*The equations describing heat transfer are complex, having some or all of the following characteristics: they are nonlinear; they comprise algebraic, partial differential and/or integral equations; they constitute a coupled system; the properties of the substances involved are usually functions of temperature and may be functions of pressure; the solution region is usually not a simple square, circle or box; and it may (in problems involving solidification, melting, etc.) change in size and shape in a manner not known in advance.*

*The equations describing heat transfer are complex, having some or all of the following characteristics: they are nonlinear; they comprise algebraic, partial differential and/or integral equations; they constitute a coupled system; the properties of the substances involved are usually functions of temperature and may be functions of pressure; the solution region is usually not a simple square, circle or box; and it may (in problems involving solidification, melting, etc.) change in size and shape in a manner not known in advance.*

The usual objective in any heat transfer calculation is the determination of the rate of heat transfer to or from some surface or object.

In conduction problems, this requires finding the temperature gradient in the material at its surface.

The error can, in general, be estimated and it can also be reduced at the price of increased effort (meaning increased computer time).

It could now be said that an approximate solution will be obtained for an exact problem.

The solution variables—temperature, velocity, etc.—are not obtained at all of the infinite number of points in the solution region, but only at the finite number of nodes of the grid, or at points within the finite number of elements.

The differential equations are replaced by set of linear (or, rarely, nonlinear) algebraic equations, which must and can be solved on a computer.Radiation is somewhat different, involving surfaces separated (in general) by a fluid which may or may not participate in the radiation.If it is transparent, and if the temperatures of the surfaces are known, the radiation and convection phenomena are uncoupled and can be solved separately.In this entry, emphasis will be given to methods for conduction and convection problems, with only a brief mention of radiation.Attention will be limited to incompressible fluids except when buoyancy is important, in which case the Boussinesq approximation will be made.These equations represent a system of five equations in three dimensions, or four equations in two dimensions.For conduction in solids, DT/Dt in (3) is replaced by ∂T/∂t, and (1) and (2) are not relevant.The heat transfer by conduction in solids can only take place when there is a variation of temperature, in both space and time.Let us consider a small volume of a solid It should be noted that Fourier law can always be used to compute the rate of heat transfer by conduction from the knowledge of temperature distribution even for unsteady condition and with internal heat generation. Due to roughness, 40 percent of the area is in direct contact and the gap (0.0002 m) is filled with air (k = 0.032 W/m K).It could be said that an exact analytical solution will be obtained for an approximate problem.The solution will, to some extent, be in error, and it will not normally be possible to estimate the magnitude of this error without recourse to external information such as an experimental result. To do this, the continuous solution region is, in most methods, replaced by a net or grid of lines and elements.

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