Title: Fast fourier transformed acoustic resonances with sonic transform
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Title: Fast fourier transformed acoustic resonances with sonic transform
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Creator: McGill, Kenneth C., 1957-
Copyright Date: 1990
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FAST FOURIER TRANSFORMED ACOUSTIC RESONANCES
WITH SONIC TRANSFORM















By

KENNETH C. MCGILL


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


1990


































Copyright 1990

by

Kenneth Charles McGill















DEDICATION



This work is dedicated to the three people whom I owe

so much: To my mother Martha Senogles who gave me life; to

my late wife Natalie McGill who gave me her life; and to my

wife Susan McGill who is giving me a new life in my first

child.















ACKNOWLEDGMENTS


I would like to thank Dr. S.O. Colgate personally for

his support and guidance during the development of this

technique.

I would also like to thank Chadin Dejsupa and Joe

Shalosky for assisting in the construction of various parts

of the apparatus, Casey Rentz for the use of his computer

and Evan House for convincing me to join the Colgators.

In addition, I would like to thank Dr. Grant Schrag for

the development of the tapered ram seal used for the

electrical feed-throughs of the transducers, Dr. Cliff

Watson for his assistance in programming the Fast Fourier

Transform and Steve Miles for his contribution on the

development of the magnetic pump.

Also, I thank my wife, Susan, for instructing me on the

use of WordPerfect so that I could perfect the format of

this dissertation.
















TABLE OF CONTENTS
page
ACKNOWLEDGMENTS..................... .................... iv

LIST OF TABLES....... ............. ......................vii

LIST OF FIGURES...................................... ..... ix

ABSTRACT..................................................... xi

CHAPTERS


1 INTRODUCTION........... ........................

2 THEORY.........................................

Theory of Design...............................
Theory of Operation...........................

3 EXPERIMENTAL......................... ..........

Interfacing ....................................
Apparatus ......................................
Spherical Cavity........................
Pump ......................................
The Bellows................................

4 DATA AND RESULTS................................

Time Domain Plots...............................
Frequency Domain Plots.........................
Sonic Domain Plots.............................
Volume and Pressure Calibration................

5 CONCLUSION .....................................


APPENDICES

A

B

C

D


FAST FOURIER TRANSFORM SOURCE CODE..............

SONIC TRANSFORM SOURCE CODE................... ..

EQUATION OF STATE FOR ARGON SOURCE CODE.........

DATA ACQUISITION SOURCE CODE....................


1

8

9
18

29

30
34
34
35
37

42

44
45
46
48

69



76

80

84

87









E DATA CONVERSION SOURCE CODE .................... 93

BIBLIOGRAPHY ................... .......................... 95

BIOGRAPHICAL SKETCH...................................... 97















LIST OF TABLES
page
Table 2-1. The values of the roots to the first
derivative of a Bessel function of the first
kind................................................. 12

Table 2-2. Reduced second virial coefficients for
the Lennard-Jones 6-12 potential.................... 20

Table 2-3. Reduced third virial coefficients and
their derivatives for the Lennard-Jones 6-12
potential........................................... 22

Table 4-1. Low temperature time domain parameters of
argon............................................... 53

Table 4-2. Low temperature frequency domain parameters
of argon........................................... 54

Table 4-3. First sonic domain parameters of argon at
low temperature..................................... 55

Table 4-4. Second sonic domain parameters of argon at
low temperature..................................... 56

Table 4-5. Third sonic domain parameters of argon at
low temperature..................................... 57

Table 4-6. Fourth sonic domain parameters of argon at
low temperature..................................... 58

Table 4-7. High temperature time domain parameters
of argon........................................... 59

Table 4-8. High temperature frequency domain parameters
of argon............................................ 60

Table 4-9. First sonic domain parameters of argon at
high temperature.................................... 61

Table 4-10. Second sonic domain parameters of argon at
high temperature.................................... 62

Table 4-11. Third sonic domain parameters of argon at
high temperature..................................... 63


vii









Table 4-12. Fourth sonic domain parameters of argon at
high temperature ............ ......................... 64

Table 4-13. Outside volume calibration.................. 65

Table 4-14. Total volume of apparatus .................... 66

Table 4-15. Bellows volume calibration.................. 67

Table 4-16. Compiled results of sonic speeds of argon at
low and high temperatures for various roots......... 68


viii















LIST OF FIGURES
Dage
Figure 3-1. Instrument rack............................. 32

Figure 3-2. Spherical cavity sections and clamping
flanges............................................. 36

Figure 3-3. Pump assembly............................... 38

Figure 3-4. The bellows and bellows chamber............. 39

Figure 3-5. Apparatus assembly.......................... 41

Figure 4-1. Theoretical ADC signal for 350 m/s speed
of sound............................................ 50

Figure 4-2. Theoretical ADC signal for 150 m/s and
350 m/s speeds of sound............................. 50

Figure 4-3. FFT of theoretical ADC signal for
350 m/s speed of sound............................... 51

Figure 4-4. FFT of theoretical ADC signal for
150 m/s and 350 m/s speeds of sound................. 51

Figure 4-5. ST of FFT of theoretical ADC signal
for 350 m/s......................................... 52

Figure 4-6. ST of FFT of theoretical ADC signal
for 150 m/s and 350 m/s speeds of sound............. 52

Figure 4-7. ADC signal of argon at low temperature...... 53

Figure 4-8. Expanded section of Figure 4-7............... 53

Figure 4-9. FFT of ADC signal of argon at low
temperature........................................... 54

Figure 4-10. Expanded section of Figure 4-9............. 54

Figure 4-11. First ST of argon at low temperature....... 55

Figure 4-12. Expanded section of Figure 4-11............. 55

Figure 4-13. Second ST of argon at low temperature....... 56









Figure 4-14. Expanded section of Figure 4-13............. 56

Figure 4-15. Third ST of argon at low temperature....... 57

Figure 4-16. Expanded section of Figure 4-15............. 57

Figure 4-17. Fourth ST of argon at low temperature...... 58

Figure 4-18. Expanded section of Figure 4-17............. 58

Figure 4-19. ADC signal of argon at high temperature.... 59

Figure 4-20. Expanded section of Figure 4-19............ 59

Figure 4-21. FFT of ADC signal of argon at high
temperature......................................... 60

Figure 4-22. Expanded section of Figure 4-21............. 60

Figure 4-23. First ST of argon at high temperature...... 61

Figure 4-24. Expanded section of Figure 4-23............ 61

Figure 4-25. Second ST of argon at high temperature..... 62

Figure 4-26. Expanded section of Figure 4-25............ 62

Figure 4-27. Third ST of argon at high temperature...... 63

Figure 4-28. Expanded section of Figure 4-27............. 63

Figure 4-29. Fourth ST of argon at high temperature..... 64

Figure 4-30. Expanded section of Figure 4-29............. 64

Figure 4-31. Outside volume calibration................. 65

Figure 4-32. Total volume of apparatus.................. 66

Figure 4-33. Bellows calibration plot................... 67
















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


FAST FOURIER TRANSFORMED ACOUSTIC RESONANCES
WITH SONIC TRANSFORM

By

Kenneth C. McGill

December 1990


Chairman: S.O. Colgate
Major Department: Chemistry


In this study, a novel approach for detecting one or

more speeds of sound was developed. By employing a Sonic

Transform (ST), the data are transformed in real time to a

domain that is directly related to the speed of sound within

a cavity. The transform is of order < n2 and is equivalent

to a Fast Fourier Transform in computation time. The study

contains a discussion of the apparatus design as well as

interfacing techniques involved in its operation. Source

code and algorithms that describe the analysis and data

acquisition in detail are also contained within the study.















CHAPTER 1
INTRODUCTION

The measurement of state variables is of interest to

researchers in thermodynamics. The two most commonly

measured state variables are temperature and pressure.

Techniques for their measurement have been developed that

have high accuracy and speed of operation and are relatively

easy to use. The equation of state for even the simplest

system, for example, a single component gas, requires at

least another variable. For whatever additional variable is

chosen, it is desirable that its measurement be performed as

quickly and as easily as those of temperature and pressure.

The most commonly measured third state variable is

volume. The measurement of volume is often done by a batch

process where a fluid substance is placed in a vessel of

calibrated volume. This process is time-comsuming and is

prone to error. Individual error can occur in recording the

measurement and it is impossible to do real time processing

of the data.

There are other state variables that could be measured,

such as entropy, enthalpy, and free energy, but these are

even more difficult to measure in a batch process or in real

time. If a reliable equation of state relating three state

variables is available, then the magnitude of the third

1











variable may be calculated after measuring the other two.

This method works well for single component gases, but it is

not very accurate for multicomponent mixtures of gases or

for any gas near its critical region.

In this work, emphasis is placed on the development of

a novel sonic speed measurement technique to facilitate the

use of this state variable along with temperature, pressure

and volume in physical relationships. An effort to make the

measurement of the speed of sound as accurate and as easy as

temperature and pressure has been made; that is, a process

has been developed that can operate in real time with high

accuracy and with little interaction from the user. The

measurements of volume and speed of sound are similar; one

way to measure the volume of a gas involves the geometry of

the vessel in which the gas is contained and, similarly, one

way to measure the speed of sound in a gas involves the

geometry of the vessel in which the gas is contained. By

knowing the geometry of the vessel, the volume can be

calculated by measuring the dimensions of the vessel. The

speed of sound can be found by measuring the acoustic

resonances within the cavity. The speed of sound is also

dependent on the density and mass of the gas being measured.

An accurate method for measuring the speed of sound

involves examination of the resonances that occur in an

acoustic cavity. The selection of the geometry of the

cavity can make a significant difference in the ease of











interpretation of the resonance frequencies. For example,

resonances in a cylindrical cavity are complicated by

problems, such as unresolved modes and viscous drag along

the longitudinal walls. These problems have been examined

in detail elsewhere.1

Another potential problem with any shaped cavity

results from a precondensation effect that occurs on the

surface of the cavity.2 This effect appears most strongly

at low frequencies in resonators with large surface-to-

volume ratios. To avoid these problems, a spherical cavity

was chosen since: 1) Viscous drag does not occur for the

radial vibrations within a spherical cavity; 2) surface-to-

volume ratio is minimized for spherical geometry; and 3) the

acoustic energy is highest at the center of the sphere.

In a study that included a treatment of the

precondensation effect in a spherical cavity, the speed of

sound of a gas was measured with an accuracy approaching

0.0005% or 5 ppm.3 Neglecting the precondensation effect,

1 J.B. Mehl and M.R. Moldover, "Precision Acoustic
Measurements with a Spherical Resonator: Ar and C2H4,"
Journal of Chemical Physics 74 (April 1981): 4062-4077;
A.R. Colclough, "Systematic Errors in Primary Acoustic
Thermometry in the Range 2-20 K," Metroloqia 9 (1973): 75.

2 J.B. Mehl and M.R. Moldover, "Precondensation
Phenomena in Acoustic Measurements," Journal of Chemical
Physics 77 (July 1982): 455-465.
3 M.R. Moldover, J.B. Mehl, and M. Greenspan, "Gas-
Filled Spherical Resonators: Theory and Experiment,"
Journal of the Acoustical Society of America 79 (February
1986): 253-271.











accuracies of 0.01% are readily obtained for physical

properties inferred from sonic speed measurements, these

include reference state heat capacities,4 thermophysical

properties of alkanes,5 and heat capacity ratios.6 In all

of these experiments, the first step is to analyze a

frequency spectrum and then select only a few of the

resonances, at most five or six depending upon the

experiment, to measure the speed of sound.7 This

interaction from the user requires an intuition as to where

the resonances occur, and locating them with confidence is

often tedious and can take a considerable amount of

experimental time. This places an added burden on the

maintenance of the system's state. The measurement of

temperature and pressure can be very accurate, but

maintaining them for long periods of time is not easy. All



4 S.O. Colgate, C.F. Sona, K.R. Reed, and A.
Sivaraman, "Experimental Ideal Gas Reference State Heat
Capacities of Gases and Vapors," Journal of Chemical and
Engineering Data 35 (1990): 1-5.


5 M.B. Ewing, A.R.H. Goodwin, and J.P.M. Trusler,
"Thermophysical Properties of Alkanes from Speeds of Sound
Determined Using a Spherical Resonator 3. n-Pentane,"
Journal of Chemical Thermodynamics 21 (1989): 867-877.
6 S.O. Colgate, K.R. Williams, K. Reed, and C. Hart,
"Cp/CV Ratios by the Sound Velocity Method Using a Spherical
Resonator," Journal of Chemical Education 64 (June 1987):
553-556.

7 M.B. Ewing, M.L. McGlashan, and J.P.M. Trusler,
"The Temperature-Jump Effect and the Theory of the Thermal
Boundary Layer for a Spherical Resonator, Speeds of Sound in
Argon at 273.16 K," Metrologia 22 (1986): 93-102.











of these methods assume that only one speed of sound is

present within the medium of interest. If multiple speeds

of sound are present within a medium, the difficulties of

the job of analysis are seriously compounded. Ideally, a

method that can identify the resonances as well as calculate

a close approximation of the speed of sound very quickly

would represent a significant advance in the art of sonic

speed measurements.

Since the number of possible resonances is of the order

of the number of molecules, it is for all practical purposes

infinite. Ideally, a broad band of resonances should be

used to determine the speed of sound within the gas. One

such attempt at measuring a truncated set of resonances was

made by Tewfik et al.8 This study modeled two dimensional

waves such as the waves on the ocean. Their method involved

a rather large calculation employing Householder routines to

solve an nXn linear matrix problem. A Householder routine9

is an operation of order n3 for which even a relatively

small set of resonances becomes costly in computation time.

Hence, although the Householder routine is capable of high

accuracy, it can not be considered useful as a real time process.


8 A.H. Tewfik, B.C. Levy, and A.S. Willsky, "An
Eigenstructure Approach for the Retrieval of Cylindrical
Harmonics," Signal Processing 13 (September 1987): 121-139.


9 G.H. Golub and C.F. Van Loan, Matrix Computations
(Baltimore: Johns Hopkins University Press, 1985), 38.











In order to overcome these boundaries, a technique was

developed in the present work to transform the Fourier

coefficients of a captured time domain signal to the sonic

domain. Once in this domain, the speed of sound is easily

determined. For the development of this technique, a

spherical cavity and a truncated set of resonances were

used. The truncated set of resonances was transformed from

a measured time domain signal to the sonic domain using a

transform operation of order nlog2n + nm, where m is the

number of resonances.

To test the method, a theoretical (computer

synthesized) frequency spectrum was created and then the

speed (or speeds) of sound were found from the spectrum and

compared to the speed (or speeds) of sound used to produce

the spectrum. Once satisfied that the method could

reproduce the speed of sound from a simulated spectrum, some

experimental spectra were analyzed. The transformed speeds

of sound obtained from these experimental spectra were then

compared to known values, which for the gas in question,

argon, have been shown to be in accord with those directly

calculated using a truncated virial equation of state. The

transformed speeds of sound may be lower than the calculated

speeds since the latter are the speeds of sound at zero

frequency and the transformed ones are an average speed of

sound over all the frequencies within the spectrum.









7

The following chapters describe the theory of design as

well as the theory of operation of this transformation

technique. The design of the apparatus is similar to other

acoustic devices with a few exceptions. The seal technology

employed allows operation over wider temperature and

pressure ranges. Another unique feature of the apparatus is

the ability to vary its volume with a specially designed

bellows assembly. This apparatus has the capability to

measure four state variables simultaneously. In addition,

the source code for all measurement techniques has been

included in the appendices to describe the operation of the

apparatus in detail.















CHAPTER 2
THEORY


Two basic theoretical constructs central to the present

novel sonic speed technique are explained in this chapter--

the theory of design and the theory of operation. The

theory of design begins with established theories of wave

phenomena and applies modern computational methodologies to

them. A new algorithm developed here facilitates the

computations.

The theory of operation is presented to reveal the

order of events that lead to the measurement of the speed of

sound with this technique. The equations and operational

bounds may seem trivial to anyone familiar with Fast Fourier

Transform (FFT) techniques, but, to the newcomer, these will

likely seem arbitrary and unbounded. They are, in fact,

very closely interrelated. The two parameters that govern

the operation of any FFT spectrometer are the buffer size

and the sample rate of the ADC; other parameters may be

deduced from them. The operation of many of the basic

theories described are transparent to the user since they

are contained mainly within the source code given in the

appendices.











Theory of Design

The dynamics associated with the acoustical field of a

nondissipating gas were first examined by Rayleigh in

1872.' Rayleigh's development revealed a basis set of

resonant frequencies of sound for a gas in a cavity.

Experimentally these frequencies have heretofore been

measured by observing the response of the gas to a slowly

varying periodic stimulus. The present work is concerned

with obtaining the information implicit in the frequency

spectrum very rapidly. Acquisition of the frequency domain

may be accomplished by a Fast Fourier Transform (FFT) of a

time domain signal from an Analog to Digital Converter

(ADC). Through a Sonic Transformation (ST) of the Fourier

coefficients, this information can be further transformed

into the sonic domain which readily reveals the speed of

sound and other features of the acoustic field.

First, assume there exists a velocity potential r such

that

V--V| Equation 2-1.


where v is the velocity of the gas. The standing wave

produced in the gas with a speed of sound (c) is related to

Sby the standard wave equation




1 J.W.S. Rayleigh, Theory of Sound (New York:
Dover, 1894), reprinted 1945, Section 331.











V2 _1 -ffiA
C 2 at2


Equation 2-2.


Assuming a time separable solution to the above equation


Equation 2-3.


where t0 is then the solution to a scaler Helmholtz equation


+*o ( ( )2*0 0,


Equation 2-4.


then the analytical expression for r0 2 is


0 ( ( ) Pfm(cos(8)) (Asin(mp) + Bcos(m(p))

Equation 2-5.



The function j, is a Bessel function of the first kind and Pm

is an associated Legendre polynomial in cos(6). Since, by

definition, a nondissipating gas is contained, the boundary

condition of the radial component is that the velocity of

the gas is zero at the rigid wall


fb
J. V'dd- 0.
Surf


Equation 2-6.


2 H.G. Ferris, "The Free Vibrations of a Gas
Contained within a Spherical Vessel," Journal of the
Acoustical Society of America 24 (January 1952): 57.


Crg 0, o) ei~










For a spherical cavity, the surface is described by

da-g2sin(8)dOdprf, Equation 2-7.


where g is a geometric factor or the radius of the spherical

cavity. Substitution of the gradient of i in Equation 2-6

yields


l P'(cos(0)) (Asin(mp)+Bcos(m())g2sin(8)d~d' j_ I- 0.
Surf
Equation 2-8.





Since this must be zero for all values of a and b, then

a- (-r) L 0. Equation 2-9.




For a given value of 1, there are an infinite number of

roots for the above relation. The lowest positive root is

denoted by n=l, the next root is n=2, the following n=3, and

so forth. These integral values represent the modes of

vibration for that given 1. The roots of the above

relations have been calculated in increasing magnitude as

shown in Table 2-1.3


3 Ferris.













Table 2-1. The values of the roots to the first
derivative of a Bessel function of the first kind.

i Ri 1 n

1 2.08158 1 1
2 3.34209 2 1
3 4.49341 0 1
4 4.51408 3 1
5 5.64670 4 1
6 5.94036 1 2
7 6.75643 5 1
8 7.28990 2 2
9 7.72523 0 2
10 7.85107 6 1
11 8.58367 3 2
12 8.93489 7 1
13 9.20586 1 3
14 9.84043 4 2
15 10.0102 8 1
16 10.6140 2 3
17 10.9042 0 3
18 11.0703 5 2
19 11.0791 9 1
20 11.9729 3 3
21 12.1428 10 1
22 12.2794 6 2
23 12.4046 1 4
24 13.2024 11 1
25 13.2956 4 3
26 13.4721 7 2
27 13.8463 2 4
28 14.0663 0 4
29 14.2580 12 1
30 14.5906 5 3
31 14.6513 8 2
32 15.2446 3 4
33 15.3108 13 1
34 15.5793 1 5
35 15.8193 9 2
36 15.8633 6 3
37 16.3604 14 1
38 16.6094 4 4
39 16.9776 10 2
40 17.0431 2 5
41 17.1176 7 3
42 17.2207 0 5










13



Table 2-1 continued.

i Ri 1 n

43 17.4079 15 1
44 17.9473 5 4
45 18.1276 11 2
46 18.3565 8 3
47 18.4527 16 1
48 18.4682 3 5
49 18.7428 1 6
50 19.2628 6 4
51 19.2704 12 2
52 19.4964 17 1
53 19.5819 9 3
54 19.8625 4 5
55 20.2219 2 6
56 20.3714 0 6
57 20.4065 13 2
58 20.5379 18 1
59 20.5596 7 4
60 20.7960 10 3
61 21.2312 5 5
62 21.5372 14 2
63 21.5779 19 1
64 21.6667 3 6
65 21.8401 8 4
66 21.8997 1 7
67 22.0000 11 3
68 22.5781 6 5
69 22.6165 20 1
70 22.6625 15 2
71 23.0829 4 6
72 23.1067 9 4
73 23.1950 12 3
74 23.3906 2 7
75 23.5194 0 7
76 23.6534 21 1
77 23.7832 16 2
78 23.9069 7 5
79 24.3608 10 4
80 24.3821 13 3
81 24.4749 5 6
82 24.6899 22 1
83 24.8503 3 7
84 24.8995 17 2











A solution to the above equation occurs when


i g-Ri Equation 2-10.




and the frequency of the standing wave within the cavity at

speed c is then

Ri c
o -2nf -- Equation 2-11.
g



where g is a geometric factor and Ri is the ith tabulated

root.

The previous equation describes the frequency basis for

all standing waves or resonant excitations in the cavity.

Experimentally, the resonant frequencies are acquired in the

Fourier format (see Appendix A) where


Equation
F(t) -C (Apsin () t) +Bcos (pt)) 2-12.
p




If multiple speeds of sound occur within the cavity medium,

each having an almost infinite number of resonant

frequencies, the job of determining the speeds of sound from

the corresponding frequencies is tedious. Even with a

truncated basis of roots (as in Table 2-1), finding the

speed is not easy and requires considerable analysis. The










15

ST developed below facilitates this task. It transforms the

coefficients of the FFT directly to the sonic speed domain.

Consider a system through which sound propagates at one

or more speeds. Let the associated frequencies be weighted

by some values ki, where


00

F s (t)-C kif(i, t)


Equation 2-13.


and


00
f (ci, t) (aijsin (gcRj t) +bijcos (gciRj t) .


Equation 2-14.








If we assume that all signals detected in the Fourier

coefficients are acoustic resonances


Equation 2-15.


then it follows that


00 00
A,-Z kia-ij (W p, gciRj)
1 3


Equation 2-16.


F F(t) -FS(t) ,











where the value of 6 is as follows,


8(O gciRj)6 1' p-gciiRj Equation 2-17.
PI gcjO pij 0, W *gciRj




The values of k, are of greater interest than the Fourier

coefficients. One method to acquire n coefficients for a

truncated sum of m roots would be to perform n truncated

least square operations of order 2m+l to obtain n functions

f(ci,t) and then perform one more least square operation of

order n to obtain the coefficients ki. Each least square

operation is approximately an n-cubed operation (FLOPs4

n3). By performing the transformation shown below, weights

that are proportional to ki can be obtained with

considerably fewer FLOPs.

Let

n o
w FAPpl, Equation 2-18.
mp




then by substituting Ap from Equation 2-16 into the above

expression,







4 FLOP is a FLoating point OPeration (see Chapter 3,
Theory of Operation).













kiaj8Pl6j8 p.m Equation 2-19.
mp i




Since 1 is fixed, then for a given m and p, the only nonzero

values occurs when i=l and j=m. This reduces the above

expression to

n oo
W E-- k kam8 plm Equation 2-20.
mp




For a given 1 and m, there is only one nonzero value p,

hence

n n
w-i knaE -k alm-ka- l Equation 2-22.
m m




where at is the average amplitude over n roots of the Ith

speed. Most importantly, this result shows that this choice

of weights is directly proportional to the sonic

coefficients k,. The relative values of w, cannot be used

for determining relative values of k,. Since there is an

overlap of different Ri values, the weights can be used to

detect the presence of resonant speed of sound within the

cavity.











Theory of Operation

For the purpose of evaluating the sonic transform

technique, its use on a gas with known properties is

required. Argon was chosen for this purpose because of its

relative simplicity and well-documented physical behavior.

The speed of sound in argon has been carefully measured and

shown to be in agreement with values calculated with the

virial equation of state.5

At moderate pressures (< 10 atm) two terms in the virial

expansion are sufficient to give sonic speeds within

experimental uncertainty. For this work the sonic speed in

argon was calculated from the virial equation of state

(truncated after the third term) using reduced virial

coefficients obtained from a Lennard-Jones 6-12 potential.

The speed of sound at zero frequency6 may be related to

either the adiabatic or isothermal partial derivative of

pressure with respect to molar density. Specifically, the

square of the speed of sound is


C 2_I, Equation 2-22.
0,M( ap M) ap






5 R. Byron Bird, "Numerical Evaluation of the Second
Virial Coefficient," The Virial Equation of State CM-599
(Madison: University of Wisconsin, May 10, 1950), 47-52.
6 J.O. Hirschfelder, C.F. Curtiss, and R.B. Bird,
Molecular Theory of Gases and Liquids (New York: Wiley and
Sons, 1954), 369.









19

where P is pressure, M is the molecular weight and p is the

molar density. Using the constant temperature form of the

above equation, where y is the ratio of heat capacities, the

speed of sound can be found by solving for the individual

values of Cp, CV and the constant temperature derivative.

There is no equation of state that can be expressed in a

single analytical expression that has high enough accuracy

for this experiment. The best possible solution is a

truncated virial equation with numerically calculated

coefficients at various temperatures. The values of the

second virial coefficients are given in Table 2-2 and the

values of the third virial coefficients are given in Table

2-3. The accuracy of this numerical solution has been

investigated by Bird.7 Using the truncated virial equation

of state in terms of reduced virial coefficients given by:


Key Terms, Symbols and Definitions for Truncated Virial
Equation of State

B = Second Virial k = Boltzmann's constant
Coefficient

C = Third Virial R = Gas constant
Coefficient

b0 = %7R3 B* = B/bo

a = Lennard-Jones 6-12 C* = C/b20
collision diameter

6 = Lennard-Jones 6-12 T* = kT/E
maximum energy attrac-
tion or depth of N = Avogadro's Number
potential well


7 Bird.













Table 2-2. Reduced second virial coefficients for
the Lennard-Jones 6-12 potential.

T* B* BB* B2* Bi*-B*


0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.80
1.85
1.90
1.95
2.00
2.10
2.20
2.30
2.40
2.50
2.60
2.70
2.80


-27.880581
-18.754895
-13.798835
-10.754975
-8.720205
-7.2740858
-6.1979708
-5.3681918
-4.7100370
-4.1759283
-3.7342254
-3.3631193
-3.0471143
-2.7749102
-2.5380814
-2.3302208
-2.1463742
-1.9826492
-1.8359492
-1.7037784
-1.5841047
-1.4752571
-1.3758479
-1.2847160
-1.2008832
-1.1235183
-1.0519115
-0.98545337
-0.92361639
-0.86594279
-0.81203328
-0.76153734
-0.71414733
-0.66959030
-0.62762535
-0.55063308
-0.48170997
-0.41967761
-0.36357566
-0.31261340
-0.26613345
-0.22358626
-0.18450728


76.607256
45.247713
30.267080
21.989482
16.923690
13.582156
11.248849
9.5455096
8.2571145
7.2540135
6.4541400
5.8034061
5.2649184
4.8127607
4.4282616
4.0976659
3.8106421
3.5592925
3.3374893
3.1404074
2.9642040
2.8057826
2.6626207
2.5326459
2.4141403
2.3056683
2.2060215
2.1141772
2.0292621
1.9505276
1.8773287
1.8091057
1.7453722
1.6857016
1.6297207
1.5275444
1.4366294
1.3552188
1.2819016
1.2155320
1.1551691
1.1000353
1.0494802


-356.87679
-189.46536
-116.36604
-78.87795
-57.33952
-43.88245
-34.91869
-28.64050
-24.06266
-20.61311
-17.94190
-15.82546
-14.11557
-12.71081
-11.53985
-10.55133
-9.70744
-8.97985
-8.34700
-7.79217
-7.30227
-6.86692
-6.47777
-6.12805
-5.81225
-5.52578
-5.26485
-5.02628
-4.80738
-4.60587
-4.41980
-4.24750
-4.08753
-3.93863
-3.79972
-3.54814
-3.32647
-3.12974
-2.95401
-2.79614
-2.65355
-2.52416
-2.40623


104.488
64.003
44.066
32.744
25.644
20.8563
17.4468
14.9137
12.9672
11.4299
10.1884
9.1665
8.3120
7.5877
6.9663
6.4279
5.9570
5.5419
5.1734
4.8442
4.5483
4.2810
4.0385
3.8174
3.6150
3.4292
3.2579
3.0996
2.9529
2.8165
2.6894
2.5706
2.4595
2.3553
2.2573
2.0782
1.9183
1.7749
1.6455
1.5281
1.4213
1.3236
1.2340













Table 2-2 continued.

T* B* B1* B2* B1*-B*


2.90
3.00
3.10
3.20
3.30
3.40
3.50
3.60
3.70
3.80
3.90
4.00
4.10
4.20
4.30
4.40
4.50
4.60
4.70
4.80
4.90
5.00
6.00
7.00
8.00
9.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
100.00
200.00
300.00
400.00


-0.14850215
-0.11523390
-0.08441245
-0.05578696
-0.02913997
-0.00428086
0.01895684
0.04072012
0.06113882
0.08032793
0.09839014
0.11541691
0.13149021
0.14668372
0.16106381
0.17469039
0.18761774
0.19989511
0.21156728
0.22267507
0.23325577
0.24334351
0.32290437
0.37608846
0.41343396
0.44059784
0.46087529
0.52537420
0.52692546
0.51857502
0.50836143
0.49821261
0.48865069
0.47979009
0.47161504
0.46406948
0.41143168
0.38012787
0.35835117


1.0029572
0.9600031
0.9202229
0.8832774
0.8488746
0.8167606
0.7867145
0.7585430
0.7300758
0.7071630
0.6836715
0.6614830
0.6404922
0.6206045
0.6017352
0.5838082
0.5667545
0.5505118
0.5350237
0.5202387
0.5061101
0.4925951
0.3839722
0.3082566
0.2524801
0.2097011
0.1758670
0.0286638
-0.0174929
-0.0393115
-0.0516478
-0.0593621
-0.0645039
-0.0680819
-0.0706470
-0.0725244
-0.0775400
-0.0765245
-0.0747534


-2.29831
-2.19920
-2.10785
-2.02340
-1.94511
-1.87231
-1.80447
-1.74108
-1.68174
-1.62605
-1.57371
-1.52441
-1.47789
-1.43394
-1.39234
-1.35291
-1.31548
-1.27991
-1.24606
-1.21381
-1.18305
-1.15367
-0.919393
-0.757930
-0.639879
-0.549792
-0.478779
-0.170403
-0.072012
-0.024109
0.003927
0.022147
0.034817
0.044056
0.051031
0.056441
0.077296
0.081397
0.082055


Source: J.O. Hirschfelder, C.F.
Bird, Molecular Theory of Gases
York: Wiley and Sons, 1954), 11


Curtiss, and R.B.
and Liquids (New


14.


1.1515
1.0752
1.0046
0.93906
0.87802
0.82104
0.76776
0.71782
0.67094
0.62684
0.58528
0.54607
0.50900
0.47392
0.44067
0.40912
0.37914
0.35062
0.32346
0.29756
0.27285
0.24925
0.06107
-0.06783
-0.16095
-0.23090
-0.28501
-0.49671
-0.54442
-0.55789
-0.56001
-0.55758
-0.55316
-0.54787
-0.54226
-0.53659
-0.48897
-0.45665
-0.43310













Table 2-3. Reduced third virial coefficients and
their derivatives for the Lennard-Jones 6-12
potential.

T* C* C,* C2


0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.80
1.85
1.90
1.95
2.00
2.10
2.20
2.30
2.40
2.50
2.60
2.70
2.80
2.90
3.00


-3.37664
-1.79197
-0.84953
-0.27657
0.07650
0.29509
0.42966
0.51080
0.55762
0.58223
0.59240
0.59326
0.58815
0.57933
0.56831
0.55611
0.54339
0.53059
0.51803
0.50587
0.49425
0.48320
0.47277
0.46296
0.45376
0.44515
0.43710
0.42260
0.40999
0.39900
0.38943
0.38108
0.37378
0.36737
0.36173
0.35675
0.35234


28.68
18.05
11.60
7.561
4.953
3.234
2.078
1.292
0.7507
0.3760
0.1159
-0.0646
-0.1889
-0.2731
-0.3288
-0.3641
-0.3845
-0.3943
-0.3963
-0.3929
-0.3858
-0.3759
-0.3643
-0.3516
-0.3382
-0.3245
-0.3109
-0.2840
-0.2588
-0.2355
-0.2142
-0.1950
-0.1777
-0.1621
-0.1482
-0.1358
-0.1247


-220.
-140.
-92.1
-62.1
-42.7
-29.8
-21.0
-14.9
-10.6
-7.52
-5.29
-3.66
-2.46
-1.57
-0.910
-0.420
-0.050
0.224
0.427
0.572
0.680
0.755
0.806
0.837
0.854
0.859
0.856
0.830
0.794
0.749
0.700
0.651
0.602
0.557
0.514
0.473
0.439













Table 2-3 continued.

T* C* C1* C2


3.10
3.20
3.30
3.40
3.50
3.60
3.70
3.80
3.90
4.00
4.10
4.20
4.30
4.40
4.50
4.60
4.70
4.80
4.90
5.00
6.00
7.00
8.00
9.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
100.00
200.00
300.00
400.00


0.34842
0.34491
0.34177
0.33894
0.33638
0.33407
0.33196
0.33002
0.32825
0.32662
0.32510
0.32369
0.32238
0.32115
0.32000
0.31891
0.31788
0.31690
0.31596
0.31508
0.30771
0.30166
0.29618
0.29103
0.28610
0.24643
0.21954
0.20012
0.18529
0.17347
0.16376
0.15560
0.14860
0.14251
0.10679
0.08943
0.07862


Source: J.O. Hirschfelder, C.F. Curtiss, and R.B.
Bird, Molecular Theory of Gases and Liquids (New
York: Wiley and Sons, 1954), 1116.


-0.1148
-0.1060
-0.09826
-0.09133
-0.08510
-0.07963
-0.07462
-0.07024
-0.06634
-0.06286
-0.05989
-0.05709
-0.05458
-0.05237
-0.05040
-0.04865
-0.04712
-0.04579
-0.04461
-0.04359
-0.03893
-0.03989
-0.04231
-0.04529
-0.04825
-0.06437
-0.06753
-0.06714
-0.06566
-0.06388
-0.06203
-0.06025
-0.05857
-0.05700
-0.04599
-0.03970
-0.03551


0.400
0.369
0.340
0.313
0.288
0.266
0.246
0.227
0.210
0.194
0.183
0.169
0.156
0.145
0.134
0.125
0.116
0.108
0.100
0.0934
0.0449
0.0258
0.0192
0.0183
0.0199
0.0502
0.0654
0.0717
0.0742
0.0750
0.0748
0.0741
0.0732
0.0722
0.0619
0.0547
0.0496













Equation 2-23.


dB*) dC*
dT' dT'


*2( dT2B dT*2_
dT*2 dT *


The constant pressure and constant volume heat capacities,

respectively, are given by


5 B2 (B'-B )2-C+C*- + 1C
C,-R5 ---+ 2
2 V* (V*)2








Cv( 3 2B*+B2 2C:+C2. Eqi
v 2 V* 2 (V*)2


Equation
2-24.


nation 2-25.


The constant temperature derivative is given by


SRT+ B* (*)2
V V* (V*)


Equation 2-26.


S-R 1+2 +3*
S1 V*" (V*)2
[jV Tv


and










A copy of the source code for the calculation of the

speed of sound using the truncated virial equation of state

for argon is given in Appendix C.

The FFT was performed using the base 2 Cooley Tukey

algorithm.8 The base 2 algorithm was chosen to optimize

the round off error. Although the base 3 algorithm has a

more efficient Floating Point Operation (FLOP) count,9 a

digital computer, which also operates normally in base 2,

preferentially accommodates calculations which use the base

2 number system. This then calls for the number of samples

to be some integral power of 2. The operational bounds of

the FFT are as follows:

The magnitude of the frequency domain is

Equation
n ns S no. of samples Equ227.on
f 2 f no. of frequencies





the Nyquist limit of maximum frequency is


f sample rate
ax Equation 2-28.






8 J.W. Cooley and J.W. Tukey, "An Algorithm for the
Machine Calculation of Complex Fourier Series,"
Mathematical Computations 19 (April 1965): 297-301.

9 G.D. Bergland, "A Fast Fourier Transform Algorithm
for Real-Valued Series," Communications of the ACM 11
(October 1968): 703-710.











the period for sampling is


Tns
(sample rate)


Equation 2-29.


the frequency resolution is


nf
res n
nlf


Equation 2-30.


and the FLOP count is


FLOP n, log2 n,.


Equation 2-31.


The ST was performed using the following algorithm


denom-2n gfre
For i-1, nm
Ci
ra ti o-
denom
w2-O
For j-l, n
index integer(ratio*Rj)
Wi wi+aindex


Equation
2-32.


The maximum speed is limited by insuring that all ST

frequencies of the roots exist within the FFT frequency

domain













Cmax 7
Rn,


Equation 2-33.


where nr = number of roots in the truncated ST basis set.

Since


c f
res f es
Cmax fmax I


Equation 2-34.


then the resolution of the algorithm and magnitude of the

sonic domain are, respectively,


max
Rmax 'f
Acres


Equation 2-35.


with a FLOP count of


FLOPs nma2n,.


Equation 2-36.


The theory of design is related to the construction of

the apparatus in that building a very precisely known

spherical cavity provides a geometry factor (g) that is

simply the radius of the sphere. Once the geometry factor

is known and an ADC is chosen, the rest of the parameters

are fixed. For a given buffer size and sample rate, the

frequency range and resolution are set; and for a given set

of roots, the sonic range and resolution are set. The


max
res









28

selection of an ADC should consider the geometry of the

apparatus as well as the sonic range of interest. In the

next chapter the interfacing employing an ADC as well as the

design of the apparatus are described.















CHAPTER 3
EXPERIMENTAL


The experimental design of this technique must address

two principal problems, the computer interfacing of the data

acquisition methods and the mechanical design of the

apparatus. In a modern laboratory, data acquisition is no

longer the tedious matter of turning dials, reading meters

and logging data. Even the most impartial researcher tends

to be inconsistent when manually measuring large amounts of

data over long periods of time. The digital computer has

taken over these more tedious tasks with much better speed

and consistency. The first part of this chapter describes

the interfacing of the computer involving data acquisition.

In addition, as more tasks are controlled by the

computer, more time remains for the scientist to evaluate

results and implement design improvements. Specifically

during this work, volume control was added for the first

time. Knowing the exact volume of the apparatus has always

been necessary, but this volume has in the past been fixed.

Many experiments, however, would benefit from a direct

measurement of the effect of volume change. For example,

V2(aP/aV), could be substituted into Equation 2-22 along

with the speed of sound at zero frequency (co) and the









30
molecular weight (M) for a direct measurement of y. To this

end, an extremely accurate variable volume control was

designed for this apparatus.



Interfacing

Data collection utilizes five basic devices and two

computers. The physical parameters measured are

temperature, equilibrium pressure, amplitude (- acoustic

pressure) and time. The first two of these measurements

were made using standard laboratory instruments.

Temperature is obtained by measuring the resistance of a

platinum Resistance Temperature Device (RTD) using a

Keithley 195a Digital Multi-Meter (DMM). The acquired

resistance was updated and sent to the 8088 Central

Processing Unit (CPU) along the National Instruments General

Purpose Interface Bus (GPIB or IEEE) every 0.1 seconds.

Resistance was then converted to temperature, in accordance

with the RTD manufacturer's specifications, by the 8088 CPU.

Pressure was read from a calibrated pressure sensitive

Beckman Digital Strain Gauge in units of Pounds per Square

Inch Absolute (PSIA). These readings were updated and sent

to the CPU along the IEEE bus every 0.5 seconds.

Amplitude and time were measured simultaneously by the

WAAG II Analog to Digital Converter (ADC). The WAAG ADC has

eight-bit resolution, a 32768 point buffer and multiple

sample rates of 40MHz, 4MHz, 400kHz, 40kHz and 4kHz. The











measurements are read and stored into the buffer

sequentially. As a new measurement is added to the buffer,

the oldest value is discarded. Once polled by the computer,

the WAAG II dumps its entire buffer to the 8088 Random

Access Memory (RAM) and then proceeds to acquire new data.

The algorithm for the acquisition (source code provided in

Appendix D) is as follows

For i-l,n
r-resistance rtd
Ti-convert(r)
P-reading for strain guage Equation 3-1.
amp-dump ADC buffer
dump amp on hard drive
dump T on hard drive
dump P on hard drive










Using Equation 2-28, a sample rate of 40kHz leads to a

maximum frequency (fax) of 20 kHz. With a buffer size of

32768, the period of sampling (7) and frequency magnitude

(nf) were found from Equations 2-29 and 2-27, respectively,

to be 0.8192 seconds and 16384. The resulting frequency

resolution (fes) from Equation 2-30 was approximately 1.22

Hz.

The excitation frequency is generated by a Hewlett

Packard HP3325b function synthesizer. When the HP3325b is

put into discrete sweep mode, it generates a frequency-































Figure 3-1. Instrument rack.

modulated-phase-consistent sinewave that sweeps from 0 Hz to

20 kHz in 0.8192 seconds, then repeats from OHz to 20kHz

with a peak-to-peak voltage of 20.0 Volts. A TTL reference

wave is sent to a Stanford Research SR510 lock-in amplifier

from the HP3325b. The return signal from the resonator is

also sent to the SR510, and all frequencies except for the

reference frequency are filtered out by the frequency

dependent band pass filter in the SR510. The resulting

signal is then amplified and sent to the WAAG II ADC. All

gain and power settings can be sent to the instruments along

the IEEE bus.

The waveform collected by the ADC is dumped to hard

drive in binary format while temperature and pressure are

stored in an array but are later dumped to hard drive in

binary format just before the program terminates. The











binary data are then sent to the DELL system 310 micro

computer, the processing computer system. The binary format

of the 8088 (8 bit) is different from the binary format of

the DELL (32 bit), so the data must be translated to a

common format. Since the data ranges in values from 0 to

255, two hexadecimal numbers can contain one datum (for

source code, see Appendix E). The binary data are

transformed to hexadecimal by the DELL then further

transformed into the frequency domain.

Because the resulting large data set was limited to

eight-bit resolution, a time correlation method was used to

reduce floating point error. This method simply doubles the

data set by adding the waveform to itself. It should be

noted that this does not increase resolution by having a

double basis set but simply lessens round off error of the

computer; the frequency domain data are unaffected. The

data are then dumped to the DELL hard drive. Data are then

transformed to the sonic domain and dumped to the DELL hard

drive in binary format (see Appendix B for source code).

Three of the data sets--time domain 8088 binary format,

frequency domain DELL binary format, and sonic domain DELL

binary format--are then stored, along with all the source

code used in the process, on tape. The process was then

repeated for different temperatures.











Apparatus

The apparatus consists of four basic parts--the

spherical cavity, the volume-controlling bellows, the

reciprocating pump and the Delta Design series 9000

environmental chamber.



Spherical Cavity

The spherical cavity was constructed from two solid

pieces of 303 stainless steel; a three-inch radius spherical

cavity was cut from the center. Excess material was removed

from the outer portion to lower the mass of the sphere

thereby making it easier to control its temperature. To

assure safe operation at the highest intended pressure (4000

PSIA), the minimum wall thickness was set at 6.4 mm (0.25

in). This dimension was based on a calculation of the

bursting pressure in a spherical shell obtained by setting

the force acting to stretch the walls equal to the tensile

strength of the stainless steel. A safety factor of 4 was

used.

The top portion of the sphere contains the two

transducer mounts. A Macor insulated electrical feed-

through was mounted by employing a customized tapered ram

seal with annealed copper gaskets. The inner threaded

portion was used to align the transducer. The transducers

were Piezoelectric lead-Zirconate lead-Titanate (PZT)

bimorphs which have high motion sensitivity. They were








35

placed as close as possible to the surface of the sphere in

order to minimize departure from the sphericity. The two

halves of the sphere were sealed together using an annealed

copper gasket with a conflat type knife-edge seal and held

together with two mild steel clamps as shown in Figure 3-2.

Inlet ports for the gas were constructed on the top and

bottom of the spherical cavity. The entire assembly was

pressure tested to 3500 PSIA at room temperature.



Pump

The pump chamber (Figure 3-3) was constructed of a 304

stainless steel tube, 13 inches long with 1.250 inch outside

diameter and 0.148 inch wall. The top portion was sealed by

brazing a 304 stainless seal plug 1/2 inch thick with a 1/16

inch bore. The bottom portion was sealed by a 304 stainless

steel plate with an annealed copper gasket on a conflat

knife-edge seal. Seven magnetic field coils aligned

concentrically on the tube create the pumping action by

successively attracting a magnetic piston free to move

inside the stainless steel tube. The bottom two coils are

switched on remotely; the third coil from the bottom is

activated as the bottom coil is turned off. This action is

repeated until the magnetic piston reaches the top of the

tube. Then a reversed action moves the magnetic piston to

the bottom of the tube to complete one pumping cycle.

Doubled-pumping action is created by use of four one-way

































Figure 3-2. Spherical cavity sections and clamping flanges.











valves placed outside the assembly. The strength of the

magnetic field as well as the frequency of field oscillation

are adjusted remotely. At the highest field strength and

frequency of oscillation, a pumping speed of 200 mL per

second at room temperature and pressure was recorded. An

aluminum mount was constructed to hold the pump in an

upright position.



The Bellows

The addition of the bellows assembly brings on-line

volume or density control to this technology for the first

time. The collapsible bellows, constructed of 0.005 inch

thick 304 stainless steel, was welded to a 1 inch thick

plate which had a 1/4 inch hole bored horizontally to

connect the adjustable volume of the bellows to the

spherical cavity.

The outer portion of the bellows is contained in a

chamber that was constructed from a solid piece of stainless

steel and sealed to the lower plate with a triangular

annealed copper seal1. The volume of the outer chamber was

isolated from the spherical cavity and maintained at

pressures slightly below (approximately 20 PSI) that of the

spherical cavity. This then maintained the bellows in an

expanded position. The volume of the bellows was controlled

by a threaded ram bolted to the top of the outer chamber.


1 Technology developed by S.O. Colgate in 1990.





















MAGNETIC FIELD COILS









Figure 3-3. Pump assembly.
PIS rON












Figure 3-3. Pump assembly.





co


















III I


AWWAMAW


vVWM W


z


Figure 3-4. The bellows and bellows chamber.











The position of the ram was externally controlled by a

customized micrometer to within 0.001 inch. The pressure

was monitored by two Sensotec pressure transducers. The

pressure transducers were not able to operate in the harsh

conditions of the environmental chamber so they were placed

outside the chamber and connected to the apparatus by two

stainless steel capillary tubes. These capillary tubes

prevented a large volume of the sample from being outside

the temperature-controlled volume. The assembled apparatus

was connected as shown in Figure 3-5.

The completely assembled apparatus was then placed into

the environmental chamber. The environmental chamber

operates over the temperature range of 1500C to -170oC and

is controlled by the manufacturer's programming language

sent along the IEEE bus. The assembled apparatus was

pressure tested up to 2800 PSIA.

The calibrated apparatus presently requires that only

one parameter, the volume, be monitored and controlled by

the user. The other three state variables, temperature,

pressure and speed of sound, are acquired automatically by

the computer. Typical results are displayed in the next

chapter.











4 TO PRESSURE TRANSDUCERS .


Signal From Rack


Signal To Rac


Figure 3-5. Apparatus assembly.


PUMP














CHAPTER 4
DATA AND RESULTS

The data and plots resulting from this experiment are

discussed in three groups. This includes a theoretical

computer synthesized set of data, an experimentally acquired

set of data for argon at low temperatures and then a

discussion of argon at high temperatures.

Figure 4-1 depicts a theoretical waveform based on

using the first 84 resonances in a spherical cavity (radius

of 3 inches) filled with a fluid medium which propagates

sound at 350 m/s. Figure 4-2 depicts a similar theoretical

waveform again using the first 84 resonances in the same

cavity but now containing a fluid medium which propagates a

speed of sound at two speeds, 350 m/s and 150 m/s. These

two waveforms simulate those which would be acquired by the

ADC under ideal conditions.

Figure 4-3 depicts the FFT of the waveform shown in

Figure 4-1 while Figure 4-4 depicts the FFT of the waveform

shown in Figure 4-2. Figure 4-5 displays the final results

of the ST of the FFT described in Figure 4-3 and Figure 4-6

displays the results of the ST of the FFT in Figure 4-4.

These six figures portray the chronological order of

acquisition and calculation for the simulated set of data.










Note that ST transforms shown in Figures 4-5 and 4-6

correctly recover the input sonic speeds (350 m/s and 150

m/s).

Figure 4-7 is an experimentally acquired waveform of

the resonances of argon at a low temperature (-31.56oC) in a

spherical cavity with a 3.000 inch radius. The experimental

conditions are given in Table 4-1. An expanded view of a

section of Figure 4-7 is given in Figure 4-8 to show the

resolution with which the waveform is acquired in other

regions. The FFT of the waveform of Figure 4-7 is shown in

Figure 4-9 and the relevant physical and computational

parameters are given in Table 4-2. As seen in Figure 4-9,

the baseline is not very stable in the region of 10,000 Hz.

An expanded view of this region is shown in Figure 4-10.

Several STs were performed on the data in Figure 4-9 using

different numbers of roots. The resulting ST weights

employing the first 21 roots are shown in Figure 4-11 with

an expanded view of the region that contains the known speed

of sound in argon shown in Figure 4-12. The experimental

and computational parameters are given in Table 4-3. Four

STs were performed on the same FFT data in which only the

number of roots used in the ST were changed. The results

are shown in Figures 4-11 through 4-18 while parameters are

listed in Tables 4-3 through 4-6. These results reveal the

important features of the technique; they are described

later in this chapter.










The same experiment was performed at a higher

temperature (50.93oC). Figure 4-19 shows the experimentally

acquired waveform with the experimental and computational

parameters given in Table 4-7. The expanded view shown in

Figure 4-20 indicates that more of the resolution of the ADC

was utilized. The baseline of the FFT shown in Figure 4-21

is considerably better than that of the low temperature

experiment (Figure 4-9). The expanded view shown in Figure

4-22 indicates that the sharp acoustic resonances are larger

than the perturbed baseline and are better resolved than

those in Figure 4-10. The four STs using the different sets

of basis functions at this temperature are shown in Figures

4-23 through 4-30 along with the corresponding parameters in

Tables 4-9 through 4-12.

Interpretations of the data and graphs presented above

are organized as follows. The first section discusses the

characteristics of the time domain signal and how it

deviates from ideality. The second discusses the

characteristics of the frequency domain while the third

section examines the sonic domain and the influence of

varying the number of roots (nr). In addition, the volume

calibration data are included at the end of the chapter.



Time Domain Plots

Figures 4-1 and 4-2 show two computer simulated ADC

signals. Figure 4-1 was generated from the sum of 84









45
sinewaves with frequencies generated from Equation 2-11 for

a sonic speed (c) of 350 m/s, a geometric factor (g) of 3

inches and assuming equal amplitudes of the resonances.

Figure 4-2 was generated from two sets of 84 sinewaves--one

for c = 150 m/s, the other for c = 350 m/s. Both waveforms

are similar in that they show no beat patterns or

interference. Figure 4-7 shows a low temperature ADC signal

where the resolution is quite low except for when the

excitation frequency corresponds closely to a resonance

frequency. This is an indication that the resonances are

decaying rapidly. Figure 4-19 shows a high temperature ADC

signal where the resolution is better since clearly the

resonances are not decaying as rapidly as in the low

temperature case. In other words, Figure 4-19 is

approaching the characteristic of Figures 4-1 and 4-2.

Ideally, an evenly distributed waveform uses the entire

resolution of the ADC as was seen in the expanded Figures 4-

8 and 4-20; that is not the case here. The resolution

acquired is less than half the ADC resolution.



Frequency Domain Plots

The baselines of the frequency domain plots in Figures

4-3 and 4-4 indicate that the amplitudes are perturbed due

to floating point calculation error. The low temperature

frequency domain plot in Figure 4-9 shows an extremely large

broad peak in the center of the frequency spectrum. The









46
expanded view shown in Figure 4-10, however, shows the sharp

gas resonances imposed on top of this large peak. As the

temperature is increased and the decaying of the resonances

decreases, the broad peak decreases in size as well as

frequency. All of these characteristics indicate that this

portion of the signal is associated with vibrations of a

solid, perhaps along the walls of the sphere or in the

transducers themselves.



Sonic Domain Plots

The two sonic domain plots in Figures 4-5 and 4-6

indicate that the amplitude perturbations of the frequency

domain do not affect the amplitudes in the sonic domain, but

that much of the floating point calculation noise is carried

through. The plots do show that the ST will resolve

multiple speeds of sound if present in the data, although

all of the low and high temperature plots shown in the

remaining figures have considerably different baselines.

The baselines are attributed to the reproducible apparatus

frequencies which are not due to normal mode vibrations of

the cavity fluid. These are called nonacoustic frequencies.

The reason that they are identifiable as being nonacoustic

is that they do not move across the baseline as the basis

set of roots is changed. The four low temperature figures

(4-11 to 4-18) as well as the four high temperature figures

(4-19 to 4-30) show that the baseline maps predominately










with respect to index and not speed. Only resonances that

are acoustic will be speed dependent and not index

dependent. As the absorption of energy by the gas increases

in the high temperature spectra, the amplitude of the speed

of sound begins to predominate as would be expected. It

should be recalled here that the time domain signal is the

same for all sonic domain plots of a given temperature; the

only thing that was changed was the number roots used to

form the basis. In addition, the size of the basis did not

seem to have a large effect on the resolution. It was not

until nr = 63 that the resolution saw any significant

increase, but this could be due to where the resonance was

with respect to the noise and does not necessarily reflect

an increase in gain.

The speeds of sound in argon calculated from the

truncated virial equation (see Appendix C) are 291.644 m/s

for the low temperature data (@ -31.56oC and 870.5 PSIA) and

350.245 m/s for the high temperature data (@ 50.93oC and

1285.5 PSIA). The ST speeds of sound are given in Tables 4-

1 through 4-12. The ST basis assumes a perfect sphere with

a radius of 3 inches. Even using this simplification, the

ST method gives sonic speeds within less than 0.5% deviation

from the calculated values. The other three physical

measurements (temperature, pressure and volume) employed

standard techniques and were calibrated as discussed in the

next section.











Volume and Pressure Calibration

The volume and pressure calibration required two

standard devices. For the pressure calibration, a Ruska

Model 2465 Dead Weight Pressure Gauge was used. The

accuracy of the Ruska gauge was 0.001 PSIA with a range

from 0.000 PSIA to 650.000 PSIA. For the volume

calibration, the Ruska gauge as well as a Ruska Model 25652

volumetric pump was used. The accuracy of the Ruska pump

was 0.01 mL. The actual calibration of the Sensotec

pressure transducers was the three point calibration

described in the Beckman 620 owner's manual. The three

pressures chosen were 0.000 PSIA, 320.000 PSIA and 640.000

PSIA. Since the accuracy of the Sensotec pressure

transducers was only 0.5 PSIA, the accuracy of the three

calibration pressures was more than necessary.

The volume calibration involved taking several volume

and pressure measurements and employing the ideal gas

equation to deduce the absolute volume as shown below


POVO Pi (V + AVi)
PiAV. P Equation 4-1.
-- V V P
Po a P





where Po is the initial pressure and Vo is the total volume

of the apparatus at that pressure. Pi and AV, are measured

by the Ruska gauge and pump, respectively. The outside









49

volume of the calibration equipment was found from the data

in Figure 4-31. The total volume of the apparatus as well

as the calibration equipment was then found from the data in

Figure 4-32. The volumes were all compared to a common

point on the Ruska pump since the pump has its own volume

that must be considered. The outside volume of the

calibration equipment was then subtracted from the combined

total to obtain the true total volume of the apparatus.

Once the total volume was found, the change in volume due to

the bellows from the same common point was found from the

data in Figure 4-33. The change in volume with respect to

the change in length of the external adjustment ram was

observed to correlate best to a second order polynomial fit.

The result (Table 4-15) was an expression for the total

volume of the apparatus as a function of the external ram

setting. The uncertainty in a total volume for a given ram

setting was 0.01%. The range of the total volume of the

apparatus was from 2350.00 to 2878.00 mL.










50





30


20


10


0


-10 -- -


-20



0 1 02 03 0.4 0.5 0.6 0.7 0.8 9

Time /s

Figure 4-1. Theoretical ADC signal for 350
m/s of speed of sound.

I .


30


20


10


0


-10


-20


-30
o 0.1 0.2 03 0.4 0.5 0.6 0.7 0.8 0.9

Time /s


Figure 4-2. Theoretical ADC signal for 150
m/s and 350 m/s speeds of sound.


































Figure 4-3. FFT of theoretical ADC signal
for 350 m/s speed of sound.


Figure 4-4. FFT of theoretical ADC sign
for 150 m/s and 350 m/s speeds of sound.
















1.80'1 ___



1. 10' .....----- .. ... ...... ........... ..... .....


0 .... .... .......


S1. 000 -........



4000 ... ---- .-- ....
2000 . --- --------- ---- ----. .. .- . .. .. .
2000


0 50 100 150 200 250 300 350 400

Speed of Sound /(m/s)

Figure 4-5. ST of FFT of theoretical ADC
signal for 350 m/s.


1.0x10'


1.40o ----- ---------------- ....................


1.20l 0' ....... ........ ... ....... .. ....... ... .....

1.0010' - ------- -------- .. ..... .. ... .. ..
m wflO4 ........................................ .....




0 '.
^ 6000

4000 *** .*- .




0 50 100 150 200 250 300 350 400

Speed of Sound /(m/s)


Figure 4-6. ST of FFT of theoretical ADC
signal for 150 m/s and 350 m/s speeds of
sound.










































Figure 4-7.


ADC signal of argon at low temperature.


Table 4-1. Low
temperature time domain
parameters of argon.

T = -31.56 .10oC

P = 870.5 .5 PSIA

ns = 32768

Sample rate = 40 kHz


250 ------------------------

200 --- --- --- -- -- ------------- ------

250 ......... ..... .. ....... ... ..... .. ....



.150 --------- --------------------



Orr 100



0
100 ---------- ------ -- ---- -------------





0 .. . . ..*. . . ... . . .. . ..
0 1.ooxo14 2.ool0 O 3.00310'

Time = X/40000 s


I Iime X/ o 40000
Figure 4-8. Expanded section
of Figure 4-7.












































Figure 4-9. FFT of
temperature.


ADC signal of argon at low


dud 7..----.----.....4--.---.--...-......--------------------



a-

OdF .. ..... ....... ..... i i

S lkl' -

0
6500 7ooo anoo a)soo K 9o50 LOnOIO'
Frequency X/O.8192 Hz

Figure 4-10. Expanded section
of Figure 4-9.


Table 4-2. Low
temperature frequency
domain parameters of
argon.

T = -31.56 .10oC

P = 870.5 .5 PSIA

nf = 16384

fmx = 20 kHz


5.000lO'



4.00xl0'



0









LOOxlO'
1.O-10


Frequency = X/0.8192 Hz


I


I









55






7.00x10

6.00x10 --- -------------- ---- ---- -----




4. --------
s.00xI1P ...........


2.OOxl0' - ------- .. -. -. .......... .. ..



0o o . : .. ........ ... ..... .....

0 100 200 300 400 500 60000 0 800

Speed of Sound /(m/s)



Figure 4-11. First ST of argon at low temperature.





Table 4-3. First sonic
3~mW domain parameters of
S......:.. ...... argon at low
.. temperature.

T = -31.56 .10oC
P = 870.5 .5 PSIA
..... c = 290.91 .05 m/s
u cmx = 788.532 m/s

uL e nr = 21
M 2 MMM "t m M, M Note: See Equations
ap of Sound /(m/s) 2-33, 2-35.


Figure 4-12. Expanded section
of Figure 4-11.










































Figure 4-13. Second ST of argon at low temperature.






7 .A . Table 4-4. Second sonic
domain parameters of
-... -.. .. .......argon at low
Temperature.

IT = -31.56 .10oC
S..... P = 870.5 .5 PSIA
c = 290.96 .03 m/s
s- .......C x = 556.016 m/s

S 2M nn = 422
peed f Sod /(m/) Note: See Equations
Figure 4-14. Expanded section 2-33, 2-35.


of Figure 4-13.
of Figure 4-13.


]










































1 Ij
Figure 4-15. Third ST of argon at low temperature.




Table 4-5. Third sonic
domain parameters of
l .. argon at low
Temperature.

T = -31.56 .10oC
SP = 870.5 .5 PSIA
c = 290.962 .027
m/s
e .c.. = 443.741 m/s
W e ......i. ... ... ..L... ... .
Snr = 63
2aM a2. .M2W 2 M 2 Note: See Equations
Spod f Sod /(m/s) 2-33, 2-35.


7.00x10*

.o10 --- ------.-- ... .--. .------ -..--- ........
.0xl0' -- -- ..

.sOxlo' -- ---- .-- . .

Sl. O ... .... -...... ........ .. .... ... ..........
.0010 - ---. ---: .. . ... .. ..........


2. ..6 .. ...

O *--- --- -----
0 ....

o 50 100 150 200 250 300 350 400 450

Speed of Sound /(m/s)


Figure 4-16. Expanded section
of Figure 4-15.


I









58






8.00x1

7001oi ----- ------ -----I
00 ...........- ---- ------- -- -- -----



50010 ......................

4.00 -0----- -------- I----- -------- -

3.00xlO -...... ..... -------- --- . ... ..... ---
2-(x10P . . . .. -, .. .. -. .. .. -. ..-. .- .-... -. ..



2 0 ...l* ... ....* ... .. .....
- -- -


0 50 100 150 200 250 300 350 400

Speed of Sound /(m/s)



Figure 4-17. Fourth ST of argon at low temperature.




Table 4-6. Fourth sonic
domain parameters of
7,i argon at low
S: temperature.

: T = -31.56 .100C
SP = 870.5 .5 PSIA
.. c = 290.962 .023
m/s
C cnx = 384.545 m/s

nr = 84
m 2M 29 aM 2a 2a a 2M9 Note: See Equations
Spd Soud/(m/s) 2-33, 2-35.


Figure 4-18. Expanded section
of Figure 4-17.


I














7


250



200



150



100


II


0 1..x.4 2.0
0 1.00x10' 2.0


Time = X/40000 s


'4


Figure 4-19. ADC signal


of argon at high temperature.


Table 4-7. High
temperature time domain
parameters of argon.

T = 50.93 .06oC

P = 1285.5 .8 PSIA

ns = 32768

Sample rate = 40 kHz


Tume = X/40000 s


Figure 4-20. Expanded section
of Figure 4-19.


---r


-----------------I -



-- ------------ -


3.00----
3.00fx10'


0x10'









60







S2.50x10 ------------------------



12.00xi0' ------- ---- - - ------ -. ---
.0.



IL oX o6 --------------- - ..-- .. .. .. ..- ..





s.OOX1OP ---


S. .. ... . .... . . .
0 4000 8000 1.20x10' L60x104

Frequency = X/0.8192 Hz

Figure 4-21. FFT of ADC signal of argon at high
temperature.





S -. .............. ............ Table 4-8. High
r ... i ........ : ..-.- l..... .. temperature frequency
: domain parameters of
Un -o~ ... .......... ........ .... .......... ..... argon .
argon.

... ..... -T = 50.93 .06oC

P = 1285.5 .8 PSIA

n = 16384
mF0 sM0ew s XO f = 20 kHz
Frequency X/0.8192 Hz max


Figure 4-22. Expanded section
of Figure 4-21.









61






5.001106




4.3 &i ............ ........ .........



4.02 -.---.-.------... ........ ..... .... F1s.

S2.0 10 ---- --.... ---- .... ... ... 1.

1.5-q 0 ( -P ------------- ---- -- --.
10OX106 -- - ------ - - - 1 - 1 .

75...... ..... -------- - - - 3 - 1 - 1 -
LO UIO . .... ....... .... .. .. .. ,


O 100 200 300 400 500 600 700 800

Speed of Sound /(m/s)




Figure 4-23. First ST of argon at high temperature.





Table 4-9. First sonic
ad' .. .... domain parameters of
argon at high
nu temperature.

4w2o T = 50.93 + .06oC
P = 1285.5 + .8 PSIA
w .. c = 348.90 .05 m/s
Sjmx = 788.532 m/s

nr = 21
3 34 34 34 7 M4 349 3 35 I 3 0 Note: See Equations
S d of Soud /(m/s) 2-33, 2-35.


Figure 4-24. Expanded section
of Figure 4-23.









62





4.50x10'


4.00x10' -------.----------............------ ----


4.5oxlo6 ................ ........ . .. .. ....... .......

3.OOxlO --- -.- ----.--- -- ----- ------- -------


^SOxlO6








342 344 346 348 350 352 354

Speed of Sound /(m/s)

Figure 4-25. Second ST of argon at high temperature.





Table 4-10. Second
*u I sonic domain parameters
of argon at high
a"me temperature.

T = 50.93 .06oC
.P = 1285.5 .8 PSIA
c = 348.79 .03 m/s
SCImax x = 556.016 m/s

--- nr = 42

30 W 34 2 M3 Note: See Equations
Spd of Sound/(ms) 2-33, 2-35.


Figure 4-26. Expanded section
of Figure 4-25.









63












1.ooxloi
6.0Q10* --- ---,,---------------- ....

5.00oxi1 ... . .... ... .... .... .... ...... ...














0 ....***********
^ D Ox ................. .. .. .... ........ .





0 50 100 150 200 250 300 350 400 450

Speed of Sound /(m/s)



Figure 4-27. Third ST of argon at high temperature.




Table 4-11. Third sonic
domain parameters of
ssli argon at high
Temperature.
ST = 50.93 .06oC
P = 1285.5 .8 PSIA
Sc = 348.806 .027
m/s
Sc .. = 443.741 m/s

Snr = 63

3a M 30 3 M M Note: See Equations
Spd of Sound Am/s) 2-33, 2-35.


Figure 4-28. Expanded section
of Figure 4-27.









64






6.Ox10M6
S34L7917a__
4.0040PO ---------------------------
: : './i :




0,.001106- -
n .010' .. .... .... .. .. ... ... ..
o 4.00x106 - -- .-... ... .... .... -..... .... .. ..





S12.00x106 --- - ----------- .--- ----.- ... --.-- .I ..
1.0010* -- ----:-.-- --.---.,- ----- ------. ...
l.OOx0l6



0 50 100 150 200 250 300 350 400

Speed of Sound /(m/s)




Figure 4-29. Fourth ST of argon at high temperature.




Table 4-12. Fourth
sonic domain parameters
of argon at high
u .temperature.
S---T = 50.93 .060C
u P = 1285.5 + .8 PSIA
C = 348.797 + .023
m/s
Scm = 384.545 m/s

.. nr = 84

3 w 34o 34 30 3 M3 n 3 3M Note: See Equations
Speed ofSund /(m/s) 2-33, 2-35.


Figure 4-30. Expanded section
of Figure 4-29.







































I P/Po

Figure 4-31. Outside volume calibration.








Table 4-13. Outside volume calibration.

slope = -235.074 mL

intercept = 235.0671 mL

V = 235.071 .007 mL

V250 = 100.953 .007 mL

correlation coefficient = 0.9999945


60


50


40


0






0
Pe





10


0


I







































I _P/Po

Figure 4-32. Total volume of apparatus.







Table 4-14. Total volume of apparatus.

slope = -2498.55 mL

intercept = 2498.517 mL

V = 2498.54 .03 mL

V250 = 2463.82 .03 mL

Vt = 2362.87 .03 mL at L = .250 inches

correlation coefficient = .999999


600


500


1 400


*> 300
P.

> 200


100


0
0."


0.8 0.85


0.9 0.95







































Length /in


I
Figure 4-33. Bellows calibration plot.









Table 4-15. Bellows volume calibration.

First order coefficient = 251.60 0.25
mL/in

Second order coefficient = -8.25 0.10
mL/in2

Vt = 2300.19 + 251.60 L 8.25 L2

Correlation coefficient = .999999













Table 4-16. Compiled results of sonic speeds of argon
at low and high temperatures for various roots.


Speed No. of Other
(m/s)"- Roots Parameters


- - - - Low Temperature- - - - - - -
290.91 .05 21 T = -31.56 .10oC
290.96 .03 42 P = 870.5 .5 PSIA
290.962 .027 63 LJ 6-12 speed of sound
290.962 .023 84 c = 291.644 m/s
% difference = 0.2


- - - - High Temperature- - - - - - -
348.90 .05 21 T = 50.93 .060C
348.79 .03 42 P = 1285.5 .8 PSIA
348.806 .027 63 LJ 6-12 speed of sound
348.797 .023 84 c = 350.245 m/s
% difference = 0.4















CHAPTER 5
CONCLUSION

From the results in Figures 4-5 and 4-6, one sees that

the ST can correctly resolve the speed of sound or speeds of

sound in an idealized spherical acoustic cavity. The

identifiable speed of sound in Figure 4-5 is 350.000 m/s

which is precisely the speed used to develop the time domain

signal. In Figure 4-6, the identifiable speeds of sound

were 150.000 m/s and 350.000 m/s which also matched

precisely the speeds used to calculate the time domain

signal. As discussed previously in the introduction, this

transform assumes that there is no frequency dependence on

the speed of sound. The speed that has thermodynamic

significance as seen in Equation 2-22 is the speed of sound

at zero frequency. This speed can be calculated by using

the speed from the ST to identify the frequencies. Once

these are identified and measured precisely, the speed at

each frequency can be calculated by rearrangement of

Equation 2-10 and a plot of speed vs. frequency can be

developed. Extrapolation of this data to zero frequency

will reveal the thermodynamically significant speed of sound

at zero frequency.











This still does not account for the precondensation

effects with the walls of the cavity.' Precondensation

effects will also show up in the frequency domain. The

actual magnitude of this effect can be very accurately

investigated once the data are acquired. Although, the most

accurate method of determining the speed of sound at zero

frequency is still not certain, the present method is the

first step to complete automation of this measurement. Even

with no analysis or calibration (see Table 4-16), the ST

speed of sound obtained from measurements on argon is within

0.5% of the calculated thermodynamic speed of sound at zero

frequency.

The ST baseline for the experimental data had

considerable noise due to the assumption made in Equation 2-

12 that all frequencies detected by the FFT are acoustic.

Clearly the baseline represents nonacoustic resonances of

some kind. There are, of course, several ways to reduce

this problem by increasing the gain of the acoustic

frequencies. One way would involve isolating the

transducers from any contact with the cavity and

acoustically insulating the outer portion of the sphere.

Another method would be to excite the acoustic frequencies

selectively; or, in other words, perform an inverse ST to

produce an arbitrary waveform that could be sent to the


1 Mehl and Moldover, "Precondensation Phenomena in
Acoustic Measurements."










driving transducer by a Digital to Analog Converter (DAC).

By coupling the ADC signal to the waveform produced by the

DAC, a sonic sweep could be performed where the arbitrary

wave is swept over a sonic range and the sonic speed

spectrum recorded. This would be analogous to the frequency

swept method used in the past.

Even without resorting to the use of methods to enhance

the baseline of the sonic spectrum, it is apparent from

consideration of Figures 4-23 through 4-30 that the speed of

sound can be expeditiously deduced with this technique. The

time of acquisition is approximately 10 seconds with the

equipment used in this experiment; thus, technically this is

not a real time measurement. Bear in mind, however, that

the acquisition was performed with an 8088 CPU (8 bit)

computer. If a larger and faster computer were used, such

as an 80386 (32 bit) computer, the total time of processing

would be slightly more than the time of acquisition or

approximately 1 second. By decreasing the sonic resolution,

even shorter acquisition times could be achieved. These

would then be comparable to the acquisition times of

temperature and pressure measurement. For the ADC used in

this experiment with an 84 root basis, a sonic resolution of

0-.023 m/s or a full scale resolution of 6 ppm was achieved.

This far exceeds state-of-the-art pressure resolution and is

comparable to the resolution of high quality temperature

measurements.










The basic device developed here has many potential

applications. For example, it has recently been discovered

that a single fluid can propagate sound at more than one

speed.2 The technique used for detecting this unexpected

phenomenon did not involve a resonance behavior, but rather

the traverse time of flight of pressure-pulse generated

waves. If the phenomenon of multiple speeds of sound in a

fluid is well-founded, there must be observable resonance

effects corresponding to those speeds. The theoretical

results in Figure 4-6 show that the ST method would be well

suited for investigating this phenomenon.

Also, with sensitive enough detection such that no

external excitation is needed, a similar device could simply

listen to the noise already in a cavity and from that deduce

the speed of sound. For a pipeline in which the fluid is

energized by the pumping action, one could detect the speed

of sound in a passing fluid by simply listening to the

fluid. The fluid motion leads to an apparent separation of

sonic speed via the Doppler effect and a ST determination of

that separation would lead to a direct measurement of the

flow velocity. Since fluid density may be related to the

sonic speed, the mass flow rate could also be determined.

Combining these with pressure and temperature measurements,


2 J. Bosse, G. Jacucci, M. Ronchetti, and W.
Schirmacher, "Fast Sound in Two-Component Liquids"
Physical Review Letters 57 (December 1986): 3277.








73

valuable information about flowing streams could be obtained

by passive noninvasive processes. Representatives of the

petroleum and pipeline industries have already shown a

strong interest in this new art. Negotiations are presently

underway to cooperate with these industries in further

development of the technique.

Measuring critical phenomena of fluids with sonic

techniques is difficult when using a frequency tracking

method. When the fluid is close to the critical temperature

and density, the mixture approaches a chaotic state and the

speed of sound approaches zero. As this occurs, the

spectrum collapses and bunches all the frequencies closer

together while the speed of sound and resonance frequency

are dropping rapidly. It is easy to lose the frequency

being tracked since it is moving very rapidly. With the ST,

all frequencies would be measured for a given basis set of

roots and then transformed automatically to the sonic domain

providing that resonances can be detected.

Another area with good potential for the utilization of

a sonic speed meter is that of reaction kinetics. The sonic

speed is highly sensitive to all changes in the structure or

composition of a material system and thus could be used to

monitor the progress of a chemical or physical

transformation. The chemical industry has again expressed

interest in this newly evolving technology as a possible











means of remotely following the kinetics of a complex

polymerization reaction in large batch reactor.

The applications that have been mentioned thus far are

only a few of the possibilities for this new technique. To

list all potential possibilities would be like listing all

of the applications of a thermometer. The most important

result of this study is the application of an ideal

numerical model of a physical phenomena to a real

experiment. The data of many phenomena can be transferred

from an arbitrary domain to a domain that communicates more

information. For example, these same principles could

relate molecular geometries to vibrational spectra or

trajectories to ion cyclotron resonance spectra. Any

phenomenon that has an ideal or reference state model could

be transformed to an ideal domain. The frequency domain

spectra are necessary for investigation of fine structure.

In fact, the transform to an ideal domain should demonstrate

these deviations readily.

The availability of fast computational processes has

facilitated this blend of theory and experiment on a

numerical level. Since modern modeling techniques generally

involved numerical solutions, it is natural that the

communication of these theories to experiments should also

be numerical. This experiment is representative of the

current influence of numerical mathematics on scientific

research, which will significantly change the perceptions








75

and interpretations of future physical experiments. In the

future, numerical mathematics should not be avoided in

applications of experimental science, but rather employed

vigorously throughout all of experimental science.















APPENDIX A
FAST FOURIER TRANSFORM SOURCE CODE


IMPLICIT REAL*8(A-H,O-Z)
IMPLICIT INTEGER*4 (I-N)
INTEGER*2 HEX(256),HIGH,LOW,TAF(16384)
CHARACTER*1 A(64)
CHARACTER*20 FILENAME,FILEOUT
DIMENSION XR(65536),XI(65536)
COMMON XMAX,PI,NU,NDP,NDPDIV2,NDPDIV4,NDPMIN1,IND
C USE FFT TRANSFORM WITH REAL DATA IN XR ARRAY
PI=2.0*ACOS(0.0)
NU=16
NDP=2**NU
NREAD=1024
NDPDIV2=NDP/2
NDPDIV4=NDP/ 4
NDPMIN1=NDP-1
IND=-1
CALL HEXGET(HEX)
C START TIME AVG
DO 20 IF=100,599
WRITE(FILENAME,'(A8,I3,A4)') 'E:\\HEX\\F',IF,'.OUT'
OPEN(10,FILE=FILENAME,STATUS='OLD')
READ(10,*) T,P
DO 30 I=1,NREAD
READ(10,300) (A(K),K=1,64)
300 FORMAT(64A1)
DO 50 K=2,64,2
HIGH=HEX(ICHAR(A(K-1)))*16
LOW=HEX(ICHAR(A(K)))
XR((I-1)*32+K/2)=FLOAT(HIGH+LOW)
50 CONTINUE
30 CONTINUE
CLOSE(10)
DO 70 I=1,NDPDIV2
XR(NDPDIV2+I)=XR(I)
70 CONTINUE
CALL BASELINE(XR,XI)
CALL BLACK(XR)
CALL FFT(XR,XI)
XMAX=0.0
DO 41 L=1,400
XR(L)=0
41 CONTINUE












DO 40 L=101,NDPDIV2
XMAX=AMAX1(XMAX,XR(L))
40 CONTINUE
DO 60 L=2,NDPDIV2,2
TAF(L/2)=INT((XR(L)+XR(L-1))/XMAX*8192)
60 CONTINUE
WRITE(FILEOUT,'(A7,I3,A4)') 'E:\\FD\\F',IF,'.FFT'
WRITE(*,200) FILEOUT
OPEN(10,FILE=FILEOUT,FORM='UNFORMATTED')
WRITE(10) T,P
WRITE(10) TAF
CLOSE(10)
20 CONTINUE
200 FORMAT(A)
END


C*********************************************************
C234567
SUBROUTINE BASELINE(XR,XI)
IMPLICIT REAL*8(A-H,O-Z)
IMPLICIT INTEGER*4 (I-N)
DIMENSION XR(1),XI(1)
COMMON XMAX,PI,NU,NDP,NDPDIV2,NDPDIV4,NDPMIN1,IND
ARX = 0.0
DO 100 I = 1 NDP
ARX = ARX + XR(I)
100 CONTINUE
ARX = ARX / FLOAT(NDP)
DO 200 I = 1,NDP
XR(I) = XR(I) ARX
200 CONTINUE
DO 300 I=1,NDP
XI(I)=0.0
300 CONTINUE
RETURN
END

C***************************************
SUBROUTINE BLACK(XR)
IMPLICIT REAL*8(A-H,O-Z)
DIMENSION XR(1)
COMMON XMAX,PI,NU,NDP,NDPDIV2,NDPDIV4,NDPMIN1,IND
DO 100 I = 1,NDP
C = 2.0*PI*FLOAT(I)/FLOAT(NDP)
A = 0.49755 COS(C)
B = 0.07922 COS(2.0*C)
XR(I) = XR(I) (0.42423 A + B)
100 CONTINUE
RETURN
END












SUBROUTINE FFT(XR,XI)
IMPLICIT REAL*8(A-H,O-Z)
IMPLICIT INTEGER*4 (I-N)
DIMENSION XR(1),XI(1)
COMMON XMAX,PI,NU,NDP,NDPDIV2,NDPDIV4,NDPMIN1,IND
DO 100 L = 1,NU
LE = 2**(NU+1-L)
LE1 = LE/2
U1 = 1.0
U2 = 0.0
ARG = PI/LE1
C = COS (ARG)
S = IND*SIN(ARG)
DO 101 J = 1,LE1
DO 102 I = J,NDP,LE
IP = I + LE1
T1 = XR(I) + XR(IP)
T2 = XI(I) + XI(IP)
T3 = XR(I) XR(IP)
T4 = XI(I) XI(IP)
XR(IP) = T3*U1-T4*U2
XI(IP) = T4*U1+T3*U2
XR(I) = T1
XI(I) = T2
102 CONTINUE
U3 = U1*C-U2*S
U2 = U2*C+Ul*S
U1 = U3
101 CONTINUE
100 CONTINUE
J = 1
DO 104 I = 1,NDPMIN1
IF (I .GE. J) GOTO 25
TEMP = XR(I)
XR(I) = XR(J)
XR(J) = TEMP
TEMP = XI(I)
XI(I) = XI(J)
XI(J) = TEMP
25 K = NDPDIV2
20 IF (K .GE. J) GOTO 30
J = J-K
K = K/2
GOTO 20
30 J = J + K
104 CONTINUE
DO 60 I = 1 NDPDIV2
XR(I) = SQRT(XR(I)*XR(I)+XI(I)*XI(I))
60 CONTINUE
RETURN
END












SUBROUTINE HEXGET(HEX)
IMPLICIT REAL*8(A-H,O-Z)
IMPLICIT INTEGER*4 (I-N)
INTEGER*2 HEX(256)
HEX(ICHAR(' '))=0
HEX(ICHAR(0 '))=0
HEX(ICHAR('1'))=1
HEX(ICHAR('2'))=2
HEX(ICHAR('3'))=3
HEX(ICHAR('4'))=4
HEX(ICHAR(5 '))=5
HEX(ICHAR('6'))=6
HEX(ICHAR( 7 ))=7
HEX(ICHAR('8'))=8
HEX(ICHAR( 9'))=9
HEX(ICHAR('A'))=10
HEX(ICHAR( 'B' )=11
HEX(ICHAR('C'))=12
HEX(ICHAR('D'))=13
HEX (ICHAR( E') )=14
HEX(ICHAR('F'))=15
RETURN
END















APPENDIX B
SONIC TRANSFORM SOURCE CODE


IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION ROOT(84),SMAG(16384),T(500),P(500)
INTEGER*2 CO(16384),C,SPO(16384)
CHARACTER*20 FILEIN,FILEOUT
TWOPI=4.0*ACOS(0.0)
CALL RTGET(ROOT)
TAVG=0.0
PAVG=0.0
NROOT=84
CMAX=200.0*TWOPI*3.0*2.54/ROOT(NROOT)
CRES=CMAX/16384.0
NMAX=16384
FRES=20000.0/16384
DENOM=TWOPI*3.0*2.54/100*FRES
DO 40 1=100,599
INDX=I-99
WRITE(FILEIN,'(A7,I3,A4)') 'E\:\\FD\\F',I,'.FFT'
WRITE(*,100) FILEIN
100 FORMAT(A)
OPEN(1,FILE=FILEIN,FORM='UNFORMATTED')
READ(1) T(INDX),P(INDX)
READ(1) CO
CLOSE(1)
XMAX=0.0
DO 10 C=1,NMAX
SPEED=FLOAT(C)*CRES
RATIO=SPEED/DENOM
SMAG(C)=0.0
DO 20 J=1,NROOT
INDEX=INT(RATIO*ROOT(J)+0.5)
IF(INDEX.GT.16384) THEN
TEMP=0.0
ELSE
TEMP=DBLE(CO(INDEX))
ENDIF
SMAG(C)=SMAG(C)+TEMP
20 CONTINUE
XMAX=AMAX1(XMAX,SMAG(C))
10 CONTINUE
DO 30 C=1,NMAX
SPO(C)=INT(SMAG(C)/XMAX*16384.0)
30 CONTINUE











WRITE(FILEOUT,'(A7,I3,A4) ') 'E\:\\SD\\F',I, .SPD'
OPEN(1,FILE=FILEOUT,FORM='UNFORMATTED')
WRITE(1) T(INDX),P(INDX)
WRITE(1) SPO
CLOSE(1)
TAVG=TAVG+T(INDX)
PAVG=PAVG+P(INDX)
WRITE(*,*) T(INDX),P(INDX)
40 CONTINUE
TAVG=TAVG/500.0
PAVG=PAVG/500.0
SDT=0.0
SDP=0.0
DO 50 1=1,500
SDT=SDT+(TAVG-T(I))**2
SDP=SDP+(PAVG-P(I))**2
50 CONTINUE
SDT=SDT/499.0/500.0
SDT=1.96*SQRT(SDT)
SDP=SDP/499.0/500.0
SDP=1.96*SQRT(SDP)
WRITE(*,200) TAVG,SDT,PAVG,SDP
200 FORMAT(F10.4,'+/-',F7.4,F10.4,'+/-',F7.4)
WRITE(*,300) CRES
300 FORMAT(' RESOLUTION OF SONIC DOMAIN=',F10.5)
END

C
C
C
C
C
C
SUBROUTINE RTGET(ROOT)
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION ROOT(1)
ROOT(1)=2.08158
ROOT(2)=3.34209
ROOT(3)=4.49341
ROOT(4)=4.51408
ROOT(5)=5.64670
ROOT(6) =5.94036
ROOT(7)=6.75643
ROOT(8)=7.28990
ROOT(9)=7.72523
ROOT(10)=7.85107
ROOT(11)=8.58367
ROOT(12)=8.93489
ROOT(13)=9.20586
ROOT(14)=9.84043
ROOT(15)=10.0102
ROOT(16)=10.6140











ROOT(17)=10.9042
ROOT(18)=11.0703
ROOT (19)=11.0791
ROOT(20)=11.9729
ROOT(21)=12.1428
ROOT(22)=12.2794
ROOT(23)=12.4046
ROOT(24)=13.2024
ROOT(25)=13.2956
ROOT(26)=13.4721
ROOT(27) =13.8463
ROOT(28) =14.0663
ROOT(29)=14.2850
ROOT(30)=14.5906
ROOT(31)=14.6513
ROOT(32)=15.2446
ROOT(33)=15.3108
ROOT(34)=15.5793
ROOT(35)=15.8193
ROOT(36)=15.8633
ROOT(37)=16.3604
ROOT(38)=16. 6094
ROOT (39)=16.9776
ROOT(40)=17.0431
ROOT(41)=17. 1176
ROOT(42)=17.2207
ROOT(43)=17.4079
ROOT(44)=17.9473
ROOT (45)=18. 1276
ROOT (46)=18.3565
ROOT(47)=18.4527
ROOT(48)=18.4682
ROOT(49)=18.7428
ROOT(50)=19.2628
ROOT(51)=19.2704
ROOT (52)=19.4964
ROOT(53)=19.5819
ROOT(54)=19.8625
ROOT(55)=20.2219
ROOT(56)=20.3714
ROOT(57)=20.4065
ROOT(58)=20.5379
ROOT (59)=20.5596
ROOT(60)=20.7960
ROOT(61)=21.2312
ROOT (62)=21.5372
ROOT(63)=21.5779
ROOT(64)=21.6667
ROOT(65)=21.8401
ROOT(66)=21.8997
ROOT(67)=22.0000
ROOT(68)=22.5781











ROOT(69)=22.6165
ROOT(70)=22.6625
ROOT(71)=23.0829
ROOT(72)=23.1067
ROOT(73)=23.1950
ROOT(74)=23.3906
ROOT(75)=23. 5194
ROOT(76)=23.6534
ROOT(77)=23.7832
ROOT(78)=23.9069
ROOT(79)=24.3608
ROOT(80)=24.3821
ROOT(81)=24.4749
ROOT(82)=24.6899
ROOT(83)=24.8503
ROOT(84)=24.8995
RETURN
END
















APPENDIX C
EQUATION OF STATE FOR ARGON
SOURCE CODE


IMPLICIT REAL*8 (A-H,O-Z)
REAL*8 B(100),B1(100),B2(100),T(100)
REAL*8 C(100),C1(100),C2(100),BD(100),N
EPK=119.8D 00
BO=49.80D-03
PCON=6.8046D-02
KEL=273.15
OPEN (10,FILE='E\:\\SOURCE\\TVC.CON',STATUS='OLD')
DO 25 1=1,74
READ(10,*)
CT(I),B(I),B1(I),B2(I),BD(I),C(I),C1(I),C2(I)
25 CONTINUE
CLOSE (10)
100 WRITE(*,*) INPUT TEMPERATURE (Celcius) AND
PRESSURE (psia)'
READ(*,*) TEMP,P
P=P*PCON
TEMP=TEMP+KEL
TS=TEMP/EPK
CALL
CQAND(BS,BS1,BS2,BSD,CS,CS1,CS2,TS,T,
CB,B1,B2,BD,C,C1,C2)
BV=BS*BO
CV=CS*BO*BO
CALL VERVOL(P,V,TEMP,BV,CV)
VS=V/BO
CALL SPEED(CAR,TEMP,BS,BS1,BS2,BSD,CS,CS1,CS2,VS)
WRITE(*,*) SPEED=',CAR
GOTO 100
END
C
C
C
C
C
C
C
SUBROUTINE
CQAND(BS,BS1,BS2,BSD,CS,CS1,CS2,TS,T,B,B1,B2
C,BD,C,C1,C2)
IMPLICIT REAL*8 (A-H,O-Z)











REAL*8 B(1),B1(1),B2(1),T(1),C(1),C1(1),C2(1),BD(1)
REAL*8 M
DO 20 1=2,74
IF (TS.GT.T(I-1).AND.TS.LT.T(I)) THEN
M=(TS-T(I-1))/(T(I)-T(I-1))
BS=B(I-1)+M*(B(I)-B(I-1))
BS1=B1(I-1)+M*(BI(I)-Bl(I-1))
BS2=B2(I-1)+M*(B2(I)-B2(I-1))
BSD=BD(I-1)+M*(BD(I)-BD(I-1))
CS=C(I-1)+M*(C(I)-C(I-1))
CS1=C1(I-1)+M*(C1(I)-C1(I-1))
CS2=C2(I-1)+M*(C2(I)-C2(I-1))
RETURN
ENDIF
20 CONTINUE
WRITE(*,*)'TSTAR OUT OF RANGE'
RETURN
END
C
C
C
C
C
C
C
SUBROUTINE VERVOL(P,V,T,B,CV)
IMPLICIT REAL*8 (A-H,O-Z)
R=8.20575D-02
TOL=1.0D-16
C INPUT P IN ATM
C INPUT T IN KELVIN
V=R*T/P
10 VN=R*T/P*(1.0D 00 + B/V + CV/V/V)
TEST=V/VN
IF(TEST.GT.1.0) THEN
TEST=1.0D 00 1.OD 00/TEST
ELSE
TEST=1.0D 00 -TEST
ENDIF
V=VN
IF(TEST.GT.TOL) GOTO 10
RETURN
END
C
C
C
C
C
C
C
SUBROUTINE SPEED(C,T,BS,BS1,BS2,BSD,CS,CS1,CS2,VS)
IMPLICIT REAL*8 (A-H,O-Z)









86

REAL*8 M,DSQRT
M=39.948D-03
R=8.31441D 00
GAMA=
C5.D0/2.ODO-BS2/VS+(BSD*BSD-CS+CS1-0.5DO*CS2)/VS/VS
GAMA=GAMA/(3.ODO/2.ODO-(2.ODO*BS1+BS2)/VS-
@(2.0DO*CS1+CS2)/2.0DO/VS/VS)
C=GAMA*R*T/M*(1. OD 00+2.0DO*BS/VS+3.ODO*CS/VS/VS)
C=DSQRT(C)
RETURN
END















APPENDIX D
DATA ACQUISITION SOURCE CODE


#include
#include
#include
#include
#include "DECL.H"
libraries */
#include
#include
#include
#include


#define
#define
#define
#define
#define
#define


PORTO
PORT1
PORT2
PORT3
COMMA
NOERR


0x178
0x179
Ox17A
0x17B
Ox2c
0


/* supplied with driver


/* default setting */
/* all switches are off */


/****************************/
/* driver library functions */
/****************************/
extern int ibfind();
extern void ibtmo();
extern void ibclr();
extern void ibeos();
extern void ibrd();
extern void ibwrt();
extern void ibcmd();
extern void ibsic();
extern void ibloc();
extern void ibrsp();
extern void ibwait();

FILE *stream;
char *dacoutput=(char*)OxD0000000;
int addr=0x20;

unsigned char io[32768];
long timeout[500];
float pout[500];
double tout[500];
long t,t0,tday;












/* sends temperature in celcius to the oven, */
/**********************************************/
void wrtoven(float setpoint)
{
int j,ovn;
char ostring[14],fp[10];
gcvt(setpoint,5,fp);
strcpy(ostring,"setpoint ");
ostring[9]=fp[0];
ostring[10]=fp[l];
ostring[ll]=fp[2];
ostring[12]=fp[3];
ostring[13]=fp[4];
ostring[14]='\0';
ovn = ibfind("oven");
ibwrt( ovn, string, 15);
printf(" %s\n",ostring);
return;
}


/*************************************************/
/* reads temp in celcius form rtd, */
/*********************************************/
double rdrtd()
{
int i,rtd;
double res,rc,aldel,ptl,pt2,pt3,pt4,t2;
double r0=99.98;
double alpha=0.0039076;
double delta=1.5205;
char rstring[16];
rtd = ibfind("kl95a");
ibrd( rtd, rstring, 17 );
for(i=0; i<4; i++) rstring[i]=' ';
for(i=15; i<17; i++) rstring[i]=' ';
res=atof(rstring);
aldel=alpha*delta;
rc=res/r0;
rc=rc-1.0;
ptl=aldel/100.0;
ptl=ptl+alpha;
pt2=ptl*ptl;
pt3=4.0*rc;
pt3=pt3*aldel;
pt3=pt3/10000.0;
pt4=2.0*aldel;
pt4=pt4/10000.0;
t2=sqrt(pt2-pt3);
t2=ptl-t2;










return (t2/pt4);
}


/'*********************************************/
/* reads the pressure transducer, */
/********************************************/

float rdpress()
{
int ptrans;
char pstring[9];
ptrans = ibfind("beckman");
ibrd( ptrans, pstring, 10 );
pstring[7]=' ';
pstring[8]=' ';
pstring[9]=' ';
return (atof(pstring));
}

/************************************/
/* gets the time in milliseconds */
/**********************************/

void get_milli()
{
-char tmp[l];
long h,m,s;

struct timeb timebuffer;
char *timeline;

ftime(&timebuffer);
timeline = ctime(&(timebuffer.time));

tmp[0]=timeline[ll];
tmp[l]=timeline[12];
h=atol(tmp);
h=h*3600;
tmp[0]=timeline[14];
tmp[l]=timeline[15];
m=atol(tmp);
m-m*60;
tmp[0]=timeline[17];
tmp[l]=timeline[18];
s=atol(tmp);
t=h+m;
t=t+s;
.t=t*1000;
t=t+timebuffer.millitm;
s=t-tO;
if(s < 0 ) {




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