Ind. Eng. Chem. Res. 2006, 45, 6007-6020
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SEPARATIONS A Method for Modeling Two- and Three-Phase Reactive Distillation Columns Rahman Khaledi and P. R. Bishnoi* Department of Chemical and Petroleum Engineering, The UniVersity of Calgary, 2500 UniVersity DriVe NW, Calgary, Alberta, Canada T2N 1N4
An algorithm for the steady-state simulation of two- and three-phase multistage reactive distillation processes with equilibrium chemical reactions is developed. In the developed algorithm, the phase stability, phase equilibrium, and chemical reaction equilibrium calculations are preformed simultaneously. The algorithm was used to simulate a variety of two- and three-phase reactive distillation processes. The simulation results were compared against the available experimental data in the literature. Good agreement was observed between the simulation results and experimental data. A multistage reactive distillation process for the production of cyclohexanol from the hydration of cyclohexene with two liquid phases present on multiple stages was simulated. A high-purity cyclohexanol product with a high cyclohexene conversion was obtained for this process. Introduction Reactive distillation is a process in which the chemical reaction and separation happen continuously in a single operation unit. It is generally used for performing equilibrium-limited chemical reactions. This technology offers significant benefits over conventional processes, such as the elimination of a separate reaction vessel, fewer separation units, high conversion of reactants, improved selectivity of products, and reduced reboiler duty in the case of exothermic reactions. The reactive distillation process has been long known in the chemical industry.1 However, it is only during the past decade that there has been a tremendous amount of interest and an increase in the number of publications on this subject.2 This process recently became very important in the production of fuel additives such as methyl tert-butyl ether (MTBE), ethyl tert-butyl ether (ETBE), and tert-amyl methyl ether (TAME), and also in the production of many other chemicals such as esters and alcohols.3 Most of the published papers on the modeling of reactive distillation primarily involve the development of methods for two-phase (vapor-liquid) reactive distillation columns. Taylor and Krishna2 have provided an excellent review on the recent developments in the modeling of reactive distillation processes. Even though there has been a considerable amount of research on the modeling of reactive distillation systems, there has been a little research done on reactive distillation processes involving liquid-phase splitting. Liquid-phase splitting may occur on one or more stages within the reactive distillation column when highly nonideal mixtures are being processed. An example of such cases is the heterogeneous azeotrope reactive systems. In this case, a low-boiling azeotrope vapor mixture is formed in the top section of the reactive distillation column. This vapor mixture splits into two liquid phases in the decanter after being condensed in the condenser. Hexyl acetate4 and butyl acetate5-8 production via reactive distillation are two examples of the * To whom correspondence should be addressed. Tel: +(403) 2206695. Fax: +(403) 282-3945. E-mail address:
[email protected].
heterogeneous azeotrope reactive distillation process. In other cases, the presence of partially miscible components in the system results in liquid-phase splitting under certain conditions. In such systems, two liquid phases might be found on several stages, depending on the operating conditions of the column. An example of such cases is the production of cyclohexanol via a reactive distillation process.9,10 In this system, the presence of organic compounds (cyclohexanol, cyclohexene, and cyclohexane) that are partially miscible in water cause the liquid to split into aqueous and organic liquid phases. In recent studies on the simulation of butyl acetate reactive distillation columns, to model the liquid-phase splitting in the column, only a liquid-liquid phase equilibrium model was considered for the decanter.5,8 Schmitt et al.4 conducted an experimental study for the production of hexyl acetate via reactive distillation. In their simulation of experiments, they did not model the decanter, and the reflux to the column was considered to be an independent feed with the conditions similar to the measured data from the experiments. Qi and Sundmacher11 and Qi et al.9 developed a systematic approach to study the feasibility of reactive distillation for the systems with liquidphase splitting using residue curve maps. The residue curve maps analysis of Steyer et al.,10 from the same research group, on a batch reactive distillation for the production of cyclohexanol, shows that the synthesis of cyclohexanol via a continuous multistage three-phase reactive distillation column is conceptually feasible. Gumus and Ciric12 proposed a bilevel optimization approach to explore the design of reactive distillation columns that may involve liquid-phase splitting on the stages. Their proposed approach includes a Gibbs free energy minimization problem that determines the number of phases present on the stages. This Gibbs free energy minimization is nested in a larger minimization problem that minimizes the annual cost of the column. Gumus and Ciric12 demonstrated the computational performance of their algorithm by solving a single-stage multiphase reactive flash example. The modeling of multistage three-phase (VLL) reactive distillation columns involves many complexities, because of the
10.1021/ie051384a CCC: $33.50 © 2006 American Chemical Society Published on Web 07/26/2006
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Figure 2. Schematic of a reactive stage in multiphase reactive distillation column (k ) 1 denotes vapor; k ) 2, ..., π denotes liquid).
developed by Gupta et al.,14 in the governing equations of the system. The developed algorithm15 is used successfully to simulate a variety of two- and three-phase reactive distillation columns. Model Equations for Multiphase Reactive Distillation Columns
Figure 1. Schematic representing a multistage three-phase reactive distillation column.
simultaneous presence of reaction, liquid-phase splitting, and separation in the system. The main difficulty in the modeling of these processes is due to the fact that the number of phases present on each stage is not known in advance. To our knowledge, there are no published articles in the open literature on the simulation of three-phase multistage reactive distillation columns. In the present work, the development of a method for the simulation of two- and three-phase multistage reactive distillation columns with chemical reaction equilibrium on reactive stages is presented. This method is an extension of the threephase distillation without reaction developed recently by Khaledi and Bishnoi.13 In the developed method, the liquid-phase splitting is allowed at each stage and a prior knowledge of the phase pattern is not required. The phase stability test, and the phase and chemical reaction equilibrium calculations, are conducted simultaneously. The simultaneous phase stability test is accomplished by including a phase stability equation,
The schematic of a multistage three-phase reactive distillation column is illustrated in Figure 1. The column consists of Ns theoretical stages. The stages are numbered from the top to the bottom of the column. The first theoretical stage is combination of a condenser and a decanter, and the last theoretical stage is a partial reboiler. The condenser can be either a partial or total condenser. Figure 2 shows the schematic of a typical equilibrium reactive stage for the column. Multiple liquid phases may exist on the stage. An independent feed stream consisting of Nc components is introduced on the stage and Nr number of chemical reactions occur on the stage. The vapor and liquid side streams may be withdrawn from the stage and heat may be withdrawn or added to the stage, using a side heater or cooler. The following assumptions were made in formulating the mathematical model equations: (1) All the phases are completely separated from each other and are at thermodynamic equilibrium when leaving the stage. (2) Chemical reaction equilibrium conditions exist on the reactive stages. (3) The process occurs under steady-state conditions. The governing equations that are required to describe a reactive distillation column are component material balance, chemical reaction equilibrium, energy balance, mole fraction summation, phase fraction summation, phase stability, and thermodynamic phase equilibrium. These equations are written for steady-state conditions in the form of discrepancy equations for each stage as follows. It is noted that the model equations for reactive distillation are similar to distillation without reaction,13 except for the component material balance (eq 1) and chemical reaction equilibrium (eq 2). Component Material Balance Equation. The component material balance around stage j for component i can be written as π
δM ij ) Fjzij -
∑nijk + (1 - wrj+1,1)ni,j+1,1 +
k)1 π
∑
k)2
Nr
νirξjr ∑ r)1
(1 - wrj-1,k)ni,j-1,k + τj
(i ) 1, ..., Nc; j ) 1, ..., Ns) (1) The last term in eq 1 corresponds to the total number of moles
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of component i generated or consumed by the chemical reaction. νi,r is the stoichiometric coefficient of component i in reaction r and ξj,r is the extent of reaction r on stage j. The stoichiometric coefficient is negative for the reactants and positive for the products. The parameter τj is set to 1 for a reactive stage and 0 for a nonreactive stage. Chemical Reaction Equilibrium Equation. In the model, it is considered that Nr number of reactions are occurring in the system. It is noted that, at thermodynamic equilibrium, the activity of a component is equal in all liquid phases, provided that the same standard state fugacity of the component is used for all the liquid phases. Under this condition, the equation for the chemical reaction equilibrium is the same for all the liquid phases. Therefore, the chemical reaction equilibrium equation written for only one liquid phase is enough to describe the chemical reaction equilibrium condition on a reactive stage. The chemical reaction equilibrium for liquid-phase k ) 2, based on liquid-phase activity for the reactive stage j, is given by
δRrj ) (
∏i aˆ ijk|ν |)reactants ir
1 Keq rj
∏i aˆ ijk|ν |)products
(
δSt jk )
Rjkθjk Rjk + θjk
(j ) 1, ..., Ns; k ) 2, ..., π)
where Krjeq is the chemical equilibrium constant for reaction r on stage j. The chemical reaction equilibrium equation is valid only for the reactive stages where the chemical reaction occurs. Hence, the total number of equations describing chemical reaction equilibrium is the product of the number of reactions and the number of reactive stages (i.e., Nrs × Nr). Energy Balance Equation. In formulating the energy balance equation, the enthalpies are calculated using the reference state of pure elements, as discussed later. As a result, the heat of reaction is embedded in the enthalpy of the component and does not appear as an explicit term in the energy balance equation. The energy balance equation for stage j is given by
Note that the reference phase (k ) 1, which denotes the vapor phase) is always stable and the phase stability variable, θjk, for this phase is always equal to zero. Hence, the phase stability equation for this phase is excluded from the model equations. Phase Equilibrium Equation. The thermodynamic phase equilibrium equation is given by
xijk ) xij1Kijkeθjk
(i ) 1, ..., Nc; j ) 1, ..., Ns; k ) 2, ..., π) (7)
where Kijk is the equilibrium ratio, which is defined as the ratio of fugacity coefficient of the reference phase and the fugacity coefficient of phase k. That is,
Kijk )
φˆ ij1 φˆ ijk
(i ) 1, ..., Nc; j ) 1, ..., Ns; k ) 2, ..., π)
nijk ) σijkmij
π
mij )
(j ) 1, ..., Ns) (3)
Mole Fraction Summation Equation. The mole fraction summation for phase k on stage j, in terms of discrepancy, is given by
(10)
nijk (11)
π
∑nijk
ni,j+1,1hhi,j+1,1 + ∑ ∑nijkhhijk + (1 - wrj+1,1)∑ i)1 k)1 i)1
∑ ∑(1 - wrj-1,k)ni,j-1,khhi,j-1,k i)1 k)2
∑nijk
k)1
Nc
Nc π
(9)
where mij and σijk are given by
) Nc π
(8)
The component molar flow rate, nijk, may be expressed in terms of the total number of moles of component i leaving stage j (mij) and variable σijk as follows:
σijk )
Hfj + Qj -
(6)
ir
(r ) 1, ..., Nr; j ) 1, ..., Ns; k ) 2) (2)
δEj
j in the form of a discrepancy equation:
k)1
By combination and manipulation of eqs 7 and 11, an expression for variable σijk is obtained as follows:
σijk )
RjkeθjkKijk (12)
π
Rjpeθ Kijp ∑ p)1 jp
Nc
δSjk )
(xijk - xij1) ∑ i)1
(j ) 1, ..., Ns; k ) 2, ..., π) (4)
where xij1 refers to the mole fraction in the reference phase (k ) 1, which denotes the vapor phase). Phase Fraction Summation Equation. The phase fraction summation equation is given as π
δPS j )
∑
Rjk - 1
(j ) 1, ..., Ns)
(5)
k)1
Phase Stability Equation. The modified form of the phase stability equation suggested by Gupta et al.14 is written for stage
The component mole fraction, xijk, can be expressed as
xijk )
mijKijkeθjk Nc
(13) π
mlj∑Rjpeθ Kijp ∑ l)1 p)1 jp
Derivation of eqs 12 and 13 is given by Khaledi and Bishnoi.13 Equations 12 and 13 implicitly incorporate the equilibrium relation given by eq 7. Equation 13 is used to update the phase composition in the outer loop. The governing equations can be expressed in terms of mij and σijk. This simplifies the form of governing equations and their derivatives. The new set of
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governing equations in terms of variables mij and σijk is given below: Component Material Balance Equation:
δM ij ) -mij + Fij + (1 - wrj+1,1)σi,j+1,1mij+1 + Nr
π
∑
νirξjr ∑ r)1
(1 - wrj-1,k)σi,j-1,kmij-1 + τj
k)2
(i ) 1, ..., Nc; j ) 1, ..., Ns) (14) Chemical Reaction Equilibrium Equation:
δRrj ) (
1
∏i aˆ ijk|ν |)reactants ir
Keq rj
(
∏i aˆ ijk|ν |)products ir
(r ) 1, ..., Nr; j ) 1, ..., Ns; k ) 2) (15) Energy Balance Equation: Nc π
δEj ) Hfj + Qj -
∑ ∑σijkmijhhijk + i)1 k)1 Nc
(1 - wrj+1,1)
σi,j+1,1mi,j+1hhij+11 + ∑ i)1
Nc π
∑ ∑(1 - wrj-1,k)σi,j-1,kmi,j-1hhij-1,k i)1 k)2
(j ) 1, ..., Ns) (16)
Mole Fraction Summation Equation:
δSjk )
1 Nc
∑ l)1
Nc
(
σijk
∑ i)1 R
mlj
-
jk
)
Figure 3. Flowchart of the “inside-out” algorithm.
σij1 Rj1
mij
(j ) 1, ..., Ns; k ) 2, ..., π)
(17)
Model Equations Solution Algorithm
Phase Fraction Summation Equation: π
δPS j )
∑Rjk - 1 k)1
(j ) 1, ..., Ns)
(18)
Phase Stability Equation:
δSt jk )
Rjkθjk Rjk + θjk
(j ) 1, ..., Ns; k ) 2, ..., π)
(Tj), phase fractions (Rjk), and phase stability variables (θjk). This results in Ns × (Nc + 2π) + Nrs × Nr number of unknown variables.
(19)
Equations 14-19 are model equations for the steady-state simulation of multistage, multicomponent, and multiphase reactive distillation columns. The variable σijk in the aforementioned model equations is calculated using the phase equilibrium relation given by eq 12. These model equations describe twoand three-phase reactive distillation columns under chemical reaction equilibrium conditions. The advantage of the aforementioned set of equations is that they easily allow for the appearance and disappearance of liquid phases during computations.13 The total number of these equations is Ns × (Nc + 2π) + Nrs × Nr. For the specified feed conditions, side heat duties, side-withdrawal ratios, and stage pressures, we are required to solve the governing equations for the overall component stage flow rates (mij), stage reaction extents (ξj,r), stage temperatures
An approach that is similar to the “inside-out” method of Boston and Sullivan16 is implemented for solution of the model equations.13 Figure 3 shows the flow diagram of the solution algorithm. In this approach, the thermodynamic properties are approximated with simple models. These models are used to evaluate the thermodynamic properties in an inner loop. When the inner loop is converged, the parameters of approximate models are updated using rigorous thermodynamic models in an outer loop. The component activity, using a liquid activity model, is expressed as
aˆ ik ) γikxik
(20)
In the aforementioned equation, the activity coefficient is assumed to be only a function of temperature in the approximate thermodynamic model. This temperature dependency is given by
ln γik(T) ) γika - γikb
(
)
1 1 T Tb
(21)
The approximate model for the K-values and the partial molar
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residual enthalpies are presented as simple temperature-dependent functions in the following forms:
ln Kik(T) ) Kika - Kikb
(
1 1 T Tb
)
(22)
hhRik ) hika + hikb(T - Tb)
(23)
It is assumed that the parameters of the approximate models γika, γikb, Kika, Kikb, hika, and hikbsare independent of temperature and composition and are kept constant in the inner loop. These parameters are evaluated from rigorous thermodynamic models in the outer loop. The material balance, chemical reaction equilibrium, energy balance, mole fraction summation, phase fraction summation, and stability equations (eqs 14-19) form the set of inner loop equations. These equations are grouped by the stage,17 as given by eqs 24 and 25:
F ) {f1, f2, ..., fj, ..., fNs}T
(24)
partial molar residual enthalpy of component i. The standard heat of formation, ∆h°i,f298, is calculated for the formation of a compound from its pure elements, which are chosen as a reference state. The enthalpy of a pure component at the ideal gas state and temperature T, relative to the reference temperature (To) of 298 K, is calculated using the ideal-gas heat-capacity data. That is,
hig i )
S S St St St , ..., δj,π , δPS δj,k j , δj,2, ..., δj,k, ..., δj,π} (25)
The independent variable vector, which consists of overall component molar flow rates, extent of reactions, inverse of temperatures, phase stability variables, and phase fractions, is given by
X ) {x1, x2, ..., xj, ..., xNs}T
(26)
where xj is the independent variable vector for stage j, in which the variables are sorted in the following order:
{
1 xj ) m1,j, m2,j, ..., mNc,j, ξj,1, ξj,2, ..., ξj,Nr, , Tj
hhRi ) -RT2
}
Keq r )
The set of Ns × (Nc + 2π) + Nrs × Nr inner loop equations, F, are solved simultaneously for Ns × (Nc + 2π) + Nrs × Nr independent variables, X, using a combination of homotopycontinuation18 and Powell’s method.19 The phase compositions are updated using eq 13 in the outer loop. Consequently, the parameters of approximate model are updated using the calculated compositions. A more-detailed description of the method of solution of the model equations and the convergence criterion are given by Khaledi and Bishnoi.13
Keq r ) a exp
hRi hhi ) ∆h°i,f298 + hig i +h
(28)
In eq 28, hhi is the partial molar enthalpy, ∆h°i,f298 the standard hRi the heat of formation, hig i the ideal gas-state enthalpy, and h
p,x
(30)
ir
∏i
(27)
The thermodynamic properties are calculated using an equation of state for the vapor phase and a liquid activity model for the liquid phase. Enthalpy Calculations. The enthalpy of a stream is calculated using the partial molar enthalpies of its components. The partial molar enthalpy of a component i is calculated using the following equation:
∂ ln φˆ i ∂T
∏i aˆ ijk|ν |)products
(
Thermodynamic Properties Calculations
( )
Chemical Equilibrium Constant Calculations. The chemical equilibrium constant can be either evaluated from standardstate Gibbs free-energy data or from chemical equilibrium experimental measurements. In the latter method, the liquid compositions are measured at the chemical equilibrium condition for a specific temperature. Using an appropriate liquid activity model, the activities of the components are evaluated in the reaction mixture, and the chemical equilibrium constant is calculated from eq 31. The temperature dependency of the equilibrium constant can be expressed by eqs 32 and 33. The parameters a, b, c, d, e, and f in these equations are determined by fitting the equation to the experimental equilibrium constant data obtained at different temperatures:
(
θj,2, ..., θj,k, ..., θj,π, Rj,1, ..., Rj,k, ..., Rj,π
(29)
o
The equations for the calculation of the component partial molar residual enthalpy hhRi in the vapor and liquid streams are derived from the expressions for the fugacity coefficients, using an equation of state for the vapor phase and a liquid activity model for the liquid phase. The following equation is used for the derivation of the component partial molar residual enthalpy, hhRi :
where fj is the discrepancy of eqs 14-19 for stage j, which are sorted in the following order: M M R R S fj ) {δ1,j , δ2,j , ..., δNMc,j, δ1,j , δ2,j , ..., δNRr,j, δEj , δj,2 , ...,
∫TT)298K Cpigi dT
(31) aˆ ijk|νir|)reactants
(Tb + c ln T + q(T))
(32)
where
q(T) ) dT + eT2 + fT3
(33)
In the Simulation Results and Discussions section, which is presented later in this paper, the expression and the parameters for chemical equilibrium constant Keq r , are given for each of the reactions involved in the reactive distillation examples. Initialization of the Iterative Variables. To begin the iterative calculations for solving the model equations, initial values for all of the iterative variables are required. The initialization procedure for two-phase reactive distillation columns begins by initializing its temperature profile. This profile is assumed to be a linear interpolation between the condenser bubble point and reboiler dew point temperatures evaluated at the average composition of all feeds introduced to the column. A linear pressure profile is considered in the column. Vapor and liquid-phase fractions are initialized by assuming constant molal overflow in the column and initial phase-stability variables for all phases are set equal to zero. To evaluate the initial values for the component overall flow rates,
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mij, first a nonreactive form of eq 14 (by dropping the reactive term from the equation) is utilized. The component overall flow rates mij are calculated by solving the tridiagonal system of linear equations formed by this nonreactive form of component material balance equation. The component phase fractions, σijk, in eq 14 are calculated using eq 12 with ideal equilibrium ratios evaluated at stage temperatures and pressures. Subsequently, the component molar flow rate, nijk, is calculated using eq 9. To estimate the initial number of moles of the components in the presence of chemical reaction equilibrium on each stage, the component material balance and chemical reaction equilibrium equations, given by eqs 34 and 35 are solved simultaneously for each reactive stage. Nr
ni ) n°i +
νi,rξr ∑ r)1
(i ) 1, ..., Nc)
(( )) (( )) ∏i
ni
|νir|
-
Nc
nl ∑ l)1
1
∏i eq
Kr
reactants
ni
|νir|
Nc
nl ∑ l)1
(34)
)0
products
(r ) 1, ..., Nr) (35)
In eq 34, the initial number of moles, n°i, are assumed to be the liquid component molar flow rates in the liquid phase leaving stage j that were calculated in the previous step. Solution of the system of nonlinear equations (eqs 34 and 35) using the Newton-Raphson method for each stage gives the number of moles of each component, ni, in the liquid phase and the extent of reactions, ξr, for each reactive stage. Note that eq 35 is a simplified form of the chemical reaction equilibrium equation (eq 15) obtained by assuming ideal solution and replacing activities of the components by mole fractions (i.e., assuming an activity coefficient equal to 1). This simplification is acceptable because the obtained extent of reactions and component molar flow rates are used only as initial estimates for these variables. Subsequently, the initial values for the component overall flow rates mij are recalculated using eq 10. The initial liquid-phase component mole fractions are calculated from liquid component mole numbers, and the vapor-phase component mole fractions are simply calculated using thermodynamic equilibrium relation with ideal K-values. The parameters for the approximate thermodynamic models then are calculated at the initial stage temperatures and compositions, using the rigorous thermodynamic models. In a three-phase reactive distillation column, iterative calculations require a good set of initial estimates. To obtain these initial estimates, the column first is initialized as a two-phase reactive distillation column. By performing one iteration of twophase reactive distillation calculations, appropriate initial temperature, composition, and flow rate profiles are obtained. To generate the composition and phase fraction for the second liquid phase on the stages, a liquid-liquid flash calculation is performed on the liquid phase that was obtained from the twophase reactive distillation calculations in the previous step. The initial K-values for the liquid-liquid flash calculation are generated using a noniterative stability analysis method.20 The resulting compositions and phase fractions from the liquidliquid flash calculation are used as initial values for the liquid phases. If the performed liquid-liquid flash calculation does not find a second liquid on a stage, the compositions and the phase fractions of a nearest stage that has two liquid phases are used to initialize the compositions and phase fractions of the liquid phases on this stage. These compositions are used to
initialize the parameters of the approximate thermodynamic models using the rigorous thermodynamic models for the first outer-loop iteration in the three-phase reactive distillation column. Simulation Results and Discussions To demonstrate the capability of the developed algorithm in modeling a variety of reactive distillation processes, the simulation results of five different reactive distillation columns examples are presented in this article. Table 1 gives the details of column specifications for the reactive distillation examples. The first and second examples are two-phase reactive distillation columns for the production of methyl acetate and ethyl tertbutyl ether (ETBE), respectively. The third example is a heterogeneous azeotropic reactive distillation column for the production of butyl acetate. The liquid-phase splitting occurs in the decanter of this column, which makes the process a threephase reactive distillation type. To examine the validity of the simulation results, the results of the first and third examples are compared with the available experimental measurements obtained from the literature. Example 4 is another heterogeneous azeotropic reactive distillation column that is used for the production of hexyl acetate. The last example is a three-phase reactive distillation column for the production of cyclohexanol. The presence of partially miscible components in the system results in the formation of two immiscible liquid phases on many of the column stages. To our knowledge, this is the first time that a multistage reactive distillation column with two liquid phases present on the stages within the column is simulated. For all the examples discussed in this work, the PengRobinson21 equation of state is used to calculate the vapor phase thermodynamic properties. For the liquid phase, liquid activity models are used as mentioned in the discussions for each example. The ideal-gas heat capacities and standard-state fugacities are taken from Prausnitz et al.22 A computer program that was written in the C++ language was used to apply the developed algorithm in this work. The program is compiled using Visual C++ Version 5.0. The required numbers of outer-loop iterations and CPU times to converge the algorithm for the examples are given in Table 2. These examples are executed on a personal computer that is equipped with a 2.4 GHz Pentium 4 processor. Two-Phase Reactive Distillation Columns. (A) Example 1: Methyl Acetate Production. Methyl acetate is used as a solvent for lacquers, celluloses, paints, resins, coatings, and perfumes. It is also used as an intermediate chemical in synthetic flavors and artificial leather manufacturing. Methyl acetate can be produced via the esterification of methanol with acetic acid in the liquid phase in the presence of an acidic catalyst. The equilibrium reaction may be written as follows: H+
CH3COOH + CH3OH 798 CH3COOCH3 + H2O (36) The catalyst can be either a homogeneous catalyst, such as sulfuric acid,23 or a heterogeneous catalyst, such as an acidic ion-exchange resin.24,25 Conventional production of high-purity methyl acetate is very difficult, because of the chemical equilibrium limitations in the reactor and also the presence of methanol-methyl acetate and water-methyl acetate minimumboiling azeotropes in the separation section of the process. The Eastman Kodak Company’s conventional process for the production of methyl acetate is a complex process that consists of one reactor followed by nine separation units.2,3 The reactive distillation process first developed and commercialized by the
Ind. Eng. Chem. Res., Vol. 45, No. 17, 2006 6013 Table 1. Column Specifications for Two- and Three-Phase Reactive Distillation Examples Feed Conditions Compositiona component
Column Specifications
FI
FII
acetic acid methanol methyl acetate water
1.0 0.0 0.0 0.0
0.0 1.0 0.0 0.0
iso-butylene 1-butene ethanol ETBE
0.073 0.545 0.091 0.291
F6: 100
butanol acetic acid butyl acetate water
0.18 0.17 0.32 0.33
F23: 18.8
butanol acetic acid butyl acetate water
0.18 0.17 0.32 0.33
F12: 18.8
hexanol acetic acid hexyl acetate water
1.0 0.0 0.0 0.0
cyclohexane cyclohexene cyclohexanol water
0.1 0.4 0.0 0.5
a
0.0 1.0 0.0 0.0
rate (kmol/h) F8: 100 F21: 100
F8: 50 F14: 50
F6: 100
conditions
number of stages, Ns
Example 1 (see refs 23-27 and 29) saturated liquid 27 P: 0.1013 (MPa) saturated liquid P: 0.1013 (MPa) Example 2 (see refs 30 and 31) liquid 10 T: 303 K P: 0.95 (MPa) Example 3a (see refs 5-8, 32, 33) liquid 47 T: 360 K P: 0.1013 (MPa) Example 3b (see refs 5-8,32,33) liquid 25 T: 360 K P: 0.1013 (MPa) Example 4 (see ref 4) saturated liquid 23 P: 0.04 (MPa) saturated liquid P: 0.04 (MPa) Example 5 (see refs 9, 10, 34) saturated liquid 20 P: 0.1013 (MPa)
reactive stages
reflux ratio
P (MPa)
B (kmol/h)
8-20
R: 2.2
0.1013
100
3-5
R: 4
0.95
21-27
RI: 99 RII: 0 (aq.)
0.1013
8.74
11-15
RI: 99 RII: 0 (aq.)
0.1013
8.74
8-13
RI: total RII: 0 (aq.)
0.04
50
2-11
RI: 9 (org.) RII: 3.6
0.1013
40
38
FI, first feed from top; FII, second feed from top.
same company can produce high-purity methyl acetate with only one reactive distillation column.23 An Eastman Kodak plant in Tennessee produces 180 000 tons/yr of high-purity methyl acetate with only a single reactive distillation column.3 This process is an example of the success of the reactive distillation technology, and it has been studied by many authors.23-27 The column specification for Example 1 is given in Table 1, and this information is similar to the specification of the pilotplant scale column studied by Bessling et al.24 For this example, the NRTL activity model with parameters taken from Gmehling et al.28 is used to calculate the liquid phase thermodynamic properties (see Table A1 in the Appendix). The esterification reaction for producing methyl acetate over a heterogeneous catalyst (using Amberlyst 15W, which is an ion-exchange resin) was studied by Song et al.29 The activity-based reaction equilibrium constant for the esterification reaction is taken from Song et al.29 and is given by
Keq ) 2.32 exp
(782.98 T )
(37)
where the temperature T is given in Kelvin. For this example, the algorithm converged within 22 iterations and a CPU time of 16.7 s, as given in Table 2. The simulation results are presented in Figures 4-6. Figures 4 and 5 show the column temperature and liquid-phase composition profiles, respectively, for a reflux ratio of 2.2. Figure 5 shows that a high-purity distillate product (97.12 mol % methyl acetate) can be obtained at this reflux ratio, whereas virtually no methyl acetate is lost through the bottom product. Figure 6 compares
Table 2. Number of Iterations and Computational Time for Examples of Table 1
a
example
outer-loop iteration
CPU time (s)a
1 2 3a 3b 4 5
22 5 13 6 6 21
16.7 18.2 57.6 68.4 17.8 88.6
Using a 2.4 GHz Pentium 4 processor.
the simulated acetic acid conversion in this work with the experimental data obtained by Bessling et al.24 It can be observed that the simulated acetic acid conversion shows good agreement with the experimental data obtained by Bessling et al.24 The results show that the conversion increases rapidly as the reflux ratio increases until it reaches a maximum value of 97.12% at the optimum reflux ratio of 2.2. A similar experimental optimum reflux ratio was observed by Bessling et al.; this value corresponded to the maximum experimental conversion value of 96.95%. In Figure 6, the deviation between simulated acetic acid conversion and experimental data at higher reflux ratios is due to a reduction in the liquid residence time in the experimental column, where the assumption of chemical equilibrium reaction is no longer a very good assumption.24,25 Nevertheless, the chemical reaction equilibrium assumption provides an excellent prediction in the range of optimal reflux ratio. (B) Example 2: Ethyl tert-Butyl Ether (ETBE) Production. ETBE is an ether that is used as an oxygenate additive for enhancing the octane number of gasoline fuel. Recently,
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Figure 4. Stage temperature profile in Example 1 for R ) 2.2; feeds are introduced to stages 8 and 21; reactive stages include stages 8-20.
Figure 7. Stage temperature profile in Example 2; feed is introduced to stage 6; reactive stages include stages 3-5.
Figure 8. Liquid-phase composition profile in Example 2. Figure 5. Liquid-phase composition profile in Example 1 (R ) 2.2).
occurs in the liquid phase. The reaction extent is only 84.7% from a stoichiometric mixture of reactants at 70 °C. The equilibrium conditions of this reaction were studied by Jensen and Datta.30 The reaction equilibrium constant for ETBE reaction (eq 39) resulting from this study was estimated as follows:
4060.59 - 2.89055 ln T T 0.0191544T + 5.28586 × 10-5T2 - 5.32977 × 10-8T3 (39)
ln Keq ) 10.387 +
Figure 6. Comparison of the simulated acetic acid conversion in this work with the experimental data obtained by Bessling et al.24 in Example 1.
much attention has been given to ETBE production, because of its higher-octane-enhancing properties and lesser fuel vaporization loss, which is due to its lower volatility. Also, this low vapor pressure of ETBE reduces the emission of volatile organic compounds of fuel. Therefore, ETBE is considered an environmentally friendly fuel additive. ETBE is the product of the etherification reaction of isobutylene with ethanol over an acidic catalyst, such as an acidic ion-exchange resin (e.g., Amberlyst 15).
(CH3)2CdCH2 + C2H5OH T (CH3)3COC2H5
(38)
The reaction is an exothermic equilibrium limited reaction that
where T is given in Kelvin. ETBE can be produced in a reactive distillation column to eliminate the equilibrium limitation and attain a high conversion of olefin feed and produce high-purity product. Example 2 is taken from Sneesby et al.31 The thermodynamic properties for the liquid phase are calculated using the UNIQUAC activity model (see Table A2 in the Appendix). For this example, the algorithm is converged within five iterations and a CPU time of 18.2 s. The simulation results for this example are presented in Figures 7 and 8, for temperature and liquid composition profile, respectively. The composition profile shows that the bottom stream is a high-purity (95.25 mol % ETBE) product. A high isobutylene conversion of 99.96 mol % is obtained by this reactive distillation column. Three-Phase Distillation Columns. (A) Example 3: Heterogeneous Azeotropic Reactive Distillation Column, Butyl Acetate Production. Butyl acetate is a widely used solvent in coatings and paints industries. It is also used in leather treatment
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Figure 9. Comparison of simulated temperature profile in this work with the experimental data obtained by Hanika et al.6 in Example 3a; feed is introduced to stage 23; reactive stages include stages 21-27.
Figure 10. Organic liquid-phase composition profile in Example 3a.
and as an intermediate solvent in the manufacture of adhesives. Butyl acetate can be produced via the liquid-phase reaction of butanol and acetic acid in the presence of a suitable acidic catalyst. The esterification reaction is given by H+
CH3COOH + C4H9OH 798CH3COOC4H9 + H2O (40) This reaction, in the presence of an ion-exchange resin as a catalyst, has been studied by Gangadwala et al.,32 and the activity-based reaction equilibrium constant is given by
Keq ) 3.8207 exp
(430.803 T )
(41)
where T is given in Kelvin. The production of butyl acetate via reactive distillation is an example of a heterogeneous azeotropic reactive distillation process. In this process, a minimum-boiling ternary azeotrop mixture (bp 90 °C) between butyl acetate, butanol, and water is formed on the top stage of the column.7,33 The azeotropic vapor mixture splits in two organic and aqueous liquid phases in the decanter when it is condensed in the condenser. The aqueous liquid phase that contains mainly water is totally withdrawn from the column and the major portion of the organic liquid phase is refluxed to the column. The bottom stream is a high-purity butyl acetate product. Hanika et al.6 studied the production of butyl acetate experimentally, using a packed reactive distillation column. Example 3a in Table 1 gives specifications for this column. The number of theoretical stages for the column reported in Table 1 was obtained by the authors,6 using the height and the number of theoretical stages per meter (NTSM) of packing. The reactive section is located in the middle of the column, and the feed is introduced into the reactive section. The UNIQUAC model, with parameters taken from Venimadhavan et al.,7 is used to model the liquid-phase thermodynamic properties (see Table A3 in the Appendix). The simulation results show the formation of two liquid phases on the first stage (condenser-decanter). Figure 9 shows the temperature profile in the column. The figure shows that the predicted temperature profile is in very good agreement with the experimental temperature profile obtained by Hanika et al.6 Figure 10 presents the liquid-phase mole fraction profile in the column (organic liquid for the first stage). Hanika et al.6 suggested that at least 20 theoretical stages in the stripping section of the column are necessary to accomplish a good separation efficiency for separating acetic acid from butyl acetate. However, as can be seen from the simulation results in Figure 10, the liquid phase becomes almost-pure butyl acetate
Figure 11. Organic liquid-phase composition profile in Example 3b. Table 3. Distillate and Bottom Product Liquid Compositions for Example 3a (47-Stage Column) and Example 3b (25-Stage Column) Liquid Mole Fraction Example 3a component
Example 3b
aqueous organic bottom aqueous organic bottom
butanol acetic acid butyl acetate water
0.0134 0.0223 0.0012 0.9631
0.3539 0.0686 0.2037 0.3738
0.0000 0.0000 1.0000 0.0000
0.0123 0.0203 0.0012 0.9662
0.3449 0.0626 0.2359 0.3566
0.0001 0.0000 0.9999 0.0000
flow rate (mol/h)
9.1138
0.9462 8.7400
9.1355
0.9245 8.7400
in the stripping section of the column immediately after the reactive zone. This observation suggests the idea that a large number of stages in the stripping section of the column for achieving a high-purity product is not actually necessary. To examine this idea, a smaller column with total of 25 theoretical stages (Example 3b in Table 1) is simulated. Under the same operating conditions as the previous 47-stage column, the composition profile for Example 3b is shown in Figure 11 and Table 3. It is noted, from the results, that the same high-purity product is achieved for the column with 25 stages as was achieved with the 47-stage column. The computation time and number of iterations for Examples 3a and 3b are given in Table 2. (B) Example 4: Heterogeneous Azeotropic Reactive Distillation Column, Hexyl Acetate Production. Hexyl acetate is a fruity-smelling substance that is used as a flavoring agent in food industries and as aroma in perfume manufacturing. Hexyl acetate is produced via the liquid-phase esterification reaction of hexanol with acetic acid over a strong acid catalyst. The liquid-phase reversible reaction is given by H+
CH3COOH + C6H13OH 798 CH3COOC6H13 + H2O (42)
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12, because of the variation of the concentrations of hexyl acetate and acetic acid (nbp 118.1 °C), which is introduced as feed on stage 14. Figure 13 shows the organic liquid-phase composition profile. This figure shows that a high-purity bottom product (99.91 mol % hexyl acetate) can be produced by this process. This example converged within six iterations and a CPU time of 17.8 s. (C) Example 5: Three-Phase Reactive Distillation Column, Cyclohexanol Production. Cyclohexanol is a saturated alicyclic alcohol that is used in the production of intermediates for nylons, plasticizers, pesticides, and detergents. Cyclohexanol can be produced via the hydration of cyclohexene with water over a suitable catalyst. The hydration reaction is given by cat
cyclohexene + water 798 cyclohexanol Figure 12. Stage temperature profile in Example 4. Feeds are introduced to stages 7 and 14; reactive stages include stages 8-13.
Figure 13. Organic liquid-phase composition profile in Example 4.
This esterification reaction in the presence of an ion-exchange resin (Amberlyst CSP2) as a catalyst has been studied by Schmitt et al.4 They estimated an activity-based reaction equilibrium constant that is independent of temperature, given by
Keq ) 31.9281
(43)
This liquid-phase equilibrium reaction can be performed in a reactive distillation column. The column specifications for the production of hexyl acetate by the reactive distillation process are given in Example 4 of Table 1. The NRTL model with parameters taken from Schmitt et al.4 is used to calculate the liquid phase thermodynamic properties (see Table A4 in the Appendix). The simulation results of this example show the formation of two liquid phases on the first stage (condenser-decanter). The aqueous phase contains mainly water (99.92 mol % H2O) and is withdrawn completely from the decanter. Figure 12 shows the stage temperature profile in the column. This figure shows that the stage temperature decreases after stage 8, where the hexanol (nbp 157.2 °C) feed stream is introduced to the column. After stage 10, the stage temperature increases as the concentration of hexanol decreases (by consumption in the chemical reaction) and the concentration of hexyl acetate, which is a compound with a higher boiling point (nbp 171.3 °C), increases via production in the chemical reaction (Figure 13). Another decline and climb in stage temperature is observed after stage
(44)
The industrial process for the production of cyclohexanol, which was developed by the Asahi Chemical Co., consists of a slurry reactor, followed by a decanter vessel and a distillation column.10 The hydration reaction occurs in the slurry reactor, where cyclohexene and water come into contact with a very fine catalyst (zeolite catalyst, H-ZSM 5). Subsequently, the reaction product is separated into organic and aqueous phases in the decanter. The catalyst is recovered in the liquid phase and recycled to the reactor, and the organic phase that contains the cyclohexanol product is sent to the distillation section for purification. The feasibility of cyclohexanol production via a reactive distillation process was examined by Steyer et al.10 and Qi et al.,9 using batch reactive distillation experiments and residue curve map studies. Their residue curve maps study shows that the production of high-purity cyclohexanol via a reactive distillation process is conceptually possible. However, as they indicated, the simulation of a multistage reactive column for this process involves many complexities, because of the simultaneous presence of reaction, distillation, and liquid-phase splitting in the system. Using the algorithm developed in the present paper, a threephase reactive distillation column for the production of cyclohexanol is simulated. The column specifications are given in Table 1. The feed is a mixture of water and hydrocarbon compounds. The hydrocarbon mixture is considered to be a mixture of cyclohexene as the reactive component and cyclohexane as an inert impurity that comes from an upstream unit. The UNIQUAC model with parameters taken from Gmehling et al.28 is used to model the liquid-phase thermodynamic properties (see Table A5 in the Appendix). The liquid-phase hydration of cyclohexene catalyzed by a strong acid ionexchange resin was studied by Panneman and Beenackers.34 The activity-based reaction equilibrium constant for this reaction is given by
Keq ) 2.37 × 10-5 exp
(3636.757 ) T
(45)
where T is given in Kelvin. Simulation results for this example are presented in Figures 14-18. The computation for this example converged within 21 iterations and a CPU time of 88.6 s. Figure 14 shows the vapor and liquid profiles in the column. As it can be seen, two liquid (organic and aqueous) phases are present from stage 1 through stage 18. The second liquid (aqueous) phase does not exist on stages 19 and 20. The organic liquid phase leaves the bottom of the column as a high-purity cyclohexanol product. This phase is partially (10% of organic liquid) withdrawn from the top of
Ind. Eng. Chem. Res., Vol. 45, No. 17, 2006 6017
Figure 14. Vapor and liquid flow rates in Example 5.
Figure 17. Effect of the organic liquid reflux ratio (RI) on the cyclohexene conversion and bottom product organic-phase cyclohexanol purity in Example 5.
Figure 15. Organic-phase (liquid I, LI) composition profile in Example 5. Figure 18. Effect of the organic liquid reflux ratio (RI) on aqueous-phase (liquid II, LII) fraction profile in Example 5.
Figure 16. Stage temperature profile in Example 5.
the column as a distillate product that contains a high concentration of cyclohexane. The organic liquid phase composition profile is shown in Figure 15. The aqueous phase contains mainly water. The temperature profile is shown in Figure 16. The cyclohexene conversion, cyclohexanol purity in the bottom product, and the phase pattern in the column are all dependent on the operating conditions of reactive distillation column. The effect of the organic phase reflux ratio on the cyclohexene conversion and the cyclohexanol purity of organic phase in the bottom product is examined and shown in Figure 17. The cyclohexene conversion increases as the organic phase reflux ratio increases until it reaches a maximum (∼98% conversion) near an organic reflux ratio of RI ) 9. It then begins to decrease with further increases in the reflux ratio. The cyclohexanol purity of the bottom product does not show a major change with increases in the organic reflux ratio until it reaches an organic reflux ratio of RI ≈ 0.5, where the phase fraction of aqueous liquid approaches to zero on the last stage.
It then increases with the disappearance of the aqueous liquid phase on the last stage with further increases in reflux ratio until it reaches a maximum at RI ≈ 9, which is the same reflux ratio for maximum cyclohexene conversion. Therefore, there is an optimum reflux ratio in which both conversion and purity are at their maximum. It is important, from an operational point of view, to keep the reflux ratio near its optimum value for high efficiency of the reactive distillation column. It is also important to know that an increase in reflux ratio does not always increase the purity and conversion. Hence, a specific consideration should be given in choosing a control strategy for this type of column. The phase pattern within the column also is dependent on the organic reflux ratio. Figure 18 shows the aqueous liquid (LII) phase fraction profile for different organic liquid reflux ratios. As it can be seen, two liquid phases are present on all stages for an organic reflux ratio of RI ) 0.5. The aqueous liquid phase begins to disappear gradually from the bottom stages of the column as the organic phase reflux ratio increases. Conclusions An algorithm for the simulation of two- and three-phase multistage reactive distillation columns was developed. In this algorithm, the phase stability, phase equilibrium, and chemical reaction equilibrium calculations in three-phase reactive distillation columns are performed simultaneously. A prior knowledge of the phase pattern in the column is not required. The resulting nonlinear model equations in the reactive distillation system are solved using an “inside-out” method. The algorithm was used to simulate a variety of two- and three-phase reactive distillation columns. The results of five
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different reactive distillation examples are presented in this work. The simulation results of two examples were compared with the available experimental data in the literature. Good agreement was observed between the simulated results and the experimental data. The algorithm was efficient and robust and converged within a reasonable number of iterations and computational time for all the examples. The algorithm was used to simulate a multistage three-phase reactive distillation for the production of cyclohexanol. It was observed that a high-purity cyclohexanol product and a high reactant (cyclohexene) conversion can be obtained using the reactive distillation process. The high-purity product was obtained when a single liquid phase existed on the last stage. The cyclohexanol reactive column performance was studied under different operating conditions. The process exhibited a maximum conversion and product purity at a specific optimum reflux ratio. The ability of the algorithm in determining the phase pattern within the reactive distillation column was shown in the cyclohexanol reactive column by changing the column operating conditions. Acknowledgment The financial support for this work, provided by the Natural Science and Engineering Research Council (NSERC) of Canada, is greatly appreciated.
Table A3. UNIQUAC Binary Parameters (aij(K)) and UNIQUAC Structural Parameters for Example 3 aij(K)
i ) butanol i ) acetic acid i ) butyl acetate i ) water
j) butanol
j) acetic acid
j) butyl acetate
j) water
0 -66.308 12.398 292.444
74.619 0 358.409 265.662
41.532 -150.178 0 232.222
34.223 -172.902 345.062 0
component
r
q
q′
butanol acetic acid butyl acetate water
3.454 2.202 4.827 0.920
3.052 2.072 4.196 1.400
3.052 2.072 4.196 1.400
Table A4. NRTL Binary Parameters (aij(K) and βij)a and NRTL rij Parameters for Example 4 j) hexanol i ) hexanol i ) acetic acid i ) hexyl acetate i ) water
j) water
-1049.70 -1489.92 0 3545.58
-690.50 119.03 998.70 0
0 0 -4.065 18.510
0 0 -5.456 1.805
3.241 4.290 0 -1.748
3.466 -0.746 -1.315 0
Combination i-j
Table A1. NRTL Binary Parameters (aij (K))a and NRTL rij Parameters for Example 1 aij(K)
a
j) acetic acid
j) methanol
j) methyl acetate
j) water
0 -78.839 733.371 1360.576
54.402 0 309.351 560.141
-398.460 76.818 0 762.181
-693.496 -203.072 240.178 0
Combination i-j component i
component j
Rij ) Rji
acetic acid acetic acid acetic acid methanol methanol methyl acetate
methanol methyl acetate water methyl acetate water water
0.3067 0.3026 0.1463 0.2968 0.3004 0.2152
Used in eq A.2 in the Appendix.
Table A2. UNIQUAC Binary Parameters (aij(K)) and UNIQUAC Structural Parameters for Example 2
component i
component j
Rij ) Rji
hexanol hexanol hexanol acetic acid acetic acid hexyl acetate
acetic acid hexyl acetate water hexyl acetate water water
0.3 0.3 0.3 0.3 0.3 0.2
Used in eq A.3 in the Appendix.
Table A5. UNIQUAC Binary Parameters (aij(K)) and UNIQUAC Structural Parameters for Example 5 aij(K) j) j) j) cyclohexane cyclohexene cyclohexanol
j) water
-49.121 0 76.514 466.350
1247.300 1024.100 159.289 0
i ) cyclohexane i ) cyclohexene i ) cyclohexanol i ) water
0 60.813 -150.907 540.360
component
r
q
q′
4.046 3.814 4.349 0.920
3.240 3.027 3.512 1.400
3.240 3.027 1.780 1.000
The NRTL equation is given as Nc
j ) iso-butene
j ) 1-butene
j ) ethanol
j ) ETBE
0 -23.894 436.034 39.215
24.245 0 404.721 42.130
-46.937 -26.930 0 -102.322
-21.484 -20.041 424.521 0
component
r
q
q′
iso-butene 1-butene ethanol ETBE
2.92 2.92 2.11 5.86
2.68 2.56 1.97 4.94
2.68 2.56 0.92 4.94
448.368 -53.743 0 128.476
cyclohexane cyclohexene cyclohexanol water
aij(K) i ) iso-butene i ) 1-butene i ) ethanol i ) ETBE
j) hexyl acetate
βij
i ) hexanol i ) acetic acid i ) hexyl acetate i ) water
The NRTL and UNIQUAC parameters are given in Tables A1, A2, A3, A4, and A5 for Examples 1, 2, 3, 4, and 5, respectively.
a
aij(K) 579.91 0 2184.61 -86.26
0 -269.12 1522.47 -3501.50
Appendix: NRTL and UNIQUAC Parameters
i ) acetic acid i ) methanol i ) methyl acetate i ) water
j) acetic acid
ln γi )
∑ j)1
( ) Nc
τjiGjixj
Nc
Glixl ∑ l)1
Nc
+
xjGij
xrτrjGrj ∑ r)1
τij ∑ N N j)1 Gljxl Gljxl ∑ ∑ l)1 l)1 c
(A.1)
c
where
τij )
aij T
(A.2)
Ind. Eng. Chem. Res., Vol. 45, No. 17, 2006 6019
or
τij )
aij + βij T
(A.3)
and
Gji ) exp(-Rjiτji)
(Rij ) Rji)
(A.4)
The UNIQUAC equation is given as
ln γi ) ln
( ) ( ) ( ) ( )∑ Φi xi
+
z
2
Θi
qi ln
Φi
+ li -
Nc
q′i ln(
∑ j)1
Φi xi
Nc
xjlj -
j)1 Nc
Θ′jτij
∑ N j)1 ∑Θ′kτkj k)1
Θ′jτji) + q′i - q′i
(A.5)
c
where
z li ) (ri - qi) - (ri - 1) 2 Φi )
(for z ) 10)
(A.6)
rixi (A.7)
Nc
rjxj ∑ j)1 Θi )
Greek Symbols
qixi (A.8)
Nc
qjxj ∑ j)1 Θ′i )
q′ixi (A.9)
Nc
q′jxj ∑ j)1 τij ) exp
( ) -aij T
Hf ) feed enthalpy (kJ) K ) equilibrium ratio Keq ) chemical reaction equilibrium constant l ) parameter given by eq A.6 L ) liquid flow rate (kmol/h) m ) total molar flow rate (kmol/h) n ) molar flow rate (kmol/h) Nc ) number of components Ns ) number of stages Nr ) number of reactions Nrs ) number of reactive stages q ) UNIQUAC parameter in eqs A.5, A.6, A.8, and A.9 Q ) stage energy input (kJ) r ) UNIQUAC parameter in eqs A.6 and A.7 R ) reflux ratio R ) universal gas constant (kJ kmol-1 K-1) Ri ) generation of component i due to reaction(s) (kmol) RL ) liquid reflux rate (kmol/h) RVL ) vapor liquid ratio T ) temperature (K) V ) vapor flow rate (kmol/h) WL ) liquid side withdrawal (kmol/h) WV ) vapor side withdrawal (kmol/h) wr ) side-withdrawal ratio x ) mole fraction or stage independent variable vector X ) independent variables vector z ) feed mole fraction
(A.10)
Nomenclature a ) coefficient in eq 32 or in eqs A.2, A.3, and A.10 aˆ i ) activity of component i in the mixture b ) coefficient in eq 32 B ) bottom product molar rate (kmol/h) BL ) bottom liquid product (kmol/h) c ) coefficient in eq 32 -1 kmol-1) Cig p ) ideal-gas heat capacity (kJ K d ) coefficient in eq 33 DL ) distillate liquid product (kmol/h) DV ) distillate vapor product (kmol/h) e ) coefficient in eq 33 f ) model equation functions or coefficient in eq 33 ˆfi ) fugacity of component i in the mixture F ) model equations vector F ) feed molar rate (kmol/h) G ) NRTL parameter given by eq A.4 hhi ) partial molar enthalpy of component i in the mixture (kJ/ kmol) hig i ) ideal gas molar enthalpy of component i (kJ/kmol) hhRi ) partial molar residual enthalpy of component i in the mixture (kJ/kmol)
R ) phase fraction or parameter in eq A.4 β ) parameter in eq A.3 ∆h°i,f298 ) standard heat of formation of component i (1 atm, 25 °C) δ ) discrepancy or function residual Φ) UNIQUAC segment fraction given by eq A.7 φˆ i ) fugacity coefficient of component i in the mixture γi ) activity coefficient of component i in the mixture ν ) stoichiometric coefficient π ) number of phases θ ) phase stability variable Θ ) UNIQUAC area fraction given by eqs A.8 and A.9 σ ) component phase fraction τ ) stage parameter or interaction parameter given by eqs A.2, A.3, and A.10 ξ ) extent of reaction Literature Cited (1) Backhaus, A. A. Continuous Processes for the Manufacture of Esters. U.S. Patent No. 1,400,849, 1921. (2) Taylor, R.; Krishna, R. Modeling Reactive Distillation. Chem. Eng. Sci. 2000, 55, 5183. (3) Sundmacher, K., Kienle, A., Eds. ReactiVe Distillation: Status and Future Directions; Wiley-VCH Verlag GmbH & Co.: Weinheim, Germany, 2001. (4) Schmitt, M.; Hasse, H.; Althaus, K.; Schoenmakers, H.; Go¨tze, L.; Moritz, P. Synthesis of n-Hexyl Acetate by Reactive Distillation. Chem. Eng. Process. 2004, 43, 397. (5) Zhicai, Y.; Xianbao, C.; Jing, G. Esterification-Distillation of Butanol and Acetic Acid. Chem. Eng. Sci. 1998, 53, 2081. (6) Hanika, J.; Kolena, J.; Smejkal, Q. Butyl Acetate via Reactive DistillationsModeling and Experiment. Chem. Eng. Sci. 1999, 54, 5205. (7) Venimadhavan, G.; Malone, M. F.; Doherty, M. F. A Novel Distillate Policy for Batch Reactive Distillation with Application to the Production of Butyl Acetate. Ind. Eng. Chem. Res. 1999, 38, 714. (8) Gangadwala, J.; Kienle, A.; Stein, E.; Mahajani, S. Production of Butyl Acetate by Catalytic Distillation: Process Design Studies. Ind. Eng. Chem. Res. 2004, 43, 136.
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(9) Qi, Z.; Kolah, A.; Sundmacher, K. Residue Carve Maps for Reactive Distillation Systems with Liquid-Phase Splitting. Chem. Eng. Sci. 2002, 57, 163. (10) Steyer, F.; Qi, Z.; Sundmacher, K. Synthesis of Cyclohexanol by Three-Phase Reactive Distillation: Influence of Kinetics on Phase Equilibria. Chem. Eng. Sci. 2002, 57, 1511. (11) Qi, Z.; Sundmacher, K. Bifurcation Analysis of Reactive Distillation Systems with Liquid-Phase Splitting. Comput. Chem. Eng. 2002, 26, 1459. (12) Gumus, Z. H.; Ciric, A. R. Reactive Distillation Column Design with Vapor/Liquid/Liquid Equilibria. Comput. Chem. Eng. 1997, 21, S983. (13) Khaledi, R.; Bishnoi, P. R. A Method for Steady-State Simulation of Multistage Three-Phase Separation Columns. Ind. Eng. Chem. Res. 2005, 44, 6845. (14) Gupta, A. K.; Bishnoi, P. R.; Kalogerakis, N. A Method for the Simultaneous Phase Equilibria and Stability Calculations for Multiphase Reacting and Non-Reacting Systems. Fluid Phase Equilib. 1991, 63, 65. (15) Khaledi, R. Simulation of Multiphase Reactive and Non-Reactive Separation Processes. Ph.D. Thesis, University of Calgary, Calgary, Alberta, Canada, 2005. (16) Boston, J. F.; Sullivan, S. L. A New Class of Solution Methods for Multicomponent, Multistage Separation Processes. Can. J. Chem. Eng. 1974, 52, 52. (17) Naphthali, L. M.; Sandholm, D. P. Multicomponent Separation Calculations by Linearization. AICHE J. 1971, 17, 148. (18) Rheinboldt, W.; Burkardt, J. A Program for a Locally Parametrized Continuation Process. ACM Trans. Math. Soft. 1983, 9, 236. (19) Powell, M. J. D. A Hybrid Method for Non-Linear Equations. In Numerical Methods for Nonlinear Algebraic Equations; Rabinowitz, P., Ed.; Gordon and Breach: New York, 1970. (20) Trebble, M. A. A Preliminary Evaluation of Two and Three Phase Flash Initiation Procedures. Fluid Phase Equilib. 1989, 53, 113. (21) Peng, D. Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. (22) Prausnitz, J. M.; Anderson, T. F.; Grens, E. A.; Eckerts, C. A.; Hsieh, R.; O’Connell, J. P. Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria; Prentice Hall: Englewood Cliffs, NJ, 1980. (23) Agreda, V. H.; Partin, L. R.; Heise, W. H. High-Purity Methyl Acetate via Reactive Distillation. Chem. Eng. Prog. 1990, 86, 40.
(24) Bessling, B.; Lo¨ning, J. M.; Ohligschla¨ger, A.; Schembecker, G.; Sundmacher, K. Investigations on the Synthesis of Methyl Acetate in a Heterogeneous Reactive Distillation Process. Chem. Eng. Technol. 1998, 21, 393. (25) Po¨pken, T.; Steinigeweg, S.; Gmehling, J. Synthesis and Hydrolysis of Methyl Acetate by Reactive Distillation Using Structured Catalytic Packings: Experiments and Simulation. Ind. Eng. Chem. Res. 2001, 40, 1566. (26) Barbosa, D.; Doherty, M. F. Design and Minimum-Reflux Calculations for Double-Feed Multicomponent Reactive Distillation Columns. Chem. Eng. Sci. 1988, 43, 2377. (27) Huss, R. S.; Chen, F.; Malone, M. F.; Doherty, M. F. Reactive Distillation for Methyl Acetate Production. Comput. Chem. Eng. 2003, 27, 1855. (28) Gmehling, J.; Onken, U. Vapor-Liquid Equilibrium Data Collection; DECHEMA Chemistry Data Series, Vol. 1, Part 1; 1977. (29) Song, W.; Venimadhavan, G.; Manning, J. M.; Malone, M. F.; Doherty, M. F. Measurement of Residue Carve Maps and Heterogeneous Kinetics in Methyl Acetate Synthesis. Ind. Eng. Chem. Res. 1998, 37, 1917. (30) Jensen, K. L.; Datta, R. Ethers from Ethanol. 1. Equilibrium Thermodynamic Analysis of the Liquid-Phase Ethyl tert-Butyl Ether Reaction. Ind. Eng. Chem. Res. 1995, 34, 392. (31) Sneesby, M. G.; Tade’, M. O.; Datta, R.; Smith, T. N. ETBE Synthesis via Reactive Distillation. 1. Steady-State Simulation and Design Aspects. Ind. Eng. Chem. Res. 1997, 36, 1855. (32) Gangadwala, J.; Mankar, S.; Mahajani, S. Esterification of Acetic Acid with Butanol in the Presence of Ion-Exchange Resins as Catalysts. Ind. Eng. Chem. Res. 2003, 42, 2146. (33) Lo¨ning, S.; Horst, C.; Hoffmann, U. Theoretical Investigations on the Quaternary System n-Butanol, Butyl Acetate, Acetic Acid and Water. Chem. Eng. Technol. 2000, 23, 789. (34) Panneman, H. J.; Beenackers, A. A. C. M. Solvent Effects in the Liquid-Phase Hydration of Cyclohexene Catalyzed by a Macroporous Strong Acid Ion Exchange Resin. Chem. Eng. Sci. 1992, 47, 2635.
ReceiVed for reView December 12, 2005 ReVised manuscript receiVed May 10, 2006 Accepted June 19, 2006 IE051384A
