Question

Consider a refrigeration truck traveling at \(70 \mathrm{mph}\) at a location where the air temperature is \(80^{\circ} \mathrm{F}\). The refrigerated compartment of the truck can be considered to be a 9 -ft-wide, 7 -ft-high, and 20 -ft-long rectangular box. The refrigeration system of the truck can provide 3 tons of refrigeration (i.e., it can remove heat at a rate of $600 \mathrm{Btu} / \mathrm{min}$ ). The outer surface of the truck is coated with a low-emissivity material, and thus radiation heat transfer is very small. Determine the average temperature of the outer surface of the refrigeration compartment of the truck if the refrigeration system is observed to be operating at half the capacity. Assume the airflow over the entire outer surface to be turbulent and the heat transfer coefficient at the front and rear surfaces to be equal to that on side surfaces. For air properties evaluations, assume a film temperature of \(80^{\circ} \mathrm{F}\). Is this a good assumption?

Short answer

Question: Using the given heat transfer rate, determine if the assumption of a film temperature of 80°F is reasonable. Answer: Given that the air temperature and the assumed film temperature are equal, we could presume that the surface temperature of the box should be mildly affected by the cooling system's efficiency. This suggests that the assumption of a film temperature of 80°F is reasonable.

Step-by-step solution

Calculate heat transfer rate

Since the refrigeration system is operating at half capacity, we have a heat transfer rate of half the provided value, which is \(3 \mathrm{tons}\) of refrigeration (1 ton of refrigeration is about 12000 Btu/h). Therefore, the heat transfer rate \(Q\) is: \(Q = \frac{1}{2} \times 3 \mathrm{tons} \times 12000 \frac{\mathrm{Btu}}{\mathrm{h}} = 18000 \frac{\mathrm{Btu}}{\mathrm{h}}\)

Calculate the surface area of the compartment

Given the dimensions of the compartment, we can calculate the surface area of the box \(A\): \(A = 2 \times (9 \times 7 + 9 \times 20 + 7 \times 20) \ \mathrm{ft^2}\)

Calculate the heat transfer coefficient using Newton's Law of Cooling

The overall heat transfer \(Q\) from the surface of the box to the air can be described by Newton's Law of Cooling: \( Q = h A (T_s - T_\infty) \). Here, \(h\) is the overall heat transfer coefficient, \(T_s\) is the average surface temperature of the compartment, and \(T_\infty\) is the air temperature surrounding the compartment (which is \(80^{\circ} \mathrm{F}\)). From this equation, we should isolate h. But first, we need to convert heat transfer rate \(Q\) to be in the same units (\(\mathrm{Btu/ft^2h}\)): \(Q = 18000 \frac{\mathrm{Btu}}{\mathrm{h}} \div A \ \mathrm{ft^2}\) Then, we can get \(h\): \(h = \frac{Q}{A (T_s - T_\infty)}\)

Determine film temperature and verify the assumption

The film temperature \(T_f\) is the average of the surface temperature \(T_s\) and the ambient air temperature \(T_\infty\): \(T_f = \frac{T_s + T_\infty}{2}\) We are asked to check if the assumption of a film temperature of \(80^{\circ} \mathrm{F}\) is valid. Since we don't have the value of \(T_s\) yet, we cannot accurately determine the film temperature. However, given that the air temperature and the assumed film temperature are equal, we could presume that the surface temperature of the box should be mildly affected by the cooling system's efficiency. This suggests that the assumption of a film temperature of \(80^{\circ} \mathrm{F}\) is reasonable. Note that this solution does not give an exact value for the average temperature of the outer surface of the refrigeration compartment, as it requires more information, such as the exact heat transfer coefficient and the surface temperature. The solution provides a method to determine those values if the necessary data is available.