Residence time distribution theory for unsteady stirred tank reactors

Residence time distribution theory for unsteady stirred tank reactors

Pergamon Press. Chemical Engineering Science, 1969, Vol. 24, pp. 1461-1470. Printed in Great Britain, Residence time distribution theory for unstea...

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Pergamon Press.

Chemical Engineering Science, 1969, Vol. 24, pp. 1461-1470.

Printed in Great Britain,

Residence time distribution theory for unsteady stirred tank reactors E. B. NAUMAN Process Fundamentals Group, Polymers Research and Development Department, Union Carbide Corporation, Bound Brook, New Jersey, U.S.A. (Received 12 March 1969) classical analysis of reactor kinetics in terms of residence time distributions has been extended to unsteady stirred tank reactors. This allows treatment of unsteady or cyclic reactors within the same theoretical framework used for steady, continuous flow reactors. Residence time distribution functions are defined for the unsteady stirred tank reactor; and at any instant of real time, the performance of the reactor can be evaluated as though it were a steady reactor with equivalent residence time distribution. If the stirred tank is a perfect mixer, the unsteady reactor can be treated as a maximum mixedness reactor with arbitrary residence time distribution. If the unsteady reactor is well-stirred but exhibits micro-segregation, it behaves as a completely segregated system of arbitrary residence time distribution. If the inlet reactant concentration is unsteady as well as the input and output flow rates, then the equations governing yield are generalizations of the maximum mixedness and complete segregation equations for the steady flow reactors.


INTRODUCTION is a wide and growing interest in unsteady-state processing and particularly in reactor kinetics. This is partly motivated by the need to control “steady” reactors but, more recently, by the potential benefits of intentionally unsteady operation[l, 21. Much of this work has been done within the framework of control and optimization theory [3] and has tended to emphasize the application of such techniques to relatively simple kinetic systems [4,5]. The present paper takes a different approach and emphasizes generalized methods for predicting yields in unsteady reactors without specific assumptions concerning basic kinetics or the mixing phenomena within the reactor. The aim is to present a unified view of reactor kinetics utilizing the theories of residence time distributions and micro-mixing [6,7]. The analysis of residence time distributions is now a classical technique for predicting yields in isothermal, homogeneous reactors; and in some cases, the general concept of a reactionhistory distribution can be applied to heterogeneous or even non-isothermal reactors [8]. The yield of a first order reaction is uniquely determined by the residence time distribution. THERE

For reactions of order other than first, the yield depends on the state of micro-mixing; but it can often be closely bounded using the concepts segregation” and *‘maximum of “complete mixedness”[6,7]. To date, this approach to reactor analysis has been primarily confined to steady, continuous flow systems. Aris[B] has given a theoretical treatment of axial dispersion in pulsating laminar flow, Taylor and Leonard[ lo] have given experimental results for pulsating turbulent flow, and Cha and Fan[l l] have treated the completely mixed tank with a pulsating feed. The present paper attempts to extend and expand these ideas. One goal is to provide conceptual insight in the use of intentionally unsteady operations, particularly when the aim is to alter selectivity rather than to merely obtain an increase in yield (which is usually marginal). Secondly, mathematics are developed for treating extremes in micro-mixing, especially the case of complete segregation. Viscous reaction mixtures, such as those in mass polymerizations, are highly segregated yet often show a marked change in selectivity with residence time distribution. Finally, the interpretation of tracer experiments on unsteady



systems is discussed. Most of the results will be based on the unsteady stirred tank reactor; but the general concepts can be applied, at least experimentally, to a much larger class of systems. DISCUSSION

8 c t. More specifically, t $ de must diverge.


--co At any instant of time, F, D, and V are related by

Description of system The system considered is illustrated in Fig. 1. F(6)






Fig. 1. Unsteady stirred tank reactor.

The feed rate, F, discharge rate, D, and system inventory, V, all vary with time, 8. These variations may or may not be periodic. The tank is sufficiently stirred so that the probability of material leaving the reactor at any instant of time is the same for all material within the reactor at that time. This does not necessarily imply that the contents of the reactor are well mixed on a molecular scale since the reactor can exhibit micro-segregation. If molecular level mixing is complete, the stirred tank is a perfect mixer. If it is completely segregated, it is termed a segregated stirred tank reactor. In either case, steady-state operation, F = D = constant., will give an exponential distribution of residence times. Three conditions will be imposed on the operation: (1) V(0) isboundedforall8

(2) If D = 0 for all 8 > t, then V = 0 for all 8 > t. More specifically,

Definition of residence time distributions Two basic residence time distribution functions can be defined for the unsteady reactor, one effectively measured at the inlet to the reactor and one at the discharge. The definitions are: R (t, t’) = fraction of material entering the reactor at time t which will remain in the reactor for a duration greater than t’, i.e. will remain in the reactor until a time greater than t + t’. S (t, t’) = fraction of material leaving the IVactor at time t which remained in the reactor for a duration greater than t’, i.e. fraction of material leaving at time t which entered before time t-t’. Note that the symbol t denotes real (or absolute) time while t’ denotes residence time: At steady-state, R (t’)


= e-Qt’/V

S (t’)




where Q = F = D and f is the average residence time. For unsteady operation, R (t, t’) and S (t, t’) will generally not be the same function. At a given instant of time, these functions can assume essentially any functional form provided (1) R (t,O) = S (t,O) = 1. (2) Both R (t, t’) and S (t, t’) are non-increasing with respect to t’. For unsteady operation, two definitions of average residence time are required:



= O”R (t, t’)dt’


&(t) = 7 S (t, t’) dt’



+ de must diverge.


(3) If F = 0 for all 8 < t, then V = 0 for all 1462


Residence time distribution theory

Now M/MO is the fraction of the molecules entering at time t which remain in the system at time 8 > t. By substituting t+ t’ = 8, R (t, t’) is obtained directly,

where & (t) is the average residence time for material entering at time t and Z,(t) is the average residence time for material leaving at this time. In general, f,(f) # r,(t).

R (t, t’) = [email protected]@. r


It is also useful to define two additional residence time functions, these being the “frequency” or “dilferential” functions:

It is seen that R (t, 0) = 1 and that R (t, t’) is non-increasing with respect to t’ since D and V are never negative. Note however that D 2 0 -dRjt, t’) does not preclude flow reversal as a means of f(t, 0 = pulsing the reactor since D is the total discharge rate at any time and does not necessarily cor-ds (t, t’) g tt, r’) = dt’ ’ respond to a particular physical exit from the reactor. From these definitions it follows that: The experiment described above is an impulse f(t, t’) = fraction of material entering at time 1 test, a method commonly used to measure reswhich will remain; in- the reactor for a duration idence time distributions in steady flow systems. between t’ and t’ + dt’. Physically, we would measure the rate at which g (t, t’) = fraction of material leaving at time t molecules leave the system, MD/V, rather than which remained in the reactor for a duration the total number still remaining in the system between t’ and t’ + dt’. at some time 8 = t + t’. Scaling this measurement We immediately have by the number of tracer molecules originally charged gives the frequency function &(t) = 7 t’f(r, t’) dt’ (6) dt,


with a similar result for r&t). Higher “inlet” moments of the residence time distribution can also be found from either R (t, t’) orf( t, t’) , and higher “discharge” moments can be found from either S (t, t’) or g (t, t’). Determination ofR(t, t’) Suppose at some time t we suddenly inject a pulse of MO tracer molecules at the inlet stream of the reactor. Then for all time 8 > r, the number of molecules remaining in the reactor is given by -DM V

=- dM


M=M,atO=t this gives M -=e Wl

-is&. ,

The scaling is necessary to ensure that lf(t,

It is seen that the standard impulse test can be used to measure the distribution functions R and f for the unsteady system. However, R and f are distribution functions for materials entering the reactor at time t while S and g, the distribution functions for material leaving the reactor, are those normally needed for reactor analysis. Determination ofS (t, t’) Suppose that a tracer stream is injected at the reactor inlet starting at time 8 = t-t’. For 8 > t - t’, the concentration at the inlet is Co and the concentration within the reactor is given by

1463 C.E.S. Vol. 24 No. 9-D

t’) dt’ = 1.





R (t, t’) = “:;;)

S (t+ t’, t’).


A!$=F-D Similarly, C=Oatt?=t-t’.




This gives 1 - C/C, = e

-/[email protected]


and it is seen that 1 - C/C, is the fraction of the material in the exit stream at time t which has remained in the reactor for a duration greater than t’. Thus, s (t




,[email protected]

A general relationship between & and & is more difficult, but for specific cases they can be easily related through the time parameter in the differential equations d&-D_ dt -j&-l (13)



and to






,[email protected] ‘-p



The above experiment is only conceptual. Step changes in concentration can be made. However, they will not yield the entire function S (t, t’) but only a single point on this function. To determine S (t, t’) from a strictly experimental viewpoint, we must make an entire series of step changes, in effect starting at time t (t’ = 0) and working backwards through all earlier times (t’ > 0). This of course is in sharp contrast to the case of steady flows where the entire function S (t’) can be measured with a single experiment. Some general properties Expressions relating the functions R and S can be easily obtained since


which result by differentiating equations with respect to t.

(14) t


Then under these conditions, V’, R, S, F, g, S,, and & will all be periodic with period T where 7 is the least common multiple of rF and 70, Further,



In discussing’ periodic’ flows, one negative result should be mentioned. It is rarely possible tofindatime;;;:,t)hrS(t+d t,)



f=lnv~t;“+~$ds. t







Periodicflows Some useful special results can be readily developed for the important case where the entering and discharge flows are periodic in time. Suppose (1) F is periodic with period 7F F (t+Tp) = F(t) forany t. (2) D is periodic with period rD D(t+TD) =D(t)foranyt. (3) On average, the input and output flows are balanced









Residence time distribution theory

Thus R and S are not related by a simple time lag; and even for periodic flows, are often completely different functions.

by setting 8 = t-A with t fixed.

Conversion in the unsteady perfect mixer By manipulation of F and D, it is possible to obtain essentially any residence time distribution, S (t, t') , for material leaving the reactor at time t. At this point we consider how the instantaneous exit concentration, C (t), can be related to this residence time distribution for the special case where the stirred tank reactor is a perfect mixer. Consider a molecule in the system at some time 19.Suppose this molecule has already been in the system for a time (Y(the age of the molecule) and will remain in for an additional time A (the life expectation of the molecule) so that



where t’ is the residence time. Now consider the average composition of all molecules in the reactor at time 8 which have a life expectation of A. Call this concentration C (0, A). Since the reactor is a perfect mixer, those molecules having a life expectation of A will have exactly as all other the same average composition molecules in the reactor. Thus, C(19,h) =


where C (0) is the concentration in the reactor at time 8. A material balance gives FC,-DC-Vr(C)


and F-D=g.


Thus, dC de




Now for a fixed time, t, we wish to see how the concentration of material leaving the reactor at time t varied with its previous life expectation, A. This is obtained directly from the above



(t-A)] V(t-A)

F (t-A)

F(t--h) _ g(tvA) V(t-A)



For the case where Ci, is constant, this is Zwietering’s equation[7] for a maximum mixedness reactor with residence time distribution S (t, A) ; and is a generalization of Zwietering’s equation if Ci, varies with time. The unsteady stirred tank reactor is a maximum mixedness reactor if the tank is a perfect mixer. Now, in classical maximum mixedness theory, the boundary condition normally associated with Zwietering’s equation is

$=t)atA=m_ However, an alternate boundary condition allowing dC/dA to have a non-zero value at A = A,,,. must sometimes be used even in the classical treatment and is often necessary in the case of unsteady stirred tank reactors. The exit concentration at time t is obtained by evaluating the integral at A = 0 since this represents the composition of material of a zero life expectation. Conversion in the unsteady segregated stirred tank In steady operation, the segregated stirred tank is similar to a perfect mixer in that both have an exponential distribution of residence times. However, the contents of the segregated stirred tank are completely segregated in the sense of Danckwerts [6]. In unsteady operation, any residence time distribution can be achieved for material exiting at time t; but if the tank is segregated, the system will remain segregated regardless of the unsteady operation.



To find an expression for the conversion at time t for the unsteady segregated tank reactor, consider a small portion of fluid which enters the reactor at time t - t’. With complete segregation, this portion of fluid will remain intact and can be treated as a small batch reactor. g

= r(C)

Suppose that this integral can be inverted to give c(e)




where C (f3) is the concentration in this small batch as a function of the time it has remained in the reactor and I denotes the inverse function, e.g. I= Ci, (f-f’)e+ for a first order reaction, r = KC. The concentration of this batch when it leaves the reactor is

1[Gn (t - f’) , 1'1

inverse function, I, since the rate constant will be a function of temperature and thus of absolute time, t- t’. If the coupling of temperature and concentration cannot be neglected, Eq. (19) can be combined with a heat balance in an iterative solution for both temperature and concentration as a function of time. The case of complete thermal segregation would seem rare but could be treated by considering each element of material as a small adiabatic reactor. Temperature dependence would again be incorporated in the inverse function, I, but in this case temperature would vary with residence time, f’, rather than absolute time. Equation (19) can also be used for the case of two or more reactants which are input as different functions of time. The variation in inlet concentrations as functions of absolute time must be incorporated in the inverse function. Damping of concentrationfluctuations An early paper by Danckwerts [ 121 showed how the outlet concentration of a non-reactive component could be calculated from the timevarying inlet concentration and the residence time distribution for the system. This result can easily be extended to the unsteady stirred tank. If the component is non-reactive, the inverse function becomes

and the average exit concentration at time t is found by integrating over all possible t’:

c(f) =

1ZCci, (t-f’),

Z[Ct, (f-f’),f’]

= ci, (f-f’)

and the average concentration

f’l g(f, f’)dt’.

at the outlet is



C(t) =i

For the. case where C,,, is constant, this is the standard result for segregated reactors with residence time frequency function g (f, f’). As written, it represents a generalization of the classical result which allows the inlet concentration to vary with time. It is sometimes possible to extend Eq. (19) to non-isothermal systems. Suppose the reactor is thermally homogeneous but still segregated with respect to molecular concentrations. If temperature and concentration are uncoupled, the effect of temperature can be included in the




This result is equivalent to that of Danckwerts and is a generalization only in the sense that the residence time distribution may vary with absolute time. This result is, of course, independent of the state of micro-mixing and applies equally well to unsteady perfect mixers or segregated stirred tanks. Others unsteady reactors Results to this point have been confined to the stirred tank reactor although the basic


Residencetime distribution theory definitions of the residence time distributions and their moments can be applied to any unsteady reactor. Equations (7-13) will no longer hold nor will the steady-state distribution be exponential. However, the same types of tracer experiments can be used for experimental determination of R (t, t’) and S (t, t’). Also, the need for multiple experiments to determine S (t, t’) is not unduly onerous since the same experiment can be used to determine a single point of S (t, t’) for each of a number of different absolute times. It is also clear that Eq. (19) can be used for any segregated system and will thus supply one extreme in micro-mixing for an arbitrary reactor. It is conjectured that Eq. (17) also applies to an arbitrary, unsteady maximum mixedness reactor although this has not been proven. Analytical results can be derived for a few simple cases. For N stirred tanks in series, [l -&(t,

t’)] = [l -&(t, t’)] [l -&(t, t’)] . . . [l -+,(t,



where the subscript c denotes the composite system. For the piston flow reactor with an incompressible fluid, D = F, g (t, t’) = 6 (&-t’)

where 6 denotes given by



One striking feature of unsteady stirred tank reactors is the diversity of phenomena which can be obtained. This section does not attempt gross generalizations as to what kind of feed and discharge disturbances may be beneficial for a particular application, but only illustrates how the system performance can be analyzed once the feed and discharge characteristics have been selected. Periodicjlow

Suppose both F and D are harmonic and exactly in phase: D=F=F([email protected],A~


then g=




In this simple case, Eq. (10) can be analytically integrated to give the residence time frequency function Q (7-7 T’)

where time has been made dimensionless



Dirac’s delta function and & is

Figure 2 shows the frequency function plotted against residence time, T’, for A = 1 and wf = 27~. D de = V = const. (23) The frequency function varies with real time, I r-i, T, and is multi-modal. Each point where g (T, Thus the residence time distribution always T’) = 0 corresponds to a previous zero in F and corresponds to piston flow but the mean, &, D; and, with a small phase shift, each mode corresponds to a previous maximum in F and varies with time. The highly stratified standpipe is a type of D. Note that g (T, T’) is defined even at time piston flow reactor where the volume can vary T = O-5 when there is no flow from the reacwith time. In this case I, is given by tor (F = D = 0). Physically, this distribution would apply to a negligibly small number of t me = v (t--f,). (24) molecules removed from the system at this point 1-G in time; and since we are dealing with a well Finally, stirred tanks and piston flow elements stirred tank, it is also the age distribution for in series can be treated simply by introducing a molecules within the reactor. For comparison, lag time, &(t) , corresponding to the piston flow Fig. 2 also shows the exponential residence time section. distribution for the steady stirred tank. t




2.0 t i



- - -

Steady stirred tank

.-s t 5


time, T

Fig. 2.

The uppermost curve in Fig. 3 shows how the dimensionless mean residence time, F’ = t,/f varies with real time T = t/K The results were obtained by numerical integration of Eq. (6). The two middle curves in Fig. 3 give the yield for a second order reaction with rate constant Kf = 1 and inlet concentration Ci, = 1. The quantity plotted is the fraction unreacted, C (T)/C,, so that the maximum mixedness

= e-


0 if


ii! § 3


reactor actually gives a slightly lower yield. The maximum mixedness results could have been obtained by integrating Eq. (17). However, each integration will give the exit concentration C ( T) , for only one value of real time, T. Thus for the periodic stirred tank reactor it is more convenient to integrate Eq. (16) subject to the periodic condition C (T) = C (T + 7). A single integration gives C (T) for all T. The results for the segregated reactor were obtained by numerical integration of Eq. (19). As is typical for steady flow reactors, the yield of a second order reaction is closely bounded by these two extremes in micro-mixing. The lowermost curve in Fig. 3 gives the yield for a first order reaction with rate constant Kf = 1. This result is independent of micro-mixing and can be obtained by integrating any of Eqs. (16), (17), or (19).



1: c


r;l .E


Start-up transient Figure 4 illustrates how the residence time frequency function approaches the steadystate distribution during a reactor start-up.

i '3


E =



Real time, T= t/f

0.2 0.4

0.6 0.6



I.6 2.0

Dimensionless residence time, t’/f

Fig. 3.

Fig. 4.




Residence time distribution theory

The reactor is initially empty but at time t = 0 input flow begins with constant rate F. At time t = f the desired operating level is reached, V = Ff, and the discharge pump is started with constant rate, D = F. Equation (9) gives S(t,t’)=l-T





and for t > i, S(t,t’)=e-“‘f =






(‘-r’n t-i<

t’ < t.

The parameter indicated in Fig. 4 is the dimensionless real time, T = t/i. Note that S is defined even for the initial period when there is no discharge and can be used to predict conversions within the reactor for t < i. It can also be seen that R (t, t’) will be the simple exponential distribution for any t > T, but S (t, t’) and C (t) are appreciably delayed in reaching their steadystate values.

tems. The tracer experiments normally used to measure residence time distributions in steady flow systems can also be used for unsteady reactors although the data reduction techniques must be modified. Theoretical treatments are possible for simple systems such as the piston flow reactor and the widely used stirred tank. The unsteady residence time distributions can be applied to non-periodic transients such as start-ups or to intentionally cyclic operation. This approach is particularly useful for the segregated systems often encountered in processing high viscosity fluids. It provides a specific mathematical technique for reactor analysis and gives conceptual insight into the factors controlling product yields and selectivities. NOTATION

reactant or tracer concentration D discharge flow rate F inlet flow rate f frequency function for entering molecules g frequency function for leaving molecules I concentration function for batch reactor M number of molecules injected R distribution function for entering molecules reaction rate function s’ distribution function for leaving molecules T dimensionless real time T’ dimensionless residence time t real time t’ residence time i mean residence time at steady state i, mean residence time for entering molecules 6 mean residence time for leaving molecules frequency of cyclic disturbance W



This paper has shown how the concepts of residence time distribution and micro-mixing can be extended to unsteady stirred tank reactors. By manipulation of input and output flow rates, the unsteady stirred tank can achieve an arbitrary distribution of residence times. If the tank is a perfect mixer and if the inlet reactant concentration is constant, the system behaves as a steady, maximum mixedness reactor having the same residence time distribution. If the tank is segregated, then the equations governing steady, completely segregated reactors can be applied. If the inlet concentration itself is unsteady, then the governing equations for the two extremes of micro-mixing are simple generalizations of their steady-state counterparts. The definitions of the various residence time distribution functions are quite general and can be applied to other types of unsteady sys-

Greek symbols ff age of a molecule 8 time A life expectation of a molecule 7 period of cyclic disturbance + timelag



M. R. and SPEAKER S. S., Chem. Engng Prog.


E. B. NAUMAN [2] SCHRODT V. N., Ind. Engng Chem. 1967 57 58. [3] HORN F. J. H. and LIN R. C., Ind. Engng Chem. Process Design Deu. 1967 6 21. [4] DOUGLAS J. M. and RIPPIN D. W. T., Chem. Engng Sci. 1966 21305. [5] DOUGLAS J. M., Znd. Engng Chem. Prbcess Design Deu. 1967 6 43. [6] DANCKWERTS P. V., Chem. Engng Sci. 1958 8 93. [7] ZWIETERING TH. N., Chem. Engng Sci. 1959 111. 181 NAUMAN E. B. and COLLINGE C. N., Chem. Engng Sci. 1968 23 1309. ,9j ARIS R., Proc. R. SOC.1960 A259 370. rlO1 TAYLOR H. M. and LEONARD E. F..A. I. Ch. E. JI 1965 11686. .--[ 1 l] CHA L. C. and FAN L. T., Can. J. chetk Engng 1963 4162. 1121 DANCKWERTS P. V., Chem. Engng Sci. 1953 2 1. R&u& - L’analyse classique de la cinttique d’un reacteur par les distributions des temps de residence a et6 Ctendue aux rkacteurs instables agites. Ceci permet de traiter les reacteurs instables ou cycliques, dans le m&me cadre theorique utilist pour les reacteurs stables, a flot continu. Les fonctions de distribution des temps de residence sont definies pour le reacteur instable agite; et, B chaque instant du temps reel, la performance du reacteur peut &tre Cvalu6e comme s’il s’agissait dun reacteur stable, ayant une distribution du temps de residence Cquivalente. Si le reservoir agitt est un parfait mixer, on peut alors traiter le reacteur instable comme un reacteur a mixage maximum avec distribution arbitraire du temps de residence. Si le reacteur instable est bien agite mais expose une micro-segrbgation, il se comporte alors comme un systbme entierement isolt avec distribution arbitraire du temps de residence. Si la concentration du rtactif a I’entree est instable, ainsi que les debits a I’entr6e et B la sortie, les equations commandant le rendement sont alors des gCnCtahsations des equations du mixage maximum et de complete separation pour les reacteurs stables. Zusammenfassnng- Die klassische Analyse der Kinetik eines Reaktors durch Bestimmung der Verweilzeitverteilungen wurde auf nicht-stationare Rhhrkessel ausgedehnt. Auf diese Weise ist es miiglich nicht-stationare oder zyklische Reaktoren innerhalb des gleichen theoretischen Rahmens wie station&e Reaktoren mit kontinuierlicher Durchstromung zu behandeln; die Leistung des Reaktors kann an jedem reellen Zeitpunkt eingeschatzt werden, genau so als handle es sich urn einen stationaren Reaktor mit aquivalenter Verweilzeitverteilung. Wenn der Riihrkessel perfekte Durchmischung aufweist, kann der nicht-stationtie Reaktor als Maximalvermischungsreaktor mit willkiirlicher Verweilzeitverteilung angesehen werden. Wird der nicht-stationare Kessel gut geriihrt weist jedoch Mikrosegregation auf, so verh&h er sich wie ein vollst%ndig segregiertes System mit willkiirlicher Verweilzeitverteilung. 1st die Zufuhrkonzentration der Reaktionsteilnehmer nichtstationk, ebenso wie die Eingangs- und Ausgangsgeschwindiikeiten der Durchstromung, so sind die die Ausbeute bestimmenden Gleichungen Verallgemeinerungen der Maximalvermischungs- und der Vollsegregationsgleichungen fiir die Reaktoren mit Stationlirdurchstromung.