Chapter8 - 57:020 Mechanics of Fluids and Transport...

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57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2007 1 Chapter 8 Flow in Conduits Entrance and developed flows Le = f(D, V, ρ , μ ) Π i theorem Le/D = f(Re) Laminar flow: Re crit 2000, i.e., for Re < Re crit laminar R e > R e crit turbulent Le/D = .06Re from experiments L e max = .06Re crit D 138D maximum Le for laminar flow
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57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2007 2 Turbulent flow: D Le 6 / 1 Re 4 . 4 from experiment Laminar vs. Turbulent Flow laminar turbulent spark photo Reynolds 1883 showed difference depends on Re = ν VD Re Le/D 4000 18 10 4 20 10 5 30 10 6 44 10 7 65 10 8 95 i.e., relatively shorter than for laminar flow
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57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2007 3 Shear-Stress Distribution Across a Pipe Section Continuity: Q 1 = Q 2 = constant , i.e., V 1 = V 2 since A 1 = A 2 Momentum: () ρ = A V u F s = ( ) 2 2 2 1 1 1 A V V A V V ρ ρ = 0 V V Q 1 2 = ρ 0 ds r 2 sin W A ds ds dp p pA = π τ α Δ + Ads W γ = Δ ds dz sin = α 0 ds r 2 ds dz Ads dsA ds dp = π τ γ ÷ Ads γ + = τ z p ds d 2 r 0 2 W r d p z ds τγ =− + τ varies linearly from 0.0 at r = 0 (centerline) to τ max (= τ w ) at r = r 0 (wall), which is valid for laminar and turbulent flow.
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57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2007 4 no slip condition Laminar Flow in Pipes () γ + = μ = μ = τ z p ds d 2 r dr dV dy dV y = wall coordinate = r o r dy dV dr dy dy dV dr dV = = γ + μ = z p ds d 2 r dr dV C z p ds d 4 r V 2 + γ + μ = γ + μ = = z p ds d 4 r C 0 r V 2 o o 2 22 0 1 4 o C rr dr Vr p z V ds r γ μ ⎛⎞ ⎡⎤ =− + = ⎜⎟ ⎢⎥ ⎣⎦ ⎝⎠ where = A d V Q = π o r 0 rdr 2 r V π = θ = 2 rdr rdrd dA Exact solution to Navier-Stokes equations for laminar flow in circular pipe 2 4 o C r d Vp z ds +
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57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2007 5 () γ + μ π = z p ds d 8 r Q 4 o 2 82 oC rV Qd Vp z Ad s γ μ ⎡⎤ == + = ⎢⎥ ⎣⎦ For a horizontal pipe, 22 44 oo C rr dp z ds L μμ Δ =− + = where L = length of pipe = ds 2 2 0 0 1 o r pr p Vr r r Lr L ⎛⎞ ΔΔ = ⎜⎟ ⎝⎠ 0 4 0 0 21 2 8 r p Dp Qr r r d r L L π = Energy equation: L 2 2 2 2 1 2 1 1 h z g 2 V p z g 2 V p + + + γ = + + γ 12 pp hz z γγ Δ= + +
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57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2007 6 friction coefficient for pipe flow boundary layer flow () 12 0 2 ()[ ][] W L pp dh L d L hz z h L p z ds ds r τ γ γγ =+ = Δ = = + = Define friction factor 2 w V 8 f ρ τ = 2 w f V 2 1 C ρ τ = 2 2 00 28 2 [][ ] 2 W Lf fV LL L V hh f rr D g ρ == = = Darcy – Weisbach Equation, which is valid for both laminar and turbulent flow.
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This note was uploaded on 12/08/2011 for the course MECHANICS 57:020 taught by Professor Fredrickstern during the Fall '10 term at University of Iowa.

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Chapter8 - 57:020 Mechanics of Fluids and Transport...

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