# ch64 - Chapter 6: Viscous Flow in Ducts 6.1 Reynolds Number...

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Chapter 6: Viscous Flow in Ducts 6.1 Reynolds Number Regimes 6.2 Internal vs External Fluid Flows 6.3 Head Loss – The Friction Factor 6.4 Laminar Fully Developed Pipe Flow 6.5 Turbulence Modeling 6.6 Turbulent Pipe Flow 6.7 Four Types of Pipe Flow 6.8 Flow in Noncircular Ducts* 6.9 Minor Losses Pipe Systems* 6.10 Multiple Pipe Systems* 6.11 Diffuser Performance* 6.12 Fluid Meters* * skim

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fully developed, steady, incompressible and laminar pipe flow
Laminar Flow Turbulent Flow

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fully developed, steady, incompressible laminar pipe flow Shear Stress Profile
• Conservation of Momentum d(m V )/dt sys = Σ F = d/dt CV V ρ dV ol + CS V ρ V n dA (3.35) If steady and fully developed τ w = ( Δ p - ρ g Δ z)R/(2L) Σ F = Δ p π r 2 - ρ g( π r 2 L) sin θ - τ (r)2 π rL = 0 Δ p/L = -dp/dx Δ z/L = dz/dx τ (r) = Δ pr/(2L) - ρ gr Δ z/(2L) = ( Δ p - ρ g Δ z)r/(2L) True for Laminar & Turbulent Flow and incompressible

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τ w (r=R) = ( Δ p + ρ g Δ z)R/(2L) Σ F = Δ p π r 2 - ρ g( π r 2 L) sin θ - τ (r)2 π rL = 0 Σ F = Δ p π r 2 + ρ g( π r 2 L) sin θ - τ (r)2 π rL = 0 τ w = ( Δ p - ρ g Δ z)R/(2L)
p+ Δ p r p L τ τ R τ = ( Δ p)r/(2L) L Δ p π r 2 - τ (r)2 π rL = 0 Δ z = 0 STEADY, FULLY DEVELOPED, FLOW r = R τ w = ( Δ p)R/(2L)

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SHEAR STRESS PROFILE τ (r) = ( Δ p)r/(2L) τ tot = τ lam TRUE FOR LAMINAR AND TURBULENT FLOW shear direction on Control Volume shear magnitude Steady, Fully Developed Cons. of Momentum τ lam = μ du/dy = - μ du/dr τ tot = τ lam + τ turb BUT R-r = y
Aside Note: shear stress, τ rx or τ yx are tensors

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White If normal to surface is in the + z-direction so positive stress is defined in the + x-direction Sign convention for stresses x If normal to surface is in the negative z-direction so positive stress is defined in the negative x-direction)
SHEAR STRESS PROFILE SIMILAR IN DUCT τ yx = μ (du/dy) y = 0 y = a For positive flow, dp/dx = negative Shear force + + + shear direction τ yx in top ½ is negative & shear force is in the – x direction τ yx in bottom ½ is positive & shear force is in the – x direction

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u(r) fully developed, steady, incompressible laminar pipe flow
p 1 r p 2 L p 1 + Δ p = p 2 τ τ LAMINAR, FULLY DEVELOPED FLOW R What is velocity profile across the pipe? 2 τ /r = Δ p/L { for lam & turb !} τ = - μ (du/dr) { for lam } du = - τ dr/ μ = -r Δ pdr/(2 μ L) u(r) = -r 2 Δ p/(4 μ L) + c

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p 1 r p 2 L p 1 + Δ p = p 2 τ τ LAMINAR, FULLY DEVELOPED FLOW R What is velocity profile across the pipe? u(r) = -r 2 Δ p/(4 μ L) + c u(R) = ? = -R 2 Δ p/(4 μ L) + c u(r) = [ Δ p/(4 μ L)][-r 2 + R 2 ] u(r) = [ Δ pR 2 /(4 μ L)][1 - r 2 /R 2 ] u(R) = 0 = -R 2 Δ p/(4 μ L) + c