Solving the Colebrook Equation for Friction Factors
Part 3 of 3 in a series of articles

Special Cases

Terminology is critical when we speak of Special Cases.  Literature is inconsistent in referring to the various areas of the Moody Diagram.  As stated previously, I will strive to use terminology consistent with the original article of Moody (1944) and the Moody Diagram shown in that paper.

The zones and special lines in the Moody Diagram are:

Laminar Zone:  This is the area of Reynolds Number less than 2000.  In this zone, the Friction Factor is defined as f = 64 / Reynolds Number.  The Colebrook Equation does not apply.

Critical Zone:  This is the area between  Reynolds Numbers greater than 2000 and less than 4000.  The Colebrook Equation is not intended for this area.

Smooth Pipe:  This is the line drawn at Relative Roughness, (e/D) equal to zero.

Dashed Line:  This is the line plotted from the relationship:
1 / f^.5 = R x e/D / 200.

Transition Zone:  Area bound by Reynolds Number greater than 4000, the Smooth Pipe Line and the Dashed Line.

Complete Turbulence, Rough Pipe:  Area to the right of the Dashed Line.

By applying a set of logical “IF Statements” consistent with these definitions, we can determine where on the Moody Diagram a set of conditions lies.  This is done in the demonstration spreadsheet, under the Tab “Special Cases”, cells E1 to I4.   Knowing the Zone, we can determined if a “Special Case” applies.

The Special Cases evaluated here are those dealing with Complete Turbulence, Rough Pipe and Smooth Pipe.  The first case is the Complete Turbulence, Rough Pipe.

Looking at Eq 1 and a Moody Diagram, it can be seen that;

1.       The fraction with Reynolds Number in denominator approaches zero as Reynolds Number becomes larger and larger.

2.       At higher Relative Roughness, the Reynolds Number has less impact on the Friction Factor.

3.       At Reynolds Number of 10⁸, all Relative Roughness curves are essentially flat and the Friction Factor is independent of the Reynolds Number.

This gives rise to a simplification of the Colebrook Equation where the Friction Factor is factorable:

            f = 1 / ( 2*log₁₀(3.7 / e/D )²

One question might be, “Does the Colebrook Equation still produce accurate results under the above condition?”  The demonstration spreadsheet, under the Tab “Special Cases” can help answer this question.  Along the Relative Roughness Curve of 0.05, examine the deviation between the Friction Factor as calculated with the full Colebrook Equation versus the Special Case:

 *  The Dashed Line intersects the e/D of .05 at approximately this point

Two observations can be made:

1.       At the Dashed Line, there is a definite deviation, (2.07… %)

2.       At Reynolds Number of 100,000,000, the deviation is extremely small

With regard to the first observation, a visual examination of the Moody Diagram will reveal that at .05 Relative Roughness and Reynolds of 15,000, the Relative Roughness curve still has some curvature to it.  Given that, some deviation should be expected.

The example used is an extreme case but it shows that the Colebrook Equation is accurate in this Special Case situation.  This leads me to conclude that the Special Case Equation was developed for ease of use and not accuracy.

The second Special Case is used for Smooth Pipe, where the Relative Roughness is zero.  Here again, simplification of the Colebrook Equation, for this condition, exists that is factorable for “f”.  Two equations are given for two ranges of Reynolds Numbers:

f = 0.3164 / Reynolds Number ^.25

            limited to Reynolds Number <10⁵

and

            f = 0.0032 + 0.221 / Reynolds Number ^.237

                        limited to 10⁵ < Reynolds Number < 3*10⁶

Deviations of 1.09…%, at Reynolds Number of 99,999 and 1.97…% at Reynolds Number of 100,001 compared to Eq 1 can be seen in the demonstration spreadsheet.  I have no standard by which to judge the deviations.  At the Reynolds Numbers used, there is a .89…% variation between the two Special Cases themselves.  I suspect, without an real data, that the Colebrook Equation is perfectly adequate for this Special Case situation.

What Should be Considered in Selecting a Method to Solve Colebrook?

With the spreadsheets available today, numerous methods exist for calculating the Friction Factor from the Colebrook Equation.  Some cover the entire range of the Moody Diagram while others are limited to only part of the Diagram.  Special Cases, while simpler in format, are limited in their application as well.  It is the writers opinion the there is an overwhelming advantage to using a method that has no limits.  This seems especially true where a spreadsheet will be shared.  While the original writer of the spreadsheet may be aware of its limitations, use by others, not familiar with these limitations, could lead to significant inaccuracies.

The issue of Ease of Use is very much as individual matter.   Some may shy away from UDF’s as too complicated but for myself, I find UDF’s very easy to incorporate in both new and existing spreadsheets.  Once they exist in a spreadsheet, I find them easier to enter than the Explicit Equations that I would consider as acceptable alternatives.

Iterations, once setup are easy to use on an individual case but are not easy to use in a piping network.  This is particularly true when “What If” scenarios are being evaluated.

Eq 3 and fEq3 do not produce a result with Relative Roughness of 0.0.  While there are ways to deal with this, such as using an extremely low Relative Roughness, they do posse a problem that isn’t an issue with many other methods.

Whats acceptable?  That’s up to the engineer.  For me, I want a solution that covers the full range of the Moody Diagram, is as accurate as the deviations between the various forms of Colebrook and is easy to use.  I find that all of the following meet these criteria:

1.       UDF’s “fEq1”, fEq2” or “fSerg”

2.       The Explicit Methods of Serghide and Zigrang

References

1.         Moody, L, 1944, “Friction Factors for Pipe Flow”, ASME Transactions Vol 66, Page 671.

2.  The American Heritage Dictionary of the English Language, Third Edition.  Copyright © 1991 by Columbia University Press.

3. T.K.Serghide’s implementation of Steffenson’s accelerated convergence technique, reportedly to have appeared in Chemical Engineering March 5, 1984.

4.  Zigrang and Sylvester’s solution reportedly to have appeared in AIChE Journal vol. 28, 1982.

5.  Swamee and Jain Equation  www.agen.okstate.edu/darcy/DarcyWiesbach/Darcy-WiesbachEq.htm

6.         2001 ASHRAE Handbook – Fundamentals p.35xx

7.  Lester, T.  “Calculating Pressure Drops in Piping Systems.”  ASHRAE Journal Sept. 2002.

8.  Lester, T.  “Solving for Friction Factor.”  ASHRAE Journal July, 2003.

By: Thomas G. Lester, P.E., Bergmann Associates
tlester@bergmannpc.com