Consider a long solid cylinder of radius \(r_{o}=4 \mathrm{~cm}\) and thermal conductivity \(k=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). Heat is generated in the cylinder uniformly at a rate of \(\dot{e}_{\text {gen }}=35 \mathrm{~W} / \mathrm{cm}^{3}\). The side surface of the cylinder is maintained at a constant temperature of \(T_{s}=80^{\circ} \mathrm{C}\). The variation of temperature in the cylinder is given by $$ T(r)=\frac{\dot{e}_{\text {gen }} r_{o}^{2}}{k}\left[1-\left[1-\left(\frac{r}{r_{o}}\right)^{2}\right]+T_{s}\right. $$ Based on this relation, determine \((a)\) if the heat conduction is steady or transient, \((b)\) if it is one-, two-, or three-dimensional, and \((c)\) the value of heat flux on the side surface of the cylinder at \(r=r_{o^{*}}\)

In a steady heat conduction process, the temperature does not change over time. To check if it's steady or transient, we need to examine if time (t) is present in the temperature equation given: $$ T(r) = \frac{\dot{e}_{\text{gen}} r_{o}^{2}}{k} \left[1-\left(1-\left(\frac{r}{r_{o}}\right)^{2}\right)\right] + T_{s} $$ No time-dependent term (t) is present in the equation. Therefore, the heat conduction is steady. (a) The heat conduction is steady.

To identify if the heat conduction is one-, two-, or three-dimensional, we need to examine how the temperature varies with different coordinates. The given temperature equation has only the radial coordinate (r) playing a role in temperature variation: $$ T(r) = \frac{\dot{e}_{\text{gen}} r_{o}^{2}}{k} \left[1-\left(1-\left(\frac{r}{r_{o}}\right)^{2}\right)\right] + T_{s} $$ Since only the radial coordinate (r) affects the temperature, it is a one-dimensional heat conduction. (b) The heat conduction is one-dimensional.

To find the heat flux at the side surface of the cylinder, we need to compute the radial heat flux at \(r = r_{o}\) using Fourier's Law of heat conduction: $$ q_r = -k\frac{dT}{dr} $$ First, we compute the derivative of the temperature function with respect to r: $$ \frac{dT}{dr} = \frac{\dot{e}_{\text{gen}} r_{o}^{2}}{k} \cdot \left(\frac{-2r}{r_{o}^2}\right) = - \frac{2 \dot{e}_{\text{gen}} r_{o} r}{k} $$ Now, substituting the values of \(k\), \(\dot{e}_{\text{gen}}\), and \(r_{o}\), and calculating the heat flux at the side surface \(r = r_{o}\): $$ q_{r_{o}} = -k \cdot \left(- \frac{2 \dot{e}_{\text{gen}} r_{o} r_{o}}{k}\right) = 2 \dot{e}_{\text{gen}} r_{o}^2 = 2 \cdot 35 \mathrm{~W/cm^3} \cdot (4\mathrm{~cm})^2 = 1120 \mathrm{~W/cm^2} $$ (c) The heat flux on the side surface of the cylinder at \(r = r_{o}\) is 1120 W/cm².