Question

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 convection heat transfer coefficient on the fuel rod is \(5000 \mathrm{~W} / \mathrm{m}^{2}\). \(\mathrm{K}\), 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: Following the steps above, we calculate the surface temperature, \(T_s\), of the cylindrical fuel rod. We then compare this with the cooling water temperature, and if the difference is too large or unrealistic, the given convection heat transfer coefficient may not be reasonable. The reasonability of the convection heat transfer coefficient can also be assessed by comparing it with typical values found in literature or real-life scenarios.

Step-by-step solution

Find the heat generation rate per unit volume of the rod

The given heat generation rate is \(150 \mathrm{MW} / \mathrm{m}^{3}\) which needs to be converted to \(\mathrm{W} / \mathrm{m}^{3}\). $$ Q_{gen} = 150 \times 10^6 \mathrm{~W} / \mathrm{m}^{3} $$

Calculate the cross-sectional area of the cylindrical fuel rod

The diameter of the fuel rod is given as \(2 \mathrm{~cm}\). We need to convert it to meters and compute the cross-sectional area, \(A\), using the formula for the area of a circle: $$ A = \pi r^{2} = \pi (0.01 \mathrm{m})^{2} $$

Compute the heat generated per unit length of the fuel rod

Multiply the heat generation rate per unit volume, \(Q_{gen}\), with the cross-sectional area, \(A\), to find the heat generated per unit length, \(q'\): $$ q' = Q_{gen} \times A $$

Use the convection heat transfer equation to find the surface temperature

The convection heat transfer equation is given by: $$ q' = h A (T_{s} - T_{\infty}) $$ Where \(h\) is the convection heat transfer coefficient, \(A\) is the surface area of the fuel rod (per unit length), \(T_{s}\) is the surface temperature of the fuel rod, and \(T_{\infty}\) is the temperature of the cooling water far from the fuel rod. We are given \(h = 5000 \mathrm{~W} / \mathrm{m}^{2} \mathrm{K}\), and \(T_{\infty}= 70^{\circ} \mathrm{C}\). Substitute the given values and the values calculated in Steps 2 and 3: $$ q' = h (2 \pi r) (T_{s} - T_{\infty}) $$ Next, we need to isolate \(T_{s}\) in the above equation and then calculate the value for \(T_{s}\): $$ T_{s} = \frac{q'}{h (2 \pi r)} + T_{\infty} $$

Discuss the reasonability of the given convection heat transfer coefficient

Compare our calculated surface temperature with the cooling water temperature. If the difference between the surface temperature and the cooling water temperature is too large or unrealistic, then the given convection heat transfer coefficient may not be reasonable. Also, compare the convection heat transfer coefficient value with typical values found in literature or real-life scenarios.