- Email: [email protected]
Diffusional characteristics of a swelling gel and its consequences for bioreactor performance
The Chemical Engineering
B35
Journal, 55 (1994) B35-B39
Diffusional characteristics of a swelling gel and its consequences for bioreactor performance Anders Axelsson* Department
of Chemical Engineering
I, University
of Lund, PO Box 124, S-22100 Lund (Sweden)
Csaba Sisak Research Institute of Chemical Engineering
of the Hungarian
Academy
of Sciences,
Veszprem
(Hunga?-&
Bengt A. Westrin Perstorp Pharmq
Perskn-p AB, 28480 Perstorp
{Sweden)
Bela Szajani Reanal Factory
of Laboratory
Chemicals,
H-1147 Budapest
(Hungaryj
(Received January 10, 1994)
Abstract A diaphragm
diffusion cell was used to determine
the effective diffusion coefficient
of L-methionine
in a
modified polyacrylamide gel at different degrees of swelling. As could be expected from theory, the value of the effective diffusion coefficient increased with increasing swelling. The combined effect of increased radius and increased effective diffusion coefficient is anaiysed in terms of the Thiele modulus and the effectiveness factor. The results can explain the reduced production rate obtained in bioreactors with immobilized
enzymes
using gel carriers
with a shrinking-swelling
1. Introduction The efficiency of reactors with immobilized biocatalysts is strongly influenced by the internal masstransfer characteristics of the support material. The parameters which must be considered, as well as the methods of calculation, have been thoroughly elaborated for numerous pore-diffusion-controlled biochemical reactions [ 11. The use of hydrogels as support material makes the topic even more complex since their properties can be affected by the reaction conditions (e-b. pH, temp.erature, ionic strength etc.) VI. In a previous investigation of amino acid resolution by immobilized amino-acylase [ 31, a polyacrylamide derivative was used as support material in a fluidizedbed reactor. The support material was produced by partial hydrolysis of commercial polyacrylamide gel filtration beads [4], in which approximately 50% of the amide groups had been transformed to carboxylic groups [ 5, 6 1. The activation of these groups by water-soluble carbodi-imides provided immobiliza*Author to whom correspondence should be addressed.
behaviour
during processing.
tion sites for the amino acylase [ 71. These hydrolysed polyacrylamide beads exhibited a significantly altered swelling-shrinking behaviour. A 35% decrease in the catalyst bed height was observed when the ammo acid concentration increased from 0.1 to 0.8 M. A decrease in the efficiency was also observed. For a complete characterization of this reactor it is necessary to quantify the effect of the swellmg-shrinking on the internal diffusion. Preliminary studies revealed that the ionic strength, rather than the pH, had the strongest influence on the swelling. The aim of the present work was to investigate the effect of ionic-strengthinduced swelling-shrinking on the internal diffusion characteristics of the modified polyacrylamide gel. By applying the results to gel beads used in immobilized enzyme reactors, the combined effect of an altered bead radius and an altered effective diffusion coefficient can be explained. 2. Theory 2.1. The eflective [email protected] coeflcient The effective diffusion coefficient D, for a solute in a gel or a porous solid is defmed by
0923-0467/94/$07.00 0 1994 Elsevier Science S.A. All rights reserved SSDI 0923-0467(94)06054-g
A. Axekwn
B36
N= -D,
et al. / Di~ional
2
where N is the diffusion flux of the solute in the x direction, and CL is the solute concentration of the liquid phase in the pores and in the surrounding solution. The concept of liquid&led pores is somewhat misleading in the case of hydrogels, since they do not have well-defined permanent porous structures [ 81. The effective diffusion coefficient is nevertheless a very useful quantity often used to describe the internal mass transfer. D, must not be confused with the diffusion coefficient D for a solute in a gel, which is defined by
(2) where C, is the solute concentration in the gel (i.e. the amount of solute per unit volume of gel). The relation between D, and D can be expressed by D, = (I-
~DP)D
(3)
where (pp is the volume fraction occupied by the polymer and hence not accessible to the solute. 2.2. Influence of swelling on the eflective
da$%sioncoeflcient There are many equations which predict the diffusion coefficients in hydrogels as a function of the polymer concentration [ 8-101. The equation of Mackie and Meares [ll] (4) (where D,, is the corresponding diffusion coefficient in water) requires a minimum of parameters, but still has a high predictive value [ 9, 121. If D, is required, eqn. (3) can be used to rewrite eqn. (4) in the form D
=
e
u-(pP)3D (1 + ‘pp)2
aq
(5)
Thus, since increased swelling means decreased polymer fraction, it is likely to lead to an increased effective diffusion coefficient, and vice versa. 2.3. Internul mass-transfer hindrance for
immobilized enzym.es The internal mass-transfer effects on an immobilized enzyme in a gel bead are most easily discussed in terms of the effectiveness factor q and the Thiele modulus Qi. The effectiveness factor describes the relation between the true reaction rate and the rate
characteristics
of a sweUing
gel
that could hypothetically be achieved if the enzyme were exposed to the bulk concentration CL. The reaction rate for a reaction following Michaelis-Menton kinetics is expressed by
where k (mol kg-’ s-‘) is the rate constant, K, (mol mW3) is the Michaelis-Menton constant and E (kg (m3 gel)- ‘) is the enzyme concentration. The effectiveness factor is thus expressed by 3 D,(KW7-),=, ’ = E [rate(CL)lrZR The Thiele modulus for Michaelis-Menton is deiined by
De(Cdr=~
(7) kinetics
(8)
This means that @ expresses the ratio between the reaction rate and the internal diffusion rate. A large value of @results in a mass-transfer-controlled process while a small value results in a kinetically controlled process. The relation between 77and Cp can be obtained from a numerical solution of a reaction-diffusion model [ 131. From these kinds of calculation it can be concluded that 77is close to unity, and independent of @, for a kinetically controlled process. For a mass-transfercontrolled process, 17is proportional to l/Q. The relative importance of internal mass-transfer hindrance increases when the bead diameter increases, when the effective diffusion coefficient decreases, and when the reaction rates are high (for example those obtained with high biocatalyst activity). For immobilized enzymes, such as the amino acylase used in the previous investigation [3], the catalytic activity is generally high enough for the process to be controlled by mass transfer. At this point it would be of interest to discuss how the swelling, which gives the combined effect of an increased bead radius and an increased effective diffusion coefficient, would influence the enzymatic reaction, assuming a constant amount of immobilized enzyme in each bead. When a certain amount of enzyme has been immobilized in a bead, the enzyme concentration in the spherical bead is expressed by
E=
amount of enzyme 4?rR3/3
(9)
If this is inserted into eqn. (6) and (8) it can be seen that the Thiele modulus is proportional to and that consequently the effectiveness l/(R.WD
A. Axelsson et al. / Di&.Gcmal characteristics of a swelling gel
factor is proportional to (RDe)ln for a mass-transfercontrolled process. This means that the effectiveness factor increases for a swelling bead and is even further increased if the effective diffusion coefficient increases due to the swelling. However, the resulting reaction rate for the single bead, obtained by multiplying the effectiveness factor by [rate(CL)]r_R in eqn. (6), will of course decrease as the same amount of enzyme is present in a larger bead (decreased enzyme concentration).
3. Materials
and methods
3.1. DiJikon numsurmts The effective diffusion coefficients were measured with a diaphragm diffusion cell according to a method described by Westrin and Axelsson [ 14, 151. A 10 mg n-t- ’ solution of L-methionine was used. The same kind of phosphate buffer as that used for storing the gel disc was used as solvent in each experiment. In this way the environment of each gel disc was the same during the diffusion experiment as during storage, except for the negligible change in ionic strength and pH caused by the amino acid. The L-methionine concentrations were analysed with the ninhydrin method [IS], using a Shimadzu UV 160 spectrophotometer. L-methionine solutions of O-10 mg ml-’ were used as a standard series. Three identical experimental set-ups were available. For each group of gel discs (see below), three identical parallel experiments were performed, and the mean value of D, was calculated. 3.2. Preparation of the gel discs A monomer solution was prepared of 8.9 wt.% acrylamide, 8.9 wt.% acrylic acid, 8.7 wt.% tris(hydroxymethyl)aminomethane (TRIS), 0.9 wt.% N,N-methylenbisacrylamide, 0.22 wt.% ammonium persulphate and 7.24 wt.% water. Nitrogen was bubbled through the solution to remove dissolved oxygen. A 5.1 wt.% solution of N,N,N’,N’-tetramethyl-ethylene-diamine (TEMED) was prepared and the solutions were mixed: 1 part TEMED solution to 5.2 parts monomer solution. This solution was then poured into rectangular vertical moulds, and the free-radical solution polymerization thus initiated was allowed to proceed at room temperature. After 1 h the gel sheets were removed from the moulds. Each was weighed and immersed in a 0.500 mol 1-l phosphate buffer solution of pH 6.9. The solution was later replaced with fresh solution. The procedure is suitable for producing gel sheets with internal properties modelling approximately those of the partially hydrolysed polyacrylamide gel
I337
beads with or without immobilized enzymes, used in the reactor studies mentioned in the introduction ]31. After storage for 3 days at room temperature, each gel sheet was again weighed. The sheets were then divided into five groups, each containing three sheets. One group was kept in the 0.500 mol 1-l phosphate buffer solution, while the others were immersed in phosphate buffer solutions of 0.250 mol l-r, 0.100 mol l-‘, 0.050 mol 1-l and 0.001 mol l- ’ respectively. All the solutions had a pH of 6.9. After another 3 days, the sheets were again weighed. They were then cut into circular diaphragms, with a radius of 30 mm, and reimmersed in their solutions. At this point, the diaphragms had thicknesses of 1.8-2.7 mm, which were suitable for the diffusion cell experiments.
4. Results
and discussion
4.1. Swelling of the gels In Table 1 it can be seen how the polymer weight fraction depends on the ionic strength of the phosphate buffer solvent. In the calculation of the ionic strengths only Nit+, HPOI- and H2P04’- ions were taken into consideration. The role of L-methionine was assumed to be negligible because of the low degree of dissociation at a pH of 6.9. 4.2. The effective [email protected] coeficients The mean values of the diffusion coefficient values for each group of experiments are presented in Table 1 and Fig. 1. The full curve in Fig. 1 represents eqn. (5) in which D,=8.2~10-‘~ m2 s-’ and y+ is approximated by the polymer weight fraction. The D, value used is that at 30 “C for norleucine [ 171, an amino acid similar to methionine in shape and size. This value was used since a literature value for Lmethionine was not available. The experimental data have a somewhat stronger dependence on the polymer concentration than predicted by eqn. (5) of Mackie and Meares [ 111, which just considers the exclusion and obstruction effects and not any other effects such as ionic interactions. However, eqn. (5) is still sufficient to give a rough estimate of the effective diffusion coefficient. 4.3. InJluence
of swelling
on biocatalyst
eY@iency The results in Table 1 are likely to be valid not only for the investigated gel discs, but also for the catalyst gel beads used in the previous investigation
A. Axelsson
B38 TABLE
1. Muence
of the ionic strength
et al, / Di~imal
characteristics
of a .weUing
on the swelling and diffusion characteristics
-
gel
experimental
results”
Phosphate buffer concentration (mol 1-l)
Ionic strength
Hydration (g water (g swollen gel)-‘)
Polymer weight fraction (g monomers (g swollen gel)-‘)
Density (g ml-‘)
Q (X 10-i’
0.001 0.050 0.100 0.250 0.500
0.002 0.094 0.189 0.472 0.945
0.974 0.961 0.954 0.943 0.936
0.026 0.039 0.046 0.057 0.064
1.00 1.01 1.02 1.05 1.09
7.85 7.49 6.33 5.64 5.10
Temperature,
m2 s-‘)
30 “C; pH 6.9.
10
F 3 b5c 0”
1 0.025
0. 0
I 0.050
0.075
Polymer weight fraction F’ig. 1. Dependence of the effective diffusion coefficient on the polymer weight function: -, representation of eqn. (5). TABLE 2. The influence of the ionic strength on the effectiveness factor for a single bead (normalized values) calculated from eqns. (6X9) Phosphate buffer concentration (mol 1-i)
Volume
0.001 0.050 0.100 0.250 0.500
1 0.66 0.55 0.43 0.37
Radius
1 0.87 0.82 0.76 0.72
D,
1 0.95 0.81 0.72 0.65
Thiele modulus
Effectiveness factor
1 1.10 1.23 1.35 1.46
1 0.91 0.81 0.74 0.68
[ 31. Based on this assumption, the combined effect of the bead radius and the effective diffusion coefficient is quantified and given in Table 2. It is assumed that the ionic strength does not affect the enzyme activity. The normalized volumes for the different gels were calculated from the values of hydration and density. The normalized volumes are normalized relative to the value for the gel exposed to the lowest ionic strength (0.001 mol I-‘). The radius and the effective diffusion coefficients are
normalized in the same way, as well as the Thiele modulus and the effectiveness factor. Since the Thiele modulus is proportional to 1/ (RLWa and the effectiveness factor is proportional to (De) In for a mass-transfer-controlled process, these can easily be calculated as normalized quantities. When the bead radius decreases by 28% as a result of the increased ionic strength, the effectiveness factor decreases by more than 30%, due to the combined effect of the decreasing radius and the decreasing effective diffusion coefficient. From these data it can be concluded that the effectiveness factor decreases considerably when the ionic strength increases. The actual reaction rate for a single bead is obtained by multiplying the reaction rate expression (eqn. (6)) by the effectiveness factor. Furthermore, the production rate P (moles of product per cubic metre of reactor per second) based on total reactor volume is given by P=(l
-+I
kE(CLL=R Kn + (GLR
(10)
where 1 - E is the volume fraction of gel in the reactor. If the total amount of enzyme in the reactor E(1 - E) is kept constant, the enzyme concentration E in the beads increases in direct proportion to the decrease of the gel volume in the reactor 1 - E. This means that the production rate P is proportional to @z&)‘n. An increased ionic strength will thus decrease the gel volume and subsequently the effectiveness factor as well as the production rate. Thus, if the total amount of enzyme is constant in the reactor, the conversion decreases when the substrate concentration increases not only due to pure reaction kinetics but also due to the decreased effectiveness factor. This was observed in the previous experimental study (31 where immobilized aminoacylase was used for the resolution of racemic amino acids. In a future study the reactor performance will be compared with computer simulations to explain the
A. Axelsson
et al. / D@~~ional
reactor behaviour in detail. The experimentally determined effective diffusion coefficients will in that study be of great importance to describe accurately the internal mass transfer.
Acknowledgments This work was Snancially supported by the Hungarian Academy of Sciences and the Royal Swedish Academy of Engineering Sciences.
characteristics
of a s-we&z&? gel
11 J.S. Ma&e and P. Meares, Prwc. R. Sot. London, Ser. A, 232 (1955) 498. 12 W. Brown and R.M. Johnson, Polymer, 22 (1981) 185. 13 Y.Y. Lee and G.T. Tsao, J. Food Sci., 39 (1974) 667. 14 B.A. Westrin and A. Axelsson, Biotechnol. Tech., 5 (1991) 303. 15 B.A. Westrin, Diffusion measurement in gels, PhD Thesis, Department of Chemical Engineering 1, University of Lund, 1991. 16 S. Moore, J. Bill. Chem., 243 (1968) 6281. 17 L.G. Longsworth, J. Am. Chern. Sot., 75 (1953) 5705.
Appendix A: Nomenclature
co
References C. Horvath and J.M. Engasser, Biotechnol. Bioeng., 16 (1974) 909. T.G. Park and A.S. Hoffman, Appl. Biochem. B&xn.g., 19 (1988) 1. Cs. Sisak, L. Boross and B. Szajani, in F. Hites (ed.), F’roc. 5th CorEf. vn Applied Chemistry Unit Operations and Processes, Vol. 2, EFCE Publication Series, No. 74, Hungarian Chemical Society, Budapest, 1989, p. 365. Reanal Product for Solid-Phase Biochemistry, Reanal, Budapest, 1991. J.K. Inman and H.M. Dintzis, B&hem., 8 (1969) 4074. J.K. Inman, Methods Enzymol., 34 (1974) 30. B. Sz&%ni, K. Ivony and L. Boross, J. Appl. Biochem., 2 (1980) 72. NA. Peppas and S.R. Lustig, in N.A. Peppas (ed.), Hydrogels in Medicine and Phurmacy, CRC Press, Boca Raton, FL, 1986, p. 57. A.H. Muhr and J.M.V. Blanshard, Polymer, 23 (1982) 1012. 10 H. Yasuda, C.E. Lamaae and L.D. Ikenberry, M&omol. Chem., 118 (1968) 19.
B39
concentration in gel (mol (m3 gel)-‘) concentration in bulk or pore liquid (mol mm3) diffusion coefficient in gel (m” s-l) D D-2 diffusion coefficient in water (m” s- ‘) DC? effective diffusion coefficient in gel (m’ s-‘) E enzyme concentration (kg (m3 gel)-‘) reaction rate constant (mol kg- ’ s- ‘) k Michaelis-Menton constant (mol mP3) Kll N molar flux in x direction (mol m-’ s-l) P production rate (moles per cubic metre of reactor per second) spherical length coordinate (m) ; bead radius (m) linear length coordinate (m) X
CL
Greek E
77 (Pp @
letters
reactor void volume effectiveness factor volume fraction of polymer in a gel Thiele modulus
B35
Journal, 55 (1994) B35-B39
Diffusional characteristics of a swelling gel and its consequences for bioreactor performance Anders Axelsson* Department
of Chemical Engineering
I, University
of Lund, PO Box 124, S-22100 Lund (Sweden)
Csaba Sisak Research Institute of Chemical Engineering
of the Hungarian
Academy
of Sciences,
Veszprem
(Hunga?-&
Bengt A. Westrin Perstorp Pharmq
Perskn-p AB, 28480 Perstorp
{Sweden)
Bela Szajani Reanal Factory
of Laboratory
Chemicals,
H-1147 Budapest
(Hungaryj
(Received January 10, 1994)
Abstract A diaphragm
diffusion cell was used to determine
the effective diffusion coefficient
of L-methionine
in a
modified polyacrylamide gel at different degrees of swelling. As could be expected from theory, the value of the effective diffusion coefficient increased with increasing swelling. The combined effect of increased radius and increased effective diffusion coefficient is anaiysed in terms of the Thiele modulus and the effectiveness factor. The results can explain the reduced production rate obtained in bioreactors with immobilized
enzymes
using gel carriers
with a shrinking-swelling
1. Introduction The efficiency of reactors with immobilized biocatalysts is strongly influenced by the internal masstransfer characteristics of the support material. The parameters which must be considered, as well as the methods of calculation, have been thoroughly elaborated for numerous pore-diffusion-controlled biochemical reactions [ 11. The use of hydrogels as support material makes the topic even more complex since their properties can be affected by the reaction conditions (e-b. pH, temp.erature, ionic strength etc.) VI. In a previous investigation of amino acid resolution by immobilized amino-acylase [ 31, a polyacrylamide derivative was used as support material in a fluidizedbed reactor. The support material was produced by partial hydrolysis of commercial polyacrylamide gel filtration beads [4], in which approximately 50% of the amide groups had been transformed to carboxylic groups [ 5, 6 1. The activation of these groups by water-soluble carbodi-imides provided immobiliza*Author to whom correspondence should be addressed.
behaviour
during processing.
tion sites for the amino acylase [ 71. These hydrolysed polyacrylamide beads exhibited a significantly altered swelling-shrinking behaviour. A 35% decrease in the catalyst bed height was observed when the ammo acid concentration increased from 0.1 to 0.8 M. A decrease in the efficiency was also observed. For a complete characterization of this reactor it is necessary to quantify the effect of the swellmg-shrinking on the internal diffusion. Preliminary studies revealed that the ionic strength, rather than the pH, had the strongest influence on the swelling. The aim of the present work was to investigate the effect of ionic-strengthinduced swelling-shrinking on the internal diffusion characteristics of the modified polyacrylamide gel. By applying the results to gel beads used in immobilized enzyme reactors, the combined effect of an altered bead radius and an altered effective diffusion coefficient can be explained. 2. Theory 2.1. The eflective [email protected] coeflcient The effective diffusion coefficient D, for a solute in a gel or a porous solid is defmed by
0923-0467/94/$07.00 0 1994 Elsevier Science S.A. All rights reserved SSDI 0923-0467(94)06054-g
A. Axekwn
B36
N= -D,
et al. / Di~ional
2
where N is the diffusion flux of the solute in the x direction, and CL is the solute concentration of the liquid phase in the pores and in the surrounding solution. The concept of liquid&led pores is somewhat misleading in the case of hydrogels, since they do not have well-defined permanent porous structures [ 81. The effective diffusion coefficient is nevertheless a very useful quantity often used to describe the internal mass transfer. D, must not be confused with the diffusion coefficient D for a solute in a gel, which is defined by
(2) where C, is the solute concentration in the gel (i.e. the amount of solute per unit volume of gel). The relation between D, and D can be expressed by D, = (I-
~DP)D
(3)
where (pp is the volume fraction occupied by the polymer and hence not accessible to the solute. 2.2. Influence of swelling on the eflective
da$%sioncoeflcient There are many equations which predict the diffusion coefficients in hydrogels as a function of the polymer concentration [ 8-101. The equation of Mackie and Meares [ll] (4) (where D,, is the corresponding diffusion coefficient in water) requires a minimum of parameters, but still has a high predictive value [ 9, 121. If D, is required, eqn. (3) can be used to rewrite eqn. (4) in the form D
=
e
u-(pP)3D (1 + ‘pp)2
aq
(5)
Thus, since increased swelling means decreased polymer fraction, it is likely to lead to an increased effective diffusion coefficient, and vice versa. 2.3. Internul mass-transfer hindrance for
immobilized enzym.es The internal mass-transfer effects on an immobilized enzyme in a gel bead are most easily discussed in terms of the effectiveness factor q and the Thiele modulus Qi. The effectiveness factor describes the relation between the true reaction rate and the rate
characteristics
of a sweUing
gel
that could hypothetically be achieved if the enzyme were exposed to the bulk concentration CL. The reaction rate for a reaction following Michaelis-Menton kinetics is expressed by
where k (mol kg-’ s-‘) is the rate constant, K, (mol mW3) is the Michaelis-Menton constant and E (kg (m3 gel)- ‘) is the enzyme concentration. The effectiveness factor is thus expressed by 3 D,(KW7-),=, ’ = E [rate(CL)lrZR The Thiele modulus for Michaelis-Menton is deiined by
De(Cdr=~
(7) kinetics
(8)
This means that @ expresses the ratio between the reaction rate and the internal diffusion rate. A large value of @results in a mass-transfer-controlled process while a small value results in a kinetically controlled process. The relation between 77and Cp can be obtained from a numerical solution of a reaction-diffusion model [ 131. From these kinds of calculation it can be concluded that 77is close to unity, and independent of @, for a kinetically controlled process. For a mass-transfercontrolled process, 17is proportional to l/Q. The relative importance of internal mass-transfer hindrance increases when the bead diameter increases, when the effective diffusion coefficient decreases, and when the reaction rates are high (for example those obtained with high biocatalyst activity). For immobilized enzymes, such as the amino acylase used in the previous investigation [3], the catalytic activity is generally high enough for the process to be controlled by mass transfer. At this point it would be of interest to discuss how the swelling, which gives the combined effect of an increased bead radius and an increased effective diffusion coefficient, would influence the enzymatic reaction, assuming a constant amount of immobilized enzyme in each bead. When a certain amount of enzyme has been immobilized in a bead, the enzyme concentration in the spherical bead is expressed by
E=
amount of enzyme 4?rR3/3
(9)
If this is inserted into eqn. (6) and (8) it can be seen that the Thiele modulus is proportional to and that consequently the effectiveness l/(R.WD
A. Axelsson et al. / Di&.Gcmal characteristics of a swelling gel
factor is proportional to (RDe)ln for a mass-transfercontrolled process. This means that the effectiveness factor increases for a swelling bead and is even further increased if the effective diffusion coefficient increases due to the swelling. However, the resulting reaction rate for the single bead, obtained by multiplying the effectiveness factor by [rate(CL)]r_R in eqn. (6), will of course decrease as the same amount of enzyme is present in a larger bead (decreased enzyme concentration).
3. Materials
and methods
3.1. DiJikon numsurmts The effective diffusion coefficients were measured with a diaphragm diffusion cell according to a method described by Westrin and Axelsson [ 14, 151. A 10 mg n-t- ’ solution of L-methionine was used. The same kind of phosphate buffer as that used for storing the gel disc was used as solvent in each experiment. In this way the environment of each gel disc was the same during the diffusion experiment as during storage, except for the negligible change in ionic strength and pH caused by the amino acid. The L-methionine concentrations were analysed with the ninhydrin method [IS], using a Shimadzu UV 160 spectrophotometer. L-methionine solutions of O-10 mg ml-’ were used as a standard series. Three identical experimental set-ups were available. For each group of gel discs (see below), three identical parallel experiments were performed, and the mean value of D, was calculated. 3.2. Preparation of the gel discs A monomer solution was prepared of 8.9 wt.% acrylamide, 8.9 wt.% acrylic acid, 8.7 wt.% tris(hydroxymethyl)aminomethane (TRIS), 0.9 wt.% N,N-methylenbisacrylamide, 0.22 wt.% ammonium persulphate and 7.24 wt.% water. Nitrogen was bubbled through the solution to remove dissolved oxygen. A 5.1 wt.% solution of N,N,N’,N’-tetramethyl-ethylene-diamine (TEMED) was prepared and the solutions were mixed: 1 part TEMED solution to 5.2 parts monomer solution. This solution was then poured into rectangular vertical moulds, and the free-radical solution polymerization thus initiated was allowed to proceed at room temperature. After 1 h the gel sheets were removed from the moulds. Each was weighed and immersed in a 0.500 mol 1-l phosphate buffer solution of pH 6.9. The solution was later replaced with fresh solution. The procedure is suitable for producing gel sheets with internal properties modelling approximately those of the partially hydrolysed polyacrylamide gel
I337
beads with or without immobilized enzymes, used in the reactor studies mentioned in the introduction ]31. After storage for 3 days at room temperature, each gel sheet was again weighed. The sheets were then divided into five groups, each containing three sheets. One group was kept in the 0.500 mol 1-l phosphate buffer solution, while the others were immersed in phosphate buffer solutions of 0.250 mol l-r, 0.100 mol l-‘, 0.050 mol 1-l and 0.001 mol l- ’ respectively. All the solutions had a pH of 6.9. After another 3 days, the sheets were again weighed. They were then cut into circular diaphragms, with a radius of 30 mm, and reimmersed in their solutions. At this point, the diaphragms had thicknesses of 1.8-2.7 mm, which were suitable for the diffusion cell experiments.
4. Results
and discussion
4.1. Swelling of the gels In Table 1 it can be seen how the polymer weight fraction depends on the ionic strength of the phosphate buffer solvent. In the calculation of the ionic strengths only Nit+, HPOI- and H2P04’- ions were taken into consideration. The role of L-methionine was assumed to be negligible because of the low degree of dissociation at a pH of 6.9. 4.2. The effective [email protected] coeficients The mean values of the diffusion coefficient values for each group of experiments are presented in Table 1 and Fig. 1. The full curve in Fig. 1 represents eqn. (5) in which D,=8.2~10-‘~ m2 s-’ and y+ is approximated by the polymer weight fraction. The D, value used is that at 30 “C for norleucine [ 171, an amino acid similar to methionine in shape and size. This value was used since a literature value for Lmethionine was not available. The experimental data have a somewhat stronger dependence on the polymer concentration than predicted by eqn. (5) of Mackie and Meares [ 111, which just considers the exclusion and obstruction effects and not any other effects such as ionic interactions. However, eqn. (5) is still sufficient to give a rough estimate of the effective diffusion coefficient. 4.3. InJluence
of swelling
on biocatalyst
eY@iency The results in Table 1 are likely to be valid not only for the investigated gel discs, but also for the catalyst gel beads used in the previous investigation
A. Axelsson
B38 TABLE
1. Muence
of the ionic strength
et al, / Di~imal
characteristics
of a .weUing
on the swelling and diffusion characteristics
-
gel
experimental
results”
Phosphate buffer concentration (mol 1-l)
Ionic strength
Hydration (g water (g swollen gel)-‘)
Polymer weight fraction (g monomers (g swollen gel)-‘)
Density (g ml-‘)
Q (X 10-i’
0.001 0.050 0.100 0.250 0.500
0.002 0.094 0.189 0.472 0.945
0.974 0.961 0.954 0.943 0.936
0.026 0.039 0.046 0.057 0.064
1.00 1.01 1.02 1.05 1.09
7.85 7.49 6.33 5.64 5.10
Temperature,
m2 s-‘)
30 “C; pH 6.9.
10
F 3 b5c 0”
1 0.025
0. 0
I 0.050
0.075
Polymer weight fraction F’ig. 1. Dependence of the effective diffusion coefficient on the polymer weight function: -, representation of eqn. (5). TABLE 2. The influence of the ionic strength on the effectiveness factor for a single bead (normalized values) calculated from eqns. (6X9) Phosphate buffer concentration (mol 1-i)
Volume
0.001 0.050 0.100 0.250 0.500
1 0.66 0.55 0.43 0.37
Radius
1 0.87 0.82 0.76 0.72
D,
1 0.95 0.81 0.72 0.65
Thiele modulus
Effectiveness factor
1 1.10 1.23 1.35 1.46
1 0.91 0.81 0.74 0.68
[ 31. Based on this assumption, the combined effect of the bead radius and the effective diffusion coefficient is quantified and given in Table 2. It is assumed that the ionic strength does not affect the enzyme activity. The normalized volumes for the different gels were calculated from the values of hydration and density. The normalized volumes are normalized relative to the value for the gel exposed to the lowest ionic strength (0.001 mol I-‘). The radius and the effective diffusion coefficients are
normalized in the same way, as well as the Thiele modulus and the effectiveness factor. Since the Thiele modulus is proportional to 1/ (RLWa and the effectiveness factor is proportional to (De) In for a mass-transfer-controlled process, these can easily be calculated as normalized quantities. When the bead radius decreases by 28% as a result of the increased ionic strength, the effectiveness factor decreases by more than 30%, due to the combined effect of the decreasing radius and the decreasing effective diffusion coefficient. From these data it can be concluded that the effectiveness factor decreases considerably when the ionic strength increases. The actual reaction rate for a single bead is obtained by multiplying the reaction rate expression (eqn. (6)) by the effectiveness factor. Furthermore, the production rate P (moles of product per cubic metre of reactor per second) based on total reactor volume is given by P=(l
-+I
kE(CLL=R Kn + (GLR
(10)
where 1 - E is the volume fraction of gel in the reactor. If the total amount of enzyme in the reactor E(1 - E) is kept constant, the enzyme concentration E in the beads increases in direct proportion to the decrease of the gel volume in the reactor 1 - E. This means that the production rate P is proportional to @z&)‘n. An increased ionic strength will thus decrease the gel volume and subsequently the effectiveness factor as well as the production rate. Thus, if the total amount of enzyme is constant in the reactor, the conversion decreases when the substrate concentration increases not only due to pure reaction kinetics but also due to the decreased effectiveness factor. This was observed in the previous experimental study (31 where immobilized aminoacylase was used for the resolution of racemic amino acids. In a future study the reactor performance will be compared with computer simulations to explain the
A. Axelsson
et al. / D@~~ional
reactor behaviour in detail. The experimentally determined effective diffusion coefficients will in that study be of great importance to describe accurately the internal mass transfer.
Acknowledgments This work was Snancially supported by the Hungarian Academy of Sciences and the Royal Swedish Academy of Engineering Sciences.
characteristics
of a s-we&z&? gel
11 J.S. Ma&e and P. Meares, Prwc. R. Sot. London, Ser. A, 232 (1955) 498. 12 W. Brown and R.M. Johnson, Polymer, 22 (1981) 185. 13 Y.Y. Lee and G.T. Tsao, J. Food Sci., 39 (1974) 667. 14 B.A. Westrin and A. Axelsson, Biotechnol. Tech., 5 (1991) 303. 15 B.A. Westrin, Diffusion measurement in gels, PhD Thesis, Department of Chemical Engineering 1, University of Lund, 1991. 16 S. Moore, J. Bill. Chem., 243 (1968) 6281. 17 L.G. Longsworth, J. Am. Chern. Sot., 75 (1953) 5705.
Appendix A: Nomenclature
co
References C. Horvath and J.M. Engasser, Biotechnol. Bioeng., 16 (1974) 909. T.G. Park and A.S. Hoffman, Appl. Biochem. B&xn.g., 19 (1988) 1. Cs. Sisak, L. Boross and B. Szajani, in F. Hites (ed.), F’roc. 5th CorEf. vn Applied Chemistry Unit Operations and Processes, Vol. 2, EFCE Publication Series, No. 74, Hungarian Chemical Society, Budapest, 1989, p. 365. Reanal Product for Solid-Phase Biochemistry, Reanal, Budapest, 1991. J.K. Inman and H.M. Dintzis, B&hem., 8 (1969) 4074. J.K. Inman, Methods Enzymol., 34 (1974) 30. B. Sz&%ni, K. Ivony and L. Boross, J. Appl. Biochem., 2 (1980) 72. NA. Peppas and S.R. Lustig, in N.A. Peppas (ed.), Hydrogels in Medicine and Phurmacy, CRC Press, Boca Raton, FL, 1986, p. 57. A.H. Muhr and J.M.V. Blanshard, Polymer, 23 (1982) 1012. 10 H. Yasuda, C.E. Lamaae and L.D. Ikenberry, M&omol. Chem., 118 (1968) 19.
B39
concentration in gel (mol (m3 gel)-‘) concentration in bulk or pore liquid (mol mm3) diffusion coefficient in gel (m” s-l) D D-2 diffusion coefficient in water (m” s- ‘) DC? effective diffusion coefficient in gel (m’ s-‘) E enzyme concentration (kg (m3 gel)-‘) reaction rate constant (mol kg- ’ s- ‘) k Michaelis-Menton constant (mol mP3) Kll N molar flux in x direction (mol m-’ s-l) P production rate (moles per cubic metre of reactor per second) spherical length coordinate (m) ; bead radius (m) linear length coordinate (m) X
CL
Greek E
77 (Pp @
letters
reactor void volume effectiveness factor volume fraction of polymer in a gel Thiele modulus