Question

A cylindrical fuel rod $(k=30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}) 2 \mathrm{~cm}$ in diameter is encased in a concentric tube and cooled by water. The fuel rod generates heat uniformly at a rate of $100 \mathrm{MW} / \mathrm{m}^{3}$, and the average temperature of the cooling water is \(75^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of $2500 \mathrm{~W} / \mathrm{m}^{2}\(. \)\mathrm{K}$. The operating pressure of the cooling water is such that the surface temperature of the fuel rod must be kept below \(200^{\circ} \mathrm{C}\) to prevent the cooling water from reaching the critical heat flux (CHF). The critical heat flux is a thermal limit at which a boiling crisis can occur that causes overheating on the fuel rod surface and leads to damage. Determine the variation of temperature in the fuel rod and the temperature of the fuel rod surface. Is the surface of the fuel rod adequately cooled?

Short answer

To answer this question, determine the surface temperature of the fuel rod and compare it to the given critical heat flux limit using the provided parameters and methodology. If the calculated surface temperature is below 200°C, then the fuel rod is adequately cooled.

Step-by-step solution

Calculate heat generation rate

To calculate the heat generation rate per unit volume, we are given q_gen=\(100 MW/m^{3}\). We need to convert this value into W/m³: \(q_{gen} = 100 \times 10^6 \, \mathrm{W}/\mathrm{m}^{3}\)

Identify the given parameters

We are given the following information: - Diameter of the fuel rod (\(D\)) = 2 cm = 0.02 m - Thermal conductivity of the fuel rod (\(k\)) = 30 W/m.K - Average temperature of cooling water (\(T_{water}\)) = 75°C - Convection heat transfer coefficient (\(h\)) = \(2500 \mathrm{W}/\mathrm{m}^{2}.K\) - Surface temperature limit = 200°C

Calculate the temperature variation within the fuel rod

Using the heat conduction equation for a cylinder: $$ q_{gen} = \frac{k(T_{max} - T(r))}{ r} $$ where, \(r\) is the distance from the center of the rod and \(T(r)\) is the temperature at a distance \(r\) from the center. Let's find \(T_{max}\), which is the temperature at the center of the fuel rod (\(r=0\)): $$ T_{max} = \frac{q_{gen} \cdot r}{k} + T(r) $$

Calculate the surface temperature of the fuel rod

Now, let's calculate the temperature at the surface of the fuel rod, which occurs at \(r = R\) (where \(R\) is the radius of the fuel rod): $$ T_{surface} = T_{max} - \frac{q_{gen} \cdot R}{k} $$ We are given radius \(R = 0.01\) m, and \(k=30\) W/m.K. We already have \(q_{gen} = 100 \times 10^6 \mathrm{W} / \mathrm{m}^{3}\). Now we can find the surface temperature, \(T_{surface}\).

Check if the surface temperature is below the critical limit

To determine if the fuel rod is adequately cooled, we need to compare the calculated surface temperature with the critical heat flux limit of 200°C: Adequate cooling: \(T_{surface} < 200°C\)

Calculate and compare the temperature values

Now we have everything we need to calculate the temperature variation within the fuel rod and the surface temperature. We just need to compare the calculated surface temperature with the critical limit to know if the fuel rod is adequately cooled: - Compare the found value for \(T_{surface}\) to the critical limit of 200°C. If the surface temperature of the fuel rod is below this critical limit, we can conclude that the surface of the fuel rod is adequately cooled.