Semi-analytical solutions in bio-process engineering

Tara Kerr

Research Assistant

Spring 2006

Project Summary

The purpose of this project was to analyse two mathematical models that have applications to bio-process engineering. The first model is applicable in describing the processes occurring within a one-cell diffusion reactor. The second model relates to a larger reactor, perhaps similar to that of the colon, where substances are allowed to pass into and pass out of the reactor, i.e. a flow reactor. The analysis of the first model comprises the majority of the report as it was the first problem addressed by this research.

Both mathematical models studied in this report were coupled sets of second order partial differential equations. This in itself very largely restricts the analytical solutions that can be obtained. As such the primary method of solution and analysis uses the semi-analytical Galerkin method. In this method, approximating functions are constructed and then used to convert the PDEs to ODEs. These approximating functions satisfy the boundary conditions of the PDE but do not satisfy the governing PDEs. it is for this reason that this method is termed semi-analytical and the solutions gained are semi-analytical solutions. It is thought that the more terms that are taken to make up the approximating functions, the more accurate the solution becomes. In taking more terms., however, the mathematics and calculations become increasingly more complex. For example, in a one-term approximation, a matrix of size two by two is obtained. In a two-term approximation this matrix is of size four by four, making calculations such as determinants and eigenvalue more involved. As a result, this report will also address the costs and benefits of using more terms in the approximating functions.

In applying the Galerkin method to the system of non-linear partial differential equations, characteristics such as steady-states and their stability are still the fundamental way in which the bio-processes occurring can be understood and explained. Thus, this will be the main analysis in the report. Further, due to the complicated mathematics and significant calculations required by the Galerkin method, this report will serve to provide Maple coding and explanations to perform the analysis. In cases where Maple cannot produce the solution, ways o solving via other means will be suggested.

By completion of this project, it is hoped that there will be a greater understanding of one-cell reaction-diffusion and flow reactor processes. Further, this report will consist of a comprehensive application of the Galerkin model with knowledge of the advantages and complexities of this method of solution discussed throughout.

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Last Updated: 2nd Decemebr 2009.