Keep Turning Up the Hot Water!

In the last two posts, we considered the cooling of the exit temperature of a hot water tank with a given flow rate and heat input rate, and thought about recharge times once the hot water withdrawal was stopped.  However, in real life you’d keep turning up the proportion of hot water as long as you could in order to keep your shower at a constant temperature.  Of course, the cooler the water came out from the hot water tank, the more you’d need to use, making it cool down even faster.

  To approximate this effect, let’s modify our tank diagram to include an automated mixing valve (black box).  


Our assumption now is that the mass flow rate coming out of the valve (ṁ_m) is constant, but the flow rate through the water heater, ṁ, and the flow rate of cool mixing water, ṁ_i, will both change with time.  Also, the water temperature leaving the mixing valve is held constant at T_m.  (That is the purpose of the mixing valve, after all).


We still are interested in calculating the tank temperature as a function of time, T(t).
We can start by doing a First Law balance on the flow through the mixing valve. After some manipulation, this yields:

According to our assumptions, everything on the right side of the equation is constant with respect to time. Returning to a First Law balance on the tank, it turns out that this results in the tank temperature decreasing linearly with time, like this:

The recharge rate, once the water is turned off, is still linear, just as before.  So now, draining the tank down to some temperature, T_A, and recharging back to the original temperature, T₀, looks like this:

Since both sides (discharge and recharge) are linear functions with constant slope, any increase in t_r results in a proportional increase in t_A, and the ratio t_r/t_A no longer depends on the size of T_A: 

and can be expressed as simply:

where N_m looks similar to the parameter N that we used before, but has the constant ṁ_m in the denominator:

This figure shows t_r/t_A as a function of N_m: