Question

A stainless steel plate is connected to a copper plate by long ASTM B98 copper-silicon bolts \(9.5 \mathrm{~mm}\) in diameter. The portion of the bolts exposed to convection heat transfer with hot gas is \(5 \mathrm{~cm}\) long. The gas temperature for convection is at \(500^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of $50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. The thermal conductivity of the bolt is known to be \)36 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The stainless steel plate has a uniform temperature of \(100^{\circ} \mathrm{C}\), and the copper plate has a uniform temperature of \(80^{\circ} \mathrm{C}\). According to the ASME Code for Process Piping (ASME B31.3-2014, Table A-2M), the maximum use temperature for an ASTM B98 copper-silicon bolt is \(149^{\circ} \mathrm{C}\). Using the finite difference method with a uniform nodal spacing of \(\Delta x=5 \mathrm{~mm}\) along the bolt, determine the temperature at each node. Plot the temperature distribution along the bolt. Compare the numerical results with the analytical solution. Would any part of the ASTM B 98 bolts be above the maximum use temperature of \(149^{\circ} \mathrm{C}\) ?

Short answer

Question: Using the finite difference method (FDM), determine the temperature distribution along a copper-silicon bolt that connects two plates. The diameter of the bolt is 9.5 mm, and its length is 5 cm. The convection heat transfer coefficient is 50 W/m²·K, and the gas temperature is given. The uniform temperatures of the connected plates are 100°C and 80°C. Check if any part of the bolt exceeds the maximum use temperature according to the ASME Code (149°C).

Step-by-step solution

Set up the Finite Difference Method (FDM) formulation

To estimate the temperature distribution, the heat equation must be discretized using FDM. The steady-state heat equation is given by: \(\frac{d}{dx}\left( k \frac{dT}{dx} \right) - hA(T-T_\infty) = 0\) where \(k\) is the thermal conductivity, \(h\) is the convection heat transfer coefficient, \(A\) is the cross-sectional area of the bolt, and \(T_{\infty}\) is the gas temperature. For simplicity, we will use the central difference method with uniform nodal spacing \(\Delta x\). The finite difference equation at node \(i\) is given by: \(\frac{k(T_{i+1}-T_i) - k(T_i-T_{i-1})}{\Delta x^2} - h(2\pi r_iL)(T_i - T_{\infty}) = 0\) The boundary conditions are given by the temperatures of the plates: \(T(0) = T_1 = 100^{\circ} \mathrm{C}\) and \(T(L) = T_n = 80^{\circ} \mathrm{C}\). Solving these equations simultaneously using a system of linear equations will yield the temperatures at each node.

Calculate the properties and parameters for the system

First, calculate the cross-sectional area and perimeter of the bolt: \(A = \frac{\pi d^2}{4} = \frac{\pi (9.5 \times 10^{-3})^2}{4}\mathrm{m^{2}}\) \(hA = 50 \cdot 2\pi (9.5 \times 10^{-3})/2 \cdot 5 \times 10^{-2}\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\) Now, prepare a table with columns for node index, nodal position, cross-sectional area, and the corresponding linear equation coefficients (A, B, C, D), and row values for each node: 1. Nodo index: 0 to n, where n is the number of nodes 2. Nodal position: \(0, \Delta x, 2\Delta x, ..., L\) (in meters) 3. Cross-sectional area: A (m^2) calculated above for all nodes 4. Coefficients A, B, C, D for each node's equation: A = k/Δx^2, B = -2k/Δx^2 - h(2πrL), C = k/Δx^2, D = -h(2πrL)T∞

Solve the linear system of equations

Using the coefficients in the table (Step 2), set up and solve the linear system of equations. Since this system can be easily solved using Gaussian elimination or any linear algebra calculator, the solution will be the temperatures at each node.

Plot the temperature distribution

Using the nodal temperatures obtained in Step 3, plot the temperature distribution along the bolt (as a function of the position), alongside the analytical solution for comparison.

Check if the bolt temperature exceeds the maximum use temperature

Compare the maximum temperature obtained from the FDM or analytical solution with the maximum use temperature for ASTM B98 copper-silicon bolts (\(149^{\circ} \mathrm{C}\)). If the maximum temperature exceeds this value, the bolt would be above its maximum use temperature. Finally, based on the plotted temperature distributions and comparison with the maximum use temperature, it can be concluded whether any part of the bolt would be above the maximum use temperature of \(149^{\circ} \mathrm{C}\).