GAS TWPERATURE MEASUREKENTS BY ULTRASONIC PULSE METHOD
by
THCAS PAUL
RONA.
M.E. Ecole d'Electricite et de Mecanique Industrielles
(Paris, France 1943)
E.E. Ecole Superieure d' Electricite
(Paris, France 1945)
M.S. Massachusetts Institute of Technology
(1953)
SUa1ITTED IN PARTIAL FULFILIMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1955
Signature of Author
-
-
.
.
.
w w-....0.... ....
Department of Electrical Engineering, May 16, 1955
Certified by
Thesis Supervisor
Accepted by
Chairman, D
ental Committee on Graduate Students
GAS TIMPERATURE MEASUREMENTS BY ULTRASONIC PULSE METHOD
by
THOKAS PAUL RONA
Subitted to the Department of Electrical Engineering on May 16, 1955 in
partial fulfillment of the requirements for the degree of Doctor of Science
ABSTRACT
An instrument measuring the phase velocity of sound has been
constructed with the purpose of gas temperature determination.
In view of the immediate goal of temperature measurements in
internal-combustion engine chambers, a broad-band electro-acoustical
system, centered on 2 Mcps was used.
The design includes crystal transducers with convenient adapters
to the engine combustion chamber, electronic power generator and signal
amplifier as well as the incorporation of conventional calibrated-sweep
oscilloscope for time-interval measurement.
Operation is synchronized with the engine cycle so that the
sound pulse may be initiated at any desired crank angle.
The interpretation of sound velocity in terms of temperature is
studied. Accounting for the classical causes of velocity dispersion and
evaluating the possible dispersion caused by heat capacity lag in some
of the gas components the signal shape is determined with particular
regard for the possible measurement error caused by those phenomena.
C
Subject to some restrictions in the macroscopic structure of the
propagation medium, it is concluded that the temperature measurement is
made in the actual conditions to better than 1% of accuracy. This conclusion
is amply supported by experimental evidence on motoring and firing
engine, as well as by tests in chemically controlled steady-state
conditions, it does however not apply in the present state of knowledge
to the flame region.
The theoretical background and some of the accessory experiments
are summarized.
The sound-velocity method for gas temperature measurements appears
convenient and satisfactory for the intended purpose.
T
S~i upt- rvi;or
Jon'dir, *T. 'Rqr''h
ACIOWLEDGEMENTS
The work repoorted here has been supDorted in its various
phases by the U.S. Army Office of Ordnance Research and the Coordinating
Research Council.
The initiative and supervision of this program was entrusted
to Professor C. Fayette Taylor, Head of the Division of Automotive
Engineering at the Massachusetts Institute of Technology. ( Sloan
Laboratories for Automotive and Aircraft Engines.)
The helpful and fiiendly cooperation and advice of the
whole staff of the Sloan Laboratories is hereby gratefully acknowledged.
Mr. James C. Livengood has been particularly active with constructive
comments and everyday's guidance. Messrs E.A. Oster and P.C. Wu
contributed in various ways to test the equipment in actual use on
combustion engines.
On the theoretical side, the help given by the Guidance
Committee of the Department of Electrical Engineering has been of
considerable importance. Professors Jordan J. Baruch, L.L. Beranek
Dr. T.F. Hueter and R.H. Bolt contributed generously their time and
advices.
In the field of theoretical physics, Professors R.M. Fano
of the Department of Electrical Engineering and L. Tisza of the
Physics Department allowed the author to draw on their vast
background.
Mr. J. Caloggero of the Sloan Laboratories must be credited
with resourcefulness in the construction of numerous special
measuring devices.
The author' s wife, Monique R. Rona, contributed to an
unusual degree in the creation of the moral and material surrounding
which was felt essential in the performance of this work.
I N D EX
Chapter 1 Generalites
1.1
Origin and Nature of the Problems
1.11
1
1
Gas Temperature Measurements in Eigine
Combustion Chamber
1
1.12
Typical End-Gas Characteristics
3
1.13
Sumary Description of the Sound Velocity
Method
4
1.14
Discussion of the Principal Sources of Error
1.2
Causes of Departure from the Ideal Behavior
12
18
1.21
Classical Dispersion and Absorption
18
1.22
Thermal Relaxation
26
1.23
Real Gas Behavior
S4
1.24
Dissociation
58
Chapter 2 Analysis of Velocity Dispersion with Particular
Reference to the Actual Sound.
2.1
Nature of the Sonic Excitation
65
65
2.11
Description of the Acoustical Circuit
65
2.12
Receiver Characteristics
67
2.2
Effects of Amplitude and Phase Distortion on
Resultant Frequency Spectnm
69
2.21 Fourier Analysis
2.22
2.23
2.3
Interpretation of Amplitude Distorsion
by Paired Echoes
72
Generalization of the Paired Echo Method
76
Absorption and Dispersion caused by the
Acoustical Phenomena
Chapter 3
69
82
2.31 Viscosity and Heat Conduction
82
2.32
85
Heat Capacity Lag
Properties of the Gas Mixtures Present in the
Test Path
Chenical Composition
3.1
3.2 Heat Capacity Lag in the various Gas
Components
95
95
98
3.21
Nitrogen and Rare Gases
98
3.22
Cxygen
99
3,23
Carbon Dioxide
3.24 Water Vapor
3.3
101
105
Dispersion Characteristics of Typical
Cylinder Charge
108
Chapter
4
Theoretical and Experimental Results
110
4.1 Shape of Amplitude and Phase Distortion
4.2
Characteristics
110
Nature of the Experirmental confinnation
113
h. 21 Engine Experiments
h.22
4.23
113
Principle and Justification of the
Experinental Procedure
11L
Interpretation of Experiments
116
Chapter 5 Conclusions
5.1
Critical Discussion of Results
5.2 Suggested Program for Future Work
135
137
5.21 Study of Macroscopic Gas Properties
137
5.22 Behavior of Reactive Mixtures
138
5.23 1ultipoint Cycle Analysis
138
5.3 Possible Uses of Sound Velocity for
Temperature Measurement
Bibliography
139
CHAPTER -I-
GENERAL IT
1.1
IES
ORIGIN AND NATURE OF THE PROBLEM
1.11
Gas temperature Measurements in Engine Combustion Chamber.
Part of the improvement of internal combustion engine
performance in recent years has been brought around by a more detailed
and more intimate knowledge of the phenomena which take place in the
fuel-air mixture before and during combustion. In addition to other
characteristics such as pressure, chemical composition and specific
volume, the accurate knowledge of specific temperature in the various
phases of the compression stroke has been recognized to be of primary
importance for further investigation of fuels and combustion chamber
designs.
More specifically, the temperature history of the
compression stroke plays a determining role in the engine "knock"
which at the present time is one of the factors limiting engine
perfonance. 'TKnock" or detonation of a portion of the fuel-air
mixture before arrival of the thomal" flame front produced by the
spark is believed to originate in the rapid pompression of the endgas (portion of the charge last to burn in normal operation) by the
pressure rise caused by combustion in the other part of the cylinder.
The instrument herein described and the experimental
program based on this instrument were aimed at the development and
evaluation of a measurement method to determine the "end-gas" temperature
changes immediately before the actual combustion.
It was necessary to
delimit geometrically the end-gas portion and find a measurement
method fast enough to yield accurate results with the high rates of
temperature change present in that region.
Temperature measurements based on the sound velocity
method were considered simultaneously with techniques based on absorption
spectra, infrared radiation and thermo-couples* among these latter,
only the absorption method has been carried through the development
stage. In despite of the fact that no direct comparison was possible
up to the present between absorption-type of measurements and the sound
velocity method, it is hoped that this comparison will be forthcamingt*
The advantages of using sound velocity as a measuring
device in comparison to thermo-couples and resistance wires are mainly
the quasi-instantaneous nature of the measurement, the non-interference
of the sound pulse with the gas and the relative ease to synchronize
the process with the engine cycle. The possibility of adapting the
sound-velocity measurement to digitalrecording is another distinct
advantage.
---------------------------------------------
( tO
) "Four Proposed Methods of measuring end-gas Properties ",
9
) Chen S.K., Beck N.J. Uyehara O.A. and Myers P.S. S.A.E.
(
Coordinating Research Council, Inc, New-York, 1953
Trans., 195,
p.503
1.12
Typical End-Gas Characteristics.
Fig.
I
shows the general aspect of the sound-velocity
device installed in the engine. The end-gas is geometrically determined
by the special sandwich-plate construction of the combustion chamber,
the test volume is typically
}"
long and has 3/8" in diameter.
The conditions prevailing in the end-gas have been
predetermined from the pressure records taken by high speed engine
indicator and by applying iso-entropic assumption to the gas evolution.
The pressure limits appear to be 15 to 700 psia with 200,000 psi/sec
maximum rate of change whereas temperature may be between
S00
and
20000 Fahrenheit Absolute with rates of change exceeding 100,0000 FA
persecond.
It will be seen that the transit time of the sound signal
in the gas is around 30 psec ; the temperature change occuring during
this time interval is of negligible magnitude. Pressure changes, which
influence the velocity of sound propagation to a smaller extent, will
have a negligible effect a fortiori.
The main components of the end-gas are nitrogen and
oxygen from the atmosphere, some C02 and water vapor remaining from the
previous cycle if operating under firing conditions and finally fuel
vapor which in the bulk of our experiments was iso-octane, normal
heptane and propane, although some commercial fuels have also been explored.
In other experiments carefully purified gases were
investigated in chemically known conditions.
1.13
Sumnary Description of the Sound Velocity Method
1.130
Note.
The design factors and justification of the
compromises accepted have been explained in our preliminary survey,
where also crystal performance and power requirements were evaluated.
Findings of this evaluation will not be repeated here ; it will be
sufficient to mention that they have been essentially supported by
experimental results gained from about two years of continuous
operation with the instruments. Some of the new developments are
reported in Appendix I and II ; but it is apparent that their impact
on the instrument design and result interpretation is of minor
consequence. The present section is merely intended to sumnarize
the instrument description in order to make this text self-contained.
1.131
Princip.
The phase velocity of sound in a fluid function
of its compressibility alone.
The compressibility of course depend on
--------------------------- I-----------------------------* ( 48 ) Measurement of Ultrasonic Propagation Velocity in Gases,
S.M. Thesis, H.I.T. Dept. of Klectrical Engineering, June 1953
**
( 4) ) See for example Lord Rayleigh, Theory of Sound, London, 1887
the thermodynamic path followed by the fluid during the sonic disturbance
and also on its ltate" i.e. the relative proportion of Kinetic and
potential energies present in the molecular field. This latter criterion
is used to be translated in "'more or less distant from the perfect
state" in which the fluid particles are believed to have translational
Kinetic energies only.
The compressibility
is in these conditions function
of the temperature only if the sonic frequency is low enough to
warrant the isothermal assumption.
When the successive compressions and expansions are so rapid
that the evolution may be thought of as essentially iso-entropic, the
compressibility is known to be function of the ratio of specific heats
and of the absolute temperature.
The propagation in a real but ideal gas is characterized
by the phase velocity"
1.13 - 1
For most gases and gas mixtures the
P/f
ratio as a
function of temperature is known from the so-called virial coefficient
can be calculated as shown in Sect. 1.22;
and
in ideal
propagation conditions, the sound velocity is therefore unequivocally
related to the gas temperature. In later chapters corrections due to
------------------------------------------------------The "perfect state" also implies the absence of intramolecular
*
and intra-atomic energies, but this distinction is immaterial at
this place.
(
5
) T.F. Hueter and R.H. Bolt
"SONICS", J.Wiley & Sons, 1954.
. mE
the non-ideal behavior will be discussed.
The instrument measures transit time
over a calibrated lenght of gas path, L.
t
of a sound pulse
The sound velocity
is then used to compute temperature from (1.13 - 1). In fact, in most
cases the "sound velocity" is given as a function of temperature
directly* Phase velocity characteristics for mixtures of known
proportions of non-reacting gases are readily constructed.
1.132
System Components
(a)
Sound Generator
A 2 Mcps Barium Titanate crystal is acoustically coupled
to the transmitter coupling rod which is part of the combustion chamber.
The rod is acoustically insulated from the combustion chamber walls.
The electrical excitation signal is as close as possible to a single
pulse ; it is obtaiein the present version by the hydrogen thyratron
( 3C45 ) discharge of a condenser. The circuit characteristics,
analyzed in Ref.
48
lead to the signal shape shown on Fig. A
where the slight oscillations are due to the purely electrical
properties of the discharge circuit.
The transmitter construction and circuitry is shown in
3 4
--------------------------------------------------------------
Figs
*
( 3R ) Keenan J.H. and Kaye J. "Gas Tables" New-York 1945
John Wiley & Sons.
P.C. Wu, M.E. Thesis, Dept of Mechanical Engineering, Jan, 1954. (63)
(b) Sound Detector
A transducer crystal, identical to the one used in the
generator is coupled to the receiver coupling bar so that the sonic
pressure wave reaching the receiver bar surface in the combustion chamber
is transmitted to the crystal.
A broad-band amplifier with low-noise input stage follows
the crystal. The signal is then amplified to about 80 dB above its input
value and fed into the synchroscope amplifier circuits.
(c) Time interval measuring device
Among the various possible electronic methods to
measure accurately short durations, the calibrated sweep oscilloscope
with variable precision delay has been chosen** The delay calibration
is claimed to be accurate to
.1
psec in absolute value so that the
time interval measurements will be accurate to at least that extent.
For reasons of convenience and simplicity, the internal repetitive
trigger source is used to initiate the sound generator when the engine
is at standstill, the internal trigger generator is also used in
connection with the engine distribution shaft breaker when synchronized
operation with the engine cycle is desired.
-------------------------------------------------------*
See Sect. 2.12
for amplifier characteristic discussion.
A.B. Du Mont Manufacturing Co, MOD 256/B and MOD/F.
Figures
5'
through
1'
show constructional details of
the system components as used on the actual engine set-up.
(d) Special Test arrangement
---------------------In addition to engine operation, the instrument was
tested in a special acoustical circuit where a known path of chemically
controlled gas content was inserted between the coupling rods.
The acoustical components are identical to those
used in the engine, except the test chamber which is made of heavy
copper for uniform temperature distribution. The inside of this test
chamber (Fig. I(o ) and the radiating and receiving surfaces of the
coupling rods are gold plated to avoid reactions by some of the corrosive
gases used in the experiments.
Special washers made out of alternating layers of
stainless steel and teflon were used to insure acoustical insulation
of the coupling rods from the test chamber walls.
The drying and
mixing tanks necessary to prepare the gas mixtures have been made out
of stainless steel and designed for operation from good vacuum
(< 1 m of mercury) up to 200 psi pressure. Tests for leakage were
made up to 350 psi. Teflon is being used as gasket material, threads
are degreased and coated with special high-vacuum grease.
Fig.
17
shows the schematic of the gas circuits, with
mixing tank, drying tanks and admission of liquids of low vapor tension
at room temperature.
A vacuum pump and two test chamber thermo-couples are
provided. All valves are stainless steel with teflon packing.
1.133
Facts and Figures
It appears necessary to complete this section by giving
a certain number of numerical data which will illuminate the design
factors to the reader. The reference or the origin of the figures is
also given.
(TABLES
1 9-
I
)
'1
T A BL E
ACCUSTICAL
T
CIRCUIT
Value
Quantity
Unit
Ref.
.02
2
~..
Barium Ttanate Crystal
Longitudinal natural frequency
Dimensions (diameter & thickness)
Barim Titanate (Material)
Acoustical Ipedance
2 1. .02
1.27 1.1
27 to 35.105
Mcps
cm
g/cmsec
Free Cutting Commercial Brass
Velocity of sound at O0 C
35,o40
Mass density
Acoustical Impedance
29.5 x 105
cm sge
g cm-se
Air at Standard Pressure & Temperature
Mass density
Iso-entropical stiffness
Velocity of sound
1.293 X 10-3
1.46 .10
3.31
10
g/cm32 2
dynes/cm
cm/sec
Gage Characteristics
Wavelenght to path lenght ratio
in air at S.T.P. for 2 Mcps
Sound frequency
1.27
.94
Wavelenght in brass at 2 Mcps
.18
Transit time in gap
Transit time in both brass bars
(typical value)
lenght
diameter
.016
20 to 4o
36.o
psec
TABL E
ELECTRICAL
I
CIRCUITS
Value
Quantity
Ref.
Unit
Barium Titanate Crystal
Longitudinal, field/strain ratio
Dielectric Constant (approx.)
Ferroelectric constant
Transformation ratio (Symetrical load)
1.38.109
1700
Coul/cm
1.21. 108
cgs
9.95. 1013
Motional stiffness
21.9
Equivalent inductance
164
Equivalent capacitance
Radiation inpedance for symetrical
brass load
Clamped Crystal resistance
2
20.8. 10~4
.358
Motional Mass
v/m
9.160.106
1.9
gr
dynes/cm
pH
p pF
cgs
m pF
Transmitter Circuit
Tube type (hydrogen thyratron)
Plate voltage (adjustable)
Instantaneous peak power at 60 0 v plate
3 C 45
200 to 1400
volts
2
approx. 70
w/cm
2.7
pV
Receiver Circuit
Thermal noise 300 0K
600
6 dB
3.8
Mcps
Midband frequency
2.1
Mcps
Bandwidth
52
-,
U
-
1
l.14
Discussion of the Principal Sources of Error.
As explained in Section 1.13, the gas temperature
measurement by the sound velocity method is essentially a transit-time
determination across the gap of given lenght, followed by an interpretation
of the phase velocity in terms of various properties of the gas,
temperature included. The total error to be expected will therefore
contain all those pertaining to the phase velocity measurement plus the
inaccuracies in the accepted interpretation theory.
Fig. LM shows schematically the relationships among the
various error sources. The "timing" has been introduced for sake of
completeness in this representation, but quite evidently its presence is
not inherent to the sound-velocity method, besides the improvement in this
respect, although a matter of straight-forward electronic technique, has
not been found necessary in the present state of the equipment. In fact the
cycle-to-cycle fluctuation is such that little or nothing would be gained
by improving the timing stability.
The principle of the sound velocity method should likewise not
be held responsible for errors caused by the "fluctuations". This term
designates cycle-to-cycle changes in the transit time apparently due to
the variations in spark timing, gas composition and residual combustion
products content. The cure may well be the substitution to the
"stroboscopic" measurement technique of the 'ultipoint"
method discussed
in Section 5.Z
and for this reason no attempt will be made here to
analyze the magnitude of this error component.
The gage-lenght deternination influences significantly the error
on velocity measurement. It is now common practice to measure each coupling
rod to
1/10000" and thus, know the gage-lenght to better than
2/10000.
The mounting method is such that only about 2 cm total rod lenght may
cause gap variations with changes in rod temperature. Water-cooling is
provided for both rods in order to protect transducer crystals from
temperature changes. It is assumed that the average rod temperature
between mounting flanges and radiating surfaces will not change by more
than
500
C with respect to calibrating conditions. An expansion coefficient
would lead to a change of about 9/10000 ".
of 18.3 X 10-
The order of magnetude of the error on velocity measurement
introduced by the gap uncertainty is thus
"P
.2% for a nominal lenght of
1.27 cm.
The other components of the error are those connected with the
transit time determination, which in turn may be separated in phase reading,
delay standardisation and delay reading errors.
Phase reading is connected with the nature of the oscillographic
screen display. The finite rise time of the signal envelope makes the
definition of transit time somewhat arbitrary unless specific conventions
are established for this purpose. After about one-and-a half year of
extensive use by various operators of the equipments available a reading
---------
-------------------------------------------
*
Ref
( 2' )
Hidnert, Scientific Paper No h10, U.S. Bureau of Standards, 1921.
error of 1/10 cycle or roughly .o5 psec appears as a reasonable estimate.
Should however be added to this value the errors due to the time calibration
of the synchroscope delay circuits (
.o5
psec)
and any additional
inaccuracy due to the calibration reference introduced when the electroacoustical delay of the system (less gap transit time) is measured by means
of a known gas in the test volume.
It is in order to mention that calibration errors will affect
absolute temperature values but not results for changes in gas temperature.
Also changes in rod temperature can be accounted for. (App. 1)
1I.
Table
shows a recapitulation of the numerical values
accepted for the various elements of the velocity measurement error.
Care should be taken, when evaluating the total error associated
with temperature measurement, to double the effect of velocity errors :
2
c
-T
1*1
T
From the preceding discussion it may be inferred that velocity
measurement errors are important in magnitude but comparatively easy to
evaluate in the light of the now available experimental evidence.
On the other hand the inaccuracies or uncertainties affecting the
sound velocity-temperature relationship are smaller in predictable
magnitude but unfortunately little experimental background has been analyzed
and for the time being it is difficult to extrapolate to gases not yet
experimented with.
I
TAB L E
f
VELOCITY MEASURMENT ERRORS
ELEMENT
GAGE LENGHT
CAUSES
MEASUREENT, THER4AL
REL.
ABS.
.2%
1/10001"
EXPANSION STRAINS
PHASE READING
RISE TIME
SPOT THICKNESS
.05 pe
max
DELAY
SYNCHROSCOPE
STANDARDISATION
CIRCUITRY
.05 isee
max
DELAY READING
DIAL
.01 psec
max
CALIBRATION
REFERENCE GAS
REFERENCE
COUPLING ROD
TPERATURE
.25%
.25%
'~
In Fig.
18
the "interpretation' error is shown to ccmprise two
categories viz.
(a)
MEDE1M i.e. errors which are results of the particular environment
chosen to perform measurements,
(b)
PROPAGATION THEORY meaning that unless a particular equation
of state is chosen to represent dynamically the actual gaseous
content of the test volume, the sound velocity and temperature
relationship can not be predicted.
Under heading (a) effects of the variable chemical composition
(due to the so-called ore-flame reactions and to the unknown proportion
of resident gases in the combustion chamber) would have to be duIscussed.
The macroscopic structure of the propagation medium, velocity,
density and temperature gradients do have important bearing on sound
transmission but the discussion of their effect is beyond our present
scope. The macroscopic non-equilibrium of the gas has been recognized
as irrelevant as far as the sound propagation is concerned except for
effects mentioned in (a). In fact, the high temperature and pressure
rates of change, mentioned in Sect. 1.12, are still very slow compared
to the rate of energy exchange between molecules ; this is a definite
fact for the so-called "active" degrees of molecular freedom whose
time constants are of the order of 10~9 sec, or less.
Under heading (b) the effects of internal lag ( thermal
relaxation),
the real-gas behavior and the so-called classical dispersion
will be discussed (Sect. 1.21 to 1.2 3)."Classical"dispersion occurs in
presence of viscosity, heat conduction and thermal radiation. Their
effect has been discussed in a previous work* where orders of magnitude
have been given ; in the present text the actual influence on temperature
measurement will be emphasized.
To include all these phenomena in the evaluation of "errors"
is not absolutely proper; they should rather be thought of as variations
with respect to the perfect-gas law and in presence of known gas
mixtures the corresponding deviation may be evaluated, approximated or
be given limit values.
R..Rnsen----------------------------------------------R~f~4&T.P. Rona, Master's Thesis, 14. I.T. E.E.Department.
IL,
1.2
CAUSES OF DEPARTURE FRCM THE IDEAL BEHAVIOR
1.21
Classical Dispersion and Absorption
1.210
Ideal Propagation
The phase velocity in ideal conditions is derived
from the equation of state, the continuity of the propagating mediun
and from Newton's first law applied to the elementary portion of the path.
Thus,
p(?)
0 being the equation of state,
the equation of continuity is written
and,
in presence of acoustical waves where
respect to
14
and
,
is negligible with
reduces to
b
Ar
From Newton's Law of motion
P?
It
rb
121 -
1
Identical equation is being obtained by elimination of between
the three conditions of state, continuity and motion
ID
121 - 2
The "propagation velocity" of any pressure or particle velocity function
generated along the rectilinear propagation axis is obtained from the
general solution of equations of type 1.21 - 1 and 1.21 -2,
this latter
is known to be any function
C
Swhere
The phase velocity of any acoustical signal will thus be essentially
) or the particular equation of state
determined by the nature of
assumed for the experimental conditions.
More specifically an ideal gas suhnitted to isothermal (slow)
acoustical cycles follows the equation of state
pv~V
=m
4%4v
RT
Constant
RTkIm
1.21 - 3
When iso-entropic evolution Ls being assumed the equation
IS r
of state should be
=
I%.
C.
-t
z
Y0
C,
PT
since
1,21
-
?.
4
The departure from the ".deal" conditions may be represented by the
introduction of additional tems either in the equation of state which in
this case may be understood as a dynamical relationship expressing the
strain-stress dependency along the (time dependent) evolution path or in
the continuity equation. This latter method is used almost exclusively
in the domain of the so-called translational dispersion when the distance
successive pressure and/or velocity maxima is of the order of magnitude
of the mean free path between molecular collisions.
The present text is explicitly limited to the study of
dynamical equations of state, it may be shown
that in the conditions
present in our experimental work, the mean free path is of the order of
lO
cm and thus the frequency region where translational dispersion
may occur is
C.
3.o
-
3o o Mcps
A
The various causes of dispersion introduced as dynamic
tems in the equation of state are studied in the following paragraphs.
1.211
Viscosity and Heat conduction
Viscous forces are assumed to act on each fluid particle
proportionally to its velocity. In general the stress produced in a point
of a viscous fluid is a tensor with components shown on Fig. 19
EC----------h-------------------------------------------------------------
E~. Skudrzyrk
Chap.I
(16A)
If
,
PX
PX
Pz
yx
PX
yy
PY
y
PZ
zx
zy
zz
i and I
are the particle displacenients of the
reference point, (origin, for example) an arbitrary point IT (xyz) will
be displaced by
+a
47
represent the longitudinal Strains in the reference
directions (usually denoted S
, S~y,
Szz) whereas the cross-terms may
be called shear strain components, and usually called
Il
_
S~
The viscous forces influencing a particle are then written in
terms of the compressional viscosity N and shear viscosity
by
analogy with the static stress-strain relationship and assming isotropic
media
IX
) 11
wave
is chrceie
sic poin,
by
(efitio
by
isretice
to th
YZ all
X~. axi,an
00
P
21
This "excess" pressure has to be added to the one caused
by the sound wave to find the inertia force acting on the fluid particle.
Similarly, the excess pressure due to heat conduction may be expressed.
Fig. 20
Q
let
shows a hypothetical layer in the propagation path,
be the heat production rate per unit volume in this portion.
Then, it can be shown that the quasi-static equilibrium of
this layer implies the equality between the heat conduction flux, the
rate of energy transformation of heat and the heat stored in the fluid
layer. The net result*, with the assumption of harmonic sound propagation
and small acoustical changes in density and pressure, is the presence
of a term
expressing a force
P(-
due to heat conduction proportional to the heat conductivity and to the
rate of change of strain.
)-
The equation of state
may then be interpreted
as
is harmonic function of time with the angular frequency
and if
which leads to the dynamical fon of
The factor
4W
1.21- 1
is seen to modify the modulus and the
phase of the propagation constant and is therefore significant with
(
)
(424 )
s------------------------------W.P, Mason, Piezo M~ectric Crystals and I-ave Filters, p.480
respect to dispersion and absorption of the propagation wave
and
_
ic
Y
thus the complex propagation
constant may be written as
n<~
1.2a- 5
OryK C,)
where Cis the isothermal stiffness of the medium.
Practically, in all the accessible frequency domain the
real
the im&Aefr part of .E
and imaginary parts of f
is small compared
to its modulus and the real
may therefore be expressed by the usual
approximation method
2)
rA 2A
The numerical tables enclosed show that the phase distortion
is small in all practical cases, whereas effects of absorption are of
the order of magnitude found experimentally.
----------------------------------------------
Y
"% So
for infinitesimal amplitudes.
1.212
Heat radiation
The role of heat radiation on sound propagation has
been studied since more than a half a century but practical conclusions
are not apparent due to the uncertainty for the numerical values of the
"emissivities" involved.
The heat flux from an elementary volume traversed
by the sound is written proportional to the temperature difference
with respect to the surroundings
C
(T--1.21-6
and thus, by various manipulations* get the equation of state
accounting for heat radiation
lf
+
q,/1~
rwP
It will be seen in the section deaing with relaxation
how this type of equation is interpreted in terms of absorption and
phase velocity. A matter of particular interest is that although the
radiation mechanism is strictly similar to the one found for thermal
relaxation ( as seen by eq. 1.21 - 6) the characteristic angular
frequencies for radiation are of the order of 1010 Cps and thus not
detectable in pur experiment.
n----------------------------------------------
(
(144
)Jo
J,, Markham, R.T, Beyer and R.B. Lindsay
1.22
Thermal Relaxation
1.221
Nature of the phenomenon
Compared to other causes of non-ideal sound propagation
thermal relaxation or "heat capacity lag" is a newcomer in the field of
ultrasonics and it is natural to find, at the present stage of the art,
various and sometimes contradictory explanations of the same physical
event.
In thermal equilibrium the entropy of any macroscopic
portion of fluid is the maximum compatible with its energy content. All
molecules of the body*
and in fact all modes of motion of the said
molecules are then part of the canonical
assembly where the Maxwell-
Boltzmann distribution law may be used to define the probability,
(and thus the concentration) of a particle in any given energy state
6
In case of monoatomic gases in ideal condition, when
E S are functions of
potential energies are negligible the only
the translational momenta \'x , Py
and
&
;
thus yn being the mass of a particle
+TPT
and the probability of finding any particle in the range
--------------------------------------------*
called "fluid" hereafter
yPA C
Ftis
4,'
-
a
- P13-6
- 3/,L
=
k.
2r vk
0
P
c010'
OAk
3
The mean kinetic energy of the assembly can be calculated from this
expression or simply by considering the momentum space and a volume of a
spherical shell( 4'tW1VdinY)to express the kinetic energy irrespective
of direction.
Then
1.
W04 z
=
/.6
"""A
k
4 %AV%C.
(2 1
T)
CA V
4nJ
Iak
V3Q
SAV
And the weighed mean is
IMV
V
6
4 A (2 it -wkT
I
V Ii.r
= 2n(2. a A WT3/Z
I
/per
3'bi~t
Ah
(-S
particle
This conclusion, known as the equipartition of the average translational
energy in the three degrees of freedom, can be extended to all types of
molecular and intramolecular motion provided however that the energy level
spacings be close enough together so that the integral expression in
place of the discrete summation is justified. It can be shown that this
is practically always the case for molecular translation but that such
assumption is far from being justified in general for intramolecular
vibrations (and sometimes rotations) unless temperatures are far above
the characteristic temperature of the particular mode ( Tc )
Care should be exercised however to include in the total
energy associated with vibrations the potential energy due to the
elastic restoring forces ; when submitted to the equipartition rule,
this cnergy is also 2
WT
per mode for each molecule.
It can be shown that for diatomic molecules, rotation
and vibration are the possible internal motions, the spacing of the two
lowest rotational levels is
A E.
=correeponding
to the
characteristic temperature
For all gases, hydrogen excepted (T. rot is of the order of a few
or a fraction of degree Abs. ; thus rotational levels will be practically
continuous at room temperatures.
The specific heat corresponding to rotational energy in
a diatomic gas may be computed from the partition function
- kt
(
--
*
(
1
- --
------
)J.C.
-----
--
-
--
--
-
Z
*
t Ca7
--------------
Slater , Introduction to Chemical Phlysics.
) "-I
where
Z
is the value
2rof
k
r Ltk
C-'
Z
The factor ( All +1 ) is introduced to account for the
space quantization, i.e. for the fact that the projection of the
angular momentum on a fixed direction should be integer multiple of
Z. n-
Tc.
>
For room temperatures, i.e.
the summation
in the partition function becomes
Z
(-+1
~')
for high values of 'A
and thus
(C ) V-
TT (
T
I
Zrot
for each molecule.
This is in perfect agreement with the equipartition since
2 rotational degrees of freedom are theoretically possible.
The nature of rotation remains unchanged for more complicated
molecules, the only difference being that three rather than two rotational
coordinates are needed to describe the motion. Thus an energy of
3 W/2
per molecule has to be assigned to rotation as soon as temperature
becomes important with respect to
( Te,)m.
; which is always the case
in experiments above a few degrees K.*
Vibration of atoms around their equilibrium position are,
on the other hand provided with much large spacings between successive
energy levels. In absence of interaction between rotation and vibration,
the vibration frequency of a diatomic molecule is
(
D = Dissociation energy
a
(
2a
Constant,
characterizing the
binding force
around the
equilibrium position
equivalent
mass
the successive energy levels differ by f9 , thus
E
and
*----------------------------- -------------------1710 K.
(Tcj)dt.'
*Hydrogen is a notable exception with
It is found that
(T).
except for K2 , Na2
are very high compared to usual temperatures
12 and Br2may be set up by
A partition function
suramation
zviV,
4
-
ho
,(.^+I)am
a, KI
e,
kv,
ie
0
(C, VwZ
hOTK
L
()
per molecule
- T
(~c~ )~,
per molecule - 5r
R= Nok
No being Avogadro's Number. It is easier to visualize the meaning of
this expression by introducing the characteristic temperature
(
(CV)
/
V1,6
1.22 - 1
This expression has the limit
by the equipartition principle.
when
-T; e
as provided
For polyatomic molecules,
( 3 N - 6 ) vibrational modes are possible,
each having its own characteristic temperature and its own fraction of
specific heat at a given temperature, such as shown by formula 1.22 - 1.
Some corrections have to be introduced in these (comparatively) simple
pictures, these are of significance for cases where interaction of
rotation and vibration can no more be neglected, when vibrations are
no more simply harmonic, and finally when potential energy may couple
rotational modes . Energy levels of a different order of magnitude
are involved in the electronic excitation and even more so in changes
occuring in the nuclear structure of individual atoms. These two
phenomena are excluded from the present discussion.
All the preceding statements were obtained from statistical
considerations on quasi equilibrium situations. The specific heats
calculated are especially those prevailing for transitions between two
infinitely close equilibrium configurations. The time variable is therefore
not apparent in the expression of these specific heats ; from this fact,
one may infer that serious modifications of these concepts are necessary
when rapid changes of internal energy are present.
This is precisely the case when high frequency acoustical
disturbance alternately comprimes and expands portions of the
propagation path.
Heat cor
relaxation is essentially the nature
of modifications affecting specific heats in presence of energy changes
of finite amplitude or duration.
TA BL E
CHARACTERISTIC
IV
TEMPERATURES FOR
DIATCMIC VIBRATIONS
H2
61400
CH
4100
NH
W4OO
OH
5360
HC1
4300
NO
2740
02
2260
N2
3380
0C
3120
C2
2370
012
810
Abs
Br2740
k2
310
Li 2
500
Na 2
230
K
140
2
51
) J.C. Slater, Introduction to Chemical Physics,
p.142
1.222
Time Dependence of Energ Exchanges
Whereas the specific heats discussed in the previous
section are uniquely defined by the number of degrees of freedom of the
molecules and the absolute temperature, the situation is deeply modified
when changes in the average translational kinetic energy are so rapid
that other modes of motion (vibration and rotation) fail to be excited.
The number of molecular collisions per second is finite and the
probability of energy transfer is in ordinary conditions, very much
smaller than 1, at least for vibration. One may therefore expect a great
number of collisions (thus a finite time interval) necessary to
establish the equilibrium between the various modes of motion.
Translation is comparatively simple to dispose of.
It can be shown that the various translational "states" are so closely
spaced that for practical purposes they may be regarded as continuous.
x, y, and z, and the momenta associated
V of dimensions
Assume a volume
with the motion of one molecule, pj, py and pz.
The quantum theory
indicates that for each dimension-pair of the phase space
K
thus,
where
W
h
=
X
k
&
4C'
is an integer multiple of
in the momentum space is then
'
corresponding to a given momentum
I
.1The point density
; the energy
6
p is
1-
thus,
~e
and the number of the states with energy less
than
E
is the number of all points in the volume
The number of states between C' and
cA-3-
--
6
n
p
=
-.--
+ M is then
')
1'
The average energy jump between states is then*
This number is seen entirely negligible for the usual macroscopic
dimensions " but care should be taken not to extrapolate to sonic
phenomena in gases at very high frequency. When the wavelenght of sound
bedomes of the order of a few mean free path in the gas (say 10-5 cm)
the energy jump between translational states may be multiplied by a
factor of 1015 on account of V and thus become significant with
respect to the total energy of the molecules. As a conclusion,
translational motion in normal conditions of pressure and at moderately
high frequencies, will be considered as continuous and thus the
probability of energy exchange identical to the one encountered in
collision of macroscopic bodies. It is known that this latter is
The time lag necessary to
equal to 1 for perfectly elastic collisions.
-------- --------------------------------------------) J.C. Slater, Introduction to Chemical Physics, Ec Graw Hill
*( 5j
N.Y. 1939, p.55
'About 10-37 ergs for 1 cm3 of helium at 10 K
establish translational equilibrium is thus at most seen equal to the
average time interval between collisions.
The average Maxwell velocity in a perfect gas (hydrogen at
300
K
is
kT
9
and thus, for a mean free path of 10-5
--
. 052
38 .10
-l6
soo
cm
eQo
sc.
The frequencies where translational dispersion may be expected are
thus definitely beyond the reach of the present experimental techniques.
For molecular rotation, following Landau & Teller
the
collision effectiveness for a certain rctational mode is determined by:
Duration of Collision
Natural
period
The collision duration has a somewhat flexible definition, for molecules
of no chemical affinity it may be thought of as the time interval of
presence of the colliding molecules in each other force-field which
3 X 10-
is essentially of the order of the Bohr radius,
* Phys. Zeitschr der Soviet Union, 10, 34, (1936)
(
cm.
on the other hand
therefore
=
to V
Close to thermal equilibrium
( r
(
(
(
Ve
distance of atom
to the center
of gravity of
the molecule
(vt = peripherial atom
-
(
t
t-
on the average, since this
limit is actually attained in equilibrium,
of the same order of magnitude, thus
velocity
and V' are certainly
as a value close to 1. The
collision effectiveness or probability of energy exchange between
translation and rotation is therefore very high ; in practice, a few
collisions will cnly be required to establish equilibrium between these
modes. Experiment,al values* show rotational relaxation times in N2 and
02 of the order of 240 Mcps and 50 Reps respectively!*
The quasi-instantaneous nature of energy conversion
between translation and rotation warrants to consider these as "active"
with respect to our comparatively slow sonic excitation.
--------------------------------------------T.F. Hueter and R.H. Bolt, "SONICS"' John Wiley and Sons, N.Y.
195h
Some controversy is present on the former figure, other sources
point to 220 Mcps.
(s)
Moleculae Vibrations.
The energy absorbed in molecular vibrations being
of predominant importance for acoustical phenomena at the
frequencies present in the reported series of experiments, it is
felt necessary to outline the pertinent background in some detail.
The problem is to find to what extent a specified
vibrational state will participate in the (periodic) changes in
translational energy content caused by the sound pressure. The
proportion of energy exchanged with the vibrational mode will
determine the actual ratio of specific heats t to be used for
the sound excitation, and hence the absorption and dispersion
affecting the propagation.
Intramolecular vibrations are "excited" for the
present purposes, exclusively by collisions with other molecules
i.e. by close-range interaction whereby energy and momentum is
being transferred from one molecule to the other.
The probability of energy transfer which depends
on the nature of the close-range force field, the geometry of
the collidingsystems and the relative kinetic energies, is one
of the factors governing the time dependence of vibrational
specific heat. The other factor is quite obviously the numiber
of binary collisions per unit time* the triple and higher order
*
( 55 ) R.A. Walker, T.D. Rossing and S. Levgold : The
role of triple collisions in excitation of molecular vibrations
in N2 0 ; N.A.C.A. TN 3210, May 1954, Washington
K)
collisions having a non-discernible part,
The linear dependence of relaxation times on 1/p
confirms the preponderant part p]red by binary collisions.
Calculation methods and detailed results are, at
the time of writing controversial and no clear-cut advantage can
be decided in favor of any particular collision model or potentialfield assumption.
Following Bethe and Teller* and others, the simplest
approximation uses a one-dimensional model to show that the
interaction probabilities are small for vibration.
Assume a harmonic vibrator at frequency
perturbation function
?
.
a
F( t )will interact strongly only if a
significant spectral component of
to
,
F(
t
) has a frequency close
The perturbation spectrum is known to be centered around
the frequency /
which was defined above as the typical
"collision duration" or time of travel of the impinging molecule
in the range of molecular forces. Thus the interaction probability
is determined again by the ratio
2 n)
/_
YV
Two ways can be used to show the orders of magnitude
involved, one is to evaluate
1 (.5)
9
in terms of vibration amplitude
P
and vibration energy (assuming the lowest quantum number) ; the other
is to assume
;
from existing physical data and use
#
given
by infrared spectroscopic measurements. Both approximations will be
used here to show the nature of the assumptions involved :
Referring to a vibrator of frequency
21
9
the
energy associated with this vibration is
vibration amplitude in the motion
A
.2
(
6
,
thus
'I
and
e ....-
-- --
VA
g/A
Iis
(Ev'
M
1.222 - 1
the ratio of the molecular diameter to the vibration
amplitude, and this ratio is of the order of 300 *, whereas the second
factor is the ratio of vibrational energy to impinging kinetic energy
or
Tc./T
ratio of the vibrational characteristic temperature to the
translational temperaturc of the gas.
TABLE
NI
will be of the order of
*
in paragraph 1.221 show that this ratio
10 for the usual diatomic gases (112 , 02 etc)
-----------------------------------------------------S.J. Lukasik, NOTES on calculation of Acoustic Relaxation times,
Feb. 23, 1955, Mass. Inst. of Tech. (Unpublished).
and for most of the polyatomic vibration modes ; however a few
important molecules have vibrations with characteristic temperatures
as low as 22900 K (water vapor, corresponding to the absorption
line at 1595,o cm-1 in the infrared spectrum) 960c K (carbon dioxide,
line 667.3 cm~1 ) and even
568 0
K (carbon disulfide, line 396.7 cm-1)
The important fact is to notice that X for molecular vibrations
will always be at least of the order of b 0 to 50 which indicates
that the energy transfer probability will be exceedingly low.
Bethe and Telleil express the vibration-translation
interaction probability as
ioe,
where P10 is the probability of transfer from one vibrational mode
a quantum h9 to the translation and
d
is a geometrical factor
accounting for the orientation of the impinging velocity vector
with respect to the vibration axis. d is indicated as having values
between 1/3 and 1/30 ; this (empirical) coefficient accounts for
practically all discrepancies between theoretical and experimental
results** To account for the different velocities present in a
canonical (equilibrium) distribution of translational energies,
the
above probability should be averaged over a Maxwellian velocity
-------------------------------------------------------
)
3
Rep. X-117, Aberdeen Proving Grounds, (
Deviations from thermal equilibrium in shock waves, by H.A. Bethe
and E. Teller.
( 49 ) R.N. Schwartz, Z.I. Slawsky and K.F. Herzfeld.
Calculation of vibrational Relaxation Times, J. Chem. Phys. L), 10, 1951
I
'1
distribution so that finally the expression valid for energy exchange
between translation and vibration for a canonical assembly of molecules
is obtained:
where
h9
2/.
3T
'/A
ibple : Assume
Then
0-.
P 0
3 (15)
p
-
2/3
(2)
.396
10
1/3
(C2
1.5 X 5.95 x 1.26
X 11.3
1.5
11.3
-11.3
1.15C
~~=
1.15 .
11.3
which gives
10-5whcgie
the average number of collisions for 1 quantum transfer
Z
Q
t/
=
87,000
To obtain a number of practical significance in sound
propagation this probability has to be referred to the actual
population in state 1. Assume to be known the probability K0 of
transition from state
1 -P 0 per second, then the rate of change
in population of state 0 is proportional to the probabilities K10
and
-Kvl affecting the populations 71,
and
Y) respectively,
thus
W
-~
and, at equilibrium
e-
IK
ot 1o
(1.222 - 2)
-I
-
h
0
N
according to
the configuration probability explained in Sect. 1.221.
Thus, close to equilibrium
- 4 0 1 WT
i01z
I
and similarly for any vibrational state
w
q-1)
- 4#/K-r
I(?-,)
1; 1
The total vibrational energy present is
10
hko
IE
V
,
thus its time differential
may be obtained by summing all expressions of the form
- 'kT
A
69
-EV}I
-
V,1)1.222
where
4./kr,
1.222 - 2:
-
is the equilibrium value of
by Einstein' s Formula for harmonic vibrations.
Ey
, as provided
3
Equation 1.222
-
3 is a somewhat oversimplified translation of the
fact that the vibrational equilibrium is approached with a
"reaction rate" proportional to the "deviation" from the equilibrium
value.
..
The factor
o
is usually denoted by
X'
/ has the dimensions
'-I
t
(Relaxation time constant.)
The correspondance between
-C and
.
Ser.
and
as defined
earlier is readily derived
t
j\J~
N
the number Zp
being the "average number of collisions" necessary to transfer 1
vibrational quantum from state (1) to the ground state.
N is the
number of collisions per second which may be determined by the
number of molecules per unit volume and the kinetic collision cross
section of the gas considered, although this latter is somewhat
uncertain. With the questionable assumption that the cross section
is essentially the one presented by rigid spheres of diameter
can be written as*
where 3
is the number of molecules per unit volume
CO
For
n
33
at S.T.P. conditions we obtain
6.0o2 , 1023
22400
----- -------------------
(
2
2.7 . 10
1
molecules/ cu.cm.
----------------------------------------------
)
E.H. Kennard, Kinetic Theory of Gases, p.ll3, Mc Graw Hill, N.Y. 1938
A
a. 8
..
5
N
.1.41
1.380
Thus, K1 0
=
!5.6
.
3.72 . 10 4 cm/sec
10- 8 c
3.14 X 2.7 . 1019 X 3.72
. lo0
.
31 . 10-16
104
collisions/sec
P10 N
is, with the above hypothesis for
1
-
1.15 .]0-5 X 1.38 .
-
1.6.
which would yield for
105
CO
10
sec-1
K=2
1.6 x 8.64., 10
1.38 . 105 sec-1
According to Bethe and Teller, the correct value
is .98 . 105 sec -1.
The discrepancy should not surprise the reader
with all the more or less plausible assumptions used to obtain the
result.
The above calculation was based on the assumption
of identical molecules with no effective action beyond the molecular
diameter
5 . Experiments on sonic absorption have shown however
that the nature of the Jelliding molecules and their relative velocity
are apt to change vibrational de-excitation probabilities by factors
of 100 and even 1000 so that the corresponding relaxation time
constants are reduced in the same proportion.
The last twenty years have seen considerable activity
towards the determination of relaxation times of gas mixtures ; both
theoretical* and experimental** results point toward the extreme
importance of the "impureties" i.e. molecules whose concentration
is not large enough to cause substantial change in the specific
heats of the "main gas" but which do .
manifest sharp increase
in PlO on account of their large average velocity or the presence
of chemical affinity which modifies essentially the nature of
the interaction during collision.
------------------------------------------------------*
H.O. Kneser and V.0. Kundsen, Ann der Physik. 21., 628, 1935 (
D.G. Bourgin, Sound Absorption and Velocity in Mixtures, Phys.
Review, 5
355, 1936.
255, 1940
* V.0. Kundsen and E. Fricke, J.A.S.A. 12
E.F. Fricke, J.A.S.A. 12,
245
R.W. Leonard, J.A.S.A. 12, 241,
F.A. Angona , J.A.S.A.
1940, (
tO
)
(
40
)
1940
25, 1-116, 1953
(
)
(
5'
)
34 )
Two conclusions should impress the reader's mind
(a)
The extreme sensitivity of
"C
to the impurety
concentration ; (see figs 29, 2Z)showing results obtained by
Knudsen and Fricke) 1/1000 molecular concentration of H20
will increase relaxation frequency of CO2 from 20 Kcps to
240 Kcps at 1 atmosphere.
(b)
No reliable calculation methods for
LP
are at the present
available in presence of several highly active impureties.
Even in the simple cases, often geometrical factors of the
order of 1 to 30 are inferred from experimental results in
acoustical measurements.
The lack of theoretical background is particularly acute
in the field of complicated polyatomic molecules, whose infrared
spectrum has not yet been entirely interpreted in terms of molecular
vibration modes.
The summary of the most important experimental results
has been reproduced in Table
V
. In particular, the
influence of water vapor on Oxygen and CO2 can be conjectured
from these data. It has to be remembered when trying to apply energy
transfer probabilities that the temperature dependence of PlO is not
an accurately known function on account of the uncertainties on
.
/I
TA8LE
V
AVERAGE NUMBER OF COLLISIONS
DE-EXCITATION
FOR VIBRATION
Cos
H2 0
Co2
400
25000
(3)
8500
20000
(3)
(3)
86000
6200
CS2
N2
100000
02
Sooooo
(3)
(3)
1400
(30)
0
100000
314
(53)
(14)
3550
3200
8
(Ir)
(iT)
(ii)
K02
Cos
215
(3d)
-
9600
(Ia)
2000
(%4)
200
(14)
00
Co
(go)
220
(se)
18
(If',
C82
Bethe and Teller
11)4
14
)
is )
(
Eucken and Becker
Fricke 1940
(
6
)
Kundsen and Fricke (1940)
(
I5'
)
Eucken and Aybar
(
bo
)
A Kantrowitz
(
53
)
VanItterbeek, Bruyn, Mariens
(
50 )
(1940)
(1947)
Sherrat and Griffith, 1934.
8700
(4)
4
According to Kuchler* the value of
.
and the ratio
predict O
,
2.
2.
changes on account of both
but the theory is not in position to
for any temperature in a given gas mixture. The order of
magnitude of V
seems to remain constant for a large range of
temperatures.
Table
V
shows the values of Z
at 293 0 K for gases
having the greatest interest from our view point.
Dipersion and Absorption of Sound causedby
1.223
Thennal Relaxation.'
The summary given hereafter has been based on the
method of irreversible thermodynamics
,
as interpreted by Markham
and al.
If we assume that the vibrational energy is function
of the "vibrational"temperature only, the relation establishing the
time variation of the vibrational energy (1.222 - 3) may be
interpreted as :
1.223 - 1
----------------
------------
-------------------
* L. Kuchler, Zeitsch. Phys. Chem. B 41 3 199,
1938
(37)
(
as
) K.F. Herzfeld and F.O. Rice, Phys. Rev. 31 691-(1928)
(
g
Mod.
) J.J. Markham, R.T. Beyer and R.B. Lindsay, Rev.
cit.
Phys. 23 - 4 - Oct. 1951 p 359 & seq.
loc.
T
being the temperature characterising the vibrational energy
( or any other energy connected with an "inertt " mode of motion)
and
T
being the equilibrium value of
T
Tc
. Obviously
is also the "external" or translational temperature, since at
adiabatic
equilibrium all the various temperatures are equal. With
assumption we have
a
a.
c4~)~~ 4IV
61 U SU~
Then following Harkham, Beyer and Lindsay, the following transformations
are justified:
UC
cA"
tC
C
e
jT e
TC2
e e
volume coefficient
of thermal expansion
+
V
7
tW
-,
1.223 - 2
-Ir
t
D
-
)
00
Te
1.223 - 3
Using the classical relationship
dvy=
(o T*V +
.
*
V/,j fN
-"--
f3 V~
V
Substituting in 2,233
#;
cv.
4.
c
+
v)
y
cy*
,
-1:
C -,eve
4'.
\vr~V~)
-r(Iv
(1.223
- 4)
writing
the non-linear teris
Neglecting
Negectngtheno-lieartefas-)
and making
0
=
(\
w
4L
k
-(v/cC)
Cut =
ith
c
~EC f7'kV
+
t
T
/ C.' )
%X/C'
tC
we obtain
I/P
I- 9
Y./
1.223 - 5
t
where
It can be shown that the dynamical equation of state represented
by 1.223 -
5
leads to an absorption per unit lenght
2.
1+
0
woo.
1.223
and a
phase
-
6
ve.ocitV:
y
0
~
L<
WV
0.
K-~
The equation of state described for heat radiation being
also of the general fom 1.223 -
5
can be also represented by
propagation characteristics given by 1.223 - 6 except for the
numerical values of
4Jr , angular relaxation frequency.
Section 2.32 will show how to apply the above result in
the case of pulse propagation.
1.23
REAL-GAS BEHAVIOR
The simple relationship which holds for perfect gases
is known to break down for high densities i.e. where the
ratio of the total potential energy stored in the intermolecular
field is no more negligibly small with respect to the total
kinetic energy.
The acoustical propagation velocity is still defined
by
as found previously, but quite obviously this factor
will have to be expressed now in terms of the real equation
of state. The ratio of specific heats has been discussed as a
function of temperature, we shall limit this section to the
changes introduced in the
ratio by the presence of additional
terms in the equation of state.
For a pure real gas it is convenient to approximate the
true equation of state by
RLT f1 ' ( 6
!r) I
*
------------------------------------------------
)e
C
J.C. Slater
-
Introduction to Chemical Physics, p. 195
(Si)
The coefficient a and b are defined by
2
3
where
4
is the interaction potential between molecules.
It is not believed that in the present state of
experiments application of the above formulas is necessary or
adviceable. As a matter of fact, curves of sound velocity for
various pressures and temperatures have been published for a
great number of gases, in particular for variable concentrations
of atmospheric air, octane, heptane and residual products of
combustion. It will be seen in chapter
,
3 that use of the actual
sound velocity data eliminates measurement errors due to
deviation from the perfect-gas law.
We reproduce in fig.
24
curves published for C02
which are of particular interest for our experiments on thermal
relaxation. Fig.
24
shows the actual sound velocity
characteristics which have been used for the study of engine
chamber temperatures.
Another formulation, based on the Leonard-Jones concept
of intermolecular potential field
t-
( - maximum energy
( Vof attraction
( rbetween
(
molecules.
The equation of state of a real gas is represented as
the infinite power series
Pv
B
-
C-/vt
B +
I
--
RIT
(called the second" virial" coefficient)can be evaluated as a
function
of temperature
(o)
5 (T )
b.
-
EO )T
( is one-half of the total
2wher
=
where
6
6
volume of
N
rigid
( spheres of radius ro
(4
4
V4
in this place designates the "gamma function"
Fortunately the determination of G(T) has been perfomed for
great number of values of the ratio
a
C- between the ranges
-------------------------------------------------------Bird R.B. and Spotz E.L. "The Virial Equation of State" Univ. of
(4)
Wisconsin, N.R.L. Report CM 599, 1950
.3 and hoo, and values of B (T) are thus available, as well as
the corresponding first, second and third temperature derivatives.
The sound velocity being for adiabatic evolution
c- =
(Yo/.."
C -
we obtain
(
)
+T/M B/V +
C/v,&
-
)
The sound velocity for dry and moist atmospheric air has been
computed* on basis of this formula.
The actual
V
can also be computed from
~
shown
~pb~/~T)as
in
( 48 )
It is desirable to give here a few orders of magnitude
of the virial correction introduced in the energy content of dry air:
Density Ratio
0.008
1.0
TOK
700
E(Ideal)
AE(Virial)
122.55 Cal/gr
3.10 -cal/gr
1000
181.50
3.10-4
3000
637.27
-4.10-5
700
122.47
4.10-2
1000
181.44
3.10~
3000
637.29
-10-2
---------------------------------------------------*
J.O.Hirschfelder and C.F.Curtiss" Thermodynamic properties of air"
N.R.L. Report CM-518, 1948 (Zia)
(
48 ) Ref: T.P. Rona - Master's Thesis, June 1953
25
700
120.57
1.0
1000
179.88
.84
3000
637.76
-.24
In this table" the density is refend to the density of
air at 1 atmosphere and 00 C.
When evaluating the possible influence on sound velocity
method for engine temperature measurements it has to be remembered
that high temperatures and high pressures occur simultaneously in
these experiments and real-behavior
ill therefore be a minor effect.
Preparation of a set of tables and curves accounting for the
real-gas behavior of water vapor and carbon dioxide in the
cylinder content is being contemplated to carry calculations further
in accuracy of interpretation.
1.24
DISSOCIATION
Although dissociation is not believed to play a
significant part in sonic dispersion at the temperatures present
in our experiments, its presence should be mentioned in order to
help future extensions of the measurement method.
Dissociation modifies the sound propagation characteristics
in two distinct ways, namely by modifying the number of
particles present in the "equivalent perfect gas"
independent
and by presenting a "lag" between the energy absorption and
restitution, just as it happened for the intramolecular vibrations.
1.241
Energy Content
In experiments pertaining to temperature measurements
in internal-combustion engines before the arrival of the flame
front both effect will have negligible magnitude especially in
view of the fact that each octane molecule present there are about
12 Oxygen molecules and about 47 Nitrogen molecules in the
combustion chamber" and thus the dissociation characteristics will
be primarly those of atmospheric air. Some caution has to be
exercised however to account for the presence of C02 and H2 0 in
small quantities due to the lack of complete scavenging of the
cylinder before admission of fresh mixture.
Theoretical fuel-air ratio
0
According to Bethe and Teller, the energy content of a
gas may be written as
where the "relative energy
contents",
are
E,
3/2
P/4
FI?
etc are the ratios of
translational, rotational and vibrational energies to
we obtain
respectively. Then the sound velocity being
7pIF
=
it is easy to see that this expression is, in particular, correct
for ideal gases ;
IL4
C
may be written as
( /D?
13/Dy)
1.24 - 1
Or in an ideal gas,
whereas
e
are independent of the density,
is clearly pressure dependent.
In adiabatic conditions, we shall obtain:
C
=-
+
f
since in adiabatic evolution
) times the
the relative temperature change
relative change in density.
cog
which has been the result found previously for perfect gases.
The form ( 1.24 - 1) is particularly useful to express sound
velocity in dissociated and possibly slowly reacting mixtures.
It is found that %
being the molecular energy
content, the same factor for a dissociated state of degree of
dissociation
od
and energy of dissociation
3
should be
written
I~o
Z
with Z
OP-
C
-
[pf
t
being the energy content of the atomic gas
'
a
(3/2) .
The degree of disso&iAation et
and energy content of gas
mixtures may be found by weighing the respective values by the
concentration , thus
where
are the volume (molecular) concentrations. Table V
shows typical values calculated by Teller for atmospheric air with
:921 rare gas content and a proportion of .o3% of CO2 . Changes in
are noticeable above 600 0 K but values of
are negligible below
3000 0 K. The final result for sound velocity is given by the
authors asI
It is easy to see that
with
varies only from 1.400 at 300 0 K to 1.382 at 1000 0 K and
less than 10
of
is
.
The variations in energy content due to dissociation
will be neglected for the time being, but the method of analysis
outlined above should be kept in mind especially to investigate
further the effects of water vapor dissociation. Most of the gas
tables calculate
at high temperatures by taking the
dissociation into account.
VI
T ABL E
ENERGY CONTENT AND DISSOCIATION CF AIR
ocq
-)
lp
0
Rare
P(1+00
Gases
C02
o.0 2 3
0.001
3.483
3.483
3.494
3.494
0.745
3.507
3.507
2.748 5
0.754
3.527
3.527
700
2.764
0.764
3.552
3.552
800
0.774
"
2.781 5
3. 58o
3.530
900
2.805
0.7835
"t
3.613
3.613
1000
2.8295
0.792 5
3.647
3.647
1250
2.8895
0.813
3.727
3.727
1500
2.950 5
0.829
3.805
3.805
1750
3.001
0.843
3.869
3.869
2000
3.044
0.855
3.924
0.0000
3.924
2500
3.115
0.8335
I"
4.024
o.oooh
14.023
3000
3.171
o.949
"t
4.145
0.0030
4.133
3500
3.2275
1.1o6
4.359
0.0121
4.307
4000
3.287
1.4oo
0.003
4.713
0.0331
4.562
4500
3.3765
1.808
0.0035
5.211
0.0687
4.875
5000
3.527
2.226
o.ook
5.780
o.1058
5.227
300
2,726
0.733
400
2.731
0.739
500
2.738
6oo
t
"
0.001
5
o.002
i
5
0.0025
"t
l.242
Time Dependence of Dissociation
The reasoning applied to vibrational relaxation
has been extended to cover time lags in the adjustments of internal
energy in presence of dissociation. Owing to the large value of
:
(energy of dissociation) the number of molecules ablis to dissociate
is excessively small* and assuming that all collisions were effective
( PM
-A
t 1) dissociation would not be instantaneous. The same
applies to the recombination process which requires double or triple
collision between atoms whose partial pressure (= concentration)
is of the order of 1% of the total pressure.
The examination of the collision efficiency will yield
as previously an estimate of the average number of collisions required
to produce dissociation. According to Teller's calculations, this
number is
where CA
is
geometrical factor taking into account the possibly unfavorable
direction of the colliding molecules, its value is usually comprised
between 10 and 800.
As it has been seen for vibration, the relaxation time
can be found by consideration the number of colliding particles
cC bW
per second
--------------------------------------------------------S-DlkT~
-11
Even at 30000K the factor me
is only lo
with D
7.35
for N2 (Slater ( 51 ))
The relaxation times for dissociation are thus seen to be
very long compared to the values found for vibration, values ranging
from .01 to .03 seconds are found in the litterature for air at
less than 3500 0 K.
It is safe to recognize that at high sonic frequencies
the relaxation of the dissociation process will not play any
noticeable part.
CHAPTER
2
ANALYSIS OF VELOCITY DISPERSION
WITH PARTICULAR REFERENCE TO THE ACTUAL SOUND
2.1
NATURE OF THE ACOUSTIC EXCITATION
2.11 Description of the acoustical circuit
Fig.
8 shows the nature of the acoustical circuit
employed. The discussion of the design factors of this type of circuit
have been thoroughly discussed in a previous paper*; it will be
sufficient to mention here that in the present experimental conditions
( i.e. high and rapidly changing pressures) the presence of metallic
coupling rods appears essential. The power levels present in the
acoustical circuit have been evaluated and shown in Fig.
/A$
8 bis
This evaluation is based on a somewhat simplified model, where all
elements were considered as frequency-independent altesdators ; its
purpose was merely to predict the order of magnitude of the signalto-noise ratio at the receiver. To give a more concrete basis to our
pulse analysis, the acoustical circuit has been tested with the
exclusion of the gas path in order to study its own amplitude and phase
------------------------------------------------------*( AB )
T.P. R~ona, M~aster's Thesis 1953, MY.I.To
characteristics. Appendix IV gives results and calculations based on
this test ; the acoustic circuit is there assimilated to an amplifier
with 3 RC coupled stages and the equivalent rise-time is computed.
When expressing the phase-distorsion of this circuit which does not
contain any gas portion, it
should be remembered that a linear phase
characteristic due to the (dispersionless) phase velocity in the
coupling =ods has to be added to the overall transfer function.
It is in order to mention the disappointing nature of
numerical calculations aiming to the accurate waveform prediction.
As mentioned elsewhere* the preparation of crystals, their coupling
and mechanical clamping may well modify the received signal-to-noise
ratio by a factor of 3 and more.
The transmitter used for the crystal excitation has been
considerably modified since the preliminary work performed on this
problem. Fig.
4
construction. The
shows the schematic and Fig.
3
the details of
# 3C45 Hydrogen thyratron has been selected for the
ultra-short ionisation time and the best-power output impedance which
allows relatively simple matching to the cable and crystal impedances.
The power supply has been provided for full power
operation at 200 pulses per second, it will be seen in Appendix III
that higher pulse recurrence frequencies are desirable for further
improvement of the equipment usefulness.
------------------------------------------*
(
52
-----------
) Sloan Laboratories, Massachusetts Institute of Technology
Report on Contract
DA - 19020 - ORD 2831
2.12
Receiver Characteristics
Two amplifiers were used in connection with the present
equipment, the first version is a six-stage 6 AK 5 preamplifier, with
the first stage mounted in triode connection, the most recent version
uses a 6 AK 5 mounted in triode followed by a grounded-grid 6U triode*
This latter receiver uses 6 AC 7ts for the following stages and has a
series of experimental circuits intended to discriminate the sound
signal in the noise and operate automatic recording equipment.
Owing to the considerable differences between individual
crystals, the gain obtained in the signal-to-noise ratio is noticeable
but unimportant. With the levels attained at present, the major part
of uncertainty on the signal position is not due to random noise but
to alternate-path signals* and to parasitic reflections in the receiver
bar, due to electromafttkradiation from the transmitter to the receiver.
It is safe to state however that with the present
combustion chamber design, transducer mounts and acoustical insulation
the above causes of signal uncertainty are of minor importance for gas
densities above 1 atmosphere.
Both receivers have bandwidths of 4 mcps centered around
2.2 Mcps. The impulse response has been tested and the rise time found
around
.12
psec.
The 6 AC 7 tube receiver is based on 8000 phos
transconductance and 15 p pF interstage capawttamitance which calls for
------------- -----------------------------------* Wallnan H. Macnee A.B. and Gadsden C.P. Proc. I.R.E. 36 p 700
(S7)
iH% The acoustic signals reaching the receiver crystal without passing
through the gas path.
5
stages* at 18 dB/stage or a total of 90 dB. The amplifier of the
synchroscope is connected to the output of each amplifier, so that
the overall amplifier gain is 125 dB. According to the 18 dB gain per
stage the equivalent plate load resistor is 1.2 K. For the purpose of
our dispersion analysis it will be assumed that the presence of this
amplifier causes the following effects :
.12
ysec
(a)
Rise time of
(b)
Constant delay, small campared to the transit time for the
total electro-acoustical circuit**,
(c)
125 dB max.
The constant delay will be included in the "calibration"
of the instrument, whereas the rise time will add its effects to the rise
time found previously for the acoustical circuit itself, so that the
total rise time will be :
From appendix IV it is then concluded that the transmitted pulse through
the canplete circuit is of the form (gas path excepted)
C S)
C-
(S
'~.
--------------------- ------------------------*
Elmore and Sands, Electronics, p. 145
Ed.l
(2 )
** This delay may be d'efined as the time abscissa of the impulse
response, it has been found to be of the order of .25 ysec.
I.
with C detemined by the amplifier gain and the transmitter power
control. Owing to the linear characteristics assumed for all the
elements of the propagation path and to the negligible reaction of
the gas path, the fact of lumping all the transfer impedances which
are not dependent on the gas characteristics in one is justified. The
signal ultimately displayed on the scope will be the signal (Obtained
in absence of gas) modified by the transfer impedance of the gas
section. This latter should, obviously include the interfaces between
coupling rods and the combustion chamber.
2.2
EFFBTS OF AMPLITUDE AND PHASE DISTORTION ON RESULTANT FREQUENCY
SPECTRUI.
2.21
Fourier Analysis
Whenever the amplitude response and the phase distortion
characteristic on a physical network are known, the output obtained
for an input signal of known spectral characteristics are readily
obtained.
Let be
A (A)
ei
the frequency spectrum of the input signal,
the amplitude response, and L (A) the phase response of the
network, then the transfer function can be written
and
being understood that
will in general have its own phase
characteristic. This result, which is straightforward consequence
of the superposition of the infinitesimal spectral components nay be
stwnarized by saying that the resulting amplitude function is the
product of the moduli of the input and response spectra, whereas the
resulting phase characteristic is the difference of the phase functions
of the input spectrum and the transfer impedance.
As usual, integration in terms of the real variable
is cumbersome and will not yield simple results unless both
and
are elementary functions. By introducing a
complex variable however, often the integration can be simplified by
the application of Cauchy's integral theorem and by choosing the
integration contour with care. Integration in the complex plane is
straightforward provided poles are simple and the function of the
complex variable
is single valuedt
This in particular, is the case for multiloop lumped-
constant passive networks where impedance function are found to be
rational fractions of polynomials in terms of
5
*
( AS
) L3.A. Guillemin, Communication Networks
,
Vol. I& II.
The study of the nature and location of the driving-point and
transfer impedance poles is the basis of modern circuit analysis.
The situation is however radically changed in presence
of distributed-constant circuits where the 'quivalent "impedance
functions turn out to be quite complicated.
A typical example is found in filter theory~'~ and in
radio-wave propagation in dispersive media!*
In some particular instances, however, the situation is
greatly simplified by the fact that the actual amplitude and phase
characteristics may be approximated by simple "reference characteristics"
and distortion in both phase and amplitude may be expressed with
respect to these reference characteristics.
--- --------------------------------------------------------------------A
62
2.22
INTERPRETATION OF AMPLITUDE AND PHASE DISTORTION by
PAIRED ECHOES.
The magnitude of amplitude and phase distortion caused
by the gas path in the acoustical signal is such that the search
for a simplified and readily graspable interpretation is warranted.
The method exposed in this section has been developed by iheeler for
low-distortion television circuits' and is based on a physical
interpretation of the amplitude and phase characteristics of the
transmission system.
Assume the input signal to be the unit impulse ; the
equivalent frequency spectrum is then essentially a constant for all
frequencies between 0 and
0 . If the received signal can be
considered as the superposition of a "main signal" impulse of
magnitude 1 and an "echo" impulse of magnitude t occuring at the
instants
to and
to + te
respectively with respect to the
transmitted pulse, the amplitude and phase response as a function
of the angular frequency can be inferred. (Fig.
2.5
)
3ince all the spectral components have equal magnitude
the main signal is the superposition of an infinite number of
infinitesimal components having all been subjected to a phase
distortion
-------------------------------------------------------
H.A. Wheeler, Proc. I.R.E.
,
27.- 6 19385359
and similarly the echo has a phase rotation with respect to the
main signal which is
W
both being considered here functions of
.
Fig.
A26
shows
the vector relations in the Y plane ; the diagram represented is
fixed in shape (for given phase relations) and rotates with the
angular velocity t when
W.is being varied from
0 to 0* .
The amplitude and phase of the resultant received
signal can be approximated by
provided
I. and
fe
is reasonably small. If attenuation is to be
expressed in nepers so that Sa
phase distortion by
'-
'o
e~
and the
then we have
These equations show that, within the limits of the above restrictions
on the magnitude of e. and
&f
, the signal transmission correspond
to an amplitude response
and a phase rotation with respect to an ideal transmission link
having linear phase characteristic
e(t,)
It is easy to realize that a symetrical pair of echoes
and tC#t) will correspond to amplitude distortion alone
(e
whereas a skew-symetrical pair
te, )
and - e (A4)) will be
produced by phase distortion alone.
The next step is obviously to decompose the actual amplitude and phase
response characteristics in a limited frequency band in fouriercomponents. It has been found expedient to refer the attenuation to
a constant-attenuation line whereas the phase characteristic should
be referred to the ideal linear phase response
If
=.
6 W
.
The input transient being then considered as the
superposition of impulses
the output transient is then reconstructed
from the undistorted main signal which has the same shape as the input
signal and is delayed by
t
Tiasntoialottebiofori-----------------------------------------------------
=
; and the positive or negative
This assu~mption is also at the basis of Fourier analysis.
echo pairs corresponding to the spectral components
AY
a(n)
and
(n) of the prescribed response characteristics.
A component of amplitude
(on the
(A)axis)
of intensity
+ if
'
a (n) having a "wavelenght"
---- on the amplitude response will correspond an echo
e
A rlh
"
(K ) =
2"
-- a(IN)
at the symetrical position
-
(W
may be conveniently chosen as the
cut-off frequency of the transient signal itself or of other
"filter" components of the transmission system.
Any component
AJ&
of "wavelenght"
---
on the
phase response will give rise to a skew-symetrical echo pair of
amplitude
located at
1
The method appears as particularly favorable for application in
the present situation since the problem is to locate the phase
characteristic by the knowledge of the transmitted and received
signals.
The application made in Section 4.12 will illuminate
the fine points in the use of this method.
2*23
GENERALISATION OF THE PAIRED ECHO1METHOD"
The paired echo method, as shown in the precedent
paragraph is an approximation valid only with the assumption that both
QA and AC
be small, i.e. a fraction of neper or radian. This is the
case for all the actual situations encountered in the engine
measurements but for the specific objactive of studying relaxation
dispersion one had to create artificially highly dispersive gases
where the phase distortion, in particular, was far from being small
or even of the order of one radian.
This is readily understandable when considering that
our phase rotations are ideally of the order of 3 to 600 radians,
rather than the values found in usual amplifiers. Sect. 2.32 shows
the numerical values likely to be encountered in presence
of
thermal relaxation.
The interpretation by means of hypothetical superposed
"echoes" can be generalized-towards possible application for
amplitude and non-linear phase distortion of arbitrary magnitude.
+4
Let the input signal be
SCJ
jL~.t
()
e.
and the response characteristic of the circuit, as previously
A()so
that
The original generalisation was noticed and developed by
Dr C.R. Bu.rrows in the discussion following Wheeler's paper.
(PROC. I.R.E. 27 - 6. 384.)
A((W)
The output signal is then
JA
(wt
+U'
-and if we can express
A (W)
and
+ 4f(W))
4
(i ) in terms of spectral
components such that
AL
0.
ck,- Cos V%%Uj
)
-
1
ct~ e
02
In these formulas A has dimensions of time and is chosen to limit
the frequency domain to the actual variations of
(0
;
:maxhighest
frequency
present in
the spec trum
Jith this choice the lowest "ccmponent" on the frequency axis
will have a wavelenght equal to 2 times the actual passing band.
Individual terms of the series for L?(Wtcan be
expanded as infinite sums of Bessel functions
YA
1 =QO@
~0@
T-
and thus the output fm ction to be represented by
-d*o
a
k=-e
C.R. Burrows has shown that this expression is the equivalent
of the response ofa series of chain-connected networks, I0 ,
...etc, so that the response is
.
:
A(w) .. A. (w) . AI(w) .A,(W).
with
and
A
AM
The fir st term is seen to represent the amplitude characteristic
lumped with the linear component of the phase characteristic, so
that the output of the first network is
o0
which is precisely a sun of echoes of the same waveshape as the
input signal.
(Sa
/
o
is then transmitted in the successive network portions
having responses A,()
will give a waveform
490
ZI
-(
I(
'
k
s(+
In order to illustrate the method we will assune a
hypothetical network having amplitude and phase characteristics
showm on fig.
27
so that
-bw
CLO
all other 0, and
b
at
%0
being zero, but
arbitrary values. The input signal 5,
unit impulse U, (A )
4
CL
and
I
having
is assuned to be the
t = 0
Then
The second network portion has the response
A,(w
---00 x" ( ,
To be noticed that in this formulation
than nepers or dB.
fa is actual ratio rather
and rill give three (infinite) series of output signals
corresponding to
(S2 ) 0
(6t)
Ck 0 L4
10.
it =-QD
4(6
-t --!i. )
-Z fW
4mb-rn
4
4-~
VA
Since only integer values of
:7x
z(-
,
k
2TW
K are being considered, we have
I)
and the
position of the "echos" is perfectly defined. It is useful to have
orders of magnitude present in mind :
b = o
b =.20
b1
b= 2
b-
5
b=10
.9900
.9385
.7652
.2239
-. 1776
-. 2459
.0995
.2423
.44ol
.5767
-.3276
.o043
.o5o0
.0306
.11h9
.3528
.005
.0002
.0026
.0196
.1289
.3715
.0000
.0002
.0025
.0340
.3853
It is seen that the notion
1
more and more confused when the phase rotation
ain signal" is becoming
1 becomes significant ;
m
',1
this was naturally to be expected since the "group velocity" or
"wave packet" or "time of arrival of the signal power" are devoid
of significance in presence of non-linear distortion applied to a
broadband signal.
As a partial conclusion the noticeable presence of "echoes"
will indicate that part of the signal spectrun is in the dispersive
region. Sect. 2.32 gives the interpretation to heat capacity lag in
acoustical propagation.
2.3
ABSORPTION AND DISPERSION CAUSED BY ACOUSTICAL PHENWCENA IN THE
GAS PATH.
2.31
VISCOSITY AND HEAT CONDUCTION
In sect. 1.21 the viscous and non-isoentropic nature
of the sound propagation has been investigated and it has been found
that propagation is governed by the attenuation
and the phase velocity
MC
For a given gas in specified physical conditions the ratio of input
and output signal is
L being the lenght of the
gas path. The phase angle of a sinusoidal signal of angular frequency
is then
A few numbers will show the possible
w
C.
orders of magnitude of attenuation and phase distortion due to these
"classical" causes.
T A B L E
ATTENUATION AND DISTORTION DATA FOR
VISCOSITY AND HEAT CONDUCTION
.'4
GAS9
C%&
s2/cm
g/cm 3
cm/sec
Air 200
1.29
3.43
1.40
1.71
1.94
.31
.67.020
Oxygen
1.33
3.28
1.40
1.90
1.90
.37
.66
Nitrogen
1.17
3.51
1.40
1.75
1.90
.33
.66
Carbon
Dioxide
1.85
2.68
1.30
1.40
1.63
.33
.63
Propane
1.90
2.48
1.114
.80
1.30
.210
.48
g/cm-s
This Table is computed with Stokes
,AC,/
assumption
<AC Heat cap/gram
j'sec 2
F
2.32
Heat Capacity Lag
The phase velocity in presence of heat capacity lag may be
written as
/J
C
A -"W
Call
dimensionless angular
frequency
c
r
oo
Call
dimensionless
phase velocity
r
)
e C1
r
*
k
t
4~
4.
1.I
coo
23.
2~~~
c.
Lc:
ca
(+2~
~
ibcanple:
J
m&t
g
reference
(velocity
=
C
Is
r
.? CoI
I-t-r1
rL
(~)
6
.
t
x
,3 1 X 1.
4ni 10
w=i
The attenuation for
oC1
)
=S
2 Mcps is for example
--
and o( for air is
CV
. The corresponding change in
aC 1
phase velocity is
The phase distorion for a transit time of 40 ysec in
the gas path would be at
cps
i
:
4o . 10-
6
X 16
X 1.1 . 10
.
106
-:
2.12 . 10-2 radians.
Noticing the fact that
4 is not affected by changes in density* and
varies slowly with temperature, it is expected that both attenuation
and phase distortion due to classical effects will bear negl igible
on the signal envelope shape and on the measured transit time.
This conclusion should be reinforced by the way the
instrument is being calibrated.
-------------------------------------------(33
)Kennard,
Kinetic theory.
I>
Curve on
P.
shows variation of
E,
~
with frequency SL between
values of .1 and 10 which covers the important portion of the
relaxation domain. (See Angona, etc)
The attenuation characteristic in the relaxation domain
can be written as*
Nepers/wavelenght .
CAJ ..........tIt
(Arni
/2
KOA)r
LJ+A.1
ec
r
W.
The attenuation constant a
r
In
i4J
is then
6
r.
+~
Ar
r
I+JL
-M
r
(
44
1~
+
.
Neper/cm
) J.J. Markham, R.T.Beyer and R.B. Lindsay, Absorption of
Sound in Fluids, Rev. Mod. Phys. Vol 23, 4, 1951, p.353 & seq.
T A B L Ei Vill
DIENSIONLESS
For
DISPERSION
r
FUNCTION
= 1.05
I--------------------------------
--- --------------------
.10
.9456
.9725
.0283
.00283
.20
.9570
.9780
.o224
.oo448
.30
.9606
.9800
.02o4
.00612
.50
.9711
.9855
.0147
.00735
.70
.9830
.9915
.0085
.00595
.90
.9947
.9972
.0028
.00252
1.00
1.0000
1.0000
.0000
.00000
1.10
1.0043
1.0022
.0022
.00241
1.20
1.0087
1.0043
.oo43
.oo514
1.30
1.0125
1.0063
.0063
.00814
1.50
1.0188
1.0094
.0093
.01397
1.70
1.0240
1.0120
.0119
.02016
2.00
1.0299
1.0150
.0148
.02956
2.50
1.0363
1.0182
.01784
.o4467
3.00
1.0403
1.0201
.0197
.05910
3.50
1.0429
1.0215
.0210
.07364
4.oo
1.0447
1.0223
.0218
.08728
4.50
1.0459
1.0230
-0225
.10120
5.oo
1.0468
1.0234
.0228
.1140
10.00
1.0499
1.0250
.0244
.2439
r
r
; thus the actual phase "rotation"
with respect to the reference characteristic
qr
1)L
e
is
r
r
Cr
The parameter
L
Ab-~
appears
therefore here as the dimensionless
transmission path lenght ; just as it was present in the attenuation
O
function. In tems of the ratio
,
the response characteristics
will have the form :
r
Application
CO) + .1% Water Vapor
Cr
r
=
-- 8.75
2.70 X 10
240 Kcps/atn
CM/sec
2.70
10 4
2.40
2
10
5
(0
= 1.12
cig
.10
p
cm
at
1 atn
It is good to have orders of magnitude in mind ;
Bbample:
r
for C02
for
* 1.o4
A
r - 30 Kcps
= 2.68 X 10 cm/sec
3 X10
:3.L4
.88
S
Fr
.08
1,o4
sec~1
.275 nepers/cm
r= .088 cm
r= 300 Kcps
o"
.
.88 cm
= 2.75 nepers/cm
It is seen that the absorption increases enormously with increase in
relaxation frequency, the dominant feature of relaxation in the
frequency band considered is a slight phase distortion and a considerable
attenuation.
These two effects may be taken separately into account.
The phase velocity expressed in dimensionless form has to be translated
in phase distortion characteristic corresponding to an equivalent
network :
,f (W~)
=
Lu
C.
I
p
Cr
An ideal transmission path, with linear phase characteristic (
phase velocity) would have a phase characteristic
constant
A
i
0~
since the relax. freq. changes
proportionally to pressure
P
4(1)
(P)
A
-
02 n ( A (1) J, (1))
1
66)
SLf.8.75 X 2r
II
rejer
Icd i-
f-0
(function shown on curve)
rAeLE Vill
phase distortion remains constant with pressure
The absolute value of tis
its maximum value is
2n 60).00735
M .4 radians
This A f will increase linearly with
f
, relaxation frequency
provided f,(l) is being used.
The attenuation coefficient is:
of L. =.
Adt
For
aid
-v-2.1 Nepers
p -
5
ata :
ior
8.75
08
51
>
'n-5 v
I
= 1.05 Nepers for
1) A=10.2 Nepers
SL
=
1/10000
1
In view of the possible application of the paired-echo method
explained in Sect. 2.22 and 2.23 the amplitude and phase response
of the gas path has to be discussed
I
is, for a given gas and a given test gap, function of TL alone
Tn
the multiplication coefficient being
a.
Values of
O
are shown in Table
a.
,
4
which are
the attenuation in Nepers at the relaxation frequency for a path
having
A
wavelenghts. To evaluate the attenuation and the "echo"
amplitudes, one has to know
(a)
Values of
2.
i.e. the relaxation frequency with
respect to the frequencies used,
(b) The value of
Or , which can be estimated very
closely when knowing the gas mixture ( r ) and the
relative path lenght
A
.
The set of curves in fig.
of
-O-
for various values of
28
shows the variation
r, . In chapter 4 a numerical
example will illustrate the procedure.
The study of the dimensionless dispersion curve can
be made in a similar way. We found previously
------------------- ~----------------------------------is the absorption at the relaxation
* With this notation ca.
frequency ur -
T ABL E
DIMENSIONLESS
ATTENUATION
(Nepers)
r2
-
1/r
r 2_1
r
A . lo
A = 25
h
=5o
1.0010
.001999
.0o628
1.0050
.009975
.03133
1.0075
.o14944
.oh694
.23
.57
1.15
1.0100
.019900
.o6251
.31
.77
1.55
1.0150
.029778
.09355
.47
1.16
2.32
1.0200
.039607
.12442
.62
1.55
3.10
1.0250
.043902
.13792
.69
1.72
3.44
1.0300
.059126
.18574
.93
2*32
4.64
1.0350
.068816
.21619
1.08
2.70
5.ho
1.ohoo
.078461
.2h649
1.23
3.08
6.16
1.O45o
.o88o62
.27665
1.38
3.96
7.93
1.0500
.109761
.34482
1.72
4.31
8.62
.03
.37
.78
9 :~
T ABL E
I% 5
AMPLITUDE RESPONSE OF GAS PATH
(RATIOS)
St
.1
.9999
.996
.992
.984
. 960
.920
.5
.982
. 905
.818
.670
.370
.135
.75
.964
.834
.695
.433
.162
.026
1.00
.952
.778
.606
.370
.082
.oo66
.933
.707
.500
.250
.0310
.00095
2.00
.923
.668
.448
.200
.0108
3.00
.913
.635
.4o5
.164
.0100
5.00
.907
.618
.381
.606
.370
10.00
.
905
.0072
.135
.0063
(
.1
The function
--
' /
...
i)
representing the
phase lag with respect to the linear phase characteristic has been
calculated and shown in Table
VIII
and fig.
29
The nature of the phase-distortion components can
then be estimated quite closely by the knowledge of
frequency range
42
A
and the
. (See chapter. 14)
It has to be noticed that whenever AT is becoming
significant, 0 r becomes so large that there is little hope to
transmit signals with the present instrumentation.
~
CHAPTER 3
PROPERTIES OF THE GAS MIXTURES
PRESENT IN THE TEST PATH
3.1
CHH4ICAL COMPOSITION
The gas mixtures used for combustion-chamber
temperature measurements originate in the following :
(a)
Atmosphere air of assumed standard compesition
78o5 c by volume
N2
"t
02
.92)
"t
Rare Gases
.3
"
rn
21.00
2
(b) Residual gases, whose composition depend on the fuel-air
ratio used, F. Assuming total combustion, the concentration in water
vapor and carbon dioxide of the exhaust gases can be computed. Then,
another estimate is necessary to evaluate the proportion of residual
gases, i.e. the fraction which has not yet been extracted from the
cylinder by the exhaust stroke or scavenging process.
From Table
)(
it
is
seen that for a given group of
fuels ( Chain hydrocarbons of the general formula
Cn(H2n + 2
the composition of the exhaust gases is remarkably constant, so that,
in view of the comparatively small proportion ( 5 to 15%) admittedly
remaining in the cylinder will contribute a quasi-constant proportion
of CO2 and H2 0.
TA BLE
A
COMPOSITION OF EXHAUST GASES
(N - Heptane)
F/F
1.66
1.43
N2
72.63
72.85
0
020
1.18
73.20
0
1.00
73.4
0
.87
.77
.71
74.o
74.6
75.1
2.64
.56
5.57
002
12.25
12.33
12.ho
12.h
lo.96
9.77
9.03
H20
13.95
14.07
14.10
14.2
12.4o
11.07
10.30
1.17
.75
.30
0
0
07H6
(c)
0
o
Fuel
In all the firing experiments, fuel premixed with the
inlet air in a mixing tank was admitted in the combustion chamber. Since
heptane was used for a great proportion, our typical fuel
molecule will be represented by open-chain saturated hydrocarbons
(propane, heptane, octane, etc). A tentative estimate of vibration
characteristics of alkane molecules has been made (See Sect.
2,z
).
Since relaxation characteristics are predominantly discussed in this
paper, we rill assume the less favorable case possible where large
proportion of fuel is admitted ( say F/Fcc = 1.66) and 20% of the
exhaust gases remain in the cylinder. In these conditions, our typical
cylinder charge can be determined as follows
N2
02
80% Fresh mixture 76.6
61.25
72.6
14.50
20% Residual Gas
20.3
0
75.75
CO2
16.25
0
0
12.25
16.25
Nitrogen 75.75,
Oxygen
H20
0
0
C7H16
3.1
0
2.45 13.95
2.79 1.17
2.5
2.79
2.48
.28
2.76
by volume
16.25
Carbon Dioxide
2.45
Water vapor
2.79
n-Heptane
2.76
100.00
This represents the volume concentration of the cylinder content
with maximum proportion of residual gases and high fuel-air ratio.
3.2
HEAT CAPACITY LAG IN THE VARIOUS GAS CGIPONETS
3.21
Nitrogen and rare gases.
The proportion of rare gases is so small that their
effect on variations of specific heat will be neglected.
Characteristic temperatures of N2 for rotation and
vibration are known:
5.780 K
(T r.)
(TC)
33800 K
v03
we will therefore consider the rotation always excited and for
the vibration the specific heat given by Einstein's formula for
the maximum temperature reached in the pre-flame region
(20000 F :
1360 0 K).
TC
-.
T
( . 2.58
2,5
3380
=2.58
1360
i
2
6.62
C._2.58
e-
-
.
13.2
(12.2)2
1
=
.59
R
The rotational relaxation frequency is known tc be very
high (See sect. 1.22), values of 220 to 240 Mcps/atm are found in
the literature* so that only vibrational relaxation may occur.
However, vibrational relaxation time is so long ( 10~1 sec) that
even at high pressures it is safe to believe that no vibration
will be actually excited at 2 Mcps. This statement is corroborated
by the fact that no "activation" effect was to be found on
N2
by C02 or H20. The ratio of specific heats for N2 will therefore
1.40 at all the temperatures, since
be accepted as 7/5
and
3.22
Cy =
5/2.
C
7/2
R.
Oxygen
Two essential differences appear with respect to N2 viz.
(a)
The vibrational characteristic temperature is 22600 K, so
at the maximm operating temperature we may have
(TC/
)
2260
T
(b)
10~
1360
1.67
-
Oxygen has a vibrational relaxation time of the order of
sec but is extremely sensitive to activation by water vapor
and by C02
Z for
02
-
02
collisions
02
-
C02
i
02
-
H20
500000
25ooo
hoo
e er-o ic , J hn W,------------------------------ad--F-H
Bo t and
-?.H.
. . Bolt
T.F. Hueter,
Sonics, John "Wiley and Sons,
1954
--
The presence of
5%water
vapor will thus have the same effect as
increasing the number of collisions per second in the ratio
5ooooo
.05
=
62.5
4oo
Relaxation time will be decreased in the same ratio and for very high
pressures, the relaxation domain may be approached. Paragraph 3.1 has
shown that the maximum concentration of H20 with respect to 02 is
2.79
26.25
or 17.1% ; the maximum partial- pressure of Oxygen is 16.25%
of say So atm . 8.1 atm,
ther1 the "equivalent" pressure
which accounts for H20 activation is 5000
1.71
x
8.1 =
1760 atm.
which in turn would correspond to :
.02 sec
1760
thus
5r
89.5
1.12 . 10-5
sec
Kcps
This value is still small compared to the sonic frequency used and
the excitation of the vibration of 02 may be neglected to the same
extent as it was done for
112
The specific heat ratio will thus be
7-2
512
of our experiments.
1.0o0
for all the range
1~
0
3,23
Carbon Dioxide
This gas is wellknown in studies of heat capacity lag ;
historically it was the first to exhibit relaxational dispersion* and
has been subjected to continuous investigation ever since. The
rotation of the linear C02 molecule does not involve any potantial
energy thus the available two degrees of rotational freedom will
contribute (at ordinary temperatures) a rotational specific heat of
(CV)
RD
R
Cal/
Mol
Adjustment to rotational to translational energy is
a high efficiency process
and will therefore be considered as
having very high relaxation frequency, although no satisfactory
numerical evidence could be found of this fact.
The vibrational characteristic temperatures are low ;
the natural frequencies of the vibrational modes are, according to
R.W. Leonard
667,5 cm~1
T K
Mode
955.,5'
J (2)
1388,4
-1
cm~
1980s
1987
2349
cm-
33704
(1)
(
.5,
--------------------- -- --------------------- -------G. Pierce,
1926
See Sect. 1.22
The contribution of the unsymetrical linear vibration can
not be detected in our experiments but the interaction of the
various modes and their harmonics must be considered. This has been
done by R.W. Leonard who sets up a partition function
where
is the "weight" assigned to each level
K
of
characteristic temperature (T.
The vibrational specific heat is found to be
1.85
cal/mol (
.935 R) for all temperatures where the unsymetrical
vibration can be neglected.
For our experiments, the following values are assumed
Cv
Low frequency specific heat
R
R
.935 R
( Translation)
( Rotation)
( Vibration)
3.435 R
Low frequency ratio of specific heats:
4.435
=1.290
3.435
= 2.5 R
High frequency specific heat
(Translation
+
Rotation only)
1.hoo
2.5o
---------------------------------------------------------------D.D. Dennisonl, Phys. Rev. 41i
310
1932
We have then
,=1.0 83
**~~
e 1.042
therefore
,
is the ratio to be used in the
calculations shown in sect. 2.32
The thoughts of workers in the field of relaxation
studies have been subjected to considerable evolution since the
first relaxation times of CO2 vibrations were announced.
It is now apparent that the early experiment were
conducted without due regard to the extraordinary "activation" role
played by water vapor. Experimental evidence indicates 2240 Kcps
shift in the relaxation frequency for each %of water vapor content
in C 02* The reason of this activation is traced back to the chemical
affinity between the C02 and H20 molecules, it has been found that
about 20 to 100 H20 collisions are required to produce the same
energy transfert from vibration to translation than 60000 to 89000
C02 collisions.
The impurity effect of HpO and other contaminants
has been extrapolated with more or less justification to 0% impurity
content and the relaxation frequency accepted for pure C02 is now
around 20 Kcps/atm.
The important conclus ion of this paragraph is that whenever C02 is
present together with a substantial amount of H2 0 (say more than 10%
H2 0/C0 2 ) the C02 vibrations will cease to be inert degrees of freedom
-----------------------------------------L.-)
L4. Bergaann
(z)
10
and will be sonically excited to frequencies considerably higher
than those used in the measuring device. In Sect, 3.1
we have seen
that the H20/002 ratio is actually always (n + 1) for saturated
n
hydrocarbon fuels and will therefore indicate that relaxation
frequencies should correspond to CO2 - H20 collisions rather than
to 002~
2 collisions. Obviously the idea of "impurety" is no more
justified but the approximative relaxation frequency can be
evaluated on basis of R. Walkerts* paper. Assume
002
-
=
H2
collisions
50000
Z0- 002 0
then in.a mixture of 605
50
120
"
+ 40% C02
the relaxation frequency will be
50000,.
20 %eps 60
2
Ncps/ati
lo0
50
This very crude but reasonable approximation shows that the relaxation
domain for this sort of mixture will be around
.2 a-tnospheres for
2 Mcps nominal frequency. It happens that this (partial) pressure
domain is
comprised in the range of engine operation. We mist therefore
accept the fact that Co2 relaxation will be present in the actual
operating conditions,
N.A.C.A.
TN
2537
Nov. 1951
(6-,)
[ II
I
3h
WATER VAPOR
Translational and rotational data are identical to those
explained for C02 . The number of rotational coordinates in now 3
rather than two since the water molecule is known to be non-linear*
In vibrational modes, the small mass of the H atoms make the
frequencies of the symetrical and anti-symetrical vibrations so high
that the respective characteristic temperatures are 52900 K and
54000 k respectively.
The only component of the vibrational specific heat which will
significantly contribute below 13000 K actually present in our engines
is the OH bond, with an infrared spectral line at 15950 cm-1 and
characteristic temperature 2290 0 K. The contribution to the total
specific heat is shown by Slater
to be .79 R.
The ratio of high to low frequency specific heat ratios
is then
/
4L/3
1.332
4.79/3.79
1.053
1*264
a
and hence
V-
1.
1.026
--------- ----------------------------------* G. Herzberg, Infrared and Raman Spectra of polyatomic
molecules. Van Nostrand Co Inc, 1946
J. C. Slater (
1
)
I(, _
3.25
F UEL S
The probabilities of energy exchange and the acoustical
dispersion characteristics are almost entirely unexplored at the
time of writing. The scarce and highly hypothetical data on propane*
locate a great number of natural frequencies in the 720 to 1470 cm~1
region would indicate that a substantial proportion of the vibrational
specific heat is actually exmited at the temperatures of operation.
In iso-octane infrared absorption spectra the predominant features
are maxima at 723, 1470 cm~1 and 2920 cm~1 ; whereas N-heptane has
the same general features plus a marked absorption, band at 930 cm~1
On basis of the above numbers, one would expect
relaxation frequencies for the lowest mode will fall in the
50
to
120 Kcps region. The picture is however completely modified in a
somewhat unpredictable way by the presence of comparatively high
oxygen and water vapor content.
To avoid the uncertainties inherent to the evaluation
of fuel relaxation frequencies in presence of these gases, we will
only attempt to set an upper limit to the error that may be present
when all the probable vibrational modes are completely excited and
when, in addition, the relaxation frequency lies right in the
frequency band used for measurement.
-----------------------------------------Herzberg
(
4
)
In complex molecular structures where widely different vibration
frequencies are simultaneously present the model explained in sect.l.22
indicates that the probability of energy transfer decreases rapidly
with
9
. This indicates that a few low frequency modes, completly
excited, absorb practically all the translational specific heat and
thus the high-frequency modes do practically not participate in sonic
excitation above a few ten cycles.
Experimental evidence corroborates this statement ;
attempts to demonstrate "absorption maxima"l for the varinus molecular
modes failed systematically. Bethe and Teller* confirm this conclusion
and explain the same in a semi-quantitative manner.
Our assumption is that only the lowest 3 modes
(completely excited) will participate in possible relaxation to a
significant extent. Then
C,
3
7
3 +1
rO
)
)
)
)
36
1.17
1.
The values for very high frequencies remain as
previously
4
1.333
so that
1.333
*
1.17
-------------------------------------------------------
*(
3
Actually, these would be "low frequency specific heats" rather than
those corresponding to " 0 frequency values".
3.3
DISPERSION CHARACTERISTICS OF THE TYPICAL
CYLINDER CHARGE
The present paragraph presupposes that there is no
significant proportion of gas molecules which react mutually.
The following composition is assmed : (Sect. 3.1)
%
CP.
Cvo
Cp
Nitrogen
75.75
3.50
2.50
3.50
2.50
0xgen
16.25
3.50
2.50
3.50
2.50
CVv,
C02
2.45
4.435
3.435
3.50
2.50
H0
2
2.79
4.790
3.790
h.oo
3.oo
n-Heptane
2.76
7.oo
6 .oo
L.oo
3.oo
The equivalent specific heats can then be computed
CPO
C
3.655
)
)
2.655 )
=
3.528 )
CP.
)
Cuy.
1 .138
1.3766
=
=
2.528
=.
1.3956
)
r
1.0069
The important conclusion of our chapter Y 3 is that the gas
mixture in the most
defavorable case (large proportion of residual
gases and very rich fuel-air ratio) is such that the ratio of high
I~
and low frequency propagation velocities is about
ir=- 1.007.
This statement implies the following assumptions
(a) No vibrational excitation of 02
(b)
Only 3 vibrational modes of the fuel molecules are excited
(c) The fuel-air ratio and the proportion of residual gases is
within the usual engineering limits.
It is believed that all four of these assumptions
are amply justified both by evidence found in previous works and in
our own experimental investigation.
I
~?
CHAP T E R 4
THEORETICAL AND
4. 1
EXPERIENTAL
RESULTS
SHAPE OF AMPLITUDE AND PHASE DISTORTION CHARACTERISTICS
The results obtained in paragraph 2.23 and 2.32 can now
be applied to the actual gas path as defined and analyzed in
Chapter 3. We will assume the gap lenght to be 1.27 cm and also
that the low-frequency sound velocity is known as a function
of temperature, taking into account all the virial and/or
dissociation terms.
C being of the order of 4.00 . 10
Then
the wavelenght is at
cm/sec
2 Mcps
.
1
4_*
2.10-2
m
2.106
The value of
is then
1.27/ 2.10-2 =
63.5
and the
maximum midband attenuation will be (assuming the midband frequency
being equal to the relaxation frequency)
1.57
x 63.5
1.007
X .0138 -1.37 Nepers
<y
12
dB
Unless the relaxation domain occurs at very low pressures,
this attenuation will not impair the use of the instrument, whose
present signal-to-noise ratio, even at atmospheric pressure, is
around 20 dB.
The phase distortion is characterized by the
function for r*
Cr
1.007, its minimum is obtained for
y, 1.007
((
+4.2
2
)(
2( .J-R+
2.5172
-
=(1-
r) 2hti
2
r)
2( 1.2638)
o99588
6.28 X 63.5
-
1.25 X 2.0138
1
1.004131
2.5276
Thus
12
.
x .04131
2
.
.823 radian
As a first approximation, this value can be accepted as the
first spectral term of the phase distortion curve (b1 ) and thus
the "echo" amplitudes would be
a
J1 (.823)
.369
2 J2 (.823)
.077
2 J 3 (.823)
Compared to the main signal
=
.011
li
J
(.823)
850
=
Thus in presence of relaxation, the signal will be preceded by
an echo (J1 ) having the same envelope shape and having
i
.369/.850
=
%
of the main peak signal value.
The absence of such echos proves that the dispersion
for r = 1.007
over the frequency band is smaller than the I
The error on temperature interpretation is definitely much
smaller than the one resulting from a velocity dispersion of
7/1000. It is believed that the absence of "echoes" can be
estimated whenever their amplitude reaches 15% of the main signal;
Arg
referring to the first "echo" this would give
J, (.1483)
AT .3 radians which in turn can be translated in r. The result
is of the order of
rmin = 1.002. The minimum velocity dispersion
susceptible of being detected is thus
.2% meaning
.h% measurement
error on temperature.
r c1.002
The attenuation for
&
SL >>
= 63.5
1
would be
ar
6.28
x 63.5 x .004
:ta.
.80 Nepers
and plainly perceptible to the operator. The closeness of relaxation
domain, whether signifying large or small ris is always detectable
by marked attenuation. The echo amplitudes and locations can
always be determined by the method explained in 2.32.
If, in particular, the sonic excitation level is set in such a
way that the (dispersionless) signal-to-noise ratio is just at
the limit that can be unambiguously detected, any marked increase
in attenuation will be followed by disappearance of the signal
on the CR screen thus making the observer aware of the necessity
of closer examination.
4.2
NATURE OF THE EXPERIMENTAL CONFIH4ATION
4.21
Engine Experiments
During the years of 1952, 1953 and 1954 the present
model of sonic temperature measuring instrument and its various
predecessors have been in operational use in the Sloan Laboratories
for Automotive and Aircraft Engines. It is the remarkably small
scattering of the results around the average values which prompted
the present investigation to give some rational basis for the degree
of confidence vested in the procedure.
The problem, in devising a series of confirmatory
experiments is to find a suitable physical surrounding so that
the properties of the gas could be determined to a reasonably high
degree of accuracy. The engine combustion chamber was obviously
not the place for this kind of experiments ; the comparatively
large volume of the cylinder, the presence of the lubricating
oils and combustion deposits on the wall made the creation of a
chemically defined atmosphere unpractical.
The experimental program called in consequence for
the construction of a special test chamber of small volume where
the desired gases could be mixed in the necessary proportions.
4.22
Principle and Justification of the Experimental Procedure.
The purpose of the experiments was to demonstrate
numerically the validity of the approach taken in chapter 2. Of
particular interest was to obtain points on the dispersion curve
of mixtures of known characteristics ; to detect the presence of
"echoes" (although measurement of echo amplitude was hopeless
under the circumstances) and, to show, directly, the accuracy
of temperature measurements.
Since the instrnent has a fixed frequency range
of .45 to 4.3 Mcps, the relaxation domain was scanned by means
of pressure variations and the gas relaxation was brought into the
available pressure range by adding controlled (small) proportions
of impureties.
Carbon Dioxide and Water vapor were chosen as
primary media, but other gases and activators have also been tried.
Nitrogen, Oxygen and propane were checked for relaxational
absorption and dispersion.
NOTE :
The great number of measurements available made the
selection of the numerical results necessary. The run number shown
on the following pages is not a serial number but refers to the
logbook page where the results were reported. Each "run" was
followed by about
results obtained.
h
to
5
check runs intended to duplicate the
116i
4.23
INTERPRETATION OF EXPERJIMTS
RUN TYPE
57
Check with atmospheric air to verify internal delay
constancy. Tn the various air temperature measurements the internal
delay was found to vary by almost 1.2 ysec as a function of
operating time. The Transmitter and CR Scope warm-up seem to be
the main factors in this variation since the temperature variations
in the coupling bars were essentially small. The careful calibration
with reference gas such as N2 or air is a must before each
measurement series, but at least once every
RUN TYPE
hour.
58
Measurement on B.D. C02, without additional drying.
This series of Measurements was made among the first and by
inadvertance about 180 psi N2 was left separated only by a needle
valve from the mixing tank. This would explain the extraordinary
high velocity values which have not been found again in subsequent
duplications.
RUN TYPE 59
Measurement on Water Pumped Nitrogen. This run has been
duplicated a great number of times both to check the instrument
11v
R U N # 57
24.30 bar temp.
Gage lenght determination:
Transmitter bar :
.449 in
Gap lenght:
.44
.009
Air temperature :
2.891
in
2.893
in
5.784
in
.449
5.784
.
1.1176
229
cm
1.1405
cm
24.30 C
Test Chanber temperature :
11412 yd
6.233
Total distance:
-,-w
.975
23
Cu-Const
mV
75,2 =
77M.
Velocity:
69.55
C2
24.50 C
273.1
X 8.31 . 107 X 297.6
28.87 = 12.00.108
,.4o2
2
sec
800 R Sweep
M z 21
X 32 + 79
X 28
=
672
2215
2887
3.46 . 104
=
C
cm/sec
Formula given by Bergmann*
c,
a
331
C :
7ii
Gas transit time:
6o x 24.5
s approx.
-0"I"c
1 t 24.5
1
(1.086)2
331.3
14.7
34i60
1.044
1.1405 m
3.46.102
-
32.95
ysec
69.55
Instrument Delay : 32.95
36.60
2,-)-L.-Bergmann-"-Der-Ultraschall"-
*-(-
*
(
2
) L. Bergmann 11Der UltraschaJ-1"
Ed.6-1952-------
Ed.6 1954,
p 502
psec
S118
R U N # 58 Calibration
Gap lenght:
Total measurement 6.172 in
Coupling Bars
5.784 in
.388 in.
(glyploted gaskets)
.388 in. . 98 S7 cm
7.620 m
Air temperature : 22.60 C (Room)
22.10 C
.875 mv
T.C. Reading
SV in Air
.6 X 22.1
331.1
13.25
344.357m/s
Transit time in gap
2.032
.203
9.57-m
3.44 x 104 cm/ sec
=
.98570
3. W43.10'4
Scope reading : 10740 yds
-+
=
sec
2.0
65.48
28070
761.2 m Hg .
po
Gage
p
B
1.018 NB
gage
MB
MBPabs
3678
( Gaskets have been
jisec
changed since Run
Scope
yds , P
tI
-L
t
57)
Obs.
cm/sec
33.23
104
2.966
Aver.
72.29
33.51
2.942
Temp.
11861
72.32
33.54
2.938
reading
2.26
11860
72.32
33.54
2.938
.87 mV
1.790
2.81
11872
72.39
33.61
2.938
. 20.7 0C
40.5
2.750
3.81
11905
72.59
33.81
2.915
52.0
3.680
4.70
1910
72.62
33.84
2.912
8.00
9.02
11990
73.11
34.33
1.02
11810
72.01
.930
1.95
n855
1.032
2.o5
18.0
1.240
26.o
C
70
Theoretical
Value
t = 9 . 20o70
c 2 = 1.292 X 8.31 X 293.8
~7
=
C
7.167 X 108 cm/sec
5292
x 104 cm/sec
20*7
273.1
2.67
(Co)
2.871
M = 44.010
= 1.292
37.2
32.0
460
44.01
11.61
Value given by Keenan:
R
2 93.8 *F
878.4 ft/sec = 265.97
1
16E
=
oo6o9761
RUN
#
59
WATER PUMPED NITROGEN
Gage lenght
.388"
Temp = .86o mV
Pgage
gage
MB
21.70 C
220 C
Tair
Psi/
(AIFCO)
Pabs
Scope
MB
y AS
t
C
cv%Is
-
1.020
10775
65.72
28.55
3.452 . 0
10
.6895
1.709
10770
65.69
28.52
3.456
15
1.034
2.054
2o
1.379
2.399
30
2.068
3.088
140
2*758
3.778
10765
65.66
28.49
3.459
50.5
3.482
4.502
61.o
4*206
5.226
10760
65.63
28 .46
3.463
70
4.826
5.846
8o
5.516
6.536
91.0
6.274
7.294
10755
65.60
28.43
3.467
100
6.895
7.915
120
8.274
9.294
28.4o
3.1470
C
10750
reproducibility and behavior of N2 at high densities. The amplitude
of the receiving signal increases markedly with pressure, before
saturation of the amplifier, the variationship is almost
Signal peak
r
0
p
PO
The change in C is in the direction and of the
magnitude predicted by the real-gas considerations.
MUN TYPE 59 B
Check on propagation velocity variation with pressure.
Same remarks as for Run 59.
RUN T'PE 60
Measurement on B.D. CO2 dried through P2 35. The
relaxation frequency being around 25 to 30 Kcps/atn (to account for
the small amount of water vapor possibly present). Even at the
highest pressure used,
4, - 12 X 30 Kcps = 360 Kcps, and no
significant relaxational dispersion is to be expected.
The abeorption is however easier to detect, at
l.15 atm.
,-a30
Kcps
A
-
2.78 -104
30
.
.94 cm
10
.9857
.94
a,
= .~23 Nepers
for
rz 1.04
112
RUN
# 59
CCMPRESSED AIR
(Lab Distribution)
T =21.4 0 c
Gap lenght = .9857 cm
E.A. Delay
37.17
Pmy
-
.835
21.10 C
psec
Scope
1 3
10738
t,
65 .5
65951
t
28 .3
28.34
C
.8cm
/ Sec
3*478,104
10740
1074o
1.402 X 8.312 X 294.2
28.87
calc. value
-
With the atmospheric air
Scope
65.51
11.875 x 108
=
sc2
4
C = 3.446 X 10
eM/see
10765 yds
65.66 pspc
C = 3.459 X lo+ C, I se
- 131
13
-
.3%4ige
higher
.37%
t: 28.49
s
I 2~
R U N 60
B.D.
Gage lenght
Calibration at
.368
21.80 C
Calibration : Air
Ca
a
331.1
13.08
CO2
-->
.850
10760 yds
10760 yds
-4
3.8.
mV
-P
65.63
28.64
36.99
Dry through
P2 05
21.80 C
psec
(The hygrometric degree of
atmospheric air must be checked)
-----------------------------------------------------------------------psi
P
P
Scope
t
t
C
gage
g MB
abs MB
C/sec.1ld
cm1
-----------------------!--------------------2
.138
1.158
1186o
72.35
35.36
2.787
15
1.034
2.054
885
72.49
35.50
2.777
24
1.655
2.675
895
72.56
35.57
2.771
31
2.137
3.157
910
72.65
35.66
2.764
44
3.033
4.053
920
72.71
35.72
2.759
55
3.792
4.812
935
72.80
35.81
2.752
74
5.102
6.122
950
72.90
35.91
2.745
88
6.067
7.087
975
73.04
36.05
2.734
loh
7.170
8.190
980
73.08
36.09
2.731
122
8.411
9.431
12.oo8
73.24
36.25
2.719
128
8.825
9.845
010
73.26
36.27
2.717
140
9.653
10.673
025
73.35
36.36
2.711
150
10.342
11.362
038
73.43
36.44
2.705
160
11.032
12.052
055
73.53
36.54
2.698 Amp
r
12.0 atm.
(
6 W .9857
.o8
Here St >I
,
Ar
360 Kcps
o8 cm
12.25
l22
erz
..j
thus the attenuation will be almost constant over
the band, and equal to
Thus we would have
2
0.,,*
S
(1-15 atm)
S
(12 atm)
W.2.75
Neper
-+
16
12
On account of the density change, the signal would increase as
10.14
1 .1
62%. The logbook indicates
10.4, so the net loss is 10.4
16
that the signal, which was reasonably observable around 80 psi ( 6 atm.
abs.) and vanishes at 12.5 atm. This is in substantial agreement with
the above findigs, since the exact water vapor content is unknown.
( .05%would make 4 , =100 Kcps/atm.)
RUN 60 B
.14,; water vapor was mixed to
Check on CO2 + H2 0
the pre-dried C02.
(Frcm Knudsen and Fricke' s data)
+ 300
Kcpsfatm
. 104
If276
3 10 5
sc
sec~1
.092 cm
at
1 atm
.0244 cm
at
3.75 atm
12 ':~
R U N 60 B
co
16o
psi
CO2
Concentration
+.14%
( B.D. Dryer)
and
H2 0
13 mm
(number of
molecules)
13
31.77 X 160
T
psi
PgageMB
H2 0
Pabs
MB
20.20 C
Scope
lots
.855
-+
t
I es
mv
C
cm/see
.4137
1.433
11865
72.37
35.59
2.770 - 10
.758
1.778
11880
72.46
35.68
2.763
1.516
2.536
11900
72.59
35.81
2.752
2.000
3.020
11915
72.68
35.90
2.746
2.758
3.778
11935
72.80
36.02
2.736
334 + 12 -
Cal : Nitrogen at 75 psi
Reading
10700
M
for
65.27
3.46 .
14
cm/sec
.388" gap
y
Pec
ta
28.49 psec
te
36.78 ysec
.
.9857
l10 7
A=
.9857
= 40.5
S.-r
here J2's
.092
*024
are of the order of
a,
, 132 Neper
a, -* 4.96 Neper
1, thus the relative signal strenghts
would be
1
2.54 Neper
12,6
ratio
and taking in account the pressure ratio,
3.75
12.6
*3
The amplitudes observed bear out this conclusion to a remarkable
extent ; the signal being perfectly observable (about 3/10 inches
peak to peak) at atmospheric pressure, and disappears around 4 atn.
abs. A REMARKABLE FACT IS THE APPARITION OF ECHOES at high pressures,
demonstrating in an illuminating way the paired-echo method.
It is unfortunate that the levels available do
not allow actual amplitude measurements on the echoes ; these latter
being
partially hidden by the noise.
The dispersion, as shown on Fig.
the one expected for the above values of
r
3o
and .A .
is perfectly
12
RUN 61 B
f
Check on C02 +
H2 0 ; .0225 H20 was admitted to pre-dried
002. The effect on dispersion is visible and corresponds to
predictions. Curve on Fig. 3o
shows the comparison with Runs 60,
60 B and the difference is marked although too small to be checked
numerically. The relaxation frequency is
r
At
2.70 . 10h
6* 10
= .45 cm
6 .104
12.0 atm
Ar .0378 cm
60 Kcps
thus
t
AZ
. -9857
t28
.45
A.=
21
Or 3.45 N
27
Signal Intensity ratio :
4r -27 N
12
X e-3.18
This amplitude ratio is experimertally demonstrated to a remarkable
degree of accuracy.
RUN 63 B
Check on B.D. Pre-dried CO2 * Results are identical to Run 60,
same remarks and conclusions are to be drawn.
RUN 62 B 2
Test on 002 + H2 0 in Nitrogen ; a mixture of 170 psi C02
saturated with water vapor at 19.2 C was prepared and carefully mixed
to N2 in variable proportions. From previous tests the relaxation
R U NS
334
11.o4
.6 x 18.4
345.04
EA
36.94 psec
WATER
Concentr.
mM/Hg
0/0
-----
2 m
.o22%
Pgage
--
CN = 3.45 . 1+4 cm/sec
tN = .9857 =
lS
Pg MB
.730 mV -- 18.40 C
65.51 psec
1074o vta
0 psi Gage
0O02+ H20
19.20 C
.388 gap
Reference Nitrogen
Velocity in N2
61B
p
Scope
mS
-------------------------------------
28.57
psec
t2
y4'-A
C
t
"'
e* As
0
-
1.020
11870
72.41
35.47
2.779. (o
10
.6895
1.709
890
72.53
35.59
2.769
20
1.379
2.399
905
72.62
35.68
2.762
30
2.068
3.088
930
72.77
35.83
2.751
40
2.758
3.778
940
72.83
35.89
2.746
55
3.792
4.812
955
72.92
35.98
2.739
70
4.826
5.846
975
73.0A
36.10
2.730
6o
4.137
5.157
990
73.13
36.19
2.723
80
5.116
6.137
12000
73.20
36.26
2.718
100
6.895
7.915
020
73.32
36.38
2.709
120
8.274
9.294
oho
73.44
36.5o
2.700
lo
9.653 1o.673
070
73.62
36.68
2.687
150
10.342 11.362
100
73.81
36.87
2.673
160
11.032 12.052
Dried
Reference as per
Gap lenght
63A
.9857 cm
R UN
63 B
B.D.
002
EA DMAY
37.17 psec
Temperature
Air
mV
Gage Pres.
psi
0
PMB
Pabs MB
Scos
t,
19.40 c
.760
t
C
cm/sec
0
1.020
11890
72.53
35.36
2.788
15
1.334
2.354
11900
72.59
35.42
2.782
26
1.792
2.812
11930
72.77
35.60
2.769
28
1.930
2.950
-
-
-
-
ho
2.758
3.778
-
-
-
-
5D
3.448
4.467
11950
72.89
35.72
2.759
60
4.137
5157
11970
73.01
35.84
2.75o
80
5.516
6.536
11990
73.14
35.97
2.740
105
7.323
8.343
12020
73.32
36.15
2.727
126
8.688
9.7o8
12040
73.44
36.27
2.718
180
12.441
13.461
12110
73.87
36.70
2.685
I
RU N 62
B 2
VARIABLE PROPORTION OF
Reference N2
:
19.20 C
N Velocity
10740 y
Scope
334
CN
-
1n.5
345.5
.6
C02+ H20 in NITROGEN
107h0 for
19,2
.9857
t
=
cm checked
.9857
3.455' . 104
-285p
iS
65.51 psec
28.53
36,98
002 +-1H20
E.A.
sat, at 170 psi/19.2 0 C
(has been checked for
-------------------------------------------------------
r
4-------
C 10
t
ti
Scope
Concentr.
C02 H20
N2
pres. psi
psi
o02+H20/N2 yds
psec
psec cm/sec
--- gage--------------------------------------------------------
X0
0
10740
65.51
28.53
3.455
0
2
.oh
10780
65.75
28.77
3.426
3.85
5
.11
lo84o
66.12
29.14
3.383
10.00
8.5
.19
10910
66.55
29.57
3.333
15.97
.34
11010
67.16
30.18
3.266
25.38
0
10735
65.48
28.50
3.458
20
.43
11110
67.77
30.79
3.201
30.07
32
.68
11320
69.o5
32.07
3.073
4o.48
41
.82
11460
69.90
32.92
2.991k
45.o6
60
1.36
unread.
50
1.08
1565
70.52
33.54
2.938
51.93
16
0
0
57.69
4o
.85
n14o
69.78
32.80
3.005
45.97
30
.62
11280
68.80
31.82
3.098
38.27
9
.20
10930
66.67
29.69
3.320
16.67
0
0
10740
65.51
28.53
3.455
0
frequency of CO2 was known. On figure
31
the results are plotted
in comparison with the ones calculated by assuming Bergmannt s
proportional mixture formula and using values of sound velocity
in CO2 found in Run 60 B, (Read from curve)
IT IS REARKABLE TO FIND THE CALCULATED POINTS more
scattered around the faired curve than those obtained experimentally.
(No attempt was made to read curve 60 B to better than normal
accuracy for experimental curves.)
RUN 63 A
Calibration run with Nitrogen for 63 B.
Results check most
TEST ON CO
accurately with previously obtained values.
Methanol
(No 70)
As known, in order to ensure perfect temperature
uniformity in the test chamber, it was decided to operate in equilibrium
with the ambient temperature. This limited obviously the maximum H2 0
content to the saturation at about
2000 or
.1% when using the
maximum C02 pressure (170 psi).
Methanol was used to overcome this difficulty. The
following is a typical check on influence of methanol:
I
# 63 A
RU N
Gage lenght:
T c 19.4 0 c
*9857 cm
my
1.,400 X 8.31 X 292.4 _ 12.142 .108
Velocity in N =
25.016
t
t1lpsec
psi
.760
.
19.30
273.1
2-92.4
-
C= 3.485. 10
psec
3.485 . 104 cm/sec
28.28
10730 fcs
.9857
=
28.28
psec
E.A.
10
37.17
(colder than
a 53 )
155
10725
180
10710
65.42
3.485 .*104
(V
28.25
(0
(
28.16
65.33
- 45 psi
C ( V15
3.489
*
10
( V180
3.500
.
104
273.1 }
Corrected to 00 C:
Cc
3.485 x (-22.)
.96478
( .930811)2
_-o 3.*36 . 104
Value given by BergmannA
. 104
3.34
Equivalents of yards in
c
cm7sec
ysec
:C
X
...- 2.9978 . 10
1 yd :
=
163.921 yds
psec
=
.006100
yds
(
C-lMSCC
2
AC
) p.516
Psec/yd
t)
cI/ tsec
91.440183 cm
cm/sec
ysec
3 d)
L
Partial pressure of sat.
CH OH
:
3
70 mm mercury
140 psi
Pressure of CO
2
70 mm
Concentration :
.77%
or
.0077
8090
.,
z
(Knudsen
420 Kcps/atm
and Fricke)
Aiery weak signal was obtained at 1 atm (about 6 dB above noise) and
then, by increasing pressure slowly, the signal idisappeared
completely at
RUN 74
2.2
atmospheres.
GAP
TESTS ON CS2
.3881"
E.A.
36.98 Jisec
The low pressure of this liquid at 200 C compelled the
use of CS2 and Nitrogen mixture, in order to have pressures accessible
to our measurements .
(a)
Dry
CS2 at 300 mn
10
'
psig
-
2
N2
1300 mm
18.75%
Dry CS2
H20
N2
at
N
N
300 mm mercury
18 rm
1300
N2
11130 yds
Scope reading
(b)
20.10 C
mercury
mm
20.10 C
81. 25
r
V
The H20 concentration with respect to CS2 is very high:
5.666% thus
the relaxation frequency is around
18/318
13.2 Mcps/atm*
or at 300 m Hg, at 5.23 Mcps.
Scope reading
11180 yds
at 20.20 C
(very small signal, barely readable)
The velocities for the mixture are thus obtained as follows
t1 (a)
67.89
pisec
tg(a)
67.89
-
C (a)
.9857
3.189 . 104
30.91 usec
ti (b)
68.20
t (b)
68.20 - 36.98
C (b)
cm/sec
Aa
.9857
31.22
31.22
ysec
cm/sec
3.14 7 . 104
Assuming the values given by Angona*
1.389
30.91 Psec
36.98
we have
(C-5j2)
r. = 1.225
(C Sa )
M (cs.) =76
and the mixture velocities are
F
C
C v;
[X
(a)
~
:5I
81.25
.
3.5 + 18.75 .3.63
81.25
.
2.5
'
1
R
A
-+-
18.75 .2.63
075R)2
.03775SRT)
----- --------------------------------------------------------------------------28 -+1875 X 76
8125 X8l.2~x8.~1RT
) Knudsen and Fricke
(
(
1
=76(
t
(b)
L 81.25
[81.25
X 3.5 + 18.75 x 5.5
X 2.5 +- 18.75 x 4.5
1
37.00
1
( .03642
C-(o
--
( 3.775
c4le)
RT)2
)
thus
(1.0365)2
3.642
1.0182
whereas the experimental value is
3.189
m1.0133
3.1 4 7
This agreement demonstrates that the 2 Mcps sound pulse does excite
the vibrational modes of CS2
when activated by water vapor.
NOTE
A rapid check was made in the last days of the program
on COS but they were discontinued since the equipment was not designed
to handle toxic gases. So no accurate date are available, but the
sound velocity was measured once ( found 2.305 . 104 cm/sec at 200 C)
and attenuation so strong that only at the lowest pressures could a
signal be obtained. Additional work will be done when the equipment
is perfected.
7
C HA P T E R
5
CONCLUSIONS
5.1
CRITICAL DISCUSSION OF RESULTS
It is believed that this program conclusively established
the reasons for the ±
1% scattering of results claimed as an
experimental fact heretofore .
The following factors contribute to this belief
(a)
The reduction on the 'Ieading edge error" by increase of
the signal-to-noise ratio ; at the present time the signal
phase is determined to better than .2 Ysec
and
(b)
.05 ysec
(for low densities)
for high densities.
The calculations and results on phase distortion caused
by classical and relaxational dispersion show that in the
worst possible experimental case (high fuel-air ratio, high
proportion of residual gases, simultaneous relaxation in
water vapor, CO2 and fuel molecules) the error on velocity
would be only
.3% and even then, the instrument would
detect the presence of dispersion by anomalous signal
envelope shape and by sharp reduction in the signal-to-noise
ratio.
Care should be exercised to notice that the figure
of reproducibility does not imply accuracy on the absolute values
of the temperature. However due to the calibration process, the
instrument measures the ratio of the sound velocity in the gas to
the one in the reference gas to at least 4 .51 accuracy and the
errors that will affect the interpretation in terms of temperature
can be taken into account. (Real-gas properties and dissociation)
Io
attempt was made in this paper to discuss
chemically reactive media, non-uniform temperature distributions
although some effort was devoted in this direction".
The experimental side of the instrument development
has been described thoroughly in other publications
(
( 48
) and
43 ), so this paper was chiefly devoted to elucidate the
transient pulse analysis in relaxing media and the chemicallycontrolled-medium experiments pertaining thereto.
The experimental procedure used can be criticized
on the grounds that no measurement was performed in steady state
gases duplicating acurately the pressures and temperatures present
in the engine cylinder.
The answer to this is that as of now, nobody really
knows exactly what are the molecules and unstable radicals
--------------------------------------------------* The experimental program toward the latter phase is underway at
the Sloan Laboratories at the Massachusetts institute of Technology.
instantaneously present
in the combustion chamber. The present
heat-capacity lag theory however is sufficiently well established
and confirmed experimentally to give to our "gas analog"
experiments a reasonable degree of reliability. The choice of
water activated CO2 was practically imposed by the (understandable)
desire of keeping the auxiliary instrumentation cost to a
reasonable minimum. Thus the use of a fixed nominal-frequency
sound system (the one which has also been in actual use on the
engine) at 2 Mcps was practically imposed. Temperature and pressure
ranges of our experiments (outside of the engine) were also severely
limited by practical considerations.
opinion
It is the author's considered/that the C02 - H2 C
series of relaxation experiments are both pertinent and conclusive
with respect to the calculation method given in Chapter 2.
5.2 SUGGESTED PROGRAM FOR FURTHER WORK
5.21 Study of Macroscopic Gas Properties
The interpretation of temperature measurements for the
purpose of engine studies could be further improved by having
some degree of knowledge of the temperature profiles and the velocity
gradients present in the path. It has also been suggested that
r.m.s. turbulence numbers could be statistically measured by
the effective sound attenuation in the path.
5.22 Behavior of reactive Mixtures.
It is known that reactive gas mixtures do have
different compressibilities for infinitesimal compression and
expansion and therefore will exhibit relaxation times similar
to those found in the heat capacity lag (
43
). The use of ultrasonic
wideband pulse transmission to detect reaction rates in rapid
chemical phenomena appears to be a very promising possibility. The
theoretical background is already explored for steady-state sonic
excitation, so it is believed that a little supplementary effort
will only be necessary to transpose our findings on pulse
interpretation inbo the domain of chemical reactions.
5.23
Multipoint Cycle Analysis
The present state of the sonic temperature measurement
method is such that the greatest factor of uncertainty is the
fluctuation in the cycle-to-noise results imposed by the "stroboscopic"
scanning method.
Considerable additional experience could be gained in the
study of engine knock for example, if particular cycles could be
analyzed in details, i.e. have at leat 20 to 30 measurement points
6n the compression stroke.
Steps are being taken to make use of the "discrete
information output" feature of the sound-velocity device. The
ellapsed-time measurement (between transmitted and received signal)
would be ideally digitalized and recorded at high speed in view of
the eventual decoding and processing. The instrument performing the
above is at present in the block diagram stage at the Sloan
Laboratories.
5.3 POSSIBLE USES OF SOUND VELOCITY for TI4PERATURE MEASUREI4ENT
The preliminary results obtained in the steady-flow
tunnel experiments indicate that there is no difficulty in obtaining
perfectly readable signals at transverse velocities of 150 to 200
ft/sec, provided the flow is reasonably non-turbulent.
The possible uses suggested by this experimental fact
areThumerous, among them the following are well worth of expending
some design effort:
/
(a )
Ehaust gas temperature measurements
(b)
Temperature measurements in gases containing small
droplets in suspension.
(c)
Flow velocity measurement by using oblique propagation
paths with respect to the flow axis.
All these, and many other methods will naturally
benefit from the experience gained with the present instrument.
In this respect, a non-dimensional study of the main design
factors appears desirable.
BIBLIOGRAPHY
1.
ANGONA F.A., Absorption of Sound in Gas Mixtures,
J.A.S.A. 25 - 6, 1116-1122, (1953)
2.
BERGMANN L., DER ULTRASCHALL, Ed. 6, 502, (1954)
3.
BETHE H.A.& TELLER, E., Deviations from Thermal Equilibrium
in Shock Waves,
Ballistic Research Lab. Aberdeen Proving Ground, (1942)
4.
BIRD R.B. & SPOT E.L., The virial equation of state,
Univ. of Wisconsin, N.R.L. Report CM
5.
BOLT R.H. & HUETER T.F.,
(1950)
Sonics,
J. Wiley and Sons Inc. New-York
6.
599
(1954)
BOURGIN D.G., Sound Absorption and Velocity in Mixtures,
Phys. Rev. 50 - h, 355-369 (1936)
7.
BUSCHMANN K.F. & SCHAFER K., Collision Excitation of Intramolecular
Vibrations in Gases and Gas Mixtures.
Zeitschr. phys. Chemie, Sec.B 50-1/2
8.
CASTELLAN G.W. & HULBURT H.M., Interchange of Translational and
Vibrational Energy in an Asymetric
Molecular Potential Field.
Jour. Chem. Phys. 18- 3
9.
312-322
(1950)
CHEN S.K., BECK N.J. UYCHARA O.A. &MYERS P.S.
S.A.E. Trans.
10.
73-99 (1941)
503
(1954)
C.R.C. Four Proposed Methods of Measuring end-gas properties.
N.Y.
(1953)
11.
DENNISON D.M. & JOHNSTON M.,
Interaction between Vibration and
Rotation for Symmetrical Molecules)
Phys. Rev.
47 - 1 93-94 (1935)
12.
ELMORE and SANDS,
13.
EUCKEN A. & BECKER R., Collision Excitation of Intramolecular
Vibrations in Gases and Gas Mixtures on the
basis of Sound Dispersion Measurements.
Electronics,
Ed. I, 145.
Zeitschr. phys. Chemie Sect.B, 27 - 3/4 235-262 (1934)
14.
EUCKEN A. & BECKER R., Transition for Translation to Vibration
Energy in the Collision of Molecules of
Different Types on the Basis of Sound
Dispersion Measurements.
Zeitschr. phys. Chemie Sect.B, 20 - 5/6
15.
EUCKEN A. & AYBAR S., Collision 11citation of Intramolecular
Vibrations in Gases and Gas Mixtures.
Zeitschr. phys. Chemie Sect.B, 46 - 4
16.
195-211
(1940)
EUCKEN A. & KUCHLER L. Activation of Intramolecular Vibrations by
Collisions.
Phys. Zeitschr. 39 - 23/24,
17.
467-474 (1933)
831-835
(1938)
FRANOCK J. & EUCKEN A., Conversion of Translational Energy into
Vibrational Energy in Molecular Collision
Processes.
Zeitschr. phys. Chemie Sect B, 2o - 5/6,
18.
460-466 (1933)
FRICKE E.F., Absorption of Sound in Five Triatamic Gases
Jour. Acous. Soc.
19.
GARDNER & BARNES,
20.
GERSHINOWITZ, H.
12 - 2.
245-254
(1940)
Transients in Linear System, No 2.121, p.346
The Transfer of Energy in Molecular Systems.
Jour. Chem. Phys.
5
- 1
54-59
(1937)
4
21.
HERBERT S. GREEN,
The Molecular Theory of Fluids,
Interscience Publ. N.Y. (1952)
22.
GRIFFITH, W.C., Relaxation Times for Exchange of Vibrational
Energy in Molecules,
PH.D. Thesis (1949)
Vol I & II.
23.
GUILLEKIN E.A., Communication Networks,
24.
HERZBERG G., Infrared and Raman Spectra of Polyatomic Molecules,
(1946)
Van Nostrand Co Inc.
25.
HERZFELD K.F. & RICE F.0., Dispersion and Absorption of
Frequency Sound,
Phys. Rev.
26.
31
-
4,
High
(1928)
691-695
HERZFELD K.F.& SCHWARTZ R.N. & al., Calculation of Vibrational
Relaxation Time in Gases,
Jour. Chem. Phys.
20 - 10,
159-1 - 159-9 (1952)
27.
HIDNERT, Scientific Paper No 410, U.S. Bureau of Standards (1921)
28.
HIRSCHFELDER J.0. & GIRTISS C.F., Thermodynamic Properties of Air,
N.R.L. Report
29.
CM 518, (1948)
HUBBARD, J.C., Ultrasonics, a Survey,
Amer. Jour. Phys.
30.
KANTROWITZ A.,
8 - h,
(1940)
Heat Capacity Lag in Gas Dynamics.
Jour. Chem. Phys. 14 - 3,
31.
207-221
KANTROWITZ A. & HUBER P.W.,
150-164
(1946)
Heat Capacity Lag in Various Gases.
Jour. Chem. Phys. l5 - 5,
275-284
(1947)
32/
KEENAN J.H. & KAYE J. Gas Tables, N.Y. Wiley and Sons (1945)
33.
KENNARD E.H.,
Kinetic Theory of Gases, Mc Graw Hill, N.Y. p.113 (1938)
34.
KNESER H.O. & KNUDSEN V.0., Ann der Physic,
35.
KNUDSEN V.0. & FRICKE E.F., Absorption of Sound in C02 and other Gases
21 , 628,
Jour. Acous. Soc. Am. 10 - 2,
36.
89-97 (1938)
KNUDSEV
V.0. & FRICKE E.F., Absorption of Sound in C02 and in CS2,
Containing Added Impurities.
Jour. Acous.Soc.Am. 12 - 2, 244-254
37.
(1935)
(1940)
KUCHLER L. Collision Excitation of Intramolecular Vibrations in
Gases and Gas Mixtures.
Zeitschr. phys. Chemie sec.B 41 - 3, 199-214 (1938)
38.
LANDAU L. & TELLER E., On the theory of Sound Dispersion,
Phys. Zeitschr. 10 -
34-43
20 - pp 1024, (1952)
39.
LENNARD-JONES L.E., J. Chem. Phys.
40.
LEONARD R.W., Absorption of Sound in 002.
Jour. Acous. Soc.Am
41.
(1936)
(1940)
12 - 2, 241-244
LUKASIK S.J., Calculation of Acoustical Relaxation Times,
Seminar M.I.T. Acoust. Lab. (2.23.1955) unpublished.
42.
NASON W.P. Piezo ELectric Crystals and Wave Filters, p.480
43.
MANES M., Relationship between Kinetics and Acoustic Phenomena
in Equilibrium Systems.
T. Chem.Phys.
44.
21 - 10,
1791-1796
(1953)
MARKHAM JJ. BEYER R.T. & LINDSAY R.B. Absorption of Sound in Fluids.
Rev. of Mod. Physics
23 - 4
353-411
(1951)
(1943)
45.
PIELMEIER W.H. & BYERS W.H., J.A.S.A. 15
46.
RHODES J.E. Jr, Velocity of Sound in Hydrogen when Rotational
Degrees of Freedom fail to be excited.
Phys. Rev.
70 - 11,12
-
932-938
p.17
(1946)
London(1887)
Theory of Sound,
47.
Lord RAYLEIGH,
48.
RONA T.P. Measurement of Ultrasonic Propagation Velocity in Gases.
M.I.T. Master's Thesis, E.E.
49.
SCIWARTZ R.A SLAWSKY Z.I. & HERZFELD K.F., Calculation of
Vibrational Relaxation Times
(1951)
20 - 10
Jour. Chem. Phys.
50.
June (1953)
Determination of Specific Heats of
Gases at High Temperature by Sound
SHERRATT G.G. & GRIFFITHS E.,
Velocity Method.
Proc. Roy.Soc. London. Ser A. 2
292
(1934)
51.
SLATER J.C., Introduction to Chemical Physics,
52.
SLOAN LABORATORIES, M.I.T. Report on Contract DA 19020 ORD 2831.
53.
VAN ITTERBEEK A. and MARIENS P.,
of Sound in CO2
Relaxation TimeA
Hunction of the
Physica
54.
_ - 2,
9 - 1
(1945)
25
TN 3210
May(1954)
WALKER R. Heat Capacity Lag in Gases,
N.A.C.A.
57.
(1938)
WALKER R.A. ROSSING T.P.& LEVGOLD S., The role of triple collisions
in excitation of Yolecular Vibrations in H2 0
N.A.C.A.
56.
153-160
VLASOV A. On the kinetic Theory on an Assembly Particles with
Collective IEnteraction.
Jour. Phys. USSR,
55.
Measurements on the Absorption
Gas. Determination of the
for the Vibrational Energy as a
Temperature.
Note 2537
Washington.
Iowa State College
(1951)
WALIMAN H. MACNEE A.B. & GADSDE1 C.P.
Proc Y.R.E.
36 , p.700
58.
WATMANN
.H. Period of Establishment of the Vibrational Energy
in CO2 as a Function of Pressure and Foreign Gases.
Ann. Phys. Ser.5
59.
WRLER
60.
WIDCM B. & BAUER S.H.
H.A.,
Proc. I.R.E.
21 - 7
27 - 6
WU, Ta-You,
(1938)
21 - 10,
1670-1685
(1953)
Excitation of Molecular Vibrations by Islectrons
Phys. Rev.
62.
359,
(1935)
Energy Exchange in Molecular Collisions
Jour. Chem. Phys.
61.
671-681
71 - 2
111-118
(1947)
POINCELOT P., Les Regimes Transitoires dans les Reseaux Electriques
Collection Technique et Scientifique du C.N.E.T.
Gauthier-Villars, (1953)
M.E. Thesis, Dept. of Mechanical Engineering, M.I.T.
(1954)
63.
WU P.C.,
64.
SKUDRZYK, E.
Die Grundlagen der Akustik,
Wien,
(1954).
APPENDIX
No
I
PROPERTIES OF THE BRASS COUPLING RODS
The protection of the Bariun Titanate crystals against changes in
tanperature requires water cooling of the brass coupling rods. The
changes in acoustical delay caused by bar temperature variations was
considered as unimportant, and so it was proven experimentally.
In another series of experiments reported elsewhere (52) the
bar temperatures were deliberately changed to investigate the gradients
caused by the bar surfaces in the gas.
The transit time variations in
the brass coupling rods havebeen found to be
important with respect
to the measurement accuracy expected.
Fig. 32
(obtained by Mr P. C. Wu of the Sloan Laboratories) shows
the numerical values found experimentally.
Tests on other metals,
(Monel, Molybdenum, Invar, etc)
are
under way in order to select the best material with respect to dimensional and elasticity changes, without making too important s acrifice
on the crystal matching on the other 9 de.
APPENDIX
NO
2
PRACTICAL CONSIDERATIONS ON THE USE OF CRYSTALS
The crystal transducer disks are supplied by the Gulton 4anufacturing
Company, Metuchen, N. J.
Crystals are individually tested and selected
for close matching in sensitivity and frequency.
It was found that
two crystals, apparently identical and having sane dimensions, may differ
by as much as 20% on natural frequency.
The c oupling of crystals to the bars represents a very important
factor in the overall performance. Silicon Stopcock grease is used at
present, since it has desirable electrical characteristics. In particular, it has been found that ordinary grease breaks often down and
carbonizes whenever a spark is arcing through the crystal or across the
crystal edge. This limited quite seriously the maxinui power input
on the crystal. At present a limit is set only by the depolarisation,
peak voltages of 40 mhlts/mil have been found acceptable for short peak
durations.
The main cause for crystal failure is mechanical, since the "clamping" of the crystal is necessary
to obtain maximum output, it is im-
possible to refrain enthousiastic operators to sqeeze the crystal in the
mount.
A fair indication is obtained by considering that we operate now
3 pairs of crystal mounts on various experiments and a supply of 10
crystals every 6 months was found adequate.
APPENDIX
NO
3
PROBLMS IN THE DE.SIGN OF MULTIPOINT RECORDING INSTRUMhITS
Assuming that all the instrumentation problems related
to the high speed digital recording of time intervals are solved, we have
to consider the limitations imposed by the sonic temperature measuring
device.
a)
Three basic factors have to be o nsidered:
Sampling Rate:
With the present acoustical circuit the tempo-
ral attenuation is such that it takes about 120 usec for a pulse
to decay to a non-discernible level.
It is thought that this
will limit our PR? to about 5,000 pulses per second.
b) Accuracy:
It is desirable not to lose on the present reading
accuracy, so that the pulse shaping and discriminating citcuits
should be able to do at least as well as does the human eye on
the calibrated-delay oscilloscope.
This problem is not as simple
as it appears, the human operator, with kis built-in feature to
account for the natural continuity of physical phenomena is
difficult to beat in this domain, especially if accidental
noise fluctuations may disturb the signal leading edge.
"Level"
type discriminators are invariably apt to be fooled by a coincidental fluctuation of the input voltage.
c) 2rystal performance : The exceptionally high specific output
obtained from the crystal is based on the very low duty cycle.
The maximum level obtained in
5,O0o cps
operation is open to
question and has not yet been investigated.
APPEN'!DIX No
A
ACOUSTICAL CIRCUIT TEST RESULTS
Test arrangement : See
1E cperimental
Commercial
Fig. 6~
Crystal Mounts are used on brass specimen F.C.
-I" diameter.
The transmitted waveform and the received waveform are studied
for various settings of the transmitter high-voltage control. Peak
amplitudes and envelope rise times are studied. The overall scope
sensitivity on the "direct" vertical plates is 71
is claimed to be linear within
in, the amplifier
1 dB from a few Kcps to 10 Mcps.
Fig. 2 & 33 shows a typical "transmitted" and "received" electrical
signals and it appears necessary to consider an "equivalent"
electroacoustical circuit which would yield the same transmission
characteristics. The reason for operating in this manner in found
in the obviously non-linear characteristics of the transmitting
crystal (which could be approximated by means of the well-known
equivalent circuit in case of small amplitudes), and also in the
somewhat unpredictable characteristics of the grease-coupling
layers. The use of the actual transmitted signal to infer a
posteriori the circuit characteristics has been found easier from
both simplicity and applicability standpoint.
The transmitted signal will be assimilated to the unit impulse
at
t
=0
and the transit characterisic is then found in terms of
a numerical factor
Qm
and the complex normalized transfer
function. The simplest approximation of the received signal is
2--
C
which has a horizontal tangent at t - 0 and at t
(e,
value of this maximu being
-C- .75
Inspection shows here
and
)%e,
= .56 -C'
Psec
; thus
2 "C
,
the
-c .37 psec.
1)
l.3v thus
C
---
The transform of C36) is* Fg-
1.3
.56
-52.35 volts
2 C(,
(S -4
which, with the
1/33
above assumption becomes the response to the unit impulse multiplied
by the numerical attenuation factor. This factor will be arbitraitr
expressed as the ratio of C to the peak voltage of the transmitted
pulse so
2.35
540
-3
------------------------------------------------------(C) ) Gardner and Barnes, Transists in Linear Systems.
* (
p. 346 N0 2.121
coaxial cable
to transmitter
inlet
transmitting
crystal
ionization ga
ga skets
end -gas zone
spark plug
pressure
indicator
brass bar
receiving
crystal,
piston with
special crown
coaxial cable
to receiver
ig. 1 - Sound Velocity Instrument installed
on Engine Head.
WMWM"6-
Fig. 3
Electronic Instruments
B
6.3/60 cps
B
-150 V ImA
4A
A
63/60cps
A
GREEN
1.8 KW PEAK PULSE GENERATOR
PROJECT 7143
25n
U3 800U-09 0
I-.-
-0V
(RC
Gas
Path
II
XRC
SY
Synchroscope
PO
S Engine distribution shaft
ER Breaker
ST Synchronizer
TR Transmitter
XTR Transmitter crystal
IRC Receiver crystal
RC Receiver
CICoupling rods
PO Calibrated delay
'Prti.ipe
of the ap-p-ratus use-
9 v
:) i
83sind
LNS9
83AI80
3dO:)S
IS3i
J/99Z
Fic-.
Acoust ic C ircuit ( Princ iple )
8
1.Test
-Pction
2.Transmitter
Coupling Rod
Coupling Rod
3.
Receiver
4.
5.
Transmitting Crystal
Dete cting Cryst al
E.
Silver
7.
Transmitter D-Wamper
8.
Detector Damper
9.
Sil
10.
Electrodes
ic one Jre ase
Layer s
( Tr ansmi tt er )
)
Silicone , rease Layers (Detector
TISO
Fig. 9
Transducer Assembly
j
3:)IA3G
ONI
8 nSV3VY
3VYII
JO 3-ldIDNIdd -01-013
-1v I cl
AV-130
Cl3 iV 8 13 1-lV3
S31VId
N0 11 D3 -1J 3 0
-1VI N0 Z 18 0 H
--
-
8OiV83N39
d33MS
-Lin o 8 1D
AV-13 G
dOiV83N30
8 3 0 0 18 1
83AV389
3NION3
VY08A
--
~
__
--
_
_
_
_
_4
Fig.
Fig.
11
12
Transducer Mount Details
Transducer Assembly
Fig.
13
Receiver
Chassis
A -b
-105
0
-los5
jjjjjj
GC-GG
LOW-NOISE VIDEO
PROJ. 7143
RECEIVER
---
200 V STAB
Jill
~~-a i
W
Fig. 15 Engine Head with Transducer Mounts
I
I
I
N
U
-
-,
~
--
Eu-
Fig. 17
Special Test Arrangement
00
occ
-L
<r r
<
w*
U
4Sf4
64
bIAT" 0d
R E MEs>
-
'2
j, 2
3
IELA1
N cc
C~t
HEASU
00
vAgtATIoOS
40
a.
S'
e
S0
'
w-4
IZ)
Ol
L44
41
EMEEE
EEE
E,.E.E..E
urn.....
III....
IIIIIIII.....
EEE..EEEEEEE-EE
*EEEEEEE.EEEEEEE
IIIIIIIII
1
*
1
-
-i
1
I
IF
..
-
1I
-
-
-
-
1 -
J
-
IF
1 I
1
IFIJ
a
--
-
L. .I
-
]1
4'
IFIt I II,
t+~~
1
-
I
[F
1
---
I~~I
I
I
I
I
I
-
FILII
]
tf t
F 4111
I
i
[~
1
-
--
I
1
-
'I
LL
I
f t111!1r
-
|.f
. . .-
II
-
- -
_Ll, ,|
- -
I bil
1.
MADE
-
-
- -
F,
-
--
I
lines accentied.-
INU. S.A.
10 x 10 to the 1, inch. 5th
Al
I
I F
sit11
-H
I
F
I
--
-
-
-
I
-1I
--
-
I
t
7-
IVGNA=
O6
-did
E.E.EEE-LEE
ME.....EE.E*
EEEEEEEEEEE
EE..E.EE.E.E
o
o
o
oomo
_
Eu.-.
urn...m...
MMMMMMMM
F/
.I8
a
-
iirat
/e
ii
Com
-rep
E.-EEE
rn-.
EEE.EME___|U
EEEEEEE-E.
EEEEEE-E
Eu-.-.-.
U.--M |
Eu...M
U-.. M
U.....-|
HE....0025
U/..-7.W/"/E// !F
*.
Fig. 33
Fig. 34
20 psi gage Received Signal
15 psi gage Received Signal
BIOGRAPHICAL
NOTE
The author, Thomas P. Rona was born in Budapest, Hungary on
January 17, 1923. Raised and educated in Paris, France, has obtained
the degree, Mechanical Engineer in 1943 (Ecole d'Electricite et
Mecanique Industrielle, Paris), the degree, Electrical Engineer in 1945
(Ecole Superieure d' Electricite) Paris, and the Certificat d' Etudes
Superieures, Electricite Generale (Sorbonne, Paris) in 1945.
His professional carrier includes the French Thompson Houston Co
(Radio transmitters) the French National Office for Aeronautical Research
(wing vibrations and Analog Computers) and a Consulting Engineer Practice
in the French Equatorial Africa (Civil Engineering and Hydro-Electric
Power Plant Sites).
* In 1951, the author came in Montreal, Canada, as Assistant
Professor of Electrical Engineering at the Ecole Polytechnique of Montreal.
He joined the Massachusetts Institute of Technology staff in 1952, where
he obtained the degree of Master of Science in Electrical Engineering
in 1953. Currently holds the appointment of Assistant Professor of
Mechanical Engineering at the Institute.
His family includes his wife, Monique R. Rona-Noel, and three
children, John-Michael (1951), Marie-Helen (1952) and Thomas Paul Jr. (1954)