Friction Factor Expressions - Implicit and Explicit

Written by quark:

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, 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. 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)]

Personal Notes/Comments

  • The friction factor ‘f’ is Darcy’s friction factor(thanks to katmar for pointing out)
  • The comparison of accuracies of these equations is done based upon the presumption that Colebrook’s equation is perfect and flawless.
  • Serghides opines that Zigrang equation and his own equation have the higher accuracies
  • Away from the critical region, the inaccuracy of any of the above equations is insignificant. I easily accepted this observation due to two facts: First - The easily available pipes have diameters in steps (i.e if our flow rate requires a pipe just bigger in size than a 3" pipe, our option is 4"), Second - We engineers generally require generous FOS for future expansion and other kind of things.
  • 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.
  • Within the critical region, where 2100<Re<3000, one should dare to take the responsibility of calculating friction factor oneself.
  • 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.
  • 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.
  • Please note that Zigrang’s equation in Serghides paper is misquoted.

Thanks to @katmar, TD2K, and Montemayor