Enzyme Adsorption in Porous Supports: Local Thermodynamic Equilibrium Model

Henrik Pedersen and Larry Furler Department of Chemical and Biochemical Engineering, Rutgers University, P. 0. Box 909, Piscatawa y, New Jersey 08854

K. Venkatasubramanian Department of Chemical and Biochemical Engineering, Rutgers University, P. 0. Box 909, Piscataway, New Jersey 08854 and H. J. Heinz Company, World Headquarters, P. 0. Box 57, Pittsburgh, Pennsylvania 15230

Jiri Prenosil and Ernst Stuker Department of Chemical Engineering, Swiss Federal Institute of Technology, CH-8092, Zurich, Switzerland

Accepted for Publication October 12, 1984

Enzyme adsorption from a finite bath (batch adsorption) onto porous spherical supports is investigated both ex- perimentally and theoretically using p-galactosidase and Duolite ion-exchange resin as a model system. Efficient numerical techniques are presented that have been used in conjunction with a parameter estimation routine to evaluate adsorption isotherm constants. Results show that even for adsorption processes lasting almost 10 h, the majority of the enzyme is confined to the outer half of the support and, for high initial enzyme concentrations in the bath, this loading takes place as a slowly moving front. Information on the enzyme distribution has practical importance in the design of immobilized enzyme reactors that in previous works have almost always been analyzed assuming a uniform catalyst distribution.

INTRODUCTION

immobilized Enzymes

To facilitate the use of enzymes as catalysts in various industrial, analytical, or medical applications, it is often convenient to immobilize the enzyme. This is generally accomplished by either entrapment or by adsorption to a solid support that may be chemically treated to promote binding. Entrapment methods may be carried out with gels or membranes. Gel entrapment is more suited for cells or organelles, however, than for the relatively small enzyme molecules.' The variety of supports capable of adsorbing enzymes include acti- vated carbon, alumina, cellulose, ion-exchange resins, and activated Sephadex or Sepharose, for example.'

In many cases, the enzyme is covalently bound or crosslinked to the carrier to stabilize the catalyst but, if relatively strong adsorptive forces are employed such as ionic or in some cases hydrophobic bonding the adsorbed enzymes may be sufficiently stable to be used as catalysts without additional covalent bind- ing.3 There are also situations where covalent binding is too harsh for the enzyme, particularly if the enzyme is a large multisubunit complex. Then, simple adsorption or entrapment is preferred.

For the vast majority of applications porous supports are superior to nonporous supports due to the higher surface area available for adsorption that is often orders of magnitude greater than the geometric surface area of the ~ a r r i e r . ~ Enzyme loading on porous supports has often been assumed to be uniform, but it has been shown that this is not always the case and may, in fact, be undesirable in many Immobilized enzymes are susceptible to internal mass transfer lim- itations which decrease their catalytic efficiency, and in such cases it proves preferable to immobilize the enzyme primarily near the surface. For example, Hor- vath and Engasser7 have demonstrated that for simple Michaelis-Menten kinetics, a given enzyme loading is better utilized, i.e., has a higher value of the ef- fectiveness factor, when arranged in a pellicular con- figuration. This has important practical consequences when the immobilized enzyme particles are used in packed-bed reactors. Similar effects have been ex- amined by several authors in other areas of hetero- geneous catalysis.8-'o Camers with activity concentrated

Biotechnology and Bioengineering, Vol. XXVII, Pp. 961-971 (1985) 0 1985 John Wiley & Sons, Inc. CCC 0006-3592/85/070961-11$04.00

near the surface generally demonstrate superior per- formance and selectivity but are also generally less stable, deactivating more rapidly because of the higher effectiveness factors. These results are based, of course, on the apparent rates and it can be assumed that the intrinsic rates are independent of the enzyme loading. The determination of an optimal profile is thus influ- enced by the specific deactivation mechanism, the rate of deactivation and the diffusional resistances of the system. Once the importance of the enzyme distribution within the carrier is established it becomes desirable to develop a model to predict this distribution. The previous work of Horvath and Engasser7 dealt with an enzyme distribution that was determined a priori.

Enzyme Adsorption

Several models describing simultaneous adsorption and diffusion in porous particles have appeared in the literature. Marcussen" investigated water adsorption from air in porous alumina. The support was modeled as spherical particles of uniform porosity surrounded by air at constant humidity. The model equations are analogous to the problem of enzyme immobilization in an infinite bath. Nere tne ik~ '~ . '~ developed a com- prehensive model for adsorption in finite baths and countercurrent flow systems that allows for surface or solid diffusion as well as pore diffusion. This model was used to investigate the adsorption of aromatics in activated carbon and to experimentally determine effective diffusion coefficients, but may be applied to other systems as well. Carleysmith and co -w~rke r s '~ demonstrated nonuniform enzyme adsorption on Am- berlite XAD-7 using a protein staining procedure to view the immobilization profile of penicillin acylase and bovine serum albumin. A model was developed describing the rapid irreversible binding of an enzyme to a porous spherical carrier of constant high capacity. Under these conditions, the immobilization occurs in a saturated boundary layer near the outer surface of the carrier. As the immobilization proceeds this boundary layer penetrates into the carrier. Expessions were developed for determining the penetration depth and loading in finite and infinite baths. B u c h ~ l z ' ~ mod- eled reversible enzyme adsorption in spheres of uniform porosity using a Langmuir adsorption isotherm. Do and co-workersI6 expanded on the work of Bucholz, increasing the applicability of the model and introducing different numerical methods of solution. Furthermore, a convenient lumped parameter model was shown to be particularly useful although not particularly easy to develop.

While most models discussed above allow that dif- fusion through the pores of the carrier occurs far slower than adsorption, permitting the assumption that all points within the carrier remain in local thermodynamic equilibrium, Do and co-workersI6 allowed for Separate

adsorption and desorption steps so that irreversible (covalent) binding could be accounted for. The rate equation used yields a Langmuir isotherm at equilibrium for the reversible process. In this work, we treat re- versible binding with a local thermodynamic equilibrium assumption. The model equations are solved using polynomial approximation and the method of orthogonal collocation. In this way, solutions are obtained with a minimum of computational effort and some simple lumped approximations are easily developed. Results are compared with experimental data using P-galac- tosidase immobilized in a porous ion-exchange resin as well as with the results from the theoretical model of Do and co-workers.I6

EXPERIMENTAL

Materials

The P-galactosidase (E.C. 3.2.1.23) from Aspergillus niger (Lactase AN) was obtained from Societe Rapidase (Seclin, France). The ion-exchange resin was type Duolite S-761, a phenol-formaldehyde matrix obtained from Dia-Prosim (Vitry, France). All reagents were analytical grade and materials of construction were glass, Teflon, or stainless steel.

Methods

The experimental methods are described in detail by Stuker.I7 Some important points are mentioned below.

Carrier Preparation

The carrier material was stored in NaCl for 12 h, washed, and used in a stirred vessel or packed into a specially prepared column. Particle sizes of 0.4-0.8 mm diameter were used in all experiments. The column arrangement is shown in Figure 1. In the column ex- periments, the enzyme solution was initially pumped from the well-mixed vessel through the bypass shunt (3). At t = 0, valves 1 and 2 were adjusted and the solution started to pass through the column. At the same time the column was drained until the entire volume of the column was removed. The drain (2) was then closed and the recirculation resumed. In this way, no dilution takes place. All operations were carried out at 22°C.

Sampling

Samples of the recirculating or bulk fluid were taken during the adsorption run for time periods up to 500 min. Sample sizes did not effect the bulk liquid volume. Catalyst samples were taken after immobilization to determine the global rates for the immobilized enzyme.

962 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 27, JULY 1985

3

COLUMN REACTOR

SAMPLE PORT

Figure 1. Schematic diagram of the apparatus used to investigate enzyme adsorption in porous carriers. The total volume is 50-70 mL, the column has dimensions of 24 cm length and 1.8 cm width, and the flow rates used were in the range 4-45 mL/min. Typically, 10-22 g carrier could be used in this apparatus. In batch experiments, the catalyst was contained in the stirred reservoir.

Enzyme Assay

The enzyme activity was followed by monitoring initial rate glucose levels with an automated analyzer (Yellow Springs Instruments, model 27, Chicago, IL). The glucose levels are a measure of the extent of lactose hydrolysis. Reactions for soluble or immobilized enzyme were carried out at 30°C, pH 7.2, in 0.1M phosphate buffer using excess lactose so that zero- order reaction kinetics were followed. Protein con- centration was also monitored by using a modification of the Lowry method."

THEORETICAL MODEL

The immobilization of enzymes on porous supports is commonly accomplished by immersing the carrier, initially free of enzyme, in a concentrated enzyme solution or by an in situ process whereby the enzyme solution is recirculated over the carrier. The adsorption process may be viewed in three steps: first, diffusion of enzyme to the carrier surface through an external boundary layer that is determined by the mixing in the bulk solution; subsequently, both internal diffusion and adsorption occur simultaneously as the enzyme is loaded onto the porous matrix.

In this analysis, the porous support particles are considered as spheres suspended in a finite bath as shown schematically in Figure 2. The pores of the

E, - initml E mentrot ion

d - dimemionless E con-

V, - solid (gewnetnc) phase

centration

volume

RESERMWR (FINITE BATH)

E, - vad f m t m POROUS ADSORBENT

(SOLID

o, - surface uea per volume

- dimensonless E concentration in the pores

E conxntmtlal on the solid

PHASE)

6: - dimensonless

Figure 2. associated nomenclature.

Pictorial representation of the adsorption system and

support initially contain solute devoid of enzyme and the initial enzyme concentration in the bulk phase is assumed to be sufficiently low to effectively eliminate net fluid convection into the particles due to diffusion. While the factors influencing enzyme adsorption and diffusion through the pores of the carrier are essentially fixed once a given combination of enzyme and carrier are chosen, the transfer rate from the bulk solution to the carrier surface is a function of the bulk mixing in the system and hence subject to ready manipulation. It is thus possible to decrease the effects of the external film resistance to mass transfer to the point where it is insignificant, subject to the constraints imposed by the stability of both the carrier and the enzyme to fluid shear. This analysis assumes sufficient mixing is possible to eliminate the. effects of the external film resistance that was also verified here by experiments with the p-galactosidaselion-exchange resin system.

Model Equations

The physical system and the associated notation are also depicted in Figure 2. The governing equation for enzyme adsorption in the porous support material is

where q is the total enzyme concentration in the sup- port, given by

E, and E:' are the liquid (pore) and solid (surface) phase concentrations, respectively, and a, is the solid area available for enzyme adsorption per unit volume.

PEDERSEN ET AL: ENZYME ADSORPTION 963

The liquid effective diffusivity is D, , the void fraction is E, and the parameter m is

1 - E, m = - 8,

( 3 )

The associated boundary conditions are

q = 0; t = O , O < r < R ( 4 )

E, = E; t > O , r = R ( 5 ) The dimensionless variables 4 are introduced by

normalizing the corresponding concentrations E to the initial concentration Eo. The dimensionless time, 8, and distance, 5, are

_ - aEs - 0 ; t > O , r = O ar

Since the bath (bulk) is finite, we also have that

dE dt

V - = - &,D,4rR2 - (7)

where V , is the bath volume. The concentration of enzyme in the bath, E, is the only observable dependent variable under typical experimental conditions. The initial concentration of enzyme at time t = 0 is just EO .

These equations are supplemented by an equilibrium relation for the solid-phase and local liquid-phase con- centrations. We choose to work here with a Langmuir adsorption isotherm.

r [ = - R

The parameters that define the particular problem are

The isotherm parameters Ei,,, and K , are functions of the support material as well as the enzyme. At this point we note that if the local thermodynamic equi- librium assumption is relaxed, the above equation is replaced by

dE:' dt (9)

that must be solved for all local values of E, . The equilibrium parameter K, = k,,/k, and eq. (8) is re- covered if local thermodynamic equilibrium is assumed so that the surface concentration tracks the local liquid phase concentration. Furthermore, if external diffusion is important then eq. (5 ) becomes

- - - ka(Ey,,, - EY)E, - kdEg

and the model is reduced to the functional form 4 =

446; NR, m 4 Yrn 0).

Numerical Techniques

It is convenient to introduce the transformation x = t2 to eliminate eq. (15) and the two-point nature of the boundary conditions. Thus, the governing equa- tions are further reduced to

and the boundary conditions given in eqs. ( 1 3 ) , (14), and (16).

These equations may be solved using the method of orthogonal collocation. Let

where kL is the mass transfer coefficient. Eqs. ( 1 ) - (4) , (6) , (7), (9) , and (10) are essentially the model of Do and co-workers.16

In dimensionless form, eqs. (1)-(8) are reduced to

be approximations to the corresponding derivatives at the locations xi that are the roots of an Nth-order Jacobi polynomial. The semidiscretization elements A , and B, that are used to approximate the first and

964 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 27, JULY 1985

second derivatives, respectively, depend on the par- ticular polynomial approximation used. The evaluation of these elements and the underlying theoretical support for the method can be found in Villadsen and Michelsen” who also provide subroutine listings that we have made use of in our work. The boundary condition is just +s ,N+l = 4 where N is the number of internal collocation points that corresponds to a particular Nth-order polynomial approximation. Sub- stituting the above relationships into the diffusion equations yields a set of first-order ordinary differential equations :

where (29)

that are easily integrated by an explicit Runge-Kutta type method” or similar technique with the initial con- ditions &(O) = 0 and +(O) = 1 .

C . . = 4 x 3 . . + 6A..

Parameter Estimation

Under conditions where the relative volumes are known and the pore diffusion coefficient can be es- timated a priori, the parameters to work with are 4 y,,, and w . The diffusion coefficient is estimated using the method described by Chantong and Massoth” that accounts for restrictive diffusion of the relatively large enzyme molecules. We thus find that D , = 1.5 x lo-’ cm’/s for diffusion of P-galactosidase in the ion- exchange resin. Alternatively, we could carry out equilibrium experiments and obtain the isotherm pa- rameters from the usual linear plots and thus estimate D, . * I However, the time to “reach” equilibrium is quite long and it is easier to work with the transient data directly. In fact, all the parameters can then be extracted from a single experiment.

The experimental data, 4 vs. 8, are compared to the model predictions by choosing parameters 4 ym and w that give a best f i t of the model to the data. A nonlinear parameter estimation technique is followed22223 that uses a weighting factor for the residuals proportional to 4 - * so that the information near equilibrium con- ditions is highlighted. We have found, in fact, that any reasonable weighting scheme, including equal weighting, gives similar results. Details are described by F ~ r l e r . ’ ~

RESULTS AND DISCUSSION

Accuracy of the Numerical Technique

The solution to eqs. (27) and (28) depends on the choice of N and the Jacobi polynomial used for the

basis functions. The choice of the Jacobi polynomial is not crucial and is simply motivated by the spherical geometry and the integral used to evaluate the average pore concentration.” The Jacobi polynomial we have used in all numerical approximations is characterized by the weighing factor ~ ‘ ’ ~ ( 1 - x ) over the interval 0 < x < 1. The effect of N , the number of internal collocation points, on the accuracy is more pronounced, however, as shown in Figure 3 for parameter values that are typical of our subsequent experimental results. The simple choice N = 1 is of some interest even though it gives a poor estimate of the bulk phase enzyme concentration (compared to the curve N = 8 that is the “exact” result). The use of N = 1 cor- responds to the lumped model for our work and leads to the set of equations

_ - _ - d4 do

Basically, the lumped model assumes a parabolic profile for the enzyme concentration +s in the support. Un- fortunately, this turns out to be a poor choice as dis- cussed later since the concentration profile actually displays a number of inflection points. Interestingly, Rice and ~ o - w o r k e r s ~ ~ have found that the above type of lumped model is much more accurate in a related problem dealing with leaching in an open system. A more suitable lumping procedure for enzyme adsorption is discussed by Do and co-workers.’6 However, their procedure does not give any information on the con- centration profile in the support and this may limit its applicability in reactor design. Also, their approach relies on the assumption that NR << 1. We have chosen to work with N = 4 that represents a compromise between extreme accuracy and computational speed. This is of some importance since the method is used internally by a parameter estimation routine.

At large 8 values a mass balance shows that all results asymptotically tend towards

where 4z = +(m) and b = NR(+y,,, + o) + w - 1. Our computational results agree exactly with this value, however, as shown in Figure 4 the results found from the model of Do and co-workers’6 are different. The primary reason for this is simply that they have chosen to neglect terms involving pore (liquid) enzyme con- centrations in the mass balance. Even for values of NR = 0.05, however, the equilibrium values of the bulk enzyme concentration are not accurately deter- mined, particularly at large w values. As either NR becomes smaller or 4 ,, becomes larger, the accuracy of their approach improves however. The nonequili- brium values are different for the two models as well

PEDERSEN ET AL: ENZYME ADSORPTION 965

- -

-

I I I I 1 I I I I OO I 2 3 4 5 6 7 8 9 0

TIME, 8

Figure 3. Effect of N on the accuracy of the numerical techniques. Parameter values are NR = 0.033, E, = 0.6, Q:,,, = 168.75, and o = 0.1.

TIME, e Figure 4. Comparison of the local thermodynamic equilibrium model (dashed lines) with the reversible kinetic model (solid lines) of Do and co-workersi6 under similar conditions at different values of the isotherm constant o. Adapted from Figure 8 in ref. 16.

since we have assumed local thermodynamic equilib- rium conditions.

Galactosidase Adsorption

The experimental concentration-time curves for p- galactosidase adsorption are shown in Figure 5 for three different initial enzyme concentrations. The results of batch as well as recycle (column) runs are presented together since it was observed that the enzyme ad- sorption profiles found were independent of the liquid Reynolds number over the range of flow rates inves- tigated," Re, = 0.1-0.3 where Re, = 2vR/u. The fluid superficial velocity is v and the kinematic viscosity is u. Furthermore, the single pass residence time in the recycle runs was on the order of 1 min that is

small compared to the time scale of the experiments and, additionally, when the catalyst was removed from the column and assayed separately there were no dif- ferences found in the activity with respect to position in the c01umn.'~

The experimental data shown were used in the pa- rameter estimation routine to find the isotherm pa- rameters. The model results with these estimates are shown as the smooth, solid curves in Figure 5 . The corresponding estimates of the parameters are found to be

(33)

(34) when NR = 0.037 and E, = 0.6. The correlation coef- ficient for all curves is r = 0.90 and the correlation

ELuY = 312 k 17g/L

K, = 0.23 2 O.OSg/L

966 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 27, JULY 1985

I I I I I I I I I I00 200 300 400

TIME, minutes

4

- -

I I I I I 1 I I I 100 200 300 400 500

TIME, minuter

Figure 5. The (0-0, A-A) experimental and (-) theoretical results for the bath enzyme concentration at three different initial enzyme levels. BR refers to the (A) batch run and RER refers to the recycle (column) (0) run. The upper right quadrant shows the calculated and experimental (estimated) adsorbed enzyme concentrations (see text). Temperature in all experiments was 22°C. pH 7.2, and 0.01M phosphate buffer was used.

between the two parameters is found to be 0.77. It is seen that the model is quite consistent over the range of initial enzyme concentrations investigated and is able to approximate all the experimental data fairly well. At Eo = 5 mg/mL, the model consistently over- estimates the extent of adsorption which is probably due to experimental errors. In the absence of further

information, however, we have decided to include all experimental data “as is.” A better indication of the utility of our approach may be surmised from a plot of the residuals as a function of the observed 4 value as seen in Figure 6. An analysis of the residuals indicates perhaps a slight bias in the model since the slope of the solid curve in Figure 6, that represents the linear

PEDERSEN ET AL: ENZYME ADSORPTION 967

30- -150 - 4

-50 F 2

- 0.4 - -

-

TIME, minutes

Figure 5. Continued

OBSERVED 9

Figure 6. Residual plot and histogram of observed and calculated bath enzyme concentrations for the 85 data points shown in Figure 5 . The open symbol (0) designates a single value whereas a solid symbol (U) designates two values located at the same point. The hatched area is inaccessible since the parameters are positive quantities. The curves correspond to least squares analysis of the (-) total data, the (---) BR data, and the (---I RER data. Class refers to increments of 0.05 in the residuals centered about 0 that is shown as the vertical dashed line in the upper left-hand quadrant.

968 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 27, JULY 1985

least squares fit of the residuals, is 0.212. Although one would expect it to be zero, the parameters have been restricted to positive values only and this ensures a negative intercept and hence a positive slope for a consistent model. Thus we conclude that the data and the model are, indeed, consistent. Regression analysis on the batch and recycle data separately are shown by the two other curves drawn in Figure 6 and indicate a similar trend. The batch (upper) curve has a slope 0.228 and the recycle column (bottom) curve has a slope 0.171.

The catalyst was also assayed for both the batch and recycle runs at times r = 120, 300, and 480 min. The results are shown in the upper right-hand quadrant of Figure 5 where an arbitrary activity scale is chosen so that all final ( t = 480 min.) values are assigned an activity of 100. The purpose of the graph is simply to indicate how the global particle activity varies with time. The model results for the trend in the global activity can be found from a plot of the average adsorbed enzyme concentration, $:, versus time assuming that the effectiveness factor is unity.7 The model is su- perimposed on the data at the t = 480 min. value. It is seen that at low initial enzyme levels in the bath, a relatively sharp increase in activity is predicted, whereas a more gradual trend towards the maximum activity is seen at higher enzyme loadings that is also suggested by the data. Further work on these reactor design aspects of the model is in progress.

Enzyme Concentration Profiles

The intraparticle enzyme concentrations have been calculated using the estimated model parameters and the results are shown in Figure 7 for three different initial enzyme concentrations at three different times. The solid curves represent the dependent variable used in the model that is approximated by a polynomial in 6. It is easily seen that a parabolic profile would be a poor approximation and this explains the need for a more detailed model than one based on the simple lumped parameter idea.

As the initial enzyme concentration increases, the distribution of adsorbed enzyme shown by the dashed lines in Figure 7 “sharpens” inside the particle and eventually approaches a wavelike front that slowly penetrates the particle. Since this wave structure is associated with a saturated particle, the global rate always increases with increased loading time. On the other hand, the effectiveness factor for the immobilized enzyme7 has been shown to be large when the front is close to the particle surface and decreases as the enzyme becomes distributed throughout the particle. Therefore, there is likely to be an optimum loading time based on the desired reaction rates, enzyme costs and enzyme stability.

Furthermore, a comparison of Figures 5 and 7 clearly shows that although the bulk enzyme concentration

profile appears to have reached a steady state the intraparticle enzyme concentration is still far from its eventual equilibrium (constant) value. This is partic- ularly true at low initial enzyme concentrations. Therefore, analysis of enzyme adsorption by an equi- librium approach may require extremely long contact times that may not even be possible due to enzyme stability problems. Additionally, quasi-steady-state models of intraparticle diffusionI6 are likely to be in error at low initial enzyme concentrations.

CONCLUSIONS

In this report, we have demonstrated how enzyme adsorption from a finite bath can be modeled by a local thermodynamic equilibrium approach. Similar equations have been put forth by other for conditions of kinetically controlled adsorption and desorption rates that will tend towards our model as a limiting case. In that sense, eqs. (11)-(17) are less general, but a comparison with extensive experimental data shows that the model is entirely adequate to handle a wide range of adsorption conditions. The numerical techniques employed are efficient and the equations can be easily incorporated into available parameter estimation routines.

The model can also be used, of course, to describe leaching of reversibly adsorbed enzyme by adjusting the initial conditions. Also, by modifying the r function it is possible to use other isotherm expressions for adsorption or to describe the simultaneous adsorption of more than one enzyme. All of these modifications have eventual application in the design of enzyme reactors and will lead to a better understanding of the

E,= 2 m g / d

04

RADIAL DISTANCE, E Figure 7. Enzyme concentration profiles in the particle calculated from the parameter values corresponding to Figures 5 and 6 at three different initial enzyme levels and at three different times. The solid lines correspond to the pore (free) enzyme concentrations shown on the left axis whereas the dashed lines correspond to the surface (adsorbed) enzyme concentrations shown on the normalized right axis.

PEDERSEN ET AL: ENZYME ADSORPTION 969

0 5, I 1 I I ,I 0

04

03-

+s

02-

01-

E, = 5mq/ml

-

-0 2

065 0 6 07 08 09 I0

RADIAL DISTANCE, (

Figure 7. Continued.

factors governing immobilized enzyme stability, yield and selectivity.

NOMENCLATURE

solid adsorption area/unit volume (m2/m3) discretization elements for the derivatives parameter defined in eq. (32) effective diffusion coefficient (m'/s) enzyme concentration (g/L, g/m', or g/m') counting indices desorption, adsorption rate constants W', g/L/s) mass transfer coefficient (m/s) isotherm equilibrium constant (g/L) parameter defined in eq. (3) number of internal collocation points capacity factor (= E $ V J V ~ )

Greek letters r

X w

9;. m

total particle enzyme concentration (g/L) particle Reynolds number ( = 2 v R / v ) radial coordinate, radius (in) time (min) superficial column velocity (m/s) bulk and solid volumes (L)

dimensionless column or bath isotherm,

solid void fraction dimensionless time ( = D,r/R') kinematic viscosity (m2/s) dimensionless radial coordinate ( = r / R ) dimensionless enzyme concentration (= EIEo) transformed radial coordinate (= 5') dimensionless isotherm parameter (= K,/Eo)

eq. (17)

970 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 27, JULY 1985

Subscripts 0 S solid sm maximum solid value

initial value ( t = 0)

Superscripts per unit area particle average asymptotic, equilibrium value (t-00)

I

References

1. B. Mattiasson, Immobilized Cells and Organelles, Volume 1 , B. Mattiason, Ed. (CRC, Boca Raton, FL, 1983), pp. 3-25.

2. 0. Zaborsky, Immobilized Enzymes (CRC, Boca Raton, FL, 1973).

3. M . Nemat-Giorgani and K. Karimian, Biotechnol. Bioeng., 25, 2617 (1983).

4. R. A. Messing, Immobilized Enzymes for Industrial Reactors (Academic, New York, 1975).

5. K. Bucholz, Enzyme Technology, R. M. Laf€erty, Ed. (Springer- Verlag, New York, 1983), pp. 9-21.

6. J. Lasch, FEBS Proc., 52, 495 (1979). 7. C. Horvath and J.-M. Engasser, Ind. Eng. Chem. Fundam.,

8. J. J. Carberry, Trans. N Y Acad. Sci., 31, 813 (1%9). 9. S . Minhas and J. J . Carberry, J . Catal., 14, 270 (1969).

12, 229 (1973).

10. W. E. Corbett and D. Luss, Chem. Eng. Sci . , 29, 1473 (1974). 1 1 . L. Marcussen, Chem. Eng. Sci., 25, 1487 (1970). 12. I . Neretnieks, Chem. Eng. Sci., 31, 107 (1976). 13. I. Neretnieks, Chem. Eng. Sci., 31, 1029 (1976). 14. S. W. Carleysmith, M. B. L. Eames, and M. D. Lilly, Biotechnol.

15. K. Bucholz, Biotechnol. Lett., 1, 18 (1979). 16. D. D. Do, D. S. Clark, and J. E . Bailey, Biotechnol. Bioeng.,

24, 1527 (1982). 17. E. Stuker, Dissertation ETH N o . 7572, Swiss Federal Institute

of Technology, Zurich, Switzerland, 1984. 18. J. V. Villadsen and W. E. Stewart, Chem. Eng. Sci., 22, 1483

( 1967). 19. J. V. Villadsen and M. Michelsen, Solution of DiffPrential

Equation Models by Polynomial Approximation, (Prentice-Hall, Englewood Cliffs, NJ, 1978).

20. J. R. Rice, Numerical Methods, Software, and Analysis (McGraw- Hill, New York, 1983), pp. 265-310.

21. A. Chantong and F. E. Massoth, AIChE. J . . 29, 725 (1983). 22. C. M. Metzler, G. L. Elfring, and A. J. McEwen, Biometrics,

30, 562 (1974). 23. J. Seinfeld and L. Lapidus, Process Modeling, Estimation and

Identification (Rentice-Hall, Englewood Cliffs, NJ, 1970), pp. 383-401.

24. L. Furler, M.S. Thesis, Department of Chemical and Biochemical Engineering, Rutgers University, New Brunswick, NJ, 1984.

25. R. G. Rice, K. C. Nadler, and F. C. Knopf, Chem. Eng. Com- mun., 21, 55 (1983).

Bioeng., 22, 957 (1980).

PEDERSEN ET AL: ENZYME ADSORPTION 97 1