The temperature u{x, t) of a long, thin rod of constant cross section and homogeneous

Chapter 12, Problem 13

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The temperature u{x, t) of a long, thin rod of constant cross section and homogeneous conducting material is governed by the one-dimensional heat equation. If heat is generated in the material, for example, by resistance to current or nuclear reaction, the heat equation becomes 3 2 u Krdu - -| = K , 0 < x < /, 0 < r. dx2 pC 3t where / is the length, p is the density, C is the specific heat, and K is the thermal diffusivity of the rod. The function r r(x, t, u) represents the heat generated per unit volume. Suppose that / = 1.5 cm. A" = 1.04 cal/cm deg s, p = 10.6 g/cm3 , C = 0.056 cal/g deg,and rix, t, u) 5.0 cal/cm3 s. Ifthe ends of the rod are kept at 0 CC, then r/(0, t) u(l, t) 0, r > 0. Suppose the initial temperature distribution is given by u(x, 0) = sin , 0 < x < I. Use the results of Exercise 17 to approximate the temperature distribution with A = 0.15 and k = 0.0225.