Friction Factor Expressions 8

[quark]

I just finished writing two FAQs on friction factor expressions as per Mr Montemayor's suggestion (but I took it as an order)

Friction Factor Expressions - Implicit and Explicit faq378-1236

Spreadsheet for Friction Factors faq378-1237

I am thankful to members Art Montemayor, Katmar and TD2K for sharing with me some excellent papers on this subject and explaining me the subtleties of the subject. I hope you will enjoy reading these. Your comments are welcome.

Best Regards,

 

[Latexman]

quark,

Please review the Wood Equation in FAQ378-1236. I think the TGML is incomplete in the f equation where it says

88(e/D)<sup>0.44 x Rea

or something.




Good luck,
Latexman</sup>

 

[quark]

Thanks for pointing out the error. While I ask the forum management for necessary correction of the FAQ, please consider the number 0.44 as exponent to e/D. The code was supposed to be

[ignore]88(e/D)^0.44x Re^a[/ignore] which results in 88(e/D)^0.44x Re^a

 

[quark]

Equation corrected.

 

[thermcool]

Thats a good work, quark and bosses,

Cheers

 

[25362]

Quark,

These two sites also refer to the thread's heading, in particular for the limiting cases, for your consideration:

http://www.eng.auburn.edu/users/tplacek/courses/3600/colebrook.pdf

http://arash.dejkam.com/software/pressure_loss/methods.php

 

[quark]

25362,

Thanks for the interesting links. It is good to have some more collection.

To all,

Though the first FAQ is just a collection of technical papers and bright ideas from other members (there is almost nothing I contributed to it myself, except writing it), I really did laborius work making those excel calculations and copying them down here. I will be satisfied, if atleast one person tries and gets benefit out of them.

 

[iainuts]

Hi Quark. Interesting FAQ, thanks for posting. Wish I'd had that 15 years ago.

One of your assumptions was:
2. The comparison of accuracies of these equations is done based upon the presumption that Colebrook’s equation is perfect and flawless.

I've always assumed the Colebrook equation was a curve fit to experimental data (in the turbulent zone). Is that incorrect? If not, how is it derived. If it is, then the Colebrook equation itself has limitations on it's accuracy.

 

[quark]

That is a good question to which I don't have any answer at the moment, except some guess work. I did search the books and internet before writing my FAQ about the Colebrook's equation but didn't succeed.

I feel that the Colebrook equation is derived from theoretical equations for the following reasons.

1. I am not sure whether we can curve fit an implicit equation(for example, the common error in excel can be circular reference).

2. Moody prepared the first chart of friction factors based on Colebrook's equation. This indicates that there were no graphical presentations of Colebrook's equation when it was written.

Theoretical derivations of friction factor formulae base on the shear stress equation. You can get a hint of Prandtl's work in this line, if you refer to either Heat Transfer by JP Holman or Incropera and DeWitt.

A classical paper by Chandra P. Verma, Solve pipe flow problems directly, August 1979, Hydrocarbon Processing (thanks again to Katmar) describes a method to write the friction factor equations.

Hope this may be of some help.

 

[Montemayor]

My sincerest compliments go to Quark for the extensive and hard work he has put into this subject and which represents so much positive input to the learning and background of all Forum members. I am only disappointed in by the lack of “Stars” next to his name which would indicate the gratitude of the many readers who obviously will profit from this work.

The fluid flow pioneering work done by such people as Prandtl, Reynolds, Froude, Darcy, Weisbach, Colebrook, et al has been of continuing interest to me as well as perhaps to many others. Allow me to share some interesting information from the information I’ve gathered through the years:

I have never succeeded in obtaining or reading Colebrook’s famous 1937 or 1939 papers on his findings and I wonder if anyone has a copy they can share. I have always been under the impression that he and White developed their findings based on the original data from Nikuradse. I stand corrected if they actually derived the equation from theoretical relationships – as Quark mentions. In any event, the whole subject of fluid flow and its description by the friction factor and related equations is a fascinating one that has always intrigued me. I have developed a better and deeper understanding of the science – as well as an appreciation of the findings – by carefully reading and researching the history of how it all came about.

My favorite expert historian on the subject is Prof. Glenn O. Brown of Oklahoma State University. His writings on the subject are extensive and extremely interesting in their historical content and findings. For a few of his insights go to:

http://biosystems.okstate.edu/darcy/DarcyWeisbach.htm

And read his interesting papers (in either English or Portuguese) and also download and read his fascinating Adobe paper on the History of the Darcy-Weisbach Equation. Pay special attention to the part on how Colebrook and White’s findings found their way into Hunter Rouse’s (What a name!) expert hands and what he did with the information. Contrary to popular belief, it was Rouse, not Lewis Moody, who constructed the first graphical depiction of the friction factor. After this exciting first act, the intrigue follows. Lewis F. Moody (late of Princeton) enters the performance as a Johnny-come-lately and upstages Hunter Rouse and all his efforts. As I understand it, Hunter never spoke to Lewis again. Hunter presumably went into seclusion and fluid flow History forgot him. You’ve gotta luv it…Hollywood couldn’t do it better.

Putting the romantic history aside, the current state of affairs with the friction factor is that present-day application of extensive and critical natural gas pipelines have led to the discovery of major areas of inaccuracy in the Colebrook-White equation. If one can appreciate the history, I suspect that we all could understand why this would come about.

To quote just one of several engineering abstracts – “A continuously increasing demand for higher pipeline transport capacity combined with increased computer simulation capacity has pursued the research for a more accurate description of the pressure loss due to friction in gas pipelines – particularly sub-sea natural gas pipelines. From research work performed by various parties it is well known that the methods for calculating the friction loss as used today are inaccurate and subject to uncertainty, especially for high Reynolds number flow.”

Also, from

http://www.ipt.ntnu.no/~jsg/undervisning/gass/abstracts/elling.html

:
Elling Sletfjerding
1996
“ABSTRAKT
The accuracy of the Colebrook-White equation for friction factor calculations for high Reynolds number flow in gas pipelines is investigated. The friction factor is essential to the deliverability of a natural gas transmission line. Using second order analysis of the fully developed turbulent flow it is shown that the friction factor may be overestimated with as much as 15 % using the Colebrook-White equation. This is due to Reynolds number dependence of the slope of the logarithmic velocity profile in turbulent pipe flow.”

I know there are a lot of concerned and interested sharp engineers out there (like Katmar, for one) who have a professional and personal interest in following up on this subject and I would draw their attention to the above in the event it can be of any assistance. I also have labored under the dark cloud of not knowing to what degree of accuracy I can subject Colebrook-White and what is the relevancy of seeking other equations that are measured to their standard of “accuracy”.

All this fortifies what I stated originally: Quark’s contribution should be of great interest because it allows us to better understand and employ the concept of frictional losses in fluid flow. By easing the pain and the effort to attain usable frictional values, he is allowing us all to keep abreast with what we are doing in our engineering efforts and always seeking better methods to achieve better results.

Thank you, Quark. May your tribe increase.

 

[Latexman]

Mr. Montemayor,

I may have a copy of Colebrook's article in my files at work. I will look Monday. If I don't, I will pay a visit to the company library during lunch one day and see if we have it. Sometimes working for the world's largest chemical company comes in handy.

Good luck,
Latexman

 

[bulkhandling]

A No-Name explicit equation I have used for years:

f=1.325/(ln(e/(3.7*D)+5.74/R^0.9))^2

I've been feeling comfortable about its accuracy but you may check it against the listed equations.

Regards,

 

[UmeshMathur]

bulkhandling:

I believe that the equation you have cited is algebraically nearly identical to Jain's explicit equation from 1976 which is:

1/f^0.5 = 1.14 - 2*LOG10(e/D + 21.25/R^0.9)

Reference: A.K. Jain: "An Accurate Explicit Equation for Friction Factor", Journal of Hydraulics Division, American Society of Civil Engineers, pp. 674-677 (May, 1976).

Also, the numerical results from your unattributed equation, Jain's equation, and Colebrook's equation are within a fraction of a percentage point from each other.

Example: With e=0.16, D=80, R=80000, the results are:

Your equation: f=0.02571
Colebrook: f=0.02546
Jain: f=0.02569

These numerical differences are far less than the uncertainty in the roughness factor, e.

 

[25362]

To bulkhandling, yours is the Swamee-Jain equation, used for Re > 4000.

Which can alternatively be written:

f = 0.25[&divide;][log₁₀([&epsilon;]/3.7D + 5.74/Re^0.9)]²​

 

[iainuts]

Hi Quark. Did you ever find out if the Colebrook equation was derived or empirical? I've kept an eye on this thread to see if anyone can answer that definitively.

 

[Latexman]

I just got Colebrook's article today. I'll study it and let ya'll know.

Good luck,
Latexman

 

[Montemayor]

Latex:

If at all possible, could you facilitate a copy to an old sentimental Chem E who dabbles in nostalgia?

Just let me know how much, where, and how.

Thanks
Art Montemayor

 

[Latexman]

Mr. Montemayor,

Check with Infotrieve at

http://www4.infotrieve.com

It cost:
Service : US$9.50
Copyright: US$37.00
Delivery: US $1.00
Total: US$47.50

Infotrieve had to contract with "The British Library" because they did not have it on record. It may be cheaper through Infotrieve now that their upfront costs have been paid, or they may have to go the same route again. If they have to go through The British Library again, you may try to get it through them directly. I'd be curious what happens.

Here's the information you need:
DocTitle: Turbulent Flow in Pipes with Particular Reference to the Transition Region
Author/s: C. F. Colebrook
Volume: 133
Year: 1939
ISSN: 03682455
Publication Name:J. of Inst. Civil Eng.


Good luck,
Latexman

 

[Latexman]

iainuts,

It’s hard to say what Colebrook did and didn’t do without having the references he refers to and seeing their work, but I’ll give you my read on it. I’ll also add present day nomenclature to the 1940 nomenclature where it may clarify the explanation.

Prandtl, von Karmen, and Taylor expressed in mathematical form the mechanism of turbulence from theory. They integrated their equation for two cases, smooth pipe flow and rough pipe flow.

Nikuradse’s experimental results showed complete agreement with the theoretical “smooth law” and “rough law” as long as the “roughness Reynolds number” [the product of the resistance-coefficient (similar to the friction factor), relative roughness, and Reynolds number] was < 3 or > 60.

Colebrook did some “dimensional reasoning” (dimensional analysis) to sort out what was important in the different flow regions, he substituted the “smooth law” and the “rough law” into this, and after a lot of calculus, rearranging and algebra he came up with his single famous equation that spans what we know today as the laminar region to the turbulent region.

So, Colebrook’s equation is derived from theory and validated with data. That’s about as good as it gets, right?


Good luck,
Latexman

 

[Montemayor]

Great work Latex.

Well expressed and explained, as well.

Thank you for the footwork and the retrival information. I'm going to follow up on it.

Art Montemayor