Anyone derive venturimeter for me

Venturimeter

• Venturimeter is a device to measure the flow of incompressible liquid.
  • It consists of a tube with a broad diameter having a larger cross-sectional area but there is a small constriction in the middle.
  • It is attached to U-tube manometer. One end of the manometer is connected to the constriction and the other end is connected to the broader end of the Venturimeter.
  • The U-tube is filled with fluid whose density is ρ.
  • A1= cross-sectional area at the broader end, v1 = velocity of the fluid.
  • A2=cross-sectional area at constriction, v2= velocity of the fluid.
• By the equation of continuity, wherever the area is more velocity is less and vice-versa.As A1 is more this implies v1 is less and vice-versa.
• Pressure is inversely ∝ to Therefore at A1 pressureP1 is less as compared to pressure P2 at A2.
  • This implies P1<P2 as v1>v2.
• As there is difference in the pressure the fluid moves,this movement of the fluid is marked by the level of the fluid increase at one end of the U-tube.

A schematic diagram of Venturimeter

Venturimeter: determining the fluid speed

• By Equation of Continuity: -A1v1=A2v2.
• This implies v2=(A1/A2)v1(Equation(1))
• By Bernoulli’s equation:- P1 + (1/2) ρ v12 + ρg h = (1/2) ρ v22 + ρg h
  • As height is same we can ignore the term ρg
  • This implies P1-P2=(1/2) ρ(v22- v12)
  • =1/2ρ(A12/A22v12- v12)(Using equation(1)
  • =1/2ρv12(A12/A22 -1)
  • =1/2ρv12(A12/A22-1)
• As there is pressure difference the level of the fluid in the U-tube changes.
• (P1-P2) = hρmgwhere ρm(density of the fluid inside the manometer).
• 1/2ρv12(A12/A22-1)=hρmg
• v1 = 2hρmg/ρ[A12/A22-1]^-1/2