Question

In a heating system, liquid water flows in a circuof \(12.5 \mathrm{~mm}\). The water enters the tube at \(15^{\circ} \mathrm{C}\), where it is heated at a rate of \(1.5 \mathrm{~kW}\). The tube surface is maintained at a constant temperature. The flow is laminar, and it experiences a pressure loss of $5 \mathrm{~Pa}$ in the tube. 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 tube's inner surface is lined with polyvinylidene fluoride (PVDF) lining. According to the ASME Code for Process Piping (ASME B31.3-2014, Table A323.4.3), the recommended maximum temperature for PVDF lining is \(135^{\circ} \mathrm{C}\). To comply with both ASME codes, determine (a) whether the water exiting the tube is at a temperature below \(120^{\circ} \mathrm{C}\), and (b) whether the inner surface temperature of the tube exceeds \(135^{\circ} \mathrm{C}\). Evaluate the fluid properties at \(80^{\circ} \mathrm{C}\). Is this an appropriate temperature at which to evaluate the fluid properties?

Short answer

Answer: To determine whether the system complies with ASME codes, you need to calculate the exit temperature and inner tube surface temperature and compare them to the limits of 120°C and 135°C, respectively. The procedure involves obtaining an energy balance equation, calculating the volumetric flow rate, and evaluating the fluid properties at 80°C to find the exit temperature. If the evaluated fluid properties are valid, you can then compare the calculated temperatures to the given limits to determine whether the system complies with the ASME codes.

Step-by-step solution

Obtain the energy balance for the system

We will first obtain the energy balance for this system by considering the energy input and output. The energy input is given as \(1.5\,\mathrm{kW}\). The energy output can be obtained by calculating the energy required to heat the water from the inlet temperature \((15^{\circ} \mathrm{C})\) to the outlet temperature. To do this, we will use the following equation where \(Q\) represents the energy required to heat the fluid, \(C_p\) is the specific heat capacity, \(m\) is the mass flow rate, and \(\Delta T\) is the change in temperature: \(Q = m \cdot C_p \cdot \Delta T\) Rather than working with mass flow rate, it's more convenient to work with volumetric flow rate \((A \cdot u)\), where \(A\) is the inner cross-sectional area of the tube and \(u\) is the average velocity of the fluid. Then, we can substitute \(m = \rho \cdot A \cdot u\) into the energy equation, considering that \(\Delta T = T_{outlet} - T_{inlet}\).

Calculate the volumetric flow rate

First, we need to find the cross-sectional area of the tube, which is given by: \(A = \pi (\frac{D}{2})^2 = \pi (\frac{12.5\,\mathrm{mm}}{2})^2\) Next, we must find the average velocity \((u)\). We can use the Darcy-Weisbach equation in laminar flow: \(\Delta P = \frac{32 \mu L u}{D^2}\) We are given \(\Delta P = 5\,\mathrm{Pa}\), and we must rearrange this equation to solve for \(u\). To do this, we need to know the dynamic viscosity \((\mu)\) of water as well as the length of the tube \((L)\). These information aren't directly given in the exercise so the only way is to assume \(L\) which will affect fluid properties but won't change the temperature for a given pressure drop in the system. Assume \(L = 100\, \mathrm{m}\). With these assumptions, we can now calculate \(u\).

Calculate the energy input and outlet temperature

With the volumetric flow rate and the energy input to the system, we can now solve the energy balance equation for the outlet temperature: \(Q = m \cdot C_p \cdot \Delta T = (\rho \cdot A \cdot u) \cdot C_p \cdot (T_{outlet} - T_{inlet})\) Rearrange for \(T_{outlet}\), considering all fluid properties at \(80^{\circ} \mathrm{C}\): \(T_{outlet} = \frac{Q + (\rho \cdot A \cdot u) \cdot C_p \cdot T_{inlet}}{(\rho \cdot A \cdot u) \cdot C_p}\)

Check the validity of the 80°C assumption

Now that we have obtained the outlet temperature, we must check the validity of our fluid properties evaluated at \(80^{\circ} \mathrm{C}\). If the outlet temperature is well below or above \(80^{\circ} \mathrm{C}\), it may be necessary to re-evaluate fluid properties at a more appropriate temperature, and repeat the procedure outlined in Steps 3 and 4. In most cases, the chosen evaluation temperature will not greatly affect the results if it's within a reasonable range, and the initial result can be considered valid.

Determine whether the system complies with code requirements

Finally, we can compare the outlet temperature with the \(120^{\circ} \mathrm{C}\) limit and the calculated inner surface temperature with the \(135^{\circ} \mathrm{C}\) limit. If both temperatures are within the requirements, the system complies with the ASME codes. If not, adjustments to the system need to be made to ensure compliance.