# Discretization methods

### From Thermal-FluidsPedia

The discretization of the computational domain is a process that divides the computational domain into many control volumes as shown in the figure. Each subdomain is bounded by faces represented by the dashed lines. The physical properties of the cell are defined at the grid points. In Practice A, the faces are always located midway between the grid points. On the contrary, the grid points are always located in the center of the control volume in Practice B. Obviously, if a uniform grid is used, the two practices result in identical grid size and therefore, any discussion on the differences between two practices are meaningful only if the grid size is not uniform. Since the faces in Practice A are located in the midway between grid points, the heat flux across the faces can be more accurately calculated. The disadvantage of Practice A is that the properties of the entire control volume are represented by a point not located in the center of the control volume, which will result in inaccuracy. On the other hand, the properties of the entire control volume in Practice B are represented by the grid point at the center which is a better representation. In addition, Practice B can easily handle discontinuity of thermophysical properties or boundary conditions.

The discretization of governing equations can be done by local or point-wise representation of the partial differential equations (finite difference method; FDM), or weighted integral of the partial differential equations (finite element method; FEM). Patankar (1980) proposed a finite volume method (FVM) involving obtaining the discretized equation by performing integration on the governing equation over the small region. While the resultant algebraic equations for FVM and FDM are often similar, the FVM can guarantee conservation of the mass, momentum and energy on each cell, regardless of the size of the cell. The FVM can also be very easily extended to convective heat transfer because mature numerical methods have been developed in the last three decades.

## 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.