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Solution of a mixing model due to van de Vusse by a simple probability method
Chemical Engineering Science, 1967, Vol. 22, pp. 517-523. Pergamon Press Ltd., Oxford.
Printed in Great Britain.
Solution of a mixing model due to van de Vusse by a simple probability method L. G. GIBILARO, H. W. KROPHOLLER and D. J. SPIKINS Department
of Chemical Engineering,
University of Technology,
Loughborough,
Leicestershire
(Received 13 May 1966; in revisedform 25 October 1966) Abstract-A revised version of a multiloop circulation model first proposed by van de Vusse is shown to fit experimental residence time distribution data obtained for a 100 1. stirred vessel. A simple and generally applicable method for the solution of models of this type is described. 1. INTR~OUOTI~N of circulation models have been proposed to describe mixing in a stirred vessel. The simplest models of this type contain a single recirculation loop : material is ‘pumped’, by the impeller, around the whole vessel in a single stream. The advantage of single loop models [Fig. l(b)] is that they can generally be easily dealt with analytically. These models, however, do not represent the known flow patterns in a stirred vessel. VAN DE VUSSE[l] proposed the model shown in Fig. l(a) which appears to represent a more realistic picture of mixing in a continuous stirred vessel. In this model each half loop consists of n completely mixed stages in series. The transfer function of the model is given by :
A NUMBER
with n completely mixed stages and recirculationFig. l(b). Thus, much of the object of the model has been lost. Figure 5 demonstrates the extent to which the above simplifications have changed the response of the model for the case where the circulatory flow is equal to the throughput. By using the method now described no simpliíìcations are necessary for determining the real time solution of the 3 loop model. In fact, the model could be further complicated by the inclusion of additional loops and of flow between loops. The same computer programme can be used to deal with al1 of these cases.
--
C(s)=
__
-_- QRA __
(1)
rlRlR,+r,RsR,+rsRsR6-(Q+q)
where Ri, (i= 1,2, 3,4, 5,6) is the transfer function of a half loop. Because Eq. (1) was considered to be too complicated for practica1 use, van de Vusse made the following simplications : --
--
--
--
R=R,R2=R,~,=~,~s=~2~, (bl
K is then the transfer function of an ‘average’ loop and -n V R= l+“J where J=n> (q+Q) ( This reduces Eq. (1) to C(s) =
Q (4 + Q>(l + sJl4 -“- 4
Q
FIG. 1 (a) Multiple loop recirculation. (b) Tanks in series with recirculation.
(2)
Solutions of this equation for various values of n were published [ll. It can be easily shown that Eq. (2) is the transfer function of a single loop model
2. MARKOV PROCESSES [2] The definitions and equation of the discrete time Markov process that wil1 be used to evaluate the 517
response ofcontinuous 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 pij. The rows of P consist of the probabilities of al1 possible transitions from a given state and so sum to 1. This matrix completely describes the Markov process Defined as the Si(M) the state probability. probability that the system wil1 be in the state i after n transitions from a given starting point the state probability vector: a line vector w composed of elements si(n). For each state i, there exists a probability that it wil1 be occupied after n transitions from a given state. The sum of these probabilities must be 1.
jl
si(n)=
1
+
l)=
i$l
si(n)
l
PI,
p=
Pi2
P13
***
PIN-
l
.. .. .. .. ..
P21
P22
P23
'**
P2N
. ..
. . .
. . .
...
. ..
PN1
PN2
PN3
**-
PNN
A pictorial representation
from which it follows that sj(n
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 wel1 mixed stage is less than 0.01 then the response at any point is within 1 per cent of the maximum value of the continuous time solution. Knowing the sizes of al1 the vessels and the magnitude of the flows connecting them, it becomes an easy matter to assign probabilities pij to al1 possible state transitions. Thus
I
of the composition of
P is given in Fig. 2. ;
n=O,
1,2;.*
(3)
and S(n+l)=S(n).
P
2.1 Application to flow models Consider a continuous tlow system consisting of a number of wel1 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 smal1 finite time interval At, one of two events wil1 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 smal1 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 we make At the closer is the approximation of the discrete time Markov 518
Probabilities
of
FK;. 2. Pictorial representation
of transition matrix.
For convenience the above matrix will be used to describe a continuous flow system containing N- 1 wel1 mixed vessels. The Nth state becomes the trapping state; an element in state Nis one that has left the system and cannot return. Thus, P,,=O,
i#N
PNN=~
The transition matrix can now be used to find the response of the system to a tracer input to any vessel.
Solution of a mixing model due to van de Vusse by a simple probability method 1
Recirculation models.
Consider the three loop model shown in Fig. 3 consisting of six completely mixed stages. 2.1.1
Gk$ cl
2
3
r
r
4
r
+r
a 1
FIG. 3. Three loop model with inflow to impeller. This ditfers from Fig. l(a), where (n= 1) in that the inflow is direct to the impeller. This configuration corresponds to one set of conditions for which experimental results have been obtained. Two assumptions have been made concerning the symmetry of the system : the stages are of equal volume and the circulatory flow (q) is divided equally between the three loops. Any other assumptions could be dealt with equally easily. In time At a molecule in vessel 1 can either remain where it is or move on to vesse12. The probabilities of these alternatives are and 1 - e(-r/“)At 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,
P21=P23=3ï+Q
’
(1 _eWWq
r+ Q (l_e(-r/v~At) P25=3r+Q P*‘%=P26=P27=0 Similar expressions can be written for al1 the elements of P as follows: 519
L. G. GIBILARO, H. W.
KROPHOLLER
Now v= V/6 and r=q/3. Let AT=At Q/V, where T is the time in normalised units. So that: ,$ -r/uMt= e( - &+)At= e( - WQW
Initially S(1) is obtained from S(1) = S(0) . P S(0) 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, wil1 be in states 1, 2, 3, 4, 5, 6 and 7 after time AT.
and $_Q.
(l-&-‘/“‘Af)=
-2qjQ)Ap
3(qq;+l)(l-e’
S(2) is then obtained from S(2)=S(l)P.
etc.
Thus to give values to al1 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 probability of leaving any state is less than 0.01. The transition matrix for this case with q/ Q = 1 and T= 04005 (dimensionless units) is
P=
0.999001
oWO999
O+MO167
0.999001
and D. J. SPIIUNS
This vector contains the state probabilities after time 2AT. In rhe same way the state probabilities are obtained after 3AT 4AT . . , nAT After each applicati:n of’iq. (3) there are two elements of S(n) that are of particular interest: these are S,(n) and S,(n). The elements S,(n), . . ) are the probabilities that the tagged (n=1,2,3,. molecule wil1 be in vessel 5 (the vessel from which
0
0 0 *000167
0
0 0 Cl0666
0
0
0
0
0
0
0
0.999001
oNlO999
0*000167
0
0 WO
0.999001
0MO666
0
0
0
0
0
0
0.996008
0*001001
0*003001
0
OGlO167
0
0 WO
0.999001
0
0
0
0
0
1
O%M_M67 0
It onlyremains to write the initial state probability vector-S(0). A labelled molecule enters the vessel at the impeller. Initially it can be in states 1,3 or 5, the respective probabilities being: r L, 3r+Q
r 3r+Q’
r+Q 3r+Q
Thus S(O)= :, [ 3r+Q
0, r, 3r+Q
0, r+Q, 3r+Q
0, 0
1
which for the case of q/Q= 1 becomes S(0) = [0~166667,0,0~166667,0,0~666667,0,0] A digital computer is now used to apply repeatedly Eq. (3): that is to postmultiply the state probability vector ( S(n) ) by the transition matrix (P).
0
0
0
material leaves the system) after time nAT. This is the impulse response of the system. The elements S,(n), (n=l, 2, 3,. . .) are the cumulative probabilities of the tagged molecule leaving the system after times AT, 2AT: . . . , nAT: they therefore give the response to a unit step. The impulse response for this example is shown in Fig. 6; Figs. 7 and 8 show results obtained at various circulation rates; the effect of increasing the number of stages in the loops is demonstrated in Fig. 9. Figures 10 to 12 show the effect of changing the feed inlet position: the feed, in these cases, is assumed to be divided equally between the two upper recirculation loops-Fig. 4. The same computer programme was used to obtain al1 these results.
520
Solution of a mixing model due to van de Vusse by a simple probability
method
Inflaw to impeller
Experimental
FIG. 4.
response
Three loop model with inflow divided quality between upper loop 1.0
0.5
2.0
1.5 T
FIG.
Inflaw
7
to impeller
C
Experimentol -
reste
Model
C
FIG. 5. Curve (a)-3 loop Fig. 1 (a) with rl=rz=rs=q/3.
recirculation model, Curve (b)-Eq. (2).
,
I
1.5
I .o
0.5
T
0
FIG. 1.0 -
Inflow 0
8
to impeller
q/Q qI 0
0 -
* Experimeniol
I,/
response
Model Inflow to
impeller
q/O=l
c
I-
0!5
l 1.5
I 1.0
0.
I 2.0
l
FIG.
FIG.
0.5
,
T
6
521
9.
I
1.0
1
1.5
I
I
2.0
25
Effect of increasing number of stages per half loop, n, of 3 loop model (Fig. 3).
L. G. GIBILARO, H. W. KROPHOLLER and D. J. SPIKINS Inflow
to upper loops
Inflow
q/Q=I 1.c
q/Q
a
to upper loops =20 Experimental Model
0
0
-
response
c 0.5
I
I
0.5
1.0
to uppr
FIG.
loops
I
1.0 Exprrimental -
response
Model
C
0.5
05
1.0 FIG.
I
2.0
T
T
FIG. 10 Inflow
I
1.5
1.5
2.0
1:
3. EXPERIMENTAL
Impulse response tests were performed on a stirred 100 1. spherical glass vessel-a commercially available standard unit. The turbine impeller has a variable speed range of 30-300 rev/min. The tracer material (Nigrosine dye solution) was injected into the inlet line as a square pulse of 5 sec duration: for practica1 purposes this can be considered as a true impulse, the system mean time being of the order of 500 sec for al1 runs. For the first set of runs, a feed dip pipe directed flow into the region of the impeller : this corresponds to the arrangement shown in Fig. 3. The dip pipe was then changed for one that caused the feed to
12
enter close to, but directed away from, the impeller and into the upper region of the vessel: this is the situation shown in Fig. 4. The output from a photocell detector on the glass outlet line from the bottom of the vessel was connected to a digital voltmeter and data logger. During runs, this output was read at 1 sec intervals and recorded on punched paper tape which was then processed on a digital computer to produce the normalised response curves. Normalised concentrations and times were used in order that the experimental response could be compared with that of the model computed as described above: concentration is divided by the initial concentration of dye assuming that the vessel is wel1 mixed; and time is divided by the system mean time (v/Q). The impeller pumping capacity was measured using the technique of MARRand JOHNSON[3]. This consists of finding the average time taken by a zero buoyancy ‘float’, a resin sealed polystyrene and cork tablet, to travel from the impeller into the body of the vessel and back to the impeller. Some 200 counts were made for each impeller speed. The pumping capacity is obtained by dividing the vessel holdup by this average circulation time. For the inlet and outlet positions used (Figs. 3 and 4), the two upper loops are identical and could be combined. This is not the case for the more genera1 system originally considered by van de Vusse; Fig. l(a). 522
Solutionof a mixingmodeldue to van de Vusse by a simple probability method
4.CONCLUSIONS The experimental impulse response curves obtained from the 100 1. stirred vessel are compared with the 3 loop model in Figs. 6-8, 10-12. At the very low stirrer speeds-Figs. 6 and l@-the internal flow patterns are not wel1 established and the fit is poor: at higher speeds, however, the model adequately accounts for these widely varying experimental responses. There are two parameters in this model: the impeller pumping capacity and the number of stages per loop. The pumping capacity can be easily determined experimentally by the flow follower technique [3], or estimated from a knowledge of the impeller geometry [4]. The number of stages per loop is obtained from response tests on the stirred vessel. On increasing the number of stages per half loop, the response of the model oscillates-Fig. 9. This effect could be expected in a large vessel with wel1 defined circulation patterns. An advantage of the method of solution described, is the ease with which models consisting of first order elements can be elaborated to describe particular systems. The calculation of probabilities can be built into the programme (using the empirical choice of time increment as described) so that only stage holdups and flows between stages need be
entered as data. Conventional numerical solutions may wel1 be more efficient and should, perhaps, be used when a large number of solutions of a particular model are required, but this method has its use when a number of different models require comparison with experimental results. Ackaowledgment-The authors thank the Science Research Council who financed this research.
NOTATION
normalised concentration: a function of T normalised concentration: a function of s nth stage, nth time increment total number of states probability transition matrix circulatory flow throughput IIow n wel1 mixed stages in series i= 1, 2, 3, 4, 5. 6 n= 1, 2, 3, . . . state probability after n transitions state probability vector after n transitions Laplace variable time Normalised time ...T= tQ/V hold-up of single vessel hold-up of system
REFERENCES
PI ::; 141
VAN DE VUSSEJ. G., Chem. Engng Sci. 1962 17 507. HOWARD R. A., Dynamic Programming and Markov Processes, Wiley, New York 1960. MARR G. R. and JOHNSONE. F., A.I.Ch.E. Jll963 9 383. RUSHTONJ. H., MACK D. E. and EVERETI.H. J., Trans. A. Inst. Chem. Engrs 1946 42 441.
Résamé-On montre une version modifiée dun modèle de circulation a boucles muhiples proposé pour la premiere fois par van der Vusse, pour tenir compte des donnt%s experimentales de la distribution du temps de contact pour un r&cipient de 100 litres avec agitateur. Une methode simple et géneralement applicable pour la solution de modèles de ce genre est dt5crite. Zusanunenfassuag-Es wird gezeigt, dass für bestimmte Warme- und Stofftbertragungsprobleme in der turbulenten Grenzschicht gleichartige Lösungen erzielt werden können, sofem die Wärmeübertragungs- und Stoffübertragungsschichten innerhalb der Schicht konstanter Scherder Fa11 ist. Geschlossene Lösungen erhält man für den Wärmeübergang von einer Fläche konstanter Temperatur, für den Stoffibergang bei endlicher Grenzflächengeschwindigkeit und fti den Wärmeübergang von einer Fläche mit bestimmter Temperaturverteilung. Ein Vergleich experimentell ermittelter Värmeübertragungsdaten bei Rohren mit Spaldingfunktionen, die aufgrund der beiden Spaldinggesetze berechnet wurden, zeigt, dass man mit einem der beiden Gesetze bedeutend bessere Ergebnisse erzielt.
523
Printed in Great Britain.
Solution of a mixing model due to van de Vusse by a simple probability method L. G. GIBILARO, H. W. KROPHOLLER and D. J. SPIKINS Department
of Chemical Engineering,
University of Technology,
Loughborough,
Leicestershire
(Received 13 May 1966; in revisedform 25 October 1966) Abstract-A revised version of a multiloop circulation model first proposed by van de Vusse is shown to fit experimental residence time distribution data obtained for a 100 1. stirred vessel. A simple and generally applicable method for the solution of models of this type is described. 1. INTR~OUOTI~N of circulation models have been proposed to describe mixing in a stirred vessel. The simplest models of this type contain a single recirculation loop : material is ‘pumped’, by the impeller, around the whole vessel in a single stream. The advantage of single loop models [Fig. l(b)] is that they can generally be easily dealt with analytically. These models, however, do not represent the known flow patterns in a stirred vessel. VAN DE VUSSE[l] proposed the model shown in Fig. l(a) which appears to represent a more realistic picture of mixing in a continuous stirred vessel. In this model each half loop consists of n completely mixed stages in series. The transfer function of the model is given by :
A NUMBER
with n completely mixed stages and recirculationFig. l(b). Thus, much of the object of the model has been lost. Figure 5 demonstrates the extent to which the above simplifications have changed the response of the model for the case where the circulatory flow is equal to the throughput. By using the method now described no simpliíìcations are necessary for determining the real time solution of the 3 loop model. In fact, the model could be further complicated by the inclusion of additional loops and of flow between loops. The same computer programme can be used to deal with al1 of these cases.
--
C(s)=
__
-_- QRA __
(1)
rlRlR,+r,RsR,+rsRsR6-(Q+q)
where Ri, (i= 1,2, 3,4, 5,6) is the transfer function of a half loop. Because Eq. (1) was considered to be too complicated for practica1 use, van de Vusse made the following simplications : --
--
--
--
R=R,R2=R,~,=~,~s=~2~, (bl
K is then the transfer function of an ‘average’ loop and -n V R= l+“J where J=n> (q+Q) ( This reduces Eq. (1) to C(s) =
Q (4 + Q>(l + sJl4 -“- 4
Q
FIG. 1 (a) Multiple loop recirculation. (b) Tanks in series with recirculation.
(2)
Solutions of this equation for various values of n were published [ll. It can be easily shown that Eq. (2) is the transfer function of a single loop model
2. MARKOV PROCESSES [2] The definitions and equation of the discrete time Markov process that wil1 be used to evaluate the 517
response ofcontinuous 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 pij. The rows of P consist of the probabilities of al1 possible transitions from a given state and so sum to 1. This matrix completely describes the Markov process Defined as the Si(M) the state probability. probability that the system wil1 be in the state i after n transitions from a given starting point the state probability vector: a line vector w composed of elements si(n). For each state i, there exists a probability that it wil1 be occupied after n transitions from a given state. The sum of these probabilities must be 1.
jl
si(n)=
1
+
l)=
i$l
si(n)
l
PI,
p=
Pi2
P13
***
PIN-
l
.. .. .. .. ..
P21
P22
P23
'**
P2N
. ..
. . .
. . .
...
. ..
PN1
PN2
PN3
**-
PNN
A pictorial representation
from which it follows that sj(n
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 wel1 mixed stage is less than 0.01 then the response at any point is within 1 per cent of the maximum value of the continuous time solution. Knowing the sizes of al1 the vessels and the magnitude of the flows connecting them, it becomes an easy matter to assign probabilities pij to al1 possible state transitions. Thus
I
of the composition of
P is given in Fig. 2. ;
n=O,
1,2;.*
(3)
and S(n+l)=S(n).
P
2.1 Application to flow models Consider a continuous tlow system consisting of a number of wel1 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 smal1 finite time interval At, one of two events wil1 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 smal1 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 we make At the closer is the approximation of the discrete time Markov 518
Probabilities
of
FK;. 2. Pictorial representation
of transition matrix.
For convenience the above matrix will be used to describe a continuous flow system containing N- 1 wel1 mixed vessels. The Nth state becomes the trapping state; an element in state Nis one that has left the system and cannot return. Thus, P,,=O,
i#N
PNN=~
The transition matrix can now be used to find the response of the system to a tracer input to any vessel.
Solution of a mixing model due to van de Vusse by a simple probability method 1
Recirculation models.
Consider the three loop model shown in Fig. 3 consisting of six completely mixed stages. 2.1.1
Gk$ cl
2
3
r
r
4
r
+r
a 1
FIG. 3. Three loop model with inflow to impeller. This ditfers from Fig. l(a), where (n= 1) in that the inflow is direct to the impeller. This configuration corresponds to one set of conditions for which experimental results have been obtained. Two assumptions have been made concerning the symmetry of the system : the stages are of equal volume and the circulatory flow (q) is divided equally between the three loops. Any other assumptions could be dealt with equally easily. In time At a molecule in vessel 1 can either remain where it is or move on to vesse12. The probabilities of these alternatives are and 1 - e(-r/“)At 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,
P21=P23=3ï+Q
’
(1 _eWWq
r+ Q (l_e(-r/v~At) P25=3r+Q P*‘%=P26=P27=0 Similar expressions can be written for al1 the elements of P as follows: 519
L. G. GIBILARO, H. W.
KROPHOLLER
Now v= V/6 and r=q/3. Let AT=At Q/V, where T is the time in normalised units. So that: ,$ -r/uMt= e( - &+)At= e( - WQW
Initially S(1) is obtained from S(1) = S(0) . P S(0) 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, wil1 be in states 1, 2, 3, 4, 5, 6 and 7 after time AT.
and $_Q.
(l-&-‘/“‘Af)=
-2qjQ)Ap
3(qq;+l)(l-e’
S(2) is then obtained from S(2)=S(l)P.
etc.
Thus to give values to al1 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 probability of leaving any state is less than 0.01. The transition matrix for this case with q/ Q = 1 and T= 04005 (dimensionless units) is
P=
0.999001
oWO999
O+MO167
0.999001
and D. J. SPIIUNS
This vector contains the state probabilities after time 2AT. In rhe same way the state probabilities are obtained after 3AT 4AT . . , nAT After each applicati:n of’iq. (3) there are two elements of S(n) that are of particular interest: these are S,(n) and S,(n). The elements S,(n), . . ) are the probabilities that the tagged (n=1,2,3,. molecule wil1 be in vessel 5 (the vessel from which
0
0 0 *000167
0
0 0 Cl0666
0
0
0
0
0
0
0
0.999001
oNlO999
0*000167
0
0 WO
0.999001
0MO666
0
0
0
0
0
0
0.996008
0*001001
0*003001
0
OGlO167
0
0 WO
0.999001
0
0
0
0
0
1
O%M_M67 0
It onlyremains to write the initial state probability vector-S(0). A labelled molecule enters the vessel at the impeller. Initially it can be in states 1,3 or 5, the respective probabilities being: r L, 3r+Q
r 3r+Q’
r+Q 3r+Q
Thus S(O)= :, [ 3r+Q
0, r, 3r+Q
0, r+Q, 3r+Q
0, 0
1
which for the case of q/Q= 1 becomes S(0) = [0~166667,0,0~166667,0,0~666667,0,0] A digital computer is now used to apply repeatedly Eq. (3): that is to postmultiply the state probability vector ( S(n) ) by the transition matrix (P).
0
0
0
material leaves the system) after time nAT. This is the impulse response of the system. The elements S,(n), (n=l, 2, 3,. . .) are the cumulative probabilities of the tagged molecule leaving the system after times AT, 2AT: . . . , nAT: they therefore give the response to a unit step. The impulse response for this example is shown in Fig. 6; Figs. 7 and 8 show results obtained at various circulation rates; the effect of increasing the number of stages in the loops is demonstrated in Fig. 9. Figures 10 to 12 show the effect of changing the feed inlet position: the feed, in these cases, is assumed to be divided equally between the two upper recirculation loops-Fig. 4. The same computer programme was used to obtain al1 these results.
520
Solution of a mixing model due to van de Vusse by a simple probability
method
Inflaw to impeller
Experimental
FIG. 4.
response
Three loop model with inflow divided quality between upper loop 1.0
0.5
2.0
1.5 T
FIG.
Inflaw
7
to impeller
C
Experimentol -
reste
Model
C
FIG. 5. Curve (a)-3 loop Fig. 1 (a) with rl=rz=rs=q/3.
recirculation model, Curve (b)-Eq. (2).
,
I
1.5
I .o
0.5
T
0
FIG. 1.0 -
Inflow 0
8
to impeller
q/Q qI 0
0 -
* Experimeniol
I,/
response
Model Inflow to
impeller
q/O=l
c
I-
0!5
l 1.5
I 1.0
0.
I 2.0
l
FIG.
FIG.
0.5
,
T
6
521
9.
I
1.0
1
1.5
I
I
2.0
25
Effect of increasing number of stages per half loop, n, of 3 loop model (Fig. 3).
L. G. GIBILARO, H. W. KROPHOLLER and D. J. SPIKINS Inflow
to upper loops
Inflow
q/Q=I 1.c
q/Q
a
to upper loops =20 Experimental Model
0
0
-
response
c 0.5
I
I
0.5
1.0
to uppr
FIG.
loops
I
1.0 Exprrimental -
response
Model
C
0.5
05
1.0 FIG.
I
2.0
T
T
FIG. 10 Inflow
I
1.5
1.5
2.0
1:
3. EXPERIMENTAL
Impulse response tests were performed on a stirred 100 1. spherical glass vessel-a commercially available standard unit. The turbine impeller has a variable speed range of 30-300 rev/min. The tracer material (Nigrosine dye solution) was injected into the inlet line as a square pulse of 5 sec duration: for practica1 purposes this can be considered as a true impulse, the system mean time being of the order of 500 sec for al1 runs. For the first set of runs, a feed dip pipe directed flow into the region of the impeller : this corresponds to the arrangement shown in Fig. 3. The dip pipe was then changed for one that caused the feed to
12
enter close to, but directed away from, the impeller and into the upper region of the vessel: this is the situation shown in Fig. 4. The output from a photocell detector on the glass outlet line from the bottom of the vessel was connected to a digital voltmeter and data logger. During runs, this output was read at 1 sec intervals and recorded on punched paper tape which was then processed on a digital computer to produce the normalised response curves. Normalised concentrations and times were used in order that the experimental response could be compared with that of the model computed as described above: concentration is divided by the initial concentration of dye assuming that the vessel is wel1 mixed; and time is divided by the system mean time (v/Q). The impeller pumping capacity was measured using the technique of MARRand JOHNSON[3]. This consists of finding the average time taken by a zero buoyancy ‘float’, a resin sealed polystyrene and cork tablet, to travel from the impeller into the body of the vessel and back to the impeller. Some 200 counts were made for each impeller speed. The pumping capacity is obtained by dividing the vessel holdup by this average circulation time. For the inlet and outlet positions used (Figs. 3 and 4), the two upper loops are identical and could be combined. This is not the case for the more genera1 system originally considered by van de Vusse; Fig. l(a). 522
Solutionof a mixingmodeldue to van de Vusse by a simple probability method
4.CONCLUSIONS The experimental impulse response curves obtained from the 100 1. stirred vessel are compared with the 3 loop model in Figs. 6-8, 10-12. At the very low stirrer speeds-Figs. 6 and l@-the internal flow patterns are not wel1 established and the fit is poor: at higher speeds, however, the model adequately accounts for these widely varying experimental responses. There are two parameters in this model: the impeller pumping capacity and the number of stages per loop. The pumping capacity can be easily determined experimentally by the flow follower technique [3], or estimated from a knowledge of the impeller geometry [4]. The number of stages per loop is obtained from response tests on the stirred vessel. On increasing the number of stages per half loop, the response of the model oscillates-Fig. 9. This effect could be expected in a large vessel with wel1 defined circulation patterns. An advantage of the method of solution described, is the ease with which models consisting of first order elements can be elaborated to describe particular systems. The calculation of probabilities can be built into the programme (using the empirical choice of time increment as described) so that only stage holdups and flows between stages need be
entered as data. Conventional numerical solutions may wel1 be more efficient and should, perhaps, be used when a large number of solutions of a particular model are required, but this method has its use when a number of different models require comparison with experimental results. Ackaowledgment-The authors thank the Science Research Council who financed this research.
NOTATION
normalised concentration: a function of T normalised concentration: a function of s nth stage, nth time increment total number of states probability transition matrix circulatory flow throughput IIow n wel1 mixed stages in series i= 1, 2, 3, 4, 5. 6 n= 1, 2, 3, . . . state probability after n transitions state probability vector after n transitions Laplace variable time Normalised time ...T= tQ/V hold-up of single vessel hold-up of system
REFERENCES
PI ::; 141
VAN DE VUSSEJ. G., Chem. Engng Sci. 1962 17 507. HOWARD R. A., Dynamic Programming and Markov Processes, Wiley, New York 1960. MARR G. R. and JOHNSONE. F., A.I.Ch.E. Jll963 9 383. RUSHTONJ. H., MACK D. E. and EVERETI.H. J., Trans. A. Inst. Chem. Engrs 1946 42 441.
Résamé-On montre une version modifiée dun modèle de circulation a boucles muhiples proposé pour la premiere fois par van der Vusse, pour tenir compte des donnt%s experimentales de la distribution du temps de contact pour un r&cipient de 100 litres avec agitateur. Une methode simple et géneralement applicable pour la solution de modèles de ce genre est dt5crite. Zusanunenfassuag-Es wird gezeigt, dass für bestimmte Warme- und Stofftbertragungsprobleme in der turbulenten Grenzschicht gleichartige Lösungen erzielt werden können, sofem die Wärmeübertragungs- und Stoffübertragungsschichten innerhalb der Schicht konstanter Scherder Fa11 ist. Geschlossene Lösungen erhält man für den Wärmeübergang von einer Fläche konstanter Temperatur, für den Stoffibergang bei endlicher Grenzflächengeschwindigkeit und fti den Wärmeübergang von einer Fläche mit bestimmter Temperaturverteilung. Ein Vergleich experimentell ermittelter Värmeübertragungsdaten bei Rohren mit Spaldingfunktionen, die aufgrund der beiden Spaldinggesetze berechnet wurden, zeigt, dass man mit einem der beiden Gesetze bedeutend bessere Ergebnisse erzielt.
523