Question

A water heater uses \(100 \mathrm{~kW}\) to heat 60 gallons of liquid water from an initial temperature of \(20^{\circ} \mathrm{C}\). The average density and specific heat of the water are $970 \mathrm{~kg} / \mathrm{m}^{3}\( and \)4.18 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}$, respectively. According to the service restrictions of the ASME Boiler and Pressure Vessel Code (ASME BPVC.IV-2015, HG-101), water heaters should not be operating at temperatures exceeding \(120^{\circ} \mathrm{C}\) at or near the heater outlet. Determine the heating duration such that the final water temperature that would exit the water heater would be in compliance with the ASME Boiler and Pressure Vessel Code.

Short answer

Answer: The required heating duration is approximately 913.55 seconds.

Step-by-step solution

Convert gallons to cubic meters

Begin by converting the volume of water from gallons to cubic meters for further calculations. 1 gallon is approximately equivalent to 0.00378541 cubic meters, so: \(60 \text{ gallons} = 60 * 0.00378541 \mathrm{~m}^{3} \approx 0.227 \mathrm{~m}^{3}\)

Calculate the mass of water

Next, calculate the mass of the water using its volume(0.227 m\(^3\)) and given density. \(m = V * \rho = 0.227 \mathrm{~m}^{3} * 970 \mathrm{~kg} / \mathrm{m}^{3} = 220.09 \mathrm{~kg}\)

Calculate the heat required

Now, calculate the heat required to heat the water to the maximum allowable temperature (120\(^{\circ} \mathrm{C}\)) using the mass, specific heat, and initial temperature (20\(^{\circ}\mathrm{C}\)): \(Q = m * c * \Delta T = 220.09 \mathrm{~kg} * 4.18 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K} * (120^{\circ} \mathrm{C} - 20^{\circ} \mathrm{C}) = 91355.124 \mathrm{~kJ}\)

Calculate the heating duration

Finally, determine the heating duration by dividing the necessary heat by the rate at which the heater provides it (100 kW). \(T = \frac{Q}{P} = \frac{91355.124 \mathrm{~kJ}}{100 \mathrm{~kW}} = 913.55124 \mathrm{~s}\) The heating duration that would ensure the final water temperature to be in compliance with the ASME Boiler and Pressure Vessel Code is approximately 913.55 seconds.