ABSTRACT
Nutrients released into soils from uncoated fertilizer granules are lost continuously due to volatilization, leaching, denitrification, and surface run-off. These issues have caused economic loss due to low nutrient absorption efficiency and environmental pollution due to hazardous emissions and water eutrophication. Controlled-release fertilizers (CRFs) can change the release kinetics of the fertilizer nutrients through an abatement strategy to offset these issues by providing the fertilizer content in synchrony with the metabolic needs of the plants. Parametric analysis of release characteristics of CRFs is of paramount importance for the design and development of new CRFs. However, the experimental approaches are not only time consuming, but they are also cumbersome and expensive. Scientists have introduced mathematical modeling techniques to predict the release of nutrients from the CRFs to elucidate fundamental understanding of the dynamics of the release processes and to design new CRFs in a shorter time and with relatively lower cost. This paper reviews and critically analyzes the latest developments in the mathematical modeling and simulation techniques that have been reported for the characteristics and mechanisms of nutrient release from CRFs. The scope of this review includes the modeling and simulations techniques used for coated, controlled-release fertilizers.
Subject(s)
Drug Liberation , Fertilizers , Models, Theoretical , Computer Simulation , Delayed-Action Preparations/chemistryABSTRACT
A mathematical model for the reaction-diffusion equation is developed to describe the nutrient release profiles and degradation of poly(lactic acid) (PLA)-coated controlled-release fertilizer. A multi-diffusion model that consists of coupled partial differential equations is used to study the diffusion and chemical reaction (autocatalytic degradation) simultaneously. The model is solved using an analytical-numerical method. Firstly, the model equation is transformed using the Laplace transformation as the Laplace transform cannot be inverted analytically. Numerical inversion of the Laplace transform is used by employing the Zakian method. The solution is useful in predicting the nutrient release profiles at various diffusivity, concentration of extraction medium, and reaction rates. It also helps in explaining the transformation of autocatalytic concentration in the coating material for various reaction rates, times of reaction, and reaction-multi diffusion. The solution is also applicable to the other biodegradable polymer-coated controlled-release fertilizers.
ABSTRACT
BACKGROUND: In the past few decades, a significant volume of work has been carried out on various aspects of the state estimation problem to estimate an optimum state vector of the power system. This problem has been focused on, in previous studies regarding the computational efficiency and numerical robustness in view to find point estimates for system state parameters. This current investigation, constructed confidence intervals for the unknown state parameters of the system. The research indicates that confidence intervals can yield addition useful information about the estimated parameters. METHODS: The feasible interval estimates for the system state parameters have been modelled in this study by considering the random uncertainty in the processing measurements. The statistical assumptions of the measurement errors have been utilized to characterize the probabilistic behavior of the estimated parameters in terms of confidence intervals. The Gauss-Newton algorithm has been adopted for maximizing the likelihood function of the processing measurements and obtaining the confidence intervals. RESULTS: The usage of the confidence intervals was demonstrated through Monte Carlo experiments on a real dataset of the 6-bus and IEEE 14-bus power systems for both small and large sample sizes. The confidence intervals were constructed for the test networks for the sample of measurements 18, 28, 44 and 68 based on the redundancy ratio R. The proposed interval estimates outperformed for the sample sizes of 28 in the 6 bus and 68 in the IEEE 14-bus systems, respectively. The poor performance for the constructed interval estimates have been reported even for the large sample sizes in the existence of contaminated measurements. CONCLUSIONS: The results of the study show that the method is effective and practically applicable in the state estimation of a power system. The constructed confidence intervals for the system state parameters adequately perform for the lager sample size. However, the existence of the gross errors in the processing measurements had severe effect on the performance of the proposed interval estimates.
