# 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.[1]

General Transport equation can be define as

where,
${\displaystyle \rho }$ is density and ${\displaystyle \phi }$ is conservative form of all fluid flow,
${\displaystyle \Gamma }$ is the Diffusion coefficient[2] and ${\displaystyle S}$ is the Source term.[3]
${\displaystyle \operatorname {div} (\rho \phi \upsilon )}$ is Net rate of flow of ${\displaystyle \phi }$ out of fluid element(convection),
${\displaystyle \operatorname {div} (\Gamma \operatorname {grad} \phi )}$ is Rate of increase of ${\displaystyle \phi }$ due to diffusion,
${\displaystyle S_{\phi }}$ is Rate of increase of ${\displaystyle \phi }$ due to sources.

${\displaystyle {\frac {\partial \rho \phi }{\partial t}}}$ is Rate of increase of ${\displaystyle \phi }$ of fluid element(transient),

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

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

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

or,

${\displaystyle {\frac {d}{dx}}(\Gamma \operatorname {grad} \phi )+S_{\phi }=0}$

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.
Dividing small domains and assingning nodal points (Figure 1)
• Create control volume using these nodal points.
Control volume and control volume & boundary faces (Figure 2)
• 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)
Steady state one-dimensional diffusion (Figure 3)

STEP 2
Discretization

Control volume width (Figure 4)
• 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)

${\displaystyle A}$ is Cross-sectional Area Cross section (geometry) of control volume face,${\displaystyle \Delta V}$ is Volume,${\displaystyle {\overrightarrow {S}}}$is average value of source S over control volume

${\displaystyle \left({\frac {d\phi }{dx}}\right)_{e}={\frac {\phi _{E}-\phi _{P}}{\delta x_{PE}}}}$
${\displaystyle \left({\frac {d\phi }{dx}}\right)_{w}={\frac {\phi _{P}-\phi _{W}}{\delta x_{WP}}}}$

• 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

## 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