Useful equations, formulae, physical constants and properties

$($i$)$ Energy equation for the total flow in a pipeline running from point $1$ to point $2$:

$\frac{V_1^2}{2}g+h_1+z_1=\frac{V_2^2}{2}g+h_2+z_2+H_L$

where $V$ is the velocity of the flow expressed in $\frac{m}{s}$

$h$ is the hydraulic pressure expressed in $m$

$z$ is the elevation of the pipe expressed in $m$ above datum

$H_L$ is the head loss expressed in $m$

$($ii$)$ Darcy$-$Weisbach Formula: $H_L=fL\frac{V^2}{2gd}$

where $f$ is a friction factor

$L$ is the length of the pipe

$d$ is the internal diameter of the pipe

$($iii$)$ Colebrook and White Formula:

$\sqrt[\phantom{2}]{\frac{1}{f}}=-2lo{g}_{10}\{\frac{k_s}{3\mathrm{.}71d}+\frac{2\mathrm{.}51}{Re\sqrt[\phantom{2}]{f}}\}$

where $k_s$ is the linear measurement of surface roughness of the pipe

$Re$ is Reynolds number $=V\frac{d}{v}$

$v$ is the kinematic viscosity of the fluid

$($iv$)$ Values to be used for design purpose: $k_s=0\mathrm{.}9mm$;$v=1{10}^{-6}\frac{m^2}{s}$;$g=9\mathrm{.}8\frac{m}{s^2}$

$($v$)$ For simplicity, the nominal diameter $($DN$)$ of the pipe can be taken as $d$ for design

purpose. Common pipe sizes include DN$600,$ DN$800,$ DN$1000$ and DN$1200.$

$($vi$)$ Friction factor $(f)$ calculated in an iterative process using $k_s=0\mathrm{.}9mm$ and different

combinations of $d$ and $Re$ values are shown in the table below. You may use the nearest

Re value to obtain $f.$Shown below is the diagrammatic illustration of a water supply system with the following

details:

A water treatment works $($WTW$)$ supplies treated water to a primary service reservoir

$($PSR$).$ The water main connecting the WTW to the PSR measures $5km$ in length.

The PSR provides treated water to two secondary service reservoirs $($SSR$1$ and SSR$2).$

SSR$1$ provides treated water to Town A$.$ Its average daily water demand is $30,000$

$\frac{m^3}{d}.$ The water main connecting SSR$1$ to the perimeter of Town A measures $8km$ in

length. The hydraulic pressure of the water main at the perimeter of Town A shall not

be less than $15m$ at all time.

SSR$2$ provides treated water to Town B$.$ Its average daily water demand is $60,000$

$\frac{m^3}{d}.$

According to the local water authority, trunk mains shall be designed to cater for $1\mathrm{.}5$

times average daily water demand, whereas distribution mains shall be designed to cater

for $3$ times average daily water demand.

$($a$)$ Work out the most economical size for the water main connecting the WTW to the PSR

$(6$ marks$)$

$($b$)$ Work out the most economical size for the water main connecting SSR$1$ to the perimeter

of Town A

$(6$ marks$)$

$($c$)$ Name three $(3)$ purposes of service reservoirs.

$(3$ marks$)$

$($d$)$ Name one advantage and one drawback of having primary service reservoirs. $(2$ marks$)$

Useful equations, formulae, physical constants and properties are provided in the following

page for reference.