Analytical solution of solute diffusion and biodegradation in spherical aggregates

Microbial degradation and transformation of substances in soils plays a crucial role in the nutrient turnover of ecosystems. To quantify these processes, a mathematical description is needed. For this purpose, an analytical solution to a model of solute diffusion and biodegradation in soil aggregates was developed. The model is a first approach toward understanding the influence of geometric arrangement of microorganisms and substrates in structured soils. These soils are considered to consist of uniformly sized and shaped aggregates surrounded by surface films of the soil solution. The model simulates transient diffusion of finite substrate amounts from the surface films into spherical aggregates. Biodegradation is considered for the special case of unlimited microbial growth, and adsorption is assumed to follow a linear Freundlich isotherm. The system is represented by a composite sphere, the outer sphere being the solution film and the inner sphere representing the soil aggregate. The diffusion equations are solved by Laplace transformation. The model solution gives a direct relationship between the initial substrate and biomass concentrations, the diffusion coefficient, the specific growth rate, and the adsorption coefficient. Good agreement between this closed form solution and numerical solutions is obtained for diffusion with and without biodegradation. Since the substrate is exhausted by organisms close to the surface, the centers of large aggregates are not reached by the diffusing substrate. These unaffected centers become larger as the growth rate is higher, the diffusion constant is lower, and adsorption of the substrate is stronger.