9.24. The term conduction refers to the transfer of heat by molecular (and sometimes electronic) interaction without any accompanying mac­roscopic displacement of matter. The flow of heat by conduction is gov­erned by the relationship known as the Fourier equation, i. e.,

«’-4 (9-6)

where q is the rate (per unit time) at which heat is conducted in the x direction through a plane of area A normal to this direction, at a point where the temperature gradient is dtidx.[3] The quantity к, defined by equa­
tion (9.6), is the thermal conductivity. In SI units, q is expressed in J/s (or W), A in m2, and dt/dx in °C/m (or K/m); hence, the units of к are (W/m2)(m/K), usually written in the form W/m • K. In English units, к will be in Btu/(hr)(ft2)(°F/ft).

9.25. The thermal conductivity A: is a physical property of the medium through which the heat conduction occurs. For anisotropic substances, the value of A: is a function of direction; although methods are available for making allowance for such variations, they are ignored in most analytical solutions of conduction problems. The thermal conductivity is also tem­perature dependent and can generally be expressed in a power series; thus,

к = c0 + cxt + c2t2 + • • • ,

which, in many cases, may be approximated to the simple linear form

к = A0(l + at).

Where considerable accuracy is desirable (and possible), allowance must be made for the variation of thermal conductivity with temperature. But very frequently, especially when the temperature range is not great, к is taken to be constant. However, when uranium oxide is used as fuel, as it is in most power reactors, the temperature gradients are quite large, and allowance must be made for the variation of the thermal conductivity with temperature (§9.46). Some values of к of interest in reactor design are given in the Appendix.

9.26. Upon integration of equation (9.6) over the x direction, it is found, for unidirectional heat flow by conduction in a slab of constant cross section, with к independent of temperature, that

q = — kA h ~ *2, (9.7)

Xi — X2

where tx and t2 are the temperatures at two points whose coordinates are

and x2. The result means that the temperature gradient at a point, i. e., dt/dx, in the Fourier equation (9.6) may be replaced by the average gradient over any distance, i. e., by (tx — t2)/(x1 — x2).

9.27. If tx — t2 is replaced by Дt, the temperature difference, and x2 — Xi by L, the length of the heat-flow path, equation (9.7) upon rearrange­ment takes the form

_ At 4 ~ L/kA’

This expression is analogous to Ohm’s law, I = E/R; hence the quantity L/kA is often called the thermal resistance for a slab conductor. The analogy between conduction of heat and electricity is the basis of the thermal circuit concept which is very useful in solving heat-transfer problems. In general, the rate of heat flow q is equivalent to the current /; the temperature difference At is the analogue of the potential difference (or EMF) E; and the thermal resistance replaces the electrical resistance.