MPEC Problem Formulations in Chemical Engineering Applications

Objective SBB CONOPT-PF IPOPT-CNFE Function CPU s. CPU s. Iterations CPU s. Iterations10 1.4738 0.242 0.047 15 0.110 17100 1.7536 50.266 0.234 78 1.250 411000 1.7864 410.195 9.453 680 28.406 782000 1.7888 >6000 35.359 1340 14.062 253000 1.7894 — 112.094 2020 106.188 844000 1.7892 — 211.969 2679 84.875 565000 1.7895 — 340.922 3342 199.391 876000 1.7898 — 468.891 3998 320.140 1157000 1.7896 — 646.953 4655 457.984 1418000 1.7898 — 836.891 5310 364.937 98Table 2: Solution times (Pentium 4, 1.8 GHz, 992 MB RAM) for differentsolution strategiesdynamic tray columns [14, 20, 23]. In these studies smoothing methods andregularization methods were applied for distillation optimization, with the refluxratio, feedtray location and tray number as decision variables. In this section, werevisit these steady state distillation optimizations for the sake of a performancecomparison, particularly with the penalty formulation, PF(ρ).Here we define distillation column models that consist of MESH (Mass balance,Equilibrium, Summation, and Heat balance) equations that are modifiedto optimize the feed tray location and total tray count. As shown in Figure4, streams for the feed and the reflux are fed to all trays as dictated by twodiscretized Gaussian distribution functions. As developed and described in [14],both distribution functions are defined by a mean and standard deviation. Forthe feed and reflux distributions, their means represent the feedtray locationand the total number of trays, respectively, and these are used as continuousdecision variables in the distillation optimization. We also choose a standarddeviation of 0.5, so that distribution flow to trays away from the mean tapersoff smoothly to negligible values. Moreover, the grayed area in Figure 4 consistsonly of vapor traffic, and consequently, each tray model must include complementaritiesthat allow for disappearance of the liquid phase. As described inSection 5.6, phase equilibrium is relaxed by the complementarity constraints:y ij = β i K ij x ijβ i = 1 − s l i + sv i(49a)(49b)0 ≤ L i ⊥ s l i ≥ 0 (49c)0 ≤ V i ⊥ s v i ≥ 0 (49d)where i = {1, 2, . . .,N max } and N max is the maximum number of equilibriumstages, j is the component index, L i and V i are the liquid and vapor flow rateon tray i, and x and y are the respective mole fractions.The resulting complementarity distillation model is an alternative nonlin-22