Question

Liquid water flows in fully developed conditions through a circular tube at a mass flow rate of \(3.5 \mathrm{~g} / \mathrm{s}\). The water enters the tube at \(5^{\circ} \mathrm{C}\), and the average convection heat transfer coefficient for the internal flow is $20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. The tube is \)3 \mathrm{~m}$ long and has an inner diameter of \(25 \mathrm{~mm}\). The tube surface is subjected to a constant heat flux at a rate of \(300 \mathrm{~W}\). 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 \(79^{\circ} \mathrm{C}\). Would the inner surface temperature of the tube exceed the recommended maximum temperature for PVDC lining? If so, determine the axial location along the tube where the tube's inner surface temperature reaches $79^{\circ} \mathrm{C}\(. Evaluate the fluid properties at \)15^{\circ} \mathrm{C}$. Is this an appropriate temperature at which to evaluate the fluid properties?

Short answer

If yes, what is the axial location where the maximum temperature is reached? Answer: To answer this question, we need to calculate the inner surface temperature of the tube and compare it with the recommended maximum temperature for PVDC lining (79°C). First, we calculate the heat transfer per unit length (q''), temperature increase along the tube (ΔT), and then the inner surface temperature (T_surface) using the formulas provided in the solution. If T_surface > 79°C, then the inner surface temperature exceeds the recommended maximum temperature for PVDC lining. The axial location where the maximum temperature is reached can be determined from the temperature distribution equation mentioned in step 3. Question: Is 15°C an appropriate temperature at which to evaluate the fluid properties? Answer: To determine if 15°C is an appropriate temperature at which to evaluate the fluid properties, we compare it with the average fluid temperature (T_fluid) calculated in the solution. If 15°C is approximately equal to T_fluid, then it is an appropriate temperature for evaluating the fluid properties.

Step-by-step solution

Calculate heat transfer per unit length

First, we need to determine the heat transfer per unit length \((q'')\). We are given the total heat transfer rate \((Q = 300 \mathrm{~W})\) and the inner circumference of the tube \((C = \pi D = \pi * 25 * 10^{-3} \mathrm{~m})\). We can then compute the heat transfer per unit length: $$ q'' = \frac{Q}{C} = \frac{300}{\pi * 25 * 10^{-3}} \mathrm{~W} / \mathrm{m} $$

Find the temperature increase along the tube

Next, we need to find how much the temperature increases along the tube due to the constant heat flux. To do so, we will calculate the heat transfer per unit length per unit mass flow rate, which is the increase in the fluid's enthalpy: $$ \Delta h = \frac{q''}{\dot{m}} = \frac{300}{3.5 * 10^{-3} * \pi * 25 * 10^{-3}} \mathrm{~W} / \mathrm{g} \cdot \mathrm{s} $$ Assuming the specific heat capacity \((c_p)\) of water remains constant, we can find the corresponding temperature increase: $$ \Delta T = \frac{\Delta h}{c_p} = \frac{300}{3.5 * 10^{-3} * \pi * 25 * 10^{-3} * c_p} \mathrm{~K} $$

Calculate the inner surface temperature of the tube

Next, we will calculate the inner surface temperature of the tube by using the average convection heat transfer coefficient \((h = 20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K})\) and the heat transfer per unit length: $$ T_\mathrm{surface} = T_\mathrm{fluid} + \frac{q''}{h A} = T_\mathrm{fluid} + \frac{q''}{h \pi D L} $$ where \(T_\mathrm{fluid}\) is the average fluid temperature and \(A = \pi DL\) is the internal surface area. From the temperature increase calculated in step 2, the exit fluid temperature \((T_\mathrm{exit})\) can be determined. Since the water enters the tube at a temperature of \(5^{\circ} \mathrm{C}\), we find the average fluid temperature along the tube: $$ T_\mathrm{fluid} = \frac{T_\mathrm{exit} - T_\mathrm{inlet}}{2} + T_\mathrm{inlet} $$

Compare the inner surface temperature with the recommended maximum temperature for PVDC lining

Now we have all the necessary information to calculate the local inner surface temperature at the axial location along the tube. We will substitute the given values and compare the calculated inner surface temperature with the recommended maximum temperature for PVDC lining \((T_\mathrm{max} = 79^{\circ} \mathrm{C})\). We can rearrange the equation for \(T_\mathrm{surface}\) and plug in the values to get: $$ T_\mathrm{surface} = T_\mathrm{fluid} + \frac{300}{3.5 * 10^{-3} \pi * 25 * 10^{-3} * c_p * 20 * \pi * 25 * 10^{-3} * 3} $$ We then compare this value with the recommended maximum temperature for PVDC lining: $$ T_\mathrm{surface} > T_\mathrm{max} $$

Determine if \(15^{\circ} \mathrm{C}\) is an appropriate temperature at which to evaluate the fluid properties

Finally, we need to evaluate if \(15^{\circ} \mathrm{C}\) is an appropriate temperature to evaluate the fluid properties. We will compare the average fluid temperature \((T_\mathrm{fluid})\) with this value: $$ 15^{\circ} \mathrm{C} \approx T_\mathrm{fluid} $$ If they are approximately equal, then \(15^{\circ} \mathrm{C}\) is an appropriate temperature at which to evaluate the fluid properties.