Mathematical modelling of controlled release fertilizer


  • Sayed Ameenuddin Irfan Universiti teknologi petronas
  • Radzuan Razali Universiti teknologi petronas



Adomian decomposition method, nutrient release, NPK, ionic diffusion, molecular diffusion


Controlled release fertilizers (CRFs) are essential for sustainable agriculture system. CRFs are designed to maintain the constant optimum release rate of nutrients from the coated granule. This increase the plant uptake of nutrients hence reduces the soil pollution and decreases the crop expenditure. In the literature, the maximum studies have been done by considering the molecular diffusion as the only phenomenon responsible for nutrient release from CRFs. The molecular diffusion model is solved mostly by using the variable separable methods and Laplace transform as well as finite difference methods by different researchers. The release of NPK (nutrient) depends on both molecular diffusions which are expressed by Fick’s second law of diffusion and ionic diffusion, due to the electrolytic behavior of NPK in the soil. In this work, an analytical solution is presented. The obtained solution helps to find the effect of granule coating thickness, nutrient release rate, pH of the soil and temperature of the soil on the nutrient release profiles.




M. E. Trenkel, Slow- and Controlled-Release and Stabilized Fertilizers: An Option for Enhancing Nutrient Use Efficiency in Agriculture. International Fertilizer Industry Association, 2010.

B. Azeem, K. Kushaari, Z. B. Man, A. Basit, and T. H. Thanh, “Review on materials & methods to produce controlled release coated urea fertilizer,” J. Control. Release, vol. 181, pp. 11–21, 2014.

A. Shaviv, “Advances in controlled-release fertilizers,” in Plant nutrition – Food security and sustainability of agro-ecosystems, 2001, pp. 1–49.

S. M. Al-Zahrani, “Controlled-release of fertilizers: modelling and simulation,” Int. J. Eng. Sci., vol. 37, no. 10, pp. 1299–1307, Aug. 1999.

A. Shaviv, S. Raban, and E. Zaidel, “Modeling controlled nutrient release from polymer coated fertilizers: Diffusion release from single granules,” Environ. Sci. Technol., vol. 37, no. 10, pp. 2251–2256, 2003.

C. Du, J. Zhou, a. Shaviv, and H. Wang, “Mathematical model for potassium release from polymer-coated fertiliser,” Biosyst. Eng., vol. 88, no. 3, pp. 395–400, 2004.

S. K. Basu and N. Kumar, “Mathematical model and computer simulation for release of nutrients from coated fertilizer granules,” Math. Comput. Simul., vol. 79, no. 3, pp. 634–646, Dec. 2008.

S. K. Basu, N. Kumar, and J. P. Srivastava, “Modeling NPK release from spherically coated fertilizer granules,” Simul. Model. Pract. Theory, vol. 18, no. 6, pp. 820–835, 2010.

S. K. Basu, N. Kumar, and J. P. Srivastava, “Modeling NPK release from spherically coated fertilizer granules,” Simul. Model. Pract. Theory, vol. 18, no. 6, pp. 820–835, 2010.

Keith J. Laidler, Chemical Kinetics, 3rd Edition. Pearson, 1987.

H. M. Lu, G. Li, Y. F. Zhu, and Q. Jiang, “Temperature dependence of self-diffusion coefficient in several liquid metals,” J. Non. Cryst. Solids, vol. 352, no. 26–27, pp. 2797–2800, Aug. 2006.

S. Irfan, R. Razali, K. KuShaari, and N. Mansor, “Reaction-Multi Diffusion Model for Nutrient Release and Autocatalytic Degradation of PLA-Coated Controlled-Release Fertilizer,” Polymers (Basel)., vol. 9, no. 3, p. 111, Mar. 2017.

S. A. Irfan, R. Razali, K. KuShaari, N. Mansor, and B. Azeem, “Analytical solution of diffusion model for nutrient release from controlled release fertilizer,” J. Phys. Conf. Ser., vol. 890, p. 12078, Sep. 2017.