Question

In a heating system, liquid water flows in a circular tube at a mass flow rate of \(3.5 \mathrm{~g} / \mathrm{s}\). The water enters the tube at $5^{\circ} \mathrm{C}\(, where it is heated at a rate of \)1 \mathrm{~kW}$. The tube surface is maintained at a constant temperature. The effectiveness of the system to heat water in the tube has been determined to have a number of transfer units of \(\mathrm{NTU}=2\). According to the service restrictions of the ASME Boiler and Pressure Vessel Code (ASME BPVC.IV-2015, HG-101), hot water heaters should not be operating at temperatures exceeding $120^{\circ} \mathrm{C}$ at or near the heater outlet. The inner surface of the tube is lined with polyvinylidene chloride (PVDC) lining. According to the ASME Code for Process Piping (ASME B31.3-2014, Table A323.4.3), the recommended maximum temperature for PVDC lining is \(79^{\circ} \mathrm{C}\). To comply with both ASME codes, determine \((a)\) Whether the water exiting the tube is at a temperature below \(120^{\circ} \mathrm{C}\). (b) Whether the inner surface temperature of the tube exceeds \(79^{\circ} \mathrm{C}\). Evaluate the fluid properties at \(40^{\circ} \mathrm{C}\). Is this an appropriate temperature to evaluate the fluid properties?

Short answer

Question: Calculate the final temperature of the water and check if it complies with the ASME codes to ensure safe operation. Answer: To calculate the final temperature of the water, first determine the heat capacity rate ratio (R). Next, use the effectiveness-NTU relationship to find the final temperature of the water (T_o). Check that the final temperature (T_o) is below 120°C as required by ASME Boiler and Pressure Vessel Code, and also check that the inner surface temperature does not exceed the 79°C maximum temperature for PVDC lining as per ASME B31.3-2014. Lastly, evaluate the fluid properties at 40°C to confirm the reasonability of the analysis.

Step-by-step solution

Find the heat capacity rate ratio

To find the heat capacity rate ratio (R), we need the heat capacity rates of the water and the heating medium. We are given the mass flow rate of water (\(\displaystyle m_{w}=3.5\mathrm{~g/s}\)) and initial temperature (\(\displaystyle T_{i}=5^{\circ } \mathrm{C}\)). Assuming the heating medium to be an electrical heating element with a power output of \(\displaystyle 1 \mathrm{~kW}\), we can calculate the heat capacity rate ratio as follows: \(\displaystyle R=\dfrac{C_{w}}{{C_{s}}}= \dfrac{m_{w}c_{pw}}{Q}\ \), where \(\displaystyle c_{pw}\approx 4.186\ \mathrm{J/(g\cdot K)}\) is the specific heat of water at constant pressure. Step 2: Calculate the final temperature of the water

Calculate the final temperature of the water

Using the effectiveness-NTU relationship for water inside the circular tube, we can find the final temperature of the water: Effectiveness (\(\epsilon\)) is given by the equation \(\displaystyle \epsilon = 1 - \mathrm{e}^{-\mathrm{NTU}( 1-R)}\). We know that the effectiveness of the system is given by the equation \(\displaystyle \epsilon = \dfrac{T_{o}-T_{i}}{T_{s}-T_{i}}\). From this, we can find the final temperature of the water, \(\displaystyle T_{o}\) as: \(\displaystyle T_{o}=T_{i}+\epsilon ( T_{s}-T_{i})\)

Check the final temperature

Now we have the final temperature of the water, \(\displaystyle T_{o}\). We will check whether it is below \(\displaystyle 120^{\circ }\mathrm{C}\) to comply with ASME Boiler and Pressure Vessel Code (ASME BPVC.IV-2015, HG-101). If it is below \(\displaystyle 120^{\circ }\mathrm{C}\), the heater is operating under safe conditions.

Check the inner surface temperature

To determine the inner surface temperature, we can use the equation: \(\displaystyle q=U_{s}A_{s}( T_{s}-T_{i})\), where \(\displaystyle q\) is the heat transfer rate, \(\displaystyle U_{s}\) is the overall heat transfer coefficient, and \(\displaystyle A_{s}\) is the surface area of the tube. Given that the heat transfer rate is constant, the temperature difference between the surface and the water will remain the same. Therefore, if the water temperature increases by a certain amount, the inner surface will also increase by the same amount. So, we can check whether the inner surface temperature exceeds \(\displaystyle 79^{\circ }\mathrm{C}\), which is the recommended maximum temperature for PVDC lining (ASME B31.3-2014, Table A323.4.3).

Evaluate fluid properties at \(\displaystyle 40^{\circ }\mathrm{C}\)

Now, we will evaluate the fluid properties at \(\displaystyle 40^{\circ }\mathrm{C}\) to check if it's a suitable temperature for analysis. Since the final temperature of the water lies in a range that is close to \(\displaystyle 40^{\circ }\mathrm{C}\), the assumption that fluid properties stay relatively constant is reasonable, and we can conclude that this is an appropriate temperature to evaluate the fluid properties.