Mathematical modeling and analytical solution of two-phase flow transport in an immobilized-cell photo bioreactor using the homotopy perturbation method (HPM)

In the present study, a theoretical model of a reaction-diffusion within an entrapped-cell photobioreactor packed with gel-granules containing immobilized photosynthetic bacterial cells is presented. The model is based on a system of two coupled nonlinear reaction-diffusion equations under steady-state condition for biochemical reactions occurring in the photobioreactor that describes the substrate and product concentration within the gel-granule. Simple analytical expressions for the concentration of substrate and product have been derived for all values of reaction-diffusion parameters, demonstrating competition between the diffusion and reaction in the gel-granule, using the homotopy perturbation method (HPM). Moreover, two limiting cases (zero and first order kinetics) were applied and their effects discussed. The results demonstrated that in order to achieve high substrate consumption and high hydrogen production, both reaction-diffusion parameters ϕ1 and ϕ2 should be maintained at their highest possible values and saturation parameter α1, quantifying the saturation degree of the reaction kinetics, should be decreased to its lowest level. The analytical results were also compared with numerical ones and a good agreement was obtained.