Question

Liquid water flows in a circular tube at a mass flow rate of $0.12 \mathrm{~kg} / \mathrm{s}\(. The water enters the tube at \)65^{\circ} \mathrm{C}\(, where it is heated at a rate of \)5.5 \mathrm{~kW}$. The tube is circular with a length of \(3 \mathrm{~m}\) and an inner diameter of $25 \mathrm{~mm}$. The tube surface is maintained isothermal. The inner surface of the tube is lined with polyvinylidene chloride (PVDC) lining. The recommended maximum temperature for PVDC lining is \(79^{\circ} \mathrm{C}\) (ASME Code for Process Piping, ASME B31.3-2014, Table A323.4.3). Is the PVDC lining suitable for the tube under these conditions? Evaluate the fluid properties at \(70^{\circ} \mathrm{C}\). Is this an appropriate temperature at which to evaluate the fluid properties?

Short answer

Provide a brief explanation for your answer.

Step-by-step solution

Calculate the final temperature of water exiting the tube

To find the final temperature, we can use the energy balance equation: \(q = m \cdot C_p \cdot \Delta T\), where \(q\) is the heat input rate, \(m\) is the mass flow rate of water, \(C_p\) is the specific heat capacity of water, and \(\Delta T\) is the change in temperature. We are given \(q = 5.5 \mathrm{~kW}\) and \(m = 0.12 \mathrm{~kg} / \mathrm{s}\). The specific heat capacity of water at \(70^{\circ} \mathrm{C}\) can be taken as \(C_p = 4.18 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\). We can rearrange the energy balance equation to find \(\Delta T\): \(\Delta T = \frac{q}{m \cdot C_p}\)

Calculate the change in temperature and the final temperature

Now, we can plug the given values into our formula: \(\Delta T = \frac{5.5 \mathrm{~kW}}{0.12 \mathrm{~kg} / \mathrm{s} \cdot 4.18 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}} = 10.85 ^{\circ} \mathrm{C}\) The initial temperature of the water is \(65^{\circ}\mathrm{C}\). Therefore, the final temperature of water exiting the tube is: \(T_{final} = T_{initial} + \Delta T = 65^{\circ}\mathrm{C} + 10.85^{\circ}\mathrm{C} = 75.85^{\circ}\mathrm{C}\)

Compare the final temperature with the maximum allowable temperature for PVDC lining

The maximum allowable temperature for PVDC lining is \(79^{\circ}\mathrm{C}\). Since the final temperature of water is lower than the maximum allowable temperature for PVDC lining (\(75.85^{\circ}\mathrm{C} < 79^{\circ}\mathrm{C}\)), the PVDC lining is suitable for the tube under these conditions.

Discuss the appropriateness of evaluating the fluid properties at \(70^{\circ}\mathrm{C}\)

We evaluated the fluid properties at \(70^{\circ}\mathrm{C}\), which is an average temperature between the initial temperature and the final temperature of the water. This is an appropriate temperature for evaluating the fluid properties, as it provides an estimation of the fluid properties during the heating process when the temperature changes between the initial and final values. This estimate should provide reasonably accurate results for this problem.