8. One Dimensional Steady State Heat Conduction through Spherical Surface



\[\begin{aligned} {2} T &= T_1 \qquad \text{at} \quad r&=a \\ T &= T_2 \qquad \text{at} \quad r&=b \end{aligned}\]

\[ \frac{d}{d r}\left(r^2 \frac{d T}{\partial r}\right) = 0 \] Intergrating , we get \[ r^2\frac{dT}{dr} = C_1 \] i.e., \[ \frac{dT}{dr} = \frac{C_1}{r^2} \tag*{(1)} \] Integrating further, we get \[ T = -\frac{C_1}{r} + C_2 \tag*{(2)} \] Using the boundary condition at r=a,b gives \begin{align*} T_1 &= -\frac{C_1}{a} + C_2 \tag*{(3)}\\ T_2 &= -\frac{C_1}{b} + C_2 \tag*{(4)} \end{align*}

Eqn.(3) \(-\) Eqn.(4) \(\Longrightarrow\) \[T_1-T_2 = \frac{C_1}{b} - \frac{C_1}{a} = \frac{aC_1-bC_1}{ab} = -\frac{b-a}{ab}C_1\] i.e., \[C_1 = -\frac{ab}{(b-a)}(T_1-T_2)\] From the definition of heat flux, \(q=-k\dfrac{dT}{dr}\), and from Eqn.(1), we get \[q = \frac{k}{r^2}\frac{ab}{(b-a)}(T_1-T_2)\] \[Q=qA=q(4\pi r^2) \quad \Longrightarrow \quad Q = \frac{4\pi k ab}{(b-a)}(T_1-T_2) = \frac{T_1-T_2}{R}\] where \[\boxed{R = \frac{b-a}{4\pi k ab}}\]