Flow in partially full conduits

Introduction

In fluid mechanics, flows in closed circular conduits are usually encountered in places such as drains and sewers where the liquid flows continuously in the pipelines. These kinds of fluid 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 as discussed below.

Mathematical Analysis

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 :

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

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:

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

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

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

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

$Q_{(full)}=K\pi$ (Equation 4)

Final Dimensionless Quantities

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

${\frac {Q}{{Q}_{full}}}={\frac {1}{\pi }}{\biggl (}\theta -{\frac {sin2\theta }{2}}{\biggr )}{{{\biggl (}1-{\frac {sin2\theta }{2\theta }}{\biggr )}}^{\frac {2}{3}}}$ (Equation 5)

Similarly for velocity (V) we can write :

${\frac {V}{{V}_{(full)}}}={{{\biggl (}1-{\frac {sin2\theta }{2\theta }}{\biggr )}}^{\frac {2}{3}}}$ (Equation 6)

The depth of flow (H) is expressed in a dimensionless form as :

${\frac {H}{D}}={\frac {1}{2}}-{\frac {1}{2}}cos\theta$ (Equation 7)

Observations

The variations of Q/Q_(full) and V/V_(full) with H/D ratio is shown in figure(b).From the equation 5 ,maximum value of Q/Q_(full) is found to be equal to 1.08 at H/D =0.94 which implies that maximum rate of discharge through a conduit is observed for a conduit partly full. Similarly the maximum value of V/V_(full) (which is equal to 1.14) is also observed at conduit partly full with H/D = 0.81.The physical explanation for these results are generally attributed to the typical variation of Chezy’s coefficient with hydraulic radius R_h in Manning’s formula.^[1]

References

^ ^a ^b Suman Chakraborty, S K Som (2004). Introduction to Fluid Mechanics and Fluid Machines. New Delhi: McGraw Hill Education. pp. 599, 600. ISBN 978-0-07-132919-4.

[:0-1] Suman Chakraborty, S K Som (2004). Introduction to Fluid Mechanics and Fluid Machines. New Delhi: McGraw Hill Education. pp. 599, 600. ISBN 978-0-07-132919-4.

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