Question

While driving down a highway early in the evening, the air flow over an automobile establishes an overall heat transfer coefficient of \(18 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The passenger cabin of this automobile exposes \(9 \mathrm{~m}^{2}\) 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) \(670 \mathrm{~W}\) (b) \(1284 \mathrm{~W}\) (c) \(2106 \mathrm{~W}\) (d) \(2565 \mathrm{~W}\) (e) \(3210 \mathrm{~W}\)

Short answer

Answer: The required cooling capacity is 2106 W.

Step-by-step solution

Calculate the temperature difference

To find the temperature difference between the cabin and the outside air, subtract the cabin temperature from the ambient temperature: \(\Delta T = T_{ambient} - T_{cabin} = 33^{\circ} \mathrm{C} - 20^{\circ} \mathrm{C} = 13 \mathrm{~K}\) (NOTE: We use Kelvins(K) for temperature differences in the formula.)

Calculate the heat transfer rate

Now that we have the temperature difference, we can use the convective heat transfer equation to find the cooling capacity required: \(Q = hA\Delta T\) where \(h = 18 \frac{\mathrm{W}}{\mathrm{m}^{2} \cdot \mathrm{K}}\), \(A = 9 \mathrm{~m}^{2}\), and \(\Delta T = 13 \mathrm{~K}\). Plugging in the given values, we get: \(Q = 18 \frac{\mathrm{W}}{\mathrm{m}^{2} \cdot \mathrm{K}} \cdot 9 \mathrm{~m}^{2} \cdot 13 \mathrm{~K} = 2106 \mathrm{~W}\) The cooling capacity required for the air conditioning system to maintain the temperature at \(20^{\circ} \mathrm{C}\) in the passenger cabin is \(2106 \mathrm{~W}\), which corresponds to the choice (c) in the possible answers.

Key concepts

Understanding Heat Transfer Coefficient

The heat transfer coefficient is a measure of a material's ability to allow heat to pass through it. In the context of our exercise, it represents the effectiveness of the air flow over the car in transferring heat away from the vehicle's passenger cabin. The larger the heat transfer coefficient, the more efficient the heat transfer process becomes. An understanding of this concept is critical for engineers and designers as it influences various decisions, from the selection of materials to the design of heating and cooling systems.

As stated in the problem, the heat transfer coefficient (\( h \) ) for the air flow over the automobile is given as 18 W/m²·K. This indicates that for each square meter of the car’s surface and for each degree Kelvin of temperature difference between the surface and the ambient air, 18 Watts of heat is transferred. It is important to note that the heat transfer coefficient is dependent on the properties of the fluid (air in this case), the flow conditions, and the surface geometry and properties of the material in question.

Temperature Difference in Heat Transfer

The temperature difference, often denoted as ∆T, is the driving force behind heat transfer by convection. It represents the difference in temperature between the surface of interest and the surrounding fluid; in this case, the car cabin and the ambient air. The greater the difference in temperature, the greater the heat transfer rate.

To calculate this, we subtract the interior temperature from the ambient temperature, which gives us a temperature gradient. If this temperature gradient did not exist, there would be no heat flow. In our exercise, the ambient air is warmer at 33°C while the desired cabin temperature is 20°C, resulting in a temperature difference of 13K (note that temperatures in such calculations are often converted to Kelvin to maintain consistency in units). Understanding this temperature difference is essential for designing thermal systems, such as the car's air conditioning system, to ensure adequate control over the environment.

Cooling Capacity: Maintaining Comfort

Cooling capacity is the amount of heat energy that an air conditioning system must remove to maintain a certain temperature level in a given space. In thermal management, this is a critical value as it determines the power that the system must have to ensure comfortable living or working conditions. The cooling capacity is often expressed in Watts (W) or British Thermal Units per hour (BTU/hr).

In our example, the air conditioning system must supply significant cooling capacity to counteract the heat transferred by convection through the car cabin’s walls. Using the heat transfer coefficient, surface area, and temperature difference, we calculated the necessary cooling capacity as 2106W. This tells us how powerful the air conditioning system needs to be to maintain the cabin temperature at a comfortable 20°C despite the higher outside temperature. Consequently, this information is pivotal for HVAC engineers when sizing and installing air conditioning systems.