AM  Vol.5 No.8 , May 2014
Central Upwind Scheme for Solving Multivariate Cell Population Balance Models
ABSTRACT

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.


Cite this paper
Rehman, S. , Kiran, N. and Qamar, S. (2014) Central Upwind Scheme for Solving Multivariate Cell Population Balance Models. Applied Mathematics, 5, 1187-1201. doi: 10.4236/am.2014.58110.
References
[1]   Roels, J.A. (1983) Energetics and Kinetics in Biotechnology. Elsevier, Amsterdam.

[2]   Slavov, N. and Botstein, D. (2011) Coupling among Growth Rate Response, Metabolic Cycle and Cell Division Cycle in Yeast. Molecular Biology of the Cell, 22, 1997-2009.
http://dx.doi.org/10.1091/mbc.E11-02-0132

[3]   Eakman, J.M., Fredrikson, A.G. and Tsuchiya, H.M. (1966) Statistics and Dynamics of Microbial Cell Populations. Chemical Engineering Progress, 62, 37-49.

[4]   Fredrikson, A.G., Ramkrishna, D. and Tsuchiya, H.M. (1967) Statistics and Dynamics of Procaryotic Cell Populations. Mathematical Biosciences, 1, 327-374.
http://dx.doi.org/10.1016/0025-5564(67)90008-9

[5]   Hulburt, H.M. and Katz, S. (1964) Some Problems in Particle Technology: A Statistical Mechanical Formulation. Chemical Engineering Science, 19, 555-574.
http://dx.doi.org/10.1016/0009-2509(64)85047-8

[6]   Randolph, A.D. and Larson, M.A. (1988) Population Balances: Theory of Particulate Processes. 2nd Edition, Academic Press, San Diego.

[7]   Miller, S.M. and Rawlings, J.B. (1994) Model Identification and Quality Control Strategies for Batch Cooling Crystallizers. AIChE Journal, 40, 1312-1327.
http://dx.doi.org/10.1002/aic.690400805

[8]   Smith, M. and Matsoukas, T. (1998) Constant-Number Monte Carlo Simulation of Population Balances. Chemical Engineering Science, 53, 1777-1786.
http://dx.doi.org/10.1016/S0009-2509(98)00045-1

[9]   Marchisio, D.L., Vigil, R.D. and Fox, R.O. (2003) Quadrature Method of Moments for Aggregation-Breakage Processes. Journal of Colloid and Interface Science, 258, 322-334.
http://dx.doi.org/10.1016/S0021-9797(02)00054-1

[10]   Barrett, J.C. and Jheeta, J.S. (1996) Improving the Accuracy of the Moments Method for Solving the Aerosol General Dynamic Equation. Journal of Aerosol Science, 27, 1135-1142.
http://dx.doi.org/10.1016/0021-8502(96)00059-6

[11]   Kumar, J. (2006) Numerical Approximations of Population Balance Equations in Particulate Systems. PhD thesis, Otto-von-Guericke University, Magdeburg.

[12]   Ramkrishna, D. (2000) Population Balances: Theory and Applications to Particulate Systems in Engineering. Academic Press, San Diego.

[13]   Nikolaos, V.M., Daoutidis, P. and Srienc, F. (2001) Numerical Solution of Multi-Variable Cell Population Balance Models: I. Finite Difference Methods. Computers & Chemical Engineering, 25, 1411-1440.
http://dx.doi.org/10.1016/S0098-1354(01)00709-8

[14]   Nikolaos, V.M., Daoutidis, P. and Srienc, F. (2001) Numerical Solution of Multi-Variable Cell Population Balance Models: II. Spectral Methods. Computers & Chemical Engineering, 25, 1441-1462.
http://dx.doi.org/10.1016/S0098-1354(01)00710-4

[15]   Nikolaos, V.M., Daoutidis, P. and Srienc, F. (2001) Numerical Solution of Multi-Variable Cell Population Balance Models: III. Finite Element Methods. Computers & Chemical Engineering, 25, 1463-1481.
http://dx.doi.org/10.1016/S0098-1354(01)00711-6

[16]   Henson, M.A. (2003) Dynamic Modeling of Microbial Cell Populations. Current Opinion in Biotechnology, 14, 460467.
http://dx.doi.org/10.1016/S0958-1669(03)00104-6

[17]   Qamar, S., Elsner, M.P., Angelov, I., Warnecke, G. and Seidel-Morgenstern, A. (2006) A Comparative Study of High Resolution Finite Volume Scheme for Solving Population Balances in Crystallization. Computers & Chemical Engineering, 30, 1119-1131.
http://dx.doi.org/10.1016/j.compchemeng.2006.02.012

[18]   Qamar, S. and Rehman, S.M. (2013) High Resolution Finite Volume Schemes for Solving Multivariable Biological Cell Population Balance Models. Industrial & Engineering Chemistry Research, 52, 4323-4341.
http://dx.doi.org/10.1021/ie302253m

[19]   Kurganov, A. and Tadmor, E. (2000) New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection-Diffusion Equations. Journal of Computational Physics, 160, 241-282.
http://dx.doi.org/10.1006/jcph.2000.6459

[20]   Kurganov, A. and Levy, D. (2000) A Third-Order Semi-Discrete Central Scheme for Conservation Laws and Convection-Diffusion Equation. SIAM Journal on Scientific Computing, 22, 1461-1488.
http://dx.doi.org/10.1137/S1064827599360236

[21]   Attarakih, M.M., Bart, H.-J. and Faqir, N.M. (2006) Numerical Solution of the Bivariate Population Balance Equation for the Interacting Hydrodynamics and Mass Transfer in Liquid-Liquid Extraction Columns. Computers & Chemical Engineering, 61, 113-123.

 
 
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