In the hydraulic system depicted in the figure, the main water tank is buried underground, and the overflow valve is open to the atmosphere. $40,000$ liters of water per hour is supplied from this tank to two separate lines, A and B$,$ located at different heights. Line A is equipped with a valve, heater, and expansion tank, while line B has a filter, valve, and expansion tank. The main line uses DN$100$ stainless steel pipes, DN$60$ pipes for line A$,$ and DN$40$ pipes for line B$.$ With this configuration:

a$)$ Determine the head losses for lines A and B$.$

b$)$ Find the head produced by the pump used.

c$)$ Calculate the power of the pump used.

e$($equivalent roughness$)=0\mathrm{.}002$ mm$,$ viscosity of water is $0\mathrm{.}00112,$ La$=45$ m$,$ Lb$=35$ m$,$ and if reynold number is bigger than $4000$ use Haaland equation for friction factor, Hl$=$f$*$L$*$V$^2($D$*2*$g$),$ g$=9\mathrm{.}81$ m$/$s$2$NOTE:

Pwater $=997k\frac{g}{m}3,KK$ elbow $=0,75,KK$ valve $=2,5,KK$ filter $=24,KK$ heater $=12,KK$ sudden expansion $=1,5$