Air Entrainment Relationship With Water Discharge of Vortex Drop
Air Entrainment Relationship With Water Discharge of Vortex Drop
Air Entrainment Relationship With Water Discharge of Vortex Drop
2011
by
Cody N. Pump
Graduate College
The University of Iowa
Iowa City, Iowa
CERTIFICATE OF APPROVAL
_______________________
MASTER'S THESIS
_______________
This is to certify that the Master's thesis of
Cody N. Pump
has been approved by the Examining Committee
for the thesis requirement for the Master of Science
degree in Civil and Environmental Engineering at the May 2011 graduation.
Thesis Committee: ___________________________________
A. Jacob Odgaard, Thesis Supervisor
___________________________________
Larry J. Weber
___________________________________
AthanasiosN. Papanicolaou
ACKNOWLEDGMENTS
I would like to thank CH2M HILL and AECOM for allowing me to use data
collected from their respective models to analyze for my research.
I would also like to thank Andy Craig and Troy Lyons for helping me with my
research and Professor Jacob Odgaard for all of his help with my academics throughout
my collegiate career.
Thank you to my parents and family for all of their support and encouragement.
ii
ABSTRACT
Vortex drop shafts are used to transport water or wastewater from over-stressed
existing sewer systems to underground tunnels. During the plunge a large amount of air
is entrained into the water and released downstream of the drop shaft into the tunnel. This
air is unwanted and becomes costly to treat and move back to the surface. Determining
the amount of air that will be entrained is a difficult task. A common method is to build a
scale model and measure the air discharge and scale it back to prototype. This study
investigated a possible relationship between the geometry of the drop structure, the water
discharge and the amount of air entrained. The results have shown that air entrainment is
still not entirely understood, however we are close to a solution. Using a relationship of
the air core diameter, drop shaft length and terminal velocity of the water, a likely
exponential relationship has been developed.
iii
TABLE OF CONTENTS
LIST OF FIGURES .......................................................................................................... VI
LIST OF SYMBOLS ....................................................................................................... VII
CHAPTER 1. INTRODUCTION TO VORTEX DROP STRUCTURES ..........................1
1.1. A brief review of drop shaft literature .......................................................1
1.1.1. Simple air entrainment ratio for a plunging jet ...............................2
1.1.2. Revised air entrainment ratio for a plunging jet ..............................3
1.1.3. Analytical and theoretical modeling of air entrainment for a
plunging jet ................................................................................................3
1.1.4. Processes of air entrainment of plunging jets ..................................5
1.1.5. Volumetric oxygen entrainment of hollow jets ...............................5
1.1.6. Air entrainment of plunge flow into a pool from a horizontal
pipe ............................................................................................................6
1.1.7. CFD modeling of air entrainment of a plunging jet ........................6
1.1.8. Theory of air entrainment around plunge ........................................6
1.1.9. Scaling effects of air entrainment ....................................................7
1.1.10. Drop shaft design criteria ..............................................................7
1.1.11. Dimensionless water discharge air entrainment relationship ........7
1.1.12. Air core diameter formula .............................................................8
1.1.13. Tangential inlet velocity ..............................................................10
1.1.14. Air entrainment relationship with water jet velocity ...................10
1.1.15. Air entrainment of low velocity plunging jets .............................10
CHAPTER 2. MODEL DESIGN AND RESEARCH SETUP..........................................12
2.1. Model Specifications ...............................................................................12
2.2. Instrumentation ........................................................................................19
CHAPTER 3. DATA COLLECTION AND METHODOLOGY .....................................21
3.1. Testing .....................................................................................................21
CHAPTER 4. DATA ANALYSIS AND UNCERTAINTIES ..........................................24
4.1. Air core and length analysis ....................................................................24
4.1.1. Analytical air core diameter ..........................................................24
4.1.2. Air core ratio formula ....................................................................24
4.2. Air core ratio analysis ..............................................................................26
4.3. Jet velocity analysis .................................................................................26
4.4. Dimensionless analysis ............................................................................28
4.5. Froude number analysis ...........................................................................29
4.6. Dimensional analysis ...............................................................................30
4.7. Uncertainty analysis.................................................................................32
CHAPTER 5. AIR ENTRAINMENT RELATIONSHIP ..................................................35
5.1. Best fit relationship ..................................................................................35
iv
LIST OF FIGURES
Figure 1-1: Photo of INDY drop shaft model .....................................................................2
Figure 1-2: Example of a plunging jet from nozzle into pool of water ..............................3
Figure 1-3: Plan view of tangential inlet.............................................................................8
Figure 1-4: Profile view of tangential inlet .........................................................................9
Figure 2-1: WS6 drop shaft layout....................................................................................13
Figure 2-2: Type H-1 outlet for vortex drop shafts...........................................................14
Figure 2-3: AS6 drop shaft layout ....................................................................................15
Figure 2-4: MPS1 drop shaft layout..................................................................................16
Figure 2-5: ADDS 3D model ............................................................................................17
Figure 2-6: INDY drop shaft layout..................................................................................18
Figure 2-7: Type H-4 outlet for vortex drop shafts...........................................................19
Figure 2-8: Hotwire calibration curve with check ............................................................20
Figure 3-1: Spring line depth in the INDY model .............................................................23
Figure 4-1: Yu and Lee data vs. results for air core ratio formula....................................25
Figure 4-2: Air discharge relationship with calculated jet velocity ..................................27
Figure 4-3: Terminal jet velocity vs. air discharge ...........................................................27
Figure 4-4: Zhao data with current data ............................................................................28
Figure 4-5: Air concentration vs. Froude number of terminal velocity ............................30
Figure 4-6: X parameter with air core diameter ................................................................31
Figure 4-7: Equivalent diameter .......................................................................................31
Figure 4-8: X parameter relationship with equivalent diameter........................................32
Figure 5-1: Best fit relationship ........................................................................................35
Figure 5-2: Standard deviations for best fit relationship ..................................................36
vi
LIST OF SYMBOLS
A=
=
=
=
C =
d=
de =
Dj, dj =
dn, d0 =
D, Ds =
=
=
Fr =
Fr =
Htot =
heqtot =
hj =
KG, K =
KLa(20) =
L=
n=
Q, Qw, QL =
Qa, QG =
=
r =
r=
VX =
VZ =
VJ =
vj =
V =
W=
Constant
Exponent or angle of plunging jet
Exponent
Air core ratio squared: d2/D2
Coefficient
Diameter of air core at throat of vortex
Equivalent diameter of water cross sectional area
Diameter of water jet at plunge point
Diameter of nozzle
Diameter of drop shaft
Surface disturbance on jet surface
Width of opening from tangential inlet into the drop shaft
Froude number
Froude number of terminal velocity
Distance from water surface in head tank to plunge point
Head of feeding tank over the vena contracta after orifice
Height of jet nozzle from plunge point
Proportionality coefficient
Volumetric oxygen transfer coefficient at standard conditions
Length of nozzle or height of drop shaft
Manning roughness factor
Water discharge
Air discharge
Angle of floor of tangential inlet
Air concentration ratio
Radius of water jet
Horizontal or tangential velocity of water in drop shaft
Vertical velocity of water in drop shaft
Stream wise velocity of water in drop shaft or jet velocity
Water jet velocity
Terminal velocity of water in drop shaft
Circumference of water jet
vii
determining the amount of air that will be entrained in many scenarios, will come the
solution to this study.
Qa
Qw
dj 2
dn
(1-1)
Figure 1-2: Example of a plunging jet from nozzle into pool of water
Bagatur and Sekerdag found that the ratio worked for the rectangular jets as well,
showing that air entrained by the jet was independent of the shape. This is a theory that
will be applied to the vortex drop shaft in the analysis of this study.
1.1.2. Revised air entrainment ratio for a plunging jet
McKeogh and Ervine (1980) studied the factors that govern the air entrainment
rate of plunging jets. Similar to Bagatur and Sekerdag (2003), they compared their
results to Henderson et al. (1970). Through their study, they found that the ratio was
related to the surface roughness of the jet ( ), rather than just the diameter. The
relationship derived from their results states:
= 1.4
+2
0.6
0.1
(1-2)
(1-3)
In the new study, this equation was transformed using more variable that include
foamy height of water around the jet known as the air tore height. The final model
developed through various modeling and reduction was:
(1-4)
This provided a better fit for their data and allowed for an air entrainment
prediction law for plunging jets based on several values of heqtot/Ds when using the values
of K=0.00425, =0.81 and =0.83.
Luca, Paolo and Guelfo (2008) used Gualtieri and Doria (2006) as a base equation
to manipulate into a better fit for their model. Using further analysis of the models
dimensionless variables, and integration of experimental data to develop a Gaussian
curve for the constant K, they concluded that the following expression represents the air
entrainment ratio:
0.25
1.28
(1-5)
entrainment. These regimes are related physically to the velocity of the jet and the height
of air through which it falls.
magnitude for air entrainment) has very fine bubbles entrained at a relatively slow rate.
In regime II (when the air cavity around the perimeter of the jet at the plunge point
becomes unstable), which occurs at a jet velocity of 1 m/s, the air entrained increases
notably. For the highest velocity jet (a velocity of 3.5 m/s or higher) in regime III (when
the air cavity along the perimeter of the jet at the plunge point is elongated and broken up
into the plunge pool), a spike was found in the number of 1 mm air bubbles entrained.
1.1.5. Volumetric oxygen entrainment of hollow jets
Deswal (2009) used the theory of oxygen transfer to determine the entrainment
rate. Using an equation from Sande and Smith (1975), the formula was used to derive a
relationship from the experimental data collected.
(1-6)
= 0.016
0.28
1.17
0.4
0
1-7
This relationship states that the entrainment of air is related to the Froude number
of the jet. This was not accurate enough for the CFD model to work correctly so a
coefficient of drag function of the bubbles entrained was introduced.
1.1.8. Theory of air entrainment around plunge
Chanson and Brattburg (1998) studied the effects of a jet impinging into a pool of
water to develop a formula for the flow region of the very-near field (a depth within 5
times the diameter of the jet).
entrainment is not well understood. However, the water surface near the jet forms an
induction trumpet in which the water is pulled under the surface with the jet. The
velocity of the water near the jet was found to be a function of the jet velocity and
velocity at which air entrainment begins.
1.1.9. Scaling effects of air entrainment
Chanson (2008) analyzed the scaling effects of various air-water interfaces. Jet
plunge flow was determined to be similar to a horizontal hydraulic jump with a different
diffusion pattern. The effects of scaling were determined to be: the fluid properties
(density, viscosity, surface tension etc.); geometry of the structure; and the plunge
properties (inflow depth, velocity, turbulent velocity, and boundary layer thickness).
Using dimensionless analysis, among the numerous ratios created for scaling, the wellknown Froude, Reynolds and Weber numbers are found and the relationship may be
expressed using only the properties of water.
1.1.10. Drop shaft design criteria
Jain (2004) investigated the hydraulic performance of different types of drop
structures. The main two drop shafts were plunge-flow and vortex-flow structures. The
vortex-flow structure was determined to have superior hydraulic performance when
considering air venting and other various criteria. A derivation of the Manning equation
is given that determines the terminal velocity (V) of water falling as an annular jet:
1 3 5
2 5
(1-8)
relationship between the two upon further experimentation. The dimensionless water
discharge is essentially a Froude number.
1.1.12. Air core diameter formula
Jain and Kennedy (1983) researched extensively the hydraulics of many drop
shafts including vortex drop shafts.
hydraulically superior to other types, experiments were done in which the air core
diameter at the throat of the core was measured. From this a formula was developed to
predict the diameter of the core given the water discharge and properties of the structure:
1 3
2
=4
2
3 16 cos 4
1 3
(1-9)
is the air core diameter ratio: d2/D2, is the width of the opening into the drop
shaft from the tangential inlet, shown in Figure 1-3, and is the angle of the floor of the
tangential inlet, shown in Figure 1-4. The angle of the horizontal approach to the tangent
is , but it is not used for this study.
Yu and Lee (2009) revised the design criteria for vortex drop shafts. While
studying the hydraulics of the inlet geometry, the air core theory was derived using a
tangential velocity from a free vortex rather than a constant to obtain:
1 3
2 2
=4
2
3 16 cos 4
1 3
(1-10)
This theory was plotted against Jain and Kennedy (1983) data and measured data
to show that the air core theoretical equations both are still not
entirely
accurate
at
predicting the actual measured diameter. While the data does tend to follow the trend of
the prediction, the error could be as much as 20% different from predicted to measured,
for the upper range of the air core ratio. The right hand side of this equation will be
referred to as RHS here after.
10
1 3
cos
4 3
(1-11)
1 0.5
2
(1-12)
= 0.00002 1
+ 0.0003 1
+ 0.0074 1 0.0058
(1-13)
Zhao also proposed an equation for the vertical velocity of the water from
continuity with the thickness of the water (t) against the wall of the drop shaft is:
= 2 (2)
(1-14)
11
base equation from kinetic energy analysis and the angle of attack, the following equation
was found to fit the data quite well once the exponents were solved for:
= 0.015
3 4
2 3 1 2
sin 1.5
(1-15)
This equation was for plunging low velocity jets with long cylindrical nozzles
where Dj is the diameter of the jet and L is the length of the nozzle. The X parameter is
the combination of variables inside of the parentheses.
Bin (1993) used Sande and Smith (1976) to establish that the equation worked for
other applications. Bin defined the X parameter as:
= 2 3 1
sin1.5
(1-16)
From there, the data from a low velocity jet with a short cylindrical nozzle was fit
to the air entrainment to develop the following relationship:
= 0.0076 0.75
(1-17)
This relationship has the same slope and is within the same ranges as the Sande
and Smith (1976) data. This theory used the diameter, velocity and length of the jet as
the main variables for air entrainment. This study showed that the X parameter can be
used for different applications and still fit experimental data very well.
12
13
The AS6 drop shaft has an H-1 type outlet that vents into the main tunnel with a
unique inlet design. Rather than the typical tangential inlet that requires one wall to
intersect the drop shaft at a tangent while the opposing wall angles into the opening, the
AS6 inlet has a widening of the approach channel into a preliminary inlet box structure
where both walls angle towards the drop shaft tangential inlet.
The design
specifications are shown in Figure 2-3. The outlet directs the water through a manhole
structure then into a down-step ramp into the tunnel.
14
An H-1 type outlet typically has a secondary tunnel perpendicular to the drop
shaft that feeds the water into the main tunnel. This tunnel is usually a constant diameter
for the entire length. The H-1 can have a vent for air that goes either back to the surface
or to the top of the tangential inlet to recycle the air. In the case of AS6 and MPS1 drop
shafts, the vent is not included so that the air is directed into the main tunnel for the
purpose of a controlled air flow direction when in operation. Jain and Kennedy (1983)
have a comparison of all 6 types of outlet configurations.
15
16
MPS1 drop shaft has a tangential inlet similar to the WS6 drop shaft and an H-1
type outlet. This drop shaft is scaled down smaller than both AS6 and WS6. The outlet
leads into the main tunnel via a long adit conduit and enters the tunnel down a stair-step
energy dissipater. The specifications are given in Figure 2-4. A perspective view of the
entire ADDS model is shown in Figure 2-5.
17
The second model labeled INDY has one drop shaft with a simple tangential
inlet with an H-4 type outlet that includes a de-aeration chamber with an air vent that
connects to a main tunnel through an adit. The model simulates the main tunnel using the
tail box with a valve-controlled outlet. The INDY model specifications are shown in
Figure 2-6.
A typical H-4 type outlet, shown in Figure 2-7, has a large secondary tunnel that
suddenly contracts into a smaller diameter tunnel that then feeds the water into the main
tunnel. The larger tunnel that the drop shaft plunges into is called the de-aeration
chamber. The chamber has a vent for the entrained air to escape back to the surface or be
recycled back into the tangential inlet. The purpose of an H-4 type outlet is to capture all
of the air before the water discharges into the main tunnel. The contraction of the tunnel
serves as an orifice and is usually designed to create a backup of water that submerges the
smaller tunnel, forcing all of the air out of the de-aeration chamber vent.
18
19
2.2. Instrumentation
The hot wire transducers used for all models were TSI Air Velocity Transducer
Model 8455 hotwire anemometers. The hotwire anemometers were calibrated in-situ by
IIHR. The hotwires were installed in the vent pipes with the ceramic heating element of
the hotwire positioned in the center of the pipe. The airflow through each standpipe was
controlled using an extraction fan at a constant speed. Upon reaching steady state, an
20
Alnor Model RVA801 vane anemometer was mounted to the vent pipe and the average
air velocity was measured over a period of 30 seconds. The average voltage output from
the hotwire was recorded simultaneously. The airflow was established at steady state for
several fan speed settings in order to bracket the expected range of air flows during
model tests. A linear calibration curve resulted for each hotwire. A representative curve,
as shown in Figure 2-8, converted voltage into velocity (m/s).The calibrations were
checked periodically to ensure repeated accuracy.
12
Velocity (m/s)
10
y = 2.490x - 0.007
R = 1
8
6
Calibration
4
Check
2
0
0
Voltage
The direction of air flow was determined with the Alnor rotating vane
anemometer. The manufacturer specifies accuracies of +/- 1% of the reading between 50
to 6,000 ft/min. The vane anemometer features a 4-inch diameter bore allowing direct
adaption to the 4-inch air outlet vents on the model.
21
The
display showed live data output allowing steady-state to be seen visually before any data
was acquired. LabVIEW records the data into tabulated text files for each test run. The
frequency can be set manually in the user interface, which was set to 100 Hz. The typical
test was recorded for 60 seconds. This gives a total of 6000 data points that can be
averaged.
For the ADDS model, each drop shaft was run independently. The stand pipes for
all drop shafts were closed except the one in use and the tunnel outlets. This kept the
tunnel from building up any pressure that would alter the air discharge from the drop
shafts. For the INDY model the air was measured downstream of the drop shaft in the
de-aeration chamber and in the tail box. The values used were the summation of the two
air vents for a total air discharged.
For each test a water discharge was set as a percentage of the design flow. To
determine the flow rate, a manometer was used in junction with an orifice plate that has a
predetermined coefficient. The orifice plate was installed upstream of the valve for each
drop shaft. Each orifice plate has its own size and coefficient. The AS6 drop shaft used
an elbow meter installed upstream of the valve in junction with the manometer. The
MPS1 drop shaft used both an elbow meter and orifice plate upstream of the valve to
ensure a broad enough range of flows could be accurately measured to an acceptable
number of significant digits.
3.1. Testing
The ADDS model tests were run to include data from a wide range of flow rates.
Each drop shaft was tested at 14 flow rates as a percentage of the design discharge. The
percentage of design discharge ranged from 10% to 140% in 10% increments. For the
22
MPS1 drop shaft, a repeated test was performed for 40%, 100% and 140% flows. For the
AS6 and WS6 drop shafts, a repeated test was performed for all even percentages (i.e.
20%, 40%, 60%, etc.). The INDY model was run at flow rates of 10%, 25%, 50%, 75%
and 100% of the design discharge. The model had a limiting peak discharge of 117% due
to geometric constraints of the approach channel. Consequently, no data was collected
above the design discharge. The 5 flow rates tested on the INDY model were repeated
numerous times using several different methods to measure the air discharge. Using
calibration and instrument error probabilities, the most accurate instrument for measuring
the air discharge was the hot wire anemometers. The final test series was performed and
used for this study.
The tunnel water depth for the ADDS model tests was self-setting based on the
discharge of the drop shaft. The tunnel discharges into a tail box that is outlet controlled.
The tunnel is elevated above the floor of the box and the outlet is located in a sub-floor
chamber. The water is required to pass through a perforated plate in the floor to enter the
chamber. This allows the valve of the outlet to be closed enough to back up the water
until the depth is above the perforate plate and below the tunnel invert. The water depth
assures that any air that may be entrained into the tail box water will be able to rise out of
solution before exiting the model. The outlet is kept fully submerged for all tests to
eliminate any possibility of air leaving through the un-monitored outlet.
The water depth in the INDY model was kept at a depth known as spring line
depth.
The spring line depth is a percentage of the adit tunnel height that was
determined to be the depth at which the main tunnel would back up water into the drop
structure in the field. In model units, the spring line depth is 4.67 inches of the 7 inch
adit inner diameter, or 67 percent. The spring line depth can be seen in Figure 3-1. This
pre-determined depth had minimal effect on the air discharge for the 2 tests in which the
adit tunnel had a free surface of water from the de-aeration chamber. For the higher
water flow rates, the adit remained submerged at the entrance from the de-aeration
23
chamber. This assured that no air was freely entering the tail box. Air measurements of
the tail box confirmed this assumption.
For all tests in the INDY model, the water leaving the tail box must also pass
through a perforated plate close to the floor of the box. The short height of the perforated
plate allowed the depth of the water in the tail box to be relatively low and still maintain a
submerged outlet. The plate was attached to a sealed chamber that disconnected any air
or water from entering other than the designed opening. The spring line depth was
significantly higher than the minimum depth required for a submerged outlet. The depth
in the tail box was measured using a point gage attached to the outside of the box in a
stilling basin that had a feed line into a low-turbulence region. The point gage setup
allowed for very steady measurements due to a large buffer that eliminated any waves or
spikes in water depth inside the tail box.
24
25
measured data. The air core diameters for the current data could not be measured directly
due to the model designs. Equation 4-1 could also be seen as an easier method to solve
for the air core ratio using the RHS of Equation 1-10, given by:
= 0.6 + 0.96
(4-1)
1.0
0.9
Current
Calculated
0.8
0.7
Yu & Lee
Calculated
0.6
0.5
Yu & Lee
Measured
0.4
0.3
0.2
Jain &
Kennedy
Measured
0.1
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
RHS
Figure 4-1: Yu and Lee data vs. results for air core ratio formula
Comparisons were made between the air concentration and the air core using
various combinations of air core diameter, drop shaft length and air core volume. These
relationships show that there is no direct correlation between the variables alone. The air
discharge was also compared to the air core diameter divided by the drop shaft length as
well as the air core volume. The results were similarly inconclusive.
26
27
0.30
0.25
Qair/L-d-
0.20
0.15
Current
0.10
0.05
0.00
0.0
0.5
1.0
1.5
2.0
2.5
Vj (m/s)
0.30
0.25
Qair/L-d-
0.20
0.15
Current
0.10
0.05
0.00
4
V (m/s)
28
This relationship is similar to the trend found by Zhao (2006). Using the jet
velocity plotted against air discharge over jet thickness (W=*D), Zhao showed an
increasing trend of air discharge with increasing velocity. Data was taken from Zhaos
plot and compared with current jet velocity and air discharge in the same manner. The
comparison is shown in Figure 4-4. This relationship has much scatter and uses a
coefficient to manipulate data by each drop shaft, if needed, to obtain a better fit. This
coefficient is considered impractical for the purpose of this study but nonetheless shows a
correlation between previous studies of vortex drop shafts and the current one.
0.060
0.050
0.040
0.030
Zhao
Current
0.020
0.010
0.000
0
10
Vj and V (m/s)
29
probable relationship. The air concentration was used often in the same manner. Trying
various combinations of lambda, drop shaft length, air core diameter and drop shaft
diameter to obtain a dimensionless parameter, the calculations were plotted against the air
concentration or a new dimensionless version of air discharge. This approach showed
many plots with tremendous scatter among the data with no leads towards a direct
relationship. The terminal velocity was also derived into a dimensionless form using the
drop shaft length, water discharge, and drop shaft diameter to obtain:
= 2
3 5
(4-2)
While the dimensionless terminal velocity showed a trend against the air
concentration, the scatter of the data was too great for a good relationship.
4.5. Froude number analysis
From the dimensionless analysis, a Froude number approach was developed. The
velocity of the jet was introduced with gravity on both axes of the plots to attempt to
obtain a definite correlation. The Froude number of the water was calculated using the
jet velocity at the throat of the air core and the depth of the water against the drop shaft
wall at the throat of the air core. While this did not give a usable relationship, a trend in
some of the data could be seen.
The Froude number was calculated using the terminal velocity as well. Plotting
against the air concentration, both types of Froude numbers showed a small trend but still
lacking in confidence. The Froude number was multiplied with the air core ratio as well
but with less success than other attempts. The Froude number analysis does show a trend
towards an air concentration of roughly 0.5 when the Froude number is above 1.0 or
super critical flow.
30
7
6
QAir/QWater
5
4
Current
Design
2
1
0
0.0
0.5
1.0
1.5
2.0
Fr
parameter. The drop shaft diameter combined with the terminal velocity and drop shaft
length showed a good trend with the data. The air core diameter was tried and gave even
better results. The data was compared to data extracted from 2 figures from Bin (1993)
and is nearly on the same trend line. The data is an order of magnitude greater than the
jet nozzle data with slightly different coefficients for the power law relationship.
31
1.0E-01
1.0E-02
Bin Fig. 10
1.0E-03
Bin Fig. 11
Current
Design Flows
1.0E-04
1.0E-05
1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01
X = d2 V 3 L0.5 (m5.5 /s3 )
A new concept was also explored in which the equivalent diameter (de) was
calculated from the air core. Using the width of the jet against the wall of the drop shaft,
the cross sectional area of the water was translated from a ring to a circle. The diameter
of the circle calculated was then used as a representative for a plunging jet.
32
The equivalent diameter was substituted in for the air core diameter in the X
parameter. The results gave more scatter of the data but showed a better slope in
comparison to the Sande and Smith data. The Bin data was then also compared to both
sets and fit the scatter wonderfully.
1.0E-01
1.0E-02
Bin Fig. 10
1.0E-03
Bin Fig. 11
Current
Design Flows
1.0E-04
1.0E-05
1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01
X = de2 V 3 L0.5 (m5.5 /s3 )
Using data from water flows greater than 40% of the design flow, the plot shows
scatter among relationship. Adding in the 40% design flows, the scatter becomes more
scattered. Using only the 100% design flows, the trend fits nearly perfect to the Bin data.
4.7. Uncertainty analysis
An analysis of possible errors in data collection was done to determine the
uncertainty of the data before analysis.
instrumentation. The hot wire anemometers have a factory error of up to 2% (or 1%)
33
for the range of velocities they were used for. Other factory errors for environmental
conditions such as temperature and humidity are considered negligible since the
calibration error accounts for the environment.
The second possible error is the calibration error of the instrument. The hot wire
anemometers were calibrated in-situ to determine the accurate slope and intercept to
translate voltage into velocity. This calibration has a different slope than the factory
setting. The difference from factory to in-situ can be considered a possible error in the
data collection process.
correctly translating the data for the environment the instrument is in, however it must be
considered in the error analysis. The calibrations for each hot wire anemometer were
different, so an average error was calculated to represent them. The average error was
2.8%.
A third error is the precision of the data collection. A moving average determines
the time required for the average of the data to stabilize. From the time required, a
sample time is established. Every time the sample time is reached during the data
collection, the data is considered an individual sample. The data was typically collected
for 60 seconds and the sample time was determined to be 20 seconds. Thus, there were 3
samples per data collection. Some data collection was acquired for 300 seconds in which
case there were 15 samples per collection. Using a t-student distribution with a 95%
confidence level, the precision limit was found for a 300 second collection. Using a
value of 2 for the coverage factor, the precision limit was calculated to be relatively
small. The precision limit was found to have an error of 1.1% when compared to the
accepted value or average of the collection.
For all errors in the equations used, the error was considered indeterminate since
there is no error analysis for most of the empirical and analytical formulas. All error
from these formulas must be considered an integral part of the studies themselves.
34
Once the main sources of error in the data collection were determined, an overall
error combining the sources was determined to be 3.6%.
35
However the X parameter with the equivalent diameter has a slope and
coefficient closer to that of the Bin data. The equation of the best fit line, shown in
Figure 5-1 is: Qa = 0.0078X0.7338, where the coefficient and exponent from Bin are
0.0076 and 0.75, respectively.
1.0E-01
1.0E-02
1.0E-03
All
1.0E-04
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
36
1.0E-01
1.0E-02
1.0E-03
All
1.0E-04
Standard
deviation = 51.9%
1.0E-05
1.0E-06
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
The next best fit relationship is the terminal velocity plotted against the air
discharge divided by the drop shaft length, diameter, and the air core ratio, shown in
Figure 4-3.
The plot shows that VQair/ (Ld) and when fit with an exponential
= 0.0012.76
(5-1)
37
The relationship has a sum of squares value of 0.775. Had the AS6 drop shaft
data not been used, the relationship would have proven much better. The data shows a
split in the relationship for AS6 around a terminal velocity of 4 m/s.
5.2. Scaling Effects
The air entrainment relationships are based on model scale measurements and
therefore needs scaling for use upon prototype designs. Scaling is still an unconfident
issue for air discharge. The scaling law says to multiply by the scale ratio to a power of
2.5. However it was determined that scaling alone will under estimate the flow by a
factor between 2 and 4 according to Ervine and Kolkman (1980).
The Jain and Kennedy (1983) study of vortex flow drop structures included an
estimate of the factor of underestimation. The estimate was obtained by comparing flow
characteristics in two models of different scales, a small-scale model and a largescale model. The large-scale model was 2.3 times larger than the small-scale model.
The comparison shows that airflow rates do not scale according to this law. The results
suggest that airflow rates that are scaled according to Froudes law must be multiplied by
an additional scaling factor equal to the prototype to model ratio raised to a power of
0.34. For the ADDS model, the prototype to model ratio is 7, so the additional scaling
factor is 70.34 = 1.94.
Another concern is not knowing what the scale should be before any design
work is done. This study should merely be used for an approximation of the air discharge
of model scale vortex drop shafts.
applications and against prototype values, a more accurate prediction method can be
developed.
5.3. Example of application
The process of determining air discharge for a model should be done as described
in the following paragraphs. All units are in SI, and all lengths are used as meters.
38
For a model scale vortex drop shaft, once the geometry is designed according to
typical criteria, the right-hand side of the Yu and Lee (2009) formula, Equation 1-10,
should be solved for the design water discharge. The air core ratio can then be solved
using iterations or a solver. Alternatively, a much easier to solve, but unproven method
of using the linear relationship of Yu and Lees data: = -0.6*RHS+0.96 could be used.
The diameter of the air core is determined from = d2/D2 to serve as the jet diameter.
The terminal velocity of the water inside the drop shaft is then calculated using Jains
(2004) Equation 1-8 with water discharge and drop shaft diameter. The roughness is the
Manning roughness. The length of the drop shaft in this study is considered to be from
the bottom of the tangential inlet to the top of the adit or de-aeration chamber. Using
Figure 4-3, the air discharge can be determined with the calculated variables.
To use the recommended X parameter method, the water area can be solved for
from the diameter using A = *(D2-d2)/4. The equivalent diameter of the water area is
then calculated by assuming a circle for the area, rather than a ring. The X parameter can
now be solved with the equivalent diameter, terminal velocity and length of the drop
shaft where X = de2 V3 L0.5. The air discharge can now be read from Figure 4-8 or
calculated with the relationship: Qa = 0.0078X0.7338. The discharge also has a standard
deviation of roughly 51.9% that must be taken into consideration for any practical
purposes.
39
CONCLUSION
The dimensional analysis of the vortex drop shaft structure is currently the best
theory for predicting the air discharge. The quantity of air is best determined using an
exponential relationship of terminal velocity, air core diameter and drop shaft length with
individual exponents to create the X parameter that is also multiplied by a coefficient and
raised to a given power.
Ultimately, the geometry of the inlet and outlet will change the amount of air that
will be entrained. This study attempted to incorporate the different types of geometric
configurations that are common among current drop shafts.
40
REFERENCES
Ahmed, A. "Aeration by Plunging Liquid Jet." Ph.D. Thesis. Loughborough University
of Technology, 1974.
Bagatur, Tamer, and Nusret Sekerdag." Air-Entrainment Characteristics in a Plunging
Water Jet System Using Rectangular Nozzles with Rounded Ends.Water SA 29.1
(2003): 35-38. Water Research Commission. Web.<www.wrc.org.za>.
Bin, Andrezej K. "Gas Entrainment by Plunging Liquid Jets. Chemical Engineering
Science 48.21 (1993): 3385-630. Print.
Chanson, H., and R. Manasseh."Air Entrainment Processes in a Circular Plunging Jet:
Void-Fraction and Acoustic Measurements. Journal of Fluids Engineering 125.5
(2003): 910-21. Print.
Chanson, H., and T. Brattberg."Air Entrainment by Two-Dimensional Plunging Jets: The
Impingement Region and the Very-Near Flow Field. ASME. Proc. of ASME Fluids
Engineering Division Summer Meeting, Washington, DC.June 1998.Web.
Chanson, H. "Turbulent AirWater Flows in Hydraulic Structures: Dynamic Similarity
and Scale Effects. Environmental Fluid Mechanics 9.2 (2008): 125-42. Print.
Cumming, I. W. "The Impact of Falling Liquids with Liquid Surfaces." Ph.D. Thesis.
Loughborough University of Technology., 1975.
Deswal, Surinder. "Oxygenation by Hollow Plunging Water Jet.
Institute of Engineering 7.1 (2009): 1-8. Print.
Journal of the
41
Kumagai, M., and H. Imai. "Gas Entrainment Phenomena and Flow Pattern of an
Impinging Water Jet." Kagaku Kagaku Ronbunshu 8 (1982): 510-13.
Luca, Ciaravino, Paola Gualtieri, and Guelfo Pulci Doria."Air Entrainment in Central Jet
Drop Shafts: Theoretical Formulation and Experimental Implementation. (2008).
Print.
Lyons, Troy C., and A. Jacob Odgaard. Hydraulic Model Study for the City Of
Indianapolis Deep Rock Tunnel Connector Drop Structures. Limited Distribution
Report No. 370 Iowa City, Iowa: IIHR Hydroscience & Engineering, The
University of Iowa, 2010. Print.
Lyons, Troy C., Marian Muste, Andy J. Craig, and A. Jacob Odgaard. Hydraulic Model
Studies for Drop Structures: Abu Dhabi Strategic Tunnel Enhancement Programme
(STEP). Limited Distribution Report No. 371. Iowa City, Iowa: IIHR
Hydroscience & Engineering, The University of Iowa, 2010. Print.
McKeogh, E. J., and D. A. Ervine."Air Entrainment Rate and Diffusion Pattern of
Plunging Liquid Jets. Chemical Engineering Science 36 (1981): 1161-172. Print.
Ohkawa, A., D. Kusabiraki, Y. Shiokawa, N. Sakai, and M. Fujii. "Flow and Oxygen
Transfer in a Plunging Water System Using Inclined Short Nozzles and Performance
Characteristics of Its System in Aerobic Treatment of Wastewater." Biotechnol.
Bioengng 88 (1986): 1845-856.
Ohkawa, A., D. Kusabiraki, and N. Sakai. "Effect of Nozzle Length on Gas Entrainment
Characteristics of Vertical Liquid Jet." J. Chem. Engng Japan 20 (1987): 295-300.
Rajaratnam, N., A. Mainali, and C. Y. Hsung." Observations on Flow in Vertical Drop
Shafts in Urban Drainage Systems. Journal of Environmental Engineering 123.5
(1997): 486-91. ASCE. Web.
Schmidtke, M., and D. Lucas."On The Modeling of Bubble Entrainment by Impinging
Jets in CFS-Simulations. 2008. Web.
Smit, Arnout. "Air Entrainment with Plunging Jets.
Technology, 2007.Print.
Toda, K., and K. Inoue." Hydraulic Design of Intake Structures of Deeply Located
Underground Tunnel Systems. Wat. Sci. Tech. 39.9 (1999): 137-44. Print.
Van De Sande, E., and John M. Smith." Jet Break-Up and Air Entrainment by Low
Velocity Turbulent Water Jets. Chemical Engineering Science 31 (1976): 219-24.
Print.
Yu, Daeyoung, and Joseph H. W. Lee." Hydraulics of Tangential Vortex Intake for
Urban Drainage. Journal of Hydraulic Engineering ASCE 135.3 (2009): 164-74.
Web.
Zhao, Can-Hua, David Z. Zhu, Shuang-Ke Sun, and Zhi-Ping Liu. "Experimental Study
of Flow in a Vortex Drop Shaft. Journal of Hydraulic Engineering ASCE January
(2006): 61-68. Web.
42
APPENDIX DATA
Drop
Shaft
L
(m)
D
(m)
e
(m)
()
MPS1
27.5
AS6
35.0
WS6
27.5
INDY
28.0
QWater
QAir
QAir /
QDesign
(m /s)
(m3/s)
QWater
RHS
d
(m)
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
110%
120%
130%
140%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
110%
120%
130%
140%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
110%
120%
130%
140%
10%
25%
50%
75%
100%
0.00092
0.00184
0.00276
0.00367
0.00459
0.00551
0.00643
0.00735
0.00827
0.00919
0.01011
0.01102
0.01194
0.01286
0.00320
0.00641
0.00961
0.01282
0.01602
0.01923
0.02243
0.02564
0.02884
0.03205
0.03525
0.03846
0.04166
0.04487
0.00572
0.01144
0.01715
0.02287
0.02859
0.03431
0.04003
0.04574
0.05146
0.05718
0.06290
0.06862
0.07434
0.08005
0.00368
0.00920
0.01840
0.02759
0.03679
0.00247
0.00203
0.00014
0.00073
0.00138
0.00174
0.00229
0.00302
0.00366
0.00452
0.00537
0.00632
0.00696
0.00758
0.00626
0.00700
0.00694
0.00951
0.01728
0.03081
0.04016
0.03963
0.03931
0.03546
0.03773
0.03817
0.03841
0.04098
0.00634
0.00741
0.00851
0.01063
0.01143
0.01179
0.01335
0.01538
0.01773
0.02048
0.02248
0.02677
0.03016
0.02819
0.02165
0.02238
0.01901
0.02270
0.02387
2.686
1.107
0.052
0.198
0.301
0.316
0.356
0.411
0.442
0.492
0.531
0.574
0.583
0.589
1.952
1.092
0.721
0.742
1.079
1.602
1.790
1.545
1.363
1.106
1.070
0.992
0.922
0.913
1.108
0.648
0.496
0.465
0.400
0.344
0.334
0.336
0.345
0.358
0.357
0.390
0.406
0.352
5.886
2.433
1.033
0.823
0.649
0.0559
0.0888
0.1164
0.1410
0.1636
0.1847
0.2047
0.2238
0.2421
0.2597
0.2767
0.2932
0.3093
0.3250
0.1465
0.2326
0.3047
0.3692
0.4284
0.4838
0.5361
0.5860
0.6339
0.6800
0.7246
0.7679
0.8100
0.8510
0.1127
0.1790
0.2345
0.2840
0.3296
0.3722
0.4124
0.4508
0.4877
0.5232
0.5575
0.5908
0.6232
0.6547
0.1647
0.3033
0.4815
0.6310
0.7644
0.926
0.907
0.890
0.875
0.862
0.849
0.837
0.826
0.815
0.804
0.794
0.784
0.774
0.765
0.872
0.820
0.777
0.738
0.703
0.670
0.638
0.608
0.580
0.552
0.525
0.499
0.474
0.449
0.892
0.853
0.819
0.790
0.762
0.737
0.713
0.690
0.667
0.646
0.626
0.606
0.586
0.567
0.861
0.778
0.671
0.581
0.501
0.193
0.190
0.189
0.187
0.186
0.184
0.183
0.182
0.181
0.179
0.178
0.177
0.176
0.175
0.202
0.196
0.190
0.186
0.181
0.177
0.172
0.168
0.164
0.160
0.156
0.153
0.149
0.145
0.243
0.237
0.233
0.229
0.225
0.221
0.217
0.214
0.210
0.207
0.203
0.200
0.197
0.194
0.189
0.179
0.166
0.155
0.144
43
44
Data Extracted from Bin (1993) Fig. 11
X
Qa
X
Qa
0.00019 0.00001 0.00297 0.00005
0.00024 0.00001 0.00224 0.00006
0.00027 0.00001 0.00202 0.00006
0.00030 0.00001 0.00173 0.00007
0.00037 0.00001 0.00164 0.00007
0.00037 0.00002 0.00175 0.00007
0.00037 0.00002 0.00184 0.00007
0.00049 0.00001 0.00204 0.00007
0.00048 0.00003 0.00209 0.00008
0.00054 0.00003 0.00238 0.00008
0.00053 0.00002 0.00235 0.00008
0.00057 0.00002 0.00229 0.00007
0.00060 0.00002 0.00279 0.00008
0.00060 0.00003 0.00294 0.00007
0.00069 0.00002 0.00305 0.00006
0.00073 0.00002 0.00422 0.00005
0.00079 0.00003 0.00356 0.00008
0.00065 0.00003 0.00321 0.00008
0.00065 0.00003 0.00313 0.00008
0.00069 0.00004 0.00301 0.00009
0.00081 0.00004 0.00264 0.00010
0.00098 0.00004 0.00271 0.00010
0.00111 0.00003 0.00290 0.00011
0.00122 0.00002 0.00293 0.00011
0.00089 0.00002 0.00286 0.00012
0.00135 0.00002 0.00317 0.00013
0.00187 0.00002 0.00347 0.00013
0.00140 0.00003 0.00356 0.00014
0.00120 0.00004 0.00411 0.00015
0.00095 0.00005 0.00444 0.00015
0.00101 0.00006 0.00438 0.00014
0.00130 0.00005 0.00421 0.00013
0.00140 0.00004 0.00356 0.00012
0.00164 0.00004 0.00366 0.00011
0.00184 0.00004 0.00338 0.00010
0.00184 0.00005 0.00334 0.00009
0.00173 0.00005 0.00356 0.00009
0.00144 0.00005 0.00380 0.00008
0.00150 0.00005 0.00400 0.00008
0.00164 0.00005 0.00427 0.00009
0.00168 0.00005 0.00416 0.00010
0.00150 0.00006 0.00395 0.00010
0.00179 0.00006 0.00462 0.00009
0.00207 0.00005 0.00400 0.00011
0.00218 0.00005 0.00438 0.00010
0.00227 0.00005 0.00462 0.00010
0.00272 0.00004 0.00480 0.00011
0.00251 0.00005 0.00450 0.00011
X
0.00486
0.00468
0.00506
0.00547
0.00568
0.00553
0.00560
0.00461
0.00480
0.00539
0.00560
0.00567
0.00605
0.00663
0.00698
0.00698
0.00613
0.00663
0.00606
0.00630
0.00699
0.00717
0.00745
0.00717
0.00775
0.00816
0.00871
0.00816
0.00905
0.00882
0.00784
0.00795
0.00837
0.00905
0.00953
0.00952
0.00940
0.01172
0.01098
0.01142
0.01203
0.01267
0.01219
0.01300
0.01334
0.01369
0.01387
0.01442
Qa
0.00012
0.00007
0.00008
0.00009
0.00011
0.00012
0.00014
0.00015
0.00016
0.00018
0.00019
0.00021
0.00023
0.00023
0.00021
0.00019
0.00017
0.00016
0.00014
0.00012
0.00012
0.00012
0.00015
0.00017
0.00017
0.00016
0.00017
0.00018
0.00019
0.00021
0.00023
0.00026
0.00029
0.00024
0.00025
0.00028
0.00034
0.00031
0.00029
0.00027
0.00029
0.00030
0.00026
0.00027
0.00029
0.00026
0.00027
0.00025
X
0.01480
0.01370
0.01251
0.01235
0.01157
0.01099
0.01173
0.01085
0.01057
0.01085
0.02021
0.01600
0.01539
0.01600
0.01752
0.02183
0.01917
0.01729
0.01917
0.01821
0.01751
0.01558
0.01558
0.01620
0.01558
0.01404
0.01351
0.01993
0.01892
0.01773
0.01941
0.02126
0.02126
0.02421
0.02685
0.02828
0.02420
0.02452
0.02482
0.02754
0.02863
0.03054
0.03176
0.03095
0.03303
0.03712
0.03389
0.03808
Qa
0.00028
0.00023
0.00022
0.00023
0.00023
0.00021
0.00020
0.00019
0.00018
0.00015
0.00021
0.00022
0.00024
0.00025
0.00027
0.00027
0.00031
0.00031
0.00034
0.00031
0.00033
0.00032
0.00034
0.00034
0.00038
0.00039
0.00036
0.00036
0.00040
0.00044
0.00047
0.00047
0.00041
0.00033
0.00033
0.00039
0.00040
0.00043
0.00057
0.00049
0.00053
0.00059
0.00052
0.00049
0.00046
0.00042
0.00049
0.00051
X
0.03663
0.04279
0.04686
0.03569
0.03757
0.03521
0.04622
0.04564
0.04010
0.04390
0.04506
0.04745
0.04998
0.05196
0.05331
0.06308
0.06998
0.08285
0.09303
0.07464
Qa
0.00049
0.00048
0.00049
0.00055
0.00065
0.00073
0.00077
0.00068
0.00060
0.00062
0.00059
0.00060
0.00056
0.00061
0.00072
0.00079
0.00078
0.00058
0.00100
0.00094
45
Current Data
X
Qa
0.00677 0.00247
0.01971 0.00203
0.03776 0.00014
0.06049 0.00073
0.06049 0.00245
0.08767 0.00138
0.11913 0.00174
0.15474 0.00229
0.19439 0.00302
0.23799 0.00366
0.28547 0.00452
0.28547 0.00552
0.33677 0.00537
0.39183 0.00632
0.45059 0.00696
0.51302 0.00758
0.51302 0.00911
0.07551 0.00626
0.24350 0.00700
0.24350 0.00776
0.49166 0.00694
0.81482 0.00951
0.81482 0.01044
1.20971 0.01728
1.67394 0.03081
1.67394 0.03094
2.20566 0.04016
2.80337 0.03963
2.80337 0.03937
3.46579 0.03931
4.19183 0.03546
4.19183 0.03427
4.98054 0.03773
5.83107 0.03817
5.83107 0.03604
6.74266 0.03841
7.71461 0.04098
7.71461 0.03811
0.10247 0.00634
0.32268 0.00741
0.32268 0.00994
0.64335 0.00851
1.05795 0.01063
1.05795 0.01028
1.56298 0.01143
2.15400 0.01179
2.15400 0.01313
2.82925 0.01335
X
3.58681
3.58681
4.42623
5.34387
5.34387
6.33959
7.41381
7.41381
8.56267
9.78670
9.78670
0.12195
0.58568
1.99351
4.12694
6.94294
Qa
0.01538
0.01796
0.01773
0.02048
0.02041
0.02248
0.02677
0.03009
0.03016
0.02819
0.02496
0.02165
0.02238
0.01901
0.02270
0.02387