Problem $4(33$ points total$)$

Consider the following pipeline consisting of a reservoir and four horizontal pipe segments

in series.

Each pipe segment has a length of $x=100[m],$ has a diameter of $D=75[cm],$ and

is made of commercial steel with Mannings roughness of $n=0\mathrm{.}010.$

The last pipe segment is open to the atmosphere and discharges at a known flow rate

of $Q_3=1[\frac{m^3}{s}].$

The depth of water in the reservoir is $z_{}=5[m]$ above the pipe entrance.

The pressure head is measured to be $\frac{p_c}{}=1[m]$ at point $($c$).$

At point $($b$),$ a service line extracts water from the pipeline at an unknown flow rate

of $Q_{\text{o}\text{u}\text{t},b}.$

At point $($d$),$ another service line extracts water from the pipeline at an unknown flow

rate of $Q_{\text{o}\text{u}\text{t},d}.$

Write the energy balance equation from $($a$)$ to $($c$),$ using Mannings equation $(h_L=$

$K{Q}^2)$ for the frictonal head loss. Neglect minor losses. Simplify as much as possible,

but do not yet substitute numeric values. $(6$ points$)$

Write the energy balance equation from $($c$)$ to $($e$),$ using Mannings equation $(h_L=$

$K{Q}^2)$ for the frictional head loss. Neglect minor losses. Simplify as much as possible,

but do not yet substitute numeric values. $(6$ points$)$

Write the mass balance equation $($continuity$)$ around junction $($b$).(2$ points$)$

Write the mass balance equation $($continuity$)$ around junction $($d$).(2$ points$)$

Compute the cross$-$sectional area of the pipe, $A.(1$ point$)$

Compute the value of $K=\frac{10\mathrm{.}3n^{2}x}{D^{0\mathrm{.}33}}$ for Mannings equation for a single pipe segment of

length points$)$

Compute flow rate $Q_2.(6$ points$)$

Compute flow rate $Q_1.(6$ points$)$

Compute the demand flow rates $Q_{\text{o}\text{u}\text{t},b}$ and $Q_{\text{o}\text{u}\text{t},d}.(2$ points$)$