Question

Consider a nuclear fuel element $(k=57 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\( that can be modeled as a plane wall with thickness of \)4 \mathrm{~cm}\(. The fuel element generates \)3 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}$ of heat uniformly. Both side surfaces of the fuel element are cooled by liquid with temperature of \(80^{\circ} \mathrm{C}\) and convection heat transfer coefficient of $8000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. Using a uniform nodal spacing of \)8 \mathrm{~mm}$, (a) obtain the finite difference equations, (b) determine the nodal temperatures by solving those equations, and (c) compare the surface temperatures of both sides of the fuel element with the analytical solution.

Short answer

Answer: The finite difference equations for the internal nodes are represented by the following equation: k * (T_{i-1} - 2 * T_{i} + T_{i+1}) / (Δx)^2 + Q_gen = 0 For the left-side node, the equation is: -k * (T_2 - T_1) / (Δx) + h * (T_L - T_1) = 0 And for the right-side node, the equation is: - k * (T_N - T_{N-1}) / (Δx) + h * (T_L - T_N) = 0 The nodal temperatures can be determined by solving these N-1 simultaneous equations numerically using various methods such as the Gauss-Seidel method or a matrix solver. The calculated surface temperatures can then be compared with the analytical solution given by the following equation: T(x) = T_L + Q_gen * (x^2 - L * x) / (2 * k) If the results from the numerical solution are accurate, the calculated surface temperatures will be close to the values obtained from the analytical solution.

Step-by-step solution

Define the Given Parameters

We are given the following information: Thermal conductivity, k = 57 W/m·K Wall thickness, L = 4 cm = 0.04 m Heat generation, Q_gen = 3 * 10^7 W/m³ Liquid temperature, T_L = 80 °C Convection heat transfer coefficient, h = 8000 W/m²·K Uniform nodal spacing, Δx = 8 mm = 0.008 m

Derive the Finite Difference Equations

Using the heat balance, we can model the heat transfer through the wall by dividing it into nodes and applying the conservation of energy to each node. For an internal node i, the heat balance is as follows: k * (T_{i-1} - 2 * T_{i} + T_{i+1}) / (Δx)^2 + Q_gen = 0 For the nodes on both sides of the wall, we also need to incorporate the convection term in the heat balance: For the left-side node (i = 1), we have: -k * (T_2 - T_1) / (Δx) + h * (T_L - T_1) = 0 For the right-side node (i = N), we have: - k * (T_N - T_{N-1}) / (Δx) + h * (T_L - T_N) = 0

Determine the Nodal Temperatures

Now we have N-1 simultaneous equations, one for each internal node and two boundary conditions. You can solve these equations numerically using various methods, such as the Gauss-Seidel method or a matrix solver. The results will give the temperatures at each node.

Compare the Surface Temperatures with the Analytical Solution

To compare our results with the analytical solution, we can calculate the surface temperature based on conduction and internal heat generation in the plane wall. This in Cartesian coordinates, can be determined as follows: T(x) = T_L + Q_gen * (x^2 - L * x) / (2 * k) We can then compare the calculated surface temperatures (T(0) and T(L)) from our numerical solution to the analytical solution using this equation to ensure our finite difference equations were solved accurately.