Process simulation in the complex domain

Lucia, Ángelo; Guo, Xinzhou

doi:10.1002/aic.690390309

Cited by 17 publications

(16 citation statements)

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“…The use of complex domain mathematics in process simulation was first proposed by Lucia et al (1990), and later Lucia and Taylor (1992) applied the complex domain calculation method proposed by Lucia and Xu (1992) and Lucia et al (1993) to vapor-liquid equilibrium (VLE) flash calculations. In those papers, Lucia and co-workers have used the complex roots of the CEOS to determine the pseudo properties of a phase, which means that the computer code for thermodynamic and process calculation models should be able to handle the complex numbers.…”

Section: Introductionmentioning

confidence: 99%

Applications of Complex Domain in Vapor−Liquid Equilibrium Calculations Using a Cubic Equation of State

Zhao

Saha

1998

Ind. Eng. Chem. Res.

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A new method is proposed to identify the spurious root (density or compressibility) in a cubic equation of state when it is used for phase property calculations and also to evaluate the pseudo root for an unfeasible phase specification based on a heuristic approach. The variation of real and real part of the complex roots with pressure for a cubic equation of state at constant temperature and composition is taken into consideration in the development. A complete set of analytical expressions is developed for the implementation of the method. The validity of the method is demonstrated through examples. The proposed method is simple and easy to use. Moreover, the computational load involved in the implementation of the proposed method is minimal because no iterative strategy is required.

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Section: Introductionmentioning

confidence: 99%

Applications of Complex Domain in Vapor−Liquid Equilibrium Calculations Using a Cubic Equation of State

Zhao

Saha

1998

Ind. Eng. Chem. Res.

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“…Figure shows the basins of attraction for the extended trust region (or dogleg) strategy for the same reactor temperature calculation. See Lucia and Xu and Lucia et al for a formal description of complex domain trust region methods. Note the similarity in overall geometric structure and, in particular, that the basin of attraction for the singular points looks very much like the Julia set for Newton's method shown in Figure , except that it is no longer fractal since norm reduction is forced.…”

Section: Julia Sets and Basin Boundariesmentioning

confidence: 99%

“…Solving the Reverse Iteration Problem. We use the complex domain dogleg strategy of Lucia and coworkers (Lucia et al 10 ) with second-order acceleration (Lucia and Liu 8 ) to solve eq 2 since it is stable with respect to all solutions. The mechanics of using any Newton-like method to solve eq 2 are not entirely transparent, especially in the multivariable case.…”

Section: Mathematical Formalitiesmentioning

confidence: 99%

Reverse Iteration in Chemical Process Simulation

Lucia

Liu

1998

Ind. Eng. Chem. Res.

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A new technique for generating approximations of Julia sets, called reverse iteration, is proposed. The concept of reverse iteration is simple and based on the idea of solving an iterative map of the general form Z k = G(Z k - 1, p) for Z k - 1 instead of Z k . It is shown that if the process of reverse iteration is initiated at any singular point, the collection of inverse images must be members of the Julia set since singular points are in the Julia set and the Julia set is closed. This sequence of reverse iterates, say {Z k - 1}, is necessarily distributed throughout the basin boundaries. It is also shown that reverse iteration can have multiple inverse images and a tree structure for the Julia set but that the associated potential combinatorial computational demand is easily resolved by exploiting the fractal nature of any Julia set. From this, practical ways generating initial values that converge to solutions to the given model equations are proposed. Several examples and geometric illustrations are used to elucidate key concepts.

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“…In a recent series of articles on the behavior of commonly used iterative maps for steady-state process simulation (i.e., direct substitution, Newton's method, and trust region methods), Lucia and co-workers (Lucia and Xu, 1992; Lucia and Taylor, 1992; Lucia et al, 1993) show that complex domain calculations offer the following advantages: (1) the ability to find real-valued solutions from starting points in the complex domain that cannot be calculated using real initial values; (2) smoothness of the conservation equations and physical property models across phase boundaries that removes the need for ad hoc model manipulation in systems in which the number of phases may change as a process parameter is varied or during iteration; (3) termination of simulations in a finite number of iterations to a real or complex-valued steady-state solution that provides some physical insight about the solution.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, Lucia and Xu (1992) and Lucia et al (1993) develop trust region methods in the complex domain that exploit Cauchy-Riemann differentiability and the Laplacian equations and avoid both the periodic/ aperiodic behavior that commonly plague direct substitution and Newton's method as well as the termination at singular points often experienced by traditional (real domain) trust region methods. They also use simple chemical process examples to illustrate the characteristics listed above while Lucia and Taylor (1992) apply some of these algorithms to flash processes.…”

Section: Introductionmentioning

confidence: 99%