How Long Should I Wait to Shower?

Last post we talked about the temperature of a hot water tank as hot water is withdrawn, and also about the temperature while it is recharging.  With the expressions that we had for those processes, it is possible to calculate the time that it would take for the tank temperature to reach a certain point, and for the tank to recharge from that point. 
 In order to keep the expressions readable, let’s call the low temperature that the tank reaches T_A, and express it in a dimensionless form:

We’ll also define the following:

Then, the time t_A for the tank temperature to reach T_A while it is discharging can be expressed:

Now, we can re-arrange the expression from the last post for the temperature while the tank is recharging to get an expression for the time it will take to recharge from the low temperature T_A all the way back to the original temperature, T₀:

This figure shows the relationship between t_A and t_r for a single discharge-recharge cycle:

With a little bit of work, we can re-arrange the expression for t_r to write it in terms of A, N, and b so that it will be in the same language that we used for t_A:

Now, we can  express both the time for a full cycle, (t_A + t_r) and the ratio of recharge time to draining time, (t_r/t_A) solely in terms of the dimensionless parameters A, N, and b.  Here are some plots showing how these are related for ranges of values that might be applicable to a residential water heater:

 

The plots above are for the case of b=1*(1/hr); other values of b would just scale the vertical axis appropriately.  For the plots below, the value of b doesn't matter because it cancels out of the ratio.

Finally, we note that the tacit assumption in all of this is that the tank temperature is dropping which is the usual case with residential water heaters.  While there is nothing that would physically require the parameter N to be less than 1, a simple energy balance shows that if N > 1, the tank temperature would increase rather than decrease.  This figure shows how the mass flow rate and heating rate are related: