BIOTECHNOLOGY AND BIOENGINEERING
VOL. XVII (1975)
COMMUNICATIONS TO THE EDITOR Nonisothermal Heterogeneous Reaction in a Denaturable Immobilized Enzyme Catalyst INTRODUCTION In recent years, considerable attention has been focused on heterogeneous catalytic reaction in immobilized enzyme catalysts.'-9 Lee and Tsao,I Ollis,* Rony,3 Sundaram e t al.,d and Vieth e t al.5 examined the effectiveness factor of immobilized enzyme catalysts of different geometries. In these works, a simplified Michaelis-Menten rate equation of zero or first order has been employed so that a closed form of effectiveness factor in terms of the Thiele Modulus and other parameters could be obtained. I n reality, such a simplified Michaelis-Menten rate equation constitutes only a special case of its general form which may not be able to display the true reaction characteristics of the immobilized enzyme catalysts. Fink et a1.,6 Miyamoto et al.,? and Moo-Young and Kobayashis investigated the effectiveness factor by using the complete form of the Michaelis-Menten rate equation. Horvath and Engasserg considered the effectiveness factor in a pellicular immobilized enzyme catalyst. A common assumption made in all the above works is that the enzymatic reaction is carried out under isothermal condition and with a constant enzyme activity during the whole reaction. This may not be true in a number of real circumstances because of the exothermic nature of enzymatic reactions.*0-I6 For a nonisothermal reaction, temperature plays an extremely important role in evaluating the effectiveness factor of an immobilized enzyme catalyst because the activity of enzyme is rather sensitive to the reaction temperature. It has been widely recognized that above a certain level of temperature, the enzyme activity decreases significantly.10-' This phenomenon, known as the thermal inactivation or denaturation, occurs frequently in many enzymatic reactions. The decreasing enzyme activity tends to place the entire enzymatic reaction in a transient state rather than a steady state as is generally considered. Thus far no work has ever taken thermal inactivation into consideration in the investigation of the effectiveness factor of an immobilized enzyme catalyst. The purpose of this work is to point out that it is important to consider the transient state when investigating the heterogeneous reactions in an immobilized enzyme catalyst.
TRANSIENT MATHEMATIC MODEL Consider the following enzymatic reaction
E
k + S Fki= ES -A E +P
k-i
1237
@ 1975 by John Wiley & Sons, Inc.
1238
BIOTECHNOLOGY AND BIOENGINEERING VOL. XVII (1975)
The unsteady state material and energy balances are given by
&,?= k , bt
(?
+ ?.>
brz
- k$E( S
be
c
- A H ) exp
+ k,
(- "> ROT
(3)
in which the enzyme concentration, E , is given by bE
=
-k.E exp
(- g)
(4)
bt
The initial and boundary conditions for the above equations are:
t
=o;
f
=o;
S
=
0,
-bS= o , br
T = To, E = Eo
-bT= o be
By imposing appropriate assumptions, eqs. (2)through (7) can be reduced to all the cases investigated by the previous authors.1-9 00
0.6
0
04
0.2
I
C
30
60
I 90
1 I20
Fig. 1. Effect of mass transfer Nusselt number on the dimensionless substrate concentration with a, = 1.0, K , = 0.35, p2 = 117.5,@, = 18.5, K . = 1.1 X 1046, a2 = 0.5,h = 0.5 and (Nu),, = 0.5.
COMMUNICATIONS TO THE EDITOR
1239
It can be noted that it is more difficult to solve eqs. (2), (3), and (4) than to solve the corresponding material balance equation for the steady state because of the highly nonlinear reaction terms involved in these equations. However, eqs. (2) and (3) can be numerically solved by the implicit Crank-Nicolson finite difference method," and the fourth order Runge-Kutta method" can be employed to integrate eq. (4). Figures 1, 2, and 3, respectively, show the transient variations of the dimensionless substrate concentration, enzyme activity, and dimensionless temperature at the catalyst center (dashed lines) and the catalyst surface (solid line) for a specific example. For steady state enzymatic reaction, the effectiveness factor is defined as the ratio of actual reaction rate to the hypothetic reaction rate in the absence of internal diffusion resistance.lsJ9 According to the definition, the effectiveness factor can be written as
I€
04
Df
8 04
0.2
C 0
30
60
90
120
r Fig. 2.
Effect of mass transfer Nusselt number on the enzyme activity.
1240
BIOTECHNOLOGY AND BIOENGINEERING VOL. XVII (1975) 1.07r
0
30
60
90
120
r
Fig. 3. Effect of mass transfer Nusselt number on the dimensionless temperature.
For the steady state reaction, the concentration gradient at the catalyst surface is an invariant; however, for the unsteady state, it is time-dependent. The substrate which has already diffused into the particle does not represent the actual reaction rate because part of the substrate is accumulated inside the particle. Therefore, eq. ( 8 ) can not be considered as the effectiveness factor for the unsteady state case and does not have specific meaning except for representing the variation of substrate concentration gradient a t the catalyst surface. Nomenclature a
C CP
D E
Eo AEi AEz h
surface area per unit volume of catalyst particle dimensionless substrate concentration, S/So heat capacity of fluid effective diffusivity enzyme concentration initial enzyme concentration activation energy for enzymatic reaction activation energy for enzyme inactivation Thiele modulus, rg dkoEo/DSo
COMMUNICATIONS TO T H E EDITOR hr ( -AH)
r ro
R R, S
so t T To
1241
heat transfer coefficient heat generation by enzymatic reaction turnover number inactivation coefficient of enzyme mass transfer coefficient Michaelis-Menten coefficient frequency factor thermal conductivity dimensionless inactivation coefficient of enzyme, k,ro2/ D dimensionless Michaelis-Menten coefficient, k,/So heat transfer Nusselt number, htro/kt mass transfer Nusselt number, kLaro/D radial coordinate radius of catalyst particle .: dimensionless radial coordinate, r/ro gas constant substrate concentration substrate concentration outside the catalyst time temperature initial temperature
Greek Letters ratio of Schmidt number to Prandtl number, kl/pCpD dimensionless heat generation parameter, ( - A H ) S O / ~ C , T O a2 dimensionless activation energy for enzymatic reaction, A E I / R , T o 81 dimensionless activation energy for enzyme inactivation, A&/R,l'u 82 P fluid density e dimensionless temperature, T I T O 'T dimensionless time, t D/ro2 enzyme activity, E/EO dimensionless parameter defined by eq. (8) 7 a1
+
References 1. Y. Y. Lee and G. T. Tsao, J . Food Sn'., 39, 667 (1974). 2. D. Y. Ollis, Biotechnol. Bioeng., 14, 871 (1972). 3. P. R. Rony, Biotechnol. Bioeng., 13, 431 (1971). 4. P. V. Sundaram, A. Tweedale, and K. J. Laidler, Can. J . Biochem., 48, 1498 (1970). 5. W. R. Vieth, A. V. Mendiratta, A. V. Morgensen, R. Saini, and V. Venkatasubramanian, Chem. Eng. Sci., 28, 1031 (1973). 6. D. J. Fink, T. Y. Na, and J. S. Schutz, Biotechnol. Bioeng., 15,879 (1973). 7. K. Miyamoto, T. Fujiii, IT.Tamaoki, M. Okazaki and Y. Miura, J . Ferment. Technol., 51, 566 (1973). 8. M. Moo-Young and T. Kobayashi, Can. J . Chern. Eng., 50, 162 (1972). 9. C. Horvath and J. M. Engasser, Ind. Eng. Chem. Fundam., 12.229 (1973). 10. K. J. Laidler, The Chemical Kinetics of Enzyme Action, Oxford Univ. Press, England, 1958. 11. I. W. Sizer, in Advances in Enzymology, Vol. 3, Interscience, New York, 1943.
1242
BIOTECHNOLOGY AND BIOENGINEERING VOL. XVII (1975)
12. 13. 15, 47 14. 15. 16. 17.
K. J. Tagawa, J . Ferment. TecAnol., 48, 730 (1970). T. Tosa, T. Sato, T. Mori, Y. Matuo, and I. Chibata, Biotechnol. Bioeng., (1973).
L. Y. Ho and A. E. Humphrey, Biotechnol. Bioeng., 12, 291 (1970). S. P. O’Neil, Biotechnol. Bioeng., 14, 473 (1972). S. H. Lin, Biophysik, 10, 235 (1973). E. E. Peterson, Chemical Reactor Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1965. 18. C. N. Satterfield, Mass Transfer in Heterogeneous Catalysis, M.I.T. Press, Cambridge, Mass., 1970. 19. L. Lapidus, Digital Computation for Chemical Engineers, McGraw-Hill, New York, 1962.
S. H. LIN Dept. of Chemical Engineering University of Melbourne Parkville, Victoria 3052, Australia Accepted for Publication February 21, 1975
VOL. XVII (1975)
COMMUNICATIONS TO THE EDITOR Nonisothermal Heterogeneous Reaction in a Denaturable Immobilized Enzyme Catalyst INTRODUCTION In recent years, considerable attention has been focused on heterogeneous catalytic reaction in immobilized enzyme catalysts.'-9 Lee and Tsao,I Ollis,* Rony,3 Sundaram e t al.,d and Vieth e t al.5 examined the effectiveness factor of immobilized enzyme catalysts of different geometries. In these works, a simplified Michaelis-Menten rate equation of zero or first order has been employed so that a closed form of effectiveness factor in terms of the Thiele Modulus and other parameters could be obtained. I n reality, such a simplified Michaelis-Menten rate equation constitutes only a special case of its general form which may not be able to display the true reaction characteristics of the immobilized enzyme catalysts. Fink et a1.,6 Miyamoto et al.,? and Moo-Young and Kobayashis investigated the effectiveness factor by using the complete form of the Michaelis-Menten rate equation. Horvath and Engasserg considered the effectiveness factor in a pellicular immobilized enzyme catalyst. A common assumption made in all the above works is that the enzymatic reaction is carried out under isothermal condition and with a constant enzyme activity during the whole reaction. This may not be true in a number of real circumstances because of the exothermic nature of enzymatic reactions.*0-I6 For a nonisothermal reaction, temperature plays an extremely important role in evaluating the effectiveness factor of an immobilized enzyme catalyst because the activity of enzyme is rather sensitive to the reaction temperature. It has been widely recognized that above a certain level of temperature, the enzyme activity decreases significantly.10-' This phenomenon, known as the thermal inactivation or denaturation, occurs frequently in many enzymatic reactions. The decreasing enzyme activity tends to place the entire enzymatic reaction in a transient state rather than a steady state as is generally considered. Thus far no work has ever taken thermal inactivation into consideration in the investigation of the effectiveness factor of an immobilized enzyme catalyst. The purpose of this work is to point out that it is important to consider the transient state when investigating the heterogeneous reactions in an immobilized enzyme catalyst.
TRANSIENT MATHEMATIC MODEL Consider the following enzymatic reaction
E
k + S Fki= ES -A E +P
k-i
1237
@ 1975 by John Wiley & Sons, Inc.
1238
BIOTECHNOLOGY AND BIOENGINEERING VOL. XVII (1975)
The unsteady state material and energy balances are given by
&,?= k , bt
(?
+ ?.>
brz
- k$E( S
be
c
- A H ) exp
+ k,
(- "> ROT
(3)
in which the enzyme concentration, E , is given by bE
=
-k.E exp
(- g)
(4)
bt
The initial and boundary conditions for the above equations are:
t
=o;
f
=o;
S
=
0,
-bS= o , br
T = To, E = Eo
-bT= o be
By imposing appropriate assumptions, eqs. (2)through (7) can be reduced to all the cases investigated by the previous authors.1-9 00
0.6
0
04
0.2
I
C
30
60
I 90
1 I20
Fig. 1. Effect of mass transfer Nusselt number on the dimensionless substrate concentration with a, = 1.0, K , = 0.35, p2 = 117.5,@, = 18.5, K . = 1.1 X 1046, a2 = 0.5,h = 0.5 and (Nu),, = 0.5.
COMMUNICATIONS TO THE EDITOR
1239
It can be noted that it is more difficult to solve eqs. (2), (3), and (4) than to solve the corresponding material balance equation for the steady state because of the highly nonlinear reaction terms involved in these equations. However, eqs. (2) and (3) can be numerically solved by the implicit Crank-Nicolson finite difference method," and the fourth order Runge-Kutta method" can be employed to integrate eq. (4). Figures 1, 2, and 3, respectively, show the transient variations of the dimensionless substrate concentration, enzyme activity, and dimensionless temperature at the catalyst center (dashed lines) and the catalyst surface (solid line) for a specific example. For steady state enzymatic reaction, the effectiveness factor is defined as the ratio of actual reaction rate to the hypothetic reaction rate in the absence of internal diffusion resistance.lsJ9 According to the definition, the effectiveness factor can be written as
I€
04
Df
8 04
0.2
C 0
30
60
90
120
r Fig. 2.
Effect of mass transfer Nusselt number on the enzyme activity.
1240
BIOTECHNOLOGY AND BIOENGINEERING VOL. XVII (1975) 1.07r
0
30
60
90
120
r
Fig. 3. Effect of mass transfer Nusselt number on the dimensionless temperature.
For the steady state reaction, the concentration gradient at the catalyst surface is an invariant; however, for the unsteady state, it is time-dependent. The substrate which has already diffused into the particle does not represent the actual reaction rate because part of the substrate is accumulated inside the particle. Therefore, eq. ( 8 ) can not be considered as the effectiveness factor for the unsteady state case and does not have specific meaning except for representing the variation of substrate concentration gradient a t the catalyst surface. Nomenclature a
C CP
D E
Eo AEi AEz h
surface area per unit volume of catalyst particle dimensionless substrate concentration, S/So heat capacity of fluid effective diffusivity enzyme concentration initial enzyme concentration activation energy for enzymatic reaction activation energy for enzyme inactivation Thiele modulus, rg dkoEo/DSo
COMMUNICATIONS TO T H E EDITOR hr ( -AH)
r ro
R R, S
so t T To
1241
heat transfer coefficient heat generation by enzymatic reaction turnover number inactivation coefficient of enzyme mass transfer coefficient Michaelis-Menten coefficient frequency factor thermal conductivity dimensionless inactivation coefficient of enzyme, k,ro2/ D dimensionless Michaelis-Menten coefficient, k,/So heat transfer Nusselt number, htro/kt mass transfer Nusselt number, kLaro/D radial coordinate radius of catalyst particle .: dimensionless radial coordinate, r/ro gas constant substrate concentration substrate concentration outside the catalyst time temperature initial temperature
Greek Letters ratio of Schmidt number to Prandtl number, kl/pCpD dimensionless heat generation parameter, ( - A H ) S O / ~ C , T O a2 dimensionless activation energy for enzymatic reaction, A E I / R , T o 81 dimensionless activation energy for enzyme inactivation, A&/R,l'u 82 P fluid density e dimensionless temperature, T I T O 'T dimensionless time, t D/ro2 enzyme activity, E/EO dimensionless parameter defined by eq. (8) 7 a1
+
References 1. Y. Y. Lee and G. T. Tsao, J . Food Sn'., 39, 667 (1974). 2. D. Y. Ollis, Biotechnol. Bioeng., 14, 871 (1972). 3. P. R. Rony, Biotechnol. Bioeng., 13, 431 (1971). 4. P. V. Sundaram, A. Tweedale, and K. J. Laidler, Can. J . Biochem., 48, 1498 (1970). 5. W. R. Vieth, A. V. Mendiratta, A. V. Morgensen, R. Saini, and V. Venkatasubramanian, Chem. Eng. Sci., 28, 1031 (1973). 6. D. J. Fink, T. Y. Na, and J. S. Schutz, Biotechnol. Bioeng., 15,879 (1973). 7. K. Miyamoto, T. Fujiii, IT.Tamaoki, M. Okazaki and Y. Miura, J . Ferment. Technol., 51, 566 (1973). 8. M. Moo-Young and T. Kobayashi, Can. J . Chern. Eng., 50, 162 (1972). 9. C. Horvath and J. M. Engasser, Ind. Eng. Chem. Fundam., 12.229 (1973). 10. K. J. Laidler, The Chemical Kinetics of Enzyme Action, Oxford Univ. Press, England, 1958. 11. I. W. Sizer, in Advances in Enzymology, Vol. 3, Interscience, New York, 1943.
1242
BIOTECHNOLOGY AND BIOENGINEERING VOL. XVII (1975)
12. 13. 15, 47 14. 15. 16. 17.
K. J. Tagawa, J . Ferment. TecAnol., 48, 730 (1970). T. Tosa, T. Sato, T. Mori, Y. Matuo, and I. Chibata, Biotechnol. Bioeng., (1973).
L. Y. Ho and A. E. Humphrey, Biotechnol. Bioeng., 12, 291 (1970). S. P. O’Neil, Biotechnol. Bioeng., 14, 473 (1972). S. H. Lin, Biophysik, 10, 235 (1973). E. E. Peterson, Chemical Reactor Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1965. 18. C. N. Satterfield, Mass Transfer in Heterogeneous Catalysis, M.I.T. Press, Cambridge, Mass., 1970. 19. L. Lapidus, Digital Computation for Chemical Engineers, McGraw-Hill, New York, 1962.
S. H. LIN Dept. of Chemical Engineering University of Melbourne Parkville, Victoria 3052, Australia Accepted for Publication February 21, 1975
