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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Finite volume method for one-dimensional steady state diffusion ： ウィキペディア英語版 Finite volume method for one-dimensional steady state diffusion Finite volume method in computational fluid dynamics is a discretization technique for partial differential equations that arise from physical conservation laws. These equations can be different in nature, e.g. elliptic, parabolic, or hyperbolic. First well-documented use was by Evans and Harlow (1957) at Los Alamos. The general equation for steady diffusion can be easily be derived from the general transport equation for property ''Φ'' by deleting transient and convective terms. General Transport equation can be define as $\backslash frac\; +\; \backslash operatorname(\backslash rho\; \backslash phi\; \backslash upsilon)\; =\; \backslash operatorname(\backslash Gamma\; \backslash operatorname\; \backslash phi)\; +\; S\_\backslash phi$ where, $\backslash rho$ is density and $\backslash phi$ is conservative form of all fluid flow, $\backslash Gamma$ is the Diffusion coefficient and $S$ is the Source term. $\backslash operatorname(\backslash rho\; \backslash phi\; \backslash upsilon)$ is Net rate of flow of $\backslash phi$ out of fluid element(convection), $\backslash operatorname(\backslash Gamma\; \backslash operatorname\; \backslash phi)$ is Rate of increase of $\backslash phi$ due to diffusion, $S\_\backslash phi$ is Rate of increase of $\backslash phi$ due to sources. $\backslash frac$ is Rate of increase of $\backslash phi$ of fluid element(transient), Conditions under which the transient and convective terms goes to zero: * Steady State * Low Reynolds Number For one-dimensional steady state diffusion, General Transport equation reduces to: ::$\backslash operatorname(\backslash Gamma\backslash operatorname\backslash phi)+\; S\_\backslash phi=0$ or, ::$\backslash frac\; (\backslash Gamma\backslash operatorname\backslash phi)\; +S\_\backslash phi=0$ The following steps comprehend one-dimensional steady state diffusion - STEP 1 Grid Generation * Divide the domain in equal parts of small domain. * Place nodal points midway in between each small domain. * Create control volume using these nodal points. * Create control volume near the edge in such a way that the physical boundaries coincide with control volume boundaries.(Figure 1) * Assume a general nodal point 'P' for a general control volume.Adjacent nodal points in east and west are identified by E and W respectively.The west side face of the control volume is referred to by 'w' and east side control volume face by 'e'.(Figure 2) * The distance between WP, wP, Pe and PE are identified by $\backslash delta\; x\_$,$\backslash delta\; x\_$,$\backslash delta\; x\_$ and $\backslash delta\; x\_$ respectively.(Figure 4) STEP 2 Discretization * The crux of Finite volume method is to integrate governing equation all over control volume, known discretization. * Nodal points used to discretize equations. * At nodal point P control volume is defined as (Figure 3) $\backslash int\_\; \backslash frac\backslash left\; (\backslash Gamma\; \backslash frac\backslash right\; )\; dV\; +\; \backslash int\_\; S\; dV\; =\; \backslash left\; (\backslash Gamma\; A\; \backslash frac\backslash right\; )\_e\; -\; \backslash left\; (\backslash Gamma\; A\; \backslash frac\backslash right\; )\_w\; +\; \backslash overrightarrow\; \backslash Delta\; V=\; 0$ where $A$ is Cross-sectional Area Cross section (geometry) of control volume face,$\backslash Delta\; V$ is Volume,$\backslash overrightarrow$is average value of source S over control volume *It states that diffusive flux Fick's laws of diffusion$\backslash phi$ from east face minus west face leads to generation of flux in control volume. *$\backslash phi$ diffusive coefficient and $\backslash frac$ is required in order to interpreter useful conclusion. *Central differencing technique () is used to derive $\backslash phi$ diffusive coefficient. $\backslash Gamma\_w\; =\; \backslash frac$ $\backslash Gamma\_w\; =\; \backslash frac$ *$\backslash frac$ gradient from east to west is calculated with help of nodal points.(Figure 4) $\backslash left\; (\; \backslash frac\; \backslash right\; )\_e=\backslash frac\; \backslash right\; )\_w=\backslash frac\; \backslash Delta\; V=\; S\_u\; +\; S\_p\backslash phi\_p$ * Merging above equations leads to $\backslash Gamma\_e\; A\_e\backslash left\; (\; \backslash frac$ STEP 3: Solution of equations * Discretized equation must be set up at each of the nodal points in order to solve the problem. * The resulting system of linear algebraic equation Linear equation is then solved to obtain distribution of the property $\backslash phi$ at the nodal points by any form of matrix solution technique. * The matrix of higher order () can be solved in MATLAB. This method can also be applied to a 2D situation. See Finite volume method for two dimensional diffusion problem. ==References== * Patankar, Suhas V. (1980), Numerical Heat Transfer and Fluid Flow, Hemisphere. * Hirsch, C. (1990), Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows, Wiley. * Laney, Culbert B.(1998), Computational Gas Dynamics, Cambridge University Press. * LeVeque, Randall(1990), Numerical Methods for Conservation Laws, ETH Lectures in Mathematics Series, Birkhauser-Verlag. * Tannehill, John C., et al., (1997), Computational Fluid mechanics and Heat Transfer, 2nd Ed., Taylor and Francis. * Wesseling, Pieter(2001), Principles of Computational Fluid Dynamics, Springer-Verlag. * Carslaw, H. S. and Jager, J. C. (1959). Conduction of Heat in Solids. Oxford: Clarendon Press * Crank, J. (1956). The Mathematics of Diffusion. Oxford: Clarendon Press * Thambynayagam, R. K. M (2011). The Diffusion Handbook: Applied Solutions for Engineers: McGraw-Hill * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Finite volume method for one-dimensional steady state diffusion」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.