Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
177
OPTIMUM HYDRAULIC AND STRUCTURAL DESIGN OF
INVERTED SIPHON
Dr. Haider K. Ammash
College of Engineering
University of Al-Qadisiya
Thulfikar R. Al- Husseini
College of Engineering
University of AL-Qadisiya
Munaf A. Al- Rammahi
College of Engineering
University of AL-Qadisiya
Abstract
In the present study, the optimum hydraulic and structural design of inverted siphon has been
studied. Modified Hooke and Jeeves method considered in the present study and some
modifications were doing. These modifications are:1) modification on the assumed initial
base points, 2) modification on the value of step length, 3) modification on the value of the
reduced step length at each trial. Many shapes of inverted siphon considered as (circular
section, square section, and rectangular section). A computer program was writing by
FORTRAN90 language depending on the method mentioned. STAAD Pro2006 software
program was considering to structural analysis and design of inverted siphon. Many
examples solved to check the accuracy of the written program.
Keywords: Optimum, hydraulic, inverted siphon, structural design, reinforced concrete
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Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
178
Nomenclature
b: span of box section (m)
D:Diameter of pipe (m)
Ca: constant for showing irrigation canal (Ca=1) or drainage canal (Ca=2).
d: height of box section (m)
f: Lacey silt factor (m)
f1:coefficient of head losses at entry (m)
F1: constant for elbows losses.
f2 : is a coefficient such that the losses of head through the barrel due to
surface friction,
F2: constant for bend losses
hf : the total head losses which equals to the difference between the
upstream and downstream water level (m).
K1:Entrance coefficient (m)
K2:Outlet coefficient (m)
Kcon:Contraction coefficient (m)
Ke: entrance coefficient.
Kel: elbows coefficient
Ko: outlet coefficient.
Kex:expansion coefficient (m)
Ks: screen coefficient
L: length of inverted siphon (m).
q:Discharge / meter width (m
3
/m.s)
Q: discharge (m
3
/sec).
R:hydraulic mean radius of the barrel (m)
R1:Depth of scour below water level (m)
S: the slope of siphon (dimensionless).
V:velocity of flow through the barrel (m/s)
Va:velocity of approach and is often neglected (m/s)
Vc: velocity of canal (m/sec)
Zcb:cost of construction for concrete box (I.D)
Zcp:cost of construction for concrete pipe (I.D)
Zs:cost of construction for steel pipe (ID).
Y: elevation of water (m).
Ye: depth of water in canal (m).
Introduction
Highways and railroads traversing the land cut across individual watersheds. To allow the
flow from each watershed across the embankment, culverts, siphons and aqueducts are used.
Although these structures simple in appearance, their hydraulic design is no easy matter. The
operation of these structures under the various possible discharge conditions presents a
somewhat complex problem that cannot be classified either as flow under pressure or as free
surface flow. The actual conditions involve both of these concepts.
Hydraulic structures employed to convey, control, measure and protect the flow of
water at various locations in an irrigation and drainage system. There are many different
types of structures can be used for these purposes though individually they may be small or
large. Structures can contribute to a large part of the overall capital cost of an irrigation
project; hence, a proper design of these facilities is a major factor in making the scheme
Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
179
efficient and keeping capital costs to a minimum. The final design must take into
consideration the practices and farming methods in vogue and should endeavor to meet the
requirements of the farmer and thereby enlist his co-operation. This will minimize
maintenance costs by minimizing mis-use and vandalism. Other factors that affect the design
and operation of irrigation structures include site conditions, the methods employed for the
conveyance of water and the availability of construction materials. Only the smaller sizes of
structure are amenable to having the design procedure standardized.
Purpose and Description of Inverted Siphons
Inverted siphons used to convey canal water by gravity under roads, railroads, other
structures, various types of drainage channels, and depressions. A siphon is a closed conduit
designed to run full and under pressure (Aisenbrey, et al., 1974)
(1)
. The structure should
operate without excess head when flowing at design capacity.
A-Application
Economics and other considerations determine the feasibility of using a siphon or another
type of structure to accomplish the previous objectives. The use of an elevated flume would
be an alternative to a siphon crossing a depression, drain channel or another manmade
channel. The use of bridge over a canal would be an alternative to a siphon under a road or a
railroad.
B-Advantages and Disadvantages of Inverted Siphons
Inverted siphons are economical, easily designed and built, and have proven a reliable means
of water conveyance. Normally, canal erosion at the ends of the siphon is inconsequential if
the structures in earth waterways have properly designed and constructed transitions and
erosion protection.
Costs of design, construction, and maintenance are factors that may make an inverted
siphon more feasible than another structure that might used for the same purpose. There may
be, however, instances where the value of the head required to operate a siphon may justify
the use of another structure such as a bridge (Aisenbrey, et al., 1974)
(1)
.
An inverted siphon may present a hazard to life, especially in high population density
areas.
Structure Components
The siphon profile is determined in such a way as to satisfy certain requirements of cover,
siphon slopes, bend angles, and submergence of inlet and outlet. Siphon cover requirements
are (Aisenbrey, et al., 1974)
(1)
:
1) At all, siphons crossing under roads other than farm roads and siphons crossing under
railroads, a minimum of (0.91 m) of earth cover should provided. Farm roads require only
(0.61 m) of earth cover and are frequently ramped using 10 to 1 slopes (10 percent grade)
when necessary to provide minimum cover requirements. If roadway ditches exist and
extended over the siphon, the minimum distance from the ditch to the top of the pipe
should be (0.61 m).
2) At siphons crossing under cross-discharge channels, a minimum of (0.91 m) of earth
cover should provided unless studies indicate more cover is required because of projected
future retrogressions of the channel.
3) At siphons crossing under an earth canal, a minimum of (0.61 m) of earth cover should
provide.
Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
180
4) At siphons crossing under a lined canal, a minimum of (0.15 m) of earth cover should
provide between the canal lining and the top of siphon.
Roadway widths and side slopes at road and railroad siphon crossings should match
existing roadway widths and side slopes, or as otherwise specified. Side slopes should not be
steeper than (1-1/2 to 1).
Siphon slopes should not be steeper than (2 to 1) and should not be flatter than a slope
of (0.005).
Hydraulic Design Considerations
Available head, economy, and allowable siphon velocities determine the size of the siphon.
Thus, it is necessary to assume internal dimensions for the siphon and compute head losses
such as entrance, friction, bend, and exit. The sum of all the computed losses should
approximate the difference in energy grade elevation between the upstream and downstream
ends of the siphon (available head).
In general, siphon velocities should range from (1.07 m/s) to (3.05 m/s), depending on
available head and economic considerations (Aisenbrey, et al., 1974)
(1)
.
The following velocity criteria may use in determining the dimensions of the siphon:
1) (1.07 m/s) or less for a relatively short siphon with only earth transitions provided at
entrance and exit.
2) (1.52 m/s) or less for a relatively short siphon with either a concrete transition or a control
structure provided at the inlet and a concrete transition provided at the outlet.
3) (3.05 m/s) or less for a relatively long siphon with either a concrete transition or a control
structure provided at the inlet and a concrete transition provided at the outlet.
Where there is reasonable, confidence that a good standard of construction will
achieve the upper limit of velocity through a siphon may take as (3 m/s), where doubts exist
as to construction quality this figure may be reduced to (2 m/s),(Jawad, Kanaan, 1983) (5) .
To avoid sedimentations, the minimum velocity that considered is (0.6 m/s).
For discharges up to about (2.5 m3/s), pipes can be used but for larger discharges a
box section is preferred, (Jawad, Kanaan, 1983)
(5)
.
For future increase in demand, it is usual to design all canal structures for (1.2 Q)
where Q is the design flow during the period of maximum demand. The minimum design
flow is usually taken as (0.7 Q), blow this value, there is the risk that sedimentation will
occur due to low velocities. For drainage structures, the maximum flow taken as (1.5) times
the design flow. Minimum flow does not apply to drainage structures, (Jawad, Kanaan, 1983)
(5)
.
The most common materials used for culverts are concrete and corrugated steel. The
roughness in both cases usually assumed constant for any flow depth. On hilly terrains where
the culvert slope expected to be relatively steep and the flow through the culvert gains
considerable energy, corrugated steel pipes offer energy-dissipating advantages. On flat
terrains, energy loss through a culvert is undesirable; hence, concrete pipes are more suitable
(Simon, 1997) (9). For design purposes in Iraq, concrete and steel pipes considered with
roughness coefficient (n) equal to (0.014) and (0.01) respectively(Jawad, Kanaan, 1983) (5).
Drainage structures must checked against piping, uplift pressure, erosion at upstream,
downstream, and protected against sulphate attack on concrete. The protection consist of
(0.4 × 0.4 × 0.2 thick m) concrete blocks over (0.1 m) gravel bedding for length (2 × D) of the
barrel for upstream. Downstream is similar for length (3 × D). Depth of scour in unlined canal
can be calculated from Lacey formula and compared with (2 × D) and (3 × D) (Jawad, Kanaan,
1983)
(5)
.
Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
181
0.33
2
1
f
q
1.35=R
×××× (1)
where:
f : Lacey silt factor can obtained from Table (1) which used for Iraq.
flow of Depth -R 1.5= D 1s ×××× (2)
Ds1.5= protection of Length ×××× (3)
To ensure inlet submergence the invert of the siphon section should be dropped so
that D greater than the elevation of water at the canal with the drop at the siphon entrance
with slope (1 to 5 (V:H)) which generally equal to (0.5 m). So the discharge will divided into
2,3,4,….etc, to satisfy the previous condition, and that means the siphon sections equal to
2,3,4,…etc.
Reinforced concrete rigid frame box culverts with square or rectangular opening are
use up to spans of (4 m). The height of the vent generally does not exceed (3 m). Adopting
thickness of slab as (100 mm/ meter span) (Simon,1997)
(9)
.
Head Losses of Inverted Siphon
Total head losses equal to the sum of the friction, entrance, exit, screens, elbows, transitions
and bends losses. The minimum overall head loss for inverted siphons is (0.2 m) (Jawad,
Kanaan,.1983)
(5)
. Losses through siphon can calculate by using the following formula (Garg,
Santosh , 1978)
(4)
:
conex hh
g
Va
g
V
KsFF
R
L
ffH ++++++++−−−−
++++++++++++××××++++++++====
22
21211
22
∆∆∆∆ (4)
A briefly, discussions about these losses can illustrate as follows:
Friction Losses
Friction losses through the barrels of siphon can calculate by using the following formula.
g
V
R
L
fh f
2
2
2
××××==== (5)
where:
++++
R
b
a= f 2 1 (6)
where the values of a and b for different materials may be taken as given in Table (2).
Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
182
Entrance and Exit Losses
Entrance losses can calculate from the following equation (Jawad, Kanaan,1983)
(5)
:
====
2g
V
K h
2
1e (7)
where:
K1: Entrance coefficient, which obtained from Table (3).
Exit losses can calculate from the following equation
(5)
:
====
2g
V
K h
2
2o (8)
where:
K2: Outlet coefficient which taken as 1.0 for most outlets.
Screen Losses
Losses through screens and trash racks are related the velocity head of the approach flow and
the geometry of the rack (spacing, thickness and shape of bars and inclination to the
horizontal). Screens used at entry and exit on inverted siphons as a safety precaution to
prevent children and unauthorized personal from gaining access. The head loss can related to
the percentage of waterway area that the bars take up as shown in Table (4) below:
====
2g
V
K h
2
ss (9)
Elbows and Bends Losses
It remains now to consider the effect of elbows and bends, on the discharge of the siphon.
The losses in an elbow or a bend in a pipe appear to be due to secondary circulation and to
contraction of the flow, which occur, in, and immediately downstream of, the cause of the
disturbance.
Weisbach gives the following formula for the loss of head in elbows
(6)
.
====
2g
V
Fh
2
1e1 (10)
where:
2
sin 2.047+
2
sin 0.9457= F
42
1
θθθθθθθθ
(11)
According to this theory, if the radius of curvature R does not change, the length ℓb of the
bent portion of the pipe, and angle ϑ , have very little effect (if any) on the total loss of head
attributable to the bend. The formula was therefore as follows
(Leliavsky, Serrge, 1979)
(6)
.
Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
183
====
2g
V
Fh
2
2b (12)
where:
90
27
θθθθ
/
R
D
0.16 + 0.13= F2 (13)
Transitions Losses
A channel transition may define, as a local change is cross section, which produces a
variation in flow from one uniform state to another. In many hydraulic structures the main
reason for constricting or fluming the flow at the inlet is to reduce, the costs of construction
of the structure and in some cases can provide an expedient device for measurement of
discharges in the main body of the structure. It is important that transitions to and from
structures is properly designed when head losses are critical. Transitions can serve several
other functions, namely:
1- To minimize canal erosion.
2- To increase the seepage path and thereby provide additional safety against piping.
3- To retain earth fill at the ends of structures.
All transitions may classify as either inlet (contraction) or outlet (expansion)
transition. Expansion and contraction losses can be finding from the following equations
(Jawad, Kanaan,1983)
(5)
:
−−−−
g
Vc
g
V
K= h exex
22
22
(For canal structures) (14)
−−−−
g
Vc
g
V
K= h concon
22
22
(For canal structures) (15)
−−−−
g
Vc
g
V
K= h exex
22
21
22
. (For drainage structures) (16)
−−−−
g
Vc
g
V
K= h concon
22
21
22
. (For drainage structures) (17)
where:
Kex & Kcon: are the expansion and contraction coefficient respectively, which can
obtained from Table (5) depending on the discharge values (depending on the reference of
[5]).
Angle of contraction at the upstream is preferred as equal as or less than 27.5
o
, and the
optimum contraction angle is 14
o
, while the angle of the expansion at the downstream is
equal or less than 22.5
o
(Varshney,1972)
(10)
.
Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
184
Optimization Method For Designing the Inverted Siphon
The purpose of optimization is to find the best possible solution among the many potential
solutions satisfying the chosen criteria. Designers often based their designs on the minimum
cost as an objective, safety and serviceability.
A general mathematical model of the optimization problem can represent in the
following form
(2)
:
A certain function (Z), called the objective function,
n iXf Z i ,.........}} 22221,1,1,1,==== ==== (18)
Which is usually the expected benefit (or the involved cost), involves (n) design
variable {X}? Such function is to be maximized (or minimized) subject to certain equality or
inequality constraints in their general forms:
(((( )))) i ib Xg iii ,.........22221,1,1,1,==== ==== (19)
(((( )))) j ib Xq jji ,.........22221,1,1,1,==== ≥≥≥≥ (20)
The constraint reflects the design and functional requirements. The vector {X} of the
design variables will have optimum values when the objective function reaches its optimum
value.
Objectives Functions
The objective function (Zs) of the present research for steel pipes siphons involves the cost of
transportation, cutting, constructions, and filling as (Zs = C × D)
(Raju,1986)
(8)
, (Zs: total cost
per unit meter, C: constant, D: diameter of siphon)
.
. The following equation estimated by (STATISTICA) program with a regression
coefficient of (R = 0.996):
0.469596
s D 248791.9 Z ××××==== (21)
In addition, the objective functions for concrete pipe and box siphons shown below.
Data for this formula obtained from the reference
(7)
.
2
cp D177798.44 Z ××××==== (22)
++++====
33
22
db
514500 Z cb (23)
The following Figure shows the relationships between the costs and the dimensions
of concrete siphon (pipe and box) and steel siphons, depending on the previous formulas.
Constraints for the optimization Technique
The following constraints that considered for the optimizations technique in this research was
as follows:
1- 0.005 ≤ S≤ 0.5
2- 0.6 ≤ V ≤ 3.0
3- hf ≥ 0.2
4- b ≤ 4.0
5- d ≤ 3.0
6- d ≤ (Y+0.5)
Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
185
Modified Hooke and Jeeves
This method dates back to 1961
(3)
but is nonetheless a very efficient and ingenious procedure.
The search consists of a sequence of exploration steps about a base point, which if successful
followed by pattern moves. The procedure is as follows:
A) Choose an initial base point b1 and a step length hj for each variable Xj, j = 1, 2… n. The
program given later uses a fixed step h for each variable, but the modification indicated can
be useful.
B) Carry out an exploration about b1. The purpose of this is to acquire knowledge about the
local behavior of the function. This knowledge used to find a likely direction for the pattern
move by which it hoped to obtain an even greater reduction in the value of the function. The
exploration about b1 proceeds as indicated.
i) Evaluate ƒ (b1).
ii) Each variable now changed in turn, by adding the step length. Thus ƒ (b1 +h1 e1 ) can be
calculated where e1 is a unit vector in the direction of the X1-axis. If this reduces the function,
replace b1 by (b1 +h1 e1). If not find ƒ (b1 -h1 e1 ) and replace b1–by (b1 -h1 e1 ) if the function–
is reduced. If neither step gives a reduction leave b1 unchanged and consider changes in X2,
i.e. find ƒ (b1 +h2 e2 ) etc. When it has considered all n variables, a new base point b2 can
obtained.
iii) If b2 = b1 i.e. no function reduction has been achieved, the exploration is repeated about
the same base point b1 but with a reduced step length. Reducing the step length(s) to one tenth
of its former value appears to be satisfactory in practice.
iv) If b2 ≠ b1 make a pattern move.
C) Pattern moves utilize the information acquired by exploration, and accomplish the
function minimization by moving in the direction of the establish "pattern". The procedure is
as follows.
i) It seems sensible to move further from the base point b2 in the direction b2 – b1 since that
move has already led to a reduction in the function value. Therefore, the evaluation of the
function at the next pattern point can do.
)b-(b 2 b P 1221 ++++==== (24)
In general:
)b-(b 2 b P i1iii ++++++++==== (25)
ii) Then continue with exploratory moves about P1 (Pi).
iii) If the lowest value at step C (ii) is less than the value at the base point b2 (bi+1 in general)
then a new base point b3 ( bi+2 ) has been reached. In this case, repeat C (i). Otherwise abandon
the pattern move from b2 (bi+1) and continue with an exploration about b2 ( bi+1).
D) Terminate the process when the step length(s) has reduced to a predetermined small value.
In this research some modifications on this method has achieved, since this method is
not able to move along the constraint and converges on the first point on the constraint that it
locates as the solution. With a certain initial point and certain step length, a certain solution
has obtained, while with another initial point or another step length another different solution
has obtained. These modifications reduce the difference between the two obtained solutions.
These modification in this research as follows:
1- Assume the initial base point within the range of constraints.
2- Reduce the step length to a lower value such as (0.00001).
3- Reducing the step length(s) to (1/1.001) of its former value appears to be more satisfactory
in practice instead of one tenth at step B (iii).
Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
186
Optimum Structural Design of Inverted Siphon
The inverted siphon consisting of two horizontal and two vertical slabs built monolithically
are ideally suited for a road or a railway bridge crossing with high embankments crossing a
stream with a limited flow. The inverted siphon are economical due to their rigidity and
monolithic action and seprate foundationa are not required since the bottom slab resting
directly on the soil, serves as raft slab.
The structural design of a reinforced concrete inveted siphon comprises the detailed
analysis of the frame and thrusts due to various types of loading. The computer program was
used for an analysis and design of inverted siphon and includes the type of loading that
applies on the reinforced concrete inverted siphon, as shown in Figure (1).
1.inverted siphon weight
The total weight of a reinforced concrete siphon per unit length, Wsiph, at a given section with
concrete can be calculated from the following simplified equations for approximate total
weight of concrete in kN per m and the density of concrete is 24 kN/m
3
, so can be obtained
Wsiph from the following equation:
)( ssiph LhW ××××==== 96
2. Fluid Loads
The weight of fluid per unit length, Wf, inside a siphont filled with fluid can be calculated
from the following simplified equations for approximate total weight of water in kN per m
and the density of water is 10 kN/m
3
, so can be obtained Wf from the following equation:
2
10 )( hLW sf −−−−====
3.Earth Loads
Earth load in kN/m is determined by multiplying the weight of the earth prism load above the
extremities of the siphon and the density of soil is 18 kN/m
3
, so the earth load on the top slab
as follows:
es HP γγγγ====1
and the lateral pressure due to the effect of imposed dead load surcharge only, as follows:
)( yHkP esss ++++==== γγγγ
where ks equal to (1/3) and the Pb equal to summation of self weight of siphon and the soil
weight and the water weight as follows:
)( hLWsiphWfP swb −−−−××××++++++++==== γγγγ
So, in this section, the STAAD Pro.2006 sofware program was used to analysis and
design the inverted siphon using the above loadings to the reinforcement steel area that
required with thickness of slabs (h=(span (Ls))/10). In the present study was used the cost of
plain concrete is (250000 ID/m
3
) and the cost of reinforced steel is (1200000 ID/ton) as
shown in Table (7).
Computer Program for Optimum Hydraulic Design
In this research, the program was writing with "Quick- Basic" Language depending on
(Modified Hook and Jeeves) method with some modifications -since (SUMT) method
cannot be used due to the fractional powers of constrains- to find the optimum hydraulic
Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
187
design for the inverted siphon. Many examples applied by using this program to check the
accuracy of it. The following example solved without the optimization technique and with the
optimization technique by using this program.
The input data for this program were:(Ca, Cs, L, Ke, K0, Ks, Cel, Kex, …)
X(i): initial dimension for siphon (i=1 for pipe siphon, i= loop from 1 to 2, for box and
rectangular siphon).
H: step length.
Many examples applied using the program and compared with the design without
optimization as follow:
The following figure shows a typical section for inverted siphon and so the following
Table (8) shows the dimensions of siphon with variable discharges, constant velocity (0.82
m/s) and variable elevation for the same canal.
The following Table (9) shows the dimensions of siphon with variable discharges,
variable velocity and constant elevation (1.5 m) for the same canal.
The following Figure (3) shows the relationship between the cost of siphon
construction and discharge.
Conclusions
1. The modification of the assumed initials base points improved the modified Hooke and
Jeeves method.
2. The modification of the value of step length also improved the modified Hooke and Jeeves
method.
3. In addition, the modifications above, the modification of the value of the reduced step
length at each trial improved the modified Hooke and Jeeves method.
4. A significant construction cost reduced due to optimum hydraulic and structural design for
inverted siphon.
References
1. Aisenbrey, A.J., Hayes, R.B., Warren, H.J., Winsett, D.L. and Young, R.B., (1974),
"Design of small canal structures ", Denver, Colorado, U.S.A.
2. Al- Husseini, Thulfikar, R.,(2004), " Analysis and evaluation of water supply
facilities for Al- Hilla city" , M.Sc. Thesis, College of Engineering, Babylon
University, Iraq.
3. Bunday, Brian D., (1985), "Basic Optimisation Methods ", School of Mathematical
Sciences, University of Bradford, London.
4. Garg, Santosh Kumar, (1978), "Irrigation engineering and hydraulic structures ",
Khanna Publishers, Delhi, India.
5. Jawad, Kanaan, A., (1983), "Design manual irrigation and drainage ", Pencol
Engineering Consultants and State Organization for Land Reclamation, Baghdad,
Iraq.
6. Leliavsky, Serrge, (1979), " Irrigation Engineering: Syphons, Weirs and Locks ",
Chapman and Hall LTD, London, New Delhi, India.
7. Ministry of Irrigation, State Commission for Irrigation at Babylon Governorate,
(2004), Annual Report.
8. Raju, N. Krishna, (1986), "Advanced reinforced concrete design", CBS Publishers
and Distributors, Delhi, India.
9. Simon, Andrew L., Korom, Scott F. (1997), "Hydraulics", Prentice Hall, U.S.A.
10. Varshney, R.S., Gupta, S.C., Gupta, R.L., (1972), "Theory and design of irrigation
structures", Roorkee University, India.
Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
188
Table (1): Silt Factor [5].
Region f
Northern 0.7-1.0
Central 0.6
Southern 0.5
Table (2): Values of a and b for different materials [4].
Materials of the surface of the
barrel
a b
Smooth iron pipe 0.00497 0.025
Encrusted pipe 0.00996 0.025
Smooth cement plaster 0.00316 0.030
Ashlars or brick work 0.00401 0.070
Rubble masonry or stone pitching 0.00507 0.250
Table (3): Entrance coefficient [5].
Description K1
For square edged inlet flush with vertical walls 0.5
For rounded inlets, radius r where r/ D ≤ 0.15 0.1
For grooved or socket ended pipes 0.15
For projecting concrete pipes 0.2
For projecting steel pipes 0.85
Table (4): Loss coefficients for screens and trash racks [5].
Bar area/waterway
area %
5 10 15 20 25 30
Loss coefficient (Ks) 0.12 0.25 0.42 0.62 0.86 1.15
Table (5): Expansion and contraction coefficients
Siphon section Discharge m
3
/s Kex Kcon
2.5-5 0.6 0.3
Box
> 5 0.2 0.1
< 0.5 1.0 0.5
Pipe
0.5-2.5 0.7 0.4
Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
189
Table(6): Data for optimum hydraulic design for inverted siphon
Q(m3/s) Vc(m/s)
Elev. Of
water(m)
Length of
siphon(m)
Kent Kext Ks Kelb
Case 1 4 0.82 1.50 41.3 0.2 0.3 0.2 0.05
Case 2 5 1.0 2.0 50 0.2 0.3 0.2 0.05
Case 3 10 1.50 2.0 70 0.2 0.3 0.2 0.05
Case 4 20 1.50 2.0 70 0.2 0.3 0.2 0.05
Table (7): Output of the above input data with and without optimization technique.
Case
No. of
Sections
Dimension (m)
Weight of
steel (kg/m)
Price of
siphon
(ID/m)
Without
optimization
1 1.45 x 1.45 261.812974 531672
1
With
optimization
1 1.15 x 1.15 265.105820 456120
Without
optimization
1 1.54 x 1.54 249.4580 545740
2
With
optimization
1 1.29 x 1.29 270.321760 366309
Without
optimization
1 2.13 x 2.13 255.984728 775776
3
With
optimization
1 1.83 x 1.83 244.387199 640956
Without
optimization
2 2.13 x 2.13 444.715580 1353708
4
With
optimization
2 1.83 x 1.83 423.398260 1116555
Table (8): Dimension of siphon and its number of sections for constant velocity.
Q(m
3
/s)
4
8
1
0
1
2
1
5
1
8
2
0
2
5
3
0
Y(m)
V(m/s)
1
.5
3
.0
3
.7
5
4
.5
5
.6
3
6
.7
5
7
.5
9
.3
8
1
1
.3
0
.8
2
1
.1
2
1
.1
2
1
.6
4
1
.6
4
1
.8
2
1
.8
2
2
.0
0
2
.0
0
2
.2
3
2
.2
3
2
.4
5
2
.4
5
2
.5
8
2
.5
8
2
.8
8
2
.8
9
2
.2
3
2
.2
4
S
e
c
ti
o
n
N
o
.
1
1
1
1
1
1
1
1
2
Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
190
He
Ps Ps
Pb
P1
Do
h
Ls
Ls
y
Table (9): Dimension of siphon and its number of sections for constant elevation.
Q(m
3
/s)
4
8
1
0
1
2
1
5
1
8
2
0
2
5
3
0
V(m/s)
Y(m)
0
.8
2
1
.6
4
2
.0
5
2
.4
6
3
.0
8
3
.6
9
4
.1
0
5
.1
3
6
.1
5
1
.5
0
1
.1
2
1
.1
2
1
.5
8
1
.5
8
1
.7
6
1
.7
7
1
.9
3
1
.9
4
1
.5
8
1
.5
8
1
.7
3
1
.7
3
1
.8
2
1
.8
3
1
.6
6
1
.6
7
1
.8
2
1
.8
3
S
e
c
ti
o
n
N
o
.
1
1
1
1
2
2
2
3
3
Figure (2): Relationships between materials cost/ unit meter length and dimensions of inverted
siphon.
0.40 0.50 0.60 0.70 0.80 0.90 1.00
Dimension of siphon (m)
25000
50000
75000
100000
125000
150000
175000
200000
225000
250000
275000
300000
325000
350000
C
o
s
t
(I
.D
)
Cost of Concrete Pipes
Cost of Concrete box
Cost of Steel Pipes
Figure (1): Distribution of Earth Pressures on Inverted
Siphon.
Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009
191
Figure (3): Typical section for inverted siphon.
Figure (4): Relationships between cost and discharge of inverted siphon.
Vc
Θ
Main Road
∆
Hint: Length of siphon= L1+L2+L3
4 6 8 10 12 14 16 18 20 22 24 26 28 30
Discharge (m3/s)
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
11.00
12.00
13.00
14.00
15.00
C
o
s
t
*
1
0
^
7
(
I.
D
)
With variable velocity and constant elevation
With constant velocity and variable elevation