The heat flow across any surface area, A, is given in Fig. A.12.1 by the heat-flow rate per unit area dQ/(Adt) in Eq. (1), a vector quantity in J m ^?2s ^?1. It is equal to the negative of the thermal conductivity, ? in J m ^?1s ^?1K ^?1, multiplied by the temperature gradient (dT/dr). Equation (2) represents the differential heat flow into the volume V and can be derived from the definition of the heat capacity dQ/dT = mc _p. The symbols have the standard meanings; ? is the density and c _p, the specific heat capacity per unit mass, so that m = V ?.

Figure A.12.1

Standard techniques of vector analysis allow to equate the heat flow into the volume V to the heat flow across its surface. This operation leads to the linear and homogeneous Fourier differential equation of heat flow, given as Eq. (3). The letter k represents the thermal diffusivity in m ² s ^?1, which is equal to the thermal conductivity ? divided by the density and specific heat capacity. The Laplacian operator is ? ² = ? ²/ ?x ² + ? ²/ ?y ² + ? ²/ ?z ², where x, y, and z are the space coordinates. In the present cylindrical symmetry, the Laplacian, operating on temperature T, can be represented as d ²T/dr ²