Question

Hollow cylinder of k = $\frac{\pi}{10}$ W/mK Find temperature gradients in SI units.

Answer 1

At the inner surface ($r=0.1$ m): $\frac{-1000}{\ln(2)}$ K/m; At the outer surface ($r=0.2$ m): $\frac{-500}{\ln(2)}$ K/m

Answer 2

The temperature gradient in a hollow cylinder under steady-state heat conduction varies radially. It is given by $\frac{dT}{dr} = \frac{T_2 - T_1}{r \ln(r_2/r_1)}$. Substituting the given temperatures and radii ($T_1=100^\circ$C, $T_2=0^\circ$C, $r_1=0.1$ m, $r_2=0.2$ m), the gradient is $\frac{-100 \text{ K}}{r \ln(2)}$. The plural "gradients" suggests reporting values at different radii, typically the inner and outer surfaces. At the inner radius ($r_1=0.1$ m), the gradient is $\frac{-1000}{\ln(2)}$ K/m. At the outer radius ($r_2=0.2$ m), the gradient is $\frac{-500}{\ln(2)}$ K/m.