Hydraulic Theory
Pipe Design


From the MIDUSS Version 2
Reference Manual  Chapter 8
(c) Copyright Alan A. Smith Inc. 
This section summarizes the hydraulic principles
which are used in MIDUSS for the analysis and design of pipes. Flow
is assumed to be uniform within each reach of pipe, so that the depth
and other crosssectional properties are constant along the length of
the pipe. It follows that the bed slope
S0, the water surface and the
slope of the energy line
Sf are all parallel.
The resistance is assumed to be represented by the Manning equation:
where
Q =
normal discharge (c.m/s or
c.ft/s)
M =
1.0 for metric units
1.49
for imperial or US customary units
n =
Manning's roughness coefficient
A =
crosssectional area
R =
hydraulic radius = Area/Wetted perimeter
S0 =
bed slope (m/m or ft/ft)
No allowance is made for any apparent variation
of 'n' with the relative depth of flow in the pipe.
Normal Depth in
Pipes
For a partfull circular section the
crosssectional properties are expressed in terms of the angle
f
subtended at the centre by the free surface as shown in Figure
8.1.
Figure
8.1  Definition sketch of a partfull pipe.
The following equations can be obtained by
considering the geometry of the triangle subtending the halfangle
f/2
at the centre of the pipe.
The value of
f can be
found in terms of the ratio of the discharge
Q to the fullbore pipe
capacity Qfull
by an iterative solution of the implicit equation [8.5].
where
Equation [8.5] is solved by a NewtonRaphson
procedure, thus:
where
and
Equation [8.6] is applied until
Df <
0.001 radians; the depth is then determined from equation [8.7].
For a crosssection
with a closed top it is usual to find that maximum normal discharge
occurs at a depth slightly below fullbore flow. For a circular pipe
this occurs at a relative depth of
y/D
= 0.93818. It follows that there must be a smaller depth which
produces a discharge equal to the fullbore flow. In a partfull pipe
this occurs when
y/D
= 0.81963.
The root finding
procedure in MIDUSS will always find a solution within the relative
depth range 0.0 < (y/D)
< 0.81963 as long as the discharge is less than the fullbore flow.
If the discharge is greater than this then MIDUSS reports that the
pipe will be surcharged and the slope of the hydraulic grade line is
reported. (See Chapter 4 Design Options Available, Surcharged
Pipe Design )
It is not possible,
therefore, to take advantage of the slightly higher carrying capacity
in the range 0.81963<(y/D)<1.0.
It is not normally good practice to design pipes for uniform flow in
this range of depth because the slightest surface disturbance will
cause the free surface to 'snap through' abruptly to a condition of
pressurized flow.
Critical Depth
in Pipes
When a pipe is
designed it is often important to know if the normal flow depth
y0
is less than or greater than the critical depth
ycr. If
y0 <
ycr then the flow is
supercritical and there is a high probability that a hydraulic jump
will occur at some point downstream. This is usually to be avoided.
The calculation of critical depth in a circular
pipe is based on the critical flow condition of minimum specific
energy which leads to the criterion of equation [8.8].
This is solved by an interval halving procedure
using a function of the form:
in which A
is obtained by combining equation [8.3] with equations [8.10] and
[8.11] below.
Convergence is assumed
when
Dy/y
< 0.00001.
Equation [8.9] cannot
be solved if the freesurface width
T
is zero. A test is therefore made to ensure that the specified
discharge is not greater than the critical discharge corresponding to
a depth of
ycr
= 0.999 D. If this condition is
violated MIDUSS assumes the critical depth to be equal to the
diameter. For further information on uniform or critical flow in
pipes see a text on Open Channel Flow such as Henderson (References ).
