COMMUNICA TI 0 NS
On the Use of the Generalized Recycle Model to Interpret Micromixing in Chemical Reactors
The use of the generalized recycle model (GRM) in interpretation of micromixing in chemical reactors is examined. It is shown that GRM is not permissible as a model of micromixing if an arbitrary residence time distribution (RTD) is to be preserved in its entirety. The only exception is the case of a perfectly mixed stirred tank reactor with exponential RTD. The model can be used to investigate micromixing effects for other RTD’s when only the first four moments of the RTD are kept constant.
Introduction A generalized recycle model (GRM), presented in Figure 1, was introduced by Gillespie and Carberry (1966) and used by Dohan and Weinstein (1973) to describe various levels of micromixing in chemical reactors of a fixed residence time distribution. G ( 0 )is the unit impulse response of the section where reaction takes place and which is assumed to be in segregated flow conditions. The model is based on the second assumption that mixing on molecular level (micromixing) occurs instantaneously and completely at the point M between flow rate q of fresh fluid and flow rate Rq of fluid of all ages which is recirculated through the recycle stream of negligible capacitance. The total net flow rate in the inflow and outflow is q. The residence time distribution (RTD) of the whole system, as detected in the outflow, is E ( 0 ) . One possible use of the model is in describing a vessel, with RTD given by G ( 0 ) ,which is subject to recirculation of the portion of the outflow in such a way that the only mixing on molecular level occurs at point M due to the action of the recirculation pump. In this situation the unknown E ( 0 ) can be found from the following relationship (Fu et al., 1971; Gibilaro, 1971).
expressed as a function of the recycle ratio, R (Dohan and Weinstein, 1973)
Now one is faced with the inverse problem to eq 1,Le., the problem of finding a probability density G(8) 2 0 on k ( 0 ,m ) for a given E ( 0 ) 2 0. Formally one can write the answer as G(8) = ( R
+ 1)n2= O (-11nRnEn+1(8)
and in general G(8) may turn out to be negative at certain intervals of 0 for some values of R. The problem is to find the range of permissible R’s, if any, for which G(8) is nonnegative everywhere. Dohan and Weinstein (1973) tried to avoid the problem of having portions of G ( 0 )negative by takingE(8) to be the response of nmintanks in series and lay keeping the variance of E(0)constant but not its higher moments. C(0) was then selected to be the response of n tanks in series. The variance for the system of n-CSTR’s in series with recycle is given by
u2
This integral equation represents the basic mass balance for the recycle system and can be solved either by iteration or by Laplace transforms for E ( 0 ) (Fu et al., 1971; Gibilaro, 1971)
where G,(0) is the n t h convolution of G ( 0 )upon itself. Thus, as G(8) 1 0 always, it is readily seen that E ( 0 ) is unique and always nonnegative for all values of 0 and R. This interpretation of the model is not useful for observation of the effect of micromixing as both the RTD and micromixing vary with R.However, this form of the model with various G(0) has been used for other purposes (Gillespie and Carberry, 1966;Rippin, 1967). The other interpretation of the model, used by Dohan and Weinstein (1973) and Yang et al. (1974), is that for a fixed E ( @ ,which is the RTD of a real reactor, one can describe the various states of micromixing at constant level of macromixing by changing simultaneously R and G(0) in such a way that E(0)stays constant. If one can find a way of accomplishing this then the degree of segregation (Danckwerts, 1958) can be
(4)
+
1 nR =n ( R 1)
+
Thus, when n = nmin,R is zero and the syst.ern is, by assumption of the model, completely segregated. Various levels of micromixing were simulated by letting n vary from n,,in to infinity while R varied simultaneously from 0 to R,,,,.
The claim was made that the errors introduced in caiculating the degree of segregation, J , were negligible, Le., less than 10%(Dohan and Weinstein, 1973). However, the error in J can be much larger since the method of Dohan and Weinstein (1973) does not preserve constant VAR(n) which depends on the third moment, p3, of the E curve. It is well known that it is difficult to measure accurately the third and higher moments of the exit age distribution and thus from the practical standpoint Dohan and Weinstein’s approach seems to represent a useful approximation. However, their approach contains an inherent difficulty and inconsistency as explained below. -For example, for an RTD of two tanks in series (nmln= 2) u2 = 0.5 and VAR(a) is equal to 0.4375 at R = 0. However, at Ind. Eng. Chem., Fundarn., Vol. 16, No. 3, 1977
385
I
I
IL ----------------__
I
extent on the variance of the internal age distribution, VAR(a), which as shown above is changing. In some cases small variations in the degree of segregation, J , caused by changes in VAR(a) will have a negligible effect on reactor performance and Dohan and Weinstein’s approach is a good approximation. However, there are situations where even small variations in J at low levels of J affect the reactor performance profoundly (DudukoviE, 1977).
I d
Figure 1. Generalized recycle flow model.
-
n m and R = R,, = 1,VAR(a) = 0.5208 which is a change of 19%. If the degree of se e ation, J , which can be calculated from eq 3, is based on V%) of the original overall RTD (0.4375), as suggested by Dohan and Weinstein’s approach, its value would be negative, J = -0.1429 and thus unacceptable. If the value is based on VAR(a) of the newly developed E ( 0 ) at R = 1one obtains J = 0.0399 which is a difference of 128% and still 45% higher than Jmin = 0.0275 which characterizes the maximum mixedness condition for the original RTD (Zwietering, 1959). The above difficulties are the result of the fact that in gereral, for arbitrary unimodal E($),it may be impossible to find a probability density G(0) from eq 1 or 4 such that G(0) 1 0 for all 0 and for a range of R’s, 0 5 R 5 R,,. Often the only value of R permitting G(0) 1 0 for &(O, -) is R = 0 when G (0) = E @ ) .For example, if one selects E ( $ )to have the form of the RTD of two equal size stirred tanks in series, as described by eq 7 E ( $ ) = 48e-28
(7)
the analytical solution of eq 1 results in the following expression for G ( 0 ) 2(R 1) G ( @= e - 2 8 sin ( 2 f l O )
+
fl
Clearly, only when R = 0, G ( 0 ) is always nonnegative and equal to E($).Otherwise for any other value of R, G ( 0 )changes signs at the points. 0
--. k.lr - 2fl’
Modification of the Model In general it is impossible to keep E($)constant and vary R and G(0) simultaneously. However, the effects of changes in the E ( $ ) on conversion can be reduced if VAR(a) is kept constant. In this manner the GRM can be used to estimate micromixing effects under small changes in E ( $ )reflected only in w4 and higher moments. Keeping VAR(a) constant reduces even further the permissible range of recycle ratio as shown below. The following approach in relating G(0) and E ( $ )and selecting R,, can be used. By taking the Laplace transform of eq 1 and by expanding E ( s ) and G(s) in power series of s containing their moments, p n and M,, respectively, as coefficients, the following relationship for the moments is obtained. (124
Mo=po=l
Mn=*-R
+1
R R
n!
+ 1 5 k ! ( n- k ) ! k=l
PkMn-k
( n = 1 , 2 , 3 , 4 ) (12b) This allows the calculation of all moments of G(8) for any R when E ( $ ) ,and therefore its moments, is given. In order for G(0) to be a probability density for Bt(0, m ) the following conditions must be satisfied (Feller, 1971)
Mn>O
( n = 1 , 2 , 3 , 4, . . . )
(13a)
and all the two-by-two and three-by-three determinants (13b) and (13c) must be positive.
k = 0, 1 , 2 , 3
G ( 0 )is always positive up to its first root, 01 = r I 2 f i . If the maximum permissible value of R is small enough, the value of G(0) a t 0 > $1 will be very small and it can be assumed without much error, that G ( $ )= 0 for 0 > 01. Tke above examples demonstrate that the GRM cannot be used to represent different levels of micromixing at a fixed E($).In general at each level of micromixing, Le., at each R the overall RTD changes and therefore its VAR(a) deviates from the original value. It has been shown here that, therefore, meaningful degrees of segregation cannot be calculated if only the variance of the RTD is kept constant. Reactor performance according to the model (Dohan and Weinstein, 1973; Yang et al. 1974) is given by
-
This leads to a very difficult infinite set of conditions which determine the permissible range of R , if any, other than R = 0, for which G(0) is nonnegative everywhere. Usually when investigating the effect of micromixing one wants to study the reactor performance a t various degrees of segregation, J , which depends only on the first four moments (po to p 3 ) of the E curve. Therefore, all that is necessary is to use eq 12 for the first four moments of the G(O),Mo to M3. The upper bound on the permissible recycle ratio is established from eq 13 which for the first 4 moments in particular result in inequalities (14) M2
where for any given rate r(Xb)
- Mi2> 0
(144
M3>0
(14b)
MlMs
and X b ( 0 ) is the reactant concentration in fluid aggregate of age 8. When conversion x is plotted against R as done by Dohan and Weinstein (1973), such a representation contains both the effects of changes in micromixing and of variations in the RTD. Clearly, reactor performance depends to some 386 I d . Eng. Chem., Fundam., Vol. 16, No. 3, 1977
- M22 > 0
(14~)
The lowest value of R resulting from the above inequalities is the maximum permissible recycle ratio Rmm.The minimum allowable degree of segregation J m i n can then be calculated from eq 3. The impulse response of the inner reactor section of the model, G (e), can now be represented by the Laguerre series (Hulburt and Katz, 1964)
~ ( 0 =) b p , a
(!)[
1
(:)I
Table I. Upper Bounds on the Recycle Ratio and Lower Bounds on the Degree of Segregation for the Case of n Tanks in Series
+ i=3 5 kLbLib
where a=M1
1 2
3 4
5
r ( b) ($)"-JM,-;
10
2 (-1); j ! r ( n+ b - j ) ( n- j ) ! n ! r ( n+ b)tn-; L,b(t) = 2 (-1);. ~ ! (-nj ) ! r ( n+ b - j ) knb=
j=O
;=0
This representation of G ( 0 )given by eq 15 is used in eq 10 to evaluate reactor performance. If only two terms of the series are used they depend only on the first four moments Mo to M3 which were evaluated from eq 12. However, if one uses two terms to represent G(0) this may result in G having negative values a t large 0. T o overcome this difficulty additional moments of G can be generated from inequalities (13) and more terms can be added to the Laguerre series until G becomes positive overall or until the negative values are small and delayed until very large times. The addition of these terms will not change the first three moments M I ,M2, M3 of G or F I , PZ, fi3 of E . However, eq 12b will not hold anymore for higher moments of G and E , but it can be safely assumed that the variation in moments higher than third of the RTD will have a negligible effect on conversion. As an example let us consider E ( 0 ) to be the response of n equal-sized stirred tanks in series
E ( 0 ) = n20n-le-nO (17) For n = 1there are no restrictions on R , which can be seen also from the analytical solution of eq 1resulting in the expression for G(0) described by eq 18, and all states of micromixing can be covered. G(0) = ( R
+ l)e-(R+l)'J
(18)
For all other n's the most severe constraint on R arises from eq 14c, which yields the upper bounds on R presented in Table I. From Table I it is apparent that the model is capable of representing all states of micromixing for a single stirred tank but falls short of reaching maximum mixedness condition for the RTD of a sequence of stirred tanks. It is of interest to notice that Jmin values in Table I correspond to the values of J at conditions of sequential mixedness (Tsai et al., 1969;Fan et al., 1970), i.e. to the condition when each tank in series is in maximum mixedness condition. Why this is so is not entirely clear at present. It is not surprising that the condition of maximum mixedness for a general E(0) cannot be reached with this model. The problem is that, with the exception of the E ( 0 ) curve for a simple stirred tank which allows mixing between molecules of all ages, the other RTD's impose a.restriction on the permissible amount of mixing between molecules of various ages and do not permit indiscriminate mixing of molecules of all ages. Yet, this single parameter model allows mixing of fresh feed with a fraction E ( 0 ) of molecules of any age 0 at a ratio of 1:R and therefore tends to violate the restriction imposed by RTD. It becomes impossible to find a G(0) 2 0 everywhere for R,, 1 R L 0. Either G ( 0 )starts having negative portions to compensate for the unpermissible mixing, or if G(0) is maintained positive R turns negative. In order to avoid this the
m
0
0
m
0.464 0.243 0.164 0.124 0.0554
0.143 0.249 0.333 0.399 0.667
0.0275 0.0831 0.1375 0.1913
0.612 0.335 0.235 0.169 ...
...
range of permissible R's has to be constrained further to Rmm > Rma, 2 R 2 0 . The above model modification was successfully employed to calculate the effect of micromixing on the performance of various reactors for the case of substrate inhibited reactions (DudukoviC and Lamba, 1975). Conclusions We have shown here that the generalized recycle model (GRM) is not permissible as a model of micromixing a t a strictly constant residence time distribution E ( 0 ) . However, if E ( 0 ) is allowed to vary in moments higher than the third, k3, than GRM can be used to investigate micromixing effects with the first four moments (goto p a ) of E ( 0 ) kept constant. This allows the calculation of the degree of segregation as a function of micromixing only, since the variance of ages of fluid elements in the system, VAR(a), is kept constant. In spite of the modification of the model presented here, GRM is not the most suitable model for micromixing. Our recent studies (Dudukovif, 1977) indicate that the consecutive segregated maximum mixedness model (CSMM), parallel maximum mixedness model (PSMM) and other two-environment models (Weinstein and Adler, 1967; Nishimura and Matsubara, 1970) are much more acceptable based both on their description of physical reality and on the computational convenience. Nomenclature a = parameter defined by eq 16a, dimensionless b = parameter defined by eq 16b, dimensionless E ( 0 ) = exit age distribution (residence time distribution of the whole system), dimensionless G(0) = impulse response, residence time distribution of the portion of the system which is in segregated flow condition, dimensionless G,(0) = nth convolution of G upon itself, dimensionless J = degree of segregation, dimensionless Jmin = degree of segregation corresponding to the maximum permissible value of R , dimensionless J,, = degree of segregation at maximum mixedness condition, dimensionless k, = constants defined by eq 16d, dimensionless L , = associated Laguerre polynomial, eq 16e, dimensionless M , = nth moment of the G curve, dimensionless p b ( 0 ) = probability density defined by eq 16c, dimensionless r = rate expression, dimensionless R = recycle ratio, dimensionless R,,, = maximum permissible recycle ratio, dimensionless R,, = recycle ratio corresponding to conditions of maximum mixedness, dimensionless t = time, s t = mean residence time, s VAR(a) = variance of ages of the fluid elements in the system (variance of the internal age distribution), dimensionless x = reactant fractional conversion, dimensionless Ind. Eng. Chem., Fundam., Vol. 16,No. 3, 1977
387
= reactant concentration in a fluid aggregate, dimensionless Xf = reactant concentration at the exit, dimensionless Xi = reactant concentration after ixing at point M prior to entrance in the reactor section, dimensionless X O = inlet reactant concentration, dimensionless Xb
Greek Letters a = age of a fluid element in the system dimensionless 0 = t h = residence time, dimensionless
= nth moment of the E curve, dimensionless 3 = variance of the E curve, dimensionless z = variable of integration, dimensionless
Literature Cited Danckwerts,P. V., Chem. Eng. Sci., 8,93 (1958). Dohen. L. A., Weinstein, H., Ind. Eng. Chem., Fundem., 12,64 (1973). Dudukovlc, M. P.. AIChE J., in press (1977a). DbdukoviC. M. P., Chem. Eng. Sci, in press (1977). DwJukoviC,M. P., Lamba. H. S., 68ih A C E Annual Meeting, Los Angeies, Nov. 1475,Microfiche No.35,Paper No. 55c, 1975. Fan. L. T., Tsai, B. I., Erickson, L. E., AIChEJ., 17(3),689 (1971).
Feller. W., “An introduction to Probability Theory and its Applications.” 2nd ed, Wiley. New York. N.Y., 1971. Fu, B. J.. Weinstein, H.. Bernstein, B., Shaffer, A. 8..lnd. Eng. Chem., Process Des. Dev., 10, 501 (1971). Gibilaro. L. G., Chem. Eng. Sci., 26,299 (1971). Olliespie, 8..Carberry, J. J.. Ind. Eng. Chem., Fundarn., 5, 164 (1966). Huibvt. H. M., Katz, S., Chem. Eng. Sci., 19, 555 (1964). Nishimwa. Y.. Matsubara. M., Chem. Eng. Sci., 25, 1785 (1970). Rippin, D. W., Ind. Eng. Chem., Fundam., 6, 488 (1967). Tsai. B. I., Erickson, L. E., Fan, L. T., Biotechnol. Bioeng., 11(2), 181 (1969). Weinstein. H., Adier, R. J.. Chem. €nu. Sci., 22. 64 (1967). Y a n T. ~ ~C.,Weinsteln, H., Bemsiein, 6,Roc. &kit. Syt$ &em. k t . Eng.,
133. 532 (1974). Zwietering, Th. N., Chem. Eng. Sci., 11, l(1959).
Chemical Reaction Engineering Laboratory Department of Chemical Engineering Washington University St. Louis, Missouri 63130
Milorad P. DudukoviC!
Received for review November 17, 1976 Accepted April 22,1977
An Efficient Numerical Solution of First-Order, Hyperbolic Differential Equations with Split Boundary Conditions
An efficient numerical method that converts a boundary value problem to an initial value problem is given for a pair or pairs of first-order hyperbolic partial differential equations. This method also enables one to convert the nonrectangular grid of characteristic lines to a rectangular grid, which is most efficient on a digital computer.
Consider a pair of first-order hyperbolic partial differential equations
dz _ -1 dt
(3)
and the solutions are given by with boundary and initial conditions: x l ( 0 , t ) = x l o ( t ) ;x p ( 1 , t ) = x z d t ) ; Xl(Z,O) = x1M)(z);xp(z,O) = xzOO(z) (0 5 2 I 1). Under appropriate assumptions, countercurrent and plug-flow processes of tubular reactors, gas absorbers, and heat exchangers can be represented by the above equations. The forcing function, u,can be either a function of z and t or a function oft only. For instance, u would be equal to zero for an isothermal reactor. Even though the equations are treated as scalar equations for the purpose of illustrating the numerical method, they can be treated as vector equations. For instance, a countercurrent reactor for the successive substitutive halogenation of hydrocarbons would require a vectorial representation. The theory for a pair of first-order hyperbolic partial differential equations based on the canonical system is well established and the solution of an initial-value problem is rather straightforward (Forsythe and Wasow, 1960). However, efficient implementation of the canonical system with split boundary conditions on a digital computer is difficult. The characteristics of the equations are: 1 Address correspondence to the author at the Department of Chemical Engineering, University of Florida, Gainesville, Fla. 32611.
388 Ind. Eng. Chem., Fundam., Vol. 16,No. 3,1977
(5) along the characteristic line of eq 3 and
dong the characteristic line of eq 4. A t each mesh point where two characteristic lines intersect, eq 5 and 6 may be solved simultaneously to obtain the solutions. Integration of a hyperbolic system along the characteristics is equivalent to following a solution as it progresses through a domain. Figure 1shows how the solution progresses. Any base line, which is not characteristic, may be chosen as a starting line. For instance, from the known information at points A and B, and B and C, values of x 1 and x p at points E and D are calculated using the finite-difference version of eq 5 and eq 6. A similar procedure provides solutions a t each mesh point on the dotted line, H-H, which is then used as a new base line for further calculation. However, there is a major obstacle in the numerical scheme given. To obtain the solution at point K, information on x l ( 1 , t ) at point J is necessary, which is not known because of split boundary conditions. This means a trial-and-error iteration in the z direction to obtain the solution at point K. A similar
