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The finite volume method for diffusion problems

General transport equations

( ) div( ) div( grad )U St

convective term diffusive term source term

0 div( grad ) S

Pure diffusion:

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0 div( grad )d dCV CV

V S V

0 n ( grad )d d

A CV

A S V

0 div( grad ) S

FVM for 1D diffusion problems

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Grid:

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Diffusive flux of

Generation

Net output

w

e

P

input output

n n x

0 n ( grad )d dA CV

A S V

n ( ) n ( )e w

d d

i A i Adx dx

e w

d dA A

dx dx

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At interface, for a uniform grid

w

e

P

input output

E

W

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Approximated by a linear function

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All points: System of linear equations

For point P

Example 1: 1D steady conductive heat transfer

Source-free

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1000 W/mKk

20.01mA

Grid: five control volumes

Thermal conductivity,

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For points 2,3,4

P EW

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For point 1,

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For point 5,

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For point 5

For point 1

For points 2,3,4

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Example 2: a large plate of thickness L=2cm,

uniform heat generation q =1000kW/m3

0.5 W/mKk

thermal conductivity,

100 C 200 CA B

T T

Assume variation only in x direction,

1D problem

Exact solution:

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Grid:

Control Volume

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FVM for 3D diffusion problems

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Source is approximated by a linear function

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