Question

The walls of a food storage facility are made of a 2 -cm-thick layer of wood \((k=0.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) in contact with a 5 -cm- thick layer of polyurethane foam $(k=0.03 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\(. If the temperature of the surface of the wood is \)-10^{\circ} \mathrm{C}$ and the temperature of the surface of the polyurethane foam is \(20^{\circ} \mathrm{C}\), the temperature of the surface where the two layers are in contact is (a) \(-7^{\circ} \mathrm{C}\) (b) \(-2^{\circ} \mathrm{C}\) (c) \(3^{\circ} \mathrm{C}\) (d) \(8^{\circ} \mathrm{C}\) (e) \(11^{\circ} \mathrm{C}\)

Short answer

Answer: (c) 3°C

Step-by-step solution

Find the thermal resistance of each layer.

To find the thermal resistance for each layer, we need their thickness, thermal conductivity, and surface area (assumed equal for both layers). The formula for thermal resistance is: \(R = \frac{L}{kA}\) Where \(R\) is the thermal resistance, \(L\) is the thickness, \(k\) is the thermal conductivity, and \(A\) is the surface area. Here we only need the ratio of the thermal resistances of both layers, so we don't actually need the surface area as it will cancel out later on. For the wood layer: \(R_\text{wood} = \frac{0.02}{0.1}\) For the polyurethane layer: \(R_\text{poly} = \frac{0.05}{0.03}\)

Calculate the temperature difference ratio between the two layers.

Now, since both layers are in series, we can calculate the temperature difference ratio between the two layers. The total temperature difference is: \(\Delta T_\text{total} = T_\text{wood} - T_\text{poly} = -10°C - 20°C = -30°C\) The temperature difference ratio between the two layers can be calculated using their thermal resistances: \(r = \frac{R_\text{wood}}{R_\text{wood} + R_\text{poly}}\) \(r = \frac{\frac{0.02}{0.1}}{\frac{0.02}{0.1} + \frac{0.05}{0.03}}\)

Calculate temperature at the surface where the two layers are in contact.

Now, we can calculate the temperature difference between the wood surface and the surface where the two layers are in contact: \(\Delta T_\text{wood} = r * \Delta T_\text{total}\) \(\Delta T_\text{wood} = \frac{\frac{0.02}{0.1}}{\frac{0.02}{0.1} + \frac{0.05}{0.03}} * (-30°C)\) Finally, we determine the temperature at the surface where the two layers are in contact: \(T_\text{contact} = T_\text{wood} + \Delta T_\text{wood}\) \(T_\text{contact} = -10°C + \Delta T_\text{wood}\) Calculating this expression gives us the contact temperature \(T_\text{contact} \approx 3°C\). Therefore, the correct option is: (c) \(3^{\circ} \mathrm{C}\)