Water flows through a $3$ cm diameter and $40$ meters long cast iron pipe inclined at an angle of $15$ degrees from the horizontal at a flow rate of $0\mathrm{.}04$ m$^3/$s$.$

a$)$ Calculate the average velocity of the water in the pipe.

b$)$ Determine the pressure drop due to friction in the pipe using the Darcy$-$Weisbach equation. You will need to find the friction factor $($f$)$ for cast iron pipes at the specific Reynolds number for this flow.

c$)$ Considering the pipe is inclined, calculate the pressure change due to the elevation difference.

d$)$ Determine the total head loss in the pipeline by combining the pressure drop due to friction $($from part b$)$ and the pressure change due to elevation $($from part c$).$

Data:

Pipe diameter $(D)=0\mathrm{.}03m$

Pipe length $(L)=40m$

Flow rate $(Q)=0\mathrm{.}04\frac{m^3}{s}$

Water density $()=1000k\frac{g}{m^3}$

Gravity $(g)=9\mathrm{.}81\frac{m}{s^2}$

Pipe angle $()=15$ degrees