Effective diffusion coefficient of sucrose in calcium alginate gel Ulkfi Mehmetoglu D e p a r t m e n t o f C h e m i c a l E n g i n e e r i n g , S c i e n c e Faculty, A n k a r a University, T a n d o g a n , A n k a r a , Turkey
The effective diffusion coefficient o f sucrose in 5% calcium alginate gel containing 41.6 g.d.c, l-L
Saccharomyces cerevisiae was investigated. Both free and immobilized S. cerevisiae in 0.175 cm and 0.3 cm diameter particles were used and the reactions were achieved in a medium containing 100 g l-1 sucrose and 0.05 M CaCI2. With the assumption that the microorganisms did not grow or die in this medium, the results were analyzed according to Michaelis-Menten kinetics and the values o f the parameters were determined as: Vm = 0.256 g m1-1 gel h -1, Kin0 = 0.097 g ml -I, Kin, = 0.125 g ml I, and Kin2 = 0.165 g m1-1. Using these values, effectiveness factors were calculated as ~1 = 0.89 and vl2 = 0.76, and effective diffusion coefficients for sucrose in calcium alginate gel were determined as De, = 4.1 × 10 -6 c m 2 s - j and De, = 4.0 × 10 6 cm 2 s 1,for the particle size involved.
Keywords:Diffusion coefficient; calcium alginate; sucrose Introduction
Theory
Recently, calcium alginate gel has received increasing attention as an immobilization material. One problem c o m m o n to all the immobilized systems, however, is the mass transfer restriction caused by the additional resistance of the gel. The pore size of the gel, together with the size and the concentration of the diffusing molecules, influences the substrate or product diffusion rate, which in turn influences the reaction rate catalysed by immobilized e n z y m e s or microorganisms. F o r this reason, the pore size is a critical parameter in the selection of a gel. It is also obvious from the a b o v e discussion that the diffusion coefficients of the substrate and product are of primary importance in the design of such fermenters. T h e r e are only a few studies in the literature employing various mathematical techniques for the determination of glucose diffusion coefficients in calcium alginate and K-carrageenan gels.J-3 H o w e v e r , there is no research which investigates the diffusion phenomena and reaction rate at the same time, and which determines the effective diffusion coefficient. In this study, the diffusion and reaction of sucrose in calcium alginate gel containing S. cerevisiae were investigated and the diffusion coefficient is determined using a novel approach.
It is a well-known fact that the apparent reaction rate decreases during diffusion in porous media. It was Thiele 4 who first theoretically investigated diffusion and reaction together and put forward the concept of an effectiveness factor. The effectiveness factor is defined as the ratio of the reaction rate with diffusion restriction to the one without. This concept was later investigated by various researchers. 5~6 Lee et al. 7 applied this c o n c e p t to spherical e n z y m e carriers. Assuming that the reaction proceeds according to Michaelis-Menten kinetics, a mass balance over a spherical porous particle yielded:
Address reprint requests to Dr. Mehmeto~lu at the Department of Chemical Engineering, Science Faculty, Ankara University, Tandogan, Ankara, Turkey Received 11 July 1988; revised 15 December 1988
124
Enzyme Microb. Technol., 1990, vol. 12, February
De \ dr 2 + -r dr
-
Vm
-~rn + ~
= 0
(1)
with the following boundary conditions: r = R r = 0
C = C~ dC - 0 dr
(2) (3)
Solving this nonlinear differential equation numerically, they obtained relationship between effectiveness factor ~/and Thiele modulus ~ for different values of/3, where /3-
Km C~
(4)
d~ = R ~/VD~-m
(5)
O
© 1990 Butterworth Publishers
Diffusion coefficient of sucrose: U. MehmetoOlu
Figure 1 shows a reproduction of their results for
0.9
/3 = 0.8.s
nO pC)
Materials and methods
0.7 0.5
P
0.4
Microorganism and media z
LU
S. cerevisiae Y-567 was obtained from the National
0.3
I.--
Regional Research Center in Peoria, Illinois. The growth medium was the one given in ref. 9 and the reaction media used in the experiments contained I00 g 1-J sucrose and 0.05 M CaC12.
UJ U-
C"
02
0.1
I 20
10
Experimental apparatus and procedures The experiments were performed in 250-ml Edenm e y e r flasks. The mixing rate and media temperature were kept constant with the aid of a constant temperature shaker set to 200 rev rain 1, a speed high enough to eliminate the external film resistance to mass transfer. After a sufficient number of microorganisms were obtained, the growth medium was centrifuged at 6000 rev min -1 for 10 min. The concentrated cell mass was added to 5% sodium alginate gel (BDH) to obtain 0.25 g.d.c./6 ml concentration. 1° This mixture was passed through hypodermic needles of different sizes by a peristaltic pump, and a direct observation method '~ was used for determining the average size of produced beads. A portion of the centrifuged mass was used without immobilization. Into three separate 100-ml reaction media, 0.25 g.d.c, of free cells or calcium alginate beads of two sizes containing immobilized cells with 6 ml total volume were placed. The temperature and p H were kept constant at 30°C and 5.0, respectively. It was assumed that no loss of microorganisms occurred during immobilization or the subsequent 24 h experimental incubation time.
Analytical methods Sucrose concentrations were determined by a spectrophotometric method n using invertase and glucose oxidase-peroxidase enzymes obtained from Sigma Chemical Co.
Results and discussion The change in the sucrose concentration of the media was measured in the experiments performed with free microorganisms and with microorganisms immobilized in particles with radius R = 0.175 cm and R = 0.3 cm. For all experiments W and initial Cb values were 0.25 g.d.c, and 100 g 1-1, respectively. Under these conditions, the reaction rate of the free microorganisms can be assumed to be the rate without the porous diffusion effects. Figure 2 shows the variation of the sucrose concentration with time. Using these curves and assuming that the immobilized microorganisms, which act as enzyme stores, cause the reaction to follow the Michaelis-Menten kinetics, one can calculate the Michaelis-Menten constants. Assuming that the external film resistance is negligible, the interface substrate
cb
[ 3.0
t
I I I i II 5.0 70 10
20
¢ : RV,/~V---~Km o De
Figure 1
~ vs ~b v a l u e s (ref. 8)
1 0
2
~
6
8
10 12 14
16
18
20 2z
2~
t (h)
Figure 2 V a r i a t i o n o f s u c r o s e c o n c e n t r a t i o n w i t h time. T 30°C, pH = 5, W = 0.25 g.d.c. ©, Free m i c r o o r g a n i s m s ; A, i m m o bilized m i c r o o r g a n i s m s , R1 - 0.175 cm; O, i m m o b i l i z e d microorganisms, R2 = 0.3 cm
concentration (Cs) can be considered equal to that of the bulk concentration (Cb). The change in the enzyme activity during 24 h test duration was neglected based on previous experiments. 13Figure 3 shows (l / V) vs ( 1/ Cb) values. The Km and Vm values were determined from the slopes and from the intercept, respectively. For the free and immobilized microorganisms in R = 0.175 cm and R = 0.3 cm particles, the MichaelisMenten constants were determined as Kin0 = 0.097 g m1-1, Km, = 0.125 g ml -I, and Km2 = 0.165 g ml -l, respectively. The maximum reaction rate, Vrn, was found as 0.256 g m1-1 gel h -1 (Table 1). Using these values, the reaction rate values can be calculated and these are also shown in Table 1. The effectiveness factor is defined as ~ = V/Vo. For the two diameters involved, the ~ values can be calculated, and using Figure 1 Thiele modulus, q5 values can also be determined. F o r /3 = 0.8, "01 and r/2 values corresponding to R = 0.175 cm and 0.3 cm are 0.89 and 0.76, respectively. F r o m Figure 1, the corresponding ~b values were determined as 2.3 and 4.0. Knowing the Enzyme Microb. Technol., 1990, vol. 12, February
125
Papers values, that is 4.05 x 10 6 cm 2, can be considered as the effective diffusion coefficient of sucrose in calcium alginate gel. This value is considerably less than the effective diffusion coefficient of sucrose in water, which is 5.6 x 10 -6 c m 2 s - I . This difference in the values of the effective diffusion coefficients is in agreement with the results of l t a m u n a o l a 2 who also reported a 30% reduction in the diffusion coefficient of glucose in 5% sodium alginate gel with respect to that in water. This a g r e e m e n t can be considered as a further confirmation of the a c c u r a c y of the method used to calculate the effective diffusion coefficient of solutes in gels.
4O
3O
•
oa
20
_2
E
Nomenclature
Cn
10
C Cb C~
0
I 01
0
I 0.2
I 03
I 04
1 0.5
I 0.6
1 t C b x 10- 2 ( g t m l ) - 1 Figure3
(V)
vs ( ~ )
values ©, Free microorganisms;
A,
immobilized microorganisms, R~ = 0,175 cm; I , immobilized microorganisms, R2 = 0.3 cm
Table 1 Michaelis-Menten constants and reaction rates for different conditions Km (g m1-1) Free microorganism
0.097
V• (g ml l g e l h
1)
Reaction rates (g ml 1 gel h 1)
0.256
Substrate concentration in gel, g ml -~ Substrate concentration in bulk, g ml -~ Substrate concentration at liquid-solid interface, g ml -l De Effective diffusion coefficient, cm 2 s Km0 Michaelis-Menten constant when there is no diffusion restriction due to the gel, g m l Kin,, Km2 Michaelis-Menten constants for particles with radius R = 0.175 cm and R = 0.3 cm, respectively r Variable radius, cm R Radius of spherical particles, cm V Reaction rate, g ml -~ gel h Vm M a x i m u m reaction rate, g ml-i gel h-J t Time, s /3 Dimensionless Michaelis-Menten constant, Km/Cs
Effectiveness factor
V0
0.256 C~ 0.097 + Ca
4, Thiele modulus, R
Immobilized microorganisms R = 0.175 cm
0.125
0.256
Vl
0.256 Cb 0.125 + Cb
Abbreviations
R = 0.3 cm
0.256
V2
0.256 Ca 0.165 + Ca
d.c. D r y cell
0.165
•/
Vm
DeKm,,
References Table 2
Effectiveness factors, Thiele modulus, and effective diffusion coefficients for different conditions r/
R = 0.175 cm R = 0.3 cm
0.89 0.76
cb 2.3 4.0
1. 2. 3.
De (cm 2 s 1) 4.1 x 10 0 4.0 x 10 -6
4. 5. 6. 7. 8.
actual Km value as K ~ = 0.097 g ml -~ from definition of 4~, the effective diffusion coefficients were calculated. F o r two sizes involved, that is, R] = 0.175 cm and R2 -- 0.3 cm, they were found to be De, = 4.1 x 10 _6 c m 2 s i a n d D~ = 4 x 10 -6 c m 2 s - ] , respectively (Table 2). T h e c l o s e n e s s o f t h e s e v a l u e s is to b e e x p e c t e d , since the diffusion coefficient does not change with the size of the particle. The arithmetic average of these
126
Enzyme Microb. Technol., 1990, vol. 12, February
9. 10. 11. 12. 13.
Tanaka, H., Matsumara, M, and Veliky, 1. A. Biotechnol. Bioeng. 1984, 124, 53-58 Itamunoala, G. F. Biotechnol. Prog. 1987, 3,2, 115-120 Nguyen, L. and Luong, J. H. T. Biotechnol. Bioeng. 1986, 28, 1261-1267 Thiele, E. W. Ind. Eng. Chem. 1939, 31, 916 Smith, J. M. Chemical Engineering Kinetics, 2d ed., 1970, McGraw-Hill, New York Marsh, D. R., Lee, Y. Y. and Tsao, G. T. Biotechnol. Bioeng. 1973, 15, 483-493 Lee, Y, Y. and Tsao, G. T. J. Food Sci. 1974, 39, 667-672 Lee, Y. Y., Fratzke, A. R., Wun, K. and Tsao, G. T. Biotechnol. Bioeng. 1976, 18, 389-413 Bazua, C. D. and Wilke, C. R. Biotechnol. Bioeng. Syrup. 7, 1977, 105-118 Kierstan, M. and Bucke, C. Biotechnol. Bioeng. 1977, 19, 387-397 Nguyan, A. L. and Luong, J. H. T. Biotechnol. Bioeng. 1986, 28, 1261-1267 Bergmeyer, H. U. Methods of Enzymatic Analysis, 2rid ed., 1965, Academic Press, New York, pp. 99-102 Kurnaz (Mehmetoglu), U., Ph.D. Thesis, Hacettepe University, Ankara, Turkey, 1984
The effective diffusion coefficient o f sucrose in 5% calcium alginate gel containing 41.6 g.d.c, l-L
Saccharomyces cerevisiae was investigated. Both free and immobilized S. cerevisiae in 0.175 cm and 0.3 cm diameter particles were used and the reactions were achieved in a medium containing 100 g l-1 sucrose and 0.05 M CaCI2. With the assumption that the microorganisms did not grow or die in this medium, the results were analyzed according to Michaelis-Menten kinetics and the values o f the parameters were determined as: Vm = 0.256 g m1-1 gel h -1, Kin0 = 0.097 g ml -I, Kin, = 0.125 g ml I, and Kin2 = 0.165 g m1-1. Using these values, effectiveness factors were calculated as ~1 = 0.89 and vl2 = 0.76, and effective diffusion coefficients for sucrose in calcium alginate gel were determined as De, = 4.1 × 10 -6 c m 2 s - j and De, = 4.0 × 10 6 cm 2 s 1,for the particle size involved.
Keywords:Diffusion coefficient; calcium alginate; sucrose Introduction
Theory
Recently, calcium alginate gel has received increasing attention as an immobilization material. One problem c o m m o n to all the immobilized systems, however, is the mass transfer restriction caused by the additional resistance of the gel. The pore size of the gel, together with the size and the concentration of the diffusing molecules, influences the substrate or product diffusion rate, which in turn influences the reaction rate catalysed by immobilized e n z y m e s or microorganisms. F o r this reason, the pore size is a critical parameter in the selection of a gel. It is also obvious from the a b o v e discussion that the diffusion coefficients of the substrate and product are of primary importance in the design of such fermenters. T h e r e are only a few studies in the literature employing various mathematical techniques for the determination of glucose diffusion coefficients in calcium alginate and K-carrageenan gels.J-3 H o w e v e r , there is no research which investigates the diffusion phenomena and reaction rate at the same time, and which determines the effective diffusion coefficient. In this study, the diffusion and reaction of sucrose in calcium alginate gel containing S. cerevisiae were investigated and the diffusion coefficient is determined using a novel approach.
It is a well-known fact that the apparent reaction rate decreases during diffusion in porous media. It was Thiele 4 who first theoretically investigated diffusion and reaction together and put forward the concept of an effectiveness factor. The effectiveness factor is defined as the ratio of the reaction rate with diffusion restriction to the one without. This concept was later investigated by various researchers. 5~6 Lee et al. 7 applied this c o n c e p t to spherical e n z y m e carriers. Assuming that the reaction proceeds according to Michaelis-Menten kinetics, a mass balance over a spherical porous particle yielded:
Address reprint requests to Dr. Mehmeto~lu at the Department of Chemical Engineering, Science Faculty, Ankara University, Tandogan, Ankara, Turkey Received 11 July 1988; revised 15 December 1988
124
Enzyme Microb. Technol., 1990, vol. 12, February
De \ dr 2 + -r dr
-
Vm
-~rn + ~
= 0
(1)
with the following boundary conditions: r = R r = 0
C = C~ dC - 0 dr
(2) (3)
Solving this nonlinear differential equation numerically, they obtained relationship between effectiveness factor ~/and Thiele modulus ~ for different values of/3, where /3-
Km C~
(4)
d~ = R ~/VD~-m
(5)
O
© 1990 Butterworth Publishers
Diffusion coefficient of sucrose: U. MehmetoOlu
Figure 1 shows a reproduction of their results for
0.9
/3 = 0.8.s
nO pC)
Materials and methods
0.7 0.5
P
0.4
Microorganism and media z
LU
S. cerevisiae Y-567 was obtained from the National
0.3
I.--
Regional Research Center in Peoria, Illinois. The growth medium was the one given in ref. 9 and the reaction media used in the experiments contained I00 g 1-J sucrose and 0.05 M CaC12.
UJ U-
C"
02
0.1
I 20
10
Experimental apparatus and procedures The experiments were performed in 250-ml Edenm e y e r flasks. The mixing rate and media temperature were kept constant with the aid of a constant temperature shaker set to 200 rev rain 1, a speed high enough to eliminate the external film resistance to mass transfer. After a sufficient number of microorganisms were obtained, the growth medium was centrifuged at 6000 rev min -1 for 10 min. The concentrated cell mass was added to 5% sodium alginate gel (BDH) to obtain 0.25 g.d.c./6 ml concentration. 1° This mixture was passed through hypodermic needles of different sizes by a peristaltic pump, and a direct observation method '~ was used for determining the average size of produced beads. A portion of the centrifuged mass was used without immobilization. Into three separate 100-ml reaction media, 0.25 g.d.c, of free cells or calcium alginate beads of two sizes containing immobilized cells with 6 ml total volume were placed. The temperature and p H were kept constant at 30°C and 5.0, respectively. It was assumed that no loss of microorganisms occurred during immobilization or the subsequent 24 h experimental incubation time.
Analytical methods Sucrose concentrations were determined by a spectrophotometric method n using invertase and glucose oxidase-peroxidase enzymes obtained from Sigma Chemical Co.
Results and discussion The change in the sucrose concentration of the media was measured in the experiments performed with free microorganisms and with microorganisms immobilized in particles with radius R = 0.175 cm and R = 0.3 cm. For all experiments W and initial Cb values were 0.25 g.d.c, and 100 g 1-1, respectively. Under these conditions, the reaction rate of the free microorganisms can be assumed to be the rate without the porous diffusion effects. Figure 2 shows the variation of the sucrose concentration with time. Using these curves and assuming that the immobilized microorganisms, which act as enzyme stores, cause the reaction to follow the Michaelis-Menten kinetics, one can calculate the Michaelis-Menten constants. Assuming that the external film resistance is negligible, the interface substrate
cb
[ 3.0
t
I I I i II 5.0 70 10
20
¢ : RV,/~V---~Km o De
Figure 1
~ vs ~b v a l u e s (ref. 8)
1 0
2
~
6
8
10 12 14
16
18
20 2z
2~
t (h)
Figure 2 V a r i a t i o n o f s u c r o s e c o n c e n t r a t i o n w i t h time. T 30°C, pH = 5, W = 0.25 g.d.c. ©, Free m i c r o o r g a n i s m s ; A, i m m o bilized m i c r o o r g a n i s m s , R1 - 0.175 cm; O, i m m o b i l i z e d microorganisms, R2 = 0.3 cm
concentration (Cs) can be considered equal to that of the bulk concentration (Cb). The change in the enzyme activity during 24 h test duration was neglected based on previous experiments. 13Figure 3 shows (l / V) vs ( 1/ Cb) values. The Km and Vm values were determined from the slopes and from the intercept, respectively. For the free and immobilized microorganisms in R = 0.175 cm and R = 0.3 cm particles, the MichaelisMenten constants were determined as Kin0 = 0.097 g m1-1, Km, = 0.125 g ml -I, and Km2 = 0.165 g ml -l, respectively. The maximum reaction rate, Vrn, was found as 0.256 g m1-1 gel h -1 (Table 1). Using these values, the reaction rate values can be calculated and these are also shown in Table 1. The effectiveness factor is defined as ~ = V/Vo. For the two diameters involved, the ~ values can be calculated, and using Figure 1 Thiele modulus, q5 values can also be determined. F o r /3 = 0.8, "01 and r/2 values corresponding to R = 0.175 cm and 0.3 cm are 0.89 and 0.76, respectively. F r o m Figure 1, the corresponding ~b values were determined as 2.3 and 4.0. Knowing the Enzyme Microb. Technol., 1990, vol. 12, February
125
Papers values, that is 4.05 x 10 6 cm 2, can be considered as the effective diffusion coefficient of sucrose in calcium alginate gel. This value is considerably less than the effective diffusion coefficient of sucrose in water, which is 5.6 x 10 -6 c m 2 s - I . This difference in the values of the effective diffusion coefficients is in agreement with the results of l t a m u n a o l a 2 who also reported a 30% reduction in the diffusion coefficient of glucose in 5% sodium alginate gel with respect to that in water. This a g r e e m e n t can be considered as a further confirmation of the a c c u r a c y of the method used to calculate the effective diffusion coefficient of solutes in gels.
4O
3O
•
oa
20
_2
E
Nomenclature
Cn
10
C Cb C~
0
I 01
0
I 0.2
I 03
I 04
1 0.5
I 0.6
1 t C b x 10- 2 ( g t m l ) - 1 Figure3
(V)
vs ( ~ )
values ©, Free microorganisms;
A,
immobilized microorganisms, R~ = 0,175 cm; I , immobilized microorganisms, R2 = 0.3 cm
Table 1 Michaelis-Menten constants and reaction rates for different conditions Km (g m1-1) Free microorganism
0.097
V• (g ml l g e l h
1)
Reaction rates (g ml 1 gel h 1)
0.256
Substrate concentration in gel, g ml -~ Substrate concentration in bulk, g ml -~ Substrate concentration at liquid-solid interface, g ml -l De Effective diffusion coefficient, cm 2 s Km0 Michaelis-Menten constant when there is no diffusion restriction due to the gel, g m l Kin,, Km2 Michaelis-Menten constants for particles with radius R = 0.175 cm and R = 0.3 cm, respectively r Variable radius, cm R Radius of spherical particles, cm V Reaction rate, g ml -~ gel h Vm M a x i m u m reaction rate, g ml-i gel h-J t Time, s /3 Dimensionless Michaelis-Menten constant, Km/Cs
Effectiveness factor
V0
0.256 C~ 0.097 + Ca
4, Thiele modulus, R
Immobilized microorganisms R = 0.175 cm
0.125
0.256
Vl
0.256 Cb 0.125 + Cb
Abbreviations
R = 0.3 cm
0.256
V2
0.256 Ca 0.165 + Ca
d.c. D r y cell
0.165
•/
Vm
DeKm,,
References Table 2
Effectiveness factors, Thiele modulus, and effective diffusion coefficients for different conditions r/
R = 0.175 cm R = 0.3 cm
0.89 0.76
cb 2.3 4.0
1. 2. 3.
De (cm 2 s 1) 4.1 x 10 0 4.0 x 10 -6
4. 5. 6. 7. 8.
actual Km value as K ~ = 0.097 g ml -~ from definition of 4~, the effective diffusion coefficients were calculated. F o r two sizes involved, that is, R] = 0.175 cm and R2 -- 0.3 cm, they were found to be De, = 4.1 x 10 _6 c m 2 s i a n d D~ = 4 x 10 -6 c m 2 s - ] , respectively (Table 2). T h e c l o s e n e s s o f t h e s e v a l u e s is to b e e x p e c t e d , since the diffusion coefficient does not change with the size of the particle. The arithmetic average of these
126
Enzyme Microb. Technol., 1990, vol. 12, February
9. 10. 11. 12. 13.
Tanaka, H., Matsumara, M, and Veliky, 1. A. Biotechnol. Bioeng. 1984, 124, 53-58 Itamunoala, G. F. Biotechnol. Prog. 1987, 3,2, 115-120 Nguyen, L. and Luong, J. H. T. Biotechnol. Bioeng. 1986, 28, 1261-1267 Thiele, E. W. Ind. Eng. Chem. 1939, 31, 916 Smith, J. M. Chemical Engineering Kinetics, 2d ed., 1970, McGraw-Hill, New York Marsh, D. R., Lee, Y. Y. and Tsao, G. T. Biotechnol. Bioeng. 1973, 15, 483-493 Lee, Y, Y. and Tsao, G. T. J. Food Sci. 1974, 39, 667-672 Lee, Y. Y., Fratzke, A. R., Wun, K. and Tsao, G. T. Biotechnol. Bioeng. 1976, 18, 389-413 Bazua, C. D. and Wilke, C. R. Biotechnol. Bioeng. Syrup. 7, 1977, 105-118 Kierstan, M. and Bucke, C. Biotechnol. Bioeng. 1977, 19, 387-397 Nguyan, A. L. and Luong, J. H. T. Biotechnol. Bioeng. 1986, 28, 1261-1267 Bergmeyer, H. U. Methods of Enzymatic Analysis, 2rid ed., 1965, Academic Press, New York, pp. 99-102 Kurnaz (Mehmetoglu), U., Ph.D. Thesis, Hacettepe University, Ankara, Turkey, 1984
