Diffusion of fertilizers from controlled-release sources uniformly distributed in soil

Controlled release of a fertilizer from capsules, uniformly distributed in the soil and its diffusion under different degrees of soil saturation are studied theoretically. Solutions are suggested for slow release with constant or first order consumption (absorption, degradation) rate of the fertilizer in the soil. The fertilizer's release rate is dominated by two consecutive processes: 1) Release from the capsule to the soil solution, and 2) Molecular diffusion and mass transport in the soil. In this study, diffusion in stagnant soil solution is considered. For a systematic examination of membrane-controlled release mechanism three classes of fertilizers are studied: (A) low, (B) medium and (C) very high solubility (although the last generally does not need a membrane cover to further impede its slow release into the soil solution). An approximated solution is suggested assuming slow release into a spherical soil domain with zero flux on its boundary. An implicit finite difference scheme is used for the numerical solution, and the results are analyzed by means of dimensional analysis. The applied simplifying assumptions allowed us to study quantitatively the interactive role of soil properties, capsules, fertilizers and sink functions, in controlled release processes. The release rate is found to be higher as fertilizers' solubility, membrane conductance, source size and soil moisture content increase. Higher number of capsules per unit volume of soil and higher encapsulated mass elongate the release process. The suggested model may be useful in testing existing or designed controlled-release devices under simulated normal, as well as extreme, environmental conditions, and thereby help improve products and application.