I n d . Eng. Chem. Res. 1987, 26, 1810-1817
1810
Examples in three dimensions and higher are at present being looked into as well as extensions to the concentration-time space.
Superscripts 0 = initial value of a variable * = value of variable after mixing
Acknowledgment
Subscripts
D.G. thanks Malcolm Baird and McMaster University for the appointment as Hooker Distinguished Visiting Professor during 1984 when much of the work reported in this paper was initiated. D.H. thanks the University of Potchefstroom for the time to pursue this work. Nomenclature A, B, C, D = chemical species A , B = sets in Concentration space a, = dimensionless rate constant ratios c = concentration vector c, = concentration of species i g = resultant vector k , = rate constant for reaction i R = recycle ratio r = rate of formation vector ri = rate of formation of species i t = time U = base of set in concentration space x, y = normalized concentrations of species
Achenie, L. E. K.; Biegler, L. T. Ind. Eng. Chem. Fundam. 1986,25, 621. Chitra, S . P.; Govind, R. AIChE J . 1985, 31, 177. Feinberg, M. In Dynamics and Modelling of Reactive Systems; Academic: New York, 1980; p 59. Glasser, D.; Horn, F. J. M. Chem. Eng. Sci. 1980, 35, 2281. Glasser, D.; Horn, F.; Meidan, R. J. Math. Anal. Appl. 1980, 73,325. Glasser, D.; Jackson, R. In 8th International Symposium on Chemical Reaction Engineering; IChE Symposium Series; Pergamon: London, 1985; Vol. 87, p 535. Horn, F. in Third European Symposium on Chemical Reaction Engineering, Pergamon: London, 1964; p 1. Jackson, R.; Glasser, D. Chem. Eng. Commun. 1986, 42, 17. Paynter, J. D.; Haskins, D. E. Chem. Eng. Sci. 1970,25, 1415. Shinnar, R. Ann. N . Y. Acad. Sci. 1983, 404, 432. Shinnar, R.; Feng, C. A. Ind. Eng. Chem. Fundam. 1985, 24, 153.
0 = initial value of normalized concentrations
Literature Cited
Greek Symbols a = mixing ratio d = boundary of the set which it precedes
Received for review April 7, 1986 Revised manuscript received May 26, 1987 Accepted June 7, 1987
Reaction Kinetics and Mass-Transfer Effects in a Fixed-Bed Biochemical Reactor with Invertase Immobilized on Alumina Christie J. Geankoplis* Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455
Edwin R. Haering and Michael C. Hut Department of Chemical Engineering, The Ohio State University, Columbus, Ohio 43210
Experimental data for a fixed-bed reactor were obtained for the reaction of sucrose to give glucose plus fructose using the enzyme invertase which was immobilized by covalently binding it to a polymer matrix adsorbed in the pores of alumina. A mathematical model, which included external film mass transfer, internal pore diffusion, axial dispersion, and enzymatic reaction with both substrate and product inhibition, was used to predict the performance of the fixed-bed reactor over a wide range of operating variables. T h e tortuosity factor for pore diffusion was measured separately for use in the model. Hence, external and internal pore diffusion effects were known and it was necessary to obtain only the activity of the immobilized enzyme from the experimental reactor data. Comparisons of the experimental data and the theoretical predictions from the model indicated a 70% loss of the native invertase activity constant resulting from the immobilization and a 32% decrease for the Michaelis constant and the product and substrate inhibition constants.
Introduction and Literature Review Knowledge of the changes that occur in enzymatic kinetic parameters due to immobilization of the enzyme is essential in the design of immobilized enzyme reactors. Due to the presence of external and internal mass-transfer effects, the intrinsic reaction kinetics of immobilized en+Presentaddress: Shell Canada, Oakville, Ontario L6J 5C7, Canada.
0888-5885/87/2626-l810$01.50/0
zymes must be studied with the aid of a mathematical model which simultaneously takes into account the mass-transfer phenomena and reaction kinetics. Theoretical treatments of the heterogeneous kinetics of immobilized enzyme systems have appeared extensively in the literature. However, experimental proof is sometimes lacking and is often limited to the case of first-order enzymatic kinetics. The improper estimation of physical parameters such as internal pore diffusion and external mass transfer and oversimplification of models have also 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987 1811 led to misinterpretation of experimental results. The purpose of this study was to experimentally observe both external and internal mass-transfer effects in a fixed-bed biochemical reactor with an immobilized enzyme catalyst and to develop a mathematical model to account for these mass-transfer effects, the intrinsic enzyme kinetics, and the nonideality of the reactor. The invertase was immobilized by covalently binding it to a polymer matrix adsorbed on the pores of the alumina. The experiments were carried out over a broad range of experimental conditions. The tortuosity factor in the pores of the particle had been experimentally determined (Hu et al., 1985), and other physical parameters such as the external mass-transfer and axial dispersion coefficients were estimated from existing correlations. Finally, the suitability of the proposed model was tested by comparison with experimental results. General Characteristics of the Immobilized Enzymes. The advantages and methods of enzyme immobilization on solid supports have been reviewed recently (Hu et al., 1985). Enzymatic properties will experience some changes during immobilization, and the specific activity of the immobilized enzyme is usually less than that of the free or native enzyme, varying approximately from a maximum of 90% for absorbed or entrapped enzymes to 30% for many which are covalently bound (Kent et al., 1978). Conformational change is the most obvious explanation for enzyme deactivation. To act as a specific catalyst, the enzyme must be in precisely the correct conformation and be able to undergo conformational changes during catalysis. Immobilization on surfaces in an asymmetric manner is likely to distort the structure and may impair catalytic activity. Furthermore, access and binding of substrate or cofactors to the active site may be restricted by the orientation of the enzyme on the surface as well as by direct chemical modification of amino acid residues in the active center of the enzyme. The immobilized enzyme matrix will also provide the immobilized enzyme with a microenvironment such as ionic strength, pH, and substrate concentration which may differ from the solvent environment in which a free enzyme is suspended. This so-called microenvironment effect may be caused by the partitioning of ionic molecules by a polyionic carried (Royer, 1980) or other specific interactions between the support and a particular solute such as hydrophobic interactions (Trevan, 1980). Finally, contrary to that of a free enzyme in solution, the overall kinetic rate of an immobilized enzyme is always influenced by the external and internal mass-transfer effects and axial dispersion, if in a packed bed. These mass-transfer resistances have complicated the evaluation of the intrinsic enzymatic kinetics, which is necessary information for design of an immobilized enzyme reactor. Effects of Mass Transfer on the Reaction Rate. If the intrinsic rate of reaction is faster than the rate of diffusion, not all the catalyst will contribute to the observed reaction rate. Then, the effectiveness factor, defined as the ratio between the actual conversion in the catalyst particle and the conversion if the entire particle was exposed to concentration of the reactants equal to that at the surface of the particles, will be less than 1. The effectiveness factor is normally expressed as the function of a dimensionless group, the Thiele modulus. Effectiveness factor expressions have been obtained for immobilized enzymes following Michaelis-Menten kinetics when the third parameter, the ratio between the Michaelis constant and external substrate concentration, has to be
introduced (Fink et al., 1973; Halwachs, 1979). For a substrate-inhibited enzymatic reaction, the effectiveness factor was found to exceed unity and display multiple steady-state behavior (Wadiak and Carbonell, 1975). In the case of a product-inhibited reaction, the effectiveness factor is always less than that which corresponds to a Michaelis-Menten relationship (Moo-Young and Kobayashi, 1972; Wadiak and Carbonell, 1975). Numerical solutions have been obtained for combined internal and external diffusion effects for MichaelisMenten kinetics (Fink et al., 1973; Horvath and Engasser, 1974; Ramachandran, 1975; Greenfield et al., 1975; Frouws et al., 1976; Narsimhan, 1981). Gonda (1977) theoretically studied the effects of external and internal mass transfer on two- and three-enzyme systems. Engasser and Hisland (1979) investigated diffusion effects on the heterogeneous kinetics of two-substrate enzymatic reactions. To overcome the disadvantages of low effectiveness factors and wastage of active catalyst, Horvath and Engasser (1973) proposed a shell-structured catalyst, which consists of an inert core where no enzyme is present and an outer shell containing the active immobilized enzyme catalyst. They carried out a theoretical analysis of such a system and obtained numerical as well as asymptotic analytical solutions for the effectiveness factors. Karanth and Patwardhan (1980) further theoretically studied the effectiveness factors in an immobilized enzyme packed-bed reactor containing a shell-structured catalyst. A review of the theoretical treatment of mass-transfer effects on immobilized enzyme systems has been given by Kasche (1983). Mass-transfer effects on immobilized enzyme systems have been studied experimentally by numerous investigators (Bunting and Laidler, 1972; Rovito and Kittrell, 1973; Regan et al., 1974; Traher and Kittrell, 1974; Dahodwala et al., 1976; Grulke et al., 1977; Swanson et al., L978a,b; Emery and Cardoso, 1978; Lee et al., 1979, 1980; Greco and Alfani, 1980; Hollo et al., 1981). However, most previous works have been limited to the substrate concentration ranges of first-order and zero-order Michaelis-Menten kinetics. In some cases, the interpretation of experimental results may be questionable due to the inaccurate estimation of mass-transfer effects and oversimplification of the kinetics.
Mathematical Modeling of Simultaneous Diffusion and Reaction in a Fixed-Bed Biochemical Reactor The mathematical model of the behavior of a fixed-bed reactor with an immobilized enzyme as the catalyst was derived from the simultaneous solution of differential mass balances outside and inside the particle, coupled with appropriate boundary conditions, and an expression describing the enzyme reaction rate. The following assumptions were made in the derivation of this model: (1) isothermal steady-state operation; (2) deviations of an ideal plug-flow reactor accounted for by axial dispersion; (3) spherically shaped pellet; (4) homogeneous distribution of enzyme activity in the pellet; (5) invariable enzyme activity during reaction; (6) distribution coefficient between the liquid phase and pores in the solid phase being 1.0. The final steady-state mass balances on the substrate in solution outside the particle (eq 1)and inside the particle (eq 2) are
1812 Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987
The boundary conditions generally used for the liquid phase are the Danckwerts boundary conditions:
(4) The boundary conditions for inside the particle are dCi,r - -0
dr
-
I
00275 0.0356
.
f
0 0502
at r = O
, , , , l , , , , l l # , ,,
, I
1
,
1
,
1
1
,
~
,
,
1
/
1
,
-
Analytical solutions for eq 1 and 2 are possible if the reaction rate term is linear. A solution for first-order kinetics with the Danckwerts boundary conditions has been given by Marrazzo et al. (1975). For more complicated models, numerical integration methods must be used.
Experimental Methods Immobilization Procedure and Tortuosity Measurement. The porous alumina carrier particle used was alumina with a pore volume of 0.599 mL/g, surface area of 225 m2/g, solid density of 3.21 g/mL, and a particle density of 1.098 g/mL. The invertase (grade VII, 400 units/mg) from bakers' yeast was obtained from Sigma Chemical Co. The invertase was immobilized in the porous alumina pellet by using the following procedure. An aqueous solution of the monomer tetraethylenepentamine was added to the alumina and was adsorbed in the pores. Then a glutaraldehyde solution was added, forming a copolymer on the surface of the pores with pendent groups extending from this copolymer. Finally, the invertase solution was added and the coupling carried out. Details are given elsewhere (Hu et al., 1985). The tortuosity factor in the pores of plain alumina was measured experimentally to be 2.0. In the presence of the immobilized invertase and polymer matrices, the tortuosity was measured to be 2.75 (Hu et al., 1985). Details on the materials used, the dinitrosalicylic analytical method for analysis of the products, and other analytical procedures have been described by Hu (1983). Homogeneous Kinetic Studies. The homogeneous kinetics of invertase were studied by both the initial rate of reaction and the time course of reaction method at 25 "C and a pH of 5.0. Initial reaction rates of invertase were determined in test tubes with initial sucrose concentrations ranging from 26.6 to 1594 mM and evaluated by analyzing the release of glucose plus fructose from sucrose during a short period. The reaction time involved depended on the enzyme loading and initial sucrose concentration. In most cases, the final conversion were less than 20% to ensure a constant rate. A t the end of the preestimated reaction time, 3,5-dinitrosalicylic reagent was added to the reaction medium to terminate the reaction. To study the time course of reaction of sucrose (integral method), the reactions were conducted in several test tubes with the same initial conditions and the reactions were stopped at a given time interval. Then, the amount of glucose and fructose produced was used to determine the fraction conversion,
X. The initial rate or initial activity data for the sucroseinvertase reaction are shown in Figure 1 and are best
Fraction 0 6 1 Converslon.
'k
/ ! /
*'
I
. I
'
20
40
'
60
80
100
120
1)O
Time, min
Figure 2. Integral rate of reaction of sucrose with homogeneous invertase, [So] = 132.9 mM. I
0 8-
Fraction 06: Converslon
X
0 4021
4''
-i
#& o!
20
'
40
60
80
100
Time rnln
120
io
Figure 3. Integral rate of reaction of sucrose with homogeneous invertase, [So] = 398.4 mM.
correlated by the substrate inhibition plus total water concentration model of Bowski et al. (1971):
,--;-I The total water concentration was calculated by (Bowski et al., 1971)
[W] = 55.33 - O.O1186[So]
(8)
The best values of ko, K,, and K I were evaluated to be 331.4 pmol/(min.mg), 41.6 mM, and 1753 mM, respectively. The experimental initial rates are compared with the predictions of the model in Figure 1. The time courses of reaction data are shown in Figures 2 and 3 for initial sucrose concentrations of 132.9 and 398.4
Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987 1813 Table I. Final Conversion of Fixed-Bed Runs for Different Amounts of Packing final mol mass-transfer flow rate, Q, fraction coeff, i05kL mass of catalyst, g mL/min W/Q, g.min/mL conversion, X, Reynolds no., NRe m/s 0.30 0.320 0.927 0.794 0.0121 0.98 0.655 0.458 0.605 0.0211 1.25 1.024 0.293 0.477 0.0329 1.45 1.984 0.151 0.246 0.0638 1.80 1.0 0.397 2.517 0.982 0.0128 ' 1.06 1.020 0.981 0.891 0.0328 1.44 2.220 0.450 0.621 0.0714 1.87 3.837 0.261 0.445 0.1234 2.25 2.26 0.434 0.1251 3.891 0.257 0.1848 2.57 5.747 0.174 0.328 2.60 0.318 0.1914 5.952 0.168 0.247 0.2616 2.89 8.137 0.123 3.0 1.622 1.849 0.989 0.0521 1.68 0.1268 2.27 3.945 0.761 0.851 6.017 0.499 0.727 0.1934 2.61 12.50 0.2401 0.454 0.4018 3.33 19.27 0.323 0.6194 3.85 0.156 5.0 4.977 1.004 0.937 0.1600 2.27 0.748 0.3146 2.84 9.785 0.511 14.29 0.350 0.610 0.4592 3.22 20.04 0.250 0.471 0.6442 3.61 8.0 8.000 1.000 0.935 0.2572 2.79 15.97 0.501 0.742 0.5135 3.51 25.06 0.319 0.566 0.8057 4.08 a
Initial sucrose concentration = 10 g/100 mL; diameter of particles = 0.359 mm.
Table 11. Final Conversion of Fixed-Bed Runs for Different Particle Diametersa final mol flow rate, fraction, Rey n o1d s particle diameter, mm Q, mL/min W/Q, g.min/mL no., NRe conversion, Xf 0.840 0.2276 5.003 0.999 0.508 10.00 0.500 0.620 0.4551 14.31 0.349 0.470 0.6511 20.00 0.250 0.364 0.9102 0.274 5.000 1.000 0.963 0.1226 0.2453 10.00 0.500 0.827 0.3514 14.33 0.349 0.717 20.08 0.249 0.589 0.4925 0.200 5.070 0.986 0.989 0.0907 0.1794 10.03 0.498 0.950 14.29 0.350 0.854 0.2556 0.3578 20.00 0.250 0.697 a
dispersion coeff, W E L , m/s 0.66 1.34 2.10 4.04 0.82 2.08 4.52 7.78 7.89 11.60 12.02 16.36 3.31 8.00 12.14 24.97 38.19 10.07 19.62 28.48 39.69 16.09 31.78 49.40
mass-transfer coeff, i05kL,m/s 2.64 3.33 3.76 4.20 3.22 4.04 4.55 5.09 2.50 3.14 3.54 5.09
dispersion coeff, i07EL,
m2/5
14.26 28.22 40.10 55.64 7.74 15.35 21.88 30.55 5.74 11.27 15.99 22.28
Initial sucrose concentration = 10 g/100 mL; mass of particles in bed = 5.0 g.
mM, respectively, and were first tested by integrating eq 7 for the substrate inhibition plus total water concentration model. The predicted curves are also shown in Figures 2 and 3. l The substrate inhibition plus total water concentration model was modified by incorporating the product inhibition and becomes
ko[Eol [SI rc =
KM I + -
+[SI+
(9)
[wol
The time course of reaction data for the initial sucrose concentration of 398.4 mM was used to evaluate the best value of Kp by setting ko,KM,and KI equal to those values used for eq 7. The best value of Kp is 92.9 mM. The calculated curves obtained by integrating eq 9 are plotted in Figures 2 and 3. Heterogeneous Kinetic Study in Fixed-Bed Biochemical Reactor. The materials of construction for the reactor and accessories were glass, Teflon, and stainless steel. The feed solution was placed in a 20-L bottle at a height of 1 m above the top of the reactor. The feed
solution first passed through a glass cooling coil immersed in a water bath a t 25 "C. The feed solution was then adjusted to the desired flow rate by a rotameter equipped with a needle valve. The fixed-bed biochemical reactor was a modified jacketed straight condenser with an inside diameter of 13.5 mm. An 80-mesh stainless steel wire cloth was held between two Teflon screens on the bottom of the reactor as the bed support. To eliminate channeling and end effects, the packed bed of immobilized enzyme particles was placed between two, 1-cm-thick layers of inert glass beads with sizes similar to those of the packed catalyst. Constant-temperature water (25 "C) was circulated through the column jacket. Consecutive samples of product were taken and analyzed until the achievement of steady state. The concentrations of sucrose ranged from 1to 20 w t %. The amount of catalyst packed in the reactor was varied from 0.3 to 8 g. Four different particle diameters were used in this study and were 0.508,0.359, 0.274, and 0.200 mm. The final conversion of sucrose, Xf, was calculated from the concentrations of sucrose and glucose in the product stream. The data for the various runs are given in Tables 1-111. Modeling and Simulation of Fixed-Bed Biochemical Reactor Results. The Gauss-Seidell with overrelaxation
1814 Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987 Table 111. Final Conversion of Fixed-Bed Runs for Different Initial Sucrose Concentrations" final mol mass-transfer initial sucrose flow rate, Q, fraction coeff, i05kL, concn, g/100 mL mL/min W / Q , gmin/mL conversion, Xf Reynolds no., NRe m/s 1 4.995 1.001 0.993 0.2014 2.50 9.823 0.509 0.955 0.3961 3.14 14.12 0.354 0.903 0.5696 3.54 19.88 0.251 0.817 0.8017 3.97 3 5.020 0.996 0.987 0.1934 2.46 9.901 0.505 0.907 0.3814 3.08 14.29 0.350 0.835 0.5503 3.48 20.00 0.250 0.718 0.7705 3.90 20 5.005 0.999 0.780 0.1196 2.00 10.00 0.500 0.536 0.2389 2.52 14.29 0.350 0.409 0.3412 2.84 19.88 0.251 0.309 0.4749 3.17
dispersion coeff, i07EL, mz/s 10.08 19.63 28.04 39.19 10.13 19.80 28.38 39.46 10.15 20.13 28.61 39.60
"Mass of particles in bed = 5.0 g; diameter of particles = 0.359 mm.
technique was employed to numerically integrate eq 1and 2. The computer program with double precision used in this work was discussed elsewhere (Hu, 1983). The diffusivity of sucrose in water at 25 "C has been given by Gosting and Morris (1949):
D, = 0.5226(1 - 0.0148F)10-9
The range of concentrations for their experiments was 0.75 Ic I5.55 g/100 mL. Henrion (1964a) further studied the diffusion of sucrose in aqueous solutions over a concentration range of 8-80 g/100 mL. Henrion's data indicated that the Gosting and Morris equation, eq 10, could be extrapolated to concentrations as high as 40 g/100 mL without any error. In a later article, Henrion (1964b) reported that there was no significant difference between the diffusivities for sucrose in water an in a 1 M KC1 solution. The diffusivity of sucrose was considered to be constant along the reactor and was evaluated from eq 10 for each run. The effective pore diffusivity of sucrose in the immobilized invertase alumina was calculated by considering the porosity and tortuosity of the particle. The tortuosity had been experimentally determined to be 2.75 (Hu et al., 1985) for immobilized invertase present in the alumina. For the packed bed, the porosity of the particle tp = 0.658, the particle density pp = 1.098 g/mL, and the bed void fraction t = 0.43. The equation proposed by Wilson and Geankoplis (1966) for liquids in packed beds was used to estimate the external mass-transfer coefficient, hL. For 0.0016 < NRe< 55, JD = (1.09/~)N,;~/~
+ 0.011~~2.48
10
30 5 0
. .
0
05
15
10
8 0
20
25
i
30
W/Q, g min/mL
Figure 4. Experimental and predicted final conversions for model I for different weights of packing.
W
= 5.09
dp mm
0 200 0 271
0 359
* l
0
04
08
W/Q
,
0508 .
,
,
.
12
,
16
20
g min/mL
Figure 5. Experimental and predicted final conversions for model I for different particle diameters.
(11)
To calculate the axial dispersion coefficient, the equation proposed by Chung and Wen (1968) was used: N~~ = 0.20
.
(10)
Fraction Final Conversion. Xf
(12)
All of the calculated mass-transfer coefficients, axial dispersion coefficients, and Reynolds numbers used for the reactor runs are given in Tables 1-111. The initial invertase concentration, [E,], in eq 9 was replaced by the amount of invertase immobilized per gram of alumina, [E',], which was experimentally determined to be 25 mg of invertase/g of alumina. Equations 1and 2 were numerically integrated by using eq 9 for the kinetic model. In order to fit the experimental conversion data, two models were used, models I and 11. In model I, a new set of kinetic constants, ho,KM, K,, and K p ,was generated by adjusting all of the constants to get a good fit. The values obtained were h, = 107.0 wmol/(min.mg), K M = 31.75 mM, KI = 979.2 mM, and KP = 59.9 mM. The simulation
W/Q,
g,min/mL
Figure 6. Experimental and predicted final conversions for model I for different initial sucrose concentrations.
results are plotted in Figures 4-6. Model I predicted the data with an average deviation of *2.7%. In model 11, the ko used was 99.4 Mmol/(min.mg). The other three contants, KM, K,, and Kp, were calculated by multiplying the homogeneous kinetic constants by the same factor of 0.68, giving KM = 28.3 mM, KI = 1192 mM,
Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987 1815 and Kp = 63.2 mM. The predictions using model I1 gave results very similar to those shown in Figures 4-6 for model I. Model I1 predicted the experimental data with an average deviation of *2.9%, which is very close to that for model I.
Discussion and Conclusions Homogeneous Kinetics of Invertase. The initial rate of sucrose reaction was obtained a t initial sucrose concentrations in the range 26-1594 mM. The concentration of invertase in the reaction mixtures was also varied in the range 0.0200-0.0502 mg/mL. The normalized initial rate data given in Figure 1 show that the initial rate is simply proportional to the invertase concentration present. The resulting data also show that the initial rate increases gradually until the sucrose concentration reaches about 260 mM, after which the apparent reaction velocity decreases with increasing sucrose concentration. Several similar results have also been reported in the literature (Nelson and Schubert, 1928; Arnold, 1965; Bowski et al., 1971; Lilly et al., 1973; Marconi et al., 1974; Combes and Monsan, 1982). This value of the sucrose concentration of 260 mM at the peak rate of this work is in reasonable agreement with the literature values. The Michaelis-Menten model is also shown in Figure 1. The prediction at high sucrose concentrations does not fit the experimental data a t all. The substrate inhibition plus total water concentration model of eq 7 gave the best fit and was used to correlate the data. The water concentration term [ W] / [Wo] means that the invertase activity is first order with respect to water concentration. Nelson and Schubert (1928) also reported that the rate was proportional to the total water concentration. The integral form of substrate inhibition plus total water concentration model, eq 7 , was plotted in Figures 2 and 3. This substrate inhibition model fails to predict the experimental data in the higher conversion region where the effect of product inhibition on the invertase activity becomes significant. With an additional adjustable constant, Kp, the integral form of the substrate plus product inhibition model, eq 9, predicted the time course of reaction data quite well for two initial sucrose concentrations, 132.9 and 398.4 mM, as shown in Figures 2 and 3. The final values of the kinetic constants are ho = 331.4 pmol/(min.mg), KM= 41.6 mM, KI = 1753 mM, and KP = 92.9 mM. Some investigators have inadequately reported their invertase kinetics due to the employment of the initial rate technique with only low initial sucrose concentrations. No simple correlation of the kinetic constants could be found with literature data. The reported kinetic constants depended greatly on the source and purity of invertase. The reaction conditions would also affect the performance of invertase. The reported ho values were between 16 (Filippusson and Hornby, 1970) and 1300 pmol/(min.mg) (Adachi et al., 1980, 1981). The reported KMvalues were between 0.448 (Mason and Weetall, 1972) and 880 mM (Kahrig et al., 1980). The reported KI and KP values had a range of 864 (Bowski et al., 1971) to 4884 mM (Dickensheets et al., 1977) and a range of 34 (Santin et al., 1982) to 128 mM (Combes and Monsan, 19821, respectively. The kinetic constants reported in this work fall well in these ranges. Fixed-Bed Biochemical Reactor Results. The external mass-transfer resistance in the fixed-bed reactor can be qualitatively investigated by operating the reactor at various packed weights of catalyst. At the same space time with the presence of external mass-transfer resistance, the final extent of conversion will be higher for the deeper bed.
As shown in Figure 4, the reactor packed with 0.3 g of immobilized invertase on alumina experienced a strong external mass-transfer resistance, especially at the region of high conversions and low flow rates. Evidently, when the packed weight of immobilized invertase on alumina reaches about 5 g, the external mass-transfer resistance can be completely neglected. From Tables 1-111, it can be concluded that for immobilized enzyme systems having the same order of activity of this work, the external masstransfer resistance is small or negligible as long as the external mass-transfer coefficient is greater than about 2.5 x m/s. The fixed-bed reactor was operated with invertase immobilized on four different particle diameters, 0.508,0.359, 0.274, and 0.200 mm, to study the effect of internal pore diffusion on the reaction rate. The results of these studies presented in Figure 5 indicate the strong dependence of the reaction rate upon the catalyst size. At the same value of W/Q,the final conversion progressively increases as the particle size of the catalyst decreases, as is expected when pore diffusion significantly affects the observed rate of reaction. This set of experiments was conducted with reactor conditions such that the external mass-transfer resistance was unimportant, as shown by the large predicted mass-transfer coefficients tabulated in Table 11. In order to investigate the intrinsic kinetics of the immobilized invertase on alumina, the reactions were studied with four different initial sucrose concentrations of 1,3, 10, and 20 g/100 mL. Since the degree of conversion is independent of substrate concentration only for a firstorder reaction, the experimental results plotted in Figure 5 were expected. For the same space time, the higher initial sucrose concentration gives a smaller final conversion. Modeling and Simulation. The diffusion and reaction model, eq 1 and 2, includes the effects of external mass transfer, internal mass transfer, and axial dispersion. According to Smith (1981), the variations of interstitial velocity and concentration radially across a packed bed are small when d / d , 2 30, where d is the reactor diameter. In this work, this condition was met. The effect of the axial dispersion depends on the length of the reactor, the effective diffusivity, and the velocity. Its importance decreases as the velocity and length increase. For low velocities (NRe< 1)in short reactors, axial dispersion cannot be neglected (Smith, 1981). As shown in Tables 1-111, the axial dispersion was significant for the experimental conditions of this study. The accuracy of the model depends greatly on the proper estimation of physical constants. As discussed previously by Hu et al. (1985), the tortuosity measured by using diffusion of a solute other than reactant into the active immobilized enzyme carrier gives a more accurate estimation of effective diffusivity in the pores. The tortuosity of immobilized invertase alumina was measured by using a low glucose concentration, 1.2 mg/mL, to eliminate the surface diffusion effects. There was some concern that with a sucrose concentration of 20 g/100 mL, the surface diffusion might become significant. According to the sucrose adsorption data on polymerized alumina for this system given by Hu et al. (1985), the capacity for sucrose adsorption was depressed by the presence of polymer matrices to only 6% surface coverage for the sucrose concentration of 20 g/100 mL. The surface diffusion can be neglected for such a small 6% surface coverage (Leyva-Ramos, 1981). Furthermore, in the reaction medium, sucrose will also be subject to competitive adsorption by the presence of glucose and fructose. Since the glucose and
1816 Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987
fructose adsorb more strongly than sucrose, the amount of sucrose adsorbed will be further depressed. Hence, surface diffusion of sucrose is quite small and can be neglected (Leyva-Ramos and Geankoplis, 1986). Some investigators (Kobayashi and Moo-Young, 1973; Toda and Shoda, 1975) did not take into account the concentration dependency of sucrose diffusivity. The sucrose diffusivities for 1, 3, 10, and 20 g/100 mL were calculated to be 0.5149, 0.4994, 0.4452, and 0.3679 X mz/s by using eq 10. Ignoring this effect may cause some error. In model I, the best set of intrinsic kinetic constants were evaluated for immobilized invertase. As shown in Figures 4-6, with one set of kinetic constants, the model predicted all the data quite well with an average deviation of only &2.7%, Compared with the homogeneous kinetic constants, the heterogeneous constants KM, KI, and Kp decreased by 2470, 4470, and 36%. The predictions of conversion by assuming no external and pore mass-transfer resistances are also plotted in Figure 5. Due to the high immobilized invertase activity, the smallest particle still experienced some internal pore diffusion resistance. It was surmised that the immobilization might induce the same degree of effect on the kinetic constants KM, KI, and Kp. Therefore, in model 11, the values of the homogeneous constants KM, K,, and Kp were multiplied by the same factor of 0.68 or a decrease of 32% for the model I1 calculations. The predictions using model I1 when plotted in a manner similar to Figures 4-6 gave results almost identical with those for model I. Models I and I1 incorporate external mass transfer, experimental internal pore mass transfer, axial dispersion in the packed bed, and a complicated enzyme reaction model for both substrate and product inhibition. Hence, it appears that the two models, which are very similar, represent the experimental results quite well and confirm the various mechanisms in the model. This study is also novel in that pore diffusion tortuosities were separately experimentally determined. As a result, only the kinetic constants needed to be determined using the model and the experimental conversion data. Many other investigators determined both the pore diffusion and kinetic constants from experimental conversion data which can lead to more uncertainties in confirming a proposed model. Heterogeneous Kinetic Constants of Invertase. Invertase is one of the most studied of all enzymes. Many carriers and immobilization techniques have been used by previous investigators. Depending on the sources of invertase, the chemistry and morphology of carriers, the mode of immobilizations, and the techniques employed for enzymatic kinetic studies, the results in the literature have shown a broad range of differences in the extent of changes on the kinetic constants due to the immobilization. Hence, a direct comparison of the kinetic constants is difficult to make. Data for chemically attaching the enzyme to the solid for porous glass show a value of k o which is 38% as large as that of the native enzyme activity (Ooshima and Harano, 1980, 1981); for chemical attachment to Sepharose, 32% (Choi et al., 1980);for polystyrene, 11% (Filippusson and Hornby, 1970); and for cellulose, 34% (Lilly et al., 1973). Hence, the value of 30% for k , when compared to the soluble or free enzyme of this work for model I1 appears reasonable. The Michaelis constant, K,, reflects the affinity between the enzyme and the substrate. After chemically attaching the enzyme to Sepharose, a value for KMwhich was 78% as large as that for the free enzyme was obtained (Choi et
al., 1980). For polystyrene, this value was 222% (Filippusson and Hornby, 1970); for cellulose, 21 % (Lilly et al., 1973); for porous glass, 100% (Mason and Weetall, 1972); for porous glass, 103% (Ooshima and Harano, 1980, 1981); for clay, 93% (Santin et al., 1982);and for Sephadex, 357% (Woodward and Wiseman, 1982). To explain the apparent increase in KM,Filippusson and Hornby (19'70) postulated that the attachment of the enzyme to the surface of a hydrophobic structure such as polystyrene might create an environment with less concentration of substrate at the surface of the polymer matrix than that measured in the bulk solution. This would ensure an increase in the observed KW For the present work, a value was obtained for KM which is 68% as large as that for the free enzyme, which is lower than most of the reported values. The reason for this is not clear, but this could possibly be explained by the fact that alumina is not hydrophobic and could attract substrate, product, or other molecules. These higher local concentrations should depress KM and the inhibitor constants KI and KP by about the same amount, which occurred in this work. Very little data are available in the literature for chemical attachment and its effect on the constants KI and Kp. In this work both constants were 68% as large as those of the soluble enzyme. Since the experiments in this work were carried out for a wide range of sucrose concentrations, final conversions, particle sizes, and flow conditions, and since the model accurately predicts the results over these wide ranges, the heterogeneous kinetic constants generated in this study should reflect quite well the true heterogeneous kinetics constants of invertase. A very important feature of the present immobilization procedure is the long-term stability, which showed only a 14% decrease in initial activity in 22 days (Hu e t al., 1985).
Nomenclature c = average concentration, g/100 mL CL,r= substrate concentration in the porous solid phase of catalyst at distance of r , mM C,, = substrate concentration at outer surface of catalyst, mM C1 = substrate concentration in liquid phase of fixed-bed reactor, mM C, = initial substrate concentration at entrance of fixed-bed reactor, mM d = reactor diameter, m d, = particle diameter, m DAB= molecular diffusivity of diffusing solute, m2/s De, = effective pore volume diffusivity, mz/s D, = diffusivity of substrate, m2/s EL = axial dispersion coefficient based on interstitial velocity, mz/s [E,] = initial enzyme concentration, mg/mL [E',] = 25 mg of invertase/g of alumina H = height of packing, m JD = ( ~ , / U , ) ( P / P D ~ ~ ) ~ / ~ k L = external mass-transfer coefficient, m/s ho = catalytic rate constant defined in eq 7, pmol/(min.mg of invertase) K I = substrate inhibition constant, mM KM = Michaelis constant, mM KP = product inhibition constant, mM Np, = Peclet number, uod,/EL N,, = Reynolds number, d p u o p / l [PI = total product concentration, mM Q = flow rate of substrate solution, mL/min r = distance in the radial direction of catalyst, m rc = rate of reaction, Fmol/(min.mL) ro = initial activity, pmol/(min.mg of invertase) R = particle radius, m R*(C,,) = rate of reaction per unit weight of catalyst, mM/(s.g of catalyst) [SI = substrate concentration = [S,](l - X),mM
Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987 1817 [So] = initial substrate concentration, mM uo = superficial average velocity, m/s
W = amount of catalyst in packed bed, g [W] = total water concentration, M [W,] = 55.33 mol of water/L at 25 "C X = mole fraction conversion Xf = final mole fraction conversion in fixed-bed reactor z = height in the axial direction of fixed-bed reactor, m Greek Symbols t = void fraction of the bed = medium viscosity, g/(m.s) p = medium density, g/m3 pp = particle density, g/m3
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