Heat is transported through solid materials from the high-temperature region to the low-temperature region. Thermal conduction is a principal mode of heat trans­fer in solid materials. In nuclear reactors, the heat is conducted away by the clad­ding materials from the fuel interior. Thermal conductivity (k) is defined by the following equation known as Fourier’s law:

dT

q = ~k dx ’

where q is the heat flux (the heat flow per unit perpendicular area per unit time) and dT/dx is the thermal gradient. This equation is applicable for the steady-state heat flow. The minus sign comes due to the heat being conducted from the hot region to cold region, that is, down the temperature gradient. The SI unit of ther­mal conductivity is Wm-1K-1. The above equation is much similar to Fick’s steady-state flow (Eq. (2.21)). Thermal diffusivity is another term that is often used. It is given by the following expression:

k

D = —, (5.87)

Cp Q

where CP is the constant pressure specific heat and q is the physical density.

Heat conduction takes place by both phonons and free electrons. Thus, the ther­mal conductivity (k) is given by

k = kl + ke, (5.88)

where kl is the phonon contribution and ke is the electronic contribution to thermal conductivity.

In high-purity metals, thermal energy transport through free electrons is much more effective compared to the phonon contribution as free electrons are readily available as they are not easily scattered by atoms and imperfections in the crystal and they have higher velocities. The thermal conductivities of metals can vary from 20 to 400 W m-1 K-1. Silver, copper, gold, aluminum, and tungsten are some of the common metals with high thermal conductivities. Metals are generally much better thermal conductors than nonmetals because the same free electrons that partici­pate in electrical charge transport also take part in the heat conduction. For metals, the thermal conductivity is quite high, and those metals that are the best electrical conductors are also the best thermal conductors. At a given temperature, the ther­mal and electrical conductivities of metals are proportional, but interestingly the temperature increases the thermal conductivity while reducing the electrical con­ductivity. This behavior can be explained with the help of Wiedemann-Franz law:

LWF = CT ’ (5.89)

where T is the temperature in K, Ce is the electrical conductivity, LWF is a constant (Lorenz number) that is ~2.44 x 10-8 V W K-2. The above relation is based on the

fact that both heat and electrical (charge) transport are associated with free elec­trons in metals. The thermal conductivity in metals increases with the average electron velocity as that increases the forward transport of energy. However, the electrical conductivity decreases with increasing electron velocity because the scattering or collisions divert the electrons from forward transport of charge. For more details on thermal properties, readers may consult an excellent text by Ziman [14]. The presence of grain boundaries and other crystal defects reduces the ther­mal conductivity. Researchers have observed a marked reduction in nanocrystalline materials (i. e., with grain size less than 100 nm).

Alloying generally acts upon the thermal conductivity of metals. The alloying atoms act as scattering centers for free electrons and thus reduce the effective ther­mal conductivity. Brass has a lower thermal conductivity compared to pure copper for that reason. Specifically, for a 70Cu-30Zn, thermal conductivity at room tempera­ture is 120 Wm-1 K-1, whereas the thermal conductivity of pure copper at the same temperature is 398Wm-1K-1. Figure 5.57 illustrates the point. For the same rea­son, stainless steels are generally poor conductors of heat compared to pure iron.

The thermal conductivity ofglass and ceramics is generally smaller compared to that of metals. They range between 2 and 50 W m-1 K-1. As free electrons are not available in these materials, phonon is the main mode of heat transport (at least at lower temperatures). Glass and other noncrystalline ceramics have much lower thermal conductivity compared to crystalline ceramics as phonons are more sus­ceptible to scattering in the materials lacking definite atomic order. With increasing temperature, thermal conductivity of materials decreases; but at higher tempera­tures, another mode of heat transport known as infrared radiant heat transport becomes active and its contribution increases as the temperature increases, espe­cially in transparent ceramics. Porosity in ceramics also contributes to the reduc­tion in thermal conductivity. Figure 5.58 shows the variation of thermal

Temperature (°С)

Figure 5.58 Dependence of thermal conductivity of different ceramic materials on temperature. From Ref. [1].

conductivity of certain ceramics as a function of temperature, and compared against that of graphite. The still air generally present in the pores has extremely low thermal conductivity, on the order of 0.02 W m-1 K-1, thus giving the porous material low thermal conductivity. That is why thermal insulating materials are made porous. Figure 5.59 shows thermal conductivity of some oxide ceramics (some of nuclear importance) as a function of temperature.

5.2.4

Summary

Thermophysical properties play an important role in the selection of materials as well as in their service performance in nuclear reactors. Here, three thermophysi­cal properties (specific heat, thermal expansion coefficient, and thermal conductiv­ity) are discussed. The effects of structure and composition on these properties are also highlighted. It also elucidates the effect of temperature on thermal conductiv­ity, specific heat, and coefficient of thermal expansion in metals and nonmetals. However, it should be noted that there are various specific exceptions where the foregoing discussion may not apply.

5.3