# Thermal Conductivity of Solids and Metals

## Thermal Conductivity of Solids

Transport of thermal energy in solids may be generally due to two effects:

• the migration of free electrons
• lattice vibrational waves (phonons)

When electrons and phonons carry thermal energy leading to conduction heat transfer in a solid, the thermal conductivity may be expressed as:

k = ke + kph

## Thermal Conductivity of Metals

Metals are solids and as such they possess crystalline structure where the ions (nuclei with their surrounding shells of core electrons) occupy translationally equivalent positions in the crystal lattice. Metals in general have high electrical conductivity, high thermal conductivity, and high density. Accordingly, transport of thermal energy may be due to two effects:

• the migration of free electrons
• lattice vibrational waves (phonons).

When electrons and phonons carry thermal energy leading to conduction heat transfer in a solid, the thermal conductivity may be expressed as:

k = ke + kph

The unique feature of metals as far as their structure is concerned is the presence of charge carriers, specifically electrons. The electrical and thermal conductivities of metals originate from the fact that their outer electrons are delocalized. Their contribution to the thermal conductivity is referred to as the electronic thermal conductivity, ke. In fact, in pure metals such as gold, silver, copper, and aluminum, the heat current associated with the flow of electrons by far exceeds a small contribution due to the flow of phonons. In contrast, for alloys, the contribution of kph to k is no longer negligible.

Wiedemann-Franz Law - Lorentz Number
At a given temperature, the thermal and electrical conductivities of metals are proportional, but raising the temperature increases the thermal conductivity while decreasing the electrical conductivity. This behavior is quantified in the Wiedemann–Franz law. This law states that the ratio of the electronic contribution of the thermal conductivity (k) to the electrical conductivity (σ) of a metal is proportional to the temperature (T).

Qualitatively, this relationship is based upon the fact that the heat and electrical transport both involve the free electrons in the metal. The electrical conductivity decreases with particle velocity increases because the collisions divert the electrons from forward transport of charge. However, the thermal conductivity increases with the average particle velocity since that increases the forward transport of energy. The Wiedemann-Franz law is generally well obeyed at high temperatures. In the low and intermediate temperature regions, however, the law fails due to the inelastic scattering of the charge carriers.

It must be noted, the general correlation between electrical and thermal conductance does not hold for other materials, due to the increased importance of phonon carriers for heat in non-metals.

## Thermal Conductivity of Nonmetals

For nonmetallic solids, k is determined primarily by kph, which increases as the frequency of interactions between the atoms and the lattice decreases. In fact, lattice thermal conduction is the dominant thermal conduction mechanism in nonmetals, if not the only one. In solids, atoms vibrate about their equilibrium positions (crystal lattice). The vibrations of atoms are not independent of each other, but are rather strongly coupled with neighboring atoms. The regularity of the lattice arrangement has an important effect on kph, with crystalline (well-ordered) materials like quartz having a higher thermal conductivity than amorphous materials like glass. At sufficiently high temperatures kph ∝ 1/T.

The quanta of the crystal vibrational field are referred to as ‘‘phonons.’’ A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, like solids and some liquids. Phonons play a major role in many of the physical properties of condensed matter, like thermal conductivity and electrical conductivity. In fact, for crystalline, nonmetallic solids such as diamond, kph can be quite large, exceeding values of k associated with good conductors, such as aluminum. In particular, diamond has the highest hardness and thermal conductivity (k = 1000 W/m.K) of any bulk material.

## Thermal Conductivity of Uranium Dioxide

Most of PWRs use the uranium fuel, which is in the form of uranium dioxide. Uranium dioxide is a black semiconducting solid with very low thermal conductivity. On the other hand the uranium dioxide has very high melting point and has well known behavior. The UO2 is pressed into pellets, these pellets are then sintered into the solid.

These pellets are then loaded and encapsulated within a fuel rod (or fuel pin), which is made of zirconium alloys due to its very low absorption cross-section (unlike the stainless steel). The surface of the tube, which covers the pellets, is called fuel cladding. Fuel rods are base element of a fuel assembly.

The thermal conductivity of uranium dioxide is very low when compared with metal uranium, uranium nitride, uranium carbide and zirconium cladding material. The thermal conductivity is one of parameters, which determine the fuel centerline temperature. This low thermal conductivity can result in localised overheating in the fuel centerline and therefore this overheating must be avoided.  Overheating of the fuel is prevented by maintaining the steady state peak linear heat rate (LHR) or the Heat Flux Hot Channel Factor – FQ(z) below the level at which fuel centerline melting occurs. Expansion of the fuel pellet upon centerline melting may cause the pellet to stress the cladding to the point of failure.

Thermal conductivity of solid UO2 with a density of 95% is estimated by following correlation [Klimenko; Zorin]:

where τ = T/1000. The uncertainty of this correlation is +10% in the range from 298.15 to 2000 K and +20% in the range from 2000 to 3120 K.

Special reference: Thermal and Nuclear Power Plants/Handbook ed. by A.V. Klimenko and V.M. Zorin. MEI Press, 2003.

Special reference: Thermophysical Properties of Materials For Nuclear Engineering: A Tutorial and Collection of Data. IAEA-THPH, IAEA, Vienna, 2008. ISBN 978–92–0–106508–7.

## Thermal Conductivity of Zirconium

Zirconium is a lustrous, grey-white, strong transition metal that resembles hafnium and, to a lesser extent, titanium. Zirconium is mainly used as a refractory and opacifier, although small amounts are used as an alloying agent for its strong resistance to corrosion. Zirconium alloy (e.g. Zr + 1%Nb) is widely used as a cladding for nuclear reactor fuels. The desired properties of these alloys are a low neutron-capture cross-section and resistance to corrosion under normal service conditions. Zirconium alloys have lower thermal conductivity (about 18 W/m.K) than pure zirconium metal (about 22 W/m.K).

Special reference: Thermophysical Properties of Materials For Nuclear Engineering: A Tutorial and Collection of Data. IAEA-THPH, IAEA, Vienna, 2008. ISBN 978–92–0–106508–7.

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