open channel flow may 14, 2015 . hydraulic radius steady-uniform flow: force balance w w sin xx a...
TRANSCRIPT
Hydraulic radiusHydraulic radius
Steady-Uniform Flow: Force Balance
W
W sin
x
a
b
c
d
Shear force
Energy grade line
Hydraulic grade line
Shear force =________
0sin xPxA o 0sin xPxA o
sin P
Ao sin
P
Ao
hR = P
AhR =
P
A
sin
cos
sin S
sin
cos
sin S
W cos
g
V
2
2
g
V
2
2
Wetted perimeter = __
Gravitational force = ________
oP x
P
x sin
Dimensional analysisDimensional analysisRelationship between shear and velocity? ___________________
Geometric parameters ___________________ ___________________ ___________________
Write the functional relationship
Geometric parameters ___________________ ___________________ ___________________
Write the functional relationship
P
ARh
P
ARh Hydraulic radius (Rh)Hydraulic radius (Rh)
Channel length (l)Channel length (l)
Roughness ()Roughness ()
Open Conduits:Dimensional Analysis
Open Conduits:Dimensional Analysis
Re, , ,ph h
lC f
R Reæ ö
= ç ÷è øF,M,WRe, , ,p
h h
lC f
R Reæ ö
= ç ÷è øF,M,W
glVFglVF
ReVlrm
=
Pressure Coefficient for Open Channel Flow?
Pressure Coefficient for Open Channel Flow?
2
2C
V
pp
2
2C
V
pp
2
2C
V
ghlhl
2
2C
V
ghlhl
2
f2C
f
V
lgSS
2
f2C
f
V
lgSS
lShl f lShl f
lhp lhp Pressure CoefficientPressure Coefficient
Head loss coefficientHead loss coefficient
Friction slope coefficientFriction slope coefficient
(Energy Loss Coefficient)(Energy Loss Coefficient)
Friction slopeFriction slope
The friction slope is the slope of the EGL. The friction slope is The friction slope is the slope of the EGL. The friction slope is the same as the bottom slope (the same as the bottom slope (SSoo) for steady, uniform flow.) for steady, uniform flow.
Dimensional AnalysisDimensional Analysis
f, ,ReS
h h
lC f
R Reæ ö
= ç ÷è øf, ,ReS
h h
lC f
R Reæ ö
= ç ÷è ø
l
RC hSf
l
RC hSf
l
R
V
lgS h
2
f2 l
R
V
lgS h
2
f2
hRgS
V f2
hRgSV f2 hRS
gV f
2
hRS
gV f
2
f,ReS
h h
lC f
R Reæ ö
= ç ÷è øf,ReS
h h
lC f
R Reæ ö
= ç ÷è øHead loss length of channelHead loss length of channel
f,Reh
Sh
RC f
l Re
læ ö
= =ç ÷è øf,Reh
Sh
RC f
l Re
læ ö
= =ç ÷è ø
2
f2f
V
lgSCS
2
f2f
V
lgSCS
(like f in Darcy-Weisbach)
2
f 2lh
l Vh S l
R gl= =
2
f 2lh
l Vh S l
R gl= =
Chezy formula
Open Channel Flow FormulasOpen Channel Flow Formulas
VAQ VAQ
2/13/21oh SAR
nQ 2/13/21
oh SARn
Q
1/2o
2/3h SR
1n
V 1/2o
2/3h SR
1n
V
hSRg
V
2 hSRg
V
2
Dimensions of n?
Is n only a function of roughness?
hSRCV hSRCV
Manning formula (MKS units!)
NO!
T /L1/3
Manning FormulaManning Formula
The Manning n is a function of the boundary roughness as well as other geometric parameters in some unknown way... ____________________ _______________________________
Hydraulic radius for wide channelsHydraulic radius for wide channels
The Manning n is a function of the boundary roughness as well as other geometric parameters in some unknown way... ____________________ _______________________________
Hydraulic radius for wide channelsHydraulic radius for wide channels
1/2o
2/3h SR
1n
V 1/2o
2/3h SR
1n
V
RAPh A bh
P b h 2
Rbhb hh 2
Channel curvature (bends)Cross section geometry
1 2
P1 < P2
Rh1 > Rh2
Why Use the Manning FormulaWhy Use the Manning Formula
Tradition Natural channels are geometrically complex and
the errors associated with using an equation that isn’t dimensionally correct are small compared with our inability to characterize stream geometry
Measurement errors for Q and h are large We only ever deal with water in channels, so we
don’t need to know how other fluids would respond
Tradition Natural channels are geometrically complex and
the errors associated with using an equation that isn’t dimensionally correct are small compared with our inability to characterize stream geometry
Measurement errors for Q and h are large We only ever deal with water in channels, so we
don’t need to know how other fluids would respond
Values of Manning nValues of Manning nLined Canals n
Cement plaster 0.011Untreated gunite 0.016Wood, planed 0.012Wood, unplaned 0.013Concrete, trowled 0.012Concrete, wood forms, unfinished 0.015Rubble in cement 0.020Asphalt, smooth 0.013Asphalt, rough 0.016
Natural ChannelsGravel beds, straight 0.025Gravel beds plus large boulders 0.040Earth, straight, with some grass 0.026Earth, winding, no vegetation 0.030Earth , winding with vegetation 0.050
Lined Canals nCement plaster 0.011Untreated gunite 0.016Wood, planed 0.012Wood, unplaned 0.013Concrete, trowled 0.012Concrete, wood forms, unfinished 0.015Rubble in cement 0.020Asphalt, smooth 0.013Asphalt, rough 0.016
Natural ChannelsGravel beds, straight 0.025Gravel beds plus large boulders 0.040Earth, straight, with some grass 0.026Earth, winding, no vegetation 0.030Earth , winding with vegetation 0.050
The worst channel has…
Roughness at many scales!
Example: Manning FormulaExample: Manning Formula
What is the flow capacity of a finished concrete channel that drops 1.2 m in 3 km?
What is the flow capacity of a finished concrete channel that drops 1.2 m in 3 km?
11
22
3 m3 m
1.5 m1.5 m
solution
Depth as f(Q)Depth as f(Q)
Find the depth in the channel when the flow is 5 m3/s
Hydraulic radius is function of depth Area is a function of depth Can’t solve explicitly Use trial and error or solver
Find the depth in the channel when the flow is 5 m3/s
Hydraulic radius is function of depth Area is a function of depth Can’t solve explicitly Use trial and error or solver
QnAR Sh o
1 2 3 1 2/ /
SummarySummary
Open channel flow equations can be obtained in a similar fashion to the Darcy-Weisbach equation (based on dimensional analysis)
The dimensionally incorrect Manning equation is the standard in English speaking countries
Open channel flow equations can be obtained in a similar fashion to the Darcy-Weisbach equation (based on dimensional analysis)
The dimensionally incorrect Manning equation is the standard in English speaking countries
Turbulent Flow Losses in Open Conduits
Turbulent Flow Losses in Open Conduits
Maximum shear stress
No shear stress
Hydraulic JumpHydraulic Jump
in in out outV y V y=in in out outV y V y=
yy11yy11
2 2
2 2in out
in out L
V Vy y h
g g+ = + +
2 2
2 2in out
in out L
V Vy y h
g g+ = + +
yy22
2 2
2 2in out
L
in out
V Vp py y h
g gg gæ ö æ ö
+ + = + + +ç ÷ ç ÷è ø è øEnergyEnergy
MassMass Per unit widthPer unit width
Unknown lossesUnknown losses
cs1cs1
cs2cs2
Hydraulic JumpHydraulic Jump
221122
212
1 ypypyVyV 221122
212
1 ypypyVyV
yy11yy11
yy22
22
22
21
22
212
1
yyyVyV
22
22
21
22
212
1
yyyVyV
gyVyy
y 12
12
112
222
gyVyy
y 12
12
112
222
MomentumMomentum
Much algebra...Much algebra...
ssxppxx FFFMMxx
2121 ssxppxx FFFMMxx
2121
21
1
yp
21
1
yp
ExampleExample
QnAR Sh o
1 2 3 1 2/ /
Q m m1
0 0129 0 927 0 00042 2 3 1 2
.. .
/ /c ha f a f
RAP
mh 0 927.A m m m m m 3 15 3 15 9 2afa f a fa f. .
P m m m m 3 2 3 15 9 712 2 2af a f a f. .
n 0 012.
Q m s 14 3 3. /
Smm0
123000
0 0004 .
.
Columbia Basin ProjectColumbia Basin Project
The Columbia Basin Project is a major water resource development in central Washington State with Grand Coulee Dam as the project's primary feature. Water stored behind Grand Coulee Dam is lifted by giant pumps into the Banks Lake Feeder Canal and then into Banks Lake. The water stored in Banks Lake is used to irrigate 0.5 million acres of land stretching 125 miles from Grand Coulee Dam.
The Columbia Basin Project is a major water resource development in central Washington State with Grand Coulee Dam as the project's primary feature. Water stored behind Grand Coulee Dam is lifted by giant pumps into the Banks Lake Feeder Canal and then into Banks Lake. The water stored in Banks Lake is used to irrigate 0.5 million acres of land stretching 125 miles from Grand Coulee Dam.
PumpsPumps
At the time of original construction the pumping plant contained six 65,000 horsepower pumps. In 1973 work began on extending the plant. The pump bay was doubled in length to the south and six 67,500 horsepower pump/generators were added (the last in 1983) providing 12 pumps in all.
Each pump lifts water from Lake Roosevelt up through a 12 foot diameter discharge pipe to the feeder canal above. For most of their length the discharge pipes are buried in the rocky cliff to the west but at the top of the hill they emerge and can be seen as 12 silver pipes leading to the headworks of the feeder canal. The original pumps can supply water to the feeder canal at a rate of 1,600 cubic feet of water a second while the newer units can supply 2,000 cubic feet of water a second. They also have the advantage of being reversible. During times of peak power need the new pumps can be reversed thus turning them into generators. Water flows back down through the outlet pipes, through the generators and into Lake Roosevelt. When operating in this mode each pump can produce 50 megawatts of electrical power.
At the time of original construction the pumping plant contained six 65,000 horsepower pumps. In 1973 work began on extending the plant. The pump bay was doubled in length to the south and six 67,500 horsepower pump/generators were added (the last in 1983) providing 12 pumps in all.
Each pump lifts water from Lake Roosevelt up through a 12 foot diameter discharge pipe to the feeder canal above. For most of their length the discharge pipes are buried in the rocky cliff to the west but at the top of the hill they emerge and can be seen as 12 silver pipes leading to the headworks of the feeder canal. The original pumps can supply water to the feeder canal at a rate of 1,600 cubic feet of water a second while the newer units can supply 2,000 cubic feet of water a second. They also have the advantage of being reversible. During times of peak power need the new pumps can be reversed thus turning them into generators. Water flows back down through the outlet pipes, through the generators and into Lake Roosevelt. When operating in this mode each pump can produce 50 megawatts of electrical power.
Grand Coulee Feeder CanalGrand Coulee Feeder Canal
The Grand Coulee Feeder Canal is a concrete lined canal which runs from the outlet of the pumping plant discharge tubes to the north end of Banks Lake. The original canal was completed in 1951 but has since been widened to accommodate the extra water available from the six new pump/generators added to the pumping plant. The canal is 1.8 miles in length, 25 feet deep and 80 feet wide at the base. It has the capacity to carry 16,000 cubic feet of water per second.
The Grand Coulee Feeder Canal is a concrete lined canal which runs from the outlet of the pumping plant discharge tubes to the north end of Banks Lake. The original canal was completed in 1951 but has since been widened to accommodate the extra water available from the six new pump/generators added to the pumping plant. The canal is 1.8 miles in length, 25 feet deep and 80 feet wide at the base. It has the capacity to carry 16,000 cubic feet of water per second.
Unsteady Hydraulics!Unsteady Hydraulics!
The base width of the feeder canal was increased from 50 to 80 feet; however, the operating capacity remained at 16,000 cubic feet per second. Water depth was reduced from 25 to about 20 feet to safely accommodate wave action when the water flow is reversed as the pump-generators are changed from pumping to generating and vice-versa.
The base width of the feeder canal was increased from 50 to 80 feet; however, the operating capacity remained at 16,000 cubic feet per second. Water depth was reduced from 25 to about 20 feet to safely accommodate wave action when the water flow is reversed as the pump-generators are changed from pumping to generating and vice-versa.