Structuring of Reactive Distillation olumns for Non-Ideal Mixtures using MINLP-Techniques Guido Sand *, Sabine Barmann and Sebastian Engell Process ontrol Laboratory, Dept. of Biochemical and hemical Engineering, Universität Dortmund, Emil-Figge-Straße 70, 44227 Dortmund, Germany Gerhard Schembecer Process Design enter B.V., J.-v.-Fraunhofer-Straße 20, 44227 Dortmund, Germany Abstract This contribution deals with the optimisation of the design of reactive distillation columns. For a highly non-ideal reacting mixture, an MINLP-model is presented and different solution approaches are discussed. Due to the efficient modelling and implementation of the superstructure and an appropriate setting of non-standard solver options, optimisation problems with up to 10,000 variables and constraints could be solved robustly without providing problem specific initial values or scaling factors. Keywords: reactive distillation, non-ideal mixtures, mathematical programming 1. Introduction A current trend in process design is towards integrated processes, i.e. the integration of reactive and separating functionalities into a single apparatus as, e.g., a reactive distillation column. ompared with the classical serial arrangement of unit operations, this advanced concept has the potential to decrease the dimensions of the equipment and to increase the degree of heat integration. Furthermore, it provides the opportunity to overcome chemical and thermodynamical boundaries, such as chemical equilibria or distillation boundaries due to azeotropes. Separations of non-ideal mixtures with simultaneous chemical reactions belong to the most difficult design problems and should be solved in an integrated fashion. The design of a reactive distillation column constitutes a constrained combinatorial optimisation problem which is amenable to MINLP-techniques. In practice, such problems often are hard to solve due to non-linear and integrality constraints. The optimisation of reactive distillation columns has previously been addressed by Grossmann and co-worers (Jacson and Grossmann, 2001) and Stichlmair and coworers (Frey and Stichlmair, 2000, Poth et al., 2003). This wor differs from the aforementioned in the modelling of the superstructure and in the applied solution methodology. A highly non-ideal chemical system is considered, and the robust solution by means of an efficient implementation of the model and a specific treatment of the integers but without a priori (heuristic) nowledge on the solution is highlighted. In * Author to whom correspondence should be adressed: guido.sand@udo.edu
Sect. 2, an MINLP-model for the production of methyl-tertiary-butyl-ether (MTBE) is presented. Then three algorithmic approaches to the solution of the problem without an elaborated initialisation and scaling strategy are compared. In Sect. 4, the (locally) optimal solution of the example problem is presented and discussed: It is demonstrated graphically that the chemical and thermodynamical boundaries are overcome. 2. Mixed-Integer Model of a Non-Ideal Reactive Separation Process The inetically controlled production of MTBE from isobutene and methanol (IB + MeOH MTBE) in the presence of n-butane at a pressure of 8 bar is considered here. The column is represented by a superstructure of trays each of which may be active or inactive. The model for the costing comprises revenues, running costs, and annualised investment costs. 2.1 Tray column The model of a tray is based upon the MESH-equations which are extended by source terms for the reaction (material/enthalpy balance). The equilibrium condition is extended by the Murphree efficiency for the vapour phase: E ideal ( y y ) ( y y ) i = i, i, 1 i, i, 1,. (1) The continuous variable y i, denotes the mole-fraction of the component i of the vapour stream from tray. The assumed efficiency E = 0.7 for each tray determines the ratio between the real change of the concentrations and the maximum possible change given by the thermodynamic equilibrium. The non-ideal liquid phase is modelled by activities which appear in the reaction inetics and in the thermodynamic equilibrium condition. The activity-coefficients are calculated by Wilson s approach, describing three binary azeotropes. The temperature dependencies of the vapour pressures are modelled by the Antoine equation. The formation and the decomposition of MTBE is heterogeneously catalysed and characterised by the following inetics: form eff form 1 Pr = H ai= IB, ai= MeOH, dec eff dec 2 Pr = H ai= MTBE, ai= MeOH,, (2). (3) The production rates Pr [mole/s] are calculated from variable efficient hold-ups H eff [ltr], the inetic constants and polynomials of activities a i,. The efficient hold-up is smaller than the total hold-up reflecting the amount and the efficiency of the catalyst on a tray. The temperature dependencies of the inetic constants are modelled by an extended Arrhenius approach. The thermodynamic properties and the inetics were taen from Beßling (1998) and Rehfinger and Hoffmann (1990). The geometry of the column is constrained by an upper bound on the efficient hold-up and upper and lower bounds on the vapour velocity. Both values do not appear explicitly but are bounded implicitly by the following constraints (Dia represents the column diameter and V represents the vapour flow rate from tray ):
eff H 2 Dia 103, 7, (4) 3 2 3 7,7 10 V Dia 19,2 10 V. (5) The right hand side in the height-constraint Eq. (4) results from the maximum weir height of a tray and the volume specific catalyst efficiency, and the constant factors in the velocity-constraints Eq. (5) are calculated from the minimum and maximum F-factor and the mean molar volume of the vapour phase. 2.2 Superstructure The superstructure consists of N serially arranged trays, each tray has a (free) efficient hold-up H eff and (free) feeds of both raw materials. A tray can be inactivated by means of a binary variable ϕ, except for the bottom and the top tray (reboiler/condenser). The eff reactive functionality of a tray is controlled by its effective hold-up H IR + (0 means no reaction). The separating functionality is controlled by switching its efficiency E IR + by means of the activation variable ϕ : E 0.7 =ϕ. (6) For deactivated trays, the efficient hold-up and the feed are forced to zero by big-m inequalities such that the vapour and the liquid streams pass a deactivated tray unaffected. Additionally, the sum of the feeds to the trays must be equal to the two given total feeds of methanol and isobutene/n-butane. Ambiguous activation patterns are avoided by the logic constraint: ϕ + 1 ϕ 0 < N 1. (7) Note, that the binary variables appear linearly throughout. 2.3 ost model The column is designed to produce MTBE with a purity of 99% (liquid mole fraction x =1 MTBE 0.99) with minimum annual cost ann [ /a]. These costs are calculated from the annualised investment costs for the shell shell, the internals inter, the reboiler reb, the condenser cond and the catalyst cat, in addition to running costs for cooling and heating (Q =n, Q =1 [W]) and the feed streams (F MeOH, F IB/but [mole/s]), reduced by revenues for the bottom and top distillate streams (D =1, D =N MeOH, D =N IB/but [mole/s]). ann 64 44 7444 8 64444444444 74444444444 8 0.81 1.03 = 13.53H Dia + 1.05 ϕ + 0.2507Q 825.1D shell reb = 1 = 1 0.6 = 1 0.6 = 1 + 0.01377Q 138.4D = N MeOH = N 2 3 ( 0.522 + 0.059Dia + 0.258Dia 0.021Dia ) cond 64 44 7444 8 64447444 8 644744 8 + 0.4761 T Q + 0.4944 T Q + 3.0 10 0,6 ln, = N + 184.6F 410.8D 0.6 = N MeOH IB/but = N inter cat + 410.8F 3 eff H IB/but (8)
H [m] denotes the column height, and T =1 [K] and T ln, =N [K] denote the (logarithmic) temperature differences of the reboiler and the condenser, respectively. The objective function is taen from ooboo (2003). 2.4 Efficient implementation Preliminary tests of different NLP-solvers (cf. Sect. 4) indicated that ONOPT3 is the most promising choice. The difficulties in the algorithmic solution process can be reduced by efficiently implementing the model equations (see rules of thumb in ONOPT, 2003): Degeneracy as well as very small or large function values and gradients should be avoided, and highly non-linear expressions should be re-written. To this end, unnecessary bounds are removed (e.g. bounds for activities are not stated explicitly but implicitly by means of Wilson s approach), algorithmic bounds are added (e.g. Q =1 1 to avoid Q =1 = 0) and auxiliary variables are introduced to simplify nonlinear functions of expressions (e.g. the reformulation of the Arrhenius approach: e -E/(RT) e aux. aux T = -E/R). A specific type of algorithmic bounds is introduced into Eqs. (2) and (3) to prevent the activities a i=meoh, from assuming a value of zero due to insufficient numerical precision: Instead of imposing small lower bounds on a i=meoh, (which would be valid globally), a small value ε = 10-2 is introduced into Eqs. (2) and (3) only, they are implemented as Pr Pr form eff form ( ai= MeOH, + ε) = H ai= IB, 2 eff dec ( a + ε) = H a dec i= MeOH, i= MTBE, 3. Solution Algorithms, (9). (10) The solution procedure is based on the decomposition of the MINLP problem into an IP-master-problem (optimisation of the number of trays) and NLP-sub-problems (optimisation of continuous variables for a fixed number of trays). For the IP-part three algorithmic approaches were compared; the NLP-sub-problems are always solved by the generalized reduced gradient method implemented in ONOPT3. 3.1 NLP-sub-problems ONOPT was adjusted to the problem at hand by iteratively setting non-standard options based on evaluations of the results of tests without applying any heuristic nowledge on good solutions. This procedure ensures a conceptual partition of the model from the solution procedure. All variables were initialised by 1 and no scaling factors were supplied. To chec the reproducibility of the results, 91 problems with different numbers of trays N (10 N 100) were solved. The models comprise 1,021 10,111 variables (8 98 binaries) and 1,051 10,411 constraints. With the standard solver options, the robustness turned out to be insufficient: Only 35% of the problems were solved to (local) optimality. This value was increased to 95% by decreasing the feasibility tolerance (rtnwma = 10-6 ), increasing the maximum number of stalled iterations (lfstal = 10 5 ), by the introduction of slac variables for the constraints in the first iteration step (lslac = true), by optimising the step-length in Newton iterations
(lmmxsf = 1) and by automatically scaling the variables and constraints in each iteration step (lfscal = 1). The solution times were 6 1,779 PU-s (Windows-P, 1,5 GHz). 3.2 IP-master-problems The first approach for the IP-part was the branch and bound-algorithm implemented in SBB (2003). The integrality requirements are totally relaxed in the root node and reinforced for one variable after the other from layer to layer. At each node of the search tree, a lower bound on the (locally) optimal solution is generated by solving an NLP (integer-relaxed MINLP). For a superstructure with N = 100 trays, a (locally) optimal solution was found after the exploitation of 58 nodes (PU-time 44.5 min). The annual cost is ann = 1,050 /a and 49 trays are active. Note that in the worst case the number of NLP-sub-problems that has to be solved is in the order of O(2 N ). Secondly, a problem specific approach was applied which completely enumerates the subspace of binary variables ϕ which is feasible wrt. Eq. (7). In fact, this is exactly the same as a solution of problems for different numbers of trays as treated in Sect. 3.1. The solution (identical with the first one) was obtained after solving 91 NLPs. However, it should be noted that in general the number of NLPs to be solved is only in the order of O(N). Under the assumption that for a fixed number of trays the minimum annual cost ann is unimodal wrt. the number of trays, the optimal number of trays can be found efficiently by an interval reduction algorithm: The search for the optimal number of trays is considered as a one-dimensional unconstrained optimisation problem, which must be solved by direct methods (without gradient information). Initially, two problems with N 0 = {10;100} were solved under the assumption that the optimal solution is within the interval [10;100]. The range of the interval is reduced iteratively by a factor of 1/3 as follows: Two new NLPs with N 1 = {40;70} are solved and the corresponding values ann are compared. From ann (40) < ann (70) it follows that the optimum must be in the interval [10;70]. This procedure is repeated iteratively until the interval range is 1, i.e. a local optimum is obtained. Here, the number of NLPs to be solved was only 17 (identical solution as above), in general it is only in the order of O(lg 3/2 N). In comparison, in the disjunctive programming approach presented by Jacson and Grossmann (2001), the number of NLPs to be solved is up to two orders of magnitude larger: For a model with 38 binary variables they solved 1,270 NLPs. Butane Feed MeOH MeOH MTBE Feed IB/Butane IB Azeotropes Separating Trays Integrated Trays Figure 1: Liquid mole fractions, chemical equilibrium lines and distillation boundary
4. Optimal Structure The (locally) optimal column has 49 trays with separating functionality with reactive functionality on trays 11 48. Feed trays (with non-marginal feed) are 48 49 for MeOH and 11 14 for IB/butane. The diameter is Dia = 0.39 m and the reflux ratio is 4,5. From the graphical representation of the liquid mole fractions on the trays in Figure 1, it can be seen that the boundaries given by the chemical equilibrium and the distillation boundary (given by the azeotropes) are overcome. 5. Summary and Perspectives An MINLP-model for the optimal structuring of a reactive distillation column for the synthesis of MTBE was presented. The superstructure is modelled by linear constraints and the highly non-linear equations are implemented efficiently. Three approaches to solve the IP-master-problems which require a relatively small number of NLP-subproblems to be solved were presented. urrent research deals with the systematic integration of heuristic nowledge into the model and the algorithm in order to increase the solution efficiency while maintaining its robustness. Acnowledgement The research reported here was supported by the Deutsche Forschungsgemeinschaft as part of the research unit Integrated Reaction and Separation Processes at the Universität Dortmund. References Beßling, B., 1998, Zur Reativdestillation in der Prozeßsynthese, Dr.-Ing.-Dissertation, Fachbereich hemietechni, Universität Dortmund (in German). ONOPT, 2003, http://www.gams.com. ooboo, 2003, Process Design enter B.V., Breda, The Netherlands. Frey, T. and J. Stichlmair, 2000, MINLP Optimization of Reactive Distillation olumns, in: ESAPE 10, omputer Aided hemical Engineering 8, 115-120, Ed. S. Pierucci, Elsevier, Amsterdam. Jacson, J.R. and I.E. Grossmann, 2001, A Disjunctive Programming Approach for the Optimal Design of Reactive Distillation olumns, omp. hem. Engg. 25, 1661-1673. Poth, N., D. Brusis and J. Stichlmair, 2003, Rigorous Optimization of Reactive Distillation in GAMS with the Use of External Functions, in: ESAPE 13, omputer Aided hemical Engineering 14, 869-874, Ed. A. Kraslawsi and I. Turunen, Elsevier, Amsterdam. Rehfinger, A. and U. Hoffmann, 1990, Kinetics of Methyl Tertiary Butyl Ether Liquid Phase Synthesis atalyzed by Ion Exchange Resin I. Intrinsic Rate Expression in Liquid Phase Activities, hem. Eng. Sci., 45, 1605-1617. SBB, 2003, Users Manual, ARKI onsulting & Development, http://www.gams.com.