AE-201UDCÄ24

OCNuu Heat Transfer Analogies

A. Bhattacharyya

AKTIEBOLAGET ATOMENERGI

STOCKHOLM, SWEDEN 1965

AE-201

HEAT TRANSFER ANALOGIES

Ajay Bhattacharyya

Abstract

This report contains descriptions of various analogues uti-

lised to study different steady-state and unsteady-state heat trans-

fer problems. The analogues covered are as follows:

1 . Hydraulic

a) water flow

b) air flow

2. Membrane

3. Geometric Electrical

a) Electrolytic-tank

b) Conducting sheet

4. Network

a) Resistance

b) R- C

A comparison of the different analogues is presented in the

form of a table. The bibliography contains 81 references,

(This report was submitted to the Royal Institute of Techno-

logy, Stockholm, in partial fulfilment of the preliminary require-

ments for the degree of Teknologie Licentiat,,}

Printed and distributed in November 1965

LIST OF CONTENTS

Page

Abstract 1

Introduction 3

I. Hydraulic Analogy 4

I, 1 Application in air-conditioning 8

I. 2 Application in design of regenerators 10

1.3 Steady-state heat exchangers 11

1.4 Transient conditions in parallel and counter flowheat-exchangers 14

1.5 Air-flow analogue 16

1.6 Practical problems 16

II. Membrane analogy 17

III. Geometric electrical analogues 19

III. 1 General description of the electrolytic tank analogue 21

III. 2 Axi- symmetric problems 23

III. 3 Application in refrigeration 23

IIIo 4 Distributed heat generation 25

III. 5 Temperature control 26

III. 6 Sources of error in the electrolytic tank analogue 2?

HI. 7 General description of the conducting sheet analogue 28

IV. Network analogues 29

IV. 1 Network analogues for steady-state problems 31

IV. 2 Advantages and disadvantages of resistance networkanalogues 33

IV. 3 Network analogues for unsteady-state problems

' (RC-network) 34

V. Comparison of analogues 37

Acknowledgement 39

Bibliography 40

- 3 -

Introduction

Two physical phenomena are considered to be analogous when

the mathematical equations describing their nature are identical in

form. This applies to both algebraic and differential equations. As

an example, the case of steady-state heat conduction in two dimen-

sions is considered. The differential equation governing the pheno-

mena is

where t = temperature,

and x, y = coordinates.

The vertical deflection of a homogenous membrane which is

stretched with uniform tension at the edge and is not subjected to

any lateral pressure gives rise to the following differential equation

-^-f = O ( 2 - 4 )Sy2

where z - vertical deflection of membrane..

It is readily seen that the differential equations governing the

two entirely different systems are same or in other words there is

complete analogy between the temperature field and the vertical

deflection of the membrane.

Differential equation governing some heat flow phenomena are

unsolvable. In some other cases even if they are solvable analyti-

cally the solutions are very tedious, complex and time consuming.

In some of the problems belonging to the two categories mentioned

above it is possible to find physical phenomena which are mathema-

tically analogous and experimentally easier to investigate. It should

be specifically mentioned that an analogy does not give any analyti-

cal solutions.

- 4

Such analogies as mentioned above exist among a large number of

natural phenomena. Many problems in elasticity, vibrations, fluid flow

and heat transfer can be solved by utilising analogies. In this paper only

analogies used in the field of heat transfer will be discussed.

L Hydraulic Analogy

Heat flow and fluid flow under laminar flow conditions are governed

by basic equations which are similar. The equations for steady state can

be written as

Quantity = Constant. Driving force

For heat flow Q = U A • AT (1-1)

where Q = heat flow

U = overall heat transfer coeff.

A s area of heat flow

AT= temp. diff.

For fluid flow Vk = K • AP (1 - 2)

where V, = volumetric flow

K = constant

AP= pressure head

To solve periodic heat flow problems with the help of the above

mentioned analogy it is necessary to take into account the effect of storage

of heat or fluid. Moor (63) and Bäckström (8) independent of each other

designed,,similar hydraulic circuits to study periodic heat flow problems.

For a clearer understanding of the basic principles involved, the hydrau-

lic circuit called HYDROCAli designed by Moor (63) is discussed below.

Ref. 63 gives the use of hydrocal in the solution of a simple heat

transfer problem - the heating of a rod from one end, no heat flow being

permitted from the other end or from the side area. As shown in the fig. 1,

the rod is divided into five incremental blocks. Thermal capacity of each

block is arbitrarily lumped at the centre of the block. Six glass tubes called

standpipes are mounted as shown in the fig, 2a and 2b. The pipes

numbering from 1 to 5 represent the five ttiermal capacities of

the five increments. Quantity of water in the standpipe represents

stored heat in the rod increment. Quantity of water needed to raise

the level of water in the standpipe per unit length, say per centi-

meter, represents the increment's heat capacity. The level of

water represents the temperature level in the increment assuming

that the initial level in the standpipe corresponded to the initial

temperature level in the increment.

Each tube beginning from 0 to 4 is connected to the next

tube on the right hand side» Connecting tubes offer constant re-

sistance to water flow because the flow is maintained in the la.-

minar region by proper choice of dimensions. This constant re-

sistance to water flow represents the resistance to heat flow from

the centre of one block to the next. The resistance between the

standpipe numbered 0 and 1 is slightly different from other re-

sistances because it represents thermal resistance due to one

half of a block plus a surface heat transfer resistance. To represent

the higher temperature source used for heating the rod the stand-

pipe numbered 0 is connected to a water tank at a higher level. The

level of water in the standpipes is adjusted to the initial level of

the temperature in the rod. Connections are opened and the water

levels in the different standpipes start changing. At any time the

levels car», be directly interpreted as temperatures of the centres of

the various blocks. (See fig. 2b).

Bäckström (8) describes various application of the analogy in

studying transient heat flow problems connected with cold storage

of food. Five examples of increasing complications are described

in the ref. 8. One of the problems considered is illustrated in the

fig. 3a and fig. 3b.

The initial level of water in B represents the initial temperature

of the body put in the cold storage room. The level of water in S is

kept constant to simulate the constant temperature of the surroun-

dings. Similarly the initial level of water in R and C denote the tern-

- 6 -

pexature of the room and the cooling coils at the beginning of a transient

phenomena. The flow of water from the standpipe C is handregulated in

such a way that a valve is opened when the water level exceeds the level

corresponding to + Z C. The water flow is stopped when the level falls

below the level corresponding to - 8 C. In this way regulation with

"pressostat" in a cold storage room is simulated manually in an ana-

logue.

Differential equations for transient heat transfer and laminar

flow are presented together for a better understanding of the principles

governing the design of a hydraulic analogue (14).

Continuity and Navier-Stokes equation for laminar flow through

a circular pipe yield, the following equation for pressure drop over a

length dx of a capillary tube of diameter D .

32 ]x V dxdP = pg dh = ^ (1 ~ 3)

c

where P = pressure drop in the capillary tubing

p = density

h = difference in hydrostatic pressure in length units

\x - viscosity

V = velocity

x = distance

g = acceleration due to gravity

D = capillary diameter

Let Q = volumetric flowc

or, Q = ̂ D 2 V (1 - 4)c 4 c c x '

Substituting for V̂ , from equation (l - 3) in equation (1 - 4} one

obtains

IT. p g D dhQ = 2 ( 1 . 5 )

128 u dx

- 7 -

L e t

o r

IT

Q

P

1

c

g

28

D

V-

1B

4c

•

- =

dhdx

1B

(1 - 6)

(1 - 7)

where B is a constant for a particular hydraulic system.

Differentiating equation (1 - 7) w. r-, t. x one obtains

dx " B , 2 V *)dx

T o standpipe cross section

capillary length

A length of tube Ax over a time increment A 6 is considered

here. Storage capacity per unit length in the standpipe for a length

Ax of the capillary tube is S A x. If Q- and Q~ are rate of volumetric

flow entering and leaving a capillary tubing of length Ax over a time

interval of A 8 then

Q_ A9 - Q, A9 = h (S Ax) (1 - 9)

Let Q2 - Qt ~ AQ c

A Q

Ax " ö A0 ^ I

Assuming that the differences are very small one obtains

^ = s | | (1 - 11)

dx dö x '

Combining equation ( 1 - 8 ) and equation (1 - 11) one obtains

^ „ J_ A , sd8 ~ SB , 2 K ' 'dx

Let ^L= a1

2dh 1 d h

o r dW = a T~2dx

The differential equation governing one dimensional heat flow

through an infinite slab is as follows

where t = temperature

9 = time

I. 1 Application in air-conditioning

In designing air-conditioning systems one needs detailed informa-

tion about the magnitude and the variation of cooling loads» A standard pro-

blem in air-conditioning is the evaluation of cooling load for a room which

has a sunlit glass or has artificial lighting» This evaluation is very compli-

cated because of the interaction of radiation, convection and conduction phe-

nomena. The energy input to the space of the room throxtgh the sunlit glass

can be classified as follows (52) (59) (60)

i) direct solar radiation

ii) sky or diffuse radiation

iii) radiant energy from the room surface of the glass to the

walls of the room

iv) convectjve energy to indoor air from the room surface

of the glass

The last of the above mentioned four posts is the only part which

immediately affects the cooling load« Radiative heat energy is not directly

absorbed by the air in the room. The floor, the ceiling and the- walls absorb

radiative heat energy and become hotter» The rate at which the temperature

of the surfaces go up is a function of the following factors?

i) thermal capacity of the walls, the floor and the ceiling

ii) thermal conductivity of the wall materials

iii) coefficient of absorption and emission of the surfaces

_ 9 -

As the surfaces get hotter heat is transmitted to the air layer

in contact with the walls by convection.

The heat transfer phenomena described above is very complex

for any analytical solution. A full-scale experiment is impractical.

Such an experiment is of doubtful value when the parameters are

changed in different design problems,

Leopold (52) and Mackey et al. (59) (60) have described some

hydraulic analogues designed under some simplifying assumptions

for different air conditioning problems.

Leopold (52) has described a hydraulic analogue which was

used to study the cooling requirements for a room which has indi-

rect illumination with load equivalent 36. 6 W/m . Two cases were

considered.

Case A. An amount of air was introduced in the room which

would be sufficient to maintain the temperature of

the air constant in the room if there were no ther-

mal storage. The air temperature variation with

time was studied.

Case B. Variation of the amount of air supplied to maintain

a constant temperature in the room.

Fig. 4 shows a sketch of the system under consideration and

the fig. 5 shows the corresponding hydraulic analogue for the case A

described above.

Mackey et al» (60) describes another case study with their

hydraulic analogue of an air-conditioning problem which was si-

multaneously solved analytically. Results from the analogue com-

pare favourably with the results from the analytical study. This

proves the reliability of the analogue systems.

Mackey et al. (59) describes another application of the same

analogue in which cooling loads and instantaneous rate of heat gain

from a sunlit glass and wall were studied over an analogous time

period of 24 hours» One singular feature of this experiment was that

one had to start with fictitious values of temperature and heat flow in

- 10 -

the beginning» After running through the whole cycle of 24 hours which

was 24 minutes in the model 4 or 5 times, one obtained an exactly re-

producible cyclic variation. The total amount of heat input due to solar

radiation should be equal to the heat removed by the air flow over a pe-

riod of 24 hours,, The rate of heat input due to solar radiation varies in

such a way that it has the same value after 24 hours. In the hydraulic ana-

logue this condition is fulfilled by ensuring that after a complete cycle the

levels of the fluid in the standpipes of the analogue should be same as they

were in the corresponding time in the previous cycle.

The heat input due to solar radiation varies cyclically over 24 hours,

Fig. 6 shows the corresponding variation in cooling load obtained with the

help of the analogue» In the analogue a periodic flow pump is used to obtain

a variation in flow to simulate a variation in radiant heat input,

I. 2 Application in design_of regenerators

Duhne (14) describes a simple analogue to study the transient heat

transfer phenomena in an infinite slabs The infinite slab is originally at

an uniform temperature t-. The temperature of the surface is raised to tC

Duhne (14) mensions that the comparison of data from the analogue and a

mathematical solution indicated a mean deviation of 0,47 percent and a

maximum deviation of - 1.20 percent.

Duhne and Garcia (15) have used a similar hydraulic analogue to

prepare charts for the solution of problems of heat storage in bodies of

parallelopiped form. These charts can be used for designing regenerators

in which the temperature rise in the gas is small.

Results from the analogue model for one-dimensional heat flow

through an infinite slab is used for calculating a 3 dimensional phenomena,

A parameter Z is defined as follows:

Q - QZ«-g ( 1 - 1 5 )

00

where Q = heat stored by the solid at any time, above its initial tempera-ture t

oQ = heat stored by the solid above its initial temperature t when

9=0 secondsoo

-li-

lt can be shown that (1 5)

Z = Z Z Z ( 1 - 1 6 )x y z \ . /

where the suffixes x, y and z denotes the cases where the heat flows

one dimensionally in the x, y and z direction respectively. The equa-

tion (1 - 16) makes it possible to evaluate Z , Z and Z separately

from the charts prepared with the help of the analogue mentioned

above.

I, 3 Steady-state heat exchangers

All the hydraulic analogies which have been discussed up till

now solved problems connected with periodic heat flow, Juhasz and

Hooper (28) have presented a method for studying steady-state heat

exchangers using a hydraulic analogue.

The analogue was developed only for parallel flow heat exchanger

problems. In a parallel flow heat exchanger as shown in the fig. 7 .the

temperature difference between the two fluids decrease towards the

outlet. In a hydraulic analogue the difference in level between two

standpipes connected with a capillary tube decreases with time. In

the analogue proposed by Juhasz et al, time in the analogue is equi-

valent to relative area of heat exchange.

Let y = relative surface in the heat exchanger, dimensionless

Then y = 0 at the inlet and y = j at the outlet to the exchanger

Let 8 = time in the model, hours

Then y = n 8y y

The following relations are valid for the system described

above;

At=n t - A, (1 - 17)

y = n y 8 ( 1 - 1 8 )

D 4

2

1 7 -

Ai, C2 p ?« St - (1 - 20)

m. C1 Pi

CP?

ÄTT

where,2A = area of parallel flow heat exchanger, m

C = volume of liquid flowing through the capillary of the

model per -unit time and unit he ad, meter /meter, hour

C = specific heat of fluid 1 at constant pressure, Kj/kg °C

C n specific heat of fluid 2 at constant pressure, Kj/kp °Cp2

D « inside diameter of capillary tube, meter

K = constant, l/meter, hour

JL = length of capillary tube, meter

m, - mass flow of fluid 1, kg/hour

m9 ~ mass flow of fluid 2, kg/hour

n Ä conversion factor between time in the model and relative-1area in the heat exchanger, hour

n = conversion factor between height in the model and theIs

temperature in the heat exchanger, C/meter2

S, = cross section area of tube 1, meter2

So " cross section area of tube 2, meterCa

U = overall heat transfer coefficient in the heat exchanger,

KJ/meter • hour • C= temperature difference between the fluid 1 and the

fluid Z, °C

= level difference in the model between levels 1 and 2, meter

'13 -

Juhasz et al. (29) discuss certain additional attachments to the

analogue described above to take into account the following conditions.

a. variation in specific heat with temperature

b. change of phase

c. heat loss to the surroundings

d. variation in overall coefficient of heat transfer with

temperature.

Variation in specific heat affects the rate at which the tempera-

ture of the fluids rise. In the analogue it will be reflected as a varia-

tion in the rate at which the height, of the liquid columns in the stand-

pipes will change for the same rate of flow through the capillary tube.

This is achieved by inserting a template in either of the standpipes to

vary the internal area of the standpipes with height. The profile of the

template corresponds to the variation of specific heat with temperature»

Change of phase affects the heat transfer rate in the following ways

1. Temperature of the fluid undergoing change of phase stays

• constant irrespective of the rate of heat transfer» It is

equivalent to an infinite specific heat. This is arranged in

the model by connecting a reservoir of a very large diameter

in parallel with the tube corresponding to the fluid undergoing

a change of phase»

2. Coefficient of heat transfer changes significantly, A different

set of capillary tubes corresponding to different values of U

can be connected in parallel.

Heat leakage to the surroundings can be simulated in the following

way. The tube representing the fluid from which the leakage takes place

is connected to a large reservoir whose level corresponds to the tem-

perature of the surroundings. The capillary tube connecting the large

reservoir to the tube is selected in accordance with the magnitude of

the over-all coefficient of heat transfer between the fluid and the

surroundings.

- 14

Variation in the over-all coefficient of heat transfer with temperature

can be taken into account in the same way as described above in connection

with change of phase. Different capillary tubes in parallel can.be used for

different levels of the liquid in the tube.

I._4_ __Transient conditions in^garallel and cpunter flow heat-exchangers

In the process industry it is very common to find heat exchangers

with automatic control. För an effective design of the control system it is

necessary to understand the transient behaviour of such heat exchangers.

Analytical solutions describing transient conditions are often very compli-

cated. Added complications like longitudinal heat conduction as in liquid

metal heat exchangers or like variable specific heats make the analytical

solutions more complicated* For an easier understanding of the problem

Juhasz and Clark (29) had proposed a hydraulic analogue to study, under

some simplifying assumptions, the transient conditions in cotinterflow and

parallel flow heat exchangers» The analogue is valid for a co-axial heat

exchanger. The inner tube carries the hot fluid and the cold fluid flows

through the annulus. The basic principles involved in designing the ana-

logue are same as they were in other analogues for periodic heat flow de-

scribed previously. The novel feature of this analogue is that the movement

of the liquids with respect to the tube and the shell is simulated with the

help of rotary valves. The principle behind the use of rotary valves is de-

scribed below. For the sake of clarity a more simple example than the one

given in ref. 29 is described here,

A body A which is divided into 4 incremental blocks is heated with a

liquid B. The volume of the liquid B flowing past the body A is also divided

into incremental blocks as shown in the fig. 8e

At 8 = 8 the whole of the body A is at the same temperature t . TheO zK

liquid B is also at an uniform temperature t .After a time interval tB from 8 , the block IB is in contact with the

o

block 1A. Heat flow takes place from the block 1B to the block 1A by con-

vection. Subsequently, heat flows from 1A to 2A, 3A and 4A by conduction.»

- 15 -

After a time interval A8 from. S + A6 , the block IB which isc. o

cooler due to heat loss to 1A in the previous interval is in contact

with the block 2A. Similarly, 2B is in contact with 1A. If the longi-

tudinal heat conduction in the liquid is not negligible then heat will

flow from 2B to 1B« The other heat flow paths are shown with arrow-

heads.

In this way as the liquid B moves relative to the body A the com-

bination of the blocks from the set A and the set B keep changing.

, In the hydraulic analogue proposed by Juhasz et al. (29) the

blocks are represented by standpipes containing water. For a time

internal A8, the standpipe 1A is connected to the standpipe 1B. In

the next time interval A 0_ the standpipe IB is connected to the stand-

pipe 2A and the standpipe 2B to 1A. These changes in the connections,

between the different standpipes at the end of each time interval is

conveniently and quickly accomplished by using rotary valves.

Since time in the analogue is equivalent to time in the heat

exchanger, when the x*otary valves are being reset there should not

be any flow between the standpipes belonging to the same set» This

is an added complication for the mechanical design of the system»

In the co-axial heat exchanger described, by Juhasz et al. (29)

one needs 4 sets of standpipes as described below.

Set

A

B

C

D

Description

Hot liquid

Inner tube wall

Cold liquid

Outer shell wall

To simulate the heat loss to the surroundings an additional

standpipe with constant level is connected to the standpipes represen-

ting the outer shell. Variable specific heat of the fluids can be taken

into account using templates as described in ref, 28. For a counter-

flow heat exchanger the standpipes belonging to the set B and the set C

will be connected in opposite order.

- 16 -

I._5 Air-flow analogue

All the analogues which have been described so far were based on

laminar flow of liquids,, Coyle (11) had designed, an analogue using air as

the flow medium* Fig» 9 shows a simple case of heating of an infinite slab .

of a constant thickness. Heating is done at the boundaries AA • and BB« ,

OO' is the line % of symmetry for the temperature profile. An air-flow ana-

logue for one-half of the slab is also shown in fig. 9»

The basic principles of an air-flow analogue is same as that of

liquid-flow analogues. The air pressure and mass of air in each standpipe

is equivalent to temperature and quantity of heat in the corresponding blocks

in the slab»

Variation in thermal conductivity was taken into account by using a

capillary the bore of which could be partially blocked for a varying pro-

portion of its length by a wire sliding within it9

With liquid-flow analogues the problem of air-locks limit the con-

venience with which cross-connections in a central exchange panel of a

large scale analogue can be made» With air or any other gas, however,

this problem does not exist. One could use a gas lighter than air to reduce

the end losses in the capillaries,, But the advantage of using air is that one

need not have a closed-system.

I. 6 Practi_c_aJ.j3roblems

Moore (63) has discussed the different practical problems associated

in constructing hydraulic analogues» The problems can b.e classified as

follows:

i) manufacturing tolerances

ii) hydraulic phenomena like meniscus effect in standpipes or

flow resistance in the rubber tubing joining the flow tubes

in series

iii) chemical phenomena like aging of rubber or corrosion

.Leopold (52) used a silicone liquid (Dow-Corning Corp. DC 20) of

20 centistokes kinematic viscosity. The change of viscosity with tempera-

ture is less than that for water. So a constant temperature bath was not

required for the conductances» The chemical problems reported by Moore

(63) were absent in the experimental" setup of Leopold (52).

- 17 -

Errors:

The errors in the system will be caus'ed by the following in-accuracies:

i) inaccuracy due to dividing a continuous system into a

system with finite increments. The larger the number

of divisions the less would be the error due to this,ii) inaccuracies in physical constants like thermal conductivity etc.

iii) errors in the physical dimension of the analogueiv) unaccounted for resistances in the flow tube paths. The errors

due to the last two inaccuracies would be of the order of 3percent, according to Moore (63).

II. Membrane analogy (7?) (81) (76) (24) (64)

Small deflections of the surface of an ideal membrane stretchedover a closed boundary and dilated by a pressure p satisfy the followingdifferential equation

- 4 s(x,y)+il 2(x,y) = -H (2 - ,)3x, 3y

An ideal membrane is one which has no mass and which is sub-jected to an uniform tension T per unit length of boundary. It is alsoassumed that the membrane deflections are sufficiently small so thatfor all points on the membrane the sine of the angle of inclination ofthe tangent plane from the horizontal may be taken equal to the angleitself, expressed in radians.

The Poisson equation for heat conduction with internal heatgeneration is as follows

2.. . 2 »»» i

where q = internal heat generation rate per unitX = thermal conductivity.

In the case of the stretched membrane if there is no dilation pressure

the governing diffential equation can be written as follows

z ( x , y) + » Z ( x , y) s o ( 2 - 3 )Sx^ 3y

Similarly-, if there is no internal heat generation the Poisson equation

becomes

= 0 ( 2 - 4 )

Equations ( 2 - 1 ) and (2 - 3) are similar to the equations (2 - 2) and

(2 - 4) respectively. The following scale factors are defined.

i) Geometrical scale factor {3

h A - length in the heat transfer problem

length in the membrane problem

ii) Temperature scale factor -y

, temperature 11»where v = *~ T~n—r= ;—:1 membrane deflection « z '

It can be shown that

,2

Using the above mentioned similarity, problems of 2-dimensional

steady-state conduction with internal heat generation can be solved with

the help of a membrane analogue. For s.olving a particular problem one

constructs the two-dimensional boundaries of the region. A soap film is

used as a membrane, The soap film is obtained (81) by using a soap solution

which contains sodium oleate, glycerin and water. The boundary tempera-

tures are duplicated by building up the edges of the model boundary in scale

to the boundary values of the problem. The deflection of the membrane

- 19 -

z (x, y) is interpreted as the temperature t at the location x, y. A

travelling micrometer gauge is used for measuring the elevations

at different locations» Handpowered machine screws with vernier

scales are used for determining x and y. According to Wilson and

Miles (81) the accuracy with which a measuring point can be located

is "t 0. 025 mm. Schneider and Cambel (77) mention that the accuracy

of these experimental solutions can be held to within - 3 per cent.

It is not possible to measure skin tension 7 per unit length of• f

boundary directly. — is indirectly calculated from an auxiliary film

stretched over a circular boundary (76). For problems in which

there is no internal heat generation this measurement is not ne-

cessary.

Aluminium is usually used as the material for the models to

reduce oxidation. The soap film is very sensitive to vibra.tion, tem-

perature variation and contamination. To prolong the life of a film

and to obtain consistent results it is necessary to perform the expe-

riment in a^ibration-free and temperature-controlled enclosure.

Durability of the film is increased by aging the fresh soap solution

in a closed vessel for a period of 24 hours. There is an aging effect

in the film. In the beginning the thickness of the film is non-uniform.

The membrane should be allowed to drain for at least one hour before

any measurements are taken.

Advantages of a membrane analogue is that it is inexpensive. For

• steady-state heat conduction problems in irregular geometries the so-

lution can be obtained in a simpler way than geometric electrical

analogues.

HE. Geometric electrical analogues

Geometric electrical analogues are mainly of two types

i) Electrolytic tank analogtie

ii) Conducting sheet analogue

- 20 -

Like the membrane analogues the geometric electrical analogues

are also continuous field analogues. For a clearer understanding of the

analogy the differential equation governing the steady-state electrical

conduction in a sheet of conducting material (either solid or liquid) is

presented below. Fig. 10 shows a differential element dx « dy of a con-

diicting material whose boundary is subjected to a given voltage distri-

bution e, . A distributed current i per unit area is supplied over its sur-

face. From the equation of continuity one obtains

\dy }. dx - i dx'y y( i +• - ~ - • dx ) • dy - i • dy -5- ( i + -r-̂ ~ * dy }V x 3x / y x ^ \ y 3 y > ' y

- i • dx • dy a 0 (3 - 1)

where

i = current per unit length of boundary in the x-direction

i = current per unit length of

From the equation ( 3 - 1 ) one obtains

i = current per unit length of boundary in the y-direction

3x

i and i can be rewritten in terms of voltage gradients

1 9eXx = " R ' I E

y R

where R = resistance between two parallel edges of a square. Substituting

equations (3 - 3) and (3 - 4) in (3 - 2) one obtains

3. 2.I® + ± ® -. - R i ( 3 - 5 )3 ^ ^

- 21 -

If no distributed current is supplied over the surface of the con

ducting material then,

2 23 e +

3 e „ n /->. _ i \

It is readily seen that a complete analogy exists between the

equation (3 - 5) and the equation (2 - 2) which is Poisson equation for

heat conduction with internal heat generation. The equation (3 - 6) is

the Laplace equation and is analogous to the equation (2 - 3) describing

steady-state conduction without generation (32).

HI. 1 _ j^eneral_descrip_tion_of_ the electrolytic J;ank_analogue_

An electrolytic tank analogue (1) (24) (2) (4) (58) consist of a

tank filled with an electrolyte. The geometric form o£ the tank is

similar to that of the heat transfer region to be investigated. The

tank should be large enough so that models of convenient size can be

used in it. It is easier to eliminate edge-effects in a large tank. If

there is no control system to maintain a constant temperature, a

large tank tends to stabilize rapid room-temperature variations.

Fig, 11 reproduced from the ref, 74 shows a sketch of an electrolytic

tank analogue.

A short description of an analogue designed by Langmuir, Adams

and Meikle (58), who were pioneers in this field, follows. Langmuir

et al. utilised the electrolytic analogue to study the heat flow pheno-

menon through a thick corner. The model consisted of an electrolytic

bath in a shallow tank» The hot and cold surfaces of the thermal cor-

ner were represented by metallic sides. The inner and the outer

surfaces i .e. , the metallic sides were connected to a power supply

to establish a potential difference across the liquid electrolyte repre-

senting the homogeneous wall material. The potential difference between

the inner and the outer surfaces was analogous to the temperature difr

ference between the two wall surfaces in the heat flow problem.

- 22 -

In the mathematical analogy described by the equations (3 - 5) and

(5 - 6) it is assumed that the conducting medium is purely resistive. Be-

cause of electrolysis effect, it is not possible to use direct current in this

analogy. With alternating current because of the polarization effect it is

difficult to obtain a purely resistive conducting medium. The junction of

the electrode and the electrolyte constitutes a capacitive impedance. The

effect of polarization is considered to be the inclusion of a reactive and

resistive impedance in parallel, in the immediate vicinity of an electrode

surface, which is in series with the electrolyte resistance. The magnitude

of the polarization impedance is determined by the electrolyte-electrode

combination and the frequency of the tank voltage,> A figure of merit F

is used for comparing different electrolyte-electrode combinations. It

is defined as,

( 3 - 7 )

where r = resistance per cubic centimeter of the electrolyte

s. = surface impedance of one square centimeter of electrode surface

The reciprocal of F is the equivalent extension of the length of the

electrolyte required to give the observed impedance. The higher the value

of F the better is the combination. The following table gives some results

from an experiment described by Amort (1),

Table 3 - 1

Combination F in l/cm

Tap water + Cu-electrode 7

Distilled water + polished brass 1 00

Distilled water + HC1 cleaned bradd 300

The frequency of the tank voltage is also to be taken into account.

- 23 -

Polarization effects are more pronounced at low frequencies.

If the frequency is too high, the effects of stray capacitances create

trouble. Amort (1) recommended frequencies from about 400 cps to

1500 cps. Anthony and Fridensohn (3) (4) and McNall, Jr . , and

Janssen (65) used 60 cps. A curve is reproduced (see fig. 12) from

the ref. 1 to show how resistance and capacitance components of

polarization for distilled water and polished brass electrodes vary

with frequency.

Nukiyama and Tanasawa (68) describes the use of an electroly-

tic analogue to study an axi- symmetric case of radial heat flow from

a finite hollow cylinder. The bases and the inner surface of the cy-

linder was at 0 C. The inner surface was kept at T °C. The percen-

tage error in calculating the heat flow by assuming the cylinder to be

infinite was evaluated with the help of the analogue. Since it was an

axi-symmetric case the tank was made in the form of a cylindrical

wedge cut by two radial planes both passing through the cylinder a%iso

When the cylinder was assumed to be finite the copper electrodes si-

mulating the bases and the inner surface of the cylinder was kept at

the same potential. To simulate the case of an infinite cylinder the

copper electrodes representing the two bases were replaced with in-

sulating plates. In an infinite cylinder with an axi-symmetric tempe-

rature distribution there cannot be any axial heat flow. By maintaining

the same potential difference in both the cases the variation in current

was measured to evaluate the error in neglecting end effects.

Pierre (74) used an electrolytic analogue to study the influence

of metallic frames on the insulating value of, and the temperature

distribution in the insulation of cold-storage chambers on board ship.

One of the problems investigated and its model is shown in fig. 13,

Pierre (74) had investigated different conducting mediums but had found

ordinary tap water to be best suited for the purpose.

- 24 -

The width of the tank between the copper electrodes representing

the external and the internal surface of the wall was increased to account

for the film resistances according to the relation given below.

\ . X. 6,8 = 6. + —i-+ — - + \ . S ^ (3-8)

int ext b

where

5 = distance between the copper plates

5. = thickness of insulationi

V = thermal conductivity of insulation

a. = thermal conductance of the film between the air in themt

cold storage chamber and the wall of the chambera , = thermal conductance of the film between the ambientext

fluid and the outer wall surface

6, = thickness of bearing structures forming part of the wall

of the cold storage chamber

X, = thermal conductivity of material forming bearing structure

in the experiment a constant voltage was applied across the copper

terminals. Decrease in the insulating value due to the presence of metal

frames was evaluated by measuring the current flows in the presence and

absence of the frame models in the bath. The following equation gave the

percentage reduction in the insulating value

ir R.y - i - R w 7/

vfr

where

i*. = the current flowing through the tank in the presence of the

frame model

i. = the current flowing through the tank in the absence of the

frame model

R. _ = the ohmic resistance of the model in the absence of the frameins

R, = the ohmic resistance of the model in the presence of the frame

- 25 -

P-Ji éPL^Iikuted *? ? §•* B^P^I ation

The electrolytic tank analogues which have been described so far

did not take into account distributed heat generation. The distributed

heat generation can be approximated by introducing current into the

electrolyte through a grid of input electrodes. The current to different

electrodes in a grid can be controlled and measured independently of

each other to simulate non-uniformly distributed heat generation.

McNall, Jr. , and Janssen (65) described an electrolytic analogue to

study the problem of heat conduction within;, a transistor. In a tran-

sistor the heat generated at the interface between the collector and

the base flows through several different materials of varying thermal

conductivity to an infinite heat sink e9 g, atmosphere. Fig. 14 shows

one of the configurations simulated in the analogue. Each piece of

material was represented by a geometrically similar shallow rectangu-

lar lucite tray containing a solution of potassium chloride in distilled

water. Gold plated copper strips were used to form the equipotential

lines in the boundary between the two different materials. Electrical

connection between adjacent trays were accomplished by soldering the

tips of the gold-plated copper strips in corresponding trays. The in-

put current to simulate the interface heat generation was fed at the

corresponding copper strips in the model» Interface resistance at the

clamped joint was represented by having a resistor between the two

trays representing the copper support and the aluminium chassis (see

fig. 14). Each strip in the interface, where heat generation was simu- •

lated, had an independent circuit which contained a 25,000 ohm resistor,,

Such a high value of resistance ensured that the effect of changing the

input current in one circuit on currents in the other circuits would be

nagligible.

The arrangement of the analogue was very flexible. Changes in

configuration were obtained conveniently by merely soldering and

unsoldering some of the strips in the.interface» The size and shape of

the lucite trays were easily changed by inserting insulating baffles0

The error due to lumping in the surface of discontinuties in the form.

of strips is discussed in detail in ref, 65. The error was of the order

- 2 6 -

o.f 0,5 C, The experimental accuracy was of the order of 6 - 8 percent.

It is mentioned that the accuracy of the analogue could have been improved

by increasing the frequency to 1500 cps from 60 cps and by using graphi-

tiaied. or platinized electrodes.

Malavard and Miroux (66) had -»jtilised an electrolytic bath with a

sloping bottom to study heat transfer problems with rotational symmetry.

Fig. 15 shov/3 sketches of such a problem and its analogue. The mercury

was contained in a heat ir.sulated glass tube with a hemispherical bottom.

The heat a our so v?ae at the centre of the surface at the position of the cathode

spot ox the arc which was assumed to be stationary. The heat was dissipated

by evaporation of mercury at the free surface, The heat transfer condition

at es-ch point of this surface caxi be expressed by tiie relation,

- \ g £ + . A . *(T)«0 (3- 10)

where

"k :s thermal conductivity of mercury

A » a constant related to the heat of evaporation

f (T) « function defining the pressure of saturated vapour, as

given by an experimental diagram»

In the analogue, the above equation gives a relation between the elec-

trode current and the potential. Correct settings were obtained by successive

approximation, starting from a plausible temperature distribution.

•P?i §.. -The electrolyte tank analogues which have been described so far did

net contain any special arrangements for temperature control of the electro-

lyte. The conductivity of most of the electrolytes is a strong function of tem-

pevatare. For example» the resistivity of a weak solution of potassium chlo-

ride in distilled watsr decreases about 1,8 per cent for each 1 °C tempera-

ture increase (65). McNall and Jans sen (65) allowed a maximum temperature

rise of 1,7 °C in their experiment. Anthony and Fridensohn (3) (4) had de-

signed an electrolytic tank with 7, 920 electrode positions. A hydraulic system

was incorporated to continuously circulate the electrolyte. The purpose of

circulation was to maintain a constant temperature and to reduce the effects

of contamination and evaporation. The hydraulic system also permitted a very

- 27 -

accurate control over the electrolyte level in the problem. Sodium

chromate was used as the electrolyte. The conductivity of sodium

chrornate solution changes 2. 5 percent per C. To keep the con-

ductivity of the solution constant within 0. 5 % it was necessary to

maintain the temperature within 0. 2 C.

The analogue was designed to handle heat transfer problems

encountered in nuclear reactor design. Distributed heat generation

due to fission or neutron absorption is common in such problems.

In the analogue the distributed heat generation was simulated by

introducing current into the electrolyte through a grid of input elec-

trodes. Each electrode had a resistance in series which was 100

times larger than the other resistances in the system. That made

the current in each electrode practically independent of all other

adjustments made in the system while the problem was being set

up. The sum of the currents fed into the system were collected

in sinks representing coolant passages in a nuclear reactor core.

Maximum number of sinks available were 20. The experimental

accuracy, verified against an analytical solution of a standard pro-

blem, was of the order of 3 per cent.

III. 6 Sources of error in the electrolytic tank analogue

A chart based on discussion of errors, presented by Amort (1)

is given below. For a more detailed information ref, 1 7 should be

consulted.

Type of error Precautions to Establishedminimize error magnitude of error

Presence of probes Experimental deter- Very smallwith finite spacing mination of best

combination of elec-trodes and electro-lyte

Meniscus, probe Less than 1 per centpenetration andcapillary action

Variation of probe Levelling of probepenetration depths carriage

- 28 -

Tank-edge meniscus

Electrolyteevaporation

Variation inelectrolyte depth

Combined tankerrors

Mechanical errors

Fill tank exactly to up-per edge of the model:add wetting agent

Constant circulationof electrolyte

Levelling of tank

Large figure of me-rit for electrod-electrolyte combi-nation, sufficienttank size for accu-rate models, tanklevelling

Minimize backlashand friction. Struc-ture must be rigid

Depends on ambienthumidity and timefor problem solution

Can be made in-significant

Can be held to 0. 2 percent (see ref» 17)

III. 7 General description of the conducting sheet analogue

The basic equations governing the potential distribution as a re-

sult of current flow in an electrically conducting sheet are same as those

governing current flow through an electrolyte. In a conducting sheet ana-

logue the sheet is given the same shape as the prototype thermal system

(75). An isothermal boundary is simulated by an equipotential boundary

by using a highly conducting material. An adiabatic boundary is represen-

ted by an electrically insulating material. Fig. 16 shows a sketch of a

simple conducting sheet analogue. In a conducting sheet analogue the power

supply can be either d-c or a-c. In a conducting sheet analogue there is

no problem, of electrolysis as in the case of electrolytic tank analogue.

The conducting material most commonly used is Teledeltos facsimile

paper (27). Stainless steel sheets were used by Fitts et al. (21).

Kayan (3 8) (39) (40) (42) has done an extensive amount of work in

the development of analogues using conducting sheets,, It was even pro-

posed that analogues of this type should be called a "Kayanalogger".

In a conducting sheet analogue, using a uniform sheet, the dimen-

sions are obtained in the same way as in an electrolytic tank analogue.

"In using uniform sheet, linear distances o.n the sheet are commensurate

29 -

of resistance to heat transfer between the boundary fluid, at its pre-

vailing temperature, and the wall itself (45). " In ref. 39 conducting

sheets were used to study problems similar to the ones studied in

ref. 74 using the electrolytic tank analogue,

Kayan (45) has utilised the conducting sheet analogue to study

a great variety of problems. Some of the more important variations

are listed below.

1. Problem of an embedded structural element within a wall (39)

2. Problem of heat transfer pipes embedded in a slab (42)

3. Problem of wall of a multiple-component material (40)

Fig. 1 7 is a description of a problem in refrigeration (40). Fig. 18

is a description of the conducting sheet analogue for the same problem.

The aim was to obtain the steady-state internal temperature lines, in

the analogue the solid wall was considered to be the basic material in

the problem. The lengths of the actual heat flow paths were directly

represented on the conducting sheet. The insulating wall, however, has

higher resistance. To account for that, the electrical characteristics

of the sheet representing the insulation was modified.

By making perforations (see fig. 1 8) in the conducting sheet the

effective electrical unit resistance of a given section was altered. This

alteration was made in progressive steps during an investigation. This

enabled one to make a progressive study of the effect of different values

of thermal resistance in insulation.

Uniformity of the sheet electrical conductivity in all directions is

an important condition for accurate results. The resistance ratio bet-

ween the solid sheet and the perforated sheet should be the same in all

directions. This is dependent largely upon the accuracy of cutting and

duplicating the mesh. Kayan (40) obtained an apparent deviation of less

than 5 percent.

IV» Network analogues

Network analogues (32) are the most important form of analogues

because of their flexibility and ease of measurement. The mathematical

- 30 -

basis for direct network analogue simulators lies in the calculus of finite

differences. The electrical analogue model is constructed by recognizing

the formal similarity between the finite difference equation applying to

any node point in a discretized system and Kirchhoff 's current-law equation

for a node formed by passive electrical elements.

A discretized system in the form of line segments is shown in the

fig. 19a. The nodes are numbered as Z. 0 and 1. The transient or steady-

state values of the dependent variable y at these three adjacent node points

are indicated by y?, y^, y. . It can be shown that

4 ^ ^cbc h 2 hT

where h = Ax.

Considering a second-order differential equation

40 (4 " 2>dx

Substituting from, the equation ( 4 - 1 ) the equation (4 - Z) can be

rewr i t ten as

2h h

( 4 - 3 )

Fig. 19b shows a node formed in an electrical circuit by the junction

of two resistors. From Kirchhoff's current law, one obtains

i 1 + i 2 = 0

o re i " e0 . e2 " e0— R - + — R —

- 31 -

A comparison of the equations (4 - 3) and (4 - 4) indicate that

a typical node of fig, . 19b is analogous to a typical node of the finite

difference grid if

The voltage e« at the node 0 is then proportional to y^, the

dependent variable.

Network analogues fox steady-state problems consist of only

resistors» Unsteady-state problems may be represented by either

pure resistance networks or resistance-capacitance networks.

iy^ l Network analogues j£pj_ steady ^

A network analogue describing a s teady-s ta te heat t ransfe r

phenomenon consis ts only of r e s i s t o r s (24) (75) (18) (21), Uniformly

or non-uniformly dis tr ibuted heat generat ion can be simulated by

the introduction of flux at the network junctions,,

Design of a simple analogue is descr ibed below (18). A fin

attached to a wall is shown in fig. 20a» It is assumed that there

is no heat t ransfer from the free end of the fin. The fin rece ives

heat from the hot gas around it. The gas is kept at t C. The wallo *?

tempera tu re is t C. The res i s t ance R~, represent ing the gas to

surface heat t ransfe r for the four sides of each fin segment, is

given by

•R ~ __JL_ (A _ K\K o " * o p f h

{ b)

where

a~ ~ heat transfer coefficient from the gas to fin

Pj. = perimeter of the fin segment receiving heat

h = width of a fin segment receiving heat

The lumped thermal resistance R, between each node point

in the fin is given by

- 32 -

where

X.£ = thermal conductivity of fin material

A,. = cross-sectional area of thin fin

From the equations (4 - 5) and (4 - 6) one obtains

Rn t \ ,A ,u _ ', c

x x :- g (gay) (4 _ 7}

In an experims ntal set up either Rn or TX. can be arbitrarily

chosen. The value of the other resistance is given by equation (4 - 7).

The value of B is determined by the geometrical and boundary con-

ditions of the problem. The superimposed voltage from the voltage

source will be directly proportional to the difference in temperature

between the gas and the wall. Fig. 20b shows the network analogue

for the problem shown in fig0 20a«

Network analogues have been used very extensively to study

steady-state heat transfer problems. Two interesting examples are

cited below. Ellerbrock, Jr. , Schum and Nachtigall (18) utilised

the network analogue to study the temperature distribution in a cooled

turbine blade. Three different network analogues were designed for

the following blades;

1. 13-fin shell-supported air-cooled blade

2. strut supported air-cooled blade

3. liquid cooled blade.

The local values of temperatures obtained from the analogue

were not in complete agreement with the values obtained by ana-

lytical calculations. The differences, however, were not anything

serious. In the midchord region experimental values obtained from

an actual heat transfer experiment indicated that the analogue values

were nearer truth. Lumping of resistances did not introduce any

- 33 -

serious error in the result. Reducing the number of subdivisions in the

blade-analogue from 55 to 31 elements changed the local temperature

differences by about - 1 per cent. Even though the local temperatures

obtained from the analogue and mathematical analysis showed some

difference, the average temperature of the blade obtained by the two

methods showed very good agreement,

Fitts, Flemons and Rogers (21) designed a combined network

and conducting sheet analogue to measure sheath temperatures of

finned fuel elements,, Stainless steel sheet was used as the conducting

medium. Lumped resistances were utilised to simulate the boundary

film. Resistive elements were attached to the boundary of the conduc-

ting sheet. Fitts et al, (21) mention that mechanical and material

tolerances introduced an error of about 4 per cent» Error due to

thermoelectric voltage was maximum 0. 1 7 C. Another possible

source of error was due to variation of the resistance of the conductors

with changing temperature. It was found to be negligible.

IV. 2 Advantages and disadvantages of resistance network analogues (55)

Advantages;

1. Any change in conductance can be easily accounted for by

changing the particular resistors

2. All the measuring difficulties inherent in the electrolytic

tank analogue are eliminated. There is no polarization

effect

3. Accuracy is very high. A large number of resistor compo-

nents gives rise to a statistical cancellation of errors

4. The problem, can be set up quickly

5. A great variety of problems can be tackled

Disadvantages:

1. The technique is cumbersome when applied to very compli-

cated geometries

2, An error is introduced because of lumping. This disadvantage

is not serious if the network is large enough

- 34 -

IV. 3 Network analojgue_s_for_ unsteady-state_probl^ms_[RC>ne_two_rk)

In solving an unsteady state heat transfer problem account rmxst

be taken of both thermal resistance and capacity. In a corresponding

network analogue the thermal resistances and capacities are simulated by

electrical resistors and capacitors respectively. The following table

indicates the different quantities which are analogous in the two systems;

Table 4 - 1

Variable Thermal Electrical

Potential temperature voltage

Flux rate of heat transfer current

Resistance resistance to heat flow ohmic resistance

Capacitance thermal capacity capacitors

Time time time

The corresponding equations are of the following form:

-. . , potential difference across a resistanceResistance = *- r;

flux_. .. net flux into a capacitanceCapacitance = —. r—-?—r *—? 1—TT—T-

time rate of change of potential

In the electrical system it is necessary that the resistance be inde-

pendent of the current flow through the resistor and that the capacitance

be the same for all capacitor voltage levels. There should not be any cur-

rent leakage from the capacitors. The network is not suitable for systems

where thermal conductance and capacitance vary with temperature.

Writing the equations governing the electrical and thermal problem

in the fig. 21

( 4 - 8 )

de2(e 2 + e , - eQ) = R e C - ^ (4 - 9)

e

- 35 -

where t = temperature

e := voltage

R= resistance

C = capacitance

T = time

Subscripts:

2. 0, 1 = node designations

e = electrical

t = thermal

Let T = D.7 (4 - tO)

t

q . .

where q» *

i

= A] e + Bj

= c t i

s= heat flux

» current

D,,, A-, B, and C. = scale factors

It can be readily shown that

R Cx z

R Ce e

(4 - 14)

The potential factors A. and B., can be chosen independently.

The requirement of similarity places no restriction on these factors.

All the quantities are chosen to get a suitable value of D., the scale

factor for time.

Practical values of C are less than 1 00 • 10 farad. Certaine

amount of leakage of current to the ground throxxgh the capacitors is

unavoidable. To make the ratio of the current through the analogue

to the leakage current as large as possible the maximum value of R

36 -

and C make the transient in the analogue very fast for any manual re cor-

ding. In almost all cases it is necessary to record the current flow auto-

matically. Amperege through the analogue is very very low because of

the high circuit resistance. Fast transients and low amperege necessitates

the use of complicared measuring instruments.

Paschkis (70) (71) (72) (73) has done an extensive amount of pioneering

work on network analysers. Using a network analyser Paschkis (72) studied

how the heat flow from the inside surface of a building wall changed when

the outside surface temperature changed periodically and the inside air

temperature was held constant. In many transient heat transfer problems

the direction of heat flow does not change in the time period being considered.

In this particular problem the direction of heat flow changed in each cycle.

The temperature - time curve for the outer-surface swung periodically

around a zero line, e.g. the constant room temperature. Fig. 22a is a

sketch of the heat transfer problem. The network analogue for the problem

is shown in the fig, 22b. The potential factors were chosen in such a way

that the daily cycle of 24 hours was represented by 768 seconds in the ana-

logue. The voltage corresponding to the outside surface temperature was

changed stepwise after every 4 seconds in the analogue, 4 seconds in the

analogue represented 15 minutes in the actual problem. The advantages of

using this analogue were:

1. Elimination of building of actual walls

2. Considerably shorter time for experiments

3. Elimination of the dependence on weather

4. Possibility of maintaining constant air temperatures

5. Possibility of investigating separately the influence of the

different variables (e.g. film conductance, etc.),

Paschkis and Baker (73) utilised the network analogue to study the

temperature distribution in the insulation round a steam pipe line as a func-

tion of time for a given time cycle of turning on and off the steam flow. Com-

parisons with actual experimental data indicated an excellent agreement,

- 37 -

Paschkis and Baker (73) also described a large permanent

electric model for general use. It consisted of 525 condensers di-

vided into 15 equal groups. The condensers ranged in size ffom

0.1 uf to 20 jif. All condensers were accurate to within .1 1 per cent.

All condensers had at least } 5, 000 megohms per microfarad insu-

lation resistance* The resistor boards contained four decade re-

sistors each, which together yielded any resistance value from

100 ohms to 1, 111, 000 ohms in steps of 100 ohms.

V. Comparison of analogues

Type of analogue

Hydraulic

Advantages

Cheap equipment,easy technique, verysuitable from heuris-tic point of view*apply to transientconditions

Dis advantage s

Limited accuracy,tedious calibrationof capillaries

Membrane Cheap equipment,applicable to comp-licated geometries

Limited accuracy,measuring difficulties,aging of membrane,very sensitive to am-bient conditions

Electrolytic tank

Conducting sheet

Applicable to compli-cated geometries

Cheap equipment,easy technique, app-licable to complica-ted geometries, notsensitive to ambientconditions

Difficult measuringtechnique, limitedaccuracy

Limited accuracy,scale distortion,2 dimension only

- 38 -

Type of analogue

Resistance network

RC-networks

Advantages

High accuracy, easytechnique, applicableto mathematicallymore complicatedproblems

Apply to transientconditions, highlyflexible

Disadvantages

Cumbersome whenapplied to complicatedgeometries and un-steady-state problems

Limited accuracy, spe-cialized applications,require simple geometries,expensive components andmeasuring instruments

Note:

In the above mentioned table the analogue computers have not been con-sidered. Only direct simulators have been duscussed. Unlike RC-networksanalogue computers can steer a system as in an on-line analogue computer.Components lika operational amplifiers in analogue computers do not havephysical counterparts. Analogue computers are extremely useful andflexible tools for studying steady and transient phenomena.

- 39 -

Acknowledgement

The author takes this opportunity to thank Professor Bo Pierre

of the Royal Institute of Technology, Stockholm, for his kind help and

keen interest in this work. The author is also indebted to the post-

graduate students of the Refrigeration and Applied Thermodynamics

Department for their kind help in procuring some of the references»

The support given by the Reactor Technology Section of AB Atomenergi,

Stockholm, is also gratefully acknowledged.

- 40 -

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- 42 -

24. GEBHART, BHeat TransferNew York McGraw-Hill 1961, 400-23.

25. HICKMAN, R SThe measurement of radiation configuration factors with para-bolic mirrorsProc Heat Transfer Fluid Mech. Inst. Seattle, Wash. 1962Stanford, Calif. Stanford Univ. Press. 1962, 89-94.

26. HUTCHEON, I C and SPALDING, D BPrismatic fin with non-linear heat loss analysed by resistancenetwork and iterative analogue computerBrit. J. Appl. Phys. 9 (1958) 185-91.

27. HOTCHKISS, GElectrosensitive recording paper for facsimile telegraph apparatusand graphic chart instrumentsWest. Un. Tech. Rev. 2 (1948) 176-87.

28. JUHASZ, S I and HOOPER, F CHydraulic analog for studying steady-state heat exchangersInd. Eng. Chem. 45 (1953) 1359-62.

29. JUHASZ, S I and CLARK, JHydraulic analogy for transient conditions in heat exchangersSwedish Association of Engineers and ArchitectsTrans, of The Heat Power Group (1957) 3, 2-13.

30. JUHASZ, S I and HOOPER, F CHydraulic analogy applied to crossflow heat exchangersNational Congress of Applied Mechanics 2. Proceedings ASMEAnn. Arbor. (1954) 805.

31. JOHNSON, KR and SUNDERLAND, J EHeat conduction through edge sections with convective boundaryconditionsAppl. Sci. Res. Sect. A, 12 (1963) 73-80.

32. KARPLUS, W JAnalog simulationNew York McGraw-Hill 1958.

33. KAY AN, C FCombined longitudinal wall-conduction and counter currentfluid-flow heat exchange via the resistance conceptBull int. Inst. Refrig. Annex 1961-3, 97-107.

34. KAY AN, C FErmittlung des zeitlichen Abkilhl- und Gefriervorganges bei platten-förmigen Körper.n mit Hilfe des elektrischen AnalogieverfahrensKältetechnik, 13 (1961) 80-84.

- 43

35. KAY AN, C F and McGAUGE, J ARheo-electric simulation of the freezing process in a slabBull. int. lust. .Refrig. Annex 1960-2, 115-28.

36. KAYAN, C FTransient behaviour of refrigerated space with a chilledbrineheat exchange system via rheo-electric simulation techniqueBull int. Inst. Refrig, Annex 1960-3, 67-74.

37. KAY AN, C FSome aspects of transient heat flow as associated with vapourmovement for refrigerated warehouse packagesBull int. Inst. Refrig. Annex 1958-4, 61-71.

3 8. KAY AN, C FHeat-transfer temperature patterns of a multicomponentstructure by comparative methodsTrans. ASME 71 (1949); 1, 9-16.

39. KAY AN, C FTemperature patterns and heat transfer for a wall containinga submerged metal memberRefrig. Eng. 46, 1946, 533-37,

40. KAYAN, C FAn electrical geometrical analogue for complex heat flowTrans. ASME 67 (1945) 713-18.

41. KAYAN, CF and McCAGUE, JATransient refrigeration load as related to energy flow conceptsASHRAE Journal 1 (1959) 77-82.

42. KAYAN, C FTemperatures and heat flow for a concrete slab with imbeddedpipesRefrig. Eng. 54 (1947) 143-51.

43. KAYAN, C FTemperature distribution in complex wall structures by geome-trical electrical analogueASRE 45 (1945) 113-17.

44. KAYAN, C FElectric analogger studies on panels with imbedded tubesAm. Soco Heat. Vent. Engrs 22 (1950) 123.

45. KAYAN, C FHeat flow temperature patterns of complex structures by re-sistance concept and electrical analogySwedish Association of Engineers and Architects. Trans, ofHeat Power Group. 2 (1956):1, 16

- 44 -

46. KETTLEBOROUGH, C FAnalogue study of temperature distribution in cooled gas-turbinebladesBrit. J. Appl. Phys. 6 (1955) 174-76.

47. KOMOSSA, HZur Ermittlung der Wärmeleistung berippter OberflächenKältetechnik 10 (1958) 92-96.

48. KOURIM, GDie elektrische Nachbildung der instationären thermischen Vor-gänge beim WärmeaustauscherRegelungstechnik 5 (1957) 163-67.

49. KRZHIZHANOVSKI, G M (ed. by)Heat transfer and thermodynamic modellingWash., D. C. , U.S. Dept. of Commerce, Office of TechnicalServices i960.

50. LAMB, J FSun effect and heat flow through brick walls as studied with anhydrocalHeating and Ventilating (1936) Aug. 32-35.

51. LAWSON, D I and McGUIRE, J HSolution of transient heat flow problems by analogous electricalnetworksProc. Inst. Mech. Eng. Ser«, A 167 (1953) 275-87.

52. LEOPOLD, C SHydraulic analogue for the solution of problems of thermal storage,radiation, convection and conductionTrans. Am. Soc. Heat. Vent. Engrs. 54 (1948) 389-406.

53. LIEBMANN, GNew electrical analog method for solution of transient heat con-duction problemsTrans. ASME 78 (1956) 655-65.

54. LIE B MANN, GSolution of transient heat-transfer problems by the resistance-network analog methodTrans. ASME 78 (1956) 1267-72.

55. LIEBMANN, GElectrical analoguesBrit. J. Appl. Phys. 4 (1953) 193-200.

56. LORETT, J A and DRUMMOND, WHydraulic analogue for condenser designEngineering 193 (1962) 733-34.

- 45 -

57. LANDIS, F and ZUPNIK, TEffectiveness of stub-fins as determined by the teledeltospaper analogASME paper 57-HT-20.

58. LANGMUIR, I, ADAMS, E Q and MEIKLE, F SFlow of heat through furnace wallsTrans. Am. Electrochem. Soc. 24 (1913} 53-84.

59. MACKEY, C O and GAY, N RCooling loads from sunlit glass and wallTrans. Am. Soc. Heat. Vent. Engrs 60 (1954) 469-86.

60. MACKEY, C O and GAY, N RCooling load from sunlit glassTrans, Am. Soc. Heat. Vent. Engrs (1952) 321-30.

61. McCANN, GD and WILTS, CHApplication of electric-analog computers to heat-transferand fluid-flow problemsTrans. ASME 71 (1949) 247-58.

62. MILLS, JAn analogue solution of the heating of a power cable buriedin the soilJ. Inst. Engrs, Australia, 30 (1958):3 91-98.

63. MOORE, A DA hydrodynamic calculating machine for solving unsteady-statehydrocals - problems in heat transfer and other types of diffusionInd. Eng. Chem. 28 (1936) 704-708.

64. MOORE, A DSoap film and sandbed mapper techniquesAppl. Mech. 17 (1950) 291-98.

65. McNALL, P E Jr and JANSSEN, J EAn electrolytic analog applied to heat conduction within transistorTrans. ASME 78 (1956) 1181-86.

66. MALAVARD, L and MIROUX, JElectrical analogies for heat transfer problemsEngineers digest 13 (1952) 417-20.

67. NEEL, C B JrInvestigation utilizing electronical analogue of cyclic de-icingof a hollow steel propeller with external blade shoe. 1952(NACA-TN-2852)

68. NUKIYAMA, S and T ANAS AW A, YOn an electric experiment upon the flow of heat, axial symmetryabout an axisJ. Soc. Mech. Engrs. Japan. 33 (193O):3 137-42.

- 46 -

69. PLATT, A and NORBURY, J FTemperature inequalities in the electrolytic tankJ. Roy. Aeron. Soc. 62 (1958) 456 (Tech. note).

70. PASCHKIS, V and HLINKA, J WElectrical analogue studies of the transient behaviour ofheat exchangersTrans. New York Acad. Sci. 19 (1957) 714-24.

71. PASCHKIS, VCombined geometric and network analog computer for transientheat flowTrans. ASME, Ser. C, J. Heat Transfer 81 (1959) 144-50.

72. PASCHKIS, VPeriodic heat flow in building walls determined by electricalanalogy methodTrans. Am. Soc. Heat. Vent. Engrs. 48 (1942) 75-90.

73. PASCHKIS, V and BAKER, H DA method for determining unsteady-state heat transfer by meansof an electrical analogyTrans. ASME 64 (1942) 105-12.

74. PIERRE, BInfluence of frames on insulation of cold storage chambers onboard shipTrans. Roy. Inst. Technol. , Stockholm, Sweden. No 50, 1951.

75. ROHSENOW, W M and CHOI, H YHeat, mass and momentum transferEnglew. Cliffs N. J. Prentice-Hall 1961, 444-68.

76. SCHNEIDER, P JThe Prandtl membrane analogy for temperature fields withpermanent heat sources or sinksJ. Aeronaut. Sci. 19 (1952) 644.

77. SCHNEIDER, P J and CAMBEL, A BMembrane apparatus for analogic experimentsRev. Sci. Instr. 24 (1953) 513-14.

78. SCHWARTZ, C H, GOLDBERG, S A and ORNING, A AThe use of visible light models for the study of radiant heattransfer in furnacesTrans. ASME, Ser. A, J. Eng. Power 84 (1962) 358-64.

79. STEPHENSON, D GMethods of determining non- steady- state heat flow through wallsand roofs of buildingsJ. Inst. Heating Ventilating Eng. 30 (1962) 64-73.

- 47 -

80. SHORTLEY, G H and WELLER, RThe numerical solution of Laplace 's equationAppl. Phys. 9 (1938) 334-48,

81. WILSON, LH and MILES, A JApplication of the membrane analogy to the solution ofheat-conduction problemsJ. Appl. Phys. 21 (1950) 532-35.

direction ofheat flow

insulation

Fig. 1 HEATING OF A ROD

initialwater level

Fig. 2 a. STAND PIPES WITH INITIAL LEVEL OF WATER

XFig. 2,b. WATER LEVEL IN STANDPIPES AFTER A TIME INTERVAL

Fig. 3 a. COLD STORAGE ROOM

B Body placed in a cold storage room

R Cold storage room

C Cooling coils

S Surroundings

B

start

R

C-8 C stop

i F

-35°C

FIg. 3 b. HYDRAULIC MODEL FOR A COLD STORAGE ROOM

Code for fig. 3 b

B Body placed in a cold storage room

R Cold "storage room

C Cooling coils

S Surroundings

t. Initial temperature of the body

t Initial temperature of the room

t-, Temperature of the cooling coil

The required resistances in the capillary tubes joining the different

standpipes are calculated by using the following relation,

c k a

wherej

wk

Kk

Ri Ki

k kWk

time factor between the actual periodic phenomena and,, , , seconds - actualthe model,

seconds - modelZ

area of a standpipe, m

thermal capacity of a body , j / C

total heat release from a body per C, j / S C

(F) Thermal storage

\

(H) 20 % total source energy (test constant)heats ceiling surface substantially in-dependent of surface temperature.

Ö

o

(Ö fl

'(C) Convectionceilingto air

(D) Convectionfloor toair

(J) Air supply vol.

(K) Air supply tem^r

Radiation leaving lamp

(A) 32, 9 % total source energy to airby convection and absorption ofr.-adiation by CO- and water vapour.

V(G) Thermal

storage

(B ) 47. 1 % total source energy heatsfloor surface substantially in-dependent of surface temperature.

Fig. 4 ANALYSIS OF ROOM LOAD

K

F i g . 5. DIAGRAMMATIC SKETCH OF ANALOGUE SIMULATINGL CONDITIONS OF FIG. 4.

Instantaneous rateof heat gain .

12 4 8 10 12 2 4 6 8 10 12

AM Time of day P M

Fig. 6. RESULTS FROM AN ANALOGUE

1 Btu/hr. sq. ft. = 3. 15 2

Saturation temperaturelevel reservoir

Tube 1steam

Starting level 1

Sat n temp, levelTube 2

Template

Roomtemp,ilevel

Capillary

Capillary

Capillary

Starting level2

e = o

(y )=0

Steam i n l

water in

9 = 9Time "(Area) (y) = 1 Total

water out

steam out

Fig. 7 DIAGRAMMATIC LAYOUT OF A PARALLEL FLOW ANALOGUE.

At e = e

i A 2 A 3 A H4 A

3 B 2 B 1 B

After 6 =

5 B 4 B HAfter e =

direction of motion

h B 2

1 A

B 1 B

K2 A 3 A

5 B 4 B 3 B

T1

21B l

2 A

H

3 A

I B

4 A

After e = A e 3 + A e 2 + A e 1 + e 0 -

1 A

5 B

2 A

4 B 3 B

3 A 4 A

2 B

Fig . 8 BLOCK DIAGRAM FOR TRANSIENT HEATING OF A BODY A

WITH A LIQUID B FLOWING OVER BODY A.

A

applied pressure changev\

A

A

r• i •

T | "1 • 11 1

!. 1

i

-8-1§

ii .I

T1 .|

i .i

ri •

1

B

liquid level h

datum pressure p

pumps

overflow

Fig. 9 A HEAT FLOW PROBLEM AND ITS AIR-FLOW ANALOGUE.

9i

X

Fig. 10. CURRENT BALANCE ON A CONDUCTING ELEMENT

B

FIG. 11. ELECTROLYTIC TANK ANALOGUE

A. Laminated wood board. B. Wooden laths.

C. Electrolyte (water). D. Copper bars.

E. Wooden laths with slots for copper bar D.

F, Copper frame model. G. Avometer,

3,2

<uoÖ(ti+->• r-iOni&O

800

600

400

200

0250 500 750 1000

Frequency, cps

1250

u •Sd

FIG. 12. RESISTANCE AND CAPACITANCE COMPONENTS

OF POLARIZATION FOR DISTILLED WATER AND

POLISHED BRASS ELECTRODES

insulation

cold storagechamber wall

plating on the ship's side

ext

6.i

copper bar

copper bar

metalframe

electrolyte

Fig. 13. SKETCHES OF A LONGITUDINAL SECTION OF WALL OF COLD STORAGE CHAMBER ON BOARD SHIPAND ITS TEST MODEL

A

Emitter (INDIUM)

Base (GERMANIUM)

Collector (INDIUM)

Support (COPPER)

Clamped joint withinterface resistance

Chassis (ALUMINIUM)

Lucite tray

Fig. 14 a. A TRANSISTOR CONFIGURATION(Heat generation along interface AA , andheat sink along BB )

Electrolyte

Gold-plated copper strips

Electrolyte

Fig. 14 b. ANALOGUE INTERFACE

Glass tube

Cathode spot

Surface ofMercyry

z-axis

Mercury

'100 VoltsAC

Axis ofrevolution

Electrodes

Insulating wall

Sloping bottom

Fig. 15. SKETCH OF A MERCURY VAPOUR RECTIFIER AND ITS ELECTROLYTIC ANALOGUE

Boundary Electrodes

T

DUCT

Voltage Divider

Probe

Fig. 16. ELECTRICAL FIELD ANALOGUE OF HEAT TRANSFERTHROUGH CORNER OF THICK-WALLED DUCT

line of symmetry

cementplasterfacing

concrete

insulation

concrete

17. CROSS SECTION OF BUILDING STRUCTURE

-AV.(air boundary) ,

(concrete floor)

(concrete slab)

(air boundary)

line of symmetry

esh (insulation)

e probe

Fig. 18. CONDUCTING SHEET ANALOGUE

h h

R R e.TT I , . , _

2 0 1

h

(a) (b)

Fig. 19. A DISCRETIZED SYSTEM AND ITS NETWORK ANALOGUE

Wall temp.t °C

w

Gas temp, t C

r - T

]

\

s s <i

/

(a) Fin illustration

Wall temp.

t °cw

-~LI Voltage source

Gas temp. t,.oC

R<=fe^ ..,. ,i

adiabaticsurface

measuringprobe

S LHi?_J.

h(b) Analogue wiring diagram

Fig. 20. ANALOGUE WIRING DIAGRAM FOR SIMPLE CASE OF FINATTACHED TO WALL

R / 2 R / 2

i

Fig. 21. A GENERAL NETWORK ELEMENT

P. S. = power supply

S. S. = selector switch taking anyvoltage from + 100 % to zero

So = switch taking voltage forconstant room air tem-perature

M = model, consisting of Rresistors, and C condensers

Fig. 22 b. CIRCUIT DIAGRAM

Outsidesurface

(periodic \variation \in temp.)

brick

concrete

plaster

room (constant temp, )

Fig» 22 a. DIAGRAM OF A WALL

LIST OF PUBLISHED AE-REPORTS

1—130. (See the back cover earlier reports.)131. Measurements of hydrodynamic instabilities, flow oscillations and bur

nout in a natural cirkulation loop. By K. M. Becker, R. P. Mathisen, O.Eklind and B. Norman. 1964. 21 p. Sw. cr. 8:—.

132. A neutron rem counter. By 1. O. Andersson and J. Braun. 1964. 14 p.Sw. cr. 8:—.

133. Studies of water by scattering of slow neutrons. By K. Sköld, E. Pilcherand K. E. Larsson. 1964. 17 p. Sw. cr. 8:—.

134. The amounts of As, Au, Br, Cu, Fe, Mo, Se, and Zn in normal and urae-mic human whole blood. A comparison by means neutron activationanalysis. By D. Brune, K. Samsahl and P. O. Wester. 1964. 10 p. Sw. cr.8:—.

135. A Monte Carlo method för the analysis of gamma radiation transportfrom distributed sources in laminated shields. By M. Leimdörfer. 1964.28 p. Sw. cr. 8:—.

136. Election of uranium atoms from UO2 by fission fragments. By G. Nilsson.1964. 38 p. Sw. cr. 8:—.

137. Personell neutron monitoring at AB Atomenergi. By S. Hagsgård andC.-O. Widell. 1964. 11 p. Sw. cr. 8:—.

138. Radiation induced precipitation in iron. By B. Solly. 1964. 8 p. Sw. cr.8:—.

139. Angular distributions of neutrons from (p, n)-reactions in some mirrornuclei. By L. G. Strömberg, T. Wiedling and B. Holmqvist. 1964. 28 p.Sw. cr. 8:—.

140. An extended Greuling-Goertzel approximation with a Pn -approximationin the angular dependence. By R. Håkansson. 1964. 21 p. Sw. cr. 8:—.

141. Heat transfer and pressure drop with rough surfaces, a literature survey.By A. Bhattachayya. 1964. 78 p. Sw. cr. 8:—.

142. Radiolysis of aqueous benzene solutions. By H. Christensen. 1964. 50 p.Sw. cr. 8:—.

143. Cross section measurements for some elements suited as thermal spect-rum indicators: Cd, Sm, Gd and Lu. By E. Sokolowski, H. Pekarek andE. Jonsson. 1964. 27 p. Sw. cr. 8:—.

144. A direction sensitive fast neutron monitor. By B. Antolkovic, B. Holm-qvist and T. Wiedling. 1964. 14 p. Sw. cr. 8:—.

145. A user's manual for the NRN shield design method. By L. Hiärne. 1964.107 p. Sw. cr. 10:—.

146. Concentration of 24 trace elements in human heart tissue determinedby neutron activation analysis. By P. O. Wester. 1964. 33 p. Sw. cr. 8:—.

147. Report on the personnel Dosimetry at AB Atomenergi during 1963. ByK.-A. Edvardsson and S. Hagsgård. 1964. 16 p. Sw. cr. 8:—.

148. A calculation of the angular moments of the kernel for a monatomic gasscatterer. By R. Håkansson. 1964. 16 p. Sw. cr. 8:—.

149. An anion-exchange method for the separation of P-32 activity in neu-tron-irradited biological material. By K. Samsahl. 1964. 10 p. Sw. cr8:—.

150. Inelastic neutron scattering cross sections of Cu'" and Cu*5 in the energyregion 0.7 to 1.4 MeV. By B. Holmqvist and T. Wiedling. 1964. 30 p.Sw. cr. 8:—.

151. Determination of magnesium in needle biopsy samples of muscle tissueby means of neutron activation analysis. By D. Brune and H. E. Sjöberg.1964. 8 p. Sw. cr. 8:—.

152. Absolute El transition probabilities in the dofermed nuclei Yb" ; andHfi". By Sven G. Malmskog. 1964. 21 p. Sw. cr. 8:—.

153. Measurements of burnout conditions for flow of boiling water in vertical3-rod and 7-rod clusters. By K. M Becker, G. Hernborg and J. E. Flinta.1964. 54 p. Sw. cr. 8:—.

154. Integral parameters of the thermal neutron scattering law. By S. N.PuroTiit. 1964. 48 p. Sw. cr. 8:—.

155. Tests of neutron spectrum calculations with the help of foil measurmentsin a D2O and in an H^O-moderated reactor and in reactor shields ofconcrete and iron. By R. Nilsson and E. Aalto. 1964. 23 p. Sw. cr. 8:—.

156. Hydrodynamic instability and dynamic burnout in natural circulationtwo-phase flow. An experimental and theoretical study. By K. M. Beck-er. 5. Jahnberg, I. Haga, P. T. Hansson and R. P. Mathisen. 1964. 41 p.Sw. cr. 8s—.

157. Measurements of neutron and gamma attenuation in massive laminatedshields of concrete and a study of the accuracy of some methods ofcalculation. By E. Aalto and R. Nilsson. 1964. 110 p. Sw. cr. 10:—.

158. A study of the angular distributions of neutrons from the Be* (p,n) B'reaction at low proton energies. By. B. Antolkovic', B. Holmqvist andT. Wiedling. 1964. 19 p. Sw. cr. 8:—.

159. A simple apparatus for fast ion exchange separations. By K. Samsahl.1964. 15 p. Sw. cr. 8 r -

160. Measurements of the Fes< (n, p) Mn5* reaction cross section in the neutronenergy range 2.3—3.8 MeV. By A. Lauber and S. Malmskog. 1964. 13 p.Sw. cr. 8:—.

161. Comparisons of measured and calculated neutron fluxes in laminatediron and heavy water. By. E. Aalto. 1964. 15. p. Sw. cr. 8:—.

162. A needle-type p-i-n junction semiconductor detector for in-vivo measure-ment of beta tracer activity. By A. Lauber and B. Rosencrantz. 1964. 12 p.Sw. er. 8:—.

163. Flame spectro photometric determination of strontium in water andbiological material. By G. Jönsson. 1964. 12 p. Sw. cr. 8:—.

164. The solution of a velocity-dependent slowing-down problem using case'seigenfunction expansion. By A. Claesson. 1964. 16 p. Sw. cr. 8:—.

165. Measurements of the effects of spacers on the burnout conditions forflow of boiling water in a vertical annulus and a vertical 7-rod cluster.By K. M. Becker and G. Hemberg. 1964. 15 p. Sw. cr. 8:—.

166. The transmission of thermal and fast neutrons in air filled annular ductsthrough slabs of iron and heavy water. By J. Nilsson and R. Sandlin.1964. 33 p. Sw. cr. 8:—.

167. The radio-thermoluminescense of CaSO^: Sm and its use in dosimetry.By B. Bjärngard. 1964. 31 p. Sw. cr. 8:—.

168. A fast radiochemical method for the determination of some essentialtrace elements in biology and medicine. By K. Samsahl. 1964. 12 p. Sw.cr. 8:—.

169. Concentration of 17 elements in subcellutar fractions of beef heart tissuedetermined by neutron activation analysis. By P. O. Wester. 1964. 29 p.Sw. cr. 8:—.

170. Formation of nitrogen-13, fluorine-17, and fluorine-18 in reac'or-irradiatedH2O and D2O and applications to activation analysis and fast neutronflux monitoring. By L. Hammar and S. Forsen. 1964. 25 p. Sw. cr. 8:—.

171. Measurements on background and fall-out radioactivity in samples fromthe Baltic bay of Tvären, 1957—1963. By P. O. Agnedal. 1965. 48 p. Sw.cr. 8:—.

172. Recoil reactions in neutron-activation analysis. By D. Brune. 1965. 24 p.Sw. cr. 8:—.

173. A parametric study of a constanl-Mach-number MHD generator withnuclear ionization. By J. Braun. 1965. 23 p. Sw. cr. 8:—.

174. Improvements in applied gamma-ray spectrometry with germanium semi-conductor dector. By D. Brune, J. Dubois and S. Hellström. 1965. 17 p.Sw. cr. 8:—.

175. Analysis of linear MHD power generators. By E. A. Witalis. 1965. 37 p.Sw. cr. 8:—.

176. Effect of buoyancy on forced convection heat transfer in vertical chann-els — a literature survey. By A. Bhattacharyya. 1965. 27 p. Sw. cr. 8:—.

177. Burnout data for flow of boiling water in vertical round ducts, annuliand rod clusters. By K. M. Becker, G. Hernborg, M. Bode and O. Erik-son. 1965. 109 p. Sw. cr. 8:—.

178. An analytical and experimental study of burnout conditions in verticalround ducts. By K. M. Becker. 1965. 161 p. Sw. cr. 8:—.

179. Hindered El transitions in Eu'« and Tb'". By S. G. Malmskog. 1965. 19 p.Sw. cr. 8:—. v

180. Photomultiplier tubes for low level Cerenkov detectors. By O. Strinde-hag. 1965. 25 p. Sw. cr. 8:—.

181. Studies of the fission integrals of U235 and Pu239 with cadmium andboron filters. By E. Hellstrand. 1965. 32 p. Sw. cr. 8:—.

182. The handling of liquid waste at the research station of Studsvik,Sweden.By S. Lindhe and P. Linder. 1965. 18 p. Sw. cr. 8:—.

183. Mechanical and instrumental experiences from the erection, commis-sioning and operation of a small pilot plant for development work onaqueous reprocessing of nuclear fuels. By K. Jönsson. 1965. 21 p. Sw.cr. 8:—.

184. Energy dependent removal cross-sections in fast neutron shieldingtheory. By H. Grönroos. 1965. 75 p. Sw. cr. 8:—.

185. A new method for predicting the penetration and slowing-down ofneutrons in reactor shields. By L. Hjärne and M. Leimdörfer. 1965. 21 p.Sw. cr. 8:—.

186. An electron microscope study of the thermal neutron induced loss inhigh temperature tensile ductility of Nb stabilized austenitic steels.By R. B. Roy. 1965. 15 p. Sw. cr. 8:—.

187. The non-destructive determination of burn-up means of the Pr-144 2.18MeV gamma activity. By R. S. Forsylh and W. H. Blackadder. 1965.22 p. Sw. cr. 8:—.

188. Trace elements in human myocardial infarction determined by neutronactivation analysis. By P. O. Wester. 1965. 34 p. Sw. cr. 8:—.

189. An electromagnet for precession of the polarization of fast-neutrons.By O. Aspelund, J. Bäckman and G. Trumpy. 1965. 28 p. Sw. cr. 8:—.

190. On the use of importance sampling in particle transport problems. ByB. Eriksson. 1965. 27 p. Sw. cr. 8:—.

191. Trace elements in the conductive tissue of beef heart determined byneutron activation analysis. By P. O. Wester. 1965. 19 p. Sw. cr. 8:—.

192. Radiolysis of aqueous benzene solutions in the presence of inorganicoxides. By H. Christensen. 12 p. 1965. Sw. cr. 8 r - .

193. Radiolysis of aqueous benzene solutions at higher temperatures. ByH. Christensen. 1965. 14 p. Sw. cr. 8:—.

194. Theoretical work for the fast zero-power reactor FR-0. By H. Häggblom.1965. 46 p. Sw. cr. 8:—.

195. Experimental studies on assemblies 1 and 2 of the fast reactor FRO.Part 1. By T. L. Andersson, E. Hellstrand, S-O. Londen and L. I. Tirén.1965. 45 p. Sw. cr. 8:—.

196. Measured and predicted variations in fast neutron spectrum when pene-trating laminated Fe-D2O. By E. Aalto, R. Sandlin and R. Fräkr. 1965.20 p. Sw. cr. 8:—.

197. Measured and predicted variations in fast neutron spectrum in massiveshields of water and concrete. By E. Aalto, R. Fräki and R. Sandlin. 1965.27 p. Sw. cr. 8:—.

198. Measured and predicted neutron fluxes in, and leakage through, a con-figuration of perforated Fe plates in D2O. By E. Aalto. 1965. 23 p. Sw.cr. 8:—.

199. Mixed convection heat transfer on the outside of a vertical cylinder.By A. Bhattacharyya. 1965. 42 p. Sw. cr. 8:—.

200. An experimental study of natural circulation in a loop with parallelflow test sections. By R. P. Mathisen and O. Eklind. 1965. 47 p. Sw.cr. 8:—.

201. Heat transfer analogies. By A. Bhattacharyya. 1965. 55 p. Sw. cr. 8:—.

Förteckning över publicerade AES-rapporter

1. Analys medelst gamma-spektrometri. Av D. Brune. 1961. 10 s. Kr 6:—.2. Beslrålningsförändringar och neutronatmosfär i reaktortrycktankar —

några synpunkter. Av M. Grounes. 1962. 33 s. Kr 6:—.3. Studium av sträckgränsen i mjukt stål. Av G. Ostberg och R. Attermo.

1963. 17 s. Kr 6:—.4. Teknisk upphandling inom reaktorområdet. Av Erik Jonson. 1963. 64 s.

Kr 8:—.5. Ågesta Kraftvårmeverk. Sammanställning av tekniska data, beskrivningar

m. m. för reaktordelen. Av B. Lilliehöök. 1964. 336 s. Kr 15:—.

Additional copies available at the library of AB Atomenergi, Studsvik,Nyköping, Sweden. Transparent microcards of the reports are obtainablethrough the International Documentation Center, Tumba, Sweden.

EOS-tryckerierna, Stockholm 1955