Analysis of Steady-State Multiplicity in Reactive Distillation Columns

Article pubs.acs.org/IECR

Analysis of Steady-State Multiplicity in Reactive Distillation Columns Jalesh L. Purohit, Sanjay M. Mahajani,* and Sachin C. Patwardhan Department of Chemical Engineering, Indian Institute of Technology, Bombay, Mumbai 400076, India ABSTRACT: We propose a relatively simple method to identify multiplicity in reactive distillation due to the interaction of reaction and distillation. The nonlinear generation term and linear removal term, for both material and energy, are isolated from the conservation equations and evaluated at different values of suitable parameters such as product ratios (distillate to bottoms). The plot thus obtained is used to identify the presence of multiplicity and the cause for the same. The method presented here is inspired by the thermal stability analysis of nonisothermal CSTRs. Using the proposed method, the occurrences of multiple steady states in reactive distillation columns with the hypothetical ternary reaction a + b ⇔ c (with inerts) and quaternary reaction a + b ⇔ c + d with ideal vapor−liquid equilibrium (VLE) are predicted. The predictions of steady-state multiplicity obtained by the proposed method are in agreement with the results of bifurcation analysis using the arc-length continuation method. The method is further extended to the industrially relevant reactive distillation processes for the synthesis of methyl acetate and methyl tert-butyl ether (MTBE). The proposed method, as compared to the bifurcation technique, makes the analysis simpler and provides more insight into the influence of different factors on the multiplicity behavior.

1. INTRODUCTION Integration of chemical reaction and distillation in a single unit, called reactive distillation (RD), has many advantages over the conventional reactor−separator sequence. These include reduced capital costs, complete conversion in equilibriumlimited reactions, heat integration through utilization of the reaction exotherm, and reduced waste generation.1−4 However, because of the complex interaction between reaction and phase equilibrium, the process is highly nonlinear and, at times, poses challenges in terms of operation and control. Kienle and Marquardt5 presented a review of the nonlinear dynamic effects such as both input and output steady-state multiplicities and sustained oscillations in reactive distillation. The presence of multiple steady states (MSS) in an RD column has been reported by many researchers6−20 and experimentally verified for the synthesis of tert-amyl methyl ether (TAME) in an RD column.21 There is no single cause of MSS, and it depends on the reacting system of interest. For an efficient design of the control system, it is thus necessary to identify the root cause of MSS and the parametric space over which it prevails. Multiple steady states are known to exist for conventional chemical reactors operated in continuous mode, particularly continuous stirred tank reactors (CSTRs), caused by heat effects or kinetic instabilities.22−26 On the other hand, in a nonreactive simple binary distillation, the presence of MSS is also known and has been investigated theoretically27,28 as well as experimentally.29,30 Furthermore, for azeotropic multicomponent systems, theoretical analysis31−36 and experimental verification37,38 of MSS have also been the subjects of research in the past two decades. Investigations of the physical phenomena that cause MSS in nonreactive and reactive distillation columns by different research groups show that there are distinct reasons for the occurrences of output multiplicity. The presence of a singularity due to nonlinear unit transformation,27 the coupling of energy balance with material balance,27 and the presence of azeotropes31 can lead to multiplicity in both nonreactive and © 2013 American Chemical Society

reactive distillation systems. In addition, the presence of reaction hysteresis19 due to the interaction between reactive and nonreactive sections can also lead to MSS behavior. For the analysis of MSS in nonreactive distillation systems, Bekiaris et al.31 developed a method to construct a bifurcation diagram for infinite reflux and infinite number of stages with the distillate flow rate as the bifurcation parameter. Based on the geometrical properties of the distillation boundaries, they derived necessary and sufficient conditions for the presence of MSS and also identified the feed composition in composition space that can lead to MSS behavior for a given mixture. Reactive distillation (RD), which combines the features of both CSTR and distillation, is expected to be more complex and highly nonlinear. Güttinger and Morari8 extended the concept of infinity/infinity analysis of Bekiaris et al.31 to find the singularities, if any, in an RD column. They assumed the reactions to be instantaneous and used the concept of transformed variables. Their analysis did not consider the effect of kinetics or the Damköhler number (Da) on the presence of MSS. Furthermore, as shown by Sneesby et al.19 for the methyl tert-butyl ether (MTBE) RD process, extension of infinity/infinity analysis to a column with a finite number of stages leads to uncertainties in MSS predictions. Chen et al.6 studied the effect of kinetics on MSS behavior for the syntheses of TAME and MTBE by reactive distillation using bifurcation analysis with Da as the bifurcation parameter. They reported MSS at high values of Da (i.e., reaction equilibrium regime) in the MTBE RD column, whereas in the case of TAME, they found MSS at low values of Da (i.e., in the kinetic regime). The homotopy continuation method has also been used to study MSS in RD columns.39−41 Ciric and Miao41 found the wide difference in the boiling points of the reactants in an ethylene Received: Revised: Accepted: Published: 5191

January 25, 2013 March 13, 2013 March 14, 2013 March 14, 2013 dx.doi.org/10.1021/ie400288r | Ind. Eng. Chem. Res. 2013, 52, 5191−5206

Industrial & Engineering Chemistry Research

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methyl acetate and MTBE in RD columns, are also investigated to demonstrate the usefulness of the proposed approach. This article is organized as follows: First, the thermal stability analysis of a nonisothermal CSTR, which forms the basis of the entire work, is briefly reviewed. The RD model considered in the present work is presented, followed by the development of the proposed approach for RD systems. Hypothetical and real RD systems are then analyzed for their possible MSS behavior using the proposed approach.

glycol (EG) RD column to be the cause of MSS behavior, whereas Singh et al.39,40 did not speculate on the cause of the MSS they observed in the MTBE case. Despite the many investigations of these interesting and important phenomena in RD, only a few efforts to understand the basic cause of MSS have been reported in the literature. Reactive flash is structurally the simplest unit that combines the attributes of reaction and vapor−liquid equilibrium. The MSS behavior of the reactive flash model was studied by few investigators to understand the MSS in RD columns.42−45 The necessary conditions for MSS are established42 when energy balance is coupled with material balance along with vapor− liquid equilibrium (VLE). The interaction of reaction kinetics with VLE properties, when energy balance is decoupled, were also studied,43 and it was demonstrated that the ratio of activation energy to latent heat of vaporization should be greater than 1 for MSS to occur. Lakerveld et al.44 employed a different approach for a binary isomerization reaction in a reactive flash in which the state variables are distinguished from system properties, such as heat of reaction or activation energy, and control parameters, such as reboiler heat input or reflux flow rate. They showed that high heat of reaction, activation energy, and reactant volatility result in an enlarged window of operating parameters in which output multiplicity is realized. Reactive flash, however, does not bring in the effect of increased dimensionality due to structural parameters such as the number of stages in a reactive distillation column. According to Waschler et al.,45 a light-boiling reactant and a sufficiently large difference in boiling points are necessary for MSS to occur. This work was extended by Katariya et al.46 to the case of nonequimolar reaction systems, and they showed that the number of steady states increases with increasing number of stages in an RD column. In most previous studies, the existence of MSS was identified through bifurcation diagrams obtained by continuation techniques without theoretical analysis to investigate the cause of MSS. In some studies,41,47 the effects of different parameters on the bifurcation diagram were studied to shed light on the possible reasons for MSS. In all of these cases, a highly efficient solver capable of giving multiple roots and performing stability analysis was necessary. Moreover, there was no certainty that the root cause of this behavior would be found. In the present work, we present a systematic yet relatively simple method to determine whether the interaction of reaction and distillation is the cause of multiplicity. Recently, Kano et al.28 proposed an interesting approach to predict MSS in nonreactive binary distillation columns. Their approach was motivated by thermal stability analysis used for predicting MSS in nonisothermal CSTRs. In this approach, an overall energy balance around the distillation column is considered, and heat input and output terms are plotted against the composition of the more volatile component in each the distillate and the bottom product. The application of this method is limited to binary systems and to nonreactive distillation; direct extension to more complex multicomponent reactive distillation columns is challenging. In the present work, we propose to extend the thermal stability analysis approach of nonisothermal CSTRs to multicomponent reactive distillation columns due to the interaction of reaction and distillation. Ternary and quaternary reaction systems with hypothetical kinetics and ideal VLE are considered as illustrative examples. Furthermore, industrial case studies, namely, the syntheses of

2. THERMAL STABILITY ANALYSES OF A NONISOTHERMAL CSTR The MSS in nonisothermal CSTR can be explained by thermal stability analysis in which heat generation (HG) and heat removal (HR) terms are obtained by combining energy balance and material balance equations48 HG(T ) = [−ΔHr(TR ) + ΔCp(T − TR )]X

HR (T ) =

UA (T − Ta) + FAO

∑ θiCp(T − To) i

(1)

(2)

where X is the conversion; TR is the reference temperature; ΔCp is the change in specific heat due to reaction; U is the heat transfer coefficient; A is the heat transfer area; FAO is the molar feed rate of reactant A; and Ta and To are ambient and coolant temperatures, respectively. Equations 1 and 2 are obtained by separating the linear and nonlinear terms from the material and energy balance equations. Heat generation HG is a nonlinear function, whereas heat removal (HR) is a linear function of the state variables such as temperature and conversion. Heat generation HG is the amount of heat generated as a result of exothermic reaction. An increase in temperature results in a higher reaction rate to release more heat, which, in turn, causes a further rise in temperature. Thus, an increase in temperature acts as a positive feedback for the heat generation term, which exhibits nonlinear sigmoidal behavior. Figure 1 shows MSS in a nonisothermal CSTR. Heat generation (HG) and heat removal (HR) intersect at three points, which indicate the occurrence of three steady states. Usually, the middle steady state is unstable, whereas the upper and lower steady states are stable. The total heat removal (HR) is the net effect of the enthalpy difference

Figure 1. Steady-state multiplicity in a nonisothermal CSTR. 5192

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Figure 2. RD column configuration for hypothetical ternary or quaternary systems.

between the material flowing in and out and the heat removed by the cooling system, if any. For an adiabatic nonisothermal CSTR, the heat removed by the cooling system is zero.

equilibrium-stage RD model based on material balance49 is considered. This model assumes constant liquid holdups, fixed heats of reaction, and fixed heats of vaporization. A constant molar overflow (CMO) exists in the nonreactive sections, whereas internal flows change only in the reactive zone as a result of heat effects associated with the reaction and nonequimolarity, if any. Steady-State Component Mole Balance Equations. The mole balance equation for the reboiler is

3. DEVELOPMENT OF THE PROPOSED APPROACH In a reactive distillation column, reactive stages act as flash reactors in which reaction and phase equilibrium occur simultaneously. To extend the concept of thermal stability analysis of nonisothermal CSTRs to reactive distillation columns, it is necessary to identify a nonlinear generation term (heat or material) and a linear removal term (heat or material) from the steady-state conservation (material and energy) equations. Because the temperature of the reactive stage is dependent on the phase equilibrium and internal flows in the RD column, unlike that in a CSTR, the heat generation and heat removal terms cannot be evaluated independently as functions of temperature. Hence, a different mathematical treatment is necessary. To begin, we present the RD model considered in this work. Further, we rearrange the model equations and separate the nonlinear and linear terms for both mass and energy. An algorithm that evaluates these terms to identify MSS is then presented. 3.1. RD Model. The RD column configuration for the hypothetical RD systems considered here for ternary and quaternary reaction systems is illustrated in Figure 2. The reactive, rectifying, and stripping sections contain NR, NRX, and NS stages, respectively. The RD column configuration is referred to here as NR/NRX/NS, where each value indicates the number of stages in the respective section of the column. Reaction occurs only in the reactive section. Pure reactant a enters the column on the first tray (NS + 1) of the reactive section, and pure reactant b enters the column on the last reactive stage (NS + NRX). The RD column has a total N stages (excluding a reboiler and total condenser), and the stages are numbered from bottom to top. Various types of models, involving different levels of complexity, can be used to simulate the dynamics of an RD column. In this work, a simple

L1x1, j − Bxb, j − VSyb, j = 0

j = 1, 2, ..., NC

(3)

where NC is the number of components, VS is vapor flow rate from reboiler, L1 is the liquid flow rate leaving bottom stage and B is the bottom product flow rate. The liquid-phase composition of L1 is x1, and that of B is xb. The vapor-phase composition of VS is yb. The mole balance equation for column stage i, 1 ≤ i ≤ N − 1, is Li + 1xi + 1, j + Vi − 1yi − 1, j − Lixi , j − Vy i i , j + Fz i i , j + υjR i , j = 0 (4)

where Li+1 is the flow rate of the liquid entering stage i having composition xi+1, Vi−1 is the flow rate of vapor entering the stage i having composition yi−1, Li is the flow rate of the liquid leaving the stage i having composition xi, Vi is the flow rate of vapor leaving stage i having composition yi, Fi is the feed rate for stage i, zi,j is the feed composition, υj is the stoichiometric coefficient of component j, and Ri,j is the reaction rate on stage i for component j. It should be noted here that Fi = 0, except for the feed stages, and Ri,j = 0 for all nonreactive stages. The mole balance equation for the top stage, i = N, is rxd, j + Vi − 1yi − 1, j − Lixi , j − Vy i i,j = 0

(5)

where r is reflux flow rate with composition xd. The mole balance equation for the condenser is VN yN , j − rxd, j − Dxd, j = 0 5193

(6)



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Vapor and Liquid Flow Rates Leaving Stage i in the Reactive Section. Ternary RD System. For a ternary RD system, the vapor flow rate on stage i is given by Vi = Vi − 1 +

ΔHr Ri λ

rate of liquid entering the reactive section from the rectifying section is equal to the reflux flow rate, and the rate of vapor entering the reactive section from the stripping section is equal to the vapor boil-up rate in the reboiler. Here, Fb is the flow rate of the heavy reactant, and Fa is the flow rate of the light reactant entering the reactive stage. In multireactive stages, b enters at the last reactive stage, and a enters at the first reactive stage. Heat is removed mainly by the vaporization of liquid on the reactive stage in both ternary (eq 8) and quaternary (eq 9) RD systems. Hence, linear and nonlinear terms can be obtained by considering the liquid flow balance (see Figure 3) around the reactive section (i.e., NRX > 1) for a general case (i.e., equilmolar and nonequimolar) as follows

(7)

where ΔHr is the heat of reaction, λ is the latent heat of vaporization, and Ri is the net reaction rate on stage i. The liquid flow rates are calculated according to the stoichiometry of the ternary reaction scheme (see Figure 2) because 2 mol of reactants produce 1 mol of the product, and reaction takes place in the liquid phase. Thus, the molar flow rate of the liquid in the reactive section would decrease due to the reaction, yielding the equation Li = Li + 1 −

ΔHr Ri − Ri λ

N

S

(8)

Li = Li + 1

S

(10)

In eq 10, F is the total feed flow rate, the liquid flow entering the reactive section is Lin = r, and the liquid flow leaving the reactive section is Lout = B + VS because of the CMO conditions in the nonreactive sections. Hence, substituting for Lin and Lout in eq 10 gives

It is assumed that the temperature difference between the liquid and vapor streams entering each stage is negligible. Quaternary RD System. The vapor flow rates in a quaternary RD system are calculated by eq 7, and the liquid flow rate on stage i is given by ΔHr − Ri λ

N

RX RX ΔHr ( ∑ R i) − ( ∑ R i) λ i=N +1 i=N +1

Lout = F + L in −

N

N

RX RX ΔHr B + VS = F + r − ( ∑ R i) − ( ∑ R i) λ i=N +1 i=N +1

(9)

S

Along with these equations, appropriate vapor−liquid equilibrium model and summation constraints are incorporated. 3.2. Linear and Nonlinear Terms. In the RD model considered here, all of the stages are adiabatic, and heat generated by the exothermic reaction, which takes place on the reaction stages (see eqs 7−8), is utilized for vaporization. It should be noted here that the vapor and liquid flow rates in the nonreactive stripping and rectifying sections remain unchanged due to CMO assumption. The liquid and vapor flow rates vary from stage to stage in the reactive section and are calculated from eqs 7 and 8 for the ternary system and from eqs 7 and 9 for the quaternary system. To identify the linear and nonlinear terms, we first consider a hybrid RD column with a single reactive stage (Figure 3). Because of the CMO conditions in the nonreactive sections, the

S

(11)

In eq 11, the product of the heat of reaction and the overall reaction rate is the total heat generation (HG) in the reactive section NRX

HG = ΔHr(

∑

R i) (12)

i = NS + 1

Now, the overall material balance around the RD column can be used to replace bottom flow rate (B) in eq 11. The overall material balance can be written as NRX

F=B+D+(

∑

R i) (13)

i = NS + 1

The term for the bottom flow rate (B) in eq 11 can be written as N

∑i =RXN + 1 R i F D S =1+ + B B B ⎛ 1 R ) ∑ i ⎜⎜ ⎝1 + i = NS + 1 NRX

B = (F −

(14)

⎞ ⎟ D ⎟ B ⎠

(15)

Hence, replacing the bottom flow rate (B) in eq 11 using eq 15 and rearranging gives NRX

ΔHr(

NRX

∑

R i) + (

i = NS + 1

∑

R i)αλ = (r − VS)λ + Fαλ

i = NS + 1

⎛ D/B ⎞ ⎟ where α = ⎜ ⎝ 1 + D/B ⎠

(16)

In eq 16, the left-hand side is a nonlinear term, and the righthand side is a linear term. It can be noted here that the linear term can be evaluated independently for fixed values of the reflux rate (r), boil-up rate (VS), total feed rate (F), and α. When plotted against the parameter α, the linear term gives a

Figure 3. Schematic of a single reactive stage. 5194



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slope of Fλ and an intercept of (r − VS)λ. At steady state, the linear and nonlinear terms are equal, and if they intersect at multiple points, then this indicates the presence of MSS. The nonlinear term (eq 16) depends on the heat of reaction and the rate, which, in turn, is governed by the temperature and the liquid-phase compositions of the reactive stage. Because of the implicit relationship between temperature and liquid compositions through phase equilibrium, the reactive-stage temperatures and liquid compositions depend on the number of stages. Hence, the curvature of the nonlinear term is influenced by the structural parameters (i.e., number of stages NS, NRX, and NR), in addition to kinetic and thermodynamic parameters. In the absence of heat effects (i.e., ΔHr = 0), eq 16 is reduced to

where aF and aB are the pre-exponential factors, EF and EB are the activation energies, Rgas is the universal gas constant, Ti is the absolute temperature on stage i, and Mi is the amount of catalyst on stage i (i.e., reactive-stage holdup). The ternary reaction scheme with inerts is an important class of reactions. The majority of commercial reactive distillation columns [e.g., those for ethylene glycol, MTBE, ethyl tert-butyl ether (ETBE), and TAME] fall into this category.49 Although the reactions involve three components, in many real cases such as MTBE and ETBE RD columns, some inert components are usually present in the reactant feed. Hence, in the case of ternary systems, the inerts are separated from either top or bottom, depending on their volatilities. In the present work, product c, being a heavy component, is separated through the bottoms, and the inert, being lighter, is removed from the top. 4.1.1. Ternary System with Inerts. The proposed method is first used for a ternary system containing inerts. The kinetics and VLE data (Table 1) and design parameters such as feed

NRX

(

∑

R i)α = (r − VS) + Fα

i = NS + 1

(17)

The left- and right-hand sides of eq 17 are nonlinear and linear terms, respectively. These terms can be plotted against parameter α to find MSS in the absence of heat effects. It can be inferred that the nonequimolarity of the reaction can also be a root cause of multiplicity in some cases, and it is not necessary that the reaction be highly exothermic. Equation 17 can also be derived by applying an overall material balance across the reactive stages. 3.3. Stepwise Procedure to Identify MSS. Based on the preceding discussion, a stepwise procedure to identify the presence of MSS can be formulated as follows: (1) Fix the values of the reflux flow rate (r) and the vapor boil-up rate (VS). (2) Specify parameter α. (3) Calculate the linear term. (4) Guess initial liquid-phase compositions on all stages and solve the nonlinear steady-state material balance in eqs 3−6, along with eqs 7−8 and appropriate VLE equations, by a numerical method (e.g., Newton−Raphson method), and calculate the nonlinear term. (5) Go back to step 2, specify another value of α, and repeat steps 3 and 4. The initial guess of the liquid composition for the new value of α can be taken from the converged solution for the previous value of α. (6) Plot the linear and nonlinear terms against the parameter α.

Table 1. Reaction Kinetics and VLE Properties of the Ternary and Quaternary Systems ternary (with inerts) activation energy (kcal mol−1) forward 30 backward 40 specific reaction rate at 366 K (kmol s−1 kmol−1) forward 0.008 backward 0.00016 chemical equilibruim constant at 366 K 50 heat of reaction (kcal mol−1) −10 heat of vaporization (kcal mol−1) 6.994 vapor pressure constants Avpj Bvpj Avpj a b c d inerts

(18)

R i , j = Mi(k Fixaix bi − k Bixcixdi)

(19)

(20)

k Bi = aB exp( −E B /R gasTi )

(21)

12.34 11.65 10.96 13.04 −

0.008 0.004 2 −10 6.994 Bvpj 3682 3682 3682 3682 −

Table 2. Design Parameters for Ternary Reactive Distillation System with Inerts parameter pressure (bar) fresh feed flow rates (mol/s) Fa Fb feed composition reactant a reactant b vapor boil-up rate (mol/s) reflux flow rate (mol/s) liquid holdup (mol) on reactive stage on nonreactive stage

The forward and backward specific reaction rate constants on stage i for the ternary and quaternary reactions are k Fi = aF exp( −E F /R gasTi )

3862 3862 3862 − 3862

30 40

rates, feed compositions, pressures, and holdups (Table 2) were taken from the model example of Luyben and Yu.49 The vapor and liquid flow rates for the ternary system are calculated using eqs 7 and 8, respectively. The bifurcation diagram (Figure 4) of the 2/2/2 RD configuration for the ternary system with inerts exhibits MSS behavior. Analysis of the nonlinear and linear

4. CASE STUDIES 4.1. Hypothetical Examples. The hypothetical cases with a nonequimolar ternary system and an equimolar quaternary system considered here are the benchmark RD systems.49 The reaction rates of component j on stage i for the ternary and quaternary systems are given by eqs 18 and 19, respectively R i , j = Mi(k Fixaix bi − k Bixci)

12.34 11.65 10.96 − 12.34

quaternary

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value 8 24.51 12.82 50% a and 50% inerts pure 65.1 70 2000 670



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Figure 4. Bifurcation diagram for ternary 2/2/2 RD.

Figure 5. Nonlinear and linear analysis for ternary 2/2/2 RD.

Figure 6. Bifurcation diagrams for ternary RD systems in the absence of heat effects.

exhibits MSS behavior even in the absence of heat effects. The proposed method in the absence of heat effects (eq 17) was used to predict MSS in a ternary system, and it successfully predicted MSS behavior as shown in Figure 7 for 3/3/3 RD. To gain more insight and understand why the nonlinear term exhibits sigmoidal behavior when the parameter α is increased, we plotted the composition and temperature profiles along the

terms (Figure 5) for ternary 2/2/2 RD at a vapor boil-up rate of 73 mol/s shows intersections at multiple points, which indicates the occurrence of output multiplicity. The steady-state behavior of a ternary system with inerts, in the absence of all heat effects, was also studied in the present work. Output multiplicity in 2/2/2 RD (Figure 6) disappears in the absence of heat effects. However, 3/3/3 RD (Figure 6) 5196



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Figure 7. Nonlinear and linear analysis for ternary 3/3/3 RD in the absence of heat effects.

Figure 8. Effect of parameter α on temperature profiles.

column height (Figures 8 and 9) against the parameter α. As the parameter α increases, the temperatures in the column (Figure 8), and especially in the reactive section, increase because of higher concentrations (Figure 9) of heavy components at the respective positions. The increases in temperature increase the forward rate constant kF (Figure 10), and because kF dominates the net reaction rate (Figure 10), the reaction rate increases with increasing α. The rate constant kF (eq 20) is a nonlinear function of temperature. The increase in temperature acts as a positive feedback that further increases kF and the net reaction rate. As a result, the nonlinear term shows characteristic sigmoidal behavior. Similar phenomena occur in the case of an exothermic reaction in a CSTR (Figure 1), wherein an increase in temperature increases the reaction rate, which further increases the temperature because of the exothermic reaction, and hence, heat generation exhibits sigmoidal behavior. In the case of a CSTR, temperature can be independently selected as a parameter, whereas in the case of an RD system, temperature depends on the VLE. Nevertheless, increasing the value of the parameter α causes the temperatures in the RD column to increase and brings

about effects similar to those seen in the case of nonisothermal CSTRs. 4.1.2. Quaternary System. In this section, the proposed method is applied to a quaternary system that is associated with an equimolar exothermic reaction. Hence, unlike the previous case, we do not encounter a nonlinear term in the absence of heat of reaction because of equimolar reaction. However, because of the exothermic reaction, the energy balance contains a nonlinear heat-generation term that can be exploited to analyze multiplicities, if any. For the considered quaternary system, the feed rates of reactants Fa and Fb are 12.6 mol/s each, the total pressure P is 9 bar, and the catalyst holdup (MRX) is 1 kmol per stage. The kinetic and VLE parameters are given in Table 1. The nonlinear and linear terms were evaluated for different RD configurations. The effects of RD configuration on the curvature of nonlinear term is shown in Figure 11. However, it is also necessary for the linear term to intersect the nonlinear term at multiple points for multiplicity to occur. It is the vapor boil-up [i.e., intercept (r − VS)λ in eq 16] that determines the position of line corresponding to the linear term. Hence, the boil-up was varied (Figure 12) in RD configurations 2/1/2 and 5197



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Figure 9. Effect of parameter α on concentration profiles.

Figure 10. Effect of parameter α on reaction terms.

Figure 11. Effects of geometric parameters (NS, NRX, and NR) on the curvature of the nonlinear term (eq 16).

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Figure 12. Effects of vapor rate on the nonlinear and linear terms (eq 16) in 2/1/2 and 2/2/2 RD configurations.

Figure 13. Comparison of MSS predictions obtained by the proposed method and the arc-length continuation method in 2/1/2 and 2/2/2 RD configurations.

2/2/2 to identify MSS, if any. The multiple intersections of the nonlinear and linear terms indicate the presence of MSS in the 2/1/2 and 2/2/2 RD configurations (Figure 12), which was also validated by constructing the bifurcation diagram produced by the arc-length method as shown in Figure 13. The numerical values of nonlinear and linear terms at three intersection points for the 2/1/2 and 2/2/2 RD configurations are reported in Table 3. An analysis similar to that for the ternary system (see Figures 8 and 9) was carried out to understand the MSS behavior in the quaternary system. Hence, to gain more insight, the temperature and concentration profiles along the RD column for the 2/1/2 RD configuration were plotted against the parameter α (Figures 14 and 15).

Here again, the concentration of heavy components increases in the column as the parameter α is increased, and as a result, the temperature in the column also increases. This further increases the reaction rate and hence the heat generation (i.e., nonlinear term), thereby changing the internal flow rates. The flow rates vary in such a way that the stage temperatures increase further. This cycle of events thus leads to positive feedback and multiplicity behavior. The large amount of heat generation is due to higher reaction rates or the large exothermicity of the reaction. Figure 16 shows the significance of heat effects on the curvature of the nonlinear term for quaternary configuration 2/1/2. The sigmoidal nature of the nonlinear term is more pronounced at the higher value of the heat of reaction (i.e., 10 kcal/mol). This indicates that the MSS behavior in 2/1/2 RD will disappear at lower heats of reaction, as was also confirmed by bifurcation analysis. From the preceding discussion, it is clear that heat effects play an important role in the MSS behavior of quaternary systems. In contrast to quaternary systems, ternary systems exhibit MSS even in the absence of heat effects. However, the explanations for the existence of MSS in the ternary and quaternary systems investigated here are similar.

Table 3. Numerical Values of Nonlinear and Linear Terms (cal/s) at the Points of Intersections (cf. Figure 13) in RD Configurations 2/1/2 and 2/2/2 RD configuration

lower point

middle point

upper point

2/1/2 2/2/2

6.78 × 104 8.155 × 104

7.434 × 104 8.8407 × 104

7.95 × 104 9.5653 × 104 5199



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Figure 14. Effect of parameter α on the temperature profile in ideal quaternary 2/1/2 RD.

Figure 15. Effects of parameter α on the concentration profiles in ideal quaternary 2/1/2 RD.

Figure 16. Effect of heat of reaction on the curvature of the nonlinear term (heat generation) in an ideal quaternary RD column.

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Figure 17. RD column for the synthesis of MTBE.

Figure 18. Bifurcation diagram and analysis of the nonlinear and linear terms in an MTBE 3/3/3 RD column.

4.2. Real Examples. The applicability of the proposed method to two industrially important RD processes, namely, the syntheses of MTBE and methyl acetate, is demonstrated here. The steady-state nonlinear RD model is the same as given before, except for the fact that the VLE descriptions are now real and nonideal. The heats of reaction and the latent heats of vaporization were assumed to be constant in both examples. The RD configurations for MTBE and methyl acetate were kept as simple as possible to understand the MSS behavior and to demonstrate the ability of the proposed method to predict MSS for RD systems with real reactions. Moreover, it should also be noted here that, in the presence of azeotropes, the range of the parameter α (eq 16) becomes limited in real cases. 4.2.1. Synthesis of Methyl tert-Butyl Ether (MTBE). MTBE is produced by the etherification of isobutene with methanol in the presence of a strong acid catalyst. Usually, the C4 feed also

consists of n-butene, which does not take part in the reaction and remains in the system as an inert component. The primary objective of the RD column is to offer high-purity MTBE product from the bottoms and a distillate product containing high-purity n-butene. It is a ternary reacting system containing inerts, which is similar to the hypothetical ternary case considered in the present work. However, the calculation procedure for the linear term was modified to take into account the fact that the mixed butene feed is in vapor form. The Wilson parameters were used to calculate the liquid-phase activity coefficients. The design parameters (feed conditions, column pressure, etc.; Figure 17) were taken from Singh et al.40 The presence of multiplicity in MTBE RD columns has been reported by many researchers.6−14,19,20,40,50 We investigated the 3/3/3 RD configuration for MSS behavior using the proposed approach (eq 16). The bifurcation 5201



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Figure 19. Effect of parameter α on the temperature profiles in an MTBE 3/3/3 RD column.

Figure 20. Effect of parameter α on the concentration profiles in an MTBE 3/3/3 RD column.

diagram (Figure 18) in the MTBE 3/3/3 RD column under consideration indicates the presence of output multiplicity. The plot of nonlinear and linear terms (Figure 18) shows multiple intersections, indicating the presence of MSS. The range of parameter α (Figure 18) in the proposed approach was kept between 0.6 and 0.67. This range of parameter α was chosen based on the bifurcation results, and thus, only the MSS region was selected. Figures 19 and 20 show the temperature and concentration profiles, respectively, as parameter α is increased in the MTBE column. It can be seen from these figures that, because of the effects of distillation (i.e., increasing parameter α), the concentrations of heavy components (i.e., methanol and MTBE) increase in the column, and hence, the temperature increases, thereby causing a positive feedback that leads to multiplicity. 4.2.2. Synthesis of Methyl Acetate. The synthesis of methyl acetate in an RD column is a classic example of a quaternary reversible reaction. Methanol and acetic acid reversibly react to

produce methyl acetate and water in an RD column with a configuration as shown in Figure 21. The occurrence of MSS in a methyl acetate RD column was reported by Al-Arfaj and Luyben15 and Kumar and Kaishtha;39 however, the cause of MSS has not been investigated. Acetic acid can be recovered from its dilute aqueous solutions using methyl acetate synthesis in a reactive distillation column.51 Singh et al.47 found MSS in this case and attributed the behavior to the nonreactive and nonideal VLE. In the present work, the proposed method was used to investigate the MSS behavior for the synthesis of methyl acetate. The nonideal UNIQUAC model was used for the VLE, and the reaction kinetics was taken from Pöken et al.52 Pure reactant feeds were introduced on the stages above and below the reactive section. The numbers of stages were taken as NS = 3, NRX = 3, and NR = 3. Because the reaction stoichiometry is the same as that of the hypothetical quaternary reversible reaction analyzed in this work, the nonlinear and linear terms 5202



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Figure 21. RD column configuration for the synthesis of methyl acetate.

Figure 22. Bifurcation diagram and analysis of the nonlinear (HG) and linear (HR) terms for the synthesis of methyl acetate in a 3/3/3 RD column.

were evaluated (Figure 22) by the proposed method using eq 16. The bifurcation diagram (Figure 22) indicates the presence of MSS in the methyl acetate 3/3/3 RD column investigated here. The parameter α in the proposed approach was varied between 0.5 and 0.53. The methyl acetate synthesis reaction is a mild exothermic reaction. Detailed analysis (Figures 23 and 24) shows that the temperatures in the RD column increase because of the increase in the concentrations of heavy components (acetic acid and water) as the parameter α is increased. Hence, the reaction rate also increases, which further increases the temperatures because of the exothermicity of the reaction. From this case study of methyl acetate, it can be concluded that a mildly exothermic reaction can also exhibit multiplicity behavior. The MSS behaviors investigated for MTBE (Figure 18) and methyl acetate (Figure 22) in the present work were observed to occur in a very small range of continuation parameter, which

is difficult to realize in practical situations. Bifurcation studies with respect to other parameters were not performed, as the objective was only to demonstrate the applicability of the proposed approach for real reactions.

5. CONCLUSIONS In the present work, a new method is proposed to predict and analyze the possible steady-state MSS behavior in RD columns due to the interaction of reaction and distillation. The proposed method was inspired by the classical thermal stability method used for nonisothermal CSTRs, in which trends in nonlinear heat generation and linear heat removal functions are viewed against the reactor temperature. In the present case of reactive distillation, the nonlinear and linear terms were separated from the conservation equations and plotted against suitable parameters such as the distillate-to-bottoms ratio. Multiple intersections of these nonlinear and linear terms indicate the 5203



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Figure 23. Effect of parameter α on the temperature profiles in methyl acetate 3/3/3 RD.

Figure 24. Effect of parameter α on the concentration profiles in methyl acetate 3/3/3 RD.

nature of the reactions, causes the flow rates to vary and the temperature to increase further.

presence of multiplicity. The proposed approach was found to work well for both cases with and without heat effects. It allows the presence of multiplicity to be determined without the need for bifurcation analysis. Further, it helps elucidate whether the multiplicity is because of reaction and distillation or some other effect. It also helps determine the effects of important structural parameters, such as feed location and numbers of reactive and nonreactive stages, and other operating parameters, such as reflux and reboil ratios. Moreover, the method is independent of the RD structure (i.e., hybrid or nonhybrid) and is applicable to multicomponent systems and all types of reactions (fast and slow). The proposed approach was illustrated for both hypothetical and real RD systems, namely, the syntheses of MTBE and methyl acetate. For nonequimolar reactions, multiplicity can occur even in the absence of all thermal effects, and for equimolar reactions, multiplicity can occur even if the reactions are only mildly exothermic (methyl acetate). In all of the cases investigated in the present work, it was found that increasing the column temperature acts as a positive feedback that further increases the reaction rates and, because of the exothermic

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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NOMENCLATURE a = light reactant component b = heavy reactant component B = bottoms flow rate (mol/s) c = light product component d = heavy product component D = distillate flow rate (mol/s) Fa = fresh feed flow rate of reactant a (mol/s) Fb = fresh feed flow rate of reactant b (mol/s) kB = specific reaction rate of the reverse reaction (kmol s−1 kmol−1) dx.doi.org/10.1021/ie400288r | Ind. Eng. Chem. Res. 2013, 52, 5191−5206


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kF = specific reaction rate of the forward reaction (kmol s−1 kmol−1) L = liquid flow rate (mol/s) MB = liquid holdup in the reboiler (mol) MD = liquid holdup in the condenser (mol) Mrx = liquid holdup on the reactive trays (mol) Mtray = liquid holdup on nonreactive tray i (mol) P = column pressure (bar) Ri,j = rate of production of component j on tray i (mol/s) r = reflux flow rate (mol/s) Ti = temperature on stage i including the reboiler (K) Vi = vapor flow rate on tray i (mol/s) VS = vapor flow rate from the reboiler (mol/s) xb,j = bottoms composition of component j (mole fraction) xd,j = liquid composition of component j in distillate d xi,j = liquid composition of component j on tray i (mole fraction) yb,j = vapor composition of component j in the reboiler (mole fraction) yi,j = vapor composition of component j on tray i (mole fraction) za = feed composition of fresh feed flow rate of reactant a zb = feed composition of fresh feed flow rate of reactant b ΔHr = heat of reaction (cal/mol) λ = heat of vaporization (cal/mol)

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Analysis of Steady-State Multiplicity in Reactive Distillation Columns

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