mélanie beauchemin - university of toronto t-space · les poules, and fantasio for the happiness...

INVESTIGATIONS OF NOZZLE DISCHARGE COEFFICIENTS IN A

COMPLIANT AIR BEARING SYSTEM

Mélanie Beauchemin

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Graduate Department of Aerospace Science and Engineering University of Toronto

Copyright @ 1999 by M é h i e Beauchemin

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INVESTIGATIONS OF NOZZLE DISCHARGE COEFFICIENTS IN A COMPLIANT AIR BEARING

SYSTEM

Mélanie Beauchemin Master of Applied Science

Graduate Department of Aerospace Science and Engineering

University of Toronto 1999

Abstract

A novel compliant air bearing is being developed for matenais handling and low speed

guided transportation systems. In a typical application the load being transported is camied on

a 1 m. square platform; this platform is supported on two compliant elements, or runners, which

move along shallow concave guideways, or rails. Cmently the runners are 1 m. long cylinders

having an 108mm. x 58 mm. section; the rails have a 152 mm. radius and are 127 mm. wide.

Air for bearing action is introduced fiom supply manifolds integrated with the underside of the

rail through 0.52 mm diameter nozzies which are spaced at 152 mm intervals along the rail

axis. With loads as high as 1500 kg the system can achieve effective coefficients of sliding

friction as low as 0.1 percent with ody modest air consumption. Although it is an extemally

pressurized compliant surface air bearing it has been developed on a trial-and-enor basis and,

as such, has controversial features. One is the geometry of the nozzles; they are inclined at

25 degrees to the tmck surface and at 45 degrees to the direction of travel. Furthemore the

gap between rail and runner is usually less than the diameter of the nozzles. In order to ailow

the use of simple fiow diagnostic techniques, a b t investigation of rhis geometry used scaied

up models and incompressible flow. However, under nomal operating conditions, the flow

is compressible and usually gasdynamically choked so that the present work complements

the previous investigation by considering this aspect. The aims are to provide data for input

to a mathematical mode1 of the system currently being developed, and to suggest alternative

geometnes which may be both simpler to manufacture and incur reduced fluid losses. The

results of previous research on the system are summarized, and the journal Iiterature on similar

geometries is reviewed. Development of an orifice-plate m a s flow measurement technique

is described. Preliminary results for the systems nozzle geometry are obtained and compared

with two alternatives. We conclude that the flow can be modelled as by using an inviscid orifice

flow with a suitable discharge coefficient that is only weakly dependent on Reynolds Number.

However, for most operating conditions choking appears to occur at the nozzle exit and where

it tums and spreads under the ninner, so that the controlling area is not that of the nozzle, but

that of the defined by the nozzle periphery and the runner-raii gap.

1 would iike to express my profound gratitude to Professor P.A. Sullivan who believed in rny

capability to fulfill my research objectives, and who has been very encouraging and supportive

throughout rny thesis. A special thank you to Charles Perez who helped me designed and

build many of my experhental apparatus, and to the faculty and staff at UTIAS for creating an

excellent research environment. I also woufd like to thank UTIAS Ph.D, student Kevin Linfield

who provided me with a lot of documentation and information about orifices, our summer

student, Philip Beatty who built a new orifice calibration apparatus. and Frank Feuchter who

participated in various flow measurement experiments. To finish, 1 thank my mother, Drobcek,

les poules, and Fantasio for the happiness and joy they provided me during the months 1 spent

in Toronto.

Abstract i

Acknowledgements iii

List of Tables vii

List of Figures viii

Nomenclature xii

1 Introduction 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Objectives and Scope 5

2 Review of Previous Research on SaiIRd 7

2.1 OveraIl Structure of Fluid mechanics . . . . . . . . . . . . . . . . . . . . . . . 7

CONTENTS v

. . . . . . . . . . . . . . . . . . . . . . . 2.2 Characteristics of the Nozzle Region 11

3 Compressible nozzie flow 19

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Theory of choked nozzie flow 19

. . . . . . . . . . . . . . . . . . . . . . . . 3.2 Dennition of discharge coefkient 20

. . . . . . . . . . . . . . . . . . . . 3.3 Dimensional analysis for SailRail nozzles 21

3.4 The influence of the nozzle length-diameter ratios (ZN/dN) on the discharge

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . coefficient 23

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 ZN/dN ratios bellow 6 23

. . . . . . . . . . . . . . . . . . . . . . 3.4.2 ZN/dN ratios between b and e 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 ZN/dN ratios above c 24

. . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Numerical values of 6 and é 26

. . . . . . 3.5 The effect of the Reynolds number on nozzles discharge coefficients 26

3.6 The influence of the nozzIe machining quality on the discharge coefficients . . 27

4 Mass flow measurements 30

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction 30

. . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Description of the orifice meter 32

4.3 Theory of subsonic rnetenng device . . . . . . . . . . . . . . . . . . . . . . . 33

. . . . . . . . . . . . . . . . . . . . . 4.4 Discharge coefficient through an orifice 34

4.4.1 Description of the vena contracta effect . . . . . . . . . . . . . . . . . 35 . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Cd expresseci as a Taylor series 36

. * * * * * . . . . . . . . . . . . . . . . . . . 4.5 Orifice meter calibration .. . . 40

4.6 Orifice rneter calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.7 Error analysis for the orifices caiibration . . . . . . . . . . . . . . . . . . . . . 48

5 Experimental investigation of the nozzles 50

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The main expriment 52

. . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Description of the apparatus 52

CONTENTS vi

5.1.2 Measurements to be taken . . . . . . . . . . . . . . . . . . . . . . . . 53

. . . . . . . . . 5.1.3 The procedure to end the nozzle discharge coefficients 55

. . . . . . . . . . . . . . . . . . . . . . 5.1.4 Reliminary experiment results 57

5.1.5 Main experiment results . . . . . . . . . . . . . . . . . . . . . . . . . 58

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Uacovered nozzles 58

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coverednozzles 62

. . . . . . . . . . . . . . . 5.2 Experiments conducted on the monorail apparatus 68

6 Conciusion 72

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Review of the main results 72

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Recornmendations 73

A Flow through a choked nozzle , 74

B Orifice meter drawings 77

C Pressure tramducers 84

C.l Pressure tramducers calibration . . . . . . . . . . . . . . . . . . . . . . . . . 85

D Cornparison between steel and aluminum orifices

E Error analysis for orifice meter calibration 94

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1 Error analysis 94

E.2 Orifice cdibration charts with error bars . . . . . . . . . . . . . . . . . . . . . 97

F defiection of a Bat plate 102

G Nozzle experiment drawings 103

4.1 Discharge coefficient for aluminum and stainiess steel orifices. First calibration method . . . . . . 44

4.2 Cornparison of the discharge coefficients of Our second steel orifices calibration with Grace and

LappIeresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

CI 1 List of pressure transducers and equipment used . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

C.2 Values of the pressure versus the measured voItage . . . . . . . . . . . . . . . . . . . . . . . . . . 85

E. 1 The uncertainties for the instruments used in the orifice calibration . . . . . . . . . . . . . . . . . . 95

1.1 Cross section of SailRail nrnner and rail, . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Nozzie patters, Pattern A is the original configuration, Pattern B is the currcnt standard adopted. . 4

Effect of track pressure on pressure distributions afkr li fi-O ff has occurred (Sullivan et al. 1985) . . 9

'Qpical measured pressure and gap profiles (Hinchey et al., 198 1) . . . . . . . . . . . . . . . . . IO

Nozzle region, The three possibIe locations of choking in the covered case at positions 1,2 and 3. . 13 Geometries used for investigation of the effect of nozzle angle on head losses at nozzle-entry and

cavity-entry regions (Sullivan et al. 1985). , . . . . . . . , . . . . , , . . , , . . , . . . , . . . . 14

Resistance chancteristics of inclined and radiai noules (Sullivan et al. 1985). . . . . . . . . . . + 15

Effect of nozzle angle on head losses in nozzle-entry region for t h e Reynolds number (Sullivan

etal. 1985). . . . . . , , . , . . . . . . . . . . . . . . . . . . , . . , . . . . . . . . , . . . . . . 15

Effect of nonle angle on cavity-entry head losses for three values of h,/d at Red = 104 (Sullivan

etd.1985) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 The volume Bow foc various nozzte geometries and for nozzles covered and uncovered by the

mnner (Sullivan et a i 1985). . . , , , , . . . . . . . . . . . . . . . , . . . . . . , . . . . . . . . 17

3-1 Sepmted flow in a sharp-edgedorifice . . . . . - . . . . . . . . . . . . . . , , . . 24

LIST OF FIGURES k

3.2 Marginally re-attached flow in a sharpedged orifice . . . . . . . . . . , . . . . . . . . . . . . . 24

3.3 Location of choking for an attached flow orifice . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

. . . . . . . . . . . . . . . . . . . . . . . . 3.4 Location of choking inside a long cyIindrical orifice 25

. . . . . 3 5 The critical dischargr coefficient versus the cylindrical orifice length over diameter ratio 26

. . . . . . . . . 3.6 The critical pressure ratio versus the cylindrical orifice iength over diameter ratio 27

. . . . . . . . . . . . . . . . 3.7 The influence of the orifice entry radius on the discharge coefficient 28

3.8 Comparison between the discharge coefficient of a sharp inlet radial nozzle and a countoured inlet

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cadialnoz.de. 29

4.1 The discharge coefficient versus the Reynolds number for different fl ratios according to Grace

andhpple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 The standad discharge coefficient graph for an axially symmetric orifice . . . . . . . . . . . . . . 34

. . . . . 4.3 The variation of Cd and CdO with the Reynolds number for orifice with B ratio of 0.1 189 37

4.4 The variation of Cd and Ca with the pressure ratio across the orifice with @ ratio of 0. 1189 . . . . 38

. . . . 4.5 The variation of Cd - Cdo with the pressure ratio across the orifice with @ ratio of 0.1189. 39

4.6 Outline of the second orifice calibration expriment . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.7 The influence of orifice shape on the discharge coefficient (Callaghan and Bowden, 1949) . . . . . 46

4.8 The upstream side of the g = 0.06 aluminum orifice, magnification = 64 . . . . . . . . . . . . . ,O 47

4.9 The downstream side of the g = 0.06 aluminum orifice, magnification = 32 (scale 1/64 inch) . . . . 47

4.10 Comparison of the error bars for the new and old calibration method for orifice diameter of

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1189" 49

5.1 The three nozzle geometries studied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2 Outline of the nozzle experïmen t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3 Discharge coefficient versus the pressure ratio for three uncovered radial nozzles of difFerent size

but similar 0 and l N / d ~ ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.4 The discharge coefficient versus the pressure ratio across the three uncovered nozzles . . . . . . . 60

5.5 The volume flow through the uncovered radial and inclined nozzles versus the pressure ratio

parm/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.6 The discharge coefficient versus the height of the cavity over the nozzie diarneter for the three

nozzle geometries, with reference am, A2, q u a i to the area defineci by the nozzle periphesy and

the cavity height (and pressure ratio pst, /pr = 0.49). ........ .. . . . . . . . . . . . . . 63

LIST OF FIGURES K

5.7 The discharge coefficient versus the height of the cavity over the nozzle diameter for the three

nozzle geomeûks. with reference area. A*. equal to the area defined by the nozzle penphery and

thecavity height(pressureratiopat,/~=037) . . . , . . . . . . . . . . . . . . . . . . . . . . 64


nozzle geomeûies. with reference area, Ai. equal to the nozzle area (and with pressure ratio

patm/m=0.49). . . . . . . . . . . . . . . . . . . . . . . . . . . . * . . . . . . . . . . . . . . . 66


nozzle geometries, with reference area, AL. equal to the nozzle area (pressure ratio patm/m =

0.37). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.10 The non dirnensional volume flow versus the track pressure for uncovered and covered radial

nomle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.1 1 The non dirnensional volume flow venus the track pressure for mcovered and covered inclined

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nozzle. 71

B.1 Assembly drawing of the orifice meter . . . . . . . . . . . . . . . . . . . . . . . . . . . .O . . . . 78

B.2 Detail drawing of pipes l and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

B.3 Detail Drawingof unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8.4 Detail drawing of the pressure transducers chamber . . . . . . . . . . . . . . . . , . . . . , . . . . 81

B.5 Detail drawing of the pressure transducers plugs . . . . . ,. . . . . . . . . . . . . . . . . . . . . 82

B.6 DetaiI drawingofone of the orifice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

C.l The calibration of the pressure uansducerE2 6R. . . . . . . . . . . . , . . . . . .o. . . , . . . 86

C.2 The calibration of the pressure transducer 47rY . . . . . . . , . . , . . . . . . . . . . . . . . . . 87

C.3 The caiibration of the pressure transducerE27R . . . . . . . . . . . . o . . . . . . . . . . - . . . 88

C.4 The calibration of the pressure transducer D 1 1 H . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

. . . . . . . . . D . 1 Orifice calibration for stainless steel and aluminum onfices with ,8 ratios of 0.06. 9 1

. . . . . . . . . D.2 Orifice calibration for stainless steel and aluminum orifices with ratios of 0.12. 92

. . . . . . . . . D.3 Orifice calibration for stainless steel and aluminum orifices with f l ratios of 0.18. 93

. . . . . . . . E-1 Emranalysis on orifice calibration for stainless steel orifices with B ratio of 0.03 16 98

. . . . . . . E.2 Error analysis on orifice calibration for stainless steel orifices with /3 ratio of 0.060. 99

E3 Error anafysis on orifice calibration for stainless steel orifices with /3 ratio of 0.1 189 . . . . . . . . 100 E-4 Enor anaiysis on orifice calibration for stainless steel orifices with /3 ratio of 0.1787. ....... 101

G . 1 Assembly drawing of the main expenment mount . . . . . . . . . . . . . . . . . . . . . . . . . . . I O 4

LIST OF FIGURES xi

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.2 DetaildtawingofIhechamber 105

. . . . . . . . . . . . . . . . . . . . . . . . . . . . G.3 Detail drawing of main experiment nozzie plate 106

. . . . . . . . . . . . . . . . . . GA Detail drawing of the preliminary expriment radial noule plates 107

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.5 Detail drawing of thecoverplate 108

Roman Symbols

speed of sound in rail manifold

cntical speed of sound

Area

compressibility correction function

intercept

siope

reference area based on the nozzle area

reference area base on the area made by the nozzle pen'phery and the cavity gap

throat area

criticaI area

horizontal width of the rai1 bearing section

dope of the vena contracta correction factor relation with pressure ratio

discharge coefficient

orifice discharge coefficient, not hciuding compressLbility effects at vena contracta

average orifice discharge coefficient, not including compressibility effects at vena contracta

cntical discharge coefficient

NOMENCLATURE xiii

nozzie discharge coefficient

theoretical discharge coefficient

diametex

plate deflection

nozzie diameter

modulus of elasticity

gravitational acceleration

fitting function

enthalpy

total enthdpy

gap height in cavity region

minimum gap height between mnner and rail, or seal height

water level in water column

maximum gap height

compressibility correction factor

correction factor to convert the ideal to the reai c2vity pressure

nozzle inIet loss coefficient

cavity inlet loss

nozzle length

length of a mnner

total Iength of pcessunzed rail

Mach number

m a s fiow

ided mass flow for unchoked condition

ideal mass flow mass Clow for choking condition

experimentai mass fiow

mass fiow rneasured by the orifice rneter

number of independent variables

pressure

partial pressure of the ;Ur inside the water column

amiospheric pressure

pressure in cavity

pressure rneasured in the chamber

NOMENCLATURE xiv

Pci

Pd

PT

Ptrana

Pu

Pu

Pwc

Po

P l

Q QT

r

R

RN

RP Red

t

t,

T

Tatm

TT

T.

To

v KI

h

v2

w

Wi

w x

W

X

ided cavity pressure

pressure downstream of orifice

pressure in rail manifold, or track pressure

pressure measured by the pressure transducers

pressure upstream of orifice

vapor pressure of the water

pressure measured in the water column

total pressure

critical pressure

volume fiow

volume flow in the rail manifold

downsûeam over upstream pressure ratio

gas constant for a specific gas

radius of nozzle

plate radius

Reynolds number based on diameter

tirne

plate thichess

absolute temperature

atrnospheric temperature

temperature of the air in the rail manifold

critical temperature

total temperature

Voltage

initial VoItage

initial volume of air inside the water column

final volume of air inside the water column

Ioad applied to the plate

uncertainties of the variables

uncertainties of the result

Weight

linear function

NOMENCLATURE xv

Greek Symbols

angle of orifice wail to flow axis

ratio of orifice diameter to internai pipe diameter

specific heat ratio

differential pressure

dynamic viscosity

dynamic viscosity inside the rail manifold

effective coefficient of sIiding friction

constant = 3.1415926.. . density

initial air density inside the water colurnn

final air density inside the water coIumn

density inside rail manifold

upstream density

water density

critical density

Angle of nozz1e versus rail surface

1.1 Background

A novel compliant air bearing system known comrnercially as "SailRail" has been developed

by the late Herbert E. Gladish (HEG). This system is capable of achieving very low effec-

tive coefficients of sliding fiction while using only modest levels of air power. However, it

has been developed on a trial-and-error basis, and, as such, has several controversial features.

These include the geometry and construction of the compiiant element, and the geometry of

the nozzles used to feed the air to the bearïng surfaces. The present work is an experimental

investigation of the properties of the nozzle geometry with a view to understanding its role in

the sy stem performance, and to developing al tematives.

The SailRail compliant air bearing system was designed for a variety of applications includ-

hg: high density warehouse storage systems for goods handled on standard industrial paiiets,

movement and transfer of materials and assemblies on industrial production lines, and Iow

speed passenger transportation systems in a protected environment such as comecting tunnels

in airport tenninds. For the last application linear induction motor propulsion has been suc-

cessfdly integrated into the system. The concept is described here using the warehouse storage

application.

Figure 1.1 depicts the main elements of the system. For the 1.22 m x 1.22 m pallet-based

storage systems. two elements known as runners, which are attached to the underside of the

palîet, move dong two shallow concave guideways, or rails, which are spaced about 1.0 m

apart. The two rails are said to form a track. Cumenly the rumers consist of ovd-sectioned

cylinders having cross-sectionai dimensions of about 108 mm x 58 mm, and the rail is a cir-

cu ls arc having a 152 mm radius and 127 mm width. The runners are constnicted fkom a

combination of matenais which provides flexibility to establish air bearing action and to adapt

to inegularities in rail alignrnent. Air for bearing action is introduced to the rail-runner inter-

face through srna11 nozzles in the rail from air supply manifolds which are integral with the rail

section. Currently the m e r s are constructed by winding cellulose fibre tissue onto a core and

then enclosing the combination in a cover made from a sheet of polyethylene having a thick-

ness of about 1 mm; the rails are extmded durninum. With air power consumption of about

0.1 kW per tonne of carried load, the system is capable of achieving effective coefficients of

sliding friction of one percent and better (Sullivan, 1997).

The noules are currently drilled inclined to the rail surface at an angle of 25' and to the

rail axis at 45'; they have a diameter of about 0.56 mm (0.022 in). Figure 1.2 shows two nozzle

layout patterns used in the SailRail system. Initially it consisted of two noules spaced at 304.8

mm intervals dong the rail axis or direction of travel, and at angular distances of -15" and

+5" fiom the center of the rail. However, following UTIAS research, this was changed to a

single row of nozzles at +5" with 152.4 mm axial spacing (Hinchey et al.198 1).

The controversial features arose fiom the first application; onginally called Sailstrip. It was

Figure 1.1 : Cross section of SailRail runner and rail.

introduced in 1971 by HEG to solve a problem in manufacninng and packaging of wound

cellulose tissue roli. The method then used to transfer the rolls to the packaging machine al-

lowed them to partially unwind, thus creating loose "tails" which had to be corrected rnanually.

To eliminate this problem, and to simpliq the traasfer machinery, HEG proposed a device in

which the roll is supported in a concave guideway or rail by a film of air (Sullivan 1997).

The material choice for the mnner in the c m n t SailRail application foIlows ftom this first

concept. Both the geomeery and the materials contrast with the more conventional cornpliant

air bearing which would consist of a block of elastomer such as a wlcanized rubber. The first

controversial features is that other materials could allow the air bearing effect, be sirnpler and

cheaper to manufacture, and be less prone to degradation from exposure to wet environments

than the current cellulose m e r . Also, it is from Sailstrip application that the use of nozzles

inclined to both the rail surface and the direction of travel was adopted. The inclined nozzles

suppressed the dynamic instability, it hparted a rotation to the rolls to prevent unwinding, and

it provided a propulsion force (Sullivan 1997). However, even if uiis nozzle geornetry has been

kept on the current SailRail system, none of these issues are relevant to the present application.

radius

Pattern A original standard

Pattern B new standard

Figure 1.2: Nozzle patters. Pattern A is the original configuration. Pattern B is the current standard adopted.

Nevertheless, HEG has suggested that this unconventional nozzle geometry plays a role in

limiting air consumption. In this regard, some applications operate with substantial sections of

track uncovered by the moving platform and its m e r s ; a possible rnechanism by which this

might occur, related to the phenornenon of gasdynamic choking, is discussed below.

1.2 Objectives and Scope

Previous research at the author's institute (UTIAS) shows that the flow through the nozdes is

compressible and usuaily gasdynamically choked (Sullivan et al. 1985). However, to enable the

use of large-scale models for diagnostic purposes, UTIAS investigations of the nozzle geometry

undertaken to date used incompressible flow which scaled Reynolds number effects. The work

described in this report is an investigation of compressibility effects, with two objectives:

(A) Obtaining data on nozzle characteristics such as discharge coefficients for use in nu-

merical modelling.

(B) Investigation of alternative geometries that may be both simpler to manufacture and

incur reduced fluid Iosses.

With regard to the structure of this report, Chapter 2 summarizes the results of previous

UTIAS research. Chapter 3 presents a dimensional analysis of the problem and reviews the

relevant journal literature on compressible nozzle flow for geometries similar to the SailRail

system and related alternatives. Chapter 4 describes the technique used to measure mass flow

rates. The results of the mass flow metenng device calibration are also presented. Chapter 5

relates to the experiments conducted on the different nozzle geometries. The description of

the apparatus, the experimental rnethodology and results are presented. Chapter 6 presents a

discussion of the results and the main conclusions. In appendk A the equation of choked mass

flow is demonstrated. The drawings of the orifice meter and orifice are included in appendix

B. Ail the data conceming the pressure transducers used in our experhents are presented in

appendix C. The orifice meter calibration charts are in appendix D. The orifice meter calibra-

tion emor analysis is explained in appendix E. The calculations for the deflection of flat plate

in included in appendix F, and finally, the nozzle main experiment drawings are in appendix G.

2.1 Overali Structure of Fluid mechanics

Figure 2.1 is a typical pressure distribution at the rail-rumer interface (Sullivan et al 1985).

It shows that the bulk of the load is supported by a plateau of essentidy constant pressure

which, above a certain critical value is independent of the pressure in the rail manifold, the

trackpressure, pr. Furthemore, with LR and BR being, respectively, the length of the runner

and horizontal width of the rail bearing section, and with W being the weight supported, one

can define an ideai cushion or foupint pressure pn as

The data in Figure 2.1 shows that the average pressure in the plateau is about 25 percent higher

Data such as that depicted in Figure 2.1 led Suilivan et al (1985) to suggest that the fiuid

mechanics of the system could be classified into three distinct regions: the nozzle, the cauity

region, and the seal region. In the nozzle region the air flows into the nozzle entrance in the

rail manifold, through the nozzle itself and, at the nozzle exit, tums, spreads outwards between

rail and ninner, and decelerates coming vimially to rest. The cavity region encompasses the

pressure plateau in Figure 2.1 which, for most purposes, cm be assurned to be at a constant

pressure p,, the cauitypresswe. The seal region is the narrow strip around the perîphery of

the cavity region where the pressure decays rapidly and the bearing air escapes to atmosphere.

The values of 6 in Figure 2.1 suggest that the flow through the nozzle region is compress-

ible and, as discussed below, usually choked. Furthemore estimates of Reynolds numbers Red

based on measured mass flow rates and the nozde diameter dN suggest Red is in the range

0.5x104 to 3x10~ for most operating conditions, so that the flow through the nozzle itself is

believed to be turbulent. However, for the cavity and seal regions, the flow is believed to be

inertiaiess and governed by the Reynolds equations for compressible lubncation (Sullivan et

ai, 1985).

In relation to compressibility phenomena, according to ideal fluid theory, for a calorically

perfect gas such as air at room temperature, the absolute pressure ratio pJpt at onset of choking

is given by

where for air with 7 = 1.4, pJpt = 0.528. Typicai operating ranges of pr and p,. are, respec-

tively, 170-250 Wag and 10-50 khg. Thus, the absolute pressure ratio pc/pr is usually less

than 0.528, so the Bow inside the nozzle is likely to be choked.

With regard to the gap (h,) between rail and runner, rneasurements of this quantity were

Shoe met thickness = 1.524 mm

PROBE NUMBER

Figure 2.1 : Effect of track pressure on pressure distributions after lift-off has occunred (Sullivan et al. 1985) .

obtained by Hinchey et. al. (1 98 l), and Figure 2.2 gives typical results. They suggest that h,

is a maximum of 0.3 mm, which is Iess than haif the value of dN then in use, namely 0.66 m.

(0.026 in.). The technique used by Hinchey et al (1981) could not resolve the minimum value

of h, in the seal region, namely h,; subsequently, Sullivan et al (1985) used measured mass

flow rates and the Reynolds lubrication equations to estimate it as about 0.017 mm. For the

present investigation, we may say that, since the nozzies are not placed in the seal region, in

the cavity region h, varies between 0.1 mm and 0.3.mm. Hence, since current practice uses dN

= 0.56 mm (0.022 in.), the cavity -to noule diameter ratio (hJdN) is in the range 0.18 to 0.54.

This parameter has an important bearing on the present investigation.

In thinking about the role of the nozzles in the SailRa. system, experiments of the type

7 W i n

Figure 2.2: 'LLpicai measured pressure and gap profiles (Hinchey et al., 1981) .

used to obtain the data in Figure 2.1 have repeatedly shown that, when operating at effective

coefficients of sliding fnction bjf of a typical design value of 1 percent, the cavity pressure

is determined almost entirely by LR, BR and W ; that is, p, = KI p, where, depending on the

match between m e r and rail geometry, Kf is in the range 1.1 to 1.5. Then the air flow rate

m is determined by p e f j or, equîvalently, the effective minimum gap h,. Consequently, the

role of the nozzles is essentially to deliver the specified Ra at p,. with a minimum energy l m .

This goal may be achieved by choosing a nozzle geornetry which, for given dN, minimizes the

required m.

In this regard, calculations suggest that, the value of a required to achieve a given pcfi

may be much less than actually used in practice; typically pr is estimated to be less than l.lp,.

(Feuchter 1999). However, in practice, the requisite 7 n is always delivered at much higher

p ~ ; apparently this is done to avoid the costs of delivering large volumes of low pressure air:

Hence the objective of nozzle design rnay also be stated as finding a geometry which delivers

the maximum m for the given pr.

2.2 Characteristics of the Nozzle Region

Figure 2.3 depicts the details of the geometry of the nozzle region. In general the length 1~ of

the novle is about 7.5 1 mm, so that l N / d N is about 13.44. This suggests that there are three

possible locations where choking could occur. The fist is at the vena contracta formed at the

nozzle entrance (point 1 on Figure 2.3). The second possibility is choking due to fiction at

the nozzle exit (point 2), commonly called Fanno choking. If choking occurs at one of these

two locations, the choking effect will lirnit the flow when the nozzle is uncovered by motion of

the m e r dong the rail, thus minimizing the air consumption. The third possibüity, however,

would not limit the flow when uncovered by the m e r , thus increasing the air 0ow consump-

tion. This is when area choking occurs as the flow leaves the nozzle, tums and spreads out

under the nimer, it is likely to happen for sufficiently srnall values of hC/& (Sullivan et al.

1985).

Rail M5lmd Figure 2.3: Nozzie region. The three possible locations of choking in the covered case at positions 1,2 and 3.

The potential importance of compressibility effects notwithstanding, the first UTIAS in-

vestigation of the nozzle region used incompressible flow. It adopted this approach in order

to allow the use of simple diagnostics in scaied-up models which matched Reynolds number.

Despite this limitation useful insights were obtained. In discussing these results it is useful

to recognize that energy losses should corne from two main sources. The first is at the nozzIe

entrance, and the second is at the nozzle exit or, as it is often termed, the cavityinlet. Losses in

the nozzle passage caused by shear forces at the nozzle wall are likely to be rnuch smaller than

these two. Considenng first the nozzie entrance, since the nozzles are incorporated by drilling

from the rail exterior surface, the nozzle entrance always has a sharp edge. Hence the entering

fiow separates, forming a vena-contracta as depicted in Figure 2.3, and ths subsequent abrupt

enlargement to n11 the nozzle passage generates considerable energy loss. At the novle exit, if

the edge is sharp, separation again occurs as the flow is forced to turn and spread out under the

m e r . The subsequent slowing of the fiow as it spreads out to filI the gap between runner and

rail will generate losses over and above that incmed by injecting a Stream of fluid into a caviy.

Sullivan et al (1985) investigated both entrance and exit losses using the geometries de-

picted in Figure 2.4. Figure 2.5 gives the static pressure p, in the delivery tube measured at the

point shown in Figure 2.4 as a function of volume flow Q for the SailRail geometry, and for

two alternatives: a radial nozzle with a sharp edged exit, and a radiai nozzle with a contoured

or rounded exit. In the context of Figure 2.4, a radial nozzle corresponds to the angle 0 = 90".

The data shows that the pressure required to drive the flow for the SailRail configuration is less

than half that of a sharpedged radial nozzle; but, the contoured radial nozzle dnving pressure is

much smaller than that for the SailRail nozzle. Figures 2.6 is a plot of data on nozzle inlet loss

coefficient KL as a function of 0 with Red as a parameter. As might be expected, KL increases

as 0 decreases from the 90" radial case to the 20' SailRail case. Figure 2.7 gives corresponding

data for a cavity inlet loss Ko with hJdN as parameter. In this case losses decrease as 6 de-

creases from the radial to the SailRail geornetry, but the amount of decrease depends criticdly

on hc/dN; this decrease is small at the upper end of the expected range of h,ldN (0.45), and

very large at the lower end (0.1 1). One may ider nom this discussion that, if compressibility

effects do aot modiQ this pattern significantly, the nozzle geometry minirnizing both nozzie

inlet and cavity inlet losses is radial with a contoured exit

Turning now to compressibility effects, the eRect of flow limitation manifests itself in two

ways; fint if the delivery pressure pr is held fixed, the mass flow m/ through the nozzle ap-

proaches a limit as the downstream pressure p, decreases. If, however, p, is fixed and pr is

increased, the volume Aow QT at pressure pr approaches a limit. In the investigation reported

by Sullivan et al (1985) the latter method of ascertaining the presence of limitation was used;

Figure 2.8 surnmarizes their data for sharp-edged radial and SailRail nozzles. Results were

obtained with the nozzles both covered by a loaded runner and uncovered; in the latter case p,

becomes p,,, the pressure of the atmosphere.

Ef fect Rail manifold

Meusureci Hem

Nonle Inlet Model

0- Atmosoheric

0 -Plessure

1.4 cm ID Noale

t Static Pressure Measured Hem

Cavity Inlet Model

Figure 2.4: Geometnes used for investigation of the effect of nozzle angle on head losses at nozzie-entry and cavity-entry regions (Sullivan et al. 1 985).

V U E FLOW Q (IOJnibi88~)

Figure 2.5: Resistance characteristics of inclined and radial noules (Sullivan et al. 1985).

Figure 2.6: Effect of nozzIe angle on head losses in nozzle-entry region for three Reynoids number (Sullivan et al. 1985).

Red = 104 - h/d 0.11 -r h/d = 0.23 ir hld = 0.45

NOZZLE ANGLE (DEG)

Figure 2.7: Effect of nozzie angle on cavity-entry head losses for three values of hJd at Red = 104 (Sullivan et al. 1985)

In d cases the results irnply that choking occurs at design values of a, which is above

50 TRACK

100 150 PRESSURE Pr

200 250

(kPo gauge 1

Figure 2.8: The volume flow for various n o d e geometries and for nozzles covered and uncovered by the runner (Sullivan et al. 1985).

150 kPag. Also, when is high enough to cause choking in the covered case, if removal of

the runner causes an increase in QT, then one may infer that choking occurs at the cavity inlet

and not at the nozzle inlet. For the SailRaiI nozzles, removai of the m e r causes only a modest

increase in QT, suggesting that choking has occurred at the nozzle inlet, In contrast, for the

radid nozzles with dN = 0.71 1 mm and 1.016 mm, substantial flow increases occur, suggesting

that choking has occurred at the cavity inlet. The 0.508 mm radial nozzles behave as if choking

has occwred at the nozzle inlet,

In concluding this review, the incompressible data of Sullivan et al (1985) suggest that

substantial reductions in n o d e losses cm be achieved by appropriate modifications of noule

geometry. However, these losses are strongly dependent on such details of the nozzle geornetry

as h,/dN and O. Furthemore their exploratory investigation of compressibility effects high-

lights the importance of choking phenomena which may significantly modw the conclusions

drawn fiom the incompressible investigation. Clearly, making design improvements is contin-

gent upon completing a corresponding investigation of these effects.

COMPRESSIBLE NOZZLE FLOW

3.1 Theory of choked nozzle flow

The ideal m a s flow corresponds to a one dimensional fnctiodess isentropic flow from stagna-

tion conditions upstream of the orifice. It is the flow that would pass through if full expansion

had occurred and the Stream lines were parallel in the plane of the orifice. According to the

one-dimensional isentropic flow theory, the unchoked mass flow mi through a nozzle having

an upstream or reservoir pressure a and ternperature TT, a throat of cross-sectional area 4,

and a downstream or exit pressure p, is given by

When the pressure downstream of a constriction in a gaseous flow is continuously reduced,

CHAPTER~, COMPRESSIBLE NOZZLE FLOW 20

while the conditions upstream are held constant, the flow rate increases until it eventudy

reaches a constant maximum value which cannot be exceeded by making M e r changes in

the downstream pressure. The limitation of the mass flow rate in this way is known as choking.

According to the one-dimensional isentropic flow theory, the choked mass flow mi' is aven by

the following equation

The proof of this equation is presented in Appendix A. The mass flow through a chocked

nozzle can be increased by raising pr. However, the volume flow at track pressure p~ is given

by QT = * /pr where pr = pT/(m). Thus, the equation of the volume flow QT becomes:

Thus, if we assume .y and R constant, the volume flow is function only of the throat area

and the track temperature. Hence to observe choking effects in the present research where the

cavity pressure, p,, is fmed and the track pressure varies, it will be ody necessary to evaluate

the volume flow (Sullivan et al 1985).

3.2 Definition of discharge coefficient

The discharge coefficient accounts for two-dimensional and real gas effect. The discharge

coefficient through an orifice is defined as the ratio of the actual mass flow m over the ideal

mass fiow 6.


B y definition, the vena contracta is the plane in the subsonic jet at which the streamlines of

the flow downstream of the orifice become pataliel. Thus. the discharge coefficient cm also be

defined as the ratio of the vena contracta area over the orifice area.

In a simple one-dimensional mode1 of compressible flow through a convergent-divergent

nozzle, choking occurs when the fiow becomes sonic at the plane of minimum cross-sectionai

area. In practice, the flow behavior is more cornplex. As will be explained later in this chapter,

the sonic condition is a necessary condition for choking, but it is not a sufficient one .

3.3 Dimensional analysis for SailRail nozzles

From the choked mass flow equation presenred earlier in this chapter (equation 3.2) where we

assurneci At to be a function of dN, ZN, hc and 8, we can Say that the rneasured choked mass

flow should depend on the following quantities:

where is the viscosity of the air at the track pressure condition.

Using the following reference quantities: = Il2, a, dNT we can non-dimensionalize

7ni' as follows with, pr = ml(=).

where the quantity is a Reynolds number based on the nozzle diameter and the

speed of sound in the reservoir.

From the Pi theorem of dimensional anaiysis, the number of variables (eleven initially)

is reduced by the nurnber of dimensions invoIved, which is four here (mass, length, t h e ,

temperature).

~ H A P T E R ~ . COMPRESSIBLE NOZZLE FLOW 22

We c m thus define a discharge coefficient Cd as

Since the SailRail system operates only with air for which 7 = 1.4, we have

This dimensional analysis was provided by professor P.A. Sullivan.

From their experiments conducted on orifices, Rohde et al (1969) amived to a sirnilar con-

clusion. They found out that the discharge coefficient through a cylindrical orifice depends on

Approach Mach number,

static pressure differential across the orifice,

inlet edge radius of the orifice,

the angle between the approaching flow and the axis of the onfice

the ratio of the orifice thickness to orifice diameter In/dN,

the viscosity of the fluid.

They also found out that the following parameters had a negligible influence on the discharge

coefficients:

a the effect of temperature and pressure levels,

a the orifice surface finish,

multiple orifice interference,

a the approach passage geometry and Iength.

in the following sections, the most important parameters of this Est will be discussed.

CHAPTER~. COMPRESSIBLE NOZZLE FLOW 23

3.4 The influence of the nozzle length-diameter ratios (ZN/dN)

on the discharge coefficient

The flow c m be classified in three distinct categories depending of the choking location inside

the nozzle. This ~Iassification depends on the value of the nozzle length over diameter ratio,

l N / d N . The value of i N / d N corresponding to the transition from an incompressible flow to a

choked rnarginally re-attached fiow is represented by the Greek letter 6. The value of lN/dnr

corresponding to the transition from a choked attached flow to a Fanno choked flow is labeled

by E.

3.4.1 EN/dN ratios bellow 6

Choking of the flow tbrough an orifice is related to the extent of the expansion of the jet within

the length of the orifice . Brain and Reid (1973) stated that for small lN/dN ratios (sharp-

edge orifices) the vena contracta foms downstream of the nozzle and because of this, the back

pressure ratio can affect its cross-sectional area even when sonic velocity exists there. As the

back pressure is decreased the maximum contraction area enlarges and the plane of maximum

contraction moves towards and eventually enters the nozzle bore. At sufnciently smal1 pressure

ratios the vena contracta foms just inside the inlet edge of the nozzle and the jet re-attaches

itself to the wall of the nozzle. When the jet is firmly attached to the wall and sonic velocity

exists at the plane of maximum contraction, the flow upstream of this plane will be isolated

from the changes in downstreârn pressure and the n o d e is said to be choked. Its performance

will remain unaffected by any further reductions in the back pressure ratio. For onfices with

t ~ / d . ~ ratios bellow 6 choking occurs at pressure ratios much lower than the theoretical value

of 0.528. In fact, choking conditions are reached at a pressure ratio close to zero or never being

reached. Ward-Smith (1979) showed that for these ZN/dN ratios bellow 6, the critical discharge

coefficient decreases with increasing l N / d N .


Figure 3.1 : Separated fiow in a sharp-edged orifice

Figure 3.2: Marginally re-attached flow in a sharp-edged orifice

3.4.2 l N / d N ratios between 6 and E

In contrast to sharp-edge orifices, those with lN/dN ratios of 2.0 and 1.0 clearly shows the

existence of area choking. In both cases the jet can forms its own nozzle within the length of

the orifice (Deckker and Chang, 1966). They also showed that for the l N / d N ratio of 2.0, the

threshold of choking is at a slightiy smaller pressure ratio than the theoretical value of 0.528.

This is caused by the recovery of pressure beyond the plane of minimum pressure in the jet.

Thus, for ZN/dN ratio between b and c, the choked condition is achieved when the velocity in

the plane of the vena contracta become sonic (Wm-srnith, 1979). For ZN/dN ratios between 6

and É, the critical discharge coefficient is constant.

For adiabatic fiow through cylindrical nozzles with large IN/dN ratios, because of fiction

effects, sonic velocity can occur well downstream of the inlet edge causing a marked aiteration

in nozzle performance (Brain and Reid, 1973). Fanno choking occurs under conditions of


figure 3.3: Location of choking for an attached flow orince

adiabatic fictional flow in a duct of constant cross-sectional area when the Mach number at

the downstream exit plane of the duct becomes equal to unity. For ZN/dN = B with decreasing

back pressure ratio, sonic condition will occur simultaneously at the vena contracta plane and

at the outlet plane of the nozzle . For ZN/dN above E, when choking is initiated, the flow

through the entire orifice is subsonic, except at the exit plane where the flow is sonic. Further

reduction of the back pressure results in the formation of an expansion wave system outside the

nozzle exit, the flow within the orifice remaining unchanged. Above e, the critical discharge

coefficient decreases as the ZN/dN ratio increases @idSrnith. 1979).

Figure 3.4: Location of choking inside a long cylindrical orifice


3.4.4 Numericd values of 6 and E

Figure 3.5: The criticai discharge coefficient versus the cylindxical orifice length over diameter ratio

Ward-Smith (1979) found experimentally the numerical values of the b and E. The ZN/dN ratio

b is approximately equal to 1 and e to 7. These values depend of the sharpness of the leading

edge of the orifice. Generally b lies between 0.3 and 1 and c between 7 and 10. The highest

value corresponds to the orifice which has the smoother bore. Wud-Smith (1 979) presented the

plot of the Mitical pressure ratios versus the ZN/dN ratios. This plot summarized the different

results obtained by different research teams. It is clea. that for Z N / & ratios lower than one the

cnticd conditions are reached at a pressure ratio below the theoretical value. For higher ziv/dN

ratios, the pressure ratios are slightly higher than the theoretical value.

0 9 ~

OB-

Q

07.

Odo

3.5 The effect of the Reynolds number on nozzles discharge

coefficients

* 1 \

1 -

-UrlYraiHiI- m

c- ---.Dm , -

urrcr i r -a-'

1

1 I . 5 7 1s 20 25

Many researches have been conducted in order to study the effect of Reynolds number on

the discharge coefficient inside orifices. Deckker and Chang (1966), and Brain and Reid

Figure 3.6: The critical pressure ratio versus the cylindrical orifice length over diameter ratio

(1973) showed that the discharge coefficients inside cylindrical sharp-edged nozzles of dif-

ferent l N / d N ratios were reasonably insensitive to Reynolds number when Red was above 104.

On the other hand, Ward-Smith (1 979) observed that for short cylindrical orifices with appre-

ciable roundhg of the upstream edge the discharge coefficients were sensitive to Reynolds

number effects.

From the study of the effect of IN/dN ratios on the discharge coefficient, we know that Cd

is decreasing with l N / d N for zN/dN higher than 10. Ward-Smith (1979) stated that the critical

value of Cd in this case will in general vary with Red and the surface roughness. Thus, for the

inclined SailRail nozzles where zN/div is equal to 13.4, Cd should vary with Red.

3.6 The influence of the nozzle machining quality on the dis-

charge coefficients

It is weII known that a light variation in the enhy radius of the nozde cm produce an important

variation on the discharge coefficients as specified by Kastner (1964). This phenomena is

attributed to the elimination of the separation at the upstream edge (Rohde, 1969). The next


figure fiom Kastner shows very weU the effect of countoured inlet on the discharge coefficient.

Only a slight entry radius cm increase drastically the discharge coefficient. Between r / d N = 0

and 0.1, there is an increase of about 12% on Cd.

Figure 3.7: The influence of the orifice entry radius on the discharge coefficient

I verified this phenornena by taking flow measurements on a radial nozde with an average

diameter of 2.22 mm (0.0873 in). When the nozzle had a sharp inlet, we measured the discharge

coefficient for different pressure ratios patm/pT . Then, we contoured the novle inlet to have a

r /dN ratio of about 0.29, where r is the curvature radius of the contoured idet and is equal to

0.64 mm (0.025 in). The cri tical Cd for the sharp inlet nozzle is 0.82 and 0.93 for the contoured

inlet. The increase in Cd is about 10% which is close to the range presented by Kastner (1964).

Also, smail imperfections like bum obstructing the orifice can have a considerable effect

by limiting the mass flow and thus reducing the discharge coefficient.

CHAPTER~ . COMPRESSIBLE NOSSLE FLOW 29

Discharge coeffident versus the pressure ratio !or radial nodes with sharp and countoured inlet.

Figure 3.8: Cornparison between the discharge coefficient of a sharp inlet radial nozzle and a countoured înlet radial nozzle.

MASS FLOW MEASUREMENTS

4.1 Introduction

Obtainuig accurate measurements of mass flow is crucial to understand the flow behaviour

inside the SailRail nozzles. To evaluate these mass flows, we decided to use an orifice meter

because of its simplicity of use, its reliability, its adaptability to a wide range of mass flows, and

its low cost. Grace and Lapple (1951) conducted researche on discharge coefficients for small

diameter orifices. We reproduced exactly their orifice meter by keeping the same proportions.

For a given Reynolds number and orifice to pipe diameter ratio B, Grace and LappIe evaluated

the discharge coefficient (see figure 3.1). Initially, the idea was to use these results to find the

discharge coefficient corresponding to a known Reynolds number in order to calculate the mass

ff ow. After reflection, we decided that it was safer to calibrate the orifice meter so that we could

be confident in our mass flow measurements.

CHAPTER~, MASS FLOW MEASUREMENTS 31

Figure 4.1 : The discharge coefficient versus the Reynolds number for different ratios according to Grace and Lapple

CHAPTER~. MASS FLOW MEASUREMENTS 32

4.2 Description of the orifice meter

Our orifice meter consists of a twenty-two inch 6061-T6 alurninum pipe of one inch intemal

diameter containhg a gasketed union in which an orifice is inserted. The orifice meter is fed by

dry air fkom a cornpressed air bottle. The air 00ws through the orifice which is located fifteen

inches from the orifice meter inlet. On both side of the orifice there is a pressure tap. The

location of the taps to the orifice does not foIlow an ASME standard; it corresponds instead to

a particular design developed by Grace and Lapple (1951). The two taps are connected to a

hermetic chamber. A 50 psi gauge piezoresistive pressure transducer, mounted in the chamber

wall measures the charnber pressure. Since the downstrearn tap is opened to the chamber, this

transducer measures the downstream pressure relatively to th$ atmosphenc pressure. A 5 psi

gauge transducer is connected to the upstream tap inside the chamber; it gives the differentid

pressure between the upstream and downstream side of the orifice plate. The drawing of the

orifice meter is presented in Appendix B.

The orifices consist of a circular plate of 1.5 inch in diameter with a thickness of 0.06 inch.

On the upstream side, a sharp-edged circular hole is machined, and on the downstrearn side

a conical section is machined. A drawing of one of these knife-edged orifices is presented in

Appendix B. The fmt set of orifices where machined from aluminum 606LT6 since it was

cheaper to manufacture. The second set of orifices where made from stainless steel series 400

which we thought would give a better reproducibility and be closer fiom Grace and Lapple

discharge coefficients. The stainless steel orifices where used in the mass Aow measurements

of the main experiment.

The description of the equipment used and the pressure transducea calibration cm be found

in Appendix C.


4.3 Theory of subsonic metering device

The mass flow equation through an orifice is given by the following equation. Except for the

k t tem, the discharge coefficient, Cd, this equation is equivalent to equation 3.1 with the

ciifference that in the latter equation we neglected the P ratio (ratio of the orifice diameter over

pipe diameter).

which can be rewritten in term of differential pressure by

where Ap = pu - p d , and r = pd/pu which gives the pressure ratio across the orifice. The

first square root term of the equation corresponds to the incompressible fiow and the last term

takes the cornpressibiüty into account.

The mass flow measurements are entirely satisfactory under srnall pressure drops across

the orifice, but as demonstrated by Kastner et al (1964) they should not be used when the

pressure difference across the orifice is large because the discharge coefficient is infiuenced by

a vena contracta effect and tends to increase steadiiy as the back pressure is reduced. Figure

3.2 provided by Linfield (1 999), presents this phenornena. Thus, in order to maintain adequate

sensitivity in the flow measurement, it is necessary to use a series of metering elements of

different diameters to cover a given range of flows. For the range of mass flows needed for my

experiments, 1 used the knife-edged orifices with f l =0.03,0.06,0.12 and 0.19.


O 02 0.4 û.8 0.6 1 l-PsbnlPo

Figure 4.2: The standard discharge coefficient graph for an axially symmeeic orifice

Even for pd/pu close to unity, there is a non negligible variation on the discharge coefficient.

Indeed, for pu/pd = 0.9 there is an increase of about 4 1 on Cd. Consequently, it is important

to take this vena contracta effect even for pressure ratios close to unity.

4.4 Discharge coefficient through an orifice

The discharge coefficient, Cd in equation 3.1 and 3.2 c m be divided in two components. The

first terni is a discharge coefficient which takes into account the Iosses due to the orifice inflow.

It is dependent of the Reynolds number only and is labeled Cd. Then, to correct for the vena

contracta effect, we introduce a correction factor K, which depends on the pressure ratio across

the orifice. The discharge coefficient is thus represented by the foilowing equation


4.4.1 Description of the vena contracta effect

The foilowing equation corresponds to the discharge coefficient which take into account the

vena contracta effect. It corresponds to the curve presented in Figure 4.2 and we will call it the

theoretical discharge coefficient, Cdt. For an incompressible and slightly compressible flow,

Cdt is given by:

For axially syrnrnetric flow with zero area ratio we have:

where cr is the orifice wail angle to the strearn(in our case 90"). The fitting finction is given by

the following equation.

These equations were provided by Linfield (1 999).

The correction factor,Kc, is given by:

where for r = 1, the theoretical discharge coefficient is Cdt = 0.5914.

To illustrate the behavior and intluence of Cdo, we took the results from Bow measurements


made with the orifice meter using the stainless steel orifice of diameter 0.1 189 in. Figure 4.3

shows the variation of the discharge coefficient (Ca and CdO) venus the Reynolds aumber. It

is clear that CdO as well as Cd Vary linearly with Red. However, the slope of Cd* linear fit

is weaker than the one for Cd. This agrees with our previous discussion since Cd represents

the total discharge coefficient including also the effect of the vena contracta at the orifice exit.

Figure 4.4 presents the discharge coefficient versus the pressure ratio across the ofice. The

same conclusion can be made for this case: a lin- behavior is observed, and the pressure ratio

has a stronger influence on Cd than on Ca. Finally, we plotted the ciifference between Cd and

Ca versus the pressure ratio in order to ver@ the linearity of the correction factor Kc.

4.4.2 Cd expressed as a Taylor series

We demonstrate that equation 4.3 is an aitemate to Taylor's series expansion. From the

equation of mass fiow 4.2, if Cd = Cd(Red,pd/pu) = Cd(Red,r) for small variation of Cd with

Red and pd/pu we can expand in a Taylors senes about palpu = r = 1 and Red = 0.

cd = a + b r + c ~ e ~ + d ~ ~ + e r ~ e ~ + f ~ e i + ...

As a result of firting data to experiments and analysis we have

CdO = AO +AlRed

and

Kc = 1 +&(1 - T )

Where Ao, Al and BI are fitted coefficients. Thus,


Figure 4.3: The variation of Cd and CdO with the Reynolds number for orifice with P ratio of 0.1 189.


Figure 4.4: The variation of Cd and CdO with the pressure ratio across the orifice with P ratio of 0.1 189.


Figure 4.5: The variation of Cd - Cd With the pressure ratio across the orifice with P ratio of 0.1 189.


Thus, by cornparison the three last equations we have,

Wth terms e and f ornitted.

4.5 Orifice meter calibration

In order to veriQ the mass flow measurements of our orifice meter, we designed an experirnent

based on the principle of a gasometer. The fint attempt consisted of expelling a given volume

of water from a plastic colurnn under the action of the air flow. We measured the time required

for the water to go fkom the upper reference level to the lower reference level of the water col-

umn. We measured the volume of air inside the water column at the upper and lower reference

level. The density of the air in the water column for the two reference levels was calculated

from a pressure equilibnum described later in this section. These parameters dlowed us to

evaluate the mass flow in the system. After each test, a vacuum purnp c o ~ e c t e d to the water

column was used to rernove the air in the column and, the later was automatically filled with

the water contained in the reservoir, The orifice meter and the water column were connected


together by a polyrner tube and were fed by the laboratory air cornpressor. To find the orifice

discharge coefficient we divided the experimental mass 0ow by the theoretical orifice mass

flow using the pressure measurements inside the orifice.

This b t experiment was limited to small mass flows. In fact, the volume of water was

too s m d (only 74 to take accurate flow measurements for larger orifices. Furthemore, the air

flowing inside the water column was controlled by a manual valve and thus, was limited to the

reflex of the operator. Despite the lack of accuracy, this experiment was a good starhg point

in order to get a first approximation of the orifices discharge coefficients.

We subsequently developed a more elaborate mass flow rneasurement experiment. The di-

agram of this expenment is presented in the Figure 4.6. This second mass flow experiment

consisted of expelling water from a water tank during a given t h e . By doing this, the bub-

bles forming in the water column due to the air flow had no more effect on the quality of Our

measurements . An electronic valve activated by a cornputer was used in order to control very

precisely the tirne of opening and closing of the valve. A d e r was mounted on the water col-

urnn, so by taking the height of the water at its initial and final level, we could determine the

volume of air in the column. A pressure transducer was c o ~ e c t e d to the top of the water col-

umn in order to take air pressure measurements at the initial and final stage of the experiment.

The air was passed through the water to ensure that the water vapor pressure (partial pressure

of water contained in the column)could be included in the pressure measurement. ffiowing

ail these parameters, we could evaiuate the mass fiow . In order to eliminate unknown effects

of the air humidity, we used a bottle of dry air to feed the system. Between the dry air bottle

and the orifice meter, the air was Ied through a copper pipe contained inside a temperature

controlIed water bath to ensure room temperature of the air.

The following h e s describe how we manage to meamre the mass flow in the h t exper-

iment. From a static equilibriurn, the air pressure inside the water colurnn and in the pipe is


Figure 4.6: Outline of the second orifice calibration expenment.


given by:

Assuming a perfect gas, the air density inside the water column is given by:

Thus, the mass flow is given by:

where 1 and 2 stand for the initial and final water level in the water column.

In the second mass flow experiment we used the exact same mass flow equation. The only

Merence is that we had a pressure transducer on the top of the water column to rneasure the

air pressure inside the column. Thus the equation to h d the air pressure in the column is:

This change contributed to increase the accuracy of the mass flow measurement. In the

previous method based on pressure equilibrium, we had to rneasure the height of the water

column in order to find the air pressure inside the water column which was not very accurate.

The charactenstics of the pressure transducer used in the second calibration expenment are

presented in Appendix B.

CHAPTER~. MASS EXOW MEASUREMENTS 44

4.6 Orifice meter calibration results

TWO sets of orifices were used. The-fht series of orifices were made of aluminum 6061-T6.

The second set was made of stainless steel series 400. Table 4.1 and the charts of Cd versus the

mass flow in Appendix D show that the choice of matenal has a direct impact on the discharge

coefficient,

Table 4.1: Discharge coefficient for aluminum and stainless steel orifices. First caiibration method.

I II AI orifices I Steel onfices I

By choosing a soft materiai like aluminum, it is impossible to machine a sharp inlet espe-

cially for small orifices. Consequentiy, the aluminum orifices have a rounded edge instead of a

sharp edge as it should be and the discharge coefficients are thus higher. For practical reasons,

we abandoned the idea of using aluminurn orifices to measure our mass 0ow. Aluminum is a

very ductile material so it is very likely to be damaged on the edges of the orifice hole thus

increasing the chances of modifj6ng the discharge coefficient.

The discharge coefficients measured by Grace and Lapple are compared with our measure-

ments made with the second mass flow expenment since it was more precise than the first

method. Table 4.2 presents the average values of Cd obtained by Grace and Lapple and our

results for the different orifices. All the charts of the measured Cd versus the experimental

mass Bow for aluminum and steel orifices cm be found in Appendix D.

In general Our results and the ones obtained by Grace and Lapple agreed well except for

the 0.03 1 diameter orifice. Many factors can cause the discrepancies between our experimental

results and the Grace and Lapple ones. The difficulty to respect the geometry of the orifice is

CHAPTER~, MASS EXOW MEASUREMENTS 45

Table 4.2: Cornparison of the discharge coefficients of our second steel orifices calibration with Grace and Lapple results.

the main source of error. The diverging shape and size of the hole, the bad machining which

includes the damages inside the hole and the contoured inlet are al l significant factors which

contribute in the variation of the discharge coefficients.

If the orifice shape is more eiliptical than circular, which can be easily seen from an optic

microscope, it would be normal to get slightly higher discharge coefficients as demonstrated

by Callaghan and Bowden (1949). The ellipse approach a slot in shape and therefore yield the

highest flow coefficient because the flow lines fiom a slot converge in only one direction as

oppose to the flow lines emerging fiom a circuiar onfice which converge fiom al1 azimuths and

hence result in the lowest flow coefficients. We can clearly see this effect on Figure 4.7 fiom

Cailaghan and Bowden.

P RQ Ca

We took pictures of the smallest alurninum onfice (Figures 4.8 and 4.9). From these pic-

tures we can see al1 the orifice imperfections. The upstream side picture shows that aluminum

particles are obstnicting the hole. On the downstream side picture we detect dirt accumulation

from machining and other extemal sources. Dirt accumulation and metallic particles are ob-

stnicting the fiow and thus reducing considerably the discharge coefficient. This could explain

the low discharge coefficients for the orifice of diameter 0.03 16 in.

UTIAS

An other important source of variation in Cd is the quality of measurernent of the orifice

hole diameter. Indeed, a slight variation of a few tens of microns in the diameter of the orifice

Grace and Lapple 0.0300 8000 0.6265

0.1787 38127 0.6227

0.0316 8079 0.5082

0.0600 22000 0.6050

0.0600 18245 0.6030

0.1189 37911 0.6113

0.1200 42000 0.6067

0.1900 46000 0.6067

CNAPTER~. MASS FLOW MEASUREMENTS 46

Figure 4.7: The influence of orifice shape on the discharge coefficient (Callaghan and Bowden, 1 949).


Figure 4.8: The upstream side of the ,û = 0.06 duminum orifice, magdicatioa = 64.

Figure 4.9: The downstteam side of the P = 0.06 aiuminum orifice, mapification = 32 (scale 1 /64 inch).

~ H A P T E R ~ , MASS FLOW MEASUREMENTS 48

hole can have a major impact on the value of Cd. The discharge coefficient is directly propor-

tional to the square of the inverse orifice diameter. AU OUI diameter measurements have been

done on a optic microscope which c&d a micron meter d e r . We took about five diameter

rneasurements for each orifices in different directions.

4.7 Error analysis for the orifices calibration

The enor analysis was conducted following the method presented by Kline and McClintock

(1953) for single-sample experiments i.e. for experiments in which uncertainties are not found

by repetition. To evaluate the enor on my results I used the second-power equation which is

an excellent approximation to calculate the uncertainty interval in the result. If X is a linear

hinction of n independent variables, each of which is norrnally distributed, then the relation

between the interval for the variables wi, and the interval for the result YX is given by

Al1 the equations used to calculate the error bars for the orifices calibration are presented

in Appendix E. A table containing the Iist of instruments used for the calibration and their un-

certainties is also put in this appendk.

Figure 4.10 shows the reduction of the error bars on Cd after using the second apparatus to

calibrate the orSces. The values are in the same range which shows that the first calibration

was a good approximation, but the error bars are now considerably reduced.

~ H A P T E R ~ . MASS FLOW MEASUREMENTS 49

Figure 4.1 0: Cornparison of the error bars for the new and old calibration method for orifice diameter of O. 1 1 89"

EXPERIMENTAL INVESTIGATION OF THE NOZZLES

This chapter is divided into two sections. The first one is about the experiment conducted on

the different noule geometries. It consists of a preliminary experiment which study the depen-

dence of the discharge coefficient on the Reynolds number, and on the main experiment which

compares the flow behaviour inside the SailRail noule with two other nozzle geornetries. The

three nozzle geometries which are going to be studied are: the current SailRail inclined nozzle (

25" to the surface of the rail), the radial nozzle (perpendicular to the surface of the rail) and the

radial novle with a diffuser exit. The three nozzle configurations are depicted in Figure 5.1.

The last section of this chapter presents the results of the experiment conducted on the mono-

rail apparatus. This is used to conduct sliding fiction coefficient experiments on the SailRail

runners. In the present research, it was used to ver@ the location of choking for inclined and

radiai nozzles.

As explained in chapter 2, because of the pressure inside the SailRail system, the flow is

CHAPTER~. ExPERIMENTAL INVESTIGATION OF THE NOZUES 51

Figure S. 1 : The three nozzle geornetries studied.

mainly compressible and likely to be choked. In the expenments described here, we rnanaged

to get pressure ratios below the critical limit of 0.528. We built an apparatus to evaluate the

discharge coefficient when the nozzles were uncovered to find out which geomew allowed

the fess inlet losses, and when they were covered to study which configuration was the rnost

restrictive in the cavity region. This experiment took into account al1 the parameters on which

the discharge coefficient depends (see chapter 3). The only variable which was not simulated

properly was the cover of the mnner. We used a thick aluminum plate as the runner. However,

according to Sullivan et al. (1985), the cover can be modelled as a rigid surface since its de-

flection Iocally around the nozzle is believed to be very small. The first section of this chapter

describe this experiment.

CHAPTERS. EXPERIMENTAL INVESTIGATION OF THE NOZSLES 52

5.1.1 Description of the apparatus

The apparatus consisted of a cylindrical aluminum chamber with an intemal diameter of 165.1

mm (6.5 in) and 152.4 mm (6.0 in) deep. On the top of this reservoir, we fixed an 25.4 mm (1

in) thick duminum plate on which the nozzles were drilled. This plate was called the nozzle

plate and was fixed to the chamber by four bolts. Six radial nozzles including one with a dif-

fuser exit were dri1Ied around the center of the plate dong a circle with a radius of 12.7 mm

(0.5 in). An Uiclined nozzle was drilled at a radial distance of 25.4 mm (1 in) fiom the center.

Three other aluminum nozzle plates were also machined to be used in the Reynolds number

experiment. Through each of them was drilled a radial noule. The first plate correspond to

two times the dimension of the current S ailRail nozzle: nozzle diameter of 1.1 2 mm (0.044

in) with a thichess of 6.35 mm (0.25 in). The second and third plate had nozzle diameters of

2.23 mm (0.088 in) and 3.35 mm (0.132 in) respectively, and thickness of 12.7 mm (0.5 in)

and 19.05 mm (0.75 in) respectively. To simulate the runner over the nozzles, we used another

alurninum plate of 25.4 mm (1 in) thickness. This plate, cded the cover plate, was put on top

of the nozzle plate. We couid vary the gap between the plates by putting spacers between them.

For the main experiment, we wanted to keep the same dimensions as the current SailRail

nozzles which are 0.56 mm (0.022 in) of diameter and 3.1 8 mm (0.125 in ) thick. The use of a

3.18 mm thick duminum plate was unacceptable because the plate would have bent due to the

high pressure in the chamber and would have generate wrong results dunng the covered test.

From the Machinery's Handbook (Oberg, E. et al. 21st ed. M l ) , we calculated the thickness

needed to prevent the deflection of the plate. We found out that a 25.4 mm ( 1 in) thick plate

would be acceptable for the pressure range used (see Appendix F for detail calculations). We

thus drilled holes of 9.53 mm (0.375 in) diameter through 22.23 mm (0.875 in) of the lower

portion of the plate, and drilleci the nozzles through the remaining thickness. B y doing this, we

CKAPTER~. ExPERIMENTAL INVESTIGATION O F THE NOZZLES 53

were getting the right nozzle length and preventing further plate deflection, and maintaining a

porosity factor of 0.003 (0.022~/0.375*). The porosity factor is the ratio of the cross-sectional

area of the constriction (in our case, the noule) over the area of the approach 00w. If this value

is smailer than 0.1, the incoming flow can be considered at the stagnation or total conditions

(Linfield, 1999).

The system was fed by compressed dry air which Bowed first through a copper coil soaking

in a water bath in order to control the air temperature. After its passage into the coil, the air

flowed inside the orifice meter. Then it went through a plastic tube which was connected to a

hole cûilled on the bottom side of the chamber. The air in the chamber escaped by one of the

nozzles drilled through the nozzle plate. Figure 5.2 shows the diagram of this experimental

mount. AU the drawings of the air chamber are shown in the Appendix G.

Before and after every test sessions, we had to take the barometric pressure and the atmospheric

temperature. During the test, we measured the temperature of the water bath in which the cop-

per tube was soaking.

To calculate the mass flows we took voltage measurements of the orifice meter pressure

transducers, as explained in detail in Chapter 4. The orifice selected had a ,û ratio of 0.03 16.

This orifice allowed us to measure mass flows in the desirable range by keeping an acceptable

pressure ratio across the orifice. A pressure transducer was mounted in the durninuin chamber

wall . This pressure transducer allowed us to measure precisely the pressure inside the cham-

ber. The characteristics of the later pressure transducer are presented in Appendix C. During

the test, the chamber pressure was adjusted in order to get similar SailRail pressure ratio p, l

m. In our experiment, the cavity pressure p, is the aunosphenc pressure p,,,. Thus, we cm

CHAPTER~. EXPERIMENTAL INVESTIGATION OF THE NOZZLES 54

Figure 5.2: Outline of the noule experiment.

CHAPTERS. EXPERIMENTAL. INVESTIGATION OF THE NOZZLES 55

operate with smaller pressure in the chamber.

As explained before, we conducted two types of tests. First, we tested the uncovered noz-

zles. We fixed the nozzle plate to the chamber by four bolts. Then, we gradually increased

the pressure inside the chamber by the help of a throttling valve. For each chamber pressure

increment of 0.1 volt, we took voltage rneasurements of the pressure transducers inside the

orifice meter.

The second case consisted of the covered nozzles. We bolted to the chamber the nozzle

plate and the cover plate on top of it. Between hem, we put the spacers. As shown in the

chapter 2, we know fiom previous researches conducted at UTIAS that the value of the cavity

height , h,, varies between 0.1 mm and 0.3 mm. We selected the spacer accordingly. For each

cavity gap, we took the mass flow measurement for a given chamber pressure. We tried to

maintain a constant pressure ratio a t m / p T for al1 measurements. This allowed us to study the

behavior of each nozzle in relation with the variation of the cavity gap only.

5.1.3 The procedure to find the nozzle discharge coefficients

The nozzle discharge coefficients were

the orifice meter by the ideal mass flow

calculated by dividing the measured mass flow from

through a constriction.

The ideal mass flow corresponds to equations 3.1 and 3.2 for unchoked and choked (at the vena

contracta formed inside the nozzle) flow respectively.

The b t step was to determine the mass flow through the orifice meter, k,, using equation

4.1. The downstream and differentid pressures in the orifice were measured by the transducers

CHAPTERS. EXPERIMENTAL INVESTIGAT~ON OF THE NOZZLES 56

and thus assuming a perfect gas, the density could be found. The orifice area was know from

microscope measurements. The only unknown was the discharge coefficient. However, we

knew how Cd vked with Red and the pressure ratio pd/pu as explained in chapter 4. For

each measment , we calculated the value of pd/pu. Then, it was possible to calculate the

correction factor Kc. Frorn the average value CdDo foi the orifice used, we started the iteration

process in order to find the values of CdO for the corresponding ratio pd/pu. We started with

this initial mass flow equation:

Then we calculated the corresponding Reynolds number:

In chapter 4 we discussed about the fact that the relation between Cd and Red could be ap-

proximated by a hear finction of the form :

For each onfice we found the value of the intercept A. and the dope Al. For the orifice used

in these fiow measurements, the values of ilo and Al are respectively 0.418435 and 1.1399 x

lW5. We introduced this new CdO inside equation 5.3, which became:

and we got a new orifice mass fiow. We introduced this new mass flow into equation 5.3 and

so on. We continued this iteration process until the value of the mass flow converged with an

CHAPTERS. EXPERIMENTAL WVESTIGATION OF THE NOZZLES 57

error less than IO-". Thus, we could divided the orifice mass flow by the ideal mass flow

through the nozzle. When the pressure ratio across the nozzle p=tm/m was over the cntical

value 0.528, equation 3.1 for unchoked flow was used. On the other hand, the equation 3.2 was

used when pat,/m was below the cntical E t .

According to the dimensional analysis presented in chapter 3, the discharge coefficient of the

flow through a SailRail nozzle is a function of the Reynolds number, the pressure ratio p,/m, the height of the cavity over the nozzle diameter hc/dN, the length of the novle over the nozzle

diameter lNldN and the angle of the nozzle to the surface of the rail B.

To study the dependence of Cd on Red, we built three nozzle plates of different sizes.

Through each of them was drilled a radial node. Each nozzle had the same ratio lNldN which

was equivalent to the the SailRail noule proportion. The three nozzles had respectively a noz-

zle diameter and length of ~ W O , four, and six times larger than the cunent SailRail nozzle.

We installed these plates one at the Ume on the top of the chamber. We let the plate uncov-

ered so the nozzle exit pressure was the atrnospheric pressure and thus we were not taking into

account the ratio hcldN. The value of 0 was the same for each nozzle and was equd to 90".

We plotted the reIation between Cd and patm/pr. Consequently, if the Reynolds number had no

effect on Cd, we should obtained similar discharge coefficients for every nozzle.

Figure 5.3 presents the plot of Cd versus the pressure ratio for the three radial plates of

different size. From this graphic, we observe hat the smallest and largest nozzles have similar

behaviours. However, there critical discharge coefficient of 0.94 and 0.90 for the smallest and

largest novles respectively are much more higher than the one predicted by the theory which

is 0.825 for cylindncai orifices with length over diameter ratio between 1 and 7 (see chapter 3).

CHAPTERS. EXPERIMENTAL INVESTIGATXON OF THE NOZZLES 58

On the other hand, the medium nozzle plate presents a cntical discharge coefficient similar to

the accepted value. It appears to us that the smallest and largest nozzles have probably lightiy

rounded nozzIe inlet. As been shown in chapter 3, it does not take to much of a rounded inlet

to cause a major increase in the value of Cd. The nozzles were drilled into alurninum plate

thus increasing the difficulty of machining s h q edges. According to these later remarks, we

can conclude that the effect of Red on Cd is weak and that we can neglect it in the following

experirnents.

5.1.5 Main experiment results

Uncovered nozzles

The results of the k t part of this experiment are not very conclusive. First it is not clear

where choking occurs. On Figure 5.4 the relation between the discharge coefficient and the

pressure ratio pamr/pr is plotted for the inclined, radial and radial with difiser exit nozzles.

The discharge coefficient of al1 b e e nozzles continues to increase even after critical condition

is reached. According to the theory, these nozzles should choke at a pressure ratio close to the

theoretical choking value of 0.528. It is not clear what causes this observed behaviour. Further

investigations should be done in order to solve this problem.

Figure 5.5 presents the plot of the volume flow versus the pressure ratio for uncovered

inclined and radial nozzles. Here again, the nozzles do not show any signs of choking which

is unusual. However, we can make this interesting observation that when uncovered and for

a given pressure, the radial nozzles deliver a higher volume flow than the inclined noules.

Which is opposite to the observation made in the previous UTIAS research discuss in chapter

2, for covered nozzles with a hJdN of 0.109.


Discharge coefficient versus the pressuib ratio amss three radlal nonles of dlfferent she but same proportion.

Figure 5.3: Discharge coefficient versus the pressure ratio for three uncovered radial noules of different size but similar B and lN /dN ratio.

CHAPTERS. EXPERIMENTAL INVESTIGATION OF THE NOZZLES 60

Figure 5.4: The discharge coefficient versus the pressure ratio across the three uncovered nozdes.


Vdume fiaw versus oie pressure falio for un#wred inclineci and radiai naales with diameter of 0.m

C

Figure 5.5: The volume Row through the uncovered radial and inclined nozzles versus the pressure ratio p a h / a .


Covered nozzles

Figures 5.6 and 5.7 presents the relation between the discharge coefficient and hc/dN. The

reference area considered is defined by the area made by the nozzle periphery and the cavity

height. For the sharp edged radial nozzle and the radial nozzle with a diffuser exit, the reference

area Arz is given by :

Where d is the diameter of the nozzle in the case of the sharp edged radiai noule, and the

dinuser diameter in the case of the radial noule with the diffuser exit. For the inclined noule,

the area is given by the following equation:

Where dN is the diameter of the inclined noule. For these cases, we are considering the situ-

ation where area choking would occur at the tight gap produced by the separation bubble and

the cover in the cavity region.

From these plots, we observe for the sharp edged radial nozzle that Cm decreases drnost lin-

early with increasing cavity gap above hc/dN of 0.3. Below that vaiue, the discharge coefficient

is scatter, but seems to oscillate around a constant vaiue of 0.55. The value of hJdN where the

area of the sharp radial nozzle and the area made by the nozzle periphery and the cavity gap

are equal is 0.25 ( jd2 = r d h, + h, I d d.25). below that value, it is the area made by the

n o d e periphery and the cavity gap which govems the flow. The experimentd results suggest

this passage between the two controlling regions.

A similar phenornenon is observed for the radial nozzle with diffuser exit and inclined noz-

de. In the case of the radiai nozzle with the diffuser exit, Cm decreases hast linearly with

increasing cavity gap above h c / d ~ of 0.2. Below that value, the discharge coefficient is scatter,


Cd2 versus hld for ltvee ride geometries with nominal dameter of O . O Z , pressure raüo 0.49.

Figure 5.6: The discharge coefficient versus the height of the cavity over the nozzle diameter for the three novle geometries, with reference area, Aî, equal to the area defined by the nozzle penphery and the cavity height (and pressure ratio pah/p, = 0.49).

CHAPTERS. EXPERIMENTAL INVESTIGATION OF THE NOZZLES a

cd2 versus Nd for three diiferent nonle geombses with minal diameter d O . O Z , presswe ratio of 0.37

Figure 5.7: The discharge coefficient versus the height of the cavity over the nozzle diameter for the three novle geometries, with reference area, il2, equal to the area defined by the nozzle periphery and the cavity height (pressure ratio p.,/m = 0.37).


but seems to oscillate around a constant value of 0.85. The value of hJdN where the area of

the radial nozzle with the diffuser exit and the area made by the nozzle periphery and the cavity

gap are equal is 0.16. The diameter of the diffuser is 0.89 mm (0.035 in). Thus,:d2 = ?r 0.89

hc + h, / d 4.16.

For the inclined nozzle, Ca decreases aimost iinearly with increasing cavity gap above

hJdw of 0.1. Below that value, the discharge coefficient drops to a constant value of 0.25. The

value of hJdN where the area of the inclined nozzle and the area made by the nozzle periphery

and the cavity gap are equal is 0.14 (qd2 = 1.81 64 n d h, -+ h, / d =0.14).

From this experiment, we conclude that the flow in the radiai novle is limited by the the

area fomed by the periphery of the nozzle and the cavity height. Thus, the flow through a

radiai nozzie is likely to choke at this location by opposition to the radial nozzle with diffiser

exit and the inclined noule which are more likely to choke at the nozzie inlet.

The last two figures (Figures 5.8 and 5.9) present the discharge coefficient plots versus the

height of the cavity over the nozzie diarneter. The reference area considered here is the area of

the nozzle for which Al = (?r dZ )/4. Tbo pressure ratios were used; patm/pT = 0.37 and 0.49.

For these pressure ratios, the flow should be choked. Thus, we could detect which nozzles

choked at the inlet of the nozzles.

These charts showed that the discharge coefficients of the three nozzles are almost constant

and similar when the cavity height over the nozzle diameter,h,/dN, is above 0.3. Thus, for the

studied nozzles of diarneter 0.56 mm ( 0.022 in), over a cavity gap of 0.17 mm (0.0066 in),

d l three nozzles present a constant maximum Cd which is around 0.77. However, the radial

nozzie with diffuser exit reaches a constant Cd at a vahe of hcldN of about 0.2, and inclined

noule at 0.15. As explained previously, these defiection points correspond approximately to

the vdue of hc where the area made by the periphery of the nozzle and the cavity gap is equal

to the nozzle area Below h,/& their respective deflection point, the discharge coefficient

CHAETER~. EXPERIMENTAL INVESTIGATION OF THE NOZZLES 66

The discharge coefiicien! veisus the cavity heigh! over tfie noule diameter for the three nonle geometrles with diameter of O . O Z , pa!m/pt 4.49

Figure 5.8: The discharge coefficient versus the height of the cavity over the nozzle diameter for the three noule geometries. with reference area, Al, equal to the nozzle area (and with pressure ratio pah/m = 0.49).


The dlçcharge coeffidentversus the hdght d Ihe caviîy over the noule diametef for thme nonle geomebies wati dlameter of o . o z , paanlp 4.37

Figure 5.9: The discharge coefficient versus the height of the cavity over the nozzle diameter for the three nozzie geometries, with reference area, Al, equal to the nozzle area (pressure ratio p d p r = 0.37).


seems to increase IinearIy with h,; the siope of the inclined nozzle being the more steep and

the radial nozzle the one with the slowest increase of Cd. Theses charts also show that choking

in the radial nozzle is controlled by the area made by the periphery of the nozzle and the cavity

height. Further more, for the two cases studied, the pressure ratios used (pat,/pr = 0.37 and

0.49) were bellow the cntical condition and showed similar flow behaviour.

5.2 Experiments conducted on the monorail apparatus

The monorail apparatus consists of a single rail mounted on a large 1-beam, with the load and

its carnage of total mass m, king slung undemeath the beam. A cabie-pulley system connects

the caniage to a falling weight of mass md, and, at the opposite end of the rail, a pneumati-

cally operated mechanism releases the caniage, thus allowing it to move under the action of

the falling weight (Sullivan 1997). This apparatus was designed to conduct a variety of tests

including investigations on the use of m e r end-caps and the coefficient of sliding fiction.

In the present research, the monorail apparatus was used to compare the behaviour of the

inclined and radial nozzles. There are two rows of nozzles drilled through the surface of the

rail. One row constitutes of inclined noules and the other of radial nozzles. To study the flow

inside one of the nozzle series, we blocked the other nozzle row with plaster. This allowed us to

test only one type of nozzle. The plaster could easily be removed with a hand drill after the test.

We measured the volume flow for the nozzles Iocated under the ninner. In all there were

seven nozzles opened. The fist measurernent consisted of measuring the volume flow when the

nozzles were uncovered by the nozde. Then, we put back the runner on top of the n o d e and

took an other series of volume flow measurements. We repeated the measurements four and

nventy four hours later to see if there was any m e r conditionhg effect. These measurements

were done for the radiai and inclined nozzles. The volume flow were taken by rotameters.


Since they never been recalibrated since 1985, we decided to non dimensionalyse the volume

flow by dividuig it by the maximum volume flow measured for each male. This would con-

serve the behaviow of the flow in each nozzle but would not provide any infornation on the

value of the volume flow.

Figure 5.10 and 5.1 1 presents respectively the behaviour of the radial and inclined novles

when uncovered and covered by the runner. For the radial nozzles, we observed that the volume

flow when nozzies are uncovered is much higher than when covered by the runner. It confirms

that the radial nozzle does not choke at the vena contracta fonned in the nozzie inlet but rather

at the area fonned by the periphery of the nozzie and the cavity gap. On the other hand, the

volume flow for the inclined nozzies does not increase when uncovered by the nozzie. Thus, it

confhns that the inclined nozzles choke at the vena contracta inside the nozzle.

CHAPTERS. EXPERMENTAL INVESTIGATION OF THE NOZZLES 70

AdlmensIonal volume flow versus the track pressure for radial nozzles

O 50 100 150 MO W ) 300 350

Track pmssure (kPa gage)

Figure 5.10: The non dimensional volume flow venus the track pressure for uncovered and covered radial nozzle.

CHARERS. EXPERIMENTAL INVEST~GATION OF THE NOZZLES 71

Adimensional votume flow vs the trac& pressure for inctined nonles

O 50 100 150 200 250 300 350

Track pceruum (kPa gage)

Figure 5.1 1: The non dimensional volume Bow versus the track pressure for uncovered and covered incfined nozzle.

6.1 Review of the main results

From the mass flow measurement instrument that we developed, we observed that aluminum

orifices gave higher discharge coefficient than stainless steel orifices. The main explmation for

this phenornenon is the difficulty to machine sharp edges with aluminum because of its inuin-

sic properties. Thus, the inlet of duminum orifices are contoured and the discharge coefficient

increases. The behaviour of our orifices confirmed previous research done on that topic. We

also c o n h e d that the shape of the onfice and the qudity of machining are other factors which

cm affect the discharge coefficient.

From the preliminary experiment where we used different sizes of radial nozzles having the

same proportions as the current SailRail nozzles, we confirmed that the discharge coefficient is

weakly dependent of the Reynolds number.

The results of the main experiment showed that when uncovered, radial nozzles are deliv-

ering more flow for a given pressure ratio than the inclined nozzles. From the test made with

cover plate on top of the nozzles, we discovered that radial nozzles are more likely to choke

in the area made by the periphery of the nozzle and the cavity gap than at the vena contracta

formed inside the nozzle. We also discovered that radial nozzles with a diffUser exit have a

similar behaviour than the inclined nozzle; in other words that they choke at the nozzle inlet.

Furthermore, the tests conducted with the nozzle plates allowed us to verify the critical

discharge coefficient for nozzles with a lN/dN ratio of 5.7 (for radial nozzles) and 13.4 (for

inclined nozzles). From the literature review, the critical discharge coefficient for sharp inlet

cylindrical orifice with IN/dN ratio has been shown to lie between 0.8 1 and 0.86 (Ward-Smith,

1979). Experimentally, we found critical discharge coefficient varying between 0.75 and 0.93.

The main cause of these discrepancies are attributable to the variation in the sharpness of the

nozzle inlet, and the inaccuracy in the nozzle diameter measurements. These two factors have

been shown to have an important impact on the value of the critical discharge coefficient

The monorail apparatus experiments allowed us to confinned that the radial nozzles are not

choking at the vena contracta formed at the nozzle inlet, but at the area made by the nozzle

periphery and the cavity gap. The results also confirmed that the inclined nozzles choke at the

nozzle inlet.

6.2 Recommendations

In the first instance, we should test the radial nozzle with the diffuser exit on the monorail

apparatus in order to get an other confirmation of its choking location. If this test agrees

with the previous results, we should recommend to replace the acmal inclined nozzle by radial

nozzles with diffuser exit The later being easier to manufacture and thus allowing savings in

time and energy.

FLOW THROUGH A CHOKED NOZZLE

Assuming an ideal gas @ = pRT) and isentropic flow (fi = const). T'us, fiom the energy

equation (h + iv2 = ho) we find the relation between the temperature ratio versus the Mach

number.

At the critical condition, M = 1 and 9 becomes:

APPENDIXA. FLOW THROUGH A CHOKED NOZZLE 75

At the choked condition the ideal mass fiow is given by:

7jtf = p.aA.

The speed of sound at the choking condition is given by:

Since we assumed the flow to be isentropic, we can find the density at choking by the following

relations:

substituting p by pRT, we get the following expression:

A P P E N D U . FLOW THROUGH A CHOKED NOZZLE 76

and the critical density can be written as:

By substituting the temperature ratio inside the previous equation, we get:

Substituthg p. and a. in the main equation of the critical mass flow, we get:

(A. 10)

(A. 1 1)

ORIFICE METER DRAWINGS

APPENDIXB. ORIFICE METER D R A ~ G S 78

Figure B.l: Assembly drawing of the orifice meter.

APPENDUCB . ORIFICE METE32 DRAWINGS 79

Figure B.2: Detail drawing of pipes 1 and 2.

mmON C-C

UNION UNION

P B /au-

SECTION I-t

UNION #3

APPENDIXB . ORIFICE METER DRAWTNGS 81

Figure B.4: Detail drawing of the pressure transducers chamber.

SECTION A-A HIGH PRESSURE SIDE

SECTION B-B LOW PRESSURE SIDE

SANE D I m m O N S A3 $ml ON A-A EXCEPT WR nrs ONE INDIcArsrri)

O et. O ?

ORIFICE PLATE

PRESSURE TRANSDUCERS

The pressure measurements made inside the orifice meter, at the top of the water column for the

mass flow experiment and inside the pressure chamber for the main experiment were done by

piezoresistive pressure transducers. These pressure transducers have an electric output propor-

tional to the pressure on theK sensing surface. The piezoresistive pressure transducer is useful

for measuring steady-state or static pressures as well as dynamic pressures (Endevco, 1990).

Table C. 1 presents the main data relative to each of the pressure transducer used in the present

work.

Table C.1: List of pressure transducers and equipment used.

pressure hnsducer AP Pd

Pwc Peh

presmre range 5 psig 50 psig 5 P% 50 psig

gain 10 10

mode1 Endevco 85 108-5 Endevco 85 10B-50 Endevco 85108-5 Endevco 85108-50

signal conditioner mode1 4423 4423

SN AW91 AW94

SN E26R 47YY E27R DL lH

4423 4423

sensitivfty 45.7mVlpsi 6,17mV/psi 48.6mVlpsi 6.34 mVIpsi

AM76 AM80

10 10

APPENDUCC. PRESSURE TRANSDUCERS 85

C.l Pressure transducers calibration

It was essential to recalibrate the pressure transducers in order to get the proper pressure value

for each given voltage. To do so, 1 plugged a pressure transducer in each pressure tap of the ori-

fice meter. The orifice meter exit was blocked and the inlet was connected to a pressure gauge.

The system was fed by the laboratory air cornpressor. 1 let the pressure inside the orifice rneter

built up and when it reached its maximum value, 1 took note of the pressure and voltage for

each transducer. I repeated the calibration for each pressure msducer. The following figures

present the calibration curve for each pressure transducer.

The following table gives the calibration equations for each pressure transducer.

Table C.2: Values of the pressure versus the measured voltage.

1 pressure transducer II Equation 1

APPENDIXC. PRESSURE TRANSDUCERS 86

Figure C. 1 : The calibration of the pressure trûnsducer E26R.

APPENDIXC. PRESSURE TRANSDUCERS 87

Figure C.2: The calibration of the pressure transducer 47YY.

APPENDLXC. PRESSURE TRANSDUCERS

Figure C.3: The calibration of the pressure transducer E27R.

APPENDIXC, PRESSURE TRANSDUCERS 89

Figure C.4: The caiibration of the pressure transducer D 1 1 H.

COMPARISON BETWEEN STEEL AND ALUMINUM ORIFICES

In the following pages, the discharge coefficient charts for the orifice in stainless steel and

aluminum are presented. The Cd are plotted versus the experimental mass fiow for orifices

with nominal p ratios of 0.06,0.12 and 0.18. From these charts. we can study the impact of

the orifice material on the discharge coefficient.

APPENDIXD. COMPARISON BETWEEN STEEL AND ALUMINUM ORIFICES 91

Figure D.1: Orifice calibration for stainless steel and alurninum orifices with P ratios of 0.06.

APPENDIXD. COMPARISON B ETWEEN STEEL AND ALUMINUM ORIFICES 92

Figure D.2: Orifice calibration for stainless steel and aluminum orifices with #3 ratios of 0.12.

APPENDIXD. COMPARISON BETWEEN STEEL AND ALUMINUM ORIFICES 93

Figure D.3: Onfice calibration for stainless steel and aluminum orifices with B ratios of 0.18.

ERROR ANALYSIS FOR ORIFICE METER CALIBRATION

The present Appendix presents the detailed equations used to evaiuate the total enor made on

the orifice calibration. First, the list of equipment used and there uncertainties is introduced.

These uncertainties are going to be used in the error calculation. In the last section of this

Appendix, the final orifice calibration charts are presented with the error bars.

Table D. 1 is a list dl instruments used in the orifice calibration and their uncenainty which we

assumed equal to the half of the smallest division.

APPENDIXE. ERROR ANALYSTS FOR ORIFICE METER CALIBRATION 95

Table E.1: The uncertainties for the instruments used in the orifice calibration.

voltme ter barorneter

1 volume of water II + 0.1251 1

I 0.0005 V or 3~ 0.005 V 3r 0.05 mmHg

thennometer d e r

1 I

pressure gauge f 2 m m ~ g I

=t0.OS0C k 0.0005rn

In the following lines, al1 the equations used to calculate the error bars on the discharge

coefficients are presented.

101300Pa pair = Patm 760mmHg - - P"

pair Pair = RT

APPENDIXE. ERROR ANALYSE FOR ORIFICE METER CALIB RATION 96

APPENDLXE. ERROR ANALYSIS FOR ORIFKE METER CALIBRATION 97

Neglecîing the porosity factor (j3 -+ O), the general mass flow equation through the orifice

becomes:

- - 7 1+1 - 2puY pu- y - 1 ~ d 7 = "

E.2 Orifice calibration charts with error bars

(E. 17)

The following figures shows the plots of the discharge coefficient versus the experimentd mass

fiow for the onfices in stainiess steel. The error bars are added and the experimental mass flows

APPENDWE. ERROR ANALYSIS FOR ORIFICE METER CALIBRATION 98

correspond to the mass flow measured by the second calibration experiment.

Figure E.1: Enor analysis on orifice calibration for stainless steel orifices with P ratio of 0.03 f 6.

APPENDLXE* ERROR ANALYSIS FOR ORIFICE METER CALIBRATION 99

Thr dlsahrrge coetfldmnt vrrrur th- mxperlmrntml masr flow for oriflœ In itrlnlsu r f m d wtth

Figure E.2: Error analysis on orifice calibration for stainless steel orifices with f l ratio of 0.0600.

APPENDUCE. ERROR ANALYSIS FOR ORIFKE METER CALIBRATION

Th* dlrcrhmrgr a#lllclrnt vamur the oxporîmmntrl m a u naw lor i n orIllor h stnlnless rtwl wlth Cntm rmtlo of O . t l 6 0

Figure E.3: Error analysis on orifice caiibration for stainless steel orifices with B ratio of 0.1 189.

APPENDIXE. ERROR ANALYSIS FOR ORIFICE METER CALIBRATION 101

The dl~harq, c-oonlalsnt vomus th* rxpmrlrnrnti) mrsr now for i n orifiam In rtmlnleu itoal wltk i ktr mtlo ot O.17û7

Figure E.4: Error analysis on orifice calibration for stainless steel orifices with ,f3 ratio of 0.1 787.

DEFLECTION OF A FLAT PLATE

From the machinery's handbook (Oberg et al. 198 l), the deflection of a circular flat plate where

the edge id fixed around the circumference, and where the Ioad is uniformly distributed over

the surface of the plate is given by:

where D is the deflection of the plate, ut is the load applied to the plate, R, is the plate radius.

E is the modulus of elasticity, and t, is the plate thickness. To use this equation, al1 the units

must be in English unit.

The maximum pressure to be used in the charnber is 330 kPa (47.89 psi). The maximum

load applied to the plate, w, is equal to the product of the maximum pressure inside the cham-

ber, p& by the plate area supponing the load, A,- Thus, w = 47.89 psi x (7.125in)* ~ 1 2 =

1909.33 Ibs. For aiuminum, E = 10.3 x 106 psi. Therefore, the deflection, D, for a one inch

aiuminum plate is equal to 3.2 pm (1.27 x 1oe4 in).

APPENDMG. NOZZLE EXPERIMENT DRAWWGS 104

\ A

Figure (3.1: Assembly drawing of the main experiment mount.

Figure (3.2: Detail drawing of the chamber.

APPENDIXG. NOZZLE EXPERIMENT D R A W G S 106

Figure G.3: Detail drawing of main experiment nozzIe plate.

APPENDIXG. NOZZLE EXPERIMENT DRAWINGS 107

Figure (3.4: Detail drawing of the preliminary experiment radial n o d e plates.

APPENDIXG. NOZZLE EXPERMENT DRAWINGS 108

Figure G.5: Detail dtawing of the cover plate.

[ l ] Alder, G.M., The numerical solution of choked and superrritical ideul gasflow thmugh orijces and conver-

gent conical nozzles Goodier, Journal o f mechanicd engineering science, 1979, pp.197-203.

[2] Amberg, B.T., Britton,C.L., Seidl,W.F.,Discharge coejjicient correlations for circular-arc ventitri)?ownre-

ters at critical (sonic)flow, Journal o f ff uids engineering, lune 1974,ppJ 1 1-123.

[3] Benson, R.S., Pool, D.E., The compressibleflow discharge coeflcients for a two-dimensional dit , interna-

tiond journal of mechanical sciences,Vol7, 1965, pp.337-353.

[4) Bragg, S.L., Effecr of compressibility on the dischatge coe$îcient of onpces and convergent noules , Journal

o f mechanical engineering science, Vol 2 No 1 , 1960, pp.35-44.

[SI Brairi, T.S.S., Reid, I., Peflonnance of small diameter cylindricul critical-fow nozles, National engineering

laboratory, report No 546, June 1973.

[6] Cdlaghan, E.E., Bowden, D.T., Invesrigation o fjo w coeBcient of circuia~ square, and ellipticci l onpces ar

high pressure ratios, National Advisory Cornmittee for Aeronautics, Technicd note 1947, September 1949.

[fl Deckker, B .EL, Chang, XE, An investigation of steady compressible~ow thm ugh thick orifices, Proc hstn

Mech Engrs 1965-1966, pp. 3 12-323-

B IB LI0 GRAPHY 110

[8] Deckker, BEL., Chang, Y.E. Transient Mects in the discharge of compressed airfium a cylinder thmugh

an ori$ce, Journal of basic engineering 1968, pp, 1-9.

[9] Endevco, Piezoresistive pressure transducers in.stnrcn'an manual, IM8500, San Juan Capistrano, California,

July 1990.

[IO] Feuchter, EJn investigation on the energy eficiency of the SaiRail cornpliant air bearing., University of

Toronto Institute for Aerospace Studies, 1999.

[IL] Grace, H.P., Lapple, CX., Discharge coe$cients of small diameter orifices andflow noales, Transactions

of the ASME, h l y 195 1, pp. 639-647.

[12] Hinchey, MJ., Sullivan, P.A., and Graham, T,A.J?esearch on the SailRail Cornpliant SuSuce Air Bearing,

Transportation Development Center, Transport Canada, Montreal, Quebec, Report D-500-599- l, Jan* 198 1.

1131 Jackson, R.A., The compressible discharge of air thmugh small thick plate orifices , Applied scientific

reseatcti- Section A , 1963, pp.241-248.

[14] Kastner, LJ., Williams, T.J., Sowden, R.A., Critical-flow noule merer and its applicaiion to the measure-

ment of massflow rate in steady and pulsating streams of gas, Journal mechanicd engineering science,

1964, pp. 88-98.

[I 51 Kline, SJ., McClintock, RA., Describing uncenainries in single-sample enperiments, Journal of Mechanical

Engineering, Januw 1953, pp.3-8.

1161 Linfield, K.W.,A study of the discharge coeflcient of jets j v m angled slors and orijices. , University of

Toronto hstitute for Aerospace Studies, Doctor of Philosophy, 1 999.

1171 Miller, R.W., Flow measurement engineering handbook, McGraw-Hill Publishine Company, second edition,

New-York, 1989.

1181 Oberg,E., lones,ED., Horton, H.L., Machinery 's handbook, Industrial press inc., twenty-first edition, New-

York, 198 1.

[ 191 Ower, E., Pankhurst, R.C., The rneasurement of airflo w, Pergarnon Press Ltd., Great-Britain, 1969.

[20] Perry, LA., Cniical Jow troligh sharp-eiiged onpces, Transaction of the ASME, October 1949, pp. 757-

764.

121 1 Spigel, M.R, Schaum 's autline series theory and pmblems of statisrics, McGraw-Hill book Company, Sec-

ond edition, 1988-

BIBLIOGRAPHY 111

[22] Sullivan, PA., Hinchey, ML, Graham, TA., Simard, M., and Yong, L.Z., A Theoretical and Experimental

Investigation of the Cornpliant Air Bearing Characteristics of the SailRail Transportafion System, ûans-

portation development Center, Transport Canada, Montreai, Quebec, Report Tp 49 16E, Jan. 1985.

[23] Sullivan, PA., A review of the developrnent of the SailRail rechnology: 1971-1 996, Aerospace Engineering

and research Consultant Limiteci, Downsview, Ontario, Report 97-01, May 1997.

[24] Sullivan, P.A., Research on the SailRail cornpliant air beanng materials handling sysrem 1994-1997, Insti-

tute for Aerospace Studies, University of Toronto, Downsview, Ontario, March 1998.

1251 Ramamurihi, k., Nandakumar, K., Characthen'stics o f f i w thmug h small sharp-edged cylindrical orijïces ,

Flow measurement and instrumentation, 1999, pp. 133-143.

[26] Rohde, LE., Hadley T. Richards, Metger, G.W.. Discharge coeficients for thick plate orifices with ap-

pmachjfowperpendicular and inclined to the orifice axis , National Aeronautics and space administration ,

Technical note D-5467, October 1969.

1271 Ward-Smith, A.J., CnXcal flowmetetfng: the characteristics of cylindrical noules with sharp upstrearn

edges, International journal of heat and flow, Vol, 1 No 3, July 1979, pp* 123-132.

[28] White, EM., Fluid Mechanics, McGraw-Hi11 Inc., 2nd Edition, New-York, 1986.

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