CONDUCTION

Conduction is the mode of heat transfer in which energy exchange takes place from the region of high temperature to that of low temperature by the kinetic motion or direct impact of molecules, as in the case of fluid at rest, and by the drift of electrons, as in the case of metals. In a solid which is a good electric conductor, a large number of free electrons move about in the lattice; hence materials that are good electric conductors are generally good heat conductors (i.e., copper, silver, etc.).
The empirical law of heat conduction based on experimental observations originates from Biot but is generally named after the French mathematical physicist Joseph Fourier who used it in his analytic theory of heat. This law states that the rate of heat flow by conduction in a given direction is proportional to the area normal to the direction of heat flow and to the gradient of temperature in that direction. For heat flow in the x direction, for example, the Fourier law is given as
Qx = -kA (dT/dx) Watt (1)
or
qx = Qx/A = -k(dT/dx) W/m2 (2)

where Qx is the rate of heat flow through area A in the positive x direction and qx is called the heat flux in the positive x direction. The proportionality constant k is called the thermal conductivity of the material and is a positive quantity. If temperature decreases in the positive x direction, then dT/dx is negative; hence qx (or Qx) becomes a positive quantity because of the presence of the negative sign in Eqs. (1) and (2). Therefore, the minus sign is included in Eqs. (1) ans (2) to ensure that qx (or Qx) is a positive quantity when the heat flow is in the positive x direction. Conversely, when he right-hand side of Eqs. (1) and (2) is negative, the heat flow is in the negative x direction.
The thermal conductivity k in Eqs. (1) and (2) must have the dimensions W/(m.C) or J/(m.s.C) if the equations are dimensionally correct. There is a wide difference in the range of thermal conductinities of various engineering materials. Beetwen gaes and highly conducting metals, such as copper or silver, k varies by a factor of about 10000. The highest value is for highly conducting pure metals, and the lowest value is for gases and vapors, excluding the evacuated insulating systems. The nonmetallic solids and liquids have thermal conductivities that lie between them. Metallic single crystals are exceptions, which may have vary high thermal conductivities; for example, with copper crystals, value of 8000 W/(m.C) and even higher are possible.
Thermal conductivity also varies with temperature. This variation, for some materials over certain temperature ranges, is small enough to be neglected; but for many cases the variation of k with temperature is signifficant. Especially at very low temperatures k varies rapidly with temperature; for example, the thermal conductivities of copper, aluminum, or silver reach values 50 to 100 times those that occur at room temperature.