My wife likes to take extremely long showers. While she takes the shower, I get to watch our son, so I used to take my showers after her. Quite frequently, the shower would be cold and uncomfortable. She was okay with this for some time, but I tried to convince her to let me take a shower first, and she could go after. Since I took a faster shower, I figured she’d still have the hot water required to get her through. However, my argument did not convince her, so I was left to take cold showers.
As this went on for some time, I decided that I should change my approach on the persuasion attempt. Therefore, I actually calculated what temperature the water would be after a given amount of time. From this, I was able to tell her that, provided I finished in 10 minutes, she would have enough water. Therefore, I wanted to share exactly what went into this calculation.
To begin with, we will need some information about showers and hot water heaters. Most of this information will be different for each shower and water heater; however, here we will use common values so that the calculations will be close for most people. Therefore, let’s assume that you have a hot water heater with a capacity of 190 liters (~50 gallons), a power rating of 4000 watts and is set at the recommended 50 degrees Celsius (~125 degrees Fahrenheit). Furthermore, we will assume that the flow rate of water out of your shower is 8 liters per minute (~2 gallons per minute). We will also assume that the water going into your hot water heater flows in at the same rate water flows out and is at the room temperature of 20 degrees Celsius (~68 degrees Fahrenheit).
In order to get to our model, we will note that if we were to drain l liters of water of the hot water heater and replace it with l liters of water at room temperature, and we assume that the result is well mixed, then we can find that the new temperature of the water in the heater will be given as the weighted average of temperature of remaining water and added water. That is
In order to find the change in temperature, we would then find the difference between the old temperature and the new temperature. This will then give us,
We can then note that l is the change in volume, therefore, if we look at this change in temperature over a change in time, we get
where T is temperature and t is time. If we then solve this differential equation with the initial condition that T(0)=50, we get that
Graphically, the temperature would look like
For reference, we note that the temperature of water would be 30 degrees Celsius (86 degrees Fahrenheit) after 27.5 minutes. Therefore, after my wife’s hour long shower, the temperature would be rather cold.
I want to point out here that this model is based off the fact that the water heater isn’t on while you are showering. If the hot water heater was on, it would be adding energy to the water and would change the rate of temperature. In particular, since the water heater is rated at 4000 watts, it is adding 4000 joules of energy per second. Since it takes 4185 joules to raise the temperature of 1 kg of water 1 degree Celsius, we note that the heater will raise the temperature of all 190 liters by .3 degrees Celsius per minute. Therefore, accounting for the water heater, we have that
If we now solve this initial value problem with T(0)=50, we get
Graphically the temperature of the water would then look like,
For reference, we note that the temperature of the water will be 30 degrees Celsius after 55 minutes.
Combining these together, we see that, had the hot water heater not turned on, it is unlikely my wife would have been in the shower for an hour. However, if the hot water heater is running, the water would be getting cold as she was getting out, thus letting her have her shower and leaving me with no hot water. On the other hand, if I were to take my 10 minutes shower, she would be able to begin her shower with the water temperature of 42.5 degrees Celsius (~110 degrees Fahrenheit) which would still be plenty warm for her, she would just likely have to get out after 50 minutes in the shower instead of an hour.
After presenting this to my wife, I finally convinced her to give me a few trial runs just to see if things would work out the way I said they would. Indeed, we have kept this schedule going since then, because she is still able to get in a warm shower after me. However, if I’m not done after the ten minutes, she does kick me out to make sure she gets her warm water.