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# Solutions Of Colebrook & White Equation ~ My Engineering World

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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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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.

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).

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

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Christos Samaras

Mechanical Engineer (Ph.D. cand.), M.Sc. Cranfield University, Dipl.-Ing. Aristotle University, Thessaloniki - Greece.

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Join the discussion

Eugene Fang 4 months ago

Thank you very much for this excel sheet

Christos Samaras 4 months agoAdmin Eugene Fang

You 're welcome, my friend!

Otieno Otieno D 9 months ago

This is a nice one, it has helped me alot. Thanks alot Christos.

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

Bkash 9 months ago

Thanks mate... Much appreciated

Christos Samaras 9 months agoAdmin Bkash

You 're welcome, my friend!

Kind Regards,

Christos

mike a year ago

how to solve the colebrook equation by just hand calculating? i find no way. any idea?

thanks.

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.

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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Drawing Circles n AutoCAD UsingExcel & VBA

Christos Samaras Hi keneshi,The

application that you want to build what will

do?It will design a ductile pipe or

Christos Samaras Good to hear that

QWT 2 Get The Unique Values FromAn Excel List

Petros Chatzipantazis Christos Thanks

for wrapping up the year with a useful

tip.My favorite tools are:1) Remove

Custom Elevation Function (VBA &Google API)

Muhammed Ali AYDIN Thanks Christos

ALSO ON MY ENGINEERING WORLD

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

jack a year ago

thanks for the great work

best regards

Christos Samaras a year agoAdmin jack

Jack. thank you very much for your kind comment!

jack a year ago

hi chris

I just want know why Colebrook equation is the most

1

Christos Samaras a year agoAdmin jack

Hi Jack,

Sorry, but I cannot understand the question. Your comment was somehow cut in

the middle.

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

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

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