Approximate analytic solutions to a nonlinear digester problem


  • A. K. Alsharidi
  • John J. Shepherd RMIT University
  • A. J. Stacey
  • A. A. Khan



chemical reactors, approximations, linearization


Biological reactors are employed in industrial applications to break down organic waste from a range of sources into components that may be used in other applications. Such reactors may involve complex processes and many components linked by complicated interrelations. These reactions are represented mathematically as nonlinear initial value problems that must be solved numerically. Even smaller systems, more amenable to analytical analysis, require numerical solution methods due to their nonlinearity. We study a simple reactor with only two interacting components—a bacteria consuming a substrate (waste), represented by a \(2\times 2\) autonomous nonlinear initial value problem not solvable analytically. We describe a process to convert this problem to an approximating linear one that can be solved exactly to provide a closed form approximate representation of the evolving system. We assess the results of this approach and show they often agree favourably with numerical computations of the original nonlinear problem, although not always.


  • J. E. Bailey and D. F. Ollis. Biochemical Engineering Fundamentals. McGraw-Hill Book Company, New York, 1966.
  • D. J. Batstone, J. Keller, I. Angelidaki, S. V. Kalyuzhny, S. G. Pavlostathis, A. Rozzi, W. T. M. Sanders, H. Siegrist, and V. A. Vavilin. The IWA anaerobic digestion model No 1 (ADM1). Water Sci. Tech., 45(10):65–73, 2002. doi:10.2166/wst.2002.0292.
  • D. T. Hill. Simplified Monod kinetics of methane fermentation of animal wastes. Agri. Wastes, 5(1):1–16, 1983. doi:10.1016/0141-4607(83)90009-4.
  • J. Monod. The growth of bacterial cultures. Ann. Rev. Microbio., 3(1):371–394, 1949. doi:10.1146/annurev.mi.03.100149.002103.
  • F. E. Mosey. Mathematical modelling of the anaerobic digestion process: Regulatory mechanisms for the formation of short-chain volatile acids from glucose. Water Sci. Tech., 15(8–9):209–232, 1983. doi:10.2166/wst.1983.0168.
  • J. Rodriguez, E. Roca, J. M. Lema, and O. Bernard. Determination of the adequate minimum model complexity required in anaerobic bioprocesses using experimental data. Chem. Tech. Biotech., 83(12):1694–1702, 2008. doi:10.1002/jctb.1990.
  • H. L. Smith and P. Waltman. The Theory of the Chemostat: Dynamics of Microbial Competition. Cambridge University Press, 1995. doi:10.1017/CBO9780511530043.

Author Biography

John J. Shepherd, RMIT University

School of Science, RMIT





Proceedings Engineering Mathematics and Applications Conference