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: List the data.
Thickness $d = 1.8\,\text{cm} = 1.8\times10^{-2}\,\text{m}$, temperature difference $\Delta\theta = 9^\circ\text{C}$, and heat flux per unit area $\dfrac{Q}{At} = 10\,\text{kcal/s}\cdot\text{m}^2$.
Step 2: State Fourier's law.
\[ \frac{Q}{t} = \frac{kA\Delta\theta}{d}. \]
Step 3: Divide by area.
\[ \frac{Q}{At} = \frac{k\Delta\theta}{d}. \]
Step 4: Rearrange for $k$.
\[ k = \frac{Q}{At}\cdot\frac{d}{\Delta\theta}. \]
Step 5: Substitute the numbers.
\[ k = 10\times\frac{1.8\times10^{-2}}{9} = \frac{18\times10^{-2}}{9}. \]
Step 6: Simplify.
\[ k = 2\times10^{-2} = 0.02\,\text{kcal/m}\cdot\text{s}\cdot^\circ\text{C}. \] That is option (A). \[ \boxed{k = 0.02\,\text{kcal/m}\cdot\text{s}\cdot{}^\circ\text{C}} \]