Bernoulli_Equation_web

Bernoulli_Equation_web - + Z 2 1 dp + 1 2 ( V 2 2-V 2 1 ) +...

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ENU 4133 – Bernoulli Equation January 29, 2010
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Figure 3.14 Control volume around streamline .
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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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Constructing Bernoulli’s Equation Put back into dF s : - Adp - ρ gAdz = ∂ρ V t Ads + d ( ˙ mV ) (9) - Adp - ρ gAdz = ∂ρ t VAds + V t ρ Ads + Vd ˙ m + ˙ mdV (10) Using continuity, RHS terms 1 and 3 cancel 0 = Adp + ρ gAdz + V t ρ Ads + ˙ mdV (11) 0 = V t ds + dp ρ + VdV + gdz (12) 0 = Z 2 1 V t ds + Z 2 1 dp ρ + 1 2 ( V 2 2 - V 2 1 ) + g ( z 2 - z 1 ) (13) Bernoulli’s equation (general case – unsteady, compressible, but frictionless flow). In general, path-dependent.
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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 ) +...

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