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Apparatus and method for combined acoustic resonance spectroscopy-density determinations

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Apparatus and method for combined acoustic resonance spectroscopy-density determinations
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Tatro, Daniel S., 1968-
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English
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vii, 113 leaves : ill. ; 29 cm.

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Subjects / Keywords:
Acoustic resonance ( jstor )
Argon ( jstor )
Average linear density ( jstor )
Buoys ( jstor )
Carbon dioxide ( jstor )
Density ( jstor )
Hydrometers ( jstor )
Signals ( jstor )
Strain gauges ( jstor )
Supersonic transport ( jstor )
Chemical apparatus ( lcsh )
Chemistry thesis, Ph. D ( lcsh )
Dissertations, Academic -- Chemistry -- UF ( lcsh )
Matter -- Properties ( lcsh )
Physical measurements ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1996.
Bibliography:
Includes bibliographical references (leaves 110-112).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Daniel S. Tatro.

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University of Florida
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University of Florida
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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.
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APPARATUS AND METHOD FOR COMBINED ACOUSTIC RESONANCE
SPECTROSCOPY-DENSITY DETERMINATIONS




















By

DANIEL S. TATRO

















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

1996




APPARATUS AND METHOD FOR COMBINED ACOUSTIC RESONANCE
SPECTROSCOPY-DENSITY DETERMINATIONS
By
DANIEL S. TATRO
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
1996


To my wife, Mary, my daughters, Danielle and Gabrielle,
and the rest of my family,
Evelyn, Laura, Gary and Christine,
who love me without end and support me without question.


ACKNOWLEDGEMENTS
The author wishes to thank all of those who have
contributed to the education he has received at the University
of Florida. His transformation from a student to a competent
researcher and instructor could not have been possible without
the support and direction of the faculty of the University of
Florida, Department of Chemistry. In particular, he extends
his greatest thanks to his committee chairman, Samuel Colgate.
The author considers it a privilege to have worked with such
an outstanding chemist and engineer.
The author wishes to thank the other members of the
Colgate research group, Johnny Evans, Evan House, Vu Thieu,
and Karl Zachary, for their continued interactions. In
particular, the author wishes to thank Troy Halvorsen, for
without his help, the development of the densimeter would not
have been possible.
Technical support of this research project was
extraordinary. The author wishes to thank the staff of both
the electronic and machine shops and would like to
acknowledge, in particular, machinist Joe Shalosky, whose
craftsmanship is second to none.
iii


Additional thanks are extended to the author's family and
friends. It has been their belief in the author that has been
his source of inspiration. Specifically, special gratitude is
extended to his wife, Mary, whose backing and support have
been unwavering. In addition, the author wishes to thank John
Magrino, Nial McGloughlin, Mitch Morrall, and Gary Tatro whose
friendships have supplied continuous moral support.
Finally, the author wishes to thank the University of
Florida, Department of Chemistry and the University of Florida
Division of Sponsored Research for financial support.
IV


TABLE OF CONTENTS
ACKNOWLEDGMENTS iii
ABSTRACT vii
CHAPTERS
1 INTRODUCTION 1
2 DENSITY MEASUREMENT 7
Fundamental Theory 7
Theory of Design 8
3 SPEED OF SOUND MEASUREMENT 12
Spherical Cavity Resonance Equation
Development 12
Resonance Frequency Identification 15
4 DENSIMETER AND SPHERICAL RESONATOR RESEARCH
AND DEVELOPMENT 21
Densimeter 22
Spherical Buoys 22
Deflection Beam and Semiconductor
Strain Gages 24
Coiled Spring and LVDT Sensor 27
Densimeter Body 29
Buoyancy Assembly and Sphere Lifting
Mechanism 31
Top Flange Assembly 37
Spherical Resonator 38
Resonator Cavity 38
Transducer Assembly 40
Final Assembly 43
5 EXPERIMENTAL 47
Gases 47
Interfacing 47
Measurement Hardware 48
Pressure and Temperature 48
v


Density 51
Speed of Sound 53
Experimental Data Collection Procedure ... 55
6 RESULTS 59
Eigenvalue Calibration Data 59
Argon 59
Carbon Dioxide 69
Experimental Data 69
Equation of State-Data and Results .... 74
Uncertainties in Measured and
Calculated Values 78
7 CONCLUSION 79
APPENDICES
A SPHERICAL BUOY MASS AND VOLUME
CALIBRATION 83
B CIRCUIT DIAGRAM FOR STRAIN GAGE SIGNAL
CONDITIONER 88
C COMPUTER PROGRAMS 90
D TEMPERATURE CONTROL 98
E REDLICH-KWONG EQUATION OF STATE AND
THERMODYNAMIC EQUATIONS 104
REFERENCES 110
BIOGRAPHICAL SKETCH 113
vi


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
APPARATUS AND METHOD FOR COMBINED ACOUSTIC RESONANCE
SPECTROSCOPY-DENSITY DETERMINATIONS
By
Daniel S. Tatro
May 1996
Chairperson: Samuel 0. Colgate
Major Department: Chemistry
A novel instrument was developed for simultaneous
measurement of temperature(T), pressure(P), speed of sound(c),
and density(p) of fluids over wide ranges of these variables.
The speed of sound measurements were performed using a
spherical acoustic resonator designed and fabricated in this
laboratory. In order to measure density, an innovative
densimeter based on application of Archimedes principle was
developed. Confirmatory experiments were performed on a near
ideal gas (argon) and a nonideal fluid (carbon dioxide) where
P, T, c, and p data were collected. The report describes the
research and development of the entire instrument as well as
the demonstration of its performance.
vii


CHAPTER 1
INTRODUCTION
History demonstrates that in order to attain a
comprehensive understanding of physical science, the
successful practice of conducting physical property
measurements must be established and practiced.1 Experimental
measurement is the foundation upon which most scientific
interpretation of material behavior is made. Often, with the
knowledge of a small but sufficient number of a system's
properties over a suitably wide range of variables, much of
the behavior of a system can be specified in terms of known
theoretical and empirical relationships among these and its
other properties. In certain cases, pertinent equations of
state and theories can be used to make correlations and
approximations. Some scientists predict that the new age of
computational chemistry is at hand, and the need for
experimental work will become obsolete. While a careful look
at the complexity of real systems of actual importance to
applied science reveals this optimistic prediction to be
exceedingly premature, even if it were correct, the need would
still persist for continual experimental confirmation of
theoretical predictions.
I


2
Generally, equations of state are based on empirical or
semiempirical models.z They tend to work well for systems
included in the supported data bank, which is to say they can
be used to fit experimental data over the range of available
observations. Intolerable deviation of calculated properties
from actual system properties commonly occurs when attempts
are made to extrapolate information to regions far removed
from the experimental data. Also, equations of state often
fail when extended to other systems of different structures or
compositions for which no experimental data are available(i.e
multicomponent or multiphase systems).
To understand the behavior of complex systems and to
develop more reliable equations of state, it is necessary to
make physical property measurements. The measurement of some
physical properties (e.g. pressure, P, and temperature, T) can
be made with acceptable accuracy and precision using
relatively simple measurement apparati or techniques. For such
measurements scientists are able to purchase state-of-the-art
devices suitable for use over a wide range of conditions.
Unfortunately, not all necessary state variables can be
measured with such ease and precision. Two important
properties, in particular density and the speed of sound, have
no simple, wide range, robust measurement methods. The
research presented here addresses this problem through the
development of a densimeter/spherical resonator apparatus
(DSRA).


3
Currently, speed of sound measurement by means of
acoustic resonance methods is performed routinely by only a
few research groups in the world, our laboratory being one of
these. The speed of sound measurements along with
simultaneously collected pressure and temperature data have
been used to characterize and identify many thermophysical
properties.3'1' The acoustic resonance technique developed to
date has significantly advanced some areas in the art of
thermodynamic property measurement,5 but it fails to address
one very important state variable, namely that of fluid
density.
Many methods have been explored for possible use as
versatile, accurate means for measuring fluid density, and it
is appropriate now to review how the best established
techniques, pycnometry, the Burnett method, and the vibrating
tube densimetry operate and to compare their strengths and
their specific limitations.
Pycnometry6, using constant volume cells, leads to
density directly from the measurement of the mass of a
confined fluid. The problem of accurately weighing a cell
filled with fluid at high pressure and at high or low
temperature is not minor and in fact has been a forbidding
deterrent to the widespread use of this technique. Here,
careful volume calibration is essential, and the volume change
as a function of pressure and temperature must also be
accounted for by either measurement or prediction.


4
The Burnett method1 for gases entails a series of
controlled expansions to collect a set of pressure-volume data
pairs at constant temperature. The expansions lead ultimately
to a sufficiently low pressure that an established equation of
state may be used reliably to calculate the density. This
gives the system mass, from which the initial density may be
determined. Drawbacks of this method include its limitation to
gas density measurements only, its dependence on volume
calibrations, and perturbations due to adsorption of gas on
the cell walls. Both of the above techniques are sampling
methods. The sample extracted for density determination is
ultimately discarded, thereby introducing problems such as
possible separation of components in the case of mixtures, and
limitation of the number of measurements which can be made for
a given system. Another disadvantage is that the time to
obtain a single measurement is very long compared to that
typically required for measurement of other variables.
Vibrating tube density meters provide an indirect
measurement of density.8 The working principle is based on the
dynamics of a mass/spring combination. As the mass, which
includes that of the fluid inside the tube, changes, the
frequency of vibration will change in concert. The vibrational
frequency of a U-tube rigidly supported at its ends is
measured when empty, when filled with calibrating fluid of
known density and when filled with the fluid under study. From
these frequencies the unknown fluid density can be determined.


5
Although appealing for its simplicity, use of the vibrating
tube densimeter is impractical for thermodynamic measurements
due to its extensive calibration requirements. Frequent
calibration over pressure and temperature ranges is required
as well as calibration over the density range to be studied.
There is also a hysteresis effect of the spring constant which
further limits the instrument's reliability.
In contrast to the above techniques, there is a class of
density measurement devices that offers both simplicity in
theory and, in some situations, experimental technique. These
devices are based on the buoyancy force of fluids on submerged
solid bodies and derive from ideas first developed centuries
ago by Archimedes. This technique has often been put to use as
a practical means of fluid density measurement, especially in
liquids at ambient conditions9, but only recently has it been
extended to dense gases.10 Later chapters will detail the
design and use of a versatile new density meter based on this
principle.
In view of this background, the focus of this work is
concentrated in two areas, 1) development of a high
performance fluid density measuring device (densimeter) for
accurate experimental density determination under ambient and
non-ambient conditions and 2) incorporation of this densimeter
in-line with an acoustic spectrometer. This coupling will
yield, for the first time, the capability of acquiring in a
single apparatus pressure, temperature, speed of sound and


6
density information simultaneously. Knowledge of these four
properties permits others to be inferred.
The decision to pursue the development of an Archimedes
type densimeter was made after careful consideration of
current state of the art density measurement techniques and
how they could be adapted to our specific experimental
requirements. Initially, efforts were directed at building a
U-tube vibrating density meter. Its straightforward design and
ease of construction along with our laboratory's experience in
precision frequency measurements made this instrument
appealing at first. A prototype was built and functioned well
in a bench top capacity, but this device was abandoned after
it was apparent that the extensive calibration requirements
and unpredictable hysteresis effects could not be
circumvented. The research then shifted to the development of
a density meter based on Archimedes' principle. Again, the
simple theory was appealing, and, unlike the U-tube density
meter, a measurement technique could be devised such that the
limitations of extensive calibration could be eliminated. It
was revealed initially that hysteresis effects would be a
problem, but could be eliminated with appropriate modification
of the density meter.


CHAPTER 2
DENSITY MEASUREMENT
In certain cases, when developing a method for physical
property measurement, the underlying theory is simple and
straightforward. Often the challenge is the design,
development and implementation of an appropriate experimental
technique. The density measurement method outlined in this
chapter is illustrative of these circumstances. The theory is
based on well established and familiar principles, while its
implementation required exploration of innovative approaches
to solving challenging experimental problems.
Fundamental Theory
First, consider a rigid body immersed in a static,
isothermal fluid of density, p. The predominant forces acting
on the body are those associated with Archimedes' principle
and Newton's second law of motion. Specifically, Archimedes'
principle states that an object immersed in a fluid is buoyed
up by a force equal to the weight of the fluid that is
displaced. Hence, the buoyancy force, FB is defined as
7


8
F = P (2-1)
where p is the density of the fluid, V is the displaced fluid
volume, g is the local acceleration due to gravity, and n is
a unit vector directed along the line from the center of mass
of the body to the center of mass of the earth.
In addition, by Newton's second law of motion, the
gravitational force exerted on the rigid body is proportional
to its mass times the acceleration due to gravity
F = m g (2-2)
where FN is the Newtonian gravitational force and m is the
mass of the submerged body.
Thus, the resultant of the two forces is,
Ftot = ( mg p Vg ) (2-3)
or in terms of magnitudes alone, since there is no ambiguity
about direction in this one dimensional case ,
Ftot = ( m P V) g. (2-4)
Equation 2-4 is the working equation from which density is
obtained and the basis for the operational development of the
densimeter design.
Theory of Design
Consider two spherical objects having volumes V2 and V2
and having masses mx and m2 suspended in an isothermal, static


9
fluid of density p. Recalling equation 2-4, the net force
acting on each sphere may be represented independently by the
following equations,
E
(tot, spherel)
II
1
P
?!
g
(2-5)
F(tot, sphere2)
= sr -
P
9
(2-6)
Now, assume that the net force on each sphere can be
measured independently. To accomplish this in practice, the
spheres are suspended by a spring as shown in figure 2-1.
Figure 2-1. Schematic diagram of fundamental densimeter
design


10
Assuming for the moment that the spring obeys Hooke's
law, the total force acting to stretch or compress the spring
is proportional to the deflection from its relaxed position
given as
F = -k X (2-7)
Where x is the deflection distance and k is the spring
proportionality constant.
If the spheres can be independently placed on or removed
from the suspension rack, four separate loadings of the spring
are possible. The resultant force for each of the four
loadings is as follows;
I. Both spheres off:
F=-kx0 (2-8)
Where x0 is the length of the spring loaded only by the
suspension rack.
II. Sphere 1 on; sphere 2 off:
Fi =~k *1 = ( m1 p Vi ) g k x0 (2-9)
III. Sphere 1 off; sphere 2 on:
F2 = ~k x2 = { mj p V2 ) g k x0 (2-10)
IV. Sphere 1 on; sphere 2 on:
F3 =-k x3 = ( m3 p V3 ) g k x0 (2-11)
where m3 = m3 + m2 and V, = V: + V,.
Equations 2-8 through 2-11 form a coupled system from
which, knowing Xi-x,,, x2-x0, x3-x0, m1( m2, V, and V2, the fluid


11
density may be determined. It is not necessary to know the
acceleration of gravity g or the spring constant k. These
features which the method useful, as the elasticity of any
practical spring material will vary with temperature, T, and
pressure, P, but should remain constant over any measurement
taken at constant T and P. In the experiments for which this
device is intended to be used, T and P will vary widely but
not during any single density measurement. Also, g varies with
position, but this is fixed through a measurement, and the
magnitude of g cancels out in the calculation of density.
Other methods previously used to make fluid density
measurements under non-ambient conditions require calibration
to account for the non-negligible temperature and pressure
discriminations. They are further compromised by the
irreproducibility of the physical properties of the
densimeters due to hysteresis effects. The present work seeks
to overcome these unwanted limitations by developing an
absolute instrument, free from the requirement of calibrations
and operating with a highly elastic force sensor.


CHAPTER 3
SPEED OF SOUND MEASUREMENT
Observation of acoustic resonance in fluid-filled
cavities has been used as an effective, practical means to
determine the speed of sound in gases and liquids. In this
laboratory and others, spherical cavity resonators have been
used to acquire sonic speed data of both high precision and
accuracy.11 The following development of the theory for
spherical cavity acoustic resonance and the subsequent
calculation of the speed of sound from resonance measurements
follows that of Reed12, McGill13 and Dejsupa14 based on the
earlier work of Rayleigh15 and Ferris.16
Spherical Cavity Resonance Equation Development
The general wave equation that describes the propagation
of pressure waves in a lossless fluid at rest, contained
within a rigid walled spherical cavity, is,
d2l¥(v)
at2
V2'P(^)
(3-1)
where T is the velocity potential and c is the speed of sound.
12


13
Assuming a time separable velocity potential such that T =
Toe1"1, equation 3-1 becomes
(3-2)
to = 0
where a is the frequency.
The solution to this wave equation is separable in terms
of the angular and radial components. Using spherical
coordinates, and invoking the usual boundary conditions, the
angular part of the solution is the known set of spherical
harmonics, and the radial part is the spherical Bessel
function of the first kind. The overall solution is,11
*1
(3-3)
where to = time-independent velocity potential
r,0,(p = spherical polar coordinates
ji = Bessel function of the first kind(radial part)
Yj1" = Spherical harmonics of the first kind (angular part)
n,l,m = integers.
The rigid spherical cavity imposes a boundary condition
that the radial component of fluid velocity must be zero at
the cavity wall. This generates an equation for the speed of
sound in terms of the normal frequencies of vibration, given
as,


14
C
fl,n 2*r
5 l.n
(3-4)
where f1|lt is the resonance frequency, r is the radius of the
spherical cavity, and the eigenvalue, ?l n, is the nth root of
the first derivative with respect to r of the 1th spherical
Bessel function of the first kind, also referred to as the
resonance frequency eigenvalue. It should be noted that
because of the oscillatory nature of the Bessel function, for
each value of 1 there exits an infinite number of roots
(eigenvalues), ?lf. The smallest 72 frequency eigenvalues,
calculated by Ferris, are given in table 3-1. These have been
confirmed by a computer program written here to evaluate ¡jlin
generally. While the values of higher order eigenvalues are
mathematically interesting, they are of little practical use
in the present experiments, as the associated resonance
frequencies lie beyond the upper range of the acquisition
hardware used in our measurements.
The practical advantage of using spherical resonators
derives from a special characteristic of the purely radial
modes of vibration.12 These, being functions of r, are pure
breathing modes and thus are not subject to viscous damping
caused by tangential gas motion with respect to the cavity
wall. In addition, the radial modes of vibration are known to
be less sensitive to perturbations caused by imperfections in
the spherical nature of the cavity.11 Therefore, we identify
and utilize frequencies associated with pure radial modes


15
(1=0) to facilitate the most accurate measurements of sonic
speed.
Resonance Frequency Identification
The scheme used here for assigning mode identities to
resonance peaks in an acoustic resonance spectrum was
developed by Dejsupa.14 In order to calculate the speed of
sound, a resonance frequency flin and its corresponding
eigenvalue must be identified. Rearranging equation 3-4
and solving for frequency yields
fl.n
c h,a
2 71 r
(3-5)
Knowing the cavity radius, r, and choosing an estimated value
of the sonic speed, c, one may calculate an approximate
frequency for each mode. One such frequency is then used as a
starting point for the identification of frequency values in
an experimental acoustic resonance spectrum. A typical
experimental frequency spectrum is shown in figure 3-1, where
at each acoustic resonance the frequency amplitude shows a
sharp increase in magnitude with relation to the baseline.
This experimental frequency spectrum is scanned in the region
of the calculated frequency and a tentative mode assignment is
made to that peak which most nearly matches the trial value.
The assignment is tested for accuracy by predicting the


16
Figure 3-1. Typical experimental frequency spectrum of
argon at 1359.3 psia and 278.15 K.


17
relative location of other modes and looking at the spectrum
for corresponding signal peaks.
Once a frequency has been identified, that is a
frequency/root pair has been determined, it serves as a
reference value used to identify other frequencies and their
corresponding eigenvalues. For example, if a reference
frequency is assigned the mode designation l,n,
(3-6)
the frequency of mode 1' ,n' should be
(3-7)
The experimental frequency spectrum is then scanned at this
new frequency value to confirm its existence and the actual
experimental frequency value is recorded.
It should be mentioned that this method serves as a self
check for the correct identification of the first reference
frequency. Incorrect identification of the reference frequency
(i. e incorrectly matching the frequency with the correct
eigenvalue) will predict other frequencies which do not match
the experimental spectrum.
To this point the development has been based on an
idealized cavity resonator. In reality there are perturbations
that account for deviations of vibrational frequencies from


18
the idealized model.18 The corrections may be modeled, but
applying the models can be computational demanding and can
require detailed knowledge of the system and its properties
which are unavailable. As mentioned before, using the purely
radial modes of vibration for identification and calculation
reduces some of the error due to higher order perturbations.
To reduce this even further, without attempting to quantify
all of the perturbations, a relative measurement technique has
been utilized for which those errors which derive from
departure of the cavity walls from perfect sphericity and
rigidity are eliminated by cancellation. A reference gas with
well known physical properties or equation of state is
required.14
Argon has been exhaustively studied and was used as the
reference system for this experiment. Its behavior over the
range of pressures and temperatures involved is well
characterized by a truncated virial equation of state13, from
which the speed of sound may be accurately calculated.
Equation 3-4 may be rewritten as
where vk is defined as
v
k
(2nz)
il-O,n-k
(3-9)


19
and k is an index that corresponds to the radial modes of
vibration only. Equation 3-8 is used to generate eigenvalues,
vk, corrected for the specific apparatus.
The experimental speeds of sound of the fluid under
investigation can be calculated by generalizing equation 3-8;
(3-10)
C* = V* fk
where ck is the speed of sound of the fluid at fk and vt .
Assuming that ck is not a function of frequency, then the
speed of sound ca was taken as the average of the speeds of
sound calculated from the radial mode frequencies obtained
experimentally with the spherical resonator apparatus. This
can be expressed by the following equation,
n
(3-11)
C.
Jc=l
a
n


endnote #19 diffusion stuff15
enote 2020 21 22 2! 2< 25 26 21 28
29 30 31 32 33 39 35


20
Table 3-1. The values of the frequency roots in ascending
order of magnitude with the purely radial modes in bold print.
Index
1
n
Index
?i,
i
n
1
2.08158
1
i
37
16.3604
14
i
2
3.34209
2
1
38
16.6094
4
4
3
4.49341
0
1
39
16.9776
10
2
4
4.54108
3
1
40
17.0431
2
5
5
5.64670
4
1
41
17.1176
7
3
6
5.94036
1
2
42
17.2207
0
5
7
6.75643
5
1
43
17.4079
15
1
8
7.28990
2
2
44
17.9473
5
4
9
7.72523
0
2
45
18.1276
ii
2
10
7.85109
6
i
46
18.3536
8
3
11
8.58367
3
2
47
18.4527
16
1
12
8.93489
7
1
48
18.4682
3
5
13
9.20586
1
3
49
18.7428
1
6
14
9.84043
4
2
50
19.2628
6
4
15
10.0102
8
1
51
19.2704
12
2
16
10.6140
2
3
52
19.4964
17
1
17
10.9042
0
3
53
19.5819
9
3
18
11.0703
5
2
54
19.8625
4
5
19
11.0791
9
1
55
20.2219
2
6
20
11.9729
3
3
56
20.3714
0
6
21
12.1428
10
1
57
20.4065
13
2
22
12.2794
6
2
58
20.5379
18
1
23
13.4046
1
4
59
20.5596
7
4
24
13.2024
11
1
60
20.7960
10
3
25
13.2956
4
3
61
21.2312
5
5
26
13.4712
7
2
62
21.5372
14
2
27
13.8463
2
4
63
21.5779
19
1
28
14.0663
0
4
64
21.6667
3
6
29
14.2580
12
1
65
21.8401
8
4
30
14.5906
5
3
66
21.8997
1
7
31
14.6513
8
2
67
22.0000
ii
3
32
15.2446
3
4
68
22.5781
6
5
33
15.3108
13
1
69
22.6165
20
1
34
15.5793
1
5
70
22.6625
15
2
35
15.8193
9
2
71
23.0829
4
6
36
15.8633
6
3
72
23.1067
9
4


CHAPTER 4
DENSIMETER AND SPHERICAL RESONATOR RESEARCH AND DEVELOPMENT
The instrumentation required to accomplish the
experimental measurements presented here consists principally
of an innovative densimeter installed in a flow loop with a
high performance spherical resonator. Creation of the
densimeter involved a complete research and development
program, including design, fabrication, and the necessary
modifications dictated by the typically erratic path along the
learning curve leading to the new technology. In contrast to
the extensive development effort that was required to produce
the densimeter, the spherical acoustic resonator was generated
using proven principles previously developed and commonly
practiced in this laboratory.
Both the densimeter and the spherical resonator were
designed and constructed to withstand high pressures (up to
3000 psia) and operate over a broad temperature range (-80C
to 150C) These subassemblies were constructed primarily from
304 stainless steel (304SS) and brass components, each sized
to meet the designed performance standards.
21


22
Densimeter
From theoretical considerations previously described, a
means is required to load and unload the spheres from the
force sensing device, and the net force must be suitably
transduced into a proportional electrical signal in order for
practical measurements to be made. Details of the lifting
mechanism depend on the actual loadings themselves, i.e the
spheres; therefore these were dimensioned first.
Spherical Buovs
The hollow spherical sinkers were designed such that
their volumes would be as closely matched as possible (V =V¡) ,
but their masses would be different by a factor of two
(2m1=m2) The spheres were fabricated from four identical
hemispheres, each with 0.6250 inch OD (outside diameter)
machined from type 304 stainless steel. These were hollowed to
a 0.437 inch ID (inside diameter), yielding the necessary wall
thickness to sustain the maximum design pressure of 3000 psia
without collapse or significant dimensional change. To achieve
the desired mass ratio, a copper bead with a mass equal to
that of two of the hemi-spheres was made. Pairs of hemi
spheres were then Heli-Arc welded together forming two
outwardly identical spheres; one, however, contained the
copper bead, shown in figure 4-1. Table 4-1 lists the final
sphere masses and their volumes determined by pycnometry.1
See appendix A for pycnometer design.


23
Table 4-1. Masses and volumes of the spheres.
Sphere
Mass (g)
Volume (cc)
1
6.9821
1.9361
0.0001
0.0002
2
13.9658
1.9365
0.0001
0.0002
Spherical Buoy
Outer Shell
Weld
Copper Bead
Figure 4-1. Spherical buoy with enclosed copper bead.


24
Deflection Beam and Semiconductor Strain Gages
Having established the total mass and the dimensions of
the spheres, a deflection and sensing device was designed and
constructed. Miniature semiconductor strain gages were
purchased from Entran Devices Inc., model ESU-025; see table
4-2 for specifications. By fixing the strain gages on a thin
cantilevered beam from which the sinkers are suspended,
deflections due to the loadings could be detected and
correlated with the strain gage signals.
A deflection beam was cut from a piece of 0.007 inch
thick 302 stainless steel sheet metal, 0.150 inch wide by
0.620 inch long. The above dimensions yielded a deflection of
0.080 inch when a load of 25 grams was placed on the beam.
This corresponds to the manufacturer's optimum operating
strain range for the gages.
The gages were fixed to the beam per the manufacturer's
instructions using the application kit (ES-TSKIT-1) supplied
by Entran Devices, Inc. and the recommended polymeric epoxy,
(M-Bond 610) supplied by M-Line Accessories Measurements
Group, Inc. Four gages were applied to the beam, two on each
side. These became the arms of a bridge circuit that was used
in conjunction with an electronic strain gage conditioning
component (model 1B31AN) manufactured by Analog Devices. An
output signal device was built that detected and conditioned
the strain gage signal.2
See appendix B for circuit diagram.


25
A wire rack was fabricated to hang from the free end of
the beam and support the spherical sinkers. For the initial
trial tests, the sinkers were manually loaded and unloaded
from the rack. Degassed distilled water and n-hexane were used
as test buoyant media. The initial results yielded relative
errors of 5% to 10% assumed to be due mainly to air drafts and
temperature fluctuations which characterized the crude bench
tests. It was believed that these would diminish considerably
under the more stable conditions to be used in the planned
experiments.
Unfortunately, a substantial portion of the error proved
to derive from unforeseen design flaws which necessitated
fundamental changes. First, due to their miniature size, the
strain gages were not very robust. The delicate electrical
leads were easily damaged or broken and could not be
reattached. Second, nonlinear hysteresis and signal drift
effects were found in the beam assembly, as indicated by the
instability of the strain gage circuit output signal. The
drift effect is illustrated in figures 4-2 and 4-3 for two of
the weight loadings. It was not known whether these effects
were characteristic of the stainless steel shim or of the
epoxy resin used to apply the strain gages to the beam, and
the supply of semiconductor strain gages was exhausted before
the problem could be solved. At this point it was decided to
pursue a different and seemingly more manageable solution.


26


27
Although the above approach, using strain gages and a
flexible beam, was not utilized further in this research
project, plans to use this sensor have not been abandoned. We
have learned a good deal about how to proceed in this
direction. For future use, the following suggestions should be
considered: 1) Polycrystalline materials such as the stainless
steel alloy used in construction of the deflection beam are
unsuitable for use as spring elements and should be replaced
by more elastic materials such as fused quartz or single
crystal silicon, for example, to eliminate the hysteresis
effects seen in polycrystalline materials. 2) The beam
assembly should be redesigned so that the electrical
connections to the semi-conductor strain gages are protected
by encapsulating the fine gold wire leads in a layer of epoxy.
Coiled spring and LVDT sensor
Unable to bring the available strain gage deflection beam
into satisfactory practice within acceptable limits, attention
was shifted to a linear displacement measurement technology
already in successful use in the laboratory. The LVDT (Linear
Variable Differential Transformer) has been used as a linear
displacement sensor in acoustic resonance experiments for
measuring volume changes by transducing piston movement.19 That
experience suggested that an LVDT could be used to follow the
deflection of an elastic member at resolutions appropriate for
densimetry.


28
A high resolution LVDT was obtained from Trans-Tek Inc.
that could measure a total displacement of 0.100 inch; see
table 4-2 for further information. The LVDT consists of a
small weakly magnetic core (0.099 inch OD by 0.492 inch long)
that is used in conjunction with a separate induction coil
sensing device. An added appealing feature of employing this
technology was that there were no electrical connections
inside the pressure chamber and therefore no need for high
pressure electrical feedthroughs; also, the instrument can be
used in conductive fluids as well as non-electrolytes. And
finally, the fluid of interest would contact only the
noncorrosive materials within the densimeter.
The LVDT measures linear displacement along a single
axis, which dictated that the cantilevered deflection beam had
to be replaced with a device that exhibits a linear vertical
displacement. A coiled spring was selected as an obvious
choice. A commercial metal spring was selected for which the
total design displacement of 0.080 inch lay within the Hookian
region (linear displacement versus weight) and occurred when
a 25 gram total mass was attached. Initial tests of the metal
spring indicated that its reproducability was not within
acceptable limits. The metal spring exhibited a small but
devastating hysteresis effect, and it became obvious that
ordinary spring materials would be unsuitable for this service
and that one of extraordinary elasticity would be required.
The hysteresis is probably due to the irreversible changes
which occur under high strain at the grain boundaries of


29
polymorphic materials such as polycrystalline metals and
alloys. This suggests that single crystal or perhaps amorphous
spring elements would perform much better. Bench tests with
hand-coiled springs fabricated from fused quartz rod proved
this idea to be correct, at least for amorphous materials.
A quartz filament 0.06 inch OD was wound in a coiled
spring fashion to approximately 0.2 5 inch OD and 0.5 inch
length with the end loops turned up to make hooks used for
connection to other densimeter parts. In order to obtain the
desired deflection of 0.080 inch for a total mass load of 25
grams, the spring was placed in an etching solution of HF
(2.5M) and was removed at measured time intervals for testing.
This procedure gave excellent control over the spring
constant, and with it the desired deflection was easily
achieved.
Densimeter Body
The densimeter body assembly is shown in figure 4-4. The
bulk of the fluid chamber is a cylindrical tube machined from
304SS with a wall thickness of 0.125 inch and a length of
5.819 inch. A lower end cap 0.500 inch thick was machined to
slip fit the ID of the tube where it was then brazed into
place. A hole was drilled and tapped 1/8-NPT in the lower end
cap to allow for connection to other system components.


30


31
The densimeter body flange, 0.500 inch thick, with an OD
of 2.125 inch and an ID of 1.260 inch was drilled and tapped
to accept twelve 10-32 bolts on a 1.688 DBC (diameter bolt
circle). The flange was brazed onto the densimeter body
cylinder leaving an offset of 0.115 inch from the top of the
cylinder body, shown in figure 4-4. This offset was necessary,
as it was an integral part of the novel high pressure seal
used in construction of the densimeter apparatus. This seal
utilizes a copper o-ring gasket and two clamping flanges, one
being the flange on the densimeter body and the other a mating
flange located on the densimeter top assembly(detailed later
in this chapter) To create this seal reliably the cross-
sectional area of the copper 0-ring gasket was required to be
95% to 98% of the cross-sectional area of the triangular gland
created by the clamping of the top and bottom flanges, see
figure 4-5. The copper 0-ring gasket used for this seal was
made by cutting a piece of 12 guage copper wire to a length of
3.92 inch. The wire was pre-formed to the diameter of the
densimeter and the ends were heli-arc welded together,
creating a continuous ring.
Buoyancy Assembly and the Sphere Lifting Mechanism
The buoyancy assembly consists of the quartz spring, the
LVDT core, the sphere suspension rack and the connecting
hardware, shown in figure 4-6. The suspension rack was
constructed by brazing two 0.062 inch diameter stainless steel


32
Figure 4-5. Copper o-ring gasket and clamping flanges used
to generate a high pressure seal.


33
Brass Posts
Bottom view detail showing
Vshape sphere holders
Figure 4-6. Suspension rack assembly.


34
rods onto a brass disc. Short pieces of 0.031 inch diameter
stainless steel wire brazed to the rods at right angles and
bent in V-shapes formed supports for the spheres (see detail
in figure 4-6). The diameter of the brass disk was chosen such
that the entire rack could hang freely within the densimeter
body. The thickness of the brass disk was adjusted to yield
the appropriate mass needed to give the proper deflection when
used in conjunction with the spring and the spheres. The brass
disk was centrally drilled and tapped 5-40. Two brass rods
were constructed to connect the suspension rack to the quartz
spring. One rod was threaded 5-40 on one end to fit the brass
disk, the other end was threaded 1-72 to fit the LVDT core,
which was manufactured with tapped 1-72 holes on both ends.
The second rod was threaded 1-72 on one end to fit the LVDT
core, while a hole was drilled through a flattened section at
the other end to allow connection to the end-coil loop of the
quartz spring.
The lifting mechanism was designed and fabricated after
the vertical displacements due to loading and unloading the
spheres were determined precisely. A magnetically coupled cam
system and two vertical slides were constructed. These were
dimensioned to provide the proper clearance required for each
of the four loadings.
The lifter holder is a tubular brass piece machined with
an OD (1.049 inch) to slip fit inside the densimeter body.
The inside diameter of the lifter holder was large enough to


35
allow the suspension rack and the spheres to hang freely
without contacting the inside walls. Two slots were machined
axially 90 apart on the cylinder wall. The slots accommodate
the lifter slides. The lifter slides are shown in figure 4-7.
These have forks which extend into the fluid chamber to pick
up the densimeter spheres when actuated by the cam. The lifter
slides, machined from brass, are 3.990 inch long by 0.185 inch
wide by 0.145 inch thick. Small brass wheels were fitted into
slots on the bottom of the slides to facilitate relative
motion of the cam assembly. Two lifters were constructed from
1/32 inch diameter stainless steel wire and designed to lift
the spheres by cupping the spheres at three points. The
lifters where brazed to the lifter slides at the appropriate


36
position so that under operation, the spheres could be raised
completely off the suspension rack.
The magnetic cam assembly that produces the movement of
the lifters consists of a two-tiered cam, an internal
permanent magnet and an external coupled magnetic driver. The
cam with an OD of 1.030 inch has two steps joined by a 45
ramp, see figure 4-8. The total rise between steps was 0.250
inch. From a top view of the cam, figure 4-8, four quadrants
Q1-Q4 are apparent and when the two lifters are placed in the
quadrants accordingly, they account for the four separate rack
loadings. The arc length around each step is large enough to
accommodate both lifters simultaneously (Q1 to Q3, Q2 to Q4).
This accounts for two of the necessary measurements; both


37
spheres raised from the rack, and both spheres supported by
the rack. The arc length from one step to the other (Q1 to Q2,
Q3 to Q4) is large enough to accommodate one lifter on each
step. This accommodates the remaining two measurements; one
sphere raised, the other supported and vice versa.
The magnetic turner, shown in figure 4-9, is a brass
piece (1.000 inch OD) with a stem at each end. The cam fits
over the top stem and is fixed into position with a set-screw.
Brass spacers with ball bearings separate the cam from the
turner as well as the turner from the bottom of the
densimeter. It is the spacer/bearing combination that allows
movement to occur. A hole was drilled through the diameter of
the turner to accommodate a ferromagnet, which was held in
position with a set-screw.
An external drive ring, figure 4-9, was placed around the
outside of the densimeter body and loaded with two
ferromagnets. Together these generate a strong magnetic field
with which the magnetically loaded cam aligns. This
configuration produced an adequate turning force.
Tod Flange Assembly
The top flange assembly, shown in figure 4-10, contains
the spring and LVDT sensing element. The spring housing is
removable to allow for servicing or replacement. The LVDT
sensing element is held in a brass case, designed and
constructed to allow translation in the vertical direction,
which allows adjustment over density ranges.


38
Spherical Resonator
The spherical resonator used in this research project was
designed and built based on knowledge and experience
previously gained in the development of related instruments in
this laboratory. The spherical resonator assembly consists
principally of the spherical cavity and the acoustic
transducer mounting hardware.
Resonator cavity
The resonator cavity was designed with the following
considerations; 1) The cavity must be able to withstand high
pressure and extreme temperatures. 2) The cavity should have


39


40
a small volume yet still produce resonant frequencies when
filled with test fluids within our measurement capabilities
(typically acoustic frequencies less than 50KHz).
The spherical cavity consists of two identical 304SS
hemispheres each cut with a 1.000 inch radius. The inside
cavity was polished to obtain a smooth acoustically reflective
surface. The outside of the hemispheres was machined to
accommodate clamping flanges, shown in figure 4-11. A minimum
wall thickness of 0.250 inch was maintained to ensure the
cavity's ability to withstand the maximum design fluid
pressure of 3000 psia. A V-groove was cut on the 2.125 inch
diameter of each cavity block to house a diamond shaped copper
o-ring gasket (figure 4-11) that operates similarly to the
gaskets described previously. A 0.650 inch diameter hole was
cut at the apex of each of the cavity hemispheres to contain
the acoustic transducers and their mounting hardware.
Transducer Assembly
The transducer assembly, shown in figure 4-12, consists
of the transducers, the transducer mounting pieces and the
electrical feedthrough end-cap. The transducers were cut from
a piezo-electric speaker element, see table 4-2 for
specifications, to a diameter of 0.680 inch. The transducers
were held in position by using stainless steel spacers, which
in turn were lightly loaded by means of co-axial wave springs.
The spacers slip fit over the feedthrough in the end-cap where
the electrical connection was made.


41


42
for the spherical resonator.


43
The end-cap was constructed to serve as the electrical
connection to the transducers. The end-cap utilizes the copper
o-ring gasket technology which allows the transducers to be
removed and serviced without compromising the spherical cavity
seal. A ceramic-insulated copper electrical feedthrough was
obtained with a stainless steel weld preform. The endcap was
drilled through to accept the weld preform which was
subsequently welded into position.
Final Assembly
Figures 4-13 and 4-14 show assembled views of the
densimeter and spherical rensonator, respectively. The
instruments were connected using thick wall 1/4 inch OD
stainless steel tubing (304SS). The necessary valves to allow
for charging of the apparatus with experimental fluid and
later servicing of the instrument were incorporated into the
experimental setup and attached in-line using the stainless
steel tubing.


44
Top Flange
Assembly
Suspension
Rack
Densimeter
Body
Internal
Magnetic
Turner
Assembly
Figure 4-13. Assembled view of the densimeter


45
spherical resonator.


Table 4-2. Summary of Measurement hardware.
Item
Manufacturer
Model
number
Range of
operation
Resolution
Function
Semiconductor
strain gage
Entran
Devices,Inc.
ES-025
o-iooone
(=strain)
0.0001
volts
Measure
deflection of
beam.
Semiconductor
strain gage
output device
University of
Florida,
Electronics
Dept.a
N/A
N/A
Accepts
signal from
strain gage
circuit
bridge.
LVDT
Trans-Tek,
Inc.
240-0015
+0.050 inch.
+ 1 micron
Measure
deflection of
spring.
Piezo-electric
speaker element
Tandy Corp.
High
Efficiency
500Hz-50kHz
N/A
Generate and
receive
acoustic
signal
see appendix B for circuit diagram used
0>


CHAPTER 5
EXPERIMENTAL
A set of confirmatory experiments was performed to test
the operation of the densimeter and to verify that the entire
apparatus, including the spherical resonator, was functioning
as expected. For these purposes studies were carried out on
a nearly ideal gas and a nonideal gas at high densities.
Acoustic resonance spectra, density, pressure and temperature
were acquired for both argon and carbon dioxide at several
isotherms over a range of densities.
Gases
The argon (supplied by Bitec) used in this experiment had
a purity greater than 99.99 mole percent. The carbon dioxide
(supplied by Scott Specialty Gases) had a purity greater than
99.999 mole percent. No further analysis or treatment was
performed on the gases.
Interfacing
All components of the densimeter/spherical resonator
apparatus (DSRA) were linked to a micro-computer where data
were collected and processed. The computer contained a 486
internal processor operating at 33 megahertz with 16 megabytes
47


48
of RAM. Figure 5-1 shows the schematic setup of the electronic
hardware used in the experiments. The interfacing protocol
used for the connection of the electronic hardware to the
micro-computer was the standardized IEEE general purpose
interface bus (GPIB) including the 8-bit i/o card and the
accompanying software. The electronic equipment used in the
experiment contained the IEEE connection interface when
purchased.
Measurement Hardware
A schematic diagram of the instrumental setup is shown in
figure 5-2. Each physical property measurement required a
specific protocol for acquisition and subsequent processing of
the data. The following sections describe the measurement
procedures, including operation of the electronic hardware,
listed in table 5-1, and give details of the computer software
used in the experiments.
Pressure and Temperature
A Sensotec model TJE/743-03 strain gage pressure
transducer was used in conjunction with a Beckmann model 610
electronic readout device to acquire the pressure data. For
the pressure calibration of the Sensotec transducer, a Ruska
model 2465 standardized Dead Weight Pressure Gage was used.
The Ruska gage had a pressure range of 0.000 psia to 650.000


Figure 5-1. Schematic diagram of the electronic hardware setup.
VO


Temperature Controlled Chamber
Figure 5-2. Schematic diagram of the apparatus setup.
ui
o


51
psia and an accuracy of 0.001 psia. The pressure calibration
results are shown graphically in figure 5-3.
A four-wire platinum resistance temperature device RTD
was used with a Keithly model K-196 digital multimeter to
obtain resistance values that were later converted to
corresponding temperatures. The platinum RTD was calibrated
versus a standardized RTD (NIST traceable) obtained from HY-
CAL engineering.
The Beckmann readout device and the Keithly multimeter
were interfaced to the computer via the IEEE general purpose
interface bus (GPIB).
Density
The displacement information of the four separate
spherical buoy loadings was linearly translated to voltage by
the LVDT and its circuit. The voltage data were acquired with
an Iotech model ADC488/8SA analog-digital converter (ADC) that
was connected to the computer by means of an IEEE interface.
The advantage of using the ADC rather than acquiring the
voltage with a digital multimeter is the speed of the former
which gives it the ability to capture a relatively large
amount of data in a short period of time. For example, in this
experiment, each recorded single-loading voltage value was in
fact an average of 16,384 independent voltage measurements
acquired at a rate of 1 kHz. This resulted in an average
aquisition time of 16.4 seconds. Four voltage values,


52
a Experimental data Regression fit
Figure 5-3. Pressure gage calibration data using argon
gas at 298 K.


53
corresponding to the separate quartz spring loadings, were
collected and stored in the computer. These four values were
then used to calculate a single density datum point by means
of equations 2-8 through 2-11.
Speed of Sound
The speed of sound of the experimental fluid was
calculated using information obtained from a measurement
technique involving generation and acquisition of acoustic
resonance frequencies within the spherical cavity. The
resonance frequencies were produced by stimulating one
transducer with a waveform generated by a Hewlett Packard
HP3325B function generator. The waveform signal was a
continuous sine-wave ramp over a frequency range of 0-50KHZ
with a time base of 0.1638 seconds. The receiving transducer
was connected to a Stanford Research pre-amplifier model SR-
530 where the signal was conditioned with a band-pass filter
and amplified. The SR-530 was connected to the Iotech ADC. The
time domain resonance data were acquired at a rate of lOOKHz
and were processed using a fast fourier transform (FFT). A
typical time domain data plot and the subsequent FFT are shown
in figures 5-4 and 5-5, respectively. The resonance frequency
data were used as explained previously in chapter three to
calculate the speed of sound.


54
0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350
Time (s)
Figure 5-4. ADC signal of argon at 278.15 K and 1417.4
psia.
Figure 5-5. FFT of ADC signal of argon at 278.15 K and
1417.4 psia.


55
Experimental Data Collection Procedure
The entire apparatus was cleaned with multiple washings
of acetone. It was then baked at 150 C, and a vacuum was
applied to remove all contaminants from the system. The system
was then charged with the gas of interest to approximately
100 psia, allowed to eguilibrate, and a vacuum was again
applied. The charging procedure was carried out a minimum of
four times before the final loading of gas was injected into
the system.
The entire apparatus was contained within an insulated
temperature-controlled chamber that had been adapted with a
novel liquid nitrogen refrigeration control valve. This valve
along with a proprtionation-integration-differentiation (PID)
control algorithm provided temperature control to better than
+ 0.01 K.
Once the system was charged to the appropriate pressure
loading and was equilibrated, data were acquired using
software developed specifically for this experiment. The
magnetically driven cam assembly of the densimeter was turned
to the first displacement position. The corresponding LVDT
reading, along with the pressure and temperature, were
recorded and stored in the computer. Consecutive turnings of
the cam assembly were performed until all four displacement
readings were obtained. An acoustic-resonance time-domain
signal was acquired between the second and third displacement


56
positions. This system of data acquisition constituted one
trial. A run consisting of a minimum of 10 trials was
performed under each set of equilibrium conditions (P,T,p).
Consecutive runs were performed along the same isotherm
but at different densities. This was achieved by bleeding a
sufficient amount of the experimental gas from the system to
attain a new lower density. On reaching the low pressure
limit, the system was recharged to the upper density range
limit, equilibrated and another isothermal run was started.
This procedure was followed until the temperature range of the
study was completed.


Table 5-1. Summary of measurement hardware.
Item
Manufacturer
Model
number
Range of
operation
Resolution
(Manufacturer's
stated value)
Function
Function
Generator
Hewlett
Packard
HP 3325B
1 /iHz to
20.99 MHz
1 /iHz
Generates a
precise input
waveform.
Low-noise
preamplifier
Stanford
Research
Systems
SR560
< 250 mV rms
input
1 % Gain
Filter and
amplify
acoustic
signal.
Digital strain
gage transducer
indicator
Beckmann
Industrial
Corp.
610
-150 m to 1
volt
0.01 % Full
Scale
Accepts
strain gage
transducer
input
Strain gage
pressure
transducer
Sensotec
TJE/743-03
0-3000 psia
+0.01% full
scale
Measure
pressure.
RTD
HY-CAL
Engineering
RTS 31-A
-200 to 280C
0.00 Ohms
Measure
temperature.
Analog/Digital
Converter
Iotech
ADC488
10 volts
(Max. scan
rate-100MHz)
53 /iVolts
Converts
analog signal
to digital
format.


Table 5-1 (Continued).
Item
Manufacturer
Model
number
Range of
operation
Resolution
(Manufacturer's
stated value)
Function
Digital
Multimeter
Keithly
Instruments
Inc.
kl96
4-terminal
resistance
measurements,
100 /iQ to
20 MQ
Range
dependent, see
manual for
details.
Measure
resistance
from Pt
temperature
probe.
Computer
interface board
National
Instruments
GPIB-PCII
Transfer rate
200 Kbytes/s
N/A
Communication
interface for
electronic
hardware.
U1
03


CHAPTER 6
RESULTS
Eigenvalue Calibration
The data for calibration of the radial mode eigenvalues
were collected using argon at 295.15 + 0.01 K over a pressure
range of 50.0 0.1 psia to 322.0 0.1 psia. The speeds of
sound, calculated using a third-order truncated virial
equation of state20, and the experimental frequencies were used
with equation 3-8 to calculate the experimental eigenvalues,
and these are listed in table 6-1. Table 6-2 shows a
comparison of the experimental eigenvalues for this apparatus
with the theoretical eigenvalues for a perfect sphere and
lists the relative percent deviations. The results illustrate
that the perturbations from ideality in a well designed and
constructed cavity are quite small.
Argon
The measurements on argon provided an initial test of the
apparatus' functionality. Argon was chosen for this test
because of its ready availability and its near ideal behavior
even at substantial pressures. Data were collected on argon
over a density range of 65.825 kg/m3 to 205.07 kg/m3 with an
59


60
Table 6-1. Experimental data and results for the spherical
eigenvalue calibration using argon at 295.15 K.
p
0.1
(psia)
ca
(m/s)
kb
1.5
(Hz)
V*
(m)
327.1
322.05
1
9066
3.551E-02
2
15589
2.0657E-02
3
22009
1.4632E-02
4
28370
1.1351E-02
5
34747
9.2683E-03
6
41117
7.8323E-03
304.5
321.88
i
9059
3.553E-02
2
15576
2.0664E-02
3
21986
1.4640E-02
4
28343
1.1356E-02
5
34733
9.2671E-03
6
41082
7.8349E-03
200.8
321.15
1
9045
3.550E-02
2
15556
2.0644E-02
3
21954
1.4628E-02
4
28305
1.1346E-02
5
34693
9.2566E03
6
41023
7.8284E03
50.7
320.24
1
9013
3.552E-02
2
15509
2.0648E-02
3
21891
1.4628E-02
4
28219
1.1348E-02
5
-
-
6
40843
7.8407E-03
a Speed of sound calculated using a truncated virial
equation of state.(Source: Hirschfelder J.O., Curtiss C.F.,
and R.B. Bird, Molecular Theory of Gases and Liquids, Wiley
and Sons, New York, 1954.)
Radial mode index


61
Table 6-2. Theoretical radial mode eigenvalues listed with the
averages of the corrected eigenvalues.
k
Average
Corrected
Eigenvalues
V
+ 0.0091E-3
(m)
Theoretical
Eigenvalues
^l-O.n-k
(m)
Relative
Deviation
(%)
1
3.552E-02
3.552E-02
0.000
2
2.0653E-02
2.0658E-02
-0.024
3
1.4632E-02
1.4635E-02
-0.021
4
1.1350E-02
1.1350E-02
0.000
5
9.2683E-03
9.2675E-03
0.009
6
7.8354E-03
7.8342E-03
0.015
average standard deviation over that range of 2.43 kg/m3.
The corresponding speed of sound range was 313.78 m/s to
344.34 m/s with an average standard deviation of 0.54 m/s.
The argon density and speed of sound data were acquired on the
following three isotherms; 278.15 K, 288.15 K and 298.15 K
with a standard deviation 0.01 K for each temperature. The
results of the measurements performed on argon using the DSRA
are listed in table 6-3. Figures 6-1 and 6-2 show graphically
the speed of sound and density versus pressure along the
isotherms for which data were collected compared to data
calculated using the truncated virial equation of state. The
argon equation of state has been tested and confirmed to yield
calculated density and sonic speed values with relative
uncertainties of < 0.10% over the above temperature and
density ranges.13 Comparison of the experimental speed of sound


62
and density data with that of the calculated virial equation
of state data gives a reflection of the accuracy of the
instrument. Figures 6-3 through 6-5 show the relative
deviation of the experimental speed of sound (100(cexp-
cvirii)/cvirii ) versus pressure. Figures 6-6 through 6-8 show
the relative deviation of the experimental density (100(pexp-
Pviri.i)/Pviriai ) versus pressure.


63
Table 6-3. Experimental argon (39.948 g/mol) data with the
average standard deviation given at each column heading.
T
P
c
p
0.01
0.5
0.54
2.34
(K)
(psia)
(m/s)
(kg/m3)
278.15
563.6
313.78
69.296
753.7
315.70
94.221
858.5
316.89
107.38
1016.8
318.84
128.26
1208.3
321.47
153.32
1359.7
323.68
172.84
1417.4
324.29
183.02
1544.5
325.94
198.80
288.15
581.6
319.92
68.552
740.7
-
88.706
887.2
-
106.39
995.3
324.82
120.52
1000.8
-
120.77
1150.0
-
138.65
1253.2
328.08
151.21
1270.0
328.41
154.19
1416.7
331.14
174.03
1471.3
332.06
181.24
1671.9
335.93
205.13
298.15
578.4
-
66.100
731.0
-
83.470
864.9
-
99.651
1014.1
-
117.23
1166.1
333.31
135.02
1304.5
335.52
151.89
1312.7
-
152.45
1412.0
-
164.04
1444.4
337.98
169.77
1614.5
340.68
189.42
1755.8
343.08
208.39


64
278.15 K o 288.15 K 298.15 K
Figure 6-1. Experimental data points and virial equation of
state curves of the speed of sound vs. pressure isotherms
for argon.


65
Figure 6-2. Experimental data points and virial equation of
state curves of density vs. pressure isotherms for argon.


66
sound compared to the calculated virial speed of sound
of argon versus pressure at 278.15 K.
0.06
0.04
C? 0.02
o-
u
-0.02

a -0.04
A A
-0.06
A
-0.08 1 + + 4 + ( + 1
400.0 600.0 800.0 1000.0 1200.0 1400.0 1600.0 1800.0
Pressure (psia)
Figure 6-4. Relative deviation of experimental speed of
sound compared to the calculated virial speed of sound
of argon versus pressure at 288.15 K.


67
sound compared to the calculated virial speed of sound
of argon versus pressure at 298.15 K.
0.60
0.40
0.20
£
c 0.00
o
1
S -0.20
TJ
I -0.40
a:
-0.60
-0.80
-1.00
400.0 600.0 800.0 1000.0 1200.0 1400.0 1600.0
Pressure (psia)
Figure 6-6. Relative deviation of experimental density
compared to the calculated virial density vs. pressure
of argon at 278.15 K.


68
compared to the calculated virial density vs. pressure
of argon at 288.15 K.
0.80
0.60
_ 0.40 -
g
c 0.20 -f
o
1
i 0.00
l -0.20
-0.80
400.0 600.0 800.0 1000.0 1200.0 1400.0 1600.0 1800.0
Pressure (psia)
Figure 6-8. Relative deviation of experimental density
compared to the calculated virial density vs. pressure
of argon at 298.15 K.


69
Carbon Dioxide
P, T, c, and p data were acquired for high pressure
carbon dioxide to extend the confirmatory measurements using
the DSRA to a non-ideal fluid. In addition to corroborating
the apparatus' functionality, carbon dioxide data afford a
means to illustrate the utility of the apparatus for physical
property determinations in general.
Experimental Data
The data obtained from the measurements performed on
carbon dioxide are listed in table 6-4. The temperature,
pressure, speed of sound, and density ranges and the average
standard deviation over those ranges were 290.15 K 0.01 K to
310.15 K + 0.01 K, 1005.5 psia to 1804.3 psia 0.43 psia,
211.30 m/s to 392.98 m/s 0.28 m/s, 584.11 kg/m3 to 889.82
kg/m3 5.92 kg/m3, respectively.
The relative deviation of the present experimental carbon
dioxide density data compared to other values found in the
literature is presented in figures 6-9 through 6-12. Due to
the scarcity of published carbon dioxide speed of sound data,
figure 6-13 simply presents the experimental speed of sound
data graphically versus pressure. The lines are drawn as a
visual aid to help identify the various isotherms.


70
Table 6-4. Experimental carbon dioxide (44.011 g/mol) data
with the average standard deviation given at each column
heading.
T
P
c
P
0.01
0.4
0.37
4.17
(K)
(psia)
(m/s)
(kg/m3)
290.15
1017.5
342.07
841.11
1068.8
348.15
843.17
1127.9
353.98
851.79
1355.3
375.33
868.24
1612.4
392.98
889.82
297.15
1048.6
299.33
770.67
1127.3
311.43
783.08
1257.5
325.70
804.19
1401.3
342.65
821.15
1573.0
358.88
839.77
300.15
1005.5
277.26
700.14
1052.8
285.17
720.59
1329.8
320.03
787.07
1581.4
343.01
819.09
1804.3
362.51
836.30
303.15
1235.4
288.12
730.14
1500.5
320.35
780.75
1769.3
343.87
814.97
307.15
1172.2
228.50
584.11
1219.0
239.10
639.18
310.15
1340.5
212.27
636.14


71
0.30
0.20
!* 0.10
; 0.00

| -0.10
i -0.20
-0.30
-0.40
1000.0 1100.0 1200.0 1300.0 1400.0 1500.0 1600.0 1700.0
Pressure (psia)
Wagner(21) v Kessel'man(22) m Duschek(23) x Holste(20)
Figure 6-9. Relative percent deviation of literature values
of density for carbon dioxide at 290.15 K from the present
measurements.
3!
s>
or
Pressure (psia)
| Wagner(21) v Duschek(23) >: Michels(32)
Figure 6-10. Relative percent deviation of literature
values of density for carbon dioxide at 297.15 K from
the present measurements.


72
[" Wagner(21) v Duschek(23) s Ely(24) s Michels(32) |
Figure 6-11. Relative percent deviation of literature
values of density for carbon dioxide at 300.15 K from
the present measurements.
0.20
0.15
| 0.10 --
Q
0.05
0.00
1200.0 1300.0 1400.0 1500.0 1600.0 1700.0 1800.0
Pressure (psia)
Wagner(21) v Duschek(23) ffl Ely(24)
Figure 6-12. Relative percent deviation of literature
values of density for carbon dioxide at 303.15 K from
the present measurements.


73
CU
CU
CL
cn
400.00
350.00
300.00
250.00
200.00
1000.0 1200.0 1400.0 1600.0 1800.0 2000.0
Pressure (psia)
- 290.15 K 297.15 K 300.15 K 303.15 K + 307.15 310.15 K
Figure 6-13. Experimental speed of sound data versus
pressure of carbon dioxide at different isotherms.


74
Equation of State-Data and Results
Experimental physical chemistry and its application to
the thermodynamics of fluids ultimately strives to provide
insight into the behavior of chemical systems over a wide
range of state variables. Important aspects of this behavior
are quantified in several different manners as specific
physical properties, enthalpy (H) entropy (S) heat
capacities (C) ... etc. To illustrate the utility of the
DSRA, which acquires T, P, c, and p data simultaneously, for
deducing other thermodynamic properties, the heat capacities
at constant pressure Cp and constant volume Cv and their ratio
y were calculated for carbon dioxide.
The experimental data were fit to the Redlich-Kwong (R-K)
equation of state
vm-b srvjvm+b) 16
and the R-K parameters a,b were determined as described in
appendix E. Figure 6-15 compares the experimental densities to
the R-K calculated densities graphically versus pressure.
Table 6-5 lists the experimental densities and the R-K
calculated densities and relative percent deviations between
them ( 100 (pexp pR-K) / pR_K ).
Using fundamental thermodynamic relationships, equation
6-1 and the experimental data for carbon dioxide, values for
Cp, Cv, and y were calculated (refer to appendix E for


75
1000.0 1200.0 1400.0 1600.0 1800.0
Pressure (psia)
- 290.15 K 297.15 K 300.15 K 303.15 K 307.15 K 310.15 K
Figure 6-15. Carbon dioxide density, experimental data
and RK calculated curves, versus pressure.


76
Table 6-5. Experimental and Redlich-Kwong calculated densities
of carbon dioxide.
Redlich-Kwong constants
a = 5.8752 0.0037 Pa m6 K1/2 mol-2
b = 2.6010E-5 0.0001E-5 m3 mol-1
T
(K)
P
(psia)
Pexp
(kg/m3)
Prk
(kg/m3)
Relative
deviation
(%)
290.15
1017.5
841.11
830.36
-1.28
1068.8
843.17
837.90
-0.63
1127.9
851.79
846.01
-0.68
1355.3
868.24
873.18
0.57
1612.4
889.82
898.50
0.98
297.15
1048.6
770.67
758.49
-0.60
1127.3
783.08
774.59
-1.08
1257.5
804.19
798.77
-0.67
1401.3
821.15
820.82
-0.04
1573.0
839.77
843.01
0.39
300.15
1005.5
700.14
_
_
1052.8
720.59
-
-
1329.8
787.07
780.63
-0.82
1581.4
819.09
819.62
0.06
1804.3
836.30
846.65
1.24
303.15
1235.4
730.14
723.87
-0.86
1500.5
780.75
780.47
-0.04
1769.3
814.97
819.65
0.57
307.15
1172.2
584.11
618.62
5.91
1219.0
639.18
649.21
1.57
310.15
1340.5
636.14
641.10
t"
o
i
thermodynamic equations used to calculate the heat
capacities) Table 6-6 lists Cp, Cv, and y determined from the
experimental data and the R-K equation of state along with
literature values. Due to the lack of published values for
carbon dioxide over the range of these experiments only a few
comparisions could be presented as shown in table 6-6.


77
Table 6-6. Comparison of experimental Cp, Cv, y values with other
experimental literature values.
T = 290.15 K P = 1612.5 psia
Source
Cp
J/(mol-K)
RDa
(%)
Cv
J/(mol-K)
RD
(%)
Y
RD
(%)
DIN33
107
1
2
50.3
1.5
5.6
2.13
0.01
-3.80
Int.
Critical
Tables34
105
5
8
48
1
10
2.2
0.1
-6.9
Vargaftik26
108.1
0.5
1.6
This work
108.91
0.37
53.14
0.41
2.049
0.009
T = 300.15 K P = 1
329.5
osia
Source
J/(mol K)
RD
(%)
c,
J/(mol K)
RD
(%)
Y
RD
(%)
DIN33
140
1
2
50.9
1.5
-2.5
2.75
4.72
Int.
Critical
Tables34
149
5
-4
46
1
8
3.2
-9.8
Vargaftik26
143.9
0.5
-0.8
This work
142.73
0.28
49.59
0.39
2.884
0.006
Relative deviation


78
Uncertainties in Measured and Calculated Values
For argon, the precision of the pressure, temperature,
speed of sound and density measurements were +0.5 psia, 0.006
K, 0.54 m/s, and 2.34 Kg/m3, respectively. The accuracy of
the pressure and temperature measurements based on the
calibration standards were 0.01 K and 0.1 psia. The
accuracies of the speed of sound and density measurements
relative to those inferred from the truncated virial equation of
state were no greater than 0.15 m/s and 0.65 kg/m3.
The precision and accuracies of the measurements of
pressure and temperature for carbon dioxide were the same as the
argon measurements. The precision of the experimental speed of
sound and density measurements for carbon dioxide were 0.37
m/s and 4.17 kg/m3. The uncertainty in the experimental
density measurements in the range of the present measurements
was less than 0.36 kg/m3.
The R-K equation of state for carbon dioxide obtained as
the best fit to these measurements yielded an average relative
deviation of 0.98 kg/m3 of the R-K calculated densities from
the experimental densities. The above uncertainties lead to
average uncertainties in Cp, Cv, and y of + 0.40 J/(mol-K), +
0.41 J/(mol-K), 0.008, respectively.


CHAPTER 7
CONCLUSION
The research effort presented in this document focused on
the development of an apparatus that has the capability of
measuring pressure, temperature, speed of sound, and density
of fluids over wide ranges of the variables. An apparatus was
designed, built and tested. It accomplished this goal by
measuring the above variables simultaneously. While reliable
pressure and temperature data acquisition are routine in many
scientific laboratories, the measurement of the sonic speed
and density are not and require deliberate effort and insight
into the experimental approach.
The apparatus employed two separate and experimentally
significant data collection instruments coupled together to
perform measurements on the same system of gas. The first
instrument, the spherical resonator, utilized technology for
the generation and acquisition of acoustic resonance
information of fluid-filled cavities which has been developed
in this laboratory and has become a staple of our laboratory
procedures. The utility of the technique derives from the high
precision and accuracy with which resonance phenomena may be
discerned. The second instrument, the densimeter, was built in
response to an effort to advance the capabilities of the
79


80
laboratory and introduce a new era of useful physical property
determination and prediction. Since a robust, versatile,
instrument for the direct measurement of density is not
commercially available, nor has such an instrument been
described in the literature, complete design, fabrication and
development of a densimeter was required and ultimately
accomplished. While the fundamental theory of the operation
of the densimeter was not advanced, the art of completing
useful measurements was. For the first time fluid density can
be accurately inferred from electrical signals acquired non-
invasively by induction in an instrument which is self
calibrating during each use.
The combination of the above instruments along with
precision devices to measure pressure and temperature resulted
in achievement of the major goal of this research, namely a
practical densimeter/spherical resonator apparatus, DSRA. The
success of the DSRA is evident from the results of the
confirmatory experiments, performed on argon and carbon
dioxide. The argon measurements were performed over a pressure
range of 560.6 psia to 1755.8 psia at three isotherms covering
a range of 278.15 K to 298.15 K that yielded a density range
of 69.296 kg/m3 to 207.06 kg/m3. The corresponding speed of
sound measurements for argon covered a range of 313.77 m/s to
344.34 m/s. The carbon dioxide data encompassed a pressure
range of 550.5 psia to 1302.3 psia at six isotherms spanning
a range of 273.15 K to 308.15 K. These ranges yielded density


81
measurements from 618.62 kg/m3 to 898.50 kg/m3 and speed of
sound measurements 304.22 m/s to 318.54 m/s.
Uncertainties in the presently measured densities are
beyond the targeted goal of + 0.1 %, but this derives not from
a fundamental flaw in the principles applied but rather from
the fact that the attainable precision in measurements of this
kind is determined in large part by the difference in density
between the fluid and that of the lightest buoy. The optimum
condition is that for which the buoy density is only slightly
higher than that of the fluid. To illustrate this fact
consider determination of the fluid density from measurements
of the three suspension rack loadings corresponding to
equations 2-8, 2-9, 2-10. Assume for simplicity that m2 = 2m2
= m and V£ = V2 which table 4-1 shows to be nearly true. The
fluid density p£ then is found to be
Pf
p'
S2-2S!
(7-1)
where p' is m/V, the net density of the lighter spherical
buoy, and Si and S2 are the net LVDT signals relative to the
unloaded suspension rack. The uncertainty in the computed
value of p depends on the uncertainties in the LVDT signals
and the magnitude of the buoy density. Taking the signal
uncertainties to be equal u3 and the uncertainty in buoy
density to be negligible these propagate to an uncertainty in
fluid density upf which is


82
UB /
u = o'
(7-2)
This shows that a
large buoy
density leads
to a
correspondingly large
uncertainty
in the measured
fluid
density. By these considerations it should be clear that the
highest precision and smallest uncertainty in pf caused by
choice of buoy density occurs when these densities are equal.
Because it is necessary for p1 to exceed pf for the instrument
to operate, the buoy density should be only slightly greater
than that of the fluid for optimum results.
In practice it is necessary to operate with buoys dense
enough to accommodate the densest fluid to be studied. This is
why it is recommended to provide future versions of this
instrument with interchangeable sets of buoys with densities
appropriate for use with a variety of fluids. In this work the
density of the lightest buoy exceeded that of the densest
fluid measured by a factor of 100. If that were reduced to say
1.5 the uncertainty in measured density would improve by about
a factor of 1/100. It should be apparent that attaining the
level of precision observed in the present test measurements
is a confirmation of the potential of the technique for
accurate fluid density determinations. The buoys fabricated
for this instrument were intended ultimately for use with
liquids and high pressure gases under conditions for which the
mismatch in fluid and buoy density is much smaller.


APPENDIX A
SPHERICAL BUOY MASS AND VOLUME CALIBRATION
As stated in the development of the operational theory of
the densimeter, chapter 2, the mass and volume of each
spherical buoy must be known accurately. The masses of the
spheres were determined using an analytical balance that was
calibrated with certified standard masses. In order to
determine the volume of each sphere, degassed, distilled water
was used with a pycnometer that was built specifically to
perform this task.
For the mass calibration of each sphere, data of the
electronic load cell balance readouts versus the standard
masses were collected in the region of mass of each sphere.
Each sphere was then weighed on the analytical balance and the
corresponding calibrated mass was calculated, see figures A-l
and A-2. Corrections for the buoyancy of air on the standard
brass masses and spheres were applied.
The pycnometer, shown in figure A-3, consisted of a
volume chamber, a spherical lid, and a frame to hold the
pycnometer in place. The volume chamber, 0.750 inch OD and
1.00 inch in length, was bored out to an ID of 0.676 inch. The
top surface of the volume chamber was machined and lapped to
83


84


85
Figure A-l. Graph showing calibration data used to
determine the mass of spherical buoy #1.
Calibration Data Regression Fit o Mass of Buoy #2
Figure A-2. Graph showing calibration data used to
determine the mass of spherical buoy #2.


86
fit exactly the contour of a precision stainless steel ball
bearing with a 0.750 inch OD. The ball-bearing was used as a
lid which could be reproducibly seated in a manner which
assured the volume was filled with liquid and there were no
trapped air bubbles. The pycnometer frame was built from
aluminum with a screw mounted at the top. This screw was used
to hold the spherical ball lid firmly in place thereby
producing a positive liquid tight seal confining a definite,
fixed volume inside the cell. A torque screw driver assured
the sealing load was reproducible.
One volume measurement consisted of filling the
pycnometer with water, drying the outside completely, seating
the lid, and measuring the mass of the assembly using an
analytical balance. A sphere was then placed in the volume
chamber displacing an equivalent volume of water and the
assembly was re-weighed. The difference in these measurements
is the mass of the sphere less the mass of the displaced
water. The latter is proportional to the volume of water that
was displaced by the sphere. By knowing the density of the
water at temperature, T, of the measurement, the volume of the
spherical buoy can be calculated using the following equation,
V = Mb M{p^ ~ (A-l)
pw(T)
where V. is the volume of the sphere, Ms is the mass of the
sphere, Mp+W.t3 is the mass of the pycnometer, the undisplaced


87
water (W) and sphere, Mp,w is the mass of the pycnometer
filled with water (w) and pw(T) is the density of the water.
The results for the volume calibrations are listed in table
A-l.
Table A-l. Data and results for the volume calibration of the
spherical buoys.
P294.65 kHzO)1 = 0.99873 g/CC Mptw = 72.4896 g
0.0002 g
Spherical
Buoy
ms (g)
Mp+w'+s (g)
Volume (cc)
1
6.9821
0.0001
77.5380
0.0002
1.9329
+ 0.0002
2
13.9658
0.0001
84.5194
0.0001
1.9331
0.0002
1 Weast R.C and M.J. Astle, CRC Handbook of Chemistry and
Physics, 1974, F-ll.


APPENDIX B
CIRCUIT DIAGRAM FOR THE STRAIN GAGE SIGNAL CONDITIONER
The block diagram of the strain gage signal conditioning
component is shown in figure B-l.28 The circuit diagram that
was followed in the construction of the signal conditioning
device is given in figure B-2.28
FUNCTIONAL BLOCK DIAGRAM
Figure B-l. Block diagram of the strain gage signal
conditioning component.
88


89
Figure B-2. Circuit diagram of the strain gage conditioning
device.


APPENDIX C
COMPUTER PROGRAMS
Listed below are the source codes for the programs that
were used to perform the data acquisition (freqdens.c) and
temperature control (tcntrl.c) with the DSRA. The programs
were written using ANSI C standard code language. The programs
were written by Mr. John Hornick and the author.
Progam Freqdens.c
/include
/include
/include
/include
/include
/include
/include
/include
/include









/define PRESSURE_NAME
/define TEMPERATURE_NAME
/define IOTECH_NAME
/define SYNTH_NAME
/define IOTECH_CHANNEL
/define MaxLoop
/define SamplingRate
/define NumTDPoints
/define IOTECH_STRING
/define NUM POINTS
typedef int BOOL;
/define TRUE 1
/define FALSE 0
"BECKMAN"
"K195"
"IOTECH"
"HP3325B"
2
21
1000001
327681
"c3r/3,2i4gllt0x"
81921
90


91
Directory[100];
IotechString[100];
SynthString[100];
SynthAddress, IotechAddress, PAddress,
char
char
char
int
TAddress;
int huge TimeDomainData = NULL;
int IotechBuffer[8192];
FILE *stream;
long num,d;
float Voltage;
BOOL GetParameters( void );
BOOL ValidSamplingRate( long Rate );
BOOL ValidNumTDPoints( long NumPts );
void CreateStrings( void );
int SampleRateToIotechNumber( long SampleRate );
void AcquireData( long LoopNumber );
BOOL AllocateMemory( void );
void WriteTimeDomainData( long LoopNumber );
void ReadExtralnstruments( float Temperature, float
Pressure );
void IEEEDelay( void );
float CDV (float Vk,float Vbl, float Vb2, float Vbh );
float ReadVoltage( void );
float RawDataToVoltage( float RawData );
int main( int argc, char *argv[] )
{
long LoopNumber = 0;
float Pressure, Temperature;
float Vk,Vbl,Vb2,Vbh,Vavg,Tk,Tbl,Tb2,Tbh,Tavg ;
float Pk,Pbl,Pb2,Ph,Pavg;
char cbuffer[20],buffer[20],filename[30];
printf("Enter directory for data storage:\n");
scanf("%s"(Directory);
gets(buffer);
num=ll;
CreateStrings();
SynthAddress = ibfind( SYNTH_NAME );
IotechAddress = ibfind( IOTECH_NAME );
PAddress = ibfind( PRESSURE_NAME );
TAddress = ibfind( TEMPERATURE_NAME );
ibwrt( SynthAddress, SynthString, strlen( SynthString )
);


Full Text
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APPARATUS AND METHOD FOR COMBINED ACOUSTIC RESONANCE
SPECTROSCOPY-DENSITY DETERMINATIONS
By
DANIEL S. TATRO
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
1996

To my wife, Mary, my daughters, Danielle and Gabrielle,
and the rest of my family,
Evelyn, Laura, Gary and Christine,
who love me without end and support me without question.

ACKNOWLEDGEMENTS
The author wishes to thank all of those who have
contributed to the education he has received at the University
of Florida. His transformation from a student to a competent
researcher and instructor could not have been possible without
the support and direction of the faculty of the University of
Florida, Department of Chemistry. In particular, he extends
his greatest thanks to his committee chairman, Samuel Colgate.
The author considers it a privilege to have worked with such
an outstanding chemist and engineer.
The author wishes to thank the other members of the
Colgate research group, Johnny Evans, Evan House, Vu Thieu,
and Karl Zachary, for their continued interactions. In
particular, the author wishes to thank Troy Halvorsen, for
without his help, the development of the densimeter would not
have been possible.
Technical support of this research project was
extraordinary. The author wishes to thank the staff of both
the electronic and machine shops and would like to
acknowledge, in particular, machinist Joe Shalosky, whose
craftsmanship is second to none.
iii

Additional thanks are extended to the author's family and
friends. It has been their belief in the author that has been
his source of inspiration. Specifically, special gratitude is
extended to his wife, Mary, whose backing and support have
been unwavering. In addition, the author wishes to thank John
Magrino, Nial McGloughlin, Mitch Morrall, and Gary Tatro whose
friendships have supplied continuous moral support.
Finally, the author wishes to thank the University of
Florida, Department of Chemistry and the University of Florida
Division of Sponsored Research for financial support.
IV

TABLE OF CONTENTS
ACKNOWLEDGMENTS iii
ABSTRACT vii
CHAPTERS
1 INTRODUCTION 1
2 DENSITY MEASUREMENT 7
Fundamental Theory 7
Theory of Design 8
3 SPEED OF SOUND MEASUREMENT 12
Spherical Cavity Resonance Equation
Development 12
Resonance Frequency Identification 15
4 DENSIMETER AND SPHERICAL RESONATOR RESEARCH
AND DEVELOPMENT 21
Densimeter 22
Spherical Buoys 22
Deflection Beam and Semiconductor
Strain Gages 24
Coiled Spring and LVDT Sensor 27
Densimeter Body 29
Buoyancy Assembly and Sphere Lifting
Mechanism 31
Top Flange Assembly 37
Spherical Resonator 38
Resonator Cavity 38
Transducer Assembly 40
Final Assembly 43
5 EXPERIMENTAL 47
Gases 47
Interfacing 47
Measurement Hardware 48
Pressure and Temperature 48
v

Density 51
Speed of Sound 53
Experimental Data Collection Procedure ... 55
6 RESULTS 59
Eigenvalue Calibration Data 59
Argon 59
Carbon Dioxide 69
Experimental Data 69
Equation of State-Data and Results .... 74
Uncertainties in Measured and
Calculated Values 78
7 CONCLUSION 79
APPENDICES
A SPHERICAL BUOY MASS AND VOLUME
CALIBRATION 83
B CIRCUIT DIAGRAM FOR STRAIN GAGE SIGNAL
CONDITIONER 88
C COMPUTER PROGRAMS 90
D TEMPERATURE CONTROL 98
E REDLICH-KWONG EQUATION OF STATE AND
THERMODYNAMIC EQUATIONS 104
REFERENCES 110
BIOGRAPHICAL SKETCH 113
vi

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
APPARATUS AND METHOD FOR COMBINED ACOUSTIC RESONANCE
SPECTROSCOPY-DENSITY DETERMINATIONS
By
Daniel S. Tatro
May 1996
Chairperson: Samuel 0. Colgate
Major Department: Chemistry
A novel instrument was developed for simultaneous
measurement of temperature(T), pressure(P), speed of sound(c),
and density(p) of fluids over wide ranges of these variables.
The speed of sound measurements were performed using a
spherical acoustic resonator designed and fabricated in this
laboratory. In order to measure density, an innovative
densimeter based on application of Archimedes principle was
developed. Confirmatory experiments were performed on a near
ideal gas (argon) and a nonideal fluid (carbon dioxide) where
P, T, c, and p data were collected. The report describes the
research and development of the entire instrument as well as
the demonstration of its performance.
vii

CHAPTER 1
INTRODUCTION
History demonstrates that in order to attain a
comprehensive understanding of physical science, the
successful practice of conducting physical property
measurements must be established and practiced.1 Experimental
measurement is the foundation upon which most scientific
interpretation of material behavior is made. Often, with the
knowledge of a small but sufficient number of a system's
properties over a suitably wide range of variables, much of
the behavior of a system can be specified in terms of known
theoretical and empirical relationships among these and its
other properties. In certain cases, pertinent equations of
state and theories can be used to make correlations and
approximations. Some scientists predict that the new age of
computational chemistry is at hand, and the need for
experimental work will become obsolete. While a careful look
at the complexity of real systems of actual importance to
applied science reveals this optimistic prediction to be
exceedingly premature, even if it were correct, the need would
still persist for continual experimental confirmation of
theoretical predictions.
I

2
Generally, equations of state are based on empirical or
semiempirical models.z They tend to work well for systems
included in the supported data bank, which is to say they can
be used to fit experimental data over the range of available
observations. Intolerable deviation of calculated properties
from actual system properties commonly occurs when attempts
are made to extrapolate information to regions far removed
from the experimental data. Also, equations of state often
fail when extended to other systems of different structures or
compositions for which no experimental data are available(i.e
multicomponent or multiphase systems).
To understand the behavior of complex systems and to
develop more reliable equations of state, it is necessary to
make physical property measurements. The measurement of some
physical properties (e.g. pressure, P, and temperature, T) can
be made with acceptable accuracy and precision using
relatively simple measurement apparati or techniques. For such
measurements scientists are able to purchase state-of-the-art
devices suitable for use over a wide range of conditions.
Unfortunately, not all necessary state variables can be
measured with such ease and precision. Two important
properties, in particular density and the speed of sound, have
no simple, wide range, robust measurement methods. The
research presented here addresses this problem through the
development of a densimeter/spherical resonator apparatus
(DSRA).

3
Currently, speed of sound measurement by means of
acoustic resonance methods is performed routinely by only a
few research groups in the world, our laboratory being one of
these. The speed of sound measurements along with
simultaneously collected pressure and temperature data have
been used to characterize and identify many thermophysical
properties.3'1' The acoustic resonance technique developed to
date has significantly advanced some areas in the art of
thermodynamic property measurement,5 but it fails to address
one very important state variable, namely that of fluid
density.
Many methods have been explored for possible use as
versatile, accurate means for measuring fluid density, and it
is appropriate now to review how the best established
techniques, pycnometry, the Burnett method, and the vibrating
tube densimetry operate and to compare their strengths and
their specific limitations.
Pycnometry6, using constant volume cells, leads to
density directly from the measurement of the mass of a
confined fluid. The problem of accurately weighing a cell
filled with fluid at high pressure and at high or low
temperature is not minor and in fact has been a forbidding
deterrent to the widespread use of this technique. Here,
careful volume calibration is essential, and the volume change
as a function of pressure and temperature must also be
accounted for by either measurement or prediction.

4
The Burnett method1 for gases entails a series of
controlled expansions to collect a set of pressure-volume data
pairs at constant temperature. The expansions lead ultimately
to a sufficiently low pressure that an established equation of
state may be used reliably to calculate the density. This
gives the system mass, from which the initial density may be
determined. Drawbacks of this method include its limitation to
gas density measurements only, its dependence on volume
calibrations, and perturbations due to adsorption of gas on
the cell walls. Both of the above techniques are sampling
methods. The sample extracted for density determination is
ultimately discarded, thereby introducing problems such as
possible separation of components in the case of mixtures, and
limitation of the number of measurements which can be made for
a given system. Another disadvantage is that the time to
obtain a single measurement is very long compared to that
typically required for measurement of other variables.
Vibrating tube density meters provide an indirect
measurement of density.8 The working principle is based on the
dynamics of a mass/spring combination. As the mass, which
includes that of the fluid inside the tube, changes, the
frequency of vibration will change in concert. The vibrational
frequency of a U-tube rigidly supported at its ends is
measured when empty, when filled with calibrating fluid of
known density and when filled with the fluid under study. From
these frequencies the unknown fluid density can be determined.

5
Although appealing for its simplicity, use of the vibrating
tube densimeter is impractical for thermodynamic measurements
due to its extensive calibration requirements. Frequent
calibration over pressure and temperature ranges is required
as well as calibration over the density range to be studied.
There is also a hysteresis effect of the spring constant which
further limits the instrument's reliability.
In contrast to the above techniques, there is a class of
density measurement devices that offers both simplicity in
theory and, in some situations, experimental technique. These
devices are based on the buoyancy force of fluids on submerged
solid bodies and derive from ideas first developed centuries
ago by Archimedes. This technique has often been put to use as
a practical means of fluid density measurement, especially in
liquids at ambient conditions9, but only recently has it been
extended to dense gases.10 Later chapters will detail the
design and use of a versatile new density meter based on this
principle.
In view of this background, the focus of this work is
concentrated in two areas, 1) development of a high
performance fluid density measuring device (densimeter) for
accurate experimental density determination under ambient and
non-ambient conditions and 2) incorporation of this densimeter
in-line with an acoustic spectrometer. This coupling will
yield, for the first time, the capability of acquiring in a
single apparatus pressure, temperature, speed of sound and

6
density information simultaneously. Knowledge of these four
properties permits others to be inferred.
The decision to pursue the development of an Archimedes
type densimeter was made after careful consideration of
current state of the art density measurement techniques and
how they could be adapted to our specific experimental
requirements. Initially, efforts were directed at building a
U-tube vibrating density meter. Its straightforward design and
ease of construction along with our laboratory's experience in
precision frequency measurements made this instrument
appealing at first. A prototype was built and functioned well
in a bench top capacity, but this device was abandoned after
it was apparent that the extensive calibration requirements
and unpredictable hysteresis effects could not be
circumvented. The research then shifted to the development of
a density meter based on Archimedes' principle. Again, the
simple theory was appealing, and, unlike the U-tube density
meter, a measurement technique could be devised such that the
limitations of extensive calibration could be eliminated. It
was revealed initially that hysteresis effects would be a
problem, but could be eliminated with appropriate modification
of the density meter.

CHAPTER 2
DENSITY MEASUREMENT
In certain cases, when developing a method for physical
property measurement, the underlying theory is simple and
straightforward. Often the challenge is the design,
development and implementation of an appropriate experimental
technique. The density measurement method outlined in this
chapter is illustrative of these circumstances. The theory is
based on well established and familiar principles, while its
implementation required exploration of innovative approaches
to solving challenging experimental problems.
Fundamental Theory
First, consider a rigid body immersed in a static,
isothermal fluid of density, p. The predominant forces acting
on the body are those associated with Archimedes' principle
and Newton's second law of motion. Specifically, Archimedes'
principle states that an object immersed in a fluid is buoyed
up by a force equal to the weight of the fluid that is
displaced. Hence, the buoyancy force, FB is defined as
7

8
F„ = p Vgra (2-1)
where p is the density of the fluid, V is the displaced fluid
volume, g is the local acceleration due to gravity, and n is
a unit vector directed along the line from the center of mass
of the body to the center of mass of the earth.
In addition, by Newton's second law of motion, the
gravitational force exerted on the rigid body is proportional
to its mass times the acceleration due to gravity
F„ = m g ñ (2-2)
where FN is the Newtonian gravitational force and m is the
mass of the submerged body.
Thus, the resultant of the two forces is,
Ftot = ( mg - p Vg ) ñ (2-3)
or in terms of magnitudes alone, since there is no ambiguity
about direction in this one dimensional case ,
Ftot = ( m - P V) g. (2-4)
Equation 2-4 is the working equation from which density is
obtained and the basis for the operational development of the
densimeter design.
Theory of Design
Consider two spherical objects having volumes V2 and V2
and having masses mx and m2 suspended in an isothermal, static

9
fluid of density p. Recalling equation 2-4, the net force
acting on each sphere may be represented independently by the
following equations,
F
(tot, spherel)
II
1
P
g
(2-5)
F(tot, spheie2)
= sr -
P
9
(2-6)
Now, assume that the net force on each sphere can be
measured independently. To accomplish this in practice, the
spheres are suspended by a spring as shown in figure 2-1.
Figure 2-1. Schematic diagram of fundamental densimeter
design

10
Assuming for the moment that the spring obeys Hooke's
law, the total force acting to stretch or compress the spring
is proportional to the deflection from its relaxed position
given as
F = -k x (2-7)
Where x is the deflection distance and k is the spring
proportionality constant.
If the spheres can be independently placed on or removed
from the suspension rack, four separate loadings of the spring
are possible. The resultant force for each of the four
loadings is as follows;
I. Both spheres off:
F„=-kx0 (2-8)
Where x0 is the length of the spring loaded only by the
suspension rack.
II. Sphere 1 on; sphere 2 off:
Fi =~k x1 = ( m1 - p V1 ) g - k x0 (2-9)
III. Sphere 1 off; sphere 2 on:
F2 =-k x2 = { m2 - p V2 ) g - k x0 (2-10)
IV. Sphere 1 on; sphere 2 on:
F3 =-k x3 = ( m3 - p V3 ) g - k x0 (2-11)
where m3 = m3 + m2 and V, = V: + V,.
Equations 2-8 through 2-11 form a coupled system from
which, knowing Xj-x,,, x2-x0, x3-x0, m„ m2, V, and V2, the fluid

11
density may be determined. It is not necessary to know the
acceleration of gravity g or the spring constant k. These
features which the method useful, as the elasticity of any
practical spring material will vary with temperature, T, and
pressure, P, but should remain constant over any measurement
taken at constant T and P. In the experiments for which this
device is intended to be used, T and P will vary widely but
not during any single density measurement. Also, g varies with
position, but this is fixed through a measurement, and the
magnitude of g cancels out in the calculation of density.
Other methods previously used to make fluid density
measurements under non-ambient conditions require calibration
to account for the non-negligible temperature and pressure
discriminations. They are further compromised by the
irreproducibility of the physical properties of the
densimeters due to hysteresis effects. The present work seeks
to overcome these unwanted limitations by developing an
absolute instrument, free from the requirement of calibrations
and operating with a highly elastic force sensor.

CHAPTER 3
SPEED OF SOUND MEASUREMENT
Observation of acoustic resonance in fluid-filled
cavities has been used as an effective, practical means to
determine the speed of sound in gases and liquids. In this
laboratory and others, spherical cavity resonators have been
used to acquire sonic speed data of both high precision and
accuracy.11 The following development of the theory for
spherical cavity acoustic resonance and the subsequent
calculation of the speed of sound from resonance measurements
follows that of Reed12, McGill11 and Dejsupa14 based on the
earlier work of Rayleigh15 and Ferris.16
Spherical Cavity Resonance Equation Development
The general wave equation that describes the propagation
of pressure waves in a lossless fluid at rest, contained
within a rigid walled spherical cavity, is,
d2l¥(v)
at2
V2'P(^)
(3-1)
where T is the velocity potential and c is the speed of sound.
12

13
Assuming a time separable velocity potential such that T =
Toe1"1, equation 3-1 becomes
(3-2)
to = 0
where a is the frequency.
The solution to this wave equation is separable in terms
of the angular and radial components. Using spherical
coordinates, and invoking the usual boundary conditions, the
angular part of the solution is the known set of spherical
harmonics, and the radial part is the spherical Bessel
function of the first kind. The overall solution is,11
+1
(3-3)
where to = time-independent velocity potential
r,0,(p = spherical polar coordinates
ji = Bessel function of the first kind(radial part)
Yt” = Spherical harmonics of the first kind (angular part)
n,l,m = integers.
The rigid spherical cavity imposes a boundary condition
that the radial component of fluid velocity must be zero at
the cavity wall. This generates an equation for the speed of
sound in terms of the normal frequencies of vibration, given
as,

14
C
fl,n
TTn
(3-4)
where f1|lt is the resonance frequency, r is the radius of the
spherical cavity, and the eigenvalue, Si.n* is the nth root of
the first derivative with respect to r of the 1th spherical
Bessel function of the first kind, also referred to as the
resonance frequency eigenvalue. It should be noted that
because of the oscillatory nature of the Bessel function, for
each value of 1 there exits an infinite number of roots
(eigenvalues), ?lin. The smallest 72 frequency eigenvalues,
calculated by Ferris, are given in table 3-1. These have been
confirmed by a computer program written here to evaluate ¡jlin
generally. While the values of higher order eigenvalues are
mathematically interesting, they are of little practical use
in the present experiments, as the associated resonance
frequencies lie beyond the upper range of the acquisition
hardware used in our measurements.
The practical advantage of using spherical resonators
derives from a special characteristic of the purely radial
modes of vibration.12 These, being functions of r, are pure
breathing modes and thus are not subject to viscous damping
caused by tangential gas motion with respect to the cavity
wall. In addition, the radial modes of vibration are known to
be less sensitive to perturbations caused by imperfections in
the spherical nature of the cavity.11 Therefore, we identify
and utilize frequencies associated with pure radial modes

15
(1=0) to facilitate the most accurate measurements of sonic
speed.
Resonance Frequency Identification
The scheme used here for assigning mode identities to
resonance peaks in an acoustic resonance spectrum was
developed by Dejsupa.14 In order to calculate the speed of
sound, a resonance frequency flin and its corresponding
eigenvalue must be identified. Rearranging equation 3-4
and solving for frequency yields
fl.n
c h,a
2 7t r
(3-5)
Knowing the cavity radius, r, and choosing an estimated value
of the sonic speed, c, one may calculate an approximate
frequency for each mode. One such frequency is then used as a
starting point for the identification of frequency values in
an experimental acoustic resonance spectrum. A typical
experimental frequency spectrum is shown in figure 3-1, where
at each acoustic resonance the frequency amplitude shows a
sharp increase in magnitude with relation to the baseline.
This experimental frequency spectrum is scanned in the region
of the calculated frequency and a tentative mode assignment is
made to that peak which most nearly matches the trial value.
The assignment is tested for accuracy by predicting the

16
Figure 3-1. Typical experimental frequency spectrum of
argon at 1359.3 psia and 278.15 K.

17
relative location of other modes and looking at the spectrum
for corresponding signal peaks.
Once a frequency has been identified, that is a
frequency/root pair has been determined, it serves as a
reference value used to identify other frequencies and their
corresponding eigenvalues. For example, if a reference
frequency is assigned the mode designation l,n,
(3-6)
the frequency of mode 1' ,n' should be
(3-7)
The experimental frequency spectrum is then scanned at this
new frequency value to confirm its existence and the actual
experimental frequency value is recorded.
It should be mentioned that this method serves as a self
check for the correct identification of the first reference
frequency. Incorrect identification of the reference frequency
(i. e incorrectly matching the frequency with the correct
eigenvalue) will predict other frequencies which do not match
the experimental spectrum.
To this point the development has been based on an
idealized cavity resonator. In reality there are perturbations
that account for deviations of vibrational frequencies from

18
the idealized model.18 The corrections may be modeled, but
applying the models can be computational demanding and can
require detailed knowledge of the system and its properties
which are unavailable. As mentioned before, using the purely
radial modes of vibration for identification and calculation
reduces some of the error due to higher order perturbations.
To reduce this even further, without attempting to quantify
all of the perturbations, a relative measurement technique has
been utilized for which those errors which derive from
departure of the cavity walls from perfect sphericity and
rigidity are eliminated by cancellation. A reference gas with
well known physical properties or equation of state is
required.14
Argon has been exhaustively studied and was used as the
reference system for this experiment. Its behavior over the
range of pressures and temperatures involved is well
characterized by a truncated virial equation of state13, from
which the speed of sound may be accurately calculated.
Equation 3-4 may be rewritten as
where vk is defined as
v
k
(2nz)
>1-0,n-k
(3-9)

19
and k is an index that corresponds to the radial modes of
vibration only. Equation 3-8 is used to generate eigenvalues,
vk, corrected for the specific apparatus.
The experimental speeds of sound of the fluid under
investigation can be calculated by generalizing equation 3-8;
(3-10)
C* = V* fk
where ck is the speed of sound of the fluid at fk and vt .
Assuming that ck is not a function of frequency, then the
speed of sound ca was taken as the average of the speeds of
sound calculated from the radial mode frequencies obtained
experimentally with the spherical resonator apparatus. This
can be expressed by the following equation,
n
(3-11)
C.
Jc=l
a
n

endnote #19 diffusion stuff15
enote 2020 21 22 2! 2i 25 26 27 28
29 30 31 32 33 39 35

20
Table 3-1. The values of the frequency roots in ascending
order of magnitude with the purely radial modes in bold print.
Index
1
n
Index
?i,„
i
n
1
2.08158
1
i
37
16.3604
14
i
2
3.34209
2
1
38
16.6094
4
4
3
4.49341
0
1
39
16.9776
10
2
4
4.54108
3
1
40
17.0431
2
5
5
5.64670
4
1
41
17.1176
7
3
6
5.94036
1
2
42
17.2207
0
5
7
6.75643
5
1
43
17.4079
15
1
8
7.28990
2
2
44
17.9473
5
4
9
7.72523
0
2
45
18.1276
ii
2
10
7.85109
6
i
46
18.3536
8
3
11
8.58367
3
2
47
18.4527
16
1
12
8.93489
7
1
48
18.4682
3
5
13
9.20586
1
3
49
18.7428
1
6
14
9.84043
4
2
50
19.2628
6
4
15
10.0102
8
1
51
19.2704
12
2
16
10.6140
2
3
52
19.4964
17
1
17
10.9042
0
3
53
19.5819
9
3
18
11.0703
5
2
54
19.8625
4
5
19
11.0791
9
1
55
20.2219
2
6
20
11.9729
3
3
56
20.3714
0
6
21
12.1428
10
1
57
20.4065
13
2
22
12.2794
6
2
58
20.5379
18
1
23
13.4046
1
4
59
20.5596
7
4
24
13.2024
11
1
60
20.7960
10
3
25
13.2956
4
3
61
21.2312
5
5
26
13.4712
7
2
62
21.5372
14
2
27
13.8463
2
4
63
21.5779
19
1
28
14.0663
0
4
64
21.6667
3
6
29
14.2580
12
1
65
21.8401
8
4
30
14.5906
5
3
66
21.8997
1
7
31
14.6513
8
2
67
22.0000
ii
3
32
15.2446
3
4
68
22.5781
6
5
33
15.3108
13
1
69
22.6165
20
1
34
15.5793
1
5
70
22.6625
15
2
35
15.8193
9
2
71
23.0829
4
6
36
15.8633
6
3
72
23.1067
9
4

CHAPTER 4
DENSIMETER AND SPHERICAL RESONATOR RESEARCH AND DEVELOPMENT
The instrumentation required to accomplish the
experimental measurements presented here consists principally
of an innovative densimeter installed in a flow loop with a
high performance spherical resonator. Creation of the
densimeter involved a complete research and development
program, including design, fabrication, and the necessary
modifications dictated by the typically erratic path along the
learning curve leading to the new technology. In contrast to
the extensive development effort that was required to produce
the densimeter, the spherical acoustic resonator was generated
using proven principles previously developed and commonly
practiced in this laboratory.
Both the densimeter and the spherical resonator were
designed and constructed to withstand high pressures (up to
3000 psia) and operate over a broad temperature range (-80°C
to 150°C) . These subassemblies were constructed primarily from
304 stainless steel (304SS) and brass components, each sized
to meet the designed performance standards.
21

22
Densimeter
From theoretical considerations previously described, a
means is required to load and unload the spheres from the
force sensing device, and the net force must be suitably
transduced into a proportional electrical signal in order for
practical measurements to be made. Details of the lifting
mechanism depend on the actual loadings themselves, i.e the
spheres; therefore these were dimensioned first.
Spherical Buovs
The hollow spherical sinkers were designed such that
their volumes would be as closely matched as possible (V =V¡) ,
but their masses would be different by a factor of two
(2m1=m2) . The spheres were fabricated from four identical
hemispheres, each with 0.6250 inch OD (outside diameter)
machined from type 304 stainless steel. These were hollowed to
a 0.437 inch ID (inside diameter), yielding the necessary wall
thickness to sustain the maximum design pressure of 3000 psia
without collapse or significant dimensional change. To achieve
the desired mass ratio, a copper bead with a mass equal to
that of two of the hemi-spheres was made. Pairs of hemi¬
spheres were then Heli-Arc welded together forming two
outwardly identical spheres; one, however, contained the
copper bead, shown in figure 4-1. Table 4-1 lists the final
sphere masses and their volumes determined by pycnometry.1
See appendix A for pycnometer design.

23
Table 4-1. Masses and volumes of the spheres.
Sphere
Mass (g)
Volume (cc)
1
6.9821
1.9361
± 0.0001
± 0.0002
2
13.9658
1.9365
± 0.0001
± 0.0002
Spherical Buoy
Outer Shell
Weld
Copper Bead
Figure 4-1. Spherical buoy with enclosed copper bead.

24
Deflection Beam and Semiconductor Strain Gages
Having established the total mass and the dimensions of
the spheres, a deflection and sensing device was designed and
constructed. Miniature semiconductor strain gages were
purchased from Entran Devices Inc., model ESU-025; see table
4-2 for specifications. By fixing the strain gages on a thin
cantilevered beam from which the sinkers are suspended,
deflections due to the loadings could be detected and
correlated with the strain gage signals.
A deflection beam was cut from a piece of 0.007 inch
thick 302 stainless steel sheet metal, 0.150 inch wide by
0.620 inch long. The above dimensions yielded a deflection of
0.080 inch when a load of 25 grams was placed on the beam.
This corresponds to the manufacturer's optimum operating
strain range for the gages.
The gages were fixed to the beam per the manufacturer's
instructions using the application kit (ES-TSKIT-1) supplied
by Entran Devices, Inc. and the recommended polymeric epoxy,
(M-Bond 610) supplied by M-Line Accessories Measurements
Group, Inc. Four gages were applied to the beam, two on each
side. These became the arms of a bridge circuit that was used
in conjunction with an electronic strain gage conditioning
component (model 1B31AN) manufactured by Analog Devices. An
output signal device was built that detected and conditioned
the strain gage signal.2
See appendix B for circuit diagram.

25
A wire rack was fabricated to hang from the free end of
the beam and support the spherical sinkers. For the initial
trial tests, the sinkers were manually loaded and unloaded
from the rack. Degassed distilled water and n-hexane were used
as test buoyant media. The initial results yielded relative
errors of 5% to 10% assumed to be due mainly to air drafts and
temperature fluctuations which characterized the crude bench
tests. It was believed that these would diminish considerably
under the more stable conditions to be used in the planned
experiments.
Unfortunately, a substantial portion of the error proved
to derive from unforeseen design flaws which necessitated
fundamental changes. First, due to their miniature size, the
strain gages were not very robust. The delicate electrical
leads were easily damaged or broken and could not be
reattached. Second, nonlinear hysteresis and signal drift
effects were found in the beam assembly, as indicated by the
instability of the strain gage circuit output signal. The
drift effect is illustrated in figures 4-2 and 4-3 for two of
the weight loadings. It was not known whether these effects
were characteristic of the stainless steel shim or of the
epoxy resin used to apply the strain gages to the beam, and
the supply of semiconductor strain gages was exhausted before
the problem could be solved. At this point it was decided to
pursue a different and seemingly more manageable solution.

26

27
Although the above approach, using strain gages and a
flexible beam, was not utilized further in this research
project, plans to use this sensor have not been abandoned. We
have learned a good deal about how to proceed in this
direction. For future use, the following suggestions should be
considered: 1) Polycrystalline materials such as the stainless
steel alloy used in construction of the deflection beam are
unsuitable for use as spring elements and should be replaced
by more elastic materials such as fused quartz or single
crystal silicon, for example, to eliminate the hysteresis
effects seen in polycrystalline materials. 2) The beam
assembly should be redesigned so that the electrical
connections to the semi-conductor strain gages are protected
by encapsulating the fine gold wire leads in a layer of epoxy.
Coiled spring and LVDT sensor
Unable to bring the available strain gage deflection beam
into satisfactory practice within acceptable limits, attention
was shifted to a linear displacement measurement technology
already in successful use in the laboratory. The LVDT (Linear
Variable Differential Transformer) has been used as a linear
displacement sensor in acoustic resonance experiments for
measuring volume changes by transducing piston movement.19 That
experience suggested that an LVDT could be used to follow the
deflection of an elastic member at resolutions appropriate for
densimetry.

28
A high resolution LVDT was obtained from Trans-Tek Inc.
that could measure a total displacement of 0.100 inch; see
table 4-2 for further information. The LVDT consists of a
small weakly magnetic core (0.099 inch OD by 0.492 inch long)
that is used in conjunction with a separate induction coil
sensing device. An added appealing feature of employing this
technology was that there were no electrical connections
inside the pressure chamber and therefore no need for high
pressure electrical feedthroughs; also, the instrument can be
used in conductive fluids as well as non-electrolytes. And
finally, the fluid of interest would contact only the
noncorrosive materials within the densimeter.
The LVDT measures linear displacement along a single
axis, which dictated that the cantilevered deflection beam had
to be replaced with a device that exhibits a linear vertical
displacement. A coiled spring was selected as an obvious
choice. A commercial metal spring was selected for which the
total design displacement of 0.080 inch lay within the Hookian
region (linear displacement versus weight) and occurred when
a 25 gram total mass was attached. Initial tests of the metal
spring indicated that its reproducability was not within
acceptable limits. The metal spring exhibited a small but
devastating hysteresis effect, and it became obvious that
ordinary spring materials would be unsuitable for this service
and that one of extraordinary elasticity would be required.
The hysteresis is probably due to the irreversible changes
which occur under high strain at the grain boundaries of

29
polymorphic materials such as polycrystalline metals and
alloys. This suggests that single crystal or perhaps amorphous
spring elements would perform much better. Bench tests with
hand-coiled springs fabricated from fused quartz rod proved
this idea to be correct, at least for amorphous materials.
A quartz filament 0.06 inch OD was wound in a coiled
spring fashion to approximately 0.2 5 inch OD and 0.5 inch
length with the end loops turned up to make hooks used for
connection to other densimeter parts. In order to obtain the
desired deflection of 0.080 inch for a total mass load of 25
grams, the spring was placed in an etching solution of HF
(2.5M) and was removed at measured time intervals for testing.
This procedure gave excellent control over the spring
constant, and with it the desired deflection was easily
achieved.
Densimeter Body
The densimeter body assembly is shown in figure 4-4. The
bulk of the fluid chamber is a cylindrical tube machined from
304SS with a wall thickness of 0.125 inch and a length of
5.819 inch. A lower end cap 0.500 inch thick was machined to
slip fit the ID of the tube where it was then brazed into
place. A hole was drilled and tapped 1/8-NPT in the lower end
cap to allow for connection to other system components.

30

31
The densimeter body flange, 0.500 inch thick, with an OD
of 2.125 inch and an ID of 1.260 inch was drilled and tapped
to accept twelve 10-32 bolts on a 1.688 DBC (diameter bolt
circle). The flange was brazed onto the densimeter body
cylinder leaving an offset of 0.115 inch from the top of the
cylinder body, shown in figure 4-4. This offset was necessary,
as it was an integral part of the novel high pressure seal
used in construction of the densimeter apparatus. This seal
utilizes a copper o-ring gasket and two clamping flanges, one
being the flange on the densimeter body and the other a mating
flange located on the densimeter top assembly(detailed later
in this chapter) . To create this seal reliably the cross-
sectional area of the copper 0-ring gasket was required to be
95% to 98% of the cross-sectional area of the triangular gland
created by the clamping of the top and bottom flanges, see
figure 4-5. The copper 0-ring gasket used for this seal was
made by cutting a piece of 12 guage copper wire to a length of
3.92 inch. The wire was pre-formed to the diameter of the
densimeter and the ends were heli-arc welded together,
creating a continuous ring.
Buoyancy Assembly and the Sphere Lifting Mechanism
The buoyancy assembly consists of the quartz spring, the
LVDT core, the sphere suspension rack and the connecting
hardware, shown in figure 4-6. The suspension rack was
constructed by brazing two 0.062 inch diameter stainless steel

32
Top
Flange
Copper
0 —ring
Gasket
Bottom
Flange
Figure 4-5. Copper o-ring gasket and clamping flanges used
to generate a high pressure seal.

33
Brass Posts
Bottom view detail showing
V—shape sphere holders
Figure 4-6. Suspension rack assembly.

34
rods onto a brass disc. Short pieces of 0.031 inch diameter
stainless steel wire brazed to the rods at right angles and
bent in V-shapes formed supports for the spheres (see detail
in figure 4-6). The diameter of the brass disk was chosen such
that the entire rack could hang freely within the densimeter
body. The thickness of the brass disk was adjusted to yield
the appropriate mass needed to give the proper deflection when
used in conjunction with the spring and the spheres. The brass
disk was centrally drilled and tapped 5-40. Two brass rods
were constructed to connect the suspension rack to the quartz
spring. One rod was threaded 5-40 on one end to fit the brass
disk, the other end was threaded 1-72 to fit the LVDT core,
which was manufactured with tapped 1-72 holes on both ends.
The second rod was threaded 1-72 on one end to fit the LVDT
core, while a hole was drilled through a flattened section at
the other end to allow connection to the end-coil loop of the
quartz spring.
The lifting mechanism was designed and fabricated after
the vertical displacements due to loading and unloading the
spheres were determined precisely. A magnetically coupled cam
system and two vertical slides were constructed. These were
dimensioned to provide the proper clearance required for each
of the four loadings.
The lifter holder is a tubular brass piece machined with
an OD (1.049 inch) to slip fit inside the densimeter body.
The inside diameter of the lifter holder was large enough to

35
allow the suspension rack and the spheres to hang freely
without contacting the inside walls. Two slots were machined
axially 90° apart on the cylinder wall. The slots accommodate
the lifter slides. The lifter slides are shown in figure 4-7.
These have forks which extend into the fluid chamber to pick
up the densimeter spheres when actuated by the cam. The lifter
slides, machined from brass, are 3.990 inch long by 0.185 inch
wide by 0.145 inch thick. Small brass wheels were fitted into
slots on the bottom of the slides to facilitate relative
motion of the cam assembly. Two lifters were constructed from
1/32 inch diameter stainless steel wire and designed to lift
the spheres by cupping the spheres at three points. The
lifters where brazed to the lifter slides at the appropriate

36
position so that under operation, the spheres could be raised
completely off the suspension rack.
The magnetic cam assembly that produces the movement of
the lifters consists of a two-tiered cam, an internal
permanent magnet and an external coupled magnetic driver. The
cam with an OD of 1.030 inch has two steps joined by a 45°
ramp, see figure 4-8. The total rise between steps was 0.250
inch. From a top view of the cam, figure 4-8, four quadrants
Q1-Q4 are apparent and when the two lifters are placed in the
quadrants accordingly, they account for the four separate rack
loadings. The arc length around each step is large enough to
accommodate both lifters simultaneously (Q1 to Q3, Q2 to Q4).
This accounts for two of the necessary measurements; both

37
spheres raised from the rack, and both spheres supported by
the rack. The arc length from one step to the other (Q1 to Q2,
Q3 to Q4) is large enough to accommodate one lifter on each
step. This accommodates the remaining two measurements; one
sphere raised, the other supported and vice versa.
The magnetic turner, shown in figure 4-9, is a brass
piece (1.000 inch OD) with a stem at each end. The cam fits
over the top stem and is fixed into position with a set-screw.
Brass spacers with ball bearings separate the cam from the
turner as well as the turner from the bottom of the
densimeter. It is the spacer/bearing combination that allows
movement to occur. A hole was drilled through the diameter of
the turner to accommodate a ferromagnet, which was held in
position with a set-screw.
An external drive ring, figure 4-9, was placed around the
outside of the densimeter body and loaded with two
ferromagnets. Together these generate a strong magnetic field
with which the magnetically loaded cam aligns. This
configuration produced an adequate turning force.
Tod Flange Assembly
The top flange assembly, shown in figure 4-10, contains
the spring and LVDT sensing element. The spring housing is
removable to allow for servicing or replacement. The LVDT
sensing element is held in a brass case, designed and
constructed to allow translation in the vertical direction,
which allows adjustment over density ranges.

38
Spherical Resonator
The spherical resonator used in this research project was
designed and built based on knowledge and experience
previously gained in the development of related instruments in
this laboratory. The spherical resonator assembly consists
principally of the spherical cavity and the acoustic
transducer mounting hardware.
Resonator cavity
The resonator cavity was designed with the following
considerations; 1) The cavity must be able to withstand high
pressure and extreme temperatures. 2) The cavity should have

39

40
a small volume yet still produce resonant frequencies when
filled with test fluids within our measurement capabilities
(typically acoustic frequencies less than 50KHz).
The spherical cavity consists of two identical 304SS
hemispheres each cut with a 1.000 inch radius. The inside
cavity was polished to obtain a smooth acoustically reflective
surface. The outside of the hemispheres was machined to
accommodate clamping flanges, shown in figure 4-11. A minimum
wall thickness of 0.250 inch was maintained to ensure the
cavity's ability to withstand the maximum design fluid
pressure of 3000 psia. A V-groove was cut on the 2.125 inch
diameter of each cavity block to house a diamond shaped copper
o-ring gasket (figure 4-11) that operates similarly to the
gaskets described previously. A 0.650 inch diameter hole was
cut at the apex of each of the cavity hemispheres to contain
the acoustic transducers and their mounting hardware.
Transducer Assembly
The transducer assembly, shown in figure 4-12, consists
of the transducers, the transducer mounting pieces and the
electrical feedthrough end-cap. The transducers were cut from
a piezo-electric speaker element, see table 4-2 for
specifications, to a diameter of 0.680 inch. The transducers
were held in position by using stainless steel spacers, which
in turn were lightly loaded by means of co-axial wave springs.
The spacers slip fit over the feedthrough in the end-cap where
the electrical connection was made.

41

42
for the spherical resonator.

43
The end-cap was constructed to serve as the electrical
connection to the transducers. The end-cap utilizes the copper
o-ring gasket technology which allows the transducers to be
removed and serviced without compromising the spherical cavity
seal. A ceramic-insulated copper electrical feedthrough was
obtained with a stainless steel weld preform. The endcap was
drilled through to accept the weld preform which was
subsequently welded into position.
Final Assembly
Figures 4-13 and 4-14 show assembled views of the
densimeter and spherical rensonator, respectively. The
instruments were connected using thick wall 1/4 inch OD
stainless steel tubing (304SS). The necessary valves to allow
for charging of the apparatus with experimental fluid and
later servicing of the instrument were incorporated into the
experimental setup and attached in-line using the stainless
steel tubing.

44
Top Flange
Assembly
Suspension
Rack
Densimeter
Body
Internal
Magnetic
Turner
Assembly
Figure 4-13. Assembled view of the densimeter

45
spherical resonator.

Table 4-2. Summary of Measurement hardware.
Item
Manufacturer
Model
number
Range of
operation
Resolution
Function
Semiconductor
strain gage
Entran
Devices,Inc.
ES-025
o-iooo fie
(€=strain)
±0.0001
volts
Measure
deflection of
beam.
Semiconductor
strain gage
output device
University of
Florida,
Electronics
Dept.a
N/A
N/A
Accepts
signal from
strain gage
circuit
bridge.
LVDT
Trans-Tek,
Inc.
240-0015
+0.050 inch.
+ 1 micron
Measure
deflection of
spring.
Piezo-electric
speaker element
Tandy Corp.
High
Efficiency
500Hz-50kHz
N/A
Generate and
receive
acoustic
signal
see appendix B for circuit diagram used
0>

CHAPTER 5
EXPERIMENTAL
A set of confirmatory experiments was performed to test
the operation of the densimeter and to verify that the entire
apparatus, including the spherical resonator, was functioning
as expected. For these purposes studies were carried out on
a nearly ideal gas and a nonideal gas at high densities.
Acoustic resonance spectra, density, pressure and temperature
were acquired for both argon and carbon dioxide at several
isotherms over a range of densities.
Gases
The argon (supplied by Bitec) used in this experiment had
a purity greater than 99.99 mole percent. The carbon dioxide
(supplied by Scott Specialty Gases) had a purity greater than
99.999 mole percent. No further analysis or treatment was
performed on the gases.
Interfacing
All components of the densimeter/spherical resonator
apparatus (DSRA) were linked to a micro-computer where data
were collected and processed. The computer contained a 486
internal processor operating at 33 megahertz with 16 megabytes
47

48
of RAM. Figure 5-1 shows the schematic setup of the electronic
hardware used in the experiments. The interfacing protocol
used for the connection of the electronic hardware to the
micro-computer was the standardized IEEE general purpose
interface bus (GPIB) , including the 8-bit i/o card and the
accompanying software. The electronic equipment used in the
experiment contained the IEEE connection interface when
purchased.
Measurement Hardware
A schematic diagram of the instrumental setup is shown in
figure 5-2. Each physical property measurement required a
specific protocol for acquisition and subsequent processing of
the data. The following sections describe the measurement
procedures, including operation of the electronic hardware,
listed in table 5-1, and give details of the computer software
used in the experiments.
Pressure and Temperature
A Sensotec model TJE/743-03 strain gage pressure
transducer was used in conjunction with a Beckmann model 610
electronic readout device to acquire the pressure data. For
the pressure calibration of the Sensotec transducer, a Ruska
model 2465 standardized Dead Weight Pressure Gage was used.
The Ruska gage had a pressure range of 0.000 psia to 650.000

Figure 5-1. Schematic diagram of the electronic hardware setup.
VO

Temperature Controlled Chamber
Figure 5-2. Schematic diagram of the apparatus setup.
ui
o

51
psia and an accuracy of ± 0.001 psia. The pressure calibration
results are shown graphically in figure 5-3.
A four-wire platinum resistance temperature device RTD
was used with a Keithly model K-196 digital multimeter to
obtain resistance values that were later converted to
corresponding temperatures. The platinum RTD was calibrated
versus a standardized RTD (NIST traceable) obtained from HY-
CAL engineering.
The Beckmann readout device and the Keithly multimeter
were interfaced to the computer via the IEEE general purpose
interface bus (GPIB).
Density
The displacement information of the four separate
spherical buoy loadings was linearly translated to voltage by
the LVDT and its circuit. The voltage data were acquired with
an Iotech model ADC488/8SA analog-digital converter (ADC) that
was connected to the computer by means of an IEEE interface.
The advantage of using the ADC rather than acquiring the
voltage with a digital multimeter is the speed of the former
which gives it the ability to capture a relatively large
amount of data in a short period of time. For example, in this
experiment, each recorded single-loading voltage value was in
fact an average of 16,384 independent voltage measurements
acquired at a rate of 1 kHz. This resulted in an average
aquisition time of 16.4 seconds. Four voltage values,

52
a Experimental data — Regression fit
Figure 5-3. Pressure gage calibration data using argon
gas at 298 K.

53
corresponding to the separate quartz spring loadings, were
collected and stored in the computer. These four values were
then used to calculate a single density datum point by means
of equations 2-8 through 2-11.
Speed of Sound
The speed of sound of the experimental fluid was
calculated using information obtained from a measurement
technique involving generation and acquisition of acoustic
resonance frequencies within the spherical cavity. The
resonance frequencies were produced by stimulating one
transducer with a waveform generated by a Hewlett Packard
HP3325B function generator. The waveform signal was a
continuous sine-wave ramp over a frequency range of 0-50KHZ
with a time base of 0.1638 seconds. The receiving transducer
was connected to a Stanford Research pre-amplifier model SR-
530 where the signal was conditioned with a band-pass filter
and amplified. The SR-530 was connected to the Iotech ADC. The
time domain resonance data were acquired at a rate of lOOKHz
and were processed using a fast fourier transform (FFT). A
typical time domain data plot and the subsequent FFT are shown
in figures 5-4 and 5-5, respectively. The resonance frequency
data were used as explained previously in chapter three to
calculate the speed of sound.

54
0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350
Time (s)
Figure 5-4. ADC signal of argon at 278.15 K and 1417.4
psia.
Figure 5-5. FFT of ADC signal of argon at 278.15 K and
1417.4 psia.

55
Experimental Data Collection Procedure
The entire apparatus was cleaned with multiple washings
of acetone. It was then baked at 150 °C, and a vacuum was
applied to remove all contaminants from the system. The system
was then charged with the gas of interest to approximately
100 psia, allowed to eguilibrate, and a vacuum was again
applied. The charging procedure was carried out a minimum of
four times before the final loading of gas was injected into
the system.
The entire apparatus was contained within an insulated
temperature-controlled chamber that had been adapted with a
novel liquid nitrogen refrigeration control valve. This valve
along with a proprtionation-integration-differentiation (PID)
control algorithm provided temperature control to better than
+ 0.01 K.
Once the system was charged to the appropriate pressure
loading and was equilibrated, data were acquired using
software developed specifically for this experiment. The
magnetically driven cam assembly of the densimeter was turned
to the first displacement position. The corresponding LVDT
reading, along with the pressure and temperature, were
recorded and stored in the computer. Consecutive turnings of
the cam assembly were performed until all four displacement
readings were obtained. An acoustic-resonance time-domain
signal was acquired between the second and third displacement

56
positions. This system of data acquisition constituted one
trial. A run consisting of a minimum of 10 trials was
performed under each set of equilibrium conditions (P,T,p).
Consecutive runs were performed along the same isotherm
but at different densities. This was achieved by bleeding a
sufficient amount of the experimental gas from the system to
attain a new lower density. On reaching the low pressure
limit, the system was recharged to the upper density range
limit, equilibrated and another isothermal run was started.
This procedure was followed until the temperature range of the
study was completed.

Table 5-1. Summary of measurement hardware.
Item
Manufacturer
Model
number
Range of
operation
Resolution
(Manufacturer's
stated value)
Function
Function
Generator
Hewlett
Packard
HP 3325B
1 |iHz to
20.99 MHz
1 /¿Hz
Generates a
precise input
waveform.
Low-noise
preamplifier
Stanford
Research
Systems
SR560
< 250 mV rms
input
1 % Gain
Filter and
amplify
acoustic
signal.
Digital strain
gage transducer
indicator
Beckmann
Industrial
Corp.
610
-150 m to 1
volt
0.01 % Full
Scale
Accepts
strain gage
transducer
input
Strain gage
pressure
transducer
Sensotec
TJE/743-03
0-3000 psia
+0.01% full
scale
Measure
pressure.
RTD
HY-CAL
Engineering
RTS 31-A
-200 to 280°C
±0.00 Ohms
Measure
temperature.
Analog/Digital
Converter
Iotech
ADC488
± 10 volts
(Max. scan
rate-100MHz)
53 /iVolts
Converts
analog signal
to digital
format.

Table 5-1 (Continued).
Item
Manufacturer
Model
number
Range of
operation
Resolution
(Manufacturer's
stated value)
Function
Digital
Multimeter
Keithly
Instruments
Inc.
kl96
4-terminal
resistance
measurements,
100 /iQ to
20 MQ
Range
dependent, see
manual for
details.
Measure
resistance
from Pt
temperature
probe.
Computer
interface board
National
Instruments
GPIB-PCII
Transfer rate
200 Kbytes/s
N/A
Communication
interface for
electronic
hardware.
Ul
03

CHAPTER 6
RESULTS
Eigenvalue Calibration
The data for calibration of the radial mode eigenvalues
were collected using argon at 295.15 + 0.01 K over a pressure
range of 50.0 ± 0.1 psia to 322.0 ± 0.1 psia. The speeds of
sound, calculated using a third-order truncated virial
equation of state20, and the experimental frequencies were used
with equation 3-8 to calculate the experimental eigenvalues,
and these are listed in table 6-1. Table 6-2 shows a
comparison of the experimental eigenvalues for this apparatus
with the theoretical eigenvalues for a perfect sphere and
lists the relative percent deviations. The results illustrate
that the perturbations from ideality in a well designed and
constructed cavity are quite small.
Argon
The measurements on argon provided an initial test of the
apparatus' functionality. Argon was chosen for this test
because of its ready availability and its near ideal behavior
even at substantial pressures. Data were collected on argon
over a density range of 65.825 kg/m3 to 205.07 kg/m3 with an
59

60
Table 6-1. Experimental data and results for the spherical
eigenvalue calibration using argon at 295.15 K.
p
± 0.1
(psia)
ca
(m/s)
kb
± 1.5
(Hz)
V*
(m)
327.1
322.05
1
9066
3.551E-02
2
15589
2.0657E-02
3
22009
1.4632E-02
4
28370
1.1351E-02
5
34747
9.2683E-03
6
41117
7.8323E-03
304.5
321.88
i
9059
3.553E-02
2
15576
2.0664E-02
3
21986
1.4640E-02
4
28343
1.1356E-02
5
34733
9.2671E-03
6
41082
7.8349E-03
200.8
321.15
1
9045
3.550E-02
2
15556
2.0644E-02
3
21954
1.4628E-02
4
28305
1.1346E-02
5
34693
9.2566E—03
6
41023
7.8284E—03
50.7
320.24
1
9013
3.552E-02
2
15509
2.0648E-02
3
21891
1.4628E-02
4
28219
1.1348E-02
5
-
-
6
40843
7.8407E-03
a Speed of sound calculated using a truncated virial
equation of state.(Source: Hirschfelder J.O., Curtiss C.F.,
and R.B. Bird, Molecular Theory of Gases and Liquids, Wiley
and Sons, New York, 1954.)
Radial mode index

61
Table 6-2. Theoretical radial mode eigenvalues listed with the
averages of the corrected eigenvalues.
k
Average
Corrected
Eigenvalues
v„
± 0.0091E-3
(m)
Theoretical
Eigenvalues
^l-O.n-k
(m)
Relative
Deviation
(%)
1
3.552E-02
3.552E-02
0.000
2
2.0653E-02
2.0658E-02
-0.024
3
1.4632E-02
1.4635E-02
-0.021
4
1.1350E-02
1.1350E-02
0.000
5
9.2683E-03
9.2675E-03
0.009
6
7.8354E-03
7.8342E-03
0.015
average standard deviation over that range of ± 2.43 kg/m3.
The corresponding speed of sound range was 313.78 m/s to
344.34 m/s with an average standard deviation of ± 0.54 m/s.
The argon density and speed of sound data were acquired on the
following three isotherms; 278.15 K, 288.15 K and 298.15 K
with a standard deviation ± 0.01 K for each temperature. The
results of the measurements performed on argon using the DSRA
are listed in table 6-3. Figures 6-1 and 6-2 show graphically
the speed of sound and density versus pressure along the
isotherms for which data were collected compared to data
calculated using the truncated virial equation of state. The
argon equation of state has been tested and confirmed to yield
calculated density and sonic speed values with relative
uncertainties of < 0.10% over the above temperature and
density ranges.13 Comparison of the experimental speed of sound

62
and density data with that of the calculated virial equation
of state data gives a reflection of the accuracy of the
instrument. Figures 6-3 through 6-5 show the relative
deviation of the experimental speed of sound (100(cexp-
cviri«i) /cviri»i ) versus pressure. Figures 6-6 through 6-8 show
the relative deviation of the experimental density (100(pexp-
Pviri.i)/Pviriai ) versus pressure.

63
Table 6-3. Experimental argon (39.948 g/mol) data with the
average standard deviation given at each column heading.
T
P
c
p
± 0.01
± 0.5
± 0.54
± 2.34
(K)
(psia)
(m/s)
(kg/m3)
278.15
563.6
313.78
69.296
753.7
315.70
94.221
858.5
316.89
107.38
1016.8
318.84
128.26
1208.3
321.47
153.32
1359.7
323.68
172.84
1417.4
324.29
183.02
1544.5
325.94
198.80
288.15
581.6
319.92
68.552
740.7
-
88.706
887.2
-
106.39
995.3
324.82
120.52
1000.8
-
120.77
1150.0
-
138.65
1253.2
328.08
151.21
1270.0
328.41
154.19
1416.7
331.14
174.03
1471.3
332.06
181.24
1671.9
335.93
205.13
298.15
578.4
-
66.100
731.0
-
83.470
864.9
-
99.651
1014.1
-
117.23
1166.1
333.31
135.02
1304.5
335.52
151.89
1312.7
-
152.45
1412.0
-
164.04
1444.4
337.98
169.77
1614.5
340.68
189.42
1755.8
343.08
208.39

64
* 278.15 K o 288.15 K ° 298.15 K
Figure 6-1. Experimental data points and virial equation of
state curves of the speed of sound vs. pressure isotherms
for argon.

65
Figure 6-2. Experimental data points and virial equation of
state curves of density vs. pressure isotherms for argon.

66
sound compared to the calculated virial speed of sound
of argon versus pressure at 278.15 K.
0.06
0.04
C? 0 02
o-
â– u
© -0.02
1o
©
a -0.04
A A
-0.06
A
-0.08 * 1 + t + 4 * 1 + + ♦ 1 t
400.0 600.0 800.0 1000.0 1200.0 1400.0 1600.0 1800.0
Pressure (psia)
Figure 6-4. Relative deviation of experimental speed of
sound compared to the calculated virial speed of sound
of argon versus pressure at 288.15 K.

67
sound compared to the calculated virial speed of sound
of argon versus pressure at 298.15 K.
0.60
0.40
0.20
£
c 0.00
o
1
S -0.20
TJ
I -0.40
a:
-0.60
-0.80
-1.00
400.0 600.0 800.0 1000.0 1200.0 1400.0 1600.0
Pressure (psia)
Figure 6-6. Relative deviation of experimental density
compared to the calculated virial density vs. pressure
of argon at 278.15 K.

68
compared to the calculated virial density vs. pressure
of argon at 288.15 K.
0.80
0.60
_ 0.40 -
g
c 0.20 -f
o
{
r\ r
l -0.20
-0.80
400.0 600.0 800.0 1000.0 1200.0 1400.0 1600.0 1800.0
Pressure (psia)
Figure 6-8. Relative deviation of experimental density
compared to the calculated virial density vs. pressure
of argon at 298.15 K.

69
Carbon Dioxide
P, T, c, and p data were acquired for high pressure
carbon dioxide to extend the confirmatory measurements using
the DSRA to a non-ideal fluid. In addition to corroborating
the apparatus' functionality, carbon dioxide data afford a
means to illustrate the utility of the apparatus for physical
property determinations in general.
Experimental Data
The data obtained from the measurements performed on
carbon dioxide are listed in table 6-4. The temperature,
pressure, speed of sound, and density ranges and the average
standard deviation over those ranges were 290.15 K ± 0.01 K to
310.15 K + 0.01 K, 1005.5 psia to 1804.3 psia ± 0.43 psia,
211.30 m/s to 392.98 m/s ± 0.28 m/s, 584.11 kg/m3 to 889.82
kg/m3 ± 5.92 kg/m3, respectively.
The relative deviation of the present experimental carbon
dioxide density data compared to other values found in the
literature is presented in figures 6-9 through 6-12. Due to
the scarcity of published carbon dioxide speed of sound data,
figure 6-13 simply presents the experimental speed of sound
data graphically versus pressure. The lines are drawn as a
visual aid to help identify the various isotherms.

70
Table 6-4. Experimental carbon dioxide (44.011 g/mol) data
with the average standard deviation given at each column
heading.
T
P
c
P
± 0.01
± 0.4
± 0.37
±4.17
(K)
(psia)
(m/s)
(kg/m3)
290.15
1017.5
342.07
841.11
1068.8
348.15
843.17
1127.9
353.98
851.79
1355.3
375.33
868.24
1612.4
392.98
889.82
297.15
1048.6
299.33
770.67
1127.3
311.43
783.08
1257.5
325.70
804.19
1401.3
342.65
821.15
1573.0
358.88
839.77
300.15
1005.5
277.26
700.14
1052.8
285.17
720.59
1329.8
320.03
787.07
1581.4
343.01
819.09
1804.3
362.51
836.30
303.15
1235.4
288.12
730.14
1500.5
320.35
780.75
1769.3
343.87
814.97
307.15
1172.2
228.50
584.11
1219.0
239.10
639.18
310.15
1340.5
212.27
636.14

71
0.30
0.20
!* 0.10
; 0.00
»
| -0.10
i -0.20
-0.30
-0.40
1000.0 1100.0 1200.0 1300.0 1400.0 1500.0 1600.0 1700.0
Pressure (psia)
Wagner(21) v Kessel'man(22) m Duschek(23) x Holste(20)
Figure 6-9. Relative percent deviation of literature values
of density for carbon dioxide at 290.15 K from the present
measurements.
3!
s>
or
Pressure (psia)
| â–  Wagner(21) v Duschek(23) >: Michels(32)
Figure 6-10. Relative percent deviation of literature
values of density for carbon dioxide at 297.15 K from
the present measurements.

72
["â–  Wagner(21) v Duschek(23) s Ely(24) % Michels(32) |
Figure 6-11. Relative percent deviation of literature
values of density for carbon dioxide at 300.15 K from
the present measurements.
0.20
0.15
| 0.10 -â– 
Q
0.05
0.00
1200.0 1300.0 1400.0 1500.0 1600.0 1700.0 1800.0
Pressure (psia)
Wagner(21) v Duschek(23) ffl Ely(24)
Figure 6-12. Relative percent deviation of literature
values of density for carbon dioxide at 303.15 K from
the present measurements.

73
CU
CU
CL
cn
400.00
350.00
300.00
250.00
200.00
1000.0 1200.0 1400.0 1600.0 1800.0 2000.0
Pressure (psia)
- 290.15 K * 297.15 K « 300.15 K * 303.15 K -+•■ 307.15 x 310.15 K
Figure 6-13. Experimental speed of sound data versus
pressure of carbon dioxide at different isotherms.

74
Equation of State-Data and Results
Experimental physical chemistry and its application to
the thermodynamics of fluids ultimately strives to provide
insight into the behavior of chemical systems over a wide
range of state variables. Important aspects of this behavior
are quantified in several different manners as specific
physical properties, enthalpy (H) , entropy (S) , heat
capacities (C) , ... etc. To illustrate the utility of the
DSRA, which acquires T, P, c, and p data simultaneously, for
deducing other thermodynamic properties, the heat capacities
at constant pressure Cp and constant volume Cv and their ratio
y were calculated for carbon dioxide.
The experimental data were fit to the Redlich-Kwong (R-K)
equation of state
vm-b jrvjv^b) 16 X)
and the R-K parameters a,b were determined as described in
appendix E. Figure 6-15 compares the experimental densities to
the R-K calculated densities graphically versus pressure.
Table 6-5 lists the experimental densities and the R-K
calculated densities and relative percent deviations between
them ( 100 (pexp - pR-K) / pR_K ).
Using fundamental thermodynamic relationships, equation
6-1 and the experimental data for carbon dioxide, values for
Cp, Cv, and y were calculated (refer to appendix E for

75
1000.0 1200.0 1400.0 1600.0 1800.0
Pressure (psia)
- 290.15 K * 297.15 K • 300.15 K * 303.15 K * 307.15 K » 310.15 K
Figure 6-15. Carbon dioxide density, experimental data
and RK calculated curves, versus pressure.

76
Table 6-5. Experimental and Redlich-Kwong calculated densities
of carbon dioxide.
Redlich-Kwong constants
a = 5.8752 ± 0.0037 Pa m6 K1/2 mol-2
b = 2.6010E-5 ± 0.0001E-5 m3 mol-1
T
(K)
P
(psia)
Pexp
(kg/m3)
Prk
(kg/m3)
Relative
deviation
(%)
290.15
1017.5
841.11
830.36
-1.28
1068.8
843.17
837.90
-0.63
1127.9
851.79
846.01
-0.68
1355.3
868.24
873.18
0.57
1612.4
889.82
898.50
0.98
297.15
1048.6
770.67
758.49
-0.60
1127.3
783.08
774.59
-1.08
1257.5
804.19
798.77
-0.67
1401.3
821.15
820.82
-0.04
1573.0
839.77
843.01
0.39
300.15
1005.5
700.14
-
_
1052.8
720.59
-
-
1329.8
787.07
780.63
-0.82
1581.4
819.09
819.62
0.06
1804.3
836.30
846.65
1.24
303.15
1235.4
730.14
723.87
-0.86
1500.5
780.75
780.47
-0.04
1769.3
814.97
819.65
0.57
307.15
1172.2
584.11
618.62
5.91
1219.0
639.18
649.21
1.57
310.15
1340.5
636.14
641.10
o
i
thermodynamic equations used to calculate the heat
capacities) . Table 6-6 lists Cp, Cv, and y determined from the
experimental data and the R-K equation of state along with
literature values. Due to the lack of published values for
carbon dioxide over the range of these experiments only a few
comparisions could be presented as shown in table 6-6.

77
Table 6-6. Comparison of experimental Cp, Cv, y values with other
experimental literature values.
T = 290.15 K P = 1612.5 psia
Source
Cp
J/(mol-K)
RDa
(%)
Cv
J/(mol-K)
RD
(%)
Y
RD
(%)
DIN33
107
± 1
2
50.3
± 1.5
5.6
2.13
± 0.01
-3.80
Int.
Critical
Tables34
105
± 5
8
48
± 1
10
2.2
± 0.1
-6.9
Vargaftik26
108.1
± 0.5
1.6
This work
108.91
± 0.37
53.14
± 0.41
2.049
± 0.009
T = 300.15 K P = 1
329.5
osia
Source
J/(mol K)
RD
(%)
c,
J/(mol K)
RD
(%)
Y
RD
(%)
DIN33
140
± 1
2
50.9
± 1.5
-2.5
2.75
4.72
Int.
Critical
Tables34
149
+ 5
-4
46
± 1
8
3.2
-9.8
Vargaftik26
143.9
± 0.5
-0.8
This work
142.73
± 0.28
49.59
± 0.39
2.884
± 0.006
Relative deviation

78
Uncertainties in Measured and Calculated Values
For argon, the precision of the pressure, temperature,
speed of sound and density measurements were +0.5 psia, ± 0.006
K, ± 0.54 m/s, and ± 2.34 Kg/m3, respectively. The accuracy of
the pressure and temperature measurements based on the
calibration standards were + 0.01 K and ± 0.1 psia. The
accuracies of the speed of sound and density measurements
relative to those inferred from the truncated virial equation of
state were no greater than ± 0.15 m/s and ± 0.65 kg/m3.
The precision and accuracies of the measurements of
pressure and temperature for carbon dioxide were the same as the
argon measurements. The precision of the experimental speed of
sound and density measurements for carbon dioxide were ± 0.37
m/s and ± 4.17 kg/m3. The uncertainty in the experimental
density measurements in the range of the present measurements
was less than ± 0.36 kg/m3.
The R-K equation of state for carbon dioxide obtained as
the best fit to these measurements yielded an average relative
deviation of ± 0.98 kg/m3 of the R-K calculated densities from
the experimental densities. The above uncertainties lead to
average uncertainties in Cp, Cv, and y of + 0.40 J/(mol-K), +
0.41 J/(mol-K), ± 0.008, respectively.

CHAPTER 7
CONCLUSION
The research effort presented in this document focused on
the development of an apparatus that has the capability of
measuring pressure, temperature, speed of sound, and density
of fluids over wide ranges of the variables. An apparatus was
designed, built and tested. It accomplished this goal by
measuring the above variables simultaneously. While reliable
pressure and temperature data acquisition are routine in many
scientific laboratories, the measurement of the sonic speed
and density are not and require deliberate effort and insight
into the experimental approach.
The apparatus employed two separate and experimentally
significant data collection instruments coupled together to
perform measurements on the same system of gas. The first
instrument, the spherical resonator, utilized technology for
the generation and acquisition of acoustic resonance
information of fluid-filled cavities which has been developed
in this laboratory and has become a staple of our laboratory
procedures. The utility of the technique derives from the high
precision and accuracy with which resonance phenomena may be
discerned. The second instrument, the densimeter, was built in
response to an effort to advance the capabilities of the
79

80
laboratory and introduce a new era of useful physical property
determination and prediction. Since a robust, versatile,
instrument for the direct measurement of density is not
commercially available, nor has such an instrument been
described in the literature, complete design, fabrication and
development of a densimeter was required and ultimately
accomplished. While the fundamental theory of the operation
of the densimeter was not advanced, the art of completing
useful measurements was. For the first time fluid density can
be accurately inferred from electrical signals acquired non-
invasively by induction in an instrument which is self¬
calibrating during each use.
The combination of the above instruments along with
precision devices to measure pressure and temperature resulted
in achievement of the major goal of this research, namely a
practical densimeter/spherical resonator apparatus, DSRA. The
success of the DSRA is evident from the results of the
confirmatory experiments, performed on argon and carbon
dioxide. The argon measurements were performed over a pressure
range of 560.6 psia to 1755.8 psia at three isotherms covering
a range of 278.15 K to 298.15 K that yielded a density range
of 69.296 kg/m3 to 207.06 kg/m3. The corresponding speed of
sound measurements for argon covered a range of 313.77 m/s to
344.34 m/s. The carbon dioxide data encompassed a pressure
range of 550.5 psia to 1302.3 psia at six isotherms spanning
a range of 273.15 K to 308.15 K. These ranges yielded density

81
measurements from 618.62 kg/m3 to 898.50 kg/m3 and speed of
sound measurements 304.22 m/s to 318.54 m/s.
Uncertainties in the presently measured densities are
beyond the targeted goal of + 0.1 %, but this derives not from
a fundamental flaw in the principles applied but rather from
the fact that the attainable precision in measurements of this
kind is determined in large part by the difference in density
between the fluid and that of the lightest buoy. The optimum
condition is that for which the buoy density is only slightly
higher than that of the fluid. To illustrate this fact
consider determination of the fluid density from measurements
of the three suspension rack loadings corresponding to
equations 2-8, 2-9, 2-10. Assume for simplicity that m2 = 2mt
= m and V2 = V2 , which table 4-1 shows to be nearly true. The
fluid density p£ then is found to be
Pf =
p'
S2-2S1
Si-S,
(7-1)
where p' is m/V, the net density of the lighter spherical
buoy, and S2 and S2 are the net LVDT signals relative to the
unloaded suspension rack. The uncertainty in the computed
value of p depends on the uncertainties in the LVDT signals
and the magnitude of the buoy density. Taking the signal
uncertainties to be equal u3 and the uncertainty in buoy
density to be negligible these propagate to an uncertainty in
fluid density upf which is

82
UB /
u. = — p'
(7-2)
This shows that a
large buoy
density leads
to a
correspondingly large
uncertainty
in the measured
fluid
density. By these considerations it should be clear that the
highest precision and smallest uncertainty in pf caused by
choice of buoy density occurs when these densities are equal.
Because it is necessary for p1 to exceed pf for the instrument
to operate, the buoy density should be only slightly greater
than that of the fluid for optimum results.
In practice it is necessary to operate with buoys dense
enough to accommodate the densest fluid to be studied. This is
why it is recommended to provide future versions of this
instrument with interchangeable sets of buoys with densities
appropriate for use with a variety of fluids. In this work the
density of the lightest buoy exceeded that of the densest
fluid measured by a factor of 100. If that were reduced to say
1.5 the uncertainty in measured density would improve by about
a factor of 1/100. It should be apparent that attaining the
level of precision observed in the present test measurements
is a confirmation of the potential of the technique for
accurate fluid density determinations. The buoys fabricated
for this instrument were intended ultimately for use with
liquids and high pressure gases under conditions for which the
mismatch in fluid and buoy density is much smaller.

APPENDIX A
SPHERICAL BUOY MASS AND VOLUME CALIBRATION
As stated in the development of the operational theory of
the densimeter, chapter 2, the mass and volume of each
spherical buoy must be known accurately. The masses of the
spheres were determined using an analytical balance that was
calibrated with certified standard masses. In order to
determine the volume of each sphere, degassed, distilled water
was used with a pycnometer that was built specifically to
perform this task.
For the mass calibration of each sphere, data of the
electronic load cell balance readouts versus the standard
masses were collected in the region of mass of each sphere.
Each sphere was then weighed on the analytical balance and the
corresponding calibrated mass was calculated, see figures A-l
and A-2. Corrections for the buoyancy of air on the standard
brass masses and spheres were applied.
The pycnometer, shown in figure A-3, consisted of a
volume chamber, a spherical lid, and a frame to hold the
pycnometer in place. The volume chamber, 0.750 inch OD and
1.00 inch in length, was bored out to an ID of 0.676 inch. The
top surface of the volume chamber was machined and lapped to
83

84

85
Figure A-l. Graph showing calibration data used to
determine the mass of spherical buoy #1.
■ Calibration Data — Regression Fit o Mass of Buoy #2
Figure A-2. Graph showing calibration data used to
determine the mass of spherical buoy #2.

86
fit exactly the contour of a precision stainless steel ball¬
bearing with a 0.750 inch OD. The ball-bearing was used as a
lid which could be reproducibly seated in a manner which
assured the volume was filled with liquid and there were no
trapped air bubbles. The pycnometer frame was built from
aluminum with a screw mounted at the top. This screw was used
to hold the spherical ball lid firmly in place thereby
producing a positive liquid tight seal confining a definite,
fixed volume inside the cell. A torque screw driver assured
the sealing load was reproducible.
One volume measurement consisted of filling the
pycnometer with water, drying the outside completely, seating
the lid, and measuring the mass of the assembly using an
analytical balance. A sphere was then placed in the volume
chamber displacing an equivalent volume of water and the
assembly was re-weighed. The difference in these measurements
is the mass of the sphere less the mass of the displaced
water. The latter is proportional to the volume of water that
was displaced by the sphere. By knowing the density of the
water at temperature, T, of the measurement, the volume of the
spherical buoy can be calculated using the following equation,
V = Mb + M{p^ ~ (A-l)
pw(T)
where V. is the volume of the sphere, Ms is the mass of the
sphere, Mp+W.t3 is the mass of the pycnometer, the undisplaced

87
water (w1) and sphere, Mp,w is the mass of the pycnometer
filled with water (w) and pw(T) is the density of the water.
The results for the volume calibrations are listed in table
A—1.
Table A-l. Data and results for the volume calibration of the
spherical buoys.
P294.65 KÍHzO)1 = 0.99873 g/CC Mp+W = 72.4896 g
± 0.0002 g
Spherical
Buoy
ms (g)
mp+„.+3 (g)
Volume (cc)
1
6.9821
± 0.0001
77.5380
± 0.0002
1.9329
+ 0.0002
2
13.9658
± 0.0001
84.5194
± 0.0001
1.9331
± 0.0002
1 Weast R.C and M.J. Astle, CRC Handbook of Chemistry and
Physics, 1974, F-ll.

APPENDIX B
CIRCUIT DIAGRAM FOR THE STRAIN GAGE SIGNAL CONDITIONER
The block diagram of the strain gage signal conditioning
component is shown in figure B-l.28 The circuit diagram that
was followed in the construction of the signal conditioning
device is given in figure B-2.28
FUNCTIONAL BLOCK DIAGRAM
Figure B-l. Block diagram of the strain gage signal
conditioning component.
88

89
Figure B-2. Circuit diagram of the strain gage conditioning
device.

APPENDIX C
COMPUTER PROGRAMS
Listed below are the source codes for the programs that
were used to perform the data acquisition (freqdens.c) and
temperature control (tcntrl.c) with the DSRA. The programs
were written using ANSI C standard code language. The programs
were written by Mr. John Hornick and the author.
Progam Freqdens.c
#include
#include
/include
/include
/include
/include
/include
/include
/include









/define PRESSURE_NAME
/define TEMPERATURE_NAME
/define IOTECH_NAME
/define SYNTH_NAME
/define IOTECH_CHANNEL
/define MaxLoop
/define SamplingRate
/define NumTDPoints
/define IOTECH_STRING
/define NUM POINTS
typedef int BOOL;
/define TRUE 1
/define FALSE 0
"BECKMAN"
"K195"
"IOTECH"
"HP3325B"
2
21
1000001
327681
"c3r/3,2i4gllt0x"
81921
90

91
Directory[100];
IotechString[100];
SynthString[100];
SynthAddress, IotechAddress, PAddress,
char
char
char
int
TAddress;
int huge ‘TimeDomainData = NULL;
int IotechBuffer[8192];
FILE *stream;
long num,d;
float Voltage;
BOOL GetParameters( void );
BOOL ValidSamplingRate( long Rate );
BOOL ValidNumTDPoints( long NumPts );
void CreateStrings( void );
int SampleRateToIotechNumber( long SampleRate );
void AcquireData( long LoopNumber );
BOOL AllocateMemory( void );
void WriteTimeDomainData( long LoopNumber );
void ReadExtralnstruments( float ‘Temperature, float
‘Pressure );
void IEEEDelay( void );
float CDV (float Vk,float Vbl, float Vb2, float Vbh );
float ReadVoltage( void );
float RawDataToVoltage( float RawData );
int main( int argc, char *argv[] )
{
long LoopNumber = 0;
float Pressure, Temperature;
float Vk,Vbl,Vb2,Vbh,Vavg,Tk,Tbl,Tb2,Tbh,Tavg ;
float Pk,Pbl,Pb2,Ph,Pavg;
char cbuffer[20],buffer[20],filename[30];
printf("Enter directory for data storage:\n");
scanf("%s"(Directory);
gets(buffer);
num=ll;
CreateStrings();
SynthAddress = ibfind( SYNTH_NAME );
IotechAddress = ibfind( IOTECH_NAME );
PAddress = ibfind( PRESSURE_NAME );
TAddress = ibfind( TEMPERATURE_NAME );
ibwrt( SynthAddress, SynthString, strlen( SynthString )
);

92
if( ! AllocateMemoryO )
{
return 1;
>
sprintf(filename,"%s\\ptddata.dat"(Directory);
if( ( stream=fopen(filename,"at") ) == NULL)
{
printf("Error opening file\n");
return(0);
}
printf(" BOTH press enter when ready ") ;
gets(buffer);
Vboth=ReadVoltage();
ReadExtralnstruments( SPressure, STemperature );
Tboth=Temperature;Pboth=Pressure;
printf(" BUOY2 ONLY press enter when ready ");
gets(cbuffer);
Vball2=ReadVoltage();
ReadExtralnstruments( SPressure, STemperature );
Tball2=Temperature;Pball2=Pressure;
AcquireData( LoopNumber );
printf(" RACK ONLY press enter when ready ");
gets(buffer);
Vrack=ReadVoltage();
ReadExtralnstruments( SPressure, STemperature );
Track=Temperature;Prack=Pressure;
printf(" BUOY1 ONLY press enter when ready ");
gets(buffer);
Vballl=ReadVoltage();
ReadExtralnstruments( SPressure, STemperature );
Tba11l=Temperature;Pballl=Pressure;
Pavg=(Prack+Pballl+Pball2+Pboth)/4.Of;
Tavg=(Track+Tballl+Tball2+Tboth)/4.Of;
Vavg=CalculateDensityV(Vrack,Vballl,Vball2,Vboth);
Vavg=(Vavg*1000.Of);
printf("Temperature[%.2f]Pressure[%.If]Density[%.4f]
\n",Tavg,Pavg,Vavg);
fprintf(stream," %ld %e
%e\n",LoopNumber,Tavg,Pavg,Vavg);
fclose(stream);

93
LoopNumber++;
} while ( num > 01);
printf("Thank you for using Densities R Us \n");
return 0;
}
void CreateStrings( void )
{
int IotechNumber;
IotechNumber = SampleRateToIotechNumber( SamplingRate
) ;
sprintf( IotechString, "c%di%dr#%d,3gllt3x",
IOTECH_CHANNEL, IotechNumber, IOTECH_CHANNEL );
sprintf( SynthString,
"FRO.01HZ;AM5V0;FUI;RF1;SMI;SP%fHZ;STO.01HZ;TI%fSE;RSW",
(float)SamplingRate/2.Of, (float)NumTDPoints/
(float)Samp1ingRate );
>
int SampleRateToIotechNumber( long SampleRate )
{
switch( SampleRate )
{
case 100000: return 0; break;
case 50000: return 1; break;
case 20000: return 2; break;
case 10000: return 3; break;
case 5000: return 4; break;
case 2000: return 5; break;
case 1000: return 6; break;
>
return 0; >
void AcquireData( long LoopNumber )
{
ibwrt(IotechAddress, IotechString,strlen( IotechString
>);
ibwrt( SynthAddress, "SS", 2 );
ibrd( IotechAddress, (char huge *)TimeDomainData,
NumTDPoints*21 );
ibclr( IotechAddress );
ibwrt( SynthAddress, "RSW", 3 );
WriteTimeDomainData( LoopNumber );
}
BOOL AllocateMemory( void )

94
{
if( (TimeDomainData=(int huge *)halloc( NumTDPoints,
sizeof(int) )) == NULL )
{
printf( "Error Allocating Memory\n" );
return FALSE;
}
return TRUE;
}
void WriteTimeDomainData( long LoopNumber )
{
char FileName[256];
FILE *File;
long i;
sprintf(FileName, "%s\\raw%ld.dat", Directory,
100001+LoopNuraber );
if( (File=fopen(FileName, "wb")) == NULL )
{
printf( "Error opening [%s] for writing\n", FileName
) ;
return;
}
for(i=0;i {
if( fwrite( &TimeDomainData[i], sizeof(int), 1024u,
File ) != 1024U )
{
printf( "Error writing to [%s]\n", FileName );
fclose( File );
return;
>
}
fclose( File );
printf( "[%s] written successfully\n", FileName );
}
void ReadExtralnstruments( float ‘Pressure, float
♦Temperature )
{
char Buffer[20];
float Pres;
ibrd( PAddress, Buffer, 17 );
Pres = (float)atof( Buffer );
*Pressure=(Pres+0.97349629f)/l.00304717f;
ibrd( TAddress, Buffer, 17 );
♦Temperature = (float)atof(&Buffer[4]);

95
}
float CDV( float Vk,float Vbl, float Vb2, float Vh )
{
float vbl=1.9361f,vb2=1.9365f,mbl=6.9821f,mb2=13.9658f;
float AvgDen;
dl=( mbl*(Vball2-Vrack) - mb2*(Vballl-Vrack) )/(
vbl*(Vball2-Vrack) - vb2*(Vballl-Vrack) );
d2=( mbl*(Vrack-Vball2) + mb2*(Vboth-Vball2) )/(
vbl*(Vrack-Vball2) + vb2*(Vboth-Vball2) );
d3=( mbl*(Vboth-Vballl) + mb2*(Vrack-Vballl) )/(
vb2*(Vrack-Vballl) + vbl*(Vboth-Vballl) );
d4=( mb2*(Vba112-Vboth) + mbl*(Vboth-Vballl) )/(
vb2*(Vball2-Vboth) + vbl*(Vboth-Vballl) );
AD =(dl+d2+d3+d4)/4.0f;
return( AD );
}
float ReadVoltage( void )
{
long i;
float Min=O.Of, Max=0.Of,volt=0.Of;
ibwrt(IotechAddress, IOTECH_STRING,
strlen(IOTECH_STRING));
IEEEDelay();
ibrd( IotechAddress,(char *)IotechBuffer, NUM_P0INTS*21
);
IEEEDelay();
ibclr( IotechAddress );
Sum = O.Of;
Min = (Max = (float)(IotechBuffer[0]));
for(i=0;i {
Sum += (float)(IotechBuffer[i]);
if( Max < (float)(IotechBuffer[i]) )
Max = (float)(IotechBuffer[i]);
if( Min > (float)(IotechBuffer[i]) )
Min = (float)(IotechBuffer[i]);
}
Sum /= (float)NUM_POINTS;
Sum = RawDataToVoltage( Sum ) ;
Min = RawDataToVoltage( Min );

Max = RawDataToVoltage( Max );
return(Sum);
96
>
float RawDataToVoltage( float RawData )
{
RawData += 30000.Of;
RawData = -5.Of + ((RawData/60000.Of)*10.Of);
return RawData;
>
Tcntrl.c
Tcntrl.c was written to operate under Microsoft® Windows®
version 3.11. Due to the extensive extraneous code necessary
with any program of this type, only the PID algorithm is given
here.
void Control(
{
float
float
int
void )
TimeDiff;
NewTime;
NegLimit;
Resistance = GetResistance();
Temperature = ApplyTemperatureConversion( Resistance );
TError = SetPoint - Temperature;
NegLimit = -1 * OutputLimit;
NewTime = (float)clock();
TimeDiff = NewTime - OldTime;
if( TimeDiff == O.Of ) TimeDiff = 0.000001f;
OldTime = NewTime;
Propor = ( PMul * TError );
Propor += 2048.Of; // Add 2048 to shift effective
range for P
if( Propor < NegLimit )
Propor = NegLimit;
if( Propor > OutputLimit )
Propor = OutputLimit;

IntegAccum += ( IMul * TError ) * TimeDiff;
if( IntegAccum > OutputLimit )
IntegAccum = OutputLimit;
if( IntegAccum < NegLimit )
IntegAccum = NegLimit;
if( fabs( TError ) > I_Cutoff )
IntegAccum = O.Of;
Deriv = DMul * (( TError - LastError ) / TimeDiff);
if( Deriv > OutputLimit )
Deriv = OutputLimit;
if( Deriv < NegLimit )
Deriv = NegLimit;
if( fabs( TError ) > D_Cutoff )
Deriv = O.Of;
Output = (int)(Propor + IntegAccum + Deriv);
if( Output > OutputLimit )
Output = OutputLimit;
if( Output < 0 )
Output = 0;
LastError = TError;
outpw( HEATER_PORT, (unsigned int)Output«4 ) ;

APPENDIX D
TEMPERATURE CONTROL
Accurate temperature measurement and control is essential
for the determination of thermodynamic state properties. The
ability to establish an isothermal environment and maintain it
precisely over long periods of time is vital. Doing this in an
air bath at above ambient temperatures is not too difficult
with electric heating elements and circulatory blowers in well
insulated enclosures. Operation at near and sub-ambient
temperatures is considerably more challenging. In these
situations a source of refrigeration is needed to counteract
the heat production by the blowers and the heat flow from the
surroundings. Enclosures which utilize closed cycle
compression/expansion refrigerators are widely used, but they
are complicated to fabricate, maintain, and control. We
purchased a state-of-the-art industrial controlled temperature
enclosure with sub-ambient capability which used liquid
nitrogen, LN2, as the refrigerant. This has great appeal, as
no compressor or heat exchange coils are required, and
operation at much lower temperatures than can be reached with
conventional refrigerants is possible. The unit worked by
connecting a feed line into the enclosure to a storage tank of
98

99
LN2 through an electrically operated solenoid valve. The valve
was open momentarily, on demand by the control computer, to
admit short bursts of LN2 into the squirrel cage blowers where
its latent heat of vaporization was quickly extracted from the
gas inside the enclosure.
Although the system worked to maintain the average
temperature to a selected set-point, the standard deviation
was poor (± lOOmK) and the hammering action of the solenoid
produced annoying noise and unwanted vibrations. It was
decided that a new delivery system for the LN2 was needed, and
to attain the highest level of control, it should be fully
proportioning for use with a custom proportionation-
integration-differentiation (PID) algorithym.
Hardware
The hardware consisted of four major components; an RTD
thermometer, a microcomputer, a temperature chamber, and the
LN2 valve. The RTD used with the temperature control assembly
was a platinum resistance type, previously documented in table
5-1. The microcomputer used was a PC compatible with an 80286-
16 MHz CPU.
The temperature chamber, Delta Design model 9670, had an
operational range of 89 K to 588 K and, using the OEM solenoid
valve, could maintain the temperature roughly to ± 0.1 K about
a given set-point. This was clearly inadequate temperature
control and prompted the design and development of the LN2

100
valve. The functional parts of the Delta Design temperature
chamber that were used with the new LN2 valve were the Delta
Design's insulated chamber and housing along with the
circulatory fans. All other temperature control components
were disabled.
The LN2 valve assembly is shown schematically in figure
D-l. The valve consists of the valve housing (with inlet and
outlet ports), the tapered seat and stem, a heat exchanger, a
linear slide, and a DC motor/potentiometer drive assembly. The
potentiometer was used as a position encoder. The computer
instructs the motor control circuit1 to drive the valve to a
position for which the encoder signal matches a reference
voltage generated by a program controlled digital to analog
convertor board (Keithly model DAC-02). The position
instruction is updated at approximately 5Hz. The slew rate of
the valve is on the order of 200ms.
Software
The software used to control the LN2 valve was based on
a custom PID algorithym. The development of the PID software
used in this work was developed by Dr. Evan House. This
algorithm was incorporated into a program designed to operate
in Microsoft® Windows® (v3.1) . The code for the PID subroutine
is printed in appendix C.
1 The control circuit was constructed by Mr. Steve Miles of
the University of Florida Electronic Shop.

101
Data and Results
To ensure that the system under study had reached an
equilibrated state, initial experiments were performed where
system temperature was recorded as a function of time. Figure
D—2 is an example of one such run where the temperature
chamber was taken from ambient temperature conditions with no
control to temperature control about a set-point of 288.15 K
(15.00 °C). The average minimum equilibration time for
temperature deviation of ± 0.1 K was i 0.5 hours, but to
attain deviation of i 0.01 K the average time was 2 1.5 hours.
Typical temperature deviation as a function of time once
equilibration had been established is illustrated in figure D-
3, where the temperature was observed to fluctuate about the
set-point with a standard deviation of 0.007 K over an
approximately 3/4 hour interval.
Figure D-l. Schematic diagram of the liquid nitrogen valve
assembly.

102
Time (hours)
Figure D-2. Typical temperature versus time plot of oven
chamber temperature control showing equilibration.

103
Time (hours)
Figure D-3. Temperature control over a 0.7 hours about a
setpoint of 288.150 Kelvin with a standard deviation of
± 0.006 Kelvin.

APPENDIX E
REDLICH-KWONG EQUATION OF STATE AND THERMODYNAMIC EQUATIONS
Ideally P, T, c, and p data on a system of interest (e.g.
C02) would be collected at close intervals over a wide range
of variables P and T and the thermophysical properties deduced
by numerical manipulations, including integration and
differentiation of this extensive data base. Often it is
necessary or convenient to operate with a coarser set of data
for which wide-spread interpolation is impractical. In these
situations excellent results may still be obtained if the
briefer data set is fit to a suitable analytical equation of
state on which subsequent manipulations are made. Fortunately
much effort has gone into the development of such equations of
state. Each of these is appropriate for some specific model of
the intermolecular interactions within a fluid. The equations
differ in complexity generally as the complexity of the
interactions. One very important use of benchmark measurements
of thermophysical properties is to guide the development of
useful equations of state, especially for fluid mixtures. It
is anticipated that future generations of the apparatus
described here will be put to that use. For the present,
however, it is appropriate to demonstrate its use to generate
104

105
parameters for established equations of state. The most
practical and widely used of these are the cubic equations of
state. These cubic equations derive principally from the first
such equation proposed by van der Waals. The van der Waals
equation is historically important as the first to introduce
the ideas of long-range attractions and short-range repulsions
between molecules. Because the model gas on which it is based
is a collection of weakly attracting hard spheres, its utility
is extremely limited. A number of workers have produced
"advanced" cubic equations in which some "softness" is
introduced into the short range potential. Like the van der
Waals equation, these equations have two adjustable parameters
per molecular species and have the capability of emulating
somewhat the behavior of fluids in both the vapor and liquid
states.
Well known equations of state of this class include the
Peng-Robinson2’, Redlich-Kwong29 and Souve-Redlich-Kwong31
equations. For purposes of illustration the Redlich-Kwong (R-
K) equation was chosen to be used as a fitting function for
the C02 data collected in this work. The R-K equation is
claimed by some to be the best of the two-parameter cubic
equations.30'35

106
Redlich-Kwona Equation of State
The RK equation of state given in terms of P, T, and Vm
(molar volume) is
p = RT - a ^\
vm-b yT Vaivm*b) 1 11
where R is the gas constant and a and b are the RK constants.
The a and b values for C02 were determined by finding the
best fit to the experimental P, T, and Vm carbon dioxide data.
Equation E-l was rearranged and solved for a:
a
RT
Vm-b
-P
frvjv^b)
(E-2)
For each set of P, T, Vm data points a trial value of a can be
calculated for any selected value of b. This a can be
represented as,
aijj, (p, T,v,)1 (E-3)
where i is an index of each P, T, Vm triplet and j is an index
of each trial b value. The average and the standard deviation
of at are calculated over the data set for each bj value. The
ratio of the standard deviation to the average are tabulated
and plotted versus b,. The best fit for the RK parameters is
assumed to occur when the above ratio is minimized, shown
graphically in figure E-l. For the range of the experimental
data, the R-K parameters a and b were determined to be 5.8752
± 0.0037 Pa m6 K1/2 mol'2 and 2.6010E-5 ± 0.0001 m3 mol'1 ,
respectively.

107
2.6006E-05 2.6008E-05 2.6010E-05 2.6012E-05 2.6014E-05
b (mA3/mol)
Figure 6-14. The ratio of the standard deviation of a to
the average of a versus trial b values (a and b are the
Redlich-Kwong constants).

108
Thermodynamics
The decision to calculate the heat capacities Cp and Cv
and their ratio, y, as opposed to other thermodynamic
properties stems directly from the fact that, in addition to
having the experimental ability to acquire P, T, and p of the
fluid under study, the DSRA can obtain the speed of sound as
well. The heat capacity functions are the simplest which
require the full set of this data.
The first significant heat capacity relationship is given
as the difference of Cp and Cv,
where a is the iobaric coefficient of thermal expansion
(E-5)
and K is the isothermal coefficient of compressibility
(E-6)
It is clearly seen from the above equations the heat
capacity difference can be obtained from P, T, and p data, but
in order to determine the individual values of Cp and C„ a
second, independent, relationship between Cp and Cv must be

109
utilized. A convenient second relationship involves the heat
capacity ratio y and the speed of sound. It is given as,
’'(t.) ‘ ,E'7'
where M is the molecular weight (g/mol).
Now the heat capacities can be evaluated using the
experimental data and equations E-4 and E-7. For this work,
the R-K equation of state with the best fit a and b parameters
was used to determine the partial derivatives in a and K
above. aR_K was determined to be
-| RjT(Vm+b) - -^-{V2 - b2
2 2 sft
(E-8)
and kr_k was evaluated to be
vfr (v2 - b2)
(E-9)
where the denominator D is
D = a + y/T P( 3 - b2 ) - R T3/2 ( 2Vm + b ) .
(E-10)

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BIOGRAPHICAL SKETCH
Daniel S. Tatro was born October 3, 1968, in Middlebury,
Vermont. Dan's mother, Laura Tatro, raised him with his
younger brother and sister in the town of Jeffersonville,
Vermont, where he attended grade school. Dan attended Lamoille
Union for his middle and high school education and in 1986 he
graduated third in his class.
In 1986, under scholarship, Dan attended the University
of Florida for a semester. Dan returned to his home state in
the spring of 1987 where he completed three semesters of
undergraduate work at the University of Vermont. Dan returned
to the University of Florida in 1988 and received his B.S. in
chemistry from the University of Florida in June of 1990.
Dan's research experience began in June of 1990 as an
undergraduate at the University of Florida where he worked in
the experimental physical chemistry laboratory of Dr. S.O.
Colgate. Dan began his graduate career in August of 1990 by
continuing his research under the direction of Dr. Colgate.
Principle areas of research include the development of
instrumentation and methods for the measurement of physical
properties.
113

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.
(CfeaÁ —
Samuel 0. Colgate/ chair
Professor of Chemistry
I certify that X 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.
44L
Syler )
R. Eyler
fessor 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.
UÜ.
Kathryjí R. Williams
Associate in 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.
\v. \
—,—
William Jones
Distinguished Service 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 Qf Doctor of Philosophy.
Barney L.' Capehart
Professor of Industrial and
Systems 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.
May 1996
Dean, Graduate School

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WÍ9ST0N
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UBi?ARr
UNIVERSITY OF FLORIDA
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