A central hole of radius r_1 is created in a uniform circular disc...

Consider a cylindrical shell of radius $r and thickness$dr$within the disc.<br><br>The surface area of this shell through which heat flows radially is$A = 2\pi r d$.<br><br>The thermal resistance of this elemental shell is given by$dR = \dfrac{dr}{KA} = \dfrac{dr}{K(2\pi r d)}$.<br><br>The total thermal resistance of the disc is obtained by integrating from$r = r_1$to$r = r_2$:<br><br>$R = \int_{r_1}^{r_2} \dfrac{dr}{2\pi K r d} = \dfrac{1}{2\pi K d} \ln \left(\dfrac{r_2}{r_1}\right)

The rate of heat flow is given by $H =$\dfrac{\Delta \theta}{R}, where $\Delta \theta = \theta_1 - \theta_2$.

$H =$\dfrac{\theta_1 - \theta_2}{\dfrac{1}{2\pi K d} \ln\left(\dfrac{r_2}{r_1}\right)} = \dfrac{2\pi K d (\theta_1 - \theta_2)}{\ln(r_2 / r_1)}