Lecture_Note_Ch_6

# Lecture_Note_Ch_6 - ME 342 Fluid Mechanics Lecture Note on...

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Lecture Note on Ch. 6: Viscous Flow in Ducts Prof. Chang-Hwan Choi Stevens Institute of Technology Department of Mechanical Engineering ME 342 Fluid Mechanics Spring 2008

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Motivation Pipe problem: – Given the pipe geometry and its added components plus the desired flow rate and fluid properties, what pressure drop is needed to derive the flow? – Given the pressure drop available from a pump, what flow rate will ensue? Pipe flows are everywhere, often occurring in groups or networks. They are designed using the principles outlined in this chapter.
Laminar: smooth and steady flow Turbulent: fluctuating and agitated flow Transition: Laminar Æ Turbulent Reynolds Number Regimes The three regimes of viscous flows: (a) laminar flow at low Re; (b) transition at intermediate Re; (c) turbulent flow at high Re. Flow issuing at constant speed from a pipe: (a) high-viscosity, low-Reynolds-number, laminar flow; (b) low-viscosity, high- Reynolds-number, turbulent flow.

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Primary parameter affecting transition: Reynolds number – 0<Re<1: highly viscous laminar “creeping” motion – 1<Re<100: laminar, strong Reynolds number dependence – 100<Re<10 3 : laminar, boundary layer theory useful –1 0 3 <Re<10 4 : transition to turbulence 0 4 <Re<10 6 : turbulent, moderate Reynolds number dependence 0 6 <Re< : turbulent, slight Reynolds number dependence These representative ranges vary somewhat with flow geometry, surface roughness, and the level of fluctuations in the inlet stream. Reynolds Number Regimes (cont.)
Reynolds Number Regimes (cont.) Experimental evidence of transition for water flow in a 0.25” smooth pipe 10’ long. Reynolds’ sketches of pipe flow transition: (a) low-speed, laminar flow; (b) high-speed, turbulent flow; (c) spark photography of condition (b). The accepted design value for pipe flow transition: Re d ,crit 2300

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Internal Viscous Flows Developing velocity profiles and pressure changes in the entrance of a duct flow. () 1/6 By dimensional analysis: =, , , R e 0.06Re for laminar 4.4Re for turbulent e e e e d L Vd LfdV g g d L d L d ρ ρµ µ  ⇒= =   An internal flow is constrained by the bounding walls, and the viscous effects will grow and meet and permeate the entire flow. Shortness can be a virtue in duct flow if one wishes to maintain the inviscid core, e.g., wind tunnel
Head Loss – Friction Factor () 12 21 22 1 By control volume analysis: Continuity: Momentum: 0 sin 2 Energy equation: x w f f VV V Fm V V pR g RL R L pV zz h gg p hz z g πρ π φ τ αα ρρ ρ == =− = =∆ +  ++ =++ +   =−+ & 2 2 2 24 2 8 where : Darcy friction factor fcn(Re , , duct shape) ww w d p p z LL L V f gR gd d g f V f d ττ ε =∆ + === = = Control volume of steady, fully developed, incompressible flow between two sections in an inclined constant-area pipe.

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Laminar Fully Developed Pipe Flow 2 max 2 2 max 24 max 2 0 2 max 2 Fully Developed Laminar Pipe Flow: 1 where , 4 12 8 28 R w r uu R dp R dp p g z u dx dx L rR p g z Q udA u rdr L R V QQ R pg z V AL R ρ µ πρ π τ  =−   ∆+ ∆ =
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## This note was uploaded on 10/03/2009 for the course ME me342 taught by Professor Choi during the Spring '09 term at Stevens.

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Lecture_Note_Ch_6 - ME 342 Fluid Mechanics Lecture Note on...

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