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.
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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.)
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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
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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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