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Flow in Pipe

Friction Factor Expressions - Implicit and Explicit by quark
Posted: 20 Apr 06 (Edited 31 Jul 09)

Friction factor is calculated, generally, by any one of the three implicit equations of Colebrook. The equation, I use, is of the form

1/(f)1/2 = -2.0log((e/D)/3.7 + 2.51/Re(f)1/2)
Note: This is applicable for Re>3000. For laminar flows, Re<2100, Poiseuille's law (f = 64/Re)is used

Engineers, who think that it is difficult to solve this equation, use Moody's chart to get values of 'f'. But, if you go through next FAQ(Spreadsheet for Friction Factors), you will find out how easy it is to solve the implicit equation by some simple calculations in Excel.

To make our life easier, some great engineers developed explicit expressions for the friction factor. As I went on reading the subject, I came to know that there were many explicit expressions that equal no. of hairs on my head (no, I am not bald FYI). Out of those, the following are the famous equations.

Serghides Equation (for Re>2100 and any e/D)
f = [A – [(B-A)2/(C-2B+A)]]-2
A = -2.0 log((e/D)/3.7 + 12/Re)
B = -2.0 log((e/D)/3.7 + 2.51A/Re)
C = -2.0 log((e/D)/3.7 + 2.51B/Re)

Moody Equation (4000<Re<107 and e/D <0.01)
f = 5.5x10-3(1+ (2x104e/D + 106/Re)1/3)

Wood Equation (Re>4000 and any e/D)
f = 0.094(e/D)0.225 + 0.53(e/D) + 88(e/D)0.44 x Rea
a = -1.62(e/D)0.134

Jain Equation (for 5000<Re<107 and 0.00004<e/D<0.05)
1/f1/2 = 1.14 – 2.0 log (e/D + 21.25/Re0.9)

Churchill Equation (for all values of Re and e/D)
f = 8((8/Re)12 + 1/(A+B)1.5)1/12
A = (-2.457ln((7/Re)0.9 + 0.27e/D))16
B = (37530/Re)16

Chen Equation (for all values of Re and e/D)
1/(f)1/2 = -2.0log((e/D)/3.7065 – 5.0452A/Re)
A = log((e/D)1.1098/2.8257 + (5.8506/Re0.8981))

Zigrang and Sylvester Equation (for 4000<Re<108 and 0.00004<e/D<0.05)
1/(f)1/2 = -2.0log ((e/D)/3.7 – 5.02A/Re)
A = log[(e/D)/3.7 – (5.02/Re)log((e/D)/3.7 + 13/Re)]


After my discussion with Art Montemayor and Katmar, I would dare putting the following comments for member perusal.

1. The friction factor 'f' is Darcy's friction factor(thanks to katmar for pointing out)

2. The comparison of accuracies of these equations is done based upon the presumption that Colebrook's equation is perfect and flawless.

3. Serghides opines that Zigrang equation and his own equation have the higher accuracies (and this is tested by Art Montemayor, Katmar and myself)

4. Away from the critical region, the inaccuracy of any of the above equations is insignificant (as observed by Katmar). I easily accepted this observation due to two facts. The first one is that, the easily available pipes have diameters in steps(i.e if our flowrate requires a pipe just bigger in size than a 3" pipe, our option is 4") . Second one is that, we engineers generally require generous FOS for future expansion and other kind of things.

5. Member Katmar developed a brilliant expression for friction factor by modifying Chruchill's equation, which is in perfect agreement with Serghides and Zigrang as far as accuracies are concerned.

6. Within the critical region, where 2100<Re<3000, one should dare to take the responsibility of calculating friction factor oneself.

7. Katmar opines that Churchill Equation (and so his version of Churchill equation, obviously) is the better one to use for all conditions as it gives a continuous curve when the data is represented graphically. This seems to be a strong point to me.

8. As pointed out by Katmar, one should be careful while using any of these equation for laminar flow or the critical zone. So, instead of using Chen equation for laminar flows, better go with Poiseuille.

9. Please note that Zigrang's equation in Serghides paper is misquoted(first pointed out by TD2K and then I checked it from Zigrang's original paper)

10. Please inform me incase of any mistakes, misinterpretations and misconduct as far as this FAQ is concerned.

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