Unsteady flow of gas in a semi-infinite porous medium: a numerical investigation by using RBF-DQM
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ORIGINAL SCIENTIFIC RESEARCH
Unsteady flow of gas in a semi-infinite porous medium: a numerical investigation by using RBF-DQM K Parand1,2, S Hashemi-Shahraki1 and M Hemami1* 1
Department of Computer Sciences, Shahid Beheshti University, G.C., 19697-64166 Tehran, Iran 2
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Canada Received: 31 December 2019 / Accepted: 17 April 2020
Abstract: The study aims to investigate how an effective numerical algorithm can be used to solve the unsteady isothermal flow of gas through a semi-infinite micro-nano porous medium. The unsteady gas equation used here is a nonlinear, second order differential equation with two points of boundary value on the semi-infinite domain. The study uses RBFs-DQ method in which the derivative value of function with respect to the point is directly approximated by a linear combination of all functional values in the entire domain. The main purpose of using this method is to determine the weight of coefficients. The study also used Gaussian (GS) function to approximate the solution of the mentioned equation. The efficiency and accuracy of this method are verified by the comparison made between our results and other numerical methods including shooting method, RBF.G and Wavelet Legendre collocation method. As a results, by comparison made to other numerical methods this showed a Meshfree (RBF-DQ) method which its validity is equal or even more valid than numerical methods. Keywords: Unsteady flow; RBF-DQ; Unsteady gas equation; Gaussian RBF; Semi-infinite problem o oP oP P ¼A ; oz oz ot
1. Introduction Gas solid processes such as adsorption are widely used in the chemical industries for separation of solutes from a fluid stream diffusion and transport in micro-nano porous materials has long been the focus of research groups. Mathematical modelling of gas flow through a porous medium is very valuable because of its importance in investigating solid gas processes. In the study of the unsteady flow of gas through a semi-infinite porous medium [1, 2] initially filled with gas at a uniform pressureP0 0, at time t ¼ 0, the pressure at the outflow face is abruptly reduced from P0 to P1 0(P1 ¼ 0 is the case of diffusion into a vacuum) and is, there after, maintained at this lower pressure. The unsteady isothermal flow of gas is described by a nonlinear partial differential equation (PDE) r2 ðP2 Þ ¼ 2A
oP ; ot
ð1Þ
where the constant A is given by the properties of the medium. In the one-dimensional medium extending from z ¼ 0 to z ¼ 1, this reduces to
with the boundary conditions Pðz; 0Þ ¼ P0 ; 0\z\1
ð2Þ
Pð0; tÞ ¼ P1 ðP0 Þ:0 t\1
To obtain a similarity solution, Agarwal and O’Regan [3] introduced the new independent variable 1 z A 2 x ¼ pffi ; t 4P0 and the dimension-free dependent variably, defined by P2 ðzÞ YðxÞ ¼ a1 1 2 ; P0 P2
where a ¼ 1 P12 . 0 According to the aforementioned variable, the problem takes the form of (unsteady gas equation) 2x y00 ðxÞ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y0 ðxÞ ¼ 0; x [ 0; 0
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