ضریب اصطکاک دارسی
جدول تقریب

جدول زیر تقریب های تاریخی رابطه کولبروک – وایت را برای جریان فشار محور فهرست می کند. معادله چرچیل (1977) تنها معادله ای است که می تواند برای جریان بسیار آهسته ارزیابی شود (عدد رینولدز <1)، اما چنگ (2008) و بلوس و همکاران. معادلات (2018) همچنین مقدار تقریباً صحیحی را برای ضریب اصطکاک در ناحیه جریان آرام برمی‌گرداند (عدد رینولدز < 2300).  بقیه موارد فقط برای جریان انتقالی و آشفته هستند.
| معادله | نویسنده | سال | محدوده | رفرنس |
| --- | --- | --- | --- | --- |
|   f=0.0055\left[1+\left(2\times 10^{4}\cdot {\frac {\varepsilon }{D}}+{\frac {10^{6}}{\mathrm {Re} }}\right)^{\frac {1}{3}}\right] | Moody | 1947 |  4000\leq \mathrm {Re} \leq 5\times 10^{8}  0\leq \varepsilon /D\leq 0.01 |  |
|   f=0.094\left({\frac {\varepsilon }{D}}\right)^{0.225}+0.53\left({\frac {\varepsilon }{D}}\right)+88\left({\frac {\varepsilon }{D}}\right)^{0.44}\cdot {\mathrm {Re} }^{-{\Psi }} که  \Psi =1.62\left({\frac {\varepsilon }{D}}\right)^{0.134} | Wood | 1966 |  4000\leq \mathrm {Re} \leq 5\times 10^{7}  0.00001\leq \varepsilon /D\leq 0.04 |  |
|   {\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon /D}{3.715}}+{\frac {15}{\mathrm {Re} }}\right) | Eck | 1973 |  |  |
|   f={\frac {0.25}{\left[\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {5.74}{\mathrm {Re} ^{0.9}}}\right)\right]^{2}}} | Swamee and Jain | 1976 |  5000\leq \mathrm {Re} \leq 10^{8}  0.000001\leq \varepsilon /D\leq 0.05 |  |
|   {\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon /D}{3.71}}+\left({\frac {7}{\mathrm {Re} }}\right)^{0.9}\right) | Churchill | 1973 |  |  |
|   {\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon /D}{3.715}}+\left({\frac {6.943}{\mathrm {Re} }}\right)^{0.9}\right) | Jain | 1976 |  |  |
|   f/8=\left[\left({\frac {8}{\mathrm {Re} }}\right)^{12}+{\frac {1}{(\Theta _{1}+\Theta _{2})^{1.5}}}\right]^{\frac {1}{12}} که  \Theta _{1}=\left[-2.457\ln \left(\left({\frac {7}{\mathrm {Re} }}\right)^{0.9}+0.27{\frac {\varepsilon }{D}}\right)\right]^{16}  \Theta _{2}=\left({\frac {37530}{\mathrm {Re} }}\right)^{16} | Churchill | 1977 |  |  |
|   {\frac {1}{\sqrt {f}}}=-2\log \left[{\frac {\varepsilon /D}{3.7065}}-{\frac {5.0452}{\mathrm {Re} }}\log \left({\frac {1}{2.8257}}\left({\frac {\varepsilon }{D}}\right)^{1.1098}+{\frac {5.8506}{\mathrm {Re} ^{0.8981}}}\right)\right] | Chen | 1979 |  4000\leq \mathrm {Re} \leq 4\times 10^{8} |  |
|   {\frac {1}{\sqrt {f}}}=1.8\log \left[{\frac {\mathrm {Re} }{0.135\mathrm {Re} (\varepsilon /D)+6.5}}\right] | Round | 1980 |  |  |
|   {\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {4.518\log \left({\frac {\mathrm {Re} }{7}}\right)}{\mathrm {Re} \left(1+{\frac {\mathrm {Re} ^{0.52}}{29}}(\varepsilon /D)^{0.7}\right)}}\right) | Barr | 1981 |  |  |
|   {\frac {1}{\sqrt {f}}}=-2\log \left[{\frac {\varepsilon /D}{3.7}}-{\frac {5.02}{\mathrm {Re} }}\log \left({\frac {\varepsilon /D}{3.7}}-{\frac {5.02}{\mathrm {Re} }}\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {13}{\mathrm {Re} }}\right)\right)\right] یا  {\frac {1}{\sqrt {f}}}=-2\log \left[{\frac {\varepsilon /D}{3.7}}-{\frac {5.02}{\mathrm {Re} }}\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {13}{\mathrm {Re} }}\right)\right] | Zigrang and Sylvester | 1982 |  |  |
|   {\frac {1}{\sqrt {f}}}=-1.8\log \left[\left({\frac {\varepsilon /D}{3.7}}\right)^{1.11}+{\frac {6.9}{\mathrm {Re} }}\right] | Haaland | 1983 |  |  |
|   {\frac {1}{\sqrt {f}}}=\Psi _{1}-{\frac {(\Psi _{2}-\Psi _{1})^{2}}{\Psi _{3}-2\Psi _{2}+\Psi _{1}}} یا  {\frac {1}{\sqrt {f}}}=4.781-{\frac {(\Psi _{1}-4.781)^{2}}{\Psi _{2}-2\Psi _{1}+4.781}} که  \Psi _{1}=-2\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {12}{\mathrm {Re} }}\right)  \Psi _{2}=-2\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {2.51\Psi _{1}}{\mathrm {Re} }}\right)  \Psi _{3}=-2\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {2.51\Psi _{2}}{\mathrm {Re} }}\right) | Serghides | 1984 |  |  |
|   A=0.11\left({\frac {68}{Re}}+{\frac {\varepsilon }{D}}\right)^{0.25} if  A\geq 0.018 then  f=A and if  A<0.018  then   f=0.0028+0.85A | Tsal | 1989 |  |  |
|   {\frac {1}{\sqrt {f}}}=-2\log \left({\frac {\varepsilon /D}{3.7}}+{\frac {95}{\mathrm {Re} ^{0.983}}}-{\frac {96.82}{\mathrm {Re} }}\right) | Manadilli | 1997 |  4000\leq \mathrm {Re} \leq 10^{8}  0\leq \varepsilon /D\leq 0.05 |  |
|   {\frac {1}{\sqrt {f}}}=-2\log \left\lbrace {\frac {\varepsilon /D}{3.7065}}-{\frac {5.0272}{\mathrm {Re} }}\log \left[{\frac {\varepsilon /D}{3.827}}-{\frac {4.657}{\mathrm {Re} }}\log \left(\left({\frac {\varepsilon /D}{7.7918}}\right)^{0.9924}+\left({\frac {5.3326}{208.815+\mathrm {Re} }}\right)^{0.9345}\right)\right]\right\rbrace  | Romeo, Royo, Monzon | 2002 |  |  |
|   {\frac {1}{\sqrt {f}}}=0.8686\ln \left[{\frac {0.4587\mathrm {Re} }{(S-0.31)^{\frac {S}{(S+1)}}}}\right] که:  S=0.124\mathrm {Re} {\frac {\varepsilon }{D}}+\ln(0.4587\mathrm {Re} ) | Goudar, Sonnad | 2006 |  |  |
|   {\frac {1}{\sqrt {f}}}=0.8686\ln \left[{\frac {0.4587\mathrm {Re} }{(S-0.31)^{\frac {S}{(S+0.9633)}}}}\right] که:  S=0.124\mathrm {Re} {\frac {\varepsilon }{D}}+\ln(0.4587\mathrm {Re} ) | Vatankhah, Kouchakzadeh | 2008 |  |  |
|   {\frac {1}{\sqrt {f}}}=\alpha -{\frac {\alpha +2\log \left({\frac {\mathrm {B} }{\mathrm {Re} }}\right)}{1+{\frac {2.18}{\mathrm {B} }}}} که  \alpha ={\frac {0.744\ln(\mathrm {Re} )-1.41}{1+1.32{\sqrt {\varepsilon /D}}}}  \mathrm {B} ={\frac {\varepsilon /D}{3.7}}\mathrm {Re} +2.51\alpha  | Buzzelli | 2008 |  |  |
|  {\frac {1}{f}}=\left({\frac {Re}{64}}\right)^{a}\left(1.8\log {\frac {Re}{6.8}}\right)^{2(1-a)b}\left(2.0\log {\frac {3.7D}{\epsilon }}\right)^{2(1-a)(1-b)} که  a={\frac {1}{1+\left({\frac {Re}{2720}}\right)^{9}}}  b={\frac {1}{1+\left({\frac {Re}{160{\frac {D}{\epsilon }}}}\right)^{2}}}   | Cheng | 2008 | All flow regimes |  |
|   f={\frac {6.4}{(\ln(\mathrm {Re} )-\ln(1+.01\mathrm {Re} {\frac {\varepsilon }{D}}(1+10{\sqrt {\frac {\varepsilon }{D}}})))^{2.4}}} | Avci, Kargoz | 2009 |  |  |
|   f={\frac {0.2479-0.0000947(7-\log \mathrm {Re} )^{4}}{(\log \left({\frac {\varepsilon /D}{3.615}}+{\frac {7.366}{\mathrm {Re} ^{0.9142}}}\right))^{2}}} | Evangelides, Papaevangelou, Tzimopoulos | 2010 |  |  |
|   f=1.613\left[\ln \left(0.234\left({\frac {\varepsilon }{D}}\right)^{1.1007}-{\frac {60.525}{\mathrm {Re} ^{1.1105}}}+{\frac {56.291}{\mathrm {Re} ^{1.0712}}}\right)\right]^{-2} | Fang | 2011 |  |  |
|   f=\left[-2\log \left({\frac {2.18\beta }{\mathrm {Re} }}+{\frac {\varepsilon /D}{3.71}}\right)\right]^{-2} ,  \beta =\ln {\frac {\mathrm {Re} }{1.816\ln \left({\frac {1.1Re}{\ln \left(1+1.1\mathrm {Re} \right)}}\right)}} | Brkić | 2011 |  |  |
|   f=1.325474505\log _{e}\left(A-0.8686068432B\log _{e}\left(A-0.8784893582B\log _{e}\left(A+(1.665368035B)^{0.8373492157}\right)\right)\right)^{-2} که  A={\frac {\varepsilon /D}{3.7065}}  B={\frac {2.5226}{\mathrm {Re} }} | S.Alashkar | 2012 |  |  |
|  f=\left({\frac {64}{\mathrm {Re} }}\right)^{a}\left[0.75\ln {\frac {\mathrm {Re} }{5.37}}\right]^{2(a-1)b}\left[0.88\ln 3.41{\frac {D}{\epsilon }}\right]^{2(a-1)(1-b)} که  a={\frac {1}{1+\left({\frac {\mathrm {Re} }{2712}}\right)^{8.4}}}  b={\frac {1}{1+\left({\frac {\mathrm {Re} }{150{\frac {D}{\epsilon }}}}\right)^{1.8}}} | Bellos, Nalbantis, Tsakiris | 2018 | All flow regimes |  |
|  {\frac {1}{\sqrt {f}}}=A-{\frac {(B-A)^{2}}{C-2B+A}} که  A=-2\log \left({\frac {\varepsilon /D}{3.7}}+{4.5547 \over \mathrm {Re} ^{0.8784}}\right)  B=-2\log \left({\frac {\varepsilon /D}{3.7}}+{2.51A \over \mathrm {Re} }\right)  C=-2\log \left({\frac {\varepsilon /D}{3.7}}+{2.51B \over \mathrm {Re} }\right) | Niazkar | 2019 |  |  |
|  f={\frac {1}{\left(0.8284\ln \left({\dfrac {\varepsilon /D}{4.913}}+{\dfrac {10.31}{\mathrm {Re} }}\right)\right)^{2}}} | Tkachenko, Mileikovskyi | 2020 | Deviation 5.36 %,  2320\leq {\mathrm {Re} }\leq 10^{9}  0\leq {\varepsilon /D}\leq 0.65 |  |
|  f=\left({\frac {8.128943+A_{1}}{8.128943A_{0}-0.86859209A_{1}\ln \left({\dfrac {A_{1}}{3.7099535\mathrm {Re} }}\right)}}\right)^{2} که  A_{0}=-0.79638\ln \left({\frac {\varepsilon /D}{8.208}}+{\frac {7.3357}{\mathrm {Re} }}\right)  A_{1}=\mathrm {Re} \left(\varepsilon /D\right)+9.3120665A_{0} | Tkachenko, Mileikovskyi | 2020 | Deviation 0.00072 %,  2320\leq {\mathrm {Re} }\leq 10^{9}  0\leq {\varepsilon /D}\leq 0.65 |  |