Question

A thick spherical shell of inner and outer radii r and R respectively has thermal conductivity $k = \frac{\rho}{x^n}$, where $\rho$ 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

For steady-state heat conduction in a spherical shell, the rate of heat flow $H$ is given by $H = -k(4\pi x^2) \frac{dT}{dx}$. Substituting the given thermal conductivity $k = \frac{\rho}{x^n}$, we get $H = - \frac{\rho}{x^n} (4\pi x^2) \frac{dT}{dx}$. Rearranging for the temperature gradient, $\frac{dT}{dx} = -\frac{H}{4\pi\rho} x^{n-2}$. For the temperature gradient to be constant, the exponent of $x$ must be zero. Thus, $n-2 = 0$, which gives $n=2$.