Bernoulli_Equation_web

# Bernoulli_Equation_web - Z 2 1 dp 1 2 V 2 2-V 2 1 g z 2-z 1...

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ENU 4133 – Bernoulli Equation January 29, 2010

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Figure 3.14 Control volume around streamline .
Conservation of Mass d dt ±Z CV ρ d V ² + ˙ m out - ˙ m in = 0 (1) ∂ρ t d V + d ˙ m 0 (2) d V = Ads (3) d ˙ m = d ( ρ AV ) = - ∂ρ t Ads (4)

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Conservation of Momentum, Streamwise Direction X dF s = d dt ±Z CV V ρ d V ² + ( ˙ mV ) out - ( ˙ mV ) in (5) ( ρ V ) t Ads + d ( ˙ mV ) (6) Assume frictionless ﬂow ( dW – diﬀerential weight, not diﬀerential work): dF s , grav = - dW sin θ = - ρ gA sin θ ds = - ρ gAdz (7) dF s , press = 1 2 dpdA - dp ( A + dA ) ≈ - Adp (8)

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Steady, Incompressible Flow Z 2 1 V t ds
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Unformatted text preview: + Z 2 1 dp + 1 2 ( V 2 2-V 2 1 ) + g ( z 2-z 1 ) = 0 (14) Z 2 1 dp + 1 2 ( V 2 2-V 2 1 ) + g ( z 2-z 1 ) = 0 (15) p 2-p 1 + 1 2 ( V 2 2-V 2 1 ) + g ( z 2-z 1 ) = 0 (16) Path independent, need only starting and ending states. p 1 + 1 2 V 2 1 + gz 1 = p 2 + 1 2 V 2 2 + gz 2 = constant (17) Restrictions on Use of Bernoulli Equation 1. Steady ow (reasonably common in reality, very common in text) 2. Incompressible ow (good assumption/simplication/idealization, much of the time) 3. Frictionless ow (poor assumption for many applications, for which friction is present at walls) 4. Flow along streamline 5. Does not account for heat or work Example Problems...
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Bernoulli_Equation_web - Z 2 1 dp 1 2 V 2 2-V 2 1 g z 2-z 1...

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