Question

A nonmetal plate \((k=0.05 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) is attached on the upper surface of an ASME SB-96 copper-silicon plate $(k=36 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$. The nonmetal plate and the ASME SB-96 plate have thicknesses of \(20 \mathrm{~mm}\) and \(30 \mathrm{~mm}\), respectively. The bottom surface of the ASME SB-96 plate (surface 1) is subjected to a uniform heat flux of \(150 \mathrm{~W} / \mathrm{m}^{2}\). The top nonmetal plate surface (surface 2) is exposed to convection at an air temperature of \(15^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of \(5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Copper- silicon alloys are not always suitable for applications where they are exposed to certain median and high temperatures. Therefore, the ASME Boiler and Pressure Vessel Code (ASME BPVC.IV-2015, HF-300) limits equipment constructed with ASME SB-96 plate to be operated at a temperature not exceeding \(93^{\circ} \mathrm{C}\). Using the finite difference method with a uniform nodal spacing of \(\Delta x=5 \mathrm{~mm}\) along the the temperature distribution as a function of \(x\). Would the use of the ASME SB- 96 plate under these conditions be in compliance with the ASME Boiler and Pressure Vessel Code? What is the lowest value of the convection heat transfer coefficient for the air so that the ASME SB-96 plate is below $93^{\circ} \mathrm{C}$ ?

Short answer

Answer: The finite difference method for heat conduction can be used to calculate the temperature distribution in the composite structure. By iterating the temperature calculation to convergence, checking for ASME compliance, and adjusting the convection heat transfer coefficient, the lowest value for compliance with the ASME Boiler and Pressure Vessel Code can be determined.

Step-by-step solution

Calculate temperature at each node

Since we have a uniform nodal spacing of Δx = 5mm, we will discretize the problem into 10 nodes, with 6 nodes representing the ASME SB-96 plate (30mm thickness) and 4 nodes representing the nonmetal plate (20mm thickness). We know the heat flux at the bottom of the ASME SB-96 plate is 150 W/m². The boundary condition at the top of the nonmetal plate is given by: q'' = h(T_surf - T_air) where h is the convection heat transfer coefficient, T_surf is the temperature at the top surface (surface 2), and T_air is the air temperature. Using the general finite difference equation for steady-state two-dimensional heat conduction: k1(T_1 - T_2) / Δx + k2(T_4 - T_2) / Δx = 0 we can calculate the temperature at each node.

Iterate temperature calculation to convergence

We can start with an initial guess for the temperature at each node, and iterate the finite difference equation until the difference in temperature between iterations is minimized. At the end of this process, we will obtain the temperature distribution in the composite structure.

Check ASME compliance

To check if the temperatures in the ASME SB-96 plate do not exceed 93°C, compare the calculated temperatures at nodes 1 through 6. If all of them are below 93°C, the structure is in compliance with the ASME Boiler and Pressure Vessel Code.

Calculate lowest convection coefficient for compliance

To find the lowest convection heat transfer coefficient that ensures the compliance with the ASME Boiler and Pressure Vessel Code, we need to: 1. Start with the given convection coefficient value (h = 5 W/m²K) and calculate the temperature distribution. 2. Increment or decrement the convection coefficient and calculate the temperature distribution again. 3. Repeat this process until the highest temperature in the copper-silicon plate is exactly 93°C. The convection coefficient value found at this point corresponds to the lowest value for which the structure remains in compliance with the ASME Boiler and Pressure Vessel Code.