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SEPARATIONS Mixed-Integer Nonlinear Programming Optimization of Reactive Distillation Processes J. Stichlmair* and Th. Frey Lehrstuhl fu¨ r Fluidverfahrenstechnik, Technische Universita¨ t Mu¨ nchen, Germany
The paper presents a conceptual method for the design of optimized processes of reactive distillation. The first part of the paper deals with the thermodynamic fundamentals of reactive distillation. Their knowledge is an indispensable prerequisite for the design of an appropriate superstructure for a given system. The second part outlines the strategy for the development of optimized processes by mixed-integer nonlinear programming (MINLP) on the basis of this superstructure. The objective function applied is the annualized total costs that contain both the investment and the operating costs. In the third part, finally, the effectiveness of the proposed method is demonstrated by two important examples of industrial reactive distillation processes, that is, the synthesis of methyl tert-butyl ether and of methylacetate. Mixed-integer nonlinear programming (MINLP) is a novel mathematical optimization method that can be applied to the development of chemical processes. This method is of particular interest for the design of novel processes where no empirical knowledge is available. In such cases, engineers rely on the availability of mathematical optimization tools for process design. Reactive distillation is such a novel separation process. Reactive distillation denotes the simultaneous implementation of a chemical reaction and a distillation in a counter-currently operated column. This simultaneous implementation of two unit operations is often superior to the conventional sequential implementation. Among the most important advantages are a total conversion, even in equilibrium reactions, a higher selectivity, a simplification of downstream processing of the products, and a direct utilization of the exothermic heat of reaction for distillation. The objectives of this paper are the thermodynamic fundamentals of reactive distillation and process optimization with MINLP. Thermodynamic Fundamentals of Reactive Distillation First, the simple reaction a + b T c is considered. Here, a denotes the low boiling, b the intermediate boiling, and c the high boiling constituent of the mixture. The reaction is presumed to take place in the liquid phase only. The chemical equilibrium is adequately described by the equilibrium constant KR. The higher the value of KR, the more convex is the equilibrium curve in the triangular composition space; see Figure 1. Besides the equilibrium of the reaction, the reaction path is of particular interest. It can easily be described * To whom correspondence should be addressed. Prof. Dr. Johann Stichlmair, Lehrstuhl fu¨r Fluidverfahrenstechnik, Technische Universita¨t Mu¨nchen, Boltzmannstr. 15, 85747 Garching, Germany. Phone: 0049-89-289-16500.
by the stoichiometry of the reaction. However, the changes of the total number of moles must be taken into account. The equation is given in Figure 1. An evaluation reveals that all stoichiometric lines emerge from the pole π whose coordinates are xπi ) vi/vt. Here, vt denotes the sum of all stoichiometric coefficients vi. By convention, the stoichiometric coefficients of the reactants and products are negative and positive, respectively. In conventional (i.e., nonreactive) distillation, a plot of distillation lines in a triangular diagram constitutes an excellent basis for a conceptual process design. Distillation lines represent, in essence, a sequence of vapor/liquid equilibrium states; see Figure 2. To an initial liquid concentration, xi0, the equilibrium concentration in the vapor y*i0 is determined. Subsequent condensation of this vapor generates a new liquid state with the concentration xil ) yi0. A sequence of several such equilibrium stages constitutes the paths of the distillation lines shown in Figure 2. At high reflux rates, the distillation lines represent the course of the concentration profiles within a distillation column. Thus, they allow the easy determination of the feasible products. This feature is the great merit of distillation line charts; see Stichlmair and Fair.1 The concept of distillation lines can also be applied to reactive distillation. Again, the vapor concentration y*i0 (point 1*) is determined, which is in an equilibrium state with an initial liquid concentration xi0 (point 1) on the chemical equilibrium line; see Figure 3. By condensation, a new liquid is generated that has the same concentration as the vapor xil ) y*i0 (point 1*). This liquid mixture no longer lies on the chemical equilibrium line. Hence, a reaction takes place in the liquid phase that changes the liquid concentration along the corresponding stoichiometric line. Eventually, point 2 at the chemical equilibrium is reached. A sequence of such distillation and reaction steps constitutes the reactive distillation line, which is, in this case, identical
10.1021/ie010324b CCC: $20.00 © 2001 American Chemical Society Published on Web 11/16/2001
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Figure 1. Depiction of chemical equilibrium and reaction path.
Figure 4. Determination of reactive azeotropes.
Figure 2. Triangular diagram with distillation lines.
Figure 3. Reactive distillation line and reactive azeotrope.
to the chemical equilibrium. The sequence of distillation and reaction steps is directed toward the low boiler a, which is marked by an arrow on the reactive distillation line. Performing the same procedure with a different starting point, for example, point 10 in Figure 3, gives a sequence that is directed toward the intermediate boiler b.
A special situation exists when this procedure is applied to the liquid state A in Figure 3. The equilibrium vapor state is point A*. After condensation, the liquid reacts back to the initial state A. Hence, the sequence of distillation and reaction does not effect any concentration changes of the liquid. The concentration change of the distillation (A f A*) is completely compensated for by the concentration change of the reaction (A* f A). Thus, point A is called a reactive azeotrope according to Barbosa and Doherty.2 The direction of the concentration change caused by distillation is described by a tangent to a liquid residuum line. The direction of the concentration change caused by the reaction is described by the stoichiometric line. Hence, at the reactive azeotrope, a stoichiometric line forms a tangent to a residuum line. This is a very important statement as it allows a very simple graphical determination of reactive azeotropes. This graphical approach is demonstrated in detail in Figure 4.3 The system considered exhibits two (nonreactive) binary minimum azeotropes with a distillation boundary in between. The residuum lines run between the a-b-minimum azeotrope and the high boiler c and the intermediate boiler b, respectively. The distillation boundary divides the concentration space into two distillation fields. In the right field there are obviously no points of tangential contact between stoichiometric lines and residuum lines. Thus, no reactive azeotropes can exist in this field. In the left field,
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however, there are two regions of tangential contact between stoichiometric and residuum lines. The points of tangential contact constitute two lines (dashed pointed lines in Figure 4) where the first condition for the formation of reactive azeotropessthe collinearity of distillation and reaction concentration changessis met. The second condition requires that reactive azeotropes be an element of the chemical equilibrium line (for instantaneous reactions). Hence, the dashed pointed collinearity line and the chemical equilibrium line intersect at the reactive azeotropes. Depending on the value of the chemical equilibrium constant KR, none, one, or two reactive azeotropes can exist in the system considered in Figure 4. This graphical approach for the determination of reactive azeotropes can be applied to all ternary and even to quaternary systems. For multicomponent systems, however, a mathematical formulation of the two conditions described above is needed. The knowledge of reactive and nonreactive distillation lines as well as of reactive and nonreactive azeotropes constitutes an excellent basis for a conceptual process design. The fundamental procedure is outlined in Stichlmair and Frey4 in several examples different with respect to type of vapor-liquid equilibrium and type of chemical reaction. These examples make clear the huge advantages reactive distillation offers in some cases. MINLP Optimization of Reactive Distillation Processes Knowledge of the course of reactive and nonreactive distillation lines allows the conceptual design of reactive distillation processes. However, one has to be aware that only the superstructure of the process can be designed in this way. The so developed processes contain many free variables whose optimal values have to be determined later on. Among the free variables are operational variables, for example, temperatures of the feed streams, reflux ratio, energy demand, and so forth, and structural variables, for example, feed stages, number of distillative and reactive stages and positions of the reactive zones within the column, and so forth. At present, only the operational variables can be determined by standard optimization techniques. Thus, all the structural details of the process have to be found by empiricism or by intuition. MINLP is the first mathematical tool that allows the simultaneous optimization of both the operational and the structural variables. The characteristic feature of MINLP is the use of integer variables besides the real variables. Generally, the integer variables are used to define the structure of the process. The integer variables can have only the values 1 or 0. A value of 1 stands for the existence and a value of 0 for the nonexistence of any process element, for instance, feedstreams, heat exchangers, column sections, or even whole columns. During the mathematical optimization, the values of the integer variables as well as the values of the real variables are changed. Thus, the structure of the process is changed during the optimization. The use of the integer variables gives, however, rise to severe discontinuities in the system of mathematical equations which make the optimization procedure very complex. In this work the commercial software DICOP++ (Viswanathan and Grossmann5) within the framework of GAMS (Brooke et al.6) has been successfully used for solving the optimization problem. An
Figure 5. VLE of quaternary mixture n-butane/isobutylene/ methanol/MTBE.
equilibrium stage model was used. A decisive precondition is an appropriate formulation of the MESH equations and a sufficient initialization of the free process variables.7 The objective function for the optimization was the annualized total costs of the process. Standard methodes have been used for estimating the costs, for example, see Douglas,8 Floudas and Anastasiadis,9 and Knight and Doherty.10 The variables in these cost functions have been adjusted to actual European prices. The operating costs have been estimated from the energy costs multiplied with an overall surcharge factor of 4. The specific costs of steam and cooling water were 100 and 40 $/kW, respectively. The estimation of the investment costs demands a rough dimensioning of the equipment, that is, columns and heat exchangers. The key parameter for column dimensioning was a superficial gas load of F ) 1.5 Pa1/2. The costs of the catalytic packing was assumed to be as high as 10000 $/m3. The key parameter for dimensioning the heat exchangers was an overall heat-transfer coefficient of K ) 800 kW/(m2‚K). The calculated yearly operation time was 8000 h. In the following pages, the effectiveness of this novel optimization technique will be demonstrated with two important processes of reactive distillation. MTBE Synthesis Methyl tert-butyl ether (MTBE) is a very important antiknock additive to gasoline. Methanol and isobutylene react to MTBE. In most cases, the feedstock contains high amounts (≈20%) of n-butane, which is inert for the reaction. The quaternary mixture forms three binary minimum azeotropes that constitute a distillation boundary area in the concentration tetrahedron shown in Figure 5. Thus, the concentration space is divided into two sections, which impedes the downstream processing of the reaction products drastically. Figure 6 shows the reaction equilibrium that establishes a section of the mantle area of a cone. All reactive distillation lines run on this mantle area from methanol (high boiler) to isobutylene and the n-butane/methanol azeotrope, respectively. In between, there exists a boundary reactive distillation line. The nonreactive distillation lines, not shown in Figure 6, start from the
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Figure 6. Chemical reaction equilibrium and column superstructure for MTBE synthesis.
Figure 8. Chemical reaction equilibrium, reactive distillation lines, and column superstructure for the methyl acetate synthesis.
Methyl Acetate Synthesis
Figure 7. Optimization results and concentration profiles for MTBE synthesis.
high boiler MTBE. The superstructure of the process consists, in principle, of a single column with an upper reactive section (e.g., equipped with catalyst) and a lower distillative section. The superstructure of the process shown in Figure 6 contains several operational and structural variables whose optimal values have to be found by MINLP optimization. The objective function for the optimization is the total annualized costs of the process. Figure 7 shows the results of the MINLP optimization. In total, the column consists of 25 equilibrium stages; 11 of them are reactive stages. The reactive section extends from stage 5 to stage 15. The energy demand is 440 kW for a feed rate of 10 mol/s. The column diameter is 600 mm. Very important is the multiple feeding of the educts. It effects a drastical reduction (more than 40%) of the total costs of the process compared to a single feed column. Even better would be a separate feeding of methanol (near the top) and isobutylene/n-butane mixture (near the bottom). However, this further reduction of the annualized costs is only small.
The second example of a reactive distillation process is the synthesis of methyl acetate.11 Acetic acid and methanol react to methyl acetate and water. Figure 8 shows the concentration space of this mixture in the form of a tetrahedron. The reaction equilibrium forms a saddle area that contains all reactive distillation lines. All reactive distillation lines run from acetic acid to the minimum azeotrope formed by methyl acetate and methanol. An important characteristic is that the desired products water and methyl acetate are no end points of distillation lines. Hence, they cannot be gained as bottoms and overhead fractions of a single feed column. The products can only be separated into pure fractions if the reactants are separately fed into the column. The column consists of three different sections, with a reactive section in the middle. In principle, chemical absorption takes place in this section. The acetic acid, fed at the top of this section, physically absorbs the vaporous methanol, which is fed below. In the liquid phase, the two reactants are converted into the products methyl acetate and water. The methyl acetate, being the low boiling product, proceeds upward in the column. The water moves downward as it is the high boiling product. Above and below the reactive section, there are nonreactive sections that purify the two products by simple distillation. On the basis of the superstructure developed in Figure 8, the optimal values of the structural and operational variables are determined by MINLP optimization. Again, the annualized costs are used as objective function in the optimization. The results are shown in Figure 9. The column consists of 45 equilibrium stages. The reactive section extends from stage 10 to stage 40. Acetic acid is fed on the sixth stage at a temperature of 349 K. Methanol is vaporously fed at 5 succeeding stages at the lower end of the reaction zone, that is, stages 36-40. The energy demand is as high as 606 kW for the specified feed rate of 20 mol/s. The column diameter is as high as 900 mm. Decisive for the process is the multiple feeding of the methanol in a vapor state. This suppresses the internal recirculation of water that is indicated by a sharp maximum in the concentration profile.11 The vaporous and multiple feeding of methanol drastically reduces the energy demand of the process.
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Figure 9. Optimization results for methyl acetate synthesis.
Summary MINLP optimization has proven to be a very powerful tool for process design. It has the potential of revolutionizing process design in chemical engineering. The available software is, however, difficult to handle. Therefore, it needs further development. Literature Cited (1) Stichlmair, J. G.; Fair, J. R. DistillationsPrinciples and Practice; Wiley-VCH: New York, 1998.
(2) Barbosa, D.; Doherty, M. F. Theory of Phase Diagrams and Azeotropic Conditions for Two Phase Reactive Systems. Proc. R. Soc. London 1987, A431, 443-458. (3) Frey, Th.; Stichlmair, J. Thermodynamic Fundamentals of Reactive Distillation. Chem. Eng. Technol. 1999, 22 (1), 11-18. (4) Stichlmair, J.; Frey, Th. Reactive Distillation Processes. Chem. Eng. Technol. 1999, 22 (2), 95-103. (5) Viswanathan, J.; Grossmann, I. E. A Combined Penalty Function and Outer-Approximation Method for MINLP Optimization. Comput. Chem. Eng. 1990, 14 (7), 769-782. (6) Brooke, A.; Kendrick, D.; Meeraus, A. A Users’s Guide, Release 2.25; The Scientific Press: Palo Alto, 1992. (7) Frey, Th.; Stichlmair, J. MINLP Optimization of Reactive Distillation Columns. European Symposium on Computer Aided Process Engineering-10, Computer Aided Chemical Engineering 8; Sauro Pierucci, Ed.; Elsevier: Amsterdam, 2000; pp 115-120. (8) Douglas, J. M. Conceptual Design of Chemical Processes; McGraw-Hill: New York, 1988. (9) Floudas, C. A.; Anastasiadis, S. H. Sythesis of Distillation Sequences with Several Multicomponent Feed and Product Streams. Chem. Eng. Sci. 1988, 43 (9), 2407-2419. (10) Knight, J. R.; Doherty, M. F. Optimal Design and Synthesis of Homogeneous Distillation Sequences. Ind. Eng. Chem. Res. 1989, 28 (5), 564-572. (11) Agreda, V. H.; Parin, L. R.; Heise, W. H. High Purity Methyl Acetate via Reactive Distillation. Chem. Eng. Prog. 1990, 86, 40-46.
Received for review April 12, 2001 Revised manuscript received August 13, 2001 Accepted August 15, 2001 IE010324B
