Modeling, Analysis, and Simulation of a Methyl tert

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2696

Ind. Eng. Chem. Res. 1996, 35, 2696-2708

Modeling, Analysis, and Simulation of a Methyl tert-Butyl Ether Reactive Distillation Column Miguel A. Isla* and Horacio A. Irazoqui Instituto de Desarrollo Tecnolo´ gico para la Industria Quı´mica, Consejo Nacional de Investigaciones Cientı´ficas y Te´ cnicas-Universidad Nacional del Litoral, Gu¨ emes 3450, 3000 Santa Fe, Argentina

All the steps needed to build a reactive column simulator for the MTBE synthesis are analyzed. The column is modeled as a sequence of stages at partial equilibrium. The physical model adopted is discussed in depth as well as the resulting mathematical model. The solution algorithm, based on a Napthali-Sandholm-type method, is simple, robust, and time efficient. The simulation module is used to perform sensitivity analysis to structural and operating variables like total catalyst load, catalyst distribution, operating pressure, and reflux ratio, among others. The analyzed 11-component case study corresponds to a situation of practical interest. Introduction Methyl tert-butyl ether (MTBE) is synthesized in the liquid phase by the reaction of isobutene (iC4) and methanol (MeOH) in the presence of a cationic exchange resin as the catalyst. A common source of iC4 consists of C4 cuts available from steam or catalytic crackers. Conventional MTBE synthesis involves two steps, chemical reaction followed by separation. In the first step, the synthesis reaction takes place in one or more fixed bed catalytic reactors. Under the usual process conditions, the iC4-to-MTBE conversion can only attain values lower than 90% at the exit of the reactor section due to chemical equilibrium limitations. The stream from the reactors is led to a separation section where unconverted reactants, inerts, and products are separated. Under certain circumstances, the unconverted reactants could be recycled to the reactors. A combination of the conventional reaction section with a reactive distillation column may be optimal in the case of MTBE synthesis (De Garmo et al., 1992). In this arrangement, the reactor outlet stream is the reactive distillation column feed, in which the iC4-toMTBE conversion is completed. In this scheme, the reactive distillation unit operates in a fashion that differs in some aspects from the typical reactive distillation operation. Continuous removal of reaction products has been quoted as the unique feature that gives reactive distillation its technical and economic advantages where it is applicable. This makes it possible to bring equilibrium-limited reactions to completion while using the heat of exothermic reactions to provide boil-up for fractionation. In this particular application, the reaction product (MTBE) essentially remains in the same phase it was produced, flowing from the reaction section downwards into the stripping section with the liquid phase. The unreacted iC4 and MeOH, which form a lowboiling-point binary azeotrope, are continuously removed from the reaction zone liquid effluent. These reactants build up its concentration in the upgoing gas stream by fractionation against the liquid reflux stream along the rectifying section. Thus, higher yields can be attained by continuously recycling unconverted reactants into the reaction section with the enriched liquid reflux stream. * Author to whom correspondence should be addressed.

A detailed description of the advantages of using a reactive finishing column can be found elsewhere (Nijhuis et al., 1993). Chemical Reactions The MTBE synthesis reaction from methanol and isobutene is reversible and catalyzed by acids

iC4 + MeOH S MTBE The extent of this exothermic reaction is limited by equilibrium. The reaction temperature should be low enough to increase the iC4 equilibrium conversion. Lower reaction temperatures also decrease the formation of byproducts. Although several acidic catalysts can be used for the MTBE synthesis reaction, current processes are based on cationic catalytic resins. Possible side reactions are the dimerization of iC4

2iC4 S C8H16 the hydration of iC4 to give tert-butyl alcohol

iC4 + H2O S C4H9OH and the methanol etherification reaction

2MeOH S (Me)2O + H2O These reactions are all favored by the presence of acidic catalysts. Chemical Equilibrium Colombo et al. (1983) estimated the equilibrium constant of the MTBE synthesis reaction as a function of temperature and pressure. Whenever available, they used tabular standard free energies of formation in the gas phase to estimate the gas-phase equilibrium constants. For those components for which the energies of formation were not found in the literature, they used a group contribution method for thermochemical quantities (Benson, 1976). The pure-component saturation fugacities needed to calculate the standard free energies of formation in the liquid phase from those in the gas phase were approximated by the corresponding saturation pressures as predicted by the Antoine equation.

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Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 2697 Table 1. UNIQUAC Parameters from Rehfinger and Hoffmann (1990) (1 ) MeOH; 2 ) iC4; 3 ) MTBE) j i

1

1 2 3

1403.5000 931.4300

2

3

-70.3003

-174.9400 103.7300

-48.9310

The equilibrium constant Ka as a function of temperature obtained with this procedure was checked against values estimated from experimental measurements of the equilibrium concentrations XE,iC4, XE,MeOH, and XE,MTBE. The activity coefficients γE,iC4, γE,MeOH, and γE,MTBE needed to estimate the equilibrium constants through

Ka ) KxKγ )

γE,MTBE XE,MTBE XE,iC4XE,MeOH γE,iC4γE,MeOH

(1)

were calculated by using UNIFAC (Fredenslund et al., 1977). Disagreement between predicted and “experimental” values of Ka became important at low methanol concentrations. This disagreement was attributed by the authors to the UNIFAC prediction of the concentration dependence of γMeOH, which they considered to be weaker than necessary to match the experimental and predicted values of the equilibrium constant. This conclusion is in agreement with the results of Gicquel and Torck (1983). These authors also used UNIFAC to estimate the activity coefficients of the components of the reacting mixture, which are necessary to calculate the reaction equilibrium constant from equilibrium conversion measurements. They observed that the trend of the ratio Ka/Kx against XE,MeOH was similar to that of γE,MeOH against XE,MeOH, thus concluding that the isothermal variation of Kx is essentially due to the isothermal variation of the activity coefficient of methanol. Izquierdo et al. (1992) also estimated the equilibrium constant of the synthesis reaction in the liquid phase from measurements of the equilibrium composition. Their method differs from that adopted by Colombo et al. (1983) in the procedure they used to estimate the standard enthalpy and entropy of reaction in the liquid phase. Rehfinger and Hoffmann (1990) used the UNIQUAC model to describe the nonideality of a liquid reaction mixture consisting of MeOH, iC4, MTBE, and 1-butene or n-butane. Based on considerations regarding the liquid-vapor equilibrium behavior of the binaries, the authors treated the C4 hydrocarbons present in the mixture collectively as if they all were iC4, thus reducing the problem to an effective three-component mixture. The values of the binary interaction parameters are listed in Table 1. In this work the expression of the equilibrium constant of the liquid-phase synthesis reaction was adjusted on the basis of the experimental equilibrium compositions measured by Izquierdo et al. (1992). This was done to keep consistency between the thermodynamic description of the nonideal liquid and gas phases, the phase equilibrium model, the kinetic model, and the chemical equilibrium model. The UNIQUAC model was used to describe the nonideal liquid phase. The aij binary interaction parameters involving pairs among MeOH, C4 hydrocarbons, and MTBE were adopted from the work by Rehfinger and Hoffmann (1990), while the remaining interaction parameters have been estimated by regres-

Figure 1. Refitting of equilibrium constant. UNIQUAC model is used for activity coefficients. Equilibrium composition data from Izquierdo et al. (1992).

sion from activity coefficients predicted by means of the UNIFAC method. This set of interaction parameters was used to predict L-V equilibrium conditions for mixtures involving different pairs among the reaction mixture components. The predicted results showed satisfactory agreement when compared with binary L-V experimental data (Churkin et al., 1978; Alm and Ciprian, 1980; Oscarson et al., 1987; Leu et al., 1989; Leu and Robinson, 1992). With this approach, the expression

ln Ka )

A +B T

(2)

with A ) 4361.23 and B ) -9.24416 reproduces satisfactorily the experimental results, as is shown in Figure 1. Thermodynamic Properties of Mixtures Pure-component vapor pressures were estimated according to Reid et al. (1987, Appendix A). The purecomponent fugacity-pressure ratios at the saturation pressure, φ°i, and the fugacity-pressure ratios of the components in the gas mixture, φi, were estimated by means of the Lee-Kesler correlation (Reid et al., 1987). The latter was also used to estimate the enthalpy of nonideal gas and liquid mixtures. Reaction Mechanism and Kinetic Model The reaction between methanol and isobutene on sulfonic catalytic resins has been studied by several research groups. Starting from the stoichiometric ratio between reactants, or with an excess of alcohol, Ancillotti et al. (1978) found that initial rates were zero order in the alcohol and first order in the olefin. The effect of the methanol concentration on the initial reaction rate, at constant isobutene concentration and constant temperature, was also studied. It was found that, at low methanol concentrations, the initial rate increases with CMeOH up to a maximum which depends on the isobutene concentration (CiC4). This maximum occurs at CMeOH values such that CiC4/CMeOH ) 10. Increasing CMeOH beyond that maximum causes the initial rate to monotonically decrease until an asymptotic value is reached. At low methanol concentrations, the initial rate is independent of CiC4 (i.e., for CiC4/CMeOH > 10). After the maximum is surpassed by increasing CMeOH (i.e., for CiC4/

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2698 Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996

CMeOH < 2.7), the kinetic order in the olefin becomes larger than zero and reaches a stable first order in the large CMeOH region, where the rate is zero order in the methanol concentration (i.e., for CiC4/CMeOH < 1.7). An ionic mechanism operating through the slow, rate-determining protonation of the olefin by the solvated proton followed by the interaction of the carbon ion with the alcohol is proposed to interpret the initial rate asymptotic behavior for CiC4/CMeOH < 1.7. This type of mechanism agrees with the first order kinetics observed for the olefin and the zero-order kinetics observed for methanol, which is in great excess over its stoichiometric ratio to the olefin. When the CiC4/CMeOH ratio becomes larger than 1.7, the initial rate of the etherification reaction increases as CMeOH decreases. This tendency can be accounted for by considering that the equilibrium reaction

SO3H + ROH S SO3- + ROH2+ which is completely shifted to the right for CiC4/CMeOH < 1.7, begins to move to the left. The isobutene can take the proton directly from the sulfonic group, which is a more acidic species than the solvated proton, thus increasing the reaction rate. When CiC4/CMeOH ratios larger than 3.5 are attained, dimerization of isobutene also occurs. This tendency is kept until the CiC4/CMeOH ratio reaches a value close to 10.0, where the maximum of the initial rate of the etherification reaction is found. At this point, the favorable effect on the reaction rate of the CMeOH reduction is suddenly reversed. For CiC4/ CMeOH ratios larger than 10.0, methanol behaves as the scarce reactant with a kinetic order equal to one, while the isobutene concentration has no influence on the reaction rate. The MTBE synthesis reaction catalyzed by Amberlyst 15 macroreticular sulfonic resin was also studied (Gicquel and Torck, 1983). This catalyst acts through its sulfonic groups bonded to the resin. The porous spherical beads contain 4.9 equiv protons/kg. At low alcohol concentrations, the resin retains a network in which each sulfonic group is linked to its neighboring groups through hydrogen bonds, competing with its bonding to the alcohol itself. This residual network of hydrogen bonds disappears as the alcohol concentration becomes large and all the protons are solvated (Gicquel and Torck, 1983). The increasing resin activity observed is related to the environment of the catalytic sites on the sulfonic resin. When the alcohol concentration is very low and most of the sulfonic groups remain interlinked by hydrogen bonds, the iC4 dimerization reaction is important. The exothermic addition of MeOH in great excess with respect to equivalent protons leads to the dissociation of the hydrogen bonds and to the solvation of the protons. Under these circumstances, specific catalysis occurs by means of protons solvated by several MeOH molecules. Between these two extreme cases, protons can be solvated to a variable extent and its catalytic activity can also be variable. The kinetics of the MTBE synthesis reaction catalyzed by the Amberlyst 15 ion-exchange resin was further studied by Rehfinger and Hoffmann (1990). To interpret their results, it must be recalled that a bead of a macroreticular resin is considered to be an ensemble of microspheres (gel), consisting of a network of crosslinked chains of a styrene-divinylbenzene copolymer.

The diameter of these gel particles is about 10-4 times the diameter of the average bead in the case of Amberlyst 15. All the experimental runs were conducted with a large excess of alcohol with respect to the amount of equivalent proton of the resin. The reaction between methanol and isobutene to give MTBE has been observed to be highly selective if the initial CiC4/CMeOH ratio is smaller than the stoichiometric one and if there is no water initially present. In order to preserve selectivity, the presence of water should be avoided. Since water is more polar than methanol, it will be preferentially adsorbed on the sulfonic groups from isobutene/methanol/water mixtures. Then, the adsorbed water can react with isobutene to give tert-butyl alcohol. Rehfinger and Hoffmann (1990) confirmed the atypical behavior that the initial rate of the etherification reaction shows when the initial CiC4/CMeOH ratio is smaller than (but close to) 10.0. As a consequence of this behavior, when diffusive transport begins to control the MeOH transfer from the bulk liquid phase, its concentration in the gel phase lowers and the reaction rate increases. The high stirring speeds used in laboratory experiments to diminish the external mass- and heat-transfer resistances caused lower reaction rates than those measured under a less intense agitation. The corresponding change on the mean temperature in the bead reinforces this tendency, since it rose as the agitation speed lowered while the bead radial temperature profile remained flat. Finely divided particles of different exchange resins with different degrees of cross-linking showed the same initial rate of etherification at high MeOH concentrations. The interpretation given to this experimental result is that the reaction rate is influenced neither by the diffusion inside the microsphere nor by the internal surface area. In light of these conclusions it can be accurately assumed that each gel microsphere contributes to the reaction with an effectiveness factor equal to unity and only macropore diffusion effects should be considered. Intrinsic forward reaction rates were determined in mixtures with an stoichiometric excess of MeOH to avoid macropore diffusion effects. The reaction rate was found to increase linearly with CiC4 at constant CMeOH. When CMeOH is increased at constant total concentration by decreasing the inert solvent concentration, a slight increase of the reaction rate was detected. In previous work (Ancillotti et al., 1978), the asymptotic value of the kinetic order of MeOH was reported to be equal to zero, probably due to the fact that experiments were not free of macropore diffusion effects. The sorption equilibrium between the reaction mixture and the catalytic active gel phase of the resin is assumed to obey Langmuir’s sorption model. Therefore, the fraction of total sites occupied by species i, θi, is given by

Ks,iai

θi ) 1+

∑j Ks,j aj

(3)

where Ks,i is the equilibrium sorption constant of species i and ai is the activity of species i in the liquid phase,

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Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 2699

measured in the rational symmetric scale,

ai ) γiXi

γi f 1; Xi f 1

(4)

The rate-controlling step is the chemical reaction between sorbed molecules of reactants

MeOH‚S + iC4‚S S MTBE‚S + S where S stands for an active site. The controlling reaction rate is

r ) (k+Ks,MeOHKs,iC4aMeOHaiC4 - k-Ks,MTBEaMTBE)/ (1 +

∑j Ks, jaj)2

(5)

Regression of kinetic and sorption constants from experimental results shows that MeOH is the dominant sorbed species and that the fraction of vacant sites is negligible. Therefore, the expression of the rate of the controlling step can be simplified to give

{

aiC4 1 aMTBE r) k aMTBE Ka (a )2 MeOH

}

(6)

where

Ks,iC4 k ) k+ Ks,MeOH

(7)

and Ka is the equilibrium constant of the etherification reaction. The expression of k as a function of temperature is

k ) 8.3939 × 1013 exp(-11 112.85/T)

(8)

where k is given in units of mol h-1 g of catalyst-1 and T is the absolute temperature (K). At MeOH mole fractions lower than 0.3, the reaction rate predicted with the intrinsic kinetic model is smaller than that from experimental measurements. At these MeOH concentrations, macropore diffusion is no longer negligible and the MeOH concentration in the active gel phase is lower than that in the bulk liquid phase, thus increasing the reaction rate. On the basis of the experimental results from different authors briefly reviewed in this section, the following assumptions were made for the reaction system under the conditions prevailing in a finishing reactive column fed with the main synthesis reactor outlet stream: (i) The main reactor is fed with a molar flow of MeOH exceeding that corresponding to its stoichiometric ratio to iC4. As a consequence of this, the CiC4/CMeOH ratio in the reactor outlet stream, which is also the column feed stream, is necessarily smaller than unity. (ii) The column operating conditions are chosen so that the amount of MeOH present in the column bottom stream is very small. This guaranties a CiC4/CMeOH ratio smaller than unity for all reactive stages if placed, as they will be, within the rectifying section. (iii) Side reactions in the finishing reactive column are negligible at the chosen operating conditions (dimerization of isobutene; formation of tert-butyl alcohol from isobutene, and water present in the column feed).

With these assumptions, the rate of iC4 conversion to MTBE can be accurately described by the expression proposed by Rehfinger and Hoffmann (1990). Column Stage at Partial Equilibrium A widely used approach for the analysis and calculation of conventional separation stages is to assume the attainment of complete phase equilibrium between streams leaving each stage. Corrections to this equilibrium stage hypothesis are introduced as a final step in the analysis or calculation (King, 1981). This is usually done by adopting an empirical overall stage efficiency for a particular separator. The variation in efficiency from one stage to another for individual components can be described by the Murphree stage efficiency. The occurrence of a chemical reaction simultaneously with the phase equilibration process in the separation stage makes it necessary to reconsider the equilibrium stage assumption based on phase equilibrium only. When the reversible chemical reaction is very fast so that chemical equilibrium is attained in a time much shorter than the phase equilibration time, the chemical reaction follows the component exchange process through a continuous succession of chemical equilibrium states. Under these circumstances, component transfer between phases is the rate-determining process, and the equilibrium stage hypothesis can be readily extended to systems with reversible chemical reactions. If the reversible chemical reaction is not instantaneous, chemical equilibrium will not be reached within the short holding time in a contacting device. The concept of equilibrium stage can still be used in this case, but Murphree efficiencies must be based on states of total equilibrium (phase and chemical). The effect of a chemical reaction of finite rate is necessarily a reduction of the stage efficiency by keeping the outgoing streams from reaching the state of total equilibrium, even in the case in which phase equilibrium has been attained. This approach unifies the design and simulation strategies for reactive and nonreactive columns introducing the chemical reaction only through its stoichiometry and its equilibrium constant. Nevertheless, total equilibrium might be too far from the actual operating conditions, especially when the reaction is slow. Besides, in the case of catalytic reactions, it gives no clue to the designer as to the amount of catalyst needed to achieve a given conversion and how to distribute it between the separation stages. The hypothesis that a separation stage operates at partial equilibrium aims at giving a more realistic description of the behavior of separation stages over a wider range of the ratio of the chemical reaction rate to the phase equilibration rate. The stage model at partial equilibrium is based on the following assumptions: The stage outgoing streams are at phase equilibrium with each other, i.e., they have achieved thermal and component transfer equilibrium during the stage holding time. The rate of the phase component transfer process is greater than the chemical reaction rate to the extent that the chemical reaction can be thought of as the ratedetermining process, preventing the outgoing streams from reaching mutual phase and chemical equilibrium. From the chemical reaction standpoint, the reactive stage can be modeled as a continuous stirred tank

+

+


where

HLj-1 ) H(Tj-1, Pj-1, Xj-1) HVj+1 ) H(Tj+1, Pj+1, Yj+1);

Mathematical Model of a Stage at Partial Equilibrium The reactive distillation column is conceived as a stack of partial equilibrium stages which are labeled with a j index running from 2 at the top stage to N - 1 at the bottom one. The total condenser is designated as stage 1 and the reboiler as stage N. Total liquid (L) and vapor (V) molar flow rates bear the label of the stage in which the corresponding stream is exiting. The same convention is adopted for the corresponding stream properties. A schematic representation of a partial equilibrium stage is shown in Figure 2. The stage mathematical model includes a set of balance equations about stage j and the phase equilibrium constraint equation: (a) One part is the component mole balance equations, allowing for possible chemical reaction as well as for the feed and withdrawal of liquid (U) and vapor (S) side streams

Li, j-1 + Vi, j+1 + Fi, j - Li, j - Vi, j - Ui, j - Si, j + ri, j ) 0; i ) 1, 2, ..., M (9) The reaction term in eq 9 is given by

ri, j ) νiwjrj

(14)

HVj ) H(Tj, Pj, Yj)

(15)

Lj )

∑i Li,j

(16)

Vj )

∑i Vi, j

(17)

Fj )

∑i Fi, j

(18)

Sj )

∑i Si, j

(19)

Uj )

∑i Ui, j

(20)

(c) The stage mathematical model is completed with the phase equilibrium constraint between outgoing streams

yi, j ) Ki, jxi, j

(10)

Xj ) (xi,j), i ) 1, 2, ..., M

is the rate of the synthesis reaction as given by eq 6, calculated at the conditions of the liquid outlet stream (at phase equilibrium with the vapor outlet stream). There are M independent component mass balance equations. (b) Another part is energy balance, allowing for stage heating or cooling (i.e., stage heat losses)

Lj-1HLj-1 + Vj+1HVj+1 + FjHFj - LjHLj - VjHVj UjHLj - SjHVj + Qj ) 0 (11)

(21)

where

Ki, j )

γi, jφ°i, jPv, j φi, jPj

(22)

is the phase equilibrium ratio of component i in stage j,

γi, j ) γi(Tj, Pj, Xj)

(23)

is the activity coefficient of component i in the liquid stream leaving stage j, measured in the rational symmetric scale,

φ°i, j ) φi(Tj, Pv, i(Tj))

(24)

is the pure-component fugacity-pressure ratio of species i at its saturation pressure corresponding to the stage temperature Tj, Pv, i(Tj), and

φi, j ) φi(Tj, Pj, Yj)

where νi is the stoichiometric coefficient of species i, wj is the amount of catalyst charged to stage j, and

rj ) r(Tj, Xj);

Yj ) (yi, j), i ) 1, 2, ..., M (13)

HLj ) H(Tj, Pj, Xj)

Figure 2. Scheme of reactive stage j.

reactor (CSTR), neglecting composition and temperature gradients within the stage contact volume. It should be noted that this model reduces itself to the total equilibrium stage model when the reaction rate is fast enough so that the chemical reaction approaches its equilibrium conversion within a narrow difference. The model also allows for the inclusion of irreversible chemical reactions.

(12)

(25)

is the fugacity-partial pressure ratio of species i in the vapor mixture leaving stage j. There are M independent phase equilibrium conditions of this type. The closure condition on mole fractions must also be included as constraint equations

∑i xi, j ) 1

(26)

∑i yi, j ) 1

(27)

∑i xi, j-1 ) 1

(28)

∑i yi, j+1 ) 1

(29)

The total number of equations is

+

+


Ne ) 2M + 5 To describe mathematically the behavior of any stage j, the values of the following variables must be known: Tj-1, Pj-1, Tj, Pj, Tj+1, Pj+1, Lj, Lj-1, Vj, Vj+1, Xj, Xj-1, Yj, Yj+1, and Qj. Therefore, the total number of unknowns is

The M (N - 1) component balance equations, the M (N - 1) phase equilibrium conditions between stage outgoing streams, and the (N - 1) stage energy balance equations can be recast in the form

Emi,1 ) Vi,2 - (R + 1)Di; Epi,1 ) Li,1 - RDi;

Nv ) 4M + 11 To solve the mathematical model for a given stage j, it is necessary to know the value of 2M + 6 independent variables. Usually, the set of variables {Tj-1, Pj-1, Pj, Tj+1, Pj+1, Lj-1, Vj+1, xi, j-1 (i ) 1, .., M - 1), yi, j+1 (i ) 1, .., M - 1), Qj} including the stage pressure and the variables associated with the inlet liquid and vapor streams is assumed to be known. The set of equations describing any stage j can be readily specialized to describe the column reboiler (j ) N). The corresponding component balance equations reduce to the following set

LN-1xi,N-1 - LNxi,N - VNyi,N ) 0;

i ) 1, 2, ..., M (30)

and the energy balance becomes L

LN-1H

N-1

- LNH

L N

- VNH

V N

+ QN ) 0

(31)

The mathematical model of the column reboiler is completed with M phase equilibrium constraint equations and three mole fraction closure equations. The total number of equations is 2M + 4, involving 3M + 8 variables: {TN-1, PN-1, TN, PN, LN, LN-1, VN, XN, XN-1, YN, QN}. Therefore, it is necessary to know or to specify a total of M + 4 independent variables. The total mole balance equation about the total condenser is

Vi,2 - Di(R + 1) ) 0;

i ) 1, 2, ..., M

(32)

where Di is the component distillate molar flow rate and R the reflux ratio. The corresponding energy balance is

V2HV2 - D(R + 1)HL1 + Q1 ) 0

(33)

where R is the reflux ratio relating the distillate mole flow rate to the corresponding downward liquid stream flow rate

Li,1 ) RDi,

i ) 1, 2, ..., M

(34)

i ) 1, 2, ..., M (35) i ) 1, 2, ..., M

Eh1 ) V2 HV2 - (R + 1)DHL1 + Q1

(36) (37)

Emi, j ) Li, j-1 + Vi, j+1 + Fi, j - Li, j - Vi, j - Ui, j Si, j + ri, j; i ) 1, ..., M, j ) 2, ..., N- 1 (38) Li, j Epi, j ) VjKi, j - Vi, j; Lj i ) 1, 2, ..., M, j ) 2, ..., N - 1 (39) h

E j ) Lj-1

HLj-1 HVj

HVj+1

+ Vj+1

HVj

HFj

HLj

+ Fj V - Lj V H j H j

HLj Qj Uj V + V - Sj - Vj; H j H j

j ) 2, ..., N - 1 (40)

Emi,N ) Li,N-1 - Li,N - Vi,N; i ) 1, 2, ..., M Epi,N ) VNKi,N

HLN-1

h

E

Li,N - Vi,N; LN

N

) LN-1

HVN

(41)

i ) 1, 2, ..., M (42)

HLN

QN - LN V + V - VN H N H N

(43)

where Emi, j; Epi, j, and Ehj are the component balance, the phase equilibrium, and the energy balance discrepancy functions, respectively. Throughout the following derivation, we will assume that the variables Ui, j and Si, j (i ) 1, ..., M, j ) 2, 3, ..., N), are either known or specified (i.e., they have fixed values). The same assumption is made for Qj, j ) 2, 3, ..., N. Solving the reactive distillation model amounts to solving the following set of equations

Emi,1(R; Di; Vi,2) ) 0;

i ) 1, ..., M

(44)

Epi,1(Li,1; Di; R) ) 0;

i ) 1, ..., M

(45)

The total number of equations is 2M + 1, involving the following set of 3M + 6 variables: {T1, P1, T2, P2, L1,1, L2,1, ..., LM,1, V1,2, V2,2, ..., VM,2, D1, D2, ..., DM, Q1, R}. To solve the total condenser, it is necessary to know or to specify a total of M + 5 independent variables. The problem can be solved when T1, P1, P2, and R are specified and V1,2, V2,2, ..., VM,2 and T2 are known from the solution of the system of equations describing the column stages and the reboiler.

Emi, j(Li, j-1; L1, j, ..., LM, j; Vi, j; Tj; Vi, j+1) ) 0; i ) 1, ..., M, j ) 2, ..., N - 1 (47)

Reactive Distillation Calculations by Linearization

Ehj(L1, j-1, ..., LM, j-1; Tj-1; L1, j, ..., LM, j; V1, j, ..., VM, j; Tj; V1, j+1, ..., VM, j+1; Tj+1) ) 0; j ) 2, ..., N - 1 (49)

The multicomponent reactive column calculation technique is based on the linearization method by Naphtali and Sandholm (1971).

Eh1(D1, ..., DM; T1; R; V1,2, ..., VM,2; T2; Q1) ) 0 (46)

Epi, j(L1, j, ..., LM, j; V1, j, ..., VM, j; Tj) ) 0; i ) 1, ..., M; j ) 2, ..., N - 1 (48)

Emi,N(Li,N-1; Li,N; Vi,N) ) 0;

i ) 1, ..., M

(50)

+

+


Epi,N(L1,N, ..., LM,N; V1,N, ..., VM,N; TN) ) 0; i ) 1, ..., M (51) EhN(L1,N-1, ..., LM,N-1; TN-1; L1,N, ..., LM,N; V1,N, ..., VM,N; TN) ) 0 (52) for known values of T1 and L1,1, ..., LM,1. To simplify the application of the linearization method, the set of discrepancy functions can be considered as a one-dimensional array of functions

F ) (Fk)

Fk(x) ) Emi, j(Li, j-1; L1, j, ..., LM, j; Vi, j; Tj; Vi, j+1); k ) i + (2M + 1)(j - 2) (54) Fk(x) ) Epi, j(L1, j, ..., LM, j; V1, j, ..., VM, j; Tj); k ) i + (2M + 1)(j - 2) + M (55) Fk(x) ) Ehj(L1, j-1, ..., LM, j-1; Tj-1; L1, j, ..., LM, j; V1, j, ..., VM, j; Tj; k ) (j - 1)(2M + 1) (56) V1, j+1, ..., VM, j+1; Tj+1); and

(57)

where

xk ) Li, j;

k ) i + (2M + 1)(j - 2)

xk ) Vi, j;

k ) (j - 1)(2M + 1)

(60)

With this notation, the Newton-Raphson method becomes

xn+1 ) xn - (∂F/∂x)-1Fn

(61)

where n stands for the iteration number and (∂F/∂x)-1 is the inverse matrix of

∂F/∂x ) (∂Fi/∂xj)

(62)

This system of linear algebraic equations can be written in the following equivalent form

(∂F/∂x)n(xn+1 - xn) ) -Fn

35 20 from 4 to 10 300 1.1 7.0 0.2 0.025 13.20

Table 3. Simulation Results for the Case Study stream temp, °C pressure, bar vapor fraction molar flow rate, kmol/h mass flow rate, kg/h enthalpy flow, GJ/h

feed

distillate

bottoms

69.00 8.00 0.4813 636.33 38084.4 -75.76

45.14 7.00 0.0000 480.49 25953.3 -41.80

131.73 8.03 0.0000 140.47 12131.1 -39.95

composition mole mole mole kmol/h fraction kmol/h fraction kmol/h fraction propane 30.96 0.0486 30.96 0.0644 0.00 propene 44.28 0.0696 44.28 0.0922 0.00 isobutane 142.95 0.2246 142.95 0.2975 0.00 isobutene 17.37 0.0273 2.00 0.0042 0.00 1-butene 88.11 0.1385 88.10 0.1834 0.01 n-butane 56.43 0.0887 56.40 0.1174 0.03 2-trans-butene 60.06 0.0944 58.62 0.1220 1.44 2-cis-butene 39.78 0.0625 39.66 0.0825 0.12 n-pentane 12.54 0.0197 0.01 0.0000 12.53 methanol 32.76 0.0515 17.39 0.0362 0.00 MTBE 111.09 0.1746 0.12 0.0002 126.34

0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0103 0.0008 0.0892 0.0000 0.8994

(58)

k ) i + (2M + 1)(j - 2) + M (59)

xk ) Tj;

no. of ideal stages feed stage reactive stages catalyst load, kg/stage reflux ratio top pressure, bar condenser pressure drop, bar stage pressure drop, bar reboiler heat duty, GJ/h

(53)

where

x ) (xk)

Table 2. Reactive Column Specifications

condenser heat duty, GJ/h isobutene conversion, %

19.19 88.48

with the set of operating and structural variables listed in Table 2 and 3 are shown in Figures 3 and 4. Because this work deals with a finishing reactive column, the composition profiles obtained are not substantially different from those obtained with the chemical reaction turned off, except for a moderately larger amount of MTBE present in the rectifying section. Methanol forms minimum boiling temperature binary azeotropes with iC4 and with n-butane (Rehfinger and Hoffmann, 1990; Leu et al., 1989). In both cases, the azeotropic composition belongs to the C4-rich region. It can be reasonably assumed that, in the absence of MTBE, methanol forms similar minimum boiling temperature binary azeotropes with each one of the other butanes and butenes present in the feed stream (Co-

(63)

The fact that the matrix ∂F/∂x is block tridiagonal greatly simplifies the solution of the linear system of algebraic equations given by eq 63. For this, LINPACK subroutines were used (Dongarra et al., 1979). Case Study The simulation program developed was used to simulate the reactive distillation column specified in Table 2. The reboiler heat duty, QN, was adjusted externally to obtain a 90% MTBE product bottom stream, expressed on a mole basis. The feed conditions and the corresponding simulation results are summarized in Table 3. Discussion of Results. The concentration profiles of both reactive and inert species obtained by simulation

Figure 3. Simulated composition profile. Reactive species.

+

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Figure 4. Simulated composition profile. Nonreactive species.

lombo et al., 1983). To simplify the present analysis, the butanes and butenes present in the feed stream can be effectively considered as a single C4 pseudocomponent. The binary MTBE-MeOH system shows a minimum boiling temperature azeotrope with xMTBE,az > 0.5 (Mullins et al., 1989; Nijhuis et al., 1993). At equal pressures, the MTBE-MeOH azeotrope has a boiling temperature higher than that of the MeOH-C4 pseudobinary azeotrope. Methanol also forms a minimum boiling temperature binary azeotrope with n-pentane, with a boiling temperature between that of the MTBE-MeOH azeotrope and that of the MeOH-C4 pseudobinary azeotrope. Chemical species and binary azeotropes can be ordered according to the decreasing values of their “pure component” bubble pressures as follows: propene > propane > MeOH-C4 azeotrope > isobutane > isobutene > 1-butene > n-butane > 2-trans-butene > 2-cis-butene > n-pentane-MeOH azeotrope > n-pentane > MTBEMeOH azeotrope > MTBE > MeOH. If the MeOH/total C4 feed ratio belongs to the C4-rich side of the pseudobinary azeotrope, most of the methanol fed will flow upwards from the feed stage, trying to build vapor and liquid phases ever richer in methanol with respect to total C4 as higher stages are considered in the column. Under these operating conditions, MeOH dies out rapidly at the stages below the feed because of the high volatility of the MeOH-C4 pseudobinary azeotrope, while different C4’s, which collectively are in excess over the azeotropic concentration, reach deeper into the stripping section as their volatilities are lower. Proceeding up the column from its bottom, the lowest stages are devoted to a fractionation of MTBE against n-pentane, 2-trans-butene, and 2-cis-butene. As these components are more volatile than MTBE, they concentrate in the vapor stream, increasing their mole fraction at the expense of MTBE. Progressing further in the upwards direction, fractionation of MTBE against n-pentane, 2-trans-butene, and 2-cis-butene is still a very active process, but from a certain stage up, fractionation between n-pentane against 2-trans-butene and 2-cis-butene begins. The mole fractions of 2-trans-butene and 2-cis-butene, which are more volatile than n-pentane, tend to increase, while that of n-pentane decreases after showing a maximum concentration point. In the next few upper stages, fractionation of MTBE against more volatile components occurs at a slower pace, while fractionation of 2-trans-butene and 2-cis-

butene against the more volatile isobutane, 1-butene, and n-butane becomes important. There is a concentration buildup of these components at the expense of both 2-trans-butene and 2-cis-butene, causing concentration profiles of the latter to show maximum points. This analysis of the stripping section has been made on the assumption that the MeOH/total C4 feed ratio belongs to the C4-rich side of the pseudobinary azeotrope. Under these operating conditions, the unreacted MeOH can be removed from the top of the distillation column, despite of the fact that it has the lowest pure component vapor pressure. A few stages below the feed are enough to deplete the MeOH concentration to a very low value. Within about 10 stages above the feed point, both MTBE and pentane drop to very low concentrations because of their fractionation against the dominant C4’s and the MeOH-C4 pseudobinary azeotrope, whose volatilities are higher. At the top stages, fractionation of 1-butene, n-butane, 2-trans-butene, and 2-cis-butene occurs against propane, propene, and isobutane. Following the upwards direction, the mole fractions of propane, propene, and isobutane tend to increase, while those of 1-butene, n-butane, 2-trans-butene, and 2-cis-butene decrease after showing maximum concentration points. If MeOH were fed in excess with respect to the C4MeOH pseudobinary azeotrope composition, part of it would tend to flow downwards to reach the lower stages in the stripping section. Because MeOH is the species with the least pure-component vapor pressure, its concentration would tend to build up in both the vapor phase and the liquid phase as it reaches lower stages in the stripping section. Let us consider the case in which MeOH is fed in excess with respect to the C4-MeOH pseudobinary azeotrope composition but where the MeOH/pentane feed ratio still falls on the pentane-rich side of the binary MeOH-pentane azeotrope, at least at those stages about which the pentane mole fraction profile shows a maximum under the original feed conditions. Most of the methanol that reached these stages will flow upwards, trying to build vapor and liquid phases with higher methanol concentrations, ever closer to that of the azeotrope. There could be a beneficial side effect, since part of the pentane that otherwise will contaminate the MTBE in the bottom stream can be recovered at the top of the column. Finally, when fed at an even larger excess, methanol may reach the bottom stages, giving rise to fractionation between MTBE and the MeOH-MTBE azeotrope, with the result of larger quantities of MTBE reaching stages located deeper into the fractionating section to finally produce MTBE losses from the top of the column. Because of the larger concentration of the product at the reactive stages, the described situation will also produce an unfavorable mass effect on the rate of the synthesis reaction, with the consequence of larger losses of unreacted isobutene. The temperature profile in the stripping section corresponding to the case study follows a trend totally consistent with the fractionation process taking place in this section in the virtual absence of MeOH, which we already discussed. As shown in Figure 5, the temperature steadily decreases in the upward direction as more volatile components increase their concentration at the expense of less volatile ones.

+

+


Figure 6. Influence of reflux ratio on iC4 conversion. Figure 5. Simulated column temperature profile.

In most of the rectifying section, the temperature profile is rather flat, owing this characteristic mainly to the slow fractionation of MTBE against C4 hydrocarbons. Finally, the temperature profile drops at the top stages due to an increase in the concentration of the C3 components present. Influence of the Reflux Ratio. The influence of the reflux ratio on the iC4 conversion to MTBE in the range 0.6 < R < 2.5 was also studied. Figure 6 shows that, for a given catalyst load, conversion vs R reaches an asymptotic region, where it cannot be increased significantly by increasing R. A rapid decrease of the iC4 conversion to MTBE with decreasing R can be observed at the lower end of the interval scanned. The predicted behavior can be linked to the extent in which MTBE, whether it comes from the column feed or the synthesis reaction, is confined in the stripping section as the result of its fractionation against more volatile components. This fractionation process becomes more efficient as R increases, depleting the MTBE concentration in the reactive zone with only a marginal modification of the iC4/MeOH ratio. This combination of effects tends to increase the rate of the synthesis reaction by increasing its departure from chemical equilibrium, thus increasing the iC4 conversion to MTBE. This is illustrated in Figure 7, where the chemical equilibrium departure function

Deq ) 1 -

1 aMTBE Ka aMeOHaiC4

(64) Figure 8. Influence of reboil ratio on iC4 conversion.

and the corresponding synthesis reaction rate

aiC4

D r)k aMeOH eq

Figure 7. Influence of product concentration. Conditions at the lower reactive stage.

(65)

are represented against the MTBE liquid mole fraction at the conditions of the lower reactive stage. As happens with conventional distillation, once the total number of theoretical stages is fixed, the heavycomponent concentrations in the rectifying section diminish only slightly when the reflux ratio is increased beyond a certain value which, in this case study, lies at about R ) 2.5. This is the reason why the impact of R on the iC4 conversion to MTBE progressively attenuates as the reflux ratio is increased to finally become negligible. This sets a limit to the possibility of compensating the unfavorable impact on the iC4 conversion of a low catalyst load by increasing the reflux ratio.

Influence of the Reboil Ratio. For a specified MTBE concentration in the bottom stream, the reboil ratio (VN/LN) necessary to operate the reactive column becomes determined once the reflux ratio has been chosen. Due to this one-to-one relationship, the dependence of the iC4 conversion to MTBE on the reboil ratio follows the same trend as its dependence on the reflux ratio, as shown in Figure 8. The attempt to increase the MTBE concentration in the bottom stream by increasing the reboil ratio, keeping the reflux ratio constant, may result in lower iC4to-MTBE conversion values. For the conditions of the case study, this situation is reached for reboil ratios larger than 4.7, as shown in Figure 9. At these reboil ratios, the MTBE concentration at the reactive stages reaches values large enough to deplete the iC4 conversion to MTBE due to the reversibility of the synthesis reaction.

+

+


Figure 9. Influence of reboil ratio on column performance at constant reflux ratio.

Figure 11. Influence of temperature on reaction rate. Conditions at the lower reactive stage.

Figure 10. Influence of column operating pressure on iC4 conversion.

Figure 12. Parametric optimization of column operating conditions.

Influence of the Operating Pressure. The impact of pressure on the performance of the finishing reactive distillation column was studied in the interval 6.0-10.0 bar (top pressure), with R ) 1.1. The simulation results are summarized in Figure 10. As pressure increases, a lowering trend of the fractionation performance was predicted because of the smaller differences between the component volatilities due to the rise of the column temperature profile. As a consequence of this, MTBE reaches stages located higher into the fractionation section above the feed stage. On the other hand, according to the Arrhenius law, the reaction rate constant becomes larger as the temperature increases in the reactive zone, thus tending to increase the rate of the MTBE synthesis reaction. But this tendency may be attenuated or even reversed due to the fact that the synthesis reaction is both reversible and exothermic, with an equilibrium constant always smaller as the temperature increases. These competing tendencies associated to a pressure-driven increasing temperature are illustrated in Figure 11 for the composition of the lower reactive stage. At constant composition, the forward reaction rate monotonically increases with temperature according to the Arrhenius law, while the value of Deq diminishes due to a decreasing equilibrium constant, Ka (see eq 64). As shown in Figure 11, these conflicting tendencies may cancel each other, causing the net reaction rate to reach a maximum. The larger amount of MTBE present in the reactive section as a consequence of a lower fractionation performance reinforces the tendency to lower or even to reverse the reaction rate. This analysis is consistent

with the simulation results, shown in Figure 10 as iC4 conversion vs top operating pressure. Optimal Operating Pressure and Reflux Ratio. The fractionation process in the rectifying section becomes more efficient as R increases, depleting the MTBE concentration in the reactive zone followed by small changes of the iC4/MeOH ratio. This combination of effects tends to increase the rate of the synthesis reaction by increasing its departure from chemical equilibrium. On the other hand, the temperature in the reactive section of the column increases with increasing pressures. A pressure-driven temperature rise influences the reaction rate in two opposite ways by simultaneously increasing the rate constant and decreasing the departure from chemical equilibrium. As pressure increases, the latter of these two effects overcomes the first one, causing the reaction rate to decrease. As a consequence of this, an optimal operating pressure can be found for which a given conversion may be achieved with a minimum R, which in turn can be associated with a minimum reboiler heat load. In Figure 12, it is shown that, for the situation adopted as the case study and a conversion value corresponding to 88.5% of the total iC4 fed to the column, a minimum R is required at operating pressures of about 7.5 bar. Catalyst Load. The effect of the total catalyst load on the iC4 conversion to MTBE was also studied. For this, a fixed amount of catalyst per reactive stage has been assigned to a variable number of reactive stages, starting from the fourth stage downwards. The number of reactive stages was varied from 0 to 12, each one bearing a load of 300 kg of catalyst.

+

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Figure 15. Influence of catalyst position on iC4 conversion. Figure 13. Influence of the total catalyst load on iC4 conversion.

Figure 14. Influence of catalyst distribution on iC4 conversion.

Simulation results are shown in Figure 13 as iC4 conversion to MTBE vs total catalyst load. This curve asymptotically tends to a maximum conversion as the total charge of catalyst is increased, providing little incentive to adopt a total catalyst load larger than 2100 kg, for the conditions of the case study. Catalyst Distribution among Reactive Stages. Starting from the fourth stage downwards, the fixed total catalyst load corresponding to the case study was uniformly distributed among several contiguous reactive stages whose number was varied from 5 to 14. The simulation results of Figure 14 show that the iC4 conversion to MTBE increases with the number of reactive stages at constant total catalyst load, due to the fact that the series of CSTR’s each time approximates better the behavior of a continuous plug flow reactor. Positioning of the Catalytic Bed. The effect of the location of the sequence of contiguous reactive stages on the iC4 conversion to MTBE was also studied. Starting each time from a different stage downwards, the fixed total catalyst load corresponding to the case study was uniformly distributed among seven contiguous stages in the rectifying section. The simulation results of Figure 15 show that conversion slightly decreases when the top reactive stage is risen from the fourth position to the second one. This structural modification eliminates the rectifying section above the top reactive stage. Therefore, there is no fractionation of the MTBE leaving the top reactive stage with the vapor stream against more volatile components.

A larger impact is predicted when the reactive stages are lowered to positions close to the feed stage. In this case, the reactive stages are immersed into the lower portion of the rectifying section, thus being reached by larger amounts of MTBE coming both from the stripping section and the feed stream. The larger amount of MTBE present in the reactive section tends to lower or even to reverse the reaction rate, diminishing the overall iC4 conversion. Conclusions Simulation results show the importance of the MeOH/ total C4 feed ratio. If this ratio belongs to the C4-rich side of the MeOH-total C4 pseudobinary azeotrope, most of the MeOH fed will flow upwards from the feed stage, trying to increase its concentration in the column upper stages. Under these feed conditions, MeOH dies out rapidly in the downward direction, reaching only a few stages below the feed position, and unreacted MeOH can be removed from the top of the distillation column despite the fact that it has the lowest pure-component vapor pressure among all components. Fractionation between MTBE against C5 and C4 hydrocarbons occurs along the stripping section, and a liquid stream of MTBE containing C5 hydrocarbons can be withdrawn from the bottom. The iC4 conversion to MTBE can be increased by increasing the reflux ratio. This behavior can be linked to the extent in which MTBE (the reaction product) can be confined in the stripping section, whether it had been supplied by the column feed or produced by the synthesis reaction. Fractionation of MTBE against more volatile components in the stripping section improves by increasing R, depleting the MTBE concentration in the reaction zone with a minor modification of the iC4/ MeOH ratio and favoring the iC4 conversion to MTBE. As R becomes larger than a certain value depending on the operating conditions, the iC4 conversion can only be marginally improved by increasing the reflux ratio. For a fixed MTBE purity in the bottom stream, the reboil ratio necessary to operate the column corresponds to the chosen reflux ratio. Under these conditions, the dependence of the iC4 conversion to MTBE on the reboil ratio follows the same trend as its dependence on the reflux ratio. A warning should be made regarding the attempt to increase the MTBE purity in the bottom stream beyond a certain value depending on the operating conditions, by increasing the reboil ratio with a fixed reflux ratio. A situation can be reached where the MTBE concentration at the reactive stages reaches values large enough to deplete the iC4 conversion to MTBE.

+

+


A pressure-induced increase of the temperature profile brings about conflicting effects on the iC4 conversion to MTBE. To the favorable raise of the reaction rate constant value is opposed the unfavorable decrease of the equilibrium constant of the exothermic synthesis reaction. The larger amount of MTBE present in the reactive section adds to the unfavorable consequences of a pressure-induced increase of the temperature. An optimal operating pressure can be found for which a given conversion may be attained with a minimum R. The total catalyst load, the catalyst weight distribution among reactive stages, and the positioning of the reactive stages in the rectifying section are design parameters with a great impact on the iC4 conversion to MTBE. In this work, the sensibility of the reactive column performance to changes on the operating and design parameters has been explored, rather than attempting an optimization of its design and operation. Further work will include the modeling of the main synthesis reactor that produces the feed stream to the finishing column and the optimization of the entire reaction section design and operation.

Financial support from Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas and Universidad Nacional del Litoral is gratefully acknowledged.

Pv, j: saturation pressure of species i at the temperature of stage j, bar Qj: rate of stage heating or cooling, J/h r: specific rate of etherification reaction, mol h-1 g of catalyst-1 R: reflux ratio, nondimensional rj: rate of etherification reaction at stage j, mol/h ri, j: component rate of reaction at stage j, mol/h Sj: total flow rate of side stream withdrawn from vapor stream exiting stage j, mol/h Si, j: component flow rate of side stream withdrawn from vapor stream exiting stage j, mol/h T: absolute temperature, K Tj: absolute temperature of stage j, K Uj: total flow rate of side stream withdrawn from liquid stream exiting stage j, mol/h Ui, j: component flow rate of side stream withdrawn from liquid stream exiting stage j, mol/h Vj: total flow rate of vapor stream exiting stage j, mol/h Vi, j: component flow rate of vapor stream exiting stage j, mol/h wj: amount of catalyst charged to stage j, g x: one-dimensional array of independent variables (Naphtali-Sandholm method) xi, j: mole fraction of component i in the liquid stream from stage j Xj: one-dimensional array of component mole fractions in the liquid stream from stage j Yj: one-dimensional array of component mole fractions in the vapor stream from stage j

Nomenclature

Greek Symbols

Acknowledgment

ai: activity of species i in the liquid phase; rational symmetric scale, nondimensional A: equilibrium constant parameter, K B: equilibrium constant parameter, nondimensional CiC4: isobutene concentration, mol/L CMeOH: methanol concentration, mol/L D: total distillate flow rate, mol/h Deq: departure from chemical equilibrium, nondimensional Di: component distillate molar flow rate, mol/h Ehj: energy balance discrepancy function for stage j, mol/h Emi, j: component balance discrepancy function for stage j, mol/h Epi, j: phase equilibrium discrepancy function for stage j, mol/h Fj: feed total flowrate at stage j, mol/h Fi, j: feed component flow rate at stage j, mol/h Fk: kth element in the one-dimensional array of balance discrepancy functions, mol/h HFj: enthalpy of feed stream at stage j, J/mol HLj: enthalpy of liquid stream from stage j, J/mol HVj: enthalpy of vapor stream from stage j, J/mol k: global kinetic constant of the rate controlling reaction, mol h-1 g of catalyst-1 k+: forward kinetic constant of the rate controlling reaction, mol h-1 g of catalyst-1 k-: backward kinetic constant of the rate controlling reaction, mol h-1 g of catalyst-1 Ka: equilibrium constant of the etherification reaction, nondimensional Ki, j: phase equilibrium ratio of component i in stage j Ks,i: equilibrium sorption constant of species i, nondimensional Lj: total flow rate of liquid stream from stage j, mol/h Li, j: component flow rate of liquid stream from stage j, mol/h M: number of components N: number of stages, including the column reboiler and total condenser Pj: absolute total pressure at stage j, bar

γi, j: activity coefficient of component i in the liquid stream from stage j φ°i: fugacity-pressure ratio of species i at its saturation pressure, nondimensional φi: fugacity-pressure ratio of component i in a vapor mixture, nondimensional φ°i, j: fugacity-pressure ratio of species i at its saturation pressure corresponding to the temperature of stage j, nondimensional φi, j: fugacity-partial pressure ratio of species i in the vapor mixture leaving stage j, nondimensional θi: fraction of total sites occupied by species i, nondimensional νi: stoichiometric coefficient of species i

Literature Cited Alm, K.; Ciprian, M. Vapor Pressures, Refractive Index at 20.0 C, and Vapor-Liquid Equilibrium at 101.325 kPa in the Methyl tert-Butyl Ether-Methanol System. J. Chem. Eng. Data 1980, 25, 100-103. Ancillotti, F.; Massi Mauri, M.; Pescarollo, E.; Romagnoni, L. Mechanism in the Reaction Between Olefins and Alcohols Catalyzed by Ion Exchange Resins. J. Catal. 1978, 4, 37-48. Benson, S. W. Thermochemical Kinetics; John Wiley & Sons: New York, 1976. Churkin, V. N.; Gorhskov, V. A.; Pavlov, S. Yu.; Levicheva, E. N.; Karpacheva, L. L. Liquid-Vapor Equilibria in C4 Hydrocarbons-Methanol Systems. Zh. Fiz. Khim. 1978, 52, 488489. Colombo, F.; Cori, L.; Dalloro, L.; Delogu, P. Equilibrium Constant for the Methyl tert-Butyl Ether Liquid-Phase Synthesis by Use of UNIFAC. Ind. Eng. Chem. Fundam. 1983, 22, 219-223. De Garmo, J. L.; Parulekar, V. N.; Pinjala, V. Consider Reactive Distillation. Chem. Eng. Prog. 1992, (March), 43-50. Dongarra, J. J.; Moler, C. B.; Bunch, J. R.; Stewart, G. W. LINPACK User’s Guide; SIAM: Philadelphia, PA, 1979. Fredenslund, A.; Gmehling, J.; Rasmussen, P. Vapor-Liquid Equilibria Using UNIFAC; Elsevier: Amsterdam, 1977. Gicquel, A.; Torck, B. Synthesis of Methyl Tertiary Butyl Ether Catalyzed by Ion-Exchange Resin. Influence of Methanol Concentration and Temperature. J. Catal. 1983, 83, 9-18.

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2708 Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 Izquierdo, J. F.; Cunill, F.; Vila, M.; Tejero, J.; Iborra, M. Equilibrium Constants for Methyl tert-Butyl Ether LiquidPhase Synthesis. J. Chem. Eng. Data 1992, 37, 339-343. King, C. J. Separation Processes; McGraw-Hill: New York, 1981. Leu, A. D.; Robinson, D. B. Equilibrium Phase Properties of the Methanol-Isobutane Binary System. J. Chem. Eng. Data 1992, 37, 10-13. Leu, A. D.; Chen, C. J.; Robinson, D. B. Vapor-Liquid Equilibrium in Selected Binary Systems. AIChE Symp. Ser. 1989, 85 (271), 11-16. Mullins, S. R.; Oehlert, L. A.; Wileman, K. P.; Manley, D. B. Experimental Vapor-Liquid Equilibria for the Methanol/Dimethylsulfide, Methanol/Methyl Tert-Butyl Ether, and NHexane/N,N-Diethylmethylamine Systems. AIChE Symp. Ser. 1989, 85 (271), 94-101. Naphtali, M. A.; Sandholm, D. P. Multicomponent Separation Calculations by Linearization. AIChE J. 1971, 17 (1), 148153. Nijhuis, S. A.; Kerkhof, F. P. J. M.; Mak, A. N. S. Multiple

Steady States During Reactive Distillation of Methyl tert-Butyl Ether. Ind. Eng. Chem. Res. 1993, 32, 2767-2774. Oscarson, J. L.; Lundell, S. O.; Cunningham, J. R. Phase Equilibria for Ten Binary Systems. AIChE Symp. Ser. 1987, 256 (83), 1-17. Rehfinger, A.; Hoffmann, U. Kinetics of Methyl Tertiary Butyl Ether Liquid Phase Synthesis Catalyzed by Ion Exchange ResinsI. Intrinsic Rate Expression in Liquid Phase Activities. Chem. Eng. Sci. 1990, 45 (6), 1605-1617. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1987.

Received for review September 25, 1995 Accepted April 23, 1996X IE9505930 X Abstract published in Advance ACS Abstracts, June 15, 1996.

Modeling, Analysis, and Simulation of a Methyl tert

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