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Models for mixing in stirredvessels
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•
LOUGHBOROUGH
UNIVERSITY OF TECHNOLOGY
LIBRARY 1 ________ _
1 AUTHOR I
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MOOELS FOR MIXING IN STlRREO VESSElS
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L.G,GIBILARO.
MODELS FOR MIXING IN STIRRED VESSELS
by
L.G. GIBILARO
A Thesis
Submitted for the degree of
Doctor of Philosophy
Loughborough University of Technology
Supervisors: Mr. H.W. Kropholler
Dr. D.J. Spikins
Department of Chemical Engineering. May, 1967.
\
Contents:
Section
Acknowledgments
1 Abstract
2 Introduction
Page
1
2
3
3 Literature Survey
4
5
6
3.1.2.
3.2.1.
Mixing in stirred vessels:· the blending of miscible liquids.
Flow Patterns
Pumping capacity determinations
Dynamic testing of linear systems
Model fitting techniques: the method of moments.
Dynamic models for stirred vessels
5
7
10
13
15
Circulation Models
18
24
28
28
4.1
4.1.1
4.1.2
Solution of equations
Single loop models
Multiloop models
A general method for the solution of mixing models.
31
34
Theory
Markov processes
Application to flow models
Recirculation models
5.3 The computer programme
34
35
36
38
42
The truncated moments 44
6.1 Sensitivity of moments to experimental 45 error
6.2
6.2.1
6.2.2
Truncated moments 46
Experimental determination of 46 truncated moments
Determination of the truncated moments 47 from the process transfer function.
Section
7
8
The truncated mean
Description of the apparatus and experimental nrocedure.
7.1.2
7.2
7.3
7.3.1
7.3.2
7.4
7.4.1
7.4.2
The mixing vessels
The 100 litre spherical vessel
The 9 litre cylindrical vessel
Tracer injection equipment
The photocell detector
Construction
Calibration
The impulse response experiments
Runs on the 9 litre vessel
Dye balances
Determination of impellerpumping capacity using the flow follower technique.
Experimental results and comparison with the proposed model
8.1
8.1.1
8.1.2
8.1.3
8.2
8.3
8.3.1
8.3.2
8.3.3
Impeller pumping capacities
9 litre cylindrical vessel system
100 litre spherical vessel system
Experimental errors in the pumping capacity determinations
The proposed model
Results of impulse response tests
100 litre spherical vessel
9 litre cylindrical vessel
Sensitivity of model to pumping capacity measurement
The truncated mean
Estimation of errors in impulse response experiments
Page
50
56
56
56
60
62
63
63
63
66
67
68
69
71
71
71
71
71
75
76
77
85
98
100
104
Section
9
Page
Discussion 106
9.1 Comparison with other pUblished 107 results.
9.2 Conversion for a first order reaction 108
9.3 Suggestions for further work 111
9.4 Conclusions 113
Appendices 114
114
126
126
A1
A2
A3
Experimental Results
Com~uter programmes
A2.1
A2.1.1
A2.1 .2
A general computer programme r~ continuous flow mixing models
Programme running instructions
Programme details
Example: n stages in series
126
129
135
A2.2 Programme for the normalised response 137 curves and truncated moments of impulse experiments
Mass tracer injector
The mass tracer injection system
Tracer response runs using the mass tracer injector
139
139
142
Bibliography 144
ACKNOWLEDGMENTS
The author wishes to thank the following:
Professor D.C. Freshwater for his interest and encouragement; Dr. D.J. Spikins and Mr. H.W. Kropholler who jointly supervised this research; Mr. B.A. Buffham for his many valuable suggestions; the Science Research Council for their financial support.
1. ABSTRACT
1. Abstract
A generally applicable model for mixing in stirred vessels
is derived; it is based on the flow patterns and internal
circulation rates produced by the impeller which behaves as a
submerged pump.
The model is solved by means of a new and powerful numerical
technique which can be applied to mixing models consisting of any
configuration of well mixed stages; it is based on a probabilistic
treatment of an ideal mixing stage and leads to a flexible, easy
to use and efficient computer programme.
The truncated moments, a set of easily measureable and
model - independent parameters, are suggested for the
characterisation of dynamic responses; the first of these
moments provides a measure of the most significant features of
the proposed model and, for certain applications, gives a,direct
indication of the optimal mode of operation.
A simplified version of the general model is shown to fit
three quite different turbine stirred systems over a wide range
of operating conditions.
~'--""--~ .. ~------~.
2
2. INTRODUCTION
I
I
I
I
2. Introduction:
This study of mixing in continuous flow stirred vessels
was financed by a special grant from the Science Research Council;
it represents one of a number of investigations carried out in
the Department of Chemical Engineering into the dynamic
behaviour of chemical process eQuipment, the intended underlying
theme being the application of the Paynter stochastic analogy
to the analysis.
The objects of the work were twofold: to arrive at a simple
and realistic mathematical model for mixing in the continuous
flow stirred vessel whose behaviour deviates signficantly from
ideality; and to study the applicability of moments methods
in general, and Paynters generalised transfer function in
particular, to the characterisation of dynamic responses, and
as a means of testing and fitting proposed models.
For reasons discussed in Sections 3 and 6, direct moments
methods \"ere abandoned and an alternative set of model
independent parameters - the truncated moments - suggested as
being more available experimentally if less attractive
theoretically.
The model fitting method eventually chosen, however,
consisted of comparing directly experimental response curves
with the model solutions; the solutions were obtained by a
new numerical technique that is described in Section 5; a
simple model, readily adaptable to different stirred vessels
systems, was well confirmed experimentally.
Parts of section 8 and most of section 5 have been
• pUblished , as has a modified version of the computer programme
described in Appendix 2.17
..... ________ J~ ______ __
• Gibilaro, L.G., Kropholler, H.W., and Spikins, D.J., 'The solution of a mixing model due to van de Vusse by a simple probability method'. Chem. Eng. Science 1967 22 517-523.
Gibilaro, L.G., Kropholler, H.W., and Spikins, D.J., 'A general computer programme for continuous flow mixing models!. Paper presented to the Institution of Chemical Engineers Symposium. Efficient computer methods for the practising chemical engineer. Nottingham University. April 1967.
4
3. LITERATURE SURVEY
3.1. Mixing in Stirred Vessels: the blending of miscible liquids.
Hixson (1), writing in 1944, defined mixing as 'a unit
operation whereby particles of the components of a mass of materiaLs,
are put in some space relation to each other, so that some desired
result may be obtained'; and the purpose' of research as being to
minimise the time and energy expended in achieving this end.
The initial problem, then, was to devise means of measuring
mixer power consumption and the time required for mixing, and to
correlate these factors with the relevant properties of the fluid
and the design and mode of operation of the mixing vessel.
Following the pioneer work of White et al (2), considerable
research effort had been devoted to producing dimensionless plots
of 'Power number' against 'Reynolds number' for a wide variety of
impeller types, tank geometry and fluid properties. These plots
enabled the power requirements for the mixing of Newtonian Fluids
to be estimated with reasonable confidence but gave little
information as to the degree of mixing that ,,,ould ensue, or of
the time that would be required before the desired space relation
of particles had been achieved.
Many invesigators had defined criteria for assessing the
performance of agitators (1) but these were by nature limited
to particular applications and, as they were not based on
fundamental concepts, Were liable to lead to apparent anomalies.
Thus an agitator that had been found to be ideal for promoting
liquid - liquid emulsification would be quite unsuitable for the
blending of miscible liquids and vice versa.
It was left to Rushton and Miller (3) to establish generally
applicable criteria by means of which the performance of impeller
mixers could be judged. They defined four basic factors that
together characterised the 'fluid regime' in the vessel.
5
First is the power requirement: most of the research on
mixing had been concerned with this characteristic to the almost
complete exclusion of the others.
Second is the impeller discharge capacity: this is the
volumetric flow rate of fluid inside the vessel, and is brought
about by the pumping action of the impeller. For the blending
of miscible liquids this characteristic was found to be of prime
importance. The streams from the impeller, flowing through the
relatively stagnant bulk fluid, result in the transfer of
momentum between the two regions; material from the bulk becomes
entrained in the flowing stream, which spreads promoting mixing
in all parts of the vessel (4).
Third, is the Velocity of Discharge: this is the velocity
of the fluid leaving the impeller. For any particular application
there will be a certain minimum discharge velocity requirement
regardless of the discharge capacity; for miscible liquid
blending, this minimum will be close to zero, but for, say,
maintaining an even suspension of sand in water, there must be
sufficient upwards velocity in all parts of the tank, to overcome
the settling velocity of the particles.
Fourth and last are the Shearing characteristics. These
are important in cases such as emulsion formation where the
mechanical action of the impeller is required to provide the very
high local shear forces required. Metzner and Taylor (5) have
shown that the shear rates in an agitated fluid are very high
at the impeller, but decrease by several orders ·.ofmagnitude
within a very short distance; in this region the degree of
mixing, however defined, is likely to be very good even, perhaps,
':dbWn to:.the molecular level, but for miscible fluid blending,
mixing on this scale is unnecessary and, as the region in which
it occurs is small, unlikely to greatly influence the overall
performance.
6
It would appear then, that for the 'blending of miscible
liquids, the power input to the impeller is best utilised in
establishing large volumetric flow rates in the vessel, and that
the design of the impeller should seek to maximise this
characteristic at the expense of the others; and also that the
performance of an agitated vessel as a blender of miscible fluids,
whether under conditions of batch or continuous operation, will
be largely determined by this characteristic. This has been
demonstrated experimentally by Rushton (6) who determined the
relationship between impeller discharge capacities and discharge
velocities, for conditions of unit power input. The technique
consisted of running different sized, but geometrically similar
impellers at different speeds, so that the power absorbed in
each case was the same. Under these conditions a large impeller
produces a large volumetric flow at low discharge velocity;
whereas a small impeller, transmitting the same power, produces
a lower volumetric flow at a higher velocity. Processes studied
in this way revealed optimum volumetric flow/discharge velocity
ratioE" which for typical blending operations were found to be
high.
3.1.1. Flow patterns.
The use of impeller mixers for a large variety of applications
in numerous industries, became established long before the
fundamental principles of their operation had been seriously
studied; this has lead to the development, along empirical lines,
of the almost unlimited variety of impeller types in use today.
Van de Vusse (7) has broadly classified them according to the
way in which they transfer momentum to the liquid: impellers
of the cone and disc type transmit momentum by means of shearing
stress, perpendicular to the direction of flow; whereas for
impellers that produce strong circulation currents in the liquid,
7
,---------------------------------------------------------------------------------- - -
the transfer of momentum is in the direction of flow. Propeller
and turbine mixers fall into this second catagory as do all
impellers suitable for miscible fluid blending.
However, the flow patterns produced by propeller and turbine
impellers, are quite different (8): a propeller draws material
from above and discharges it vertically downwards; whereas a
turbine impeller acts essentially as an uncased centrifugal
pump, drawing material into its centre and discharging it radially.
In unbaffled cylindrical vessels, the flow patterns for quite
different impeller types become similar at high speeds: the
angular velocity component predominates and leads, eventually,
to vortex formation at the liquid surface. That this effect is
detrimental to most mixing applications has been demonstrated by
Van de Vusse (7) who derived an expression for the radial pumping
c~pacity for turbomixers. For the case of straight bladed
turbines, this expression reduced to:-
q = 2 2 2
". nd w (1 - b) . . . . . . . . . . . where n, d and ware the impeller speed, . diameter and blade width
respectively, and b is the ratio of the angular velocity of the
liquid to that of the impeller; so that the angular velocity
component, by increasing the ratio b, decreases the pumping
action of the impeller.
Figure 3.1 shows the basic flow pattern produced by turbines
in baffled cylindrical vessels (8): the angular velocity component
is virtually eliminated, the baffles effectively dividing the vessel
into discrete sectors.
8
; ....
: . . . . .....
Figure 3.1
It has been suggested in 3.1 that the most significant single
factor contributing to the performance of a stirred vessel as a
miscible liquid blender, is the mixing that occurs due to the
high velocity streams of liquid from the impeller, flowing
through the slow moving bulk. This mixing, due to momentum
transfer across the jet/bulk liquid interface, has been the
subject of considerable study for the case of a single jet of
liquid issuing from a nozzle into a large vessel. It has been
shown (9) that in this situation the jet can be treated as an
ideal jet - which emerges from a point source into an infinite
fluid; and a theoretical treatment by Folsorn (10), which
yielded results in good agreement with all the available
experimental evidence, showed the diameter of such a jet and the
entrainment flow (that is the quantity of liquid picked up by the
jet) to be simple linear functions of the distance from the
nozzle, both increasing with distance of travel of the jet; and
that all the fluid induced into the stream could be assumed to be
well mixed.
The complexity of the flow patterns in an agitated vessel
precludes such direct treatment of the problem,but the entrainment
flow has been observed and measured in more detailed studies of
the flow patterns in turbine (11) and propeller (12) agitated
vessels. Figure 3.2 shows the basic induced flow patterns for
9
these two systems: in each case the regions of induced flow are
completely surrounded by the main impeller-to-impeller streams
from which they derive their energy.
:::.:-:-: ':-:, :: " .:
,::y:: L}' . : : :
'. : •.
Ca) Propeller
Figure 3.2 Cb) Turbine
: :: ". " .....
. .
For the case of the propeller system the induced flow was
found to be a function of tank geometry, the ratio of induced to
impeller flow being dependent, to a considerable degree,on the
tank/impeller diameter ratio. No explanation to this phenomenon
was offered but, as the greater induced flows corresponded to
the relatively large vessels, the results are in qualitative
agreement with those on jet mixing reported above; the pure
pumping capacity of the impeller, on the other hand, was found to
be independent of vessel geometry.
For thin fluids the basic flow patterns in baffled vessels
are independent of impeller speed; this much is implied in the
above discussion and has been well established by Aiba (13) for
a variety of impeller types, including turbines and propellers,
operating in the turbulent range.
3.1.2. Pumping capacity determinations:-
Although a simple theoretical treatment of turbomixers (7)
leads to an expression for the pumping capacity of the impeller,
it is clear that in a true physical system a number of ill-defined
and intractable factors can greatly influence this effect; and
the same is true of propeller mixers, for which an even simpler
10
theoretical treatment is possible.
In order to make quantitative use of flow pattern studies,
realistic estimates of impeller pumping capacities ar~ required,
and once the importance of this parameter had been recognised a
number of experimental investigations followed.
Rushton et al (14) used a specially constructed double tank,
so arranged that the fluid was pumped by the impeller from an
inner to an outer vessel and so measured; the design of the
system sought to minimise the disturbance to the normal flow
patterns inherent in such a technique, and for propeller mixers
the pumping capacity was found to be proportioned to the impeller
speed and the square of its diameter, over the range considered.
The technique was also applied to turbine impellers: the results
indicated that the flow was again proportional to impeller speed
and that in one revolution the quantity of fluid pumped in the
radical direction was of the same order of magnitude as that pumped
in the axial direction by a propeller of the same diameter. The
disturbance to the turbine flow patterns, however, must have been
considerable as the method involved the physical separation of
the fluid leaving the impeller into supposedly equal, upper and
lower streams.
That the radial flow is proportional to turbine speed
was confirmed by Sachs and Rushton (15) by means of a photographic
technique that entailed no disturbance to the flow pattern:
illuminated particles in the fluiu were photographed, the exposure
time being such as to result in a particle appearing as a streak
on the developed film; measurement of a large number of these
streaks from different views of the vessel, enabled the magnitude
and direction of point velocities to be obtained throughout the
system. The results showed that in the annular space between
the turbine blades and the vessel wall, the radial velocity at
11
any point was proportional to impeller speed, as was the radial
flow at allannulL; the vertical velocity was found to increase
with radial distance from the impeller as indicated by the flow
pattern shown in Figure 3.2b.
Other reported methods for turbine pumping capacity
measurement (16, 17) involve the use of velocity measuring probes
of the pitot tube and thermistor type; an accurate traverse
of the impeller with such a device enables the total flow to be
obtained by an integration procedure; however, results obtained
in this way are influenced to some extent by the presence of the
probe in'the discharge stream.
A simple technique that can be used for any impeller type
and which does not disturb the flow patterns in the system being
studied, has been described by Marr and Johnson (18). It consists
essentially of finding the average time taken by a zero buoyancy
float to travel from the impeller, into the body of the vessel,
and back to the impeller; the principle of the method being that
for any closed continuous flow system the mean residence time of
fluid elements is given by the fluid holdup volume divided by the
throughput flow rate regardless of the mixing patterns in the
system (19). The only apparatus required, in addition to the
'flow follower' is a stop clock and, perhaps, a tape recorder.
This method was used to measure the pumping capacities of a
number of square pitch marine propellers in the turbulent range of
operation; the results were found to be best fitted by the
following expression for the pumping capacity,(q).
3 q = .61nD • • • • • • • • • • • • • • • • • •
where n is the impeller speed and D its diameter; and in another
paper (12) the flow follower technique was used to measure
entrainment as well as impeller flow for similar impeller types.
12
3.2.1.Dynamic testing of linear systems
Dynamic testing techniques (20) were firmly established
in the field of automatic control long before the attention of
the chemical engineer had been drawn to the possible application
of the theory of servomechanisms to the behaviour of chemical plant;
but onCe the advantages of an understanding of chemical process
dynamics became apparent - particularly with regard to improved
contro~ability - considerable research effort has been devoted
to the development of dynamic process models by mathematical
analysis and dynamic testing procedures.
The dynamic tests considered in this section are of the
stimulus response class, in which a black box approach to the
system under study enables its dynamic behaviour to be described
independently of the factors that govern it: a system, operating
in the steady state is disturbed and its·time varying response
to this disturbance, measured. The disturbance can take any
form, impulse, pulse, and step forcing being examples of one-shot
techniques, while continuous sinosiodal forcing is used in the
popular frequency response method (21); continuous disturbances
of a random nature have also·found favour in certain applications
(20, 22).
An impulse disturbance is one that for practical purposes
may be treated as a true Dirac delta function, its duration being
negligible compared with that of the system response that it
produces; a pulse disturbance is similar except that its
duration, as well as its shape, significantly affect the system
response; and a step function input is one in which a sudden
disturbance is sustained over the period of the experiment.
For linear systems all disturbances give the same
information; the results are interchangeable and the choice of
forcing function is dictated by considerations of
13
which would, for example, preclude the use of impulse forcing
on a system whose response time is very small, or of arbitrary
pulse forcing in the absence of adequate data-processing facilities.
Frequency response testing overcomes most of the practical
difficulties: two easily measured parameters are obtained from
the forced system when it has attained a new steady state and,in
certain control applications, direct use can be made of these
results (obtained over a range of frequencies), without the need
for any further processing (21); but this convenience has been
gained at the expense of a considerable increase in experimentation
time which becomes increasingly difficult to justify as the
availability of data logging and high speed processing
equipment becomes more general; especially when the frequency
response can be so readily computed from the response to an
arbitrary pulse (23, 24).
The interpretation of the response of continuous flow systems
to step and impulse tracer-concentration disturbances was
presented by Danckwerts (19) in 1953. These responses, were
shown to have probabilistic significance being closely related to
the internal and external age distribution functions respectively,
and following the publication of this work a considerable amount
of research has been devoted to the study of non-ideal mixing in
continuous flow equipment (25). Kramers and Alberda (26) drew
attention to the counterpart of this study in the theory and
testing of servomechanisms and considered the application of
frequency response techniques to chemical process equipment in
general, and to the study of axial mixing of fluid flowing
through packed beds, in particular} and Gutoff (41) used
sino.s61'dal concentration fluctuations both to analyse mixing
in ideal stirred vessels, and as a means of measuring departures
from ideality in real plant.
14
3.2.2 Model fitting techniques: the method of moments.
The term model is used for a set of equations that seeks to
describe the dynamic response of a system to an impulse disturbance;
the equations may be in terms of time, normalised time or the
:lLaplace variable Cs), and in later sections they will be
represented by the flow diagrams their physical interpretation
dictates. •
Given an experimental impulse response curve for a process,
the obvious way of determining whether or not it is adequately
represented by a particular model is to solve the equations and
compare curves; but because this can be a tedious and time
consuming exercise - particularly when more than one parameter of
the model must be determined from the comparisons - alternative
methods have been the subject of some study.
The frequency response technique, while in some cases
overcoming the analytical difficulties inherent in the solution
of the model, still leads to curve matching procedures in the
evaluation of the unknown parameters and so offers little
advantage over the more direct method - particularly when efficient
computational facilities are available for the model solution.
The use of moments for characterismg experimental impulse
response curves has received considerable attention during the
last ten or so years. Moments provide a convenient method for
characterising a probability distribution without making any
assumptions as to its nature; and as the impulse response for a
continuous flow system has been shOlm to be such a distribution (25)
characterisation by its moments would ap~ sensible; and other
systems where the probabilistic significance of the response may
be obscure - if, indeed, it exists - may be treated as analogous
to those for which this interpretation is apparent (27).
15
The first moment, or mean, which is taken about the origin,
locates the distribution on the time axis, while other moments,
which are taken about the mean, provide measures of its spread,
skewness and other, less apparent,features (28); so that given
the response curVe of a system forced by an impulse, any number
of characterising parameters can be obtained.
By definition the nth moment about the origin is given by:
M n
= l;}GCt)dt
j,GCt)dt
= (say) a n
a o • • • • • • •
and the nth moment about the mean - which can be easily computed
from the moments about the origin - by:':
T n
= ~t~M1)n G(t)dt
jooG(t)dt o
• • • • • • • • •
Ratios of these moments have been used for the single
parameter characterisation of systems for certain applications
(29,30), but the real attraction of the method lies in the ease
with which the moments can often be obtained from the model, thus
providing an explicit method for parameter matching without the
need for curve fitting or the numerical solution of the model
equations.
The method derives from the definition of the Laplace
transform. For linear systems, and systems whose response over
a limited range may be treated as linear, the Laplace transform has
proved an invaluable tool, the ratio of the transform of a system
response to that of the disturbance that forces 'it being known,
in the language of automatic control, as the system transfer
function - GCs); and because the Laplace transform of a Dirac
delta function is unity, G(s) also represents the transformed
response of the system to a unit impulse.
16
For an impulse response function G(t), the system transfer
function is by definition given by:
and
G(s):c ja<t) o
Limit GCs) s~o
exp(-st)dt
- fi<t)dt o
Differentiating '.5 w.r.t. s givesr
and
G' (s) = jc;G(t) exp(-st)dt o
Limit G' ~s) = -'l~Ct)dt s---.d 0
• •
=
=
• • •
a o •
• • • • •
•
• • • • • • • • •
• • • • • •
Further differentiation yields the relationship
a = (_1)n Limit Gn(s) n s ---1"' 0'
. . . . . . . . . . . . 3·9
So that to obtain the nth moment of the impulse response for a
model it is only necessary to differentiate the transfer function
n times with respect to the Laplace variable, and then set this
variable equal to zero.
The application of this analysis to a simple model fitting
exercise has been illustrated on; and the flexibility of the
method increased by a treatment that extends its applicability
to responses forced by arbitrary pulses (32). A more extensive
treatment by Paynter (27) results in a generalised expression
for the system transfer function in terms of the cumulants of
the impulse response curve - which are closely related to the
moments, (28). And in a later paper (3) this analysis is used
to fit dynamic models to heat exchanger responses; however due to
the practical difficulty of measuring the higher moments it was
only possible to match the low frequency region of the curves.
This inability to measure the higher moments to any degree
of accuracy is not surprising; the tail of the distribution affects I
considerably even the first moment and completely dominates the
higher ones: thus for a tracer impulse response test on a well
17
mixed vessel, the contributions to the first, second and third
moments of material that has resided in the vessel for more than
three times the mean residence time are respectively 19·9,42·4
and 64·8, per cent; and any inaccuracy in the measurement of
the: response at long time is correspondingly magnified in
evaluating the moments. For flow in porous beds the situation
can be even worse; it has demonstrated (34) that for .such systems,
truncation of the response curve after 99 per cent of the tracer
has been recovered, can cause a 50 per cent (or more) error in the
first moment, and that even with extreme care being taken, the
higher moments are virtually inaccessible.
Most of the published work on the moments of tracer responses
have been concerned with mixing models that account for small
departures from plug flow, the dispersion model having received
considerable attention (19, 25, 35-39) together with its discrete
space analogue the stages-in-series-with-backflow model (40); but
although these model responses decay more rapidly than, ·for
instance, that of the first order system, a consideration of the
precision with which the moments can be obtained from the
experimental curves is deserving of more attention than it receives
and weighs heavily against the elegance of the analysis.
3.3 Dynamic models for stirred vessels.
The information obtained from tracer response experiments on
stirred vessels reveals only the macroscopic quality of mixing
that occurs in the system and tells nothing of the homogeneity
on a micro or molecular scale. The distinction between these
degrees of mixedness has been thoroughly dealt with by Levenspiel
(25) but in many publications on this subject the scale of
scrutiny considered is not clear from the immediate context in
which quality of mixing is discussed: the following discussion
concerns only mixing on a macroscopic level except where
18
specifically stated to the contrary.
For the time Danckwerts (19) drew attention to the
interpretation of tracer response experiments in terms of the
concepts of residence time distribution, a number of experimental
investigations has indicated that the departure from ideality of
the mixing produced by a continuous stirred vessel can be very
considerable; and models that seek to describe and predict the
behaviour of real physical systems have been the subject of
considerable attention (43).
These models may be graded according to the extent to which
their parameters are determined by theoretical considerations:
thus at one extreme is the model that makes no attempt to explain
the mechanism that results in the observed behaviour but contents
itself with describing the response cur.ves solely in terms of
empirical parameters; and at the other is the model whose
parameters are based purely on a hydrodynamic study and can be
determined without recourse to tracer response experiments. The
models that have been considered represent progressions through
this grading the incentive being that, although the complexity
of the physical system precludes a completely theoretical study,
the more realistic models can be applied with greater confidence
to the prediction of system behaviour at the design stage, and to
the assessment of the performance of the vessel as a chemical
reactor; they also point the way to improved vessel design and
optimal mode of operation.
Probably the first systematic attempt to model the behaviour
of non-ideal stirred vessels in continuous flow systems was made by
Cholette and Cloutier in 1959 (47), They considered the
deviations from ideality to be due to three factors: stagnant
regions in the vessel, bypassing of a fraction of the feed
directly to the outlet, and the presence of regions through which
19
material flows but no mixing takes place'. Methods were presented
whereby the parameters providing measures of these effects could
be obtained graphically from t'racer response experiments; the
'effective volume' giving a measure of stagnant fluid; the
region of plug flow being measured as a fraction of the total
volume, in series with the perfectly mixed remainder (first-order
and-dead-time); the bypass flow being measured directly as a
fraction of the throughput. Experimental tests showed these
effects, as measured by the proposed techniques, to decrease to
zero with increasing agitation.
The first-order-and-dead-time model has wide application in
all manner of dynamic systems and is probably the most widely used
'model of this type for stirred vessel responses (23,48); but
these 'mixed models' - which can be combined together and further
complicated by the introduction of additional factors (49, 50) -
are quite unrelated to even the most elementary study of mixing
patterns and impeller characteristics, and can therefore be used
only for extracting certain parameters. from experimental results
as a means of avoiding numerical computation; and even when
these parameters have been determined the model gives no
indication of the effect of changing the mode of operation.
It was seen in 3.1 that of all the fundamental characteristics
of impeller performance as a miscible fluid blender, the pumping
capacity was of prime importance; and it is, therefore, not
surprising that in the next stage in the development of a realistic
model, this factor should appear as a key parameter. But before
considering such models, two methods of characterising the axial
spreading of flowing fluid will be considered as they have been
used, together with the pumping capacity factor, in the
development of these 'circulation models' for stirred vessels.
20
The first, a descriptive model which has had many varied
applications, is the tanks-in-series model (25, 26, 44): the
mixing of fluid as it passes through the system is characterised
by a cascade of n well mixed regions of equal volume, so that with
n equal to 1 and infinity it represents respectively perfect
mixing and plug flow; and for intermediate values of n it
provides a convenient measure of mixing in systems whose
behaviour falls between that of these two ideal extremes.
A. somewhat similar characterisation is by means of the
dispersion model that attributes the axial spreading of material
as it passes through the system to fluctuations of a stoch~tic
nature in the point velocities of the flowing fluid (19); this
treatment results in the characterisation of local fluid mixing
by a parameter, D, analogous to molecular diffusivity; and mixing
in the system becomes dependent on the dimensionless Peclet
number - D/uL, where u and L are the fluid mean velocity and mean
flow path, respectively.
The most simple model that can be devised incorporating
the impeller pumping capacity contains a single circulation loop:
material pumped by the impeller sweeps through the whole vessel
before returning to the impeller. This basic model has been
presented a number of times in the literature, variety being
introduced in the characterisation of the mixing in the
recirculation loop, and in the assumptions regarding the relative
sizes of the impeller and bulk regions.
Weber (51) suggested the model, with zero impeller volume.
and plug flow recirculation, as a cautious design criterion for
fluid blenders. Marr and Johnson (52) proposed that for propeller
mixers the small region of the impeller accounts for most of the
mixing (presumably on a micro-scale) and that the dispersion in
21
the single recirculation loop can be characterised by the tanks-
in-series model; observations made during bat.ch mixing
experiments suggested a value of 2 for the number of tanks in th.e
cascade. Norwood and Metz~er (17) presented the mode~ as a
means of acco;unting f9r mixing on a sufficiently microscopic scale . ,
,to promote a neutralisation reaction: their results offered
confirmation of the proposal that all this mixing takes place in
t,he immediate vicinity of the impeller the pumping action of which , ,
served only to transport the contents of the vessel to this region.
Gibilaro (31) suggested that this same model could be used for
describing the dynamic behaviour of continuous stirred vessels,
the volume of the well mixed region being determined from the
residence time distribution curves. Holmes et al (45) in a study
of batch mixing in turbine agitated vessels, also proposed a
single loop circulation model with a negligibly small impel1er
region; they chose to characterise the mixing in the loop by
means of the dispersion model and found by experiment that this
indicated a surprisingly small amount of mixing - equivalent to
more than 15 well mixed stages in series. In a later paper (46)
the model was applied to continuous stirred systems and the
effect of the feed stream on the dispersion in the loop
examined by impulse response tests; the results were inconclusive
but suggested that the feed stream could either reinforce the
circulatory flow - in which case the dispersion in the loop
remains as for the batCh case - or, if oppositely directed to.the
internally produced stream, result in an increase in dispersion;
after 5 times the mean circulation time, the response became the
same as for an ideal stirred vessel.
These single loop models can only rarely be said to represent
the flow patterns in agitated vessels; the inflow and outflow
22
are more usually located on separate circulation loops and the
introduction of the throughput stream upsets the symmetry in a way
that is indescribable in terms of a single re circulating stream.
Van de Vusse (53) proposed a model consisting of three
circulation loops, which appeared to represent a more realistic
picture of mixing in a continuous stirred vessel; however, as
will be seen in section 4, the simplifications employed to make
the model manageable analytically, completely change its character
by reducing it to a single loop model of the type discussed in
the previous paragraph.
23
4. CIRCULATION MODELS
4. Circulation Models
The path of interest of an element in a continuous flow
system begins at the inlet and ends at the outlet; there are an
infinite number of such paths and the distribution of the times
taken by elements to pass through the system is given directly
by the impulse response curve. A realistic model, on the other
hand, gives more information:· it explains why the residence
times are so distributed by describing the path of elements
through the system. This enables the model to be used for
predictive purposes:: the effect of changing the operating
conditions - throughput, stirrer speed, inlet/outlet positioning,
etc. - can be quantitatively assessed, and the limits between
which the conversion of a chemical reaction, with non-linear
kinetics; will lie, can be determined; the optimal mode of
operation for any application can be estimated.
It follows from the discussion in Section 3 that a realistic
model for the blending of miscible Newtonian liquids in stirred
vessels must be based on the pumping capacity of the impeller
and the flow patterns established by the circulating streams.
However, in formulating the model, the nature of the physical
system demands that some compromise be reached between simplicity
and accuracy; not so much in the interests of ease of solution
of the equations ,which with digital computers and efficient
programmes need not too seriously affect the issue, but mainly in
order to allow for some flexibility in adapting the model to new
situations for which only superficial information is available: a
general model broadly fitting a class of systems being more
useful than a more complex one accurately describing the
behaviour of one of the class,
All but one of ~he circulation models mentioned in 3.3
achieve this compromise by combining the circulation streams
24
together into a single loop. This considerably simplifies
the solution of the equations and can provide for flexibility by
leaving a single parameter - the degree of axial mixing in the
loop, or the volume of the impeller region - to cover descriptive
inadequacies. All these single loop mOdels are special cases
ofageneral form shown in figure 4.1; table 4.1 summarieses the
relaionships. v is the volume of the well mixed region around m
the impeller, q the impeller pumping capacity, n the number of
well mixed stages of equal volume in the recirculation loop, and
b the backflow between these stages.
m
q
~ q.b IJ+b 1 2 ~ 3 ~
~ f--b b
n
b'----l Figure 4.1
Table 4.1. Relationship between pUblished single
loop models and Figure 4.1.
Reference Restrictions adjustable parameter
(i) 17 n=G'J , b = 0 -
I (h) 31 n=oo , b = 0 v m
I' (hi) 45, 46 n=cD , v = 0 b m
(iv) 51 n = 00 , v = 0, -m b = 0
(v) 52 n = 2, v = 0 -b 0 m
=
(vi) 53 b = 0, v = 0 n m
25
In practice, for continuous flow systems, the conditions
under which a single circulation loop can be said to account for
the overall flow pattern are very restrictive, even when care is
taken to preserve the system symmetry for reasons of analytical
convenience. Consider the turbine agitated system shown in
Figure 4.2: the impeller is rent rally placed in the vessel so that
under batch operation conditions the flow pattern in the lower half
of the vessel can be assumed a mirror image of that above.
!
(b)
Figure 4.2
An impulse of tracer material injected at the impeller of the
batch system would distribute itself equally between aliissuing
streams, and the probability of a fluid element returning to the
impeller in a certain timeiS.independent of the stream to which
the impeller directs it; the system is quite· symmetrical and
combination of the streams into a single loop is in order.
For continuous operation, however, this symmetry is upset:
material leaving from the bottom of the vessel may be drawn
equally from the lower circulation loops, if the outlet port is
positioned centrally, but the behaviour in the lower region can
no longer mirror that of the upper. For the inflow position
shown in figure 4.2 there will certainly be more flow in the
bottom half of the vessel as can be seen by considering the flow
pattern when the stirrer is stopped (Figure 4.2b); and although
it would not be completely true, for the continuous system, simply
26
to superimpose the throughput flow pattern for the unstirred
vessel on to that of the stirred batch system, this would appear
to be a reasonable approximation in cases where the inflow enters
either with little momentum or in such a manner as to reinforce an
impeller to impeller circulation loop; so that for the inflow
position of Figure 4.2, the flow in the lower loops would be half
the impeller pumping capacity plus the through-put flow, together
with the flow induced by this combined stream.
For the more usual geometry in which the impeller is situated
lOl.er in the vessel, the lack of symmetry will be more pronounced
and the inadequacy of the single loop model more serious.
The mixing in the loops of a multiloop model could be
described in the same way as for the single loop model of
figure 4.1; however, the generality suffers little by restricting
the characterisation of dispersion in the loops to be tanks-in-
series model. The general mUlnloop model on this basis is given
in Figure 4.3.
Vn .............. -----
2 2 2
Figure 4.3
The vessel is divided into the well mixed region around the
impeller (vm) and n other discrete regions of volume v1
' v2 •• , v ;
n
these regions are flushed by fluid pumped by the impeller at rates
r • n' the mixing in the loops being characterised by
27
m1 , m2
, •••• mn well mixed stages in series. The throughput
flow must be superimposed on this batch model as dictated by the
system geometry.
With. potentially, an infinity of parameters this model
can certainly be made to fit experimental responses, but it is
unlikely to be of practical use unless nearly all of them can
be ascribed values independently of the dynamic response results.
Ideally only one parameter should be obtained from a comparison
of ~heoretical and experimental res~s; this allows for some
flexibility in dealing with unaccounted for secondary effects.
while excluding models based on inadequate assumptions.
The geometry of the system may suggest values for certain
of these parameters: the number of discrete loops (n) and their
volumes (v1 ' v2 ' "t, vn» may be· apparent from a consideration
of the flow patterns, and the volume of the impeller region is
likely to be negligible except in relatively small vessels; but
for the chara.cterisation of dispersion in the loops (m1
• m2 •••••
m ), tracer response experiments are almost certainly required. n
4.1 Solution of equations:
Models of the types shown in figures 4.1 and 4.3 consist
of sets of first order. linear differential equations that are
obtained from dynamic mass balances on each of the stages. The
Laplace transforms of these equations usually yield the system
transfer function without too much difficulty; the analytical
problems arise· in the inversion procedure. These, difficulties
can often be avoided by using either frequency response. or
moments techniques for characterising the experimental responses
but, as discussed in 3.2 and 6.1, the problems thereby introduced
are likely to be greater than those they circumvent •
. 4.1.1 Single loop mOdels:
Consider the completely general single loop model of
figure 4.4.
28
, .. O+q : : , , ,
:.: ..
0; - __ --jA
q
Figure 4.4 General single loop circulation model.
The loop ls dlvlded by the lnlet and outlet streams and the
two reglons are characterlsed by the transfer functions F1(s)
and F2
(s) as shown.
The system transfer functlon -G(s)- can be obtained from
transformed material balances about points A and B as follows:
from which:
G(s) C = 0 =
Cl
(q + Q) C x • • • • • • • • • • •
. . . . . . . . . . .
• • • • • •
This transfer function can usually be inverted by a binomial
expansion as follows:
G(s) = Q. F1 (s) q F1(s).F2(s) ! -1
Q 1 - Q + q + q
Q [F1 (s)] j
= Q +
4.1
4.2
,t{ [Q 1 q' F2(S)r-1}
q J = 1 • • • 4.4
For models of the type considered, the products of powers of
F1
(s) and F2
(s) can usually be inverted directly ,so that 'the
real tlme solution can be obtained from the term by term lnversion
of equation 4.4.
29
The terms of this binomial expansion have a physical
significance which Can be illustrated by deriving the system
transfer function somewhat differently as follows.
Following the injection of an impulse of tracer into the
inflow, the response of the outflow will be made up of an infinite
number of pulses; the first is produced by material that has
only passed through that region of the vessel characterised by
F1 (s); the shape of the second pulse will have been modified
twice by F1 (s) and once by F2
(S); the third pulse three times
by F1 (s) and twice by F2(s) and so on. The transformed equations
of the first, second and jth pulse are:
g1 (s) = ~ F1(s) Q + q • • . • • • • • • • • . • • • 4.5
( ) Q g2 s = q +: Q • • • • · . • 4.6
j-1 •
q +: Q • • •
and the system transfer function G(s) is given by the sum of the
pulses:'
= Q j
q + Q
Equation 4.8 is identical to equation 4.4, the first, second
and jth pulse corresponding to the first, second and jth term of
the binomial expansion. It follows from the physical sigruficance
of the terms of the expansion, that the effect of truncating the
series will be most felt at long time: material ignored will be
that which circulates many times around the system. In this
respect the method is complementary to that of summing the
residues at the poles, which, with very few terms, gives a good
approximation of the long time response. This latter method is
considered in section 6 for estimating the
30
contribution to the moments.of the tail of the model response.
4.1.2 Multiloop Models
The complexity of the general multiloop model precludes its
treatment in the manner used above for the general single loop.
Taken individually the models may be capable of simplification but
the example that follows illustrates the danger of mathematical
simplification. the physical significance of which is obscure.
As mentioned in 3.3. the only multiloop circulation model
to receive attention in the literature has been that proposed
by Van de Vusse (figures 4.5); this is a special case of the
general model of figure 4.3.
a a
Figure 4.5
R1 • R2 • •••• R6 represent half loops in which the mixing
is' characterised by m well mixed stages in series. The inflow
enters halfway along one loop and leaves halfway along another;
all other streams from the impeller are lumped together to
form the third loop; the pumping capacity, q, is divided among
these three streams.
Transformed mass balances on the six regions. lead to the
following transfer function:
G(s) = == == •• • 4.9
31
Because equation 4.9 Was considered to be too complicated for
practical use, Van de Vusse made the following simplifications:
R is then the transfer function of an 'average' loop and was
defined as follows:
-m R ;. (1 + Js/m)
V , where J = q+Q
V being the total volume of the system.
This reduces equation 4.9 to:
( ) Q G s = (q+Q); (1 + Js/mlm _ q • • • • • • • • • • •
However comparison of equation 4.10 with the general transfer
4.10
function of the single loop model (equation 4.3) shows that the
simplifications employed to make equation 4.9 managable have had
the effect of reducing the model of figure 4.5 to one containing
a single recirculation loop - figure 4.6.
Q Q 1 2 3 f- ....... - m
q
Figure 4.6
The whole character of the model has been changed: the three
loop model indicates that at low impeller speeds a significant
fraction of the inflow bypasses most of the vessel - a phenomenon
often observed in practice - whereas the single loop model indicates
that, even with zero impeller pumping capacity, no bypassing
occurs.
The inversion of equation 4.9, however, would involve some
32
considerable labour, and it seems likely that, had solutions
been obtained, a confrontation with experimental curves would
suggest modifications to the original form of the model, which
would require different methods of solution.
The method of solution presented in the next section
overcomes all these analytical difficulties and will handle
models far more complex than those considered here; the same
computer programme can be used, without modification, for all
cases.
33
5. A GENERAL METHOD FOR THE SOLUTION OF MIXING MODELS
5. A general method for the solution of mixing models.
The mixing models discussed in section 4 consist of networks
of well mixed stages. Although some of the more simple cases
considered, can be handled analytically, this is frequently
laborious, particularly when a number of different models require
comparison. Even apparently minor alterations to a model can
necessitate a quite different analytical approach and for mOdels
containing a number of circulation loops analytical solutions
may be quite out of the question.
The method to be described is based on a probabilistic
treatment of an ideal mixing stage, and enables the time domain
solution of any flow model, consisting of a finite number of
ideal mixing stages, to be readily computed.
Full details of the computer programme are given in
Appendix 2.
5.1. Theory
Consider an ideal mixing stage of volume v flushed
continuously with a flow q (Figure 5.1)
Figure 5.1
The probability (p.) of an element remaining in the vessel during r
a very small time interval at is given by:
p = v/(v + qat)', •• r
. . . . . . 5.1.
The probability (p ) of it remaining in the vessel for a larger v
time interval ~t where ~t = not will be:
p = v f. v ~n _ ( l' \n \' + qat) 1 + \~ t')
34
In the limit as ot ---> 0" n_oo
Lmt f. 1 \n = n--->oo \1 + q v~t)
exp(-q .:It/v) • • • • 5.2
And the probability (Pq) of it leaving the vessel during the
interval .:lt will be:
Pq = 1 - exp(-q .:It/v) • • • • • • • . . • . • 5.3
If this vessel (i) is now considered as part of a network of N
vessels, then the probabilities of an element remaining in
vessel i will be N
Pii = exp (- ~ • • • • • . • . . . 5.4
and the probability of transferring to any other vessel (j) will be
given by:
Pij = qij N • • • • • • • • • • • •
L: qik k=-1
These probabilities (of remaining in a stage and transferring
to another stage) are independent of the past history of the
element and therefore the process may be considered to be a
simple Markov Process and treated in the manner to be now described.
5.2 Markov Processes
The definitions and equations of the discrete time Markov
Process that will be used to evaluate the response of continuous
flow systems to input disturbances are as follows:
Pij The probability of a transition from state
i to state j.
p The transition matrix, having elements p ij'
rows of P consist of the probabilities of all
The
possible transitions from a given state and so sum
to 1. This matrix completely describes the Markov
Process.
35
s. (n) J.
The state probaality. Defined as the probability
that the system will be in the state i after n
transitions from a given starting point.
S(n) The state probability vector:: a line vector
composed of elements s.(n). J.
For each state i, there exists a probability that it will be
occupied after n transitions from a given state. The sum of
these probabilities must be 1.
s. (n) J.
from which it follows that
=
and S(n + 1) = S(n).P
= 1
Application to Flow Models
n=O,1,2, ••••
••••••••.••••••••.•••••••••••• 5.6
Consider a continuous flow system consisting of a number of
well mixed vessels, connected together to any fashion and numbered
from 1 to N. Labelling one fluid element in the system makes it
possible to define the state of the system as the number of the
vessel occupied by the fluid element at the time of observation;
thus the state of the system may be 1,2,3, •••••••• N. In a small
finite time interval At, one of two events will be seen to occur:
the tagged element remains in the same vessel or it moves to
another. In either case it is convenient to say that a state
transition has been made; from state i to state i or from state i
to state j.
Two assumptions have been made: first, At is small enough
to render the probability of two state transitions occurring in
this interval extremely small, and, secondly, the transition itself
is instantaneous.
Two factors must be considered when choosing the size of At:
the smaller ~le make At the closer is the approximation of the
36
discrete time Markov Process to the Continuous time process
under study; on the other hand the computation time increases
proportionally with decreasing At. In practice, it has been
found that if the probability of an element leaving a well mixed
stage is less than .01 then the response at any point is within
1% of the maximum value of the continuous time solution.
Knowing the sizes of all the vessels and the magnitude of the
flows connecting them, it becomes an easy matter to assign
probabilities Pij to all possible state transitions.
Thus
P11 P12 P13 ... P1N
P21 P22 P23 ... P2N
P = . .. . .. ... ... . .. . . . .. . . . . .. . . .. PN1 PN2 PN3 ... PNN
A pictorial representation of the composition of P is
given in Figure 5.2
'-'.
probabilities of leaving vessel i
Figure 5.2. Pictorial representation of transition matrix.
37
For convenience the above matrix will be used to describe
a continuous flow system containing N - 1 well mixed vessels.
The Nth state becomes the trapping state; an element in state N
is one that has left the system and cannot return.
Thus, = 0
= 1',
i
The transition matrix can now be used to find the response
of the system to a tracer input to any vessel.
Recirculation models
Consider the three loop model shown in Figure 5.3 consisting
of six completely mixed stages.
Q
2 3 r
4
6 5
Q
7
Figure 5.3 Thre~ loop model with inflow to impeller.
38
In time ~t an element in vessel 1 can either remain "here
it is or move on to vessel 2. The probabilities of these
alternatives are
exp(-r ~t/v)
and
1 - exp(-r At/V) respectively.
The probability of moving to any other state (3,4,5,6 or 7)
is zero. (State 7 is the trapping state). Similarly an
element in state 2 (i.e. vessel 2) can remain in that state
during time At or transfer, via the impeller, to states
1,3 or 5.
Thus
P22 =
=
=
P24 ='
exp(-r at/v);
='
r + Q. 3r + Q
P26 =
r (1-exp(-r at/v» 3r + Q
(1-exp(-r At/v));
=' o.
Similar expressions can be written for all the elements
of p as follows:
where
r a = -::-~-;:-3r + Q'
39
•
~ <J '9 0 ~ 0 0 0 0 ~ -a>
I I
--=----«>
...-;-. '$ .SI ~ Cl <i ,
0 0 0 0 ~. ~ 0 Tb' '.,
I ~
I-
~ ~
~ ~ 'S-~ <::J <g
~
~ IQ) I., 'Q) 0 I 0 I ,a> I 0
---=--- ---=--- ........:::,... ~ ~ <n.
~
~ ~., 0 0 I ~ 0 0 0 - '",
--l' ~ 'S > <d <g - ~ IQ) IQ)
I'" I I - I 0 0 0 I", - -.....=...- ........:;::... -----...-" " "
>
" ~ ~ '''' <:J
0 0 I - 0 0 0 I", -~ ~ ~ -s ~ <I ~ ~
--;:1 I", I'" ,,,, I 0 0 I 0 I", -.....=...- ........:;::... ........:;::...
" " d
" Cl..
40
Now v = v/6 and r = q/3.
Also in normalised time units, the mean time is by definition
equal to 1: i.e. V/Q = 1 ; and T = t.Q/V. So that:
exp(-r At/v) = exp(-2q M/V) = exp(-2q AT/Q)
and 3r : Q (l-exp(-r 6t/v» _ 9/2
- 3{q/Q+1) .(1-exp(-2q6T/Q»
etc.
Thus to give values to all the elements of P, it is only
necessary to know the value of q/Q - the ratio of circulatory
to throughput flow. For reasons already mentioned, a value is
then given to AT such that the largest pnbability of leaving
any state is less than .01. The transition matrix for this
case with q/Q = 1 andAT = .0005 (dimensionless units) is
.999001 .000999 0 0 0 0 0
.000167 .999001 .000167 0 .000666 0 0
0 0 .999001 .• 000999 0 0 0
p = .000167 0 .000167 .999001 .000666 0 0
0 0 0 0 .996008 .001001 .003001
.000167 0 .000167 0 .000666 .999001 0
0 0 0 0 0 0 1
It only remains to write the initial state probability vector
s(o). A labelled molecule enters the vessel at the impeller.
Initially it can be in states 1, 3 or 5, the respective
probabilities being:
r r r + 2 3r + Q 3r + Q 3r + Q.
Thus
S(O) = [3r : 0 , r 0 r + ~ , o , oJ
Q 3r + Q 3r + Q
which for the case of q/Q = 1 becomes
S(O) = [.166667, 0, .166667, 0, .666667, 0, OJ
41 L-______________________________________ _
A digital computer is now used to apply repeatedly equation 5.6;
that is to postmultiply the state probability vector ( S(n) } by
the transition matrix (P). Initially S(1) is obtained from
= S(O) .F
S(O) and P being entered as data. S(1) is the state probability
vector after one transition; it has as elements the probabilities
that an element that entered with the feed, at time 0, will be in
states 1,2,3,4,5,6 and 7 after time ~ T.
S(2) is then obtained from
S(2) = S(1) P.
This vector contains the state probabilities after time 2 /::,. T.
In the same way the state probabilities are obtained after 3/::"T,
4AT, ... , nA T.
After each application of equation 5.6 there are two elements
of S(n) that are of particular interest:. these are S5(n} and S7(n).
The elements S5(n), (n = 1,2,3, ••• ) are the probabilities that
the tagged molecule will be in vessel 5 (the_vessel from which
material leaves the system) after time n~T. This is the impulse
response of the system. The elements S7(n), (n = 1,2,3, ••• ,)
are the cumUlative probabilities of the tagged molecule leaving
the system after times~T, 2~T, ... n~T:: they therefore give
the response to a unit step.
5.3 The computer programme
The programme computes the elements of P from the volumes
of the stages and the flows between them; the value of~T is
computed such that the maximum leaving probability is .01;
equation 5.6 is then applied repeatedly and the specified response
printed out: this makes the programme very simple to use, the
data being obtained directly from the model diagram. The version
of the progrs.mme presented in Appendix 2 was written for the
Argus 100 process control computer; to make efficient use of the
42
storage locations of this small machine, only the non-zero
elements of the matrix are stored. This programme can be very
easily adapted for computation of the truncated moments of the
impulse response curve for models that do not lend themselves
to the analytical treatment described in 6.2.2
Another, more efficient, programme for use on larger
machines has been presented elsewhere (Reference 541; this
makes use of equation 5.7, which follows from equation 5.6. n
3(il) = 3(0) p . . . . . . . . . . . . . . The elements of P are computed in the same way and then P is
raised to a high power so that equation 5.7 need only be applied
once for each required point on the response curve; this
procedure greatly reduced the computation time but requires
the full matrix to be stored twice.
43
6. THE TRUNCATED MOMENTS
6. Single parameter characterisation: the truncated moments
The mixing models discussed in Section 4 can be readily
solved by the method presented in Section 5; and comparison
of these solutions with the experimental response curves leads
to the selection of the model that best describes the behaviour
of the system. A realistic model of this type gives the
designer all the information that a macroscopic mi~ study can
provide, but considerations of computational convenience and
simplicity have directed attention towards single parameter
characterisation of non ideal flow.
A number of such parameters have been suggested (see 3.2. )
but these have generally been based on the moments of the impulse
response curve which whilst being easily determinable for the
model, are liable to enormous errors when measured from
experimental response curves.
This latter point, already discussed in 3.2. is now
further considered, and a technique for considerably reducing
the experimental error in the moments measurements is presented:
this consists of truncating the response curve at some arbitrary
point along the time axis; the contribution of the response,
beyond this point is then ignored in computing the moments about
the origin.
The experimental and theoretical determination of these
'truncated moments' is considered in 6.2.; and in 6.3 the first
of these moments - the truncated mean - is shown to be a simple
and effective single parameter for characterising the performance
of systems that behave broadly in accordance with the mOdels
discussed in section 4.
6.1. Sensitivity of moments to experimental error.
The practical difficulty of measuring the moments of the
impulse response curve for a process whose behaviour approaches
that of a first order system, can best be illustrated as follows:
consider a process having a normalised impulse response f(T),
(Figure 6.1).
The mean is at T equal to 1 and the area under the curve is unity.
The nth moment about the origin is given by:
d
M = I Tnf(T)dT
o
. . . . . . . . . . . . . 6.1.
where d is the value of T at which the curve returns to the T
axis. In practice, for a first order system, this upper limit
occurs at approximately ten times the system mean time:
i .• e. d,: 10.
If, in a test on a true first order system, the experimental
values of f(T) are liable to some error e , then equation 6.1.
becomes:
= r Tn(e-T +t) dt o
Equation 6.2 may be used to estimate the possible error
• 6.2
in the measured moments. For example, if during an experiment
the base line value of f(T) were to drift by 0.5% of full scale,
then the error in the second moment, as calculated from equation
6.2, can be in excess of 8~~; and for higher moments,
considerably more than this.
45
6.2 Truncated moments
A solution to the measurement problem is to let d in
equation 6.1. represent some arbitrary time (say 2 or 3 normalised
time units) beyond which point the contribution of the response
to the moments is ignored. N n
/ and M n
in equations 6.1 and 6.2,
then represent the 'truncated moments' of the response -
truncated at T equal to d. This has the effect of reducing
the measurement error very considerably: in the example quoted
above, the error in the second moment is reduced to below 5%.
Also, the truncated moments only characterise the shape of that
part of the response curve which, for most practical purposes,
gives the most useful information on how the system will behave.
An advantage of the use of moments, is the ease, with which
they can often be obtained from the system transfer function
(see' 3.2. ); an advantage held in common with the frequency
response technique. Although the truncated moments of a model
can be easily determined from the time domain solution, this
route would appear to destroy much of the usefulness of the method.
As will be seen in 6.2.2, the truncated moments of the models of
section 4, Can often be obtained, without too much difficulty,
directly from the transfer function, and so could be considered
as alternatives to the frequency response' parameters, for
characterising dynamic behaviour; a possible advantage of this
being that the characterisation is achieved using only a few
truncated moments, whereas the frequency response method requires
that the gain and phase lag be specified over a whole spectrum
of frequencies.
6.2.1 Experimental Determination of truncated moments
The normalised response curve, which by definition encloses
unit area with the normalised time axis, is obtained from the
experimental response curve of tracer concentration verses time.
46
6.2.2
The total area under this latter curve is required in order to
achieve the desired normalisation. This involves either
continuing the experiment until the base line has been regained,
(Equation 6.2 indicates that, with reasonable precision in
measurement, the error to be expected by this procedure is small);
or by terminating the experiment at some point and making some
assumption about the area under the remainder of the curve. The
latter procedure was adopted, the assumption being made that
beyond the truncation point the response curve behaves as if the
vessel were ideally mixed. That this assumption was reasonable,
was amply confirmed experimentally and, in any case, the area
under the tail represents so small a fraction of the total
(less than 5% for a truncation at3 times the mean time), that a
very considerable departure from its assumed area would be
required to significantly affect the normalised response. This
procedure has the additional virtue of considerably reducing
the duration of the experiment.
Determination of the truncated moments from the process
transfer function.
The truncated means of the multiloop model that are
compared with experimental results in 8.3.4, were computed from
the time domain solutions obtained using the method presented in
Section 5. The method now to be described, however, need not
involve any digital computation.
The method consists of first obtaining the moments about
the origin, by differ<!ntiating the process transfer function as
described in 3.2. ) and then subtracting from these moments
the contribution made to them by the tail of the most significant
residues. In many cases one residue will completely account
for the response beyond the truncation point.
47
For example, consider the simple circulation model, shown in
figure 6.7. A dynamic material balance on the well mixed region
yields the following transfer function::
G(s) _ k . . . . . . . . . . . . . 6.3. s + b1 - b2exp(-b3s)
where:
b1 = (q + Q) Cv +: v )/Q,.v • m p m
b2 = qCv + v )/Q.v • m p m
b3 = Q.v /q(v + v ) .
p m p
The inversion of equation 6.3 can be accomplished by finding the
roots of the denominator and summing the residues at the poles.
The roots can be found by setting s = x +;. iy and solving the
simultaneous equations for x and y as follows::
x +: b1 - b2 exp(-b3x). cos b3y
y + b2
exp (:"'b3
X). Sin b3
y = 0 • •
• • • • • • •
. . . . . . . One solution to equation 6.5 is y = 0, and substituting this
value in equation 6.4 enables the one real root (xr ) to be
determined graphically.
The residue at this real pole is then given by:
k exp(x T) r '" k 1 exp(x T)
r • • • • • • •
and the contribution of the tail of this residue to the nth
moment about the origin by:
k 1jOOTnexp(x T) dT Cl r
• • • • • • • • • • • • • • • • •
where d is ehe truncation point
6.4
6.6
Figure 6.2 shows that for v =. v and q = 5"1., the response" m p
of this model for T greater than 0.4 is effectively due to this
single residue; and it is clear that beyond the truncation point
of T equal to 2 or 3, the sole contribution to the moments will
48
c
1· 0
·8
·6
o
\ I I
" I I I I
• I I I • I I I I I I
• I \ I I I
• I I I I I I I I
! \~ /
v =V m p
exact response
1 residue
/ '.
3 residues
·2
' . • ".
q=5Q,
·4
Figure 6.2: Approximation to exact inversion of equation 6.3 by the most significant residues.
A9
T
- - - --------------------
come from this source.
Intuitively, this would suggest a likely generalisation.
The long time behaviour of any real process is generally accepted
to be exponential in form)as is suggested by consideration of
generalised transfer function of equation 6.8.
G(s) 1 = 6.8. + ••••. • • • • • • •
At low frequencies (small s) this can be approximated by the
first order expression::
G(s) • • • • • • • • • • • • • • • • •
For models in which more than one residue is significant
beyond the truncation point, this feature will be apparent in
the graphical determination of the roots. For a physical system
of the type considered, however, oscillations in the response
will certainly have died away well before the truncation point,
so that, if the model is at all realistic, only real roots need
be considered as having any influence on the tail.
6.3 The truncated mean.
It will now be shown that the most significant features of
the models discussed in section 4 can be broadly described by
means of a simple mixed model which measures departures from
ideality in terms of two pseudo physical characteristics: the
fraction of incoming material that bypasses the vessel, and the
fraction of the vessel volume in which no mixing occurs as
material flows through it. The truncated mean will be shown
to provide a measure of both these characteristics which for
circulation models are primarily functions of the position of
the fluid inlet relative to the outlet, and the impeller
pumping capacity.
50
M1 ·8
- - - - -- - --------------------~
Consider the simple model for a stirred vessel shown in
Figure 6.3.
Q ,.
Figure 6.3.
This is one of the mixed models discussed in 3.3. , and consists
of a well mixed region which is flushed by a fraction, r, of
the throughput flow; the remainder of the inflow bypasses
directly to the outlet. The effect of the bypass flow on the
truncated mean of this model is easily obtained: with no bypass
flow, the model reduces to that of an ideal stirred vessel, so
that the truncated mean, for a truncation point of T equal to 3,
is, by definition, given by:-
Ml = f3 Texp(-T)dT
o
With bypass flow this becomes:-
Ml = 2f3 .
r Texp(-rT)dT o
where r = Q/Q
. . . . . . . . . . . .
• • • • • • • • • • • • •
Figure 6.4 shows this effect: for this simple model, the
6.9
6.10
truncated mean, which is never greater than for an ideal stirred
vessel, gives a direct measure of the bypass flow.
ideal
·7~ ______________ -L ______________ ~
Figur e 6.4 '8 ·9 1
r
51
Another of the simple models that has been much used for
describing the dynamic behaviour of stirred vessels, is shown
in Figure 6.5.
Figure 6.5. This consists of a well mixed stage, of volume v , in series
m
with a plug flow region of volume v • p
The truncated mean is
given by:
MI 1 = 1 - D
where D = v IQ.. p
~ T exp (-; - ~) dT •••••••••• 6.11 o
Figure 6.6 shows the effect ort the truncated mean of decreasing
the dead time (D). In this case the truncated mean,which is
always greater than for an ideal stirred vessel, gives a measure
of the plug flow region.
MI
·85
·8L-________________ ~ ______________ ~
·9 Figure 6.6 ·8 1-0 1
6.2.1 Application to circulation mOdels
Consider the simple circulation model shown in Figure 6.7.
q
52 Figure 6.7.
One region of volume v , is perfectly mixed. m
Material, pumped
out of this region by the impeller,circulates through the
remainder of the vessel, in which no mixing takes place, returning
to the region of the impeller. Superimposed on this basic
circulation pattern, is the effect of the throughput flow ('1),
which enters and leaves the well mixed region.
QUalitatively, the response of this model to an impulse of
tracer material in the inflow, can be easily visualised. the
most striking difference between this response and that of an
ideal stirred vessel, being the high initial value of the outlet
stream concentration: a quantity of material effectively bypasses
the vessel and this quantity will decrease as the impeller
pumping" action is increased; at very high pumping rates the
system will behave essentially as a well mixed vessel.
Figure 6.9. shows the effect on the truncated mean of
increasing the impeller pumping action. As predicted, the
effective bypass flow, as measured by the truncated mean,
decreases to zero with increasing impeller speed.
Consider now the same basic model for the stirred vessel,
but with the inlet repositioned as shown in Figure 6.B.
Q vm
q Q
'--l
Figure 6.B.
The most significant effect of this modification on the
system response, will be to delay the initial response by a time
equal to the dead time in the re circulation loop. This time
53
delay will be relatively large at low pumping rates, and will
decrease with increasing impeller speed. Figure 6.10 shows the
anticipated effect on the.truncated mean: at low impeller
pumping rates, the truncated mean is above that ofawellmixed
vessel, and the ideal value is ap~roached at high pumping rates.
Although the above discussion has been restricted to one,
rather limited, single loop circulation model, the effects
measured by the truncated mean are common to all the circulation
models discussed in section 4. When the outlet is located soon
after the inlet on the same loop or, more generally, when a
significant fraction of the inflow short circuits a significant
fraction of the vessel, the truncated mean will tend to be less
than for the well mixed vessel; and when, in the absence of
significant bypassing, the bulk of the inflow passes through a
large number of stages before reaching the leaving stage, the
truncated mean will tend to be high. Clearly, some interaction
between these two effects is possible,particularly for multiloop
models, and it will be seen in 8.3.4. that the truncated
mean of the model that fits all the experimental results reported
in section 8 can have a clearly defined maximum for a particular
value of the impeller pumping capacity; and in 9.2 this value
of impeller pumping capacity is shown to lie very close to that
at which the optimum steady state conversion for a first order
reaction occurs.
54
----------------------------------------------------------------------------------l
'80
·78
'86
'82
o 10
Figure 6.9: Truncated mean for model of Figure 6.7, vs q/Q.
ideal
o 10 20
Figure 6.10: Truncated mean for model of Figure 6.8, vs q/Q.
55
I
I
I
I
I
I
I
--------- - -- - - - - - - - -- - - - - - - - - -- - - - - - -
7. DESCRIPTION OF THE APPARATUS AND EXPERIMENTAL PROCEDURE
7. Description of the Apparatus ana-experimental procedure
Tracer response tes~s were carried out on two, continuous
flow, agitated systems: a 100 litre spherical vessel and a
9 litre cylindrical vessel. The impellers, although of different
construction in the two cases, both fell broadly into the turbine
classification.
The effect of the following variables on the impulse
response was studied. For the 100 litre vessel: impeller
speed and fluid inlet position; and for the 9 litre vessel:
impeller speed, fluid inlet position and degree of baffling.
In addition, an estimate of the two impeller pumping
capacities was made using the flow follower technique.
7.1. The mixing vessels.
7.1.1.The 100 litre spherical vessel.
This is a standard Q.V.F. Spherical Vessel of 100 litres
nominal capacity (Figure 7.1, Cat. No. vs 100/E4). Around the
central 6 inch diameter neck of the vessel, through which passed
the stirrer shaft, are fitted four side necks. Two are 4 inches
in diameter and these were blanked off. The other two are l~
o inches inldiameter and are angled at 10 to the vertical. One
served as a vent and the other was fitted with a 1 inch diameter
dip pipe adaptor through which fluid entered the vessel close
to the stirrer blades.
The stirrer was of the 'Vortex' type with a hollow shaft
that extended to the bottom outlet (Figure 7.2a). It was driven
by a t H.P. motor that was mounted on a base plate above the
vessel along with a speed regulator and reduction box that
enabled the stirrer speed to be varied from 34 to 300 rpm.
The connection from the reduction box to the stirrer chuck was
by means of a flexible drive shaft. The combined Chuck Steady
Bearing was mounted directly on the glass adaptor that fitted
on the top neck of the vessel. The seal comprised a P.T.F.E.
56
100 L. Vessel
dye - to waste
photocell
water in
FIGURE 7.1
57
FIGURE 7.2
a:
9" , ,
"mpeller 'vortex I
I" 10"
H E coi I b: ..
fe ed dip pipes c:
---7·5"
t· l"dia·U
"d' / '75 la.
58
16 '5"
--------------~---------------------------~
bellows having a glass loaded face which ran on the ground glass
static face of the seal plate. The bellows were secured to the
stirrer 'shaft and the seal plate clamped to the neck adaptor.
A water cooling ring, fitted to the top of the seal plate,
removed the heat generated at the seal face.
The 9 inch bottom outlet of the vessel was fitted with an
immersion heat exchanger coil (Figure 7.2b). A pipe reducer
on the 1~ inch outlet from the cooler was connected to a 1 inch
three way cock that enabled the outlet fluid to be directed
either into the vertical line alongside the vessel that carried
the.photocell detector, or directly to waste. Fluid leaving the
photocell passed a syphon breaker and screw down valve befor~
discharging to waste.
Feed water was obtained from a mains header tank on the
roof of the building. It passed through a globe. valve and a
10-100 g.p.h. Rotameter mounted on the vessel support structure.
The Rotameter was connected, by means of 1 inch polythene tubing,
to a vertical section of glass pipe. The nozzle of the tracer
injection value was inserted in a tee piece that was connected
to this vertical pipe section, and a further length of polythene
tubing joined the tee section to the dip pipe adaptor on the top
of the stirred vessel.
Two dip pipes were constructed: one consisted of a length
of open ended glass pipe which directed the inflow into the
impeller region; and the other of a similar length of pipe the
bottom end of which was sealed, so that liquid entered the vessel
through a circular hole in the side of this pipe close to the
sealed end: this enabled the inflow to enter close to·, but
directed away from the impeller and into the upper region of the
vessel (Figure 7.2c).
59
Except for the two lengths of polythene tubing refered to
above, all sections of pipework were of one inch I.D.,Q..V.F.
glass.
7.1.2.The 9 litre cylindrical vessel.
This was constructed from a length of 9 inch I.D. Keebush
pipe (Figure 7.3). The vessel was flat bottomed with an outlet
port, adapted to connect directly to the one inch I.D. glass
outlet line, the first section of which carried the photocell
detector.
A perspex lid, held in place by two locating pins, supported
the glass inlet lines, the impeller shaft passing through a hole in
its centre. Slots in the side of this lid enabled three
equispaced steel baffles to be positioned against the vessel
walls when required, perspex blOcks cemented around these slots
ensuring a rigid fit. The baffles were .9 in. in width.
The two inlet lines were of .4 in. I.D. glass tubing. They
were shaped and positioned as shown in figure 7.3 both being in
place for all runs.
The 2.5 in. dia. straight, six bladed turbine impeller is
shown in figure 7.3'1. It was driven by means of a .25 II.P.
motor, through a variable speed transmission unit mounted
directly above the vessel. This enabled the impeller speed to
be varied between 0 and 1,000 rpm.
Water from the header tank, flowed through a needle valve,
a 'Metric 7' Rotameter and a .;;. in. B.S.P. tee piece, the leg of
which was fitted with a rubber, 'Subaseal' cap; from here the
connection to one or other of the glass inlet lines mounted on the
vessel lid, was by means of rubber tubing.
Fluid leaving the vessel, after passing through the glass
line carrying the photocell detector, a length of flexible hosing,
60
a: General layout.
I' .75" "1
'-.1 -11-11 ....,;,,1 ! ---,--I I--LII~I t ·3"
11 2.5" J
61
-
; ';";; : : :; . . ~ ~.'
":) .... " ..
T I I
b: Turbine impeller.
FIGURE 7.3
9"
and a syphon breaker, flowed to waste. The vessel holdup
could be varied by adjusting the height of the syphon breaker.
7.2. Tracer Injection Equipment
For the 100 litre vessel, nigrosene dye solution was
introduced into the fluid inlet line by means of a servo
mechanism, designed and manufactored by Gloster Equipment Ltd.,
for the programmed injection of tracer fluids.
It consists essentially of a valve with linear flow/
displacement characteristics. The valtre position can be varied
by applying an external d.c. signal to the control unit: a zero
volt signal holds the valve shut whereas a positive ten volt
signal opens it wide; the signal/displacement relationship is
linear over most of the range so that an electrical pulse of
any shape fed ... into the control unit, will result in a similar
pulse of tracer material being discharged through the valve.
A brief description of the hardware, abstracted in the main
from material obtained from the designers, is included in
Appendix 3.
The dye is contained in a pressure controlled reservoir
upstream of the valve, and for the runs on the 100 litre vessel,
a Servomex LF51 signal generator was used to initiate the pulse.
This injection system proved to be unnecessarily complicated
for these response; experiments,and for the 9 litre vessel. the
dye solution was injected into the ~ piece on the fluid inlet
line through the 'subaseal' cap, by means of a graduated 5 cc.
hyperdermic syringe. It was possible to inject 3 or 4 ccs of
dye solution in less than 2 seconds. This arrangement is
perfectly adequate for systems of this type in which the mean
residence time in the vessel is so much larger than the
injection time that the input may be assumed to be a true impulse.
62
- ------ -------------------------------,
7.3. The Photocell Detector.
7.3.1 Construction.
The concentration of dye in the outflow from the stirred
vessels, was measured by means of a photocell detector built
around an 11 inch section of the glass outlet line.
The detector contained two Mullard 90AV photo emissive
cells, and four resistors (one variable) arranged to form the
simple Wheatstone bridge circuit shown in Figure 7.4. The cells
were located on either side of a six watt filament bulb, and a
section of one inch diameter glass pipe, clamped firmly to the
detector housing, was located between each cell and this light
source. The cells were shielded $0 that virtually all the light
falling on the sensitive cathode surfaces, first passed through
one or other of the glass pipes (Figure 7.5).
One of these pipes constituted a section of the outlet
line and so contained the solution whose concentration was to
be measured·; the other was for the reference· fluid, which in
this case was distilled water.
Power was supplied from two transistorised power packs
which provided stabilised volt ages of 8 and 30 volts to the bulb
and bridge circuit respectively. The inbalance of the
Wheatstone bridge provided a signal for a digital voltmeter which
could be zeroed by means of the variable resistance incorporated
in the circuit.
Two later modifications considerably improved the stability
of the instrument: the light bulb was ventilated to prevent
overheating, and the circuit was thermally insulated with glass
fibre wadding.
7.3.2.Calibration
The photocell detector was calibrated each time a series of
runs was carried out. This was accomplished by disconnecting
63
PHOTOCELL DETECTOR signal
FIG.7·4
'-----------_ 30V. +
FIG. 7·5
r-~ ~I?::::::::: ~e
glass wool
,
1"QVF pipe
64
C 9/1
·02
·01
o
FIG.7·6
65
2 v
l I
I
the section of pipe to which the detector was fitted, and clamping
it to a suitable support. A standard solution, containing
exactly 1 g/l of nigrosene was then made up and quantities of
this solution were diluted with ~ater in standard flasks and
introduced into the detached pipe section, the bottom of which
was sealed with a rubber bung.
digital voltmeter.
The output was measured on the
Figure 7.6 shows a typical calibration curve. Although
the relationship between voltage output and nigrosene concentration
is linear over a considerable range, deviations occur at very
low, as well as high, concentrations. The very low concentration
region was effectively eliminated, in the impulse response runs,
by terminating the experiment before the base line had been
regained (see 6.2.1), but in order to obtain maximum sensitivity
over most of the response curve, it was sometimes necessary for
the linear calibration region to be exceeded early in a run. For
this reason the computer programme for the normalised response
curves and the truncated moments of these curves, (Appendix 2),
employs a linear interpolation between pairs of points from the
calibration curve in order to convert logged voltages into
concentration units.
7.4 The impulse response experiments.
The experimental procedures for the impulse response
tests on the two systems were virtually identical except for the
mode of injection of the tracer. For runs on the 100 litre
vessel, the setting up of the Mass Tracer Injector was somewhat
involved and is described in Appendix 3. The procedure for the
9 litre vessel was considerably simplified by the use of a
hyperdermic syringe for the dye injection,and is described below.
The rotameters and vessel volumes were calibrated for both
systems: the rotameters by collecting and weighing water which
66
•
had flowed through them at a constant setting, for a measured
period of time; and the volumes by filling the vessels from
standard litre flasks.
The same photocell detector was used for both systems.
7.4.1 Runs on the 9 litre vessel •
. The photocell detector was allowed to warm up overnight.
It was then calibrated, as described in 7.3.2., and connected
in the outlet line directly below the vessel which was then filled
with water.
The desired operating conditions were then established:
the water flow rate by means of the feed rotameter and needle
valve; the impeller speed by adjustment of the micrometer control
on the variable speed transmission unit; and the vessel holdup
by adjustment of the syphon beaker position. The system was
then allowed to steady out under the desired conditions, small
adjustments being made-where necessary. In the meantime provision
was made for the photocell detector output to be logged on
punched paper tape, at one second intervals. A library
programme enabled the Argus 100 digital computer to be used for
this purpose while at the same time being available for off line
computation. This programme was then read in and the digital
voltmeter, which has been continuously indicating the base line
output from the photocell detector, was put on to manual control.
The desired logging interval (one second) was then set up on a
Dekatron counter,the output of which is linked to the computer.
Three or four ccs. of a 5 ell solution of nigrosene dye were
then injected, through the subaseal cap, into the fluid inlet
line by means of the hyperdermic syringe. At the same time the
logger was started by providing the initiating pulse to the
computer from an external trip switch, mounted close to the
vessel.
67
When the run had progressed for over 2~ times the mean
holding time, logging wFs discontinued and the vessel drained
and flushed out; and after establishing new operating conditions
the above procedure was repeated.
The logged data tapes were processed on the Argus 100
with the programmes described in Appendix 2, to provide the
normalised response curves and the truncated moments. presented
in section 8.
7.4.2 Dye Balances:
Measurement of the area under the experimental curve of
tracer concentration against time, enables the total quantity
of dye, as measured by the photocell detector, to be evaluated;
and comparison of this quantity with the amount of dye injected,
provides some check on the reliability of the results. The
trouble with this check was found to be the difficulty of
measuring precisely how much dye had been injected: the
reproducability of the volume~ic discharge during a 5 second
square pulse from the Mass Tracer Injector was about 5 per cent
under the conditions described in Appendix 3, and although this
was improved upon in the simple syringe injection technique used
for the smaller vessel, the"impreoision in this measurement still
appeared to outweigh the other experimental errors of which the
dye balance was expected to provide a measure.
The real purpose of the dye balance is to test the
calibration curve and so instead of attempting to improve the
precision of the injection, which only affects the dye balance
result, one additional run was performed each time a new
calibration curve was used. For this run, steady conditions
were established exactly as described above, but instead of
injecting the dye solution into the feed line, a quantity of
nigrosene, accurately weighed out and dissolved in water, was
68
added to the stirred vessel from a beaker, which was then rinsed
into the vessel with a little more water.
The shape of the response curve thus produced defies
interpretation, but the dye balance obtained on processing the
logged tape is completely free from error due to input imprecision
and so provides a meaningful check on the calibration results.
In every case the results showed a dye balance of between 98 and
102 per cent.
7.5 Determination of Impeller pumping capacity using the Flow Follower technique.
The circulation models discussed in section 4 have one
parameter in common - the pumping capacity of the impeller.
Although this can be determined from the impulse response
experiments, it would simplify the selection of a suitable model
if some independent estimate of this parameter was availaple.
The Flow Follower technique used by Marr and Johnson (18) for
measuring marine propeller pumping capacities, and discussed in
section 3.1, was used for this purpose.
Flow Followers were made by machining small tablets of
polystyrene (about the size and shape of small asprin tablets)
from a block, drilling out, a cavity in the centre and sealing
in this cavity a small piece of cork; so that when the seal
had dried, the flow follower would just sink in water. The
buoyancy was then adjusted by carefully shaving polystrene from
the edge of the tablet.
The results were never perfect, but it was eventually
possible to produce tablets that would remain suspended in a
beaker of stagnant water for about a minute before finding
their way to the surface or the bottom.
The following modifications were made before the pumping
69
capacity determinations were carried out: - the 9 litre Keebush
vessel was replaced by a similar one of glass; and the bottom
section of the 100 litre spherical vessel, containing the heat
exchanger coil, was substituted for one without a coil, which,
it was found, tended to trap the flow follower.
Preliminary tests also showed that the flow follower tended
to stick to the air-water interface: this was prevented by
reducing the surface tension with a little Teepo1 solution.
The experimental procedure was then very simple. The flow
follower was introduced into the stirred vessel; a stop watch
was started when it first passed through the impe11er, and
subsequent entries of the flow follower into the region of the
impe11er were counted. When some 150-200 counts had been made
the clock was stopped and the average circulation time computed.
The experiment ~tas repeated over a range of stirrer speeds for
the two systems. Two additional runs were later performed
on the 100 litre vessel, in which. the actual times when the
flow follower entered the impeller region were recorded. This
enabled a frequency distribution of circulation times to be
plotted, from which some assessment of the reliability of the
technique can be made. (Section 8.1.3.)
70
8. EXPERIMENTAL RESULTS AND COMPARISON WITH PROPOSED MODEL
r- - - - - - - -- - - - - -- --- - --------------_____________ ----,
8.1. Impeller pumping capacities:-
Results are presented of flow follower experiments in
the two vessels. Details of all runs are given in Appendix 1.
8.1.1. 9 litre cylindrical vessel system
Ten runs were carried out in this vessel, with a liquid
holdup of 9 litres. It can be seen from Figure 8.1. that
the pumping capacity, as measured by this method, varies
linearly with stirrer speed over the renge tested, and that
the best line through the points passes through the origin.
8.1.2. 100 litre spherical vessel system.
The results of 14 runs on this system, with a liquid
holdup of 55 litres, are shown in Figure 8.2.
Although the relationship between pumping capacity and stirrer
speed was found to be linear over most of the range covered,
the best line through the experimental points did not pass
through the origin. Some observations on the reliability of
these results are made below.
8.1.3. Experimental errors in the pumping capacity determinations:-
In order to obtain some indication of the reliability of
the pumping capacity determinations by the flow follower
technique, frequency distributions (Figures 8.3 and 8.4) were
produced for two of the runs reported in 8.1.2.
The problem is that of assessing how good an estimate"
the mean of this sample distribution is likely to be of the true
distribution from which it is drawn; e.nd this involves
as summing a theoretical distribution against which the sample
can be compared.
To do this, it is necessary to anticipate the conclusions
drawn from the residence time distribution tests reported in 8.3.
The multiloop models which were found to fit all the situations
71
Cl
60
40
20
o rpm
Figure 8.1: 9 litre system: pumping capacity vs impeller speed.
300
200
100
o o rpm
Figure 8.2: 100 litre system: pumping capacity vs impeller speed.
72
-------------------~.-----~
f
f
30 -
IT: 10 P
1\:.
o
13
, ':'.
. . , .,::;.<
I. J1. 16·2
30
'" ", •. : :'ii.:<::\ " ..
. . 60
t
Figure 8.3: 100 litre system: frequency distribution of flow follower counts; impeller speed 150 rpm.
, .. ;
30
_11·3
I. J1.=11·3
10
o 30 60 t
Figure 8.4: 100 litre system: frequency distribution of flow follower counts; impeller speed 200 rpm.
73
tested, reduce to the simple model shown in Figure 8.5afor the
case of batch mixing: i.e. two well mixed stages in series,
with recirculation.
V/2 V/2 q Figure 8. 5a
I
The response of this model to flow follower tests is the
same as that of the two stages without the recirculating stream.
In this respect, these tests differ from conventional tracer
experiments in that an element is only timed once around the
loop, whereas a dye injection technique would result in 'elements'
being counted as many times as they completed the loop.
The distribution against which the flow follower
distributions are to be compared is therefore given by:
. • • • . • . . . . . • .. 8.1 ..
Integrating this distribution over discrete intervals,
enables the model to be compared directly with the experimental
distribution, and a z(goodness-of-fit test performed.
The I! test indicated that the fit Was poor and a direct
comparison of the experimental and theoretical distributions
(Figures 8.3 and 8.4) indicates why this was so: the mean of
the experimental distribution was much influenced by the tail
which was considerably in excess of that predicted by the model.
If the model is correct, and there is abundant evidence from the
impulse response results to suggest that it is, then it would
appear that the flow follower measurements could considerably
overestimate the mean circulation time, and hence underestimate
the impeller pumping capacity.
74
- - - - - - - - - - - - - - - -- --------------------------
Figure 8.3 also shows the theoretical distribution having
a mean some 20% lower than that of the experimental distribution.
Apart from the tail, this is much clocer to the observed
distribution and suggests that the mean had been overestimated
by about 25"/0.
Qualitatively, it is easy to see how these errors occur.
Small density differences between the flow follower and the
fluid introduce gravity effects which tend to delay the flow
follower, either high in the vessel if the density is low, or at
the bottom if the density is high. These effects are likely to
be felt more at low impeller speeds, when the drag forces are
relatively weak.
8.2. The Proposed Model
The simplest model of those discussed in Section 4 that
adequately describes the behaviour of both vessels, under all
conditions tested, contains n equal recirculation loops. The
mixing in each loop can be described by two well mixed stages in
series, and the circulatory flow from the impeller is divided
equally between each loop; all stages are of equal volume.
(Figure 8.5)
2 n
Figure 8.5
For the case of batch mixing, all these loops can be lumped
together to give a single re circulation loop model; but for
7S
the case of continuous mixing, the presences of inlet and outlet
streams upsets the symmetry of the system. The results
presented in the succeeding sections suggested that the value of
n and the manner in which the loops may be grouped together,
is solely dependent on the geometry of the system and may be
predicted from qualitative considerations of the flow pattern.
8.3. Results of Impulse Response tests
The normalised response curves for all runs are presented
in Appendix I.
In this section, experimental responses are compared with
those predicted by the proposed model for each situation, and
over the range of stirrer speeds, tested.
The impeller pumping capacities (q), for all these
comparisons, were obtained from the flow follower experiments.
It l>las initially intended to use these values as first
approximations and to modify them on the basis of the residence
time distribution results; however, it was found that the best
values for this parameter lay very close to those obtain~d from
the flow follower results. These points are further discussed
in 8.3.3.
The computer programme described in Appendix 2 was used to
obtain all solutions to the proposed model included in this
section. Modifications to this same programme were used to
produce the truncated means of this model for all the situations
tested. These latter results are compared with experimental
values in 8.3.4.
Full details of the experimental results used in this
section are included in Appendix I.
76
8.3.1 100 Litre Spherical Vessel
Runs were carried out in this vessel for a range of
impeller speeds under the following conditions.
Runs SI Runs SIX Runs SO
Liquid holdup 55 55 55 (1)
Flow rate 7.56 4.54 7.56 (l/min.)
Inlet Position Into impeller Into impeller Away from impeller
Model for vessel with inflow to impeller (Runs SI, SIX)
The flow follower tests on this vessel gave a good
qualitative indication of the flow pattern. A fluid element
either circulates in the main spherical region of the vessel in
the direction of rotation of the impeller, with seemingly random
vertical and radial velocity componants, until it becomes
entrained in a stream that enters the impeller; or it enters
the lower cylindrical region, at the bottom of which is situated
the outlet pipe. In this latter case the subsequent behaviour
of the element is quite predictable: it flows in a spiral, down
the outside of the cylinder until it reaches the bottom of the
vessel; here it either leaves with the outflow, or becomes
drawn into the impeller suction, either through the extended
hollow impeller shaft or by spiralling up around it.
There are then two main circulation loops:: an upper one
which leads back to the impeller, and a lower one, halfway
along which is situated the exit pipe.
For a liquid holdup of 551., the ratio of the upper to the
lower volume has been taken as 2 : 1, although the somewhat
complicated geometry of the vessel makes this only an approximation.
77
Refering to the proposed model (8.2) the smallest value of n
required to describe the geometry of this system is 3. The
model is shown in Figure~8.6.
--...",~=----- Q
Figure 8.6
Q
The upper loops, being identical, can be combined to give the
2 loop model of Figure 8.7.
2v 2\1
2r Q
r+Q
v v Figure 8.7
Q
Figures 8.8 to 8.11 show how the response of this model compares
with the experimental responses obtained for Runs SI. At very
low stirrer speeds the fit is tolerable, and for values of q/Q
• above 5, the model fits the experimental responses reasonably
well. In particular, it predicts the most significant feature
78
c
1
o
c
1
o
o
o o o o o o
Figure 8.8: run SI 2; q/Q = 1
1 2 T
Figure 8.9: run SI 4; q/Q = 5
o o
1 2 T
79
c
1 Figure 8.10: run SI 5; q/Q = 9
o 1 2 T
c
A 1 Figure 8.11: run SI 7; q/Q = 22
o 1 2 T
80
of these responses, which is the fraction of material that
effectively short circuits the system.
Figures 8.12 and 8.13 show similar responses obtained for
Runs SIX. The only difference between these runs and Runs SI
being the throughput flow rate.
c
o Figure 8.12: run SIX 2; q/Q = 2.1
1
o
o 1 T
c
1 Figure 8.13: run SIX 3; q/Q = 5,5
o 1 T
81
110del for vessel with inflow away from impeller (Runs SO)
The basic flow pattern for this case remains unaltered.
The inflow, however, enters close to the impeller, but directed
into the upper part of the vessel, so that the momentum of the
entering stream reinforces the predominant angular velocity
component in that region.
The model for this case is shown in Figure 8.14. This is
identical to Figure 8.7 except for the fluid inlet position
which conforms to the situation just described.
however, is quite different.
Q. 2v 2v
Q.+ r
r Q.+r
v v Figure 8.14
Q.
The response,
Figures 8.15 to.8.18 compare the experimentally obtained
responses with those predicted by the model. Except for very
low impeller speeds, the fit is quite good and the effect of
increased agitation on the position of the peak response is well
demonstrated.
82
c
1
o
c
o
o o 0 00
o o
o
Figure 8.15: run SO 2; q/Q ~ 1
o o
o
1
o o
o
2
Figure 8.16: run SO 4; q/Q ~ 5
1 2
o o
T
T
c
1
o
c
1
o
Figure 8.17: run SO 5; q/Q ; 11
2
Figure 8.18: run SO 6; q/Q ; 20
1 84
2
T
T
9 litre Cylindrical Vessel
Runs were performed on this system for a range of impeller
speeds, under the following conditions: unbaffled vessel,
inflow to impeller; unbaffled vessel, inflow into upper region
of vessel; baffled vessel, inflow to impeller; baffled vessel,
inflow into upper region of vessel.
Runs UI Runs UO Runs BI Runs BO
Liquid holdup 9 9 9 9 (1)
Flow rate 1.1 1.1 1.1 1.1 (l/min. )
~nlet Position Into Away from Into Away from Impeller Impeller Impeller Impeller
!3a f fling Unbaffled Unbaffled Baffled Baffled
Model for unbaffled vessel, with inflow to impeller (Runs UI)
The basic flow patterns set up by an agitator in unbaffled
cylindrical vessels are well known, and were in part confirmed
during the course of the flow follower experiments. As for the
spherical vessel, the predominant velocity component in most of
the vessel, is angular and in the direction of rotation of the
impeller; the radial and vertical components being ill defined and
apparently random. In the horizontal plane through the impeller,
however, radial velocities were found to be significant, as were
vertical components close to the axis of the vessel (Figure 8.19).
85
- -------------------------------------------~
[a] Figure 8.19
These local variations in the overall pattern, imply that
the presences of the inlet pipes, and the skin friction at the
walls of the vessel, provide some degree of baffling; in-an
ideal unbaffled vessel all elements would rotate with the same
angular velocity as the impeller, which would then cease to act
as a circulation pump. Vortex Formation at the liquid surface
was only just evident at the highest impeller speeds tested.
As for the 100 litre spherical vessel, the ratio of the
upper to the lower volume is 2:1 and the outflow is from the
bottom of the vessel. This enables the same value of n to be
used in the proposed model; and the same considerations of symmetry
further reduce this to the 2 circulation loop model, with inflow
into the impeller, of Figure 8.7.
In Figures 8.20 to 8.24, the experimental results are shown
to be in reasonable agreement with the model, over a large range
of impeller speeds; as before, the effective bypass flow
predicted by the model is well confirmed. These figures also
show experimental results obtained for the baffled vessel under,
otherwise, identica~ conditions. It will be shown that inspite
of the different flow patterns in the baffled vessel, the proposed
model reduces to this same 2 loop model under these conditions.
86
c
Figure 8.20: runs BI 7, UI 2; q/Q = 4.5
1
o 1 T
c Figure 8.21: runs BI 6, UI 3; q/Q = 9
1
o 1 T
87
c
+ Figure 8.22: runs BI 4, UI 4; q/Q = 14.1
o T
c Figure 8.23: runs BI 3, UI 5; q/Q = 20.4
o T
88
c 8 24: Figure • runs Bl 2, IQ - 27.3 Ul 6; q -
o 1 T
89
~odel for unbaffled vessel with inflow into upper region (Runs UO)
The unusual geometry of the 100 litre spherical vessel, made
it possible to direct the inflow into the upper circulation loop
from quite clcse to the impeller. For the 9 litre vessel,
however, it was necessary to raise the inlet pipe some 3 inches
above the.impeller in order to prevent a fraction of the
incoming stream from entering the bottom circulation loop. The
inlet pipe was arranged so that the stream entered the vessel
with the minimum amount of disturbance to the flow pattern
produced by the impeller.
Under these conditions, the model becomes identical to that
for the spherical vessel with inflow into the upper loop
(Figure 8.14).
Figures 8.25 to 8.29, show how well the experimental results
for Runs ub, agree with those predicted by the model: even at
the lowest speeds tested the fit is tolerable,and for values of
q/Q above about 8, it is remarkably good.
c
1
o 1 T
Figure 8.25: run UO 2; q/Q = 2.3
90
----------------- - - --- - -- -- - - -------------- - - -
c
Figure 8.26: run UO 5, q/Q ~ 9.1
o 1 T
c
Figure 8.27: run UO 6, q/Q ~ 12.7
1
o 1 T
91
~---------------------------------- -- - -
c
Figure 8.28: run UO 8, q/Q = 28
1
o 1 T
c
Figure 8.29: run UO 9, q/Q = 33
1
o
o T
92
Model for baffled vessel, with inflow into impeller (Runs Bl)
The effect of baffles on the flow pattern in a turbine
stirred cylindrical vessel has been discussed in section 3.1.1.,
and observed, by means of the flow follower. The 3 baffles
used for these runs,. effectively divide the vessel into 6 regions,
with 3 above and 3 below the impeller; the volumes of the upper
region, being twice that of those below. (Figure 8.30).
---... /) ~*
\\ .. . . . :;.: . ... "
, ... "
Figure 8.30
The proposed model reduces for this system to that shown in .
Figure 8.31~.
2v 2v 2V 2v 2v 2v
2r~ ~~~ r ~,
r
v v v v V V
Figure 8.31
For the case of inflow to the impeller and outflow from all
bottom loops, this further reduces to the same 2 loop model as
for the unbaffled vessel (Figure 8.7)
As predicted by the proposed model, the baffles have no
noticable effect on the impulse response of the system. Figures
8.20 to 8.24 show that the experimental responses for the baffled
and unbaffled cases agree very well with each other, and
reasonably well with with model - particularly at the higher
impeller speeds. 93
Model for baffled vessel with inflow into upper loop (Runs BO)
The basic model of the baffled system, shown in Figure 8.31
still applies. However, the upper loops are quite distinct in
this case, and it is to one of them that the inflow is directed.
The other two remain identical and can be combined together,
as can the bottom 3 loops.
Thus, the proposed model reduces to the 3 loop model of
Figure 8.32.
Q 2v 2v
4v 4v
3v 3v
Figure 8.32
Q
Figures 8.33 to 8.37 show the model to fit the experimental
curves extremely well at values of q/Q above 9. At lower
impeller speeds, the results suggest that the tendency of
incoming material to short circuit the vessel is rather more
pronounced when baffles are present; it may be that with low
internal circulation rates, material striking a baffle is able
to flow down along the baffle sides, into the bottom of the
vessel.
94
c
c
o
1
o
o o
Figure 8.33: run BO 1; q/Q ~ 2.3
T
Figure 8.34: run BO 4; qlQ ~ 9.1
T
95
c
Figure 8.35: run BO 5; q/Q = 12.7
o T
c
Figure 8.36: run BO 7; q/Q = 28
1
o 1 T
96
c
Figure 8.37: run BO 8; q/Q = 38
1
o 1 T
97
Sensitivity of Model to pumping capacity measurement.
As stated in 8.3, the comparisons between the experimental
responses and the proposed model, presented in that section,made
use of the pumping capacity measurements obtained using the flow
follower. The reliability of these results has been discussed
in 8.1.3 and it was concluded that they should be treated as
first approximations in suggesting the form of a suitable model,
and that the parameter, q, should then be obtained from the tracer
response curves.
It was not even certain that the true impeller pumping
capacity was the best value that could be assigned to q: a
pseudo pumping capacity that took into account the flow induced by
the main circulating streams could have been preferable, and this
could only be obtained from the impulse response results.
It is therefore rather surprising that the model fits
these very different systems so well; although it is clear that
for some of the lower impeller speeds, a mOdified value for q
would improve the fit. For example, Figure 8.9 shows the
comparison between experimental results and the model for
q!Q = 6. Figure 8.38 shows these same experimental points,
compared with the model for q!Q = 3 and 7: . i.e. 4~~ smaller
and greater than the value of Q/q obtained from the flow follower
runs. It is clear that a value of about 4 for q!Q would give the
best fit, SO that this parameter has been overestimated by about
20%.
The same procedure was used to find the best values of q!Q
for a number of tracer response runs performed on the spherical
vessel, for both inlet positions. These are shown in Figure 8.39.
The line about which these points lie, is the best line through
the values of q!Q obtained from the flow follower runs, and is
clearly very close to the best line through the points shown.
98
c
o 1 T
Figure 8.38: sensitivity of model to pumping capacity measurement; run SI 4 (qjQ = 5), compared with model (q/Q = 3 and 7).
30
20
10
o 50 100 150 rpm.
Figure 8.39: best line through flow follower results compared with best value for q.
99
The Truncated Mean
Figures 8.40 to 8.44 compare the Truncated Mean of the
experimental impulse response curves with those of the relevant
model.
For the case of the 100 litre spherical vessel, the mean has
been truncated at 3V/Q whereas for runs on the 9 litre cylindical
vessel a truncation point of 2V/Q has been used.
When the inflow is directed into the impeller, the model
predicts the quantity of material that short circuits part of the
vessel. For the reasons discussed in Section 6, the truncated
mean provides a measure of this effect,and Figures 8.40 and 8.42
(Runs SI, UI and BI) show the experimental values of this
parameter to be in good agreement with those predicted by the
model. At high circulation rates the truncated mean approaches
the value for an ideal stirred vessel.
Figures 8.41 and 8.43 compare experimentally determined
values of the truncated mean with the model for the unbaffled
vessels-with the inflow into the upper circulation loops.
(Runs SO and UO respectively). Under these conditions the
model predicts the truncated mean to be greater than for an ideal
stirred vessel and to decrease to the ideal value as the impeller
speed is increased. These effects are confirmed, although
Figure 8.41 shows the experimental values for Runs SO, to be
consistantly some 6% higher than predicted by the model.
For the case of the baffled vessel, with the inflow into
one of the upper loops (Runs BO), the effect on the truncated
mean of increasing q/Q is shown in Figure 8.44. The model,
for this situation (Figure 8.32), predicts a peak value
occurring at q/Q = 7. The experimental points, are in reasonable
agreement ,dth the model for q/Q < 10, and suggest the presences
of a peak occurring near predicted point. Unfortunately, the
fit of the model was poor at low values of q/Q, so that this
predicted feature remains unconfirmed. 100
M1
o ideal
·6 ::>
·4
I I I I
0 20 40 q/Q Figure 8.40: runs SI; M1 - truncated at T = 3 -,VS qJQ
M1 (D ·9 I-
0
1-0 0
0 0 0 0
·8 ideal
o 20 40
Figure 8.41: runs SO; M1 - truncated at T = 3 - vs q/Q
101
M1
·6
·5
M1
·65
·6
'-
( 0
0 0
I-
0
0
Figure
o
0 0
0
8.42: runs UI
0
"'U
0 0
0
0- runs UI
Q- runs BI
, 20 40
and BI; Ml - truncated at T = 2 -
o
o
20 40
Figure 8.43: runs UO; Ml - truncated at T = 2 -, vs q/Q
102
q/Q vs q/Q
M1 '64
0 a a a a ·6 I- a -ideal\ a
a
J ,
o 20 40
Figure 8.44: runs BO; Ml - truncated at T = 2 - vs q/Q
103
Estimation of errors in the impulse response experiments.
An advantage of model fitting by the direct comparison
of experimental with theoretical response curves, is that the
method is relatively insensitive to slight base line drifts in
the measuring instruments; and as no trouble was experienced
in maintaining the flow rate from the header tank to within
1% of the desired rotameter setting, there would appear to be
every reason for confidence in the results.
Two points, however, require further examination: the dye
injection and the degree of mixing in the inlet/outlet lines;
in computing the normalised responses, true impulse inputs and
plug flow in the lines were assumed.
For the 100 litre vessel the dye was injected as a square
pulse of 5 seconds duration. This represents approximately
one per cent of the mean holding time for runs SI and SO. The
volume of the inlet/outlet lin~s, from the injection point to
the photocell deteptor, was .82 litres (.46 litres of which was
contributed by the inlet lines), which is 1.5 per cent of the
vessel holdup for these runs.
Visual observation of the dye pulse as it passed through
the transparent inlet line indicated a quite appreciable amount
of axial mixing. The likely effect of this departure from the
assumption of plug flow, can be gauged by ·assuming complete
mixing in the lines: this enables the upper limit of the error
due to this factor to be evaluated and the combined effect of
this and the square wave input pulse to be assessed.
The response of a well mixed vessel of volume V, having
well mixed inlet and outlet lines of volumes v. and v 1. 0
respectively, to a square wave input of duration d seconds is
given by:
104
and if the twin assumptions of a true impulse input and plug
flow in the lines are made to estimate the response of the vessel
alone, then its 'transfer function will be found to be:
() _ ________ ~(~l~-~e-_d_s~)L/~dg~ ________ _
G s '" . (v. + v ) s/Q;
fie ~ 0
(Vs/et +> 1) (v s/et + 1) o . . . . . .
Figure 8.45 compares the normalised time domain solution of
equation 8.2 with the true response of the well mixed vessel.
Bearing in mind that the error indicated by Figure 8.45 is
representative of considerably more dispersion in the lines
than occurs in practice, it will be seen that the assumptions
will not significantly affect the experimental response curves.
For the 9 litre vessel, both the duration of the pulse
relative to the system mean time, and the volume of the
8.2
inlet/outlet lines, relative to the vessel holdup, were considerably
less than for the 100 litre vessel, and so the error due to
these causes may be assumed to be even less significant.
c
1
o
Figure 8.45: maximum possible error in impulse response results due to mixing in outlet/inlet lines and pulse input.
T
105
9. DISCUSSION
9 Discussion
The most significant factor contributing to the behaviour
of the continuous stirred vessel systems reported in Section 8,
was found to be the positioning of the feed inlet port; the
proposed model, by predicting quantitatively the very considerable
change in the response brought about by changing the direction
of the inflow, represents an advance on the single loop
circulation models previously considered. The ability of the
model to predict the behaviour of quite different systems
over a Io[ide range of operating conditions provides strong
evidence to support assumptions upon which it is based; and,
as only a qualitative knowledge of the flow patterns and an
approximate measure of the impeller pumping capacity are
required for applying the model to a particular system, it
would appear to be of considerable practical use, both at the
design stage for predicting the behaviour of alternative
proposals, and during operation for suggesting likely methods
of improving performance.
In 9.1 the experimental results of section 8 are compared
with those of other experimenters.
The model may be used for predicting the steady state
conversion for a first order reaction, and by making the
assumptions of micro mixing and complete segregation, the
conversion limits of other reactions can be calculated; the
first order reaction case is considered in 9.2.
106
9.1 Comparison with other published results.
Qualitatively the model is in good agreement with the
known features of non ideal stirred vessel behaviour: it
predicts the bypassing and dead time effects (3.3) which are
both known to depend on the inlet/outlet positioning and to
decrease with increasing agitation. However, in order to make
quantitative comparisons between the proposed model and other
• published results it is necessary to have some idea of the"
geometry of the system in which the results were obtained, and
the pumping capacity of the impeller used; this effectively
limits the comparisons to the few results presented in support
of single loop circulation models.
For the case of batch mixing in propeller agitated vessels,
Marr and Johnson (52) found that the dispersion in the
circulation loop could be characterised by two stages-in-series:
this agrees exactly with the results presented here for turbine
impellers. The results of Van de Vusse (53) for a continuous
system are also reasonably consistent with the proposed model,
although with no details of the inlet-outlet positions other
than that they were located on separate recirculation loops, it
is not possible to make a detailed comparison. The single
loop model (Figure4.6) reported as best fitting the results
contained 4 stages-in-series in the recirculation loop; this is
equivalent to somewhat fewer stages per loop in a multiloop
model so that the results can be assumed to have some measure
of agreement with those presented here.
The results of Rolmes et. al (45, 46) however, appear to be
quite contradictory to those obtained in this study. For batch
systems, very similar in many respects to the 9 litre vessel
used here, the dispersion in the single recirculation loop, was
found to be equivalent to between 15 and 20 stages-in-series;
107
this represents remarkably little axial mixing and, indeed,
the method used for measuring the dispersion depended on this
being the case: the distance between successive peaks and
valleys of a tracer injeoted and measured close to the impeller
being used to calculate the pumping capacity, and the normalised
concentrations at these points being used to characterise the
dispersion. The results obtained for the continuous flow system
were inconclusive, but the batch results showed exoellent
reproductability and consistancy, the dispersion for a fixed
geometry, being independant of the stirrer speed over the range
tested. The impeller speeds at which dispersion measurements
were made however, were considerably higher (up to 1,500 rpm)
than those used for the runs reported here, and this would
suggest a likely explanation of the apparent discrepancy in the
two sets of results. For the continuous system (46) the
oscillations detected in the outlet pipe following a tracer
injection in the feed, were of such high frequenoy (approximately
one per second), and so short lived, that the response may, for
most practical purposes be assumed ideal; but their presence
confirms, at least qualitatively, the dispersion measurements,
on the batch system and suggests, in the light of the results
of section 8 of this work, that the axial mixing in the loops
decreases at high impeller pumping rates; the deviations from
ideality at which this effect is significant are considerably
less than those observed in this present study.
9.2 Conversion for a first order reaction
The proposed model may be used for predicting the steady
state conversion for a first order reaction carried out in a
continuous flow, non-ideal stirred reactor. Steady state mass
balances on each stage yield a set of simultaneous algebraic
108
equations which may be solved to give the ratio of outlet to
inlet concentration of the reactant. For an ideal stirred
vessel with the concentrations expressed in normalised units the
per cent conversion, R, is given by equation 9.1, where k is the
reaction rate constant:
R = (1 k :.,. ) .100 • • • • • • • • • • •
With a value of 4 for the rate constant, the steady state
conversion for the ideal system is 80%. This value for k will
now be used to obtain conversions for the proposed model under
the conditions considered in section 8.
For the case of inflow to the impeller (Figure 8.7 ), the
per cent conversion is given by equation 9.2.
R =" 100 [, . 2 + 2r3 (r+vk+9) r (2,+ 3r )(r+vk+2,)
r+vk (r+vk)2 CQ+r) Q+r • 9.2
Figure 9.1 shows the effect of the pumping capacity: at very
low impeller speeds the conversion is well below that of the
ideal system, the ideal behaviour being rapidly approached at
higher speeds; the truncated mean. which is also presented on
figure 9.1 shows the same effect.
For the case of the unbaffled vessel with the inflow into
the upper loops (Figure 8.14), the per cent conversion is given
by equation 9.3. and the effect of increased agitation shown in
Figure 9.2.
R 100 [, - g.;y: .. ,] Or+g-2r;y:) (g"+r+vk) - r r+Q r:i-Vk • • • • 9.3
where y .... ( 2r+9 y 2vk+2r+Q
109
R
80
60
o
R
90
Figure 9.1
Ml
ideal ·6 "'-- - - - - - --- -- - --- ---=..-=--:...-=--=---=--:.....----=--=-..::::..-=.=..=.-=-=--~ -------
I I I ,
I
I
/ -" /Ml
/
~-
----------\
ideal
--
20 40
Figure 9.2
---------------------------
·55
Ml
·63
·6
80 \
o
R
80
70 o
20 40
Figure 9.3 Ml
R .~
~--'-~~----~id~e:al~--------------------------J
,
-----..Ml / =--. I - __ _
( , --, --I I
-------- ---------·6
I ideal . -~----------------------------------------
I , I· I
20 40 110
This time the maximum conversion occurs at zero pumping capacity;
the truncated mean again predicts this effect.
For the case of the baffled vessel with the inflow into one
of the upper loops (Figure 8.32), the conversion is given by
equation 9.4.
Q.y 1. 9.4 4r
3 2 - 2ry ) 3r+3kv+Q
(r+kv) 3r+Q
Figure 9.3 shows that once again the truncated mean gives a direct
indication of the optimal impeller pumping capacity for this
application; this occurs at q/Q = 5. At very low speeds the
model predicts poor conversion due to the bypassing effect; the
conversion rapidly increases to a maximum value in excess of that
for an ideal system, and then gradually approaches the ideal
value as the pumping action is further increased.
The experimental results reported in Section 8 show that
the model does not apply at very low impeller speeds; at values
of q/Q>9, however, the fit is excellent so that for all cases
where the inflow is directed away from the circulation loop from
which material leaves the vessel, the best value of q/Q for a
first order reaction system would be about 10; for the cases
considered in figures 9.2 and 9.3. this gives an increase in
conversion OVer the ideal stirred vessel system of between
2 and 3 per cent.
9.3 Suggestions for further work:
The experimental investigation reported in Section 8 could
• be extended in a number of ways:· other impeller types and vessel
geometries require investigation as does the effect of varying
the fluid properties; tests on much larger industrial equipment
would reveal whether the proposed model of figure 8.5 is as
111
generally applicable as these results suggest, or whether another
simplification of the general multiloop model of figure 4.3
would be more suitable.
The results of Holmes et al. discussed in 9.1, require
further attention in the light of the multiloop model presented
here: does the axial mixing in the loops decrease at high pumping
rates?; testing this for continuous systems poses some proolems
as the deviation from perfect mixing is likely to be very small,
and apparently random effects could mask the very small initial
oscillations which would require precise measurement.
For most industrial applications, continuous stirred vessels
are required to perform other functions in addition to
distributing the residence times of material elements in a
desirable manner; mixing on a microscopic scale occurs mainly
in the immediate vicinity of the impeller so that the optimal
design for a particular application must take into account both
shearing and pumping characteristics; this has already been
discussed in 3.1 but the effect of the distribution of residence
times on the optimal pumping/shearing ratio could, in some
circumstances, be significant and deserves some attention.
The Markov process method for solving the model equations
warrants further study, particularly with regard to its extension
to deal with piecewise linear systems and partial differential
equations; the matrix is as simple to set up as for the Euler
method, and some preliminary tests suggest that it is as efficient
in computer time as the Crank-Nicholson; it is absolutely
stable, regardless of the time increment used, and so could be
particularly useful for solving problems where physical
instabilities are expected.
The truncated moments could also repay further attention;
112
with precise measuring instruments several of these parameters
are accessible and they can be tailored to characterise just
that part of the response which is important for the application
considered; in both these respects they are superior to the
moments of the complete response which receive, so much attention
in the literature.
9.4 Conclusions:-
A general multiloop circulation model is derived for mixing
in stirred vessels.
A single simplified version of this model was found to fit
the behaviour of quite different turbine stirred, vessels under
widely varying conditions of operation; the most significant
variable was found to be the inlet/outlet configuration and
its effect was predicted quantitatively by the proposed model.
The model indicates that the conversion of a first order
reaction can, according to the conditions of operation, be higher
or lower than for an ideal stirred vessel; a measure of this
effect is provided by the 'truncated mean', which is the first
of a new set of model-independent parameters which can be used
for characterising impulse response curves and which largely
overcome the practical difficulty of measuring the moments of
experimental curves.
A probabilistic treatment of the mixing models considered,
leads to a new and efficient numerical method of solution which
is potentially extendable to other, more complex, problems.
113
APPENDICES
- -------------------------
Al. EXPERIMENTAL RESULTS
Run No.
rpm.
q/Q
Run No.
rpm.
q/Q
1
0
0
1
o
o
Impulse Response Runs
100 Litre Spherical Vessel.
2 3
34 52
1 1.4
Run No.
rpm.
q/Q
2
34
1
\14
Runs SI
4 5
66 78
5 9
Runs SIX
1 2
34 43
1.65 2.1
Runs SO
3 4
52 64
1.5 5
5
86
II
,/
6
105
13
3
59
5.5
7 8 9
124 150 300
22 29 72
6 7 8 9
116 156 202 300
20 30.5 44 71
9 Litre Cylindrical Vessel
Runs Ul
Run No. 1 2 3 4 5 6
rpm 14 24 45 66 95 140
q/Q 2.7 4.5 6.3 14.1 20.4 27.3
Runs UO
Run No. 1 2 3 4 5 6
rpm 0 12 22 38 47 65
q/Q 0 2.3 4 7.3 9.1 12.7
Runs Bl
Run No. 1 2 3 4 5 6
rpm 171 140 95 66 45 34
q/Q 33 27.3 20.4 14.1 9 6.3
Runs BO
Run No. .1 . . 2 3 4 5 6
rpm. 11 22 38 48 64 103
q/Q 2.3 4 7.3 9.1 12.7 20.4
lIS
7
100
21.5
7
104
20.4
7
24
4.5
7
142
28
8 9
17 o 200
3 3 42
8 9
1 43 176
28 32.7
8 9
14 o
2. 7 0
8
19 2
3 8
Normalised Response Curves
Spherical Vessel: Inflow to Impeller.
Runs SI
C
T 1 2 3 4 5
0.00 0.000 0.000 0.000 0.009 0.003 0.05 5.611 5.218 4.907 3.955 3.980 0.10 1.438 1.501 1.369 1.127 1.034 0.15 1.288 0.898 0.745 0.796 0.774 0.20 0.965 0.545 0.567 0.701 0.698 0.25 0.672 0.'+6/i 0.'+79 0.635 0.656 0.30 0.468 0.388 0.447 0.597 0.606 0.35 0.386 0.336 0.431 0.577 0.589 0.40 0.288 0.295 0.401 0.554 0.562 0.45 0.255 0.277 0.379 0.522 0.543 0.50 0.223 0.273 0.3~3 0.506 0.515 0·55 0.212 0.256 0.363 0.490 0.492 0.60 0.197 0.248 0.350 0.461 0.477 0.65 0.182 0.240 0.329 0.440 0.455 0.70 0.177 0.236 0.324 0.422 0.438 0.75 0.167 0.231 0.321 0.'+10 0.'+16 0.80 0.158 0.222 0.305 0.394 0.396 0.85 0.153 0.220 0.292 0.371 0.384 0.90 0.149 0.211 0.285 0.359 0.366 0.95 0.145 0.213 0.273 0.348 0.356 1.00 0.136 0.206 0.2~7 0.331 0.343 1.05 0.133 0.202 0.264 0.317 0.331 1.10 0.130 0.201 0.245 0.307 0.312 1.15 0.125 0.193 0.243 0.286 0.304 1.20 0.125 0.185 0.227 0.280 0.290 1.25 0.123 o .1/i2 0.21/i 0.269 0.272 1.30 0.114 0.177 0.214 0.258 0.268 1.35 0.112 0.177 0.210 0.248 0.257 1.40 0.103 0.170 0.200 0.239 0.245 1.50 0.098 0.158 0.189 0.216 0.222 1.60 0.096 0.152 0.173 0.202 0.204 1.70 0.093 0.141 0.163 0.190 0.188 1 •. 80 0.088 0.138 0.150 0.175 0.172 1.90 0.086 0.129 0.145 0.161 0.160 2.00 0.084 0.128 0.137 0.153 0.145 2.10 0.0/i0 0.119 0.127 0.1'+0 0.13'+ 2.20 0.078 0.112 0.119 0.131 0.123 2.30 0.075 0.106 0.111 0.119 0.114 2.40 0.070 0.100 0.101 0.114 0.107 2.50 0.068 0.095 0.095 0.109 0.096 2.60 0.067 0.0($($ 0.0($7 0.101 0.091 2.70 0.065 0.086 0.084 0.096 0.085 2.80 0.064 0.086 0.079 0.087 0.077 2.90 0.062 0.081 0.078 0.077 0.073 3.00 0.058 0.074 0.076 0.077 0.066
116
6 7 8 9
0.000 0.000 0.001 0.000 2.388 1.072 0.990 0.928 0.927 0.917 0.962 0.915 0.789 0.855 0.913 0.869 0.732 0.805 0.860 0.824 0.693 0.764 0.817 0.780 0.654 0.734 0.775 0.736 0.621 0.697 0.737 0.693 0.589 0.661 0.695 0.662 0.567 0.634 0.665 0.632 0.5'+'+ 0.597 0.62/i 0.602 0.511 0.569 0.596 0.572 0.492 0.526 0.566 0.545 0.467 0.501 0.532 0.519 0.452 0.478 0.498 0.496 0.'+22 0.45~ 0.475 0.475 0.410 0.436 0.454 0.453 0.395 0.416 0.435 0.433 0.381 0.398 0.410 0.413 0.366 0.375 0.393 0.395 0.350 0.360 0.371 0.378 0.329 0.340 0.357 0.359 0.314 0.331 0.342 0.344 0.304 0.313 0.325 0.327 0.289· 0.302 0.310 0.312 0.276 0.2/i8 0.297 0.297 0.269 0.275 0.281 0.283 0.255 0.265 0.270 0.270 0.241 0.250 0.257 0.256 0.222 0.228 0.234 0.233_ 0.203 0.209 0.213 0.211 0.188 0.189 0.195 0.191 0.169 0.169 0.176 0.173 0.155 0.156 0.159 0.158 0.144 0.141 0.145 0.144 0.131 0.12/i 0.132 0.131 0.120 0.118 0.120 0.121 0.112 0.107 0.109 0.110 0.104 0.097 0.100 0.101 0.095 0.089 0.092 0.0_~2
0.087 0.082 0.084 0.0($5 0.082 0.075 0.078 0.076 0.075 0.069 0.072 0.069 0.069 0.063 0.066 0.063 0.065 0.056 0.062 0.058
Spherical Vessel: Inflow to impeller.
Runs SIjX
C T
1 2 3
0.00 0.000 0.002 0.000 0.02 1.610 3.556 4.005 0.04 5.056 2.656 1.872 0.06 2.684 1.710 1.208 0.08 1.623 1.152 0.992 0.10 1.059 1.004 0.850 0.12 0.i)33 0.i)lJ.5 0.767 0.14 0.750 0.738 0.724 0.16 0.648 0.721 0.686 0.18 0.648 0.678 0.675 0.20 0.572 0.643 0.657 0.25 0.506 0.583 0.628 0.30 0.486 0.558 0.597 0.35 0.445 0.530 0.574 0.40 0.433 0.508 0.554 0.45 0.398 0.474 0.528 , 0.50 0.416 0.465 0.510 0.55 0.378 0.450 0.490 0.60 0.364 0.425 0.470 0.65 0.365 0.405 0.444 0.70 0.}40 0.}84 0.427 0.75 0.334 0.371 0.409 0.80 0.311 0.352 0.386 0.85 0.312 0.339 0.366 0.90 0.313 0.326 0.349 0.95 0.292 0.315 0.340 1.00 0.281 0.305 0.323 1.05 0.265 0.29i) 0.311 1.10 0.261 0.283 0.300 1.15 0.254 0.272 0.283
. 1.20 0.243 0.264 0.268 1.25 0.241 0.253 0.257 1.30 0.221 0.242 0.251 1.35 0.222 0.234 0.239 1.40 0.210 0.225 0.225 1.45 0.203 0.214 0.216 1.50 0.201 0.212 0.202 1.55 0.193 0.202 0.196 1.60 0.180 0.195 0.187 1.65 0.181 0.187 0.179 1.70 0.176 0.180 0.176 1.75 0.169 0.178 0.167 1.80 0.162 0.169 0.164 1.85 0.152 0.163 0.156 1.90 0.155 0.157 0.150 1.95 0.145 0.150 0.141 2.00 0.11f3 0.148 0.138
117
Spherical Vessel: Inflow away from Impeller
Runs sO
e-T
1 2 3 4 5 6 7 8 9
0.00 0.008 0.006 0.000 0.001 0.002 0.003 0.008 0.006 0.003 0.05 0.054 0.772 0.518 0.335 0.317 0.504 0.632 0.813 0.891 0.10 0.348 0.570 0.558 0.593 0.747 0.847 0.892 0.894 0.889 0.15 0.536 0.675 0.717 0.798 0.863 0.878 0.870 0.856 0.849 0.20 0.654 0.781 0.811 0.841 0.869 0.849 0.835 0.816 0.808 0.25 0.766 0.1$1:> 0.1$01$ 0.i53lJ. 0.i5lJ.1 0.1$11 0.793 0.779 0.773 0.30 0.792 0.812 0.791 0.803 0.797 0.772 0.755 0.742 0.738 0.35 0.809 0.787 0.760 0.768 0.758 0.735 0.720 0.706 0.702 0.40 0.799 0.763 0.722 0.733 0.720 0.699 0.684 0.673 0.670 0.45 0.784 0.729 0.689 0.700 0.688 0.667 0.652 0.641 0.639 0.50 0.752 0.~9lJ. 0.651$ 0.670 0.~48 0.b36 0.b22 0.b11 0.609 0.55 0.728 0.656 0.628 0.633 0.614 0.604 0.595 0.581 0.580 0.60 0.689 0.623 0.594 0.607 0.590 0.574 0.567 0.552 0.553 0.65 0.657 0.592 0.563 0.573 0.561 0.547 0.542 0.524 0.529 0.70 0.623 0.558 0.537 0.542 0.537 0.522 0.516 0.500 0.504 0.75 0.5t17 0.531 0.509 0.517 0.511 0.lJ.9lJ. 0.lJ.93 0.lJ.7i5 0.lJ.79 0.80 0.561 0.506 0.480 0.495 0.482 0.471 0.469 0.457 0.456 0.85 0.532 0.482 0.461 0.471 0.458 0.450 0.456 0.436 0.434 0.90 0.508 0.457 0.437 0.448 0.435 0.431 0.431 0.416 0.414 0.·95 0.479 0.436 0.419 0.428 0.414 0.408 0.409 0.397 0.395 1.00 0.453 0.417 0.398 0.410 0.395 . 0.392 0.392 0.379 0.377 1.05 0.428 0.394 0.379 0.378 0.375 0.368 0.375 0.363 0.360 1.10 0.406 0.376 0.364 0.360 0.358 0.349 0.358 0.347 0.345 1.15 0.387 0.355 0.346 0.343 0.340 0.335 0.340 0.332 0.329 1.20 0.368 0.337 0.330 0.325 0.320 0.335 0.325 0.316 0.314 1.25 0.3lJ.1:I 0.319 0.313 0.307 0.307 0.305 0.311 0.300 0.300 1.30 0.326 0.306 0.300 0.294 0.295 0.287 0.298 0.287 0.287 1.35 0.313 0.292 0.285 0.279 0.280 0.274 0.283 0.275 0.274 1.40 0.293 0.276 0.271 0.266 0.269 0.261 0.270 0.262 0.261 1.50 0.264 0.221 0.245 0.240 0.243 0.236 0.247 0.240 0.2~_ 1.60 0.235 0.221$ 0.225 0.216 0.219 0.215 0.225 0.21/$ 0.217 1.70 0.213 0.204 0.203 0.197 0.199 0.196 0.206 0.199 0.199 1.80 0.193.0.185 0.184 0.178 0.175 0.178 0.187 0.183 0.181 1.90 0.171 0.165 0.167 0.163 0.161 0.162 0.173 0.165 0.165 2.00 0.155 0.151 0.152 0.146 0.147 0.146 0.157 0.152 0.150 2.10 0.139 0.135 0.139 0.132 0.13lJ. 0.133 0.1lJ.3 0.139 0.137 2.20 0.126 0.123 0.126 0.119 0.121 0.121 0.133 0.128 0.125 2.30 0.113 0.111 0.115 0.109 0.109 0.111 0.120 0.116 0.115 2.40 0.103 0.100 0.105 0.099 0.096 0.100 0.111 0.106 0.105 2.50 0.093 0.092 0.097 0.088 0.088 0.091 0.100 0.097 0.095 2.60 0.Otl6 0.01:13 O.O~I:I 0.Otl1 0.Otl1 0.01$3 0.091 0.01$9 O.o~tI 2.70 0.077 0.075 0.082 0.072 0.073 0.076 0.084 0.083 0.080 2;80 0.070 0.069 0.076 0.066 0.066 0.070 0.080 0.076 0.073 2.90 0.064 0.062 0.071 0.060 0.059 0.064 0.073 0.071 0.068 3.00 0.059 0.057 0.065 0.054 0.053 0.059 0.068 0.066 0.062
118
Cylindrical Vessel: Unbaffled with inflow to Impeller.
Runs UI
C T
1 2 3 4 5 6 7 8 9
0.000 0.000 1.154 0.369 0.000 0.000 0.023 0.000 0.235 0.000 0.025 1.850 2.806 1.746 1.127 1.341 1.185 1.081 1.108 1.011 0.050 1.264 1.300 1.360 1.078 1.054 1.036 1.023 0.978 0.978 0.075 0.830 1.073 1.145 0.993 1.016 0:988 0.957 0.955 0.914 0.100 0.807 0.958 1.013 0.957 0.965 0.964 0.907 0.931 0.895 0.125 0.919 0.900 0.980 0.933 0.927 0.940 0.880 0.909 0.875 0.150 0.919 0.888 0.954 0.908 0.915 0.915 0.860 0.889 0.858 0.175 0.874 0.853 0.840 0.872 0.880 0.891 0.837 0.868 0.834 0.200 0.841 0.830 0.819 0.850 0.857 0.867 0.818 0.848 0.813 0.225 0.1,)30 0.b18 0.799 0.1,)39 0.1,)'+6 0.1,)'+6 0.797 0.b27 0.794 0.250 0.818 0.795 0.768 0.807 0.823 0.824 0.778 0.817 0.772 0.275 0.796 0.783 0.737 0.796 0.800 0.802 0.760 0.797 0.755 0.300 0.785 0.760 0.747 0.785 0.766 0.781 0.739 0.776 0.735 0.325 0.762 0.736 0.737 0.752 0.766 0.759 0.722 0.750 0.716 0.350 0.762 0.725 0.717 0.730 0.732 0.737 0.702 0.743 0.696 0.375 0.729 0.701 0.676 0.709 0.715 0.726 0.681 0.727 0.679 0.400 0.717 0.678 0.665 0.698 0.699 0.705 0.668 0.711 0.667 0.425 0.706 0.67b 0.655 0.6b2 0.691 0.6b6 0.646 0.696 0.6'+'+ 0.450 0.695 0.655 0.635 0.659 0.674 0.670 0.631 0.680 0.626 0.475 0.684 0.643 0.635 0.659 0.658 0.655 0.614 0.664 0.611 0.500 0.650 0.619 0.614 0.635 0.649 0.639 0.596 0.649 0.589 0.550 0.639 0.584 0.594 0.611 0.617 0.608 0.581 0.618 0.578 0.600 0.605 0.549 0.553 0.580 0.584 0.577 0.552 0.586 0.576 0.650 0.583 0.514 0.532 0.549 0.551 0.561 0.521 0.563 0.517 0.700 0.538 0.503 0.502 0.525 0.526 0.522 0.496 0.539 0.492 0.750 0.504 0.456 0.471 0.502 0.510 0.499 0.471 0.508 0.467 0.800 0.482 0.444 0.450 0.478 0.477 0.476 0.444 0.485 0.440 0.850 0.448 0.421 0.420 0.447 0.452 0.452 0.421 0.461 0.418 0.900 0.426 0.397 0.410 0.431 0,428 0.429 0.396 0.438 0.395 0.950 0.404 0.362 0.379 0.408 0.411 0.413 0.376 0.414 0.373 1.000 0.392 0.351 0.358 0.384 0.386 0.390 0.357 0.391 0.354 1.050 0.370 0.327 0.325 0.368 0.362 0.366 0.33? 0.367 0.335 1.100 0.348 0.304 0.317 0.345 0.345 0.351 0.316 0.344 0.315 1.150 0.336 0.292 0.297 0.321 0.329 0.327 0.301 0~328 0.301 1.200 0.303 0.281 0.287 0.306 0.312 0.312 0.284 0.313 0.286 1.250 0.280 0.257 0.266 0.290 0.288 0.288 0.268 0.289 0.265 1.300 0.269 0.234 0.256 0.274 0.271 0.273 0.253 0.274 0.251 1.350 0.247 0.222 0.225 0.259 0.255 0.257 0.239 0.258 0;237 1.400 0.235 0.210 0.215 0.243 0.247 0.242 0.226 0.242 0.222 1.450 0.224 0.199 0.205 0.227 0.222 0.234 0.212 0.227 0.210 1.500 0.202 0.175_ 0.184 0.220 0.206 0.210 0.201 0.211 0.198 1.550 0.179 0.164 0.174 0.204 0.197 0.203 0.189 0.195 0.185 1.600 0.179 0.152 0.154 0.188 0.189 0.187 0.177 0.188 0.175 1.650 0.157 0.140 0.143 0.172 0.173 0.172 0.168 0.180 0.163 1.700 0.146 0.129 0.133 0.165 0.156 0.164 0.156 0.164 0.154 1.750 0.135 0.117 0.123 0.157 0.148 0.148 0.147 0.149 0.144 1.800 0.123 0.105 0.113 0.141 0.140 0.140 0.139 0.141 0.136 1.850 0.112 0.094 0.102 0.133 0.132 0.133 0.129 0.133 0.128 1.900 0.101 0.082 0.092 0.125 0.123 0,117 0.122 0.125 0.119 1.950 0.090 0.070 0.082 0.110 0.107 0.109 0.114 0.109 0.113 2.000 0.078 0.070 0.082 0.102 0.099 0.101 0.106 0.102 0.105
119
Cylindrical Vessel: Unbaffled with Inflow away from Impeller
Runs UO
C T
'i
1 2 3 4 5 6 7 8 9
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.025 0.000 0.000 0.000 0.000 0.271 0.040 0.098 0.359 0.375 0.050 0.000 0.254 0.657 0.732 0.431 0.423 0.815 0.919 0.949 0.075 0.646 0.588 0.682 0.824 0.726 0.702 0.954 0.939 0.973 0.100 0.376 0.676 0.061 0.835 0.782 0.842 0.954 0.939 0.961 0.125 0.793 0.676 0.167 0.928 0.876 0.909 0.934 0.909 0.937 0.150 0.769 0.779 0.959 0.875 0.897 0.929 0.924 0.899 0.917 0.175 0.761 0.820 0.959 0.885 0.897 0.929 0.904 0.870 0.896 0.200 0.769 0.841 0.898 0.865 0.897 0.919 0.884 0.861 0.875 0.225 0.932 0.779 0.1:)61:) 0.1:)65 0.1:)1:)6 0.900 0.1:)75 0.1:)32 0.b54 0.250 0.009 0.820 0.827 0.865 0.865 0.890 0.845 0.822 0.833 0.275 0.963 0.800 0.807 0.835 0.844 0.861 0.825 0.793 0.812 0.300 0.912 0.831 0.807 0.824 0.834 0.842 0.805 0.783 0.792 0.325 0.942 0.820 0.799 0.804 0.813 0.822 0.787 0.758 0.773 0.350 0.932 0.820 0.782 0.794 0.792 0.803 0.770 0.750 0.757 0.375 0.902 0.810 0.766 0.773 0.774 0.783 0.754 0.734 0.741 0.400 0.853 0.800 0.741 0.755 0.758 0.766 0.729 0.726 0.725 0.425 0.~73 0.771 0.732 0.74b 0.742 0.750 0.721 0.702 0.717 0.450 0.823 0.763 0.716 0.724 0.734 0.734 0.705 0.686 0.693 0.475 0.823 0.747 0.707 0.709 0.718 0.718 0.688 0.670 0.678 0.500 0.785 0.739 0.691 0.693 0.694 0.694 0.680 0.662 0.662 0.550 0.720 0.707 0.657 0.662 0.662 0.670 0.639 0.630 0.630 0.600 0.703 0.676 0.623 0.623 0.638 0.638 0.615 0.598 0.598 0.650 0.679 0.636 0.591 0.600 0.606 0.606 0.582 0.574 0.574 0.700 0.613 0.604 0.558 0.576 0.566 0.574 0.557 0.550 0.550 0.750 0.597 0.572 0.533 0.553 0.551 0.542 0.533 0.518 0.518 0.800 0.556 0.540 0.499 0.514 0.519 0.519 0.500 0.494 0.494 0.1:)50 0.523 0.517 0.474 0.491 0.41:)7 0.41l7 0.475 0.471 0.470 0.900 0.491 0.493 0.458 0.460 0.471 0.463 0.451 0.447 0.446 0.950 0.466 0.469 0.424 0.436 0.439 0.439 0.426 0.423 0.422 1.000 0.442 0.421 0.408 0.413 0.415 0.415 0.410 0.407 0.407 1.050 0.417 0.405 0.383 0.389 0.391 0.383 0.377 0.383 0.383 1.100 0.384 0.389 0.358 0.366 0.375 0.375 0.361 0.359 0.359 1.150 0.360 0.366 0.341 0.343 0.351 0.343 0.344 0.343 0.343 1.200 0.335 0.334 0.316 0.327 0.327 0.327 0.320 0.327 0.327 1.250 0.311 0.318 0.300 0.312 0.311 0.303 0.303 0.303 0.303 1.300 0.286 0.302 0.283 0.288 0.295 0.287 0.279 0.287 0.287 1.350 0.270 0.286 0.266 0.273 0.271 0.263 0.270 0.271 0.271 1.400 0.254 0.262 0.250 0.257 0.255 0.255 0.254 0.255 0.255 1.450 0.229 0.246 0.233 0.234 0.239 0.231 0.229 0.247 0.239 1.500 0.213 0.231 0.216 0.226 0.223 0.223 0.221 0.223 0.223 1.550 0.204 0.215 0.200 0.210 0.207 0.199 0.205 0.215 0.207 1.600 0.180 0.199 0.191 0.195 0.199 0.199 0.197 0.207 0.191 1.650 0.172 0.191 0.175 0.179 0.184 0.176 0.180 0.191 0.175 1.700 0.155 0.175 0.166 0.164 0.168 0.168 0.164 0.183 0.167 1.750 0.139 0.167 0.150 0.156 0.160 0.152 0.156 0.167 0.151 1.800 0.131 0.151 0.133 0.140 0.144 0.144 0.148 0.152 0.143 1.850 0.123 0.135 0.133 0.132 0.128 0.128 0.131 0.152 0.136 1.900 0.106 0.127 0.117 0.117 0.120 0.120 0.123 0.136 0.120 1.950 0.090 0.111 0.108 0.109 0.112 0.112 0.115 0.128 0.112 2.000 0.082 0.111 0.100 0.101 0.096 0.104 0.107 0.120 0.104
120
Cylindrical Vessel: Baffled with inflow to impeller.
Runs Bl
C T
1 2 3 4 5 6 7 8 9
0.000 0.109 0.039 0.046 0.516 0.212 0.011 0.410 0.310 0.000 0.025 1.157 1.256 1.407 1.068 1.046 1.758 1.879 2.370 1.972 0.050 1.012 0.999 1.126 1.293 1 .31+4 1.128 1.200 0.943 1.333 0.075 0.976 0.963 0.985 1.036 1.085 1.112 1.050 0.843 0.758 0.100 0.952 0.939 0.950 0.940 0.953 0.994 0.917 0.765 0.686 0.125 0.92~ 0.915 0.915 0.917 0·940 0.925 0.~41 0.~09 0.790 0.150 0.904 0.891 0.891 0.906 0.913 0.880 0.830 0.865 0.852 0.175 0.880 0.856 0.868 0.871 0.887 0.868 0.830 0.865 0.821 0.200 0.857 0.834 0.844 0.849 0.874 0.822 0.798 0.843 0.800 0.225 0.835 0.812 0.813 0.837 0.860 0.800 0.787 0.821 0.779 0.250 0.813 0.802 0.802 0.814 0.821 0.777 0.766 0.809 0.779 0.275 0.802 0.780 0.781 0.803 0.794 0.765 0.744 0.787 0.748 0.300 0.781 0.758 0.760 0.780 0.781 0.742 0.722 0.743 0.727 0.325 0.74~ 0.737 0.739 0.745 0.76~ 0.731 0.701 0.721 0.717 0.350 0.737 0.715 0.718 0.734 0.741 0.720 0.701 0.710 0.707 0.375 0.716 0.704 0.697 0.722 0.728 0.685 0.669 0.721 0.696 0.400 0.694 0.678 0.676 0.711 0.715 0.697 0.658 0.687 0.686 0.425 0.678 0.662 0.660 0.688 0.702 0.674 0.658 0.687 0.675 0.450 0.663 0.654 0.653 0.688 0.688 0.651 0.625 0.665 0.644 0.475 0.647 0.639 0.630 0.654 0.675 0.628 0.615 0.654 0.623 0.500 0.639 0.623 0.622 0.631 0.662 0.617 0.604 0.632 0.623 0.550 0.608 0.600 0.592 0.6<?8 0.609 0.605 0.571 0.599 0.5~2 0.600 0.577 0.569 0.569 0.585 0.596 0.560 0.561 0.577 0.571 0.650 0.554 0.545 0.547 0.550 0.556 0.548 0.528 0.532 0.530 0.700 0.530 0.522 0.516 0.527 0.516 0.514 0.507 0.532 0.499 0.750 0.499 0.491 0.493 0.505 0.490 0.491 0.485 0.499 0.478 0.800 0.476 0.467 0.471 0.470 0.477 0.457 0.453 0.455 0.457 0.850 0.452 0.444 0.448 0.447 0.450 0.434 0.431 0.444 0.436 0.900 0.437 0.421 0.425 0.424 0.410 0.411 0.410 0.410 0.416 0.950 0.405 0.405 0.402 0.401 0.397 0.400 0.388 0.388 0.395 1.000 0.382 0.389 0.:280 0.378 0.371 0.366 0.367 0.3_66 0.395 1.050 0.367 0.358 0·364 0.355 0.357 0.354 0.356 0.355 0.353 1.100 0.343 0.343 0.342 0.333 0.331 0.343 0.334 0.333 0.332 1.150 0.328 0.327 0.326 0.310 0.318 0.308 0.323 0.310 0.332 1.200 0.312 0.312 0.311 0.298 0.291 0.297 0.302 0.288 0.312 1.250 0.296 0.296 0.296 0.275 0.278 0.274 0.280 0.277 0.291 1.300 0.273 0.280 0.281 0.264 0.265 0.263 0.270 0.266 0.291 1.350 0.257 0.265 0.258 0.252 0.252 0.251 0.259 0.244 0.260 1.400 0.242 0.257 0.250 0.229 0.238 0.240 0.237 0.222 0.239 1.450 0.226 0.241 0.235 0.218 0.212 0.217 0.226 0.211 0.229 1.500 0.211 0.226 0.220 0.206 0.199 0.206 0.205 0.200 0.208 1.550 0.203 0.218 0.213 0.195 0.1~5 0.194 0.205 0.1~9 0.197 1.600 0.187 0.203 0.197 0.183 0.172 0.171 0.183 0.177 0.187 1.650 0.179 0.187 0.182 0.172 0.159 0.171 0.173 0.155 0.177 1.700 0.172 0.179 0.175 0.161 0.159 0.160 0.173 0.144 0.177 1.750 0.156 0.171 0.159 0.149 0.146 0.137 0.162 0.144 0.156 1.800 0.148 0.164 0.152 0.138 0.132 0.137 0.151 0.133 0.145 1.850 0.140 0.156 0.144 0.126 0.119 0.126 0.140 0.122 0.135 1.900 0.125 0.148 0.137 0.115 0.106 0.114 0.129 0.111 0.125 1.950 0.117 0.140 0.129 0.103 0.106 0.103 0.119 0.100 0.114 2.000 0.101 0.125 0.114 0.103 0.093 0.103 0.119 0.100 0.114
121
Cylindrical Vessel: Baffled with inflow away from Impeller.
Runs EO
C T
1 2 3 4 5 6 7 8
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.025 0.293 0.325 0.776 0.456 0.326 0.504 0.495 0.662 0.050 1.263 0.570 1.020 0.672 0.857 0.948 0.934 0.977 0.075 1.187 0.937 1.087 0.865 0.931 0.969 0.986 0.987 0.100 0.956 0.885 0.997 0.934 0.963 0.959 0.976 0.967 0.125 0.846 0.906 . 0.975 0.958 0.952 0.938 0.955 0.957 0.150 0.879 0.906 0.941 0.958 0.931 0.928 0.934 0.926 0.175 0.890 0.916 0.919 0.934 0.910 0.896 0.913 0.915 0.200 0.868 0.916 0.897 0.911 0.889 0.886 0.893 0.885 0.225 0.846 0.896 0.874 0.888 0.868 0.855 0.872 0.864 0.250 0.825 0.875 0.841 0.876 0.847 0.844 0.851 0.843 0.275 0.814 0.854 0.830 0.853 0.825 0.823 0.830 0.823 0.300 0.803 0.834 0.798 0.819 0.805 0.793 0.799 0.802 0.325 0.782 0.803 0.787 0.807 0.774 0.782 0.789 0.782 0.350 0.760 0.783 0.754 0.784 0.763 0.762 0.769 0.762 0.375 0.739 0.763 0.732 0.761 0.743 0.742 0.749 0.742 0.400 0.717 0.753 0.710 0.738 0.722 0.721 0.718 0.722 0.425 0.696 0.723 0.684 0.721 0.702 0.691 0.708 0.702 0.450 0.678 0·703 0.668 0.705 0.681 0.681 0.678 0.682 0.475 0.662 0.693 0.660 0.697 0.660 0.660 0.668 0.652 0.500 0.655 0.662 0.644 0.672 0.653 0.643 0.648 0.642 0.550 0.624 0.621 0.613 o. M·7 0.616 0.b14 0.611 0.b13 0.600 0.593 0·599 0.589 0.605 0.594 0.592 0.589 0.584 0.650 0.562 0.570 0.566 0.580 0.564 0.563 0.560 0.563 0.700 0.539 0.549 0.534 0.547 0.542 0.533 0.531 0.526 0.750 0.524 0.520 0.503 0.514 0.519 0.504 0.509 0.505 0.800 0.485 0.491 0.479 0.489 0.490 0.482 0.480 0.476 0.850 0.462 0.476 0.456 0.464 0.467 0.460 0.466 0.454 0.900 0.439 0.447 0.432 0.439 0.438 0.438 0.437 0.433 0.950 0.416 0.426 0.409 0.423 0.423 0.416 0.415 0.411· 1.000 0.393 0.397 0.385 0.398 0.401 0.395 0.393 0.397 1.050 0.370 0.383 0.369 0.381 0.378 0.373 0.371 0.3bli 1.100 0.354 0.361 0.354 0.357 0.356 0.358 0.349 0.353 1.150 0.331 0.339 0.330 0.340 0.341 0.336 0.335 0.332 1.200 0.316 0.325 0.314 0.323 0.326 0.314 0.320 0.310 1.250 0.300 0.296 0.306 0.307 0.304 0.300 0.298 0.296 1.300 0.285 0.289 0.283 0.282 0.289 0.285 0.284 0.281 1.350 0.270 0.267 0.267 0.265 0.275 0.270 0.262 0.267 1.400 0.254 0.260 0.251 0.249 0.252 0.256 0.255 0.245 1.450 0.239 0.238 0.236 0.232 0.237 0.234 0.233 0.238 1.500 0.223 0.231 0.212 0.216 0.223 0.227 0.226 0.224 1.550 0.216 0.217 0.1?b 0.207 0.215 0.219 0.211 0.209 1.600 0.200 0.202 0.189 0.191 0.200 0.197 0.196 0.195 1.650 0.193 0.188 0.181 0.182 0.193 0.190 0.182 0.180 1.700 0.177 0.180 0.165 0.166 0.178 0.175 0.167 0.173 1.750 0.162 0.166 0.157 0.158 0.163 0.1615 0.160 0.159 1.800 0.162 0.159 0.149 0.141 0.156 0.153 0.153 0.151 1.850 0.146 0.144 0.134 0.133 0.141 0.146 0.138 0.137 1.900 0.131 0.137 0.126 0.124 0.134 0.139 0.131 0.130 1.950 0.131 0.123 0.110 0.116 0.126 0.124 0.116 0.115 2.000 0.123 0.115 0.110 0.108 0.119 0.117 0.116 0.115
, 22
Impulse Response Runs
Truncated first and second moments
Runs SI: Truncated at T = 3.
Run No. 1 2 3 4 5 6 7 8 9
Truncated .449 .612 .704 .786 .801 .809 .822 .819 .839 1st moment
Truncated .696 1.001 1.113 1.196 1.212 1.216 1.206 1.182 1.227 2nd moment
Runs SO: Truncated at T:= 3
Run No. 1 2 3 4 5 6 7 8 9
Truncated .911 .877 .893 .875 .869 .866 .869 .865 .856 1st moment
Truncated 1.299 1.263 1.324 1; 268 ,1 ~ 264 1.270 1.288 1.291 1.271 2nd moment
Runs UI: Truncated at. T = 2
Run No. 1 2 3 4 5 6 7 8 9
Truncated .565 .518 .531 ;586 .576 .583 .589 .586 .586 1st moment
Truncated .581 .525 .542 .618 .605 .614 .620 .617 .613 2nd moment
Runs UO: Truncated at T = 2
Run No. 1 2 3 4 5 6 7 8 9
Truncated 6 1st moment .629 .637 • 03 .614 .616 .615 .607 .614 .602
Truncated 2nd moment .640 .671 .631 .643 .645 .643 .639 .657
123
Runs Bl: Truncated at T = 2
Run No. 1 2 3 4 5 6 7 8 9
Truncated .586 .597 .589 .572 .567 .562 .572 .558 ·575 1st moment
Truncated .621 .646 .632 .599 .590 .591 .614 .583 .614 2nd moment
Runs BO: Truncated at T = 2
Run No. 1 2 3 4 5 6 7 8
Truncated .607 .614 .592 .604 .609 .606 .600 .597 1st moment
Truncated .650 .655 .624 .635 .650 .647 .637 .634 2nd moment
•
124
Besults of Flow Follower experiments
1. 9 litre Cylindrical Vessel
Average Pumping Stirrer Number of Total Time Circulation Capacity
Speed Counts (secs) time. ( secs) (l/min) (rpm) N T TAV = T/N P '" V x 60/TAV
8§. 192 7360 38.33 14.1
105 180 5649 31.40 17.2
133 165 3124 18.96 28.5
138 165 2701 16.38 33.0
165 170 2280 13.40 40.3
187 180 2412 13.40 40.3
222 180 2138 11.87 45.5
233 180 2194 12.20 44.2
267 180 1703 9.47 57.0
320 182 1482 8.15 66.2
2. 100 litre Spherical Vessel
Average Pumping Stirrer Number of Total Time Circulation" Capacity
Speed Counts (secs) Time. (secs) ( 1/min) (rpm) N T TAV " T/N P = V x 60/TAV
34 150 65,520 437.0 7.6
62 170 18,724 110.0 30.0
75 170 10,286 60.5 54.6
81 145 7,290 42.9 76.8
91 160 5,262 32.9 100·3 104 150 3,410 22.75 145.2
116 170 3,218 18.91 175.2
132 140 2,398 17.14 192.8
135 165 2,909 17.64 187.1
150 165 2,508 15.22 217.0
150 143 2,310 16.17 204.0
170 167 2,011 12.05 274.2
200 167 1,900 11.39 292.5
220 165 1,443 8.74 378.0
125
A2. COMPUTER PROGRAMMES
A2.1. A GENERAL COMPUTER PROGRAMHE FOR CONTUnJOUS
FLOW MIXING MODELS
The programme to be described computes the response of
continuous flow models that comprise a finite number of ideal
mixing stages. A simple probability method is used. The
volumes of the stages and the magnitudes of the flows between
stages, are entered as data and the response of the model to
impulse Or step inputs is obtained.
Models consisting of large numbers of stages generally
have zero flow between most of the stages. To save computer
time and storage, the matrix containing these flows is stored
in a compressed form.
A2.1 .1 • Programme running instructions
After compilation of the programme, data (in any consistent
units) are required in the following order:
1 • the throughput flow • • Q
2. the total volume . V
3. the number of states • N
4. the required response Nr
5· the system matrix • . M
6. the initial state vector S(o)
7. the print out interval h
8. the last point . . . Tmax
These are obtained, from the model to be tested, as follows:
1. The throughput flow (Q)
The units must be the same as for the flow elements of
the system matrix.
2. The total volume M
The units must be the same as the volume elements of M
They need not, however, be consistent with those of Q.
126 .
3. The number of states (N)
The stages of the model are numbered from 1 to N - 1 in
any fashion. The number N is used to represent material that
has left the system: thus for the model shown in FigureA2,l,N = 5.
et 12
Q Q ~ 1 2 5
et 21
et'3 Q'2
et3• ,
3 4
et'3
Thus N = number of stages in model + 1.
4. The required response Nr
This is an integer in the range 1 to N. Usually it is
the response of the whole system that is requ~red. If the
integer N is entered here the response of the system to a unit
step function will be computed, regardless of the manner in which
the stages are numbered. Alternatively, the impulse response
of any stage can be obtained by entering the appropriate stage
number at this point. The impulse response of the whole system
is the same as that of the stage from which material leaves the
system: thus for the impulse response of the model in Figure
A2.1, the integer 2 would be entered at this p oint.
5. The system matrix M
This is an N x N square matrix.
are the volumes (vi) of the stages and
are the flows (q .. ) between stages. ~J
127
The d iagonal elements
the non diagonal elements
For computational
convenience the flows from a given stage (i) are entered in the
ith column. The system matrix for the model in Figure AZ.1 is::
v1 QZ1 0 0 0
q1Z v2 0 q42 0
M = q13 0 v3 q43 0
° 0 q34 v4 0
o o o
This matrix is read in rows: i.e.
v1 , Q21' 0, 0, 0, q12' v2 ' 0, Q42' 0, Q13' etc.
The final element of M has little meaning at this stage.
Arbitrarily a value of 1 is assigned to it.
On receiving this matrix some computation takes place before
the next data are called for.
6. The 'initial state vector' 8(0)
This is an N element vector which indicates the stage
subjected to the tracer disturbance to time zero.
When the response of the whole system is required, the
element of this vector corresponsing to the stage into which
material enters the system, is given a value of 1 and the other
elements a value of zero. Thus for the above example:
S(o) = [1, 0, 0, 0, OJ
The initial disturbance need not be restricted to one stage: it
could be divided into any number of stages.
however, must sum to 1. For example:
8(0) = [0, .5. .5, 0, oJ 7.· The print out interval h
8. The last point Tmax
The elements of S
For convenience the response curves are normalised: the
time scale is expressed in dimensionless units T, where T = tQ/v
Thus with h =.1 and Tmax = 2,the specified response
curve will be printed out at intervals of .1 of the mean time up
to twice the mean time.
128
A2.1.2 Programme Details
The programme is executed in 3 stages which may be summarised
as follows:
1.
2.
Efficient storage of the system matrix.
Calculation of transition probabilities (Pij
) and
replacement of the system matrix (M) by the system
transition matrix (P).
3. Repeated multiplication of the Transition Matrix by
the state vector (S) to produce the required system
response.
1. System Matrix Storage (Figure A2.2)
The system matrix (M) consists of 3 types of element: stage
volumes (vi) which lie on the diagonal, flows between stages,and
zeros.
It is stored as 2 matrices: a compressed version of the
system matrix, consisting of all the non-zero elements of M, and
integers representing a string of zeros; and a code matrix (C)
which enables the different types of element of M to be
distinguished.
Elements of M are designated H(n) (reading across the rows),
and the corresponding element of C is C(n).
The following code has been adopted:
When M(n) is a volume element,
" H(n) is a flow element,
C(n) = .5
C(n) = 0
II M(n) refers to a string of zeros C(n) = M(n).
This means that if the original system matrix is read
1 2 0 0 0
3 1 0 3 0
M = 3 0 1 1 0
0 0 4 1 0
0 4 0 0 1
129
in as
'"' o
READ Q,V,N,Nr
n=l 10=0 1e=1 Ir=l
READ M(n)
? . yes
r---+----------------------------;1r=Ir+l
10=10+1
C(n)=1o n=n+l M(n)=M(n-1) M(n-1)=10 10=0
M(n)=Io C(n)=Io n=n+1 10=0
yes
? o no
FIGURE A2.2 System Matrix Storage
------------------------------------------------------------------------------------------
--------------------------------------------------------------------------------------
It is stored as M and C where:
M • 1, 2, 3, 3, 1, 1, 3, 1, 3, 1, 1, 1, 1, 2, 4, 1, 1, 1, 4, 2, 1.
C •• 5. 0, 3. 0, .5, 1, 0, 1, 0. 1, .5. 0. 1, 2, 0, .5, 1, 1, 0. 2,.5.
2. The Transition Matrix (Figure A2.3.)
Three operations are performed to convert the matrix M to
the transition matrix P. Each operation is performed on the
columns of M which means that, as M is in a compressed form,
with elements stored sequentially row after row, it is necessary
to scan and identify each element in turn; the code matrix (C)
is used for this identification.
When the first element in the column to be operated on,
has been identified, a count is kept as subsequent elements are
scanned; volume and flow elements increment this count by 1 and
elements representing m zeros by m. When this count reachesN
the next element in the column has been reached. This procedure
enables the following three operations to be performed on the
columns of M
(i) The time increment (~t) is computed to make the
minimum probability of remaining in a stage equal to
.99. This appears to give a sufficiently accurate
solution for most practical purposes: for models
whose analytical solutions were available, agreements
to within about 1% of the maximum response were
obtained over the whole curve. Greater accuracy,
at the expense of increased computation time, can be
obtained by increasing this minimum 'staying'
probability.
A normalised form of equation 5.4 is used to find
theLlT that satisfies this condition. The
normalisation is achieved by multiplying the vessel
volumes (v.) by Q/V in equations 5.4 and 5.5. This ~
has the effect of setting the nominal system mean
131
u ....
Tm=V/Q
OP=l 4T=1 NORM=V
NORM· NORM-TmxVOL
Nr=N-l ?
-FLOW x.6T =exp v VOL =l-p
q v LOWA = FLOW
n=l 1e=0 1r=0 10=0
2
yes
no
A=VOL X.OlO06
no
Op=
'-_______ ot-n_o--< Ir=N
yes
FIGURE A2.3
M(n)=M(n)xp FLOWA q
Transition Matrix from
System Matrix
time equal to 1.
(i.e. V/Q = 1)
(ii) Having foundL\T as describ.ed above, the staying and
leaving probabilities for the first vessel are
computed using the normalised form of equations
5.4 and 5.5.
(iii) The elements of the first column are located once
again, and the volume and flow elements are replaced
by the appropriate transition probabilities as
calculated above.
Operations (ii) and (iii) are repeated for each column of M
up to the (N-1)th column; the Nth column is left unchanged.
The system matrix has now been replaced by the transition
matrix (p») stored in compressed form; the elements representing
strings of zeros remained unchanged and the last stored element of P
-FNN-retains a value of 1.
As the elements of the transition matrix are probabilities,
they have values between 0 and 1. Any element in the compressed
matrix that is equal to 1 is either PNN or represents a zero;
all elements greater than 1 represent a string of zeros. As no
distinction need be made between staying and leaving probabilities,
and as zeros can now be easily identified, the code matrix (C) is
no longer required and can be overwritten by the state vectors in
the third and final stage of the execution.
3. The System Response (Figure A2.4.)
The intial state vector is now read into locations
C(le), le = 1, 2, "" N. Multiplication of P by this vector
yields the state vector after time L\T. This new state vector
is stored in locations C(le2), Ie2 = N + 1, N + 2, • •• , N + N.
The reason for storing M (and hence p») in the transposed
form will now be apparent; multiplication of the compressed
133
I~------------ - - - -
-'" ..
Fl~M(Nr) r----------------~-1Ie~1,Ie2~N+l
Ie~Ie+1
F~ [M(Nr) -Fi] [TP-T+ h
F~ [F+M(Nr)]. NORM
no
T)Tmax ?
yes
Ie~N
?
PRINT T, F
yes
no
no
Ie2~2N
?
n=l 1e=1 Ir=l
Ie2~N+1
TP=h T~O
READ C(Ie)
READ
SUM~O
1e=1
2
1e=le+1
FIGURE A2.4 The System Response
M(n) > ?
no
X=M(n)xC(Ie) SUM~SUM + X
no
n=ll+l Ie~Ie+1
no ,-__________ -,
/"-; ........ Ie~Ie+M(n)
yes
C(Ie2)~SUM
Ie2~le2+1
n~n+M(n)
---- -----------------------------------------------------------
matrix P by Scan nOli be performed in a minimum number of
operations.
The state vector thus formed is transferred to locations
Cere) re = 1, 2, ••• , N and the multiplication repeated. Such
operations yield the state vector after time n~T,and when nLlT
becomes larger than the print out interval the required response
is computed and printed out. The system step response at time
n~T is simply the last (Nth) element of the state vector after
n matrix mUltiplications. Similarly the impulse response of the
ith stage is obtained from the ith element of this vector; in this
case, however, the element must be multiplied by the ratio. of
the system volume to the stage volume (V/v.) in order to produce J.
the desired normalisation. A linear interpolation between the
appropriate vector elements after n-1 and n multiplications is
employed to obtain the response at the required time.
The whole procedure is repeated until the full time range
has been covered.
A2.1.3 Example: n stages in series
The data for the impulse responses of elev~n stages in
series are as follows
1 11 12 11
100 0 0 000 0 0 0 0 1 100 0 0 0 0 000 0 o 1 1 000 0 0 0 0 0 0 001 1 000 0 0 0 0 0 000 1 100 0 0 0 0 0 o 0 001 100 0 000 o 0 0 001 1 0 0 0 0 0 o 0 0 0 001 1 000 0 o 0 0 0 0 001 100 0 o 0 0 0 0 0 001 100 o 0 0 0 0 0 0 0 0 1 1 0 o 0 0 0 0 0 0 0 0 0 1 1
• 1 0 0 0 0 0 0 0 000 0 .1 2
135
c
1
o
Figure A2.5 shows the analytical solution for this model
and points from the computed response.
1 2 T
----------e----------
136
A2.2 Programme for the normalised response curves and truncated
moments of impulse experiments.
The experimental procedure, described in 7.4, resulted in
the output from the photocell detector being logged at one
second intervals on punched paper tape during an impulse response
experiment; the print out was typically as follows:
logged data: +1 +1 +1 +1 +1 +1 +27 +87 +.99 +149 etc.
As the linear portion of the calibration curve was usually
exceeded early in a run, pairs of voltage/concentration points
from this curve were also lequired so that the logged voltages
could be converted, by means of a linear interpolation, into
concentration units. A typical calibration tape read as follows:
calibration tape: v 1
100 152.5 199.9 235.1 256.5
c o 4.2 8
12 16 20
A third data tape contained the following:
run data tape: run number throughput flow vessel holdup truncation point print out interval dead time in lines
N Q V d h D
The computation is very simple. The system mean time, and hence
the number of points on the logged data tape that are required
- d.V/Q - are calCUlated when the run data tape is read; the
calibration tape is then read in, followed by the logged data
tape; the first D points on this latter tape are ignored, the
remainder being read in one at a time and converted into
concentration units; the area between successive pairs of
points is computed using the trapezoidal rule; the moments of
these areas about the origin are also computed and a cumulative
137
record is kept of these areas and moments until the truncation
point is reached; the total area is then computed assuming that
the tail behaves as for a well mixed vessel·(6.2.1); the
moments are then printed out. The logged data.tape must then
be fed in again so that after D + h, D * 2h, •••• , D + nh, points
have been read the normalised concentrations can be computed -
using the previously calculated area and system mean time - and
printed out.
138
A3. MASS TRACER INJECTOR
Appendix 3
A3.1 Mass Tracer Injection System
This has been designed and manufactured by Gloster Equipment
Ltd" for the programmed injection of tracer fluids.
Provision is made for the injection of the trace fluid at
predetermined intervals at a controlled rate of flow for
preselected periods.
The main units are:
1. Control unit.
2. Electro-hydraulic servo valve.
3. Trace injector jack with flow control valve, and
feedback potentiometer.
4. Hydraulic power pack.
5. Tracer reservoir.
Control Unit
The control unit comprises an amplifier, an electrical
circuit for converting the signal to a mark space ratio and an
output circuit. The variable mark space circuit consists of a
multivibrator whose frequency of operation is governed by a
separate timing circuit. In the steady state, with zero system
error, the output is a 50:50 mark space square wave which
corresponds to a no-output position in the electro-hydraulic
servo valve. The demand and feedback signals are mixed in a
difference network the output of which is amplified and converted
to a mark space signal. The feedback signal is in opposition
to the demand signal, so that when the correct operating position
of the tracer injector jack is reached, the demand minus the
feedback signal prov~es a 50:50 mark space ratio which will hold
the jack in this position. The amplifier is a single push pull
stage which is fed by the variable mark space signal generated by
the multivibrator. The output stage has as its load the relay
coils of the electo-hydraulic servo valve.
1>9
Electro-hydraulic Servo Valve.
The electro-hydraulic servo valve consists of a pilot
valve and main valve. The pilot valve is a combined differential
relay and hydraulic valve contained in a body with the main
valve and comprising two separate coils arranged axially around
two hydraulic jets. Between the jets is a diaphragm free to
vibrate and control the flow of fluid from the jets. The outputs
from the mark space generator are applied to the coils of the
pilot valve, producing movement of the diaphragm in the same
relation as that of the electrical signals. The effective
hydraulic orifices are thus modified by the period of dwell in
either direction, dictated by the mark space signal. The main
valve is a standard four way type valve with feedback slots cut
in the ends of the spool. When e~ual mark space signals are
applied to the coils of the pilot valve the effective area of the
jet orifices are the same and there are e~ual pressures on each
end of the main valve spool which remains stationary in mid
position. Varying the mark space signal causes the diaphragm
to flex in one direction or the other, creating a build up of
pressure in one line from the main valve and a reduction of
pressure in the other line.
As a result of this unbalancing of pressures and flows the
spool of the main valve will move, uncovering the main ports.
One port is thus free to deliver pressure oil direct to
the trace injector jack. Movement of the main valve spool
results in a reduction of the effective discharge area of th.
slots at the higher pressure end of the spool, since these slots
are of tapered form, and, correspondingly an increase of the
discharge area of the slots at the lower pressure end of the
spool. This results in a balance being restored to maintain the
main valve spool in its new position with e~ual hydraulic
140
pressures on each end of the spool. Variation of the mark
space signal in the opposite ratio causes the main valve spool
to move in the reverse direction, thus opening the other outlet
port to the jack.
Trace Injector Jack Unit
The trace injector jack with trace flow control valve and
feedback potentiometer are mounted as one unit with the valve
spool connected to one end of the jack piston rod and the
potentiometer wiper attached to the other end. Thus rigid control
with no backlash is obtained. Linear flow/displacement
characteristics are achieved by the use of long, narrow slots
in the hollow valve spool, so that orifice area open to flow
varies linearly with displacement. Complete leakproof valve
shut-off between injections is ensured by the use of a rubber seal
in the face of the valve and by slightly biasing the 50:50 mark
space ratio.
Hydraulic Power Pack
The hydraulic power pack is a self contained electrically
powered unit to provide hydraulic power for the operation of the
servo valve/trace injector jack circuit.
is driven by a 5 h.p. three phase motor.
The hydraulic pump
A 5 micron filter
and a pressure gauge are incorporated in the unit. A water
cooler on the delivery line enables the jack to be held
indefinitely in the closed position without causing overheating
of the oil.
Tracer Reservoir
The tracer fluid is contained in a cylindrical 2 gallon
stainless steel pressure vessel. Pressure is maintained by means
of compressed air, from a Taylor Air Unit, which enters at the
top. 1-a-" from the bottom of the vessel are two tappings : one is
141
connected directly to the Injector flow control valve and the
se.qond to a 0 30 psig pressure gauge. An easily removable
top plate and a drain valve facilitate charging and cleaning.
A3.2. Tracer response runs using the Mass Tracer Injector.
The following procedure was adopted for setting up the
equipment.
1) The photocell, which had been left connected to the
power supplies overnight, was calibrated as described
in 7.3.2.
2) Cooling water flow was established to the hydraulic
pOl1er pack cooler.
3)' The hydraulic power pack was started up with the oil
circulating through the pressure relief valve. When
the pressure had built up to 1,000 psig, the outlet
valve was opened allowing the oil to circulate through
the servo valve.
4) The control unit was switched on and, after five minutes
warming up time, the jack was set at the fully open
position for a few seconds and then closed. This was
repeated a few times to ensure that any air in the oil line
had been displaced. The jack was then left in the
closed position.
5) The dye reservoir was washed out and charged with water;
the flow control valve and dye line were flushed by
pressurising this water and setting the jack to the open
position. The glass line below the injection tee was
broken to enable this wash water to be drained.
The dye line and valve were then blown clear with air.
The reservoir was dried and charged with dye solution.
The control valve was then opened again and, when about
300 mls of dye had been discharged, closed.
is discarded. 142
This dye
6) A Servomex LF51 signal generator was set up to provide
a 0 - 10V square pulse of five seconds duration.
To do this a 5V potential was applied across the
common and earth terminals of the generator. A dry cell
and potentiometer was used for this purpose and a
digital voltmeter was used for the fine adjustments.
The output was connected to the control unit.
7) The Taylor air unit was adjusted to give a dye pressure
of 5 psig at the outlet of the reservoir. A five second
pulse was then triggered from the signal generator, the
dye discharging into a measuring cylinder. This was
repeated 5 or 6 times, the quantity discharge being
noted.
8) The inlet line was reconnected and flushed out.
Water feed was established at the desired rate by means
of the valve below the feed rotameter.
The required vessel level was adjusted by means of the
screw down valve in the outlet line.
9) The stirrer was started and the speed of rotation
adjusted to the desired rate.
The lower speeds were measured by counting shaft
revolutions in a given time.
strobascope was used.
For higher speeds a
10) When conditions were steady the data logger was reset,
the punch drive unit started up, and the dye pressure
adjusted to 6.2 psig. This gave a 5 psi pressure drop
across the control valve.
11) A run was started by tripping the data logger and
initiating a 5 sec pulse from the signal generator.
143
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144
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1--- _ .. _ I, O\ ____ ~-Cl-:ooq9----------~~L.-. ----------.. --.. i ________ ,:() 'L.. ____ ~-=-~u0{,------ b ____ .___ 0 --u 2- ------
_._ _____ ___ ,. 03 . . ____ O~-020 I . ... \'-- ., . \ u-<_ ------_____ . ____ ._:..'_ ' __ 0 .... 4-. _ _. __ O-~-.t2.3i?I}... I ¥-- ... .. -.-------{!)~\(--
. ______ L-()s'-- 0 . 0 If 7b-- .-- :J..<f______---o· otjo ...
____ ~ ... ...Q.{)- .. ~. __ o_'_J2..5bb- _--- S-v_ .. g---o~LI}---._. __ -.J~v'7 .. __ ______ Q..:....-oJ,.5q.. . ___________ 2-~---------- o· ob5_ ..... .
, __ (,) ~ 0 ' If''V 0 'os g
.~ ..• ~ 1~~i) ·-~_Qd~::~--~ =::.= ~-CO?:. .••..
.. __ . [.:....i()_ __ . o:00OCJ-----.-----------· .. 1..f'-----.-------o.', It?j ___ -
. __ ___J":'k9--- .. ...._<?_'-..!.b (:,1:> _________ . _____ . ____ .Jt--------e --J 2--.-:{---.---.-
... _______---L}~-------(): .. 23,<>_",) --I ~ .. __ . ___ ___ o--:..El) .. -----
_ .. _________--L...5£.Q---------Q ...• 2-~S...7~---------L~- 0 -
057 ___ -
.------ _._ ... __ /.:.5- _____ 0_:3.33 __ ;)._ Lt. ." () ·o<f b ______ _ 33 0-03(,
.® •.•• _=-_ .•• = .... -- •. ~~Yf!J-@;:~~~~:=------ 4-~~_=-~~=- O.~} ~==-~ ~I It--'6' D ' 02-6
'(.5;;.-::-.. ~~- ~i£ -= ~-~~--~~~_~__ _____ '_c ___ ~--------~-.
. ...... _ 10 @ __ .. , .. :~==_~~~=~~~=--~~=-~-;;.--;~ .. () ~ --=~~-=~ ..•.• ~==-=~ -. ~----1~---"; ~~ f--- ......... -- ...---- -'---.--~.: r ····~-bj?---···-----·----·-.. -----
___ _____ '-_'J,.... ________ ... __ O_I_' -.-~----------------'--.'- ---... ------------- - .----------
~) -lo --c:-;f..!......--------------~,9b---Q-'--'·t~_____ ..-------~-' ._------- ------------ - .-.-----_.--_ .. -~-.. "------------------- -_.'--- -------- .. --- - ----". ------_._------- _ .. ,."- ~ ---- .. ----_.-
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146
1
,I
,--------------------- ------ -----