On the Steady-State Multiplicities for an Ethylene Glycol Reactive

Rodrigo López-Negrete de la Fuente and Antonio Flores Tlacuahuac. Industrial & Engineering Chemistry Research 2007 46 (7), 2092-2111. Abstract | Full ...
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Ind. Eng. Chem. Res. 1999, 38, 451-455

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On the Steady-State Multiplicities for an Ethylene Glycol Reactive Distillation Column Rosendo Monroy-Loperena† and Jose Alvarez-Ramirez*,‡ Area de Ana´ lisis Aplicado and Area de Ingenierı´a Quı´mica, Universidad Auto´ noma MetropolitanasIztapalapa, Apartado Postal 55-534, Me´ xico D.F. 09000, Me´ xico

An ethylene glycol column is studied in this paper, where the main objective is to describe the bifurcation diagram with respect to the reboiler boilup ratio. The bifurcation analysis reveals the existence of a unique equilibrium point at low reboiler heat inputs and three equilibrium points (two stable and one unstable) at high reboiler heat inputs. Moreover, the existence of input multiplicities at moderate and high product purity is also revealed. 1. Introduction The importance of multiplicity of solutions in separation processes stems from the fact that one set of design parameters can yield several operating conditions. A consequence of this situation is that control actions must to be taken to regulate the operation of the process. Multiplicity of solutions in separation processes has been a largely studied problem. While some results state the uniqueness of solution for binary distillation,1-3 other results come to the conclusion that multiple steady state can arise in binary distillation if the flow rates are given on a mass basis instead of a molar basis4 or if multicomponent mixtures are separated in processes with an increase of the number of stages.5 Regarding reactive distillation, several works have reported the existence of multiple steady states.6-13 Both input and output multiplicities have been described for the cases methyl tert-butyl ether and ethyl tert-butyl ether reactive distillations.9,10 Karpilovskiy et al.13 derived a qualitative criterion for the existence of multiple solutions (output multiplicity) in single-product reactive distillation, given the main features of a distillation line diagram and reaction stoichiometry. The nature of Karpilovskiy et al.’s approach (see also Pisarenko et al.6) has a lot of similarities with that used to analyze multiplicity of steady states in chemical reactors. The singularity theory approach has also been used to look for the origin of the multiplicities in reactive distillation.11,12 Gehrke and Marquardt11 used a onestage reactive distillation process for the identification of possible sources of steady-state multiplicities related to different chemical species. Gu¨ttinger and Morari12 reported an ∞/∞ analysis for the prediction of multiple steady states in multicomponent azeotropic distillations. Although both works substantiate the importance of multiplicities in reactive distillation processes, they conclude that more systematic research based on rigorous models is required to identify the physical reasons for the phenomenon leading to multiple steady states. Despite the large amount of reported results on multiplicity in reactive distillation, only recently have * To whom correspondence should be addressed. E-mail: [email protected]. Fax: +52-5-7244900. Phone: +52-5-7244649. † Area de Ana ´ lisis Aplicado. ‡ Area de Ingenierı ´a Quı´mica.

their causes and implications for operation and control been discussed.4,9,10,14 One of the first systematic approaches to the problem of controlling reactive distillation was addressed by Kumar and Daoutidis.14 The main control objective is to control the product purity using the reboiler heat input. As a preliminary study, a bifurcation analysis was performed, which revealed a region of output multiplicity and a transition from minimum-phase behavior at moderate product purity to a nonminimum-phase behavior at high product purity. For the case of an ethylene glycol reactive distillation column, a nonlinear controller was designed that yields good performance in setpoint tracking. Ciric and Gu15 have presented a reactive distillation column for manufacturing ethylene glycol. Using homotopy methods, Ciric and Miao8 demonstrated the existence of output multiplicities for a cost-optimal column configuration (i.e., total reflux and optimal distributed feed). The aim of this short paper is to describe the bifurcation diagram of the ethylene glycol distillation column with respect to the reboiler heat input. The main task is to demonstrate the existence of both input and output multiplicities with respect to the reboiler heat input which in the cases of columns with total reflux is the most commonly used manipulated variable to regulate the product purity. In Kumar and Daoutidis’s work,14 some preliminary results on the multiplicities of ethylene glycol reactive distillation columns were reported. Basically, the work by Kumar and Daoutidis focuses mainly on the design of a control law and uses the bifurcation diagram to illustrate the existence of unstable operating points. In this work, we take a step ahead by showing that these columns may present both input and output multiplicities. Moreover, the origin of the multiplicities and the main variables affecting the bifurcation maps are discussed. On the one hand and in the spirit of Pisarenko et al.’s work,6 the main idea to explain the origin of the multiplicities is that intermediate products are recycled via the separation stages, which leads to an increase of the reaction rate and consequently to an increase in the heat production due to exothermic reactions. On the other hand, the main variables which may affect multiplicities are the reboiler boilup ratio and the feed composition. It must be pointed out that this work must be seen as a preliminary study toward addressing the control problem of ethylene glycol distillation columns. In fact, Jacobsen and Skogestad4 have remarked on the impor-

10.1021/ie980424q CCC: $18.00 © 1999 American Chemical Society Published on Web 01/14/1999

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tance of manipulated input/regulated output bifurcation diagrams in obtaining a priori information about the operability of the process and possible control performance limitations. These issues will be discussed in this paper. 2. Base Case Study Ciric and Gu15 have presented a reactive distillation column for manufacturing ethylene glycol (EG) and diethylene glycol (DEG) from ethylene oxide and water.

C2H4O + H2O f C2H6O2 (EG) C2H40 + C2H6O2 f C4H10O3 (DEG)

(1)

The following assumptions were adopted to develop a rigorous model (see Ciric and Miao8): (1) the vapor and liquid phases are in equilibrium on each tray; (2) the vapor-liquid equilibrium can be considered ideal (ideality existing between polar molecule interactions cancel out); (3) the reactions only occur in the liquid phase; (4) the liquid phase is homogeneous; (5) the enthalpy of the liquid streams is negligible relative to the heat of vaporization and the heats of reaction; (6) the heat of vaporization and the heats of reaction are constant; (7) the reactions only occur on the trays within the reaction zone, and (8) the column operates at total reflux. The steady-state model of the column is given by material and energy balances, reaction rate equations, and summation equations. The model we use in this work was taken from refs 8 and 15. For completeness in presentation, a brief description is given in the Appendix. Ciric and Gu15 used a mixed-integer nonlinear programming approach to obtain a cost-optimal column design (see Figure 1). Because ethylene oxide is the common reactive to both reactions (see reaction scheme (1)), the idea is to optimize its feeding along the column. The cost-optimal column contains 10 trays. Separation and reaction occur on trays 5-10, and separation without reaction occurs on trays 1-4. Reaction volumes are distributed unevenly among trays 5-10. Water in the amount of 26.3 kmol/h is fed onto the top tray, and a total feed of 27.56 kmol/h of ethylene oxide is distributed among trays 6-10. The distribution of ethylene oxide and the liquid holdup volumes among the reactive trays was taken as in Table 2 in Ciric and Miao.8 In addition, physical properties for chemical reactions were taken as in Table 1 in Ciric and Miao.8 It must be remarked that the physical realization of the optimal column with the feed distribution presented by Ciric and Gu15 is potentially hard as the ratio of the liquid holdup on two consecutive stages is roughly 125 (see Table 4 in Ciric and Gu15). In fact, the two-feed column is significantly more balanced. Although this case could be a better base case study, in the next section we will show that the feed distribution scheme does not have a significant effect on the qualitative behavior of the bifurcation maps (structural stability). Ciric and Miao8 used homotopic continuation methods to describe multiple steady-state solutions. Using the liquid holdup on reactive trays as a homotopy parameter, they demonstrated the existence of three steady states for the optimal value of the boilup ratio β ) 0.958. Because all of the states of a reactive distillation column are bounded, one can conclude that the steady state corresponding to the middle product purity is unstable.

Figure 1. Optimal distributed feed reactive distillation column for ethylene glycol synthesis (Ciric and Miao8).

Using eigenvalue analysis, one can also conclude that the other two steady states (high and low product purities) are stable. Therefore, the cost-optimal operating point corresponds to the (stable) high product purity. In reactive distillation columns with total reflux, the reboiler heat duty, say Qh, is taken as the manipulated variable to regulate the product purity.14 In our case (see the model in the Appendix), the boilup ratio (RBR) β is the portion of liquid stream leaving the reboiler tray. Thus, β is proportional to Qh and can be taken as the manipulated variable. To explore the effects of the distributed ethylene oxide feed in the bifurcation map, we will also consider a suboptimal case. Here, the distributed feed was replaced with a single ethylene oxide feed located on tray 5. In this suboptimal column, all of the 27.56 kmol/h ethylene oxide is fed to tray 5, rather than distributed among trays 5-10, and reaction volumes are setting on each reaction stage to 0.549 m3. Although we have called suboptimal to this feed distribution configuration, it must be remarked that the column structures studied are both cost-optimal but for a different number of maximum feed points (for details, see Ciric and Gu15). 3. Computational Aspects The model defined in the Appendix leads to a set of 70 algebraic equations, which is highly nonlinear and ill-conditioned (the corresponding Jacobian presents high conditioning numbers). The bifurcation study has been conducted using AUTO, a software package developed by Doedel.16 Among other capabilities, AUTO generates one- and two-parameter bifurcation diagrams for systems of nonlinear equations in the steady state.

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Figure 2. Bifurcation diagram for the cost-optimal configuration. The symbols s and ××× denote stable and unstable steady states, respectively.

The set of algebraic equations was solved with a Newton-Raphson method. To reduce round-off effects due to ill-conditioning and sparsity of the Jacobian matrix, matrix inversion was made with the subroutine MA28.17 Moreover, initial conditions to track steadystate branches were taken as the (unstable) middle product purity solution corresponding to β ) 0.958. This solution was obtained via the homotopic continuation method suggested by Ciric and Miao.8 4. Results The bifurcation diagram between the EG product purity (output variable) and the RBR β (input variable; recall that 0 e β e 1) for the cost-optimal and the suboptimal cases are shown in Figures 2 and 3, respectively. Note that the bifurcation diagram corresponding to the cost-optimal case is very similar to the bifurcation diagram corresponding to the suboptimal case. This show that the steady-state structure displayed by the cost-optimal configuration is structurally stable. This means that small changes in the configuration and/or operating conditions of the cost-optimal reactive distillation column do not drastically change the bifurcation diagram. Moreover, structural stability of the bifurcation diagram in Figure 2 provides further evidence that the multiple steady-state phenomenon is a result of neither the modeling techniques nor optimized configurations. Regarding the bifurcation diagram presented in Figures 2 and 3, the following comments are in order: (i) Input Multiplicities. Second-order input multiplicities were obtained for EG product purities higher than about 0.27. This implies that the same EG product purity is obtained at two different values of the RBR. However, the operating conditions corresponding to the higher RBR are open-loop unstable while the other ones corresponding to the lower RBR are stable. Hence, open-

Figure 3. Bifurcation diagram for the suboptimal configuration.

loop operation of the EG reactive distillation column is feasible at operating conditions corresponding to low energy consumption rates. This is not the case in the methyl tert-butyl ether (MTBE) reactive distillation column studied by Sneesby et al.9,10 where low and high energy consumption operations are possible because of a third-order input multiplicity. In such a case of reactive distillation, the manipulated variable is the reflux rate from a total condenser. The same MTBE product purity is obtained at three different values of the reflux rate. However, the steady states corresponding to low and high reflux rates are stable, while the steady state corresponding to the middle reflux rate is unstable. (ii) Output Multiplicities. Output multiplicity occurs at high RBRs. At RBRs between 0.76 and 1.0, there are three separate unique solutions to exactly the same value of the RBR β. Each solution corresponds to a different product purity of EG. When only one steady state is present (no output multiplicity), such a steady state is stable. On the other hand, when three steady states are present, the ones corresponding to low and high product purities are stable, while the other one corresponding to middle product purities is unstable (saddle type). Along the branch containing unstable steady-state solutions, open-loop operation is not possible. In fact, an arbitrarily small disturbance acting on the process moves the operating point toward either the high or the low product purity operating points. Along this unstable branch, a counterintuitive phenomenon can be observed: Increasing the product purity results in decreased RBR. From an operating viewpoint, it is not hard to conclude that this phenomenon is inherently unstable.4 Moreover, because of the fact that for each unstable steady state there exists a stable steady-state solution with a lower reboiler boilup ratio (second-order input multiplicity), closed-loop (controlled) operation cannot be realized along the unstable branch. That is, for a given setpoint of the product purity and under a

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stabilizing feedback control, the manipulated variable converges to the stable branch. (iii) Startup of the Reactive Distillation Column. If operating conditions are chosen such that input multiplicity is present, the startup process can be complex. In fact, this is because the initial regime usually corresponds to a high concentration of reagent and a low concentration of product and because the column is first filled with the initial mixture. Thus, the conditions are created for the process trajectory to be in the region of attraction of the steady state with the lower reaction extent. To avoid these startup problems, the structure of the bifurcation diagram can be exploited to obtain the following startup guideline. Note that the bifurcation diagram can be seen as composed of two branches (see Figure 2). Branch A is defined for all β and corresponds to stable steady states. Branch B is defined only for high values of the RBR (β > 0.85) and contains both stable (low product purity) and unstable (middle product purity) steady states. The idea is to start up at operating conditions (sufficiently low values of the RBR β) where only one steady state is present. Once the reactive distillation column has reached the corresponding stable steady state (belonging to branch A), the roboiler ratio must be increased slowly up to its design value β*. Thus, the reactive distillation column will track branch A to attain the steady state with high product purity corresponding to the design value β*. (iv) Cost-Optimal Operating Conditions. The bifurcation diagram in Figure 2 corresponds to the costoptimal configuration. The optimal value of the reboiler boilup ratio obtained by Ciric and Miao8 is β ) 0.958, whose corresponding steady states are shown in Figure 2. Under cost-optimal operating conditions, the EG reactive distillation column presents three steady states (output multiplicities). However, the cost-optimal steady state corresponds to the higher product purity (xD,EG = 0.965), which is stable and the higher possible purity that can be obtained with the lower possible reboiler boilup ratio. Note that if β increases, the energy consumption also increases while the product purity decreases. (v) Mechanisms. Input and output multiplicities arise because of the very complicated link between separation and reaction processes or more formally because of the considerable nonlinearity of the equations describing these processes. Hauan et al.18 developed a qualitative understanding of this phenomenon for the case of MTBE production. The main idea of their qualitative understanding is that intermediate products are recycled via the separation stages, which leads to an increase of the reaction rate and consequently to an increase in the heat production due to exothermic reactions. In this sense, multiplicity of steady states is the way the reactive distillation column conciliates its internal competing mechanisms, namely, heat extraction via the vapor flow rate (separation mechanism) and heat production via the exothermic reactions (reaction mechanisms). As the internal recycling increases, more reagents are incorporated into the reaction stages, thus leading to more stressed competing mechanisms. Evidence of this phenomenon is presented in Figure 4 where the internal recycling rate (backmixing) defined by F ) L10/FT as a function of the RBR β is presented. Here FT is the total fed flow rate. The internal recycling rate F represents the average number of times a pack

Figure 4. Recycling backmixing rate (F) as a function of the reboiler boilup ratio (β). This diagram corresponds to the costoptimal bifurcation diagram in Figure 2.

of reagents is incorporated into the separation/reaction path. For low values of β, F is on the order of 3-5,19 which are typical values for nonreactive distillation columns. For high values of β, F is on the order of 50 and larger, which evidences the fact that the reactive distillation column behaves like a tubular reactor with high rates of recycled reagents.20 Within a qualitative framework, our findings agree well with those described in Figure 1b of Karpilovskiy et al.’s work.13 (vi) General Reactive Distillation Configurations. It is our belief that the input and output multiplicity structure and competing mechanisms described in this work are typical of general reactive distillation columns. This means that input and output multiplicities are very likely to occur in reactive distillation columns under high internal recycling rates. We based our conclusion on the result in this work and previously reported ones.6,8,9,10,14,18 In our case, output multiplicity occurs at high RBRs. In Sneesby et al.’s work9,10 for the case of MTBE reactive distillation columns, output multiplicity occurs at high reflux rates. Both mechanisms, high RBRs and high reflux rates, have the effect of increasing the incorporation of reagents into the reactive stages, thus leading to a highly stressed operation. 5. Conclusions This short paper discussed the steady-state structure displayed by an ethylene glycol reactive distillation column. The cost-optimal reactive distillation configuration reported by Ciric and Miao,8 which is a total reflux distillation column, was taken as the base case study. The following conclusions were obtained: 1. Very similar bifurcation diagrams were obtained for the cost-optimal and suboptimal cases. This evidences that the steady-state phenomenon is a result of neither the modeling techniques nor optimized configurations.

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2. Input multiplicities occur at moderate and high EG product purity. The same EG product purity is obtained at two different values of β. The steady state corresponding to low energy consumption (low β) is stable, so then open-loop operation of the EG reactive distillation column with low energy consumption rates is possible. 3. Output multiplicities occur at high RBRs (β > 0.76). Generically, either one or three steady states can be present, which agrees with previously reported results for other reactive distillation columns.9,10 Appendix Assume that the column has N trays (counted from bottom to top), I chemical species, and J chemical reactions and operates at total reflux. The material balance for chemical species i on tray k in the middle of the column is given by

Fi,k + Vk-1xi,k-1 + Lk+1xi,k+1 - Lkxi,k - VkKi,kxi,k + J

νi,jξj,k ) 0 ∑ j)1

(A.1)

where i, j, and k denote the chemical species in the reactive system, the chemical reaction, and the tray in the column, numbered from bottom to top, respectively. Fi,k is the feed rate of the chemical species i to the tray k, Ki,k is the vapor-liquid equilibrium coefficient for component i on tray k (computed from Antoine’s equation), νi,j is the stoichiometric coefficient of component i in reaction j, and ξj,k is the extent of reaction j on tray k. The vapor stream VN off the top tray is totally condensed and returned to the tray: J

Fi,N + VN-1Ki,N-1xi,N-1 - LNxi,N +

νi,jξj,1 ) 0 ∑ j)1

(A.2)

A portion β (the reboiler boilup ratio) of the liquid stream leaving the bottom tray is vaporized and fed back to the tray: J

L2xi,2 - (1 - β)L1xi,1 - V1Ki,1xi,1 +

νi,jξj,1 ) 0 ∑ j)1

(A.3)

The extent of reaction j is calculated from the kinetic equation

ξj,k ) Wkfj(xi,k,Tk)

(A.4)

where Wk is the liquid holdup volume on tray k and fj is the kinetic rate of reaction j. The energy balance for tray k becomes J

Hvap(Vk-1 - Vk) -

∆Hjξj,k ) 0 ∑ j)1

(A.5)

where Hvap is the heat of vaporization and ∆Hj is the heat of reaction j. Finally, the mole fractions of chemical species in each liquid or vapor stream must sum to unity: I

I

xi,k ) 1, ∑Ki,kxi,k ) 1 ∑ i)1 i)1

(A.6)

Equations A.1-A.6 contains N(I + J +3) + 1 variables (N temperatures, JN extents of reactions, N liquid streams, N vapor phase streams, IN liquid compositions, and the RBR β). There are N(I + J + 3) equations in the model. Therefore, the model (A.1)-(A.6) has a degree of freedom, which we take as the RBR β. Note that 0 e β e 1. Literature Cited (1) Doherty, M. F.; Perkins, J. D. On the dynamics of distillations process: IV. Uniqueness and stability of the steady state in homogeneous continuous distillation. Chem. Eng. Sci. 1982, 37, 381. (2) Lucia, A. Uniqueness of solutions to single-staged isobaric flash process involving homogeneous mixtures. AIChE J. 1986, 11, 1761. (3) Shridar, L. N.; Lucia A. Analysis and algorithms for multistage separation processes: fixed temperature and pressure profiles. Ind. Eng. Chem. Res. 1989, 28, 793. (4) Jacobsen, E. W.; Skogestad, S. Multiple steady states and instabilities in distillation. Implications for operation and control. Ind. Eng. Chem. Res. 1995, 34, 4395. (5) Gani, R.; Jorgensen, S. B. Multiplicity in numerical solution of nonlinear models: separation processes. Comput. Chem. Eng. 1994, 18, S55. (6) Pisarenko, Y. A., Epifanova, O. A.; Serafimov, L. A. Steady states for a reaction-distillation column with one product stream. Theor. Found. Chem. Eng. 1988, 4, 281. (7) Nijhuis, S. A.; Kerkhof, F. P. J. M.; Mak, A. N. S. Multiple steady states during reactive distillation of methyl tert-butyl ether. Ind. Eng. Chem. Res. 1993, 32, 2767. (8) Ciric, A. R.; Miao, P. Steady-state multiplicities in an ethylene glycol reactive distillation column. Ind. Eng. Chem. Res. 1994, 33, 2738. (9) Sneesby, M. G.; Tad, M. O.; Smith, T. N. Implications of steady-state multiplicity for operation and control of esterification columns. Inst. Chem. Eng. Symp. Ser. 1997, No. 142, 205. (10) Sneesby, M. G.; Tad, M. O.; Datta, R.; Smith, T. N. ETBE synthesis by reactive distillation. 2. Dynamics simulation and control aspects. Ind. Eng. Chem. Res. 1997, 36, 1870. (11) Gehrke, V.; Marquardt, W. A singularity theory approach to the study of the reactive distillation. Comput. Chem. Eng. 1997, Suppl. 21, S1006. (12) Gu¨ttinger, T. E.; Morari, M. Predicting multiple steady states in distillation: Singularity analysis and reactive systems. Comput. Chem. Eng. 1997, Suppl. 21, S1001. (13) Karpilovskiy, O. L.; Pisarenko, Y. A.; Serafinov, L. A. Multiple solutions in single-product reactive distillation. Inst. Chem. Eng. Symp. Ser. 1997, No. 142, 685. (14) Kumar, A.; Daoutidis, P. Nonlinear control of a high-purity ethylene glycol reactive distillation column. Proceedings of IFACADCHEM, Banff, Canada, July 1997; p 371. (15) Ciric, A. R.; Gu, D. Synthesis of nonequilibrium reactive distillation processes via mixed integer nonlinear programming. AIChE J. 1994, 40, 1479. (16) Doedel, E. J. AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations; Computer Science Department, Concordia University: Montreal, Canada, 1986. (17) Duff, I. S. MA28sA set of Fortran subroutines for sparse unsymmetric linear equations; Harwell Report AERE-R.8730; 1980. (18) Hauan, S.; Hertzberg, T.; Lien, K. M. Why methyl tertbutyl ether production by reactive distillation may yield multiple solutions. Ind. Eng. Chem. Res. 1995, 34, 987. (19) Holland, C. D. Fundamentals of Multicomponent Distillation; McGraw-Hill: New York, 1981. (20) Matros, Y. S. Catalytic Process under Unsteady-State Conditions; Elsevier: Amsterdam, The Netherlands, 1989.

Received for review July 2, 1998 Revised manuscript received November 6, 1998 Accepted November 9, 1998 IE980424Q