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FULLY DEVELOPED TURBULENT PIPE FLOW CLASS 2 - REVIEW

FULLY DEVELOPED TURBULENT PIPE FLOW CLASS 2 - REVIEW. FULLY DEVELOPED , STEADY, NO BODY FORCES , LAMINAR PIPE FLOW. = 0. = 0. = 0. F Sx + F Bx = /t ( cv udVol )+  cs u V d A Eq. (4.17).  = (r/2)(dp/dx) Eq 8.13a.  = (r/2)(dp/dx).  yx = (du/dy)+u’v’

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FULLY DEVELOPED TURBULENT PIPE FLOW CLASS 2 - REVIEW

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  1. FULLY DEVELOPED TURBULENT PIPE FLOW CLASS 2 - REVIEW

  2. FULLY DEVELOPED, STEADY,NO BODY FORCES, LAMINAR PIPE FLOW = 0 = 0 = 0 FSx + FBx = /t (cvudVol )+ csuVdA Eq. (4.17)  = (r/2)(dp/dx) Eq 8.13a

  3.  = (r/2)(dp/dx) yx = (du/dy)+u’v’ uavg = UC/L(1-r/R)1/n Q = 0 uldy;V = Q/A V/UC/L = 2n2/(2n+1)(n+1) yx = (du/dy) u = - (R2/4)(dp/dx) x [1 – (r/R)2] = UC/L[1-(r/R)2] Q = 0 uldy;V = Q/A V/UC/L = 1/2 (empirical) a a turbulent laminar

  4. u(r)/Uc/l = (y/R)1/n = ([R-r]/R)1/n= (1-r/R)1/n n=6-10 Laminar Flow u/Uc/l = 1-(r/R)2 u(r)/Uc/l = (y/R)1/n

  5. Eq. 8.30 “one of the most important and useful equations in fluid mechanics” Fox et al. ENTER ENERGY EQUATION

  6. V12/ (2) + p1/() + gz1=V22/ (2) + p2/() + gz2 + hlT hlT has units of enery per unit mass [V2] V12/ (2g) + p1/(g) + z1=V22/ (2g) + p2/(g) + z2 + HlT HlT has units of enery per unit weight [L] from hydraulics during 1800’s “one of the most important and useful equations in fluid mechanics” Fox et al. Allows calculations of capacity of an oil pipe line, what diameter water main to install or pressure drop in an air duct, ……

  7. V12/ (2g) + p1/(g) + z1=V22/ (2g) + p2/(g) + z2 + HlT A ½ V(r)2 V(r)dA = (dm/dt)½ V2  = [A  V(r)3dA ]/ {(dm/dt)V2} Turbulent Flow: V(r)/Uc/l = (1-r/R)1/n Laminar Flow: V(r)/Uc/l = 1 – (r/R)2

  8. V12/ (2g) + p1/(g) + z1=V22/ (2g) + p2/(g) + z2 + HlT • = [A  V(r)3dA ]/ {(dm/dt)V2} • = 1 for potential flow • = 2 for laminar flow •  1 for turbulent flow V(r)/Uc/l = (y/R)1/n • = (Uc/l/V)3 2n2 / (3 + n)(3 + 2n)* • = 1.08 for n = 6;  = 1.03 for n = 10

  9. V1avg2/ (2g) + p1/(g) + z1 = V2avg2/ (2g) + p2/(g) + z2 + Hl (Eq. 8.30) + Hl (Eq. 8.34) V = Q/Area Hl

  10. BREATH (early 20th Century turbulent pipe flow experiments)

  11. fF = wall /{(1/2) V2} Similarity of Motion in Relation to the Surface Friction of Fluids Stanton & Pannell –Phil. Trans. Royal Soc., (A) 1914

  12. fF = wall /{(1/2) V2} ~1914 fF = wall /{(1/2) V2} fD = (p/L)D/{(1/2) V2} =(p/L)2R2/2{ ½ V2} = 4wall/{(1/2) V2} = 4 fF

  13. BREATH (rough pipe turbulent flow experiments)

  14. Original Data of Nikuradze p  U? Stromungsgesetze in Rauhen Rohren, V.D.I. Forsch. H, 1933, Nikuradze

  15. aside Sir Isaac Newton (1642 – 1727) p  uavg2 Newton believed that drag  uavg2 arguing that each fluid particle would lose all their momentum normal to the body. Drag = Mass Flow x Change in Momentum Drag = dp/dt  (UA)U  U2A Drag/Area U2

  16. Fully rough zone where have flow separation over roughness elements and p ~ V2 k* = u*/; k* < 4: hydraulically smooth 4 < k*< 60 transitional regime; k* > 60 fully rough (no  effect) White 1991 – Viscous Fluid Flow

  17. Curves are from average values good to +/- 10%

  18. BREATH (Moody Diagram)

  19. Hl= f (L/D)V2/(2g) laminar t u r b u l e n t f = 64/Re and is proportional to  in laminar flow f is not a function of /D in laminar flow f = const. and is not a function of  at high enough Re turbulent flows in a rough pipe f is usually a function of /D in turbulent flows

  20. Curves are from average values good to +/- 10% fD = (p/L)D/{(1/2) V2} Darcy friction factor ReD = UD/ For new pipes, corrosion may cause e/D for old pipes to be 5 to 10 times greater.

  21. fF = -2.0log([e/D]/3.7 + 2.51/(RefF0.5)] If first guess is: fo = 0.25[log([e/D]/3.7 + 5.74/Re0.9]-2 should be within 1% after 1 iteration

  22. For turbulent flow in a smooth pipe and ReD < 105, can use Blasius correlation: f = 0.316/ReD0.25 which can be rewritten as: wall = 0.0332  V2 (/[RV])1/4)

  23. For turbulent flow and Re < 105 can use Blasius correlation: fD = 0.316/Re0.25 Which can be rewritten as: wall =0.0332  V2 (/[RV]) PROOF fD = 4 fF 0.316 1/4/ (V1/4 D1/4) = 4wall/(1/2 V2) wall = (0.0395 V2) [1/4/ (V1/4 (2R)1/4) wall = (0.0332 V2) [  / (VR)]1/4 QED

  24. Question? Looking at graph – imagine that pipe diameter, length, viscosity and density is fixed. Is there any region where an increase in V results in an increase in pressure drop?

  25. Question? Looking at graph – imagine that pie diameter and kinematic viscosity and density is fixed. Is there any region where an increase in V results in an increase in pressure drop? Everywhere!!!!!!! Turbulent flow Instead of non-dimensionalizing p by ½ V2; use D3  /( 2L) transition pD3 /(2L) Laminar flow From Tritton

  26. Some history ~ “Moody Diagram”

  27. f = function of V, D, roughness and viscosity f is dimensionless Hl Antoine Chezy ~ 1770: for channels: V2P = AS extrapolate this for pipe: Hl = (4/C2)(L/D)V2 Gaspard Riche de Prony (1800) Hl = (L/D)(aV + bV2) C; a and b are not dimensionless C; a and b are not a function of roughness Antoine Chezy

  28. Hl f = function of V, D and roughness f is dimensionless

  29. Traditional to call f the Darcy friction factor although Darcy never proposed it in that form Hl f is a function of  and D better estimates of f Could be dropped for rough pipes

  30. Hl   = w/( ½ Vavg2) prob 8.83 Combined Weisbach’s equation with Darcy and other data, compiled table for f but used hydraulic radius.

  31. Hl Eq. 8.34

  32. 4000< ReR < 80000 Full range of turbulent Reynolds numbers

  33. Re f 1/f Re/f “ These equations are obviously too complex to be of practical use. On the other hand, if the function which they embody is even approximately valid for commercial surfaces in general, such extremely important information could be made readily available in diagrams or tables.”

  34. Hl f = [p/(g)]D2g/(LV2) f = {[p/L]D}/{1/2V2} “The author does not claim to offer anything particularly new or original, his aim merely being to embody the now accepted conclusion in convenient form for engineering use.”

  35. Hl

  36. T H E E N D

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