# 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

Conditions under which the transient and convective terms goes to zero:

For one-dimensional steady state diffusion, General Transport equation reduces to:

$\operatorname {div} (\Gamma \operatorname {grad} \phi )+S_{\phi }=0$ or,

${\frac {d}{dx}}(\Gamma \operatorname {grad} \phi )+S_{\phi }=0$ The following steps comprehend one dimensional steady state diffusion -

STEP 1
Grid Generation

• 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)

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)

$\left({\frac {d\phi }{dx}}\right)_{e}={\frac {\phi _{E}-\phi _{P}}{\delta x_{PE}}}$ $\left({\frac {d\phi }{dx}}\right)_{w}={\frac {\phi _{P}-\phi _{W}}{\delta x_{WP}}}$ • In practical situation source term can be linearize
• Merging above equations leads to
• Re-arranging
• Compare and identify above equation with

STEP 3:
Solution of equations