Home Improvement Stack Exchange is a question and answer site for contractors and serious DIYers. It only takes a minute to sign up.
Sign up to join this community
Given a 40 gallon natural gas water heater, and a shower head with a flow rate of 2.5 gallons per minute. How long should the shower be able to maintain 105°F water temperature?
Assume the average cold water supply temperature is 58.7°F. The tank will be set at 140°F, to avoid Legionella. The tank recovery is typical of a natural gas heater, so it can recover its volume in an hour (40 gph in this case).
Model the water heater as a continuously stirred tank reactor (CSTR), so it is always at a uniform temperature. Assume the recovery time is not dependent on temperature and completely accounts for insulation losses and the like. Neglect losses in pipes and assume the operator controls the shower temperature to 105°F perfectly. Taking the stopping criterion from the question, the shower is over when the water in the tank becomes 105°F.
The recovery rate raises 140 gallons of water by 81.3°F in one hour. We'll say this is a constant heat input of 9118 W.
The key is that a constant-temperature shower removes a constant rate of heat from the tank, that which is associated with raising 2.5 gpm of water from 58.7°F to 105°F. This is 16986.5 W.
The difference is 7868.5 W. With constant heat, you don't need any complicated integrals.
The time for the tank to drop from 140°F to 105°F is 1568 s or 26 minutes.
The (maybe counterintuitive) fact that the variable flow rate from the water heater does not influence the rate of heat removal from the water heater comes from the fact that the incoming cold water is the same temperature at the shower and at the water heater.
Note that, because 
does not appear in the expression for 
, is now a constant.
Hot Water % = (Mixed Water Temp. - Cold Water Temp.) / (Hot Water Temp. - Cold Water Temp.)might be helpful. You can also assume the showerer, is constantly adjusting the knobs to maintain a 105°F water temperature for as long as possible. Once the cold knob is all the way off, and the hot is all the way on. The showerer will stay in the shower until the water temperature drops below 80°F, at which point they'll get out of the shower.