Who Used All the Hot Water?

You may have had the wonderful experience of starting a nice, hot shower, only to find that somebody has mostly drained the hot water heater tank.  This leaves you desperately turning the tap to more and more hot water while the shower gets cooler and cooler.  You lather, rinse, (no time for repeat) with feverish speed, but eventually wind up rinsing off in lukewarm (or ice-cold) water.  If so, this post is for you.

In addition to supplying hot water for your home, hot water heaters and other mixing tanks are used in all kinds of industrial applications, so it is useful to have a model for evaluating and/or predicting their performance.

 

For the purposes of analysis, we’ll consider the tank to be well-mixed and at steady-state, meaning that the water in the tank is all at the same temperature, and water enters and leaves at the same mass flow rate.  As depicted in this sketch, cold water comes into the tank at temperature T_i and mass flow rate ṁ, and leaves at the same mass flow rate.  Heat is added to the water in the tank at the rate of Q(dot).  The water temperature of the tank is assumed to initially be T_o.  The total mass of water in the tank is m and the specific heat of the water is c. We are interested in examining the temperature of the water in the tank as a function of time: T(t). 

 We can begin with the first law of thermodynamics for an open system:

  which leads to this expression for the tank temperature as a function of time, T(t):

Now, as time increases to t-> very large, it is obvious that:

We can look at the temperature as a function of time in two different ways.  The top figure shows a dimensionless temperature as a function of time, and the bottom figure depicts the actual temperature, (in say, degrees F or degrees C), as a function of time.  In both cases, the temperature which is approached by the tank after a very long time is indicated on the lower right side.

Now, when hot water isn’t being drawn off, and the tank heater is still running, the tank temperature recharges linearly, like this, where T_A is the initial temperature of the tank when the hot water withdrawal quits:

 As before, the temperature as a function of time is shown in both dimensionless and dimensional coordinates.  In this case, the recharge rate is proportional to the heating rate, and inversely proportional to the size of the tank (m is the mass of the water in the tank).

OK, as a last thought, is there some hot water withdrawal rate that would just balance the cold water coming in so that the temperature of the tank doesn’t change at all?  Of course, this is easy to see by imagining: 

and substituting this expression into the first equation for T(t).  This immediately leads to: 

and it is clear that for this mass flow rate, the temperature, T(t), just stays stationary at the original temperature, T₀.

Next post we'll talk about how long it takes to discharge and recharge the hot water tank.