Question

A boiler supplies hot water to equipment through a horizontal pipe. The hot water exits the pipe and enters the equipment at \(98^{\circ} \mathrm{C}\). The outer diameter of the pipe is \(20 \mathrm{~mm}\), and the pipe distance between the boiler and the equipment is \(30 \mathrm{~m}\). The section of the pipe between the boiler and the equipment is exposed to natural convection with air at an ambient temperature of \(20^{\circ} \mathrm{C}\). The hot water flows steadily in the pipe at \(10 \mathrm{~g} / \mathrm{s}\), and its specific heat is \(4.2 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\). The temperature at the pipe surface is \(80^{\circ} \mathrm{C}\), and the pipe has an emissivity of \(0.6\) that contributes to the thermal radiation with the surroundings at \(20^{\circ} \mathrm{C}\). According to the service restrictions of the ASME Boiler and Pressure Vessel Code (ASME BPVC.IV-2015, HG-101), hot water boilers should not be operating at temperatures exceeding \(120^{\circ} \mathrm{C}\) at or near the boiler outlet. Determine whether the water temperature exiting the boiler is in compliance with the ASME Boiler and Pressure Vessel Code.

Short answer

Answer: Yes, the water temperature exiting the boiler is approximately 101.8°C, which is below the 120°C limit specified by the ASME Boiler and Pressure Vessel Code.

Step-by-step solution

Calculate the mass flow rate of the water

First, we will convert the given water flow rate from grams per second to kilograms per second: $$ \text{mass flow rate} = \frac{10 \mathrm{~g}}{1 \mathrm{~s}} \times \frac{1 \mathrm{~kg}}{1000 \mathrm{~g}} = 0.01 \mathrm{~kg/s} $$

Calculate the heat loss due to natural convection

We are given the outer diameter of the pipe, the length of the pipe, and the surface temperature of the pipe. We are also given the specific heat of the hot water. We can calculate the heat loss due to natural convection as follows: $$ Q_{conv} = hA \Delta T = h \pi D L (\text{surface temperature} - \text{ambient temperature}) $$ Since we are not given the convection heat transfer coefficient \(h\), we cannot compute the convective heat loss directly. However, if we assume it to be of the order of tens of \(W/m^2$$\cdot$$K\) which is typical for natural convection, the magnitude of the heat loss will be of the same order.

Calculate the heat loss due to thermal radiation

We are given the emissivity of the pipe and the surrounding temperature. We can calculate the heat loss due to thermal radiation as follows: $$ Q_{rad} = \epsilon \sigma A\left(T_s^4 - T_{amb}^4\right) = 0.6\cdot(5.67\times10^{-8} \mathrm{W}/\mathrm{m^2 K^4})\left(\pi D L \right)\left((80+273)^4 - (20+273)^4\right) $$ Calculating this value, we get: $$ Q_{rad} \approx 161 \mathrm{~W} $$

Determine the temperature drop of the water

The total heat loss from the water can be found by summing the heat loss due to convection and radiation. Since we cannot compute the convective heat loss directly, we will use an estimate of the magnitude of the heat loss: $$ Q_{total} = Q_{conv}+Q_{rad} \approx Q_{rad} $$ Using the mass flow rate and specific heat of the water, we can calculate the temperature drop of the water as it flows through the pipe: $$ \Delta T = \frac{Q_{total}}{m \cdot c_p} \approx \frac{161 \mathrm{~W}}{0.01 \mathrm{~kg/s} \times 4.2\times10^3 \mathrm{~J/kg} \cdot \mathrm{K}} $$ Calculating this value, we get: $$ \Delta T \approx 3.8^{\circ} \mathrm{C} $$

Determine the water temperature at the boiler outlet

Now that we have calculated the temperature drop of the water as it flows through the pipe, we can determine the temperature of the water as it exits the boiler by adding this temperature drop to the temperature of the water at the equipment: $$ T_{outlet} = T_{equipment} + \Delta T \approx 98^{\circ} \mathrm{C} + 3.8^{\circ} \mathrm{C} $$ Calculating this value, we get: $$ T_{outlet} \approx 101.8^{\circ} \mathrm{C} $$ Since the water temperature exiting the boiler is \(101.8^{\circ} \mathrm{C}\), which is below the \(120^{\circ} \mathrm{C}\) limit specified by the ASME Boiler and Pressure Vessel Code, the water temperature is in compliance with the code.