Physical principles

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Thermal conductivity (λ) is the intrinsic physical property of a substance describing its ability to conduct heat, whereas thermal diffusivity (a) indicates how fast the heat is conducted.

These thermal properties, which are involved in all thermal phenomena, can be illustrated by the two following examples:

 If a spoon is left in a cup of hot soup, the spoon handle will get hot, even if it is outside the cup. Heat from the soup is conducted along the spoon handle.  If you put your finger on a piece of wood and then onto a piece of metal which is at the same temperature, you nevertheless feel that the metal is colder than the wood. This is because the metal conducts heat from your fingers more effectively because of its higher thermal conductivity.

Why are these properties important?

Thermal conductivity and thermal diffusivity are essential properties since transfer of heat is involved in many industrial applications and processes.

Thermal conductivity is important for heat exchange calculations and also for evaluation of thermal insulation performance.

Thermal diffusivity is important for thermal stress calculations caused by fast temperature changes (thermal shock) and transient heat flow calculations generally.

How are they used?

An accurate knowledge of these thermal properties is often necessary, to check the adequacy of materials within a specific application, to solve problems of thermal transfer or to calculate temperature distributions in more or less complex systems.

Conductivity is used to calculate heat transfer by conduction, using the Fourier heat equation.

Diffusivity is used for calculation of the speed of heat diffusion (or rate of spread of heat) in a material.

How is heat conducted?

Conduction is the process of heat transfer by molecular motion, supplemented in some cases by the flow of free electrons, within a medium (solid, liquid or gas) from a region of high temperature to a region of low temperature.

The mechanism of heat conduction in liquids and gases is based on the transfer of kinetic energy of the molecules situated in the high temperature region to molecules in the low temperature region, by successive collisions between molecules.

For an ideal gas, the thermal conductivity λ is proportional to the average molecular velocity v, the mean free path l and the molar heat capacity C. In solid materials, heat is transported by both lattice vibration waves (phonons) and free electrons. Free electrons in a hot region migrate to colder areas, where some of their kinetic energy is transferred to the atoms as a consequence of collisions with phonons. For metals, where the density of free electrons is high, energy transferred by free electrons is predominant compared to energy transferred by molecular and lattice vibrations, while for insulating materials the majority of heat transfer is by phonons.

 The French mathematician Jean-Baptiste Joseph Fourier (right) gave in 1822 a general expression for heat transfer by conduction. This law states that the rate at which heat is conducted through unit area of a homogeneous body is proportional to the negative of the temperature gradient in the direction perpendicular to the area. The general heat flow or Fourier equation is given by:   where Qn is heat flux or power (units W) in direction n, A is the area through which the heat is flowing, λ is thermal conductivity in direction n and dT/dn is the temperature gradient in direction n. The negative sign in the above equation denotes that the heat flow is from a higher to a lower temperature.

How is thermal conductivity related to electrical conductivity?

For metals, there is a relationship between thermal conductivity and electrical conductivity. The Wiedemann-Franz law states that the ratio of the thermal conductivity λ (W.m-1.K-1) to the electrical conductivity λ (Ω-1.m-1) of a metal is proportional to the absolute temperature T (K). with where L is the Lorentz number, k is the Boltzmann constant and e is the electronic charge.

This relationship is based on the fact that in metals both heat and electricity are transported primarily by free electrons. Thermal conductivity increases with average particle velocity - which leads to increased transport of heat energy - whereas electrical conductivity decreases because collisions divert the electrons from transporting charge in a particular direction.

The Wiedemann-Franz law can be applied only in cases where the electron contribution is much higher than the phonon contribution - and the latter is the only transport mechanism in insulating materials. This happens for all pure metals, and for some alloys, but not for semiconductors.

This law allows the thermal conductivity of a metal to be determined via an electrical measurement, which is often simpler and more convenient.

EQUATIONS

In the table below are summarised some equations relating to thermal conductivity and thermal diffusivity.

 Title Equation Terms Thermal diffusivity a [m2.s-1] λ = thermal conductivity (Wm-1 K-1)ρ = density (kg.m-3)cp = specific heat at constant pressure (J.kg-1.K-1) Thermal conductivity of gases λ [W.m-1.K-1] λ = thermal conductivity (W.m-1 .K-1)C = heat capacity per mole (J.mol-1.K-1)v = average molecular velocity (m.s-1)l = mean free path of the molecule (m) Thermal effusivityb [W.s-0.5.m-2.K-1] λ = thermal conductivity (Wm-1 K-1)ρ = density (kg.m-3)cp = specific heat at constant pressure (J.kg-1.K-1) Thermal conductivity of solids in linear coordinatesλ [W.m-1.K-1] Q = heat flux (W)A = area (m2)L = thickness (m)ΔT = temperature difference (K) Thermal conductivity of solids in radial coordinatesλ [W.m-1.K-1] Q = heat flux (W)L = thickness or length (m)ΔT = temperature difference (K)r1 = inner radius of the cylinderr2 = outer radius of the cylinder Fourier's law Q= heat flux (W)A = area (m2)λ = thermal conductivity (W.m-1.K-1)dT/dx = temperature gradient (K.m-1) Wiedemann-Franz law σ = electrical conductivity (Ω-1.m-1)L = Lorentz number (2.45x10-8 W.Ω.K-2)T= absolute temperature (K)

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