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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vii, 113 leaves : ill. ; 29 cm.
Tatro, Daniel S., 1968-
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Physical measurements   ( lcsh )
Chemistry thesis, Ph. D   ( lcsh )
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Thesis (Ph. D.)--University of Florida, 1996.
Includes bibliographical references (leaves 110-112).
Statement of Responsibility:
by Daniel S. Tatro.
General Note:
General Note:

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University of Florida
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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.


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.

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.







Fundamental Theory . .
Theory of Design . .


Spherical Cavity Resonance Equation
Development . .
Resonance Frequency Identification .


Densimeter . .
Spherical Buoys . .
Deflection Beam and Semiconductor
Strain Gages . .
Coiled Spring and LVDT Sensor .
Densimeter Body . .
Buoyancy Assembly and Sphere Lifting
Mechanism .....
Top Flange Assembly .. .....
Spherical Resonator . .
Resonator Cavity . .
Transducer Assembly . .
Final Assembly . .


Gases . . .
Interfacing . . .
Measurement Hardware . .
Pressure and Temperature . .


S. ii
. vii

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







REFERENCES . . ... 110


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



Daniel S. Tatro

May 1996

Chairperson: Samuel O. 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.


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.' 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.


Generally, equations of state are based on empirical or

semiempirical models.2 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



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.3'4 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


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.


The Burnett method' 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.' 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.


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 conditions', but only recently has it been

extended to dense gases.1 Later chapters will detail the

design and use of a versatile new density meter based on this


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 densimeterr) 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


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.


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, F is defined as

F, = p vg (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

S= 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 Vg ) A (2-3)

or in terms of magnitudes alone, since there is no ambiguity

about direction in this one dimensional case ,

Fot = (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 V, and V2

and having masses m, and m2 suspended in an isothermal, static


fluid of density p. Recalling equation 2-4, the net force

acting on each sphere may be represented independently by the

following equations,

F(ot,spherel) = mi g p V, g (2-5)

F(tot,sphere2) = 2 g P V2 g (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.

Suspension Rack



- Sphere 1

- Sphere 2

Figure 2-1. Schematic diagram of fundamental densimeter


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 = -kx (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 =-k xo (2-8)

Where x0 is the length of the spring loaded only by the

suspension rack.

II. Sphere 1 on; sphere 2 off:

F, =-k x, = ( m p V) g k x (2-9)

III. Sphere 1 off; sphere 2 on:

F2 =-k x2 = ( m2 p V ) g k x0 (2-10)

IV. Sphere 1 on; sphere 2 on:

F3 =-k x = ( m p V3 ) g k x0 (2-11)

where m3 = mi + m2 and V, = V, + V1.

Equations 2-8 through 2-11 form a coupled system from

which, knowing x1-xo, x,-xo, x3-x0, m,, m,, V, and V2, the fluid


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

discrimination. 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.


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." 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 Reed"1, McGill13 and Dejsupa1 based on the

earlier work of Rayleigh'1 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,

w i(t2 ot(ti) a (3-1)
where is the velocity potential and c is the speed of sound.(3

where T is the velocity potential and c is the speed of sound.


Assuming a time separable velocity potential such that T =

oe"1t, equation 3-1 becomes

V2 + O* =0 (3-2)

where w 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,"

S +1
*o (r,', p) = jE (f1,nr)Y1m(')) (3-3)
n=O m-1 ,0

where 1t = time-independent velocity potential

r,O,p = spherical polar coordinates

ji = Bessel function of the first kind(radial part)

Yim = 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


fl,n 2Xr
c = 2r (3-4)

where f,,n is the resonance frequency, r is the radius of the

spherical cavity, and the eigenvalue, (1,,, is the n" 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), 1,n. 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 4,,

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." 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." Therefore, we identify

and utilize frequencies associated with pure radial modes


(1=0) to facilitate the most accurate measurements of sonic


Resonance Frequency Identification

The scheme used here for assigning mode identities to

resonance peaks in an acoustic resonance spectrum was

developed by Dejsupa." In order to calculate the speed of

sound, a resonance frequency f1,n and its corresponding

eigenvalue 4,,, must be identified. Rearranging equation 3-4

and solving for frequency yields

f c 4'n (3-5)
,n 2 c r

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


6.00E+06 -




0 10000 20000 30000
Frequency (Hz)

40000 50000

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


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,

c- (3-6)

the frequency of mode l',n' should be

S) = ref ( n) (3-7)
,n) ref

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


the idealized model." 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


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 state", from

which the speed of sound may be accurately calculated.

Equation 3-4 may be rewritten as

Vk k (3-8)

where vk is defined as

Vk (3-9)


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;

Ck= vk k (3-10)

where c, is the speed of sound of the fluid at f, and vk .

Assuming that ck is not a function of frequency, then the

speed of sound c, 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,

Vk k (3-11)
C. J-- n

endnote #19 diffusion stuff19

enote 2020 21 22 2

29 30 31 32 33 14

Table 3-1. The values of the frequency roots in ascending
order of magnitude with the purely radial modes in bold print.
Index i, 1 n Index t ,n 1 n

1 2.08158 1 1 37 16.3604 14 1
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 11 2
10 7.85109 6 1 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 11 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


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 1500C). These subassemblies were constructed primarily from

304 stainless steel (304SS) and brass components, each sized

to meet the designed performance standards.


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 Buoys

The hollow spherical sinkers were designed such that

their volumes would be as closely matched as possible (V1=V2),

but their masses would be different by a factor of two

(2ml=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.

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

Figure 4-1. Spherical buoy with enclosed copper bead.

Spherica Buoy
Outer Shell


Copper Bead

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.'

See appendix B for circuit diagram.


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


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.

Figure 4-2. Output signal of the strain gage sensor versus
time of the suspension rack with no spheres loading.








7.7840 ---- r--
0 3 6 9 12 15 18 21
Time (min)

Figure 4-3. Output signal of the strain gage sensor versus
time of the suspension rack and both spheres loading.

-7.6950 .






0 3 6 9 12 15 18 21
Time (min)


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." That

experience suggested that an LVDT could be used to follow the

deflection of an elastic member at resolutions appropriate for



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


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 0.25 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


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.

O-ring Offset---

Clamping Flange



I -

Copper O-ring

Densimeter Body

End Cap

Figure 4-4. Densimeter body assembly.


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 O-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 O-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





D )2



0.98( L2

.26D for 98% fil

LVDT Core Brass Posts

e Holders ; Bottom view detail showing

V-shape sphere holders

Figure 4-6. Suspension rack assembly.




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


4-7. Spherical buoy

allow the suspension rack and the spheres to hang freely

without contacting the inside walls. Two slots were machined

axially 900 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


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 450

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 (Ql to Q3, Q2 to Q4).

This accounts for two of the necessary measurements; both

Figure 4-8. Detail of cam showing step and step quadrants,
Q1 through Q4.

3 Q4



spheres raised from the rack, and both spheres supported by

the rack. The arc length from one step to the other (Ql 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.

Top 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.

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

Copper Gasket

Translational /
LVDT Housing

Quartz Spring

LVDT Sensor



Figure 4-10. Densimeter top flange assembly.


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.

Transducer Housing

Spherical Cavity I

V-groove for
O-ring gasket 12_ I
I .! --

Figure 4-11. Spherical cavity and clamping flanges.

Clamping Flanges

transducer assembly


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.

Top Flange




r~ur 4li ~ef.Le iwo.tednsmtr


Figure 4-13. Assembled view of he

Wave Spring

Spherical Cavity



of the

Table 4-2. Summary of Measurement hardware.

Item Manufacturer Model Range of Resolution Function
number operation
Entran ES-025 0-1000Me 0.0001 Measure
Semiconductor Devices,Inc. (e=strain) volts deflection of
strain gage beam.

Semiconductor University of N/A N/A Accepts
strain gage Florida, signal from
output device Electronics strain gage
Dept.a circuit
LVDT Trans-Tek, 240-0015 0.050 inch. 1 micron Measure
Inc. deflection of

Piezo-electric Tandy Corp. High 500Hz-50kHz N/A Generate and
speaker element Efficiency receive

a see appendix B for circuit diagram used.


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.


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.


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



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


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.

Figure 5-2. Schematic diagram of the apparatus setup.


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).


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 lotech 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

acquisition time of 16.4 seconds. Four voltage values,




400.0 -


P(sensotec)= 1.0030 P(ruska) + 0.97349
100.0 R = 0.9999998

0.0 I Ii I
0.000 100.000 200.000 300.000 400.000 500.000 600.000 700.000
Ruska gage pressure (psia)

A Experimental data Regression fit

Figure 5-3. Pressure gage calibration data using argon
gas at 298 K.


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 lotech ADC. The

time domain resonance data were acquired at a rate of 100KHz

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.

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




2.00E+06 I. i

0.00E+00 I
0 10000 20000 30000 40000 50000
Frequency (Hz)

Figure 5-5. FFT of ADC signal of argon at 278.15 K and
1417.4 psia.

Experimental Data Collection Procedure

The entire apparatus was cleaned with multiple washings

of acetone. It was then baked at 150 OC, 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 equilibrate, 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


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 Range of Resoutio
Item Manufacturer o (Manufacturer's Function
number operation stated value)
stated value)
Function Hewlett HP 3325B 1 pHz to 1 pHz Generates a
Generator Packard 20.99 MHz precise input

Low-noise Stanford SR560 < 250 mV rms 1 % Gain Filter and
preamplifier Research input amplify
Systems acoustic
Digital strain Beckmann 610 -150 m to 1 0.01 % Full Accepts
gage transducer Industrial volt Scale strain gage
indicator Corp. transducer
Strain gage Sensotec TJE/743-03 0-3000 psia 0.01% full Measure
pressure scale pressure.

RTD HY-CAL RTS 31-A -200 to 2800C 0.00 Ohms Measure
Engineering temperature.

Analog/Digital Iotech ADC488 10 volts 53 AVolts Converts
Converter (Max. scan analog signal
rate-100MHz) to digital

Table 5-1 (Continued).
Model Range of Resolution
Item Manufacturer aton (Manufacturer's Function
number operation stated value)

Digital Keithly k196 4-terminal Range Measure
Multimeter Instruments resistance dependent, see resistance
Inc. measurements, manual for from Pt
100 MQ to details, temperature
20 MO probe.
Computer National GPIB-PCII Transfer rate N/A Communication
interface board Instruments 200 Kbytes/s interface for


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.


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


Table 6-1. Experimental data and results for the
eigenvalue calibration u K


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.)

b Radial mode index.


P Ca fk Vk
0.1 (m/s) kb 1.5 (m)
(psia) (Hz)
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 1 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


Table 6-2. Theoretical radial mode eigenvalues listed with the
averages of the corrected eigenvalues.

Average Theoretical Relative
k Corrected Eigenvalues Deviation
Eigenvalues Ri0,n.k
Vk (m) (%)
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


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-

Cvirial)/Cvirial ) versus pressure. Figures 6-6 through 6-8 show
the relative deviation of the experimental density (100(pexp-

Pviriai)/Pvirili ) versus pressure.

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





' 325.00

m 320.00



600.0 800.0 1000.0 1200.0
Pressure (psia)

1400.0 1600.0 1800.0

S278.15 K 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.


d /P






m 160.00




60.00 I
400.0 600.0

800.0 1000.0 1200.0 1400.0 1600.0
Pressure (psia)

S278.15 K 288.15 K 298.15 K

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


Figure 6-3. Relative deviation of experimental speed of
sound compared to the calculated virial speed of sound
of argon versus pressure at 278.15 K.


0.04 -


- o

g -0.02-

o -0.04 -


-0.08 I A ,
400.0 600.0 800.0 1000.0 1200.0
Pressure (psia)

1400.0 1600.0 1800.0

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.


0 0

" "

-0.04 A

-0.06 I
400.0 600.0 800.0 1000,0 1200.0 1400.0 1600.0
Pressure (psia)



0 02


1100.0 1200.0 1300.0 1400.0 1500.0 1600.0 1700.0 1800.0
Pressure (psia)

Figure 6-5. Relative deviation of experimental speed of
sound compared to the calculated virial speed of sound
of argon versus pressure at 298.15 K.


0.40 -







-1.00 I -
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.


0.40 "







-1.00 I I
400.0 600.0 800.0 1000.0 1200.0 1400.0 1600.0 1800.0
Pressure (psia)

Figure 6-7. Relative deviation of experimental density
compared to the calculated virial density vs. pressure
of argon at 288.15 K.






-0.20 m a



-0.80 I I I I
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.

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.

Table 6-4. Experimental carbon dioxide (44.011 g/mol) data

with the average

standard deviation given at each column

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



0.10 *





-0.40 I I I
1000.0 1100.0 1200.0 1300.0 1400.0 1500.0 1600.0 1700.0
Pressure (psia)

I Wagner(21) v Kessel'man(22) 3 Duschek(23) Holste(20)

Figure 6-9. Relative percent deviation of literature values
of density for carbon dioxide at 290.15 K from the present



o 0.10



1100.0 1200.0 1300.0 1400.0 1500.0
Pressure (psia)

i Wagner(21) v Duschek(23) x Michels(32)


Figure 6-10. Relative percent deviation of literature
values of density for carbon dioxide at 297.15 K from
the present measurements.

-0.20 4-







-0.30 I T
1000.0 1200.0 1400.0 1600.0 1800.0 2000.0
Pressure (psia)

u Wagner(21) v Duschek(23) a Ely(24) x Michels(32)

Figure 6-11. Relative percent deviation of literature
values of density for carbon dioxide at 300.15 K from
the present measurements.







-0.05 i I I I t
1200.0 1300.0 1400.0 1500.0 1600.0 1700.0 1800.0
Pressure (psia)

Wagner(21) v Duschek(23) si Ely(24)

Figure 6-12. Relative percent deviation of literature
values of density for carbon dioxide at 303.15 K from
the present measurements.

1200.0 1400.0 1600.0
Pressure (psia)

1800.0 2000.0

Figure 6-13. Experimental speed of sound data versus
pressure of carbon dioxide at different isotherms.

400.00 -


I 300.00


* K-


20U.U0 -

200.00 -


* 290.15 K -

297.15 K 300.15 K 303.15 K 307.15 x 310.15 KI

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 C, and their ratio

y were calculated for carbon dioxide.

The experimental data were fit to the Redlich-Kwong (R-K)

equation of state

RT a 1
V,-b vr V.(v+b)

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, C,, and y were calculated (refer to appendix E for




< 800.00


8 700.00





Pressure (psia)



290.15K 297.15 K 300.15K 303.15 K 307.15K 310.15 K

Figure 6-15. Carbon dioxide density, experimental data
and RK calculated curves, versus pressure.



Table 6-5. Experimental and Redlich-Kwong calculated densities
of carbon dioxide.

Redlich-Kwong constants
a = 5.8752 0.0037 Pa m6 K12 mo1-2
b = 2.6010E-5 0.0001E-5 m3 mol-'

T P Pexp PRK Relative
(K) (psia) (kg/mn) (kg/m3) 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
S1612.4 889.82 898.50 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 -0.77


equations used to calculate the

capacities). Table 6-6 lists C,, 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

comparisons could be presented as shown in table 6-6.


Table 6-6. Comparison of experimental C,, C,, y values with
experimental literature values.



a Relative deviation

T = 290.15 K P = 1612.5 psia

Source C, RDa C, RD Y RD
J/(mol-K) (%) J/(mol-K) (%) (%)

DIN" 107 2 50.3 5.6 2.13 -3.80
____ 1 1.5 0.01
Critical 105 8 48 10 2.2 -6.9
Tables34 5 +1 + 0.1
Vargaftik26 108.1 1.6
This work 108.91 53.14 2.049
0.37 0.41 0.009

T = 300.15 K P = 1329.5 psia

Source Cp RD C, RD 7 RD
J/(mol K) (%) J/(mol K) (%) (%)

DIN33 140 2 50.9 -2.5 2.75 4.72
__ 1 1.5
Critical 149 -4 46 8 3.2 -9.8
Tables" 5 + 1

Vargaftik26 143.9 -0.8
This work 142.73 49.59 2.884
0.28 0.39 + 0.006

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.


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



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

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 = 2m,

= m and Vi = V2 which table 4-1 shows to be nearly true. The

fluid density pf then is found to be

= Ps-2-S (7-1)

where p' is m/V, the net density of the lighter spherical

buoy, and S, 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 u, and the uncertainty in buoy

density to be negligible these propagate to an uncertainty in

fluid density up, which is

p U p (7-2)
up S1-S

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 p, caused by

choice of buoy density occurs when these densities are equal.

Because it is necessary for p' 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.


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-1

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







SClosure Screw

Figure A-3. Pycnometer and frame assembly.








6.9770 I
6.9770 6.9790 6.9810 6.9830
Standard Mass (g)

Calibration Data Regression Fit o Mass of Buoy #1

Figure A-i. Graph showing calibration data used to
determine the mass of spherical buoy #1.





g /i



13.9590 13.9610 13.9630 13.9650 13.9670
Standard Mass (g)

SCalibration Data Regression Fit o Mass of Buoy #2

Figure A-2. Graph showing calibration data used to
determine the mass of spherical buoy #2.


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,

Ma + M(P) Mp+'s) (A-l)
S pw(T)

where V, is the volume of the sphere, M, is the mass of the

sphere, M,.w.,, is the mass of the pycnometer, the undisplaced


water (w') and sphere, M,,p 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


Table A-i. Data and results for the volume calibration of the
spherical buoys.

P294.65 (H20)1 = 0.99873 g/cc M,, = 72.4896 g
0.0002 g

phBuoy M (g) M,+w, (g) Volume (cc)

6.9821 77.5380 1.9329
0.0001 0.0002 0.0002

2 13.9658 84.5194 1.9331
0.0001 0.0001 0.0002

1 Weast R.C and M.J. Astle, CRC Handbook of Chemistry and
Physics, 1974, F-l1.


The block diagram of the strain gage signal conditioning

component is shown in figure B-1.28 The circuit diagram that

was followed in the construction of the signal conditioning

device is given in figure B-2.28


Figure B-1. Block diagram of the strain gage signal
conditioning component.


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


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 Freadens.c


#define SYNTH NAME
#define MaxLoop
#define SamplingRate
#define NumTDPoints

#define NUM POINTS

typedef int BOOL;
#define TRUE 1
#define FALSE 0



char Directory[100];
char IotechString[100];
char SynthString[100];
int SynthAddress, IotechAddress, PAddress,
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 ReadExtraInstruments( float *Temperature, float
*Pressure );
void IEEEDelay( void );
float CDV (float Vk,float Vbl, float Vb2, float Vbh );
float ReadVoltage( void );
float RawDataToVoltage( float RawData );

main( int argc, char *argv[] )


LoopNumber = 0;
Pressure, Temperature;

printf("Enter directory for data storage:\n");


SynthAddress = ibfind( SYNTH NAME );
IotechAddress = ibfind( IOTECH NAME );
PAddress = ibfind( PRESSURE NAME );
TAddress = ibfind( TEMPERATURE_NAME );

ibwrt( SynthAddress, SynthString, strlen( SynthString )

if( AllocateMemory( )
return 1;


if( ( stream=fopen(filename,"at") ) == NULL)
printf("Error opening file\n");
printf(" BOTH --- press enter when ready ");
ReadExtraInstruments( &Pressure, &Temperature );

printf(" BUOY2 ONLY --- press enter when ready ");
ReadExtraInstruments( &Pressure, &Temperature );
AcquireData( LoopNumber );
printf(" RACK ONLY --- press enter when ready ");
ReadExtraInstruments( &Pressure, &Temperature );

printf(" BUOY1 ONLY --- press enter when ready ");
ReadExtraInstruments( &Pressure, &Temperature );


fprintf(stream," %ld %e


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