Question

While driving down a highway early in the evening, the airflow over an automobile establishes an overall heat transfer coefficient of $25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The passenger cabin of this automobile exposes \(8 \mathrm{~m}^{2}\) of surface to the moving ambient air. On a day when the ambient temperature is \(33^{\circ} \mathrm{C}\), how much cooling must the air-conditioning system supply to maintain a temperature of $20^{\circ} \mathrm{C}$ in the passenger cabin? (a) \(0.65 \mathrm{MW}\) (b) \(1.4 \mathrm{MW}\) (c) \(2.6 \mathrm{MW}\) (d) \(3.5 \mathrm{MW}\) (e) \(0.94 \mathrm{MW}\)

Short answer

Based on the given information and calculations, the cooling required by the air-conditioning system to maintain a temperature of \(20^{\circ} \mathrm{C}\) inside the passenger cabin is approximately \(2.6 \mathrm{kW}\) (or \(0.0026 \mathrm{MW}\), which is closest to option (c) \(2.6 \mathrm{MW}\)).

Step-by-step solution

Convert temperatures to Kelvin

First, we need to convert the given temperatures from Celsius to Kelvin. To do this, we need to add 273.15 to each temperature: \(T_a = 33^{\circ} \mathrm{C} + 273.15 = 306.15 \mathrm{K}\) \(T_c = 20^{\circ} \mathrm{C} + 273.15 = 293.15 \mathrm{K}\)

Calculate the cooling required (Q)

Now that we have the temperatures in Kelvin, we can use the heat transfer formula to calculate the cooling required by the air-conditioning system: \(Q = hA(T_a - T_c)\) Where \(h = 25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), \(A = 8 \mathrm{~m}^{2}\), \(T_a = 306.15 \mathrm{K}\), and \(T_c = 293.15 \mathrm{K}\). \(Q = 25 \cdot 8(306.15 - 293.15)\)

Evaluate and find the answer

Now we can evaluate the expression to find the cooling required. \(Q = 25 \cdot 8 \cdot 13 = 2600 \mathrm{W} = 2.6 \mathrm{kW}\) From the given options, we see that \(2.6 \mathrm{kW}\) is closest to (c) \(2.6 \mathrm{MW}\), but we need to adjust the unit. Note that \(2.6 \mathrm{kW} = 0.0026 \mathrm{MW}\), which isn't one of the provided options. There might be a typographical error in the options, so we will go with the closest option which is (c) \(2.6 \mathrm{MW}\).