Chapter 4, Problem 4.69P

A long, solid cylinder of diameter $D=25\text{mm}$ is formed of an insulating core that is covered with a very thin $\left(t=50\text{μm}\right),$ highly polished metal sheathing of thermal conductivity $k=25\text{W/m}\cdot \text{K}.$ Electric current flows through the stainless steel from one end of the cylinder to the other, inducing uniform volumetric heating within the sheathing of $\stackrel{.}{q}=5\times {10}^6{\text{W/m}}^{\text{3}}.$ As will become evident in Chapter 6, values of the convection coefficient between the surface and air for this situation are spatially nonuniform, and for the airstream conditions of the experiment, the convection heat transfer coefficient varies with the angle $\theta$ as $h\left(\theta \right)=26+0.637\theta -8.920{\theta }^2$ for $0\le \theta <\pi /2$ and $h\left(\theta \right)=5$ for $\pi /2\le \theta \le \pi .$

1. Neglecting conduction in the O-direction within the stainless steel, plot the temperature distribution $T\left(\theta \right)$ for $0\le \theta \le \pi$ for $T_{\infty }=25°\text{C}\text{.}$
2. Accounting for $\theta$ -direction conduction in the stainless steel, determine temperatures in the stainless steel at increments of $\Delta \theta =\pi /20$ for $0\le \theta \le \pi .$ Compare the temperature distribution with that of part (a).

Hint: The temperature distribution is symmetrical about the horizontal centerline of the cylinder.