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Kennedy Theory
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DESIGN OF HYDRAULIC STRUCTURES
ALLUVIAL CHANNEL DESIGN
KENNEDY’S THEORY
R. G. Kennedy – 1895
Non-silting non-scouring reaches for 30 years in Upper Bari Doab Canal (UBDC) system.
Vertical eddies generated from the bed are responsible for keeping silt in suspension.
Critical velocity
Mean velocity which keeps the channel free from silting and scouring.
Vo = 0.55 D0.64
which can be written in general form as,
Vo = C Dn
where, Vo = critical velocity, D = depth of water
C = constant and n = index number
Later on realizing the channel material (sandy silt in UBDC), he modified the equation as
Vo = 0.55 m D0.64
or Vo = C m Dn
where,
m = C.V.R = Critical Velocity Ratio = V/Vo ; V = actual velocity
m = 1.1 – 1.2 coarser sand
m = 0.7 – 0.9 finer sand
m = 0.85 Sindh canals
Values of C (the constant in Kennedy’s Eq.) and m (the Critical Velocity Ratio, CVR) for various grades of silt
Type of silt grade C m
Coarser silt 0.7 1.3
Sandy loam silt 0.65 1.2
Coarse light sandy silt 0.59 1.1
Light sandy silt 0.53 1.0
Rugosity coefficient
Kennedy used Kutters equation for determining the mean velocity of flow in the channel
Where N depends upon the boundary material
Channel condition N
Very good 0.0225
Good 0.025
Indifferent 0.0275
Poor 0.03
Discharge (cumec) N (in ordinary soil)
14 – 140 0.025
140 – 280 0.0225
> 280 0.02
RS
R
N
S
SNV
00155.0
231
00155.0123
Water Surface Slope
No relationship by Kennedy.
Governed by available ground slope.
Different sections for different slopes.
Wood’s normal design table for B/D ratio.
Silt Carrying Capacity of Channel
Qt = K B Vo0.25
where
Qt = total quantity of silt transported
B = bed width
Vo = critical velocity
K = constant, whose value was not determined by Kennedy
Modification for Sindh Canals
In 1940, while designing Guddu Barrage project canals, K. K. Framji
proposed B/D ratio for Sindh canals as:
5.15.3 61 QD
B
Discharge
m3/sec
B/D ratio for
standard section
B/D ratio for
limiting section
100 6.0 4.0
1000 8.4 5.0
5000 13.0 8.0
Design Procedure
Case I : Given Q, N, m and S (from L-section)
1. Assume D
2. Calculate velocity from Kennedy’s equation, VK = 0.55 m D0.64
3. Calculate area, A = Q / VK
4. Calculate B from A = B D + z D2 ; assume side slope 1(V) : ½(H), if not given.
5. Calculate wetted perimeter and hydraulic mean depth from;
6. Determine mean velocity from Chezy’s equation, Vc =C √(RS)
if Vc = Vk then O.K.
otherwise repeat the above procedure with another value of D until Vc = Vk.
Note: increse D if Vk < Vc
decrease D if Vk > Vc
DBP 5
DB
DBD
P
AR
5
5.0 2
Problem:
Design an irrigation channel for the following data using Kennedy’s
theory:
Full Supply Discharge (F.S.Q) = 14.16 cumec
Slope, S = 1/5000
Kutter’s rugosity coefficient, N = 0.0225
Critical velocity ratio, m =1
Side slope, z = ½
Solution:
1. Assume D = 1.72 m
2. Vk = 0.55 m D0.64 =0.55(1)(1.72)0.64 = 0.778 m
3. A = Q/Vk = 14.16/0.778 = 18.2 m2
4. A = B D + 0.5 D2 for z =1/2 or 0.5
18.2 = 1.72 B + 0.5(1.72)2
B = 9.72 m
5.R = A / P = 18.2 / 13.566 = 1.342 m
6.
Vc = 0.771 m
≈ 0.778 m
Result:B = 9.72 m
D = 1.72 m
m 566.13)72.1(572.95 DBP
RS
R
N
S
SNVc
00155.0
231
00155.0123
50001342.1
342.1
0225.0
50001
00155.0231
50001
00155.0
0225.0
123
cV
EXAMPLE PROBLEM
Q = 80 m3/sec
S = 1:5500 = 0.00018 m/m
m = 1
DATA
Assume D = 2.5 m
Vk = 0.55 D0.64 = 0.989 m/sec
A = 80.918 m2
Side Slope = 1V:1.5H
n = 0.0225
A = B D+ 1.5D2
B = 28.617 m
P = 32.223 m
R = A/ P = 2.511 m
Using Kutter’s Formula in S.I. Units
C = 52.479
Vc = C√RS = 1.121 m/sec
Keep on trailing till Vc = Vk
D
B
1
1.5
1.803
PROBLEM
Design an irrigation channel using Kennedy’s theory that irrigates
a cultral commanded area of 50,000 hectares. Given that delta of
the crops is 0.12 m/month, slope of country is 1 in 4000 and
Kutter’s rugosity coefficient is 0.023. Assume the missing data.
222 5.05.0 DyDDBDA
)1()5.0(2 yDA
)55.0).(5.0( 64.02 mDyDQ
64.21
5.055.0
ym
QD
DB
DBDR
5
5.0 2
Problem:
Using Kennedy’s theory design an irrigation channel to carry a discharge of 56.63 cumec. Assume N = 0.0225, m = 1.03 and B/D = 11.3.
Solution:
1. B/D = 11.3, therefore B = 11.3 D
A = B D + 0.5 D2 =11.3 D2 + 0.5 D2 = 11.8 D2
2. V = 0.55 m D0.64 = 0.55 (1.03) D0.64 = 0.5665 D0.64
3. Q = A V
56.63 = (11.8 D2 ) (0.5665 D0.64 )
D = 2.25 m
4. B = 11.3 (2.25) = 25.43 m
5. R = A / P
A = B D + 0.5 D2 = (25.43)(2.25) + 0.5 (2.25)2 = 59.75 m2
P = B + √5 D = 25.43 + √5 (2.25) = 30.46 m
R = 59.75 / 30.46 = 1.96 m
6. V = 0.55 m D0.64 = 0.55 (1.03) (2.25)0.64 = 0.95 m/sec
7.
Simplifying, we get;
67.44 S3/2 – 0.93 S + 1.55x10-3 S1/2 = 1.68x105
Solving by trial and error, we get
S = 1 in 5720
Results:B = 25.43 m
D = 2.25 m
S = 1 / 5720
RS
R
N
S
SNV
00155.0
231
00155.0123
S
S
S )96.1(
96.1
0225.000155.0231
00155.0
0225.0
123
95.0
Case III : Given S, N, m and B/D
1. From the B/D ratio, determine B in terms of D.
2. Determine A, P and R in terms of D.
3. From Kennedy’s equation, determine velocity (Vk) in terms of D.
4. Putting values of N, S and R in the Chezy’s equation and Kutter’s
formula, determine velocity (Vc). Simplify the expression, and solve it
by trail and error for D.
5. Knowing D, calculate B, A and Vk.
6. Using continuity equation, determine the discharge (Q).
Problem:
Design a section by Kennedy’s theory, given B/D = 5.7, S = 1/(5000+X)
and N = 0.0225. Also determine the discharge carried by the channel.
Solution:
B/D = 5.7, B = 5.7 D
Assuming z = ½
Since V = 0.55 m D0.64
Assuming m =1
V = 0.55 D0.64 ---------- (1)
Also
DD
D
DD
DD
DB
DBDR 78.0
94.7
2.6
57.5
5.07.5
5
5.0 2222
RS
R
N
S
SNV
00155.0
231
00155.0123
2783.0
939.05000178.0
78.0
0225.0
50001
00155.0231
50001
00155.0
0225.0
123
D
DD
D
V
Equating equation (1) and (2)
0.55 D1.14 – 0.939 D + 0.43 D0.64 = 0
By trial and error
D = 2.1 m
B = 5.7 x 2.1 = 11.97 m
A = B D + z D2 = (11.97 x 2.1) + 0.5 (2.1)2 = 14.175 m2
V = 0.55 (2.1)0.64 = 0.884 m/sec
Q = A V = (14,175)(0.884) = 12.53 m3/sec.
Results:
B = 11.97 m
D = 2.1 m
Q = 12.53 cumec
D
DD
783.0
939.055.0 64.0
Shortcomings of Kennedy’s theory
1. The method involves trial and error.
2. Shape of section i.e. B/D is not known inadvance.
3. Kutter’s equation is used instead of Manning’sequation. Therefore limitations of Kutter’sformula are also incorporated in Kennedy’stheory. Moreover it involves morecomputations.
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