Let T (x, y,z, t) denote the temperature at the point (x, y,z) of a solid object D at

Chapter 7, Problem 9

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Let T (x, y,z, t) denote the temperature at the point (x, y,z) of a solid object D at time t. We define the heat flux density H by H = kT . (The constant k is the thermal conductivity. Note that the symbol denotes differentiation with respect to x, y,z, not with respect to t.) The vector field H represents the velocity of heat flow in D. It is a fact from physics that the total heat contained in a solid body D having density and specific heat is D T dV. Hence, the total amount of heat leaving D per unit time is D T t dV. (Here we assume that and do not depend on t.) We also know that the heat flux may be calculated as D H dS. Exercises 610 concern these notions of temperature, heat, and heat flux density(a) If , , and k are constant and the temperature T of the solid D is independent of time, show that the (net) heat flux of H across the boundary of D must be zero. (b) Let D be the solid region between two concentric spheres of radii 1 and 2. Suppose that the inner sphere is heated to 120 C and the outer sphere to 20 C. Use the result of part (a) to describe the rate of heat flow across the spheres.