Environmental Engineering Reference
In-Depth Information
When a head in the manhole is between the
ground level and the water level on the surface
(Fig. 13.13b) the water still flows from the surface
to the pipes, and may best be represented using an
orifice equation. However, it is stressed that the
selectionof the appropriate discharge coefficient is
uncertain.
Finally, if the below-ground system becomes
surcharged and the head in the manhole is greater
than that created by the water level on the street
(Fig. 13.13c), water from the below-ground system
will issue from the manhole onto the street
surface. Here, the most appropriate formula to
describe the flow out of the inlet is that of an
orifice equation, but again it has to be recognized
that the selection of the appropriate discharge
coefficient is uncertain.
What is clear is that the mathematical equa-
tions used to describe the change in flow regime,
from (a) to (b) to (c) and vice versa, and their
discharge coefficients, have to form a continuum
with a smooth transition in the head discharge
relationships between inflow and outflow
values and vice versa. Research is ongoing to
address this issue.
f
j
þ1
i
f
j
þ1
i
f
i
þ1
f
i
Þ
x
q
ð
þ1
Þ
þð1qÞ
ð
q
f
ð
13
:
5
Þ
D
x
D
x
q
f
q½y
f
j
þ1
i
þ1
þð1yÞ
f
j
þ1
i
þð1qÞ½y
f
i
þ1
þð1yÞ
f
i
ð
13
:
6
Þ
where f
¼
any function,
y
,
q¼
spatial and temporal
weighting coefficients, respectively, and i, j
¼
space and time indices, respectively. Commonly,
y¼
0.5 and
q¼
0.67. Where f is a product or a ratio
of variables there is no separation and the pro-
ducts/ratios are discretized as such. Where any
variable
j
is in front of the differentiation operator,
the discretization is as follows:
x
q
ð
w
j
þ1
þ
w
j
þ1
i
f
j
þ1
i
f
j
þ1
i
w
q
f
q
þ1
Þ
ð
þ1
Þ
i
2
D
x
þð1qÞ
ð
w
i
þ
w
i
þ1
Þ
2
ð
f
i
þ
1
f
i
Þ
D
ð
13
:
7
Þ
x
Substitution of Equations 13.4 to 13.7 into
Equations 13.2 and 13.3 and linearization leads to:
a
i
Q
j
þ1
i
þb
i
z
j
þ1
i
þ c
i
Q
j
þ1
i
þ1
þd
i
z
j
þ1
þ1
¼ e
i
ð
13
:
8
Þ
i
a
0
Q
j
þ1
i
b
0
i
z
j
þ1
c
0
i
Q
j
þ1
þ
þ
Numerical solution procedure for the
below-ground drainage system
i
þ1
d
0
i
z
j
þ1
þ1
¼ e
0
i
þ
ð
13
:
9
Þ
i
The numerical procedure described here utilizes
the general algorithm for solving finite difference
problems originally introduced byFriazinov (1970).
It is based on the idea of temporary elimination of
variables at internal cross-sections and thus reduc-
ing all equations to a system of unknown water
levels at network nodes. This is an elegant algo-
rithm, and, combined with different numerical
methods and various matrix solvers, has been used
in several commercial models, for example, Info-
Works CS and MOUSE.
The St-Venant equations are solved by a variant
of the Preissmann implicit finite-difference
method where:
where
a
i
; b
i
; ...; e
i
¼
abbreviations, most of which
include variables Q
j
þ1
or z
j
þ1
. Equations 13.8
and 13.9 for all subreaches of a single pipe/channel
form the system of algebraic equations (for
i
¼ 1; 2; ...;
N
1
, where N
¼
number of cross-
sections; see Fig. 13.14).
By eliminating the unknowns at
internal
cross-sections (from i
N - 1), this system
of 2N - 2 equations is reduced to the equivalent
system of two equations:
¼
2toi
¼
Q
j
þ1
1
¼ f
1
z
j
þ1
1
þg
1
z
j
þ1
þh
1
ð
13
:
10
Þ
N
Q
j
þ1
N
¼ f
N
z
j
þ1
1
þg
N
z
j
þ1
þh
N
ð
13
:
11
Þ
N
f
j
þ1
i
f
i
þ1
Þ
f
j
þ1
i
f
i
Þ
t
y
ð
þ1
q
f
þð1yÞ
ð
where f
1
,g
1
...
,h
N
¼
abbreviations. Substitution of
free-outflow conditions
ð
13
:
4
Þ
and/or
the
energy
D
t
D
t
q
