Question

A series of ASME SA-193 carbon steel bolts are bolted onto the upper surface of a metal plate. The metal plate has a thickness of \(3 \mathrm{~cm}\), and its thermal conductivity is \(15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The bottom surface of the plate is subjected to a uniform heat flux of $5 \mathrm{~kW} / \mathrm{m}^{2}$. The upper surface of the plate is exposed to ambient air with a temperature of \(30^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The ASME Boiler and Pressure Vessel Code (ASME BPVC.IV-2015, HF-300) limits the maximum allowable use temperature to \(260^{\circ} \mathrm{C}\) for the SA-193 bolts. Formulate the variation of temperature in the metal plate, and determine the temperatures at \(x=0,1.5\), and \(3.0 \mathrm{~cm}\). Would the SA-193 bolts comply with the ASME code?

Short answer

Answer: No, the SA-193 bolts do not comply with the ASME code, as the highest temperature on the plate (530°C) is higher than the maximum allowable use temperature for the bolts (260°C).

Step-by-step solution

Understand the problem and note the given information

The given information is: Thickness of the plate: \(h = 3 \mathrm{~cm}\) Thermal conductivity: \(k = 15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) Heat flux: \(q_s = 5 \mathrm{~kW} / \mathrm{m}^{2}\) Ambient temperature: \(T_\infty = 30^{\circ} \mathrm{C}\) Convection heat transfer coefficient: \(h_c = 10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) Maximum allowable use temperature for SA-193 bolts: \(T_{max} = 260^{\circ} \mathrm{C}\)

Calculate the steady-state heat conduction equation for the metal plate

To find the steady-state heat conduction equation for the metal plate, we can use Fourier's Law: \(\frac{dT}{dx} = -\frac{q_s}{k}\), where \(dT\) is the temperature change across a distance \(dx\), and \(q_s\) is the heat flux. By integrating Fourier's Law, we get the temperature distribution equation: \(T(x) = -\frac{q_s}{k}x + C1\), where \(C1\) is the constant of integration.

Apply boundary conditions

We have two boundary conditions: 1. At the upper surface (ambient air), where \(x = 0,\) the temperature is: \(T(0) = T_\infty = 30^{\circ} \mathrm{C}.\) 2. At the lower surface (where heat flux is applied), \(x = 3 cm,\) and the heat flux can be related to the convection coefficient and temperature difference by \(q_s = h_c(T(3) - T_\infty).\) Plugging in the numbers and solving for \(T(3),\) we find that: \(T(3) = T_\infty + \frac{q_s}{h_c} = 30^{\circ} \mathrm{C} + \frac{5\times 10^3 \mathrm{~W} / \mathrm{m}^{2}}{10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}} = 530^{\circ} \mathrm{C}.\)

Calculate the temperature distribution equation with the boundary conditions

Substituting \(x = 0\) and \(T_\infty = 30^{\circ} \mathrm{C}\) into the temperature distribution equation, we get: \(30^{\circ} \mathrm{C} = -\frac{5 \times 10^3 \mathrm{~W} / \mathrm{m}^{2}}{15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}}(0) + C1.\) This implies that \(C1 = 30^{\circ} \mathrm{C}.\) Thus, the temperature distribution equation becomes: \(T(x) = -\frac{5 \times 10^3 \mathrm{~W} / \mathrm{m}^{2}}{15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}}x + 30^{\circ} \mathrm{C}.\)

Calculate the temperatures at \(x = 0, 1.5\) cm, and \(3.0\) cm

We can now calculate the temperature at each point by plugging the respective x values into the temperature distribution equation: \(T(0) = 30^{\circ} \mathrm{C}\) (already known from boundary condition 1) \(T(1.5) = -\frac{5 \times 10^3 \mathrm{~W} / \mathrm{m}^{2}}{15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}}(0.015 \mathrm{~m}) + 30^{\circ} \mathrm{C} \approx 145^{\circ} \mathrm{C}\) \(T(3) = 530^{\circ} \mathrm{C}\) (already known from boundary condition 2)

Determine if the SA-193 bolts comply with the ASME code

According to the ASME code, the maximum allowable use temperature for the SA-193 bolts is 260°C. As the highest temperature on the plate is 530°C, which is higher than the maximum allowed temperature, the bolts do not comply with the ASME code.