Heat Production

Heat conduction theory enables us to determine the distribution of temper ature in a region given information about the temperatures or heat fluxes on the boundaries of the region and the sources of heat production in the region. In general, we can also use the theory to determine time variations in the temperature distribution. We first develop the theory for the simple situation in which heat is transferred in one direction only and there are no time variations (steady state) in the temperature or heat flow. The basic equation of conductive heat transfer theory is a mathematical statement of conservation of energy; the equation can be derived as follows. Consider a slab of infinitesimal thickness δy, as sketched in Figure 4–5. The heat flux out of the slab q(y + δy) crosses the face of the slab located at y + δy, and the heat flux into the slab q(y) crosses the face located at y. The net heat flow out of the slab, per unit time and per unit area of the slab’s face, is q(y + δy) − q(y).

Since δy is infinitesimal, we can expand q(y + δy) in a Taylor series as q(y + δy) = q(y) + δy dq dy + · · · . (4.9) 250 Heat Transfer Thus we find q(y + δy) − q(y) = δy dq dy = δy d dy [ −k ( dT dy )] = δy [

−k (

d2T

dy2

)]

, (4.10)

where we have used Equation (4–1) (Fourier’s law) for q, and we have assumed a constant thermal conductivity. The right side of Equation (4–10) is the net heat flow out of a slab of thickness δy, per unit time and per unit area. It is nonzero only when there is curvature in the temperature profile. If there is a nonzero net heat flow per unit area out of the thin slab, as given by Equation (4–10), this heat flow must be supplied, in steady state, by heat generated internally in the slab. If H is the heat production rate per unit mass, the amount of heat generated in the slab per unit time and per unit area of the slab face is ρHδy, (4.11) where ρ is the density of the slab. By equating (4–10) and (4–11), one obtains 0 = k d2T

dy2 + ρH. (4.12) his equation can be integrated to determine temperature as a function of position y once the region of interest and appropriate boundary conditions have been specified.