14 - Conduction - Critical Radius of Insulation

Spherical Surface:

\[Q = \frac{T_i-T_{\infty}}{R} = \frac{T_i-T_{\infty}}{R_{\text{ins}}+R_o}\] where \[R_{\text{ins}} = \frac{r_o-r_i}{4\pi k r_or_i} \qquad \text{ and } \qquad R_o = \frac{1}{h4\pi r_o^2}\] \[Q = \frac{(T_i-T_{\infty})4\pi k}{\left(\dfrac{r_o-r_i}{r_or_i}\right) + \dfrac{k}{hr_o^2}}\]

\[\frac{dQ}{dr_o} = 0 \qquad \Longrightarrow \quad \frac{d}{d r_o}\left[\left(\dfrac{r_o-r_i}{r_or_i}\right) + \dfrac{k}{hr_o^2} \right] = \frac{d}{d r_o}\left[\left(\dfrac{1}{r_i}-\dfrac{1}{r_o}\right) + \dfrac{k}{hr_o^2} \right]\] Equating the above to zero (at \(r_o=r_{oc}\)), gives \[\frac{1}{r_{oc}^2} - \frac{2k}{hr_{oc}^3} = 0 \qquad \Longrightarrow \quad \boxed{r_{oc} = \frac{2k}{h}}\]