To measure the flow rate in a gas pipeline a narrowing is made in it and the pressure difference between the wide and the narrow parts of the pipe is measured (Fig.). Find the gas flow rate, if its density is `1.4 kg//m^3`, the diameter of the pipe is 50 mm, the diameter of the narrowing is 30 mm and the pressure difference is 18 mm of water. The compressibility of the gas is to be neglected.
 

Answer

`mu=(piD^2d^2)/(4)sqrt(2rhorho_0gh)/(D^4-d^4)=1.7xx10^(-2)kg//5`

Explanation

Applying the Bernoulli equation for an incompressible fluid to both cross sections and expressing the velocity in the narrowing in terms of the gas velocity in the pipeline, we obtain
`Delta p = (rho v^2)/2 ((D^4)/(d^4) - 1)`
where D and d are the diameters of the pipeline and of the narrowing. The flow rate is
`mu = rho S v = 1/4 pi rho v D^2`
and the pressure drop `Delta p = rho_0 gh`, where `rho_0` is the density of water. After some simple transformations we obtain the final expression for the flow rate.