Phase stability problems are crucial in the computation of phase equilibria and thus play a significant role in various chemical engineering applications such as extraction and distillation. The problem involves determining whether a given phase with certain composition, pressure and temperature is stable or will split into multiple phases. Phase stability is frequently tested using the well known Tangent Plane Criterion (Baker et al., 1982). The criterion formulates the tangent plane distance function (TPDF), defined as the vertical distance between the molar Gibbs free energy surface and the tangent plane at the given composition, as the objective function. The problem can be solved using two approaches namely: solving a system of non-linear equations for stationary points (Michelsen, 1982) and the direct minimization of TPDF function. The former approach is a conventional approach where the solution obtained may be trivial or local, and is mainly dependent on the initial guess. The later approach employs the global minimization techniques because of the high non-linearity associated with the objective function. The presence of comparable minima (i.e., function value at the local minimum will be nearest to that at the global minimum) in TPDF poses a computationally challenging problem to many of the global optimization methods. The complexicity in the TPDF is mainly due to the thermodynamic models that are used to describe the non-ideality in the Gibbs free energy function.  

Problem Formulation            

        The molar Gibbs energy of a system, g at a given temperature and pressure is the summation of the product of mole fraction and partial molar Gibbs energy G_i^p for all components:   

            g = ∑x_iG_i^p                                   (i = 1, 2, 3, ... , N)                                            (1)  

where x_i is the mole fraction of i^th component and N is the total number of components in the system. Tangent plane, t at the specified composition x^* = (x₁^*, x₂^*,…, x_N^*) can be written as:

            t = ∑x_iG_i^p*                                (i = 1, 2, 3, ... , N)                                            (2)  

Thus the objective function (TPDF) can be expressed as  

            F = g - t = ∑x_i (G_i^p - G_i^p*)            (i = 1, 2, 3, ... , N)                                            (3)  

subject to the equality constraint  

             ∑x_i = 1                                     (i = 1, 2, 3, ... , N)                                            (4)  

and the non-negative conditions:  0 ≤ x_i ≤ 1.

            With the recent advances in global optimization methods, several authors addressed this problem using different methods (Sun and Seider, 1995; McDonald and Floudas, 1995; Hua et al., 1998; and Zhu and Xu, 1999) but not yet examined by the promising stochastic methods such as Differential Evolution (Storn and Price, 1997; and Babu et al., 2004) and Tabu Search (Chelouah and Siarry, 2000 and Teh and Rangaiah, 2003).  

            DE is simple, robust and requires few control variables. The method is a population based search and consists of 3 steps namely: mutation, crossover and selection. The mutation generates a new individual by adding the weighted difference between two individuals to a third individual in the population. Crossover is performed mainly to increase the diversity during the search. The selection step determines whether or not the new individual is allowed into the next generation using some greedy criterion.  

            TS is meta-heuristic method that guides and improves the search in the solution space. The method escapes from the local minimum by generating solutions that differs in various ways from those seen in the previous generations. The algorithm avoids the repeated visits to the same place during the search thus increasing computational efficiency. After a number of iterations several promising areas are identified for further in depth search known as intensification.  

            In this work both DE and TS are implemented and tested for benchmark problems involving 2 to 20 variables and a few to hundreds of local minima. The results show that the methods are better or comparable to several other methods reported in the literature. The potential of DE and TS is then examined for the phase stability problems involving several components with different feed compositions and thermodynamic models. The results show that both DE and TS are reliable in solving phase stability problems and are computationally efficient than other stochastic methods such as genetic algorithms. All these results will be reported and discussed in the conference presentation.

References

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Teh, Y.S. and Rangaiah, G.P. Tabu search for global optimization of continuous functions with application to phase equilibrium calculations. Computers and Chemical Engineering, 27, pp.1665-1679. 2003.

Zhu, Y. and Xu, Z. A reliable prediction of the global phase stability for liquid-liquid equilibrium through the simulated annealing algorithm: Application to NRTL and UNIQUAC equations. Fluid Phase Equilibria, 154, pp.55-69. 1998.