
Solving the
Colebrook Equation for Friction Factors
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: 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 10^{8},
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*log_{10}(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 <10^{5} and
f = 0.0032 + 0.221 / Reynolds Number ^{.237}
limited to 10^{5}
< Reynolds Number < 3*10^{6} 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 3. T.K.Serghide’s implementation of Steffenson’s
accelerated convergence technique, reportedly to have appeared in Chemical Engineering 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/DarcyWiesbachEq.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 
