International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 9, September 2013)
Kinetic Energy and Momentum Correction Coefficients for a
Small Irrigation Channel
Umaru Garba Wali
National University of Rwanda, Faculty of Applied Sciences, Department of Civil Engineering, Water Resources and
Environmental Management Programme, P. O. Box 117, Huye, Southern Province, Rwanda.
Continuity equation
Abstract-- The main aim of this study was to determine the
kinetic energy and momentum coefficients of a small
trapezoidal irrigation canal. A laboratory setup with a cyclic
flow of water was used. The flow cross section was divided
into ten equal sections with nine verticals in between. Pitot
static tube was used to measure static and stagnation pressure
heads at 1, 2, 3, 4, 5, or 6 point on the verticals. These
measurements were conducted at a depth corresponding to
0.1h, 0.2h, 0.4h, 0.6h 0.8h and 0.9h from the free surface of the
water in the canal. The different between stagnation head and
static head was used to compute the local instantaneous
velocity at each measured point. Three different discharges
ware used 3.5, 9.5, and 11.4 l/s. The results of the study
reaffirms the non equal to unity of energy and momentum
correction factors and that for regular canal the value of α
range from 1.10 to 1.20 and β from 1.03 to 1.07. It is
recommended that the energy and momentum correction
factors be taken into consideration even for simple hydraulic
analysis in open channel flow.
Energy equation
∑
Momentum equations
[
]
Where: Q- discharge, A1 & A2 and U1 & U2 cross
sectional areas and mean velocities at point 1 and 2. As
shown in all the equations 1, 2 and 3 above the mean
velocity U is used, this assumed that the flow is steady
uniform and non-varying vertically across the flow cross
section. However, it is well known that the flow velocity
distribution across any flow cross-section of open channel
is not uniform [3, 4, 5 & 6]. The nature of flow velocity
distribution is shown in figure 1 as reported by Chaudhry
[7].
This assumption introduced an error in the final results
for the energy and momentum and their related output
calculations [8 & 9]. The error introduced by this
assumption is not significant if the flow is steady or nearly
uniform and in such case corrections to energy and
momentum equations may be neglected (α=1 and β=1).
Keywords-- kinetic energy coefficient, momentum
coefficient, open channel, Pitot static tube, dynamic head,
static head, mean velocity
I. INTRODUCTION
Open channels are important infrastructures which are in
continuous application in our day to day water conveyance
activities. The two most important characteristics of open
channel flows are free surface with the pressure equals to
atmospheric pressure and the flow powered by gravity [1 &
2]. Some of the application of open channels includes:
irrigation and drainage, hydropower, water supply and
sanitation, urban and highways drainages etc. Because of
the important of open channel flow in civil engineering
activities it is necessary to continuously investigate it to
enable us to upgrade the design of this type of
infrastructures toward achieving more effective design. It is
necessary that the design of open channel satisfy technical,
socioeconomic and esthetic conditions. The technical
design of open channels involves the application of the
three basics laws of conservations. These are Law of
conservation of mass, Law of conservation of energy and
Law of conservation of momentum and for open channel
applications these laws are express in term of continuity
equation, energy (Bernoulli’s) equation and momentum
equation respectively as shown in equations 1, 2 and 3 [2].
Figure 1. Velocity distribution in open channel flow
But it is clear that the channel boundary resistances alter
the velocity distribution along the cross-section, with
varying influence that reduces from the wall of the channel
to the water free surface [1].
315
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Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 9, September 2013)
In most of these cases the introduced error is significant.
This non-uniform velocity distribution needs to be taken
into consideration for appropriate design of the related
structures. The energy and momentum coefficients are used
to correct these effects in term of energy and momentum
equations respectively as shown in equations 2 and 3 [8 &
9]. The energy correction coefficient is equal to the ratio of
the kinetic energy of mass of the fluid flowing within a unit
time through a given cross section to the kinetic energy of
fluid mass estimated with the assumption that velocity at
every point in the liquid equal to the mean velocity of the
flow [10 and 11]. This definition can be express as in
equations 4 and 5 below, [1, 6 and 11].
Energy coefficient or energy correction factor can be
estimated as follows:
∫
It measures velocity using the differential pressure
principles. The used tube has “L” shape with one opening
at the tip and two openings at the side. The openings are
connected to the scale through flexible tubes at the end of
the tube a vacuum hand pump is connected that is used to
remove air in the tube and raise the level of water so that it
can be conveniently read on the scale. The opening at the
tip records the stagnation pressure which is the sum of
static and dynamic pressures equation (6), while the side
openings record the static pressure. The different of the two
measurements gives the dynamic pressure which is then
converted to the measured velocity equation (7).
Stagnation pressure:
∑
√
Momentum correction coefficient
∫
∑
[
]
Writing equation (7) in term of measured depth we
obtained:
√
Most irrigation canals are design with consideration that
velocity across the section is unity; this may not be
acceptable when the variability of velocity distribution is
high. The main objective of this study is to determine the
kinetic energy and momentum coefficients of a small
irrigation trapezoidal canal.
[
]
Transforming equation (8) results in equation (9) that
was used to compute the actual local velocity at a point on
the vertical.
√
II. MATERIALS AND METHODS
The author conducted this study in the hydrological
laboratory of the All Russian Scientific Institute of
Hydrotechnic and Reclamation Moscow, Russia. The
apparatus used has cyclic water flow that allows continuous
water flow with constant discharge figure 2. A steel sheet
of 2 mm thick was used to construct the model trapezoidal
canal, section 9 of figure 2, used for the experiments. The
control cross section of the canal was imaginary divided
into ten sections each of 48 mm, with nine velocity
verticals in between them. The local velocities were
measured in five replicates using Pitot static tube at 1, 2, 3,
4, 5 or 6 points depending on the depth of water (h) at the
vertical. The velocity was measured at the depths
corresponding to 0.1h, 0.2h, 0.4h, 0.6h, 0.8h and 0.9h, as
shown in figure 3. Where h is highest water depth in the
control cross section at a given discharge.
Pitot static tube was selected for velocity measurement
in this study because (i) it allows minimal disturbance of
the flow, (ii) it is simple to use, (iii) it is accurate and (iv) it
is relatively inexpensive compared to other velocity
measurement instruments.
Figure 2. Laboratory setup for
316
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Column 6: Area of the velocity distribution diagram at the
vertical. The velocity distribution graphs were plotted
manually on graph papers and the areas computed
geometrically.
Column 7: Mean velocity on the vertical computed as the
ratio of the area of the velocity distribution diagram and the
depth of water at the vertical.
Column 8: Area around velocity vertical with half
contribution from both sides.
Column 9, 10 & 11: The products of column 7 and column
8 in accordance to UdA, U2dA and U3dA.
The mean velocity of the entire cross section was
computed using equation (10) and the flow across the
section was determined using equation (11).
Mean velocity in the cross section
[
Figure 3. Cross section of the trapezoidal canal with velocity
measurement locations
The analysis of results was conducted in a tabular form
Table 1, Table 2 and Table 3 for three discharges used
11.407 Ls-1, 9.504 Ls-1and 3.514 Ls-1 respectively. The
information in these tables is described as follows [12]:
]
[
]
Discharge in the cross section
[
]
[
]
The energy and momentum correction coefficients were
then calculated using equations 4 and 5.
Column 1: Serial number of the verticals which were
imaginary fixed in accordance with the shape of the
experimental canal and does not change with change in
water flow.
Column 2: Depth of water at a given vertical depends on
the flow.
Column 3 &4: Mean values of stagnation and statics heads
each from five measurements conducted at a point within
one to two minutes.
Column 5: Local velocity computed from equation 9.
III. RESULTS AND DISCUSSIONS
The main results of the experiments was summarized in
a tabular form Table 1, Table 2 and Table 3, corresponding
to discharges 11.407 Ls-1, 9.504 Ls-1and 3.514 Ls-1
respectively. The numerical values of velocity distribution
for the three conducted experiments in this study are
presented in Figure 4 to Figure 6. While the computed
value of α and β from those velocities are shown in
Table 4.
317
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TABLE 1.
SUMMARY OF RESULTS WITH Q=11.407 L/S
S/N of
velocity
vertical
1
1
2
Depth of
water at
the
vertical h,
m
2
0.04
0.09
3
0.14
4
0.19
5
0.19
6
0.19
7
0.14
8
0.09
9
0.04
Mean value of head
Stagnation,
Statics,
hD
hS
cm
cm
3
10.44
10.51
10.49
10.52
10.52
10.54
10.62
10.61
10.54
10.57
10.64
10.64
10.89
11.02
10.52
10.55
10.57
10.66
10.69
11.04
10.53
10.56
10.61
10.67
10.69
11.03
10.53
10.56
10.61
10.63
10.51
10.52
10.47
10.44
4
10.33
10.31
10.34
10.42
10.29
10.32
10.35
10.38
10.31
10.30
10.36
10.36
10.57
10.84
10.30
10.29
10.35
10.35
10.41
10.83
10.30
10.31
10.35
10.41
10.39
10.83
10.31
10.31
10.35
10.37
10.35
10.36
10.39
10.37
Local
velocity,
-1
Ms
5
0.1408
0.1882
0.1630
0.1276
0.2027
0.1965
0.2154
0.2036
0.2053
0.2162
0.2250
0.2242
0.2350
0.1815
0.1974
0.2162
0.2000
0.2335
0.2203
0.1956
0.2000
0.2121
0.2162
0.2170
0.2289
0.1910
0.1992
0.2087
0.2146
0.2170
0.1694
0.1673
0.1234
0.1048
Area of the
velocity
distributio
n diagram
at the
vertical
,
m2s-1
6
0.005633
Mean
velocity at
the
vertical,
ms-1
7
0.14083
0.013980
Area
around
velocity
vertical, fi
m3s-1
m4s-1
m5s-1
8
0.001820
9
0.000256
10
0.000036
11
0.0000051
0.15534
0.004207
0.000654
0.000102
0.0000158
0.028661
0.20473
0.006707
0.001373
0.000281
0.0000576
0.041569
0.21879
0.008895
0.001946
0.000426
0.000932
0.040527
0.21330
0.009207
0.001964
0.000419
0.0000893
0.040651
0.21395
0.008895
0.001903
0.000407
0.0000871
0.029634
0.21168
0.006707
0.001420
0.000301
0.0000636
0.013646
0.15163
0.00427
0.000638
0.000098
0.0000149
0.002200
0.05501
Sum:
0.001820
0.052463
0.000100
0.010253
0.000006
0.002075
0.0004268
0.0004268
318
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TABLE 2.
SUMMARY OF RESULTS WITH Q=9.504 L/S
S/N of
velocity
vertical
1
1
2
Depth of
water at
the
vertical h,
m
2
0.023
0.073
3
0.12
4
0.173
5
0.173
6
0.173
7
0.123
8
0.073
9
0.023
Mean value of head
Dynamics,
Statics,
hD
hS
cm
cm
3
11.07
11.12
11.13
11.11
11.34
11.34
11.30
11.28
11.39
11.40
11.39
11.39
11.37
11.38
11.43
11.44
11.45
11.48
11.51
11.53
11.45
11.41
11.44
11.40
11.37
11.35
11.25
11.27
11.31
11.31
11.23
11.19
11.14
11.14
4
10.97
10.93
10.95
10.99
11.11
11.11
11.10
11.14
11.13
11.14
11.14
11.11
11.13
11.16
11.19
11.20
11.20
11.21
11.29
11.34
11.22
11.19
11.15
11.17
11.15
11.16
11.00
11.02
11.08
11.12
11.00
11.00
11.00
10.98
Local
velocity,
m/s
5
0.1304
0.2000
0.1805
0.1445
0.1992
0.2009
0.1882
0.1586
0.2129
0.2129
0.2104
0.2227
0.2079
0.1965
0.2036
0.2070
0.2096
0.2187
0.1983
0.1825
0.2027
0.2009
0.2242
0.1983
0.1974
0.1853
0.2112
0.2129
0.2053
0.1844
0.2000
0.1853
0.1574
0.1248
Area of the
velocity
distributio
n diagram
at the
vertical
,
m2s-1
6
0.000357
Mean
velocity at
the
vertical,
ms-1
7
0.01552
0.012755
Area
around
velocity
vertical, fi
m3s-1
m4s-1
m5s-1
8
0.000641
9
0.000010
10
0.0000002
11
0.0000000
0.17473
0.003091
0.000540
0.000094
0.0000165
0.022697
0.18452
0.005716
0.001055
0.000195
0.0000359
0.036633
0.21170
0.008541
0.001808
0.000383
0.0000810
0.035516
0.20530
0.009066
0.001861
0.000382
0.0000784
0.035113
0.20297
0.008541
0.001734
0.000352
0.0000714
0.024854
0.20207
0.005716
0.001155
0.000233
0.0000472
0.013196
0.18076
0.003091
0.000559
0.000101
0.0000183
0.000327
0.01423
Sum:
0.000641
0.045044
0.000009
0.008731
0.0000001
0.001740
0.0000000
0.0003487
319
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TABLE 3.
SUMMARY OF RESULTS WITH Q=3.514 L/S
S/N of
velocity
vertical
Depth of
water at
the
vertical h,
m
1
2
3
0.05
4
0.1045
5
0.1045
6
0.1045
7
0.0545
Mean value of head
Dynamics,
Statics,
hD
hS
cm
cm
Local
velocity,
m/s
3
8.76
8.74
8.71
4
8.64
8.64
8.66
5
0.1433
0.1383
0.0941
8.78
8.80
8.80
8.81
9.50
9.69
9.95
9.98
10.13
10.18
10.21
10.25
9.91
9.88
9.85
9.82
9.78
9.73
8.73
8.73
8.73
8.64
8.64
8.65
8.65
9.38
9.61
9.78
9.84
9.98
10.03
10.10
10.17
9.75
9.73
9.71
9.67
9.70
9.63
8.62
8.64
8.66
0.1597
0.1673
0.1619
0.1673
0.1458
0.1205
0.1735
0.1608
0.1641
0.1630
0.1370
0.1220
0.1683
0.1608
0.1574
0.1641
0.1234
0.1317
0.1408
0.1262
0.1065
Area of the
velocity
distributio
n diagram
at the
vertical
,
m2s-1
6
Mean
velocity at
the
vertical,
ms-1
Area
around
velocity
vertical, fi
m2
m3s-1
m4s-1
m5s-1
7
8
9
10
11
0.006143
0.11271
0.003022
0.000341
0.000038
0.0000043
0.016350
0.15645
0.004741
0.000780
0.000116
0.0000182
0.016146
0.15451
0.005047
0.000780
0.000120
0.0000186
0.015795
0.15114
0.004741
0.000717
0.000108
0.0000164
0.006095
0.11184
0.003022
0.000338
0.000038
0.0000042
0.02573
0.002917
0.000421
0.000062
Sum:
Figure 5. Velocity distribution at Q=9.504 Ls-1
Figure 4. Velocity distribution at Q=11.407 Ls-1
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V. NOTATIONS
A= cross-sectional area, m2
dA= elemental cross-sectional area, m2
f= cross-sectional area between verticals, m2
g=acceleration due to gravity, ms-2
h=head, m
P= pressure, Nm-2
Q= discharge, m3s-1
U= mean velocity, ms-1
V= local velocity, ms-1
α= energy or Coriolis correction coefficient
β=momentum or Boussinesq correction coefficient
ρ=density, kgm-3
1…n= serial number of cross sections
Figure 6. Velocity distribution at Q=3.514 Ls-1
TABLE 4.
DETERMINED VALUE OF α AND β
S/N
1
2
3
Average
α
1.12
1.10
1.08
1.10
β
1.05
1.04
1.04
1.043
Acknowledgment
The author is grateful the Russian Government for PhD
fellowship during which this study was conducted,
Moscow state University of environmental Engineering for
PhD training and the All Russian Scientific Institute of
Hydrotechnic and Reclamation (BINGIIM) Moscow,
Russia for allowing the work to be conducted in their
hydrologic laboratory.
Numerous sources confirmed the practical consideration
of Coriolis and Boussinesq coefficients (energy and
momentum correction factors respectively) as unity in
hydraulic problem analysis [3, 5, 9 and 13], though the
same sources indicate that the values are not equal to unity.
This study uses the traditional approach of determining
Coriolis and Boussinesq coefficients using numerical
integration, equations 4 and 5 and still demonstrates that
even for a small prismatic canal the values of α and β are
different from unity. The findings presented here are in
accord with what was reported by Chow, [14] which stated
that for a regular canal the value of α range from 1.10 to
1.20 and β from 1.03 to 1.07. However, this values can
reach α=2.0 and β=1.33 for a River valley. Fenton,
reported that neglecting to consider these coefficients in
hydraulic calculations of pipe and open channel flow could
introduced up to 5-10% error even in simple problems [9],
as a result he encourages the use of the integral forms of
the energy and momentum equations in practical problems
of hydraulics to ensure the proper consideration of these
factors.
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Thandaveswara B.S., Hydraulics. Indian Institute of Technology
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[4]
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IV. CONCLUSIONS
This study reaffirms the non equal to unity of energy and
momentum correction factors and that for regular canal the
value of α range from 1.10 to 1.20 and β from 1.03 to 1.07.
It is recommended that the energy and momentum
correction factors be taken into consideration even for
simple hydraulic analysis in open channel flow. Doing this
could avoid the possible errors that could be associated
with it nonconsideration.
[10] Djelezniakov, G.B., 1981. Hydrology and Hydrometry. Moscow
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321
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 9, September 2013)
[11] Luchsheva A.A., 1972. Practical
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