Question

A plane wall of thickness $0.1\text{ m}$ and thermal conductivity $50\text{ W/m}\cdot\text{K}$ has a surface area of $2\text{ m}^2$. The temperature difference across the wall is $100\text{ K}$. What is the rate of heat transfer through the wall?

• A. $200\text{ kW}$
• B. $100\text{ kW}$
• C. $50\text{ kW}$
• D. $25\text{ kW}$

Hint:

Be careful with units when performing calculations.
Ensure the final answer is converted from watts ($\text{W}$) to kilowatts ($\text{kW}$) by dividing by 1,000 as required by the options.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This problem involves calculating steady-state, one-dimensional heat conduction through a plane wall with known dimensions, thermal conductivity, and boundary temperatures.

Step 2: Key Formula or Approach:
Fourier's Law of Heat Conduction for a plane wall is expressed as:
\[ Q = \frac{k A \Delta T}{L} \] Where:
$Q$ is the rate of heat transfer (W).
$k$ is the thermal conductivity of the wall ($\text{W/m}\cdot\text{K}$).
$A$ is the heat transfer area ($\text{m}^2$).
$\Delta T$ is the temperature difference across the wall (K or $^\circ\text{C}$).
$L$ is the thickness of the wall (m).

Step 3: Detailed Explanation:

• Let's list the given parameters:
Thickness, $L = 0.1\text{ m}$
Thermal conductivity, $k = 50\text{ W/m}\cdot\text{K}$
Surface area, $A = 2\text{ m}^2$
Temperature difference, $\Delta T = 100\text{ K}$

• Substitute these values into Fourier's Law:
\[ Q = \frac{50 \times 2 \times 100}{0.1} \]
• Simplify the numerator:
\[ 50 \times 2 \times 100 = 10,000\text{ W} \]
• Divide by the wall thickness:
\[ Q = \frac{10,000}{0.1} = 100,000\text{ W} \]
• Convert the heat transfer rate into kilowatts ($\text{kW}$):
\[ Q = \frac{100,000}{1,000}\text{ kW} = 100\text{ kW} \]

Step 4: Final Answer:
The rate of heat transfer through the plane wall is $100\text{ kW}$, which corresponds to option (B).