Q$5.$ A discharge meter to be used in a $40cm$ pipeline carrying oil $,$

$\{$:$=860\frac{kg}{m^3}),$ is to be calibrated by a means of a model $(1$:$5)$ scale carrying

water $(T=10C$;$v={10}^{-6}\frac{m^2}{s},=1000\frac{kg}{m^3}).$ The model is operated at a

velocity of $1\mathrm{.}5\frac{m}{s},$ the pressure drop in the model was measured at $5\mathrm{.}0$kPa.

Assuming that the pressure drop, $P(\frac{M}{L}{T}^2)$ is function of the diameter $(L),$

the velocity $(\frac{L}{T}),$ the density $(\frac{M}{L^3}),$ the viscosity $(\frac{M}{L}T)$ of the fluids, or

$P=f(V,D,,)$ Find:

a$)$ The $$ groups for the system using dimensional analyses.

b$)$ Estimate the velocity and pressure difference in the prototype based on the

information obtained from the model?

c$)$ The flow time in the model is measured to be $20sec.$ What is the

corresponding flow time in the prototype?