Answer
$\mathcal{V}=1.54\ m/s$
$\Delta t=7.21\ h$
Work Step by Step
$\mathcal{V}=\sqrt{\frac{2gz}{1.5+fL/D}}$
Constant $g=9.81\ m/s,\ f=0.015,\ L=100m,\ D=0.10\ m$
$\mathcal{V}=1.0905\sqrt{z}=k\sqrt{z}$
Initially $z_1=2\ m$: $\mathcal{V}=1.54\ m/s$
Mass balance:
$\frac{dm}{dt}=-\dot{m}_e$
$\rho\frac{dV}{dt}=-\rho\frac{\pi D^2}4\mathcal{V}$
$\frac{\pi D_0^2}4 \frac{dz}{dt}=-\frac{\pi D^2}4k\sqrt{z}$
Defining $k^*=k\frac{D^2}{D_0^2}=1.0905\times10^{-4} s^{-1}$
$\frac{dz}{dt}=-k^*\sqrt{z}$
Integrating from beginning to end:
$[2z^{1/2}]_{z_0}^0=-k^*\Delta t$
$\Delta t=25938\ s=7.21\ h$
