Fanning Friction Factor Equation

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Fanning Equation

For a single phase, fully developed flow in a pipe, the shear stress at the fluid-solid boundary is

balanced by the pressure drop (seeFigure 1). A one-dimensional force balance equation of this

flow can be written as:

(1)

where S is the pipe cross-sectional area and A is the pipe surface area. Here, wis the wall shear

stress which is dependent upon the following parameters

fluid velocity

fluid properties, namely, density and viscosity

pipe diameter

surface roughness of the interior pipe wall

The first two parameters are due to the nature and the flow characteristics of the fluid itself. Thelast two depend on the physical geometry of the pipe. The stress can be expressed as

(2)

where f is the Fanning friction factor.

Friction Factor (f)

The friction factor, f, is a dimensionless factor that depends primarily on the velocity u, diameter

D, density , and viscosity . It is also a function of wall roughness which depends on the size ,

spacing ' and shape of the roughness elements characterized by ''. and ' have the dimension

of length whereas '' is dimensionless. Since the friction factor is dimensionless, the quantities

that it depends upon should appear in the dimensionless form. In this case, the terms u, D, and

can be rearranged as uD/ which is theReynolds Number, Re. For the

characteristic roughness factors( and '), it may be made dimensionless by dividing these terms

by D (the term /D is called the relative roughness). Hence, the friction factor can be written in a

general form as:

(3)

From this we see that the friction factor of pipes will be the same of their Reynolds number,

roughness patterns, and relative roughness are the same. For a smooth pipe, the roughness term

is neglected and the magnitude of the friction factor is determined by fluid Reynolds number

alone.
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Figure 1. Forces acting on the fluid during single phase steady flow in a pipe.

Fanning Friction Factor

The friction factor is found to be a function of the Reynolds number and the relative roughness.

Experimental results of Nikuradse (1933) who carried out experiments on fluid flow in smooth

and rough pipes showed that the characteristics of the friction factor were different for laminar

and turbulent flow. For laminar flow (Re < 2100), the friction factor was independent of the

surface roughness and it varied linearly with the inverse of Reynolds number. In this case, the

friction factor of the Fanning equation can be calculated using the Hagen-Poiseuille equation

(seePoiseuille Flow).

(4)

For turbulent flow, both Reynolds number and the wall roughness influence the friction factor.

At high Reynolds number, the friction factor of rough pipes becomes constant, dependent only

on the pipe roughness. For smooth pipes, Blasius (1913) has shown that the friction factor (in a

range of 3,000 < Re < 100,000) may be approximated by:

(5)

However, for Re > 105, the following equation is found to be more accurate:

(6)

and this was used by Taitel and Dukler (1976).

Karman-Nikuradze Equation

Nikuradse (1933) measured the velocity profile and pressure drop in smooth and rough pipes

where inner surfaces of rough pipes were roughened by sand grains of known sizes. He showed

that the velocity profile in a smooth pipe is given by:

(7)

where (Von Karman constant) and B are 0.4 and 5.5.

u* is the friction velocity (u* = w/ ) = / is the kinematic viscosity
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The friction factor can be related to mean flow velocity and Reynolds number by employing

equation (1) and the relationship

(8)

The friction factor for smooth pipe is then expressed as:

(9)

which is called the Karman-Nikuradse equation.

For the rough pipes, the velocity distribution is defined as:

(10)

Here the constant ' depends on the geometric characteristic of the roughness elements.

Nikuradse classified the characteristics of the rough surface into three regimes based on the

value of the dimensionless characteristic roughness, u*/, where is the equivalent roughness

height. The three roughness regimes are as follows:

1.

Dynamically smooth: 0 u*/ 5

2.

Transition: 5 < u*/ 70

3.

Completely rough: u*/ > 70

For the completely rough regime, the value of ' is equal a constant of 8.48. The friction factor for

rough pipes can be expressed in a form similar to that for smooth pipe as:

(11)

Colebrook-White Formula

Friction factor of commercial pipes can be calculated using equation (5) if the pipe roughness is

in the completely rough region. In the transition region where the friction factor depends on both

Reynolds number and the relative roughness (/D), the friction factor of the commercial pipe isfound to be different from those obtained from the sand roughness used by Nikuradse

(seeFigure 2). This may be because the roughness patterns of commercial pipes are entirely

different from, and vary greatly in uniformity compared to the artificial roughness. However, the

friction factor of the commercial pipe in this zone can be calculated using an empiricism equation

which is known as the Colebrook-White formula:

(12)
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Figure 2. The difference between the Nikuradse sand roughness and the commercial

roughness.

The formula can also be used for the smooth or rough pipes where it gets similar values to the

Karman-Nikuradse equation when 0 or Re .

Table 1. Average roughness of commercial pipes, Streeter and Wylie (1983)

Material mmCast iron 0.26

Galvanised iron 0.15

Asphalt cast iron 0.12

Commercial steel or

wrought iron0.046

Drawn tubing 0.0015

Glass Smooth

Moody Chart

In engineering applications there is a wide range of pipe wall roughness due to the different

materials and methods of manufacture used to produce commercial pipes. Although the

Colebrook-White formula can be used to calculate the value of the friction factor accurately from

given value of the relative pipe roughness, the use of the formula is not practicable because of

the complicated structure of the equation itself. Moody (1944) used the Colebrook-White formula

to compute the friction factor of commercial pipes of different materials and summarised the

data in the graph showing the relationship between friction factor, Reynolds number and

relative roughness (Figure 3which is known as theMoody Chartor Diagram). Typical values of

the roughness size of different pipe material are given inTable 1.

It is important to note that the value of friction factor obtained from the Moody Chart is equal

four times of the Fanning friction factor.

(13)

A useful explicit equation that applies to turbulent flow (10 4> Re > 4 108) in both smooth and

rough pipes has been presented by Chen (1979):

(14)
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Figure 3. Moody diagram. (Adapted from Streeter V.L. et. al. (1985) with permission.)

References

1.

Vennarf J. K. and Street R. L. Elementary Fluid Mechanics, 5th Ed., John Wiley and Sons Inc,

USA.

2.

Streeter V. L. and Wylie E. B. (1985) Fluid Mechanics, McGraw-Hill Inc, USA.

3.

Nikuradse, J. (1950) Stromungsgesetze in rauhen rohren, VDI-Forschungsheft, 361, 1933, see

English Translation NACA TM 1292.

4.

Blasius, H. (1913) Forschungsarbeiten auf dem Gebiete des Ingenieusersens, 131.

5.

Taitel, Y. and Dukler, A. E. (1976) A model for predicting flow regime transition in horizontal

and near horizontal gas-liquid flow, AIChE J, vol. 22 pp. 47-55. DOI:10.1002/aic.690220105

6.

Moody, L. F. (1944) Friction Factors for Pipe Flow, Trans. A. S. M. E., vol. 66.

7.

Douglas J. F. Gasiorek, J. M. and Swaffield, J. A. (1985) Fluid Mechanics, 2nded., Pitman

Publishing Ltd.
http://dx.doi.org/10.1002/aic.690220105http://dx.doi.org/10.1002/aic.690220105http://dx.doi.org/10.1002/aic.690220105http://dx.doi.org/10.1002/aic.690220105


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8.

Chen N. H. (1979) An explicit equation for friction factor in pipes, Int. Eng. Chem. Fundam.,

18(3), 296. DOI:10.1021/i160074a020
http://dx.doi.org/10.1021/i160074a020http://dx.doi.org/10.1021/i160074a020http://dx.doi.org/10.1021/i160074a020http://dx.doi.org/10.1021/i160074a020

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