The Chemical Engineering Journal, 34 (1987) B35 - B47

Effects of Nonuniform Immobilized Enzyme Distribution in Porous Solid Supports on the Performance of a Continuous Reactor

MD. M. HOSSAIN and D. D. DO*

Department of Chemical Engineering, University of Queensland, St. Lucia, Queensland 4067 (Australia)

(Received August 14, 1986)

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ABSTRACT

A comprehensive model for an immobilized enzyme reactor is developed to study the performance of a nonuniform biocatalyst in a continuous stirred basket reactor. Numerical simulations have been carried out to study the influence of particle size on enzyme stability and effectiveness. The advantages of non- uniform profiles have been shown for moderate to large values of the Thiele modulus for systems obeying Michaelis- Menten kinetics. Experimental results [ 1 ] for hydrogen peroxide-immobilized enzyme on controlled-pore glass and Si-Al particles were re-analyzed using the in formation obtained for the nonuniform distribution of immobi- lized enzyme. It is demonstrated that the deactivation data obtained in a continuous stirred basket reactor are followed much more closely by the nonuniformly than by the uniformly distributed enzyme model.

1. INTRODUCTION

The efficient utilization of immobilized enzyme catalysts in continuous reactors is very important in expanding their industrial application. The efficiency depends on various parameters such as transport processes of reactants and products, heterogeneous reaction kinetics, catalyst distribution and deactivation.

So far, most of the existing literature on diffusion and reaction phenomena in immobi- lized enzyme reactors has dealt with uniform activity distribution. Recently there has been increased interest in nonuniformly active catalysts because of their improved effective-

*Author to whom correspondence should be addres- sed.

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ness and significant yield advantage over uniform catalysts [ 2 - 71.

A few studies have been reported on immobilized enzyme catalysis with non- uniform distribution of biocatalyst in the support matrix. However, to complete the definition of the problem, a detailed simula- tion had to be undertaken, taking into account all the key parameters - mass transfer resistances, deactivation phenomena and variable reaction rate.

Horvath and Engasser [8] found better enzyme effectiveness for pellicular biocatalysts (active exterior layer) following Michaelis- Menten kinetics. Lasch [ 91 demonstrated similar results for a shell-like enzyme distribu- tion fitted by a fourth-order polynomial. Park et al. [6] reached the same conclusions, i.e. that a “shell” type distribution is the most effective for positive-order kinetics, and for negative-order kinetics the “core” type configuration is more effective. In the analysis by Borchert and Buchholz [lo], the adsorbed enzyme profiles were obtained from a very simple model, which excluded the restricted diffusion of enzymes into the carrier which is prevalent in many practical immobilization processes; moreover, enzyme deactivation was not considered. Juang and Weng [ 111 studied the effectiveness of deactivating nonuniform biocatalysts, but they did not present any immobilization model for enzyme profiles in practical enzyme-support systems.

It is difficult directly to determine the experimental active enzyme profiles in porous solid supports. Hence, the analysis of the immobilized enzyme reactor must include either theoretical profiles obtained from an appropriate immobilization model or approxi- mate profiles obtained from kinetic deactiva- tion experiments. Recently, the authors have developed a mathematical model for covalent immobilization processes, which takes into

0 Elsevier Sequoia/Printed in The Netherlands

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account restricted diffusion phenomena and chemical coupling of the enzymes. The internal enzyme profiles have been obtained for a real enzyme-carrier system at various hours of contacting. In the first part of our study, we examine the effects of such distribu- tions on effectiveness-time and conversion- time relationships in a continuous stirred basket reactor (CSBR).

Active enzyme concentration profiles from deactivation studies of immobilized catalase in a CSBR have been determined [12], and we show that the nonuniform profile of the immobilized enzyme fits the practical conver- sion-time data better than does the uniform profile, if the intrinsic parameters obtained from continuous runs are used in the numerical calculations.

2. MATHEMATICAL MODEL

2.1. Physico-chemical fundamentals A schematic diagram of the reactor system

is shown in Fig. l(a), with a total mass mp of immobilized enzyme particles suspended in a well-stirred basket reactor (CSBR) of volume V. The substrate is added continuously at a flow rate F. In the reactor the following processes take place in sequence.

(1) Substrate diffusion from the external solution to the surface of the catalyst pellets through the solution-pellet interphase.

(2) Diffusion into the pores, which are initially filled with buffer to avoid convective flow. For the calculation of the diffusional

tal

tb,

Fig. 1. Schematic diagram of immobilized enzymatic reaction in a continuous stirred basket reactor (CSBR): (a) CSBR; (b) biocatalyst pellet.

flux, we use a Fickian type equation with the substrate’s effective diffusivity correlated t137141 by

Des(r) = fp(‘-)Dt,Kp(r)Kr(r)

7

where D,, K,, K,, E and 7 are the molecular diffusivity, equilibrium partition coefficient, drag coefficient for pore diffusion, particle voidage and support tortuosity factor respec- tively.

The equations for estimating E,, K,, K, in terms of effective pore radius are presented in Appendix A. It is to be noted that due to the adsorption of enzyme in the support pores, the pore dimension is reduced (effective pore radius) as shown in Fig. 2. This modified pore structure causes a variation in the effective diffusivity of the substrate molecules in the catalyst pellet [ 151. In this paper, we are considering this fundamental aspect of the transport phenomena of substrate into the intracatalyst pores.

(3) Simultaneous reaction of the diffusing substrate molecules with active enzymes distributed nonuniformly. For the reaction model, the Michaelis-Menten scheme of kinetics is assumed. The rate of immobilized enzyme (IME) reaction at any point in the pellet is given by

Vi = ke&b K,+s

where k, is the reaction rate constant, e,(r) is the active immobilized enzyme concentration at a radial distance r, s is the substrate con- centration and K, is the Michaelis constant. This kinetic form represents the local relation- ship between the reaction rate and the con- centrations of active enzyme and substrate in the support’s microenvironment.

Immobilized

Restricted Pore Area

Fig. 2. Modification of support pores due to immo- bilization of enzymes in a nonuniform fashion.

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(4) As the reaction time progresses, the immobilized enzyme (IME) loses its activity and the reaction rate decreases. The deactiva- tion rate is assumed to be of the Michaelis type (i.e. the same as the main reaction) as proposed by Do and Wetland [16]

de -8 =

kdes(rb

dt’ (&I + s) (3)

where kd is the deactivation rate coefficient of the enzyme catalyst.

For the convenience of analysis, the other assumptions are as follows.

(5) Temperature and density remain constant throughout the system.

(6) A quasi-steady state holds in the reactor. (7) The support particles are spherical and

uniform in size. (8) The deactivation reaction is slow com-

pared to the main reaction.

2.2. Nonuniform immobilized enzyme concen- tration profiles

Nonuniform IME concentration profiles can be produced intentionally by employing a high concentration of activation agent, incomplete or controlled adsorption on the support, nonuniform porosity in the support and restricted diffusion of macromolecules into the support [ 17 - 221.

The IME concentration profiles examined were obtained from the diffusion- chemical coupling model of Hossain et al. [17] for glucose oxidase on porous glass particles of pore size 475 A. Typical profiles are displayed in Fig. l(b), which shows the IME concentra- tion as a function of particle radius. However, the model predicts the total adsorbed concen- tration e,(tot), of which only a portion could be active. This is converted to the active IME concentration by e, = ye,(tot), where r is the active fraction of total enzyme initially avail- able in the support. It is to be noted that the previous kinetic and diffusion studies are based on the total adsorbed concentration.

Next, we employed the following equations for the nondimensional active enzyme profile

E,(X) = 3.0 exp{-6.4(1 -x)} (4a)

E,(x) = 25.3 exp{--87( 1 - x)} (4b)

These equations were determined from experimental data on hydrogen peroxide- immobilized catalase on a porous support in a continuous reactor [1] by applying the

analytical solution developed by Do and Hossain [ 121. Equations (4a) and (4b) describe the active enzyme concentrations on Si-Al (150 pm size) and controlled-pore glass particles (450 pm size) respectively.

In order to formulate the complete model, we derive the material balance equations for a small control volume (single particle) and a large control volume (the reactor) and relate the two sets of balance equations using the inter-facial boundary conditions.

2.3. Reactor model and parameters The mass balance of a substrate inside an

immobilized enzyme catalyst of spherical shape, which deactivates according to the rate form (3) is

a’ epW z = _Z ,’ $ -/r’D.,(r)-& 1 kp,e,(rb - + kdPp%(r)s

(Kn + s) VLl + s) (5)

Since the deactivation reaction rate is slower than the main reaction rate (assumption (8)), the last term in the right-hand side of eqn. (5) can be neglected compared with the other terms.

The substrate material balance in the reactor can be represented by the following

v!$ = F(s,-se)

- (:I ($1 tQ42..,,) t6) where V is the volume of the reacting system, s, is the inlet substrate concentration, s, is the substrate concentration in the reactor, F is the flow rate and R, is the radius of enzyme particle.

The reaction boundary conditions are

t’ = 0; e,(r) = e,(r) (7a)

as r=O; -_=()

ar (7b)

r= R - D,$ PI l-‘Rp

= km&. -ship) (7~)

where k, is the mass transfer coefficient based on exterior surface area.

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For simplicity and in order to study the effects of process parameters, eqns. (3), (5) and (6) can be cast into dimensionless form by using the following dimensionless variables and parameters

se= s”, SE _f- ,=L SO so ’ RP

t’= ;, t = (%),t’ %2Wo

G= kdso(e,)O

O!= %2Wo~

ksesapp ’ (De, )oV

knR, Bi= -

(D,,), ’

N = 3(%lP,)(%)O

V

e,(r) E,(x) = -y ,

K,= K, sa SO

$2 = JwsaRp2Pp D,,(r)

so(Des)o ’ f1(x) = (D,,),

@a)

(8b)

(8c)

(se)

W)

Equations (5)) (3) and (6) are then trans- formed to

as la as(x) -=-- at x2 ax

f,(x)x2 - ax

- G2 wx)s(x) Kl + s(x)

aEs(x) - = -$(#p Edx)s(x) at Kl+ s(x)

subject to

t= 0; Es(x) = Go(x)

as x=0; -_=o ax

x= 1; fi(x) g = Bi(S, - SI i)

(9)

(10)

(11)

(124

(12b)

(12c)

where Bi is the Biot number for mass transfer and can be calculated as outlined in ref. 23. The Thiele modulus $I for the reaction has been defined in terms of the average active enzyme concentration in the particle.

In eqn. (9)) we have dropped the third term of eqn. (5) because of assumption (8) in our model formulation. Also in the process of non- dimensionalization, we have used the empty- pore diffusivity value (Des)es=O which is normally used in the analysis of diffusion- biochemical reaction problems.

Therefore the function f i(x) in our model describes the reduction in substrate diffusivity in the porous catalyst due to the presence of immobilized enzyme with respect to the ordinary diffusivity used by previous workers.

The parameter $ represents the ratio of the rate of deactivation reaction to that of the main reaction and N represents the ratio of pore volume to that of the reacting system.

In the nondimensionalization, we have used the volumetric mean active enzyme concen- tration esa as the characteristic concentration. So in the case of uniformly distributed catalyst

E,(x) = 1

For the case of nonuniformly distributed catalyst (as in this case), the mean concentra- tion is given by

RP s r2e,(r) dr 0

esa = _ KP s r2dr

0 3 RP

=- J G3 0

r2es(r) dr

WW

Cl=)

This gives the average of the total internal enzyme distributed heterogeneously, of which a fraction y is catalytically active to enhance enzymatic reaction. This fraction can be found experimentally for serine proteases by an active-site titration method [ 241.

2.4. Calculation of the effectiveness factor The performance of the nonuniformly

distributed enzyme catalyst can be estimated by the effectiveness factor which is defined [ll] as

rl=

(actual reaction rate in the biocatalyst pellet)

(reaction rate in a biocatalyst pellet of uniform activity distribution based on the condition of bulk fluid (i.e. without diffusion resistance and enzyme deacti- vation))

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In the case of negligible film diffusion, 7 truly indicates the extent of internal diffusion limitation.

For Michaelis- M enten type kinetics, the expression for Q is

1

q = 3(1 + K,) _I- dx (14) 0

9 depends on the activity of IME catalyst and varies with time, as the catalyst deactivates. Thus the effectiveness-time relationship should quantitatively determine the effect of diffusion on the operational stability of the deactivating system. Strictly speaking, we should define 17 in terms of time-dependent bulk reactant concentration instead of constant initial bulk reactant concentration to achieve a rigorous solution of the coupled particle-reactor problem [ 251. However, for the qualitative assessment of the nonuniform biocatalyst in utilizing the enzyme activity we employ eqn. (14) to calculate the effective- ness factor.

3. ANALYSIS OF THE MODEL

In order to observe the transient decay behaviour of the immobilized enzyme system described by eqns. (9) - (ll), we must introduce the following slow time scale [26]

i-=$t $I<1 (15)

The parameter $J is indeed very small (i.e. of the order of 10m4 - 10v5 as reported by Do and Hossain [ 231, which is actually justified for many practical enzyme systems (otherwise the system would be uneconomical).

With respect to the time scale (15), eqns. (9) - (11) are transformed to a new set of equations

(16)

(17)

(1 --Se) = a-1 jf,(x)NE I,;, (18)

Note that eqn. (18) represents the substrate conversion for the present CSBR system and

we can calculate the conversion in the reactor by knowing the substrate concentration gradient at the particle surface.

To solve the coupled partial differential eqns. (16) - (18), we apply a combination of numerical techniques - orthogonal collocation and Runge-Kutta. The numerical procedure is fairly straightforward [27] and produces accurate, reliable results with computational advantages over other techniques [ 28 - 301.

The governing equations are transformed to a set of coupled ordinary differential equations by applying the orthogonal collocation method. A computer program has been developed for the simultaneous solution of the discretized equations with the aid of the subroutine listings given by Villadsen and Michelsen [ 271. This calculates the effective- ness factor of the biocatalyst particles as a function of process time. It also calculates the intracatalyst substrate concentration profiles and substrate conversions in the basket reactor for various operating condi- tions.

The immobilized catalase- hydrogen peroxide system follows first order kinetics under the conditions described by Tai and Greenfield [ 11. The detailed derivation of the model equations is presented elsewhere [ 231. Next we solve the governing equations by the techniques mentioned earlier and compare the numerical results with the experimental data on immobilized cat&se in a CSBR.

4. RESULTS AND DISCUSSION

The quasi-steady-state model (eqns. (16) - (18)) was solved numerically using a PDP 11/44 computer. The required physical characteristics of the biocatalyst and enzyme reactor are presented in Table 1. The kinetic and mass transfer constants used for numerical computation are listed in Table 2. In Table 3, some of the typical values for nondimensional parameters calculated from Tables 1 and 2 are shown. The initial total enzyme profiles have been calculated from the immobilization model and experimental conditions presented by Hossain et al. [ 171.

The profiles have been normalized with respect to their average value and modified to provide the active concentration by using a value of 7 = 0.3, as reported for immobilized

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TABLE 1

Physical parameters related to immobilized enzyme catalysta and reactorb

es, (mg g-’ ) d, (pm) rpo (A)

? I” Cm-3) V’icom3) F (cm3 mine1 ) mp (8)

23.9 150,300,450 475 0.67 0.69 900.0 1.0 1.0

* Hossain et al. [ 17 1. bTai [37].

TABLE 2

Typical values of parameters for immobilized enzyme reaction in porous pellet

(Des)0 (cm2 s-l 1 5x10-6 [38] ks (s-l 1 120.0 [39] so (pm01 ml-‘) 0.5 [40] kd (s-l) 5.0 x10-6 [41] Y 0.3 [24] K, (pm01 ml-’ ) 0.05 [41] Bi 50,100,150

TABLE 3

Values of nondimensional parameters employed in the simulation studies

Parameter Particle size (pm)

150 300 450

$ x 106 1.07 1.07 1.07 01x10s 0.86 3.45 7.76 NX103 3.43 3.43 3.43 @ 6.06 12.07 18.10

enzymes in porous glass by Ford et al. [24]. The active enzyme profiles thus obtained are illustrated in Fig. 3, where (a) and (b) refer to the active IME profiles at 1.1 and 4.4 h of coupling time respectively, for a pellet size of 150 pm. For other pellet sizes, the time for immobilization can be estimated from the relation

t” = WWot’

For simulation of various particle sizes, we have used the same enzyme loading and so the enzyme profiles used are different in each case.

.8 r-- -------I

.2 .6 1 X

Fig. 3. Initial distributions of active immobilized enzyme concentration (Es) in the catalyst particle: (a), (b), nonuniform; (c) uniform.

Next, we have applied the analytical procedure of Do and Hossain [12] and determined the intrinsic parameters and active enzyme distribution for immobilized catalase on controlled-pore glass and S-Al particles. The detailed analysis can be found elsewhere [ 311. The profiles can be fitted by the eqns. (4a) and (4b) presented earlier and

TABLE 4

Characteristics of immobilized catalase-hydrogen peroxide system on controlled-pore glass particles (450 pm)a

rpo (8) 330 pp (g cme3) 0.89

(fp)O 0.55

mp W 1.12

(&JO (cm2 s-l 1 3.0 x10-6 (kE,) ( cm3 g-’ s-l ) 63.1 kd (IIlOl-’ S-l) 1.10 N 2.31 x10-3 $ 1.08 x 1O-4 (Y 0.153 Bi 106.13

aSee refs. 31 and 37.

TABLE 5

Characteristics of immobilized catalase-hydrogen peroxide system on Si-Al particle@

dp Mm) 150 Pp (g cmp3 1 1.13 (fp)o 0.55 mp (g) 1.64 UW0 (cm2 s-l 1 3.5 x 10-h (k&J (cm3 g-l s-l) 87.94 k, (mOl-’ S-I) 0.49 N 2.66 x 1O-3 II/ 2.7 1 x 10-s cr 1.70 x 10-s Bi 22.0

Wee refs. 31 and 37.

the corresponding parameters for the reaction systems are listed in Tables 4 and 5 respec- tively .

4.1. Variation of biocatalyst effectiveness with process time

The effect of particle size on the effective- ness factor-process time relationship is compared in Fig. 4 for both uniform and non- uniform catalysts; for the nonuniform profile ((a) of Fig. 3) and for the uniform catalyst the same amount of enzyme is distributed in the pellet (profile (c) of Fig. 3).

As shown in the figure, the nonuniformly distributed catalyst offers better effectiveness n for a process time up to 100 - 150 h. Beyond this period, there is no significant difference in rl between the two types of distribution for large particle sizes. The initial value of n is low for large particle size (large 4) and changes slowly over a long period. This is due to a lower deactivation rate at high values of # (slow diffusion of substrate).

For a small particle (small @), the initial value of 7) is high; it changes moderately up to a certain process time and then drops sharply in the case of the nonuniform catalyst. This is due to accelerated substrate diffusion, as 4 is small, causing faster deactivation of the enzyme catalyst, which is concentrated near the surface. In contrast the catalyst with the uniform profile can continue the enzymatic reaction until the substrate reaches the particle’s centre. Thus the uniform catalyst maintains its effectiveness until all the enzyme in the entire particle is deactivated.

01 0 100 200 300

t; h

Fig. 4. Effectiveness factor v us. process time (h) at various particle sizes in pm (i.e. various f#~): -, nonuniform; - - -, uniform.

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4.2. Effect of particle size on enzyme deacti- vation

The substrate conversion-time relationships for catalysts with various particle sizes having nonuniform and uniform enzyme profiles are given in Fig. 5. The enzyme particle sizes considered are 150 pm, 300 pm and 450 pm. It is quite clear why the nonuniform catalyst particles yield better conversion for relatively short periods of operation.

For a small particle, the initial conversion is high, but it drops to a very low value after this period because of faster deactivation of the nonuniform enzyme. The uniform catalyst particles of small size maintain the improved conversion because the deactivation of the enzyme is gradual.

In large particles, the diffusional resistance is stronger and it improves the enzyme’s stability towards deactivation. Therefore reactor conversion does not vary significantly from nonuniform to uniform distribution in long-term performance.

.8

.8

XC

A

.2

0 100 200 300

t:h

Fig. 5. Effect of particle size on immobilized enzyme stability. The values in the figure show the average diameter in Mm.

4.3. Effect of intraparticle diffusion on catalyst stability

It is evident from Figs. 4 and 5 that for a nonuniform enzyme distribution, the influence of intraparticle diffusion stabilizes the large catalyst particles (i.e. 300 E.trn and 450 pm) relative to small particles (150 pm). For the time range considered in the simulation, the stabilization effect seems to be significant in the size range 150 E.trn - 300 pm. The influence

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of stronger diffusion can be explained by the time variation of the substrate concentration profile in the pellet.

The distribution of the substrate concen- tration was calculated for particle sizes of 300 pm and 150 pm. The results are illustrated in Figs. 6 and 7 respectively, for both uniform and nonuniform distribution of immobilized enzyme.

The time variation of the substrate con- centration profiles becomes significant with increase in particle size for nonuniform catalysts (i.e. increase in 4). At high values

.4 -

b

.4 .7 1 X

Fig. 6. Dimensionless intraparticle substrate concen- tration (S) profile within 300 pm particle size: (a) nonuniform; (b) uniform.

S

.8

b

.4 t i

0 d .4

.7 x 1

Fig. 7. Dimensionless intraparticle substrate concen- tration (S ) profile within 150 pm particle size during reaction process: (a) nonuniform; (b) uniform.

of 4, the enzymatic reaction is initially limited near the external layer of the particle where substrate concentration is high. This produces a sharp profile within the reaction zone and a quick drop in concentration from the surface to its centre (Figs. 6 and 7).

As the operating time elapses, the immobi- lized enzyme deactivates, allowing more substrate to diffuse into the interior of the particle. So, with continued use of the catalyst, the diffusional resistance progres- sively diminishes for large particles (because of enzyme deactivation) and results in a moderate substrate concentration distribution in the particle. Under this condition, enzymes in the interior of the particle can gradually play their role in the reaction process and may cause an improvement in operational stability.

For small particle sizes, there is a different pattern of change in the substrate concentra- tion profile for nonuniform (Fig. 7(a)) and uniform distribution (Fig. 7(b)). For small particles (low c$), substrate diffusion is com- paratively fast and reaction occurs in a wider zone near the surface of the particle. Thus, in the case of nonuniformly distributed enzyme in a particle of small size, the sub- strate concentration profile is moderate up to a certain process time and then gradually becomes flat throughout the particle (Fig. 7(a)). This is due to the diminishing reaction rate in the particle as the catalyst localized in the outer layer deactivates to a low con- centration level. In contrast, the uniform catalyst contains more enzyme in the interior of the pellet, allowing the reaction to proceed. So the substrate concentration profile in this case remains moderately nonuniform until the entire particle is deactivated. Thus the availability of active enzyme in a uniform catalyst of small size provides better effective- ness in long-term operation.

4.4. Effect of deactivation on immobilized enzyme concentration profile

Figure 8 (a - d) displays the time variation of internal enzyme profiles for deactivation catalysts of two particle sizes. In the case of a nonuniform distribution, profiles in both particle sizes are sigmoidal (Figs. S(a) and 8(b)) in shape with a rapid drop in maximum concentration for the smaller one. This con- centration drops to a low value (Fig. 8(a)) at

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.6 ,

.5 .75 1 .75 1 X

.‘r------

01 0 lO0 200 :

t’, h

Fig. 8. Time variation of dimensionless immobilized enzyme concentration (Es) profile within support particles: (a) nonuniform in 150 pm; (b) nonuniform in 300 pm; (c) uniform in 150 pm; (d) uniform in 300 pm.

f = 0.09 (approx. 150 hours) to cause a sudden fall of effectiveness and conversion for the small particle size (Figs. 4 and 5). But for the same process time, an appreciable concentra- tion of catalyst is available for the large particle size (i.e. large 4) and so the deactiva- tion region is smaller (Fig. 8(b)).

On the other hand, the uniform catalyst does not show any such sudden event for the particle sizes and process times considered. The catalyst concentration is at a maximum in the interior and at a minimum at the surface (Figs. 8(c) and 8(d)). As expected, the deactivation zone is more extensive for small particle sizes and causes poor operational stability [30] by comparison with large support pellets.

4.5. Effect of nonuniform enzyme content on effectiveness factor and reactor conversion

Figure 9 shows the variation of q with process time for particle sizes 150 pm and 300 pm respectively, having different non- uniform profiles. The profiles considered are those in Figs. 3(a) and 3(b) which describe the active enzyme concentrations with average content of 12.2 mg g-l and 16.8 mg g- ’ respectively. It is shown in Fig. 9 that loading more enzyme into the support pellet (i.e. obtaining profile (b) of Fig. 3) reduces the effectiveness for both sizes. But for the smaller particle size, the higher enzyme content brings about a significant enhance- ment in the stability of the enzyme catalyst.

Figure 10 shows the influence of higher enzyme concentration on reactor conversion for the particle sizes used in Fig. 9. For small

Fig. 9. Effects of nonuniform distribution on effectiveness factor rj of catalyst particle of sizes 150 pm (---) and 300 pm ( -) respectively. (a) and (b) are the profiles at ?= 0.011 and 0.044 respectively.

0 100 200 300

t’, h

Fig. 10. Effects of nonuniform distribution on reactor conversion X, for particle sizes of 150 pm (- - -) and 300 pm ( -) respectively. (a) and (b) are the same as those in Fig. 9.

particles, where diffusional resistance is less, the higher the enzyme content the better is the conversion (Fig. lO( b)). But in the case of large particles, no such significant improve- ment in conversion is achieved, as the process is diffusionally limited.

4.6. Examination of the experimental data of immo biked ca talase on con trolled-pore glass and Si-Al particles

In Fig. 11 we compare experimental conversion-time data for the immobilized catalase reaction on controlled-pore glass [l] with the calculated values obtained for non- uniform and uniform distribution. The reaction parameters, deactivation constant, diffusivity and nonuniform enzyme profile were

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oL._ I \

0 100 200

t’,min

Fig. 11. Comparison of experimental reactor conver- sion X, data (0) for immobilized catalase on controlled- pore glass particles (450 pm) with those predicted by nonuniform (- ) and uniform (- - -) distribu- tions of the same amount of catalyst.

determined from quasi-steady-state and deactivation analysis of the CSBR data. For the uniform catalyst weassume the same amount of enzyme distributed uniformly. It is clearly shown in Fig. 11 that the nonuniform internal profile fits the experimental data much better than its uniform counterpart does. Therefore the use of intrinsic parameters from batch study and the assumption of uniform distribu- tion of enzyme lead to an incorrect analysis of the continuous reactor data. Moreover, the enzyme size is approximately 106 A and if immobilized on porous glass of pore size 330 A, the profiles would be severely non- uniform [ 17,32,33].

In Fig. 12 we compare the experimental conversion data of immobilized catalase on Si-Al particles with numerical values obtained from the model. Clearly the nonuniform profile gives closer agreement with the experi-

0 400 700

t’, min

Fig. 12. Comparison of experimental conversion-time data (0) for immobilized catalase on Si-Al particles (150 pm) with those predicted by nonuniform (-) and uniform (- - -) distributions of the same amount of catalyst.

mental results than the uniform profile does. However the particle size of the Si-AI support is small (150 pm) and so diffusion effects are not as severe as in the case of glass particles. Thus the active enzyme profile is moderately nonuniform and yields reactor conversions, not as widely deviating from those yielded by the uniform enzyme profile, as in the case of glass particles (Fig. 11).

5. CONCLUSIONS

A mathematical simulation has been presented for the performance of deactivating spherical immobilized enzyme particles with radially non-uniform activity distribution. The solutions of the performance equations for CSBRs can be used to evaluate the effectiveness factors and the intraparticle substrate and biocatalyst concentration profiles of the nonuniform enzyme system. It has been shown that for short-term opera- tion, effectiveness factors improve with non- uniformity in enzyme distribution for moderate to large values of the Thiele modulus. The operational stability of the biocatalyst can be improved by choosing either nonuniformly distributed catalyst particles of large size, or moderately non- uniform particles of small size.

The effect of nonuniformity is more significant in catalyst supports of small size, as displayed by the time variation of the effectiveness factor, reactor conversion, intra- support substrate and immobilized enzyme concentration profiles.

It has been demonstrated that the intrinsic kinetic parameters from continuous reactor data and nonuniform catalyst distribution in the pellet give good agreement with experi- mental results for the immobilized catalase- hydrogen peroxide system.

The results of the present investigation can be utilized to analyze and design more effective and stable enzyme catalyst systems following positive-order reaction kinetics and a parallel deactivation mechanism.

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B46

36

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APPENDIX A

For K,, the following formula for cylindrical pores proposed by Ferry [ 341 can be used

(Al)

where a, is the substrate molecular radius and Reff is the effective pore radius of the catalyst pellet. This pore radius is related to the initial pore radius by the approximation [35]

Red9 = rpo \(l- +J

where rpb, is the initial pore radius, a, is the effective enzyme molecular radius, e, is the immobilized enzyme concentration and esm is the maximum capacity for immobilization in the support.

To calculate K,, the following formula proposed by Renkin [36] can be used

K, = iI- 2.104[&j + 2.09[&j3

(A3)

The local porosity of the catalyst particle can be calculated from the equation

This has been derived by Hossain et al. [ 171 assuming a monomolecular layer coverage of enzyme catalyst inside the void spaces of the particle and has been found valid for most enzyme support systems (i.e. up to ae/rpo < 0.2). This expression suggests a nonuniform porosity distribution inside the pellet because of heterogeneous distribution of immobilized enzyme concentration in it.

APPENDIX B: NOMENCLATURE

a,

as

aR

Bi Db

Dez(r)

D,,(r)

dP e,(r)

e,(tot)

e e, Sat

e,

E,(x)

F

kn

K,

ks

effective radius of the diffusing enzyme molecule (A) solute radius of the diffusing substrate molecule (A) relative activity of the enzyme pellet Biot number for mass transfer bulk diffusivity of substrate in free solution ( cm2 s-l) effective diffusivity of enzyme defined similarly as in eqn. (1) (cm2 s-l) effective diffusivity of substrate in catalyst pellet defined in eqn. (1) (cm2 s-l) diameter of the particle (pm) active immobilized enzyme concen- tration (pm01 g-r) total immobilized enzyme concentra- tion (pm01 g-l) mean active enzyme concentration; defined in eqn. (13b) maximum immobilized enzyme con- centration, mg g-’ nondimensional immobilized enzyme concentration volumetric flow rate of substrate to the reactor (cm3 s-r) mass transfer coefficient based on exterior surface area (cm s-l) nondimensional Michaelis constant for the reaction reaction rate constant (pm01 substrate s-l pmol-l of active catalyst)

B47

kd

Klll

K,(r) K,(r)

mP

N

rP’p0

R* S

SO

s

Se

deactivation rate constant (pm01 substrate s-l pmol-’ of active catalyst) Michaelis constant for the reaction (pm01 ml-‘) equilibrium partition coefficient fractional reduction of diffusivity of substrate within pores total mass of support in the baskets (g) ratio of support volume to that of reactor Avogadro’s number, 6.023 X 10z3 radial position in the pore effective pore radius of the pellet at any loading; defined in eqn. (A2) (A) initial pore radius of the particle (A) radius of the particle (pm) intracatalyst substrate concentration (pm01 ml-l) inlet substrate concentration (pm01 ml-‘) reactor substrate concentration (pm01 ml-r) nondimensional intracatalyst sub- strate concentration nondimensional reactor substrate concentration

t’ t

2: V

K

X

XC

dimensional time (h) nondimensional time defined in eqn. (8b) slow time = $t defined in eqn. (15) volume of continuous stirred basket reactor ( cm3 ) rate of immobilized enzyme reaction per g of support nondimensional spatial variables = r/R, reactor conversion

Greek symbols o! parameter defined in eqn. (8~) e&9 pore volume fraction of support

defined in eqn. (A4)

$1

parameter defined in eqn. (8~) parameter defined in immobilization model of Hossain et al. [17]; = eQebo /Pp esm

&,,0 tortuosity of the support initial porosity of the support pellet

PP density of the immobilized enzyme particles (g cmV3)

d Thiele modulus defined in eqn. (8f) V effectiveness factor defined in

eqn. (14)