Model Approximation and Simulations of a Class of Nonlinear Propagation Bioprocesses

It is well known that the biotechnology is one of the fields that over the last three decades has a very high and quick development. Therefore, due to their advantages, the control of industrial bioprocesses has been an important practical problem attracting wide attention. The main motivation in applying control methods to such living systems is to improve their operational stability and production efficiency. The operation in Stirred Tank Reactors (STR) has been and it is still a widely used technology in fermentation bioprocesses. But, other new technologies such as fixed bed, fluidized bed or air lift reactors, are considered for bioprocesses operation. These reactors present several advantages over the “classical” STRs. For instance, the fixed bed and fluidized bed reactors are characterized by higher production performance, i.e. larger production capacity and higher productivity (Bastin & Dochain, 1990; Bastin, 1991; Bouaziz & Dochain, 1993). From mathematical point of view, the dynamics of these processes are characterized by partial differential equations and therefore are classified as distributed parameter systems (Bastin & Dochain, 1990; Bouaziz & Dochain, 1993; Christofides, 2001; Dochain et al., 1992). For instance, the concentrations of the reactants and products are not anymore homogeneous in the whole reactor, like in STRs, but are characterized by a spatial profile along the reactor. It is clear that the distributed parameter feature of these systems makes the control problem even more difficult (Bouaziz & Dochain, 1993; Christofides, 2001; Dochain et al., 1992; Petre & Selisteanu, 2007a; Slotine & Li, 1991). Therefore, the modelling and simulation of them, which are the objectives of this chapter, are associated with the formulation of the process model using partial differential equations (PDEs), in general nonlinear, and with the computation of the solution. Since in the most of cases for this kind of systems, with the exception of a few simple cases, there are no analytical methods for finding the solution of the involved equations, the following alternatives are commonly employed (Christofides, 2001; Petre & Selisteanu, 2007a; Slotine & Li, 1991; Vilas, 2008): To assume that these processes behave like lumped parameter systems (the states are only time dependent). To use classical numerical methods like finite differences, finite elements or finite volumes. These methods are based on discretization techniques which allow us to approximate the infinite set of numbers that represent a continuous function by means of a finite set of