Citation
Acoustic determination of phase boundaries and critical points of gases

Material Information

Title:
Acoustic determination of phase boundaries and critical points of gases CO2, CO2-C2H6 mixture, and C2H6
Creator:
Dejsupa, Chadin
Publication Date:
Language:
English
Physical Description:
xiv, 203 leaves : ill., photos (some col.) ; 29 cm.

Subjects

Subjects / Keywords:
Acoustic resonance ( jstor )
Carbon ( jstor )
Carbon dioxide ( jstor )
Critical points ( jstor )
Critical temperature ( jstor )
Fluids ( jstor )
Isochores ( jstor )
Pistons ( jstor )
Supersonic transport ( jstor )
Temperature ( jstor )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Matter -- Properties ( lcsh )
Thermochemistry ( lcsh )
Ultrasonic waves ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1991.
Bibliography:
Includes bibliographical references (leaves 194-202)
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Chadin Dejsupa.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
026891018 ( ALEPH )
25541142 ( OCLC )

Downloads

This item has the following downloads:


Full Text












ACOUSTIC DETERMINATION OF PHASE BOUNDARIES
AND CRITICAL POINTS OF GASES:
CO,, C02-C2H6 MIXTURE, AND C2H6


















By

CHADIN DEJSUPA


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1991

































To my mother and father

who eternally love me -

and to my sister Narisa,

my brothers Bordin and Pubordee















ACKNOWLEDGEMENTS


The author expresses his sincere gratitude to his

advisor, professor Samuel O. Colgate for his indispensable

intellectual and moral support, and especially when

bottlenecks were encountered. It has been privilege working

with such an outstanding teacher and chemist. His dedication

to excellence and clarity of thought and lucid insights will

perpetually serve as an example.

This work would also have been impossible without the

expertise of Drs. A. Sivaraman, Kenneth C. McGill, and V. Evan

House of the acoustic research group which made colossal

barriers become insignificant. Thanks also to Joseph Shalosky

for his marvelous machining dexterity. Special thanks are

extended to professors David Micha and Martin Vala and to Paul

Campbell for their valuable moral support and to all of peers

of the author in this institution, particularly Casey Rentz

and Michael Clay for their truly meaningful friendships.

Last but not least the author wishes to warmly thank all

of his teachers who shared their knowledge and understanding

with him.


iii

















TABLE OF CONTENTS


ACKNOWLEDGEMENTS . .

LIST OF TABLES . .

LIST OF FIGURES . .


ABSTRACT

CHAPTER 1

CHAPTER 2


CHAPTER 3











CHAPTER 4



CHAPTER 5


iii


viii

xiii


. . .


INTRODUCTION


THEORETICAL BACKGROUND .
Acoustics ..
Angular Part .
Radial Part .
Overall Solutions .


. 6
. 6
. 7
. 8
. 11
0 0 0 8
o : 1


Searching for Resonance Frequencies


APPLICATION OF SPEED OF SOUND TO LOCATE
CRITICAL POINTS .
Phase Behavior of Fluids and Fluid
Mixtures . .
Determinations of Critical Point .
Isotherm Approach . .
Isobar Approach . .
Isochore Approach . .
Speed of Sound as a Probe for the
Critical Point .

EXPERIMENT . .
Instruments . .
Resonance Frequency Measurements .

RESULTS AND DISCUSSIONS .
Carbon Dioxide. .. .
A Mixture of Carbon Dioxide and Ethane .
Ethane . . .
Analysis of Ethane Data .
Observation 1 . .
Observation 2 . .
Observation 3 . .
Observation 4 . .
Curve Fitting . .


19

19
24
24
25
28

30

32
32
51

60
60
70
90
105
106
107
107
112
112






rr

r


. .


* .










Three Dimensional Phase Diagram .

CHAPTER 6 CONCLUSION . .

APPENDIX A CHARGING PRESSURE CALCULATIONS .

APPENDIX B QUANTITATIVE ANALYSIS OF GAS MIXTURE

APPENDIX C SIMPLEX ... .

APPENDIX D LABORATORY STANDARD PRESSURE GAUGE .

APPENDIX E COMPUTER PROGRAMS . .

REFERENCES . . .

BIOGRAPHICAL SKETCH . .


S 116

S 120

S 126

S 128

S 136

S 147

S 163

194

203















LIST OF TABLES


Table 2.1. The values of the roots E in ascending order
of magnitude. . .. . 18

Table 4.1. Summary of electronic hardware. 48

Table 4.2. Gas Specifications. .. 51

Table 4.3. Charging Pressure of gas systems. 51

Table 5.1. Comparison of Dewpoint Pressures of Carbon
Dioxide. . . 67

Table 5.2. Values of critical point parameters from the
literature. ... . .. 68

Table 5.3. Angles between pressure-temperature lines
before and after phase boundary of supercritical
density fluid of carbon dioxide and ethane. 79

Table 5.4. Experimental results for construction of
phase diagrams of a mixture of carbon dioxide and
ethane. . .. ... 81

Table 5.5. Comparison of Dewpoint Pressures of CO2 + C, H
Mixture (X02 = 0.7425). . 89

Table 5.6. Sixth order polynomial fit statistics of
GRAPHER program applied to experimental data points
in figure 5.25. . . 93

Table 5.7. Summary of results for ethane measurements. 98

Table 5.8. Comparison of Bubble-Point Pressures of
Ethane. . ... 101

Table 5.9. Comparison of experimental vapor pressures
and those calculated with an equation of state of
ethane proposed by Sychev.("0 . 102

Table 5.10. Chronological collection of critical point
parameters of ethane. . 103









Table 5.11. Coefficients Obtained by Simplex
Optimization Method. . 115

Table 6.1. Comparison of Critical Temperatures and
Pressures. . . 123

Table B.1. Data for calibration curve of ethane. 134

Table D.1. Experimental Test of Dead Weight Pressure
Gauge. . . 158


vii















LIST OF FIGURES


Figure 2.1. The first six orders of the spherical Bessel
function. Points where slope equals zero yield
eigen values, . . 12

Figure 3.1. Figure 3.1. The p-9-T behavior of pure
fluid. In the center is sketched the surface p =
p(9,T). [From Hirschfelder Joseph 0., Curtiss
Charles F., and Bird R. Byron. Molecular Theory of
Gases and Liquids. Copyright 0 1954 by John Wiley &
Sons, Inc. Reprinted by permission of John Wiley &
Sons, Inc.]. . .. 21

Figure 3.2 Pressure-Temperature-Mole fraction
relationship of system of carbon dioxide and
ethane. . . 23

Figure 3.3. Pressure-temperature projection of the
system carbon dioxide-ethane in the critical
region. Lines Z K and Z K are vapor pressure curves
of pure carbon dioxide and of pure ethane
respectively. . ... 23

Figure 3.4. Schematic diagram of pressure and volume
relationship of carbon dioxide with several
isotherms in a broad region. . 25

Figure 3.5. Pressure and density relationship of carbon
dioxide with several isotherms around critical
point(47) 26
point47 . . 26

Figure 3.6. Schematic diagram of graphical determination
of critical temperature of carbon dioxide. 26

Figure 3.7. Schematic diagram of critical point
determination from measurements of the liquid
density p, and the vapor density p according to the
method of Cailletet and Mathias. .. 27

Figure 3.8. Schematic diagram of isochoric method of
critical point determination. . 28

Figure 4.1. Observed sonic speed versus temperature for
a North Sea natural gas mixture. Rf""r" 53 33


viii









Figure 4.2. A side view of the spherical resonator
equipped with the transducers. . 35

Figure 4.3. The spherical resonator. . 36

Figure 4.4. The transducer (cross-sectional view). 38

Figure 4.5. The mixing control unit. . 40

Figure 4.6. The Instrumental Setup. . 43

Figure 4.7. The experimental setup for CO2 and CO,-C2,H
mixture. . . 44

Figure 4.8. The first experimental setup for C2 H. 45

Figure 4.9. The second experimental setup of C H 46

Figure 4.10. Series of scanning routine. 55

Figure 4.11. Two experimental approaches of resonance
frequency determination: the maximum amplitude
approach and the voltage phase change approach. 58

Figure 4.12. Flow chart of experimental scheme and data
processing. The broken-line boxes represent the
experimental systems of interest. The thick-line
boxes represent the computer programs. 59

Figure 5.1. Relationship between resonance frequency and
temperature of carbon dioxide for an isochore near
its critical density . 61

Figure 5.2. Relationship between first radial mode
resonance frequency and temperature of carbon
dioxide for an isochore near the critical
density. . . .... 62

Figure 5.3. Relationship between the speed of sound and
temperature of carbon dioxide for an isochore near
the critical density. . 62

Figure 5.4. First derivative of resonance frequency
versus temperature. . .. 64

Figure 5.5. Enlarged temperature scale of figure 5.4
showing the critical temperature to be at 304.215
K. .. . . 64

Figure 5.6. Pressure and temperature behavior of carbon
dioxide for isochore near the critical density. 66









Figure 5.7. Experimental vapor pressure curve compared
with the NIST model. ...... . 66

Figure 5.8. Typical behavior of resonance frequency
versus temperature of the carbon dioxide-ethane
mixture at a supercritical density 72

Figure 5.9. Typical behavior of resonance frequency
versus temperature of the carbon dioxide-ethane
mixture at a subcritical density. . 73

Figure 5.10. Typical behavior of resonance frequency
versus temperature of the carbon dioxide-ethane
mixture near critical density. ... 74

Figure 5.11. Typical experimental results of near-
critical density gas of the carbon dioxide-ethane
mixture. Curve 1 shows a forced cooling run. Curve
2 shows a forced warming run. Curve 3 is a slow
naturally warming run. . 76

Figure 5.12. Other resonance frequency (top curve)
results in same critical temperature as first
radial resonance frequency (bottom curve.) 77

Figure 5.13. A sequence of supercritical pressure versus
temperature isochores of the carbon dioxide-ethane
mixture. . . 78

Figure 5.14. Relationship between the angles 0 and the
charging pressures of the carbon dioxide-ethane
mixture. . . 80

Figure 5.15. Relationship between the starting pressures
and the temperatures of phase change of the carbon
dioxide-ethane mixture. ..... ..... 85

Figure 5.16. Relationship between the starting pressures
and the speed of sound at a phase change of the
carbon dioxide-ethane mixture. . 86

Figure 5.17. Coexistence curve of the carbon dioxide-
ethane mixture near azeotrope composition. 86

Figure 5.18. The sonic speed versus temperature of the
carbon dioxide-ethane mixture for an isochore near
critical density. . . 87

Figure 5.19. Pressure and temperature behavior of the
carbon dioxide-ethane mixture charged near its
critical density. . . 87









Figure 5.20. Pressure and temperature relationship of a
set of several isochores of the carbon dioxide-
ethane mixture. .. 88

Figure 5.21. Resonance frequency (Is mode) and
temperature relationship of ethane near the
critical density. . ... 91

Figure 5.22. Resonance frequency (Is mode) and pressure
relationship of ethane near the critical density. 91

Figure 5.23. Pressure and temperature relationship of Is
mode resonance frequency of ethane near the
critical density. . .. 92

Figure 5.24. Speed of sound and temperature of is mode
resonance frequency of ethane near the critical
density. . . 92

Figure 5.25. Phase diagram of pure ethane showing
temperature of phase changes as a function of
starting pressure. . .. 94

Figure 5.26. Phase diagram of ethane showing pressure
and temperature behavior of 23 different
isochores. . ... .. 95

Figure 5.27. Trajectory plot of temperature, pressure,
and sonic speed of ethane at different densities.
The one nearest the critical density reaches the
lowest sonic speed. . ... 96

Figure 5.28. Resonance frequency and temperature
relationship revealing some phenomena observed in
the experiment on pure ethane. .. 106

Figure 5.29. Movement of tracked peak and its neighbor
below the critical temperature. These curves are
numbered in the proper time sequence but many
curves showing only one strong peak have been
omitted. . .. ... 108

Figure 5.29. Continued movement of tracked peak and its
neighbor below the critical temperature. 109

Figure 5.30. Possible routes of movement of two neighbor
peaks (peak one and peak two) below the critical
temperature. . .. 110

Figure 5.31. Experimental curve of ethane showing
dynamic behavior of speed of sound near critical
point. . . .. 113









Figure 5.32. Comparison of experimental data for the
ethane and curve generated by renormalization group
theory equation. . .. 114

Figure 5.33. Three dimensional phase diagram of a
mixture of carbon dioxide and ethane. 118

Figure 5.34. Three dimensional phase diagram of
ethane. . . 119

Figure B.1. Block diagram of the gravimetric method. 129

Figure B.2. The hexagon reaction vessel for a
gravimetric analysis. . .. 130

Figure B.3. The high pressure gas sampling container. 131

Figure B.4. Calibration curve of ethane for gas
chromatography. .. . 135

Figure D.1. Conventional dead weight pressure gauge: A =
piston; B = cylinder; C = steel U tube; D = oil
injector; N, N' = indicator contact needles. 148

Figure D.2. The present deadweight pressure gauge. 149

Figure D.3. The arrangement of the deadweight pressure
gauge. .. . 150

Figure D.4. An electroformed nickel bellow pressure
sensor. . . 160

Figure D.5. The calibration curve for dead weight
pressure gauge. . ... 162


xii















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

ACOUSTIC DETERMINATION OF PHASE BOUNDARIES
AND CRITICAL POINTS OF GASES:
CO,, CO,-C2H, MIXTURE, AND C2H'

By

Chadin Dejsupa

December 1991

Chairperson: Samuel O. Colgate
Major Department: Chemistry

The acoustic resonance technique developed in this

laboratory is a remarkable tool for the determination of the

thermophysical properties of materials. The measured

frequencies of radially symmetric modes of vibration are

simply related to the sonic speed in fluids enclosed in the

spherical cavity. To minimize experimental perturbations a

relative measurement against argon gas used as a standard

reference was employed. The sonic speed in turn is utilized as

a sensitive probe for detecting phase boundaries and critical

points of substances. Location of the boundary is revealed by

a discontinuity in the variation of sonic speed with

temperature for a sample confined isochorically in the

resonator. The corresponding temperature at which this

discontinuity occurs was taken as a phase boundary

temperature. This indicator is very pronounced, especially

xiii









when the system is near the critical condition for which

strong local density fluctuations disperse sound and cause the

sonic speed to dip sharply toward zero. This research reports

on such studies for two pure gases CO2, and C2,H and one

mixture of these two. For each system, data on resonance

frequency, amplitude and pressure as a function of temperature

were collected under computer control for a series of

different loading densities around the critical value.

Portions of the phase diagrams revealed by the measurements

were recorded in tabular and graphical formats.


xiv















CHAPTER 1
INTRODUCTION





Natural gas and gasoline are primarily mixtures of the

lighter hydrocarbons with varying amounts of nonhydrocarbons

such as water, carbon dioxide, and hydrogen disulfide. Heavier

fossil fuel mixtures such as crude oil consist of a myriad of

higher boiling hydrocarbons and various compounds containing

sulfur, nitrogen, and oxygen. In designing production,

processing, transport, and handling systems for these

materials reliable knowledge of their physical properties is

crucial. For example, in the important activities of enhanced

petroleum recovery by carbon dioxide injection into a

resevoir' and transport of natural gas mixtures containing

carbon dioxide, precise knowledge of the equilibrium phase

behaviors of the systems is important. Additionally, the need

exists for knowledge of critical point parameters, hydrate

formation conditions, density, enthalpy, dew points, bubble

points, etc. Information of these kinds about select pure


'Under the right condition of pressure, temperature, and
oil composition, carbon dioxide will mix with the crude in the
reservoir to form a single phase liquid which is much lighter
than the original oil and consequently easier to bring to the
surface.









2

components and mixtures is further needed to support the

development of semiempirical equations of state and other

correlations useful for predicting the behavior of a wide

range of systems. For this reason, much research has been done

in both the measurement (experimental method) and prediction

(theoretical method) of hydrocarbon fluid properties. The

great variety of chemical systems of practical interest,

combined with the experimental difficulties and time

investment involved with direct measurements, is a powerful

driving force behind the development of useful correlations.

Numerous empirical equations of state have been developed

over the years. The pioneering work of van der Waals focused

attention on the possible causes for nonideal gas behavior.

Although his efforts remain historically instructive, his

equation of state"') is little used today having been

superseded by a succession of others ranging in complexity

from those of Benedict, Webb and Rubin,(2' Redlich and Kwong,O)

and Peng and Robinson"4) to much more sophisticated ones

containing forty or more coefficients such as the AGA8

equation,(5) for example. Also a number of semi-theoretical and

theoretical equations based on molecular models have been

devised (see for examples references 6-11.) They were invented

to rectify for non-ideality both pure and multicomponent gases

and, in general, they work remarkably well for predictions of

gas properties in the noncritical region. They generally fail,

however, to describe the unique behavior of fluids in the









3

critical region, but progress toward overcoming this

shortcoming is being made on some fronts."12'' Thus far all

models developed to work in the critical region require

knowledge of the critical parameters; therefore the prediction

abilities of these models rely partly on the accuracy of these

parameters which in turn must be obtained experimentally. This

is typically done only for pure components, and the critical

behavior of mixtures is deduced using various combining

schemes. The chance for introducing uncertainty increases then

as the complexity of the mixture and is generally substantial

for most mixtures of importance to the gas industry. One

popular combining scheme uses Kay's mixing rules'"16: P =ZY.P .

and T =EY.T. where P and T are the so-called pseudocritical

pressure and temperature, respectively, for the mixture, and

Yi, Pc, Tci is the mole fraction, pressure, and temperature,
respectively, of component i in the mixture. Alternatively,

pseudocritical parameters may be obtained from a correlation

based on a collection of natural gas data''7'"8. The values for

a specific mixture in general depend on the combining scheme

used to predict them.

The importance of accurate values of the critical point

parameters of both pure and mixed fluids is established. Good

values for many pure components are available, but very few

mixtures have received careful experimental study. The higher

the number of system components the fewer the number of

tabulated values are generally available. This scarcity is









4

partly due to the difficulty in location of the critical point

in mixtures by conventional methods. The most widely used

method is based on direct observation of the appearance and

disappearance of a meniscus, indicating a boundary line

between gas and liquid phases when a system enters the

critical condition.("' This method relies on the visual acuity

of the operator, and clearly has some subjectivity.

Acoustic determination of thermodynamic properties such

as heat capacity, virial coefficients, vapor pressure, etc. by

spherical resonator techniques has been very successful in the

past, especially in this research group.("-26) This success

stems mainly from the unusual sensitivity of the cavity

resonator. Resonance frequencies can be readily measured to

precisions of one part in 106 or better compared to one part

in 104 or 105 for pressure, temperature, volume density etc.

The acoustic resonator also benefits thermodynamic

measurements by operating at low frequencies. At high

frequencies the speed of sound is affected by

irreversibilities resulting principally from the delay of

energy flow into and out of internal energy modes,

particularly vibrational modes.(27)

This work is a first attempt to bring the superiority of

the spherical acoustic resonance technique to bear on

development of a new approach to critical point determination.

Experiments were performed according to logical scientific

approach. To start with, verification of this new technique









5

was made using a gas system for which the critical point

parameters are well-established. That gas was pure carbon

dioxide, which has the richest history of prior critical

behavior studies. Choosing this gas is beneficial not only

because of its well-known values of critical point parameters

but also for its recognized importance in the fuel industry as

a near-critical-state solvent in enhanced oil recovery and as

a supercritical extractant. Next, the complexity of the gas

system was increased by selecting a binary component gas

mixture (CO2-C2H6) for study. The lessons learned in working

with these first two systems were applied to a follow-up study

of the other component of the mixture, ethane, to further

confirm the reliability of the technique. Chapter two will

briefly review some relevant equations of acoustics and

explain how to identify individual resonance frequencies,

which are the key parameters leading to values of the speed of

sound. Chapter three will review some aspects of the phase

behavior of both pure and mixed gases. The critical point

determination will subsequently be discussed, starting from a

background on conventional approaches to the newly proposed

approach, which utilizes speed of sound as a critical point

indicator. Chapter four explains comprehensively the

experimental procedures used in this work. Chapter five

contains results including discussions and observations. The

final chapter is a conclusion.















CHAPTER 2
THEORETICAL BACKGROUND


Acoustics



This derivation follows that given by references 28, 29,

30, and 31. The propagation of pressure waves in a lossless

fluid at rest satisfy the general wave equation'28


oV- 1 82 (2.1)
C2 at2
where V' = Laplacian operator, 0 is the velocity potential and

C is the wave velocity (speed of sound).

Assuming that 4 = goei't equation (2.1) becomes


V-2 + d-0 0 (2.2)
C2
or


70 + k24 0 (2.3)


where k = w/C = wave number. If we consider only the time-

independent wave, equation (2.3) becomes


V20 + k20 0 (2.4)









7

If the variables are separable 0o may be written as o a

f(r)g(O)h(p), and equation (2.4) becomes:(29'


r2d2f + 2rdf + (k2r2-M2)f -0
dr2 dr

s1 d +sin g + M2 -g -0 (2.5)
sin9 UgB\ -a6 sin 2() 1
d2h
d2h + q2h 0.
do
where M2 and q2 are separation constants. The first of these

three equations gives the radial part and the other two the

angular part of the solution.

Angular Part

The last equation in equations (2.5) is a linear

homogeneous second-order differential equation. Its solutions

are h = sine(qq) and cosine(qp). In order that h be single-

valued [h(r,O,(+2n) = h(r,0,p)] values of q must be integers,

i.e. q = 0, 1, 2,.... Denoting this integer by m the second

of equations (2.5) becomes


1 d sin dg+ M2 m2 (2.6)
-asin0 + M2 g 0. (2.6)
sin8 (O8 tB sin2O}) -


This equation has finite solutions for 0 = 0 & n when

M2 = 2(2+1) ; 2 =0,1,2,...

Then
1 -- dsin dg) + e(+1) M2 g 0. (2.7)
si"T Ud \9 ( ) sin n


The general solution of this equation is











8 A,P,(cosO) + BQ(cose). (2.8)


where P1'(cos0) = The associated Legendre polynomial of the

first kind of degree 2 and order m

Q,"(cos0) = The associated Legendre polynomial of the

second kind of degree 2 and order m.

Since Q'm(cosO) becomes infinite at 0 = 0 & rr, it does not

apply to this physical situation(28'. Consequently,

0 APm(cosO)

where Pm(cosO) (sinO) 2P(cos) (2.9)
d(cos0)"
if m = 0, P,(cos0) is the Legendre polynomial.

Finally, the general solution of the angular part can be

written as follows:


Y"(e,p) P.(cos9) n(mp). (2.10)


These are the spherical harmonics of the first kind.


Radial Part

Equation 2.5a is:


r2d + 2r df + (k22 M2)f 0. (2.11)
dr2 cE
Using the above result for the angular part:

M2 = (+1) ; I = 0,1,2,... equation (2.11) becomes


r2d2 + 2rdf + (k2r2 e(1+l))f 0. (2.12)
dr2 c











Dividing both sides by k2r2 gives



kr2 dr2 k k2r2
Changing the form of the function from f(r) to F(Z) where Z =

kr and rearranging yields:


d2F(Z) + 2dF(Z) 1 (+ 1))F(Z) 0. (2.14)
dz2 Z dZ Z
Particular solutions are the spherical Bessel function of

three kinds:

First kind,
F(z) j,(z) 7 J + (Z). (2.15)
2
Second kind,
F(z) y,(z) Y (Z). (2.16)
2
Third kind,
F(z) h;"(z) j,(z) + iy,(z) H' (Z)
2 (2.17)
F(z) h2)(z) j,(z) iy (z) H2 (Z)
(2+-)
2

where J(Z), Y(Z), and H(Z) is Bessel function of the first,

second, and third kind, respectively.

With, for example,


j0(z) sin)
z
j1(z) [sin(z) z-cos(z)]
(2.18)
S cos(z)
yo --
z
l [cos(z) + z-sin(z)]
z2









10

For higher orders of 1 the solutions may be obtained from the

following recurrence relation:


f( +,, (z) (2 + l)f,(z) f, -1_ (2.19)

The general solution may be written as a linear combination of

the first and second kind solutions as follows:

F(kr) Aj,(kr) + By,(kr). (2.20)
F, however, must be finite at r=0. Spherical Bessel functions

of the second kind do not satisfy this constraint, so the

equation (2.20) becomes

F(kr) = Aj,(kr). (2.21)

A second condition imposes that the normal velocity must

vanish at the rigid wall of the sphere, i.e.

F(kr) 0. (2.22)
\ r r-a

This is a transcendental equation whose roots give the normal

frequencies of vibration, (. For example if

j0(kr) sin(kr)
kr
Then djo(kr) sin(ka) + cos(ka) 0
dr ra ka2 a (2.23)
sin(ka) cos(ka)
ka2 a
Hence, tan(ka) ka (,.

For each value of I there exists an infinite number of E

values designated as ,n. The first few points corresponding

to roots of equation (2.23) for the first six orders are shown

in figure 2.1.











Overall Solutions

Combining the results of the radial and angular parts

yields the solution of the time-independent monochromatic wave

equation in the spherical polar coordinate system as

M +e M
00(r,8,p) j j,(k,,r)Ym(8,,(). (2.24)
n-0 m--2 2-0




It is clear that in order to determine fully any particular

mode of vibration, we must specify the values of three

characteristic numbers: n, 9, and m.

The roots of the radial part n correspond to the

frequency through the relation


n ka a 2rnfna
-n C C (2.25)
fln "1a

where C = speed of sound

a = the radius of the spherical cavity

fn = resonance frequencies corresponding to An

S= root of the spherical Bessel function of the first

kind

= eigenvalues.

Lord RayleighO30) was the first to study this problem. In

1872 he solved for the lowest 23 eigenvalues (sometimes called

natural frequencies). In 1952 H. G. Ferris"31' revised and

extended the list to the lowest 84 eigenvalues as shown in



















































Figure 2.1. The first six orders of the spherical Bessel
function. Points where slope equals zero yield eigen values,
ti,n"









13

table 2.1. Even more complete tabulations can be found in the

literature32'.




Searching for Resonance Frequencies

(n C
From the relationship f -2- it is obvious that in
Sln Tna
order to find the resonance frequency corresponding to the

normal mode of vibration of an eigenvalue, ,,, the speed of

sound and the radius of the sphere must be known. The latter

may be obtained precisely from geometric considerations on the

resonator cavity. The former may be approximately calculated

from an assumed equation of state. In this study we used the

American Gas Association equation called AGA80'' which requires

as input only temperature, pressure, and composition. Using

the predicted sonic speed together with the measured cavity

radius and the tabulated eigenvalue, the corresponding

resonance frequency f ,n may be approximately calculated and

then experimentally searched by scanning frequency around that

value. When excited at a resonance frequency the amplitude of

the detected acoustic signal sharply increases. The series of

resonance peaks arranged by frequency defines a zeroth order

acoustic spectrum of resonances of gas enclosed in the

spherical cavity.

Once a resonance frequency has been found, it can serve

as an internal reference point to search for another, as

follows:












f C
firn I TTra
fn C (2.26)

g

If g is assumed to be a constant, this enables us to calculate

a resonance frequency corresponding to each root ,n from the

following relationship:


(f2,n)ref fref
,n) ref ref
Sg (2.27)
"- n at any other values of (1,n).

Consequently,


f,n (E,,n)g. (2.28)


Note that this method also serves as a self check routine for

the identity of the first reference frequency. If the observed

frequency is not correctly identified, it is unlikely that we

will find another resonance frequency in the expected region

by using the above equation.

So far we have discussed a zeroth order perturbation

relationship:


fn ,n- (2.29)

In reality there are higher perturbations to this relation.

Moldover has suggested that these effects should be added to

the above equation in the following manner133,34):












fn "- ,) (C) + Af. + Af.1 + Aft + Afgeo*** (2.30)

The first correction term is due to the thermal boundary layer

effect at the wall of the resonator. The second correction

term is due to the finite elastic compliance of the sphere

which is not infinitely rigid. The third term corrects for

effects of the gas entrance and gas exit tubes which

contribute departures from sphericity. The last term corrects

for imperfection of the cavity geometry. For absolute

measurements the exact treatment could be computationally and

experimentally demanding.

These problems, however, can be greatly reduced by

utilizing a relative measurement technique as follows:

equation (2.30) may be written as


f 2,n- (vn,,)(C) (2.31)
f---7Tr

where vln is an effective eigenvalue which depends on the

unperturbed eigenvalue and all perturbations for the

particular mode (l,n). If it is assumed to be independent of

gas properties, then


., (2na)(f'n) (2.32)
V1,n C

Consequently, if we choose some system whose speed of sound is

known or can be calculated precisely from a reliable equation

of state, we will get a relationship:

or












2na( cJ 2rrafi (2.33)



CI CAf (2.34)
-fAr/ ,n

where I refers to the system under investigation. The speed of

sound of argon is calculated from the virial equation

truncated after the third term as follows'35':


C yRT A(T)A(T) (2.35)
MAr V 2


where y = heat capacity ratio (Cp/C )

MA = Molecular mass of argon

R = Gas constant

T = Absolute Temperature

9 = molar volume

AI(T) = Second acoustic virial coefficient

A,(T) = Third acoustic virial coefficient.

The effective eigenvalue determined this way is not exact, but

the effect on sonic speed measurements is small and of no

consequence for the accurate determination of phase

boundaries.

The practical advantages of spherical resonators is

realized for the purely radial modes of vibration, which

involve no tangential motion of the gas with respect to the

resonator wall, i.e. no viscous drag effect. In addition, the









17

radial mode resonance peaks typically have narrow half-width,

(high Q value2). Furthermore, resonance frequencies of the

radial modes are sensitive to imperfections of non-sphericity

only to second order"36). Consequently, from all points of view

the most accurate measurements of the speed of sound should be

obtained by utilizing the radial modes of vibration. The first

few radial modes are mostly used because of their associated

low frequencies. A more complete account of these effects can

be found in references 33 and 34.





























2Q is usually defined as:
SCenter Frequency of Resonance
Frequency Width 3dB Points










Table 2.1. The values of the roots ,, in ascending order of
magnitude. D=degeneracy

S,n n 1 D NAME ,n n 1 D NAME
2.08158 1 1 3 ip 12.2794 6 2 13 2i
3.34209 2 1 5 id 12.4046 1 4 3 4p
4.49341 0 1 1 is 13.2024 11 1 23 In
4.51408 3 1 7 if 13.2956 4 3 9 3g
5.64670 4 1 9 ig 13.4721 7 2 15 2j
5.94036 1 2 3 2p 13.8463 2 4 5 4d
6.75643 5 1 11 lh 14.0663 0 4 1 4s
7.28990 2 2 5 2d 14.2580 12 1 25 io
7.72523 0 2 1 2s 14.5906 5 3 11 3h
7.85107 6 1 13 li 14.6513 8 2 17 2
8.58367 3 2 7 2f 15.2446 3 4 7 4f
8.93489 7 1 15 Tj 15.3108 13 1 27 lq
9.20586 1 3 3 3p 15.5793 1 5 3 5p
9.84043 4 2 9 2g 15.8193 9 2 19 21
10.0102 8 1 17 1k 15.8633 6 3 13 3i
10.6140 2 3 5 3d 16.3604 14 1 29 ir
10.9042 0 3 1 3s 16.6094 4 4 9 4g
11.0703 5 2 11 2h 16.9776 10 2 21 2m
11.0791 9 1 19 11 17.0431 2 5 5 5d
11.9729 3 3 7 3f 17.1176 7 3 15 3j
12.1428 10 1 21 Im 17.2207 0 5 1 5s
17.4079 15 1 31 it 21.6667 3 6 7 6f
17.9473 5 4 414h 21.8401 8 4 17 4k
18.1276 11 2 23 2n 21.8997 1 7 3 7p
18.3565 8 3 17 3k 22.0000 11 23 3n
18.4527 16 1 33 lu 22.5781 6 5 T3 5i
18.4682 3 5 7 5f 22.6165 20 1 41 ly
18.7428 1 6 3 6p 22.6625 15 2 31 2t
19.2628 6 4 13 4i 23.0829 4 6 9 6g
19.2704 12 2 25 2o 23.1067 9 4 19 41
19.4964 17 1 35 lv 23.1950 12 3 25 3o
19.5819 9 3 19 31 23.3906 2 7 5 7d
19.8625 4 5 9 5g 23.5194 0 7 1 7s
20.2219 2 6 5 6d 23.6534 '1 1 43 Iz
20.3714 0 6 1 6s 23.7832 16 T 33 2u
20.4065 13 2 27 2q 23.9069 7 5 15 5j
20.5379 18 1 37 1w 24.3608 10 4 21 4m
20.5596 7 4 15 4j 24.3824 13 3 27 3q
20.7960 10 3 21 3m 24.4749 5 6 11 6h
1.2312 5 5 11 5h 24.6899 22 1 45 la
21.5372 14 2 29 2r 24.8503 3 7 7 7
21.5779 19 1 39 ix 24.8995 17 2 35 2v















CHAPTER 3
APPLICATION OF SPEED OF SOUND TO LOCATE CRITICAL POINTS



Phase Behavior of Fluids and Fluid Mixtures


The critical point of a fluid, whether it be a pure

substance or a mixture, is a property of considerable

practical as well as theoretical importance. This is because

the critical point identifies the temperature, pressure and

density at which the liquid and gaseous phases have identical

properties and is, therefore, a key point in the construction

of the phase diagram. Also, a knowledge of critical

temperature and critical pressure makes possible, through the

principle of corresponding states and an appropriate equation

of state, the prediction of the thermodynamic properties of

the compound when these properties have not yet been

experimentally determined. The need for information about

phase behavior has substantially increased in recent years.

For example, the petroleum industry has given much attention

to the phase behavior of fluid mixtures at high pressures to

support useful strategies to increase the yield of petroleum

from oil wells. To have a clear picture of the phase behavior

at high pressure and temperature of a fluid mixture, the









20

critical point, the maximum pressure and the maximum

temperature for the heterogeneous region must be known.

Evidence for the existence of a critical point was first

presented in 1823 by de La Tour'37), who observed that a liquid

when heated in a hermetically sealed glass tube is reduced to

vapor in a volume from two to four times the original volume

of the sample. However, it was not until the quantitative

measurements of Andrews on carbon dioxide(38'39' in 1869 that the

nature of the transition was understood. He was the first to

coin the term "critical point" for the phenomenon associated

with this liquid-vapor transition. For fluid mixtures the

first reliable experimental investigation of the critical

state began with the work of Kuenen(4'41) in 1897. Interest in

the critical region in the period 1876-1914 was peaked by the

experimental and theoretical studies of pressure-volume-

temperature (p-9-T) relationship for both pure gases and

gaseous mixtures of van der Waals and his associates at the

Universities of Amsterdam and Leiden.

Figure 3.1 shows the p-9-T relationship of the liquid and

gas phases of a pure fluid. The upper-right projection shows

several isotherms on a p-9 diagram, the upper-left projection

is a p-T plot showing several isochores and the bottom

projection shows several isobars on a T-9 plot. The tongue-

shaped region bounded by points A, D, C, E, B is the

coexistence curve (or vapor pressure curve.) This curve may be

considered as consisting of two curves: the bubble-point (the
































Figure 3.1. The p-9-T behavior of pure fluid. In the center is
sketched the surface p = p(9,T). [From Hirschfelder Joseph O.,
Curtiss Charles F., and Bird R. Byron. Molecular Theory of
Gases and Liquids. Copyright 1954 by John Wiley & Sons, Inc.
Reprinted by permission of John Wiley & Sons, Inc.]

point of initial vaporization when the pressure of the liquid

is reduced) curve, ADC, and the dew-point (the point of

initial condensation when the pressure of the gas is

increased) curve, BEC. These two curves meet at the critical

point C which, belonging to both curves, indicates the

identity of the liquid and vapor phases. At this point,

viewing from the isothermal perspective,


Pi P 0. (3.1)
T T v2T









22

In pure fluid systems the phenomenon of condensation is

associated with lowering the temperature and raising the

pressure and vice versa for the phenomenon of vaporization.

In a fluid mixture system we have besides the external

parameters: temperature and pressure, the internal parameters

which identify the composition. The phase behavior of the

system can be more complicated than that of a pure fluid

system due to many factors such as the possibility of various

kinds of retrograde phenomena, the occurrence of azeotropic

mixtures of positive or negative types etc. In a binary

system, for example, on the basis of an analysis of the phase

diagrams by means of the van der Waals equation of state, nine

major types of phase diagram may be obtained"42-44. The complete

phase behavior of a binary system is represented by the four

dimensional surface, p = p(v,t,x) where x is the mole fraction

of one of the two components. Consequently, the critical point

of a mixture has to be redefined as the point where liquid and

vapor become identical, subject to the constraint of equality

of composition.

Since this work deals with a binary mixture of carbon

dioxide and ethane, attention throughout will be mainly

focused on the relevant information for this mixture. Carbon

dioxide and ethane form an azeotropic mixture at the

appropriate composition. This binary system deviates

positively from the ideal solution, which obeys Raoult's law.

In this case a critical point locus follows a curve between




























Figure 3.2 Pressure-Temperature-Mole fraction relationship
of system of carbon dioxide and ethane.


80

KA





260 6



Iw A


20
10 20 30 t,C 40
Figure 3.3. Pressure-temperature projection of the system
carbon dioxide-ethane in the critical region. Lines Z K
and Z K are vapor pressure curves of pure carbon dioxide
and of pure ethane respectively.









24

the critical point of pure carbon dioxide and that of pure

ethane. Figures 3.2 and 3.3 illustrate the pressure-

temperature-mole fraction diagram and critical locus of this

mixture, respectively. In figure 3.3 at point B the maximum

azeotrope is tangent to the critical curve (K BK,), which has

a minimum in temperature.




Determinations of Critical Point


Conventionally, there are three principal methods of

locating the critical point. These are described briefly

below. (41

Isotherm Approach

Critical temperature may be estimated to within a few

hundredths of a kelvin by an analysis of the geometry of the

isotherms"46'. The isotherms in the immediate neighborhood of

the critical temperature are measured with sufficient accuracy

to allow the critical temperature to be determined from the

inflection points of the isotherms. The minimum values of the

derivative of pressure with respect to volume at constant

temperature, () found graphically are plotted as a

function of temperature (or pressure or molar volume.) The

intersection of this line with the horizontal axis gives the

critical parameters (To, Pc, v'.) Figures 3.4-3.6 illustrate

this approach.











Isobar Approach

This method is sometimes called the rectilinear diameter

method of Cailletet and Mathias."48 If p, and p are the

densities of liquid and of saturated vapor in equilibrium with

it (so called orthobaric densities), their mean is a linear

function of temperature.


1(P + Pg) Po + at (3.2)

where po = mean density of liquid and its saturated vapor at

0 C

t = temperature in celsius

a = constant.


[ I Volume j1

Figure 3.4. Schematic diagram of pressure and volume
relationship of carbon dioxide with several isotherms in
a broad region.






























t825 19 20 21 22 23 24 25 2.6
VP (cm3/g)--
Figure 3.5. Pressure and density relationship of carbon
dioxide with several isotherms around critical point(47'.





12



8







0-
2 ........ .......I i.......... .. .I

-2 .... ... .....
31 31.04 31.08 31.12 31.16 312
Temperature, 'C
Figure 3.6. Schematic diagram of graphical determination of
critical temperature of carbon dioxide.









27

Thus, if Pg and p, are separately plotted against the

temperature t, the locus of the points bisecting the joins of

corresponding values of p, and pg is a straight line (see

figure 3.7.) The point where this straight line cuts the

coexistence curve is the critical point. In many cases this

empirical law of rectilinear diameters holds very well, but

sometimes over a large range of temperature the actual

behavior shows a slight curvature. Furthermore, very near the


Density --

Figure 3.7. Schematic diagram of critical point
determination from measurements of the liquid density p
and the vapor density p according to the method of
Cailletet and Mathias.









28

critical point (within several mK of T ) real systems show a

deviation from linearity.



Isochore Approach

There are two slightly different ways for taking this

approach. The first one is to study the discontinuity of the

isochore curve.(") This method begins with loading a bomb of

constant volume with a series of known weights of the

substance and then studying the behavior of the pressure as


Figure 3.8. Schematic diagram of isochoric method of
critical point determination.


TEWERATURE-









29

the temperature changes. The following possible kinds of

isochoric curves could result if the charging density p,is

greater than the critical density pc, the isochore curve shows

an upward bend as in C, or if p is less than pc the curve

shows downward bend as in A. However, if p is equal to pc the

isochore exhibits no break as in B. This method is less

demanding in terms of skillful technique and special

apparatus, but the accuracy is poor. Figure 3.8 shows data

illustrating all three possibilities described above.

The second established isochoric method is to study the

appearance and disappearance of the meniscus between the

phases near the critical temperature.('"9 The procedure is the

same as above except that, instead of measuring pressure as

temperature changes, the behavior of the meniscus separating

the liquid and gaseous phases is observed. There are also

three possible phenomenon:

If p < pc the meniscus falls until the entire container

is filled with gas.

If p > pc the meniscus rises until the entire container

is filled with liquid.

If p = pc the meniscus approximately located at a point

halfway up the container will flatten, then

become very faint and finally disappear.

The temperature and density at which the meniscus disappears

are taken to be the critical parameters.










Speed of Sound as a Probe for the Critical Point

By conventional methods the critical point can be located

indirectly as the zero point of the first derivative of

pressure with respect to volume or of volume with respect to

temperature on an isotherm or isobar, respectively, or by

analyzing the slope of lines plotted as the first derivative

of pressure with respect to temperature versus temperature for

isochores. The technique involving use of the principle of

appearance and disappearance of a meniscus is to some degree

subjective due to the heuristic effect. A better alternative

would be to use some thermodynamic property which shows a

significant change at the critical point. In 1962 Bagatski and

his coworkers (50,5,s experimentally observed the asymptotic

behavior of heat capacity at constant volume of argon in the

immediate vicinity of the critical point. As expected Cp falls

off rapidly and it appeared that this could be used to detect

the critical point. Measuring heat capacity, however, is not

an easy task. On the other hand, the speed of sound similarly

shows an abrupt change toward zero at the critical point since

it is inversely proportional to Cv which diverges weakly and

is directly proportional to Cp which diverges strongly and to

the first derivative of pressure with respect to molar volume

at constant temperature, which also vanishes at the critical

state. The appropriate equation is:

1/2
c2 -i / (3.3)
o M (T d)









31

where CO is the sonic speed in the limit of zero frequency.

Because the sonic speed is directly proportional to the

frequencies of normal mode resonance vibrations in a filled

cavity resonator, one can expect to detect the approach to the

critical state by observing changes in resonance frequency.

Spherical cavities yield especially sharp resonances and are

therefore well suited for this purpose. In the first phase of

this work the isochoric approach was employed. However, unlike

the conventional method of using the straightness of the

pressure-temperature plot or using the appearance and

disappearance of the meniscus, the speed of sound or a

resonance frequency was utilized as a probe of a critical

point.















CHAPTER 4
EXPERIMENT



Instruments


The development of a spherical acoustic resonator as a

tool to the study speed of sound in gases such as argon,

butane, isobutane etc. which, in turn, leads to values of some

thermodynamic and equation of state properties has been

successful in the past by workers in this research group.(20-26)

There are, however, some limitations on the previous work due

mainly to the construction of the spherical resonator. It was

made from aluminum alloy and designed to operate near

atmospheric pressure. These constraints limit its applications

substantially. For example, if one wishes to study natural gas

mixtures, one would likely encounter the presence of some

corrosive, acidic or basic gases such as hydrogen disulfide,

carbon dioxide etc., and these, especially in the presence of

water, attack aluminum to some extent. In addition,

experiments may call for the measurements at much higher

pressures than atmospheric.

The present research was undertaken to extend the

applicability of the acoustic resonance technique to the study

of natural gas mixtures at high pressures. To fulfil this goal










33

a new high performance stainless steel resonator assembly was

designed and built, and new data acquisition software was

developed."52' The new apparatus was used to measure the sonic

speed in some carefully blended gas mixtures. Because the

mixtures contained small concentrations of condensibles such

as hexane for example, it was possible to cool them below

their dew points and it was observed that the sonic speed

faithfully revealed the precise location of the phase


Figure 4.1. Observed sonic speed versus temperature for a
North Sea natural gas mixture. (Refrence 53)


430
-


420


S410


Q 400-
rd
vl
S390
0

S380


370


360 -
-40


0 20
TEMPERATURE /C









34

change'53'. This behavior was soon recognized as a valuable

tool for phase equilibrium studies. Figure 4.1 shows the

typical behavior in the variation of sonic speed with

temperature for such a mixture. The dew point is clearly

evident. Following this discovery it was decided to evaluate

the use of the acoustic resonator to locate the most

interesting feature of the phase diagram, namely the critical

point. The work reported in this dissertation deals primarily

with this difficult and challenging task.

Spherical resonator

The resonator cavity is the heart of the apparatus. To

permit its use at high pressure and in the environment of

corrosive gases, the spherical resonator was built from

stainless steel. Its side view is shown in figure 4.2.

The resonator cavity was fashioned from a welded

spherical, type 304 stainless steel tank approximately 0.203

m.(8 in.) in diameter with a 0.24 cm.(3/32 inch.) thick wall.

The tank is not perfectly spherical. This imperfection,

however, does not significantly degrade the data integrity as

long as appropriate measurement techniques are used as

discussed in the pervious section and proven to be valid by

the previous investigators.20'21 The tank was obtained from the

manufacturer (Pollution Measurement Corp.,Chicago, IL) with

only a single 0.6 cm.( in.) hole located at one pole (8 = 0

in spherical polar coordinates). A matching hole was drilled

through the opposite pole (8 = n) and two 0.953 cm (% in.)





















































Figure 4.2. A side view of the spherical resonator equipped
with the transducers.






















































Figure 4.3. The spherical resonator.









37

holes drilled through the wall at 8 = n/4 with # = 0 and n,

respectively. No holes were located on the equatorial weld

bead (8 = r/2). Figure 4.3 shows the overall view of the

spherical resonator.

Transducers

The active element of the acoustic transducer is a 0.750

in. diameter piezoelectric ceramic bimorph (Vernitron,

Bedford, Ohio) made of PZT-5A material (lead zirconate-lead

titanate.) This ceramic was found to produce a signal of high

volume when driven with a sinusoidal waveform of 10 volts

peak-to-peak. It also has a high maximum operating temperature

of 250 oC.(54) A transducer assembly was designed and

fabricated. It is shown in figure 4.4. The electrical

feedthrough made from brass was held in a Delrin insert to

prevent a short circuit. A cross-sectional view of the

transducer mounted on the resonator is also shown in figure

4.2. The transducer assemblies were mated to the resonator

through short % in. diameter tubes.

Two identical transducers located at right angles to each

other were used. This orientation has proved to yield a better

resolution of the radial modes of vibration due to diminishing

in intensity of some non-radial modes, p modes for example.

One transducer functions as a speaker. It is an input

transducer which mechanically deforms when a voltage is

applied. The other transducer functions as a microphone. It is

an output transducer which operates on a reverse mechanism of































Thru Guide


Figure 4.4. The transducer (cross-sectional view).









39

the input transducer: producing an electric signal when

mechanically deformed by pressure waves.

Mixing control system

A circulating pump is needed to promote mixing of system

components and (or) phases and to assist with thermal

equilibration by eliminating stratification. It was made from

stainless steel and consists of two parts: the liquid

collector and the pump. Figure 4.5 shows the design of the

unit. It consists of a vertically mounted stainless steel

cylinder 3.18 cm (1.25 in.) in diameter with a freely sliding

internal piston 2.10 cm (0.851 in.) in diameter. The piston

has a built in reed valve which is normally open under gravity

but is forced closed by viscous drag as the piston descends

causing the fluid to flow from top to bottom through the pump

on descending motion of the piston. Piston motion is driven by

interaction between an iron slug attached to the piston and an

external magnet driven in turn by linkage to a motorized bell

crank.

Apparatus assembly

Tubular fittings to the spherical resonator to other

parts of the system were provided with 0.6 cm (k in.) diameter

pilot tubulations which slip into the polar holes and when

pressed against the tank were sealed by elastomeric O-rings

confined to glands comprised of circular grooves in the

fittings around the pilot tubes and the ring like areas of the

adjacent tank wall surrounding the holes. These fittings were




























MAGNET ASSEMBLY









STIrllLE- ; STEEL
CYLINDER



FREELY SLIDIIIG
IRON PISTON






TO RESONATOR TOP



FLLI-) FLOW
(PISTON MOVING LF'P '-PD)


SHIM VALVE

TO RES,:r ,\TOR BOTTOM


Figure 4.5. The mixing control unit.









41

brazed to stainless steel tubes used for charging the

resonator and circulating its contents through an external

loop containing the magnetically driven pump. The two

identical PZT bimorph piezoelectric transducer assemblies were

similarly mounted to the tank at the 0.953 cm (% in.) hole

positions.

The assembly support frame consists of two parallel 0.6

cm (h in.) thick stainless-steel plates between which an array

of 2.54 cm (1 in.) thick stainless-steel blocks were attached

with bolts to support the tube fittings and transducer

assemblies and provide a means of applying clamping forces to

the O-ring seals. The clamping force acting along the polar

axis is applied by a hallow jam screw which slips over the

charging tube and pushes against the brazed O-ring gland. The

block which supports the lower tube fitting is rigidly fixed

and provides a definite reproducible location for the tank

body. The transducer assemblies were clamped against the

resonator by set screws threaded through two of the blocks

bolted to the mounting frame. Set screws in two additional

blocks located opposite the transducer mounting holes apply

counter forces against the tank to stabilize the assembly

further.

The circulating pump is connected to the upper and lower

tube fittings by demountable compression fittings (Swagelok.)

The normal pumping rate of 4 liters per minute proved adequate

for these purposes. A 0.6 cm (h in.) tube connected to the









42

fluid circulation loop by a tee leads to a diaphragm-type

pressure gauge and a system shut-off valve.

The resonator assembly and circulating pump were

suspended from a horizontal plate which forms the cover of a

stirred liquid bath in which the system is mounted. The bath

was housed in a well-insulated container and connected by

insulated tubing to a computer-controlled heat exchanger

capable of operating from 233 to 373 K. Bath temperature was

measured with a four-wire platinum resistance thermometer

(RTD.) Figure 4.6 shows the instrumental setup.

Electronic hardware

The automated systems employed in this work are shown in

figures 4.7, 4.8, and 4.9. Figure 4.7 shows the setup used in

the experiments on pure carbon dioxide and on a mixture of

carbon dioxide and ethane. Figure 4.8 shows the setup for the

experiments on pure ethane. In this latter setup we have

explored the capability of using a fast-fourier transform

(FFT) technique. Unfortunately, the data collected by this

technique was unreliable due to leaks in the system found

after the experiment was finished. While the leaks were being

fixed, the instruments were transferred to another project.

Consequently, the FFT exploration was discontinued for ethane

and the setup shown in figure 4.9 was built to finish this

project.













Service Valve


Hood


Figure 4.6. The Instrumental Setup.








SYSTEM CONTROLLER


OUTPUT SIGNAL ANALYZER SYSTEM


4.7. The experimental setup for CO2 and CO2-CH mixture.


_


Figure
















































Figure 4.8. The first experimental setup for C2H6.

u'
















I --- --------
D AC VOLTAGE I PRESSURE
CONTROL GUAGE




I_ ELECTRONICS
I I

I RI




FUNCTION
GENERATOR










PRE APLIFIERI
COUNTER




LOCK-IN DIGITAL
AMPLIFIER VOLTMETER
L - I _


RESONATOR TEMPERATURE
CONTROL

BATH






S- INPUT UNIT

-------- OUTPUT SIGNAL ANALYSER UNIT


Figure 4.9. The second experimental setup of C2 H.









47

Even though there were different setups, the basic

principles are the same and can be divided into two parts: the

input signal generation part and the output signal analysis

part. These are shown by dashed-line boxes in figures 4.7 and

4.9. Commands and data were transferred among the instruments

and computer over the General Purpose Interface Bus (GPIB)

cables. Each component on the GPIB was equipped with an IEEE-

488 standard interface. The computer was always a controller-

in-charge sending commands and acquiring data according to the

written computer program. Each instrument was a listener

and/or a talker depending on a currently executed line of the

program. Table 4.1 summarizes electronic hardware components

used in this work and their specifications.



Gases

Carbon dioxide and ethane gases used were high purity

grade. No further analysis was performed on carbon dioxide.

Table 4.2 summarizes the pure gas qualities.

For the experiments on a mixture of CO2-C 2H quantitative

analyses were performed since the results of speed of sound

measurements depend on the precise composition. Both

gravimetric and gas chromatography analyses were used to

fulfil this task. Details of these analyses are presented in

appendix B.





STable 4.1. Summary of electronic hardware.

Item Manufacturer Model Range of Accuracy Function
number operation (Manufacture
quote)
Temperature Brinkmann RKS-20D -40 to 150 0.03 OC Control
controller OC temperature.
Pressure Sensotec TJE/743-03 0-3000 psia 0.1% of full Measure pressure.
Transducer with scale
excitation
voltage = 10
volts
Digital Beckman 600 Series -150 my to 1 0.01 % of Accepts a strain
Strain gage industrial volt full scale, gage type
transducer corp. 1 digit. Full transducer input
indicator scale = 3000 to measure
psia. pressure.
Lock-in Stanford SR510 0.5 Hz to 10 volt AC Signal
amplifier research Single 100 KHz output recoveries :
system, Inc. phase corresponds measure signals
to full scale down to 10 nv full
input scale while
rejecting
interfering
signals up to 105
times larger by
the performance of
two lines notch
filters and
autotracking
bandpass filter.
Frequency Sencore FC71 10 Hz to 1 2 digits on Measure frequency
counter GHz frequency of periodic
less than 100 signal.
KHz





Table 4.1. (Continued)


Item Manufacturer Model Range of Accuracy Function
number operation (Manufacture
quote)
Pre- Stanford SR550 Up to 250 my 1 % Gain Reduces input
Amplifier research rms input noise and extends
systems, the full scale
Inc. sensitivity to 10
nv.
Voltmeter Keithley 177 200 my to 0.04 % Measure Ac or Dc
instruments, 1200 volts Reading + 2 Voltage.
Inc. digits for
minimum range
or 0.035 %
rdg. + 1
digit for
max. range
Digital Keithley 195A 2-Terminal & Depends on Measure resistance
multimeter instruments, 4-terminal range. See from Pt
Inc. resistance manufacture's temperature probe.
measurements manual for
in the range details.
of 100 PQ to
20 M _
Function Wavetex San 182A 0.004 Hz to Dial accuracy Generates a
generator Diego, Inc. 4 MHz = 5 % of precise sine
full scale. /triangle/square
__waveform.





Table 4.1. (Continued)


Item Manufacturer Model Range of Accuracy Function
number operation (Manufacture
quote)
Waveform Hewlett- 3325B 1 pHz to 21 5X10-6 of Generates sine,
synthesizer Packard MHz for sine selected triangular, or
and function wave with value in square wave.
generator amplitude of temperature
1 mv to 10 v range 20 to
p-p 30 C
Oscilloscope Leader 1020 Input Vertical Display waveform
signals section : 3
higher than % (5% mag X
400 volts 10.)
(Acp-p + Dc) Horizontal
may damage section : 3
circuit. %
COMPONENTS
Dynamic Hewlett- HP35660A 488 pHz to No Wave signal
signal Packard 102.4 KHz information source. Also
analyzer for single- available. contains FFT
channel algorithm to
spectrum or convert an analog
half of the input signal(time
this range domain) to a
for two- signal displayed
channel in the frequency
spectrum. domain.












Table 4.2. Gas Specifications.

Gas Manufacturer Grade Purity
Carbon dioxide Scott Research 99.99 Mole %
specialty Grade
gases
Ethane Scott Research 99.9 %
Specialty Grade
Gases
Argon Matheson gas Research 99.9995 Volume %
Products Grade

Table 4.3. Charging Pressure of gas systems.


Gas system Starting Charging pressure
temperature (OC) (psia)
Carbon dioxide 36.0 1200
CO2 + C2H6 30.0 630 for C2,H and
make up to 1300
with CO
CH, 37.0 808


Resonance Frequency Measurements


Measurements were made on isochore samples introduced

into the spherical resonator which was initially rinsed with

argon gas and subsequently evacuated overnight at high

temperature (40-50 OC) with a mechanical pump connected to a

liquid nitrogen trap. The apparatus was then brought to a

temperature above the known literature critical temperature.

After that gas was introduced slowly into the resonator to

bring the pressure to the calculated value (see table 4.3).

The apparatus was then sealed off. The charging pressure value









52

was calculated using the AGA8 program. Details of the

calculation are given in appendix A. In the case of the

mixture of CO2 and C2H, ethane was first introduced. Pressure

was monitored roughly by a Bourdon pressure gauge and more

precisely by a digital pressure gauge previously calibrated to

yield an absolute pressure reading. The magnetic circulating

pump was turned on to assure adequate mixing. Once the system

was at stable conditions in temperature and pressure (which

took approximately a few hours), searching for resonance

frequencies began.

Fluids in the resonator were stimulated acoustically by

driving one of the bimorph transducers with a 40 volt peak-to-

peak sine-wave signal generated under microprocessor control

(PC-AT) of a programmable signal synthesizer, HP3325B in the

case of pure carbon dioxide and of the of CO2 and CH, mixture

or of a wave function generator (WAVETEX) controlled through

a programmable digital-to-analog converter (DAC) in the case

of pure ethane. The fluid response was determined at the other

transducer which feeds its signal to a lock-in amplifier. The

response of the speaker transducer in terms of voltage output

signal was monitored closely to changes in frequency of the

microphone transducer. At resonance, the output amplitude

increased dramatically to maximum enabling a precise frequency

measurement. Using an oscilloscope as a visual aid to locate

the vicinity of this maximum signal and a digital voltmeter to

find the precise position of the signal peak, resonance









53

frequency could be found with high accuracy. Usually, the

first radial mode of vibration was selected as the tracked

peak throughout this work.

The approximate location of the desired peak, first

radial mode for example, was initially calculated from the

AGA8 equation using pressure, temperature, and composition as

input. Then the actual location of the peak was experimentally

searched using a program called SCAN. Scanning was performed

in a reasonable frequency range around the center frequency

identified as the output value of the AGA8 calculation. Once

the peak(s) was found, location of another mode was calculated

as described in the theoretical background section. The

purpose of this routine was to increase the degree of

confidence that each experimentally found peak was indeed

correctly identified. Similarly, if more than one peak

appeared together in the range while scanning, each was

checked against the other radial modes to sort out the best

candidate for labeling as a resonance mode based on the

hypothesis that if it is a genuine system peak, its frequency

should reveal the correct position of the other peaks. Figure

4.11 shows an example of a series of scans performed

chronologically. The figure in the first row is an acoustic

spectrum scanned in a wide frequency range. The figure in the

second row is the acoustic spectrum scanned in a narrower

frequency than the one in the previous figure. In this

incident there were three peaks near the predicted frequency









54
of the first radial mode peak (peak 1 in figure 4.11). Each of

these was investigated closely as shown in the subsequent four

rows. Each figure in the third row is a spectrum of a single

peak of row two scanned in a much narrower frequency range.

Frequencies found for each peak in this row were used to

calculate the second, third, and forth radial frequencies.

Peaks corresponding to these calculated frequencies were

searched as shown the figures in the forth, fifth, and sixth

row. Apparently, the third peak (peak 3 in figure 4.10) is not

the correct peak as it failed to predict the location of

higher radial modes. Peaks 1 and 2 give similar predictions

since they lie very close together; about 5 Hz apart.

Predictions of overtones based on peak 1 were in better

agreement with experiment than from peak 2 therefore peak 1 is

correctly interpreted as the first radial mode peak. Note that

the SCAN program was used for pure carbon dioxide and for the

CO2-C H, mixture experiments. For pure C2 H6 experiment this

process was performed manually via a voltage-control-generator

(VCG) box since the instrumental setup was altered.

Having established the location of the peak to be

tracked, data collection was begun. For pure CO2 and for the

mixture of CO2-C2 H experiments programs called VIKING and

VIKAN were used. In this version only data corresponding to

the maximum amplitude were recorded for each sweep frequency

range. For the pure C2 H6 experiment, however, data of the

entire sweep frequency range were recorded (via the program



















































FREQUENCY, Hz -+

Figure 4.10. Series of scanning routine.









56

called MAX) since it was believed the more detailed

information would be instructive. This change was possible

because of advances in the available data acquisition

hardware. In both cases the programs locked onto the tracked

peak for the entire process as bath temperature was lowered

(or raised) at a rate of a few millikelvin per minute over a

period of several hours. Pressure and temperature measurements

were simultaneously collected along with frequency and

amplitude. Upon completion of the experiment the temperature

of the system was brought back to the starting value. A small

amount of gas was removed to change the density of the system,

and then another data collection was begun. Typically, the

density of the system was initially at a supercritical value

(p > pc) and was finally at a subcritical value (p < pc). The

usual number of runs for a system was about fifty.

Note also that the resonance frequency may also be

determined accurately by observation of the phase change of

the output voltage relative to the excitation. This approach

is necessary for the most accurate determination of absolute

sonic speeds, but great care must be taken since the phase

meter is extremely sensitive and responds to peaks at even the

noise level. All programs were written in FORTRAN 77. They are

shown in appendix E. Figure 4.11 shows typical output response

of these two approaches.

Once data collection was complete the corresponding speed

of sound were calculated with a computer program called









57
GETSPG. This program utilizes the principle of relative

measurement mentioned earlier in the theory section. Also the

theoretical speeds of sound were calculated from the AGA8

equation through a computer program call TOTDAT for comparison

purposes. Finally the results were manipulated and graphically

displayed using popular software including: LOTUS-123,

QUATTRO, GRAFTOOL, and GRAPHER. Figure 4.12 shows a schematic

diagram of the entire process of data collection.









50


40


50


? 20





10
o *:---------------- : =.....----------









23.1 23.42 23.4s 29 23. 23i: 23: 21.7
(Thousands)
FREQUENCY. HZ 10
Figure 4.11. Two experimental approaches of resonance frequency determination: the
maximum amplitude approach and the voltage phase change approach.


































Figure 4.12. Flow chart of experimental scheme and data processing. The broken-line
boxes represent the experimental systems of interest. The thick-line boxes represent the
computer programs.















CHAPTER 5
RESULTS AND DISCUSSIONS



Carbon Dioxide


As mentioned in the introduction, due to its well-

established values, carbon dioxide was chosen to be the first

candidate used to test this new acoustic resonance technique

for locating critical point parameters. Learning how best to

do this required considerable effort, often involving trial

and error procedures. Many adjustments had to be made to both

instruments and programs to identify the optimum conditions

for performing this experiment.

Figure 5.1 shows the typical relationship between the

first radial mode resonance frequency (hereafter abbreviated

resonance frequency) and the temperature of carbon dioxide in

the resonator charged near its critical density. It shows the

results of four runs of this same isochore. For each run the

curve may be roughly divided into two parts for the sake of

discussion. The first one is a curve where the system

temperature is greater than or equal to the critical

temperature, Tc. In this region the speed of sound is directly

proportional to the system temperature, i.e. as temperature








W 1l1


16-

1.-





12-

t-


1. I I I I I I I
29.4 29.8 302 30.6 31 314 318
TEMPBUTURE (C)


132-
13-
128-
126-
124-

122-

|w
1-
4te-
I-
im-
106-
we-
104-


305


307 30E9 3t
TEAfERPB RE (


3t 31.5


I I I I I I I I I I
30 302 304 306 308 31
TEMPER tE (C)


I 3 It I 31
312 34 318


TEMPERALUE (C)


Figure 5.1. Relationship between resonance frequency and temperature of carbon dioxide o
for an isochore near its critical density.


uu ..


15-

1.4-



12-

11-


29.8


I I ., ,,,i,, ,,- .,, ,,.,


1 .


`J










62


17-


1.6-


15-








12-



28.8 29.2 29.6 30 30.4 30.8 31.2 3.6
TEMPERATURE (C)
Figure 5.2. Relationship between first radial mode resonance
frequency and temperature of carbon dioxide for an isochore
near the critical density.


240-

230-



S210-

200-

190-

iMo



160-
28.8 29.2 29.6 30 304 308 '31.2 31.6
TEMPERATURE C)
Figure 5.3. Relationship between the speed of sound and
temperature of carbon dioxide for an isochore near the
critical density.









63

decreases, resonance frequency and, hence speed of sound,

decreases. This represents a single phase region of the fluid.

The second part is a curve where the system temperature is

less than the critical temperature. Unlike the first region,

the speed of sound is inversely proportional to the system

temperature, i.e. as temperature decreases, resonance

frequency increases. This represents a two-phase region where

both gas and liquid are present in equilibrium. Each of the

experimental plots has a broken tip which occurs in the

critical region indicating loss of signal and hence loss of

precise track of the resonance peak in this vicinity. This is

understandable in view of the extremely fast dynamic behavior

of the resonance frequency in the critical vicinity and the

simultaneous dramatic reduction in signal amplitude. The best

experimental result was obtained during a run in which the

temperature of the well insulated system was initially set at

a value above the estimated critical temperature, and the

system was then allowed to cool down naturally to room

temperature (around 296 K.) The cooling time was approximately

17 hours during which approximately 1000 data points were

collected. Figures 5.2 and 5.3 represent results of this

particular run showing the relationships between system

temperature and resonance frequency and sonic speed

respectively. Results of this run clearly show the extremely

sensitive dependence of sonic speed on temperature in the

critical region. Figure 5.4 shows a plot of the first






























Figure 5.4. First derivative of resonance
temperature.


frequency versus


30418 3042 30422 304.24
TEMPERATURE (K)
Figure 5.5. Enlarged temperature scale of figure 5.4 showing
the critical temperature to be at 304.215 K.


1.5 -
15-

1

05-

0-

-0.5-


-15-

-2-
2-25 .


302.4 302.8 303.2 303.6 304
TEMPERATURE. K


~I


302


304.4 304.8









65

derivative of resonance frequency with respect to temperature

versus system temperature for the data of figure 5.3. It gives

a precise value of the critical temperature as seen in figure

5.5 which is a portion of figure 5.4 with an expanded

temperature scale. The critical temperature obtained from this

plot is 304.2150.007 kelvin. Figure 5.6 displays the

relationship between the system pressure and temperature for

this same run. It shows a nearly straight line with no

evidence of phase separation.

Experiments were also run with subcritical density

charges. These show the phase boundary location (dew points)

as changes in slope of resonance frequency versus system

temperature. Table 5.1 gives a comparison of results found by

the present method with available literature values. The

present results are in good agreement with the other

experimental values and with predictions based on an equation

of state'55~ and a correlation model developed at the National

Institute of Standards and Technology (NIST, formerly NBS) 56'.

The acoustic data and NIST predictions are shown graphically

in figure 5.7. Table 5.2 gives a chronological collection of

experimental values of critical point parameters (T, and P )

reported by several authors. The value obtained by this sonic

method is only 0.0016 percent different from the best value

obtained in the NIST laboratories (Morrison and Kincaid, 1984.

Reference 96).

















1085

1080
1005 .------ -------- ...... .......-......------- -------------

1075 .. .... .... .. ... .... ....

1070 .. .. ... .... .. ..

1065 ....

i 1060 ..... ..

1050 -... ..



1050 .
29.8 30 302 30.4 306 30.8 31 31.2 314

TEMPERATURE. CELSIUS

Figure 5.6. Pressure and temperature behavior of carbon
dioxide for isochore near the critical density.



r


Figure 5.7. Experimental vapor pressure curve compared with
the NIST model.


26 28 30
TEMPERATURE (C)
- MST MODEL A PRESENT















Table 5.1. Comparison of Dewpoint Pressures of Carbon Dioxide.


No. T (K) Dewpoint pressure, psia Percent difference
Present Equation Burnett NIST I II III
sonic of state method Model"
method (I) (II) (III)
1 299.905 971.46 971.045 971.10 971.53 +0.0427 +0.037 -0.0072
2 301.012 995.60 995.418 996.56 996.41 +0.0183 -0.096 -0.081
3 301.915 1015.30 1015.632 1017.10 1017.16 -0.0327 -0.177 -0.183
4 303.046 1041.74 1041.379 1044.23 1043.75 +0.0347 -0.239 -0.192


a Reference 55.
b Reference 57.
c Reference 56.









68


Table 5.2. Values of critical point parameters from the
literature.


Temperature Pressure Density Author(s)*
K
atm. psia. Kg/m3
304.07 73.0 1072.77 Andrews (58)
304.15 Hautefeuille
Cailletet (59)
305.05 77.0 1131.55 Dewar (60)
304.50 72.9 1071.30 Amagat (61)
304.55 Chappuis (62)
304.85 Villard (63)
304.15 Verschaffelt (64)
304.25 73.26 1076.59 Kuenen (65)
304.55 De Been (66)
304.10 Von Wesendonck
(67)
304.13 72.93 1071.74 Keesom (68)
304.27 Brinkmann (69)
304.135 Onnes & Fabius
(70)
304.41 Bradley, Brown, &
Hale (71)
304.25 73.00 1072.77 Dorsman (72)
304.15 72.85 1070.56 Cardoso & Bell
(73)
304.12 Hein (74)
304.35 Dieterici C. (75)
304.10 Pickering S.F.
(76)
304.25 72.95 1072.03 Meyers & Van
Dusen (77)
304.110.01 Kennedy H.T.,
Cyril H., and
Meyer (78)











Table 5.2. (Continued)


Temperature Pressure Density Author(s)*
K
atm. psia. Kg/m3
304.11 Kennedy (79)
304.19 Michael A.
Blaisse S.
Michael C. (80)
304.19 Lorentzen and Han
Ludvig (81)
304.19 72.80 1069.83 Tielsch H. (82)
304.16 75.20 1105.10 Ernst S. and
Thomas W. (83)
304.21 72.87 1070.86 Wentorf R.H. (84)
304.16 Ambrose D. (85)
0.03
304.15 75.282 1106.30 Ernst S. and
0.01 Traube K. (86)
304.2 72.83 1070.34 468 Matthews J.F.
(87)
304.150 72.79 1069.68 467.8 Moldover (88)
0.004
304.19 72.86 1070.78 468 Altunin V. (89)
304.13 Krynicki (90)
304.16 Lesnevskaya (91)
304.13 72.79 1069.62 467 Sengers J.V. (92)
304.20 Morrison G. (93)
304.13 467.4 Adamov (94)
304.13 72.79 1069.66 467.8 Shelomentsev (95)
304.206 72.87 1070.96 468.248 Morrison Kincaid
0.001 0.01 0.01 0.008 (96)
304.215 73.28 1076.88 This work
0.007 3.00 3.00

Numbers in parenthesis refer to references in bibliography.









70

Encouraged by the apparent success of this approach to

the location of phase boundaries, including critical points,

for a pure substance, we opted to proceed to the next stage of

the study and perform similar tests an a simple fluid mixture.

The properties of mixtures are not so well documented as those

of carbon dioxide, and there are few previously well

characterized binary systems to choose from. Carbon dioxide-

ethane is one of these, and a mixture of these two was

selected for this study. The results obtained are presented in

the next section.


A Mixture of Carbon Dioxide and Ethane


Since an acoustic resonance technique has been

successfully used to detect phase changes along the

coexistence curve of a single component system, it is a

naturally logical forward step to investigate its suitability

for use with more complex systems. What is needed are specific

mixtures with well characterized phase behavior. A binary

mixture of 74.25 mole percent carbon dioxide and ethane was

selected for this purpose. The carbon dioxide-ethane system

has been studied by several researchers beginning in 1897. The

first critical measurements were carried out by Kuenen in

1897"'" and 1902(98' followed by Khazanova and Lesnevsicaya in

1967'""', Fredenslund and Mollerup in 1974("00) and Morrison and

Kincaid in 1984(96). Phase boundary studies have been performed

by Clark and Din in 1953(101), Jensen and Kurata in 1971(102),









71

Gugnoni and Eldridge in 197410'), Nagahama et al. in 197404"),

Dabalos in 1974(105', and Stead and Williams in 1980106'. In 1978

Moldover and Gallagher(107' presented a correlation for this

mixture in an analogy with pure fluids. The specific mixture

selected for this study is one which was studied by Khazanova

and Lesnevsicaya('9'. Figures 5.8, 5.9, and 5.10 show the

behavior of the resonance frequency versus temperature of this

mixture at three different gas densities: supercritical

density (Pga, > Pc), subcritical density (Pga < Pc), and near

critical density (pg,. = PC). Note the sharper turning point in

the curve around the critical region in figure 5.10 compared

to the turning points of the curves of figure 5.8 and 5.9. The

former is a distinctive characteristic of a system at its

critical conditions (To, Pc, and pc.) The latter are

characteristic of a system reaching a coexistence boundary at

other conditions (bubble points or dew points) than the

critical. Figure 5.11 displays typical experimental results

for a near-critical-density gas mixture. Curve 1 represents a

run for which the bath temperature was ramped downward from

-30 to -12 OC. Curve 2 shows data for an upward ramp from -10

to -22 OC. In both cases the tracked peak (first radial mode

resonance frequency) was lost over a small temperature

interval in the critical vicinity due to the extremely fast

change of sonic speed and drop in amplitude. Curve 3 is like

curve 2 except that the temperature of the well-insulated





1900


1800


1700


1600


S 1500


1400


1300


1200 *. .
10 15 20 25 30

Temperature, Celsius
Figure 5.8. Typical behavior of resonance frequency versus temperature of the carbon
dioxide-ethane mixture at a supercritical density.






1. 46

1. 45 -

1.44 -

1.43 -

1. 42 -

1.41 -

1.4 -
N
^ 1. 39-

C> 1.38
ze
uJ 1. 37
Or




1. 34

1. 33

1. 32

1.31 -

1.3 -

1.29
9 11 13 15 17 19 21 23 25 27 29 31

TEMPERATURE, C

Figure 5.9. Typical behavior of resonance frequency versus temperature of the carbon
dioxide-ethane mixture at a subcritical density.




























16 17 18 19 20


Temperature, celsius


Figure 5.10. Typical behavior of resonance frequency
dioxide-ethane mixture near critical density.


versus temperature of


the carbon


1500


1400


1300


1200


1100


1000


N


&
U
U



900









75

system was allowed to raise naturally from 15 OC to room

temperature. This procedure was similar to the one utilized in

the experiments with carbon dioxide (and also with ethane as

will be seen later) except that the run went from low to high

temperature as the critical temperature of this particular

mixture is lower than room temperature. Again, the peak is

easier to track if the system temperature changes very slowly.

The speed of sound is very sensitive to temperature change in

the critical region. Note that a sharp turning point gives a

quite accurate value of the transition temperature but not

necessarily of speed of sound. Figure 5.12 is similar to

figure 5.11 but illustrates how other modes of vibration

reveal the same critical point temperature. In principle every

normal mode resonance exhibits a discontinuity at the critical

point, so there are very many possible indicators which are

equally capable of locating the critical point.

Figure 5.13 shows the behavior of pressure versus

temperature for several isochoric runs representing different

fluid densities. The upward direction of the breaks in these

curves shows them each to result from approaching the phase

boundary on cooling from a supercritical density. The breaks

mark individual bubble points on the phase envelope.

Approaches from subcritical densities would break downward at

the corresponding dewpoints, and the critical isochore would

exhibit no break at all. These facts are helpful in searching








250

240

230 -

220 -

o 210 -

E 200 1

190 -

o 180 -

170 -

160 -
1
150 -

140 -

130 -
3
120 -

1 1 0 I I I I I I I I I I I I I I I I I
10 12 14 16 18 20 22 24 26 28 30

Temperature, C
Figure 5.11. Typical experimental results of near-critical density gas of the carbon
dioxide-ethane mixture. Curve 1 shows a forced cooling run. Curve 2 shows a forced
warming run. Curve 3 is a slow naturally warming run.
























N
WI-

Z
- *
Oa
z *
D o
O=

u.


10 12 14 16 18 20 22 24 28 28 30


TEMPERATURE, C


Figure 5.12. Other resonance frequency (top curve) results in
as first radial resonance frequency (bottom curve.)


same critical temperature











78

for the critical point of a system when it is not previously

known. For example the starting pressure at 30 OC for the

critical isochore could be approximately calculated from the

information in figure 5.13 in the following fashion. For each

isochore the two branches, supercritical vapor above the

bubble point and two phase equilibrium below it, were fit to

straight lines by linear regression. The angle 0 between the

two branches is directly related to the slopes of these lines.

A plot of initial charging pressure versus 8 indicates by

extrapolation the approximate critical density charging

pressure for which 0 = 0. The results for the data of figure

5.13 are tabulated in table 5.3, and the plot of charging






1.4


1.3


1.2



w-S

0. 9 -
I0

S-
O.9


0.
0 7 T -l l


9 11 13 15 17 19 21 23 25 27 29 31
TEMPERATURE (C)
Figure 5.13. A sequence of supercritical pressure versus
temperature isochores of the carbon dioxide-ethane
mixture.









79

pressure versus 0 is shown in figure 5.14. This technique

revealed the magnitude of the charging pressure to be used

when data were to be collected in search of the critical

point.



Table 5.3. Angles between pressure-temperature lines before

and after phase boundary of supercritical density fluid of

carbon dioxide and ethane.


File Slope of line Slope of line Angle between

# before phase after phase two lines

boundary boundary (degree)

729 34.637 18.307 14.733

731 34.074 18.293 14.479

801 33.562 18.371 14.091

802 32.842 18.444 13.594

807 27.882 18.690 10.356

811 25.149 18.607 7.992

815 24.701 18.846 7.191


This prediction was important here since the system volume is

fixed. The only way to change the system density was to


































Figure 5.14. Relationship between the angles 8 and the
charging pressures of the carbon dioxide-ethane mixture.

release some of the material, and this process was potentially

troublesome because of the possibility of inducing unwanted

composition changes. Care was taken to release gas only from

the system in the single phase condition and then only after

it had been blended thoroughly by continued mixing. Typically,

when the system was close to the critical density, only small

amounts of gas were released for each successive run. The drop

in charging pressure at 30 OC was typically one or two psia.

For this work the total number of runs was fifty.

Table 5.4 lists the experimental results needed to

construct phase diagrams of this system. Figures 5.15 to 5.19

show various types of phase diagram created from different

combinations of the parameters in table 5.4. Figure 5.15 shows

the relationship between starting pressure at 30 OC and the


1320.0

S1300.0

1280D
W
- 1280.0
cr)

1240.0
0.
Q 1220.0 -
z
o 1200.0

S1180.0

1160.0 5


0 100 11.0 12.0 130 14.0 15.0

ANGLE, DEGREE





Table 5.4. Experimental results for construction of phase diagrams of a mixture of

carbon dioxide and ethane.


FILE NAME Starting point Point of minimum frequency
T Pressure Frequency T (oC) Frequency Sonic Pressure
(C) psia Hz Hz speed psia
m/sec
TDA729 30.004 1303.6 1976.38 17.006 1554.82 216.72 857.0

TDA731 29.993 1292.9 1951.08 17.200 1533.44 213.70 860.7

TDA801 30.002 1283.3 1925.92 17.485 1507.68 210.11 865.8

TDA802 30.002 1271.9 1895.52 17.632 1470.04 204.87 869.4

TDA807 30.003 1200.2 1669.91 19.228 1203.10 167.71 901.2

TDA820 30.013 1156.6 1535.02 19.860 935.06 130.33 914.5

TDA907 30.007 1135.3 1460.87 19.950 802.36 111.83 915.9

TDA914 30.011 1114.8 1425.97 19.964 934.84 130.30 916.3

TDA917,925 30.060 1107.1 1420.32 19.920 1030.75 143.66 915.6

TDA926 30.039 1101.2 1426.42 19.885 1073.42 149.61 914.9

TDA928 30.020 1098.6 1416.55 19.851 1086.70 151.46 914.1
I-------- ----







Table 5.4. (Continued)


FILE NAME Starting point Point of minimum frequency
T Pressure Frequency T (OC) Frequency Sonic Pressure
(C) psia Hz Hz speed psia
m/sec
TDA929 30.020 1098.6 1416.55 19.835 1087.73 151.61 913.5

TDA1003 30.018 1095.7 1416.57 19.800 1105.21 154.04 913.2

TDA1004 30.006 1090.1 1415.82 19.737 1132.31 157.86 911.3

TDA1006 30.015 1083.6 1417.73 19.596 1158.91 161.58 909.2

TDA1007 30.005 1077.6 1419.66 19.476 1182.36 164.80 906.3

TDA1009 30.007 1066.1 1425.68 19.181 1217.97 169.81 899.8

TDA1010 30.001 1057.3 1431.13 18.899 1239.56 172.76 893.8

TDA1011 29.997 1047.7 1437.77 18.532 1258.82 175.44 886.3

TDA1012 30.000 1037.9 1444.67 18.043 1274.82 177.67 876.8

TDA1013,14 29.992 1027.6 1452.33 17.800 1289.80 179.75 870.7

TDA1015(1) 30.029 1017.0 1459.64 17.200 1300.55 181.24 859.2







Table 5.4. (Continued)


FILE NAME Starting point Point of minimum frequency
T Pressure Frequency T (C) Frequency Sonic Pressure
(C) psia Hz Hz speed psia
m/sec
TDA1015(2) 30.029 1017.0 1459.64 17.174 1300.34 181.22 858.8

TDA1016,17 30.047 1006.7 1468.85 16.750 1313.48 183.04 850.1









84

temperature of phase change. Experimental results are

represented by points. The smooth curve is a result of a third

order polynomial curve fitting procedure. The corresponding

equation is as follows:

T = a + b*p + c*p2 + d*p3

where T = temperature, celsius

p = pressure, psia

a = -8.137773 X 102

b = 2.020012

c = -1.616095 X 10-3

d = 4.260451 X 10-7

r = correlation coefficient

= 0.99749.

Figure 5.16 shows the relationship between starting pressure

and the speed of sound at the phase change. A sharp change in

the speed of sound at the critical point is evident compared

to the slow change of temperature shown figure 5.15. Figure

5.17 shows a coexistence curve. Note that the curve is so slim

that the bubble-point line almost overlaps the dew-point line.

Recall that a similar plot for a pure fluid yields a nearly

straight line (see figure 5.23 of ethane.) This clearly

reveals a property of this mixture as being nearly azeotropic.

It behaves much as if it were a pure fluid. The literature

value of the azeotrope composition of this binary mixture is

reported to be 0.255 mole fraction of ethane"'08'. This is very

close to the composition of the mixture studied in this work











20.00


19.50


19.00


18.50


18.00


17.50


17.00


16.50
1000 1050 1100 1150 1200 1250 1300


STARTING PRESSURE,


1350


PSIA


Figure 5.15. Relationship between the starting pressures and
the temperatures of phase change of the carbon dioxide-ethane
mixture.













220.00 -


Figure 5.16. Relationship between the starting pressures and
the speed of sound at a phase change of the carbon dioxide-
ethane mixture.


0aa.0

910.0


aoo.o -
800.0



O 8.0


[ 870.0

880.0

850.0
16.5 17.0 17.5 18.0 18.5 18.0 19.5 20.0

TEMPERATURE, CELSIUS

Figure 5.17. Coexistence curve of the carbon dioxide-ethane
mixture near azeotrope composition.


S200.00


E 180.00





0 140.00

U-
o 120.00
03



100.00


1050 1100 1150 1200 1250 1300

STARTING PRESSURE, PSIA




Full Text
o n n n o o
174
SWEEP?
ISTART=ICENT-N/2
CALL VOLTSO(IFLIP(ISTART))
Write(*,*) WHAT IS THE INITIAL TEMPERATURE VOLTAGE?
READ( * ) TBB
ITB=INT(TBB)
CALL VOLTS1(IFLIP(ITB))
Write( * ) WHAT IS THE FINAL TEMPERATURE VOLTAGE?'
READ( * ) TF
Write(*,*) HOW MANY HOURS FOR THIS TEMPERATURE
READ( * ) TIM
TIM=TIM*3600000.0
TMM=(TF-TBB)/TIM
CALL TLOOK(T)
T0=T
c
c
c
WRITE( * ) WHAT IS THE SENSITIVITY ID NUMBER?
READ( * ) IG
OPEN(15,FILE=SENSE',STATUS='NEW')
CALL TEMPGET(TEMP)
TSHIFT=TEMP-22.0
ITSHIFT=TSHIFT
200
WRITE(5,200) TSHIFT
FORMAT(IX, 'TEMPERATURE SHIFT=',F8.3)
CALL PRESGET(P)
PSHIFT=P-3000.0
300
WRITE(5,300) PSHIFT
FORMAT(IX,PRESSURE SHIFT=,F10.3)
CALL FREQGET(FREQ)
FSHIFT=FREQ-10000
400
C
C
WRITE(5,400) FSHIFT
FORMAT(IX,FREQUENCY SHIFT=,I6)
CALL PHASEGET(AMPL)
CALL LOCKIN(AMPL,ANGL)
ASHIFT=ANGL-80.0
C
C 700
C
WRITE(5,700) ASHIFT
FORMAT(IX,'ANGLE SHIFT=',F8.3)
WRITE(*,*) AMPL=,AMPL
C
500
WRITE(*,*) 'ANGL=',ANGL
WRITE(5,500) N
FORMAT(IX, LOOP RANGE=',I4)
CLOSE(5)
KBLOCK=0
1000
KBLOCK=KBLOCK+1
LARGE=-20000
COUNT=0
ISTART=ICENT-N/2
ISTOP=ISTART+N
CALL VOLTSO(IFLIP(ISTART))
CALL SECS(T,TO)
TEMP=FLOAT(T)*TMM+TBB
ITEMP=INT(TEMP)
CALL VOLTS1(IFLIP(ITEMP))


30
Speed of Sound as a Probe for the Critical Point
By conventional methods the critical point can be located
indirectly as the zero point of the first derivative of
pressure with respect to volume or of volume with respect to
temperature on an isotherm or isobar, respectively, or by
analyzing the slope of lines plotted as the first derivative
of pressure with respect to temperature versus temperature for
isochores. The technique involving use of the principle of
appearance and disappearance of a meniscus is to some degree
subjective due to the heuristic effect. A better alternative
would be to use some thermodynamic property which shows a
significant change at the critical point. In 1962 Bagatski and
his coworkers (50'51) experimentally observed the asymptotic
behavior of heat capacity at constant volume of argon in the
immediate vicinity of the critical point. As expected Cp falls
off rapidly and it appeared that this could be used to detect
the critical point. Measuring heat capacity, however, is not
an easy task. On the other hand, the speed of sound similarly
shows an abrupt change toward zero at the critical point since
it is inversely proportional to Cy which diverges weakly and
is directly proportional to Cp which diverges strongly and to
the first derivative of pressure with respect to molar volume
at constant temperature, which also vanishes at the critical
state. The appropriate equation is:
1/2
C2 V -1
0 M
T
(3.3)


18
Table 2.1. The values of the roots i in ascending order of
magnitude. D=degeneracy
n
1
D
NAME
n
1
D
NAME
2.08158
1
1
3
IP
12.2794
6
2
13
2i
3.34209
2
1
5
Id
12.4046
1
4
3
4p
4.49341
0
1
1
Is
13.2024
11
1
23
In
4.51408
3
1
7
If
13.2956
4
3
9
3g
5.64670
4
1
9
ig
13.4721
7
2
15
5.94036
1
2
3
2p
13.8463
2
4
5
4d
6.75643
5
1
11
lh
14.0663
0
4
1
4s
7.28990
2
2
5
2d
14.2580
12
1
25
lo
7.72523
0
2
1
2s
14.5906
5
3
11
3h
7.85107
6
1
13
li
14.6513
8
2
17
2k
8.58367
3
2
7
2f
15.2446
3
4
7
4f
8.93489
7
1
15
lj
15.3108
13
1
27
iq
9.20586
1
3
3
3p
15.5793
1
5
3
5p
9.84043
4
2
9
2g
15.8193
9
2
19
21
10.0102
8
1
17
lk
15.8633
6
3
13
3i
10.6140
2
3
5
3d
16.3604
14
1
29
lr
10.9042
0
3
1
3s
16.6094
4
4
9
4g
11.0703
5
2
11
2h
16.9776
10
2
21
2m
11.0791
9
1
19
11
17.0431
2
5
5
5d
11.9729
3
3
7
3f
17.1176
7
3
15
12.1428
10
1
21
lm
17.2207
0
5
1
5s
17.4079
15
1
31
It
21.6667
3
6
7
6f
17.9473
5
4
11
4h
21.8401
8
4
17
4k
18.1276
11
2
23
2n
21.8997
1
7
3
7p
18.3565
8
3
17
3k
22.0000
11
3
23
3n
18.4527
16
1
33
lu
22.5781
6
5
13
5i
18.4682
3
5
7
5f
22.6165
20
1
41
iy
18.7428
1
6
3
6p
22.6625
15
2
31
2t
19.2628
6
4
13
4i
23.0829
4
6
9
6g
19.2704
12
2
25
2o
23.1067
9
4
19
41
19.4964
17
1
35
lv
23.1950
12
3
25
3o
19.5819
9
3
19
31
23.3906
2
7
5
Id
19.8625
4
5
9
5g
23.5194
0
7
1
7s
20.2219
2
6
5
6d
23.6534
21
1
43
lz
20.3714
0
6
1
6s
23.7832
16
2
33
2u
20.4065
13
2
27
2q
23.9069
7
5
15
~T
20.5379
18
1
37
lw
24.3608
10
4
21
4m
20.5596
7
4
15
4j
24.3824
13
3
27
3q
20.7960
10
3
21
3m
24.4749
5
6
11
6h
21.2312
5
5
11
5h
24.6899
22
1
45
la
21.5372
14
2
29
2r
24.8503
3
7
7
li
21.5779
19
1
39
lx
24.8995
17
2
35
2v


119
Figure 5.34. Three dimensional phase diagram of ethane.


22
In pure fluid systems the phenomenon of condensation is
associated with lowering the temperature and raising the
pressure and vice versa for the phenomenon of vaporization.
In a fluid mixture system we have besides the external
parameters: temperature and pressure, the internal parameters
which identify the composition. The phase behavior of the
system can be more complicated than that of a pure fluid
system due to many factors such as the possibility of various
kinds of retrograde phenomena, the occurrence of azeotropic
mixtures of positive or negative types etc. In a binary
system, for example, on the basis of an analysis of the phase
diagrams by means of the van der Waals equation of state, nine
major types of phase diagram may be obtained'42'44. The complete
phase behavior of a binary system is represented by the four
dimensional surface, p = p(v,t,x) where x is the mole fraction
of one of the two components. Consequently, the critical point
of a mixture has to be redefined as the point where liquid and
vapor become identical, subject to the constraint of equality
of composition.
Since this work deals with a binary mixture of carbon
dioxide and ethane, attention throughout will be mainly
focused on the relevant information for this mixture. Carbon
dioxide and ethane form an azeotropic mixture at the
appropriate composition. This binary system deviates
positively from the ideal solution, which obeys Raoult's law.
In this case a critical point locus follows a curve between


24
the critical point of pure carbon dioxide and that of pure
ethane. Figures 3.2 and 3.3 illustrate the pressure-
temperature-mole fraction diagram and critical locus of this
mixture, respectively. In figure 3.3 at point B the maximum
azeotrope is tangent to the critical curve (KftBKB) which has
a minimum in temperature.
Determinations of Critical Point
Conventionally, there are three principal methods of
locating the critical point. These are described briefly
below.(45)
Isotherm Approach
Critical temperature may be estimated to within a few
hundredths of a kelvin by an analysis of the geometry of the
isotherms'46. The isotherms in the immediate neighborhood of
the critical temperature are measured with sufficient accuracy
to allow the critical temperature to be determined from the
inflection points of the isotherms. The minimum values of the
derivative of pressure with respect to volume at constant
temperature, found graphically are plotted as a
function of temperature (or pressure or molar volume.) The
intersection of this line with the horizontal axis gives the
critical parameters (Tc, Pc, vc. ) Figures 3.4-3.6 illustrate
this approach.


Table 5.5. Comparison of Dewpoint Pressures of C02 + C2H6 Mixture (Xco2 = 0.7425).
No.
Temperature
(K)
Dewpoint pressures, psia
Percent
Difference
Present
sonic
method
Equilibrium
cell of the
vapor
recirculation
type
(interpolated)1
(I)
NIST
DDMIX
Model2
(II)
I
II
1
290.156
850.88
849.950
862.296
-0.109
-1.34
2
290.350
854.55
853.500
866.068
+0.123
-1.35
3
290.635
859.62
858.620
871.580
+0.116
-1.39
1 A. Fredenslund and J. Mollerup, J. Chem. Soc. Faraday Trans. I, 1_0_, 1653 ( 1974).
2 DDMIX model developed by NBS (1988) uses Peng-Robinson equation of state for
coexisting-phase composition calculations and NBS corresponding states model with shape
factors and van der Waals one-fluid mixing rules for phase properties. The uncertainties
in DDMIX model are principally due to the uncertainties associated with the mixing
rules.
oo
VO


Table 5.8. Comparison of Bubble-Point Pressures of Ethane.
No.
T (K)
Bubble-point Pressures, psia
Percent Difference
Present
sonic method
Compressibility
apparatus1 (I)
NIST DDMIX
Model (II)2
I
II
1
304.718
697.67
697.645
697.497
+0.003
+0.024
2
304.949
700.83
701.020
700.978
-0.027
-0.021
3
305.141
703.71
703.830
703.735
-0.017
-0.004
4
305.161
704.30
704.120
704.025
+0.026
+0.039
5
305.193
704.70
704.600
704.605
+0.014
+0.013
6
305.240
705.40
705.450
705.330
-0.008
+0.009
D R* Douslin and R. H. Harrison, J. Chem. Thermodynamics, 5, 491, 1973.
2 See footnote of table 5.5 for explanation.
o


23
Figure 3.2 Pressure-Temperature-Mole fraction relationship
of system of carbon dioxide and ethane.
Figure 3.3. Pressure-temperature projection of the system
carbon dioxide-ethane in the critical region. Lines Z K
and ZbKb are vapor pressure curves of pure carbon dioxide
and of pure ethane respectively.


169
CH(3) = '1'
CH(4) = 'A'
CH(5) = 'M'
CH(6) = 'O'
CH(7) = 11
CH ( 8 ) = 'V
CH(9) = 'O'
CH(10) = 'S'
CH (11) = T"
WRITE(STARTF,(F9.3)') SF
READ(STARTF,'(Al)') CH(12)
READ(STARTF,'(IX,Al)) CH(13)
READ(STARTF, (2X,Al) ) CH(14)
READ(STARTF,'(3X,A1)') CH(15)
READ(STARTF,(4X,A1)') CH(16)
READ(STARTF,'(5X,A1)') CH(17)
READ(STARTF,'(6X,A1)') CH(18)
READ(STARTF,'(7X,Al)') CH(19)
READ(STARTF,'(8X,Al)') CH(20)
READ(STARTF,'(9X,Al)') CH(21)
CH(22) = 'H'
CH(23) = 'Z'
CH(24) = S'
CH(25) = 'P'
WRITE(STOPF,'(F9.3)') SPF
READ(STOPF,'(Al)') CH(26)
READ(STOPF,'(1X,A1)') CH(27)
READ(STOPF,'(2X,Al)') CH(28)
READ(STOPF,'(3X,Al)') CH(29)
READ(STOPF,'(4X,Al)') CH(30)
READ(STOPF,'(5X,A1)) CH(31)
READ(STOPF,'(6X,Al)') CH(32)
READ(STOPF,'(7X,Al)') CH(33)
READ(STOPF,'(8X,A1)') CH(34)
READ(STOPF,'(9X,A1)') CH(35)
CH(36) = 'H'
CH(37) = Z'
CH(38) = 'T'
CH(39) = I'
READ(SWEEP,'(Al)') CH(40)
READ(SWEEP,'(IX,Al)') CH(41)
CH(42) = 'S'
CH(43) = E'
CH(44) = 'S'
CH(45) = 'S
CH(46) = S'
CH(47) = 'S'
N=12
M=4 7
DO 30 1=1,N
WRT(I) = 0
DO 40 J=1,4


31
where CQ is the sonic speed in the limit of zero frequency.
Because the sonic speed is directly proportional to the
frequencies of normal mode resonance vibrations in a filled
cavity resonator, one can expect to detect the approach to the
critical state by observing changes in resonance frequency.
Spherical cavities yield especially sharp resonances and are
therefore well suited for this purpose. In the first phase of
this work the isochoric approach was employed. However, unlike
the conventional method of using the straightness of the
pressure-temperature plot or using the appearance and
disappearance of the meniscus, the speed of sound or a
resonance frequency was utilized as a probe of a critical
point.


56
called MAX) since it was believed the more detailed
information would be instructive. This change was possible
because of advances in the available data acquisition
hardware. In both cases the programs locked onto the tracked
peak for the entire process as bath temperature was lowered
(or raised) at a rate of a few millikelvin per minute over a
period of several hours. Pressure and temperature measurements
were simultaneously collected along with frequency and
amplitude. Upon completion of the experiment the temperature
of the system was brought back to the starting value. A small
amount of gas was removed to change the density of the system,
and then another data collection was begun. Typically, the
density of the system was initially at a supercritical value
(Pg > Pc) and was finally at a subcritical value (pg < pc) The
usual number of runs for a system was about fifty.
Note also that the resonance frequency may also be
determined accurately by observation of the phase change of
the output voltage relative to the excitation. This approach
is necessary for the most accurate determination of absolute
sonic speeds, but great care must be taken since the phase
meter is extremely sensitive and responds to peaks at even the
noise level. All programs were written in FORTRAN 77. They are
shown in appendix E. Figure 4.11 shows typical output response
of these two approaches.
Once data collection was complete the corresponding speed
of sound were calculated with a computer program called


95
26 28 30 32 34 36 38
TEMPERATURE, CELSIUS
Figure 5.26. Phase diagram of ethane showing pressure and
temperature behavior of 23 different isochores.


106
Figure 5.28. Resonance frequency and temperature relationship
revealing some phenomena observed in the experiment on pure
ethane.
memory size and resolution of the display monitor etc. At
present only limited observations were practical.
Figure 5.28 is used in conjunction with the following
observations. Note that each curve in this figure does not
necessarily represent a system having the same density as the
others. They are considered together for the purpose of the
following comparison.
Observation 1
At temperatures above the critical temperature the first
radial mode of vibration (the Is peak) and the first nonradial
mode of vibration (the If peak) were in close proximity (about
15-20 hz apart). Consequently, the If peak tended to interfere
with tracking of the Is peak. The data acquisition program
selected the signal of highest amplitude in the observation


55
t
tr
o
>
FREQUENCY. Hz ->
figure 4.10. Series of scanning routine


TEMPERATURE, C
Figure 5.9. Typical behavior of resonance frequency versus temperature of the carbon
dioxide-ethane mixture at a subcritical density.
OJ


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ACOUSTIC DETERMINATION OF PHASE BOUNDARIES
AND CRITICAL POINTS OF GASES:
co2, co2-c2h6 mixture, and c2h6
By
Chadin Dejsupa
December 1991
Chairperson: Samuel 0. Colgate
Major Department: Chemistry
The acoustic resonance technique developed in this
laboratory is a remarkable tool for the determination of the
thermophysical properties of materials. The measured
frequencies of radially symmetric modes of vibration are
simply related to the sonic speed in fluids enclosed in the
spherical cavity. To minimize experimental perturbations a
relative measurement against argon gas used as a standard
reference was employed. The sonic speed in turn is utilized as
a sensitive probe for detecting phase boundaries and critical
points of substances. Location of the boundary is revealed by
a discontinuity in the variation of sonic speed with
temperature for a sample confined isochorically in the
resonator. The corresponding temperature at which this
discontinuity occurs was taken as a phase boundary
temperature. This indicator is very pronounced, especially
Xlll


CHAPTER 4
EXPERIMENT
Instruments
The development of a spherical acoustic resonator as a
tool to the study speed of sound in gases such as argon,
butane, isobutane etc. which, in turn, leads to values of some
thermodynamic and equation of state properties has been
successful in the past by workers in this research group. (20'26)
There are, however, some limitations on the previous work due
mainly to the construction of the spherical resonator. It was
made from aluminum alloy and designed to operate near
atmospheric pressure. These constraints limit its applications
substantially. For example, if one wishes to study natural gas
mixtures, one would likely encounter the presence of some
corrosive, acidic or basic gases such as hydrogen disulfide,
carbon dioxide etc., and these, especially in the presence of
water, attack aluminum to some extent. In addition,
experiments may call for the measurements at much higher
pressures than atmospheric.
The present research was undertaken to extend the
applicability of the acoustic resonance technique to the study
of natural gas mixtures at high pressures. To fulfil this goal
32


86
STARTING PRESSURE, PSIA
Figure 5.16. Relationship between the starting pressures and
the speed of sound at a phase change of the carbon dioxide-
ethane mixture.
TEMPERATURE, CELSIUS
Figure 5.17. Coexistence curve of the carbon dioxide-ethane
mixture near azeotrope composition.


26
Figure 3.6. Schematic diagram of graphical determination of
critical temperature of carbon dioxide.


Three Dimensional Phase Diagram ... 116
CHAPTER 6 CONCLUSION 120
APPENDIX A CHARGING PRESSURE CALCULATIONS .... 126
APPENDIX B QUANTITATIVE ANALYSIS OF GAS MIXTURE 128
APPENDIX C SIMPLEX 136
APPENDIX D LABORATORY STANDARD PRESSURE GAUGE . 147
APPENDIX E COMPUTER PROGRAMS 163
REFERENCES 194
BIOGRAPHICAL SKETCH 203
v


when the system is near the critical condition for which
strong local density fluctuations disperse sound and cause the
sonic speed to dip sharply toward zero. This research reports
on such studies for two pure gases C02, and C2H6 and one
mixture of these two. For each system, data on resonance
frequency, amplitude and pressure as a function of temperature
were collected under computer control for a series of
different loading densities around the critical value.
Portions of the phase diagrams revealed by the measurements
were recorded in tabular and graphical formats.
xiv


29
the temperature changes. The following possible kinds of
isochoric curves could result if the charging density p,is
greater than the critical density pc, the isochore curve shows
an upward bend as in C, or if p is less than pc the curve
shows downward bend as in A. However, if p is equal to pc the
isochore exhibits no break as in B. This method is less
demanding in terms of skillful technique and special
apparatus, but the accuracy is poor. Figure 3.8 shows data
illustrating all three possibilities described above.
The second established isochoric method is to study the
appearance and disappearance of the meniscus between the
phases near the critical temperature.(19) The procedure is the
same as above except that, instead of measuring pressure as
temperature changes, the behavior of the meniscus separating
the liquid and gaseous phases is observed. There are also
three possible phenomenon:
If p < pc the meniscus falls until the entire container
is filled with gas.
If p > pc the meniscus rises until the entire container
is filled with liquid.
If p = pc the meniscus approximately located at a point
halfway up the container will flatten, then
become very faint and finally disappear.
The temperature and density at which the meniscus disappears
are taken to be the critical parameters.


ABSOLUTE PRESSURE GAUGE CALIBRATION
162


129
Figure B.l. Block diagram of the gravimetric method.
A block diagram of the experimental setup is shown in
figure B.l. A gas sampling container was made from a stainless
steel tube provided with high pressure needle valves A and B.
The experiment started with pre-conditioning the system by
flowing argon gas to the reaction vessels (hexagonal absorber
array 5, see figure B.2) through valves 4 and 5 for one-half
hour. After that valves 4 and 5 were closed followed by
opening valve B. The pressure of gas in the gas sampling
container was shown on gauge 3 (typical pressure was around
1200 psia.) Then valve 5 was opened carefully to bleed gas
5The hexagon absorber arrays of vertical tubes each
having a wash board shape were designed to prolong contact
time of gas with absorbant to assure completeness of the
reaction. Each section of the first five tubes was filled with
1M NaOH. The last tube was filled with drierite (moister
absorbent.)


52
was calculated using the AGA8 program. Details of the
calculation are given in appendix A. In the case of the
mixture of CO, and C,H, ethane was first introduced. Pressure
was monitored roughly by a Bourdon pressure gauge and more
precisely by a digital pressure gauge previously calibrated to
yield an absolute pressure reading. The magnetic circulating
pump was turned on to assure adequate mixing. Once the system
was at stable conditions in temperature and pressure (which
took approximately a few hours), searching for resonance
frequencies began.
Fluids in the resonator were stimulated acoustically by
driving one of the bimorph transducers with a 40 volt peak-to-
peak sine-wave signal generated under microprocessor control
(PC-AT) of a programmable signal synthesizer, HP3325B in the
case of pure carbon dioxide and of the of C02 and C2H6 mixture
or of a wave function generator (WAVETEX) controlled through
a programmable digital-to-analog converter (DAC) in the case
of pure ethane. The fluid response was determined at the other
transducer which feeds its signal to a lock-in amplifier. The
response of the speaker transducer in terms of voltage output
signal was monitored closely to changes in frequency of the
microphone transducer. At resonance, the output amplitude
increased dramatically to maximum enabling a precise frequency
measurement. Using an oscilloscope as a visual aid to locate
the vicinity of this maximum signal and a digital voltmeter to
find the precise position of the signal peak,
resonance


71
Gugnoni and Eldridge in 1974<103), Nagahama et al. in 19 7 4(104),
Dabalos in 1974(105)/ and Stead and Williams in 19 8 0(106). In 1978
Moldover and Gallagher(10?) presented a correlation for this
mixture in an analogy with pure fluids. The specific mixture
selected for this study is one which was studied by Khazanova
and Lesnevsicaya(99). Figures 5.8, 5.9, and 5.10 show the
behavior of the resonance freguency versus temperature of this
mixture at three different gas densities: supercritical
density (pga8 > pc) subcritical density (pgaa < pc) and near
critical density (pga8 = pc) Note the sharper turning point in
the curve around the critical region in figure 5.10 compared
to the turning points of the curves of figure 5.8 and 5.9. The
former is a distinctive characteristic of a system at its
critical conditions (Tc, Pc, and pc. ) The latter are
characteristic of a system reaching a coexistence boundary at
other conditions (bubble points or dew points) than the
critical. Figure 5.11 displays typical experimental results
for a near-critical-density gas mixture. Curve 1 represents a
run for which the bath temperature was ramped downward from
-30 to -12 C. Curve 2 shows data for an upward ramp from -10
to -22 C. In both cases the tracked peak (first radial mode
resonance frequency) was lost over a small temperature
interval in the critical vicinity due to the extremely fast
change of sonic speed and drop in amplitude. Curve 3 is like
curve 2 except that the temperature of the well-insulated


202
138. Ley-Koo M., and M. S. Green, Phvs. Rev. A. 23, 5, 2650,
1981.
139. McGill K. C., Fast Fourier Transformed Acoustic
Resonance With Sonic Transform., Ph.D. Dissertation,
The University of Florida, 1990.
140. Skoog Douglas A. and West Donald M., Fundamentals of
Analytical Chemistry, 4th edition, Saunders College
Publishing, New York, 1982.
141. Harris Daniel C., Quantitative Chemical Analysis, 2nd.
ed., W. H. Freeman Company, New York, 1986.
142. Shoemaker David P., Garland Carl W., and Joseph W.
Nibler, Experiments in Physical Chemistry, 5th ed. ,
McGraw-Hill Publishing Company, New York, 1989.
143. Harland Philip W. Pressure Gauge Handbook, U. S. Gauge
Division, Ametek Inc., Sellersville, Pennsylvania,
1985.
144. Stull Daniel R., Ind. Eng. Chem., 39, 540, 1947.
145. Gilkey W. K., Gerald Frank W., and Milo E. Bixler, Ind.
Eng. Chem., 23, 4, 364, 1931.


63
decreases, resonance frequency and, hence speed of sound,
decreases. This represents a single phase region of the fluid.
The second part is a curve where the system temperature is
less than the critical temperature. Unlike the first region,
the speed of sound is inversely proportional to the system
temperature, i.e. as temperature decreases, resonance
frequency increases. This represents a two-phase region where
both gas and liquid are present in equilibrium. Each of the
experimental plots has a broken tip which occurs in the
critical region indicating loss of signal and hence loss of
precise track of the resonance peak in this vicinity. This is
understandable in view of the extremely fast dynamic behavior
of the resonance frequency in the critical vicinity and the
simultaneous dramatic reduction in signal amplitude. The best
experimental result was obtained during a run in which the
temperature of the well insulated system was initially set at
a value above the estimated critical temperature, and the
system was then allowed to cool down naturally to room
temperature (around 296 K.) The cooling time was approximately
17 hours during which approximately 1000 data points were
collected. Figures 5.2 and 5.3 represent results of this
particular run showing the relationships between system
temperature and resonance frequency and sonic speed
respectively. Results of this run clearly show the extremely
sensitive dependence of sonic speed on temperature in the
critical region. Figure 5.4 shows a plot of the first


97
the present experimental bubble point pressures and available
literature values. These include the experimental values
measured in the compressibility apparatus of Douslin and
Harrison(109) and the values predicted by DDMIX(5S), a
correlation model developed at the National Institute of
Standards and Technology (NIST, formerly NBS). Table 5.9
compares the present vapor pressures with those of the
predictive equation proposed by Sychev et. al.(110) This
equation derived from a compilation of vapor pressures of
ethane by Goodwin et. al.(111) The Sychev equation is:
In ps = a + bx + cu + du2 + eu3 + fu(l-u)6
where x(T) = [ 1-(Ttr/T) ] /[ 1-(Ttr/Tcr) ]
U(T) = [T-Ttr]/[Tcr-Ttr]
Ttr = Temperature at the triple point
= 90.348 0.010 K
Tcr = Critical Temperature
= 305.33 0.02 K
a = -11.38996
b = 18.84523
c = -7.635425
d = 5.428443
e = -1.362327
f = 0.7692493
e =1.30.


41
brazed to stainless steel tubes used for charging the
resonator and circulating its contents through an external
loop containing the magnetically driven pump. The two
identical PZT bimorph piezoelectric transducer assemblies were
similarly mounted to the tank at the 0.953 cm (% in.) hole
positions.
The assembly support frame consists of two parallel 0.6
cm (h in.) thick stainless-steel plates between which an array
of 2.54 cm (1 in.) thick stainless-steel blocks were attached
with bolts to support the tube fittings and transducer
assemblies and provide a means of applying clamping forces to
the O-ring seals. The clamping force acting along the polar
axis is applied by a hallow jam screw which slips over the
charging tube and pushes against the brazed O-ring gland. The
block which supports the lower tube fitting is rigidly fixed
and provides a definite reproducible location for the tank
body. The transducer assemblies were clamped against the
resonator by set screws threaded through two of the blocks
bolted to the mounting frame. Set screws in two additional
blocks located opposite the transducer mounting holes apply
counter forces against the tank to stabilize the assembly
further.
The circulating pump is connected to the upper and lower
tube fittings by demountable compression fittings (Swagelok.)
The normal pumping rate of 4 liters per minute proved adequate
for these purposes. A 0.6 cm (k in.) tube connected to the


193
data ATN/16/,TACS/8/,LACS/4/,DTAS/2/,DCAS/1/
data EDVR/O/,ECIC/1/,ENOL/2/,EADR/3/,EARG/4/
data ESAC/5/,EABO/6/,ENEB/7/,EOIP/10 /,ECAP/11/
data EFSO/12/,EBUS/14/,ESTB/15/,ESRQ/16/
data BIN/4096/,XEOS/2048/,REOS/1024/
data TNONE/0/,T10us/1/,T30us/2/,T100us/3/,T300us/4/
data Tlms/5/,T3ms/6/,TlOms/7/,T30ms/8/,Tl00ms/9/
data T300ms/10/,Tls/ll/,T3s/12/Tl0s/13/,T3Os/14/
data Tl00s/15/,T300s/16/,T1000s/17/
data S/08/,LF/10/


78
for the critical point of a system when it is not previously
known. For example the starting pressure at 30 C for the
critical isochore could be approximately calculated from the
information in figure 5.13 in the following fashion. For each
isochore the two branches, supercritical vapor above the
bubble point and two phase equilibrium below it, were fit to
straight lines by linear regression. The angle 0 between the
two branches is directly related to the slopes of these lines.
A plot of initial charging pressure versus 0 indicates by
extrapolation the approximate critical density charging
pressure for which 0=0. The results for the data of figure
5.13 are tabulated in table 5.3, and the plot of charging
TEMPERATURE ( C)
Figure 5.13. A sequence of supercritical pressure versus
temperature isochores of the carbon dioxide-ethane
mixture.


148
manometer. Higher pressures require the use of pressure gauges
of the dead weight variety143.
This paper reports on the construction and use of a
simple dead weight gauge for absolute pressure measurements
and a variation of that gauge which works well but requires
calibration. Figure D.l illustrates the basic features of a
conventional dead weight pressure gauge. The pressure to be
measured is transmitted to the floating piston through a null
detector such as a mercury U-tube with adjustable electric
contacts, for example. The null detector is used to isolate
the system gases from the lubricating oil which surrounds the
piston. This oil is continually lost by viscous flow and must
Figure D.l. Conventional dead weight pressure gauge: A =
piston; B = cylinder; C = steel U tube; D = oil injector; N,
N' = indicator contact needles.


69
Table 5.2. (Continued)
Temperature
Pressure
Density
Author(s)*
K
atm.
psia.
Kg/m3
304.11
-
-
-
Kennedy (79)
304.19


Michael A.
Blaisse S.
Michael C. (80)
304.19
-
-
-
Lorentzen and Han
Ludvig (81)
304.19
72.80
1069.83
-
Tielsch H. (82)
304.16
75.20
1105.10
-
Ernst S. and
Thomas W. (83)
304.21
72.87
1070.86
-
Wentorf R.H. (84)
304.16



Ambrose D. (85)
0.03
304.15
0.01
75.282
1106.30
-
Ernst S. and
Traube K. (86)
304.2
72.83
1070.34
468
Matthews J.F.
(87)
304.150
72.79
1069.68
467.8
Moldover (88)
0.004
304.19
72.86
1070.78
468
Altunin V. (89)
304.13
-
-
-
Krynicki (90)
304.16
-
-
-
Lesnevskaya (91)
304.13
72.79
1069.62
467
Sengers J.V. (92)
304.20
-
-
-
Morrison G. (93)
304.13
-
-
467.4
Adamov (94)
304.13
72.79
1069.66
467.8
Shelomentsev (95)
304.206
72.87
1070.96
468.248
Morrison Kincaid
0.001
0.01
0.01
0.008
(96)
304.215
0.007
73.28
3.00
1076.88
3.00
-
This work
Numbers in parenthesis refer to references in bibliography.


39
the input transducer: producing an electric signal when
mechanically deformed by pressure waves.
Mixing control system
A circulating pump is needed to promote mixing of system
components and (or) phases and to assist with thermal
equilibration by eliminating stratification. It was made from
stainless steel and consists of two parts: the liquid
collector and the pump. Figure 4.5 shows the design of the
unit. It consists of a vertically mounted stainless steel
cylinder 3.18 cm (1.25 in.) in diameter with a freely sliding
internal piston 2.10 cm (0.851 in.) in diameter. The piston
has a built in reed valve which is normally open under gravity
but is forced closed by viscous drag as the piston descends
causing the fluid to flow from top to bottom through the pump
on descending motion of the piston. Piston motion is driven by
interaction between an iron slug attached to the piston and an
external magnet driven in turn by linkage to a motorized bell
crank.
Apparatus assembly
Tubular fittings to the spherical resonator to other
parts of the system were provided with 0.6 cm (h in.) diameter
pilot tubulations which slip into the polar holes and when
pressed against the tank were sealed by elastomeric O-rings
confined to glands comprised of circular grooves in the
fittings around the pilot tubes and the ring like areas of the
adjacent tank wall surrounding the holes. These fittings were


TABLE OF CONTENTS
ACKNOWLEDGEMENTS iii
LIST OF TABLES vi
LIST OF FIGURES viii
ABSTRACT xiii
CHAPTER 1 INTRODUCTION 1
CHAPTER 2 THEORETICAL BACKGROUND 6
Acoustics 6
Angular Part 7
Radial Part 8
Overall Solutions 11
Searching for Resonance Frequencies . 13
CHAPTER 3 APPLICATION OF SPEED OF SOUND TO LOCATE
CRITICAL POINTS 19
Phase Behavior of Fluids and Fluid
Mixtures 19
Determinations of Critical Point .... 24
Isotherm Approach 24
Isobar Approach 25
Isochore Approach 28
Speed of Sound as a Probe for the
Critical Point 30
CHAPTER 4 EXPERIMENT 32
Instruments 32
Resonance Frequency Measurements .... 51
CHAPTER 5 RESULTS AND DISCUSSIONS 60
Carbon Dioxide 60
A Mixture of Carbon Dioxide and Ethane 70
Ethane 90
Analysis of Ethane Data 105
Observation 1 106
Observation 2 107
Observation 3 107
Observation 4 112
Curve Fitting 112
IV


Table 5.4. (Continued)
FILE NAME
Starting point
Point of minimum frequency
O
o n
Pressure
psia
Frequency
Hz
T (C)
Frequency
Hz
Sonic
speed
m/sec
Pressure
psia
TDA1015(2)
30.029
1017.0
1459.64
17.174
1300.34
181.22
858.8
TDA1016,17
30.047
1006.7
1468.85
16.750
1313.48
183.04
850.1
05
Co


133
curve for ethane was made first using pure ethane. Both
calibration and sample analysis were performed on the same day
basis to minimize errors. Details of the equipment and
optimized conditions, which were determined by trial and error
methods, are summarized as follows:
Gas chromatograph: Varian model 3700
Syringe: Pressure-Lock gas syringe series A with side
port. Dynatech Precision Sampling Corporation.
Column: Fused silica megabore column GSQ, 0.53 mm i.d.,
30 m long, open tubular. J&W Scientific.
Optimized conditions:
Temperature(C) Injector: 100
- Flame Ionization Detector: 100
- Column : 80
Flow rate(ml/min) Helium : 6
- Hydrogen : 30
- Air : 300
Count rate on the integrator : 12,000 cpm.
Sample results of calibrations are summarized in table B.l.
Figure B.3 shows the calibration curve for ethane.


85
STARTING PRESSURE, PSIA
Figure 5.15. Relationship between the starting pressures and
the temperatures of phase change of the carbon dioxide-ethane
mixture.


152
Figure D.3 shows the arrangement of the gauge elements.
System pressure is delivered to the top surface of the
follower and transmitted through it to the top of the piston
which slides freely in the mating cylinder. The resultant
force is countered by the upward force of the loaded balance
beam. The travel of the piston is limited by a stop pin
extending into an oversize hole in the beam. Total vertical
displacement of the piston and angle between the beam and true
horizontal influence the net downward force. For this reason
the balance point is always taken as the median position, for
which the beam is horizontal and the relationship P=Ff/A
holds, where Ft is the upward force transmitted by the piston
to the follower. This gauge was built for the specific purpose
of measuring pressures and standardizing secondary gauges used
in an ongoing research effort on sonic speeds in gases at
pressures up to about 103 pascal (- 150 psi.) The stresses
generated on the gauge elements at these pressures are
relatively small so ordinary materials could be used. The
piston, cylinder and gauge body were machined from type 303
stainless steel. For use at high pressures these elements
should be made from heat treatable alloys and hardened in the
conventional manner. Many materials are available for use as
follower. A choice may be made principally on the basis of
compatibility with the fluids to be measured and flexibility.
Low permeability is desirable, of course, but since the
follower is supported by dense solid surfaces, loss of gas


149
Figure D.2. The present deadweight pressure gauge.


200
103. Gugnoni R. J., Eldridge J. W. Okay V. C., and T. J.
Lee, A. I. Ch. E. Journal.. 17, 357, 1974.
104. Nagahoma Kunio, Konishi Hitoshi, Hoshino Daisuke, and
Mitsuho Hirata, J. chem. Eng. (Japan), 7, 323, 1974.
105. Davalos Juan, Anderson Wayne R., Phelps Robert E., and
Arthur J. Kidnay, J. Chem. Eng. Data. 21, 81, 1976.
106. Stead K. and J. M. Williams, J. Chem. Thermodynamics.
12, 265, 1980.
107. Moldover M. R. and J. S. Gallagher, A. I. Ch. E.
Journal, 24, 2, 267, 1978.
108. Khazanova N. E. and E. E. Sominskaya, Russ. J. Phys.
Chem.. 51, 4, 548, 1977.
109. Douslin D. R. and R. H. Harrison, J. Chem.
Thermodynamics. 5, 491, 1973.
110. Sychev V. V., Vasserman A. A., Kozlov A. D.,
Zagoruchenko V. A., Spiridonov G. A., and V. A.
Tsymarny, Theodore B. Seloves Jr. English-Language
edition editor, Thermodynamics Properties of Ethane,
Hemisphere publishing corporation, Washington, 1987, p.
34.
111. Goodwin R. 0., Roder H. M., and G. C. Stragy,
Thermophysical Properties of Ethane from 90 to 600 K at
pressure to 700 bar. NBS Technical Note Number 684, U.
S., Department of Commerce, NBS 1976.
112. Wiebe R., Hubbard K. H., and M. J. Brevoort, J. Am.
Chem. Soc.. 52, 611, 1930.
113. Sage B. H., Webster D. C., and W. N. Lacey, Ind. Eng.
Chem.. 29, 658, 1937.
114.
Kav W. B., Ind. Enq. Chem..
30, 459, 1938.
115.
Beattie J. A., Su G.-J., and
Soc., 61, 924, 1939.
G. L. Simard, J.
Am.
Chem.
116.
Lu H., Newitt D. M., and M.
London 1. 178A. 506. 1941.
Rubemann, Proc.
Rov.
Soc.
117.
Kay W. B. and D. B. Brice,
1953.
Ind. Enq. Chem..
in
615,
118.
Whiteway S. G. and S. G. Mason, Can. J. Chem.
1953.
, 31,
569,


Figure 4.6. The Instrumental Setup.
co


CHAPTER 3
APPLICATION OF SPEED OF SOUND TO LOCATE CRITICAL POINTS
Phase Behavior of Fluids and Fluid Mixtures
The critical point of a fluid, whether it be a pure
substance or a mixture, is a property of considerable
practical as well as theoretical importance. This is because
the critical point identifies the temperature, pressure and
density at which the liquid and gaseous phases have identical
properties and is, therefore, a key point in the construction
of the phase diagram. Also, a knowledge of critical
temperature and critical pressure makes possible, through the
principle of corresponding states and an appropriate equation
of state, the prediction of the thermodynamic properties of
the compound when these properties have not yet been
experimentally determined. The need for information about
phase behavior has substantially increased in recent years.
For example, the petroleum industry has given much attention
to the phase behavior of fluid mixtures at high pressures to
support useful strategies to increase the yield of petroleum
from oil wells. To have a clear picture of the phase behavior
at high pressure and temperature of a fluid mixture, the
19


7
If the variables are separable (po may be written as #o =
f(r)g(0)h( (2.5)
where M2 and q2 are separation constants. The first of these
three equations gives the radial part and the other two the
angular part of the solution.
Angular Part
The last equation in equations (2.5) is a linear
homogeneous second-order differential equation. Its solutions
are h = sine(qq>) and cosine (qtp). In order that h be single
valued [ h (r, 0, (p+2rr) = h(r,0,tp)] values of q must be integers,
i.e. q = 0, 1, 2,.... Denoting this integer by m the second
of equations (2.5) becomes
(2.6)
This equation has finite solutions for 0=0 & n when
M2 = £(£ +1) ; £ =0,1,2,...
/
Then
(2.7)
The general solution of this equation is


66
Figure 5.6. Pressure and temperature behavior of carbon
dioxide for isochore near the critical density.


LIST OF FIGURES
Figure 2.1. The first six orders of the spherical Bessel
function. Points where slope equals zero yield
eigen values, n 12
Figure 3.1. Figure 3.1. The p-v-T behavior of pure
fluid. In the center is sketched the surface p =
p(v,T). [From Hirschfelder Joseph 0., Curtiss
Charles F., and Bird R. Byron. Molecular Theory of
Gases and Liquids. Copyright 1954 by John Wiley &
Sons, Inc. Reprinted by permission of John Wiley &
Sons, Inc.] 21
Figure 3.2 Pressure-Temperature-Mole fraction
relationship of system of carbon dioxide and
ethane 23
Figure 3.3. Pressure-temperature projection of the
system carbon dioxide-ethane in the critical
region. Lines ZftKft and ZK are vapor pressure curves
of pure carbon dioxide and of pure ethane
respectively 23
Figure 3.4. Schematic diagram of pressure and volume
relationship of carbon dioxide with several
isotherms in a broad region 25
Figure 3.5. Pressure and density relationship of carbon
dioxide with several isotherms around critical
point*47 26
Figure 3.6. Schematic diagram of graphical determination
of critical temperature of carbon dioxide 26
Figure 3.7. Schematic diagram of critical point
determination from measurements of the liquid
density pL and the vapor density p according to the
method of Cailletet and Mathias. 9. 27
Figure 3.8. Schematic diagram of isochoric method of
critical point determination 28
Figure 4.1. Observed sonic speed versus temperature for
a North Sea natural gas mixture.(Reference 53) .... 33
viii


non
187
Q***********************************************************
WRITE(*,400)TI,TF,VI,VF,M,F,T,JRMAX,JFMAX,RATIO
400 FORMAT(IX,2F8.3,2F8.1,16,F10.3,18,214,F9.3)
DT=TF-TI
DF=FLOAT(JRMAX-JFMAX)/FLOAT(M)*WIDTH
C USED ONLY AFTER CENTERING
C***********************************************************
F=F+DF*1.5
ENDIF
30 CONTINUE
IF(ABS(DT).GT.0.02) READS=READS+1
IF(ABS(DT).GT.0.03) READS=READS+1
IF(ABS(DT).GT.0.04) READS=READS+1
IF(ABS(DT).GT.0.0 5) READS=READS+1
IF(ABS(DT).GT.0.06) READS=READS+1
IF(ABS(DT).GT.0.07) READS=READS+1
IF(ABS(DT).GT.0.08) READS=READS+1
IF(ABS(DT).LT.0.05.AND.READS.GT.2) READS=READS-1
IF(ABS(DT).LT.0.04.AND.READS.GT.2) READS=READS-1
IF(ABS(DT).LT.0.03.AND.READS.GT.2) READS=READS-1
IF(ABS(DT).LT.0.02.AND.READS.GT.2) READS=READS-1
IF(ABS(DT).LT.0.015.AND.READS.GT.2) READS=READS-1
IF(ABS(DT).LT.0.01.AND.READS.GT.2) READS=READS-1
IF(ABS(DT).LT.0.0 0 5.AND.READS.GT.2) READS=READS-1
WRITE(*,380) VOLT
380 FORMAT (IX,'SET POINT VOLTAGE = ',F9.4)
c Call scan(F,IG)
OPEN(30,File='c:Peak.buf',STATUS='NEW')
WRITE(30,*) F,TF
CLOSE(30)
if (tf.lt.10.0) goto 169
IF(TF.GT.100.0) GOTO 169
IF(VOLT.GE.VOLTE) THEN
VOLT=VOLT+STP
OPEN(60,File='c:VOLT.BUF',STATUS='NEW')
WRITE(60,*) VOLT
CLOSE(60)
GOTO 100
ELSE
IF(VOLT.GT.VOLTE-0.4) THEN
VOLT=VOLTE-0.5
OPEN(60,File='c:VOLT.BUF',STATUS='NEW)
WRITE(60,*) VOLT
CLOSE(60)
GOTO 100
ENDIF
GOTO 101
ENDIF
169 OPEN(70,File='c:STOP.RUN',STATUS='NEW')
CLOSE(70)
CLOSE(20)


BIOGRAPHICAL SKETCH
Chadin Dejsupa was born in Bangkok, Thailand where he
attended primary and secondary school. He received a B.Sc.
degree in chemistry from Kasetsart University in 1981. He
worked as a laboratory instructor in general and physical
chemistry at kasetsart University for two years. In 1986 he
completed a M.S. degree in physical chemistry specializing in
X-ray crystallography from the University of Hawaii. He joined
the department of chemistry at the University of Florida in
1986 in order to pursue a Ph.D. There he studied under the
supervision of professor Samuel 0. Colgate in the physical
chemistry division. In 1989 he received the chemistry
department's outstanding teaching assistant award.
203


3
critical region, but progress toward overcoming this
shortcoming is being made on some fronts. <12'15) Thus far all
models developed to work in the critical region require
knowledge of the critical parameters; therefore the prediction
abilities of these models rely partly on the accuracy of these
parameters which in turn must be obtained experimentally. This
is typically done only for pure components, and the critical
behavior of mixtures is deduced using various combining
schemes. The chance for introducing uncertainty increases then
as the complexity of the mixture and is generally substantial
for most mixtures of importance to the gas industry. One
popular combining scheme uses Kay's mixing rules(16): Ppc=£Y.Pci
and Tpc=SYiTci where Ppc and Tpc are the so-called pseudocritical
pressure and temperature, respectively, for the mixture, and
Y.., Pci, Tci is the mole fraction, pressure, and temperature,
respectively, of component i in the mixture. Alternatively,
pseudocritical parameters may be obtained from a correlation
based on a collection of natural gas data(17,18). The values for
a specific mixture in general depend on the combining scheme
used to predict them.
The importance of accurate values of the critical point
parameters of both pure and mixed fluids is established. Good
values for many pure components are available, but very few
mixtures have received careful experimental study. The higher
the number of system components the fewer the number of
tabulated values are generally available. This scarcity is


38
Delrin FeedThru Guide
0-Ring
Brass FeedThru
Delrin Insulator
Spacer
Wave Spring
Spacer
Transducer
0Ring
figure 4.4. The transducer (cross-sectional view).


80
ANGLE, DEGREE
Figure 5.14. Relationship between the angles 0 and the
charging pressures of the carbon dioxide-ethane mixture.
release some of the material, and this process was potentially
troublesome because of the possibility of inducing unwanted
composition changes. Care was taken to release gas only from
the system in the single phase condition and then only after
it had been blended thoroughly by continued mixing. Typically,
when the system was close to the critical density, only small
amounts of gas were released for each successive run. The drop
in charging pressure at 30 C was typically one or two psia.
For this work the total number of runs was fifty.
Table 5.4 lists the experimental results needed to
construct phase diagrams of this system. Figures 5.15 to 5.19
show various types of phase diagram created from different
combinations of the parameters in table 5.4. Figure 5.15 shows
the relationship between starting pressure at 30 C and the


Table 4.1. (Continued)
Item
Manufacturer
Model
number
Range of
operation
Accuracy
(Manufacture
quote)
Function
Waveform
synthesizer
and function
generator
Hewlett-
Packard
3325B
1 pHz to 21
MHz for sine
wave with
amplitude of
1 mv to 10 v
P-P
5X10-6 of
selected
value in
temperature
range 20 to
30 C
Generates sine,
triangular, or
square wave.
Oscilloscope
Leader
1020
Input
signals
higher than
400 volts
(Acp-p + Dc)
may damage
circuit.
COMPONENTS
Vertical
section : 3
% (5% mag X
10. )
Horizontal
section : 3
%
Display waveform
Dynamic
signal
analyzer
Hewlett-
Packard
HP35660A
488 pHz to
102.4 KHz
for single
channel
spectrum or
half of the
this range
for two-
channel
spectrum.
No
information
available.
Wave signal
source. Also
contains FFT
algorithm to
convert an analog
input signal(time
domain) to a
signal displayed
in the frequency
domain.
U1
o


noon
188
Call scan(F,IG)
Call tempset(O.O)
STOP
END
c
c
c
Subroutine scan(F,IG)
CHARACTER*2 SWEEP
Character*3 name
Character*8 fname
Sweep='60'
c SWEEP='15'
OPEN(80,FILE='INDEX.BUF',STATUS='OLD')
READ(80,*) INDEX
CLOSE(80)
INDEX=INDEX+1
OPEN(80,FILE='INDEX.BUF',STATUS='NEW')
WRITE(80,*) INDEX
CLOSE(80)
WRITE(NAME,'(13)') INDEX
Write (fname,(Al, A3, A4)) 'F1,NAME,'.DAT'
Open (50,file=fname,status='new')
WIDTH = 100.0
IF (IG.LT.24) IG=IG+1
CALL GAINSET(IG)
WRITE(50,*) IG,F
c the following line is for avoiding too much signal
c at the end of the run.
f=f+200
SF = F-WIDTH/2.0
SPF = F+WIDTH/2.0
XMAX=SPF
XMIN=SF
YMAX=-1.0E32
YMIN=-YMAX
IXT=3
C IXT=1
NA=15/1XT
NP=2790/NA/IXT
SLOPE=WIDTH/FLOAT(NP)
CALL HP(SF,SPF,SWEEP)
IDELY = 14
DO 42 ID = 1,IDELY
CALL KEITH(MV)
42 CONTINUE
DO 40 J=1,NP
AVG=0.0
DO 41 JA=1,NA
300 CALL KEITH(MV)
IF(MV.GT.20000.0R.MV.LT.100) GOTO 300
AVG=AVG+FLOAT(MV)


xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EY1CZWG6B_Z76QPF INGEST_TIME 2013-10-24T23:10:48Z PACKAGE AA00017679_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES


161
thousand dollars and are generally beyond the means of many of
the laboratories which could benefit from their use. Perhaps
gauges based on the technology presented here will bridge the
present gap between conventional secondary gauges and high
cost absolute ones. They are at least working well in this
laboratory by increasing the confidence in the determination
of pressure sensitive acoustic and thermodynamic properties.


Table 5.10. (Continued)
Temperature
K
Pressure
Density
Author(s) *
Year
atm
psia
kg/m3
305.35
48.112
707.04
204.3
Miniovich and Sorina
(128)
1973
305.3627
-
-
205
Shmakov (129)
1973
305.330
48.0805
706.589
206.581
Douslin and Harrison
(109)
1973
305.368
(0.005)
-
-
205.5
Strumpf et. al. (130)
1974
305.37
48.117
707.30
202.7
Goodwin (131)
1975
305.340
48.076
706.50
206.9
Moldover and Gallagher
(107)
1978
305.360
-
-
-
Berestov et. al. (132)
1979
305.379
-
-
-
Morrison (93)
1981
305.393
0.001
47.16
0.01
692.99
0.01
205.860
0.005
Morrison and Kincaid
(96)
1984
305.370
0.007
48.16
3.00
707.78
3.00
-
This work
1990
Numbers in parenthesis refer to references in bibliography.


92
TEMPERATURE, CELSIUS
Figure 5.23. Pressure and temperature relationship of Is mode
resonance frequency of ethane near the critical density.
TEMPERATURE, CELSIUS
Figure 5.24. Speed of sound and temperature of Is mode
resonance frequency of ethane near the critical density.


155
where g = local acceleration of gravity,
m. = added load mass
2.
n,. = position of load mass and
0 = the angle between the top of the beam and the
horizontal.
In order to use the simpler relationship
Ff = g Z n.m. (D.2)
it is necessary to adjust the position of the gauge body
relative to the plane of the bearing pad so that the balance
point occurs when the beam is precisely horizontal (i.e. cos
0 = 1.) To accomplish this the gauge body is mounted in a
grooved way and adjusted vertically by a fine lead screw. A
precision spacer is positioned between the lower end of the
cylinder and the flange on the piston during the adjustments.
This spacer assures that the piston can extend into the
cylinder only to the median position. The beam is loaded
sufficiently to hold the piston against the spacer and the
gauge body adjusted vertically by means of the lead screw so
that a precision level shows the beam to be horizontal.
Following the adjustments the gauge body is clamped securely
in place.
While the piston is held in the median position by use of
the spacer, the balance scale and indicator are set to "zero".
Although a conventional scale and pointer combination can be
used with this apparatus, greater sensitivity is easily
achieved by using an optical indicator consisting of a He-Ne


115
Temperature and speed of sound at the intersection of these
two smooth curves were treated as turning-point parameters of
the corresponding experimental data set. Table 5.11 shows
results of searching coefficients of experimental data shown
in figure 5.31 by the simplex optimization program. Figure
5.32 compares the experimental data and fitting curve
generated by this method for ethane at the critical density.
Note that in this figure only one-forth of the experimental
points were plotted to preserve picture clarity. Clearly, this
equation represents very well the behavior of the speed of
sound in the vicinity of the critical point. Furthermore, it
also yields an equally satisfactory result for the mixture of
carbon dioxide and ethane described in the previous section.
Table 5.11. Coefficients Obtained by Simplex Optimization
Method.
Coefficients
High Temperature
Side
Low Temperature
Side
A,
11.95399178
4.4228669
A,
-897.98691534
360.1419335
A,
-3400.8303361
2448.9833655
A<
2976.6473117
-1666.3105786
A,
7325.7816766
-7444.4800181
\
-56033.5889588
52720.3509156
T
_ __ e __
305.3979294
305.9057717
a
0.1250001
0.1250000
P
0.3333333
0.3333333
A
0.5000000
0.5000000


ACKNOWLEDGEMENTS
The author expresses his sincere gratitude to his
advisor, professor Samuel 0. Colgate for his indispensable
intellectual and moral support, and especially when
bottlenecks were encountered. It has been privilege working
with such an outstanding teacher and chemist. His dedication
to excellence and clarity of thought and lucid insights will
perpetually serve as an example.
This work would also have been impossible without the
expertise of Drs. A. Sivaraman, Kenneth C. McGill, and V. Evan
House of the acoustic research group which made colossal
barriers become insignificant. Thanks also to Joseph Shalosky
for his marvelous machining dexterity. Special thanks are
extended to professors David Micha and Martin Vala and to Paul
Campbell for their valuable moral support and to all of peers
of the author in this institution, particularly Casey Rentz
and Michael Clay for their truly meaningful friendships.
Last but not least the author wishes to warmly thank all
of his teachers who shared their knowledge and understanding
with him.
in


Table 5.7. Summary of results for ethane measurements.
File
Number
Conditions at start
of run (T = 37 C)
Sonic
speed at
phase
boundary
(m/sec.)
Conditions at minimum resonance frequency
Frequency
(Hz)
Pressure
(psia)
T (C)
P
(psia)
Frequency
(Hz)
Sonic
speed
(m/sec)
Amp.
(mv)
919
1616.2
800.5
179.89
31.568
697.65
1287.0
179.37
254
920
1563.8
796.0
172.19
31.799
700.83
1227.3
171.05
4958
924
1500.9
791.4
162.40
31.991
703.71
1154.6
160.92
1799
9284
1485.0
789.8
156.15
32.011
704.30
1119.6
155.69
1891
1002
1477.5
789.1
154.15
32.043
704.70
1103.1
153.74
718
1007
1458.9
787.5
147.93
32.090
705.40
1067.2
148.73
862
1008
1449.6
786.4
143.99
32.116
706.09
1036.2
144.41
741
1010
1440.5
785.2
141.75
32.149
706.39
1020.7
142.25
506
1012
1436.1
784.9
139.19
32.151
706.59
1004.6
140.01
274
10141
1436.6
784.0
139.01
32.163
706.79
998.1
139.10
279
vo
oo


Table 5.4. (Continued)
FILE NAME
Starting point
Point of minimum frequency
T
(C)
Pressure
psia
Frequency
Hz
i-3
o
n
Frequency
Hz
Sonic
speed
m/sec
Pressure
psia
TDA929
30.020
1098.6
1416.55
19.835
1087.73
151.61
913.5
TDA1003
30.018
1095.7
1416.57
19.800
1105.21
154.04
913.2
TDA1004
30.006
1090.1
1415.82
19.737
1132.31
157.86
911.3
TDA1006
30.015
1083.6
1417.73
19.596
1158.91
161.58
909.2
TDA1007
30.005
1077.6
1419.66
19.476
1182.36
164.80
906.3
TDA1009
30.007
1066.1
1425.68
19.181
1217.97
169.81
899.8
TDA1010
30.001
1057.3
1431.13
18.899
1239.56
172.76
893.8
TDA1011
29.997
1047.7
1437.77
18.532
1258.82
175.44
886.3
TDA1012
30.000
1037.9
1444.67
18.043
1274.82
177.67
876.8
TDA1013,14
29.992
1027.6
1452.33
17.800
1289.80
179.75
870.7
TDA1015(1)
30.029
1017.0
1459.64
17.200
1300.55
181.24
859.2


SYSTEM CONTROLLER
Figure 4.7. The experimental setup for C02 and C02-C2H6 mixture


198
66. De Heen, Bull. Acad. Roy. Bilfique, 31, 147, 379, 1896.
67. Von Wesendonck, Verhandel Deut. Physik. Ges.. 5, 238,
1903.
68. Keesom W. H., Verhandel Akad. Amst.. 321, 533, 616,
1903; Comm. Leiden. No. 88, 1903.
69. Quinn Elton L. and Jones Charles L., Carbon dioxide,
Reinhold Publishing Corporation, New York, 1936, p. 59.
70. Kammerlingh-Onnes H. and G. H. Fabius, Verhandel Akad.
Amst., 44, 1907; Comm. Leiden, No. 98, 1907; Proc. kon.
Akad. Amst.. Vol. 10, 215, 1907.
71. Bradley W. P., Brown A. W., and C. F. Hale Phvs. Rev. .
26, 470, 1908.
72. Meyers C. H. and M. S. Van Dusen, Bur. Standards J.
Research. 10, 381, 1933.
73. Cardoso E. and R. Bell, J. Chim. Phvs.. 10, 500, 1912.
74. Hein Paul, Z. Phvsik. Chem.. 86, 385, 1913-1914.
75. Dieterici C., Ann. Phvsik.. 62, 75-94, 1920.
76. Pickering S. F., J. Phys. Chem.. 28, 97-124, 1924.
77. Meyers C. H. and M. S. Van Dusen, Refrigerating Eng..
13, 180, 1926.
78. Kennedy H. T., Cyril H., and Meyer S., Bur. Standards.
Refrigerating Eng.. 15, 125-30, 1928.
79. Kennedy H. T., J. Am. Chem. Soc.. 51, 1360, 1929.
80. Michael A., Blaisse S., and Michael C., Proc. Roy. Soc.
I London). 160A, 358, 1937.
81. Lorentzen and Han Ludvig, Acta Chem. Scand.. 7, 1335-
1346, 1953.
82. Tielsch H., and Tanneverger H., Z. Phvsik.. 137, 256-
264, 1954.
83. Ernst S., and Thomas W., Forsch. Gebiete Ingenierw..
2OB, 161-170, 1954.
84. Wentorf R. H., J. Chem. Phvs.. 24, 607-615, 1956.
85. Ambrose D., Trans. Faraday Soc.. 52, 772-781, 1956.


116
Three Dimensional Phase Diagram
Conventionally the phase diagram in three dimensions is
represented by using parameters of temperature, pressure, and
molar volume leading to three well-known two dimensional
projections: isotherm, isobar, and isochore curves. Figure 3.1
illustrates an example of this type of plot for a typical pure
fluid. The critical point is located at the top of a
coexistence curve. It can be found by a null change of slope
in the critical isotherm or isobar or by a smooth curve
without break in the critical isochore. These are among the
traditional indicators, but their response to the critical
point is not highly sensitive.
For more accurate measurements a more sensitive indicator
is needed. The sonic speed or cavity resonance frequency
serves nicely for this purpose due to their dramatic change at
the critical point. Replacing molar volume by speed of sound
in the conventional phase diagram plot leads to a pyramid
shape curve as shown in figure 5.33 for a mixture of carbon
dioxide and ethane. Exploring in a much narrower range of
temperature around the critical point leads to an outstanding
tornado shape curve as shown in figure 5.34 for ethane. These
two curves were constructed from experimental data. The
critical point is apparently easily located as an extrumum
point on these surfaces. It corresponds to the point of
minimum speed of sound, which theoretically speaking, should


134
Table B.l. Data for calibration curve of ethane.
No.of
repetitions
Attenuation
factor
No.of
counts
No.of
adjusted
counts
Volume of
gas
injected,
5
32
41452
132
5.0
5
256
1929
482
10.0
6
256
3997
1002
15.0
7
1024
1315
1315
20.0
6
1024
1882
1882
25.0
5
1024
252 + 4
252 + 4
30.0
6
1024
3102
3102
35.0
6
1024
361 2
3672
40.0
6
1024
4232
423 + 2
45.0
6
1024
4832
4832
50.0


184
OPEN(60,File='c:DATA.BUF',STATUS='OLD')
READ(60,*) VOLT,IG
CALL GAINSET(IG)
CLOSE(60)
VOLTE = (10.O-TZERO)*0.01 + VOLT
write (*,*) volte
CLOSE(10)
DTEMP='c:DATA.'
OPEN(70,File='c:NAME.NUM',STATUS='OLD')
READ(70,301) DNAME
301 FORMAT(IX,A10)
CLOSE (70)
READ(DNAME,' (7X,13) ) NUM
NUM=NUM+1
OPEN(70,File=c:NAME.NUM',STATUS='NEW')
WRITE(70,302) DTEMP,NUM
WRITE(*,302) DTEMP,NUM
302 FORMAT(IX,A7,13)
CLOSE (70)
OPEN(20,FILE=DNAME,STATUS='NEW')
OPEN(30,File='c:Peak.buf',STATUS='OLD')
READ(30,*) F,TI
TI=TI+273.15
CALL KEITHA(TF)
TF=TF+273.15
F=SQRT(TF/TI)*F
CLOSE(30)
IXT=3
c IXT=1
NA = 21/IXT
IDELY=2*NA
q* *********************** -k -k ****************************** *
c M=1000/NA/IXT-IDELY/NA
C USED ONLY FOR CENTERING
q**********************************************************
M=2 9
C USED AFTER CENTERING
q**********************************************************
XMAX=FLOAT(M)
1001 CALL TIME(10,TSTR)
READ(TSTR,*(12)*,ERR=1001) HR
READ(TSTR,'(3X,I2)',ERR=1001) MIN
READ(TSTR,'(6X,I2)',ERR=1001) SEC
TOO = HR*3600+MIN*60+SEC
100 CALL TEMPSET(VOLT)
101 DO 30 1=1,READS
500 SF=F-WIDTH/2.0
SPF=F+WIDTH/2.0
INDEX=1
POINTS=M
ISTART=1
200 CALL HP(SF,SPF,SWEEP)


40
MAGNET ASSEMBLY
SPRING
STAINLESS STEEL
CYLINDER
FREELY SLIDING
IRON PISTON
TO RESONATOR TOP
FLUID FLOW
(PISTON MOVING UPWARD)
SHIM VALVE
TO RESONATOR BOTTOM
Figure 4.5. The mixing control unit.


201
119. Palmer H. B., J. Chem. Phvs.. 22, 625, 1954.
120. Schmidt E. and W. Thomas, Forsch. Gebiete Inq., 20B,
161, 1954.
121. Kudchadker A. P., Alani G. H., and B. J. Zwolinski,
Chem. Rev.. 68, 659, 1968.
122. Sliwinski P., Z. Phys. Chem. (Frankfurt). 68, 91, 1969.
123. Bulavin P. A., Ostanevich Yu. M. Simkina A. P., and A.
V. Strelkov, Ukr. Fiz. Zh., 16, 90, 1971.
124. Minivich V. M. and G. Sorina, Russ. J. Phvs. Chem., 45,
306, 1971.
125. Khazanova N. E. and E. E. Sominskaya, Russ. J. Phvs.
Chem., 45, 88, 1971.
126. Veronel' A. I., Gorbunova B. G., and V. A. Smirnov et.
al., Zh. Eksp. Teor. Fiz.. 63, 964, 1972.
127. Berestov A. G., Giterman M. Sh., and N. G. Shmakov, Zh.
Eksp. Tekhn. Fiz., 64, 2232, 1973.
128. Miniovich V. M. and G. A. Sorina, Teplofiz. Svoista
Veshchestv. Mater.. 6, 134, 1973.
129. Shmakov N. G. Teplofiz. Svoistva Veshchestv Mater. 7,
155, 1973.
130. Strumpf H. J., Collings, A. F., and C. J. Pings, J.
Chem. Phvs., 60, 309, 1974.
131. Goodwin R. D., J. Res. NBS, 79A, 71, 1975.
132. Berestov T. A. and S. B. Kiselev, Teplofiz. Vvs. Temp.,
17, 1202, 1979.
133. Stell G. R. and S. Torquoto, Ind. Enq. Chem. Fundam..
21, 202, 1982.
134. Sivaraman A., Magee J. W. and R. Kobayashi, Fluid
Phase Equilibria. 16, 1, 1984.
135. Sivaraman A., Kragas T., and R. Kobayashi, Fluid Phase
Equilibria. 16, 275, 1984.
136. Wegner F. J., Phvs. Rev. B. 5, 4529, 1972.
137. Ley-Koo M., and M. S. Green, Phvs. Rev. A. 16, 2483,
1977.


APPENDIX B
QUANTITATIVE ANALYSIS OF GAS MIXTURE
Reported values of critical point parameters for gas
mixtures are meaningless unless accompanied by their
quantitative compositions. For this particular mixture of
carbon dioxide and ethane two methods were used for
quantitative analysis: a gravimetric method for carbon dioxide
and a gas chromatography method for ethane. Carbon dioxide was
not analyzed by the latter method due to an unavailability of
a suitable detector during the course of analysis. The
assumption used through out this analysis was that there were
no significant impurities since the mixture was carefully
prepared in this laboratory from highly pure gases.
Gravimetric Method
Carbon dioxide reacts with base to form carbonate salt
which readily dissolves in water as shown in the following
chemical equation:
C02 + 2 OH' = C032- + H20
The difference between weights of reaction vessels before and
after carefully passing carbon dioxide is the weight of the
gas. The weight of ethane was calculated by subtracting the
measured weight of carbon dioxide from the weight of the
sample.
128


Table 5.7. (Continued)
File
Number
Conditions at start
of run (T = 37 C)
Sonic
speed at
phase
boundary
(m/sec.)
Conditions at minimum resonance frequency
Frequency
(Hz)
Pressure
(psia)
T (C)
P
(psia)
Frequency
(Hz)
Sonic
speed
(m/sec)
Amp.
(mv)
1110
1434.2
761.2
N/A
31.682
698.54
1296.2
180.63
1079
1111
1430.0
758.7
N/A
31.475
696.16
1314.3
183.16
1124
The sonic speed at the phase boundary was determined by a Simplex routine from the
experimental data. See curve fitting section.
o
o


42
fluid circulation loop by a tee leads to a diaphragm-type
pressure gauge and a system shut-off valve.
The resonator assembly and circulating pump were
suspended from a horizontal plate which forms the cover of a
stirred liquid bath in which the system is mounted. The bath
was housed in a well-insulated container and connected by
insulated tubing to a computer-controlled heat exchanger
capable of operating from 233 to 373 K. Bath temperature was
measured with a four-wire platinum resistance thermometer
(RTD.) Figure 4.6 shows the instrumental setup.
Electronic hardware
The automated systems employed in this work are shown in
figures 4.7, 4.8, and 4.9. Figure 4.7 shows the setup used in
the experiments on pure carbon dioxide and on a mixture of
carbon dioxide and ethane. Figure 4.8 shows the setup for the
experiments on pure ethane. In this latter setup we have
explored the capability of using a fast-fourier transform
(FFT) technique. Unfortunately, the data collected by this
technique was unreliable due to leaks in the system found
after the experiment was finished. While the leaks were being
fixed, the instruments were transferred to another project.
Consequently, the FFT exploration was discontinued for ethane
and the setup shown in figure 4.9 was built to finish this
project.


o o o
167
C IF(ABS(DT).GT.0.04) READS=READS+1
C IF(ABS(DT).GT.0.05) READS=READS+1
C IF(ABS(DT).GT.O.O 6) READS=READS+1
C IF(ABS(DT).GT.0.07) READS=READS+1
C IF(ABS(DT).GT.0.08) READS=READS+1
C IF(ABS(DT).LT.0.05.AND.READS.GT.2) READS=READS-1
C IF(ABS(DT).LT.0.04.AND.READS.GT.2) READS=READS-1
C IF(ABS(DT).LT.0.03.AND.READS.GT.2) READS=READS-1
C IF(ABS(DT).LT.0.02.AND.READS.GT.2) READS=READS-1
C IF(ABS(DT).LT.0.015.AND.READS.GT.2) READS=READS-1
C IF(ABS(DT).LT.0.01.AND.READS.GT.2) READS=READS-1
C IF(ABS(DT).LT.0.005.AND.READS.GT.2) READS=READS-1
WRITE(*,380) VOLT
380 FORMAT (IX,'SET POINT VOLTAGE = ',F9.4)
OPEN(30,File='c:Peak.buf',STATUS='NEW')
WRITE(30,*) F,TF
CLOSE(30)
IF(TF.GT.100.0) GOTO 169
IF(VOLT.LE.VOLTE) THEN
VOLT=VOLT+STP
OPEN(60,File='c:VOLT.BUF',STATUS='NEW')
WRITE(60,*) VOLT
CLOSE(60)
GOTO 100
q* **************** * * ******** **************** * * * ***** *
C ELSE
C IF(VOLT.GT.VOLTE-0.4) THEN
C VOLT=VOLTE-0.5
C OPEN(60,File='c:VOLT.BUF',STATUS='NEW')
C WRITE(60,*) VOLT
C CLOSE(60)
C GOTO 100
C ENDIF
C GOTO 101
C USED IN NORMAL SCAN
Q*************************************************** **********
ENDIF
169 OPEN(70,File='c:STOP.RUN',STATUS='NEW')
CLOSE(70)
CALL TEMPSET(0.0)
STOP
END
SUBROUTINE KEITHA(T)
common /ibglob/ ibsta,iberr,ibcnt
'COMMON GROUP 1.
integer*4 cmd(10),rd(512),wrt(512)
character*8 bname,bdname,TEMP
COMMON GROUP 2.'
character*50 flname


165
ENDIF
IF(KOUNT.GT.3*NA) THEN
KOUNT=0
GOTO 500
ENDIF
IF(MV.GT.20000.0R.MV.LT.100) GOTO 300
KOUNT=0
AVG=AVG+FLOAT(MV)
41 CONTINUE
AMP=AVG/FLOAT(NA)
DX(J)=FLOAT(J)
DY(J)=AMP
IF(AMP.GT.AMAX) THEN
AMAX=AMP
JMAX=J
ENDIF
IF(AMP.LT.XMIN) XMIN=AMP
40 CONTINUE
IF(INDEX.EQ.1) THEN
NPLOT=l
ELSE
NPLOT=0
ENDIF
YMAX=AMAX
C CALL PLOT(M,DX,DY,XMAX,XMIN,YMAX,YMIN,NPLOT)
C ONLY USED FOR CENTERING
Q*******************************************************
IF(INDEX.EQ.1) THEN
INDEX=-INDEX
TEMP=SF
SF=SPF
SPF=TEMP
ISTART=POINTS
POINTS=l
JFMAX=JMAX
AVGA=AMAX
CALL KEITHA(TI)
IF(TI.LT.25.0) GOTO 169
CALL PRES(VI)
GOTO 200
ELSE
CALL KEITHA(TF)
CALL PRES(VF)
JRMAX=JMAX
FO=FLOAT(JFMAX+JRMAX)/2.0
CALL MAX(F,WIDTH,FO,M)
F=FO
AMP=(AMAX+AVGA)/2.0
IF(JFMAX.EQ.1.OR.JFMAX.EQ.M.OR.JRMAX.EQ.1.OR.
CJRMAX.EQ.M) THEN
OPEN(30,File='c:Peak.buf',STATUS='OLD')


96
the critical density reaches the lowest sonic speed.


Table 5.1. Comparison of Dewpoint Pressures of Carbon Dioxide.
No.
T (K)
Dewpoint pressure, psia
Percent difference
Present
sonic
method
Equation
of state
(I)
Burnett
method13
(II)
NIST
Model0
(HI)
I
II
III
1
299.905
971.46
971.045
971.10
971.53
+0.0427
+0.037
-0.0072
2
301.012
995.60
995.418
996.56
996.41
+0.0183
-0.096
-0.081
3
301.915
1015.30
1015.632
1017.10
1017.16
-0.0327
-0.177
-0.183
4
303.046
1041.74
1041.379
1044.23
1043.75
+0.0347
-0.239
-0.192
a Reference 55.
b Reference 57.
Reference 56.


70
Encouraged by the apparent success of this approach to
the location of phase boundaries, including critical points,
for a pure substance, we opted to proceed to the next stage of
the study and perform similar tests an a simple fluid mixture.
The properties of mixtures are not so well documented as those
of carbon dioxide, and there are few previously well
characterized binary systems to choose from. Carbon dioxide-
ethane is one of these, and a mixture of these two was
selected for this study. The results obtained are presented in
the next section.
A Mixture of Carbon Dioxide and Ethane
Since an acoustic resonance technique has been
successfully used to detect phase changes along the
coexistence curve of a single component system, it is a
naturally logical forward step to investigate its suitability
for use with more complex systems. What is needed are specific
mixtures with well characterized phase behavior. A binary
mixture of 74.25 mole percent carbon dioxide and ethane was
selected for this purpose. The carbon dioxide-ethane system
has been studied by several researchers beginning in 1897. The
first critical measurements were carried out by Kuenen in
1897<97) and 1902<98) followed by Khazanova and Lesnevsicaya in
1967(99), Fredenslund and Mollerup in 19 7 4(100) and Morrison and
Kincaid in 1984(96). Phase boundary studies have been performed
by Clark and Din in 1953<101), Jensen and Kurata in 1971(102>,


LIST OF TABLES
Table 2.1. The values of the roots £ n in ascending order
of magnitude 18
Table 4.1. Summary of electronic hardware 48
Table 4.2. Gas Specifications 51
Table 4.3. Charging Pressure of gas systems 51
Table 5.1. Comparison of Dewpoint Pressures of Carbon
Dioxide 67
Table 5.2. Values of critical point parameters from the
literature 68
Table 5.3. Angles between pressure-temperature lines
before and after phase boundary of supercritical
density fluid of carbon dioxide and ethane. ... 79
Table 5.4. Experimental results for construction of
phase diagrams of a mixture of carbon dioxide and
ethane 81
Table 5.5. Comparison of Dewpoint Pressures of C02 + C2H6
Mixture (Xco2 = 0.7425) 89
Table 5.6. Sixth order polynomial fit statistics of
GRAPHER program applied to experimental data points
in figure 5.25 93
Table 5.7. Summary of results for ethane measurements. 98
Table 5.8. Comparison of Bubble-Point Pressures of
Ethane 101
Table 5.9. Comparison of experimental vapor pressures
and those calculated with an equation of state of
ethane proposed by Sychev.(110) 102
Table 5.10. Chronological collection of critical point
parameters of ethane 103
vi


15
",n
if |(Cl
- ^iin : + Afs + Afel + Aft + Af .
Tna. 8 el 1 geom
(2.30)
The first correction term is due to the thermal boundary layer
effect at the wall of the resonator. The second correction
term is due to the finite elastic compliance of the sphere
which is not infinitely rigid. The third term corrects for
effects of the gas entrance and gas exit tubes which
contribute departures from sphericity. The last term corrects
for imperfection of the cavity geometry. For absolute
measurements the exact treatment could be computationally and
experimentally demanding.
These problems, however, can be greatly reduced by
utilizing a relative measurement technigue as follows:
equation (2.30) may be written as
l n
K,n)(C)
Tna
(2.31)
where vL n is an effective eigenvalue which depends on the
unperturbed eigenvalue and all perturbations for the
particular mode (l,n). If it is assumed to be independent of
gas properties, then
v
e.n
(2na) (fJfB)
C
(2.32)
Consequently, if we choose some system whose speed of sound is
known or can be calculated precisely from a reliable equation
of state, we will get a relationship:
or


172
common /ibglob/ ibsta,iberr,ibcnt
'COMMON GROUP 1.'
integer*4 cmd(10),rd(512),wrt(512)
character*8 bname,bdname,TEMP
1 COMMON GROUP 2.'
kl = IBFIND ('kl77 ')
i24 = 2**24
il6 = 2**16
i8 = 2**8
1006 call ibrd (KL,RD,6)
11 = rd(l)/i24
12 = (rd(1)-il*i24)/il6
13 = (rd(l)-il*i24-i2*il6)/i8
14 = rd(1)-l*i24-i2*il6-i3*i8
15 = rd(2)/i24
16 = (rd(2)-i5*24)/il6
17 = (rd(2)-i5*i24-i6*il6)/i8
i88 = rd(2)-5*i24-i6*il6-i7*i8
WRITE(TEMP,'(8A1)') CHAR(14),CHAR(13),CHAR(12),
CCHAR(Il),CHAR(188),CHAR(17),CHAR(16),CHAR(15)
READ(TEMP,'(16),ERR=1006) MV
RETURN
end
c
c
c
SUBROUTINE PLOT(N,DX,DY,XMAX,XMIN,YMAX,YMIN,NPLOT)
REAL*4 DX(1),DY(1)
REAL*4 XMAX,XMIN,YMAX,YMIN
INTEGER*2 SYMBOL,PEN,M,DEVTYPE,ADDR
INTEGER*2 XAXIS,YAXIS
LOGICAL*2 LIN
CHARACTER*40 XLBL,YLBL,TITLE
C* ***************************************************** *
C GRAPHING PERAMETERS
C eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
M=16
C eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
C 1 IS FOR SCREEN 2 IS FOR PLOTTER
DEVTYPE=1
INTERFACE=#C400
ADDR=5
C****************************** **********************
XAXIS=1
YAXIS=0
YLBL=' AMP¡
XLBL=' INDEX VARIABLE¡'
TITLE= AMP VS INDEX¡'
LIN='0'
SYMBOL=0
PEN=10
LEFT=0.0


157
pressure was determined using a mercury barometer noting the
usual corrections for thermal expansions of mercury and brass.
The measured pressure was calibrated from the relationship
P = Patm + (g Z n.m.)/A (D.3)
where A = nr2 = cross sectional area of the piston
The local acceleration of gravity must be known or measured.
In this laboratory it is 979.284 cm/sec.2 and the resulting
gauge constant is 0.0200584 psig/g.
Table D.l gives the results of these tests at two
temperatures, along with the corresponding vapor pressures
which may be inferred from the precise measurements of Gilkey
et. al.(144' 145) The close agreement shows that the present gauge
works well. These results are not intended to serve as a
calibration of the gauge but rather as a check of its
performance.
The freon used was manufactured for refrigeration service
and was of unknown purity. The consistency of the gauge
readings indicates that the absolute pressure gauge is
accurate to within the limits of the smallest mass difference
which may be reproducibility discerned. Using a 30 pm polyimid
film follower (DuPont Kapton) the loading mass could be
measured easily to within 0.5 g, which corresponds to the
limits of uncertainty stated in the table. The gauge is a
little more responsive if a 15 pm latex follower is used (ca.
0.1 g) but the lubricating oils used thus far permeate or
react with the latex causing operational difficulties. Thin


Figure 4.8. The first experimental setup for C2H6.


25
Isobar Approach
This method is sometimes called the rectilinear diameter
method of Cailletet and Mathias.<48) If p: and pg are the
densities of liquid and of saturated vapor in equilibrium with
it (so called orthobaric densities), their mean is a linear
function of temperature.
(3.2)
where po = mean density of liquid and its saturated vapor at
0 C
t = temperature in Celsius
a = constant.
i
\ T
V
Volume
Figure 3.4. Schematic diagram of pressure and volume
relationship of carbon dioxide with several isotherms in
a broad region.


CHAPTER 2
THEORETICAL BACKGROUND
Acoustics
This derivation follows that given by references 28, 29,
30, and 31. The propagation of pressure waves in a lossless
fluid at rest satisfy the general wave equation*28
V2# JL§l. (2.1)
c2 at2
where V2 = Laplacian operator, $ is the velocity potential and
C is the wave velocity (speed of sound).
Assuming that (p = or
V +

C2
(2.2)
V2^ + k2

(2.3)
where k = o/C = wave number. If we consider only the time-
independent wave, equation (2.3) becomes
V2^ + k20o 0 (2.4)
6


197
49. Ipatieff V. N., and G. S. Monroe, Ind. Eng. Chem.
(Anal. Ed.). 14, 171, 1942.
50. Bagatskii M. I., Voronel' A. V. and V. G. Guak, J.
Exptl. Theoret. Phvs. (U.S.S.R.), 43, 728, 1962.
English translation: Soviet Phvs. JETP. 16, 517, 1963.
51. Voronel' A. V., Chashkin Yu R., Popov V. A., and V. G.
Simkin, J. Exptl. Theoret. Phvs. (U.S.S.R.), 45, 828,
1963.
52. Colgate S. O., Sivaraman A., Dejsupa C., and K. C.
McGill, Rev. Sci. Instrum.. 62, 1, 198, 1990.
53. Colgate S. 0., Sivaraman A., Dejsupa C., and K. C.
McGill, J. Chem. Thermodynamics. 23, 23, 1991.
54. Piezoelectric Technology Data for Designers;
Vernitron: Bedford, Ohio, p. 14.
55. Huang Feng-Hsin, Li Meng-Hui, Lee Lloyd L., and Kenneth
E. Starling, J. Chem. Eng. (Japan). 18, 6, 490, 1985.
56. Ely J., NBS Reference Database number 14, "DDMIX",
National Institute for Science and Technology,
Supercritical Fluid Properties Consortion, Boulder,
Colorado, May 1988..
57. Holste J. C., Hall K. R. Eubank P. T., Esper G.,
Watson M. Q., Warowny W., Bailey D. M., Young J. G. ,
and M. T. Bellomy, J. Chem. Thermodynamics. 19, 1233,
1987.
58. Andrews J. W., Trans. Roy. Soc. (London). 159, II, 575,
1869.
59. Hautefeuille L. and L. P. Cailletet, Compt. Rend.. 92,
840, 1881.
60. Dewar J., Phil. Mag.. 18, 210, 1884.
61. Amagat E. H., Compt. Rend.. 114, 1093, 1332, 1892.
62. Chappuis J., Compt. Rend.. 118, 976, 1894.
63. Villard P., J. Phvs.. 3, 441, 1894.
64. Verschaffelt J. E., Verhandel Akad. Wetenschappen
Amsterdam. 94, 1896; Comm. Leiden. No. 28.
65. Kuenen J. P., Phil. Mag.. 5, 44, 179, 1897.


FREQUENCY. HZ 10
Figure 4.11. Two experimental approaches of resonance frequency determination: the
maximum amplitude approach and the voltage phase change approach.
cn
00


non non non
178
C END IF
CH(1) = 'Q'
WRT(1)=ICHAR(CH(1))
CALL IBWRT(LK,WRT,1)
1005 CALL IBRD (LK,RD,12)
1(4) = RD(1)/12 4
1(3) = (RD(1)1(4)*124)/I166
1(2) = (RD(1)-I(4)*I24-I(3)*I166)/I88
1(1) = RD(1)-I(4)*I24-I(3)*I166-I(2)*I88
1(8) = RD(2)/I24
1(7) = (RD(2)-I(8)*I24)/I166
1(6) = (RD(2)1(8)*1241(7)*1166)/I88
1(5) = RD(2)-I(8)*I24-I(7)*I166-I(6)*I88
1(12) = RD(3)/I24
1(11) = (RD(3)-1(12)*12 4)/116 6
1(10) = (RD(3)-I(12)*l24-I(ll)*I166)/I88
1(9) = RD(3)-I(12)*I24-I(11)*I166-I(10)*I88
DO 20 K=1,12
IF (I(K).EQ.CR) THEN
KK=K-1
GOTO 40
END IF
20 CONTINUE
40 IF (KK.EQ.10) THEN
WRITE(TEMPO,'(10A1)') CHAR(I(1)),CHAR(I(2)),
CCHAR(1(3)),CCHAR(1(4)),CHAR(1(5)),
CCHAR(1(6)),CHAR(1(7)),CHAR(I(8)),
CCHAR(1(9)),CHAR(1(10))
READ(TEMPO,'(F10.2)',ERR=1005) AMPL
IF (1(10).EQ.51) AMPL=AMPL/10.0**3.0
IF (1(10).EQ.54) AMPL=AMPL/10.0**6.0
IF (1(10).EQ.57) AMPL=AMPL/10.0**9.0
ELSE IF (KK.EQ.9) THEN
WRITE(TEMPO,'(9A1)') CHAR(I(1)),CHAR(I(2)),
CCHAR(1(3)), CCHAR(1(4)),CHAR(I(5)),
CCHAR(1(6)),CHAR(1(7)),CHAR(I(8)),CHAR(I(9))
READ(TEMPO,(F9.2)',ERR=1005) AMPL
IF (1(9).EQ.51) AMPL=AMPL/10.0**3.0
IF (1(9).EQ.54) AMPL=AMPL/10.0**6.0
IF (1(9).EQ.57) AMPL=AMPL/10.0**9.0
ELSE IF (KK.EQ.8) THEN
WRITE(TEMPO,'(8A1)) CHAR(I(1)),CHAR(I(2)),
CCHAR(1(3)), CCHAR(1(4)),CHAR(1(5)),
CCHAR(1(6)),CHAR(1(7)),CHAR(1(8))
READ(TEMPO,'(F8.2),ERR=1005) AMPL
IF (1(8).EQ.51) AMPL=AMPL/10.0**3.0
IF (1(8).EQ.54) AMPL=AMPL/10.0**6.0
IF (1(8).EQ.57) AMPL=AMPL/10.0 * 9.0
ELSE IF (KK.EQ.7) THEN
WRITE(TEMPO,'(7A1)') CHAR(I(1)),CHAR(I(2)),
CCHAR(I(3)), CCHAR(1(4)),CHAR(I(5)),CHAR(I(6)),CHAR(I(7))
READ(TEMPO,'(F7.2),ERR=1005) AMPL


4
partly due to the difficulty in location of the critical point
in mixtures by conventional methods. The most widely used
method is based on direct observation of the appearance and
disappearance of a meniscus, indicating a boundary line
between gas and liquid phases when a system enters the
critical condition.(19) This method relies on the visual acuity
of the operator, and clearly has some subjectivity.
Acoustic determination of thermodynamic properties such
as heat capacity, virial coefficients, vapor pressure, etc. by
spherical resonator techniques has been very successful in the
past, especially in this research group.(20-26) This success
stems mainly from the unusual sensitivity of the cavity
resonator. Resonance frequencies can be readily measured to
precisions of one part in 106 or better compared to one part
in 104 or 105 for pressure, temperature, volume density etc.
The acoustic resonator also benefits thermodynamic
measurements by operating at low frequencies. At high
frequencies the speed of sound is affected by
irreversibilities resulting principally from the delay of
energy flow into and out of internal energy modes,
particularly vibrational modes.(27)
This work is a first attempt to bring the superiority of
the spherical acoustic resonance technique to bear on
development of a new approach to critical point determination.
Experiments were performed according to logical scientific
approach. To start with, verification of this new technique


CHAPTER 6
CONCLUSION
A new approach using an acoustic resonance technique for
phase boundary detection may be summarized as follows:
1. Measure several resonance frequencies of standing
acoustic waves excited inside a fluid filled spherical
acoustic cavity.
2. Identify each resonance frequency.
3. Track an assigned resonance frequency as system
parameters such as temperature, pressure, or density are
varied.
4. Locate phase equilibria by abrupt changes in resonance
frequency (hence speed of sound) as the system state traverses
a phase boundary.
5. Locate critical points as minima on (P,T,C) or (p,T,C)
diagrams.
The principal results of this work are tabulated and
compared with the available literature values in table 6.1.
Apparently, the present results are in good agreement with
established literature values, proving the viability of this
new technique. The largest discrepancy is in the system
120


N
&
s
0
CL)
l-H
tin
1900
1800
1700
- 1600
1500
1400
1300
1200
10
15
20
25
30
Temperature, Celsius
Figure 5.8. Typical behavior of resonance frequency versus temperature of the carbon
dioxide-ethane mixture at a supercritical density.


156
laser beam reflected off a mirror attached to the balance beam
onto a suitable scale. We have used a sheet of precision graph
paper taped to a wall of the laboratory. The path length
between the mirror and the scale is about 3 meters.
The unloaded beam is first balanced by using the
adjusting weights added as necessary to restore the balance of
forces on the piston. To reduce the effects of friction, which
can lead to weighing errors, the piston is coated with oil and
provided with a horizontal lever arm which is used to
oscillate it back and forth through a small angle while it
settles into its vertical equilibrium position. Freedom from
sticking is evident when the balance point is constant on
repeated tries including those with both positive and negative
initial offsets.
Test Results
The absolute dead weight pressure gauge was tested for
accuracy by using it to determine the pressure of gaseous
CF2C12 (freon-12) in equilibrium with its liquid phase at
constant temperature. This system has been thoroughly studied
by Gilkey et. al.(144' 145) using more elaborate methods. The
freon container was placed in an adjustable subambient
constant temperature bath and connected to the gauge inlet.
Air was purged from the gauge and lines by vacuum pumping and
repeated flushing with small aliquots of freon. After
equilibrium was reached the beam was balanced and the
requisite masses and their positions noted. The atmospheric


TEMPERATURE, C
Figure 5.12. Other resonance frequency (top curve) results in same critical temperature
as first radial resonance frequency (bottom curve.)


107
window, and this was prone to shift between these two on
occasion. This situation caused an unsmooth experimental curve
as shown in the curve labeled observation 1 of figure 5.28.
Observation 2
On cooling below the critical temperature in some runs
the tracking program locked onto some peak other than the
desired Is peak and showed aberrant behavior on continued
cooling. The curve labeled observation 2 of figure 5.28 shows
a behavior of this sort. No further investigation was done to
identify the spurious signal.
Observation 3
If the situation in observation 2 did not occur, the peak
emerging after the critical condition is the Is peak3. At
still lower temperatures, however this is lost intermitentally
to a companion peak which grows in amplitude beyond that of
the Is peak. The partial history of the scanned signal is
depicted in the plots of figures 5.29 and 5.30 which show the
splitting of the tracked peak. The scans are shown in proper
time order but those for which the peak is single are mainly
omitted. Curves 4 and 5 of figure 5.29 show the onset of
splitting. As the system moved away from the critical point
3 Based on the technique used in this work one cannot say
confidently what the emerged peaks are. The only sure way to
solve this is to collect all peaks in the entire frequency
range (from 0 to 10,000 khz) and observe the relationships
among them. This, however, was beyond the scope of this work.
Other work in this laboratory, however, is directed to this
problem.


33
a new high performance stainless steel resonator assembly was
designed and built, and new data acquisition software was
developed.(52) The new apparatus was used to measure the sonic
speed in some carefully blended gas mixtures. Because the
mixtures contained small concentrations of condensibles such
as hexane for example, it was possible to cool them below
their dew points and it was observed that the sonic speed
faithfully revealed the precise location of the phase


110
the separation of these two peaks increased (curves 6 through
11 of figure 5.29.) The right hand peak (peak one) was bigger
and moved faster than the left peak (peak two.) It moved so
fast that peak two was eventually out of the sweep range (see
curve 12 of figure 5.29.) This single peak remained strong
until the temperature was around 31 C. At that temperature
the other peak (possibly peak two) reappeared in the sweep
range (curves 15 and 16 of figure 5.29.) It was at this time
1700
O
\
1200
1400
1300
- EXPERIMENTAL CURVE
PEAK TWO ROUTE
-PEAK ONE ROUTE
\ '
S
V
1100 11 1111111i i 111 i i i i 1111 111 11i 11
28 28.5 29 29.5 30 30.5 31 31.5 32 32.5
TEMPERATURE, CELSIUS
Figure 5.30. Possible routes of movement of two neighbor peaks
(peak one and peak two) below the critical temperature.


145
A6R=A(6)
TCR=A(7)
ALPHAR=A(8)
BETAR=A(9)
DELTAR=A(10)
A1L=B(1)
A2L=B(2)
A3L=B(3)
A4L=B(4)
A5L=B(5)
A6L=B(6)
TCL=B(7)
ALPHAL=B(8)
BETAL=B(9)
DELTAL=B(10)
WRITE(*,*) 'INPUT TR & TL IN KELVIN'
READ( * ) TR, TL
T=(TR+TL)/TWO
P2L = BETAL + DELTAL
P3L = 1.0-ALPHAL+BETAL
P2R = BETAR + DELTAR
P3R = 1.0-ALPHAR+BETAR
10 TRL = DABS((T-TCL)/TCL)
TRR = DABS((T-TCR)/TCR)
C write (*,*) trl,trr,p21,p31,p2r,p3r
S=(DSQRT(TCL))*(A1L*TRL* *BETAL+A2L*TRL**P2L
C+A3L*TRL* *P3L+A4L*TRL+A5L*TRL* *2+A6L*TRL* 3)
R=(DSQRT(TCR))*(AlR*TRR**BETAR+A2R*TRR**P2R
C+A3R*TRR* *P3R+A4R*TRR+A5R*TRR* *2+A6R*TRR** 3)
DEL=R-S
IF (DABS(DEL).LE.1.OD-16) THEN
GOTO 100
C ELSE IF (R.GT.S) THEN
ELSE IF (DEL.GT.ZERO) THEN
TR=T
T=(TR+TL)/TWO
DELE=T-E
IF (DELE.EQ.ZERO) THEN
GOTO 90
END IF
E=T
WRITE (*,*) DEL,T,TR
GOTO 10
C ELSE IF (R.LT.S) THEN
ELSE IF (DEL.LT.ZERO) THEN
TL=T
T=(TR+TL)/TWO
DELF=T-F
IF (DELF.EQ.ZERO) THEN
GOTO 95
END IF
F=T


n n o
173
BOTTOM=0.0
RIGHT=1.0
TOP=0.8
IF(NPLOT.EQ.1) THEN
CALL SCREENMODE (M)
CALL PLOTTER (DEVTYPE,INTERFACE,ADDR)
CALL SCALE (XMIN,YMIN,XMAX,YMAX)
CALL WINDOW (LEFT,BOTTOM,RIGHT,TOP)
CALL GRAPHXY(DX,DY,N,XAXIS,XLBL,YAXIS,
CYLBL,TITLE,LIN,SYMBOL,PEN)
ELSE
CALL GRAPHXY(DX,DY,N,XAXIS,XLBL,YAXIS,
CYLBL,TITLE,LIN,SYMBOL,PEN)
ENDIF
RETURN
END
PROGRAM MAX3
INTERFACE TO SUBROUTINE TLOOK[C](T)
INTEGER*4 T [NEAR,REFERENCE]
END
COMMON 11,18,116,124
INTEGER*4 11,18,116,124
INTEGER*4 NDELY,T,TO,TB,TL
INTEGER AMP,COUNT,FREQ,FSHIFT
REAL*8 ASUM,PHSUM
C REAL*8 AGSUM
INTEGER*2 AM(120),PH(120),PR(120)
INTEGER*2 FM(120),TM(120),LARGE
C INTEGER*2 AG(120)
CHARACTER*20 FILERAW,FILEFORM
N=120
NDELY=1000
11=1
I8=2**8
116=2**16
124=2**24
Write(*,*) 1 WHAT IS THE DATA OUTPUT FILE NAME?'
READ(*,100) FILERAW
Write(*,*) WHAT IS THE FORMAT OUTPUT FILE NAME?'
READ(*,100) FILEFORM
100 FORMAT(A)
open(3,file='block',status='new')
OPEN(5,FILE=FILEFORM,STATUS='NEW')
OPEN (10, FILE=FILERAW,FORM='UNFORMATTED' ,STATUS ='NEW' )
write(*,*) WHAT IS THE RESONANCE DAC SETTING?'
READ(*,*) ICENT


51
Table 4.2. Gas Specifications.
Gas
Manufacturer
Grade
Purity
Carbon dioxide
Scott
specialty
gases
Research
Grade
99.99 Mole %
Ethane
Scott
Specialty
Gases
Research
Grade
99.9 %
Argon
Matheson gas
Products
Research
Grade
99.9995 Volume %
Table 4.3. Charging Pressure of gas systems.
Gas system
Starting
temperature (C)
Charging pressure
(psia)
Carbon dioxide
36.0
1200
co2 + c2h6
30.0
630 for C2Hg and
make up to 1300
with CO,
c2h6
37.0
808
Resonance Frequency Measurements
Measurements were made on isochore samples introduced
into the spherical resonator which was initially rinsed with
argon gas and subsequently evacuated overnight at high
temperature (40-50 C) with a mechanical pump connected to a
liquid nitrogen trap. The apparatus was then brought to a
temperature above the known literature critical temperature.
After that gas was introduced slowly into the resonator to
bring the pressure to the calculated value (see table 4.3).
The apparatus was then sealed off. The charging pressure value


11
Overall Solutions
Combining the results of the radial and angular parts
yields the solution of the time-independent monochromatic wave
equation in the spherical polar coordinate system as
4>o(r,6, n-0 ml £-0
It is clear that in order to determine fully any particular
mode of vibration, we must specify the values of three
characteristic numbers: n, i, and m.
The roots of the radial part n correspond to the
frequency through the relation
5if
f
£,n
ka
C
5JfnC
2na
2nfi.*a
(2.25)
where C = speed of sound
a = the radius of the spherical cavity
n = resonance frequencies corresponding to £e
$ = root of the spherical Bessel function of the first
kind
= eigenvalues.
Lord Rayleigh'30 was the first to study this problem. In
1872 he solved for the lowest 23 eigenvalues (sometimes called
natural frequencies). In 1952 H. G. Ferris'31 revised and
extended the list to the lowest 84 eigenvalues as shown in




Table 4.1. (Continued)
Item
Manufacturer
Model
number
Range of
operation
Accuracy
(Manufacture
quote)
Function
Pre-
Amplifier
Stanford
research
systems,
Inc.
SR550
Up to 250 mv
rms input
1 % Gain
Reduces input
noise and extends
the full scale
sensitivity to 10
nv.
Voltmeter
Keithley
instruments,
Inc.
177
200 mv to
1200 volts
0.04 %
Reading + 2
digits for
minimum range
or 0.035 %
rdg. + 1
digit for
max. range
Measure Ac or Dc
Voltage.
Digital
multimeter
Keithley
instruments,
Inc.
195A
2-Terminal &
4-terminal
resistance
measurements
in the range
of 100 pfi to
20 MQ
Depends on
range. See
manufacture's
manual for
details.
Measure resistance
from Pt
temperature probe.
Function
generator
Wavetex San
Diego, Inc.
182A
0.004 Hz to
4 MHz
Dial accuracy
= 5 % of
full scale.
Generates a
precise sine
/triangle/square
waveform.


CHAPTER 1
INTRODUCTION
Natural gas and gasoline are primarily mixtures of the
lighter hydrocarbons with varying amounts of nonhydrocarbons
such as water, carbon dioxide, and hydrogen disulfide. Heavier
fossil fuel mixtures such as crude oil consist of a myriad of
higher boiling hydrocarbons and various compounds containing
sulfur, nitrogen, and oxygen. In designing production,
processing, transport, and handling systems for these
materials reliable knowledge of their physical properties is
crucial. For example, in the important activities of enhanced
petroleum recovery by carbon dioxide injection into a
resevoir1 and transport of natural gas mixtures containing
carbon dioxide, precise knowledge of the equilibrium phase
behaviors of the systems is important. Additionally, the need
exists for knowledge of critical point parameters, hydrate
formation conditions, density, enthalpy, dew points, bubble
points, etc. Information of these kinds about select pure
*Under the right condition of pressure, temperature, and
oil composition, carbon dioxide will mix with the crude in the
reservoir to form a single phase liquid which is much lighter
than the original oil and consequently easier to bring to the
surface.
1


190
CBD(I),C(I),C1(I),C2(I)
25 CONTINUE
CLOSE (10)
OPEN (3 0,FILE='NAT.GAS 1,STATUS=OLD')
OPEN (40,FILE='SPEED.GAS',STATUS='NEW')
c open (50,file='a.gas',status='new')
c open (60,file=b.gas,status='new')
c open (70,file=c.gas',status=new')
N=ZERO
500 READ(30,*,END=300) Tl,T2,Pi,P2,AMP,FREQ,TIM,IN
c write (60,*) amp,in
c write (50,*) freq
N=N+ONE
IF(Pi.EQ.0.0.OR.P2.EQ.0.0) GOTO 500
TEMP=(T1+T2)/TWO
P=(P1+P2)/TWO
DT=T2-T1
DP=P2-P1
IF(TA(1).LT.TEMP) GOTO 500
IF(TA(M).GT.TEMP) GOTO 500
IF(AMP.GT.19000.0D0) IN=IN-1
IF(AMP.LT.7000.0D0) IN=IN+1
c write (60,*) freq
CALL AMPGET(AMP,IN)
c write (70,*) freq
DO 20 1=1,M
ISTOP=0
IF (TA(I).LT.TEMP) GOTO 400
ISTOP=l
20 CONTINUE
400 IF (ISTOP.EQ.l) GOTO 500
FAR=(FR(I)-FR(I-1))/(TA(I)-TA(I-1))*
C(TEMP-TA(1-1))+FR(1-1)
PAR=(PA(I)-PA(I-1))/(TA(I)-TA(I-l))*
C(TEMP-TA(I1))+PA(1-1)
TEMP=TEMP+KEL
TS=TEMP/EPK
CALL QAND(BS,BS1,BS2,BSD,CS,CS1,
CCS2,TS,T,B,Bl,B2,BD,C,C1,C2)
C WRITE(*,*) QAND'
BV=BS*B0
CV=CS*B0*B0
CALL VERVOL(PAR,V,TEMP,BV,CV)
C WRITE(*,*) VERVOL'
VS=V/B0
CALL SPEED(CAR,TEMP,BS,BS1,BS2,BSD,CS,CS1,CS2,VS)
C WRITE(*,*) SPEED'
CG=FREQ*CAR/FAR
TEMP=TEMP-KEL
WRITE(40,*) N,DP,DT,P,TEMP,AMP,FREQ,CG,TIM,CAR,FAR
GOTO 500
close(70)


124
comparable to that at the major standards laboratory of the
nation. Given equivalency of thermometric accuracy, the new
method of critical point determination is clearly superior and
is a good candidate to become the method of choice in future
studies. Even so improvements are possible and some of these
are being incorporated in the next generation version of this
apparatus. The major change will be the inclusion of a volume
change capability so that data can be taken isothermally or
isobarically as well as isochorically.
The spherical resonator is a remarkably accurate and
convenient tool for the measurement of thermophysical
properties of fluids. The speed of sound in a gas of interest
can be measured with high accuracy merely by measuring the
frequencies of the radial normal modes of vibration. A
relative measurement against a reference gas such as argon
further simplified the procedure. This technique offers a
proven alternative experimental approach to valuable physical
properties including phase boundaries, thermodynamic
properties and equation of state parameters, all of which are
useful to both pure and applied science. Applications are
numerous and only now beginning to be explored. One important
application is indirectly determining the intermolecular
potentials of fluids from acoustic virial coefficients via an
inversion technique without using the parameter optimization
techniques for model potentials such as Lennard-Jones.
Accurate sonic speed data as a function of temperature of both


28
critical point (within several mK of Tc) real systems show a
deviation from linearity.
Isochore Approach
There are two slightly different ways for taking this
approach. The first one is to study the discontinuity of the
isochore curve.(49) This method begins with loading a bomb of
constant volume with a series of known weights of the
substance and then studying the behavior of the pressure as
J
%
f
SUBCRmCAL DENSfTY
4
>
*
. MEAR-CRITICAL DENSITY
. i
*
a SUPERCRITICAL DENSITY
&
>
y
*
s
*
y
y
S'
wf
TEMPERATURE
Figure 3.8. Schematic diagram of isochoric method of
critical point determination.


Table 5.4. Experimental results for construction of phase diagrams of a mixture of
carbon dioxide and ethane.
FILE NAME
Starting point
Point of minimum frequency
T
(C)
Pressure
psia
Frequency
Hz
T (C)
Frequency
Hz
Sonic
speed
m/sec
Pressure
psia
TDA729
30.004
1303.6
1976.38
17.006
1554.82
216.72
857.0
TDA731
29.993
1292.9
1951.08
17.200
1533.44
213.70
860.7
TDA801
30.002
1283.3
1925.92
17.485
1507.68
210.11
865.8
TDA802
30.002
1271.9
1895.52
17.632
1470.04
204.87
869.4
TDA807
30.003
1200.2
1669.91
19.228
1203.10
167.71
901.2
TDA820
30.013
1156.6
1535.02
19.860
935.06
130.33
914.5
TDA907
30.007
1135.3
1460.87
19.950
802.36
111.83
915.9
TDA914
30.011
1114.8
1425.97
19.964
934.84
130.30
916.3
TDA917,925
30.060
1107.1
1420.32
19.920
1030.75
143.66
915.6
TDA926
30.039
1101.2
1426.42
19.885
1073.42
149.61
914.9
TDA928
30.020
1098.6
1416.55
19.851
1086.70
151.46
914.1


158
films of synthetic elastometers such as neoprene, viton,
silicone rubbers etc. should work well, but have not thus far
been available for testing.
The tests included comparison of the readings of a
precision Bourdon gauge connected to the pressure line. Table
D.l. shows that this particular gauge is in error or requires
calibration.
Deadweight Gauge with Bellows Sensor
The basic pressure gauge herein described was designed to
accommodate interchangeable sensing elements. It can, for
Table D.l. Experimental Test of Dead Weight Pressure
Gauge.
Temperature, K
Vapor Pressure R-12, psia
(in parenthesis is % deviation
from literature value)
Gilkey
(1947)
This work
Bourdon
Gauge
271.92
43.04
42.9710.01
(0.16%)
42.47
(1.3%)
287.45
69.87
69.66
(0.31%)
68.90
(1.4%)
example, be fitted with piston/cylinder combinations of
different diameters or with entirely other types of sensors.
In particular it was intended to investigate the suitability
of a thin walled metallic bellows for use as a balance
detector. An electroformed nickel bellows with 0.0018 inch


o
a>
o
00
*0
a>
a>
Temperature, C
Figure 5.11. Typical experimental results of near-critical density gas of the carbon
dioxide-ethane mixture. Curve 1 shows a forced cooling run. Curve 2 shows a forced
warming run. Curve 3 is a slow naturally warming run.
i
CT>


Figure 5.7. Experimental vapor pressure curve compared
with the NIST model 66
Figure 5.8. Typical behavior of resonance frequency
versus temperature of the carbon dioxide-ethane
mixture at a supercritical density 72
Figure 5.9. Typical behavior of resonance frequency
versus temperature of the carbon dioxide-ethane
mixture at a subcritical density 73
Figure 5.10. Typical behavior of resonance frequency
versus temperature of the carbon dioxide-ethane
mixture near critical density 74
Figure 5.11. Typical experimental results of near-
critical density gas of the carbon dioxide-ethane
mixture. Curve 1 shows a forced cooling run. Curve
2 shows a forced warming run. Curve 3 is a slow
naturally warming run 76
Figure 5.12. Other resonance frequency (top curve)
results in same critical temperature as first
radial resonance frequency (bottom curve.) .... 77
Figure 5.13. A sequence of supercritical pressure versus
temperature isochores of the carbon dioxide-ethane
mixture 78
Figure 5.14. Relationship between the angles 0 and the
charging pressures of the carbon dioxide-ethane
mixture 80
Figure 5.15. Relationship between the starting pressures
and the temperatures of phase change of the carbon
dioxide-ethane mixture 85
Figure 5.16. Relationship between the starting pressures
and the speed of sound at a phase change of the
carbon dioxide-ethane mixture 86
Figure 5.17. Coexistence curve of the carbon dioxide-
ethane mixture near azeotrope composition 86
Figure 5.18. The sonic speed versus temperature of the
carbon dioxide-ethane mixture for an isochore near
critical density 87
Figure 5.19. Pressure and temperature behavior of the
carbon dioxide-ethane mixture charged near its
critical density 87
x


APPENDIX D
LABORATORY STANDARD PRESSURE GAUGE
Introduction
Pressure gauges are essential instruments of science and
industry and are available in a variety of types. Most of the
common types are favored for possessing one or more of the
following desirable features: low cost, ease of use, rugged
construction, suitability for remote sensing. The attainment
of these features is generally facilitated by the use of
gauges which measure pressure indirectly. Such gauges require
calibration and are, therefore, not suitable for use as
primary standards. Unless their calibrations are checked
frequently, their readings are always somewhat suspect.
While for most purposes the common gauge types give
sufficiently reliable readings, there are occasions which call
for absolute pressure measurements made by gauges which
measure directly the average normal force acting over a known
area. The pressure range from about 10'3 to 10"5 pascal is
served adequately at the low end by the McLeod gauge if
suitably operated to avoid the errors associated with the
mercury pumping effect142 and at the upper end by the U-tube
147


164
OPEN(70,File=c:NAME.NUM',STATUS='NEW')
WRITE(70,302) DTEMP,NUM
WRITE(*,302) DTEMP,NUM
302 FORMAT(1X,A7,13)
CLOSE (70)
OPEN(20,FILE=DNAME,STATUS=NEW')
OPEN( 30,Filete:Peak.buf ,STATUS=OLD' )
READ(30,*) F,TI
TI=TI+273.15
CALL KEITHA(TF)
TF=TF+273.15
F=SQRT(TF/TI)*F
CLOSE(30)
IXT=3
c IXT=1
NA = 21/IXT
IDELY=2*NA
C M=1000/NA/IXT-IDELY/NA
C USED ONLY FOR CENTERING
0**********************************************************
M=61
C USED AFTER CENTERING
XMIN=1.0
XMAX=FLOAT(M)
1001 CALL TIME(10,TSTR)
READ(TSTR,'(12)',ERR=1001) HR
READ(TSTR,(3X,I2)',ERR=1001) MIN
READ(TSTR,'(6X,I2)',ERR=1001) SEC
TOO = HR*3600+MIN*60+SEC
100 CALL TEMPSET(VOLT)
101 DO 30 1=1,READS
500 SF=F-WIDTH/2.0
SPF=F+WIDTH/2.0
INDEX=1
POINTS=M
ISTART=1
200 CALL HP(SF,SPF,SWEEP)
AMAX=0.0
XMIN=200000.0
DO 42 ID=1,IDELY
CALL KEITH(MV)
42 CONTINUE
DO 40 J=ISTART,POINTS,INDEX
AVG=0.0
DO 41 JA=1,NA
300 CALL KEITH(MV)
KOUNT=KOUNT+1
IF(KOUNT.EQ.2*NA) THEN
IG=IG+1
CALL GAINSET(IG)


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Timothy J. Anderson
Professor of Chemical
Engineering
This dissertation was submitted to the Graduate Faculty
of the Department of Chemistry in the College of Liberal Arts
and Sciences and to the Graduate School and was accepted as
partial fulfillment of the requirements for the degree of
Doctor of Philosophy.
December 1991
Dean, Graduate School


151
be replenished by means of an injection pump. The upward force
on the piston is balanced by the sum of the external pressure,
gravity acting on the piston and an additional masses added as
required to keep the piston afloat. Although this type of
pressure gauge is capable of high accuracy, its use is
tedious, and commercial units are expensive, being beyond the
means of many small laboratories. The present gauges were
developed to avoid these problems.
Dead Weight Pressure Gauge
The present gauge is shown in figure D.2. It differs
principally from the conventional designs in the following
respects:
1. The piston is inverted being forced vertically
downward by the system pressure.
2. The loading weight is generated by standard masses
suspended from accurately spaced hooks on a pivoted balance
beam.
3. The load is transferred to the piston by point contact
with a hardened precision ball accurately located on the beam
with respect to the hooks.
4. A thin flexible membrane across the top of the
cylinder acts as a piston follower thereby separating the
piston from the system gases and eliminating the need for a
null detector or oil injection pump.


13
table 2.1. Even more complete tabulations can be found in the
literature132'.
Searching for Resonance Frequencies
£ C
From the relationship it is obvious that in
order to find the resonance frequency corresponding to the
normal mode of vibration of an eigenvalue, n, the speed of
sound and the radius of the sphere must be known. The latter
may be obtained precisely from geometric considerations on the
resonator cavity. The former may be approximately calculated
from an assumed equation of state. In this study we used the
American Gas Association equation called AGA8(5) which requires
as input only temperature, pressure, and composition. Using
the predicted sonic speed together with the measured cavity
radius and the tabulated eigenvalue, the corresponding
resonance frequency fL n may be approximately calculated and
then experimentally searched by scanning frequency around that
value. When excited at a resonance frequency the amplitude of
the detected acoustic signal sharply increases. The series of
resonance peaks arranged by frequency defines a zeroth order
acoustic spectrum of resonances of gas enclosed in the
spherical cavity.
Once a resonance frequency has been found, it can serve
as an internal reference point to search for another, as
follows:


17
radial mode resonance peaks typically have narrow half-width,
(high Q value2). Furthermore, resonance frequencies of the
radial modes are sensitive to imperfections of non-sphericity
only to second order(36). Consequently, from all points of view
the most accurate measurements of the speed of sound should be
obtained by utilizing the radial modes of vibration. The first
few radial modes are mostly used because of their associated
low frequencies. A more complete account of these effects can
be found in references 33 and 34.
2Q is usually defined as:
n Center Frequency of Resonance
Frequency Width 3dB Points


137
CLOSE(5)
50 IF (SSRSD(l).LT.1.0E-17) THEN
C WRITE(*,*) FIN CON 1'
GOTO 100
END IF
CALL SHELL(B,SSRSD,WORST)
CHECKER=CHECKER+SSRSD(1)
KOUNT=KOUNT+1
C WRITE( * ) *************************KOUNT=,KOUNT
KKOUNT=KKOUNT+1
IF (KKOUNT.EQ.3 0 0) THEN
C IF (KKOUNT.EQ.200) THEN
KKOUNT=0
CHECKER=(CHECKER/300.0)-SSRSD(1)
C CHECKER=(CHECKER/200.0)-SSRSD(1)
IF (CHECKER.LT.1.00000E-17) THEN
C IF (CHECKER.LT.1.0E-16) THEN
GOTO 100
END IF
CHECKER=0.0
END IF
IF (SSRSD(11).EQ.WORST) THEN
C WRITE(*,*) 'FLAG RAISED FLAG RAISED FLAG RAISED'
Z=SSRSD(10)
SSRSD(10)=SSRSD(11)
SSRSD(11)=Z
DO 140 J=l,10
ZZ=B(J,10)
B(J,10)=B(J,11)
B(J,11)=ZZ
140 CONTINUE
END IF
C WRITE(*,*) 'START REFLEX'
CALL REFLEX (SSRSD,B,L,T,C,W)
C WRITE(*,*) 'PASS REFLEX'
IF (SSRSD(13).LT.SSRSD(1)) THEN
CALL EXPAND(B,SSRSD,C,T,L,W)
C WRITE(*,*) 'PASS EXPAND'
GOTO 50
ELSE IF (SSRSD(13).GT.SSRSD(1).AND.
CSSRSD(13).LT.SSRSD(10)) THEN
SSRSD(11)=SSRSD(13)
DO 70 1=1,10
B(I,11)=B(1,13)
70 CONTINUE
C WRITE(*,*) 'PASS BETWEEN'
GOTO 50
ELSE IF (SSRSD(13).GT.SSRSD(10)) THEN
IF (SSRSD(13).LT.SSRSD(11)) THEN
CALL CNTR(B,SSRSD,C,T,L,W)
C WRITE(*,*) 'PASS CNTR'
ELSE IF (SSRSD(13).GT.SSRSD(11)) THEN


53
frequency could be found with high accuracy. Usually, the
first radial mode of vibration was selected as the tracked
peak throughout this work.
The approximate location of the desired peak, first
radial mode for example, was initially calculated from the
AGA8 equation using pressure, temperature, and composition as
input. Then the actual location of the peak was experimentally
searched using a program called SCAN. Scanning was performed
in a reasonable frequency range around the center frequency
identified as the output value of the AGA8 calculation. Once
the peak(s) was found, location of another mode was calculated
as described in the theoretical background section. The
purpose of this routine was to increase the degree of
confidence that each experimentally found peak was indeed
correctly identified. Similarly, if more than one peak
appeared together in the range while scanning, each was
checked against the other radial modes to sort out the best
candidate for labeling as a resonance mode based on the
hypothesis that if it is a genuine system peak, its frequency
should reveal the correct position of the other peaks. Figure
4.11 shows an example of a series of scans performed
chronologically. The figure in the first row is an acoustic
spectrum scanned in a wide frequency range. The figure in the
second row is the acoustic spectrum scanned in a narrower
frequency than the one in the previous figure. In this
incident there were three peaks near the predicted frequency


n o
182
'COMMON GROUP 2.'
BMAD=IBFIND('BM600 ')
1006 call ibrd (BMAD,RD,10)
11 = rd(l)/i24
12 = (rd(1)-il*i24)/il6
13 = (rd(1)-il*i24-i2*il6)/i8
14 = rd(1)-l*i24-i2*il6-i3*i8
15 = rd(2)/i24
16 = (rd(2)-i5*i24)/il6
17 = (rd(2)-i5*i24-i6*il6)/i8
i88 = rd(2)-i5*i24-i6*il6-i7*i8
WRITE(Pres,'(7A1)) CHAR(I4),CHAR(13),CHAR(12),
CCHAR(Il),CHAR(188),CHAR(17),CHAR(16)
READ(PRES, '(F7.1) ,ERR=1006) P
RETURN
end
SUBROUTINE SENSET(IG)
common /ibglob/ ibsta,iberr,ibcnt
INTEGER*4 11,188,1166,124
DIMENSION 1(20)
'COMMON GROUP 1.'
integer CR
integer cmd(10),rd(512),wrt(512),IP(4)
CHARACTER*1 CH(20)
CHARACTER*2 GAIN
character*8 bname,bdname
CHARACTER*20 TEMP
'COMMON GROUP 2.'
data CR/13/
IP(4) = 2**24
IP(3) = 2**16
IP(2) = 2**8
IP(1) = 1
124 = 2**24
1166 = 2**16
188 = 2**8
11=1
J=0
K=1
L=0
20 LK = IBFIND ('LOCKIN ')
CH(1) = 'Y'
CH(2) = '4'
WRT(1)=ICHAR(CH(1))+ICHAR(CH(2))*IP(2)
CALL IBWRT(LK,WRT,2)
CALL IBRD (LK,RD,12)
1(4) = RD(1)/I24
1(3) = (RD(1)-I(4)*I24)/I166
1(2) = (RD(1)-I(4)*I24-I(3)*I166)/I88
1(1) = RD(1)-I(4)*I24-I(3)*I166-I(2)*I88
1004


Figure 5.20. Pressure and temperature relationship of a
set of several isochores of the carbon dioxide-
ethane mixture 88
Figure 5.21. Resonance frequency (Is mode) and
temperature relationship of ethane near the
critical density 91
Figure 5.22. Resonance frequency (Is mode) and pressure
relationship of ethane near the critical density. 91
Figure 5.23. Pressure and temperature relationship of Is
mode resonance frequency of ethane near the
critical density 92
Figure 5.24. Speed of sound and temperature of Is mode
resonance frequency of ethane near the critical
density 92
Figure 5.25. Phase diagram of pure ethane showing
temperature of phase changes as a function of
starting pressure 94
Figure 5.26. Phase diagram of ethane showing pressure
and temperature behavior of 23 different
isochores 95
Figure 5.27. Trajectory plot of temperature, pressure,
and sonic speed of ethane at different densities.
The one nearest the critical density reaches the
lowest sonic speed 96
Figure 5.28. Resonance frequency and temperature
relationship revealing some phenomena observed in
the experiment on pure ethane 106
Figure 5.29. Movement of tracked peak and its neighbor
below the critical temperature. These curves are
numbered in the proper time sequence but many
curves showing only one strong peak have been
omitted 108
Figure 5.29. Continued movement of tracked peak and its
neighbor below the critical temperature 109
Figure 5.30. Possible routes of movement of two neighbor
peaks (peak one and peak two) below the critical
temperature 110
Figure 5.31. Experimental curve of ethane showing
dynamic behavior of speed of sound near critical
point 113
xi


112
Observation 4
An attempt at running the experiment with ascending
temperature was made. The same uncertainties about phase
distribution obscures the identify of the peak selected for
tracking. The possibility of selecting a peak which evolves
irregularly or fades into the background exists. The curve
labeled observation 4 in figure 5.28 shows this type of
observation.
Curve Fitting
Over a small temperature interval in the critical
vicinity the task of single peak tracking is quite severe, and
the peak is often lost. This is due to two main effects.
First, in terms of dynamics, the rate of decrease in resonance
frequency, hence speed of sound, with respect to temperature
is extremely high approaching infinity, i.e. the speed of
sound exhibits a mathematical singularity at the critical
point. Second, the amplitude of the tracked peak decreases
dramatically toward zero or at least below the noise level of
the system. As a consequence, an ambiguity of the critical
point arises. In order to solve this problem a mathematical
model describing the relationship between the speed of sound
and temperature near the critical was applied. This model was
originally derived by Stell et. al.(133) to fit data of latent
heat of vaporization and temperature and later modified to
satisfy a corresponding states principle by Sivaraman et.
a.1. and applied to speed of sound data by the same


47
Even though there were different setups, the basic
principles are the same and can be divided into two parts: the
input signal generation part and the output signal analysis
part. These are shown by dashed-line boxes in figures 4.7 and
4.9. Commands and data were transferred among the instruments
and computer over the General Purpose Interface Bus (GPIB)
cables. Each component on the GPIB was equipped with an IEEE-
488 standard interface. The computer was always a controller-
in-charge sending commands and acquiring data according to the
written computer program. Each instrument was a listener
and/or a talker depending on a currently executed line of the
program. Table 4.1 summarizes electronic hardware components
used in this work and their specifications.
Gases
Carbon dioxide and ethane gases used were high purity
grade. No further analysis was performed on carbon dioxide.
Table 4.2 summarizes the pure gas qualities.
For the experiments on a mixture of C02-C2H6 quantitative
analyses were performed since the results of speed of sound
measurements depend on the precise composition. Both
gravimetric and gas chromatography analyses were used to
fulfil this task. Details of these analyses are presented in
appendix B.


94
Initial Pressure, psia
Figure 5.25. Phase diagram of pure ethane showing temperature
of phase changes as a function of starting pressure.
software, are listed in table 5.6. Figure 5.26 shows a
collection of pressure and temperature data of 23 isochores
which cover a broad range of densities from supercritical to
subcritical values. These curves coincide within experimental
uncertainty in the liquid-vapor equilibrium region. Figure
5.27 shows a trajectory plot of a collection of 23 pressure-
temperature-speed of sound data sets of different densities.
It clearly shows the pronounced trend of decreasing sonic
speed as the system approaches critical conditions. Table 5.7
is a compilation of all necessary experimental data to
construct the phase diagram in different views as shown in the
above listed figures. Tables 5.8 gives the comparison between


oon non nno ono
CALL IBWRT(LK,WRT,9)
RETURN
end
171
SUBROUTINE GAINSET(IG)
common /ibglob/ ibsta,iberr,ibcnt
'COMMON GROUP 1.'
integer*4 cmd(10),rd(512),wrt(512),IP(4)
CHARACTER*1 CH(20)
CHARACTER*2 GAIN
character*8 bname,bdname
'COMMON GROUP 2.'
LK = IBFIND ('LOCKIN )
IP(4) = 2**24
IP(3) = 2**16
IP(2) = 2**8
IP(1) = 1
WRITE(GAIN,'(12)') IG
CH(1) = 'G'
READ(GAIN,'(Al)') CH(2)
READ(GAIN,'(1X,Al)') CH(3)
WRT(1)=ICHAR(CH(1))+ICHAR(CH(2))*
CIP(2)+ICHAR(CH(3))*IP(3)
CALL IBWRT(LK,WRT,3)
RETURN
end
SUBROUTINE PRES(P)
character*14 a
open(10,file='coml',status='old')
read(10,100) a
READ(a,'(3X,F8.3)',ERR=200) P
100 format(al4)
RETURN
CLOSE(10)
GOTO 300
end
SUBROUTINE MAX(F,W,FO,M)
FO=W/FLOAT(M)*FO+F-W/2.0
RETURN
END
SUBROUTINE KEITH(MV)


CHAPTER 5
RESULTS AND DISCUSSIONS
Carbon Dioxide
As mentioned in the introduction, due to its well-
established values, carbon dioxide was chosen to be the first
candidate used to test this new acoustic resonance technique
for locating critical point parameters. Learning how best to
do this required considerable effort, often involving trial
and error procedures. Many adjustments had to be made to both
instruments and programs to identify the optimum conditions
for performing this experiment.
Figure 5.1 shows the typical relationship between the
first radial mode resonance frequency (hereafter abbreviated
resonance frequency) and the temperature of carbon dioxide in
the resonator charged near its critical density. It shows the
results of four runs of this same isochore. For each run the
curve may be roughly divided into two parts for the sake of
discussion. The first one is a curve where the system
temperature is greater than or equal to the critical
temperature, Tc. In this region the speed of sound is directly
proportional to the system temperature, i.e. as temperature
60


62
Figure 5.2. Relationship between first radial mode resonance
frequency and temperature of carbon dioxide for an isochore
near the critical density.
Figure 5.3. Relationship between the speed of sound and
temperature of carbon dioxide for an isochore near the
critical density.


199
86. Ernst S., and Traube K., Progr. Intern. Res. Thermodyn.
Transport Properties Papers Symp. Thermophys.
Properties, 2nd., Priceton, New Jersey, 193-205, 1962.
87. Matthews, J. F., Chem. Rev.. 72, 71-100, 1972.
88. Moldover, M. R., J. Chem. Phys., 61, 5, 1766-1778,
1974.
89. Altunin, V. V., The Thermophysical Properties of C02
[in Russian], Moscow, 1975.
90. Krynicki K., Neragi A. L. and J. G. Powles, Ber.
Bunsenqes. Phys. Chem.. 85, 12, 1153-1154, 1981.
91. Lesnevskaya M. S., Domracheva T. I., and M. B.
Nikiforova, Zh. Fiz Khim.. 55, 2671-2672, 1981.
92. Sengers J. V. Basu K. S., and Sengers J. M. H. Levett,
NASA Contractor Report 3424, NASA Scientific and
technical Information Branch, p. 59, 1981.
93. Morrison G., J. Phys. Chem.. 85, 759, 1981.
94. Adamov Sh. P., Anisimov, M. A., and V. A. Smirnov, The
Thermophysical Parameters of Substances and Materials
[in Russian], Moscow, No. 18, pp. 7-73, 1983.
95. Shelomentsev A. M., Teplofiz. Svoistva Uqlevo Dorodov
Nefteprod.. 139-154, 1983.
96. Morrison G., and J. M. Kincaid, AICHE. 30, 2, 257-262,
1984.
97. Kuenen J. P., Phil. Mag.. 44, 174, 1897.
98. Kuenen J. P. and W. G. Robson, Phil. Mag. 4, 116,
1902.
99. Khazanova N. E. and L. S. Lesnevskaya, Russ. J. Phvs.
Chem.. 41, 9, 1279, 1967.
100. Fredenslund A. and J. Mollerup, J. Chem. Soc. Faraday
Trans. I. 70, 1653, 1974.
101. Clark A. M. and F. Din, Disc. Faraday Soc.. 15, 202,
1953.
102.Jensen R. H. and F. Kurata, A. I. Ch. E. Journal.. 17,
357, 1971.


121
consisting of a C02~C2H6 mixture. This is not surprising
inasmuch as the literature value was obtained by extrapolation
of the reported experimental data and thus is subject to
greater uncertainty. This comparison is intended principally
to show the capability of the acoustic method to detect the
critical point of the more complicated system and should be
judged more on the precision than the accuracy of the outcome
since the basis for judging the latter is not sufficiently
strong.
Generally speaking, considering the results for pure
gases, even though the values of this work are not all within
the reported uncertainty of the literature values, the work
should not be judged a failure. In the first place the maximum
disparity between the different methods is only 1 mK, and an
increase in the estimated limits of uncertainty of only 1 mK
would bring the divers results into accord. There are reasons,
however, to suspect that the present results may be more
reliable. Consider, for example, the conventional method of
detecting the critical point. This involves the direct
observation of the transition from two phases to one, with the
critical state taken to be the single condition for which the
meniscus disappears when the volumes of liquid and vapor are
just the same. Ideally, critical points would be determined by
the direct observation of this phenomena at the exact middle
of the cylindrical sample cell. Realistically, such a
measurement is experimentally impracticable'96 and the


o o
186
CALL PRES(VI)
GOTO 200
ELSE
CALL KEITHA(TF)
CALL PRES(VF)
JRMAX=JMAX
FO=FLOAT(JFMAX+JRMAX)/2.0
CALL MAX(F,WIDTH,FO,M)
F=FO
AMP=(AMAX+AVGA)/2.0
IF(JFMAX.EQ.1.OR.JFMAX.EQ.M.OR.JRMAX.EQ.1.OR.
CJRMAX.EQ.M) THEN
OPEN(30,File='c:Peak.buf,STATUS= OLD')
READ(30,*) F,TI
CLOSE(30)
TI=TI+273.15
TF=TF+273.15
F=SQRT(TF/TI)*F
GOTO 500
ENDIF
1003
CALL TIME(10,TSTR)
READ(TSTR, (12 ) ,ERR=1003) HR
READ(TSTR,(3X,I2)',ERR=1003) MIN
READ(TSTR,(6X,I2)',ERR=1003) SEC
T = HR*3600+60*MIN+SEC
T=T-T00
IF(T.LT.O.O) THEN
T00=T00-24*3600
T = HR*3600+60*MIN+SEC
T=T-T00
ENDIF
IF(AMP.LT.7000) THEN
IG=IG-1
IF(IG.GE.7) CALL GAINSET(IG)
ENDIF
IF(AMP.GT.19000) THEN
IG=IG+1
IF(IG.LE.24) CALL GAINSET(IG)
ENDIF
RATIO=F/SQRT((TI+TF+273.15*2.0)/2.0)
WRITE(20,*) TI,TF,VI,VF,AMP,F,T,IG
Q* ********************************************************* *
C
C
DO 18 IF=1,19
WRITE(*,*)
c 18
CONTINUE
WRITE(*,390) SPF,SF
c
c 390
FORMAT(IX,'FREQUENCY3 ,F10.3,48X,F10.3)
WRITE(*,*)
c
c
c
IF (JRMAX .GT. JFMAX ) M=M-1
IF (JRMAX .LT. JFMAX ) M=M+1
DF=0.0
USED ONLY FOR CENTERING


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Samuel 0.Colgate/ Chairman
Associate Professor of Chemistry
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
^ Cxt ,
Martin Vala
Professor of Chemistry
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
David A. Micha
Professor of Chemistry and
Physics.
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Willis B. Person
Professor of Chemistry


37
holes drilled through the wall at 0 = n/4 with

respectively. No holes were located on the equatorial weld
bead (0 = n/2) Figure 4.3 shows the overall view of the
spherical resonator.
Transducers
The active element of the acoustic transducer is a 0.750
in. diameter piezoelectric ceramic bimorph (Vernitron,
Bedford, Ohio) made of PZT-5A material (lead zirconate-lead
titanate.) This ceramic was found to produce a signal of high
volume when driven with a sinusoidal waveform of 10 volts
peak-to-peak. It also has a high maximum operating temperature
of 250 C.(54) A transducer assembly was designed and
fabricated. It is shown in figure 4.4. The electrical
feedthrough made from brass was held in a Delrin insert to
prevent a short circuit. A cross-sectional view of the
transducer mounted on the resonator is also shown in figure
4.2. The transducer assemblies were mated to the resonator
through short % in. diameter tubes.
Two identical transducers located at right angles to each
other were used. This orientation has proved to yield a better
resolution of the radial modes of vibration due to diminishing
in intensity of some non-radial modes, p modes for example.
One transducer functions as a speaker. It is an input
transducer which mechanically deforms when a voltage is
applied. The other transducer functions as a microphone. It is
an output transducer which operates on a reverse mechanism of


no noon onnon noon
175
C...FORWARD SWEEP
DO 20 I=ISTART,ISTOP
CALL VOLTSO(IFLIP(I))
CALL SECS(T,TO)
TB=T
ASUM=0.0
PHSUM=0.0
C AGSUM=0.0
KOUNT=0
3000 CALL SECS(T, TO)
CALL Kel77b(AMP)
C CALL PHASEGET(AMPL,ANGL)
CALL PHASEGET(AMPL)
ASUM=ASUM+FLOAT(AMP)
PHSUM=PHSUM+AMPL
AGS UM=AGS UM+ANGL
WRITE(*,*) 'PHSUM=',PHSUM
WRITE(*,*) 'ASUM=',ASUM
WRITE(*,*) 'AGSUM=',AGSUM
KOUNT=KOUNT+1
TL=T-TB
IF(TL.LT.NDELY) GOTO 3000
CALL TEMPGET(TEMP)
CALL FREQGET(FREQ)
CALL PRESGET(P)
COUNT=COUNT+1
AM(COUNT)=INT2(ASUM/FLOAT(KOUNT))
PH(COUNT)=INT2((PHSUM/FLOAT(KOUNT))* 1.0E6)
AG(COUNT)=INT2(((AGSUM/FLOAT(KOUNT))
C-ASHIFT)*100.0)
WRITE (*,*) 'AM=',AM(COUNT)
WRITE (*,*) 'PH=',PH(COUNT)
WRITE (*,*) 'AG=',AG(COUNT)
FM(COUNT)=INT2(FREQ-FSHIFT)
TM(COUNT)=INT2((TEMP-TSHIFT)* 1000.0)
PR(COUNT)=INT2((P-PSHIFT)* 10.0)
LARGE=MAX0(LARGE,AM(COUNT))
IF (LARGE.EQ.AM(COUNT)) ICENT=I
CALL ADJUST(IG)
ISEN(COUNT)=INT1(IG)
20 CONTINUE
WRITE(10) AM,PH,PR,FM,TM,AG
WRITE(10) AM,PH,PR,FM,TM
CALL SENSET(IG)
WRITE(*,30) AM(120),PH(120),PR(120),FM(120),
CTM(120)+ITSHIFT*1000
WRITE(15,*) IG
C,AG(120)
30 FORMAT(IX,5110)
CALL SECS(T,TO)
IF(T.GT.INT(TIM)) GOTO 2000
Write(*,*) 'BLOCK NUMBER',KBLOCK


117
be zero. In that case P-T-C phase diagram curve would have
much more prominent tornado-in-action shape.


114
Figure 5.32. Comparison of experimental data for the
ethane and curve generated by renormalization group
theory equation
Figure 5.33. Three dimensional phase diagram of a
mixture of carbon dioxide and ethane 118
Figure 5.34. Three dimensional phase diagram of
ethane 119
Figure B.l. Block diagram of the gravimetric method. 129
Figure B.2. The hexagon reaction vessel for a
gravimetric analysis 130
Figure B.3. The high pressure gas sampling container. 131
Figure B.4. Calibration curve of ethane for gas
chromatography 135
Figure D.l. Conventional dead weight pressure gauge: A =
piston; B = cylinder; C = steel U tube; D = oil
injector; N, N' = indicator contact needles. . 148
Figure D.2. The present deadweight pressure gauge. 149
Figure D.3. The arrangement of the deadweight pressure
gauge 150
Figure D.4. An electroformed nickel bellow pressure
sensor 160
Figure D.5. The calibration curve for dead weight
pressure gauge 162
xii


65
derivative of resonance frequency with respect to temperature
versus system temperature for the data of figure 5.3. It gives
a precise value of the critical temperature as seen in figure
5.5 which is a portion of figure 5.4 with an expanded
temperature scale. The critical temperature obtained from this
plot is 304.2150.007 kelvin. Figure 5.6 displays the
relationship between the system pressure and temperature for
this same run. It shows a nearly straight line with no
evidence of phase separation.
Experiments were also run with subcritical density
charges. These show the phase boundary location (dew points)
as changes in slope of resonance frequency versus system
temperature. Table 5.1 gives a comparison of results found by
the present method with available literature values. The
present results are in good agreement with the other
experimental values and with predictions based on an equation
of state<55) and a correlation model developed at the National
Institute of Standards and Technology (NIST, formerly NBS)(56).
The acoustic data and NIST predictions are shown graphically
in figure 5.7. Table 5.2 gives a chronological collection of
experimental values of critical point parameters (Tc and Pc)
reported by several authors. The value obtained by this sonic
method is only 0.0016 percent different from the best value
obtained in the NIST laboratories (Morrison and Kincaid, 1984.
Reference 96).


34
change(53). This behavior was soon recognized as a valuable
tool for phase equilibrium studies. Figure 4.1 shows the
typical behavior in the variation of sonic speed with
temperature for such a mixture. The dew point is clearly
evident. Following this discovery it was decided to evaluate
the use of the acoustic resonator to locate the most
interesting feature of the phase diagram, namely the critical
point. The work reported in this dissertation deals primarily
with this difficult and challenging task.
Spherical resonator
The resonator cavity is the heart of the apparatus. To
permit its use at high pressure and in the environment of
corrosive gases, the spherical resonator was built from
stainless steel. Its side view is shown in figure 4.2.
The resonator cavity was fashioned from a welded
spherical, type 304 stainless steel tank approximately 0.203
m.(8 in.) in diameter with a 0.24 cm.(3/32 inch.) thick wall.
The tank is not perfectly spherical. This imperfection,
however, does not significantly degrade the data integrity as
long as appropriate measurement techniques are used as
discussed in the pervious section and proven to be valid by
the previous investigators.20'21 The tank was obtained from the
manufacturer (Pollution Measurement Corp.,Chicago, IL) with
only a single 0.6 cm.(^ in.) hole located at one pole (0=0
in spherical polar coordinates). A matching hole was drilled
through the opposite pole (0 = n) and two 0.953 cm (% in.)


84
temperature of phase change. Experimental results are
represented by points. The smooth curve is a result of a third
order polynomial curve fitting procedure. The corresponding
equation is as follows:
T = a + b*p + c*p2 + d*p3
where T = temperature, Celsius
p = pressure, psia
a = -8.137773 X 102
b = 2.020012
c = -1.616095 X 10-3
d = 4.260451 X 10'7
r = correlation coefficient
= 0.99749.
Figure 5.16 shows the relationship between starting pressure
and the speed of sound at the phase change. A sharp change in
the speed of sound at the critical point is evident compared
to the slow change of temperature shown figure 5.15. Figure
5.17 shows a coexistence curve. Note that the curve is so slim
that the bubble-point line almost overlaps the dew-point line.
Recall that a similar plot for a pure fluid yields a nearly
straight line (see figure 5.23 of ethane.) This clearly
reveals a property of this mixture as being nearly azeotropic.
It behaves much as if it were a pure fluid. The literature
value of the azeotrope composition of this binary mixture is
reported to be 0.255 mole fraction of ethane(108). This is very
close to the composition of the mixture studied in this work


APPENDIX A
CHARGING PRESSURE CALCULATIONS
Since in this work data were collected in the isochoric
mode, the density of the system was constant throughout each
run as well. To locate the critical without further addition
of gas required that the initial runs be made on samples at
supercritical densities. Density was not directly measured but
was infered from the AGA8 equation of state<5); therefore the
starting conditions were specified by composition,
temperature, and pressure. The procedures followed:
Pure Fluid lCarbon dioxide or Ethane)
1. Obtain literature values of the critical point
parameters (Tc, Pc, pc. )
2. Use the AGA8 equation to calculate a series of
densities corresponding to varying pressures in the vicinity
of the critical pressure at a charging temperature which must
be higher than the critical temperature by about five to seven
degrees Celsius. This set of densities should span the
critical density.
3. Plot the calculated densities as a function of the
corresponding charging pressures, then apply a suitable curve
fit regression, generally polynomial of the nth order, to
these data.
126


75
system was allowed to raise naturally from 15 C to room
temperature. This procedure was similar to the one utilized in
the experiments with carbon dioxide (and also with ethane as
will be seen later) except that the run went from low to high
temperature as the critical temperature of this particular
mixture is lower than room temperature. Again, the peak is
easier to track if the system temperature changes very slowly.
The speed of sound is very sensitive to temperature change in
the critical region. Note that a sharp turning point gives a
quite accurate value of the transition temperature but not
necessarily of speed of sound. Figure 5.12 is similar to
figure 5.11 but illustrates how other modes of vibration
reveal the same critical point temperature. In principle every
normal mode resonance exhibits a discontinuity at the critical
point, so there are very many possible indicators which are
equally capable of locating the critical point.
Figure 5.13 shows the behavior of pressure versus
temperature for several isochoric runs representing different
fluid densities. The upward direction of the breaks in these
curves shows them each to result from approaching the phase
boundary on cooling from a supercritical density. The breaks
mark individual bubble points on the phase envelope.
Approaches from subcritical densities would break downward at
the corresponding dewpoints, and the critical isochore would
exhibit no break at all. These facts are helpful in searching


108
700
600
500
400
300
200
100
0
1090 1100 1110 1120 1130 1100 1110 1120 1130 1140 1110 1120 1130 1140 1150
Frequency, Hi
Figure 5.29. Movement of tracked peak and its neighbor below
the critical temperature. These curves are numbered in the
proper time sequence but many curves showing only one strong
peak have been omitted.


Figure 4.2. A side view of the spherical resonator
equipped with the transducers 35
Figure 4.3. The spherical resonator 36
Figure 4.4. The transducer (cross-sectional view). . 38
Figure 4.5. The mixing control unit 40
Figure 4.6. The Instrumental Setup 43
Figure 4.7. The experimental setup for C02 and C02~C2H6
mixture 44
Figure 4.8. The first experimental setup for C2Hg. . 45
Figure 4.9. The second experimental setup of C2Hg. . 46
Figure 4.10. Series of scanning routine 55
Figure 4.11. Two experimental approaches of resonance
frequency determination: the maximum amplitude
approach and the voltage phase change approach. 58
Figure 4.12. Flow chart of experimental scheme and data
processing. The broken-line boxes represent the
experimental systems of interest. The thick-line
boxes represent the computer programs 59
Figure 5.1. Relationship between resonance frequency and
temperature of carbon dioxide for an isochore near
its critical density 61
Figure 5.2. Relationship between first radial mode
resonance frequency and temperature of carbon
dioxide for an isochore near the critical
density 62
Figure 5.3. Relationship between the speed of sound and
temperature of carbon dioxide for an isochore near
the critical density 62
Figure 5.4. First derivative of resonance frequency
versus temperature 64
Figure 5.5. Enlarged temperature scale of figure 5.4
showing the critical temperature to be at 304.215
K 64
Figure 5.6. Pressure and temperature behavior of carbon
dioxide for isochore near the critical density. 66
ix


195
15. Chen Z. Y., Abbaci A., Tang S., and J. V. Sengers,
Phvs. Rev. A. 42, 8, 1990.
16. Kumar Sanjay, Gas Production Engineering. Gulf
Publishing Company, Houston, Texas, 1987, pp. 46, 51-
52.
17. Brown G. G., Katz D. L., Oberfell G. B., and R. C.
Alden, Natural Gasoline and Volatile Hydrocarbons NGAA,
Tulsa, Oklahoma.
18. Thomas L. K., Hankinson, R. W., and K. A. Philips, J.
Pet. Tech.. 22, 889, 1970.
19. White D., Freidman A., and H. Johnson, J. Am. Chem.
Soc.. 72, 3565, 1950.
20. Sona C. F., An Acoustic Study of Gases and Vapors,
Ph.D. Dissertation, The University of Florida, 1986.
21. Reed Kyle, Thermophysical Properties of Hydrocarbons
Determined Using a Spherical Resonator., M.S. Thesis,
The University of Florida, 1990.
22. Colgate S. 0., Sivaraman A., and Reed K., Acoustic
Determination of The Thermodynamic Reference State Heat
Capacity of n-Heptane Vapor., Research Report RR-109,
Project 831-83 thru 86, Gas Processors Association,
Tulsa, Oklahoma, 1987.
23. Colgate S. 0., Sivaraman A., and Reed K., Reference
State Heat Capacities of Three c-8 Compounds, Research
Report RR-123, Project 831, Gas Processors Association,
Tulsa, Oklahoma, 1989.
24. Colgate S. 0., Sona C. F., Reed K. R. and A.
Sivaraman, J. Chem. Eng. Data. 35, 1, 1990.
25. Colgate S. 0., Sivaraman A., and K. R. Reed, J. Chem.
Thermodynamics., 22, 245, 1990.
26. Colgate S. 0., Sivaraman A., and K. R. Reed, Fluid
Phase Equilibria. 60, 191, 1990.
27. Colclough A. R., Metrologa. 9, 75, 1973.
28. Temkin Samuel, Elements of Acoustics, John Wiley & Sons
company, New York, 1981.
29. Levine Ira N., Quantum Chemistry, 3rd. Ed., Allyn and
Bacon Inc., Boston, Massachusette, 1983.


Table 6.1. Comparison of Critical Temperatures and Pressures
No.
System
Critical
Temperature (K)
Percent
Different
Sources
This
work
Literature
1
Carbon Dioxide
304.215
0.007
304.206
0.001
0.003
Morrison &
Kincaid
(1984)
2
Ethane
305.370
0.007
305.379
-0.003
Morrison
(1981)
3
74.25 mole % C02
25.75 mole % C2Hg
293.134
0.007
293.072*
0.021
Morrison &
Kincaid
(1984)
Critical Pressure
Psia
1
Carbon dioxide
1069.23
3.00
1070.96
0.01
-0.01
Morrison &
Kincaid
(1984)
2
Ethane
707.78
3.00
707.8
0.1
-0.002
Khazanova &
Sominskaya
(1971)
3
74.25 % mole C02
25.75 % mole C,Hc
916.76
3.00
N/A
N/A
N/A
Extrapolated value


n o o
168
KLA = IBFIND ('K195A ')
124 = 2**24
116 = 2**16
18 = 2**8
1004 CALL IBRD (KLA,RD,12)
15 = RD(2)/I24
16 = (RD(2)-I5*l24)/I16
17 = (RD(2)-I5*l24-l6*I16)/I8
188 = RD(2)-I5*l24-l6*I16-I7*l8
19 = RD(3)/I24
110 = (RD(3)19*124)/I16
111 = (RD(3)-I9*I24-I10*I16)/l8
112 = RD(3)-I9*I24-I10*I16-I11*I8
WRITE(TEMP,'(8A1))char(188),char(17),char(l6),char(I5),
CCHAR(112),CHAR(Ill),CHAR(I10),CHAR(19)
READ(TEMP,'(F8.3)',ERR=1004) R
R0=99.99
ALPHA=0.0039042
DELTA=1.5205
c R0=100.06
c ALPHA=.0039046
c DELTA=1.5205
ALDEL=ALPHA*DELTA
RC=R/R0-1.0
PART1=ALPHA+ALDEL/10 0.0
PART2=(ALDEL/100.O+ALPHA)**2
PART3=4.0*RC*ALDEL/10000.0
PART4=2.0*ALDEL/10000.0
TEMP 2=PART1-SQRT(PART2-PART 3)
T=TEMP 2/PART4
RETURN
end
SUBROUTINE HP(SF,SPF,SWEEP)
common /ibglob/ ibsta,iberr,ibcnt
COMMON GROUP 1.
integer*4 cmd(10),rd(512),wrt(512),IP(4)
character*1 ch(40)
CHARACTER*9 STARTF,STOPF
CHARACTER*2 SWEEP
character*8 bname,bdname
'COMMON GROUP 2.'
call ibinit(ibsta)
IP(1) = 1
IP(2) = 2**8
IP(3) = 2**16
IP(4) = 2**24
IHP = IBFIND ('hp3325 ')
CH(1) = 'F'
CH(2) = 'U'


57
GETSPG. This program utilizes the principle of relative
measurement mentioned earlier in the theory section. Also the
theoretical speeds of sound were calculated from the AGA8
equation through a computer program call TOTDAT for comparison
purposes. Finally the results were manipulated and graphically
displayed using popular software including: LOTUS-123,
QUATTRO, GRAFTOOL, and GRAPHER. Figure 4.12 shows a schematic
diagram of the entire process of data collection.


ACOUSTIC DETERMINATION OF PHASE BOUNDARIES
AND CRITICAL POINTS OF GASES:
co2, co2-c2h6 mixture, and c2h6
By
CHADIN DEJSUPA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1991


9
Dividing both sides by k2r2 gives
_1 d^f + _2_ df + L 1(1 +1)'
k2 dr2 k2r ^
k2r2
f 0
(2.13)
Changing the form of the function from f(r) to F(Z) where Z =
kr and rearranging yields:
d2F(Z> + idF(Z) + L £(£+1)\F(Z) o. (2.14)
dZ2 Z^Z^ \ Z I
Particular solutions are the spherical Bessel function of
three kinds:
First kind,
F(z) jf(z) -
. J i (Z).
\ 2z (+-)
2
Second kind,
F(z) y.(z)
" Y
\ "2"z (i + i)
2
(Z)
(2.15)
(2.16)
Third kind,
F(z) h*1 (z) j.(z) + iy.(z)
H<1> i (Z)
N ZZ (£ + -)
2
F (z) h2)(z) jf(z) iy,(z) -
(2.17)
H<2> i (Z)
N ZZ <* + _)
2
where J(Z), Y(Z), and H(Z) is Bessel function of the first,
second, and third kind, respectively.
With, for example,
o ( z)
jl(z)
y0 -
Yi ~
sin(z)
z
[sin(z) z-cos(z)]
z2
cos(z)
z
[cos(z) + z- sin(z) ]
(2.18)


176
GOTO 1000
2000 Write(3,600) KBLOCK
600 FORMAT(IX,' NUMBER OF BLOCKS OF DATA='/I5)
CLOSE(10)
CLOSE(3)
c CLOSE(15)
END
C
C
C
FUNCTION IFLIP(INPUT)
IFLIP=-INPUT+4095
RETURN
END
C
C
C
SUBROUTINE VOLTS0(ITEMP)
INTEGER*2 ADAPT,DEVICE,CTRL,STAT,VDACHl
VDACHl = INT2(ITEMP)
ADAPT=0
DEVICE = 9
CTRL = 0
STAT = 0
CALL AOUS (ADAPT,DEVICE,0,CTRL,VDACHl,STAT)
RETURN
END
C
C
C
SUBROUTINE VOLTS1(ITEMP)
INTEGER*2 ADAPT,DEVICE,CTRL,STAT,VDACHl
VDACHl = INT2(ITEMP)
ADAPT=0
DEVICE = 9
CTRL = 0
STAT = 0
CALL AOUS (ADAPT,DEVICE,1,CTRL,VDACHl,STAT)
RETURN
END
C
C
C
SUBROUTINE PHASEGET(AMPL)
common /ibglob/ ibsta,iberr,ibcnt
INTEGER*4 11,188,1166,124
DIMENSION 1(20)
'COMMON GROUP 1.'
integer CR
integer cmd(10),rd(512),wrt(512),IP(4)
CHARACTER*1 CH(20)
character*8 bname,bdname


160
Figure D.4. An electroformed nickel bellow pressure sensor,
over the range of pressures tested. The effective area
inferred by the calibration is 0.0734 inch2.
The disadvantage of this gauge relative to the piston
type is the need for calibration. The advantage is the
intrinsic greater sensitivity available with the bellows. The
present sensor responds to mass differences of 0.05 gram which
corresponds to pressure differences of 0.0015 psi.
Conclusion
This work shows that even small laboratories can have
reliable pressure standards at relatively low cost. The gauges
described here were constructed from readily available
materials at a total cost of less than $300 (excluding the
laser.) Commercial dead weight pressure gauges cost several


Ill
stronger than peak one leading to switching of the tracked
peak from peak one to peak two. The window then shifted from
peak one which became lost. The peak in curve 16 is peak one
but that of curve 17 is peak two. This is apparently so since
the peak in curve 17 is lower in frequency than the peak in
curve 16.) The reverse of this phenomenon happened around 29.5
C where peak one re-entered the window as the dominant
signal. Finally, these two peaks merged together at around 29
C. Data were not collected below about 29 C, so the
subsequent behavior of these peaks was not observed. This type
of behavior is interesting and unexpected, but it is not
central to the objective of the present research. It is
probably associated with the fact that below the critical
temperature there are two phases present and their relative
proportions in the resonator cavity change according to the
effects of the magnetically driven circulating pump.
The plausible routes of these two peaks based on the
above situations are depicted in figure 5.30. A full
interpretation of these phenomena would require further
experimentation to characterize the contents of the cavity in
the two phase region. In the early stage of phase separation
the bubbles or dew drops are believed to be small and
entrained in the dominant phase as foam or fog, so the
location of the phase transition is not obscured by massive
phase separations which occur well away from the phase
boundary.


n o o
181
'COMMON GROUP 1.'
integer*4 cmd(10),rd(512), wrt(512)
character*8 bname,bdname,TEMP
'COMMON GROUP 2.'
K195AAD=IBFIND('K195A ')
1004 CALL IBRD (K195AAD,RD,12)
15 = RD(2)/I24
16 = (RD(2)-I5*l24)/I16
17 = (RD(2)-I5*l24-I6*I16)/I8
188 = RD(2)-15*124-16*116-17*18
19 = RD(3)/12 4
110 = (RD(3)-l9*l24)/ll6
111 = (RD(3)-l9*l24-I10*I16)/l8
112 = RD(3)-I9*I24-I10*I16-I11*I8
WRITE(TEMP,'(8A1)')char(188),char(I7),
Cchar(16),char(15), CCHAR(I12),CHAR(I11),
CCHAR(I10),CHAR(19)
READ(TEMP,'(F8.3)',ERR=1004) R
R0=99.98
ALPHA=0.0039076
DELTA=1.5205
ALDEL=ALPHA*DELTA
RC=R/R0-1.0
PART1=ALPHA+ALDEL/100.0
PART2=(ALDEL/100.0+ALPHA)**2
PART3=4.0*RC*ALDEL/10000.0
PART4=2.0*ALDEL/10000.0
TEMP 2=PART1-SQRT(PART2-PART3)
T=TEMP2/PART4
RETURN
end
c
c
c
subroutine error
common /ibglob/ ibsta, iberr, ibent
write (*,100) ibsta,iberr,ibent
100 format (' Error',i6,i6,i6)
return
end
SUBROUTINE PRESGET(P)
common /ibglob/ ibsta,iberr,ibent
COMMON 11,18,116,124
INTEGER*4 11,18,116,124
'COMMON GROUP 1.
integer BMAD
integer*4 cmd(10),rd(512),wrt(512)
character*8 bname,bdname
character*7 Pres


APPENDIX C
SIMPLEX
There are three programs for searching for minimum points
of speed of sound (points along coexistence curve). The first
is the simplex optimization program used to search for eleven
coefficients in an equation (5). The second program is used to
calculate speed of sound with coefficients found in program
one. The last program finds the common point of two curves
generated from programs one and two.
Program I. Simplex Optimization
C234567
IMPLICIT REAL*8 (A-H,0-Z)
DIMENSION C(4000),T(4000),B(20,20)
DIMENSION SSRSD(500)
WRITE(*,*) 'INPUT MOLECULAR WEIGHT IN KG/MOLE'
READ (*,*) W
OPEN (10,FILE='DATA.RAW',STATUS='OLD')
L=1
20 READ(10,*,END=30) T(L),C(L)
L=L+1
GOTO 20
30 CLOSE(10)
L=L-1
WRITE(*,*) 'L=',L
WRITE(*,*) 'W=',W
OPEN (40,FILE='INITIAL.DAT',STATUS='OLD')
READ(40,*) ((B(I,J),1=1,10),J=1,11)
CALL INITLCOOR (B,SSRSD,L,T,C,W)
OPEN (5,FILE='START.DAT',STATUS='NEW')
DO 80 J=l,ll
DO 90 1=1,10
WRITE (5,*) B(I,J)
90 CONTINUE
WRITE (5,*) 'SSRSD',J,'=',SSRSD(J)
WRITE (5,*) '
80 CONTINUE
136


191
close(60)
close(50)
300 CLOSE(40)
CLOSE(30)
END
SUBROUTINE QAND(BS,BS1,BS2,BSD,
CCS,CS1,CS2,TS,T,B,B1,B2,BD,C,C1,C2)
IMPLICIT REAL*8 (A-H,0-Z)
REAL*8 B(l),Bl(l),B2(1),T(1),C(1),C1(1),C2(1),BD(1)
REAL*8 M
DO 20 1=2,74
IF (TS.GT.T(I-1).AND.TS.LT.T(I)) THEN
M=(TS-T(I-1))/(T(I)-T(I-l))
BS=B(1-1)+M*(B(I)-B(1-1))
BSl=Bl(1-1)+M*(B1(I)-Bl(1-1))
BS2=B2(1-1)+M*(B2(I)-B2(I1))
BSD=BD(I1)+M*(BD(I)-BD(I -1))
CS=C(1-1)+M*(C(I)-C(1-1))
CS1=C1(1-1)+M*(Cl(I)-Cl(1-1))
CS2=C2(1-1)+M*(C2(I)-C2(I-1))
RETURN
ENDIF
20 CONTINUE
WRITE(*,*)'TSTAR OUT OF RANGE'
RETURN
END
C
SUBROUTINE VERVOL(P,V,T,B,CV)
IMPLICIT REAL*8 (A-H,0-Z)
R=8.20575D-02
TOL=l.OD-16
C INPUT P IN ATM
C INPUT T IN KELVIN
V=R*T/P
10 VN=R*T/P*(1.0D 00 + B/V + CV/V/V)
TEST=V/VN
IF(TEST.GT.1.0) THEN
TEST=1.0D 00 1.0D 00/TEST
ELSE
TEST=1.0D 00 -TEST
ENDIF
V=VN
IF(TEST.GT.TOL) GOTO 10
RETURN
END
C
SUBROUTINE SPEED(C,T,BS,BS1,BS2,BSD,CS,CS1,CS2,VS)
IMPLICIT REAL*8 (A-H,0-Z)
REAL*8 M,DSQRT
M=39.948D-03
R=8.31441D 00


130
Figure B.2. The hexagon reaction vessel for a gravimetric
analysis.


21
Figure 3.1. The p-v-T behavior of pure fluid. In the center is
sketched the surface p = p(v,T). [From Hirschfelder Joseph 0.,
Curtiss Charles F., and Bird R. Byron. Molecular Theory of
Gases and Liquids. Copyright 1954 by John Wiley & Sons, Inc.
Reprinted by permission of John Wiley & Sons, Inc.]
point of initial vaporization when the pressure of the liquid
is reduced) curve, ADC, and the dew-point (the point of
initial condensation when the pressure of the gas is
increased) curve, BEC. These two curves meet at the critical
point C which, belonging to both curves, indicates the
identity of the liquid and vapor phases. At this point,
viewing from the isothermal perspective,
/ap\ -
' '
a2P
Mt
CN
>
T
0.
(3.1)


Injected volume, ul
Figure B.4. Calibration curve of ethane for gas chromatography.
h-
CO
cn


N
K
£
§
g
cu
(X|
1500
1400
1300
1200
1100
1000
900
Temperature, Celsius
Figure 5.10. Typical behavior of resonance frequency versus temperature of the carbon
dioxide-ethane mixture near critical density.
4=.


non onn noo
179
IF (1(7).EQ.51) AMPL=AMPL/10.0**3.0
IF (1(7).EQ.54) AMPL=AMPL/10.0**6.0
IF (1(7).EQ.57) AMPL=AMPL/10.0 * 9.0
END IF
IF (ABS(AMPL).GT.20000.0) GOTO 1005
RETURN
end
SUBROUTINE Ke177b(MV)
common /ibglob/ ibsta,iberr,ibcnt
COMMON 11,18,116,124
INTEGERS 11,18,116,124
COMMON GROUP 1.'
integer*4 cmd(10),rd(512),wrt(512)
character*8 bname,bdname,TEMP
'COMMON GROUP 2. '
K177BAD=IBFIND('K177B ')
1006 call ibrd (K177BAD,RD,6)
11 = rd(l)/i24
12 = (rd(1)-il*i24)/il6
13 = (rd(1)-il*i24-i2*il6)/i8
14 = rd(1)-l*i24-i2*il6-i3*i8
15 = rd(2)/i24
16 = (rd(2)-i5*i24)/il6
17 = (rd(2)-i5*i24-i6*il6)/i8
i88 = rd(2)-5*i24-i6*il6-i7*i8
WRITE(TEMP,(8A1)') CHAR(I4),CHAR(I3),CHAR(I2),
CCHAR(Il),CHAR(188),CHAR(17),CHAR(16),CHAR(15)
READ(TEMP,'(16),ERR=1006) MV
IF (ABS(MV).GT.20000) GOTO 1006
RETURN
end
SUBROUTINE FREQGET(FREQ)
common /ibglob/ ibsta,iberr,ibcnt
COMMON 11,18,116,124
INTEGER*4 11,18,116,124
INTEGER FREQ
COMMON GROUP 1.'
integer SEAD
integer*4 cmd(10),rd(512),wrt(512)
character*8 bname,bdname
CHARACTER*7 TEMP
'COMMON GROUP 2.'
i3 = ichar(3')
iO = ichar(O)
SENAD=IBFIND('SENCORE ')
1006 call ibrd (SEAD,RD,17)


n o o
140
C 600 CONTINUE
C GOTO 350
C ELSE IF (I.EQ.13.AND.TC.GT.294.15) THEN
C WRITE(*,*) UPPER BOUNDARY UPPER BOUNDARY
C TC=294.15
C FACTOR=(TC-B(7,11))/(B(7,13)-B(7,11))
C DO 700 J=l,10
C B(J,13)=FACTOR*(B(J,13)-B(J,11))+B(J,ll)
C 700 CONTINUE
C GOTO 350
C END IF
CALL RESIDUAL(Al,A2,A3,A4,A5,A6,TC,ALPHA,BETA
C ,DELTA,C,T,L,ST,W)
SSRSD(I)=ST
C WRITE( * ) 'SSRSD',1,=',SSRSD(I)
200 CONTINUE
RETURN
END
SUBROUTINE SHELL(B,SSRSD,WORST)
IMPLICIT REAL*8 (B,D-H,0-Z)
DIMENSION SSRSD(500),B(20,20)
INTEGER A,C
WORST=SSRSD(11)
C WRITE(*,*) 'SHELL'
J=1
N=11
M=N
90 IF (M.GE.l) THEN
J=1
M=M/2
K=N-M
50 IF (J.EQ.l) THEN
J=0
DO 60 1=1,K
L=I+M
IF (SSRSD(I).GT.SSRSD(L)) THEN
V=SSRSD(I)
SSRSD(I)=SSRSD(L)
SSRSD(L)=V
DO 70 C=l,10
W=B(C,I)
B(C,I)=B(C,L)
B(C,L)=W
70 CONTINUE
J=1
END IF
60 CONTINUE
GOTO 50
END IF


n n n n non
138
CALL CNTW(B,SSRSD,C,T,L,W)
C WRITE(*,*) 'PASS CNTW'
END IF
GOTO 50
END IF
CLOSE(40)
100 OPEN (60,FILE='COOR.OUT',STATUS='NEW')
DO 800 J=l,ll
DO 900 1=1,10
WRITE (60,*) B(I,J)
900 CONTINUE
WRITE (60,*) 'SSRSD',J,'=',SSRSD(J)
WRITE (60,*) '
800 CONTINUE
WRITE(60,*) 'NUMBER OF ITERATIONS = ',KOUNT
CLOSE(60)
STOP
END
SUBROUTINE INITLCOOR (B,SSRSD,L,T,C,W)
IMPLICIT REAL*8 (A-H,0-Z)
DIMENSION B(20,20),T(4000),C(4000),SSRSD(500),A(10)
DO 10 1=1,11
DO 20 J=l,10
A(J)=B(J,I)
20 CONTINUE
A1=A(1)
A2=A(2)
A3=A(3)
A4=A(4)
A5=A(5)
A6=A(6)
TC=A(7)
ALPHA=A(8)
BETA=A(9)
DELTA=A(10)
CALL RESIDUAL(Al,A2,A3,A4,A5,A6,TC,ALPHA,BETA
C,DELTA,C,T,L,ST,W)
SSRSD(I)=ST
WRITE(*,*) 'SSRSD',I,'=',SSRSD(I)
10 CONTINUE
RETURN
END
SUBROUTINE RESIDUAL(Al,A2,A3,A4,A5,A6,TC,ALPHA,BETA
C,DELTA,C,T,L,ST,W)
IMPLICIT REAL*8 (A-H,0-Z)
REAL*8 DABS,DSQRT


132
slowly to the hexagon reaction vessels. Opening of valve 5 was
judged from the rate of bubbles rising up in the absorber
tubes. If valve 5 was opened too wide, carbon dioxide gas
would not have time to react efficiently with sodium hydroxide
in the absorber. The completeness of the reaction was checked
by passing exiting gas from the second absorber array through
a lime water, saturated Ca(OH)2 solution. Turbidity of the
solution due to forming of CaC03 indicates incompleteness of
the reaction. Once bubbles stopped rising up (about 2-3 hours)
argon was delivered to the system through valve A to flush out
all sample gas remaining in the lines for one-half hour. After
that, screw clamps of the entrance and exit tygon tubes to the
absorber array were tightly closed. The vessels were
disconnected from the system and weighed on an analytical
balance. During the entire course of these experiments the
lime solution remained clear. The total absorption time was
about 3-4 hours. Note that before analyzing the sample gas a
blank test was performed by continuously flowing argon through
the system for 3-4 hours. No significant change in weight
(comparable with the uncertainty of the balance, 1 mg. ) of the
- absorbers before and after passing argon was observed. Despite
its tedious procedure this method worked reasonably well,
provided that sufficient care was exercised.
Gas Chromatography Method
Description of standard procedures for performing gas
chromatography can be found elsewhere. (140'141) A calibration


12
Figure 2.1. The first six orders of the spherical Bessel
function. Points where slope equals zero yield eigen values,
£l,n*


185
AMAX=0.O
XMIN=200000.O
DO 42 ID=1,IDELY
CALL KEITH(MV)
42 CONTINUE
DO 40 J=ISTART,POINTS,INDEX
AVG=0.0
DO 41 JA=1,NA
300 CALL KEITH(MV)
KOUNT=KOUNT+1
IF(KOUNT.EQ.2*NA) THEN
IG=IG+1
CALL GAINSET(IG)
ENDIF
IF(KOUNT.GT.3*NA) THEN
KOUNT=0
GOTO 500
ENDIF
IF(MV.GT.20000.0R.MV.LT.100) GOTO 300
KOUNT=0
AVG=AVG+FLOAT(MV)
41 CONTINUE
AMP=AVG/FLOAT(NA)
DX(J)=FLOAT(J)
DY(J)=AMP
IF(AMP.GT.AMAX) THEN
AMAX=AMP
JMAX=J
ENDIF
IF(AMP.LT.XMIN) XMIN=AMP
4 0 CONTINUE
IF(INDEX.EQ.1) THEN
NPLOT=l
ELSE
NPLOT=0
ENDIF
YMAX=AMAX
Q* ************************************************** 1t * *
c CALL PLOT(M,DX,DY,XMAX,XMIN,YMAX,YMIN,NPLOT)
C ONLY USED FOR CENTERING
Q*******************************************************
IF(INDEX.EQ.l) THEN
INDEX=-INDEX
TEMP=SF
SF=SPF
SPF=TEMP
ISTART=POINTS
POINTS=l
JFMAX=JMAX
AVGA=AMAX
CALL KEITHA(TI)
IF(TI.LT.10.0) GOTO 169


16
i
(2.33)
(2.34)
where I refers to the system under investigation. The speed of
sound of argon is calculated from the virial equation
truncated after the third term as follows*35:
(2.35)
where y = heat capacity ratio (Cp/Cv)
MAr = Molecular mass of argon
R = Gas constant
T = Absolute Temperature
v = molar volume
AX(T) = Second acoustic virial coefficient
A2(T) = Third acoustic virial coefficient.
The effective eigenvalue determined this way is not exact, but
the effect on sonic speed measurements is small and of no
consequence for the accurate determination of phase
boundaries.
The practical advantages of spherical resonators is
realized for the purely radial modes of vibration, which
involve no tangential motion of the gas with respect to the
resonator wall, i.e. no viscous drag effect. In addition, the


Table 5.11. Coefficients Obtained by Simplex
Optimization Method 115
Table 6.1. Comparison of Critical Temperatures and
Pressures 123
Table B.l. Data for calibration curve of ethane. . 134
Table D.l. Experimental Test of Dead Weight Pressure
Gauge 158
vii


36
Figure 4.3. The spherical resonator.


UNIVERSITY OF FLORIDA
3 1262 08556 6999


113
TEMPERATURE, C
Figure 5.31. Experimental curve of ethane showing dynamic
behavior of speed of sound near critical point.
author. It has a root from rigorous mathematical derivations
by Wegner<136) and further by Green and Ley-koo(137,138) based on
renormalization group theory.
M
- + A2T^+A + A3T3_a+/3 + A4Tr
+ A,T2 + A,T3
5 r 6 r
(5)
where C = Speed of sound, m/sec
R = Gas constant, J/sec
Tc = Critical Temperature, K
M = Molecular mass, Kg/mole
Ai = Equation coefficients
Tr = Reduced Temperature
a, 13, A = Critical exponents.


146
WRITE (*,*
GOTO 10
END IF
) DEL,T,TL
90
WRITE (*,*)
GOTO 97
'****',T,E
95
WRITE (*,*)
GOTO 97
****fT,F
97
WRITE (*,*)
'>UNABLE TO REFINE FURTHER'
100
WRITE (*,*)
T,T-273.15,DEL
R=R*(SQRT(8
.3144/0.03007))
S=S*(SQRT(8
.3144/0.03007))
WRITE (*,*)
STOP
END.
R,S


nnnnnn non n nn
139
DIMENSION C(4000),T(4000)
GAS=8.3144
W=40.539/4000.0
W=0.040441
CON=GAS*TC/W
ST=0.0
DO 10 1=1,L
TR=DABS( (T (I)-TC)/TC)
RS=(C(I)/DABS(DSQRT(CON)))-(Al*TR**BETA+A2*TR**(BETA
C+DELTA)+A3 *TR** (1.0-ALPHA+BETA) + (A4*TR) + (A5*TR**2.0)
C+(A6*TR**3.0))
WRITE(*,*) 'RS=',RS
RS=RS**2
ST=ST+RS
10 CONTINUE
RETURN
END
SUBROUTINE REFLEX (SSRSD,B,L,T,C,W)
IMPLICIT REAL*8 (A-H,0-Z)
REAL*8 DSQRT
DIMENSION B(20,20),C(4000),T(4000),SSRSD(500),A(10)
DO 10 J=l,10
CUM=0.0
DO 50 1=1,10
CUM=CUM+B(J,I)
50 CONTINUE
B(J,12)=CUM/10.0
B(J,13)=B(J,12)+(B(J,12)-B(J,11))
10 CONTINUE
DO 200 1=12,13
350 DO 300 J=l,10
A(J)=B(J,I)
300 CONTINUE
Al=A(1)
A2=A(2)
A3=A(3)
A4=A(4)
A5=A(5)
A6=A(6)
TC=A(7)
ALPHA=A(8)
BETA=A(9)
DELTA=A(10)
IF (I.EQ.13.AND.TC.LT.292.15) THEN
WRITE(*,*) 'LOWER BOUNDARY LOWER BOUNDARY'
TC=292.15
FACTOR=(TC-B(7,11))/(B(7,13)-B(7,ll))
DO 600 J=l,10
B(J,13)=FACTOR*(B(J,13)-B(J,11))+B(J,11)


54
of the first radial mode peak (peak 1 in figure 4.11). Each of
these was investigated closely as shown in the subsequent four
rows. Each figure in the third row is a spectrum of a single
peak of row two scanned in a much narrower frequency range.
Frequencies found for each peak in this row were used to
calculate the second, third, and forth radial frequencies.
Peaks corresponding to these calculated frequencies were
searched as shown the figures in the forth, fifth, and sixth
row. Apparently, the third peak (peak 3 in figure 4.10) is not
the correct peak as it failed to predict the location of
higher radial modes. Peaks 1 and 2 give similar predictions
since they lie very close together; about 5 Hz apart.
Predictions of overtones based on peak 1 were in better
agreement with experiment than from peak 2 therefore peak 1 is
correctly interpreted as the first radial mode peak. Note that
the SCAN program was used for pure carbon dioxide and for the
C02-C2H6 mixture experiments. For pure C2H6 experiment this
process was performed manually via a voltage-control-generator
(VCG) box since the instrumental setup was altered.
Having established the location of the peak to be
tracked, data collection was begun. For pure C02 and for the
mixture of C02-C2H6 experiments programs called VIKING and
VIKAN were used. In this version only data corresponding to
the maximum amplitude were recorded for each sweep frequency
range. For the pure C2H6 experiment, however, data of the
entire sweep frequency range were recorded (via the program


127
4.Charging pressure corresponding to literature critical
density is now calculated from the best curve fitting equation
found in step 3. This value plus about 10 psia is taken as a
charging pressure of fluid at the starting temperature. The
extra 10 psia is added as a safety factor due to the
possibility of error in the literature critical density.
Fluid Mixture
1. Decide on the composition of carbon dioxide-ethane
mixture to be studied.
2. Approximate charging pressure at starting temperature
from closest literature critical point parameters available.
3. Calculate the density of the gas mixture corresponding
to the charging pressure estimated in step 2 at the starting
temperature and mole fractions of carbon dioxide and ethane
using the AGA8 equation.
4. Calculate the density of ethane from the density of
the gas mixture obtained from step 3.
5. Follow procedures in step 2 to 4 for a pure fluid. The
calculated density obtained in step 3 for this procedure is
taken as the literature critical density called for in step 4
of pure fluid. The resulting pressure is taken as the target
charging pressure of ethane.
6. Ethane was charged first to the pressure calculated in
the previous step. Carbon dioxide was subsequently introduced
to the resonator while mixing until the desired total pressure
was obtained (estimated value in step 2.)


REFERENCES
1. Atkins P. W., Physical Chemistry, 3rd. ed., W. H.
Freeman and Company, New York, 1986.
2. Benedict M., Webb G. B., and L. C. Rubin, J. Chem.
Phvs., 8, 334, 1940.
3. Redlich 0. and J. N. S. Kwong, Chem. Reviews. 44, 233,
1949.
4. Peng, D.-Y., and D. B. Robinson, Ind. Eng. Chem. Fund..
15, 59, 1976.
5. Starling K. E. editor, AGA8 "Compressibility and
supercompressibility for natural gas and other
hydrocarbon gases", AGA Transmission measurement
committee report number 8 (AGA8), American Gas
Association, 1986.
6. Vimalchand P. and Marc D. Donohue, Ind. Eng. Chem.
Fundam.. 24, 246, 1985.
7. Vidal J., Ber. Bunsenqes. Phvs. Chem.. 88, 784, 1984.
8. Vera J. H. and J. M. Prausnitz, Chem. Eng. J.. 3, 1,
1972.
9. Prausnitz J. M., Int. Chem. Eng.. 19, 3, 401, 1979.
10. Inomata H., Arai K., and Saito S., J. Chem. Eng.
(Japan). 22, 6, 695, 1989.
11. Kreglewski Aleksander, Equilibrium Properties of Fluids
and Fluid Mixtures, Texas A&M University Press, College
Station, 1984, chapter 6.
12. Albright P. C., Edwards T. J., Chen Z. Y., and J. V.
Sengers, J. Chem. Phvs.. 87, 3, 1717, 1987.
13. Olchowy G. A. and J. V. Sengers, Phvs. Rev. Lett.. 61,
1, 15, 1988.
14. Chen Z. Y., Albright P. C., and J. V. Sengers, Phvs.
Rev. A. 41, 6, 3161, 1990.
194


non
170
IF((I1)*4+J.LEM) THEN
WRT(I) =WRT(I) +ICHAR(CH((I-l)*4+J))*IP(J)
ENDIF
40 CONTINUE
30 CONTINUE
CALL IBWRT(IHP,WRT,M)
RETURN
end
c
c
c
subroutine error
common /ibglob/ ibsta, iberr, ibcnt
write (*,100) ibsta,iberr,ibcnt
100 format (' Error',i6,i6,i6)
return
end
SUBROUTINE TEMPSET(VS)
common /ibglob/ ibsta,iberr,ibcnt
'COMMON GROUP 1.'
integer* 4 cmd(10),rd(512),wrt(512),IP(4)
CHARACTER*1 CH(20)
CHARACTER*6 VOLTS
character*8 bname,bdname
'COMMON GROUP 2.*
LK = IBFIND ('LOCKIN )
IP(4) = 2**24
IP(3) = 2**16
IP(2) = 2**8
IP(1) = 1
WRITE(VOLTS,(F6.3)') VS
CH(1) = X
CH(2) = '6'
CH(3) =
READ(VOLTS,'(Al)') CH(4)
READ(VOLTS,'(IX,Al)') CH(5)
READ(VOLTS,'(2X,Al)') CH(6)
READ(VOLTS,'(3X,Al)') CH(7)
READ(VOLTS,'(4X,Al)') CH(8)
READ(VOLTS,'(5X,Al)') CH(9)
DO 10 1=1,3
WRT(I) = 0
DO 20 J=1,4
K=(1-1)*4+J
IF(K.LE.9) THEN
WRT(I) = WRT(I) + ICHAR(CH(K))*IP(J)
ENDIF
20 CONTINUE
10 CONTINUE


131


35


122
appearance of the meniscus elsewhere is used as a to determine
the critical parameters. In practice these are determined by
interpolation of data taken when the meniscus appears just
above and just below the middle of the cell. This method is
then subject to an heuristic effect. Besides, near the
critical condition the phenomena of critical opalescence
(light scattering effect) may obscure the observation to some
extent. In the acoustic method, the instrument is generally
capable of measuring frequencies with an accuracy of at least
1 mHz. In the course of these experiments data were taken over
a 4 K range around the critical temperature over a period of
twelve hours. The recorded data show that nearest the observed
critical state the temperature gradient of resonance frequency
exceeds 2000 Hz/K on both sides of the frequency minimum for
which the corresponding temperature was taken as a critical
temperature. A remarkably abrupt change in frequency is
evident. The sharpness of this turning point together with the
stated precision of frequency measurements illustrates how the
present technique can locate the extremum to a resolution of
about one part per million in the temperature. Demonstrating
this fact was indeed the principal goal of this research. That
the acoustic resonance method is vastly more sensitive (ca.
three orders of magnitude) than the visual method is far more
important than whether of not the accuracy of temperature
measurement at a small poorly funded university laboratory is


non no
142
SUBROUTINE CNTR(B,SSRSD,C,T,L,W)
IMPLICIT REAL*8 (A-H,0-Z)
DIMENSION B(20,20),C(4000),T(4000),SSRSD(500),A(10)
DO 10 1=1,10
B(I,14)=B(I,12)+0.5*(B(I,12)-B(1,11))
10 CONTINUE
DO 30 K=1,10
A(K)=B(K,14)
30 CONTINUE
A1=A(1)
A2=A(2)
A3=A(3)
A4=A(4)
A5=A(5)
A6=A(6)
TC=A(7)
ALPHA=A(8)
BETA=A(9)
DELTA=A(10)
WRITE(*,500) Al,A2,A3,A4,A5,A6,TC,ALPHA,BETA,DELTA
500 FORMAT (IX,7F8.2,3F4.2)
CALL RESIDUAL(Al,A2,A3,A4,A5,A6,TC,ALPHA,BETA
C ,DELTA,C,T,L,ST,W)
SSRSD(14)=ST
SSRSD(11)=SSRSD(14)
DO 20 J=l,10
B(J,11)=B(J,14)
20 CONTINUE
RETURN
END
SUBROUTINE CNTW(B,SSRSD,C,T,L,W)
IMPLICIT REAL*8 (A-H,0-Z)
DIMENSION B(20,20),C(4000),T(4000),SSRSD(500),A(10)
DO 10 1=1,10
B(I,14)=B(I,12)-0.5*(B(I,12)-B(I,11))
10 CONTINUE
DO 30 K=1,10
A(K)=B(K,14)
30 CONTINUE
A1=A(1)
A2=A(2)
A3=A(3)
A4=A(4)
A5=A(5)
A6=A(6)
TC=A(7)
ALPHA=A(8)
BETA=A(9)
DELTA=A(10)


91
TEMPERATURE, CELSIUS
Figure 5.21. Resonance frequency (Is mode) and temperature
relationship of ethane near the critical density.
PRESSURE, PSIA
Figure 5.22. Resonance frequency (Is mode) and pressure
relationship of ethane near the critical density.


14
? c
^'n7a
C
2 na
g
(2.26)
If g is assumed to be a constant, this enables us to calculate
a resonance frequency corresponding to each root n from the
following relationship:
"ref
(f*,n)ref
( ^i,n) ref Vref
- 9
<*,.->
TIT^T
Consequently,
(2.27)
at any other values of (£,n)
f£,n (5Jfn)g-
(2.28)
Note that this method also serves as a self check routine for
the identity of the first reference frequency. If the observed
frequency is not correctly identified, it is unlikely that we
will find another resonance frequency in the expected region
by using the above equation.
So far we have discussed a zeroth order perturbation
relationship:
f
i,n
c
2 na
(2.29)
In reality there are higher perturbations to this relation.
Moldover has suggested that these effects should be added to
the above equation in the following manner*33'34:


68
Table 5.2. Values of critical point parameters from the
literature.
Temperature
K
Pressure
Density
Author(s)*
atm.
psia.
Kg/m3
304.07
73.0
1072.77
-
Andrews (58)
304.15
-
-
-
Hautefeuille
Cailletet (59)
305.05
77.0
1131.55
-
Dewar (60)
304.50
72.9
1071.30
-
Amagat (61)
304.55
-
-
-
Chappuis (62)
304.85
-
-
-
Villard (63)
304.15
-
-
-
Verschaffelt (64)
304.25
73.26
1076.59
-
Kuenen (65)
304.55
-
-
-
De Heen (66)
304.10
-
-
-
Von Wesendonck
(67)
304.13
72.93
1071.74
-
Keesom (68)
304.27
-
-
-
Brinkmann (69)
304.135
-
-
-
Onnes & Fabius
(70)
304.41
-
-
-
Bradley, Brown, &
Hale (71)
304.25
73.00
1072.77
-
Dorsman (72)
304.15
72.85
1070.56
-
Cardoso & Bell
(73)
304.12
-
-
-
Hein (74)
304.35
-
-
-
Dieterici C. (75)
304.10
-
-
-
Pickering S.F.
(76)
304.25
72.95
1072.03
-
Meyers & Van
Dusen (77)
304.1110.01



Kennedy H.T.,
Cyril H., and
Meyer (78)


79
pressure versus 0 is shown in figure 5.14. This technique
revealed the magnitude of the charging pressure to be used
when data were to be collected in search of the critical
point.
Table 5.3. Angles between pressure-temperature lines before
and after phase boundary of supercritical density fluid of
carbon dioxide and ethane.
File
Slope of line
Slope of line
Angle between
#
before phase
after phase
two lines
boundary
boundary
(degree)
729
34.637
18.307
14.733
731
34.074
18.293
14.479
801
33.562
18.371
14.091
802
32.842
18.444
13.594
807
27.882
18.690
10.356
811
25.149
18.607
7.992
815
24.701
18.846
7.191
This prediction was important here since the system volume is
fixed. The only way to change the system density was to


Table 5.7. (Continued)
File
Number
Conditions at start
of run (T = 37 C)
Sonic
speed at
phase
boundary
(m/sec.)
Conditions at minimum resonance frequency
Frequency
(Hz)
Pressure
(psia)
T (C)
P
(psia)
Frequency
(Hz)
Sonic
speed
(m/sec)
Amp.
(mv)
10170
1430.7
783.4
138.76
32.173
705.89
935.0
139.39
64
1021
1428.5
782.4
136.72
32.175
707.08
979.2
136.47
92
10262
1408.4
781.4
136.51
32.176
707.08
987.8
137.66
144
10291
1399.4
778.9
138.00
32.203
707.78
947.4
141.47
163
1031
1393.0
776.3
137.19
32.210
707.78
993.6
138.47
108
1102
1393.8
771.6
N/A
32.158
706.49
1155.0
160.97
2929
1105
1399.5
769.0
N/A
32.095
705.40
1199.4
167.16
1707
1107
1403.0
767.7
N/A
32.033
704.50
1233.5
171.92
1686
1108
1408.5
765.9
N/A
31.946
703.11
1248.3
173.98
1949
1109
1417.6
762.8
N/A
31.812
700.53
1280.2
178.42
1305


Table 4.1. Summary of electronic hardware.
Item
Manufacturer
Model
number
Range of
operation
Accuracy
(Manufacture
quote)
Function
Temperature
controller
Brinkmann
RKS-20D
-40 to 150
C
0.03 C
Control
temperature.
Pressure
Transducer
Sensotec
TJE/743-03
0-3000 psia
with
excitation
voltage = 10
volts
0.1% of full
scale
Measure pressure.
Digital
Strain gage
transducer
indicator
Beckman
industrial
corp.
600 Series
-150 mv to 1
volt
0.01 % of
full scale,
1 digit. Full
scale = 3000
psia.
Accepts a strain
gage type
transducer input
to measure
pressure.
Lock-in
amplifier
Stanford
research
system, Inc.
SR510
Single
phase
0.5 Hz to
100 KHz
10 volt
output
corresponds
to full scale
input
AC Signal
recoveries :
measure signals
down to 10 nv full
scale while
rejecting
interfering
signals up to 10s
times larger by
the performance of
two lines notch
filters and
autotracking
bandpass filter.
Frequency
counter
Sencore
FC71
10 Hz to 1
GHz
2 digits on
frequency
less than 100
KHz
Measure frequency
of periodic
signal.
oo


114
To fit this non linear equation to experimental data of
speed of sound and temperature, eleven system-dependent
coefficients must first be searched. This task was
accomplished by the simplex mathematical optimization
technique. (A computer program written by the author for this
purpose is given in appendix C.) In general the experimental
data of speed of sound and corresponding system temperature
were divided into two parts according to temperature. The
first one is all data at temperatures above the critical
temperature (roughly estimated from the literature value.) The
second one is all data at temperature below the critical
temperature. Each data set was independently fitted to the
above equation resulting in a set of eleven coefficients.
TEMPERATURE, CELSIUS
FITTING CURVE + EXPERIMENTAL POINTS
(PLOTTED EVERY 4 POINTS)
Figure 5.32. Comparison of experimental data for the ethane
and curve generated by renormalization group theory equation.


177
C
C
C
C 1004
C
C
C
C
C
C
C
C
C
C
C
C
C
CC
C
c
c
c
C 10
C 30
C
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
CHARACTER*20 TEMPO,TEMP
'COMMON GROUP 2.1
data CR/13/
LK = IBFIND ('LOCKIN ')
IP(4) = 2**24
IP(3) = 2**16
IP (2) = 2**8
IP(1) = 1
124 = 2**24
1166 = 2**16
188 = 2**8
11=1
CH(1) = 'P'
WRT(1)=ICHAR(CH(1))
CALL IBWRT(LK,WRT,1)
CALL IBRD (LK,RD,12)
1(4) = RD(1)/12 4
1(3) = (RD(1)-I(4)*I24)/I166
1(2) = (RD(1)-I(4)*I24-I(3)*I166)/I88
1(1) = RD(1)-I(4)*I24-I(3)*I166-I(2)*I88
1(8) = RD(2)/12 4
1(7) = (RD(2)-I(8)*I24)/I166
1(6) = (RD(2)-I(8)*I24-I(7)*I166)/I88
1(5) = RD(2)-I(8)*I24-I(7)*I166-I(6)*I88
1(12) = RD(3)/12 4
1(11) = (RD(3)-I(12)*124)/I166
1(10) = (RD(3)-I(12)*I24-I(11)*I166)/I88
1(9) = RD(3)-I(12)*I24-I(11)*I166-I(10)*I88
DO 10 J=1,12
WRITE (*,*) I(J), CHAR(I(J))
IF (I(J).EQ.CR) THEN
JJ=J-1
GOTO 30
END IF
CONTINUE
IF (JJ.EQ.7) THEN
WRITE(TEMP,'(7A1)') CHAR(1(1)),CHAR(1(2)),
CCHAR(1(3)), CCHAR(1(4)),CHAR(I(5)),
CCHAR(1(6)),CHAR(1(7))
READ(TEMP,(F7.2)',ERR=1004) ANGL
ELSE IF (JJ.EQ.6) THEN
WRITE(TEMP,(6A1)') CHAR(1(1)),CHAR(1(2)),
CCHAR(1(3)),CHAR(1(4)),CHAR(1(5)),CHAR(1(6))
READ(TEMP,1(F6.2)*,ERR=1004) ANGL
ELSE IF (JJ.EQ.5) THEN
WRITE(TEMP,'(5A1)) CHAR(1(1)),CHAR(1(2)),
CCHAR(1(3)),CHAR(1(4)),CHAR(I(5))
READ(TEMP,(F5.2)',ERR=1004) ANGL
ELSE IF (JJ.EQ.4) THEN
WRITE(TEMP,(4A1)1) CHAR(1(1)),CHAR(1(2)),
CCHAR(1(3)),CHAR(1(4))
READ(TEMP,'(F4.2)',ERR=1004) ANGL


U1
Figure D.3. The arrangement of the deadweight pressure gauge.


105
The predictive and experimental pressures are in good
agreement. Notice that this equation is indeterminate when
temperature is greater than 305.33 K since the u(T) term is
more than unity. Table 5.10 chronologically collects the
literature values of the critical parameters of ethane and the
value obtained in this work.
Analysis of Ethane Data
As mentioned earlier in the section on experimental
procedure, the ethane experiment was performed differently
from those on carbon dioxide and the carbon dioxide-ethane
mixture. This resulted from advances in the experimental
technique as the research progressed. In the case of ethane
all data points along the entire frequency sweep range were
saved on memory discs. These data can, in turn, be examined
and analyzed later to obtain more understanding of the
behavior of the gas under acoustic excitation. The analysis,
however, is still very difficult and time consuming due in
part to the large number of data points. Graphic methods are
appropriate for this purpose. For example, in the ethane
experiment for each isochoric density there are approximately
400 data files each containing 120 sets of four parameters:
temperature, pressure, resonance frequency, and amplitude. For
two dimensional observations one would have to pick two out of
four parameters to plot each data set for the total of 120X400
plots. The practicability of doing this is limited by many
factors such as the available graphics capability, computer


192
GAMA=5.0D0/2.0D0-BS2/VS+
C(BSD*BSD-CS+CS1-0.5D0*CS2)/VS/VS
GAMA=GAMA/(3.0D0/2.0D0-(2.0D0*BS1+BS2)/VS-
C(2.0D0*CS1+CS2)/2.ODO/VS/VS)
C=GAMA*R*T/M*(1.OD 00+2.0D0*BS/VS+3.ODO*CS/VS/VS)
C=DSQRT(C)
RETURN
END
PROGRAM TOTDAT
IMPLICIT REAL*8 (A-H,0-Z)
OPEN(10,FILE='SPEED.GAS',STATUS='OLD')
OPEN(20,FILE='TOTDAT.OUT',STATUS='NEW')
100 READ(10,*,END=200) X,X1,X2,P,T,AMP,F,EXPC,IT,ARGC,ARGF
CALL SPEEDZ(T,P,RHO,Z,Xl,X2,X3,X4,THEOC,X5,X6)
T=T/1.8DO-273.15
WRITE(*,*) IT
THEOC=THEOC/3.2808399D0
WRITE(20,300) P,T,AMP,F,ARGF,IT,ARGC,EXPC,THEOC
300 FORMAT(IX,F7.1,F8.3,F9.1,2F9.2,18,3F9.3)
GOTO 100
200 CLOSE (10)
CLOSE (20)
END
COMMON GROUP 1
integer UNL, UNT, GTL, SDC, PPC, GET, TCT
integer LLO, DCL, PPU, SPE, SPD, PPE, PPD
integer ERR, TIMO, END, SRQI, RQS, CMPL, LOK
integer REM, CIC, ATN, TACS, LACS, DTAS, DCAS
integer EDVR, ECIC, ENOL, EADR, EARG, ESAC, EABO
integer ENEB, EOIP, ECAP, EFSO, EBUS, ESTB, ESRQ
integer BIN, XEOS, REOS
integer TNONE, TlOus, T30us, TlOOus, T300us
integer Tims, T3ms, TIOms, T30ms, TIOOms
integer T300ms, Tls, T3s, TIOs, T30s
integer TlOOs, T300s, TlOOOs
integer S,LF
COMMON GROUP 2
character*50 flname
integer*4 bd,dvm,v,cnt,mask
integer*4 spr,ppr
data UNL/63/,UNT/95/,GTL/01/,SDC/04/,PPC/05/
data GET/08/,TCT/09/,LLO/17/,DCL/20/,PPU/21/
data SPE/24/,SPD/25/,PPE/96/,PPD/112/
data TIMO/16384/,END/8192/,SRQI/4096/
data RQS/2048/,CMPL/256/,LOK/128/,REM/64/,CIC/32/


BES. O
PRESSURE
Figure 5.33. Three dimensional phase diagram of a mixture of carbon dioxide and ethane.
oo
-FreIjenCy


2
components and mixtures is further needed to support the
development of semiempirical equations of state and other
correlations useful for predicting the behavior of a wide
range of systems. For this reason, much research has been done
in both the measurement (experimental method) and prediction
(theoretical method) of hydrocarbon fluid properties. The
great variety of chemical systems of practical interest,
combined with the experimental difficulties and time
investment involved with direct measurements, is a powerful
driving force behind the development of useful correlations.
Numerous empirical equations of state have been developed
over the years. The pioneering work of van der Waals focused
attention on the possible causes for nonideal gas behavior.
Although his efforts remain historically instructive, his
equation of state'11 is little used today having been
superseded by a succession of others ranging in complexity
from those of Benedict, Webb and Rubin,(2) Redlich and Kwong,<3)
and Peng and Robinson<4) to much more sophisticated ones
containing forty or more coefficients such as the AGA8
equation,(5) for example. Also a number of semi-theoretical and
theoretical equations based on molecular models have been
devised (see for examples references 6-11.) They were invented
to rectify for non-ideality both pure and multicomponent gases
and, in general, they work remarkably well for predictions of
gas properties in the noncritical region. They generally fail,
however, to describe the unique behavior of fluids in the


109


27
Thus, if pg and px are separately plotted against the
temperature t, the locus of the points bisecting the joins of
corresponding values of px and pg is a straight line (see
figure 3.7.) The point where this straight line cuts the
coexistence curve is the critical point. In many cases this
empirical law of rectilinear diameters holds very well, but
sometimes over a large range of temperature the actual
behavior shows a slight curvature. Furthermore, very near the
Density
Figure 3.7. Schematic diagram of critical point
determination from measurements of the liquid density p
and the vapor density p according to the method of
Cailletet and Mathias.


n n n n o
189
41 CONTINUE
AMP=AVG/FLOAT(NA)
DY=AMP
DX=SLOPE*FLOAT(J)+SF
WRITE(50,*) DX,DY
IF(AMP.GT.YMAX) THEN
YMAX=AMP
IMAX=J
ENDIF
IF (AMP.LT.YMIN) YMIN=AMP
40 CONTINUE
C CALL PLOT(NP,DX,DY,XMAX,XMIN,YMAX,YMIN)
CALL KEITHA(T)
CALL PRES(P)
WRITE(50,*) T,P
IG=IG-1
CLOSE(50)
return
End
PROGRAM GETSPG
$LARGE
IMPLICIT REAL*8 (A-H,0-Z)
REAL*8 PA(10000),TA(10000),FR(10000),CA,DSQRT
REAL*8 B(100),B1(100),B2(100),T(100)
REAL*8 C(100),C1(100),C2(100),BD(100),N
INTEGER TIM
ZERO=0.0D 00
ONE=1.0D 00
TWO=2.0D 00
EPK=119.8D 00
B0=49.80D-03
CON=6.8046D-02
KEL=273.15
OPEN (10,FILE=*ARGON.DAT1,STATUS='OLD')
1=1
200 READ(10,*,END=100) TI,T2,Pi,P2,AMP,FR(I),TIM,IN
IF(Pl.EQ.O.ODO) GOTO 200
IF(P2.EQ.0.0D0) GOTO 200
IF(Tl.EQ.O.ODO) GOTO 200
IF(T2.EQ.0.0D0) GOTO 200
TA(I)=(T1+T2)/TWO
PA(I)=(Pl+P2)*CON/TWO
1=1+1
GOTO 200
100 CLOSE(10)
M=I-1
OPEN (10,FILE='TVC.DAT',STATUS='OLD')
DO 25 1=1,74
READ(10,*) T(I),B(I),Bl(I),B2(I),


183
WRITE(TEMP,'(Al)') CHAR(I(1))
READ(TEMP,'(11),ERR=1004) ISTATUS
L=L+1
IF (L.LT.3) GOTO 20
IF (J.EQ.1.AND.ISTATUS.EQ.O) GOTO 30
IF (J.EQ.1.AND.K.EQ.O.AND.ISTATUS.EQ.O) GOTO 30
IF (ISTATUS.EQ.1) THEN
IG=IG+2
IF (IG.GT.24) IG=24
IF (IG.LT.l) IG=1
J=1
GOTO 10
ELSE IF (ISTATUS.EQ.O) THEN
IG=IG-1
K=0
IF (IG.GT.24) IG=24
IF (IG.LT.l) IG=1
GOTO 10
END IF
10 WRITE(GAIN,'(12)') IG
CH(1) = 'G
READ(GAIN,'(Al)') CH(2)
READ(GAIN,'(1X,A1)') CH(3)
WRT(1)=ICHAR(CH(1))+ICHAR(CH(2))*IP(2)+
CICHAR(CH(3))*IP(3)
CALL IBWRT(LK,WRT,3)
GOTO 20
30 RETURN
end
PROGRAM VIKAN2
$large
INTERFACE TO SUBROUTINE TIME (N,STR)
CHARACTER*10 STR [NEAR,REFERENCE]
INTEGER*2 N [VALUE]
END
REAL*4 DX(3000),DY(3000),XMAX,XMIN,YMAX,YMIN
CHARACTER*1 KOL
CHARACTER*2 SWEEP
CHARACTER*10 DNAME
CHARACTER*? DTEMP
CHARACTER*10 TSTR
INTEGER*2 HR,MIN,SEC
INTEGER READS,T,TC,TO,TOO,POINTS
SWEEP='15
WIDTH =30.0
READS = 3
KOUNT = 0
STP = -0.00125
CALL KEITHA(TZERO)


20
critical point, the maximum pressure and the maximum
temperature for the heterogeneous region must be known.
Evidence for the existence of a critical point was first
presented in 1823 by de La Tour'37, who observed that a liquid
when heated in a hermetically sealed glass tube is reduced to
vapor in a volume from two to four times the original volume
of the sample. However, it was not until the quantitative
measurements of Andrews on carbon dioxide<38,39) in 1869 that the
nature of the transition was understood. He was the first to
coin the term "critical point" for the phenomenon associated
with this liquid-vapor transition. For fluid mixtures the
first reliable experimental investigation of the critical
state began with the work of Kuenen(40,41) in 1897. Interest in
the critical region in the period 1876-1914 was peaked by the
experimental and theoretical studies of pressure-volume-
temperature (p-v-T) relationship for both pure gases and
gaseous mixtures of van der Waals and his associates at the
Universities of Amsterdam and Leiden.
Figure 3.1 shows the p-v-T relationship of the liquid and
gas phases of a pure fluid. The upper-right projection shows
several isotherms on a p-v diagram, the upper-left projection
is a p-T plot showing several isochores and the bottom
projection shows several isobars on a T-v plot. The tongue
shaped region bounded by points A, D, C, E, B is the
coexistence curve (or vapor pressure curve.) This curve may be
considered as consisting of two curves: the bubble-point (the


64
Figure 5.4. First derivative of resonance frequency versus
temperature.


125
pure fluids and mixtures would be useful source of this
information. The properties of bulk matter derive ultimately
from the interactions between its molecules, and information
about these interactions supports the development of useful
correlations and of statistical mechanics in general.
It is apparent from this work and others that the natural
vibrations of fluid masses convey a vast storehouse of encoded
information. Learning to acquire and interpret that
information is the long range goal of this laboratory.4
4
Some work has already begun. See reference 139.


Figure 5.1. Relationship between resonance frequency and temperature of carbon dioxide
for an isochore near its critical density.


153
through it is generally negligible. Some suitable follower
materials are thin elastomeric sheets, plastic films and metal
foils if not stressed beyond their elastic limits. The balance
beam was machined from brass bar stock. The crucial features
on it are the location of the contact ball and the weight
suspension hook pins with respect to the knife edge. These
features were located on 2.000 inch centers with maximum
uncertainty of 0.0005 inch. The contact ball sits in a 60
tapered center drill hole on top of the beam, and the hardened
0.1250 inch hook pins are pressed into mating holes located
along the lower edge of the beam. The S hooks fit over the
pins and pivot freely in a groove machined into the bottom of
the beam. The pins are numbered consecutively one to ten
beginning with the one nearest the fulcrum. The beam is
pivoted on a knife edge. Conventional balances have quartz or
other hard crystalline knife edges and bearing pads. Because
such components are expensive and not readily available,
especially for supporting heavy loads, it was decided to make
use of a common low cost material which seemed well suited for
the purpose. Cemented carbide inserts produced by powder
metallurgical techniques for use as cutting tools in machining
operations are available in many standard compositions and
geometries. These are harder than quartz and are manufactured
to exacting dimensions with close tolerances. We used a
standard insert 0.500 X 0.500 X 0.125 inch which was set into
a seat cut at a 45 angle with respect to the bottom of the


159
walls, 0.375 inch o.d., 0.25 inch i.d. and 24 convolutions was
brazed to fittings which permitted it to be utilized as a re
entrant pressure sensor (see figure D.4.) System pressure is
applied externally around the bellows causing it to contract
or shorten. A push rod extending through the bellows is acted
upon by the contact ball as before. The beam is loaded to
counteract the downward force due to the system pressure. The
bellows are very flexible axially but stiff in the radial
direction. The downward force is then equal to the magnitude
of the system pressure times the active cross sectional area,
A. or
p = Va.,, V. + qZmLnL/AM. (D.4)
Aeff is given approximately by the bellows manufacturer as
0.0723 inch2, but lack of accuracy in this value and
uncertainty in the effects of the brazing operations on it
require that it be determined by calibration. Freon-12 was
used for calibrating the bellows. The weight required to
balance the system pressure was measured versus temperature of
the vapor-liquid equilibrium. If the bellows does behave as a
piston of fixed active cross section, a plot of system
pressure versus the mass required to balance the beam should
be a straight line with slope of g/Aeff. Figure D.5 shows a
plot of the calibration data. The slope was determined by a
linear regression and is equal to 0.03000273 psig/g. The
correlation coefficient of the linear fit is 0.99995 which
shows the gauge to perform in accordance with equation D.4


To my mother and father
who eternally love me -
and to my sister Narisa,
my brothers Bordin and Pubordee


87
Figure 5.18. The sonic speed versus temperature of the carbon
dioxide-ethane mixture for an isochore near critical density.
Figure 5.19. Pressure and temperature behavior of the carbon
dioxide-ethane mixture charged near its critical density.


144
C+A6*TR**3.0)
WRITE(40,*) T(I), C(I)
C WRITE(*,*) T(I), C(I)
50 CONTINUE
CLOSE(40)
STOP
END
Program III. Crossing-Point Searching
C234567
program min
CHARACTER*20 U,V
DOUBLE PRECISION
DOUBLE PRECISION
DOUBLE PRECISION
DOUBLE PRECISION
DOUBLE PRECISION
DOUBLE PRECISION
DOUBLE PRECISION
DOUBLE PRECISION
DIMENSION A(20),
ZERO=0.0D00
TWO=20.0D-1
Q,T,S,R,E,F
DEL,DSQRT,DABS,ZERO,TWO,DELE
P2L,P3L,TCL,TL,TRL
P2R,P3R,TCR,TR,TRR,DELF
AIL,A2L,A3L,A4L,A5L,A6L
ALPHAL,BETAL,DELTAL
A1R,A2R,A3R,A4R,A5R,A6R
ALPHAR,BETAR,DELTAR
B (20)
WRITE(*,*) 'RIGHT SIDE FILE NAME'
READ(*,50) U
50 FORMAT (A20)
OPEN (20,FILE=U,STATUS='OLD')
DO 40 1=1,10
READ (20,*) A(I)
WRITE(*,*) A(I)
40 CONTINUE
CLOSE(20)
WRITE(*,*) 'LEFT SIDE FILE NAME
READ(*,50) V
OPEN (30,FILE=V,STATUS='OLD')
DO 60 1=1,10
READ (30,*) B(I)
WRITE(*,*) B(I)
60 CONTINUE
CLOSE(30)
AlR=A(1)
A2R=A(2)
A3R=A(3)
A4R=A(4)
A5R=A(5)


10
For higher orders of 1 the solutions may be obtained from the
following recurrence relation:
f(i +i,(z) (2£ + l)f,(z) f(i.ir (2.19)
£
The general solution may be written as a linear combination of
the first and second kind solutions as follows:
F(kr) Ajf(kr) + BYi(kr). (2.20)
F, however, must be finite at r=0. Spherical Bessel functions
of the second kind do not satisfy this constraint, so the
equation (2.20) becomes
F(kr) = Aj^kr) (2.21)
A second condition imposes that the normal velocity must
vanish at the rigid wall of the sphere, i.e.
j = 0. (2.22)
This is a transcendental equation whose roots give the normal
frequencies of vibration, £. For example if
Then
(-
j0(kr)
i(kr))
HF
sin(ka)
ka2
Hence, tan(ka)
sin(kr)
' ki
sin(ka) A cos(ka)
kl2 5
cos(ka)
a
- ka 5JfB.
(2.23)
For each value of SL there exists an infinite number of $
values designated as n* The first few points corresponding
to roots of equation (2.23) for the first six orders are shown
in figure 2.1.


102
Table 5.9. Comparison of experimental vapor pressures and
those calculated with an equation of state of ethane proposed
by Sychev.(110)
Experimental
Predicted
pressure
psia
Percent
difference
T (K)
P (psia)
304.72
697.65
697.36
0.0410
304.95
700.83
700.79
0.0053
305.14
703.71
703.66
0.0062
305.16
704.30
703.96
0.0481
305.19
704.70
704.45
0.0360
305.24
705.40
705.16
0.0338
305.27
706.09
705.55
0.0764
305.30
706.39
706.05
0.0475
305.30
706.59
706.08
0.0713
305.31
706.79
706.27
0.0735
305.32
705.89
706.42
-0.0747
305.32
707.08
706.45
0.0896
305.33
707.08
706.47
0.0874
305.35
707.78
N/A*
N/A
305.36
707.78
N/A*
N/A
305.31
706.49
706.19
0.0421
305.24
705.40
705.23
0.0231
305.18
704.50
704.30
0.0292
305.10
703.11
702.99
0.0176
304.96
700.53
700.98
-0.0648
304.83
698.54
699.05
-0.0726
304.62
696.16
695.99
0.0242
Equation diverges at this temperature


noon non oonnnn
141
GOTO 90
END IF
DO 900 1=1,11
WRITE(*,*) SSRSD(l)
900 CONTINUE
RETURN
END
SUBROUTINE EXPAND(B,SSRSD,C,T,L,W)
IMPLICIT REAL*8 (A-H,0-Z)
DIMENSION B(20,20),C(4000),T(4000),SSRSD(500),A(10)
DO 10 1=1,10
B(I,14)=B(I,13)+(B(I,12)-B(I,11))
10 CONTINUE
DO 30 K=1,10
A(K)=B(K,14)
30 CONTINUE
Al=A(1)
A2=A(2)
A3=A(3)
A4=A(4)
A5=A(5)
A6=A(6)
TC=A(7)
ALPHA=A(8)
BETA=A(9)
DELTA=A(10)
WRITE(*,500) Al,A2,A3,A4,A5,A6,TC,ALPHA,BETA,DELTA
500 FORMAT (IX,7F8.2,3F4.2)
CALL RESIDUAL(A1,A2,A3,A4,A5,A6,TC,ALPHA,BETA
C ,DELTA,C,T,L,ST,W)
SSRSD(14)=ST
WRITE(*,*) 'SSRSD',I,'=',SSRSD(I)
IF (SSRSD(14).LT.SSRSD(13)) THEN
SSRSD(11)=SSRSD(14)
DO 20 J=l,10
B(J,11)=B(J,14)
20 CONTINUE
ELSE IF (SSRSD(14).GT.SSRSD(13)) THEN
SSRSD(11)=SSRSD(13)
DO 50 J=l,10
B(J,11)=B(J,13)
50 CONTINUE
END IF
RETURN
END



PAGE 1

$&2867,& '(7(50,1$7,21 2) 3+$6( %281'$5,(6 $1' &5,7,&$/ 32,176 2) *$6(6 FR FRFK PL[WXUH DQG FK %\ &+$',1 '(-683$ $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$

PAGE 2

7R P\ PRWKHU DQG IDWKHU ZKR HWHUQDOO\ ORYH PH DQG WR P\ VLVWHU 1DULVD P\ EURWKHUV %RUGLQ DQG 3XERUGHH

PAGE 3

$&.12:/('*(0(176 7KH DXWKRU H[SUHVVHV KLV VLQFHUH JUDWLWXGH WR KLV DGYLVRU SURIHVVRU 6DPXHO &ROJDWH IRU KLV LQGLVSHQVDEOH LQWHOOHFWXDO DQG PRUDO VXSSRUW DQG HVSHFLDOO\ ZKHQ ERWWOHQHFNV ZHUH HQFRXQWHUHG ,W KDV EHHQ SULYLOHJH ZRUNLQJ ZLWK VXFK DQ RXWVWDQGLQJ WHDFKHU DQG FKHPLVW +LV GHGLFDWLRQ WR H[FHOOHQFH DQG FODULW\ RI WKRXJKW DQG OXFLG LQVLJKWV ZLOO SHUSHWXDOO\ VHUYH DV DQ H[DPSOH 7KLV ZRUN ZRXOG DOVR KDYH EHHQ LPSRVVLEOH ZLWKRXW WKH H[SHUWLVH RI 'UV $ 6LYDUDPDQ .HQQHWK & 0F*LOO DQG 9 (YDQ +RXVH RI WKH DFRXVWLF UHVHDUFK JURXS ZKLFK PDGH FRORVVDO EDUULHUV EHFRPH LQVLJQLILFDQW 7KDQNV DOVR WR -RVHSK 6KDORVN\ IRU KLV PDUYHORXV PDFKLQLQJ GH[WHULW\ 6SHFLDO WKDQNV DUH H[WHQGHG WR SURIHVVRUV 'DYLG 0LFKD DQG 0DUWLQ 9DOD DQG WR 3DXO &DPSEHOO IRU WKHLU YDOXDEOH PRUDO VXSSRUW DQG WR DOO RI SHHUV RI WKH DXWKRU LQ WKLV LQVWLWXWLRQ SDUWLFXODUO\ &DVH\ 5HQW] DQG 0LFKDHO &OD\ IRU WKHLU WUXO\ PHDQLQJIXO IULHQGVKLSV /DVW EXW QRW OHDVW WKH DXWKRU ZLVKHV WR ZDUPO\ WKDQN DOO RI KLV WHDFKHUV ZKR VKDUHG WKHLU NQRZOHGJH DQG XQGHUVWDQGLQJ ZLWK KLP LQ

PAGE 4

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

PAGE 5

7KUHH 'LPHQVLRQDO 3KDVH 'LDJUDP &+$37(5 &21&/86,21 $33(1',; $ &+$5*,1* 35(6685( &$/&8/$7,216 $33(1',; % 48$17,7$7,9( $1$/<6,6 2) *$6 0,;785( $33(1',; & 6,03/(; $33(1',; /$%25$725< 67$1'$5' 35(6685( *$8*( $33(1',; ( &20387(5 352*5$06 5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+ Y

PAGE 6

/,67 2) 7$%/(6 7DEOH 7KH YDOXHV RI WKH URRWV e Q LQ DVFHQGLQJ RUGHU RI PDJQLWXGH n 7DEOH 6XPPDU\ RI HOHFWURQLF KDUGZDUH 7DEOH *DV 6SHFLILFDWLRQV 7DEOH &KDUJLQJ 3UHVVXUH RI JDV V\VWHPV 7DEOH &RPSDULVRQ RI 'HZSRLQW 3UHVVXUHV RI &DUERQ 'LR[LGH 7DEOH 9DOXHV RI FULWLFDO SRLQW SDUDPHWHUV IURP WKH OLWHUDWXUH 7DEOH $QJOHV EHWZHHQ SUHVVXUHWHPSHUDWXUH OLQHV EHIRUH DQG DIWHU SKDVH ERXQGDU\ RI VXSHUFULWLFDO GHQVLW\ IOXLG RI FDUERQ GLR[LGH DQG HWKDQH 7DEOH ([SHULPHQWDO UHVXOWV IRU FRQVWUXFWLRQ RI SKDVH GLDJUDPV RI D PL[WXUH RI FDUERQ GLR[LGH DQG HWKDQH 7DEOH &RPSDULVRQ RI 'HZSRLQW 3UHVVXUHV RI & &+ 0L[WXUH ;FR f 7DEOH 6L[WK RUGHU SRO\QRPLDO ILW VWDWLVWLFV RI *5$3+(5 SURJUDP DSSOLHG WR H[SHULPHQWDO GDWD SRLQWV LQ ILJXUH 7DEOH 6XPPDU\ RI UHVXOWV IRU HWKDQH PHDVXUHPHQWV 7DEOH &RPSDULVRQ RI %XEEOH3RLQW 3UHVVXUHV RI (WKDQH 7DEOH &RPSDULVRQ RI H[SHULPHQWDO YDSRU SUHVVXUHV DQG WKRVH FDOFXODWHG ZLWK DQ HTXDWLRQ RI VWDWH RI HWKDQH SURSRVHG E\ 6\FKHYf 7DEOH &KURQRORJLFDO FROOHFWLRQ RI FULWLFDO SRLQW SDUDPHWHUV RI HWKDQH YL

PAGE 7

7DEOH &RHIILFLHQWV 2EWDLQHG E\ 6LPSOH[ 2SWLPL]DWLRQ 0HWKRG 7DEOH &RPSDULVRQ RI &ULWLFDO 7HPSHUDWXUHV DQG 3UHVVXUHV 7DEOH %O 'DWD IRU FDOLEUDWLRQ FXUYH RI HWKDQH 7DEOH 'O ([SHULPHQWDO 7HVW RI 'HDG :HLJKW 3UHVVXUH *DXJH YLL

PAGE 8

/,67 2) ),*85(6 )LJXUH 7KH ILUVW VL[ RUGHUV RI WKH VSKHULFDO %HVVHO IXQFWLRQ 3RLQWV ZKHUH VORSH HTXDOV ]HUR \LHOG HLJHQ YDOXHV Q )LJXUH )LJXUH 7KH SY7 EHKDYLRU RI SXUH IOXLG ,Q WKH FHQWHU LV VNHWFKHG WKH VXUIDFH S SY7f >)URP +LUVFKIHOGHU -RVHSK &XUWLVV &KDUOHV ) DQG %LUG 5 %\URQ 0ROHFXODU 7KHRU\ RI *DVHV DQG /LTXLGV &RS\ULJKW k E\ -RKQ :LOH\ t 6RQV ,QF 5HSULQWHG E\ SHUPLVVLRQ RI -RKQ :LOH\ t 6RQV ,QF@ )LJXUH 3UHVVXUH7HPSHUDWXUH0ROH IUDFWLRQ UHODWLRQVKLS RI V\VWHP RI FDUERQ GLR[LGH DQG HWKDQH )LJXUH 3UHVVXUHWHPSHUDWXUH SURMHFWLRQ RI WKH V\VWHP FDUERQ GLR[LGHHWKDQH LQ WKH FULWLFDO UHJLRQ /LQHV =IW.IW DQG =. DUH YDSRU SUHVVXUH FXUYHV RI SXUH FDUERQ GLR[LGH DQG RI SXUH HWKDQH UHVSHFWLYHO\ )LJXUH 6FKHPDWLF GLDJUDP RI SUHVVXUH DQG YROXPH UHODWLRQVKLS RI FDUERQ GLR[LGH ZLWK VHYHUDO LVRWKHUPV LQ D EURDG UHJLRQ )LJXUH 3UHVVXUH DQG GHQVLW\ UHODWLRQVKLS RI FDUERQ GLR[LGH ZLWK VHYHUDO LVRWKHUPV DURXQG FULWLFDO SRLQWrf )LJXUH 6FKHPDWLF GLDJUDP RI JUDSKLFDO GHWHUPLQDWLRQ RI FULWLFDO WHPSHUDWXUH RI FDUERQ GLR[LGH )LJXUH 6FKHPDWLF GLDJUDP RI FULWLFDO SRLQW GHWHUPLQDWLRQ IURP PHDVXUHPHQWV RI WKH OLTXLG GHQVLW\ S/ DQG WKH YDSRU GHQVLW\ S DFFRUGLQJ WR WKH PHWKRG RI &DLOOHWHW DQG 0DWKLDV )LJXUH 6FKHPDWLF GLDJUDP RI LVRFKRULF PHWKRG RI FULWLFDO SRLQW GHWHUPLQDWLRQ )LJXUH 2EVHUYHG VRQLF VSHHG YHUVXV WHPSHUDWXUH IRU D 1RUWK 6HD QDWXUDO JDV PL[WXUH5HIHUHQFH f YLLL

PAGE 9

)LJXUH $ VLGH YLHZ RI WKH VSKHULFDO UHVRQDWRU HTXLSSHG ZLWK WKH WUDQVGXFHUV )LJXUH 7KH VSKHULFDO UHVRQDWRU )LJXUH 7KH WUDQVGXFHU FURVVVHFWLRQDO YLHZf )LJXUH 7KH PL[LQJ FRQWURO XQLW )LJXUH 7KH ,QVWUXPHQWDO 6HWXS )LJXUH 7KH H[SHULPHQWDO VHWXS IRU & DQG &a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

PAGE 10

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f UHVXOWV LQ VDPH FULWLFDO WHPSHUDWXUH DV ILUVW UDGLDO UHVRQDQFH IUHTXHQF\ ERWWRP FXUYHf )LJXUH $ VHTXHQFH RI VXSHUFULWLFDO SUHVVXUH YHUVXV WHPSHUDWXUH LVRFKRUHV RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH )LJXUH 5HODWLRQVKLS EHWZHHQ WKH DQJOHV DQG WKH FKDUJLQJ SUHVVXUHV RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH )LJXUH 5HODWLRQVKLS EHWZHHQ WKH VWDUWLQJ SUHVVXUHV DQG WKH WHPSHUDWXUHV RI SKDVH FKDQJH RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH )LJXUH 5HODWLRQVKLS EHWZHHQ WKH VWDUWLQJ SUHVVXUHV DQG WKH VSHHG RI VRXQG DW D SKDVH FKDQJH RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH )LJXUH &RH[LVWHQFH FXUYH RI WKH FDUERQ GLR[LGH HWKDQH PL[WXUH QHDU D]HRWURSH FRPSRVLWLRQ )LJXUH 7KH VRQLF VSHHG YHUVXV WHPSHUDWXUH RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH IRU DQ LVRFKRUH QHDU FULWLFDO GHQVLW\ )LJXUH 3UHVVXUH DQG WHPSHUDWXUH EHKDYLRU RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH FKDUJHG QHDU LWV FULWLFDO GHQVLW\ [

PAGE 11

)LJXUH 3UHVVXUH DQG WHPSHUDWXUH UHODWLRQVKLS RI D VHW RI VHYHUDO LVRFKRUHV RI WKH FDUERQ GLR[LGH HWKDQH PL[WXUH )LJXUH 5HVRQDQFH IUHTXHQF\ ,V PRGHf DQG WHPSHUDWXUH UHODWLRQVKLS RI HWKDQH QHDU WKH FULWLFDO GHQVLW\ )LJXUH 5HVRQDQFH IUHTXHQF\ ,V PRGHf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f EHORZ WKH FULWLFDO WHPSHUDWXUH )LJXUH ([SHULPHQWDO FXUYH RI HWKDQH VKRZLQJ G\QDPLF EHKDYLRU RI VSHHG RI VRXQG QHDU FULWLFDO SRLQW [L

PAGE 12

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n LQGLFDWRU FRQWDFW QHHGOHV )LJXUH 7KH SUHVHQW GHDGZHLJKW SUHVVXUH JDXJH )LJXUH 7KH DUUDQJHPHQW RI WKH GHDGZHLJKW SUHVVXUH JDXJH )LJXUH $Q HOHFWURIRUPHG QLFNHO EHOORZ SUHVVXUH VHQVRU )LJXUH 7KH FDOLEUDWLRQ FXUYH IRU GHDG ZHLJKW SUHVVXUH JDXJH [LL

PAGE 13

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

PAGE 14

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

PAGE 15

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r8QGHU WKH ULJKW FRQGLWLRQ RI SUHVVXUH WHPSHUDWXUH DQG RLO FRPSRVLWLRQ FDUERQ GLR[LGH ZLOO PL[ ZLWK WKH FUXGH LQ WKH UHVHUYRLU WR IRUP D VLQJOH SKDVH OLTXLG ZKLFK LV PXFK OLJKWHU WKDQ WKH RULJLQDO RLO DQG FRQVHTXHQWO\ HDVLHU WR EULQJ WR WKH VXUIDFH

PAGE 16

FRPSRQHQWV DQG PL[WXUHV LV IXUWKHU QHHGHG WR VXSSRUW WKH GHYHORSPHQW RI VHPLHPSLULFDO HTXDWLRQV RI VWDWH DQG RWKHU FRUUHODWLRQV XVHIXO IRU SUHGLFWLQJ WKH EHKDYLRU RI D ZLGH UDQJH RI V\VWHPV )RU WKLV UHDVRQ PXFK UHVHDUFK KDV EHHQ GRQH LQ ERWK WKH PHDVXUHPHQW H[SHULPHQWDO PHWKRGf DQG SUHGLFWLRQ WKHRUHWLFDO PHWKRGf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n LV OLWWOH XVHG WRGD\ KDYLQJ EHHQ VXSHUVHGHG E\ D VXFFHVVLRQ RI RWKHUV UDQJLQJ LQ FRPSOH[LW\ IURP WKRVH RI %HQHGLFW :HEE DQG 5XELQf 5HGOLFK DQG .ZRQJf DQG 3HQJ DQG 5RELQVRQf WR PXFK PRUH VRSKLVWLFDWHG RQHV FRQWDLQLQJ IRUW\ RU PRUH FRHIILFLHQWV VXFK DV WKH $*$ HTXDWLRQf IRU H[DPSOH $OVR D QXPEHU RI VHPLWKHRUHWLFDO DQG WKHRUHWLFDO HTXDWLRQV EDVHG RQ PROHFXODU PRGHOV KDYH EHHQ GHYLVHG VHH IRU H[DPSOHV UHIHUHQFHV f 7KH\ ZHUH LQYHQWHG WR UHFWLI\ IRU QRQLGHDOLW\ ERWK SXUH DQG PXOWLFRPSRQHQW JDVHV DQG LQ JHQHUDO WKH\ ZRUN UHPDUNDEO\ ZHOO IRU SUHGLFWLRQV RI JDV SURSHUWLHV LQ WKH QRQFULWLFDO UHJLRQ 7KH\ JHQHUDOO\ IDLO KRZHYHU WR GHVFULEH WKH XQLTXH EHKDYLRU RI IOXLGV LQ WKH

PAGE 17

FULWLFDO UHJLRQ EXW SURJUHVV WRZDUG RYHUFRPLQJ WKLV VKRUWFRPLQJ LV EHLQJ PDGH RQ VRPH IURQWV nf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nV PL[LQJ UXOHVf 3SF e<3FL DQG 7SF 6
PAGE 18

SDUWO\ GXH WR WKH GLIILFXOW\ LQ ORFDWLRQ RI WKH FULWLFDO SRLQW LQ PL[WXUHV E\ FRQYHQWLRQDO PHWKRGV 7KH PRVW ZLGHO\ XVHG PHWKRG LV EDVHG RQ GLUHFW REVHUYDWLRQ RI WKH DSSHDUDQFH DQG GLVDSSHDUDQFH RI D PHQLVFXV LQGLFDWLQJ D ERXQGDU\ OLQH EHWZHHQ JDV DQG OLTXLG SKDVHV ZKHQ D V\VWHP HQWHUV WKH FULWLFDO FRQGLWLRQf 7KLV PHWKRG UHOLHV RQ WKH YLVXDO DFXLW\ RI WKH RSHUDWRU DQG FOHDUO\ KDV VRPH VXEMHFWLYLW\ $FRXVWLF GHWHUPLQDWLRQ RI WKHUPRG\QDPLF SURSHUWLHV VXFK DV KHDW FDSDFLW\ YLULDO FRHIILFLHQWV YDSRU SUHVVXUH HWF E\ VSKHULFDO UHVRQDWRU WHFKQLTXHV KDV EHHQ YHU\ VXFFHVVIXO LQ WKH SDVW HVSHFLDOO\ LQ WKLV UHVHDUFK JURXSf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f 7KLV ZRUN LV D ILUVW DWWHPSW WR EULQJ WKH VXSHULRULW\ RI WKH VSKHULFDO DFRXVWLF UHVRQDQFH WHFKQLTXH WR EHDU RQ GHYHORSPHQW RI D QHZ DSSURDFK WR FULWLFDO SRLQW GHWHUPLQDWLRQ ([SHULPHQWV ZHUH SHUIRUPHG DFFRUGLQJ WR ORJLFDO VFLHQWLILF DSSURDFK 7R VWDUW ZLWK YHULILFDWLRQ RI WKLV QHZ WHFKQLTXH

PAGE 19

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f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

PAGE 20

&+$37(5 7+(25(7,&$/ %$&.*5281' $FRXVWLFV 7KLV GHULYDWLRQ IROORZV WKDW JLYHQ E\ UHIHUHQFHV DQG 7KH SURSDJDWLRQ RI SUHVVXUH ZDYHV LQ D ORVVOHVV IOXLG DW UHVW VDWLVI\ WKH JHQHUDO ZDYH HTXDWLRQrf 9 -/iO‹ f F DW ZKHUH 9 /DSODFLDQ RSHUDWRU LV WKH YHORFLW\ SRWHQWLDO DQG & LV WKH ZDYH YHORFLW\ VSHHG RI VRXQGf $VVXPLQJ WKDW S 3RH[RW HTXDWLRQ f EHFRPHV RU 9I! f§S & f 9A NS f ZKHUH N R& ZDYH QXPEHU ,I ZH FRQVLGHU RQO\ WKH WLPH LQGHSHQGHQW ZDYH HTXDWLRQ f EHFRPHV 9A NR f

PAGE 21

,I WKH YDULDEOHV DUH VHSDUDEOH SR PD\ EH ZULWWHQ DV R IUfJfKSf DQG HTXDWLRQ f EHFRPHVnn f ZKHUH 0 DQG T DUH VHSDUDWLRQ FRQVWDQWV 7KH ILUVW RI WKHVH WKUHH HTXDWLRQV JLYHV WKH UDGLDO SDUW DQG WKH RWKHU WZR WKH DQJXODU SDUW RI WKH VROXWLRQ $QJXODU 3DUW 7KH ODVW HTXDWLRQ LQ HTXDWLRQV f LV D OLQHDU KRPRJHQHRXV VHFRQGRUGHU GLIIHUHQWLDO HTXDWLRQ ,WV VROXWLRQV DUH K VLQHTT!f DQG FRVLQH TWSf ,Q RUGHU WKDW K EH VLQJOHn YDOXHG > K U SUUf KUWSf@ YDOXHV RI T PXVW EH LQWHJHUV LH T s s 'HQRWLQJ WKLV LQWHJHU E\ P WKH VHFRQG RI HTXDWLRQV f EHFRPHV f 7KLV HTXDWLRQ KDV ILQLWH VROXWLRQV IRU t Q ZKHQ 0 ee f e f f f 7KHQ f 7KH JHQHUDO VROXWLRQ RI WKLV HTXDWLRQ LV

PAGE 22

$L3fFRVf %UHVHf f ZKHUH 3IPFRVf 7KH DVVRFLDWHG /HJHQGUH SRO\QRPLDO RI WKH ILUVW NLQG RI GHJUHH % DQG RUGHU P 4AFRV2f 7KH DVVRFLDWHG /HJHQGUH SRO\QRPLDO RI WKH VHFRQG NLQG RI GHJUHH % DQG RUGHU P 6LQFH 4LPFRVf EHFRPHV LQILQLWH DW t Q LW GRHV QRW DSSO\ WR WKLV SK\VLFDO VLWXDWLRQf &RQVHTXHQWO\ $-3fFRVf ZKHUH 3FRVf VLQfP f GA FRVf GFRVfP LI P 3ArFRVf LV WKH /HJHQGUH SRO\QRPLDO )LQDOO\ WKH JHQHUDO VROXWLRQ RI WKH DQJXODU SDUW FDQ EH ZULWWHQ DV IROORZV f
PAGE 23

'LYLGLQJ ERWK VLGHV E\ NU JLYHV B GAI BB GI / fn N GU NU A NU I f &KDQJLQJ WKH IRUP RI WKH IXQFWLRQ IURP IUf WR )=f ZKHUH = NU DQG UHDUUDQJLQJ \LHOGV G)=! LG)=f / B eef?)=f R f G= =A=A ? = 3DUWLFXODU VROXWLRQV DUH WKH VSKHULFDO %HVVHO IXQFWLRQ RI WKUHH NLQGV )LUVW NLQG )]f MI]f L =f ? ] f 6HFRQG NLQG )]f \]f < ? ] L Lf =f f f 7KLUG NLQG )]f Krf ]f M]f L\]f +! L =f 1 == e f ) ]f Kff]f MI]f L\]f f +! L =f 1 == r Bf ZKHUH -=f <=f DQG +=f LV %HVVHO IXQFWLRQ RI WKH ILUVW VHFRQG DQG WKLUG NLQG UHVSHFWLYHO\ :LWK IRU H[DPSOH ’ R ]f MO]f \ VLQ]f ]FRV]f@ ] FRV]f ] >FRV]f ] VLQ]f @ f

PAGE 24

)RU KLJKHU RUGHUV RI WKH VROXWLRQV PD\ EH REWDLQHG IURP WKH IROORZLQJ UHFXUUHQFH UHODWLRQ IL L]f f se OfI]f ILLU f e£ 7KH JHQHUDO VROXWLRQ PD\ EH ZULWWHQ DV D OLQHDU FRPELQDWLRQ RI WKH ILUVW DQG VHFRQG NLQG VROXWLRQV DV IROORZV )NUf $MINUf %
PAGE 25

2YHUDOO 6ROXWLRQV &RPELQLQJ WKH UHVXOWV RI WKH UDGLDO DQG DQJXODU SDUWV \LHOGV WKH VROXWLRQ RI WKH WLPHLQGHSHQGHQW PRQRFKURPDWLF ZDYH HTXDWLRQ LQ WKH VSKHULFDO SRODU FRRUGLQDWH V\VWHP DV !RUSf f e  e MLNLQUf
PAGE 26

)LJXUH 7KH ILUVW VL[ RUGHUV RI WKH VSKHULFDO %HVVHO IXQFWLRQ 3RLQWV ZKHUH VORSH HTXDOV ]HUR \LHOG HLJHQ YDOXHV eOQr

PAGE 27

WDEOH (YHQ PRUH FRPSOHWH WDEXODWLRQV FDQ EH IRXQG LQ WKH OLWHUDWXUHn 6HDUFKLQJ IRU 5HVRQDQFH )UHTXHQFLHV e & )URP WKH UHODWLRQVKLS LW LV REYLRXV WKDW LQ RUGHU WR ILQG WKH UHVRQDQFH IUHTXHQF\ FRUUHVSRQGLQJ WR WKH QRUPDO PRGH RI YLEUDWLRQ RI DQ HLJHQYDOXH Q WKH VSHHG RI VRXQG DQG WKH UDGLXV RI WKH VSKHUH PXVW EH NQRZQ 7KH ODWWHU PD\ EH REWDLQHG SUHFLVHO\ IURP JHRPHWULF FRQVLGHUDWLRQV RQ WKH UHVRQDWRU FDYLW\ 7KH IRUPHU PD\ EH DSSUR[LPDWHO\ FDOFXODWHG IURP DQ DVVXPHG HTXDWLRQ RI VWDWH ,Q WKLV VWXG\ ZH XVHG WKH $PHULFDQ *DV $VVRFLDWLRQ HTXDWLRQ FDOOHG $*$f ZKLFK UHTXLUHV DV LQSXW RQO\ WHPSHUDWXUH SUHVVXUH DQG FRPSRVLWLRQ 8VLQJ WKH SUHGLFWHG VRQLF VSHHG WRJHWKHU ZLWK WKH PHDVXUHG FDYLW\ UDGLXV DQG WKH WDEXODWHG HLJHQYDOXH WKH FRUUHVSRQGLQJ UHVRQDQFH IUHTXHQF\ I/ Q PD\ EH DSSUR[LPDWHO\ FDOFXODWHG DQG WKHQ H[SHULPHQWDOO\ VHDUFKHG E\ VFDQQLQJ IUHTXHQF\ DURXQG WKDW YDOXH :KHQ H[FLWHG DW D UHVRQDQFH IUHTXHQF\ WKH DPSOLWXGH RI WKH GHWHFWHG DFRXVWLF VLJQDO VKDUSO\ LQFUHDVHV 7KH VHULHV RI UHVRQDQFH SHDNV DUUDQJHG E\ IUHTXHQF\ GHILQHV D ]HURWK RUGHU DFRXVWLF VSHFWUXP RI UHVRQDQFHV RI JDV HQFORVHG LQ WKH VSKHULFDO FDYLW\ 2QFH D UHVRQDQFH IUHTXHQF\ KDV EHHQ IRXQG LW FDQ VHUYH DV DQ LQWHUQDO UHIHUHQFH SRLQW WR VHDUFK IRU DQRWKHU DV IROORZV

PAGE 28

" F AnQD & QD J f f ,I J LV DVVXPHG WR EH D FRQVWDQW WKLV HQDEOHV XV WR FDOFXODWH D UHVRQDQFH IUHTXHQF\ FRUUHVSRQGLQJ WR HDFK URRW Q IURP WKH IROORZLQJ UHODWLRQVKLS UHI IrQfUHI ALQf UHI 9UHI r! 7,7A7 &RQVHTXHQWO\ f DW DQ\ RWKHU YDOXHV RI eQf IeQ -IQfJ f 1RWH WKDW WKLV PHWKRG DOVR VHUYHV DV D VHOI FKHFN URXWLQH IRU WKH LGHQWLW\ RI WKH ILUVW UHIHUHQFH IUHTXHQF\ ,I WKH REVHUYHG IUHTXHQF\ LV QRW FRUUHFWO\ LGHQWLILHG LW LV XQOLNHO\ WKDW ZH ZLOO ILQG DQRWKHU UHVRQDQFH IUHTXHQF\ LQ WKH H[SHFWHG UHJLRQ E\ XVLQJ WKH DERYH HTXDWLRQ 6R IDU ZH KDYH GLVFXVVHG D ]HURWK RUGHU SHUWXUEDWLRQ UHODWLRQVKLS I LQ F QD f ,Q UHDOLW\ WKHUH DUH KLJKHU SHUWXUEDWLRQV WR WKLV UHODWLRQ 0ROGRYHU KDV VXJJHVWHG WKDW WKHVH HIIHFWV VKRXOG EH DGGHG WR WKH DERYH HTXDWLRQ LQ WKH IROORZLQJ PDQQHUrnf

PAGE 29

Q LI _&O ALLQ f§ $IV $IHO $IW $I 7QD HO JHRP f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f PD\ EH ZULWWHQ DV O }Q .Qf&f 7QD f ZKHUH Y/ Q LV DQ HIIHFWLYH HLJHQYDOXH ZKLFK GHSHQGV RQ WKH XQSHUWXUEHG HLJHQYDOXH DQG DOO SHUWXUEDWLRQV IRU WKH SDUWLFXODU PRGH OQf ,I LW LV DVVXPHG WR EH LQGHSHQGHQW RI JDV SURSHUWLHV WKHQ Y HQ QDf I-I%f & f &RQVHTXHQWO\ LI ZH FKRRVH VRPH V\VWHP ZKRVH VSHHG RI VRXQG LV NQRZQ RU FDQ EH FDOFXODWHG SUHFLVHO\ IURP D UHOLDEOH HTXDWLRQ RI VWDWH ZH ZLOO JHW D UHODWLRQVKLS RU

PAGE 30

L f f ZKHUH UHIHUV WR WKH V\VWHP XQGHU LQYHVWLJDWLRQ 7KH VSHHG RI VRXQG RI DUJRQ LV FDOFXODWHG IURP WKH YLULDO HTXDWLRQ WUXQFDWHG DIWHU WKH WKLUG WHUP DV IROORZVrf f ZKHUH \ KHDW FDSDFLW\ UDWLR &S&Yf 0$U 0ROHFXODU PDVV RI DUJRQ 5 *DV FRQVWDQW 7 $EVROXWH 7HPSHUDWXUH Y PRODU YROXPH $;7f 6HFRQG DFRXVWLF YLULDO FRHIILFLHQW $7f 7KLUG DFRXVWLF YLULDO FRHIILFLHQW 7KH HIIHFWLYH HLJHQYDOXH GHWHUPLQHG WKLV ZD\ LV QRW H[DFW EXW WKH HIIHFW RQ VRQLF VSHHG PHDVXUHPHQWV LV VPDOO DQG RI QR FRQVHTXHQFH IRU WKH DFFXUDWH GHWHUPLQDWLRQ RI SKDVH ERXQGDULHV 7KH SUDFWLFDO DGYDQWDJHV RI VSKHULFDO UHVRQDWRUV LV UHDOL]HG IRU WKH SXUHO\ UDGLDO PRGHV RI YLEUDWLRQ ZKLFK LQYROYH QR WDQJHQWLDO PRWLRQ RI WKH JDV ZLWK UHVSHFW WR WKH UHVRQDWRU ZDOO LH QR YLVFRXV GUDJ HIIHFW ,Q DGGLWLRQ WKH

PAGE 31

UDGLDO PRGH UHVRQDQFH SHDNV W\SLFDOO\ KDYH QDUURZ KDOIZLGWK KLJK 4 YDOXHf )XUWKHUPRUH UHVRQDQFH IUHTXHQFLHV RI WKH UDGLDO PRGHV DUH VHQVLWLYH WR LPSHUIHFWLRQV RI QRQVSKHULFLW\ RQO\ WR VHFRQG RUGHUf &RQVHTXHQWO\ IURP DOO SRLQWV RI YLHZ WKH PRVW DFFXUDWH PHDVXUHPHQWV RI WKH VSHHG RI VRXQG VKRXOG EH REWDLQHG E\ XWLOL]LQJ WKH UDGLDO PRGHV RI YLEUDWLRQ 7KH ILUVW IHZ UDGLDO PRGHV DUH PRVWO\ XVHG EHFDXVH RI WKHLU DVVRFLDWHG ORZ IUHTXHQFLHV $ PRUH FRPSOHWH DFFRXQW RI WKHVH HIIHFWV FDQ EH IRXQG LQ UHIHUHQFHV DQG 4 LV XVXDOO\ GHILQHG DV Q B &HQWHU )UHTXHQF\ RI 5HVRQDQFH )UHTXHQF\ :LGWK G% 3RLQWV

PAGE 32

7DEOH 7KH YDOXHV RI WKH URRWV L f LQ DVFHQGLQJ RUGHU RI PDJQLWXGH GHJHQHUDF\ Q 1$0( Q 1$0( ,3 L ,G S ,V ,Q ,I J LJ S G OK V G OR V K OL N I I OM LT S S J ON L G OU V J K P G I OP V ,W I K N Q S N Q OX L I L\ S W L J R OY R ,G J V G O] V X T a7 OZ P M T P K K OD U OL O[ Y

PAGE 33

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

PAGE 34

FULWLFDO SRLQW WKH PD[LPXP SUHVVXUH DQG WKH PD[LPXP WHPSHUDWXUH IRU WKH KHWHURJHQHRXV UHJLRQ PXVW EH NQRZQ (YLGHQFH IRU WKH H[LVWHQFH RI D FULWLFDO SRLQW ZDV ILUVW SUHVHQWHG LQ E\ GH /D 7RXUnf ZKR REVHUYHG WKDW D OLTXLG ZKHQ KHDWHG LQ D KHUPHWLFDOO\ VHDOHG JODVV WXEH LV UHGXFHG WR YDSRU LQ D YROXPH IURP WZR WR IRXU WLPHV WKH RULJLQDO YROXPH RI WKH VDPSOH +RZHYHU LW ZDV QRW XQWLO WKH TXDQWLWDWLYH PHDVXUHPHQWV RI $QGUHZV RQ FDUERQ GLR[LGHf LQ WKDW WKH QDWXUH RI WKH WUDQVLWLRQ ZDV XQGHUVWRRG +H ZDV WKH ILUVW WR FRLQ WKH WHUP FULWLFDO SRLQW IRU WKH SKHQRPHQRQ DVVRFLDWHG ZLWK WKLV OLTXLGYDSRU WUDQVLWLRQ )RU IOXLG PL[WXUHV WKH ILUVW UHOLDEOH H[SHULPHQWDO LQYHVWLJDWLRQ RI WKH FULWLFDO VWDWH EHJDQ ZLWK WKH ZRUN RI .XHQHQf LQ ,QWHUHVW LQ WKH FULWLFDO UHJLRQ LQ WKH SHULRG ZDV SHDNHG E\ WKH H[SHULPHQWDO DQG WKHRUHWLFDO VWXGLHV RI SUHVVXUHYROXPH WHPSHUDWXUH SY7f UHODWLRQVKLS IRU ERWK SXUH JDVHV DQG JDVHRXV PL[WXUHV RI YDQ GHU :DDOV DQG KLV DVVRFLDWHV DW WKH 8QLYHUVLWLHV RI $PVWHUGDP DQG /HLGHQ )LJXUH VKRZV WKH SY7 UHODWLRQVKLS RI WKH OLTXLG DQG JDV SKDVHV RI D SXUH IOXLG 7KH XSSHUULJKW SURMHFWLRQ VKRZV VHYHUDO LVRWKHUPV RQ D SY GLDJUDP WKH XSSHUOHIW SURMHFWLRQ LV D S7 SORW VKRZLQJ VHYHUDO LVRFKRUHV DQG WKH ERWWRP SURMHFWLRQ VKRZV VHYHUDO LVREDUV RQ D 7Y SORW 7KH WRQJXHn VKDSHG UHJLRQ ERXQGHG E\ SRLQWV $ & ( % LV WKH FRH[LVWHQFH FXUYH RU YDSRU SUHVVXUH FXUYHf 7KLV FXUYH PD\ EH FRQVLGHUHG DV FRQVLVWLQJ RI WZR FXUYHV WKH EXEEOHSRLQW WKH

PAGE 35

)LJXUH 7KH SY7 EHKDYLRU RI SXUH IOXLG ,Q WKH FHQWHU LV VNHWFKHG WKH VXUIDFH S SY7f >)URP +LUVFKIHOGHU -RVHSK &XUWLVV &KDUOHV ) DQG %LUG 5 %\URQ 0ROHFXODU 7KHRU\ RI *DVHV DQG /LTXLGV &RS\ULJKW k E\ -RKQ :LOH\ t 6RQV ,QF 5HSULQWHG E\ SHUPLVVLRQ RI -RKQ :LOH\ t 6RQV ,QF@ SRLQW RI LQLWLDO YDSRUL]DWLRQ ZKHQ WKH SUHVVXUH RI WKH OLTXLG LV UHGXFHGf FXUYH $'& DQG WKH GHZSRLQW WKH SRLQW RI LQLWLDO FRQGHQVDWLRQ ZKHQ WKH SUHVVXUH RI WKH JDV LV LQFUHDVHGf FXUYH %(& 7KHVH WZR FXUYHV PHHW DW WKH FULWLFDO SRLQW & ZKLFK EHORQJLQJ WR ERWK FXUYHV LQGLFDWHV WKH LGHQWLW\ RI WKH OLTXLG DQG YDSRU SKDVHV $W WKLV SRLQW YLHZLQJ IURP WKH LVRWKHUPDO SHUVSHFWLYH DS? n n D3 0W &1 7 f

PAGE 36

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nnf 7KH FRPSOHWH SKDVH EHKDYLRU RI D ELQDU\ V\VWHP LV UHSUHVHQWHG E\ WKH IRXU GLPHQVLRQDO VXUIDFH S SYW[f ZKHUH [ LV WKH PROH IUDFWLRQ RI RQH RI WKH WZR FRPSRQHQWV &RQVHTXHQWO\ WKH FULWLFDO SRLQW RI D PL[WXUH KDV WR EH UHGHILQHG DV WKH SRLQW ZKHUH OLTXLG DQG YDSRU EHFRPH LGHQWLFDO VXEMHFW WR WKH FRQVWUDLQW RI HTXDOLW\ RI FRPSRVLWLRQ 6LQFH WKLV ZRUN GHDOV ZLWK D ELQDU\ PL[WXUH RI FDUERQ GLR[LGH DQG HWKDQH DWWHQWLRQ WKURXJKRXW ZLOO EH PDLQO\ IRFXVHG RQ WKH UHOHYDQW LQIRUPDWLRQ IRU WKLV PL[WXUH &DUERQ GLR[LGH DQG HWKDQH IRUP DQ D]HRWURSLF PL[WXUH DW WKH DSSURSULDWH FRPSRVLWLRQ 7KLV ELQDU\ V\VWHP GHYLDWHV SRVLWLYHO\ IURP WKH LGHDO VROXWLRQ ZKLFK REH\V 5DRXOWnV ODZ ,Q WKLV FDVH D FULWLFDO SRLQW ORFXV IROORZV D FXUYH EHWZHHQ

PAGE 37

)LJXUH 3UHVVXUH7HPSHUDWXUH0ROH IUDFWLRQ UHODWLRQVKLS RI V\VWHP RI FDUERQ GLR[LGH DQG HWKDQH )LJXUH 3UHVVXUHWHPSHUDWXUH SURMHFWLRQ RI WKH V\VWHP FDUERQ GLR[LGHHWKDQH LQ WKH FULWLFDO UHJLRQ /LQHV = DQG =E.E DUH YDSRU SUHVVXUH FXUYHV RI SXUH FDUERQ GLR[LGH DQG RI SXUH HWKDQH UHVSHFWLYHO\

PAGE 38

WKH FULWLFDO SRLQW RI SXUH FDUERQ GLR[LGH DQG WKDW RI SXUH HWKDQH )LJXUHV DQG LOOXVWUDWH WKH SUHVVXUH WHPSHUDWXUHPROH IUDFWLRQ GLDJUDP DQG FULWLFDO ORFXV RI WKLV PL[WXUH UHVSHFWLYHO\ ,Q ILJXUH DW SRLQW % WKH PD[LPXP D]HRWURSH LV WDQJHQW WR WKH FULWLFDO FXUYH .IW%.%f ZKLFK KDV D PLQLPXP LQ WHPSHUDWXUH 'HWHUPLQDWLRQV RI &ULWLFDO 3RLQW &RQYHQWLRQDOO\ WKHUH DUH WKUHH SULQFLSDO PHWKRGV RI ORFDWLQJ WKH FULWLFDO SRLQW 7KHVH DUH GHVFULEHG EULHIO\ EHORZf ,VRWKHUP $SSURDFK &ULWLFDO WHPSHUDWXUH PD\ EH HVWLPDWHG WR ZLWKLQ D IHZ KXQGUHGWKV RI D NHOYLQ E\ DQ DQDO\VLV RI WKH JHRPHWU\ RI WKH LVRWKHUPVnf 7KH LVRWKHUPV LQ WKH LPPHGLDWH QHLJKERUKRRG RI WKH FULWLFDO WHPSHUDWXUH DUH PHDVXUHG ZLWK VXIILFLHQW DFFXUDF\ WR DOORZ WKH FULWLFDO WHPSHUDWXUH WR EH GHWHUPLQHG IURP WKH LQIOHFWLRQ SRLQWV RI WKH LVRWKHUPV 7KH PLQLPXP YDOXHV RI WKH GHULYDWLYH RI SUHVVXUH ZLWK UHVSHFW WR YROXPH DW FRQVWDQW WHPSHUDWXUH IRXQG JUDSKLFDOO\ DUH SORWWHG DV D IXQFWLRQ RI WHPSHUDWXUH RU SUHVVXUH RU PRODU YROXPHf 7KH LQWHUVHFWLRQ RI WKLV OLQH ZLWK WKH KRUL]RQWDO D[LV JLYHV WKH FULWLFDO SDUDPHWHUV 7F 3F YF f )LJXUHV LOOXVWUDWH WKLV DSSURDFK

PAGE 39

,VREDU $SSURDFK 7KLV PHWKRG LV VRPHWLPHV FDOOHG WKH UHFWLOLQHDU GLDPHWHU PHWKRG RI &DLOOHWHW DQG 0DWKLDVf ,I S DQG SJ DUH WKH GHQVLWLHV RI OLTXLG DQG RI VDWXUDWHG YDSRU LQ HTXLOLEULXP ZLWK LW VR FDOOHG RUWKREDULF GHQVLWLHVf WKHLU PHDQ LV D OLQHDU IXQFWLRQ RI WHPSHUDWXUH f ZKHUH SR PHDQ GHQVLW\ RI OLTXLG DQG LWV VDWXUDWHG YDSRU DW r& W WHPSHUDWXUH LQ &HOVLXV D FRQVWDQW L ? 7 9 9ROXPH )LJXUH 6FKHPDWLF GLDJUDP RI SUHVVXUH DQG YROXPH UHODWLRQVKLS RI FDUERQ GLR[LGH ZLWK VHYHUDO LVRWKHUPV LQ D EURDG UHJLRQ

PAGE 40

)LJXUH 6FKHPDWLF GLDJUDP RI JUDSKLFDO GHWHUPLQDWLRQ RI FULWLFDO WHPSHUDWXUH RI FDUERQ GLR[LGH

PAGE 41

7KXV LI SJ DQG S[ DUH VHSDUDWHO\ SORWWHG DJDLQVW WKH WHPSHUDWXUH W WKH ORFXV RI WKH SRLQWV ELVHFWLQJ WKH MRLQV RI FRUUHVSRQGLQJ YDOXHV RI S[ DQG SJ LV D VWUDLJKW OLQH VHH ILJXUH f 7KH SRLQW ZKHUH WKLV VWUDLJKW OLQH FXWV WKH FRH[LVWHQFH FXUYH LV WKH FULWLFDO SRLQW ,Q PDQ\ FDVHV WKLV HPSLULFDO ODZ RI UHFWLOLQHDU GLDPHWHUV KROGV YHU\ ZHOO EXW VRPHWLPHV RYHU D ODUJH UDQJH RI WHPSHUDWXUH WKH DFWXDO EHKDYLRU VKRZV D VOLJKW FXUYDWXUH )XUWKHUPRUH YHU\ QHDU WKH 'HQVLW\ )LJXUH 6FKHPDWLF GLDJUDP RI FULWLFDO SRLQW GHWHUPLQDWLRQ IURP PHDVXUHPHQWV RI WKH OLTXLG GHQVLW\ S DQG WKH YDSRU GHQVLW\ S DFFRUGLQJ WR WKH PHWKRG RI &DLOOHWHW DQG 0DWKLDV

PAGE 42

FULWLFDO SRLQW ZLWKLQ VHYHUDO P. RI 7Ff UHDO V\VWHPV VKRZ D GHYLDWLRQ IURP OLQHDULW\ ,VRFKRUH $SSURDFK 7KHUH DUH WZR VOLJKWO\ GLIIHUHQW ZD\V IRU WDNLQJ WKLV DSSURDFK 7KH ILUVW RQH LV WR VWXG\ WKH GLVFRQWLQXLW\ RI WKH LVRFKRUH FXUYHf 7KLV PHWKRG EHJLQV ZLWK ORDGLQJ D ERPE RI FRQVWDQW YROXPH ZLWK D VHULHV RI NQRZQ ZHLJKWV RI WKH VXEVWDQFH DQG WKHQ VWXG\LQJ WKH EHKDYLRU RI WKH SUHVVXUH DV b I f 68%&5P&$/ '(16I7< r 0($5&5,7,&$/ '(16,7< L r D 683(5&5,7,&$/ '(16,7< t \ r V r \ \ 6n ZI 7(03(5$785( )LJXUH 6FKHPDWLF GLDJUDP RI LVRFKRULF PHWKRG RI FULWLFDO SRLQW GHWHUPLQDWLRQ

PAGE 43

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f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

PAGE 44

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nf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f

PAGE 45

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

PAGE 46

&+$37(5 (;3(5,0(17 ,QVWUXPHQWV 7KH GHYHORSPHQW RI D VSKHULFDO DFRXVWLF UHVRQDWRU DV D WRRO WR WKH VWXG\ VSHHG RI VRXQG LQ JDVHV VXFK DV DUJRQ EXWDQH LVREXWDQH HWF ZKLFK LQ WXUQ OHDGV WR YDOXHV RI VRPH WKHUPRG\QDPLF DQG HTXDWLRQ RI VWDWH SURSHUWLHV KDV EHHQ VXFFHVVIXO LQ WKH SDVW E\ ZRUNHUV LQ WKLV UHVHDUFK JURXS nf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

PAGE 47

D QHZ KLJK SHUIRUPDQFH VWDLQOHVV VWHHO UHVRQDWRU DVVHPEO\ ZDV GHVLJQHG DQG EXLOW DQG QHZ GDWD DFTXLVLWLRQ VRIWZDUH ZDV GHYHORSHGf 7KH QHZ DSSDUDWXV ZDV XVHG WR PHDVXUH WKH VRQLF VSHHG LQ VRPH FDUHIXOO\ EOHQGHG JDV PL[WXUHV %HFDXVH WKH PL[WXUHV FRQWDLQHG VPDOO FRQFHQWUDWLRQV RI FRQGHQVLEOHV VXFK DV KH[DQH IRU H[DPSOH LW ZDV SRVVLEOH WR FRRO WKHP EHORZ WKHLU GHZ SRLQWV DQG LW ZDV REVHUYHG WKDW WKH VRQLF VSHHG IDLWKIXOO\ UHYHDOHG WKH SUHFLVH ORFDWLRQ RI WKH SKDVH

PAGE 48

FKDQJHf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f LQ GLDPHWHU ZLWK D FP LQFKf WKLFN ZDOO 7KH WDQN LV QRW SHUIHFWO\ VSKHULFDO 7KLV LPSHUIHFWLRQ KRZHYHU GRHV QRW VLJQLILFDQWO\ GHJUDGH WKH GDWD LQWHJULW\ DV ORQJ DV DSSURSULDWH PHDVXUHPHQW WHFKQLTXHV DUH XVHG DV GLVFXVVHG LQ WKH SHUYLRXV VHFWLRQ DQG SURYHQ WR EH YDOLG E\ WKH SUHYLRXV LQYHVWLJDWRUVn 7KH WDQN ZDV REWDLQHG IURP WKH PDQXIDFWXUHU 3ROOXWLRQ 0HDVXUHPHQW &RUS&KLFDJR ,/f ZLWK RQO\ D VLQJOH FPA LQf KROH ORFDWHG DW RQH SROH LQ VSKHULFDO SRODU FRRUGLQDWHVf $ PDWFKLQJ KROH ZDV GULOOHG WKURXJK WKH RSSRVLWH SROH Qf DQG WZR FP b LQf

PAGE 50

)LJXUH 7KH VSKHULFDO UHVRQDWRU

PAGE 51

KROHV GULOOHG WKURXJK WKH ZDOO DW Q ZLWK S DQG Q UHVSHFWLYHO\ 1R KROHV ZHUH ORFDWHG RQ WKH HTXDWRULDO ZHOG EHDG Qf )LJXUH VKRZV WKH RYHUDOO YLHZ RI WKH VSKHULFDO UHVRQDWRU 7UDQVGXFHUV 7KH DFWLYH HOHPHQW RI WKH DFRXVWLF WUDQVGXFHU LV D LQ GLDPHWHU SLH]RHOHFWULF FHUDPLF ELPRUSK 9HUQLWURQ %HGIRUG 2KLRf PDGH RI 3=7$ PDWHULDO OHDG ]LUFRQDWHOHDG WLWDQDWHf 7KLV FHUDPLF ZDV IRXQG WR SURGXFH D VLJQDO RI KLJK YROXPH ZKHQ GULYHQ ZLWK D VLQXVRLGDO ZDYHIRUP RI YROWV SHDNWRSHDN ,W DOVR KDV D KLJK PD[LPXP RSHUDWLQJ WHPSHUDWXUH RI r&f $ WUDQVGXFHU DVVHPEO\ ZDV GHVLJQHG DQG IDEULFDWHG ,W LV VKRZQ LQ ILJXUH 7KH HOHFWULFDO IHHGWKURXJK PDGH IURP EUDVV ZDV KHOG LQ D 'HOULQ LQVHUW WR SUHYHQW D VKRUW FLUFXLW $ FURVVVHFWLRQDO YLHZ RI WKH WUDQVGXFHU PRXQWHG RQ WKH UHVRQDWRU LV DOVR VKRZQ LQ ILJXUH 7KH WUDQVGXFHU DVVHPEOLHV ZHUH PDWHG WR WKH UHVRQDWRU WKURXJK VKRUW b LQ GLDPHWHU WXEHV 7ZR LGHQWLFDO WUDQVGXFHUV ORFDWHG DW ULJKW DQJOHV WR HDFK RWKHU ZHUH XVHG 7KLV RULHQWDWLRQ KDV SURYHG WR \LHOG D EHWWHU UHVROXWLRQ RI WKH UDGLDO PRGHV RI YLEUDWLRQ GXH WR GLPLQLVKLQJ LQ LQWHQVLW\ RI VRPH QRQUDGLDO PRGHV S PRGHV IRU H[DPSOH 2QH WUDQVGXFHU IXQFWLRQV DV D VSHDNHU ,W LV DQ LQSXW WUDQVGXFHU ZKLFK PHFKDQLFDOO\ GHIRUPV ZKHQ D YROWDJH LV DSSOLHG 7KH RWKHU WUDQVGXFHU IXQFWLRQV DV D PLFURSKRQH ,W LV DQ RXWSXW WUDQVGXFHU ZKLFK RSHUDWHV RQ D UHYHUVH PHFKDQLVP RI

PAGE 52

'HOULQ )HHGf§7KUX *XLGH 5LQJ %UDVV )HHGf§7KUX 'HOULQ ,QVXODWRU 6SDFHU :DYH 6SULQJ 6SDFHU 7UDQVGXFHU f§5LQJ ILJXUH 7KH WUDQVGXFHU FURVVVHFWLRQDO YLHZf

PAGE 53

WKH LQSXW WUDQVGXFHU SURGXFLQJ DQ HOHFWULF VLJQDO ZKHQ PHFKDQLFDOO\ GHIRUPHG E\ SUHVVXUH ZDYHV 0L[LQJ FRQWURO V\VWHP $ FLUFXODWLQJ SXPS LV QHHGHG WR SURPRWH PL[LQJ RI V\VWHP FRPSRQHQWV DQG RUf SKDVHV DQG WR DVVLVW ZLWK WKHUPDO HTXLOLEUDWLRQ E\ HOLPLQDWLQJ VWUDWLILFDWLRQ ,W ZDV PDGH IURP VWDLQOHVV VWHHO DQG FRQVLVWV RI WZR SDUWV WKH OLTXLG FROOHFWRU DQG WKH SXPS )LJXUH VKRZV WKH GHVLJQ RI WKH XQLW ,W FRQVLVWV RI D YHUWLFDOO\ PRXQWHG VWDLQOHVV VWHHO F\OLQGHU FP LQf LQ GLDPHWHU ZLWK D IUHHO\ VOLGLQJ LQWHUQDO SLVWRQ FP LQf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f GLDPHWHU SLORW WXEXODWLRQV ZKLFK VOLS LQWR WKH SRODU KROHV DQG ZKHQ SUHVVHG DJDLQVW WKH WDQN ZHUH VHDOHG E\ HODVWRPHULF 2ULQJV FRQILQHG WR JODQGV FRPSULVHG RI FLUFXODU JURRYHV LQ WKH ILWWLQJV DURXQG WKH SLORW WXEHV DQG WKH ULQJ OLNH DUHDV RI WKH DGMDFHQW WDQN ZDOO VXUURXQGLQJ WKH KROHV 7KHVH ILWWLQJV ZHUH

PAGE 54

0$*1(7 $66(0%/< 635,1* 67$,1/(66 67((/ &
PAGE 55

EUD]HG WR VWDLQOHVV VWHHO WXEHV XVHG IRU FKDUJLQJ WKH UHVRQDWRU DQG FLUFXODWLQJ LWV FRQWHQWV WKURXJK DQ H[WHUQDO ORRS FRQWDLQLQJ WKH PDJQHWLFDOO\ GULYHQ SXPS 7KH WZR LGHQWLFDO 3=7 ELPRUSK SLH]RHOHFWULF WUDQVGXFHU DVVHPEOLHV ZHUH VLPLODUO\ PRXQWHG WR WKH WDQN DW WKH FP b LQf KROH SRVLWLRQV 7KH DVVHPEO\ VXSSRUW IUDPH FRQVLVWV RI WZR SDUDOOHO FP K LQf WKLFN VWDLQOHVVVWHHO SODWHV EHWZHHQ ZKLFK DQ DUUD\ RI FP LQf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f 7KH QRUPDO SXPSLQJ UDWH RI OLWHUV SHU PLQXWH SURYHG DGHTXDWH IRU WKHVH SXUSRVHV $ FP N LQf WXEH FRQQHFWHG WR WKH

PAGE 56

IOXLG FLUFXODWLRQ ORRS E\ D WHH OHDGV WR D GLDSKUDJPW\SH SUHVVXUH JDXJH DQG D V\VWHP VKXWRII YDOYH 7KH UHVRQDWRU DVVHPEO\ DQG FLUFXODWLQJ SXPS ZHUH VXVSHQGHG IURP D KRUL]RQWDO SODWH ZKLFK IRUPV WKH FRYHU RI D VWLUUHG OLTXLG EDWK LQ ZKLFK WKH V\VWHP LV PRXQWHG 7KH EDWK ZDV KRXVHG LQ D ZHOOLQVXODWHG FRQWDLQHU DQG FRQQHFWHG E\ LQVXODWHG WXELQJ WR D FRPSXWHUFRQWUROOHG KHDW H[FKDQJHU FDSDEOH RI RSHUDWLQJ IURP WR %DWK WHPSHUDWXUH ZDV PHDVXUHG ZLWK D IRXUZLUH SODWLQXP UHVLVWDQFH WKHUPRPHWHU 57'f )LJXUH VKRZV WKH LQVWUXPHQWDO VHWXS (OHFWURQLF KDUGZDUH 7KH DXWRPDWHG V\VWHPV HPSOR\HG LQ WKLV ZRUN DUH VKRZQ LQ ILJXUHV DQG )LJXUH VKRZV WKH VHWXS XVHG LQ WKH H[SHULPHQWV RQ SXUH FDUERQ GLR[LGH DQG RQ D PL[WXUH RI FDUERQ GLR[LGH DQG HWKDQH )LJXUH VKRZV WKH VHWXS IRU WKH H[SHULPHQWV RQ SXUH HWKDQH ,Q WKLV ODWWHU VHWXS ZH KDYH H[SORUHG WKH FDSDELOLW\ RI XVLQJ D IDVWIRXULHU WUDQVIRUP ))7f WHFKQLTXH 8QIRUWXQDWHO\ WKH GDWD FROOHFWHG E\ WKLV WHFKQLTXH ZDV XQUHOLDEOH GXH WR OHDNV LQ WKH V\VWHP IRXQG DIWHU WKH H[SHULPHQW ZDV ILQLVKHG :KLOH WKH OHDNV ZHUH EHLQJ IL[HG WKH LQVWUXPHQWV ZHUH WUDQVIHUUHG WR DQRWKHU SURMHFW &RQVHTXHQWO\ WKH ))7 H[SORUDWLRQ ZDV GLVFRQWLQXHG IRU HWKDQH DQG WKH VHWXS VKRZQ LQ ILJXUH ZDV EXLOW WR ILQLVK WKLV SURMHFW

PAGE 57

)LJXUH 7KH ,QVWUXPHQWDO 6HWXS FR

PAGE 58

6<67(0 &21752//(5 )LJXUH 7KH H[SHULPHQWDO VHWXS IRU & DQG &&+ PL[WXUH

PAGE 59

)LJXUH 7KH ILUVW H[SHULPHQWDO VHWXS IRU &+

PAGE 61

(YHQ WKRXJK WKHUH ZHUH GLIIHUHQW VHWXSV WKH EDVLF SULQFLSOHV DUH WKH VDPH DQG FDQ EH GLYLGHG LQWR WZR SDUWV WKH LQSXW VLJQDO JHQHUDWLRQ SDUW DQG WKH RXWSXW VLJQDO DQDO\VLV SDUW 7KHVH DUH VKRZQ E\ GDVKHGOLQH ER[HV LQ ILJXUHV DQG &RPPDQGV DQG GDWD ZHUH WUDQVIHUUHG DPRQJ WKH LQVWUXPHQWV DQG FRPSXWHU RYHU WKH *HQHUDO 3XUSRVH ,QWHUIDFH %XV *3,%f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

PAGE 62

p 7DEOH 6XPPDU\ RI HOHFWURQLF KDUGZDUH ,WHP 0DQXIDFWXUHU 0RGHO QXPEHU 5DQJH RI RSHUDWLRQ $FFXUDF\ 0DQXIDFWXUH TXRWHf )XQFWLRQ 7HPSHUDWXUH FRQWUROOHU %ULQNPDQQ 5.6' WR r& r& &RQWURO WHPSHUDWXUH 3UHVVXUH 7UDQVGXFHU 6HQVRWHF 7-( SVLD ZLWK H[FLWDWLRQ YROWDJH YROWV sb RI IXOO VFDOH 0HDVXUH SUHVVXUH 'LJLWDO 6WUDLQ JDJH WUDQVGXFHU LQGLFDWRU %HFNPDQ LQGXVWULDO FRUS 6HULHV PY WR YROW b RI IXOO VFDOHs GLJLW )XOO VFDOH SVLD $FFHSWV D VWUDLQ JDJH W\SH WUDQVGXFHU LQSXW WR PHDVXUH SUHVVXUH /RFNLQ DPSOLILHU 6WDQIRUG UHVHDUFK V\VWHP ,QF 65 6LQJOH SKDVH +] WR .+] s YROW RXWSXW FRUUHVSRQGV WR IXOO VFDOH LQSXW $& 6LJQDO UHFRYHULHV PHDVXUH VLJQDOV GRZQ WR QY IXOO VFDOH ZKLOH UHMHFWLQJ LQWHUIHULQJ VLJQDOV XS WR V WLPHV ODUJHU E\ WKH SHUIRUPDQFH RI WZR OLQHV QRWFK ILOWHUV DQG DXWRWUDFNLQJ EDQGSDVV ILOWHU )UHTXHQF\ FRXQWHU 6HQFRUH )& +] WR *+] s GLJLWV RQ IUHTXHQF\ OHVV WKDQ .+] 0HDVXUH IUHTXHQF\ RI SHULRGLF VLJQDO RR

PAGE 63

7DEOH &RQWLQXHGf ,WHP 0DQXIDFWXUHU 0RGHO QXPEHU 5DQJH RI RSHUDWLRQ $FFXUDF\ 0DQXIDFWXUH TXRWHf )XQFWLRQ 3UH $PSOLILHU 6WDQIRUG UHVHDUFK V\VWHPV ,QF 65 8S WR PY UPV LQSXW b *DLQ 5HGXFHV LQSXW QRLVH DQG H[WHQGV WKH IXOO VFDOH VHQVLWLYLW\ WR QY 9ROWPHWHU .HLWKOH\ LQVWUXPHQWV ,QF PY WR YROWV b 5HDGLQJ GLJLWV IRU PLQLPXP UDQJH RU b UGJ GLJLW IRU PD[ UDQJH 0HDVXUH $F RU 'F 9ROWDJH 'LJLWDO PXOWLPHWHU .HLWKOH\ LQVWUXPHQWV ,QF $ 7HUPLQDO t WHUPLQDO UHVLVWDQFH PHDVXUHPHQWV LQ WKH UDQJH RI SIL WR 04 'HSHQGV RQ UDQJH 6HH PDQXIDFWXUHnV PDQXDO IRU GHWDLOV 0HDVXUH UHVLVWDQFH IURP 3W WHPSHUDWXUH SUREH )XQFWLRQ JHQHUDWRU :DYHWH[ 6DQ 'LHJR ,QF $ +] WR 0+] 'LDO DFFXUDF\ s b RI IXOO VFDOH *HQHUDWHV D SUHFLVH VLQH WULDQJOHVTXDUH ZDYHIRUP

PAGE 64

7DEOH &RQWLQXHGf ,WHP 0DQXIDFWXUHU 0RGHO QXPEHU 5DQJH RI RSHUDWLRQ $FFXUDF\ 0DQXIDFWXUH TXRWHf )XQFWLRQ :DYHIRUP V\QWKHVL]HU DQG IXQFWLRQ JHQHUDWRU +HZOHWW 3DFNDUG % S+] WR 0+] IRU VLQH ZDYH ZLWK DPSOLWXGH RI PY WR Y 33 s ; RI VHOHFWHG YDOXH LQ WHPSHUDWXUH UDQJH WR & *HQHUDWHV VLQH WULDQJXODU RU VTXDUH ZDYH 2VFLOORVFRSH /HDGHU ,QSXW VLJQDOV KLJKHU WKDQ YROWV $FSS 'Ff PD\ GDPDJH FLUFXLW &20321(176 9HUWLFDO VHFWLRQ s b sb PDJ ; f +RUL]RQWDO VHFWLRQ s b 'LVSOD\ ZDYHIRUP '\QDPLF VLJQDO DQDO\]HU +HZOHWW 3DFNDUG +3$ S+] WR .+] IRU VLQJOHn FKDQQHO VSHFWUXP RU KDOI RI WKH WKLV UDQJH IRU WZR FKDQQHO VSHFWUXP 1R LQIRUPDWLRQ DYDLODEOH :DYH VLJQDO VRXUFH $OVR FRQWDLQV ))7 DOJRULWKP WR FRQYHUW DQ DQDORJ LQSXW VLJQDOWLPH GRPDLQf WR D VLJQDO GLVSOD\HG LQ WKH IUHTXHQF\ GRPDLQ 8 R

PAGE 65

7DEOH *DV 6SHFLILFDWLRQV *DV 0DQXIDFWXUHU *UDGH 3XULW\ &DUERQ GLR[LGH 6FRWW VSHFLDOW\ JDVHV 5HVHDUFK *UDGH 0ROH b (WKDQH 6FRWW 6SHFLDOW\ *DVHV 5HVHDUFK *UDGH b $UJRQ 0DWKHVRQ JDV 3URGXFWV 5HVHDUFK *UDGH 9ROXPH b 7DEOH &KDUJLQJ 3UHVVXUH RI JDV V\VWHPV *DV V\VWHP 6WDUWLQJ WHPSHUDWXUH r&f &KDUJLQJ SUHVVXUH SVLDf &DUERQ GLR[LGH FR FK IRU &+J DQG PDNH XS WR ZLWK &2 FK 5HVRQDQFH )UHTXHQF\ 0HDVXUHPHQWV 0HDVXUHPHQWV ZHUH PDGH RQ LVRFKRUH VDPSOHV LQWURGXFHG LQWR WKH VSKHULFDO UHVRQDWRU ZKLFK ZDV LQLWLDOO\ ULQVHG ZLWK DUJRQ JDV DQG VXEVHTXHQWO\ HYDFXDWHG RYHUQLJKW DW KLJK WHPSHUDWXUH r&f ZLWK D PHFKDQLFDO SXPS FRQQHFWHG WR D OLTXLG QLWURJHQ WUDS 7KH DSSDUDWXV ZDV WKHQ EURXJKW WR D WHPSHUDWXUH DERYH WKH NQRZQ OLWHUDWXUH FULWLFDO WHPSHUDWXUH $IWHU WKDW JDV ZDV LQWURGXFHG VORZO\ LQWR WKH UHVRQDWRU WR EULQJ WKH SUHVVXUH WR WKH FDOFXODWHG YDOXH VHH WDEOH f 7KH DSSDUDWXV ZDV WKHQ VHDOHG RII 7KH FKDUJLQJ SUHVVXUH YDOXH

PAGE 66

ZDV FDOFXODWHG XVLQJ WKH $*$ SURJUDP 'HWDLOV RI WKH FDOFXODWLRQ DUH JLYHQ LQ DSSHQGL[ $ ,Q WKH FDVH RI WKH PL[WXUH RI &2 DQG &+ HWKDQH ZDV ILUVW LQWURGXFHG 3UHVVXUH ZDV PRQLWRUHG URXJKO\ E\ D %RXUGRQ SUHVVXUH JDXJH DQG PRUH SUHFLVHO\ E\ D GLJLWDO SUHVVXUH JDXJH SUHYLRXVO\ FDOLEUDWHG WR \LHOG DQ DEVROXWH SUHVVXUH UHDGLQJ 7KH PDJQHWLF FLUFXODWLQJ SXPS ZDV WXUQHG RQ WR DVVXUH DGHTXDWH PL[LQJ 2QFH WKH V\VWHP ZDV DW VWDEOH FRQGLWLRQV LQ WHPSHUDWXUH DQG SUHVVXUH ZKLFK WRRN DSSUR[LPDWHO\ D IHZ KRXUVf VHDUFKLQJ IRU UHVRQDQFH IUHTXHQFLHV EHJDQ )OXLGV LQ WKH UHVRQDWRU ZHUH VWLPXODWHG DFRXVWLFDOO\ E\ GULYLQJ RQH RI WKH ELPRUSK WUDQVGXFHUV ZLWK D YROW SHDNWR SHDN VLQHZDYH VLJQDO JHQHUDWHG XQGHU PLFURSURFHVVRU FRQWURO 3&$7f RI D SURJUDPPDEOH VLJQDO V\QWKHVL]HU +3% LQ WKH FDVH RI SXUH FDUERQ GLR[LGH DQG RI WKH RI & DQG &+ PL[WXUH RU RI D ZDYH IXQFWLRQ JHQHUDWRU :$9(7(;f FRQWUROOHG WKURXJK D SURJUDPPDEOH GLJLWDOWRDQDORJ FRQYHUWHU '$&f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

PAGE 67

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f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

PAGE 68

RI WKH ILUVW UDGLDO PRGH SHDN SHDN LQ ILJXUH f (DFK RI WKHVH ZDV LQYHVWLJDWHG FORVHO\ DV VKRZQ LQ WKH VXEVHTXHQW IRXU URZV (DFK ILJXUH LQ WKH WKLUG URZ LV D VSHFWUXP RI D VLQJOH SHDN RI URZ WZR VFDQQHG LQ D PXFK QDUURZHU IUHTXHQF\ UDQJH )UHTXHQFLHV IRXQG IRU HDFK SHDN LQ WKLV URZ ZHUH XVHG WR FDOFXODWH WKH VHFRQG WKLUG DQG IRUWK UDGLDO IUHTXHQFLHV 3HDNV FRUUHVSRQGLQJ WR WKHVH FDOFXODWHG IUHTXHQFLHV ZHUH VHDUFKHG DV VKRZQ WKH ILJXUHV LQ WKH IRUWK ILIWK DQG VL[WK URZ $SSDUHQWO\ WKH WKLUG SHDN SHDN LQ ILJXUH f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f ER[ VLQFH WKH LQVWUXPHQWDO VHWXS ZDV DOWHUHG +DYLQJ HVWDEOLVKHG WKH ORFDWLRQ RI WKH SHDN WR EH WUDFNHG GDWD FROOHFWLRQ ZDV EHJXQ )RU SXUH & DQG IRU WKH PL[WXUH RI &&+ H[SHULPHQWV SURJUDPV FDOOHG 9,.,1* DQG 9,.$1 ZHUH XVHG ,Q WKLV YHUVLRQ RQO\ GDWD FRUUHVSRQGLQJ WR WKH PD[LPXP DPSOLWXGH ZHUH UHFRUGHG IRU HDFK VZHHS IUHTXHQF\ UDQJH )RU WKH SXUH &+ H[SHULPHQW KRZHYHU GDWD RI WKH HQWLUH VZHHS IUHTXHQF\ UDQJH ZHUH UHFRUGHG YLD WKH SURJUDP

PAGE 69

W WU R )5(48(1&< +] ILJXUH 6HULHV RI VFDQQLQJ URXWLQH

PAGE 70

FDOOHG 0$;f VLQFH LW ZDV EHOLHYHG WKH PRUH GHWDLOHG LQIRUPDWLRQ ZRXOG EH LQVWUXFWLYH 7KLV FKDQJH ZDV SRVVLEOH EHFDXVH RI DGYDQFHV LQ WKH DYDLODEOH GDWD DFTXLVLWLRQ KDUGZDUH ,Q ERWK FDVHV WKH SURJUDPV ORFNHG RQWR WKH WUDFNHG SHDN IRU WKH HQWLUH SURFHVV DV EDWK WHPSHUDWXUH ZDV ORZHUHG RU UDLVHGf DW D UDWH RI D IHZ PLOOLNHOYLQ SHU PLQXWH RYHU D SHULRG RI VHYHUDO KRXUV 3UHVVXUH DQG WHPSHUDWXUH PHDVXUHPHQWV ZHUH VLPXOWDQHRXVO\ FROOHFWHG DORQJ ZLWK IUHTXHQF\ DQG DPSOLWXGH 8SRQ FRPSOHWLRQ RI WKH H[SHULPHQW WKH WHPSHUDWXUH RI WKH V\VWHP ZDV EURXJKW EDFN WR WKH VWDUWLQJ YDOXH $ VPDOO DPRXQW RI JDV ZDV UHPRYHG WR FKDQJH WKH GHQVLW\ RI WKH V\VWHP DQG WKHQ DQRWKHU GDWD FROOHFWLRQ ZDV EHJXQ 7\SLFDOO\ WKH GHQVLW\ RI WKH V\VWHP ZDV LQLWLDOO\ DW D VXSHUFULWLFDO YDOXH 3J 3Ff DQG ZDV ILQDOO\ DW D VXEFULWLFDO YDOXH SJ SFf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

PAGE 71

*(763* 7KLV SURJUDP XWLOL]HV WKH SULQFLSOH RI UHODWLYH PHDVXUHPHQW PHQWLRQHG HDUOLHU LQ WKH WKHRU\ VHFWLRQ $OVR WKH WKHRUHWLFDO VSHHGV RI VRXQG ZHUH FDOFXODWHG IURP WKH $*$ HTXDWLRQ WKURXJK D FRPSXWHU SURJUDP FDOO 727'$7 IRU FRPSDULVRQ SXUSRVHV )LQDOO\ WKH UHVXOWV ZHUH PDQLSXODWHG DQG JUDSKLFDOO\ GLVSOD\HG XVLQJ SRSXODU VRIWZDUH LQFOXGLQJ /2786 48$7752 *5$)722/ DQG *5$3+(5 )LJXUH VKRZV D VFKHPDWLF GLDJUDP RI WKH HQWLUH SURFHVV RI GDWD FROOHFWLRQ

PAGE 72

)5(48(1&< += f )LJXUH 7ZR H[SHULPHQWDO DSSURDFKHV RI UHVRQDQFH IUHTXHQF\ GHWHUPLQDWLRQ WKH PD[LPXP DPSOLWXGH DSSURDFK DQG WKH YROWDJH SKDVH FKDQJH DSSURDFK FQ

PAGE 73

)LJXUH )ORZ FKDUW RI H[SHULPHQWDO VFKHPH DQG GDWD SURFHVVLQJ 7KH EURNHQOLQH ER[HV UHSUHVHQW WKH H[SHULPHQWDO V\VWHPV RI LQWHUHVW 7KH WKLFNOLQH ER[HV UHSUHVHQW WKH FRPSXWHU SURJUDPV YR

PAGE 74

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f DQG WKH WHPSHUDWXUH RI FDUERQ GLR[LGH LQ WKH UHVRQDWRU FKDUJHG QHDU LWV FULWLFDO GHQVLW\ ,W VKRZV WKH UHVXOWV RI IRXU UXQV RI WKLV VDPH LVRFKRUH )RU HDFK UXQ WKH FXUYH PD\ EH URXJKO\ GLYLGHG LQWR WZR SDUWV IRU WKH VDNH RI GLVFXVVLRQ 7KH ILUVW RQH LV D FXUYH ZKHUH WKH V\VWHP WHPSHUDWXUH LV JUHDWHU WKDQ RU HTXDO WR WKH FULWLFDO WHPSHUDWXUH 7F ,Q WKLV UHJLRQ WKH VSHHG RI VRXQG LV GLUHFWO\ SURSRUWLRQDO WR WKH V\VWHP WHPSHUDWXUH LH DV WHPSHUDWXUH

PAGE 75

)LJXUH 5HODWLRQVKLS EHWZHHQ UHVRQDQFH IUHTXHQF\ DQG WHPSHUDWXUH RI FDUERQ GLR[LGH IRU DQ LVRFKRUH QHDU LWV FULWLFDO GHQVLW\

PAGE 76

)LJXUH 5HODWLRQVKLS EHWZHHQ ILUVW UDGLDO PRGH UHVRQDQFH IUHTXHQF\ DQG WHPSHUDWXUH RI FDUERQ GLR[LGH IRU DQ LVRFKRUH QHDU WKH FULWLFDO GHQVLW\ )LJXUH 5HODWLRQVKLS EHWZHHQ WKH VSHHG RI VRXQG DQG WHPSHUDWXUH RI FDUERQ GLR[LGH IRU DQ LVRFKRUH QHDU WKH FULWLFDO GHQVLW\

PAGE 77

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f 7KH FRROLQJ WLPH ZDV DSSUR[LPDWHO\ KRXUV GXULQJ ZKLFK DSSUR[LPDWHO\ GDWD SRLQWV ZHUH FROOHFWHG )LJXUHV DQG UHSUHVHQW UHVXOWV RI WKLV SDUWLFXODU UXQ VKRZLQJ WKH UHODWLRQVKLSV EHWZHHQ V\VWHP WHPSHUDWXUH DQG UHVRQDQFH IUHTXHQF\ DQG VRQLF VSHHG UHVSHFWLYHO\ 5HVXOWV RI WKLV UXQ FOHDUO\ VKRZ WKH H[WUHPHO\ VHQVLWLYH GHSHQGHQFH RI VRQLF VSHHG RQ WHPSHUDWXUH LQ WKH FULWLFDO UHJLRQ )LJXUH VKRZV D SORW RI WKH ILUVW

PAGE 78

)LJXUH )LUVW GHULYDWLYH RI UHVRQDQFH IUHTXHQF\ YHUVXV WHPSHUDWXUH

PAGE 79

GHULYDWLYH RI UHVRQDQFH IUHTXHQF\ ZLWK UHVSHFW WR WHPSHUDWXUH YHUVXV V\VWHP WHPSHUDWXUH IRU WKH GDWD RI ILJXUH ,W JLYHV D SUHFLVH YDOXH RI WKH FULWLFDO WHPSHUDWXUH DV VHHQ LQ ILJXUH ZKLFK LV D SRUWLRQ RI ILJXUH ZLWK DQ H[SDQGHG WHPSHUDWXUH VFDOH 7KH FULWLFDO WHPSHUDWXUH REWDLQHG IURP WKLV SORW LV s NHOYLQ )LJXUH GLVSOD\V WKH UHODWLRQVKLS EHWZHHQ WKH V\VWHP SUHVVXUH DQG WHPSHUDWXUH IRU WKLV VDPH UXQ ,W VKRZV D QHDUO\ VWUDLJKW OLQH ZLWK QR HYLGHQFH RI SKDVH VHSDUDWLRQ ([SHULPHQWV ZHUH DOVR UXQ ZLWK VXEFULWLFDO GHQVLW\ FKDUJHV 7KHVH VKRZ WKH SKDVH ERXQGDU\ ORFDWLRQ GHZ SRLQWVf DV FKDQJHV LQ VORSH RI UHVRQDQFH IUHTXHQF\ YHUVXV V\VWHP WHPSHUDWXUH 7DEOH JLYHV D FRPSDULVRQ RI UHVXOWV IRXQG E\ WKH SUHVHQW PHWKRG ZLWK DYDLODEOH OLWHUDWXUH YDOXHV 7KH SUHVHQW UHVXOWV DUH LQ JRRG DJUHHPHQW ZLWK WKH RWKHU H[SHULPHQWDO YDOXHV DQG ZLWK SUHGLFWLRQV EDVHG RQ DQ HTXDWLRQ RI VWDWHf DQG D FRUUHODWLRQ PRGHO GHYHORSHG DW WKH 1DWLRQDO ,QVWLWXWH RI 6WDQGDUGV DQG 7HFKQRORJ\ 1,67 IRUPHUO\ 1%6ff 7KH DFRXVWLF GDWD DQG 1,67 SUHGLFWLRQV DUH VKRZQ JUDSKLFDOO\ LQ ILJXUH 7DEOH JLYHV D FKURQRORJLFDO FROOHFWLRQ RI H[SHULPHQWDO YDOXHV RI FULWLFDO SRLQW SDUDPHWHUV 7F DQG 3Ff UHSRUWHG E\ VHYHUDO DXWKRUV 7KH YDOXH REWDLQHG E\ WKLV VRQLF PHWKRG LV RQO\ SHUFHQW GLIIHUHQW IURP WKH EHVW YDOXH REWDLQHG LQ WKH 1,67 ODERUDWRULHV 0RUULVRQ DQG .LQFDLG 5HIHUHQFH f

PAGE 80

)LJXUH 3UHVVXUH DQG WHPSHUDWXUH EHKDYLRU RI FDUERQ GLR[LGH IRU LVRFKRUH QHDU WKH FULWLFDO GHQVLW\

PAGE 81

7DEOH &RPSDULVRQ RI 'HZSRLQW 3UHVVXUHV RI &DUERQ 'LR[LGH 1R 7 .f 'HZSRLQW SUHVVXUH SVLD 3HUFHQW GLIIHUHQFH 3UHVHQW VRQLF PHWKRG (TXDWLRQ RI VWDWHp ,f %XUQHWW PHWKRG ,,f 1,67 0RGHO +,f ,, ,,, D 5HIHUHQFH E 5HIHUHQFH r 5HIHUHQFH

PAGE 82

7DEOH 9DOXHV RI FULWLFDO SRLQW SDUDPHWHUV IURP WKH OLWHUDWXUH 7HPSHUDWXUH 3UHVVXUH 'HQVLW\ $XWKRUVfr DWP SVLD .JP $QGUHZV f +DXWHIHXLOOH &DLOOHWHW f 'HZDU f $PDJDW f &KDSSXLV f 9LOODUG f 9HUVFKDIIHOW f .XHQHQ f 'H +HHQ f 9RQ :HVHQGRQFN f .HHVRP f %ULQNPDQQ f 2QQHV t )DELXV f %UDGOH\ %URZQ t +DOH f 'RUVPDQ f &DUGRVR t %HOO f +HLQ f 'LHWHULFL & f 3LFNHULQJ 6) f 0H\HUV t 9DQ 'XVHQ f f f§ f§ .HQQHG\ +7 &\ULO + DQG 0H\HU f

PAGE 83

7DEOH &RQWLQXHGf 7HPSHUDWXUH 3UHVVXUH 'HQVLW\ $XWKRUVfr DWP SVLD .JP .HQQHG\ f f§ f§ 0LFKDHO $ %ODLVVH 6 0LFKDHO & f /RUHQW]HQ DQG +DQ /XGYLJ f 7LHOVFK + f (UQVW 6 DQG 7KRPDV : f :HQWRUI 5+ f f§ f§ f§ $PEURVH f s s (UQVW 6 DQG 7UDXEH f 0DWWKHZV -) f 0ROGRYHU f s $OWXQLQ 9 f .U\QLFNL f /HVQHYVND\D f 6HQJHUV -9 f 0RUULVRQ f $GDPRY f 6KHORPHQWVHY f 0RUULVRQ .LQFDLG s s s s f s s s 7KLV ZRUN 1XPEHUV LQ SDUHQWKHVLV UHIHU WR UHIHUHQFHV LQ ELEOLRJUDSK\

PAGE 84

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f DQG f IROORZHG E\ .KD]DQRYD DQG /HVQHYVLFD\D LQ f )UHGHQVOXQG DQG 0ROOHUXS LQ f DQG 0RUULVRQ DQG .LQFDLG LQ f 3KDVH ERXQGDU\ VWXGLHV KDYH EHHQ SHUIRUPHG E\ &ODUN DQG 'LQ LQ f -HQVHQ DQG .XUDWD LQ !

PAGE 85

*XJQRQL DQG (OGULGJH LQ f 1DJDKDPD HW DO LQ f 'DEDORV LQ f DQG 6WHDG DQG :LOOLDPV LQ f ,Q 0ROGRYHU DQG *DOODJKHU"f SUHVHQWHG D FRUUHODWLRQ IRU WKLV PL[WXUH LQ DQ DQDORJ\ ZLWK SXUH IOXLGV 7KH VSHFLILF PL[WXUH VHOHFWHG IRU WKLV VWXG\ LV RQH ZKLFK ZDV VWXGLHG E\ .KD]DQRYD DQG /HVQHYVLFD\Df )LJXUHV DQG VKRZ WKH EHKDYLRU RI WKH UHVRQDQFH IUHJXHQF\ YHUVXV WHPSHUDWXUH RI WKLV PL[WXUH DW WKUHH GLIIHUHQW JDV GHQVLWLHV VXSHUFULWLFDO GHQVLW\ SJD SFf VXEFULWLFDO GHQVLW\ SJDD SFf DQG QHDU FULWLFDO GHQVLW\ SJD SFf 1RWH WKH VKDUSHU WXUQLQJ SRLQW LQ WKH FXUYH DURXQG WKH FULWLFDO UHJLRQ LQ ILJXUH FRPSDUHG WR WKH WXUQLQJ SRLQWV RI WKH FXUYHV RI ILJXUH DQG 7KH IRUPHU LV D GLVWLQFWLYH FKDUDFWHULVWLF RI D V\VWHP DW LWV FULWLFDO FRQGLWLRQV 7F 3F DQG SF f 7KH ODWWHU DUH FKDUDFWHULVWLF RI D V\VWHP UHDFKLQJ D FRH[LVWHQFH ERXQGDU\ DW RWKHU FRQGLWLRQV EXEEOH SRLQWV RU GHZ SRLQWVf WKDQ WKH FULWLFDO )LJXUH GLVSOD\V W\SLFDO H[SHULPHQWDO UHVXOWV IRU D QHDUFULWLFDOGHQVLW\ JDV PL[WXUH &XUYH UHSUHVHQWV D UXQ IRU ZKLFK WKH EDWK WHPSHUDWXUH ZDV UDPSHG GRZQZDUG IURP WR r& &XUYH VKRZV GDWD IRU DQ XSZDUG UDPS IURP WR r& ,Q ERWK FDVHV WKH WUDFNHG SHDN ILUVW UDGLDO PRGH UHVRQDQFH IUHTXHQF\f ZDV ORVW RYHU D VPDOO WHPSHUDWXUH LQWHUYDO LQ WKH FULWLFDO YLFLQLW\ GXH WR WKH H[WUHPHO\ IDVW FKDQJH RI VRQLF VSHHG DQG GURS LQ DPSOLWXGH &XUYH LV OLNH FXUYH H[FHSW WKDW WKH WHPSHUDWXUH RI WKH ZHOOLQVXODWHG

PAGE 86

1 t V f &/f O+ WLQ 7HPSHUDWXUH &HOVLXV )LJXUH 7\SLFDO EHKDYLRU RI UHVRQDQFH IUHTXHQF\ YHUVXV WHPSHUDWXUH RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH DW D VXSHUFULWLFDO GHQVLW\

PAGE 87

7(03(5$785( & )LJXUH 7\SLFDO EHKDYLRU RI UHVRQDQFH IUHTXHQF\ YHUVXV WHPSHUDWXUH RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH DW D VXEFULWLFDO GHQVLW\ 2-

PAGE 88

1 e i J FX ;_ 7HPSHUDWXUH &HOVLXV )LJXUH 7\SLFDO EHKDYLRU RI UHVRQDQFH IUHTXHQF\ YHUVXV WHPSHUDWXUH RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH QHDU FULWLFDO GHQVLW\

PAGE 89

V\VWHP ZDV DOORZHG WR UDLVH QDWXUDOO\ IURP r& WR URRP WHPSHUDWXUH 7KLV SURFHGXUH ZDV VLPLODU WR WKH RQH XWLOL]HG LQ WKH H[SHULPHQWV ZLWK FDUERQ GLR[LGH DQG DOVR ZLWK HWKDQH DV ZLOO EH VHHQ ODWHUf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

PAGE 90

R D! R r D! D! 7HPSHUDWXUH & )LJXUH 7\SLFDO H[SHULPHQWDO UHVXOWV RI QHDUFULWLFDO GHQVLW\ JDV RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH &XUYH VKRZV D IRUFHG FRROLQJ UXQ &XUYH VKRZV D IRUFHG ZDUPLQJ UXQ &XUYH LV D VORZ QDWXUDOO\ ZDUPLQJ UXQ f§L &7!

PAGE 91

7(03(5$785( & )LJXUH 2WKHU UHVRQDQFH IUHTXHQF\ WRS FXUYHf UHVXOWV LQ VDPH FULWLFDO WHPSHUDWXUH DV ILUVW UDGLDO UHVRQDQFH IUHTXHQF\ ERWWRP FXUYHf

PAGE 92

IRU WKH FULWLFDO SRLQW RI D V\VWHP ZKHQ LW LV QRW SUHYLRXVO\ NQRZQ )RU H[DPSOH WKH VWDUWLQJ SUHVVXUH DW r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f )LJXUH $ VHTXHQFH RI VXSHUFULWLFDO SUHVVXUH YHUVXV WHPSHUDWXUH LVRFKRUHV RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH

PAGE 93

SUHVVXUH YHUVXV LV VKRZQ LQ ILJXUH 7KLV WHFKQLTXH UHYHDOHG WKH PDJQLWXGH RI WKH FKDUJLQJ SUHVVXUH WR EH XVHG ZKHQ GDWD ZHUH WR EH FROOHFWHG LQ VHDUFK RI WKH FULWLFDO SRLQW 7DEOH $QJOHV EHWZHHQ SUHVVXUHWHPSHUDWXUH OLQHV EHIRUH DQG DIWHU SKDVH ERXQGDU\ RI VXSHUFULWLFDO GHQVLW\ IOXLG RI FDUERQ GLR[LGH DQG HWKDQH )LOH 6ORSH RI OLQH 6ORSH RI OLQH $QJOH EHWZHHQ EHIRUH SKDVH DIWHU SKDVH WZR OLQHV ERXQGDU\ ERXQGDU\ GHJUHHf 7KLV SUHGLFWLRQ ZDV LPSRUWDQW KHUH VLQFH WKH V\VWHP YROXPH LV IL[HG 7KH RQO\ ZD\ WR FKDQJH WKH V\VWHP GHQVLW\ ZDV WR

PAGE 94

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r& ZDV W\SLFDOO\ RQH RU WZR SVLD )RU WKLV ZRUN WKH WRWDO QXPEHU RI UXQV ZDV ILIW\ 7DEOH OLVWV WKH H[SHULPHQWDO UHVXOWV QHHGHG WR FRQVWUXFW SKDVH GLDJUDPV RI WKLV V\VWHP )LJXUHV WR VKRZ YDULRXV W\SHV RI SKDVH GLDJUDP FUHDWHG IURP GLIIHUHQW FRPELQDWLRQV RI WKH SDUDPHWHUV LQ WDEOH )LJXUH VKRZV WKH UHODWLRQVKLS EHWZHHQ VWDUWLQJ SUHVVXUH DW r& DQG WKH

PAGE 95

7DEOH ([SHULPHQWDO UHVXOWV IRU FRQVWUXFWLRQ RI SKDVH GLDJUDPV RI D PL[WXUH RI FDUERQ GLR[LGH DQG HWKDQH ),/( 1$0( 6WDUWLQJ SRLQW 3RLQW RI PLQLPXP IUHTXHQF\ 7 r&f 3UHVVXUH SVLD )UHTXHQF\ +] 7 r&f )UHTXHQF\ +] 6RQLF VSHHG PVHF 3UHVVXUH SVLD 7'$ 7'$ 7'$ 7'$ 7'$ 7'$ 7'$ 7'$ 7'$ 7'$ 7'$

PAGE 96

7DEOH &RQWLQXHGf ),/( 1$0( 6WDUWLQJ SRLQW 3RLQW RI PLQLPXP IUHTXHQF\ 7 r&f 3UHVVXUH SVLD )UHTXHQF\ +] L R Q )UHTXHQF\ +] 6RQLF VSHHG PVHF 3UHVVXUH SVLD 7'$ 7'$ 7'$ 7'$ 7'$ 7'$ 7'$ 7'$ 7'$ 7'$ 7'$f

PAGE 97

7DEOH &RQWLQXHGf ),/( 1$0( 6WDUWLQJ SRLQW 3RLQW RI PLQLPXP IUHTXHQF\ 2 R Q 3UHVVXUH SVLD )UHTXHQF\ +] 7 r&f )UHTXHQF\ +] 6RQLF VSHHG PVHF 3UHVVXUH SVLD 7'$f 7'$ &R

PAGE 98

WHPSHUDWXUH RI SKDVH FKDQJH ([SHULPHQWDO UHVXOWV DUH UHSUHVHQWHG E\ SRLQWV 7KH VPRRWK FXUYH LV D UHVXOW RI D WKLUG RUGHU SRO\QRPLDO FXUYH ILWWLQJ SURFHGXUH 7KH FRUUHVSRQGLQJ HTXDWLRQ LV DV IROORZV 7 D ErS FrS GrS ZKHUH 7 WHPSHUDWXUH &HOVLXV S SUHVVXUH SVLD D ; E F ; G ; n U FRUUHODWLRQ FRHIILFLHQW )LJXUH VKRZV WKH UHODWLRQVKLS EHWZHHQ VWDUWLQJ SUHVVXUH DQG WKH VSHHG RI VRXQG DW WKH SKDVH FKDQJH $ VKDUS FKDQJH LQ WKH VSHHG RI VRXQG DW WKH FULWLFDO SRLQW LV HYLGHQW FRPSDUHG WR WKH VORZ FKDQJH RI WHPSHUDWXUH VKRZQ ILJXUH )LJXUH VKRZV D FRH[LVWHQFH FXUYH 1RWH WKDW WKH FXUYH LV VR VOLP WKDW WKH EXEEOHSRLQW OLQH DOPRVW RYHUODSV WKH GHZSRLQW OLQH 5HFDOO WKDW D VLPLODU SORW IRU D SXUH IOXLG \LHOGV D QHDUO\ VWUDLJKW OLQH VHH ILJXUH RI HWKDQHf 7KLV FOHDUO\ UHYHDOV D SURSHUW\ RI WKLV PL[WXUH DV EHLQJ QHDUO\ D]HRWURSLF ,W EHKDYHV PXFK DV LI LW ZHUH D SXUH IOXLG 7KH OLWHUDWXUH YDOXH RI WKH D]HRWURSH FRPSRVLWLRQ RI WKLV ELQDU\ PL[WXUH LV UHSRUWHG WR EH PROH IUDFWLRQ RI HWKDQHf 7KLV LV YHU\ FORVH WR WKH FRPSRVLWLRQ RI WKH PL[WXUH VWXGLHG LQ WKLV ZRUN

PAGE 99

67$57,1* 35(6685( 36,$ )LJXUH 5HODWLRQVKLS EHWZHHQ WKH VWDUWLQJ SUHVVXUHV DQG WKH WHPSHUDWXUHV RI SKDVH FKDQJH RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH

PAGE 100

67$57,1* 35(6685( 36,$ )LJXUH 5HODWLRQVKLS EHWZHHQ WKH VWDUWLQJ SUHVVXUHV DQG WKH VSHHG RI VRXQG DW D SKDVH FKDQJH RI WKH FDUERQ GLR[LGH HWKDQH PL[WXUH 7(03(5$785( &(/6,86 )LJXUH &RH[LVWHQFH FXUYH RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH QHDU D]HRWURSH FRPSRVLWLRQ

PAGE 101

)LJXUH 7KH VRQLF VSHHG YHUVXV WHPSHUDWXUH RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH IRU DQ LVRFKRUH QHDU FULWLFDO GHQVLW\ )LJXUH 3UHVVXUH DQG WHPSHUDWXUH EHKDYLRU RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH FKDUJHG QHDU LWV FULWLFDO GHQVLW\

PAGE 102

f+ X X 7HPSHUDWXUH &HOVLXV )LJXUH 3UHVVXUH DQG WHPSHUDWXUH UHODWLRQVKLS RI D VHW RI VHYHUDO LVRFKRUHV RI WKH FDUERQ GLR[LGHHWKDQH PL[WXUH

PAGE 103

7DEOH &RPSDULVRQ RI 'HZSRLQW 3UHVVXUHV RI & &+ 0L[WXUH ;FR f 1R 7HPSHUDWXUH .f 'HZSRLQW SUHVVXUHV SVLD 3HUFHQW 'LIIHUHQFH 3UHVHQW VRQLF PHWKRG (TXLOLEULXP FHOO RI WKH YDSRU UHFLUFXODWLRQ W\SH LQWHUSRODWHGf ,f 1,67 ''0,; 0RGHO ,,f ,, $ )UHGHQVOXQG DQG 0ROOHUXS &KHP 6RF )DUDGD\ 7UDQV BB f ''0,; PRGHO GHYHORSHG E\ 1%6 f XVHV 3HQJ5RELQVRQ HTXDWLRQ RI VWDWH IRU FRH[LVWLQJSKDVH FRPSRVLWLRQ FDOFXODWLRQV DQG 1%6 FRUUHVSRQGLQJ VWDWHV PRGHO ZLWK VKDSH IDFWRUV DQG YDQ GHU :DDOV RQHIOXLG PL[LQJ UXOHV IRU SKDVH SURSHUWLHV 7KH XQFHUWDLQWLHV LQ ''0,; PRGHO DUH SULQFLSDOO\ GXH WR WKH XQFHUWDLQWLHV DVVRFLDWHG ZLWK WKH PL[LQJ UXOHV RR 92

PAGE 104

f FRQILUPLQJ WKH H[SODQDWLRQ )LJXUHV DQG VKRZ WKH UHODWLRQVKLSV QHDU WKH FULWLFDO GHQVLW\ EHWZHHQ WHPSHUDWXUH RI SKDVH FKDQJH DQG VSHHG RI VRXQG DQG WKH SUHVVXUH UHVSHFWLYHO\ $V H[SHFWHG WKH VRQLF VSHHG ILJXUH f VKRZV D SURQRXQFHG FKDQJH LQ WKH FULWLFDO UHJLRQ ZKLOH WKH SUHVVXUH ILJXUH f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s 7KH RQO\ DYDLODEOH OLWHUDWXUH H[SHULPHQWDO YDOXH IRU WKLV FRPSRVLWLRQ ZDV IRXQG WR EH 6f 7KHVH YDOXHV GLIIHU E\ RQO\ SHUFHQW (WKDQH 7KH JUDSKV VKRZQ LQ ILJXUHV WR UHSUHVHQW WKH LVRFKRULF EHKDYLRU RI WKH V\VWHP FKDUJHG ZLWK SXUH HWKDQH DW QHDU FULWLFDO GHQVLW\ )LJXUH VKRZV D W\SLFDO UHODWLRQVKLS EHWZHHQ D UHVRQDQFH IUHTXHQF\ DQG WKH V\VWHP

PAGE 105

7(03(5$785( &(/6,86 )LJXUH 5HVRQDQFH IUHTXHQF\ ,V PRGHf DQG WHPSHUDWXUH UHODWLRQVKLS RI HWKDQH QHDU WKH FULWLFDO GHQVLW\ 35(6685( 36,$ )LJXUH 5HVRQDQFH IUHTXHQF\ ,V PRGHf DQG SUHVVXUH UHODWLRQVKLS RI HWKDQH QHDU WKH FULWLFDO GHQVLW\

PAGE 106

7(03(5$785( &(/6,86 )LJXUH 3UHVVXUH DQG WHPSHUDWXUH UHODWLRQVKLS RI ,V PRGH UHVRQDQFH IUHTXHQF\ RI HWKDQH QHDU WKH FULWLFDO GHQVLW\ 7(03(5$785( &(/6,86 )LJXUH 6SHHG RI VRXQG DQG WHPSHUDWXUH RI ,V PRGH UHVRQDQFH IUHTXHQF\ RI HWKDQH QHDU WKH FULWLFDO GHQVLW\

PAGE 107

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f WHPSHUDWXUH ([SHULPHQWDO GDWD SRLQWV ZHUH ILWWHG WR D SRO\QRPLDO RI WKH VL[WK RUGHU ,WV FRHIILFLHQWV DORQJ ZLWK VRPH VWDWLVWLFDO YDOXHV JHQHUDWHG IURP WKH GDWD E\ *5$3+(5

PAGE 108

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

PAGE 109

7(03(5$785( &(/6,86 )LJXUH 3KDVH GLDJUDP RI HWKDQH VKRZLQJ SUHVVXUH DQG WHPSHUDWXUH EHKDYLRU RI GLIIHUHQW LVRFKRUHV

PAGE 110

WKH FULWLFDO GHQVLW\ UHDFKHV WKH ORZHVW VRQLF VSHHG

PAGE 111

WKH SUHVHQW H[SHULPHQWDO EXEEOH SRLQW SUHVVXUHV DQG DYDLODEOH OLWHUDWXUH YDOXHV 7KHVH LQFOXGH WKH H[SHULPHQWDO YDOXHV PHDVXUHG LQ WKH FRPSUHVVLELOLW\ DSSDUDWXV RI 'RXVOLQ DQG +DUULVRQf DQG WKH YDOXHV SUHGLFWHG E\ ''0,;6f D FRUUHODWLRQ PRGHO GHYHORSHG DW WKH 1DWLRQDO ,QVWLWXWH RI 6WDQGDUGV DQG 7HFKQRORJ\ 1,67 IRUPHUO\ 1%6f 7DEOH FRPSDUHV WKH SUHVHQW YDSRU SUHVVXUHV ZLWK WKRVH RI WKH SUHGLFWLYH HTXDWLRQ SURSRVHG E\ 6\FKHY HW DOf 7KLV HTXDWLRQ GHULYHG IURP D FRPSLODWLRQ RI YDSRU SUHVVXUHV RI HWKDQH E\ *RRGZLQ HW DOf 7KH 6\FKHY HTXDWLRQ LV ,Q SV D E[ FX GX HX IXOXf ZKHUH [7f > 7WU7f @ > 7WU7FUf @ 87f >77WU@>7FU7WU@ 7WU 7HPSHUDWXUH DW WKH WULSOH SRLQW s 7FU &ULWLFDO 7HPSHUDWXUH s D E F G H I H

PAGE 112

7DEOH 6XPPDU\ RI UHVXOWV IRU HWKDQH PHDVXUHPHQWV )LOH 1XPEHU &RQGLWLRQV DW VWDUW RI UXQ 7 r&f 6RQLF VSHHG DW SKDVH ERXQGDU\ PVHFf &RQGLWLRQV DW PLQLPXP UHVRQDQFH IUHTXHQF\ )UHTXHQF\ +]f 3UHVVXUH SVLDf 7 r&f 3 SVLDf )UHTXHQF\ +]f 6RQLF VSHHG PVHFf $PS PYf YR RR

PAGE 113

7DEOH &RQWLQXHGf )LOH 1XPEHU &RQGLWLRQV DW VWDUW RI UXQ 7 r&f 6RQLF VSHHG DW SKDVH ERXQGDU\ PVHFf &RQGLWLRQV DW PLQLPXP UHVRQDQFH IUHTXHQF\ )UHTXHQF\ +]f 3UHVVXUH SVLDf 7 r&f 3 SVLDf )UHTXHQF\ +]f 6RQLF VSHHG PVHFf $PS PYf 1$ 1$ 1$ 1$ 1$

PAGE 114

7DEOH &RQWLQXHGf )LOH 1XPEHU &RQGLWLRQV DW VWDUW RI UXQ 7 r&f 6RQLF VSHHG DW SKDVH ERXQGDU\ PVHFf &RQGLWLRQV DW PLQLPXP UHVRQDQFH IUHTXHQF\ )UHTXHQF\ +]f 3UHVVXUH SVLDf 7 r&f 3 SVLDf )UHTXHQF\ +]f 6RQLF VSHHG PVHFf $PS PYf 1$ 1$ 7KH VRQLF VSHHG DW WKH SKDVH ERXQGDU\ ZDV GHWHUPLQHG E\ D 6LPSOH[ URXWLQH IURP WKH H[SHULPHQWDO GDWD 6HH FXUYH ILWWLQJ VHFWLRQ R R

PAGE 115

7DEOH &RPSDULVRQ RI %XEEOH3RLQW 3UHVVXUHV RI (WKDQH 1R 7 .f %XEEOHSRLQW 3UHVVXUHV SVLD 3HUFHQW 'LIIHUHQFH 3UHVHQW VRQLF PHWKRG &RPSUHVVLELOLW\ DSSDUDWXV ,f 1,67 ''0,; 0RGHO ,,f ,, 'm 5r 'RXVOLQ DQG 5 + +DUULVRQ &KHP 7KHUPRG\QDPLFV f 6HH IRRWQRWH RI WDEOH IRU H[SODQDWLRQ R

PAGE 116

7DEOH &RPSDULVRQ RI H[SHULPHQWDO YDSRU SUHVVXUHV DQG WKRVH FDOFXODWHG ZLWK DQ HTXDWLRQ RI VWDWH RI HWKDQH SURSRVHG E\ 6\FKHYf ([SHULPHQWDO 3UHGLFWHG SUHVVXUH SVLD 3HUFHQW GLIIHUHQFH 7 .f 3 SVLDf 1$r 1$ 1$r 1$ (TXDWLRQ GLYHUJHV DW WKLV WHPSHUDWXUH

PAGE 117

7DEOH &KURQRORJLFDO FROOHFWLRQ RI FULWLFDO SRLQW SDUDPHWHUV RI HWKDQH 7HPSHUDWXUH 3UHVVXUH 'HQVLW\ $XWKRUVf r
PAGE 118

7DEOH &RQWLQXHGf 7HPSHUDWXUH 3UHVVXUH 'HQVLW\ $XWKRUVf r
PAGE 119

7KH SUHGLFWLYH DQG H[SHULPHQWDO SUHVVXUHV DUH LQ JRRG DJUHHPHQW 1RWLFH WKDW WKLV HTXDWLRQ LV LQGHWHUPLQDWH ZKHQ WHPSHUDWXUH LV JUHDWHU WKDQ VLQFH WKH X7f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

PAGE 120

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f DQG WKH ILUVW QRQUDGLDO PRGH RI YLEUDWLRQ WKH ,I SHDNf ZHUH LQ FORVH SUR[LPLW\ DERXW K] DSDUWf &RQVHTXHQWO\ WKH ,I SHDN WHQGHG WR LQWHUIHUH ZLWK WUDFNLQJ RI WKH ,V SHDN 7KH GDWD DFTXLVLWLRQ SURJUDP VHOHFWHG WKH VLJQDO RI KLJKHVW DPSOLWXGH LQ WKH REVHUYDWLRQ

PAGE 121

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f DQG REVHUYH WKH UHODWLRQVKLSV DPRQJ WKHP 7KLV KRZHYHU ZDV EH\RQG WKH VFRSH RI WKLV ZRUN 2WKHU ZRUN LQ WKLV ODERUDWRU\ KRZHYHU LV GLUHFWHG WR WKLV SUREOHP

PAGE 122

)UHTXHQF\ +L )LJXUH 0RYHPHQW RI WUDFNHG SHDN DQG LWV QHLJKERU EHORZ WKH FULWLFDO WHPSHUDWXUH 7KHVH FXUYHV DUH QXPEHUHG LQ WKH SURSHU WLPH VHTXHQFH EXW PDQ\ FXUYHV VKRZLQJ RQO\ RQH VWURQJ SHDN KDYH EHHQ RPLWWHG

PAGE 124

WKH VHSDUDWLRQ RI WKHVH WZR SHDNV LQFUHDVHG FXUYHV WKURXJK RI ILJXUH f 7KH ULJKW KDQG SHDN SHDN RQHf ZDV ELJJHU DQG PRYHG IDVWHU WKDQ WKH OHIW SHDN SHDN WZRf ,W PRYHG VR IDVW WKDW SHDN WZR ZDV HYHQWXDOO\ RXW RI WKH VZHHS UDQJH VHH FXUYH RI ILJXUH f 7KLV VLQJOH SHDN UHPDLQHG VWURQJ XQWLO WKH WHPSHUDWXUH ZDV DURXQG r& $W WKDW WHPSHUDWXUH WKH RWKHU SHDN SRVVLEO\ SHDN WZRf UHDSSHDUHG LQ WKH VZHHS UDQJH FXUYHV DQG RI ILJXUH f ,W ZDV DW WKLV WLPH 2 ? (;3(5,0(17$/ &859( f 3($. 7:2 5287( 3($. 21( 5287( ? n 6 9 ‘ ‘‘L ‘ L ‘ ‘ ‘ ‘ ‘ L L ‘ ‘‘ ‘ L ‘ L L 7(03(5$785( &(/6,86 )LJXUH 3RVVLEOH URXWHV RI PRYHPHQW RI WZR QHLJKERU SHDNV SHDN RQH DQG SHDN WZRf EHORZ WKH FULWLFDO WHPSHUDWXUH

PAGE 125

,OO VWURQJHU WKDQ SHDN RQH OHDGLQJ WR VZLWFKLQJ RI WKH WUDFNHG SHDN IURP SHDN RQH WR SHDN WZR 7KH ZLQGRZ WKHQ VKLIWHG IURP SHDN RQH ZKLFK EHFDPH ORVW 7KH SHDN LQ FXUYH LV SHDN RQH EXW WKDW RI FXUYH LV SHDN WZR 7KLV LV DSSDUHQWO\ VR VLQFH WKH SHDN LQ FXUYH LV ORZHU LQ IUHTXHQF\ WKDQ WKH SHDN LQ FXUYH f 7KH UHYHUVH RI WKLV SKHQRPHQRQ KDSSHQHG DURXQG r& ZKHUH SHDN RQH UHHQWHUHG WKH ZLQGRZ DV WKH GRPLQDQW VLJQDO )LQDOO\ WKHVH WZR SHDNV PHUJHG WRJHWKHU DW DURXQG r& 'DWD ZHUH QRW FROOHFWHG EHORZ DERXW r& VR WKH VXEVHTXHQW EHKDYLRU RI WKHVH SHDNV ZDV QRW REVHUYHG 7KLV W\SH RI EHKDYLRU LV LQWHUHVWLQJ DQG XQH[SHFWHG EXW LW LV QRW FHQWUDO WR WKH REMHFWLYH RI WKH SUHVHQW UHVHDUFK ,W LV SUREDEO\ DVVRFLDWHG ZLWK WKH IDFW WKDW EHORZ WKH FULWLFDO WHPSHUDWXUH WKHUH DUH WZR SKDVHV SUHVHQW DQG WKHLU UHODWLYH SURSRUWLRQV LQ WKH UHVRQDWRU FDYLW\ FKDQJH DFFRUGLQJ WR WKH HIIHFWV RI WKH PDJQHWLFDOO\ GULYHQ FLUFXODWLQJ SXPS 7KH SODXVLEOH URXWHV RI WKHVH WZR SHDNV EDVHG RQ WKH DERYH VLWXDWLRQV DUH GHSLFWHG LQ ILJXUH $ IXOO LQWHUSUHWDWLRQ RI WKHVH SKHQRPHQD ZRXOG UHTXLUH IXUWKHU H[SHULPHQWDWLRQ WR FKDUDFWHUL]H WKH FRQWHQWV RI WKH FDYLW\ LQ WKH WZR SKDVH UHJLRQ ,Q WKH HDUO\ VWDJH RI SKDVH VHSDUDWLRQ WKH EXEEOHV RU GHZ GURSV DUH EHOLHYHG WR EH VPDOO DQG HQWUDLQHG LQ WKH GRPLQDQW SKDVH DV IRDP RU IRJ VR WKH ORFDWLRQ RI WKH SKDVH WUDQVLWLRQ LV QRW REVFXUHG E\ PDVVLYH SKDVH VHSDUDWLRQV ZKLFK RFFXU ZHOO DZD\ IURP WKH SKDVH ERXQGDU\

PAGE 126

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f WR ILW GDWD RI ODWHQW KHDW RI YDSRUL]DWLRQ DQG WHPSHUDWXUH DQG ODWHU PRGLILHG WR VDWLVI\ D FRUUHVSRQGLQJ VWDWHV SULQFLSOH E\ 6LYDUDPDQ HW D Ln! DQG DSSOLHG WR VSHHG RI VRXQG GDWD E\ WKH VDPH

PAGE 127

7(03(5$785( & )LJXUH ([SHULPHQWDO FXUYH RI HWKDQH VKRZLQJ G\QDPLF EHKDYLRU RI VSHHG RI VRXQG QHDU FULWLFDO SRLQW DXWKRU ,W KDV D URRW IURP ULJRURXV PDWKHPDWLFDO GHULYDWLRQV E\ :HJQHUf DQG IXUWKHU E\ *UHHQ DQG /H\NRRf EDVHG RQ UHQRUPDOL]DWLRQ JURXS WKHRU\ 5f 7Ff 0 $7A$ $7BD $7U $7 $7 U U f ZKHUH & 6SHHG RI VRXQG PVHF 5 *DV FRQVWDQW -VHF 7F &ULWLFDO 7HPSHUDWXUH 0 0ROHFXODU PDVV .JPROH $L (TXDWLRQ FRHIILFLHQWV 7U 5HGXFHG 7HPSHUDWXUH D $ &ULWLFDO H[SRQHQWV

PAGE 128

7R ILW WKLV QRQ OLQHDU HTXDWLRQ WR H[SHULPHQWDO GDWD RI VSHHG RI VRXQG DQG WHPSHUDWXUH HOHYHQ V\VWHPGHSHQGHQW FRHIILFLHQWV PXVW ILUVW EH VHDUFKHG 7KLV WDVN ZDV DFFRPSOLVKHG E\ WKH VLPSOH[ PDWKHPDWLFDO RSWLPL]DWLRQ WHFKQLTXH $ FRPSXWHU SURJUDP ZULWWHQ E\ WKH DXWKRU IRU WKLV SXUSRVH LV JLYHQ LQ DSSHQGL[ &f ,Q JHQHUDO WKH H[SHULPHQWDO GDWD RI VSHHG RI VRXQG DQG FRUUHVSRQGLQJ V\VWHP WHPSHUDWXUH ZHUH GLYLGHG LQWR WZR SDUWV DFFRUGLQJ WR WHPSHUDWXUH 7KH ILUVW RQH LV DOO GDWD DW WHPSHUDWXUHV DERYH WKH FULWLFDO WHPSHUDWXUH URXJKO\ HVWLPDWHG IURP WKH OLWHUDWXUH YDOXHf 7KH VHFRQG RQH LV DOO GDWD DW WHPSHUDWXUH EHORZ WKH FULWLFDO WHPSHUDWXUH (DFK GDWD VHW ZDV LQGHSHQGHQWO\ ILWWHG WR WKH DERYH HTXDWLRQ UHVXOWLQJ LQ D VHW RI HOHYHQ FRHIILFLHQWV 7(03(5$785( &(/6,86 ),77,1* &859( (;3(5,0(17$/ 32,176 3/277(' (9(5< 32,176f )LJXUH &RPSDULVRQ RI H[SHULPHQWDO GDWD IRU WKH HWKDQH DQG FXUYH JHQHUDWHG E\ UHQRUPDOL]DWLRQ JURXS WKHRU\ HTXDWLRQ

PAGE 129

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

PAGE 130

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

PAGE 131

EH ]HUR ,Q WKDW FDVH 37& SKDVH GLDJUDP FXUYH ZRXOG KDYH PXFK PRUH SURPLQHQW WRUQDGRLQDFWLRQ VKDSH

PAGE 132

%(6 2 35(6685( )LJXUH 7KUHH GLPHQVLRQDO SKDVH GLDJUDP RI D PL[WXUH RI FDUERQ GLR[LGH DQG HWKDQH RR )UH,MHQ&\

PAGE 133

)LJXUH 7KUHH GLPHQVLRQDO SKDVH GLDJUDP RI HWKDQH

PAGE 134

&+$37(5 &21&/86,21 $ QHZ DSSURDFK XVLQJ DQ DFRXVWLF UHVRQDQFH WHFKQLTXH IRU SKDVH ERXQGDU\ GHWHFWLRQ PD\ EH VXPPDUL]HG DV IROORZV 0HDVXUH VHYHUDO UHVRQDQFH IUHTXHQFLHV RI VWDQGLQJ DFRXVWLF ZDYHV H[FLWHG LQVLGH D IOXLG ILOOHG VSKHULFDO DFRXVWLF FDYLW\ ,GHQWLI\ HDFK UHVRQDQFH IUHTXHQF\ 7UDFN DQ DVVLJQHG UHVRQDQFH IUHTXHQF\ DV V\VWHP SDUDPHWHUV VXFK DV WHPSHUDWXUH SUHVVXUH RU GHQVLW\ DUH YDULHG /RFDWH SKDVH HTXLOLEULD E\ DEUXSW FKDQJHV LQ UHVRQDQFH IUHTXHQF\ KHQFH VSHHG RI VRXQGf DV WKH V\VWHP VWDWH WUDYHUVHV D SKDVH ERXQGDU\ /RFDWH FULWLFDO SRLQWV DV PLQLPD RQ 37&f RU S7&f GLDJUDPV 7KH SULQFLSDO UHVXOWV RI WKLV ZRUN DUH WDEXODWHG DQG FRPSDUHG ZLWK WKH DYDLODEOH OLWHUDWXUH YDOXHV LQ WDEOH $SSDUHQWO\ WKH SUHVHQW UHVXOWV DUH LQ JRRG DJUHHPHQW ZLWK HVWDEOLVKHG OLWHUDWXUH YDOXHV SURYLQJ WKH YLDELOLW\ RI WKLV QHZ WHFKQLTXH 7KH ODUJHVW GLVFUHSDQF\ LV LQ WKH V\VWHP

PAGE 135

FRQVLVWLQJ RI D &a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s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nf DQG WKH

PAGE 136

DSSHDUDQFH RI WKH PHQLVFXV HOVHZKHUH LV XVHG DV D WR GHWHUPLQH WKH FULWLFDO SDUDPHWHUV ,Q SUDFWLFH WKHVH DUH GHWHUPLQHG E\ LQWHUSRODWLRQ RI GDWD WDNHQ ZKHQ WKH PHQLVFXV DSSHDUV MXVW DERYH DQG MXVW EHORZ WKH PLGGOH RI WKH FHOO 7KLV PHWKRG LV WKHQ VXEMHFW WR DQ KHXULVWLF HIIHFW %HVLGHV QHDU WKH FULWLFDO FRQGLWLRQ WKH SKHQRPHQD RI FULWLFDO RSDOHVFHQFH OLJKW VFDWWHULQJ HIIHFWf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f WKDQ WKH YLVXDO PHWKRG LV IDU PRUH LPSRUWDQW WKDQ ZKHWKHU RI QRW WKH DFFXUDF\ RI WHPSHUDWXUH PHDVXUHPHQW DW D VPDOO SRRUO\ IXQGHG XQLYHUVLW\ ODERUDWRU\ LV

PAGE 137

7DEOH &RPSDULVRQ RI &ULWLFDO 7HPSHUDWXUHV DQG 3UHVVXUHV 1R 6\VWHP &ULWLFDO 7HPSHUDWXUH .f 3HUFHQW 'LIIHUHQW 6RXUFHV 7KLV ZRUN /LWHUDWXUH &DUERQ 'LR[LGH s s 0RUULVRQ t .LQFDLG f (WKDQH s 0RUULVRQ f PROH b & PROH b &+J s r 0RUULVRQ t .LQFDLG f &ULWLFDO 3UHVVXUH 3VLD &DUERQ GLR[LGH s s 0RUULVRQ t .LQFDLG f (WKDQH s s .KD]DQRYD t 6RPLQVND\D f b PROH & b PROH &+F s 1$ 1$ 1$ ([WUDSRODWHG YDOXH

PAGE 138

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

PAGE 139

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

PAGE 140

$33(1',; $ &+$5*,1* 35(6685( &$/&8/$7,216 6LQFH LQ WKLV ZRUN GDWD ZHUH FROOHFWHG LQ WKH LVRFKRULF PRGH WKH GHQVLW\ RI WKH V\VWHP ZDV FRQVWDQW WKURXJKRXW HDFK UXQ DV ZHOO 7R ORFDWH WKH FULWLFDO ZLWKRXW IXUWKHU DGGLWLRQ RI JDV UHTXLUHG WKDW WKH LQLWLDO UXQV EH PDGH RQ VDPSOHV DW VXSHUFULWLFDO GHQVLWLHV 'HQVLW\ ZDV QRW GLUHFWO\ PHDVXUHG EXW ZDV LQIHUHG IURP WKH $*$ HTXDWLRQ RI VWDWHf WKHUHIRUH WKH VWDUWLQJ FRQGLWLRQV ZHUH VSHFLILHG E\ FRPSRVLWLRQ WHPSHUDWXUH DQG SUHVVXUH 7KH SURFHGXUHV IROORZHG 3XUH )OXLG O&DUERQ GLR[LGH RU (WKDQHf 2EWDLQ OLWHUDWXUH YDOXHV RI WKH FULWLFDO SRLQW SDUDPHWHUV 7F 3F SF f 8VH WKH $*$ HTXDWLRQ WR FDOFXODWH D VHULHV RI GHQVLWLHV FRUUHVSRQGLQJ WR YDU\LQJ SUHVVXUHV LQ WKH YLFLQLW\ RI WKH FULWLFDO SUHVVXUH DW D FKDUJLQJ WHPSHUDWXUH ZKLFK PXVW EH KLJKHU WKDQ WKH FULWLFDO WHPSHUDWXUH E\ DERXW ILYH WR VHYHQ GHJUHHV &HOVLXV 7KLV VHW RI GHQVLWLHV VKRXOG VSDQ WKH FULWLFDO GHQVLW\ 3ORW WKH FDOFXODWHG GHQVLWLHV DV D IXQFWLRQ RI WKH FRUUHVSRQGLQJ FKDUJLQJ SUHVVXUHV WKHQ DSSO\ D VXLWDEOH FXUYH ILW UHJUHVVLRQ JHQHUDOO\ SRO\QRPLDO RI WKH QWK RUGHU WR WKHVH GDWD

PAGE 141

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f

PAGE 142

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n & + 7KH GLIIHUHQFH EHWZHHQ ZHLJKWV RI UHDFWLRQ YHVVHOV EHIRUH DQG DIWHU FDUHIXOO\ SDVVLQJ FDUERQ GLR[LGH LV WKH ZHLJKW RI WKH JDV 7KH ZHLJKW RI HWKDQH ZDV FDOFXODWHG E\ VXEWUDFWLQJ WKH PHDVXUHG ZHLJKW RI FDUERQ GLR[LGH IURP WKH ZHLJKW RI WKH VDPSOH

PAGE 143

)LJXUH %O %ORFN GLDJUDP RI WKH JUDYLPHWULF PHWKRG $ EORFN GLDJUDP RI WKH H[SHULPHQWDO VHWXS LV VKRZQ LQ ILJXUH %O $ JDV VDPSOLQJ FRQWDLQHU ZDV PDGH IURP D VWDLQOHVV VWHHO WXEH SURYLGHG ZLWK KLJK SUHVVXUH QHHGOH YDOYHV $ DQG % 7KH H[SHULPHQW VWDUWHG ZLWK SUHFRQGLWLRQLQJ WKH V\VWHP E\ IORZLQJ DUJRQ JDV WR WKH UHDFWLRQ YHVVHOV KH[DJRQDO DEVRUEHU DUUD\ VHH ILJXUH %f WKURXJK YDOYHV DQG IRU RQHKDOI KRXU $IWHU WKDW YDOYHV DQG ZHUH FORVHG IROORZHG E\ RSHQLQJ YDOYH % 7KH SUHVVXUH RI JDV LQ WKH JDV VDPSOLQJ FRQWDLQHU ZDV VKRZQ RQ JDXJH W\SLFDO SUHVVXUH ZDV DURXQG SVLDf 7KHQ YDOYH ZDV RSHQHG FDUHIXOO\ WR EOHHG JDV 7KH KH[DJRQ DEVRUEHU DUUD\V RI YHUWLFDO WXEHV HDFK KDYLQJ D ZDVK ERDUG VKDSH ZHUH GHVLJQHG WR SURORQJ FRQWDFW WLPH RI JDV ZLWK DEVRUEDQW WR DVVXUH FRPSOHWHQHVV RI WKH UHDFWLRQ (DFK VHFWLRQ RI WKH ILUVW ILYH WXEHV ZDV ILOOHG ZLWK 0 1D2+ 7KH ODVW WXEH ZDV ILOOHG ZLWK GULHULWH PRLVWHU DEVRUEHQWf

PAGE 144

)LJXUH % 7KH KH[DJRQ UHDFWLRQ YHVVHO IRU D JUDYLPHWULF DQDO\VLV

PAGE 146

VORZO\ WR WKH KH[DJRQ UHDFWLRQ YHVVHOV 2SHQLQJ RI YDOYH ZDV MXGJHG IURP WKH UDWH RI EXEEOHV ULVLQJ XS LQ WKH DEVRUEHU WXEHV ,I YDOYH ZDV RSHQHG WRR ZLGH FDUERQ GLR[LGH JDV ZRXOG QRW KDYH WLPH WR UHDFW HIILFLHQWO\ ZLWK VRGLXP K\GUR[LGH LQ WKH DEVRUEHU 7KH FRPSOHWHQHVV RI WKH UHDFWLRQ ZDV FKHFNHG E\ SDVVLQJ H[LWLQJ JDV IURP WKH VHFRQG DEVRUEHU DUUD\ WKURXJK D OLPH ZDWHU VDWXUDWHG &D2+f VROXWLRQ 7XUELGLW\ RI WKH VROXWLRQ GXH WR IRUPLQJ RI &D& LQGLFDWHV LQFRPSOHWHQHVV RI WKH UHDFWLRQ 2QFH EXEEOHV VWRSSHG ULVLQJ XS DERXW KRXUVf DUJRQ ZDV GHOLYHUHG WR WKH V\VWHP WKURXJK YDOYH $ WR IOXVK RXW DOO VDPSOH JDV UHPDLQLQJ LQ WKH OLQHV IRU RQHKDOI KRXU $IWHU WKDW VFUHZ FODPSV RI WKH HQWUDQFH DQG H[LW W\JRQ WXEHV WR WKH DEVRUEHU DUUD\ ZHUH WLJKWO\ FORVHG 7KH YHVVHOV ZHUH GLVFRQQHFWHG IURP WKH V\VWHP DQG ZHLJKHG RQ DQ DQDO\WLFDO EDODQFH 'XULQJ WKH HQWLUH FRXUVH RI WKHVH H[SHULPHQWV WKH OLPH VROXWLRQ UHPDLQHG FOHDU 7KH WRWDO DEVRUSWLRQ WLPH ZDV DERXW KRXUV 1RWH WKDW EHIRUH DQDO\]LQJ WKH VDPSOH JDV D EODQN WHVW ZDV SHUIRUPHG E\ FRQWLQXRXVO\ IORZLQJ DUJRQ WKURXJK WKH V\VWHP IRU KRXUV 1R VLJQLILFDQW FKDQJH LQ ZHLJKW FRPSDUDEOH ZLWK WKH XQFHUWDLQW\ RI WKH EDODQFH PJ f RI WKH DEVRUEHUV EHIRUH DQG DIWHU SDVVLQJ DUJRQ ZDV REVHUYHG 'HVSLWH LWV WHGLRXV SURFHGXUH WKLV PHWKRG ZRUNHG UHDVRQDEO\ ZHOO SURYLGHG WKDW VXIILFLHQW FDUH ZDV H[HUFLVHG *DV &KURPDWRJUDSK\ 0HWKRG 'HVFULSWLRQ RI VWDQGDUG SURFHGXUHV IRU SHUIRUPLQJ JDV FKURPDWRJUDSK\ FDQ EH IRXQG HOVHZKHUH nf $ FDOLEUDWLRQ

PAGE 147

FXUYH IRU HWKDQH ZDV PDGH ILUVW XVLQJ SXUH HWKDQH %RWK FDOLEUDWLRQ DQG VDPSOH DQDO\VLV ZHUH SHUIRUPHG RQ WKH VDPH GD\ EDVLV WR PLQLPL]H HUURUV 'HWDLOV RI WKH HTXLSPHQW DQG RSWLPL]HG FRQGLWLRQV ZKLFK ZHUH GHWHUPLQHG E\ WULDO DQG HUURU PHWKRGV DUH VXPPDUL]HG DV IROORZV *DV FKURPDWRJUDSK 9DULDQ PRGHO 6\ULQJH 3UHVVXUH/RFN JDV V\ULQJH VHULHV $ ZLWK VLGH SRUW '\QDWHFK 3UHFLVLRQ 6DPSOLQJ &RUSRUDWLRQ &ROXPQ )XVHG VLOLFD PHJDERUH FROXPQ *64 PP LG P ORQJ RSHQ WXEXODU -t: 6FLHQWLILF 2SWLPL]HG FRQGLWLRQV 7HPSHUDWXUHr&f ,QMHFWRU )ODPH ,RQL]DWLRQ 'HWHFWRU &ROXPQ )ORZ UDWHPOPLQf +HOLXP +\GURJHQ $LU &RXQW UDWH RQ WKH LQWHJUDWRU FSP 6DPSOH UHVXOWV RI FDOLEUDWLRQV DUH VXPPDUL]HG LQ WDEOH %O )LJXUH % VKRZV WKH FDOLEUDWLRQ FXUYH IRU HWKDQH

PAGE 148

7DEOH %O 'DWD IRU FDOLEUDWLRQ FXUYH RI HWKDQH 1RRI UHSHWLWLRQV $WWHQXDWLRQ IDFWRU 1RRI FRXQWV 1RRI DGMXVWHG FRXQWV 9ROXPH RI JDV LQMHFWHG s s s s s s s s s s s s s s s s s

PAGE 149

,QMHFWHG YROXPH XO )LJXUH % &DOLEUDWLRQ FXUYH RI HWKDQH IRU JDV FKURPDWRJUDSK\ K} &2 FQ

PAGE 150

$33(1',; & 6,03/(; 7KHUH DUH WKUHH SURJUDPV IRU VHDUFKLQJ IRU PLQLPXP SRLQWV RI VSHHG RI VRXQG SRLQWV DORQJ FRH[LVWHQFH FXUYHf 7KH ILUVW LV WKH VLPSOH[ RSWLPL]DWLRQ SURJUDP XVHG WR VHDUFK IRU HOHYHQ FRHIILFLHQWV LQ DQ HTXDWLRQ f 7KH VHFRQG SURJUDP LV XVHG WR FDOFXODWH VSHHG RI VRXQG ZLWK FRHIILFLHQWV IRXQG LQ SURJUDP RQH 7KH ODVW SURJUDP ILQGV WKH FRPPRQ SRLQW RI WZR FXUYHV JHQHUDWHG IURP SURJUDPV RQH DQG WZR 3URJUDP 6LPSOH[ 2SWLPL]DWLRQ & ,03/,&,7 5($/r $+=f ',0(16,21 &f7f%f ',0(16,21 6656'f :5,7(rrf n,1387 02/(&8/$5 :(,*+7 ,1 .*02/(n 5($' rrf : 23(1 ),/( n'$7$5$:n67$786 n2/'nf / 5($'r(1' f 7/f&/f / / *272 &/26(f / / :5,7(rrf n/ n/ :5,7(rrf n: n: 23(1 ),/( n,1,7,$/'$7n67$786 n2/'nf 5($'rf %,-f ff &$// ,1,7/&225 %6656'/7&:f 23(1 ),/( n67$57'$7n67$786 n1(:nf '2 OOO '2 :5,7( rf %,-f &217,18( :5,7( rf n6656'n-n n6656'-f :5,7( rf n &217,18(

PAGE 151

&/26(f ,) 6656'Of/7(f 7+(1 & :5,7(rrf f),1 &21 n *272 (1' ,) &$// 6+(//%6656':2567f &+(&.(5 &+(&.(56656'f .2817 .2817 & :5,7( r r f frrrrrrrrrrrrrrrrrrrrrrrrr.2817 f.2817 ..2817 ..2817 ,) ..2817(4 f 7+(1 & ,) ..2817(4f 7+(1 ..2817 &+(&.(5 &+(&.(5f6656'f & &+(&.(5 &+(&.(5f6656'f ,) &+(&.(5/7(f 7+(1 & ,) &+(&.(5/7(f 7+(1 *272 (1' ,) &+(&.(5 (1' ,) ,) 6656'f(4:2567f 7+(1 & :5,7(rrf n)/$* 5$,6(' )/$* 5$,6(' )/$* 5$,6('n = 6656'f 6656'f 6656'f 6656'f = '2 O == %-f %-f %-f %-f == &217,18( (1' ,) & :5,7(rrf n67$57 5()/(;n &$// 5()/(; 6656'%/7&:f & :5,7(rrf n3$66 5()/(;n ,) 6656'f/76656'ff 7+(1 &$// (;3$1'%6656'&7/:f & :5,7(rrf n3$66 (;3$1'n *272 (/6( ,) 6656'f*76656'f$1' &6656'f/76656'ff 7+(1 6656'f 6656'f '2 %,f %f &217,18( & :5,7(rrf n3$66 %(7:((1n *272 (/6( ,) 6656'f*76656'ff 7+(1 ,) 6656'f/76656'ff 7+(1 &$// &175%6656'&7/:f & :5,7(rrf n3$66 &175n (/6( ,) 6656'f*76656'ff 7+(1

PAGE 152

Q Q Q Q QRQ &$// &17:%6656'&7/:f & :5,7(rrf n3$66 &17:n (1' ,) *272 (1' ,) &/26(f 23(1 ),/( n&225287n67$786 n1(:nf '2 OOO '2 :5,7( rf %,-f &217,18( :5,7( rf n6656'n-n n6656'-f :5,7( rf n &217,18( :5,7(rf n180%(5 2) ,7(5$7,216 n.2817 &/26(f 6723 (1' 68%5287,1( ,1,7/&225 %6656'/7&:f ,03/,&,7 5($/r $+=f ',0(16,21 %f7f&f6656'f$f '2 '2 O $-f %-,f &217,18( $ $f $ $f $ $f $ $f $ $f $ $f 7& $f $/3+$ $f %(7$ $f '(/7$ $f &$// 5(6,'8$/$O$$$$$7&$/3+$%(7$ &'(/7$&7/67:f 6656',f 67 :5,7(rrf n6656'n,n n6656',f &217,18( 5(7851 (1' 68%5287,1( 5(6,'8$/$O$$$$$7&$/3+$%(7$ &'(/7$&7/67:f ,03/,&,7 5($/r $+=f 5($/r '$%6'6457

PAGE 153

QQQQQQ QRQ Q QQ ',0(16,21 &f7f *$6 : : &21 *$6r7&: 67 '2 / 75 '$%6 7 ,f7&f7&f 56 &,f'$%6'6457&21fff$Or75rr%(7$$r75rr%(7$ &'(/7$f$ r75rr $/3+$%(7$f $r75f $r75rrf &$r75rrff :5,7(rrf n56 n56 56 56rr 67 6756 &217,18( 5(7851 (1' 68%5287,1( 5()/(; 6656'%/7&:f ,03/,&,7 5($/r $+=f 5($/r '6457 ',0(16,21 %f&f7f6656'f$f '2 O &80 '2 &80 &80%-,f &217,18( %-f &80 %-f %-f%-f%-ff &217,18( '2 '2 O $-f %-,f &217,18( $O $f $ $f $ $f $ $f $ $f $ $f 7& $f $/3+$ $f %(7$ $f '(/7$ $f ,) ,(4$1'7&/7f 7+(1 :5,7(rrf n/2:(5 %281'$5< /2:(5 %281'$5
PAGE 154

Q R R & &217,18( & *272 & (/6( ,) ,(4$1'7&*7f 7+(1 & :5,7(rrf f833(5 %281'$5< 833(5 %281'$5
PAGE 155

QRRQ QRQ RRQQQQ *272 (1' ,) '2 :5,7(rrf 6656'Of &217,18( 5(7851 (1' 68%5287,1( (;3$1'%6656'&7/:f ,03/,&,7 5($/r $+=f ',0(16,21 %f&f7f6656'f$f '2 %,f %,f%,f%,ff &217,18( '2 $.f %.f &217,18( $O $f $ $f $ $f $ $f $ $f $ $f 7& $f $/3+$ $f %(7$ $f '(/7$ $f :5,7(rf $O$$$$$7&$/3+$%(7$'(/7$ )250$7 ,;))f &$// 5(6,'8$/$$$$$$7&$/3+$%(7$ & '(/7$&7/67:f 6656'f 67 :5,7(rrf n6656'n,n n6656',f ,) 6656'f/76656'ff 7+(1 6656'f 6656'f '2 O %-f %-f &217,18( (/6( ,) 6656'f*76656'ff 7+(1 6656'f 6656'f '2 O %-f %-f &217,18( (1' ,) 5(7851 (1'

PAGE 156

QRQ QR 68%5287,1( &175%6656'&7/:f ,03/,&,7 5($/r $+=f ',0(16,21 %f&f7f6656'f$f '2 %,f %,fr%,f%ff &217,18( '2 $.f %.f &217,18( $ $f $ $f $ $f $ $f $ $f $ $f 7& $f $/3+$ $f %(7$ $f '(/7$ $f :5,7(rf $O$$$$$7&$/3+$%(7$'(/7$ )250$7 ,;))f &$// 5(6,'8$/$O$$$$$7&$/3+$%(7$ & '(/7$&7/67:f 6656'f 67 6656'f 6656'f '2 O %-f %-f &217,18( 5(7851 (1' 68%5287,1( &17:%6656'&7/:f ,03/,&,7 5($/r $+=f ',0(16,21 %f&f7f6656'f$f '2 %,f %,fr%,f%,ff &217,18( '2 $.f %.f &217,18( $ $f $ $f $ $f $ $f $ $f $ $f 7& $f $/3+$ $f %(7$ $f '(/7$ $f

PAGE 157

R R :5,7(rf $O$$$$$7&$/3+$%(7$'(/7$ )250$7 ,;))f &$// 5(6,'8$/$O$$$$$7&$/3+$%(7$ & '(/7$&7/67:f 6656'f 67 6656'f 6656'f '2 %-f %-f &217,18( 5(7851 (1' 3URJUDP ,, 6SHHG RI 6RXQG &DOFXODWLRQV & ,03/,&,7 5($/r $+=f 5($/r '$%6'6457 ',0(16,21 &f7f$f 23(1 ),/( n&225287n67$786 2/'nf '2 5($' rf $,f & :5,7(rrf $,f &217,18( &/26(f $ $f $ $f $ $f $ $f $ $f $ $f 7& $f $/3+$ $f %(7$ $f '(/7$ $f 23(1 ),/( n'$7$5$:n67$786 n2/'nf 23(1 ),/( n7&'$7n67$786 n1(:nf / 5($'r(1' f 7/f F :5,7(rrf 7/f / / *272 &/26(f / / *$6 & : : &21 *$6r7&: '2 / 75 '$%67,f7&f7&f & :5,7(rrf n75 n75 &,f '6457&21fr$r75r r%(7$$ r75rr%(7$ &'(/7$f$r75rr$/3+$%(7$f$r75$r75rr

PAGE 158

&$r75rrf :5,7(rf 7,f &,f & :5,7(rrf 7,f &,f &217,18( &/26(f 6723 (1' 3URJUDP ,,, &URVVLQJ3RLQW 6HDUFKLQJ & SURJUDP PLQ &+$5$&7(5r 89 '28%/( 35(&,6,21 '28%/( 35(&,6,21 '28%/( 35(&,6,21 '28%/( 35(&,6,21 '28%/( 35(&,6,21 '28%/( 35(&,6,21 '28%/( 35(&,6,21 '28%/( 35(&,6,21 ',0(16,21 $f =(52 7:2 4765() '(/'6457'$%6=(527:2'(/( 3/3/7&/7/75/ 35357&575755'(/) $,/$/$/$/$/$/ $/3+$/%(7$/'(/7$/ $5$5$5$5$5$5 $/3+$5%(7$5'(/7$5 % f :5,7(rrf n5,*+7 6,'( ),/( 1$0(n 5($'rf 8 )250$7 $f 23(1 ),/( 867$786 n2/'nf '2 5($' rf $,f :5,7(rrf $,f &217,18( &/26(f :5,7(rrf n/()7 6,'( ),/( 1$0(f 5($'rf 9 23(1 ),/( 967$786 n2/'nf '2 5($' rf %,f :5,7(rrf %,f &217,18( &/26(f $O5 $f $5 $f $5 $f $5 $f $5 $f

PAGE 159

$5 $f 7&5 $f $/3+$5 $f %(7$5 $f '(/7$5 $f $/ %f $/ %f $/ %f $/ %f $/ %f $/ %f 7&/ %f $/3+$/ %f %(7$/ %f '(/7$/ %f :5,7(rrf n,1387 75 t 7/ ,1 .(/9,1n 5($' r r f 75 7/ 7 757/f7:2 3/ %(7$/ '(/7$/ 3/ $/3+$/%(7$/ 35 %(7$5 '(/7$5 35 $/3+$5%(7$5 75/ '$%677&/f7&/f 755 '$%677&5f7&5f & ZULWH rrf WUOWUUSSSUSU 6 '64577&/ffr$/r75/r r%(7$/$/r75/rr3/ &$/r75/r r3/$/r75/$/r75/r r$/r75/r r f 5 '64577&5ffr$O5r755rr%(7$5$5r755rr35 &$5r755r r35$5r755$5r755r r$5r755rr f '(/ 56 ,) '$%6'(/f/(2'f 7+(1 *272 & (/6( ,) 5*76f 7+(1 (/6( ,) '(/*7=(52f 7+(1 75 7 7 757/f7:2 '(/( 7( ,) '(/((4=(52f 7+(1 *272 (1' ,) ( 7 :5,7( rrf '(/775 *272 & (/6( ,) 5/76f 7+(1 (/6( ,) '(//7=(52f 7+(1 7/ 7 7 757/f7:2 '(/) 7) ,) '(/)(4=(52f 7+(1 *272 (1' ,) ) 7

PAGE 160

:5,7( rr *272 (1' ,) f '(/77/ :5,7( rrf *272 nrrrrn7( :5,7( rrf *272 frrrrfI7) :5,7( rrf nf§!81$%/( 72 5(),1( )857+(5n :5,7( rrf 77'(/ 5 5r6457 ff 6 6r6457 ff :5,7( rrf 6723 (1' 56

PAGE 161

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n WR SDVFDO LV VHUYHG DGHTXDWHO\ DW WKH ORZ HQG E\ WKH 0F/HRG JDXJH LI VXLWDEO\ RSHUDWHG WR DYRLG WKH HUURUV DVVRFLDWHG ZLWK WKH PHUFXU\ SXPSLQJ HIIHFW DQG DW WKH XSSHU HQG E\ WKH 8WXEH

PAGE 162

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n LQGLFDWRU FRQWDFW QHHGOHV

PAGE 163

)LJXUH 7KH SUHVHQW GHDGZHLJKW SUHVVXUH JDXJH

PAGE 164

8 )LJXUH 7KH DUUDQJHPHQW RI WKH GHDGZHLJKW SUHVVXUH JDXJH

PAGE 165

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

PAGE 166

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f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

PAGE 167

WKURXJK LW LV JHQHUDOO\ QHJOLJLEOH 6RPH VXLWDEOH IROORZHU PDWHULDOV DUH WKLQ HODVWRPHULF VKHHWV SODVWLF ILOPV DQG PHWDO IRLOV LI QRW VWUHVVHG EH\RQG WKHLU HODVWLF OLPLWV 7KH EDODQFH EHDP ZDV PDFKLQHG IURP EUDVV EDU VWRFN 7KH FUXFLDO IHDWXUHV RQ LW DUH WKH ORFDWLRQ RI WKH FRQWDFW EDOO DQG WKH ZHLJKW VXVSHQVLRQ KRRN SLQV ZLWK UHVSHFW WR WKH NQLIH HGJH 7KHVH IHDWXUHV ZHUH ORFDWHG RQ LQFK FHQWHUV ZLWK PD[LPXP XQFHUWDLQW\ RI LQFK 7KH FRQWDFW EDOO VLWV LQ D r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r DQJOH ZLWK UHVSHFW WR WKH ERWWRP RI WKH

PAGE 168

EHDP 7KH LQVHUW KDV HLJKW XVHDEOH r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f

PAGE 169

ZKHUH J ORFDO DFFHOHUDWLRQ RI JUDYLW\ P DGGHG ORDG PDVV Q SRVLWLRQ RI ORDG PDVV DQG WKH DQJOH EHWZHHQ WKH WRS RI WKH EHDP DQG WKH KRUL]RQWDO ,Q RUGHU WR XVH WKH VLPSOHU UHODWLRQVKLS )I J = QP 'f LW LV QHFHVVDU\ WR DGMXVW WKH SRVLWLRQ RI WKH JDXJH ERG\ UHODWLYH WR WKH SODQH RI WKH EHDULQJ SDG VR WKDW WKH EDODQFH SRLQW RFFXUV ZKHQ WKH EHDP LV SUHFLVHO\ KRUL]RQWDO LH FRV f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

PAGE 170

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f LQ HTXLOLEULXP ZLWK LWV OLTXLG SKDVH DW FRQVWDQW WHPSHUDWXUH 7KLV V\VWHP KDV EHHQ WKRURXJKO\ VWXGLHG E\ *LONH\ HW DOn f XVLQJ PRUH HODERUDWH PHWKRGV 7KH IUHRQ FRQWDLQHU ZDV SODFHG LQ DQ DGMXVWDEOH VXEDPELHQW FRQVWDQW WHPSHUDWXUH EDWK DQG FRQQHFWHG WR WKH JDXJH LQOHW $LU ZDV SXUJHG IURP WKH JDXJH DQG OLQHV E\ YDFXXP SXPSLQJ DQG UHSHDWHG IOXVKLQJ ZLWK VPDOO DOLTXRWV RI IUHRQ $IWHU HTXLOLEULXP ZDV UHDFKHG WKH EHDP ZDV EDODQFHG DQG WKH UHTXLVLWH PDVVHV DQG WKHLU SRVLWLRQV QRWHG 7KH DWPRVSKHULF

PAGE 171

SUHVVXUH ZDV GHWHUPLQHG XVLQJ D PHUFXU\ EDURPHWHU QRWLQJ WKH XVXDO FRUUHFWLRQV IRU WKHUPDO H[SDQVLRQV RI PHUFXU\ DQG EUDVV 7KH PHDVXUHG SUHVVXUH ZDV FDOLEUDWHG IURP WKH UHODWLRQVKLS 3 3DWP J = QPf$ 'f ZKHUH $ QU FURVV VHFWLRQDO DUHD RI WKH SLVWRQ 7KH ORFDO DFFHOHUDWLRQ RI JUDYLW\ PXVW EH NQRZQ RU PHDVXUHG ,Q WKLV ODERUDWRU\ LW LV FPVHF DQG WKH UHVXOWLQJ JDXJH FRQVWDQW LV SVLJJ 7DEOH 'O JLYHV WKH UHVXOWV RI WKHVH WHVWV DW WZR WHPSHUDWXUHV DORQJ ZLWK WKH FRUUHVSRQGLQJ YDSRU SUHVVXUHV ZKLFK PD\ EH LQIHUUHG IURP WKH SUHFLVH PHDVXUHPHQWV RI *LONH\ HW DOn f 7KH FORVH DJUHHPHQW VKRZV WKDW WKH SUHVHQW JDXJH ZRUNV ZHOO 7KHVH UHVXOWV DUH QRW LQWHQGHG WR VHUYH DV D FDOLEUDWLRQ RI WKH JDXJH EXW UDWKHU DV D FKHFN RI LWV SHUIRUPDQFH 7KH IUHRQ XVHG ZDV PDQXIDFWXUHG IRU UHIULJHUDWLRQ VHUYLFH DQG ZDV RI XQNQRZQ SXULW\ 7KH FRQVLVWHQF\ RI WKH JDXJH UHDGLQJV LQGLFDWHV WKDW WKH DEVROXWH SUHVVXUH JDXJH LV DFFXUDWH WR ZLWKLQ WKH OLPLWV RI WKH VPDOOHVW PDVV GLIIHUHQFH ZKLFK PD\ EH UHSURGXFLELOLW\ GLVFHUQHG 8VLQJ D SP SRO\LPLG ILOP IROORZHU 'X3RQW .DSWRQf WKH ORDGLQJ PDVV FRXOG EH PHDVXUHG HDVLO\ WR ZLWKLQ s J ZKLFK FRUUHVSRQGV WR WKH OLPLWV RI XQFHUWDLQW\ VWDWHG LQ WKH WDEOH 7KH JDXJH LV D OLWWOH PRUH UHVSRQVLYH LI D SP ODWH[ IROORZHU LV XVHG FD s Jf EXW WKH OXEULFDWLQJ RLOV XVHG WKXV IDU SHUPHDWH RU UHDFW ZLWK WKH ODWH[ FDXVLQJ RSHUDWLRQDO GLIILFXOWLHV 7KLQ

PAGE 172

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b GHYLDWLRQ IURP OLWHUDWXUH YDOXHf *LONH\ f 7KLV ZRUN %RXUGRQ *DXJH bf bf bf bf H[DPSOH EH ILWWHG ZLWK SLVWRQF\OLQGHU FRPELQDWLRQV RI GLIIHUHQW GLDPHWHUV RU ZLWK HQWLUHO\ RWKHU W\SHV RI VHQVRUV ,Q SDUWLFXODU LW ZDV LQWHQGHG WR LQYHVWLJDWH WKH VXLWDELOLW\ RI D WKLQ ZDOOHG PHWDOOLF EHOORZV IRU XVH DV D EDODQFH GHWHFWRU $Q HOHFWURIRUPHG QLFNHO EHOORZV ZLWK LQFK

PAGE 173

ZDOOV LQFK RG LQFK LG DQG FRQYROXWLRQV ZDV EUD]HG WR ILWWLQJV ZKLFK SHUPLWWHG LW WR EH XWLOL]HG DV D UHn HQWUDQW SUHVVXUH VHQVRU VHH ILJXUH 'f 6\VWHP SUHVVXUH LV DSSOLHG H[WHUQDOO\ DURXQG WKH EHOORZV FDXVLQJ LW WR FRQWUDFW RU VKRUWHQ $ SXVK URG H[WHQGLQJ WKURXJK WKH EHOORZV LV DFWHG XSRQ E\ WKH FRQWDFW EDOO DV EHIRUH 7KH EHDP LV ORDGHG WR FRXQWHUDFW WKH GRZQZDUG IRUFH GXH WR WKH V\VWHP SUHVVXUH 7KH EHOORZV DUH YHU\ IOH[LEOH D[LDOO\ EXW VWLII LQ WKH UDGLDO GLUHFWLRQ 7KH GRZQZDUG IRUFH LV WKHQ HTXDO WR WKH PDJQLWXGH RI WKH V\VWHP SUHVVXUH WLPHV WKH DFWLYH FURVV VHFWLRQDO DUHD $f RU S 9D r 9 T=P/Q/$0 'f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

PAGE 174

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f &RPPHUFLDO GHDG ZHLJKW SUHVVXUH JDXJHV FRVW VHYHUDO

PAGE 175

WKRXVDQG GROODUV DQG DUH JHQHUDOO\ EH\RQG WKH PHDQV RI PDQ\ RI WKH ODERUDWRULHV ZKLFK FRXOG EHQHILW IURP WKHLU XVH 3HUKDSV JDXJHV EDVHG RQ WKH WHFKQRORJ\ SUHVHQWHG KHUH ZLOO EULGJH WKH SUHVHQW JDS EHWZHHQ FRQYHQWLRQDO VHFRQGDU\ JDXJHV DQG KLJK FRVW DEVROXWH RQHV 7KH\ DUH DW OHDVW ZRUNLQJ ZHOO LQ WKLV ODERUDWRU\ E\ LQFUHDVLQJ WKH FRQILGHQFH LQ WKH GHWHUPLQDWLRQ RI SUHVVXUH VHQVLWLYH DFRXVWLF DQG WKHUPRG\QDPLF SURSHUWLHV

PAGE 176

$%62/87( 35(6685( *$8*( &$/,%5$7,21

PAGE 177

$33(1',; ( &20387(5 352*5$06 6LQFH PRVW RI WKH SURJUDPV DUH OHQJWK\ RQO\ SURJUDPV 9,.,1* DQG 0$; DUH VKRZHG LQ IXOO 7KH RWKHUV DUH OLVWHG RQO\ PDLQ SURJUDPV 0RVW RI WKHVH SURJUDPV ZHUH ZULWWHQ LQ IRUWUDQ E\ 'U 0F*LOO 7KH UHPDLQLQJ E\ WKH DXWKRU 352*5$0 9,.,1* ODUJH ,17(5)$&( 72 68%5287,1( 7,0( 1675f &+$5$&7(5r 675 >1($55()(5(1&(@ ,17(*(5r 1 >9$/8(@ (1' 5($/r ';f'
PAGE 178

23(1)LOH fF1$0(180n67$786 n1(:nf :5,7(f '7(03180 :5,7(rf '7(03180 )250$7;$f &/26( f 23(1),/( '1$0(67$786 f1(:nf 23(1 )LOHWH3HDNEXI n 67$786 f2/'n f 5($'rf )7, 7, 7, &$// .(,7+$7)f 7) 7) ) 64577)7,fr) &/26(f ,;7 F ,;7 1$ ,;7 ,'(/< r1$ & 0 1$,;7,'(/<1$ & 86(' 21/< )25 &(17(5,1* rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr 0 & 86(' $)7(5 &(17(5,1* ;0,1 ;0$; )/2$70f &$// 7,0(7675f 5($'7675nfn(55 f +5 5($'7675f;,fn(55 f 0,1 5($'7675n;,fn(55 f 6(& 722 +5r0,1r6(& &$// 7(036(792/7f '2 5($'6 6) ):,'7+ 63) ):,'7+ ,1'(; 32,176 0 ,67$57 &$// +36)63)6:((3f $0$; ;0,1 '2 ,' ,'(/< &$// .(,7+09f &217,18( '2 ,67$5732,176,1'(; $9* '2 -$ 1$ &$// .(,7+09f .2817 .2817 ,).2817(4r1$f 7+(1 ,* ,* &$// *$,16(7,*f

PAGE 179

(1',) ,).2817*7r1$f 7+(1 .2817 *272 (1',) ,)09*7509/7f *272 .2817 $9* $9*)/2$709f &217,18( $03 $9*)/2$71$f ';-f )/2$7-f '<-f $03 ,)$03*7$0$;f 7+(1 $0$; $03 -0$; (1',) ,)$03/7;0,1f ;0,1 $03 &217,18( ,),1'(;(4f 7+(1 13/27 O (/6( 13/27 (1',) <0$; $0$; & &$// 3/270';'<;0$;;0,1<0$;<0,113/27f & 21/< 86(' )25 &(17(5,1* 4rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr ,),1'(;(4f 7+(1 ,1'(; ,1'(; 7(03 6) 6) 63) 63) 7(03 ,67$57 32,176 32,176 O -)0$; -0$; $9*$ $0$; &$// .(,7+$7,f ,)7,/7f *272 &$// 35(69,f *272 (/6( &$// .(,7+$7)f &$// 35(69)f -50$; -0$; )2 )/2$7-)0$;-50$;f &$// 0$;):,'7+)20f ) )2 $03 $0$;$9*$f ,)-)0$;(425-)0$;(4025-50$;(425 &-50$;(40f 7+(1 23(1)LOH nF3HDNEXIn67$786 n2/'nf

PAGE 180

R Q 5($'rf )7, &/26(f 7, 7, 7) 7) ) 64577)7,fr) *272 (1',) &$// 7,0(7675f 5($'7675n,fn(55 f +5 5($'7675n;,fn(55 f 0,1 5($'7675n;,ff(55 f 6(& 7 +5rr0,16(& 7 77 ,)7/722f 7+(1 7 7r 7 +5rr0,16(& 7 77 (1',) ,)$03/7f 7+(1 ,* ,* ,),**(f &$// *$,16(7,*f (1',) ,)$03*7f 7+(1 ,* ,* ,),*/(f &$// *$,16(7,*f (1',) 5$7,2 )64577,7)rff :5,7(rf 7,7)9,9)$03)7,* & '2 ,) :5,7(rrf F &217,18( F :5,7(rf 63)6) F )250$7,;n)5(48(1&< n);) F :5,7(rrf F ,) -50$; *7 -)0$; f 0 0 F ,) -50$; /7 -)0$; f 0 0 F 2 ,, R f R & 86(' 21/< )25 &(17(5,1* rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr :5,7(rf7,7)9,9)0)7-50$;-)0$;5$7,2 )250$7,;))))f &rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr '7 7)7, ') )/2$7-50$;-)0$;f)/2$70fr:,'7+ & 86(' 21/< $)7(5 &(17(5,1* ) )')r (1',) &217,18( ,)$%6'7f*7f 5($'6 5($'6 ,)$%6'7f*7f 5($'6 5($'6

PAGE 181

R R R & ,)$%6'7f*7f 5($'6 5($'6 & ,)$%6'7f*7f 5($'6 5($'6 & ,)$%6'7f*722 f 5($'6 5($'6 & ,)$%6'7f*7f 5($'6 5($'6 & ,)$%6'7f*7f 5($'6 5($'6 & ,)$%6'7f/7$1'5($'6*7f 5($'6 5($'6 & ,)$%6'7f/7$1'5($'6*7f 5($'6 5($'6 & ,)$%6'7f/7$1'5($'6*7f 5($'6 5($'6 & ,)$%6'7f/7$1'5($'6*7f 5($'6 5($'6 & ,)$%6'7f/7$1'5($'6*7f 5($'6 5($'6 & ,)$%6'7f/7$1'5($'6*7f 5($'6 5($'6 & ,)$%6'7f/7$1'5($'6*7f 5($'6 5($'6 :5,7(rf 92/7 )250$7 ,;n6(7 32,17 92/7$*( n)f 23(1)LOH nF3HDNEXIn67$786 n1(:nf :5,7(rf )7) &/26(f ,)7)*7f *272 ,)92/7/(92/7(f 7+(1 92/7 92/7673 23(1)LOH nF92/7%8)n67$786 n1(:nf :5,7(rf 92/7 &/26(f *272 Tr r rrrrrrrrrrrrrrrr r r r r r rrrrrrrr r rrrrrrrrrrrrrrrr r r r r r r r rrrrr r & (/6( & ,)92/7*792/7(f 7+(1 & 92/7 92/7( & 23(1)LOH nF92/7%8)n67$786 n1(:nf & :5,7(rf 92/7 & &/26(f & *272 & (1',) & *272 & 86(' ,1 1250$/ 6&$1 4rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrr (1',) 23(1)LOH nF6723581n67$786 n1(:nf &/26(f &$// 7(036(7f 6723 (1' 68%5287,1( .(,7+$7f FRPPRQ LEJORE LEVWDLEHUULEFQW n&20021 *5283 f LQWHJHUr FPGfUGfZUWf FKDUDFWHUr EQDPHEGQDPH7(03 f &20021 *5283 n FKDUDFWHUr IOQDPH

PAGE 182

Q R R ./$ ,%),1' n.$ nf rr rr rr &$// ,%5' ./$5'f 5'f, 5'f,rOf, 5'f,rOOr,f, 5'f,rOOr,,rO 5'f, 5'ff§rf, 5'f,r,,r,fO 5'f,r,,r,,r, :5,7(7(03n$fffFKDUfFKDUfFKDUOfFKDU,f &&+$5f&+$5,OOf&+$5,f&+$5f 5($'7(03n)fn(55 f 5 5 $/3+$ '(/7$ F 5 F $/3+$ F '(/7$ $/'(/ $/3+$r'(/7$ 5& 55 3$57 $/3+$$/'(/ 3$57 $/'(/2$/3+$frr 3$57 r5&r$/'(/ 3$57 r$/'(/ 7(03 3$5764573$573$57 f 7 7(03 3$57 5(7851 HQG 68%5287,1( +36)63)6:((3f FRPPRQ LEJORE LEVWDLEHUULEFQW f &20021 *5283 f LQWHJHUr FPGfUGfZUWf,3f FKDUDFWHUr FKf &+$5$&7(5r 67$57)6723) &+$5$&7(5r 6:((3 FKDUDFWHUr EQDPHEGQDPH n&20021 *5283 n FDOO LELQLWLEVWDf ,3f ,3f rr ,3f rr ,3f rr ,+3 ,%),1' nKS nf &+f n)n &+f n8n

PAGE 183

&+f nn &+f n$n &+f n0n &+f n2n &+f n &+ f n9 &+f n2n &+f n6n &+ f f7 :5,7(67$57)f)fnf 6) 5($'67$57)n$Ofnf &+f 5($'67$57)n,;$Offf &+f 5($'67$57)f ;$Of f f &+f 5($'67$57)n;$fnf &+f 5($'67$57)f;$fnf &+f 5($'67$57)n;$fnf &+f 5($'67$57)n;$fnf &+f 5($'67$57)n;$Ofnf &+f 5($'67$57)n;$Ofnf &+f 5($'67$57)n;$Ofnf &+f &+f n+n &+f n=n &+f f6n &+f n3n :5,7(6723)n)fnf 63) 5($'6723)n$Ofnf &+f 5($'6723)n;$fnf &+f 5($'6723)n;$Ofnf &+f 5($'6723)n;$Ofnf &+f 5($'6723)n;$Ofnf &+f 5($'6723)n;$fff &+f 5($'6723)n;$Ofnf &+f 5($'6723)n;$Ofnf &+f 5($'6723)n;$fnf &+f 5($'6723)n;$fnf &+f &+f n+n &+f f=n &+f n7n &+f f,n 5($'6:((3n$Ofnf &+f 5($'6:((3n,;$Ofnf &+f &+f n6n &+f f(n &+f n6n &+f n6f &+f f6n &+f n6n 1 0 '2 1 :57,f '2

PAGE 184

QRQ ,),f§fr-/(f0f 7+(1 :57,f :57,f ,&+$5&+,Ofr-ffr,3-f (1',) &217,18( &217,18( &$// ,%:57,+3:570f 5(7851 HQG F F F VXEURXWLQH HUURU FRPPRQ LEJORE LEVWD LEHUU LEFQW ZULWH rf LEVWDLEHUULEFQW IRUPDW n (UURUnLLLf UHWXUQ HQG 68%5287,1( 7(036(796f FRPPRQ LEJORE LEVWDLEHUULEFQW n&20021 *5283 n LQWHJHUr FPGfUGfZUWf,3f &+$5$&7(5r &+f &+$5$&7(5r 92/76 FKDUDFWHUr EQDPHEGQDPH n&20021 *5283 r /. ,%),1' n/2&.,1 n f ,3f rr ,3f rr ,3f rr ,3f :5,7(92/76f)fnf 96 &+f f;f &+f nn &+f 5($'92/76n$Ofnf &+f 5($'92/76n,;$Ofnf &+f 5($'92/76n;$Ofnf &+f 5($'92/76n;$Ofnf &+f 5($'92/76n;$Ofnf &+f 5($'92/76n;$Ofnf &+f '2 :57,f '2 fr,)./(f 7+(1 :57,f :57,f ,&+$5&+.ffr,3-f (1',) &217,18( &217,18(

PAGE 185

RRQ QRQ QQR RQR &$// ,%:57/.:57f 5(7851 HQG 68%5287,1( *$,16(7,*f FRPPRQ LEJORE LEVWDLEHUULEFQW n&20021 *5283 n LQWHJHUr FPGfUGfZUWf,3f &+$5$&7(5r &+f &+$5$&7(5r *$,1 FKDUDFWHUr EQDPHEGQDPH n&20021 *5283 n /. ,%),1' n/2&.,1 ff ,3f rr ,3f rr ,3f rr ,3f :5,7(*$,1nfnf ,* &+f n*n 5($'*$,1n$Ofnf &+f 5($'*$,1n;$Ofnf &+f :57f ,&+$5&+ff,&+$5&+ffr &,3f,&+$5&+ffr,3f &$// ,%:57/.:57f 5(7851 HQG 68%5287,1( 35(63f FKDUDFWHUr D RSHQILOH nFRPOnVWDWXV nROGnf UHDGf D 5($'Dn;)fn(55 f 3 IRUPDWDOf 5(7851 &/26(f *272 HQG 68%5287,1( 0$;):)20f )2 :)/2$70fr)2): 5(7851 (1' 68%5287,1( .(,7+09f

PAGE 186

FRPPRQ LEJORE LEVWDLEHUULEFQW n&20021 *5283 n LQWHJHUr FPGfUGfZUWf FKDUDFWHUr EQDPHEGQDPH7(03 &20021 *5283 n NO ,%),1' nNO nf L rr LO rr L rr FDOO LEUG ./5'f UGOfL UGfLOrLfLO UGOfLOrLLrLOfL UGfOrLLrLOLrL UGfL UGfLrfLO UGfLrLLrLOfL L UGfrLLrLOLrL :5,7(7(03n$fnf &+$5f&+$5f&+$5f &&+$5,Of&+$5f&+$5f&+$5f&+$5f 5($'7(03nff(55 f 09 5(7851 HQG F F F 68%5287,1( 3/271';'<;0$;;0,1<0$;<0,113/27f 5($/r ';f'
PAGE 187

Q Q R %27720 5,*+7 723 ,)13/27(4f 7+(1 &$// 6&5((102'( 0f &$// 3/277(5 '(97<3(,17(5)$&($''5f &$// 6&$/( ;0,1<0,1;0$;<0$;f &$// :,1'2: /()7%277205,*+7723f &$// *5$3+;<';'<1;$;,6;/%/<$;,6 &&@7f ,17(*(5r 7 >1($55()(5(1&(@ (1' &20021 ,17(*(5r ,17(*(5r 1'(/<7727%7/ ,17(*(5 $03&2817)5(4)6+,)7 5($/r $6803+680 & 5($/r $*680 ,17(*(5r $0f3+f35f ,17(*(5r )0f70f/$5*( & ,17(*(5r $*f &+$5$&7(5r ),/(5$:),/()250 1 1'(/< rr rr rr :ULWHrrf :+$7 ,6 7+( '$7$ 287387 ),/( 1$0("n 5($'rf ),/(5$: :ULWHrrf n :+$7 ,6 7+( )250$7 287387 ),/( 1$0("n 5($'rf ),/()250 )250$7$f RSHQILOH nEORFNnVWDWXV nQHZnf 23(1),/( ),/()25067$786 n1(:nf 23(1 ),/( ),/(5$:)250 n81)250$77('n 67$786 n1(:n f ZULWHrrf n :+$7 ,6 7+( 5(621$1&( '$& 6(77,1*"n 5($'rrf ,&(17

PAGE 188

R Q Q Q R R 6:((3"f ,67$57 ,&(171 &$// 92/762,)/,3,67$57ff :ULWHrrf n :+$7 ,6 7+( ,1,7,$/ 7(03(5$785( 92/7$*("f 5($' r r f 7%% ,7% ,177%%f &$// 92/76,)/,3,7%ff :ULWH r r f n :+$7 ,6 7+( ),1$/ 7(03(5$785( 92/7$*("n 5($' r r f 7) :ULWHrrf n +2: 0$1< +2856 )25 7+,6 7(03(5$785( 5($' r r f 7,0 7,0 7,0r 700 7)7%%f7,0 &$// 7/22.7f 7 7 F F F :5,7( r r f n :+$7 ,6 7+( 6(16,7,9,7< ,' 180%(5"f 5($' r r f ,* 23(1),/( f6(16(n67$786 n1(:nf &$// 7(03*(77(03f 76+,)7 7(03 ,76+,)7 76+,)7 :5,7(f 76+,)7 )250$7,; n7(03(5$785( 6+,)7 n)f &$// 35(6*(73f 36+,)7 3 :5,7(f 36+,)7 )250$7,;f35(6685( 6+,)7 f)f &$// )5(4*(7)5(4f )6+,)7 )5(4 & & :5,7(f )6+,)7 )250$7,;f)5(48(1&< 6+,)7 f,f &$// 3+$6(*(7$03/f &$// /2&.,1$03/$1*/f $6+,)7 $1*/ & & & :5,7(f $6+,)7 )250$7,;n$1*/( 6+,)7 n)f :5,7(rrf f$03/ f$03/ & :5,7(rrf n$1*/ n$1*/ :5,7(f 1 )250$7,;f /223 5$1*( n,f &/26(f .%/2&. .%/2&. .%/2&. /$5*( &2817 ,67$57 ,&(171 ,6723 ,67$571 &$// 92/762,)/,3,67$57ff &$// 6(&6772f 7(03 )/2$77fr7007%% ,7(03 ,177(03f &$// 92/76,)/,3,7(03ff

PAGE 189

QR QRRQ RQQRQ QRRQ &)25:$5' 6:((3 '2 ,67$57,6723 &$// 92/762,)/,3,ff &$// 6(&6772f 7% 7 $680 3+680 & $*680 .2817 &$// 6(&67 72f &$// .HOE$03f & &$// 3+$6(*(7$03/$1*/f &$// 3+$6(*(7$03/f $680 $680)/2$7$03f 3+680 3+680$03/ $*6 80 $*6 80$1*/ :5,7(rrf n3+680 n3+680 :5,7(rrf n$680 n$680 :5,7(rrf n$*680 n$*680 .2817 .2817 7/ 77% ,)7//71'(/
PAGE 190

*272 :ULWHf .%/2&. )250$7,;n 180%(5 2) %/2&.6 2) '$7$ n,f &/26(f &/26(f F &/26(f (1' & & & )81&7,21 ,)/,3,1387f ,)/,3 ,1387 5(7851 (1' & & & 68%5287,1( 92/76,7(03f ,17(*(5r $'$37'(9,&(&75/67$79'$&+O 9'$&+O ,17,7(03f $'$37 '(9,&( &75/ 67$7 &$// $286 $'$37'(9,&(&75/9'$&+O67$7f 5(7851 (1' & & & 68%5287,1( 92/76,7(03f ,17(*(5r $'$37'(9,&(&75/67$79'$&+O 9'$&+O ,17,7(03f $'$37 '(9,&( &75/ 67$7 &$// $286 $'$37'(9,&(&75/9'$&+O67$7f 5(7851 (1' & & & 68%5287,1( 3+$6(*(7$03/f FRPPRQ LEJORE LEVWDLEHUULEFQW ,17(*(5r ',0(16,21 f n&20021 *5283 n LQWHJHU &5 LQWHJHU FPGfUGfZUWf,3f &+$5$&7(5r &+f FKDUDFWHUr EQDPHEGQDPH

PAGE 191

& & & & & & & & & & & & & & & & & && & F F F & & & & F F F F F F F F F F F F F F &+$5$&7(5r 7(0327(03 n&20021 *5283 GDWD &5 /. ,%),1' n/2&.,1 nf ,3f rr ,3f rr ,3 f rr ,3f rr rr rr &+f n3n :57f ,&+$5&+ff &$// ,%:57/.:57f &$// ,%5' /.5'f f 5'f f 5'f,fr,f, f 5'f,fr,,fr,f, f 5'f,fr,,fr,,fr, f 5'f f 5'f,fr,f, f 5'f,fr,,fr,f, f 5'f,fr,,fr,,fr, f 5'f f 5'f,frf, f 5'f,fr,,fr,f, f 5'f,fr,,fr,,fr, '2 :5,7( rrf ,-f &+$5,-ff ,) ,-f(4&5f 7+(1 -*272 (1' ,) &217,18( ,) --(4f 7+(1 :5,7(7(03n$fnf &+$5ff&+$5ff &&+$5ff &&+$5ff&+$5,ff &&+$5ff&+$5ff 5($'7(03f)fn(55 f $1*/ (/6( ,) --(4f 7+(1 :5,7(7(03f$fnf &+$5ff&+$5ff &&+$5ff&+$5ff&+$5ff&+$5ff 5($'7(03)fr(55 f $1*/ (/6( ,) --(4f 7+(1 :5,7(7(03n$fff &+$5ff&+$5ff &&+$5ff&+$5ff&+$5,ff 5($'7(03f)fn(55 f $1*/ (/6( ,) --(4f 7+(1 :5,7(7(03f$ff &+$5ff&+$5ff &&+$5ff&+$5ff 5($'7(03n)fn(55 f $1*/

PAGE 192

QRQ QRQ QRQ & (1' ,) &+f n4n :57f ,&+$5&+ff &$// ,%:57/.:57f &$// ,%5' /.5'f f 5'f f 5'ff§frf, f 5'f,fr,,fr,f, f 5'f,fr,,fr,,fr, f 5'f, f 5'f,fr,f, f 5'ff§frf§frf, f 5'f,fr,,fr,,fr, f 5'f, f 5'ffr f f 5'f,frO,OOfr,f, f 5'f,fr,,fr,,fr, '2 ,) ,.f(4&5f 7+(1 .. *272 (1' ,) &217,18( ,) ..(4f 7+(1 :5,7(7(032n$fnf &+$5,ff&+$5,ff &&+$5ff&&+$5ff&+$5ff &&+$5ff&+$5ff&+$5,ff &&+$5ff&+$5ff 5($'7(032n)fn(55 f $03/ ,) f(4f $03/ $03/rr ,) f(4f $03/ $03/rr ,) f(4f $03/ $03/rr (/6( ,) ..(4f 7+(1 :5,7(7(032n$fnf &+$5,ff&+$5,ff &&+$5ff &&+$5ff&+$5,ff &&+$5ff&+$5ff&+$5,ff&+$5,ff 5($'7(032f)fn(55 f $03/ ,) f(4f $03/ $03/rr ,) f(4f $03/ $03/rr ,) f(4f $03/ $03/rr (/6( ,) ..(4f 7+(1 :5,7(7(032n$fff &+$5,ff&+$5,ff &&+$5ff &&+$5ff&+$5ff &&+$5ff&+$5ff&+$5ff 5($'7(032n)ff(55 f $03/ ,) f(4f $03/ $03/rr ,) f(4f $03/ $03/rr ,) f(4f $03/ $03/ r r (/6( ,) ..(4f 7+(1 :5,7(7(032n$fnf &+$5,ff&+$5,ff &&+$5,ff &&+$5ff&+$5,ff&+$5,ff&+$5,ff 5($'7(032n)ff(55 f $03/

PAGE 193

QRQ RQQ QRR ,) f(4f $03/ $03/rr ,) f(4f $03/ $03/rr ,) f(4f $03/ $03/ r r (1' ,) ,) $%6$03/f*7f *272 5(7851 HQG 68%5287,1( .HE09f FRPPRQ LEJORE LEVWDLEHUULEFQW &20021 ,17(*(56 f&20021 *5283 n LQWHJHUr FPGfUGfZUWf FKDUDFWHUr EQDPHEGQDPH7(03 n&20021 *5283 n .%$' ,%),1'n.% nf FDOO LEUG .%$'5'f UGOfL UGfLOrLfLO UGfLOrLLrLOfL UGfOrLLrLOLrL UGfL UGfLrLfLO UGfLrLLrLOfL L UGfrLLrLOLrL :5,7(7(03f$fnf &+$5,f&+$5,f&+$5,f &&+$5,Of&+$5f&+$5f&+$5f&+$5f 5($'7(03nff(55 f 09 ,) $%609f*7f *272 5(7851 HQG 68%5287,1( )5(4*(7)5(4f FRPPRQ LEJORE LEVWDLEHUULEFQW &20021 ,17(*(5r ,17(*(5 )5(4 f&20021 *5283 n LQWHJHU 6(“$' LQWHJHUr FPGfUGfZUWf FKDUDFWHUr EQDPHEGQDPH &+$5$&7(5r 7(03 n&20021 *5283 n L LFKDUfnf L2 LFKDUf2ff 6(1$' ,%),1'n6(1&25( nf FDOO LEUG 6(“$'5'f

PAGE 194

QRQ R Q R ,&f UGOfL ,&f UGOf,&OfrLfLO ,&f UGOf,&OfrL,&frLOfL ,&f UGOf,&OfrL,&frLO,&frL ,&f UGfL ,&f UGf,&frLfLO ,&f UGf,&frL,&frLOfL ,&f UGf,&frL,&frLO,&frL ,&f UGfL ,&f UGf,&frLfLO ,&OOf UGf,&frL,&frLOfL ,&f UGf,&frL,&frLO,&OOfrL ,&f UGfL ,&f UGf,&frLfLO ,&f UGf,&frL,&frLOfL ,&f UGf,&frL,&frLO,&frL ,),&f(4f 7+(1 :5,7(7(03n$fnf &+$5,&ff&+$5,&ff &&+$5,&ff &+$5,&ff&+$5,&ff &&+$5,&ff&+$5,&ff 5($'7(03nfn(55 f )5(4 (1',) ,),&f(4L2f 7+(1 :5,7(7(03f$fnf &+$5,&ff&+$5,&ff &&+$5,&ff&+$5,&ff&+$5,&ff &&+$5,&ff&+$5,&ff 5($'7(03n,fn(55 f )5(4 (1',) 5(7851 HQG 68%5287,1( 6(&6772f ,17(*(5r 772'$< '$< rr &$// 7/22.7f ,)7*7'$<257/7f *272 7 77 ,)7/72f 7+(1 7 7'$< 7 7'$< (1',) 5(7851 (1' 68%5287,1( 7(03*(77f FRPPRQ LEJORE LEVWDLEHUULEFQW &20021 ,17(*(5r

PAGE 195

Q R R n&20021 *5283 n LQWHJHUr FPGfUGf ZUWf FKDUDFWHUr EQDPHEGQDPH7(03 n&20021 *5283 n .$$' ,%),1'n.$ nf &$// ,%5' .$$'5'f 5'f, 5'f,rOf, 5'f,rO,r,f, 5'frrr 5'f 5'fOrOfOO 5'fOrO,r,fO 5'f,r,,r,,r, :5,7(7(03n$fnfFKDUfFKDU,f &FKDUfFKDUf &&+$5,f&+$5,f &&+$5,f&+$5f 5($'7(03n)fn(55 f 5 5 $/3+$ '(/7$ $/'(/ $/3+$r'(/7$ 5& 55 3$57 $/3+$$/'(/ 3$57 $/'(/$/3+$frr 3$57 r5&r$/'(/ 3$57 r$/'(/ 7(03 3$5764573$573$57f 7 7(033$57 5(7851 HQG F F F VXEURXWLQH HUURU FRPPRQ LEJORE LEVWD LEHUU LEHQW ZULWH rf LEVWDLEHUULEHQW IRUPDW n (UURUnLLLf UHWXUQ HQG 68%5287,1( 35(6*(73f FRPPRQ LEJORE LEVWDLEHUULEHQW &20021 ,17(*(5r n&20021 *5283 f LQWHJHU %0$' LQWHJHUr FPGfUGfZUWf FKDUDFWHUr EQDPHEGQDPH FKDUDFWHUr 3UHV

PAGE 196

Q R n&20021 *5283 n %0$' ,%),1'n%0 nf FDOO LEUG %0$'5'f UGOfL UGfLOrLfLO UGfLOrLLrLOfL UGfOrLLrLOLrL UGfL UGfLrLfLO UGfLrLLrLOfL L UGfLrLLrLOLrL :5,7(3UHVn$fff &+$5,f&+$5f&+$5f &&+$5,Of&+$5f&+$5f&+$5f 5($'35(6 n)f n (55 f 3 5(7851 HQG 68%5287,1( 6(16(7,*f FRPPRQ LEJORE LEVWDLEHUULEFQW ,17(*(5r ',0(16,21 f n&20021 *5283 n LQWHJHU &5 LQWHJHU FPGfUGfZUWf,3f &+$5$&7(5r &+f &+$5$&7(5r *$,1 FKDUDFWHUr EQDPHEGQDPH &+$5$&7(5r 7(03 n&20021 *5283 n GDWD &5 ,3f rr ,3f rr ,3f rr ,3f rr rr rr / /. ,%),1' n/2&.,1 nf &+f n
PAGE 197

:5,7(7(03n$Ofnf &+$5,ff 5($'7(03nff(55 f ,67$786 / / ,) //7f *272 ,) -(4$1',67$786(42f *272 ,) -(4$1'.(42$1',67$786(42f *272 ,) ,67$786(4f 7+(1 ,* ,* ,) ,**7f ,* ,) ,*/7Of ,* *272 (/6( ,) ,67$786(42f 7+(1 ,* ,* ,) ,**7f ,* ,) ,*/7Of ,* *272 (1' ,) :5,7(*$,1nfnf ,* &+f n*f 5($'*$,1n$Ofnf &+f 5($'*$,1n;$fnf &+f :57f ,&+$5&+ff,&+$5&+ffr,3f &,&+$5&+ffr,3f &$// ,%:57/.:57f *272 5(7851 HQG 352*5$0 9,.$1 ODUJH ,17(5)$&( 72 68%5287,1( 7,0( 1675f &+$5$&7(5r 675 >1($55()(5(1&(@ ,17(*(5r 1 >9$/8(@ (1' 5($/r ';f'
PAGE 198

23(1)LOH nF'$7$%8)n67$786 n2/'nf 5($'rf 92/7,* &$// *$,16(7,*f &/26(f 92/7( 27=(52fr 92/7 ZULWH rrf YROWH &/26(f '7(03 nF'$7$n 23(1)LOH nF1$0(180n67$786 n2/'nf 5($'f '1$0( )250$7,;$f &/26( f 5($''1$0(n ;f f f 180 180 180 23(1)LOH fF1$0(180n67$786 n1(:nf :5,7(f '7(03180 :5,7(rf '7(03180 )250$7,;$f &/26( f 23(1),/( '1$0(67$786 n1(:nf 23(1)LOH nF3HDNEXIn67$786 n2/'nf 5($'rf )7, 7, 7, &$// .(,7+$7)f 7) 7) ) 64577)7,fr) &/26(f ,;7 F ,;7 1$ ,;7 ,'(/< r1$ Tr rrrrrrrrrrrrrrrrrrrrrrr N r N rrrrrrrrrrrrrrrrrrrrrrrrrrrrrr r F 0 1$,;7,'(/<1$ & 86(' 21/< )25 &(17(5,1* Trrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr 0 & 86(' $)7(5 &(17(5,1* Trrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr ;0$; )/2$70f &$// 7,0(7675f 5($'7675rfr(55 f +5 5($'7675n;,fn(55 f 0,1 5($'7675n;,fn(55 f 6(& 722 +5r0,1r6(& &$// 7(036(792/7f '2 5($'6 6) ):,'7+ 63) ):,'7+ ,1'(; 32,176 0 ,67$57 &$// +36)63)6:((3f

PAGE 199

$0$; 2 ;0,1 2 '2 ,' ,'(/< &$// .(,7+09f &217,18( '2 ,67$5732,176,1'(; $9* '2 -$ 1$ &$// .(,7+09f .2817 .2817 ,).2817(4r1$f 7+(1 ,* ,* &$// *$,16(7,*f (1',) ,).2817*7r1$f 7+(1 .2817 *272 (1',) ,)09*7509/7f *272 .2817 $9* $9*)/2$709f &217,18( $03 $9*)/2$71$f ';-f )/2$7-f '<-f $03 ,)$03*7$0$;f 7+(1 $0$; $03 -0$; (1',) ,)$03/7;0,1f ;0,1 $03 &217,18( ,),1'(;(4f 7+(1 13/27 O (/6( 13/27 (1',) <0$; $0$; 4r rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr W r r r F &$// 3/270';'<;0$;;0,1<0$;<0,113/27f & 21/< 86(' )25 &(17(5,1* 4rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr ,),1'(;(4Of 7+(1 ,1'(; ,1'(; 7(03 6) 6) 63) 63) 7(03 ,67$57 32,176 32,176 O -)0$; -0$; $9*$ $0$; &$// .(,7+$7,f ,)7,/7f *272

PAGE 200

R R &$// 35(69,f *272 (/6( &$// .(,7+$7)f &$// 35(69)f -50$; -0$; )2 )/2$7-)0$;-50$;f &$// 0$;):,'7+)20f ) )2 $03 $0$;$9*$f ,)-)0$;(425-)0$;(4025-50$;(425 &-50$;(40f 7+(1 23(1)LOH nF3HDNEXIf67$786 r 2/'nf 5($'rf )7, &/26(f 7, 7, 7) 7) ) 64577)7,fr) *272 (1',) &$// 7,0(7675f 5($'7675 n f n (55 f +5 5($'7675f;,fn(55 f 0,1 5($'7675f;,fn(55 f 6(& 7 +5rr0,16(& 7 77 ,)7/722f 7+(1 7 7r 7 +5rr0,16(& 7 77 (1',) ,)$03/7f 7+(1 ,* ,* ,),**(f &$// *$,16(7,*f (1',) ,)$03*7f 7+(1 ,* ,* ,),*/(f &$// *$,16(7,*f (1',) 5$7,2 )64577,7)rff :5,7(rf 7,7)9,9)$03)7,* 4r rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr r & & '2 ,) :5,7(rrf F &217,18( :5,7(rf 63)6) F F )250$7,;n)5(48(1&< f);)f :5,7(rrf F F F ,) -50$; *7 -)0$; f 0 0 ,) -50$; /7 -)0$; f 0 0 ') 86(' 21/< )25 &(17(5,1*

PAGE 201

QRQ 4rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr :5,7(rf7,7)9,9)0)7-50$;-)0$;5$7,2 )250$7,;))))f '7 7)7, ') )/2$7-50$;-)0$;f)/2$70fr:,'7+ & 86(' 21/< $)7(5 &(17(5,1* &rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr ) )')r (1',) &217,18( ,)$%6'7f*7f 5($'6 5($'6 ,)$%6'7f*7f 5($'6 5($'6 ,)$%6'7f*7f 5($'6 5($'6 ,)$%6'7f*7 f 5($'6 5($'6 ,)$%6'7f*7f 5($'6 5($'6 ,)$%6'7f*7f 5($'6 5($'6 ,)$%6'7f*7f 5($'6 5($'6 ,)$%6'7f/7$1'5($'6*7f 5($'6 5($'6 ,)$%6'7f/7$1'5($'6*7f 5($'6 5($'6 ,)$%6'7f/7$1'5($'6*7f 5($'6 5($'6 ,)$%6'7f/7$1'5($'6*7f 5($'6 5($'6 ,)$%6'7f/7$1'5($'6*7f 5($'6 5($'6 ,)$%6'7f/7$1'5($'6*7f 5($'6 5($'6 ,)$%6'7f/7 $1'5($'6*7f 5($'6 5($'6 :5,7(rf 92/7 )250$7 ,;n6(7 32,17 92/7$*( n)f F &DOO VFDQ),*f 23(1)LOH nF3HDNEXIn67$786 n1(:nf :5,7(rf )7) &/26(f LI WIOWf JRWR ,)7)*7f *272 ,)92/7*(92/7(f 7+(1 92/7 92/7673 23(1)LOH nF92/7%8)n67$786 n1(:nf :5,7(rf 92/7 &/26(f *272 (/6( ,)92/7*792/7(f 7+(1 92/7 92/7( 23(1)LOH nF92/7%8)n67$786 n1(:ff :5,7(rf 92/7 &/26(f *272 (1',) *272 (1',) 23(1)LOH nF6723581n67$786 n1(:nf &/26(f &/26(f

PAGE 202

QRRQ &DOO VFDQ),*f &DOO WHPSVHW22f 6723 (1' F F F 6XEURXWLQH VFDQ),*f &+$5$&7(5r 6:((3 &KDUDFWHUr QDPH &KDUDFWHUr IQDPH 6ZHHS nn F 6:((3 nn 23(1),/( n,1'(;%8)n67$786 n2/'nf 5($'rf ,1'(; &/26(f ,1'(; ,1'(; 23(1),/( n,1'(;%8)n67$786 n1(:nf :5,7(rf ,1'(; &/26(f :5,7(1$0(nfnf ,1'(; :ULWH IQDPHf$O $ $fff n)1$0(n'$7n 2SHQ ILOH IQDPHVWDWXV nQHZnf :,'7+ ,) ,*/7f ,* ,* &$// *$,16(7,*f :5,7(rf ,*) F WKH IROORZLQJ OLQH LV IRU DYRLGLQJ WRR PXFK VLJQDO F DW WKH HQG RI WKH UXQ I I 6) ):,'7+ 63) ):,'7+ ;0$; 63) ;0,1 6) <0$; ( <0,1 <0$; ,;7 & ,;7 1$ ;7 13 1$,;7 6/23( :,'7+)/2$713f &$// +36)63)6:((3f ,'(/< '2 ,' ,'(/< &$// .(,7+09f &217,18( '2 13 $9* '2 -$ 1$ &$// .(,7+09f ,)09*7509/7f *272 $9* $9*)/2$709f

PAGE 203

Q Q Q Q R &217,18( $03 $9*)/2$71$f '< $03 '; 6/23(r)/2$7-f6) :5,7(rf ';'< ,)$03*7<0$;f 7+(1 <0$; $03 ,0$; (1',) ,) $03/7<0,1f <0,1 $03 &217,18( & &$// 3/2713';'<;0$;;0,1<0$;<0,1f &$// .(,7+$7f &$// 35(63f :5,7(rf 73 ,* ,* &/26(f UHWXUQ (QG 352*5$0 *(763* /$5*( ,03/,&,7 5($/r $+=f 5($/r 3$f7$f)5f&$'6457 5($/r %f%f%f7f 5($/r &f&f&f%'f1 ,17(*(5 7,0 =(52 21( 7:2 (3. % &21 .(/ 23(1 ),/( r$5*21'$767$786 n2/'nf 5($'r(1' f 7,73L3$03)5,f7,0,1 ,)3O(422'2f *272 ,)3(4'f *272 ,)7O(422'2f *272 ,)7(4'f *272 7$,f 77f7:2 3$,f 3O3fr&217:2 *272 &/26(f 0 23(1 ),/( n79&'$7n67$786 n2/'nf '2 5($'rf 7,f%,f%O,f%,f

PAGE 204

&%',f&,f&,f&,f &217,18( &/26( f 23(1 ),/( n1$7*$6 67$786 f2/'nf 23(1 ),/( n63(('*$6n67$786 n1(:nf F RSHQ ILOH nDJDVnVWDWXV nQHZnf F RSHQ ILOH fEJDVfVWDWXV nQHZnf F RSHQ ILOH fFJDVnVWDWXV fQHZnf 1 =(52 5($'r(1' f 7O73L3$03)5(47,0,1 F ZULWH rf DPSLQ F ZULWH rf IUHT 1 121( ,)3L(4253(4f *272 7(03 77f7:2 3 33f7:2 '7 77 '3 33 ,)7$f/77(03f *272 ,)7$0f*77(03f *272 ,)$03*7'f ,1 ,1 ,)$03/7'f ,1 ,1 F ZULWH rf IUHT &$// $03*(7$03,1f F ZULWH rf IUHT '2 0 ,6723 ,) 7$,f/77(03f *272 ,6723 O &217,18( ,) ,6723(4Of *272 )$5 )5,f)5,ff7$,f7$,ffr &7(037$ff)5f 3$5 3$,f3$,ff7$,f7$,Offr &7(037$,f§ff3$f 7(03 7(03.(/ 76 7(03(3. &$// 4$1'%6%6%6%6'&6&6 &&6767%%O%%'&&&f & :5,7(rrf n 4$1'n %9 %6r% &9 &6r%r% &$// 9(592/3$597(03%9&9f & :5,7(rrf n 9(592/n 96 9% &$// 63(('&$57(03%6%6%6%6'&6&6&696f & :5,7(rrf f 63(('n &* )5(4r&$5)$5 7(03 7(03.(/ :5,7(rf 1'3'737(03$03)5(4&*7,0&$5)$5 *272 FORVHf

PAGE 205

FORVHf FORVHf &/26(f &/26(f (1' 68%5287,1( 4$1'%6%6%6%6' &&6&6&6767%%%%'&&&f ,03/,&,7 5($/r $+=f 5($/r %Of%OOf%f7f&f&f&f%'f 5($/r 0 '2 ,) 76*77,f$1'76/77,ff 7+(1 0 767,ff7,f7,Off %6 %f0r%,f%ff %6O %Of0r%,f%Off %6 %f0r%,f%,f§ff %6' %',f§f0r%',f%', ff &6 &f0r&,f&ff &6 &f0r&O,f&Off &6 &f0r&,f&,ff 5(7851 (1',) &217,18( :5,7(rrfn767$5 287 2) 5$1*(n 5(7851 (1' & 68%5287,1( 9(592/397%&9f ,03/,&,7 5($/r $+=f 5 72/ O2' & ,1387 3 ,1 $70 & ,1387 7 ,1 .(/9,1 9 5r73 91 5r73r' %9 &999f 7(67 991 ,)7(67*7f 7+(1 7(67 ' 7(67 (/6( 7(67 7(67 (1',) 9 91 ,)7(67*772/f *272 5(7851 (1' & 68%5287,1( 63(('&7%6%6%6%6'&6&6&696f ,03/,&,7 5($/r $+=f 5($/r 0'6457 0 5

PAGE 206

*$0$ ''%696 &%6'r%6'&6&6'r&6f9696 *$0$ *$0$'''r%6%6f96 &'r&6&6f2'29696f & *$0$r5r70r2' 'r%6962'2r&69696f & '6457&f 5(7851 (1' 352*5$0 727'$7 ,03/,&,7 5($/r $+=f 23(1),/( n63(('*$6n67$786 n2/'nf 23(1),/( n727'$7287n67$786 n1(:nf 5($'r(1' f ;;;37$03)(;3&,7$5*&$5*) &$// 63(('=735+2=;O;;;7+(2&;;f 7 7'2 :5,7(rrf ,7 7+(2& 7+(2&' :5,7(f 37$03)$5*),7$5*&(;3&7+(2& )250$7,;)))))f *272 &/26( f &/26( f (1' &20021 *5283 LQWHJHU 81/ 817 *7/ 6'& 33& *(7 7&7 LQWHJHU //2 '&/ 338 63( 63' 33( 33' LQWHJHU (55 7,02 (1' 654, 546 &03/ /2. LQWHJHU 5(0 &,& $71 7$&6 /$&6 '7$6 '&$6 LQWHJHU ('95 (&,& (12/ ($'5 ($5* (6$& ($%2 LQWHJHU (1(% (2,3 (&$3 ()62 (%86 (67% (654 LQWHJHU %,1 ;(26 5(26 LQWHJHU 7121( 7O2XV 7XV 7O22XV 7XV LQWHJHU 7LPV 7PV 7,2PV 7PV 7,22PV LQWHJHU 7PV 7OV 7V 7,2V 7V LQWHJHU 7O22V 7V 7O222V LQWHJHU 6/) &20021 *5283 FKDUDFWHUr IOQDPH LQWHJHUr EGGYPYFQWPDVN LQWHJHUr VSUSSU GDWD 81/817*7/6'&33& GDWD *(77&7//2'&/338 GDWD 63(63'33(33' GDWD 7,02(1'654, GDWD 546&03//2.5(0&,&

PAGE 207

GDWD $717$&6/$&6'7$6'&$6 GDWD ('952(&,&(12/($'5($5* GDWD (6$&($%2(1(%(2,3 (&$3 GDWD ()62(%86(67%(654 GDWD %,1;(265(26 GDWD 7121(7XV7XV7XV7XV GDWD 7OPV7PV7O2PV7PV7OPV GDWD 7PV7OVOO7V7OV72V GDWD 7OV7V7V GDWD 6/)

PAGE 208

5()(5(1&(6 $WNLQV 3 : 3K\VLFDO &KHPLVWU\ UG HG : + )UHHPDQ DQG &RPSDQ\ 1HZ
PAGE 209

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t 6RQV FRPSDQ\ 1HZ
PAGE 210

/RUG 5D\OHLJK 3URF /RQGRQ 0DWK 6RF LY )HUULV +RUDFH $FRXVW 6RF $P 5R\DO 0DWKHPDWLFDO 6RFLHW\ 7DEOHV 9ROXPH %HVVHO )XQFWLRQV 3DUW ,,, =HURV DQG $VVRFLDWHG 9DOXHV &DPEULGJH 8QLYHUVLW\ 3UHVV /RQGRQ 0HKO -DPHV % DQG 0ROGRYHU 5 &KHP 3K\V 0HKO -DPHV % $FRXVW 6RF $P (ZLQJ 0 % 0F*ODVKDQ 0 / DQG 7UXVOHU 3 0 0ROHFXODU 3K\VLFV &DPSEHOO $FRXVWLFD GH OD 7RXU &DJQDLG $QQDOHV GH &KLPLH $QGUHZV 7 3KLO 7UDQV $QGUHZV 7 3URF 5R\ 6RF /RQGRQf .XHQHQ 3 &RPPXQ 3K\V /DE 8QLY /HLGHQ 1R % .XHQHQ 3 .RQLQNO $NDG :HWHQVFKDSS $PVWHUGDP 9HUVO 9DQ 9HUTUDG 6FRWW 5 / DQG 3 + YDQ .RQ\QHQEXUJ 'LVFXVV )DUDGD\ 6RF 6FRWW 5 / %HU %XQVHQTHV 3K\V &KHP :LFKWHUOH ,YDQ )OXLG 3KDVH (TXLOLEULD +LUVFKIHOGHU -RVHSK &XUWLVV &KDUOHV ) DQG 5 %\URQ %LUG 0ROHFXODU 7KHRU\ RI *DVHV DQG /LTXLGV -RKQ :LOH\ t 6RQV 1HZ
PAGE 211

,SDWLHII 9 1 DQG 6 0RQURH ,QG (QJ &KHP $QDO (Gf %DJDWVNLL 0 9RURQHOn $ 9 DQG 9 *XDN ([SWO 7KHRUHW 3KYV 8665f (QJOLVK WUDQVODWLRQ 6RYLHW 3KYV -(73 9RURQHOn $ 9 &KDVKNLQ
PAGE 212

'H +HHQ %XOO $FDG 5R\ %LOILTXH 9RQ :HVHQGRQFN 9HUKDQGHO 'HXW 3K\VLN *HV .HHVRP : + 9HUKDQGHO $NDG $PVW &RPP /HLGHQ 1R 4XLQQ (OWRQ / DQG -RQHV &KDUOHV / &DUERQ GLR[LGH 5HLQKROG 3XEOLVKLQJ &RUSRUDWLRQ 1HZ
PAGE 213

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

PAGE 214

*XJQRQL 5 (OGULGJH : 2ND\ 9 & DQG 7 /HH $ &K ( -RXUQDO 1DJDKRPD .XQLR .RQLVKL +LWRVKL +RVKLQR 'DLVXNH DQG 0LWVXKR +LUDWD FKHP (QJ -DSDQf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/RQGRQ $ 5XEHPDQQ 3URF 5RY 6RF .D\ : % DQG % %ULFH ,QG (QT &KHP LQ :KLWHZD\ 6 DQG 6 0DVRQ &DQ &KHP

PAGE 215

3DOPHU + % &KHP 3KYV 6FKPLGW ( DQG : 7KRPDV )RUVFK *HELHWH ,QT % .XGFKDGNHU $ 3 $ODQL + DQG % =ZROLQVNL &KHP 5HY 6OLZLQVNL 3 = 3K\V &KHP )UDQNIXUWf %XODYLQ 3 $ 2VWDQHYLFK
PAGE 216

/H\.RR 0 DQG 0 6 *UHHQ 3KYV 5HY $ 0F*LOO & )DVW )RXULHU 7UDQVIRUPHG $FRXVWLF 5HVRQDQFH :LWK 6RQLF 7UDQVIRUP 3K' 'LVVHUWDWLRQ 7KH 8QLYHUVLW\ RI )ORULGD 6NRRJ 'RXJODV $ DQG :HVW 'RQDOG 0 )XQGDPHQWDOV RI $QDO\WLFDO &KHPLVWU\ WK HGLWLRQ 6DXQGHUV &ROOHJH 3XEOLVKLQJ 1HZ
PAGE 217

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nV RXWVWDQGLQJ WHDFKLQJ DVVLVWDQW DZDUG

PAGE 218

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

PAGE 219

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

PAGE 220

81,9(56,7< 2) )/25,'$


Table 5.10. Chronological collection of critical point parameters of ethane.
Temperature
K
Pressure
Density
Author(s) *
Year
atm
psia
kg/m3
305.2
-
-
212
Wiebe et. al. (112)
1930
306.7
-
-
-
Sage et. al. (113)
1937
305.4
48.5
712.1
-
Kay (114)
1938
305.41
(0.01)
48.2
(0.02)
707.8
203 (2)
Beattie et. al. (115)
1939
305.25
48.6
713.6
213
Lu et. al. (116)
1941
305.11
48.12
707.2
-
Kay and Brice (117)
1953
305.31
-
-
-
Whiteway and Mason (118)
1953
305.47
-
-
-
Palmer (119)
1954
305.33
48.180
708.05
-
Schmidt and Thomas (120)
1954
305.42
-
-
-
Khazanova et. al. (99)
1967
305.43
-
-
-
Kudchadker et. al. (121)
1968
305.33
-
-
-
Sliwinski (122)
1969
305.35
-
-
206.2
Bulavin et. al. (123)
1971
305.35
48.112
707.04
205.8
Miniovich and Sorina
(124)
1971
305.340
48.2
(0.1)
707.8
203.9
Khazanova and Sominskaya
(125)
1971
305.360
-
-
-
Voronel et. al. (126)
1972
305.3633
-
-
-
Berestov et. al. (127)
1973
103


5
was made using a gas system for which the critical point
parameters are well-established. That gas was pure carbon
dioxide, which has the richest history of prior critical
behavior studies. Choosing this gas is beneficial not only
because of its well-known values of critical point parameters
but also for its recognized importance in the fuel industry as
a near-critical-state solvent in enhanced oil recovery and as
a supercritical extractant. Next, the complexity of the gas
system was increased by selecting a binary component gas
mixture (C02-C2H6) for study. The lessons learned in working
with these first two systems were applied to a follow-up study
of the other component of the mixture, ethane, to further
confirm the reliability of the technique. Chapter two will
briefly review some relevant equations of acoustics and
explain how to identify individual resonance frequencies,
which are the key parameters leading to values of the speed of
sound. Chapter three will review some aspects of the phase
behavior of both pure and mixed gases. The critical point
determination will subsequently be discussed, starting from a
background on conventional approaches to the newly proposed
approach, which utilizes speed of sound as a critical point
indicator. Chapter four explains comprehensively the
experimental procedures used in this work. Chapter five
contains results including discussions and observations. The
final chapter is a conclusion.


8
0 AiP(cos0) + Brese) .
(2.8)
where Pfm(cos0) = The associated Legendre polynomial of the
first kind of degree B and order m
Q^cosO) = The associated Legendre polynomial of the
second kind of degree B and order m.
Since Qim(cos0) becomes infinite at 0 = 0 & n, it does not
apply to this physical situation(28). Consequently,
0 AJP(cos0)
where P"(cos0) (sin0)m/2
(2.9)
d"^ (cos0)
d(cos0)m
if m = 0, P^(cos0) is the Legendre polynomial.
Finally, the general solution of the angular part can be
written as follows:
(2.10)
Y (0, These are the spherical harmonics of the first kind.
Radial Part
Equation 2.5a is:
r2 + 2r + (k2r2 M2)f 0. (2.]
dr2 dr
Using the above result for the angular part:
M2 = B(B+1) ; B = 0,1,2,... equation (2.11) becomes
(2.12)


o n
166
READ(30,*) F,TI
CLOSE(30)
TI=TI+273.15
TF=TF+273.15
F=SQRT(TF/TI)*F
GOTO 500
ENDIF
1003 CALL TIME(10,TSTR)
READ(TSTR,'(I2)',ERR=1003) HR
READ(TSTR,'(3X,I2)',ERR=1003) MIN
READ(TSTR,'(6X,I2),ERR=1003) SEC
T = HR*3600+60*MIN+SEC
T=T-T00
IF(T.LT.O.O) THEN
T00=T00-24*3600
T = HR*3600+60*MIN+SEC
T=T-T00
ENDIF
IF(AMP.LT.7000) THEN
IG=IG-1
IF(IG.GE.7) CALL GAINSET(IG)
ENDIF
IF(AMP.GT.19000) THEN
IG=IG+1
IF(IG.LE.24) CALL GAINSET(IG)
ENDIF
RATIO=F/SQRT((TI+TF+273.15*2.0)/2.0)
WRITE(20,*) TI,TF,VI,VF,AMP,F,T,IG
C DO 18 IF=1,19
WRITE(*,*)
c
18
CONTINUE
c
WRITE(*,390) SPF,SF
c
390
FORMAT(IX,'FREQUENCY=
',F10.3,48X,F10
c
WRITE(*,*)
c
IF (JRMAX .GT. JFMAX )
M=M-1
c
IF (JRMAX .LT. JFMAX )
M=M+1
c
O
II
o

o
C USED ONLY FOR CENTERING
0***********************************************************
WRITE(*,400)TI,TF,VI,VF,M,F,T,JRMAX,JFMAX,RATIO
400 FORMAT(IX,2F8.3,2F8.1,16,F10.3,18,214,F9.3)
C***********************************************************
DT=TF-TI
DF=FLOAT(JRMAX-JFMAX)/FLOAT(M)*WIDTH
C USED ONLY AFTER CENTERING
F=F+DF*1.5
ENDIF
30 CONTINUE
IF(ABS(DT).GT.0.02) READS=READS+1
IF(ABS(DT).GT.0.03) READS=READS+1


88
H
0
<
u
3
0
0

u
5 10 15 20 25 30 35
Temperature, Celsius
Figure 5.20. Pressure and temperature relationship of a set of
several isochores of the carbon dioxide-ethane mixture.


196
30. Lord Rayleigh, Proc. London Math. Soc., 1, iv 93, 1872.
31. Ferris Horace G., J. Acoust. Soc. Am., 24, 1, 57, 1952.
32. Royal Mathematical Society Tables, Volume 7: Bessel
Functions, Part III: Zeros and Associated Values,
Cambridge University Press, London, 1960.
33. Mehl James B., and Moldover R., J. Chem. Phys., 74, 7,
4062, 1981.
34. Mehl James B., J. Acoust. Soc. Am.. 71, 5, 1109, 1982.
35. Ewing M. B., McGlashan M. L., and Trusler J. P. M. ,
Molecular Physics. 60, 3, 681-690, 1987.
36. Campbell I. D., Acoustica, 5, 145, 1955.
37. de la Tour Cagnaid, Annales de Chimie, 3, 21, 127,
1822; 22, 410, 1823.
38. Andrews T., Phil. Trans.. 159, 575, 1869.
39. Andrews T., Proc. Roy. Soc. (London). 18, 42, 1869.
40. Kuenen J. P., Commun. Phys. Lab. Univ. Leiden. No. 4B,
7, 1892.
41. Kuenen J. P., Koninkl. Akad. Wetenschapp. Amsterdam.
Versl. Van Verqrad.. 1, 15, 1892.
42. Scott R. L., and P. H. van Konynenburg, Discuss.
Faraday Soc., 49, 87, 1970.
43. Scott R. L., Ber. Bunsenqes. Phys. Chem.. 76, 296,
1972.
44. Wichterle Ivan, Fluid Phase Equilibria. 1, 161, 1977.
45. Hirschfelder Joseph 0., Curtiss Charles F., and R.
Byron Bird, Molecular Theory of Gases and Liquids, John
Wiley & Sons, New York, 1954.
46. Michels A., Blaisse B., and C. Michels, Proc. Roy. Soc.
(London). A160, 358, 1937.
47. Wentorf R. H. and C. A. Boyd, J. Chem. Phys.. 24, 607,
1956.
Cailletet L., and E. Mathias, Comptes Rendus De
L'Academie Des Sciences. 104, 563, 1887.
48.


90
(0.256) confirming the explanation. Figures 5.18 and 5.19 show
the relationships near the critical density between
temperature of phase change and speed of sound and the
pressure respectively. As expected the sonic speed (figure
5.18) shows a pronounced change in the critical region while
the pressure (figure 5.19) shows a smooth passage from the
one-phase to the two-phase region. Figure 5.20 shows a
composite plot of pressure and temperature data representing
23 separate isochoric runs. Those loadings with supercritical
densities show phase change by upward breaks at the
corresponding bubble points. Those loadings at subcritical
densities show downward breaks at the corresponding dew
points. Loading at the critical density leads to no break at
all. Table 5.5 compares dew point pressures observed in this
work with those available in the literature. Clearly, the
agreement is quite good. The critical temperature for this
mixture obtained in this work was 293.1340.007 K. The only
available literature experimental value for this composition
was found to be 293.072 K (9S). These values differ by only
0.021 percent.
Ethane
The graphs shown in figures 5.21 to 5.24 represent the
isochoric behavior of the system charged with pure ethane at
near critical density. Figure 5.21 shows a typical
relationship between a resonance frequency and the system


non o n o
180
IC(1) = rd(l)/i24
IC(2) = (rd(l)-IC(l)*i24)/il6
IC(3) = (rd(l)-IC(l)*i24-IC(2)*il6)/i8
IC(4) = rd(l)-IC(l)*i24-IC(2)*il6-IC(3)*i8
IC(5) = rd(2)/i24
IC(6) = (rd(2)-IC(5)*i24)/il6
IC(7) = (rd(2)-IC(5)*i24-IC(6)*il6)/i8
IC(8) = rd(2)-IC(5)*i24-IC(6)*il6-IC(7)*i8
IC(9) = rd(3)/i24
IC(10) = (rd(3)-IC(9)*i24)/il6
IC(ll) = (rd(3)-IC(9)*i24-IC(10)*il6)/i8
IC(12) = rd(3)-IC(9)*i24-IC(10)*il6-IC(ll)*i8
IC(13) = rd(4)/i24
IC(14) = (rd(4)-IC(13)*i24)/il6
IC(15) = (rd(4)-IC(13)*i24-IC(14)*il6)/i8
IC(16) = rd(4)-IC(13)*i24-IC(14)*il6-IC(15)*i8
IF(IC(14).EQ.3) THEN
WRITE(TEMP,'(7A1)') CHAR(IC(1)),CHAR(IC(8)),
CCHAR(IC(7)), CHAR(IC(5)),CHAR(IC(12)),
CCHAR(IC(11)),CHAR(IC(10))
READ(TEMP,'(17)',ERR=1006) FREQ
ENDIF
IF(IC(14).EQ.iO) THEN
WRITE(TEMP,(7A1)') CHAR(IC(1)),CHAR(IC(8)),
CCHAR(IC(7)),CHAR(IC(6)),CHAR(IC(5)),
CCHAR(IC(12)),CHAR(IC(10))
READ(TEMP,'(I7)',ERR=1006) FREQ
ENDIF
RETURN
end
SUBROUTINE SECS(T,TO)
INTEGER*4 T,TO,DAY
DAY=3600*24*1000
100 CALL TLOOK(T)
IF(T.GT.DAY.OR.T.LT.0) GOTO 100
T=T-T0
IF(T.LT.O) THEN
T0=T0-DAY
T=T+DAY
ENDIF
RETURN
END
SUBROUTINE TEMPGET(T)
common /ibglob/ ibsta,iberr,ibcnt
COMMON 11,18,116,124
INTEGER*4 11,18,116,124


Figure 4.12. Flow chart of experimental scheme and data processing. The broken-line
boxes represent the experimental systems of interest. The thick-line boxes represent the
computer programs.
vo


APPENDIX E
COMPUTER PROGRAMS
Since most of the programs are lengthy only programs
VIKING and MAX3 are showed in full. The others are listed only
main programs. Most of these programs were written in fortran
77 by Dr. McGill. The remaining by the author.
PROGRAM VIKING
$large
INTERFACE TO SUBROUTINE TIME (N,STR)
CHARACTER*10 STR [NEAR,REFERENCE]
INTEGER*2 N [VALUE]
END
REAL*4 DX(2000),DY(2000),XMAX,XMIN,YMAX,YMIN
CHARACTER*1 KOL
CHARACTER*2 SWEEP
CHARACTER*10 DNAME
CHARACTER*7 DTEMP
CHARACTER*10 TSTR
INTEGER*2 HR,MIN,SEC
INTEGER READS,T,TC,TO,TOO,POINTS
SWEEP='30'
WIDTH =8.0
READS = 124
KOUNT = 0
STP = 0.0025
CALL KEITHA(TZERO)
OPEN(60,File='c:DATA.BUF',STATUS='OLD')
READ(60,*) VOLT,IG
CALL GAINSET(IG)
CLOSE(60)
0* ************** ****************************** * * * * * *
C VOLTE = (25.0-TZERO)*0.01 + VOLT
C TO BE USED IN NORMAL SCAN
Q* ********************************************************* *
VOLTE = 0.0175
CLOSE(10)
DTEMP='c:DATA.1
OPEN(70,File='c:NAME.NUM',STATUS='OLD)
READ(70,301) DNAME
301 FORMAT(IX,A10)
CLOSE (70)
READ(DNAME,'(7 X,13) ') NUM
NUM=NUM+1
163


o o
143
WRITE(*,500) Al,A2,A3,A4,A5,A6,TC,ALPHA,BETA,DELTA
500 FORMAT (IX,7F8.2,3F4.2)
CALL RESIDUAL(Al,A2,A3,A4,A5,A6,TC,ALPHA,BETA
C ,DELTA,C,T,L,ST,W)
SSRSD(14)=ST
SSRSD(11)=SSRSD(14)
DO 20 J=1,10
B(J,11)=B(J,14)
20 CONTINUE
RETURN
END
Program II. Speed of Sound Calculations
C234567
IMPLICIT REAL*8 (A-H,0-Z)
REAL*8 DABS,DSQRT
DIMENSION C(4000),T(4000),A(12)
OPEN (60,FILE='COOR.OUT',STATUS=1 OLD')
DO 70 1=1,10
READ (60,*) A(I)
C WRITE(*,*) A(I)
70 CONTINUE
CLOSE(60)
A1=A(1)
A2=A(2)
A3=A(3)
A4=A(4)
A5=A(5)
A6=A(6)
TC=A(7)
ALPHA=A(8)
BETA=A(9)
DELTA=A(10)
OPEN (10,FILE='DATA.RAW',STATUS='OLD')
OPEN (40,FILE='TC.DAT',STATUS='NEW')
L=1
30 READ(10,*,END=20) T(L)
c WRITE(*,*) T(L)
L=L+1
GOTO 30
20 CLOSE(10)
L=L-1
GAS=8.3144
C W=0.040441
W=0.03007
CON=GAS*TC/W
DO 50 1=1,L
TR=DABS((T(I)-TC)/TC)
C WRITE(*,*) 'TR=',TR
C(I)=DSQRT(CON)*(A1*TR* *BETA+A2 *TR**(BETA
C+DELTA)+A3*TR**(1.0-ALPHA+BETA)+A4*TR+A5*TR**2.0


93
temperature. Figure 5.22 is similar to figure 5.21 but with
pressure plotted instead of temperature. These two figures
have the same characteristics implying a linear relationship
between temperature and pressure as expected for this
condition. This is even more evident in figure 5.23. Figure
Table 5.6. Sixth order polynomial fit statistics of GRAPHER
program applied to experimental data points in figure 5.25.
Degree
Polynomial
Coefficients
Sums of Squares
of Residuals
0
2.15029E+007
0.952964
1
-166034
0.909732
2
533.701
0.0191329
3
-0.914178
0.00389193
4
0.000880105
0.0019246
5
-4.51543E-007
0.00192265
6
9.64558E-011
0.00192195
Total points = 23
Points in fit interval = 23
5.24 shows plot of the speed of sound as a function of
temperature. It is similar to figure 5.21 since the speed of
sound is directly proportional to the resonance frequency.
Figure 5.25 shows the relationship between the initial
charging pressure and the coexistence (liquid-vapor
equilibrium) temperature. Experimental data points were fitted
to a polynomial of the sixth order. Its coefficients, along
with some statistical values generated from the data by GRAPHER


154
beam. The insert has eight useable 90 knife edges. A second
insert was used as the bearing pad. It fits in a seat machined
in a support member and securely fastened to the gauge frame.
When not in use the beam may be elevated to lift the knife
edge off of the bearing pad to prolong the life of the knife
edge by supporting it on the hardened pins.
The cylinder was machined by drilling and reaming and
finished by ballizing. The piston was machined on a precision
lathe and polished with rouge paper. The diameter was measured
with a Pratt and Whitney supermicrometer. Ten readings taken
at different positions around the piston gave the average
diameter 0.37383 inch with a standard deviation of 0.00001
inch.
Because in operation the contact ball moves on a circular
arc and the piston is constrained to move only vertically,
there is some relative motion between them resulting in some
unwanted sliding friction. To avoid this problem a small
thrust bearing was used between the lower surface of the
piston and a hardened chip to convert the sliding motion to
the less restricted rolling motion.
The gauge is mounted on a rigid beam which is securely
bolted to the laboratory floor. The mounting plate is
adjustable to allow the bearing pad to be precisely leveled.
The vertical component of upward force exerted by the contact
ball on the gauge piston is equal to
F. = g cos 0 E n.m.
3 11
(D.l)