Describe the construction and working of venturimeter and obtain an equation for the volume of liquid flowing per second through a wider entry of the tube.

Explanation

Venturimeter :
This device is used to measure the rate of flow (or say flow speed) of the incompressible fluid flowing through a pipe. It works on the principle of Bernoulli's theorem.
It consists of two wider tubes A & A' (with cross sectional area A) connected by a narrow tube B (with cross sectional area a). A manometer in the form of u - tube is also attached between the wide and narrow tubes. The manometer contains a liquids of density `'rho_(m)'`
Let `P_(1)` be the pressure of the fluid at the wider region of the tube A. Let us assume that the fluid of density `'rho'` flows from the pipe with speed `'v_(1)'` and into the narrow region, its speed increases to `'v_(2)'`. According to the Bernoulli's equation, this increase in speed is accompanied by decrease in the fluid pressure `P_(2)` at the narrow region of the tube B. Therefore, the pressure difference between the tubes A and B is noted by measuring the height difference
`(DeltaP P_(1) - P_(2))` between the surfaces of the manometer liquid.
From the equation of continuity, we can say that `A v_(1) = a v_(2)` which means that
`v_(2) = (A)/(a)v_(1)`
Using Bernoulli's equation
`P_(1) + rho(v_(1)^(2))/(2) = P_(2) + rho(1)/(2)((A)/(a)v_(1))^(2)`
From the above equation, the pressure difference
`Delta P = P_(1) - P_(2) = rho(v_(1)^(2))/(2)((A^(2) - a^(2))/(a^(2)))`
Thus, the speed of flow of fluid at the wide end of the tube A
`v_(1)^(2) = (2(Delta P)a^(2))/(rho(A^(2) - a^(2))) rArr v_(1) = sqrt((2(DeltaP)a^(2))/(rho(A^(2) - a^(2)))`
The volume of the liquid flowing out per second is
`V = Av_(1) = Asqrt((2(DeltaP)a^(2))/(rho(A^(2) - a^(2)))) = aA sqrt((2(DeltaP))/(rho(A^(2) - a^(2))))`