Acoustic determination of phase boundaries and critical points of gases

MISSING IMAGE

Material Information

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

Subjects

Subjects / Keywords:
Matter -- Properties   ( lcsh )
Thermochemistry   ( lcsh )
Ultrasonic waves   ( lcsh )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Genre:
bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

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

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 026891018
oclc - 25541142
System ID:
AA00017679:00001


This item is only available as 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
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