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Ind. Eng. Chem. Res. 2002, 41, 1754-1762
Nonideal Liquid-Phase Intraparticle Transport and Reaction Faisal H. Syed† and Ravindra Datta*,‡ Chemical Market Resources, Inc., Houston, Texas 77058-3320, and Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, MA 0160-22809
The effect of liquid-phase nonideality on intraparticle transport with chemical reaction is explored theoretically and experimentally. Generalized transport equations for the multicomponent diffusion and flow of n species participating in q reactions are developed on the basis of the dusty-fluid model, in which the diffusional driving force of species i is its chemical potential gradient. This is appropriate owing to the nonideality of reaction mixture and is consistent with the thermodynamics of irreversible processes. Nonideality also impels the use of species activities as a measure of composition in the reaction rate expression. The constitutive equations are generally supplemented by n material balances, but because of stoichiometry, only q of these are independent. The n “species” fluxes in the material balance and constitutive equations are, therefore, replaced by an independent set of q “reaction” fluxes, resulting in a simpler and more direct description. This transport and reaction model is applied to the case of liquid-phase synthesis of ethyl tert-butyl ether (ETBE) under isothermal conditions. The model is experimentally validated for a variety of conditions, providing good agreement between theory and experiments without the use of any adjustable parameters. Introduction Although the literature is replete with the treatment of molecular diffusion in multicomponent mixtures,1-4 the role of liquid-phase nonideality in diffusion-reaction systems is not frequently addressed. The transport and reaction in the liquid phase is complicated by thermodynamic nonidealities that affect both the diffusion and reaction rates.5-8 In highly nonideal mixtures, the thermodynamic component of the diffusion coefficient is a strong function of the mixture composition that vanishes in the region of the critical point.9 However, most problems of interest are far from this region, thus requiring transport models to adequately account for nonideality. Furthermore, this also requires that rate expressions be written in terms of species activities, rather than mole fractions or concentrations, to provide consistency with thermodynamics and rate constants that are independent of composition.7,10-15 The generalized Stefan-Maxwell (GSM) formulation is increasingly used to model multicomponent diffusion in nonideal mixtures.4,5,7,16,17 Fick’s law, although substantially simpler, is not accurate for multicomponent systems and is incapable of predicting some aspects, e.g., osmotic and reverse diffusion phenomena.18 The alternate formulation of the Onsager19 relations for multicomponent diffusion is more satisfactory, but its diffusion coefficients are not predictable. For nonideal mixtures, the diffusional driving force, in the absence of forces other than composition gradients, is best described by chemical potential gradients, which is also consistent with irreversible thermodynamics.4,20,21 Although the GSM diffusion coefficients are dependent on the mixture composition and can be estimated, for instance, by using interpolating equations for the binary Vignes equation, they are frequently assumed to be * To whom correspondence should be addressed: rdatta@ wpi.edu † Chemical Market Resources, Inc. ‡ Worcester Polytechnic Institute.
constant and are evaluated at mean or surface conditions.8,17,22 However, as shown here, for many nonideal liquid systems, the composition dependence is significant and, thus, cannot a priori be assumed to be constant.5 Here, we examine the role of nonideality in liquidphase intraparticle diffusion and reaction systems by utilizing the dusty-fluid model3 (DFM), which provides a more complete transport description than GSM and incorporates ordinary diffusion and matrix (Knudsen, for the case of gases) diffusion accounting for friction between species and porous medium, as well as convection. Further, the order of the system is reduced by replacing the n “species” fluxes by q independent “reaction” fluxes in both the material balance and constitutive equations. These generalized equations are applied to the case of the highly nonideal liquid-phase synthesis of ethyl tert-butyl ether (ETBE). ETBE, the ethanol counterpart of the widely used oxygenate methyl tert-butyl ether (MTBE), provides similar air quality benefits but is less water-soluble and thus is currently being investigated as a substitute for MTBE in gasoline blends. Sundmacher et al.23 are the only investigators so far to study intraparticle diffusion effects in ETBE synthesis. However, their analysis is based on the GSM equations. Further, the rate constant and the tortuosity factor in their study were used as adjustable parameters, resulting in the adoption of a low value for the tortuosity factor to provide reasonable agreement with experiments. All parameters utilized in this study, on the other hand, are either independently experimentally determined or theoretically estimated, resulting in a model that is completely free of fitted parameters. Theoretical predictions are compared with experimental results under a variety of conditions and for different particle sizes and are found to provide good agreement. Theory The most general form of the dusty-fluid model for multicomponent mass transport in a porous medium can
10.1021/ie010407s CCC: $22.00 © 2002 American Chemical Society Published on Web 03/09/2002
Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1755
be written as2,24
-
cxiF*i
)
RT
Ni
n
+
e DiM
cxiB0 e ηDiM
Heij ≡ δij
∑ j)1
1 Deij
(xjNi - xiNj) +
cxi∇Tµi
)
RT
e DiM
n
+
∑
xh
cRTxiB0
-
j* i
n
[∇p -
cxjFjR)] ∑R (∑ j)1
(i ) 1, 2, ..., n) (1)
Ni
n
+
e DiM
∑ j)1
δij )
∇p (i ) 1, 2, ..., n) (2)
e ) where c is the total mixture concentration, DiM K0DiM is the effective matrix diffusion coefficient accounting for the friction between species i and the porous matrix, and K0 ) 2a/3τ is the DFM constant for matrix diffusion written in terms of the porosity , tortuosity τ, and mean pore radius a of the porous medium.2 The effective binary diffusion coefficients are defined as Deij ) K1Dij, where Dij represents the binary diffusion coefficients and K1 ) /τ is the DFM structural constant for the molecular diffusion coefficient. In the absence of other driving forces, B0 ) a2/8τ, is the d’Arcy permeability. Summing over all n species, eq 2 yields
∇p ) -
n
∑ W j)1
Nj
e DiM
+
Deij
cRTxiB0 e e ηWDiM DjM
}
(6)
{
0 (j * i) 1 (j ) i)
}
(7)
q
∇ ‚Ni )
∑ νFirF
(i ) 1, 2, ..., n)
(8)
F)1
where the rate of Fth reaction is rF(a1, a2, ..., an). In general, for a catalyst particle, eqs 5 and 8 need to be solved subject to the split boundary conditions
xi(R) ) xis and Ni(0) ) 0 (i ) 1, 2, ..., n) (9)
j* i
RT
{
xi
These constitutive relations are generally supplemented with the n material balance equations involving reactions
(xjNi - xiNj) + Deij
e ηDiM
+
where δij is the Kronecker delta function
1
cxiB0
}
e 2 ηW(DiM )
e h)1D h* i ih
(δij - 1)
where F*i is the generalized driving force including chemical potential gradients as well as other forces, such as electric field, centrifugal forces, etc. The terms on the right-hand side describe matrix (Knudsen) diffusion, Stefan-Maxwell terms, and bulk flow induced by pressure gradients and other driving forces (e.g., electroosmotic flow). Further refinements of DFM include viscous selectivity,25 which is unnecessary here. Because the other driving forces are not of interest here, eq 1 can be rewritten in terms of only the gradient of chemical potential to yield
-
{
1
which presents some numerical difficulty. Further, only q of n relations in eq 8 are independent, the other fluxes being related to these through stoichiometry. Thus, taking a cue from Kaza and Jackson,26 we define q independent fluxes as follows. Integrating eq 8 over a volume V enclosed by surface S of area A within the particle, we obtain
∫
q
(∇ ‚Ni) dV ) V
∫∑ F)1 ( V
q
νFirF) dV )
∑ νiF∫VrF dV
(10)
F)1
Application of the Gauss-Ostrogradskii divergence theorem then yields
∫V(∇ ‚ Ni) dV ) ∫ANi ‚ dS ) IA(Ni ‚ n) dA
(11)
which relates the total divergence of the flux of species i within a volume to the net flow of species i emerging from its surface. From eqs 10 and 11
(3) q
∫ANi ‚ dS ) ∑ νFi∫VrF dV
(12)
F)1
where
W≡1+
B0cRT η
n
∑ i)1
xi
e DiM
Let the net flow associated with reaction F be defined by
(4)
Substitution of eq 3 into eq 2 to eliminate the pressure gradient provides flux relations of the form
-
c RT
∫AΛF ‚ dS ≡ ∫VrF dV
where ΛF is the reaction flux associated with reaction F. Equation 13 is a generalization of the 1-D slab case considered by Kaza and Jackson.26 From eqs 12 and 13
n
xi∇Tµi )
HeijNj ∑ j)1
(i ) 1, 2, ..., n)
(5)
where the effective frictional coefficients Heij are defined as
(13)
q
Ni )
∑ νFiΛF
(14)
F)1
Replacing the species fluxes Ni in eqs 5 and 8 by the independent set of reaction fluxes by using eq 14 results
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Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002
in the following constitutive and material balance equations
-
c RT
q
∑ F)1
xi∇Tµi )
1
e DiF
Table 1. Physical Properties of Amberlyst-15 Catalyst28 average particle diameter, mm surface area, m2/g porosity, % average pore diameter, nm apparent density, g/cm3 packing density, g/cm3 ion-exchange capacity, equiv/g swelling ratio in methanol swelling ratio in ethanol
νFiΛF (i ) 1, 2, ..., n) (15)
and
∇ ‚ ΛF ) rF (F ) 1, 2, ..., q) where
1 e DiF
≡
1 e DiM
n
+
∑ j)1
1
(
Deij
j* i
νFj
)
(16)
cRTxiB0
x j - xi e νFi ηWDiM
n
∑ j)1
1 νjF
e ν DjM iF (17)
which have to be solved subject to the boundary conditions
xi(R) ) xis (i ) 1, 2, ..., n) Λi(0) ) 0 (F ) 1, 2, ..., q)
(18)
Further, if the effect of intraparticle pressure on the gradient of the chemical potential is small, then
h i∇p + ∇T,Pµi ≈ RT ∇ ln ai ) RT ∇ ln(γixi) ∇ T µi ) V (19) where the species activity ai ) γixi, and thus the lefthand side of eq 15 can be rewritten in terms of species mole fractions as
-
c RT
n-1
xi∇Tµi ≈ -cxi ∇ ln(γixi) ) -c
Γij∇Txi ∑ j)1
(20)
where the thermodynamic correction factor Γij accounting for reaction mixture nonideality is defined as4,5
Γij ) δij + xi
∂ ln γi | ∂xj T,P,xk,k*j,n
(21)
The effectiveness factor of a catalyst particle for reaction F is
ηF )
∫A ΛFs ‚ dS p
VprFs
(22)
which for the isotropic case becomes
ηF )
Ap ΛFs Vp rFs
(23)
where Ap/Vpis the particle surface area-to-volume ratio and ΛFs and rFs refer to the surface conditions. Finally, the observed rate of production of species i is q
ri,obs )
∑ ηFνFirFs F)1
(24)
Experimental Section Apparatus. The kinetics were measured in a differential packed-bed reactor setup similar to that of Zhang and Datta.27 The desired amounts of ethanol and
a
Rehfinger and Hoffmann.22
b
0.74 42.5 32 28.8 1.01 0.61 4.3 × 10-3 0.50a 0.12b
This study.
isobutylene were premixed in a stainless steel cylinder and then transferred under pressure to a 1000-mL syringe pump (Isco, model 1000D). The preheater and reactor were of double-pipe configuration and were heated to the desired temperature by circulating water (Fisher Isotemp 9101 water circulator) on the shell side. The reactants were pumped in a pulse-free manner through the preheater before entering the packed-bed reactor. The catalyst in the reactor was held in place with glass-wool plugs. The temperature was monitored with a thermocouple (Omega type K) placed in the middle of the catalyst bed. The system pressure was kept above the mixture bubble point in the range of 200-250 psig [(13.5-16.9) × 105 Pa) using a backpressure regulator (Upchurch Scientific, P738) to ensure liquid-phase operation. Materials. Amberlyst-15 ion-exchange resin catalyst (obtained from Sigma) was treated by washing the resin with ethanol, followed by 1.0 N HNO3 and then ethanol again to remove any excess free acid remaining on the resin beads. After being dried for 12 h in an oven under vacuum at 105 °C, the intact catalyst beads were sieved into four different sizes having average diameters of 0.39, 0.485, 0.915, and 0.986 mm. Crushed catalyst having an average diameter of 0.188 mm was also used in some experiments to confirm intrinsic kinetics in accordance with previous studies on MTBE and ETBE synthesis.11,27 Other physical properties of Amberlyst15 are provided in Table 1.28 Dehydrated 200-proof ethanol was obtained from Pharmco. CP-grade isobutylene was obtained from Specialty Gas Concepts with purity greater than 99.933%. These chemicals were used without any further pretreatment. Procedure. Differential kinetic measurements were conducted by varying the weight hourly space velocity (WHSV), defined as the total mass flow rate per hour divided by the mass of the catalyst in the reactor, to obtain conversions of 5% or less. The observed rate ri,obs could thus be calculated simply from the composition difference between the reactor inlet and outlet. The effluent from the reactor was analyzed via gas chromatography to determine the overall conversion. An online liquid sampling valve (Valco, model CL4WE) and a Perkin-Elmer AutoSystem gas chromatograph were utilized for this purpose. A capillary column (Supleco, SPB-1, 0.25-mm i.d., 1.0-µm film thickness, 60-m length) with a flame ionization detector (FID) was used. The gas chromatograph oven was held at 40 °C, with helium as the carrier gas at a pressure of 50 psig (3.4 × 105 Pa). The FID and injector temperatures were held at 150 °C. The side products [i.e., tert-butyl alcohol (TBA), diisobutene] formed in the reactor were insignificant and were, therefore, neglected from further consideration.
Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1757
7.418 × 1012 mol/(h g) and 60.4 × 103 J/mol, respectively. The ETBE synthesis equilibrium constant is given as34
4060.59 - 2.890 55 ln T T 1.915 44 x 10-2T + 5.285 86 x 10-5T2 5.329 77 x 10-8T3 (27)
lnK ) 10.387 +
The rate expression utilizes the most abundant adsorbed species assumption for ethanol with the adsorption equilibrium constant of ethanol on the basis of liquid-phase activities expressed as10
1 )] [11R000(T1 - 303
KA ) 27 exp
(28)
The catalyst mass-based rate expression in eq 26 can be written in terms of unit volume, as needed for the material balance equation, using Figure 1. Species activities as a function of mole fraction for binary ethanol-isobutylene system calculated using UNIFAC method at T ) 343 K.
Results and Discussion Application of Model to ETBE Synthesis. ETBE is industrially synthesized in the liquid phase from ethanol and isobutylene over an acidic ion-exchange resin catalyst at temperatures ranging from 313 to 353 K.11,29
The interactions between the strongly polar alcohol and the nonpolar olefin result in an ETBE reaction mixture that is highly nonideal, as shown in Figure 1, thus requiring the use of species activities in the rate expression.11,15,30-32 The most common catalyst used for fuel ether production is Amberlyst-15. It is a solid acid ion-exchange resin catalyst based on a styrene divinylbenzene backbone with bound sulfonic acid groups. Its macroreticular structure allows for a high surface area and easy accessibility for reactants. However, at higher temperatures, the etherification reaction can become diffusion-limited.11,22,23,27,33 The kinetics of the ETBE reaction system have been accurately determined by Jensen11 using the Langmuir-Hinshelwood-Hougen-Watson (LHHW) formalism in which the transition-state theory was applied to the elementary steps of adsorption, surface reaction, and desorption involved in the overall catalytic reaction, resulting in the following rate expression
r′ )
ksiaA2aB (1 + KAaA)
3
(
1-
aC aAaBK
)
(26)
where ai ) xiγi and the activity coefficients are calculated using the UNIFAC method, which has been found to provide reliable estimates of activity coefficients in such mixtures.10,11 The rate constant, ksi, in eq 26 is written in the Arrhenius form, with the preexponential factor and effective activation energy having values of
r)
(
)
1- F r' 1 + SA C
(29)
where the porosity , catalyst density FC, and swelling ratio in ethanol SA are provided in Table 1. Amberlyst15 swells to varying degrees depending on the nature of the solvent. Its swelling in ethanol was experimentally determined by measuring the catalyst particle size under dry and saturated conditions and calculated to be approximately 12%, significantly lower than that of methanol (50%) reported by Rehfinger and Hoffmann.33 Sundmacher et al.23 erroneously assumed the swelling ratio of ethanol to be equal to that of methanol, thus potentially affecting the volumetric rate by 25%. For the case of a single reaction in a spherical particle, eq 15 results in
-
c dµi Rp x ) νΛ RT i dy De i
(30)
i
i.e., the Fickian form, along with
1 d 2 (y Λ) ) Rpr y2 dy
(31)
The resulting species conservation relations for diffusion and reaction in a spherical pellet of radius Rp were solved as an initial-value problem by guessing the reaction flux at the surface Λs and integrating using a fourth-order Runge-Kutta routine.35 The value of Λs was adjusted to meet the boundary condition in eq 18. Numerical estimates were used for the thermodynamic correction factor in accordance with the UNIFAC method. Estimation of Stefan-Maxwell Diffusion Coefficients. The Stefan-Maxwell diffusion coefficients Dij for multicomponent liquid mixtures were calculated using the generalized Vignes equation4 n
Dij ) (Doij)xj(Doji)xi
(Dij,x f1)x ∏ k)1 k
k
(32)
k*i,j
where Doij represents the diffusion coefficients of species i infinitely diluted in species j and Dij,xkf1 represents the limiting values of the Stefan-Maxwell diffusivities
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Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002
Figure 2. Simulated concentration profile for stoichiometric feed ratio, ΘA ) 1, T ) 343 K, and dp ) 0.915 mm.
in a concentrated solution of k estimated using the model proposed by Kooijman and Taylor36
Dij,xkf1 ) (DoikDoki)1/2
Figure 3. Simulated rate and affinity of reaction profiles for stoichiometric feed ratio, ΘA ) 1, T ) 343 K, and dp ) 0.915 mm. Normalized to surface values of rs ) 244 mol/(s m3) and As/RT ) 8.1.
The affinity of the reaction, defined as
(33)
n
AF ≡ Doij
The infinite-dilution binary diffusion coefficients were calculated using the Tyn and Calus method for organic liquids.37 The effective binary diffusion coefficients were calculated by multiplying Dij by K1 ) /τ ) q, where the porosity, given in Table 1, is equal to 0.32 and the Bruggeman exponent38,39 q ) 1.5. In accordance with previous studies on Amberlyst-15, the matrix diffusion coefficients are negligible because of the macroporous structure of the resin.23,33 However, unlike previous studies on MTBE and ETBE, the diffusion coefficients here were allowed to vary with changing mixture composition across the catalyst diameter. Simulated Intraparticle Profiles. Simulated concentration and rate profiles for a stoichiometric feed mixture (molar ratio of ethanol to isobutylene ΘA ) 1) at a temperature of 343 K and a particle diameter of 0.915 mm are shown in Figures 2 and 3, respectively. Figure 2 shows that the mole fractions of (A) ethanol and (B) isobutylene increase with increasing radial distance from the center, while that of (D) ETBE decreases, corresponding to bulk values at the surface (y ) 1). It is apparent that the intraparticle transports of ethanol and isobutylene occur at markedly different rates (xA ≈ 0.025 and xB ≈ 0.31 at y ) 0), with ethanol being the more transport limited species. The rate of reaction (Figure 3) is highest at the surface of the pellet and decreases to zero toward the center, thus clearly indicating that the reaction is diffusion-limited under these conditions. It is interesting to observe, however, that the rate does not vary with radius in the expected manner (Figure 3). This behavior is a direct result of thermodynamic nonideality and the nonlinear rate expression. Because of the inverse dependence of the rate on ethanol and the steeper decline of the ethanol mole fraction in the particle interior, the rate of change of the reaction rate is nonmonotonic, initially declining and then increasing as equilibrium is approached at the particle center.
∑ i)1
n
νFiµi ) RT(ln KF -
νFi ln ai) ∑ i)1
(34)
where KF is the equilibrium constant for the Fth reaction (eq 27 for ETBE synthesis), is also plotted in Figure 3; it represents how far the reaction is from equilibrium and is the thermodynamic driving force of the reaction. As can be seen, the affinity is highest at the surface, where the reverse reaction is essentially negligible, and declines to zero at the center, indicating that the center of the catalyst is at equilibrium, even though the concentration of the reactants are finite. It is interesting to write alternate forms of the constitutive and material equations (eqs 15 and 16) in terms of the affinity of the reaction by defining RF ≡ AF/RT. This results in
∇RF )
1
q
∑ c F)1
() 1
DeF
ΛF
(35)
and
r F(1 - e-Rr) (F ) 1, 2, ..., q) ∇ ‚ ΛF ) b
(36)
where
1 DeF
n
≡
∑ i)1
()
νiF2 1 xi De iF
(37)
and b rF is the rate of the forward Fth reaction. Thus, for the case of ETBE synthesis (eq 26)
b rF )
ksiaA2aB (1 + KAaA)3
(38)
Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1759
Figure 4. Simulated concentration and rate profile for substoichiometric feed ratio, ΘA ) 0.25, T ) 343 K, and dp ) 0.915 mm.
Figure 5. Simulated rate and affinity of reaction profiles for substoichiometric feed ratio, ΘA ) 0.25, T ) 343 K, and dp ) 0.915 mm. Normalized to surface values of rs ) 296 mol/(s m3) and As/ RT ) 8.4.
and
e-RF )
1 aC K aAaB
(39)
Because of the reversibility of the reaction, the shellcore model is rather appropriate in describing the intraparticle behavior, in which the main etherification reaction occurs in the outer shell, with the core of the catalyst being at equilibrium.17,33 The higher the diffusion limitation, the larger the volume of the unutilized core. For the case considered in Figures 2 and 3, however, the core volume VC f 0. For substoichiometric feed conditions, e.g., ΘA ) 0.25, ethanol is even more limiting. However, the concentrations of all species (shown in Figure 4) remain constant within the core to about y ) 0.6. Thereafter, ethanol and isobutylene increase while ETBE decreases toward the catalyst surface. However, the rate of reaction (Figure 5) increases toward the center before reaching a maximum at y ) 0.8 and then precipitously declines
Figure 6. Variation of effective diffusivity coefficient of ethanol across catalyst radius, ΘA ) 1, T ) 343 K, and dp ) 0.915 mm.
to zero at y ≈ 0.6. The initial increase in the rate of reaction is due to the inverse dependence of ethanol in kinetic expression 26. The rate then declines by virtue of ethanol exhaustion and approach to equilibrium. It is evident that such behavior can result in a value of η > 1, despite diffusional limitations, for particle sizes smaller than that shown in Figure 5, even though there is a substantial core (VC/V ≈ 0.21) where there is virtually no reaction. In comparison to stoichiometric conditions (Figure 3), the rate and affinity of the reaction for substoichiometric conditions become zero much before the catalyst center, indicating a substantial core volume (VC/V ≈ 0.21). The effective diffusivity DeA for ethanol as a function of catalyst radial distance is shown in Figure 6 and has an average value of 1.75 × 10-9 m2/s in Amberlyst-15 at 343 K, compared to 3.5 × 10-9 m2/s for methanol as reported by Rehfinger and Hoffmann.22 For relatively dilute systems, the variation with changing mixture composition is less significant, and thus, the mass transport coefficients can be assumed to be constant.5 However, for concentrated systems, model predictions can be improved by accounting for their composition dependence.23 Experimental Validation. Experimental η values as a function of temperature for a stoichiometric feed mixture (ΘA ) 1) and a particle diameter of 0.915 mm are shown in Figure 7. For T e 333 K, η is essentially unity, indicating the absence of diffusion limitations. As T increases beyond 333 K, η decreases quickly to reach roughly 0.54 at T ) 353 K. Thus, the reaction becomes substantially diffusion-limited for T > 333 K. The model predictions for four different cases are also shown in Figure 7 for the purpose of comparison. Case 1 assumes constant effective diffusion coefficients evaluated at the surface conditions for an ideal mixture; case 2 assumes constant effective diffusion coefficients evaluated at the surface conditions for a nonideal mixture; case 3 assumes composition-dependent effective diffusion coefficients and an ideal mixture; and case 4 is general, with composition-dependent effective diffusion coefficients as well as a nonideal reaction mixture. It is evident that accounting for nonideality in the transport and reaction terms is important for increasing the
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Figure 7. Predicted and experimental effectiveness factor as a function of temperature for stoichiometric feed ratio, ΘA ) 1 and dp ) 0.915 mm.
Figure 9. Effectiveness factor η versus Thiele modulus φ for various feed compositions and particle diameters.
f 0) or ethanol (xAs f 1), η f 0. The model provides good agreement and is capable of predicting the maximum observed in the experiments, although the experimental peak is sharper. The model also does not predict η > 1 under these conditions. Under industrial conditions, a slight excess of ethanol is used (i.e., ΘA ) 1.051.2), corresponding to η ≈ 0.75 at 343 K. However, typically, olefin feedstock can contain up to 50 wt % inerts, which would decrease the rate of reaction and result in a reduction of the diffusion limitations. Intraparticle Diffusional Resistances. The importance of diffusion and reaction in a catalyst pellet is typically characterized by the generalized Thiele modulus, φ, defined as5
φ)
Figure 8. Predicted and experimental effectiveness factor as a function of ethanol mole fraction in feed xAs at T ) 343 K and dp ) 0.915 mm.
accuracy of predictions. Although excluding the composition dependence of the effective diffusion coefficients provides reasonable predictions (case 2), better agreement is obtained when composition dependence as well as thermodynamic nonideality are included. Vanni et al.8 arrived at similar conclusions for interfacial mass transfer with chemical reaction in liquid-liquid systems. Predicted η values as a function of the ethanol mole fraction in the feed, xAs, for a particle diameter of 0.915 mm and a temperature of 343 K are shown in Figure 8, along with experimental data. Because of the nonlinear nature of the kinetic expression (eq 26), η goes through a maximum (xAs ) 0.25) for intermediate values, corresponding to the concentration and rate profiles depicted in Figures 4 and 5, respectively. Thus, for substoichiometric conditions, the rate of reaction inside the pellet can exceed that at the surface, or in other words, η can be greater than unity. For conditions corresponding to a large excess of either isobutylene (xAs
( )
Vp rs Ap cDe
1/2
(40)
The effective diffusivity De in eq 40 was defined on the basis of the alcohol species and set equal to DeA. A similar approach was taken by Rehfinger and Hoffmann,33 Berg and Harris,5 and Zhang and Datta27 for MTBE synthesis. Figure 9 shows the variation of the effectiveness factor η with the Thiele modulus φ for three different feed compositions. In general, for a given feed composition, φ is primarily a function of the catalyst radius. Thus, for low catalyst particle radii, the rate is kinetically controlled, and η approaches unity (i.e., diffusion-free regime). For higher values, η decreases with increasing φ, approaching the asymptotic solution of 1/φa, where40
φa )
Vp rs [ Ap x2
∫xx
As
A,eq
cDeAr dxA]-1/2
(41)
As discussed above, under certain conditions, η can exceed unity because of the inverse dependence of ethanol in the rate expression. Furthermore, the nonlinear nature of the LHHW mechanism can conceivably give rise to steady-state multiplicity,5 although this was not observed under the conditions investigated. Conclusions Generalized equations for multicomponent diffusion and reaction for a nonideal liquid mixture in a porous
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catalyst are derived using the dusty-fluid model. The specific case of nonideal liquid-phase ETBE synthesis over Amberlyst-15 is considered in detail. The liquidphase nonideality requires the use of species activities to correctly describe the kinetic behavior, which is furthermore consistent with reaction thermodynamics. The nonideality also calls for the use of chemical potential gradients in the diffusion model. Replacing the n species fluxes by q independent reaction fluxes in both the material balances and the constitutive equations reduces the order of the system. The dusty-fluid model, along with the LHHW rate expression, provides reasonable agreement between theory and experiments for the case of ETBE synthesis. Because of the inverse dependence of the ethanol species, effectiveness factors greater than unity are theoretically and experimentally realized. Nomenclature Ai ) preexponential factor of reaction i, mol/(h g) AF ) affinity of reaction F aj ) activity of species j ≡ γjxj B0 ) d’Arcy permeability Dij ) binary diffusion coefficient, m2/s Deij ) effective binary diffusion coefficient, m2/s DiM ) matrix diffusion coefficient, m2/s e DiM ) effective matrix diffusion coefficient, m2/s e DiF ) effective diffusion coefficent of species i in reaction F, m2/s Ei ) activation energy of reaction i, kJ/mol Heij ) effective frictional coefficient, s/m2 ksi ) effective rate constant, mol/(h g) KiF ) thermodynamic equilibrium constant of reaction F KA ) ethanol adsorption equilibrium constant K0 ) DFM constant for matrix diffusion Ni ) molar flux of species i, mol/(s m2) n ) total number of chemical species q ) total number of chemical reactions R ) gas constant, 8.3143 J/(mol K) rF ) rate of reaction F, mol/(s m3) r′F ) rate of reaction F, mol/(s gcat) rFs ) rate of reaction F at surface conditions, mol/(s m3) T ) temperature, K V h i ) partial molar volume of species i, m3/mol xj ) mole fraction of species j y ) dimensionless radial distance Greek Letters γj ) activity coefficient of species j ∇Tµi ) chemical potential gradient of species i ΛF ) Fth reaction flux, mol/(s m2) Γij ) thermodynamic correction factor Θj ) molar feed ratio of species j with respect to isobutylene (B), Njo/NBo φ ) Thiele modulus νFi ) stoichiometric coefficient of species i in reaction F ) porosity of catalyst particle δij ) Kronecker delta function ηF ) effectiveness factor for reaction F FC ) catalyst particle density, g/cm3 τ ) tortuosity of catalyst particle Subscripts A ) alcohol B ) olefin D ) ether j ) species j
Abbreviations DFM ) dusty-fluid model EtOH ) ethanol ETBE ) ethyl tert-butyl ether LHHW ) Langmuir-Hinshelwood-Hougen-Watson MEOH ) methanol MTBE ) methyl tert-butyl ether WHSV ) weight hourly space velocity ) total hourly mass feed flow rate divided by the mass of the catalyst, h-1
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Received for review May 4, 2001 Revised manuscript received August 31, 2001 Accepted September 5, 2001 IE010407S
