Abstract An analytical expression is derived for the optimal design of a series of CSTR's performing reversible. Michaelis-Menten kinetics in the liquid phase.
Bioprocess Engineering 15 (1996) 257-264 9 Springer-Verlag 1996
Optimal design for CSTR's in series using reversible Michaelis - Menten reactions I.M. Abu-Reesh
257 Abstract An analytical expression is derived for the optimal design of a series of CSTR's performing reversible Michaelis-Menten kinetics in the liquid phase. The optimal design is based on minimum overall volume of N reactors in series required to achieve a certain degree of substrate conversion. The reversible Michaelis-Menten equation is also able to explain competitive product inhibition and irreversible Michaelis-Menten kinetics. The reversible Michaelis-Menten kinetics covers three types of enzymatic reactions depending on the values of the rate constant for the forward (k,) and reverse (k;) reactions. An optimum design is obtained in the three cases of Ks=Kp, K,>Kp and Ks Km the volume of individual reactor becomes progressively larger with increasing the reactor number (Table 4). While in the case of Ks < g p the volume of reactor becomes smaller (Table 5). This result can be explained by inspecting the variation of reaction rate with reduced substrate concentration g in the three cases: 1) In the case of Ks = Kp, the reaction is considered a pseudofirst order. The reaction rate is a linear function of the reduced substrate concentration which results in equal-sized reactors. 2) In the case of K,>Kp, the rate-concentration curve is concave (reaction order > 1). Decreasing Sresults in a decrease
Table 2. The optimum intermediate substrate concentrations represented as (:r c~e)t(C~o--ar in cascades of N CSTR's performing reversible Michaelis-Menten kinetics assuming substrate conversion of 90% of the equilibrium. i
1 1 2 3 4 5 6 7 8
0.100 0,316 0,464 0.562 0.631 0.681 0.720 0,750
9
0.774
10
0.794
2
3
4
5
6
7
8
9
10
0.100 0.215 0.316 0.398 0.464 0.518 0.562 0.599 0.631
0.100 0.178 0.251 0.316 0.373 0.422 0.464 0.501
0.100 0.158 0.215 0.268 0.316 0,359 0,398
0.100 0.147 0.i93 0.237 0.278 0.316
0,100 0.139 0,178 0,215 0,251
0.100 0,133 0,167 0.200
0.100 0.129 0.518
0,100 0.126
0.i00
261
Bioprocess Engineering 15 (1996) Table 4. The optimum holding times in cascades of N CSTR's performing reversible Michaelis Menten kinetics in the case of K,= 1.5 Kp assuming substrate conversion of 90% of the equilibrium.
262
zi(h) N
1
2
3
1 2 3 4 5 6 7 8 9 10
3.232 0.748 0.389 0.258 0.191 0.151 0.125 0.107 0.093 0.082
0.776 0.406 0.269 0.199 0.158 0.130 0.110 0.096 0.085
4
0.415 0.276 0.205 0.162 0.133 0.113 0.098 0.087
5
0.279 0.208 0.165 0.136 0.115 0.100 0.088
6
0.210 0.167 0.138 0.117 0.101 0.090
7
0.168 0.139 0.118 0.102 0.091
8
0.140 0.119 0.103 0.091
9
0.120 0.104 0.092
0.105 0.093
10
qot(h)
"ctotlrp
0.093
3.232 1.524 1.210 1.082 1.013 0.970 0.941 0.920 0.903 0.891
4.111 1.939 1.539 1.376 1.289 1.234 1.197 1.170 1.149 1.133
~p= 0.786 h
Table 5. The optimum holding times in cascades of N CSTR's performing reversible Michaelis Menten kinetics in the case of Kp = 1.5 1(, assuming substrate conversion of 90% of the equilibrium ri(h) N
1
2
1 2 3 4 5
0.953 0.237 0.130 0.088 0.067
3
4
5
Tto t (h)
. . . . . . . .
:
:
: : :
i
i
i
i
i
:
: ::
i
KS=I.5Kp
: ::
"ctot/'cp ,,10
0.229 0.125 0.085 0.065
0.122 0.083 0.064
0.082 0.063
0.062
0.953 0.466 0.376 0.340 0.520
3.727 1.823 1.473 1.330 1.253
~6 :N2-2 .... N = 3
1
rp = 0.256 h
......
.....
:
:
::::
i::
: i
"
.......... i Ks=Kp
:
:
:
: ::
........
: :
N=oo
0,01
....
:
!::!
}
0.1 1-X/Xe
1
Fig. 2. Comparison of the volume of optimally-designed N CSTR's in series with the volume of plug flow reactor over a range of substrate conversion for reversible Michaelis-Menten kinetics in the case of /(,=1.5 Kp
:.
~N=I ~1o
e
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
" ~ :
...... i : : : :N=I
"6
I
i
: : i i i
:: !!
Kp-l.5Ksi i
' i i !
o10
~a
1 0.01
0.1 1 -X/Xe
"6
N=a
Fig. 1. Comparison of the volume of optimally-designed N CSTR's in series with the volume of plug flow reactor over a range of substrate conversion for reversible Michaeiis-Menten kinetics in the case of
K,=Kp
in the numerator and an increase in the denominator of the rate equation. This decreases the reaction rate more rapidly. T h u s i n c r e a s i n g t h e size o f d o w n s t r e a m r e a c t o r s . 3) I n t h e c a s e o f K , < K p , t h e r a t e - c o n c e n t r a t i o n c u r v e is c o n v e x a n d t h e r e a c t i o n o r d e r is b e t w e e n z e r o a n d first w i t h r e s p e c t to S. T h e v o l u m e o f r e a c t o r b e c o m e s s m a l l e r w i t h increasing the reactor number. This result agrees with that of irreversible Michaelis-Menten kinetics reported in the litera-
N=~ i
....
0.01
!
iii~i
.......... . . . . i iiiil ! i
! :: ii i ilii
0.1 1 -X/Xe
Fig. 3. Comparison of the volume of optimally-designed N CSTR's in series with the volume of plug flow reactor over a range of substrate conversion for reversible Michaelis Menten kinetics in the case of
G=I.SK, t u r e [1]. S i m i l a r r e s u l t s in t h e t h r e e c a s e s a r e g i v e n b y S z e p e a n d L e v e n s p i e l [18] f o r r e a c t i o n o f o r d e r n u s i n g t w o m i x e d reactors in series.
I.M. Abu-Reesh: Optimization of N CSTR's in series Table 6. The intermediate substrate concentrations represented as ( e ~ - - ~ ) / ( c G - G ) that corresponds to equal-sized mixed reactors assuming K,= 1.5 Kp and substrate conversion of 90% of the equilibrium
i
1
2
0.100 0.312 0.455 0.550 0.618 0.667 0.705 0.736 0.760 0.781
0.100 0.212 0.309 0.388 0.451 0.503 0.546 0.583 0.614
3
4
5
6
7
8
9
Trot
10
zi(h)
"c,ot(h)
-~p
0.100
3.232 0.763 0.403 0.271 0.203 0.162 0.134 0.115 0.100 0.089
3.232 1.527 1.210 1.082 1.014 0.971 1.941 0.920 0.904 0.891
4.111 1.942 1.540 1.377 1.289 1.235 1.197 1.I70 1.150 1.133
N 1 2 3 4 5 6 7 8 9 10
0.100 0.175 0.245 0.307 0.362 0.408 0.449 0.485
0.100 0.156 0.211 0.261 0.307 0.348 0.384
0.100 0.145 0.189 0.231 0.270 0.306
0.100 0.137 0.174 0.210 0.244
0.100 0.132 0.164 0.195
0.100 0.128 0.156
0.100 0.125
rp = 0.786 h
Table 7. The intermediate substrate concentrations represented as Kp = 1.5 K, and substrate conversion of 90% of the equilibrium
i
1
2
3
4
5
(~i--:Xe)/(O:o--~:,)that
6
corresponds to equal-sized mixed reactors in the case of
7
8
9
10
r~(h)
Ztot(h)
Trot
--
~p
N 1 2 3 4 5 6 7 8 9 10
zp=
0.100 0.320 0.473 0.573 0.643 0.694 0.732 0.762 0.786 0.806
0.100 0.218 0.323 0.408 0.476 0.531
0.100 0.180 0.257 0.325 0.383 0.434 0.478 0.516
0.577 0.614 0.646
0.100 0.160 0.220 0.275 0.325 0.370 0.410
0.100 0.148 0.197 0.243 0.286 0.326
0.100 0.140 0.181 0.221 0.258
0.100 0.135 0.170 0.204
0.100 0.130 0.161
0.100 0.127
0.100
0.953 0.233 0.125 0.085 0.064 0.051 0.043 0.037 0.032 0.028
0.953
0.466 0.376 0.340 0.320 0.308 0.300 0.294 0.289 0.285
3.727 1.823 1.473 1.330 1.253 1.205 1.173 1.149 1.131 1.117
0.2557 h
Table 8. The optimum holding times for cascades o f N CSTR's performing reversible Michaelis-Menten kinetics in the case of Kp = 1.5 K, assuming enzyme deactivation and substrate conversion of 90% of the equilibrium
Table 9. Percent reduction in total volume using m i n i m u m volume design as compared to equal-volume reactors for two, three and four reactors in series
zi(h)
Relative conversion, Xr (% of equilibrium)
N= 2
90 99 99.9
0.163 1.128 11.259
1
2
3
4
5
.Ctot(h) Ztot -c7
N 1 2 3 4 5
0.983 0.239 0.130 0.089 0.067
0.232 0.126 0.086 0.065
0.124 0.084 0.064
0.083 0.063
0.983 0.472 0.380 0.342 0.062 0.322
3.830 1.837 1.479 1.333 1.256
0
.
'/o reduction=
Teq - - "['opt
x 100 "Ceq N= 3 0.148 0.875 8.060
N=4 0.140 0.762 6.589
zv = 0.2567 h
I n t h e c a s e s o f Ks > Kp a n d Ks < Kp w h e n t h e o p t i m u m d e s i g n is r e a c t o r s o f v a r i a b l e size, t h e l a r g e s t d i f f e r e n c e i n v o l u m e b e t w e e n t w o s u b s e q u e n t r e a c t o r s is b e t w e e n t h e first two. T h i s difference becomes smaller in downstream reactors. Using reversible Michaelis-Menten equation the average r e a c t i o n r a t e is h i g h e r i n p l u g - f l o w r e a c t o r c o m p a r e d to C S T R
o f t h e s a m e size. T h e p l u g - f l o w r e a c t o r v o l u m e is s m a l l e r t h a n t h e C S T R r e q u i r e d to a c h i e v e t h e s a m e d e g r e e o f c o n v e r s i o n . Figs. 1, 2 a n d 3 s h o w t h e p e r f o r m a n c e o f N C S T R ' s i n s e r i e s c o m p a r e d to p l u g - f l o w r e a c t o r o v e r a r a n g e o f s u b s t r a t e c o n v e r s i o n in t h e t h r e e c a s e s s t u d i e d [i.e. K,=Kp, Ks= 1.5 Kp, Kp = 1.5 Ks]. It is c l e a r f r o m t h e s e f i g u r e s t h a t p l u g - f l o w r e a c t o r p e r f o r m a n c e is s u p e r i o r to C S T R a n d t h e r a t i o "C,ot/'cpa p p r o a c h e s 1 f o r l a r g e n u m b e r o f r e a c t o r s a n d at s m a l l c o n v e r s i o n . C o m m e r c i a l l y t h e c o n v e r s i o n o f g l u c o s e to f r u c t o s e is
263
BioprocessEngineering~5(1996)
264
carried out in immobilized packed bed reactors which n o r m a l ly approach plug-flow behavior. C o m p a r i s o n between the design criteria of m i n i m u m overall v o l u m e a n d equal-volume reactors shows that the total holding times in both design criteria are almost the same u n d e r practical operating conditions of glucose isomerization. The two design criteria give exactly the same results in the case of Ks =Kp. In the other two cases (Table 6 a n d 7), the reduction in total v o l u m e as compared to equal-sized reactors is very small (less t h a n 1% in the case of Ks= 1.5Kp) a s s u m i n g glucose conversion of 90% of the equilibrium. The reduction in total volume becomes appreciable only at very high conversion a n d for small n u m b e r of reactors. Table 9 shows the percent reduction in total v o l u m e as compared to equal-sized reactors for 2, 3 a n d 4 reactors at different degrees of conversion. The highest reduction is observed at very high conversion for 2 reactors. Therefore, one can conclude that it is more c o n v e n i e n t to operate in equal-volume reactors, since the other design criteria of using variable v o l u m e reactors is valid for specified substrate conversion only. The effect of glucose isomerase deactivation on the o p t i m u m design is investigated. C o m p a r i s o n of the holding times of the reactors in the three cases studied above a s s u m i n g with a n d without enzyme deactivation shows very small difference in the total reactors holding time specially at low temperatures such as 61 ~ where the deactivation rate of the enzyme is relatively low. The difference is small even at relatively high temperature such as 80 ~ in the case ofKp= 1.5 Ks (Table 8). It is i m p o r t a n t to note that the condition for negligible enzyme deactivation used by [7] is satisfied for glucose isomerase even at 80 ~ (i.e. l/~/0tot>> 1). This temperature is about 20 ~ higher t h a n practical operating temperature for HFCS i n d u s t r y [16].
References 1. Luyben, K.C.; Tramper, l.: Optimal design for continuous stirred tank reactors in series using Michaelis Menten kinetics. Biotechnol. Bioeng. 24 (1982) 1217-1220 2. Gooijer, C.D.; Hens, H.I.H.; Tramper, ].: Optimum design for a series of continuous stirred tank reactors containing immobilized biocatalyst beads obeying intrinsic Michaelis-Menten kinetics. Bioprocess Eng. 4 (1989) 153-158 3. Wail, J.B.; Hill, G.A.: Optimum CPST bioreactor design: Experimental study using batch growth parameters for S. cerevisiae producing ethanol. Can. J. Chem. Eng. 70 (1992) 148-152 4. HiU, G.A.; Robinson, C.W.: Minimum tank volumes for CFST bioreactors in series. Can. J. Chem. Eng. 67 (1989) 818 824 5. Malcata, F.N.: A heuristic approach for the economic optimization of a series of CSTR's performing Michaelis-Menten reactions. Biotechnoi. Bioeng. 33 (1989) 251 255 6. Maicata, F.X.: Optimal design on an economic basis for continuous stirred tank reactors in series using Michaelis Menten kinetics for Ping-Pong reactions. Can. ]. Chem. Eng. 66 (1988) 168 172 7. Lopes, T.I.; Malcata, F.X.: Optimal design of a series of CSTR's for biochemical reactions in the presence of enzyme deactivation. I. Chem. Eng. Japan. 26(1) (1993) 94-98 8. Malcata, F.X.: On the maximum conversion of substrate during biochemical reactions performed by a series of CSTR's in the presence of enzyme deactivation. I. Chem. Eng. Japan 23(3) (1990) 372-375 9. Malcata, F.X.: The effect of the level of micromixing on the optimal design of CSTR's performing Michaelis-Menten reactions. Can. J. Chem. Eng, 68 (1990) 330-336 10. Maicata, F.X.; Cameron, D.C.: Optimal design of a series of CSTR's performing reversible reactions catalyzed by soluble enzymes: A theoretical study. Biocatalysis 5 (1992) 233-248 11. Yang, S.T.; Okos, M.R.: Effects of temperature on lactose hydrolysis by immobilized fi-glactosidase in plug-flow reactor. Biotechno]. Bioeng. 33 (1989) 873-885 12. Van Tilburg, R.: Enzymatic isomerization of corn starch-based glucose syrups In: Starch conversion technology. Edited by G.M.A. Van Beynum and J.A. Rods, Marcell Dikker 1985 13. Bailey, J.E.; OUis, D.F." Biochemical engineering fundamentals, second ed, McGraw-Hill. 1986 14. Levenspiet, O.: Chemical reaction engineering, Wiley, New York, 1972 15. Abu-Reesh, I.M.; Faqir, N.M.: Simulation of glucose isomerase reactor: optimum operating temperature. Bioprocess Eng. 14(4) (1996) 205510 16. Venkatasubramanian, K.: Enzyme reactor design, Kinetics and performance. In Food Process Engineering: Enzyme engineering in food processing. Edited by P. Linko and J. Larinkari, Vol. 2, Applied Science, London (1979) 162 174 17. Illanes, A.; Zuniga, M.E.; Contreras, S.; Guerrero, A.: Reactor design for the enzymatic isomerization of glucose to fructose. Bioprocess Eng. 7 (1992) 199-204 18. Szepe, S.; Levenspiel, O.: Optimization of backmix reactors in series for a single reaction. Ind. Eng. Chem. Process Design Develop (3) (1964) 214 217
