Question

A solid sphere has characteristic length $L_c = 0.02\text{ m}$, thermal conductivity $K = 40\text{ W/mK}$, and convective heat transfer coefficient $h = 10\text{ W/m}^2\text{K}$. Find the correct statement.

• A. Lumped system valid
• B. Lumped system not valid
• C. Boundary layer negligible
• D. Steady state condition

Hint:

A low Biot number ($Bi < 0.1$) signifies high thermal conductivity and/or a small size, allowing heat to distribute rapidly throughout the solid, maintaining spatial temperature uniformity.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This heat transfer question asks us to evaluate the validity of the lumped heat capacity method for transient heat conduction in a solid sphere under the given conditions.

Step 2: Key Formula or Approach:
The lumped parameter analysis assumes the temperature within the solid is spatially uniform at any instant during transient heat transfer.
The criterion for the validity of lumped system analysis is based on the Biot number ($Bi$):
\[ Bi = \frac{h \cdot L_c}{K} \]
The lumped system approximation is considered valid if:
\[ Bi < 0.1 \]

Step 3: Detailed Explanation:

• Identify the given parameter values:
- Characteristic length, $L_c = 0.02 \text{ m}$
- Thermal conductivity of the solid, $K = 40 \text{ W/(m K)}$
- Convective heat transfer coefficient, $h = 10 \text{ W/(m}^2\text{K)}$

• Calculate the Biot number ($Bi$):
\[ Bi = \frac{h \cdot L_c}{K} \]
\[ Bi = \frac{10 \times 0.02}{40} = \frac{0.2}{40} = 0.005 \]

• Compare the computed Biot number with the limit of $0.1$:
\[ 0.005 < 0.1 \]

• Since the Biot number is much less than $0.1$, the resistance to heat conduction within the solid is negligible compared to the resistance to heat convection at the surface.

• Therefore, the temperature within the sphere can be assumed uniform, making the lumped parameter system model valid.

Step 4: Final Answer:
The lumped system is valid.