$(30$ points$)$ Consider the piping system shown below. The cross$-$sectional area of the pipe is $5\mathrm{.}10{10}^{-4}{m}^2.$ Water enters and leaves the piping system at the same speed. The goal of the pump is to increase the water pressure at the outlet by a factor of $1\mathrm{.}1($when compared to the pressure at the inlet.$)$ The head loss in the piping system is given by the following equation:

$h_L=(0\mathrm{.}107\frac{s^2}{m})V^2$

where $V$ is the fluid velocity in the pipe, in $\frac{m}{s}.$ Note that the outlet is at a lower height than the inlet $($in the figure, gravity applies a force downward.$)$ The density of water is $1000k\frac{g}{m^3},$ and the acceleration due to gravity $(g)$ is $9\mathrm{.}81\frac{m}{s^2}.$

If the mass flow rate through the pipe is $2\mathrm{.}5k\frac{g}{s},$ how much power must the pump provide $($in W $)$ to increase the pressure from $100$ kPa to $110$ kPa $?$