Generalized Recycle Reactor Model for Micro-Mixing Luc A. Dohanl and Herbert Weinstein* Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Ill. 60616
A general model i s proposed which represents an infinite number of states of micro-mixing at any given level of macro-mixing. The model consists of a segregated internal vessel with a recycle stream. A simple relation between degree of segregation and recycle ratio is found. The model is used with Michaelis-Menten kinetics to demonstrate the effects of micro-mixing. lt i s shown that an intermediate level of macro-mixing with the maximum permissible micro-mixing gives maximum conversion for this example. This application also brings some insight into the problem of the prediction of conversion at intermediate degrees of micro-mixing in any chemical reactor system.
M i x i n g effects in chemical reactors are usually separated into two components, macro-mixing and micro-mixing. I n many cases of simple homogeneous reactions, micro-mixing has a limited effect on conversion. However, this effect usually cannot be neglected in the design of heterogeneous reactors or in homogeneous reactors in which some complex or autocatalytic reactions take place. I n certain cases, a description of micromixing is certainly a fundamental part of the modeling of the reactor system. One-parameter residence time distribution (RTD) models are unable to describe micro-mixing over its entire permissible range continuously and independently from the macromixing. I s examples, the dispersion model (cf. Levenspiel, 1962) describes macro-mixing, but i t implies also a fixed state of micro-mixing for each state of macro-mixing. I n the same way, the recycle reactor, Le., plug flow with recycle, can model macro-mixing (Gillespie and Carberry, 1966a) but implies also a maximum mixedness state of micro-mixing (Rippin, 1967). The equal sized tanks in series model can be used to describe, in a discrete manner, macro-mixing. Using sequential mixing (Tsai, et al., 1969; Zwietering, 1959) with this model allows a limited number of states of micro-mixing. Two-parameter models can represent a continuous range of micro-mixing for a given state of macro-mixing. Previously proposed two-parameter models (Ng and Rippin, 1964; Weinstein and ildler, 1967) have, however, been difficult to visualize and apply. In this work, a two-parameter, deterministic model is proposed t o describe an infinite number of states of micromixing a t any given level of macro-mixing. This model is a generalization of the recycle reactor. It is readily visualized and applied. An example is presented using the model with Michaelis-Menten kinetics. This rate expression is chosen because of its essentially autocatalytic behavior. Recycle and Macro-Mixing
Macro-mixing takes into account the variation of residence times of elements of fluid passing through the reactor, and is entirely described by the RTD. Let us consider a recycle system as shown in Figure 1. This flow system has a n K T D , E ( t ) , and consists of a n internal vessel, whose R T D is G ( t ) , and a recycle stream, p'. We define R = q ' / p as the recycle Present address, European Technical Center, Proctor and Gamble Corp., Brussels, Belgium. 64 Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
ratio. The introduction of the recycle does not change the mean residence time in the system. The first moment is always f = V/q. However, some recycled elements of fluid will spend more time in the system and nonrecycled elements less time; therefore the resulting R T D , E ( t ) ,will be broader (larger variance) than G(t).This is a macro-mixing effect. Fu, et al. (1971), showed that for this system, E(t) car, be expressed in terms of a Volterra integral equation of the second kind whose solution is the sum of a series of convolution integrals. The particular case where G(t) is the R T D of n equal sized tanks in series of volume V/'neach, xhich they treated, is repeated here since it is used later.
The corresponding E(t)is given by
with first moment i = V ,q and dimensionless valiance
a2 = (I
+ n R ) / n ( R + 1)
(3)
When the number of tanks, n, goes to 0 3 , rye have :Lplug flow reactor with recycle. The results for a plug flow reactor were first shown by Rippin (1967) and can be obtained from eq 1-3 by taking limits properly. For the special case of n = 1, E(t) = (l/t)ed"forall R. Recycle and Micro-Mixing
I n order to describe the environment of a n element of fluid during its passage through the reactor, we use the notion of a "point." Danckwerts (1958) defined a point as a very small element of fluid, just big enough to have the average value of some intensive property (such as concentration) independent of random molecular variations. Inside a point just entering the system all molecules have the same age, zero. Inside a point leaving the system, all molecules have the same life expectation, zero. Weinstein and Adler (1967) defined micro-mixing as the transition from a segregation by age to a segregation by life expectation. It is clear that the extent of micro-mixing is bounded by the macro-mixing level. T o indicate the extent of micro-mixing, Danckwerts (1958)
proposed the use of a parameter, which he called the degree of segregation, J .
where Var(a,) is the variance of the age distribution between the points inside the reactor and V a r ( a )is the variance of the age distribution of all the molecules in the system. I n this discussion, all ages are made dimensionless on the i of the reactor. Zwietering (1959) showed that
Var(a) = Bar(ai)
+ Var(a,)
Figure 1 . Recycle system
Zwietering showed t h a t V a r ( a ) can be expressed in term of the second and third moment of E @ ) .
\diere Var(,,) is the variance of t h e age distribution inside a point. Therefore, J can be also written with
(4) taE(t)dt We can now visualize the two extremes of the range of micromixing. In the first, case, molecules stay grouped by age, as if the boundary of the points were impermeable. Var(ai) is then equal to zero and we have J = I , or segregated flow. I n the other extreme, the points are mixed on a molecular level as early as possible and \\-e have the maximum mixediiess state which Zwietering (1959) analyzed. Considering the recycle system of Figure 1, we impose tlvo conditions: (a) a t point .I,complete mixirig on a molecular level occurs; (b) the internal reactor operates under a completely segregated flow condition. 3lolecules of different ages vi11 meet and mix a t point 1.This is a micro-mixing effect. It is clear also t'hat increasing R will increase this effect. The degree of segregation can then be evaluated for this description. If Z ( a ) is the age distribution at point ;1after mixing on a molecular level but before entrance into the reactor, a mass balance gives the fraction of entering fluid which has an age of a1
The first' term on the right of eq 5 corresponds to the fraction iii t'he recycle and the second term to the fraction in the incoming feed, where 6(aJ is a Dirac function 6 a t zero. The mean age of fluid eiiteriiig the reactor is simply
Siiice the internal reactor is fully segregated, the points are formed a t the entrance of the reactor and do not exchange with other points. Therefore
Varja,) =
S,=
a z l ( a ) d a - E'
B
=
krn
tZE(t)dt
(11)
If the macro-mixing level is kept constant as R varies, b z and Var(a) are also constant and we can write by taking the derivative of eq 9
We have
+
dJ 1 3 -
