Steady-state conditions exist within the one-dimensional composite cylinder shown. The three materials have thermal conductivities of \(k_{\mathrm{A}}, k_{\mathrm{B}}=2 k_{\mathrm{A}}\), and \(k_{\mathrm{C}}=k_{\mathrm{A}}\), respectively. Uniform volumetric energy generation, \(\dot{q}\), occurs within Material B and contact resistances exist at \(r_{\mathrm{A}}\) and \(r_{\mathrm{B}}\). The cylinder is placed within a vacuum chamber so that only radiation losses occur at \(r=r_{\mathrm{C}}\).

(a) Write the appropriate form of the heat equation for each of the three materials.

(b) Write an expression for the temperature \(T\left(r_{\mathrm{C}}\right)\) in terms of relevant quantities provided above, the temperature of the surroundings \(T_{\text {sur, }}\), and the emissivity of the exposed surface.

(c) On \(T-r\) coordinates, sketch the radial temperature distribution \(T(r)\) for \(0 \leq r \leq r_{\mathrm{C}}\) for the case when \(r_{\mathrm{A}}=r_{\mathrm{B}}-r_{\mathrm{A}}=r_{\mathrm{C}}-r_{\mathrm{B}}\).

(d) On \(q^{\prime \prime}-r\) coordinates, sketch the radial heat flux for \(0 \leq r \leq r_{\mathrm{C}}\).

Transcribed Image Text:

TB A B