Flow in open channel i sem_13_14 [compatibility mode]
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Transcript of Flow in open channel i sem_13_14 [compatibility mode]
10/12/2013
1
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Flow in Open Channel
By
Dr. Ajit Pratap Singh
Civil Engineering Department
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
CE F312: Flow in Open Channel
by Dr. Ajit Pratap Singh
Brays Bayou
Concrete Channel
Uniform Open Channel FlowUniform Open Channel FlowUniform Open Channel FlowUniform Open Channel Flow Open Channel FlowOpen Channel FlowOpen Channel FlowOpen Channel Flow1. Uniform flow - Manning’s Eqn in a prismatic
channel - Q, v, y, A, P, B, S and roughness are all
constant as discussed in last classes
2. Critical flow - Specific Energy Eqn (Froude No.)
3. Non-uniform flow - gradually varied flow (steady
flow) - determination of floodplains
4. Unsteady and Non-uniform flow - flood waves
CE F312: Flow in Open Channel
by Dr. Ajit Pratap Singh
CE F312: Flow in Open Channel
by Dr. Ajit Pratap Singh
Open Channel Flow• Liquid (water) flow with a ____ ________
(interface between water and air)
• relevant for
– natural channels: rivers, streams
– engineered channels: canals, sewer lines or culverts (partially full), storm drains
• of interest to hydraulic engineers
– location of free surface
– velocity distribution
– discharge - stage (depth) relationships
– optimal channel design
free surface
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Properties of Open Channels
• Free water surface
– Position of water surface can change in space and time
• Many different types
– River, stream or creek; canal, flume, or ditch; culverts
• Many different cross-sectional shapes
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CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
TYPES OF CHANNELS
• Prismatic and Non-prismatic ChannelsA channel in which the cross-sectional shape and sizeand also the bottom slope are constant is termed as aprismatic channel. Most of the man-made (artificial)channels are prismatic channels over long stretches.The rectangle, trapezoid, triangle and circle are some ofthe commonly-used shapes in man-made channels. Allnatural channels generally have varying cross-sectionsand consequently are non-prismatic.
• Rigid and Mobile Boundary ChannelsOn the basis of the nature of the boundary openchannels can be broadly classified into two types: (i) rigidchannels and (ii) mobile boundary channels.
Definition of Geometric Elements
R = hydraulic radius = A/P
D = hydraulic depth = A/T
y = depth of flow
d = depth of flow section
T = top width
P = wetted perimeter
A = flow area
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh Common Geometric Properties Cot α = z/1
α
z
1
Classification of Flows• Steady and Unsteady
– Steady: velocity at a given point does not change with time
• Uniform, Gradually Varied, and Rapidly Varied
– Uniform: velocity at a given time does not change within a given length of a channel
– Gradually varied: gradual changes in velocity with distance
• Laminar and Turbulent
– Laminar: flow appears to be as a movement of thin layers on top of each other (if Re < 500)
– Turbulent: packets of liquid move in irregular paths ((if Re >1000)
(Temporal)
(Spatial)
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• Based on the relative importance of inertia and gravity forces, flow may be critical, sub-critical and super-critical flow:
– Fr = 1 for critical flow
– Fr < 1 for subcritical flow
– Fr > 1 for supercritical flow
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Flow Classification
• Uniform (normal) flow: Depth is constant at every section along length of channel
• Nonuniform (varied) flow: Depth changes along channel
– Rapidly-varied flow: Depth changes suddenly
– Gradually-varied flow: Depth changes gradually
10/12/2013
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CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
CLASSIFICATION OF FLOWS
• Steady and Unsteady Flows
A steady flows occurs when the flow properties, such asthe depth or discharge at a section do not change withtime. As a corollary, if the depth or discharge changeswith time the flow is termed unsteady.
• Flood flows in rivers and rapidly-varying surges in canalsare some example of unsteady flows. Unsteady flowsare considerably more difficult to analysis than steadyflows.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• Uniform and non-uniform Flows
• If the flow properties, say the depth of flow, in an openchannel remain constant along the length of channel, theflow is said to be uniform. As a corollary of this, a flow inwhich the flow properties vary along the channel istermed as non-uniform flow or varied flow.
• A prismatic channel carrying a certain discharge with aconstant velocity is an example of uniform flow (Fig. (a)).
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• Flow in a non-prismatic channel and flow with varyingvelocities in a prismatic channel are examples of variedflow. Varied flow can be either steady or unsteady.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• Gradually-varied and Rapidly –varied Flows
• If the change of depth in a varied flow is gradual so thatthe curvature of streamlines is not excessive, such a flowis said to be a gradually –varied flow (GVF). Thepassage of a flood wave in a flood wave in a river is acase of unsteady GVF (Fig. (b)).
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• A hydraulic jump occurring below a spillway or a sluicegate is an example of steady RVF. A surge , moving upa canal (Fig. (c)) and a bore traveling up a river areexamples of unsteady RVF.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• Spatially-varied flow
• Varied flow classified as GVF and RVF assumes that noflow is externally added to or taken out of the canalsystem. The volume of water in a known time interval isconserved in the channel system. In steady-varied flowthe discharge is constant at all sections. However, ifsome flow is added to or abstracted from the system theresulting varied flow is known as a spatially varied flow(SVF).
• SVF can be steady or unsteady. In the steady SVF thedischarge while being steady-varies along the channellength. The flow over a bottom rack is an example ofsteady SVF (Fig (d)).
10/12/2013
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CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• The production of surface runoff due to rainfall, known as overland flows, is a typical example unsteady SVF.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• Classification Thus open channel flows are classified for purposes of identification and analysis.
• Fig. 1.1(a) through (d) shows some typical examples of the above types of flows
21
Classification of Channel Flow: An Example
CE F312: Flow in Open Channel
by Dr. Ajit Pratap Singh
22
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
State of Flow
• Flow in open channels is affected by viscous and gravitational effects
• Viscous effects described by Reynolds number, Re = VR/ν
• Gravitational effects described by Froude number, Fr = V/(gD)1/2
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Viscous Effects
• For Re < 500, viscous forces dominate and flow is laminar
• For Re > 2000, viscous forces are weak and flow is turbulent
• For Re between 500 and 2000, there is a transition between laminar and turbulent flow
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CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Gravitational Effects
• Critical flow is the point where velocity is equal to the speed of a wave in the water
• For F = 1, flow is critical
• For F < 1, flow is subcritical
– Wave can move upstream
• For F > 1, flow is supercritical
– Wave cannot move upstream
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Equations of Motion
• There are three general principles used in solving problems of flow in open channels:
– Continuity (conservation of mass)
– Energy
– Momentum
• For problems involving steady uniform flow, continuity and energy principles are sufficient
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Conservation of Mass
• Since water is essentially incompressible, conservation of mass (continuity) reduces to the following: discharge in = discharge out
• Stated in terms of velocity and area:
Q = V1A1 = V2A2
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Velocity ProfilesVelocity Profiles
Water SurfaceWater Surface
Maximum VelocityMaximum Velocity
Maximum VelocityMaximum Velocity
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Open Channel Flow: Uniform Flow
• Uniform Flow
– Discharge-Depth relationships
• Channel transitions
– Control structures (sluice gates, weirs…)
– Rapid changes in bottom elevation or cross section
• Critical, Subcritical and Supercritical Flow
• Flow Mesurements
normal depth
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Normal depth is function of flow rate, and
geometry and slope. One usually solves for normal
depth or width given flow rate and slope information
B
b
10/12/2013
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CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Normal depth implies that flow rate, velocity, depth,
bottom slope, area, top width, and roughness remain
constant within a prismatic channel as shown below
Q = C
V = C
y = C
S0 = C
A = C
B = C
n = C
UNIFORM FLOW
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• Given a long channel of constant slope and cross section find the relationship between discharge and depth
• Assume
– Steady Uniform Flow - ___ _____________
– prismatic channel (no change in _________ with distance)
• Use Energy, Momentum, Empirical or Dimensional Analysis?
• What controls depth given a discharge?
• Why doesn’t the flow accelerate?
Open Channel Flow: Discharge/Depth Relationship
P
no acceleration geometry
Force balance
A
l
dhl
40
γτ −=
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Open Conduits:Dimensional Analysis
• Geometric parameters
• Does Fr affect shear?
P
AR =Hydraulic radius (Rh)
Channel length (l)
Roughness (k)
No!gy
VFr =
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• If the suitable form of the Reynolds no. can be found, theresults obtained in pipe analysis can be transformed forChannel Analysis.
• Characteristic length dimension in pipe case : Diameter
• Equivalent Characteristic length dimension for channels:Hydraulic radius
• Re for channel = ρ V R/µ
• For a pipe flowing full, R = D/4
• Re for channel = Re for pipe/4
• Therefore for laminar channel flow Re < 500 and forturbulent channel flow Re > 1000
• Note that upper limit of Re is not so well defined forchannels and is normally taken to be 2000.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• For pipe case, Darcy-Weisbach formula forpipe friction wasintroduced and Moody’sdiagram is developedaccordingly.
• A similar diagram forchannels may beobtained
(I) V
8gRSf
4R2g
fVS
L
h
4R2g
fLVh
pipefor 2gD
fLVh
2
0
2
0f
2
f
2
f
=⇒
×==⇒
×=
=
CE F312: Flow in Open Channel
by Dr. Ajit Pratap Singh
Uniform Flow
• Equations are developed for steady-state conditions
– Depth, discharge, area, velocity all constant along channel length
• Rarely occurs in natural channels (even for constant geometry) since it implies a perfect balance of all forces
• Two general equations in use: Chezy and Manning formulas
10/12/2013
7
Reach of an Open-channel flow
CE F312: Flow in Open Channel
by Dr. Ajit Pratap Singh
37
Steady-Uniform Flow: Force Balance
Shear force =________
0sin =∆−∆ xPxwA oτθ
sin θ P
Awτ o =
R = P
A
θθ
θsin
cos
sin≅=S
θ
W
θ
W sin θ
∆x
a
b
c
d
Shear force
Energy grade line
Hydraulic grade line
W cos θ
g
V
2
2
Wetted perimeter = __
Gravitational force = ________
Hydraulic radius
Relationship between shear and velocity? ___________
τoP ∆ x
P
ρgA ∆x sinθ
Turbulence
(II) wRSτo =
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
f
8g
fρ
8w C where,
RSCV
RSfρ
8wV
wRSρV8
fτ
ρV8
fτ
2
o
2
o
=×
=
=⇒
×=⇒
==⇒
=
CE F371: Flow in Open Channel
by Dr. Ajit Pratap Singh
• Let a state of rough turbulent flow
(IV) t coefficien sChezy' is C where
RSCV RSf
ρgV
(III) and (II) From
×=⇒×=
το ∝ V2 or τo = KV2 (III)
11/2T L ofdimension with the
f
8gC Thus
(IV) and (I) From
−
=
CE F312: Flow in Open Channel
by Dr. Ajit Pratap Singh
Chezy Equation (Summary)• Introduced by the French engineer Antoine Chezy in
1768 while designing a canal for the water-supply system of Paris
• Balances force due to weight of water in direction of flow with opposing shear force
• Note: V is mean velocity, R is hydraulic radius(area/wetted perimeter), S is the slope of energygradeline, and C is the Chezy coefficient
• Chezy’s coefficient C is a depends on Reynolds no.and boundary roughness
V = C RS
CE F312: Flow in Open Channel
by Dr. Ajit Pratap Singh
Manning Equation
• Similarly, the Manning equation which is anempirical relationship can be expressed as:
• Note: V is mean velocity (m/sec), R ishydraulic radius (m), S is the slope ofthe energy gradeline (m/m), and n is theManning roughness coefficient
2/13/21SR
nV =
10/12/2013
8
CE F312: Flow in Open Channel
by Dr. Ajit Pratap Singh
Manning Equation (1891)
• Similarly, Manning’s equation in SI (metric) units
n Manning' isn where
SRn
1V 1/22/3=
1/6R
n
1C and =
Ganguillet-Kutter Formula
• Where n is roughness coefficient known as Kutter’s ‘n’. This is different from Manning’s n
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
R
n
S
0.00155231
n
1
S
0.0015523
C
++
++
=
CE F312: Flow in Open Channel
by Dr. Ajit Pratap Singh
Manning’s Roughness (n)
• Roughness coefficient (n) is a function of:
– Channel material
– Surface irregularities
– Variation in shape
– Vegetation
– Flow conditions
– Channel obstructions
– Degree of meandering
Manning’s n
Lined Canals n
Cement plaster 0.011
Untreated gunite 0.016
Wood, planed 0.012
Wood, unplaned 0.013
Concrete, trowled 0.012
Concrete, wood forms, unfinished 0.015
Rubble in cement 0.020
Asphalt, smooth 0.013
Asphalt, rough 0.016
Natural Channels
Gravel beds, straight 0.025
Gravel beds plus large boulders 0.040
Earth, straight, with some grass 0.026
Earth, winding, no vegetation 0.030
Earth , winding with vegetation 0.050
n = f(surface roughness, channel irregularity, stage...)
d = median size of bed material
min is d if 038.0
ftin is d if 0.031dn
6/1
1/6
dn =
=
CE F312: Flow in Open Channel
by Dr. Ajit Pratap Singh
+−=
−
fRe
0.6275
14.8R
k2log
f
1
channelsfor formula equivalent WhiteColebrook
s
+−=
=
0
s0
2
0
8gRSRe
0.6275V
14.8R
klog8gRS2V
equations above Combining
V
8gRS f But,
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• f-Re diagram channels may be derived using above equation. However,
• The application of these equations to a particular channel more complex than the pipe case due to extra variables involved say R changes with depth and channel shape.
• Validity is questionable because of the presence of free surface has considerable effect on the velocity distributions.
10/12/2013
9
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Discharge using Manning’s Equation
• Using continuity equation (Q=VA), Manning’s equation for English units can be written as
• And for metric units
Q =1.486
nAR
2 / 3S
1/ 2
Q =1
nAR
2 / 3S
1/ 2
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Conveyance
• For uniform flow, A, R and n are constant thus
Q = KS1 / 2
3/21AR
nK =
• The term K is conveyance, given as
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Normal Section Factor
• Based on Manning’s equation, define a normal section factor for the portion dependent on geometry, namely
Z n = AR2 / 3
• For simple geometries, Zn can be written as a function of yn
• For complex geometries, use tabulated values
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Computing Normal Flow
Three cases to consider:
�Find Q for known values of yn and S
�Find S for known values of Q and yn
�Find yn for known values of Q and S
• From Manning’s equation, see that
2/11SZ
nQ n=
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Computing Normal Flow
2/11SZ
nQ n=
• For known values of yn and S, can use Manning’s equation directly
• Based on relationship between yn and Zn, determine the value of Zn
• Compute discharge from Manning’s equation, namely
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Computing Normal Slope
2
1
=
n
Z
QS
n
• For known values of yn and Q, can use Manning’s equation directly
• Based on relationship between yn and Zn, determine the value of Zn
• Rearrange Manning’s equation, and compute slope as
10/12/2013
10
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Computing Normal Depth
• For known values of Q and S, can compute Zn as
2/1S
nQZn =
• Normal depth (yn) is then determined from the relationship between yn and Zn
Note
• For each regular channel section, we can construct the curves R versus y and AR2/3
versus y:
(a) For example Circular with diameter of 2.0 m
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
AR2/3 versus y Curve
• We can construct the curves R versus y and AR2/3
versus y for Trapezoidal, with bottom width = 3.0m and side slopes H:V 2.5
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Channel Design
• Previously, we have assumed that channel dimensions and slope are known
• In design situations, usually discharge is and need to determine geometry and slope
• In an open channel, usually have allowable velocity range
– Minimum of 2-3 ft/s to move grit
– Maximum velocity to prevent scouring
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Channel Design
• For circular pipes, can determine minimum slope needed to achieve a minimum velocity of 2 ft/s
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Channel Design
• From basic principles, we know that:
– For a given slope and roughness, Q increases with increase of section factor (Zn=AR2/3)
– For a given area, Zn is maximum for a minimum wetted perimeter (P)
• Based on these principles, we can develop guidelines for efficient hydraulic sections for given channel shapes
10/12/2013
11
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Design Procedure
• For a specified Q, select S and estimate n Compute section factor from Q, n and S
• Select a shape, and express section factor in terms of y; solve for y
• Try different values of S, n and shape and compute cost for each
• Verify that velocity exceeds 2-3 ft/s
• Add freeboard if open section
Rectangular Channel
CE F312 Flow in Open Channel by Dr. Ajit Pratap Singh
2y Bor 2
202y
AdP
0dy
dP condition, P minimumFor
area, sectional cross-given x aFor
22
y
APB2y PBut
y
ABBy A
2
2
2
==⇒
==⇒=+−=⇒
=
⇒
+=+=⇒+=
=⇒=
By
ByyAdy
y
yAy
Rectangular Channel…
• Rectangular channel section will be mosteconomical when either the depth of flow isequal to half of bottom width of the channel orhydraulic radius al to the half of the depth offlow.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
24
22
2/
2
2
2
R Radius, Hydraulic
Similarly,
2
yB
B
B
B
BB
yy
A
By
P
A==
×+
×
=
+
==
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Trapezoidal Channel
• Derive P = f(y) and A = f(y) for a trapezoidal channel
• How would you obtain y = f(Q)?
z1
B
y( )
zyy
Az12yP
Bz12y PBut
zyy
AByzyBA
y2
2zyBB A
2
2
−++=⇒
++=
−=⇒+=
++=
2/13/21
oh SAR
n
Q =
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
A being constant, differentiating P with respect to y
( )( )
( )
2
y
P
A R and
z12y 2zyB
z12zy
y zyBy zyBABut
z12zy
A
0zy
Az12
dy
dP
2
2
2
2
2
2
2
==
+=+
+=++
⇒+=
+=+
=−−+=
Problem
• A channel of trapezoidal section has to convey a discharge of 7 m3/sec of water at a velocity of 1.75 m/sec. The sides of the channel have a slope of 1 vertical to 2 horizontal. Find the sectional dimensions of the most economical channel. Also find the slope of the channel.
• For conveying the same discharge if a rectangular channel 1.25 m wide and 3.6m deep is provided what would be the saving in power in km length of the channel? Take C = 60.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
10/12/2013
12
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Flow in Round Conduits
• Circular sewers are very easily manufactured• A circular section gives the maximum area for a given
perimeter and thus gives greatest H.M.D. when runningfull or half full.
• Most efficient section at these flow conditions• The circular section has uniform curvature all round and
hence it offers less opportunities for deposits• Minimum grades are enough so long as circular sewer
flow more than half full• Both the wetted area as well as the wetted perimeter
varies with the depth of flow. Thus the condition of areaof flow section being constant can not applied all thetime.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
(i) Condition for maximum discharge
( )sinθ-θ2
r
2
2
θrcos
2
θ2rsin
θ2π
πr
OAB ∆ of AreaOADBO arc of AreaA Area Wetted
2
2
=
×
−=
−=
rθ PPerimeter Wetted =
rθ PPerimeter Wetted =
r
C
y
T
A θ
O
D
B
0P
A
dθ
d
S and Cgiven for P
A when maximum be willDischarge
SP
ACRSACAVQ
3
0
3
0
3
0
=
⇒
===
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
( ) ( )( )
o
22
2
32
308θ
0sinθ3.cosθ-2θ
0rsinθθ2
rcosθ1
2
r3rθ
0dθ
dPA
dθ
dAP.3
P
dθ
dPA
dθ
dAP.3A
=
=+
=−−
−
=
−
=
−
⇒
( )
( )
0.5733r0.29DR
308θ and sinθθ2θ
r
P
AR
,Similarily
diameter is d whereD 0.95 y or
1.8988rcos261ry
2
308-360rcosry
o
==⇒
=−==
=
=+=⇒
+=
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Circular Sections
• For partially full circular sections (with diameter do), the geometric elements are a function of depth (y)
• Values have been tabulated in non-dimensional form using do as a scaling parameter
• Note that maximum flow occurs at a depth of 0.94do
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Geometric Elements for Circular Pipes
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Efficient Hydraulic Sections
Section Most efficient
Trapezoidal Base < depth
Rectangular Width = 2 x depth
Triangular No specific relationship
Circular Semicircle if open Circle if closed
10/12/2013
13
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh Optimal Channels - Max R and Min P CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• For other most economical sections, refer class notes and text book.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Design Procedure
• For a specified Q, select S and estimate n Compute section factor from Q, n and S
• Select a shape, and express section factor in terms of y; solve for y
• Try different values of S, n and shape and compute cost for each
• Verify that velocity exceeds 2-3 ft/s
• Add freeboard if open section
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Computation of uniform flow
section. channel theofcapacity
carrying of measure is andsection channel
theof conveyance is and RACK where
SRSACAVQ
=
=== K
2/3
1/22/3
ARn
1K where
SARn
1AVQ
OR
=
==
S
QnAR OR
ARK Conveyance
2/3
2/3
=
=⇒=∴ KnS
Q
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• L.H.S. depends only on the geometry of the wetted area
• Note that for a given value of n, Q and S, there is only one possible depth of flow at which the uniform flow will be maintained in any channel section. This depth is known as normal depth (yn)
• Similarly, for a given value of n, S and the depth of flow, there is only one possible discharge for maintaining a uniform flow through any channel section. This discharge is known as normal discharge (Qn)
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Momentum and Energy Equations
• Conservation of Energy
– “losses” due to conversion of turbulence to heat
– useful when energy losses are known or small
• ____________
– Must account for losses if applied over long distances
• _____________________________
• Conservation of Momentum
– “losses” due to shear at the boundaries
– useful when energy losses are unknown
• ____________
Contractions
Expansion
We need an equation for losses
10/12/2013
14
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Open Channel Flow: Energy Relations
2g
V2
11α
2g
V2
22
α
xSo∆
2y
1y
x∆
L fh S x= D
______
grade line
_______
grade line
velocity head
Bottom slope (So) not necessarily equal to EGL slope (Sf)
hydraulic
energy
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Control Volume for Open Channels
• Conservation of energy applied to control volume results in the following:
L
2
2222
2
1111 h
2g
Vαyz
2g
Vαyz +++=++
where z1,z2 are elevations of the bed,
y1, y2 are depths of flow,
V1, V2 are velocities,
α1, α2 are kinetic energy corrections, and
hL is the frictional loss.
Energy Relationships: Conservation of Energy
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Energy Relationships: Conservation of Energy
2 2
1 1 2 21 1 2 2
2 2L
p V p Vz z h
g ga a
g g+ + = + + +
2 2
1 21 2
2 2o f
V Vy S x y S x
g g+ D + = + + D
Turbulent flow (α ≅ 1)
z - measured from
horizontal datum
y - depth of flow
Pipe flow
Energy Equation for Open Channel Flow2 2
1 21 2
2 2o f
V Vy S x y S x
g g+ + D = + + D
From diagram on previous slide...
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Energy Coefficient
• The term associated with each velocityhead (α) is the energy coefficient
• This term is needed because we are usingthe average velocity over the depth tocompute the total kinetic energy
• Integrating the cubed incrementalvelocities is not equal to the cube of theintegrated incremental velocities
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Specific Energy
• Specific energy is defined as the energy in a channel section measured w.r.t. channel bed
• If slope of channel is relatively flat, we can let z1 = z2 = 0
• For energy coefficient (α) = 1, specific energy (E) is thus
g
VyE
2
2
+=
10/12/2013
15
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Specific Energy… Note
The syllabus of Tutorial III will be up to this Portion. Next Classes, I’ll discuss about critical flow equations (for prismatic channels) and applications of specific energy diagrams and discharge diagrams.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Significance of Specific Energy…
• The sum of the depth of flow and the velocity head is the specific energy:
g
VyE
2
2
+=
If channel bottom is horizontal and no head loss
21EE =
y - _______ energy
g
V
2
2
- _______ energy
For a change in bottom elevation
1 2E y E- D =
xSExSE o ∆+=∆+ f21
y
potential
kinetic
+ pressure
P
A
Critical Flow
T
dy
y
T=surface width
Find critical depth, yc
2
2
2gA
QyE +=
0=dy
dE
dA =0dE
dy= =
3
2
1
c
c
gA
TQ=
Arbitrary cross-section
A=f(y)
2
3
2
FrgA
TQ=
22
FrgA
TV=
dA
AD
T= Hydraulic Depth
2
31
Q dA
gA dy-
0
1
2
3
4
0 1 2 3 4
E
y
yc
Tdy
More general definition of Fr
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
( )
1Fr1gD
V1
gD
V
T
AD D
g
V
T
A
g
VA
T
A
g
Q
T
ADDepth hydraulc and AVQ
2
2
3232
=⇒=⇒=∴
==∴
=⇒=
==
Q
T
A
g
Q
satisfy must surfaceat water T width topandA area sec- xthe
Q), dischargegiven a(for flow of state criticalin is flow theif Thus
32
=
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Critical Depth
• For higher values of specific energy there are two values that give the same specific energy
• If Q is fixed, for the smaller depth there will be a higher velocity (supercritical) and for the larger depth there will be a lower velocity (subcritical)
10/12/2013
16
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Critical depth is used to characterize channel flows --
based on addressing specific energy E = y + v2/2g :
E = y + QE = y + QE = y + QE = y + Q2222/2gA/2gA/2gA/2gA2 2 2 2 where where where where Q/A = q/y and q = Q/bQ/A = q/y and q = Q/bQ/A = q/y and q = Q/bQ/A = q/y and q = Q/b
Take Take Take Take dE/dy = (1 dE/dy = (1 dE/dy = (1 dE/dy = (1 –––– qqqq2222/gy/gy/gy/gy3333) and set ) and set ) and set ) and set = 0= 0= 0= 0.... qqqq = = = = constconstconstconst
E = y + qE = y + qE = y + qE = y + q2222/2gy/2gy/2gy/2gy2222
y
E
Min E Condition, q = C
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Solving Solving Solving Solving dE/dy = (1 dE/dy = (1 dE/dy = (1 dE/dy = (1 –––– qqqq2222/gy/gy/gy/gy3333) and set ) and set ) and set ) and set = 0= 0= 0= 0....
For a rectangular channel bottom width b,
1.1.1.1. EEEEminminminmin = 3/2Y= 3/2Y= 3/2Y= 3/2Ycccc for critical depth for critical depth for critical depth for critical depth y = yy = yy = yy = ycccc
2.2.2.2. yyyycccc/2 = V/2 = V/2 = V/2 = Vcccc2222/2g/2g/2g/2g
3. yc = (Q2/gb2)1/3
Froude No. = v/(gy)1/2
We use the Froude No. to characterize critical flows
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Y vs E E = y + qE = y + qE = y + qE = y + q2222/2gy/2gy/2gy/2gy2222
q = constq = constq = constq = const
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
In general for any channel shape, B = top width
(Q(Q(Q(Q2222/g) = (A/g) = (A/g) = (A/g) = (A3333/B) /B) /B) /B) at y = yat y = yat y = yat y = ycccc
Finally Fr = v/(gy)1/2 = Froude No.
Fr = 1 for critical flowFr = 1 for critical flowFr = 1 for critical flowFr = 1 for critical flowFr < 1 for subcritical flowFr < 1 for subcritical flowFr < 1 for subcritical flowFr < 1 for subcritical flowFr > 1 for supercritical flowFr > 1 for supercritical flowFr > 1 for supercritical flowFr > 1 for supercritical flow
Critical Flow in Open Channels
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Critical Flow & its Computation• When the depth of flow of water
over a certain reach of a givenchannel is equal to the criticaldepth yc, the flow is described ascritical flow
• Zc is a section factor for thechannel.
• For a prismatic channel sectionfactor is a function of depth of flow
• When Discharge is given, fromabove equation, Zc can beobtained. Z is f(y) and a curvebetween Z and y can be obtainedand with the help of this curve,corresponding to the known valueof Z=Zc, the value of depth of flowcan be obtained which will becritical depth yc
c
c
Zg
Q
T
AA ==
When the depth of flow isgiven, from above equation,corresponding value of thesection factor Z can beobtained from the same andfrom this value of Z, thecritical discharge Qc can beobtained Qc =Z g
10/12/2013
17
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Critical Flow…
• From necessary condition for minimum specific energy, can define section factor for critical flow (Zc):
g
QDAZc ==
where Zc is the critical section factor,
A is the area of flow,
D is the hydraulic mean depth, and
Q is the discharge
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Computing Critical Flow
• If yc is known, can compute Zc for specific geometry. Then critical discharge is given as
gZQ cc =
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Computing Critical Depth
• If Q is known, can compute Zc as
• For simple geometries, Zc can be written as a function of yc For complex geometries, use tabulated values
g
QZc =
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Characteristics of Critical State of Flow
• Specific energy is a minimum for givendischarge
• Discharge is a maximum for a given specificenergy
• Velocity head is equal to the half of the hydraulichead in a channel of small slope
• Froude no. is equal to unity
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Critical Flow: Rectangular channel
yc
T
Ac
(I) gA
TQ1
3
c
c
2
=
3
cgyqor =
Only for rectangular channels!
cTTB ==
Given the depth we can find the flow!
Let q be the discharge per unit width of the channel section i.e. Q = Bq
Corresponding to a critical depth of flow yc, the area of rectangular channel section i.e. Ac = byc
Substituting the value of Q and A in eq. (I)
T
yBBq3
c
322
=g
1/32
c
3
c
2
g
q yOr
yg
q
=
=⇒
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
( )( )
( ) (IV) yy-E2gqOr
(III) 2gy
qyE
By2g
qBy
2gA
QyE
2
2
2
2
2
2
2
=
+=⇒
+=+=
For a given specific energy E, the condition for discharge per unit
width q to be maximum may be obtained by
( )( )
(V) y 2
3E
yyE2
3y2ye2g0
dy
dq
2
2
=⇒
−
−==
Substituting the value of E for maximum discharge in eq. (IV)
g
q2
3
32
y
gyyy-y2
32gq
=⇒
=
=
3 roots (one is negative)
How many possible depths given a
specific energy? _____2
10/12/2013
18
Critical Flow Relationships:Rectangular Channels
3/12
=
g
qyc cc yVq =
=
g
yVy
cc
c
22
3
g
Vy
c
c
2
=
1=gy
V
c
cFroude number
velocity head =
because
g
Vy cc
22
2
=
2
c
c
yyE += Eyc
3
2=
forcegravity
forceinertial
0.5 (depth)
g
VyE
2
2
+=
Kinetic energy
Potential energy
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Applications
• Use of sp. Energy and discharge diagrams areused to analyze channel section in the case oftransition
• A portion of channel with varying x-section whichconnects one uniform channel to another whichmay or may not have same x-sectional area.These transitions may be either sudden orgradual transition
• Variation of channel may be caused either byreducing or increasing the width or by raising orlowering the bottom of the channel.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Applications …
• The transitions are being used for metering flow.The main devices used here are Venturi flume,Parshall flume, standing wave flume etc.
• The transitions are being used for dissipation ofenergy by providing a sudden drop in the channelbottom or by providing gradual expansion orcontraction with well rounded corners.
• The transitions are being used for reduction invelocities in irrigation channels in order to preventscouring velocities or increase in velocities in orderto prevent shoaling for navigation channels.
• Often channel x-section is reduced, if a bridge hasto be constructed across it.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Reduction in x-section: An Example
• By decrease in width
• By decrease in depth or raised bottom
• Or by combination of both
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Case I: Decrease in width
• (a) If width at u/s is B1 and at transition it is B2 which isbasically reduced from B1 to B2 but it is more than Bc
• (b) If width at u/s is B1 and at transition it is B2 which isbasically reduced from B1 to B2 but it is equal to Bc
• (c) If width at u/s is B1 and at transition it is B2 which isbasically reduced from B1 to B2 but it is less than Bc
• Note that analysis of above cases depends upon thetype of flow whether it is sub-critical, critical orsupercritical
• For the analysis in these cases, mainly dischargediagram is used.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Case I (a)
(1) For given data, calculate specificenergy at the u/s
( )222
2
221
2
yB2g
QyEE
equation usingerror & by trial y Calculate (2)
+==
10/12/2013
19
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Case I (b)
(1) For given data, calculate specific energy
at the u/s
( )3/21
c
c
Eg
Q84.1B
equation using B width critical Calculate (2)
=
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Case II: Decrease in depth
• (a) If depth at u/s is y1 and at transition it is y2 which is basically reduced from y1 to y2 but it is more than yc
• (b) If width at u/s is y1 and at transition it is y2 which is basically reduced from y1 to y2 which is equal to yc
• (c) If width at u/s is y1 and at transition it is y2 which is basically reduced from y1 to y2 but it is less than yc
• Note that analysis of above cases depends upon thetype of flow whether it is subcritical, critical orsupercritical.
• In the analysis mainly specific energy diagram is used.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Principle
2
2
1
2
22
++=
++
g
Vyz
g
Vyz
2
22
22 gy
qy
g
VyHE +=+==
zHH ∆=− 21
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
(1) In a rectangular channel 3.5 m wide,laid at a slope of 0.0036, uniform flowoccurs at a depth of 2.0 m. Find howhigh can a hump be raised on thechannel bed without causing a changein the upstream depth. If the upstreamdepth is to be raised to 2.4 m whatshould be the height of the hump?Assume Manning’s n = 0.015.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Problems
(2) Water flows in a 4.0 m wide rectangularchannel at a depth of 2.50m and a velocityof 2.25 m/sec. If the width of the channel isreduced to 2.50 m and bed of channel israised by 0.20 m at a section downstreamside, will the water surface in the channelupstream be disturbed? Why? Alsodetermine the depth of flow over the raisedbed if it is disturbed and show how will thelevel of water surface in the channel beaffected?
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
(3) Water flows in a rectangular channel 3 m wide at avelocity of 3 m/s at a depth of 3 m. There is anupward step of 0.61 m. What expansion in widthmust take place simultaneously for this critical flowto be possible?
10/12/2013
20
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Problems
(4) Water flows in a 4.0 m wide rectangularchannel at a depth of 2.0m and avelocity of 1.5 m/sec. Determine (i) thewidth at contraction which just causescritical flow without a change in the u/sdepth (ii) the depth in the contractionwhen the width at the throat is 50%more than the above value.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Problems..
(5) Water flows in a 4.0 m wide rectangularchannel at a depth of 2.0m and avelocity of 1.5 m/sec. Determine (i) theheight of hump required to producecritical flow without affecting the u/sdepth (ii) the depth over the humpwhen the height of the hump is half theabove value.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
METERING FLUMES
• Critical depth of flow may be obtained at certain sections
in an open channel where the channel bottom is raised by
the construction of low hump or the channel is
constructed by reducing its width.
• Since at critical state of flow the relationship between the
depth of flow and discharge is definite and is independent
of the channel roughness and other uncontrollable
factors, provides a theoretical basis for the measurement
of discharge in open channels. As such various devices
which have been developed for the flow measurement are
based on principle of critical flow.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
FLUMES
Flume is a term applied to the devices in which theflow is locally accelerated due to a
• a stream lined lateral contraction in the channel
sides
• the combination of lateral contraction togetherwith a stream lined hump in the invert channel bed.
Flumes are usually designed to achieve criticallyflow in the narrowest section together with smallafflux. And are applicable where deposition ofsolids must be avoided.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
VENTURI FLUME:
• The venturi flume is a structure in a channel which
has a contracted section called throat,
downstream of which follows a flared transition
section designed to restore the stream to its
original width.
• It is an open channel counterpart of a Venturi-
meter, which is for measuring discharge in open
channels.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Plan of venturi flume:
Elevation of venturi flume:
10/12/2013
21
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• The velocity at the throat section is always
less than the critical velocity. Therefore the
discharge passing through it will be function
of the deference between the depth of the
flow upstream of the entrance section and at
the throat.
• At the throat section there will be a drop in
the water surface and this drop in the water
surface can be related to the discharge.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• Further since the velocity of the flow at the throat is
less than the critical velocity, standing wave or
Hydraulic jump will not be formed at any section in
the venturi flume.
• Let B, H, V be normal breadth, depth of the flow
and velocity at the entrance of the flume.
• Let b, h, v be the breadth, depth of the flow and
velocity at the throat.
• So, at the throat the velocity v>V
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Hence there will be drop in water level at the throat as
the total energy remains constant
Due to continuity of flow
Q = BHV = bhv
Let A = BH , a = bh => V = (a/A)v
By Bernoulli's equation
H + V2/2g = h + v2/2g
v2 – (a2/A2)V2 = 2g(H – h)
by rearranging we get
v2[(A2 – a2)/A2] = 2g(H – h)
Discharge Q = avCE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
In the actual case the discharge obtained is slightly less than the above value
due to the losses in the flume. So in order to get the actual discharge multiply
the above equation with coefficient of discharge Cd .
Although the float bed of venturi flume is simpler to construct , it is sometimes
necessary to raise the invert in the throat to attain critical conditions.
( )hH2g
aA
Aa Q Discharge Theretical
22−
−
=∴
( )
discharge oft coefficien theis C where
hH2g
aA
AaC Q Discharge Actual
d
22d −
−
=∴
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
STANDING WAVE FLUME OR CRITICAL DEPTH FLUME
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• It is a structure in a channel which has narrowed throat having a
raised floor (or hump) at the bottom as shown which act as broad
crest weir.
• The downstream of the throat section is followed by the flared
transition section to restore the stream to its original width.
• For any discharge flowing in the channel the velocity of the flow at
the throat of the flume is greater than critical velocity.
• As such a standing wave or hydraulic jump is formed at or near
the down stream end of raised floor.
• Since the velocity at the throat is greater than the Critical velocity.
The depth of flow at a section on upstream of the entrance of the
flume remains unaffected by variations in the down stream depth
until the downstream depth of submergence becomes greater
than about 0.7 of the up-stream.
10/12/2013
22
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
• Therefore as long as the downstream depth
of the flow is kept below the limiting value,
the discharge passing through a standing
wave flume will be a function of only the
depth of the flow ‘H’ (above the raised floor)
at a section well upstream of the entrance
section.
• This is similar to the Venturi-flume, except
that in this case a standing wave is formed.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Considering an upstream section and a section at the throat, if
the depths of flow at these sections are ‘H’ and ‘h’ respectively
( )hH2gv
2
vh
2
VH
1
1
22
−=∴
=+=+∴ Hgg
( )
( ) ( )1/2321
321
1
hhH2gbhhH2gbQ
hH2gbhbhv Q Discharge Theretical
−=−=
−==∴
For discharge to be maximum the quantity (H1h2 – h3)
should be maximum
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Thus we find that for the condition of maximum discharge
the depth of flow at the throat should be 2/3 the total
energy head. For this condition
( )
1
21
321
H3
2h
03hh2H
0hhHdh
d
=⇒
=−⇒
=−
( )
ghhh2
32gv
hH2gv 1
=
−=⇒
−=∴
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Which means for the condition of maximum
discharge the depth of flow at throat is equal to
critical depth.
substituting h = (2/3) H1 in discharge expression
Q = 1.705 bH11.5
=> Q = 1.705b(H + V2/2g)1.5
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Problem
• A rectangular channel is 1.20 m wide and itswidth is narrowed to 0.60 m to form a throatregion of venturiflume. The depths of flow in thetwo sections being 0.6m and 0.55m. Neglectinglosses, determine the discharge through thechannel assuming its bed to be horizontal.
If a hump of 0.18 m height is provided at thethroat region so as to produce a standing waveon the downstream side, find the depth of flowon the upstream side for the same discharge.
Problem 2
• A 1.0 m wide rectangular section channel is considerded to a depth of 0.5m, the depth of water ahead of the flume being 0.4. If a hump of 0.15m height is place at the throat so that standing wave flume occurs on d/s, determine the discharge through the channel assuming its bed to be horizontal and cd=0.95. apply correction for velocity of approach.
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
10/12/2013
23
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Metering of Stream flow
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Flumes• Flumes include various
specially shaped and stabilized channel sections that are used to measure flow.
• Use of flumes is similar to use of weirs in that flow is related to flow depths at specific points along the flume.
Parshall FlumeCE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Measuring flow using orifice plates
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Weirs
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Weir Shapes and FormulasWeir Shapes and Formulas
10/12/2013
24
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Open Ditches Flumes
Flume Upstream Measure
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
RCB Flume with
Pressure Transducer
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
RCB Flume in Drain Ditch
Notice Drop
CE F312 Flow in Open Channel
by Dr. Ajit Pratap Singh
Furrow Irrigation