Biodiesel is an alternative diesel fuel chemically defined as the mono-alkyl esters of long chain fatty acids derived from vegetable oils or animal fat. It is becoming more attractive as an alternative fuel due to the depleting fossil fuel resources. A mathematical model for the synthesis of biodiesel from vegetable oils and animal fats is presented in this study. Numerical solutions of the model are found using a spectral relaxation method. The method, originally developed for boundary value problems, is an iterative scheme based on the Chebyshev spectral collocation method developed by decoupling systems of equations using Gauss-Seidel type of techniques. The effects of the reaction rate constants and initial concentrations of the reactants on the amount of the final product are being investigated. The accuracy of the numerical results is validated by comparison with known analytical results and numerical results obtained using ode45, an efficient explicit 4th and 5th order Runge-Kutta method used to integrate both linear and nonlinear differential equations. 1. Introduction Due to the continuous uncertainty and increasing environmental impact associated with the utilization of petroleum-based diesel fuel, the demands for alternative fuels have increased considerably in recent years [1]. Methyl and ethyl esters derived from vegetable oil or animal fat, known as biodiesel, have good potential as alternative diesel fuel [2]. Biodiesel is synthesized through a chemical process called transesterification. During this chemical process, triglycerides in vegetable oils or animal fats react with short chain alcohols such as methanol and ethanol in the presence of homogeneous basic catalysts such as sodium hydroxide and potassium hydroxide to form fatty acid methyl esters (biodiesel) and glycerin [3, 4]. Biodiesel has many advantages over petroleum-based diesel fuels. It is biodegradable and nontoxic and produce less particles, smoke, and carbon monoxide [2]. It is also renewable and usable in a variety of diesel engines with minimum or no modification necessary [3, 5]. It burns clean, and do not form engine deposits or generate harmful pollutants which results in a significant reduction of the types of pollutants that contribute to pollution and global warming and releases up to 85% fewer cancer-causing agents [6–10]. Experiments have been carried out to investigate different aspects in the production of biodiesel [3, 4, 11–15]. In this work, we carry out a numerical study of the transesterification process for the synthesis of biodiesel from vegetable oils.
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