058 0160 Professor Fred Stern Chapter 6 part1 1 Fall 2009 Chapter 6 Viscous Flow in Ducts 6 1 Laminar Flow Solutions Entrance developing and fully developed flow Le f D V L theorem f Re f Re from AFD and EFD D e i Laminar Flow Recrit 2000 L D 06 Re e L e max 06 Re D 138D crit Re Recrit laminar Re Recrit unstable Re Retrans turbulent Max Le for laminar flow 058 0160 Professor Fred Stern Chapter 6 part1 2 Fall 2009 Turbulent flow Re 4000 104 105 106 107 108 Le D 18 20 30 44 65 95 L D 4 4 Re 1 6 e Relatively shorter than for laminar flow Laminar vs Turbulent Flow Hagen 1839 noted difference in p p u but could not explain two regimes Laminar Turbulent Spark photo Reynolds 1883 showed that the difference depends on Re VD 058 0160 Professor Fred Stern Chapter 6 part1 3 Fall 2009 Laminar pipe flow 1 CV Analysis Continuity 0 V dA Q1 Q2 const CS i e V1 V2 sin ce A1 A2 r const and V Vave Momentum b2V2 b1V1 Fx 14p12 4p32 p R 2 t w 2p RL g1p2R32 L sin J m 1 44 2 4 43 W Dz L 0 Dp p R 2 RL R z 0 2 L p z 2 2 w w R 058 0160 Professor Fred Stern Chapter 6 part1 4 Fall 2009 Dh h1 h2 D p g z 2t w L g R or w R h R dh 2 L 2 dx R d p z 2 dx For fluid particle control volume r d p z 2 dx i e shear stress varies linearly in r across pipe for either laminar or turbulent flow Energy p1 1 p V1 z1 2 2 V2 z 2 hL 2g 2g h h L 2 L R w once w is known we can determine pressure drop In general roughness V D w w i Theorem 8 w V 2 f friction factor f Re D D where VD Re D 058 0160 Professor Fred Stern Chapter 6 part1 5 Fall 2009 h hL f LV2 D 2g Darcy Weisbach Equation f ReD D still needs to be determined For laminar flow there is an exact solution for f since laminar pipe flow has an exact solution For turbulent flow approximate solution for f using log law as per Moody diagram and discussed late 2 Differential Analysis Continuity V 0 Use cylindrical coordinates r z where z replaces x in previous CV analysis 1 1 r r z 0 r r r z where V r er e z ez Assume 0 i e no swirl and fully developed flow z 0 which shows r constant 0 since r R 0 z V z ez u r ez 058 0160 Professor Fred Stern Chapter 6 part1 6 Fall 2009 Momentum DV p z 2 V Dt z equation u V u p z 2u z t 1 u 0 p z r r r z r f z f r both terms must be constant u p r r r r z u 1 p 2 r r A r 2 z u 1 p r A r 2 z 1 p 2 u r A ln r B 4 z finite u r 0 u r R 0 r R dp u r 4 dz 2 2 p p A 0 R dp B 4 dz 2 R dp u u 0 4 dz 2 max 058 0160 Professor Fred Stern Chapter 6 part1 7 Fall 2009 u u r fluid shear stress r r z r p 2 z u u R p w y r R r r R 2 z As per CV analysis y R r p R4 d p 1 Q u r 2p r dr umaxp R 2 8m dz 2 0 R Vave Q 1 2 umax pR 2 R d p 8m dz 2 Substituting V Vave f 8 w V 2 w R 8 Vave 4 Vave 8 V 2 R2 R D f or h hL f Cf w 1 V 2 2 64 64 DV Re D f 16 4 Re D LV2 64 L V 2 32 LV D 2 g DV D 2 g gD 2 V 058 0160 Professor Fred Stern Chapter 6 part1 8 Fall 2009 for z 0 p V Both f and Cf based on V2 normalization which is appropriate for turbulent but not laminar flow The more appropriate case for laminar flow is P0cf C f Re 16 Poiseuil e P0 P0 f f Re 64 for pipe flow Compare with previous solution for flow between parallel plates with p x 2 y u u 1 h max 4 2h q hu p 3 3 3 max x h u p 2 2 max x 058 0160 Professor Fred Stern q h 2 v p u 2h 3 3 Chapter 6 part1 9 Fall 2009 2 x w 3 V max h f 24 48 96 Vh Re 2 h Re 4h Re D h Cf f 4 Cf 6 12 24 Vh Re 2 h Re 4h Re Dh P0cf C f ReDh 24 Poiseuil e P0 P0 f f ReDh 96 Same as pipe other than constants P0 c f pipe P0c f channel based on Dh P0 f pipe P0 f channel based on Dh 16 64 2 24 96 3 058 0160 Professor Fred Stern Chapter 6 part1 10 Fall 2009 Exact laminar solutions are available for any arbitrary cross section for laminar steady fully developed duct flow BVP u 0 x 0 p u u x YY ZZ u h 0 h U p Related umax 2 Re only enters y y h through stability and transition z z h u 1 2 u u U x Poisson equation u 1 0 Dirichlet boundary condition Can be solved by many methods such as complex variables and conformed mapping transformation into Laplace equation by redefinition of dependent variables and numerical methods 058 0160 Professor Fred Stern Fall 2009 Chapter 6 part1 11 058 0160 Professor Fred Stern Fall 2009 Chapter 6 part1 12 058 0160 Professor Fred Stern Fall 2009 Chapter 6 part1 13 058 0160 Professor Fred Stern Fall 2009 Chapter 6 part1 14
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