Question

An ASTM B209 5154 aluminum alloy plate is connected to an insulation plate by long metal bolts \(4.8 \mathrm{~mm}\) in diameter. The portion of the bolts exposed to convection heat transfer with air is \(5 \mathrm{~cm}\) long. The thermal conductivity of the bolt is known to be $15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(. The ambient condition of the air is at \)20^{\circ} \mathrm{C}\( with a convection heat transfer coefficient of \)20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. According to the ASME Code for Process Piping (ASME B31.3-2014, Table A-1M), the maximum use temperature for the ASTM B209 5154 aluminum alloy plate is \(65^{\circ} \mathrm{C}\). If the temperature of the bolt at midlength ( \(2.5 \mathrm{~cm}\) from the upper surface of the aluminum plate) is \(50^{\circ} \mathrm{C}\), determine the temperature \(T_{b}\) at the upper surface of the aluminum plate. What is the rate of heat loss from each bolt to convection? Is the use of the ASTM B209 5154 plate in compliance with the ASME Code for Process Piping?

Short answer

In this exercise, we were asked to find the temperature at the upper surface of the aluminum plate, the rate of heat loss from each bolt to convection, and check if the ASTM B209 5154 aluminum alloy plate is in compliance with the ASME Code for Process Piping. Following the five steps outlined above, we analyzed heat transfer through the bolt by conduction, calculated heat loss through convection, found the temperature at the upper surface of the aluminum plate, and checked for compliance with the ASME Code for Process Piping requirements.

Step-by-step solution

Calculate heat transfer by conduction through the bolt

Firstly, we need to determine the temperature distribution along the bolt. To do this, we can use the one-dimensional steady-state conduction equation with constant thermal conductivity as follows: \(q = k A \frac{dT}{dx}\) Where: - \(q\) is the heat transfer rate (\(\mathrm{W}\)) - \(k\) is the thermal conductivity of the bolt (\(15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\)) - \(A\) is the cross-sectional area of the bolt (\(\pi (0.0048 \mathrm{~m} / 2)^{2}\)) - \(\frac{dT}{dx}\) is the temperature gradient along the bolt length (\(\mathrm{K} / \mathrm{m}\)) We are given the temperature at midlength is \(50^{\circ}\mathrm{C}\), and we need to find the temperature at the upper surface of the aluminum plate, \(T_{b}\). Due to the symmetrical nature of the problem, we consider half the bolt length, and the temperature on the other end of that half-length is \(T_{b}\). \(T_{midlength} = 50^{\circ} \mathrm{C}\) \(x_{midlength} = 2.5\mathrm{cm} = 0.025\mathrm{m}\) We can calculate the temperature gradient using the given temperatures and length: \(\frac{dT}{dx} = \frac{T_{midlength} - T_{b}}{x_{midlength}}\)

Calculate the rate of heat loss through convection

Now, we will calculate the heat loss from the bolt to air by convection using Newton's Law of Cooling: \(q = h A_{s} (T_{b} - T_{\infty})\) Where: - \(q\) is the heat transfer rate (\(\mathrm{W}\)) - \(h\) is the convection heat transfer coefficient (\(20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\)) - \(A_{s}\) is the surface area of the bolt exposed to convection (\(\pi D L\)) - \(T_{b}\) is the temperature at the upper surface of the aluminum plate (\(\mathrm{K}\)) - \(T_{\infty}\) is the ambient temperature (\(20^{\circ} \mathrm{C}\))

Find the temperature at the upper surface of the aluminum plate, \(T_{b}\)

As heat transfer through the bolt by conduction provides the heat lost through convection, we can equate the two equations from Step 1 and Step 2 to find the temperature at the upper surface of the aluminum plate, \(T_{b}\): \(k A \frac{dT}{dx} = h A_{s} (T_{b} - T_{\infty})\) Plug in the known values and solve for \(T_{b}\).

Calculate the rate of heat loss from each bolt to convection

Using the found value for \(T_{b}\), we can now calculate the rate of heat loss from each bolt to convection using either the conduction equation from Step 1 or the convection equation from Step 2: \(q = k A \frac{dT}{dx}\) or \(q = h A_{s} (T_{b} - T_{\infty})\)

Check compliance with the ASME Code for Process Piping

Finally, we need to check if the use of the ASTM B209 5154 aluminum alloy plate is in compliance with the ASME Code for Process Piping requirements. According to the code, the maximum use temperature for the ASTM B209 5154 aluminum alloy plate is \(65^{\circ}\mathrm{C}\). We compare the found value of \(T_{b}\) with this limit. If \(T_{b}\) is less than or equal to the limit (\(65^{\circ}\mathrm{C}\)), then the use of the plate is in compliance with the ASME Code for Process Piping.