Originally Posted by **tom89** **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.** |

I get the same result, but a different conclusion, because you need to consider the dependence of $\displaystyle K$ on $\displaystyle \dot{n}$.

If you rearrange the new formula for $\displaystyle t$ and substitute for $\displaystyle K$, you get

$\displaystyle t = \frac{n}{\dot{n}} \ln \left(\frac{1}{1 - \frac{\dot{n}RA}{\dot{q}}\left(T - T_0\right)}\right)$

This formula is the product of two factors, one which is proportional to $\displaystyle 1/\dot{n}$ and one which is proportional to the logarithm term. The two factors compete with each other to be the main factors affecting the result. When I plotted $\displaystyle t$ as a function of $\displaystyle \dot{n}$, I got a slowly increasing slope, so time to heat

**increases** with increasing $\displaystyle \dot{n}$. This was because the logarithm term tends to win out with the choice of parameters I used, but I need to plot it with the particular parameters you're adopting to check whether your conclusion is consistent.