Question

A cylindrical fuel rod of \(2 \mathrm{~cm}\) in diameter is encased in a concentric tube and cooled by water. The fuel generates heat uniformly at a rate of \(150 \mathrm{MW} / \mathrm{m}^{3}\). The convection heat transfer coefficient on the fuel rod is \(5000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and the average temperature of the cooling water, sufficiently far from the fuel rod, is \(70^{\circ} \mathrm{C}\). Determine the surface temperature of the fuel rod and discuss whether the value of the given convection heat transfer coefficient on the fuel rod is reasonable.

Short answer

Answer: The formula used to find the surface temperature of the fuel rod is: $$ T_{surface} = \frac{150\cdot 10^6 \mathrm{~W}/\mathrm{m}^3}{5000 \mathrm{~W}/\mathrm{m}^2 \cdot \mathrm{K} \cdot 2 \pi (0.01) h} + 70 $$ where \(T_{surface}\) is the surface temperature of the fuel rod, \(h\) is the height of the fuel rod, and 70 is the average temperature of the cooling water in degrees Celsius.

Step-by-step solution

Calculate the heat generation

The heat generation rate in the fuel rod is given as \(150 \mathrm{MW} / \mathrm{m}^{3}\). We need to convert this to watts per cubic meter: $$ Q_{generation} = 150 \cdot 10^6 \mathrm{~W} / \mathrm{m}^{3}$$

Determine the surface area of the fuel rod

The diameter of the fuel rod is given as \(2 \mathrm{cm}\). We can calculate the radius and then find the surface area (A) of the fuel rod using the formula for the surface area of a cylinder: $$ A = 2 \pi r h $$ First, convert the diameter to meters and find the radius: $$ r = \frac{d}{2} = \frac{0.02 \mathrm{m}}{2} = 0.01 \mathrm{m} $$ As the height (h) is not given, we cannot find the numerical value of the surface area. However, we can rewrite the formula in terms of height, which will be useful later. $$ A = 2 \pi (0.01) h $$

Calculate the heat transfer rate using convection

We can use Newton's law of cooling to calculate the rate of heat transfer (Q) by convection: $$ Q = hA(T_{surface} - T_{water}) $$ where \(h\) is the convection heat transfer coefficient, which is given as \(5000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), \(T_{surface}\) is the surface temperature of the fuel rod, and \(T_{water}\) is the average temperature of the cooling water, given as \(70^{\circ} \mathrm{C}\).

Set up an equation and solve for the surface temperature of the fuel rod

Since the fuel rod is being cooled by convection, the heat generated in the fuel rod should equal the heat transferred by convection: $$ Q_{generation} = Q $$ Replace Q with the convection heat transfer equation: $$ 150 \cdot 10^6 \mathrm{~W} / \mathrm{m}^{3} = 5000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K} \cdot (2 \pi (0.01) h) (T_{surface} - 70) $$ Now solve for \(T_{surface}\): $$ T_{surface} = \frac{150\cdot 10^6 \mathrm{~W}/\mathrm{m}^3}{5000 \mathrm{~W}/\mathrm{m}^2 \cdot \mathrm{K} \cdot 2 \pi (0.01) h} + 70 $$ Since we don't have the height of the fuel rod, we Cannot find the numerical value. However, the above equation can be used to find the surface temperature if the height was given.

Discuss the reasonability of the given convection heat transfer coefficient

The given convection heat transfer coefficient is \(5000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). This value is reasonable for water cooling in a high-heat flux situation, such as in nuclear reactors. However, it would also depend on factors such as the flow rate of the water, the roughness of the surfaces of the fuel rod, and the tube, and the presence of any fouling or debris that might affect the heat transfer. While we cannot provide a specific answer without more information, the given value seems reasonable given the context of the problem.

Key concepts

Convection Heat Transfer

Convection heat transfer is a mode of heat transfer that occurs between a solid surface and a fluid (like water or air) that is in motion. This process can either involve the fluid being heated by the solid or the solid being cooled by the fluid. In the context of the fuel rod, convection heat transfer is responsible for carrying the heat away from the rod into the water cooling system.

• Newton's Law of Cooling states that the rate of heat transfer is proportional to the temperature difference between the solid surface and the fluid.
• The equation used is: \[ Q = hA(T_{surface} - T_{fluid}) \]where \( Q \) is the heat transfer rate, \( h \) is the convection heat transfer coefficient, \( A \) is the surface area, \( T_{surface} \) is the temperature of the surface, and \( T_{fluid} \) is the temperature of the fluid.

In essence, the higher the temperature difference, the higher the heat transfer rate. Similarly, a larger surface area and a higher convection coefficient increase the heat transfer capability.

Fuel Rod Cooling

Fuel rods in nuclear reactors generate substantial amounts of heat. Therefore, efficient and effective cooling methods are critical to maintaining safe and stable operations. The cooling process typically involves immersing the fuel rod in water to take advantage of convective heat transfer.

• The water, acting as a coolant, flows around the fuel rod and absorbs the generated heat.
• By circulating the water, the system effectively dissipates the heat from the rod, preventing it from overheating.
• Maintaining an adequate flow rate of water is crucial, as it dictates how well the heat is carried away from the fuel rod.

By efficiently cooling, the water helps keep the fuel rod's temperature within safe limits, protecting the structural integrity of the rod and ensuring the efficient generation of energy.

Surface Temperature Calculation

Knowing how to calculate the surface temperature of a cylindrical object, such as a fuel rod, is important for evaluating thermal safety and efficiency. The calculation involves understanding the balance between heat generated and the heat that's transferred away.

• First, you need to quantify the heat generated within the rod: \[ Q_{generation} = 150 imes 10^6 ext{ W/m}^3 \]
• The heat transfer rate by convection is then determined through the equation derived from Newton's Law of Cooling. This involves the heat transfer coefficient, surface area, and temperature difference between the rod and coolant.
• The equation for the surface temperature is:\[ T_{surface} = \frac{Q_{generation}}{hA} + T_{water} \]

This relationship reveals how much the surface temperature of the rod exceeds the coolant temperature, given the conditions and parameters involved.

Convection Coefficient Reasonability

The convection coefficient, \( h \), is a measure of how effectively heat is transferred between a solid surface and a fluid. In this exercise, the convection coefficient is given as \( 5000 \text{ W/m}^2 \cdot \text{K} \). Let's evaluate its reasonability.

• This value is typical for situations involving high heat flux, such as those found in nuclear reactors, where efficient cooling is crucial.
• The coefficient depends on various factors including:
  • Flow rate and turbulence of the cooling water
  • Surface characteristics of both the rod and the cooling tube
  • The presence of fouling or impurities that could impede heat transfer
• Without further details, especially concerning water flow dynamics or surface conditions, it's hard to make a precise judgment.

However, given the context of the problem—intensive heat generation in a nuclear setting—a convection coefficient of this magnitude is generally considered a reasonable and likely necessary value to achieve adequate cooling. This coefficient ensures efficient heat extraction, reducing the risk of overheating.