Analysis of Pervaporation-Aided Esterification of Organic Acids

8490

Ind. Eng. Chem. Res. 2007, 46, 8490-8504

Analysis of Pervaporation-Aided Esterification of Organic Acids Satish J. Parulekar* Department of Chemical & Biological Engineering, Illinois Institute of Technology, Chicago, Illinois 60616

Reactors coupled with membrane separation, such as pervaporation, can help enhance the conversion of reactants for thermodynamically or kinetically limited reactions via selective removal of one or more product species from the reaction mixture. Efficacy of pervaporation in driving esterification of lactic and succinic acids with ethanol, which are both reversible reactions, to near completion by stripping water has been established in the author’s previous experimental studies. A detailed analysis of reversible condensation reactionsswhere one of the products (water) is removed via membrane separationsis presented in this work for different reactor-separator configurations. Appropriate numerical illustrations are provided to supplement the analysis. The role of resistances for reaction and membrane separation is investigated, including comparison to perfect membrane, and performance of the reactor-separator systems is compared. 1. Introduction Recently, there has been an increasing effort to combine downstream/upstream separation with reaction to improve process performance. In this regard, membrane technology has emerged as one of the viable separation processes. Since membranes allow selective permeation of a component from a multicomponent mixture, these can help enhance the conversion of reactants for thermodynamically or kinetically limited reactions via selective removal of one or more product species from the reaction mixture. When multiple reactions are involved, the yield or selectivity of a desired product, usually an intermediate, can be enhanced by controlled addition of one or more reactants and/or removal of one or more intermediates. Membrane reactors have been extensively investigated both experimentally and theoretically for gas-phase reactions (see Benedict et al.1 and Benedict2 for citations). Studies on liquid-phase reactions have been far fewer, because of the lack of suitable membranes with good permselectivity and solvent resistance. Pervaporation is used to separate a liquid mixture by partially vaporizing it through a porous inorganic membrane (transport through pores) or a nonporous permselective membrane (solution-diffusion mechanism).1-8 The “feed” liquid mixture is allowed to flow along one side of the membrane, and a fraction of it (the “permeate”) is recovered in the vapor state on the other side of the membrane. The permeate is kept under vacuum by continuous pumping or is purged with a stream of carrier gas.8-10 Maintaining a low vapor pressure on the permeate side, eliminating thereby the effect of osmotic pressure, induces mass transport through the membrane in this process. The permeate is finally obtained in the liquid state after condensation. The permeate is enriched in the more rapidly permeating component of the feed mixture, whereas the remainder of the feed that does not permeate through the membrane (the “retentate”) is depleted in this component.1-3,8 Applications of pervaporation reported in the literature include dehydration of organic solvents,11 separation of aromatic/ aliphatic hydrocarbon mixtures,12,13 and removal of water from solutions of organic acids and alcohols.14-18 Removal of water from mixtures (dehydration) is the most widely used application of pervaporation. Hydrophilic polymeric membranes, such as those based on poly(vinyl alcohol) and polysaccharides such * To whom correspondence should be addressed. Tel.: (312) 5673044. Fax: (312) 567-8874. E-mail address: [email protected].

as cellulose and chitosan, show a stronger affinity for water and, hence, have been a proper choice for dehydration purposes.11,15,18 Pervaporation is ideal for enhancing conversion in reversible condensation reactions, such as esterification of carboxylic acids and alcohols, generating water as a product. To achieve a high ester yield, it is typical to use a large excess of alcohol, or follow the reaction by distillation to remove in situ product(s) to drive the equilibrium to the ester side.19 The use of a large excess of one reactant leads to a higher cost of subsequent separations to recover the unused reactant, and operations involving reaction followed by distillation have several pitfalls.4,14,15 Pervaporationaided reactors are attractive in this regard, because, being a ratecontrolled separation process, the separation efficiency in pervaporation is not limited by relative volatility as in distillation, the energy consumption in pervaporation is low, and it can be performed at a temperature that is optimal for reaction; this a feature that is especially important for enzymatic esterification. Pervaporation-aided reactors have been used for esterification of carboxylic acids, such as acetic, erucic, lactic, oleic, propionic, succinic, tartaric, and valeric acids, and alcohols, such as benzyl alcohol, butanol, cetyl alcohol, ethanol, methanol, and propanol, with various acids or enzymes as catalysts.1,3,20-31 Substantial acceleration of these reactions can be achieved using appropriate commercially available solid catalysts.1,3 The increasing interest in pervaporation-aided esterification is also revealed by theoretical studies that have been performed over the past few years4,22,25,27,31-34 and the availability of rate expressions for various esterification systems.1,33 Lactic and succinic acids are important intermediates, which have a significant role in glucose metabolism in most living systems. Diethyl succinate and ethyl lactate are important chemical intermediates. Esterification of each acid with ethanol is a reversible reaction and, hence, thermodynamically limited. Efficacy of pervaporation in driving esterification of the two acids with ethanol almost to completion, with almost-total conversion of the stoichiometrically limiting reactant, by stripping water, which is a reaction product, has been established in author’s previous experimental studies in well-mixed reactors using solid catalyst (Amberlyst XN-1010).1,3 A detailed analysis of the reversible reactions, where one of the products is removed by membrane separation, is presented here. The reactions of interest are condensation reactions, with one of the products

10.1021/ie061157o CCC: $37.00 © 2007 American Chemical Society Published on Web 10/16/2007

Ind. Eng. Chem. Res., Vol. 46, No. 25, 2007 8491

(water) to be removed by pervaporation. The reactor-separator configurations considered are (1) a semi-batch perfectly mixed reactor with complete recycle of pervaporation retentate, (2) a plug flow reactor (PFR) with in situ pervaporation (tubular membrane reactor), (3) a continuous stirred tank reactor (CSTR) with in situ pervaporation, and (4) a CSTR/PFR with external pervaporation and recycle. The analysis is supplemented by appropriate numerical illustrations, based on the esterification of lactic acid with ethanol.1,3 The effects of resistances for reaction and membrane separation on the performance of the reaction-pervaporation systems are studied. Performance of membranes with finite resistance for mass transfer is compared with that of perfect membranes, which are membranes with negligible mass-transfer resistance. The outcome of problem formulation for each reactor configuration is presented in Section 2, with essential details being reported in the Appendix. This is followed by numerical illustrations in Section 3. 2. Summary of Problem Formulation The reactions under consideration are esterification of an organic acid (A) and alcohol (B) to generate ester (C) and water (D) as per the stoichiometry:

aA + bB h cC + dD

(1)

Conservation of mass requires that

∑J νJMJ ) (cMC + dMD - aMA - bMB) ) 0

(2)

per reaction stoichiometry (where MJ is the molecular weight of component J). For the esterification of monovalent acids, such as acetic acid (CH3COOH) and lactic acid (C3H6O3), with alcohols, such as methanol (CH3OH) and ethanol (C2H5OH), a ) b ) c ) d ) 1, whereas for the esterification of divalent acids, such as succinic acid (C4H6O4), with these alcohols, a ) c ) 1 and b ) d ) 2. These reactions are usually moleconserving, i.e., (a + b) ) (c + d), as revealed by the specific examples previously mentioned. The stoichiometry of the reaction in eq 1 implies the following linear relations among the rates of formation of the four species:

RJ )r νJ

(for J ) A, B, C, D)

(3)

where νA ) -a, νB ) -b, νC ) c, and νD ) d. Here, we consider esterification that has been conducted in perfectly mixed reactors or tubular flow reactors with no axial mixing and perfect radial mixing. In case a solid catalyst is used for accelerating esterification, the catalyst is considered to be packed uniformly in the tubular reactor and either freely suspended and retained or contained in spinning baskets in the case of perfectly mixed reactor. When one of the reaction products (water, in the case of esterification reactions) is removed preferentially, the perfectly mixed reactor can be operated in semi-batch or continuous mode. The formulation for each reactor configuration, summarized here, with necessary details being relegated to the Appendix, is applicable irrespective of which of the two reactants is stoichiometrically limiting. Several commercial pervaporation membranes exhibit very high selectivity for permeation of water (D). In a binary mixture of species A and D, the selectivity of the more rapidly permeating component D, Sm, is defined as

Sm )

(CD/CA)p (CD/CA)r

)

(xD/xA)p (xD/xA)r

with the superscripts p and r referring to the permeate and retentate streams, respectively. The definition of selectivity can be readily extended to multicomponent mixtures with A representing the remainder of the mixture (mixture with exclusion of the rapidly permeating component D). Even when the feed to the pervaporation unit is lean in D, and, therefore, (xD)r , 1, the permeate is comprised almost exclusively of D [(xD)p f 1]. The very high selectivity, with respect to water, that these membranes have makes them very suitable for pervaporation-aided esterification of fermentation-derived organic acids, such as acetic, formic, lactic, and succinic acids. For this reason, we consider only D to be transported across the pervaporation membrane. The volume of the reaction mixture (semi-batch reactor) or volumetric flow rate of the reaction mixture (continuous flow reactors) can be calculated if information on density of the reaction mixturesparticularly, its variation with the composition of the reaction mixturesis available. For many liquid-phase esterification reactions, including the esterification of lactic and succinic acids with ethanol,1,3 variation in density of the reaction mixture with variation in composition of the reaction mixture in batch, semi-batch, and continuous flow reactor operations has been found to be insignificant. This is considered to be the case in the analysis that follows. 2.1. Semi-Batch Reactor. The operation of a perfectly mixed semi-batch reactor (see Figure 1a) is completely described by

MDFDpCA0 dV )dXA aFV0rV

(4a)

where

FDp ) AmnD

(V(0) ) 1)

(4b)

C A0 dt ) dXA arV

(t(0) ) 0)

(5)

with the reaction rate r being

r)

{() r′

W r′′ V

(for homogeneous reaction) (for heterogeneous catalytic reaction)

(6)

and being expressed exclusively in terms of XA and V, given eq A6. Equation 4 is independent of t, the solution of which generates a phase portrait of XA and V. With the right-hand side of eq 5 being independent of t, this equation can be integrated explicitly to yield t, in terms of XA and V. For a perfect membrane, V is related to XA as

d V ) 1 - β ΘD + XA a

(

)

(7)

To identify the trajectory of the reaction mixture in the case of a perfect membrane, one must solve eq 5 in conjunction with eqs 7 and A6 (ND = 0). In this limiting case of dehydrated reaction mixture, the reaction is essentially irreversible and the semi-batch reactor will permit attainment of maximum conversion of the two reactants. 2.2. Tubular Membrane Reactor. Next, we consider a continuous flow tubular (plug flow) reactor with a pervaporation

8492

Ind. Eng. Chem. Res., Vol. 46, No. 25, 2007

2.3. Perfectly Mixed Reactor with In Situ Membrane Separation. Next, we consider the steady-state operation of a CSTR with in situ removal of D by membrane separation (see Figure 1c). The master reactor design equation is

τ)

CA0XA

(11)

ar

with r being a function of FJ (J ) A, B, C, D) and q, the former being expressed in terms of dimensionless variables XA and z in eq A13, and the latter being expressed via the overall mass balance for the CSTR, as

Fq0 ) Fq + MDFDp w p )

q ) 1 - βz q0

(12)

The master design equation (eq 11) provides one relation involving XA and z. The other relation, which involves the two variables, is provided by the kinetics of pervaporation. For a perfect membrane, z is related to XA as

z ) Θ D + XA

Figure 1. Reactor-separator configurations under consideration: (a) semibatch reactor, (b) tubular (plug flow) membrane reactor, (c) continuously stirred tank reactor (CSTR) with in situ separation, and (d) continuously stirred tank reactor/plug flow reactor (CSTR/PFR) with an external separator and recycle.

membrane being placed on the reactor wall (Figure 1b). Operation of this reactor is completely described by

MDRmDCA0

dp )dXA

(8a)

aFr

where

RmD )

Am n V D

C A0 dτ ) dXA ar

(p(0) ) 1) (τ(0) ) 0)

(8b)

(9)

with r being defined in eq 6 and expressed exclusively in terms of dimensionless variables XA and p, given eq A12. Equation 8 is independent of τ, solution of which generates a phase portrait of XA and p. The right-hand side of eq 9 being independent of τ, this equation can be integrated explicitly to yield τ, in terms of XA and p. For a perfect membrane, p is related to XA as

d p ) 1 - β ΘD + X A a

(

)

(10)

To identify the trajectory of the reaction mixture in the case of a perfect membrane, one must solve eq 9 in conjunction with eqs 10 and A12 (FD = 0). The reaction is essentially irreversible, and the tubular membrane reactor will permit the maximum conversion of the two reactants to be attained.

(13)

2.4. CSTR/PFR with External Membrane Separation and Recycle. Finally, we consider steady-state operation of a continuous flow reactor coupled to a pervaporation unit, with the two being physically segregated (see Figure 1d). The reaction rate terms in molar balances for CSTR and PFR, in eqs A18 and A19, respectively, are functions of variables F′Js (F′J ) FJ for CSTR (J ) A, B, C, D)) and q. The relations in eqs A17 and A23 permit us to express the rate terms exclusively in terms of dimensionless variables XA, ζ, and z. Working with the mass balance for A in eqs A18 and A19 and the relations in eqs 3, A21, and A22, the master equation for evaluating the reactor performance is obtained as eq 11 for a CSTR and

τ)

C A0

∫X a(1 - ζ) ζX

A A

dX′A r(X′A,ζ,z)

(14)

for a PFR. The master reactor design equation in eq 11 or eq 14 provides one relation involving XA, z, and ζ. The other relation involving the three variables is provided by kinetics of pervaporation. In reactor systems where there is feedback from the pervaporator to the reactor via recycle of a portion of the pervaporator retentate, removal of all D fed and generated due to reaction [z ) ΘD + (dXA/a)] does not require CD to be zero (reaction mixture not dehydrated in the case of esterification) as FD ) (1 + R)FA0[ΘD + (dXA/a)](1 - ζ) > 0 in this case (eq A24 with F′D ) FD). Therefore, a benefit of recycling a portion of the pervaporator retentate is that a membrane that is not perfect (very rapid removal of D) can perform as well as a perfect membrane. The same design equation (eq 11) applies for CSTR-separator configurations in Sections 2.3 and 2.4. 3. Results and Discussion For the sake of numerical computations, the kinetics of reaction of lactic acid (C3H6O3) with ethanol (C2H5OH) (eq 1, a ) b ) c ) d ) 1, D ) C5H10O3), reported by Benedict et al.,1 is adopted here. The operating temperature for the reactor and peravporator is considered to be 95 °C. A model GFT1005 membrane (Deutsche Carbone AG) was used for the pervaporation of water. The suggested maximum temperature for long-duration operation of a GFT membrane is 100 °C. Both


heterogeneous esterification and pervaporation are promoted at higher temperatures. Therefore, the choice of an operating temperature closer to, and less than, 100 °C in experiments that use lactic acid is well-justified. The rate expression for homogeneous esterification is

(

r′ ) k1 CACB -

1 C C K C D

)

(15)

with k1 ) 8.527 × 10-6 L/(mol s) and K ) 2.78177 at 95 °C. The rate expression for esterification catalyzed by Amberlyst XN-1010 catalyst (Rohm and Haas) is

r′′ )

k[CACB - (CCCD/K)] CB + K1CCCD

(16)

with k ) 0.6014 L/(kg catalyst min) and K1 ) 0.201429 at 95 °C. The rates of both homogeneous and heterogeneous reactions increase with increasing concentrations of both reactants (each reaction positive ordered, with respect to reactants) and decrease with increasing concentrations of both products (each reaction negative ordered, with respect to products). For a particular XA, a PFR should require lower τ (vis-a-vis a CSTR), and, for a particular τ, a PFR should yield higher XA (vis-a-vis a CSTR). At 95 °C, the flux of water (nD) was correlated to the concentration of water on the feed side of the pervaporation membrane (i.e., in the reaction mixture, CD) as1,3

JD ) nDMD ) R1CDβ1

(17)

where R1 ) 0.508 and β1 ) 1.1242, with JD given in units of kg/(m2 h) and CD given in units of mol/L. The nonlinear relation in eq 17 indicates that the pervaporation of water is promoted increasingly with increasing water concentration on the feed side of the pervaporation module, all other parameters remaining unaltered. Such an acceleration of water removal has also been reported by the membrane manufacturer (Deutsche Carbone AG) for ethyl acetate-water and butyl acetate-water mixtures. Equation 17 provides an unstructured kinetic representation of the pervaporation of water, which is a multistep process, and does not explicitly account for contributions of individual steps to the overall rate and effects of concentration polarization and osmotic pressure. Formulation of a structured kinetic representation of pervaporation that accounts for these, although beyond the scope of this article, is currently in progress. High water flux through the pervaporation membrane can be obtained by (i) maintaining high recirculation rate on the feed side of the pervaporation unit, (ii) higher operating temperature for the same (better adsorption of water by the membrane), and (iii) low permeate pressure (increased driving force for water transport across the membrane). Strategy (i) can minimize concentration polarization, whereas strategy (iii) can eliminate the effect of osmotic pressure, inducing thereby mass transport through the membrane. The lumped kinetic expression in eq 17 represents the experimental data very well (Figure 4) and, therefore, is appropriate for use in simulations. For all model simulations, we consider the following feed (steady-state continuous flow reactor) or initial (semi-batch reactor) composition of the reaction mixture: CA0 ) 6 mol/L, CB0 ) 7.2 mol/L, CC0 ) 0 mol/L, and CD0 ) 7.1 mol/L. Water must be present at a significant level in the initial (feed) reaction mixture to keep it in a single fluid phase. For this composition, the density of the initial (feed) reaction mixture is equal to density of water (1 kg/L). Therefore, the removal of water by

pervaporation should not alter the density of the pervaporation retentate (vis-a-vis, the feed to the pervaporator). The equilibrium conversion of A (XAe), which represents the upper limit on the conversion of A attainable in a batch reactor, a CSTR, or a PFR without employing membrane separation, for this composition is 0.521785. Esterification reaction and pervaporation being rate processes in series, the resistance of the reaction-pervaporation process is decided by the resistances of reaction and membrane separation processes. When the resistance of one of the two rate processes is the dominant one, that rate process dictates the kinetics of the overall process. The results to be presented next will repeatedly reveal this. The performance of the reactionseparation systems is analyzed next in terms of pseudo-firstorder kinetic coefficients for reaction (kr) and pervaporation (km). 3.1. Semi-Batch Reactor. In view of the relations in eq A6 and eqs 15-17, eqs 4 and 5 are restated as

dV ) -σf1(XA,V)g1(XA,V) dXA

(

for σ )

f1(XA,V) dt ) dXA kr

km , V(0) ) 1 kr

)

(for t(0) ) 0)

(18) (19)

with

R1Am(CA0)β1 km ) , g1(XA, V) ) FV0 1 1-V Θ + XA ν D β

[{

}]

β1

(20)

and

kr ) k1CA0, f1(XA, V) )

V φ(XA,V)

(21a)

where

φ(XA,V) ) (1 - XA)(ΘB - XA) -

[

]

(1 - V) 1 (21b) (Θ + XA) ΘD + XA K C β for homogeneous reaction and

kr )

kW , V0 f1(XA,V) )

{

1 V(ΘB - XA) + K1CA0(ΘC + XA) φ(XA,V) (1 - V) ΘD + X A (22) β

[

]}

for heterogeneous reaction. The pseudo-first-order kinetic coefficients associated with membrane separation and reaction (km and kr, respectively) strongly influence the behavior of the overall process. If km , kr or σ , 1, then membrane separation is the rate-limiting (bottleneck) process and if km . kr or σ . 1, then the reaction is the rate-determining process. For a perfect membrane (km f ∞), given eq 7, the solution of eq 19 is expressed as t)

{

}

XA 1 -β ln(1 - XA) + [β(ΘD + 1) - 1] kr (1 - XA)

(if ΘB ) 1)

8494

t)


{

1 [β(1 + ΘD) - 1] ln(1 - XA) + kr(ΘB - 1) ΘB - XA [1 - β(ΘB + ΘD)] ln ΘB

(

)}

(if ΘB * 1)

for homogeneous reaction and

t)

1 [βXA + {β(1 + ΘD) - 1} ln(1 - XA)] kr

for heterogeneous reaction. Portraits of t and XA for a semi-batch reactor are presented in Figure 2a for homogeneous esterification and in Figure 3a for solid-catalyzed esterification. The corresponding portraits of XA and V are presented in Figure 2b for homogeneous esterification and in Figure 3b for solid-catalyzed esterification. Based on the experimental studies in the past,1-3 the base values of Am, W, and V0 were chosen to be 0.0182 m2, 80 g, and 2 L, respectively. Therefore, the pseudo-first-order kinetic coefficient for esterification (kr) is 0.00307 min-1 for the homogeneous reaction and 0.024 min-1 for the solid-catalyzed reaction. The curves denoted by 1, 2, and 3 in Figure 2 correspond to Am values of 0.002, 0.006, and 0.0182 m2, respectively, or σ values of 0.021, 0.062, and 0.188, respectively. The curves denoted by 1, 2, and 3 in Figure 3 correspond to Am values of 0.0182, 0.04, and 0.08 m2, respectively, or σ values of 0.024, 0.053, and 0.106, respectively. The curves denoted by P in these figures represent the performance of reactionseparation process with a perfect membrane (σ f ∞). The dashed curves in Figures 2a and 3a display trajectories of reaction-only operations. An increase in the membrane area leads to faster removal of water from the reaction mixture, accelerating conversion of the two reactants (Figures 2a and 3a). For both homogeneous and heterogeneous esterification, the operation of the reactorseparation system becomes more sluggish as the membrane area is reduced. For the base value of Am ) 0.0182 m2, the deviation of the performance of the pervaporator from that of a perfect membrane is much more significant for solid-catalyzed esterification (curves 1 in Figure 3) than for homogeneous esterifi-

Figure 3. Portraits of (a) t (min) and XA and (b) XA and V for heterogeneous esterification in a semi-batch reactor-separator system. The curves labeled 1, 2, 3, and P correspond to σ ) 0.024, 0.053, 0.106, and ∞ (perfect membrane), respectively. Dashed curve in panel a represents reaction-only operation (σ ) 0).

Figure 4. Portraits of (a) t (min) and XA and (b) t and V for heterogeneous esterification in a semi-batch reactor-separator system. The solid curves represent model simulation, and the data points marked by an asterisk (*) represent experimental data.1-3 The curves labeled NP and P in panel a represent reaction-only operation and operation with a perfect membrane, respectively. The experiment was conducted at 95 °C with W ) 62.7 g, V ) 2 L, CA0 ) 5.76 mol/L, CB0 ) 6.93 mol/L, CC0 ) 0 mol/L, CD0 ) 7 mol/L, and F ) 963.2 g/L.

Figure 2. Portraits of (a) t (min) and XA and (b) XA and V for homogeneous esterification in a semi-batch reactor-separator system. The curves labeled 1, 2, 3, and P correspond to σ ) 0.021, 0.062, 0.188, and ∞ (perfect membrane), respectively. Dashed curve in panel a represents reaction-only operation (σ ) 0).

cation (curves 3 in Figure 2). The use of a solid catalyst accelerates esterification or reduces the resistance for the same, thereby leading to increased importance of resistance to pervaporation. As a result, σ is substantially lower for heterogeneous reaction (0.024) than that for homogeneous reaction (0.188). Pervaporation clearly is the rate-limiting step in the case of heterogeneous esterification for this km value. Despite the range of km values considered for homogeneous esterification being substantially lower than that for the solid-catalyzed reaction, the deviations in membrane performance from that of a perfect membrane are qualitatively similar (Figures 2 and 3), and the ranges of σ are comparable. For membranes with finite km values, the time required to attain XA in excess of XAe increases tremendously with increasing


XA. When compared to this required time, the equilibrium conversion is attained rather rapidly. The tremendous increase in t with increasing XA for XA in excess of XAe is due to the diminishing driving force for reaction and not solely due to performance of an actual membrane (vis-a-vis a perfect membrane). An actual membrane is less effective in water removal (vis-a-vis a perfect membrane), with the difference in the performance of the two membranes reducing with increasing XA and vanishing as XA f 1 (see Figures 2b and 3b). The profiles of (i) t and XA and (ii) t and V predicted by the model are compared with experimental data on heterogeneous esterification1-3 in Figure 4. Assuming the variation in density of a reaction mixture in an experiment to be insignificant, the reduction in reactor volume was estimated using data based on the amount of permeate collected. This information was used in conjunction with measurements of composition of the reaction mixture to obtain fractional conversions of the two reactants. The permeate measurements were more frequent than measurements of the reaction mixture composition, hence, the difference in the number of data points in parts a and b of Figure 4. The membrane is obviously less effective in water removal and the resulting acceleration of reaction (vis-a-vis a perfect membrane). The model simulations predict the experimental results very well. This was also the case with homogeneous esterification (data not shown). The close agreement between the experimental data and model simulations makes the rate expressions in eqs 15-17 suitable for simulation of continuous flow reactor operations. 3.2. Tubular Membrane Reactor. In view of the relations in eq A12 and eqs 15-17, eqs 8 and 9 are restated as km dp ) -σf2(XA,p)g1(XA,p), σ ) dXA kr

f2(XA,p) dτ ) dXA kr

(for p(0) ) 1)

(for τ(0) ) 0)

(23)

(24)

with

km )

R1Am(CA0)β1 FV

and

kr ) k1CA0, f2(XA,p) )

p2 φ(XA,p)

for homogeneous reactions and

kr )

kW , f2(XA,p) ) f1(XA,p) V

for heterogeneous reactions [g1(XA,p) defined in eq 20, φ(XA,p) in eq 21, and f1(XA,p) in eq 22]. For a perfect membrane (km f ∞), given eqs A12 and 10, the solution of eq 24 is expressed as

[

krτ ) β2XA -

{(βΘD - 1)2 + β2(1 + 2ΘD) - 2β} (ΘB - 1) [(βΘD - 1) + ΘB{β2(ΘB + 2ΘD) - 2β}] 2

ln(1 - XA) -

(1 - ΘB)

ln

[

(

)]

Θ B - XA ΘB

(if ΘB * 1)

krτ ) β2XA - {2β - β2(1 + ΘB + 2ΘD)} ln(1 - XA) {(βΘD - 1)2 + β2(1 + 2ΘD) - 2β}

XA

]

(1 - XA) (if ΘB ) 1)


1 τ ) [βXA + {β(1 + ΘD) - 1} ln(1 - XA)] kr for heterogeneous reaction. Portraits of τ and XA for a tubular membrane reactor are presented in Figure 5a for homogeneous esterification and in Figures 6a and 7a for solid-catalyzed esterification. The results in Figure 6 are for a catalyst loading of W/V ) 0.04 kg/L, which is the same loading as that considered for the semi-batch reactor operation (Figures 2 and 3), whereas the results in Figure 7 are for W/V ) 1 kg/L, which is typical of a packed-bed reactor. At the former loading, the tubular reactor is dilute, in terms of catalyst. The corresponding portraits of XA and p are presented in Figure 5b for homogeneous esterification and in Figures 6b and 7b for solid-catalyzed esterification. Therefore, the values of kr for the results in Figures 5, 6, and 7 are 0.00307, 0.024, and 0.601 min-1, respectively. The curves denoted as 1, 2, and 3 in Figure 5 correspond to km ) 6.346 × 10-5, 1.904 × 10-4, and 5.775 × 10-4 min-1, respectively. The corresponding values of σ are 0.021, 0.062, and 0.188, respectively. The curves denoted as 1, 2, and 3 in Figure 6 correspond to km ) 5.775 × 10-4, 5.775 × 10-3, and 0.023 min-1, respectively. The corresponding values of σ are 0.024, 0.24, and 0.95, respectively. The curves denoted as 1, 2, and 3 in Figure 7 correspond to km ) 5.775 × 10-4, 5.775 × 10-3, and 0.023 min-1, respectively. The corresponding values of σ are 9.602 × 10-4, 9.602 × 10-3, and 0.038, respectively. The curves denoted by P in these figures represent the performance of the reaction-separation process with a perfect membrane (km f ∞, σ f ∞). The dashed curves in Figures 5a, 6a, and 7a display trajectories of reaction-only operations (σ ) 0). An increase in σ leads to faster removal of water from the reaction mixture, accelerating conversion of the two reactants (Figures 5a, 6a, and 7a). For both homogeneous and heterogeneous esterification, the operation of the reactor-separation system becomes more sluggish as σ is reduced. For the base value of km (5.775 × 10-4 min-1), the deviation of the performance of the actual membrane from that of a perfect membrane is much more significant for solid-catalyzed esterification (curves 1 in Figures 6 and 7) than for homogeneous esterification (curves 3 in Figure 5). This deviation is further enhanced for the higher catalyst loading (Figure 7). The use of solid catalyst accelerates esterification or reduces the resistance for the same, thereby leading to increased importance of the resistance to pervaporation. Pervaporation clearly is the ratelimiting step in the case of heterogeneous esterification for this km value. This is the reason why, despite the range of km

8496


Figure 5. Portraits of (a) τ (min) and XA and (b) XA and p for homogeneous esterification in a tubular membrane reactor. The curves labeled 1, 2, 3, and P correspond to σ ) 0.021, 0.062, 0.188, and ∞ (perfect membrane), respectively. Dashed curve in panel a represents reaction-only operation (σ ) 0).

Figure 7. Portraits of (a) τ (min) and XA and (b) XA and p for heterogeneous esterification in a tubular membrane reactor with W/V ) 1 kg/L. The curves labeled 1, 2, 3, and P correspond to σ ) 9.602 × 10-4, 9.602 × 10-3, 0.038, and ∞ (perfect membrane), respectively. Dashed curve in panel a represents reaction-only operation (σ ) 0).

with increasing XA and vanishing as XA approaches a value of 1 (see Figures 5b, 6b, and 7b). 3.3. Perfectly Mixed Reactor with In Situ Membrane Separation. Recognizing that, given eq 17,

nDAm R1CβD1Am z) ) ) FA0 FA0 MDFA0 FDp

and using eqs A13, A17, and A23, after some rearrangement, we rewrite the working correlation for the pervaporation membrane (eq 17) as

z)

Figure 6. Portraits of (a) τ (min) and XA and (b) XA and p for heterogeneous esterification in a tubular membrane reactor with W/V ) 0.04 kg/L. The curves labeled 1, 2, 3, and P correspond to σ ) 0.024, 0.24, 0.95, and ∞ (perfect membrane), respectively. Dashed curve in panel a represents reaction-only operation (σ ) 0).

considered for homogeneous esterification being substantially lower than that for the solid-catalyzed reaction, the deviations in membrane performance from that of a perfect membrane are qualitatively similar, as can be seen upon comparison of the results in Figure 5 with those in Figures 6 and 7. A 25-fold increase in W/V from 0.04 to 1 kg/L does not, as a result, yield proportional dividends, in terms of reduced reactor space time. For membranes with finite km values, the space time required to attain XA in excess of XAe increases tremendously with increasing XA. When compared to this required space time, that required for attaining the equilibrium conversion is rather low. The tremendous increase in τ with increasing XA for XA in excess of XAe is due to the diminishing driving force for reaction and not solely due to performance of an actual membrane (vis-avis a perfect membrane). An actual membrane is less effective in water removal (vis-a-vis a perfect membrane), with the difference in the performance of the two membranes reducing

(

)

km ΘD + XA - ζz τ β 1 - ζβz

β1

(25)

The aforementioned relation applies for CSTR/PFR with external pervaporation (0 < ζ < 1) (section 2.4) and also for CSTR with in situ pervaporation with ζ ) 1 (section 2.3). The design equation for a CSTR with in situ membrane separation is provided in eq 11, with r being expressed as (ζ ) 1 and y ) XA here)

krφ(y,ζ,z) r(y,ζ,z) ) C A0 (1 - ζβz)2

(26a)

k r ) k 1C A 0

(26b)

and

1 φ(y,ζ,z) ) (1 - y)(ΘB - y) - (ΘC + y)(ΘD + y - ζz) K (26c)

[

]


r(y,ζ,z) ) C A0 krφ(y,ζ,z) [(ΘB - y)(1 - ζβz) + K1CA0(ΘC + y)(ΘD + y - ζz)] (27a)


where

where

kW kr ) V

(27b)

for heterogeneous reaction. The rate expressions are written in this form because these are also applicable for continuous-flow reactors with external pervaporation and recycle. The CSTRpervaporator system is fully described by eqs 11 and 25, two nonlinear AEs in XA and z. One is not at liberty to choose XA and z independently: choosing one fixes the other. For homogeneous esterification (kr ) 0.00307 min-1), portraits of XA and τ are presented in Figure 8a and XA and z are given in Figure 8b. The curves labeled C and CP represent an actual membrane (km ) 5.775 × 10-4 min-1, σ ) 0.188) and a perfect membrane (km f ∞), respectively. The corresponding profiles for a tubular membrane reactor, reintroduced from Figure 5, are indicated by the profiles labeled P and PP, respectively. It was stated earlier that, based on the kinetics of the homogeneous and heterogeneous reactions, a PFR is better than a CSTR and the results in Figure 8a are consistent with this expectation. An actual membrane is less effective in water removal (vis-a-vis a perfect membrane), with the difference in the performance of the two membranes reducing with increasing XA and vanishing as XA approaches a value of 1 (see Figure 8b). The required space times for CSTR and PFR, as well as the difference between these, increase with increasing XA, which is due to the diminishing driving force for the reaction. 3.4. PFR/CSTR with External Membrane Separation and Recycle. The rate expressions in eqs 26 and 27 also apply for a CSTR/PFR with external pervaporator and recycle with 0 < ζ < 1 and y ) XA for a CSTR and ζXA ey eXA for a PFR. The rate expressions are simple enough that eq 14 for a PFR with recycle is amenable to analytical solution. The resulting expression for τ is

τ)

[

]

(1 - ζβz)2 I kr(1 - ζ)

[

I1 ) K1CA0(1 - ζ)XA +

(q1y1 + q2)

(

) (

ζXA - y1 + XA - y1 (y2 - y1) XA - y2 (q1y2 + q2) ln ζXA - y2 (y2 - y1) ln

)]

q1 ) K1CA0(y1 + y2 + ΘC + ΘD - ζz) - (1 - ζβz) q2 ) K1CA0[ΘC(ΘD - ζz) - y1y2] + (1 - ζβz)ΘB for heterogeneous reaction. The design equations for the reactor and pervaporator (eqs 11 and 25 for CSTR, with the rate expression given in eq 26 or 27 (y ) XA), and eqs 25 and 28 or 29 for a PFR) involve XA, z, and ζ as variables. Assigning one of these, one can estimate the other two by simultaneous solution of the design equations, which are nonlinear AEs. This is done here for fixed XA and fixed ζ. In a steady-state operation, the rate of D removed by pervaporation must be less than or equal to the sum of feed rate of D and the rate of generation of D by the reaction. Therefore, an upper limit on FDp is FA0(ΘD + XA). Furthermore, because q must be positive (eq A17) and for ester production, the driving force for the reaction must be positive (φ(XA,ζ,z) > 0 in eq 26), it follows that

zmin e z e zmax

(30a)

where zmax ) min

{

[

{ζβ1 ,(Θ

D

}

+ XA)

K(1 - XA)(ΘB - XA) 1 zmin ) min 0, (ΘD + XA) 1 ζ (ΘC + XA)(ΘD + XA)

]}

(30b)

One can observe that zmin is zero for XA e XAe and positive for XAe < XA < 1 (XAe is the equilibrium conversion in a reactor

(28)

where

I)

[

(

)

(XA - y2)(ζXA - y1) K ln (K - 1)(y2 - y1) (ζXA - y2)(XA - y1)

s ) ΘB + 1 +

y1 )

K (s - x∆) 2(K - 1)

y2 )

K (s + x∆) 2(K - 1)

1 (Θ + ΘD - ζz) , K C (K - 1) 1 ΘB - ΘC(ΘD - ζz) ∆ ) s2 - 4 K K

]

[

]


τ)

[

]

K I kr(1 - ζ)(K - 1) 1

(29)

Figure 8. Portraits of (a) XA and τ (min) and (b) XA and z for homogeneous esterification in a CSTR with in situ pervaporation with kr ) 0.00307 min-1. Solid and dashed curves (labeled C and CP, respectively) represent (i) CSTR with a membrane with km ) 5.775 × 10-4 min-1 (σ ) 0.188) and (ii) CSTR with a perfect membrane (km f ∞), respectively. For comparison, the corresponding results for a tubular membrane reactor are displayed in part a by the curves labeled P (actual membrane, km ) 5.775 × 10-4 min-1 (solid curve)) and PP (perfect membrane (dashed curve)).

8498


Figure 9. Profiles of ζm (curve eq) and ζr (curves C (CSTR) and P (PFR)) for homogeneous esterification in a continuous flow reactor with external pervaporation and recycle. kr ) 0.00307 min-1 and km ) 5.775 × 10-4 min-1.

operating in solo). For the feed composition under consideration, zmax ) ΘD + XA. The reactor-separator system must operate under the constraints previously listed. For certain XA and ζ, zmin may exceed zmax. Operation of a reactor with an external membrane separation and recycle is then not permissible. When zmin > zmax, it follows from eq 30, after some rearrangement, that

1 φ(XA,ζ,z) ) (1 - XA)(ΘB - XA) - (ΘC + XA)(ΘD + K

[

]

XA - ζz) < 0

(for z ) (ΘD + XA))

In this situation, the driving force for the esterification reaction is negative for the highest permissible value of z, and one can then deduce that this will also be the case for every z < (ΘD + XA). Operation of the reactor-separator system then is not feasible for any z. The boundary for the feasible operating region is then given by zmin ) zmax, i.e.,

[

ζm ) 1 -

K(1 - XA)(ΘB - XA)

]

(ΘC + XA)(ΘD + XA)

(31)

One can observe that ζm increases monotonically with XA, ζm < 0 for XA < XAe, ζm ) 0 for XA ) XAe, ζm > 0 for XA > XAe, and ζm f 1 as XA f 1. For any XA, the driving force for the reaction is positive for ζ > max(0,ζm). It therefore follows that steady-state operation of the reactor-separator system is feasible for 0 < ζ < 1 when XA e XAe and for ζm < ζ < 1 when XA > XAe. For XA > XAe, operation with ζ ) ζm corresponds to reaction equilibrium in the entire reactor (CSTR) or at the reactor exit (PFR), and, therefore, operation of the reactor-separator system is not feasible for 0 < ζ < ζm. The profile of ζm for XA > XAe is displayed in Figure 9 by the curve labeled “eq”. For systems with in situ membrane separation, it was observed earlier that the upper bound on the performance of an actual membrane was that of the perfect membrane, which corresponds to the absence of D in the reactor. An interesting feature of systems with recycle of the retentate from the separator is that an actual membrane can perform as well as a perfect membrane, i.e., with z ) ΘD + XA without CD being zero under certain conditions. One can determine when this occurs by solving the appropriate design equations for CSTR and PFR (eqs 11 and 25 for CSTR, with the rate expression in eq 26 or 27 (y ) XA), and eqs 25 and 28 or 29 for a PFR) with z ) ΘD + XA. The two design equations involve ζ and XA as variables, and the solution of these will yield a profile ζr(XA) for which z ) ΘD + XA. These profiles are displayed in Figure 9 for homogeneous reaction in a CSTR and a PFR with km ) 5.775 × 10-4 min-1 by curves labeled C and P, respectively. The profiles for

Figure 10. Portraits of (a, c) XA and τ (min) and (b) XA and z for homogeneous esterification in a continuous flow reactor with external pervaporation and recycle (ζ ) 0.5, kr ) 0.00307 min-1). The curves labeled C and CP represent CSTR with a membrane with km ) 5.775 × 10-4 min-1 (σ ) 0.188) and CSTR with a perfect membrane (km f ∞), respectively. The corresponding curves for PFR are labeled P and PP, respectively. In part b, line PM represents the profile of z for CSTR and PFR with a perfect membrane. In part c, the results from part a for PFR are displayed, together with those from Figure 8a for PFR with in situ membrane separation (P (dashed curve) represents the actual membrane; PP (solid curve) represents a perfect membrane).

heterogeneous reaction are obtained using the same procedure but are not displayed here. For ζ > max{0,ζr(XA)}, z < (ΘD + XA) and the membrane performance is less than perfect. For ζ ) ζr(XA), z ) (ΘD + XA) and the membrane performs similar to a perfect membrane. For max{0,ζm(XA)} < ζ < ζr(XA), the solution of reactorseparator design equations predict z > (ΘD + XA), which is not permissible. In this region, the solution of the design equations for a perfect membrane must be used with z ) ΘD + XA. In this region also, the actual membrane is fully effective and performs similar to a perfect membrane. The results pertaining to this transition are presented in Figure 10 for ζ ) 0.5 and Figure 11 for XA ) 0.6. As anticipated, for fixed ζ, an increase in the desired conversion XA leads to an increase in the required space time for CSTR and PFR with a membrane separator that is less than perfect or perfect (Figure 10, ζ ) 0.5). Operation of the reactorseparator system is feasible only up to XA ) 0.633 as ζm(0.633) ) 0.5 and ζm < 0.5 for 0 < XA < 0.633 (see Figure 9). An actual membrane separator is less than perfect, up to XA ) 0.421 in the case of CSTR and up to XA ) 0.4935 in the case of PFR as ζ > max{0,ζr(XA)} (see Figure 9). For greater XA values (up to 0.633), the solution of the reactor-separator design equations predict z > ΘD + XA, which is not permissible. In these conversion ranges, the solution of design equations for a perfect


Figure 11. Portraits of (a) ζ and τ (min) and (b) ζ and z for homogeneous esterification in a continuous flow reactor with external pervaporation and recycle (XA ) 0.6, kr ) 0.00307 min-1). The curves labeled C and CP represent CSTR with a membrane with km ) 5.775 × 10-4 min-1 (σ ) 0.188) and CSTR with a perfect membrane (km f ∞), respectively. The corresponding curves for PFR are labeled P and PP, respectively. In part b, the dashed line PM represents the profile of z for CSTR and PFR with perfect membrane. The results for a perfect membrane are displayed by dashed curves/lines for better illustration.

membrane must be used with z ) ΘD + XA (see Figure 10). The actual membrane then is fully effective and performs similar to a perfect membrane. As XA f 0.633, there is a tremendous increase in the required space times for both CSTR and PFR, as well as the difference between these, which is due to the decreasing driving force for the reaction. The difference in the performance of the actual membrane and a perfect membrane decreases as XA increases and the actual membrane becomes more effective. In Figure 10c, the performance of PFR with external separation and recycle (Figure 10a) is compared with that of PFR with in situ membrane separation (see Figure 8a). Although there is an upper limit on attainable fractional conversion (equilibrium limitation) in the former operation for a given ζ, the latter operation is feasible for any conversion. Furthermore, for the conversion range where both operations are feasible, in situ membrane separation is superior to external membrane separation with recycle, with regard to minimization of reactor space time (Figure 10c). The same conclusions apply for a CSTR, although the corresponding comparison is not shown here, because a CSTR requires a higher τ value than that for a PFR. The results presented in Figure 11 for XA ) 0.6 reveal similar features. Operation of the reactor-separator system is feasible over the range 0.38 < ζ e 1 as ζm(0.6) ) 0.38 (see Figure 9). For a CSTR, ζ ) ζr(0.6) at ζ ) 0.797 and for a PFR, ζ ) ζr(0.6) at ζ ) 0.723. As a result, the operation of the membrane separator is less than perfect for ζ > 0.797 for a CSTR and for ζ > 0.723 for a PFR (see Figure 11b). For 0.38 < ζ e 0.797 for a CSTR and for 0.38 < ζ e 0.723 for a PFR, the solution of the reactor-separator design equations predict z > ΘD + XA, which is not permissible. In these ranges of ζ, the solution of design equations for a perfect membrane must be used with z ) ΘD + XA because the actual membrane is fully effective (see Figure 11). As ζ f 0.38, there is a tremendous increase in the required space times for both CSTR and PFR, as well as the difference between these, which is due to the decreasing driving force for the reaction. The actual membrane becomes

more effective with decreasing ζ. As ζ f 1, the reactor with an external pervaporator is equivalent to a CSTR with in situ separation. As a result, the profiles for CSTR and PFR in Figure 11 converge at ζ ) 1. A beneficial effect of using recycle is observed in minimization of the reactor space time for PFR at ζ ) 0.723. For the reasons mentioned earlier, a PFR should be more effective in terms of space time requirement (lower τ, visa-vis CSTR). The results in Figures 10 and 11 are in agreement with this expectation. It is interesting to compare the results in Figure 11a with those for CSTR and PFR with in situ membrane separation (see Figure 8a). The space times required to attain XA ) 0.6 with in situ pervaporation are (i) 810.5 (CSTR, actual membrane: C), (ii) 529.5 (CSTR, perfect membrane: CP), (iii) 370 (PFR, actual membrane: P), and (iv) 250.6 (PFR, perfect membrane: PP). In situ membrane separation is clearly superior to external membrane separation with recycle, with regard to minimization of the cost of a particular reactor type; between the two reactor types under consideration, PFR is the reactor of choice. In conjunction with the results in Figure 10c, one can conclude that, among the continuous flow reactor configurations under consideration for reversible esterification, a once-through PFR with in situ removal of product D is the preferred choice, because it requires the least space time and permits operation over the entire conversion range. The real-time analog of this operation is the semi-batch reactor-separator system (sections 2.1 and 3.1). The results for heterogeneous esterification are qualitatively similar to the results presented in Figures 8-11 and are not displayed here, for the sake of brevity. Technical feasibility of pervaporation-aided esterification for almost-total conversion of lactic acid to ethyl lactate has been demonstrated via model simulations in this article. It is anticipated that the benefits of increased production of the desired product (ester) in the separation-aided reaction process considered here will outweigh the costs associated with the additional equipment required. The target conversion will be dependent on an economic analysis, which takes into account cost of downstream processing, with the focus on ester recovery. However, such economic analysis is beyond the scope of this article. As mentioned earlier, a structured kinetic representation of pervaporation that explicitly accounts for contributions of individual steps to the permeation rate and effects of concentration polarization and osmotic pressure is currently in progress. It will enable the identification of bottleneck step(s), provide a better understanding of the separation process, and aid in improved design of pervaporation membrane and module. However, the results obtained by replacing eq 17 with a structured kinetic expression for the membrane separation will not be qualitatively different from those presented and discussed in this article. 4. Conclusions Behavior of reversible homogeneous and solid-catalyzed reactions where one of the products is removed by membrane separation was analyzed in detail in the present work. The reactor configurations considered involved perfectly mixed reactors and a plug flow reactor (PFR), with membrane separation occurring in situ or externally as a part of a recycle loop. For numerical illustrations, esterification of lactic acid with ethanol was considered as the example reaction and pervaporation was considered as an example membrane separation. The performance of the reaction-separation systems was analyzed in terms of pseudo-first-order kinetic coefficients for reaction (kr) and pervaporation (km). Although the former has an

8500


important role in reactor sizing, the latter is an important determinant of the membrane performance. The performance of the reactor-pervaporator assembly is decided by how km and kr compare to each other. The higher the km/kr value, the more similar the performance of a membrane to a perfect membrane and the more effective the membrane in regard to the removal of product D. For configurations where reaction and separation occur together, the membrane is most effective when the reaction mixture is almost entirely depleted in the permeated product. In configurations involving recycle of the pervaporator retentate, where there is feedback from the separator to the reactor, a membrane can perform most effectively, even when the permeated product is present in the reaction mixture. A distinct benefit of using recycle is that, over certain ranges of desired fractional conversion (XA) and extent of recycle (ζ), the pervaporator can operate in the most effective fashion. A membrane with finite mass-transfer resistance performs poorly (vis-a-vis, a perfect membrane) as the resistance for reaction is reduced (reaction is accelerated). As a result, promotion of reaction does not yield proportional dividends, in terms of reduced reactor space time. For the reaction under consideration. a once-through PFR with in situ membrane separation (tubular membrane reactor) or its real-time analog, a semi-batch reactor-separator system, is the preferred choice.

The numerical effort involved in identification of the reaction trajectory can be reduced substantially if one solves only two component balances or a component balance and total mass balance, two ordinary differential equations (ODEs), in conjunction with the two stoichiometric relations, linear algebraic equations (AEs), in eq A3. The balances for A and total mass are restated in semi-dimensionless form as

V)

dNJ ) RJV - FJp dt FJp ) AmnJ, NJ(0) ) NJ0

{

0 (for J ) A, B, C) nJ ) >0 (for J ) D) d(FV) ) -MDFDp (for V(0) ) V0) dt

a

NB0 - NB b

)

NC - NC0 c

(A1b)

(A1c) (A2)

) 1 (N - ND0 + d D

) at every t.

N A0 X A a

)

NB0XB b

(A4b)

MD F D p dV )dt FV0

(for V(0) ) 1)

∫0t FD

(A5)

The number of moles of water removed by pervaporation, ∫t0FDp dt, can be computed via integration of the total mass balance in eq A5. With this, the stoichiometric relations in eq A3 can be restated as

()

(A6a)

where

ΘJ )

(A1a)

with RJ being defined as in eq 3, with the reaction rate r being defined in eq 6. The total mass balance can alternately be obtained from the species molar balances by summing over all J, MJ multiples of the molar balance for J in eq A1, and reduces to the total mass balance in eq A2, given eq 2 and after recognizing that FV ) m, m ) ∑J MJNJ. Because the reaction rate, which is an intensive variable, is expressed in terms of concentrations CJ of various species (CJ ) NJ/V), identification of the trajectory of the reaction mixture requires knowledge of variations in NJ and V. Considering the appropriate binary linear combinations of molar balances of the four species in eq A1 to eliminate reaction rate terms in these, using eq 3 and integrating the resulting equations, we obtain the stoichiometric relations

)

V (XA(0) ) 0) V0

νJ NJ ) ΘJ + X N A0 a A

A.1. Semi-Batch Reactor. For the semi-batch operation of a perfectly mixed reactor (Figure 1a), the conservation equations (components and total mass) can be expressed as

N A0 - N A

(A4a)

where

Appendix: Problem Formulation

where

dXA arV ) dt CA0

NJ0

(for J ) A, B, C) N A0

(A6b)

and

νD ND (1 - V) ) ΘD + XA NA0 a β β)

N D0 M DC A0 , ΘD ) F NA0

(A6c)

(A6d)

The stoichiometric relations in eq A6 permit us to express the right-hand sides in eqs A4 and A5 exclusively in terms of XA and V. Instead of solving eqs A4 and A5, one may solve an equivalent set, obtained from these, which is listed in eqs 4 and 5. In the limiting case, where the resistance to pervaporation of water is negligible (perfect membrane), the concentration of water (CD) in the reactor will be essentially zero at every t and, therefore, so will be dND/dt; thus, it follows from the molar balance for D in eq A1 that FDp = dr V at every t. Any D present in the initial reaction mixture is instantaneously removed. This will lead to an instantaneous reduction in V from V0 to V(0+) and a corresponding reduction in V from 1 to V(0+). The latter must be used as the initial condition for eq 4. Identification of the initial condition proceeds as

FV(0+) ) FV0 - MDND0 w V(0+) ) (1 - βΘD) p

dt)

(A3)

Using this relation, one obtains eq 7 upon integrating eq 4. A.2. Tubular Membrane Reactor. If a solid catalyst is used for conducting esterification, the catalyst is considered to be distributed uniformly through the reactor; as a result, the catalyst loading is uniform in the entire reactor, i.e.,


()

FJ νJ ) ΘJ + X F A0 a A

dW ∆W W ) ) dV ∆V V with ∆ referring to a small section of the reactor. The conservation equations (components and total mass) can be expressed as

dFJ ) RJ - RmJ dV

where

ΘJ ) (A7a)

nJ )

{

0 >0

(for J ) A, B, C) (for J ) D)

d(Fq) ) -MDRmD dV

(q(0) ) q0)

a

)

FB 0 - F B b

)

F C - FC 0 c

ΘD ) (A7c) (A8)

) 1 (F - FD0 + d D

)

FA0XA a

)

FB0XB

The numerical effort involved in identification of the reaction trajectory can be reduced substantially if one solves only two component balances or a component balance and total mass balance, two ODEs, in conjunction with the two stoichiometric relations, linear AEs, in eq A9. The balances for A and total mass are restated, in semi-dimensionless form, as

dXA V ar , τ) ) dτ C A0 q0

(A12b)

(XA(0) ) 0)

(A10)

MDRmD dp q (p(0) ) 1) ), p) dτ F q0

(A11)

The molar rate of water removal by pervaporation, ∫V0 RmD dV, can be computed via integration of the total mass balance in eq A8. With this, the stoichiometric relations in eq A9 can be restated as

FD 0

(A12c)

(A12d)

F A0

with β defined in eq A6. The right-hand sides in eqs A10 and A11 are functions of variables FA, FB, FC, FD, and q. The stoichiometric relations in eq A12 permit us to express these right-hand sides exclusively in terms of XA and p. Instead of solving eqs A10 and A11, one may solve the equivalent combination, stated in eqs 8 and 9, obtained from these. When the resistance to pervaporation of water is negligible (perfect membrane), the concentration of water (CD) in the reactor will be essentially zero at every V and, therefore, so will dFD/dV, and it follows from the molar balance for D in eq A7 that RmD = dr at every V. Any D present in the reactor feed is instantaneously removed. This will lead to an instantaneous reduction in q from q0 to q(0+) and a corresponding reduction in p from 1 to p(0+). The latter must be used as the initial condition for eq 8. Identification of the initial condition proceeds as

Fq(0+) ) Fq0 - MDFD0 w p(0+) ) (1 - βΘD) Using this, one obtains eq 10 upon integrating eq 8. A.3. Perfectly Mixed Reactor with In Situ Membrane Separation. The molar species balances are

FJ0 - FJ + RJV ) 0

∫0VRmDdV) (A9)

b

(for J ) A, B, C)

F A0

νD FD (1 - p) ) ΘD + XA F A0 a β

(A7b)

with RJ being defined in eqs 3 and 6. For a tubular reactor with a cylindrical cross section with a diameter D, ∆Am/∆V ) Am/V ) 4/D. Therefore, the volume-specific membrane area is inversely proportional to the reactor diameter. The total mass balance can alternately be obtained from the species molar balances by summing over all J, MJ multiples of the molar balance for J in eq A7, and reduces to the total mass balance in eq A8, given eq 2 and after recognizing that Fq ) G, G ) ∑J MJFJ. Because the reaction rate is expressed in terms of concentrations CJ of various species (CJ ) FJ/q), identification of trajectory of the reaction mixture requires knowledge of variations in FJ and q. Considering appropriate binary linear combinations of molar balances of the four species in eq A7 to eliminate reaction rate terms in these using eq 3 and integrating the resulting equations, we obtain the following stoichiometric relations: at every V.

F A 0 - FA

FJ 0

and

where

∆Am RmJ ) n (FJ(0) ) FJ0) ∆V J

(A12a)

(for J ) A, B, C)

F D 0 - F D - FD p + R D V ) 0 The following stoichiometric relations follow, given eq 3:

F A 0 - FA a

)

FB 0 - F B b

)

FC - FC0 c

) F D + FD p - F D 0 d

)

FA0XA a

The molar flow rates of all four species can then be expressed in terms of XA as

(

νJ FJ ) Θ J + XA F A0 a

)

(for J ) A, B, C) (A13a)

and

[ ()

νD FD ) ΘD + X -z F A0 a A where

]

(A13b)

8502


z)

(for J ) A, B, C)

F′J(0) ) FJf

(A19b)

F′J(V) ) FJ

(A19c)

with the RJ terms being defined in eqs 3 and 6. It then follows that the reactor trajectory must lie on the hyperplane

FAf - F′A a


)

FBf - F′B b

)

F′C - FCf c

)

F′D - FDf d

with

F′J ) FJ


for a CSTR. Alternately, we can also obtain the following for a PFR:

F′A -FA F′B - FB FC - F′C FD - F′D ) ) ) (A20) a b c d

(A14)

FD ) FDp + FDe + FDR ) FDp + (1 + R)FDe (A15) FJf ) FJ0 + FJR ) FJ0 + RFJe

(A19a)

F A0

The simultaneous solution of the mass balances for the four species (four nonlinear AEs) is equivalent to solving the mass balance for one speciesfor example, Astogether with the linear relations in eqs A13, the latter requiring substantially less effort. The mass balance for A yields the master reactor design equation in eq 11. When the resistance to pervaporation of water is negligible (a perfect membrane), the concentration of water (CD) in the CSTR will be essentially zero and, therefore, FD also will be zero. From the mass balance for D for the overall process or from eqs A13, it follows that the rate of removal of D by pervaporation must be equal to the rate of generation of D by reaction, which is leads to eq 13. A.4. CSTR/PFR with External Membrane Separation and Recycle. The effluent from the reactor serves as the feed to the pervaporation unit (see Figure 1d). The retentate from the pervaporation unit is split into a final effluent stream and a recycle stream in the ratio of 1:R. The species mole balances around the mixing point for the reactor feed and splitting points at and after pervaporator lead to the following.

FJ ) FJe + FJR ) (1 + R)FJe

dF′J ) RJ dV

F Dp

Considering acid A as the basis specie, an indicator of the performance of the reactor-pervaporator assembly is the fractional conversion of A, which is defined as

(A16) XA ) 1 -

F Ae

)1-

FA

(A21)

The densities of the fresh feed, the reactor effluent, and the retentate stream exiting the pervaporator are considered to be essentially equal and are denoted as F. Therefore, the volumetric feed rate will be uniform throughout the reactor. The total mass balance around the mixing point for the reactor feed and splitting points at and after pervaporator and the integrated process lead to the following:

because FA ) (1 + R)FAe. The aforementioned relationships suggest that the basis for defining the fractional conversion of A anywhere in a CSTR or a PFR should be (1 + R)FA0. Working with eq A16 for J ) A, it can be established that

q ) q0 + Rqe

XAf ) ζXA

1 q ) (1 + R)qe + (MDFDp) F

X Af ) 1 -

1 q0 ) qe + (MDFDp) F Eliminating qe in the aforementioned relations, one obtains

R q ) (1 + R)q0 - (MDFDp) F

F A 0 - FA e a

ζ)

R 1+R

(0 e ζ < 1) and β being defined in eq A6. The molar species balances for a CSTR are

F J f - F J + RJ V ) 0 and those for a PFR are

(for J ) A, B, C, D) (A18)

(A22a)

F Af

(A22b)

(1 + R)FA0

)

FB0 - FBe b

)

FCe - FC0 c

)

FDe + FDp - FD0

(A17)

with ζ being the fraction of the pervaporation retentate that is recycled

(1 + R)FA0

with XA being defined in eq A21. In view of the reaction stoichiometry, the following stoichiometric relations must be satisfied for the reactor-separator assembly:

or, alternately,

q ) (1 + R)q0(1 - ζβz)

F A0

d

)

FA0XA a

It then follows from the aforementioned relations, and eqs A14 and A15, that

F Je F A0

)

FJ (1 + R)FA0

) ΘJ +

νJ X a A

(for J ) A, B, C) (A23a)

FDe F A0

) ΘD +

()

νD X -z a A

(A23b)


FD (1 + R)FA0

) ΘD +

(ad)X

A

- ζz

(A23c)

Substituting the relationships in eq A23 into eq A20, we obtain the following relationships, which are applicable everywhere in a PFR.

F′J (1 + R)FA0

) ΘJ +

()

νJ X′ a A

F′D (1 + R)FA0

(

(for J ) A, B, C)

) ΘD +

()

νD X′ - ζz a A

)

(A24a)

(A24b)

Identification of the trajectory of the reaction mixture requires prediction of FJ at various values of V. Because the variation in density of the reaction mixture in the reactor is considered to be insignificant, q is uniform in the continuous flow reactor. The numerical effort in identification of the reaction trajectory can be reduced substantially if one solves one component balance, one nonlinear AE or ODE (for example, for A), in conjunction with the relations in eqs A17 and A23. In the absence of recycle (R ) 0, ζ ) 0), eq 14 represents, as anticipated, the design equation for once-through operation of PFR. The concentration gradients of the four species in the reactor are reduced with increasing extent of recycle (increasing ζ). In the limiting case of near total recycle of pervaporator retentate (R f ∞, ζ f 1), there are no concentration gradients in the reactor and upon application of L‘Hospital’s rule, eq 14 reduces to eq 11, which is the design equation of a CSTR. Nomenclature A ) organic acid Am ) membrane area (m2) B ) alcohol C ) ester CJ ) concentration of component J (mol/L) D ) water D ) diameter of a tubular reactor (m) FJ ) molar flow rate of component J (mol/min) FJ0 ) molar feed rate of component J for the overall continuous process (mol/min) FJe ) effluent molar flow rate of component J in a reactorpervaporator loop (mol/min) FJf ) molar feed rate of component J for the reactor in a reactorpervaporator loop (mol/min) FJp ) molar flow rate of component J in pervaporator permeate (mol/min) FJR ) molar flow rate of J in the recycle stream of a reactorpervaporator loop; FJR ) RFJe (mol/min) G ) flow rate of total mass, mass of the reaction mixture, in a continuous flow reactor (kg/min) j ) stoichiometric coefficient of component J (for J ) A, B, C, D) k ) kinetic parameter in eq 16 for solid-catalyzed reaction (L/ (kg catalyst min)) K1 ) kinetic parameter in eq 16 for solid-catalyzed reaction (L/mol) k1 ) kinetic parameter in eq 15 for homogeneous esterification km ) pseudo-first-order kinetic coefficient for membrane separation (min-1) kr ) pseudo-first-order kinetic coefficient for reaction (min-1) K ) equilibrium coefficient for esterification

m ) total mass, mass of the reaction mixture, in a semi-batch reactor (kg) nJ ) flux of component J through pervaporation membrane (mol/(m2 min)) MJ ) molecular weight of component J (g/mol) NJ ) number of moles of component J in a semi-batch reactor (mol) NJ0 ) initial number of moles of component J in a semi-batch reactor (mol) p ) defined in eq A11; p ) q/q0 q ) volumetric flow rate of the reaction mixture (L/min) q0 ) volumetric feed rate for the overall continuous process (L/min) qe ) effluent volumetric flow rate in a reactor-pervaporator loop (L/min) qR ) volumetric flow rate of the recycle stream of a reactorpervaporator loop; qR ) Rqe (L/min) r ) reaction mixture volume-specific reaction rate, defined in eq 6 (mol/{L.min}) r′ ) rate of homogeneous reaction (mol/(L min)) r′′ ) rate of solid-catalyzed reaction (mol/(kg catalyst min)) RJ ) reaction mixture volume-specific rate of formation of J (mol/(L min)) RmJ ) volume-specific rate of removal of component J by separation (mol/(L min)) Sm ) membrane selectivity t ) real time (min) V ) defined in eq A4; V )V/V0 V ) reactor volume (L) V0 ) initial volume of a semi-batch reactor (L) W ) catalyst mass (kg) xJ ) mole fraction of component J XAe ) fractional equilibrium conversion of A in a reaction-only operation XAf ) defined in eq A22 XJ ) fractional conversion of component J (J ) A, B) z ) defined in eq A13 Greek Nomenclature R ) spilt ratio for recycle and effluent streams in a reactorpervaporator loop R1, β1 ) parameters defined in eq 17 β ) parameter defined in eq A6 νJ ) parameter defined in eq 3 F ) density of the reaction mixture (kg/L) σ ) km/kr τ ) reactor space time, defined in eq A10 (min) ΘJ ) for a batch reactor (eq A6), ΘJ ) NJ0/NA0; for a steadystate continuous flow reactor (eq A12), ΘJ ) FJ0/FA0 ζ ) parameter defined in eq A17 Literature Cited (1) Benedict, D. J.; Parulekar, S. J.; Tsai, S. P. Esterification of Lactic Acid and Ethanol With/Without Pervaporation. Ind. Eng. Chem. Res. 2003, 42, 2282-2291. (2) Benedict, D. J. Heterogeneous Reaction Kinetics and PervaporationAided Esterification of Organic Acids Derived from Alternative Feedstocks, M.S. Thesis, Illinois Institute of Technology, Chicago, IL, 1998. (3) Benedict, D. J.; Parulekar, S. J.; Tsai, S.-P. Pervaporation-Assisted Esterification of Lactic and Succinic Acids With Downstream Ester Recovery. J. Membr. Sci. 2006, 281, 435-445. (4) Feng, X.; Huang, R. Y. M. Studies of a Membrane Reactor: Esterification Facilitated by Pervaporation. Chem. Eng. Sci. 1996, 51, 46734679.

8504


(5) Fleming, H. L.; Slater, C. S. In Membrane Handbook; Ho, W. S. W., Sirkar, K. K., Eds.; Van Nostrand Reinhold: New York, 1992; Chapters 7-10, pp 105-159. (6) Huang, R. Y. M.; Rhim, J. W. Separation Characteristics of Pervaporation Membrane Separation Processes. In PerVaporation Membrane Separation Processes; Huang, R. Y. M., Ed.; Elsevier: Amsterdam, The Netherlands, 1991; pp 111-180. (7) Neel, J. Introduction to Pervaporation. In PerVaporation Membrane Separation Processes; Huang, R. Y. M., Ed.; Elsevier: Amsterdam, The Netherlands, 1991; pp 1-109. (8) Neel, J. Pervaporation. In Membrane Separations Technology, Principles and Applications; Noble, R. D., Stern, S. A., Eds.; Elsevier: Amsterdam, The Netherlands, 1995; pp 143-211. (9) Sullivan, D. M.; Bruening, M. L. Ultrathin, Cross-Linked Polyimide Pervaporation Membranes Prepared from Polyelectrolyte Multilayers. J. Membr. Sci. 2005, 248, 161-170. (10) Tuan, V. A.; Li, S.; Falconer, J. L.; Noble, R. D. Separating organics from water by pervaporation with isomorphously-substituted MFI zeolite membranes. J. Membr. Sci. 2002, 196, 111-123. (11) Van, Baelen, D.; Van der Bruggen, B.; Van den Dungen, K.; Degreve, J.; Vandecasteele, C. Pervaporation of Water-Alcohol Mixtures and Acetic Acid-Water Mixtures. Chem. Eng. Sci. 2005, 60, 1583-1590. (12) Schwarz, H. H.; Malsch, G. Polyelectrolyte Membranes for Aromatic-Aliphatic Hydrocarbon Separation by Pervaporation. J. Membr. Sci. 2005, 247, 143-152. (13) Yang, L.; Kang, Y.; Wang, Y.; Xu, L.; Kita, H.; Okamoto, K.-i. Synthesis of Crown Ether-Containing Copolyimides and Their Pervaporation Properties to Benzene/Cyclohexane Mixtures. J. Membr. Sci. 2005, 249, 33-39. (14) Ahn, S. M.; Ha, J. W.; Kim, J. H.; Lee, Y. T.; Lee, S. B. Pervaporation of Fluoroethanol/Water and Methacrylic Acid/Water Mixtures Through PVA Composite Membranes. J. Membr. Sci. 2005, 247, 51-57. (15) Dubey, V.; Pandey, L. K.; Saxena, C. Pervaporative Separation of Ethanol/Water Azeotrope Using a Novel Chitosan-Impregnated Bacterial Cellulose Membrane and Chitosan-Poly(Vinyl Alcohol) Blends. J. Membr. Sci. 2005, 251, 131-136. (16) Gorri, D.; Urtiaga, A.; Ortiz, I. Pervaporative Recovery of Acetic Acid from an Acetylation Industrial Effluent Using Commercial Membranes. Ind. Eng. Chem. Res. 2005, 44, 977-985. (17) Kittur, A. A.; Kulkarni, S. S.; Aralaguppi, M. I.; Kariduraganavar, M. Y. Preparation and Characterization of Novel Pervaporation Membranes for the Separation of Water-Isopropanol Mixtures Using Chitosan and NaY Zeolite. J. Membr. Sci. 2005, 247, 75-86. (18) Mohammadi, T.; Aroujalian, A.; Bakhshi, A. Pervaporation of Dilute Alcoholic Mixtures Using PDMS Membrane. Chem. Eng. Sci. 2005, 60, 1875-1880. (19) Reid, E. E. Esterification. In Unit Processes in Organic Synthesis, 4th Edition; Groggins, P. H., Ed.; McGraw-Hill: New York, 1952; pp 596-642. (20) David, M. O.; Gref, R.; Nguyen, T. Q.; Neel, J. PervaporationEsterification Coupling. I. Basic Kinetic Model. Trans. Inst. Chem. Eng. 1991, 69, 335-340.

(21) David, M. O.; Gref, R.; Nguyen, T. Q.; Neel, J. PervaporationEsterification Coupling. II. Modeling of the Influence of Different Operating Parameters. Trans. Inst. Chem. Eng. 1991, 69, 341-346. (22) Domingues, L.; Recasens, F.; Larrayoz, M. Studies of a Pervaporation Reactor: Kinetics and Equilibrium Shift in Benzyl Alcohol Acetylation. Chem. Eng. Sci. 1999, 54, 1461-1465. (23) Keurentjes, J. T. F.; Janssen, G. H. R.; Gorissen, J. J. The Esterification of Tartaric Acid with Ethanol: Kinetics and Shifting the Equilibrium by Means of Pervaporation. Chem. Eng. Sci. 1994, 49, 46814689. (24) Kita, H.; Sasaki, S.; Tanaka, K.; Okamoto, K. I., Yamamoto, M. Esterification of Carboxylic Acid with Ethanol Accompanied by Pervaporation. Chem. Lett. 1988, 2025-2028. (25) Krupiczka, R.; Koszorz, Z. Activity-Based Model of the Hybrid Process of an Esterification Reaction Coupled with Pervaporation. Sep. Purif. Technol. 1999, 16, 55-59. (26) Kwon, S. J.; Song, K. M.; Hong, W. H.; Rhee, J. S. Removal of Water Product from Lipase-Catalyzed Esterification in Organic Solvent by Pervaporation. Biotechnol. Bioeng. 1995, 46, 393-395. (27) Liu, Q. L.; Chen, H. F. Modeling of Esterification of Acetic Acid with n-Butanol in the Presence of Zr(SO4)2‚4H2O Coupled Pervaporation. J. Membr. Sci. 2002, 196, 171-178. (28) Ni, X.; Xu, Z.; Shi, Y.; Hu, Y. Modified Aromatic Polyimide Membrane Preparation and Pervaporation Results for Esterification System. Water Treat. 1995, 10, 115-120. (29) Nijhuis, H. H.; Kemperman, A.; Derksen, J. T. P.; Cuperus, F. P. Pervaporation Controlled Bio-Catalytic Esterification Reactions. In Proceedings of Sixth International Conference on PerVaporation Processes in the Chemical Industry; Bakish, R., Ed.; Bakish Materials Corp.; Englewood, NJ, 1992; pp 368-379. (30) Okamoto, K. L.; Yamamoto, M.; Otoshi, Y.; Semoto, T.; Yano, M.; Tanaka, K.; Kita, H. 1993, Pervaporation-Aided Esterification of Oleic Acid. J. Chem. Eng. Jpn. 1993, 26, 475-481. (31) Ronnback, R.; Salmi, T.; Vuori, A.; Haario, H.; Lehtonen, J.; Sundqvist, A.; Tirronen, E. Development of a Kinetic Model for the Esterification of Acetic Acid with Methanol in the Presence of a Homogeneous Acid Catalyst. Chem. Eng. Sci. 1997, 52, 3369-3381. (32) Eliceche, A. M.; Davioua, A. M. C.; Hocha, P. M.; Uribeb, I. O. Optimization of Azeotropic Distillation Columns Combined with Pervaporation Membranes. Comput. Chem. Eng. 2002, 26, 563-573. (33) Tang, Y.-T.; Chen, Y.-W.; Huang, H.-P.; Yu, C.-C.; Hung, S.-B.; Lee, M.-J. Design of Reactive Distillations for Acetic Acid Esterification. AIChE J. 2005, 51, 1683-1699. (34) Xuehui, L.; Lefu, W. Kinetic Model for an Esterification Process Coupled by Pervaporation. J. Membr. Sci. 2001, 186, 19-24.

ReceiVed for reView September 1, 2006 ReVised manuscript receiVed August 3, 2007 Accepted August 5, 2007 IE061157O

Analysis of Pervaporation-Aided Esterification of Organic Acids

Recommend Documents