Question

The temperature difference between two sides of an iron plate, $1.8\text{ cm}$ thick is $9^\circ\text{C}$. Heat is transmitted through the plate $10\text{ kcal/s}\cdot\text{m}^2$ at steady state. The thermal conductivity of iron is

• A. $0.02\text{ kcal/m}\cdot\text{s}\cdot^\circ\text{C}$
• B. $0.04\text{ kcal/m}\cdot\text{s}\cdot^\circ\text{C}$
• C. $0.05\text{ kcal/m}\cdot\text{s}\cdot^\circ\text{C}$
• D. $0.004\text{ kcal/m}\cdot\text{s}\cdot^\circ\text{C}$

Hint:

Always convert all distance units to meters immediately before solving thermal equations. Leaving the plate thickness as $1.8\text{ cm}$ without multiplying by $10^{-2}$ is the most common error made under time pressure, causing answers to be off by a factor of 100.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem concerns steady-state heat conduction through a uniform iron plate.
We are given the physical thickness of the barrier ($d = 1.8\text{ cm} = 1.8 \times 10^{-2}\text{ m}$), the temperature gradient difference ($\Delta \theta = 9^\circ\text{C}$), and the steady-state heat current density or heat flux per unit surface area ($\frac{Q}{A \cdot t} = 10\text{ kcal/s}\cdot\text{m}^2$). We need to determine the thermal conductivity constant ($k$) of the metal.

Step 2: Key Formula or Approach:
Fourier's Law of Heat Conduction states that the rate of heat flow through a material is given by:
$$\frac{Q}{t} = \frac{k A \Delta \theta}{d}$$ Rearranging this relationship to isolate the heat flux per unit area yields:
$$\frac{Q}{A \cdot t} = \frac{k \Delta \theta}{d} \implies k = \left(\frac{Q}{A \cdot t}\right) \times \frac{d}{\Delta \theta}$$

Step 3: Detailed Explanation:
Let's substitute our known numeric parameters directly into the rearranged Fourier equation:
$$10 = \frac{k \times 9}{1.8 \times 10^{-2}}$$ Isolate the thermal conductivity constant $k$:
$$k = \frac{10 \times 1.8 \times 10^{-2}}{9}$$ Simplify the values in the numerator:
$$10 \times 1.8 = 18$$ Substitute this back into the equation:
$$k = \frac{18 \times 10^{-2}}{9}$$ Divide 18 by 9:
$$k = 2 \times 10^{-2} = 0.02\text{ kcal/m}\cdot\text{s}\cdot^\circ\text{C}$$

Step 4: Final Answer:
The thermal conductivity of iron is $0.02\text{ kcal/m}\cdot\text{s}\cdot^\circ\text{C}$, matching option (A).