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7/25/2014 Solutions Of Colebrook & White Equation ~ My Engineering World
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Solutions Of Colebrook & White EquationWritten by Christos Samaras on Thursday, 19 May 2011 at 19:26
Problem description
Colebrook & White equation is widely used for calculating the friction factor in turbulent flows. The equation is difficult to
be solved by hand, but with nowadays computers we can overcome this difficulty very fast. The equation is written
below:
Where:
f: friction factor [-]
: roughness [m]
Dh: hydraulic diameter - for pipes is equal to the internal diameter [m]
Re: Reynolds number [-]
The Reynolds number is calculated by the following formula:
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7/25/2014 Solutions Of Colebrook & White Equation ~ My Engineering World
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Where:
V: mean velocity of the flow in the pipe [m/s]
L: pipe length [m]
: dynamic viscosity of the fluid [kg/(ms)]
: kinematic viscosity ( = /) [m/s]
: density of the fluid (kg/m)
Solutions
Here we will discuss about five different approaches to the above equation. The degree of accuracy differs for each
approach, as well as the level of difficulty.
1. Iteration process
If we rewrite the above equation, it will look like this:
By changing the values of friction factor we can find the solution using the trial and error method. However, this can
easily be done in an Excel worksheet using goal seek.
2. Swamee Jains equation
The Swamee Jains equation is used in order to find directly the friction factor in a full-flowing circular pipe. It is an
approximation of the Colebrook & Whites equation.
3. Haaland's equation
Haaland in 1983 simplified the Colebrook & Whites equation to the following equation:
4. Serghides' equations
The last two solutions are more complicated since they approach the problem with a set of equations. Serghides'
solution was based on Steffensen's method. The solution involves the calculation of three intermediate values first and
then substituting those values into a final equation.
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7/25/2014 Solutions Of Colebrook & White Equation ~ My Engineering World
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5. Goudar Sonnad's equations
Goudar Sonnads set of equations is the most complicated of all the solutions since there are a lot of intermediate
values that should be calculated before the final equation. This set of equations is given below:
Considering that the iteration process gives the most accurate results, I have developed an Excel workbook in order to
experiment on the other four solutions/methods. So, with this workbook you can try it yourself and see which approach
gives results that are closer to the iteration process results.
How to use it
7/25/2014 Solutions Of Colebrook & White Equation ~ My Engineering World
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In the first sheet (named Solutions) fill all the green cells and press the Solve Equation button. If the flow in the pipe
is turbulent, the sheet will automatically calculate the friction factor with the five different methods that described above.
If you press one of the buttons named Roughness, Diameter, Velocity or Viscosity you will be transferred to the
corresponding sheet. There you can play (loop calculation) with the corresponding parameter and see how the results
of five methods are affected from the parameters change. Furthermore, in each sheet there is also a chart which
depicts the results of your loops.
Downloads
This file can be opened with Excel 2007 or newer. For best results, go to File -> Options -> Formulas tab and put
Maximum Iteration = 30000 (max value is 32767) and Maximum Change = 0.00001 (or even smaller).
Read also
10 + 1 Ways To Find The Friction Factor In Pipes
Pressure Losses In Piping Systems
Friction Factor Function
Head And Pressure Losses In Pipes
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Christos Samaras
Mechanical Engineer (Ph.D. cand.), M.Sc. Cranfield University, Dipl.-Ing. Aristotle University, Thessaloniki - Greece.
Communication: tel. +30-6973513308, e-mail , Facebook , Twitter , Google+ and Linkedin. Full CV here.
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Eugene Fang 4 months ago
Thank you very much for this excel sheet
Reply
Christos Samaras 4 months agoAdmin Eugene Fang
You 're welcome, my friend!
Reply
Otieno Otieno D 9 months ago
This is a nice one, it has helped me alot. Thanks alot Christos.
Reply
Christos Samaras 9 months agoAdmin Otieno Otieno D
You 're welcome, Otieno! It's good to know that it was useful to someone.
Best Regards,
Christos
Reply
Bkash 9 months ago
Thanks mate... Much appreciated
Reply
Christos Samaras 9 months agoAdmin Bkash
You 're welcome, my friend!
Kind Regards,
Christos
Reply
mike a year ago
how to solve the colebrook equation by just hand calculating? i find no way. any idea?
thanks.
Reply
Christos Samaras a year agoAdmin mike
Mike, all these equations, as well the equations on the other post are in reality
approximations of Coolebrook-White formula. You cannot solve Coolebrook-
White formula by hand, but you can solve the other formulas by hand. And as
Wikipedia writes The Colebrook equation used to be solved numerically due to
its apparent implicit nature.
Reply
Vincent Carpenter 6 months ago Christos Samaras
Christos. Kindly note you can solve CW by hand, it just takes a while.
Typically three iterations are required if the first guesstimate is
reasonable.
Christos Samaras 6 months agoAdmin Vincent Carpenter
Vincent, thats true, but, what I was implying in the previous
answer is that there is no DIRECT solution; its not like f
=something, it is friction factor as a function of friction factor (both
sides of equation include the unknown parameter f). So, of course
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7/25/2014 Solutions Of Colebrook & White Equation ~ My Engineering World
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Christos Samaras Hi keneshi,The
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Muhammed Ali AYDIN Thanks Christos
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Reply
you can do the iterations by hand and get an acceptable accuracy
after 3-4 iterations, but, again, you can use any of the alternative
equations
Reply
jack a year ago
thanks for the great work
best regards
Reply
Christos Samaras a year agoAdmin jack
Jack. thank you very much for your kind comment!
Reply
jack a year ago
hi chris
I just want know why Colebrook equation is the most
1
Reply
Christos Samaras a year agoAdmin jack
Hi Jack,
Sorry, but I cannot understand the question. Your comment was somehow cut in
the middle.
Reply
Christos Samaras a year ago
Hi ma65p,
A short answer will be that the most accurate equation depends on the input
parameters.
I think that the iterative process gives the best results. Then are probably the Goudar
Sonnad and Serghides' equations. Excluding the iterative process, the results are
strongly depended on input parameters (roughness, diameter, fluid velocity etc.), so in
some cases maybe the Goudar Sonnad is better, while on others the Serghides.
Because this is unclear, you can run the other sheets of this workbook in order to
perform a parametric analysis and see which equation is better for your case.
You can check also this workbook that has more equations.
Kind Regards,
Christos
Reply
ma65p a year ago
Thank you for the spreadsheet. However, what was your conclusion as to which
approximation was the most accurate?
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