A venturi meter is a device for measuring the speed of a fluid within a pipe. Th

Question

A venturi meter is a device for measuring the speed of a fluid within a pipe. The drawing shows a gas flowing at a speed v2 through a horizontal section of pipe whose cross-sectional area A2 = 0.0600 m2. The gas has a density of ? = 1.30 kg/m3. The Venturi meter has a cross-sectional area of A1 = 0.0300 m2 and has been substituted for a section of the larger pipe. The pressure difference between the two sections is P2 - P1 = 160 Pa.

(a) Find the speed v2 of the gas in the larger original pipe.

(b) Find the volume flow rate Q of the gas.

Explanation / Answer

Bernoulli's Equation gives :

Delta ( fe ) + Delta ( ke ) + Delta ( pe ) = 0.0

Delta ( fe ) = P2 - P1 / dens

Delta (ke ) = V2^2 - V1^2 / 2

Delta ( pe ) = 0.0 since horizontal

( P2 - P1) / dens + ( V2^2 - V1^2) / 2 + 0.0 = 0.0

Flow Rate Continuity Equation gives :
---------------------------------------...

m2dot = m1dot

( V2 ) ( AF2 ) ( dens1 ) = ( V1 ) ( AF1 ) ( dens2 )

dens1 ~ dens2 since pressure differential has a small magnitude

V1 = ( V2 ) ( AF2 / AF1 )

V1^2 = ( V2^2 ) ( AF2 / AF1 )^2

Now Bernoulli Equation becomes :
---------------------------------------...

( P2 - P1 ) / dens + [ V2^2 - ( V2^2 ) ( AF2 / AF1 )^2 ]/ [ 2 ]

( P2 - P1 ) / dens + ( V2^2 ) ( 1 - AF2/AF1^2 ) ( 1/2 ) = 0.0

V2 = SQRT [ ( P2 - P1 ) ( 2 ) / ( dens )( ( AF2 /AF1 )^1 - 1 ) ]

V2 = SQRT [ ( 160 ) ( 2 ) / ( 1.30 ) ( 0.0600 / 0.0300 )2 - 1 ]

V2 = SQRT [ (320 ) / ( 1.30 ) ( 0.0600/0.0300 )^2 - 1 ]

V2 = SQRT [ ( 320 ) / ( 1.30 ) ( 3 ) ]

V2 = 9.0582 m/s <---------------------------------------...

V1 = ( V2 ) ( AF2 / AF1 )

V1 = ( 9.0582 ) ( 0.06000 / 0.03000 ) = 18.1164 m/s

QF = ( V2 ) ( AF2 )

QF = ( 9.0582 ) ( 0.0600 ) = 0.54349 cu m/s <---------------------------------------...

Delta ( ke ) = ( V1^2 - V2^2 )( 1/2 )

Delta ( ke ) = ( 9.0582^2 - 18.1164^2 ) /2 = - 123.0769 N - m / kg

Delta ( fe ) = P2 - P1 / dens

Delta ( fe ) = 160 / 1.3 = 123.0769 N - m / kg

Delta ( fe ) + Delta ( ke ) ~ 0.0 ---> [ OK!! ]

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