Colebrookvergelyking

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

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

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] (normaalweg 0.045 mm of 0.000045 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

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

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

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

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

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

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

Neem kennis dat die Churchillvergelyking (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
$\lambda =.0055(1+(2\times 10^{4}\cdot {\frac {\varepsilon }{D}}+{\frac {10^{6}}{Re}})^{\frac {1}{3}})$	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	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
${\frac {1}{\sqrt {\lambda }}}=\alpha -[{\frac {\alpha +2\log({\frac {\mathrm {B} }{Re}})}{1+{\frac {2.18}{\mathrm {B} }}}}]$ where $\alpha ={\frac {(0.744\ln(Re))-1.41}{(1+1.32{\sqrt {\frac {\varepsilon }{D}}})}}$ $\mathrm {B} ={\frac {\varepsilon }{3.7D}}Re+2.51\alpha$	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

• Pypvloei
• Moodygrafiek
• Darcy–Weisbach
• Manningvergelyking vir vloei in oop kanale