Chapter8_summary - Summary continuity, momentum, and energy...

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Summary continuity, momentum, and energy equations for pipe flow Continuity: 12 QQ = i.e., VV = Use continuity and momentum: () 2 rd p z ds τγ =− + Energy: 22 11 2 2 L pV p V zz h gg γγ ++ =++ + 21 L pp hz z h h h γ =+ = Δ = L dh hL ds ⎛⎞ ⎜⎟ ⎝⎠ where L ds == length of pipe p L Ld hp z ds ⎡⎤ + ⎢⎥ ⎣⎦ Combine Energy with Momentum evaluated at rR radius of pipe 2 w Rd p z ds + 2 w L L h R τ = Lf hh loss due to pipe friction Define friction factor, i.e., nondimensional wall shear stress 2 8 w f V ρ = 2 2 L V f D g Darcy-Weisbach equation
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Laminar Flow: Re 2000 VD ν =≤ () 22 4 Rr d ur p z ds γ μ =− + 4 8 Rd Qp z ds π ⎡⎤ + ⎢⎥ ⎣⎦ 2 max 0 82 2 u QR d Vp z Ad s = == + = = 2 32 Lf LV hh V D i.e. 64 Re f = friction factor for laminar pipe flow Turbulent Flow: Re 2000 ~ 3000 Re, f fk D = k = roughness f determined from 0 2 2 R r d r Q V AR with from log-law including pipe roughness. * * 1 ln uy B u κ ++ + * yu y + = , yR r , * w u τ ρ = , * 1 5l n 1 0 . 3 B k + + 12 9.35 2log 1.14 Re k f Df + + Moody Diagram Exact solution Navier-Stokes
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Three Cononical Types of Problems 1. Determine The Head Loss
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Chapter8_summary - Summary continuity, momentum, and energy...

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