Question

Two components in cryogenic equipment are held together by stainless steel (ASTM A437 B4B) bolts with diameter of \(25 \mathrm{~mm}\). The bolts have a thermal conductivity of \(23.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), a specific heat of \(460 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and a density of \(7.8 \mathrm{~g} / \mathrm{cm}^{3}\). When the cryogenic fluid flows between the two components, the bolts are submerged in the cold fluid. The length of each bolt submerged in the cryogenic fluid is \(5 \mathrm{~cm}\), and the initial temperature of the bolts is \(10^{\circ} \mathrm{C}\). The cryogenic fluid has a temperature of \(-40^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of \(40 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The ASME Code for Process Piping limits the minimum suitable temperature for ASTM A437 B4B bolts to \(-30^{\circ} \mathrm{C}\) (ASME B31.3-2014, Table A-2M). If the bolts are exposed to the cold fluid for \(12 \mathrm{~min}\), will they still comply with the ASME code? How long will it take for the bolts to reach the minimum suitable temperature?

Short answer

If not, how long will it take for the bolts to reach the minimum suitable temperature? Answer: Yes, the temperature of the stainless steel bolts will be above the minimum suitable temperature after 12 minutes. They will be at -11.433°C, which is above the minimum suitable temperature of -30°C. It will take approximately 21.75 minutes for the bolts to reach the minimum suitable temperature of -30°C.

Step-by-step solution

Calculate the mass of the bolts

First, let's calculate the mass of the submerged section of the bolts. The volume of a cylinder is given by \(V = \pi r^2h\), where \(r\) is the radius, and \(h\) is the height. The mass is given by \(m = V \cdot \rho\), where \(\rho\) is the density. For one bolt, we have: \(r = \frac{25\ \text{mm}}{2} = 12.5\ \text{mm} = 1.25\ \text{cm}\) \(h = 5\ \text{cm}\) \(\rho = 7.8\ \text{g}/\text{cm}^3\) \(V = \pi * (1.25\ \text{cm})^2 * 5\ \text{cm} = 24.543\ \text{cm}^3\) \(m = V \cdot \rho = 24.543\ \text{cm}^3 \cdot 7.8\ \text{g}/\text{cm}^3 = 191.136\ \text{g}\)

Calculate the heat transfer

Now, we need to calculate the heat transfer from the bolts to the fluid, which is given by \(q = hA\Delta T\), where \(h\) is the convection heat transfer coefficient, \(A\) is the surface area of the submerged section of the bolts, and \(\Delta T\) is the temperature difference between the initial temperature of the bolts and the fluid. For one bolt, we have: \(A = 2\pi rh = 2\pi (1.25\ \text{cm})(5\ \text{cm})=39.27\ \text{cm}^2 = 0.003927\ \text{m}^2\) \(h = 40\ \text{W}/\text{m}^2 \cdot \text{K}\) \(\Delta T = (10 - (-40))^\circ\text{C} = 50^\circ\text{C}\) \(q = hA\Delta T = 40\ \text{W}/\text{m}^2 \cdot \text{K} \cdot 0.003927\ \text{m}^2 \cdot 50^\circ\text{C} = 7.883\ \text{W}\)

Calculate the temperature change in the bolt

Next, we need to find the temperature change in the bolt. The equation to find the temperature change is \(\Delta T = \frac{q\Delta t}{mc_p}\), where \(q\) is the heat transfer, \(\Delta t\) is the time exposed, \(m\) is the mass of the bolt, and \(c_p\) is the specific heat of the bolt. For one bolt, we have: \(\Delta T = \frac{7.883\ \text{W} \cdot 720\ \text{s}}{191.136\ \text{g} \cdot 460\ \text{J}/\text{kg} \cdot \text{K}} = -21.433^\circ\text{C}\)

Check if the temperature is above the minimum after 12 minutes

We can now check if the temperature of the bolt will be above the minimum suitable temperature after 12 minutes. \(T_{final} = T_{initial} + \Delta T = 10^\circ\text{C} - 21.433^\circ\text{C} = -11.433^\circ\text{C}\) The temperature of the bolt after 12 minutes will be \(-11.433^\circ\text{C}\), which is above the minimum suitable temperature of \(-30^\circ\text{C}\). Therefore, the bolts will still comply with the ASME code.

Find how long it will take for the bolts to reach the minimum suitable temperature

To find how long the bolts will take to reach the minimum suitable temperature, we can rearrange the temperature change equation to solve for time: \(\Delta t = \frac{mc_p\Delta T}{q} = \frac{191.136\ \text{g} \cdot 460\ \text{J}/\text{kg} \cdot \text{K} \cdot (10 - (-30))^\circ\text{C}}{7.883\ \text{W}}\) \(\Delta t = 1304.99\ \text{s} = 21.75\ \text{mins}\) It will take approximately 21.75 minutes for the bolts to reach the minimum suitable temperature of \(-30^{\circ}\text{C}\).