Biotechnol. Prog. 1990, 6, 205-209
205
ARTICLES Experimental and Theoretical Evidence for Convective Nutrient Transport in an Immobilized Cell Support V. Bringit and B. E. Dale**' Department of Agricultural and Chemical Engineering, Colorado State University, Fort Collins, Colorado 80523
Even though immobilized-cell reactors possess several engineering advantages over free-cell reactors, their full potential has not been realized because mass transfer often limits the rate of nutrient supply and product removal from immobilized cell supports. We studied the interaction between mass transfer and reaction kinetics in the anaerobic conversion of glucose t o C02 and ethanol by yeast immobilized in a porous rotating disk on the agitator shaft of a conventional CSTR. A Sherwood number correlation was used t o show t h a t external mass-transfer resistances were negligible under typical operating conditions. T h e modulus of Weisz based on observable reaction parameters was used to gauge the importance of pore diffusion limitations. Under conditions for which significant pore diffusion effects and hence low effectiveness factors (77 = ca. 0.1) would be predicted, the observed reaction rates were much higher than expected (7 = ca. l),suggesting that pore diffusion limitations were a t least partially relieved by convective transport of glucose into the support. Two possible mechanisms of convective transport are discussed. We hypothesize t h a t gas evolution was responsible for the convective enhancement of glucose supply.
Introduction Immobilized-cell reactors offer several engineering advantages over free-cell reactors, including higher volumetric productivities, continuous cell reuse, and simplified downstream processing ( I ) . However, the full potential of immobilized-cell reactors has not been realized, in part because mass transfer often limits the rate of nutrient supply to, and product removal from, immobilizedcell supports (2). Other workers have successfully operated immobilizedcell reactors without mass-transfer limitations to study the effect of immobilization on the intrinsic kinetics of anaerobic glucose conversion by immobilized yeast cells (3-7). The object of our study was to determine whether conventional reaction-diffusion models could quantitatively predict reactor behavior when diffusion limitations were deliberately imposed for prolonged periods. Further, whereas previous studies used gel-entrapped or surface-immobilized cells in a variety of configurations (CSTR, recycle reactor, packed bed), this study was conducted with another reactor configuration eminently suited for studying mass-transfer effects, viz., the rotating-disk reactor. Materials and Methods Microorganisms and Media. Saccharomyces uuarum NRRL Y1347 cultures were maintained on YM
* To whom correspondence should be addressed. t Current address: School of Chemical Engineering, 120 Olin Hall, Cornel1 University, Ithaca, NY 14853. Current address: Department of Chemical Engineering, Texas A&M University, College Station, TX 77843.
*
agar slants (10 g/L glucose, 5 g/L yeast extract, 3 g/L malt extract, 2 g/L peptone) a t 4 OC and subcultured every 60 days. For inoculum, cells were grown in the same medium without agar at 30 "C in aerobic shake flasks. Reactor experiments used medium containing 2e180 g/L glucose, 5 g/L yeast extract, 2 g/L KHzP04, and 1 g/L MgS04.7H20. Assay. Glucose concentrations were measured by a Beckman glucose analyzer with use of the glucose oxidasebased enzyme assay. Immobilization. Immobilization details have been reported elsewhere (8). Porous compressed glass fiber mat supports fabricated into disks were provided by Manville Service Corp., R&D Center, Denver, CO 80217. The immobilization strategy involved (i) initial cell seeding by glutaraldehyde-facilitated adsorption onto gelatincoated support, (ii) starvation in a marine environment for 24 h to strengthen binding, and (iii) final loading by aerobic cell proliferation and growth. After immobilization was complete, the support was comparable to a dense yeast cake structurally reinforced by glass fibers. The support characteristics determined a t the end of the reactor run were as follows: disk diameter = 9 cm; thickness when wet = 0.25 cm (uniform); total support volume = 15.9 cm3; cell loading = 2.1 g total dry weight. Reactor Setup and Operation. The disk with immobilized cells was supported by stainless steel meshes (2-mm openings) and mounted on the agitator shaft of a Marubishi Model 2.6MD benchtop fermentor (Bioengineering Associates, Newton, MA 02164) as shown in Figure 1. The temperature was maintained a t 30 "C, pH was controlled at 5.0 0.1, the agitator speed was set at 90 rpm,
*
8756-7938/90/3006-0205$02.50/00 1990 American Chemical Society and American Institute of Chemical Engineers
Biotechnol. Prog., 1990, Vol. 6,No. 3
200
z 0
k r
61
TlRRlNG DEVICE EFLON WASHERS PADDLE
:I,,,
DRIVE
Figure 1. Schematic diagram of the immobilized-cell rotating-
disk reactor.
and the working volume was 1 L. Steady-state reactor and exit glucose concentrations were always the same, confirming CSTR behavior. Data were collected over 50 days of continuous operation. The concentration of free cells in the reactor was typically 0.05 g dry wt/L. Freecell numbers were always less than 10% of the immobilized cells. Dilution rates above washout (0.7-1.5 h-l) were employed to minimize the contribution of free cells to the overall reaction rate. Perhaps due to the low freecell concentration, wall growth was not observed to be significant. External Mass Transfer. A Sherwood number correlation for mass transfer from the bulk fluid to the surface of a rotating disk was calculated from Levich (9).
Sh = 0.54(Rei)o.5(Sc)0.33 valid for Rei < lo5 Impeller Reynolds number:
Re = sNd2/u
Schmidt number: Sc = u/D where d is the disk diameter (9 cm), N is the disk rpm, Y is the kinematic viscosity, and D is the diffusivity of glucose in the liquid medium (assumed to be 7 x lo+ cm2/s (IO)). At reactor operating conditions (90 rpm agitator speed); the Sherwood number is 1202. However, two numerical modifications need to be made to this value. First, since the appropriate characteristic length for external mass transfer is the disk diameter whereas internal mass transfer is characterized by the disk half-thickness, a geometry factor, i.e., disk halfthickness/disk diameter, modifies the calculated Sherwood number. Second, since external and intraparticle resistances are present simultaneously, the Sherwood number should be based on Deff, the effective diffusivity (3 X lo* cm2/s, see below) rather than on the molecular diffusivity. The modified Sherwood number or the Biot number is defined as (D/Deff)X Sh. Its value is 39, which is high enough for external mass-transfer limitations to be considered negligible for many biological systems ( I I ) . A rotating disk provides a high relative velocity at the disk-liquid interface. High Sherwood numbers and consequently high rates of interfacial mass transfer are achievable even a t low agitator speeds. This feature is one of the advantages of the rotating-disk-reactor system. Measurement of Effective Diffusivity. The effective diffusivity of glucose in the immobilization support was measured in situ under typical reactor conditions of 90 rpm agitator speed and 30 "C. A disk loaded with immobilized cells was submerged for 2 days in medium containing 30 g/L glucose and 2% Tergiquat (National Laboratories, Montvale, NJ 07645) to kill the cells. The disk presaturated with glucose was placed in the reactor containing glucose-free medium, and the transient appearance of glucose in the bulk liquid was monitored. The equations of Crank (12) were used to calculate the effective diffusivity from these data. This procedure has been
0
0
M a
100
200
GLUCOSE CONCENTRATION (giL)
v)
Figure 2. Steady-state reaction rate data from the rotatingdisk reactor. widely employed to measure effective diffusivities in immobilized cell supports (e.g., ref 13). The average effective diffusivity was (3.0 f 0.5) X lo4 cm2/s or 43% of the diffusivity in water. For comparison, the correlation of Axelsson and Persson (IO),developed for alginate-immobilized yeast cells, would have yielded a value of 2.6 X lo4 cm2/s a t a comparable volume fraction (assumed as 52.4%,the value for cubic closepacked spheres) of cells. This comparison is not unreasonable since the alginate gel itself does not significantly hinder diffusion of sugars (e.g., see ref 13). The agreement between the experimental and predicted values is reasonable.
Results and Discussion Discrepancy between Predicted and Observed Rates. Figure 2 shows steady-state reaction rate data obtained from the reactor. Flow rates and inlet glucose concentrations were manipulated so as to impose varying levels of diffusion limitations. The modulus of Weisz (14) based on observable reaction parameters was used to gauge the importance of diffusion limitations on the intrinsic reaction rate. Weisz's modulus is applicable to steady-state situations when diffusion is the sole transport mechanism into the immobilized cell support. The advantage of Weisz's modulus over the more commonly used Thiele modulus is that no knowledge of the intrinsic kinetic parameters (i.e., K,, the saturation constant, and V,,,, the maximal rate for immobilized cells) is required a priori. It is thus an extremely useful tool to assess the importance of mass-transfer limitations on reaction kinetics in cases where experimental intrinsic kinetic data are not available, as was the case for this system. Weisz's modulus is defined as
"
WJeff
where Robs is the observed reaction rate per unit volume of catalyst, L is the characteristic length (here the disk half-thickness, b ) , cb is the steady-state bulk glucose concentration, and D,ff is the effective diffusivity of glucose in the support. Weisz's general criteria are as follows: if @pw < 0.3, no gradients likely, kinetic control probable if @W > 3, gradients likely, diffusion control probable However, for the specific case of a zero-order reaction in flat-slab geometry, an explicit criterion can be analyti-
207
Biotechnol. hog., 1990, Vol. 6, No. 3
Table I. Weisz’s Modulus (h) and Expected Zero-Order Effectiveness Factors ( q ) for Representative Reactor Data Points Shown in Figure 2
specific glucose consumption rate, g / g h 2.9 2.7 4.6 4.9
Cbt’
g/L 18
27 60
75 4.0 89 3.6 123 4.0 171 a Cb = bulk glucose concentration.
Qw 30.8 19.1 14.7 12.5 8.6 5.6 4.5
2/*w 0.06 0.10 0.14 0.16 0.23 0.36 0.44
9=
cally derived following an analysis similar to that of Wheeler (15). The final result is presented here without proof for @W < 2, effectiveness factor, q = 1 for QW > 2, effectiveness factor, q = 2/9w The above result is presented in graphical form in the effectiveness factor versus Weisz’s modulus charts found in refs 1, 14, and 16. Table I summarizes the calculated values of +W and q for the various data points shown in Figure 2. Weisz’s analysis suggests that severe diffusion limitations were present, with effectiveness factors ranging from 0.06 to 0.44 at the various steady-state data points. The numerical value of the observed specific glucose uptake rate, however, is in the same range as the free-cell uptake rate for this strain (maximal rate = ca. 6 g/g.h, data not shown). It also compares well with intrinsic immobilizedcell uptake rates of other yeast strains (Table 11). A discrepancy is clearly evident-observed reaction rates are much higher than would be expected according to the reaction/diffusion analysis. Another way of stating the discrepancy is as follows: The inevitable consequence of assuming that diffusion is the sole transport mechanism is that the intrinsic rate a t which these cells are capable of consuming glucose must be, by definition, Robs/q, Le., from ca. 9 to ca. 48 g/g.h. Galazzo and Bailey (3) have shown that, in short-timecourse experiments, immobilized Saccharomyces cereuisiae are capable of metabolizing glucose a t two times the rate of free cells. Even so, the required rate of 48 g/g.h is absurd and is far above that reported for Saccharomyces. By the logic of reductio ad absurdum, diffusion could not have been the sole transport mechanism, suggesting that convective transport into the porous support was also significant. How could convective transport occur in a tightly packed yeast volume? Two possible hypotheses may be summoned. External Flow-Induced Convection. Nir and Pismen (13,and, more recently, Stephanopoulous and Tsiveriotis (181, have shown that external flow-induced pressure gradients can cause convective flow of bulk liquid into a porous particle. For a rotating disk, centrifugal forces provide an additional driving force for convection. The case of convective transport into porous rotating objects has not been analyzed, making a quantitative estimate of its extent difficult for this reactor system. However, one experimental result, viz., the experimental determination of the effective diffusivity of glucose, serves to repudiate this hypothesis. The effective diffusivity was measured in situ under conditions closely approximating an actual reactor run (see Measurement of Effective Diffusivity). The presence of external flow-induced convective flow through the immobilization support would have increased the rate of glucose efflux out of the disk. Since the model used for data analysis ascribes the total efflux to diffusion,
the presence of convective flow would simply be reflected in a higher value for Deff. Rodrigues and QuintaFerreira (19) have drawn attention to a case in which the measured effective diffusivity in a porous support increased with the particle Reynolds number and have attributed the increase to pressure-driven convective transport. In our system, a doubling of the efflux rate compared to the observed value would have resulted in Deft = 1.3 X 10-5 cm2/s (i.e., a 4-fold increase compared to the observed value). A tripling of the efflux rate would have resulted in de^ = 3.6 X lW6 cmz/s (a 12-fold increase). The fact that increases in D,ff were not observed, and that the rate of glucose efflux in the diffusivity measurements could be attributed to diffusion alone, tends to diminish the possible importance of external flowinduced convection in this reactor system. Entrained Flow Hypothesis. One physical situation that is often not considered in theoretical kinetic analyses of immobilized yeast systems is the vigorous evolution of gaseous C02 from within the immobilization support. Though gas production rates were not measured for our system, they may be readily estimated from the reaction stoichiometry: CGHIZO, = 2CzHSOH + 2C02 Assuming that the product yield was 90% of theoretical, which is not unreasonable for yeast, the rate of COZ production must have been 3.6 g/h (1860 mL of COz/h) a t the observed maximal glucose consumption rate of 4 g/g.h. To estimate what fraction of the COZmust have left the support as gas, and what fraction must have diffused into the bulk medium as dissolved COz, it is instructive to compare the characteristic times for COZproduction and for the rate of C02 diffusion. The rate of COP production was ca. 117 volumes/volume of support/h, which yields a characteristic time (defined as the reciprocal of the rate) of ca. 30 s. The characteristic time for diffusion (defined as b2/&) is ca. 1500 s. Since the local rate of C02 production was much higher than its rate of diffusion out of the support, the majority of COZ must have left the support in the gaseous form. We in fact did observe a continuous and vigorous stream of gas bubbles escaping from many well-distributed sites on the disk surface during reactor operation. We hypothesize that convective transport of liquid medium into the support must have been engendered by the evolving gas. A physically analogous situation occurs in certain electrochemical reactions, Le., a component in the bulk liquid reacts on the interior surface of a porous electrode and a gaseous byproduct is released. Electrochemists have long recognized that gas evolution enhances rates of mass transfer. The current view is that gas bubbles decrease external mass-transfer resistances by disrupting the concentration boundary layer, thereby causing convective liquid flow a t the electrode surface (20). However, Russian researchers have earlier recognized that gas evolution enhances mass transfer of liquid-dissolved components into porous electrodes as well (21,22). Feoktistov et al. (22) have suggested that such enhanced transfer should depend on the rate of gas evolution as well as on the pore structure. A similar enhancement of glucose transport may have been responsible for the increased reaction rates in our immobilized yeast system. For a mechanistic description of how convective transport could occur, it is instructive to visualize the tortuous interconnected pore network in the immobilized cell support. The yeast cells themselves would constitute the pore walls. At steady state, part of the pore volume must
Biotechnol. Pmg., 1990,Vol. 6, No. 3
200
Table 11. Summary of Intrinsic Kinetic Parameters of Immobilized Yeast Cells from the Literature Km,
Vmam
yeast
g/L
g/g.h
Saccharomyces formosensis Saccharomyces cereuisiae Saccharomyces cereuisiae (SC 4126) Saccharomyces uvarum (NRRL Y1347) Saccharomyces cereuisiae (ATCC 18790)
3.3
0.54
polyacrylamide gel,
5.4
1.10
calcium alginate,
immobilization
method/reactor CSTR
CSTR
4.2
5.90
8.0
4.0
calcium alginate, recycle packed bed rotating-disk
CSTR
?
1.16
have been occupied by gas and the rest by liquid medium. Of the possible two-phase flows encountered in capillaries, gas evolution probably occurred via “slug-flow” (23). In this flow pattern, gas bubbles, separated by slugs of liquid, move together in tandem. As the stream of bubbles evolved from the support, the liquid slugs between the bubbles would also probably be entrained into the bulk medium. To maintain constancy of the steadystate volume fraction of liquid inside the support, it follows that the bubble-entrained spent liquid medium must have been continuously replenished with bulk medium. Because of the heterogeneous nature of the support, liquid could have seeped in via cracks on the surface. Alternatively, there could have been a net movement of medium from the underside of the disk to the top surface as the bubbles rose through the pore network. Two observations recorded by Korovin et al. (21) serve to support our contention. These workers constructed a two-dimensional network of interconnected vertical and horizontal glass capillaries in order to simulate a porous medium. The vertical capillaries contained wire electrodes, from which gas bubbles evolved continuously as a byproduct of the electrochemical reaction. First, it was experimentally confirmed that the steady-state volume fractions of gas and liquid in the capillary were constant and, furthermore, were almost independent of the gas evolution rate. Second, Korovin et al. (21) observed that “Gas was expelled from the capillaries in pulses; once the plug has been expelled, one could clearly see that liquid and fine bubbles were taken up in the lattice at a depth commensurate with the capillary dimensions” (quoted from the English translation). Finally, one may estimate the fraction of gas volume that must be entrained and subsequently replaced by bulk medium in order to account for the enhanced glucose transport into the support. Assuming for simplicity that the diffusive contribution is negligible and that all the glucose is consumed before the medium is expelled, the mass balance over the support reduces to total observed uptake (g/h) = liquid entrainment rate (mL/h) X Cb (g/mL) A t Robs = 4 g / g h and c b = 100 g/L, the liquid entrainment rate would be ca. 82 mL/h. Since the volumetric gas flow rate under these conditions is ca. 1860 mL/h, the liquid equivalent to ca. 4.4% of the evolved gas volume would have to be entrained to account for the total glucose flux into the disk. At R o b = 2.6 g / g h and c b = 20 g/L, the liquid equivalent to an estimated 21 7% of the evolved gas volume would have to be entrained to account for the glucose flux into the disk. Since much higher, and perhaps unattainable, levels of entrainment were required a t lower glucose concentrations, diffusion limitations may have still played a role at low glucose con-
calcium alginate, recycle packed bed
comments nongrowing, resting cells resting cells initial rates
ref 7 6 5
this study
initial rates,
3
complete kinetics not reported
centrations and may have accounted for the high value of the saturation constant encountered in this study (e.g., see ref 24). In summary, our entrained flow hypothesis suggests that convective enhancement of nutrient supply may be expected to depend on 3 factors: (1)the volumetric gas evolution rate, (2) the microstructure of the porous support, and (3) the concentration of the reactant in the bulk medium.
Conclusions The rotating-disk reactor is a useful device to study the effects of mass transfer on the kinetics of immobilized-cell reactions. The results of this study suggest that convection-enhanced substrate supply to immobilized cells may have a t least partially circumvented diffusion limitations. Immobilized-cell reactors may hence be designed to operate a t higher effectiveness factors and consequently a t higher overall volumetric productivities. b cb
d
D Deff Km
I N Robs Vmax @W
17 Y
Rei sc Sh
Notation half-thickness of disk steady-state bulk reactant concentration disk diameter diffusivity in water effective diffusivity in the porous support saturation constant characteristic length agitator speed (rotations per minute) observed rate of reaction (based on unit support volume) maximal reaction rate Weisz’s modulus effectiveness factor kinematic viscosity impeller Reynolds number Schmidt mmber Sherwood number
Acknowledgment Support of this work by the Manville Corp. and Colorado Cooperative State Research Service under Project 1-53831 is gratefully acknowledged. Literature Cited (1) Bailey, J. E.; Ollis, D. F. Biochemical Engineering Fundamentals, 2 ed.; McGraw-Hill: New York, 1986,p 595.
(2) Efthymiou, G. S.; Shuler, M. L. Elimination of Diffusion Limitations in a Membrane Entrapped Cell Reactor by Pressure Cycling. Biotechnol. Prog. 1987,3, 259. (3) Galazzo, J. L.; Bailey, J. E. In vivo Nuclear Magnetic Resonance Analysis of Immobilizationeffects on Glucose Metab-
209
Biotechnol. Prog., 1990, Vol. 6, NO. 3
olism of Yeast Saccharomyces cereuisiae. Biotechnol. Bioeng. (4) Doran, P. M.; Bailey, J. E. Effects of Immobilization on
(15) Wheeler, A. Reaction Rates and Selectivity in Catalyst Pores. Adu. Catal. 1951,3, 245 (see Appendix on page 323). (16) Roberts, G. W.; Satterfield, C. N. Effectiveness Factors
Growth, Fermentation Properties, and Macromolecular Composition of Saccharomyces cerevisiae attached to gelatin. Biotechnol. Bioeng. 1986, 28, 73. (5) Ryu, D. D. Y.; Kim, H. S.; Taguchi, H. Intrinsic Fermentation Kinetic Parameters of Immobilized Yeast Cells. J. Ferment. Technol. 1985,62,255. (6) Furusaki, S.; Seki, M. Effect of Intraparticle Mass Transfer Resistance On Reactivity of Immobilized Yeast Cells. J . Chem. Eng. Jpn. 1985,18,389. (7) Furusaki, S.; Seki, M.; Fukumura, K. Reaction Characteristics of an Immobilized Yeast producing Ethanol. Biotechnol. Bioeng. 1983, 25, 2921. (8) Bringi, V.; Dale, B. E. Enhanced Yeast Immobilization by Nutrient Starvation. Biotechnol. Lett. 1985, 7, 905. (9) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962; pp 62-71. (10) Axelsson, A.; Persson, B. Determination of Effective Diffusion Coefficients in Calcium Alginate Gel Plates with varying Yeast Cell Content. Appl. Biochem. Biotechnol. 1988, 18, 231. (11) Buchholz, K. Characterization of Immobilized Biocatalysts; DECHEMA Monograph No. 1724-1731; Verlag Chemie: Weinheim, FRG, 1979; Vol. 84, p 232. (12) Crank, J. The Mathematics of Diffusion, 2 ed.; Oxford University Press: London, 1975; pp 57-62. (13) Tanaka, H.; Matsumura, M.; Veliky, I. A. Diffusion Characteristics of Substrates in Calcium Alginate Gel Beads. Biotechnol. Bioeng. 1984, 26, 53. (14) Weisz, P. B. Diffusion and Chemical Transformation-An Interdisciplinary Excursion. Science 1973, 179, 433.
Accepted April 20, 1990. Registry No. Glucose, 50-99-7.
1989,33, 1283.
for Porous Catalysts: Langmuir-Hinshelwood Kinetic Expressions. Znd. Eng. Chem. Fundam. 1965,4,288. (17) Nir, A.; Pismen, L. M. Simultaneous Intraparticle Forced Convection, Diffusion and Reaction in a Porous Catalyst. Chem. Eng. Sei. 1977, 32, 35. (18) Stephanopoulous, G.; Tsiveriotis, K. The Effect of Intraparticle Convection on Nutrient Transport in Porous Biological Pellets. Chem. Eng. Sei. 1989, 44, 2031. (19) Rodrigues, A. E.; Quinta-Ferreira, R. M. Convection, Diffusion and Reaction in a Large-Pore Catalyst Particle; AIChE Symp. Ser. 266; American Institute of Chemical Engineers: New York, 1988; Vol. 84, p 80. (20) Ismail, M. I. Ed. Electrochemical Reactors, Their Science and Technology, Part A: Fundamentals, Electrolysers, Batteries, and Fuel Cells; Elsevier: Amsterdam, 1989; pp 167-176. (21) Korovin, N. V.; Chudinov, A. S.; Feoktistov, A. F. A Study of the Liquid-Gas Electrode 3. A Two-Dimensional Capillary System. Sou. Electrochem. (Engl. Transl.) 1969,5,430. (22) Feoktistov, A. F.; Maksimov, P. N.; Korovin, N. V. Investigation of the Liquid-Gas Electrode 6. Pulsation Model. Sou. Electrochem. (Engl. Transl.) 1974, 10, 1592. (23) Taitel, Y.; Bornea, D.; Dukler, A. E. Modelling Flow Pattern Transitions for Steady Upward Gas-Liquid Flow in Vertical Tubes. AZChE J . 1980, 26, 345. (24) Hamilton, B. K.; Gardner, C. R.; Colton, C. K. Effect of Diffusional Limitation on Lineweaver-Burk Plots for Immobilized Enzymes. AIChE J . 1974, 20, 503.
205
ARTICLES Experimental and Theoretical Evidence for Convective Nutrient Transport in an Immobilized Cell Support V. Bringit and B. E. Dale**' Department of Agricultural and Chemical Engineering, Colorado State University, Fort Collins, Colorado 80523
Even though immobilized-cell reactors possess several engineering advantages over free-cell reactors, their full potential has not been realized because mass transfer often limits the rate of nutrient supply and product removal from immobilized cell supports. We studied the interaction between mass transfer and reaction kinetics in the anaerobic conversion of glucose t o C02 and ethanol by yeast immobilized in a porous rotating disk on the agitator shaft of a conventional CSTR. A Sherwood number correlation was used t o show t h a t external mass-transfer resistances were negligible under typical operating conditions. T h e modulus of Weisz based on observable reaction parameters was used to gauge the importance of pore diffusion limitations. Under conditions for which significant pore diffusion effects and hence low effectiveness factors (77 = ca. 0.1) would be predicted, the observed reaction rates were much higher than expected (7 = ca. l),suggesting that pore diffusion limitations were a t least partially relieved by convective transport of glucose into the support. Two possible mechanisms of convective transport are discussed. We hypothesize t h a t gas evolution was responsible for the convective enhancement of glucose supply.
Introduction Immobilized-cell reactors offer several engineering advantages over free-cell reactors, including higher volumetric productivities, continuous cell reuse, and simplified downstream processing ( I ) . However, the full potential of immobilized-cell reactors has not been realized, in part because mass transfer often limits the rate of nutrient supply to, and product removal from, immobilizedcell supports (2). Other workers have successfully operated immobilizedcell reactors without mass-transfer limitations to study the effect of immobilization on the intrinsic kinetics of anaerobic glucose conversion by immobilized yeast cells (3-7). The object of our study was to determine whether conventional reaction-diffusion models could quantitatively predict reactor behavior when diffusion limitations were deliberately imposed for prolonged periods. Further, whereas previous studies used gel-entrapped or surface-immobilized cells in a variety of configurations (CSTR, recycle reactor, packed bed), this study was conducted with another reactor configuration eminently suited for studying mass-transfer effects, viz., the rotating-disk reactor. Materials and Methods Microorganisms and Media. Saccharomyces uuarum NRRL Y1347 cultures were maintained on YM
* To whom correspondence should be addressed. t Current address: School of Chemical Engineering, 120 Olin Hall, Cornel1 University, Ithaca, NY 14853. Current address: Department of Chemical Engineering, Texas A&M University, College Station, TX 77843.
*
agar slants (10 g/L glucose, 5 g/L yeast extract, 3 g/L malt extract, 2 g/L peptone) a t 4 OC and subcultured every 60 days. For inoculum, cells were grown in the same medium without agar at 30 "C in aerobic shake flasks. Reactor experiments used medium containing 2e180 g/L glucose, 5 g/L yeast extract, 2 g/L KHzP04, and 1 g/L MgS04.7H20. Assay. Glucose concentrations were measured by a Beckman glucose analyzer with use of the glucose oxidasebased enzyme assay. Immobilization. Immobilization details have been reported elsewhere (8). Porous compressed glass fiber mat supports fabricated into disks were provided by Manville Service Corp., R&D Center, Denver, CO 80217. The immobilization strategy involved (i) initial cell seeding by glutaraldehyde-facilitated adsorption onto gelatincoated support, (ii) starvation in a marine environment for 24 h to strengthen binding, and (iii) final loading by aerobic cell proliferation and growth. After immobilization was complete, the support was comparable to a dense yeast cake structurally reinforced by glass fibers. The support characteristics determined a t the end of the reactor run were as follows: disk diameter = 9 cm; thickness when wet = 0.25 cm (uniform); total support volume = 15.9 cm3; cell loading = 2.1 g total dry weight. Reactor Setup and Operation. The disk with immobilized cells was supported by stainless steel meshes (2-mm openings) and mounted on the agitator shaft of a Marubishi Model 2.6MD benchtop fermentor (Bioengineering Associates, Newton, MA 02164) as shown in Figure 1. The temperature was maintained a t 30 "C, pH was controlled at 5.0 0.1, the agitator speed was set at 90 rpm,
*
8756-7938/90/3006-0205$02.50/00 1990 American Chemical Society and American Institute of Chemical Engineers
Biotechnol. Prog., 1990, Vol. 6,No. 3
200
z 0
k r
61
TlRRlNG DEVICE EFLON WASHERS PADDLE
:I,,,
DRIVE
Figure 1. Schematic diagram of the immobilized-cell rotating-
disk reactor.
and the working volume was 1 L. Steady-state reactor and exit glucose concentrations were always the same, confirming CSTR behavior. Data were collected over 50 days of continuous operation. The concentration of free cells in the reactor was typically 0.05 g dry wt/L. Freecell numbers were always less than 10% of the immobilized cells. Dilution rates above washout (0.7-1.5 h-l) were employed to minimize the contribution of free cells to the overall reaction rate. Perhaps due to the low freecell concentration, wall growth was not observed to be significant. External Mass Transfer. A Sherwood number correlation for mass transfer from the bulk fluid to the surface of a rotating disk was calculated from Levich (9).
Sh = 0.54(Rei)o.5(Sc)0.33 valid for Rei < lo5 Impeller Reynolds number:
Re = sNd2/u
Schmidt number: Sc = u/D where d is the disk diameter (9 cm), N is the disk rpm, Y is the kinematic viscosity, and D is the diffusivity of glucose in the liquid medium (assumed to be 7 x lo+ cm2/s (IO)). At reactor operating conditions (90 rpm agitator speed); the Sherwood number is 1202. However, two numerical modifications need to be made to this value. First, since the appropriate characteristic length for external mass transfer is the disk diameter whereas internal mass transfer is characterized by the disk half-thickness, a geometry factor, i.e., disk halfthickness/disk diameter, modifies the calculated Sherwood number. Second, since external and intraparticle resistances are present simultaneously, the Sherwood number should be based on Deff, the effective diffusivity (3 X lo* cm2/s, see below) rather than on the molecular diffusivity. The modified Sherwood number or the Biot number is defined as (D/Deff)X Sh. Its value is 39, which is high enough for external mass-transfer limitations to be considered negligible for many biological systems ( I I ) . A rotating disk provides a high relative velocity at the disk-liquid interface. High Sherwood numbers and consequently high rates of interfacial mass transfer are achievable even a t low agitator speeds. This feature is one of the advantages of the rotating-disk-reactor system. Measurement of Effective Diffusivity. The effective diffusivity of glucose in the immobilization support was measured in situ under typical reactor conditions of 90 rpm agitator speed and 30 "C. A disk loaded with immobilized cells was submerged for 2 days in medium containing 30 g/L glucose and 2% Tergiquat (National Laboratories, Montvale, NJ 07645) to kill the cells. The disk presaturated with glucose was placed in the reactor containing glucose-free medium, and the transient appearance of glucose in the bulk liquid was monitored. The equations of Crank (12) were used to calculate the effective diffusivity from these data. This procedure has been
0
0
M a
100
200
GLUCOSE CONCENTRATION (giL)
v)
Figure 2. Steady-state reaction rate data from the rotatingdisk reactor. widely employed to measure effective diffusivities in immobilized cell supports (e.g., ref 13). The average effective diffusivity was (3.0 f 0.5) X lo4 cm2/s or 43% of the diffusivity in water. For comparison, the correlation of Axelsson and Persson (IO),developed for alginate-immobilized yeast cells, would have yielded a value of 2.6 X lo4 cm2/s a t a comparable volume fraction (assumed as 52.4%,the value for cubic closepacked spheres) of cells. This comparison is not unreasonable since the alginate gel itself does not significantly hinder diffusion of sugars (e.g., see ref 13). The agreement between the experimental and predicted values is reasonable.
Results and Discussion Discrepancy between Predicted and Observed Rates. Figure 2 shows steady-state reaction rate data obtained from the reactor. Flow rates and inlet glucose concentrations were manipulated so as to impose varying levels of diffusion limitations. The modulus of Weisz (14) based on observable reaction parameters was used to gauge the importance of diffusion limitations on the intrinsic reaction rate. Weisz's modulus is applicable to steady-state situations when diffusion is the sole transport mechanism into the immobilized cell support. The advantage of Weisz's modulus over the more commonly used Thiele modulus is that no knowledge of the intrinsic kinetic parameters (i.e., K,, the saturation constant, and V,,,, the maximal rate for immobilized cells) is required a priori. It is thus an extremely useful tool to assess the importance of mass-transfer limitations on reaction kinetics in cases where experimental intrinsic kinetic data are not available, as was the case for this system. Weisz's modulus is defined as
"
WJeff
where Robs is the observed reaction rate per unit volume of catalyst, L is the characteristic length (here the disk half-thickness, b ) , cb is the steady-state bulk glucose concentration, and D,ff is the effective diffusivity of glucose in the support. Weisz's general criteria are as follows: if @pw < 0.3, no gradients likely, kinetic control probable if @W > 3, gradients likely, diffusion control probable However, for the specific case of a zero-order reaction in flat-slab geometry, an explicit criterion can be analyti-
207
Biotechnol. hog., 1990, Vol. 6, No. 3
Table I. Weisz’s Modulus (h) and Expected Zero-Order Effectiveness Factors ( q ) for Representative Reactor Data Points Shown in Figure 2
specific glucose consumption rate, g / g h 2.9 2.7 4.6 4.9
Cbt’
g/L 18
27 60
75 4.0 89 3.6 123 4.0 171 a Cb = bulk glucose concentration.
Qw 30.8 19.1 14.7 12.5 8.6 5.6 4.5
2/*w 0.06 0.10 0.14 0.16 0.23 0.36 0.44
9=
cally derived following an analysis similar to that of Wheeler (15). The final result is presented here without proof for @W < 2, effectiveness factor, q = 1 for QW > 2, effectiveness factor, q = 2/9w The above result is presented in graphical form in the effectiveness factor versus Weisz’s modulus charts found in refs 1, 14, and 16. Table I summarizes the calculated values of +W and q for the various data points shown in Figure 2. Weisz’s analysis suggests that severe diffusion limitations were present, with effectiveness factors ranging from 0.06 to 0.44 at the various steady-state data points. The numerical value of the observed specific glucose uptake rate, however, is in the same range as the free-cell uptake rate for this strain (maximal rate = ca. 6 g/g.h, data not shown). It also compares well with intrinsic immobilizedcell uptake rates of other yeast strains (Table 11). A discrepancy is clearly evident-observed reaction rates are much higher than would be expected according to the reaction/diffusion analysis. Another way of stating the discrepancy is as follows: The inevitable consequence of assuming that diffusion is the sole transport mechanism is that the intrinsic rate a t which these cells are capable of consuming glucose must be, by definition, Robs/q, Le., from ca. 9 to ca. 48 g/g.h. Galazzo and Bailey (3) have shown that, in short-timecourse experiments, immobilized Saccharomyces cereuisiae are capable of metabolizing glucose a t two times the rate of free cells. Even so, the required rate of 48 g/g.h is absurd and is far above that reported for Saccharomyces. By the logic of reductio ad absurdum, diffusion could not have been the sole transport mechanism, suggesting that convective transport into the porous support was also significant. How could convective transport occur in a tightly packed yeast volume? Two possible hypotheses may be summoned. External Flow-Induced Convection. Nir and Pismen (13,and, more recently, Stephanopoulous and Tsiveriotis (181, have shown that external flow-induced pressure gradients can cause convective flow of bulk liquid into a porous particle. For a rotating disk, centrifugal forces provide an additional driving force for convection. The case of convective transport into porous rotating objects has not been analyzed, making a quantitative estimate of its extent difficult for this reactor system. However, one experimental result, viz., the experimental determination of the effective diffusivity of glucose, serves to repudiate this hypothesis. The effective diffusivity was measured in situ under conditions closely approximating an actual reactor run (see Measurement of Effective Diffusivity). The presence of external flow-induced convective flow through the immobilization support would have increased the rate of glucose efflux out of the disk. Since the model used for data analysis ascribes the total efflux to diffusion,
the presence of convective flow would simply be reflected in a higher value for Deff. Rodrigues and QuintaFerreira (19) have drawn attention to a case in which the measured effective diffusivity in a porous support increased with the particle Reynolds number and have attributed the increase to pressure-driven convective transport. In our system, a doubling of the efflux rate compared to the observed value would have resulted in Deft = 1.3 X 10-5 cm2/s (i.e., a 4-fold increase compared to the observed value). A tripling of the efflux rate would have resulted in de^ = 3.6 X lW6 cmz/s (a 12-fold increase). The fact that increases in D,ff were not observed, and that the rate of glucose efflux in the diffusivity measurements could be attributed to diffusion alone, tends to diminish the possible importance of external flowinduced convection in this reactor system. Entrained Flow Hypothesis. One physical situation that is often not considered in theoretical kinetic analyses of immobilized yeast systems is the vigorous evolution of gaseous C02 from within the immobilization support. Though gas production rates were not measured for our system, they may be readily estimated from the reaction stoichiometry: CGHIZO, = 2CzHSOH + 2C02 Assuming that the product yield was 90% of theoretical, which is not unreasonable for yeast, the rate of COZ production must have been 3.6 g/h (1860 mL of COz/h) a t the observed maximal glucose consumption rate of 4 g/g.h. To estimate what fraction of the COZmust have left the support as gas, and what fraction must have diffused into the bulk medium as dissolved COz, it is instructive to compare the characteristic times for COZproduction and for the rate of C02 diffusion. The rate of COP production was ca. 117 volumes/volume of support/h, which yields a characteristic time (defined as the reciprocal of the rate) of ca. 30 s. The characteristic time for diffusion (defined as b2/&) is ca. 1500 s. Since the local rate of C02 production was much higher than its rate of diffusion out of the support, the majority of COZ must have left the support in the gaseous form. We in fact did observe a continuous and vigorous stream of gas bubbles escaping from many well-distributed sites on the disk surface during reactor operation. We hypothesize that convective transport of liquid medium into the support must have been engendered by the evolving gas. A physically analogous situation occurs in certain electrochemical reactions, Le., a component in the bulk liquid reacts on the interior surface of a porous electrode and a gaseous byproduct is released. Electrochemists have long recognized that gas evolution enhances rates of mass transfer. The current view is that gas bubbles decrease external mass-transfer resistances by disrupting the concentration boundary layer, thereby causing convective liquid flow a t the electrode surface (20). However, Russian researchers have earlier recognized that gas evolution enhances mass transfer of liquid-dissolved components into porous electrodes as well (21,22). Feoktistov et al. (22) have suggested that such enhanced transfer should depend on the rate of gas evolution as well as on the pore structure. A similar enhancement of glucose transport may have been responsible for the increased reaction rates in our immobilized yeast system. For a mechanistic description of how convective transport could occur, it is instructive to visualize the tortuous interconnected pore network in the immobilized cell support. The yeast cells themselves would constitute the pore walls. At steady state, part of the pore volume must
Biotechnol. Pmg., 1990,Vol. 6, No. 3
200
Table 11. Summary of Intrinsic Kinetic Parameters of Immobilized Yeast Cells from the Literature Km,
Vmam
yeast
g/L
g/g.h
Saccharomyces formosensis Saccharomyces cereuisiae Saccharomyces cereuisiae (SC 4126) Saccharomyces uvarum (NRRL Y1347) Saccharomyces cereuisiae (ATCC 18790)
3.3
0.54
polyacrylamide gel,
5.4
1.10
calcium alginate,
immobilization
method/reactor CSTR
CSTR
4.2
5.90
8.0
4.0
calcium alginate, recycle packed bed rotating-disk
CSTR
?
1.16
have been occupied by gas and the rest by liquid medium. Of the possible two-phase flows encountered in capillaries, gas evolution probably occurred via “slug-flow” (23). In this flow pattern, gas bubbles, separated by slugs of liquid, move together in tandem. As the stream of bubbles evolved from the support, the liquid slugs between the bubbles would also probably be entrained into the bulk medium. To maintain constancy of the steadystate volume fraction of liquid inside the support, it follows that the bubble-entrained spent liquid medium must have been continuously replenished with bulk medium. Because of the heterogeneous nature of the support, liquid could have seeped in via cracks on the surface. Alternatively, there could have been a net movement of medium from the underside of the disk to the top surface as the bubbles rose through the pore network. Two observations recorded by Korovin et al. (21) serve to support our contention. These workers constructed a two-dimensional network of interconnected vertical and horizontal glass capillaries in order to simulate a porous medium. The vertical capillaries contained wire electrodes, from which gas bubbles evolved continuously as a byproduct of the electrochemical reaction. First, it was experimentally confirmed that the steady-state volume fractions of gas and liquid in the capillary were constant and, furthermore, were almost independent of the gas evolution rate. Second, Korovin et al. (21) observed that “Gas was expelled from the capillaries in pulses; once the plug has been expelled, one could clearly see that liquid and fine bubbles were taken up in the lattice at a depth commensurate with the capillary dimensions” (quoted from the English translation). Finally, one may estimate the fraction of gas volume that must be entrained and subsequently replaced by bulk medium in order to account for the enhanced glucose transport into the support. Assuming for simplicity that the diffusive contribution is negligible and that all the glucose is consumed before the medium is expelled, the mass balance over the support reduces to total observed uptake (g/h) = liquid entrainment rate (mL/h) X Cb (g/mL) A t Robs = 4 g / g h and c b = 100 g/L, the liquid entrainment rate would be ca. 82 mL/h. Since the volumetric gas flow rate under these conditions is ca. 1860 mL/h, the liquid equivalent to ca. 4.4% of the evolved gas volume would have to be entrained to account for the total glucose flux into the disk. At R o b = 2.6 g / g h and c b = 20 g/L, the liquid equivalent to an estimated 21 7% of the evolved gas volume would have to be entrained to account for the glucose flux into the disk. Since much higher, and perhaps unattainable, levels of entrainment were required a t lower glucose concentrations, diffusion limitations may have still played a role at low glucose con-
calcium alginate, recycle packed bed
comments nongrowing, resting cells resting cells initial rates
ref 7 6 5
this study
initial rates,
3
complete kinetics not reported
centrations and may have accounted for the high value of the saturation constant encountered in this study (e.g., see ref 24). In summary, our entrained flow hypothesis suggests that convective enhancement of nutrient supply may be expected to depend on 3 factors: (1)the volumetric gas evolution rate, (2) the microstructure of the porous support, and (3) the concentration of the reactant in the bulk medium.
Conclusions The rotating-disk reactor is a useful device to study the effects of mass transfer on the kinetics of immobilized-cell reactions. The results of this study suggest that convection-enhanced substrate supply to immobilized cells may have a t least partially circumvented diffusion limitations. Immobilized-cell reactors may hence be designed to operate a t higher effectiveness factors and consequently a t higher overall volumetric productivities. b cb
d
D Deff Km
I N Robs Vmax @W
17 Y
Rei sc Sh
Notation half-thickness of disk steady-state bulk reactant concentration disk diameter diffusivity in water effective diffusivity in the porous support saturation constant characteristic length agitator speed (rotations per minute) observed rate of reaction (based on unit support volume) maximal reaction rate Weisz’s modulus effectiveness factor kinematic viscosity impeller Reynolds number Schmidt mmber Sherwood number
Acknowledgment Support of this work by the Manville Corp. and Colorado Cooperative State Research Service under Project 1-53831 is gratefully acknowledged. Literature Cited (1) Bailey, J. E.; Ollis, D. F. Biochemical Engineering Fundamentals, 2 ed.; McGraw-Hill: New York, 1986,p 595.
(2) Efthymiou, G. S.; Shuler, M. L. Elimination of Diffusion Limitations in a Membrane Entrapped Cell Reactor by Pressure Cycling. Biotechnol. Prog. 1987,3, 259. (3) Galazzo, J. L.; Bailey, J. E. In vivo Nuclear Magnetic Resonance Analysis of Immobilizationeffects on Glucose Metab-
209
Biotechnol. Prog., 1990, Vol. 6, NO. 3
olism of Yeast Saccharomyces cereuisiae. Biotechnol. Bioeng. (4) Doran, P. M.; Bailey, J. E. Effects of Immobilization on
(15) Wheeler, A. Reaction Rates and Selectivity in Catalyst Pores. Adu. Catal. 1951,3, 245 (see Appendix on page 323). (16) Roberts, G. W.; Satterfield, C. N. Effectiveness Factors
Growth, Fermentation Properties, and Macromolecular Composition of Saccharomyces cerevisiae attached to gelatin. Biotechnol. Bioeng. 1986, 28, 73. (5) Ryu, D. D. Y.; Kim, H. S.; Taguchi, H. Intrinsic Fermentation Kinetic Parameters of Immobilized Yeast Cells. J. Ferment. Technol. 1985,62,255. (6) Furusaki, S.; Seki, M. Effect of Intraparticle Mass Transfer Resistance On Reactivity of Immobilized Yeast Cells. J . Chem. Eng. Jpn. 1985,18,389. (7) Furusaki, S.; Seki, M.; Fukumura, K. Reaction Characteristics of an Immobilized Yeast producing Ethanol. Biotechnol. Bioeng. 1983, 25, 2921. (8) Bringi, V.; Dale, B. E. Enhanced Yeast Immobilization by Nutrient Starvation. Biotechnol. Lett. 1985, 7, 905. (9) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962; pp 62-71. (10) Axelsson, A.; Persson, B. Determination of Effective Diffusion Coefficients in Calcium Alginate Gel Plates with varying Yeast Cell Content. Appl. Biochem. Biotechnol. 1988, 18, 231. (11) Buchholz, K. Characterization of Immobilized Biocatalysts; DECHEMA Monograph No. 1724-1731; Verlag Chemie: Weinheim, FRG, 1979; Vol. 84, p 232. (12) Crank, J. The Mathematics of Diffusion, 2 ed.; Oxford University Press: London, 1975; pp 57-62. (13) Tanaka, H.; Matsumura, M.; Veliky, I. A. Diffusion Characteristics of Substrates in Calcium Alginate Gel Beads. Biotechnol. Bioeng. 1984, 26, 53. (14) Weisz, P. B. Diffusion and Chemical Transformation-An Interdisciplinary Excursion. Science 1973, 179, 433.
Accepted April 20, 1990. Registry No. Glucose, 50-99-7.
1989,33, 1283.
for Porous Catalysts: Langmuir-Hinshelwood Kinetic Expressions. Znd. Eng. Chem. Fundam. 1965,4,288. (17) Nir, A.; Pismen, L. M. Simultaneous Intraparticle Forced Convection, Diffusion and Reaction in a Porous Catalyst. Chem. Eng. Sei. 1977, 32, 35. (18) Stephanopoulous, G.; Tsiveriotis, K. The Effect of Intraparticle Convection on Nutrient Transport in Porous Biological Pellets. Chem. Eng. Sei. 1989, 44, 2031. (19) Rodrigues, A. E.; Quinta-Ferreira, R. M. Convection, Diffusion and Reaction in a Large-Pore Catalyst Particle; AIChE Symp. Ser. 266; American Institute of Chemical Engineers: New York, 1988; Vol. 84, p 80. (20) Ismail, M. I. Ed. Electrochemical Reactors, Their Science and Technology, Part A: Fundamentals, Electrolysers, Batteries, and Fuel Cells; Elsevier: Amsterdam, 1989; pp 167-176. (21) Korovin, N. V.; Chudinov, A. S.; Feoktistov, A. F. A Study of the Liquid-Gas Electrode 3. A Two-Dimensional Capillary System. Sou. Electrochem. (Engl. Transl.) 1969,5,430. (22) Feoktistov, A. F.; Maksimov, P. N.; Korovin, N. V. Investigation of the Liquid-Gas Electrode 6. Pulsation Model. Sou. Electrochem. (Engl. Transl.) 1974, 10, 1592. (23) Taitel, Y.; Bornea, D.; Dukler, A. E. Modelling Flow Pattern Transitions for Steady Upward Gas-Liquid Flow in Vertical Tubes. AZChE J . 1980, 26, 345. (24) Hamilton, B. K.; Gardner, C. R.; Colton, C. K. Effect of Diffusional Limitation on Lineweaver-Burk Plots for Immobilized Enzymes. AIChE J . 1974, 20, 503.
