# Numerical solution of multi-dimensional unsteady-state conduction

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## Two-Dimensional Transient Conduction

The energy equation for a two-dimensional transient conduction problem with temperature-dependent thermal conductivity and internal heat source can be written as

To discretize eq. (1), integrating eq. (1) with respect to *t* in the interval of (*t*,*t* + *t*) and over the control volume P (shaded area in Fig. 1) gives (Tao, 2001)

Assuming the heat fluxes are uniform on all faces of the control volume and employing a fully-implicit scheme, the above equation becomes

which can be rearranged to obtain (Tao, 2001)

where

## Three-Dimensional Transient Conduction

The discretized equation for a three-dimensional transient conduction problem with temperature-dependent thermal conductivity and an internal heat source can be obtained by integrating the energy equation with respect to *t* in the interval of (*t*,*t* + *t*) and over the three-dimensional control volume P (formed by considering two additional neighbors at top, T, and bottom, B). The final form of the discretized equation is (Patankar, 1980)

where

## References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, *Advanced Heat and Mass Transfer*, Global Digital Press, Columbia, MO.

Patankar, S.V., 1980, *Numerical Heat Transfer and Fluid Flow*, Hemisphere, Washington, DC.

Tao, W.Q., 2001, *Numerical Heat Transfer*, 2^{nd} Ed., Xi’an Jiaotong University Press, Xi’an, China (in Chinese).