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Biotechnol. Prog. 1990, 6,98-103
Multimolecular Process in a Packed-Bed Immobilized Enzyme Reactor: Numerical Simulation and Back-Mixing Effects S. Guzy,? G. M. Saidel,’ and N. Lotan*9t Department of Biomedical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel, and Department of Biomedical Engineering, Case Western Reserve University, Cleveland, Ohio 44106
I n a previous report, we presented a new analytical model describing the performance of a packed-bed catalytic unit, where the reaction between two cosubstrates is catalyzed by an enzyme immobilized on a porous carrier. T h e model explicitly takes into account t h e changes in concentrations of both cosubstrates along the reactor, as well as t h e hydrodynamic regimen (i.e., back-mixing) prevailing in the packed bed. I n t h e present report, and on t h e basis of the procedures developed, we present a detailed analysis of t h e performance of t h e reactor. With numerical simulations, the effects of internal diffusion limitations, the depth of the pores, t h e substrates’ concentration in the feed, a n d kinetic parameters are evaluated. Particular attention is also given here t o t h e back-mixing effects prevailing in t h e reactor. An experimental procedure for assessing their extent is described.
1. Introduction In a previous report ( I ) , we have developed a new analytical model for describing the performance of a packedbed reactor, in which the enzyme is immobilized on a porous carrier and operates with two cosubstrates. Instead of using the convective-dispersive model in the direction of mean flow, we used a simpler model that consists of a series of n identical stages. Two approaches were considered, depending on whether, for each stage, the input (approach I) or output (approach 11) concentration of the substrates was taken to represent the substrate concentration in the bulk phase. The model resulted in a two-point boundary-value problem of coupled, nonlinear differential equations, and these were solved with use of a well-documented algorithm based on the Newton method (2). Back-mixing phenomena in packed-bed enzymic reactors were explicitly considered before for singlesubstrate reactions (3). These effects were also taken into account in more recent analyses but only in an indirect manner, as part of consideration of overall dispersion phenomena and also for single-substrate reactions (4-9).
Axial dispersion was also analyzed for a related system, namely, a plate-pulsed countercurrent liquidliquid extraction column (IO,11). The present report gives a detailed analysis of the characteristics of the model, and this is carried out by numerical simulation in terms of actual physical parameters. In addition, an experimental procedure is described for determination of n. Finally, comparison between predicted reactor performance and results of related experimental studies is made.
2. Results and Numerical Simulation Examples of results of numerical simulation are illustrated in Figures 1-10, and they present the performance of continuously operated reactors of various types. Technion-Israel Institute of Technology.
* Case Western Reserve University.
* Author to whom correspondence should be addressed.
All calculations were performed with representative values of the operational parameters, as indicated in Table I. In these calculations, we consider the situation where the concentration of substrate 1 is higher than or equal to that of substrate 2. However, in no case is substrate 1 so much in excess over substrate 2 that the concentration of the former could be considered as constant when the reaction takes place. Therefore, all calculations explicitly consider that the concentration of both substrates changes during the reaction. For convenience, however, results are presented in terms of the conversion of substrate 2 only. In Figures 1-6, calculations were performed for two values of n: the case n = 1represents the perfectly mixed reactor; the case n = 10 represents the packed-bed reactor with a behavior approaching the plug flow regimen, the latter being characterized by n = m. For both values of n, approaches I and I1 were used. The performance of the reactors was studied with use of as variables, the following parameters: input concentration of substrate 2, i.e., S2,0(Figure 1);depth of pores, i.e., yo,which is related to the diameter of the beads (Figure 2); maximal reaction rate, i.e., V,, representing the catalytic constant or total enzyme load in the reactor (Figure 3); bulk diffusion coefficient of substrate 1, i.e., D, (Figure 4); Michaelis constant for substrate 1, i.e., (Figure 5); and volumetric flow rate through the reactor, i.e., Q (Figure 6). It can be seen that, in all cases considered (i.e., Figures 1-6), (a) approach I and approach I1 give significantly different predictions for the perfectly mixed reactor ( n = 1 type) (this difference is much smaller for the n = 10 reactor) and (b) the difference between predictions of approaches I and I1 increases with increasing conversion. It can also be seen that the conversion per pass increases with increasing either V , or D, (Figures 3 and 4, respectively) and decreases with increasing yo,Km,l,and Q (Figures 2, 5 , and 6, respectively). The conversion also decreases with increasing S2,0(Figure l),as it can be seen from the fact that the product concentration curves depart more and more from the “100% conversion” line. Figure 7 shows the conversion of substrate 2 as a func-
8756-7938/90/3006-0098$02.50/0 0 1990 American Chemical Society and American Institute of Chemical Engineers
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Biotechnol. Prog., 1990,Vol. 6, No. 2
Table I. Numerical Values of the Operational Parameters value
parameter depth of pores (cm) bulk diffusion coeff of substrate 1 (cm2/s) bulk diffusion coeff of substrate 2 (cm2/s) feed concn of substrate 1 (mM) feed concn of substrate 2 (mM) packing density porosity of beads maximal reaction rate (mM/s) Michaelis const for substrate 1 (mM) Michaelis const for substrate Km.2 2 (mM) V total vol of the reactor (mL) volumetric flow rate (mL/h) 8 n no. of stages
-
15 r
N
10
1
in the in the analysis numerical of reactor simulation performance 0.1 0.05 10-5 10-5 10-5
10-5
0.25 0.06 0.6 0.95 4.0 1.5
0.25 0.06 0.6 0.95 0.8 1.5
2.0
2.0
2.5 290 1-10
2.5 150-450 4-7
3. Comparison with Results of Experimental Studies 3.1. Determination o f n . 3.1.1. Theoretical Considerations. The number of stages, n, representing the reactor is dependent only on the transport processes taking place in the bulk-phase solution. For a particular reactor, the value of n can be determined from the system's response to a pulse-applied nonreacting and nonpenetrating tracer. Thus, for a series of identical and perfectly mixed stages, the number of stages, n, and the variance of the response curve, a, are related (12) by eq 1, where T is the mean residence time in the reactor.
r
/' ;
with shallow pores (depth = 0.1 mm), the reaction occurs a t every position into the pores. It should be pointed out that in the present report we define the position along the pore in a relative manner, starting from the pore entrance. This definition differs from the one used in our pervious report (1). There, the position was measured from the end of the pore toward its entrance, and this was done for the convenience of the mathematical analysis. Figure 10 shows the calculated conversion of substrate 2, as it occurs along the reactor. Simulations were performed for two values of n ( n = 5 and 10) and with approach I1 only. The stepped lines represent the results of numerical simulation, one step for each elementary stage. On the other hand, the continuous curves describe the actual behavior of the reactor.
/
n = T2/a2 0
I
I
1
'
I
I
I
10
20
30
0
10
20
30
I n p u t conc
of
Substrate 2 , x102
(1)
The value of T is calculated by eq 2, where V = the total volume of the reactor and p = the packing density.
(mM)
Figure 1. Simulated performance of an enzymic reactor, as a function of input concentration of substrate 2, namely S2,,.Cal-
culations were performed for n = l and 10, with approaches I (- -) and I1 (-). T h e dotted line represents the "100% conversion" line. (e-)
tion of n, as calculated by both approaches I and 11. Calculations were performed for different values of the flow rate, Q, while all other parameters are as indicated in Table I. As a general feature, we observe convergence between results calculated by the two approaches when n increases. Moreover, the number of stages, n, necessary to essentially reach the above convergence decreases with increasing Q. In order to emphasize the effect of n on the performance of the reactor, results from Figure 7 are replotted in Figure 8, this time in terms of the conversion of substrate 2 as a function of the volumetric flow rate a t different values of n, only with approach 11. It can be seen that large changes in conversion occur when 250 mL/h < Q < 500 m L / h and much smaller changes when Q > 500 m L / h and, for a given Q, most of the changes in the conversion are between n = 1 and 5, while for the cases of n > 5, there is essentially no difference in the conversion versus Q curves. Figure 9 presents the simulated concentration profile of substrate 2 in the pore phase of the first elementary stage of the reactor. Calculations were made with approach I1 only. It can be seen that, depending on the particular values of the parameters involved, different concentration profiles are obtained. Thus, for example, when deep pores are used (depth = 1 mm), the reaction occurs essentially a t the entrance of the pores. On the other hand,
The variance is calculated ( 3 ) by eq 3, where p (i = 0, 1, and 2) is the ith-order moment of the pulse response curves, such as the ones shown in Figure 11. The values a 2 = Pq l ! 5 )
2
(3) of p i are calculated from data in Figure 11 and by eq 4, where t and C are the coordinates of the points on the pulse response curve, and t = 0 corresponds to the pulse injection time.
(4) It should be pointed out that, under the conditions employed, the concentration C is directly proportional to the measured optical density, OD, shown in Figure 11. Also, in eq 3, only ratios of pi are involved. Therefore, in eq 4, the quantity C can be replaced by the measured OD. For the actual calculation of the performance of the reactor, the value of n thus obtained is to be rounded to the closest integer. This is so in view of the conceptual characteristics that n involves ( I ) , namely, the number of elementary stages. 3.1.2. Experimental Details. a. The System. The experimental system for the determination of n is similar to the one used for carrying out the enzymic process. The only difference is that, in this case, the enzymic reac-
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Figure 2. Simulated performance of an enzymic reactor, as a function of the depth of the pores, yo. Calculations were performed for n = 1 and 10, with approaches I (- -) and I1 (-).
0
5
10
15 0
Maximal reaction r o t e
5
10
15
(mM/sec)
Figure 3. Simulated performance of an enzymic reactor, as a function of the maximal reaction rate, V . Calculations were performed for n = 1 and 10, with approactes I (- -) and I1 (--).
tor was replaced with a column geometrically identical with the former but packed with noncatalytic beads. The protein tyroglobulin was used as a tracer, since its high mollecular weight ( M , = 669 000) minimizes to a large extent the band-spreading effects produced by diffusion of the protein in the bulk phase and also prevents penetration in the pore phase. The experiments were carried out a t various flow rates Q. For calculation of T ~ only , the interparticle volume that is actually available to the tracer is considered. b. Determination of n. At time t = 0, a pulse of protein (about 0.1 mL of a solution of 2 mg/mL) was injected into the column, and the absorbance of the output stream was monitored a t 260 nm. The procedure was repeated using various volumetric flow rates. In all cases, the injection time was about 2 s. It should be pointed out that the system considered is linear with respect to the tracer concentration. Therefore, it is not necessary that all these experiments be carried out with pulses containing identical amounts of protein. Representative results are presented in Figure 11. From these data, the corresponding number of stages was cal-
culated as described above. The actual values of n thus obtained, as well as the rounded values, are collected in Table 11. 3.2. Analysis of Reactor Performance. As an example, we studied t h e rection between 1-chloro-2,4dinitrobenzene (CDNB) and glutathione (GSH), as carried out by the enzyme glutathione S-transferase (GST) (13). The experimental values of the operational parameters are shown in Table I. Using the model developed ( I ) , we compare the calculated characteristics of the reactor with the ones determined experimentally a t two initial concentrations of the haloaromatic substrate (see Figure 12). In both cases, no good agreement was reached between calculated and experimentally determined performance. This aspect is analyzed in the Discussion. 4. Discussion
4.1. Analytical Model. The analytical model used for describing the performance of the reactor, as well as the procedures for numerical solution of the differential equations involved, is extensively described in our previous report ( I ) . Here, we would like only to point out that approaches I and I1 differ in their basic assumptions and, while approach I1 is the valid one for all types of reactors, approach I is valid for plug flow conditions only. However, we rely here on both approaches I and I1 concomitantly and use this procedure as a useful tool for assessing the actual mode of operation of the reactor. As expected from the above, approaches I and I1 should, in principle, give different results, and this is clearly shown in Figures 1-7. The discrepancy is more pronounced for the n = 1 (i.e., the perfectly mixed) type reactor since, in this case, the basic assumptions of approaches I and I1 differ most. Moreover, it is for the same reason that, for all values of n, the above discrepancy is also pronounced when the expected conversion of substrates is high. 4.2. Effect of Operation Parameters. The expected conversion in the reactor decreases with increasing Szs0 (Figure 1). This is due to the fact that, in enzyme-catalyzed processes, the reaction rate is proportional to the substrate concentration only a t very low values of the latter and is less and less so as this concentration increases.
Biotechnol. Prog., 1990, Vol. 6,No. 2
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i
Diffusion c o e f f i c i e n t o f
0
0.5
Substrate 1
,
x
lo5
1.5
1
2
(cm2/sec)
Figure 4. Simulated performance of an enzymic reactor, as a function of the diffusion coefficient of substrate 1. Calculations were performed for n = 1 and 10, with approaches I (- -) and I1 (-). 100
2o
r
r
t
t
100 r
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Flow r a t e 4
Krn,, ( m M ) Figure 5. Simulated performance of an enzymic reactor, as a function of the Michaelis constant for substrate 1. Calculations were performed for n = 1 and 10, with approaches I (- -)
and I1 (-).
As a matter of fact, this is the outcome of the saturationtype mechanism governing these processes, as expressed in the Michaelis-Menten formalism. The performance of the reactor is seriously affected when the depth of pores is increased, i.e., when large catalytic particles are used (Figure 2). This effect is the expression of diffusion limitations of both substrates. 4.3. Effect o f n . Figure 7 emphasizes the effect that the reactor-type characteristics have on the conversion achieved. Thus, as the throughput rate increases (i.e., the mean residence time decreases), the behaviors of the plug flow- and perfectly mixed-type reactors approach one another. Accordingly, results of calculations made with approaches I and I1 are more and more similar. Figure 8 provides a broader understanding on how various operational parameters affect the performance of the reactor, and, even more valuable, it provides the key to procedures for improving it. The main effect of decreasing conversion with increasing flow rate is, obviously, due to shortening of the residence time in the reactor. How-
150
0 (ml/hr
300
I
Figure 6. Simulated performance of an enzymic reactor, as a function of the volumetric flow rate, Q. Calculations were performed for n = 1 and 10, with approaches I (- -) and I1 (-).
ever, it can be also seen that, for any given flow rate, conversion increases with increasing n, and this is so mostly for 1. C n C 5. Therefore, in order to increase the n value of the reactor while keeping all other operational parameters constant, we rely on the understanding of the meaning of n, as well as of the various factors affecting it (see section 3.1). Accordingly, we note that, for constant Q and V , decreasing the mean linear velocity of flow in the reactor will bring about a decreased hydrodynamic interaction between the fluid and the packed bed. In turn, this leads to decreased back-mixing, Le., to increased n. In practice, this is achieved by using a shorter reactor with a correspondingly larger cross section. However, implementation of this approach can be extended only to the limit where additional factors, detrimental to the reactor performance (e.g., channeling in the shallow bed packing), are operative. 4.4. Determination o f n . Figure 11shows the results of pulse tracer experiments. From these data, the values of n, a t various volumetric flow rates, are calculated. The results are collected in Table 11. We first note that n is not a monotonic function of Q. We understand this in view of the fact that there are two main processes that affect n, namely, axial diffusion and hydrodynamic mix-
Biotechnol. Prog., 1990,Vol. 6,No. 2
102 80 r
\ -
N
\
- '006
Flow Rate, 0
\
w
\
Depth o f pores (mm)
t 0
Relative distance from pore entrance 5
IO
Number of s t a g e s ,
15
20
n
Figure 7. Simulated performance of an enzymic reactor, as a function of the number of stages in the reactor and for chosen values of the flow rate, Q. Calculations were performed with approaches I (- -) and I1 (-).
:I
Figure 9. Simulated profiles for the concentration of substrate 2 along the pores of the catalytic pellet, in the first elementary stage of the reactor. Calculations were performed for n = 10 and V,,, = 1 mM/s and for the indicated values of the depth of the pores. The results shown are normalized, relative to the concentration of substrate 2 in the reactor feed.
I-
N
50
201
; I , ,
,
,
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,
F r a c t i o n a l reoc'3r
n 10 5
t L TTL
c
I
I
400
tw 2000
Flow r o t e ( m l / h r )
Figure 8. Simulated performance of an enzymic reactor, as a function of volumetric flow rate, for selected values of n. Cal-
culations were performed with approach I1 only.
ing. However, the effect of changing Q on the two processes is opposite. Thus, by increasing Q, we do not allow time for axial diffusion to be operative. On the other hand, the same increase in Q enhances the hydrodynamic mixing. T h e end result will express the combined effects of the two processes. It is on this ground that, as indicated above, one can improve the performance of a reactor by taking advantage of understanding its flow characteristics. It should be pointed out that, in the data obtained at the highest flow rates used, a possible artifact may be involved. This is related to the time actually required for applying the pulse sample onto the column. Thus, the 2-s injection time is not negligible compared to the 12-s average residence time achieved at Q = 240 mL/h, and this may affect the calculated value of n. However,
, I
Figure 10. Simulated profile describing conversion of substrate 2 along the reactor. Calculations were performed for V = 1 mM/s and for two values of n: n = 5 (light and dark soli2 lines) and n = 10 (light and dark broken lines). The stepped lines represent results of numerical simulation, while the curves represent the continuous-type behavior of the reactor. Flow r a t e ( m l / h r )
E
0
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A
O2t 2 00
,
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,
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40
60
80
0
20
40
60
80
n
-
0
002
c
100
120
140
6
E l u l i o n time ( s e t )
Figure 11. Results of tracer pulse experiments, as obtained at
the indicated volumetric flow rates. For more experimental details, see section 3.1.2.
as the flow rate is increased (corresponding to a mean residence time of 24-480 s), this effect is less and less important.
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Biotechnol. Prog., 1990,Vol. 6, No. 2
Table 11. Determination of the Parameter na
Q,mL/h
7,s
240
12 24 48 480
120 60 6
2
exptl 40
160 217
V
n rounded value
3.6 3.6
4 4 10 1
10.5 0.7
330000 a Symbols: Q = flow rate; 7 = mean residence time; 2 = variance; and n = number of stages.
vm
Yo Fi P
2 7
--
0
100
200 0
100
200
Flow r a t a I m l / h r )
Figure 12. Performance of an enzymic reactor containing immobilized glutathione S-transferase, where the reaction between CDNB and GSH takes place. The results of theoretical predic-) are compared to those obtained experimentally tions (- -0(-x-), at the indicated initial concentrations of CDNB (Le.,S2,,J.
4.5. Analysis of Reactor Performance. A comparison between the calculated performance of an enzymic reactor and experimental results is shown in Figure 12. T h e latter were obtained with immobilized GST as the catalyst and CDNB and GSH as substrates (13). It can be seen that calculations indeed predict the general trend of reactor characteristics. However, no full agreement is seen between experimental and calculated data. This is due to the fact that our present model accounts for the enzymic reaction only, while in the actual process considered, additional factors, such as inhibition of the enzyme by the reaction product (13) are also operative. A better agreement was noted when the inhibitory effect of the product was also included in the analytical model; these results will be reported in a subsequent communication.
5. Conclusions From this study, it can be concluded that from the proposed model, the performance of the reactor in terms of operational parameters can be assessed and a deeper insight into its functioning can be reached and the extent of back-mixing prevailing in the reactor can be evaluated by a simple experimental procedure.
Notation C
concentration of the tracer in the pulse experiment (mM), replaced by the measured optical density (OD) in Figure 11 and defined in eq 4 internal (pore) diffusion coefficient for subD,,D, strates 1 and 2, respectively (cm2/s) Michaelis constants for substrates 1and 2, respecK , ,, L , Z tively (mM) n total number of elementary stages in the reactor volumetric flow rate throughout the reactor (mL/ Q h) Sl,o,Sz,oconcentration of substrates 1 and 2, respectively, in the reactor feed (mM) elution time (s), defined in eq 4 t
total volume of the reactor (mL) maximal reaction rate in terms of volumetric units (mM/s) depth of pores (cm) ith-order moment of the pulse response curves (i = 0, 1, and 2) volumetric packing density of catalytic particles (dimensionless) variance of pulse response curves (s'), defined in eq 3 mean residence time in the reactor (s),defined in eq 2 porosity of the catalytic particles (dimensionless)
Acknowledgment Partial support for this research was provided by the
M. R. Oxman, M.D., Memorial Fellowship (to S.G.), as well as by the MEP Group of the Women's Division of the American Technion Society, New York, NY, and by the Pearl S. Milch Fund for Biomedical Engineering. G.M.S. was a Henry Goldberg Visiting Professor of Biomedical Engineering. All are gratefully acknowledged. Literature Cited (1) Guzy, S.; Saidel, G. M.; Lotan, N. Packed-bed immobilized
enzyme reactors for complex processes: Modelling of twosubstrate processes. Bioprocess Eng. 1989,4, 239. (2) Pereyra, V.; PASVA 3: An adaptive finite difference Fortran program for first order non-linear ordinary boundary problems. Lect. Notes-Comput. Sci. 1978, 76, 67. (3) Kobayashi, T.; Moo-Young,M. Backmixing and mass transfer in the design of immobilized enzyme reactors. Biotechnol. Bioeng. 1971, 13, 893. (4) Morf, W. E. Theoretical evaluation of the performance of enzyme electrodes and enzyme reactors. Mikrochim. Acta 1980,2, 317. (5) Lee, G. K.; Lesch, R. A,; Reilly, P. J. Estimation of intrinsic kinetic constants for pore diffusion-limited immobilized enzyme reactions. Biotechnol. Bioeng. 1981, 232, 487. (6) Park, J. M.; Choi, C. Y.; Seong, B. L.; Han, M. H. The production of 6-aminopenicillanicacid by multistage tubular reactor packed with immobilized penicillin amidase. Biotechnol. Bioeng. 1982, 24, 1623. (7) Patwardhan, V. S.; Karanth, N. G. Film diffusional influences on the kinetic parameters in packed-bed immobilized enzyme reactors. Biotechnol. Bioeng. 1982, 24, 763. (8) Moser, A. Imperfectly mixed bioreactor systems. In Comprehensive Biotechnology; Moo-Young, M., Ed.; Pergamon Press: Oxford, 1985; Vol. 2, pp 77-98. (9) Godia, F.; Casa, G.; Sola, G. Mathematical modelization of a packed-bed reactor performance with immobilized yeast for ethanol fermentation. Biotechnol. Bioeng. 1987, 30, 836. (10) Prochazka,J.; Landau, J.; Souhrada, F.; Heybenger, A. Reciprocating plate extraction column. Br. Chem. Eng. 1971, 16, 42. (11)Srinikethan, G.; Prabhakar, A.; Varma, Y. B. G. Axial dispersion in plate-pulsed columns. Bioprocess Eng. 1987, 2,
161. (12) Lauer, H. H.; Rozing, G. P. Selection of optimum condi-
tions in HPLC: determination of external band spreading in LC instruments. Chromatographia 1981,14(11),641. (13)Grynspan, E. Enzymic reactors for blood purification. MSc. Thesis, Department of Biomedical Engineering, TechnionIIT, Haifa, Israel, 1986. Accepted September 11, 1989. Registry No. CDNB, 97-00-7; GSH, 70-18-8; GST, 5081237-8.
