I was thinking: if I change the mass flow rate (by opening the tap when I'm going to take a shower) the water will take more or less time to heat?

I've proceeded as follows:

by making the heater my control volume:

\(\displaystyle \dot{Q}=\frac{d(nU)}{dt} + \dot{n}(H^{out}-H^{in}) \)

since the mass, and consequently the moles, inside the heater does not change with time,

\(\displaystyle \dot{Q}=n\frac{dU}{dt}+\dot{n}(H^{out}-H^{in}) \)

stating the hypothesis of incompressible fluid,

\(\displaystyle \dot{Q}=nC_P\frac{dT}{dt}+\dot{n}\int\limits _{T_0}^{T} C_Pd\tau \)

where \(\displaystyle T_0 \) is the initial temperature, let's say 293.15 K;

it is common to express, for a liquid,

\(\displaystyle \frac{C_P}{R}=A+BT+CT^2 \)

in this case I'll consider only the first term (since my goal here is not to get an A+ in differential equations):

\(\displaystyle \dot{Q}=nRA\frac{dT}{dt}+\dot{n}RA(T-T_0) \)

just making \(\displaystyle \theta (t)=T(t)-T_0 \), and my initial condition being \(\displaystyle \theta (0)=0 \),

\(\displaystyle \frac{\dot{Q}}{nRA}=\dot{\theta}+\frac{\dot{n}}{n}\theta \)

then,

\(\displaystyle T(t)=T_0+\frac{\dot{Q}}{\dot{n}RA}\left[1-\exp\left(-\frac{\dot{n}t}{n}\right)\right] \)

So, after some quick research I've found that \(\displaystyle A=8.712 \) and that a usual heater provides a maximum power of \(\displaystyle \dot{Q}=28 \) kW for 21 L/min of water flowing trough it. At this point I've made other hypothesis: that the ratio \(\displaystyle \frac{\dot{Q}}{\dot{n}} \) is constant - since there must be a controller inside the heater in some way that it will not provide the maximum power when the flow rate is small.

All right, so I can finally write,

\(\displaystyle T(t)=T_0+K\left[1-\exp\left(-\frac{\dot{n}t}{n}\right)\right] \)

where \(\displaystyle K \) is constant. Also, \(\displaystyle \lim_{t\to \infty} T(t)=T_0+K \), which makes sense.

So, looking for the equation above I can tell that if I raise the mass flow rate (by opening the tap) it will take less time to the water reach some temperature.

**I want to ask you: is there any mistake that I've committed? For me, the final result doesn't make sense - I was expecting that a lower flow rate would make my time waiting for shower decrease.**

PS. I'm really sorry if you can't get something, I'm not a native speaker. Thanks everybody!