Chemical Engineering Science, 1970, Vol. 25, pp. 15951603.
Pergamon Press.
Printed in Great Britain.
Maintenance of product quality during transients in stirred tank reactors Process Fundamentals
E. B. NAUMAN Group, Research and Development Department, Bound Brook, N.J. 08805, U.S.A. (Received
27 February
Union Carbide Corporation,
1970)
AbstractA system of one or more stirred tank reactors in series can be subjected to changes in working volume and throughput while maintaining a stationary residence time distribution. This in turn will ensure constant product quality with arbitrary reaction kineticsover a broad range of operating conditions. The results are directly applicable to demand changes and shutdown operations. They can be used to generate a controlled cycle in the discharge flow rate while maintaining constant product composition. If product quality depends on the exact form of the residence time distribution, there is no way to achieve uniform quality throughout a startup. However, the total amount of substandard material can be held to an arbitrarily small limit.
INTRODUCTION
reaction systems often consist of one or more stirred tanks in series. Certainly, it is the usual intent to operate such systems at steady state. However, few plants are immune to occasional upsets which require changes in throughput. Also, it is frequently true that some part of the plant other than the reaction system is rate limiting. A typical example is where a recovery system dictates some important product characteristic. Depending on the final product specifications, this fact may allow (or demand) different production rates. The present paper considers how changes in throughput can be made while maintaining constant product quality in the discharge stream from the reactors. For simple kinetic systems, product quality can merely mean constant yield. However, in complex systems such as consecutive reactions, the yield of a single component may not be an adequate criterion of quality. We assume arbitrary reaction kinetics and will equate constant product quality with a stationary residence time distribution (RTD). A previous paper[ l] introduced the concept of residence time distributions for unsteady reactors. The residence time frequency function, g(8, t’), for material leaving the reactor at time 0 is stationary INDUSTRIAL
if it is a function only of residence time t’ and notofrealtimefI,g(8,t’) =g(t’). Now, g(0, t’) uniquely determines the yield for a first order reaction. For reactions of order other than first, the yield depends on the state of micromixing; but it can often be closely bounded using the concepts of “complete segregation” mixedness” [2,3]. In many and “maximum practical situations, a stationary RTD may, in itself, be an adequate criterion of product quality. For more complicated situations such as copolymerizations, the state of micromixing can have a dramatic effect [4,5]. However, if the vessel remains “wellstirred” (so that the steady state RTD is exponential), it is likely that changes in working volume and throughput will have little effect on the degree of micromixing. Ordinarily, low viscosity systems tend to be maximum mixed and will remain so at reasonable throughputs. In polymerizing systems, appreciable segregation can be caused by high viscosities. However, the viscosity itself will be primarily dictated by the RTD and will remain essentially constant if the RTD is constant. From a practical viewpoint, the techniques outlined in this paper should give constant product quality if the changes in working volume are within the design limits of the agitation system.
1595
E. B. NAUMAN STATIONARY RESIDENCE DISTRIBUTIONS
rate to bring the system to some new steady state with D = F, = VT/i.Imposing Eq. (1) gives
TIME
A necessary condition that g(0, t’) be stationary is that the mean residence time, ?, be constant. From a previous paper [ l]
dV =
de
V i
D
(4)
which has the solution or
so that V/F must be time invarient. V/F constant, g(e
f)
=
9
ce
F(ef’) ,6,
v(etr)
1
P/f=
But with
Governing
to
V f
g(t’)
at some time 7 > 0. The volume V must be a continuous function of time on physical grounds; and since the feed rate, F, is coupled to V by Eq. (1) then F must also be continuous. The restrictions on D depend on such factors as how fast pump speeds can be changed and on the size of upsets which can be tolerated by the downstream portions of the plant. In most of this paper, D will be allowed to undergo step changes since these give particularly simple mathematics and also minimize the time needed to reach the new steady state. However, for the moment suppose that D is continuous. Equations (1) and (4) can be combined to give ig=
FD.
If both F and D are continuous then
TANK
equations
For a single stirred tank the volume, feed rate, and discharge rate are related by g=
(5)
‘=F
Thus constancy of i is both necessary and sufficient to give a stationary RTD. Further, the only stationary RTD is the simple exponential distribution which corresponds to steady state operation. For several stirred tanks in series, it is obviously sufficient that Eq. (1) hold for each individual tank. It is unclear whether this is also a necessary condition. However, we retain it throughout this paper since it ensures a constant quality product from each reactor taken individually. Thus, if Eq. (1) is satisfied, final product quality will be maintained constant even if the reactors operate at different temperatures or if there are intermediate feed streams between reactors. STIRRED
.
The discharge flow rate, D 3 0, is selected give the desired final result
FIVdr
i
SINGLE
1
(1)
i =  = constant F
Suppose that the tank is initially operated at steady state with F, = D,, = V,,/i. Beginning at time 0 = 0, we wish to manipulate the discharge
at time e = 0,
(8) From Eq. (7) we have
FD.
(7)
or in general
1596
Stirred tank reactors
Now, if F is to increase, its lowest order derivative which is nonzero must be positive. This in turn means that the lowest order nonzero derivative of D must be negative. Thus if F is to increase with a constant RTD, then D must initially decrease. This same result is true when D undergoes a step change at time 0 = 0. Step changes
in D
Suppose that D has the form D=D,,, D = aDo, D=D,,
e
Then Eq. (5) becomes
and F = (l)FOee’c+aFo.
Evaluating Eq. (11) at time 0 = 7 gives
amaX&, an immediate step change to this value will minimize 7. The minimum time trajectory to the new steady state also minimizes the production “loss” for increases in F and V. The loss is defined as L=D;~f;D(0)de.
Applying D, = F, and Eq. (7) gives FdB.
(15)
Of this, the term f(F, F,) is constant wish to minimize the term
and we
L=i(F,FF,)+F,+;
I = F,r
J; Fd0.
(16)
This term would vanish if the feed rate could be immediately increased to the new steady state value, F,. However, this approach does not give a stationary RTD. Referring to Fig. 1, we seek to minimize the shaded area, I; and it is clear (11) that the minimum time trajectory gives the minimum I. (12) If it is desired to lower the reactor throughput, then the minimum time trajectory maximizes the “loss”. In other words, the fewest pounds are produced prior to reaching the new (13) steady state. Reactor
A valid solution, T > 0, for Eq. (13) exists if LY> 1 for
(14)
shutdown
and startup
A completely orderly reactor shutdown can be made while holding product quality constant. The simplest means of doing this is to raise the discharge rate while keeping the feed rate pro
V, < V,
or (Y< 1 for V, > V,.
Minimum time
It is seen that the initial change in discharge flow must be in the opposite direction to the desired change in volume. A step change in D gives the time optimal trajectroy to the new steady state. This fact follows directly from Eq. (7). If F is to increase, the absolute maxima in dFld0 for all 8 are given by D=aDO= 0. If we choose to decrease F, there is no absolute minimum for 7 since, in the absence of physical constraints, (Y can be made arbitrarily large. However, if the discharge flow rate is restricted to some maximum value, D,,, = 1597
”
.Wd”
Fig. 1. Minimization of I.
E. B. NAUMAN
portional to volume. Suppose at time 8 = 0 we raise the discharge rate to D = 1.10,. Then OL= 1.1 and I’, = 0. Equation (13) gives 7 = 2.4f so that the reactor goes empty after 2.4 mean residence times. During this time the discharge rate is 10 per cent higher than normal but product quality is constant. A similar kind of startup is impossible. To increase volume at constant RTD requires an initial decrease in D but D < 0 is impossible. Alternately, the steady state RTD is exponential and this cannot be achieved in a finite time. If the product quality depends on the exact form of the RTD, there is no way to ensure good product throughout the startup. Instead, product quality will asymptotically approach its steady state value as the RTD approaches an exponential distribution. Suppose that all material produced before some time 4 is substandard. In general, there is no way to reduce 4 below some minimum multiple of the mean residence time, say C#J = 5f. However, the amount of substandard material can be reduced by startingup the reactor with a low throughput and a proportionately low volume. Product quality becomes adequate after the same absolute time C#J but the amount of material produced during this initial period will be much smaller. After product quality is established, the throughput is gradually raised to its desired, steady state value using the methods outlined earlier. If the reactor can be operated at half its steady state volume, this startup procedure will halve the amount of substandard product. There is, of course, a penalty. Achievement of the full production rate for good product will be delayed. Obviously, we need an economic balance between the desire to maximize good production and the conflicting desire to minimize substandard production. Periodicflows
with a stationary
fashion. Suppose D = 0, 0<8<4 D=cuD,,, 4<0<7 D(~+T) = D(e) 4 = r/2 = f ln 2.
Requiring V to return to V, at 0 = 7 gives (Y= 1.5. In this case the volume varies from I’,, to 2V,,, D is a square wave, but the product quality is constant. The choice for (Y is critical. If (Y> 1.5, the vessel will eventually empty while (Y< 1.5 will cause the volume to increase without bound. In practice, a controller is needed to stabilize the system. This controller can either adjust cv or the time interval ~4 to ensure that V returns to V,,at the end of each period. STIRRED
TANKS
IN SERIES
Generalformulation
Figure 2 shows the system under consideration and indicate the nomenclature. For the nth tank (counted from the discharge end of the series), z=
F,Do=
F,F~_,+M,_,
(17)
where n = 1,2, . . . , N; Fnel = D, = D(e) and M,_, = 0 for n = 1. In order to maintain a stationary RTD. V  n_ tn. F,
(18)
We also require that the intermediate feed streams, M,, be ratio controlled with the discharge streams: M,=+DD,,,=y Yn
nF n
RTD
SO far, attention
has been confined to nonperiodic changes in throughput; but periodic changes are also possible and may be desirable to intentionally cycle the downstream equipment. As one example, the step changes discussed earlier can be repeated in a periodic 1598
Fig. 2. Stirred tanks in series.
(19)
Stirred tank reactors
where yn is a constant of proportionality. tion (17) becomes
(24)
n
(20)
1
f
This set of simultaneous equations can be solved to give nth order linear differential equations of the form (J+).
= no)
so that
+~(l,,_,,*.
(Si)
Cl Yl)(l  Yz)*   (17kw(e)
Equa
. . (+$V;=f.(e)
(21)
where
(0)
=
n
(wwv 
t1tz . . . f,_l
(25)
.
Two tanks, equal time constants This case will be treated in some detail since it is relatively simple from an algebraic viewpoint yet illustrates the general phenomena. Suppose i, = i, = i. Then the general solutions to Eq. (2 1) are
f,(e) =
(
(1r1) (lh)
I)“[
fl
f,
VI = eejS(C, + I eO’yI(0) d(j))
* * * (1Id . . .
fn_1
D(O)
1
and
The general solution of Eq. (21) can be immediately written down as soon as we know which of the time constants, &, tz, ..a, i n, are distinct. The initial conditions associated with Eq. (21) are V,(O=O) = (V&l diV J$ = 0,
(22)
V, = ee”(C,+cSe+JJ
where the C are constants of integration. As in the case of a single tank, there are any number of D(e) which satisfy Eq. (23). We shall use the form
i= 1,2 )..., nl.
The discharge flow rate, D, is piecewise continuous and must satisfy the equations
V,(T)  Vi(T)
V,(O)
V,(O)
o
D(e) = PO,
+ce<7
V,=V,(O)
V,=
e=o
at
(ly)V,(O)=V~(O)
!a
de,
at
e=o
=o
.
Additionally, VI, V,, and dVJd0 must all be continuous at time ~9= c$.The results are #=
1599 C.E.S. Vol. 25 No. IOG
D(O) = CYD,,,
where DO represents the discharge rate at the initial steady state. The initial conditions are
(23)
at some time r > 0. At this time, D is set equal to F1(7) and the system is in equilibrium at the new, steady state conditions. The RTD for each tank individually will have been stationary throughout this period; and with the intermediate feed streams ratio controlled, the composition at both the feed and discharge of each tank will have remained constant. It is worth noting that the only effect of intermediate feeds is to change the forcing function, L(0). Also,
ee’yz(e) d(P)
(l_(y) [email protected]+a,
o
= ( 1  a) eel?+ (a  0) e([email protected])‘f + p,
+
E. B. NAUMAN
and
!.O
v,(e) _ v,(e) = (1a) VL?/,(O)(1y)V1/1(0) X
eelt + a
= (1a)
7
o
[lf
(I =I.04 fl ~0.56
1
K = 0.5 I.5 T = 3.5t3f
I.0
[I$]eR”+(a_P)
j.5
+P,
c_$
0
Equation (23) becomes
Fig. 3. Volume change for two tanks in series with equal time constants.
V2(7)_ Vi(T) I/,(O)
0.5 B/r
Vi(O)
and applying this condition gives
(26) Let 4 = Kr where 0 < K < 1. Then Eq. (26) can be substituted into the expression for VI (T) to give
In this equation, the factor V,(T)/V,(O) represents the volumes and throughputs after the transient compared to their values before the transient. This ratio will normally be given; and we seek values for (Y,/3, and K which will achieve the change, i.e. which satisfy Eq. (27). With (Y,/3, and K known, 4 and r = c#dK can be found from Eq. (26). Except for a few special cases, Eq. (27) must be solved numerically. To increase the final volumes, VI decreases during the time interval 0 < 8 < 4 and then increases for 4 ‘< 8 < T while V, monotonically increases. For increases in volume and throughput, (Y> 1 and p < 1. Figure 3 illustrates a case where the throughput is doubled. The parameters are (Y= 1.04, p = 0.56, and K = O5. These
values satisfy Eq. (27) and give T = 3.58f. In this case VI falls to 80 per cent of its initial value at time 13= 4 = r/2. For volume decreases, the situation is reversed with (Y< 1 and /3 > 1. VI first increases, then decreases, while V, monotonically decreases. Minimization of T
For volume increases in a single tank, r has an absolute minimum which corresponds to cy= p = D = 0. For two tanks in series and any given value of K, there is a similar minimum for T which corresponds to /3 = D = 0. Associated with rmin (for a fixed K > 0) is some finite value for (Y> 1. This (Y can be found from Eq. (27) by setting p = 0, and then r,rn can be found from Eq. 26. Figure 4 shows a plot of rmin VS. K for V,(T)/V,(O) = 2. Now, Tminitself has a minimum in the limit as K approaches zero. Approaching this limit of course requires arbitrarily large (Y. Physically, the minimum time traje,ctory corresponds to a very short period of high discharge, (Y% 1, followed by a relatively long period with zero discharge, p = 0. In the real world (Ywill have some maximum possible value, and it may also be necessary to limit p to some minimum value. Numerical simulation indicates that the constrained minimum for r occurs at the smallest possible K which satis
1600
Stirred tank reactors
tive if the system is initially operating near full capacity. Two tanks, unequal time constants If I, # &, the equation for V,(O) is
s=
0<8<4
(l(Y) eelSI+,,
which has exactly the same form as for equal time constants. The equation for V,(O) is
I 0.5
I.0 0
I.0
K Fig. 4. Minimization of 7 for two tanks with equal time constants.
fies Eq. 27 for the extreme values of (Yand p. This approach also minimizes the production “loss”. For two tanks in series, another physical constraint may correspond to V,(6). For volume increases, V,(O) has a minimum at 13= C#Jwhich may have to be limited. Using Eq. 26 to evaluate V,(4) gives
V,(4) _ VI(O)

P(lK)
+fh.
(28)
The curve labeled V, = O75 in Fig. 4 represents a constrained minimization of 7 which forces V, (4) 2 0.75. For this curve, p > 0 for all K up to about O74. For K values above 0.74, V,(4) never falls below O751/, even if p = 0. Therefore, the curves marked Tminand V, = O75 become identical for K > O74. The minimum time trajectory for decreases in volume and throughput is essentially the reverse of that for increases. 7 approaches an absolute minimum as K approaches 1.0. To achieve 7min we want (Y= 0 for a relatively long period followed by a very short interval with /3 9 1. It will usually be necessary to impose a limit on VI(+). Obviously, this limit can be quite restric
4<8<7.
+P,
Applying the condition that
gives
E=
e(Twfi

e(rC)/h
e’/”  e7/t2
(29)
which is the counterpart of Eq. (26). Equation (29) plus the condition that V,(T) has some desired value form the mathematical constraints on (Y,p, 7 and 4. Solutions do exist although the allowable ranges for the parameters may be quite small. CONCLUSIONS
This paper has shown that a system of stirred tank reactors can be subjected to changes in working volume and throughput while maintaining a stationary residence time distribution. This in turn will ensure constant product qualitywith arbitrary reaction kineticsover a broad
1601
E. B. NAUMAN
range of operating conditions. It is necessary to assume that the degree of micromixing is unaffected by the changes in working volume and flow rate. Intermediate feed streams can be allowed provided that they are ratio controlled to the mainstreams. Individual tanks in a series system need not have the same temperature although the individual temperatures must be held constant during the transients. This final condition is not especially difficult to achieve since the feed flow rates and working volumes are coupled in a manner which tends to maintain a constant heat balance. In particular, the reaction exotherm and the entering feed enthalphy will have a constant ratio; and for volume changes in a fixed geometry, the heat transfer area will often be proportional to the working volume. In any event, the volume and throughput transients can be extended over a time interval which is long compared to the response time of a temperature controller. The basic method for maintaining a stationary residence time distribution is to maintain constant proportionality between feed flow rate and working volume. With this done, the discharge rate can be subjected to relatively arbitrary manipulations. The minimum time trajectory for volume changes results from a type of “bangbang” control on the discharge stream. This fact is hardly unexpected in light of optimal control theory for linear systems. What we have described is a form of optimal controller which gives zero error in product composition and the nature of which is completely independent of the reaction kinetics. For throughput changes in a single stirred tank, the discharge rate must be initially changed in the direction opposite to the final, desired throughput. Thus, the discharge rate cannot monotonically change to its final value although the feed rate and volume can be monotone. For
two tanks in series, the discharge rate must go through at least two relative extremes prior to reaching its final value. The volume of the last (discharge end) tank will go through at least one relative extreme while the volume of the first tank can be monotone. In a system of N tanks, the discharge rate must go through at least N extreme values prior to reaching the new steady state. It is apparent that the allowable D(e) become very restrictive for large N. NOTATION
C D f F
g I K
L M n N t t’ f V
constant of integration discharge flow rate forcing function feed flow rate residence time frequency function defined by Eq. (16) fraction of time interval production loss, defined by Eq. (14) intermediate feed stream refers to nth tank counted from the discharge end number of tanks in series real time residence time mean residence time working volume
Greek symbols CY,p
y 8 4 7
discharge rate expressed as a fraction of the initial value proportionality constant for intermediate feed streams real time switching time for the forcing function time required for a transient
Subscripts 0
1,2, n 7
REFERENCES NAUMAN E. B.,Chem. EngngSci. 196924 1461. DANCKWERTS P. V., Chem. Engng Sci. 1958 8 93. ZWfETERING TH. N.,Chem. EngngSci. 1939 111. SZABOT.T.andNAUMAN E. B.,A.I.Ch.E.J1196915575. O’DRISCOLL K. F. and KNORR R., Macromolecules 1969 2 507.
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refers to initial steady state refers to nth tank in series refers to final steady state
Stirred tank reactors Resume  Un systbme dun ou plusieurs reactems agites places en s&tie peut etre sujet a des changements de volume et de regime de travail, tout en gardant une repartition fixe du temps de residence. Ceci assurera la qualitd constante du produitavec des reactions de cinetique arbitraire sur une gamme etendue de conditions operationnelles. Les r&hats sont directement applicables pour entrainer des changements et interrompre les operations. IIs peuvent &tre employ& pour engendrer un cycle control& du debit de sortie, tout en maintenant constante la composition du produit. Si la qualite du produit depend de la forme exacte de la distribution du temps de residence, il n’y a aucun moyen de produire une qualid uniforme pendant le dtmarrage. Cependant, la quantite totale de materiel de qualite inf&iiure peut &tre maintenue a une faible limite arbitraire. Zusammenfassung Ein System aus einem oder mehreren hintereinander geschalteten Riihrkesseln kann, unter Einhaltung einer stationSiren Verweilzeitverteilung, Vednderungen im Arbeitsvolumen und im Durchsatz unterworfen werden. Dadurch wiederum wird bei wilkiirlicher Reaktionskinetikeine konstante Qualitat des Produktes innerhalb eines weiten Bereiches von Betriebsbedingungen sichergestellt. Die Ergebnisse sind unmittelbar auf geforderte Vetinderungen und Stilllegungsvorg%nge anwendbar. Sie kijnnen zur Aufstellung eines gesteuerten Rythmus der Ausstromungsgeschwindigkeit bei gleichzeitiger Aufrechterhaltung konstanter Produktzusammensetzung verwendet werden. Wenn die QualitPt eines Produktes auf der genauen Form der Verweilzeitverteilung beruht, gibt es kein Mittel wlhrend eines gesamten Anfahrvorganges eine gleichmassige Qualimt zu erzielen. Es ist jedoch mdglich die Gesamtmenge des qualitativ unzureichenden Materiales auf einen willkiirlich festlegbaren niedrigen Grenzwert zu beschranken.
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