Question

Liquid water flows in a tube with the inner surface lined with polytetrafluoroethylene (PTFE). The inner diameter of the tube is $24 \mathrm{~mm}\(, and its wall thickness is \)5 \mathrm{~mm}$. The thermal conductivity of the tube wall is $15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(. The water flowing in the tube has a temperature of \)50^{\circ} \mathrm{C}$, and the convection heat transfer coefficient with the inner tube surface is \(50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The outer surface of the tube is exposed to convection with superheated steam at \(600^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of $50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. According to the ASME Code for Process Piping (ASME B31.3-2014, A323), the recommended maximum temperature for PTFE lining is \(260^{\circ} \mathrm{C}\). Formulate the temperature profile in the tube wall. Use the temperature profile to determine if the tube inner surface is in compliance with the ASME Code for Process Piping.

Short answer

Question: Determine if the tube inner surface with PTFE lining is in compliance with ASME Code for Process Piping based on the given parameters and derived temperature calculations. Answer: To check if the tube inner surface is in compliance, we must calculate the temperature at the inner boundary of the tube wall, \(T_2\), and verify if this temperature is below the maximum allowable temperature for PTFE lining of \(260^{\circ} \mathrm{C}\). Calculate \(T_2\) using the derived equations and then compare it with the limit to determine compliance.

Step-by-step solution

1. Analyze Given Parameters

List down the given parameters in the problem: - Tube inner diameter: \(D_i = 24 \mathrm{~mm}\) - Tube wall thickness: \(t_w = 5 \mathrm{~mm}\) - Tube wall thermal conductivity: \(k_w = 15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) - Water temperature (\(T_1\)): \(50^{\circ} \mathrm{C}\) - Convection heat transfer coefficient of inner surface (\(h_i\)): \(50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) - Superheated steam temperature (\(T_3\)): \(600^{\circ} \mathrm{C}\) - Convection heat transfer coefficient of outer surface (\(h_o\)): \(50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\)

2. Calculate Areas and Resistance Values

First, let's find the cross-sectional area and resistance values for both convection and conduction in the tube wall: - Tube inner area: \(A_i = \pi (D_i/2)^2 = \pi ((24\times10^{-3})/2)^2\) - Tube outer diameter: \(D_o = D_i + 2t_w = 24 + 2\times5 = 34 \mathrm{~mm}\) - Tube outer area: \(A_o = \pi (D_o/2)^2 = \pi ((34\times10^{-3})/2)^2\) Now, calculate the resistance values for both convection and conduction: - Convection resistance inside: \(R_{c,i} = \frac{1}{h_i A_i}\) - Conduction resistance: \(R_{cd} = \frac{t_w}{k_w A_m}\) where \(A_m = \frac{A_i + A_o}{2}\) - Convection resistance outside: \(R_{c,o} = \frac{1}{h_o A_o}\)

3. Calculate Temperature Profile Using Electrical Analogy

Use the electrical analogy for the temperature profile calculation where the total heat flow through the tube wall is the same in every section: - Total resistance: \(R_t = R_{c,i} + R_{cd} + R_{c,o}\) - Heat flow: \(Q = \frac{T_3 - T_1}{R_t}\) Now, calculate the temperature at inner boundary of the tube wall (\(T_2\)): - Heat flow through inner convection resistance: \(Q_{c,i} = \frac{T_2 - T_1}{R_{c,i}}\) Solving for \(T_2\), we get: \(T_2 = T_1 + Q \times R_{c,i}\)

4. Check Compliance with ASME Code for Process Piping

To determine if the tube inner surface is in compliance with ASME Code for Process Piping, check if the temperature \(T_2\) is below the recommended maximum temperature for PTFE lining (\(260^{\circ} \mathrm{C}\)): - If \(T_2 \leq 260^{\circ} \mathrm{C}\), then the tube inner surface is in compliance. - If \(T_2 > 260^{\circ} \mathrm{C}\), then the tube inner surface is not in compliance. Now, calculate the temperature \(T_2\) using the equations above and check for compliance.