(Solved): 3.47 A composite cylindrical wall is composed of two materials of thermal conductivity \( k_{\math ...

3.47 A composite cylindrical wall is composed of two materials of thermal conductivity \( k_{\mathrm{A}} \) and \( k_{\mathrm{B}} \), which are separated by a very thin, electric resistance heater for which interfacial contact resistances are negligible. Liquid pumped through the tube is at a temperature \( T_{\infty, i} \) and provides a convection coefficient \( h_{i} \) at the inner surface of the composite. The outer surface is exposed to ambient air, which is at \( T_{\infty, o} \) and provides a convection coefficient of \( h_{o} \). Under steady-state conditions, a uniform heat flux of \( q_{h}^{\prime \prime} \) is dissipated by the heater. (a) Sketch the equivalent thermal circuit of the system and express all resistances in terms of relevant variables. (b) Obtain an expression that may be used to determine the heater temperature, \( T_{h} \). (c) Obtain an expression for the ratio of heat flows to the outer and inner fluids, \( q_{0}^{\prime} / q_{i}^{\prime} \). How might the variables of the problem be adjusted to minimize this ratio?