Summary

Many chemical reactions are intrinsically coupled with transport phenomena. In realistic technical systems, both chemical reactions as well as transport phenomena are therefore functions of space and time [Xue et al, 2006]. Thus, it is essential that distributed system modeling and analysis takes into account its inherent multi-scale nature. A catalytic pellet fixed-bed reactor is distributed in nature since the reaction rate varies spatially and temporally and is mathematically represented by Partial Differential Equations (PDEs). The catalytic pellets are porous to allow reactant-diffusion, consequently creating a concentration gradient with respect to its radius [Weisz and Hicks, 1962]. If the chemical reaction is exothermic, significant temperature gradient also develops within the pellet. The entire reactor could be described by the multi-scale model (two-dimensional reactor and one-dimensional pellet) based on first principles. In addition, the catalytic pellets are capable of demonstrating multiple steady states for specific realizations of the bulk concentrations and temperatures along the reactor. Knowledge about these steady states is absolutely essential in quantifying the overall reaction rate. This presentation proposes to identify all pellet steady-states (microscopic level) and analyze its impact on the bulk concentration and temperature realizations (macroscopic level).

Methodology

We use orthogonal collocation over finite elements (OCFE) [Villadsen and Michelsen, 1978] to discretize the non-linear partial differential equations describing the concentration/temperature profiles in the spherical catalytic pellets. The resulting non-linear algebraic equation system has multiple solutions for specific bulk concentration and temperature realizations. We propose to use the global terrain methods [Lucia and Feng, 2002] and sequential niche techniques based on genetic algorithms [Beasley et al, 1993, Zhang et al, 2006] to successively identify all possible solutions to this system. The global terrain methodology is based on intelligent movement along the valleys and ridges of an appropriate objective function using downhill, local minimization calculations defined in terms of a trust region method and uphill integration of the Newton-like vector field combined with intermittent SQP corrector steps. The sequential niche technique is a modified genetic algorithm that uses the concept of fitness sharing to avoid re-searching regions in the parameter space that are already explored. This drastically increases the likelihood of discovering a new solution at each iteration. Thus, we are able to identify all possible solutions to the non-linear algebraic system for pellet concentration/temperature profiles. The impact of multiplicity at the microscopic level on the macroscopic bulk properties is evaluated by considering the effect of multiple effectiveness factor values corresponding to different pellet-profiles. We generate an effectiveness-factor map on a fixed cartesian grid in the space of realizable bulk concentration/temperatures. The effectiveness factor(s) at each point on this grid is computed using interval bisection and a Weisz-Hicks type ‘shooting' method. The effectiveness factor(s) corresponding to the actual realization of bulk concentration/temperatures is evaluated by two-dimensional Lagrangian interpolation on the generated effectiveness factor data.

Significance

The proposed methodology advances mathematical programming techniques to sequentially identify all physically relevant solution profiles satisfying the non-linear partial differential equations representing mass and energy balances in the spherical catalytic pellet. At the same time, it attempts to quantify the impact of microscopic pellet-profile multiplicity on the bulk macroscopic properties of the catalytic pellet reactor. The goal of this research is to systematically quantify the impact of state multiplicity of catalytic pellets on the bulk properties of the catalytic pellet reactor. The successive identification of all possible pellet profiles in a catalytic pellet reactor using sequential niche techniques and/or global terrain methods is a first to the best of our knowledge.

References:

Beasley, D., Bull, D., and Martin, R. A sequential niche technique for multimodal function optimization, Evolutionary computing, 1, 101-125, 1993.

Lucia, A. and Feng, Y., Global terrain methods, Computers and Chemical Engineering, 26, 529-546, 2002.

Villadsen, J. and Michelsen, M., Solution of differential equation models by polynomial approximations, Prentice-Hall Inc., NJ, 1978.

Weisz, P, and Hicks, J., The behavior of porous catalyst particle in view of internal mass and heat diffusion effects, Chemical Engineering Science, 17, 265-275, 1962.

Xue, C. Zhang, L., Malcolm, A. and Kulkarni, K. Distributed system design under uncertainty, Industrial and Engineering Chemistry Research, 45, 8352-8360, 2006.

Zhang, J., Huang, D., Lok, T., and Lyu, M. A novel adaptive sequential niche technique for multimodal function optimization, Neurocomputing, 69, 2396-2401, 2006.

Extended Abstract Status: File Uploaded