Environmental Engineering Reference
In-Depth Information
The model solves all flow equations simulta-
neously such that the solution procedure does not
distinguish between the flows on the catchment
surface and theflows in the below-ground drainage
system. The only distinction between the surface
flow paths and the underground pipes relates to
differences in the characteristics of the individual
elements; for example, sewer pipes have closed
cross-sections (circular, egg-shaped, etc.) whereas
surface links are open channels, typically with
irregular geometry, which can be approximated
by a regular cross-section (e.g. trapezoidal). Hence,
any pipe, channel, inlet, weir or pump is seen by
themodel as a linkwithin one integrated network.
(surface runoff, waste water, etc.), M
¼
number of
links joining the node and Q
m
¼
discharges flow-
ing from the link to the node or vice versa.
Free surface flow in links
One-dimensional free-surface flow in a pipe or a
channel may be described by the complete
St-Venant equations, which can be written in the
form:
z
q
t
þ
1
Q
q
x
¼ 0
q
B
q
ð
13
:
2
Þ
þ
Q
2
A
Q
q
z
q
t
þ
q
q
gA
x
þ
S
f
¼ 0
ð
13
:
3
Þ
x
q
q
Governing equations and notes on specific
parameters
where z
¼
cross-sectional water level, B
¼
water
¼
discharge, x
¼
space coordi-
surface width, Q
A mathematical model of flow in a network con-
sists of a system of equations that describe all
forms of free-surface and surcharged flows.
nate, A
¼
cross-sectional area, g
¼
gravitational
constant and S
f
¼
friction slope.
Surface/subsurface links
Continuity at nodes
There are three basic cases of flow through sur-
face/subsurface links, as shown in Figure 13.13.
When a hydraulic head in the manhole is below
the ground level (Fig. 13.13a), it does not influence
the flow through the inlet. Thus the inflow can be
described by either a weir equation for shallow
depths or an orifice equation when the area of
opening of the inlet is submerged.
At nodes, the continuity equation can be written
as:
X
M
F
dZ
dt
¼
þ
¼1
ð
13
:
1
Þ
q
Q
m
m
where F
¼
node horizontal area, Z
¼
water level at
the node, t
¼
time, q
¼
external inflow to the node
a)
b)
c)
H
H
H
Fig. 13.13
Basic cases of flow through equivalent inlet: (a) free inflow, inlet as a weir (H, water depth on the surface);
(b) submerged inflow, inlet as an orifice (H, difference between water level on the surface and hydraulic head in the
manhole); (c) outflow (H, difference between hydraulic head in the manhole and water level on the surface).
