Fluid Dynamics (Bernoulli, Continuity, Flow Rate) — A/L Physics

How do I practise Fluid Dynamics (Bernoulli, Continuity, Flow Rate)?

1 question with a worked solution

1. A horizontal Venturi meter is installed in a pipeline carrying liquid of density ρ. The main pipe cross-section is A₁ and the throat is constricted to area A₂ (where A₁ > A₂). If the manometer indicates a static pressure difference ΔP between the inlet and the constriction, what is the volumetric flow rate Q through the pipe?
   1. A) Q = A₁ A₂ √[ ΔP / (2 ρ (A₁² - A₂²)) ]
   2. B) Q = √[ 2 ρ ΔP / (A₁² - A₂²) ]
   3. C) Q = (A₁ + A₂) √[ 2 ΔP / ρ ]
   4. D) Q = A₁ A₂ √[ 2 ΔP / (ρ (A₁² - A₂²)) ]
   Check my answer

   Correct answer: D) Q = A₁ A₂ √[ 2 ΔP / (ρ (A₁² - A₂²)) ]

   Combining the continuity equation Q = A₁ v₁ = A₂ v₂ with Bernoulli's equation for horizontal flow P₁ + ½ ρ v₁² = P₂ + ½ ρ v₂² gives: ΔP = P₁ - P₂ = ½ ρ (v₂² - v₁²) = ½ ρ Q² (1/A₂² - 1/A₁²) = ½ ρ Q² [(A₁² - A₂²) / (A₁² A₂²)]. Solving for Q: Q = A₁ A₂ √[ 2 ΔP / (ρ (A₁² - A₂²)) ].

   Show worked solution

   Step 1: Continuity: v₁ = Q / A₁, v₂ = Q / A₂. Step 2: Bernoulli: ΔP = ½ ρ (v₂² - v₁²) = ½ ρ Q² (1/A₂² - 1/A₁²). Step 3: Rearrange: Q² = 2 ΔP / [ρ (A₁² - A₂²) / (A₁² A₂²)] => Q = A₁ A₂ √[ 2 ΔP / (ρ (A₁² - A₂²)) ]. Answer: Q = A₁ A₂ √[ 2 ΔP / (ρ (A₁² - A₂²)) ].
   Modelled on: the A/L Physics examination pattern

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Next in the syllabus

Where does Fluid Dynamics (Bernoulli, Continuity, Flow Rate) fit in the syllabus?

Unit 10.6 · Year 2 · Term 3