For a single phase, fully developed flow in a pipe, the shear stress at the fluid-solid boundary is balanced by the pressure drop (see Figure 1). A one-dimensional force balance equation of this flow can be written as:
where S is the pipe cross-sectional area and A is the pipe surface area. Here, τ w is the wall shear stress which is dependent upon the following parameters
fluid properties, namely, density and viscosity
surface roughness of the interior pipe wall
The first two parameters are due to the nature and the flow characteristics of the fluid itself. The last two depend on the physical geometry of the pipe. The stress can be expressed as
where f is the Fanning friction factor.
The friction factor, f, is a dimensionless factor that depends primarily on the velocity u, diameter D, density ρ, and viscosity η. It is also a function of wall roughness which depends on the size ε, spacing ε' and shape of the roughness elements characterized by ε''. ε and ε' have the dimension of length whereas ε'' 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, ρ and η can be rearranged as uDη/η which is the Reynolds Number, Re. For the characteristic roughness factors (ε and ε'), it may be made dimensionless by dividing these terms by D (the term ε/D is called the relative roughness). Hence, the friction factor can be written in a general form as:
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.
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 (see Poiseuille Flow).
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:
However, for Re > 105, the following equation is found to be more accurate:
and this was used by Taitel and Dukler (1976).
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:
where κ (Von Karman constant) and B are 0.4 and 5.5.
|u* is the friction velocity (u* = √τ w/ρ )||ν = η/ρ is the kinematic viscosity|
The friction factor can be related to mean flow velocity and Reynolds number by employing equation (1) and the relationship
The friction factor for smooth pipe is then expressed as:
which is called the Karman-Nikuradse equation.
For the rough pipes, the velocity distribution is defined as:
Here the constant Β' 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*ε/ν, where ε is the equivalent roughness height. The three roughness regimes are as follows:
Dynamically smooth: 0 ≤ u*ε/ν ≤ 5
Transition: 5 < u*ε/ν ≤ 70
Completely rough: u*ε/ν > 70
For the completely rough regime, the value of Β' 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:
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 (ε/D), the friction factor of the commercial pipe is found to be different from those obtained from the sand roughness used by Nikuradse (see Figure 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:
The formula can also be used for the smooth or rough pipes where it gets similar values to the Karman-Nikuradse equation when ε → 0 or Re → ∞.
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 3 which is known as the Moody Chart or Diagram). Typical values of the roughness size of different pipe material are given in Table 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.
A useful explicit equation that applies to turbulent flow (10 4 > Re > 4 × 10 8) in both smooth and rough pipes has been presented by Chen (1979):
Vennarf J. K. and Street R. L. Elementary Fluid Mechanics, 5th Ed., John Wiley and Sons Inc, USA.
Streeter V. L. and Wylie E. B. (1985) Fluid Mechanics, McGraw-Hill Inc, USA.
Nikuradse, J. (1950) Stromungsgesetze in rauhen rohren, VDI-Forschungsheft, 361, 1933, see English Translation NACA TM 1292.
Blasius, H. (1913) Forschungsarbeiten auf dem Gebiete des Ingenieusersens, 131.
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.
Moody, L. F. (1944) Friction Factors for Pipe Flow, Trans. A. S. M. E., vol. 66.
Douglas J. F. Gasiorek, J. M. and Swaffield, J. A. (1985) Fluid Mechanics, 2nd ed., Pitman Publishing Ltd.
Chen N. H. (1979) An explicit equation for friction factor in pipes, Int. Eng. Chem. Fundam., 18(3), 296.