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Bernoulli’s Equation

Bernoulli’s Equation. Equation of Continuity. Δ L. A. 1. 2. (mass/time) 1 = (mass/time) 2. ( ρ V/t) 1 = ( ρ V/t) 2 . ( ρ A Δ L/t) 1 = ( ρ A Δ L/t) 2. ρ 1 A 1 v 1 = ρ 2 A 2 v 2. If the fluid is incompressible (constant ρ ). A 1 v 1 = A 2 v 2 =. V 1 /t = V 2 /t.

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Bernoulli’s Equation

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  1. Bernoulli’s Equation

  2. Equation of Continuity ΔL A 1 2 (mass/time)1 = (mass/time)2 (ρV/t)1 = (ρV/t)2 (ρAΔL/t)1 = (ρAΔL/t)2 ρ1A1v1 = ρ2A2v2

  3. If the fluid is incompressible (constant ρ) A1v1 = A2v2 = V1/t = V2/t (volume rate of flow) A1ΔL1 = A2ΔL2

  4. Air moves through a duct at 3.0 m/s. If air is replenished every 15 min in a room with a volume of 300 m3, what is the size of the duct? 0.11 m2

  5. (ρ constant) D2 D1 v1 speed and volume rate of flow of the fluid at 2? v2 = (D1/D2)2 v1 V/t = (πD12/4)v1 v α 1/A v α 1/D2

  6. Conservation of energy pressure PE + grav PE + KE = constant P + ρgy + ½ ρv2 = constant P1 + ρgy1 + ½ ρv12 = P2 + ρgy2 + ½ ρv22

  7. Speed of fluid at point 2? v2 = √(2gh) 1 h 2 Patm + ρgh + 0 = Patm + 0 + ½ ρv22

  8. Pressure Head – equivalent to having water at a higher point (even if water at ground level) What is the volume rate of flow of water through a hose with diameter D if the pressure head is H? (πD2/4) √(2gH)

  9. To what height can water be sprayed if the pressure of the water main is P ? P + 0 + 0 = Patm + ρgh + 0 P – Patm = ρgh h = (P – Patm)/(ρg)

  10. P+ ρgy + ½ ρv2 = constant If no change in height: P + ½ρv2 = constant slower speed faster speed more pressure less pressure

  11. As the speed of a fluid increases, the pressure in the fluid decreases. P α 1/v

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