# 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

${\displaystyle {\frac {\partial \rho \phi }{\partial t}}+\operatorname {div} (\rho \phi \upsilon )=\operatorname {div} (\Gamma \operatorname {grad} \phi )+S_{\phi }}$

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)
• The distance between WP, wP, Pe and PE are identified by ${\displaystyle \delta x_{WP}}$,${\displaystyle \delta x_{wP}}$,${\displaystyle \delta x_{Pe}}$ and ${\displaystyle \delta x_{PE}}$ respectively.(Figure 4)

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)

${\displaystyle \int _{\Delta V}{\frac {d}{dx}}\left(\Gamma {\frac {d\phi }{dx}}\right)dV+\int _{\Delta V}SdV=\left(\Gamma A{\frac {d\phi }{dx}}\right)_{e}-\left(\Gamma A{\frac {d\phi }{dx}}\right)_{w}+{\overrightarrow {S}}\Delta V=0}$

where

${\displaystyle A}$ 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

• It states that diffusive flux Fick's laws of diffusion${\displaystyle \phi }$ from east face minus west face leads to generation of flux in control volume.
• ${\displaystyle \phi }$ diffusive coefficient and ${\displaystyle {\frac {d\phi }{dx}}}$ is required in order to interpreter useful conclusion.
• Central differencing technique [1] is used to derive ${\displaystyle \phi }$ diffusive coefficient.

${\displaystyle \Gamma _{w}={\frac {\Gamma _{W}+\Gamma _{P}}{2}}}$

${\displaystyle \Gamma _{w}={\frac {\Gamma _{P}+\Gamma _{E}}{2}}}$

• ${\displaystyle {\frac {d\phi }{dx}}}$ gradient from east to west is calculated with help of nodal points.(Figure 4)

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

• In practical situation source term can be linearize

${\displaystyle {\overrightarrow {S}}\Delta V=S_{u}+S_{p}\phi _{p}}$

• Merging above equations leads to

${\displaystyle \Gamma _{e}A_{e}\left({\frac {\phi _{E}-\phi _{P}}{\delta x_{PE}}}\right)-\Gamma _{w}A_{w}\left({\frac {\phi _{P}-\phi _{W}}{\delta x_{PE}}}\right)+(S_{u}+S_{p}\phi _{p})}$

• Re-arranging

${\displaystyle \left({\frac {\Gamma _{e}}{\delta x_{PE}}}A_{e}+{\frac {\Gamma _{w}}{\delta x_{WP}}}A_{w}-S_{p}\right)\phi _{P}=\left({\frac {\Gamma _{w}}{\delta x_{WP}}}A_{w}\right)\phi _{W}+\left({\frac {\Gamma _{e}}{\delta x_{WP}}}A_{e}\right)\phi _{E}+S_{u}}$

• Compare and identify above equation with

${\displaystyle a_{P}\phi _{P}=a_{W}\phi _{W}+a_{E}\phi _{E}+S_{u}}$

where

${\displaystyle a_{W}}$ ${\displaystyle a_{E}}$ ${\displaystyle a_{P}}$
${\displaystyle {\frac {\Gamma _{w}}{\delta x_{WP}}}A_{w}}$ ${\displaystyle {\frac {\Gamma _{e}}{\delta x_{PE}}}A_{e}}$ ${\displaystyle a_{W}+a_{E}-S_{P}}$

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 ${\displaystyle \phi }$ at the nodal points by any form of matrix solution technique.
• The matrix of higher order [2] 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
1. ^ "Navier-Stokes Equations in Fluid Mechanics". Efunda.com. Retrieved 2013-10-29.
2. ^ "Diffusion – useful equations". Life.illinois.edu. Retrieved 2013-10-29.
3. ^ "SSCP: Programming Strategies". Physics.drexel.edu. Retrieved 2013-10-29.