If buoyancy is important, density must be considered as a variable and an equation of state is required.A common assumption for low speed flow (small Mach number, together with other restrictions) is to invoke the , in which the variation of density is neglected except in the body force term of the momentum equations, namely the last term of (2).
Calculate the time required for the temperature to drop to 150°C when h = 25 W/m2K and density p = 7800 kg/m3.
Numerical heat transfer is a broad term denoting the procedures for the solution, on a computer, of a set of algebraic equations that approximate the differential (and, occasionally, integral) equations describing conduction, convection and/or radiation heat transfer.
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.
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.
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.
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.
Thus analytical methods, leading to exact, closed form solutions, are almost always not available. In the first, the equations are simplified — for example, by linearization, or by the neglect of terms considered sufficiently small, or by the assumption of constant properties, or by some other technique until an equation or system of equations is obtained for which an analytical solution can be found.