Prof. Byoung-Kwon Ahn. College of Engineering, Chungnam National University. flow in pipes and the analysis of fully developed flow.

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1 Chapter 8. Flow in Pipes Prof. Byong-Kwon Ahn ac kr http//fincl.cn.ac.krcn Dept. of Naval Architectre & Ocean Engineering College of Engineering, Chngnam National University Objectives 1. Have a deeper nderstanding of laminar and trblent flow in pipes and the analysis of flly developed flow.. Calclate the major and minor losses associated with pipe pp flow in pp piping networks and determine the pmping power reqirements. 3. Understand the different velocity and flow rate measrement techniqes and learn their advantages and disadvantages. Ship Hydrodynamics

2 8.1 Introdction Friction force of wall on flid 1) Average velocity in a pipe 1 ecall - becase of the no-slip condition, the velocity at the walls of a pipe or dct flow is zero We are often interested only in V avg, which we sally call jst V (drop the sbscript for convenience) 3 Keep in mind that the no-slip condition cases shear stress and friction along the pipe walls Ship Hydrodynamics Introdction 1) For pipes of constant diameter and incompressible flow ) V avg stays s the same V avg V avg down the pipe, even if the velocity profile changes 3) Why? Conservation of Mass same same same Ship Hydrodynamics 4

8.1 Introdction For pipes with variable diameter,

V 1 V D 1 D V 1 m V m 1 Ship Hydrodynamics 5 8.

born at Belfast, Ireland 1867: gradate at Qeens

of Cambridge, in mathematics 1868: 1 st professor

`Improvements in Machinery for Propelling Ships or

3 8.1 Introdction For pipes with variable diameter, m is still the same de to conservation of mass, bt V 1 V D 1 D V 1 m V m 1 Ship Hydrodynamics Introdction Osborne eynolds (184 ~ 191) 184: born at Belfast, Ireland 1867: gradate at Qeens College, Univ. of Cambridge, in mathematics 1868: 1 st professor of engineering at Owens College, Manchester 1878: `Improvements in Machinery for Propelling Ships or Vessels e eynolds nmber Intertial force Viscos force FI ρv L ρvl FV μvl μ VL υ Ship Hydrodynamics 6

eynolds nmber 1) Critical eynolds nmber (e

laminar 300 e 4000 transitional e > 4000

4 8.1 Introdction Ship Hydrodynamics 7 8. Laminar and Trblent Flows Definition of eynolds nmber 1) Critical eynolds nmber (e cr ) for flow in a rond pipe e < 300 laminar 300 e 4000 transitional e > 4000 trblent ) Note that these vales are approximate. 3) For a given application, e depends pon 1 Pipe roghness Vibrations 3 Upstream flctations, distrbances (valves, elbows, etc. that may distrb the flow) Ship Hydrodynamics 8

5 8. Laminar and Trblent Flows 1) For non-rond pipes, define the hydralic diameter D h 4A c /P A c cross-section area P wetted perimeter ) Example: open channel A c m P m Don t cont free srface, since it does not contribte to friction along pipe walls! D h 4A c /P 40.06/ m What does it mean? This channel flow is eqivalent to a rond pipe of diameter 0.3m (approximately). Ship Hydrodynamics The Entrance egion Consider a rond pipe of diameter D. The flow can be laminar or trblent. In either case, the profile develops downstream over several diameters called the entry length L h. L h /D is a fnction of e. L h Ship Hydrodynamics 10

8.3 The Entrance egion Lh L L,laminar h,trbent

πrdrp) + ( πrdxτ) ( πrdxτ) 0 x x+ dx r r+ dr ( rτ ) (

6 8.3 The Entrance egion Lh L L,laminar h,trbent h,trbent 0.05De 1.359D e 10D 1/4 Ship Hydrodynamics Laminar Flow in Pipes Average velocity: ( π rdrp) ( πrdrp) + ( πrdxτ) ( πrdxτ) 0 x x+ dx r r+ dr ( rτ ) ( rτ ) Px+ dx Px r+ dr r + dx dr r 0 dx, dr 0 ( ) r dp d rτ 0 dx + dr d τ μ dr μ d d dp r rdr dr dx Ship Hydrodynamics 1

7 8.4 Laminar Flow in Pipes r dp r () + C1ln r+ C dp τ w 4μ dx dx dp r r () 1 4μ dx dp r dp Vavg () r rdr 1 rdr 4μ dx 8μ dx 0 0 r r () Vavg 1 at r 0 max V avg V ρrda () ρr ()πrdr c A 0 c avg ρ A c ρπ 0 The average velocity in flly developed laminar pipe flow is one half of the maximm velocity () r rdr Ship Hydrodynamics Laminar Flow in Pipes 1) There is a direct connection between the pressre drop in a pipe and the shear stress at the wall ) Consider a horizontal pipe, flly developed, and incompressible flow τ w Take CV inside the pipe wall P 1 V P L 1 Ship Hydrodynamics 14

8.4 Laminar Flow in Pipes Pressre Drop: V dp P P1 dx L 8 μ LVavg 3 μ LV Δ P P1 P D L ρv avg Δ PL f D 8τ w f : Darcy friction factor ρ Vavg 64μ 64 f ρdv e ρdv avg V avg dp 8μ dx avg In laminar flow,

8 8.4 Laminar Flow in Pipes Pressre Drop: V dp P P1 dx L 8 μ LVavg 3 μ LV Δ P P1 P D L ρv avg Δ PL f D 8τ w f : Darcy friction factor ρ Vavg 64μ 64 f ρdv e ρdv avg V avg dp 8μ dx avg In laminar flow, the friction factor is a fnction of the eynolds nmber only and is independent of the roghness of the pipe srface Ship Hydrodynamics Laminar Flow in Pipes Head Loss: eqivalent flid colmn height Δ P ρgh h L Δ P L V f ρg D g avg W V Δ P V ρgh mgh pmp L L Poiseille s law: pressre drop and reqiring power is proportional to the length of the pipe and the viscosity of the flid, inversely proportional to the forth power of the radis of the pipe ( P1 P) ( P1 P) D ΔPD Vavg 8μL 3μL 3μL 4 4 ( P1 P) ( P1 P) D P D V π Δ π V avg A c π 8μL 18μL 18μL 18μL Δ P Vavg Ac 4 π D Ship Hydrodynamics 16

5 Trblent Flow in Pipes τ w shear stress at the wall,

9 8.5 Trblent Flow in Pipes Eddy Flctation + Instantaneos profiles Ship Hydrodynamics Trblent Flow in Pipes τ w shear stress at the wall, acting on the flid w Laminar Trblent τ w τ τ lam d μ dr < τ tr τ w Ship Hydrodynamics 18

5 Trblent Flow in Pipes Joseph Bossinesq: τ trb ρv μt y μ : eddy viscosity or trblent viscosity t L.

10 8.5 Trblent Flow in Pipes Trblent shear stress: Tangential(Shear) Force: ( )( ) δ F ρvda ρvda δ F ρ v da τ ρ v trb eynolds stress or Trblent stress 0, v 0, v 0, v 0 d τ total τlam + τtr μ + ρv dy Ship Hydrodynamics Trblent Flow in Pipes Joseph Bossinesq: τ trb ρv μt y μ : eddy viscosity or trblent viscosity t L. Prandtl: τ trb μ l t ρ m y y l : mixing length τ toal ( μ+ μt ) ρ( ν + νt ) y y ν μ / ρ : kinematic eddy viscosity t t kinematic trblent viscosity eddy diffsivity of momentm Ship Hydrodynamics 0

11 8.5 Trblent Flow in Pipes Ship Hydrodynamics Trblent Flow in Pipes Viscos sb-layer: d τ w ν τ w μ ρν dy y ρ y τw / ρ : friction velocity y law of the wall ν 5ν 5ν y y δ sblayer 0 5 δ ν The thickness of the viscos sb-layer is proportional to the kinematic viscosity and inversely yproportional p to the friction velocity Ship Hydrodynamics

12 8.5 Trblent Flow in Pipes Nondimensional variables: ν : viscos lenght y y + ν + ν + + y Nomalized law of the wall: Ship Hydrodynamics Trblent Flow in Pipes Overlap layer: the logarithmic law 1 y l κ ν ln + + B ( κ 0.4, B 5.0) y + ν.5ln lny Ship Hydrodynamics 4

13 8.5 Trblent Flow in Pipes Oter trblent layer: velocity defect law: max 5ln.5ln r power-law velocity profile: max max y 1/ n r 1 1/ n one-seventh power-law Ship Hydrodynamics Moody chart Colebrook: 1 ε / D.51.0log + f e f Haaland: ε / D 1.8log + f e Ship Hydrodynamics 6

5 Moody chart 1 ε / D.51 1 ε / D.51 0l.0log g( f ).

14 8.5 Moody chart Types of Pipe Flow Problems L ρ Δ P f D 4 P D V Δ π 18μL V avg Ship Hydrodynamics Moody chart 1 ε / D.51 1 ε / D.51 0l.0log g( f ).0log 0l 0 f e f f e f ε 10, D 10, e Ship Hydrodynamics 8

15 8.8 Flow ate and Velocity Measrement 1) Pitot and Pitot-Static Probes ) Obstrction Flow-meters: 1 Orifice Ventri Meter 3 Nozzle Meter 3) Positive Displacement Flow-meters 4) Trbine Flow-meters 5) Paddlewheel Flow-meters 6) Variable Area Flow-meters 7) Ultrasonic Flow-meters 8) Doppler-Effect Ultrasonic Flow-meters 9) Electromagnetic Flow-meters 10) Vortex Flow-meters 11) Thermal (Hot-wire) Anemometers 1) Laser Doppler Velocimetry (LDV) 13) Particle Image Velocimetry (PIV) Ship Hydrodynamics 9 In Chap.5: Bernolli Eqation F s ma s F PdA ( P+ dp) da Wsinθ s dz dpda ρgdads ρdads V ds ds dp ρgdz ρvdv dp dv ( ) + gdz 0 dp dv g dz C ρ ρ + + an V / an 0 ( if ) dv W ρ gdads P V + + gz C ρ Ship Hydrodynamics 30

16 In Chap. 5: Eler & Bernolli Eqation 1) The Bernolli eqation was first stated in words by the Swiss mathematician Daniel Bernolli (1700~178) in a text written in It was later derived in general eqation from by Leonhard Eler (1707~1783) 1783) in ) The Bernolli eqation is derived assming incompressible & inviscid flow, and ths is shold not be sed fro flows with significant compressibility effects. 3) Unsteady, Compressible Flow: dp V V + ds gz C ρ + + t 4) Steady, Compressible Flow: dp V gz C ρ + + 5) Steady, Incompressible Flow: P V gz C [J] ρ + + The sm of the kinetic, potential, and flow energies of a flid particle is constant along a streamline dring steady flow Ship Hydrodynamics 31 In Chap. 5: Static, Dynamic & Stagnation Pressres 1) Ke and Pe can be converted to flow energy (vice versa) dring flow: V P+ ρ + ρgz C [Pa] 1 Static pressre ( 정압 ) Dynamic pressre ( 동압 ) 3 Hydrostatic pressre ( 정수압 ) ) Stagnation pressre ( 정체압 ) 정압 (static pressre)+ 동압 (dynamic pressres) 1) 유동압력측정기구 : Pitot tbe, Piezometer, U type manometer, Pitot-static probe etc. V Pstag P+ ρ Ship Hydrodynamics 3

FLUID MECHANICS. Dynamics of Viscous Fluid Flow in Closed Pipe: Darcy-Weisbach equation for flow in pipes. Major and minor losses in pipe lines.

FLUID MECHANICS. Dynamics of Viscous Fluid Flow in Closed Pipe: Darcy-Weisbach equation for flow in pipes. Major and minor losses in pipe lines. FLUID MECHANICS Dynamics of iscous Fluid Flow in Closed Pipe: Darcy-Weisbach equation for flow in pipes. Major and minor losses in pipe lines. Dr. Mohsin Siddique Assistant Professor Steady Flow Through