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Contraction, Velocity and Discharge Coefficients of A Rectangular Sharp Edeged Slot
Contraction, Velocity and Discharge Coefficients of A Rectangular Sharp Edeged Slot
Contraction, Velocity and Discharge Coefficients of A Rectangular Sharp Edeged Slot
Coefficien t of discharge
contraction Cc
In Figure 1. it is shown that the flow
contracta area is
vena contracta.
The velocity profile of the vena contracta
Q A vc v vc
where
A vc Cc A geom
and
v ideal 2 g h
v vc C v videal
(Eq. a)
(Eq. b)
(Eq. f)
thus
Q Cc A geom C v v ideal
Q Cc C v A geom v ideal
A vc
A geom
(Eq. c)
0.611
2
and for sharp edged slots
C c 0.61 0.62
is defined as
C D Cc C v
(Eq. d)
(Eq. e)
(Eq. g)
therefore
Q C D A geom v ideal
and with
p h
p
h
(Eq. h)
2 g h
C
A
D o
h
1 Q
2 g C D Ao
p h
h
p
1 Q
2 g C D Ao
1 Q
p
2 g C D Ao
g 1 Q
p
2 g C D Ao
1 Q
p
2 C D Ao
and with
1 Q
p 2
C D 2 Ao
h : pressure height m
m3
0.62
Cv =
0.98
Cc =
CD =
0.61
A=
Avc =
A=
0.02
h=
40
Flow rate
Q=
Ideal velocity
vid =
(2 * g * h)^0.5
h=
40
m
vid =
28.0
m/s
Avc =
v=
Q=
Case of a:
Flow rate
Q=
CD =
A=
vid =
Q=
CD * A * vid
Cc =
0.61
0.02
28.0
0.34
m/s
m/s
0.98
vid =
v=
28.0
27.45
m/s
m/s
a geometrical area
A=
1
Q
Cc A geom
v=
Cc =
Q=
Ageom =
v=
(Eq. k)
arge
Flow rate
Fluid level
level
Q C D A geom v ideal
e vena
(Eq. h)
(Eq. i)
From (Eq. h)
Q C D A geom v ideal
(Eq. j)
v ideal
and with
(Eq. f) v ideal 2 g h
Q C D A geom 2 g h
m3
Q : Flowrate
s
C D:discharge coeffici ent
ideal
h:pressure height
Ao = Ageom
Q
C D A geom
vvc Cv v ideal
vvc Cv
Ageom:geometric al area m 2
ideal
Do
m
s2
vvc
Q
C D A geom
Cv
Q
C D A geom
and with
C D Cc C v
1
C
v
Cc
CD
cient C D
(Eq. g)
contracta is
1
Q
vvc
Cc A geom
(Eq. h)
vvc
1
Q
Cc A geom
(Eq. k)
(Eq. k)
coefficien t of a slot k s
Q
p slot k slot
2 Ageom
k slot
1
C 2D
(Eq. m)
with
Pa
(Eq. n)
coefficien t of a slot k s
Q
p slot k slot
2 Ageom
k slot
1
C 2D
(Eq. m)
p slot k slot
2 Ao
with
1
k slot 2
CD
and
C D 0.61
k slot 2.69
(Eq. n)
(Eq. n)
1
2.69
C 2D
Pa
Pa
with
k slot
Q
as slot velocity
Ageom
Q
Ageom
p slot k slot
2
v slot
2
From (Eq.h)
Q C D A geom v ideal
Pa
(Eq. h)
Q
C D v ideal
Ageom
Thus, the slot velocity is
v slot C D v ideal
(Eq. p)
and the flow rate can be writen as
Q A geom (C D v ideal )
Q A geom v slot
(Eq. q)
Pressure loss
0.62
0.02
0.0124
p slot k slot
2
A
o
Pa
(Eq. n)
Ao = Ageom = Aslot
0.0124
27.4
0.34
m
m/s
m/s
0.62
0.02
ometrical area
1
Q
Cc A geom
(Eq. k)
(1 / Cc) * (Q / Ageom)
0.62
0.34
m/s
0.02
27.45
m
m/s
Fluid level
level
h
A = Avc
vi
Ao = Ageom
vi
Vena
contracta
: ideal velocity, calculated according
Torricelli
Figure 1
Cc =
0.62
1/ Cc =
1.613
and
Thus, the average velocity at
the vena contracta is
vvc =
1.613 * Q / Ageom
(m / s)
Page 2
slot is
2
Pa
(Eq. n)
Q A geom vslot
(Eq. q)
with
v
C v
(Eq. p)
slot is
2
Pa
(Eq. n)
Q A geom vslot
(Eq. q)
with
v slot C D v ideal
(Eq. p)
Q A geom C D v ideal
as slot velocity
Q A vc v vc
(Eq. f)
with
A vc Cc A geom
Pa
and
Cv
(Eq. h)
(Eq. f1)
v vc
v ideal
(Eq. b)
v vc C v v ideal
thus
Q Cc A geom C v v ideal
s
(Eq. p)
e writen as
(Eq. q)
Q Cc C v A geom v ideal
and with
Cc C v C D
Q C D A geom v ideal
(Eq. r)
Page 3
vslot =
Q /A
(Eq. n)
so designated
0.248
m/s
Aslot=
0.0183
vslot =
13.5
m/s
Q=
from sheet 3
hv_slot =
/ 2 ) * vslot^2
0.84
kg/m
13.5
m/s
hv_slot =
76.8
Pa
Kslot =
(1 / CD )^2
CD =
0.61
Kslot =
2.69
P =
Kslot =
Kslot * hv_slot
2.69
hv_slot =
76.8
Pa
P =
206
Pa
g=
9.81
m/s
[1]
where:
h = h1 - h2, differential head
Cd = 0.61, as determined experimentally.
The discharge, when velocity of approach is negligible, may be computed using equation 9-1b. T
1
The prefix "A" denotes tables that are located in the appendix.
The equation for computing the discharge of the standard submerged rectangular orifice is:
(9-1b)
where:
Q = discharge (ft3/s)
Cc = coefficient of contraction
Cvf = coefficient of velocity caused by friction loss
Cva = coefficient to account for exclusion of approach velocity head from the equation
A = the area of the orifice (ft2)
g = acceleration caused by gravity (ft/s2)
h1 = upstream head (ft)
h2 = downstream head (ft)
The coefficient of contraction, Cc, accounts for the flow area reduction of the jet caused by the flo
Cvf accounts for the velocity distribution and friction loss. The product, CcCvf, is sometimes called
for using the water head only and does not fully account for the velocity head of approach. This c
met. The effective discharge coefficient, Cd, is the product CcCvfCva, which has been determined
coefficient of contraction has the most influence on the effective coefficient discharge. Because
will increase rapidly after reaching some low velocity. Thus, the equation should not be used for
devices. The difference between upstream and downstream heads or water surface elevations is
can be rewritten as:
(9-1b)
where:
h = h1 - h2, differential head
Cd = 0.61, as determined experimentally.
The discharge, when velocity of approach is negligible, may be computed using equation 9-1b. T
1
The prefix "A" denotes tables that are located in the appendix.
The coefficient of contraction, Cc, accounts for the flow area reduction of the j
by the flow curving and springing from the orifice edges.
The coefficient Cvf accounts for the velocity distribution and friction loss.
The coefficient Cva accounts for using the water head only and does not fully
the velocity head of approach. This coefficient is near unity if all the requireme
section 4 are met.
The coefficient of contraction has the most influence on the effective coefficie
Because Cc must approach unity as velocity approaches zero, its value will in
rapidly after reaching some low velocity.
Thus, the equation should not be used for heads less than 0.2 ft even with ve
head measuring devices.
The difference between upstream and downstream heads or water surface el
is sometimes called the differential head, and equation 9-1a can be rewritten
http://www.usbr.gov/pmts/hydraulics_lab/pubs/wmm/chap09_05.html
http://www.usbr.gov/pmts/hydraulics_lab/pubs/wmm/
e computed using equation 9-1b. Table A9-21 was prepared for orifice areas from 0.25 to 2.0 ft2.
duction of the jet caused by the flow curving and springing from the orifice edges. The coefficient
roduct, CcCvf, is sometimes called the coefficient of discharge, Cd. The coefficient Cva accounts
e velocity head of approach. This coefficient is near unity if all the requirements of section 4 are
C , which has been determined experimentally to be 0.61 for rectangular irrigation weirs. The
vf va
e coefficient discharge. Because Cc must approach unity as velocity approaches zero, its value
e equation should not be used for heads less than 0.2 ft even with very precise head measuring
eads or water surface elevations is sometimes called the differential head, and equation 9-1a
e computed using equation 9-1b. Table A9-21 was prepared for orifice areas from 0.25 to 2.0 ft2.
mm/chap09_05.html
[2]
http://www.usbr.gov/pmts/hydraulics_lab/pubs/manuals/WMM_3rd_2001.pdf
Here, the discharge coefficient has been named as contraction coefficient. Since the giv
is the total coefficient, it means that it corresponds to the discharge coefficient.
3rd_2001.pdf
[3]
nver, Colorado
Note
For a rectangular orifice
CD =
0.61
The Roman engineer Frontinus, who was in charge of the water supply under Augustus, used
This was purely empirical, since the effects of pressure, or "head," and orifice size were not kn
was given by Vi = 2gh.
We still calculate the velocity from Bernoulli's principle, that h + p/g + V2/2, is a constant alon
We'll consider here the case of zero initial velocity, as at the surface of a liquid in a container w
We assume that a streamline starts at the surface, a distance h above the orifice, and neglect
The streamline then leads somehow to the orifice, and out into the jet that issues from it. We c
and find that the velocity there is Vi = 2gh, as given by Torricelli's theorem.
A jet surrounded only by air (or another fluid of small density) is called a
If the fluid is the same as that of the jet, then buoyancy eliminates the effect of gravity on it. A
We shall consider here only free jets of water, and neglect the viscosity of water, which is sma
A cross section of a circular orifice of diameter D o is shown. The thickness of the wall is assum
of the streamlines approaching the orifice, the cross section of the jet decreases slightly until
This point of minimum area is called the vena contracta. Beyond the vena contracta, friction w
This divergence is usually quite small, and the jet is nearly cylindrical with a constant velocity.
The area A of the vena contracta is smaller than the area Ao of the orifice because the velocity
For a sharp-edged, or "ideal" circular orifice, A/Ao = Cc = /( + 2) = 0.611. Cc is called the co
For a sharp - edged, or circular orifice,
A
Cc
Ao
where Cc is called contractio n coefficien t
Cc
0.611
2
For a sharp orifice, is usually estimated to be 0.62, a figure that can be used if the exact value
For an orifice that resembles a short tube, Cc = 1, but then there are turbulence losses that af
The average velocity V is defined so that it gives the correct rate of discharge when it is assum
Then, we can write V = CvVi, where Cv is the coefficient of velocity.
The coefficient of velocity is usually quite high, between 0.95 and 0.99.
Combining the results of this paragraph and the preceding one, the discharge Q = VA = CvViC
Experiments
[4]
Our apparatus consists of a tomato juice can with the top removed, and a hole near the bottom
The first experiment, to measure Cd, is performed by measuring the time required for the cont
The corrugations in the can make convenient reference points for the liquid level. For my expe
The second experiment measures Cv. Water was allowed to run from the tap into the reservoi
Other experiments and demonstrations suggest themselves. The discharge coefficient could a
The Roman engineer Frontinus, who was in charge of the water supply under Augustus, used
This was purely empirical, since the effects of pressure, or "head," and orifice size were not kn
was given by Vi = 2gh.
We still calculate the velocity from Bernoulli's principle, that h + p/g + V2/2, is a constant alon
We'll consider here the case of zero initial velocity, as at the surface of a liquid in a container w
We assume that a streamline starts at the surface, a distance h above the orifice, and neglect
The streamline then leads somehow to the orifice, and out into the jet that issues from it. We c
and find that the velocity there is Vi = 2gh, as given by Torricelli's theorem.
A jet surrounded only by air (or another fluid of small density) is called a
If the fluid is the same as that of the jet, then buoyancy eliminates the effect of gravity on it. A
We shall consider here only free jets of water, and neglect the viscosity of water, which is sma
A cross section of a circular orifice of diameter D o is shown. The thickness of the wall is assum
of the streamlines approaching the orifice, the cross section of the jet decreases slightly until
This point of minimum area is called the vena contracta. Beyond the vena contracta, friction w
This divergence is usually quite small, and the jet is nearly cylindrical with a constant velocity.
The area A of the vena contracta is smaller than the area Ao of the orifice because the velocity
For a sharp-edged, or "ideal" circular orifice, A/Ao = Cc = /( + 2) = 0.611. Cc is called the co
For a sharp - edged, or circular orifice,
A
Cc
Ao
where Cc is called contractio n coefficien t
Cc
0.611
2
For a sharp orifice, is usually estimated to be 0.62, a figure that can be used if the exact value
For an orifice that resembles a short tube, Cc = 1, but then there are turbulence losses that af
The average velocity V is defined so that it gives the correct rate of discharge when it is assum
Then, we can write V = CvVi, where Cv is the coefficient of velocity.
The coefficient of velocity is usually quite high, between 0.95 and 0.99.
Combining the results of this paragraph and the preceding one, the discharge Q = VA = CvViC
er supply under Augustus, used short pipes of graduated sizes to meter water delivered to different users.
ad," and orifice size were not known quantitatively until Torricelli, in 1643, showed that the velocity of efflux
+ p/g + V2/2, is a constant along a streamline in irrotational flow, which is equivalent to the conservation of energ
s called a free jet, and is acted upon by gravity. A jet surrounded by fluid is called a submerged jet
ates the effect of gravity on it. A submerged jet is also subject to much greater friction at its boundary.
viscosity of water, which is small, but finite.
e thickness of the wall is assumed small compared to the diameter of the orifice. Because of the convergence
the jet decreases slightly until the pressure is equalized over the cross-section, and the velocity profile is nearly r
nd the vena contracta, friction with the fluid outside the jet (air) slows it down, and the cross section increases perf
ndrical with a constant velocity. The jet is held together by surface tension, of course, which has a stronger effect t
e, the discharge Q = VA = CvViCcAo = CdAoVi. Cd, the coefficient of discharge, allows us to use the ideal velocity
oved, and a hole near the bottom. With this can, a scale, and a timing device, we can measure the coefficients of d
g the time required for the container to empty between levels h 1 and h2 through the orifice. To find the rate at whic
for the liquid level. For my experiment, Ao = 0.09932 cm2, Ac = 84.95 cm2, h1 = 10.5 cm, and h2 = 1.5 cm. Using th
n from the tap into the reservoir, keeping h constant at 16 cm. The height of the orifice was y = 10.0 cm, and the h
he discharge coefficient could also be found by keeping the head constant and measuring the water discharged i
er supply under Augustus, used short pipes of graduated sizes to meter water delivered to different users.
ad," and orifice size were not known quantitatively until Torricelli, in 1643, showed that the velocity of efflux
+ p/g + V2/2, is a constant along a streamline in irrotational flow, which is equivalent to the conservation of energ
s called a free jet, and is acted upon by gravity. A jet surrounded by fluid is called a submerged jet
ates the effect of gravity on it. A submerged jet is also subject to much greater friction at its boundary.
viscosity of water, which is small, but finite.
e thickness of the wall is assumed small compared to the diameter of the orifice. Because of the convergence
the jet decreases slightly until the pressure is equalized over the cross-section, and the velocity profile is nearly r
nd the vena contracta, friction with the fluid outside the jet (air) slows it down, and the cross section increases perf
ndrical with a constant velocity. The jet is held together by surface tension, of course, which has a stronger effect t
e, the discharge Q = VA = CvViCcAo = CdAoVi. Cd, the coefficient of discharge, allows us to use the ideal velocity
to different users.
he velocity of efflux
[4]
Page 1
merged jet.
its boundary.
Contents
i. Theory of Discharge from an Orifice
ii. Experiments
iii. References
us to use the ideal velocity and the orifice area in calculating the discharge.
Page 2
to different users.
he velocity of efflux
[4]
Page 3
merged jet.
its boundary.
Contents
i. Theory of Discharge from an Orifice
ii. Experiments
iii. References
us to use the ideal velocity and the orifice area in calculating the discharge.
[1]
[2]
[3]
http://www.usbr.gov/pmts/hydraulics_lab/pubs/manuals/WMM_3rd_2001.pdf
http://www.ferc.gov/CalendarFiles/20110928144931-Day1-part-2.pdf
https://mysite.du.edu/~jcalvert/tech/fluids/orifice.htm
[4]
https://mysite.du.edu/~jcalvert/tech/fluids/orifice.htm#Intr
Carlos J. Cruz
cjcruz@vtr.net
3rd_2001.pdf