Question

A stainless steel plate $(k=13 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, 1 \mathrm{~cm}\( thick) is attached to an ASME SB-96 coppersilicon plate ( \)k=36 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, 3 \mathrm{~cm}$ thick) to form a plane wall. The bottom surface of the ASME SB-96 plate (surface 1) is subjected to a uniform heat flux of \(750 \mathrm{~W} / \mathrm{m}^{2}\). The top surface of the stainless steel plate (surface 2 ) is exposed to convection heat transfer with air at \(T_{\infty}=20^{\circ} \mathrm{C}\), and thermal radiation with the surroundings at \(T_{\text {surr }}=20^{\circ} \mathrm{C}\). The combined heat transfer coefficient for convection and radiation is \(h_{\mathrm{comb}}=7.76 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). 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 plate thicknesses, determine the temperature at each node. 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 highest heat flux that the bottom surface can be subjected to such that the ASME SB-96 plate is still operating below \(93^{\circ} \mathrm{C}\) ?

Short answer

Question: Determine the temperature at each node using the finite difference method and check if the temperature does not exceed the limit specified by the ASME code for the first plate. Also, find the highest heat flux that keeps the temperature below the limit.

Step-by-step solution

Set up the equation for the finite difference method

First, we need to apply the finite difference method to the one-dimensional heat conduction equation: $$ q = -k \frac{dT}{dx}, $$ where \(q\) is the heat flux, \(k\) is the thermal conductivity, \(T\) is the temperature, and \(x\) is the distance along the plate thickness. For the nth node, we can approximate the temperature change by using a central difference scheme: $$ \frac{T_{n + 1} - 2T_n + T_{n - 1}}{\Delta x^2}, $$ where \(T_n\) is the temperature at the nth node.

Apply boundary conditions

We have two boundary conditions: one at the bottom surface of the first plate, where \(q_1 = 750 W/m^2\) and another one at the top surface of the second plate, where the heat flux is due to both convection and radiation. For the convection and radiation heat transfer, the heat flux at the top surface can be written as: $$ q_2 = h_{comb}(T_2 - T_{\infty}), $$ where \(h_{comb} = 7.76 W/m^2 K\) and \(T_{\infty} = T_{surr} = 20^{\circ} C\).

Create equations for each node

Now that we have our difference equation and boundary conditions, we can create equations for each node in the system by applying the difference equation to each plate separately. For each plate, we use the respective thermal conductivity, \(k\), in the equation. Write the equations in matrix form. Denote the nodal temperature in the stainless steel plate with \(T_n^s\) and the nodal temperature in the ASME SB-96 plate with \(T_n^c\), with \(n = 1, 2, 3\), as \(\Delta x = 5 mm\) for both plates.

Solve the system of equations

Once the equations are set up in a matrix form, use either a numerical method or a software tool to solve for the unknown nodal temperatures, \(T_n^s\) and \(T_n^c\). This will give the temperature profile across both plates.

Check compliance with the ASME code

Now that we have the temperature profile across the plates, assess if the temperature at the first plate (ASME SB-96 plate) does not exceed the temperature limit of \(93^{\circ} C\) specified by the ASME code. If all the nodal temperatures of the first plate are below this limit, then the plate is in compliance with the code.

Determine the highest heat flux

To find the highest heat flux that the bottom surface can be subjected to while having a temperature of \(93^{\circ} C\) at the ASME SB-96 plate, we can either use trial and error or a numerical method. Repeat steps 1-5 with different heat flux values until we find the highest heat flux that keeps the ASME SB-96 plate at a temperature below \(93^{\circ} C\).