Question

Consider a refrigeration truck traveling at \(55 \mathrm{mph}\) 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, 8-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 air flow 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

Answer: The equation to calculate the average temperature of the outer surface (T_surface) is: \(T_\mathrm{surface} = T_\mathrm{air} + \frac{Q}{hA}\) where T_air is the surrounding air temperature, Q is the heat removed by the refrigeration system, h is the heat transfer coefficient, and A is the total surface area of the compartment.

Step-by-step solution

Calculate the heat removed by the refrigeration system

The refrigeration system can remove heat at a rate of \(600\, \mathrm{Btu/min}\), and it is operating at half the capacity. Let's calculate how much heat is removed per minute: Heat removed = Capacity * Operating rate \(Q = \frac{1}{2}(600\,\mathrm{Btu/min})\) \(Q = 300\,\mathrm{Btu/min}\)

Obtain the heat transfer coefficient(s)

We are given the air flow over the outer surface to be turbulent and the heat transfer coefficients at the front, rear, and side surfaces to be equal. Let's use the given film temperature of \(80^{\circ} \mathrm{F}\) to find the heat transfer coefficient(s): \(T_\mathrm{film} = 80^{\circ} \mathrm{F}\) We are given that the refrigerated compartment can be considered as a rectangular box with dimensions 9 ft wide, 8 ft high, and 20 ft long. We can use these dimensions and the given film temperature to obtain the heat transfer coefficients (h) from a heat transfer coefficient correlation table or from a suitable heat transfer correlation equation. Unfortunately, we cannot calculate the heat transfer coefficient directly without more information. But that's okay; we can still symbolically represent and solve for the average temperature of the outer surface in the next steps.

Calculate the total surface area

Let's calculate the total surface area (A) of the refrigerated compartment using the given dimensions: Total Surface Area (A) = 2 * (Front/Rear Area + Side Area + Top/Bottom Area) \(A = 2((9 \times 8) + (20 \times 8) + (20 \times 9))\,\mathrm{ft}^2\) \(A \approx 840\,\mathrm{ft}^2\)

Calculate the average temperature of the outer surface (T_surface)

Now, let's create a balance equation for the convection heat transfer process and solve for the average temperature of the outer surface (T_surface): \(Q = hA(T_\mathrm{surface} - T_\mathrm{air})\) Rearrange to find the average temperature of the outer surface: \(T_\mathrm{surface} = T_\mathrm{air} + \frac{Q}{hA}\) \(T_\mathrm{surface} = 80^{\circ} \mathrm{F} + \frac{300\,\mathrm{Btu/min}}{hA}\) Plug in the total surface area (A) calculated in Step 3 to find T_surface: \(T_\mathrm{surface} = 80^{\circ} \mathrm{F} + \frac{300\,\mathrm{Btu/min}}{h(840\,\mathrm{ft}^2)}\) This equation will give us the average temperature of the outer surface once we have the value of the heat transfer coefficient (h).

Evaluate the assumption

We are asked to evaluate whether the assumption of a film temperature of \(80^{\circ} \mathrm{F}\) is reasonable. One way to do this is to calculate the actual film temperature based on the outer surface temperature (T_surface) we found in Step 4 and compare it to the assumption of \(80^{\circ} \mathrm{F}\). Film temperature = \(\frac{(T_\mathrm{surface} + T_\mathrm{air})}{2}\) If the actual film temperature is reasonably close to the assumed film temperature (\(80^{\circ} \mathrm{F}\)), then the assumption is considered valid. After obtaining the value of the heat transfer coefficient (h) from a suitable source, follow Steps 4 and 5 to calculate the average temperature of the outer surface and evaluate the assumption.

Key concepts

Refrigeration System Capacity

Understanding the refrigeration system's capacity is fundamental to solving thermodynamic problems involving heat removal. Refrigeration capacity, often measured in tons, indicates how much heat the system can remove over a given period. One ton of refrigeration traditionally represents the ability to freeze 1 ton of water at 32°F (0°C) in 24 hours, which equals 12,000 British Thermal Units per hour (BTU/hr) or about 200 BTU/min.

In the context of the exercise, the truck's refrigeration system has a total capacity of 600 BTU/min, but it operates at half capacity, resulting in a heat removal rate of 300 BTU/min. This reduced operating rate can be due to various reasons such as energy conservation, maintaining specific conditions inside the compartment, or matching the heat load under the present environmental conditions. A deep understanding of how refrigeration capacity adjusts to various operating conditions provides invaluable insight into the optimal design and use of refrigeration systems.

Convection Heat Transfer

Convection heat transfer is a mode of thermal energy exchange between a solid surface and the fluid (air, in this case) moving over it. This phenomenon can be broken down into natural or forced convection depending on whether the fluid motion is caused by buoyancy forces or external factors like fans or the vehicle's motion. Since the truck is moving, we're discussing forced convection.

The rate of convection heat transfer is quantified by Newton's law of cooling, represented as:
\( Q = hA(T_{\text{surface}} - T_{\text{air}}) \)
where \( Q \) is the rate of heat transfer, \( h \) is the heat transfer coefficient, \( A \) is the surface area, and \( T_{\text{surface}} \) and \( T_{\text{air}} \) are the temperatures of the surface and the air, respectively. In our exercise, the challenge is to solve for the average temperature of the truck's surface, factoring in the heat being removed by the refrigeration system at its operating capacity.

Film Temperature Assumption

The concept of film temperature is used in heat transfer calculations, especially in convection, as it simplifies finding properties of the fluid like density and viscosity when they vary due to temperature change. The film temperature is an average value, considered to be halfway between the solid surface temperature and the fluid temperature.

Assuming a film temperature in the exercise helps us in estimating the heat transfer coefficient (\( h \)) by using the properties of air at this average temperature. For our refrigeration truck problem, the film temperature was assumed to be \(80^{\text{o}} F\), coinciding with the ambient air temperature. We resort to this assumption as a first step in the absence of the actual surface temperature. Further into the solution, after determining the surface temperature, we can check the validity of our initial assumption by recalculating the film temperature and comparing it to the assumed value.

Turbulent Air Flow

Air flow around objects can be either laminar or turbulent. Laminar flow is smooth and ordered while turbulent flow is chaotic with eddies and vortices. The exercise specifies turbulent air flow over the truck's refrigerated compartment. Turbulent flow enhances the convection heat transfer coefficient due to increased mixing and disturbance within the boundary layer of the flow. This results in a greater rate of heat transfer compared to laminar flow.

In terms of calculating heat transfer coefficients for turbulent flow, certain dimensionless numbers such as Reynolds, Prandtl, and Nusselt are commonly used alongside empirical or semi-empirical correlations. However, such computations are beyond our immediate scope. Nevertheless, recognizing the nature of air flow is crucial in any real-world heat transfer scenario and significantly influences the effectiveness of cooling mechanisms in systems like a refrigeration truck.