Determine the force $P$ needed to hold the $13-m-$wide gate in theIn the supply system $(Q=0\mathrm{.}89cfs)$ shown in the figure, the pressure

required at the delivery end is $65.$ Consider the entrance and exit

losses and a water temperature of $70F,$ determine: The pumping

head and The power delivered by the pump $($efficiency $0\mathrm{.}62)$

Solution:

Velocity of the flow in pipe is $v=f\frac{t}{s}$;

Major loss $h_f$ :

Darcy$-$Weisback equation method:

Roughness of commercial steel is $e=2{10}^{-4}ft$;

At $70F,$ kinematic viscosity of water

is $v=1\mathrm{.}06{10}^{-5}f\frac{t^2}{s}$;

Relative roughness is $=$

Reynolds number is $Re=$

$x{10}^5$

From Moody diagram, frictional factor $f=$

Thus $h_f=$

$ft.$

Hazen$-$Williams equation $(C=105)$ :

$h_f=$

ft$.$

Chezy$-$Manning equation $(n=0\mathrm{.}011)$:

$h_f=$

ft$.$

Minor loss for

hydraulic fittings

$$

$h_m=$

$ft$;

Energy equation for the pump head: $Hp=$

$=$

$ft$;

Power delivered by the pump $Wp=$

hp$.$

position shown in the figure.

Solution:

Set the origin $O$ at the location

; $x$ is

horizontal pointing into the screen and $y$ is

Unit weight of water is

Height of the gate is

Area of the gate is

$m^2$;

$y_c=$

$.$

$m^2$;

$sina=$

The magnitute of the resultant force is

$kN$;

$kN$;

Moment of inertia of the gate with respect to its own centroid is I

$=,m^4$;

Location of the single resultant force is $y^{\text{'}}=$

$m$ from $O$;

To figure out the force $P$ required, the gate is in equilibrium, take

the moment of

The force $P$ is

$kN.$