Question

A composite wall is made of stainless steel $\left(k_{1}=13 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \quad 30 \mathrm{~mm}\right.$ thick), concrete $\left(k_{2}=1.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, 30 \mathrm{~mm}\right.\( thick \))\(, and nonmetal \)\left(k_{3}=0.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.\(, \)15 \mathrm{~mm}$ thick) plates. The concrete plate is sandwiched between the stainless steel plate at the bottom and the nonmetal plate at the top. A series of ASTM B21 naval brass bolts are bolted to the nonmetal plate, and the upper surface of the plate is exposed to convection heat transfer with air at \(20^{\circ} \mathrm{C}\) and $h=20 \mathrm{~W} / \mathrm{m}^{2}$.K. At the bottom surface, the stainless steel plate is subjected to a uniform heat flux of $2000 \mathrm{~W} / \mathrm{m}^{2}$. The ASME Code for Process Piping (ASME B31.3-2014, Table A-2M) limits the maximum use temperature for the ASTM B21 bolts to \(149^{\circ} \mathrm{C}\). Using the finite difference method with a uniform nodal spacing of \(\Delta x_{1}=10 \mathrm{~mm}\) for the stainless steel and concrete plates, and \(\Delta x_{2}=5 \mathrm{~mm}\) for the nonmetal plate, determine the temperature at each node. Plot the temperature distribution as a function of \(x\) along the plate thicknesses. Would the ASTM B21 bolts in the nonmetal plate comply with the ASME code?

Short answer

Question: Determine if the ASTM B21 bolts used in the nonmetal plate of a composite wall comply with the ASME code, which sets a limit for the maximum use temperature. Given the boundary conditions, use the finite difference method to find the temperature distribution along the composite wall made of stainless steel, concrete, and nonmetal plates.

Step-by-step solution

Identify the boundary conditions

The problem states that the bottom surface of the stainless steel plate is subjected to a uniform heat flux of 2000 W/m^2, and the upper surface of the nonmetal plate is exposed to convection heat transfer with air at 20°C and h=20 W/m^2.K.

Set up finite difference equations

Start by labeling the nodes in each section of the composite wall. Since we have three plates, we will have three separate equations representing each layer: - Nodes in Stainless Steel: \(T_{1,1}, T_{1,2}, T_{1,3}\) - Nodes in Concrete: \(T_{2,1}, T_{2,2}, T_{2,3}\) - Nodes in Nonmetal: \(T_{3,1}, T_{3,2}, T_{3,3}\) Apply the finite difference method to set up the equations for each node. They will help us create a system of linear equations, which can be solved for the temperature at each node.

Solve for temperatures

After setting up the equations, solve for the temperatures at each node using techniques like the Gaussian elimination method or any other method that is preferred for solving a system of linear equations.

Plot the temperature distribution

Once the temperatures at the nodes are found, create a plot that represents the temperature distribution along the thicknesses of each layer. This will visually aid in analyzing the temperature distribution along the composite wall.

Compare with ASME code

Check the temperature at the node corresponding to the position of the ASTM B21 bolts in the nonmetal plate. Compare this temperature to the maximum limit provided by the ASME code (\(149^{\circ} \mathrm{C}\)). If the temperature is less than or equal to the limit, then the ASTM B21 bolts comply with the ASME code. By following these steps, you can determine the temperature distribution along the composite wall and verify if the ASTM B21 bolts comply with the ASME code.