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 feedthroughs 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 21. The values of the roots to the first
derivative of a Bessel function of the first
kind................................................. 12
Table 22. Reduced second virial coefficients for
the LennardJones 612 potential.................... 20
Table 23. Reduced third virial coefficients and
their derivatives for the LennardJones 612
potential........................................... 22
Table 41. Low temperature time domain parameters of
argon............................................... 53
Table 42. Low temperature frequency domain parameters
of argon........................................... 54
Table 43. First sonic domain parameters of argon at
low temperature..................................... 55
Table 44. Second sonic domain parameters of argon at
low temperature..................................... 56
Table 45. Third sonic domain parameters of argon at
low temperature..................................... 57
Table 46. Fourth sonic domain parameters of argon at
low temperature..................................... 58
Table 47. High temperature time domain parameters
of argon........................................... 59
Table 48. High temperature frequency domain parameters
of argon............................................ 60
Table 49. First sonic domain parameters of argon at
high temperature.................................... 61
Table 410. Second sonic domain parameters of argon at
high temperature.................................... 62
Table 411. Third sonic domain parameters of argon at
high temperature..................................... 63
vii
Table 412. Fourth sonic domain parameters of argon at
high temperature ............ ......................... 64
Table 413. Outside volume calibration.................. 65
Table 414. Total volume of apparatus .................... 66
Table 415. Bellows volume calibration.................. 67
Table 416. Compiled results of sonic speeds of argon at
low and high temperatures for various roots......... 68
viii
LIST OF FIGURES
Dage
Figure 31. Instrument rack............................. 32
Figure 32. Spherical cavity sections and clamping
flanges............................................. 36
Figure 33. Pump assembly............................... 38
Figure 34. The bellows and bellows chamber............. 39
Figure 35. Apparatus assembly.......................... 41
Figure 41. Theoretical ADC signal for 350 m/s speed
of sound............................................ 50
Figure 42. Theoretical ADC signal for 150 m/s and
350 m/s speeds of sound............................. 50
Figure 43. FFT of theoretical ADC signal for
350 m/s speed of sound............................... 51
Figure 44. FFT of theoretical ADC signal for
150 m/s and 350 m/s speeds of sound................. 51
Figure 45. ST of FFT of theoretical ADC signal
for 350 m/s......................................... 52
Figure 46. ST of FFT of theoretical ADC signal
for 150 m/s and 350 m/s speeds of sound............. 52
Figure 47. ADC signal of argon at low temperature...... 53
Figure 48. Expanded section of Figure 47............... 53
Figure 49. FFT of ADC signal of argon at low
temperature........................................... 54
Figure 410. Expanded section of Figure 49............. 54
Figure 411. First ST of argon at low temperature....... 55
Figure 412. Expanded section of Figure 411............. 55
Figure 413. Second ST of argon at low temperature....... 56
Figure 414. Expanded section of Figure 413............. 56
Figure 415. Third ST of argon at low temperature....... 57
Figure 416. Expanded section of Figure 415............. 57
Figure 417. Fourth ST of argon at low temperature...... 58
Figure 418. Expanded section of Figure 417............. 58
Figure 419. ADC signal of argon at high temperature.... 59
Figure 420. Expanded section of Figure 419............ 59
Figure 421. FFT of ADC signal of argon at high
temperature......................................... 60
Figure 422. Expanded section of Figure 421............. 60
Figure 423. First ST of argon at high temperature...... 61
Figure 424. Expanded section of Figure 423............ 61
Figure 425. Second ST of argon at high temperature..... 62
Figure 426. Expanded section of Figure 425............ 62
Figure 427. Third ST of argon at high temperature...... 63
Figure 428. Expanded section of Figure 427............. 63
Figure 429. Fourth ST of argon at high temperature..... 64
Figure 430. Expanded section of Figure 429............. 64
Figure 431. Outside volume calibration................. 65
Figure 432. Total volume of apparatus.................. 66
Figure 433. 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 timecomsuming 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 surfaceto
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) surfaceto
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): 40624077;
A.R. Colclough, "Systematic Errors in Primary Acoustic
Thermometry in the Range 220 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): 455465.
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): 253271.
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): 15.
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. nPentane,"
Journal of Chemical Thermodynamics 21 (1989): 867877.
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):
553556.
7 M.B. Ewing, M.L. McGlashan, and J.P.M. Trusler,
"The TemperatureJump 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): 93102.
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): 121139.
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
VV Equation 21.
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 22.
Assuming a time separable solution to the above equation
Equation 23.
where t0 is then the solution to a scaler Helmholtz equation
+*o ( ( )2*0 0,
Equation 24.
then the analytical expression for r0 2 is
0 ( ( ) Pfm(cos(8)) (Asin(mp) + Bcos(m(p))
Equation 25.
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 26.
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
dag2sin(8)dOdprf, Equation 27.
where g is a geometric factor or the radius of the spherical
cavity. Substitution of the gradient of i in Equation 26
yields
l P'(cos(0)) (Asin(mp)+Bcos(m())g2sin(8)d~d' j_ I 0.
Surf
Equation 28.
Since this must be zero for all values of a and b, then
a (r) L 0. Equation 29.
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 21.3
3 Ferris.
Table 21. 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 21 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 gRi Equation 210.
and the frequency of the standing wave within the cavity at
speed c is then
Ri c
o 2nf  Equation 211.
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)) 212.
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 21), 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 213.
and
00
f (ci, t) (aijsin (gcRj t) +bijcos (gciRj t) .
Equation 214.
If we assume that all signals detected in the Fourier
coefficients are acoustic resonances
Equation 215.
then it follows that
00 00
A,Z kiaij (W p, gciRj)
1 3
Equation 216.
F F(t) FS(t) ,
where the value of 6 is as follows,
8(O gciRj)6 1' pgciiRj Equation 217.
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 ncubed 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 218.
mp
then by substituting Ap from Equation 216 into the above
expression,
4 FLOP is a FLoating point OPeration (see Chapter 3,
Theory of Operation).
kiaj8Pl6j8 p.m Equation 219.
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 220.
mp
For a given 1 and m, there is only one nonzero value p,
hence
n n
wi knaE k almka l Equation 222.
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 welldocumented 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 LennardJones 612 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 222.
0,M( ap M) ap
5 R. Byron Bird, "Numerical Evaluation of the Second
Virial Coefficient," The Virial Equation of State CM599
(Madison: University of Wisconsin, May 10, 1950), 4752.
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 22 and the
values of the third virial coefficients are given in Table
23. 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 = LennardJones 612 C* = C/b20
collision diameter
6 = LennardJones 612 T* = kT/E
maximum energy attrac
tion or depth of N = Avogadro's Number
potential well
7 Bird.
Table 22. Reduced second virial coefficients for
the LennardJones 612 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 22 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 23. Reduced third virial coefficients and
their derivatives for the LennardJones 612
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 23 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 223.
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 )2C+C* + 1C
C,R5 + 2
2 V* (V*)2
Cv( 3 2B*+B2 2C:+C2. Eqi
v 2 V* 2 (V*)2
Equation
224.
nation 225.
The constant temperature derivative is given by
SRT+ B* (*)2
V V* (V*)
Equation 226.
SR 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 228.
8 J.W. Cooley and J.W. Tukey, "An Algorithm for the
Machine Calculation of Complex Fourier Series,"
Mathematical Computations 19 (April 1965): 297301.
9 G.D. Bergland, "A Fast Fourier Transform Algorithm
for RealValued Series," Communications of the ACM 11
(October 1968): 703710.
the period for sampling is
Tns
(sample rate)
Equation 229.
the frequency resolution is
nf
res n
nlf
Equation 230.
and the FLOP count is
FLOP n, log2 n,.
Equation 231.
The ST was performed using the following algorithm
denom2n gfre
For i1, nm
Ci
ra ti o
denom
w2O
For jl, n
index integer(ratio*Rj)
Wi wi+aindex
Equation
232.
The maximum speed is limited by insuring that all ST
frequencies of the roots exist within the FFT frequency
domain
Cmax 7
Rn,
Equation 233.
where nr = number of roots in the truncated ST basis set.
Since
c f
res f es
Cmax fmax I
Equation 234.
then the resolution of the algorithm and magnitude of the
sonic domain are, respectively,
max
Rmax 'f
Acres
Equation 235.
with a FLOP count of
FLOPs nma2n,.
Equation 236.
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 222 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 MultiMeter (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
eightbit 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 il,n
rresistance rtd
Ticonvert(r)
Preading for strain guage Equation 31.
ampdump ADC buffer
dump amp on hard drive
dump T on hard drive
dump P on hard drive
Using Equation 228, 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 229 and 227, respectively,
to be 0.8192 seconds and 16384. The resulting frequency
resolution (fes) from Equation 230 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 31. Instrument rack.
modulatedphaseconsistent sinewave that sweeps from 0 Hz to
20 kHz in 0.8192 seconds, then repeats from OHz to 20kHz
with a peaktopeak voltage of 20.0 Volts. A TTL reference
wave is sent to a Stanford Research SR510 lockin 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
eightbit 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 setstime domain 8088 binary format,
frequency domain DELL binary format, and sonic domain DELL
binary formatare 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 partsthe
spherical cavity, the volumecontrolling 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 threeinch 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 leadZirconate leadTitanate (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 knifeedge seal and held
together with two mild steel clamps as shown in Figure 32.
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 33) 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
knifeedge 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.
Doubledpumping action is created by use of four oneway
Figure 32. 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 online
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 33. Pump assembly.
PIS rON
Figure 33. Pump assembly.
co
III I
AWWAMAW
vVWM W
z
Figure 34. 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 temperaturecontrolled volume. The assembled apparatus
was connected as shown in Figure 35.
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 35. 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 41 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 42 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 43 depicts the FFT of the waveform shown in
Figure 41 while Figure 44 depicts the FFT of the waveform
shown in Figure 42. Figure 45 displays the final results
of the ST of the FFT described in Figure 43 and Figure 46
displays the results of the ST of the FFT in Figure 44.
These six figures portray the chronological order of
acquisition and calculation for the simulated set of data.
Note that ST transforms shown in Figures 45 and 46
correctly recover the input sonic speeds (350 m/s and 150
m/s).
Figure 47 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 41. An expanded view of a
section of Figure 47 is given in Figure 48 to show the
resolution with which the waveform is acquired in other
regions. The FFT of the waveform of Figure 47 is shown in
Figure 49 and the relevant physical and computational
parameters are given in Table 42. As seen in Figure 49,
the baseline is not very stable in the region of 10,000 Hz.
An expanded view of this region is shown in Figure 410.
Several STs were performed on the data in Figure 49 using
different numbers of roots. The resulting ST weights
employing the first 21 roots are shown in Figure 411 with
an expanded view of the region that contains the known speed
of sound in argon shown in Figure 412. The experimental
and computational parameters are given in Table 43. 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 411 through 418 while parameters are
listed in Tables 43 through 46. 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 419 shows the experimentally
acquired waveform with the experimental and computational
parameters given in Table 47. The expanded view shown in
Figure 420 indicates that more of the resolution of the ADC
was utilized. The baseline of the FFT shown in Figure 421
is considerably better than that of the low temperature
experiment (Figure 49). The expanded view shown in Figure
422 indicates that the sharp acoustic resonances are larger
than the perturbed baseline and are better resolved than
those in Figure 410. The four STs using the different sets
of basis functions at this temperature are shown in Figures
423 through 430 along with the corresponding parameters in
Tables 49 through 412.
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 41 and 42 show two computer simulated ADC
signals. Figure 41 was generated from the sum of 84
45
sinewaves with frequencies generated from Equation 211 for
a sonic speed (c) of 350 m/s, a geometric factor (g) of 3
inches and assuming equal amplitudes of the resonances.
Figure 42 was generated from two sets of 84 sinewavesone
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 47 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 419 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 419 is
approaching the characteristic of Figures 41 and 42.
Ideally, an evenly distributed waveform uses the entire
resolution of the ADC as was seen in the expanded Figures 4
8 and 420; 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
43 and 44 indicate that the amplitudes are perturbed due
to floating point calculation error. The low temperature
frequency domain plot in Figure 49 shows an extremely large
broad peak in the center of the frequency spectrum. The
46
expanded view shown in Figure 410, 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 45 and 46
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
(411 to 418) as well as the four high temperature figures
(419 to 430) 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 412. 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 41.
 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 431. The total volume of the apparatus as well
as the calibration equipment was then found from the data in
Figure 432. 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 433. 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 415) 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 41. 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 42. Theoretical ADC signal for 150
m/s and 350 m/s speeds of sound.
Figure 43. FFT of theoretical ADC signal
for 350 m/s speed of sound.
Figure 44. 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 45. 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 46. ST of FFT of theoretical ADC
signal for 150 m/s and 350 m/s speeds of
sound.
Figure 47.
ADC signal of argon at low temperature.
Table 41. 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 48. Expanded section
of Figure 47.
Figure 49. 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 410. Expanded section
of Figure 49.
Table 42. 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.O10
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 411. First ST of argon at low temperature.
Table 43. 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) 233, 235.
Figure 412. Expanded section
of Figure 411.
Figure 413. Second ST of argon at low temperature.
7 .A . Table 44. 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 414. Expanded section 233, 235.
of Figure 413.
of Figure 413.
]
1 Ij
Figure 415. Third ST of argon at low temperature.
Table 45. 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) 233, 235.
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 416. Expanded section
of Figure 415.
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 417. Fourth ST of argon at low temperature.
Table 46. 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) 233, 235.
Figure 418. Expanded section
of Figure 417.
I
7
250
200
150
100
II
0 1..x.4 2.0
0 1.00x10' 2.0
Time = X/40000 s
'4
Figure 419. ADC signal
of argon at high temperature.
Table 47. 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 420. Expanded section
of Figure 419.
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 421. FFT of ADC signal of argon at high
temperature.
S . .............. ............ Table 48. 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 422. Expanded section
of Figure 421.
61
5.001106
4.3 &i ............ ........ .........
4.02 ...... ........ ..... .... F1s.
S2.0 10  ....  .... ... ... 1.
1.5q 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 423. First ST of argon at high temperature.
Table 49. 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) 233, 235.
Figure 424. Expanded section
of Figure 423.
62
4.50x10'
4.00x10' ............. 
4.5oxlo6 ................ ........ . .. .. ....... .......
3.OOxlO  . .    
^SOxlO6
342 344 346 348 350 352 354
Speed of Sound /(m/s)
Figure 425. Second ST of argon at high temperature.
Table 410. 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) 233, 235.
Figure 426. Expanded section
of Figure 425.
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 427. Third ST of argon at high temperature.
Table 411. 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) 233, 235.
Figure 428. Expanded section
of Figure 427.
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 429. Fourth ST of argon at high temperature.
Table 412. Fourth
sonic domain parameters
of argon at high
u .temperature.
ST = 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) 233, 235.
Figure 430. Expanded section
of Figure 429.
I P/Po
Figure 431. Outside volume calibration.
Table 413. 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 432. Total volume of apparatus.
Table 414. 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 433. Bellows calibration plot.
Table 415. 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 416. 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 612 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 612 speed of sound
348.797 .023 84 c = 350.245 m/s
% difference = 0.4
CHAPTER 5
CONCLUSION
From the results in Figures 45 and 46, 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 45 is 350.000 m/s
which is precisely the speed used to develop the time domain
signal. In Figure 46, 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 222 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 210 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 416), 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 423 through 430 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 stateoftheart 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 pressurepulse generated
waves. If the phenomenon of multiple speeds of sound in a
fluid is wellfounded, there must be observable resonance
effects corresponding to those speeds. The theoretical
results in Figure 46 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 TwoComponent 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(AH,OZ)
IMPLICIT INTEGER*4 (IN)
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=NDP1
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(K1)))*16
LOW=HEX(ICHAR(A(K)))
XR((I1)*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(L1))/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(AH,OZ)
IMPLICIT INTEGER*4 (IN)
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(AH,OZ)
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(AH,OZ)
IMPLICIT INTEGER*4 (IN)
DIMENSION XR(1),XI(1)
COMMON XMAX,PI,NU,NDP,NDPDIV2,NDPDIV4,NDPMIN1,IND
DO 100 L = 1,NU
LE = 2**(NU+1L)
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*U1T4*U2
XI(IP) = T4*U1+T3*U2
XR(I) = T1
XI(I) = T2
102 CONTINUE
U3 = U1*CU2*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 = JK
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(AH,OZ)
IMPLICIT INTEGER*4 (IN)
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 (AH,OZ)
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=I99
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+(TAVGT(I))**2
SDP=SDP+(PAVGP(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 (AH,OZ)
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 (AH,OZ)
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=49.80D03
PCON=6.8046D02
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 (AH,OZ)
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(I1).AND.TS.LT.T(I)) THEN
M=(TST(I1))/(T(I)T(I1))
BS=B(I1)+M*(B(I)B(I1))
BS1=B1(I1)+M*(BI(I)Bl(I1))
BS2=B2(I1)+M*(B2(I)B2(I1))
BSD=BD(I1)+M*(BD(I)BD(I1))
CS=C(I1)+M*(C(I)C(I1))
CS1=C1(I1)+M*(C1(I)C1(I1))
CS2=C2(I1)+M*(C2(I)C2(I1))
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 (AH,OZ)
R=8.20575D02
TOL=1.0D16
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 (AH,OZ)
86
REAL*8 M,DSQRT
M=39.948D03
R=8.31441D 00
GAMA=
C5.D0/2.ODOBS2/VS+(BSD*BSDCS+CS10.5DO*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=rc1.0;
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(pt2pt3);
t2=ptlt2;
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);
mm*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=ttO;
if(s < 0 ) {
