A Pitot tube (figure) is mounted along the axis of a gas pipeline whose cross-sectional area is equal to S. Assuming the viscosity to be negligible, find the volume of gas flowing across the section of the pipe per unit time, if the difference in the liquid columns is equal to `Deltah`, and the densities of the liquid and the gas are `rho_0` and `rho` respectively.

Explanation

Applying Bernoulli's theorem for the point A and B,
`p_A=p_B+1/2rhov^2` as, `v_A=0`
or, `1/2rhov^2=p_A-p_B=Deltalrho_0g`
So, `v=sqrt((2Deltahrho_0g)/(rho))`
Thus, rate of flow of gas, `Q=Sv=Ssqrt((2Deltahrho_0g)/(rho))`
The gas flows over the tube past it at B. But at A the gas becomes stationary as the gas will move into the tube which already contains gas.
In applying Bernoulli's theorem we should remember that `p/rho+1/2v^2+gz` is constant along a streamline. In the present case, we are really applying Bernoulli's theorem somewhat indirectly. The streamline at A is not the streamline at B. Nevertheless the result is correct. To be convinced of this, we need only apply Bernoull's theorem to the streamline that goes through A by comparing the situation at A with that above B on the same level. In steady conditions, this agrees with the result derived because there cannot be a transverse pressure differential.