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.
For a Boussinesq fluid, it is assumed that ρ = f(T).
For conduction in solids, DT/Dt in (3) is replaced by ∂T/∂t, and (1) and (2) are not relevant.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.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).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 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.Thus, a full solution of the energy equation and perhaps also the equations of motion is required.These are partial differential equations, possibly coupled.