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  • 7/25/2014 Solutions Of Colebrook & White Equation ~ My Engineering World

    http://www.myengineeringworld.net/2011/05/solutions-of-colebrook-white-equation.html 1/7

    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

    http://www.myengineeringworld.net/2011/05/solutions-of-colebrook-white-equation.html 4/7

    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

    Did you like this post? If yes, then share it with your friends. Thank you!

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  • 7/25/2014 Solutions Of Colebrook & White Equation ~ My Engineering World

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    Newer Post Older Post

    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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    16 Comments My Engineering World Login

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    Reply

    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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    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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