Question

In a piece of cryogenic equipment, two metal plates are connected by a long ASTM A437 B4B stainless steel bolt. Cold gas, at \(-70^{\circ} \mathrm{C}\), flows between the plates and across the cylindrical bolt. The gas has a thermal conductivity of \(0.02 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), a kinematic viscosity of \(9.3 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\), and a Prandtl number of \(0.74\). The diameter of the bolt is \(9.5 \mathrm{~mm}\), and the length of the bolt exposed to the gas is \(10 \mathrm{~cm}\). The minimum temperature suitable for the ASTM A437 B4B stainless steel bolt is \(-30^{\circ} \mathrm{C}\) (ASME Code for Process Piping, ASME B31.3-2014, Table A-2M). The temperature of the bolt is maintained by a heating mechanism capable of providing heat at \(15 \mathrm{~W}\). Determine the maximum velocity that the gas can achieve without cooling the bolt below the minimum suitable temperature of \(-30^{\circ} \mathrm{C}\).

Short answer

In this problem, we are asked to determine the maximum velocity of a cold gas flowing between two metal plates and over a cylindrical steel bolt in cryogenic equipment. We have been given the gas properties, the dimensions of the bolt, and the minimum suitable temperature for the steel bolt. The solution involves analyzing heat transfer between the flowing gas and the steel bolt. Given information: - Thermal conductivity of the gas, \(k = 0.02 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) - Kinematic viscosity of the gas, \(\nu = 9.3 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) - Prandtl number of the gas, \(Pr = 0.74\) - Diameter of the bolt, \(d = 9.5 \mathrm{~mm}\) - Length of the bolt exposed to the gas, \(l = 10 \mathrm{~cm}\) - Minimum temperature suitable for the steel bolt, \(T_{min} = -30^{\circ} \mathrm{C}\) - The temperature of the bolt is maintained by a heating mechanism capable of providing heat at, \(Q = 15 \mathrm{~W}\) To find the maximum gas velocity, we must first calculate the Nusselt number using the Roshko correlation: \(Nu = (0.62)(Re)^{1/2}(Pr)^{1/3}\) The Reynolds number (Re) is defined as: \(Re = \frac{u d}{\nu}\) Next, we can calculate the heat transfer coefficient (h) using the Nusselt number: \(h = \frac{Nu \cdot k}{d}\) Now, we can use the equation for heat transfer by convection to determine the maximum gas velocity, ensuring the temperature of the bolt does not go below \(T_{min}\): \(u_{max} = \dfrac{Q \cdot \nu}{h A d}\) With the given information, calculate the maximum gas velocity in this setting.

Step-by-step solution

List the given information

- Thermal conductivity of the gas, \(k = 0.02 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) - Kinematic viscosity of the gas, \(\nu = 9.3 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) - Prandtl number of the gas, \(Pr = 0.74\) - Diameter of the bolt, \(d = 9.5 \mathrm{~mm}\) - Length of the bolt exposed to the gas, \(l = 10 \mathrm{~cm}\) - Minimum temperature suitable for the steel bolt, \(T_{min} = -30^{\circ} \mathrm{C}\) - The temperature of the bolt is maintained by a heating mechanism capable of providing heat at, \(Q = 15 \mathrm{~W}\)

Calculate the Nusselt number using Roshko correlation

First, we need to find the Nusselt number (Nu) for the flow over the cylindrical bolt using the Roshko correlation: \(Nu = (0.62)(Re)^{1/2}(Pr)^{1/3}\) Reynolds number (Re) can be defined as: \(Re = \frac{u d}{\nu}\) Where \(u\) is the gas velocity and \(d\) is the diameter of the bolt.

Calculate the heat transfer coefficient

Once we have the Nusselt number, we'll calculate the heat transfer coefficient (h) using the following formula: \(h = \frac{Nu \cdot k}{d}\)

Determine the maximum gas velocity

Now we know the heat transfer coefficient, we can calculate the maximum gas velocity using the equation for heat transfer by convection: \(Q = h A(T_{bolt} - T_{gas})\) Here, \(A\) is the surface area of the bolt exposed to the gas, calculated using \(A = \pi d l\), and \(T_{bolt}\) and \(T_{gas}\) are the temperatures of bolt and gas at the location where the heat transfer happens. Since we are aiming to keep the bolt at its minimum suitable temperature, we will have \(T_{bolt} = T_{min}\). Plug in the values and solve for the gas velocity (\(u_{max}\)): \(u_{max} = \dfrac{Q \cdot \nu}{h A d}\) We have all the values required in the formula. Calculate the maximum gas velocity, ensuring that the temperature of the bolt does not go below \(T_{min}\).