B$2.$ As shown in Figure B$2,$ two pipelines in parallel connect two reservoirs that have a constant water surface of $400m$ and $300m$ above datum, respectively. Pipe $AB$ has a diameter of $100mm$ and is $1km$ long while pipe $CD$ has a diameter of $200mm$ and is $2km$ long. Water can be drawn from Pipe CD through a valve $O$ that is $1km$ away from $C.$ The effective roughness size for both pipes is $k_s=0\mathrm{.}03mm$ and the kinematic viscosity of water is $1{10}^{-6}\frac{m^2}{s}.$ Ignore the local losses.

$($a$)$ Derive the combined Colebrook$-$White and Darcy$-$Weisbach Equation $V=-2\sqrt[\phantom{2}]{2gD{S}_f}log(\frac{k_s}{3\mathrm{.}7D}+\frac{2\mathrm{.}51v}{D\sqrt[\phantom{2}]{2gD{S}_f}})$

$[5$ marks$]$

$($b$)$ Calculate the total discharge entering the downstream reservoir if no water is drawn from $O$;

$[10$ marks$]$

$($c$)$ Calculate the total discharge entering the downstream reservoir if $50\frac{l}{s}$ is drawn from $O($if a trial and error method is used, $Q_{OD}=92\mathrm{.}5\frac{l}{s}$ may be used as an initial guess$).$

$[10$ marks$]$

need urgent solution