THE EFFECT OF GIBBS ADSORPTION
ON MARANGONI INSTABILITY
by
John Richard Ross
S.B., Massachusetts Institute of Technology
(1967)
Submitted in Partial Fulfillment
of the Requirements
for the
Degree of Doctor of Philosophy
at the
Massachusetts Institute of Technology
March, 1974
Signature of Author
De.partmentof Chemical
Enaineerinac
Certified by
--Pro/essor
Accepted by
(4
ja(,
J. E. Vivian
..
Chairman, Departmental
Committee on Graduate Theses
AB STRACT
THE EFFECT OF GIBBS ADSORPTION
ON MARANGONI INSTABILITY
by
John Richard Ross
Submitted to the Department of Chemical Engineering
on March 22, 1974 in partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
An investigation has been performed to determine the
conditions at the onset of surface tension-driven instability,
in gas-liquid systems, as characterized by the critical value
of a dimensionless Marangoni number. A theoretical analysis,
for the case in which a surface tension-lowering solute
transfers from a liquid according to penetration theory, shows
that adsorption of the solute in the Gibbs layer, at the gasliquid interface, has a strong ability to retard convective
instability. Theories which ignore Gibbs adsorption predict
the onset of convection at Marangoni numbers as much as ten
thousand times higher than the values found experimentally.
With Gibbs adsorption included in the new theory, the disdrepancy is very substantially reduced, to a factor of ten or
less.
Frequently, the residual disagreement has been blamed on
the presence of minute amounts of impurities, adsorbed in the
Gibbs layer, in the experimental liquids. The revised theory
confirms that this influence can be strong under some circumstances. However, new experimental determinations of the
critical Marangoni number, during triethylamine desorption from
water, show that in typical systems the presence of trace
contaminants is inconsequential.
It is suggested that further efforts, to resolve the
data-theory discrepancy, should focus on the relation between
predictions of the critical Marangoni number and assumptions
made in the stability
theory concerning
the size of the
convective disturbance cells.
Thesis Supervisors:
Professors P. L. T. Brian and J. E. Vivian
Professors of Chemical Engineering
Department of Chemical Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts
02139
March 22, 1974
Professor David B. Ralston
Secretary of the Faculty
Massachusetts Institute of Technology
Cambridge, Massachusetts
02139
Dear Professor Ralston:
In accordance with the regulations of the Faculty,
I herewith submit a thesis, entitled "The Effect of
Gibbs Adsorption on Marangoni Instability," in partial
fulfillment
of the requirements
for the degree of
Doctor of Philosophy in Chemical Engineering at the
Massachusetts Institute of Technology.
Respectfully submitted,
John R. Ross
2
ACKNOWLEDGEMENTS
P. L. T. Brian first suggested taking on this thesis.
His guidance and encouragement became truly invaluable, especially during the
of this work.
development of the theoretical aspects
J. E. Vivian
deserves
equal recognition for
supervising my efforts; though he was an advisor from the
beginning, his task doubled in magnitude when Brian left
MIT, for private industry, in the summer of 1972.
The National Science Foundation supported this thesis
financially.
The MIT Information Processing Center provided
computational facilities, as did the Chemical Engineering
Computing Laboratory.
I am indebted to many people who aided me in this work
at various times.
Stan Mitchell gave valuable advice on
experimental techniques and also drew the figures shown in
the text.
Paul Bletzer performed expert machine shop work,
while Larry Ryan of the RLE glass shop helped to keep my
short, wetted-wall column usable.
Ann Rosenthal added her
excellent typing skills to the final effort.
A doctoral thesis is quite a strain on the psyche, and
I am grateful for kind words from Jack Donahue who was unable
to finish his own thesis.
Innumerable other friends, close
relatives, and especially my parents require additional
recognition.
With little doubt, however, my wife Myrna deserves the
Without her support and love,
most profound acknowledgement.
this thesis might still be only a dream.
3
TABLE OF CONTENTS
PAGE
1.
2.
SUMMARY
11
1.1
Descriptive Mechanism
12
1.2
Basic Theoretical Results
14
1.3
Comparison of Theory and Experiment
15
1.4
Gibbs Adsorption
16
1.5
Gibbs Adsorption of Impurities
16
1.6
Gibbs Adsorption of STL Solute
18
1.7
New Theoretical Treatment
19
1.8
Comparison of New Theory with Experiment
21
1.9
Gibbs Adsorption of Surface Active Impurities
24
1.10 Experimental Program
25
1.11 Explanation of Residual Discrepancy
27
INTRODUCTION
29
2.1
Descriptive Mechanism
31
2.2
Basic Theoretical Treatment
24
2.3
Basic Theoretical Results
41
2.4
Comparison of Theory and Experiment
43
2.5
Gibbs Adsorption
45
2.6
Gibbs Adsorption of Impurities
46
2.7
Gibbs Adsorption of STL Solute
50
2.8
Additional Experimental Work on Marangoni
Instability
52
Additional Theoretical Work on Marangoni
Instability
57
2.9
4
PAGE
3.
2.10
Thesis Objectives
63
2.11
Detection and Removal of Surfactants
64
GIBBS ADSORPTION OF A TRANSFERRING SOLUTE
67
3.1
Theoretical Model
67
3.2
Solution for Finite Liquid Depth
71
3.3
Solution for Infinite Liquid Depth
78
3.4
Description
85
3.5
Application to Penetration Mass Transfer
3.6
Effect of Surface Convection in the Gibbs
3.7
4.
5.
of Results
87
Layer
95
Comparison with Experiments
96
GIBBS ADSORPTION OF SURFACE ACTIVE IMPURITIES
103
4.1
Theoretical Model
104
4.2
Basic
107
4.3
Comparison with Experiments
109
4.4
Implications
114
Results
for Experiments
APPARATUS AND PROCEDURE
118
5.1
The ShortWetted-Wall Column
118
5.2
Flow Systems
120
5.3
Operating Procedure
122
5.4
Sampling Procedure
124
5.5
Chemical Analysis for Mass Transfer
127
5.6
Surfactant Analysis and Removal
127
5
PAGE
6.
7.
8.
EXPERIMENTAL RESULTS
129
6.1
Propylene Desorption
129
6.2
Enhanced Desorption
133
6.3
Methylene Blue Analysis
136
6.4
Experiments with Purified Water
137
6.5
Interpretation of Results
140
FUTURE WORK
142
7.1
Theoretical Estimates of Surface Elasticity
142
7.2
Explanation of Residual Discrepancy
144
7.3
Conclusions
147
CONCLUSIONS AND RECOMMENDATIONS
153
8.1
Conclusions
153
8.2
Recommendations
154
APPENDIX
9.
DETAILS OF THEORETICAL DERIVATIONS
155
9.1
Linearized Equations for a Perturbed System
155
9.2
Effect of a Surface Active Impurity
172
9.3
Derivation of Gibbs Profile
176
9.4
Calculation of Penetration Depth
179
9.5
Effect of Surface Diffusion
184
10. DETAILS OF COMPUTER CALCULATIONS
186
11. DETAILS OF APPARATUS AND PROCEDURE
214
11.1
The Glass Column and Mounting
6
214
PAGE
12.
11.2
Liquid Flow System
217
11.3
Gas Flow System
219
11.4
Procedure - A Typical Run
220
11.5
Analysis for Triethylamine
223
11.6
Analysis for Propylene
224
11.7
Analysis
228
11.8
Removal of Surfactants
11.9
Sources and Grades of Materials and Services235
for Surfactants
233
SAMPLE CALCULATIONS
237
12.1
Rate of Propylene Desorption
237
12.2
Marangoni Number
241
12.3
Surfactant Analysis
247
13.
TABULATION OF DATA
249
14.
SOURCES OF EXPERIMENTAL ERROR
255
15.
NOMENCLATURE
257
16.
LITERATURE
262
17.
AUTOBIOGRAPHICAL NOTE
CITATIONS
268
7
LIST OF FIGURES
FIGURE
NUMBER
PAGE
NUMBER
TITLE
2.1
Mechanism for Marangoni Convection
32
2.2
Unperturbed Concentration Profile
35
2.3
Curve of Marginal Stability
40
2.4
Predicted and Measured Values of Critical
Marangoni Number
42
3.1
Critical Marangoni Number for Insulating
Case
74
3.2
Effect of A on Mc for Insulating Case
75
3.3
Effect of Liquid Depth on M
Case
for Insulating
76
Effect of Liquid Depth on M
for Conducting
3.4
Case
3.5
3.6
77
Critical Marangoni Number for Infinite
Liquid Depth
Experimental Paths for Triethylamine
Desorption
3.7
84
90
Onset of Instability for Infinite Liquid
Depth
92
3.8
Onset
94
3.9
Revised Onset of Instability at Infinite
Depth for TEA
98
3.10
Onset of Instability for Ether
99
3.11
Onset of Instability for Acetone
100
4.1
Effect of Elasticity Number
108
4.2
Effect of Elasticity Number on TEA Instabilitylll
4.3
Effect of Elasticity Number on Ether
Instability
of Instability
for Finite
8
Depth
115
FIGURE
NUMBER
TITLE
PAGE
NUMBER
Effect of Elasticity Number on Acetone
Instability
116
5.1
Short Wetted-Wall Column Assembly
119
5.2
Overall Schematic Diagram of Apparatus
121
5.3
Liquid Feed Sampling System
125
5.4
Outlet Gas Sampling System
125
6.1
Rate of Propylene
Desorption
131
6.2
Critical Marangoni Number for TEA Experiments
134
6.3
Critical Marangoni Number with Filtered Water
139
7.1
Effect of Wave Number Constraint
148
7.2
Effect of Constrained Wave Number - TEA
4.4
149
Desorption
7.3
Effect of Constrained Wave Number - Ether
150
De sorption
Effect on Constrained Wave Number - Acetone
Desorption
151
11.1
Calibration for Surfactant Analysis
232
12.1
Surface Tension of TEA Solutions
245
7.4
9
LIST
TABLE
NUMBER
2.1
OF TABLES
PAGE
TITLE
NUMBER
Dimensionless Equations for Pearson (1958)
Analysi s
37
3.1
Triethylamine Desorption from Water at 298°K
88
3.2
Desorption from Water at 298°K
97
3.3
Comparison of Mayr's Experimental Results with
New Theroetical Predictions
101
11.1
Operating Conditions for Chromatograph
226
13.1
Desorption Data
250
13.2
Surfactant Analysis
254
10
1.
SUMMARY
Over the last twenty years much attention has been
given to the study of mass transfer in the presence of
interfacial convection driven by surface tension gradients.
This phenomenon, named "Marangoni instability" after an
early Italian physical chemist, is potentially of great
importance to chemical engineers.
In gas-liquid systems,
rates of mass transfer may be considerably increased by the
existence of this interfacial turbulence.
However, direct attempts to take advantage of this
enhancement of mass transfer have been rare.
In part, this
is because there is no dependable theory or empirical rule
with which to predict the precise conditions which cause
Marangoni instability.
Theoretical work on the problem has consisted of
hydrodynamic stability analyses to predict the point of
initiation of convective instability, as measured by the
critical value of a Marangoni number.
by Brian et al.
(1971) of the critical
However, measurements
Marangoni
number,
in
desorbing surface tension-lowering solutes from water, were
found to lie two to four orders of magnitude above theoretical
predictions.
The primary objective of this author's work has been to
explain and to reduce the disparity between measurements and
11
predictions of the conditions at the onset of Marangoni
convection.
Following proposals of previous investigators,
attention has been focused on the influence of Gibbs adsorption at a gas-liquid interface.
Both by theoretical
and experimental means, significant progress has been made.
It will be demonstrated that, by considering Gibbs adsorption of a transferring solute, the discrepancy between
theory and experiment can be reduced very substantially.
On the other hand, Gibbs adsorption of impurities, often
blamed for the discrepancy, will be demonstrated to be
unimportant.
A new suggestion is offered to help explain
the yet remaining disparity.
1.1
Descriptive Mechanism
Pearson (1958) performed the first theoretical analysis
of surface tension-driven convective flows.
He considered
a pure liquid subject to temperature-induced surface tension
variations.
An exact analogy exists for concentration-
induced variations in liquid solutions containing a volatile,
surface tension-lowering solute.
Consider, in Figure 2.1,
a solution resting below a gas phase.
As the solute desorbs,
a diffusion-controlled concentration gradient is created in
the liquid.
Then, if a random disturbance creates, for
instance, a slight raising of solute concentration at a point
in the liquid surface, the consequent lower tension of the
interface at that point will cause spreading or expansion of
12
the surface.
Since continuity requires that fluid be drawn
up from the bulk to replace what "spreads" away, even more
concentrated fluid will reach the surface, reinforcing the
initial disturbance.
In this way convective roll cells are
initiated.
Pearson applied the usual equations of motion to a
simplified form of the model just described.
He arrived at
a set of relations, like those given in dimensionless form
in Table 2.1, in which the perturbed velocity and concentration differences are the unknowns.
The equations have been
linearized by omitting second-order and higher terms in the
perturbation quantities.
Equations 2.1 and 2.2 are momentum
and solute material balances, respectively, on the bulk
liquid phase.
In the second equation aD/Dz represents the
initial concentration gradient.
in the list of nomenclature.
The other symbols are defined
The first two of the boundary
conditions are by far the most important.
Equation 2.3
expresses a force balance at the liquid surface and introduces the Marangoni number,
M
-
chAc
The Marangoni number represents the strength of surface
tension forces relative to viscous forces.
Equation 2.4 is
an interfacial material balance; it includes the dimensionless
group,
Hk
R
=
G
G
13
h
which is the ratio of the liquid phase resistance to mass
transfer compared to the gas phase resistance.
In the mathematical procedure, as suggested by experiments,
w and c are assumed to take on a spatially repeating pattern
in the x - y plane, with wavelength d and dimensionless wave
number
=
2rh/d.
Then, for convective instability to be possible, a solution
must exist for equations 2.1 to 2.8.
In fact for specified
values of R, l/h, and D¢/az, solutions exist only when M
assumes certain values, as an explicit function of a.
By
examining a range of values of a, one can determine the
minimum or critical value of M, and the corresponding wave
number.
The lower the value of M, the weaker the influence
of surface forces.
Hence, for a given real system, the
critical Marangoni number, M c , represents the condition
below which surface forces cannot maintain convection.
Conversely, if M in a real system exceeds Mc , then instability
is expected to occur.
1.2
Basic Theoretical Results
Pearson computed values of the critical Marangoni
number for various values of R, assuming the unperturbed
gradient
is linear,
liquid X = t/h = 1.
D@/az = 1, and extends
throughout
the
His results are given by the curve
14
labeled
X = 1 in Figure
2.4.
Conditions
predicted stable, those above unstable.
below
the curve are
Pearson's assumption
of a full-depth, linear concentration gradient is quite
restrictive.
Vidal and Acrivos (1968) examined linear and
non-linear profiles extending part of the way into a liquid
phase
for which
X > 1. A sample curve at X = 10.0, shown in
Figure 2.4 along with a limiting curve for X =
, shows that
the deeper the liquid pool, relative to the concentration
gradient, the less stable the system.
1.3
Comparison of Theory and Experiment
In mass transfer systems, measurements of the critical
Marangoni number have diverged profoundly from the predictions of Pearson or Vidal and Acrivos.
Brian et al. (1971)
report on the work of Mayr (1970), who measured critical
Marangoni numbers in aqueous solutions of acetone, diethyl
ether, and triethylamine (TEA). For all of Mayr's systems,
X was approximately 10.
His results, plotted in Figure 2.4,
show that for the three systems Mc appears to be 60, 250,
and 2200 times higher respectively than theory predicts.
In a parallel set of experiments, Clark and King (1970) found
similar disagreement with theory.
Figure 2.4 vividly illustrates the poor understanding
of interfacial convection which was available when this
study was undertaken.
It was obvious that certain influences,
not yet accounted for, had to be elucidated in order to
understand the onset of Marangoni convection.
15
Two effects
stood apart, as being potentially of major importance.
Each
was based on the convection-inhibiting effects of Gibbs
adsorption.
1.4
Gibbs Adsorption
J. Willard Gibbs is credited with first explaining that
any surface active solute (that is, a surface tension-lowering
one) adsorbs
solution.
preferentially
at the free surface of a
Mathematically, Gibbs modeled this situation as
if the interface were infinitesimally thin, but contained
a certain excess solute concentration, r, adsorbed on the
interface.
For dilute solutions, Gibbs found that, r = 6ci,
where 6 is a proportionality constant with dimensions of
length, and 6 = o/RT.
The ideas of Gibbs adsorption are now accepted as one
of the foundations of modern surface physics.
Additional
details are available from, for instance, Adamson (1967).
1.5
Gibbs Adsorption of Impurities
Berg and Acrivos (1965) extended Pearson's work to
allow for the presence of small amounts of a very highly
Their assumptions
surface active, non-volatile impurity.
allow the interfacial stress balance to be modified to
a2w
V2w
2c
M
a2c
+
-
NE
where the "Elasticity number" is defined as
16
(2.3a)
a C.
N
with a, Ci , and
EL
h
=
It is
all evaluated for the impurity.
required that the ratio h/ 6 IMP be approximately zero, where
6IMP is the "Gibbs depth" for the impurity.
Certain materials commonly used in household detergents,
substances called "surfactants" (for surface active agents),
are non-volatile and possess very large values of a in
aqueous media.
These materials are often found in trace
quantities in domestic water and as little as a few parts
5
per million may imply a value of NEL exceeding 105
Significantly, Berg and Acrivos found that such a value
would theoretically raise the critical Marangoni number by
three orders of magnitude.
Various investigators have suggested a connection between certain of their own puzzling experimental results
and the apparently powerful effects of insoluble surfactants.
Specifically,
Brian et al.
(1971, p. 83), describing
Mayr's
work, speculate that the data-theory discrepancy "may be due
to surface impurities."
Clark and King (1970) make a
similar suggestion, as do several authors who studied
thermally driven instability.
However, experimental evidence
to confirm these suggestions or verify the quantitative
theory of Berg and Acrivos is minimal.
Furthermore, the
Berg and Acrivos analysis is not applicable to certain experimental situations, because it employs the simplifying
assumption that the convection-inducing temperature or
17
concentration gradient is linear and persists through the
liquid layer.
A major objective of this author's work was to evaluate
the role of surface active impurities in concentrationdriven Marangoni convection.
To this end, the Berg-Acrivos
theory required the addition of realistic assumptions
concerning the unperturbed gradient.
Also, it was decided
to perform experiments on "contaminant free" systems.
Specifically, certain of Mayr's experiments were repeated,
after any surface active impurities had been removed from the
desorption solution.
That these could be fully removed was
assured after a review of the published techniques for
adsorption of surfactants on activated charcoal.
1.6
Gibbs Adsorption of STL Solute
Brian (1971) suggested, as an explanation for the
discrepancy between theories and experiments on Marangoni
instability, that, even in the absence of impurities, there
would be adsorption of the transferring surface tensionlowering (STL) solute in the Gibbs layer.
Mathematically,
Brian's innovation was to alter the surface material
balance to become
ac
az
aw
d D
a 2c
atz
h
a
= Rc - A-
2 c\
+-
ay
+-
h at
where A, the Adsorption number, is defined by
r
A
=
h Ac
18
ac
(2.4a)
Using the simplified Pearson model, with allowance for Gibbs
adsorption, Brian found that increasing A tends dramatically
to raise the Marangoni number required for the onset of
convection.
Indeed, if A exceeds a certain value, Marangoni
instability is completely precluded.
Brian's effort established the importance of Gibbs
adsorption, but his work, like Pearson's and that of Berg
and Acrivos (1965), was valid only for an un-perturbed
concentration gradient which was linear and extended fully
through the liquid solution.
The Gibbs adsorption ideas
were not collected into a single theoretical model, modified
for concentration-driven convection in liquids of arbitrary
depth.
However, precisely that treatment is required to
understand properly the data of Mayr (1970), whose measurements of the critical Marangoni numbers in mass transfer are
the most extensive to date.
Therefore, a main objective of
this thesis became reinterpreting Mayr's data in the light
of a newly integrated, theoretical analysis.
1.7
New Theoretical Treatment
The equations 2.1 and 2.2 were solved, by separation of
variables, subject to the boundary conditions 2.3, 2.4a
(which includes Gibbs adsorption), and 2.5 to 2.8.
The
critical Marangoni number was studied as a function of A,
the Adsorption number, M/C,
the solute concentration profile,
X, the dimensionless liquid layer depth, and R, the mass
transfer resistance ratio.
19
An initial, useful result was the discovery that values
of Mc computed for the liquid depth relevant to Mayr's
experiments, X = 10, are rather close to the values of Mc
found when X is infinite.
Since the latter assumption is
easier to treat mathematically, most of the present results
are for the case of infinite depth.
Figure 3.5 exhibits typical new findings, in a graph of
M
c
versus A.
The curves are based on the linear concentra-
tion profile of Figure 2.2b and also on a non-linear profile,
typical of penetration mass transfer.
For the latter analysis,
the definition of the penetration depth, h, is illustrated
in Figures 2.2c and 2.2d; h is the thickness of a linear
profile which has the same solute deficiency as the assumed
penetration profile.
It is seen in Figure
3.5 that Gibbs adsorption
has a
very strong stabilizing effect; the larger the value of the
adsorption number, the higher the value of M .
In fact, if
A in a deep, real system exceeds a critical value of 0.50,
complete stability is predicted.
This last result is a
clear change from Brian's finding that A
shallow liquid pool, X = 1.
c
0.083 for a very
Further analysis shows that the
critical value of A = 0.5 is attained if the amount of solute
in the Gibbs layer just equals the amount of solute removed
from the rest of the liquid phase.
A related, interesting conclusion is that the product
of A and M is a useful guide to the importance of Gibbs
adsorption.
Whereas the factors A and M individually depend
20
on the penetration depth and the mass transfer driving
force, the product does not;
AM = a2Ci/PDRT and depends only
on the intrinsic properties of the liquid phase.
Represen-
tative lines of constant AM are given in Figure 3.5.
It is
apparent that for a given value of R, a line of constant AM
roughly divides an Mc vs. A curve into both a vertical leg
for which Gibbs adsorption is controlling, and also a
horizontal leg for which Gibbs adsorption is of little
consequence.
1.8
Comparison of New Theory with Experiment
In order to judge properly the full importance of
Gibbs adsorption of the STL solute, it is necessary to
determine values of M, A, R, and
systems.
for typical desorption
This in turn requires knowledge of h and C.
While
these quantities are difficult to measure, an innovation in
this author's work was to calculate h and C. using a
penetration theory model of the liquid phase.
Note that h
and AC = CB - C i both increase with contact time; thus the
physical values of M and R will increase over time and those
of A and X will decrease.
The onset of instability will
occur when M exceeds the value of Mc corresponding to the
instantaneous values of R, A, and
.
A complication arises if storage in the Gibbs layer is
included in the penetration theory model, since the inventory,
r0,
present in the Gibbs layer at contact time
specified.
Two limiting cases may be visualized.
= 0 must be
For a "new"
surface, formed fresh at the start of the contact interval,
21
r° = 0.
liquid r
For an "aged" surface in equilibrium with the bulk
=
6 CB.
The calculation of an experimental path, over a finite
contact time, may be illustrated by the TEA desorption experiments of Mayr (1970).
Figure 3.6 shows the two limiting
cases calculated for one of Mayr's typical runs. Interes0
tingly, when r = 6CB, and R is small, both C i and r vary
slowly with
increasing
.
Therefore, as h(CB - C i) increases with
, M increases and A decreases but the product
AM remains nearly constant, and the upper path in Figure
3.6 approximates the straight line AM = 2200.
In contrast,
when r°0 is zero, the initial solute transfer is largely from
the liquid phase to the Gibbs layer, with little transfer
to the gas phase.
Thus r approximately equals the solute
deficiency, and A remains nearly 0.5 as M increases.
Later,
as solute transfer to the gas phase becomes appreciable, A
decreases.
At long times, the experimental paths for these
two cases approach each other.
When a path like those in Figure 3.6 intersects a
curve of M
versus A such that the values of M, A,
all match, the point of instability is reached.
, and R
For the
case of an infinite liquid depth, these intersections are
shown in Figure 3.9.
The two experimental paths are similar
to those shown in Figure 3.6.
The solid curve of Mc vs. A
was obtained using the broken-line concentration profile
and choosing a value of R which roughly corresponds to the
22
experimental paths at the points of intersection.
For either experimental path, surface movement of species
adsorbed in the Gibbs layer has a profound effect on the predicted point of onset of instability.
Without surface adsorp-
tion the critical Marangoni number would be equal to 4.
With
surface adsorption in the Gibbs layer, the critical Marangoni
This
number is approximately 5000 for the cases illustrated.
thousand-fold increase in Mc results from the strong surface
activity of triethylamine, as measured by its AM product.
The experimental paths of Figure 3.9 correspond to the
conditions at which Mayr (1970) determined that Marangoni
convection in TEA solutions was barely evident.
In Mayr's
wetted-wall column, the contact time was e = 0.15 seconds;
thus his experimental, critical Marangoni number was approximately
40,000.
This differs by a factor of 8 from the values of Mc
predicted by this author's new theory, in Figure 3.9.
However,
the experimental results are 10,000 times higher than the
value of Mc
4, predicted in the absence of Gibbs adsorption.
Graphs similar to Figure 3.9 have been constructed for
Mayr's studies of acetone and ether desorption.
These sys-
tems differed from theory by two to three orders of magnitude;
the new development again brings closer agreement.
acetone the discrepancy
in Mc becomes
For
a factor of 17, for
ether a factor of 50.
This author's theoretical treatment has thus confirmed
the suspected importance of Gibbs adsorption.
Comparisons
with experiments lead to much smaller and more uniform
23
differences than previously.
Discrepancies which were as
large as 10,000 for TEA, now are in the range of 10 to 50.
The resolution of the residual discrepancy requires
additional research; for instance, the importance of surface
active impurities will be considered next.
In any event, all
new treatments of concentration-induced instability must
surely include the ideas presented here, concerning the
major effects of Gibbs adsorption.
1.9
Gibbs Adsorption of Surface Active Impurities
In order to study the theoretical importance of impurities
in concentration-driven convection, and in order to guide
experimental tests of supposed contaminant effects, it is
necessary to blend the procedures and assumptions of the
previous sections with those of Berg and Acrivos (1965).
Such an analysis has been carried out, and it is confirmed
that, theoretically, large values of the Elasticity number
imply substantially raised values of the critical Marangoni
number.
To consider specific cases, Figure 4.2 shows curves of
M c vs. A, at different values of NEL, superimposed on an
experimental path for TEA from Figure 3.9.
It is to be noted
that if surfactant impurities were present in the triethylamine solution such that NEL = 2000 and h/IM
would be predicted
p
=
0, then Mc
to be 35,000 and, for all practical
purposes, the data-theory discrepancy would be explained.
24
This result is quite interesting since h/cIM
p
is indeed very
small for most surfactants, and Mayr's feed water was a possible source of contaminants.
As little as 30 parts per
billion (ppb) of surface active impurity in the water could
have brought NEL close to 2000.
However, the Elasticity num-
ber is very hard to estimate exactly, and, in any case, Mayr
did not attempt to measure the level of impurities in his
feed solutions.
than 2000.
Conceivably, NEL might have been much lower
In order to settle definitely the role of sur-
factants in interfacial convection, new experiments were
required.
1.10
Experimental Program
First, to prove the reliability of this author's pro-
cedures, certain results of Mayr (1970) were duplicated; the
critical Marangoni number was determined for instability
during desorption of triethylamine (TEA) in a short wettedwall column.
This required measurements of the rate of mass
transfer of an inert solute, propylene, both in the absence
of and in the presence of TEA.
Next, the level of surfactant
present in the column feed water was estimated, by a well
accepted chemical test.
Water was then filtered through
activated charcoal in order to prepare a feed stock from
which surface active contaminants had been removed.
Once
again, the critical Marangoni number was measured, and, as a
consequence, a conclusion was reached concerning the importance
of surface active impurities on interfacial instability.
25
By performing a series of desorption runs at several
different triethylamine concentrations, the results shown in
Figure 6.2 were determined.
The abscissa in the graph is
the maximum Marangoni number existing in the wetted-wall
column,
path.
for a given run, as computed
from an experimental
The ordinate is the enhancement,
, the ratio of the
measured propylene transfer rate to that found when no TEA
was present.
Comparing Mayr's results, using a distilled
water feed stock, with this author's findings for an identical
feed, the two are quite similar.
With the present desorption
procedures thus proved satisfactory, a surfactant analysis
was performed next.
Samples of distilled water were subjected to the
methylene blue dye test for surface active agents.
low value of 5 ppb of anionic surfactant was found.
A rather
Even
this small amount could possibly affect interfacial convection, according to the theoretical treatment presented
earlier.
Therefore, batches of distilled water were filtered
through activated charcoal, in order to remove these impurities.
It was confirmed that the surfactant content was reduced
thereby by a factor of ten or more.
If the Elasticity number actually were close to 2000,
in this author's initial desorption runs, then it can be seen
in Figure
4.2 that a factor-of-ten
decrease
in NEL, to 200,
should lead to a factor-of-three decrease in the critical
Marangoni number, to 12,000.
Such a change did not take
26
place. Figure 6.3 shows the final results; at most,removal
of surfactants caused a slight decrease in M
from 45,000
to 41,000.
To generalize, surface active impurities did not significantly affect the instability studies of this author or
Mayr.
Contrary to speculation, Gibbs adsorption of contam-
inants is not a major influence on typical systems exhibiting
Marangoni convection.
1.11
Explanation of Residual Discrepancy
Even though this author's theoretical work has helped to
reduce markedly the disparity between theoretical and measured
critical Marangoni numbers, a significant discrepancy remains.
On occasion, the lack of agreement between predictions and
experiments has been attributed to deficiencies in the theory
of Marangoni convection.
One particularly compelling ques-
tion about linearized theory, which has as yet received little
attention, is whether it properly predicts the critical wave
number,
c , and the corresponding wavelength, d
In a typical case from the present work,
be of the same order
adopted
in the theory
a c is 0.014 and d c
Several physical ar-
is 450 times the penetration depth.
guments suggest that d
= (2i7/a)h.
should not be so large, but should
as h.
Perhaps
this idea should be
as a constraint,
so that,
for instance,
one might exclude from consideration values of d c greater
than 3h or values of a less than 2.5.
This has the effect
of substantially raising the predicted value of M c .
27
In fact,
the agreement between predicted and experimental values of Mc
becomes surprisingly good.
The discrepancy for TEA and
acetone desorption is less than a factor of two, while that
for ether
is a factor of six.
The agreement reached between theory and experiment, as
a result of restricting the value of the wave number, is
quite interesting.
Nevertheless, it is hard to accept this
arbitrary constraint without rigorous justification.
Possibly,
some as yet undiscovered insight into the physical model of
Marangoni instability is required.
In any case, future
investigations of concentration-driven Marangoni instability
will be able to stand on the two most important findings
determined by this author.
First, it has been established
that Gibbs adsorption of the convection-inducing solute has
a major effect, which must be considered in theories of surface
tension-driven instability.
Second, it has been found that
adsorption of trace impurities is by contrast an exaggerated
concern, which may safely be ignored in typical experimental
situations.
28
2.
INTRODUCTION
Over the last twenty years much attention has been
given to the study of mass transfer in the presence of
interfacial convection driven by surface tension gradients.
This phenomenon, generally called "Marangoni instability,"
is potentially of great importance to chemical engineers.
In gas-liquid systems, rates of mass transfer may be
considerably increased by the existence of this interfacial
turbulence.
In liquid-liquid systems, convection may be-
come so intense that otherwise immiscible fluids may partially mix.
In painting and coating processes, the presence
of interfacial convection during drying may cause surface
irregularities in fully dried films.
The use of additives to minimize Marangoni instability
where it is harmful, as in paint films, has been largely
successful.
The existence of interfacial convection has
been established in industrially important gas absorption
systems, while its unexpected presence has seriously confused some laboratory studies of mass transfer.
Nevertheless,
direct attempts to exploit Marangoni instability, as in the
enhancement of rates of mass transfer, have been rare.
In
part, this is because there is no dependable theory or
empirical rule with which to predict the precise conditions
which cause Marangoni instability.
Experimental observations
have shown that convection may not set in until surface
29
forces are thousands of times stronger than required by
seemingly applicable theories.
The primary objective of this author's work has been
to explain and to reduce that disparity between measurements
and predictions of the conditions at the onset of Marangoni
convection.
In the main, attention has been focused on the
influences of Gibbs adsorption at a liquid surface.
Both
by theoretical and experimental means, significant progress
has been made.
On one hand, it will be demonstrated that,
by considering some aspects of Gibbs adsorption, the discrepancy between theory and experiment can be reduced very
substantially.
On the other hand, additional influences of
Gibbs adsorption, which have been suggested as reasons for
the residual discrepancy, will be demonstrated to be
unimportant.
A new speculation is offered to help explain
the yet remaining disparity.
In the rest of this chapter, the accepted mechanism of
Marangoni convection, the various theoretical treatments,
and experimental results are reviewed.
The motivation
for the exact nature of this author's program is also
described.
30
2.1
Descriptive Mechanism
Pearson (1958) performed the first theoretical analysis
of surface tension-driven convective flows.
He considered
a liquid resting on a heated surface, and therefore prone to
temperature-induced surface tension variations.
An exact
analogy to this scheme exists for liquid solutions containing
a volatile, surface tension-lowering solute.
Consider, in
Figure 2.1, just such a solution resting below a gas phase.
As the solute desorbs, a diffusion-controlled concentration
gradient is created in the liquid.
Then, if a random dis-
turbance creates, for instance, a slight raising of solute
concentration at a point in the liquid surface, the consequent lower tension of the interface at that point will
cause spreading or expansion of the surface.
Since con-
tinuity requires that fluid be drawn up from the bulk to
replace what "spreads" away, even more concentrated fluid
will reach the surface, reinforcing the initial disturbance.
In this way convective roll cells are initiated, the size
and intensity of which are limited primarily by viscous
damping in the liquid.
As implied in the figure, such
convection may ultimately form a pattern over the entire
liquid surface.
Before proceeding to consider this model in detail,
it is well to place the phenomenon of Marangoni convection
in the proper perspective.
The motive force, surface
tension, is a physical attribute well described in many
31
5
C
_J
0)
N
g3
-J
C
a:
0
U
>
d)
c
-J
0
u
c
0
L
-J
L
0
0 .'
I
E
c
L+
ci
_c
u
*1
fly
U
1
I
U
0 (
0
C
.
L
:3
I.r
I!
I
iI
32
textbooks, for instance Adamson (1967). Gross effects
caused by surface tension, including Marangoni convection,
have been reviewed by Berg (1972), Levich and Krylov (1969),
and Kenning (1968), the last in a succinct, very readable
form.
Possibly the broadest and most informative review
of Marangoni convection in gas-liquid systems is by Berg,
et al. (1966a). For liquid-liquid systems, which are not
treated in the present thesis, one may consult Berg (1972),
Mayr (1970), and Sternling and Scriven (1959). A very
interesting historic review by Scriven and Sternling (1960),
traces interfacial turbulence to the works of the Italian
Carlo Marangoni and the Englishman James Thomson, each in
the latter part of the nineteenth century.
Berg, et al. (1966a) also discuss convection driven by
thermally induced, density gradients, a phenomenon widely
recognized in atmospheric and oceanographic physics.
Indeed,
this so-called buoyancy-driven convection has been investigated far more extensively than surface tension-driven
movement.
Theoretical techniques for both types of convection
are often similar and many results in Marangoni instability
have followed from analogous buoyancy-driven cases.
The
classic work on the mathematics of buoyancy-driven instability
and on more general problems of stability is by
Chandrasekhar (1961).
33
2.2
Basic Theoretical Treatment
Most theoretical studies of surface tension-driven
convection have been attempts to predict the least severe set
of conditions under which convection can persist.
For this
purpose, a linearized stability analysis is usually performed;
Chandrasekhar (1961, pp. 1-7) gives a reasonably lucid
introduction to such analyses.
The initial effort of this
type for surface tension-based effects was by Pearson (1958).
To review his work, although casting it in terms of concentration-induced rather than thermally induced convection,
consider a broad expanse of a thin layer of a quiescent
liquid, containing a single volatile solute.
Assume that
increasing the surface concentration of the solute lowers
the surface tension of the solution.
(Most organic solutes,
for instance, will so influence water solutions.)
Assume
further that there is a linear, time-invariant gradient in
solute concentration as one proceeds from the solution
surface (z' = 0) to its interior.
The geometry of such a
system is shown in Figures 2.1 and 2.2b.
In addition, for
the Pearson model, assume that the concentration gradient
extends throughout the entire liquid layer.
I = h in Figure
(That is,
2.2b).
The analysis now proceeds; one formulates the usual
equations of continuity, motion, rate of mass transfer, and
appropriate boundary conditions.
It is then assumed that
the liquid is disturbed so that an internal fluid circulation
34
~U
U
)U
4)
0
c
L
0
C.
N
n)
I
C
0
0)
O
Iw
.©
O
0
L
v_
o
n
U~~~~~~
U
0
0
u
L
c)
U
m
CD
UC
C:
U
0O
(9
D
_L
u
cC
4)
.'_
0L
L
c)
4
N
L
N
0
O
N
C
4)
4)
CD
Ct-
0
D
r,
I--
n~~~d
C
U
U
U
35
L
appears and the concentration distribution changes.
propriate equations are again written.
Ap-
The initial equation
set is subtracted from the second, yielding a new set of
equations in which the perturbed velocity and concentration
differences are the unknowns.
Specifically, the relationships
are given in linearized, dimensionless form in Table 2.1.
(Further details of this and other derivations are in
Chapter 9.)
Note that "linearizing" the equations means
omitting second-order and higher terms in the perturbation
quantities.
Strictly speaking, this simplification is valid
only if the velocity and concentration changes are
infinitesimal.
Examining Table 2.1 in some detail, equations 2.1 and
2.2 are a momentum balance and solute material balance,
respectively, on the bulk liquid phase.
equation,
In the second
is the dimensionless solute concentration,
evaluated prior to the onset of convection, as are the
other physical parameters of the liquid solution.
The first
two of the boundary conditions are by far the most
important.
Equation 2.3 expresses a force balance at the
liquid surface and introduces the Marangoni number,
M
a hAc
-
The new symbols are defined in the list of nomenclature.
Possibly the most important one is a, which is the degree
to which surface tension is reduced by a unit amount of
36
TABSLE 2. 1
Dimensionless Equations for Pearson
/
V 2)
a-
2w
=
(1958) Analysis
0
(2. 1)
(ia
=-W
3a
At
z =
(2.2)
\9z/
0
22w
aS
//2c
M 2
=
z
2
++ Yc
(2.3)
ac
= R
(2.4)
c
az
w
=
0
(2.5)
w
=
0
(2.6)
-= 0
(2.7)
At z = t/h
c
=
0
(conducting case)
(2.8a)
-
=
0
(insulating case)
(2.8b)
or
37
solute.
The Marangoni number represents the strength of
surface tension forces relative to viscous forces.
Equation 2.4 is an interfacial material balance; it includes
another new dimensionless group
HkG h
R
=
which is the ratio of the liquid phase resistance to mass
transfer compared to the gas phase resistance.
The exact
hydrodynamics of the gas phase have been ignored.
As to
the remaining boundary conditions, equation 2.5 results from
assuming that the free surface remains perfectly flat.
Equation 2.6 expresses the no-slip condition at the bottom
wall.
Equation 2.7 follows from the no-slip condition and
the continuity equation.
Equations 28a
and 2.8b describe
a conducting wall and an insulating wall respectively, in
Pearson's heat transfer terminology.
Brian (1971) discusses
the physical meaning for mass transfer of each condition.
The conducting case implies that the bottom wall is permeable
and that a constant solute concentration is maintained at
the wall.
The insulating case implies that any flux through
the wall is unchanged by the presence of the concentration
perturbation.
This is perhaps the more useful assumption
in most situations.
To continue the mathematical analysis, assume as
suggested by experiments, that w and c take on a spatially
repeating pattern in the x - y plane, with wavelength d
38
and dimensionless wave number a, defined as
2"rh
(Note that the convection cells are of finite size, even
through the velocity and concentration are assumed to
differ infinitesimally from the unperturbed values.)
Additionally, assume that w and c have an exponential
dependence on time.
Solutions to equations 2.1 and 2.2 may
decay steadily with time, so that an arbitrary disturbance
would disappear, or they may increase with time so that a
disturbance would intensify.
be considered in Section 2.9.)
(Oscillatory solutions will
If decaying solutions are
termed stable and increasing ones unstable, then separating
these two sets will be a locus of time-invariant solutions,
describing states which neither decay nor grow.
These so-
called stationary states are considered to possess marginal
or neutral stability.
If, arbitrarily, a, R, and
/3z are
assumed specified, it is found that a unique value of M
solves the time-invariant form of the equations in Table 2.1.
By choosing a range of values of a, a curve of marginal
stability may be developed as in Figure 2.3.
The minimum
in that curve indicates the lowest value of M, and the
corresponding wave number, for which an unstable state
exists, with the other parameters as specified.
This minimum
or critical Marangoni number, Mc , is the main object of
attention.
Evidently, in a real system, if M is measured
39
250
200
M
150
100
0
1.0
2.0
3.0
Figure 2.3 Curve of Marginal Stability
40
4.0
to be less than M c , then only stable states of the real
system exist.
Conversely, if M exceeds Mc, then unstable
states may exist and it is usually assumed that instability
will be observed.
It should be understood, however, that
linearized theory, as presented,need not necessarily apply
to real disturbances, of finite intensity.
Usually, based
on limited experimental evidence, one assumes that M
and
ac from linear theory do correspond to measurable reality;
but it remains only an assumption.
(Compare with Sections
2.8 and 2.9.)
2.3
Basic Theoretical Results
Pearson computed values of the critical Marangoni
number for various values of R, assuming the unperturbed
gradient is linear, DD/Dz = 1, and extends throughout the
liquid X = /h = 1.
His results, for the insulating case,
are given in Figure 2.4 - the curve labeled X = 1 - as a
plot of M
c
versus R.
Conditions below the curve are
predicted stable, those above unstable.
Pearson's mechanism is a highly simplified, illustrative
one; even within the confines of linear thoery it ignores
many variables - for instance surface viscosity and surface
deformation.
Still, his work has been widely accepted and
it will be seen in Section 2.9 that most of his simplifying
assumptions are generally valid.
Nevertheless, the
assumption of a full-depth concentration gradient is quite
restrictive.
In 1968, Vidal and Acrivos examined linear,
41
-1U
4
10
3
10
M
2
10
10
1
-2
10
1
1
Figure 2.4 Predic ted and Measured
Critical
M arangoni Number
42
10
10 2
Val ues of
unperturbed profiles extending part of the way into a
liquid phase of arbitrary depth, X > 1.
Although only the
case R = 0 was considered, both Mayr and this author have
computed additional results over other values of R.
A
sample curve at X = 10.0 is shown in Figure 2.4, along with a
limiting curve for X =
.
As is evident, the deeper the
liquid pool, relative to the concentration gradient, the
less stable the system.
the results for X = 10.0
infinite depth.
Note also that when R exceeds 1.0,
are the same as those for
Even at low values of R, these two assump-
tions about the depth imply very similar results.
In the limit of infinite depth, Vidal and Acrivos were
able to consider non-linear, time-varying profiles typical
of penetration mass transfer.
They employed the "frozen-
profile" method, in which a slowly changing gradient is
considered invariant at an instant in time, and a stability
analysis is performed on it.
Some authors consider the
method of doubtful validity, these criticisms will be
considered in section 2.9.
In any event, the results were
very similar to those for linear, time-invariant profiles.
2.4
Comparison of Theory and Experiment
In mass transfer systems, measurements of the critical
Marangoni number have diverged profoundly from the
predictions of Pearson or Vidal and Acrivos.
Brian et al.
(1971) report on the work of Mayr (1970), who measured
critical Marangoni numbers in aqueous solutions of acetone,
43
diethyl ether, and triethylamine (TEA). For all of Mayr's
systems, X was roughly 10.
Figure
His results are plotted in
2.4; for the three systems M
appears
to be 60,
250, and 2200 times higher respectively than theory predicts.
In a parallel set of experiments Clark and King (1970), studied
a variety of organic solutes in n-tridecane; all of the
systems exhibited roughly the same critical Marangoni
number, 12,000.
The value of X was near 100, so that the
experimental result is 750 times above theory.
In a very real sense, Figure 2.4 represents the embarkation point for this author's own work.
The graph, in fact,
first appeared, in a modified form, as Figure 16 of Brian,
et al. (1971), very shortly after this thesis was commenced.
More to the point, the figure vividly illustrates the poor
understanding of interfacial convection which was common
at that time.
Marangoni instability is unlikely to be
exploited until it is predictable, and order-of-magnitude
errors, like those displayed, are hardly acceptable in this
vein.
Worse, the various data points do not even suggest
a dependable, empirical relationship.
As this author
commenced his efforts, it was obvious that certain influences,
not yet accounted for, had to be elucidated in order to
understand the onset of Marangoni convection.
Later in this chapter, in Sections 2.8 and 2.9, is a
detailed review and evaluation of various theories and
experiments concerning interfacial instability.
But even
in the absence of a complete review, two effects stood
44
apart, when this thesis began, as being of potentially
major importance.
One was the possible, convection-inhibiting
presence of an adsorbed Gibbs layer of surface active impurities, in Mayr's solutions.
The second was the definite
existence of a similar form of Gibbs adsorption, in this
case adsorption of the very solute whose concentration
gradient promoted Marangoni instability.
2.5
Gibbs Adsorption
J. Willard Gibbs is credited with first explaining
that any surface active solute (that is, a surface tensionlowering one) adsorbs preferentially at the free surface of
a solution.
Physically, Gibbs noted that what is called a
gas-liquid "interface" is more reasonably a very thin region
over which the liquid phase concentrations of a solvent and
any solute usually decrease drastically, to take on their
respective gas phase values.
Further, he deduced from
energy considerations that the concentration of a surface
tension-lowering solute would diminish more slowly than
that of its solvent, making the solute seem more "concentrated"
at the surface.
Mathematically, Gibbs successfully modeled
this situation as if the interface were in fact infinitesimally
thin, but that adsorbed on the interface was a certain
excess solute concentration, r.
For dilute solutions Gibbs
found that
r
= Ci,
45
where 6 is a proportionality constant with dimensions of
length, and that
6-
RT
(R is the gas constant and should not be confused with R,
the resistance ratio.)
The ideas of Gibbs adsorption are now accepted as one
of the foundations of modern surface physics.
Additional
details are available from, for instance, Adamson (1967).
2.6
Gibbs Adsorption of Impurities
In 1965, Berg and Acrivos extended Pearson's work to
allow for the presence of small amounts of a very highly
surface active, non-volatile impurity.
Thus, in concentra-
tion-driven convection, not one but two substances would
affect the surface tension, prompting an additional term
in the surface stress balance, equation 2.3.
To simplify
their analysis, Berg and Acrivos made the restrictive assumption that the impurity was insoluble, present only at
the solution surface.
More exactly, they assumed that
diffusion of the impurity along the liquid surface was
large, compared to diffusion from the bulk phase beneath.
This assumption allows equation 2.3 to be modified to
V2w
a2w
v
z
(/a2c2
Mt
46
2
a)
+
v)
-N
T
(2.3a)
where the "Elasticity number" is defined as
a C.
N
EL
-
=
h
1
with a, Ci, and D all evaluated for the impurity.
It was
discovered that non-zero values of NEL, theoretically, make
systems more stable.
Berg and Acrivos spoke of this in
terms of a surface elasticity imparted by a contaminant to
a system.
Certain materials commonly used in household detergents,
substances called "surfactants" (for surface active
gents),
are non-volatile and possess very large values of a in
aqueous media.
These materials are often found in trace
quantities in water and as little as a few parts per million
may imply a value of NEL exceeding 10.
More significantly,
Berg and Acrivos found that such a value would raise the
predicted critical Marangoni number by an impressive three
orders of magnitude.
As already mentioned, Berg and Acrivos considered only
insoluble impurities.
Intuitively the stabilizing influence
of an impurity should be weaker if the impurity is soluble
and less concentrated at the surface.
Directly applying
the Berg-Acrivos model to a soluble impurity, then, would
erroneously over-state the importance of the material.
Palmer and Berg (1972) verified this idea, in an extension
of the earlier analysis.
However, their work left vague
the question of how to determine if a given surfactant is
acting as a soluble or insoluble impurity.
47
None too surprisingly various investigators have
suggested a connection between certain of their own puzzling
experimental results and the supposedly powerful effects of
insoluble surfactants.
Specifically, Brian et al. (1971,
p. 83), describing Mayr's work, speculate that the datatheory discrepancy "may be due to surface impurities."
Clark and King (1970) make a similar suggestion, as do
Palmer and Berg (1973), whose thermal convection experiments
are detailed below.
Blair and Quinn (1969), Kayser and
Berg (1973), and Zeren and Reynolds (1972) likewise invoke
surfactant effects, in analyzing their separate results,
which are described in Section 2.8.
Experimental evidence to confirm these thoughts - much
less verify the quantitative theories of Berg and Acrivos
and Berg and Palmer - has been minimal.
study was by Berg et al. (1966b).
One systematic
It was observed optically
that intentionally added surfactants did indeed minimize, or
even prevent, both thermally-induced and concentrationinduced convection in evaporating pools of liquid.
However,
no Elasticity numbers or critical Marangoni numbers were
measured.
Palmer and Berg (1973) and Palmer (1971) provide some
very interesting and yet equivocal results.
Critical
Marangoni numbers were determined, by heat transfer measurements, for temperature-induced convection in glycerol-water
and in ethylene glycol.
The results are a factor of two or
48
three above theory, unlike the close agreement in instability
in silicone oils, reported in Palmer and Berg (1971).
Section 2.8.)
(See
The authors attribute the difference to sur-
factant contamination, noting the supposed "difficulty of
. high surface tension liquids to yield clean
purifying .
.
surfaces."
(Palmer and Berg, 1973, p. 1083).
Indeed, Palmer
(1971, pp. 74, 89) does report surface tension values of 47
and 65 dynes/cm for glycol and glycerol-water, obviously
higher than the 20 dynes/cm value of silicone oil.
Never-
theless, Palmer and Berg (1973, p. 1082) also state that in
"each experiment [with glycol and glycerol-water] all materials were meticulously purified to guard against spurious
contamination.
The organic compounds were doubly distilled
under vacuum using a 30-stage spinning band column, and the
water was triply distilled."
It is difficult to reconcile
this extraordinary effort at purification with the suggested
presence of impurities.
Moreover, while Mayr's work, with
its theory-data discrepancy, was also done with a solvent of
high surface tension (water, 72 dynes/cm), the work of
similar disparity by Clark and King (1970) employed a low
surface tension
organic solvent (25 dynes/cm).
In an additional set of experiments Palmer and Berg
(1973) added the surface active solutes butanol and octanol
to the glycerol-water and glycol pools respectively.
Qualitative agreement with their own theory for soluble
surfactants is claimed,
49
Mayr had a somewhat unusual justification for believing
surfactants were present in his system.
He noticed the
persistent build-up of a layer of stagnant liquid at the
outlet from his wetted-wall column.
Drawing on the con-
clusions of earlier workers (for instance, Brian, et al.,
1967), he attributed this to the influences of both fluid
dynamics and surfactant accumulation.
pp. 192-194.
(See Mayr, 1970,
Compare with Byers and King, 1967, p. 643,
Cook and Clark, 1973, and especially Kenning, 1968, pp.
1107-1108.)
Because of the common but unproven belief in the importance of low levels of surfactants, and because Mayr
did strongly suspect the presence of surfactants in his
systems, one major objective of this author's work was to
re-evaluate and attempt to settle the role of surface active
impurities in concentration-driven Marangoni convection.
2.7
Gibbs Adsorption of STL Solute
Brian (1971) suggested a second plausible explanation
for the discrepancy between theories and experiments on
Marangoni instability.
He noted that, even in the absence
of impurities, there would be adsorption of the transferring
surface tension-lowering (STL) solute in the Gibbs layer.
Brian demonstrated that, at least according to the basic
Pearson model, movement of the solute in the Gibbs layer has
a strong stabilizing effect on Marangoni instability.
The
mechanism is similar to that of Berg and Acrivos, but no
50
mention of "surface elasticity" is made.
Rather, one notes
that the presence of a surface layer of adsorbed solute,
in the un-perturbed liquid, constitutes a large inventory
of material; if a small excess of solute is pushed to the
surface, the very surface expansion it will engender could
carry away more solute than the initial excess.
This would
then cause a contraction at the surface, opposing Marangoni
convection.
Mathematically, Brian's main innovation was to
alter the surface material balance, equation 2.4, to become
ac
-
=
Rc -
az
22c
D /a
a2
s
+
2
aw
A--
az
-+y\x
h D
c
+
-
(2.4a)
h at
where A, the Adsorption number, is defined by
A -
(The parameters
Ar C
in A are evaluated
for the un-perturbed
situation.)
Brian studied Gibbs adsorption for the simplified,
Pearson model.
He found that the final term in 2.4a, which
accounts for accumulation in the Gibbs layer, and the preceding term, for surface diffusion, may both be neglected.
However, he discovered that the third term, which accounts for
convective movement along the surface, is extremely important.
Increasing A tends dramatically to raise the Marangoni
number required for the onset of convection.
Indeed, if
A exceeds a certain value, Marangoni instability is completely
51
precluded.
Thus Brian established the importance of Gibbs
adsorption; however, his work, like Pearson's, was valid
only for an un-perturbed concentration gradient which was
linear and extended fully through the liquid solution.
His results could not properly be compared to experiments
like Mayr's, in which a non-linear, partial profile prevailed.
In addition, Brian observed that his own analysis
could be interpreted to predict stability for systems that
Mayr had found were unstable.
Therefore, on one hand, Gibbs
adsorption of the STL solute provided an important new
influence which could raise theoretical predictions of Mc
closer to experimental values.
On the other hand, it yet
remained to incorporate this effect into practically useful
theories of convection.
For this reason, it became a prime
objective of this author to develop an experimentally pertinent theory of instability, which included Gibbs adsorption.
2.8
Additional Experimental Work on Marangoni Instability
In this section and in the following one, a review is
provided of the fundamental experimental and theoretical
work on Marangoni instability which has not yet been
described.
Many of these efforts are of modest import and
the reader may wish on first reading to skip to section 2.10,
to preserve continuity.
Concentration-induced convection.
There have been
relatively few investigations of concentration-induced
Marangoni instability in gas-liquid systems.
52
Matiatos (1965),
whose results were published as Brian, et al. (1967), established the existence of interfacial convection in such
systems.
The industrially important, but poorly understood,
process of absorbing carbon dioxide into a reactive amine
solution was studied.
Moslen (1971) unexpectedly discovered
interfacial turbulence, due to heats of solution, when absorbing chlorine into ortho-dichlorobenzene.
Mayr (1970)
determined critical Marangoni numbers by following desorption of three STL solutes from water.
In each of these
investigator's efforts the presence or absence of instability
was determined by monitoring the overall rate of mass transfer of an inert, tracer-material, during counter-current
flow in a short, wetted-wall column.
This method is feasible
because the existence of Marangoni convection noticeably
increases liquid-side
mass transfer coefficients.
Bautista (1973) recently used similar methods to confirm the presence of Marangoni convection during desorption
on a sieve tray apparatus.
Aqueous solutions of acetone
and methyl ethyl ketone were employed; qualitative confirmation is reported for the importance of Gibbs adsorption of
the transferring solute.
As mentioned previously, Clark and King (1970) studied
instability during desorption of several organic solutes
from n-tridecane.
The contacting apparatus was a co-current
flow, horizontal channel; the critical Marangoni number was
12,000.
53
In an early study, Berg et al. (1966b) observed convection and surfactant effects optically, during desorption
of dioxane from water.
Blair and Quinn (1969) similarly
studied convection during desorption of ether from monochlorobenzene.
No critical Marangoni numbers were reported
by either of these two sets of authors.
Thermal instability.
In contrast to the study of con-
centration-driven convection, quantitative experiments on
thermally-driven convection have been extensive.
However,
thermal experiments usually have involved linear temperature
gradients extending throughout a liquid film, whereas all
of the concentration-based work has involved non-linear partial gradients.
This may partly explain why very close
agreement with Pearson's predicted critical Marangoni number
was achieved by Palmer and Berg (1971) and Koschmieder (1967).
In each instance, a viscous silicone oil was heated from
below to create a full, linear temperature gradient.
Palmer
and Berg inferred the onset of convection from changes in
the rate of heat transfer.
Koschmieder used an optical
technique.
Additional experiments based on changes in heat transfer
rate have shown discrepancies between theory and data.
The
work by Palmer and Berg (1973), with water-glycerol and
glycol, was described in Section 2.6.
Zeren and Reynolds
(1972) heated (two-phase) pools of benzene and water.
They
report no interfacial convection, even with the Marangoni
number as much as five times above the predicted critical.
54
As mentioned in Section 2.6, the two preceding pairs of
authors cite surfactants to explain their results.
port and King (1974b) take a different tack.
Daven-
Their own
experiments, on liquid pools with a time-dependent, nonlinear temperature profile, show no Marangoni convection
even when M exceeds the predicted critical by 1000 times.
Butanol and octanol were the liquids employed.
They suggest
that random surface disturbances were totally absent from
their fluids and, hence, that surface tension-driven instability could not be initiated.
However, this would not
explain the discrepancy evident in experiments where surface
disturbances did exist, presumably including the flow
system studies of Mayr (1970) and Clark and King (1970).
Moreover, the previously noted results of Palmer and Berg
(1971) and Koschmieder (1967), for initially quiescent pools
of silicone oil, would seem to confirm that gross disturbances are not necessary for the onset of convection.
Admittedly, in those two efforts, the temperature profiles
were full-depth and constant in time, unlike those of
Davenport and King.
Optical studies.
A number of optical studies have been
performed, both to determine onset conditions for Marangoni
convection and also to observe cell size and shape.
Berg
et al (1966a) review the several experimental techniques
commonly used.
The work of Berg et al. (1966b) and Blair
and Quinn (1969) on surfactant effects has already been
55
noted in this section and in Section 2.6, as has Koschmieder's
measurement of the critical Marangoni number to agree with
Pearson's prediction.
Less successfully, Vidal and Acrivos
(1968) observed convection in very shallow, evaporating
pools of propanol.
The evaporation prompted a partial,
time-dependent thermal gradient and values of the critical
Marangoni number were high.
For a representative run with
X near 2.0, Mc (according to the present author's definition)
was 130 while theory predicts 30.
Kayser and Berg (1973) measured convection cell sizes,
while observing the transition from surface tension-driven
to buoyancy-driven flows, in pools of heated silicone oil
and water-glycerol.
Cell size was of the same order as total
liquid depth; values were reasonably close to Pearson's
predictions for disturbance wavelength.
Koschmieder (1967)
observed a hexagonal pattern of cells on the surface of
heated silicone oil.
Again cell sizes were similar to
Pearson's predictions.
In each of the last two efforts, the temperature gradient
extended throughout the liquid film.
Bakker et al. (1966),
investigating liquid-liquid mass transfer with partial-depth
concentration profiles, found that the size of a convective
cell was of the same order as the penetration depth.
Brian
et al. (1971) argued that this idea was consistent with the
discovery that, in their own liquid-gas desorption experiments,
liquid instability in no way affected the gas-side mass
transfer rate.
56
2.9
Additional Theoretical Work on Marangoni Instability
Basic results.
As mentioned previously, Pearson (1958)
did the first theoretical analysis of surface tension-driven
instability.
Only liquid phase dynamics were considered
explicitly; the work was intended for gas-liquid systems.
Sternling and Scriven (1959) published independently a more
general analysis, in which the hydrodynamics were treated
for both phases of a two-phase system.
liquids could be conisdered.
Thus immiscible
Scriven (1960) provided a
stronger foundation for both efforts by deriving a detailed
mathematical description of the stresses present at a
Newtonian interface.
Scriven and Sternling (1964) relaxed
the Pearson assumption of a non-deformable, flat interface
and found that in some cases instability was intrinsic;
there was no minimum Marangoni number.
However, Smith (1966)
added to the problem the stabilizing effects of surface
gravity waves and re-established the validity of the simplified, Pearson analysis.
When body forces due to gravity
are strong enough to prompt buoyancy-driven instability, as
well as the surface tension-driven form, the exposition by
Nield (1964) applies.
Zeren and Reynolds (1972) extended
Nield's work to include a deformable interface and Smith's
gravity waves.
Scriven and Sternling (1964) asserted that the presence
of surface-active impurities in a liquid pool would inhibit
convection, by directly increasing the surface viscosity.
57
Berg and Acrivos (1965) showed that, for insoluble surfactants, surface viscous effects were negligible, compared to
the elastic effects associated with Gibbs adsorption.
Palmer
and Berg (1972) extended this conclusion to soluble surfactants.
They also examined solutions in which there was an
assumed energy barrier to adsorption; that is, when the Gibbs
equilibrium relation, r =
ci,
was not satisfied.
Their
mathematical scheme followed that of England and Berg (1971).
Nevertheless, England and Berg (1971), Farley and Schecter
(1966), and Defay and Hommelen (1959) all agree that in
gas-liquid systems there are usually no such energy barriers.
The mathematics of an interfacial material balance,
including Gibbs adsorption, has been treated most generally
by Palmer and Berg (1972), England and Berg (1971), and
Berg (1972). Brian (1971) gives a clear physical interpretation of the mass balance equation, while Levich and
Krylov (1969) and Kenning (1968) provide slightly earlier,
lucid derivations.
Oscillatory modes.
In Section 2.2, little attention
was given to oscillatory solutions of the linearized stability
analysis.
To consider these, recall that the velocity and
concentration disturbances are assumed to have an exponential time dependence, of the form ePt.
If p is real, when
p equals zero a stationary, stable state is described.
However, if p is complex, and if it happens that the real
part of p becomes zero while the imaginary part remains
finite, an oscillatory, stable state develops.
58
(Note that
by Euler's formula ePt has a sinusoidal real part even though
p is imaginary.
513.)
See, for instance,
Hildebrand,
1962, p.
Presumably, if instability developed from an oscillatory
state it might, in some situations, by physically indistinguishable from that originating in a stationary state.
Several persons have searched for oscillatory solutions to
the various treatments of Marangoni instability.
Vidal and
Acrivos (1966) found no such modes in Pearson's model.
Brian
and Smith (1972) also found none in the Gibbs adsorption
treatment of Brian (1971). However, Linde et al. (1967)
do report the existence of oscillations in experiments and
in a treatment based on Sternling and Scriven (1959). These
results have rarely been given serious evaluation, perhaps
because all of Linde's work is in German.
Mitchell and
Quinn (1968) observed oscillating temperature disturbances in
shallow pools, heated by a point source.
Palmer and Berg
(1972) predict oscillations associated with adsorption of
surface active impurities.
Notwithstanding these last few
results, it should be understood that the existence of
oscillatory modes can decrease, but never increase, the
predictions of minimum Marangoni numbers.
If the lowest
oscillatory Marangoni number exceeds the lowest stationary
value, then the latter describes the overall minimum
condition.
Therefore, in the present circumstances, where
it seems important to justify theoretically higher values
of Mc, to reach agreement with experiments, the question of
oscillatory modes is unlikely to be fruitful.
59
Finite amplitude disturbances.
A similar comment
applies to analysis of finite scale disturbances, as performed for buoyancy-driven convection by Homsy (1973). In
his work, and in that of several predecessors, the assumption of infinitesimal perturbations is relaxed; but the main
goal becomes determining whether strong, external disturbances can be maintained, below the value of M
linear theory.
derived from
Again, a positive finding would enlarge the
theory-experiment discrepancy, not minimize it.
A different type of finite amplitude study was performed
by Scanlon and Segel (1967) to predict the hexagonal shape
of the cells exhibited in Marangoni convection.
Interestingly,
it is assumed that the cell size and wave number in finite
intensity convection are equal to those values predicted by
linearized theory, for convection of infinitesimal intensity.
The experiments by Koschmieder (1967) and Kayser and Berg
(1973) are consistent with this.
Also Berg et al. (1966a)
review several reasonably successful studies of buoyancydriven convection, in which the same assumption is made.
Frozen profile.
The frozen-profile method mentioned
in Section 2.3, has often been attacked as leading to gross
underestimates of minimum stability conditions.
Brian and
Ross (1972) refute this contention for the present work,
and the following discussion is adapted from that paper.
In the context of buoyancy-driven convective instability,
a number of investigators have assessed the adequacy of the
60
frozen-profile analysis.
It appears (Mahler et al., 1968)
that the frozen-profile analysis does give a reasonably
accurate prediction of the point at which a perturbation
just begins to grow, but the initial growth rate is slow,
and considerable time elapses before the perturbation has
grown by a substantial amount and is then growing at a fast
rate.
Thus experimental measurements (Spangenberg and
Rowland, 1961; Foster, 1965b; Blair and Quinn, 1969; Nielson
and Sabersky, 1973; Davenport and King, 1974a and 1974b)
reveal the onset of measurable convection at substantially
longer times than predicted by the frozen-profile analysis.
This has led several investigators to perform a transient
analysis of the problem, in which the rate of change of the
unperturbed profile with time is considered.
These results
have shown better agreement with the experiments, but their
interpretation is clouded by the necessity to make subjective judgements regarding the nature of the initial perturbations and the amount of growth required before the
convection is experimentally visible.
(Compare with Foster,
1965a and 1968; Mahler et al., 1968; Gresho and Sani, 1971;
and Davenport and King, 1974a).
Since the focus of the present work is upon the effect
of Gibbs adsorption, it seems appropriate to employ the
frozen-profile simplification, at least initially.
Com-
parison of the present results with those of Vidal and
Acrivos (1968) will then reveal the effect of Gibbs adsorption without the complications inherent in the complete
transient analysis.
61
Another compulsion to adopt the frozen-profile simplification stems from a desire to apply the results to
steady-state experiments in a wetted-wall column, such as
those of Mayr (1970). In these experiments the unperturbed
concentration profile is steady in time but is developing
in downstream distance, or "contact time."
In this context,
the frozen-profile assumption corresponds to neglecting the
rate of change of the unperturbed concentration profile with
downstream distance.
For this system, it is shown in
Chapter 9, by inspection of the linearized differential
equations, that the unperturbed velocity profile has no
effect on that disturbance mode which is invariant with
downstream distance, that is, for stationary roll cells with
axes aligned in the flow direction.
By analogy to buoyancy-
driven instability this is expected to be the least stable
mode in the presence of the mean flow, and thus the critical
Marangoni number is unaltered by the flow.
(Compare with
the theory of Gallagher and Mercer, 1965 and Deardorff,
1965 and with the theory and experiments of Hung and Davis,
1974.
See also the experiments of Chandra, 1938 for the
"Type I" convection.
Chandra's anamalous "Type II" con-
vection was recently reviewed by Scanlon and Segel, 1973.)
Furthermore, this disturbance mode cannot propagate
downstream, and thus the time required for such a disturbance
to grow to a "visible" magnitude is of no consequence;
convection will be observed if the point of inception of
instability is reached within the wetted-wall column.
62
In this context it is expected, therefore, that the frozenprofile analysis is more suitable than a complete analysis
of a transient penetration problem.
2.10
Thesis Objectives
To summarize the conclusions to be drawn from Sections
2.2 through
2.9, the theoretical
model
of Pearson
(1958) is
considered valid for both concentration-driven and thermal
instability; the effects of surface deformation, surface
viscosity, gas phase dynamics, buoyancy, oscillatory convection, and finite intensity disturbances are all discounted.
The deep pool, frozen-profile analysis of Vidal and Acrivos
(1968) and the Gibbs adsorption influence of Brian (1971)
are both accepted as useful modifications, where applicable.
Similarly, if adsorbed impurities are present, some aspects
of Berg and Acrivos (1965) and Berg and Palmer (1972) may
apply.
Prior to the present author's work, none of the Gibbs
adsorption models had been collected into a single theoretical
model, modified for concentration-driven convection in
liquids of arbitrary depth.
Precisely that treatment,
however, is required to understand properly the data of
Mayr (1970), whose measurements of the critical Marangoni
numbers in mass transfer are the most extensive to date.
Therefore, the first objective of this thesis was to reinterpret Mayr's data in the light of a newly integrated,
theoretical analysis.
63
Experimentally, studies of the onset of thermal convection, in shallow pools of silicone oil, are the only ones
to show clear agreement with the Pearson model.
investigations
All other
have failed to produce agreement with theory,
including studies with fluids of higher surface tension or
lower viscosity, partial-depth or time-varying gradients,
and any form of concentration-induced disturbance.
Most frequently, surfactant impurities have been cited
to explain this state of affairs.
Yet direct, experimental
proof of such claims is virtually nil.
For this reason,
a serious effort to prove or disprove these suppositions
was proposed.
Very likely, this could have taken a variety
of forms; but, mass transfer enhancement is of interest
uniquely to chemical engineering and Mayr's experiments were
of course the ones best known to this author and his
advisors.
Therefore, it was decided to duplicate certain
of Mayr's experiments, after having removed with certainty
all surface active impurities.
That these could be fully
removed, with certainty, was assured after a review of the
published techniques for adsorption of surfactants on
activated charcoal.
2.11 Detection and Removal of Surfactants
Swisher (1970, pp. 1-6, 19-37) gives a useful, broad
introduction to the chemistry and occurrence of surfactants.
Frequently, these substances are classified by the ionic
charge of their water-repelling or hydrophobic portion.
64
That is, they may be anionic (negative charge) cationic
(positive charge), or nonionic.
Of these, anionics are by
far the most commonly produced in industry, with nonionics
Anionics and nonionics may also be created by degrada-
next.
tion of waste materials in nature.
Cationics are manufac-
tured in very small quantity and, further, they do not
co-exist with anionics in a surface active form.
Because surfactants are a main constituent of detergents, they are found in significant amounts in sewage;
domestic and industrial water may contain up to 500 parts
per billion of anionic surfactant.
Nonionic concentrations
in water are not well known, but are thought to be quite
small.
Removing small amounts of any material from a dilute
solution is typically a difficult problem.
Aqueous solu-
tions are often considered especially intractable.
to Mysels and Florence, 1970.)
(Compare
However, activated charcoal
filtration is well accepted as a method for complete removal
from water of all types of surfactants.
Oehme and Martinola, 1973.)
(See for instance
Sallee et al. (1956) gave an
impressive demonstration of this.
A solution containing
100 ppb of a radioactive, anionic surfactant was passed
through a charcoal column.
The water effluent was analyzed
and found to contain 0.1 ppb of residual surfactant.
In
other words, the concentration was reduced by a factor of
1000.
Similarly, quantitative removal was reported for
nonionic surfactants by Melpolder et al. (1964).
65
Possibly, activated charcoal could add trace impurities
to a water filtrate.
However, it wold be most unlikely
that such contaminants would be surface active; and only
surface active materials could interfere with subsequent
experiments on instability in a wetted-wall column.
There-
fore, charcoal filtration was adopted, as the method for
completely removing surfactants from the feed water used
in repeating Mayr's work.
The procedure is straightforward
(see Chapters 5 and 11), and it is reasonable to assume that
surfactant removal may be virtually complete.
To confirm qualitatively the extent of the purification,
water samples can be subjected to the "methylene blue" test,
before and after filtration.
Jones (1945) first developed
this widely-used method for detection of anionic surfactants.
An unknown is mixed with aqueous methylene blue and the
resulting complex is extracted into chloroform.
The
intensity of color developed by the chloroform is a measure
of the initial surfactant concentration.
The scheme is
quite sensitive but relatively nonspecific; positive results
occur for nearly all anionic surfactants.
Additional details
are given in Chapters 5 and 11.
In summary, charcoal filtration was chosen as an
appropriate means for completely removing surfactant impurities from water.
Methylene blue testing was available
to monitor the effectiveness of the procedure.
66
3.
GIBBS ADSORPTION OF A TRANSFERRING SOLUTE
This chapter consists of an original, theoretical
treatment of concentration-driven Marangoni instability,
in liquids of arbitrary depth.
Gibbs adsorption of the
transferring solute is prominently considered.
A detailed
comparison is made with the experiments of Mayr (1970).
(Most of the material which follows was published in a
nearly identical form in Brian and Ross, 1972.)
3.1
Theoretical Model
The physical system in its initial state is that
discussed in Chapter 2 and illustrated in Figure 2.2b.
A
surface tension-lowering (STL) solute is diffusing from a
horizontal, stagnant liquid layer into a gas phase above.
Buoyancy effects and thermal effects on desorption are
assumed negligible.
The fluid mechanics of the gas phase
are not considered; the transfer of solute from the gasliquid interface to the bulk gas phase is modeled by a
gas phase mass transfer coefficient, kG, assumed to be
constant and independent of convection within the liquid
phase.
Also, deformation of the gas-liquid interface from
its initial planar state is neglected, as is surface
viscosity.
No contaminating impurities are present in
the solution.
The infinitesimal perturbations in the z'-direction
velocity and in the solute concentration are assumed to
67
be of the form
w
w
C
F(x,y) f(z) ePt
=
C
'
i=
=BC
cB
F(x,y) g(z) e
=
(3.1)
t
.
(3.2)
The function F(x,y) describes a periodic variation in the
x = x'/h and y = y'/h directions, which lie in the plane
of the gas-liquid interface.
The function is defined by
the differential equation
a2F
-
2
2F
+
+
ax
ay
2
a2
F
=
0
(3.3)
in which a
is the dimensionless wave number for the periodic
variation.
Inserting equations 3.1, 3.2, and 3.3 into the
linearized equations of motion and the diffusion-convection
equation for transport of the solute (see Table 2.1 and
Chapter 9) yields the differential equations defining f
and g:
[p - Sc (D2 -
[D2
2)]
(D2
a2) - p]g
2)f
=
f D
(3.4)
(3.5)
where D represents differentiation with respect to the
dimensionless depth coordinate, z.
The unperturbed con-
centration profile is expressed in dimensionless form as
C - C.
CB
-
Ci
=
68
(z).
(3.6)
At a dimensionless depth
, corresponding to the
bottom wall, the no-slip velocity condition requires that
f(X)
=
0
(3.7)
Df(X)
-
0.
(3.8)
The concentration boundary condition at the bottom wall
will correspond either to a constant concentration at a
conducting wall,
g(X)
=
0
(3.9a)
or to a constant solute flux at an insulating wall,
Dg(X)
as discussed
in Section
=
(3.9b)
0
2.2.
The normal velocity must vanish at the non-deformable
gas-liquid interface:
f(0)
=
0.
(3.10)
The tangential shear stress condition at the free surface
introduces the Marangoni number into the problem:
D2f(0)
=
M
g(O).
(3.11)
It should be noted that the Marangoni number is defined here
in terms of the overall concentration difference in the
liquid phase and h, the thickness of the liquid phase
concentration boundary layer.
h will be given shortly.
69
A more precise definition of
The effect of Gibbs adsorption enters the problem
through the concentration boundary condition at the free
surface:
Dg(O)
=
R g(O) + pG g(O) + a 2 S g(O) - A Df(O).
(3.12)
It is assumed that the quantity of solute adsorbed at the
surface, r, is related to the concentration at the surface,
C i, by the Gibbs adsorption isotherm,
r
=
Ci/RT.
The
derivation of equation 3.12 is given in Chapter 9.
The
term on the left-hand side represents diffusion of solute
to the surface from the liquid beneath.
The terms on the
right-hand side represent transfer to the gas phase, accumulation within the Gibbs layer, surface diffusion within
the Gibbs layer, and surface convection within the Gibbs
layer, respectively.
In seeking solutions for the case of neutral stability,
in which an infinitesimal perturbation neither grows nor
decays, p will be taken to be zero.
This corresponds to
assuming that no oscillatory modes of instability exist,
as discussed
in Section
2.9.
With p = 0, G drops out of
equation 3.12, and accumulation within the Gibbs layer has
no influence on the neutrally stable state.
The statement of the problem is now complete, and a
solution can be obtained for a specified function
(z),
representing the shape of the unperturbed concentration
profile.
70
3.2
Solution for Finite Liquid Depth
When the value
X is finite,
of
it is useful to simplify
the problem by considering only the case of a broken-line
unperturbed concentration profile, as shown in Figure 2.lb
where the definition of h is apparent.
The normalized
slope of the unperturbed concentration profile is given by
1
for
0
<
z
<
1
=
D
.
0
1
for
<
<
z
(3.13)
X
With p taken to be zero, the solution
to equation
3.4,
subject to the condition of equation 3.10, may be written
f(z)
=
--~~-
A 1 sinh az + A 2 z sinh az + A 3 z cosh az. (3.14)
Employing equations 3.7 and 3.8 yields
A3
(sinh ax) 2
=
A1a
1
(sinh aX) (cosh ax)
2
-o.
(3.16)
The concentration function is written in the form
gl(z)
g(z)
for
0
<
z
<
=
1
(3.17)
g 2 (z)
for
1
<
z
<
Employing equation 3.13 and taking p = 0, the solution to
equation
3.5 is
71
gl(z)
=
B 1 sinh az + B2 cosh
g2 (Z)
=
B 3 sinh az + B4 cosh
where the function
(z)
(3.18)
(3.19)
az
is defined as
-
(z)-
z +
z sinh a
AA~l A1
.2a 4aJ
+
-4a~
] z
+
z cosh
z2 cosh az.
sinh az +
z
(3.20)
The constants in equations 3.18 and 3.19 are determined from
the previously stated boundary conditions plus two additional
conditions requiring the continuity of the function g and
its
first derivative at z = 1:
(1)) g
Dg
(3.21)
=
(1)
Dg2 (1).
(3.22)
Employing equations 3.12, 3.21, 3.22, and 3.9a or 3.9b, the
constants are determined as:
{ (R +
S) [aGY(1)
-
B
=
(sinha
- D
(l)]
- G cosh a) [D'T(0)
+ A(aA1 + A 3)] }
{(R + a S) (cosh a- G sinh a)
2
+ c2 (sinh a -
G cosh a)}
(3.23)
aGT(1) - D(1)
B2
- Ba (cosh a - G sinh a)
=
B1 sinh c + B2 cosh a +
B
(3.24)
a(sinh a - G cosh a)
3osinh
sinh
1
-
aX
h )
sosh
(1)
(3.25)
-c cosh a
72
/sinh
aX\
(_ s
B4
B3
(3.26)
in which G represents the function
cosh a -\/ csh
sinh a27
G
=
)
sinh In these equations,
for which
equation
(3.27)
cosh a
= 1 for the conducting bottom wall,
3.9a applies,
and
v = -1
for the in-
sulating bottom wall, for which equation 3.9b applies.
The
Marangoni number corresponding to neutral stability then
follows from equation 3.11:
2A 2
M
=
The solution is now complete.
proportional to A;
.aB
cB 2
(3.28)
All of the constants are
thus the value of A
does not affect
the result and can be conveniently chosen to be 1.
For
specified values of R, A, S, and X, these equations determine M as a function of a, referred to as the curve of
neutral stability.
The minimum in that curve corresponds
to the critical Marangoni number.
(see Chapter
10) was written
A digital computer program
to compute
the values
of the
critical Marangoni number presented in Figures 3.1 through
3.4.
This required D(1)
given.
The function D(z)
in addition to the functions
is readily obtained by dif-
ferentiating equation 3.20 with respect to z.
73
- 4
1U
3
10
Mc
2
10
10
1
0-2
10
10-1
Figure
3.1 Critical
Case , A
10,4
1
Marangoni
0
74
10
Number for Insulating
6
10
5
10
4
10
3
10
2
10
10
1
10 - 3
10 - 2
10
-1
1
A
Figure
3.2. Effect
Insulat ing Case,
of
=10
75
A
on
J-.= 0
Mc
for
5
lU
A
e
4
10
3
10
Mc
2
10
10
1
10 -3
_e
10--1
10 -
1
A
Figure 3.3 Effect of Liquid Depth on
Mc for Insulating Case, R =0 ,=0
76
1I
10
10
10
1
10
-,
10- z
10
A
Figure 3.4 Effect of Liquid Depth on
Mc
fo r
C onduct ing
77
Case,
IR =O,
-=
3.3
Solution for Infinite Liquid Depth
A limiting case of considerable importance is that in
which the liquid layer depth is much greater than the thickness of the concentration boundary layer, that is for X
approaching infinity.
For this case, equations 3.7 and 3.8
require that f(z) and its first derivative must vanish as
z approaches infinity.
Taking p to be zero, the solution to
equation 3.4 which satisfies this condition and also
satisfies equation 310
is
f(z)
=
A ze-aZ
(3.29)
Adopting the definition
Q(z)
_
f(z)DO(z)
and taking p to be zero, equation
(D2
a- )g
(3.30)
3.5 becomes
=
Q(z)
(3.31)
A general solution to this equation (compare Hildebrand,
1962, pp. 25-29) can be expressed in the form
z
B5 sinh az + B6 cosh az
g(z) =
e
z
din
a
Q ()
+
e
(3.32)
dn
b
in which
n
is a dummy
variable
of integration
B 5 , and B6 are arbitrary constants.
and a, b,
In order that equation
3.9a or equation 3.9b might be satisfied as X approaches
78
infinity, it is necessary that B 6 = -B5 and also that
b = .
Equation 3.32 then becomes
g(z)
=
Z
-az
2a
B 5 (sinh az - cosh az) -
r
Q(n)
e
dn
a
oo
J
e
2a
Q()
eq
drn
(3.33)
z
From this equation the following relationship is readily
derived:
-
Dg(O) + a g(O)
Ql) e-n
dn
I.
(3.34)
Employing equations 3.11 and 3.12, the grouping on
the left-hand side of equation 3.34 can be related to the
Marangoni number:
Dg(0) + a g(O)
=
(R + aS2 + a)
I =
D f(0)3.35)
ADf(O) . (3.35)
Rearranging yields
M[I + f
(R + a2 S + a)D2f(O)
(3.36)
[I + ADf(0)] a
Using equation 3.29, this equation becomes
- 2A 4 (R+ a2S+ a)
M
=
a
2
(3.37)
(I + A A4 )
and the integral I is given by
co
I
=
- A
nD(n)
O
79
e 2
dn.
(3.38)
It is apparent from equations 3.37 and 3.38 that the value
of the constant A4 has no effect upon the result and can
therefore be conveniently chosen to be -1; equations 3.37
and 3.38 then become
R + aS + 1)
2
M
=
(3.39)
I-
I =
ne-2nD(n)dn
+
A
h(C -Ci)
B
z
'/
. (3.40)
c
0
The second equality in equation 3.40 follows from the
definitions of
and
z; it is understood in the second
integral that the relationship between z' and C is that
corresponding to the unperturbed concentration profile.
For any concentration profile in the unperturbed
state, I can be evaluated by use of equation 3.40;
equation 3.39 then yields the curve of neutral stability.
General solution for
R = 0.
Of particular interest
is the case in which the mass transfer is gas-phase controlled, that is the limit in which
R = 0.
In this case,
the neutral stability curve has the minimum value of M
at a = 0.
This is apparent from equations 3.39 and 3.40
because I is greatest at that point.
Taking a = 0,
equations 3.39 and 3.40 become
Mc
=
80
2
I-A
(3.41)
c
o
3 .
42 )
(CB-C)dz'.(
Ci B
zdC
h(CB
_ Ci)
It is seen that the integrals in equation 3.42 are equal
to the solute deficiency in the liquid phase concentration
boundary layer, corresponding to the shaded area in Figure
This result applies to any unperturbed concentration
2.2c.
profile, provided of course that C approaches CB
suf-
ficiently rapidly at large values of z' so that the
integrals in equation 3.42 will converge.
Until now the quantitative definition of h,
the
characteristic thickness of the concentration boundary
layer in the liquid phase, has been left unspecified
except for the case of the broken-line profile of Figure
Equation 3.42 suggests the general definition of h
2.2b.
which will render the critical Marangoni number independent of the shape of the unperturbed concentration profile
when
R = 0 .
The definition adopted for h is given by
C
h
z'dC
(CB-C ) i
2Ci
Therefore h
co
(C
=
-C)dz'
o
C.
0
1
is seen to be the thickness of a broken-line
profile with the same solute deficiency as the actual
concentration profile in the unperturbed state, as shown
in Figure 2.2d, in which the two shaded sections are of
equal area.
(3.43)
With this definition of h, equation 3.42
yields
81
I
(3.44)
=
and equation 3.41 becomes very simply
M
1
c
2
(3.45)
2
This result is general, independent of the shape of the
unperturbed concentration profile, when
R = 0.
It shows
that the "critical" value of A is 0.5; at this value,
surface convection in the Gibbs layer completely stabilizes
the system, raising the critical Marangoni number to
infinity.
Solution for R > 0.
When
R is greater than zero,
the minimum in the curve of M versus a will generally
not occur at
=
, and Mc will not be independent of
the shape of the unperturbed concentration profile.
Nevertheless, equation 3.43 will be maintained as the
definition of
h.
With this choice,
I = 0.5
when
a = 0,
and I decreases with increasing a, as is apparent from
equation 3.40.
Therefore, as A approaches 0.5, the
critical value of a
decreases toward zero and the critical
Marangoni number approaches infinity.
Thus the critical
value of A is equal to 0.5; this corresponds to an inventory in the Gibbs layer exactly equal to the solute
deficiency in the liquid phase, as is apparent from the
definitions of A and
h.
It is interesting to note that
at this point surface convection completely stabilizes
82
the system, irrespective of the value of R and irrespective of the shape of the unperturbed concentration profile.
For values of A less that 0.5 and for values of R
greater than zero,
Mc will depend upon the shape of the
unperturbed concentration profile.
Perhaps the simplest
profile which might be investigated is the broken-line
profile given by equation 3.13 and shown in Figure 2.2b.
For this profile, equation 3.40 yields
=
I
ne
2
dn
[1
=
[1
-
e2a
(1 + 2a
(3.46)
and the curve of neutral stability becomes
2
M
(R + a
+ 1)
1 - e-2a (1 + 2a)]- A
(3.47)
The minimum in this curve corresponds to the critical
Marangoni number.
Equation 3.47 was used to compute the
solid curves in Figure 3.5, except for the bottom curve
for R = 0, which was computed from equation 3.45.
In order to determine the effect of different shapes
of the unperturbed concentration profile, it is useful to
compare the results for the broken-line profile with those
obtained from the profile corresponding to a liquid phase
penetration theory with a constant gas phase mass transfer
coefficient, as given by Crank (1956, p. 34):
83
1
1
1(
1'
1
C
10 - 3
102-
101
1
10
A
Figure 3.5 Critical Marangoni Number
for Infinite Liquid Depth,A -=0
84
B C*2=
erfc [ [HkG
m2
-exp
HkGiV
( G - erfc
Hk 7
(3.48)
In this equation 8 represents the penetration theory
contact time, and C* is the liquid phase solute concentration in equilibrium with the bulk gas phase partial pressure.
For this profile, equation 3.43 yields the result
h
=
1 - exp
1Hk
exp
-
L
erfc
erfc
(3.49)
G
Employing equations 3.48 and 3.49 to characterize the
unperturbed concentration profile, equations 3.40 and 3.39
can be used to compute the curve of neutral stability and
thus the critical Marangoni number.
The results obtained
(see Chapters 9 and 10),using numerical evaluation of the
integral in equation 3.40, are shown as the dashed curves
in Figure
3.4
3.5.
Description of Results
As in the paper by Brian (1971), it is shown in
Chapter 9 that surface diffusion in the Gibbs layer has a
negligible effect on Mc .
Therefore it will not be considered
further, and all the results to be presented are for
S = 0.
85
Figure 3.1 illustrates the strong effect of Gibbs
adsorption on the problem considered.
Where previously,
in Figure 2.4, there was only one curve for X = 10 (and
A = 0), now there is a whole family of curves for arbitrary
values of A.
Re-plotting these curves as M
c
versus A
results, with additional calculations, in Figure 3.2.
It
is apparent that Mc increases with increasing R and with
increasing A, and indeed Mc approaches infinity as A
approaches a critical value which is independent of R.
Figures
3.3 and 3.4 show the effect of X for the insulating
and conducting cases, respectively, for gas phase controlled
mass transfer, R = 0.
For
X = 1,
these results corres-
pond to those presented by Brian (1971).
Increasing values
of X represent increasingly deep liquid layers, relative
to the mass transfer penetration depth.
At large values
of X, the results for the insulating and conducting cases
approach each other, in agreement with the obvious requirement that conditions at the bottom wall lose significance when the liquid is very deep.
The lowest
curves in Figures 3.3 and 3.4 are for X = a, as given
by equation 3.45.
The effect of the shape of the unperturbed concentration profile has been investigated only for the case of an
infinitely deep liquid layer.
Figure 3.5 compares results
for the penetration theory profile, equation 3.48, with
those for the broken-line profile.
When
R exceeds zero,
the curves are quite close, indicating the low sensitivity
86
of the result to the shape of the profile,
When R = 0,
the curves are identical and are given by equation 3.45.
3.5
Application to Penetration Mass Transfer
In order to judge properly the full importance of
Gibbs adsorption of the STL solute, it is necessary to
determine values of M, A, R, and
for real systems.
in turn requires knowledge of h and C..
1
This
While difficult
to measure, these may be calculated for a time-varying
system, using the well accepted ideas of penetration
theory.
Note that in a desorption system, h and AC = C B -C
i
will both increase with contact time; thus the physical
values of M and R will increase over time and those of A
and X will decrease.
The onset of instability will occur
when M exceeds the value of Mc corresponding to the instantaneous values of R, A, and A.
The application of the present theory to the prediction
of the onset of instability may be illustrated by the example of triethylamine desorption from water, as in the short
wetted-wall column experiments of Mayr (1970). The physical
parameters employed for this purpose in Brian and Ross
(1972) are listed in Table 3.1; they are derived in more
detail in Chapter 12.
87
TABLE 3.1
Triethylamine Desorption from Water at 2980 K
(Brian and Ross, 1972)
1.23x106(dyne) (sq.cm)/g-mole
a=
6 = 0.49x10
0.89x10
=
D
= 0.82x10 5sq.cm/sec
cm
HkG
2.21x10l cm/sec
(dyne)(sec)/sq.cm
CB
= 3x106g-moles/cu.cm
C*
=
0
Equation 3.48 represents the unperturbed concentration
profile for a penetration theory model of the liquid phase
with a constant gas phase resistance, but neglecting solute
storage in the Gibbs adsorption layer.
When storage in the
Gibbs layer is included in the analysis, assuming 6 to be
constant, the result as shown in Chapter 9 is:
Y
_
CB -C
C -C*
=
erfc
eB
LT-T
Jz'
I
exp
+
+
erfc
L-
exp
-
2
+
L6
L6
Jej
JDO2
+
-~~~~~
erfc
[
2/V-
6
erfc
+ F
6
~~~~FvJ
J.
(3.50)
This equation involves several new parameters, including
CB
O
Yi
B
CB
88
-
r/6
(3.51)
C*
in which r0 represents the inventory in the Gibbs adsorption layer at the beginning of the penetration theory
contact interval.
Employing equation 3.43, it is shown in
Chapter 9 that, for this profile, h is given by
2'i
-
2f
+L/J
-HkG
+L/
y]
- - FJ
/F
i
+
2
F
2Jexp
1erfc( 5 . e1
[6
[6
[exp
2
(3.52)
in which Yi is obtained by substituting z' = 0 into the
right-hand side of equation 3.50.
Consider two experimental cases.
In one case r0 = 0,
which corresponds to an absence of solute in the Gibbs
layer at the beginning of the contact interval.
This might
be thought to approximate the entrance conditions in short
wetted-wall column experiments, in which the liquid surface
has been freshly formed at 8 = 0.
r°
=
6C B ,
In the other case,
which corresponds to the Gibbs layer being in
equilibrium with the bulk liquid concentration at the start
of the contact interval.
This might represent an experiment
in which a liquid phase was equilibrated with a gas phase
containing triethylamine, and then at
= 0 the gas phase
was suddenly replaced by fresh gas containing no
triethylamine.
Figure 3.6 shows the experimental paths calculated using
equation 3.50 for these two limiting cases, as represented
89
4
10
M
3
10
10
0.1
1
10
A
Figure 3.6
Desorption ,
Paths for Triethylamine
Experimental
= 0,
90
in a graph of M versus A.
When r
both C i and r vary slowly with
increases with increasing
.
6CBI and
=
R is small,
Therefore as h(CB - Ci)
, M increases and A decreases
but the product of M and A,
or
ac.
2C
A M
(3.53)
X0
D
u RT
X1D
remains approximately constant, and the experimental path
in Figure
3.6 approximates
a straight
line of slope -1.
In
contrast, when r0 and C* are both zero, the initial solute
transfer is largely from the liquid phase to the Gibbs
layer, with little transfer to the gas phase.
Thus r
approximately equals the solute deficiency, and A remains
nearly 0.5 as M increases.
Later, as solute transfer to the
gas phase becomes appreciable, A decreases.
At long times,
the experimental paths for these two cases approach each
other, as shown in Figure 3.6.
When a path like those in Figure 3.6 intersects a
curve of Mc versus A such that the values of M, A,
all match, the point of instability is reached.
, and R
For the
case of an infinite liquid depth, these intersections are
shown in Figure 3.7.
The two experimental paths are the
same as those shown in Figure 3.6.
The solid curves of
Mc vs. A were obtained using the broken-line unperturbed
concentration profile and choosing the values of R which
correspond to the experimental paths at the points of
intersection.
The dashed curves of Mc
C versus A were
91
5
10
4
10
M
l3
2
10
0.45
0.46
0.48
0.47
0.49
0.50
A
Figure 3.7 Onset of Instability
Infinite
Liquid Depth, .=0
92
for
0.51
obtained using the profiles actually prevailing in the
experimental paths at the points of intersection, as given
by equation 3.50 with the physical parameters for triethylamine.
It is seen again that the result is insensitive
to the choice of unperturbed profile.
Figure 3.8 shows the intersection points when the liquid
depth is 0.025 cm, as in the experiments of Mayr (1970).
The experimental paths are the same as those in Figures
3.6 and 3.7, based on equation 3.50; this should be an
adequate approximation even for this finite liquid
depth.
The curves of Mc versus A have been obtained employing
the broken-line profile.
In view of the insensitivity of
M c to profile shape, seen in Figures 3.5 and 3.7, this
should also be a suitable approximation.
It is seen in Figures 3.7 and 3.8 that the result for
= 0.025 cm is close to that for
hand,
has a significant effect.
=
.
On the other
The contact times at
the onset of instability for the two values of r
to differ by factors of 28 and 45.
r0
are seen
This sensitivity to
will introduce considerable uncertainty in predicting
the point of instability in many experimental situations.
For example, when liquid mixing occurs periodically, as
in a packed tower, each mixing may not completely destroy
the Gibbs layer, and the value of r ° will be obscure.
93
10
M
102
0.45 0.46
0.47
0.48
0.49
0.50
A
Figure 3.8 Onset
of Instability for Finite
Depth, Inst1lating Case
94
=0.025 cm .. -=0.
0.51
3.6
Effect of Surface Convection in the Gibbs Layer
For either experimental path, surface convection in
the Gibbs layer has a profound effect on the predicted
point of inception of instability.
convection, the curves of M
be horizontal lines at M
versus A for these cases would
4, which would intersect the
experimental paths at e = 6x10
= 2x10
-10
sec for
f
=
Neglecting surface
0.
-5
sec for
These values
r
o
of
6CB and
are lower
than the ones shown in Figures 3.7 and 3.8 by factors of
four hundred and several million for the respective values
of
°
Without surface convection, the critical Marangoni
number would be equal to 4; with surface convection in the
Gibbs layer, the critical Marangoni number is approximately
4,000 for the cases illustrated.
This thousand-fold increase
in Mc results from the strong surface activity of triethylamine, as measured by its AM product of 2200.
Consider cases with r
varies little with 0.
=
6 CB,
for which the AM product
When the solute is very weakly sur-
face active, the AM product will be small, and the experimental path will be similar to the line for AM = 0.02
shown in Figure 3.5.
sect the curve of M
Assuming R
versus A at M
0, this path will inter4, the result obtained
without surface convection in the Gibbs layer.
On the
other hand, when the solute is strongly surface active,
for example with AM = 200, instability occurs essentially
95
when
A
falls below 0.5, and the critical value of
A
more significant than the critical Marangoni number.
R - 0, the line for AM = 2
is
When
is seen in Figure 3.5 to divide
the region in which surface convection is negligible from
that in which surface convection is dominant.
3.7
Comparison with Experiments
In the short, wetted-wall experiments of Mayr (1970),
the mass transfer contact time was 0.15 seconds.
For a
series of runs with triethylamine solutions, Mayr found
that mass transfer enhancement was measurable when the TEA
bulk concentration exceeded (very approximately) 3x10
g-moles/cu.cm.
6
In other words, the value of CB used in the
example calculation corresponds roughly to the conditions
at the onset of convection in Mayr's experiments.
following the experimental paths of Figure 3.6 to
By
= 0.15
seconds, one determines that Mayr's experimental value of
Mc was 30,000.
This is 8 times the values of Mc predicted
by the intersections in Figures 3.7 and 3.8, but it is
8000 times the value Mc = 4, predicted in the absence of
surface convection along the Gibbs layer.
In might be
noted parenthetically that Brian, et al. (1971) did not
suggest that Mc = 4 applied to Mayr's experiments.
Rather,
maximum values of M and R were calculated and plotted, as
in Figure 2.4.
From that graph, Mc appears to be 17 and
the discrepancy is a factor of 2200.
However, an experimen-
tal path, with A assumed to be zero, would show instability
96
possible
at R = 0 and X =
discrepancy
;
hence M c = 4 and
the
is 8000.
Since the publication of Brian and Ross (1972), in
which the parametric values of Table 3.1 were first
utilized, this author has re-examined Mayr's data and
concluded that several of the values used in Table 3.1
might properly be modified (see also Chapter 12).
A
revised set of figures for TEA is given in Table 3.2.
HkG
(dyne)(cm2 )
g-mole
TEA
10.610 5
Ethyl Ether
4. 49x10
Acetone
0. 809x10
Table 3.2
5
5
cm
cm 2
cm
sec
sec
CB
g-moles
cm
4.28x10 -5
0.82x10-
2.27X10 -3 5.00xlO6
1. 81x10 -5
0.96x10
1.42x10-2 1.61XlO
I5 28x10-4
0.327x10-5 1.i.27x10-5
5. 28xlO
27x10
5.15XlO-0
Desorption from Water at 2980 K.
= 0.894X10-
For all solutes,
C
(dyne) (sec)/cm2
-0
Experimental paths for TEA, corresponding to Table 3.2,
are shown in Figure 3.9, with an intersecting curve of Mc
vs. A.
For simplicity,
the latter curve is for a broken-
line, infinite depth profile.
The change from the
previous results is seen to be modest; for the case of
r°
3
= 0, for instance, the theory-experiment discrepancy
becomes approximately a factor of eleven.
For the case
r
310
= 6C B it is a factor of seven.
97
Figures
11-
a
~0
0.
V:
0)
O
0
r-
4-J
0
-
To
to
o
o z:
O
Int
D~(
- o
98
C
_cS
o
©
0)
a)
-C
0o
4-
3
0)
tn
0
0
0
Ir-
1t0
T-
99
j
C
c:
0o
a)
o
u
-
C0
c3
0
-I,
u)
0
0)LO
4-
a)a
(1)
I
¢I
100
Q) C
~
m~1~r
-H
0 > U..d
II
oo
o
aE
O
co
,H
U
rI4u
rd
-
:H
0
0d 0
r-r--
C)
LO
[r-
C)H
L)
H
O
O
LI
t
C-I
,ci
.P
"d
-,
rd
H
43
0
O
o
0
0
0o
N
m
-I
cr
u
Ir
4-3 I4-4
rd 0
U) H
x
o
0
C)
Hf
U)
H
o
0
U
P
C'q
H
0t
ai)
II
o
(g)
-,
(43
o
0(D)
w
4
C4
0
r-la
0
43
..-r
Ap
43
0uu
E--
101
-
and 3.11 describe the onset of instability in Mayr's experiments
with aqueous solutions of ethyl ether and acetone.
parameters are again provided in Table 3.2.
The
(In the graphs,
the case of rF = 0 is not shown explicitly since, for these
less surface active solutes, it is virtually identical to the
case of r ° =
6 CB.)
For ether the disagreement between
prediction and observation is seen to be a factor of 50, for
acetone a factor of 17.
These results and the previous one
are listed in Table 3.3.
3.8
Conclusions
To summarize this chapter, a completely new influence -
adsorption of a transferring solute - has been examined for
its importance
to the theory of Marangoni
instability.
is confirmed that the effect is profound.
It
Furthermore,
comparisons of the new theory with experiments lead to much
smaller and more uniform differences than previously.
Dis-
crepancies which had ranged from a factor of 140 for acetone
to almost 10,000 for TEA, now are as low as 7 and no higher
than 50.
The resolution of the residual discrepancy requires
additional research; for instance, the importance of surface
active impurities will be considered next.
In any event,
all new treatments of concentration-induced instability
must surely include the ideas presented in this chapter,
concerning the major effects of Gibbs adsorption.
102
-
-
4.
GIBBS ADSORPTION OF SURFACE ACTIVE IMPURITIES
In the previous chapter, Gibbs adsorption of a transferring solute was employed to improve understanding of the
onset of Marangoni instability.
However, a gap remains be-
tween theory and reality, and, in this chapter, a possible
new influence is studied:
Gibbs adsorption of undetected,
surface active contaminants.
As indicated in Chapter 2, the
importance of these materials has often been claimed.
The
theory of Berg and Acrivos (1965) suggests that even very
tiny amounts of "insoluble" surfactants can inhibit the
onset of instability.
Still, well-controlled experiments
have never been done to prove or deny this prediction.
Also,
for studies of desorption in a wetted-wall column, like those
of Mayr (1970).,the Berg-Acrivos model does not apply; the
theory not only presumes that concentration gradients extend throughout the liquid film, it also neglects Gibbs
adsorption of the transferring solute.
In order to study
the theoretical importance of impurities in concentrationdriven convection, and in order to guide experimental tests
of supposed contaminant effects, it is necessary to blend
the procedures and assumptions of the previous chapter with
those of Berg and Acrivos (1965). The theory of Palmer and
Berg (1972), which generalizes the results of Berg and
Acrivos, must also be considered.
The remainder of this
chapter is devoted to those efforts.
103
-
4.1
-
Theoretical Model
Consider a system of finite depth as already encountered
in Figure 2.2b.
A surface tension-lowering (STL) solute is
desorbing such that h/l = X
1 and the unperturbed concen-
tration varies linearly between CB and Ci .
Once again,
buoyancy effects and thermal effects on desorption are ignored,
the gas phase fluid mechanics are not considered, and deformation and viscosity along the liquid surface are
neglected.
However, it is postulated that a surface active
impurity is present.
If this substance were volatile, but
occurred in minute amounts, its influence would add inconsequentially to that of the main solute.
Instead, assume
that the contaminant is non-volatile, so that its unperturbed
bulk concentration is uniform and the substance can in no
way maintain convection.
Nevertheless, through adsorptive
effects, the material may be able to inhibit instability.
As a simplifying, but undoubtedly valid condition, energy
barriers to adsorption are ignored.
(Compare with Section
2.9.)
In the usual manner infinitesimal disturbances are
hypothesized, the one for the impurity being given by
IM
iM
P
CIMP
CI
F(xy)j(z)e pt .
.(4.1)
(4)
iIMP
According to a material balance at marginal stability, analogous to equation 3.5 for the perturbation of the surface
tension-lowering solute, the function j(z) must satisfy
104
-
-
(D2
-
2 )j
0
=
(4.2)
Again in parallel with Chapter 3, either a conducting or
insulating bottom wall requires
j(X)
=
0
(4.3a)
Dj(X)
=
0.
(4.3b)
An interfacial mass balance, similar to that for the STL
solute and also similar to that employed for the case X = 1
by Palmer and Berg (1972), results in
h
-~
Dj(0) =
2 (V2
2
j(s)
/IMP
6IMP
v
IMP\
Df(0) ,
M
where D S IMP is a surface diffusivity and
STL solute.
(4.4)
refers to the
It should be noted that the first term in
equation 4.4 represents the flux of the impurity from the
bulk liquid to the interface.
Berg and Acrivos (1965)
ignored this effect for an effectively insoluble contaminant.
Put another way, their assumption is equivalent to setting
The solution of equation 4.1, sIMPubject
to its boundary
The solution of equation 4.1, subject to its boundary
conditions, is rather straightforward and is performed in
Chapter
9.
Once
j(z) is known,
it may be employed
in a
revised tangential stress balance:
D 2 f(0)
=
a2 Mg(0) + a2NL
105
i (0),
(4.5)
-
-
which embodies the fact that both the STL solute and the
contaminant may influence the solution surface tension.
More exactly, the Marangoni number, M = ah(CB -C
i ) / p D,
describes the strength of the destabilizing forces, while
the Elasticity number NEL =
IMhCi,iMP/psiM
of the inhibiting effect of the impurity.
P
i
a measure
The symbol
is the degree of surface tension decrease caused by a unit
amount of solute.
With the new relation replacing the
single-component stress balance, equation 3.11, the mathematical procedure of the previous chapter may be repeated.
It is shown in Chapter 9 that the curve of marginal
stability
J
_ a _ _ _by
_ _ _ _is_ described
_ ^_ _ svstem
_ _ _ deth
_ _ aI finite
_ _ _ _ __ for
M=
2
sinhaX
+ A
2A2
A2 (SoshX ( + A3
_
NEL
__
aB 2
(4.6)
aB 2
IMP
~~~~~~~~~
~ ~IMP
~ ~~~~~~~~i
a
+
The constants A 2, A 3 , and B 2 have the values assigned by
equations 3.15, 3.16, and 3.24, with A1 = 1 in those relations.
The critical Marangoni number is determined from
equation 4.6 by computing the minimum value of M with
respect to a.
To simplify the computation slightly, it
may be assumed that
S
SIMP
M
IMP
' as justified in Berg
(1972).
Infinite Depth.
For a very deep liquid pool, X = a,
the procedure of Section 3.3 may be combined with equations
4.1 through 4.5.
Considering only the case in which the
106
_ ____ ____
volatile, STL solute has a linear, broken-line profile
and again letting
SIMP
=
DIMP'
it is shown in Chapter
9
that marginal stability exists when
R
(i+
cS++
S + 1)(2 +
M
4.2
=
NEL
+h
.M-- .P
(4.7)
I- A
Basic Results
Figure 4.1 shows results typical of those which are
predicted for infinitely deep systems containing surface
active impurities.
It is evident that when the Elasticity
number is large and when the impurities are insoluble,
h/SIMP = 0, there is a strong stabilizing effect.
By
contrast, the dashed curve at NEL = 10 is representative
of the results for a soluble impurity with h/SIMP = 10;
the stabilizing influence is rather slight.
Inspection of
equations 4.6 and 4.7 shows that, as h/6IMP increases, the
effects of surface elasticity diminish even further.
For a system of finite depth,
in which X = h/
is as
large as 10, results are plotted with NEL = 100 and h/SIMP
= 0 (insoluble surfactant).
The curve is nearly identical
to the corresponding one for infinite depth.
Calculations
justify a similar conclusion for a soluble surfactant,
h/SIMP
> 0.
107
-IqIp-P
~C~""Ca
6
10
5
10
4
10
M
3
10
2
10
10
1
10
10
-22
1
O1
1.0
A
Figure 4. 1 Effect of Elasticity
Linear Profile ,(R =0.10, ? =c
108
Number
4.3
Comparison with Experiments
For mass transfer experiments like those of Mayr
(1970), there are two likely sources of surface active
impurities.
One is the water which makes up the main part
of the feed to the wetted-wall column, and the second is
the surface tension-lowering substance which is mixed
with the water and later desorbed.
Also, as will be
discussed in succeeding paragraphs, two probable types of
contaminant may be distinguished; those for which hIM
P
is nearly zero and those for which it is approximately 10.
Thus, there would appear to be four conceivable combinations of sources of contaminant and type of contaminant.
However, in the remainder of this section all but a single
possibility will be eliminated.
First, consider one class of impurity, the surfactants.
These are very strongly surface active agents, used primarily in detergents, and often found in trace amounts in
domestic water.
Glassman (1948) compiled data on surface
tensions of very dilute, aqueous solutions of twenty-five,
broadly diverse surfactants.
Virtually all his results
suggest that their specific degree of surface tension
decrease
IMP' is 109 (dyne/cm)/(g-mole/cu.cm).
The
penetration depth, h, at the bottom of Mayr's wetted-wall
column was 2x10
cm.
Reasonable values for DM
IMPP and 1I
are respectively 10- 5 sq.cm/sec and 0.9x10
2
dyne-sec/sq.cm.
On this basis, the elasticity number is given by
109
-
NEL
EL
(2 x 10 13)(C
(4.7)
i, IMP
For a typical surfactant, like an alkyl benzene sulfonate
of molecular weight near 300, a surface concentration of
30 parts per billion yields the very substantial value of
NEL = 2000.
Also, recalling that dIMP =a IMP/RT, it is
easily verified that h/6IMP
0.05; in calculating the
critical Marangoni number it becomes apparent that this
is only slightly different than h/M
P
= 0.
In other words,
the Berg-Acrivos model for an insoluble substance applies
equally well to a soluble, very surface active contaminant.
The critical Marangoni number for the case just developed
is plotted in Figure 4.2, along with the experimental path
given in Figure 3.9, for Mayr's work with triethylamine
(TEA). It is assumed that the liquid surface is not freshly
formed at the start of gas-liquid contact; rather the interfacial concentrations of the STL solute and the contaminant
initially equal the bulk values.
Note that this assumption
leads to the strongest impurity effect, but its validity may
have to be reviewed later.
As indicated in Figure 4.2, if
surfactant impurities were present in the triethylamine
solution, such that NEL = 2000, then Mc would be predicted
to be 35,000 and, for all practical purposes, the datatheory discrepancy would be explained.
This is a very in-
teresting result and will be pursued later in this section.
For now, it is useful to consider the three remaining ways
in which impurities might have affected Mayr's results.
110
---C
-
11111(-··1(111
-
0
en
0 E.
I+
Ua)
10,
0
0
~
d]
i11
0
_1
LL c 3
To continue the example of triethylamine solutions,
could the TEA itself have contained a meaningful amount of
detergent-like material?
It is most unlikely.
Because
Mayr's desorption solutions were very dilute, CB
5 x 10
g-moles/cu.cm, in order to account for an Elasticity number
of 2000, it would be necessary to have had 60 parts of surfactant in every million parts of undiluted TEA.
This author
found that even a 1 ppm solution was beyond the solubility
limit of a typical surfactant in TEA.
Moreover, triethyl-
amine is a synthetic chemical, produced over a catalyst at
high temperatures and pressures, and subject to a number of
purification procedures.
and Kirk-Othmer, 1963).
(Compare with Union Carbide, 1970
Significant exposure to materials
dissimilar to itself, like surfactants, is very improbable.
Similar remarks, incidentally, apply to Mayr's other solutes,
acetone and ether.
Having examined surfactants, consider now another
possibly important type of contaminant; namely, moderately
surface active non-volatile substances, similar to the STL
solutes.
Assuming as an initial case that these enter with
the main solute, note that triethylamine was by far Mayr's
least pure additive.
Manufacturer's literature for TEA (Union
Carbide, 1970) suggests that up to 0.1 per cent of the impurities in the undiluted product could be both surface active
and non-volatile.
For these contaminants, a value of
may be assumed which is similar to the values for many
112
IMP
-
organic substances, including Mayr's solutes - 104 to 10
(dynes/cm)/(g-moles/cu.cm).
To be more specific, let cIMP
6
have a maximal value of 5 x 106
Assume that the impurity
constitutes an unusually large 0.5 per cent of the undiluted
If C
TEA.
for TEA is 5 x 10 - 6 g-moles/cm
values of DIM P and
3
, and if the
are the same as previously, it is found
that the Elasticity number is again close to 2000, but
6
h/
iMP
10.
The predicted critical Marangoni number for this example
is shown in Figure 4.2.
Even though
IMP' CiIMP,
and thus
NEL were assigned quite large values, this case is obviously
of minor consequence because of the high value of h/6dIMp
The final combination of contaminant source and contaminant type is that of non-volatile, moderately surface
active impurities in the feed water.
is easily ruled out.
In fact, this possibility
Again assuming a value of
IMP
=
5
10 ,
the feed must contain 2 parts impurity per million parts of
water,
just to reach the levels of NEL = 2000 and h/%IMP = 10.
These values, of course, were seen to be insignificant in the
last case.
More to the point, 2 ppm of organic materials
far exceeds allowable water quality standards in the United
States (see Hann, 1972, for instance).
Indeed, Mayr's
water feed, being once-distilled must have contained far
less.
It seems apparent that of all the conceivable means
for contamination considered, the only plausible one is the
existence in the feed water of roughly 30 parts per billion
113
-
-
of a strong surfactant, such that NEL = 2000.
However, even
this possibility, which impressively explains the measured
Marangoni number in TEA studies, seems contradicted by
Mayr's other results.
For acetone desorption, in Figure
4.3, the Elasticity number need only reach 50 to bring
agreement with experimental results.
The curve for NEL = 2000
is orders of magnitude too high for agreement.
an intermediate
case; as shown in Figure
Ether provides
4.4, NEL = 500 would
suffice to make theory and data match.
4.4
Implications for Experiments
Evidently, even with the new analysis of previous work,
the importance of surfactants in desorption studies remains
cloudy.
Perhaps
the assumption
that r
IMP
= 6C
is at
CB, IMP
fault; but, the opposite assumption that rF = 0 cannot
necessarily be justified.
Additional experiments appear
necessary; for instance, Mayr did not attempt to measure
the level of surfactant impurities in his feed water.
Perhaps
it was far below
30 ppb, or far above it.
Also,
he
did not remove any of the surfactant which may have been
present, an exercise which could definitely decide whether
trace impurities had inhibited convection.
Assume, as an
example, that the feedstock exhibits an Elasticity number
of 2000.
Then presume that in a new experiment the sur-
factant concentration were somehow reduced by at least a
factor of 10 so that NEL would be 200 or less.
If surfactants
were an important influence, the critical Marangoni number
f-"
4-
£D
In
L
cL
Q0
£E
4Z
,
s
> oD
I
a)
LC)
LL_
0o
115
·--
·--
I
¢)
0
U
o_
cn
._
E
)
nC
l
c-L
LLJ
x, .c
25
I
u
-O
0
o~~0
0-
0
°
116
C
LL--
I_ __ ___
would be much reduced.
111
Quantitatively, the theory presented
in this chapter predicts, and Figure 4.2 shows, that the
critical Marangoni number for TEA desorption would go down
substantially, from 40,000 to 12,000.
Of course, total re-
moval of surfactant (NEL = 0) would suggest Mc equal 5500.
As an alternate method of testing surfactant importance,
it has been suggested that considerable contaminant be added
to the feed and any gross effects observed.
If the critical
Marangoni number were obviously raised, one would conclude
that surfactants can stabilize against interfacial convection.
However, no definite conclusion could be drawn as to the
effect of lesser amounts of surfactant in experiments like
Mayr's.
Also, to interpret data taken in the presence of
substantial amounts of surfactant would require new surface
tension measurements; these could be quite difficult to perform accurately, because of the slow surface tension dynamics
which are typical of surfactant solutions.
In order to settle the questions which have been
raised, concerning the role of surfactants in interfacial
convection, this author commenced a new experimental program
which is described in the next two chapters.
First, certain
of Mayr's runs, those for triethylamine, were duplicated.
Then, a quantitative analysis was performed to find the level
of surfactant naturally present in the water feed.
Next,
the contaminant was removed and the initial measurements
repeated.
Finally, a conclusion was drawn, indicating clearly
whether surface active impurities have an effect on the onset
of Marangoni instability.
117
5.
APPARATUS AND PROCEDURE
The desorption experiments performed in this work were
modeled after those of Mayr (1970). Much of the description
which follows and the figures are adapted directly from his
thesis.
In addition, procedures are outlined for the detec-
tion and removal of surfactant impurities, work not attempted
by Mayr.
More detailed descriptions of all of the procedures
in this chapter may be found in Chapter 11.
5.1
The Short Wetted-Wall Column
The central part of the desorption apparatus was a
short, wetted-wall column (SWWC), used for contacting the
liquid and gas streams.
The round column assembly is shown
in cross-section in Figure 5.1.
Liquid flows through a side
arm into the outer section of the assembly.
The liquid com-
pletely fills this part and ultimately spills over the center
tube, forming a laminar film flowing down the inner,
"wetted-wall".
The liquid is collected at the bottom of the
center tube.
Two hollowed teflon rods and two stainless steel cover
plates were the remaining basic parts of the assembly.
The
teflon rods, threaded into the steel plates, provided inlet
and outlet channels for the up-flowing gas stream.
Also,
the upper teflon rod was maintained close to the upper part
of the wetted glass wall, thus creating a narrow "slot" through
118
TEFLON
TEFLON
O-RING
STEEL
CENTERING
SCREW
LEVELIN_
SCREW-_
TABLE
FIGURE 5.1
SHORT WETTED WALL COLUMN ASSEMBLY
119
which all liquid flowed.
The lower teflon rod contained
longitudinal grooves to allow the liquid to flow out to the
drain.
The steel cover plates, fitted with O-rings, cen-
tering and leveling screws, and tie rods, sealed the liquidcontaining unit and held it firmly to a wooden table.
The wetted-wall column used for this author's work had
an inside diameter of 2.633±0.010 cm.
The height (from top
of lower teflon piece to top of inner glass wall) was
4.719±0.010 cm.
5.2
Flow Systems
A schematic diagram of the flow apparatus is shown in
Figure 5.2.
The liquid feed to the column was made up in 15
gallon batches in a stainless steel tank and contained varying
concentrations of triethylamine (TEA) together with the
propylene tracer.
From the tank the liquid was forced
through the lines by a twenty psig nitrogen pressure.
It
flowed, in succession, through a ball valve (for fine control
of the flow rate), a rotameter (for measurement of the flow),
a screw clamp (used to damp out vibration in the liquid
itself) and then to the column.
The liquid leaving the column flowed through a flow
control valve and then to the drain
'"Tee"connections in
the liquid line just before the rotameter and just after the
column permitted sampling of both the feed and outlet
streams.
The latter sampling point was rarely used.
The
liquid temperature was measured in the storage tank, and at
120
4)
a)>
CY - aC:
r-
>1
E
L
L E >
3g )
0
,a E3 0o
C~C
4-i
0
Y
y
c3-(D
o -04
4i
LnE r ) c
L
v a 5--u
>)
-a E0
>
=uo u C
_
)W0 0VII
II
I
II
II
10
0
I
o_
_
v
L
e
0
0
L
Cr
c
4)
0
E
u
Ln
>
LF
zC
4)
L.
:3
U)
121
the column outlet.
The liquid lines were primarily half
inch stainless steel tubing.
As to the gas flow system, pure nitrogen flowed from a
gas cylinder fitted with a pressure regulator (set at ten
psig) through a needle valve for flow control, a rotameter
for flow measurement, and then, via teflon bellows and a
glass calming section, on up through the short wetted wall
column.
The calming section was loosely insulated with
fibrous material to minimize thermal currents in the gas.
A thermometer and a mercury manometer were inserted in the
line just after the rotameter.
The gas leaving the top of the column was discharged
into the hood via a two foot length of teflon bellows and
50 feet of large diameter rubber tubing.
The teflon was
fixed with a baffle to ensure efficient mixing of the gas.
A thermometer and sample line were located just beyond the
teflon bellows.
The gas lines consisted primarily of 1/2
inch brass tubing.
5.3
Operating Procedure
In this thesis the term "run" refers to the process of
obtaining 3-6 gas samples and appropriate liquid samples
from a feed of constant TEA and propylene composition.
During a run only the liquid flow rate was varied.
The operating procedure included the following.
gallons of water were added to the feed tank.
a desired volume of TEA was measured and added.
122
Fifteen
If required,
Propylene
was bubbled through the solution for about 15 minutes.
The
feed tank was then closed, nitrogen pressure applied, and
the feed solution allowed to run through the liquid lines
and column.
Bubbles in the liquid line were removed by
sweeping them out through a vent in the line, which was then
closed.
The column was readied for operation as described
in Chapter 11.
Five to ten minutes were allowed for
equilibration of the liquid phase composition.
The gas flow
was also turned on and allowed the same equilibration period.
The liquid and gas flow rates were adjusted to the
desired value for the first datum point.
The outlet liquid
valve was then adjusted so that the liquid level in the
outlet chamber of the short wetted wall column was all the
way up to the top of the liquid exit "slot".
In this manner,
the exit slot was maintained full of liquid.
Since the
entrance slot was also full of liquid, contact between the
liquid and gas streams occurred only on the wetted wall.
After about 2-3 minutes the outlet gas was sampled.
The
time required for the hold-up volume between the column and
the sampling point to be swept out was perhaps ten seconds.
After the gas sample was collected, the liquid flow rate was
then either changed to a new value or changed and returned
to the same value and the whole procedure repeated.
Samples
of the inlet liquid were collected between gas samples.
Taking 3 gas samples and associated liquid samples
consumed 2-3 gallons of liquid.
In order to conserve the
water feed (especially the filtered water described in 5.6),
123
in TEA desorption experiments the liquid tank was de-pressurized
after 3 gas samples had been taken.
More TEA was added, more
propylene bubbled through, and the tank re-pressurized.
In
this way, as many as 12 gas samples at 4 TEA levels were obtained from a 15 gallon batch of water.
Since only six gas
sample vials were on hand, chromatographic analysis was done
after two groups of three samples had been collected.
Typically 4-5 hours were required to collect and analyze each
set of six samples.
The data readings taken, as gas samples were trapped,
included the liquid temperatures both in the tank and beyond
the column, each of which was noted at the start and end of
the run.
The gas temperatures before and after the column
were similarly noted.
5.4
The barometric pressure was also read.
Sampling Procedure
As stated in the previous section, samples were taken
of the liquid feed entering the column and of the nitrogen
gas exiting from the column.
The position of the sampling
points is indicated in Figure 5.2.
The sampling devices
used are depicted in Figure 5.3 and Figure 5.4.
A known volume of the feed liquid was collected by
opening the stop cocks in the sampling line, and allowing
the liquid to run through the sampling flask shown in Figure
5.3 for about three minutes.
The magnetic stirrer was used
to displace air bubbles adhering to the sides of the flask
124
To Drain
top Cock
ampling Vial
30 cm' Capacity
From Liquid
Feed Line
Rubber Septurn
.Teflon Covered Magnet
Figure 5.3 Liquid Feed Sampling
System
To Hood
Gas
Line
--To
Aspirator
i
From
Col umn
Bubbler
Rubber
Septum
Sampling
Vic
140cm Capc
Figure 5.4 Outlet
Gas Sampling System
125
and was then switched off to promote plug flow through the
flask.
At least twenty times the flask volume was displaced
before the stop cocks were closed and a 30 ml sample of
liquid trapped.
The sampling flask was made from a standard 25 ml
Erlenmeyer flask to which a pair of two millimeter stop
cocks were attached.
The flareout blown it its side, fitted
with a small silicone rubber stopper, served as an injection
septum.
A separate sample was taken in a 500 ml Erlenmeyer
flask for use in the TEA analysis procedure.
Gas samples were collected by mercury displacement.
The gas sampling vial was securely clamped in place and
filled with mercury by raising the leveling bulb.
The
aspirator, running continuously, then sucked outlet gas
from the wetted wall column through the entire sampling line.
This flushing was continued for about two minutes before the
sampling procedure was initiated.
With the aspirator
running, the leveling bulb was lowered, allowing the mercury
to run from the sampling vial back into the bulb.
The
leveling bulb was raised again to flush out the first sample,
lowered, and raised a third time.
allowed to drain out once more.
Finally the mercury was
The lower and the upper
stop cock of the sampling vial were closed, and a sample of
the
outlet gas from the wetted wall column considered
captured.
Each gas sampling vial was made from a standard 125 ml
gas vial to which an additional 2 mm stop cock and a small
126
rubber septum had been added.
The stop cock, which minimized
leakage after the septum was (later) punctured, was kept
closed during the collection and storage of the sample.
5.5
Chemical Analysis for Mass Transfer
The relative propylene concentration of the gas and
liquid samples was determined by injecting known volumes of
each sample into a gas chromatograph.
Peak areas were deter-
mined by a digital integrator; the peak area ratio led
directly to the liquid phase mass transfer coefficient.
It
may be noted that Mayr used a more involved analytical
technique, requiring injection of "standards" into each
sample vial.
This author found his own technique comparable
in accuracy and much more rapid.
Also Mayr measured the
water content of the gas leaving the wetted-wall column; no
such analysis was made in the present work.
Triethylamine, if present, raised the pH of the liquid
feed to about 10-12.
A simple titration with HC1 was used
to determine its concentration.
5.6
Surfactant Analysis and Removal
The amount of surfactant material naturally present in
the experimental water supply was estimated by using the
"methylene blue" test.
The procedure was that given in
Standard Methods (1971). Briefly, a water sample of known
volume with added methylene blue dye, is extracted several
times with chloroform.
The chloroform, diluted to a stan-
dard volume, develops a blue tinge whose intensity is
127
characteristic of the weight of anionic surfactant present
in the original sample.
measured at 652 m
The absorbance of the chloroform is
and the anionic surfactant concentration
determined by comparison with a calibration curve, prepared
using samples of known surfactant level.
(A standard solu-
tion of surfactant, linear alkyl benzene sulfonate, was
obtained through the U.S. Environmental Protection Agency.)
Surfactant material was removed from water to be used
in mass transfer experiments, by adsorption on activated
charcoal.
A 2-inch by 2 foot glass tube, tapered to about
1/4 inch at the bottom, was filled with granular activated
charcoal.
After the charcoal had been washed and settled,
water was passed through, using the water line pressure, at
a rate of roughly 10 gallons/hour.
The filter tube was
emptied and re-filled with fresh charcoal after 15 gallons
of water had been treated.
Each 15 gallon batch of water
was then used in desorption experiments, as described in
5.3.
Samples of filtered water were also subjected to the
methylene blue test.
128
6.
EXPERIMENTAL RESULTS
This chapter describes the results of a series of
experiments, studying the alleged effects of surfactants
on Marangoni instability.
First, to prove the reliability
of the apparatus, certain results of Mayr (1970) were
duplicated; the critical Marangoni number was determined for
instability during desorption of triethylamine (TEA) in a
short wetted-wall column.
This required measurements of the
rate of mass transfer of a "tracer" material, propylene, both
in the absence of and in the presence of TEA.
Next, the
level of surfactant present in the column feed water was
determined, and water was prepared from which this material
had been removed.
Once again, the critical Marangoni number
was measured, and, as a consequence, a conclusion was reached
concerning the importance of surface active impurities on
interfacial instability.
6.1
Propylene Desorption
The first step in this author's experimental program
was to measure the rate of propylene desorption in the short
wetted-wall column apparatus.
An overall, average mass
transfer coefficient KL was determined by analyzing both the
wetted-wall column inlet liquid and outlet gas for propylene
content.
Since propylene description is almost totally liquid
phase-controlled (see Mayr, 1970, p. 168), the value of KL
129
virtually equals that of a liquid-side coefficient,
kL.
Figure 6.1 shows both predicted and measured rates of
transfer of propylene, from water into nitrogen, at a number
of liquid flow rates.
(All experimental measurements are
tabulated in Chapter 13.)
Besides the mass transfer co-
efficient, the plotted parameters are:
h
=
the length of the wetted-wall column (cm).
This is not to be confused with the mass
transfer penetration depth, h.
liquid phase diffusivity of propylene (sq.cm/sec).
E
=
liquid flow rate per unit length of column
perimeter (g/cm-min).
=
the product of the liquid viscosity and density,
evaluated at the liquid temperature, divided by
the same product evaluated at 250 C.
The reasons for grouping the variables as shown in Figure
6.1 are noted by Brian et al. (1971) and relate to corrections for temperature and column dimensions.
The upper
line in Figure 6.1 represents a prediction from penetration
theory, assuming laminar liquid flow.
1970, p. 71.)
(Compare with Mayr,
The lower line represents a least-squares
regression for Mayr's experimental data.
The intermediate
line is a least-squares fit of this author's measurements,
which are individually exhibited as the plotted points.
Equations for the three lines are as follows:
130
10.0
9.0
8.0
7.0
cm
U
6.0
a)
L
E
5.0
-
4.0
U
3.0
2.0
10
20
/Fe
30
(g /cm - min)
Figure 6.1 Rate of Propylene
131
40
Desorption
50
Penetration Theory:
kL
77p
=
0. 3 3
2.035 (/)
3
(6.1)
Mayr:
k v/7pV =
354
1.759(//)
(6.2)
This author:
k=
1. 683 (Z/)0.3705
(6.3)
As is evident from Figure 6.1, this author's regression line
and Mayr's line are very close.
Also, the standard deviation
of the experimental points for the present work is 2.5 percent, compared to Mayr's 3.0 per cent.
Slight differences
between the two investigations are not surprising.
Different
wetted-wall columns were used and the gas phase Reynolds
number for this author's data was 525 while for Mayr's work
it was 510.
Most of the variation,
however,
as well as the
larger gap between the two regression lines and penetration
theory, probably results from regions, of indeterminate
area, at the entrance and exit channels in the wetted-wall
column where non-laminar flow persists.
1970, pp. 194-195.)
In particular,
(Compare with Mayr,
this author found,
in a
series of preliminary runs, that small variations in the
height of liquid at the column exit slot could change mass
transfer rates by several per cent.
Overall, Figure 6.1 clearly indicates that the present
author's procedures were appropriate and adequate.
132
Propylene
desorption rates could be measured over a range of liquid
flows and the results agreed very well with previous work.
6.2
Enhanced Desorption
The second stage of the present work was to measure, as
Mayr did, rates of propylene desorption during the simultaneous desorption of triethylamine.
The objective was to
find at what Marangoni number liquid side mass transfer
enhancement, and presumably instability, was first detectable.
The following procedure was adopted for each run.
First, the
liquid phase propylene and TEA concentrations were determined.
Next, three gas phase samples, which had been collected under
nearly identical conditions, were analyzed for propylene
content and three values of k L were computed.
For each
value, a comparable, predicted coefficient was calculated,
using equation 6.3, and the enhancement ratio
= kL/(kL)eq 6.3 was determined; the average of the three
values of
was taken as representative of the run.
Finally
M, the highest Marangoni number attained in the run was
computed from an experimental path.
empty Gibbs layer,
r
The case of an initially
= 0, was assumed.
Note, to avoid
confusion, that M connotes properties based on triethylamine, while
connotes mass transfer rates of the tracer
component, propylene.
Figure 6.2 is a graph of
runs by this author and by Mayr.
tially similar.
versus M for a number of
The results are substan-
By plotting an arbitrary, smooth curve
133
0
0
0
o
00
0
0O
U)
E
L.
O
O
0
CT
(0
0
O
0
0o
LJ
Ld
s-
L
a)
E
z
C
C
0
O
(9
O
0to
C
u
0-
0
(D
Q
TO
03
O
O
O
O
O
134
O
O
O
0
C
(c
G)
- 6
C
through each set of data points, an approximate critical
Marangoni number is arrived at.
M
c
For this author's work
45,000; for Mayr's observations, M
c
43,000.
It is
wise to note the degree of scatter of the data shown in the
figure; the uncertainty in the extrapolated value of M c is
at least 10 per cent, perhaps slightly more.
Certain qualitative observations made in the present
work may be compared to Mayr's.
First, during this author's
desorption studies with TEA solutions, small, wave-like
disturbances were plainly visible in the liquid flowing
down the wetted-wall column.
In solutions of high TEA
concentration, these "ripples" were more intense than in
solutions of lower concentration.
However, they ceased
abruptly when the gas flow was shut off and mass transfer
was stopped.
Also, no such disturbances were evident when
propylene solutions, devoid of TEA, were studied.
The visual phenomena just mentioned were seen by Mayr
as well as by this author.
thesis.)
(Compare pp. 157-158 of his
Surprisingly, however, the present author did not
see any evidence of one of Mayr's most persistent observations, a stagnant fluid layer at the exit from the wettedwall column.
As discussed earlier in Section 2.6, Mayr
attributed the stagnant band to the presence of surfactant
impurities in his water feedstock; but, this author utilized
exactly the same source of distilled water as his predecessor.
No explanation for the differing results has yet been
forthcoming.
135
Notwithstanding the absence of a stagnant band of fluid,
the present measurements of the critical Marangoni number for
TEA desorption are essentially the same as Mayr's.
With the
procedures thus proved satisfactory, again, the next step
was to analyze quantitatively the level of surfactant in the
feed water.
6.3
Methylene Blue Analysis
Samples of laboratory tap water, as well as distilled
water, were subjected to the methylene blue dye test for
anionic surfactants.
materials.
Both sources were quite low in such
For tap water a concentration of 10 parts per
billion was detected; for distilled water from the source
used for the present desorption experiments, the level was
5 ppb.
An alternate source of distilled water showed less
than 1 ppb of anionic material.
As explained in Section 2.11, cationic and nonionic
surfactants are unlikely to appear in domestic or distilled
water.
Indeed, a series of preliminary, qualitative tests
discovered no such materials.
Section 11.7.)
(Details are provided in
However, these tests may not be capable of
detecting concentrations as low as the parts per billion
range.
Having found small, but potentially significant, amounts
of anionic surfactant in the experimental feed water, the
next task was to remove these impurities.
was employed.
136
Charcoal filtration
6.4
Experiments with Purified Water
Water Purification.
Samples of both tap water and
distilled water showed no measurable anionic surfactant,
after being filtered through activated charcoal.
Of course
it should be understood that there are constraints on the
precision of the spectrophotometer used in the methylene
blue tests.
Surfactant levels below 1 ppb were difficult
to measure, while concentrations below 0.6 ppb were clearly
undetectable.
Still, even assuming the conservative upper
limit on precision, the surfactant concentration in the tap
water was demonstrably reduced by a factor of ten, as a
result of charcoal filtration.
Presumably, the distilled
water sample, with its similar initial amount of contaminant,
was also purified by at least that degree.
In fact, surfac-
tant removal was probably far more complete than measured,
since this author's filtration scheme closely followed that
of Sallee et al. (1956), who achieved a thousand-fold
reduction, from 100 ppb to 0.1 ppb.
As a further precaution in the present procedure, no
more than 15 gallons of water were filtered at a time, at
the slow flow rate of 10 gallons per hour suggested by Sallee
et al. (1956). To ensure that no capacity limitation was
reached, in the ability of the activated charcoal to adsorb
surfactants, the methylene blue test was applied to samples
of water collected both at the beginning and at the end of
the filtration process.
No surfactant was detected in either
137
case.
The methylene blue test was also performed on filtered
water which had flowed through the entire wetted-wall column
apparatus.
Again, no surfactant was found, demonstrating
that the desorption equipment was not itself a source of
contamination.
As noted earlier, the methylene blue dye test is
specific only for anionic materials.
However, these are
by far the most common type of surface active agent; oppositely
charged surfactants, cationics,are rare and could not have
persisted in the feed water in the presence of anionics
(compare with Section 2.11).
In any case, cationic agents
would have been filtered out by the charcoal, along with any
nonionics.
The latter, in fact, adsorb on charcoal even
more strongly than anionic materials.
Davies, et al., 1973.
(See, for instance,
Consider also that Melpolder et al.,
1964 report serious difficulty in removing nonionics from
charcoal onto which they had been adsorbed; this is contrary
to the experience of Sallee et al., 1956 with anionics.)
New mass transfer experiments.
Having satisfactorily
purified the water feedstock for studies of Marangoni
instability, a new series of experiments was commenced.
results are shown in Figure 6.3.
The
It appears that, at most,
removal of surfactants caused a slight decrease in the
critical Marangoni number, from 45,000 to 41,000.
Even this
difference may result more from difficulty in interpreting
scattered data, than from a physically real effect.
at high values of
Still,
, this author's two sets of measurements
138
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00
0
000C
o
o "
0
LL
r
f-
E
O~
0
Z
O
O
0
.0
a,,
o'
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n
L)
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i
-5
U
oo
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)
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.
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0O
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0
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c))
~d
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U
0()
OV
L0
LL
139
seem to define clearly separate plots; by extension, the less
precise data, at lower values of
, probably continue to rep-
resent distinct curves, with differing values of Mc.
6.5
Interpretation of Results
The experiments described in this chapter were performed
to answer the simple question, "Did surfactant impurities
significantly affect previous measurements of critical Marangoni numbers, during triethylamine desorption?"
the answer is no.
Clearly,
To use the simplest terms, in the presence
of surfactants a certain value of Mc was found; without
surfactants virtually the same value was determined.
Hence,
the presence or absence of surfactants was of no consequence.
To take a more quantitative view, the 5 ppb of surfactant found in the feed water imply an Elasticity number of
350, based on the parameters used in Chapter 4.
As discussed
previously, a much higher value, NEL = 2000, is required
to justify the claim that contaminant effects largely explain
For this reason alone, one is inclined to
Mayr's results.
minimize the importance of surfactants.
However, the
parameters, like aIMP and DIMP , used in calculating the
Elasticity number are order-of-magnitude estimates.
a value of NEL
=
Possibly,
2000 can be justified, even with a 5 ppb
surfactant concentration.
In that case, recalling Figure
4.2, a ten-fold decrease of surfactant content, and thus NEL,
would be expected to cause at least a three-fold decrease
in the critical Marangoni number.
140
No such reduction in Mc
occurred.
The obvious conclusion is that other influences,
besides surface active impurities, must be found to explain
the experimental data for instability during triethylamine
desorption.
For Mayr's other solutes as well, comparison with
Figures 4.3 and 4.4, shows that assuming NEL = 350 grossly
overestimates the critical Marangoni number measured for
acetone and underestimates the value for ether.
Clearly,
surfactant effects cannot alone explain these experimental
results.
To generalize further, it may be concluded that, for
typical systems exhibiting concentration-driven Marangoni
instability, an order-of-magnitude discrepancy remains between the predictions of theory and the observations of
experiment.
Contrary to speculation, Gibbs adsorption of
surface active contaminants is not responsible for a major
part of that disparity; continued reliance on elusive "surface impurities," to explain the disagreement between theory
and data, is unwarranted.
Future research should explore
other, more fruitful avenues for enlightenment; some of these
are suggested
in the next chapter.
141
7.
FUTURE WORK
In this chapter, a number of suggestions are discussed
for future work which could possibly complete current
understanding of Marangoni instability.
First, however, the
failure of surface elasticity to add materially to that
understanding is reviewed.
7.1
Theoretical Estimates of Surface Elasticity
In view of the experimental findings presented in this
thesis, one may ask if surfactant effects are properly
treated by available theories.
With a measured water con-
tamination level of 5 parts per billion of surfactant, the
Elasticity number was estimated to be at least 350 and
conceivably as large as 2000.
Such values suggest a strong
elasticity effect on convection during TEA desorption, an
effect not confirmed experimentally.
Therefore, the usual
theoretical treatment of instability with contaminants seems
questioned.
Quite possibly, however, the fault may lie not with the
theory but in its application.
Specifically, it is difficult
to evaluate accurately the individual factors which make up
the Elasticity number, especially the parameters C
and aIMP.
IMP
For instance, until now it has been assumed,
B IMP, as for any aged
i,
IMP = CB,
conservatively, that C i
surface.
A priori, it is impossible to state whether the
142
solution surface in a wetted-wall column is aged or new.
If,
in fact, the surface is formed fresh at the column inlet,
then Ci IMP would be zero at the top of the column and would
increase down the column, according to an experimental path.
Thus, the maximum value of CiIMP
some fraction of C B
that C I
I,IMP
MP
in the column would be
To be precise, it can be calculated
might not exceed 1/40 of CBIMP
,IMP'
or Mayr's apparatus.
for this author's
Then the Elasticity number, due to
feed-water contaminants, would be less than 10, rather than
350 or higher as previously estimated.
Such a value for
NEL would be so low that, applying the theory presented in
Chapter 4, there would be no measurable surfactant influence
for any of Mayr's systems.
In other words, the slow movement
of contaminants, from the bulk of a solution to an initially
clean surface, is consistent with the absence of any surface
elasticity effect.
As an additional complication in estimating the Elasticity
number, aIMP =
example,
if
y/aCIMP is not known very precisely.
IMP = 10 9 dyne-sq.cm/g-mole,
For
as has been as-
sumed, the addition of 10 ppb of surfactant to pure water
would lower the surface tension by only 0.05 per cent.
Even
in careful laboratory work, it is difficult to measure such
a small change exactly.
Worse, for computing NEL to match
desorption experiments in a wetted-wall column, one properly
should use a value of Y/aCIM P evaluated in the presence of
an additional, surface tension-lowering solute.
143
No relevant
measurements of that nature are known to this author.
Indeed, such measurements could be quite formidable;
precision remains a serious problem and questions of surface
tension dynamics, that is aged surfaces versus new surfaces,
may be very hard to resolve.
pp. 37-41, 191-195.)
(Compare with Adamson, 1967,
Nevertheless, successful investigations
into the surface chemistry of solutions of two surface active
agents would improve current understanding of surface
elasticity, and also its effects on Marangoni instability.
7.2
Explanation of Residual Discrepancy
Even though this author's theoretical work has helped
to reduce markedly the disparity between theoretical and
measured critical Marangoni numbers, a significant discrepancy remains.
On occasion, the lack of agreement be-
tween predictions and experiments has been blamed on
deficiencies in the theory of Marangoni convection.
Linearized
stability theory predicts only the conditions under which an
infinitesimally weak disturbance may grow stronger.
That
the disturbance necessarily will intensify, or the rate at
which it will do so, or the size and strength which it will
ultimately attain are not predicted.
It is simply assumed
that when M exceeds the predicted critical value,measurable
convection will commence.
Possibly, linearized theory is
intrinsically inadequate and should be replaced by a general,
non-linear theory, for finite intensity disturbances.
Unfortunately, the mathematics of such an effort are
144
forbidding and few advances have been made in that direction.
(Compare with Section 2.9.)
Alternatively, it has been sug-
gested that convection begins just when M exceeds the predicted Mc, but that the motion is initially so weak that it
has a negligible effect on mass transfer.
This hypothesis
suggests that optical methods would be more appropriate than
mass transfer measurements in determining the point of onset
of instability.
However, it should be recalled from Chapter
2 that some optical studies have been performed in the past
on thermally induced convection; the results have been similar
to concentration-driven work, namely, convection has set in
at Marangoni numbers well above the predicted critical.
Disturbance Wavelength.
One particularly compelling
question about linearized theory, which has as yet received
little attention, is whether it properly predicts the critical
wave number, ac, and the corresponding wavelength, dc = 2h/ac
Assume, as is usually done, that the wavelength of any real,
convective disturbance will equal the value predicted by
linear theory.
For Mayr's experiments with triethylamine,
the conditions at the onset of instability were illustrated
in Figures 3.7-3.9.
Specifically focussing on a system of
finite depth, in Figure 3.8, it is predicted that instability
will set in at Mc
X =
/h = 101.2.
is 0.014.
3600 for a liquid film in which
The predicted
critical
wave number,
ac,
This low value is typical of the present author's
work which considers Gibbs adsorption, but it is unusually
145
small compared to values encountered in other studies of
Marangoni convection.
dc
c
Moreover, if
c = 0.014, then
c
450h; that is, the characteristic surface dimension of
each convection cell is predicted to be 450 times the penetration depth.
In geometric terms, this might lead to
skepticism concerning the assumption of infinite surface
expanse postulated for all the systems treated in this work
theoretically.
However, the dimensions of this author's
wetted-wall contactor are such that d
c
= 0.125 cm is only one
or two per cent of the length or perimeter of the wetted
area.
A more unsettling result appears if one argues that,
intuitively, a convective cell ought to be symmetric; its
width and depth should be similar.
depth must also be equal to d
ness of the liquid film is
In that case, the cell
= 450h; but the total thick-
= Xh = lOlh.
Thus, each con-
vective cell would have to be 41 times deeper than the liquid
phase in which it resides, an obvious contradiction.
If considering very small values of a may lead to
problematical situations, consider a contrasting case.
Bakker et al. (1966) and Brian et al. (1971) argue on
physical grounds, quite apart from linearized theory, that
the disturbance cell dimensions should be of the same order
as the penetration
depth,
that is d
= O(h).
Conceivably,
this idea should be adopted as a constraint in the linearized
stability theory.
That is, dc would not be allowed to
exceed (say) 5h, or ac = 2h/dc
146
would not be allowed below
1 or 2.
If the minimum Marangoni number would otherwise
correspond to a lower value of ac, this constraint would
have the effect of raising Mc .
(Compare with Figure 2.3,
for instance.)
For a modest number of cases, the critical Marangoni
number has been computed, subject to a limit on the maximum
value of the wavelength and a minimum value of the wave
number.
Representative results are plotted in Figure 7.1.
(The curves in the graph are for a system of infinite depth;
however, results for deep, finite depth films, X > 10, are
nearly identical.)
It is apparent that, theoretically, a
liquid layer in which ac must exceed 2.5, for instance, is
far more stable than a system in which the wave number is
unconstrained.
To test the wave number restriction more
exactly, curves of Mc versus A, constrained to ac > 2.5,
are plotted in Figures 7.2-7.4 along with experimental paths
for each of Mayr's systems.
Agreement between predictions
and experiments is reasonably good; the discrepancy for
triethylamine and acetone desorption is less than a factor
of two, while for ether it is a factor of six.
7.3
Conclusion
The agreement reached between theory and experiment,
as a result of restricting the value of the wave number, is
quite interesting.
Nevertheless, it is hard to accept such
an arbitrary constraint without more justification.
Possibly,
after some modification of the physical model of Marangoni
147
5
10
10
2
10
10
1.0
13
i
2
1
Figure 7.1 Effect of Wave Number Constraint.
Infinite Depth,,Broken -Line Profile R0.10,
NEL -0
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instability, linearized theory might necessarily require
larger values of a.
Also, optical studies could provide
experimental evidence, presently lacking, concerning the
proper value of the wave number.
Mathematical advances
in non-linear stability theory could determine the appropriate wave number for finite intensity convection, and the
corresponding value for use in a linearized approximation.
In any case, future investigations of concentrationdriven Marangoni instability will be able to stand on the
two most important findings determined by this author.
First, it has been established that Gibbs adsorption of the
convection-inducing solute has a major effect, which must
be considered in theories of surface tension-driven
instability.
Second, it has been found that adsorption of
trace impurities is by contrast a largely exaggerated
concern, which may safely be ignored in typical experimental
situations.
152
8.
8.1
CONCLUSIONS AND RECOMMENDATIONS
Conclusions
1.
A linearized stability analysis has been performed
for arbitrarily deep liquid systems susceptible to Marangoni
instability.
In particular, Gibbs adsorption effects of a
transferring solute have been studied and are found to have
a strong ability to inhibit the onset of convection.
2.
If Gibbs adsorption is ignored, theoretically
predicted values of the critical Marangoni number are two to
four orders of magnitude higher than experimental values.
With Gibbs adsorption included in the new theory, the discrepancy
3.
is reduced
to as little as a factor of seven.
For deep systems undergoing penetration mass transfer,
certain approximations are adequate in the new stability
analysis.
The concentration profile can be considered frozen
in time and linear, and the liquid depth may be considered
infinite.
4.
The stability theory has been extended to include
adsorption effects of surface active contaminants.
These
effects may be significant, especially if the Elasticity
number is large and the ratio h/BIMP is small for a given
physical situation.
5.
However, for typical experiments, like those of
Mayr, impurities were not an important influence.
New
measurements of Mc , made in the absence of contaminants, do
153
not differ significantly from earlier measurements in the
presence of impurities.
6.
The remaining data-theory disagreement can largely
be eliminated if, in linearized theory, the convection cell
wavelength
is assumed
to be similar in magnitude
to the
penetration depth.
8.2
Recommendations
1.
Commercial applications of Marangoni convection may
be pursued, with reasonable confidence in linearized theory,
and without fear of complications in real systems because of
trace impurities.
2.
Optical studies of Marangoni instability should be
started, both to determine convection cell wavelengths and
to determine if convection is visible before mass transfer
enhancement is measurable.
3.
New theoretical work should be commenced, possibly
with fewer physical simplifications than employed in the
present work.
Restrictions on the value of the cell wavelength
and wave number should be evaluated.
4.
A non-linear stability analysis should be attempted
in a further attempt to specify proper values of the wavelength and wave number.
154
APPENDIX
9.
DETAILS OF THEORETICAL DERIVATIONS
This chapter contains a number of detailed derivations
of theoretical results presented earlier in this thesis.
In
the first section, the linearized equations controlling
Marangoni instability are presented.
9.1
Linearized Equations for a Perturbed System
Basic relations.
The system is shown in Figure 2.2a,
with infinite lateral dimensions; assume that there are no
effects of temperature, gravity, surface viscosity, or surface deformation.
an initial,
Assume only one solute is present.
unperturbed
velocity
U = U (z)i exist.
Let
Assuming
that the liquid density and viscosity are not affected by
changes in temperature, pressure, or solute concentration,
the basic conservation equations are
Continuity:
V
U
=
0
(9.1)
aU
p -
Momentum:
at
+ pU · VU
=
-VP + pV 2 U + pX
aC
Solute Mass:
-
at
(9.2)
2
+ U
VC
155
=
V C.
(9.3)
In the above, underlined symbols are vectors and, notwithstanding the nomenclature of Chapter 3, all symbols are
dimensioned.
In equation 9.2, P is the pressure and X is
the vector sum of external body forces, like gravity; it
will be assumed henceforth that X = 0.
Derivations of the
conservation equations and sample applications are given by
Bird et al.
(1960, especially,
pp. 75, 80, 557).
Perturbations are assumed to be given by
where
U
=
U(z) + u(x,y,z,t)
(9.4)
C
=
u(Ze,) + c(x,y,z,t)
(9.5)
is the contact time parameter.
conservation law in turn.
Now, examine each
By continuity, in a disturbed
state, aUu/ax = 0 and
au
av
+ -- +
ax
ay
aw
.- =
0
(9.6)
az
where u, v, and w are the x, y, and z components respectively
of u.
(Again, all symbols here have dimensions.)
Considering next the Navier - Stokes equations, 9.2,
apply the curl operation to arrive at
p
avxU
at
+ pV
x
[U
-
VU]
-
=
-V
x VP +
2
V x V U.
To simplify subsequent work, note two identities
156
(9.7)
V x V
V x
(V
x
V)
=
=
0
(9.8)
YV) -
V(V
Y
V
(9.9)
where B is any scalar and Y is any vector.
(These identities
may be compared with Brodkey, 1967, p. 24.)
the pressure
be used to eliminate
Equation 9.8 may
9.7.
term in equation
To
eliminate the vorticity, V x U, in equation 9.7 apply the curl
again:
a [VxVxU]
p
x V x
+ p
at
=
[U. VU]
lV
2
x V X [V U.
(9.10)
Dividing through by the density and using equation 9.9, this
relation becomes
a [V (V U) -V 2U
+ V(V. [U-VU]) -
-[V(V.V
2
U)
2
-
V 2 [U. VU]
=
2
V U].
(9.11)
P
Now, to minimize complexity, determine only the zcomponent of each term of 9.11.
2
V U
V
*
2
VU
=
-
First note that,
V (U +u)i
u
-
+
2(
V2vj
_
+
2
V 2w
_
(9.12)
as on page 42 of Brodkey (1967); page 16 of Brodkey may be
consulted for a definition of the tensor VU.
With the aid
of equations 9.6 and 9.12, the z-component of the first term
in 9.11 becomes
-
at
Vw.
157
(9.13)
To evaluate the second term in equation 9.11, observe
that
U
VU
=
i
-
u
Du
au
-+ w u + j
z x
au
+ i
-
av+
u
au-
aw
-+
ax
-
W
X-x
u
av
Dx
aw
-
av
v -
-+
y
-
k
av
+ v -+
9x
-_+
+ w
9y
Dz
awl
L- 9y 9z
y
+
w a+
(9.14)
All of the terms inside the braces in equation 9.14 are
second-order; that is, they involve the product of two small
perturbations, and will hereafter by neglected.
V. L>VUI
U
=
U
au
+
Then
a v
aw
u _+
-- xu axza
+
ux
3x
au
++
aLw
ay
y
aw
(9.15)
+ U
+
u
U axaz
az ax
Rearranging yields
9w
Vo [Uvu]
=
U
--
-
u 3ax
9x
+ 2
ayy
Dz
ua
aw
u
9z
,
(9.16)
Dx
The parenthesized term in equation 9.16 is zero, by
continuity.
The z-component of the gradient of
equation
9.16 is
a
r
aUu aW1
{V(V-[U-VU]) z
az
=
2
az
ax
rau
-ax
z
w
158
92U
+
3z
u w
Dz 2
(9.17)
Equation 9.17 provides the z-component of the second term
of equation 9.11.
The third term in 9.11 is found by applying the
Laplacian
operator,
v 2 (u.vu)
I·C z
as in equation
9.12, to equation
9.14.
V2 (U VU)
=
a
a
-w+2
+V2w
+
ax az
U ax
-
au
/a2u
Considering the fourth
aw\
z
term in equation
a
(9.18)
a
9.11, it is
seen that
V · V2U
=
V2U +
ay
ax
V2v + V2w.
az
(9.19)
Then
a
=
V2U)}z
{V(V
2
V 2U)
-(V
=
0.
(9.20)
az
This last result is obtained by expanding equation 9.19
and the first part of 9.20, collecting (factored) sums of
au/ax, av/ay, and aw/az, and applying continuity, equation
9.6.
Finally, to end the evaluation of equation 9.11, observe
that
{V2V2U}
- z
=
V2{V2U}
-
z
=
V 2 V 2 {U}
-
z
=
V2V2w.
(9.21)
This follows from repeated application of the definition of
{V2U}z, as implied by equation 9.12.
Collecting the results of equations 9.11, 9.13, 9.17,
159
9.18, 9.20, and 9.21, it is found that
a
-
t
2
V2w +
a2U
aw
a
u Uu
-
2 Dx
x
V2w =
(
_ _ v2(V2w)
(9.22)
px
or, re-arranging
[- + U
+
U a
vvw =
P 1
aw
Dx
2U
u
(9.23)
az
Equation 9.23 is the desired result, a linearized momentum
balance in which the pressure and vorticity do not appear.
Consider again the solute mass balance, equation 9.3:
ac
-- +
2
· VC
at
=
V C.
In the absence of a perturbation,
C = C (z,e), the middle
(9.3)
with U = U (z)i and
term in 9.3 is zero.
-U
VC u
=
(9.24)
0
Or re-writing equation 9.3:
u
=
2C
"I
In the presence
of perturbations
-c
of the form of equations
9.4 and 9.5, equation 9.3 becomes
160
.
(9.25)
a(Cu+
U
at
ac\
ac
+ U u (ax
W
ac
az
ac
ac
2
U
v - ++ w
ay
az
ax
=
V (Cu+c)
(9.26)
Ignoring the second-order terms, in braces in equation
9.26, and subtracting equation 9.25 from 9.26, one finds
+ U at
D V)
=
U ax
-w ac
az
(9.27)
This result is the second main governing equation.
It is apparent that the solution of equations 9.23
and 9.27 requires four boundary conditions on w and two
on c.
First considering the bottom wall, the "no-slip"
condition requires
w(z = l)
=
0
(9.28)
Similarly, u and v must vanish at the wall for all x and
y, so that au/ax = av/ay = 0 at z =
.
From continuity,
then,
a w(t)
=
0
(9.29)
az
Note that, by the usual convention, 9.29 implies first
differentiating w, then evaluating the result at z =
1.
As discussed in Chapters 2 and 3,
c(1)
161
=
0
(9.30a)
at a conducting wall, and
ac (M)
=
z
0
(9.30b)
at an insulating wall.
At the free surface, it is required that
w(0)
if the interface
=
0
is not to be vertically
(9.31)
deformed.
Two
more conditions, a stress balance and a solute material
balance, are available.
These deserve special attention
since, without doubt, they are the heart of the "physics"
in the description of Marangoni instability.
Considering
first the interfacial material balance, a general formulation may be written, based on the works of Levich and
Krylov (1969, p. 296), Palmer and Berg (1972), and Brian
(1971).
(Take note of differing sign conventions among
other authors and the present work.)
21D+
U)=r = DsV2+Vc--- + Vs(U)
+ DVC- HkG(C- C*)
(.22
(9.32)
3t
The symbols V
directions.
2
s
and V
s imply evaluation only in the x and y
Obviously, equation 9.32 applies at z = 0; the
several terms represent, in order, accumulation, convection,
and diffusion in the Gibbs layer and transfer from bulk
liquid to the surface and from the surface to the gas
phase.
In the unperturbed case, with no adsorption
162
barriers, r =
= 6C (0) and it is found that
u
u
ac (0)
6u
at
ac (0)
=
u
- HkG[Cu(O) - C*1.
(9.33)
az
In the presence of perturbations, r = 6[Cu(0) + c(0)] and
6
ac
ac
ac
at
ax
u + 6 - +6u
at
ac
+ 6u
+
ac
v
ay
au
av
6C
+
u ax
-+
u ax
2o
2c
[ap2o
a2c
-LaxJ+ a
Cu2
a(c +c)
az
+
- HkG(Cu + c - C*)
where evaluation at z = 0 is assumed.
(9.34)
Subtracting equation
- -++--c
9.33 from 9.34, employing continuity, and dropping the
second order terms, in braces in 9.34, results in
ac
6 -
at
ac
+ 6u
-
u ax
aw
-
f2c
=
6
u az
a2c
ac
ay
az
Hk
ax2
(9.35)
Equation 9.35 evaluated at z = 0 is the desired solute
balance boundary condition.
The stress boundary condition is somewhat complicated.
Scriven (1960) and Scriven and Sternling (1964) provide very
general forms for a flat interface of negligible mass;
these may be written
163
-X
=
V
S
S_
s)V (V .U)
s +
+ (K +
S-
x V(k.VxU).
(9.36)
SS
In 9.36, X is the net force on the surface,
y is the surface
tension, K is the surface dilational viscosity, and
the surface shear viscosity.
S is
Evaluation of equation 9.36
is assumed to be at z=0; in fact such evaluation
is assumed
for the remainder of the equations in this paragraph.
Now,
again drawing on Scriven (1960) and Scriven and Sternling
(1964), the surface force and rate of strain are related
by
=
X
-Pk +
+ (VU)T ]
k_[V
T
is the transpose of the
where P is the pressure and (VU)
matrix VU.
(9.37)
Expanding 9.37 for the perturbed case results
in
X = -P + I/aw -au + au +
ax
az
az
aw
av\
+-)+
k
az/
\ay
-
aw\
az/j
(9.38)
To eliminate the pressure term in equation 9.38, take the
surface divergence and rearrange, finding
a au
VS X
av
az \ax
u
+
+
a2 w
au\
+ -
ax
ax
ay
a2w
.
2 +
(9.39)
y
But au /ax = 0; using continuity and the definition of V s,
it is seen that
Vs x=
(2
w -
164
32w
(9.40)
The negative of this result will be equated to the surface
divergence of equation 9.36, to be evaluated now.
-v S
=
(K + ps)V (VS U)
V2s +
+ 1sVs [xvs (Ek U) ]
(9.41)
To simplify the third term in equation 9.41, observe that
in the presence of a perturbation and employing continuity
-
VS*U
+
-
---
(9.42)
For the final term in equation 9.41, observe that
k
· V x
=
U
(9.43)
[V x U]
Dx
a av
Vs- (k
V X u)
-
-ax _
=
au\
/av
_ a
+
a_ _y - i
ay ax
ay
VxU) : - -' a av
x V s (k
ay
ay \ax
au\
a
a
(9.44)
v
a
, 1(9.45)
Applying the surface divergence to equation 9.45, it is
found that
[k x V(
vs
·
k
V x U)]
=
O
(9.46)
Combining equations 9.41, 9.45, and 9.46
-Vs
X
=
2
VsY 165
2 aw
K + US)VS
(9.47)
Now the dilational shear viscosity,
K,
may be ignored (see
Berg, 1972, for instance) or it may be assumed that the
symbol
S actually embodies (K + )pS .
Then combining
equation 9.47 with equation 9.40 yields
-V W.
Vy =P- - +
az
s
=- -
(9.48)
'
s
If y is a function only of solute concentration, then a
power series expansion for y implies
ay
¥u
+
a2r
(C -
+
C )
2
(C -
C)
+.
(9.49)
Noting that c - C - Cu and dropping non-linear terms
ay
=
Yu
+
C
-
(9.50)
8c
Letting a _
y, and noting that a is a physical property
and not a function of x or y, equation 9.48 may be rewritten
as
2
2 \ 2
~2/
-a V s c+
8w
P VS
Generally, surface viscosity
(9.51)
may be ignored (compare with
Chapters 2, 3, and 4, and Berg and Acrivos, 1965), so that
the final, linearized stress balance is
2
a VS
=
V
2
a2w
w-
2'
(9.52)
az
The stability problem is now well formulated; a solution is
required for equations 9.23 and 9.27, subject to the
166
conditions 9.28, 9.29, and 9.30 at z =
9.31, 9.35, and 9.52 at z = 0.
be made dimensionless.
and the conditions
First the equations
should
Imagine that all the dimensioned
variables and differential operators, used in this chapter,
are marked with a "prime," as was the convention in Chapter
3.
Then let new, dimensionless, un-primed variables be
given by
x, y, z =
(x',
y',
z')/h
w
=
w'h/D
Uu
=
(Uu )' h/D
t
=
t'D/h 2
c
=
c'/(CB - C i)
(
=
(C'u - Ci)/(CB -C i)
(9.53)
where h is the characteristic length, defined in Chapter 3,
and CB and Ci are dimensioned and evaluated for the unperturbed case.
With this transformation the governing
equations become
a
a
at
a
aw
2
+ U -- -
V
xD
Vw
u
2
ao
(- + U- - V )c = -wat
(9.54)
az 2
pax
a
U
=
az
Ux
167
(9.55)
0
=
w(X)
(9.56)
aw(X)
= 0
(9.57)
az
c(A)
=
0
(9.58a)
=
0
(9.58b)
0
(9.59)
ac(X)
az
w(0)
6
ac
-
az
=
R
+-
C
h at
2
Sw
=
- S
$-
-
+
a2w
-2-
6
a2c
_2c
=
M
2
h
ac
U
-
U ax
aw
- A
(9.60)
az
c.
(9.61)
(Equations 9.60 and 9.61 are, of course, to be evaluated at
z = 0.)
The dimensionless groups
, R, S, A, and Mwere
all
defined in Chapters 2 and 3.
To summarize the derivation presented to this point,
note that for the case of an initially quiescent system,
U u = 0, equations 9.54 through 9.61 are identical to the
governing equations presented in Table 2.1.
(To be more
exact, replace equation 2.4 with 2.4a and note that
Sc =
/pD.)
That is, the basic results of Chapter 2 have
been derived.
Solution.
Solving the set of euqations 9.54-9.61
requires that a functional form of solution be assumed, as
168
for any partial differential equations.
Separation of
variables may be used; Chandrasekhar (1961, pp.3-5, 43-44)
discusses the reasons for assuming the particular forms
w
=
f(z)F(x,y)eP t
c
=
g(z)F(x,y)e
P
(9.62)
(9.63)
,
where f, g, and F are dimensionless functions of position,
and F is given by the real part of
F
=
exp(ia Xx) x exp
(ia yy).
(9.64)
The last equation is justified by noting that the physical
system is infinite in the x and y directions; equation 9.64
ensures that the corresponding mathematical solution is
"well behaved" at large values of x and y.
Furthermore,
consistent with experimental observations, F connotes a
repeating, cellular disturbance since
=
Re [exp (iXx)]
sin(
x'.
(9.65)
Now, values of the sine function repeat for values of the
argument which are multiples of 2.
A wavelength, d,
may
be defined to be the minimum distance over which the function in equation 9.65 repeats itself.
x = 0 the argument
At x' = dx, by definition
is zero.
d
h
That is, at some
=
x
169
2r
(9.66)
or re-writing
2Th
ax=
d
X
(9.67)
d
Similarly, define dy by
y
=
(9.68)
d
y
=
(9.69)
-_2.
With this result it is easily verified that
2
2
It may be noted that a is interpreted as the wave number for
a disturbance whose characteristic lengths, in perpendicular,
planar directions, are d
and d.
With the assumed solutions, 9.62, 9.63,and 9.64, and
using 9.70, each of the governing equations, 9.54-9.61 may
be transformed.
[p + Ua
ux
The results are
- Sc(D
[p + U a
- a2)][D2 _ a2]f
-
(D
a
-
)g
=
a ifDU
x
-fDD
u
(9.71)
(9.72)
f(X)
=
0
(9.73)
Df(X)
=
0
(9.74)
170
Dg(0)
=
g(A)
=
0
(9.75a)
Dg(k)
=
0
(9.75b)
f(0)
=
0
(9.76)
~6~2
Rg(0) + - pg(0) + a Sg(0)
h
+ - U axig(0)
a
(Note that D
2
ADf(0)
-
=
=
f(0) + D f(0)
(9.77)
2
aMg(0).
(9.78)
d/dz; also the first term in the final equa-
tion vanishes, by virtue of 9.76.)
Once again, if U
= 0,
for a quiescent system, the above equations reduce to a
previously encountered set, equations 3.4 to 3.12, which were
used as the basis of the development in Chapter 3.
Thus the
controlling equations in both Chapters 2 and 3 have been justified.
In remains, however, to discuss the effects of a
finite velocity, a problem initially raised in Section 2.9.
Effect of Finite Velocity.
Examination of equations
9.71, 9.72, and 9.77 shows that these are the only relations
in the governing set which are affected by the existence of
a finite
initial vleocity,
functional dependence of
U = U (z)i.
Observe
that the
u is quite general for a prevailing,
one-direction velocity; if U
were a function of x or y, then
171
Cu could be also, contrary to hypothesis.
In addition, the
direction of U is totally arbitrary; after one knows the
direction of U, the x-direction of the coordinate system is
intentionally chosen to coincide with it.
Looking closely at 9.71, 9.72, and 9.77, it is apparent
that if Uu = 0 or if a=
0, the equations reduce to the
counterpart forms of Chapter 3.
In other words, a quiescent
system is controlled by the same equations as a flowing
one for which ax =0.
seriously restrictive.
But limiting ax to zero is not
In fact, based on studies of buoyancy-
driven instability, allowing ax and U
both to be non-zero
will have a stabilizing effect; the lowest, critical
Marangoni number, for instability in the presence of a mean
velocity, is expected when a
x
is zero.
Therefore, as
stated in Section 2.9, in the presence of a finite flow it
is anticipated that perturbations will appear as long rolls
(ax =
and
=
), these being governed by a stability
analysis identical to that for a quiescent system.
9.2
Effect of a Surface Active Impurity
The purpose of this section is to justify the theoretical
treatment of Chapter 4, for instability in the presence of
a surface tension-lowering, non-volatile impurity.
Modified Equations.
Reviewing the development of the
previous chapter, no modifications are required up to the
point of the interfacial stress balance.
With two solutes
affecting the surface tension, equation 9.49 becomes
172
ay
Yv(c', cM)
=
+u -
(C' - C'
9y
+
((CM
P C u)
IMCp
IMP
neglecting non-linear terms.
r
=
Yu -
Thus
IMP C'
c' -
must replace equation 9.50.
(9.79)
INMP,U
(9.80)
However, it is additionally
necessary to determine c'IMP as a function of time and position.
Digressing to do just that, observe that the im-
purity must satisfy mass balance equations analogous to
equations 9.27, 9.30 and 9.35.
Letting cIM P = CMP/C
IMP
and non-dimensionalizing, it is found that
+ Uu -
- V
IMP
c M(X)
=
0
(9.81)
0
(9.82a)
IMP
0
(9.82b)
3z
IMP
D
cIMP
9z
+
IMP
h
IMP
IMP
at
6IMP
-
h
u
U
SIY P
h D
cIMP
u
173
ax
IIu
h
2
s
z tIMP
w
az
(9.83)
Note that 9.83 is evaluated at z = 0 and that both 9.81 and
= 0 and RIM
9.83 are simplified since DC MpU/Z
a non-volatile solute.
CIMP
= 0 for
Assuming a perturbation of the form
=
j(z) F(x,y) ePt
(9.84)
equations 9.81 and 9.82 may be transformed.
The results,
letting ax = 0 and anticipating the marginal stability
requirement
p = 0, are
(D2 _
Dj(0)
=
2
IP
2)j
=
0
(9.85)
j(X)
=
0
(9.86a)
Dj(X)
=
0
(9.86b)
IMP
h
6
(0)
()
IMP
IMP
IMP
h DIM
D
Df(O)
(9.87)
P
This result was presented in Section 4.1.
With a solution for j(z) available from equations 9.85
to 9.87, one can return to modify the stress balance, equation 9.52.
In terms of c and w the result is
2
aIMPhCi IMP
2
M Vsc(0) +
IMP
c
sD
vIMP
up s
174
(0)
=
V2W()
s
a w(O)
-(9.88)
2 (988)
at~~~
With the aid of equations 9.84 and 9.87 this becomes
2
2
D f(0)
This equation
=
DIMP
2
a M g(0) + a NEL
j(0).
(9.89)
was also presented in Chapter 4.
Solution.
To determine the conditions for instability,
one must now solve equations 9.71 to 9.77, 9.81 to 9.83, and
9.89.
After finding j(z), the procedure is very similar to
that of Chapter 3.
Considering first a finite depth system,
from 9.85 and 9.86,
coshaxz
E 1[sinhaz
=
j
where E 1 is an arbitrary constant.
(9.90)
But, using 9.87 and
3.14 implies
E
=
a A1 + A 3
1
-
cxaE
2
·
(9.91)
3\coshaX/
where E 2 = hD MP/(6D) and E 3 = D S
.
Equations 9.90
and 9.91 provide a solution for j(z).
Substituting into
equation 9.89 it is found that
2A2
M
=
aB2
/sinhax\_
v
\coshaX) (aA1 + A 3 )
0B2
NEL
.... cos/(9.92)
+
DS imp.sinhaxX\'I
/
h
_---
+
Tar
-T-...
I Vi
175
\coshakI
'
'
lvik
'
With A1 = 1, this is the result given originally as
equation 4.6.
For a system of infinite depth, X =
,
for j(z) is
j(z)
A4
=
a~
-
hV IMP
S
2 DS,IMP
a
and a solution
_~~~~~~
_
+
_
IMP
e
-az
.
(9
(9.93)
Following the procedure of Chapter 3, starting just after
equation 3.34, and replacing equation 3.11 by 9.89 and 9.93,
yields the marginally stable Marangoni number as previously
given in equation 4.7.
This completes the delineation of
results for systems with impurities.
9.3
Derivation of Gibbs Profile
In this section, a derivation is presented for an un-
perturbed concentration profile which includes storage in
the Gibbs layer.
in equation
The final result was earlier presented
3.50.
The basic equation to be solved is
3
D
C
=
2
subject to r =
(9.94)
(9.94)
=
2
ae
C i and
C'
=
CB
176
at e = 0, z' > 0
(9.95)
C'
=
r°/6
at e = o,
C'
=
CB
at
ac
acC
-
HkG(C*- C')
+ 6-
ae
az'
z'
= ,
z'
=
0
(9.96)
> 0
(9.97)
at z' = 0, e8>0
(9.98)
Adopting the definitions
Y
C B -C'
=
CB -C*
z'
z
=
Z
(9.99)
D8
62
one finds that
(9.100)
atz 2
Y
=
at t = 0, z >0
0
C - r°/6
1
Y -1
(9.101)
at t = 0,
z=0
(9.102)
at z = 0,
t> 0
(9.103)
C BB - C*
=
L az
L at
where L = HkG6/D.
The solution to equation 9.100 may be found by Laplace
transforms.
(Compare with Hildebrand,
177
1962 and Crank, 1956.)
Letting an overbar denote a transform and using s as the
transformation variable,
a2Y
az
s Y.
=
(9.104)
Solving 9.104 and applying the transform of 9.101 yields
Y
=
E 4 exp(-z/s)
(9.105)
Transforming 9.103 and
where E4 is an arbitrary constant.
using it to determine E4 provides
Yi + L/s
Y
=
s+/i +L
exp(-z/Y).
(9.106)
First, fac-
Inverting equation 9.106 is somewhat involved.
tor the denominator, with the aid of the quadratic formula,
to find
iS+ L
=
(
2
-
1 +
+
42)
+
2
+ 1 -2
/
4L
(9.107)
2
Letting F = 1/2(1 + /1T- 4L) and J = 1/2(1 - VTl- 4L)
equation 9.106 becomes
Y. + L/s
Y
=
1
(/ + F) (s + J)
exp(-z V/).
Expanding the denominator by partial fractions, observe
that
178
(9.108)
(
.........
+ F)(/s+ J)
J - F(/z+ F
_
.=
-(9.109)
v +J
Using 9.109 in 9.108 and separating the terms in the
numerator of 9.108, one finds
L exp(-zAS)
(J - F) (
L exp(-zi)
(J - F)(Yi + J)s
+ F)s
Y.0 exp(-zvi)
+
1
(J - F) (
+ F)
Y.0 exp(-zVi)
1
(J - F)(v
+ J)
(9.110)
Equation 9.110 can be inverted term by term, using, for
instance, transforms number 12 and 14 in Crank (1956, p.
327).
The result is the equation given as 3.50, after having
re-substituted dimensioned variables and making use of the
identity
JL
F
(9.111)
.
=
1/F - 1/J
9.4
Calculation of Penetration Depth
In Chapter 3, results are presented for the penetration
depth, h, as a function of contact time,
concentration profiles.
section.
, in two different
Those results are derived in this
Also, the exact method of their use is explained.
Penetration Profile.
For the penetration theory
profile, equation 3.48, the penetration depth is found by
direct substitution into the definition of h, equation
3.43,
179
o
I
2
(CB - C) dz'
(3.43)
h
(CB -
C i)
To evaluate
(CB - C i ) for 3.43, simply set z' = 0 in equa-
tion 3.48.
Performaing
involved.
into 3.43.
ately.
the integration
in 3.43 is much more
First, substitute (CB - C) from equation 3.48
Then, integrate the two resulting terms separ-
The first term
f
erfc
-
(9.112)
dz'
L2/V7
is a standard form for the integral of the complement of
the error function.
The solution is given by Carslaw and
Jaeger (1959, pp. 483-484).
For the second integral which
results after substituting 3.48 into 3.43
.
erfc
0
G+
x exp
+
G dz'
(9.113)
2
integration by parts is required.
Consider the complement
of the error function as the main part of the integral, with
the exponential and dz' as the differential part.
Note that
derivatives and limits of the complement of the error function are given by Carslaw and Jaeger (1959, pp. 482-483).
Also a standard table of integrals may be necessary to
arrive
at the final result for h, originally given as
equation 3.49.
180
Once having
and
determined
h as a function
of H, kG, D,
, one could choose arbitrary values of these para-
meters and r, determine the resulting values of R and A,
and then compute the critical Marangoni number from equations 3.39 and 3.40.
Unfortunately, it would be largely
coincidence if "nice" values of R and A were encountered
by this scheme.
Since it seems more useful to choose
convenient values of R and A first, and then determine h,
0, and so on, a simplified form of such a procedure was
adopted here.
To do this, one must re-arrange 3.49 to
determine h as a function of R.
As a first step, assume
that, similar to conventional penetration theory, h is of
the form
h
and then determine
.
k v-',
=
(9.114)
Dividing equation 3.49 by
t-, it
is found that
h
k
2k
4/AT
-
=
vACV
(9.115)
R
1 - exp[R 2/2]erfc[R/k]
where it should be understood that R = Hk GkZre/D.
9.115 provides an implicit solution for
R.
Equation
as a function of
With this result, the following scheme may be adopted
for computing the critical Marangoni number based on the
penetration theory profile, equation 3.48.
rary values of R, a, and A are chosen.
Then
First, arbitis determined
from equation 9.115, with the aid of a computer program for
181
solution of implicit equations.
The integral, I,is deter-
mined, also numerically, after combining equations 3.40 and
3.48 which, with the definition of
R e R2/k
2
x
22
I =
1 - eR /
, yield
erfc[R/k]
i0n
exp
nexp
(R- - 2)
[(R
2a)T erf
r-
O
L2
+ - dn.
R1
(9.116)
E
Finally, M is calculated from equation 3.39.
The critical
Marangoni number, M , is found by repeating the process for
different choices of a, guided perhaps by a golden-section
search scheme.
Note that it is unnecessary to determine
8 explicitly in order to find Mc.
Values of k are of some special interest when comparing
this author's work with that of others who use different
definitions of h, R, and M.
pp. 160-164.)
(Compare with Mayr, 1970,
Computed values of k at several values of
R include
R
k
0.0
v
0.01
1.760
0.10
1.785
1.0
1.877
10
1.880
100
2.212
= 1.772
4/F = 2.257.
182
The first and last values above may be determined from
equation 9.115 by using limiting forms of the exponential
function and the complement of the error function.
Gibbs Profile.
The method just described for evaluating
h and k in the penetration theory profile may also be applied to the "Gibbs profile" of equation 3.50.
Once again,
determining h by performing the integration in equation
3.43 is complicated.
However, exactly the same forms and
methods used for the previous profile may be used again.
The result was given earlier as equation 3.52.
(The obser-
vant reader may note that equation 3.52 differs slightly
from the corresponding, erroneous equation (52) in Brian
and Ross, 1972.)
With the aid of equation 3.52, a new
relation for k may be determined
2 22
k
=
Y
1i
L/J2 +
k
-- -
rl/f
x
R
J-
[J2 /G2k32
exp
1i
J- F
where G - 6/h and Y
F
]erfc[J/Gk]
2 + yO
L/F
+ Gk
°
G
exp [F2 /G2 k2 ]erfc[F/Gk
(9.117)
is given by setting z' = 0 in equation
3.50
=
Y
1 +
i
[L/F + Y
I f
exp
[J2 /G2 k2 ] x erfc[J/Gk]
x ex
]x
exp[F2/G2k2erfc[[F/Gk
-,..'rF
183
]
(9.118)
Values of k from equations 9.117 and 9.118 are usually
somewhat larger than analogous values for a penetration
profile.
With k known, the relation for I may be determined
as
17
ne
I
i
2an
exp[-k2n 2/4]
i
O
L/J + Y
J-
J
J
x -
F
L/F + Y
F
+
F
x
F
-
2
x exp[Fn/G + F2/G2k
x
]
x
G
erfc[kn/2 + F/Gk]
9.5
+ J /G2k2
+ J/Gk]
erfc[kn/2
J-
x exp[Jn/G
G
d.
(9.119)
Effect of Surface Diffusion
It was stated in Chapter 3 that surface diffusion in
the Gibbs layer is of little consequence to Marangoni
instability.
tions of M,
That conclusion may be justified by calculausing many values of S.
Alternatively, Brian
(1971) derives the conclusion mathematically for a linear
profile with A = 1.
In this section, a similar justifica-
tion is shown for a penetration theory profile with X = .
Consider that, by definition,
Assuming that D s
S = (6/h)(ps/D).
p and using the definition of A
184
AC
S
=
A -.
(9.120)
Ci
Now for a penetration theory profile, equation 9.33 applies
with
6 = 0 and it is evident
that
C-
C.
=
|
R
z
-C
(9.121)
'
C
- Ci
z=0
But the left-hand-side of 9.121 may be approximated by unity,
the precise value for a linear profile and the value of the
penetration theory derivative when it is evaluated across
the full,
finite interval
0 < z' < h and C i < C < CB.
Therefore
C.
C* + AC/R.
(9.122)
Using 9.122 in 9.120, one finds
R
S
(9.123)
A
1 + RC*/AC
If C* is large, then clearly S approaches zero.
itself zero, then S
'
AR.
If C* is
Comparison of this result with
equation 3.39 shows that S appears there only as part of
a2S,
which acts as an increment to R.
Now a remains quite
small for most cases encountered in this thesis, and when
a is as large as 1 or 2 it is usually with very low values
of
A.
Thus
2 AR
is a small fraction
will be a very small increment to R.
from simply assuming that S = 0.
185
of R and
2S
=
2AR
Little error results
10.
DETAILS OF COMPUTER CALCULATIONS
In this chapter the last versions of the various
computer programs used in this thesis are listed.
It
should be noted that most of the programs were run at
various times on both the IBM 1130 and the IBM 370 computers.
The listings provided here are primarily for the
370, but, in any case, the programming differences between
the two machines are minor.
Specifically, standard word
sizes are not identical, some library functions have different names, declarations of multiple precision items are
handled in distinct ways, and input-output conventions are
not identical.
A few additional comments about the program listings
are in order.
First, several, potentially puzzling sub-
program names appear; these are routines extracted from
the standard IBM Scientific Subroutine Package.
Second,
the routines for computing Mc may seen rather involved;
this is because of the bookkeeping involved in doing a
"golden-section" search for the minimum value of M.
Third,
the use of multiple precision numbers was found to be
quite important for the programs presented; frequently,
small differences between large numbers had to be calculated.
In fact it sometimes was necessary to choose the sequence
of a calculation very carefully to maximize precision.
186
The
main example of this is the evaluation of equation 3.46,
which should not be done as written in Chapter 3, but
rather as
I
=
2ae
(1 - e 2
2
)/4 2
Another problem relating to the need for precision was
that, for some values of the argument, the library version
of the Error function could not calculate what was required;
special limiting forms were used in such instances.
A final comment about the computer programs deals with
their nomenclature.
Some of the names used in the programs
are based on the nomenclature of this author's predecessors.
The following list may help to clarify the situation.
Name in Computer
Programs Here
Symbol Used
Elsewhere in
This Thesis
B
M
EL, L
R
EPS, EP
6/h
GAM, GAMMA
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11.
DETAILS OF APPARATUS AND PROCEDURE
A significant fraction of what follows is adapted
directly from Mayr (1970).
11.1
The Glass Column and Mounting
The glass column was fabricated by joining three pieces
of 30 mm, 38 mm, and 100 mm medium wall Pyrex glass tubing.
(These and other glass tubing dimensions, if not further
described, are outside diameters.)
Overall, the glass
assembly was 4 1/4 inches high and 2 1/2 inches in diameter.
As noted in Chapter 5, the wetted wall section was
2.633 ± 0.010 cm in inside diameter and 4.719 ± 0.010 cm
high.
The top of the inner glass section and the top and
bottom of the outer section were ground as perfectly flat
as possible.
A side-arm was attached through the outer glass
wall and fitted with a 4 mm teflon stop-cock and an 18/9
glass ball joint.
Each teflon tube was machined from a two-inch diameter,
solid teflon rod.
The outer diameter was trimmed to 1 7/8
inches; the hollowed inner diameter was 2.63 cm, to match
that of the wetted wall.
Each piece was threaded over a
2 inch area at 20 threads/inch.
Some details of the shape
of the tubes are evident in Figure 5.1.
The upper rod was
about 4 inches long and had a 30 mm diameter cut near the
top, in which was held a glass tube for the exit gas.
A
32 mm cut held the glass calming section in the bottom
teflon piece.
In addition the lower piece was cut with 4
214
vertical, 5/16 inch grooves.
The lower teflon piece was 9
inches long, about 2 1/2 inches of that length above the
threads.
Mayr, (1970, p. 200) notes that early investigators
using wetted-wall columns burdened themselves with nonhorizontal inlet slots.
In his work and in this effort,
the inlet slot, that is the top of the glass wetted-wall and
the bottom of the upper teflon piece, were maintained as
flat and horizontal as possible.
The two cover plates were machined from 304 stainless
steel.
Both plates were 6 1/4 inches in diameter.
bottom plate was 1/2 inch thick.
The
The thickness of the top
cover plate was reduced to 3/16 inches to reduce its weight,
but it was left with a 1/2 inch thick collar around the central, threaded, 1 7/8 inches diameter hole to ensure that
sufficient threaded surface remained to provide a good
liquid seal.
The cover plates were clamped firmly to the
glass section by means of three 1/4 inch diameter tie rods,
with wing nuts and washers.
The tie rods were also used to
fasten the column assembly to a 3/16 inch base plate, which,
in turn was bolted firmly onto a wooden table, as indicated
in Figure 5.1.
The holes in the cover plates, through which
the tie rods passed, were unthreaded and were 3/8 inch in
diameter to provide ample clearance for column alignment.
Each cover plate had a 4 1/8 inch outside diameter by
1/4 inch wide by 1/8 inch deep groove to hold a number 342
teflon O-ring (nominally 4 inches o.d. by 3/16 inch thick).
215
Three leveling screws and three centering screws were fitted
into the lower and upper steel plates respectively.
Four
tiny vent holes, with screw-in plugs, were also present in
the top cover.
Brass or stainless steel hardware was used,
to minimize corrosion.
(TEA solutions may corrode brass;
all parts of the apparatus in continual contact with the
liquid feed were stainless steel.)
The liquid film was easily disturbed by vibrations.
Mayr used great precaution to minimize building vibrations
and this author duplicated most of his efforts.
The table
with the SWWC was mounted on five-foot long tubular steel
legs; these rested on a steel plate which in turn was on 1/2
inch thick sponge rubber.
weighed down the table.
About 200 lbs. of lead bricks
Connections to the column were by
flexible, corrugated teflon tubing (bellows) and by flatwall teflon tubing.
Whether all of these precautions truly
minimized vibrations is open to question.
frankly skeptical.
added disturbances.
This author is
The corrugated tubing may well have
However, no problems were encountered
which could be attributed to external vibration.
The column was assembled and aligned in the following
way.
The lower cover plate and the tie rods were placed in
position.
The latter were pulled tight after the base plate
was levelled with the three leveling screws.
The lower tef-
lon piece was then screwed into place from below and the
gas calming section installed.
With the upper teflon piece
in position, the upper cover plate was placed upside down on
216
a table.
plate.
The glass section was then inverted over the cover
By viewing down the length of the glass section it
was possible to gauge whether or not the upper teflon piece
and the glass section were properly aligned.
The centering
screws were then adjusted until the two sections were aligned.
The glass section, together with the upper cover plate and
teflon piece, was then slipped over the already assembled
lower teflon piece thus automatically aligning the lower
slot.
The top teflon piece was turned almost against the
glass section, leaving a tiny slot.
The wing nuts were then
tightened in rotation.
During the course of many preliminary runs, this author
concluded that proper centering of the column was crucial
for gaining good liquid circulation down the wetted-wall.
To ensure comparability of all experiments, the centering of
the column was not disturbed during the set of runs reported
in this work.
The column assembly and, indeed, all important parts
of the apparatus were cleaned initially with sulfuric aciddichromate cleaning solution.
During frequent use the column
was, in the words of Mayr (1970, p. 205), "self-cleaning".
11.2
Liquid Flow System
The liquid feed tank was made of stainless steel.
The
tank fittings were used to provide access for the nitrogen
pressure line, the propylene line, a safety valve (set to
blow at 20-25 psig), a needle valve for use as a vent, the
217
liquid flow line, a drain line, a coil for circulating hot
or cold water, and a thermometer well.
Feed water was added
through a large filling port, which could be sealed by a cap
and clamp arrangement.
Feed solution was conveyed to the SWtC via medium wall,
1/2 inch o.d. stainless steel tubing, coupled by stainless
steel Swagelock fittings.
Teflon tape was used in the liquid
line couplings and indeed everywhere in the apparatus, where
leaks were ever evident or suspected.
It might be noted that
plastics, other than teflon, were not allowed to touch the
feed solution before the liquid passed through the wettedwall column.
Matiatos (1965, p. 182) suggested that some
plastics could act as a source of surfactant impurities.
The liquid flow rate was measured by a Matheson rotameter,
622 PSV with a no. 604 tube, which had been calibrated with
distilled water, using a stopwatch and graduated cylinder.
Just before entering the SWWC, the liquid passed through a
short piece of flat wall, 1/2 inch i.d. teflon tubing.
A
common laboratory screw clamp was screwed onto the tubing
to prevent ripples in the fluid flowing down the wettedwall
(see 11.4).
The feed passed into the outer part of the SWWC, up
over and down the inner, wetted wall, and through an exit
tube in the lower steel plate.
After passing a thermometer
port, going through a one foot stretch of 1/4 inch teflon
bellows, and passing through a fine needle valve, the liquid
flowed to a building drain.
Much of the final path to the
218
drain followed an upward incline, in order to allow satisfactory flow control by means of the needle valve.
11.3
Gas Flow System
Pre-purified nitrogen was conveyed from a cylinder through
1/2 inch o.d. brass tubing and fittings toward the glass calming section.
A rotameter (Matheson PSV 701, no. 74 tube),
which had been calibrated against a dry test meter, monitored
the gas flow rate, a thermometer measured the gas temperature,
and a mercury manometer the line pressure (effectively
atmospheric).
Unlike Mayr, this author found it unnecessary
to humidify the gas or warm it electrically.
The gas
entered the insulated calming section via a 2-foot length of
1 inch teflon bellows.
The calming section was a two foot
length of 32 mm glass tubing.
After passing through the
SWWC assembly, the gas went through another glass tube, more
teflon bellows, an additional glass piece, and a 3/4 inch
i.d. rubber hose to the hood.
The corrugated teflon bel-
lows apparently mixed the exit gases well, but after preliminary runs a small baffle was added to the exit gas line
as a precaution.
A second thermometer monitored the exit
gas temperature.
The sampling line for the exit gas was
primarily 4 feet of 1/8 inch o.d. stainless steel tubing.
Glass and plastic tubing and Swagelock fittings completed the
sample line, which was connected to an aspirator as noted in
Figure 5.4.
219
11.4
Procedure - A Typical Run
Feed water was taken from a Chemical Engineering building
distilled water tap or from the charcoal filter, conveyed in
a five gallon glass jug, and poured into the storage tank.
If TEA was to be added, this was usually done after 10 gallons of the 15 gallon water charge were in the tank.
Pre-
sumably, adding the final five gallons of water mixed the
solution.
Propylene was then bubbled vigorously through the
water for about 15 minutes.
A fan was used to help dissipate
the potentially poisonous and explosive propylene vapor.
A
more satisfactory arragnement would have carried the excess
propylene to the hood.
During this absorption period, the
SWWC cover plate and the column itself, held by the centering
screws, were together taken off the mounting table and the
feed from the previous run was poured out of the column.
Also, the gas and liquid sample vials were drained, rinsed,
and/or flushed with (vacuum-drawn) air.
Any TEA sample
flasks were filled with water, HC1, and indicator, and then
weighed.
Also at this point, on very rare occasions, hot
or cold water was run through the storage tank coil to bring
the water to room temperature.
(De-carbonating the feed by
heating was once planned, but never found necessary.)
After the propylene was turned off, the liquid tank was
sealed and pressurized to 18-20 psig under pre-purified
nitrogen.
(Pressurizing with air would have posed an ex-
plosion hazard.)
The liquid valves were opened and feed
allowed to reach the column.
To minimize flow disturbances,
220
it was highly desirable to keep the feed lines devoid of air
bubbles.
To eliminate these in the downward stretch of tubing
near the SWWC, the following procedure was used.
When the
outer part of the SWWC was nearly full, the flow rate was
set very low and the stop-cock on the glass assembly was
closed.
Liquid filled the incoming line and backed up into
the adjacent vent line.
(See Figure 5.2.)
When the vent line
was filled, its stop-cock was closed and that on the SWWC
re-opened.
The incoming line then remained reasonably
bubble-free.
As liquid continued to fill the outer part of the SWWC,
the upper teflon piece was screwed down tightly to minimize
flow onto the inner glass wall.
The vent holes in the cover
plate were left open until all air bubbles were displaced and
water spurted through.
Then the vents were plugged and the
teflon piece opened two turns.
Since the liquid would not
immediately spread over all of the inner glass wall, the
glass exit tube was briefly removed from the upper teflon
piece, and a clean wire inserted which was then used to
draw fluid all over the glass surface.
Like Mayr (1970, p. 203) this author found that, to
minimize rippling of the liquid on the wetted-wall, it was
necessary to place a clamp on the liquid line.
After the
outer part of the glass assembly had been filled in the manner described, the liquid flow rate was turned up to about
twice the maximum value to be used in the run.
The clamp
was then tightened, on the smooth teflon tubing near the
221
column inlet, until the flow rate diminished to a value about
30% above the intended maximum.
A few minutes later - after
the pressure regulator on the feed tank nitrogen cylinder had
adjusted to the new pressure loss in the flow line - the
liquid rate was set, using the rotameter needle valve, to its
first operating value.
The liquid film was now quite smooth.
At this stage the nitrogen gas flow was turned on and
set to its operating value, 165.8 cubic cm/sec.
Also, the
liquid level at the outlet from the SWWC was set so that
only the upper-most millimeter or so of the lower teflon
piece was untouched by liquid.
Mayr actually kept all of
the teflon piece wetted, with the liquid held above it in a
meniscus.
However, this author found that the liquid level
drifted with time, so that spillover was likely if the liquid
level was very high to begin with.
The liquid and gas rates were allowed a several minute
equilibration period while the various thermometer readings
were taken.
The atmospheric pressure was noted, either from
an aneroid device in an adjoining room or from a recorded
weather report.
(The exact value of the atmospheric pressure
was relatively unimportant, being used only in the calculation of the degree of propylene saturation in the feed.)
Finally, a gas sample was taken, in the manner described in
Section 5.4.
The liquid flow rate was then adjusted to a
new value (or at least changed sharply up or down and then
re-set to the same value) and the sample-gathering process
repeated.
Liquid samples were taken - one for TEA, if present,
and one or two for propylene - between gas samples.
222
Throughout the run, the gas flow rate and the liquid
height at the SWWC exit were monitored.
Adjustments were
made when necessary, but not within the last minute or so
before finally capturing a sample.
If spillovers occurred,
they were dried by using tissue paper held on a flexible,
hard plastic rod.
The rod was inserted through the bottom
of the calming section, which was normally plugged by a
rubber stopper.
If TEA was present in the feed, the run was interrupted
after three gas samples had been taken.
The liquid tank was
de-pressurized, more TEA was added, propylene was bubbled
for five minutes, the SWWC was emptied and re-filled, and
a new run commenced.
In any case, after six gas samples,
the run was stopped and analysis begun.
11.5
Analysis for Triethylamine
The analysis for TEA was restricted to liquid samples
and was patterned after Mayr (1970, p. 227).
Periodically,
0.2 N HC1 and NaOH were prepared from 1 N standard solutions.
The diluted solutions were titrated against each other with
one arbitrarily assumed to be exactly 0.2 N.
the following procedure was followed.
During a run,
First, 10-20 ml of
distilled water, a stirring bar, and a "pinch" of methyl
red crystals were added to a 500 ml Erlenmeyer flask.
About
20 ml of 0.2 N HC1 were accurately added from a burette.
The flask and contents were weighed.
Later, 200 to 300 ml
of the (high pH) feed solution were added to the flask and
223
it was re-weighed.
Finally, 0.2 N NaOH was used to complete
the neutralizing of the HC1.
After checking the endpoint by
adding an extra drop or two of the reagents, the total volume of each one used was noted.
The TEA concentration was
calculated from the number of equivalents of TEA associated
with the weight of collected sample.
11.6
Analysis for Propylene
Liquid and gas samples were analyzed for propylene
content by means of a gas chromatograph.
Mayr (1970, pp.
209-212) gives useful background material on the choice of
such an instrument and its column packing.
For the present
work, like Mayr, this author used a Hewlett-Packard Series
700 gas chromatograph, fitted with a hydrogen flame ionization
detector, and containing a single 6 foot by 1/8 inch,
Poropak Q packed column.
A column oven temperature controller,
a 1 mv strip recorder, and a Varian Aerograph Model 477
digital peak integrator were also employed.
Effective, continued use of the gas chromatograph was
one of the most difficult, drawn out problems encountered
in this thesis.
The following comments represent a distil-
lation of many months of experience.
It was found that the
flame detector response was very sensitive to carrier gas
flow rate changes, caused by temperature fluctuations in
the detector block heater.
Four to six hours were apparently
required for the flow rates to stabilize after turning on
or changing the detector temperature.
224
Therefore the in-
strument, including the column oven fan, was left on constantly.
Only the electrometer (which transformed the detector signal
and sent it to the integrator and recorder), and the oven
temperature controller, the recorder, and the integrator
were turned on within two hours before analysis began.
The
flow rates (measured at ambient temperature) and settings
used are summarized in Table 11.1.
As noted in the table,
two column oven temperatures were used.
For liquid samples,
the lower temperature and the consequent slower sample movement prevented overlap of the small water "peak" (actually
a transient cooling of the detector flame) with the propylene
peak.
In the gas samples, so little water vapor was present
that the higher temperature, with its shorter sample retention was feasible.
(Propylene retention time at 850 C was
about 3 minutes, versus 5-6 minutes at 700 C.)
No significant
pause was necessary in order for the instrument to equilibrate
at a new column oven temperature.
When a set of samples was to be analyzed, the column
was usually operated first, for about an hour, at 150-160°C
to "bake out" impurities.
(Among other sources, a new in-
jection port septum always carried impurities, and the septum
was changed, to minimize leakage, at the end of each day's
analysis.)
For liquid samples, 2.75
wp of
sample was
injected from a Hamilton 10 pi syringe, fitted with a Chaney
Adaptor.
(The syringe was calibrated by weighing it, before
and after repeated water injections into a hot chromatograph
injection port.)
Propylene peak area was read directly from
225
TABLE 11.1 - OPERATING CONDITIONS FOR CHROMATOGRAPH
Helium (carrier gas):
45 cm3/min
Hydrogen:
40 cm /min @ 15 psig
3
Air:
400 cm3/min
Injection port temp:
200-220 0°C
Detector block:
2700 C
Electrometer attenuation:
@ 45 psig delivery
@ 15 psig
20 (Range = 1)
Integrator settings:
Attenuation
Peak width, half height
1
30
seconds
Slope sensitivity
4
1/2
Baseline corrector rate
5
1/2
Count rate
1000 per millivolt
Column temperature:
Liquid samples
70 0°C
Gas samples
850 C
Column packing - Poropak Q, 80/100 mesh
226
the integrator.
Usually, three injections were made from a
single liquid sampling flask and the results averaged.
For
gas samples, 1.70 ml from each gas vial were injected, using
a 2 ml gas-tight syringe.
(The injected volume was read di-
rectly from the syringe barrel.)
For each gas sample a mass
transfer coefficient, kL could then be determined by
computing
k
-
L
VG
AMT
AGAS
ALIQ
VLIQ
VGAS
where VG equals the gas flow rate when the sample was taken,
AMT is the mass transfer area in the SWWC, AGAS and ALIQ are
the areas of the propylene peaks from the gas and liquid
samples, and VLIQ equals 2.75 pi and VGAS equals 1.70 ml.
Further details of these computations are in Chapter 12.
If
TEA had been present when the sample was taken, an enhancement value would also have been computed by comparing (an
adjusted value of) kL to the value predicted from leastsquares analysis of the runs made in the absence of TEA.
Since, usually, three gas samples were taken for a given
TEA concentration, three enhancement values were computed
and the average enhancement reported.
It should be noted, by way of contrast to the procedure
just described, that Mayr used a "relative" chromatographic
technique.
He injected known amounts of, for instance,
acetone into each sample vial and used the resulting acetone
peak as a standard for his chromatographic analysis.
227
This
author found such a procedure overly long and prone to mixing
problems in the liquid samplers,
Vials had to be stirred
vigorously on a magnetic stirrer for 15 minutes and more.
Also, the gas samplers had to be heated to vaporize the
injected acetone; leaks often developed at the teflon stopcocks on the gas samplers.
Overall, the author found his
own method seemingly equivalent in accuracy and much quicker.
However, it was always valuable to repeat the evaluation
(Mayr, 1970, p. 228) for the acetone/propylene response
factor (using a saturated propylene solution), in order to
ensure that the chromatograph was working properly.
11.7
Analysis for Surfactants
Rosen and Goldsmith (1972) served as an excellent guide
to the chemical analysis of surfactants.
Swisher (1970)
was also useful, expecially pp. 1-6, 19-37, which give an
overview of the importance of surfactants, and their various
physical and chemical classifications.
For the present
work, brief qualitative tests were carried out for initial
surfactant detection.
In particular, an antagonist dye
test for all surfactants (Rosen and Goldsmith, 1972, p. 21)
and a Dragendorff reagent test (Rosen and Goldsmith, 1972,
p. 256) for non-ionic surfactants were both carried out.
(For the latter a common laboratory centrifuge tube was
employed and spun at 4000 rpm for 5-10 minutes.)
The
Department distilled water gave negative results in each of
these tests.
228
By far the greatest attention in surfactant analysis
was given to determining anionic surfactants by the Methylene
Blue test.
(See Rosen and Goldsmith, 1972, pp. 399-407 and
Swisher, 1970, pp. 47-53.)
Preliminary runs of this test
were made using the simplified method first suggested by
Jones (1945). Exactly 20 ml of sample is placed in a 100 ml
separatory funnel; a drop of HC1 is added to make the sample
acid.
One ml of 0.1% (by weight) methylene blue aqueous
solution is added.
Exactly 20 ml chloroform is added, the
combination shaken for a minute and allowed to settle for
several minutes.
The chloroform is drained into a second
funnel, washed by extracting with 20 ml of "pure" water, and
drained into a spectrophotometer cuvette.
is then measured at 652 m,
The absorbance
against a pure chloroform blank.
For meaningful quantitative tests, a more elaborate
method, was employed, from Standard Methods (1971, pp.
339-342).
In this scheme a carefully standardized total
chloroform volume is used, and an acid buffer is added to
the methylene blue and the wash solution.
The acid prevents
extraction of certain impurities from the methylene blue and
the wash minimizes organic and inorganic salt interference.
The complete procedure as adapted by this author, follows.
(1) Prepare methylene blue reagent by dissolving 100 mg
methylene blue in 100 ml distilled water.
of solution to a 1000 ml volumetric flask.
Transfer 30 ml
Add 500 ml
water, 6.8 ml conc. H2SO4, and 50 g NaH 2 PO4 H20.
stand (overnight) to dissolve.
229
Shake, let
Dilute to 1000 ml mark.
(2) Prepare wash solution by repeating above procedure
without using methylene blue.
(3) Add exactly 500 ml of
unknown to a 1000 ml separatory funnel.
(This and other
critical glassware should be pre-cleaned with HNO,
3
prepared
by mixing conc HNO 3 with an equal amount of water.)
(4) Add
10 ml spectral grade chloroform and 25 ml methylene blue
reagent, shake for 20-30 seconds, and allow to settle one
minute or more.
Draw off chloroform layer into a 250 ml
separatory funnel.
(5) Add 10 ml chloroform to first funnel
and repeat step (4), but without any additional methylene
blue.
Continue, for a third and fourth extraction.
The
250 ml funnel should contain nearly 40 ml chloroform.
(6) Add 50 ml wash solution to the 250 ml funnel.
vigorously and settle again.
100 ml volumetric flask.
form to 250
Draw off chloroform into a
(7) Twice more, add 10 ml chloro-
ml funnel, shake, settle, and drain.
flask to 100 ml mark.
Shake
(8) Fill
(9) Fill each of two sample cells,
10 mm in light path length, with solution from 100 ml
flask and measure the average transmittance at 652 m,
with pure chloroform blanks considered to provide 100%
transmittance.
For the present work, a Perkin-Elmer Coleman 111
UV/VIS Spectrophotometer was employed.
Its operation was
very simple; one turned it on, inserted the sample cells,
placed a chloroform cell in the light path and set the
100% transmittance point, and then moved an unknown cell in
the path and read its transmittance.
230
Absorbance, the true
object of the measurement, was calculated as
ABS
=
-log1 0 (TR/TR, PURE).
TR, PURE was the transmittance reading found when virtually
no surfactant had been present in the original sample; such
a condition was simulated by doing the MBAS test with a 100
ml distilled water sample rather than a 500 ml one.
To relate absorbance to anionic surfactant level, the
methylene blue test was run on samples of known surfactant
Specifically, a 4.80 wt. per cent solution
concentration.
of linear alkyl benzene sulfonate (average molecular weight
-
316) was obtained from the U.S. Environmental Protection
Agency.
Two grams of this solution were accurately weighed
and diluted to 100 ml.
Two ml of this solution was then
diluted to 250 ml to provide a standard solution of (using
exact figures) 9.06
g "actives" per ml of solution.
For
analysis purposes, standard solution samples of 11, 8, 6,
3, 1 and zero ml were accurately pipetted into a graduate
and diluted to 100 ml of solution.
Each was then analyzed
by the methylene blue test, as already described.
(For
these analyses only, a total sample of 100 ml, not 500 ml,
was employed.)
The absorbance of these samples was then
plotted against the known surfactant content, as shown in
Figure 11.1.
A calibration curve was determined by perfor-
ming a "least-squares" analysis under the assumption that
the curve passed through the origin.
231
(This assumption is
0
0
0
>1
-5
F!-
z
O
LI
O
0
LL
0
O
U
0
L
.
0
)
d
0
c'M
co
0
0
3DNVtOSEtV
232
0
C1
.
a conservative one, leading to slightly higher surfactant
level predictions than if it were dropped.)
The resulting
least-squares line is given by
ABS
=
0.00276
x (g
surfactant)
For an unknown sample, then, surfactant concentration was
found by determining the sample's methylene blue absorbance,
calculating the corresponding surfactant level from the
calibration curve equation, and then dividing the result by
the sample volume.
11.8
Removal of Surfactants
Surfactant material was adsorbed on a bed of granular,
activated carbon, Nuchar WV-G, 12x40 mesh, made by Westvaco,
Inc.
Sallee, et al. (1956) had used Nuchar C190 to filter
anionic surfactants, but that grade was discontinued in
1971.
The material used here is the recommended replacement.
The manufacturer indicates that its surface area is 1100
sq. m/g which compares well with the value cited by Oehme
and Martinola (1973) for a very efficient, surfactant-removing
carbon.
(See also Davies, et al., 1973 for a discussion of
the factors determining surfactant adsorption ability.)
The filtration unit was a 51 mm glass tube, 2 ft. long
and drawn closed at one end, save for a 7 mm outlet tube.
About 400 g of charcoal were charged in four or five
separate batches; each batch being washed first, with
233
distilled water, on a Buchner funnel and vacuum flask.
As
each batch was added to the filter tube, the charcoal was
settled by adding water to exceed the height of the charcoal.
The water was then sucked out by vacuum.
carbon "fines".
This also removed
Glass wool plugs were inserted in the tube
before and after all the charcoal.
The column was mounted vertically and the distilled
water line introduced through a large rubber stopper, which
otherwise sealed the upper, open end.
The bottom of the
column was first plugged and the charcoal fully covered with
water.
The bottom was then un-sealed and the water flow,
under pressure, was adjusted by a valve in the water line to
about 10 gallons/hour (roughly 600 ml/min).
The filtrate
was collected in a five gallon glass jug and then charged
to the SWWC feed tank.
After 15 gallons of water had been
filtered, a new charcoal column was prepared.
Filtered water samples were taken directly from the
filter apparatus, after most of the water had been passed
through it.
Also, filtered water was passed through the
entire wetted-wall column apparatus and samples taken.
These 500 ml samples always showed no measurable surfactant
content by the methylene blue test.
Thus purification was
complete, both for spot samples during a filtration and for
the aggregate.
Also, the wetted-wall column assembly did
not add contaminants.
234
11.9
Sources and Grades of Materials and Services
Obtaining suitable materials and services for this thesis
was a continuing effort.
As an aid to future investigators
and as a record of what was done by this author, Table 11.2
lists sources and grades for the most important items required
in this study.
TABLE
11.2
Materials and Services to Support Experimental Work
ITEM
GRADE
SOURCE
Chemicals
Triethylamine
Eastman Grade,
Chemical no. 616
Eastman Organic Chemicals, Rochester, NY
(Manufactured by
Union Carbide)
Propylene
C.P.
Matheson Gas Products
Gloucester, Mass.
Activated
Charcoal
Nuchar
WV-G, 12x40 mesh
We stvaco
Covington, Virginia
Surfactant
Standard
Linear alkyl
benzene sulfonate
EPA
Analytical Quality
Control Lab.
Cincinnati, Ohio 45268
Methylene Blue
Reagent
(Basic Blue 9)
Color Index 52015
MIT Lab Supplies;
J.T. Baker Chem. Co.
Phillipsburg, N.J.
Chloroform
Spectrophotometric
Quality
MIT Lab Supplies;
Matheson Coleman and Bell
East Rutherford, N.J.
Packed with
Poropak Q
Hewlett-Packard
Lexington, Mass.
Hardware
Chromatograph
Columns
235
ITEM
GRADE
SOURCE
housing
Matheson Gas Products
Gloucester, Mass.
Gas Syringe
"Pressure-Lok"
Series A
Precision Sampling
Baton Rouge, La.
Teflon O-rings
No. 342
I. B. Moore
Cambridge, Mass.
Fabrication
Ryan Velluto and
Anderson
Cambridge, Mass.
Repairs
RLE Glass Shop
Cambridge, Mass.
Rotameter
No. 604 tube and
Services
Wetted Wall
Column
236
12.
SAMPLE CALCULATIONS
This chapter describes the calculations used to
interpret this author's experimental data.
In addition,
procedures and computations required to re-interpret the
data of Mayr (1970) are illustrated.
12.1
Rate of Propylene Desorption
The calculation of the propylene desorption coefficient
is illustrated with the data of Run VI-7.
Raw Data
Atmospheric pressure, PATM
Liquid temperature, TL
Gas temperature, TG
Gas flow rate, VG
Water flow rate for gas vial
Number 3
Number 4
Number 2
30.23 in.Hg
23.40 C
22.5°C
165.8 cm 3 /sec
250.0 cm 3/min
253.0 cm 3 /min
253,0 cm 3 /min
Column diameter, DIA14
Column height
2.633 cm
4.719 cm
Volume of each liquid injection into chromatograph,
2.75 pIZ
VLIQ
Volume of each gas injection,
VGAS
1.70 mZ
Area of propylene peak for
liquid sample,ALIQ,
Trial A
Trial B
Trial C
Trial D
237
13572
13276
13433
13138
counts
counts
counts
counts
Raw Data (cont'd)
Area of propylene peak for
gas sample, AGAS, from
vial
Number
Number
Number
18015 counts
18576 counts
18067 counts
3
4
2
Water viscosity,
25 0°C
23.4°C
Water density, p
25.0°C
23.40 C
/cm
0. 8904x10 - 2 (dyne) (sec)
0.9240x10-2
2
(dyne)(sec)/cm2
0.9970 g/cm3
0.9974 g/cm3
Perimeter for Mass Transfer, PMT
=
PMT
3.1416
x 2.633
=
8.272
cm
Height of Mass Transfer Section, h
h
=
4.719 + (2 turns) (0.05 in./turn) (2.54 cm/in)
=
4.973 cm
Liquid Flow,
Z/i
Flow x p/PMT
250 x 09974/8.272
Z
=
Water
Z
=
30.14 g/cm-min.
=
30.14 x [(0.9240/0.8904)/(0.9974/0.9970)] /2
=
29.58 g/cm-min.
=
For water flow of 253 cm 3 /min.,
Z//v
=
29.94 g/cm-min.
Film Thickness,
=
3/(p
2
(see Bird et al., 1960, p.4 0 )
x GA)
238
3
x
30.14
x
0.9240
x
10
-2
0.0242 cm
980.2
x 60
x
(0.,9974)2
For the higher water flow,
= 0.0243 cm
Area for Mass Transfer, AMT
AMT
D)IAM - 2
2l
=
h x PMT
=
4,973 x 8.272 x 2.633 -
x-
DIAM
0.0484
40.37
2.633
cm 2
Same value found for larger
Average Propylene Analysis in Liquid, ALIQ
=
ALIQ
=
4
(13572 + 13276 + 13433 + 13138)
13355 counts
Propylene Diffusivity, Dp
From Stokes equation, Di/T = constant
DP
=
(1,44 x 10 - 5) (0.8904/0.9240) (296.6/298.2)
=
1.38 x 10 - 5 cm 2 /sec
Desorption Coefficient, kLV
k
L
D~P
VG x AGAS x VLIQ
AMT x VGAS x ALIQ
165.8
165.
xx 18015
801
1
xx
x 2.75
40.37 x 1.70
5.925 (cm/sec)/2
239
10-3
x,1 103
13355
'F~I.I4Z_=._973
1.38 x
0'5
For gas vials 4 and 2, results are 6.110 and 5943,
respectively.
°
at 250 C,t (kL)
Coefficiente a
50
L 225
(kL)
L 25'
-
5,925
=
1.008
444. 97310-5
x 10 -2
cm/sec
Per Cent Propylene Desorbed from Feed
% Desorbed -
o
o
_~~n
(kL)2 5 0 x AMT x 60/Water flow
1.008 x 10 - 2
x 40.37 x 60/250
9.8%
Approximate Saturation of Propylene in Feed
Based on chromatograph peak = 19300 counts, for
saturated sample.
=
% Saturation
(ALIQ/19300) (29.92/PATM)
=
62%
Propylene Concentration in Liquid
Based on 0.567 x 10- 5 g-moles/cm3 at saturation.
Prop. Conc.
=
(0.62) (0.567 x 10 - 5 )
=
0.35 x 10 - 5 g-moles/cm
240
3
Mass Transfer Enhancement,
equation
Evaluating
P= kL /[L
P
=
CL
=
P
5.925/5.904
6.3
1. 004
For other two gas vials,
=
= 29. 5 8,
/
6.3 at
is 1.030 and 1.002.
Average is
1.012, for this run.
For runs in which no TEA was present and no enhancement
occurred,
E//
was not calculated; but values of kLVhi7p and
were tabulated and used to develop the regression rela-
tion, equation 6.3.
12.2
Marangoni Number
Raw Data
Weight of
Volume of
Volume of
Normality
liquid sample
neutralizing acid
neutralizing base
of acid and base
258.8
9.65
0.20
Concentration of TEA, CB
CB
(17.60 - 9.65) (0.2 x 10-3)/258.8
6.13 x 10 -6
g-moles/cm
241
g
17.60 mt
3
ml
Weight Per Cent TEA
Weight
% = CB x Molec. Wt. x 100
6.13 x 10-6 x 101.19
x 100
0.0622%
The next several calculations use
data of Mayr (1970)
to determine H and k G for triethylamine.
By definition the
Graetz number is
NGz
=
VG/hDG
But for gas flow, the Reynolds number is
NRe
=
4VGp/(7r
x
DIAM
x
)
and the Schmidt number is
=
Sc
I/P9 G
Therefore
r
NGz
DIAM
4h
Re x
Sc.
Now Mayr measured values of k G for acetone and reported them
as equivalent Sherwood numbers
NSh
=
kGRT(DIAM - 2)/DG'
A correlation is usually expected between NGz and NSh; Mayr
(1970, p. 140) found that NSh was proportional to the cube
242
root of NGz.
With Mayr's specific result (pp. 119-120 of his
10.8 when N
thesis) that NSh
10.8
NSh
- 323, it is clear that
(NG ) 1/3
(323)1/3
1.574 (N ) /3
Gz
Several newphysical properties must be supplied before
From Mayr (1970, pp. 296-297) at 250 C and
continuing.
1 atm:
pG (cm2 /sec)
Also,
g/cm
3
TEA
0.074
Acetone
0.104
Ether
0.088
assuming
an ideal gas
H
159
29.5
897
(as did Mayr)
, and it may be verified
that
(cm 3 -atm/g-mole)
p = 1.16 x 10-3
= 1.77 x 10 -4
dyne-sec/cm2 .
To proceed then,
4(165.8) (1.16 x 10 - 3 )
NRe
(3.1416) (1.77 x 10 - 4 ) (2.633)
=
525
1.77 x 10 -
4
Sc
(1.16 x 10-3)
=
(0.074)
2.11
(3.1416) (2.633) (525) (2.11)
NGz
(4) (4.973)
=
461
243
NSh
=
(1.574) (461)1/3
=
12.1
(12.1)(0.074)
kG
(82.06) (298.2) (2.633 - 0.048)
HkG
=
1.416 x 10-5 g-mole/(cm
=
(159) (1.416 x 10- 5 )
=
2.25 x 10-3 cm/sec
2)
(sec)(atm)
New calculations for Mayr's experiments were performed
in an identical manner.
However, for Mayr's apparatus
DIAM/h = 2.614/4.91 = 0.532.
Also, the gas phase Reynolds
number differed slightly among his several systems (Mayr,
1970, pp. 108, 128, 156); for acetone, NRe
515, for ether
505, and for TEA, 513.
Before computing the Marangoni number, a was required.
A graph of a versus weight per cent TEA was constructed from
the data given by Mayr (1970, pp. 285-286).
Specifically,
at each of eight points on Mayr's graphs a tangent was
constructed and a slope determined.
To compute the value
of a, it was necessary to multiply the slope by the molecular
weight of TEA and the surface tension of water at 250 C.,
72.0dynes/cm (Drost-Hansen, 1965).
The resulting values of
a and an interpolated smooth curve are given in Figure 12.1.
For any given run,
was to be evaluated at Ci; however,
C i was to be calculated by computer (see the next paragraph)
244
0
016
l
U)
0
4-,
To
t<
o
li )
L
z
L
I
0
I
H
LW
o
'I"
LI
u
0cr
0
-n
u
o
L
(I
Ul)
a)
L
U)
0
,0U
0
~0
( alowu-6/ Z- aup )
245
0
e
LL
after 6 =
/RT was known.
Therefore a trial-and-error scheme
was used; a was chosen first, to correspond to an arbitrary
value of C i , typically twenty-five per cent below CB.
Ci was computed and a new value of a chosen.
Then
An additional
computation of C i was performed if necessary.
With a, H, kG, CB, and C i all specified, the computer
program for experimental path calculations, based on
equation 3.50, was used.
(Compare with Chapter 10.)
It was
assumed that C* = 0 and Ci = 0; also the properties
, D,
and 6 were all evaluated at 250 C., a simplification of
negligible inaccuracy.
The contact time,
, was computed,
for the typical flow rate Z = 30, by finding first the SWWC
interfacial velocity (see Bird et al., 1960, pp. 39-40).
9 x GA x ()2
Ui
=8p1
8
p
1/3
,t
3
e
=
32.55 cm/sec
=
h/U.i
=
4.97/32.55
=
0.1527
sec.
For Mayr's slightly shorter column,
246
= 0.1508 sec.
For Run IV-7, the computed value of C i is 4.55 x 10-6
g-moles/cm
3
and h = 2.33 x 10 - 3 cm, with a = 9.50 x 105
= 0.008904 dyne-sec/cm2 , and
(dynes/cm)/(g-moles/cm3 ),
cm 2 /sec.
= 0.82 x 10 - 5
Recalling
that CB = 6.13 g-moles/cm 3,
it is found that
M
=
M
=
Corresponding values of
oh(CB - C i)
47,600
R = HkGh/D, 6 = a/RT, and
A = 6Ci/h(CB - Ci) are respectively
computed
to be 0.64,
3.84 x 10 - 5 cm, and 0.048.
Mayr's results.
The critical conditions in the exper-
iments of Mayr (1970) were re-computed by the procedure just
described.
in Table
Most of the parameters used were listed earlier
3.2.
Note that the values
of C
given in Table
3.2
were determined by visual inspection of plots of Mayr's
results, shown on pp. 108 and 128 of his thesis and in
Figure 6.2 of the present work.
12.3
Surfactant Analysis
The calculation of the anionic surfactant content of an
aqueous sample is illustrated with the analysis of tap water.
Raw Data
Volume of water sample
Transmittance of pure chloroform
Transmittance of chloroform extracted
against 100 me pure water, TRP
Transmittance of chloroform, extracted
against water sample, TRS
247
0.50 liter
1.000
0.996
0.966
Adjusted Transmittance, TR
TR
=
TRS/TRP
=
0.966/0.996
=
0.970
Absorbance, ABS
ABS
=
-log 1 0 TR
=
-log 1 0
=
0.0133
(0.970)
Mass of Anionic Surfactant in Water Sample
pg surfactant
(Figure 11.1)
=
ABS/0.00276
=
0.0133/0.00276
=
4.81
Concentration of Anionic in Water Sample
Conc.
=
g surfactant/Volume water sample
=
4.81/0.50
=
9.62
g/liter
=
248
10 ppb.
13.
TABULATION OF DATA
This chapter contains a compilation of the present
author's experimental data.
Table 13.1 provides the re-
sults of the desorption experiments.
Note that runs V-1
through V-3 are the basis for the regression line, equation
6.3.
(Earlier runs were preliminary in nature and are not
reported here.)
Runs VI-3 through VI-12 were performed
using distilled water; the results were earlier presented
in Figure 6.2.
Runs VII-2 through VII-8 were performed
using filtered water; these results are plotted in Figure
6.3.
Runs M-1 through M-5 were done by Mayr.
In examining Table 13.1, note that the final value of
P, for each of the Runs VI-3 through VII-8, is an average
of the values found in the run.
values of
For runs M-1 through M-5,
were estimated from Figures 5.38 and 5.39 in
Mayr's thesis.
For all the runs V-1 through VII-8, the gas phase velocity
was 165.8 cm3/sec.
The results of this author's analysis for surfactants
are given in Table
13.2.
249
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253
TABLE 13.2.
SURFACTANT ANALYSIS
pg
(mk)
ADJ.
TRANS.
100
0.630
0.2006
76.1
100
0.698
0.1563
56.0
100
0.823
0.08444
30.0
100
0.923
0.03494
10.4
100
0.518
0.2860
101.8
500
0.970
0.01328
4.81
500
0.980
0.008818
3.19
WATER
500
0.986
0.006167
2.23
FILTERED
WATER
500
>0.998
0.000869
<0. 31
VOL
CALIBRATION
TAP WATER
ABSORB.
SURFACT.
DISTILLED
WATER
DISTILLED
254
14.
SOURCES OF EXPERIMENTAL ERROR
The reproducibility of the experimental data in this
thesis indicates that random errors were not a serious
problem.
As noted in Chapter 6, this author's measurements
of propylene desorption rates exhibit a standard deviation
of 2.5 per cent.
Admittedly, the measurements of mass
transfer enhancement, in Figures 6.2 and 6.3, show some
scatter, especially near
certainty in
= 1.0 where the estimated un-
may reach ten per cent.
However, numerous
data points were collected and the reproducibility appears
acceptable.
The calibration curve for surfactant analysis, Figure
11.1, also suggests no consequential random error.
Mayr (1970, pp. 344-347) discusses and discards several
conceivable sources of systematic errors in his desorption
experiments.
Considering the close agreement of his data
and this author's, it is improbable that systematic errors
affected the desorption experiments described here.
The analysis for surfactants and their removal from
water are novel features in the study of Marangoni convection.
As such, systematic errors in these areas may seem
to deserve special attention.
However, the standard
procedures adopted for this work are relatively simple and
have been in widespread use, in waste water analysis, for
over fifteen years.
Indeed, various aspects of the
255
procedures minimize the chance for systematic errors.
For
instance in the test for anionic surfactants, materials
from the water sample which might interfere with the analysis
were removed by multiple washes with an acid phosphate
solution.
Also, contamination of water samples, by residue
from previous analyses, was minimized by cleaning glassware with nitric acid and by using samples of large volume.
In the removal of surfactants by activated charcoal
filtration, systematic errors were similarly avoided.
The very effective purification procedure of Sallee et
al. (1956) was followed as closely as possible.
Analyses
were made to ensure that the carbon adsorptive capacity
was not exceeded.
Also it was confirmed that the wetted-
wall column assembly did not add new contaminants to
water which had already been purified.
In view of the above discussion, it is unlikely
that systematic errors were a significant problem in this
thesis.
256
15.
Al, A2
.
.
NOMENCLATURE
Arbitrary constants in Chapter 3
A
r/h(cB -C i), the Adsorption number,
represents surface convection in the Gibbs
layer
a
Arbitrary constant in equation 3.32
ABS
Absorbance of sample in methylene blue test
AGAS
Area of chromatograph peak for a gas sample
AMT
Area available for mass transfer in wettedwall column apparatus
ALIQ
Area of chromatograph peak for a liquid
sample
B1 , B 2
'
Arbitrary constants in Chapter 3
'
A general scalar in Chapter 9
b
=
Arbitrary constant in equation 3.32
Solute concentration in liquid phase,
g-mole/cu.cm
C
Value of C in equilibrium with bulk gas,
g-mole/cu.cm
c
c'/(CB-Ci) , dimensionless perturbation in
solute concentration
CIMP
cMp/Ci
c
I
iMP,
dimensionless perturbation in
impurity concentration
C- CU, perturbation in solute concentration,
g-mole/cu. cm
D
Differentiation with respect to z
d
Diffusion coefficient for solute in liquid
phase, sq.cm/sec
d
Disturbance wavelength, cm
257
DIAM
=
Diameter of wetted-wall column
E1
=
Arbitrary constant in equation 9.90
E2-
6 IMP
E3
6 MPDMp/hD,
F(x,y)
=
F
s,IMp/hDIMP,
in Chapter
in Chapter
9
9
Function representing perturbation dependence
on x and y
~1
+ V1- 4L
2
f(z)
=
Function representing variation of w with z
G
=
Group defined by equation 3.27
G
=
6/h
g(z)
=
Function representing variation of c with z
=
Portions
GA
=
Gravitational acceleration, 980.2 cm/sec 2
H
=
Henry's law constant for solute, (atm)
g91'
2
of g(z), defined
in equation
3.17
(cu.cm)/g-mole
h
=
Height of mass transfer area in wetted-wall
column, cm
h
=
Mass transfer penetration depth in liquid
phase, cm; defined by equation 3.43
I
=
Integral defined by equation 3.34
i
=
/-J, base of imaginary numbers
i,j,k
=
Unit vectors in x, y, and z directions
J
=
j(z)
=
Perturbation
KL
=
Overall mass transfer coefficient, cm/sec
kL
=
Liquid phase mass transfer coefficient, cm/sec
kG
=
Gas phase mass transfer coefficient,
g-mole/(sec) (atm)(sq.cm)
1-
1
- 4L
2
function
258
for impurity
k
=
Constant defined by equation 9.114
L
=
HkG6/D
=
Depth of liquid phase, cm
o(CB - C i )h
M
, the Marangoni number
ijD
IMPCi,IMPh
, the Elasticity number
-CjII
NEL
IMP
NGz
=
The Graetz number, in Chapter 12
NSh
=
The Sherwood number, in Chapter 12
P
=
Pressure
p
=
Growth rate constant for perturbations
PATM
=
Experimental pressure of atmosphere, in.Hg
PMT
=
Perimeter of mass transfer section of
wetted-wall column, cm
Q(z)
=
f(z) x D(z)
R
=
Gas constant, 8.314 x 107 erg/(g-mole)(°K)
R
=
HkGh/D, liquid-to-gas-phase resistance ratio
Ds 6/h,
S
in Chapter
in Chapters
9, dyne/sq.cm
3 and 9
s
=
Variable for Laplace transform, Chapter 9
Sc
=
p/pD, the Schmidt number
T
=
Absolute temperature,
t'D/h2
t
t'
°K
=
Time of growth or decay of perturbation,
sec
t
=
ED/62
TG
=
Experimental gas temperature, °K
TL
=
Experimental liquid temperature,
TR
=
Adjusted transmittance of sample in methylene
blue test
258a
OK
=
TRP
Transmittance of chloroform
extracted
against pure water, in methylene blue analysis
TRS
-
Raw value of sample transmittance
U
=
U'h/D, dimensionless velocity
U'
=
Velocity in liquid phase, cm/sec
u, v, W
-
(u', v',
=
Perturbation velocity in x, y, z directions,
cm/sec
VG
=
Experimental flow rate of gas stream,
165.8 cu.cm/sec
VGAS
=
Volume of gas sample injected into
chromatograph, 1.70 m£
VLIQ
=
Volume of liquid sample injected into
u',
v', WI
w')h/D
chromatograph,
2.75
Z
x
=
General body force in Chapter 9, dynes
x
=
Net surface force, dyne/sq.cm
(x',
x, y, z
X I, y
=
y, z)/h
Coordinates along liquid surface, cm
(CB - C)/(CB
Y
- C*)
z
=
Coordinate perpendicular to liquid surface,
measured down from gas-liquid interface, cm
z
=
z/6
a
-
2-h/d
r
=
Surface concentration in Gibbs layer,
g-mole/sq.cm
Y
=
Surface tension, dynes/cm
Greek Letters
259
TI
=
F/Ci , the Gibbs depth, cm
=
Dummy variable in definite integral,
equation 3.32
=
Time of contact for mass transfer, sec
=
Surface dilational viscosity, in Chapter 9,
dyne-sec/cm
-
Liquid viscosity, dyne-sec/sq.cm or g/sec-cm
=
Surface shear viscosity, dyne-sec/cm
=
Exponent
6
K
X
1-'
Ps
constant;
v = 1 for conducting
liquid layer, v = -1 for insulating layer
-
(P) /( P) 25 oc
z
=
Liquid density, g/cu.cm
E
=
Liquid flow rate per unit length of column
p
perimeter, g/cm-min
a
=
-ay/9C, evaluated at C i, dyne-sq.cm/g-mole
a (z)
=
(C -
Ci)/(CB
- Ci)
-
Enhancement ratio in equation 6.3
-
Function defined in equation 3.20
B
=
Bulk liquid
c
=
Critical value
G
=
Gas phase-related
IMP
=
For impurity in liquid phase
i
-
Gas-liquid interface
L
=
Liquid phase-related
E)
=
Propylene
i(z)
Subscripts
260
s
=
Evaluated along liquid surface, i.e., the
x and y directions
u
=
Unperturbed situation
x, y, z
=
Component
=
Value
=
Dimensioned variable
=
Matrix transpose
in corresponding
direction
Superscripts
o
T
Note:
at e = O
In the first part of Chapter 9, the convention of
using a prime (') to denote dimensioned variables is
suspended.
See especially equation 9.53.
261
16.
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17.
AUTOBIOGRAPHICAL NOTE
The author of this thesis, John Richard Ross, was born
in Philadelphia, Pennsylvania on November 17, 1945.
He
lived throughout childhood in Drexel Hill, Pennsylvania, a
suburb of Philadelphia, and attended the public schools of
Upper Darby, Pennsylvania.
Upon graduation from high school,
the author entered MIT, in September, 1963.
Specializing in
chemical engineering, the author also did considerable work
in computer science and dabbled occasionally in political
science and economics.
A Bachelor of Science degree was
awarded in June, 1967.
After a year of graduate study at MIT and a year of
research work with the Computer-Aided-Design Group of the
MIT Electronic Systems Lab and MIT's Project MAC, the author
returned to the Chemical Engineering Department.
He was
appointed a Teaching Assistant and given responsibility for
managing the Department's computer facility.
The following
autumn, in 1970, the author was promoted to Instructor, and
in the ensuing three years taught courses in computer programming and in extraction and distillation.
The author's doctoral efforts commenced in 1969, but it
was not until February, 1971 that the present thesis was
selected and started.
Since then, thesis work, teaching, and
direction of the computer facility have all continued in
parallel.
Also, during this period, P. L. T. Brian and the
268
author jointly wrote an article, on Marangoni instability,
which was the basis for Chapter 3 of this thesis.
Presently, the author is employed by the Cabot Corporation
in Billerica, Massachusetts and is performing research and
analysis in the design of chemical processes.
Finally, but most importantly, the author has been
married since June, 1970 to the former Myrna Sloan of
Lynn, Massachusetts.
269