Question

The 35-cm-thick roof of a large room made of concrete $\left(k=0.79 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=5.88 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)$ is initially at a uniform temperature of \(15^{\circ} \mathrm{C}\). After a heavy snowstorm, the outer surface of the roof remains covered with snow at \(-5^{\circ} \mathrm{C}\). The roof temperature at \(12 \mathrm{~cm}\) distance from the outer surface after $2 \mathrm{~h}$ is (a) \(13^{\circ} \mathrm{C}\) (b) \(11^{\circ} \mathrm{C}\) (c) \(7^{\circ} \mathrm{C}\) (d) \(3^{\circ} \mathrm{C}\) (e) \(-5^{\circ} \mathrm{C}\)

Short answer

Answer: (d) 3 °C

Step-by-step solution

1. Convert all units to SI system

To solve the problem, we need to convert all units to the SI system before we start the calculations. Thickness of the roof = 35 cm = 0.35 m Distance from the outer surface = 12 cm = 0.12 m Temperature Initial = 15°C = 288 K Temperature outer surface = -5°C = 268 K Time = 2 hours = 7200 seconds

2. Heat diffusion equation

To find the temperature T(x,t), we will use the heat diffusion equation: T(x,t) = Ti + (T0 - Ti)(1 - erf(x/(2*√(α*t)))) where Ti = initial uniform temperature (288 K) T0 = outer surface temperature (268 K) α = thermal diffusivity of concrete (5.88 × 10^(-7) m^2/s) erf is the error function.

3. Calculate erf(x/(2*√(α*t)))

First, we calculate x/(2*√(α*t)) = 0.12/(2*√(5.88 × 10^(-7) * 7200)) ≈ 0.251 Now, we need to find the error function value erf(0.251). We can find this value using a numerical calculator or a table of values for the error function. After calculating, we get: erf(0.251) ≈ 0.2234

4. Calculate the temperature using the heat diffusion equation

Now, we plug in the values into the heat diffusion equation: T(x,t) = 288 + (268-288)*(1 - 0.2234) T(x,t) ≈ 288 - 20 * 0.7766 T(x,t) ≈ 277.47 K

5. Convert the final temperature back to Celsius and choose the answer

Now, we convert the final temperature back to Celsius: T(x,t) = 277.47 K - 273.15 = 4.32 °C Based on the options given, the closest value to 4.32 °C is 3 °C. Therefore, the answer is: (d) 3 °C