Multivariable cell population balance models are commonly used to explain complicated biological phenomena associated with product formation and cell growth in microbial populations. Such models typically consist of a partial integrodifferential equation for describing cell growth and an ordinary integrodifferential equation for representing substrate consumption. Due to their mathematical complexities, the numerical solutions of such models are hard tasks for numerical schemes. In this article, semidiscrete high resolution flux-limiting finite volume schemes are applied to solve single-variate and bivariate cell population balance models. The schemes have the abilities to achieve narrow peaks and resolve sharp discontinuities in the solutions on coarse meshes. These schemes are cheaper due to their short and reliable computational coding for complex problems. Several case studies are carried out. The numerical results of the schemes are compared with each other in terms of CPU time and accuracy. After consideration of different growth rate functions and incorporating equal and unequal partitioning, the suggested schemes were found to be more reliable and effective.
The population balance modeling is regarded as a universally accepted mathematical framework for dynamic simulation of various particulate processes, such as crystallization, granulation and polymerization. This article is concerned with the application of the method of characteristics (MOC) for solving population balance models describing batch crystallization process. The growth and nucleation are considered as dominant phenomena, while the breakage and aggregation are neglected. The numerical solutions of such PBEs require high order accuracy due to the occurrence of steep moving fronts and narrow peaks in the solutions. The MOC has been found to be a very effective technique for resolving sharp discontinuities. Different case studies are carried out to analyze the accuracy of proposed algorithm. For validation, the results of MOC are compared with the available analytical solutions and the results of finite volume schemes. The results of MOC were found to be in good agreement with analytical solutions and superior than those obtained by finite volume schemes.
Microbial cultures are comprised of heterogeneous cells that differ according to their size and intracellular concentrations of DNA, proteins and other constituents. Because of the included level of details, multi-variable cell population balance models (PBMs) offer the most general way to describe the complicated phenomena associated with cell growth, substrate consumption and product formation. For that reason, solving and understanding of such models are essential to predict and control cell growth in the processes of biotechnological interest. Such models typically consist of a partial integro-differential equation for describing cell growth and an ordinary integro-differential equation for representing substrate consumption. However, the involved mathematical complexities make their numerical solutions challenging for the given numerical scheme. In this article, the central upwind scheme is applied to solve the single-variate and bivariate cell population balance models considering equal and unequal partitioning of cellular materials. The validity of the developed algorithms is verified through several case studies. It was found that the suggested scheme is more reliable and effective.
Here, the oscillatory behavior of Saccharomyces cerevisiae (baker’s yeast) was investigated during the operation of a continuous bioreactor as it is detrimental to the stability and productivity of such a system. An unstructured segregated model was employed to study this phenomenon. The mathematical model couples a biological cell population balance model (PBM), representing the dynamics of cell mass distribution, with the mass balance of the rate-limiting substrate. High resolution flux limiter finite volume schemes have been proposed for approximating model equations efficiently and accurately. Moreover, analytical solution of a simplified yeast cell PBM was derived and the accuracy of proposed numerical schemes was analyzed by comparing analytical and numerical solutions. Good agreements in results and error analysis proved the accuracy of the proposed numerical schemes. Finally, the Globally Linearizing Control (GLC) was used for obtaining the total cell mass per unit volume. The GLC damps oscillations in substrate concentration by controlling the total cell number per unit volume. The ability of this controller to stabilize the steady-state and periodic solutions was analyzed through numerical simulations
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