How Can You Calculate the Maximum Flow Rate for a Hot Shower?

[frequentPeriod]

Homework Statement

My youngest son likes a hot shower, and wants to stay in the shower as long as possible. We do not have an on-demand water heater. He is a small boy, so he doesn't need that much water. The incoming water supply is at 45 deg F and our water heater is rated at 45,000 BTU/hour. Compute the maximum flow rate of the shower (in cubic feet per minute) so that my son can stay in the shower all day in 95 deg F water.

Initial temperature: 45F
Final Temperature: 95F
Work: 45 000 BTU/hour
Density of water:
Time: 24 hours
Heat Capacity of water = 1
densityWater = 62.428;

Homework Equations

energy = work*time
mWater = Q/(cWater*differenceInTemp)
volumeWater = mWater/densityWater;
maxFlowWater = volumeWater/(time*60);

The Attempt at a Solution

differenceInTemp = 95 - 45
work = 45000
time = 24
energy = work*time
(This is the energy used by the water heater)
cWater = 1
The energy used by the water heater is equal to the energy received by the water, so: energy = Q = mWater*cWater*differenceInTemp

mWater = energy/(cWater*differenceInTemp);
densityWater = 62.428;
volumeWater = mWater/densityWater;

maxFlowWater = volumeWater/(time*60);

I was told there was a problem with my formula, but I can't find it.

 

[Alloymouse]

By "heat capacity" am I right to assume you are using "specific heat capacity"? I'm assuming you're right with the constants - imperial units are pretty alien to me hahaha

Also, in your very last step (funding flow rate), I think the "error" came about when you divided by (time*60) - there seems to be no reason to have an additional "time" in there (dividing by 60 minutes will do, since it's what you're looking at to find cubic metres per minute).

 

[frequentPeriod]

By "heat capacity" am I right to assume you are using "specific heat capacity"? I'm assuming you're right with the constants - imperial units are pretty alien to me hahaha

Also, in your very last step (funding flow rate), I think the "error" came about when you divided by (time*60) - there seems to be no reason to have an additional "time" in there (dividing by 60 minutes will do, since it's what you're looking at to find cubic metres per minute).

I'm bad with imperial units too, they do not make any sense to me. I was just asked to use them, so I'm just 80% sure about the values.
I multiplied the time by 60 to get the flow in cubic feet per minutes, rather than cubic feet per hour. I understood that I was computing the energy for 24 hours, so I should divide it back to one hour, then one minute.

 

[Alloymouse]

I'm bad with imperial units too, they do not make any sense to me. I was just asked to use them, so I'm just 80% sure about the values.
I multiplied the time by 60 to get the flow in cubic feet per minutes, rather than cubic feet per hour. I understood that I was computing the energy for 24 hours, so I should divide it back to one hour, then one minute.

Ah, understood.

Hmmm can't find anything else that might cause and error. Have you used your formulae to work things out? If so, what is the correct answer?

 

[Chestermiller]

I think the results should be reported in gallons per minute.

 

[Alloymouse]

I think the results should be reported in gallons per minute.

The question did specify cubic feet per minute though

 

[Chestermiller]

The question did specify cubic feet per minute though

True. But, in practice, home water flows are virtually always expressed in gpm.