Question $3$: A drip irrigation system uses perforated pipes to provide water to the surrounding area. If water enters the piping system $(0\mathrm{.}2m$ in diameter$)$ with a constant velocity of $7\frac{m}{s}$ and exits at discharge velocity $($v$)$ through the outlet. Water also leaves through the walls of the pipes in a $360$ pattern. The flow is considered steady and discharge through the wall of the pipe is linear as seen in the figure below. Determine the discharge velocity $($v$).$

A$.$ Draw a $3$D figure of the system.

B$.$ Write all necessary assumptions needed to solve this problem.

C$.$ Write the mass conservation equation for this system.

Hint: The volumetric flowrate at the walls is

${\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{0}{\overset{L}{}}}v(z)Rddz$

D$.$ Determine the discharge velocity $($v$).$

Note: The exit velocity at the wall is

$v(z)=-\frac{v}{36ft}x+\frac{v}{3}$

Extra credit. Derive the equation for the exit velocity at the wall shown above.