Low Order Approximations of Continuously Stirred Biofilm Reactors with Monod Kinetics

Design of controllers and optimization of plants using biofilm reactors often require dynamic models and efficient methods of simulation. Continuously stirred biofilm reactors (CSBRs) are useful model units in modeling of a variety of different biofilm reactors. Often the reaction kinetics in the biofilm is described by a Monod expression. Using standard modeling assumptions the equations describing the fast dynamics of a CSBR will then, for each substrate, be one nonlinear partial differential equation coupled with one linear ordinary differential equation. Here it is shown how a few nonlinear ordinary differential equations, which may be solved with standard integration methods, can be used as a close approximation. The approximations are derived using two common MWR methods, the Galerkin method and the orthogonal collocation method. Uniqueness and convergence properties are discussed. The stationary approximations can either be used as they are or to generate initial values for iterative methods to find solutions to more complicated models used, for example, in studies of the long term dynamics. Simulations of steps, impulses and random input responses using the approximations are compared to high accuracy simulations using the finite element method. The simulations show that a second order state space model is enough to describe the system for biofilms and reactors with low reaction rates and where the bulk volume is large compared to the liquid volume in the biofilm. A sixth order approximation, with a maximum error of 0.5%, should however be sufficient in almost all applications. The orthogonal collocation method turns out to have some advantages compared to the Galerkin method. It is easy to expand to higher orders, the errors are somewhat smaller, and the model expressions are simpler. Further, it is investigated how a liquid boundary layer on the biofilm surface affects the accuracy of the approximations. Simulations imply that the errors, compared to the FEM simulations, are smaller when this boundary layer is considered in the model. Finally, a collocation method where the collocation points are not chosen to be roots of orthogonal polynomials is shown to be less accurate than the standard orthogonal collocation method, where the collocation points are the roots of Legendre polynomials