Ind. Eng. Chem. Res. 2006, 45, 2261-2271
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PROCESS DESIGN AND CONTROL Global Bounds on Optimal Solutions for the Production of 2,3-Dimethylbutene-1 Jignesh Gangadwala,† Achim Kienle,*,†,‡ Utz-Uwe Haus,§ Dennis Michaels,§ and Robert Weismantel§ Max-Planck-Institut fu¨r Dynamik Komplexer Technischer Systeme, Sandtorstrasse 1, D-39106 Magdeburg, Germany, Lehrstuhl fu¨r Automatisierungstechnik/ Modellbildung, Otto-Von-Guericke-UniVersita¨t, UniVersita¨tsplatz 2, D-39106 Magdeburg, Germany, and Institut fu¨r Mathematische Optimierung, Otto-Von-Guericke-UniVersita¨t, UniVersita¨tsplatz 2, D-39106 Magdeburg, Germany
This paper is concerned with computer-aided optimal design of combined reaction-distillation processes. The production of solvent 2,3-dimethylbutene-1 by isomerization of 2,3-dimethylbutene-2 is considered as an innovative benchmark problem. Possible process candidates are a reactive distillation column, a reactor coupled to a nonreactive distillation column, or a reactive reboiler with a nonreactive distillation column on top. Suitable mathematical models of the different processes are formulated, and the reaction kinetics of the isomerization over an Amberlyst 15 catalyst is determined. Local mixed-integer nonlinear optimization indicates that reactive distillation has the lowest total annualized costs. However, because of the nonconvexity of the underlying optimization problem, better solutions for the other process candidates cannot be excluded with the local approach. Therefore, a new approach is presented which provides a global lower bound for the second best solution and therefore proves that reactive distillation is the best option. The new approach is based on some suitable polyhedral approximation of the underlying model equations leading to a mixedinteger linear optimization problem. 1. Introduction Reactive distillation combines chemical reaction with distillation within a single processing unit.1 Optimal design of such an integrated process can be difficult because of increased complexity. Further, reactive distillation is not the only possibility to combine reaction and separation. Depending on the application to be considered, other process configurations involving reactors, side-reactors, and nonreactive distillation columns can also be attractive.2,3 The optimization of a total annualized cost of different suitable process alternatives can provide a suitable platform to determine an overall optimal configuration. However, this class of optimization problems is difficult to solve because it includes nonlinearly connected continuous variables and discrete decision variables, leading to a mixed-integer nonlinear program (MINLP). Continuous variables are usually related to the operating conditions in such a plant, whereas discrete decisions are concerned with the number and position of feed flows, the number and position of reaction zones or reactors, and the number of column trays. In addition, some suitable column hardware has to be designed.3 However, this is usually done in a separate step and will, therefore, not be considered in this paper. For the solution of the resulting optimization problems, local MINLP optimization methods are available,4 which have been * Corresponding author. Fax: +49-391-6110515. E-mail: kienle@ mpi-magdeburg.mpg.de. † Max-Planck-Institut fu¨r Dynamik Komplexer Technischer Systeme. ‡ Lehrstuhl fu¨r Automatisierungstechnik/ Modellbildung, Otto-vonGuericke-Universita¨t. § Institut fu¨r Mathematische Optimierung, Otto-von-GuerickeUniversita¨t.
applied with some success to reactive distillation processes before.5-7 The main advantage of this approach is its ability to efficiently handle large problems. However, because of the nonlinearity of the problem, only local optima can be found. To overcome this problem in engineering applications, often stochastic optimization methods such as genetic algorithms are applied.8 These are usually computationally very expensive. Further, because of the probabilistic approach, they can also not give any guarantee for global optimality. To overcome the problems associated with both approaches, a new combined strategy is proposed in this paper. In a first step, standard local MINLP optimization methods are applied to obtain a preliminary ranking of suitable process candidates. Afterward, the result is checked by a new global approach. It is based on a suitable approximation of the underlying mathematical models and leads to a mixed-integer linear problem (MILP), which can be solved rigorously. The approximation gives global bounds on the objective function value, allowing the comparison of known local optima for different designs. As an innovative benchmark problem, isomerization of 2,3dimethylbutene-2 (DMB-2) to 2,3-dimethylbutene-1 (DMB-1) is considered. DMB-1 is used as a key component in the production of musk fragrances and insecticides.9 The difference in boiling point temperatures of the two isomers is significant. Further, the rate of reaction is sufficiently high for the operating conditions usually employed in a distillation column, so that reactive distillation seems to be a suitable process candidate. Suitable heterogeneous catalysts are sodium/potassium on alumina support or strongly acidic macroporous cation-exchange resins, like Amberlyst 15; see for example ref 10.
10.1021/ie050584j CCC: $33.50 © 2006 American Chemical Society Published on Web 02/28/2006
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Ind. Eng. Chem. Res., Vol. 45, No. 7, 2006
Table 1. Side Products and Their Normal Boiling Points (Tb)a
a
components
Tb (K)
2-penten-4-methyl (cis) 2-penten-4-methyl (trans) 2-penten-2-methyl 2-penten-3-methyl 2-hexene 2-butanone-3,3-dimethyl 2-butanol-2,3-dimethyl pinacolyl alcohol 3-butenol-2,3-dimethyl
329.6 331.8 340.5 340.9 342.0 379.0 392.0 393.0 NA
Values taken from Reid et al.22 and http://webbook.nist.gov/chemistry/.
The outline of the paper is as follows: First, reaction kinetics over an Amberlyst 15 catalyst is determined experimentally and the vapor-liquid equilibrium (VLE) is studied. Afterward, suitable process candidates are presented and a preliminary ranking is obtained using local MINLP optimization methods. The results are afterward confirmed with the new global approach described above. It is based on the polyhedral approximation of the underlying model equations, which will be discussed in some detail for the nonlinearities involved in this study. These are characteristic for many other reaction, separation, and combined reaction-separation processes.
Figure 1. Effect of stirrer speed on the reaction rate at 333.15 K and 0.6 gm/gmol catalyst loading.
2. Reaction Kinetics and Phase Equilibrium The reaction kinetics are studied in a batch reactor in the temperature range of 50-70 °C, catalyst loadings between 0.3 and 1.82 gm/gmol DMB-2, and stirrer speeds between 500 and 1500 rpm. The widely known Amberlyst 15 is used as a catalyst in the experiments. 2.1. Materials. DMB-1 (>98%) is obtained from the Fluka Chemica AG, and DMB-2 (>98%) is obtained from the Aldrich Chemica AG. The catalyst Amberlyst 15 is obtained from Rohm & Hass. The catalyst was used in the supplied form for the kinetic experiments. 2.2. Experimental Procedure and Analysis. The reaction was performed in a 100 mL batch-mode autoclave provided with a magnetic stirrer, sampling valves, and temperature and pressure sensors. Nitrogen pressure is applied to maintain the reactor at 4 bar in order to keep all the reacting components in the liquid phase. The reaction temperature was controlled by immersing the reactor in a water or oil bath. This provides very good temperature control within (1 °C. Once the desired reaction temperature was attained, the samples were withdrawn through the sampling valves at regular time intervals. The composition of the reaction mixture is analyzed by gas chromatography using FID/TCD (flame ionization/thermal conductivity) combined detectors with a 30m × 250 µm × 0.25 µm INNOWAX column (Hewlett-Packard 6890). The column temperature was maintained at 50 °C. The reaction mixture was also analyzed to check the presence of any side product in a separate gas chromatography analyzer using a GC/MSD detector with 60 m × 250 µm × 0.1 µm DB5ms column (Agilent 6890N). 2.3. Side Product Analysis. The side products detected using the GC/MSD column are shown in Table 1 with their normal boiling points. The cumulative amount of these substances, however, is very small (185 000. We aim to create a methodology in which each single subproblem is solvable using the CPLEX 7.1 mixed-integer solver within
