Numerical Analysis for the Synthesis of Biodiesel Using Spectral Relaxation Method

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

Numerical Investigation of the Effect of Unsteadiness on Three-Dimensional Flow of an Oldroyb-B Fluid.

A spectral relaxation method used with bivariate Lagrange interpolation is used to find numerical solutions for the unsteady three-dimensional flow problem of an Oldroyd-B fluid with variable thermal conductivity and heat generation. The problem is governed by a set of three highly coupled nonlinear partial differential equations. The method, originally used for solutions of systems of ordinary differential equations is extended to solutions of systems of nonlinear partial differential equations. The modified approach involves seeking solutions that are expressed as bivariate Lagrange interpolating polynomials and applying pseudo-spectral collocation in both independent variables of the governing PDEs. Numerical simulations were carried out to generate results for some of the important flow properties such as the local skin friction and the heat transfer rate. Numerical analysis of the error and convergence properties of the method are also discussed. One of the benefits of the proposed method is that it is computationally fast and gives very accurate results after only a few iterations using very few grid points in the numerical discretization process

Spectral Local Linearisation Approach for Natural Convection Boundary Layer Flow

The present work introduces a spectral local linearisation method (SLLM) to solve a natural convection boundary layer flow problem with domain transformation. It is customary to find solutions of semi-infinite interval problems by first truncating the interval and subsequently applying a suitable numerical method. However, this gives rise to increased error terms in the numerical solution. Carrying out a transformation of the semi-infinite interval problems into singular problems posed on a finite interval can avoid the domain truncation error and enables the efficient application of collocation methods. The SLLM is based on linearising and decoupling nonlinear systems of equations into a sequence or subsystems of differential equations which are then solved using spectral collocation methods. A comparative study between the SLLM and existing results in the literature was carried out to validate the results. The method has shown to be a promising efficient tool for nonlinear boundary value problems as it gives converging results after very few iterations

Spectral Local Linearisation Approach for Natural Convection Boundary Layer Flow

The present work introduces a spectral local linearisation method (SLLM) to solve a natural convection boundary layer flow problem with domain transformation. It is customary to find solutions of semi-infinite interval problems by first truncating the interval and subsequently applying a suitable numerical method. However, this gives rise to increased error terms in the numerical solution. Carrying out a transformation of the semi-infinite interval problems into singular problems posed on a finite interval can avoid the domain truncation error and enables the efficient application of collocation methods. The SLLM is based on linearising and decoupling nonlinear systems of equations into a sequence or subsystems of differential equations which are then solved using spectral collocation methods. A comparative study between the SLLM and existing results in the literature was carried out to validate the results. The method has shown to be a promising efficient tool for nonlinear boundary value problems as it gives converging results after very few iterations

The Spectral Homotopy Analysis Method Extended to Systems of Partial Differential Equations

The spectral homotopy analysis method is extended to solutions of systems of nonlinear partial differential equations. The SHAM has previously been successfully used to find solutions of nonlinear ordinary differential equations. We solve the nonlinear system of partial differential equations that model the unsteady nonlinear convective flow caused by an impulsively stretching sheet. The numerical results generated using the spectral homotopy analysis method were compared with those found using the spectral quasilinearisation method (SQLM) and the two results were in good agreement

On New Numerical Techniques for the MHD Flow Past a Shrinking Sheet with Heat and Mass Transfer in the Presence of a Chemical Reaction

We use recent innovative solution techniques to investigate the problem of MHD viscous flow due to a shrinking sheet with a chemical reaction. A comparison is made of the convergence rates, ease of use, and expensiveness (the number of iterations required to give convergent results) of three seminumerical techniques in solving systems of nonlinear boundary value problems. The results were validated using a multistep, multimethod approach comprising the use of the shooting method, the Matlab bvp4c numerical routine, and with results in the literature

Variation of the residual error, <i>Res</i>(<i>g</i>) with <i>ξ</i>.

<p><i>β</i> = 1, <i>β</i>₁ = 0.1, <i>β</i>₂ = 0.1, <i>ϵ</i> = 0.1, <i>Pr</i> = 0.8, <i>S</i> = 0.3.</p

Effect of <i>β</i> on <i>f</i>′′(0, <i>ξ</i>).

<p><i>β</i>₁ = 0.1, <i>β</i>₂ = 1, <i>ϵ</i> = 0.4, <i>Pr</i> = 2, <i>S</i> = 0.2.</p

The Spectral Homotopy Analysis Method Extended to Systems of Partial Differential Equations

The spectral homotopy analysis method is extended to solutions of systems of nonlinear partial differential equations. The SHAM has previously been successfully used to find solutions of nonlinear ordinary differential equations. We solve the nonlinear system of partial differential equations that model the unsteady nonlinear convective flow caused by an impulsively stretching sheet. The numerical results generated using the spectral homotopy analysis method were compared with those found using the spectral quasilinearisation method (SQLM) and the two results were in good agreement

Effect of <i>β</i>₁ on <i>g</i>′′(0, <i>ξ</i>).

<p><i>β</i> = 0.5, <i>β</i>₂ = 1, <i>ϵ</i> = 0.4, <i>Pr</i> = 2, <i>S</i> = 0.2.</p