Question

The critical heat flux (CHF) is a thermal limit at which a boiling crisis occurs whereby an abrupt rise in temperature causes overheating on fuel rod surface that leads to damage. 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 average temperature of the cooling water, sufficiently far from the fuel rod, is \(80^{\circ} \mathrm{C}\). The operating pressure of the cooling water is such that the surface temperature of the fuel rod must be kept below \(300^{\circ} \mathrm{C}\) to avoid the cooling water from reaching the critical heat flux. Determine the necessary convection heat transfer coefficient to avoid the critical heat flux from occurring.

Short answer

Based on the given information and the calculations performed, the necessary convection heat transfer coefficient to avoid the critical heat flux from occurring in the nuclear fuel rod is approximately 4.17 W/cm² °C.

Step-by-step solution

Identify given information

We are given the following information: - Diameter of the fuel rod (D) = 2 cm - Heat generation rate (Q_gen) = 150 MW/m³ - Temperature of cooling water far from the fuel rod (T_ambient) = 80°C - Operating pressure limit to avoid CHF (T_surface) = 300°C

Calculate the area of the fuel rod

First, we need to calculate the area of the fuel rod's surface which will be in contact with the cooling water. Given the diameter of the fuel rod, we can find its radius (r) and then calculate its area (A) using the formula for the surface area of a cylinder: \( r = \frac{D}{2} = \frac{2}{2} = 1 \mathrm{~cm} \) For the surface area, since it is a cylinder: \( A = 2\pi rh \) In our case, since we only need the convection heat transfer coefficient per unit length, we can use: \( A_1 = 2\pi r = 2\pi(1\mathrm{~cm}) = 2\pi \mathrm{~cm}^2\)

Calculate the generated heat per unit length

Next, let's calculate the amount of heat generated by the fuel rod per unit length. We are given the heat generation rate in MW/m³, and we'll need to convert this to W/cm to be consistent with the area units: \( Q_{gen_1} = 150\mathrm{~MW} / \mathrm{m}^3 \times 10^6 \mathrm{~W} / \mathrm{MW} \times \frac{1 \mathrm{~m}}{100 \mathrm{~cm}} = 1500 \mathrm{~W} / \mathrm{cm} \) Now, we find the generated heat per unit length, Q_gen_length: \( Q_{gen\_length} = Q_{gen_1} \cdot \pi r^2 = 1500 \mathrm{~W/cm} \cdot \pi (1 \mathrm{~cm})^2 = 1500\pi \mathrm{~W/cm} \)

Determine the necessary convection heat transfer coefficient

Now, we can determine the required convection heat transfer coefficient (h) using the formula Q = h * A * (T_surface - T_ambient), where Q represents the amount of heat that must be removed per unit length to keep the fuel rod below the critical heat flux: \( h = \frac{Q_{gen\_length}}{A_1 (T_{surface} - T_{ambient})} = \frac{1500\pi \mathrm{~W/cm}}{2\pi \mathrm{~cm}^2 (300^{\circ} \mathrm{C} - 80^{\circ} \mathrm{C})} \) Simplify and solve for h: \( h = \frac{1500}{2 (300 - 80)} \mathrm{~W/cm^2 ^{\circ} C} \approx 4.17 \mathrm{~W/cm^2 ^{\circ} C} \) Therefore, the necessary convection heat transfer coefficient to avoid the critical heat flux from occurring is approximately 4.17 W/cm² °C.

Key concepts

Critical Heat Flux (CHF)

The concept of Critical Heat Flux (CHF) is central to understanding the limits of heat transfer in boiling processes. CHF represents the maximum heat flux that can be accommodated by a surface undergoing boiling before a boiling crisis occurs. At this point, a sudden reduction in the efficiency of heat transfer leads to a rapid increase in surface temperature, potentially causing damage to the surface, such as the fuel rod in a nuclear reactor. In the provided exercise, it's crucial to maintain the fuel rod surface temperature below 300°C to prevent reaching CHF. A well-designed cooling system, indicated by the necessary convection heat transfer coefficient, must be in place to avoid this scenario.

To put it simply, imagine CHF as the boiling point of water - but in this context, it's the 'boiling point' where the surface can no longer effectively boil away heat, leading to overheating. Ensuring the fuel rod remains below the CHF limits allows for safe operation and maximizes the lifespan of the equipment.

Boiling Crisis

A boiling crisis, also known as 'Departure from Nucleate Boiling' (DNB), is a thermal phenomenon where a liquid in contact with a surface suddenly changes its cooling behavior. During nucleate boiling, vapor bubbles form on the heat transfer surface and help to carry away heat efficiently. However, if the heat flux exceeds CHF, these bubbles can coalesce into a vapor film, significantly reducing heat transfer and causing a steep temperature rise. This condition is detrimental and can lead to material failure. The problem details the avoidance of a boiling crisis by determining the necessary convection heat transfer coefficient to maintain surface temperatures below critical levels. In a practical sense, avoiding a boiling crisis is akin to preventing a pot of water on a stove from 'drying out' and scorching the pot's bottom.

Heat Generation Rate

The heat generation rate in a material, such as the fuel rod in the exercise, indicates the amount of heat produced per unit volume per unit time and is typically expressed in watts per cubic meter (W/m³). In thermal analysis, it is crucial to know this rate to predict how much heat a cooling system must dissipate to avoid overheating. The exercise specifies a heat generation rate of 150 MW/m³, a high value indicative of a powerful energy source, such as nuclear fuel. Managing this heat is paramount, as excessive accumulation can lead to a boiling crisis.

To contextualize, imagine a space heater in a room— the heat generation rate would be how much heat the heater emits, which a cooling system (like an air conditioner) needs to offset to maintain a comfortable temperature.

Thermal Analysis

Thermal analysis involves studying how heat moves through systems and materials and how materials respond to changes in temperature. It encompasses a broad range of calculations, such as determining heat transfer coefficients, which are necessary for the design and analysis of cooling systems. In the example, thermal analysis is used to calculate the convection heat transfer coefficient, ensuring heat is removed efficiently from the fuel rod to prevent reaching CHF. A correct coefficient will allow for the safe operation of the nuclear reactor, accounting for the high heat generation rate and mitigating the risks of a boiling crisis.

In a broader education context, understanding thermal analysis equips students with the fundamentals of how to manage heat in various engineering applications, from designing electronic devices to ensuring the safe operation of power plants.