Friction factor correlation/dimensionless quantity? 4

[LaSalle1940]

Everyone knows the Colebrook equation for the (Darcy) friction factor (OK, multiplication by a constant yields the Fanning friction factor, but it's the same thing). Many would probably recognize the correlation between the Karman number and the friction factor: the Karman number is defined as Re(f^½), which quantity can be defined/rewritten in terms of quantities that don't involve the fluid's velocity.

Now: does anyone know of a NAME for the dimensionless quantity defined as Re(f^0.2) ? This can be written in terms of quantities that don't involve the pipe radius/diameter, so that if you know flow and pressure drop, you can determine the required pipe size.

 

[thermcool]

what are you trying to calculate?

 

[LaSalle1940]

What I'm interested in doing is following up on a brief mention in a fluids text. I'm working on establishing a correlation, similar to the Colebrook equation, for the Darcy/Fanning (take your pick) friction factor as a function of Re(f^0.2). As I mentioned earlier, this last quantity can be expressed in terms that do not involve the pipe diameter: thus, given known flow specifications, one can derive a line size.

I don't know if anyone has established such a correlation already: if they have, likely the quantity Re(f^0.2)goes by some name I don't know, and the article would be listed under that name. I'm trying to avoid unnecessary work in re-establishing something already published.

 

[Assumptions]

See "Chemical Engineering Fluid Mechanics", by Ron Darby. In the second ed., it is explained on page 218, Unknown Diameter. However, no name is given to the dimensionless number.

Although, it remains a trial and error approach.

 

[LaSalle1940]

First, thanks for the reference (Darby's name is familiar; he's published quite a few articles in Chemical Engineering).

Second, it's interesting that no name has yet been assigned to this quantity, and that a trial/error approach is still the case: that says to me that nobody has yet established a spreadsheet-friendly analytical correlation.

Thanks.

 

[UmeshMathur]

LaSalle1040:

Actually, there are correlations published by Dr. A. K. Jain in Section 11.5, page 518-520 of his book "Fluid Mechanics" (Khanna Publishers, 8th Edition, 1995) that eliminate trial and error for solving for ALL the main cases:
(1) pressure drop, (2) flow, and (3) pipe diameter.

This book must be one of the best kept secrets in engineering. However, Jain's correlations have been published in the respectable research journals but appear to have escaped the notice of most professors writing books on the subject. I have used them extensively in my professional work and, I assure you, they work extremely well.

Also, they are eminently suitable for putting into a spreadsheet as (unlike some other methods, e.g., by R. P. King in "Introduction to Practical Fluid Flow", Butterworth-Heinmann, 2002) there are absolutely no graphical correlations required.

 

[LaSalle1940]

Thank you, UmeshMathur: you're right that this text must be a very well-kept secret. I did a search on Amazon and found nothing; likewise, a search of the library holdings at Rutgers turned up nothing. I assume it wasn't published in either the U.S. or the U.K., so it's probably escaped notice.

By the way, I clicked on the "spreadsheet" link (?) you included and was sent to a page regarding Quantrix, but I didn't see how to find the spreadsheet you presumably wanted to link to.

 

[UmeshMathur]

Sorry, LaSalle1940: The "link" is inserted automatically by the EngTips website when you use certain buzzwords; I did not intend that you should go to the Quantrix site.

Professor Jain was at one of India's elite I.I.T.s (at Roorkee) when he wrote the book. His book (cost under $5.00) is published in India and can be ordered from the publisher:

Khanna Publishers
2-B, Nath market
Nai Sarak
Delhi-110006
India
Phones: +(91) 11 2912380 or +(91) 11 7224179 (listed in an older catalog, may be out of date).

The shipping cost, even by surface mail, will most certainly exceed the cost of the book. The book has 926 pages and would cost a bundle to airmail - just check with Khanna. You must be ready for less than perfect book binding and printing - such books are used by very poor engineering students all over India, in relation to our situation here. However, their quality is not to be judged by their external appearances. In this instance, I would say that Jain's book is a masterpiece.

 

[UmeshMathur]

Hi, LaSalle1940:

Upon further checking, I found that Jain's explicit equations have been quoted, with solved numerical examples and highly supportive comments, in the latest edition of a standard U.S. textbook:

Streeter, V. L., and E. Benjamin Wylie: pages 239-242 in "Fluid Mechanics" (First SI Metric Edition, McGraw-Hill Ryerson Ltd., 1983). My copy is an International Edition published in Singapore. The ISBN Number is 0-07-066578-8.

Streeter quotes the relevant equations from the following original paper: Swamee, P.K. and A. K. Jain, "Explicit Equations for Pipe-Flow Problems", J. Hydr. Div., Proc. ASCE, pp. 657-664, May 1976. The Rutgers library should be able to locate this paper for you easily.

On a personal note, this discovery is interesting to me, as I studied Streeter - 3rd edition, if I recall correctly - in undergraduate school (I.I.T. Delhi) in 1963!

 

[LaSalle1940]

Many thanks again, UmeshMathur. I've already checked with Khanna concerning the purchase price of the text, and I've inquired (but haven't received an answer yet) about shipping to the U.S.; we'll see.

And I'm confident I can get Streeter and Wylie's text in the interim from one of the nearby university libraries.

Looks like I sure came to the right place to ask these questions!

 

[katmar]

It seems that the mantle of explicit piping equations has been taken over by BB Gulyani, also a professor at Roorkee. I have an article written by Gulyani and Agarwal from PTQ, Autumn 2000. I don't remember how I downloaded it - it seems you have to pay to join the PTQ site now. Anyway, these authors also use the f^0.2 factor, without giving it a name.

But I don't see the benefit of these short-cut methods any more. 25 years ago when we were calculating these things on our brand-new programmable pocket calculators saving programming steps and avoiding iteration was important. These days personal computers are so fast and have so much memory I have simply reverted to the Colebrook-White equation. I do not even test for convergence. I let the equation iterate 5 times and the friction factor will have converged to something like 7 significant places. Even if this iteration is inside an outer loop calculating the diameter by trial and error the result is instantaneous.

Unless you are building these equations into massive pipe network calculations where they are used millions of times, I doubt the value of the clever math.

Katmar Software
Engineering & Risk Analysis Software

http://katmarsoftware.com

 

[UmeshMathur]

It's a few days since katmar's last post, so maybe this thread is dead. Nevertheless, I wanted to make a few small historical points concerning the parameter first mentioned by LaSalle1940, Re(f^½). (This was erroneously rewritten as Re(f^0.2) in the same thread, and repeated later by katmar.)

The Re(f^½) factor originated, of course, in Colebrook's original equation for the friction factor. The square of this quantity is significant in the following discussion.

A parameter D*, called the "Dimensionless Pipe Diameter" in R. P. King, "Introduction to Practical FLuid Flow", (Butterworth-Hinemann, 2002, p. 16), is related to f*(Re^2) as follows:

(D*)^3 = f*(Re^2) = D^3*rho*DP/2/dynvis^2,

where D=pipe I.D., rho=fluid density, DP=friction loss, dynvis=dynamic viscosity, all in consistent units.

As explained by King, when solving for flow with a given diameter, use of D* can help make the Colebrook equation explicit.

Of course, when using the Colebrook friction factor equation, the calculated Re in any case must be above the laminar region (i.e., above Re=2300), otherwise the original formula is invalid. Jain's explicit equations, mentioned in my previous post of 16 March 2006, also suffer from this defect.

In those cases where the flow or diameter is unknown, the flow regime may slip into or out of the laminar region when iterating as part of a larger problem (e.g., in a pipe network). Anticipating such cases, Churchill published an ingenious, universal friction factor equation valid for both laminar and turbulent flow. See:

Churchill S W, "Friction factor equation spans all fluid regimes", Chem. Eng. 84: 91–92 (1977)

Finally, to katmar's last point, avoiding iterative calculations is generally considered to be more elegant than use of brute force; however, in fluid flow, the discontinuity in the transitional zone (between laminar and turbulent flow mentioned above) can lead to disastrous non-convergence unless you take precautions and use Churchill's equation. This equation is admittedly not explicit, but contains no discontinuities between the laminar and turbulent regions and, therefore, would not drive a decent non-linear search routine crazy.

 

[katmar]

Kudos to UmeshMathur for highlighting the need for a continuous equation when using computerised methods. The iterative use of the Colebrook equation fits very nicely into Churchill's method however. Churchill uses 8/Re for the friction factor in the laminar regime, and then he defines two approximations (termed "A" and "B") for the transition and fully turbulent friction factors.

The genius of Churchill's method is the way he combines these three equations into one continous equation. In my calculations I simply use the iterative solution of Colebrook in place of Churchill's "A" (taking the 1/16th power into account of course).

The differences are probably negligible, and the only reason I have done it this way is that the published curves are usually based on the Colebrook equation and I wanted my results to match any check calculation that is done by hand using these curves. But nobody does hand calcs any more, so why do I worry?

 

[joerd]

There is also an article by G. Gregory and M. Fogarasi worth reading on this topic in Oil&Gas Journal, April 1, 1985, p. 127. They did a comparison of 12 explicit equations including Churchill's for Re = 4000 - 4.E8 (only turbulent flow).
One of their conclusions is that the error introduced by the uncertainty in pipe roughness usually exceeds the error from the explicit equation.

Cheers,
Joerd

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[UmeshMathur]

I believe that kudos are due first to katmar for his insightful comments. I once did a very extensive evaluation of Churchill v/s the other contenders and found that, at least when using computers, the stability gained by a smooth handling of the transition region was priceless. This problem gets really acute with polymers and other viscous fluids (bitumen, anyone?).

My own graduation from the school of computational hard knocks was initiated back in 1973 when I learned about 2-phase flow from Prof. Jim Brill at U. of Tulsa. Brill emphasized the role of discontinuities in 2-phase flow regimes, and showed how much chaos they cause in computer codes. The single phase laminar-to-turbulent transition was just one aspect of this major set of headaches.

Finally, joerd's point about uncertainty in the pipe roughness ratio overshadowing the differences between the competing correlations is very pertinent.