Ind. Eng. Chem. Res. 1993,32, 1323-1327
1323
Phase-Plane Modeling of a Liquid-Liquid Phase Transfer Catalyzed Reaction Ho-Sheng Wu Department of Chemical Engineering, Yuan-Ze Institute of Technology, Taoyuan, Taiwan, ROC
This paper presents a new phase-plane model for the dynamics of a liquid-liquid phase transfer catalyzed batch reaction system. The mathematical problem based on the two-film theory was analyzed. Simulation results reveal that by the proposed model one can successfully make a correct judgment as to whether the quaternary onium salts in the two phases are in extractive equilibrium. Also, the respective contributions of the reaction and the mass transfer to the overall rate can explicitly be understood. A phase transfer catalyzed reaction might be described by a pseudofirst-order process. However, the consumption rate of the nucleophilic catalyst (QY) in the organic phase is not kept constant. Moreover, the reaction rate of the slow reaction is affected by the mass-transfer resistance between the two phases.
Introduction Phase transfer catalysis (PTC)has been considered to be one of the most effective tools for the synthesis of organic chemicals from two immiscible reactants. Usually, the reactivity of reaction by phase transfer catalysis is controlled by (i) the reaction rate in the organic phase, (ii) the mass-transfer steps between the organic and aqueous phases, and (iii) the partition equilibrium of the catalysts between the two phases (Weber and Gokel, 1977;Starks and Liotta, 1978; Dehmlow and Dehmlow, 1983). In general, it is thought to be reasonable to assume that the resistance to mass transfer can be neglected for a slow organic-phase reaction by phase transfer catalysis. So far, few papers have been appeared on fundamental mathematical modeling in the reaction of phase transfer catalysis. Evans and Palmer (1981)considered an interphase catalyzed reaction. Melville and Goddard (1985), as well as Melville and Yortsos (19861,described the solidliquid phase reaction. Wang et al. (Wang and Wu, 1991; Wang and Yang, 1991) discussed a pseudo-steady-state hypothesis for the phase transfer reaction. Such mathematical modeling appears to be desirable and needed in view of the widespread interest in phase transfer catalysis in the chemical literature and industry where liquid-liquid phase transfer and triphase catalysis are the most common processes. Very little has been done in the way of mathematical analysis of the phenomena, and such an analysis is especially desirable for large-scale applications. The complicated nature of the PTC reaction system can be attributed to the two mass-transfer steps and two reaction steps in the organic and aqueous phases. In addition, the equilibrium partitions of the catalysts between two phases also affect the reaction rate. On the basis of the above factor and the two-film theory, in this paper a phase-plane model for describing the dynamics of a liquid-liquid PTC reaction can be derived. The model is formulated as a system of coupled nonlinear differential and algebraic equations in which the differential equations describe the reactions occurring in the organic and aqueous phases whereas the algebraic ones describe the mass balances of species. This differential-algebraic system can be solved by Runge-Kutta method. The parameters in the proposed model, such as the reaction rate constants, the overall mass-transfer coefficients, the distribution coefficients, and the equilibrium constant of the chemical reaction in the aqueous phase, are the major factors that govern the overall dynamic behavior of a liquid-liquid PTC reaction system. Representative and physically
meaningful parameters in a dimensionless form are obtained. The proposed model can not only cover a wide range of operation conditions but also can illustrate well several important phenomena involved in a liquid-liquid PTC reaction system.
Mat hematical Modeling Starks (1978)offered a classic diagram of the phase transfer catalytic cycle: RX
+
Q'Y-
-
Q'X-
MX + Q+Y- ==z Q'X-
+ RY
organicphase
+
aqueousphase
MY
Consider an irreversible s N 2 reaction between two species RX and QY and assume the following: (a) Both RX and RY are insoluble in the aqueous phase. (b) QX and QY at the interface are in extractive equilibrium. (c) No resistance is offered to the transfer of the diffusing component across the interface. (d) The chemical reactions take place in the well-mixed organic phase and aqueous phase, respectively. (e) MX and MY can be all dissociated to M+ and X- as well as M+ and Y- in th aqueous phase. (0The activities are constant in the organic and aqueous phases. (g) There is no change in volume. (h) The reactions are conducted isothermally. On the basis of mechanism 1 and these assumptions, the species balance equations for an isothermal batch reactor are
-d[RX1o - -k,[RXl,[QYl, dt -d[QY1o - -k,[RXl,[QYl, + dt
Qsas-5ss5/93/2632-1323$04.00/0 0 1993 American Chemical Society
1324 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 I O
o= d[QX1 ko[RXlo[QYlodt
0.0
1
0.4'
The equilibrium constant Ka of the chemical reaction in the aqueous phase is (3)
I
(7) 0.0
The mass balances for Qi
Qi,
Y-, and X- are given by
vo([QYlo + [QXol) + va([QYl, + [&XI,) (8)
MY, = Vo([QYlo+ [RXli- [RXI,)
+
+ [QYI,)
V,([Y-l, MX, + Vo[RXlo+ Qi = Vo([RXlo+ [&XIo)+ V,CIX-la
dY0
Yo
50
(9)
+ [&XI,> (10)
Where MXi, MYi, and [RX]i represent the initial amounts of reactants MX, MY, and RX, respectively. A two-film theory is employed to consider the mass transfer of catalysts between the two phases. The dynamic behavior of a liquid-liquid PTC reaction is described by eqs 2-10, and by eliminating the time variable, eqs 2-6 can be further simplifield to the following system of differential equations:
- 1)
-=PI-dYl0 +QY( o mQ >:
la0 150 2w Time (rnin) Figure 1. Fraction concentration profiles of the catalysts. (1) ylo; (2) yla;(3) yb; (4) yh; ( 0 )experimental data (Wang and Yang, 1991). 0
Vo[RXIi
MY,
, Pz=-Qi PI= Qi
9
MXi p3p-
Qi
(11)
Ylo
The physical meaning of parameters is defined in the Nomenclature section. Before reaction, the species QX, QY ,Y-, and X- are in the equilibrium state in both phases, while the concentration of RX in the organic phase is [RXli.
Equations 7-lo can be further simplified to the following system of algebraic equations:
Results and Discussion In the present paper, the dynamics of two cases of this model are examined in detail: (i) a slow phase transfer reaction which is described by a pseudo-first-order hypothesis and (ii) a mass-transfer-controlled process with instantaneous phase transfer reaction. In the following, the simulation results are presented and discussed. (A) Slow Phase Transfer Reaction. In order to examine the accuracy of the proposed phase-plane modeling analytical solution, it was compared with the experimental data from the study by Wang and Yang (1991). On the basis of experimental observation, the implicit constant concentration of intermediate product QY was measured. The physically meaningful parameters calculated from the Appendix were LY = 1,~ Q =Y 300,&ex = 117, mgy = 12, -x.= 0.071, PI= 6.21, Pz = 9.73, P3 = 0, and K = 8.15. The simulation results and experimental data are presented in Figure 1,which shows the amounts of QY and QX in the organic and aqueous phases. The reactivity of phase transfer catalytic reaction is principally controlled by the amount of QY in the organic phase which can vary during the reaction period and change the phenomena of the reaction. The reaction system of Wang and Yang (1991) can be described by a pseudofirst-order reaction although, in the past, it was assumed
Ind. Eng. Chem. Res., Vol. 32, No. 7,1993 1325 I M ,
X
U .+-
IO1
t I
\
1
I
2t )
'
00.0
0.2
0.4
0.8
I
0.8
.o
Time (rnin)
Figure 2. Fraction of RXw time. The straightline is the simulation, and the symbols are experimental data (Wangand Yang, 1991).
that the concentration of QY in the organic phase was constant. On the basis of the experimental results, the fractions of the catalysts in the equilibrium state were ylo = 0.92, yla = 0.077, yz0 = 5.5 X 10-8, and y% = 7 X 10-7. After reaction, ylowas increased from 0.82 to 0.92; yla was increased from 0.0681 to 0.077; yz0 was decreased from 0.046to 5.5 X 10-8;y a was decreased from 0.069 to 7 X W. As shown in Figure 1, the fraction of QY was not constant and changed by (0.92 - 0.82)/0.92 = 0.11. In Figure 2, the logarithmic value of yovs time was a straight line. The average value of ylo calculated from the pseudofirst-order rate constant was 0.87 by the method of least squares. The r value was 0.9995. Thus, it is illustrated that although the concentration of QY in the organic phase was not kept constant, the analysis of the experimental data could be dealt with using a pseudo-first-order method. From these results, it is clear that the mass-transfer rate of anion between the organic and aqueous phases affected the whole reaction rate. It must be emphasized that the concentrations of anion (QY, QX) cannot be maintained constant in spite of the reaction rate being slow. Likewise, the assumptions that the phase transfer catalysts in the two phases are in extractive equilibrium and that the masstransfer resistance between phases can be completely neglected are wrong. The mass-transfer resistance between phases depends on (i) transfer of QY from the aqueous phase to the organic phase to react with the organic reactant and (ii) transfer of QX from the organic phase to the aqueous phase after reaction. Which step is the rate-determining step? New definitions are used to describe mass transfer between phases:
If the phase transfer catalysts in the two phases are in extractive equilibrium and the mass-transfer resistance can be completely neglected, $QY and $QX are equal to 1. From the simulation results in Figure 3, $QY was nearly equal to 1. $QX was decreased by decreasing the amount of organic reactant. The slope of the line (4.43)was constant by the method of least squares (r = 0.9997). Thus, the pseudo-first-order reaction was affected by masstransfer resistance of QX from the organic phase into the aqueous phase. When yo was decreased, dylo/dyowas dramatically decreased and dyzJdyo was slightly increased
Fraction of R X
Figure 3. Capabilityof mass transfer between phases. (1)+ey; (2) +ax*
.I. I
".tu,
- ------.--.-.-...-- ....-.. ........__.._...*.........-........... ~
0.0
0.2
0.4
0.6
0.8
I .o
Fraction of RX
Figure 4. Consumption rate of the catalysts ((1) QY; (2) QX) in the organic phase vs consumption rate of the organic reactant; RX.
(Figure 4). The meaning of the latter was that the ratio of the consumption rate of QX to that of RX in the organic phase was almost constant (=0.06). On the basis of Figures 3 and 4, it is illustrated that the main mass-transfer limitation is the transport of QX from the organic phase to the aqueous phase. The reaction system of Wang and Yang (1991) was described by a pseudo-first-order hypothesis. The physically meaningful parameters ~ Q and X ~ Q are Y the ratios of the mass-transfer rates of QX and QY, respectively, between phasesto the chemical reaction rate in the organic X ~ Q are Y equal to 117 and 300,respectively, phase. ~ Q and indicating that the mass-transfer resistance of QX is larger than that of QY. (B)Mass-Transfer-Controlled Reaction. Is the pseudo-first-order approximation applicable in the limit of mass-transfer rate? The reaction condition must have higher concentrations of salta (MOH or MY) in the aqueous phase to have a higher thermodynamic equilibrium constant (m- = mqy = 100). The ratio of the chemical reaction resistance in the organic phase to the mass transfer resistance between phases is equal to 2 ( ~ Q Y= ~ Q = X 2). The chemical reaction in the aqueous solution is a reversible reaction, and the equilibrium constant is equal to 104. The other parameters are a = 1, PI = PZ= 20, Ps = 0,and K = 100. The simulation results are a h o y in Figure 5. The concentrations of QY and QX in the organic phase have
1326 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 .......
,_,C_.__
(2)
\
-10 -
I
0.01
00
0.2
0.4
0.6
0.8
I
.o
;
(2)
LV
0.0
0.2
Fraction of R X
0.4
0.6
0.8
i I .o
Fraction of R X
Figure 5. Fraction concentration profile of the catalysta.
(1)yl,;
(2) Y20.
Figure 7. Consumption rate of the catalysta ((1)QY; (2) QX)in the organic phase vs consumption rate of the organic reactant; RX. limitation is the transport of QX from the organic phase to the aqueous phase. The results of optimum physically meaningful parameters in the phase transfer reaction systems will be presented in a future paper.
Acknowledgment The author would like to acknowledgefinancial support from the National Science Council, Taiwan, Republic of China.
0.0’ 0.0
’
0,z
0.4
0.6
0.8
‘0 I .o
Fraction of R X
Figure 6. Capability of mass transfer between phases. (1) $qy; (2) +ex.
larger fluctuations during the reaction period. The reaction system cannot be described by a pseudo-firstorder hypothesis. The capability of mass transfer is presented in Figure 6. The slopes of lines #QY and #QX vs yo are equal to 9.9 (r = 0.9999) and 1015 (r = 0.9998) by the method of least squares. These results show that , mass-transfer resistance although ~ Q isY equal to ~ Q Xthe of QX is significant. When yo is decreased, dyloldyois decreased and dyddy0 is increased. dyloldyois negative and dyaoldyois positive when yois less than 0.92. dyloldyo is symmetrical with dyzoldyo along the horizontal axis (Figure 7). The ratio of the consumption rate of QX to that of RX in the organic phase is positive. The results reveal that the rate-determining step of the reaction is the mass transport of QX from the organic phase to the aqueous phase. These results are consistent with those of Wang and Wu (1991).
Conclusion A phase-plane model, formulated as a system of coupled nonlinear differential and algebraic equations, has been presented describing the dynamics of a liquid-liquid phase transfer catalyzed reaction system. The model contains all the factors that influence the behavior of a phase transfer catalyzed reaction system. The proposed “physically meaningful parameters” are very useful for judging the reaction phenomenon. Theoretical results reveal that mass-transport resistances can have a significant effect on the response of the system. The main mass-transfer
Nomenclature [ la = concentration in the aqueous phase, M [ If = concentration at the interface between phases, M [ 3, = concentration in the organic phase, M A = interfacial area between the organic and aqueous phase, m2 kl = forward reaction rate constant of the aqueous phase, M-1min-1 k-l= backward reaction rate constant of the aqueous phase, M-1 min-1 k, = reaction rate constant of the organic phase, M-1 min-1 K, = equilibrium constant in the aqueous phase KQX= overall mass-transfer coefficient of QX, m min-1 KQY = overall mass-transfer coefficient of QY, m min-1 mgx = distribution coefficient of QXC[QXld[QXl,) mgy = distribution coefficient of QY ([QYI,f/[QYIS M = concentration, m0llm3 PI, Pz, P 3 = initial mole ratio of reactant to onium salts Qi = total moles of onium salts QX, QY = onium salts RX = chemical reactant RY = chemical product t = time, min Va = volume of aqueous phase, m3 V , = volume of organic phase, m3 X-, Y- = anions in the aqueous phase a = volume ratio (=VJV.J K = ratio of reaction rate constant in the aqueous phase to that in the organic phase ~ Q = X Damkohler number of QX $JQY = Damkohler number of QY Subscripts a = aqueous phase o = organic phase
Appendix The physically meaningful parameters in this paper are from the experimental data of Wang And Yang (1991),for
Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 1327 an allylation of 2,4,64ribromophenol in a phase transfer catalytic reaction. The reaction conditions are the following: 2,4,6-tribromophenol = 0,0091 mol, allyl bromide = 0.0058 mol, KOH = 0.025 mol, tetrabutylammonium bromide (PTC) = 0.00093 mol, H2O = 50 mL, chlorobenzene = 50 mL, temperature = 50 OC; the distribution coefficients are mgx = 0.071 and mqy = 12; the overall mass-transfer coefficients between phases are KQXA= 0.1345 L/min and KQYA = 0.342 L/min; the reaction constants are k, = 1.23 (M.min)-l and kl = 10 (M-min)-1,
Literature Cited Dehmlow, E. V.; Dehmlow, S. S. Phase Transfer Catalysis; Verlag Chemie: Weinheim, 1983. Evan, K. J.; Palmer, H. J. The Importance of Interphase Transport Resistanceson Phase Transfer Catalyzed Reaction. AZChE Symp. Ser. 1981, 77 (202), 104-113.
Melville, J. B.; Goddard, J. D. Mass-Transfer Enhancement and Reaction-Rate Limitations in Solid-Liquid Phase-Transfer Catalysis. Chem. Eng. Sci. 1985,40 (12), 2207-2215. Melville, J. B.; Yortsos, Y. C. Phase Transfer Catalysis, with Rapid Reaction. Chem. Eng. Sci. 1986, 41 (ll),2873-2882. Starks,C. M.; Liotta, C. Phase Transfer Catalysis, Principles and Techniques; Academic Press: New York, 1978. Wang, M. L.; Wu, H. S. Kinetic and Mass Transfer Studies of a Sequential Reaction by Phase Transfer Catalysts. Chem. Eng. Sci. 1991,46 (2), 509-517. Wang, M. L.; Yang, H. M. Dynamics of Phase Transfer Catalyzed Reaction for the Allylation of 2,4,6Tribromophenol. Chern. Eng. Sci. 1991, 46 (2), 619-627. Weber, W. P.; Gokel, G. W. Phase Transfer Catalysis in Organic Syntheses; Springer Verlag: New York, 1977.
Receiued for reuiew October 20, 1992 Revised manuscript receiued March 26, 1993 Accepted April 8, 1993
