Homotopy continuation

Another implementation of homotopy-continuation methods is the use of problem-dependent homotopies that exploit some physical aspect of the problem. Vickeiy and Taylor [AIChE J., 32, 547 (1986)] utilized thermodynamic homotopies for K values and enthalpies to gradually move these properties from ideal to ac tual values so as to solve the MESH equations when veiy nonideal hquid solutions were involved. Taylor, Wayburn, and Vickeiy [I. Chem. E. Symp. Sen No. 104, B305 (1987)] used a pseudo-Murphree efficiency homotopy to move the solution of the MESH equations from a low efficiency, where httle separation occurs, to a higher and more reasonable efficiency. 

Chang YA, Seader JD. Simulation of continuous reactive distillation by homotopy continuation method. Computers Chem Eng 1988 12 1243-1255. 

Reliable and fast equilibrium calculations (or so-called flash calculations) are the mechanism by which thermodynamic properties are used in industry. This area has received much attention in the past. Algorithms include successive substitution with acceleration and stability analysis,Inside-Out and Interval methods, Homotopy continuation methods with application to three-phase systems, and systems with simultaneous physical and chemical equilibrium. An area of recent focus is the flash algorithm for mixtures containing polydisperse polymers. However, many challenging problems remain. 

High-pressure systems in the vicinity of critical points, such as synthesis gas and air separation systems, remain a challenge. Our flash algorithm has difficulty in identifying the correct phase state, or converging to the correct vapor-Uquid solutions. This problem may be exacerbated by the difficulty in obtaining the equation of state volume root in the vicinity of the critical points. Further work to improve the algorithm and the equation of state volume root determination is required. It is believed that the homotopy continuation methods are probably better suited for calculations near the critical points. 

The NEQ model requires thermodynamic properties, not only for calculation of phase equilibrium but also for calculation of driving forces for mass transfer. In addition, physical properties such as surface tension, diffusion coefficients, and viscosities, for calculation of mass (and heat) transfer coefficients and interfacial areas are required. The steady-state model equations most often are solved using Newton s method or by homotopy-continuation. A review of early applications of NEQ models is available [5]. 

Lee and Dudukovic [18] described an NEQ model for homogeneous RD in tray columns. The Maxwell-Stefan equations are used to describe interphase transport, with the AIChE correlations used for the binary (Maxwell-Stefan) mass-transfer coefficients. Newton s method and homotopy continuation are used to solve the model equations. Close agreement between the predictions of EQ and NEQ models were found only when the tray efficiency could correctly be predicted for the EQ model. In a subsequent paper Lee and Dudukovic [19] presented a dynamic NEQ model of RD in tray columns. The DAE equations were solved by use of an implicit Euler method combined with homotopy continuation. Murphree efficiencies calculated from the results of an NEQ simulation of the production of ethyl acetate were not constant with time. 

Y. A. Chang, J. D. Seader, Simulation of Continuous Reactive Distillation by Homotopy Continuation Method,... 

Kovach in, I.W., and W J). Seider, Heterogeneous Azeotropic Kslil-lation Homotopy-Continuation Methods, Comput. Chem. Eng., 11,595 (1987). 

The relaxation, inside-out, and homotopy-continuation methods are extensions of whole or part of the first four methods in order to solve difficult systems or columns. The nonequilibrium models are rate- or transport phenomena-based methods that altogether do away with the ideal-stage concept and eliminate any use of efficiencies, They are best suited for columns where a theoretical stage is difficult to define and efficiencies are difficult to predict or apply. 

Wayburn T.L. and Seader J.D. (1987). Homotopy continuation methods for computer-aided process design. Computers and Chemical Engineering 11 (1), 7-25. 7.2.3... 

A predictor-corrector type of continuation method as implemented in PITCON (Ref 40) is used with acting as the independent continuation parameter. Homotopy continuation methods specialised for the nonsymmetric eigenproblem as described in Ref 41 and 42 can also be adapted. 

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