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Catalytic Distillation: A Three-Phase Nonequilibrium Model for the Simulation of the Aldol Condensation of Acetone Yuxiang Zheng, Flora T. T. Ng,* and Garry L. Rempel Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
A nonequilibrium model (MECRES equations) for the simulation of catalytic distillation (CD) processes developed by Huang et al. (Chem. Eng. Sci. 1998, 53, 3489; 2000, 55, 5919) was improved so as to provide a more generalized three-phase model for the simulation and optimal design of a CD process. Instead of using the overall mass transfer coefficients for the reaction zone determined for our column, the vapor and liquid mass transfer equations in the MECRES model were modified to take into account the effect of multicomponent mass transfer according to multicomponent mass transfer theory. In addition, temperature gradients among the vapor, liquid, and solid (catalyst) phases were considered. The heat transfer rates in the heat balances were calculated according to multicomponent heat transfer theory. The CD process for the aldol condensation of acetone to diacetone alcohol was simulated using this improved model. The model predictions for the product yield and selectivity are in excellent agreement with experimental CD data. The simulation profiles of the temperature along the column and the composition in the reboiler are in better agreement with the experimental data than those simulated by the MECRES model. This improved model is more generalized and will be used to simulate the new CD process for the oligomerization of butenes developed in our laboratory. 1. Introduction A unit operation combining chemical reaction and separation processes into a single column is known as reactive distillation (RD) or catalytic distillation (CD). In this paper, CD is defined as a process in which a heterogeneous catalyst is used, whereas a homogeneous catalyst is used in a RD process. The distinction between RD and CD is important, especially in a comparison of mathematical models, because the models for CD are more complex because of the existence of the solid catalyst phase. Many advantages are associated with the combination of reaction and separation in a single unit operation. The advantages include enhanced product yields and selectivities, significant energy savings for exothermic processes because the reaction heat is used in situ to provide the energy to vaporize the liquid for distillation, and reductions of the capital and operating costs through the combination of two unit operations into one unit operation. In addition, the lifetime of the heterogeneous catalyst is enhanced by the reduction of hot spots, fouling, and coke deposition. In summary, CD is uniquely poised as a tool for the development of new “green” processes for the 21st century. The CD technology has attracted attention from both academia and chemical industries because of the successful production of methyl tert-butyl ether (MTBE), which was used extensively as an octane enhancer. It should be noted that MTBE was recently found in groundwaters and that its usage is being phased out in the U.S. However, there are many more possible applications of CD.1 Indeed, the total numbers of patents and publications on new process development and novel design of CD packings2-4 are increasing very rapidly. It should be noted that theoretical models, which provide profiles of the vapor and liquid compositions and * Author to whom correspondence should be addressed. E-mail address:
[email protected].
temperature, are very important for the design and optimization of new processes. Most models, however, are extensions from the equilibrium model (EQ) based on the MESH equations5,6 and the nonequilibrium model (NEQ)7-9 for conventional distillation operations. Because the column rarely operates at equilibrium in an actual process, the concept of stage efficiency is generally introduced to correct the difference between the equilibrium and real stages for trays. For a packed column, the equilibrium model requires the value of the HETP (height equivalent theoretical plate). In reality, however, the stage efficiency and HETP can not be predicted for multicomponent mixtures.10 In addition, a CD process is more complex than a RD process and should be described by a three-phase model. The most recent and comprehensive review of the modeling of reactive/catalytic distillation is by Taylor and Krishna.11 Zheng and Xu7 have developed a NEQ model to simulate CD processes for the synthesis of MTBE from methanol and mixed C4 species catalyzed by macroporous cation-exchange resins. However, a quasi-homogeneous model was used to treat the heterogeneous reaction because of the inherent complexities of CD processes. Sundmacher and Hoffmann12 presented a detailed NEQ model for CD processes to simulate the production of MTBE, but they assumed that the vapor and liquid streams are in thermal equilibrium with the catalyst, which implies that heat transfer among the three phases is ignored. In addition, although microkinetics was used to model the reaction, a catalyst effectiveness factor was employed to take into account the mass transfer resistances inside the catalyst. However, the mass transfer between the liquid phase and the external catalyst surface was ignored. More recently, Higler et al.13 used the dusty fluid model to take into account mass transport inside the catalyst. However, the determination of catalyst tortuosity in the dusty model is very difficult, and no reliable method can be used to deter-
10.1021/ie001104l CCC: $20.00 © 2001 American Chemical Society Published on Web 09/28/2001
Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5343
mine the diffusion coefficients and thermodynamic behavior inside a catalyst. The present study is focused on improving the MECRES model (including mass, energy, and component balances; rate expressions; and equilibrium and summation equations) that was developed by Huang et al.14,15 The effect of mass transfer and kinetics on the product yield and selectivity was discussed.15 The MECRES model was validated by comparison with experimental data obtained for the CD process of the aldol condensation of acetone (Ac). Compared with the conventional equilibrium-based model, the main feature of the MECRES equations is that the reaction rate is equated to the mass transfer rate between the solid catalyst and the bulk liquid, and that the mass transfer rates between the vapor and liquid phases are considered for both the reactive and nonreactive sections in modeling the CD process. Because the actual rates for transport and reaction are used, the MECRES model does not require the use of stage efficiency and HETP. Because of the complexity of the mass transfer superimposed by the reaction kinetics in a CD process, the effect of multicomponent mass and energy transfer in the rate expressions was ignored in the previous MECRES model, and the empirical overall vapor-liquid and solid-liquid mass transfer correlations determined in our laboratory for our CD column16 were used. Thus, the previous MECRES model is useful only when the empirical mass transfer coefficient correlations associated with the specific process are available. In addition, it was assumed in the previous MECRES model that the liquid phase is at its bubble point and that the vapor phase at its dew point. This assumption is reasonable only for processes where the heat of the reaction is relatively small as is in the case of the aldol condensation of Ac. The heats for the formation of diacetone alcohol (DAA) and mesityl oxide (MO) were found to be -27 kJ mol-1 and + 25 kJ mol-1 respectively.17 Hence, the MECRES model only deals with nonequilibrium behavior with respect to the mass transfer processes. However, for a highly exothermic reaction process, the liquid- and vapor-phase temperatures have to be determined from the heat transfer rates within the two phases. Our newly improved model takes into account the heat transfer between the two phases and is more generalized. Figure 1 is a schematic diagram of the CD column used for the aldol condensation of Ac to produce DAA using Amberlite IRA-900 anion-exchange resin (Aldrich Chemical Co., Milwaukee, WI) as a catalyst.2 This is a reversible reaction. In addition, DAA undergoes dehydration to produce MO and water. A simplified reaction scheme can be represented by the equation k1
k2
2Ac y\ z DAA 98 MO + H2O k -1
(1)
The experiments were carried out at atmospheric pressure in a 1-in. packed column under total reflux. The Amberlite IRA-900 anion-exchange resin catalyst was placed inside a fiberglass bag that was wrapped with demister wire and arranged in the reaction zone of the column. This method of packing catalyst for a CD column is similar to that developed by Smith.18 It provides sufficient voids for vapor flow so that the pressure difference is minimized. A drawback of this packing method is that the influence of diffusion through the catalyst bag cannot be avoided.2,19 The packing used
Figure 1. Flowchart of a catalytic distillation process.
Figure 2. Three-phase model of a nonequilibrium section.
in the rectifying zone and stripping zone was 1/4-in. Intalox saddles. The packing height was 18 ft including the rectifying zone, the reaction zone, and the stripping zone. 2. A Nonequilibrium Three-Phase Model To provide a more generalized model for the optimal design of a CD process, we have now developed an improved model that accounts for the nonequilibrium behavior of both the mass and heat transfer processes. A physical model of our three-phase nonequilibrium section is shown in Figure 2. A generalized mass transport process for the reactants in the reaction zone of a CD column includes the following steps: (1) mass transfer from the vapor phase to the vapor-liquid interface, (2) mass transfer from the vapor-liquid
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interface to the liquid phase, (3) mass transfer from the liquid phase to the liquid-solid (catalyst) interface, (4) diffusion into the pores of the catalyst particle, (5) adsorption of the reactive agents on the surface of the catalyst, and (6) chemical reaction. For the current model, steps 4-6 are included in the macrokinetics. The processes for desorption of the products are repeated in the reverse sequence except that step 5 is replaced by desorption of products and step 4 is replaced by diffusion of the products out of the catalyst particle. The heat transfer processes are similar to those described for mass transfer in steps 1-3. Mass and heat are exchanged across the vapor-liquid and liquid-solid interface. The column is divided into a number of sections. The liquid and vapor phases in the condenser and reboiler are considered to be at equilibrium as no reactions occur in these sections and it is generally accepted that the liquid and vapor phases in these sections are in equilibrium in conventional distillation models. Sections are numbered starting at the overhead with a total condenser and ending with the reboiler. The pressure along the column is taken to be constant because the pressure drop in our experiment is very small (less than 0.5 psig). In addition, assuming that the operation reaches the steady state, the interface of vapor-liquid is in equilibrium for the vapor and liquid phases and the vapor and liquid bulks on each side of it are mixed perfectly so that mixing effects can be neglected. The mathematical model for the newly developed three-phase nonequilibrium section shown in Figure 2 is described below. 2.1. Mass and Heat Balances.
, respectively, where k is the multicomponent mass transfer coefficient (kmol m-2 s-1); x and y are liquid and vapor compositions, respectively; xI and yI are liquid and vapor compositions at the vapor-liquid interface, respectively; a is the effective interfacial area (m2 section-1); and the Nij is the vapor-liquid mass transfer rate for component i (kmol s-1). In addition, the vapor and liquid mass transfer rates through the vapor-liquid interface are equated by
NVij ) NLij
The multicomponent mass transfer coefficients kik are computed using the generalized Maxwell-Stefan equations for multicomponent transport in a film.20 According to multicomponent mass transfer theory, kik can be derived directly from the binary mass transfer coefficients κik as follows
vapor phase
(1 +
- vi,j+1 -
fVij
+
NVij
)0
yi
BVii )
[kLik] ) [BLik]-1[Γik]
(10)
BLii )
L L L S (1 + SLj )LjHLj - Lj-1Hj-1 - FLj HLF j - ej + Qj + ej ) 0 (5)
c-1
)
∑ k)1 c-1
NLij )
c
kVikjaj(ykj
-
yIkj)
+ yij
NVij ∑ k)1
(
1 1 κVik κVik
kLikjaj(xIkj - xkj) + xij ∑ NLij ∑ k)1 k)1
)
i ) 1, 2, ..., c - 1
(11)
i * k ) 1, 2, ..., c - 1 (12)
∂ ln γi i, k ) 1, 2, ..., c - 1 ∂xk c
+
κLik
∑
k)1 k*i
(
xk κLik
1 1 - L L κik κic
)
i ) 1, 2, ..., c - 1
(13)
(14)
i * k ) 1, 2, ..., c - 1 (15)
The binary mass transfer coefficients for a packed column have been developed by Onda et al.21 For the rectifying and stripping sections, the vapor-film mass transfer coefficients are
( )
κVik ) R
WV atµVm
0.7
(ScVik)1/3(atdp)-2
( ) atDVikP RgTV
(16)
where R is 2.0 for our 1/4-in. Intalox saddles packings.21 The liquid-film mass transfer coefficients are calculated by
κLik (7)
xi
BLik ) -xi
(6)
c
κVik
and
(4)
In these equations, S is the ratio of the side product withdrawal rate to the internal flow rate in the column; v and V are the vapor-component and the vapor flow rates, respectively; N and e are the vapor-liquid or liquid-solid mass and heat transfer rates, respectively; f and F are the feed-component and feed flow rates, respectively; H is the enthalpy; Q is the heat duty; and l and L are the liquid-component and liquid flow rates, respectively. 2.2. Rate Equations. Vapor-Liquid Mass Transfer Rates. Accounting for the effects of multicomponent mass transfer, the mass transfer rates for the vapor and liquid phases are given by
∑ k)1 k*i
Γik ) δik + xi
(1 + SLj )lij - li,j-1 - fLij - NLij + NSij ) 0
yk
c
+
κVic
BVik ) -yi
liquid phase
NVij
(9)
where
(2)
V V V - FVj HVF (1 + SVj )VjHVj - Vj+1Hj+1 j + ej + Q j ) 0 (3)
[kVik] ) [BVik]-1 liquid phase
vapor phase SVj )vij
(8)
( )
WL ) 0.0051 awµLm
2/3
(ScLik)-0.5(atdp)0.4
( ) gµLm FLm
1/3
FLm (17)
where aw is the wetted area of packing and can be
Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5345
estimated by
{
[ ( )( )
WL aw ) at 1 - exp -1.45 atµLm
0.1
Liquid-Solid Mass Transfer and Reaction Rates. Mass transfer rates for the liquid phase at the liquidsolid interface can be obtained from
-0.05
at(WL)2 g(FLm)2
( ) ( ) ]} L 2
(W )
0.2
atσmFLm
σm σc
c-1
NSij )
-0.75
(18)
In addition, the effective interfacial area is evaluated by the empirical correlation developed by Bravo and Fair22 as
( )(
a ) 0.498at
)
σm0.5
6WVµLmWL
Z0.4
atµVm FLmσmg
0.392
(19)
The binary mass transfer coefficients for the reactive section packed with catalyst inside fiberglass bags developed by Zheng and Xu23 are used for the calculations. The vapor-film mass transfer coefficients are given by
κVika ) 1.072 × 10-3
( ) ( )
atDVik 4WV dpRgTV atµVm
0.92
4WL atµLm
0.24
(ScVik)0.5 (20)
and the liquid-film mass transfer coefficients are obtained from
( )
atDLik 4WL κLika ) 0.149 d p a µL t m
0.3
(ScLik)0.5
(21)
Vapor-Liquid Heat Transfer Rates. The thermal equilibrium assumption that the liquid phase is at its bubble point and the vapor phase is at its dew point in the previous MECRES model is abandoned. The temperature gradients in the vapor and liquid phases are considered and determined by multicomponent heat transfer theory.20 The heat transfer rates for vapor and liquid phases are provided, respectively, by
Vj
c
∑
(TVj - TIj ) + NVkj HVkj (22) eVj ) hVj aj V k)1 exp j - 1 c
eLj ) hLj aj(TIj - TLj ) +
NLkj HLkj ∑ k)1
(24)
(25)
for the vapor phase and
hL ) kLav CLpm(LeL)1/2 for the liquid phase.
∑ NSij
(27)
k)1
where kS is the liquid-solid mass transfer coefficient. The generalized Maxwell-Stefan equations are also used for the description of mass transfer at the liquidsolid interface. The binary mass transfer coefficient has to be calculated from an appropriate correlation for liquid-solid mass transfer. For our catalyst packing, the correlations developed by Zheng and Xu23 are employed. These are
κSika
( ) ( )
atWL 4WV ) 0.586 L F atµVm
-0.27
4WL atµLm
-0.28
(ScLik)-2/3
(28)
We used experimental data for the aldol condensation of Ac as an example to validate our model. The reaction kinetics model determined by Podrebarac et al.25 is
RDAA ) k1CAc2 - k-1CDAA
(29)
RMO ) RH2O ) k2CDAA
(30)
where R is the reaction rate, k is the rate constant, and C is the concentration. We will use concentrations in the rate equations as reported25 rather than activities as Ac and DAA form an ideal solution26 and their activity coefficients were calculated to be 1.00 and 0.99, respectively. To describe the reaction rate in a CD column, both the kinetic rate and the liquid-solid mass transfer rate have to be considered. In this model, the mass transfer rate and the reaction rate through the liquid-solid interface have to be continuous and are assumed to be equal
NSij ) νiRijGj
(31)
where ν is the stoichiometric coefficient and G is the weight of catalyst in section j. Liquid-Solid Heat Transfer Rates. The heat transfer rate through the liquid-solid interface can be calculated from c
The heat transfer coefficients hj are estimated on the basis of the Chilton-Colburn analogy,24 namely, JH ) JD. Therefore
hV ) kVav CVpm(LeV)2/3
k)1
(23)
In addition, the vapor and liquid heat transfer rates through the vapor-liquid interface are equated by
eVj ) eLj
∑
c
kSikjaj(xkj - xSkj) + xij
(26)
eSj ) hSj aj(TLj - TSj ) +
NSkj HLkj ∑ k)1
(32)
In addition, the energy flux across the liquid film around the catalyst is assumed to be equal to the reaction heat
eSj ) Qrj
(33)
2.3. Equilibrium Relations. At steady state, equilibrium between the vapor phase and the liquid phase is assumed to exist only within the vapor-liquid interface. Thus
KIij xIij - yIij ) 0
(34)
where K is vapor-liquid equilibrium ratio, which can
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Table 1. Structural Size and Area Parameters Calculated for the Modified UNIQUAC Model parameter
acetone
DAA
MO
water
r q q′
2.57 2.34 2.34
4.06 3.22 1.78
4.36 3.86 3.86
0.92 1.40 1.00
be calculated by
KIij
foijγij ) ψijPj
(35)
where the activity coefficient γ is computed by the modified UNIQUAC method and the vapor fugacity coefficient is calculated by the virial equation.27 2.4. Summation Equations. The mole fractions of the components in the vapor, liquid, and solid phases as well as at the vapor-liquid interface are given by the following summation equations c
yij ) 1 ∑ i)1
(36)
c
xij ) 1 ∑ i)1
(37)
(38)
c
xIij ) 1 ∑ i)1
(39)
c
xSij ) 1 ∑ i)1
Table 2. Interaction Parameters Calculated for the Modified UNIQUAC Model A12 A21 temp range component 1 component 2 (kJ/kmol) (kJ/kmol) (K)
c
yIij ) 1 ∑ i)1
Figure 3. Profiles of liquid composition along the column (catalyst, 4 bags; reflux, 22.9 g/min; feed, 152 mL/h).
(40)
The above equations, which describe the mass and heat balances, rate equations, phase equilibrium relations at the vapor-liquid interface and summation equations for the CD processes, are solved simultaneously by the Newton-Raphson method. Zheng and Xu’s algorithm7 is adopted to obtain an accurate solution for the CD column. The number of model sections in the CD column was enhanced gradually in our computations until no significant change in the simulation results was found with a further increase in the number of model sections. 3. Simulation Results 3.1. Simulated Profiles for Temperature, Concentration, and Vapor-Liquid Mass Transfer Rate. Using the model described above, a simulation of the aldol condensation of Ac was implemented using the kinetic data determined by Podrebarac et al.25 In the calculation of the thermophysical constants, the commonly used correlations and equations recommended by Reid et al.28 and Danbert and Danner29 were employed. The UNIQUAC model was used to describe the liquidphase nonideality. The parameters for the UNIQUAC model listed in Tables 1 and 2 were calculated according to the UNIFAC method.27 Our computations showed that a minimum of 18 model sections were required so that no significant changes would be observed in the simulation results. Therefore, the column was divided
acetone acetone acetone DAA DAA MO
DAA MO water MO water water
-107.44 471.81 386.79 348.69 1103.36 1920.78
193.89 -349.46 2859.26 -326.61 4287.79 2935.00
320-400 320-400 320-400 320-400 320-400 320-400
into 18 nonequilibrium sections, with 3 sections in the rectifying zone, 4 sections for the reaction zone, and 11 sections for the stripping zone in our calculations. The total condenser and the reboiler are considered as equilibrium sections. Using our column geometry, i.e., a total condenser, 18 ft packings, a reboiler, and a feed inlet at 10 ft, combined with total reflux, detailed information on the concentration, temperature, and mass transfer inside the CD column was obtained. Figure 3 shows the liquid composition profiles for Ac, DAA, MO, and water. Ac is concentrated at the top and the products are found at the bottom of the column as expected from the boiling point differences. A sharp change in the component flow rates occurs only at the 4-ft height of the stripping section. We can see that the section above 6 ft does not provide much separation between the components. This suggests that such a long column might not be required for this process. The mass transfer rates of the components along the column are shown in Figure 4. Except for the feed point, the mass transfer rates of Ac along the column are all negative, which indicates that Ac is being transferred to the liquid phase. Between 10 and 17 ft of the column, the mass transfer rates of the components are close to zero, which also suggests that the column might be much longer than needed. However, further simulations using shorter columns will be required to confirm the optimum length of the CD column for this process. The profiles of the temperatures along the column are shown in Figure 5. We can see notable differences in temperature between the vapor and liquid phases in the 2-8 ft section of the column. The dashed line indicates the interfacial temperature, which shows that the temperature difference between the vapor phase and the
Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5347 Table 3. Comparison of DAA and MO Productivities between the Experimental Data and the Model Predictions DAA productivity [g (mL of catalyst)-1 h-1]
MO productivity [g (mL of catalyst)-1 h-1]
number of catalyst bags
feed rate (mL/h)
reflux flow rate (g/min)
measured data
previousa prediction
current prediction
measured data
previousa prediction
current prediction
2 2b 2b 4 6 6
152 152 152 152 152 153
22.00 25.40 15.60 22.90 23.60 16.30
0.76 0.78 0.63 0.49 0.41 0.29
0.75 0.77 0.61 0.50 0.41 0.30
0.77 0.76 0.61 0.50 0.43 0.30
0.16 0.21 0.21 0.14 0.15 0.15
0.16 0.22 0.22 0.14 0.14 0.14
0.15 0.20 0.21 0.14 0.14 0.14
a
Huang et al.14
b
Another batch of catalyst.
Figure 4. Profiles of vapor-liquid mass transfer rate along the column (catalyst, 4 bags; reflux, 22.9 g/min; feed, 152 mL/h).
Figure 5. Profiles of temperature along the column (catalyst, 4 bags; reflux, 22.9 g/min; feed, 152 mL/h).
interface is larger than that between the liquid phase and the interface. The interface temperature was found to be higher than the bulk liquid temperature, which indicates that heat is transferred to the bulk liquid. It should be noted that this nonequilibrium model could be used to design and simulate other CD processes as it utilizes generalized mass transfer coefficients. Although a nonequilibrium model does not use efficien-
cies, the efficiencies of each component and the HETP can be calculated from a nonequilibrium model according to Wesselingh.10 3.2. Experimental Validation. Table 3 shows that the productivities of DAA and MO predicted by the current model are in excellent agreement with the experimental results. It can be seen that the composition in the reboiler is also in excellent agreement with the experimental results (Figure 3). The temperature profiles predicted by the present model, the measured data, and those predicted previously by the MECRES model are shown in Figure 5. Clearly, the predictions provided by the present model are in very good agreement with the experimental data except for the region around the feed point (8-11 ft of the column). Unfortunately, the feed temperature was not measured in our experiments so we assumed it to be 20 °C in the simulation shown in Figure 5 because our previous MECRES model also used a feed temperature of 20 °C. It can also be seen in Figure 5 that the predictions of the temperature profiles by the present model are in better agreement with the experimental data than those of the previous MECRES model. We believe a feed temperature of 20 °C is much lower than the actual experimental feed temperature. Thus, we also simulated the temperature profile using feed temperatures of 30 and 54 °C, the latter being close to the boiling point of acetone. We found that an increase in the feed temperature results in better agreement between the simulated temperature profiles and the experimental data around the feed point in the column (8-11 ft), whereas the other simulated temperatures along the column are not affected by the changes in the feed temperature. Hence, these results indicate that this improved three-phase nonequilibrium model provides a better simulation of a CD process than the MECRES model. It should be noted that, although this improved threephase nonequilibrium model is successful in simulating the CD process for the aldol condensation of Ac, this model does not explicitly take into account the effect of the fluid mixing and catalyst contact time in the reaction zone of the CD column, although changes in the reflux flow rate could be related to the catalyst contact time. Recent advances in computational fluid dynamic modeling of randomly packed distillation column30 and hydrodynamics of sieve tray31 could have an impact in the development of future CD models. 4. Conclusions The MECRES model was modified and improved to take into account multicomponent mass and heat transfer according to Maxwell-Stefan theory. The assumption of thermal equilibrium between the vapor and liquid phases that was used previously in the MECRES
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model was abandoned. The newly improved model was validated on the basis of a successful simulation of the composition in the reboiler. The model predictions for the temperature profile along the column are in better agreement with the experimental data than those of the previous MECRES model. This is attributed to the consideration of heat transfer between the different phases rather than the assumption of thermal equilibrium as in the previous MECRES model. The modified model is a more generalized three-phase nonequilibrium model. The previous MECRES model is a simpler model because it assumes thermal equilibrium between the liquid and vapor phases; however, it is valid only for reaction systems where the heat of reaction is relatively small. The MECRES model also requires the determination of the mass transfer parameters specific to the column, and hence, it cannot be used to design CD columns. However, the newly modified model uses generalized mass and heat transfer coefficients, so it could be used for the design and simulation of other CD processes even when the reaction is very exothermic. We are currently using this model to simulate a CD process for the oligomerization of butenes developed in our laboratory. Nomenclature a ) effective interfacial area, m2 section-1 at ) specific surface area of packing, m2 m-3 aw ) wetted area, m2 m-3 A12, A21 ) interaction energy parameters, kJ kmol-1 c ) total number of components C ) molar concentration, kmol m-3 Cp ) heat capacity, kJ kmol-1 K-1 D ) binary diffusion coefficient, m2 s-1 dp ) equivalent diameter of packing, m e ) heat transfer rate, kJ s-1 f ) feed rate of component, kmol s-1 fo ) fugacity at saturated vapor pressure, kPa F ) feed rate, kmol s-1 g ) acceleration of gravity G ) weight of catalyst in section j, kg h ) heat transfer coefficient, kJ m-2 s-1 H ) molar enthalpy, kJ kmol-1 J ) J factor k1 ) DAA formation forward rate constant, L mol-1 min-1 k-1 ) DAA formation reversible rate constant, min-1 k2 ) MO and H2O formation rate constant, mol L-1 min-1 k ) multicomponent mass transfer coefficient, kmol m-2 s-1 K ) equilibrium ratio l ) liquid flow rate for components, kmol s-1 L ) liquid flow rate, kmol s-1 Le ) Lewis number, (λMF-1Cp-1D-1) M ) molecular weight, kg kmol-1 N ) mass transfer rate, kmol s-1 P ) pressure, kPa q ) structural area parameter q′ ) modified structural area parameter Q ) heat duty, kJ s-1 r ) structural size parameter R ) macro kinetic rate, kmol (kg of catalyst)-1 s-1 Rg ) universal gas constant, 8.314 kJ kmol-1 K-1 S ) ratio of withdrawal to flow rate of intersection Sc ) Schmidt number, (µF-1D-1) T ) absolute temperature, K v ) vapor-component flow rate, kmol s-1 V ) vapor flow rate, kmol s-1 W ) mass flow rate, kg m-2 s-1 x ) liquid composition, mole fraction
y ) vapor composition, mole fraction Z ) packed height of each section Greek Letters δ ) Kronecker delta, equal to 1 if i ) j; otherwise, equal to 0 ) thermodynamic factors γ ) liquid-phase activity coefficient Γ ) thermodynamic matrix κ ) binary mass transfer coefficient, kmol m-2 s-1 µ ) viscosity, Pa s ν ) stoichiometric coefficient F ) density, kg m-3 σ ) surface tension, N m-1 σc ) critical surface tension of packing, N m-1 ψ ) fugacity coefficient Subscripts av ) average value D ) mass transfer H ) energy transfer i ) component index j ) section index k ) alternative component index m ) property of mixture Superscripts F ) feed I ) vapor-liquid interface L ) liquid phase LF ) liquid feed r ) reaction S ) catalyst or solid phase V ) vapor phase VF ) vapor feed
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Received for review December 14, 2000 Revised manuscript received July 30, 2001 Accepted July 30, 2001 IE001104L
