This article is about flow in partially full conduits.
In
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
, 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 up to a certain depth. Typical examples of such flows are flow in circular and Δ shaped channels.
Closed conduit flow differs 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
In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress,
such as the interface between two homogeneous fluids.
An example of two such homogeneous fluids would be a body of water (liquid) and the air ...
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
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
relations are used for the mathematical calculation of such closed channel flows.
Mathematical analysis for flow in circular channel
Consider a closed circular conduit of
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
D, partly full with liquid flowing inside it. Let 2θ be the angle, in
radians, 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 :
(Equation 1)
Now, the wetted
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
(P) is given by:
Therefore, the
hydraulic radius (R
h) is calculated using
cross-sectional area
In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. The ...
(A) and wetted perimeter (P) using the relation:
(Equation 2)
The rate of discharge may be calculated from
Manning’s equation The Manning formula or Manning's equation is an empirical formula estimating the average velocity of a liquid flowing in a conduit that does not completely enclose the liquid, i.e., open channel flow. However, this equation is also used for calculat ...
:
.
(Equation 3)
where the constant
Now putting
in the above
equation yields us the rate of discharge for conduit flowing full (Q
full))
(Equation 4)
Final dimensionless quantities
In dimensionless form, the rate of discharge Q is usually expressed in a dimensionless form as :
(Equation 5)
Similarly for
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
(V) we can write :
(Equation 6)
The depth of flow (H) is expressed in a dimensionless form as :
(Equation 7)
Flow characteristics
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
Chézy's coefficient with hydraulic radius R
h in Manning’s formula.
However, an important assumption is taken that Manning’s Roughness coefficient ‘n’ is independent to the depth of flow while calculating these values. Also, the dimensional curve of Q/Q(full) shows that when the depth is greater than about 0.82D, then there are two possible different depths for the same discharge, one above and below the value of 0.938D
In practice, it is common to restrict the flow below the value of 0.82D to avoid the region of two normal depths due to the fact that if the depth exceeds the depth of 0.82D then any small disturbance in water surface may lead the water surface to seek alternate normal depths thus leading to surface instability.
References
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Fluid mechanics
Hydraulics