Question

A row of 3 -ft-long and 1-in-diameter used uranium fuel rods that are still radioactive are buried in the ground parallel to each other with a center-to- center distance of 8 in at a depth of \(15 \mathrm{ft}\) from the ground surface at a location where the thermal conductivity of the soil is \(0.6 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\). If the surface temperature of the rods and the ground are \(350^{\circ} \mathrm{F}\) and \(60^{\circ} \mathrm{F}\), respectively, determine the rate of heat transfer from the fuel rods to the atmosphere through the soil.

Short answer

Answer: The rate of heat transfer from the fuel rods to the atmosphere through the soil is approximately 9.42 Btu/h.

Step-by-step solution

Identify known values

Here are the known values that we have: - Length of the fuel rods: 3 ft - Diameter of the fuel rods: 1 in (0.0833 ft) - Depth from the ground surface: 15 ft - Thermal conductivity (k) of the soil: 0.6 Btu/h·ft·F - Surface temperature of the rods (T1): 350°F - Ground temperature (T2): 60°F

Calculate the temperature difference between the fuel rods and the ground

We have to find the temperature difference between the surface of the rods and the ground. The calculation is as follows: ΔT = T1 - T2 ΔT = 350°F - 60°F ΔT = 290°F

Calculate the surface area of the fuel rods

Now we need to calculate the surface area of the fuel rods that are in contact with the soil for the heat transfer. The calculation is as follows: Surface Area = π × D × L Surface Area = π × 0.0833 ft × 3 ft Surface Area ≈ 0.78 ft²

Calculate the rate of heat transfer

Now we can calculate the rate of heat transfer from the fuel rods to the atmosphere through the soil. We will use the following formula: q = k × (Surface Area) × (ΔT / Distance) q = 0.6 Btu/h·ft·F × 0.78 ft² × (290°F / 15 ft) q ≈ 9.42 Btu/h Thus, the rate of heat transfer from the fuel rods to the atmosphere through the soil is approximately 9.42 Btu/h.

Key concepts

Thermal Conductivity

Thermal conductivity is a fundamental property that measures a material's ability to conduct heat. It's often denoted by the symbol \( k \) and is generally expressed in units of \( \text{Btu/h}\cdot\text{ft}\cdot{}^\circ\text{F} \). This property tells us how much heat will be transferred through a material per unit time for a given temperature difference per unit thickness.
For example, in our fuel rods exercise, the thermal conductivity of the soil is given as \( 0.6 \text{ Btu/h} \cdot \text{ft} \cdot {}^\circ F \). This indicates that for every one degree Fahrenheit temperature difference across one foot of soil, 0.6 Btu of heat energy is transferred in one hour.
Understanding thermal conductivity is essential when dealing with heat transfer in materials because it influences how fast or slow heat moves through a substance. A higher thermal conductivity means that the material is better at conducting heat, which is important in applications where heat dissipation is crucial.

Temperature Difference

The temperature difference, often represented by \( \Delta T \), is a principal factor in determining the rate of heat transfer between two substances or locations. It's simply the difference in temperature between one point and another, driving the flow of heat. In our exercise, the temperature difference is between the surface of the uranium fuel rods and the surface of the ground, which are \( 350^{\circ} \text{F} \) and \( 60^{\circ} \text{F} \) respectively.
The way we calculate this difference is straightforward: \( \Delta T = T_1 - T_2 \), where \( T_1 \) is the higher temperature and \( T_2 \) is the lower temperature. For our scenario, \( \Delta T = 350^{\circ} \text{F} - 60^{\circ} \text{F} = 290^{\circ} \text{F} \).
Temperature difference plays a critical role in heat transfer because it's the driving force behind the movement of heat. The larger the temperature difference, the greater the potential for heat flow, which is why calculating it accurately is so vital in any thermal analysis.

Surface Area Calculation

Calculation of surface area is a key component in determining how much heat will be transferred between a solid object and its surrounding environment. In many heat transfer situations, the surface area through which the heat is being transferred can greatly affect the rate of heat flow.
For the fuel rods, the relevant surface area is the lateral surface area in contact with the surrounding soil. Assuming the rods are cylindrical, this area is calculated using the formula for the lateral surface area: \( \text{Surface Area} = \pi \times D \times L \), where \( D \) is the diameter, and \( L \) is the length of the cylinder.
In this exercise, \( D = 0.0833 \) ft (converted from 1 inch), and \( L = 3 \) ft. Plugging these values into the formula gives \( \text{Surface Area} = \pi \times 0.0833 \times 3 \approx 0.78 \text{ ft}^2 \).
Accurate surface area calculation ensures that we correctly estimate how much heat is being transferred. The greater the surface area, the higher the rate of heat exchange, assuming all other factors are constant.

Rate of Heat Transfer

The rate of heat transfer indicates how quickly heat energy moves from one place or object to another. It's a crucial metric in thermal management and is influenced by several factors, including thermal conductivity, temperature difference, and surface area.
To calculate the rate of heat transfer, the following formula is utilized:

• \( q = k \times \text{(Surface Area)} \times \left(\frac{\Delta T}{\text{Distance}}\right) \)

Where:

• \( q \) represents the rate of heat transfer in Btu/h,
• \( k \) is the thermal conductivity of the material,
• \( \Delta T \) is the temperature difference,
• Distance refers to the path through which the heat operates.

Substituting our values gives \( q = 0.6 \times 0.78 \times \frac{290}{15} \approx 9.42 \text{ Btu/h} \).
This result means that approximately 9.42 Btu of heat is being transferred every hour from each rod to the surroundings through the soil. Understanding this rate helps engineers design systems that can adequately manage heat transfer, whether they need to minimize heat loss or enhance cooling efficiency.