers the Reynolds number range

Re > 260. It is characterized by a constant value of the loss coefficient, which is independent of Reynolds number and pressure drop. This shows that the flow is turbulent.

2. The second section corresponds to Re < 260. It is characterized by the changes of ζ with R e ; this indicates the onset of laminar flow. Re = 260 is the critical Reynolds number for spool valve ports.

5

I *

& / Λ < 5 ν

7

im/T** ~ 3

\

'4 !

i 0 50 100 150 200 250 300 350 Re WO

F I G . 4. Curves of the local loss coefficient £, as a function of Reynolds number.

3. The third section lies in the range ~ 150 < Re < 260, and is charac-terized by the presence of the turbulent flow at a Reynolds number below the critical. The existence of this section may be explained by the destruction of the laminar layer at high velocities (it corresponds to Δρ = 10 atm).


4. The fourth section is characterized by the decrease of loss coefficient. It pertains to small gaps.

5. The fifth section corresponds to the closure of the annular ports, and is characterized by a sharp increase in hydraulic resistance.

2. D I S C H A R G E C O E F F I C I E N T

For a number of problems, for example the calculation of the output shaft

velocity of a hydraulic motor, it is more convenient to use the discharge co-

efficient:

Q_

Qo ' μ

instead of the hydraulic loss coefficient ζ. Here Q is the flow rate of a viscous (real) fluid through a passage, and Q0 the flow rate of a non-viscous (ideal) fluid through the same passage, at the same pressure drop.

-0-06 -0-Oi -0-02 0 002 0-04- 0Ό6 008 010 012 014 016 018x,mm

F I G . 5. Discharge coefficient curves for spool valve ports, as a function of the

opening κ.

The relationship between these coefficients can be found by obtaining the flow rate from the mean velocity and the geometrical flow area, and the mean velocity, in turn, from the pressure drop:

-'im

H Y D R A U L I C L O S S C O E F F I C I E N T S 339


1. The article gives diagrams of hydraulic loss coefficient and discharge co-

efficient as functions of the port opening of a spool valve. These are based on

the experimental data from Ref. 5.

2. It is shown that the critical Reynolds number for spool valve ports is

260.

3. The results obtained can be used for investigations in the force and

velocity characteristics of hydraulic control systems and follow-up devices.

R E F E R E N C E S

1. N . G . K I S E L E V , Manual of Hydraulic Calculations (Spravochnik po gidravlicheskim ras-

chotam). Gosenergoizdat, 1 9 5 0 .

2. Z . S. B L O K H , O n the theory of hydraulic servo-motors ( K teorii servomotorov gidro-

privodov). Bulletin of OTN AN (Academy of Sciences) U . S . S . R . N o . 2 , 1947 .

3. L . A . A R K I N , Collection of Works on the Regulation of Steam Turbines, ed. B . M . Y a k u b

and A.B.Shcheg lyayev . O N T I , 1 9 3 6 .

4 . I . I . A G R O S K I N , G . T. D M I T R I Y E V and F . I . P I K A L O V , Hydraulics. Gosenergoizdat, 1 9 5 0 .

5. V . A . L E S H C H E N K O , Stanki i Instrumenty, N o . 3 , 1952 .

6. G . P . V O V K , Experimental investigation of labyrinth seals. Dissertation. Stankin, 1 9 4 6 .

Then, in this case

Thus, the discharge coefficient is determined by the hydraulic loss coefficient.

The changes in discharge coefficient can obviously be obtained using

equation (5) from the data given in Fig. 3. Curves of μ as a function of the

opening are shown in Fig. 5. From these curves, it follows that with an in-

crease in opening, the discharge coefficient approaches a constant value

μ = 0-57. The figures are for the mineral oil "Turbine L " .

NETWORK ANALYSIS APPLIED TO HYDRAULIC CONTROL SYSTEMS

L . A . Z A L M A N Z O N

1. I N T R O D U C T I O N

In automatic control systems we often find complicated hydraulic or pneu-

matic networks, with local resistances in the form of restrictors, adjustable

throttles, valves, etc.

As an example of a control device with a complex system of hydraulic re-

sistances, one may consider the controller of the Stromberg injection-car-

burettor for aircraft engines (Fig. 1).

Diaphragm regulator

Fuel tö VzzM. the nozzle jj^Jlffl

1

Fuel from the pump

3 4 0

F I G . 1.

A N A L Y S I S A P P L I E D T O H Y D R A U L I C C O N T R O L S Y S T E M S 341

2. P R E L I M I N A R Y D E F I N I T I O N S

A calibrated orifice placed in a channel of much larger cross-section is called a restrictor.j As the ratio of fluid velocity in the channel and in the restrictor is small, the pressure drop p0 — ργ is due only to the resistance of the restrictor.

Usually, the discharge rate can be calculated by considering local resis-tances as simple restrictors. A plurality of restrictors, arranged in a definite pattern, is called "a system of elementary hydraulic resistances", or, briefly, "a network" or system. The system shown in Fig.2,c is called a series net-work, and that in Fig.2,ba parallel one. A mixed system has restrictors connected both in series and in parallel. In a mixed system, several restrictors connected in series are called a series set, and those in parallel a parallel set. A mixed system, consisting of a series connexion of any number of parallel sets, is called an elementary network or system (Fig.2,d). Series and parallel systems are particular cases of elementary systems.

A restrictor which can replace the whole system without altering its dis-

f The term "jet", as in a carburettor, is not suitable for more general use, in view of

possible confusion with "jet" meaning a narrow stream (Translator) .

When investigating control processes, it is often necessary to calculate the discharge rate through a hydraulic system of this type. The main problems which arise are the influence of a particular element on the discharge rate through the complete system, and the calculation of the pressure drop across a given part of the system, or calculating the pressure at a particular point.

There are many references on fluid flow through distinct restrictions, but to date, to the author's knowledge, nothing has been published on the subject of flow and pressure analysis in complete systems. References on the cal-culation of complex water supply systems (see, for example, Ref. 1) are con-cerned with specific problems of water pipelines. A similar remark would apply to references dealing with hydraulic-electric analogies. Problems con-nected with branched hydraulic lines in automatic control systems are somewhat different in character. In this case, it is essential not only to de-velop a method, but also to derive general formulae in order to simplify the solution of particular problems.

With these aims in mind, we shall investigate the turbulent flow of a homo-geneous incompressible liquid for a widely used type of complex hydraulic system.

As a result of this investigation, general formulae for equivalent passage areas and for pressures are derived. The notation at first appears to be cum-bersome, but the practical applications of these formulae are straightforward, and certainly simplify the calculations substantially.


charge rate (at a given initial and final pressures) will be called an equivalent

restrictor, and its passage area an equivalent area of a system. Equivalent areas for sets and combinations of sets can be defined in a similar way.

Pa (d)

F I G . 2.

3. R E D U C T I O N O F A M I X E D S Y S T E M

T O E L E M E N T A R Y S Y S T E M S

In the present article only parallel and series systems are considered. Such systems can be reduced to elementary ones by replacement of their sets, or combinations of sets, by equivalent restrictors.

An elementary system, obtained from a given system by the minimum of replacements of its sets (or set combinations) by equivalent restrictors, will be called a first group of the given system. Considering each combination of sets, replaced by an equivalent restrictor when forming a first group, as an independent system, and applying to it the same reasoning, it is possible to reduce each combination, in turn, to an elementary system, which will be called a second group of the given initial system.


F I G . 3.

relative to a given element or point is the serial number of the highest group, which includes the given element or point. The degree of a system relatively to a given element is lower than the overall degree of the system, or equal to it.

12 Aizerman I

In an analogous way, third groups, etc., may be obtained. Groups with a higher serial number will be termed higher, relative to those with a lower serial number. It may happen that in forming a first group, one or several set combinations replaced by equivalent restrictors are elementary systems. Then they are also second groups having no equivalent restrictors, and not subject to further transformations. Any further operations will be performed only with other set combinations (if there are any). The same may pertain to the formation of second or third groups, etc.

The serial number of the highest group, obtained by this breakdown of a given system into a number of elementary systems (or reduction to elemen-tary systems), will be called the degree of a system. The degree of a system


The principle of reduction to groups, and the définitions connected with it, are explained on the example, shown in Fig. 3. Replacing the parts A and B,

enclosed by broken lines, by equivalent restrictors, we obtain the first group. Further, as the part A is an elementary system (a series one, in this particular case), it is also one of the second groups of the system. Considering now part Β as an independent system, and replacing the part C, enclosed by broken lines by an equivalent restrictor, we obtain another second group.

The part of the system B, which is replaced by the equivalent restrictor C, is an elementary system, and therefore is already a third group, without any further transformations.

Thus the system is divided into one first group (obviously any system can incorporate only one first group), two second groups and one third group. Accordingly, it is a third degree system. Relative to the element xl — x2, or to any of the points xt and x2, it is also a third degree system, because these points are contained in the third group. But relative to points y or ζ it is a second degree system.

4. N O T A T I O N S

Restrictors of a series system are denoted by their number in the system, that is by 1, 2 , . . . / , . . . η (Fig .2 ,c) . Likewise, restrictors of a parallel system are denoted 1, 2 , . . . 7 , . . . m (Fig .2 ,b) . Restrictors of an elementary system (Fig .2 ,d) are denoted by two indices i,j, where / is the number of the parallel set incorporating a given restrictor and j is the number of the restrictor within that set.

Each group is defined by its number q in an r-th row of groups, e.g. the third from the left in a row of second groups. As each group is an elementary system, any restrictor within a q group in a row of groups r can be identified as 1,7, qr.

Similarly, any restrictor—equivalent or primary—in the r — k group will be identified as z',7, qr_k. For restrictors of the first group it is sufficient to write z, 7, as there is only one first group.

The equivalent area of the whole system is denoted by F, and by F f that of the z-th parallel set of an elementary system. The equivalent area of a por-tion of a series system, which includes restrictors (or parallel sets in an ele-mentary system) with numbers from c to k, is denoted by Fc+k.

The equivalent area of the first group of any system is, obviously, the equi-valent area of the whole system (denoted by F). For the equivalent area of any other group, the index is that of the equivalent restrictor, replacing this group in its nearest lower group. For example, the area of the z, 7, qr_2 equivalent restrictor of (r — 2 ) group, which represents in it the (r — 1) group, would be denoted as FitJtQr 2.

The pressure after the z-th restrictor in a series system (or after an z-th parallel set in an elementary system) will be denoted as /?,·, and the pressure


Comparing the latter equation with equation (2), we obtain

before an /-th restrictor in a series system (or before an /-th parallel set in an

elementary system) as pi_i.

The suffix for the pressure after any restrictor of a mixed system is the

same as the suffix for that restrictor. It is unnecessary, however, to indicate

the position of a restrictor within a parallel set, because in that case the pres-

sure is the same for all restrictors of the set. Therefore, the pressure after the

restrictor /, j, qr will be denoted as pi9 qr9 and the pressure before this re-

strictor as Pi-\,qr>

5. C A L C U L A T I O N O F E Q U I V A L E N T A R E A S

The discharge rate through a restrictor is

ρ = μΡ J^L(Po-Pl)^ (D where Ο is the flow per second, p0 and ργ the pressures before and after the

restrictor, F the area of the restriction aperture, γ the specific weight of the

fluid, g the acceleration due to gravity, and μ the discharge coefficient.

In all the following operations, it must be understood that F stands for

μΡ\ this applies also to the equivalent areas. Formulae for the equivalent

area of a series system can be derived from equation (1) applied to every

restrictor. As a result, we shall have η equations:

Pi-1 -Pi = ^-^r (f= 1 * 2 , . . . , « ) . (2)

2g Ft

Adding these equations, we obtain

Po - Pn = Σ 4 τ · ( 2 ' ) Ig i=i Fi

On the other hand, from the definition of the equivalent area, we have

η η γΡ 1

Po - Pn= — " Γ · 2e F

2

(3)


The equivalent area of a parallel system is equal to the sum of all m restrictor

areas, that is:

(4) J=i

Accordingly, the equivalent area of an i-th parallel set of mt restrictors

(Fig.2,d) is:

( 4 ' )

If we replace all parallel sets of an elementary system by equivalent restric-

tors, we shall obtain a series system. Therefore the equivalent area of an

elementary system can be obtained from equation (3), with equivalent areas

obtained from (4), i.e., in complete form:

F = 1

(5)

Henceforth, when applying formula (5), we shall write simply m instead of

mi9 not forgetting, however, that for each parallel set we must take its own

number of restrictors m.

Having derived the formula for the equivalent area of an elementary sys-

tem, and from the method of reducing a complex system to elementary ones,

it is not difficult to find the equivalent area of a complex system.

Proceeding from higher groups to lower ones, and substituting equivalent

areas for each higher group into a respective lower one, we can obtain the

equivalent area of a complex system. For a system of r-th degree we require

to perform the operation not more than r times:

1

l ϊ Γ

r te) J Let us illustrate this by a real example.

6. C A L C U L A T I O N O F T H E F L O W C H A R A C T E R I S T I C S

O F T H E S T R O M B E R G C A R B U R E T T O R ( E X A M P L E )

Referring to the control unit of this carburettor (Fig. 1), we shall consider the metering unit, connected to fuel chambers D and C of the diaphragm controller.


The diaphragm controller regulates the pressures before and after the

metering unit, depending on the pressure difference between air chambers A and B, which is a function of the air consumption.

The main elements of the metering unit are: idling needle ( / ) (controllable

restrictor), connected with an air throttle; the jet for adjustment of weak

mixture ( / / ) ; the take-off jet ( / / / ) ; jet (IV) for additional fuel, to "enrich" the

mixture to its normal composition; and an economizer (V).

F I G . 4.

Representing these metering elements as simple restrictors, each marked by the same roman numeral as the actual element in Fig. 1, we obtain a hydraulic net, shown in the upper part of Fig. 4.

For development and maintenance of a carburettor, it is important to know its adjustment characteristics—that is, curves of fuel discharge, as functions of the passage area of one or other metering elements. The ex-perimental determination of such characteristics is very laborious.

These characteristics can be calculated, using the diagram of Fig. 4. Pres-sures pD and pc (before and after the metering unit) are known, as well as the passage areas of all the five elements of adjustment: restrictors (F , , Fn, Flu,

Fiy and Fy).

Let us determine the equivalent area F of the metering unit, which is a system of the second degree. Replacing the combinations of restrictors I I I , IV and V by equivalent restrictors, we obtain the first group and one second group; both are shown in the lower part of Fig.4.

In accordance with the notation of Section 4,


Applying formula (5), we have the equivalent area of the second group:

1 ^ 2 , 2 =

L Γ

1 , 1 ,12

1

( F 2 > 1, l 2 + F 2 , 2 , l 2)2_

and the equivalent area of the whole system :

1 F =

V ι

_ r

1,1 +

1

(F2A + F2,2y

Substituting into the last equation the value of F22 from the previous one,

and returning to the original notation of roman numerals, we have the

following formula for the equivalent area of the metering unit:

F = 1

(6)

F] +

1 +

1

(FIY + F v )2

The mass flow rate of fuel, in kg/hr, will be

GT = 3600Fj[2gy(pD - P c ) ]

(all other units are expressed in kg cm sec). Substituting the area F obtained

by means of equation (6), and taking the area of a given restrictor as a vari-

Ό 10 20 30 W Effective passage a rea of economizer valve Fy,mm

2

F I G . 5.


able, we can obtain the respective adjustment characteristics. For example,

if F, = 78 mm2, Fn = 25 mm

2, Ful = 14 mm

2, Fiy = 5 mm

2, pD — pc

= 1200 mm of water, and γ = 0-73 g/cm3, we obtain the characteristic for

the variable F v shown in Fig. 5.

7. P R E S S U R E S I N A N E L E M E N T A R Y S Y S T E M

A series system, as shown in Fig. 2,c, is a convenient starting point. Let us de-

termine the pressure droppc_ x — pk for a part of system which includes restric-

tors numbered from c to k. The initial pressurep0 and final onepn are given.

According to (2), we have

2g

Dividing equation (2") by (2 ' ) , we obtain

Λ - . - Α - ^ - Σ τ τ · (2") 2g i=c F(

K 1

y — Pk t=c Fi

and consequently

Po ~ Pn y J _

K 1

Σ — t-c F

2

Pc-i — Pk = (Po - Pn) ——-1- · (7)

y — l \ Ff

Formula (7) can be written in a more convenient form, using the equivalent area for a series system, according to formula (3 ) :

( 8 )

Substituting these into (7), we have

F2

Pc-l - Pk = (Po - Pn) — - (9) F c + k

It follows that the pressure drop in any part of a series system is inversely proportional to the square of the equivalent area of this part.

Formula (7) , or its variation (9), can be used to determine the pressure drop in any part of a series system. In the particular case c = 1, formula (9)


Po - Pk = (Po - Pn) —2— ^ 1 +k

Pk = Po - (Po - Pn) —2— · (10) F 1+k

According to formula (10), the pressure after a certain restrictor of a series

system is equal to the initial pressure minus the pressure drop in the whole

system, multiplied by the square of the ratio of overall effective area to the

effective area of the portion, from the first restrictor to the given one.

Considering now an elementary system (Fig. 2,d), if all its parallel sets were

replaced by equivalent restrictors we would have a series system. Therefore,

formulae (9) and (10) are applicable to any elementary system, with sub-

stitution of equivalent areas, to replace parallel sets.

becomes

and

8. P R E S S U R E S I N A M I X E D S Y S T E M

Our aim here is to obtain equations for the pressure difference between any

two points belonging to the same r-th group of a system, for example be-

tween parallel sets from c, qr to k, qr.

Every higher group is included in its nearest lower group as an equivalent

restrictor. Therefore, the difference of pressures before and after a parallel

set of a lower group, which includes the said equivalent restrictor, will be the

full pressure difference for a given higher group.

Using formula (9), we can determine the pressure drop in each parallel set

(which includes the points we are concerned with) as a function of the full

pressure drop in a given group:

A - i . f I- Α . · , = ( A - i . . , - . - A , . , - , ) F2%i?-X

( ' = U , . . . , r ) . ( 9( / )

)

In the first of these equations

Pi-\,qt-X — Pi,q,-i =

Po — Pn> F

i j t Q l_ l = Γ

Replacing in each of the subsequent equations the full pressure drop

Pi-i,qt-i — Pi,qi-i by t r ie

expression from the preceding equation, we obtain

for the /, qr-th parallel set of the r-th group:

pl . p2 p

2 p

2

Pi-l.Qr - Pi,qr - Pn) 2 2 — — 2 —jT2 ·

As the pressure differences between the parts of an elementary system are

inversely proportional to the squares of the equivalent areas of these parts,


Pi-i.tr-* = Po - (Po - Pn) x

F2 F

2 · Ffj F2 · Ffj · F

+ —, ; + 2 ÎJ.Q2

Fi + u-D F l - F l Q 2 + i i_ 1 ) Q2 Fi - FiQ2 - Fl ,„3 + (i-d,03 p2 . p2 r-2 m p2 I Γ j j ··· Γ j j t q r _ 3 Γ ij Q r2

+ F

2 - F

2 F

2 · F

2

ri r

i,q2 "' r l,0r_2

Γ i,qr-i + U- l),qr-i

The required pressure after the k, qr-th parallel set in the i,y, <7r_!-th group (of r groups), according to (10) is:

f2 .

Pk,qr = Pi-l.Qr-i — (Pi-\,qr_x — Pi,qr-X) 2' 5 ^ Uqr + k,qr

12 a Aizerman I

the formula quoted above may be re-written in the form:

Pc-u<r - ft.., = (Po - Pn) " ^ "' · (ID

* i Γ i,q2 * i,ql * c,qr + k,qr

Formula (11) can be used to find the pressure drop in any part of a mixed

system.

T o find the pressure drop in a part of an r-th group, the overall pressure

drop of the system must be multiplied by the square of the equivalent area of

all the lower groups, including the r-th group itself, and divided by the square

of the equivalent area of the parallel set of every lower group (which contain

the r-th group), and by the square of the equivalent area of the part of the

r-th group under consideration.

Let us now derive the formula for the absolute pressure at any point of a

mixed system. The unknown quantity will be the pressure after the k-th

parallel set in the r-th group.

Singling out all these lower groups that contain the r-th group, we write

for each / of these groups the equation for pressure before the /, qrth

parallel set, which contains in one of its equivalent restrictors the r-th group.

As the pressure before the /, qrth parallel set is also the pressure after the

i — l , <7/-th parallel set, we have, according to formula (10):

Pi-Lq, = Pi-Uq^ - (Pi-Uqt-i - J2^

91-' , (10

(° )

1 , β | + ( ί -

where / = 1, 2, r — 1.

Proceeding from higher groups to lower ones, and with step-by-step re-

placement in the equation for each /-th group for pressure pi Uqi_l by its

expression according to equation ( 1 0( / _ 1 )

) , and / ? , _ i , < ? , _ , by its expression

from the equation ( Î ^ - d ) , we obtain the following expression for the pres-

sure Pi_u Qr_x :


χ I fl

ι F 2

'F f

>j

\ V2 F

2 · F

2

F2F.-F

F ι ' Fi,Q2 ' FltQi + a- l ) Qi

F2 ' Ffj ··· Ffj9Qr_3 · Ffj,qr__2 F

2 · F

2j-" Flj,qr_2 · F

2

t _|_ \2± i,j,Hr — 3 ' . J ' 4 r - 2 _

Fi 'F i t Q lF i t Q r_ 2 - ir

l f i r_ 1 + ( /_ i),qr_t Ft - Fiq2 ··· FitQr_l · F l t q r + k t Qt

(12)

Formula (12) enables us to find the pressure in any point of the mixed sys-

tem. It will be discussed in detail.

T o determine the pressure pk9Qr after a certain restrictor (or a parallel set),

incorporated into an r-th group of a mixed system, we must subtract from

the inlet pressure of the system the product of the overall pressure drop

(caused by the whole system), and the sum of r terms, which appear as frac-

tions. Both the numerator and denominator of each fraction are products of

factors the number of which is equal to the serial number of the term (frac-

tion). The numerator of each fraction is equal to the product of the squares

of equivalent areas of all the groups containing the given restrictor (from the

first to the group having a serial number equal to that of the fraction). For

all terms of the sum except the last one, one term of the denominator is the

square of the equivalent area of the groups which partly contain all parallel

sets from the first to the set which includes all the higher groups with the

given restrictor. In the last term of the sum this factor of the denominator is

replaced by the square of the equivalent area of the part of the r-th group,

which contains all parallel sets from the first to the set after which the pres-

sure is to be determined. Each of the other factors of the denominator is the

square of the effective area of a parallel set (of a certain group) containing

all higher groups with the given restrictor. In every term of the sum, except

the first, the above factors are introduced, representing all groups beginning

with the first one and ending with a serial number equal to that of the term,

minus one. In the first term of the sum, this factor is 1.

9. C A L C U L A T I O N O F P R E S S U R E S I N T H E M E T E R I N G

U N I T O F T H E S T R O M B E R G C A R B U R E T T O R ( E X A M P L E )

As can be seen from Fig. 1, the economizer Fis designed as a poppet valve.

T o test this valve separately from the rest of the metering unit, it is necessary

to know the pressure difference which causes the flow of fuel through this

valve when the carburettor is working.

or, substituting instead of pi_ltqr_i its expression obtained above, and re-

placing (pi_ltQr_l - pitQr_x) according to (11),

Pk,qr = Po - (Po - Pn) x

2 Γ2 . Γ2 Γ 2 Γ

2 . F

2


For example, it may be required to calculate this pressure difference at a

certain working condition, for which we know all passage areas FL9 FN, FLU,

F I V and FY, inlet pressure pD and outlet pressure pc.

Let us divide this system into groups, in the same way as in Section 6

(see the diagrams in the lower part of Fig. 4), and denote the passage areas

à 1000, Ν

0 10 20 30 W Effective passage area of economizer valve fy,mm

2

F I G . 6.

by FUL; F2TL; F l e l. l 2; F 2 t l e l 2; F2T2TL2 and pressures p0, ρί9 ρίΛ29ρ2, in-

stead of pD, pf9 pH, pc. From equation (11) we find the pressure difference:

Pi.ii - Pi = (Po - Pi) F2'FJ2

F22'F2

2,L2

( H O

Substituting into (11) the equivalent area determined by formulae (4), (5)

and 6 and re-introducing the original notation of Fig. 1 (or upper part of

Fig. 4), we obtain the following expression for the pressure difference at the

economizer valve :

Ph - Pc = (PD - Pc) 1

1 + ι

_F2

m ( F , v + FyyjJ

+ FV]2

+ /Γ ι , ι i l [FÎU + (FIV + Fvy~\lFlv

v U n i C i v + ^v) 2JJ (13)


If we required to calculate the absolute values of the pressures p1 and piti2

(or what amounts to the same, but in the original notation, pf and pn) in order to solve some other problems, equations (12) would be used. Then:

In this way, the pressure at any point of the metering unit can be deter-mined.

In conclusion, the results of numerical calculations may be illustrated by a curve of the pressure difference at the economizer valve (Fig. 6). The para-meters of the carburettor were the same as in Section 6.

1 ,1

(12')

Pu\2 = Po (12")

R E F E R E N C E

1. B . A . B A K H M E T E V , On the Uniform Motion of Fluid in Channels and Tubes. K U B U C h ,

Leningrad, 1929 .

DIAGRAMS FOR PARAMETERS OF STEADY-STATE AIR FLOW THROUGH

SYSTEMS OF ORIFICES I N PNEUMATIC CONTROLLERS

L . A . Z A L M A N Z O N

P N E U M A T I C controllers usually incorporate systems of calibrated restrictors separated by chambers of much larger cross-section. Such apertures will be called here restrictors. When designing pneumatic controllers, follow-up devices and other pneumatic devices (clamping fixtures, pneumatic tools, etc.) it is necessary to determine the pressure in chambers separated by re-strictors, and the flow of air through systems of restrictors.

Such calculations, for steady-state processes, do not present any funda-mental difficulties, as the determination of the pressure drop across a single restrictor has been thoroughly studied and treated in detail in recent litera-ture (see, for example, Refs. 1-3). Nevertheless, the application of these known methods is very laborious, because it is not known which flow con-dition—sub-critical or trans-critical—exists in each restrictor until the pres-sures in the chambers are determined, and consequently it is not known which of the basic equations are applicable. Also, the calculations are cumbersome since the equations for sub-critical conditions include members with fractio-nal exponents.

Problems of gas-flow through systems of restrictors have been investigated in several works (for example, Ref. 4). The importance of Ref. 4 is, however, much reduced by not considering the basic case of two different restrictors in series, the scope being narrowed to the series system of equal restrictors. Solutions for that particular case are given in a form not sufficiently con-venient for practical use.

Diagrams are given below for the more usual systems of restrictors. These may be used for the determination of the flow conditions for air (or any di-atomic gas) in a particular restrictor, and to find the pressures in inter-mediate chambers. These diagrams also simplify the calculations for more complicated systems.

In Section 1 are given the equations on the basis of which the diagrams were plotted. Readers interested only in the solution of definite practical prob-

355


lems may limit themselves to Sections 2, 3, and 4. In the process of con-structing these diagrams, certain fundamental problems of gas-flow through orifices are discussed, mainly in footnotes.

1. B A S I C E Q U A T I O N S

Equations used to plot the diagrams are derived from the equality of mass flow of air through all restrictors of a series system; steady-state flow is con-sidered.

It has been assumed that the flow through each restrictor is adiabatic, and the changes of gas state from one intermediate chamber to another are iso-thermal.f

The weight flow of air through a restrictor, at sub-critical conditions, is:

G =/,· I(IgY^P^ - ± - [rVk - rrUlk]) (D \ ' \ A: — I /

and for trans-critical flow:

The transition from the sub-critical to the trans-critical domain is indicated when ricrit = 2\{k + l )

f c/

( f c _ 1 ). In the above equations, / = index, denoting

a particular restrictor; / V i and yt_x = absolute pressure, and density of air, in the chamber before the restrictor; P f = absolute pressure at the exit from the restrictor, assumed to be equal to that in the chamber after the restrictor;

t The latter follows from the reason that during an adiabatic process, if the kinetic

energy of a gas is constant, its enthalpy is the same at the beginning and at the end of ex-

pansion. For an ideal gas, the invariability of temperatures follows. For real gases, the

temperature may change at a constant heat content, but this variation is so small that it

may be neglected for our purposes. If we proceed from the assumption that real gases follow

the Van der Waal s equation, then it is possible to derive from the differential equation for

expansion (after certain simplifications) the expression which gives the relationship be-

tween changes of pressure and of temperature. Numerical examples, at the initial tempera-

ture Τ corresponding to standard atmospheric conditions give the following changes of

temperature,expanding over 1 k g / c m2: for oxygen — 0-31 ° C ; for hydrogen -f-0-02°C; for

air —0-275°C; it is somewhat larger for multi-atom gases, being —0-77°C for carbon di-

oxide [1-3] .

The assumption of isothermal conditions in the intermediate chamber is based on the

fact that gas velocities in chambers are negligibly small when compared with the velocities

in the restrictors. The question of the limiting values for the local restriction coefficients in

pipelines, at which the initial and final gas conditions during expansion can still be assumed

isothermal, is discussed in Ref. 6.

D I A G R A M S F O R P A R A M E T E R S O F S T E A D Y - S T A T E A I R F L O W 357

rt = Λ/Λ-ι5 & = r a t

i ° ° f specific heats at constant pressure and constant volume; g = gravity acceleration;/) = passage area of the restrictor.|

For the system of two restrictors with different areas in series, with the pressure as indicated in the upper diagram of Fig. 1, we obtain from (1) and (2), taking into account γχ = γ0Ρι/Ρο = 7o

ri > the following basic equations

for sub-critical flow in both restrictors:

f \ 2 2/fc „ ( f c + D / f c

JU_\ Jj fx = r2,k _ rU + l ) / f c ( 3)

For trans-critical flow in restrictor 7, and sub-critical in restrictor 2:

r * * - r *+ i i lk

k — \ ( 2 V ' " - " \f2

2

r\ k + I \k + I J ίρ£

For trans-critical flow in restrictor 2, and sub-critical in restrictor 1 :

r m _ r( » + „ / . k — \ ( 2 \2 , ( k

-l ) 1

(4)

t "Passage area" ft must be understood as effective passage area, equal to the product

of geometric area of aperture and the discharge coefficient, which accounts for hydraulic

losses. The discharge coefficient, and consequently the effective area, can be considered

virtually constant, as for liquids, when Reynolds number R e > 3000. ( F o r throttling of

liquids see Ref. 5.) For gases it is convenient to express R e as the function of weight flow

G. A s the viscosity coefficient of gases does not depend upon pressure, but only on tem-

perature, it is possible to transform the usual expression for R e into R e = GI0-7$5gdß,

where G = weight flow, kg/sec ; d = diameter of restrictor, cm ; g = gravity acceleration ;

μ—viscosity. Substituting the numerical values for air at 20°C, R e = 7-44 χ 106G/d. In

order to ensure R e > 3000, it is necessary to satisfy the condition G > 0-4 χ \0~3d. A s a

rule, flow in restrictors of pneumatic controllers is at R e > 3000, and the problem of dis-

charge coefficient variability does not arise. F o r example, the system of restrictors quoted

in Section 4 has in all cases R e > 20,000.

For trans-critical flow in both restrictors :

(5)

(6)

In the case of η equal restrictors in series, at sub-critical flow conditions, the basic system of equations, used for plotting the diagrams, is:

(7)

where, in analogy with notations used for two-restrictor systems,


and

Pi-

In order to be able to determine the pressure distribution in a series system also in trans-critical conditions (at trans-critical flow out of its outlet restric-tor), for the region of sub-critical flow, auxiliary curves were built, using the equation :

2/k

Po)

(k+l)/k (8)

and replacing PijpQ by its expression through ρη/ρ0, obtained from equa-tions (7). In equation (8), fe = passage area of a restrictor, equivalent to the system in question;/ = passage area of every restrictor of the system; Poandpi = pressures before and after the first restrictor ;/?„ = pressure at the exit, after the n-th restrictor. Parts of auxiliary characteristics, corresponding to such values of pn/p0 when the equivalent restrictor has a trans-critical flow, and all restrictors of the initial system when sub-critical, were calculated by equation!

f_

fe

k + 1 \k + 1

2 < k - 1)

2/k ( * + 1 ) / * (8a)

For all conditions of flow—sub-critical as well as trans-critical—it is con-venient to calculate the mass flow of air by the same equation in both cases:

G = 9K/ ( V t ^ - i / ^ - i O - O ] (9)

introducing a suitable correcting coefficient 3R. In equation (9) G = weight flow of air , /) = passage area of a restrictor, y a n d pt_x = specific weight and pressure of air before a restrictor, rt = ratio of pressures before and after a restrictor.

A t the sub-critical flow :

" - ^ ( ϊ τ τ ^ - τ ΐ τ - ) · ( 1 0 )

t Equations (8) and (8a) are derived from the condition of equal mass flow through the

first restrictor of a system, and through an equivalent to that system restrictor.


k + ι k + 1 (10a)

All diagrams are constructed for k = 1-4. A t this value of k, the critical

pressure ratio r c r it = 0-53.

2. D E S C R I P T I O N O F D I A G R A M S

(a) Diagram for Determining the Conditions of Flow in the Series System of Two Restrictors with Different Passage Areas. A series system is shown sche-matically in the upper part of Fig. 1. On the diagram, abscissae represent the

3.

Pa 0-8

0-6

01,

0-2

0

I b e

'a m

1m

Ic d

I 3 fx F I G . 1. Regions of flow conditions: /—sub-critical for both restrictors; / /—trans-

critical for 7, and sub-critical for 2; ///—trans-critical for 2, and sub-critical for 1 ;

IV— trans-critical for both restrictors.

and at trans-critical flow:


ratios of passage areas / x to f2 (discharge coefficient having been taken into consideration). Ordinates represent the ratios of outlet pressure p2 to inlet pressure p0. The whole field of the diagram is divided into four regions, cor-responding to various combinations of flow conditions in restrictors 1 and 2.

r-Λ to 09 08 07 06 05

r

ό

KL —• _ ^

—o GL — —o

0-5 M OJOfîO <0f 03 04 0*5 Oti 07 F i g . 2.

Οδ

El

If the point, determined by given fxjf2 and p2IPo>is in the region / , then the flow in both restrictors is sub-critical; if in the region 77, trans-critical in restrictor / , and sub-critical in 2; if in the region / / / , sub-critical in 1 and trans-critical in 2, and if in region IV, trans-critical in both.f The diagram is limited by fx\f2 = 3; with further increase offxjf2 the curve which divides regions I and III approaches to a horizontal line p2/p0 = rcr = 0-53.

t The curve ab on Fig. 1 corresponds to equation (3) or (4) at rt = rctit, and accord-

ingly at rCTiir2 = i V ^ o i curve ae to equation (3) or (5) at r2 = rcrlt and accordingly at

fV'crit = PilPo\ curve ac corresponds to equation (6) , transformed into Ρ2ΙΡς> =

rifilfi*

substituting into this equation r2 = r c r i t. If we substitute into (3) r x = r c r i t, and r 2 = r c r It

g i v e s / i / / 2 = r c r i t. A t this value of / ι / / 2 , when we reach Ρ2/Ρ0 = rcr i t (point a on the dia-

gram) , the critical condition appears simultaneously in both restrictors. A t / i / / 2 > r c r it flow

through the restrictor 1 will be sub-critical at any Ρ 2 / Ρ 0 · This follows from equation (5) ,

which is converted into identity by substituting into it fY\f2 = r c r it and = rCTlt. W h e n

filfi > >'crit »the left-hand part of equation (5) is smaller than at / i / / 2 = r c r i t. A s

r2'k r(k+l)jk

diminishes with increase of ι\, at fx\f2 > rcrit only i\ > r c r it is possible.


04 0-8 1-2" 1-6 20

F I G . 3. Abscissae: (filfiMiPilPo) for region / / , and f2jfi—for region Ordinates:

r2 = Pi\P\ for region / / , and rx = Pilp0 for region / / / .

R\ = P\IPI- The curve of Fig.3 corresponds to equation (4) if the flow under consideration belongs to region II, and to equation (5) if it belongs to region 777.

When the flow is trans-critical in both restrictors (region IV of Fig. 1), we have rt = Pi/po — f\\fi> As p0 and p2 are given, in all cases pl becomes known when r x or r2 are found.

(b) Diagram for Determining Pressure px in the Intermediate Chamber of a System, Consisting of Two Restrictors in Series. Ratios of pressures p0 and p2, and passage areas/ x and f2, are known.

Figure 2 gives the dependence of rl = PjPo from P2/Po a

* various f±\f2

in the case of sub-critical flows in both restrictors (region 7 of Fig. 1). The curves of Fig. 2 are plotted according to equation (3). The straight

line on the left corresponds to the onset of critical flow in restrictor 2; the equation of this straight line is rx = ( l / r c r i t) (p2/p0), which follows from rtr2

= PilPo and r2 = r c r i t. The limiting straight line on the right, the equation of which is p2/p0 = r l 9 represents the limiting value r2 = 1, to which r2 ap-proaches at small f\/f2. On the top the curves are limited by the straight line ri = 1, to which the characteristics a* ! = f(p2jp0) approach, when/J/2 0 0 . The lower limiting straight line corresponds to rt = r c r i t, that is to the onset of critical condition in restrictor 7.

Figure 3 has two sets of scales. If it is required to find px at transcritical flow in restrictor 7, and at sub-critical in 2 (region 77 of Fig. 1), then abscissae mean (fi^liPilPo), and ordinates r2 = p2llpi ·

If it is required to find pl at sub-critical flow in restrictor 7, and trans-critical in 2 (region 777 of Fig. 1), then abscissae mean / 2 / / i , and ordinates

10

0-9

0-8

0-7

0-6


(c) Diagrams for Pressures in Intermediate Chambersfor Systems with Several

Identical Restrictors in Series, at Sub-critical Flow Conditions. The diagrams of Fig. 4 are constructed for systems with the number of restrictors from 2 to 7. On every diagram, the curves represent systems of equations (7), with a cer-tain n. Abscissae are the ratios of outlet pressure pn to inlet pressure p0, and ordinates ratios of pressure after each restrictor to pressure before it. For

02 Π 06 Od Μ P0 02 04 06 08 W P0

all ratios pn/p0 greater than the value marked by a vertical broken line on the left, the flow in all restrictors is sub-critical. When pn/p0 corresponds to the broken line, the flow becomes critical in the outlet («- th) restrictor of the system.


(d) Diagramsfor Determining the Passage Area of a Restrictor, Equivalent to

a System of Several Equal Restrictors in Series, at Sub-critical Flow in the Latter.

(This diagram is also used as an auxiliary one for finding pressures in inter-

mediate chambers of systems with equal restrictors in series at trans-critical

flows.)

Abscissae of Fig. 5 represent the ratio of pressure pn after the /2-th restrictor

to the pressure p0 at the inlet into the system. Ordinates are the ratios of

06 (H OU

b d

>!<-! a

I ι

I ι ' 1 ι ι ι ι 1 1

ι i I ι I ι

1 1 1 1 1 1

1 1 1 1 1 1 04 05 \ Οΰ crit

n=3

07 0-8 OS H}-F I G . 5.

the passage area of the equivalent restrictor to the a rea /o f each restrictor in a series system.

Figure 5 shows the dependence of fe/f from pjp0 for η = 2, 3, and 4 . | These curves determine the ratio fe/f at given η and pn/p0-

In addition to the above characteristics, the curve cd which is used for cal-culations of trans-critical flow conditions is also plotted. Trans-critical flow can occur only in the outlet restrictor of a system. From Fig. 4 it can be seen that, with the reduction of pnjp0, the critical condition arises first in the outlet restrictor. This also follows directly from equation (7 ) : as r{_i < 1 , then/*?

7* — r<*

+ 1>/

f c > r2 / * _ rJ*"J

1)/fc, to which corresponds rt < rt_x. There-

fore, on reduction of prJp0 the condition r — r c r il arises first in the outlet re-strictor. N o further decrease of pn/p0 can cause the critical flow in other restrictors (this might be expected to occur in the preceding restrictor, but according to Fig. 1, the trans-critical flow in two equal restrictors atfl\f1 = 1 is impossible).

For the determination of the pressures in intermediate chambers when the flow is trans-critical, different enumeration must be used; the total number of

t Range of pJPo, for which these curves are given, covers the whole region of sub-

critical flow for each system. Sections shown with solid lines correspond to equations (8)

and (7) , and those shown with broken lines to the trans-critical flow in the equivalent re-

strictor, calculated by equations (8a) and (7). The limiting points which lie on the curve ab

correspond to the onset of the critical condition in the w-th (that is the last one) restrictor

of a system.


W OS 06 04 02 0p[.

F I G . 6.

3. E X A M P L E S O F A P P L I C A T I O N O F T H E D I A G R A M S

Example 7. For the system as shown in the upper part of Fig. 1 are given:

ratio of passage areas (a) fjf2 = 0-2; (b)fljf2 = 2. It is required to find

PilPo at p2/p0 = 0-67, 0-30 and 0-05. (a) From Fig. 1 we find that for / , / / 2 = 0-2, at p2jp0 = 0-67 the flow is

sub-critical in both restrictors; at p2/p0 = 0-30 transcritical in restrictor 1 and sub-critical in 2; at p2lp0 = 0-05 trans-critical in both. Accordingly, we find from Fig.2 for fjf2 = 0-2, at p2lp0 = 0-67, Pilp0 = 0-687. At p2[p0

t The above method is based on replacing the initial system by an equivalent one, con-

sisting of two restrictors. The outlet restrictor of the new system is the last, (n + l)-th,

restrictor of the original one, and the rest is replaced by an equivalent restrictor. This

equivalent two-restrictor system can be dealt with by the aid of Fig. 3 (it must be noted

that and f2 of Fig. 3 are r e s p e c t i v e l y a n d / o f Fig. 5). Dependence between pjp0

and felf according to the case of sub-critical flow in 1 and trans-critical in 2 (region / / / )

is shown on Fig. 5 as curve cd. Intersection of this curve with each basic curve ( / e / / as

function of pjp0) gives for a respective η the value of p„lp0.

restrictors is taken to be η + 1, and the last but one restrictor, in which the flow is sub-critical, is considered as the «-th one. Accordingly, the pressurepn

is not the outlet pressure, but the pressure in the last intermediate chamber. Then at the intersection of the curve for the new η (which is the total number of restrictors, minus one) with the curve cd (Fig. 5) we find pn/p0 for the sys-tem. As p0 is given, this determines pn. The pressures in all other chambers can be found from Fig. 4 for the given « . |

(e) Diagram for the Coefficient $)l in Equation (9), for Calculations of Air Dis-

charge. This diagram gives the coefficient 9JÎ as a function of the ratio of pres-sure Pi after a restrictor to pressure pt_Y before it. Part of the curve, for Pi/Pi-i >

rc r i t > corresponds to equation (10), and the part for /?///?,·_ ι < rCTit

to equation (10a).

m

f-0

09

03

0-7

06

OS


= 0-30, the ratio (fJ^KPilPo) = 0-20/0-30 = 0-67; from Fig. 3 we find forifjfàKpi/po) = 0-67 andp2lpi = 0-903, that ispjpo = (PilPoViPilPi)

= 0-333. For p2[p0 = 0*05 we have, from equation (6), pjpo = fx\f2

= 0-2.

(b) From Fig. 1 we find that the flow is sub-critical in both restrictors at / i / / 2 = 2 and p2jp0 = 0-67, and trans-critical in 2, sub-critical in 1 at P2IP0 — 0*30 and 0*05. For p2/p0 = 0-67, we find from Fig. 2, pijpQ

= 0-95. From Fig. 3, a t / 2/ / i = 0-5 ( o r / ^ = 2) we find pjpo = 0-94. This value remains unchanged for the whole region of trans-critical flow in re-strictor 2, independently from p2/p0, and therefore applies to p2/p0 = 0-30, as well as 0*05.

Example 2. The system consists of four equal restrictors in series. It is required to determine the pressure distribution in the separate intermediate chambers at the following ratios of outlet/inlet pressures pjp0: (a) 0-46; (b) 0-20.

(a) From the respective Fig. 4 we find, at pjp0 = 0-46, Pijp0 = 0-90; p2/Pl = 0-88;/>3/ />2 = 0-83; p4/p3 = 0-70.

(b) The value p^/p0 — 0-20 lies outside the field of curves of Fig. 4. This means that the flow in the outlet restrictor is trans-critical. Referring to Fig. 5, at the intersection of curve cd with curve fejf = φ(ρη/ρο) for η = 3 (i.e. less by 1 than the number of restrictors), we find p3lp0 = 0-63. Accordingly P*IPZ = (pJPo)l(P3lPo) = 0-20/0-63 = 0-32. The pressure ratios in the other intermediate chambers can be found from Fig. 4, for η = 3: at p3jpQ = 0-63 we have px\p0 = 0-90; p2\px = 0-87; p3!p2 = 0-81.

Û (H 0* 0-3 (Ν* 0<δ M 07 &Ô (W H)P0

F I G . 7. Comparison of calculated and experimental resu l t s . / i / / 2 = 1-12. A t each

Po experiments conducted for p2 = 200, 270, 335, 460, 600 and 740 mm mercury.

O — A ) = 840mm mercury; Δ—950; •—1150; V—1350; #—1550; A—1750;

•—1950. Continuous line calculated.

The above examples include, in principle, all the possible cases of air flow in simple systems of restrictors in series. Figures 7 and 8 give results of cal-culations, compared with experimental results, for a system of restrictors shown in the upper part of Fig. 1. Numerical data are given with the dia-grams. Curves were obtained by calculations (using Figs. 1, 2 and 3), and the points experimentally.


*0 ι

0-9

07

06

06

Λ

ο I Λ

• Γ π

•

3 "

2 S -

• <*

f

6*

7 Β-

0ν? # 4 tfv5 W ^

F I G . 8. Comparison of calculated and experimental results; p2 = 0-27 k g / c m2;

/?o = 0-47; 0-57; 0-67; 0-87; 1-07; 1-27 and 1-47 kg /cm2, i — Λ / / 2 = 4-00; 2—2-24;

5—1-14; 4—1-00; 5—0-735; 6—0-591; 7—0-475.

4. T H E A P P L I C A T I O N O F D I A G R A M S

F O R M O R E C O M P L E X S Y S T E M S

Figures 1-6 can also assist in investigations of various more complex

combinations of restrictors. Let us consider some typical applications. As in

the preceding paragraphs, we shall assume that it is required to find the dis-

tribution of pressures, and discharge of air, while the inlet and outlet pres-

sures, and the areas of restrictors, are known.

We shall consider first the series systems—three or four restrictors with different passage areas; then a system of η equal restrictors, connected with m restrictors, which are equal to each other, but different from the first set of « , and a system of η equal restrictors connected with two others, differing between themselves, as well as from the first set of n.

Any of these systems can be divided into two sub-systems, each of them belonging to a type already discussed above. Obviously, the only new prob-lem is finding the pressure px in the chamber dividing these two sub-systems. For this purpose, we must take several arbitrary values of pi9 and find for each the respective air discharge G (using Fig. 6 to find the coefficient 9K) for both sub-systems. A graph can be drawn, with two curves showing the de-pendence of G from pt for each sub-system. The intersection of these curves gives the actual value of px and G. (The curves must necessarily intersect at a certain value of P i , because with the increase of arbitrary /?, the discharge G from the first sub-system diminishes or remains constant and for the second one increases.)

D I A G R A M S FOR P A R A M E T E R S OF S T E A D Y - S T A T E A I R F L O W 367

The above method can be applied also to more complex systems—for example, incorporating not two, but three sub-systems, either of the type "two different restrictors" or "n equal restrictors". Denoting the pressure in the chamber between the second and third sub-system by pj9 and taking for it several arbitrary values, we calculate separately G for the third sub-system and, using the graphical method, for the first two sub-systems. Then we draw another graph, and in the same way find the actual pj and G.

This technique can be extended to deal also with series-parallel systems, consisting of a series chain of several sets of restrictors in parallel. Each parallel set is replaced by an equivalent restrictor simply by adding the passage areas. Still more complex systems can be reduced to series-parallel type, replacing in several steps their parts by equivalent restrictors.

The following examples illustrate the applications of the above methods. Example 1. The system of four restrictors is shown on Fig.9,a (upper

part). Absolute pressure at the inlet p0 = 4 kg/cm2, at the outlet piy

= 0-3 kg/cm2. Diameters of restrictors are: dx = 1-68 mm; dn = 2-06 mm;

ta)

ι ι mm

/ 2

ι ' 2

(C)

Ibi 3-W

2Ί0

M0%

J y > ( M » — A

/ f ι

ι

ι

Id)

06 H? ft

F i g . 9.

dm = 3-02 mm; dlv = 4-71 mm. It is known that the discharge coefficient at Re > 3000 can be assumed, with a very small error, to be constant,// = 0-9. Air temperature is 15°C. It is required to find the flow conditions in each restrictor, the pressure in the intermediate chambers, and the air discharge through the system.


Effective passage areas at the above diameters and discharge coefficient

are:

/ , = 2-00 χ Ι Ο "2 c m

2, / „ = 3-00 χ 10~

2 c m

2,

/ „ , = 6-42 χ 1 0 -2c m

2, / I V = 15-65 χ 10"

2 c m

2.

The system is divided into two sub-systems A and B. For each of them, we introduce the same notation as used for Figs. 1, 2 and 3. We take the fol-lowing arbitrary values of pu: 0-6; 1-0; 1-4; 1-8 kg/cm

2.

For the sub-system A with pn = p2 the respective p2/Po are 0-15; 0-25; 0-35; 0-45. For these p2/p0 we find from the diagrams at fx\f2 = fxjfn

= 0-667, values of Pi/p0 = 0-642; 0-642; 0-642; 0-675, and respectively pv

= 2-57; 2-57; 2-57 and 2-70 kg/cm2. From equation (9) we find the discharge

GA through restrictor 1 ( / ) , substituting into this equation 9ft, determined for each pjpo from Fig. 6. Replacing in equation ( 9 ) ^ = γ0 byyatmp0lpatm9

where / ? a tm = 1 kg/cm2, and y a tm for 15 °C is 1-18 χ 10~

6 kg/cm

3, it is con-

venient to write equation (9) as

G = V (2

^ a t m ) / i » î J(l -y) = O 0 4 8 / Ä O J(l - yj.

For given values ofp 2 = pn and corresponding/^//^, we obtain 9JI = 0-775; 0-775; 0-775; 0-798, and respectively

GA = 1-78- Ι Ο "3; 1-78 χ 10~

3; 1-78 χ 10"

3; 1-75 χ 10"

3 kg/sec.

For the sub-system Β at fxjf2 = / m / / i v = 0-41, for the samep0 = pn = 0-6; 1-0; 1-4; 1-8 kg/cm

2, and p2/p0 = 0-5; 0-3; 0-214; 0-167 we find, in the

same way, Pllp0 = 0-58; 0-435; 0-41; 0-41, and SR = 0-732; 0-643; 0-630; 0-630. From equation (9) we calculate GB = 0-88 χ 10"

3; 1-49 χ 10"

3;

2-09 χ 10-3; 2-69 χ 10~

3 kg/sec.

Plotting the graphs for GA and GB as functions of pn, we obtain at the intersection of curves (Fig. 9,b)/?„ = 1-2 kg/cm

2 and G = 1-78 χ 10~

3 kg/sec.

Using this value of pu = 1-2 kg/cm2 and again considering separate sub-

systems A and B, we find from Figs. 1 and 3 that the flow in restrictors / and I Vis sub-critical and in / /and ///trans-critical, and obtain pY = 2-56 kg/cm

2

and pm = 0-49 kg/cm2.

Example 2. The system is shown in Fig. 9,c./?i = 2-0kg/cm2;/? lv = OTkg/cm

2,

temperature is 15°C, passage areas / , = 8-80 mm2, fu = 6-35 m m

2;

= 11-30 mm2, fiy = 25-40 mm

2, fw = 4-41 mm

2. It is required to find the

pressures in the intermediate chambers, and the flow through each restrictor.

The method is similar to that used for the first example. W e take several arbitrary values of pu (e.g. 0-6; 0-8; 1-0 kg/cm

2) . For each of them we find

the flow through restrictor / , through the sub-system of restrictors / / , / / /and IV (restrictors / / and / / / are considered as one restrictor with the area



1. Diagrams (Figs. 1, 2 and 3) completely determine the parameters for air (or any diatomic gas) flowing through a system of two restrictors in series as often used in pneumatic controllers and other devices of automatic control.

2. Such a system has the following properties: (a) Conditions of flow are determined by the ratio of restrictor areas f^f2,

and the pressure ratio p2jp0.

(b) When fxjf2 > r c r it (for air = 0-53), the flow in restrictor 7 cannot be trans-critical in any circumstances. Only sub-critical flow in both restrictors (region 7, Fig. 1), or trans-critical in 2 and subcritical in 7 (region 777) can occur, depending on the ratios.

When fx\f2 < rCTit, three combinations of flow conditions are possible: sub-critical in both, trans-critical in 7 and sub-critical in 2 (region 77), and trans-critical in both (region IV).

(c) For regions 7 and 77 the ratio pjp0 is determined by fxjf2 and p2lp0, (Figs.2 and 3), while for regions 777 and IV it depends only f r o m / i / / 2; for region 7Kthis is simply pjpo = filfi- The latter conclusion is important for control engineering, as it can be used for proportional pressure reduction.

3. For the system of η equal restrictors in series, pressures in intermediate chambers can be found, at sub-critical flow in all restrictors, with the aid of diagrams in Fig. 4, which cover η from 2 to 7. The limits of sub-critical flow are shown on these diagrams. As the trans-critical flow can exist only in the last restrictor, the pressure distribution in the chambers can be easily obtained also for trans-critical conditions. In that case, this can be done by intro-ducing the notion of equivalent restrictor area, and using also the additional diagram (Fig. 5), as well as Figs. 1 and 2. It is noteworthy that (as distinct from

= fn + / in = 17-65 m m2) and through restrictor V. The first curve of the

graph represents the flow through 7, and the second sum of flows through 77, 777, IV and V. The results of further calculations show that the flow in 7, IV

and V is trans-critical, and in 77 and 777 sub-critical. pu = 0-83 kg/cm2, pul

= 0-53 kg/cm2. Discharges through I, II, III and IV and V are, respectively,

4-71 χ Ι Ο "3; 1-19 χ 10~

3; 2-12 χ 10"

3; 3-31 χ 10"

3; 0-86 χ 10~

3 kg/sec.

Example 3. General method of calculation for the system Fig.9,d for which are known the areas of restrictors (all different), pressures pa and pb,

gas density.

We take several arbitrary values of pt. For each of them we calculate the flow through 7, through the sub-system 77, 777, IV, F (as in Example 1), and through VI, VII, VIII, IX, and X (as in Example 2). Having plotted a graph with a curve for flow through I, and another curve for flows through all other restrictors, we obtain at the intersection the actual /?,. Other quantities can be obtained in the same way as in Examples 1 and 2.


the case of incompressible liquids), the equivalent restrictor area depends on

the pressure ratio.

4. T o facilitate calculations, it is recommended that the same formula (9)

be used for sub-critical as well as for trans-critical conditions, by introducing

the correcting coefficient 9JÏ, which can be found from Fig. 6.

5. The method as developed can also be applied to more complex systems

of restrictors; this is illustrated by the examples of Section 4.

6. The results of calculation, using the diagrams, are well confirmed by ex-

periments. This can be seen from the examples at the end of Section 3 (Figs. 7

and 8).

R E F E R E N C E S

1. A . S . Y A S T R Z H E M B S K I I , Technical Thermodynamics, V o l . 2 , N o . 1. V V I A , 1947.

2 . A . M . L I T V I N , Technical Thermodynamics. Gosenergoizdat, 1947.

3. V . V . S U S H K O V , Technical Thermodynamics. Gosenergoizdat, 1946 .

4. R O B I N S O N , / . Appl Mech., V o l . 15 , N o . 4 , 1 9 4 8 .

5. L . A . Z A L M A N Z O N , Avtomatika i Telemekhanika, V o l . X I I , N o . 6 , 1 9 5 1 .

6. Ν . M . M A R K E V I C H , Trans. Leningrad Univ., N o . 17 , 1949 .

L A M I N A R FLOW OF AIR AT HIGH VELOCITIES I N FLAT CAPILLARY

CHANNELS

A . V . B O G A C H E V A

C A P I L L A R Y channels are used in many elements of pneumatic systems. In ad-

dition to capillaries of circular cross-section, channels of rectangular cross-

section are used, and amongst them so-called flat channels which have a

width much greater than their height. In the literature, generally, the data

are given for flat channels of comparatively large sections, or for circular sec-

tion capillaries at moderate flow velocities [1]. In the present work the results

of experimental investigation on the laminar flow of air in flat capillary

channels at great velocities are presented. The experimental data are treated

in accordance with the theory of models [2].

T H E T E S T R I G

The flow of air in flat capillaries was investigated by means of the test rig shown diagrammatically on Fig. 1. Air from the atmosphere enters through a silica-gel filter drier 1 into a flat channel 2 and is then exhausted by a va-cuum pump 5. The flow and pressures were adjusted by means of the flow-meter 4 and the valves 5 and 6. The pressures in the chambers 7 and 8, as well as the pressures at various points of the flat channel, were measured by a battery of mercury manometers 9.

A vacuum pump, and not a compressor, has been used for the test rig, as all tests were run at pressures below atmospheric. This completely excluded oil vapour, which is usually present in the air supplied by a compressor.

The flat channel has been constructed in a special detachable arrangement, which made it possible to obtain accurate finishes on the surfaces forming the channel, and the correct geometry for its inlet and outlet edges and the pressure tappings. The flat capillary channel was formed by the surfaces of steel plates. The plates were subjected to a natural ageing process during six months, and afterwards finally lapped and completed. The smoothness of the working surfaces corresponded to the class 12B GOST 2789-51, the micro-roughness, measured by an electrodynamic profilometer, not exceeding 003-0-04 μ.

371


The length of channel could be altered by changing the main plate 11 (see

Fig. 2). The height was adjusted by shims 12 made of thin sheet steel. For

measurements of static pressure at various points of the channel a number of

holes 0-35-0-45 mm diameter were drilled in the upper plate 13. The pressure

at any cross-section of the channel was assumed to be equal either to the

pressure on the axis of the channel, or was calculated as the mean of pres-

sures, measured at several points. The location of the pressure measurement

points is shown in Fig. 2.

A set of flowmeters had been calibrated with air from a measured volume.

The suction at the entry to a flowmeter, checked by a manometer 10 (see

Fig. 1), did not exceed 4 mm mercury.

F I G . 1. Arrangement of the test rig. 1—Silica-gel drier and filter; 2—flat channel;

3—vacuum pump; 4—flowmeter; 5, 6—throttle cocks; 7—inlet chamber; 8—

outlet chamber; 9—battery of manometers; 10—manometer.

All experiments were conducted without heat insulation of the device and with the temperature in the chamber 7 equal to the ambient temperature (T0 = 20-30°C). Air velocity in chambers 7 and 8 was small, as their cross-

L A M I N A R F L O W O F A I R A T H I G H V E L O C I T I E S 373

sectional area was more than a hundred times the cross-sectional area of the

channel.

In order to establish whether there is heat exchange between the air flowing

in the channel and the ambient medium, additional tests, with and without

F I G . 2. Construction of the flat channel. 11—main plate; 12—shims of thin sheet

steel; 13—upper plate; h—height of the channel; b—width of the channel;

/—length of the channel.

heat insulation, were conducted. Temperatures TQl (in chamber 7) and T02

(in chamber 8) were measured. The heat insulation consisted of a layer of asbestos 5 mm thick, then felt 20 mm thick, and again asbestos 5 mm thick. In both cases (with or without heat insulation), no difference be-tween temperatures T0i and T02 has been observed at low air velocities. This indicates the absence of heat exchange. A t sonic velocities of efflux from the channel, both at critical and trans-critical pressure differences, the tempera-ture T02 in chamber 8 was higher by 1*5-1-6°C than T0i in chamber 7. As this temperature rise was the same in both cases, it means that it was caused by irreversible losses and not by heat exchange. The temperatures T0l and T02 were measured by mercury thermometers of T L N type, with scale divi-sions 0T °C. Experiments carried out with and without heat insulation have shown no difference in air pressures measured at various points of the channel.


E X P E R I M E N T A L R E S U L T S

Experiments were carried out with channels having the length / = 0-5, 30, 70, 110 and 150 mm, and height h = 0*200 mm. In addition, tests were run with / = 30 mm and h = 0Ό95 mm, and / = 150 mm and h = 0-309 mm. The width of the channel b was in all cases 30 mm, with deviations not ex-ceeding 0-01 mm.

By adjusting the valves 5 and 6 (Fig. 1) the absolute pressures in cham-bers 7 and 8 were varied within following limits : in chamber 7 ( P 0 i ) from 750 to 100 mm mercury and in chamber 8 P02 from 750 to 20 mm mercury. The mass flow of air G was varied from 0-223 χ 1 0

- 5 to 0-106 χ 1 0

- 5 kg/sec.

T w o methods were adopted for the experiments. By the first method, a certain pressure P01 was established by adjusting valves 5 and 6, and the pressure P02 was regulated so as to maintain a constant (for a given series of tests) mass flow of air G. Figure 3 shows the relationship between the static pressure Ρ and distance χ (from the entry into the channel), obtained by

P, mm mercury :

20 W 60 SO 100 120 1V0 X,mm

F I G . 3. Distribution of static pressure Ρ = P(x) along the channel, at G = 14-32

x 1 05k g / s e c . o—l = 150 mm; %—l= 110mm; a—/ = 7 0 m m ; o — / = 30 mm.

this method. All curves shown on Fig. 3 were obtained at G = 14-32 χ 10~5 kg

/sec, for channels with / = 30, 70, 110, and 150 mm, at h = 0-200 mm. All the experimental points obtained for channels of different lengths, but cor-responding to the same value of P 0 l , lay on the same curve.

Figure 3 gives the results of one series of tests. Altogether twenty such series were conducted, each with its own G.

By the second method, the pressure P0l was again established by adjusting valves 5 and 6, but the pressure P02 was set at such a level that its further


reduction would not increase the mass flow G. The ratio of this limiting

value of P02 to the respective Pol will be denoted as β** = ( Λ ) 2 / Λ μ ) * * î t ne

corresponding mass flow G^ is then the maximum possible one for a given

pressure P0 {. Figure 4 shows the distribution of static pressure Ρ = P(x)

along the channel, obtained by experiments which were conducted according

P,mm mercury

F I G . 4. Distribution of static pressure Ρ = P(x) along the channel for G = G * * .

to the second method. The channels had / = 30; 70; 110 and 150 mm, and

h = 0*200 mm. These curves were obtained at various values of P 0 l, and

each of them corresponds to a definite value of mass flow G = G ^ ; but

on Fig. 4 instead of are given respective Reynolds numbers Re + 4.

= (6 + φ/ [#£μ(1 + h/b)], where g is the acceleration of gravity and μ the dy-

namic viscosity; the characteristic dimension was taken as twice the hydraulic

mean radius of the channel section.

It may be mentioned that curves J, 2, 3 and 4 in Fig. 3 were also obtained

in experiments, conducted by the second method, but at such values of Pox

that for them G = 14-32 χ 10"5 kg/sec =

T H E G E N E R A L I Z A T I O N O F E X P E R I M E N T A L D A T A

The Resistance Coefficient. Taking as determining criteria Re, λ, b/h, we can

find the resistance coefficient ζ as :

Î = i ( R e , A , A j , (1)

where ζ = {APj[(yw2I2g) (///?)] is the resistance coefficient, Re = G|[gbμx

χ (1 + hjb)], λ = wja^ the relative velocity; b the width of the channel; h the

13 Aizerman I


height of the channel; = yj(2kjk + 1) gRT0l the critical air velocity; w = Gjhby the mean air velocity; G the mass flow; γ = PjRTthe air density; k = Cp/Cv the adiabatic exponent; R the gas constant; T0i the temperature in the chamber 7 (see Fig. 1); Ρ the mean static pressure in a given section of channel; and Γ the air temperature in the channel.

Values of Re, λ9 bjh, and corresponding resistance coefficients ζ, were cal-culated from the experimental data, obtained with channels of various /. The magnitude of Δ Ρ was determined from an experimental curve Ρ = P(x), as the pressure difference at the limits of a given length (lengths Δχ of 1-3 mm

c.

F I G . 5. Relationship ζ = ζ(λ, R e ) for b\h = 150.

were taken). When considering curves Ρ = P(x), the entry and exit lengths of channel were excluded. An entry length has been assumed to be IH

= 0-045 Re A, in accordance with Ref. 3. An exit length was taken as several tenths of a millimetre. As the additional experiments have demonstrated that T0l = T02, the temperature at a given point in the channel was deter-mined as Τ = T0l(p/p0ly

k-

l)Ik.

Figure 5 shows the relationship ζ = ζ(λ, Re) for bjh = 150, which illus-trates the effect of compressibility, as demonstrated by the fact that, at a constant Re, the resistance coefficient ζ does not remain constant but in-creases with increase of λ. Figure 6 shows the same quantities plotted in logarithmic coordinates, as ζ = C(Re, λ) dit bjh = 150. From Fig. 6 it can be seen that the relationship (1) can be represented as:

£ R e = / ^ , A y (2)

In Fig. 7 is shown the relationship (2) obtained for three values of the para-meter bjh. The curves for bjh = 150 represents all the points obtained for channels of various lengths.


F I G . 6. Relationship ζ = f ( R e , λ) for b\h = 150.

F I G . 7. Relationship ς R e =fßb/h).


JO 1δ0 2S0 JSO ¥60 550 650 700 ï/h

F I G . 8. Relationship μρ = μρ(ΙΙΗ, R e , β) for b\h = 150.

flow calculated by Saint-Venant-Wanzel formula for a Laval nozzle, or a convergent nozzle having its cross-sectional area F = bh equal to that of the investigated channel; pressures P0i and P02 and temperature T01 are the same as those during the experiments. For sub-critical conditions of flow

GT = F J ^ J P o i Y o i [ß2,k - 0«+1>/*]).

For critical and trans-critical conditions (ß < ßj

The Discharge Coefficient. Taking now as determining criteria, Re, b/h, llh9

we can find the discharge coefficient as:

μρ = μρ | R e , / ? , j > j j > 0)

where μρ = GjGT is the discharge coefficient; γ = P02j'Λη the ratio of pressures; G the mass flow of air, determined experimentally; GT the mass


wherç = (2/k + l )f c / c

* ~ J> is the critical pressure ratio. For air, with the adia-

batic exponent k = 1-405, βφ = 0-528.

Values of Re, ß, b/h, Ijh, and corresponding^ were calculated on the basis

of experimental data. The resulting graphical representation of the func-

tion (3), at bjh = 150, is shown on Fig. 8. Experimental data, obtained at

0-9 > β > 0-03, / = 0-5, 30, 70, 110 and 100 mm, and h = 0-200 mm. Each

surface (1-10) on Fig. 8 represents a function μρ = μρ(1/η) at Re = const.

Ο Of 0-2 0-3 0-9 0-5 OS 07 0'3 OS PO ρ

F I G . 9. Dependence of coefficient a, which accounts for compressibility, on β and

on the discharge coefficient of incompressible liquid μΡΗ.

The family of curves / corresponds to the function μρ = μρ(1[η) at Re = const, and β = 0-9. The family of curves II corresponds to the function

μρ = μρ(β) at Re = const and l/h = const. The curves III and IV correspond to μρ = / i p m ax = μρ > The surface formed by these curves is given by the


relationship

β * * — β * * ( Re, —-h

For practical calculations, it is convenient to consider the discharge coeffi-cient μ ρ in relation to the coefficient of discharge for an incompressible liquid μρΗ, calculated by the formula

a + Σζηχ + ζ -V h)

where ζηι are the local resistance coefficients, and a the coefficient of kinetic energy of the flow; for laminar flow in the flat channel a = 1-54.

Accounting for compressibility by a correction coefficient oc, we have

μ Ρ = *μΡΗ· (5)

This proved to be particularly convenient, because the experiments have shown that oc depends only on β and μρΗ. Figure 9 shows the relationship

oc = oc(ß, μ ρ Η) , (6)

plotted according to our experimental data. It also contains the values of oc, arrived at analytically in Refs.4 and 5. Curves A, B, C and D pertain to various values of μρΗ from 0-058 to 0*358.


1. T o determine the resistance coefficient ζ of flat channels, the relation-ship

{ R . - , ( i . i ) .

represented graphically on Fig. 7, can be used. 2. Formula μρ = ο^μρΗ gives the discharge coefficient for a gas, if the dis-

charge coefficient μρΗ for an incompressible liquid is known ; the coefficient oc accounts for compressibility. The graphs of oc = oc(ß, μρΗ) are given on Fig. 9.

R E F E R E N C E S

1. E . P . D Y B A N and I . T . S H V E T S , Bull. Acad. Sei. U.S.S.R. ("Izv. A N S S S R , O T N ) N o . 2,

1956.

2. L . I . S E D O V , Methods of Similarity and Dimensions in Mechanics (Metody podobiya i

razmernosti ν mekhanike). G I T T L , 1954.


3. Ν . A . S L E Z K I N , Dynamics of Viscous Incompressible Fluid. (Dinamika vyazkoi neszhi-

mayemoi zhidkosti). G I T T L , 1955 .

4. A . S . C H A P L Y G I N , The jets of gases. Collected Works, V o l . II ( O gazovykh struyakh,

Sobr. soch., V o l . I I ) . G I T T L , 1948 .

5. F . I . F R A N K L , Rep. Acad . Sei. U . S . S . R . (Dokl. AN SSSR) Vol . 5 8 , N o . 3 , 1947.

SILTING OF SMALL RESTRICTIONS

I . N . K l C H I N

I T IS well known that the flow of a liquid through small passages (such as are often found in hydraulic control elements) tends to diminish with time, and even to cease altogether. This phenomenon, called silting-up'f [1, 2], occurs even with most carefully filtered fluids. It is usually explained by adsorption of polarized molecules by the walls of the capillary clearances. This does not take into consideration the possibility of accumulation in the channel of small active particles, present in the working fluid either in a colloidal state, or in suspension, and also the possible combination of this process with ad-sorption. Available references on the thickness of the adsorption layer [3] suggests that blockage of very small clearances (0-02 mm and less) can be explained by adsorption alone, but this does not apply to the observed blockage of relatively large passages—for example, channels 0-1-0-2 mm in diameter. As to the practical means to combat silting-up—these have not been sufficiently investigated.

The present work deals with the propensity of various working fluids to silting-up, with the minimum dimensions of passages, at which the flow rate still remains constant, and with some practical means of avoiding silting.

I N V E S T I G A T I O N S O F T H E E F F E C T S

O F V A R I O U S F L U I D S O N S I L T I N G

Test Rig and Experimental Methods

Silting has been investigated on the test rig shown on Fig. 1. The working fluid from a tank 1 with a pressure gauge 2 was fed by air pressure* to the throttling device, a bush 9, mounted in a holder 6" of a movable arm 7. Mechanical impurities were removed from the fluid by a mesh filter 5 and a ceramic filter 6, which does not pass particles exceeding 0-01 mm. For some experiments an electromagnetic filter was added. In the investigation the

t This term is used here (and in some British and American works on the subject) almost

as a metaphor, rather than a direct analogy with silting-up of rivers, etc. by solid particles.

Russians use the word "obliteration" (Translator). φ The air line was equipped with an air filter 4 and reducing valve 3 in order to maintain

a constant pressure.

382

S I L T I N G O F S M A L L R E S T R I C T I O N S 383

arm 7 with the holder 8 could be turned and clamped in such a way that the

jet of fluid enters either the sump 10, or one of the measuring cylinders 11

(each having a volume 2-5 c m3) .

The sump 10 was made of Plexiglass. When the jet of fluid was directed

into it, it was possible to observe through its transparent walls the changes in

the shape of the jet issuing from the orifice. The changes of discharge rate

with time were quantitatively evaluated by directing the jet into the bank of

F I G . 1. Layout of the test rig.

measuring cylinders, which could be moved in guides 12 and fixed for a certain time by a pawl with a spring. During the tests, the channel in the bush could be inspected and photographed with the aid of microscope 13. It was observed during the tests that if silting-up occurs while the fluid is flowing, it remained silted when the passage became filled with stagnant fluid. As in the latter case, the visual observations and photographs gave a clearer picture and a series of experiments was conducted without a pressure applied to the fluid at the entry to the channel.

The channel of the bush 9 had a length 0-2 mm and diameter 0-12 mm. The bush was made of brass, except for certain experiments, when bushes of steel, glass and Plexiglass were tested; these exceptions are mentioned in every case.

The channel was illuminated by a special device.

T o observe the clearance of the bush 9, a microscope MBI-1 , with magni-


fication from 250 to 1200, was used. The microscope was provided with a photo-attachment—a microphoto camera M F N - 1 .

The majority of tests were conducted with ordinary transformer oil (GOST 982-53), and the temperature was always 20°C, unless otherwise specified. In addition, some tests were made with the transformer oil filtered through silica-gel, or with ionolf added to it; also with toluene, white spirit, white Vaseline oil, refined kerosene, cetane, water, mixture of white spirit and ordinary transformer oil, and spindle oil N o . 3.

The propensity of a fluid to cause silting-up was evaluated by the number of complete blockages of the channel during the passage of 101. of fluid through it. A constant pressure before the bush was maintained. In all cases, except where otherwise stated, the pressure drop at the bush was maintained at 2 kg/cm

2. For the tests with transformer oil, the pressure was exerted by

nitrogen. The channel was cleared by a Nichrome wire0O9mm diameter after each blockage, and the flow of fluid resumed, until the flow again ceased, and so forth. The formation of silt in the channel was photographed from time to time. At each change of fluid, the apparatus was carefully rinsed and blown through with compressed nitrogen.

R E S U L T S O F E X P E R I M E N T S W I T H T R A N S F O R M E R O I L

(1) Condition of the Channel before Silting Occurs. Figure 2,a shows the photograph of the channel in the bush before any silting appeared. The channel is filled with clean transformer oil, and there are no foreign bodies.

(2) Condition of the Channel in the Process of Silting. After several seconds semi-transparent, light or dark-brown particles were observed (Fig.2,b). These particles were retarded near the walls of the channel, and formed ac-cretions, some of which broke and were removed by the stream, although this decreased with time. Observations proved that new particles joined the primary accretions near the walls, and rapidly filled the passage. This pro-cess was usually completed in 5-30 sec.

(3) Changes of the Shape of the Jet during the Silting Process. As the particles accumulated, the instability of flow became noticeable. Also, changes in the shape of the jet were observed; the jet became shortened, altered its direc-tion, split and occasionally reverted for a short time to its original form (when accretions were partially broken off and carried away by the flow). After a certain time, corresponding to complete blockage, the jet was super-seded by drops, and soon afterwards the flow stopped completely.

(4) The Conditions in the Channel Cross-section after Complete Blockage. When the process of silting-up was completed the channel was dammed by an agglomeration of particles. The photograph of the section in this con-dition is shown in Fig.2,c. The photograph shows (and this can be seen more

t A n alkylphenolic anti-oxidizing additive.

S I L T I N G O F S M A L L R E S T R I C T I O N S

F I G . 2. Photographs of the sections of the restrictions.

385


distinctly by visual observation) that in some cases a semblance of a lattice is formed in the passage. In other cases local coagulations were observed, dark brown in colour, or sometimes including larger transparent bodies. In some cases black inclusions, sometimes much larger in size than the particles mentioned above, were observed.

A similar picture was also obtained with spindle oil.

(5) Change of Discharge Rate during the Silting Process. Figure 3 shows the reduction in discharge rate during the silting process. The pressure drop in

s

1 ; |

β ; 8 1 β I I i 1?

•\ to •\ to •\ to

α - C ο to

Q 2

0

2

0 10 20 30 ¥0 δ Ο 60 70 80 30 100110

c i m e , s e c .

F I G . 3. Variation of discharge rate during the silting process.

the bush was maintained at 1 kg/cm2. The time during which the jet was di-

rected to each mesuring cylinder was 10 sec. During the time for the passage of 10 1., about thirty complete stoppages occurred. When the pressure drop was increased to 2 kg/cm

2, this number did not change.

(6) Silting of the Passage without Flow. The agglomeration of particles in the channel also occurred when there was no pressure drop, but this process was much slower, as compared with silting with flow. A t the beginning of an experiment (preceded every time by a thorough cleaning of the channel), separate particles were observed in the field of view (their agglomeration can be seen on Fig.2,d) which were easily shifted by a slight shaking of the bush. After several hours, static and durable accretions were found at the walls of the channel (Fig.2,e) . These grew in volume as new particles joined them. The particles formed a series of interlocking "bridges" across the section of channel. After a certain time the orifice was completely blocked.


(7) The Influence of Bush Material on the Silting Process. The observations given above relate to bushes made of brass or steel. Later, tests with trans-former oil were carried out using bushes of organic glass (Plexiglass) and ordinary glass. In the first case the propensity of transformer oil to induce silting was not changed substantially, and in the second case was markedly reduced.

(8) The Influence of Additional Filtration by an Electro-magnetic Filter. For certain additional experiments, the transformer oil was passed through an electro-magnetic filter, a concentric gap 0*3 mm wide and 10 mm long. This did not prevent the active silting-up.

(9) The Influence of a Magnetic Field. Silting in the presence of a magnetic field (produced by a permanent magnet) was observed. It has been established that the presence of a magnetic field, or a change of its intensity, does not influence silting.

(10) The Strength of the Silt Layer in a Restriction. As distinct from small (less than 0*02 mm) clearances in valves, the silt layer of a channel 0*12 mm diameter did not have any substantial strength. It was destroyed by vibra-tions of the test rig. During the formation of a layer of silt, breakdown can be caused either by vibration of the test rig, or by pressure pulsations. When the silt layer was destroyed by these means, or when the channel was cleared by a small diameter needle, the particles forming the layer were carried away by the flow, leaving no traces.

(11) Effect of Temperature. The tests quoted were conducted at a fluid temperature (transformer oil) of 20 °C. T o study the influence of temperature, additional tests were made with oil heated to 80 °C. The particles which form the agglomerations were similar to those observed at 20°C. The process of silting was now much more active. The number of complete blockages during the passage of 101. of heated oil was substantially increased.

(12) Filtration of Oil through Silica-gel. It is well known that silica-gel can effectively remove from oil asphaltic or tar-like substances. This is due to the good absorbent and adsorbent properties of silica-gel [3]. Oil filtered through silica-gel became almost completely colourless. The propensity for silting with this oil was greatly reduced (the accumulation of particles proceeded at a reduced rate). During the passage of 10 1., only six complete blockages were recorded. Also, after the early stages of silting, further progress was retarded (practically stopped), and for a substantial time a stable flow through a partially closed channel could be observed. The external appearance of particles and the structure of their agglomerations (see Fig. 2,f) remained unchanged.

(13) The Effects of Adding Ionol. It is known that special additives can sub-stantially retard or reduce the process of oil oxidation [3]. One of these sub-stances, which neutralizes active acid compounds, is ionol (an alkylphenolic anti-oxidizer). Experiments were carried out with transformer oil, containing ionol (0-2 per cent by weight). The results obtained were similar to those


with oil filtered through silica-gel. Only three complete blockages were re-corded for 10 1. of fluid.

Results of Experiments with Other Fluids

Following the same experimental methods, and using the same bush with 0· 12 mm channel, a number of other fluids were tested, namely toluene, white spirit, cetane, white Vaseline oil, refined kerosene, and water. None of these has caused any silting; the clearances of the channel remained fully open, as seen in Fig.2,a.

When a mixture of white spirit and transformer oil (in proportion 2 to 1 by weight) was passed, silting has been observed, but it took much longer than with neat transformer oil. During the passage of 10 1. of mixture, only six complete blockages were recorded. In some cases, when the orifice was partially closed (for example, with three-quarters of the cross-sectional area blocked), a small stable discharge was observed for a long period of time.

I N V E S T I G A T I O N S I N T O T H E S I L T I N G

O F R E S T R I C T I O N S W I T H C L E A R A N C E S

O F V A R I O U S S H A P E S

(1) Test Methods. Tests were conducted with bushes (Fig. 4,a), needle valves (Fig.4,b), and elements of the nozzle-baffle type (Fig.4,c). The start of silting was detected by the change in discharge rate and by visual observation of the

a b e F I G . 4. Throttling devices: a—bush; b—needle valve; c—element of the nozzle-

baffle type.

et. Al l tests were carried out with thoroughly filtered transformer oil, the temperature being maintained at 2 0 ° C . Pressure drops across the throttling devices were measured over a range 0-5-2-0 kg/cm

2. The flow rates were

determined volumetrically.

(2) Results of Tests with Throttling Bushes (Fig.4,a). Forty different bushes were tested, with diameters d = 0-12-1-2 mm, lengths / = 0-2-40 mm, ljd = 0-13-114. In the course of the experiments it was established that the tendency towards silting increased with a reduction in diameter, and with the increase in pressure drop.


Complete cessation of flow was observed for bushes with any length /, if d < 0-2 mm. The time from the start to complete blockage was not consis-tent, but it could be measured. For example, the minimum time for complete blockage of a channel 0-2 mm diameter was approximately 40 sec at a pres-sure drop of 2-0 kg/cm

2, and 70 sec at 0-5 kg/cm

2. For a channel 0-12 mm

diameter the respective figures were 10 and 20 sec. For channels of 0-3 mm diameter a partial reduction in flow rate with time was noted; it amounted to 3 per cent at a pressure drop 0-5 kg/cm

2, and 8 per cent at 2Ό kg/cm

2. For

channels of 0-4 mm diameter no noticeable flow reduction was observed; the discharge rate through such a channel was 90 cm

3/min with / = 1-0 mm, and

6 cm3/min with / = 40 mm, the pressure drop being 1 kg/cm

2. When the

body containing a throttling element was lightly tapped (vibration), or when the pressure rapidly fluctuated (pulsating flow), silting never occurred.

(3) Test Results with Needle Valves (Fig.4,b). The main dimensions are shown on Fig.4,b. With b = 0-1 mm (h = 0-0-026 mm) complete blockage occurred after a certain time. With b = 0-5 mm (h = 0-013 mm), and a pressure drop 2-0 kg/cm

2 the flow completely ceased after 5 sec, and with

b = 1-0 mm (h = 0-026 mm) and the same pressure drop after 15 sec. The reduction of pressure drop to 0-5 kg/cm

2 resulted, for a valve with the di-

mensions given above, in the increase of time to 15 and 40 sec respectively, that is, in a reduced rate of silting. In the latter case, during repeated tests, the flow occasionally did not dry up.

Within the range b = 1-2 mm (h = 0-026-0Ό52 mm), the blockage was incomplete. The reduction in flow rate amounted to 25-12 per cent respec-tively of the initial value. With b = 2-5 mm (h = 0-065 mm), and Ρ

= 1-0 kg/cm2 the discharge rate did not vary with time, and was constant at

110 cm3/min.

(4) Results of Tests with Nozzle-baffle Elements (Fig .4,c). The dimensions of the element were varied within the following range: nozzle diameter d = 0-5, 1-0, 1-5 and 2-0 mm; gap between nozzle and baffle // = 0-02, 0-04 and 0-07 mm; diameterD for main experiments was 1-3-3-3 times the diameter d\

and tests were also made with D = d, i.e. with a sharp-edged nozzle. In-dependent of the dimensions d and Z>, with h = 0-02 mm, complete blockage occurred after a certain time. With the pressure drop 2-0 kg/cm

2 and

d = 0-5 mm this time was 10 sec, and with d = 2-0 mm was 5 sec. With a pressure drop 0-5 kg/cm

2 the respective periods were 20 and 12 sec.

When h was increased from 0-02 to 0-05 mm, only partial reduction of dis-charge rate occurred after a period. The time for stabilization then increased with increase in h. The greater the distance h from nozzle to baffle, the less tenacious was the silting-up layer.

For the case h = 0-06 mm, the discharge rate of oil remained practically constant (with d = 0-5 mm, h = 0-06 mm, and a pressure drop 1-0 kg/cm

2,

the discharge rate was 60 cm3/min).

(5) General Remarks. The results quoted above were taken into considéra-


tion when choosing the dimensions of passages in adjustable multi-stage re-sistances for small flows, developed in the Laboratory of Pneumo- and Hydro-Automation I A T A N U . S . S . R . (see the next article). These results also made

possible a basic approach to the size of restrictions in other throttling ele-ments of hydraulic control devices [ 4 ] . |

S A F E G U A R D S A G A I N S T S I L T I N G

Well-known methods for this purpose are: provision for relative motion between parts forming a restriction; vibration of elements [2] ; and pressure pulsations. Practical recommendations, however, are given only for sliding

F I G . 5. Means of eliminating silting in nozzle-baffle elements, α—by rotary oscilla-

tions of the baffle; b—by axialvi brations of the baffle; c—by providing a pulsating

flow.

valve pairs, while in hydraulic control systems other elements are often used— for example, nozzle-baffle elements.

Figure 5,a and b shows a method of eliminating silting in nozzle-baffle elements. In the method of Fig. 5, a, silting is eliminated by rotating or by angular oscillation of the baffle and in the latter case oscillations of 5-10° are

t See also the article by B. M.Dvoretski i in tl is bcok, p. 165.


used, with h = 0*04-0-02 mm. In the method of Fig. 5,b axial vibration of the

baffle is used with an amplitude not exceeding 0-015 mm. Figure 5,c shows a

method for producing pulsating flow.

Tests of these devices have proved their effectiveness (stable discharges of

the order a few cm3/min were obtained at Ρ = 1 kg/cm

2) . Detailed descrip-

tions and test results of these devices are given in the following article.


1. The phenomenon of silting-up in small clearances has been described in the literature for the case of polarized molecular layers (clearance not ex-ceeding 0-02 mm). The experiments quoted above prove that a partial or complete obstruction of a channel by particles, segregated out of the working fluid, can also occur with much larger passages (experiments were carried out with a channel 0-12 mm dia.). Visual observations and photographs gave the picture of the build-up of the silting-layer in such channels.

2. The experiments prove that ordinary transformer oil, and also spindle oil 3, both widely used in hydraulic control systems, have a high tendency to silting. When 10 1. of transformer oil were passed through a channel 0-12 mm diameter and 0-2 mm long, at a pressure drop 2 kg/cm

2, some thirty

complete blockages of the channel were observed (after each blockage the channel was cleaned). It has been found that silting-up also occurs when the channel is filled with static oil.

3. N o silting-up has been observed during experiments with toluene, white spirit, cetane, white Vaseline oil, refined kerosene, and water. It must be noted that some of these fluids—for example, toluene, water, and kerosene [2] — contain active molecules, which can be adsorbed on a surface, and conse-quently block a clearance, commensurate with the thickness of the adsorp-tion layer. This fact, and also the considerations (given in paragraph 1 of these conclusions) relating to the experiments with transformer oil or spindle oil, prove that silting-up can be caused not only by adsorption of fluid molecules to the walls of channels, but also by the accumulation of large particles in a restriction. These particles apparently consist of asphaltic, bituminous and other active compounds.! This conclusion enables us to gain a wider under-standing of silting, and explains the instability of flow rate, observed in hydraulic control elements using comparatively large passages (e.g. 0-2-0-3 mm diameter) and working with mineral oils.

4. Experiments with a channel 0*2 mm diameter have shown that an in-crease of oil temperature from 20 to 8 0 ° C intensifies the rate of silting. The filtration of transformer oil through silica-gel, or the addition of ionol, reduces the tendency to silting.

t The present investigation was not intended to include a detailed physical-chemical

study of the process.

14 Aizerman I


5. The methods of investigation (evaluation of the intensity of silting-up by the number of complete stoppages, photographing of sections, etc.) which have been developed can be used for the further study of silting.

6. It has been established that the flow of transformer oil through cylindri-cal channels from 0-2 to 40 mm long causes practically no silting, if d>0-4 mm.

For needle valves (Fig.4,b) constancy of discharge rate occurs with b

2-5 mm (h = 0Ό65 mm). For nozzle-baffle elements (Fig.4,c) with a nozzle diameter in the range 0-5-2-0 mm, no noticeable decrease of discharge rate with time has been observed, if the gap between nozzle and flapper h

> 0Ό6 mm. These data were obtained with pressure drops of 0-5-2Ό kg/cm2.

It has been found that the constancy of discharge rate can also be main-tained at reduced values of the dimensions given above, providing the throttling element is made to vibrate, or if the flow is pulsated.

7. Methods for producing relative displacement of one of the surfaces in nozzle-baffle elements (Fig.5,a and b) , and also for producing a pulsating flow (Fig. 5,c), have been made and tested, and their effectiveness measured.

R E F E R E N C E S

1. A . S . A K H M A T O V , Rep. Acad. Sei. U . S . S . R . , Vol . X X X , N o . 2 , 1941 .

2. T. M . B A S H T A , Aircraft Hydraulic Drives and Systems (Samolyotnye gidravlicheskiye pri-

vody i agregaty). Oborongiz, 1951 .

3. Motor fu^ls, lubricants and fluids (Motornye topliva, masla i zhidkosti), Collection of

Works, Ed . K . K . P a p o k and E. G . Semenido. Vol . 2 . Gosnefteizdat, 1957.

4. Ε . Μ . N A D Z H A F O V , V . A . K H O K H L O V and I . N . K I C H I N , Report N o . 7 2 7 / 2 I A T A N

U . S . S . R . 1955 .

SOME METHODS OF COMBATING SILTING A N D ENSURING CONTROLLED

FLOW THROUGH SMALL RESTRICTIONS I N HYDRAULIC CONTROL

ELEMENTS

I . N . K l C H I N

I T IS often necessary to have a small fluid flow, either controlled or constant, but in either case not susceptible to reduction with time, due to silting of restriction. Such small flows are typical in a number of applications—for example, machine tools at small feed rates, or hydraulic controllers with high values of integral action time, etc. The difficulties arising due to silting are well known, and there is a widespread body of opinion that they should be avoided by using restrictions only with large sectional areas. This, however, raises substantially the lower limit of flow rate. In Ref. 7 it is recommended that these be obtained by using small pressure differences, but this is not al-ways possible.

Experimental investigations of silting, and some observations as to its physical nature, are discussed in the preceding article. One peculiar pheno-menon should be mentioned; the time for silting-up was substantially short-ened (sometimes to 5 sec-10 min) if the experiment with a given restriction was conducted after an interval of 12-48 hr. The explanation of this fact requires additional investigation.

1. M E C H A N I C A L S A F E G U A R D S A G A I N S T S I L T I N G

Under this heading come various alternatives for producing relative mo-tion between elements, forming a restriction (in this particular case, a nozzle-baffle pair) and provisions for pulsating the flow (pressure pulsations).

(a) Rotating Baffle

Figure 1 shows an arrangement of a nozzle-element with a rotating stop, serving as a "baffle". The nozzle 3 is located in the slide 2, which can be moved (for adjustment of the gap) in the bore of the support 7 by a micro-

M a Aizerman I 393


metric screw 5. The stop 4 is placed in the bush 6 with a radial clearance 0Ό15 mm, and located axially by the spring 7. The axial play of the stop shoulder did not exceed 0Ό02 mm. The gap h = 0Ό2,0*04 and 0*06 mm was set with the aid of gauge blocks, and checked before and after tests. For each of these gaps, tests were carried out at pressure differences A Ρ = 0-5, 1, 1-5 and 2 kg/cm

2 with nozzle diameters d = 0-5, 1 and 2 mm. The stop was rotated

through a reduction gear with the speed η = 1-5, 10, 50,100 and 150 rev/min. Transformer oil with the specific weight γ = 0-876 g/cm

2 and kinematic

viscosity 0*27 cm2/sec at 25 °C was used.

F I G . 1.

The flow characteristics Q = f(h) shown on Fig. 2 are plotted from the ex-perimental data; broken lines represent silting conditions, and solid lines the silt layer removed by rotation of the stop. The lowest measured discharge rate was 3 cm

3/min, with the gap 0-02 mm, nozzle diameter 0-5 mm and pressure

difference 1 kg/cm2. The maximum relative error ÔQ in measuring the flow

in this particular case was 5-1 per cent.

It was found that as soon as the stop was rotated with η > 1 rev/min, the silt layer disappeared almost instantaneously. The increase of rotational speed did not noticeably affect the discharge rate, but the appearance of the jet changed at 1-5 rev/min oil was spread over the end surface of the stop, but at higher speeds, particularly at η — 100-150, the jet was wound on to the rotating shaft, with subsequent splashing.

During the course of these experiments it was also found that a limited rotation of the stop by an angle exceeding 5-10°, at any velocity, also re-moved the silt layer.

(b) Vibrating Flapper

Figure 3 shows an arrangement of a nozzle-baffle element with a vibrating baffle or stop. The nozzle 3 is mounted in a fixed body 7, and the stop 4 is mounted on the shaft 2, which can move axially. The shaft, with the coil 9 attached to it, is mounted on the flexible diaphragm 10 in the magnetic coil 6 by means of nuts 11 and spacers 8. The magnetic coil unit, with its stationary

S O M E M E T H O D S O F C O M B A T I N G S I L T I N G 395

QfCn^/sec 2h

22

2Ό\

1β

16\

H

12

W

08

0-6

02\

/

/ / / /

/ /

M r

/ 1

/

1 / / / 1 ' 1 / è 'ί A '' Â 1 1

ι J

/ / /

1 I / /

/ / , w /

/ / / y

/ / / /

/ s

1 i

I

1/ /*' // / ' s //

1 1

ί / / 'X s?

S?

/ /A Ko/ y^ s HZ*

' v T

m 0 001 0Ό2 0-03 0-0* 0Ό5 0-06 h, m m

F I G . 2. F l o w characteristics with the rotating stop.

N o . of curve 7 2 4 5 6 7 5 9

AP, k g / c m2

2-0 1-0 2-0 0-5 1-0 2-0 0-5 1-0 0-5

d, mm 2-0 2-0 1-0 2-0 1-0 0-5 1-0 0-5 0-5

coil 7 and casing 5, is fastened to the body 1. A direct current of 0-15 A ,

24 V, passes through the coil 7. Control of the gap h, from 0 to 0-3 mm, ac-

cording to a linear characteristic h = / ( / ) , is brought about by direct current

input 0-30 mA, fed into the coil 9. The addition of an alternating current


/ = 0-1-5 mA, frequency 50 c/s, causes vibration of the stop with an amplitude 0-15 μ.

Constant mean gaps h = 0-01, 0-02, 003, 0-04 and 0-06 mm were main-tained during each experiment, and were checked with the aid of a micro-scope MIR-1 (magnification up to 180) and an optical attachment AM-9-2 . The amplitude was measured as the width of the "blur" of the stop, as registered by the microscope.

For each of the gaps listed above, tests were run with pressure differences A Ρ = 0-5, 1,1-5 and 2 kg/cm

2 with nozzle diameters d = 0-5, 1 and 2 mm.

5 7 2

F I G . 3.

Flow characteristics Q = f(h) with and without silting were analogous to those shown in Fig. 2, for gaps h = 0-02-0-06 mm. The minimum discharge rate was 1-2 cm

3/min at h = 0-01 mm, d = 0-5 mm and Ρ = 1 kg/cm

2. The

maximum relative error in measuring flow ÔQ for this particular experiment was 10 per cent.

It was found that when the stop was vibrated with a certain optimum am-plitude A , silting (which existed before vibration) disappeared almost in-stantaneously. T o remove a more complete blockage, a larger amplitude A was required. Vibrations with a smaller-than-optimum amplitude also re-duced the silting, but it usually re-appeared at an amplitude of less than 0-5 A .

(c) Pulsating Flow

Figure 5 shows a device for producing pulsating flow. The fluid is fed to the inner chamber of the rotor 4, which has a diameter D. The rotor is fitted into a stationary liner 3 with a clearance 0-02 mm. When the rotor turns, oil passes through a replaceable throttling bush 5 to the drain, when the passage

F I G . 5.


F I G . 4. Dependence of the optimum amplitude A on the diameter of the nozzle d.

: Ρ = 0*5 k g / c m2; : P= 1 k g / c m

2; : Ρ = 2 kg /cm

2.


in the bush 5 coincides with a cut-out in the liner. The liner is attached by

the yoke 2 to the stand / , its cut-out being directed towards the drain passage

in the stand. The diameters d of the bush 5 were 0-5, 1 and 2 mm, the length /

of the cut-out in the liner 6, 12 and 20 mm, and its width / 2 and 4 mm. At

QfcmS/sec 70

60

50\

40

JO

2-0

10

r

i r—7

/ / / / Τ 7

W I

! It

/ /

/

/ '

/ '

ΔΡ·1 Okg/cm2 1 / / —

/ / / / ' /

1 / / •7 ' /

y

//

h /A / / /

g/cm2

ii i ! y

i '//////

ψ W γ

F I G . 6. F l o w characteristics for pulsating flow. : η = 50 rev/min ;

: η = 150rev/min; : η = 600rev/min.

each combination of these dimensions, the device was tested at pressure dif-

ferences Δ Ρ = 0-5, 1,1-5 and 2 kg/cm2, f and η = 50, 150 and 600 rev/min.

On the basis of these experiments, flow characteristics Q = f(U) shown in

Fig. 6 were obtained. Here U = Fl^nD represents the product of the passage

t At standstill, with the bush coinciding with cut-out.


area F of the bush 5, and the relative duration of opening (ratio of cut-out length / to the circumference πΌ), which is independent of n.

The minimum discharge rate, obtained at d = 0-3 mm, 1 = 6 mm, η = 50 rev/min and Ρ = 1 kg/cm

2, amounted to 2-2 cm

3/min. The maximum

relative error in measurement ÔQ = 5-3 per cent.

In this process of impulse flows, the discharge rate of the fluid is equal to mlY\nD FyJ[(lglyk) AP], where A: is a coefficient of overall hydraulic losses at η = 0 (orifice fully open), and m is a coefficient accounting for the pressure rise in the rotor during the time that the orifice is closed, and also for the inertial effects, and for the change in k due to rotation.

According to the experiments, the discharge rate increased on average by 25 per cent as the speed increased from 50 to 600 rev/min. The coefficient m increased respectively from 1 -65 to 2-06 for the case d = 0-5 mm, Ρ=2 kg/cm

2,

k = 2, and from 2-03 to 2-5 for d = 2 mm, Ρ = 0-5 kg/cm2, k = 3-8.f

Changes in the cut-out width /, which was always larger than d, did not result in substantial variations in discharge rate. Small discharges were con-trolled by altering the speed of rotation. The same effect can be achieved by an axial shift of the liner, having a tapering cut-out—that is, by varying the effective length lx > d.

2. H Y D R A U L I C S A F E G U A R D S A G A I N S T S I L T I N G

Silting can be avoided by using a large number of sufficiently large throt-tling elements connected in series. Flow control can be readily effected by altering the number of restrictors. One of the variable resistances of this type is shown in Fig. 7.

F I G . 7.

The working fluid enters the inlet a, and is conveyed via the passage c to a stack of perforated washers 3, with orifices of diameter d and length /.

t The coefficient k of hydraulic losses in the orifice of bush 7 are taken from Ref. 8.


The washers are threaded on the tubular stem 5, which is provided with a

number of holes e communicating with the spaces between the washers.

When these holes are closed by the fitted pin 6, fluid can only pass through

all the orifices in series, but when the pin 6 is screwed out holes e are

uncovered, and a number of the orifices can be by-passed. The maximum

discharge rate takes place when all the holes e are uncovered, and fluid passes

through only one orifice. The minimum discharge corresponds to all the

Section 00

F I G . 8.

holes e, except the last one, being closed (pin 6 fully screwed in). T o avoid leakage the washers are fitted on the stem 5 with a clearance less than 0-02 mm, and tightened axially by the nut 9, together with spacer rings 4. The pin is lapped to the central bore of the stem with a clearance not exceeding 0Ό3 mm. The stack of washers is fitted into the sleeve 2 with a clearance 0T mm, and tightened by nuts 10 and 7. The sleeve is screwed into the holder 7, which is fastened (by means not shown) to the body 77. Sealing of faces is ensured by rubber rings 8. The holes e in the stem 5 have the diameter 1 mm, and are spaced alternately at 90°. They are never completely covered by the washers, which are 0-3 mm thick. The protruding part of the pin 6 is marked to give the number of orifices in use.

For our experiments, the device had the following parameters: maximum number of washers η = 60, diameter of the throttling holes d = 0-6 mm and 0-8 mm, length / = 0-3 mm, distance between washers L = 0-3 and 0-6 mm. Both sides of the washers were chamfered 45° χ 0-05 mm.


The flow characteristics Q = f(n) at Ρ = 1*1 and 2-2 kg/cm2 are shown in

Fig. 10.

It was found that the distance between the washers L affects the discharge rate substantially (see Fig. 10).

Figure 8 shows an adjustable rotary throttle of the "face" type. The work-ing fluid enters the annular channel b, from which it passes through the single hole c in the rotary disc 1 and one of the holes h in the front stationary disc 2, into recesses / and e (in the front disc 2 and rear disc 3), which are separated by the plate 4 with throttling holes d. Recesses / and e are milled to a depth and width 2 mm and length 2-5 mm 4- 2d. They are positioned to overlap with their rounded ends, so as to form, together with the orifices d, a zigzag labyrinth channel. Oil passes from the recess in one disc, through a hole d, and into the recess in the other disc, etc., until it reaches the outlet hole g in one of the recesses e of the disc 3. The amount of throttling is con-trolled by turning the disc 1 relatively to disc 3, so that the hole e coincides with one or other of holes h, which are drilled in each recess / of the disc 2. In intermediate positions the hole c, which has a diameter of 4 mm, always overlaps one or two holes h of 2 mm diameter. The surfaces of discs 2 and 3, plate 4 and rings 6, 7 and 10 are lapped flat. Discs are tied together by screws 9, and secured from relative rotation by the dowel 5. The rotary disc 1 has an axial clearance of 0-004 mm, which is ensured by the spacer ring 10. The complete assembly is held together tightly by the nut 7, and the rotary disc 1 is pressed to the disc 2 by a spring 8.

T o obtain very small flows, several plates 4 may be used. They must be separated by spacer discs of the thickness L, with holes 2 mm diameter, co-axial with holes d and h.

A dial, indicating the number of holes in use, is fastened to the body of the device. The following dimensions were used: maximum number of ori-fices η = 100 (one plate); diameter of these orifices d = 0-6 mm; lengths of throttling passage / = 1-5 mm and 0-3 mm; distance L between centres of orifices = 2-5 mm. Holes d at both sides of the plate were chamfered 45° χ 0-05 mm. The flow characteristics Q = f(n) at Δ Ρ = 1-1 and 2-2 kg/cm

2

are shown in Fig. 10.

Figure 10 also illustrates the influence of fluid temperature on the dis-charge rate at the maximum number of orifices in use, and with A Ρ = 2-2 kg/cm

2. It was found that the discharge was greatly affected by tem-

perature, when the length / was 1-5 mm ( / = 2-5d). For example, a change in temperature from 16 to 40 °C more than doubled the discharge rate points c' and c" at η = 100, Fig. 10). But with / = 0-3 mm ( / = 0-5d) the corre-sponding variation in discharge rate did not exceed 8-10 per cent (points q' and q" at η = 100).

Figure 9 shows the arrangement of a similar rotary throttle, differing from the previous one by the location of its control element on a cylindrical sur-face instead of on a flat face. Oil is admitted through a passage a into annular


groove b in the body 7, and then into the channel c (one only), of the sleeve

5, and into one (or two) of the holes h in the body 7. Milled recesses e and/,

and a plate or plates (two are shown) with throttling holes are analogous to

those of the device of Fig. 8. After passing a certain number of orifices, oil

flows through the channel g, which is 2 mm square in section, into the outlet

drilling /, 2 mm diameter. Control is effected by rotating the outer sleeve 5.

The recess k, through which the channel c communicates with holes h, is

dimensioned to ensure that one or two holes are covered. The mating surfaces

Section orsp

F I G . 9.

of the elements are lapped, the clearance between the body 7 and sleeve 5 not exceeding 0Ό5 mm; there are also rubber rings 6, which have a "squeeze" of 0-3 mm. The disc 2, plates and spacer rings are fastened to body 7 by screws. Several plates can be used to obtain very small discharge rates. The end sur-face of the sleeve carries a dial, showing the number of throttling holes in use.

The following parameters were used in the experiments: maximum num-ber of orifices η = 75 (one plate), 150 (two plates), and 225 (three plates), diameter d = 0-6 and 0-8 mm, length / = 0-3 mm, distance between centres of holes L = 2-5 mm.

The flow characteristics Q = f(n) at AP = M and 2-2 kg/cm2 are shown

in Fig. 10. They prove that the curves of discharge rate obtained for different numbers of plates coincide when the total number of holes is the same. The minimum discharge rate, obtained with 3 plates (n = 225, d = 0-6, Ρ

= 1-1 kg/cm2) was 4cm

3/min. The relative error of flow measurement ÔQ

was in that case 2-6 per cent. A change of temperature from 16 to 40 °C alter-ed the discharge rate by less than 5 per cent (points r' and r" at η = 150, / = 0-38J, Fig. 10).


R E S U L T S O F E X P E R I M E N T S

During experiments with all three devices no silting-up was observed, and discharge rates were constant with time over 4-8 hr of continuous work. A comparison of flow characteristics, taken for three different designs at the same working conditions, shows certain differences between them. This can be explained by the different hydraulic resistances of the channels leading to the orifices (and from them), which were all of 2 mm diameter. These pas-sages were designed to suit specific installation requirements, of a particular hydraulic system.

Generally, the design of any series system of adjustable resistances should take into consideration the following points:

(a) The distance L between resistances should be not less than the dia-meter of a throttling hole (orifice).

(b) The length of the throttling passage must be kept to a minimum (/ < 0-5d), to reduce the effects of oil temperature on the resistance.

(c) The inlet and outlet passages must be as short as possible, and not less than 2 mm diameter.

(d) Since the relative effect of adding more holes is small for η > 50-80, it is advisable to choose small diameters for orifices (less than 0-6 mm but not less than 0-4 mm, to exclude the possibility of silting, when a high resistance is required).

(e) Good filtration of the fluid is essential for these devices. Incidentally, when mechanical obstruction occurs, the latter two devices can be dismantled and cleaned much more easily than the first type.

I f all the orifices have the same dimensions, the discharge through a series

set is determined by the equation :

from which it can be seen that the curve Q = f(n) is a hyperbola. The dis-charge rate alters drastically in the early stages as the first few resistances are brought in—and only slightly at a later stage, when η is already large (see Fig. 10). For example, the experiments show that when the first ten resis-tances are added one after another, the discharge rate diminishes 3-12 times, and with the addition of the following twenty 2-3 times.t

These uneven changes of discharge rate make it difficult to adjust them accurately within the range of the first 20-30 resistances.

t According to Refs. 6 and 8 for "fixed" (not provided by means for continuous adjust-

ment) stacks of resistances the respective figures are 3-2-4 times, and 1-5-1-8 times.

Calculations of Multi-stage Series Resistances

(1)


For better control the function Q = f(n) can be linearized or given any de-sired form. For this, it is only necessary to choose the change of diameters of the orifices according to a certain law. On the basis of the Bernoulli equation, the total pressure drop across a throttling device with η resistances (having unequal diameters) in series can be given as :

/ = ! i = l a,

kl ( %QÎy

7l2g

(2)

where ZIP, is the pressure drop at the j'-th resistance with the diameter di9 and k" the overall coefficient of hydraulic losses in these resistances (the number of passages η is here an index, not an exponent). It is assumed that the velo-cities in other ducts are negligible; velocity in the throttling passage Vt

The diameter of each resistance can be found, according to Ref. 2, from the equation:

V *2gàP,

Using formula (2), we obtain

d, =

n2g

1

i- 1

AP - J AP, i= 1

LSQfrk', k\

k\ k\ — H \ + d\ d\

- M l

(3)

(4)

The analysis of experimental results for controllable multi-stage resistances proves that their coefficient of overall hydraulic loss (similarly to that of "fixed" sets, described in Ref. 8) is a definite function of the quantity //0-05J R e f where / = length of the throttling passage, Re = 4Q/ndv the Reynolds number.

For a series system of passages having the same /, we have

R e ^ ! = Rt2d2 = ·•· = Re,di = •·• = Re„d„.

Hence k\ = k\ = = k"t = — = k"K.

Introducing these results into (3) and (4), we obtain a formula for the cal-culation of variable diameters:

1

\\ àPn2g /J_

1

dt

1

dt

APn2g 1 1

(5)

83' lQfk\ QlMZl t For example,if L> d (curves 6, 8, 9 on Fig. 10) the dependence k = / ( / /0 -05 i /Re)

can be represented by curve 10 of Fig. 5 in [8] and in the case L < d (curves 2, 4 on

Fig. 10) by curves 7, 8 (Fig . 5 in Ref. 8).



1. The use of small passages (slots with h < 0-06 mm, holes with d < 0-4 mm) in order to obtain small discharge rate is accompanied as a rule by silting.

2. The intensity of silting increases with increased flow rate of the fluid (experiments pertain to transformer oil) due to the higher pressure differences, and with the reduction of the characteristic dimension of the channel. The tenacity of the silt layer is reduced if the process is recent or incomplete.

3. Mechanical elimination of silting is brought about by rotation (n

> 1 rev/min), turning (by an angle > 5°), and vibration (with an amplitude > A) of the baffle in the elements of the nozzle-baffle type. The discharge rates can then be controlled by altering the gap h between the nozzle and baffle.

4. T o obtain permanent control of small flow rates, pulsating flow devices may be used, in which the discharge is controlled by altering the number of pulsations, or the length of the slot (affecting the relationship between the "on" and "off" periods).

5. Throttling devices with a number of larger (not liable to silt-up) pas-sages in series are suitable for a constant small discharge rate of the order

For a particular case, we take a linear relationship between flow rate and the

number of resistances:

Qi = Qi - tan ψη, (6)

where Qt is the flow rate through the stack, when / resistances are used ; the required maximum flow with one resistance only; and φ the required

slope of the linear characteristic Q = f(n).

Below is given an example to calculate the dimensions of a resistance to obtain a linear characteristic Q = f(n), with discharge rates ranging from Qi = 2-5 cm

3/sec to Qn = 1 cm

3/sec. Taking tan φ = 0Ό3, we obtain from

(6) η = 50. Taking dx = 0-06 cm, / = 0-15 cm.

The required pressure drop APt = 950 g/cm2 can be found from the for-

mula (2) with η = 1, with k\ = 2-69 (from the diagram k = f(lvn/0-2Q) [8]). With two throttles in use, Q2 = 2-47 cm

3/sec, k2 = 2-71, d2 = 0-182 cm

from formula (5). With three throttles in use,Q3 = 2-44 cm

3/sec, k

l u = 2-73, d3 = 0-165cm,

etc. When all fifty resistances are brought in, Q = 1 cm3/sec, k^0 = 3*72,

dso = 0-058 cm. With these calculations, a plate was made for a controllable resistance of

the Fig. 8 type (with a flat face), and a flow characteristic was obtained (Fig. 10, curve 7a). The deviation from the corresponding theoretical charac-teristic (curve 7b) did not exceed 12 per cent.


6-4cm3/min and less. The control of flow is accomplished by altering the

number of passages in use.

6. It is possible to linearize the relation between discharge rate and the number of resistances by suitably choosing their diameters.

R E F E R E N C E S

1. A . S . A K H M A T O V , Rep. Acad . Sei., U . S . S . R . , Vo l . X X X , N o . 2 , 1941 .

2 . T. M . B A S H T A , Aircraft Hydraulic Drives and Systems (Samolyotnye gidravlicheskiye

privody i agregaty). Oborongiz , 1951 .

3. G . P . V O V K , Dissertation. Institute of Machine Tools and Instruments, Moscow, 1946 .

4 . L . S . B R O N , A hydraulic copying device (Collection), Automation of Technological

Processes (Avtomatizatsiya tekhnologicheskikh protsessov). Mashgiz, 1 9 5 1 .

5. L . A . Z A L M A N Z O N and Β. A . C H E R K A S O V , Control of Jet and Ram-jet Engines (Reguliro-

vaniye gazoturbinnykh i pryamotochnykh vozd. reaktivnykh dvigatelei). Oborongiz,

1956 .

6. A . G . S H A S H K O V , Dissertation, Ι Α Τ A N (Academy of Sciences) U . S . S . R . , 1955 .

7. T . M . B A S H T A , Vestnik Mashinostroyeniya, N o . 5, 1956 .

8. I . N . K I C H I N , Avtomatika i Tekmekhanika, V o l . X V I I I , N o . 1, 1957.

AUTHOR INDEX

Afanasyev, V . V . 311-19

Aizerman, M . A . vii

Andreyeva, Y e . A . 247-56

Babushkin, S . A . 223-34

Berends, T . K . 20-41

Berezovets, G . T . 14-19

Bogacheva, A . V . 371-81

Braverman, E . M . 157-64

Bron, L . S . 122-34

Dmitriyev, V . N . 197-216, 272-84

Dvoretskii, B . M . 165-9

Ivlichev, Y u . I . 42-58

Khokhlov, V . A . 333-9

Kichin, I . N . 382-406

Kozlov, I . F . 79-86

Krementulo, Y u . V . 187-96

Mach , Y u . L . 320-30

Nadzhafov, E . M . 42-58

Ostrovskii, Y u . I . 3-13

Podgoyetskii, M . L . 157-64

Prusenko, V . S . 237-^4

Rukhadze, V . A . 170-86

Semikova, A . I . 59-78, 87-106

Shashkov, A . G . 272-84

Shneyerov, M . A . 111-21

Shumskii, Ν . Ρ . 257-71

Stepanov, G . P . 320-30

Stupak, B . F . 135-52

Tal' , A . A . 20^1

Temnyi, V . P . 219-22

Zalmanzon, L . A . 59-78, 87-106, 340-70

Zasedatelev, S. M . 170-86

407

SUBJECT INDEX

Actuators (hydraulic) 148-52

Aggregate System ( A U S ) 9, 14, 23, 114

Aggregate System ( K B - T s M A ) 111-21

Amplifiers

hydraulic 135-40, 219-20

pneumatic 114-15, 178

Automatic re-adjustment according to load

20-26, 39-40

Baffle see Nozzle-baffle elements

Bellows 173-6

Carburettors as systems of orifices 340,

346-9, 352-4

Cetane (silting-up experiments) 384, 388

Clamping devices (hydraulic) 129-30

Compact instruments 79-86, 165

Compressibility (influence on control per-

formance) 224-33

Compressor

automatic plant 237^4

"centrifugal" 237-8

Continuous-to-digital converter (pneumat-

ic) 38-9

Controllers

hydraulic 165-9

pneumatic 14-26, 79-86, 157-64

Dehydrator for compressed air 238-42

Delay operation 34-8

Derivative action 81-2, 119, 157-8

Diaphragms 161, 311-30

effective area 161, 311-19

hysteresis 321-6

materials 116, 183, 323-9

metallic 329

overload protection 181-3

Differential pressure transducers 170-86

Ejector nozzle 179

Elastic deformation of pipelines (influence

on control performance) 224-34

Extremum controllers 3-13, 41-2

Feed mechanisms (hydraulic) for machine

tools 122-7

Filters for oil 143-4

Flat capillary channels 371-81

F low chamber (pneumatic) 14-15

F low control valves (hydraulic) 123-5

Fol low-up mechanisms 219-22

Force balance principle 116, 170-86

Furnace control, example of optimization

4-7

Hydraulic devices 122-55, 165-9, 213-34,

247-308, 333-54, 382-406

see also under detailed headings

Hydraulic equipment for machine tools

123-34, 135, 154

Hysteresis

in diaphragms 321-6

in multiplying device 55, 57

Inertia of fluid in pipelines—influence on

control performance 224-34

Integral action 20-6, 39, 80, 119, 157-9,

166-8, 316-17

Ionol 384, 387-8

Jacks (cylinders) 148-50

Jet action on the baffle 272-84

Jet-tube elements 59-78, 87-106

Kerosene 257, 268

silting-up experiments 388

Linearity

of multiplying device 55-6

of primary instruments 176-9

Linkages of pneumatic instruments 171,

176-7

Logical operations 29, 32-8

Measuring devices, hydraulic (e.g. checking

of drilled holes) 131-2

409

410 S U B J E C T I N D E X

Memorizing of maximum 7-11

Multiplying-dividing device 42-58

Networks of hydraulic resistances 340-70

Neuron (analogy) 37-8

Non-linear transformations 59-78

Non-primitive switching schemes 36-8

Nozzle-baffle elements 197-216, 247-308

Optimum controllers 3-13, 41-2

Overload protection (of diaphragms)

181-3

Parallel systems of restrictors 341-2

Pipelines

elasticity effects 224-34

plastic 121

Pneumatic devices see under detailed

headings

Positioning device 120-1

Pressure distribution

across a jet 59-65, 88-9, 92-105

on a baffle 251-5, 2 8 2 ^

Pressure transducers 170-86, 318-19

Primary (measuring-detecting) instruments

79-86, 111-22, 170-87

Primitive switching schemes 29, 32-6

Proportional band adjustment 23-6

Pulsating flow (as means against silting-up)

390, 396-9

Pumps 144-5

Radial flow in a gap 248-75

Ratio controllers 14-19

Recording instruments (pneumatic) 82-3

Regulating final mechanisms 120-1, 165,

219-34

Relays (with constant pressure differential,

etc.) 197-216

Relief valves 146-7

Ring computing schemes 34-6

Safety devices (protection of diaphragms)

181-3

Series systems of restrictors 341-2,359-68

Silica-gel 239-42, 387-8

Silting-up 382^106

influence of filtration 387

influence of temperature 387

Slot nozzles 95, 101-2

Solenoid valves 141-2

Spindle oil 222

Spool valves 137-43

discharge and loss coefficients 333-9

Square-rooting device 42-58

Sub-critical flow parameters 202-3,

212-15, 355-70

Summator 65-70, 85

Switching circuits (pneumatic) 27-41

Systems of orifices 340-70

diagrams for air flow parameters 355-7

Temperature-caused errors in instruments

181

Three-term controllers 118-19, 157-66

Throttles (hydraulic) 152-4

multiple orifice type 399-405

Toluene, silting-up experiments 388

Transducers

electro-hydraulic 137, 219-22

electro-pneumatic 187-96

pressure 116-18, 170-86

Transfer drives, reciprocating 127-8

Transformer oil 222

silting-up experiments 384-8

Variable displacement pumps—control of

135-6

Vaseline oil, silting-up 388

Vibration of valves (as means against silting-

up) 390, 394-6

Vibrators (hydraulic) 132-3

Water

silting-up experiments 388

used as hydraulic fluid for controllers

169

White spirit, silting-up experiments 388

M A D E I N GREAT B R I T A I N

pneumatic and hydraulic control systems. seminar on pneumohydraulic automation (first session)

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