Pipe friction coefficient

Physical key figure
Surname	Pipe friction coefficient
Formula symbol	${\displaystyle \lambda }$
dimension	dimensionless
definition	${\displaystyle \lambda ={\frac {2dp}{dx}}~{\frac {D}{\rho \cdot v^{2}}}}$
${\displaystyle \textstyle {\frac {dp}{dx}}}$ Pressure gradient in the pipe ${\displaystyle D}$ Pipe diameter ${\displaystyle v}$ average speed ${\displaystyle \rho }$ density
scope of application	Pipe flows

The pipe friction diagram ( Moody diagram ) shows the dependence of the pipe friction coefficient on the Reynolds number and the roughness k.

The pipe friction coefficient λ (lambda) is a dimensionless number for calculating the pressure drop with a flow in a straight pipe.

definition

The resistance of pipe flows could also be written as the pressure loss coefficient  ζ ( zeta ), but it can be further resolved:

${\displaystyle \zeta =\lambda {\frac {L}{D}}}$

With

• length ${\displaystyle L}$
• Inside diameter . ${\displaystyle D}$

Laminar flow

For the laminar , fully developed flow in a circular pipe, the pipe coefficient of friction is determined according to the Hagen-Poiseuille law :

${\displaystyle \lambda ={\frac {64}{Re}}}$

with the Reynolds number . ${\displaystyle Re}$

Turbulent flow

In the case of turbulent flow, there are several approximation formulas for determining the pipe friction coefficient , which are used depending on the roughness of the pipe:

• Hydraulically smooth tube , d. H. the unevenness of the pipe wall is completely covered by a viscous sub- layer. The value of is calculated iteratively using the Prandtl formula . The following can be used as the start value : ${\displaystyle \lambda }$ ${\displaystyle \lambda =0{,}02}$

${\displaystyle {\frac {1}{\sqrt {\lambda }}}=2{,}0\log _{10}\left(Re{\sqrt {\lambda }}\right)-0{,}8=-2\log _{10}\left({\frac {2{,}51}{Re{\sqrt {\lambda }}}}\right)}$

An explicit formulation can also be specified using the Lambertian W function :

${\displaystyle \lambda =\left({\frac {\ln 10}{2}}\right)^{2}\cdot \left[W\left({\frac {\ln 10}{2}}\cdot \exp \left(-0{,}8\cdot {\frac {\ln 10}{2}}\right)\cdot Re\right)\right]^{-2}={\frac {1{,}32547}{\left[W\left(0{,}458338\cdot Re\right)\right]^{2}}}}$

A simple correlation that is frequently used for the approximate calculation of the pressure loss behavior of the smooth pipe in the area is that of Blasius : ${\displaystyle Re<10^{5}}$

${\displaystyle \lambda ={\frac {0{,}3164}{Re^{0{,}25}}}}$

• Hydraulically rough pipe , d. H. the unevenness of the wall of the pipe is no longer covered by a viscous sub- layer. The value of is calculated using Nikuradze's formula : ${\displaystyle \lambda }$

${\displaystyle {\frac {1}{\sqrt {\lambda }}}=-2\log _{10}\left({\frac {k}{3{,}71D}}\right)}$

With

the absolute roughness (in mm) ${\displaystyle k}$

• Transition area between the states listed above. According to Colebrook and White :

${\displaystyle {\frac {1}{\sqrt {\lambda }}}=-2\log _{10}\left({\frac {2{,}51}{Re{\sqrt {\lambda }}}}+{\frac {k}{3{,}71D}}\right)}$

This formula can also be used approximately for the hydraulically smooth area and the hydraulically rough area . ${\displaystyle (k\to 0)}$ ${\displaystyle (k\to \infty )}$

According to Moody, the border between transitional and rough areas runs at

${\displaystyle Re{\sqrt {\lambda }}\ {\frac {k}{D}}=200\Leftrightarrow {\frac {1}{\sqrt {\lambda }}}={\frac {Re}{200}}\ {\frac {k}{D}}}$.

Explanations

Roughness

The table below gives examples of absolute roughness.

Material and type of pipe	Condition of the pipes	${\displaystyle k}$ in mm
absolutely smooth tube	theoretically	0
new rubber pressure hose	technically smooth	approx. 0.0016
Pipes made of copper, light metal, glass	technically smooth	0.001 ... 0.0015
plastic	New	0.0015 ... 0.007
Cast iron pipe	New	0.25 ... 0.5
	rusty	1.0 ... 1.5
	encrusted	1.5 ... 3.0
Steel pipes	even rust scars	approx. 0.15
	new, with mill skin	0.02 ... 0.06
	slight incrustation	0.15 ... 0.4
	strong incrustation	2.0 ... 4.0
Concrete pipes	new, smooth line	0.3 ... 0.8
	new, rough	2.0 ... 3.0
	after several years of operation with water	0.2 ... 0.3
Asbestos cement pipes	New	0.03 ... 0.1
Stoneware pipes	new, with sleeves and joints	0.02 ... 0.25
Clay pipes	new, burned	0.6 ... 0.8

To compare different roughness, one can use the equivalent sand roughness .

The loss coefficients can be calculated or taken from tables or diagrams.

Loss coefficients for partially filled pipes or any channel cross-section

Corresponding to the calculation of the loss coefficients for fully filled pipes, loss coefficients can also be determined for partially filled pipes or any channel cross-sections. The hydraulic diameter is used in the calculation instead of the inner pipe diameter : ${\displaystyle D}$ ${\displaystyle d_{h}}$

${\displaystyle d_{h}={\frac {4\cdot A}{U}}}$

With

• the cross-sectional area ${\displaystyle A}$
• the wetted perimeter . ${\displaystyle U}$

The use of the pipe coefficient of friction for the calculation of the runoff in open channels has not yet established itself and is only used to calculate the runoff in pipes. To calculate the runoff in open channels , the empirically obtained flow formula according to Strickler (in the English-speaking world according to Manning ) is used.

See also

• Bernoulli's equation

swell

1. ↑ Wolfgang Kalide: Introduction to technical fluid mechanics . 7th, revised edition. Hanser, Munich / Vienna 1990, ISBN 3-446-15892-8 , pp. 58 .
2. ^ Heinrich Blasius (1883–1970), dglr.de (PDF)
3. ^ Lewis F. Moody, Professor of Hydraulic Engineering, Princeton University : "Friction Factors for Pipe Flow" Trans. ASME , vol. 66, 1944.
4. ↑ Wolfgang Kalide: Introduction to technical fluid mechanics . 7th, revised edition. Hanser, Munich / Vienna 1990, ISBN 3-446-15892-8 , pp. 237 .
5. ↑ Walter Wagner: Flow and pressure loss: with collection of examples . 5th, revised. Edition. Vogel, Würzburg 2001, ISBN 3-8023-1879-X , p. 79 .
6. ↑ Buderus Heiztechnik (Ed.): Manual for heating technology. Working aid for daily practice . 34th edition. Beuth, Berlin / Vienna / Zurich 2002, ISBN 3-410-15283-0 , pp. 696 .
7. ↑ Section head of the Federal Office for Water Management, Albert Strickler (1887 - 1963) Contributions to the question of the velocity formula and the roughness index for flows, channels and closed pipes. Notices from the Federal Office for Water Management, Bern, 1923.
8. ↑ also called Philipe Gaspard Gauckler (1826–1905) antiquated