The finite volume method for diffusion problems

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The finite volume method for diffusion problems Introduction Here we develop the numerical method based on, the finite volume (or control volume) method , by considering the simplest transport process of all: pure diffusion in the steady state. The governing equation of steady diffusion can easily be derived from the general transport equation for property  by deleting the transient and convective terms. This gives div(  grad ) + S = 0 The control volume integration, which forms the key step of the finite volume method that distinguishes it from all other CFD techniques, yields the following form: ∫ div(  grad )dV + ∫ S dV CV CV = ∫n . (  grad )dA + ∫ S dV = 0 A CV The approximation techniques that are needed to obtain the so -called discretised equations are introduced. Application of the method to simple one - dimensional steady state heat transfer problems is illustrated through a series of worked examples, and the accuracy of the method is gauged by compar- ing numerical results with analytical solutions. Finite volume method for one-dimensional steady state diffusion Consider the steady state diffusion of a property  in a one-dimensional domain defined in the following Figure. The process is governed by where  is the diffusion coefficient and S is the source term. Boundary values of  at points A and B are prescribed. An example of this type of process, one-dimensional heat conduction in a rod, is studied in detail in the following sections. Step 1: Grid generation The first step in the finite volume method is to divide the domain into di screte control volumes. Let us place a number of nodal points in the space between A and B. The boundaries (or faces) of control volumes are positioned mid -way between adjacent nodes. Thus each node is surrounded by a…