Question

A series of long stainless steel bolts (ASTM A437 B4B) are fastened into a metal plate with a thickness of \(4 \mathrm{~cm}\). 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 \(7.8 \mathrm{~g} / \mathrm{cm}^{3}\). For the metal plate, the specific heat, thermal conductivity, and density are $500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, 16.3 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(, and \)8 \mathrm{~g} / \mathrm{cm}^{3}$, respectively. The upper surface of the plate is occasionally exposed to cryogenic fluid at \(-70^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of $300 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The lower surface of the plate is exposed to convection with air at \(10^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of \(10 \mathrm{~W} / \mathrm{m}^{2}\). K. The bolts are fastened into the metal plate from the bottom surface, and the distance measured from the plate's upper surface to the bolt tips is \(1 \mathrm{~cm}\). The ASME Code for Process Piping limits the minimum suitable temperature for ASTM A437 B4B stainless steel bolt to \(-30^{\circ} \mathrm{C}\) (ASME B31.3-2014, Table A-2M). If the initial temperature of the plate is \(10^{\circ} \mathrm{C}\) and the plate's upper surface is exposed to the cryogenic fluid for \(9 \mathrm{~min}\), would the bolts fastened in the plate still comply with the ASME code? Using the explicit finite difference formulations with a uniform nodal spacing of \(\Delta x=1 \mathrm{~cm}\), determine the temperature at each node for the duration of the upper surface being exposed to the cryogenic fluid. Plot the nodal temperatures as a function of time.

Short answer

To determine whether the bolts comply with the ASME code, we must check the temperature at node 4 (the location of the bolts) at the end of the 9-minute exposure time. According to the ASME code, the minimum suitable temperature for this type of bolt is -30°C. By following the step-by-step solution and implementing the finite difference method, we can find the temperature at node 4 at the end of the exposure time. If the temperature is above -30°C, the bolts will comply with the ASME code.

Step-by-step solution

Set up the finite difference grid

First, we need to set up the finite difference grid. Since the plate is 4 cm thick and the nodal spacing is 1 cm, we will have 4 nodes, each 1 cm apart. Additionally, we need to keep track of the time steps, since we're given 9 minutes to analyze the problem.

Set up the explicit finite difference equations

For each node, we can write an explicit finite difference equation relating the temperature at the current time step to the temperature at the next time step. For node 1 (the upper surface of the plate): $$ T_1^{n+1} = T_1^n + \frac{\Delta t}{\rho c_p \Delta x^2} \alpha[T_2^n - T_1^n - h(T_1^n - T_{\infty})] $$ For all other nodes (except the last one): $$ T_i^{n+1} = T_i^n + \frac{\Delta t}{\rho c_p \Delta x^2} \alpha[T_{i-1}^n -2T_i^n + T_{i+1}^n] $$ For node 4 (the lower surface of the plate): $$ T_4^{n+1} = T_4^n + \frac{\Delta t}{\rho c_p \Delta x^2} \alpha[T_3^n - T_4^n + h(T_{\infty} - T_4^n)] $$ Where \(T_i^n\) is the temperature at node \(i\) and time step \(n\), \(\Delta t\) is the time step size, \(\rho\) is the density, \(c_p\) is the specific heat, \(\Delta x\) is the nodal spacing, \(\alpha\) is the thermal diffusivity, and \(h\) is the convection heat transfer coefficient.

Determine initial conditions

The initial temperature of the entire plate is 10°C. Therefore, we can set the initial temperature values as follows: $$ T_1^0 = T_2^0 = T_3^0 = T_4^0 = 10^{\circ} \mathrm{C} $$

Implement the finite difference method

We will now implement the explicit finite difference method using the equations from step 2 and the initial conditions from step 3. We will do this iteratively until we reach the desired exposure time of 9 minutes.

Check compliance with ASME Code

After we determine the temperature at each node, we need to check if the ASTM A437 B4B stainless steel bolts comply with the ASME code. According to the code, the minimum suitable temperature for this type of bolt is -30°C. Therefore, we need to check the temperature at node 4 (the location of the bolts) at the end of the 9-minute exposure time. If the temperature is above -30°C, the bolts will comply with the code.

Plot the nodal temperatures as a function of time

Finally, we will plot the temperature values at each node as a function of time using the results obtained from the finite difference method. From this plot, we can analyze how the temperature evolves over time for each node and ultimately how the bolt's temperatures are affected by exposure to the cryogenic fluid.