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Andreas93 [3]
3 years ago
6

A venturi meter used to measure flow speed in the pipe. Derive an expression for the flow speed "H1" interns of the crossectiona

l areas "A1" and "A2" and the difference in height "h" of the liquid levels in the two vertical tubes ?
​

Physics
1 answer:
Helga [31]3 years ago
6 0

Answer:

v₁ = √[ 2gh / ((A₁ / A₂)² − 1) ]

Explanation:

Use Bernoulli's equation:

P₁ + ½ ρ v₁² + ρgz₁ = P₂ + ½ ρ v₂² + ρgz₂

Since there's no elevation change between points 1 and 2, z₁ = z₂.

P₁ + ½ ρ v₁² = P₂ + ½ ρ v₂²

Assuming incompressible fluid, the volumetric flow rate is the same at points 1 and 2.

Q₁ = Q₂

v₁ A₁ = v₂ A₂

v₂ = v₁ A₁ / A₂

Substituting:

P₁ + ½ ρ v₁² = P₂ + ½ ρ (v₁ A₁ / A₂)²

P₁ + ½ ρ v₁² = P₂ + ½ ρ v₁² (A₁ / A₂)²

P₁ − P₂ = ½ ρ v₁² (A₁ / A₂)² − ½ ρ v₁²

P₁ − P₂ = ½ ρ v₁² ((A₁ / A₂)² − 1)

v₁² = 2 (P₁ − P₂) / (ρ ((A₁ / A₂)² − 1))

v₁² = 2 (ρgh) / (ρ ((A₁ / A₂)² − 1))

v₁² = 2gh / ((A₁ / A₂)² − 1)

v₁ = √[ 2gh / ((A₁ / A₂)² − 1) ]

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