Thermal Conductivity of Metals | Electrical Engineering

In this article we will discuss about:- 1. Meaning of Thermal Conductivity 2. Wiedemann Franz Lorentz Relation of Thermal Conductivity 3. Heat Developed 4. Thermionic Emission.

Meaning of Thermal Conductivity:

We know that in a solid, atoms vibrate about their equilibrium position at ordinary temperature and the amplitude of vibration decreases to a minimum at absolute zero of temperature. This vibration of atoms brings about elastic distortion in the lattice and therefore the waves associated with the vibrations are the elastic waves of varying frequencies.

The waves superimpose to form a wave packet that moves through the solid. The quantum of elastic energy is known as phonon, whose energy is given by E = hv; v being frequency of vibration. The increasing number of vibration of atoms with temperature is represented by increasing number of phonon passing through the crystal.

It is the phonons together with valence electrons which are responsible for the transfer of thermal energy from high temperature region to low temperature region. The property that characterizes the ability of a material to transfer thermal energy (or hence the heat) is known as thermal conductivity.

The thermal conductivity of a solid is proportional to its specific heat and to the mean free path of phonons and electrons. The mean free path of electrons is very large in comparison to the mean free path of phonons (10 to 100 Å), as a result, the electrons undergo less number of collisions.

Since in metals the free electrons are sufficiently mobile, metals have shown high thermal conductivity due to the mobility of electrons and phonons as well. On the other hand, in insulators, phonons are the only particles to conduct heat, as the valence electrons are not present (ionic solids or covalent solids). In case of alloys, electrons remain scattered due to the presence of impurity elements; the thermal conductivity of alloys is reduced.

It has been observed that:

(i) The thermal conductivity of metals decreases at higher temperatures.

(ii) The thermal conductivity of polycrystalline metals is lower than that of single crystal due to scattering of electrons and phonons along grain boundaries.

(iii) The thermal conductivity of metals is lower than their alloys due to scattering of electrons and phonons with the atoms of alloying element.

(iv) The thermal conductivity of semiconducting materials increases with increasing temperature due to flow of excited electrons.

(v) The thermal conductivity of metals, alloys, semiconductors and dielectrics are in the descending order.

Wiedemann Franz Lorentz Relation of Thermal Conductivity:

Suppose a homogenous isotropic material is subjected to a temperature gradient dT/dx. The flow of heat will result in the direction opposite to the temperature gradient through the conducting medium. The heat flux Q (heat flow per unit time per unit area) will be proportional to the temperature gradient i.e. Q ∝ dT/dx

or Q = -K dT/dx

where, K is the proportionality constant and is known as coefficient of thermal conductivity. If Q is expressed in W/m² and dT/dx in °K/m, the units of K will be W/mK.

The transformation of heat in solids is due to phonons and free electrons. Hence, the coefficient of thermal conductivity K can be written as-

K = K_phonon + K_electron

In order to derive the expression for K, let us consider the heat flow from high temperature to low temperature in a metal slab having temperature gradient dT/dx.

Let C_v be the heat capacity, the heat transfer per unit area per second will be;

Where, v is the velocity of electrons and λ being mean free path of collisions.

Also, heat flux Q = K dt/dx

The energy of free electron is given by-

Specific heat at constant volume for an ideal gas is-

Above expression implies that the thermal conductivity of solids depends upon- (i) specific heat, (ii) mean free path of collisions (iii) velocity of electrons. Now consider the electrical conductivity σ.

Therefore, the ratio of thermal conductivity K to electrical conductivity σ is-

which indicates that the ratio K/σ is some for all metals and is a function of temperature only. This empirical law is known as Weidmann-Franz Lorentz relation. Hence, we can say that best electrical conductor will be a best thermal conductor.

Heat Developed in Current Carrying Conductor:

Joule observed that the heat developed in a conducting wire is given by I²R; where I is current flowing through the wire having resistance R. If p be the electrical resistivity of wire, l being length of wire and area of cross-section is A, then-

Where, V is the applied potential and E is the electric field developed across the wire of length l. Therefore, the heat developed per unit volume (lA) per second is-

W = σE² = JE (∵ J = σE)

Where, J is the current density. If J is in Ampere per m² and E is in volts per m; the units of W will be Watts per m³.

The electrons in a conductor exert an acceleration ‘a’ on application of an electric field E and is given by-

a = (e/m) E

The acceleration increases the velocity of the electron by (l/m) Et. Let v_x, v_y and v_z are components of velocity v along x, y and z direction at t = 0 i.e. electrons are not get collided. At t > 0, the velocity components will be v_x + e/m Et, v_y and v_z if the electric field is applied along x-axis.

The increase in energy of the electron over the period t is-

When averaged over a large number of electrons, the above expression becomes;

Because v_x is zero for such a group of electrons. Here we have considered no collisions between electrons.

The average energy increase of the electrons during the period between two collisions is given by-

The term dt/τ represent the probability that an electron will suffer a collision during a time dt.

Let n be the density of electrons and assuming that these electrons transfer their excess energy to the lattice; the total energy dissipated per unit volume per second will be-

This is the energy which is converted into heat during collision process.

Thermionic Emission from Metals:

We know that the moving electrons in a metal exert a pressure similar to that of a gas and these moving electrons cannot escape to the outer space because of the attractive forces at the surface that tend to keep the electrons within the metal. The attractive forces are greater than the electron gas pressure.

In order to escape the electron from the surface of metal, an electron has to do a certain amount of work called work function φ of the surface and this is different for different metals. At ordinary temperature, the kinetic energy of the electrons is much less than the work function and hence the electrons cannot escape from the surface.

With the increase in temperature of the metal surface, the kinetic energy of the electrons increases and if kinetic energy exceeds the work function, electrons can escape from the surface of the metal. This phenomenon is called thermionic emission and electrons are called thermo-electrons or thermions. The thermionic current I, is given by-

I = AT² exp [-(φ₀/K_BT)]

where A is a constant, T is the absolute temperature of the metal, φ₀ is the work function of the surface at absolute zero and K_B is Boltzmann constant. I measures the thermionic current in amperes per square meter of the emitting surface. The most commonly used emitters are – (i) Tungsten W, (ii) thoriated tungsten and (iii) oxide coated cathodes containing barium or strontium.

The above equation is known as Richardson’s equation for thermionic emission. The electron emitters are heated directly or indirectly. In case of direct heating, the current is passed through the filament which itself serves as the cathode. The indirectly heated electron emitters consist of a heater wire surrounded by a metal sleeve in the form of a cylinder whose surface is coated with electron emitting materials. The operating temperature for tungsten, thoriated tungsten and oxide coated emitters are respectively 2200-3000 K, 1900 K and 1000-1150 K.