COMPUTATIONAL AND EXPERIMENTAL STUDY
OF ORIFICE DISCHARGE COEFFICIENTS AFFECTED
BY THE PRESENCE OF A RIGID WEB
By
CLEMENT CHIHWEI TANG
Bachelor of Science
Oklahoma State University
Stillwater, Oklahoma
2000
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
May, 2003
COMPUTATIONAL AND EXPERIMENTAL STUDY
OF ORlFICE DISCHARGE COEFFICIENTS AFFECTED
BY THE PRESENCE OF A RIGID WEB
. ,n J.
Thesis Approved:
11
ACKNOWLEDGEMENT
I would like to express my most sincere gratitude to my advisor, Dr. Y. B. Chang,
for his excellent guidance, consistent encouragement, and dedication to teaching and
research; I am greatly indebted to him for his invaluable inspiration. I would also like to
express my appreciation to Dr. A. J. Ghajar, Dr. P. M. Moretti, and Dr. J. 1. Shelton; their
helpful comments, guidance, and encouragements are invaluable and inspiring.
Furthennore,. I would like to thank Adam Martin and Lyon Hong for their help and
encouragement during the course of this study.
My gratimde extends to my parents, whose support and encouragement are
priceless. Their prayers for me during the course of this study have brought many
blessings into my life. Also, I would like to thank Sue Rataree for her love and comfort.
Above all, I thank my Heavenly Father for establishing the work of my hands.
This work was supported by the Web Handling Research Center (WHRC) at
Oklahoma State University.
1 Ul

Chapter
TABLE OF CONTENTS
Page
1. INTRODUCTION "" " 1
1,,1 Background " 1
1.2 Objectives and Scope of Study .4
1.3 Methods of Study 5
1.4 Brief Outline of this Study 5
II. LITERATU'RE REVIEW 7
2.1 AirWeb Interaction on Air Supporting Devices 7
2.2 Discharge Coefficient of Orifice 8
2.3 Jet Impingement. 10
III. THEORIES " 12
3.1 Fundamenta~s of Discharge Coefficient.. 12
Pipe Flow with Constriction 12
Flow through an Orifice and Impinges on Flat Surface 14
3.2 Flow Characteristics ofJet Impingement .16
IV. COMPUTATIONAL MODEL 20
4.1 Description of Computational Model 20
4.2 Configurations of Orifice Considered 23
4.3 Mesh Considerations 24
4.4 Turbulence Models and NearWall Treatments 26
Turbulence Models 26
N,earWall Treatments 28
V. COMPUTATIONAL RESULTS , 31
5.1 Computational R:esuhs of Orifice A 31
Pressure Distributions of Orifice A. 33
Velocity Contours of Orifice A 43
Mass Flow Rates and Discharge Coefficients ofOrifice A 45
Comparison of NearWall Treatments 48
5.2 Computational Results of Orifice B .50
Pressure Distributions of Orifice B 51
Velocity Contours of Orifice B 58
Mass Flow Rates and Discharge Coefficients of Orifice B 60
IV
Chapter Page
Comparison of Turbulence Models 62
5.3 Computational Results of Orifice C 64
Pressure Distributions of Orifice C 65
Velocity Contours of Orifice C 72
Mass Flow Rates and Discharge Coefficients of Orifice C 74
Effect of Supply Pressure on Discharge Coefficient of Orifice C 76
5.4 Computational Results of Orifice D 76
Pressure Distributions of Orifice D 77
Velocity Contours of Orifice D 85
Mass Flow Rates and Discharge Coefficients of Orifice D 87
Effect of Supply Pressure on Discharge Coefficient of Orifice D 89
5.5 Comparison ofTurbulence and Laminar Models 91
5.6 Influence of Supply Pressure on Discharge Coefficient.. 93
5.7 Consideration of the d, h, and I Parameters 94
5.8 Correlation Equation for Discharge Coefficient 95
5.9 Influence ofReynolds Number on Discharge Coefficients 100
5.1 0 Closing Remarks for Chapter V .1 04
VI. EXPERIMENTAL STUDy 107
6.1 Experimental Setup 107
6.2 Experimental Procedure 11 0
6.3 Sample Calculation of Experimental Data 111
6.4 Estimation ofthe Uncertainty of C 114
6.5 Experimental Results 117
Experimental Results for Orifice C with lid = 1.03 118
Experimental Results for Orifice C with lid = 0.94 124
6.6 Closing Remarks for Chapter VI 128
VII. C,ONCLUSION·S 130
REFERENCES 132
APPENDICES , 135
APPENDIX ACOMPUTATIONAL RESULTS OF' ORIFICE A 135
APPENDIX BCOMPUTATIONAL RESULTS OF ORIFICE B 136
APPENDIX CCOMPUTATIONAL RESULTS OF ORIFICE C 137
APPENDIX DCOMPUTATIONAL RESULTS OF ORIFICE D 138
APPENDIX EEXPERfMENTAL RESULTS OF ORlFICE C 139
v
Table
LIST OF TABLES
Page
I. Summary of orifice configurations considered 23
II. Comparison of C for orifice C computed at different Po 76
III. Comparison ofRSM and laminar flow solution 92
IV. Influence of supply pressure, Po' on discharge coefficient, C 93
V. Influence ofthe parameters d, h, and I on the discharge coefficient 95
VI. Mean and standard deviation of error between Ceq and C comJl 97
VII. Summary of C and Rej at different hid and Po for orifice D .1 04
VIII. Summary of C and Rej for orifices A, B, C, and D 104
IX. Summary of the 'asymptotic values of discharge coefficient 105
X. Summary of experiment conditions for orifice C with lid = 1.03 118
XI. Mean and standard deviation of error between Cmeas and Ceq forlld = 1.03 120
XII. Summary of experiment conditions for orifice C with lid =0.94 124
XIII. Mean and standard deviation of error between Cmeas and Ceq for lid =0.94 125
XIV. Summary of computed discharge coefficients for orifice A 135
:XV. Summary of computed discharge coefficients for orifice B 136
XVI. Summary of computed discharge coefficients for orifice C 137
XVII. Summary of oomputed dmscharge coefficients for orifice D 138
XVIII. Smnmary of measured quantities and discharge coefficients for lid = 1.03 139
XIX. Summary of measured quantities and discharge coefficients for lid = 0.94 140
VI
, " T
LIST OF FIGURES
Figure Page
1. Schematic of a web floated over an air reverser 2
2. Sketch of web path supported by air reversers 3
3. Schematic of standard square edge orifice 9
4. Schematic ofconical entrance orifice 10
5. Schematic offlow through an orifice in a pipe 13
6. Schematic of flow through an orifice impinging on a plate 15
7. Flow regions in an impinging jet 18
8. Schematic of computational model.. 21
9. Threedimensional computation domain 22
10. Node pattern on a hexahedral cell .24
11. The orifice region of a meshed computational model 26
12. The dimensions of orifice A 32
13. Static pressure contours oforifice A on the xzplane 35
14. Static pressure contours oforifice A on impingement surface 36
15. Static pressure contours oforifice A on plenum surface 37
16. Static pressure profiles on impingement surface for orifice A 40
17. Static pressure profiles on plenum surface for orifice A .41
18. Relationship of static pressure on impingement surface with radial velocity .42
19. Velocity contours of orifice A on the xzplane 44
20. Effect of hid on the mass flow rate of orifice A 46
2i . Computed discharge coefficient of orifice A .48
22. Comparison ofC for orifice A computed with different nearwall treatrnents .49
23. The dimensions of orifice B 50
Vll
Figure
24.
25.,
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
Page
Static pressure contours of orifioe,B 011 the xzplane \ :d S2
Static pressure contours of orifice B on impingement surface.:..l 53
Static pr,essure contours of orifice B on pleaum surface ,.54
Static pressure profiles on impingement surface for orifice B :.56
Static pressure profiles on plenum surface for orifice B. 57
Velocity contours of orifice B.on the xzplane 59
Effect of hid on the mass flow rate of orifice B 60
Computed discharge coefficient of orifice B, :, : 62
Comparison of C for orifice B computed with different turbul€;nce models ., 63
The dimensions of orifice C~ , 64
Static pressure contours of orifice G on the xzplane 66
• .. • ~ I ,
Static pressure contours of orifice C on impingement surface 67
Static pressme contours of orifice C on plenum surface 68
Static pressure profiles on impingement surface for orifice C 70
Static pressure profiles on plenum surface for orifice c. 71
Vdocity contours of orifice C on the xzplane 73
Effect of hid on the mass flow rate of orifice C 74
Computed discharge coefficient of orifice C 75
The dimensions of orifice D ." , 77
Static pressure contours of orifice D on the xzplane 79
Static pressure contours of orifice D on impingement surface 80
Static pressure contours of orifice D on plenum surface 81
Static pressure profiles on impingement surface for orifice D 83
Static pressure profiles on plenum surface for orifice D 84
Velocity contours of orifice D on the xzplane 86
Effect of hid on the mass flow rate of orifice D 88
,
Computed discharge coefficient of orifice D 89
The effect of supply pressure on the discharge coefficient of orifie,e D 90
Vlll
Figure Page
52. The comparison between correlation equation and the computed C 99
53. Comparison of discharge coefficients for conical orifice and orifice B 101
54. Discharge coefficient versus Reynolds number for orifice C 102
55. Discharge coefficient and Reynolds number for orifice 0 .1 03
56. Schematic ofthe test section .1 08
57. The discharge coefficient measurement setup 110
58. Schematic for the configuration of orifice C 118
J.... I
59. Measured C, computed C, and Equation (5.83) for orifice C with lId = 1.03 121
. 60. Measured plenum pressure, p~, affeeted by flotation height,' h, for lId =1'.03 122
61. Measured ~ and Equat~on (5.83) for orifice C with lId = 0.94 126
62. Measured plenum pressure, Po' affected by flotation height, h, for lId = 0.94 .. .128
..
I •
IX
i' I 1 il tel' udI JJ or
NOMENCLATURE
A Ar,ea of orifice
C Discharge coefficient
Ccomp Computed discharge coefficient
Ceq Discharge coefficient determined from correlation equation, Equation (5 ..83)
Cmeas Measured discharge coefficient
d Diameter of orifice
h Orificetoimpingement surface distance (simulated flotation height)
l Length of orifice
mact Actual mass flow rate
mcornp Computed mass flow rate
mmeas Measured mass flow rate
m Theoretical mass flow rate theo
Pamb Ambient pressure
Po Supply pressure or pressure upstream of orifice
PI Pressure downstream of orifice
I1p Pressure difference across the orifice
Q Volumetric flow rate
R Universal gas constant for air
x
Rej Jet Reynolds number based on orifice diameter, pud/j.J or 4pQ/j.J trd
s Depth of straight portion for countersink orifice
~oom Room temperature
u Velocity of fluid
x, y x, ydirection (or radial direction)
z zdirection (or axial direction)
e Conical angle of orifice
fl Dynamic viscosity
p Density. of fluid
Xl
CHAPTER I
J
INTRODUCTION
1.1 Background
Webs are defined as continuous, stripformed, flexible materials such as paper,
metal foils and polymer films. Drying of a coated web material generally requires noncontact
support of the web. To accomplish noncontact support of the web, an air
reverser may be used. An air reverser is a hollow, semicylindrical, porous drum. Air
cushion is formed in the clearance between the web and! the air reverser surface by
ejecting air through the holes of the air reverser. The web is floated over the air reverser
on this air cushion (se,e Figure 1).
In some dryers, the web goes through serpentine paths, where the coated side of
the web is floated by air and the uncoated side is in contact with support rollers, as shown
in Figur,e 2. The serpentine web path increases drying capacity for limited floor space.
The coated side of each vertical span is impinged by hot air which provides both drying
and noncontact support, while the other side is in contact with multiple rollers. The
lower curved part of the air support is called the air reverser. The cushion pressure that
floats the web over the air reverser is the ratio of the web tension to the radius of the air
reverser.
1
Air reverser
Figure 1. Schematic of a web floated over an air reverser
2
Low pressure
plenum
i
Figure 2. Sketch ofweb path supported by air reversers
One prevalent problem of such drying systems is that the web tends to oscillate in
the lateral direction with increasing amplitudes as it moves downstream. This
phenomenon is called weaving or weave amplification. Design of stable air reverser
systems and troublefree operation require an understanding of the lateral dynamics of the
web.
Another type of noncontact support device is called airtum bar. Airtum bars
are circular tubes having airemitting holes, and they are widely used for changing the
direction of a moving web without contact. One of the problems of circulartube airturn
bars is their tendency to flutter violently accompanied with loud buzzing noise at the
locations where the web approaches and leaves the airturn bar (Moretti & Chang, 1997).
3
1.2 Objectives and Scope of Study I I ') 0
To develop an analytical model which can predict the behavior of airweb
interaction on air reversers, correct values of discharge coefficient for the holes of the air
reverser are needed. Presently, there ,is no open literature available that discusses the
discharge coefficient ofthe holes ofthe air reverser affected by the presence of a web.
In the present study, the discharge coefficient for the holes of the au reverser
affect~d by the presence of web is investigated. To do so, the values of discharge
coefficient for a single hole (orifice) affected by the presence of a rigid web
(impingement surface) are evaluated. The main variable in this study is the clearance
between the orifice and the impingement surface (simulated flotation height, h).
Understanding how the discharge coefficient of an orifice is affected by the
I
simulated flotation height, h, would give valuable insights on the discharge coefficient for
the holes of the air reverser while floating a web. The main objectives of this study are
listed as follows:
(1) To computationally determine the values of discharge coefficient for various
orifice configurations affected by the presence of a rigid web using a
commercial computational fluid dynamics software, FLUENT (V.6.0).
(2) To develop a correlation equation that predicts the discharge coefficients for
different orifice configurations.
(3) To experimentally verify the computational results obtained from FLUENT
(V.6.0) and the results from the correlation equation.
4
u . The study undertaken here does not attempt to I investigate the fluid mechanics of
an entire air reverser. Rather, this study is focused on a single hole of the air reverser to
evaluate how the.flotation height affects the discharge coefficient.
1.3 Methods of Study
t I I
The methods of study employed here involved both computation and experiment.
The computation was done using FLUENT (V.6.0), a commercial computational fluid
,
dynamics software. FLUENT (V.6.0) was used to compute the discharge coefficients of
four different orifice configurations at various values of flotation height, h.
The computational results for the four different orifice configurations were used
to develop a correlation equation. The development of the correlation equation involved
curvefitting of the computed data and the trialanderror method. The computed results
and the correlation equation were verified with experimental results to check their
validity.
1.4 Brief Outline ofthis Study
A brief review of the open literature that is somewhat relevant to this study is
presented in Chapter II. The literature relevant to the present study may be classified into
three categories: airweb interaction on air supporting devices, discharge coefficient of
orifice, and jet impingement. In Chapter III, the fundamentals of discharge coefficient
and the flow characteristics of jet impingement are discussed. The threedimensional
computational model of a rigid web over an airemitting orifice is discussed in Chapter
IV. In this chapter, the description for the setup of the computational model, the
5
discussions on the different orifices considered, the mesh of the computational domain,
and the description of the turbulence models were presented.
The computational results for various orific·e configurations are presented in
Chapter V. Here, the comparison between turbulence and laminar models is examined.
The influences of the supply pressure and the orifice geometric parameters on the
discharge coefficients are evaluated in this chapter also. The experimental study is
discussed in Chapter VI. In this chapter, the discussion on the experiment undertaken for
verification of the computed results and the correlation equation, Equation (5.83), ar~
presented. Finally, the conclusion ofthis study is given in Chapter VII.
6
CHAPTER II
LITERATURE REVIEW
The literature relevant to the present study can be classified into three categories:
airweb interaction in air supporting device, discharge coefficient of orifice, and jet
impingement.
2.1 AirWeb Interaction on Air Supporting Devices
The fundamentals of lateral dynamics of a moving web were established by
Shelton (1968). In his work, a web was treated as a flat beam, resisting lateral curvature
with a lateral bending moment. Although his work was on the lateral dynamics of a
moving web supported by rollers, it is still applicable for the understanding of lateral
dynamics of a web supported by air supporting devices.
Muftii, et al. (1998) analyzed the fluid mechanics of the air cushion of the air
reversers used in web handling systems. They used a twodimensional model of the air
.1
flow obtained by averaging the equations of conservation of mass and momentum over
the clearance between the web and the reverser surface (flotation height, h). Their model
was solved numerically, and the results were compared with a onedimensional analytical
solution, an empirical formula, and the onedimensional airjet theory developed for
hovercraft. Miiftii et al. (1998) assumed a value for the discharge coefficient of the air
7
reverser holes. Their twodimensional solution showed better agreement, when the
I "
discharge coefficient is greater than 0.65.
A mathematical model for the steadystate fluidstructure interaction between a
web and the aircushion generated by an air reverser was presented by Muftti and Cole
(1999). In their mathematical model, the web was modeled as a thin flexible cylindrical
shell. The web as a thin flexible cylindrical shell in its strainfree reference state is self
,
adjusting according to the interaction between the aircushion and pulldown pressures.
Pulldown pressure occurs in an air supporting device when a web is wrapped under
tension over a drum radius.
Chang (2001) attempted to obtain a closedfonn solution for the amplitude of
lateral deflection of a web over an air reverser. His attempt on the analysis was based on
a twodimensional aerodynamics model which includes the air flow in the cross machine
direction but excludes the machine direction.
2.2 Discharge Coefficient of Orifice
According to Rouse (I 946), the discharge coefficient for an irrotational efflux
from a plane orifice in an extremely large container would have the magnitude of 0.611.
This finding was obtained using the mathematical principles of which the flow net is a
graphic representation for twodimensional boundary forms (Rouse, 1946).
Orifice meter has been frequently used as a device for measuring the flow of
fluids. The standard square edge orifice (se,e Figure 3) is tbe most common restriction for
clean liquids, gases, and lowvelocity vapor flow (Miller, 1983). The discharge
coefficient for the standard square edge orifice to measure the flow from a large space
8

. "'r 7 
into another large space (i.e., d/D= 0) separated by;a partition is 0.596at a minimum '" ..
t/ /'
pipe Reynolds number of 105 (Ower & PankhuJt',1977).
>"'/,A .
;'~..j
. r
.f .....
D
.... ;j, ~ Go
/
~. _..,
d 4 I •
Figure 3. Schematic of standard square edge orifice , , ,
When the pipe Reynolds number is below 1041
, a conical entrance orifice (see
Figure 4) gives more constant and predictable discharge coefficient at lower Reynolds
numbers': At low Reynolds numbers the discharge coefficient of a square edge orifice
may change as much as 30 percent, but for acomcal entrance orifice the effect is only 1
to 2 percent (Miller, 1983). For diD = 0.1, the discharge coefficient for a corneal
,entrance orifice is 0.734 f0r 25 ~ oRe ~ 500, and 0.79 for 500 ~ Re ~ 2 x 104 (Miller,
1983).
9
· I
2d
J .!
O.021d ~I
D
I ,
Figure 4. Schematic of corneal entrance orifice
2.3 Jet Impingement
Gauntner et al. (1970) summarized numerous studies on flow characteristics of
single jets impinging on flat surfaces by various researchers. Jambunathan et at (1992)
later gave a r,eview of heat transfer data for singIe circular jet impingement. Although,
the study herein does not include heat transfer, nevertheless their fmdings gave useful
insight on the flow characteristics of a jet impingement.
The experiments undertaken by Baydar (1999) on confined impinging air jet at
low Reynolds numbers showed that a subatmospheric region occurred on the
impingement plate at small orificetoimpingement surface spacing. These Subatmospheric
regions were aIso observed by Hwang and Liu (1989) in their numerical
study of twodimensional impinging jet flowfields..
10
The observations of MeNaughton and Sinclair (1966) on submerged jets in short
cylindrical flow vessel had characterized four main types of jets based on their Reynolds
numbers: dissipatedlaminar jets, fully laminar jets, semiturbulent jets, and fully
turbulent jets. Their observations. were made through a flow visualization setup by using
an aqueous blue tracer solution in conjunction with transparent cylindrical tanks.
11
CHAPTERIU
THEORIES
3.1 Fundamentals of Discharge Coefficient
Pipe Flow with Constriction
When a fluid flows through a constriction, be it a venturi, a flow nozzle, or an
orifice, the actual mass flow rate is almost always less than the theoretically calculated
value. For a venturi or flow nozzle, where tbe reduction of flow crosssectional area is
gradual, the agreement is wi¢in approximately 1 to 3 percent (Miller, 1983). But for an
orifice, the abrupt area reduction causes the minimum flow area, so called vena
contracta, to occur downstream ofthe orifice (see Figure 5).
With the occurrence of the vena contracta, the actual mass flow rate is expected
to be smaller than the theoretical mass flow rate. This is because the actual mass flow
rate is determined by the crosssectional area of the vena contracta and not by the crosssectional
area of the orifice. AJ'so, the effects of swirl and turbulence are not accounted
for by BemouUi's equation. These two factors cause the actual mass flow rate to be
approximately 60 percent of the theoretically calculated value (Miller, 1983).
12
(3.13)
(3.11)
(3.12)
11 , 41
(1) (2) vena contracta
Figure 5. Schematic of flow through an orifice in a pipe
Z Z
Uz  ul _ PI  Pz
2 P
13
p U
Z
 + = constant,
p 2
The equation for calculating the theoretical mass flow rate of an incompressible
flow through an orifice in a pipe can be derived from Bernoulli's equation. Starting with
and applying it with steady flow from plane (1) to plane (2), as shown in Figure 5,
Bernoulli's equation without the effect of elevation,
or
Substituting Equation (3.13) in Equation (3.12) and solving for Uz gives
From continuity, the mass flow rate across section (1) in Figure 5 is equal to that across
section (2):
.\ ! I
.• (10
(3.14)
and the theoretical mass flow rate is given by
, . I I
(3.15)
for an incompressible fluid flowing through an orifice in a pipe.
14
coefficient is defined as the ratio of actual mass flow rate to theoretical mass flow rate:
(3.16)
(3.17)
c = ~aet •
m'lheo
Now, consider an incompressible fluid contained in a large cylindrical plenum
The inaccuracy of the theoretical equation for calculating mass flow rate is
Flow through an Orifice and Impinges on Flat Surface
corrected with a correction factor called the discharge coefficient, C. The discharge
calculated with Equation (3.15).
The actual mass flow rate is determined empirically, and the theoretical mass flow rate is
modified from Equation (3.15) to be
flowing through a small orifice and impinging on a flat surface (see Figure 6). The fluid
orifice diameter, D» d, therefore the tbeoretical mass flow rate equation may be
in the plenum is practically motionless and the plenum diameter is much larger than the
Equation (3.17) is the appropriate equation for calculating theoretical mass flow rate. as
all the work presented here assumes air flowing from a plenUm with a supply pressure.
Po. through an orifice, impinges on a fllat surface and finally'reaches ambient condition.
D
"""""" """ """"'" " .l·l.d 1'\:
~'
""
~ (
~
1'\
~ Plenum with
~; supply pressure. po ""~I ""1'\
~ I'.
1\
~ 1\
~
~ 1\
1\
1\
ll. l>.
15
KSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS~
Impingement surface \ ( Th
Plenum surface ss::J1 \t:SS I
Figure 6. Schematic of flow through an orifice impinging on at plate
For a fluid flowing from a plenum through an orifice and impinging on a flat
the actual mass flow rate may be considerably different than the theoretical mass flow
rate, especially when h < d. As h decreases. the actual mass flow rate is expected to
turbulence effects from Bernoulli's equation. the gap between the plenum surface and the
impingement surface, h. also affects the actual mass flow rate. The theoretical mass
flow rate equation, Equation (3.17). does not take into account the ef£ects of h; hence
surface. besides the unknown diameter of the vena contracta and the unaccounted
16
3.2 Flow Characteristics of Jet Impingement
(3.18)
(3.19)
c = 4mact = 4mact
1ld2 ~2p (po  Pamb) 1ld2J2p (tJ.p) .
impingement problems.; however, due to the similarity of this work with a jet
1992). These four regions are iUustrated in Figure 7 and are described as follows:
The discharge coefficient in this case is defined similarly as it is by Equation
The objective of the work presented here is not involved directly in solving jet
,
impingement problem, an overview on the flow characteristics of jet impingement is
discussed in this section. The flow in an air jet impinging normally on a soHd plane
surface is commonly divided into four regions (Gauntner et a/., 1970; Jambunathan et al.,
in Equation (3.] 8) with a computed mass flow rate, mcomp ' Hence, the discharge
calculating the discharge coefficient from computational result, one may replace the mac,
coefficient may be detennined computationaHy with
discharge coefficient is given as
The actual mass flow rate, macl is generally determined empirically. However, when
flowing through the small gap between the plenum surface and the impingement surface.
impingement surface decreases, viscous effects increase, resulting in less amount of fluid
(3.16). Now by substituting Equation (3.17) into Equation (3.16), the equation for
decrease also. This is because as the gap between' "the plenum swfac·e' and the
(l) Region I is the r,egion of flow establishment, wbic!fu e~t!,nds from the nozzle
exit to the tip of the potential core. The potential core is the central portion of
the jet in which the jet velocity remains almost equal to the nozzle exit velocity.
The tip of the potential core is commonly defined to be the point where the
centerline jet velocity diminishes to 95% of the nozzle exit velocity. Gauntner
et at. (1970) suggested that the length of the potential core is approximately six
times the nozzle diameter. The potential core is formed by turbulent m.ixing
that originates near the nozzle exit, where fluid from the surroundings is
entrained into the jet. This turbulent mixing and shear layer surrounds the
potential core and grows in width with downstream distance until it reaches the
jet centerline, which is the tip of the potential core.
(2) Region 2 is the region downstream of the potential core, where the jet behaves
as a free jet. As the distance from the nozzle increases, the axial )et velocity
continues to reduce while the jet spreads in the radial direction. Hence,
dissipation of axial jet velocity and turbulent mixing increases with increasing
distance downstream of the nozzle.
(3) Region 3 is the region near the impingement surface in which the jet is
deflected from the axial direction to the radial direction. Consequently, this
region is often referred to as the deflection zone or the stagnation region. In
this region, the axial jet velocity decreases rapidly with a corresponding
increase in static pressure.
(4) Region 4 is known as the wall jet region, in which the jet impinges on the
surface and spreads out in the radial direction to fonn a waH jet. The fust
17
attempt to describe the theory. of a wall j,et formed by.riet impingement was
undertaken by Glauert (1956).
• l
h
Figure 7. Flow regions in an impinging jet
The distinction between laminar and turbulent jets is of fundamental importance
in the study in any jets. To characterize the distinction between laminar and turbulent
jets, the Reynolds number describing a jet is taken to be
Re = pud = 4pQ.
] j.1 pJrd
Based on the observations of McNaughton and Sinclair (1966) on liquidtoliquid jets
using methylene blue dye, four types ofjet may be characterized as follows:
(1) Dissipated laminar jet, Rej < 300 , where the jet flows for a short distance and is
quickly dissipated into the surrounding fluid due to small inertial forces.
18
(2) Fully laminar jet, 300 < Rej <1000, in which there is no noticeable diffusion of
the jet into tbe surrounding fluid for a significant distance.
(3) Semiturbulent jet, 1000 < Rej < 3000, where jet flow near the exit is laminar,
but farther downstream, turbulent eddies become apparent and the surrounding
fluid is well mixed with the jet.
(4) Fully turbulent jet, Rej > 3000, wbere immediate turbulent mixing takes place
at the extt, and j,et remained turbulent farther downstream.
The jet Reynolds number obtained from the computational results presented
herein are within the range of 5000 < Re; < 21200. Based on the types of jet
characterized from the observations of McNaughton and Sinclair (1966), it is appropriate
to characterize all the jets considered in this study as fully turbulent jets. The turbulence
models chosen to be used in the computational fluid dynamics method are discussed in
Chapter IV.
19
CHAPTERlV
COMPUTATIONAL MODEL
The threedimensional computational model of a rigid web over an airemitting
orifice is discussed iti. this chapter. The setup of this computational model is mainly to
study tbe effects ~of the gap between the web and the air reverser surface, h, on the
discharge coefficient, C. Chapter 4 is divided into four sections: Section 4.1 describes
the setup of the computational model, Section 4.2 discusses th~ different orifice
configurations considered in the computation, Section 4.3 focuses on the mesh generated
in the computational domain, and fmally Section 4.4 gives the description of the
turbulence models and nearwall treatments used in the computation.
4.1 Description of Computational Model
The threedimensional computational model simulates a circular rigid web placed
against a circular plate with an airemitting orifice in its center. The numerical
simulation of this computational model has been conducted using a commercial
computational fluid dynamics codecaUed FLUENT (V.6.0). Figure 8 illustrates the twodimensional
schematic diagram of the computational model viewed on the xzplane. Air
under a constant supplied pressure (Po =10 inH20) flows through an orifice and
20
impinges on the circular rigid web perpendicularly. The gap between the web and. the air
Figure 8. Schematic of computational model
II • I It t 1
Rigid web (circular late)
l~
x,y
h
reverser surface, h is varied between 0.5 mm and 10 rom.
The threedimensional computational domain for the simulation using FLUENT
(V.6.0) is shown in Figure 9. The threedimensional computational domain is essentially
cylindrical in shape, and Figure 9 illustrates the computational domain viewed on the xzplane.
The boundary conditions. defined in the computational domain are characterized
as pressure itnlet, pressure outlet and wall. Constructed on a Cartesian coordinate system,
the cylindrical computational domain has a diameter of 100 mm, and the height of the
computational domain depends on h (i.e., 20 mm + h). The centerline of the orifice is
along the zaxis, hence giving a length of 50 rom from the centerline of orifice to the
21
pressure outlet. A length of 50 mID between the centerline of orifice and the pressure
outlet provides adequate space for the simulation of airflow at the orifice region to reach
a nearly ambient condition at the pressure outlet.
"
Pressure outlet Pressure outlet
________________~~~pre~s~~~
(50,0,0) (0,0,0) (50,0,0)
Wall
h
Rigid web (wall)
;
Orifice region
Plenum surface (wall)
aU
Orifice region
I,.d ~I
~ CIz
Figure 9. Threedimensional computation domain
Airflow in the computational domain is steady and assumed to behave as an ideal
gas. The dynamic viscosity of air is set constant at 1.79x 105 kg/ms. The default wall
material in FLUENT (V.6..0) is aluminum, and it is treated as smooth wall without
roughness height..
22
4.2 Configurations of Orifice Considered
Four different configurations of orifice are considered for the computational
studies. Orifice A is conical in shape with a sharp edge configuration. Orifice B is also
conical in shape, but with a countersink. Orifice C and D are straight orifices, each with
different thickness values, I. Table I summarizes these four orifice configurations.
Table L Summary of orifice configurations considered
SSSSSSSSSSSS\S\S
1 I' d~ Ih ssss~ ~K~S~S~S~S
[T
23
Orifice A
d= 3 rom (0.118 in.)
h = 0.5  10.0 nun
(0.0197  0.394 in.)
I = 3 mm (0. 118 in.)
8= 33.7°
Orifice B
d= 3 mm (0.118 in.)
h = 0.5  7.0 mm
(0.0197  0.276 in.)
1=3 mm (0.118 in.)
s = 1.5 mm (0.059 in.)
8= 59°
Orifice C
d= 3 mrn (0.118 in.)
h =0.5 5.0 mm
(0.0197  0.197 in.)
1=3 mm (0.118 in.)
Orifice D
d= 5mm (0.197 in.)
h = 0.5  5.0 nun
(0.0197  0.276 in.)
1= 1mm (0.0394 in.)
J
,"
4.3 Mesh Considerations t
All of the computational models for the different orifice configurations are
meshed by similar hexahedral cells. All of the computational models for the different
orifice configurations are geometrically cylindrical; therefore each computational model
may be easily meshed with hexahedral cells. An illustration of a hexahedral cell and its
nodes pattern is shown in Figure 10. Each hexahedral cell has eight nodes, each node is
situated at each ofthe eight comers the cell
I
I
l ",
",
•  node on element comer
Figure 10. Node pattern on a hexahedral cell
In order to reduce numerical truncation error in the computation and to ensure the
flow near the orifice and impingement region is appropriately resolved, it is necessary to
mesh a computational model with reasonably fine grid. On the other hand, overrefining
a region of a computational domain can result in adverse effects. An overrefmed region
24
solution (FLUENT 6 User's Guide. 2001).
causes very large gradients in cell volume, which can adversely affect the accuracy of the
"'.t t U \
To mesh a computational model with reasonably fine grid, especially near the
orifice region, the entire computational domain is meshed with approximately 750,000 to
1,000,000 hexahedral cells. The number of cells meslied in a computational model varies
from one to the other, because of the difference in the simulated flotation height and
orifice configuration. Hence, it is possible to have a computational model meshed with
200,000 cells more than another computational model. An example of the meshed
computational model for orifice B configuration is illustrated in Figure 11. In this
particular computational model, the entire domain is meshed with 803,496 hexahedral
cells. Figure 11 however, only shows the twodimensional planar view near the orifice
region of the computational domain.
25
1=3mm
s 1.5mm
~·····/i
/ .."/ /
.////:1 "
Rigid web (circular plate)
d 3mm
\, >". ''t'o..
Gtk!
...............·1 ·_ _. _.''  .. ~._ ._._._._  _ _ ' _. .:.. "' ._.  .
.: T
_._. _ ..11 1  ,""":1 ._. 1"  +    
Alg~1.roQ2
FLUENT6D pd. egegsl&::l. RSM)
Figure 11. The orifice region of a meshed computational model
4.4 Turbulence Models and NearWall Treatments ~I
Turbulence Models
The primary turbulence model used for the computational studies here is the
R,eynolds stress model (RSM). The RSM is the most elaborate turbulence model that
FLUENT (V.6.0) provides (FLUENT 6 User's Guide, 2001). To obtain solutions for
turbulent flow, the Reynoldsaveraged NavierStokes (RANS) equations are solved. The
RANS equations may be written in Cartesian tensor foan as:
'.1
O+'P (0pu)=o ot ax. I '
I
26
(4.41)
p (au'+U au.) ap a [ (au. au 2 au)] a ( ) j .( =+ J1 '+)__8ij_k + pu;u~ . (4.42) at ax} aXi ax} ax} aXi 3 axk. ax}
Note that Equation (4.42) is presented in indicial notation with i,j,k = 1,2,3. Equations
(4.41) and (4.42) resemble the general form of the instantaneous NavierStokes
equations, with the velocities and pressure now representing timeaveraged values.
The additional Reynolds stresses, pu;u~, appeared in Equation (4.42) that
represent the effects of turbulence make the crucial difference between RANS equations
and the general instantaneous NavierStokes equations. Consequently, the appearance of
the Reynolds stresses have posted the closure problem (i.e., more unknown quantities
than available equations) when solving Equations (4.41) and (4.42). In addition to three
velocity quantities and pressure quantity, there are the Reynolds stresses; resulting in four
equations containing more than four unknown quantities. Hence, the Reynolds stresses,
PU;U~, have to be model,ed in order to close Equations (4.41) and (4.42). In the RSM,
differential transport equations are solved for the six individual Reynolds stresses,
pu;u~. The individual Reynolds stresses are then used to obtain closure for the RANS
equations (FLUENT 6 User's Guide, 200 I).
Another turbulence model available in FLUENT (V.6.0), the realizable k  &
model, is also used as a means of comparison with the results obtained from the RSM.
Like the standard k  & model, the realizable k  & model employs the Boussinesq
hypothesis to detennine the Reynolds stresses, pu;u~, and close the RANS equations.
According to the Boussinesq hypothesis, the Reynolds stresses are expressed in relation
with the mean rate of strain,
27
t
(4.43)
Notice that the Boussinesq hypothesis has introduced a new unknown quantity referred as
the turbulent viscosity, /It' The introduction of the turbulent viscosity, IJt' into the
RANS equations does not achieve closure, but only exchanges the Reynolds stresses for
another unknown turbulent viscosity, P" Hence, in the k  £ models available in
FLUENT (V.6.0), two additional differential transport equations, the turbulent kinetic
energy, k, and the turbulent dissipation rate, £, are solved, and the turbulent viscosity,
J.11 ' is computed as a function of k and £ (FLUENT 6 User's Guide, 2001).
The realizable k£ model in FLUENT (V.6.0) proposed by Shih et al. (1995)
was intended to improve the deficiencies of the standard k  £ model by using a new
model dissipation rate, &, equation and a new realizable turbulent viscosity, J.11 '
formulation. The new realizable turbulent vtscosity, PI' formulation satisfies the
realizability of certain mathematical constraints, so that it is consistent with the physics of
turbulent flow (Shih et al., 1995).
NearWall Treatments
The presence of wans significantly affects turbulent flow. Very near the wall,
viscous stress is dominant, while away from the wall, turbulent shear stresses become
dominant. The nearwaH region can be subdivided into three layers: viscous sublayer,
buffer layer, and outer layer. In the viscous sublayer, molecular viscosity plays a major
role in momentum, heat and mass transfer, as the Reynolds shear stresses are negligible
28
•(...
~
"'.:
compared with the viscous stress.. In the buffer layer, the region between the viscous
sublayer and the outer layer, both effects of molecular viscosity and turbulence are
equally important. Finally, in the outer layer, direct effect of molecular viscosity is
negligible, as the Reynolds stresses become dominant. Since the turbulence models
provided in FLUENT (V.6.0) are primarily valid for turbulent flow somewhat away from
wall regions, nearwall treatments are necessary to make the turbulence models suitable
for wallbounded flow (FLUENT 6 User's Guide, 2001).
FLUENT (V.6.0) provides two approaches to treat the nearwall region: wall
functions approach and nearwall modeling approach. The key difference between wall
functions approach and nearwall modeling approach is that wallfunctions approach does
not resolve the viscositydominated region, instead it uses semiempirical functions to
bridge it, whereas nearwall modeling approach resolves the viscosity dominated region
all the way down to the wall with modified turbulence models (FLUENT 6 User's Guide,
2001). In the wall functions approach, FLUENT (V.6.0) provides the standard wall
functions and the nonequilibrium wall functions options. In the nearwaH modeling
approach, FLUENT (V.6.0) uses the twolayer zonal model.
The primary nearwall treatment approach used in the computational modeling is
the nonequilibrium wall functions. The nonequilibrium wall functions use twolayer
concept, where the cells adjacent to the wall are assumed to consist of a viscous sublayer
and a fully turbulent layer. The nonequilibrium wall functions are recommended to be
used in flow involving separation, reattachment, and impingement where the mean flow
and turbulence are subjected to rapid pressure gradients change (FLUENT 6 User's
Guide, 2(01).
29
The twolayer zonal model is also used for selected computational models as a
means of comparison. Wall functions are completely abandoned in the twolayer zonal
model; instead, the whole computational domain is divided into a viscositydominated
region and a turbulencedominated region. The viscositydominated region is resolved
all the way down to the wall with modified turbulence models (FLVENT 6 User 's Guide,
2001).
30
1
,~
fl
"1
j'I
"Ill
(.., ...
r
i
CHAPTER V
COMPUTATIONAL RESULTS
The computational results for orifices A, B, C, and D are discussed in this chapter.
Sections 5.1, 5.2,5.3, and 5.4 of this chapter discuss the computational results of orifices
A, B, C, and D, respectively. The discussion in these four sections includes pressure
distributions, velocity contours, mass flow rate and discharge coefficient of each orifice.
Section 5.5 compares the results solved with turbulence and laminar models.
Section 5.6 discusses the influence of supply pressure on the discharge coefficient.
Following tbat, Section 5.7 considers how the geometric parameters d, h, and I may affect
the discharge coefficient. The development of a correlation equation for the discharge
coefficient is presented in Section 5.8. The influence of Reynolds number on discharge
coefficients is discussed in Section 5.9. Finally, the conclusions drawn from this chapter
are briefly summarized in Section 5.10.
5.1 Computational Results of Orifice A
Orifice A is basically conical in shape, with a sharp edge configuration. Major
dimensions of orifice A are illustrated in Figure 12. The configuration of orifice A may
be difficult to manufacture, because the tool needed to bore this particular conical
entrance angle, e=33.7°, is not standard or easily available on the shelves of any
31
hardware store. Although the configuration of orifice A may not seem very practical on
perforateddrum type air reversers, the computational results of this orifice configuration
are informative.
Rigid web
\:;>\SS\SS\SSSS\S\SS\S\\
Impingement surface
~...............~l,mrn Plenum surface I I I
~......................................
L> z 8= 33.7° Lx,y
Figure 12. The dimensions of orifice A
h
1=3mm
,!
The computational results of orifice A were obtained from FLUENT (V.6.0). In
this simulation of orifice A, the main variable was the simulated flotation height, h. The
variation of the simulated flotation height was within the range of 0.5 mm ~ h ~ 10 rom.
The supply pressure in the plenum was set constant at Po =10 inH20 (2490 Pa). The
primary turbulence model and nearwaH treatment cbosen for the simulation were
Reynolds stress model (RSM) and nonequilibrium wall functions, respectively. In
32
addition, the twolayer zonal model was used as an alternative nearwall treatment to
compare the results obtained from the nonequilibrium wall functions.
Pressure Distributions of Orifice A
The pressure profiles of interest in this study are the pressure distributions along
the impingement surface and the plenum surface. By setting a constant supply pressure
of Po = 10 inH20 in the plenum, the changes in pressure profiles with respect to the
simulated flotation height were observed. Pressure contours of orifice A on the xzplane
for the simulated flotation heights of h = 1mm and 4 mm are illustrated in Figure 13.
From the observation of the pressure contours shown in Figure 13, when hid < I
(see Figure 13a), pressure gradient in the radial direction along the gap between the
impingement surface and the plenum surface is considerably larger than that when hid>
1 (see Figure 13b). When hid < 1, the smaller simulated flotation height forces the jet to
deflect in the radial direction immediately after exiting the orifice; hence giving a larger
radial velocity magnitude. The larger radial velocity magnitude causes a larger pressure
gradient to occur near the orifice region, when h is smaller. On the other hand, when hid
> 1, the larger simulated flotation height allows the jet to flow axially for a short distance
before it is deflected by the impingement surface, resulting in a smaller radial velocity
magnitude. When h is larger, the smaller radial velocity magnitude causes a smaller
pressure gradient to occur near the orifice region.
The pressure contours shown in Figure 14 and Figure 15 are the pressure contours
on the impingement surface and the plenum surface respectively. Similarly, a larger
pressure gradient near the orifice region can be seen in the case where h is smaller (see
33
I, ':
,I;!
'I ~.
Figure I4a and Figure I5a). When h is larger, the pressure gradient near the orifice
region is smaller (see Figure 14b and Figure ISb).
34
f
I,
c
00108, 2002
FLUENT 6D (Sd. ~f't;lgeikld. RSM)
(a) Orifice A with h = 1 mm (h/d= 1/3)
1000
~OO
250(}
1500
soo
2500
1000
1500
~oo
500
(b) Orifice A with h = 4 mm (h/d= 4/3)
Figure 13. Static pressure contours of orifice A on the xzplane
Comoul'Sol S1l>lic> P... ~UJ'<l ~,
35
00108,2OQ2
FlUENT eD (Sd, ~f't;lglldcad. RSMJ
....
2500
0000
1500
1000
!il'J0
o
soo
_1000
1500
01000
2500
Oc>108,01002
FLUENT 6D (Sd, $GglQ9lll1<¥!, FlS M)
(a) Orifice A with h = 1mm (hid = 1/3)
;:so0
0000
1500
1000
soo
00112.01002
FLUENT 80 (Sd, $Ggtcqllll<¥!, FlSM)
(b) Orifice A with h = 4 mm (hid = 4/3)
Figure 14. Static pressure contours of orifice A 011 impingement surface
36
CoI0!,2002
I=LUENT aD (Sd. :5oQSl'GglldQCj, RSt.l)
2000
(a) Orifice A with h = I mm (hid = 1/3)
2500
2500
2000
1500
1000
ruo
0
ruo
1000
1503
2000 L 2500
toDD
1500
o
500
1000
1500
2500
2DOO
suo
Col 12, 201l:!
RUI:NT eo (Sd, ""'9~lld<od. RStd)
(b) Orifice A with h = 4 mm (hid =4/3)
Figure 15. Static pressure contours of orifice A on plenum surface
37
The pressure profiles on the impingement surface and on the plenum surface are
shown in Figure 16 and Figure 17, respectively. From both Figure 16 and Figure 17, the
pressures on the impingement surface and on the plenum surface reach .approximately
atmospheric condition at about 10 mm from the center of the orifice. The pressure
variation occurs drastiJcally near the orifice region, where the jet exits from the orifice and
then is deflected by the impingement plate. Near the orifice region, there exists a subatmospheric
region (see Figure 16 and Figure 17). The subatmospheric region becomes
stronger with decreaswng simulated flotation height as shown in Figure 16a and Figure
17a. As h increases, the fluwd velocity decreases due to the jet spreading, therefore the
subatmospheric region becomes weaker, as shown in Figure 16b and Figure 17b. The
subatmospheric region was also observed by Baydar (1999) in his experiment on
confined impingement air jet.
The existence of the subatmospheric regIOn is due to the acceleration and
deceleration of the fluid in the radial direction. As h decreases, the acceleration of the
fluid in the radial direction increases. At sman h, the air jet does not have the axial
distance to spread and dissipate. Instead, much of the jet momentum is deflected in the
radial direction by the impingement plate, resulting in a stronger radial acceleration.
Figure 18 shows the relationship between the static pressure and the radial velocity on the
impingement surface. In Figure 18, the static pressure on the impingement surface is
plotted together with the radial velocity along the centerline of the gap between the
orifice and the impingement surface (i.e., O.Sh).
Comparing the radial velocity profiles shown in Figure 18, the radial velocity at
hid = 1/6 reaches a higher peak than that at hid = 1/3. As a result, a stronger sub
38
4
~ •
Ill,,,!
j
atmospheric region is observed at a smaller hid, as seen in Figure 16. The radial velocity
profiles in Figure 18 also show that the fluid accelerates in the radial direction as soon as
it is deflected by the impingement surface. The fluid accelerates until a peak velocity is
reached, and then it starts to decelerate until it finally levels off at a constant velocity. It
is due to the acceleration and deceleration of the fluid that the pressure on the
impingement surface is decreased to a subatmospheric region and then increased back to
the atmospheric condition where it is leveled off.
39
,I
2.5
Po = 2.49 kPa (10 in~O)
d= 3 rom (0.118 in)
1= 3 mm (0.118 in)
()= 33.7°
o hid = l/6 (h = 0.5 mm)
)( hid = 1/3 (h = La rom)
o hld= 1/2 (h = 1.5 mm)
1.5
15 20 25
Po = 2.49 kPa (10 in~O)
d= 3 mm (0.118 in)
I = 3 mm (0.118 in)
()= 33.7°
o hid=2/3 (h = 2.0 mm)
)( hid = 4/3 (h =4.0 mm)
o hid = 2 (h = 6.0 mm)
10
(a) For h at 0.5, ] .0, and 1.5 mm
5
Distance from center of orifice, x (mm)
a
1.0
=o 0.0
~
~
~== 0.5
~
~
~
2.5
';'
~ 2.0
~
~
~
=ell =~. a~
CD .g, 0.5
a ..
1.0
a 5 10 15 20 25
Distaftce from center of orifice, x (mm)
(b) For hat 2,4, and 6 rom
Figure 16. Static pressure profiles on impingement surface for orifice A
40
11
I
!
20 25
o h/d= 1/6
)( h/d= 1/3
o h/d= 1/2
o hid = 2/3
)( hid=4/3
o hid = 2
15
Po = 2.49 kPa (10 inH20)
d= 3 mm (0.118 in)
l = 3 mm (0.118 in)
()= 33.7°
Po = 2.49 kPa (10 inH20)
d= 3 mm (0.118 in)
l = 3 mm (0.118 in)
()= 33.7°
10
(a) For hat 0..5, 1.0, and 1.5 mm
5
Distance from center of orifice, x (mm)
o
0.0
0.2
0.2
0.0
0.6
0.8
0.4
0.8
0.6
1.0
0.4
1.2
1.0
0.2
0.2
1.2
o 5 10 15 20 25
Distance from center of orifice, x (mm)
(b) For hat 2,4, and 6 mm
Figure 17. Static pressure profiles on plenum surface for orifice A
41
3 50 C':l ~ d=3mm
'' h = 0.5 mIll
~ 40 ~
e,.) 1=3mm ~ 2 =Co
Jo ....
'=' (j= 33.7 0 = ..(.I..). < Cl Po = 2.49 kPa 30 e.
~ 0 e ~
] ....
~ 4 e1l )( Radial velocity
..C...l = 20 ....
eCo 0 Pressure u=. ..... =r
Cl 0 0  t ~ ~ 10 I l. {Il I, = ""
II.l
II.l
~
l.
~
1 0
0 5 10 15 20
Distance from center of orifice, x (mm)
I(
1, '
(a) Pressure and vdocity profiles for h = 0.5 mm (hid = 1/6) .:1• ·c,I
3 50
II
' ...'.";
C':l '. ~ d=3mm 1(, '. "" h=lmm Ill'
Q,l 40 ~ II.
e,.) IIJI'
t 2 1= 3 mm ~ ''11
.=... ,
= B= 33.7 0
fI) . .(..I.l ~ /I Cl Po = 2.49 kPa 30 e. ..
Q,l 0 f· 8 ~
1
.... Ill"
Q,l ~
.=e1l )( Radial velocity = 20 .... I Co 0 Pressure = ) 8 ...... t=irl =  0 0 ~. Q,l l. 10 = ""
{Il
{Il
~
l.
~
1 ! 0
0 5 10 15 20
Distance from center of orifice, x (mm)
(b) Pressure and velocity profiles for h = 1 mm (hid = 1/3)
Figure 18. Relationship of static pressure on impingement surface with radial velocity
42
Velocity Contours of Orifice A
The velocity contours of orifice A are illustrated in Figure 19. Figure 19a shows
the velocity contours for the case when hid = 1/3, and Figure 19b shows the velocity
contours for the case when hid = 4/3. At hid = 1/3, the sman simulated flotation height
causes the air j'et to deflect immediately after exiting the orifice. Hence, a strong radial
velocity magnitude is seen around edge of the orifice (see Figure 19a). At hid = 4/3, a
larger simulated flotation height allows the air jet to exit axially before it is deflected. by
the impingement surface. Hence, an axial velocity magnitude can be seen at the exit of
the orifice (see Figure 19b).
43
i
[
....
00108,2002
FLUENT 60 "d. ~K1gldQd.RSM)
(a) Orifice A with h = I mOl (hid = 1/3)
45
25
70
15
10
5
20
50
70
65
60
55
65
60
S5
50
O,I0{l,3J02
FLUENT6D (Sd.~K1gool~.RSM!
(b) Orifice A with h = 4 mm (hid = 4/3)
Figure 19. Velocity contours of orifice A on the xzplane
44

Mass How Rates and Discharge Coefficients of Orifice A
The main interest of this srody is to evaluate the effect of the simulated flotation
height on the discharge coefficient of the orifice. As discussed in Chapter III, the
discharg,e coefficient is defined as the ratio of the actual mass flow rate to the theoretical
mass flow rate. For computational solutions, the actual mass flow rate is taken as the
computed mass flow rate. Therefore, the appropriate definition of discharge coefficient
for the computational solutions is C =rhcompjmtheo .
The value of the simulated flotation height, h, affects the amount of flow through
the orifice. When h is very small, the flow through the orifice is limited. Therefore, the
value of mass flow rate is expected to decrease when the simulated flotation height, h
decreases. As h increases, more flow is pennitted through the orifice; hence, the mass
flow rate is expected to increase. The mass flow rate through the orifice continues to
increase with h until an asymptotic constant value of mass flow rate is reached. When
the asymptotic value is reached, further increment of h would no longer affect the value
of mass flow rate.
Figure 20 shows the trend of the mass flow rate of orifice A versus the
dimensionless flotation height, hid. The increment of hid from 1/6 to 1 is the region
where the mass flow rate of orifice A increases with the simulated flotation height, h.
From hid ~ I onwards, the value of mass flow rate remains at an asymptotic constant
value. As shown in Figure 20, the asymptotic constant mass flow rate of orifice A is
close to the value ofmass flow rate for an air jet flowing through orifice A freely, without
the presence of an impingement plate. This asymptotic constant value of mass flow rate
for orifice A is approximately 0.43 gis, for the given conditions.
45
0.5
o Orifice A
    ~ Orifice A (free jet)
d= 3 mm (0.118 in)
1=3 mm (0.118 in)
B= 33.7°
Po = 2.49 kPa (10 in~O)
0.5 LO 1.5 2.0 2.5 3.0 3.5
0.2
0.0
hid
Figure 20. Effect of hid on the mass flow rate of orifice A
The equation used for calculating the discharge coefficient is Equation (3.19),
which was presented in Chapter III:
(5.11)
where
A =area of the orifice,
mcomp = computed mass flow rate at the orifice exit,
Parnb = 0 (ambient gage pressure),
Po =2490 Pa (supply gage pressure),
46
/)"p =Po  P.mb =2490 Pa, and
p = air density in the plenum.
Using Equation (5.11) with mcomp computed by FLUENT (V.6.0), the values of
discharge coefficients for orifice A at different simulated flotation heights were
determined.
Figure 21 shows the effect of the dimensionless flotation height, hid, on the
discharge coefficient, C. The trend of discharge coefficient is similar to the trend of mass
flow rate shown in Figure 20. The discharge coefficient increases with hid in the region
where hid < 1. For hid ~ 1, further increment of hid would not affect the discharge
coefficient. The discharge coefficient for orifice A reaches its asymptotic value when hid
= 1. The value of C for hid ~ 1 is approximately 0.79, which is close to the value of C for
an air jet flowing through orifice A freely, without the presence of the impingement plate
(see Figure 21). The summary of the computed values of discharge coefficient for orifice
A is shown in Table XIV of Appendix A.
47
1
•
.=u.....
.IC.. ,
!~
~.....
2.5 3.0 3.5
o Orifice A
     Orifice A (free jet)
d= 3 mm (0.118 in)
1=3 mm (0.118 in)
8= 33.7°
p{) = 2.49 kPa (10 in~O)
2.0
hid
0.5 1.0 1.5
Figure 21. Computed discharge coefficient of orifice A
0.9
0.8
U
...s 0.7 1:1
..~.. ~ IE 0.6 ~
0
~
~ ~ 0.5
~'"'
..c:I
~ ..~... 0.4
~
0.3
0.2
0.0
Comparison of NearWall Treatments
The turbulence model chosen primarily to solve the discharge coefficient for
orifice A was the Reynolds stress model (RSM) with nonequilibrium wall functions as
the nearwall treatment. Until this point, an the results presented previously were solved
by the RSM with nonequilibrium wall functions as the nearwall treatment. To evaluate
how a different nearwall treatment would affect the discharge coefficient for orifice A,
the twolayer zonal model was chosen as a comparison.
The key difference between the nonequilibrium wall functions and the twolayer
zonal model is that the nonequilibrium wan functions treatment utilizes semiempirical
functions to bridge the viscositydominated region with the turbulencedominated region,
48
whereas the twolayer zonal model uses a modeling approach to estimate the viscositydominated
region. The twolayer zonal model uses turbulence models modified for the
viscositydominated region to predict the solution near the wall.
Figure 22 shows the discharge coefficient of orifice A computed with the
nonequilibrium waH functions and tbe twolayer zonal model. The turbulence model
used in this comparison is again the RSM. The comparison presented in Figure 22 shows
that the difference in the computed results is very small. The agreement in the results
computed by the two different nearwall treatments is quite satisfactory.
1.5 2.0 2.5 3.0 3.5
1::10 Nonequilibrium wall functions
*)( Twolayer zonal model
d= 3 mm (0.118 in)
1= 3 mm (0.118 in)
B= 33.70
Po =2.49 kPa (10 inH20)
0.5 1.0
SSSSSSS:S$$SSSSSfh
W:'~l
L/ e
0.9
0.8
U_r
= 0.7 ~ .'.... c..I e! 0.6 ~
0u
~ t)Jl 0.5
"(o":l ..c= u
..v....l 0.4
~
0.3
0.2
0.0
hid
Figure 22. Comparison of C for orifice A computed with different nearwall treatments
49
5.2 Computational Results of Orifice B
Orifice B is geometrically similar with orifice A, except that orifice B has a
countersink depth. Major dimensions of orifice B are shown in Figure 23. For orifice B,
the simulated flotation height was varied within the range of 0.5 mm ~ h ~ 7 mDl. The
RSM were also compared with results obtained from the realizable k  £ model with the
h
1=3mm
Rigid web
11~3m~1
Plenum surface I
~~ ~~
~SSSSSS\S\SSSSSSSSSSS
Impingement surface
Reynolds stress model (RSM) was chosen as the primary solver along with the
nonequibbrium wall functions for nearwan treatment.
nonequihbrium wall functions for nearwall treatment. The results obtained from the
supply pressure in the plenum was set constant at Po = 2490 Pa (10 inH20). The
Figure 23. The dimensions of orifice B
50
Pressure Distributions of Orifice B
Pressure contours of orifice B on the xzplane for the simulated flotation heights
of h = 1 mm and 4 mm are illustrated in Figure 24. From the observation of the pressure
contours shown in Figure 24,. when hid = 1/3 (see Figure 24a), the pressure gradient in
the radial direction along the gap between the impingement surface and the plenum
surface is considerably larger than that when hid = 4/3 (see Figure 24b).
The pressure contours shown in Figure 25 and Figure 26 are the pressure contours
on the impingement surface and the plenum surface, respectively. Similarly, a larger
pressure gradient near the orifice region can be seen in the case where h is smaller (see
Figure 25a and Figure 26a). When h is larger, the pressure gradient near the orifice
region is smaller (see Figure 25b and Figure 26b). The behavior in the pressure contours
of orifice B is quite similar to that observed in orifice A, as discussed in the previous
section.
51
2500
aJOO
1500
1000
2500
2000
1500
1000
500
(a) Orifice B with h = 1 rom (hid = 1/3)
t.II>./aJ.20OS
RU9\lT ell (9d. 00<0l3 t<ogOll<od. RSId)
(b) Orifice B with h =4 mm (hid = 4/3)
Figure 24. Static pressure contours of orifi.ce B on the xzpJane
52
r
2500
2000
1500
1000
&10
0
&10
_1000
1500
2000 L 2500
(a) Orifice B with h = 1 mm (hid = 1/3)
2500
2000
1500
1000
500
o
1000
1500
2000
2500
116,20.~03
RUENT 6D (ad. ~""3IOl<ld. RStd)
(b) Orifice B with h = 4 mm (hid = 4/3)
Figure 25, Static pressure contours of orifice B on impingement surface
53 
2'500
2000
1500
1000
500
o
_1000
1500
2000
2'500
(a) Orifice B with h = 1mm (hid = 1/3)
2'500
roOD
1500
1000
roo
'.....,20.2003
FLUENr 8.0 (Sd. ~"Ol3..t<od. RS'd)
(b) Orifice B with h = 4 rnm (hid = 4/3)
Figure 26. Static pressure contours of orifice B on plenum surface
54 
The pressure profiles on the impingement surface and on the plenum surface are
shown in Figure 27 and Figure 28, respectively. From both Figure 27 and Figure 2&, the
pressures on the impingement surface and on the plenum surface reacb approximately
atmospheric condition at about 10 rom from the center of the orifi·ce. The pressure
variation occurs drastically near the orifice region, where the jet is exiting through the
orifice and then deflected by the impingement plate. Near the orifice region, there exists
a subatmospheric region (see Figure 27 and Figure 28). The subatmospheric region
becomes stronger with decreasing h, as shown in Figure 27a and Figure 28a. As h
increases, the fluid velocity decreases due to the jet spreading, therefore the subatmospheric
region becomes weaker, as shown in Figure 27b and Figure 28b.
55
. 2.5
~
~ 2.0
~
c.J
~
~= 1.5 rIl ..... =~ 1.0 e~
t)J) ..=.. 0.5 Q. e .=0 0.0
~='"' rIl 0.5 rIl
~
~'"'
1.0
0
d= 3 rom (0.118 in)
/=3 mm (0.118 in)
s = 1.5 rom (0.059 in)
B= 59°
Po = 2.49 kPa (10 in~O)
o hid = 1/6 (h = 0.5 rom)
4)~( hid = 1/3 (h = 1.0 mm)
o hld= 1/2 (h = 1.5 nun)
5 10 15 20 25
Distance from center of orifice, x (mm)
(a) For h at 0.5, 1.0, and 1.5 mID
fO:;" hid = 2/3 (h = 2.0 mm)
)( hid = 4/3 (h = 4.0 mm)
fD=l hid = 2 (h = 6.0 mm)
d= 3 mm (0.118 in)
1=3 mIll (0.118 in)
s = 1.5 mm (0.059 in)
B= 59°
Po = 2.49 kPa (10 inH20)
5 10 15 20 25
. 2.5
~
~
'" 2.0
~
c.J
~ 1.5 =0 ..... =~ 1.0 I e~
t)J) .= 0.5 Q. e....
=0 0.0
~'="' rIl 0.5 0
~~'"'
1.0
0
Distance from center of orifice, x (mm)
(b) For hat 2, 4, and 6 mm
Figure 27. Static pressure profJles on impingement surface for orifice B
56
0.2
20 25
o h/d= 2/3
)( hid = 4/3
o hid = 2
o hld= 1/6
)( hid = 1/3
o hid = 1/2
15
d= 3 mm (0.118 in)
1=3 mm (0.118 in)
s = 1.5 mm (0.059 in)
0= 59°
Po = 2.49 kPa (10 inH20)
d= 3 rnm (0.118 in)
/=3 mm(0.1l8 in)
s= 1.5 mm (0.059 in)
()= 59°
Po = 2.49 kPa (10 inH20)
10
(a) For hat 0.5, 1.0, and 1.5 rom
5
Distance from center of orifice, x (mm)
o
0.0
0.0
0.2
1.0
0.8
0.2
0.4
0.8
0.6
1.0
1.2
0.6
0.4
0.2
1.2
o 5 10 15 20 25
Distaace from center of orifice, x (mm)
(b) For hat 2,4, and 6 mm
Figure 28. Static pressure profiles on plenum surface for orifice B
57
Velocity Contours of Orifice B
Figure 29 illustrates the velocity contours of orifice B. The velocity contours for
the case when hid = 1/3 are shown in Figure 29a, and the velocity contours for the case
when hid = 4/3 are shown in Figure 29b. At hid = 1/3, the small simulated flotation
height causes the air jet to deflect immediately after exiting the orifice. Hence, a strong
velocity magnitude is seen near the edge of the orifice and the axial velocity is weak: due
to the immediate deflection of the jet by the impingement surface (see Figure 29a). At
hid = 4/3, the larger simulated flotation height allows the air jet to exit axially before it is
deflected by the impingement surface. Hence, an axial velocity magnitude can be seen at
the orifice exit (see Figure 29b). Here, a similar behavior in the velocity contours is seen
as was shown in the velocity contours of orifice A. Although the behavior is quirt
similar, there is a difference in the velocity contours due to the difference in the
geometric configuration of each orifice.
58
>~~
. . .. '~
Co nlOUt!l 011 V"boly ~nllld" f!o1/1» tdat 20 , 2003
FLUENT 8.0 "'do ~o'QQedQd, RS'"')
(a) Orifice B with h = 1. rom (hid = 1/3)
70
65
61)
55
5'0
4S
Ml>.t 20 0 2000
FLU~.NF 8D (3d, ~ o'QQll:lCld, RS Iof)
(b) Orifice B with h = 4 rom (hid = 413)
Figure 29. Velocity contours of orifice B on the xzplane
59
Mass Flow Rates and Discharge Coefficients of Orifice B
The mass flow rates were computed by FLUENT (V.6.0) for different values of
simulated flotation height. Figure 30 shows the effect of the dimensionless flotation
height, hid, on the mass flow rate of orifice B. The increment of hid from 1/6 to 1/2 is
the region where the mass flow rate of orifice B increases with the simulated flotation
height, h. From hid ~ 1/2 onwards, the values of mass flow rate remain at an asymptotic
constant value. As shown in Figure 30, the asymptotic constant value of mass flow rate
of orifice B is similar to the value of mass flow rate for a free jet. This asymptotic
constant mass flow rate of orifice B is approximately 0.40 gls.
0.5
o Orifice B
     Orifice B (free jet)
d= 3 mm (O.118 in)
1=3 mm (0.118 in)
s = 1.5 mm (0.059 in)
8= 59°
Po = 2.49 kPa (10 inH20)
0.2
0.0 0.5 1.0
hid
1.5 2.0 2.5
Figure 30. Effect of hid on the mass flow rate of orifice B
60
Using Equation (5.11) with the mcomp computed ~y FLUE T (V.6.0), the
discharge coefficients of orifice B for different simulated flotation heights were
determined. It should be noted that the supply pressure here is set at 2490 Pa (lOinH20)
gage pressure, which gives !¥J =Po  Pamb =2490 Pa. Since the ambient pressure is
Pamb = 0 gage pressure.
Figure 31 shows the trend of the discharge coefficient, C, versus tbe
dimensionless flotation height, hid. The trend of the discharge coefficient is similar to
the trend of mass flow rate shown in Figure 30. The discharge coefficient increases with
hId in the region where hid < 1/2. For hid 2: 1/2, increasing the hid would no longer
affect the discharge coefficient. Figure 31 shows that at hid = 1/2, the discharge
coefficient of orifice B reaches its asymptotic value. This asymptotic constant value for
the discharge coefficient of orifice B is approximately 0.75, which is similar to the value
of C for a free jet flowing through orifice B, without the presence of an impingement
plate (see Figure 31). The summary of the computed values of discharge coefficient for
orifice B is shown in Table XV of Appendix B.
61
.',...
'..
~:
:~~ '. '' ~. ..

s SSSSS\S:SSSSSS$S I"
1 . '0. I·I . ~i ~l
Lo>
o Orifice B
     Orifice B (free jet)
d= 3 rom (0.118 in)
1= 3 mm (D.1l8 in)
s = 1.5 mm (0.059 in)
B= 59°
Po = 2.49 kPa (lOinH20)
0.5 1.0 1.5 2.0 2.5
0.8
0.7
W
..s =QJ 0.6 Y IEQJ
~ ~ 0.5
QJ
bJl
l. =.cl 0.4 y
~ .. Q
0.3
0.2
0.0
hid
Figure 31.. Computed discharge coefficient of orifice B
Comparison ofTurbulence Models
The turbulence model chosen primarily to solve tbe discharge coefficient, C for
orifice B was the Reynolds stress model (RSM). To evaluate how a different turbulence
model would affect the discharge ooefficient for orifice B, the realizable k  & model was
chosen as a comparison. The realizable k  & model in FLUENT (V.6.0) was proposed
by Shih et at. (1995) to improve the deficiencies, of the standard k  & model by using a
new model dissipation rate, &, equation and a new realizable turbulent viscosity, )1t >
fonnulation. The difference between the RSM and the realizable k  & model is that in
RSM, the Reynolds stresses,  pu;u~ > are solved individually by differential transport
62
equations. Whereas in the realizable k  e model, the Boussinesq hypothesis is used to
determined the Reynolds stresses, pu:u~ .
Figure 32 shows the discharge coefficient of orifice B computed with the RSM
and the realizable k  E modeL Both of these turbulence models were used with the
same nearwall treatment, which is the nonequilibrium wall functions treatment. The
comparison presented in Figure 32 shows that the difference in the computed results is
very small. The agreement between the results computed by the RSM and the realizable
k  e model was very well.
o Reynolds stress model
)( Realizable ke model
d = 3 mm (0.118 in)
1= 3 mID (0. ~ 18 in)
s = 1.5 rom (0.059 in)
B= 59°
Po = 2.49 kPa (10 in~O)
0.5 1.0 1.5 2.0 2.5
0.8
0.7
U
...s .=~ 0.6 ~ IS~
Q 0.5 ~
~
.b.Jl ~
~ 0.4 ~
.rI'l Q
0.3
0.2
0.0
hid
Figure 32. Comparison of C for orifice B computed with different turbulence models
63
5.3 Computational Results of Orifice C
Orifice C is simply a straight orifice and can be easily manufactured by drilling a
hole with the appropriate diameter on a plate with the appropriate thickness. The
configuration of orifice C was set such that the plate thickness to the orifice diameter
ratio was 1 (lId = 1). Major dimensions of orifice C are illustrated in Figure 33. For
orifice C, the simulated flotation height was varied within the range of 0.5 mm ~ h ~ 5
mID. The supply pressure in the plenum was set constant at Po = 2490 Pa (to inH20).
The numerical solver chosen to obtain the results of orifice C was the RSM with the
nonequilibrinm wall functions for nearwall treatment. To evaluate the effect of supply
pressure on the discharge coefficient, computations with the supply pressure of Po
4980 Pa (20 inH20) were also considered.
Rigid web
ssss~SSSSSSSSSSSSSSSS\
Impingement surface
1=3mm
h .~~r~3Im~1 ~~ Plenum ~ace ____
Figure 33. The dimensions oforifice C
64
Pressure Distributions ofOrifice C
Pressure contours of orifice C on the xzplane for the simulated flotation heights
of h = 1 rom and 4 mm are illustrated in Figure 34. When hid = II3 (see Figure 34a), the
pressure gradient in the radial direction along the gap between the impingement surface
and the plenum surface is considerably larger than that for h/d= 4/3 (see Figure 34b).
The pressure contours shown in Figure 35 and Figure 36 are the pressure contours
on the impingement surface and the plenum surface, respectively. Similarly, a larger
pressure gradient near the orifice region can be seen in the case where h is small (see
Figure 35a and Figure 36a). When h is large, the pressure gradient near the orifice region
is smaller (see Figure 35b and Figure 36b). The behavior in the pressure contours of
orifice C is quite similar to that observed in orifices A and B, as discussed in previous
sections.
65
I,
2500
alOO
1500
1000
roo
.. ':' •• :' ••• > ~ I: . '.:: :~.."... ....I'.
2500
2000
lsoa
1000
o
roo
1000
1500
_2000
2500
(a) Orifice C with h = 1 mm (hid = 1/3)
t&., 00 , ooee
FLUI:NT 8D (Sd, ~NgafQCf, RSr.!)
(b) Orifice C with h = 4 mm (hid = 413)
Figure 34. Static pressure contours of orifice C on the xzplane
66
::500
3:100
1500
1000
soo
o
so0
1000
1500
3:100
2>00
MI.. 3:1 , 3:103
I=LUENT eD (3d, ~ICo9Il1QCl. RS'"')
(a) Orifice C with h = 1mm (hId = 1/3)
::500
2000
1500
10M
soo
o
soo
1000
1500
2000
::500
,/6,3:1.3:1(6
I=LUB'ST 6.0 (3d. ""&ICo9Il1QCl, RS M)
(b) Orifice C with h = 4 mm (hId = 4/3)
Figure 35. Static pressure contours of orifice C on impingement surface
67
t&..,20, aore
FLUENiT eo (3d. :><Ig K(ll!dad, RS M)
(a) Orifice C with h = 1mm (hid = 1/3)
1500
2000
2500
2500
<!l00
1500
1000
SOD
o
SOD
1000
1500
2000
2500
··· ...... ~ ~~
~ .~ ·'" ~ ..~
·•·..t~. : :l~
'"..
...
'"
II
t,o,o.,3J,20if:I
FLUENT 6.0 (3<1, :><Ig'''''9l!dad, RStd)
(b) Orifice C with h = 4 mm (hid = 413)
Figure 36. Static pressure contours of orifice C 011 plenum surface
68
The pressure profiles on the impingement surface and on the plenum surface are
shown in Figure 37 and Figure 38, respectively. Both Figure 37 and Figure 38 show that
the pressures on the impingement surface and on the plenum surface reach approximately
atmospheric condition at about 10 mm from the center of the orifice. Drastic pressure
variation occurs near the orifice region, where the jet is exiting through the orifice and
then is being deflected by the impingement plate. Near the orifice region, there is a subatmospheric
region (see Figure 37 and Figure 38). The subatmospheric region becomes
stronger with decreasing simulated flotation height, h, as shown in Figure 37a and Figure
38a. As h increases, the fluid velocity decreases due to the jet spreading, therefore the
subatmospheric region becomes weaker, as shown in Figure 37b and Figure 38b.
69
I
....
.....
i· ~....~.fa.
: :~~ · " ·: ~,:..... · '.'.... · ::~
25
25
15 20
d= 3 mm(O.118 in)
1= 3 mm (0.118 in)
Po = 2.49 kPa (10 inH20)
d= 3 rom (0.118 in)
1= 3 mm (0.118 in)
Po = 2.49 kPa (10 inH20)
o hid = 2/3 (h = 2.0 mm)
)( hid = 1 (h = 3.0 mm)
o hid = 4/3 (h = 4.0 mm)
o hid = 1/6 (h = 0.5 mm)
4)'( hid = 1/3 (h = 1.0 mm)
o hid = 1/2 (h = 1.5 mm)
10
(a) For hat 0.5, 1.0, and 1.5 rom
5
5 10 15 20
Distance from center of orifice, x (mm)
. 2.5 =~ 2.0
Q,)
CJ
~ 1.5 =..lO..ll
ij 1.0 isQ,)
OJ)
Cl .... 0.5 c. ..e..
=Cl 0.0
Q,)
"='" lOll 0.5 flO}
Q,)
).0
~
1.0
0
2.5 . =~ 2.0
~
CJ 'C 1.5 =.f.'..l
CI
~ 1.0 is
~
CJ) ..=.. 0.5 c.
..l3..
CI 0.0 Cl
~
"='" lOll 0.5 lOll
Q,) ~'"
1.0
0
Distance from center of orifice, x (mm)
(b) For hat 2,3, and 4 rom
Figure 37. Static pressure profiles on impingement surface for orifice C
70
0.2
(a) For hat 0.5, 1.0, and 1.5 rom
25
o h/d= 2/3
)( hid = 1
o h/d= 4/3
d= 3 mm (0.118 in)
1= 3 mm (0.118 in)
Po = 2.49 kPa (10 inH20)
o h/d= 1/6
)( h/d= 1/3
o h/d= 1/2
d= 3 mm (0.118 in)
1= 3 nun (0.118 in)
Po = 2.49 kPa (10 in~O)
5 10 15 20
Distance from center of orifice, x (mm)
a
1.0
0.2
0.0
0.0
0.8
0.4
0.6
0.2
1.0
1.2
0.8
0.6
0.4
0.2
1.2
a 5 10 15 20 25
Distance from center of orifice, x (mm)
(b) For h at 2,3, and 4 rom
Figure 38. Static pressure profiles on plenum surface for orifice C
71
Velocity Contours of Orifice C
The velocity contours of orifice C are illustrated in Figure 39. The velocity
contours for hid = 1/3 is shown in Figure 39a, and the velocity contours for hid = 4/3 is
shown in Figure 39b. Unlike orifices A and B, at small h (hid = 1/3), strong velocity
magnitude near the edge of the orifice is not observed in orifice C (see Figure 39a). In
this case, a relatively longer length of the orifice (lId = 1) created a mini pipe flow effect.
Hence, the axial velocity appears to be preserved by the wall of the orifice, and is
observed at about 0.5/. Although the axial velocity is preserved here, it is not extended
beyond the orifice because the air jet is still being deflected as soon as it exits the orifice.
At hid = 4/3, the larger h allows the axial velocity to extend beyond the orifice before it is
deflected by the impingement surface (see Figure 39b).
72
'. I~
j~~
00,,''" 0,.'
"...
"o.t '.:~: ~
'.. ' ,.~
I..J
:~~
• ~I
:..~.~
70
65
60
55
50
40
35
30
25
20
15
10
5
o
.,._~~
tiIa,20, 2000
FLUBI1i 6.0 ~d, ~1'QQ1II<ld, RSM)
70
55
50
45
4J)
35
30
25
20
t~
10
5
o
(a) Orifice C with h = 1 rom (hId = 1/3)
1&.,20 , <DOO
FLU8I1T 6D (Sd, ~ I'QQlllcod. RS MJ
(b) Orifice C with h = 4 mm (hId = 413)
Figure 39. Velocity contours of orifice C on the xzplane
73
Mass Flow Rates and Discharge Coefficients of Orifice C
Figure 40 shows the effect of the dimensionless flotation height, hid, on the mass
flow rate of orifice C. From hid = 1/6 to hid = 1/2, the mass flow rate of orifice C
increases with the simulated flotation height. From hid 2: 1/2 onwards, the value of mass
flow rate remains at an asymptotic constant value of 0.39 gls (see Figure 40). As shown
in Figure 40, the asymptotic constant mass flow rate of 0.39 gls is similar to the value of
mass flow rate for a free jet flowing through orifice C.
0.5
....
..~..L' ~. • l<
H
~~
~ I!'.
o Orifice C
     Orifice C (free jet)
d= 3 rom (0.118 in)
1= 3 mm (0.118 in)
Po = 2.49 kPa (10 in~O)
r.:..,:f::r.:"_==~==e=..::er..eee     
SSSSSS:S:SSSSSSSS Ih
~ ~}
0.2
0.0 0.5 1.0
hid
1.5 2.0
Figure 40. Effect of hid on the mass flow rate of orifice C
74
~
Figure 41 shows the.effect of the dimensionless flotation height on the discharge
coefficient. The discharge coefficient increases with hid in the region where hid < 0.5.
For hid 2: 0.5, further increment of hid would no longer affect the discharge coefficient.
The discharge coefficient of orifice C reaches. its asymptotic value when hid = 0.5. The
value of C for hid 2: 0.5 is approximatdy 0.73, which is equal to the value of C for a free
jet flowing through orifioe C witbout the presence of an impingement plate (see Figure
41). Table XVI in Appendix C summarized the computed values of discharge coefficient
for orifice C.
0.8
0.7 ,
U
..,J'
..c=.u=. 0.6 CJ facu
0 0.5 CJ'
cu
QI)
lo.
to: .c: 0.4 CJ
..C.I..l
~
0.3
0.2
0.0 0.5 1.0
hid
o Orifice C
     Orifice C (free jet)
d= 3 mm (0.118 in)
I =3 mm (0.118 in)
Po = 2.49 kPa (10 inH20)
1.5 2.0
Figure 41. Computed discharge coefficient of orifice C
75
Effect of Supply Pressure on Discharge Coefficient of Orifice C
Table II summarizes the discharge coefficient of orifice C computed at different
values of supply pressure, Po for selected values of hid. The comparison shows that
doubling the supply pressure would not affect the discharge coefficient. Hence. this
suggests that the discharge coefficient remains unchanged when the supply pressure
increases up to 4980 Pa (20 inH20).
Table II. Comparison of C for orifice C computed at different Po
hid
Discharge coefficient, C Deviation
Po =2490 Pa Po =4980 Pa (%)
0.33 0.6793 0.6760 0.5
0.67 0.7306 0.7239 0.9
1.33 0.7262 0.7190 1.0
1.67 0.7251 0.7179 1.0
5.4 Computational Results of Orifice D
The configuration of orifice D is quite similar to the configuration of orifice C,
except that orifice D has a smaller lid (lid = 0.2). Major dimensions of orifice Dare
illustrated in Figure 42. For orifice D, the simulated flotation height was varied within
the range of 0.5 mm :s h :s 7 mm. Similar to the previous orifice configurations, the
supply pressure is set constant at Po = 2490 Pa (10 inH20), and the RSM with the
nonequilibrium wall functions for nearwall treatment was chosen as the primary
numerical solver. The results obtained with the supply pressure set at Po = 2490 Pa (10
76
inH20) are also compared with the results computed at the supply pressure of Po = 4980
(20 inH20). The purpose of this comparison is to determine the effect of supply pressure
on the discharge coefficient.
Rigid web ssss=»sssssssssssssssssssss
Impingement surface
Plenum surface~. \"
~~~
d=5mm ~I
h
l=lmm
IL
x,y
Figure 42. The dimensions of orifice D
Pressure Distributions of Orifice D
Pressure contours of orifice D on the xzplane for the simulated flotation heights
of h = 1 mm and 4 mrn are illustrated in Figure 43. When hid = 0.2 (see Figure 43a), tbe
pressure gradient in the radial direction along the gap between the impingement surface
and the plenum surface is considerably larger than that when hid = 1.4 (see Figure 43b).
The pressure contours shown in Figure 44 and Figure 45 are the pressure contours
on the impingement surface and the plenum surface, respectively. Similarly, a larger
77
pressure gradient near the orifice region can be seen when h is smaller (see Figure 44a
and Figure 45a). When h is larger, the pressure gradient near the orifice region becomes
smaller (see Figure 44b and Figure 45b). The behavior of the pressure contours of orifice
D is quite similar to that obs,erved in orifices A, B, and C, as discussed in previous
sections.
78
2500
2000
1500
1000
soo
2500
2000
1500
1000
o
soo
1000
1500
2000
2500
(a) Orifice D with h = 1mm (hid = 0.2)
1m,20,2003
FLUENT 6D (Sd, ~~"'od, RSM)
(b) Orifice 0 with h= 7 mm (hfd= 1.4)
Figure 43. Static pressure contours of orifice D on the xzplane
79
<!SOD
2000
1500
1000
soo
Conlou~ol S1l<Iio PIQ~UIQ ~I
Stagnation
~ressure
(a) Orifice Dwithh= 1 mm(hld=O.2}
2500
2000
1500
1000
soo
o
SOD
1000
1500
2000
2500
Co nlou ~ of SYal io Pl<ltiUIQ ~""'l
t....., a:r ,2l)(X3
I=LUI;NT' BD (3d, ~""9tl1lGd, RSM)
(b) Orifice D with h = 7 mm (hid = 1.4)
Figure 44. Static pressure contours of orifice 0 on impingement surfa;ce
80
2500
alOO
1500
1000
500
0
500
_1000
1500
alOO L _2500 Orifice
Ma., 20,2003
FLUENT 6.0 (3d. ~'GS..rocl, RSM)
(a) Orifice D with h = 1 mm (hid = 0.2)
2500
2000
1500
1000
o
500
1000
1500
2.000
2500
t/c., al. 2il03
I=LUINf 6.0 (3d, ""'il'GSI>:l<od. RSM'J
(b) Orifice D with h = 7 rnm (hid = 1.4)
Figure 45. Static pressure contours of orifice 0011 pl,enum surface
8]
The pressure profiles on the impingement surface and on the plenum surface are
shown in Figure 46 and Figure 47, respectively. From both Figure 46 and Figure 47, the
pressures on the impingement surface and on the plenum surfaee reach approximately
atmo~pheric condition at about 10 mm from the center of the orifice. Drastic pressure
variation occurs near the orifice region, where the jet is exiting through the orifice and
then is being deflected by the impingement plate. Near the orifice region, there exists a
subatmospheric region (see Figure 46 and Figure 47). The subatmospheric region
becomes stronger with decreasing simulated flotation height, h, as shown in Figure 46a
and Figure 47a. As h increases, the fluid velocity decreases due to the jet spreading,
therefore the subatmospheric region becomes weaker, as shown in Figure 46b and
Figure 47b.
82 
.(
U
~ 2.0
§ 0.0
Q,i
~
== ~ 0.5
~
l<
~
1.0
d= 5 mm (0.197 in)
1= 1 mm (0.0394 in)
Po = 2.49 kPa (10 in~O)
o hld= 0.2 (h = 1 mm)
)( hid = 0.4 (h = 2 mm)
o hid = 0.6 (h = 3 mm)
"..
a 5 10 15 20 25
Distance from center of orifice, x (mm)
(a) For hat 1, 2, and 3 mm
2.5
§ 0.0
Q,i
l<
== ~ 0.5
Q,i
l<
~
SSSSS$SSSSSS$$SS d= 5 rom (0.197 in)
1= 1 mm (0.0394 in)
Po = 2.49 kPa (10 in~O)
o hid = 0.8 (h =4 mm)
)( hid = 1.0 (h = 5 mm)
o hid = 1.2 (h =6 mm)
1.0
o 5 10 15 20 25
Distance from center of orifice, x (mm)
(b) For hat 4,5, and 6 nun
Figure 46. Static pressure profiles on impingement surface for orifice D
83 
o h/d= 0.2
)( h/d= 0.4
o h/d=O.6
d= 5 nun (0.197 in)
1= 1mm (0.0394 in)
Po = 2.49 kPa (10 inH20)
0.2
.
~ ~ 0.0
'"
<:Ij
..Csol 0.2 I. ::r ~a 0.4
::r
1:1
~  0.6 Q" =0
~ I. 0.8
::I
<Ll
rG
=I... 1.0
1.2
0 5 10 15 20 25
Distance from center of orifice, x (mm)
(a) For h at 1, 2, and 3 mm
25


.  ,  ..
1/
I
20
   ~ =
0 h/d= 0.8 
)( h/d= 1.0
0 hid = 1.2 '
I
I
15
d= 5 rom (0.197 in)
1= 1 mm (0.0394 in)
Po = 2.49 kPa (10 inH20) _
I
I
5 10
I
I
SSSSSSSS\SSSSS,s
1 I. ri ~I Ih
SSSS"SI rsssss
l
~~~ ~~ ~
A~~':) ~,,~
0.2 ~ 0.0  ~<:Ij
Col 0.2 ~ ..s l. ::r {Il
S 0.4 I
::I =~  0.6  Q" =0
~ 0.8  I. ::r {Il
{Il
<:Ij
l. 1.0 f
~
1.2
0
Distance from center of orifice, x (mm)
(b) For h at 4,5, and 6 mm
Figure 47. Static pressure profiles on plenum surface for orifice D
84
Velocity Contours ofOrifice D
The velocity contours of orifice D are illustrated in Figure 48. The velocity
contours of hid = 0.2 is shown in Figure 48a, and the velocity contours of hid = 1.4 is
shown in Figure 48b. At hid = 0.2, the small simulated flotation height, h causes the air
jet to deflect immediately after exiting the orifice. Hence, a strong radial velocity
magnitude is seen around the edge of the orifice (see Figure 48a). At hid = 1.4, a larger
simulated flotation height allows the air jet to exit axially before it is deflected by the
impingement surface. Hence, an axial velocity magnitude can be seen beyond the orifice
(see Figure 4gb). The behavior of the exit velocity for this orifice resembles the exit
velocity of the sharp edge orifice, Orifice A, due to the short orifice length, I (lId = 0.2).
85
70
65
60
55
SO
30
25
20
15
10
'5
o
70
65
60
55
35
30
25
20
15
10
5
o
(a) Orifice D with h = I mm (hid =0.2)
/'&.,2O.2OClS
FIlUI;,NT eD ('d, ~1ColJ""..:l.RSMJ
(b) Orifice D with h = 7 mm (hid = 1.4)
Figure 48. Velocity contours of orifice D on the xzplane
86
Mass Flow Rates and Discharge Coefficients of Orifice D
The mass flow rates were computed by FLUENT (V.6.0) for different values of
simulated flotation height. Unlike orifices A, B, and C, which have the same exit
diameter of d = 3 mm, the exit diameter of orifice D is d = 5 mm. Therefore, the mass
flow rate of orifice D is expected to be larger. Figure 49 shows the trend of mass flow
rate of orifice D versus the dimensionless flotation height, hid. The increment of hid
from 0.1 to 0.3 is the region where the mass flow rate of orifice D increases with the
simulated flotation height. From hid ~ 0.3 onwards, the values of mass flow rate remain
close to the asymptotic value. The mass flow rate for a free jet flowing through orifice D
without the presence of the impingement plate is mcomp = 1.03 g/s. As shown in Figure
49, the values of mass flow rate for hld~ 0.3 are within 3% of 1.03 gis, which is the mass
flow rate for a free jet flowing through orifice D.
87 
1.2
1.0
0.8
0.6 I
 
o Orifice D
     Orifice D (free jet)
d= 5 mm (0.197 in)
1= 1mm (0.0394 in)
Po = 2.49 kPa (10 inH2O)
1 SSSS\~~s:ss:\\\SS Ih
~ SSSSSI ISS5SS
II
0.4
0.0 0.5
hid
1.0 1.5
Figure 49. Effect of hid on the mass flow rate of orifice D
Using Equation (5.11), the discharge coefficients, C of orifice D for different
simulated flotation heights, h were determined. Again, similar to the computations in
orifice A, B, and C, the supply pressure here was set at 2490 Pa (10 inH20) gage
pressure, which gives /)"P = Po  Pamb =2490 Pa . Figure 50 shows the trend of the
discharge coefficient, C versus the dimensionless flotation height, hid. The discharge
coefficient, C increases with hid in the region where hId < 0.3. For hid ~ 0.3, the
discharge coefficient, C no longer increases with the increment of hId. The discharge
coefficient, C for a free jet flowing through orifice D is C = 0.68. As shown in Figure 50,
the values of discharge coefficient, C for hid ~ 0.3 are within 3% of 0.68, which is the
88
discharge coefficient for a free jet flowing through orifice D. Table xvn in Appendix 0
summarized the computed values ofdischarge coefficient for orifice D

o Orifice D
     Orifice D (free jet)
d= 5 rom (0.197 in)
1= 1mm (0.0394 in)
Po = 2.49 kPa (10 in~O)
0.5 1.0 1.5
SSSSSS\SSSSSS\SS
0.8
0.7
U
...s ..=~.. 0.6 uS
~
Q 0.5 (J
~
bJ)
10
~ = 0.4 (J
~ ....
~
0.3
0.2
0.0
hid
Figure 50. Computed discharge coefficient of orifice D
Effect of Supply Pressure on Discharge Coefficient of Orifice D
As mentioned earlier, the supply pressure was initially set at 2490 Pa (10 inH20)
gage pressure. To evaluate how the supply pressure affects the discharge coefficient of
orifice D, a different value for supply pressure was chosen. Figure 51 shows tbe
comparison of the values of discharge coefficient for orifice D computed with supply
pressures of 2490 Pa (10 inH20) and 4980 Pa (20 inH20). It should be noted that the
89
ambient pressure is Pamb = 0 gage pressure. Therefore, the supply pressures of 2490 Pa
(10 inH20) and 4980 Pa (20 inH20) also imply that the !1p are 2490 Pa (10 inH20)
and 4980 Pa (20 inH20), respectively. Based on the results shown in Figure 51, the
difference in tbe discharge coefficient computed with different supply pressures was
within 4%. Hence, it can be concluded that doubling the supply pressure does not affect
the discharge coefficient of orifice D very much.
1.0 1.5
o Po = 2.49 kPa (10 inH20)
)( Po = 4.98 kPa (20 in~O)
d= 5 mm (0.197 in)
1= 1 mm (0.0394 in)
0.5
1 SS;>SSI~s~SS~S$SS I},
If S'SiS' fiSSSS
0.8
0.7
U
....r = 0.6 ..~...
~ IS~0
0.5 ~
~
OJ)
loo
~ .c 0.4 ~
.."...l
~
0.3
0.2
0.0
hid
Figure 51. The effect of supply pressure on the discharge coefficient of orifice D
90
5.5 Comparison ofTurbulence and Laminar Models
Questions may arise whether the solution solved with laminar flow equations
agrees with the solution solved with Reynolds stress model (RSM) in FLUENT (V.6.0).
The discussion in this section is for the comparison of the results obtained from the RSM
with nonequilibrium wall functions and the laminar flow solver available in FLUENT
(V.6.0). The laminar flow solver available in FLUENT (V.6.0) solves the general
continuity equation and the general instantaneous NavierStokes equations.
To compare the results obtained from the RSM with the laminar flow solver, a
case from each of the four orifice configurations is chosen. All of the cases chosen for
the comparison are such that hid = 1, and the supply pressure is fixed at Po = 2.49 kPa
(10 inH20).. The results obtained from the RSM and the laminar flow solver are
summarized in Table Ill. The difference between the solution from RSM and the
solution from laminar flow is shown less than or equal to 3 percent.
91 _z.
Table III. Comparison ofRSM and laminar flow solution
c C Difference
(RSM) (Laminar) (%)
Orifice A
;~sr~~!~ d=3mm
h=3mm
1=3mm 0.7857 0.7976 1.5 %
B= 33.7°
L/ hid = 1
()
Orifice B
s ssssssss;sssssss!h d=3mm
h=3mm
~'i'~l 1=3mm 0.7472 0.7662 2.5%
s=l.5mm
()= 59°
~ hld= 1
Orifice C
;~ss;s~;!; d=3rnm
h=3mm 0.7290 0.7175 1.6 %
l=3mm
h/d= 1
Orifice D
1 sssss:ss;s:sssssIh d=5mm
h=5mm 0.6966 0.6758 3.0%
l= 1mm
SSSS~ KSSS$ [T h/d= 1
92 
5.6 Influence of Supply Pressure on Discharge Coefficient
To further evaluate the influence of the supply pressure, PO) on the discharge
coefficient, the discharge coefficient for orifice C was determined for different values of
supply pressure, Po' It should be noted that the pressure difference across the orifice,
I'1p, is equal to the supply pressure, Po' since the ambient pressure is zero at gage
pressure. Table IV summarizes the values of discharge coefficient for orifice C
computed at different supply pressures. It is seen that the discharge coefficient does not
deviate much with the supply pressure or the pressure difference across the orifice. The
deviation in the values of the discharge coefficient summarized in Table IV is within 6%
for a relatively wide range of /).p (62 Pa ~ /).p ~ 2490 Pa).
Table IV. Influence of supply pressure, Po' on discharge coefficient, C
Po or I'1p (Pa) C
Orifice C
62 (0.25 inH2O) 0.6908
;j~S~S~;I: d=3mm
h=3mm 100 (0.40 inH2O) 0.7108
[=3 mm
h/d= 1 747 (3.0 inH2O) 0.7303
2490 (10 inH2O) 0.7290
Note: 1). I'1p = Po' since Pamb = 0 at gage pressure
2). Results computed from RSM with nonequilibrium wan functions
93 
5.7 Consideration ofthe d, h, and 1Parameters
The three basic geometric parameters for the four orifice configurations
considered in this study are taken as follow: the exit diameter of orifice, d, the simulated
flotation height, h, and the length of orifice, 1. In this section, the influence of the three
geometric parameters d, h, and I on the discharge coefficient is examined. The influence
of d, h, and I on the discharge coefficient was evaluated with orifice C, because of the
simplicity of its geometric configuration. For orifice C, its exit diameter, d and length, I
are equal in length (d = 3 rom and I = 3 mm). The interest here is to examine how the
discharge coefficient is influenced by d and / for lid = 1. The influence of d and h is also
examined for hid = 1.
Table V summarizes the discharge coefficients computed at various values of d, h,
and I, while hid = lid = 1 was maintained. The deviation in the computed values of
discharge coefficient listed in Table V was less than 0.5%. The discharge coefficient
remains at approximately 0.73 for the different values d, h, and 1 considered. This
finding is useful for developing a correlation equation that predicts the discharge
coefficient for the four orifice configurations considered in this study.
94
Table V. Influence ofthe parameters d, h, and I on the discharge coefficient
Orifice C
d=h=I=3mrn
d=h=I=6mm
d=h=l= lOmm
d=h=l= 14mm
Discharge coefficient, C
0.7290
0.7313
0.7299
0.7279
Note: 1). I1p =Po =2490 Pa (10 inH20)
2). Results computed from RSM with nonequilibrium wall functions
5.8 Correlation Equation for Discharge Coefficient
The discussion in this section involves the development of a correlation equation
that describes the discharge coefficient for all four orifice configurations considered in
this study. The relationship between the discharge coefficient and hid follows a similar
trend for aU four orifice configurations (see Figure 21, Figure 31, Figure 41, and Figure
50). The trend of the discharge coefficient, C versus hid, may be expressed as
where
C
l = the asymptotic constant value for the discharge coefficient,
C
2 = the rate of the exponential,
d = diameter ofthe orifice, and
h = simulated flotation height.
95
(5.81)

The discharge coefficient for each orifice configuration has an asymptotic
constant value (C1) and a rate of exponential (c2 ). Each of the constants c, and c2 may
be expressed by an equation, such that the values of c1 and c2 can be predicted for each
orifice configuration. To do so, the equations for c1 and c2 are expressed in terms of the
geometric parameters that describe each orifice configuration: d, h, I, s, and B. Through a
process that involves curve fitting and trialanderror, the equations for c1 and c2 were
determined:
(
c I 11.7s ) l =O.681.01+0.06
d
+O.15 d O,and
(5.82)
where
d =diameter of the orifice,
h = simulated flotation height,
I = length ofthe orifice,
s =depth of straight portion for countersink orifice, and
0= conical angle of the orifice (in radians).
Having the equations for c1 and c2 determined, the appropriate values of c1 and c2 for
each orifice configurations considered in this study can be predicted. Hence, a
correlation equation for the discharge coefficient was obtained as
(5.83)
where
96 
(
c\ =0.68 1.01+0.06 dI +0.15 11d.7se), and
The graphs shown in Figure 52 illustrate how well the equation represents the
computed values of discharge coefficient for orifices A, B, C, and D. The mean and
standard deviation of the percentage of error of the prediction of equation compared to
the computed results are summarized in Table VI.
Table VI. Mean and standard deviation of error between Ceq and Ccomp
Orifice D 2.10
Orifice A
Orifice B
Orifice C
Mean of% error
0.53
0.74
0.50
Standard deviation of% error
0.59
0.82
0.33
3.25
97 
1.0
o Orifice A
Eq. (5.83)
d= 3 rom (0.118 in)
1= 3 nun (0.118 in)
8= 33.70 (0.59 rad.)
Po = 2.49 kPa (10 inH20)
0.0
0.0 0.5 1.0 1.5
hid
2.0 2.5 3.0 3.5
(a) Comparison ofthe correlation equation with the computed C for orifice A
0.8
U 0.6
.J' ..=~... Cj
S~
0 0.4 Cj
~
bJ) ..cu ...c: Cj
..<..'.l
Q 0.2
o Orifice B
Eq. (5.83)
d= 3 mm (0.118 in)
1=3 mm (0.118 in)
s = 1.5 rom (0.059 in)
8= 59° (1.03 rad.)
Po = 2.49 kPa (10 in~O)
0.0
0.0 0.5 1.0
hid
1.5 2.0 2.5
(b) Comparison of the correlation equation with the computed C for orifice B
98
~
0.8
o Orifice C
Eq. (5.83)
d= 3 mm (0.118 in)
1= 3 nun (0.118 in)
Po = 2.49 kPa (10 inH20)
0.0
0.0 0.5 1.0
hid
1.5 2.0
(c) Comparison ofthe correlation equation with the computed C for orifice C
0.8
U
.J' 0.6
..=~...
("j e~8
0.4 I
Q,j
bill
J~
..=
("j
fI) S 0.2
0.0
0.0
o
SSSSSSSSS\S$S$.$S
0.5
hid
o Orifice D
Eg. (5.83)
d= 5 mm (0.197 in)
1= 1 mm (0.0394 in)
Po = 2.49 kPa (10 inH20)
l.0 1.5
(d) Comparison of the correlation equation with the computed C for orifice D
Figure 52. The comparison between correlation equation and the computed C
99
~
5.9 Influence of Reynolds Number on Discharge Coefficients
The conventional definition of the Reynolds number describing a circwar jet may
be expressed as
R pud 4pQ 4m
ej ===_.
Jl Jltrd Jltrd (5.91)
Similarly, the jet Reynolds number for the computational results can be expressed as
where
4m Re. =c"om'p
J Jltrd'
d =diameter of the orifice,
mcomp =computed mass flow rate, and
Jl =1.79x 105 kg/ms (viscosity of air defined in FLUENT).
(5.92)
Equation (5.92) shows that the Reynolds number, Rej , is linearly proportional to the
mass flow rate, mcomp (Rej oc mcomp ). Therefore, when the Reynolds number is plotted as
a function of hid, the trend of the Rej versllsh curve is expected to be quite similar
with the Cversush and the mversush curves.
The configuration of orifice B is slightly similar to the conical entrance orifice
(see Figure 4 on page 10). The conical entrance orifice bas a discharge coefficient of
0.73 (Miller, 1983), and the discharge coefficient of orifice B for 5 is 0.747. Figure 53
shows the comparison between the discharge coefficients of the conical entrance orifice
and orifice B. The agreement in the discharge coefficient of orifice B with the discharge
coefficient of the conical entrance orifice is within 2.3%.
100

0.8
o
0.5
101
o Orifice B for hid> 0.5
Conical entrance orifice (Miller, 1983)
105
Figure 53. Comparison ofdischarge coefficients for conical orifice and orifice B
The discharge coefficient of orifice C for hid = 1 is plotted with various values of
Reynolds number (see Figure 54). For the range of Reynolds number in Figure 54, 1400
::; Re
j
::; 9300, the values of discharge coefficient for orifice C is within the range 0.69 ::;
C::; 0.73. Hence, the deviation in the discharge coefficient of orifice C for 1400::; Rej ::;
9300 is within 6%.
101
0.8
o 0
Orifice C:
hld=l
d= 3 rom (O.ll8 in.)
h = 3 mm (0.118 in.)
1=3 mm (O.U8 in.)
Re.
J
Figure 54. Discharge coefficient versus Reynolds number for orifice C
Figure 55 shows the discharge coefficient, C and the Reynolds number, Re.
J
computed for orifice D. As mentioned earlier, the trends of Rej versus hid are similar
to that of C versus hid, since C ex: mand Rej oc m(see Figure 55a). Note that Po =I1p ,
since the ambient gage pressure, Pamb is zero. The discharge coefficient versus the
Reynolds number is plotted for various values of hid in Figure 55b. Table VII
summarizes the values ofC and Rej computed at different hidand Po for orifice D.
102
Rej (Po = 2.49 kPa)
Rej (Po = 4.98 kPa)
A C (Po = 2.49 kPa)
o C (Po = 4.98 kPa)
•
Orifice D:
d= 5 rom (0.197 in.)
1= 1 rum (0.039 in.)
~ : : : : : : : : : : :
0.4
0.5
0.7
0.8
0.9
1.0 ,,r.,.....r.....,
0.6
c
102
1.0 1.5
0.3 "";:_L.__~.l.l.LJ
0.0 0.5
hid
(a) Discharge coefficient and Reynolds number versus hid
Orifice D:
d= 5 mm (0.197 in.)
1= 1mm (0.039 in.)
$SSS$$$$$$$$S$S$
1 ,. d _, Ih
S$SSSI KSSSS
zT
&.Ab
G _
EJ
o hid = 0.1
o hid = 0.2
A hid> 0.3
...0
0.8
0.7
U
...r ..=~.. 0.6 CJ If: ~C=J 0.5
~
~
lo.o
~ = 0.4 CJ 0 .fI.l ~
0.3
0.2
8000 10000 12000 14000 16000 18000 20000 22000
Re.
J
(b) Discharge coefficient versus Reynolds number for various hid
Figure 55. Discharge coefficient and Reynolds number for orifice D
103
_.~
Table VII. Summary of C and Re at different hid and p for orifice D J 0
Po =!Y.p =2.49 kPa Po =!Y.p =4.98 kPa
hid Rej C Rej C
0.1 8200 0.383 11800 0.384
0.2 12900 0.598 17700 0.575
> 0.3 14900 0.692 20900 0.679
Table VIn summarizes the values of C and Rej for orifices A, B, C, and D. The
discharge coefficients listed in Table VIn are the asymptotic values for orifices A, B, C,
and D. The values of Reynolds number listed in Table VIII are the Rej for orifices A, B,
C, and D at the range of hid where the asymptotic values of C are reached. Note that
the results listed in Table VIII were computed with the supply pressure of Po =2.49 kPa.
Table VIII. Summary ofC and Rej for orifices A, B, C, and D
Orifice A
Orifice B
hid
2:1
2: 0.5
c
10100 0.789
9600 0.747
Orifice C 2: 0.5 9300 0.727
Orifice D 2: 0.3 14900 0.692
Note: The above results were computed with Po =!Y.p =2.49 kPa
5.10 Closing Remarks for Chapter V
The effect of the simulated flotation height on orifice discharge coefficients was
studied computationally. The trends of C versus hid curves appeared to be similar
104
_.~
''I'", •
regardless of the orifice configurations. It was found that the discharge coefficient
increases with the flotation height when the flotation height is small, but it becomes a
constant when a certain value of flotation height is reached. Table IX is a summary of
the asymptotic values of discharge coefficient for different orifice configurations.
Table IX. Summary of the asymptotic values of discharge coefficient
hid Asymptotic value of C
Orifice A 2:1 0.789
Orifice B 2:0.5 0.747
Orifice C ~O.5 0.727
Orifice D 2: 0.3 0.692
Note: The above results were computed with Po == tip == 2.49 kPa
Orifice B bas similar configuration with a conical entrance orifice (see Figure 4
on page 10), which has a discharge coefficient of 0.73 (Miller, 1983). As listed in Table
IX, orifice B bas a discharge coefficient of 0.747 when hid 2: 0.5, which is only 2.3%
larger than the discharge coefficient of the conical entrance orifice, C = 0.73.
The comparison between the solutions solved with turbulence model and the
solutions solved with laminar flow equations showed good agreement. The difference
between the two was within 3% (see Table III on page 92).
The discharge coefficient for orifice C was not strongly affected by the supply
pressure when it was increased from 2490 Pa to 4980 Pa for 0.33 :s hid :S1.67 (see Table
II on page 76). The discharge coefficients for orifice D were computed with the supply
pressures of 2490 Pa and 4980 Pa at various hid values. The difference in the discharge
105
_.~
coefficient for orifice D computed with the two supply pressures was less than 4% (see
Figure 51 on page 90)..
The influence of the supply pressure, Po' on discharge coefficient was further
evaluated for various values of supply pressure. The deviation in the values of the
discharge coefficient within the range of 62 Pa:S Po :S 2490 Pa was less than 6% (see
Table IVan page 93). Note that the supply pressure was equal to the pressure difference
across the orifice, Ixp. Since 6p =Po  Pamb' and the ambient pressure, Pamb' was at zero
gage pressure.
The discharge coefficient for orifice C was further evaluated for different values
of geometric parameters: d, h, and I. It was found that the discharge coefficient
calculated for hid = lid = 1 does not strongly depend on individual values of d, h, and 1
(see Table V on page 95).
A correlation equation, Equation (5.83), that predicts the discharge coefficient
for different orifice configurations was established. The correlation equation was
established based on the computed resu~ts for orifices A, B, C, and D. The standard
deviation in the percentage of error for the correlation equation was found to be less than
3.5% when compared with the computed results (see Table VI on page 97). The
correlation equation should perfonn adequately within the range of 62 Pa :S fJ.p :S 4980
Pa, and for various values of d, h, and I.
106
CHAPTER VI
EXPERIMENTAL STUDY
In this chapter, the discussion on the experiment undertaken for verification of the
computational results and the correlation equation, Equation (5.83), is presented. This
chapter is organized in four sections. Section 6.1 presents the experimental setup,
specifically designed for the study of discharge coefficient. Section 6.2 discusses the
procedure of the experiment. In Section 6.3, a sample calculation of the experimental
data is shown. Here, the procedure involved for calculating the discharge coefficients
from the measurement data is described. Section 6.4 describes tbeestimation of
uncertainty of the experimental data. The experimental results are presented in Section
6.5. Finally, the closing remarks for this chapter are given in Section 6.6.
6..1 Experimental Setup
The mam objective of this experiment was to verify the validity of the
computational results and the correlation equation. A schematic diagram of the test
section is illustrated in Figure 56. The test section was designed such that the orifice and
the impingement plates were situated inside an 8 in. diameter transparent plexiglass
cylinder. Most components for the test section were made of aluminum. The transparent
cylinder was covered at its base and top with circular aluminum plates. At the base of the
107
cylinder, compressed air was supplied into the plenum through an inlet. The compressed
air was regulated by a pressure regulator before being supplied into the plenum. At the
inlet, a small perforated cylinder was used to spread the inlet air flow radially. This was
done to make the pressure inside the plenum as unifonn as possible. At the top, air flow
exiting the test section through the outlet was measured. This was done by connecting a
variablearea flowmeter (rotameter) to the outlet of the test section. The reason for this
was to collect and measure the mass flow rate of the air downstream of the impingement.
The basis of this experiment was that continuity was satisfied, such that the mass flow
rate at the orifice was equal to the mass flow rate at the outlet.
12 in.
Pressure
taps
Circular top
plate
Circular
impingement plate
Circular orifice
plate
Transparent nylon
cylinder
73/4 in.
...4.+ Small perforated
cylinder
Circular base
plate
Inlet
Figure 56. Schematic of the test section
108 
The orifice and impingement plates were made of aluminum also. The circular
orifice and impingement plates were 6 inches and 5 inches in diameter, respectively. The
gap between the orifice and the impingement plates (simulated flotation height, h) was
accurately adjusted with precision feeler gauge spacers. The surfaces of the orifice and
the impingement plate were machined carefully so that they were perfectly flat and
smooth. AU components of the test section were precisely machined and properly
assembled to avoid leakage.
Figure 57 illustrates the discharge coefficient measurement setup. The inlet was
connected to a Parker 8AWl13YA pressure regulator. Attached to the pressure regulator
were two Norgren F55001LOTO and F08000AITO filters. Compressed air was
supplied to the pressure r,egulator, flowing through the filters and into the test section.
Air flowed through the orifice and impinged on the impingement plate inside the test
section. The outlet was connected to a Dwyer VA20438 variablearea flowmeter
(rotameter), where the mass flow rate was measured. Pressure taps were attached for
pressure measurement at upstream and downstream of the orifice.
109
',"
Figure 57. The discharge coefficient measurement setup
6.2 Experimental Procedure
Before the experiment was undertaken, the diameter and the length of the orifice
were determined. The size of the reamer used for reaming the orifice was taken as
diameter of the orifice. The diameter of the orifice was then verified by a Mitutoyo
BH305 coordinate measuring machine. The diameter measured by the Mitutoyo BH305
coordinate measuring machine was slightly larger than the size of the reamer, but the
difference was less than 0.3%. The length of the orifice was measured by a dial caliper.
110
The quantities measured in this experiment were m. P P A... and T
meas' 0' amb'~' room"
The main variable of this experiment was the simulated flotation height, h, which was
accurately adjusted with precision feeler gauge spacers. At an adjusted h, the quantities
mmeas' PO' Pamb' I1p, and Troom were measured. The mass flow rate was measured by a
Dwyer VA20438 variablearea flowmeter. The measurable range for the Dwyer
VA20438 rotameter was 996 to' 16737 std. ml/min (1.66 x 105 to 2.79 X 104 std. m3/s).
The rotameter was connected to the outlet at the top of the test section to measure the
amount air flow exiting through the outlet.
The plenum pressure (or pressure upstream of the orifice), Po' was measured by a
Dwyer 123036WM vertical manometer [0 to 36 inH20 (0 to 8967 Pa»). The ambient
pressure, Pamb' and the room temperature, ~oom' in the laboratory were measured by a
ColeParmer aneroid barometer. The pressure difference across the orifice, !1p
(!1p =Pc>  PI)' was measured by a Dwyer 215 inclined manometer. The measurable
range for the Dwyer 215 inclined manometer was 0 to 0.3 inH20 (0 to 75 Pa).
6.3 Sample Calculation ofExperimental Data
To illustrate the process involved in determining the discharge coefficient from
the measured quantities mmeas' Po. Pamb' !1p, and ~oom' the following sample
calculation is presented. The equation used to determine the discharge coefficient from
the measured quantities is similar to that described by Equation (3.18) in Chapter III,
(6.31)
111 
The standard design and calibration conditions of the Dwyer flowmeter are
where the experiment was undertaken were slightly different from the design and
calibration conditions; therefore a correction factor is necessary to obtain the actual flow
. F:' ( (6.32) r&:] . m aCl = YAmmeas = Vp:: mmeas'
ruVA = ~P,.. .
Ps!Jj
Substitution ofEquation (6.32) into Equation (6.31) yields
mass flow rate, mmeas:
Hence, the actual mass flow rate, maC1 ' can now be obtained by correcting the measured
Psrd. = 1 atm (101325 Pa) and J:ld =70 of (21.1 °C). However, the conditions in the lab
rate. The correction factor, FyA , is defined as
(6.33)
In this analysis, the air density is assumed to obey the ideal gas law,
p = Po + Pm>b
RTroom
(6.34)
Substituting Equation (6..34) into Equation (6.33) yields,
(6.35)
where
d =diameter of the orifice,
mmeas =measured flow rate,
Il2
~_.
Pamb = measured lab ambient pressure,
Po = measured gage pressure in plenum (upstream of orifice),
Pstd = 101325 Pa (standard calibration conditions of flowmeter),
fj,p = pressure drop across the orifice,
R = 287.03 J/kgK (universal gas constant for air),
T.oom = measured room temperature, and
p = air density in the plenum (upstream of orifice).
Now, consider a sample experiment where the gap between plenum surface and
impingement surface was h = 3.048 mm (0.120 in.), the orifice diameter was d = 3.175
mm (0.125 in.), and the length of the orifice was 1= 3.277 mm (0.129 in.). The measured
quantities were mmeas =7.466 x 105 kg/s, Po = 1654 Pa, Pamb = 99600 Pa, fj,p = 62.27 Pa,
and I;.oom = 293.15 K. Hence, the discharge coefficient for d = 3.175 mm (0.125 in.) and
h = 3.048 mm (0.129 in.), making hid = 0.96, could be determined as
= 4(7.466 x l0
s kg~S)( 1__
;r(0.003175 m)
=0.7637.
(287.03 J/kgK)(293.15 K)
2(99600 Pa +1654 Pa)(62.27 Pa)
Thus, for hid = 0.96, the discharge coefficient was experimentally determined to be C =
0.7637.
113 
6.4 Estimation of the Uncertainty ofC
The experimental discharge coefficient is a function of six independent measured
variables: C(d,mmeas,Pamb,Po,Ap,Troom )' Each of these measured variables includes
some uncertainty, and these uncertainties will lead to an uncertainty in the calculated
result, which is the discharge coefficient, C. To fmd the uncertainty of the discharge
coefficient, the propagation of uncertainty approach was used. This approach was
established by Kline and .McClintock (1953).
As mentioned earlier, the discharge coefficient, C, is a function of six independent
measured variables:
RTroom (6.41)
where
d = diameter ofthe orifice,
m =measured mass flow rate, meas
Pamb =measured lab ambient pressure,
Po =measured gage pressure in plenum (upstream of orifice),
P =101325 Pa (standard calibration conditions of flowmeter),
sId
!J.p =pressure drop across the orifice,
R =287.03 J/kgK (universal gas constant for air),
T =measured room temperature, and
room
p = air density in the plenum (upstream of orifice).
114 
Usi.ng the propagation of uncertainty approach, the uncertainty of discharge coefficient,
c' , is assumed to behave much like standard deviation:
c' =[(aacd d*)2 +(aa.c ri't J2 +(~ . J2 meas a Pamb
mmeas 'Pamb
(6.42)
( J
ae. 2 ( ac ~ •J2 ( ac T' J2 ]~ + a Po + f)~. p + ~ room· ,
'Po P room
where d• ,mm. •eas' Pa•mb' p0' ' 'A¥n' * , and Tr*oom are the uncertainties of the measured
variables. The derivatives in Equation (6.42) are
ac =[4mmeas ( ~J
ad 1r vP:
ae 4 ( ~J ,_RT~room
ammeas = 1rd2 ~p::; 2(po + Pamb)~P'
ae 4· mmeas [(Piafmb] RTroom [ 0.5 J
apo = 1rd2
PSld 2~p (Po + Pamb).Yz '
115 
To illustrate the process involved in detennlning the uncertainty of the discharge
coefficient using the propagation of uncertainty approach, the following sample of
calculation is presented. The similar case shown in the sample calculation of
experimental data presented in Section 6.3 is used here as an example to show the
estimation of the uncertainty: d = 3.175 mm, h = 3.048 mm, 1= 3.277 mm, mroeas = 7.466
x 105 kg/s, Po = 1654 Pa, Pamb = 99600 Pa, ~ = 62.27 Pa, and Troom = 293.15 K.
The uncertainties of the six measured variables, d, m:eas ' P:'b' P:, !1p., and
r· for this case (where hid = 0.96 and lid = 1.03) are given as follow: room
d±d· =3.175±0.00254 mm,
m +m· =7.466xlO5 ±0.2317xl05 kg/s, meas meas
Pamb ±P:mb = 99600±lOO Pa,
Po ±p: = 1654±25 Pa,
!1p ± ~ = 62.27 ± 1.25 Pa, and
T +T· =293.1S±1K. room  room
With the given values listed above, the derivatives III Equation (6.42) could be
calculated:
ac_481 mI,
ad
ac =10229 s/kg,
ommeas
~=6.2626xl08FaI,
oPamb
116 
ae
=3.77l3xlO6 PaI
aJ70 '
ac
a~ =6.1323xlO3 Paol
, and
ac
= 1.3026xlO3 K1
aT,oom
Substituting the values shown above into Equation (6.42) yields, C· =0.025. Thus, the
uncertainty in the measur,ed discharge coefficient for hid = 0.96 and lid = 1.03 was
estimated to be C = 0.7637 ± 0.025.
6.5 Experimental Results
The configuration of the orifice used in this experimental study was similar to that
of orifice C. Figure 58 shows the schematic for the configuration of orifice C. The
experimental study undertaken here was done on two different dimensions of orifice C:
lid = 1.03 with d = 3.18 mm, and lid = 0.94 with d = 3.45 mm. The discharge
coefficients for the two orifi.ces with similar configurations but different dimensions were
determined experimentally at different values of simulated flotation height, h. The
discussion in the following subheadings of this section presents the measured results for
the two orifices with different dimensions: lid = 1.03 and lid = 0.94.
117
 z...
Rigid web
sss~sssssssssssssssss
Impingement surface
Figure 58. Schematic for the configuration of orifice C
Experimental Results for Orifice C with lid = 1.03
h
I
The test conditions for orifice C with lid = 1.03 are summarized in Table X. As
mentioned earlier, the dimensions for orifice C with lid = 1.03 were d = 3.18 mm and I =
3.28 mm. The pressure drop across the orifice was set as constant at 62 Pa. The main
variable in this experiment was the simulated flotation height, h, and it was varied from 0
to 7.49 rom (0 ~ hid ~ 2.36). The summary of the measured quantities and discharge
coefficient for this experiment is shown in Table XVIII of Appendix E.
Table X. Summary of experiment conditions for orifice C with lid = 1.03
Pressure drop across orifice, !1p
Orifice diameter, d
Orifice length, I
Simulated flotation height, h
118
62 Pa (0.25 inH20)
3.18 rom (0.125 in.)
3.28 nun (0.129 in.)
oto 7.49 mm (0 to 0.295 in.)
Figure 59a shows the measured discharge coefficient for orifice C for lid = 1.03.
The measured results illustrated in Figure 59a show that the values of discharge
coefficient were equal to the free jet discharge coefficient at hid 2: 0.8. The measured
free jet discharge coefficient for orifice C was 0.7518.
Figure 59b shows the measured results, the computed results, and the results of
correlation equation, Equation (5.83). The computed results were calculated under
similar conditions as the experiment conditions. The errors of the measured results from
the computed results at hid 2: 0.8 were within 8%. The largest error in the comparison
between the measured and the computed results was 14% at hid = 0.48. This error of
14% between the measured and the computed results was observed at the peak of the
bump in the measured data. The occurrence of the hump in the measured data is
unexplained. It is suspected that the computational model was unable to capture such
occurrence. Note that the error of the measured results from the computed results was
calculated with
The agreement between the measured results and results of Equation (5.83)
appears to be satisfactory (see Figure 59b). For hid 2: 0.8, the errors between the
measured results and Equation (5.83) were witbin 5%. The smallest error between the
measured results and Equation (5.83) was 3.1% at hid = 2.36. The largest error in the
comparison between the measured results and Equation (5.83) was 11.6% at hid =0.40.
The mean and the standard deviation of the error between the measured discharge
coefficients and Equation (5.83) are summarized in Table Xl. The mean and the
standard deviation of the errorbeween the measured discharge coefficients and Equation
119
(5.83) are 5.9% and 2.6%, respectively. Hence, the agreement between the measur,ed
results and the results of Equation (5.83) is quite satisfactory. Note that the error in the
measured discharge coefficients from the results of Equation (5.83) was calculated with
% error = l(Cmeas  Ceq )/CeqlxlOO.
Table Xl. Mean and standard deviation of error between Cmcas and Ceq for lid = 1.03
r
Mean of% error
Standard deviation of% error
120
5.9%
2.6%
1.0
\.,) 0.8
..s ..=~... y 0.6 e~
Q
y
~
OJ) ... 0.4 =..cl y
..r.I..l
~
0.2
e Measured
     Measured free jet
d= 3.18 mrn (0.L25 in.)
1= 3.28 mm (0.129 in.)
!1p = 62 Pa (0.25 inH20)
0.4 0.8 1.2 1.6 2.0 2.4 2.8
0.0 ~L...L._..L.LL.._'LL.L_...LL...l_~
0.0
hid
(a) Measured discharge coefficient, C
1.0
2.4 2.8
x
2.0
Measured (!1p =62 Pa)
Computed (!1p = 62 Pa)
Eq. (5.83) (!1p = 2490 Pa)
d= 3.18 mm (0.125 in.)
1= 3.28 mm (0.129 in.)
1.6
o
x
0.8 1.2
x x
0.4
0.2
0.0 ~I_.....L.._~.L._....LL_....1_L.L._.J...L_...JJL.......J
0.0
hid
(b) Comparison of measured data, computed data and Equation (5.83)
Figure 59. Measured C, computed C, and Equation (5.83) for orifice C with lid = 1.03
121
For a constant pressure drop across the orifice !1p = 62 Pa, it was necessary to
adjust the plenum pressure, Po' when h was varied. Figure 60 shows how the plenum
pressure was changed with h for t:.p = 62 Pa. Notice that h and Po in Figure 60 are
presented in dimensionless form, hid and Pol!1p. Note the similarity between the Poversus
h curve and the Cversllsh curve (Figure 60 and Figure 59). Similarly to the
discharge coefficient, the plenum pressure is also affected by the mass flow rate;
therefore such trend in Po versus h was observed in Figure 60.
30
25
d= 3.18 rom (0.125 in.)
1= 3.28 mm (0.129 in.)
!1p = 62 Pa (0.25 inH20)
20
15
0.0 0.4 0.8 1.2
hid
1.6 2.0 2.4 2.8
Figure 60. Measured plenum pressure, Po' affected by flotation height, h, for lid = 1.03
122 
The influence of mass flow rate on Po can be qualitatively explained. To satisfy
continuity, the mass flow rate at the orifice is equal to the mass flow rate at the outlet of
the test section (see schematic cftest section in Figure 56):
mm m  orifice  outlet
=(pAu) =(pAu) orifice outlet
Hence, the mass flow rate at the orifice and the outlet of the experiment setup may be
expressed as
I
I
I
I!
II
II
(6.51)
At gage pressure, the ambient pressure, Pamb' is zero; hence Equation (6.51) becomes
(6.52)
Notice that in Equation (6.52), Aorifice' Aout'et' p, and (Po  PI) may be treated as
constants. The relationship between mand PI in Equation (6.52) may be written as
• 2 rn
PI = A2 (6.53)
2p Mllet
As m increases with h, Equation (6.53) shows that PI increases with rh, where
PI oc m2
• Hence, as h affects m,min tum affects PI .
The relationship of PI and Po shown in Equation (6.52) is
Po =Pl[1 +( AO~llel J2].
Aonfice
Substitution of Equation (6.53) into Equation (6.54) yields
123
(6.54)
m
2
P = [t+ (A~)2]
o 2 A2 . . • P outlet Aorifice
(6.55)
Likewise, Equation (6.55) shows that Po ce fn2. Hence, as m increases with h, Po
increases with m. Because h affects m, at the same time m affects PI and Po'
therefore the variation of the plenum pressure, Po' with the simulated flotation height, h,
is observed in Figure 60. The maximum measured plenum pressure, Po shown in Figure
60 was 1780 Pa (7.2 inH20).
Experimental Results for Orifice C with lid = 0.94
The test conditions for orifice C with lid = 0.94 (see test conditions summarized
in Table XII) were set quite similarly as the test conditions for lid = 1.03. The main
difference in the test conditions here was the dimensions ofthe orifice: d = 3.45 mm and 1
= 3.25 mm. The main variable in this experiment was the simulated flotation height, h,
and it was varied from 0 to 7.11 rum (i.e.. 0 :s hid :s 2.06). Similar to previous test
conditions, the pressure drop across the orifice, !:J.p (i.e.. 6.p =Po  PI) was set as
constant at 62 Pa. The summary of the measured quantities and discharge coefficient for
this experiment is listed on Table XIX in Appendix E.
Table XII. Summary of experiment conditions for orifice C with lid = 0.94
Pressure drop across orifice, !:J.p
Orifice diameter, d
Orifice length, I
Simulated flotation height, h
124
62 Pa (0.25 inH20)
3.45 mm (0.136 in.)
3.25 mm (0.128 in.)
oto 7.11 rom (0 to 0.295 in.)

Figure 61a shows the measured discharge coefficient for orifice C with lid = 0.94.
The measured results illustrated in Figure 61a show that at hid ~ I, the values of
discharge coefficient are equal to the free jet discharge coefficient. The measured free jet
discharge coefficient of orifice C with lid = 0.94 was 0.7449.
Figure 6lh shows the comparison of the measured results with Equation (5.83).
The agreement between the measured results and results of Equation (5.83) appears to be
satisfactory. For hid ~ I, the errors between the measured results and Equation (5.83)
were within 5.5%. The smallest error between the measured results and Equation (5.83)
was 2.5% at hid = 1.76. The largest error in the comparison between the measured
results and Equation (5..83) was 11.6% at hid = 0.59.
The mean and the standard deviation of the error between the values of measured
discharge coefficient and Equation (5.83) are summarized in Table XIII. The mean and
the standard deviation of the error between the values of measured discharge coefficient
and Equation (5.83) were 5.9% and 3.6%, respectively. Similar to the experiment for
orifice C with lid = 1.03, the agreement between the measured results and Equation
(5.83) in this case for lid = 0.94 was satisfactory also. Note that the error in the
measured discharge coefficient from the results of Equation (5.83) was calculated with
% error =1(Cmeas Ceq)/CeqlxIOO.
Table XUI. Mean and standard deviation of error between Cmeas and Ceq for lid =0.94
Mean of% error
Standard deviation of% error
125
5.9%
3.6%

1.0
0.2
e Measured
     Measured free jet
d= 3.45 mm (0.136 in.)
1= 3.25 mm (0.128 in.)
/1p = 62 Pa (0.25 in~O)
1.2 1.6 2.0 2.4
hid
0.4 0.8
o.0 ~L._..L...L._.L.....I_L.....1_L.lJL.......J.....J
0.0
(a) Measured discharge coefficient, C
1.0
1.6 2.0 2.4
Measured (/1p = 62 Pa)
Eq. (5.83) (/1p = 2490 Pa)
d = 3.45 mm (0.136 in.)
1= 3.25 mm(0.128 in.)
1.2
hid
<>
0.4 0.8
0.0 ~_L...L._~_..J..._'L._~_...L..I'1Ll
0.0
0.2
0.8
(b) Comparison of measured data with Equation (5.83)
Figure 61. Measured C and Equation (5.83) for orifice C with IId = 0.94
126 
Sllnilar to the test for lid = 1.~3, the pressure drop across the orifice in this test for
lid = 0.94 was set at t¥J = 62 Pa. Figure 62 shows the effect of the varying simulated
flotation height, h on the plenum pressure, Po' The h and the Po in Figure 62 are
presented in dimensionless parameters, hid and Polt¥J, respectively. Note that ~ and d
were 62 Pa and 3.45 mm, respectively.
The trend of the plenum pressure, Po' with h observed in the test for lid = 1.03
(see Figure 60) was observed in this test for lid = 0.94 also (see Figure 62). The values of
Po observed in this test for lid = 0.94 were higher than those observed in the test for lid =
1.03 (compare Figure 60 with Figure 62). This is because the orifice diameter, d in this
test for lid = 0.94 was approximately 9% larger than the orifice diameter, d,. in the test for
lid = 1.03. As indicated by Equation (6.51),
the larger A.~ in this test for lid = 0.94 had caused an increase in the mass flow rate,
on:l~ce
m. With higher m, Equation (6.55) indicates that the plenum pressure, Po' also should
increase. The maximum measured plenum pressure, Po' shown in Figure 62 was 2240
127
iII

40
35
30
25
d = 3.45 nun (0.136 in.)
1= 3.25 mm (0.128 in.)
/1p = 62 Pa (0.25 inH20)
20
15
0