Question

A nonmetal plate is connected to a stainless steel plate by long ASTM A437 B4B stainless steel bolts \(9.5 \mathrm{~mm}\) in diameter. The portion of the bolts exposed to convection heat transfer with a cryogenic fluid is \(5 \mathrm{~cm}\) long. The fluid temperature for convection is at \(-50^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of $100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. The thermal conductivity of the bolts is known to be \)23.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. Both the nonmetal and stainless steel plates maintain a uniform temperature of \(0^{\circ} \mathrm{C}\). According to the ASME Code for Process Piping (ASME B31.3-2014, Table A-2M), the minimum temperature suitable for ASTM A437 B4B stainless steel bolts is \(-30^{\circ} \mathrm{C}\). 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. Compare the numerical results with the analytical solution. Plot the temperature distribution along the bolt. Would any part of the ASTM A437 B4B bolts be lower than the minimum suitable temperature of \(-30^{\circ} \mathrm{C}\) ?

Short answer

Question: Determine the temperature distribution along ASTM A437 B4B bolts using the finite difference method and compare it with the analytical solution. Also, find out if any part of the bolts would be lower than the minimum suitable temperature of -30°C. Answer: To find the temperature distribution along ASTM A437 B4B bolts and compare it with the analytical solution, follow these steps: 1. Calculate the bolt's surface area using the formula: A = 2πrL. 2. Determine the number of nodes along the length of the bolt with the given nodal spacing. 3. Set up the finite difference equations for the heat conduction equation in cylindrical coordinates for a cylinder with no heat generation. 4. Solve the finite difference equations using numerical methods, such as the Gauss-Seidel method or matrix inversion method. 5. Compare the numerical results with the analytical solution using the one-dimensional heat conduction equation for a cylinder with no heat generation. 6. Plot the temperature distribution along the bolt using the numerical and analytical solutions. 7. Check the temperature at each node obtained from the numerical method to determine if any part of the bolt would be lower than the minimum suitable temperature of -30°C. By following these steps, you will have the temperature distribution along the bolt, a comparison with the analytical solution, and an assessment of whether any part of the bolt is below the minimum suitable temperature.

Step-by-step solution

Determining the Bolt's Surface Area

The first thing we need to do is find the surface area of the bolt, which is exposed to the cryogenic fluid. Since it's a bolt, we'll model it as a cylinder. The surface area of the cylinder (ignoring the ends) can be calculated using the following formula: $$ A = 2 \pi r L $$ where \(r\) is the radius of the bolt, and \(L\) is the length of the exposed part. The bolt has a diameter of \(9.5 \mathrm{~mm}\), so its radius is \(r = \frac{9.5}{2} \mathrm{~mm}\). The exposed part is \(5 \mathrm{~cm}\) long, so \(L = 50 \mathrm{~mm}\). Now we can calculate the surface area of the bolt: $$ A = 2 \pi \left(\frac{9.5}{2} \mathrm{~mm}\right) (50 \mathrm{~mm}) = 299.24 \mathrm{~mm}^2 $$

Calculating the Number of Nodes

We are given a uniform nodal spacing of \(\Delta x = 5 \mathrm{~mm}\). We will use this to determine the number of nodes along the length of the bolt. The length of the bolt exposed to convection is \(50 \mathrm{~mm}\), so there will be \(50 \mathrm{~mm} / 5 \mathrm{~mm} = 10\) nodes, including the nodes at the beginning and end of the bolt.

Setting up the Finite Difference Equations

We will use the following finite difference form of the heat conduction equation in cylindrical coordinates for a cylinder with no heat generation: $$ \frac{\Delta x T_{i-1}-2(1+\Delta x)T_i+\Delta x T_{i+1}}{(\Delta x^2)} $$ where \(T_i\) is the temperature at node \(i\). Since there are 10 nodes in total and we assume no heat generation within the bolt, we can set up the following system of equations to represent the heat conduction equation for each of the nodes: $$ \begin{cases} T_{2}-T_{1}=0 \\ \Delta x T_{1}-2(1+\Delta x)T_{2}+\Delta x T_{3}=0 \\ \Delta x T_{2}-2(1+\Delta x)T_{3}+\Delta x T_{4}=0 \\ \cdots \\ \Delta x T_{9}-2(1+\Delta x)T_{10}+\Delta x T_{11}=0 \\ h A_f (T_{11}-T_{\infty}) = k A \frac{T_{10}-T_{\infty}}{\Delta x} \end{cases} $$ where \(h\) is the convection heat transfer coefficient, \(A_f\) is the area of the final node, \(T_{\infty}\) is the fluid temperature, \(k\) is the thermal conductivity, and \(A\) is the surface area of the bolt. We are given \(h = 100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), \(k = 23.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(T_{\infty} = -50^{\circ} \mathrm{C}\).

Solving the Finite Difference Equations

We can solve the above system of equations using various numerical methods, such as the Gauss-Seidel method or the matrix inversion method. The temperatures at each of the nodes can be calculated by implementing these methods in a computer program like Matlab or Python.

Comparing with the Analytical Solution

To compare the numerical results with the analytical solution, we can use the one-dimensional heat conduction equation for a cylinder with no heat generation. $$ T(x) = T_1 + \frac{(T_{\infty}-T_1)}{\ln \left(\frac{r_2}{r_1}\right)} \ln \left(\frac{r}{r_1}\right) $$ where \(T(x)\) is the temperature at location \(x\), \(T_1\) is the temperature at \(x=0\) (the nonmetal plate), \(r_1\) and \(r_2\) are the inner and outer radii of the bolt, and \(r\) is the radial distance from the center of the bolt. Since both the nonmetal and the stainless steel plates maintain a uniform temperature of \(0^{\circ}C\), we can use \(T_1 = 0^{\circ}C\). We can calculate the analytical values of the temperature at each node along the bolt and compare them with the numerical results obtained in Step 4.

Plotting the Temperature Distribution

Using the numerical and analytical solutions, we can plot the temperature distribution along the bolt on a graph. The x-axis represents the length of the bolt and the y-axis represents the temperature. By plotting both the numerical and analytical solutions, we can visually compare their accuracy.

Checking the Minimum Suitable Temperature

To determine whether any part of the ASTM A437 B4B bolts would be lower than the minimum suitable temperature of \(-30^{\circ}C\), we can check the temperature at each node obtained from the numerical method. If any of the node temperatures are below the minimum suitable temperature, we can conclude that part of the bolt would be in unsuitable conditions. After completing these steps using a suitable programming tool or software, we will have the temperature distribution along the bolt, a comparison with the analytical solution, and an answer to whether any part of the bolt would be lower than the minimum suitable temperature of \(-30^{\circ}C\).