Ind. Eng. Chem. Res. 1997, 36, 5467-5473
5467
Mathematical Analysis of Unsteady-State Dynamics of a Liquid-Membrane-Encapsulated Urease System R. Chowdhury and P. Bhattacharya* Chemical Engineering Department, Jadavpur University, Calcutta 700032, India
In the present investigation, a mathematical model describing the transient behavior of a water in oil type liquid-membrane-immobilized enzyme system has been presented. A deterministic model (as opposed to a stochastic one) has been presented in an attempt to elucidate the effect of mass-transfer limitation due to immobilization on the kinetics of hydrolysis of urea using an unstructured Michaelis-Menten equation and a mechanistic approach toward diffusion phenomena. An immobilized urease system has been selected to test the model within the initial substrate (urea) concentration range 0.001-4 M. With the help of concentration distribution curves of the product-carrier complex as a function of dimensionless length along each globule of emulsion with time as a parameter, an attempt has been made to show how mass-transfer resistance affects the reaction engineering behavior of the system. It has been observed that the product distribution and reaction velocity as a function of substrate concentration as predicted from the proposed model explains reality quite satisfactorily. Introduction Increasing attention has been focused on the immobilization of enzymes over the last few decades. The insolubilized enzymes have the practical advantage of being easily recovered at the end of the reaction, a property of considerable importance for their industrial application. Although numerous methods for enzyme immobilization processes have been widely explored, microencapsulation of enzymes within liquid membranes which are permeable to substrate and products but not to the enzymes has attracted much interest by different research groups (May and Li, 1972; Mohan and Li, 1974, 1975; Scheper et al., 1984, 1987; Makryaleas et al., 1985; Meyer et al., 1988; Chang and Lee, 1993) only in the recent past. Immobilization of the enzyme on a solid matrix suffers several drawbacks. For example, the insoluble matrix must be chosen carefully so as to ensure proper flow characteristics. Also proper activation conditions for the matrix must be selected so as not to significantly alter the physical characteristics of the matrix particles. In addition, the use of cyanogen bromide activation of polysaccharide carriers, the most common method employed for ligand attachment to those polymers, frequently gives rise to the operational problem of enzyme leakage as a result of inherently unstable bonds formed between the ligand and the carrier. Microencapsulated enzyme in a liquid membrane is almost free from those operational troubles, and its application is promising. A literature survey, however, shows that only a few attempts (Chang and Lee, 1993) have been made to explain the transient behavior of such a system, while it is understood that mathematical modeling of transient behavior is very much needed to explain the dynamics of the enzyme emulsion liquid membrane (EELM) system. Semipermeable emulsion membranes are usually prepared by first forming an emulsion between two immiscible phases and then dispersing the emulsion in a third (continuous) phase by agitation. The specially formulated membrane phase is the liquid phase that separates the encapsulated internal droplets in the emulsion from the external (continuous) phase. In * To whom all correspondence should be addressed. S0888-5885(97)00162-0 CCC: $14.00
general, the internal encapsulated phase and the external phase are miscible. The membrane phase should not be miscible with either of these phases in order to be stable. Effectiveness of the liquid membrane is a result of two facilitated mechanisms (Matulevicius and Li, 1975; Li, 1978, 1981) which can maximize both the extraction rate, i.e., the flux through the membrane phase, and the capacity of the receiving phase (the internal phase in the case of an external feed phase and vice versa). These two types of mechanisms are as follows: Type 1: In this type of facilitation, the reaction in the receiving phase maintains a solute concentration of effectively zero in that phase. Examples of the type 1 mechanism are extraction of weak acids or bases (Ho et al., 1982; Bunge and Noble, 1984; Chan and Lee, 1987), etc. Type 2: This is also called carrier facilitated/mediated transport. This type of carrier is incorporated in the membrane phase to carry the diffusing species across the membrane phase. The carrier is typically a complexing agent or an ion-exchange compound. Complexation reaction between the carrier and the desired species occurs either at internal (between internal and membrane phases) or external (between external and membrane phases) interfaces (Protsch and Marr, 1983; Lorbach and Marr, 1987; Draxler et al., 1988). A typical example of a type 2 system may be recovery of metal ions (Teramoto et al., 1983a; Lorbach et al., 1986). Over the last decade several mathematical models have been developed to describe these two mechanisms (Ho and Li, 1984; Teramoto et al., 1983a,b; Lorbach et al., 1986; Kataoka et al., 1990). Enzyme emulsion liquid membrane (later on abbreviated as EELM) systems involve both type 1 and type 2 mechanisms of facilitation. Permeation of substrate follows the type 1 mechanism because its concentration in the receiving phase (internal phase containing an aqueous enzyme solution) becomes effectively zero due to enzymatic reaction occurring there. On the other hand, the permeation of the product of the enzymatic reaction across the membrane follows the type 2 mechanism. With carrier the product usually forms a complex at the internal interface, the complex gets permeated through the membrane, and at the external © 1997 American Chemical Society
5468 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997
interface, decomplexation reaction occurs to make the product free. The product diffuses through the external interfacial film to the external phase (Chang and Lee, 1993). When either cationic, anionic, or zwitterionic amine is used as a carrier, the anionic, cationic, or both cationic and anionic parts of the product, respectively, make a complex with it. For decomplexation reaction to occur at the external interface, strong counterions are supplied from the external (continuous) phase to replace and release the corresponding ionic part of the product. From the studies of EELM reactors (May and Li, 1972) it is observed that the Michaelis-Menten constants obtained from these systems are many times greater than that obtained in the case of the free enzyme system. It is apparent that this activity loss (Bollmeier and Middleman, 1979; May and Li, 1972) may occur due to mass-transfer resistance (with respect to permeation of substrate and product) exterted on the reaction due to the presence of liquid membrane (May and Li, 1972). Moreover, the restricted mass transfer in the membrane may change (pH change, etc.) the local environment (microenvironment) surrounding the enzyme. This may cause a change in the activity of the enzyme. An interesting mathematical model has recently been developed by Chang and Lee (1993) describing the reaction engineering behavior of liquid-membraneimmobilized R-chymotrypsin. In their pioneering work they have presented a transient model to predict the concentration profile of product in the external phase. Their work is, however, limited to a particular initial concentration of substrate. As a result, it has not been possible for them to infer whether the MichaelisMenten kinetics still holds good in the case of the EELM system. In the present study, a mathematical model has been proposed to describe the transient behavior of an EELM reactor in which hydrolysis of urea [(NH2)2CO + H2O urease
ssf 2NH3 + CO2] is taking place in the presence of liquid-membrane-immobilized urease. The substrate (urea) concentration has been varied in the range of 0.001-4 M. Amberlite LA-2, a zwitterionic amine, has been used as the carrier at a concentration of 0.3 M. This concentration of amine has been found to be optimum for the stability of the liquid membrane. Sodium chloride at a concentration of 2 M has been used to supply counterions. The theoretical concentration profile of ammonia in the external phase against time has been shown and has been compared successfully with the experimental results for substrate concentrations of 0.001 and 3 M (Figure 2) up to a certain time period. The correlation indices (always calculated using the method of Kafarov (1976)) have been calculated to be 0.9995 for 0.001 M and 0.9697 for 3 M substrate concentrations. However, at later times the experimental values of ammonia concentration lie much below the predicted ones. This may be due to the change in the microenvironment (pH change, etc.) in the vicinity of the enzyme, i.e., in the internal phase (Bollmeier and Middleman, 1979).
Figure 1. Schematic diagram of the EELM system.
Substrate. (i) Solubilization of the substrate (urea) from the external phase in the organic membrane phase by physical solubility following the relationship
[S]organic ) R[S]aqueous (ii) Diffusion of solubilized urea through the membrane phase and the two interfaces to the internal phase. (iii) Consumption of substrate in the internal phase by enzymatic reaction given by (NH2)2CO + H2O urease
ssf 2NH3 + CO2 Products. (i) Products, namely, ammonia and carbon dioxide, form in the internal phase. (ii) After formation the products follow the following ionic equilibria:
CO2:
NH3:
CO2 + H2O / H2CO3
(I)
H2CO3 / H+ + HCO3-
(II)
HCO3- / H+ + CO32-
(III)
NH3 + H2O / NH4+ + OH-
(IV)
The dissociation constants of ionic reactions II and III are very low in comparison to ionic reaction IV (Vogel, 1978; Bollmeir and Middleman, 1979). The dissociation constant of carbonic acid is 3.0199 × 10-7 (acid exponent value pKa ) 6.52) and that of ammonia is 1.7782 × 10-5 (basic exponent value pKb ) 4.75) (Vogel, 1978; Latscha and Klein, 1996). There will be ample quantity of ammonium and hydroxyl ions present in the internal phase. Ammonium (NH4+) and hydroxyl (OH-) ions get associated to the carrier amine (Q) at the internal interface to form a complex by replacing the common ions, namely, Na+ and Cl-, from the complex of these ions and the carrier amine formed at the external interface.
Theoretical Analysis The schematic diagram of an EELM system is shown in Figure 1. Mathematical Model. In the proposed scheme the enzymatic reactions involving EELM have been assumed to occur through several elementary steps. These are as follows:
NH4+ + OH- + QNaCl / Na+ + Cl- + QNH4OH (V) (iii) Diffusion of QNH4OH takes place from the internal interface to the external interface. (iv) Ammonium and hydroxyl ions will get released from the complex at the external interface by the
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5469
following reaction:
internal interface is given by
( )(
(v) Diffusion of ammonium and hydroxyl ions take place from the external interface to the external phase. Assumptions. (i) The emulsion liquid membrane system is a monodisperse collection of spherical emulsion globules in an external continuous phase (Teramoto et al., 1983a). (ii) Sauter mean diameter is sufficient to characterize globule size, and it is not necessary to use the size distribution (Teramoto et al., 1983a,b). (iii) There is no coalescence of emulsion globules. (iv) There is no internal circulation in the globules. This assumption is justified in view of the strong presence of surfactant in the membrane phase (Rumscheidt and Mason, 1961). (v) For diffusion in the emulsion globule the composite nature of the emulsion may be disregarded and the emulsion may be treated as if it were a continuum. (vi) Breakage of the internal phase is neglected. (vii) The mass-transfer area is proportional to the volume of emulsion under a given mixing condition for contact with the external phase. This is justified with assumption ii and the fact that under a given mixing condition the Sauter mean diameter does not change significantly as a result of varying the emulsion holdup of a given EELM system (Ho, 1986). (viii) Equilibrium between both the internal and external phases with the membrane phase prevails (Chang and Lee, 1993). (ix) As the amine is zwitterionic, both the cations (NH4+, Na+, etc.) and the anions (OH-, Cl-, etc.) can be attached to it and form complexes. (x) There is a drag of the common ions, namely, Na+ and Cl-, toward the internal phase from the external phase due to its presence in excess in the latter. Development of Model Equations. (i) The mass balance equation for the substrate, urea (later on designated as “S”), in the external phase is given by
-Ve
dSe 3 - (Vi + Vm)keS(Se - Se*) dt R
(1)
(ii) The mass balance equation for the substrate in the membrane phase is given by
(
)
∂Sm DeS ∂ 2 ∂Sm Vi + Vm ) 2 r ∂t ∂r Vm r ∂r
(2)
)
∂Cim Vi ∂Pi DeC ∂ 2 ∂Cm Vi + Vm ) - 2 r ∂t Vm ∂t ∂r Vm r ∂r
QNH4OH + Na+ + Cl- a QNaCl + NH4+ + OH(VI)
(5)
(vi) The mass balance equation of “C” in the membrane phase is given by
( )(
)
DeC ∂ 2 ∂Cm Vi + Vm ∂Cm )- 2 r ∂t ∂r Vm r ∂r
(6)
(vii) The mass balance equation of the carriercommon ion complex, QNaCl (later on designated as “B”) in the membrane phase is given by
( )(
)
∂Bm DeB ∂ 2 ∂Bm Vi + Vm ) 2 r ∂t ∂r Vm r ∂r
(7)
(viii) The mass balance equation of the product in the external phase is given by
Ve
dPe 3 ) (Vi + Vm)keP(Pe* - Pe) dt R
(8)
The initial and boundary conditions are given by
Se ) Seo; Pe ) 0; Ie ) Ieo Sm ) Si ) 0
for t ) 0
(9)
for 0 < r < R and t ) 0
(10)
Pi ) Cm ) Cim ) 0; Bm ) Bim ) Qo ) Bo for 0 < r < R and t ) 0 (11) ∂Sm/∂r ) 0
for r ) 0 and t > 0
∂B/∂r ) ∂C/∂r ) 0 DeS
for r ) 0 and t > 0
∂Sm ) keS(Se - Se*) ∂r
-DeC
(13)
at r ) R and t > 0 (14)
∂Cm ) keP(Pe* - Pe) ∂t Sm ) RSe*
(12)
at r ) R and t > 0 (15)
at r ) R and t > 0
(16)
Chemical equilibrium constant at the external interface is given by
Ke )
[Cm][Ie] [Pe*][Bm]
(17)
(iii) The mass balance equation of the substrate in the internal phase is given by
The relationship between the common ion compound, NaCl (designated as “I”), and the product in the external phase is given by
vmaxSi ∂Si )∂t km + Si
[Ie] ) [Ieo] - [Pe]
(3)
(iv) The mass balance of the product, ammonia (later on designated as “P”), in the internal phase is given by
∂Pi 2vmaxSi ) ∂t km + Si
(4)
(v) The mass balance equation of the carrier-product complex, QNH4OH (later on designated as “C”), at the
(18)
Equations 1-8 have been solved numerically using the initial and boundary conditions (9)-(18). The numerical solution of partial differential equations was made by the technique of “Method of Lines” (Constantinides, 1987). In this method the partial differential equations are solved by converting them into a set of ordinary differential equations by discretizing only the spatial derivatives using finite differences and leaving the time derivative unchanged. The second-order central finite difference method is used for the spatial discretization.
5470 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 Table 1. Values of Physicochemical Parameters Used in the Model parameter
values
sources
RP RS DeB
0.78 1.97 6.6 × 10-12 m2/s
DeC DeS d32
5.8 × 10-12 m2/s 1 × 10-11 m2/s 0.4 mm
Ke keS
0.46 6.6 × 10-6 m/s
keP km N vmax
9.8 × 10-5 m/s 0.005 M 5 rps 0.012 M/min
experimental experimental Wilke and Chang, 1955; Perry and Chilton, 1973 do do photographical and by standard correlation (Rautenbach and Machhammer, 1988) experimental by standard correlation (Perry and Chilton, 1973; Skelland and Lee, 1981) do experimental experimental experimental
The number of discrete points was taken as 21. Thus, the mesh spacing in the spatial discretization is specified as 0.005 mm for this study. The time step size is set at 10-4 s. The derived set of ordinary differential equations was solved using a fourth-order Runge-Kutta method (Constantinides, 1987). Values of other parameters used for the numerical solution of the system equations are given in Table 1 along with their sources. Experimental Section Materials: crude urease pellets (BDH); urea (Glaxo India Limited); light-grade paraffin (main oil phase); Span-80 (surfactant); Amberlite LA-2 (amine); sodium chloride. Kinetics in Free Solution. Enzymatic reactions are often observed to obey the unstructured MichaelisMenten model in which reaction rate is given by
v)
vmaxS km + S
(19)
where vmax ) maximum velocity of reaction and km ) Michaelis-Menten constant. The Michaelis-Menten parameters were measured in this study from initial rates of reaction carried out at constant temperature, 25 °C, in a well-stirred batch reactor. A urea solution was prepared by adding it to 100 mL of buffcr (0.1 M tris(hydroxymethyl)aminomethane (TRIS)). Each solution was then diluted by 0.1 M TRIS with 1 mM EDTA. To this solution was added 1% ethanolic phenolpthalein. An aqueous urease solution (5 mg/mL) was added to the resulting solution. Absorbances of samples withdrawn from the reactor at different time intervals were determined using a Perkin Elmer spectrophotometer at 550 nm. Using the standard curve of absorbance versus known ammonia concentration (plot of absorbance (at 550 nm) as the ordinate and known ammonia concentration in an aqueous solution of 0.1 M TRIS with 1 M EDTA and 1% ethanolic phenolpthalein), the reactor ammonia concentrations were determined. Several experimental runs were conducted while varying the initial substrate concentration (0.001-4 M). For each concentration of urea the initial slope of the ammonia versus time plot (not shown) was determined. From these slopes the initial rates of depletion of urea were determined. These rates were plotted against the initial substrate concentration, and the maximum velocity, vmax, and Michaelis-Menten constant, km, were determined.
Preparation of Emulsion. The emulsion was prepared by adding an aqueous urease solution (pH 6.75) dropwise to the membrane-forming solution with vigorous agitation (350 rpm). The final weight ratio of the aqueous to the oil phase (i.e., Vi/Vm ) 0.8) was maintained at 0.8. The enzyme solution was composed of 5 mg/mL of urease in distilled water. The membraneforming solution consisted of 2% Span-80 (surfactant), 95% S-100N (light paraffin), and 3% amine, Amberlite LA-2 (zwitterionic). Observed Immobilized Urease Kinetics. To 60 mL of an aqueous urea solution in 0.1 M TRIS with 1 mM EDTA containing 2 M NaCl was added 1% ethanolic phenolphthalein. A total of 10 mL of emulsion containing immobilized urease (as described above) was then added to the resulting aqueous solution (i.e, Ve/ (Vm + Vi) ) 6). The contents of the reactor was stirred vigorously. The color of the external phase started changing with the propagation of the reaction due to the formation of ammonia. After a certain interval of time, stirring was stopped and the absorbance of the reaction sample was determined using the same spectrophotometer as in the case of the free solution. Reaction samples were taken at different time intervals, and their absorbances were determined using the spectrophotometer (550 nm). From the standard curve of absorbance versus ammonia concentration the concentrations of ammonia in the external aqueous phase were determined. Determination of Different Physicochemical Parameters. (1) Phase Partition Coefficient, r, of Urea. Over the entire range of concentration of urea, the aqueous solution of urea was allowed to remain in contact with an oil phase having the same composition as the membrane phase until equilibrium was attained (determined by measuring the concentration of urea in the aqueous phase. At equilibrium the aqueous phase concentration of urea was determined. By material balance the concentration of urea in the oil phase was also determined. The ratio of the concentration was then calculated to obtain the value of R. (2) Determination of Equilibrium Constant, Ke. The distribution ratio, R, of ammonia between the aqueous and oil phases has been determined by the usual method as in the case of substrate. By carrying out a batch reaction, the equilibrium rate constant, Ke, of the reaction
NH4+ + OH- + QNaCl / QNH4OH + Na+ + Cl(V) has been determined. Equimolar quantities of QNaCl and NH4OH have been allowed to react in an oil phase having the same composition as that in the liquid membrane for 48 h to ensure that equilibrium has been reached. By measuring the concentration of QNaCl at equilibrium and thereby calculating concentrations of QNaCl, QNH4OH, and NaCl, the equilibrium constant has been determined. (3) Effective Diffusivities. Effective diffusivities were obtained using the expression (Ho and Sirkar, 1992) given by
(
Deff ) Dm
)
4(1 + p)2 - π 4(1 + 2p)2
+
(
)
(1 + 2p)DADM π (20) 2 4(1 + 2p) Dm + 2pDA
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5471
where
DA )
[
( )
) ]
Di/R Di/R 2(Di/R)Dm ln -1 Di/R - Dm Di/R - Dm Di/R - Dm p ) 0.403
(
Vi Vm + Vi
(21)
-1/3
- 0.5
(22)
The values of Di and Dm for all the components were determined using the Wilke-Chang correlation (Wilke and Chang, 1955). The effective diffusivities of productamine complex (C) and common ion-amine complex (B) were assumed to be the same (Chang and Lee, 1993). The external phase mass-transfer coefficients (keS, keP) were estimated from the following equation (Skelland and Lee, 1981):
()
ke
ds ) 2.932 × 10-7(1 - φe)-0.508 1/2 T (ND)
0.548
Re1.371 (23)
(4) Determination of Emulsion Globule Diameters. The diameters of emulsion globules were measured photographically, and Sauter mean diameters, d32, were calculated using the following expression: n
d32 )
∑ i)1
n
nidi2 ∑ i)1
nidi3/
(24)
Sauter diameters have also been determined using the following expression (Rautenbach and Macchammer, 1988):
( ) ( [ (
d32 Feηem ) 0.11 ds Femηe
0.32
) ) ]
Vm + Vi Ve + Vm + Vi
0.1
We-0.7
Vi ηem ηm ) exp 5.32 - 0.1 Fem Fm Vm + Vi
(25)
(26)
The differences in the determined Sauter mean diameters between the photographic method and the Rauteubach-Machhammer correlation are within (5%. Results and Discussion Concentration Profile of Product, Ammonia. In Figure 2, the simulated concentrations of product, ammonia, in the external phase have been plotted against reaction time for two different initial substrate concentrations (0.001 and 3 M). The experimental data (each represents the mean of four experimental results taken under identical conditions) have also been presented on the same figure. For both values of initial substrate concentrations, it is observed that, as in the case of the free enzyme system, the concentration of product ammonia goes on increasing in the external phase with the progress of reaction time, indicating that the mass transport of ammonia takes place from the internal to the external phase. The figure evidently shows that the simulated profiles agree reasonably with the experimenta1 data (correlation indices are, for 0.001 M, 0.9995 and, for 3 M, 0.9697) (Kafarov, 1976). This indicates that the mechanism proposed for developing the present model is justified. However, as the reaction proceeds beyond 180 s, the simulated concentrations of ammonia deviate to a greater extent compared to the experimental values in
Figure 2. Simulated (lines) and experimental (points) product concentration in the external phase against time for initial substrate concentrations of 0.001 M (-‚-, 0) and 3 M (s, O).
the case of an initial substrate concentration of 3 M. The magnitude of simulated values is found to be larger than the experimental one. This may be due to the fact that with the progress of reaction there will be partial accumulation of ammonia in the internal phase, causing the pH in the microenvironment to shift from the optimum value, which for urea-urease system is nearly 6.75 (Bollmeier and Middleman, 1979). This behavior is not, however, observed in the case of the lower initial substrate concentration (0.001 M) system. Possibly, in this case the accumulation of ammonia in the internal phase is small enough to cause any shift of pH from the optimum value. The authors strongly feel that stochastic models incorporating both microenvironmental effects and mass-transfer resistance may give better results for high concentration (3 M, etc.) systems, which, however, is out of the scope of the present investigation. Otherwise, an attempt may be made to use a modified form of an unstructured Michaelis-Menten equation taking into account product inhibition. Radial Concentration Profiles of the ProductCarrier Complex. In the case of EELM systems following a carrier-facilitated transport (type 2) mechanism regarding mass transfer of product from the reacting phase to the external phase, the process is solely governed by the rate of formation of the productcarrier complex at the internal interface, its propagation through the membrane phase, and extraction of product from the same complex at the external interface. Thus, the radial profile of complex within an emulsion globule is an important criterion in the case of EELM systems. Following the proposed model, simulated concentrations of the product-carrier complex have been plotted against a dimensionless radial position (ζ) with reaction time as a parameter for initial substrate concentrations of 0.001 and 3 M, respectively. This is shown in Figures 3 and 4. From the study of Figures 3 and 4, it is evident that the complex (C) concentration is maximum at the internal interface and its value gradually diminishes toward the external interface. As time progresses, radial profiles become steeper for both the initial
5472 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997
Figure 3. Simulated radial profiles of product carrier complex with ζ as a parameter for 0.001 M.
Figure 5. Simulated (line) and experimental (points) MichaelisMenten plot in the [S]o range of 0.001-4 M.
From Figure 5, the value of the Michaelis-Menten constant has been determined to be 0.27 M. This value is about 54 times as large as the value of 0.005 M obtained for the free enzyme system. A similar phenomenon was observed by May and Li (1972). This increase in the km value may be due to the fact that diffusional resistance to the overall rate is predominantly high because of the presence of liquid membrane (May and Li, 1972). Since the effects of mass transfer on the reaction rate have been properly incorporated in the model, the simulated Michaelis-Menten plot represents reality. Conclusion
Figure 4. Simulated radial profiles of product carrier complex with ζ as a parameter for substrate concentration of 3 M.
substrate concentration. The very pattern of profile may be explained by taking into account the antagonistic effect of mass transfer on the overall rate of reaction. With the progress of time the complex concentration at the internal interface increases with the continuous generation of product, while the diffusional resistance opposes its propagation toward the external interface. Michaelis-Menten Plot. The Michaelis-Menten plot has been constructed using both experimental and simulated values. The experimental data represent the mean of four replicate experimental observations. The rectangular hyperbolic pattern of the curves indicates that here also the system follows the unstructured Michaelis-Menten mechanism. Comparison of the simulated curve and experimental data is evidently quite satisfactory (correlation index ) 0.9989).
A mathematical model has been presented to analyze the transient behavior of an W/O/W EELM reactor using urease as the enzyme and urea as the substrate. The model is capable of predicting the Michaelis-Menten behavior of the EELM system. Parameters required for modeling could easily be obtained either by experiments or by using standard theoretical expressions. Thus, the main advantage of the proposed model is that it is almost free from any adjustable parameter. It is expected that the present model may be used to study both the Michaelis-Menten behavior and the overall transient behavior of any EELM reactor within an appreciable length of reaction time. Nomenclature B or [B] ) concentration of the carrier-common-ion compound complex, kmol/m3 C or [C] ) concentration of the carrier-product compiex, kmol/m3 D ) molecular diffusivity, m2/s De ) effective diffusivity, m2/s ds ) diameter of the propeller, m d32 ) Sauter mean diameter, m or mm I or [I] ) concentration of common-ion compound, kmol/ m3 Ke ) equilibrium constant, dimensionless km ) Michaelis-Menten constant of the free enzyme system, M
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5473 ke ) external phase mass-transfer coefficient, m/s N ) stirring speed, rps P or [P] ) concentration of product, kmol/m3 R ) Sauter mean radius of the emulsion globule, m r ) radial coordinate, m or mm Re ) Reynolds number in eq 23 (FcND2/ηc), dimensionless S or [S] ) concentration of substrate, kmol/m3 t ) time, s or min T ) diameter of the vessel, m V ) volume, m3 v ) reaction velocity, kmol/m3‚min We ) Weber number, dimensionless Greek Symbols R ) partition coefficient between the membrane and the internal phase, dimensionless φe ) volume fraction or external phase to total volume, Ve/ (Ve + Vi + Vm), dimensionless η ) viscosity of liquid, P F ) density of liquid, kg/m3 ζ ) dimensionless radial coordinate, (r - Ri)/(R - Ri) Superscript * ) at equilibrium condition Subscripts B ) of “B” C ) of “C” c ) of/in continuous phase e ) of/in external phase i ) of/in internal phase im ) of/in internal interface m ) of/in membrane phase max ) maximum P ) of product S ) of substrate
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Received for review February 20, 1997 Revised manuscript received August 8, 1997 Accepted August 20, 1997X IE970162C
X Abstract published in Advance ACS Abstracts, October 15, 1997.
