Question

A thick spherical shell of inner and outer radii r and R respectively has thermal conductivity $k = \frac{p}{x^n}$, where p is a constant and x is distance from the center of the shell. The inner and outer walls are maintained at temperature $T_1$ and $T_2 (<T_1)$. The value of number n (call it $n_0$) for which the temperature gradient remains constant throughout the thickness of the shell will be:-

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

2

Answer 2

The rate of heat flow $Q$ through a spherical shell is given by $Q = -k(x) A(x) \frac{dT}{dx}$. Given $k(x) = \frac{p}{x^n}$ and $A(x) = 4\pi x^2$. For steady-state heat flow, $Q$ is constant. So, $Q = -\left(\frac{p}{x^n}\right) (4\pi x^2) \frac{dT}{dx}$. Rearranging for the temperature gradient: $\frac{dT}{dx} = -\frac{Q}{4\pi p} x^{n-2}$. For the temperature gradient to be constant, the exponent of $x$ must be zero, i.e., $n-2 = 0$, which gives $n=2$.