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ABSTRACT

NONLINEAR TEMPERATURE DISTRIBUTIONS

BY

Marshall P. Cady, Jr.

Pure, one-component fluids can exhibit large
nonlinear temperature distributions in nonequilibrium
situations in which nearly linear distributions are
usually expected. This dissertation includes (a) a
theoretical analysis of molecular energy transport mechanisms
which can create nonlinear temperature distributions and
(b) a Bryngdahl interferometric study of the quantitative
aspects of temperature nonlinearities in liquids. The
nonequilibrium thermodynamic regimes studied include
time-independent and nonconvecting, time-independent and
convecting, and time—dependent and convecting.

The possibility that molecular vibrational degrees
of freedom can create or contribute to spatial nonlinear,
time-independent temperature distributions is explored with
a stochastic diffusional energy transport model. Computations
indicate that liquids may exhibit temperature jumps at the

liquid-boundary interface but that the diffusional mechanism

 

Marshall P. Cady, Jr.
of energy tranSport cannot contribute to nonlinearities of
significant spatial extent. However, the diffusional
mechanism can produce gas phase nonlinearities which extend

2

0.01 cm from boundaries provided that P Q 1 x 10- atm.

Computations also show the nonlinear magnitude to be 0.03 K
for gases when 1 vibrational degree of freedom in 10‘1 at
the boundary fails to be described by a Boltzmann distribution.
A temperature jump of 0.03 K is expected for liquids when 1
vibrational degree of freedom in 102 belongs to the
nonequilibrium distribution. The diffusion model has
applications in describing energy transport away from
catalytic surfaces, near boundaries of polyatomic gases in
the "temperature jump" regime, and near boundaries of
strongly emitting and absorbing liquids.

The diffusional energy tranSport model is
characterized by energy tranSport via diffusion of fluid
molecules, the Landau-Teller transition probabilities for
resonant exchange of vibrational energy during bimolecular
collisions, and the assumption of non-Boltzmann distribution
of molecular vibrational degrees of freedom at boundaries.
The non-Boltzmann distribution is the result of boundary-
Iluid interactions. Two parameters describing temperature
nonlinearity associated with diffusion are deduced. They
provide a measure of the spatial extent of the nonlinearity
from boundaries and of the magnitude of the nonlinearity.

Bryngdahl interferometry is used to examine

temperature distributions in liquids bound by horizontal,

 

Marshall P. Cady, Jr.
parallel, silver plates which are maintained at different
temperatures. For nonconvecting states the linear
temperature distributions eXpected for pure thermal
conduction are found at distances greater than 0.3 cm from
boundaries for ethyl acetate, benzene, and carbon tetra-
chloride. However, at shorter distances the distributions
are highly nonlinear. As the metal-liquid interface is
approached, the temperature gradient increases 4.5% for
ethyl acetate, 12.5% for benzene, and 30% for carbon
tetrachloride. Full details of the observed temperature
distributions and gradient distributions are reported. In
addition, we find that the ratio of integrated absolute
deviation from linearity(1/l_ SEIT-T3.33ldz where T3.33
is the solution of V-kVT = 0) is 1/2.4/4.7/14.6 for ethyl
acetate, benzene, carbon tetrachloride, and water,
respectively. The very large integrated absolute deviation
exhibited by water is caused by temperature jump phenomena.
It appears that the temperature jump is a property of the
boundary-liquid system and is not caused by plate temperature
control and temperature measurement.

Knowledge of liquid nonlinear temperature
distributions makes possible an increase in the accuracy of
experimentally determined Soret coefficients, thermal
conductivities, and nonisothermally determined refractive
index temperature derivatives.

Finally, experimental studies of nonlinear

temperature distributions in convecting liquids are presented.

Marshall P. Cady, Jr.

The Rayleigh numbers for the critical flow transitions are

found to be R.I = 1667 1 137, RIII = 3.3 x 1011

4

i 12%, and

R = 4.42 x 10 i 14%. At RI a transition from nonconvection

IV
to steady state convection occurs; at RIII there is a
transition from three-dimensional steady flow to three-
dimensional time-dependent flow; and at RIV there is a
transition from time-dependent flow to time-dependent flow
of increased frequency. Bryngdahl interferometry is
uniquely suited to the study of the time-dependent phenomena.
Frequencies of temperature oscillations are determined via
Fourier transform of the Bryngdahl interferometric image
autocorrelation function. Four major bands are observed at
approximately 10, 30, 90, and 140 min-1. The band intensity
is proportional to e = <AT>IV - AT and decays exponentially

with angular frequency 75.8 min-1.

NONLINEAR TEMPERATURE DISTRIBUTIONS

BY

A
Marshall P2 Cady, Jr.

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Chemistry

1976

To Sharmaine

ii

ACKNOWLEDGMENTS

I thank Professors F.H. Horne and R.I. Cukier for
their efforts to improve the manuscript. In addition,

I thank my wife, Sharmaine, whose typing, encouragement,
and prodding made this dissertation possible.

Finally, I express recognition of the influence
Professor F.H. Horne has had upon my scientific deve10pment.
The excellence of his lectures in mathematical physics,
thermodynamics, and nonequilibrium thermodynamics is admired

and has provided the fundamental basis of this dissertation.

iii

TABLE OF CONTENTS

LI ST OF TABLES O O O O O O O O O O .4 O O O O O 0

LIST OF FIGURES O O O O O O O O O O O O O 0 O 0

Chapter

1.

3.

INTRODUCTION . . . . . . . . . .
1.1 Apercu . . .

1.2 The Interior Conducibility.
1. 3 The Energy Source . . . . .

O O O O
O O O O
O O O O
O O O O

MOLECULAR VIBRATIONAL CONTRIBUTIONS TO NONLINEAR

TEMPERATURE DISTRIBUTIONS NEAR WALLS
2.1 Introduction . . . . . .
2. 2 Model 1: The Diffusion Model
2. 3 Estimates of c1 and c2 . . .
2.4 Model 2 . . . . . . . . . . .
2.5 Discussion . . . . . . . . .

O O O O O O
O O O O O O
O O O O O O

BRYNGDAHL INTERFEROMETRY . . . . . . . . .
Introduction . . .

Bryngdahl Interferometry: Apparatus,
cell, Data 0 O O O O
Refined Data; Refraction of Light . .

Temperature Distribution . . . . . .

2 WWW WNW
o

O ISOTHERMAL, NONCONVECTING LIQUID STATES
Introduction . . . .

NOnlinear Temperature Distributions:

o
P W MHZ @Ulob “NH

Index of water . . . . . . . . . . .
Discussion . . . . . . . . . . . . .

41' 1!? it's]?
0

iv

0 O O O

Bryngdahl Interferometry: Applications

Refractive Index: Temperature Dependence

Ethyl Acetate, Benzene, Carbon Tetrachloride
Temperature Dependence of the Refractive

O O O O O

Page
vi

viii

hnmth

19
19
25
44

53
56

59

59
61

67
78
89
9o

92
92

101

147
164

Chapter

5. NONISOTHERMAL, CONVECTING LIQUID STATES.

5.1
5.2

5.3
5.4

APPENDIX

LIST OF REFERENCES . . . .
Chapter 1 References
Chapter 2 References
Chapter 3 References
Chapter 4 References
Chapter 5 References

IntrOduCtiono o o o o o o o o o o 0
Temperature Distributions and the

First Rayleigh Transition . . . . .
Fluctuations and Frequency Spectrum
sumary o o o o o o o o o o o o o o

O O O O O C O O O C O O O O O O O O

O O O O O O
O O O O O O
O O O O O O
O O O O O O
O O O O O O
O O O O 0 0
O O O O O O
O O O O O O
O O O O O O
O O O O O 0

Page

174
174

178
191
200

207

210.
210
213
217
223
227

LIST OF TABLES

Tabl e Page

2.1 Gas phase parameters c and c of the
molecular fundamental irequengy D. . . . . . . . . 48

2.2 Liquid phase parameters c and c of the
molecular fundamental frequency 5.. . . . . . . . 51

pc (v)Hw(v) values for water at 300 K and
vagious boundary conditions. . . . . . . . . . . . 58

3.1 Derivative estimates of M.(Equation 3.16b) for
water in a 10 deg/cm temperature gradient. . . . . 85

4.1 Temperature dependence of refractive indices:
n = a + bT + 0T2 at 632.8 nm, 1 atm. . . . . . . . 103

4.2 Temperature dependence of thermal conductivity:
106k=a+bT+cT2, d$T$eat1atm. . . . . . . 105

41.3 Isothermal and nonisothermal image gata. . . . . . 111

4.4 Nonisothermal fringe number: N8 = 2 a.x1

i=0 1
Where X = (2.00254 cm)/Oo254 cm. 0 o o o o o o o o 125

‘4.5 Experimentally determined refractive index
temperature derivatives: dn/dT = A + BT. A and
B differ from literature values of Table 4.1
because of nonisothermal interferometric image
diStortj-on O O O O O O O O O O O O O O O O O O O O 132

4.6 Temperature dependence of the refractive index
of water as determined by isotherma1(I) or
nonisothermal(N) methods:

-(dn/dT)104 = a + bT, where céTéd. The

parameters a and b are reported in the literature

or evaluated with a least squares analysis of
reported experimental data at 1 atm. . . . . . . . 15o

vi

Table
4.7

Page

Nonisothermal experimental water data. N(l/2)

is the fringe number at image center. 8 is the

angle from vertical of fringe in deg. min and
corrected for deviant verticality of the

isothermal image. Angles of experiments of

identical AT and T have been averaged. The

The isotherm 1 paraEeters are H = 4.713 cm,

2D = 0.463 cm, and 6 = 1.239 cm with standard
deviations of 0.023 cm, 0.018 cm, and 0.018 cm,
respectively . . . . . . . . . . . . . . . . . . . 153

Experimental evaluations of the first critical
Rayleigh number. . . . . . . . . . . . . . . . . . 188

LIST OF FIGURES

Figure Page

1.1 Qualitative features of the interior
conducibility, k(T), of amorphous silica . . . . . 9

1.2 Energy transport by translational degrees of
freedom. The indicated positions of molecules
labeled 1 and 2 are those at successive collisions.
(A) TranSport far from walls. (B) Transport in
the presence of a reflecting boundary. . . . . . . 15

2.1 The horizontal plate arrangement . . . . . . . . . 22

2.2 The probability distribution {xn(§)} for various g
When0=1andxq(0)=1............33

2.3 Negative of H(z) for 0 1, x0(0) = 1, and
various 01 . . . . . . . . . . . . . . . . . . . . 38

2.4 H(z) for 0 = 1, x1(0)

1, x4(0)

2.6 H(z) fort? = 1, w1(_l_) = 1, x0(0) = 1, and
various values of 01 . . . . . . . . . . . . . . . 46

1, and various 01‘ . . . . 40

2.5 H(z) for 0

1, and various c1. . . . . 42

3.1 (A) The interferometer. (B) The cell-water
jaCketassembly.......o..........69

3.2 (A) Best quality image obtainable with interfero-
meter, the slit image. (B) Isothermal cell-water
jacket assembly image. Z is the vertical image
coordinate measured from the upper, bounding plate.
2D is the shear parameter. H is the image cell
height. 8 is the distance between fringes in the
isothermal cell configuration. . . . . . . . . . . 75

3.3 Refraction of light in a one dimensional gradient.
z(y) is the path of a photon pencil of infinitesimal
cross section which enters the liquid cell at

z = 20, 2f is the exit coordinate. TU)TL. . . . 80

viii

_———i

Figure Page

4.1 EA) Reduced temperature gradient,
dT/dz)/(dT/dz)1 2 versus z/l for carbon

tetrachloride when 1 = 1.5 Cm, e = 0.05, and

ATO = 2.00 c. From-G. Schodel and U. Grigull.10

(B) Deviation from linearity, T — T3 33, of CCl4
in a cell 0.810 cm high. TU = 26.4000,
T = 23.6o°c. From J. D. Olson.12 . . . . . . . . 100

L

4.2 Light refraction in a nonisothermal experimental
configuration; TU) TL' The pencil of light shown

enters the li uid at z = 0. It leaves at
coordinate zf O) and a3g1e af(0). This pencil

travels through the glass cell at angle Bf(0),
refracts in air, and travels at angle Yf(0) until

direction is altered by the Bryngdahl optical
train. 0 O O O I 0 O 0 O 0 O O O O O O O O O O O O 108

4.3 Nonisothermal Bryngdahl interferometric image of
water at several temperature differences between
upper and lo er silveroplates. l = 3.349 cm;
ATO = 10.028 C, 13.047 C, and 16.691 C . . . . . . 113

4.4 Nonisothermal Bryngdahl interferometric image of
ethyl acetate at several temperature differences
between upper and lower silver pla es.
1 = 1.349 cm; ATo = 2.244 C, 6.261 C, and
10.019 C C O O O O O O O O O O O O 0 0 O O O O O l 115

4.5 Nonisothermal Bryngdahl interferometric image of
benzene at several temperature differences between
upper and 18wer silvgr plates.O l = 1.349 cmé
ATO = 2.399 C, 7.256 C, 10.529 C, and 13.063 C . . 117

4.6 Nonisothermal Bryngdahl interferometric image of
carbon tetrachloride at several temperature
differences between upperoand lowea silver plates.

1 = 1.349 cm; ATO = 2.637 C, 5.976 C, 7.729 C, and
12.713 C O I O I O O O O O O O O O O 0 O O O O O O 119

4.7 Nonisothermal Bryngdahl interferometric image of
methanol at several temperature differences
between upper and lower sglver plates. 0
l = 1.349 8m; ATO = 4.739 C, 9.297 C, 12.125 C,
and 13.488 C I I O O O O O O O O O O I O I O O O O 121

ix

Figure

4.8

4.9

4.12

Page

Tem erature deviation from linearity,
T(z - T3 33(z), for benzene at several

temperature differences between upper and

lower silver plates. The refractive index
temperature derivatives of Table 4.1 are used

in the method of Section 3.6 to compute T(z)

from the refined experimental data.

1 = 1.349 cm. 0 I O C O O O I O O I O O O 0 O O O 127

Tem erature deviation from linearity,
T(z - T3 33 z , for carbon tetrachloride at

several temperature differences between upper

and lower silver plates. The refractive index
temperature derivatives of Table 4.1 are used

in the method of Section 3.6 to compute T(z)

from the refined experimental data.

1 = 1.349 cm. I 0 l O O O O O O I O O O O O O I O 129

Tem erature deviation from linearity,
T(z - T3 33(z), for ethyl acetate at several

temperature differences between upper and lower
silver plates. These curves are corrected for

image distortion by using the refractive index
temperature derivatives of Table 4.5 in the

method of Section 3.6 to compute T(z) from the
refined experimental data. 1 = 1.349 cm. . . . . 134

Tem erature deviation from linearity,

T(z - T3 33(z), for benzene at several

temperature differences between upper and lower
silver plates. These curves are corrected for

image distortion by using the refractive index
temperature derivatives of Table 4.5 in the

method of Section 3.6 to compute T(z) from the
refined experimental data. l = 1.349 cm. . . . . 136

Tom erature deviation from linearity,
T(z - T3 33 z , for carbon tetrachloride at

several temperature differences between upper

and lower silver plates. These curves are

corrected for image distortion by using the
refractive index temperature derivatives of

Table 4.5 in the method of Section 3.6 to compute
T(z) from the refined experimental data.

A = 1.349 cm. 0 O O O O O O O I O I O O O O O 0 O 138

Figure

4.13

4.18

Page

Reduced temperature gradient,
(dT/dz)/(dT/dz)1/2, for ethyl acetate,

benzene, and carbon tetrachloride. Corrections
have been made for interferometric image
distortion. 1 = 1.349 cm I o o o o o o o o o o o 143

Integrated absolute deviation from linearity,
l
(1/l)SOIT - T3.33ldz, as a function of ATO.

Corrections have been amde for interferometric
image distortion. The placement of the water
curve is discussed in Section 4.3. l = 1.349 cm

for ethyl acetate, benzene, and carbon
tetrachloride. l = 0.474 cm for water. . . . . . 145

Per cent deviation of dn(l,T)/dT from values
reported by Tilton and Taylor for water at 1 atm. 149

Temperature dependence of the refractive index.
The raw experimental data is refined with the
method of Section 3.4, and dn/dT mapped with the
method of Section 3.5 assuming that T(O) = TU

and T(l) = TL. Equation 4.38(solid line) is the

best linear fit of the resulting temperature
derivatives 0 O O O O O O O O O O O O O O I I O O 157

f(T) = -(dn/dT)102/(-d2n/dT2)1/2. The data are
computed by refining the raw data with the
method of Section 3.4, and mapping dn/dT and

d2n/dT2 with the method of Section 3.5 assuming

that T(O) = TU and T(l) = TL. Equation 4.40

(solid line) is the best linear fit of the above
ratio. The dashed line is predicted by
Equation 4. 38 I O I O O O O O O O O O O O O O O O 160

(A) Molecule 1 emits an "average" photon which

is absorbed by molecule 2. This energy transport
process heats the horizontal layer of molecule 2
because the photon originates within a layer of
higher temperature. (B) Placement of a perfect
mirror between molecules 1 and 3 alters photon
energy transport processes. The photon absorbed
by molecule 2 now originates from molecule 3.
Since the horizontal layer of molecule 3 is of
lower temperature than the layer of molecule 1,
energy flux differs from the above figure . . . . 170

xi

Figure Page

5.1 Nonisothermal Bryngdahl cell images.
I = 1.440 cm, a = 5.961 cm, H = 3.175 cm,
2D = 0.138 cm, 8 = 0.353 cm. a a o o o o o a u o o 180

5.2 Temperature distributions for various
impressed temperature gradient . . . . . . . . . . 183

5.3 Experimentally determined boundary(z = 1)
temperature gradient as a function of
Rayleigh number. Curve 1 is the linear least
squares analysis of data(Equation 5.8); Curve 2
is the prediction of the Laplace equation
description of temperature . . . . . . . . . . . . 186

5.4 Time sequential temperat e distributions
for R(TM,AT) = 1037 x 10 o o o o o a o o c o o o o 190

5.5 Intenisty fluctuations at image center oi
Bryngdahl image for R(TM,AT) = 8.37 x 10 . . . . . 193

5.6 Intensity fluctuations at image center of
Bryngdahl image for various impressed
temperature gradients. . . . . . . . . . . . . . . 195

5.7 Bryngdahl image intensity fluctuation rate
at cell center versus impressed temperature
gradient and versus Rayleigh number. . . . . . . . 198

5.8 Autocorrelation function of AI(t) for 5

AT = —6.849°C and R(TM,AT) = 3.71 x 10 , . . . . . 202
5.9 Fourier transform of autocorrelation function

of AI(t) divided by e = IATI - 0.72700 . . . . . . 204

CHAPTER 1

INTRODUCTION

1.1 Apergu

The study of nonlinear temperature distributions in
pure, one-component fluids provides macroscopic, continuum
data on which molecular theories of energy transport processes
may be based. Clearly, all temperature distributions in
nonequilibrium media are the consequence of molecular energy
transport via vibrational, rotational, translational, and
electronic degrees of freedom. Because of this relationship
between molecular mechanism and macroscopic observation,
nonequilibrium thermodynamic studies are an important

1’2 to catalog molecular

complement to the modern effort
energy levels and transition probabilities.

Particulars of this work include both the conceptual
illumination of molecular mechanisms which can cause nonlinear,
Steady state temperature distributions and Bryngdahl interfer-
ometric evaluation of temperature distributions in pure,
one-component liquids. The possibility that vibrational
degrees of freedom play a contributing role in the establish-

ment of nonlinear, steady state temperature distributions in

nonconvecting fluid media is explored in Chapter 2. On the

IIIIIl--..____

2

assumption that the temperature distribution far from the
walls of the fluid container is linear, mechanisms are
presented which invalidate the steady state Fourier-Laplace
equation description of the temperature distribution when wall
effects are important. Molecular diffusion and energy flux
models of vibrational energy transfer are used to analyze the
problem quantitatively. Fluid-dependent parameters characterizing
the magnitude of the temperature nonlinearity and its extension
from the container-fluid interface are deduced. Energy trans-
port by diffusion is shown to contribute negligibly to non-
linear temperature distributions observed within the liquid
phase, but this mechanism may contribute significantly to the
establishment of nonlinearities within low pressure gases.

Before experimental evidence of nonlinear temperature
distributions in both nonconvecting liquids(Chapter 4) and
convecting liquids(Chapter 5) is presented, it is necessary
to discuss details of the Bryngdahl interferometer which is
used as a temperature probe. This is the purpose of Chapter 3.
After a survey of the applications of Bryngdahl interferometry
in Section 3.2, details of the optical apparatus, the horizontal
parallel plate-cell composite, and experimental procedure are
described in Section 3.3. The horizontal parallel plate
arrangement heats the liquid cell at either the upper or lower
plate and cools it from the opposite plate. This results in a
continuous vertical temperature distribution across the liquid
‘which causes refraction of the interferometric light beam.
Beam refraction must be accounted for when spatial derivatives

of the refractive index are mapped from the unrefined Bryngdahl

5
interferometric data. The mapping is
dw/dz ——9 dn/dz (1.1)

where w, n, and z are the optical path, refractive index,
and vertical coordinate, respectively. The left side of
Equation 1.1 is called the unrefined data; the right is
called the refined data. The mathematical mapping procedures
are established in Section 3.4. Once the refined data have
been mapped from the unrefined Bryngdahl interferometric data,
one can either assume knowledge of the temperature gradient
and compute the temperature derivative of the refractive
index with the chain rule expression

dn/dT = (dn/dz)/(dT/dz) (1.2a)
or assume knowledge of the temperature derivative of the
refractive index and compute the temperature gradient with
the equation

dT/dz = (dn/dz)/(dn/dT). (1.2b)
The alternative mathematical prodecures are established in
Sections 3.5 and 3.6, respectively.

Direct interferometric observations of nonlinear,
steady state temperature distributions in nonconvecting
liquids are reported in Chapter 4. After an introductory
discussion of linear and nonlinear distributions in Section
4.1, the qualitative and quantitative features of nonlinearities
exhibited by ethyl acetate, benzene, and carbon tetrachloride
are presented in Section 4.2. Besides a thorough detailing of
problems inherent in nonequilibrium experimental studies,
Section 4.2 includes data which lead to (1) correlation of the

dependence of the nonlinearity upon the impressed temperature

4
difference between the liquid boundaries, (2) Specification of
the Spatial domain of the nonlinearity, and (3) comparison of
the relative magnitudes of exhibited nonlinearities between
liquids. All measurements are at ambient pressure. Mean
temperatures are between 24°C and 29°C, and boundary
temperature differences are between 2°C and 13°C for the
1.349 cm high liquid cell. In Section 4.3 the temperature
dependence of the refractive index of water between 24°C and
40°C at 632.8 nm and atmOSpheric pressure is obtained.
Temperature gradients range from 4 deg cm"1 to 16 deg cm_1.
Results comparable to isothermal determinations of the
temperature dependence are found when the data analysis does
not include the measured temperature difference between the
upper and lower plates of the horizontal parallel plate
system. The presence of temperature jumps would explain
discrepancies which occur when the temperature difference is
used in the analysis.

Finally, the Bryngdahl interferometric examination of
convecting thermodynamic states exhibited by water is presented
in Chapter 5. There are four major features. These include the
determination of the critical Rayleigh number for two types
of transitions: the transitions from nonconvection to steady
state convection and from steady state convection to time-
dependent, turbulent convection. The nonlinear temperature
distributions between these transitions are mapped.

Frequencies of aperiodic motions in the turbulent states are

determined.

5
1,2 The Interior Conducibility
The nonequilibrium description of temperature(T)
in a one-component, pure, nonreacting fluid depends upon
two fundamental statements. The first is the law of
conservation of energy:

E + V~gi = oi (1.3)
where E = E(g,t), g1 = gi(r,t), and 01 = oi(r,t) are the
Specific internal energy per unit volume, the i'th form of
heat flux, and the i'th form of energy source per unit
volume per unit time, respectively. The heat flux and energy
source have been given the superscript i because differing
physical interpretations and mathematical descriptions of
the energy source are associated with nonequivalent heat flux.
The i'th form of the energy source is associated with the
i'th form of the heat flux.

The second important statement is a phenomenological
relationship between heat flux and temperature gradient.
For isotropic media it is

J1 = -kiVT (1.4)

i = k1(r) is the i'th form of the conductivity.

where R
For temperature fields which are uniform in the
horizontal plane and for which E = 0, Equations 1.3 and 1.4
simplify to
(d/dz)ki(dT/dz) = -01. (1.5)
The solution of this differential equation subject to the
boundary conditions

T(o) = T and T(l) = TL (1.6)

U
is the temperature field. l,is the distance between

 

6
horizontal boundaries. Equation 1.5 is also used in the

experimental evaluation of k1. One integration yields:
1 i
-k (z) = (JT + Io (z)dz)/(dT(z)/dz) (1.7)

where JT is the total heat flux in the z direction and will be
called the heat flux. The heat flux, dT(z)/dz, and T(z) are
experimentally determined; then, 01(2) is assumed and ki(z)
computed with Equation 1.7. A final mapping ki(z) -—9 ki(T)
yields the i'th form of the conductivity.
A particularly important form of the conductivity

is given by Fourier's3 "interior conducibility," k(T),
which is defined by

Jz = —k(dT/dz). (1.8)

This constitutive conductivity form is recognized by many

modern authors!"11

of nonequilibrium thermodynamics. There
are two criteria to help with the deduction of the energy
source form(hereafter called the energy source) which is
conjugate to the interior conducibility. The first is due
to Truesdell(as reported by Petroskia): the energy source is
determined by given functions E and g and Equation 1.3.
Secondly, we wish to construct the interior conducibility

to be a property of the media being studied. Therefore, the
interior conducibility must describe energy tranSport by all
possible modes of energy transport solely characteristic of
that media. Energy transport modes associated wtih container
walls and the external world are not described by the

interior conducibility. Very little is known about the

7

energy source,8 but with these criteria the necessary
concepts will be developed in the next section.

The multitude of possible energy transport mechanisms
is illustrated by the interior conducibility of amorphous
silica in Figure 1.1. At very low temperatures k~vT3.

This dependence is the result of phonon processes for which
k ~ CVL (1.9)
where C is the heat capacity per unit volume, V an averaged
phonon velocity, and L an averaged phonon mean path. At
low temperatures L is constant because of crystal boundary
and grain size restrictions; the temperature dependence of
Equation 1.9 is then determined by the heat capacity. At
low temperatures this dependence is the famous T3-law of the
Debye theory of heat capacity of crystals. With increasing
temperature, a second region of extreme complexity is found
which is called "the knee." The knee is apparently caused

12

by resonance scattering of phonons by local defects. At

still higher temperature a very broad region exists in which
the interior conducibility is very weakly dependent upon
temperature. In this region k is limited by phonon-phonon
processes and by the relative inability of many of the
vibrational modes, due to high spatial localization, to
transmit energy. Finally, at very high temperature there

is a fourth region of strong T3 dependence;13"16 energy
transport by photon processes have become very important.

.A photon or radiation transport process is to be understood

as either the induced or spontaneous emission of a photon

Figure 1.1

Qualitative features of the interior conducibility,

k(T), of amorphous silica.

._. 6.29.368.

 

 

 

:35. of... 29.
MP 2 Hx

 

 

 

>1 ‘Mugqgonpuog JO!J6.IU|

10
by a degree of freedom of the medium and the absorption of
that photon by another degree of freedom. Vibrational
degrees of freedom are major contributors to this mechanism
via infrared photons.

The second major conductivity form is called the
"thermal conductivity." In this form the conjugate energy
source contains all radiation tranSport processes, both
those in which the photons originate within the medium and
those in which thephotonsoriginate in the external world.
Energy transport processes due solely to thermal motion of
the elementary particles of the media are described by the

17.21 Our objection to this division

thermal conductivity.
of conduction and radiation is based on the presupposed
ability to write mathematically and nonphenomenologically a
term describing all photon processes. This term is
necessary in the Equation 1.7 evaluation of the thermal
conductivity. By this procedure each photon model yields

a characteristic thermal conductivity and those of different
laboratories are impossible to compare properly unless full
computational details are published. We believe that once
any model of radiation tranSport for a particular medium
has been experimentally proven, then the above division of
conduction and radiation becomes a valid procedure. This,
luneever, is not the usual practice. For the history of the
separation of conduction, radiation, and convection see

22 for the first suggestion that conductivity theory

3

Brush;

0
can also include radiation transport, see Stokes.”

 

 

11

1. The Ener Source

Because the interior conducibility describes all
energy transport processes characteristic of a given media,
the energy source is zero far from boundaries and the
external world. In this domain Equation 1.5 becomes the
Fourier-Laplace equation:

(d/dz)k(dT/dz) = 0. (1.10)

The interior conducibilities of gases and liquids are very
much like kIII(T) of Figure 1.1; they exhibit a very weak
temperature dependence in a broad temperature region which
may be called the "ordinary temperatures." For example,
for gaseous carbon monoxide at 1 atm: 9 x 10"3 K"1 3
(1/k)(dk/dT) 2 6 x 10’“ K‘1 when 90 K s T s 1100 K.24’25
For carbon tetrachloride at 1 atm: 1.7 x 10-3K-1é -(1/k)(dk/dT)$

2.2 x 10"3K‘1 when 255 K5 Ts 378 K.26

This demonstrates that
for liquids and gases subjected to temperature differences
of the order of 10 deg cm-a thermal conducibilities are
essentially constant at ordinary temperatures. Thus,
the temperature of fluids far from boundaries is expected
to be the solution of the Laplace equation:

T(z) at a + bz (1-11)
where a and b are constants. This linear relationship is a
consequence of the absence of an energy source term in
Equation 1.10. Thus, Equation 1.11 emphasizes that the study
of nonlinear temperature distributions in nonconvecting,
steady state fluids must be the theoretical and experimental
pursuit of the concepts behind the existence of non-zero energy

source, source magnitude, and spatial domain of the source.

 

 

 

12

There are a number of nonlinear temperature
distribution studies at ordinary temperatures and pressure
(~1 atm). For a review of nonlinear observations within
liquids see Section 4.1; see Figure 4.1 for an example of
the nonlinearity within carbon tetrachloride. Nonlinearities
at ordinary temperatures and pressure have also been found
within some gases. In 1964 Gille and Goody27 used horizontal
aluminum boundary plates spaced 2 cm apart and a Michelson
interferometer to study temperature distributions in dry
air and ammonia. Linear temperature profiles were found at
all distances from boundaries for dry air. Ammonia, however,
displayed nonlinear temperature profiles. The nonlinearities
were found to be antisymmetric about the intermediate
vertical coordinate with a negative deviation from linearity
on the half closest to the heated boundary and a positive
deviation on the half closest to the cooled boundary. A
maximum in the deviation from linearity appeared at about
0.4 cm from the boundaries. It was 0.1800 at 0.7 atm and
0.2100 at 1.3 atm. The nonlinear temperature profiles are
explained on the basis of absorption and re-emission of
radiation from plates and molecular diffusion. Schimmel
gt_§128 have used Mach-Zehnder interferometry to deduce
boundary temperature gradients in pure gaseous 002, N20, and
mixtures of CO2-CH4 and CO2-N2O in a cell bound by horizontal
aluminum plates 2.55 cm apart. They report that for N20 at
a temperature difference between plates of 10.700 and a mean

temperature of 33°C the ratios of boundary temperature

 

13

gradient to the gradient predicted by Equation 1.11 are
1.09, 1.13, 1.17, 1.20, and 1.20 at pressures of 0.25 atm,
0.50 atm, 1.00 atm, 2.00 atm, and 3.00 atm, respectively.
The ratios for C02 behave similarly with pressure, but are
somewhat smaller. These nonlinearities are attributed to
radiation transport processes. Experimental studies have
shown the nonlinear phenomena to be important in vertical

29

plate systems containing pure ammonia and nitrogen-

ammonia mixtures.30

The final example of nonlinear temperature distribution
is provided by the observations in gases at low pressure
($10—2 atm) of what is called "temperature jump" and by the
association of temperature jump with gas-wall thermal
accommodation. For the history and early development of
the concept of temperature jump see References 31 and 32;
for the association with thermal accommodation see References
33-36; for the Boltzmann equation approach to the problem
see References 37 and 38. We present a new conceptualization
of the mechanism behind this nonlinearity.

Consider a gas bound by two parallel plates of
infinite extent. These plates are maintained at different
temperatures with the "lower" wall having lower temperature,
and they are far enough apart that at intermediate distances
from them all energy transport is due solely to the
characteristics of the gas. Figure 1.2A depicts energy
tranSport by thermal motion at these intermediate distances.

The indicated particle positions are those of successive

 

 

14

Figure 1.2

Energy transport by translational degrees of freedom.
The indicated positions of molecules labeled 1 and 2 are
those at successive collisions.

(A) Transport far from walls.

(B) Transport in the presence of a reflecting boundary.

See text.

 

 

T T, Vertical Direction G

 

 

B

Q Q)

\/ Mirror

 

 

 

 

 

 

16

gas molecule-molecule collisions. These translational
motions and collisions transport thermal energy. For
example, the motion and subsequent collision of molecule 1
results in the effective heating of the horizontal layer

of the collision; the motion and subsequent collision of
molecule 2 effectively cools the horizontal layer of the
collision. All such energy transport processes are described
by the interior conducibility. Suppose a perfect mirror
which reflects molecules specularly and elasticly is placed
between molecules 1 and 2,andthen energytranSportprocesses
are altered. Molecule 1 is now reflected by the mirror to
the collision indicated in Figure 1.2B. But the interior
conducibility being ignorant of the mirror's presence
describes the origin of the colliding molecule as corresponding
to the initial position of molecule 2. This is the mirror
image of the initial position of molecule 1. The interior
conducibility is assigning less energy to the colliding
molecule than it actually has. Thus, the temperature
distribution near the lower boundary displays a positive
deviation from linearity; the energy source is positive in
this domain. Now, considering the reflection of molecule 2,
molecule 2 is reflected into a collision but carries less
energy into that collision than predicted by the interior
conducibility. The interior conducibility describes the
reflected molecule as having originated at the initial
position of molecule 1. This position corresponds to a

horizontal layer of higher temperature and, therefore,higher

 

 

17
energy than the horizontal layer of the true initial
position of the reflected molecule. Thus, the temperature
distribution near the upper boundary must display a
negative deviation from linearity; the energy source is
negative.

This conceptualization has assumed that the linear
temperature distribution observed in the gas far from
boundaries is maintained by some unspecified mechanism.

On this basis it analyzes the effect of perfect mirrors
which have a zero thermal accommodation coefficient(no
energy is transferred from mirror to molecule during
collision). This does not correspond to any possible
laboratory experiment. However, within the framework of
the model it can be easily reasoned that (1) the negative
(positive) deviation from linearity prediction of this
model at the upper(lower) boundary represents the maximum
obtainable and (2) negative(positive) deviation from
linearity at the lower(upper) boundary is not possible for any
boundary-gas system which has a thermal accommodation
coefficient between 0 and 1. The thermal accommodation
coefficient is defined as being equal to (Ti-Tr)/(Ti-Tw)
where Ti corresponds to the temperature of molecules
incident upon a wall, Tr corresponds to the temperature of
molecules reflected by a wall, and Tw is the temperature of
the wall.

Of course this conceptualization can be made

phonsically realistic through the inclusion of adsorption,

 

 

 

18

desorption, and inelastic scattering of molecules by
boundaries. The prescription for the mathematical develop—
ment of the energy source is clear, however. It is the sum
of energy transport processes which are occurring because
of the presence of walls minus those energy transport
processes which are described by the interior conducibility
but are not occurring because of the presence of walls.

Nonlinear temperature distributions result from
radiation energy transport via mechanisms which are analogous
to the above translational model. This is discussed in
Section 4.4. An analogous mechanism of phonon reflection
and bulk phonon-grain surface phonon interaction may also
be responsible for the belief that the Fourier heat law is
not valid at very low temperatures in amorphous solids.16
In this case the energy source grows with decreasing

temperature.

 

 

 

 

CHAPTER 2

MOLECULAR VIBRATIONAL CONTRIBUTIONS TO NONLINEAR
TEMPERATURE DISTRIBUTIONS NEAR WALLS

2.1 Introduction

In this chapter we examine quantitatively the
possibility that vibrational degrees of freedom of a gas or
a liquid contribute to spatial nonlinear temperature distri-
butions within a single component fluid in the nonflowing
steady state. A model in which molecular vibrational energy is
transported during diffusion of molecules is considered in
detail. Computations indicate that the diffusional mechanism
of energy transport cannot contribute to nonlinearities of
significant spatial extent within dense media. However, the
diffusional mechanism can produce gas phase nonlinearities
which extend 0.01 cm from boundaries provided that P s 1 x 10"2
atm. 0n the assumption of a non-Boltzmann distribution of
molecular vibrational degrees of freedom at boundaries,
cemputations show the nonlinear magnitude to be 0.03 K for
gases when 1 vibrational degree of freedom in 1011 at the
boundary fails to be described by a Boltzmann distribution.
A temperature jump at a liquid-boundary interface is expected

0
When 1 vibrational degree of freedom in 10“ belongs to the

19

IIIIIIIlI-—___________________________________________nnnnnnnnnn______

 

20
nonequilibrium distribution.

To explore the possibility that the interaction
between vibrational degrees of freedom of the fluid and
the wall can result in nonlinear temperature distributions,
consider Figure 2.1. The coordinate 2 measures distance
from the upper plate, which is considered to extend to (i)
infinity in the (x,y) plane. The distance between the two
infinite plates is l. The upper plate is maintained at
temperature TU while the lower is at temperature TL, with
T T

U L’
each layer having width z and the ith layer being associated

Now imagine the fluid as being divided into layers,

with the temperature T(i).
Let the 1th layer be far enough away from the wall
that wall effects are negligible. The transfer of energy

on the molecular level can be envisioned as follows: Energy

is continually being transferred from the 1th layer to the
layers j+1 and j-i. For example, when a molecule goes from
the 1th layer to the layer j+1, it effectively heats layer
j+1 because its translational, rotational, and vibrational
degrees of freedom are distributed at T(j), which is
greater than T(j+1). Once in layer j+1, this excess energy
is dissipated. Conversely, when a molecule undergoes the
layer transition j+1-9j, the jth layer is cooled. Energy
is also transferred via emission and absorption of photons

'between layers. Far from walls Fourier's interior conduci-

21

Figure 2.1

The horizontal plate arrangement.

 

_ _
_ _
_ _

_ _
1| 2 3
_ _

r

“N... _

L_ _
_ _

///

I]

l/l/l/I/ // //
////// //
xxx/xx //
/////// //
///// ///
/ /////
2 /_.IU _

1

1+-

.1—
_
_

//

/////////

//

TL

Lower Plate

 

23

bility incorporates all such processes, and Equation 1.10
describes the temperature distribution. There is no
experimental evidence to the contrary.

Near the wall, however, the Fourier-Laplace equation
may fail. For example, a molecule of the first layer is
prevented from entering the "zeroth" layer by the presence
of the wall. When reflected back to the first layer, it
may have a translational energy corresponding to TU but have
its original vibrational energy distributed at temperature
T1. Since this reflected "reentry molecule" does not have
the vibrational temperature TU, where TU) T1, layer 1
exhibits a temperature lower than that predicted by V2T = 0
in the steady state. Next to the lower plate, a reflected
molecule may have its vibrational energy distributed at a
temperature higher than TL’ As a consequence, the layers
next to the lower wall will exhibit temperatures higher than

that predicted by V2T = 0. This is one way in which walls

might interact with vibrational degrees of freedom to
produce a nonlinear temperature distribution. Analogously,
reflection by the walls of photons which have been emitted
by vibrational degrees of freedom of the fluid results in
nonlinear temperature distributions qualitatively identical
to the temperature distributions which result from molecular
reflection. Also, if the upper wall emits and absorbs

photons at rates and extents different from the fluid at

24

the same temperature, it is possible that there is a
difference in the net number of photons absorbed by the
vibrational degrees of freedom in layers near the wall
beyond that described by V2T = 0 in the steady state. One
final mechanistic example is that of a wall which catalyzes
the production of molecules in higher vibrational states.
Upon reentry into the layer next to the wall these molecules
heat the layer with a resulting nonlinear temperature
distribution.

In the following sections, the interaction between
the fluid and the wall is not directly analyzed. Rather, it
is proposed that these interactions, whatever they may be,
result in a non-Boltzmann distribution of vibrational energy
in the layer adjoining the wall. Furthermore, it is assumed
that this nonequilibrium distribution is known. Temperature
distributions can then be calculated once the energy source
term which causes deviation from the Fourier-Laplace
temperature description has been evaluated. Specific models

of vibrational energy transfer are used to evaluate the

energy source. The number of nonequilibrium molecules at
the boundary needed to achieve deviations from linearity of
10-2 K at ordinary temperatures and pressures is estimated,
and parameters which determine the thickness of the

nonlinear boundary layer are deduced.

25

The concepts used to develop the above nonequilibrium
problem are stochastic: the quantities of interest are
probabilities and averages rather than classical deterministic
quantities. Much of the mathematics and many of the
stochastic concepts developed by Rubin, Shuler, and Montroll
are applied directly.1”S The models developed belong to
class of models which include first-passage time problems,
harmonic oscillator relaxation in a heat bath, relaxation of
two interacting systems of harmonic oscillators, relaxation
of Rayleigh and Lorentz gases, reaction kinetics, and
nucleation theory. Reviews of these stochastic processes

are available.6-13

2.2 Model 1: The Diffusion Model
The time—independent temperature distribution is

described by Equation 1.5:

k(d2T/dz2) = -o (2.1)
where k is the temperature-independent interior conducibility
and o is the energy source which vanishes far from the
‘boundaries. In order to describe temperature distributions
in.all domains, including that in which walls play an

:hmportant role, we assert that we may divide the one

26

component fluid into two systems, the first of which is
the "heat bath" (HB) system. The heat bath system of the
1th layer has all degrees of freedom, vibrational included,
distributed in a Boltzmann distribution at temperature T(i).
Thus, the heat bath system is associated with the measured
temperature T.

The second system is the system of interest(SI).
The system of interest of the ith layer does not have its
vibrational degrees of freedom distributed in a Boltzmann
distribution at temperature T(i). Rather, the distribution
is a nonequilibrium distribution which depends on the nature
of the interaction between fluid and wall. The system of
interest also interacts with the heat bath system via
molecular collisions and perhaps radiative transfer, 1:3,,
exchange of photons between HB and SI. This interaction
results in a source of energy for the heat bath system,
0(2). In order to evaluate the energy source, models must
be developed which enable one to describe the number density
of the vibrational states of the system of interest as a
function of the spatial coordinate subject to a knowledge
of the vibrational energy distribution at 2:0 and 2:1, iég.,
sxflaject to boundary conditions on the distributions. Once
the vibrational distributions have been completely determined,
it is possible to write the energy source term of the bath
system and solve Equation 2.1 for the temperature distribution
Stflxject to the boundary conditions T(z) = TU at z = 0 and

T(z) = T t z = 1.

La

27

The first model, the diffusion model, has the

following characteristics:

(a)

(b)

(C)

Both the system of interest(SI) and the heat bath
system(HB) are composed of harmonic oscillators of
frequency v. pk is the number of SI molecules per unit
volume in the 5th vibrational state, and pk = pk(£,t).
The density of SI molecules is such that SI molecules
collide only with EB molecules and never with other SI
molecules. In these SI-HB molecular collisions,

resonant transfer of vibrational energy is possible.
14

Furthermore, we assume the Landau—Teller transition
probabilities per collision:
Pn,m = P1,0((n+1)6m,n+1 + n5m,n-1) = Pm,n’ (2'2)

where Pn,m is the probability per collision that a
molecule originally in the 3th harmonic state will
undergo transition to the mth harmonic state, and 6m,n
is the Kronecker delta,
0 if m i n
5 = 2.3)
11fm=n
See References 7 and 9 for a more detailed discussion
of Equation 2.2. For more recent developments, see
References 15-17.
SI molecules in the nth vibrational state diffuse at the
rate DV2pn, where D is the diffusion coefficient and is

temperature independent.

28
Because of these characteristics, the rate of
change of pn at position‘s can be decomposed into the sum
of a diffusion term and an SI-HB interaction term. The
SI-HB interaction term is given in Reference 4. The result
is
(apn/at) = (apn/at)diffusion * (apn/at)SI-HB (2°43)

where

= DV2p (2.4a)

(apn/at)diffusion n

by characteristic (c), and
(apn/at)SI_HB = K(1-e‘”)‘1(ne‘”pn_1 + (n+1)pn+1
- ((n+1)e-0 + n)pn) (2.4c)

by characteristic (b). In the last equation, 0 = (hv/kT),
where h and k are the Planck and Boltzmann constants,
respectively, v is the vibrational frequency of the molecules,
and T is the temperature at 5. Both the diffusion constant
and the interaction constant, K, are assumed to be temperature

independent. For the steady-state case, Equation 2.4 becomes
2 -0 -1 -0
DV pn + K(1-e ) (ne pn-i + (n+1)pn+1
- ((n+1)e'9 + n)pn) = 0. (2.5)

Thezsolution of this equation requires two boundary conditions:
'thervibrational distribution of $1 at z = 0 and that at

z ==‘l. In order to apply Montroll's and Shuler'sq Fourier
series solution to the problem, SI must first be separated

itho two systems. The first system is the nonequilibrium

29
distribution associated with the presence of the upper wall.
Its vibration number density distribution is designated as

{3.}. The second nonequilibrium distribution is that

J
associated solely with the presence of the lower wall. Its
vibrational number density distribution is designated as
)rj}. The set )sj} satisfies Equation 2.5 in the coordinate
system originating from the upper wall and also satisfies
the distribution boundary condition on the upper wall.
Similarly, the set )rj} satisfies Equation 2.5 in the
coordinate system originating from the lower wall and also
satisfies the distribution boundary condition on the lower
wall. When the nonequilibrium distribution associated with
both walls relaxes to a Boltzmann distribution within the
distance ;, SI is the sum of these two distributions. We
now make the assumption that SI is this simple sum. This is
not a very restrictive assumption when SI is a small fraction
of HB. For instance, suppose that SI is 1% of H3 at the
upper surface and that SI-HB interactions are negligibly
‘weak. In such a case the nonequilibrium distribution
associated with the upper wall is altered by 1% at the lower
'wall due to the presence of the lower wall. The fraction of
inolecules involved is then a mere 0.01% of HB, a number
‘which is negligible relative to the number of molecules in
the nonequilibrium distribution associated with the lower
'wallx Later it will be shown that either the nonequilibrium
(tistributions relax well within the distance I, or that SI

i1; a small fraction of HB. With the two systems of SI

 

30

clearly defined, Equation 2.5 becomes

DV2sn + K(1-e-0)-1(ne-0s + (n+1)s

n-i n+1

- ((n+1)e'9 + n)sn) = o (2.6)

which can be solved subject to the boundary condition

sj(z) = Sj(0) at z = 0. The set {r3} satisfies an identical
equation. To put this into dimensionless form, we make the
following definitions:

c1Z/l. XJ-(C) = sj(§)/§si(§). wJ-(L) = rj(§)/2iiri(c).

C

c1 (hf/D)“2 (2.7)

Furthermore, we constrain 2si(§) and 21‘1“) to equal
1 i

Es.(0) and 2r.(c ), respectively; 5x.) and {w.} are then
1 1. i. 1 1 1 1

probabilities. In accordance with the assumed form of SI,
these probabilities must relax to a Boltzmann distribution

far from their walls of origin. Equation 2.6 then becomes
-0 2 2 -6
(1-e )(d xn/dg ) + (ne xn_1 + (n+1)xn+1
- ((n+1)e-6 + n)xn) = 0. (2.8)

It is easily shown that the variation of 0 with L in
Equation 2.8 is negligible to within 1% for IS 1 cm when
AT/_1_.$ 5 deg/cm at ordinary temperature. Since this
corresponds to the usually experimental condition, we
hereafter restrict ourselves to this case. With these

conditions the Fourier solution of Equation 2.8 is(see

Appendix)

 

31

co _{1/2
xnm -—- zafiuhnew C (2.9a)
=0
.1....(v) = F(—n. n+1. 1; 1-5") = e‘MflEOU-e”) (3H3) (2.91:)
-0 w 0
aAU) = (1-6 )mEdlk(m)xm(O)' (“'90)

Comparison of this solution with the Appendix 1 solution of
Montroll and Shulerk shows that only the exponential power

is different, in that the square root is now required.

This difference is a consequence of the diffusion differential
equation being second order. In these equations, F(a, b, c; z)
is the hypergeometric function,18 and the binomial coefficients

have the usual meaning,

nJ/v!(n-v)! rfll>v
(3) = 0 if n<v or v40. (2.10)

1 if I): O

xm(0) is the boundary condition imposed on the reduced
distribution function at g: 0. _1_n(u) is the Gottlieb

polynomial.19

Figure 2.2 is a representation of the steady-state
solution of the nonequilibrium distribution associated with
the wall at g = 0. By assumption, this distribution is
independent of any effects related to the physical presence
of the wall at g = Cl. Figure 2.2 shows that the delta
distribution center upon n = 4 at L = 0 rapidly broadens
and shifts toward the equilibrium distribution with

increasing g. Intuitively this result is expected since

32

Figure 2.2

The probability distribution {xn(§)) for various L when

0 = 1 and x4(0) = 1.

33

 

 

 

no

 

 

 

 

 

 

® @ @
@e
m 1

247 mm

OOOIZFW.

.. 3. game I.
rarbrbrsrbrs

©®©©©©

6 _ new. @ @n ®\_®n
6 5 4 3 2 I 0
o. o o. o. o. o. 0

assumes

Harmonic State , n

 

34

the layers farther removed from the wall indicate the
presence of the wall to a lesser degree. However, at
L = 3 the deviation from the equilibrium distribution is
still significant. The figure indicates that molecules of
the system leave the surface at g = O with far more vibrational
energy than the average heat bath molecules possess, gradually
transfer this energy to heat bath molecules in collisions
as they diffuse away from the surface, and approach the
equilibrium distributions of heat bath layers far from the
wall.

The solution of the distribution {wj} is analogous

to that of Equation 2.9, viz.,

wn(t) = kgoaéL)ln(k)e'k1/2(°1‘§) (2.11a)
aAL) = (1'9-0) E lk(m)wm(c1) (2.11b)
m=0

wm(ci) is the boundary condition imposed upon the nonequilibrium
distribution associated with the lower wall at g = 01.
When the energy of the nth harmonic state is
En = hv(n+1/2), the steady—state source term of Equation 2.1
is given by

a = _nEoEn((arn/at)HB—SI + (aSn/at)HB-SI)

2
n=0

En((arn/at)diffusion + (aSn/at)diffusion)

2 2
DnEOEn(V rn + V sn)

DnEOEn((JErj(c1))V2wn + (ij(0))V2xn). (2.12)

35
Integration of Equations 2.1 and 2.12, with the boundary

conditions T(g) = TU at g = O and T(g) = T at g = c

L 1’

yields the temperature equation:

T(g) = TU - (TU-TL)(§/c1) + c2J2sj(o)H(£) (2.13a)
where
H(§) = 420(n+1/2)(xn(0) - Xn(c) - (xn(0) - xn(cl))L/c1
+ (wh(0) - wn(§) - (wn(0) - wn(cl))L/ci) -
- (§rj(01))/(§sj(0))) (2.1313)
and
c2 = Dhu/k, (2.130)

where k is the interior conducibility and not the Boltzmann
constant. The first two terms on the right-hand side of
Equation 2.13a represent the linear terms which would be
present even in the absence of a source. The third term
accounts for the deviation from linearity of the temperature
due to interaction between the walls and the vibrational
degrees of freedon of the fluid. When the distributions
{rj(ci)} and §sj(0)§ are zero, the expected linear steady-
state distribution is obtained. In addition, it is easily
shown by Equations 2.4c and 2.12, that whenever {rj(g)] and
{sj(§)} are Boltzmann distributions, the source term
vanishes, and the temperature distribution becomes spatially
linear.

Figures 2.3, 2.4, and 2.5 indicate the importance of

 

36
the constant cl. The graphs represent the temperature

.(0) in the z

deviation from linearity divided by 022s]

J
coordinate system for 3 = 1, and various boundary conditions

{xn(0)} . It is necessary to divide by c2283.(0) because the
J

deviation from linearity is proportional to this process

dependent quantity and elucidation of 2sj(0) requires a
3

body of theory not explored. Later developments will relate

the magnitude of 2sj(0) and that of the deviation of
J

linearity. The choice 0 = 1 is made for convenience and
plays no role in the relaxation of SI. If 0 were larger,
however, the energy transferred between SI molecules and HB
molecules would be larger thereby causing larger deviations
from linearity of the temperature when SI is a non-Boltzmann
distribution. In order to study the wall effects, the

graphs are plotted for the distribution {wh(;)}= 0 only,

and the effect of only one of the walls is examined.

Figure 2.3 shows that when the nonequilibrium distribution
at the wall contains less energy than the Boltzmann distri-
bution at the temperature of the wall, the deviation from
linearity is negative. The temperature near the wall is

then less than that predicted by the Laplace equation.V2T = 0.
Furthermore, as the ratio of the rate of HB-SI interactions
to the rate of diffusion increases at constant I, 01
increases and the deviation from linearity approaches
-0.582cgso(0) at z = 0. This asymptotic value when x0(0) = 1

represents the maximum deviation from linearity for this

¥ ___

 

37

Figure 2.3

Negative of H(z) for 0 = 1, x0(0) = 1, and various c1.

0.|

0.0l

 

38

 

 

lLllI

 

CDCI: I
® Cl” 2

 

0.0

 

 

39

Figure 2.4

H(z) for 6 = 1, x1(0) = 1, and various 01’

 

 

110
IO

J @Cl3l
_ @Cl= 2
I @0|" 3
_ @C|= 6
— ©c.=l0

@c.=40
_ ®c|=w

 

0.!

 

 

 

0.0!

0.0 0.2 0.4 0.6 0.8

39

Figure 2.4

H(z) for 0 = 1, x1(0) = 1, and various 01’

1.0

 

QOI

 

 

 

@C'=I
®C|' 2
®C|= 3
@Q‘ 6
©C|=IO

 

 

 

 

0.0 '

 

41

 

Figure 2.5

H(z) for 0 = 1, x4(0) = 1, and various 01.

 

I0.0

OJ

 

 

 

 

 

 

 

 

 

 

 

 

43
nonequilibrium boundary distribution. The physical reasoning
is that for very large values of 01 the HB-SI interaction
term is so much larger than the diffusion term that all SI
molecules leaving the surface immediately absorb energy
from HR and relax to a Boltzmann distribution. Thus,
non—Boltzmann SI molecules do not exist further away from
the surface and no further alteration of the temperature
distribution can occur; the Laplace equation then describes
the temperature. However, the energy absorbed by SI from HB
in relaxing to a Boltzmann distribution lowers the temperature
of the system. The consequence is that for very large values
of 01 the Laplace equation must be solved subject to the
boundary conditions T(l) = TL and T(O) = TU - 0.582c230(0).
The boundary temperature jump thus represents the largest
deviation form linearity of temperature possible.

Figures 2.4 and 2.5 both have nonequilibrium
distributions at the wall which contain more energy than the
Boltzmann distribution, and the graphs indicate a deviation
from linearity which is a positive function of z. The
temperature in these two cases is greater than that predicted
by the Laplace equation. It is seen that, as 01 increases,
the limiting values of the temperature deviation are
0.41802sl(0) and 3.418c2s,1(0) for the cases x1(0) = 1 and
x4(0) = 1, respectively. The general trend is that the
greater the energy difference between §xn(0)} and the
Boltzmann distribution at the surface, the greater is the

‘temperature deviation from linearity divided by c2EZsj(0).
J

 

44

When equilibrium distributions are associated with
both walls, curves such as those in Figure 2.6 result. In

this figure, the distribution {wh(l)} 0 , w1(l) = 1

nil =
is associated with the wall at z = l. The distribution
{:sn(0)}m£0 = 0, 120(0) = 1 is associated with the wall at

z = 0. The total number of molecules in each of these
boundary conditions is assumed to be identical simply for
convenience. The curves result from the superposition of the
curves given in Figures 2.3 and 2.4 and exhibit the same
characteristics. One observes that in the central region
between the plates the temperature distribution is linear,
and hence Fourier's law applies. In the limit of very large

values of c the temperature distribution also appears to be

1:
linear. The source term for this limit is a simple delta
function at the boundary which results in the appearance of
a "temperature jump," or a discontinuity in temperature at
the fluid-solid interface when a temperature gradient is
present. This temperature jump is entirely analogous to

20-23

those discussed in many other works; see Section 1.3

for additional discussion and references.

2,3 Estimates of c1 and c2

 

The parameters 0 and c2 determine the Spatial extent

1
of the temperature nonlinearity from boundaries and the

magnitude of the nonlinearity, respectively. The ratio

_1_/e1 = (1)/ml” (2.11.)

has the dimension of length and measures the spatial domain

45

Figure 2.6

H(z) for 0 = 1, w1(l) = 1, x0(0) = 1, and various

values of 01.

 

46

 

0.6

 

 

 

 

 

 

 

 

2//

47

of the nonlinearity. Likewise,

p02 = pth/k (3.15)

has the dimension of temperature and measures the magnitude

of the nonlinearity; p is the number density of oscillators

of frequency v. By Equation 2.13 the product of p02, the
number fraction of SI vibrational degrees of freedom, and

the function H~1 is the nonlinear magnitude. Equations

2.14 and 2.15 provide a convenient test of the diffusion
mechanism's contribution to any observed nonlinear temperature
distribution. The properties ( ), k, and D may be

extracted from empirical tables, theoretical equations, or
semiempirical equations. The rate of interaction between

HB and SI, K, can be estimated by

K = z P (2.16)

where ZC is the number of collisions per molecule per unit
time and P1,O(v,T) is the probability per collision of the
harmonic state transition 1—¥O. P1,0 values for a number of
gas phase systems have been determined experimentally by
impact tube methods, infrared fluorescence photometry, and

17,24-26

chemiluminescence spectrophotometry. Fourier

O
transform of infrared and Raman vibrational band contours27’“8

and picosecond spectroscopy29-31 yield information concerning
vibrational relaxation times and pathways within the liquid
phase.

Table 2.1 contains estimates of _l_/c1 and 02 for gas phase

inolecules. The computations utilize the P1 0(v,T) tabulations
’

48
Table 2.1
Gas phase parameters c1 and 02 of the molecular fundamental
frequency 5. p, the number density of the normal mode 5,
is the product of the degeneracy of the symmetry species

and the molecular number density. T = 300 K; P = 1 atm.

 

 

 

 

 

 

 

Gas _ Symmetry _
Molecules fi/cm1 Species 103(l/c1)/cm 10 2pc2/K
(36. 37)
N2 2331 870 9.3
NO 1904 0.30 7.4
02 1554 610 6.1
F2 892 2.0 3.4
C12 557 0.63 2.0
C02 667 Wu 0.77 4.4
COS 520 w 0.28 3.2
SO2 518 a1 0.077 1.6
NH3 950 a1 0.022 3.2
PH3 992 a1 0.11 3.3
ASH 906 a 0.069 2.5
3 1
CH4 1306 f2 0.65 13.0
CH301 732 a1 0.11 2.3
CH Br 611 a 0.058 1.9
3 1

02H2 612 fig 0.11 3.5
Cqu 826 b2n 0.10 2.3
C2H6 289 alu 0.013 0.77

 

49
of Stevens,26 the ideal equation of state, the kinetic

theory collision rate, and first order kinetic theory thermal

conductivities and self-diffusion coefficients:32
zC = 4Pr§(ka/w)”1/2, (2.17)
D = 2.628 x 10“3(T3/M)1/2/(Pr§0(1'1)*(T*)), (2.18)
and
k = 6.066 x 10-5R(T/M)1/2(HOV/9R)+1)/(r§9(2’2)*(T*)).

(2.19)
M is the molecular weight, T* = kT/e, 9(1’1)*(T*) and
9(2’2)*(T*) are collision integrals tabulated by Hirschfelder
£3 21,32 and ro(X) and e are the Lennard-Jones(12-6)

potential parameters defined by
2 6
¢(r) = 4e<<rO/r)1 - (ro/r) ). (2.26)

The Table 2.1 computations utilize the r0 and e tabulations
of Stevens,26 Ibele,33 and Sherwood et al.34 In Equations
2.17-2.19 D is in cm2/sec when P is in atm; k is in

cal/cm sec deg when R is in cal/mole deg. The term
(3/5)((4CV/9R)+1) in Equation 2.19 is called the Eucken
correction. It takes into account approximately the
transfer of energy between translational and internal
degrees of freedom. Heat capacities from the JANAF

Thermochemical Tables35

are used in the computation of this
term.
Table 2.2 contains estimates of l/c1 and p00 for

liquid phase systems. The self-diffusion coefficients are

 

 

 

50

computed with Houghton's equation:38

D = (an/6nm)(m/dN)2/3 (2.21)

where d,n, and N are the mass density, viscosity, and
Avogadro's number, respectively. Values of d,1}, and k are
extracted from standard thermophysical tables.39’l10 The
collision rate within liquids is calculated with the cell

model equation of Bartoli and Litovitz:41

Zc = (8kT/nm)1/2(21/6p‘1/3- ro)‘1. (2.22)

This equation is valid because (a) it is consistent with the
model of vibrational energy transfer via isolated binary

30,31

molecular collisions, and (b) Calaway and Ewing have

demonstrated that the product of Equation 2.22 and P1,0
values determined via gas phase experiments is comparable
to vibrational relaxation times exhibited by liquid phase
systems( 1 psec). However, Equation 2.22 cannot be taken
too seriously because it fails to deal properly with the
concept of "collision" and is an inaccurate formula for the
computation of collision frequenciesf‘g’l‘i3
Table 2.1 reveals that the diffusional transport
mechanism can produce nonlinear temperature distributions
at 1 atm which extend 0.87 cm, 0.61 cm, and 0.002 cm from
boundaries for gaseous N2, 02, and F2, respectively.
Furthermore, when SI contains 1 molecular vibrational degree
of freedom out of 104, the nonlinear magnitude is 0.093 K,

0.061 K, and 0.034 K for gaseous N2, 02, and F2, respectively.

 

 

51

Table 2.2
Liquid phase parameters c1 and c2 of the molecular fundamental
frequency 0. p, the number density of the normal mode 5,
is the product of the degeneracy of the symmetry species
and the molecular number density. T = 300 K; P = 1 atm.

 

 

Liquid N _1 Symmetry 8
Molecules v/cm Species 10 (l/c1)/cm pc2/K
(36,37)

Br2 321 16 0.25
H20 1595 a1 2.7 4.8
CS2 397 ”u 25 3.4
CH2C12 282 a1 5.1 0.95
CHCl3 261 e 0.90 1.1
CClq 217 e 0.86 0.41
CHBOH 1030 a' 0.86 6.4
C6H6 410 6211 2.9 2.0

 

 

 

 

 

 

 

 

52
For the other gases l/cl< 10—3 cm at 1 atm; therefore, the
diffusional mechanism operating within these gases can only
produce temperature jumps at the gas-boundary interface.
The magnitude of the temperature jump is approximately
0.03 K when 1 vibrational degree of freedom in 104 is
contained within SI. However, the extent increases as
pressure decreases. From Equations 2.14 through 2.19, it

is found that for gases

;/61~:1/P and pc2~IP°. (2.23)

For example, a N20 pressure decrease from 1 atm to 0.1 atm
increases the nonlinearity extent from 3.5 x 10-4 cm to
3.5 x 10-3 cm; at 0.1 atm the nonlinear extent approaches
measurable dimensions while the magnitude remains unchanged.
Considering the typical gas parameters to be 1 01:11 x 10-4 cm
and pc2=£3 x 102 K at 1 atm, the diffusional mechanism can
cause most gases to exhibit nonlinear temperature distri-
butions which extend 0.01 cm from boundaries at 1.0 x 10'2 atm
and have a magnitude of 0.03 K provided 1 vibrational
degree of freedom in 104 belong to SI.

The diffusional energy transport mechanism can
contribute only a temperature jump to observed nonlinear
temperature distributions within liquids. Table 2.2

8 and 10"7 cm, and the

values of l/c1 lie between 10
ratio has a very weak dependence upon pressure. The jump

magnitude is typified by CS2. It is 0.034 K when 1 molecular

 

 

53

vibrational degree of freedom(? = 397 cm-1) in 102

belongs

to SI.

2.4 Model 2
As a final example, we again consider the situation

in which the fluid between the plates can be divided into
two systems: the system of interest(SI) and the heat bath
system(HB). In SI the vibrational degrees of freedom
cannot always be described by a Boltzmann distribution
while those in BB can be described by a Boltzmann distribution.
In addition, it is assumed that molecules of SI are of such
a density and relative position that resonant transfer of
vibrational energy occurs between molecules of SI but no
diffusion of molecules occurs. Model characteristics (a)
and (b) are then identical to those of Model 1, and the
HB-SI interaction term is given in Equation 2.40. However,
characteristic (0) for Model 2 is
(c) The propagation, or velocity-probability, of vibrational

energy transfer, V, determines the SI-SI interactions

according to the expression
(apn/at)SI_SI = v((arn/az) - (asn/az)), (2.24)

where {rn} is the number density distribution of SI

associated with the wall at z = l

and {sn} is the number
density distribution associated with the wall at z = 0.

With this model, then, we have

(apn/at) = (arn/at) + (asn/at) (2.256)

 

54

where

(arn/at) = V(arn/az) + 2(1-6'0)‘1(ne‘9rn_1 + (n+1)rn+1
-((n+1)e‘9 + n)rn), (2.251

(asn/at) = -V(asn/az) + K(1-e‘9)‘1(ne“9sn_1 + (n+1)sn+1

-((n+1)e-0 + n)sn).(2.25¢

The signs of the gradient terms are chosen so that V is
the velocity-probability in the direction 9321 from the
plate in question.

The steady-state solution of Equation 2.25 is easily
obtained from Appendix I of Montroll and Shuler4(see
Appendix). With the following definitions,

;' = ciz/_l_, c; = (Kl/V), wn = rn/Ern(°1)’ xn = sn/Esnw),

(2.26)
the solution of Equation 2.25, subject to the boundary
conditions {rn(c1)} and {sn(0)}, and the constraint

ISW' = 1 = 23 is
n n Hg)"

sn(§') = k30a£U)_1_n(k)e'-kv, (2.272
rh(g') = Ega§L1;n(k)e'k(°i‘4'), (2.27:
1,,(u) = e”n”B§O(1-e”) (2)04). (2.27.
aéU) = e-k0(1-e-0)£20eneln(k)sn(0), (2.27:
aéL) = e'k”(1-e"9) 2 en?;n(k)rn(ei). (2.27.

n=0

55
The first of Equation 2.12 can now be used in Equation 2.1
to determine the temperature distribution. Double integration
and evaluation of the two resulting constants with the
boundary conditions T(g') = TU at g' = o and T(g') = TL at
g' = oi yields, upon insertion of Equation 2.27,

T(L') = TU - (TU-TLHd/cp + cg §0(n+1/2)k§1k'1ln<k)< (U) .
- (1-e'k4' - <;'/ci)(1-e'k°i)) 2 sum)
m=0

+ al({L)(1-ek4' — (L'/ci)(1-ek°i))e'kcimform(°i))'

(2.28a)

where
65 = (hsz/kK) . (2.28b)

The first two terms to the right of the equality are the
linear terms; the third is the nonlinear term. In Equation
2.28b, k is the interior conducibility not the Boltzmann
constant.

Model 2 corresponds physically to propagation of
vibrational energy along large chains or local structures
of molecules which might form in liquids due to the presence
of walls. This mechanism would transfer energy far more
rapidly than diffusion and would result in larger nonlinear
temperature deviations. The mechanism is included as a
Speculative possibility which should be pursued experimentally.

For example, estimating the SI-SI interaction term

 

56
to be of the order of the velocity of sound in solids

( 105 cm/sec) we find
’

l/ciN 10-3cm and pcéNiO6 K (2.29)

for liquids.

2.§ Discussion

It indeed appears possible for vibrational degrees
of freedom to contribute to nonlinear temperature distri-
butions near walls. Model 1, the diffusion model, indicates
that for gases with typical thermal conductivities and
diffusion constants, only 1 in 104 molecules at the wall
need be in variance with the Boltzmann distribution to
achieve a temperature deviation of 10'2 degrees from
linearity. The mere presence of walls may cause a variance
of this magnitude by the reflection mechanism discussed in
Section 1.3. Surface-catalyzed reactions and the absorption
of reflected radiation by strongly emitting absorbing gases
are also capable of producing variances of this magnitude.
Weakly absorbing gases are not included in Model 1 because
the effect of the wall has been treated as a boundary-
value problem, which excludes direct, long-range interactions
between fluid and wall via radiation. Liquids are expected
to give nonlinear temperature deviations of 10.2 degrees
when 1 in 102 molecules do not satisfy the Boltzmann
distribution.

The strongly absorbing and emitting case of water

 

57

provides a specific application of the diffusion model.

For water _l__/c1 is the order of a molecular spacing. Thus,
the model predicts that only a temperature jump can be
exhibited by the liquid. This agrees with experiment

(see Section 4.3). The order of magnitude of this
temperature jump is indicated by the Table 2.3 tabulation
of pcz(u)Hw(v) at 300 K for various assumed delta distri-
butions of the upper boundary SI molecules and for the v2
bending mode, the ”L librational mode, and the hindered
translational modes, ”T and vT2.44’45 Hw(vL'Which is H(v) in the
limit as ciacn, is evaluated at z = 0(i&g,, ignoring the SI
distribution associated with the lower boundary). For
example, if xn(v2,0) = on’i and s1(v2,0) = 0.01p, then the
contribution of v2 to the temperature jump is 0.049 deg.
Also, if xn(vT,0) = 8n,2 and s1(vT,0) = 0.01p, then the
expected contribution to the temperature jump is 0.008 deg.
Temperature jumps of these magnitudes are important in many
experimental studies and must be considered. The weakness
of models 1 and 2 is the boundary value nature of these
models. This excludes the case of weakly absorbing and
emitting gases and liquids which can interact with the walls
via photon exchange at distances which are an appreciable

fraction of l.

 

 

58
Table 2.3

p02(v)Hw(v)'values for water at 300 K.and various boundary

 

 

conditions. p is the molecular number density, 81 j is the
9
Kronecker delta, and table units are deg.
,J N "' AI
x11(0) ”2 ”L UT UT2
1595 cm” 685 cm.1 193 cm- 60 cm"1
n,0
811,1 4.9 2.0 0.21 -0.36
an,2 9.8 4.1 0.80 -0.18
.2 . .
an’3 15 6 1 4 o 0053

 

 

 

 

 

 

CHAPTER 3

BRYNGDAHL INTERFEROMETRY

3.1 Introduction

Ingelstam-Bryngdahl type wavefront shearing
interferometry utilizes Savart plates to shear a plane
polarized wavefront of monochromatic light after passage
through a test object of non-uniform refractive index.

1’2 in 1955 and

3-8

The method was first described by Ingelstam
subsequently developed by Bryngdahl and coworkers.
Section 3.2 is devoted to a survey of the application
of Ingelstam-Bryngdahl interferoemtry to problems of
nonequilibrium heat and mass transport phenomena in liquids.
Section 3.3 introduces the particular Bryngdahl interferometer
which we used to study (a) the temperature dependence of the
refractive index of water; (b) temperature distributions in
nonequilibrium, steady state experiments on ethyl acetate,
benzene, carbon tetrachloride, and water; (0) the possibility
of temperature jumps at the water-metal interface; and (d)
the critical Rayleigh numbers associated with the Bénard

problem.

59

 

60
Equations for the unrefined data,(dmw(Z)/dZm)m=1,2 ..’
where W(Z) is the optical path associated with the vertical
image coordinate Z, are also presented in Section 3.3.
Unfortunately, the unrefined data are functions of the applied
temperature field and the cell dimensions. Section 3.4
explores this problem and describes a program for mapping
spatial derivatives of the refractive index from unrefined
data. The spatial derivatives of the refractive index,

(dmn/dzm) , are called the refined data. In the

m=1,2,.
process of refining the data, the discussion is restricted

to pure liquids which do not absorb the monochromatic
radiation of the interferometer. From the refined data
either the temperature derivatives of the refractive index

or the spatial derivatives of the temperature(the temperature
distribution) may be mapped. Equations which relate either

(dmn/dTm)m=1’g..

are developed in Sections 3.5 and 3.6, respectively.

or (me/dzm) to the refined data

m=1’2,..

61
3.2 Bgzggdahl Interferometgy: Applications

Gustafsson and coworkersg-z3 have been major

pioneers in the application of wavefront shearing interferometry

to problems of measurement of Soret coefficientsg’la,

10,13,14

thermal conductivity coefficients , refractive

11,12,17,21
9

indices interdiffusion coefficients of transparent

9’15’16'18’23, and interdiffusion coefficients of

19,20,22

liquids
molten salts. In 1965 Gustafsson 23.21-9 measured
the Soret coefficient for water solutions of CdSOA, AgNOB,
and KCl by observing the time and space evolution of a
Bryngdahl interferometric image of a "sandwich" type cell.
The procedure involves application of a temperature
gradient to an equilibrium system and analysis of the

thermodiffusional demixing process; or, alternatively,

removal of the temperature gradient from a nonequilibrium

system and observation.of the mixing process. In 1970 wallin
and Wallin17 increased the sensitivity of the interferometer
with a double exposure technique which is capable of
measuring the Soret effect of water solutions a few

18

thousandths molal in Cdsoq. The second exposure is made

after a 1800 rotation of one of the Savart plates about
the optical axis.

In 1967 and 1968 Gustafsson et a110'13'1“

used a plane source of heat in a transient method to

measure the thermal conductivity of water, KNO3, LiNO3,

62
NaN03, RbN03, and CsN03. The Bryngdahl interferometer was
used to diSplay the time dependent temperature distribution
close to the plane heat source. The average deviation of

the measured thermal conductivity of water from the

recommended values of McLaughlinzq was 0.37%.

An isothermal method of measuring absolute refractive

indices has been developed by Gustafsson's research

11.12.17.21

group. The method has been used to measure

the temperature dependence of the refractive index of

21 6

with an accuracy of about 11 x 10' , and that

12

water
of the molten alkali nitrates with an accuracy of about
:3 x 10-5. The wavefront shearing interferometer of

this method combines two light beams, one passing through
the liquid and the other through a rotatable quartz plate
inside the liquid. The refractive index of the quartz
plate serves as a reference. The rotatable reference plate
technique is well known, and the equations which describe
the optical path differences as a function of rotation
angle were developed by Adams25 in 1915.

Finally, Gustafsson's research group has developed
an isothermal, bottom layer technique for the measurement
of interdiffusion coefficients of transparent liquids.

A crystal of the diffusing substance is dropped to the
bottom of the liquid container in initiation of a bottom
layer. Bryngdahl interferometry is used to observe the

transient diffusion process. The procedure requires that the

density of the crystal be much greater than that of the

 

63
liquid. In this way the amount of material being dissolved

during the "drap" is negligible. The diffusion of KCl and

15,16,23

CdSOA in pure water at 22°C and the diffusion of

silver ions15 into the alkali nitrates NaNOZ, KN03, and RbNO3

from 310°C to 380°C has been studied with the "drop", plane

source technique. The plane source diffusion technique has

also been used to measure the diffusion coefficients of
F‘ and Br‘ in LiN03; F‘, Br”, I”, 003'2, and 804-2 in NaN03;

and I” and 003-2 in RbN03.2O The temperature range of
melting point to 100°C above the melting point was studied.
The interdiffusion coefficient of thallous ion(Tl*) in
LiNO has been determined in the temperature range of

3

260°C to 355°C; Van der Waals interactions are shown to
contribute significantly to the diffusion rate.22

Thomas and coworkers have used Bryngdahl interferometry

to measure the diffusion coefficients of mono-, di-,
and tri-ethanolamines in aqueous solutions at infinite
dilution26'27 and have made a comparison with existing
semi-empirical equations. The results demonstrate the
inadequacy of these equations and suggest that hydration

of the ethanolamines plays an important role in the diffusion
process. Thomas gt_gl. have also used a special interferometric

method with increased vertical shear in conjunction with

 

64

28,29 30

a pressure transducer to study absorption and desorption
of gas at the gas-liquid interface. The absorption
experimental systems include C02-water, 02H2-water,
NH3-water, S02-water, and 002-propylene carbonate. The
desorption experimental systems include carbon dioxide-water
acetylene-water, ammonia-water, sulphur dioxide—water, and
acetone-water. Anomalously large transport of solute is
explained in terms of convective disturbances, eddies,

and microflows at the interface. The cellular structure

of the disturbances is smaller than that which is directly
observable by interferometric technique. Onset times of

the disturbances were reported. Temperature effects at

the interface which could produce such disturbances and could
even be responsible for the observed fringe shift were
reportedly eliminated by the proper choice of systems for

30

examination, and by using dilute solutions. The studies

are then limited to buoyancy and surface tension effects.

31 have developed an isothermal

Pepela 91 El.
diffusiometer which features Bryngdahl-Ingelstam interferometry.
They have determined the diffusion coefficients at 25°C
of aqueous solutions of sucrose, n-butyl alcohol, magnesium
sulfate, tetra-n-propylammonium bromide, tetra-n-butylammonium
bromide, thiourea, glycine anhydride, é-caprolactam, and

mannitol. The results for MgSOA-water, n-PquBr-water, and

thiourea-water are compared to Gouy interferometric and

 

6S

conductance methods. Data obtained from the three methods
are in excellent agreement. The authors believe that the
diffusion coefficients may be measured with their technique
to a precision of 20.3%. Staker and Dunlop32 have also
used this isothermal technique to measure the diffusion
coefficients at 25°C of the benzene-carbon tetrachloride
binary liquid over the whole concentration range. The
results agree with those of Gouy interferometry to within 1%.

Anderson and Home33 have used a nonisothermal,
wavefront shearing method very similar to the demixing-mixing

9

experiments of Gustafsson to determine concentration and
temperature dependence of the thermal diffusion factor
and diffusion coefficient for the CClq-C6H12 system
between 20°C and 35°C. The standard error of the thermal
diffusion factor is about 1% and that of the diffusion
coefficient about 3%. This interferometer was also used

by Olson and Horneak

to measure the temperature dependence

of the refractive indices of water, carbon tetrachloride,
cyclohexane, and benzene at 25°C with a nonisothermal
technique. They report uneXpected parabolic steady state
fringe patterns which are attributed to nonlinear temperature
distributions associated with metal boundaries.

Mitchell and Tyrrell35 use a Bryngdahl interferometer to
measure the diffusion coefficients of dilute solutions of benzene,
phenol, and resorcinol in propane-1,2-diol. The application
of a temperature gradient creates a gradient of solute.

The observation of the transient decay of the concentration

gradient upon removal of the temperature gradient allows

66

the calculation of diffusion coefficients. Results show

the importance of solute-solvent interactions in

diffusion. The authors claim that diffusion

coefficients as low as 5 x 10'"8 cm,2sec-1 can be

measured. In a later paper Skipp and Tyrrell36

report interdiffusion coefficients for carbon tetrachloride,

carbon tetrabromide, methyl pivalate, chlorobenzene, and

phenol in propane-1,2-diol. The results show that spherical

molecules(CC14, Cqu, methyl pivalate) diffuse at a rate

comparable to the prediction of the Stokes-Einstein

relationship D = kT/4wnr5where the molecular radius(r) is

calculated from molar volume data, and n is the viscosity.

However, planar molecules diffuse about twice as fast as

expected. The evidence suggests that free rotation of

planar solutes during diffusion is prevented in propane-1,2-

diol. In more mobile solvents free rotation is possible.
The final major research group included in this

37-42 They have used

survey is that of Porsch and Kubin.
Bryngdahl interferometry and isothermal, flowing-boundary
cells to measure the diffusion coefficients of

37939941 39,111

biphenyl-benzene, sucrose-water, and monodisperse

polystyrene-toluene39’41 at 25°C. Further, they have
determined up to four average diffusion coefficients38’40
for the polydiSperse polyisobutylene-heptane system, and
unfractionated polystyrenes in toluene. The average

diffusion coefficients are related to the moments of the

diffusion coefficient. Porsch

 

67

zxnd Kubin have also measured the concentration dependence
()1 the first four average diffusion coefficients of a
I)olystyrene-toluene system42 having a broad molecular
tveight distribution.

Bryngdahl wavefront shearing interferometry is
(Inc of a number of interferometric methods which have
lleen used successfully to study heat and mass transport.
()ther methods include Gouy interferometry, Rayleigh
iliterferometry, wedge-interference interferometry, Jamin,
hhach-Zehnder, and Michelson interferometry, and multiple-beam
iriterferometry. Useful reviews of these methods and

tlieir applications are availablefgu51

. B dahl Interferomet : A aratus Cell Data
The Bryngdahl interferometer, cell-water jacket
zassembly, and temperature measurement technique used in

ttlis work were developed by Anderson52 53

and Olson.
TTle interferometer features a Spectra Physics Stabilite
120 helium-neon laser with 5.0 mw output power at 632.8 nm
811d a Spectra Physics 336 multiwavelength collimating
lxans of collimation within 1/8 wavelength over a 4.6 cm
zxperture. The laser is oriented so that the cell is
illuminated with radiation polarized in the direction
vdiich bisects the x,z axes shown in Figure 3.1. After
passage through a cell of height I and length a, the
L1-L2 lens combination reduces the cell image to a size

compatable with the modified Savart plates QI and Q2.

68

Figure 3.1
(A) The interferometer.
1. collimator
2. cell-water jacket assembly
3. lens, L1
4. lens, L2
5. modified Savart beam splitter, Q1
6. lens, L3
7. modified Savart beam splitter, Q2
8. analyser, A

9. lens (optional)
10. camera

(B) The cell-water jacket assembly.

1. lower assembly
2. upper assembly
3. cell

x...

69

 

 

 

70

The first beam splitter, Q1, splits the wavefront into
two identical, vertically sheared wavefronts. Before
entering Q2 the double image passes through the converging
lens L3. This introduces a small shear angle between the
wavefronts which produces constant path length differences.
After Q2 the radiation passes through a polarization
analyser A which is crossed with the laser polarization.
The resulting image is sharply focused and recorded with
a Polaroid MP-3 Land Camera using Polaroid Type 42 black
and white film. The modified Savart plates were manurfactured
by Valpey Co., Halliston, Mass. Each plate consists of
three 2 cm x 2 cm x 0.7 cm quartz subplates bonded without
cement, flat to 1/20 wavelength, and parallel to 2 seconds
of arc. The principal plane of the second subplate, which
serves as a half-wave plate, is oriented 45° from the
principal axis of the first plate; the principfl. axis of
the third plate is oriented 900 from the principal axis
of the first plate. This arrangement results in symmetrical
light paths through the whole plate.8

The cell (see Figure 3.1) is made of four pieces
of % inch thick glass cemented together
with Chemgrip epoxy adhesive (Chemical Rubber Co., Cleveland,
Ohio). 7 cm is a typical width and length (a),and the
height (_l_) is typically 4 mm to 15 mm. The cell is
sandwiched between two highly polished, silver plated,
% inch copper plates. Each of these plates is secured

to a magnesium water jacket which serves to control the

 

fri—

71

temperature of the plate. The volume of a water jacket
is 230 ml. In the isothermal configuration water is fed
to both water jackets from a single 10 liter Neslab water
bath (Neslab Instruments, Durham, N.H.). In the nonisothermal
configuration each water jacket is fed separately from
45 liter Neslab water baths. The temperature of each
bath is controlled to i0.003 deg with Versa—Therm proportional
electronic temperature controllers (Cole-Palmer Instrument
Co.) balanced against coils containing cold water feed
from a 25 gallon Lab-Line heat bath (Lab-Line Instruments
Inc., Melrose Park, Ill.) maintained at about 14°C.
Water is pumped from the heat baths to the water jackets
at a flow rate of 2.8 l/min.

Temperature measurement is made with 40 gauge
copper-constantan thermocouples soldered to matched
20 gauge copper-constantan leads. The 20 gauge leads are
placed in mercury containing tubes and inserted well
within an air saturated ice-water slush in accordance
with standard procedure.54’55 Copper leads connect the
mercury tubes with the potentiometer. Thermocouple
electromotive force is measured with a Leeds and Northrup
K-3 Universal Potentiometer. The Eppley unsaturated
cadmium sulfate standard cell used in the potentiometer
was calibrated in terms of the U.S. legal volt effective
January 1, 1969,by Leeds and Northrup Co. prior to
experimentation. Uncertainty in null detection is 10.1 av,

whichcorresponds to i 0.003 deg. Thermocouple electromotive

72
force is standardized in the 1968 International Practical
Temperature Scale (IPTS-68)54’56 with the secondary
reference points: the ice point, the sodium decahydrate
sulfate—monohydrate sulfate solution point, the pressure
corrected56 boiling point of water. The resulting calibration
is

2E - 5.2922 10'7E2 (5.1)

T = 2.5577 10‘
where E is electromotive force measured in nv, and T is
temperature in 0C. This calibration is in good agreement
with National Bureau of Standard recommended (March,1974)
reference tables based on the IPTS-68.57
To determine the temperature of a silver plate, the temperatures
of two thermocouples (one on each side of the cell, out

of the optical path, and in intimate contact with the silver)

are averaged.

52 53

In the technique practiced by Anderson and Olson ,
the cell was greased, top and bottom, with silicone

or fluorosilicone lubricant and loosely clamped between

the silver plates. The cell was then filled through

a filling tube which is part of the upper plate and was
subsequently pushed away from the filling hole. This
procedure results in a fluid tight seal but has the
disadvantage that excess grease prevents the interferometric
examination of the fluid near the top and bottom walls.
Weeliminated.this difficulty by hand grinding the sealing
surfaces of the glass cell to 0.001 inch flatness with

progressively finer grades of the silicon carbide Crystolon

73
(Norton Abrasives, Worcester, Mass.). The cell is firmly
clamped between the silver plates with four securing nuts. An
interferometric check is made to insure that the glass cell is
not distorted. Evaporation of liquids such as carbon tetrachloride
and ethyl acetate through the silver-glass interface
is apparent one day after filling. This is a very long
time relative to the duration of the pure fluid experiments
of this work. Additionlly, to improve the optical quality
of the cell, the optical faces are ground with ferric
oxide powder. Figure 3.2 demonstates the quality of the
polished cell by comparing the best quality image obtainable
with the interferometer, the image of a slit, with the
image of the assembled polished cell-silver plate-water jacket
assembly in the isothermal configuration. The comparison shows
that the shear parameter, 2D, is blurred in the cell
image and fringes are not as sharp. Blurring of the shear
parameter is due to roughnessof the cell edge. Degradation
of the fringe shape is caused by optical face, surface,
roughness. The quality of the cell image is satisfactory,
however, and the cell height parameter, H, the shear
parameter, 2D, and the distance between fringes 8 can be
measured within 1% accuracy.
The experimental procedure is:
1. Establish the isothermal configuration by determining
the temperature difference
ATO = TU - TL
to be applied between the top and bottom silver plates.

 

74

Figure 3.2

(A) Best quality image obtainable with interferometer, the
slit image. (B) Isothermal cell-water jacket assembly
image. Z is the vertical image coordinate measured from
the upper, bounding, plate. 2D is the shear parameter.

H is the image cell height. 8 is the distance between

fringes in the isothermal cell configuration.

 

 

76
Adjust both plates to the mean temperature; i.e. TL + ATo/2.
TL is the temperature of the lower plate in the nonisothermal
configuration; TU is the temperature of the upper plate.
2. Fill cell with liquid, wait for temperature equilibration,
focus front face of cell onto camera.
3. Take photograph of isothermal cell image for evaluation
of 2D, H, and 5 (see Figure 3.2).
4. Apply Amo and record number of fringes which evolve
at image center in establishment of steady state.
5. Focus front face of cell. Take photogragh of
steady state image.
For further details of the temperature switching device,
see Anderson;52 for details of the photomultiplier
arrangement used in fringe counting, see Olson.53 The
nonisothermal image photograph is used to determine the
exact number of fringes, N(Z), which evolve in the establishment
of the steady state.
According to Bryngdahl8 and Olson,34’53 the image
fringe pattern is described by
W(Z+D) — w(z—D) = iAN(z) (3.2)
where W(Z) is the optical path of the light pencil associated
with the vertical image coordinate Z which is optically
conjugate to the cell vertical coordinate 2; 2D is the
shear parameter (see figure 3.2); A is the wavelength of
the interferometric light source (632.8 nm) and N(Z) is

the number of fringes which have evolved at A. The sign

of Equation 5.2 is altered if Q2 is rotated 1800 or if

77
the sign of ATO is altered. To write Equation 3.2 in a
snare useful form, expand W(ZiD) in a Taylor series about Z.
T116 even terms vanish leaving

iN(Z) = (2/l)2 (Dn/n!)(an(Z)/dZn). (3.3)

n=1,3, ..
Fixrthermore, let X(Z) be the horizontal image coordinate
off a particular fringe. Then
X(Z) = 8N(Z) + arbitrary constant. (3.4)
ngfferentiation of Equation 3.3 m times with respect to Z

yields the matrix equation

N8 _ "I3 0 133/3! 0 05/5! ".7 dW/dZ—
dX/dZ 0 D 0 133/3! 0 . . . . d2W/dZ2
d2X/dZ2 0 0 D o D3/5! . . . . d3w/dz3

: d3x/dz3 = (223/1) 0 0 o D 0 dQW/dzl’ .(3.5)
(fix/dz“ 0 o 0 0 D . . . . d5w/dz5
__ . _J _._ . . . . _ . _

 

 

 

 

 

The parameters 5 and D are known from the isothermal cell

Photographs. N(Z) and (de/dZm) are measurable

m=1,2..
from the nonisothermal image photographs. Thus, the inverse
of Equation 3.5 serves to determine the derivatives of the
Optical path with respect to the vertical coordinate; i.e.
(de/dzm)m=1,2... These derivatives are called the
"unrefined data" for reasons which will become apparent in the
next section. Assuming that the higher order fringe

derivatives, (de/dzm)m_5 6 , are equal to zero and that the
"',00

 

 

 

78

D matrix is non-singular, inversion of Equation 3.5 yields

dw/dz 7f 0 —D2/6 0 7D4/360 ‘7h6
0
d“w/dz2 0 1 0 —D2/6 0 dK/dZ
i d3w/dz3 = (A/2D6) 0 0 1 0 -D2/6 d2X/d52 . (3.6)
4. .4 3. ”3
d W/dZ 0 0 0 1 0 d x/dn
dSW/dhéd 0 0 0 0 1 dl‘X/dzl1

All quantities on the right hand side of the equality
are experimentally determined. Equation 3.6 is the
working equation for the determination of the unrefined data.
3.4 Refined Data; Refraction of Light

Defining the optical path to be the path experienced by
a radiation pencil traversing the cell, spatial derivatives of
the optical path are not properties of the liquid. The optical
path depends upon the length of the cell,a, and the magnitude of
the applied temperature gradient. Relative to the isothermal
liquid state, a temperature gradient causes an increase in the
optical path experienced by the interferometer photon
beam transversing the liquid. The beam curves toward the
cooler temperatures due to the drag effect of greater
density. This is illustrated in Figure 3.3. In radial
temperature fields this effect is termed "thermal blooming."66
To remove experimental configurational aspects of the
unrefined data, it is necessary (a) to determine the light
Path, z(y) in Figure 3.3, of a photon beam of infinitesimal

cross section which enters the liquid at z (b) to determine

0;
the dependence of the unrefined data upon cell length, and

upon spatial derivatives of the refractive index; and (c) to

 

 

79

Figure 3.3

Refraction of light in a one dimensional gradient. z(y) is
the path of a photon pencil of infinitesimal cross
section which enters the liquid cell at z = Z0' 21. is

the exit coordinate. TU ) TL.

 

 

 

9; ///// ////TJ//////////

 

 

 

 

 

m
(n
2 IIIII
O"

 

/:54///////// :17//////;/Z

81
determine eventually either the temperature dependence
of the refractive index or the spatial derivatives of the
temperature. The spatial derivatives of the refractive
index are called the "refined data."
Many of the equations derived below have been studied

50’58'62 We discuss

and published with slight modification.
these equations in detail here for the purpose of establishing
an iterative method for the numerical mapping of refined data

from unrefined. The analysis is restricted to one component

fluids in one dimensional refractive index fields, but it _
could be extended to multicomponent systems if the variation
of the optical path with mole fraction were included.

The possiblity of absorption of interferometric

light by the liquid is totally neglected. This neglect

is justified for the one component, weakly absorbing

liquids examined in this work because there is no experimental
observation of significant thermal blooming in these

'7 l . .
6J’6l However, in cases for which the liquid

liquids.
is moderately or strongly absorbing, the temperature
field within the liquid is altered. This results in
significant thermal bloomingGE-67 and necessitates the
inclusion of absorption effects.

Abnormally large thermal diffusion coefficients may render
thermal blooming very important in binary mixtures near

the consolute critical point.64’68’69 This is a Soret effect
(diffusion in a temperature gradient) response to the

temperature gradient associated with absorbed interferometric

82

radiation. Concentration differences across the beam of
a few parts in 107 give appreciable thermal blooming at
very low powers of the illumination radiation.64

Consider a beam of light of infinitesimal cross
section and frequency v which enters the liquid at the
coordinate (20,0) indicated in Figure 3.3. The path, 8,
traveled by this beam through the liquid is classically
determined by Fermat's variational principle70 for the
optical path;

5W(zo) = 0In(z)ds = O (3.7)

where W(zo) is the optical path of the photon pencil
entering the liquid at zo, and n(z) is the refractive index.
In a one dimensional refractive index field, the path
differential is

1/2az. (3.8)

ds = ((dy/dz)2 + 1)

Substitution into Equation 3.7 yields
0In(z)((dy./dz)2 + 1)1/2dz = 0. (3.9)
It is well known71 that this variation statement implies
that the integrand satisfies Euler's equation,
(a/az)<a/a<ay/az))(n(1 + (ay/az)2)1/2>
- (a/ay)<n<1 + (ay/az)2>1/2) = o. (3.10)

The second term is zer0,and the first is equivalent to

(d/az)(n(ay/dz>(1 + (dy/dz)2)'1/2)= 0. (3.11)
Integration, evaluation of the integration constant at

z = z and use of the experimental criterion dz(0)/dy = 0

0’
Yields

«n(z)/neon2 - 1)(dy/dz)2 = 1; (3.12)

83

or,
ay = :az/<(n<z)/n(zo>)2 - 1>1/2. (3.13)

Because y and z must not be imaginary,
n(Z)/n(zo) > 1. (3.14)

and the light beam must curve toward the higher refractive

indices; that is, toward the lower temperatures. Thus,

if the temperature decreases with increasing 2 in a

particular experimental configuration, then the positive

sign is used in Equation 3.13. If temperature increases

with increasing 2, then the negative sign must be retained.
We now restrictihe discussion to the case of heating

from above with the z coordinate originating from the

upper wall. The resulting equations apply to the

case of heating from below but with the coordinate system

originating from the lower wall. To facilitate the integration

of Equation 3.13, expand n(z)/n(zo) in a Taylor series

about z = z . Then,

0
n(z)/n(zo) = msgmmzfl (3.15a)
where
Nm = (1/m!n(zo))(dmn(zo)/dzom) (3.15b)
and
A2 = z - zo. (3.15c)
Furthermore,
(n(z)/n(zo))2 evsgzvmzfl (3.16a)
Ivhere
M =gum'NV'm. (3.16b)
maO

lVith Equations 3.15c, 3.16a, and the requirement y(zo)=0,

84

integration of Equation 3.13 yields

Aj(v 21M Wl/gdw. (3.17)

Table 3.1 shows that for water in a temperature gradient
of 10 deg/cm, (Mv)v_ _3 4 are of negligible magnitude
"" ’0.

relative to M They can, therefore, be neglected.

2.
The refractive indices of most liquids have first order
temperature variations which are less dependent upon
temperature than those of water; for them, (Mv)v=3 4..
are less important. Since Am ( 10m, the third order term
of Equation 3.17 is less than 0.1% of the second order
term, and can safely by discarded. Equation 3.17 becomes

.Az

y = $(M1w + M2 wz) 1/2dw (3.18a)
= (-M2)'1/2cos’1(2M2M;tAz + 1). (3.18b)

Inversion yields the desired equation,
AZ = M1(2M2)-1(cos((-M2)1/2y) - 1). (3.19)

This describes z(y) for a photon beam which enters the
liquid at 20 normal to the cell face.

It is now possible to write the Optical path as a
function of the spatial derivatives of the refractive index
and the length of the cell, a. Using the path integral
expression for the Optical path (Equation 3.7). Equations
3.8, 3.12, and 3.16a, we find

a
W(zo) = n(zo);E0Mv éQSz)vdy. (3,20)

85

Table 3.1

Derivative estimates of Mv (Equation 3.16b) for

water in a 10 deg/cm temperature gradient.

 

 

 

 

 

 

 

v Mv de/dz dsz/dz2
1 10"3cm'1 -3x10“‘cm'2 6x10"7cm"3
2 -1O-qcm-2 --10"7cm-3

3 -1o"7cm'3 _—
4 10'8cm'“ —————- ___-—

 

 

 

 

 

 

86
The indicated integration requires the use of Equation

3.19. The integrated form is
w(zo>/n(zo) = (1 + mf/(8<—M2)))a
-(Mi/(16(-M2)3/2))sin(2(-M2)1/2a). (3.21)
Expansian in powers of a, yields
w(zo)/n(zo) = a + (Mi/12)a3 - (Mf(-M2)/6o)a5
+ (Mf(—M2)2/630)a7 - (Mf(-M2)3/11340)a9

+ . . . . (3.22)

. m m
The refined data, (d n(zo)/dzo)m=1’2. , are now mapped

from (dmw(z )/dzm) by taking m derivathms with
0 O

m=1,2,..

respect to zo. The result is

a(dmn(zo)/dz§) = amw(zo)/az§ - iEOB§a2i+3 (3.23a)

where

(1/12)(dm(an)/dz§)

0335
l I

{11
ll

m (1/6o)(dm(anM2)/dz§)
(3.23b)

Bm (1/63o)(dm(anMg)/dz§)

N)
II

C!
II

m (1/11340)(dm(anMg)/dz§).

Additional equations are needed to associate the
m)

o m=1,2..‘ To flnd these

unareiined data with (de(zo)/dz
eQHations, let (a) Z(zo) be the image coordinate associated
Witfli the cell face coordinate zo, (b) yI be the plane of

fOClis within the cell, (c) zI(zo) be the image plane

 

 

 

87
z coordinate of the photon pencil associated with 20, and
(d) note that H/l is the magnification factor of the
interferometer. Then,
lz(zo)/H = 21(20) - 21(0) (3.24)
where 21(0) = 0 when the front face of the cell is focused.

Use of Equation 3.19 in Equation 3.24 yields

lZ(zo)/H = z + (M1(zo)/(2m2(zo))>(cos(<-M2(zo))1/2y1) - 1)

0

- (M1(o)/(2M2(o>))(cos((-M2(o))1/2y1) - 1).
(3.25)

Taking m derivatives with respect to 20, we find
(l/H)(dmZ(zo)/dzm) = am,1+ (yi/4)<amml(zo)/dz§) + - - (3.26)

where m = 1,2.. Equation 3.25 is used to determine

the conjugate image coordinate, Z, of a particular cell
face coordinate, zo; and Equation 3.26 is used to determine
derivatives of the optical path with respect to cell

face coordinate 20.

The difficulty with using Equations 3.25, 3.26, and
m)

. . m
3.23a is that the refined data, (d n(zo)/dzo m=1,2.

., appears

to the right of all equalities. A simple iteration procedure is
possible, however. The 1th order iteration proceeds as follows:
1. Assume an ith order approximation to the set

(dmn(zo)/dzm)

o m=1,2.. and calculate M1(zo) and M2(zo) in

the ith order with Equation 3.16b. Use these in the right
hand side of Equation 3.25 to determine z(zo) in the

ith order. The unrefined data, (de/dzm)m_1 2 , are now
_ , to

 

88

associated with 20. A convenient ch order approximation to

M1 18 zero; i.e., (M1)ch order = O.

2. Use the ith order approximation to M1(zo) in the

right hand side of Equation 3.26 to calculate (dmZ(zO)/dz§)m_1 2 .
"' , o.

3. Calculate (de(zo)/dz:) with the chain rule equations

m=1,2..

dW(zo)/dzo = (dz/dzo)(dw(z)/dz)

d2w(zo)/dz§ = (dZZ/dz§)(dW(Z)/dz)
(3.27)
+ (dZ/dzo)2(d2W(Z)/dzz)

etc.

4. Use the ith order approximation of (de(z )/dz _ ,
— O 0 Ill—1,2...

M M and the derivatives of M1 and M2 in the right

1’ 2’
hand side of Equation 3.23 to calculate the iiith order
approximation of (dmn(zo)/dz§)m=1’2 .’
5. Iterate until two successive approximations agree within
the desired accuracy.

The refined data of any particular nonequilibrium
experimental configuration are now unambiguiusly determined.
From these data it is possible to map either the temperature
derivatives of the refractive index or the spatial derivatives
of the temperature. Section 3.5 and 3.6 establish the
mathematical procedures for calculation of (dun/dTm)m=1’2...
and (me/dzm)m=1’2.. from the refined data, respectively.

89
3.5 Refractive Index: Temperature Dependence
For a one component liquid the temperature
derivatives of the refractive index are simply related to

the refined data, (dmn(z)/dzm) by the chain rule

m=1,2..’
expressions
dn/dT = (dz/dT)(dn/dz)

d2n/dT2 = (dzz/dT2)(dn/dz)
(3.28)

+ (dz/dT)2(d2n/d22)
etc.

These equations can be used to evaluate the temperature
derivatives of the refractive index provided the temperature
distribution is known. In the absence of unusual energy
sources or anomalous energy fluxes, the temperature
distribution is easily determined in the nonflowing, steady
state by72-74

(d/dz)k(dT/dz) = 0 (3.29)
where h is the thermal conductivity. Equation 3.29
must be solved subject to the boundary conditions

T(z) = T at z = O

U (3.30)

T(z) = TL at z = l

where TU is the temperature of the upper plate, and
TL is the temperature of the lower plate.

The temperature dependence of the thermal conductivity

is adequately described by the second order, truncated Taylor

seriequ

k(T) = k(TM) + (dk(TM)/dT)(T - TM)

+ (1/2)(d2k(mM)/dT2)(T - TM)2
where TM is the mean temperature defined by

(3.31)

89
3.5 Refractive Index: Temperature Dependence
For a one component liquid the temperature
derivatives of the refractive index are simply related to
the refined data, (dmn(z)/dzm)m=1’2.., by the chain rule
expressions

dn/dT = (dz/dT)(dn/dz)

d2n/dT2 = (d2z/dT2)(dn/dz)
(3.28)

+ (dz/dT)2(d2n/d22)
etc.

These equations can be used to evaluate the temperature
derivatives of the refractive index provided the temperature
distribution is known. In the absence of unusual energy
sources or anomalous energy fluxes, the temperature
distribution is easily determined in the nonflowing, steady
state by72'74

(d/dz)k(dT/dz) = 0 (3.29)
where k is the thermal conductivity. Equation 3.29
must be solved subject to the boundary conditions

T(z) = TU at z = o

T(z) T at z = 1

L
where TU is the temperature of the upper plate, and

TL is the temperature of the lower plate.
The temperature dependence of the thermal conductivity

is adequately described by the second order, truncated Taylor

serieszq

k(T) = k(TM) + (dk(TM)/dT)(T - rM)

2 2 2 . 1
+ (1/2)(d k(TM)/dT )(T - TM) (3 3 )
where TM is the mean temperature defined by

90
T = TU -.ATo/2 = TL + AH0/2 (3.32a)
and
as; = TU - TL. (3.32b)
With Equations 3.31 and 3.30, the integrated form of

Equation 3.29 is

2
012 =£Eé1/(n+1)!)(dnk(TM)/dTn)((T - TM)n+1 - osTo/2)n+1)
h (3.33a)
W ere
c1 = -1‘1(k(TM)ATO + (1/3)(a2k<TM)/dw2)(mo/2)3>. (3.331s)

T(z) is a root of Equation 3.33a (a cubic equation) and
is defined to be T3.33(z). It is numerically computed
from the above equations.
From the refined data associated with z0 in a
particular steady state experiment, Equation 3.33a is used
to determine T(zo). The temperature derivatives of
Equation 3.33a, evaluated at T(zo), are used in Equation 3.28
to calculate (dmn(T)/dTm)m=1’2...
3.6 Temperature Distributigg
When the temperature dependence of the refractive
index is known, it is possible to map the temperature
distribution from the refined data. By the chain rule,
dT/dz = (dn/dz)(dT/dn)
d2T/dz2 = (d2n/dz2)(dT/dn)
+ (dn/dz)(dT2/dn2) (3.34)

etc.

91
On the right hand side of these equations, the

(amn(zo)/dzԤ) are known, and the (me(zo)/dnm)

m=10200 m=1’2'°

can be calculated from the empirical cubic equation

n(T) = A + BT + 0T2 + DT3 (3.35)
provided T(zo) is known. Equation 3.35 is adequate for
visible wavelengths in the ordinary temperature range 20°C
to 40°C. The coefficient of the cubic term is usually
vanishingly small. Because T(zo) is an unknown, an
iteration procedure is necessary. A suitable ith order
iteration is:

1. Assume an ith order approximation to T(zo); calculate
the ith order approximation to (me(zo)/dz§)m=1’2..

with Equations 3.34 and 3.35. A convenient 9th order

approximation to T(zo) is
(T(zo))9th = TU - (ATo/2)zo. (3.36)

2. Calculate the i+1th order approximation to T(zo)

from the Taylor expansion

T(zo) = §O(me(zo-€)/dzm)(ém/m!). (3.37)

where E is some small distance, and TU is known.

 

CHAPTER 4

NONISOTHERMAL, NONCONVECTING LIQUID STATES

4.1 Introduction
As temperature increases, liquid density decreases.
Consequently, when a horizontal parallel plate apparatus
is heated from above, the liquid density increases as the
lower plate is approached. This results in a liquid
layer which is stable to convective motion(provided that
cell end effects are unimportant). A11 liquid motion decays
to stable, time-independent, nonconvecting thermodynamic
states. Far from boundary or interface, measurements of
liquid temperature distribution in these nonequilibrium,
nonconvecting states have shown that the temperature
is described by the Fourier-Laplace equation
V-kVT = o. (4.1)
For horizontal parallel plates in the absence of end
effects, Equation 4.1 becomes
(d/dz)k(dT/dz) = 0 (4.2)
where the temperature distribution, T(z), is a function
of the vertical coordinate, 2, which is measured from the
upper plate.
The parameter k of Equation 4.2 is experimentally
found to be a liquid property. At ordinary temperatures
92

 

 

 

 

93

k depends to a small degree upon temperature; it is
independent of temperature gradient. Highly nonlinear
temperature distributions which may be found in.regions
for which Equation 4.2 may not be valid(i&g,, near walls
or interfaces) are described by the experimentally found
temperature, T(z), minus the temperature predicted by

Equation 4.2, T Equation 3.33 is the solution to

3.33’
Equation 4.2 subject to the boundary conditions which are
taken to be the temperature of the upper and lower silver
plates; these are TU and TL, respectively. The function

T(z) - 3(z) is called the deviation from linearity

T
3.3
because the temperature dependence of k is so small at
ordinary temperatures that the solution of Equation 4.2
is essentially linear in the vertical coordinate.

1 called k(T) the "interior

Joseph Fourier
conducibility." He recognized that Equation 4.2 describes
all energy flux in the absence of external effects through
the Fourier heat flux relationship,

Jz = -de/dz, (4.3)

where Jz is the total heat flux in the z direction.

We concur with Fourier but restate the external effects

provision:
Equation 4.2 describes T(z) in the nonflowing,
steady state provided that z is far enough away
from any boundary interface. The interface may
be solid-liquid, solid-solid, solid-gas, liquid-
gas, etc.

{This statement is valid because there is no experimental

evidence to the contrary. Equation 4.2 describes all

 

 

94

temperature distributions which have been observed far
from boundaries.

In this chapter we present quantitative, experimental
temperature distributions for ethyl acetate, benzene, and
carbon tetrachloride. The nonlinear temperature distributions
which are found near the bounding silver walls of these pure
liquids demonstrate the failure of Equation 4.2 as the
liquid-solid interface is approached. The temperature
distributions are determined with the Bryngdahl interferometer
and mathematical analysis described in Chapter 3. Analysis
of the dependence of the nonlinearity upon the temperature
difference

ATO = T - TL > O (4.4)

U
is made, and the variation of the exhibited magnitude of the
nonlinearity from liquid to liquid is established. Evidence
is presented which suggests that Equation 4.2 describes
the temperature distribution in water at all distances
from the solid-water interface but with a temperature
jump or discontinuity at the interface. Such temperature
discontinuity may be viewed as a degenerate nonlinearity
in the sense that the nonlinearity is over a distance
too small to be observed with the Bryngdahl interferometric
probe.

Irregular temperature distributions near liquid-
solid interfaces have been reported by a number of authors.

2

In 1933 Bates found that "temperature drops" across the

solid-liquid interfaces resulted in significant variations

 

95

in the experimentally determined thermal conductivities
of water and red oil when the solid boundary was either
copper or Duco lacquered copper. He reported that a
surface effect exists which must be considered different
from the effects of the.boundary layers associated with
convecting fluids.

In 1957 Longsworth3 used a nonisothermal technique
and Rayleigh interferometry to study the Soret coefficients
of KCl solutions. The Rayleigh cell was sandwiched between
horizontal parallel p1ates(;_¢ 1 cm, ATo d 10°C). Pure water
was found to exhibit highly nonlinear temperature distributions
near bounding surfaces of either silver or stainless steel.
Longsworth reported that the "temperature drop" in water
is 2.4% to 3% of ATo with silver plates and 7% for stainless
steel. He takes these temperature draps into account in the
evaluation of Soret coefficients.

Neglect of nonlinear temperature distribution near
walls can cause systematic error in experimental thermal
conductivity calculations. Recognizing this, Poltz has

developed the concept of "effective thermal conductivity"

for liquidsf‘"7 He defines an effective thermal conductivity,li
keff’ such that
km = marrow, = (gnome + Jr) (4.5a)
= kc + (l/ATO)Jr (4.5h)

‘where l is the distance between the horizontal parallel
‘plates, and J2 is the total heat flux(W/cm2) in the z

direction. The conductive heat flux, JC, is defined to

96

be the total heat flux, Jz, minus the radiative heat flux, Jr
A11 fluxes are in the vertical, 2, direction. The parameter
kc defined in Equation 4.5b is called the "thermal
conductivity" by Poltz. From these definitions it is
apparent that Poltz uses keff as the parameter in Equation
4.2. Thus, the validity of Equation 4.2 in describing

the temperature distribution is extended to all domains
whether near or far from walls. Consequently, the effective
thermal conductivity depends upon (1) cell geometry and

(2) the nature and magnitude of the solid-liquid interaction.
Because it is not generally a property of the liquid,

the effective thermal conductivity of Poltz and the

interior conducibility of Fourier are not identical.
Analyzing what he perceives to be the radiative heat

flux, Poltz finds

k = kc + (16n2aT3/3E)Y(e,t) (4.6)

eff
where Y(e,t) is a function such that

lim Y(e,t) = 1 (4.7)
£*”

and where n and g are the average refractive index and
average absorption coefficient(cm-1) of the liquid media,
respectively. e is the emissivity of the plate surfaces,

and o is the Stefan-Boltzmann constant:

a = 5.66910 x 10'9 Wcm'2k‘“. (4.8)
The optical density,t , is defined by
r = 1g. (4.9)

Taking the limit of Equation 4.6 as law , making the

97
fundamental assumption that kc is a liquid property which
is independent of cell geometry, and employing Equation
4.7 we find

lim k

= kc + 16n20T3/3g. (4.10)
l7)”

eff

Because the right-hand side of Equation 4.10 is the sum
of two properties, the limit as l+°° of keff must be a
property. Being both a liquid property and the thermal
parameter of Equation 4.2, the limiting value of keff

must be equivalent to Fourier's interior conducibility:

112° keff = k. (4.11)

Poltz's experimental program for determination of
the interior conducibility is: (1) measure keff(l) with
Equation 4.5a at several cell heights and (2) extrapolate
the results to l = «L This yields the interior conducibility
by Equation 4.11. Using copper plates(e = 0.04), Poltz
finds an increase in keff for toluene of 7% when l is
increased from zero to infinity at a mean temperature
of 80°C.4 When the mean temperature is 20°C, keff increases
by 3.6% but the increases is 85% complete for the cell
height of 3 mm. Similar increases of 4-7% are found
for the weak infrared absorbing liquids: benzene, m-xylene,
carbon tetrachloride, and paraffin.5’6 The effective
thermal conductivity of the strong infrared absorbers
n-propanol, iso-propanol, n-butanol, sec-butanol, and
iso-butanol increases by less than 1% with increasing ;.7

In related work Novotny and co-workerss'9 have

98

evaluated the radiative contribution to energy transport
for carbon tetrachloride using experimental, frequency
dependent, absorption spectra. This seems to be the only
work of its kind for a liquid.

Direct interferometric observation of temperature
distribution near and far from walls has been reported.
In 1970 Schodel and Grigull10 used horizontal parallel
plates heated from above in conjunction with a Mach-
Zehnder interferometer to establish and determine temperature
distribution. They found nonlinear temperature distributions
for carbon tetrachloride, paraffin, and carbon disulfide,
while water and methanol exhibited linear temperature
distributions. Figure 4.1A reproduces their data for
carbon tetrachloride. The magnitude of the temperature
gradient shown in Figure 4.1A increases by 25% as the
solid-liquid interface is approached from the cell center.
There is also a central region(0.25 é z/l s 0.75) in
which the temperature gradient is constant. The cell
height is 1.5 cm; therefore, the nonlinearity lies within
0.4 cm of the liquid-silver interface(e = 0.05).

Olson and Horne have used nonisothermal Bryngdahl

11

interferometry to study the refractive indices and thermal

12 of pure liquids in a horizontal parallel plate

conductivities
arrangement. Finding parabolic steady state fringe shapes for
carbon tetrachloride, cyclohexane, and benzene,11 they suggest
that the observed parabolas are caused by slight nonlinearities

in the steady state temperature distribution. Further,

 

 

 

 

 

 

99

Figure 4.1

(A) Reduced temperature gradient, (dT/dz)/(dT/dz)l/2,
versus z/l for carbon tetrachloride when l = 1.5 cm,
e = 0.05, and ATo = 2.0000. From G. Schodel and

U. Grigull.10

(B) Deviation from linearity, T - T3 33, of C014 in a

cell 0.810 cm high. T = 26.40°c, TL = 23.6o°c.

U
From J. D. Olson.12

 

100

 

 

 

 

 

 

 

I. 2
dT/dz
I. I
I0
0.02
o
2.
\
8 000
*r'” '
[.—
__ i I 1 1
0'020 0.2 0.4 0.6 0.8 I.0

101
they speculate that the unexpected nonlinearities are
due to anomalous interactions between liquid and metal
boundaries. Olson12 has computed the temperature distribution
necessary to explain the parabolic fringe shape exhibited
by carbon tetrachloride between silver plates(l = 0.810 cm,
ATo = 2.8000C). The resulting deviation from linearity
is plotted versus z/l in Figure 4.1B. Near the upper
plate the temperature is a maximum of 0.0150C less than
the prediction of Equation 4.2. Near the lower wall the
temperature is a maximum of 0.0300C greater than the
prediction of Equation 4.2. In the central region
(0.2 cm s 2 $ 0.55 cm) the temperature distribution is
essentially linear.

Gurenkova gt all; have used a double beam diffraction
interferometer to evaluate temperature distributions. They
found distributions qualitatively similar to those of
Figure 4.1 for toluene, hexane, and octane. The nonlinearities

for water and liquid alcohols were negligibly small.

4.2 Nonlinear Temperature Distributiggg: Ethyl Acetate,
Benzene, Carbon Tetrachloride
All liquids studied are high purity. Methanol
is "acetone free" Matheson, Coleman, and Bell A.C.S.
Analyzed Reagent. Benzene is Matheson, Coleman, and
Bell Spectroquality. Carbon tetrachloride is Baker
Analyzed "Spectrophotometric" Reagent with 0.00 absorbance

at 400 nm. Ethyl acetate is Mallinckrodt A.C.S. Analytical

102

Reagent. These organic reagents are freshly opened.
Doubly distilled water with an electrical conductivity
of 0.4umho/cm is degassed by boiling just prior to use.
This eliminates troublesome air bubbles within the liquid
cell.

When the large temperature gradients needed to
study nonlinear temperature distributions are applied to
the pure organic liquids, strong refraction of the inter-
ferometric light beam causes the beam to be refracted
completely out of the optical train. This makes it
impossible to count photometrically the number, N(H/2), of
fringes which evolve at the image center, and the experimental
procedure of Section 3.3 must be altered. All isothermal
experimental aSpects remain unchanged he nonisothermal
aspects are modified as follows: once the nonequilibrium,
steady state temperature distribution is established, lens
L1 is lowered until (1) the interferometric light beam
traverses the optical train, and (2) the cell face is
sharply focused. The fringe shape is measured from a
photograph of the steady state image. Assuming T(H/2) is
the mean of TU and TL, N(H/2) is calculated from Equation 3.6.
The sum of the fringe shape and N(H/2) is the fringe number,
N(Z).

Table 4.1 contains the indices of refraction at
632.8 nm which are used with the methods of Section 3.6
to evaluate the temperature distribution, T(z). In

computing these functions, the temperature dependent

 

103

 

 

 

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104

parameters A(T), B(T), and C(T) of Cauchy's formula,20

n(T,1) = A(T) + B(T)/l2 + C(T)/14, (4.12)
are evaluated by the method of least squares from reported
refractive indices at specific temperature and variable
wavelength. A(T), B(T), and C(T) are then fitted to the
best quadratic temperature equation with the least squares
method. Table 4.1 lists the final results evaluated at
632.8 nm. The standard deviation of n(T,l) determined in
this manner agrees with the experimental data within the
reported experimental uncertainty. Table 4.2 lists thermal
conductivity functions used in the calculation of T3.33.

Two aspects of the data reduction must be justified:
(1) neglect of the pressure dependence of refractive indices
and (2) Equation 3.24. Concerning the pressure dependence,

‘1 for benzene at 24.800C,

1

(an/8P)T = 5.057 x 10'5 atm

1 atm, and 643.9 nm;14 (an/8P)T = 1.462 x 10.5 atm- for

water at 23.10C, 1 atm, and 589.3 nm;21

-1

and (an/3P)T =

4.056 x 10"5 atm for methanol at 22.80C, 1 atm, and

1

589.3 nm.2 Thus a fluctuation of 0.1 atm causes a
3

refractive index fluctuation which is smaller than the
experimental uncertainties reported in Table 4.1. Furthermore,

since (an/3T)P = -1.071 x 10-4 o -1

C for water at 1 atm and
632.8 nm, the refractive index of water decreases by about
10-3 units during a temperature increase of 10°C, a typical
temperature difference between plates; but the refractive
index increases by about 5 x 10"6 units during a pressure

increase of 0.1 atm. Even at these large variations of

 

 

105

 

 

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106
pressure, the pressure induced refractive index variation
is less than 1% of a typical temperature induced variation.
These points justify the neglect of pressure variation.
Equation 3.24 asserts that when the cell face is
focused:
Z(zo) = (H/_1_)z0 (4.13)
where Z is the vertical image coordinate of a photon
pencil which enters the liquid at the vertical coordinate

20; l is the cell height; and H is the isothermal image

 

height. In general this simple relationship is incorrect.
Alterations in relative lens position during the isothermal-
nonisothermal experimental sequence cause changes in the
magnification factor which is H/l in Equation 4.13. To
prove the validity of Equation 4.13 for our experimental
procedure, the nonisothermal image height predicted by
Equation 4.13 is compared to the experimental, nonisothermal
image height. Isothermal and nonisothermal image hieghts
differ because light is refracted toward the lower
temperatures, which causes a portion of the interferometric
radiation to strike the lower silver plate. This portion is

reflected out of the optical train, thereby reducing the

 

image height in the nonisothermal, experimental configuration.

It is easily seen from Figure 4.2 that the cross section of
the interferometric beam leaving the cell-plate composite

will have the effective height(l

-eff) given by

1 1 (4.14)

—eff = - - zend(0)

         

 

 

 

107

Figure 4.2

Light refraction in a nonisothermal experimental
configuration; TU) TL. The pencil of light shown
enters the liquid at z0 = 0. It leaves at coordinate
zf(0) and angle af(0). This pencil travels through
the glass cell at angle Bf(0), refracts in air, and
travels at angle Yf(0) until direction is altered by

the Bryngdahl optical train.

 

W/m/W/fl/l

 

    

\
--’ \

#2

hr

 

—-.

WWW/7, 4. ’///7/77/K/7Z

 

 

 

 

 

 

 

109
where zend(0) is the vertical coordinate of the light
pencil, which enters the cell at 20 = 0, passing beyond
the silver plate. From the definition of zend(0)’ we
have the trigonometric relationship

zend(0) = zf(0) + L tan flf(0) + K tan Yf(0) (4.15)

where the distances L and K are defined in Figure 4.1.
L and K are 0.635 cm and 2.450 cm, reSpectively. The angles

af(0), Bf(0), and 7f(0) are also defined in Figure 4.2.

They are related by Snell's refraction law by22
nf(0)sin af(0) = nglassSin Bf(0)
= natmsin vf(o). (4.16)

nf(0) is the liquid refractive index experienced by the
upper most light pencil leaving the liquid. The refractive

index of glass, is 1.52; the refractive index of

nglass’
air, natm’ is 1.000276.23 From Equations 4.14, 4.15, and

4.16, we have

1,- leff = zf(0) + L tan sin-1((nf(0)/ng1ass(0))sin af(0))
+ K tan sin-1((nf(0)/natm)sin af(0)). (4.17)

To find af(0), we have
tan a = dz/dy = ((n(z)/n(zo))2 - 1)1/2 (4.18)

by Equation 3.12. Furthermore, combining Equations 3.16a,
3.18b, and 4.18 and evaluating the results at y = a =

5.986 cm, we have

2
af(0) = tan-1(12 MV(M1(2M2)-1(cosh(M:/2a) - 1 ))V).

v=1
(4.19)

 

110
Equations 4.17 and 4.19 can now be used to calculate
leff‘ Equation 4.13 is correct if (H/_l_)leIf = Heff
equals the experimentally determined, nonisothermal,
image height. Table 4.3 includes the isothermal shear
parameter(2D), the interfringe distance(6), the isothermal
and nonisothermal experimental image heights, and the

computed effective height, H The last column is the

eff“
per cent deviation of the calculated height from the
experimentally determined height. In all instances, the
deviation is 1% or less. Because deviations of this
magnitude are within the uncertainty of the measured
value of H, the validity of Equation 3.24 is established.
Figures 4.3 through 4.7 are photographic prints
of nonisothermal configuration images of water, ethyl
acetate, benzene, carbon tetrachloride, and methanol.
The fringe shapes exhibited at different ATO may be
qualitatively discussed if we temporarily neglect the
bending of light in a temperature gradient. Then the
fringe number, N(z), is simply related to the temperature
gradient by Equation 3.6:

N(z) 2‘ (2Da_l_/}.H)(dn/dT)(dT/dz). (4.20)
dn/dT data of Table 4.1 are linear functions of temperature
in the temperature range of interest for all liquids.

Thus, if dT/dz is a constant independent
of z, N(z) is expected to be a linear function of z by
Equation 4.20. This expectation is confirmed at all AT0

from the Figure 4.3 nonisothermal data for water. Water,

 

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112

Figure 4.3

Nonisothermal Bryngdahl interferometric image of water at
several temperature differences between upper and lower
silver plates. 1 = 1.349 cm; ATO = 10.028°c, 13.047°c,
and 16.69100.

 

was .3 u [g

 

113

8.8.2 . ,2 8.8.9 n .2

 

1 lllllll‘ll

 

 

114

Figure 4.4
Nonisothermal Bryngdahl interferometric image of ethyl

and lower silver plates. 1 = 1.349 cm; ATO = 2.24400,

1
acetate at several temperature differences between upper
6.261%, and 10.019°c. 1

 

0.73.0.3 u F4

 

115

ohms u [rd ousNN u F4

 

3138.51.41.21 .5:

 

 

116

Figure 4.5

Nonisothermal Bryngdahl interferometric image of benzene
at several temperature differences between upper and
lower silver plates. 1 = 1.349 cm; ATo = 2.3990C, 7.25600,

10.52900, and 15.06300.

050.2 N LLQ

    

 

117

o. wNN u FQ

 

 

“sing mcmmcmm

 

118

Figure 4.6

Nonisothermal Bryngdahl interferometric image of carbon
tetrachloride at several temperature differences between
upper and lower silver plates. 1 = 1.349 cm; ATo = 2.63700,
5.976%, 7.729%, and 12.71300.

119

u. K. Na” oommx u 64

I a,

o. mom u L14 u.¢m.N u L14

I ma. T

A 38 ascozasfir :85

 

 

120

Figure 4.7

Nonisothermal Bryngdahl interferometric image of methanol
at several temperature differences between upper and
lower silver plates. I = 1.349 cm; ATo = 4.7390C,
9.297°C, 12.125°C, and 13.4880C.

 

121

rr< u.§.§ . .54

 

Egg page:

50H

 

 

122

therefore, seems to satisfy Equation 3.33; Equation 3.33
describes the temperature distribution within observable
distances from the silver-water interfaces. The temperature
dependence of dn/dT for ethyl acetate, benzene, carbon
tetrachloride, and methanol is only 1/10th to 1/5th that

of water. Because dn/dT is essentially constant for these
liquids, N(z) is proportional to the temperature derivative,
dT/dz. From the nonisothermal images of ethyl acetate in
Figure 4.4, N(z) is nearly independent of 2 at ATo = 2.240C;
therefore, dT/dz is essentially a constant. However, at

ATo = 6.2600, N(z) develops curvature near both the upper
and lower plates. This indicates nonlinear temperature
gradients in these domains. At ATO = 6.2600 and intermediate
values of the vertical coordinate, the fringe shape is

seen to be a linear function of the vertical coordinate.
This is the expected shape when dn/dT depends slightly

upon temperature, and dT/dz is a constant. At ATo = 10.0200
very pronounced fringe curvature is seen near both upper
and lower silver-ethyl acetate interfaces; the central
region still displays the expected linear fringe shape.
These qualitative observations indicate (1) nonlinear
temperature distribution near the silver-ethyl acetate
interface, (2) the magnitude of the nonlinearity increases

with increasing AT and (3) there is a central region

0,
in which dT/dz is constant.
Qualitatively comparable results are found for

benzene and carbon tetrachloride in Figures 4.5 and 4.6,

 

123

respectively. Only the magnitude of the nonlinearity

and its extension from the metal-liquid interface vary
between ethyl acetate, benzene, and carbon tetrachloride.
Methanol exhibits a characteristically different fringe
pattern in Figure 4.7. In the methanol nonisothermal
images, the expected vertical fringe pattern has small
amplitude wiggles superimposed upon it. The wiggles

are time independent with amplitude which grow slightly
with increasing ATO. By placing a rubber bulb over the
cell fill holes and "plunging," the wiggles could be made
to oscillate frantically with large amplitude at high ATO.
These time dependent fringe shapes are characteristic of
turbulent convection which strongly suggests that the
irregular, time independent wiggles are caused by

laminar convection. Observed laminar and turbulent motion
experimentally proves the importance of end effects in
the present apparatus, and the characteristic wiggles
provide a convenient test for such liquid motion. For
example, ethyl bromide is found to exhibit steady state
fringe shapes identical to those of methanol, but the
time dependent behavior can be induced at much lower ATO.
Thus, both methanol and ethyl bromide are unsuitable for
studies of temperature distribution in nonconvecting
media. The characteristic fringe pattern of methanol is
also observed for ethyl acetate at ATO above 1200,
indicating a critical, liquid dependent ATO above which

end effects become important through their production of

124
convection.

To analyze quantitatively for temperature
distribution, the fringe numbers are fitted in a least
squares sense to spatial polynomials of degree 3 to 6.

The method employed uses orthogonal Chebyshev polynomials

in the discrete range.24 Results are reported in Table 4.4.

The raw data are processed by methods described in Sections

3.3 and 3.4. Temperature distributions are mapped by the method
of Section 3.6 with the refractive index temperature derivatives
given in Table 4.1. The temperature at z = 1/2 is assumed

to be given by Equation 3.33 subject to the boundary

conditions T(O) = TU, T(l) = TL' T3.33(z) is computed

from Equation 3.33 subject to the calculated boundary
temperatures. The resulting deviation from linearity,

T(z) - T3.33(z), for benzene and carbon tetrachloride are
recorded in Figures 4.8 and 4.9, respectively.

Certain-features of these figures are clear: (1) a
negative deviation from linearity exists near the upper
silver-liquid interface, (2) a positive deviation exists
near the lower silver-liquid interface, (3) the deviation
magnitude depends upon ATO, and (4) a central region
exists in which the deviation from linearity is linear
in the vertical coordinate. There are some disturbing
aspects of these figures. First, because both upper and
lower plates are identical, we expect a high degree of
symmetry with respect to inversion through the cell center.

Figures 4.8 and 4.9 have far larger negative deviations

125

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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ensues... H.930
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.n
0.0 wmnwe

0

 

 

 

126

Figure 4.8

Temperature deviation from linearity, T(z) - T3.33(z),
for benzene at several temperature differences between
upper and lower silver plates. The refractive index
temperature derivatives of Table 4.1 are used in the

method of Section 3.6 to compute T(z) from the refined

experimental data. I = 1.349 cm.

 

 

 

127

 

 

 

 

 

 

.l2 - CeHe
.. G) ATO= 2.390%:
.03- @ATO= 7.256°C
.. ©ATO=I0529°C
o .04- @ATO=I3.063°C
O
z .. A
5” 0 K
- 0
PM
I
}_
-.04
-.08~ ®
" 69
—.I2 ~
- @
L l l l l l 1
0 0.2 0.4 0.6 0.8 |.0

 

 

 

 

128

Figure 4.9

Temperature deviation from linearity, T(z) - T3.33(z),
for carbon tetrachloride at several temperature differences
between upper and lower silver plates. The refractive
index temperature derivatives of Table 4.1 are used in

the method of Section 3.6 to compute T(z) from the refined

experimental data. 1 = 1.349 cm.

 

T-T333 /1°C

.l2

.08

129

 

CCI4

- CD ATO= 2.637%
.. ® ATO= 5.976°C

.. ® AT = 7.729°c

 

 

 

 

 

 

 

 

130
from linearity than positive; other curves are found
with far larger positive deviation than negative. Second,
we expect a regular increase in deviation with increasing
ATO; yet, near the lower plate an irregularity is found
for benzene(ATo = 7.2600). Systematic error, due to lens
aberrations, is the cause of these disturbing points.
Lowering lens L1 to bring the interferometric light beam
back into the optical train and to refocus the cell face
causes a repositioning of the light beam upon each lens.
For example, in the isothermal configuration the light
beam passes through the center of each lens. The need to
lower L1 in the nonisothermal configuration, however,
causes the light beam to traverse L1 just below center,
L2 just above center, etc. This results in systematic
distortion of the cell image. Correction for the distortion
is made as follows.

Choose two points, z and Z2, a distance m apart

1
such that

21 = (l_~ m)/2, and 22 = (l + m)/2. (4.21)

Furthermore, choose the distance m such that

dT(zl)/dz = dT(z2)/dz = dT(l/2)/dz, (4.22)

and make the following definitions:

dn(T)/dT a A + BT, (4.23)
- z2

I1 = i (dn/dz)dz, and (4.24)
1

12 E f2 (d2n/dz2)dz. (4.25)
Z

1

 

 

131
The integrals 11 and 12 are numerically evaluated from
the refined, experimental data. Equations 4.21 to 4.25

can be combined and manipulated to show that
2
B = 12/m(dT(l/2)/dz) (4.26)

where B is defined by Equation 4.23. Equation 4.26 is
used to estimate B by approximately the temperature
derivative as -ATo/l. Equations 4.21 to 4.25 can also be
combined and manipulated to show that

aA2 + bA + 0 = 0 (4.27a)

where

a = T(l/2)12 (4.27b)

0"
fl

n(dn(_l_/2)/dz)2 - 11dn(;/2)/dz + 2T(;/2)12B

(4.270)
(123(T(;/2))2 — Ildn(l/2)/dz)BT(l/2). (4.27s)

C

The final parameter of Equation 4.23, A, is the root of

Equation 4.27. In this manner, the parameters A and B

have been effectively adjusted so as to satisfy the

condition stated in Equation 4.22; image distortions are

thereby contained within these parameters. Choosing

m = 0.2;, the resulting parameters are reported in Table 4.5.
With the refractive index temperature derivatives of

Table 4.5 and the methods of Section 3.6, the temperature

distributions shown in Figure 4.10 through 4.12 are mapped.

These figures are temperature deviation from linearity

corrected for interferometric image distortion. They show

132

 

 

 

 

 

 

 

 

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0.0 0:39

 

 

Figure 4.10

Temperature deviation from linearity, T(z) - T3.33(z),

for ethyl acetate at several temperature differences

between upper and lower silver plates. These curves

are corrected for image distortion by using the refractive
index temperature derivatives of Table 4.5 in the method

of Section 3.6 to compute T(z) from the refined experimental

data. ; = 1.349 cm.

 

 

1311

 

CH3COOC2H5
.0 3 .. G) ATO= 2.244%:
(3 ATO= 6.26l°C
.0 2 _ ® ATO=IO.Ol9°C

OI-

 

'Dl-

'02P

T-T3 33/1<’<:
0
e

 

 

 

-00 3 L l J J l

 

 

135

Figure 4.11

Temperature deviation from linearity, T(z) - T3.33(z),

for benzene at several temperature differences between

upper and lower silver plates. These curves are corrected

for image distortion by using the refractive index temperature
derivatives of Table 4.5 in the method of Section 3.6 to

compute T(z) from the refined experimental data. 1 = 1.349 cm.

 

 

.l2

.08

136

 

 

 

CeHs
(D ATO= 2.399%
® ATO= 7.256°C
© ATO= no.529°c
@ ATO= l3.063°C

 

 

 

 

0 .04
2.
\
a /
.3" 0 <9 /
I
[.—
-.04 ®
0
-.08
@
-.|2
I l L l l l l
O 02 0.4 0.6 0.8 |.O

z/e

 

 

 

 

Figure 4.12

Temperature deviation from linearity, T(z) - T3.33(z),

for carbon tetrachloride at several temperature differences
between upper and lower silver plates. These curves are
corrected for image distortion by using the refractive

index temperature derivatives of Table 4.5 in the method

of Section 3.6 to compute T(z) from the refined experimental

data. l = 1.349 cm.

 

 

138

 

.l2'

.08 -

 

T -T3.33 /1°C
0

 

__ CC|4

CD ATO= 2.637%
© ATO= 5.976%
@ ATO= 7.729%
(4) ATO=12713°C

 

 

 

d “.04
elm;

-.08- ®

“.I 2 " ®

" . l6 "

@
1 1 1 1 1 1 1
O O 2 0.4 0.6 0.8 LO

139
(1) regular increase in negative deviation with increasing
ATO near the upper plate, (2) regular increase in positive
deviation with increasing ATo near the lower plate, and (3)
a central region in which the deviation is a linear function
of the vertical coordinate. The maximum observed deviations
are -o.028°c at z = 0.243 cm and 0.025°C at l-z = 0.236 cm
for ethyl acetate(AT0 = 10.019°c); -0.107°c at z = 0.216 cm
and 0.079°c at $02 = 0.297 cm for benzene(AT0 = 13.063°c);
-o.160°c at z = 0.233 cm and 0.12400 at l—z = 0.270 cm for
carbon tetrachloride(ATO = 12.71300).

Comparison of ATO and the interferometrically
determined difference between the temperature of the liquid
at the upper plate and the temperature of the liquid at
the lower plate, AT, as reported in Table 4.5, reveals
discrepancy. AT0 is the temperature difference between
upper and lower silver plates as measured by thermocouples
placed just outside the cell. ATo is consistently smaller
than AT when Table 4.5 temperature derivatives are used
to compute AT; however, ATO and AT are identical when
Table 4.1 temperature derivatives are used. This latter
fact is a computational artifact, just as the similarity of
ATO/l and -dT(l/2)/dz(see Table 4.5) is a computational
artifact in the former case. Linear least squares analysis
shows

AT - ATO = 1.00 x 10-2AT (4.28)

O

2 0C

a 0.9 x 10-

for ethyl acetate;

 

 

 

140

AT — ATO 2.43 x 10’2ATO (4.2 )

2 o

0:306X10- C

for benzene; and for carbon tetrachloride

5.67 x 10‘2ATo (4.30)

AT - ATo

2 0C

a = 3.8 x 10-
where a is the standard deviation.
The hypothesis that these discrepancies are a true
reflection of temperature control and measurement technique
is tested by replacing the liquid cell with a white pine block
(5.900 cm x 5.876 cm x 1.915 cm). Twenty-two nonisothermal
experiments in which AT is measured by thermocouples placed at
the center of the wood-silver interfaces and ATO is measured

by thermocouples placed just outside the wood cell show

(4.31)

AT — ATO 2.37 x 10-2AT

O

1.0 x 10“2 °C

0
where -4.7°04A'ros 8.1°c. ATO is smaller than AT. The
discrepancy is due to the water flow in upper and lower
water jackets which is effectively from a position
corresponding to the metal-liquid(or wood) interface
center toward the outer edges. Because heat flow is

away from the upper plate, the heat bath water cools in
flowing toward the outer edges. In addition, heat flow is
toward the lower plate; therefore, heat bath water in the
lower water jacket heats in flowing toward the outer edges.
This results in a measured temperature difference, ATO,

which is smaller than the temperature difference AT.

141

The reduced temperature gradient(RTG) is defined
as the ratio of temperature gradient to temperature
gradient at z/l = 0.5. Several important features are
apparent in the RTG of Figure 4.13. First, no systematic
behavior of RTG is found with increasing ATO; this ratio
seems to be independent of the applied temperature gradient,
ATO. Consequently, RTG's of differing AT0 are averaged
for each particular liquid. A second characteristic of
RTG curves is the exhibited nonlinearity near the solid-
liquid interfaces. At the upper interface, the temperature
gradient is 4.5% greater than (dT/dz)l/2 for ethyl acetate,
12.5% greater for benzene, and 30% greater for carbon
tetrachloride. These represent extremely large increases
in temperature gradient as interfaces are approached.
Central regions of RTG curves demonstrate the validity of
the Fourier-Laplace description of temperature far from
interfaces. Centrally, the temperature gradient is
constant. A final characteristic feature of RTG curves is
the relative magnitude of temperature gradient increase
as the lower plate is approached; the temperature gradient
increases as the lower plate is approached, but the
magnitude of increase is smaller than that observed as
the upper plate is approached. We believe these relative
differences to be due to data extrapolation problems, not
to physical differences between upper and lower silver
plates. At the lower plate a portion of the interferometric

light beam has struck and been reflected by the plate;

 

 

 

142

Figure 4.13

Reduced temperature gradient, (dT/dz)/(dT/dz)l/2, for
ethyl acetate, benzene, and carbon tetrachloride.
Corrections have been made for interferometric image

distortion. l = 1.349 cm.

 

dT/dz
(dT/dZ)f/2

 

L4

L3

|.2

LG

143

 

 

 

® 011300002145

 

 

 

@ CeHe
@0014
®
..\ 0. A
l l l J L l
o 0.2 04 0.6 0.8 1.0

 

144

 

Figure 4.14

Integrated absolute deviation from linearity,

1
(1/1) I [T - T ldz, as a function of AT . Corrections
- 0 3.33 0
have been made for interferometric image distortion. The
placement of the water curve is discussed in Section 4.3.
l = 1.349 cm for ethyl acetate, benzene, and carbon

tetrachloride. l = 0.474 cm for water.

145

IO

 

9:-
H20
3" 0014
'0 7-
Q9
'0
“3.2 6"
x c H
g 51- 6 6
:3. 4"
m
31—. 3-
“:9 2-
§ . 01-13000021-15

 

    

l l I l

 

 

O 2 4 6 8
Arc/1%

IO l2 I4

l6

 

 

 

 

146

this portion provides no experimental data. The interval
not observed is l-leff(Equation 4.17). Temperature in
this interval is computed by extrapolating the observed
fringe shapes of Table 4.4. This procedure underestimates
the growth in N(z) as the lower plate is approached and
thereby underestimates the growth in temperature gradient.
The extent of the unobserved interval can be calculated
from the data of Table 4.3 by taking the image height(H)
at ATo = O, subtracting Heff at a particular value of

AT and dividing by H/l. For benzene at ATo = 13.0630C,

0,
the unobserved interval extends 0.177 cm from the lower
plate. For smaller ATO, the interval is smaller.

Figure 4.14 is another useful representation of
the observed nonlinear temperature distributions. The
curves are the integrated absolute deviation from linearity.
They are functions of ATO, and the magnitude of their
first derivatives orders liquids as to the magnitude of
exhibited nonlinearity. Where a is the standard deviation,

we find

1
(1/;) 50 IT - T3.33|dz = 1.655 x 10 3ATO (4.32)

o 1.1 x 10“3 00

for ethyl acetate,

N
H

1
- _ -3
(1/1)‘(0 IT T3.33ld 4.059 x 10 ATO (4.33)

0 = 3.9 x 10-3 0C

for benzene, and

147

7.839 x 10'3AT0 (4.34)

1
(1/1) 50 IT - T3.33ldz

a = 8.8 x 10"3 00
for carbon tetrachloride. As discussed above, the temperature
distribution in water is described by the Fourier-Laplace
equation. Thus, neglecting the possibility of temperature
jump at the silver-liquid interface, an integrated absolute
deviation from linearity of zero is expected for water.
Figure 4.14, however, assigns water an entirely unexpected

position. This placement is discussed in Section 4.3.

4.3; Temperature Dependence of the Refractive Index of Water
Interferometric determinations of the temperature

derivatives of the refractive index of water seem to

depend upon whether an isothermal or a nonisothermal

experiment is utilized. This is illustrated in Figure 4.15

where

A1 = 100((dn/dT)i - (dn/dT)TT)/(dn/dT)TT. (4.35)

(dn/dT)i is the first derivative of the refractive index
with reSpect to the temperature, as determined by research
group i at 1 atmosphere pressure. The absolute refractive
index measurements of Tilton and Taylor(TT)25 are generally
considered most reliable and their thirteen parameter
equation for n(A,T) is used as a standard of reference.
Calculations of (dn/dT)i in Figure 4.15 are based upon the
best linear least square fit of the temperature variation,

11

except for the value reported by Olson and Horne which

is for a single temperature only(see Table 4.6). All

 

 

 

.. ___—ewe-W

Per

by Tilton and Taylor for water at 1 atm. See Equation 4.35.

I =
(1)
(2)
(3)
(4)
(5)
(6)
(7)

(8

v

148

Figure 4.15

cent deviation of dn(l,T)/dT from values reported

 

 

 

isothermal determination, N = nonisothermal determination.

I. Hawkes and Astheimer.26

l

I. Eight wavelengths, Waxler gt al.14’27
I. Andreasson gt g;,28
I. Dobbins and Peck.29
N. Bryngdahl . 3O

11
N. Olson and Horne.
N. Equation 4.38: Data of this work analyzed with
methods of Section 3.5 and T(O) = TU’ T(l) = TL

assumption.

N. Equation 4.41; Implied by the ratio of Equation 4.40.

 

149

 

 

 

 

 

 

 

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®

 

 

36

 

 

 

 

 

 

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151
interferometric studies which utilize isothermal states

1 -
of waterlk’ 25 29

agree with TT to within 4% in the
temperature range for which they are valid. The positive
curvature which results in a maximum positive 4% deviation
is due to the slight nonlinearity between dn/dT and T
which is fully described in the values of TT. However,

11'30 both of which

the two nonisothermal determinations,
employ Bryngdahl interferometry, markedly differ from the
results of TT. It is possible that temperature jumps

in the nonisothermal, nonconvecting steady state may
contribute to or be the cause of this discrepancy. The
hypothesis of the existence of temperature jumps at the
metal-water interface is a natural extrapolation of observed
nonlinear temperature distributions(Section 4.2) in which
the deviation from linearity is negative at the upper
interface and positive at the lower interface. However,

in the case of temperature jump, the spatial interval of
nonlinearity would be smaller than that observable with

the Bryngdahl interferometer(less than 0.107 cm for the
present apparatus). Such physical phenomena would appear

as a temperature jump: the interferometrically measured
temperature difference between the liquid at the upper
metal-water interface and the liquid at the lower metal-
water interface, AT, would be less than the measured
temperature difference between upper and lower metal

plates, ATO, while the observed temperature distribution

is linear.

 

 

 

152

To test this hypothesis we have examined water
in the temperature range 24-4000 with the experimental
procedure of Section 3.3. The cell height(;) and length
(a) are 0.474 cm and 6.771 cm, respectively. The raw
dataanx3reported in Table 4.7. The fringe number associated
with the vertical coordinate l/2 times the interfringe
distance,5 , has an uncertainty of $0.003 cm, and the
angle of fringe from the vertica1,13, has an uncertainty
of :10 min of arc. B is corrected for deviant verticality
of the isothermal image by subtracting the tangent of
the isothermal angle from the tangent of the nonisothermal.
This equals the tangent of the corrected angle. The
isothermal angle is typically 10-20 min of arc. The
temperature at z/l = 0.5 is assumed to be given by Equation
3.33 subject to the boundary conditions T(O) = TU and
T(l) = TL. First and second derivatives of the refractive
index with respect to temperature are mapped from the
refined data at z/l = 0.5 with the methods of Section 3.5.
The first derivatives are then fitted by least squares to the
best linear temperature equation. The second derivatives are
simply averaged and standard deviation calculated. In the
evaluation of the regression coefficients a and b for a
function such as

f(T) = a + bT (4.36)

by the method of least squares, the standard error of estimate
of f on T(o) is computed along with the standard error of the
regression coefficient a(aa), the standard error of the

regression coefficient b(ob), and the coefficient of

 

153
Table 4.7

Nonisothermal experimental water data. N(l/2) is the
fringe number at image Center. B is the angle from
vertical of fringe in deg.min and corrected for deviant
verticality of the isothermal image. Angles of experiments
of identical ATO and TL have been averaged. The isothermal

parameters are H = 4.713 cm, 2D = 0.463 cm, and 6 = 1.239 cm
with standard deviations of 0.023 cm, 0.018 cm, and 0.018 cm,
respectively, for 67 isothermal experiments.

l = 0.474 cm, a = 6.771 cm.

 

 

Auk/00 Tl /OC N(l/2) B/deg.min
2.306 23.042 2.232 1.46
2.306 23.042 2.248 1.46
2.667 22.961 2.605 2.29
2.667 22.961 2.588 2.29
3.137 22.818 3.023 3.33
3.137 22.818 3.013 3.33
3.137 22.818 3.021 3.33
3.137 22.818 3.042 3.33
4.199 22.934 4.153 6.10
4.390 22.880 4.310 6.59
5.360 23.026 5.370 10.34
5.360 23.026 5.394 10.34
5.627 23.364 5.761 11.25
5.627 23.364 5.724 11.25
5.695 23.292 5.754 11.00
5.695 23.292 5.792 11.00
6.596 23.340 6.841 14.59
6.596 23.340 6.849 14.59
6.732 23.415 7.020 15.35
3.435 30.393 3.952 3.29
3.214 29.278 3.788 3.41
3.214 29.278 3.754 3.41
3.764 29.152 4.409 4.60
3.764 29.152 4.448 4.60
3.554 29.214 4.121 4.33
4.112 29.142 4.836 5.59
4.112 29.142 4.793 5.59
4.115 29.187 4.880 6.12
4.115 29.187 4.876 6.12
5.091 29.209 6.080 8.57
5.364 29.198 6.415 9.45
5.364 29.198 6.431 9.45
5.195 29.353 6.192 8.59
5.195 29.353 6.180 8.59
5.774 29.377 6.902 11.22

 

 

 

 

 

 

 

Table 4.7(cont'd.)

154

 

 

 

ATo/° TL/°C N(1/2) B/deg.min
5.774 29.377 6.904 11.22
6.151 29.391 7.387 12.55
6.151 29.391 7.412 12.55
4.365 31.388 5.401 6.34
4.365 31.388 5.372 6.34
4.413 31.525 5.444 6.26
4.413 51.525 5.435 6.26
4.498 31.679 5.685 7.03
4.498 31.679 5.701 7.03
5.597 31.650 7.050 10.39
5.597 31.650 7.055 10.59
4.569 32.424 5.802 7.06
4.569 32.424 5.752 7.06
5.481 32.400 6.967 10.13
5.481 32.400 6.949 10.13
5.726 32.412 7.337 10.33
5.174 50.986 6.393 9.27
5.174 30.986 6.393 9.27
5.604 31.064 6.934 10.27
5.604 31.064 6.935 10.27
5.916 31.069 7.360 11.49
7.781 31.188 9.889 20.06
7.443 31.189 9.450 18.39
6.519 31.292 8.141 14.10
5.750 32.868 7.384 11.18
6.02 32.932 7.788 12.18
6.029 32.932 7.816 12.18
6.315 32.964 8.153 13.54
6.813 33.000 8.875 16.17
6.813 33.000 8.876 16.17
5.805 34.304 7.638 10.12
5.805 34.304 7.688 10.12
5.412 34.738 7.166 9.17
5.023 35.180 6.712 8.37
4.910 35.282 6.540 8.25
4.631 35.566 6.186 7.34
4.116 36.164 5.567 5.48
4.116 36.164 5.507 5.48
4.419 36.310 5.980 6.45
4.987 36.466 6.802 8.28
5.415 36.549 7.402 9.49

 

 

 

 

 

determination(r2). 0, 0a, ab, and r2 are defined for m

experiments by31'32
o = ((m - 2)‘1 2(11 - 1)2)1/2, (4.37a)
ca = (m'1( 21‘:- - m'1(2<ri)2)'121§)1/2, (4.371»)
ab (21% - m'1(211)2)‘1/2, (4.37c)
and

(21? - m'1(211)2)‘1. (4.37d)

The coefficient of determination, r2, is interpreted as
the proportion of total variation of f explained by
regression; 0~$r2£-1. If r2 = 1 the fit is perfect.
The above analytical procedure is illustrated in
Figure 4.16. With it, we find
-(dn/dT)104 = 0.2822 + 0.02671T (4.38)
a = 0.0161400’1

0.013490c‘1

0a =
ab = 0.00042°c‘2
r2 = 0.983

where T is in 0C. The standard error of about 1.5% and
the coefficient of determination demonstrate that Equation
4.38 fits the data very well. We also find

-(d2n/d1‘2)10‘l = 0.02486°c‘2 (4.39)
a = 0.00098°c'2.

The second derivative of Equation 4.38 is about 7% higher

than Equation 4.39. This is a bit large to be explained

 

156

Figure 4.16

Temperature dependence of the refractive index. The
raw experimental data is refined with the method of
Section 3.4, and dn/dT mapped with the method of
Section 3.5 assuming that T(O) = TU and T(l) = TL.
Equation 4.38(solid line) is the best linear fit of the

resulting temperature derivatives.

 

 

 

 

_ _ _
3 2 I

62.0. x P35-

 

|.4
IO -
0.92

T/°C

158

by the regression coefficients a of Equation 4.38 and

b
o of Equation 4.39. More disturbing is the discrepancy
between Equation 4.38 and the values of Tilton and Taylor;
Figure 4.15 shows Equation 4.38 to be 8-9% lower, a
magnitude which is much larger than the 1.5% standard
error.

Quite another result is found when the experimental
first derivatives divided by the square root of the

experimental second derivatives are fitted to a linear

temperature equation. Then

-(dn/dT)102/(-d2n/dT2)1/2 = 1.62619 + 0.17455T (4.40)

0 = 0.1206
0a = 0.1008

0
0b — 0.0031 C
r2 = 0.977.

This analysis is illustrated in Figure 4.17. Equation

4.40 implies that

~(dn/dT)104 = 0.2838 + 0.03047T (4.41)
where the estimated uncertainty in the calculated value
of the derivative is 2%. Figure 4.15 shows Equation 4.41
to be in very good agreement with Tilton and Taylor.

The failure of Equation 4.38 to agree with
isothermal determinations of dn/dT in the experimental
temperature range and the success of Equation 4.41 can be
qualitatively discussed by neglecting both the bending of

light in temperature gradients and small temperature

 

 

 

 

Figure 4.17

f(T) = -(dn/dT)102/(-d2n/dT2)1/2. The data are computed

by refining the raw data with the method of Section 3.4,

and mapping dn/dT and d2n/dT2 with the method of Section 3.5
assuming that T(O) = TU and T(l) = TL. Equation 4.40(solid
line) is the best linear fit of the above ratio. The

dashed line is predicted by Equation 4.38.

 

 

160

 

 

 

 

 

/
: . /
' -. ' /
o - /
../
r /
.§ /
.‘ /
1- :-//
/
/
_ ;//
/
/
l l l l l J
22 26 3O 34 38 42

 

 

161

dependence of the thermal conductivity. Then Equations
3.6 and 3.28 yield two important equations for media
which exhibit straight line fringe patterns in the
nonisothermal state. The first equation is

—dn/dT = D1N/AT (4.42)

where dn/dT is the refractive index temperature derivative
of the cell center, N is the number of fringes which

evolve at the image coordinate corresponding to cell
center, AT is the temperature difference between liquid at
upper and lower interfaces, and D1 is a constant determined
from the isothermal configuration cell image. The second

equation of importance is

-d2n/dT2 = D D2tanB /(AT)2 (4.43)

1
where d2n/dT2 is second derivative at cell center, B is the
fringe angle from the vertical in the nonisothermal, steady
state configuration, and D2 is a constant determined from
the isothermal configuration cell image. The difficulty
with Equation 4.42 and 4.43 is AT. .AT is a quantity

which is not directly measured in the present experimental
arrangement. Assumptions regarding AT may lead to erroneous
results. For example, suppose that

AT = ATO = T - T (4.1-£4)

U L‘
In this case all quantities in Equations 4.42 and 4.43

are experimentally determined and the desired derivatives

can be calculated. Equation 4.44 is assumed by Bryngdahl,30

11

Olson and Horne, and in the data analysis of this work

162
which culminates in Equation 4.38 and 4.39. Suppose,
however, thatwaterexhibits a temperature jump.
In such a case AT will be less than ATO and use of AT0
will yield computed derivatives which are smaller than the
true derivatives. This is the trend of Bryngdahl's
refractive index temperature derivatives above 20°C(see
Figure 4.15), of Olson and Horne's at 25°C, and of Equation
4.38 between 24°C and 40°C. Fortunately, it is possible
to avoid the AT problem by eliminating AT from Equations

4.42 and 4.43. This yields
-(an/aT)/(-a2n/d12>1/2 = (Di/D2)1/2N/(tans)1/2. (4.45)

Because the evaluation of dn/dT via Equation 4.45 does not
involve AT, results should agree wtih isothermal determinations.
This explains the success of Equation 4.40.

It may be that the temperature measurement technique,
not a temperature jump, is responsible for the discrepancy
between Equation 4.38 and 4.41; however, the "wood cell"
experiments described in Section 4.2 strongly indicate
that the technique is slightly compensating and not creating
an apparent temperature jump. Further, the variance
between Equation 4.38, Bryngdahl, and Olson and Horne is
expected because the magnitude of any temperature jump
will depend upon the interactions between the solid
boundaries and the liquid. This interaction is a function
of composition, structure, and condition of the solid

surface.

 

 

 

 

163
The functional dependence of AT upon ATo can be
estimated with Equation 4.42. Substituting Equation 4.38

for D1N and Equation 4.41 for dn/dT, we find

T = 0.994((1 + 0.0947T)/(1 + 0.1074T))ATO (4.46)

where 240C 6 T S 40°C, 2°C 5 AT0 é 8°C. Averaging this

relationship over the applicable temperature range, yields

AT - ATo = -9.671 x 10’2AT0. (4.47)

Further, since

as = (AT - ATO)(%- - Z/l). (4.48)

" T133

the integrated absolute deviation is
l -2
(1/1) 50 IT - T3.33ldz = 2.414 1 10 ATO. (4.49)

Figure 4.14 illustrates that

H20 > 001,1 > C6116 > 011300002115 (4. 50)

is the observed liquid order for integrated absolute
deviation.

The physical literature contains several reports
of unexplained water density fluctuation in the neighborhood
of 35°C. Hawkes and Astheimer26’33 have encountered small,
time-dependent fringe wiggle during isothermal Jamin
interferometric studies of the refractive index of water.
These wiggles correspond to density fluctuation of Ap/p =
6 x 10"6 and appear in the 34°C to 45°C interval. Varying
the temperature during 3062 isothermal Michelson interfero-

metric evaluations of the refractive index of water,

 

 

 

164

Dobbins and Peck29

found no evidence of any abrupt, large
refractive index change. Possible irregularity is,
however, indicated in their plot of mean deviation of the
experimental refractive index versus temperature. The

plot reveals rapid change at 34.50C. Andaloro gt $134
report a subtle transition in water between 30°C and

400C. Their study of the weakly absorbed 1.2 u combination
band of water yields Arrhenius plots of integrated component
intensity ratios which show neatly defined breaks occurring
in the 30°C to 40°C temperature interval. Outside this
interval, the experimental points are well aligned. The
Bryngdahl interferometric fringe pattern of this work does
not directly indicate the presence of this anomalous
phenomena. Yet, the increased deviation in the ratio of
(dn/dT)/(d'2n/dT2)1/2 above 35°C, shown in Figure 4.17, is

further indirect evidence of its existence.

4.4 Discussion

New computations, experimental observations, and
experimental deficiencies have been reported. First,
computed nonisothermal image heights have been confirmed
experimentally, proving the validity of data analysis with
respect to the experimental procedures. To our knowledge,
this is the first time that the essential correctness of
the mathematical analysis of nonisothermal data has been
experimentally proven, not simply assumed. Characteristic

fringe patterns for laminar and turbulent flows, driven by

 

 

 

 

165
end effects, have also been observed. This is important
because absence of these characteristic patterns implies
absence of convective liquid flow. Thus, convective flow
does not contribute or cause the observed nonlinear
temperature distributions. Corrections, however, are
found to be necessary for nonisothermal image distortion
which results from lens aberrations. Finally, the thermo-
couple measurement ATO has proven to be an imperfect
representation of AT. This demonstrates the need for
improved temperature control and measurement. Both image
distortion and temperature control problems must be
considered in future nonisothermal, cell system and optical
train design.

After consideration of the above nonisothermal
experimental problems, the temperature distribution
predicted by the Fourier-Laplace temperature equation for
a nonconvecting liquid is found to fail near the silver-
liquid interface for ethyl acetate, benzene, and carbon
tetrachloride. The observed nonlinear temperature
distributions are reported in three forms: (1) temperature
deviation form linearity(Figures 4.10—4.12), (2) reduced
temperature gradient(RTG, Figure 4.13), and (3) integrated
absolute deviation form linearity(Figure 4.14). Each form
has an advantage. The temperature deviation from linearity
curves directly demonstrate the magnitude of the temperature
deviation from that predicted by the Fourier-Laplace equation.

They are functions of both applied temperature gradient and

 

166

vertical coordinate. RTG curves indicate that the
temperature gradient divided by the temperature gradient
at cell center is independent of applied temperature
gradient within experimental uncertainty. Furthermore,
RTG curves reveal dramatic increases in temperature gradient
as the metal-liquid interface is approached. The magnitude
of the increase is liquid dependent and found to be 4.5%
for ethyl acetate, 12.5% for benzene, and 30% for carbon
tetrachloride when the metal is highly polished silver.
Curves of integrated absolute deviation from linearity
(functions of the applied temperature gradient) conveniently
order liquids according to the relative magnitude of their
exhibited deviation from linearity. A nonzero curve is
associated with liquids which exhibit temperature jumps
at the metal—liquid interface. Observed nonlinear temperature
distributions are found to extend 0.2-0.3 cm from the
interface for ethyl acetate, benzene, and carbon tetrachloride.
Evidence strongly suggests that water exhibits a temperature
jump.

Knowledge of liquid nonlinear temperature distributions
makes possible an increase in the accuracy of experimentally
determined Soret coefficients,

a = (~T/C1C2)(dCl/dz)/(dT/dz), (4.51)

nonisothermally determined refractive index temperature
derivatives,
dn/dT = (dn/dz)/(dT/dz), (4.52)

and interior conducibilities,

 

167
k = -JZ/(dT/dz). (4.53)

Because nonlinear temperature distributions associated
with interface effects have temperature gradients smaller
than AT/l at cell center, use of the common gradient
approximation
dT(l/2)/dz fl -AT/l

in Equation 4.51-4.53 yields computed values of a, dn/dT,
and k which are lower than the true liquid properties.
Table 4.5 shows that the linear assumption yields a 1%
systematic error for ethyl acetate, a 3% error for benzene,
and a 6% error for carbon tetrachloride. Figure 4.15
suggests an 8% error for water. Errors of this magnitude
are much larger than typical experimental inaccuracy of
1-2% reported by independent research groups for both
temperature derivatives of refractive indices and thermal
conductivities of carbon tetrachloride, benzene, and water.19

The nonisothermal, Bryngdahl interferometric image
also provides criteria for the establishment of liquid
thermal conductivity standards. Any liquid which exhibits
both linear fringe pattern and absence of a temperature
jump problem similar to that observed for water will be
an excellent standard. The temperature distribution in
such a liquid is independent of cell size and boundary
material, making the thermal conductivity measurements
laboratory independent and highly reproducible.

Nonlinear temperature distributions may prove very

useful in testing the validity of radiation energy transport

 

168

models for the liquid phase. Taking the viewpoint that
Fourier's interior conducibility incorporates all energy
transport processes characteristic of the liquid far
from walls, the time-independent, nonconvecting temperature
equation becomes

(d/dz)k(dT/dz) = -o (4.54)
where k is the interior conducibility. 0(z) is the
energy source term. It contains (1) all energy transport
processes occurring within the liquid but not described by
the interior conducibility and (2) all energy transport
processes described by the interior conducibility but not
occurring within the liquid. To conceptualize and understand,
consider the averaged photon events of Figure 4.18. Photon
events far from walls are depicted in Figure 4.18A. When
molecule 1 emits a photon and molecule 2 absorbs that
photon, the horizontal layer of molecule 2 is heated
because the horizontal layer of molecule 1 has higher
temperature. Likewise, the emission and absorption of a
photon between molecules 3 and 4 is also a photon energy
transport event. All such events are described by the
interior conducibility.(0ther energy tansport processes
contained within the interior conducibility include
vibration-vibration, translation-translation, vibration-
translation, and vibration-rotation energy transfer during
molecular collisions.) Placement of a perfect mirror
between molecules 1 and 3 alters the energy transport

processes. The photon emitted by molecule 1 is now

 

(A)

(B)

169

Figure 4.18

Molecule 1 emits an "average" photon which is absorbed
by molecule 2. This energy transport process heats the
horizontal layer of molecule 2 because the photon

originates within a layer of higher temperature.

Placement of a perfect mirror between molecules 1 and

3 alters photon energy transport processes. The photon
absorbed by molecule 2 now originates from molecule 3.
Since the horizontal layer of molecule 3 is of lower
temperature than the layer of molecule 1, energy flux

differs from the above figure. See text.

 

>rbei

ts the

 

 

 

 

 

 

 

TT. Vertical Direction 9
8
Mirror
/’/ \ \
e ‘x
\
\
\

 

 

 

171

reflected and abosrbed by molecule 4. But, Fourier's

heat flux law, ignorant of the mirror's presence, says

that the photon absorbed by molecule 4 originates from

molecule 3 whose horizontal layer is lower in temperature

than that of molecule 1. The actual photon absorbed has

more energy on the average. Consequently, positive deviation

from linearity is found near the lower wall. Similarly,

Fourier's heat flux law says that the photon absorbed by

molecule 2 originates from molecule 1. This is wrong.

Because of the mirror's presence, the photon has originated

from molecule 3 whose horizontal layer is lower in energy.

This results in negative deviation from the temperature

predicted by the Fourier-Laplace temperature equation

near the upper wall. By these arguments the extent of

the nonlinearity from the liquid-metal interface is roughly

the inverse Lambert absorption coefficient of the lowest

energy vibrational mode. For water this is 7 x 10-4 cm

(5 = 200 cm-l);35 for carbon tetrachloride it is 0.13 cm

(5 = 320 cm-i).8
The simplest mathematical model for the energy

source term: (1) assumes temperature nonlinearity is due

solely to photon events, (2) treats the metal boundary as

a non-emitting, non-absorbing, perfect mirror, (3) forbids

multiple reflection, and (4) treats the liquid as a non-

photon-scattering, isotropic media. Summing all reflection

associated emission-absorption events per unit time and

subtracting all emission—absorption events which due to

 

172

the presence of walls cannot occur yields the energy source.

We find
(E
6(z) = S a(z,v)dv (4.55)
0
where

o(z,u)/2wa(u) = S; dz1 )0 duB(z1,v)e(a(v)/u)(z1+z)
0 -1
1 duB(zl,v)e<a(v>/U><z-z1>

5

+5; dz1 31 duB(zl,v)e-(a(v)/u)((l_z1)+(l-z))
0
3.

0
-13., °d..(.1,.).<a<v>/u)<z.—z>, (.56)
u = cos¢ , (4.57)
a<v) = a<z,v) = a<z.v)(1 - e'”), (4.58)
9 = hv/kT, (4'59)
B(Z,U) = B(T(z),v)
= (2hv3/c2)/(ea -1). (4.60)

v, 9, h, k, c, and B(T,v) are the photon frequency, vertical
angle, Planck's constant, Boltzmann's constant, speed of
light, and black body radiation intensity, respectively.
a(z,v) is the Lambert absorption coefficient at the
temperature T(z). More complicating features may be built
into the model. These include emission and absorption of
photons by the metal boundaries, spectral reflection, and
photon scattering by the liquid media. Once a particular

model has been shown to accurately predict the observed

 

 

173

nonlinear temperature distributions, the model is proven.

Liquid radiation events may then be substracted from
the interior conducibility, leaving all energy transport
due to non-radiation processes. This will be an
unambiguous liquid pr0perty which may be called the thermal
conductivity.

Many review papers exist dealing with the counting

- 4
of photon events in gaseous atmospheres,36 41 ’5

and solids.112 The work of Pomraninqu’qq

liquids,
may prove particularly

useful in pursuing the above program.

 

 

CHAPTER 5

NONISOTHERMAL, CONVECTING LIQUID STATES

5.1 Introduction

As temperature increases, liquid density decreases.
Consequently, when a horizontal parallel plate apparatus
is heated from below, the liquid density decreases as the
lower plate is approached. This results in a liquid layer
which is unstable to convective motion,provided the
resulting buoyant force exceeds the viscous dissipation
force. The transition from nonconvection to convection as
the buoyant force is increased is generally called "Bénard
convection" in recognition of Bénard's1 1901 observations
of hexagonal convection cells in very thin liquid layers
with a free surface. However, it is now generally believed
that surface tension effects played an important role in
Bénard's experiments.2'4 In this chapter and the relevant
references, the layers are relatively deep and confined
within rigid boundaries. Thus, surface tension plays no
role. The transition is also called the "Rayleigh-Jeffreys
instability" in recognition of their theoretical contributions.

The important parameter describing the relative

magnitude of buoyant and viscous force is the dimensionless

174

 

Rayleigh number defined by

R - 2” 3
_ -gap CV-J‘ AT/kn, (5.1)

where p, 0V, k, and n are the mass density, specific heat,

thermal conductivity, and viscosity, respectively. g is
the gravitational constant(980.2 cm sec-2) and a is the
coefficient of expansion. When heating from below,AT is
negative and R is positive. Linear stability analysis of
the hydrodynamic equations describing p(5,t), T(5,t), and
the barycentric velocity, v(;,t), show that in the
Boussinesq approximation a transition from nonconvection
to time-independent convection occurs at the Rayleigh

number
RI = 1707.76 (5.2)

for a fluid between rigid boundaries. This theoretical
result has been reviewed by Chandrasekhar.5 For R4<RI,
the fluid is at rest and energy is transported by conduction
alone; for R ) RI’ steady state convection of the fluid
contributes to energy transport.

It has been found experimentally that several
additional transitions occur with increasing Rayleigh
number.6-12 These are: (a) a two-dimensional steady flow
to three—dimensional steady flow transition at RII' (b)

a three—dimensional steady flow to three-dimensional time-

dependent flow transition at R (c) a time-dependent

III’
flow to time-dependent flow of increased frequency transition

at R and (d) a time-dependent flow to turbulent flow

IV’

 

 

 

176

10’11 has compiled evidence

transition at RV. Krishnamurti
of these transitions and has shown that the transitions
are dependent upon the dimensionless Prandtl number defined
by
N
Pr = ch/k. (5.3)

This is a ratio of viscous energy dissipation to thermal
conductivity. Generally, the critical Rayleigh numbers
increase toward a limiting value with increasing Prandtl
number.

Detailed reviews of the Benard problem are avail-

ab1e.5’13-18

Works which relate the concepts and mathematics
of the Benard instability with the Taylor problem, the
Landau phase transition, lasing, and oscillating chemical
reactions are also available.19-21
In this chapter we explore the application of
Bryngdahl interferometry to the analysis of the Rayleigh
instabilities. Consequently, the thrust of the chapter is
experimental. All measured quantities are compared with
relevant experimental studies which have been reported in the
literature. Questions include: What transitions can be
observed with the Bryngdahl interferometer? What temperature
distributions are observed between transitions? Do time-
periodic states exist in the fluid? If so, what is the
period? Are aperiodic states a possibility? If so, for
what Rayleigh numbers and what is the observed frequency
spectrum? To answer these questions, we have studied water

with the experimental techniques anddataanalysis described

in Chapter 3 but with an experimental arrangement in which

 

 

 

177

ATo = TU - TL < 0. (5.4)

The cell used has a height, 1, of 1.440 cm and measures
5.961 cm along the optical path. It is 7.0 cm wide. The
mean temperatues, TM, of individual experiments varies
between 22.300 and 32.00C. AT0 values vary between
-0.02°C and -12.4°c. Furthermore, using the density,

coefficient of expansion, coefficient of compressibility,

and heat capacity compilation of Kell,22 the thermal

23

conductivity evaluation of McLaughlin, and viscosity

reported in the Handbook of Chemistry 229 Physics,24 we

find that for water

1 1

-(R/_l_3AT)10‘li cm3 00 —9.6914 x 10‘ + 1.7429 x 10“ T

-5.3120 x 10‘5T2 + 1.5528 x io‘l‘T3

-1.4803 x 10‘6T4, (5.5)

1T

Pr = 13.044 - 4.4926 x 10'
+ 9.1933 x 10'3T2 - 1.1055 x 10"‘T3

+ 5.9166 x 10‘714. (5.6)

T is in 00, 20°CST$35°C, and P = 1 atm.

178

512 Temperature Distributions and the First Rayleigh

Transition

 

 

Figure 5.1 reproduces the nonisothermal Bryngdahl
images of thirty-two experiments in the order of increasing
Rayleigh number. It is apparent that even at the lowest
Rayleigh number(9.3 x 103) the fringe pattern is essentially
vertical but with curvature near the upper and lower
boundaries characteristic of nonlinear temperature distri-
bution. Vertical fringe patterns of negligible shift are
characteristic of isothermal regions. Up to R = 2.9 x 104
the curvature increases gradually, and comparison of
Figure 5.1 with Figure 4.3 shows that time-independent
fluid flow is the predominant mode of energy transport.
The straight, tilted fringe characteristic of pure thermal
conduction ofenergy'is absent.

Between R = 2.9 x 10‘1 and 3.3 x 10“, the time-
independent boundary layers become well-formed and distinctly
contrast the central isothermal region in which dT/dz = O.
The boundary layers extend approximately 0.2; into the fluid.
The isothermal region is 0.6; in extent. The thermal
boundary layers grow in strength with increasing Rayleigh
number. However, at R = 5.40 x 104 a photomultiplier at
image center indicates the appearance of small amplitude,
time—dependent fluctuations which are superimposed upon the
time-independent pattern. The amplitude of the fluctuations
grow with increasing Rayleigh number and are visually

recognized in the image at R = 8.4 x 104. Finally, between

 

 

 

   

179

Figure 5.1

Nonisothermal Bryngdahl cell images. I = 1.440 cm,
a = 5.961 cm, H a 3.175 cm, 2D = 0.138 cm, 6 = 0.353 cm.

Equation 4.47 is used to evaluate AT for given ATO.

 

 

 

 

 

 

 

 

 

Exp. 10'42(TM,AT) Pr Exp. 10’42(TM,AT) Pr
1 0.929 6.30 17 7.670 6.14
2 1.433 6.28 18 8.366 6.14
3 1.796 6.27 19 9.224 6.13
4 1.998 6.28 20 9.822 6.11
5 2.268 6.28 21 10.49 6.11
6 2.577 6.27 22 11.18 6.10
7 2.888 6.27 23 12.00 6.10
8 3.321 6.26 24 12.74 6.081
9 3.808 6.25 25 13.73 6.07
10 4.136 6.25 26 14.20 6.06
11 4.426 6.24 27 21.41 6.53
12 4.859 6.24 28 30.56 6.34
13 5.365 6.20 29 37.08 6.22
14 5.700 6.19 30 43.43 6.10
15 6.377 6.17 31 57.44 5.90
16 7.347 6.15 32 72.15 5.72]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

VN mm mm _N ON 9 m. \L

g :13; a?

 

 

 

 

 

 

 

 

 

 

 

 

 

181

R = 5.7 x 105 and 7.2 x 105 the fluctuations become so
strong as to destroy the characteristic fringe pattern of
isothermal layers and boundary layers. A transition to
turbulence is evident.

The temperature distributions of Figure 5.2 are
mapped from the refined data using the method of Section
3.6. They are considered to be the average temperature
experienced by a photon pencil along the optical path.

The thermal boundary layers and isothermal region discussed
above are quantitiatively displayed in this figure.

An important aspect is the absence of "temperature
inversion" in the domain 0.2< z/lg<0.8. In their evaluation
of horizontally averaged temperature distributions,
Deardorff and Willis7 have used a resistance wire probe
to measure temperature gradient. They report distributions

6, and 1.0 x 107 which qualitatively

at R = 6.3 x 105, 2.5 x 10
agree with Figure 5.2 However, Gille's8 measurement of the
horizontally averaged temperature distribution with a
Michelson interferometer at R = 2.73 x 101i reveals temperature
inversion in the central region. The reversal is about

1.5% of the impressed temperature difference. Finally, the
Mach-Zehnder interferometrically determined temperature

profiles of Farhadieh and Tankin12 provide direct evidence

4

of a 6% temperature inversion at R = 1.57 x 10 and

2.36 x 104. This result verifies numerical computations
2
at high Rayleigh numbers of Veronis. 5 We believe that the

inversion has not been observed with the Bryngdahl interfer—

 

 

182

Figure 5.2

Temperature distributions for various impressed temperature

gradient.

 

 

—ATo/1°C 10'4R(TM,AT)
0.299 1.43
0.417 2.00
0.689 3.32
0.854 4.14
1.000 4.86

 

 

 

 

 

 

183

 

 

 

l

h

 

 

I.

0.009 -uoe<@ -
0268.0-qu @

6.86.01. 2 ® -
0.8.6.0-. e<® -
6.68.0-" Z 8 -

 

 

mwmm

QMN

Q¢N

NvN

vém

Dot/l

184
ometer because of the very small total fringe shift at
image center caused by such small temperature gradients.
At R ~I2 x 104 a 6% inversion results in a total fringe
shift of only 0.01 cm for the optical train used in this
series of experiments. Furthermore, the expected fringe
shift at larger Rayleigh numbers are masked by time-
dependent fringe fluctuation. These observations indicate
that the direct temperature measurement techniques provided
by Michelson and Mach-Zehnder interferometry are to be
preferred in the analysis of very small temperature
phenomena for which temperature gradients are nearly
constant.

Figure 5.3 compares the experimentally found
temperature derivatives at z = l with the values expected
for pure conduction. The experimental gradients are far
larger. In addition, the sudden increase in the standard
deviation at B(TM,ATO) = 3.7 x 104 indicates a transition to
time-dependent flow. This is the critical Rayleigh number

RIII within an uncertainty of 112%. With the correction

of Equation 4.47 we find

RIII(TM’AT) = 3.3 X 101* _‘L’ 12%. (507)

This agrees well with Krishnamurti's11 value of

4

3.59 x 10 I 15%. However, Schmidt and Saunders27

report

a transition from laminar to turbulent flow at Rc'4.7 x 104

while Malkus6 claims that this transition occurs between

1 x 104 and 3 x 104.

 

 

185

Figure 5.3

Experimentally determined boundary(z = 1) temperature
gradient as a function of Rayleigh number. Curve 1 is

the linear least squares analysis of data(Equation 5.8);
Curve 2 is the prediction of the Laplace equation description
of temperature.

 

 

186

 

 

-4
l R(T .ATO)

 

 

IO

 

oo\ Eo— x ~U\$V._.U

 

 

187

Linear least squares analysis of the data

R(Tb ATO)$ 4.2 x 104 yields

‘9

dT(l)/dz = -7.177 x 10‘2 + 5.075 x 10‘SR(TM,ATO) (5.8)

0 = 0.133
0a = 4.42 x 10-2
ab = 1.94 x 10'6
r2 = 0.957.

The gradient expected for pure thermal conduction is

dT(l)/dz = ~ATo/l (5.9)

-R(TM,ATO)/(lR(<TM),ATO)/ATO)
= 1.3138 x 10‘5R(TM,ATO).
The intersection of Equation 5.8 and 5.9 is the critical

Rayleigh number for the transition from nonconvection to

steady state convection. It is

RI(TM,ATO) = 1908 i 137. (5.10)
This value is significantly higher than the theoretical
value of 1708. However, the experimental result is

considerably improved with the correction of Equation 4.47.

Applying the correction, Equation 5.7 becomes
dT(l)/dz = -7.177 x 10‘2 + 5.619 x 10’5R(TM,AT). (5.11)

The intersection of Equations 5.9 and 5.11 is

RI(TM,AT) = 1667 i 137. (5.12)

 

 

188
TflfleSJ
Experimental evaluations of the first critical Rayleigh

number. a is the standard deviation or uncertainty.

 

 

 

Reference RI 0 Fluid
Schmidt and 1770 140 water
Milverton(1935)26
Schmidt and 1750 100 water
Saunders(1938)27
Malkus(1954)6 1700 80 water
Silveston(1958)28 1700 51 water, glycol
Gille and 1786 16 air
Goody(1964)29
Thompson and 1793 80 argon, air, CO2
Sagin(1966)30
Rossby(1969)9 1750 130 water, silicon

oil, Hg
Tarhadieh and 1700 30 water
Tankin(1974)12
This work 1667 137 water

 

 

 

 

 

 

189

Figure 5.4

Time sequential temperature distributions for

R(TM,AT) = 1.37 x 105.

 

 

 

 

Curve t/1 sec c/1°c
1 0 0.0
2 15 0.1
3 30 0.2
4 45 0.3
5 60 0.4
6 75 0.5
7 90 0.6
8 105 0.7

 

 

 

 

190

LC

 

 

 

 

 

 

 

C
o
3
6
6.
2
_
I
TO
A
_ _ _ . p . _ _ O.
W 6 5 My 3. 2 M Roy W O
5 5 5 5
2 2 2 2 2 2 2 2 2

0:20... .5

z/t

l—_' “V——

191
The agreement of this value with the theoretical value is
further evidence for the existence of the temperature jump
discussed in Section 4.3. It also justifies the use of
Equation 4.47 in converting B(TM,
R(TM,AT) values reported in many of the figures of this

ATO) values to the

chapter.

Table 5.1 compares experimentally determined
values of the first critical Rayleigh number. A final
example of temperature distribution is provided by the time-
sequential distributions at B(TM,AT) = 1.37 x 105 shown in

Figure 5.4.

5.3 Fluctuations and Frequency Spectrgm

For Rayleigh numbers not much higher than approximately
4 x 104, the intensity fluctuations of the Bryngdahl image
can be very periodic. The output of a photomultiplier
placed at image center reveals that at R = 8.37 x 104 the
fringe oscillate with a period of 0.5 min(Figure 5.5).
Dividing this period by the basic time unit p0Vl2/k(i;g.,
1.425 x 103 sec at 24.500 for water) yields the dimensionless

period t = 2.11 x 10-2. This is intermediate between the

-0 .—
shortest period oscillations(t = 1.6 x 10 “ and 2.6 x 10 2)

reported by Krishnamurti.11
Both the intensity fluctuation rate and amplitude
increase with Rayleigh number, and the photomultiplier

output is visually aperiodic for the Rayleigh numbers of

Figure 5.6. The fluctuations provide a convenient method

 

 

192

Figure 5.5

Intensity fluctuations at image center of Bryngdahl

1
image for B(TM,AT) = 8.37 x 10‘.

 

;’

193

58:5

 

 

 

 

 

 

 

 

 

om 6.. m. 6.0 to 0.0
- - q 4
M
0. I
m. N
o l
m. m.
m. A
e i
c 6.86....» .2
6.806728

 

 

 

 

194

Figure 5.6

Intensity fluctuations at image center of Bryngdahl

image for various impressed temperature gradients.

 

 

-ATo/1°C 10'5R(TM,AT)
4.088 2.93
4.878 2.14
6.511 3.06
7.582 3.71

 

 

 

 

 

 

195

2E?)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N._ GO 0.0 Md 0 N._ QC Q0 Md 0
4 a _ u _ _ _ _ ..O
a 1 _
im
..m
in
be
. O . 0 1w
Uonleu F4 UoZOQIH FAN I OCH
0
12: 2: L
z 1N
EomH -m
1?
Oomfim.ql Uommo..—VI Hm
"oz .62 .

 

 

 

196
for the determination of the critical Rayleigh transition
RIV‘ A plot of the fluctuation rate as a function of ATo

(Figure 5.7) indicates the linear relationship

no. fluctuations per min

3.970(IATOI - 0.805) (5.13)

O = 2.625
0a = 1.064
0b = 0.174
r2 = 0.9700

The standard error of the regression coefficients improve
when the functional dependence upon Rayleigh number is

considered. Then

no. fluctuations per min =

6.457 x 10'5(R(TM,ATO) - 4.898 x 104) (5.14)

o = 1.568
0a = 0.631
ab = 1.676 x 10"6
r2 = 0.989.

Equation 5.13 states that an apparent transition from
small amplitude fluctuations(not detectable with the
photomultiplier) to large amplitude fluctuations occurs

at (AT6)IV.= -0.805°c. Equation 5.14 establishes the

4

critical Rayleigh number RIV(TM’AT0) = 4.90 x 10 :,14%.

Applying the temperature jump correction, the result is

T 4

M,AT) = 4.42 x 10

RIV( 1 14%. (5.15)

197

Figure 5.7

 

Bryngdahl image intensity fluctuation rate at cell center

versus impressed temperature gradient and versus Rayleigh

number. The solid lines are Equations 5.13 and 5.14.

 

No. Fluctuations per min

 

198

 

 

 

 

 

 

 

I0‘5R(T .ATO)

 

 

199

This is in excellent agreement with Krishnamurti's

report of RIV744.4 x 104. Malkus6 also observed a

11

transition from one turbulent mode to another but at
R'95.5 x 104.

The frequency spectrum of the fluctuations is the
Fourier transform of the photomultiplier output autocor-
relation function. It is very important in developing an
understanding of aperiodic motions.31’32 Let AI(t) be the
photomultiplier output at time t minus the time averaged

output. The autocorrelation function, R(t), is then

defined to be

R(t) = lim (1/tx|)f: AI(t1)AI(t1 + t)dt1. (5.16)
X—boo

This is an even function; therefore, the Fourier transform

of R(t), F(w), is

F(w) = lim 5 : R(t)cos(wt)dt. (5.17)

Xv”

The importance of F(w) is demonstrated by the Wiener—

Khintchine theorem:33
F(w) = lFourier transform of AI(t)]2 (5.18a)
2
= lwA(w)‘ (5.18b)
where

1(1) = 5w A(w)cos(wt)dw. (5.18c)
0

 

 

 

 

 

 

 

 

200

Thus, F(w) is proportional to the square of the harmonic
frequency amplitude. Applying this correlation analysis

to the intensity fluctuations of Figure 5.6, the integrals
of Equation 5.16 and 5.17 are numerically evaluated with the

trapezoidal rule.34

Furthermore, the time increment

At = 0.01 min and the time interval X = 5 min are used.
Figure 5.8 is the autocorrelation function for

AT

indicates that F(w)/e where

-6.849°C and R(TM,AT) = 3.71 x 105. Figure 5.9

6 = <AT)IV - AT (5.19)

is approximately independent of the impressed temperature
difference. Four harmonics in the ratio w1/w2/w3/w I»:

1/3/9/14 are distinctly visible and the best exponential

description of the maxima is

-0.01320)

F(w)/€’V e (5.20)

where w has the dimension min-1. The only comparable

observations are those of Krishnamurti.11

Her reported
oscillation periods of light scattering from aluminum
flakes seem to be components of the two lowest frequency

peaks shown in Figure 5.9.

514 Summary

Bryngdahl interferometrically determined temperature
derivatives have proven very useful in the detection and
quantitative evaluation of critical transitions between

different modes of flow. Experimental determination of

 

 

201

Figure 5.8

Autocorrelation function of AI(t) for AT = ~6.849°C
and R(TM,AT) = 3.71 x 105.

 

‘IIII-lllllll‘

Om._

EE < ..
ON._ OQO O®.O OmO
_ _ _

 

202

 

 

[
1

db

Esmeouoe

25 moo . me .I.

58 _mo. _e .III.

 

 

 

 

 
 

 

203

Figure 5.9

Fourier transform of autocorrelation function of AI(t)
divided by e = lATl - 0.7270C. The solid line of curve D
is proportional to Equation 5.20; the experimental points

are the maxima of curves A, B, and C. T is in 0C.

 

 

-AT TM 10‘5R(TM,AT)
A(O) 3.693 31.633 2.93
B(CI) 5.881 23.342 3.06

0(1) 6.849 24.033 3.71

 

 

 

 

 

 

 

 

 

204

 

 

 

0.2 _
A
0.: -
0
B
0.2 -
0.1 _
F(w)/e
0
C
0.2
0.l —

 

 

 

 

 

O 30 60 90
l

(1)/1min-

IZO I50

 

 

205
boundary temperature gradients as a function of Rayleigh
number has yielded the critical Rayleigh transitons

R 4

1667 1 137 and R I = 3.3 x 10 1_12% for water.

I II
It has not proven possible to detect temperature inversion
at intermediate distances from boundaries because of the
small, nearly constant, temperature gradients involved.
Fluctuations in the Bryngdahl interferometer image
have proven to be extremely valuable in the study of time-
dependent properties. First, the fluctuations provide
strong evidence of the existence of a fourth Rayleigh

4

transition at RIv = 4.42 x 10 1 14%. Second, autocor-

relation analysis of intensity fluctuations reveals four
major frequency bands at w~10, 30, 90, and 140 min-1. The
intensity of the bands seems to be pr0portional to

6 = (AT)Iv - AT and they decrease exponentially with the
angular frequency 75.8 min-1. Both theoretical and further
experimental work are needed to determine whether these
frequency bands are the manifestation of buoyancy effects
driven by thermodynamic fluctuations within the liquid or
whether one or more of the bands are due to instabilities
driven by room and plate vibrations. Because of the unique
nature of the Bryngdahl interferometric observations, no
relevant literature comparison is possible. Lack of a
theoretical description provides a good, although extremely
difficult, field of research. Numerical computations35’36

show that temperature fluctuations can be conceptualized as

thermal elements or "plumes" which break away from the

206
lower thermal layer and rise in the cooler, denser
surroundings.

Bryngdahl interferometry cannot detect the transition
from two-dimensional flow to three-dimensional flow at RII
because of the absence of both temperature fluctuations
and nonlinear variation of the boundary temperature gradient
with Rayleigh number. Busse's37 theoretical study of two-
dimensional convection demonstrates that such a transition
is possible at R = 2.3 x 104 for high Prandtl number fluids.
However, in the limit of small Prandtl number, Busse finds
that two-dimensional rolls become unstable to oscillatory
three-dimensional disturbances when the amplitude of the
convective motion exceeds a finite critical value.38 It
may be possible to detect this transition with the Bryngdahl
interferometer.

Finally, the successful evaluation of R111 and
RIv for water suggests that Bryngdahl interferometry can
be used to determine the dependence of the critical Rayleigh
numbers upon Prandtl number. This dependence is believed
to be very strong for Prandtlnumberslower than that of

water.11

APPENDIX

 

 

 

APPENDIX

FOURIER POWER SERIES SOLUTION OF

THE VIBRATIONAL RELAXATION EQUATION

The general differential form of the vibrational

relaxation models of Chapter 2 is

(A.1)

k -0 k k -0
(~1) (1-e )d xn/dg + ne xn_1 + (n+1)xn+1
-0
+ ((n+1)e + n)xn = 0.
Model 1 has k = 2 and g = (K/D)1/22; Model 2 has k = 1

and L = (K/V)z. This equation must be solved subject to

the conditions

xn(g) = xn(0) at g = O and (A.2)

lim xn(;) = (1-e-0) -n6

{-900

Further, it is assumed that the variation of 0 with g is
negligible.
Taking the solution to have the form

w 1/k,

_ -11
xnm — 11509.4(“ ,

207

e . (A.3)

(A.4)

 

 

 

208

we find from Equation A.1 that

(1-e-0)uln + ne-ol 1 + (n+1);n+1 + ((n+1)e-0 + n)_l_'n = 0.

—n—
(A.5)
Gottlieb(Am. J. Math., g9, 453(1938)) has shown that the

polynomials which satisfy this recurrence formula are

1 (u) = e'ni 5 (1-e”)”(n)(“) (A 6)
-n v=0 v v . .

Montroll and Shuler(J. Chem. Phys., 26, 454(1957)) have shown

that

ln(u) = F(-n, u+1, 1; 1-e-0)

(A.7)

where F(a, b, c; z) is the hypergeometric function. The

Gottlieb polynomials have the following properties:

en91n(u) = euiiu(n). (1.8)

n
otd8

v e‘””1n<v)1m(u) = e”n”(1-e‘”)'15n,m. (1.9)

and

Eoenaln(v)ln(u) = eU0(1-e-0)-18v, (A.10)

u
n:

where 8n m is the Kronecker delta. To evaluate the functions
7

an, left operate on Equation A.5 with the summation operator

m
2 eneln(v); then, evaluate the results at L = 0. Using
n=0

Equations A.8-A.10, it is found that

- "“0(1- ‘0) E enfll (u)x (0) (A 11)
an - e e n=0 __n n . o

 

 

209

Finally, by Equations A.4, A.6, and A.11

. _ _ -n0
11m x - aol (0) — aoe

{400 n

e-no(1-e'0) 2 x (0)
n=0 n
e-n0(1—e-0)

The last equation follows from the normalization of
probability.

In summary, the solution of Equation A.1 is

xn(§) = “igloetwimho«1"“1/1‘g (1.12)
where

_1_.,,(u) = «amigo-«1043(3) (1.13)
and

au = e-u0(1-e-0) 02° e“0;n(u)xh(0). (A.14)

n=0

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T.L. Cottrell and J.C. McCoubrey, Molecular Ener
Transfer in Gases, (Butterworths, 1961).

G.M. Burnett and A.M. North (ed.), Transfer and
Storage of Energy by Molecules, Vol. 2, (Wiley, 1969).

B.6S§evens, Collisional Activation in Gases, (Pergamon,
19 7 .

 

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

‘10.

1‘1.

215

P.C.M. van WOerkom, J. de Bleijser, M. de Zwart,
P.M.J. Burgers,and J.C. Leyte, Ber, Bunsegges. Physik.
Chem,, 12, 1303 (1974). Vibrational Relaxation in
Liquids; Some Applications of the IsotOpic Dilution
Method.

W.G. Rothschild, G.J. Rosasco, and R.G. Livingston,

2, Chem. Phys., 62, 1253 (1975). Dynamics of Molecular
Reorientational Motion and Vibrational Relaxation in
Liquids. Chloroform.

A. Laubereau and W. Kaiser, Ann. Rev. Phys. Chem., 29,
83 (1975). Picosecond Spectroscopy of Molecular
Dynamics in Liquids.

W.F. Calaway and G.E. Ewing, Chem. Phys, Letters, 29,
485 (1975). Vibrational Relaxation in Liquid Nitrogen.

W.F. Calaway and G.E. Ewing, J, Chem. Phys., 63, 2842
(1975). Vibrational Relaxation of Small Molecules in
the Liquid Phase: Liquid Nitrogen Doped with 02, CO,

and CH .

J.O. Hirschfelder, C.F. Curtiss, and R.B. Bird,
Molecular Theory of Gases and Liquids, (Wiley, 1954).

W. Ibele, in the Handbook of Heat Transfer, ed. by
W.M. Rohsenow and J.P. Hartnett, (McGraw-Hill, 1973).
Thermochemical Properties.

T.K. Sherwood, R.L. Pigford, and C.R. Wilke, Mass
Transfer, (McGraw-Hill, 1975).

JANAF Thermochemical Tables, 2ng ed., NSRDS-NBS 37,
(June, 1971).

 

G. Herzberg Infrared and Raman S ectra of Polyatomic
Molecules, D. Van Nostrand, 1945).

T. Shimanouchi, DSRDS-NBS 39, (June, 1972). Tables of
Molecular'Vibrational Frequencies.

G. Houghton, J, Chem, Phys., 29, 1628 (1964). Cubic
Cell Model for Self-Diffusion in Liquids.

N.B. Vargaftik, Tables on the Thermo hysical Properties
of Liquids and Gases, (Hemisphere, 1975).

K. Raznjevic, ggpdbook of Thermod amic Tables and
Charts, (McGraw-Hill, 1976).

F.J. Bartoli and T.A. Litovitz, J. Chem. Ph 8., 2g,
413 (1972). Raman Scattering: Orientational Motions
in Liquids.

 

 

 

42.

43.

44.

45.

216

P.K. Davis and I. Oppenheim, J. Chem. Phys., 51, 505
(1972). Vibrational Relaxation in Liquids.

P.K. Davis, J, Chem, Phys., 51, 517 (1972). Collision
Frequencies in Liquids.

D. Eisenberg and W. Kauzmann, The Structure and
Pr0perties of Water, (Oxford University Press, 1969).

C.W. Robertson, B. Curnutte, and D. Williams, M01,
Phys., 26, 183 (1973). The Infrared Spectrum of water.

 

 

 

10.

11.

Chapter 3 References

E. Ingelstam, Ark. Fys., 2, 197 (1955). Measurements
Of Optical Path Gradients By Means Of Birefringence
Interferences.

E. Ingelstam, J. Opt. Soc. Am., 31 536 (1957).
Some Quantitative Measurements Of Path Differences
And Gradients By Means Of Phase Contrast And New
Interferometric Devices.

0. Bryngdahl, Acta Chem. Scand., 1;, 1017 (1957).
A New Interferometric Method For The Determination Of
Diffusion Coefficients Of Very Dilute Solutions.

0. Bryngdahl, Acta Chem. Scand., 12, 684 (1958).
Interferometric Studies Of Boundary Formation And
Deviations From Ideality In Diffusion Experiments.

0. Bryngdahl and S. Ljunggren, J. Phys. Chem., 63,

1264 (1960). A New Refractive Index Gradient Recording
Interferometer Suitable For Studying Diffusion,
E1ectrophoresis,And Sedimentation.

0. Bryngdahl, Ark. Fys., 21, 289 (1961). Genaue
Bestimmungen Der Warmeleiteigenschaften In Flussigkeiten
Mit Einer Scherinterferometrischen Methode.

O. Bryngdahl and S. Ljunggren, Acta Chem. Scand., 1Q,
2162 (1962). Neuartige Berechnungsmethoden Und
Forderungen An Scherinterferometrische Prazisions-
bestimmungen Von Diffusionskoeffizienten.

0. Bryngdahl, J. Opt. Soc. Am., 53, 571 (1963).
Wavefront Shearing Interferometer For Direct Recording
0f Refractive Index Gradient In Cartesian Coordinates.

S.E. Gustafsson, J.G. Becsey, and J.A. Bierlein,
J. Phys. Chem., 62, 1016 (1965). Thermal Diffusion
Measurements By Wavefront Shearing Interferometry.

5.2. Gustafsson, z. Naturforsch., 22A, 1005 (1967).
A Non-Steady—State Method Of Measuring The Thermal
Conductivity Of Transparent Liquids.

L.W. Wendelov, L.-E. Wallin, and 8.8. Gustafsson,

Z. Naturforsch., 222, 1180 (1967). Refractive Index
Measurements Of Molten Salts With Wavefront Shearing
Interferometry. I: Test Of Apparatus And Procedure.

217

12.

13.

14.

15.

16.

17o

18.

19.

20.

21.

22.

23.

218

L.W. Wendelov, S.E. Gustafsson, N.-O. Halling, and
H.A.E. Kjellander, Z. Naturforsch., 22A, 1363 (1967).
Refractive Index Measurements Of Molted Salts With
Wavefront Shearing Interferometry. II: LiNO3, NaNO
KNO3, RbNO And CsNO .

3’ 3
S.E. Gustafsson, N.-O. Halling and R.A.E. Kjellander,
Z. Naturforsch., 23A, 44 (1968). Optical Determinations
Of Thermal Conductivity With A Plane Source Technique.

3’

S.E. Gustafsson, N.-O. Halling, and R.A.E. Kjellander,
Z. Naturforsch., 235, 682 (1968). Optical Determination
Of Thermal Conductivity With A Plane Source Technique.
II: Molten LiNO3, RbNOB, And CsNOs.

S.E. Gustafsson, L.-E. Wallin, and T.E.G. Arvidsson,
Z. Naturforsch., 222, 1261 (1968). A Plane Source
Method For Measuring Interdiffusion Coefficients Of
Transparent Liquids.

L.-E. Wallin, and S.E. Gustafsson, Z. Naturforsch.,
24A, 436 (1969). On Non-Ideal Conditions In Plane
Source Diffusion Experiments.

L.-E. Wallin, and K. Wallin, Optica Acta, 11, 381 (1970).
The Savart Interferometer Modified For High Sensitivity
In Measurements Of Refractive Index Gradients.

L.-E. Wallin, J. Chem, Phys., 52, 552 (1970).
Thermal Diffusion Measurements With A Modified Savart
Interferometer.

T.E.G. Arvidsson, S.-A. Afsenius, and S.E. Gustafsson,
J. Chem. Phys., 53, 2621 (1970). Interdiffusion
Measurements In Fused Alkali Nitrates By Wavefront
Shearing Interferometry.

T.E.G. Arvidsson, S.-A. Afsenius, and S.E. Gustafsson,
z. Naturforsch., 26A, 752 (1971). Interdiffusion
Studies In Molten Alkali Nitrates.

S.D.H. Andreasson, S.E. Gustafsson, and N.-0. Halling,
J. Opt. Soc. Am., él, 595 (1971). Measurement Of
The Refractive Index Of Transparent Solids And Fluids.

I. Okada, and S.E. Gustafsson, Electrochim. Acta,
;§, 275 (1973). Interdiffusion'of'Thallous Ion In
Molten Lithium Nitrate Studied By Optical Interferometry.

S.E. Gustafsson, E. Karawacki and I. Okada,

2, Naturforsch, 192, 35 (1975). A Method For Measuring
Interdiffusion Coefficients In Liquids By Wavefront
Shearing Interferometry.

 

24.

25.

26.

27.

28.

29.

30.

31.

33.

311.

35.

36.

219

E. McLaughlin, Chem. Rev., fig, 389 (1964). The
Thermal Conductivity Of Liquids And Dense Gases.

L.H. Adams, J. Wash. Acad. Sci., 5, 265 (1915).
Some Notes On The Theory Of The Rayleigh-Zeiss
Interferometer.

W. J. Thomes,w Md E. McK. Nich011,Appl. Opt.,
4, 823 (1965). The Application Of The Wavefront Shearing
Optical Interferometer To Diffusion Measurements.

W. J. Thomas, and E. McK. Nicholl, J. A 1.Chem., 11,
251 (1967). Diffusion Measurements For Ethanolamine-
Water Systems With A Wavefront Shearing Interferometer.

W. J. Thomas, R. Khanna, and E.W. Palmer, Chem. Eng. J.,
5, 112 (1972). Physical Absorption Into A Pool Of
Liquid COZ-Water).

W. J. Thomas, R. Khanna, and E.W. Palmer, Chem. Eng. J.,
Q, 165 (1973). Convective Disturbances In Static
Liquid Pools.

W.J. Thomas, and M.S. Ray, Chem. Eng. J., 5, 71 (1975).
A Study Of Gas Desorption From Static Liquid Pools.

C. N. Pepela, B. J. Steel, and P. J. Dunlop,o J. Am. Chem. Soc.,
22, 6743 (1970). A Diffusion Study At 250 C With A
Shearing Diffusiometer. A Comparison With The Gouy

And Conductance Methods.

G. R. Staker, and P. J. Dunlop, J. Chem. Eng. Data, 18,

61 (1973). Use Of A New Cell To Measure Diffusion—
Coefficients For The Sygtems Benzene-Carbon Tetrachloride
And Sucrose-Water At 25 C.

T. G. Anderson, and F. H. Horne, J. Chem. Phys., 22,

2831 (1971). Pure Thermal Diffusion. II: Experimental
Thermal Diffusion Factors And Mutual Diffusion
Coefficients For CClq-C6H12.
J.D. Olson, and F.H. Horne, J. Chem. Phys., 55, 2321 (1973).
Direct Determination Of Temperature Dependence Of
Refractive Index Of Liquids.

M. Mitchell, and H.J.V. Tyrrell, J. Chem. Soc. (London),
68 F2, 385 (1972). Diffusion Of Benzene, Phenol, And
Resorcinol In Propan-1,2—diol, And The Validity Of

The Stokes-Einstein Equation.

C. J. Skipp, and H. J. V. Tyrrell, J. Chem. Soc. (London),
21 F1, 1744 (1975). Diffusion In Viscous Solvents.

II. Planar And Spherical Molecules In Propane- -1, 2- diol
At 15, 25, and 35 C.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47o

[18.

220

B. Porsch, and M. Rubin, Coll. Czech. Chem. Commgg.,
21, 1028 (1968). Limits Of Use Of A Polarization
Interferometer For The Measurement 0f Diffusion
Coefficients At Very Low Concentrations.

M. Kubin, and B. Porsch, Eur. Pol er J., g, 97 (1970).
Estimation Of Polymer Polydispersity By Diffusion
Measurements At Low Concentration.

B. Porsch, and M. Kubin, Coll. Czech. Chem. Commun.,
2Q, 4046 (1971). Flowing—Boundary Diffusion Cell And
The Determination Of Zero-Time Correction In Free
Diffusion Measurements In Very Dilute Polymer Solutions
With A Polarization Interferometer.

B. Porsch, and M. Rubin, Coll. Czech. Chem. Commun.,
31, 824 (1972). Determination 0f Criteria 0f Polymer
Polydispersity Based On The Free Diffusion Measurements
With A Polarization Interferometer.

B. Porsch, Coll. Czech. Chem. Commun., 51, 3426 (1972).
A Simple And Exact Calculation Method For Binary
Diffusion Coefficients From The Data 0f Polarization
Interferometer.

B. Porsch, and M. Kubin, Eur. Polymer J., 5, 1013 (1973).
Concentration Dependence Of The Diffusion Coefficient
0f Polydisperse Polystyrene In Dilute Solution.

L.J. Costing, Adv. Protein Chem., 429 (1956).
Measurement And Interpretation Of Diffusion Coefficients
0f Proteins.

0. Bryngdahl, Pro . O t., 5, 39 (1965). Applications
Of Shearing Interferometry.

L.G. Longsworth, Phys. Tech. Biol. Rg§., IIA, 85 (1968).
Diffusion In Liquids.

R.J. Goldstein, Measurement Techniques In Hggt Transfer,
ed. by E.R.G. Eckert and R.J. Goldstein, Ch.4, 177
(AGARDograph 130, 1970). Optical Measurement 0f
Temperature.

W. Hauf, and U. Grigull, Adv. Heat Transfer, Q, 134 (1970).
Optical Methods In Heat Transfer.

R.N. O'Brien, Phys. Methods Chem., IIIA, ed. by
A. Weissberger and R.W. Rossiter, 1 (John Wiley,
1972). Interferometry.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

61.

62.

221

P.J. Dunlop, B.J. Steel, and J.E. Lane, Phys. Methods
Chem., 11, ed. by A. Weissberger and B.W, Rossiter,
205 (John Wiley, 1972). Experimental Methods For
Studying Diffusion In Liquids, Gases, And Solids.

R.H. Muller, Adv. Electrochgm. And Electrochem. Eng.,
2, 281 (1973). Double Beam Interferometry For
Electrochemical Studies .

H. Ertl, R.H. Ghai, and F.A.L. Dullien, AIChE J.,
29, 1 (1974). Liquid Diffusion Of Nonelectrolytes.

T.A. Anderson, Ph.D. Thesis, Michigan State University,
1968. Pure Thermal Diffusion.

J.D. Olson, Ph.D. Thesis, Michigan State University,
1972. Interferometric Examination Of Temperature
Distributions In Liquids: Refractive Index And Thermal
Conductivity.

NBS Special Publication 300, V. 2 (August 1968).
Precision Measurement And Calibration Temperature.

T.J. Quinn, and J.P. Compton, Rep. Prog. Phys., 22,
151 (1975). The Foundations Of Thermometry.

NBS Monograph 126 (April 1972). Platinum Resistance
Thermometry.

NBS Monograph 125 (March 1974). Thermocouple Reference
Tables Based On The IPTS-68.

H. Svensson, Optica Acta, 1, 25 (1954). The Second
Order Aberrations In The Interferometric Measurement
Of Concentration Gradients.

R. Forsberg, and H. Svensson, O tica Acta, 2, 90 (1954).
The Second Order Aberrations In The Interferometric
Measurement Of Concentration Gradients. II: Experimental
Verification Of Theory.

H. Svensson, Optica Acta, 5, 164 (1956). The Third
Order Aberrations In The Interferometric Measurement
Of Concentration Gradients.

K.W. Beach, R.H. Muller, and C.W. Tobias, J, Opt. Soc. Am.,
22. 559 (1973). Light Deflection Effects In The
Interferometry Of One Dimensional Refractive Index

Fields.

D.T. Moore, 5, Opt. Soc. Am., 55, 451 (1975).
Ray Tracing In Gradient Index Media.

63.

64.

65.

66.

67.

68.

69.

70.

71.

72.

73.

74.

222

E.A. McLean, L. Sica, and A.J. Glass, Appl. Phys. Letters,
17, 369 (1968). Interferometric Observation 0f
Absorption Induced Index Change Associated With

Thermal Blooming.

M. Giglio and A. Vendramini, Appl. Phys. Letters, 25,
555 (1974). Thermal Lens Effect In A Binary Liquid

Mixture: A New Effect.

J.P. Gordon, R.G.C. Leite, R.S. Moore, S.P.S. Porto,
and J.R. Whinnery, J. Appl. Phys., 35, 3 (1965).

Long Transient Effects In Lasers With Inserted Liquid
Samples.

J.R. Whinnery, Acc. Chem. Res., 1, 225 (1974).
Laser Measurement Of Optical Absorption In Liquids.

L.B. Bissonnette, Appl.0pt., 12, 719 (1973). Thermally
Induced Nonlinear Propagation Of A Laser Beam In
An Absorbing Fluid Medium.

G. Thomaes, J. Chem. Phys., 25, 32 (1956). Thermal
Diffusion Near The Critical Solution Point.

M. Giglio and A. Vendramini, Phys. Rev. Letters, 55,
561 (1975). Thermal Diffusion Measurements Near A
Consolute Critical Point.

W. Pauli, Optics A29 The Theory Of Electrons, ed. by
C.F. Enz (MIT Press, 1973).

F.H. Hildebrand, Methods Of A lied Mathematics, 2nd
Edition (Prentice-Hall, 1965).

S.R. De Groot, and P. Mazur, Non-Equilibrigm Thermod amics,
(North-Holland, 1969).

R. Haase, Thermod amics 0f Irreversible Processes,
(Addison—Wesley,1929).

L.D. Landau, and E.M. Lifshitz, Fluid Mechanics,
(Pergamon Press, 1959 .

 

 

 

10.

11.

12.

13.

Chapter 4 References

J. Fourier, Theorie Analytigue de la Chaleur, 1822;
translated by A. Freeman, 1878; Dover, 1955.

0.x. Bates, Ind. Eng. Chem., 25, 431 (1933).
Thermal Conductivity of Liquids.

L.G. Longsworth, J. Phys. Chem., 21, 1557 (1957).
The Temperature Dependence of the Soret Coefficient
of Aqueous Potassium Chloride.

H. Poltz, Int. J. Heat Mass Transfer, 2, 515 (1965).
Die Warmeleitfahigkeit von FlusSigkeiten II.

H. Poltz, Int. J. Heat Mass Transfer, 2, 609 (1965).
Die Warmeleitfahigkeit von Flussigkeiten III.

H. Poltz and R. Jugel, Int. J, Heat Mass Transfer,
19, 1075 (1967). Thermal Conductivity of Liquids IV.

H. Poltz, NBS Special Publication 302 (September 1968).
Some Aspects Concerning Thermal Conductivity Data of
Liquids and Proposals for New Standard Reference
Materials.

J.L. Novotny and J. C. Bratis, J. Quant. Spectrosc.

Radiat, Transfer, 12, 901 (1972). Radiative—Trahsfér

in Liquid C014.
J.L. Novotny, D.E. Negrelli, and T. Van den Driessche,
J. Heat Transfer, 25, 27 (1974). Total Band Absorption
Models for Absorbing-Emitting Loquids: CClq.

 

G. Schodel and U. Grigull Heat Transfer, 5, R2.2
(Elsevier, Amsterdam, 1970). Kombinierte Warmeleitung
und Warmestrahlung in Flussigkeiten.

J.D. Olson and F.H. Horne, J. Chem. Phys., 8,2321
(1973). Direct Determination of Temperature Dependence
of Refractive Index of Liquids.

J.D. Olson, Ph.D. Thesis, Michigan State University,
1972. Interferometric Examination of Temperature
Distributions in Liquids: Refractive Index and Thermal
Conductivity.

T.V. Gurenkova, P.A. Norden, and A.G. Usmanov, J. Eng.
Phys.(USSR), 25, 940 (1973). On the Question of the
Influence of Radiation on the Thermal Conductivity

of Plane Layers of Liquids.

223

 

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

224

R.M. Waxler, C.E. Weir, H.W. Schamp, Jr., J. Research
555, 555, 489 (1964). Effect of Pressure and
Temperature Upon the Optical Dispersion of Benzene,
Carbon Tetrachloride, and Water.

E. Reisler, H. Eisenberg, A.P. Minton, J.C,S.
Faraday II, 22, 1001 (1972). Temperature and Density
Dependence of the Refractive Index of Pure Liquids.

B.L. Johnson and J. Smith, Refractive Indices and
Densities of Some Common Polymer Solvents, p. 27,

in Li ht Scatterin From Pol er §blutions, ed. by
M.B. Huglin (Academic Press, 1972).

E. McLaughlin, Chem. Rev., 25, 389 (1964). The Thermal
Conductivity of Liquids and Dense Gases.

N.V. Tsederberg, Thermal Conductivity of Gases and
Liguids, (M.I.T. Press, 1965).

R.W. Powell, C.Y. Ho, and P.E. Liley Thermal

Conductivit of Selected Materials, (NSRD§:NBS 8,
November 1966).

R.W. Wood, Physical Optics, (MacMillan, 1934).

E. Reisler and H. Eisenberg, J. Chem. Phys., 33,
3875 (1965). Refractive Indices and Piezo-Optic
Coefficients of Deuteruim Oxide, Methanol, and Other
Pure Liquids.

E. Hecht and A. Zajac, O tics, (Addison-Wesley, 1975).

Handbook of Chemist and Ph sics, 49th Edition
(Chemical Rubber Co., 1968).

U.W. Hochstrasser, in Handbook of Mathematical Functions,
ed. by M; Abramowitz and I.A. Stegun Dover, 19 8 .
Orthogonal Polynomials.

L.W. Tilton and J.K. Taylor, J. Research NBS, 22,
419 (1938). Refractive Index and Dispersion of
Distilleg Water For Visible Radiation at Temperatures
O to 60 C.

J.B. Hawkes and R.W. Astheimer, 2,_Opt. Soc. Am.,
8, 804 (1948). The Temperature Coefficient of the
Refractive Index of Water.

R.M. Waxler and C.E. Weir, J. Research NBS, 6 A,

163 (1963). Effect of Pressure and Temperature on

the Refractive Indices of Benzene, Carbon Tetrachloride,
and Water.

28.

29.

30.

31.

32.

33.

34.

36.
37.

38.

39.

40.

41.

1*2.

225

S.D.H. Andreasson, S.E. Gustafsson, and N. Halling,

J. Opt. Soc. Am., 61, 595 (1971). Measurement of
the Refractive Index of Transparent Solids and Fluids.

H.M. Dobbins and E.R. Peck, J. Opt. Soc. Am., 25,
318 (1973). Change of Refractive Index of Water as
a Function of Temperature.

0. Bryngdahl, Ark. Eys., 21, 289 (1961). Genaue
Bestimmungen der Warmeleiteigenschaften in Flussigkeiten
mit einer Scherinterferometrischen Methode.

P.R. Bevington, Data Reduction and Error Anal sis
For the Physical Sciences, (McGraw-Hill, 1969;.

C.W. Snedecor and W.G. Cochran, Statistical Methods,
(Iowa State University Press, 19 7).

J.B. Hawkes and R.W. Astheimer, J. Opt. Soc. Am.,
25, 105 (1974). Optical Properties of Water in the
Neighborhood of 35 C.

G. Andaloro, M.B.P. Vittorelli, and M.U. Palma,
J. Soln, Chem., 4, 215 (1975). Anomalies in the
Temperature Dependence of the 1.2u Absorption of
Liquid Water.

C.W. Robertson, B. Curnutte, and D. Williams,
Mol. Phys., 26, 183 (1973). The Infra-Red Spectrum
0 8 er.

S. Chandrasekhar, Radiative Transfer, (Dover, 1960).

R.M. Goody, Atmos heric Radiation 1. Theoretical
Basis, (Oxford, 1964).

R.D. Cess, Ad. Heat Transfer, 1, 1 (1964). The
Interaction of Thermal Radiation with Conduction
and Convection Heat Transfer.

D.H. Sampson, Radiative Contributions to Ener
and Momentum Transport in a Gas, (Interscience, 1965).
C.L. Tien, Ad. Heat Transfer, 5, 254 (1968).

Thermal Radiation Properties of Gases.

R.D. Cess and S.N. Tiwari, Ad. Heat Transfer, 2,
229 (1972). Infrared Radiative Energy Transfer in
Gases.

R. Viskanta and E.E. Anderson, Ad. Heat Transfer,
11, 318 (1975). Heat Transfer in Semitransparent Solids.

226

43. G.G. Pomraning, Prog. High Temp. Phys. and Chem., 5,
1 (1971). High Temperature Radiative Transfer and
Hydrodynamics.

44. G.C. Pomraning, The E nations of Radiation H dro—
dynamics, (Pergamon Press, 1973).

10.

11.

12.

13.

14.

Chapter 5 References

H. Bénard, Ann, Chim, Phys., 25, 62 (1901). Les
Tourbillons Cellulaires dans une Nappe Liquide
Transportant de la Chaleur par Convection en Regime
Permanent.

J.R.A. Pearson, J. Fluid Mech., 5, 489 (1958).
On Convection Cells Induced by Surface Tension.

D.A. Nield, J. Fluid Mech., 12, 341 (1964). Surface
Tension and Bouyancy Effects in Cellular Convection.

D.A. Nield, J. Fluid Mech., 1, 441 (1975). The
Onset of Transient Convection Instability.

S. Chandrasekhar, Hydrodynamic and Hydromagnetic
Stability, (Oxford, 1961).

W.V.R. Malkus, Proc. Roy. Soc., A 225, 185 (1954).
Discrete Transitions in Turbulent Convection.

J.W. Deardorff and G.E. Willis, J. Fluid Mech., 25,
675 (1967). Investigation of Turbulent Thermal
Convection between Horizontal Plates.

J. Gille, J. Fluid Mech,, O, 371 (1967). Interferometric
Measurement of Temperature Gradient Reversal in a
Layer of Convecting Air.

H.T. Rossby, J. Fluid Mech., 6, 309 (1969). A Study
of Benard Convection withrand without Rotation.

R. Krishnamurti, J, Fluid Mech., 42, 295 (1970).
On the Transition to Turbulent Convection. I. The
Transition from Two-to Three-Dimensional Flow.

R. Krishnamurti, J. Fluid Mech., 52, 309 (1970).
On the Transition to Turbulent Convection. II. The
Transition to Time-Dependent Flow.

R. Farhadieh and R.S. Tankin, J. Fluid Mech., 55,
739 (1974). Interferometric Study of Two-Dimensional
Benard Convection Cells.

L.A. Segel, in Nonequilibrium Thermodypamics, Variational
Techniqueg, and Stability, ed. by Donnelly, Herman,
and Prigogine, (UniverSIty of Chicago Press, 1966).

R.L. Koschmieder, Advan, Chem. Phys., 22, 177 (1973).
Benard Convection.

227

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

228

J.S.)Turner, Buoyancy Effects in Fluids, (Cambridge,
1973 .

P.E. Roberts, Advan, Chem, Phys., 52, 17 (1975).
Concepts in Hydrodynamic Stability Theory.

 

J.A. Whitehead, Jr., in Fluctuations, Instabilitiep1
and Phase Transitions, ed. by T. Riste, (Plenum, 1975).
A Survey of Hydrodynamic Instabilities.

M.G. Velarde, in H drod amics, ed. by R. Balian,
(Gordon and Breach, 197E).

R. Graham, 2pringer Tracts Mod. Phys., 22,1 (1973).
Statistical Theory of Instabilities in Stationary
Nonequilibrium Systems with Applications to Lasers
and Nonlinear Optics.

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