Colebrookvergelyking

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Die Colebrookvergelyking word gebruik om die wrywingsfaktor (f' of f) te bereken vir vloei in 'n pyp.

Die Moodygrafiek kan ook gebruik word om die wrywingsfaktor te bepaal as berekeninge met die hand gedoen word. Sagteware pakkette of Excel sigblaaie gebruik egter eerder die Colebrookvergelyking (of benaderings daarvan) om die wrywingsfaktor outomaties te bereken.

Verskillende wrywingsfaktore[wysig | wysig bron]

Neem kennis van die verskillende wrywingsfaktore:

f' = f_{Darcy/Moody} = 4f_{Fanning} = 8f_{Stanton \ en \ Pannel}

Maak altyd seker die regte faktor gebruik word. In hierdie blad word met die Darcy/Moody wrywingsfaktor gewerk.

Colebrookvergelyking[wysig | wysig bron]

Die Colebrookvergelyking is die akkuraatse om die wrywingsfaktor te bereken:

\frac{1}{\sqrt{f'}} = -2log \left(\frac{\varepsilon /D}{3.7} + \frac{2.51}{\text{Re} \sqrt{f'}} \right)

Waar:

  • f' - Darcy/Moody wrywingsfaktor [dimensieloos]
  • \epsilon - Ruheidsfaktor [m]
  • D - Pyp binnediameter [m]
  • \varepsilon/D - Hierdie term se eenhede moet dimensieloos wees.
  • \text{Re} - Reynoldsgetal [dimensieloos]

Die nadeel van hierdie vergelyking is dat dit deur 'n iteratiewe metode opgelos moet word of die "Goal seek" funksie in Excel moet gebruik word.

'n Algemene duimreël is om f' = 0.02 as eerste raaiskoot te neem.

Benaderings vir die Colebrookvergelyking[wysig | wysig bron]

Omdat dit nie maklik is om f' uit die Colebrookvergelyking te bepaal nie (die formule kan nie herrangskik word sodat f' alleen staan nie), is daar andere wat benaderingsformules opgestel het die vir Colebrookvergelyking:

Swamee–Jainvergelyking[wysig | wysig bron]

Hierdie is 'n goeie benadering om die Darcy of Moody wrywingsfaktor te bepaal en is makliker om te gebruik omdat dit f' slegs aan die een kant van die vergelyk het en is dit dus nie nodig om gelyktydig op te los nie:

Die volgende aanname word gemaak:

\frac{2.51}{\text{Re} \sqrt{f'}} \approx \frac{5.74}{\text{Re}^{0.9}}

Dus is:

\frac{1}{\sqrt{f'}} = -2log \left(\frac{\varepsilon /D}{3.7} + \frac{5.74}{\mathrm{Re}^{0.9} \sqrt{f'}} \right)

En daarom:

f' = 0.25\left[\log_{10} \left(\frac{\varepsilon/D}{3.7} + \frac{5.74}{\mathrm{Re}^{0.9}}\right)\right]^{-2}

Churchill se vergelykings[wysig | wysig bron]

Churchill se vergelyking kan ook gebruik word om die Darcy of Moody wrywingsfaktor te bepaal en het, soos die Swamee–Jain vergelyking, ook die voordeel dat f' makliker opgelos kan word as in die geval van die Colebrookvergelyking.

f' = 8 \left[\left(\frac{8}{\text{Re}}\right)^{12} + \frac{1}{(A+B)^{1.5}}\right]^{1/12}
A = \left(-2.457\ln\left[\left(\frac{7}{Re}\right)^{0.9} + 0.27\left(\frac{\varepsilon}{D}\right)\right]\right)^{16} \qquad B=\left(\frac{37530}{\text{Re}}\right)^{16}


Lae Reynoldsgetalle[wysig | wysig bron]

Indien die Reynoldsgetal minder as 2000 is kan die volgende benadering gebruik word:

Darcy of Moody wrywingsfaktor: f' \approx \frac{64}{Re} \quad as\ Re<2000

Fanning wrywingsfaktor: f_{Fanning} \approx \frac{1}{4}f_{Darcy/Moody} = \frac{16}{Re} \quad as\ Re<2000

Tabel van benaderings[wysig | wysig bron]

Die volgende tabel gee 'n lys van benaderings vir die Darcy/Moody wrywingsfaktor:[1]

  • Re: Reynoldsgetal (dimensieloos)
  • λ: Darcy/Moody wrywingsfaktor (dimensieloos)
  • ε: Pypruheidsfaktor (dimensie=lengte)
  • D: Pyp dinnediameter

Neem kennis dat die Churchillvergelyking [2] (1977) die enigste is wat 'n korrekte waarde vir die wrywingsfaktor gee in die laminêre vloeigebied (Reynoldsgetal < 2300). Al die ander vergelykings is opgestel vir turbulente vloei alleen.

Tabel van Colebrookvergelyking benaderings
Vergelyking Outeur Jaar Verwysing


\lambda = .0055 (1 + (2 \times10^4 \cdot\frac{\varepsilon}{D} + \frac{10^6}{Re} )^\frac{1}{3})

Moody 1947


\lambda = .094 (\frac{\varepsilon}{D})^{0.225} + 0.53 (\frac{\varepsilon}{D}) + 88 (\frac{\varepsilon}{D})^{0.44} \cdot {Re}^{-{\Psi}}

where
\Psi = 1.62(\frac{\varepsilon}{D})^{0.134}
Wood 1966


\frac{1}{\sqrt{\lambda}} = -2 \log (\frac{\varepsilon}{3.715D} + \frac{15}{Re})

Eck 1973


\frac{1}{\sqrt{\lambda}} = -2 \log (\frac{\varepsilon}{3.7D} + \frac{5.74}{Re^{0.9}})

Jain and Swamee 1976


\frac{1}{\sqrt{\lambda}} = -2 \log ((\frac{\varepsilon}{3.71D}) + (\frac{7}{Re})^{0.9})

Churchill 1973


\frac{1}{\sqrt{\lambda}} = -2 \log ((\frac{\varepsilon}{3.715D}) + (\frac{6.943}{Re})^{0.9}))

Jain 1976


\lambda = 8[(\frac{8}{Re})^{12} + \frac{1}{(\Theta_1 + \Theta_2)^{1.5}})]^{\frac{1}{12}}

where
\Theta_1=[-2.457 \ln[(\frac{7}{Re})^{0.9} + 0.27\frac{\varepsilon}{D}]]^{16}
\Theta_2 = (\frac{37530}{Re})^{16}
Churchill 1977


\frac{1}{\sqrt{\lambda}} = -2 \log [\frac{\varepsilon}{3.7065D} - \frac{5.0452}{Re} \log(\frac{1}{2.8257}(\frac{\varepsilon}{D})^{1.1098} + \frac{5.8506}{Re^{0.8981}})]

Chen 1979


\frac{1}{\sqrt{\lambda}} = 1.8\log[\frac{Re}{0.135Re(\frac{\varepsilon}{D}) +6.5}]

Round 1980


\frac{1}{\sqrt{\lambda}} = -2 \log (\frac{\varepsilon}{3.7D} + \frac{5.158log(\frac{Re}{7})} {Re(1 + \frac{Re^{0.52}}{29} (\frac{\varepsilon}{D})^{0.7} }

Barr 1981


\frac{1}{\sqrt{\lambda}} = -2 \log [\frac{\varepsilon}{3.7D} - \frac{5.02}{Re} \log(\frac{\varepsilon}{3.7D} - \frac{5.02}{Re} \log(\frac{\varepsilon}{3.7D} + \frac{13}{Re}))]

or


\frac{1}{\sqrt{\lambda}} = -2 \log [\frac{\varepsilon}{3.7D} - \frac{5.02}{Re} \log(\frac{\varepsilon}{3.7D} + \frac{13}{Re})]

Zigrang and Sylvester 1982


\frac{1}{\sqrt{\lambda}} = -1.8 \log \left[\left(\frac{\varepsilon}{3.7D}\right)^{1.11} + \frac{6.9}{Re}\right]

Haaland[lower-alpha 1][3] 1983


\lambda = [\Psi_1 - \frac{(\Psi_2-\Psi_1)^{2}}{\Psi_3-2\Psi_2+\Psi_1}]^{-2}

or


\lambda = [4.781 - \frac{(\Psi_1-4.781)^{2}}{\Psi_2-2\Psi_1+4.781}]^{-2}

where
\Psi_1 = -2\log(\frac{\varepsilon}{3.7D} + \frac{12}{Re})
\Psi_2 = -2\log(\frac{\varepsilon}{3.7D} + \frac{2.51\Psi_1}{Re})
\Psi_3 = -2\log(\frac{\varepsilon}{3.7D} + \frac{2.51\Psi_2}{Re})
Serghides 1984


\frac{1}{\sqrt{\lambda}} = -2 \log(\frac{\varepsilon}{3.7D} + \frac{95}{Re^{0.983}} - \frac{96.82}{Re})

Manadilli 1997


\frac{1}{\sqrt{\lambda}} = -2 \log \lbrace \frac{\varepsilon}{3.7065D}-\frac{5.0272}{Re}\log[\frac{\varepsilon}{3.827D} - \frac{4.657}{Re} \log ((\frac{\varepsilon}{7.7918D})^{0.9924} + (\frac{5.3326}{208.815 + Re})^{0.9345})] \rbrace

Monzon, Romeo, Royo 2002


\frac{1}{\sqrt{\lambda}} = 0.8686 \ln[\frac{0.4587Re}{(S-0.31)^{\frac{S}{(S+1)}}}]

where:
S = 0.124Re \frac{\varepsilon}{D} + \ln (0.4587Re)
Goudar, Sonnad 2006


\frac{1}{\sqrt{\lambda}} = 0.8686 \ln[\frac{0.4587Re}{(S-0.31)^{\frac{S}{(S+0.9633)}}}]

where:
S = 0.124Re \frac{\varepsilon}{D} + \ln (0.4587Re)
Vatankhah, Kouchakzadeh 2008


\frac{1}{\sqrt{\lambda}} = \alpha - [ \frac {\alpha + 2\log(\frac{\Beta}{Re})}{1 + \frac{2.18}{\Beta}}]

where
\alpha = \frac{(0.744\ln(Re)) - 1.41}{(1+ 1.32\sqrt{\frac{\varepsilon}{D}})}
\Beta = \frac{\varepsilon}{3.7D}Re + 2.51\alpha
Buzzelli 2008


\lambda = \frac{6.4}{(\ln(Re) -\ln(1+.01Re\frac{\varepsilon}{D}(1+10\sqrt{\frac{\varepsilon}{D}})))^{2.4}}

Avci, Kargoz 2009


\lambda = \frac{0.2479 - 0.0000947(7-\log Re)^{4}}{(\log(\frac{\varepsilon}{3.615D} + \frac{7.366}{Re^{0.9142}}))^{2}}

Evangleids, Papaevangelou, Tzimopoulos 2010

Kyk ook[wysig | wysig bron]

Verwysings[wysig | wysig bron]

  1. Beograd, Dejan Brkić (Maart 2012). “Determining Friction Factors in Turbulent Pipe Flow”. Chemical Engineering: 34–39.
  2. Churchill, S.W. (7 November 1977). “Friction-factor equation spans all fluid-flow regimes”. Chemical Engineering: 91–92.
  3. BS Massey Mechanics of Fluids 6th Ed ISBN 0-412-34280-4


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