ABSTRACT
A material is said to conduct heat if in the presence of a temperature gradient there is a heat flux; their ratio specifies the thermal conductivity (Section 32.1). Thus a body of high thermal conductivity transmits, for a given temperature gradient, a larger heat flux than one of low conductivity. The sign of the thermal conductivity is determined by the condition that heat flows from the hotter to the cooler bodies. In a steady state, that is, time-independent conditions, there is no accumulation or rarefaction of heat, and the heat flux out of closed region must be: (i) zero if it contains no heat sources or sinks; (ii) match the output of heat sources or sinks when they exist. It follows that the steady state temperature satisfies the Laplace (Poisson) equation in the absence (presence) of heat sources or sinks (Section 32.1); thus solutions in terms of complex functions may be obtained, for example, heat source/sinks in a corner with isothermal/adiabatic walls (Section 32.4). The latter are two among more choices of boundary conditions (Section 32.2) that apply also to other irrotational or solenoidal potential fields (Section 32.3). The solution with cylindrical symmetry (Section 32.5) corresponds to the potential field due to a line source, and the two constants of integration are determined from boundary conditions (Section 32.2) at the surface of a body, specifying: (i) the temperature, if it is contact with a heat reservoir; (ii) the temperature gradient, if it is subjected to a heat influx or outflux; (iii) a linear combination of the two in the case of convection, that is, transfer of heat from a solid to a fluid, the latter at rest (in motion) for free (forced) convection (Section 32.7). These boundary conditions apply to: (i) a solid cylinder (Section 32.5); (ii) a cylindrical cavity in an unbounded medium (Section 32.5); (iii) the hollow cylindrical tube with thin or thick walls (Section 32.6); (iv) concentric tubes of different materials (Section 32.8). The latter (iv) corresponds to a thermal conductivity with finite discontinuities at the interfaces between different materials; the thermal conductivity may also be continuously nonuniform in an inhomogeneous material (Section 32.9), just as the heat source/sinks. The applications of steady heat conduction include the heating of wires, the cooling of shafts and heat exchangers using banks of tubes. The steady conduction of heat is analogous to the steady diffusion of electric charges in a resistive medium also (also of a chemical species into another).
