Application of a collocation method for simulation of distributed parameter bioreactors

An orthogonal collocation method which belongs to the class of functional approximation methods of partial differential equations (PDEs) is applied in two distributed parameter bioreactor models. The models represent microbial growth and substrate consumption in a continuous-flow tube reactor. The growth is described by the Contois model. The points of the spatial discretisation method, the collocation points, are zeros of a Jacobi polynomial. These polynomials form a weighted set of orthogonal functions. The effect of the location and number of the collocation points is studied via simulations. It is concluded that in general relatively good accuracy is obtained already with a low number of points. In the case of the nondiffusional model, the approximation became unstable when the number of points was increased. The approximations of the diffusional model behaved smoothly and more and more accurately when the number of points was increased. Theoretically, the number of points needed for a certain accuracy level can be studied on the basis of the order reduction parameter obtained from the steady-state solution of the original PDE. Simulations for varying values of this parameter confirm the result that the decrease in the absolute value of the parameter lowers the error level of the approximation for the same number of points.