Flow in partially full conduits

In fluid mechanics, flows in closed circular conduits are usually encountered in places like drains and sewers. These kinds of fluid flow are generally governed by the principles of channel flow as the liquid flowing possesses free surface inside the conduit. However, the convergence of the boundary to the top imparts some special characteristics to the flow as discussed below.

Consider a closed circular conduit of diameter D, partly full with liquid flowing inside it. Let 2θ be the angle subtended by the free surface at the centre of the conduit as shown in figure 1.

The area of the cross-section (A) of the liquid flowing through the conduit is calculated as :

${\begin{aligned}A=&{{D^{2}\theta } \over {4}}-2{\biggl (}\left({\frac {1}{2}}\right){\frac {D}{2}}sin\theta {\frac {D}{2}}cos\theta {\Biggr )}\\&={\frac {D^{2}}{4}}{\biggl (}\theta -{\frac {1}{2}}sin2\theta {\biggr )}\\\end{aligned}}$

Now, the wetted perimeter (P) is given by :

$P=D\theta$

Therefore, the hydraulic radius (R_h) is calculated using cross-sectional area (A) and wetted perimeter (P) using the relation:

$R_{h}={\frac {A}{P}}={\frac {D}{4}}{\biggl (}1-{\frac {1}{2}}{\frac {sin2\theta }{\theta }}{\biggr )}$

The rate of discharge may be calculated from Manning’s equation :

$Q={\frac {D^{2}}{4}}{\biggl (}\theta -{\frac {1}{2}}sin2\theta {\biggr )}{\biggl (}{\frac {1}{n}}{\biggr )}S^{\frac {1}{2}}{{{\biggl (}{\frac {D}{4}}{\biggl (}1-{\frac {1}{2}}{\frac {sin2\theta }{\theta }}{\biggr )}{\biggr )}}^{\frac {2}{3}}}$

$=K{\biggl (}\theta -{\frac {sin2\theta }{2}}{\biggr )}{{{\biggl (}1-{\frac {sin2\theta }{2\theta }}{\Biggr )}}^{\frac {2}{3}}}$

where the constant $K={\frac {{D}^{\frac {8}{3}}}{{4}^{\frac {5}{3}}}}{\frac {{S}^{\frac {1}{2}}}{n}}$

Now putting $\theta =\pi$ in the above equation yields us the rate of discharge for conduit flowing full (Q_full))

$Q_{(full)}=K\pi$

In Dimensionless form ,the rate of discharge Q is usually expressed in a dimensionless form as :