Flow formula

Flow formulas are used to roughly calculate the average speed of a flow . A distinction is made between open channels and pipes with gravity or pressure drainage. The formulas depend on the hydraulic radius and the flow gradient of the water level and take into account all flow resistances in the form of empirical coefficients . These are different for each flow formula.

The discharge , which is usually to be calculated , is then obtained by multiplying the mean flow velocity found by the cross-sectional area : ${\displaystyle Q}$ ${\displaystyle v_{m}}$ ${\displaystyle A}$

${\displaystyle Q=v_{\mathrm {m} }\cdot A}$

Open channels

See also : Currents in open channels

Flow formula according to Brahms and de Chézy (oldest formula)

It was named after Albert Brahms and Antoine de Chézy .

${\displaystyle v=C{\sqrt {R\,I}}}$

With

• the flow velocity in m / s ${\displaystyle v}$
• the Chézy coefficient in m ^½ / s ${\displaystyle C}$
  • after cutter or
  • to Bazin
• the hydraulic radius in m (corresponds roughly to the water depth for very wide, shallow flow cross-sections ) ${\displaystyle R=A/U}$
  • the cross-section flowed through in m² ${\displaystyle A}$
  • the wetted circumference in m ${\displaystyle U}$
• the flow gradient in m / m ${\displaystyle I=h_{\mathrm {f} }/L}$
  • the height in m ${\displaystyle h_{\mathrm {f} }}$
  • the length in m. ${\displaystyle L}$

Flow formula according to Gauckler-Manning-Strickler

The flow formula according to Gauckler-Manning-Strickler (GMS formula, according to Philippe Gaspard Gauckler , Robert Manning and Albert Strickler ) is a strongly empirical development of the formula according to Brahms and de Chézy. It applies to the usual conditions in open rivers with good accuracy:

${\displaystyle {\begin{aligned}v_{\mathrm {m} }&=k_{\mathrm {st} }\cdot R^{\frac {2}{3}}\cdot I^{\frac {1}{2}}\\&=k_{\mathrm {st} }\cdot {\sqrt[{3}]{R^{2}}}\cdot {\sqrt {I}}\end{aligned}}}$

with the roughness coefficient according to Strickler in m ^1/3 / s for the Gerinnerauheit ${\displaystyle k_{\mathrm {st} }}$

or in the Anglo-Saxon area

${\displaystyle v_{\mathrm {m} }={\frac {1}{n}}\cdot R^{\frac {2}{3}}\cdot I^{\frac {1}{2}}}$

with the roughness coefficient according to Manning . ${\displaystyle n=1/k_{\mathrm {st} }}$

American Literature and calculations are not based on possibly SI units [m], but on the unit feet  [ft] ( English foot ).

Strickler roughness coefficient

The Strickler coefficient is to be selected depending on the surface properties , vegetation and cross-sectional shape and basically changes with the runoff depth, since the influence of the slope roughness decreases with increasing flow depth. In this way, all loss and friction influences are summarized . ${\displaystyle k_{st}}$

The Strickler coefficient was determined experimentally by Strickler both in the laboratory and in nature. Its strange unit has no physical meaning, but was determined in such a way that the equation is dimensionally true . ${\displaystyle \mathrm {{\sqrt[{3}]{m}}/s} }$

Typical river bed values:

surface	k st in m 1/3 / s
Smooth concrete	100
Straight flowing water	30-40
Meandering river bed with vegetation	20-30
Torrent with rubble	10-20
Torrent with undergrowth	<10

Sample calculation

The Rhine flows from Cologne, about 50 m above  sea ​​level , about 300 km to the mouth (0 m above sea level); so has a gradient of . It is approx. 8 m deep ( ) and has a washed out river bed . Then the flow velocity according to Gauckler-Manning-Strickler is : ${\displaystyle I\approx 0{,}167\,\mathrm {Promille} \approx 0{,}000\,167}$ ${\displaystyle R\approx 8\,\mathrm {m} }$ ${\displaystyle k_{st}\sim 30\,\mathrm {m^{1/3}/s} }$

${\displaystyle v=1{,}5\,\mathrm {m/s} =5{,}4\,\mathrm {km/h} }$, in good agreement with the measured mean speed of . ${\displaystyle 4\,\mathrm {km/h} }$

Pipe flows

See also : Flows in pipelines

Flow formula according to Darcy-Weisbach

By transforming the Darcy-Weisbach equation (after Henry Darcy and Julius Weisbach ) we get:

${\displaystyle v_{\mathrm {m} }={\sqrt {\frac {8\cdot g\cdot R\cdot I}{\lambda }}}}$

With

• the acceleration due to gravity in m ² / s ${\displaystyle g}$
• the hydraulic radius in m ${\displaystyle R=D/4}$
  • the inside diameter of the pipe in m ${\displaystyle D}$
• the pipe friction coefficient (unlike the Strickler coefficient, it is dimensionless ). ${\displaystyle \lambda }$

With one parameter , this formula corresponds to the Chézy formula . ${\displaystyle C={\sqrt {\frac {8\,g}{\lambda }}}}$

Flow formula from Prandtl-Colebrook

The formula according to Ludwig Prandtl and Cyril Frank Colebrook applies to drainage in circular or non-circular profiles with full or partial filling. It is based on the Chézy formula and has additional parameters for the viscosity of water and the roughness of the pipe.

For circular, completely filled pipes the formula is:

${\displaystyle v_{\mathrm {m} }=-2\cdot lg\,\left(2,51\cdot {\frac {\nu }{D}}+{\frac {k_{Pr}\cdot {\sqrt {2\cdot g\cdot I_{\mathrm {E} }\cdot D}}}{3,71\cdot D}}\right)}$

With

• the logarithm of ten ${\displaystyle lg}$
• the kinematic viscosity of water in m² / s ${\displaystyle {\nu }}$
• the roughness coefficient according to Prandtl-Colebrook (hydraulically effective roughness of the inner pipe wall) in m ${\displaystyle k_{Pr}}$
• the energy line gradient in m / m. ${\displaystyle I_{\mathrm {E} }}$

There is also a formula for non-circular profiles in which the pipe radius is replaced by the hydraulic radius (with other factors).

Further flow formulas

In addition to these actual flow formulas, there are others for other cases:

• The outflow formula according to the law of Torricelli is a formula for the outflow from a container or in a weir under the contactor through:

${\displaystyle v=\alpha \,{\sqrt {2\,g\,h}}}$

with the discharge or loss coefficient . ${\displaystyle \alpha }$

• The Poleni formula is used to calculate the discharge in the event of a complete overflow of weirs . It is called the raid formula and not a flow formula.
• Darcy's law is a flow formula for seepage .

See also

• Flow law
• Flow curve

Individual evidence

1. ↑ or according to other sources Gaspar-Philibert Gauckler; "Philibert Gaspard" are also the middle names of Henry Darcy
2. ↑ Introduction to hydromechanics: Gerhard H. Jirka: Introduction to hydromechanics. KIT Scientific Publishing, 2007, ISBN 978-3-86644-158-3 , p. 212 ( limited preview in Google book search).
3. ^ Open-channel hydraulics / Ven Te Chow. - New York [u. a.]: McGraw-Hill, 1959
4. ↑ DWA worksheet DWA-A 110: Hydraulic dimensioning and performance verification of sewer pipes and sewers, as of October 2012

literature

• Albert Strickler: Contributions to the question of the speed formula and the roughness figures for flows, channels and closed lines . In: Eidg. Office for Water Management (Ed.): Communications from the Office for Water Management . No. 16 . Bern 1923, p. 357 ( in the ETH library ).
• Albert Strickler: Theory of the water shock . In: Schweizerische Bauzeitung . No. 63 , 1914, pp. 25 .
• Albert Strickler: Experiments on pressure fluctuations in iron pipelines . In: Schweizerische Bauzeitung . No. 64 , 1914, pp. 85-87,123 .
• Helmut Martin, Reinhard Pohl: Technical hydromechanics . In: Hydraulic and Numerical Models . tape 4 . Berlin 2009, ISBN 3-345-00924-2 , pp. 85-87,123 .
• Willi H. Hager: Swiss contribution to water hammer theory . In: Journal of hydraulic research . 1st edition. tape 4 , no. 39 , 2001 (English, online ( Memento from February 6, 2005 in the Internet Archive ) [PDF]).
• Robert Freimann: Hydraulics for civil engineers. Hanser, 2009, ISBN 978-3-446-41054-1 , p. 121 ( limited preview in Google book search).
• Wilhelm Hosang: wastewater technology. Vieweg + Teubner Verlag, 1998, ISBN 978-3-519-15247-7 , p. 86 ( limited preview in the Google book search).
• Thomas Vetter: Flood-accompanying bed dynamics of a large flatland river (Vereinigte Mulde, Saxony-Anhalt) with special consideration of disturbed transport conditions . Ed .: Reinhard Lampe. Ernst Moritz Arndt University, Greifswald 2008, ISBN 978-3-86006-311-8 , p. 31-32 .

Web links

• 10.3.2 Flow formulas. (PDF; 590 kB) In: Chapter 10 Channel flow. Institute for Hydromechanics Karlsruhe, p. 181 , accessed on July 13, 2016 . 
• Cape. 6 hydraulics. (PDF; 392 kB) Wastewater. Jansen AG, CH Oberriet , April 2003, pp. E-69 - E-86 , archived from the original on June 24, 2011 ; accessed on July 13, 2016 . 

• Small retention basin - hydraulic calculations (PDF; 2.4 MB)
• History of the Chézy formula