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 a fuel rod surface that leads to damage. A cylindrical fuel rod \(2 \mathrm{~cm}\) 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 prevent the cooling water from reaching the critical heat flux. Determine the necessary convection heat transfer coefficient to prevent the critical heat flux from occurring.

Short answer

#tag_title#Step 2: Calculate the fuel rod's cross-sectional area#tag_content# First, let's find the cross-sectional area (A) of the fuel rod: A = (pi * D^2) / 4 A = (3.14 * (0.02 m)^2) / 4 A = 0.000314 m^2 #tag_title#Step 3: Calculate the required heat transfer (Q)#tag_content# Now we can plug in the values to find Q: Q = q * A Q = 150 MW/m^3 * 0.000314 m^2 Q = 0.0471 MW (since 1 MW = 10^6 W, Q = 47,100 W) #tag_title#Step 4: Calculate the convection heat transfer coefficient (h)#tag_content# Next, we can find the required convection heat transfer coefficient (h) using the formula: Q = h * A * (T_surface - T_cooling water) We need to rearrange the formula to solve for h: h = Q / (A * (T_surface - T_cooling water)) We are given the following values: T_surface (the temperature that must be kept below) = 100°C T_cooling water (average temperature) = 75°C Now we can plug in the values to find h: h = Q / (A * (T_surface - T_cooling water)) h = 47,100 W / (0.000314 m^2 * (100°C - 75°C)) h = 47,100 W / (0.000314 m^2 * 25°C) h = 6,000 W/(m^2 * K) Therefore, the required convection heat transfer coefficient is 6,000 W/(m^2 * K) to prevent a boiling crisis. Short Answer: The required convection heat transfer coefficient to prevent a boiling crisis is 6,000 W/(m^2 * K).

Step-by-step solution

Calculate the heat transfer (Q) required to prevent the critical heat flux

We can first find the required heat transfer (Q) by using the given heat generation rate of the fuel rod and the fuel rod cross-sectional area. The formula to find Q is given by: Q = (heat generation rate) * (fuel rod cross-sectional area) We are given the following values: Diameter of the fuel rod (D) = 2 cm = 0.02 m Heat generation rate (q) = 150 MW/m^3 First, let's find the cross-sectional area (A) of the fuel rod: A = (pi * D^2) / 4 Now we can plug in the values to find Q: Q = q * A