Question

Two thin metallic spherical shells of radii \(r_1\) and \(r_2\) (\(r_1<r_2\)) are placed with their centres coinciding. A material of thermal conductivity K is filled in the space between the shells. The inner shell is maintained at temperature \(\theta_1\) and the outer shell at temperature \(\theta_2\) (\(\theta_1<\theta_2\)). The rate at which heat flows radially through the material is :

• A. \(\frac{4\pi K r_1 r_2 (\theta_2 - \theta_1)}{r_2 - r_1}\)
• B. \(\frac{K(\theta_2 - \theta_1)(r_2 - r_1)}{4\pi r_1 r_2}\)
• C. \(\frac{K(\theta_2 - \theta_1)}{r_2 - r_1}\)
• D. \(\frac{\pi r_1 r_2 (\theta_2 - \theta_1)}{r_2 - r_1}\)

Hint:

Thermal resistance formulas to remember:
Slab: \(R = \frac{L}{KA}\)
Cylinder: \(R = \frac{\ln(r_2/r_1)}{2\pi KL}\)
Sphere: \(R = \frac{r_2 - r_1}{4\pi K r_1 r_2}\)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Heat flow in a spherical shell is radial. To calculate the rate of heat flow, we can use the concept of thermal resistance, which is analogous to electrical resistance.
Step 2: Key Formula or Approach:
The rate of heat flow is given by Fourier's Law: \(H = \frac{\Delta \theta}{R_{th}}\).
For a thin spherical shell of radius \(x\) and thickness \(dx\), the thermal resistance \(dR_{th}\) is:
\[ dR_{th} = \frac{dx}{K \cdot A} = \frac{dx}{K(4\pi x^2)} \]
Step 3: Detailed Explanation:
To find the total thermal resistance (\(R_{th}\)) between radii \(r_1\) and \(r_2\), we integrate \(dR_{th}\):
\[ R_{th} = \int_{r_1}^{r_2} \frac{dx}{4\pi K x^2} = \frac{1}{4\pi K} \left[ -\frac{1}{x} \right]_{r_1}^{r_2} \]
\[ R_{th} = \frac{1}{4\pi K} \left( \frac{1}{r_1} - \frac{1}{r_2} \right) = \frac{1}{4\pi K} \left( \frac{r_2 - r_1}{r_1 r_2} \right) \]
Now, calculate the heat flow rate \(H\):
\[ H = \frac{\theta_2 - \theta_1}{R_{th}} \]
Substitute the value of \(R_{th}\):
\[ H = \frac{\theta_2 - \theta_1}{\frac{r_2 - r_1}{4\pi K r_1 r_2}} \]
\[ H = \frac{4\pi K r_1 r_2 (\theta_2 - \theta_1)}{r_2 - r_1} \]
Step 4: Final Answer:
The rate of heat flow is \(\frac{4\pi K r_1 r_2 (\theta_2 - \theta_1)}{r_2 - r_1}\).