Uniform Flow Equations

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Uniform Flow Equations

Uniform Flow Equations

uniform flow equations

A number of equations (semi-empirical and empirical) have been proposed to describe the relationship between discharge and slope (friction loss) during steady flow in pipes and channels. All are of similar form and contain essentially the same terms. Expressed in the form usually applied to open channel flows, the uniform flow equations are :

Uniform Flow Equation "1" - Chezy equation :

\[v = C{\left( {RS} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}}}\]

Uniform Flow Equation "1" - Darcy-Weisbach equation :

\[v = {\left( {\frac{{8g}}{f}} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}}}{\left( {RS} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}}}\]

Uniform Flow Equation "1" - Manning equation :

\[v = \frac{1}{n}{S^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}}}{R^{{\raise0.5ex\hbox{$\scriptstyle 2$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 3$}}}}\]

Uniform Flow Equation "1" - Hazen-Williams equation :

\[v = 0.849{C_{HW}}{S_f}^{0.54}{R^{0.63}}\]

where :

R : Hydraulic radius of the channel (meters) and is equal to the cross sectional area divided by the wetted perimeter

S : Gradient of energy loss due to friction

In the case of steady uniform or normal flow, the energy slope Sf, the water surface slope Sw and the bed slope So are parallel. Hence bed slope So is used in the uniform flow equations. For any non-uniform or unsteady flow, these slopes are not parallel and only the energy slope Sf should be used in the equations.

The coefficients C, f, n and CHW all describe the roughness of the channel wetted perimeter.

The Chezy equation can be viewed as the general form of uniform flow equation, encompassing all others. It is not widely used in practice. The manning equation is the most commonly used open channel equation, largely because of the large body of the empirical experience accumulated and the relative ease (not accuracy) of estimating the coefficient n.

In changing from the more usual form of the Darcy-Weisbach equation :

\[{h_f} = \frac{{fL}}{D} \times \frac{{{v^2}}}{{2g}}\]

to the listed above it is necessary to replace the energy head loss hf and diameter D with terms more appropriate to open channels. These transformations are :

\[{S_f} = \frac{{{h_f}}}{L}\& R = \frac{D}{4}\]

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