| Two new models are developed in this thesis, the Minimum-Time Microbial Growth Kinetic Model, MTGKM, and the Mass Transfer Model for k{dollar}sb{lcub}rm L{rcub}{dollar}a and {dollar}phi{dollar}. The MTGKM is based on the postulate that the growth of a microbial cell population occurs so as to minimize the duration of the growth cycle, comprising an Adjustment Phase, AP, and a Dynamic Phase, DP. The growth system is described by six state variables, growth factor G, substrate S, cell mass X, toxins, T, primary product P{dollar}sb1{dollar} and secondary product, P{dollar}sb2{dollar}. The Adjustment Phase is postulated to be a period of acclimatization for the cell-substrate system, its dynamics being governed by a core subsystem comprising the first four state variables. No cell growth occurs and the main event is the accumulation of the growth factor to a critical level G{dollar}sb{lcub}rm crit{rcub}{dollar}, which triggers the onset of the Dynamic Phase. The DP includes all subsequent events of the growth cycle. The model is developed by constructing two optimal control functions, u{dollar}sb{lcub}rm G{rcub}{dollar} for the growth factor, and u{dollar}sb{lcub}rm S{rcub}{dollar} for the substrate, which minimize the cycle time. The MTGKM requires nine system parameters, {dollar}rm a in Rsp9{dollar}, and four critical parameters, {dollar}rm Gsb{lcub}crit{rcub}, Gsb{lcub}max{rcub}, Xsb{lcub}crit{rcub}{dollar}, and T{dollar}sb{lcub}rm crit{rcub}{dollar}, in addition to the initial conditions for the six system states. Optimization is done by a generalized Newton-Raphson procedure. The model successfully simulates the dynamics of the growth cycle but cannot be employed in simple everyday biochemical engineering calculations because (1) the input data for specific cell species-substrate systems are not directly available, and (2) the optimization procedure is a major computation in itself.; The mass transfer models have been developed by solving the gas bubble dissolution problem for a single bubble in free rise, with a time-dependent boundary condition for the gas concentration within the bubble. From the gas bubble density, N{dollar}sb{lcub}0{rcub}{dollar}, the results for one bubble are related to the bulk system parameters k{dollar}sb{lcub}rm L{rcub}{dollar}a, and {dollar}phi{dollar}. The problem comprises a parabolic partial differential equation for the concentration of the gas species in the liquid phase, coupled nonlinearly to an ordinary differential equation for the bubble radius. The problem was solved by the method of lines, and the integral boundary condition was handled by applying the trapezoidal rule. |
