2634
Ind. Eng. Chem. Res. 1999, 38, 2634-2640
A Dynamic Nonisothermal Model for Triphase Catalysis in Organic Synthesis Sridhar Desikan† and L. K. Doraiswamy* Department of Chemical Engineering, Iowa State University, Ames, Iowa 50011
Phase-transfer catalysis has been in use for over 3 decades, but triphase catalysis, where the phase-transfer catalyst is immobilized on a solid support, is of much more recent origin. Despite its significant advantages over soluble phase-transfer catalysts, triphase catalysis has not attracted industrial attention. One of the main reasons for this lack of industrial interest is the insufficient understanding of the complex diffusion-reaction processes involved. Significant insights into this problem can be gained by mathematical modeling of these reaction systems. Unfortunately, while a few studies have been reported on the mathematical modeling of triphase catalysis, none of them address the important problem of nonisothermal effects. In the present paper we develop a dynamic model for triphase catalytic systems that includes intraparticle heat-transfer effects. An important conclusion is that the catalyst exhibits maximum effectiveness (with an effectiveness factor greater than 1) at a particular reaction time and that it can be tailored to physically realize this enhanced conversion. 1. Formulation of the Mathematical Model In our previous paper in this area, we presented a dynamic model of triphase catalysis that takes into account the reversibility of the aqueous phase ionexchange reaction (Desikan and Doraiswamy, 1995). This paper extends the model to include nonisothermal effects within the catalyst particles. Equations 1-26 of our previous paper are retained. The new model includes the heat effects due to nonlinear behavior of rate constants with respect to temperature and the heat of reaction of the organic phase reaction. Because the aqueous phase species are ionized before the triphase reaction starts, the heats of ionization will not be considered in our mathematical model. The heat of reaction of the organic phase reaction is included in the model through a heat balance expression inside the catalyst. At any point within the catalyst, the heat generated or absorbed due to reaction balances out the heat transferred to or from the bulk phase liquid. This, in essence, is a modification of the Prater formulation used in traditional nonisothermal analysis of the reaction (Carberry, 1976). A pseudo-steady-state approximation of heat transfer is superimposed on the dynamic model for mass transfer within the catalyst. This approximation is, in general, valid because the heattransfer rates are usually much higher than the masstransfer rates in the case of solid-liquid reactions. In our model, the bulk phases are assumed to be controlled at a specified temperature as is normally done under experimental conditions. The heat balance across a catalyst particle can be written as
ke(T - Ts) ) DRX(-∆HR)(CRX,s - CRS)
swollen solid support on which the phase-transfer catalyst is immobilized with pores filled with liquid, Ts is the temperature at the surface, DRX is the effective diffusivity of the organic reactant within the solid, and -∆HR is the heat evolved due to the reaction (positive for an exothermic reaction). Equation 1 describes the relationship between the temperature T at any point within the catalyst to the concentration of the organic phase reactant CRX. The left-hand side of eq 1 corresponds to the heat transferred to the surface of the catalyst, and the right-hand side represents the heat released due to reaction. The heat balance term developed above is coupled with species mass balances through the rate constant of the organic phase reaction. The expression for the Thiele modulus modified to account for nonisothermality is then given by
φ2 ) R2(Fsk2,sCQX,0/DRX)(k2/k2,s)
[ (
k2/k2,s ) exp Rs 1 -
* To whom correspondence should be addressed. E-mail:
[email protected]. † Current address: Chemical and Analytical Development Department, Novartis Pharmaceuticals Corporation, 59 Route 10, East Hanover, NJ 07936.
)]
(3)
where Rs ) E/RTs and T ˆ ) T/Ts. From eq 1
ˆ - 1) ) DRX(-∆HR)(CRX,s - CRX) keTs(T T ˆ )1+
(4)
DRX(-∆HR)FsCQX,0γRX (C ˆ RX,s - C ˆ RX) (5) keTs ˆ RX,s - C ˆ RX) T ˆ ) 1 + βm(C
(1)
where ke is the effective thermal conductivity of the
1 T ˆ
(2)
βm )
(6)
DRX(-∆HR)FsCQX,0γRX keTs
(7)
Combining eqs 3 and 7 with eq 2, we have
[
]
(C ˆ RX,s - C ˆ RX) φ2 ) φs2 exp Rsβm 1 + βm(C ˆ RX,s - C ˆ RX)
10.1021/ie980030z CCC: $18.00 © 1999 American Chemical Society Published on Web 06/11/1999
(8)
Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 2635 Table 1. Parameter Values Used in the Simulations and Their Physical Significance parameter
definition
values used
R1
k1/k2,s
10.0
R2
k1/k2,sKeq
0.01-10.0
γY
CY,0/FsCQX,0
0.5
γRX
CRX,0/FsCQX,0
0.25
θQX θorg θaq θ′org θ′aq ψ
1/(org + aq) org/(org + aq) aq/(org + eaq) (Vorg/Vcat) [1/(org + aq)] (Vaq/Vcat) [1/(org + aq)] DY/DRX
10.0 0.5 0.5 500.0 500.0 1.0
κ
kY/kRX
1.0
Rs βm
E/RTs DRX( - ∆HR)FsCQX,0γRX/k′eTs
10, 20 0.2, 0.05, 0.0, -0.05, -0.2
φs
R(Fsk2,sCQX,0/DRX)1/2
0.01-10.0
Bim
kRXR/DRX
100.0
where φs2 ) R2(F3k2,sCQX,0/DRX) is the Thiele modulus based on surface temperature. The dimensionless mass and energy balance equations from our isothermal model (Desikan and Doraiswamy, 1995) are retained with the new expression for the Thiele moduls from eq 8 incorporated in the model. In addition to the assumptions for the isothermal model, the following additional assumptions have been made: 1. Aqueous phase reactants are instantaneously ionized. Hence, the heat of dissolution of the reactant is ignored. The reaction mixture is assumed to be stabilized to the desired temperature before the start of the triphase-catalyzed reaction. 2. Film heat transfer is negligible. In general, for liquid phase reactions, the film heat transfer is indeed negligible in comparison to the intraparticle temperature gradient (Carberry, 1976). The temperature at the surface of the catalyst is the same as the bulk phase temperature. 3. Bulk phases are well mixed. There are no temperature nonuniformities in the bulk. This can be achieved experimentally by vigorous agitation of the reaction mixture and proper heat regulation. 2. Results and Discussion The mathematical model presented in the previous section involves many parameters that depend on the catalyst support structure, experimental conditions, and the particular reaction system. For a given reaction system, if these parameters are known or can be estimated, this model can be used to study the effect of different parameters on the overall conversion. In addition to predicting the conversion, the model can be used to optimize the reaction conditions for highest conversion. The model can also be used for the design of reactors. To study the effect of different parameters, the model equations were solved numerically. A finite difference approximation was used to discretize the equations with respect to the space variable, and the resulting system of ordinary differential equations was solved using the backward differentiation formula method with NAG subroutines. The ranges of parameter values
physical significance ratio of the rate of forward ion-exchange reaction to the organic reaction ratio of the rate of reverse ion-exchange reaction to the organic reaction ratio of the initial concentration of aqueous reactant to that of the catalyst ratio of the initial concentration of organic reactant to that of the cataylst reciprocal of porosity of the catalyst fraction of the pore volume filled by the organic phase fraction of the pore volume filled by the aqueous phase extent of catalyst loading with respect to the organic phase extent of catalyst loading with respect to the aqueous phase ratio of effective diffusivities of inorganic and organic reactants into the catalyst ratio of mas-transfer coefficients of inorganic and organic reactants from the bulk to the surface of the catalyst dimensionless activation energy extent of nonisothermality of reaction; positive for exothermic and negative for endothermic reactions Thiele modulus at the surface temperature; measure of the rate of reaction to diffusion mass Biot number; measure of the rate of mass transfer in the film to diffusion within the catalyst
Figure 1. Bulk concentration of the reactants as a function of dimensionless time. Keq ) 10.0, 5.0, 0.2, and 20.0. Values of other parameters are given in Table 1.
used in the simulations and their physical significances are given in Table 1. Figure 1 shows the concentrations of the organic and inorganic reactants in the bulk phases. We can see that the isothermal model leads to different concentration profiles compared to the nonisothermal model. Dimensionless temperature profiles within the catalyst (eq 6) at different reaction times are shown in Figure 2. The parameter values corresponding to this simulation represent an irreversible exothermic reaction at diffusion-limited reaction conditions (Keq ) 10.0; βm ) 0.2; φs ) 5.0). For a reaction which is faster and more exothermic, this model predicts larger temperature effects. As would be expected for exothermic reactions, dimensionless temperatures greater than unity are obtained. As the reaction proceeds, the temperature within the catalyst increases to a maximum and then decreases because of depletion of the organic reactant within the catalyst. Similar temperature profiles are also obtained for other parameter values. The magnitude of the temperature increase depends on the exothermicity of the reaction, and the reaction time at
2636 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999
Figure 2. Dimensionless temperature (T/Ts) as a function of the radial position at different dimensionless times. Keq ) 10.0, 5.0, 0.2, and 20.0. Values of other parameters are given in Table 1.
which the maximum occurs depends on the extent of pore diffusional limitation. As in conventional heterogeneous catalysis, we can study the role of diffusion in triphase catalysis systems by defining an intraparticle effectiveness factor. We base this definition on the organic phase reaction because it is controlling:
ηi )
∫Vk2CRX(CQX,0 - CQX) dV k2CRX,s(CQX,0 - CQX,s)V
(9)
For the isothermal case, the volume integral in the numerator of eq 9 reduces to a single integral, and in dimensionless form this becomes
ηiso )
∫01Cˆ RX(1 - Cˆ QX)ω2 dω
3
C ˆ RX,s(1 - C ˆ QX,s)
(10)
(10)
However, in the case of the nonisothermal model, the temperature dependence and hence the concentration dependence through eqs 2-8 should be included in the expression for the intraparticle effectiveness factor. Equation 9 can be rewritten for the nonisothermal model as
ηnoniso ) 3
[
(C ˆ
∫01exp Rsβm 1 + β RX,s (C ˆ m
-C ˆ RX) RX,s
]
-C ˆ RX)
C ˆ RX(1 - C ˆ QX)ω2 dω
C ˆ RX,s(1 - C ˆ QX,s)
(11) Using eq 11, the effect of nonisothermality of the reaction was studied by varying the parameters Rs and βm, which respectively represent the dimensionless Arrhenius parameter and the maximum temperature rise across the catalyst. The numerical value of βm signifies the extent of nonisothermality. For exothermic reactions, βm is positive, and for endothermic reactions, it is negative. For gas-solid reactions the values of βm can be very high. For liquid phase reactions, because of lower diffusivities of reactants, βm values are typically smaller. For triphase-catalyzed reactions, these values range from -0.2 to 0.2. Figure 3 shows the nonisothermal intraparticle effectiveness factor as a function of the Thiele modulus
Figure 3. Intraparticle effectiveness factor as a function of the Thiele modulus at the surface temperature for a reversible ionexchange reaction (Keq ) 0.1). Values of other parameters are given in Table 1.
at the surface temperature at different values of βm for two different values of Rs. The reaction condition for this plot represents the case where the aqueous phase ionexchange reaction is reversible. We can see that the effectiveness factors vary for different values of βm. This indicates the importance of including the nonisothermal effects in modeling a triphase-catalyzed reaction. Neglecting nonisothermal effects leads to incorrect predictions of intraparticle effectiveness factors. We can also see that the nonisothermal effects are more prominent as Rs is increased from 10 to 20. The parameter Rs represents the dimensionless Arrhenius group (E/RTs) at the surface temperature. Its value mainly depends on the activation energy for the reaction. Hence, for systems with high activation energies, the nonisothermal effects play a significant role and should be included in the models. Figure 4 shows the corresponding plot for the case where the aqueous phase ion-exchange reaction is considered irreversible (Keq ) 10.0). The effect of nonisothermality is more pronounced in this case than for the previous case. For highly exothermic reactions, represented by βm ) 0.2, effectiveness factors greater than unity were observed. Hence, for a given reaction system, we can choose reaction conditions such that the catalyst effectiveness is maximized. For the conditions studied in this simulation where physically realistic values of model parameters were chosen, we did not observe multiplicities in effectiveness factors. The nonisothermal model developed in this paper is dynamic in nature. The concentrations of reactants inside the catalyst and the bulk change with time. Hence, the intraparticle effectiveness factor evaluated from eq 11 also changes with time. Figure 5 shows the behavior of the effectiveness factor as a function of the
Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 2637
Figure 6. Intraparticle effectiveness factor as a function of dimensionless time at different Thiele moduli for an irreversible ion-exchange reaction (Keq ) 10.0). Values of other parameters are given in Table 1.
Figure 4. Intraparticle effectiveness factor as a function of the Thiele modulus at the surface temperature for an irreversible ionexchange reaction (Keq ) 10.0). Values of other parameters are given in Table 1.
mality on triphase catalysis systems, a case study was developed. The reaction chosen for the study is the phase-transfer esterification of benzyl chloride with sodium acetate using polymer-supported tributylmethylammonium chloride as the triphase catalyst. The reaction was assumed to be carried out in a slurry reactor with bulk temperature maintained constant. The nonisothermal model predicts the concentration of the reactants in the bulk as a function of time at different reaction conditions. This model, in essence, studies the effect of intraparticle heat-transfer effects due to the reaction taking place within the catalyst phase. While the reactor is maintained at isothermal conditions, there are temperature gradients within the catalyst particle which may affect the overall conversion. The reaction conditions studied in this simulation study are as follows: volume of the organic phase ) 500 mL volume of the aqueous phase ) 500 mL initial concentration of the organic reactant ) 3 mol/L initial concentration of the inorganic reactant ) 5 mol/L concentration of the catalyst ) 5% w/v reactor temperature ) 60 °C
The polymer-supported catalyst used in this study has the following physical properties:
Figure 5. Intraparticle effectiveness factor as a function of the Thiele modulus at the surface temperature at different reaction times for an irreversible ion-exchange reaction (Keq ) 10.0). Values of other parameters are given in Table 1.
Thiele modulus at different dimensionless reaction times. We can see that the effectiveness factor goes through a maximum, with the exact position of the maximum being different for different reaction times. Figure 6 shows the effectiveness factor as a function of dimensionless time for varying values of Thiele moduli. Figures 5 and 6 indicate that, for a given reaction system, we can determine the optimal reaction time after which the catalyst effectiveness decreases. This could be of significant practical importance. 2.1. Application of Nonisothermal Model: A Case Study. To understand the effect of nonisother-
radius of the catalyst ) 200 µm porosity ) 0.3 density ) 0.9 g/cm3 thermal conductivity ) 0.8 W/(m K)
2.1.1. Estimation of Parameters. The nonisothermal model developed in this paper uses many dimensionless parameters. These parameters describe the influence of different variables in a concise way. For example, the Thiele modulus is a measure of the rate of diffusion of reactants relative to the intrinsic kinetics of the reaction, and the Biot number is a measure of the rate of external mass transfer relative to the rate of diffusion within the catalyst. Definition of these and other dimensionless variables and their physical significances are listed in Table 1. Rate Constants. The rate constant for the organic phase reaction was obtained from kinetic studies conducted in our laboratory (Desikan and Doraiswamy,
2638 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999
1999). The expression for rate constant is as follows:
kapp ) k2FsCQX,0 ) 2.77 × 1010e-11136/T 1.3055 × 1015e-15202/T s-1 where CQX,0 is the initial concentration of the catalyst, mmol/g of catalyst, and Fs is the density of the catalyst, g/cm3. Tomoi et al. (1983) have studied the kinetics of the ion-exchange reaction (aqueous phase reaction) between sodium acetate and polymer-bound tributylmethylphosphonium chloride. In the absence of kinetic data for the ion-exchange reaction for the polymer-supported tributylmethylammonium catalyst used in this study, the values reported by Tomoi et al. will be used here as an estimate. It has been observed in our previous experimental work that the polymer-supported phosphonium catalyst has about the same activity as that of the ammonium catalyst for the esterification reaction of benzyl chloride with sodium acetate. The rate constant for the forward ion-exchange reaction is k1 ) 5.2 × 10-5 s-1. The equilibrium constant for the ion-exchange reaction was estimated from the heat of reaction. Ionexchange reactions have heats of reaction typically in the range 2-5 kcal/mol.
d(ln Keq) ∆Hion-exchange ) dT RT2 From the above thermodynamic relationship, the equilibrium constant can be estimated to be 20.5. The value of Keq ) 20.5 suggests that the aqueous phase ionexchange reaction can be considered to be irreversible. Diffusion Coefficients. The binary diffusion coefficient for benzyl chloride in toluene was estimated using the Wilke-Chang correlation as 2.49 × 10-5cm2/s. The diffusion coefficient for sodium acetate in water was estimated by using the Nernst-Haskell equation for diffusion of electrolytes as 1.93 × 10-5 cm2/s (Reid et al., 1988). The effective diffusivity of a species into a catalyst is given as
DAB ) DAB/τ where is the porosity of the catalyst and τ is the tortuosity. For a typical catalyst, the ratio of porosity to tortuosity is about 0.1. Hence, the diffusivities of the organic and inorganic reactants can be estimated as
DRX ) DRX/τ ) 2.49 × 10-6 cmorg3/(cmcatalyst s) DY ) DY/τ ) 1.925 × 10-6 cmaq3/(cmcatalyst s) Heat of Reaction. The organic phase heat of reaction was calculated from estimated values of ideal gas heats of formation, ideal gas heat capacities, and heats of vaporization. Ideal gas heats of formation were obtained by the Benson’s group contribution method (Reid et al., 1988). Ideal gas heat capacities and heats of vaporization were obtained from chemical property database DIPPR (Daubert et al., 1996).
∆HR ) ∆Hf + ∆Cp(T - 298.15) + ∆Hv,benzyl chloride - ∆Hv,benzyl acetate -∆HR ) 334.49 kJ/mol
Figure 7. Comparison of predicted values of dimensionless concentration of benzyl chloride obtained from nonisothermal and isothermal models. Table 2. Estimates of Parameter Values Used in the Simulation, Which Correspond to the Esterification of Benzyl Chloride with Sodium Acetate with Tributylmethylammonium Chloride as the Phase-Transfer Catalyst dimensionless variable
estimated value
dimensionless variable
estimated value
R1 R2 γY γRX φs Bim κ ψ
0.81 0.04 7.02 3.51 0.10 100.0 1.0 0.48
θQX θorg θaq θ′org θ′aq Rs βm
5.0 0.6 0.4 66.6 66.6 33.43 0.0094
Mass Transfer Coefficients. Mass-transfer coefficients for three-phase systems are very difficult to measure. There are no reported data in the literature for triphase catalysis systems. In the absence of data, simulation studies were done at conditions corresponding to a high mass Biot number (Bim), which represents a system that does not have external mass-transfer limitations. Masstransfer coefficients for both the organic and inorganic reactants were assumed to be equal and high. The dimensionless variables used in the model were estimated using the parameters described in the previous section. The values of these parameters are listed in Table 2. It has to be noted here that the Thiele modulus at the surface temperature φs ) 0.1 corresponds to the chemically controlled regime where diffusional limitations are not significant. The result from the simulation study is shown in Figure 7. The dimensionless concentration of the organic reactant is plotted as a function of reaction time. The plot shows that the predicted values of dimensionless concentration using the nonisothermal model are different from those of the isothermal model. The difference is not appreciable in the present case. As mentioned earlier, the current model takes into account the intraparticle mass and heat transfer and external mass transfer. From the estimate of the Thiele modulus, we find that the esterification of benzyl chloride with sodium acetate is kinetically controlled. The value of the parameter βm, which is a measure of nonisothermality, is very small in this case. However, for highly exothermic reactions which are diffusion controlled, the nonisothermal model prediction is expected to be significantly different from that of the isothermal model.
Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 2639
3. Summary and Conclusions The dynamic model presented in this study is an extension of our previous isothermal model for triphase catalysis that includes the nonisothermality within the catalyst support. The heat transfer at any point within the radius of the catalyst support is considered to be at steady state and is coupled to the unsteady-state mass transfer within the pellet and the bulk phases. Simulation studies using this model for values of parameters in the ranges of practical interest have led to the following interesting conclusions: (1) Nonisothermal effectiveness factors and conversions obtained from simulation studies are different from those for the isothermal model. (2) For an exothermic organic phase reaction with high activation energy, effectiveness factors greater than unity can be obtained. For the conditions studied in our simulation that pertain to physically realistic values of model parameters, multiplicities in the effectiveness factor for a given Thiele modulus were not observed. (3) There is a maximum effectiveness of the catalyst at a particular time for systems at conditions corresponding to low internal mass-transfer resistances. The time at which the maximum occurs depends on the Thiele modulus and so is a function of the interplay between diffusion and reaction within the catalyst. For a given reaction system, we can choose a catalyst particle size and support structure so that this maximum in the effectiveness factor is physically realized. Nomenclature CQX ) concentration of the active form of the catalyst, mol/g of catalyst CRX ) concentration of the organic reactant, mol/cmorg3 CY ) concentration of the inorganic reactant, mol/cmaq3 CQY ) concentration of the active form of the catalyst, mol/g of catalyst C ˆ QX ) dimensionless concentration of the active form of the catalyst, C ˆ QX ) CQX/CQX,0 C ˆ RX ) dimensionless concentration of the organic reactant, C ˆ RX ) CRX/CRX,0 C ˆ Y ) dimensionless concentration of the inorganic reactant, C ˆ Y ) CY/CY,0 CQX,0 ) initial concentration of the active form of the catalyst (PTC), mol/g of catalyst CY,0 ) initial concentration of the inorganic reactant, mol/ cmaq3 CRX,s ) concentration of the organic reactant on the surface of the catalyst, mol/cmorg3 DAB ) binary diffusion coefficient of solute A in solvent B, cm2/s DRX ) effective diffusivity of the organic reactant in the catalyst, cmorg3/(cmcat s) DY ) effective diffusivity of the inorganic reactant in the catalyst, cmaq3/(cmcat s) -∆HR ) heat of reaction (positive for exothermic reaction), J/(mol K) k1 ) rate constant for the forward reaction in the aqueous phase, cmaq3/(mol s) k-1 ) rate constant for the reverse reaction in the aqueous phase, cmaq3/(mol s) k2 ) rate constant for the organic phase reaction, cmorg3/ (mol s) k2,s ) rate constant for the organic phase reaction at the surface temperature, Ts, cmorg3/(mol s)
ke ) thermal conductivity of the solid support, W/(m K) Keq ) equilibrium constant for the aqueous phase reaction kRX ) mass-transfer coefficient of the organic reactant in the bulk organic phase, cmorg3/(cmcat2 s) kY ) mass-transfer coefficient of the inorganic reactant in the bulk aqueous phase, cmaq3/(cmcat2 s) QX(s) ) phase-transfer catalyst in the pellet, active form participating in the aqueous phase reaction QY(s) ) phase-transfer catalyst in the pellet, inactive form participating in the organic phase reaction R ) radius of the catalyst pellet, cm RX(org) ) organic reactant in the organic phase RY(org) ) organic product T ˆ ) dimensionless temperature, T/Ts Ts ) surface temperature of the catalyst, K Vcat ) volume of the catalyst phase, cmcat3 Vorg ) volume of the organic phase, cmorg3 Vaq ) volume of the aqueous phase, cmaq3 Greek Symbols R1 ) ratio k1/k2,s R2 ) ratio k1/k2,sKeq Rs ) dimensionless Arrhenius parameter, E/RTs βm ) dimensionless adiabatic temperature rise, as defined in eq 21a γRX ) ratio CRX,0/FsCQX,0 γY ) ratio CY,0/FsCQX,0 org ) volume fraction of the organic phase in the catalyst pore aq ) volume fraction of the aqueous phase in the catalyst pore ηi ) intraparticle effectiveness factor ηiso ) intraparticle effectiveness factor, isothermal ηnoniso ) intraparticle effectiveness factor, nonisothermal φ ) Thiele modulus (based on the organic phase reaction) φs ) Thiele modulus at the surface temperature Ts (based on the organic phase reaction) θQX ) ratio 1/(org + aq) θorg ) ratio org/(org + aq) θaq ) ratio aq/(org + aq) θ′org ) ratio (Vorg/Vcat) [1/(org + aq)] θ′aq ratio (Vaq)/Vcat) [1/(org + aq)] ψ ) ratio of effective diffusivities, DY/DRX κ ) ratio of mass-transfer coefficients, kY/kRX Fs ) density of the catalyst support solid τ ) dimensionless time, DRXt/(org + aq)R2 ω ) dimensionless radius, r/R Subscripts aq ) aqueous phase b ) bulk org ) organic phase cat ) catalyst phase s ) within the solid (catalyst)
Literature Cited Carberry, J. J. Chemical and Catalytic Reaction Engineering; McGraw-Hill: New York, 1976. Daubert, T. E., Danner, R. P., Sibul, H. M., Eds. Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation; Design Institute for Physical Property Data, AIChE: New York, 1996. Desikan, S.; Doraiswamy, L. K. The Diffusion Reaction Problem in Polymer Supported Phase Transfer Catalysts. Ind. Eng. Chem. Res. 1995, 34 (10), 3524-3537. Desikan, S.; Doraiswamy, L. K. Enhanced Activity of Polymer Supported Quaternary Ammonium Salts. Chem. Eng. Sci. 1999, submitted for publication.
2640 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 Reid, R. C.; Prausnitz, J. M.; Poling, B. E. Properties of Gases and Liquids; McGraw-Hill: New York, 1988. Tomoi, M.; Hosokawa, Y.; Kakiuchi, H. Phase-Transfer Reactions Catalyzed by Polymer-Supported Phosphonium Salts. Rate of Ion-Exchange and Catalytic Activity. Makromol. Chem. Rapid. Commun. 1983, 4, 227-230.
Received for review January 20, 1998 Revised manuscript received April 20, 1999 Accepted May 6, 1999 IE980030Z
