The maximum flow rate of standard shower heads is about 3.5 gpm (13.3 L/min) and can be reduced to 2.75 gpm (10.5 L/min) by switching to low-flow shower heads that are equipped with flow controllers. Consider a family of four, with each person taking a 5-min shower every morning. City water at 15°C is heated to 55°C in an electric water heater and tempered to 42°C by cold water at the T-elbow of the shower before being routed to the shower heads. Assuming a constant specific heat of 4.18 kJ/kg·°C for water, determine (a) the ratio of the flow rates of the hot and cold water as they enter the T-elbow and (b) the amount of electricity that will be saved per year, in kWh, by replacing the standard shower heads with the low-flow ones.
(a)
The ratio of flow rates of hot and cold water as they enter the T-elbow.
Answer to Problem 168RP
The ratio of flow rates of hot and cold water as they enter the T-elbow is
Explanation of Solution
Here, the two streams (comparatively hot and cold) of fluids are mixed in a rigid mixing chamber and operates at steady state. Hence, the inlet and exit mass flow rates are equal.
Write the energy rate balance equation for two inlet and one outlet system.
Here, the rate of heat transfer is
The system is at steady state. Hence, the rate of change in net energy of the system becomes zero.
Neglect the heat transfer, work transfer, kinetic and potential energies.
The Equation (II) reduced as follows.
Substitute
The change in enthalpy is expressed as follow.
Here, the specific heat is
Substitute
Conclusion:
Substitute
Thus, the ratio of flow rates of hot and cold water as they enter the T-elbow is
(b)
The amount of electricity that will be saved per year, in
Answer to Problem 168RP
The amount of electricity that will be saved per year, in
Explanation of Solution
Here, the volume flow rate in the shower heads is lowered by installing low-flow shower heads equipped with flow controllers. This reduces the volume flow rate of water that is consumed per person.
The volume flow rated saved is expressed as follows,
The total volume flow rate saved by the family per year is expressed as follows.
Write the formula for total mass of water saved per year.
Write the formula for energy saved due to the installation of lower shower heads.
Refer Table A-3 (a), “Properties of common liquids, solids, and foods”.
The specific heat at constant pressure
The density of the water
Conclusion:
Substitute
Substitute
Substitute
Thus, the amount of electricity that will be saved per year, in
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