On the Blasius correlation for friction factors

Transcription

1 O the Blasius correlatio for frictio factors Trih, Khah Tuoc Istitute of Food Nutritio ad Huma Health Massey Uiversity, New Zealad Abstract The Blasius empirical correlatio for turbulet pipe frictio factors is derived from first priciples ad exteded to o-newtoia power law fluids. Two alterative formulatios are obtaied that both correlate well with the experimetal measuremets of Dodge, Bogue ad Yoo. Key words: Blasius, turbulet frictio factor, power law fluids Itroductio I a previous paper (Trih, 00a) logarithmic correlatios for turbulet pipe flow of Ostwald de Waele fluids have bee derived from a master curve for istataeous frictio factors (Trih, 009a). Blasius (93) has also proposed a empirical power law correlatio τ w f = = () ρ / 4 V Re where = ( DVρ µ ) Re is the Reyolds umber, D the pipe diameter, ρ the fluid desity, V the average velocity, µ the fluid viscosity ad τ w the time-averaged wall shear stress. Nikuradse (93) proposed that the Blasius correlatio correspods with a power law represetatio of the velocity profile ( y ) p U = A ()

2 where the expoet p is a fuctio of the Reyolds umber, y is the radial distace from the wall, U = U u* ad y = yu * ρ µ = yu* ν have bee ormalised with the frictio velocity u * = τ w ρ. A alterative form of the power law profile is U U p y = (3) m R Nikuradse (op.cit.) ad Pradtl (935) also proposed a alterative represetatio of the velocity profile U =. 5l y 5. 5 (4) Where At the wall Pradtl postulated the existece of a lamiar sub-layer described by = y U (5) which applies up to y 5. There is a log ad o-goig debate as to whether the power law represeted by equatio () or the so called logarithmic law of the wall, log law i short, best describes experimetal measuremets of velocity profiles i turbulet flow e.g. (McKeo et al., 004). Dodge ad Metzer (959) have exteded the Blasius correlatio to purely viscous o-newtoia fluids α f = (6) β Re g where ρ D V Re g = (7) 3 K8 4 Is called the Metzer-Reed (955)geeralised Reyolds umber ad the rheological parameters K, are defied by the lamiar equatio 8V τ w = K D = K γ w (8) For Ostwald de Waele power law fluids

3 = 3 K = K 4 I this paper, we will derive the Blasius ad Dodge-Metzer empirical equatios from theoretical cosideratios. (9) Theory There are two possible ways to derive a power law relatioship for o-newtoia fluids which must of course also apply to Newtoia fluids whe =.. Extesio of the Blasius empirical correlatio I this approach we begi with the observatio that all pipe frictio factors ca be described by a uique master curve whe expressed i terms of the critical istataeous shear stress (critical shear stress for short) at the poit of burstig i the wall layer process (Trih, 009a). The Blasius correlatio ca the be writte as: Where 0.04 e = (0) Ree f /4 = τ e ρv f e DV = = DV Ree ν e K τ e ( ) are the frictio factor ad Reyolds umber based o the critical istataeous wall shear stress τ e. For high Reyolds umbers, the wall layer is thi (Trih, 009b) ad the radius of curvature eglected. The () () τ = ( ) τ (3) w e

4 ( ) ( ) 5( ) 3 Re f Re e = g (4) 4 Substitutig equatios (3) ad (4) ito (0) gives α f = Re ( 3 ) g (5) Where ( ) ( ) α = (6) 3. Matchig the power law profile with the wall layer The most glarig weakess of the power law velocity profile is its ifiite gradiet at the wall which is physically urealistic. The problem ca be circumveted by observig that the ejectios of wall fluids that begi outside the wall layer at a time averaged distace y = 30 which is the time-averaged value of the wall layer thickess δ ν. We force equatio (3) through the edge of the wall layer U U ν m V δν = R = φuν p (7) We use the value p = 7 to keep i the same rage of Reyolds umber as the Blasius equatio. Substitutig for V = f (8) ad R ad rearragig 5 8 / 3 g f = Re (9) 4 f g α = Re ( 3 ) (0)

5 The power idex for the Reyolds umber i equatios (5) ad (0) are the same which is to be expected. But ow 4 7 α = ( 3 ) ( 3 ) ) 3 φ δ ν 7 () ( 3 ) ) 3 3 Uν 4 The ratio φ ca be calculated by itegratig equatio (3) φ = = () ( p )( p ) The problem boils dow to the a estimate of U ν ad δ ν. For a Newtoia fluid, =, these values are well kow (Trih, 009b, Trih, 00b): δ = 64. 8, U ν = Theα = ν For large Reyolds umbers, the wall layer thickess is thi compared to the pipe radius δ << R ν, the radius of curvature ca be eglected ad the wall layer ca be aalysed like a viscous sub-layer o a flat plate (Trih, 009a). The the wall layer thickess ad velocity are give as ( ) = = ( ) δν (3) δν ( ) ( ). 08 ( ) Uν = = (4) / /. 08 The ( 3 ) ( 3 ) ) 3 φ ( ) 7 3 α =. 08 (5) 3 3 4

6 3 Results ad discussio The derivatios preseted here have bee compared with the experimetal of Dodge (959), Bogue (96) ad Yoo (974). The values of α estimated from equatios (6) ad (5) differ by approximately %. A example usig equatio (6) is show i Figure 0. f 0.0 =.00 B.00 B 0.7 B D 0.76 D Y.00 D =0.700 = ,000 0,000 00,000,000,000 Reg Figure Compariso of predictios from equatio (5 ad 6) with measured frictio factors. Data of Dodge (959) Bogue (96) ad Yoo (974). Equatios (5) ad (0) both correlate 69 data poits with a stadard deviatio of about 4.9% ad a stadard error of 0.03%. A compariso betwee the measured frictio factors with the predictios by equatio (5) is show i Figure.

7 fpred f Figure Compariso of measured frictio factors ad with calculatio from equatio (5). Same data as Figure. However, Blasius type correlatios ted to uderestimate the experimetal data for 5 Re g > 0 ad overestimate them for Re This is show more clearly i g < Figure 3. Thus the low stadard errors give a slightly distorted view of predictio accuracy. fpred/f Re g Figure 3 Effect of Reyolds umber o the predictio of frictio factors usig equatio (5).

8 While the predictios of frictio factors from the exteded Blasius correlatio of Dodge-Metzer ad i this work are similar, the depedece of α ad β o are quite differet as see i Figure α β ' Dodge This work Dodge This work Figure 4 Compariso betwee experimetal values of (5). α, β of Dodge ad equatio The variatios of α ad β i this work are much stroger, especially at small values of. The relatio betwee p ad β ca be formally established (Skellad, 967, Trih, 99, Trih, 009b). Equatio (5) ca be solved for V as τ ρv w α = D V K8 ρ 3 4 β (6) Solvig for V Where V β β ( ) β ( = A ) τ R (7) w

9 A = α Equatio (3) is rearraged as β 3β β β ( ) ) β ( ) 4 β ( ) β ( ) ρ 3 β (8) p y U = Vφ (9) R Substitutig for V from (7) p w R β p β ( ) β ( u = A φ y τ ) (30) Withi the wall layer, the velocity distributio is idepedet of the radius (Trih, 009b, Skellad ad Sampso, 973) which correspods to a zero expoet for R. The β p = β ( ) (3) p β = (3) p p I the particular case whe p = 7, equatio (3) gives β = (33) 3 as show i equatios (5) ad (0). I fact the idex p ca be further expressed i terms of the Reyolds umber by matchig the logarithmic ad power law correlatios of the velocity profile (Trih, 99, Trih, 009b). This approach makes exteds the rage of applicatio of Blasius type correlatios to a larger rage of Reyolds umber. Details will be preseted i a separate paper. 4 Coclusio The Blasius empirical correlatio has bee successfully developed ad exteded to power law o-newtoia fluids from theoretical cosideratios.

10 5 Refereces BLASIUS, P. R. H. 93. Das Aehlichkeitsgesetz bei Reibugsvorgage i Flüssigkeite. Forschugsheft 3, -4. BOGUE, D. C. 96. Velocity profiles i turbulet o-newtoia pipe flow, Ph.D. Thesis, Uiversity of Delaware. DODGE, D. W Turbulet flow of o-newtoia fluids i smooth roud tubes, Ph.D. Thesis, US, Uiversity of Delaware. DODGE, D. W. & METZNER, A. B Turbulet Flow of No-Newtoia Systems. AICHE Joural, 5, MCKEON, B. J., LI, J., JIANG, W., MORRISON, J. F. & SMITS, A. J Further observatios o the mea velocity distributio i fully developed pipe flow. Joural of Fluid Mechaics, 50, METZNER, A. B. & REED, J. C Flow of No-Newtoia Fluids - Correlatio of the Lamiar, Trasitio, ad Turbulet-Flow Regios. Aiche Joural,, NIKURADSE, J. 93. Gesetzmäßigkeit der turbulete Strömug i glatte Rohre. Forsch. Arb. Ig.-Wes. N PRANDTL, L The Mechaics of Viscous Fluids. I: W.F, D. (ed.) Aerodyamic Theory III. Berli: Spriger. SKELLAND, A. H. P No-Newtoia Flow ad Heat trasfer, New York, Joh Wiley ad Sos. SKELLAND, A. H. P. & SAMPSON, R. L Turbulet o-newtoia boudary layers o a flat plate. The Chemical Egieerig Joural, 6, TRINH, K. T. 99. Turbulet trasport ear the wall i Newtoia ad o- Newtoia pipe flow, Ph.D. Thesis, New Zealad, Uiversity of Caterbury. TRINH, K. T. 009a. The Istataeous Wall Viscosity i Pipe Flow of Power Law Fluids: Case Study for a Theory of Turbulece i Time-Idepedet No- Newtoia Fluids. arxiv.org v [phys.fluid-dy] [Olie]. TRINH, K. T. 009b. A Theory Of Turbulece Part I: Towards Solutios Of The Navier-Stokes Equatios. arxiv.org v [physics.flu.dy.] [Olie].

11 TRINH, K. T. 00a. Logarithmic Correlatios For Turbulet Pipe Flow Of Power Law Fluids. arxiv.org [phys.fluid-dy] [Olie]. Available: TRINH, K. T. 00b. A Zoal Similarity Aalysis of Velocity Profiles i Wall- Bouded Turbulet Shear Flows. arxiv.org [phys.fluid-dy] [Olie]. YOO, S. S Heat trasfer ad frictio factors for o-newtoia fluids i turbulet flow, PhD thesis, US, Uiversity of Illiois at Chicago Circle.