Flow in partially full conduits

In fluid mechanics, flows in closed conduits are usually encountered in places such as drains and sewers where the liquid flows continuously in the closed channel and the channel is filled only upto a certain depth.Typical examples of such flows are flow in circular and Δ shaped channels.Closed conduits flow differ from open channel flow only in the fact that in closed channel flow there is a closing top width while open channels have one side exposed to its immediate surroundings.Closed channel flows 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 like closed channel flows have a finite depth at which maximum discharge occurs. For computational purposes,flow is taken as uniform flow .Manning's Equation,Continuity Equation (Q=AV) and channel's cross-section geometrical relations are used for the mathematical calculation of such closed channel flows.

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 (a).

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

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

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