Dynamics of Vapor Explosions: Rapid Evap

Dynamics of Vapor Explosions: Rapid Evaporation and Instability of Butane Droplets Exploding at the Superheat Limit - CaltechTHESIS
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Dynamics of Vapor Explosions: Rapid Evaporation and Instability of Butane Droplets Exploding at the Superheat Limit
Citation
Shepherd, Joseph Emmett
(1981)
Dynamics of Vapor Explosions: Rapid Evaporation and Instability of Butane Droplets Exploding at the Superheat Limit.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/E3KR-3X92.
Abstract
A preliminary experimental investigation of the vapor explosion of a single droplet (~ 1 mm diameter) of liquid butane at the superheat limit has been completed. These experiments provided the first detailed look at rapid evaporation taking place under conditions such that departures from equilibrium, evaporative fluxes and fluid accelerations are orders of magnitude larger than observed under ordinary circumstances. Single short-exposure photographs and fast-response pressure measurements were used to obtain a description of the complete explosion process within a superheated drop immersed in a bubble-column apparatus. Emphasis was placed on the early (microsecond-time-scale) evaporative stage. Despite the apparant simplicity of the vapor explosion of a single superheated droplet, the present experiments revealed a wide range of phenomina of varying complexity occurring at different stages of the explosion.
The explosion is initiated by the spontaneous formation within the drop of a single vapor bubble, which grows until the drop liquid is completely evaporated. The resulting vapor bubble undergoes volume oscillations and eventually breaks up via Taylor instability. Several new and unusual features of the early evaporative stage of the explosion have been observed, three of which are remarkably repeatable. First, photographs of the evaporative surface show a highly roughened and disturbed interface for most of the evaporative stage. At the earliest observed times (8 [mu]sec) the roughening appears to begin as a rather regular pattern on an otherwise spherical surface, suggestive of a fundamental instability due to evaporative mass flux. Second, due to the asymmetric location of the initial nucleus within the drop, a portion of the evaporating surface contacts the surrounding fluid first and becomes nonevaporating. As the bubble grows, a unique, axisymmetric structure of circumferential waves terminated by a spherical cap appears on this nonevaporating surface. Apparently, these waves are driven by the impinging jet of vapor coming from the opposing evaporating surface. Third, nucleation and initial development of the bubble in the first 10[mu]sec is accompanied by a characteristic two-step pulsating pressure signal, suggesting that a fundamental and repeatable unsteadiness, perhaps connected with the above mentioned instability, is taking place at this stage.
A preliminary estimation of the evaporative mass flux has been made from photographically-determined bubble volumes and pressure signals measured in the first 30 [mu]sec. As might have been expected in view of our observations of the highly roughened surface, the inferred mass flux (~ 400 gm/cm[superscript 2]-sec) is two orders of magnitude larger than that predicted by the classical, diffusion-limited theory of bubble growth. We propose that the interface roughening is due to an inertial instability of the evaporative surface. A preliminary calculation for the Landau mechanism of instability, supplemented by an ad hoc correction for sphericity indicates that, indeed, the classical mode of bubble growth would be unstable under the conditions found in the present experiment. An explanation of the present observations that is consistent with this theoretical prediction is that the actual instability does occur in the first 1-2 [mu]sec of bubble growth and the instability has developed well into the nonlinear stage by 8 [mu]sec, the earliest time at which bubbles have been observed in the present experiment.
The present observations are completely different than what might be predicted from previous experiments and analyses of near-equilibrium evaporation. The generality of the present results needs to be verified in detail, but they clearly indicate that evaporation at the superheat limit can be much more complex than previously expected.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
(Applied Physics) ; vapor explosions
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Sturtevant, Bradford
Group:
Explosion Dynamics Laboratory
Thesis Committee:
Unknown, Unknown
Defense Date:
12 September 1980
Record Number:
CaltechETD:etd-11122003-143525
Persistent URL:
DOI:
10.7907/E3KR-3X92
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Shepherd, Joseph Emmett
0000-0003-3181-9310
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DYNAMICS OF VAPOR EXPLOSIONS:
RAPID EVAPORATION AND INSTABILITY OF

BUTANE DROPLETS EXPLODING AT THE SUPERHEAT LIMIT

Thesis by

Joseph Emmett Shepherd

In Partial Fulfillment of the Requirernents
for the Degree of

Doctor of Philosophy

California Institute of Technology

Pasadena, California

198]

(Submitted September 12, 1980)

li
ACKNOWLEDGEMENTS

I would like to thank my advisor, Professor Bradford Sturtevant,
for his patience, encouragement and advice throughout the course of
our association.

The assistance of all those individuals who contributed to my
practical education in graduate school is gratefully acknowledged. In
particular, I would like to thank Mr. Harry Hamaguchi for his assis-
tance in photography and the staff of the Aeronautics shop for their
help and advice.

I would like to thank Mrs. Karen Valente for her expert typing
of this manuscript under sometimes trying circumstances. My thanks
to Mrs. Betty Wood for graciously drafting many of the figures on
such short notice. The editorial assistance provided by Professor
Bradford Sturtevant during the preparation of this manuscript was
invaluable.

The author personally benefited from an Earl C. Anthony Fellow-
ship during his first year of graduate school. The investigation de-
scribed in the present work was supported by the Department of
Energy.

Personally, my stay at Caltech was enriched by the friendship
of many people, especially Mark Kushner. Finally, and most im-
portant, my wife Donna provided the support, love and encouragement

without which, the task would have been insurmountable.

iil
ABSTRACT

A preliminary experimental investigation of the vapor explosion
of a single droplet (~ 1 mm diameter) of liquid butane at the super-
heat limit has been completed. These experiments provided the first
detailed look at rapid evaporation taking place under conditions such that
departures from equilibrium, evaporative fluxes and fluid accelerations
are orders of magnitude larger than observed under ordinary circum-
stances. Single short-exposure photographs and fast-response pressure
measurements were used to obtain a description of the complete explosion
process within a superheated drop immersed in a bubble-column apparatus.
Emphasis was placed on the early (microsecond-time-scale) evaporative
stage. Despite the apparant simplicity of the vapor explosion of a single
superheated droplet, the present experiments revealed a wide range of
phenomina of varying complexity occurring at different stages of the
explosion.

The explosion is initiated by the spontaneous formation within the
drop of a single vapor bubble, which grows until the drop liquid is com-
pletely evaporated. The resulting vapor bubble undergoes volume oscil-
lations and eventually breaks up via Taylor instability. Several new and
unusual features of the early evaporative stage of the explosion have
been observed, three of which are remarkably repeatable. First, photo-
graphs of the evaporative surface show a highly roughened and disturbed
interface for most of the evaporative stage. At the earliest observed
times (8 usec) the roughening appears to begin as a rather regular
pattern on an otherwise spherical surface, suggestive of a fundamental
instability due to evaporative mass flux. Second, due to the asymmetric

location of the initial nucleus within the drop, a portion of the evaporating

iv
surface contacts the surrounding fluid first and becomes nonevaporating.
As the bubble grows, a unique, axisymmetric structure of circumfer-
ential waves terminated by a spherical cap appears on this nonevap-
orating surface. Apparently, these waves are driven by the imping-
ing jet of vapor coming from the opposing evaporating surface. Third,
nucleation and initial development of the bubble in the first 10 usec
is accompanied by a characteristic two-step pulsating pressure signal,
suggesting that a fundamental and repeatable unsteadiness, perhaps
connected with the above mentioned instability, is taking place at this
stage.

A preliminary estimation of the evaporative mass flux has been
made from photographically-determined bubble volumes and pressure
signals measured in the first 30 wsec. As might have been expected
in view of our observations of the highly roughened surface, the in-
ferred mass flux (~ 400 gm/cm*-sec) is two orders of magnitude
larger than that predicted by the classical, diffusion-limited theory of
bubble growth. We propose that the interface roughening is due to an
inertial instability of the evaporative surface. A preliminary calcu-
lation for the Landau mechanism of instability, supplemented by an
ad hoc correction for sphericity indicates that, indeed, the classical
mode of bubble growth would be unstable under the conditions found
in the present experiment. An explanation of the present observations
that is consistent with this theoretical prediction is that the actual
instability does occur in the first 1-2 psec of bubble growth and the
instability has developed well into the nonlinear stage by 8 usec, the
earliest time at which bubbles have been observed in the present

experiment.

The present observations are completely different than what
might be predicted from previous experiments and analyses of near-
equilibrium evaporation. The generality of the present results needs
to be verified in detail, but they clearly indicate that evaporation at

the superheat limit can be much more complex than previously ex-

pected.

TABLE OF CONTENTS

ACKNOWLEDGEMENTS

ABSTRACT

TABLE OF CONTENTS

i. INTRODUCTION
1.1 Motivation

1.2 Basic Theory and Previous Related Work
1.2.1 Superheat Limit and Metastability

1.2.2 Experimental Technique
1.2.3 Bubble Growth and Evaporation
1.2.4 Underwater Explosion and Bubble
Oscillations
1.3 Outline of the Present Work
References

Figures

i. EXPERIMENTAL FACILITY AND INSTRUMENTATION

2.1 Introduction

2.2 Apparatus

2.3 Operation

2.4 Instrumentation and Data Acquisition
2.4.1 Pressure Measurements
2.4.2 Photography

2.5 Conditions in the Drop at the Time of the

Explosion

References

Tables

Figures

UI. SURVEY OF OBSERVATIONS

3.1 Introduction

3.2 Overview

3.3 Detailed Discussion
3.3.1 Nucleation and Initial Growth
3.3.2 Developing Bubble Structure
3.3.3 Structure of Evaporating

Interface

3.3.4 Summary of Evaporation Stage
3.3.5 Gas Bubble Oscillations and

Breakup
3.4 Pressure Signals
References
Tables
Figures

Iv. BUBBLE GROWTH AND EVAPORATION RATE
4.1 Introduction
4.2 Principles of Mass Flux Calculation

53
57
61
63
66

89
89
90

vii

TABLE OF CONTENTS (Continued)

4.3 Data Reduction
4.3.1 Photographs
4.3.2 Pressure Signals
4.4 Results
4.5 Comparison to Classical Model
References
Tables
Figures

Vv. LIQUID-VAPOR INTERFACE INSTABILITY
5. Introduction

Review of Possible Instabilities

Physics of Landau Instability

Application to the Classical Model

of Bubble Growth

5.5 Image Processing and Experimental
Evidence of Instability

5.6 Summary

References

Tables

Figures

WHE

5.
5.

VIL A ONE-DIMENSIONAL ANALOG OF THE CLASSICAL
MODEL AND SOME LIMITATIONS AT THE SUPER-
HEAT LIMIT

6.1 Introduction
6.2 One-Dimensional Version of the Classical
Model
6.2.1 Heat Transfer
6.2.2 Liquid Dynamics

6.2.3 Summary of the Model and
its Approximate Solution
6.3 Limitations of the Approximate Heat
Transfer Solution
References
Figures

Vil. CONCLUSIONS

Appendix A. PRELIMINARY MEASUREMENTS
A.1 Shadowgraphy
A.2 Extinction Meter
References
Figures

100
105
106
109

122
122
123
125

129

132
134
137
138
139

143
143

144
147
148

149

15]
156
157

158

163
163
163
165
166

vili

TABLE OF CONTENTS (Contents)

Appendix B. STARTUP OF AN AIR JET INTO WATER
Figures

Appendix C. STABILITY CRITERIA FOR OSCILLATING
BUBBLES
References

Figures

173
180
181

I. INTRODUCTION
1.1 Motivation

Destructive industrial accidents and violent natural processes
very often involve the sudden contact of two liquids at very different
temperatures or the rapid depressurization of a container of heated
liquid. In the last 30 years, many examples have been hypothesized
or documented in such varied situations as foundries (Witte et. al.
1970), the paper industry, LNG spills into seawater (Reid 1978),
nuclear reactor cooling system failures (Board and Caldarola 1977),
railroad tank car explosions (Reid 1979) and molten lava-water inter-
actions (Colgate and Sigurgierson 1973). This last example is the
most speculative and spectacular, possible cases being the cata-
strophic explosion of Krakatoa in 1886 and more recently the erup-
tion of Mt. St. Helens (Kerr 1980). A common feature of all these
events is their explosive nature; they occur very rapidly and produce
destructive blast waves which propagate into the surroundings.

Physically, the mechanisms of all these explosions is the ex-

tremely rapid evaporation of the superheated liquid resulting from

depressurization or contact with a hotter fluid. The rapidly evolved
vapor then displaces the surrounding fluid to produce the blast waves
similar to the action of a conventional explosive. However, this

vapor explosion differs in several crucial ways from a conventional

explosion: itis endothermic rather than exothermic, there is no
chemical reaction other than the change in phase and the specific
energy release is smaller.

Currently, a qualitative and detailed theory of vapor explosions

does not exist. Present understanding is based on observations of

medium and large scale experiments consisting of injecting or
pouring various combinations of molten metal and liquid into one
another (Anderson and Armstrong 1974). Such interactions are very
complex events involving fluid mixing and fragmentation (usually
turbulent), heat transfer (often film boiling), multiphase flow and

of course, highly non-equilibrium evaporation. Very often the only
result reported from such an experiment is that a particular com-
bination of liquids does or does not explode. Invariably, it is found
that explosions will always occur if the temperature of the cooler

liquid can be raised to the superheat limit. This is the highest

temperature at a given pressure to which a superheated liquid can
be elevated before spontaneously evaporating.

In contrast to vapor explosions, the homogeneous nucleation
theory of the superheat limit is relatively well developed and sup-
ported by extensive experimentation. Most of those experiments
have been carried out in a particularly simple geometry in which a
single droplet of liquid immersed in a host liquid is brought to the
superheat limit in a controlled way. Despite the widespread use of
this technique since its introduction 20 years ago (Wakeshima and
Takata 1958, Moore 1959) the fundamental details of the evaporation
process were never investigated. Experimenters were simply inter-
ested in the physical chemistry of the superheat limit and the ex-
plosive aspect of the vaporization was ignored. Rudimentary facts
such as the time scale and overpressure characterizing the explosion
were completely unknown and the subject of speculation.

The challenge of the unknown dynamics in such a simple

system made it an ideal and obvious choice for the application of

high speed diagnostic techniques (spark photography, piezoelectric
pressure transducers) which have been used extensively in gas-
dynamics and conventional explosion research. Despite the apparent
simplicity of the vapor explosion of a single superheated droplet,
the current investigation revealed a wide range of phenomena of
varying complexity occuring at different stages of the explosion. A
number of unique and previously unreported features of the early
stages of explosive evaporation of the droplet have been observed
along with some well known features, at the later stages, of bubble
dynamics. Emphasis has deliberately been placed on the most non-
equilibrium and therefore least understood process of rapid evapora-

tion occurring in the earliest stages of the explosion.

1.2 Basic Theory and Previous Related Work

In this section, a brief summary of relevant background infor-
mation is presented. The thermodynamics and kinetic theory of the
superheat limit is discussed and the experimental technique of super-
heating explained. Previous theories and experiments on the growth
of bubbles from nuclei are mentioned and key differences with the
present observations pointed out. The connection of the later stages
of droplet vapor explosion and underwater explosions is mentioned

and some features of the motion discussed.

1.2.1 Superheat Limit and Metastability

The physical basis for superheating is that it is possible for
a liquid to exist at a thermodynamic state where its chemical poten-
tial gw is higher than that of its vapor at the same state. Such

liquid states are metastable and can only be produced if all sources

of gas nuclei (rough container walls, free surfaces and dissolved gas)
are eliminated. The ultimate thermodynamic limit to the existance of
such a state is the violation of mechanical or thermal stability. In

practice, it is the former requirement which is observed to hold;

c)
(Fe), =O. (1-1)

The locus of points where this is satisfied is known as the spinodal and
can be visualized as a line in the pressure-volume plane (Figure 1.1) or
pressure-temperature plane (Figure 1.2). The unstable portion of the
isotherm, where ,, > 0, is shown as a dashed line on Figure 1.1.
The measured limit of superheat lies near but usually somewhat below
the limit predicted by equation 1-1 using typical equations of state ex-
trapolated into the metastable region (Reid 1978).

A mechanistic approach to the superheat limit which is the
basis for a more detailed description of the evaporative process is
the homogeneous nucleation theory. From the point of view of
statistical mechanics, there are constantly fluctuations in phase
occurring within the liquid. These fluctuations take the form of
microscopic gas bubbles with a spectrum of sizes. The growth or
decay of such a bubble is determined by the balance between the
work required in generating new surface area (due to surface tension)
and the lower free energy of the vapor phase. There is, therefore,
a critical size below which a bubble will collapse and above which a

bubble will grow. The classical expression for this critical radius

R, is

where PY is the pressure in the vapor, Po the ambient pressure
and o the surface tension. When the spectrum of bubbles contains
a large number with radii greater than critical, it is concluded that
liquid is unstable since these nuclei will grow and ultimately cause
a change of phase.

The spectrum of sizes is calculated on a probabilistic basis
using the fluctuation theory of statistical mechanics. The probability
of a fluctuation of a given size is proportional to

SW(R)
kT

(1-3)

where AW(R) is the change in energy necessary to create the
bubble of size R. An extensive review of the effort to calculate

the superheat limit using this approach and a number of experimental
results can be found in Skripov (1974).

The connection to the thermodynamic limit of stability is
through the fluctuation theory result of statistical mechanics (Landau
and Lifshitz 1969) which relates fluctuations in density Ap of a
volume of fluid V to the isothermal compressibility

(pk = p? “S (22) (1-4)
ve VT
Since fluctuations in phase are nothing more than fluctuations in
density in a liquid-vapor system,as (2) - 0 the fluctuations in
phase become large and the probability of the creation of a greater
than critical sized nucleus rapidly increases.

Thus, in conclusion, the absence of heterogeneous nucleation

sites leads to the existence of metastable liquid states. These

states are possible until the temperature is increased to the vicinity

of the thermodynamic limit of stability where fluctuations in phase

result in homogeneous nucleation and subsequent evaporation.

1.2.2 Experimental Technique

A number of methods have been used to produce superheated
liquids. From Figure 1.2 it can be seen that metastable liquid
states can be reached from stable states by some combination of
increasing the temperature and/or reducing the pressure. The
heterogeneous nucleation is suppressed by containing the liquid with
a microscopically smooth surface such as another liquid or a freshly
prepared glass capillary. Alternately, the experiment is performed
on a transient basis, superheating the liquid sufficiently rapidly to
achieve the desired state before heterogeneous nucleation sets in.

In the current investigation, the apparatus used is the bubble
column first devised by Wakeshima and Takata (1958). In this
device, a vertical column of host liquid is maintained hotter at the
top than at the bottom. The substance to be studied is introduced
at the bottom in the form of a small (~ 1 mm dia.) liquid drop.
The host liquid has been selected so that the drop liquid is both
immiscible and buoyant in the host. In addition, the host is chosen
so that it remains in the stable liquid state for a temperature range
encompassing the boiling and superheat limit temperatures of the
drop substance.

The droplet liquid is initially in the stable state and as it
slowly rises up the column under the action of gravity, it is super-
heated at essentially constant pressure due to the heat transfer from
the surrounding host. By selecting sufficiently small drops, the

velocity of rise will be slow enough so that the drop temperature

is reasonably uniform and close to the host temperature at that
point in the column. In this way, the drop temperature can be
increased to the limit of superheat at the top of the column where
homogeneous nucleation occurs and the drop evaporates with an
explosive pop.

In the past 20 years, a large number of substances (particularly
hydrocarbons and hydrocarbon mixtures) have been investigated by
this technique and for many of these substances, the superheat
limit at one atmosphere pressure is 0.9 TO. where T. is the
critical temperature (Blander and Katz 1975).

The explosive pop at the moment of evaporation has been re-
ported by many investigators and represents the "blast wave" radi-
ated by the mini-vapor explosion that has taken place. However,
until the current investigation, no instrument other than the human
ear has been used to measure this pressure wave. Thus the first
measurement made in this investigation was a determination of the
pressure waveform via a fast piezoelectric transducer. The wave-
form had a completely unexpected and complex appearance with
significant structure or ''bumpiness'' on a microsecond time scale
which was remarkably repeatable for different drops (Sturtevant
and Shepherd 1977). Much of the subsequent flow visualization was
motivated by the desire to understand the physical processes respon-

sible for the unique pressure waveform.

1.2.3. Bubble Growth and Evaporation

The implication of the homogeneous nucleation theory is that
evaporation at the superheat limit is effected by the growth of greater

than critical size nuclei. That is, nuclei grow to form macroscopic

bubbles and the vapor inside each bubble is provided by the evapora-
tion of liquid at the liquid-vapor interface comprising the bubble sur-
face. In the current investigation, it is found that one such bubble
is responsible for the evaporation of a drop. Thus theory and ex-
periments on the growth of vapor bubbles in superheated liquids are
very relevant to understanding the results of the current investigation.
Starting with the theoretical work of Plesset and Zwick (1954)
and the experiments of Dergarabedian (1953) there have been numerous
investigations of this problem and the closely related phenomenon of
cavitation. Experimental work has been mainly visualization studies
with high speed movies of bubble growth in bulk liquid at relatively
low superheat (some examples are Dergarabedian 1960, Florschuetz
et. al. 1969, Hooper et. al. 1970, Kosky 1968, Hewitt and Parker
1968). Results relevant to the current investigation are: the
bubbles have a smooth surface, are very spherical and grow radially
with the square root of time. This behavior and in particular the
dependence of growth rate on superheat and substance is in accord
with the theory, the most recent version of which can be found in
Prosperetti and Plesset (1978), In this theory, the vapor is assumed
to be in equilibrium and at saturated conditions as determined by
the interface temperature. The interface temperature is determined
by the heat flux into the interface which provides the latent heat
necessary to support the mass flux through the interface (evaporation).
Approximating this heat transfer process as a thin thermal
boundary layer in the liquid next to the bubble surface and determin-
ing the bubble motion by incompressible dynamics results in a

simple, complete description which has to be solved numerically in

the general case. The corresponding linear version (planar inter-

face) of this model using compressible dynamics has been developed

and the simple analytical solution presented here for comparison.
Cavitation research has emphasized a very different aspect of
bubble dynamics than the experiments mentioned above. Physically,
the dynamics of cavitation bubbles are different since the heat trans-
fer associated with evaporation is usually negligible (Plesset and
Prosperetti 1977) and the fluid dynamics of the surrounding liquid is
the main factor in determining bubble motion. There has been an
extensive treatment (beginning with Kornfeld and Suvorov (1944)) of
the role of surface instabilities in collapsing cavitation bubbles as
part of the effort to understand the mechanism of cavitation damage.
Typical examples of the experimental work can be found in Benjamin
and Ellis (1966) and Lauterborn (1974); both theoretical work and the
experiments are reviewed in detail by Plesset and Prosperetti (1977).
The importance and relevance of these results to the present work
is the possibility of such surface instabilities existing on the vapor
bubbles formed in the droplet explosion. In fact, two distinct types
of surface instabilities were observed in the present experiment; one
occurs on the evaporating bubble surface (for different reasons than
the instabilities occur on cavitation bubbles) and is discussed below
and in Chapter 5; the other occurs at a later stage on the oscillating
gas bubble (this is more closely related to the instabilities observed
in cavitation bubble collapse) and is discussed in the next section.
The conditions in the current investigation differ from those in
the previous experiments and theories mentioned above in two im-

portant ways. First,the amount of liquid superheated is small and

therefore the interaction of the growing bubble with the surrounding
non-evaporating fluid is important. This interaction was observed
in the present experiments in the form of a unique and repeatable axisym-
metric disturbance on the vapor-host interface which gives the
evaporating droplets a completely unexpected appearance. This dis-
turbance can be conceived of as being produced by the ''jet'’ of
vapor evaporated from the opposing surface and impinging on

the vapor-host interface. Second, the degree of superheat is much
larger, resulting in a system far from equilibrium where theory is
on much less firm ground and no observations have previously been
made. Present observations show that indeed, the evaporation pro-
cess appears completely different than has been previously observed
or theorized. Photographs of the evaporating surface show a highly
roughened and disturbed interface for most of the evaporative stage,
a fundamental difference which must drastically influence the mass
transfer processes occurring there. More quantitative differences
are revealed by calculation of the evaporation rates at early times
from experimental data and comparison to the Prosperetti-Plesset
theory. The origin of the interface roughness as an instability of
the evaporation process is considered and evidence that this has

occurred is presented.

1.2.4 Underwater Explosions and Bubble Oscillations

The evaporation of superheated droplets in the current inves-
tigation and associated acoustic radiation is a microscopic analog of
the explosion of a conventional charge underwater (for a complete
discussion of underwater explosions see Cole (1948)). Initial phases

of the two processes are similar but differ greatly in intensity.

Detonation of a conventional explosive results in the almost instan-
taneous creation of a high pressure gas bubble whose impulsive
motion produces a shock wave of large amplitude and short rise
time. Evaporation proceeds much more slowly than detonation and
the pressure wave produced is characterized by its slow rate of in-
crease.

Both processes result in the creation of a high pressure gas
bubble in the surrounding host fluid. Such a bubble-liquid system is
a nonlinear oscillator with the inertia of the surrounding liquid
acting as an effective mass and the pressure-volume relation of the
gas providing the spring action. For an incompressible liquid both
Cole (1948) and Heuckroth and Glass (1961) have given solutions, and
the compressible case has been treated by Epstein and Keller (1971)
and Keller and Kolodner (1956). In both cases, the result is that
radiation of pressure waves occurs primarily at the minima of the
oscillations in bubble volume. These waves are known as "bubble
pulses'' in underwater explosions and can be seen in all the pressure
traces obtained in the current investigation.

The oscillatory motion of the initially spherical bubble is un-
stable with respect to the creation of surface waves on the bubble
(Plesset 1954, Birkhoff 1956, Strube 1971). This especially occurs
near the minima in volume where the accelerations are directed
from the light vapor to the heavy liquid - a Taylor unstable motion.
It is this instability which is a dominating feature of the motion of
the bubble and leads to its eventual breakup. Very clear photographs
of this process have been obtained and the initial length scales of
the instability have been compared to those predicted by the linear

theory.

The material in this section is included in this report since it does
provide a description of the final stage of droplet vapor explosions
which has not been previously observed. However, since this aspect
of the problem is well understood in comparison with the evaporative
stage of the process, it is not further investigated or emphasized in

the present work.

1.3 Outline of the Present Work

The present work is an experimental investigation of the vapor
explosion of single liquid butane droplets at the superheat limit.

This experiment was of an exploratory nature and at the outset the
time scales, mode of evaporation and blast wave strength were com-
pletely unknown. In the course of obtaining some of this fundamental
information, it became clear that every droplet undergoes a well-de-
fined, repeatable sequence of events which divides the process into
stages.

Using high-speed photography and pressure measurements a
general survey of all stages was made and an overview of the results
is presented in Chapter 3. Briefly, a single bubble forms within
the drop and grows until the drop liquid is completely evaporated,
producing a vapor bubble which oscillates in volume and eventually
breaks up into smaller bubbles via Taylor instability. Two new and
unexpected features of the evaporation have been observed. The
evaporating bubble surface displays a regularly wrinkled appearance
at the earliest observed times (~ 8 usec) which rapidly develops into
a very rough and highly non-uniform surface persisting for the
remainder of the evaporative stage. The invariably observed asym-

metric location of the bubble within the drop results in the interaction

of a portion of the evaporating interface with the non-evaporating
host fluid near the beginning of the evaporative stage. This inter-
action results in a unique axisymmetric ridged structure on the
surface of the bubble terminated by a spherical cap.

Consequences of these unusual features of the motion are ex-
plored by using the photographically-determined volume growth rates
and measured pressures to deduce the evaporative mass flux at very
early times. The calculated mass flux is many times larger than
that predicted by the standard theories of bubble growth and has a
complex time behavior reflecting its dependence on measured pres-
sures. Possible instability of the evaporating interface is considered
as a mechanism for producing the observed roughness and wrinkling
of the bubble surface. Growth rates and the most unstable wave-
lengths are calculated for the Landau mechanism of instability and
compared to the regular wave-like features on the bubble surface
at the earliest times.

Finally, as a method for illustrating the implication of applying
the standard model of bubble growth to problems with large super-
heat, a linear one-dimensional version of bubble growth is developed
and solved. Some of the limitations at large superheats of the
approximations used are pointed out, in particular the exclusion of

a class of steady-state solutions.

REFERENCES

Anderson, R.P. and Armstrong, D.R. 1974 ''Comparison Between
Vapor Explosion Models and Recent Experimental Results",

AIChE Symposium Series 70(138), 31.

Benjamin, T.B. and Ellis, A.T. 1966 ''The Collapse of Cavitation
Bubbles and the Pressures Thereby Produced Against Solid
Boundaries", Phil. Trans. Roy. Soc. A260, 221.

Birkhoff, G. 1956 ''Stability of Spherical Bubbles", Q. Appl. Math.
13, 451.

Blander, M. and Katz, J.L. 1975 ''Bubble Nucleation in Liquids",
AIChE Journal 21, 833.

Board, S.J, and Caldarola, L. 1977 ''Fuel-Coolant Interactions in
Fast Reactors", in Symposium on the Thermal and Hydraulic
Aspects of Nuclear Reactor Safety. Vol. 2: Liquid Metal Fast
Breeder Reactions. (ed. O.C. Jones, Jr.and S.G. Bankoff),
American Society of Mechanical Engineers, New York, New York, 195.

Cole, R.H. 1948 Underwater Explosions, Princeton University Press,
Princeton, New Jersey.

Colgate, S.A. and Sigurgeirson, T. 1973 ‘Dynamic Mixing of
Water and Lava'', Nature 244, 552.

Dergarabedian, P. 1953 "The Rate of Growth of Vapor Bubbles in
Superheated Water’, J. App. Mech. 20, 537.

Dergarabedian, P, 1960 ‘Observations on Bubble Growth in Various
Superheated Liquids", J. Fluid Mech. 9, 40.

Epstein, D. and Keller, J.B. 1971 "Expansion and Contraction of
Planar, Cylindrical, and Spherical Underwater Gas Bubbles",
J. Acous. Soc. Am. 52, 975.

Florschuetz, L.W., Henry, C.L. and Khan, A. Rashid 1969
"Growth Rates of Free Vapor Bubbles in Liquid at Uniform
Superheats Under Normal and Zero Gravity Conditions'', Int.
J. Heat Mass Transfer 12, 1465.

Heuckroth, L.E. and Glass, I.I. 1961 'Low-Energy Underwater
Explosions", Can. Aero. J 7, 2095.

Hewitt, H.C, and Parker, J.D. 1968 ''Bubble Growth and Collapse
in Liquid Nitrogen'', J. Heat Transfer 90, 22.

Hooper, F.C., Eidlitz, A. and Faucher, G. 1970 "Bubble Growth
and Pressure Relationships in the Flashing of Superheated
Water", Vols. 1-3, Technical Publication 6904, University of
Toronto, Department of Mechanical Engineering.

REFERENCES (Continued)

Keller, J.B. and Kolodner, I.I. 1956 "Damping of Underwater
Explosion Bubble Oscillations’, J. App. Phys. 27, 1152.

Kerr, R. 1980 "Mt. St. Helens: An Unpredictable Foe'', Science
208, 1447.

Kornfeld, M. and Suvorov, L. 1944 "On the Destructive Action of
Cavitation", J. App. Phys. 15, 495.

Kosky, P.G. 1968 ''Bubble Growth Measurements in Uniformly
Superheated Liquids", Chem. Eng. Sci. 23, 695.

Landau, L.D. and Lifshitz, E.M. 1969 Statistical Mechanics,
Pergamon, New York.

Lauterborn, W. 1974 ''General and Basic Aspects of Cavitation",
in Finite-Amplitude Wave Effects in Fluids (ed. L. Bjérnd), IPC
Science and Technology Press, Ltd., Surrey, England, 195.

Moore, G.R. 1959 ''Vaporization of Superheated Drops in Liquids",
AIChE Journal 5, 458.

Plesset, M.S. 1954 "On the Stability of Fluid Flow with Spherical
Symmetry", J. App. Phys. 25(1), 96.

Plesset, M. and Prosperetti, A. 1977 ''Bubble Dynamics and
Cavitation", in Ann. Rev. Fluid Mech (9) (ed. van Dyke, et al),
Annual Revies Inc, Palo Alto, California, 145.

Plesset, M.S. and Zwick, S.A. 1954 "The Growth of Vapor
Bubbles in Superheated Liquids", J. App. Phys 25(4), 493.

Prosperetti, A. and Plesset, M.S. 1978 "Vapor Bubble Growth in
a Superheated Liquid", J. Fluid Mech. 85, 349.

Reid, R.C. 1978 'Superheated Liquids: A Laboratory Curiosity
and, Possibly, an Industrial Curse", Chem. Eng. Ed. 12, 60.

Reid, R.C. 1979 ''Possible Mechanism for Pressurized-Liquid Tank
Explosions or BLEVE's", Science 203, 1263.

Skripov, V.P. 1974 Metastable Liquids, Wiley, New York.

Strube, H.W. 1971 ''Numerische Untersuchungen zur Stabilitat
nichtspharisch schwingender Blasen", Acustica 25, 289.

Sturtevant, B. and Shepherd, J. 1977 "The Dynamics of Vapor
Explosions'', APS Bull. 22, 1274.

Wakeshima, H. and Takata, K. 1958 "On the Limit of Superheat",
J. Phys. Soc. Japan 13, 1398.

REFERENCES (Continued)

Witte, L.C., Cox, J.E. and Bouvier, J.E. 1970 ''The Vapor
Explosion", Journal of Metals 22, 39.

ere atte!
i ee ar he ane

unin: Metastable States

T=const.
a Critical Point

Spinodal

EEN Coexistence Curve

Ce ee Sein,
Ce ee a

owe
oat ete tat te
ee tata

Poteet ate

FIG. 1.1 PRESSURE-VOLUME PLOT SHOWING LIQUID,
VAPOR AND METASTABLE STATES

Pressure, P

Stable
Liquid

Stable
Vapor

Metastable
Vapor

Metastable
Liquid
Temperature,T
F |
‘ |
‘ H
. \
\ /
\ /
\ /
\ /
NL LU?

FIG. 2 PRESSURE-TEMPERATURE PLOT
SHOWING LIQUID, VAPOR AND

METASTABLE STATES

19
tl. EXPERIMENTAL FACILITY AND INSTRUMENTATION

2.1 Introduction

In this chapter, the equipment and operating procedure used in
the experiment is described. The basic apparatus in which ex-
plosions are produced is a vertical column of ethylene glycol heated
at the top and cooled at the bottom. Droplets of liquid n-butane
(~ 1 mm dia.) are introduced at the bottom and are superheated as
they drift up the column. Near the top of the column the droplets
reach their superheat limit (105° C) and explode. In principle, this
is the same technique of superheating that was first used by
Wakeshima and Takata (1958). The explosion occurs in an instru-
mented test section where the pressure produced by the explosion is
detected by fast piezoelectric transducers and digitally recorded,

The explosion is photographed at a preset delay time after the de-
tection of the pressure signal. Illumination is provided by a triggered
spark gap and both transmitted and scattered light are imaged simul-

taneously.

2.2 Apparatus

A bubble column designed for continuous operation with auto-
matic temperature control was the basic facility used in the current
investigation. A test section at the top of the column was equipped
with access ports into which different transducers and optical windows
could be inserted according to the type of experiment being conducted.

The choice of substance in the current investigation was dic-
tated by convenience and the past experience of other investigators.
Ethylene glycol was used as the host liquid since it is an easily

available, relatively non-toxic substance which has been used succes sfully

before, for example, see Porteous and Blander (1975). Butane,

the middle member of the alkane family of hydrocarbons, was used
for the superheated droplets since it has convenient boiling (-0.5° C)
and superheat limit (105° C) temperatures for use with pure glycol

as host. An important point is that the superheat limit is low

enough so that heat-sensitive pressure transducers can survive in

the test section without special precautions. Representative properties
of n-butane and ethylene glycol can be found in Tables 2.1 and 2.2
respectively.

Unlike many experiments with superheated liquids, no special
precautions were taken with regard to purity. Both butane and
glycol used were of technical grade and discoloration and particulates
in the glycol were quite evident after months of use. However, the
superheat limit observed was within the range reported by other ex-
perimenters (Blander and Katz 1975) and no premature nucleation
was noticeable. Contrary to some reports (Reid 1978), every drop
which contained only pure liquid reached the superheat limit and
exploded.

The column proper (see Figure 2.1) consists of a 12" section
of Pyrex 4'' diameter pipe which together with a 6" high, 4" square
anodized aluminum test section contains the host fluid. At the
bottom, the pipe is seated onto a refrigerated aluminum base plate
which contains the drop injector and a drain valve. At the top the
test section is sealed with an anodized aluminum top plate to which
is attached an expansion volume for the host fluid, an electrical
heater (500 W) and a thermocouple which can be traversed along

the centerline of the column.

Refrigeration is a two step process, with the primary unit an
ordinary freon (R-12) charged compressor system cooling an
ethylene glycol-water bath (Figure 2.2). The secondary loop is
powered by a peristaltic pump which circulates an ethylene glycol-
water solution through the bath, the base plate and the drop injection
mechanism. The bath and secondary loop are contained in an in-
sulated compartment underneath and supporting the base plate. The
test section and column are also insulated. To compensate for the
vastly different thermal conductivities of the test section and the re-
mainder of the column, two aluminum strips (labeled heat sinks in
Figure 2.1) are connected between the bottom of the test section
and the base plate. The size of strips is adjusted so that the tem-
perature gradient measured along the centerline of the column in the
test section is ~ 1.5° C/cm; a value which yielded an acceptable
minimal scatter in the location of the explosion.

Thermocouples located in the bath, base plate and top plate
are connected to on-off controllers of conventional design which
independently regulate the primary and secondary refrigeration
systems and the heater. All thermocouples used are of chromel-
constantan type and both legs of each circuit are referenced to a
water-ice bath, the voltages are read by a digital multimeter
(Keithley 163b) which can be switched in parallel to the inputs of
the controllers. Typical operating temperature ranges (determined
by the setting and hysteresis of the controller) are: top plate, 118
to 121° C; test section center, 104.5 to 105.5° C; base plate, - 8 to
- 9.5°C; bath, - 26 to - 30° C.

A stainless steel holding tank (see Figure 2.2) immersed in

the glycol-water bath is used to liquefy and store the butane.
Butane is transferred into the tank from the commercial high
pressure bottle (at room temperature) after the tank is evacuated
by a roughing pump and sealed off. The high pressure bottle and
tank are isolated and the butane allowed to condense. A small
amount of helium is then admitted to pressurize (5-10 psig) the
liquid butane.

Butane is injected into the bottom of the column through a
shortened and squared off No. 19 (0.008'' i.d.) hypodermic needle
attached via standard LUER-LOK fittings and tygon tubing to a glass
5 cc syringe. The syringe is fitted with a cooling jacket through
which the secondary coolant is circulated and a micrometer screw
which is used to translate the plunger. Approximately 1/4 turn of
the micrometer screw will produce a single ~ 1 mm dia. droplet.
At the beginning of a series of experiments (and whenever the butane
in the syringe is depleted), the syringe is filled by backing off the
plunger micrometer and cracking open the valve to the pressurized
holding tank, allowing the butane to enter and push back the plunger.

Once a drop is formed at the end of the hypodermic needle,
often surface tension forces prevent it from escaping up the column.
A jet of host fluid directed upward from a tube concentric with the
needle gives the necessary impulse to start these drops drifting up-
ward. The jet is merely a short squirt produced by a manually
operated syringe located beneath the base plate. An insulated view-
ing port equipped with a magnifying lens is used to view the opera-

tion of the drop injector.

2.3 Operation

Due to the large mass of fluid and metal involved, the column
takes approximately 10 hours to reach a steady-state condition after
turn on. Between explosions of individual droplets a certain period
of time (5-10 minutes) must be allowed to elapse during which the
mixing induced by the explosion subsides and the quiescent conditions
are re-established. These circumstances compel the actual experi-
ments to be conducted in a series of runs, for each of which the
apparatus is left on continuously for several days. During each run
between 50 and 100 individual explosions would be successfully re-
corded. The data presented in this work are the cumulation of
numerous runs over a period of three years. Many involved testing
various diagnostic techniques and thus did not contribute to the
data bank.

The course of a typical run is as follows: the primary and
secondary cooling loops are turned on and the temperature in the
bath and base plate reduced below the butane boiling point. Butane
in the holding tank is replenished if necessary and the syringe
filled with butane; after testing the drop injector and checking the
level of host fluid the heater is turned on. Monitoring the test
section temperature profile to determine when steady state conditions
are reached, the heater controller is then adjusted to obtain the
desired explosion location, i.e., position of superheat limit in the
test section. During the initial equilibration period the electronics
and optics have been set up and aligned so that recording explosions

can begin at this point.

2.4 Instrumentation and Data Acquisition

The observations which form the basis of this work are
simultaneous recordings of the pressure waves measured by fast
piezoelectric transducers and single short-exposure photographs of
the exploding drops. In addition, some preliminary measurements
using spark shadowgraphy and optical extinction methods were first
steps in understanding the problem and are documented in Appendix

2.4.1 Pressure Measurements

The utility of fast pressure transducers depends on the exis-
tence of significant structure in the acoustic radiation of the ex-
plosion on a time scale short compared to the acoustic transit time
in the test section. This was in fact verified by the first experi-
ment in the apparatus; the sound speed in ethylene glycol at 108° C
is 1.5 mm/ypsec and the characteristic scale of the test section is
100 mm yielding a transit time of 66 usec, much larger than the
observed structure (Figure 3.19) time scale of 10 usec.

Transducers used were Model PCB 113A21 manufactured by
Piezotronics and were mounted directly in aluminum or brass plugs
placed in one of three locations in the test section shown in Figure
2.3. All transducers were mounted so the face of the sensitive
element (flush with the face of the plug) was normal to and concentric
with the nominal origin of the explosions, the center of the test
section.

Acoustically, the explosion can be characterized as a compact
volume source producing, at distances large compared to the source

size, outwardly propagating spherical waves. The pressure measured

by the transducer is the sum of the pressure produced by primary
waves directly propagated from the explosion and secondary waves
indirectly propagated through reflection and diffraction from the
interior of the test section. Due to the large difference in acoustic
impedance of the glycol and the mounting plug-transducer combination,
any wave incident on the transducer will undergo reflection, momen-
tarily doubling the signal strength. Similar reflection processes
occur at the other boundaries of the test section and result in sec-
ondary waves arriving at the transducer sometime later than the
primary wave. The presence of these secondary waves and others
produced by diffraction at the edge of the transducer plug, etc.,
make quantitative interpretation of the measured pressures difficult.
This is especially true at very long times (1-10 msec) where the
test section acts like a lossy acoustic resonator driven by the
oscillations of the bubble resulting from the explosion.

Overall, the qualitative appearance of the pressure trace is
similar to that produced in the absence of these deleterious effects
since the secondary waves have traveled much farther to reach the
transducer and hence, due to their spherical nature, are considerably
attenuated relative to the primary waves. Transducer configurations
A and B of Figure 2.3 were used primarily to obtain a reference
signal to trigger the light source used for photography. Configuration
C was used to obtain quantitative information at the earliest stages
(< 30 psec) of the explosion before the secondary waves first reached
the transducer.

The voltages produced by the transducers were directly re-

corded by a Nicolet Explorer Ill digital oscilloscope and stored as

digital ‘data on a minifloppy magnetic disk. The Explorer II could
be interfaced (via IEEE 488 bus} to a calculator-based computer
system (HP 9825) to whose floppy disk the raw data were permanently
transferred at the end of a run. The scaling, plotting and further
processing of the data presented in the present work were all carried
out with this system.

A very important feature of the Explorer II for this experi-
ment was the 'pretrigger'' option; a consequence of the digital
memory, this enables a portion of a waveform received before the
trigger signal to be saved. In this case, the trigger signal was the
pressure waveform itself and the use of this feature allowed the
beginning of this signal to be recorded and used as a reference

('zero time'') for succeeding events.

2.4.2 Photography

The technical accomplishment which made the present investiga-
tion successful was the photography of evaporating vapor bubbles
within the liquid droplet, Preliminary investigations (Appendix A)
with parallel light revealed that the basic difficulty was the focusing
action of the droplet due to the greatly differing indices of refraction
of host (n = 1.43) and drop liquid (n = 1.33). Parallel light incident
on the drop is refracted and emerges as a cone of light spreading
out so that only a small fraction enters the aperture of the camera
and is imaged on the film. Therefore, with parallel illumination,
drops photographed as black spots on a white background and nothing
could be learned about the processes in the interior of the drop.

The solution of this problem was the use of a source of very

diffuse illumination behind the drop. This is not as efficient a use

of the available light as parallel illumination but it does permit
photography of the interior of the drop. Initially, Xenon flash
lamps behind ground glass diffusers (Figure 2.4) were used for
jllumination but it was found that there was considerable blur in the
image due to the intrinsic long duration of the flash (5-10 psec).
However, some results of this preliminary method are unique and
are briefly discussed in Chapter 3.

To obtain a short duration light pulse (< 1 psec}, the primary
light source was changed to a high pressure (1 atmosphere) spark
gap discharge. A conventional ''point source'' design used for shadow-
graphy was extensively modified to increase repeatability, holdoff
and light output. The final configuration utilized most of the light
from a linear spark (0.25'' long) between 2 hemispherical electrodes
in an air/argon mixture (90%/10%). Typically the gap was operated
at 7 kV (with a 0.1 pf capacitor this yields ~2.5 J energy input
per shot) with triggering through a tertiary electrode driven by a
low inductance trigger transformer (EG&G). To be able to use con-
ventional photographic film it proved necessary to collect and use as
much of the light from the spark gap as possible. A reflector
behind the spark and a collimator-condenser system were used to
collect the light and focus it onto the entrance of a high efficiency
diffuser. The diffuser was simply a section of tubing lined with a
diffuse reflector (the dull side of Reynolds wrap), one end of which
was closed except for an entrance slit and the other end covered
by ground glass or lucite. A schematic of the basic arrangement
is shown on Figure 2.5 together with the test section and cameras.

As shown in Figure 2.5 the drop could be photographed

simultaneously from two views using the transmitted light with
camera A and the perpendicularly scattered light with camera B,.
Each camera was of the same type, a 4 X 5 format view camera
with a 135 mm focal length lens, modified by attaching a 35 mm
film transport in place of the usual sheet film holder. The lens-
film distance was adjusted to give a magnification of two on the
negative; for a typical f-number of 16 this gave an acceptable com-
promise between resolution and depth of field for the available illu-
mination. A 35 mm roll type film was used instead of the usual
sheet film for economy (the smallest images only occupied < 1% of
even the 35 mm format) and convenience in camera operation and
film processing. Kodak Tri-X (49@ ASA) film was used in both
cameras, for camera A the film was processed in Kodak Microdol-X
developer using standard development time plus two minutes; for
camera B the film was ''push-processed'' to a nominal 4990 ASA
using a fine grain developer (ETHOL UFG). Aperture settings of
each camera determined the exposure. For a particular spark gap-
diffuser combination the correct settings would be determined by a
series of test exposures at the different possible apertures.

The triggering signal used to fire the spark gap was derived
from the pressure signal generated by the explosion itself. An in-
herent delay corresponding to the time required for the pressure
signal to travel from the droplet to the transducer determined the
earliest possible time any photographs could be taken (the smallest
of these inherent delays was about 8 psec corresponding to a path
length of ~ 5 mm). An auxillary oscilloscope connected in parallel

with the pressure recording oscilloscope triggered a delay generator

immediately upon the appearance of the explosion pressure signal.
The delay generator produced a signal a preset length of time fol-
lowing the trigger which fired the spark gap. Prior to the explosion
(after the drop had been injected into the column), the room had been
darkened and the camera shutters opened so that the film is exposed
when the spark gap fires. Following the explosion the camera
shutters are closed, the films advanced one frame and electronics
reset in anticipation of the next explosion.

By reading off the time elapsed from the beginning of the
pressure signal to the glitch on the pressure trace caused by electro-
magnetic noise from the spark gap circuitry and adding the drop-
transducer acoustic transit time, a unique time could be assigned to

every photograph. It is this time, the time elapsed since the pressure

wave first left the drop, which is used to label all the photographs

appearing in the present work.

2.5 Conditions in the Drop at the Time of the Explosion

As the droplet translates up the column, heat is constantly
being transferred to the drop liquid from the host. This heat trans-
fer takes place at a finite rate; hence the temperature in the interior
of the drop lags behind the ambient temperature of the glycol. Fur-
ther, the droplet liquid is circulating due to the shear stress on the
surface of the drop that is generated by the translational flow field.

Both the circulation and finite rate of heat transfer are im-
portant in determining the final temperature field in the droplet at
the time of the explosion. This problem is very complex for the
range of conditions encountered in the bubble column and no general
solution exists. In this section, the relevant physical parameters
are pointed out and an estimate given for the temperature lag of the
drop at the time of explosion.

Physically, the most important parameters in determining the
heat transfer rate are the Reynolds number Re _ of the droplet
translation, or equivalently the Peclet number Pe = RePr, and the
viscosity ratio K, K =p drop/u host. Droplets rising up the column
accelerate since the drag force decreases as the glycol temperature
increases and its viscosity falls. Glycol viscosity rapidly decreases
from a value of 60 cP at 0° C to 7 cP at 50° C and the final value
in the test section is 2 cP at 100° C. In comparison the butane
viscosity varies only slightly, from 0.2 cP at 0° C to 0.1 cP at
100° C,

Therefore, typically Re ranges from ~ 0.1 at the bottom of
the column to 50-200 in the test section (the range in the final Re

corresponds to a range of 0.5-1.0 mm in drop radius). Those

Reynolds numbers were estimated from the terminal velocity cor-
relation given in Clift et al (1978) and the measured temperatures
along the centerline of the column.

At all times during the motion of the drop the butane viscosity
is much smaller than that of glycol. Neglecting the possible in-
fluence of impurities in the glycol, this implies that the butane
will be freely circulating inside the drop. Initially, when Re §& 1,
the creeping flow solution of Hadamard (discussed in Clift, et al
1978) will be valid and the maximum circulatory velocity will occur
at the drop center and is U/2, where U is the translational vel-
ocity. As the drop rises and Re increases, the circulation pattern
becomes more complex and its characteristic velocity decreases.

The heat transfer into the drop can be described in terms of
an average Nusselt number Nu, which for large Re but laminar

el /%p,! 3 To obtain some idea of what

flow has the form Nu~R
the temperature lag in the drop was, an ad hoc model of the heat

transfer process was constructed:

a) Circulation within the drop was assumed to be sufficient
to keep the drop fluid well mixed, so that the entire
volume of the drop participated in the heat transfer
simultaneously.

b) The Nusselt number was assumed to be constant and
given by empirical formula for solid spheres (White 1974),

Nu, = 2+ 0.3 pr!/3Re% 6

c) Assuming a constant Reynolds number and temperature

gradient, the overall thermal energy balance,

gazec at
qa = ©.

into the drop was used to determine the drop temperature as a

function of time.

Using the conditions characteristic of the test section,

ees = 1.5° C/cm and U = 10 cm/sec for a 0.5 mm radius drop,

and the temperature lag is found to be ~ 10° C,

Obviously, this calculation is quite crude and the assumptions
are rather drastic. Circulation is not that effective in keeping the
drop temperature uniform and the transport of heat across the closed
streamlines of laminar flow must occur by diffusion. Generally, the
outer part of the drop and the core of circulation along the center-
line will be hotter than the interior of the circulating regions.

Circulation, while difficult to properly treat, is nevertheless
of utmost importance. If only diffusive transport is effective within
the drop, then under the same conditions as used above the tem-

perature lag in the interior of the drop would be ~ 50° C.

REFERENCES

Blander, M. and Katz, J.L. 1975 ''Bubble Nucleation in Liquids",
AIChE Journal 21, 833.

Clift, R., Grace, J.K. and Weber, M.E. 1978 Bubbles, Drops
and Particles, Academic Press, New York.

Porteous, W. and Blander, M. 1975 '‘''Limits of Superheat and
Explosive Boiling of Light Hydrocarbons and Hydrocarbon
Mixtures", AIChE Journal 21, 560.

Reid, R.C. 1978 "Superheated Liquids: A Laboratory Curiosity
and, Possibly, an Industrial Curse", Chem. Eng. Ed. 12, 60.

Wakeshima, H. and Takata, K. 1958 "On the Limit of Superheat",
J. Phys. Soc. Japan 13, 1398.

White, F.M. 1974 Viscous Fluid Flow, McGraw-Hill, New York.

TABLE 2.1

REPRESENTATIVE PHYSICAL PROPERTIES OF LIQUID ETHYLENE GLYCOL

Critical Temperature T. 372 °C
Boiling Temperature (1 Atm) Ty 197.6 °C
Freezing Temperature (1 Atm) Ty -13 °C
Density (20°C) p> 1.11 g/cm*
Specific Heat (20°C) CF 2.34 J/gm°C
Thermal Conductivity (20°C) kK 2.8 x 107% W/cm? C

Viscosity (20°C) Vee 20.9 cP

TABLE Z.2

REPRESENTATIVE PHYSICAL PROPERTIES OF SATURATED N-BUTANE LIQUID

Critical Temperature

Superheat Limit Temperature (1 Atm)
Boiling Temperature (1 Atm)
Freezing Temperature (1 Atm)
Density (20° C)

Specific Heat (20°C)

Thermal Conductivity (20°C)

Viscosity (20°C)

Tor

Ty,

C20
12°

156
105
-0.5

- 138.3
0.579
2.39

1.18 x 10°

°C
°C
°C
°C
gm/cm*
J/gm°C
W/cm? C

C D

he T

To Secondary
Cooling Loop

Butane In

FIG.2.1 BUBBLE COLUMN AND TEST SECTION

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FIG. 2.3 TRANSDUCER LOCATIONS WITHIN THE TEST

TRANSDUCER FACE)

SECTION (T

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tj. SURVEY OF OBSERVATIONS

3,1 Introduction

This chapter contains an exposition of selected photographs and
pressure traces of individual explosions arranged to illustrate the
general sequence of events and particularly interesting features dis-
covered in the present investigation. It should be made clear that
only one photograph (or two different views taken simultaneously) was
taken for each explosion and that all the time sequences shown are
composites, each print having been obtained from a different explosion.

A large catalog of individual explosions consisting of pressure
traces and photographs taken at varying time delays was built up
during the course of the experimental investigation. These records
represented a large spectrum of drop sizes and delay times and were
sorted into subsets according to the similarity of pressure traces,
stage of explosion seen and presence of unusual features. Out of
nearly 500 photographs approximately 70 were selected as being of
best quality and greatest interest and are presented here. The
majority of these selected prints are arranged in a series of 7 plates
which form a photographic history of a ''typical'' explosion. All of
the prints in this series represent explosions of comparable strength
as determined by the similarity of the pressure traces.

This chapter is organized as follows: First, an overview of
the results is illustrated by displaying selected prints from the main
series in combination with a pressure trace on a single plate. Second,
a detailed discussion of the explosion utilizing the main series of
plates follows, and particular points are illustrated with additional
plates. Third, a selection of pressure traces is presented demon-

strating the effect of initial drop size and variety of time scales

characterizing the acoustic radiation.

3.2 Overview

An overview of the various stages in the explosion up to 2
msec is shown in Figure 3.1. The bar underneath each print in-
dicates a scale of one mm (this is the convention used throughout;
in Table 3.1 the experimental run number and delay times are given
for the individual prints in each figure). Prints are in sequence of
increasing time as though they were frames excerpted from a movie
of a single droplet exploding.

The explosion begins 13 wsec before print (a) with the emission
of the first pressure signal from a spontaneously created nucleus
of vapor (the critical radius given by equation 1-2 is only 4 nm!).
The bubble, which has grown from this nucleus, can be seen in
print (a) located asymmetrically within the butane drop, which in
turn, is surrounded by the host liquid (the background is the uniform
diffuse light, which in most prints is pure white). The darkening at
the edge of the drop and the light ring just inside are optical effects
due to the strong scattering of light rays at grazing incidence, and a
similar darkening of the bubble periphery can also be seen. Of par-
ticular note is the regular pattern seen on the bubble, suggestive of
a wrinkled surface. This surface is the evaporating liquid-vapor
interface of the growing vapor bubble and the idea that interface
wrinkling is due to an evaporative instability is explored in Chapter
5.

As the bubble grows, a portion of its surface contacts the host
liquid and thus becomes nonevaporating. This relatively smooth non-

evaporating surface can be seen bulging out of the drop at the top in

print (b) (56 psec). Other views of the bulge more clearly show a
unique axisymmetric wave-like structure, the nodes appearing in some
cases to be very sharp corners (one such node can be seen in print (b)).
Simultaneously, the evaporating interface had developed a very rough
appearance and scatters light strongly so that the bubble appears
black. The non-evaporating interface spreads over the surface of the
drop as evaporation proceeds and in print (c) (88 psec) the roughness
of the evaporating interface can be seen still imprinted on this surface.
In print (c), only a small amount of the butane in the drop is still
liquid and appears as a lighter crescent-shaped area at the bottom of
the drop.

When the liquid butane has been completely evaporated (occurring
between 40-160 usec depending on the initial drop size), the resulting
gas bubble is at a higher pressure than ambient due to the nonequilibrium
nature of the evaporation process and so continues to expand. Surface
disturbances prominent in print (c) smooth out during the expansion
process and result in the smooth bubble seen in print (d) (610 usec).
Due to the large difference between the indices of refraction of the
vapor and liquid the bubble appears black except for a centrally
located white spot of transmitted light. At this point, the bubble has
actually overexpanded to a pressure less than ambient due to the inertia
of the surrounding fluid and begins to collapse. As the gas inside the bubble
is compressed, ultimately halting the collapse, the light vapor is accelerated
into the heavier host fluid, a Taylor-unstable motion. Surface distortions
induced by this instability at the maximum radius of the oscillation give
the bubble the dramatic appearance in print (e) (1.05 msec).

This instability is catastrophic in that the bubble never recovers

the smooth surface seen at the first volume maximum (print (d)). As

the bubble re-expands from the first minimum, the surface distor-
tions change scale as seen in print (f) (1.5 msec). The bubble con-
tinues to oscillate in volume and Taylor instability at successive
minima progressively breaks down the bubble surface to produce
smaller bubbles. In all experiments the final state visible to the
human observer is a cloud of small bubbles rising toward the surface
of the host liquid.

The pressure trace shown in Figure 3.1 was obtained from a
transducer 2.5 cm from the drop (configuration C of Figure 2.3).
Letters labeling particular points on the pressure trace refer to the
previously discussed photographs on this figure and show the stage
of the explosion occurring at that time.

The initial pressure rise (between a and c on Figure 3.1) is
due to the evaporation process. Increasing pressure in the vapor
bubble and displacement of surrounding fluid by the specific volume
increase in the liquid-vapor transition results in the radiation of
compression waves. The broad peak just after point c corresponds
to the completion of the evaporation process. While still expanding,
the surface of the gas bubble is decelerating and radiating expansion
waves into the surrounding liquid, so the pressure falls. At the

maximum volume (point d), the minimum pressure occurs. As the

bubble collapses, the gas inside compresses and radiates compression

waves, so the pressure rises to a peak at minimum volume (point e).

Continuing to oscillate in volume, the bubble radiates pulses at the
minima, the oscillations diminishing in amplitude as its energy is
radiated away and otherwise dissipated, and as disintegration into

smaller bubbles occurs.

3,3 Detailed Discussion

The main series of plates is presented as a group in Figures
3,2-3.8, arranged in order of increasing time. In Figure 3.9, the
particular stage of the explosion covered by each plate is indicated
on two representative pressure traces. Discussion of these plates
is found in the succeeding sections of this chapter and serves as a
framework for a more detailed exposition of various aspects of the

explosion.

3,3,1 Nucleation and Initial Growth

The earliest stage at which the explosion has been photographed
is ~ 8 psec after the first pressure signal has left the drop. In this
time, nucleation has occurred and from the microscopic nucleus, a
bubble of ~ 200 um diameter has grown. At large superheats this
growth process is highly nonequilibrium and may possibly involve
noncontinuum phenomena; the actual details are unknown. The only
information obtained in the present experiment on this process is
contained in the pressure signal, which at these times exhibits a
characteristic ''bumpy'' structure that is remarkably repeatable from
explosion to explosion. Examples of these pressure signals are
presented in Section 3.4 and a detailed discussion of the implications
of this structure and comparison to the predictions of the standard
theory is found in Chapter 4.

Photographs taken at the earliest times (using transducer con-
figuration B for triggering, Figure 2.3) are seen in plates 3.2 (main
series) and 3.10 (somewhat smaller initial drop sizes). Two features
stand out; first, all of the bubbles are very close to the drop surface

and in many cases (prints (c)-(f) of Figure 3.2) a small bulge can be

seen protruding out of the drop; second, the bubble surface appears
wrinkled, in some cases regularly and in others with a spectrum of
length scales.

The appearance of bubbles so close to and in many cases inter-
acting with the drop surface raises the inevitable question: Is the
nucleation truly homogeneous? It is reasonable that bubbles should
form close to the drop surface since there is a hot thermal layer
jn the outer part of the drop (as discussed in Section 2.5) and the
nucleation rate is strongly dependent on temperature. Unfortunately,
because no bubbles younger than 8 psec could be observed in these
experiments, all observed bubbles were of finite radius and it was
impossible to determine definitely whether nuclei formed within or on
the surface of the drop. Whether the nucleus actually forms on the
surface or in the interior of the drop depends on the relative size of
the interfacial forces acting on the emerging nucleus. Jarvis et al
(1975) in an amplification of an idea of Moore (1959) have delineated
three mechanisms of nucleation for superheated droplets immersed in
a host liquid. These three mechanisms are depicted in Figure 3.12:

a) homogeneous nucleation, occurring when the droplet liquid wets

(spreads on) the host; b) surface nucleation, occurring when neither

host nor droplet liquid wets the other; c) ''bubble blowing'', occurring

when the host wets the droplet liquid. Jarvis et al measured the
interfacial tensions at room temperature for a pentane droplet-ethylene
glycol system and concluded from extrapolation to the superheat limit
that nucleation would be homogeneous with a large margin left for
error. Butane is very similar to pentane (they are adjacent members

of the alkanes) and the interfacial tensions that are known for butane-

glycol are close enough to those of pentane-glycol to suggest that
the same conclusion can be drawn in the present experiment.

There is conclusive evidence that in at least one case nucleation
did occur in the interior of the drop; this was obtained in the course
of experimenting with dual flash lamp photography. In Figure 3.11,
four sets of prints are shown which are simultaneous views (''A’' and
"B'') taken at right angles to one another; set (a) clearly shows a
bubble completely within a drop. In summary, the evidence is strong

that nucleation occurs in the interior of the drop near the surface and

it is during the growth of the resulting bubble that interaction with

the surface occurs (typically before 10 fusec has elapsed).

3.3.2 Developing Bubble Structure

As the bubble grows from the nucleus within the drop, evap-
orating and displacing the surrounding butane and host liquid, a com-
plex and distinctive structure develops which is schematically in-
dicated in Figure 3.13. The off-center location of the bubble results
in a portion of the bubble surface closest to the drop surface contact-
ing the host fluid and becoming nonevaporating before any other portion
does so, This nonevaporating interface area grows axisymmetrically
with time, its boundary (the triple intersection of host, butane liquid
and vapor) taking, for a spherical drop, the form of a circle on the
surface of the drop. The axis of symmetry passes through the
Original drop center, the centroid of the bubble and the point where
the bubble surface first contacted the host. Random orientation of
this axis (due to the random nature of nucleation) with respect to the
camera results in a variety of perspectives on the developing bubble.

Circumferential bulges or waves culminating in a spherical cap

progressively appear on the nonevaporating surface as it covers the
drop. The first stage in this process can be seen as the single pro-
truding bulge present in many of the prints in Figures 3.2 and 3.10.
Subsequent growth of the single bulge into a spherical cap and the
addition of surrounding circumferential waves can be seen in all the
prints of Figure 3.3; in particular, the cap is pointed directly at the
viewer in print (e) and the symmetry of the waves is striking.
Striking proof of how axisymmetric these features are is found in
print sets (b), (c) and (d) of the dual flash lamp photographs in
Figure 3.11; with a certain amount of effort the viewer can visualize
the axis of symmetry and corresponding features on each print of
each set. Print set (a) of the dual view spark gap photographs
(Figure 3.23), also vividly shows these features.

Once the basic structure of the developing bubbles was deter-
mined, some highly unusual appearing explosions could be recognized
as merely peculiar perspectives of the standard configuration. The
best example is a series of prints seen in Figure 3.14 which shows

the development of the bubble from the back side; that is, the cap

is on the far surface of the drop from the viewer and only its out-
line is seen through the evaporating interface as a definite ring.

A certain amount of variation of the basic nonevaporating struc-
ture exists between individual explosions. Examples of this are
seen most clearly in the later stages of evaporation in Figure 3. 4;
prints (b) and (d) show the cap and several annuli facing the viewer
while (a), (c), (e) and (f) show profile views. In these views, an
area of irregular wrinkling of the nonevaporating bubble surface above

its boundary (Figure 3.13) can also be seen; this is a result of the

evaporating interface wrinkling which will be discussed in connection
with that phenomenon. The cap is indistinct in print (e) and no
annuli are visible; in contrast, print (f) shows very clear and sharp
edged annuli surrounding a prominent spherical cap; prints (a) and
(c) represent intermediate cases. Two of the most plausible causes
of this individual variation are: random variations of the nucleus!
distance from the drop surface and the dynamic nature (i.e. traveling
waves or oscillating standing waves) of the surface disturbances, To
actually pinpoint the mechanism, it would be necessary to take very
high speed (microsecond framing rates) movies of individual explosions.
Peculiar bubble shapes and surface waves have been seen pre-
viously in studies of cavitation bubble collapse (Ellis, 1965) and inter-
acting bubbles (Hooper et al, 1970), but in the present context this
phenomenon is completely unexpected and rather bizarre in appearance.
It is conjectured that the surface distortions and the protrusion into
the host liquid are both driven by the "jet'' of evaporated vapor coming
from the opposing interface and impinging on the nonevaporating inter-
face. Thus, the appearance of surface disturbances should be modeled
by the start-up of a gas jet into a liquid; some exploratory flow visu-
alization in a simple apparatus verified this and is described in Ap-
pendix B. However, the phenomenon is not completely understood and

the similarity of the two flows may be superficial.

3.3.3 Structure of the Evaporating Interface

The unusual shape of the nonevaporating bubble surface is inter-
esting, but a much more important discovery is the tremendous
wrinkling and extreme roughness of the evaporative interface. This

wrinkling begins on the smooth and spherical interfaces of the

50
earliest stages as a rather regular pattern covering the entire
bubble (prints (a) and (b) of Figure 3.2; prints (a), (b) and (c) of
Figure 3.10). At the same time that the bubble is interacting with
the drop surface, the regular pattern is developing into a larger scale
folding of the surface (prints (c), (d), (e) and (f) of Figure 3.2;
prints (e) and (f) of Figure 3.10). Very rapidly (within ~ 5 psec)
the evaporating surface becomes roughened to the extent that light
no longer passes through, resulting in the opaque bubbles seen in
Figure 3.3. This rapid transition from regular wrinkling suggests
the presence of a fluid dynamic instability driven by evaporation and
manifested by distortion of the interface. Further development of
this idea is found in Chapter 5.

Fuzzy outlines of the evaporating bubble surface (Figure 3.3)
and the grainy appearance of the interface in the back views (Figure
3.14) demonstrate the extent of the disruption of the interface.
Photographs of perpendicularly scattered light show that the evaporat-
ing interface is an extremely good diffuse reflector, an idea consistent
with the presence of fine scale roughening. As the bubble grows,
the evaporating surface remains in the mean symmetrical, although
at later times some large scale disturbances can be seen (prints (a),
{e) and (f) of Figure 3.4). These large scale disturbances, along
with a spectrum of other scales, can also be seen impressed on the

nonevaporating surface between the boundary of the nonevaporating

region and the more regular features near the axis of symmetry.
The overall impression is that, after a period of development,
the wrinkling of the interface saturates and persists at a constant

level for the remainder of the evaporation process. The persistent

roughening of the interface appears random in orientation and occurs
on many length scales, suggesting a fundamental instability of the
evaporation process. The significance of the roughening to the
dynamics of the evaporation process lies in the tremendous increase
of the effective area of the evaporating surface. This effective area
increase yields a proportional increase in the evaporative mass flux
per mean surface area. Calculations performed in Chapter 4 show

that this mass flux is orders of magnitude larger than the standard

diffusion-limited theory predicts. Analogous effects in other fluid
dynamic contexts are well known, for example, the vastly greater
effectiveness of turbulent over laminar mixing.

Disturbances on evaporating surfaces have been previously
observed (Hickman 1972) and possible instabilities of evaporating
interfaces have been previously proposed (Palmer 1976, Miller 1973).
However, none of the previous observations of bubble growth have
shown unstable evaporating interfaces; all of the growing vapor bubbles
observed had smooth and regular surfaces. A list of these visualiza-
tion studies and the experimental conditions in each is given in Table
3.2. Important points are that all of these experiments were done
at very low superheats (none at or even near the superheat limit) and
nucleation and growth occurred entirely in the bulk liquid. There have
also been some investigations of slightly nonequilibrium droplet evapora-
tion (Tochitani et al 1977, Simpson et al 1974) and bubble growth on
electrically pulse heated wires (Faneuff et al 1958) which do not show
any wrinkling or instability of the evaporating surface.

In comparison to all these previous visualizations, the present

experiment is unique. The degree of superheat and consequently the

52
evaporative mass flux is much larger than in any of the experiments
in Table 3.2. Therefore, it is not surprising that new phenomena
should appear. Particularly, it is important to note that the present
observations are completely different than those predicted by the mere
extrapolation of the near-equilibrium results and theories. The ex-
tent of the generality of the present results needs to be verified in
detail, but clearly, the present observations indicate that evaporation
at the superheat limit can be much more complex than previously

envisioned.

3.3.4 Summary of Evaporation Stage

The final disappearance of the liquid butane in the drop cor-
relates very well with the occurrence of the first peak in the pressure
signal. At this point, the evaporating interface appears to have con-
sumed all the liquid butane and the obvious conclusion is that evap-
oration has ceased. However, the possibility also exists that the
bubble still contains a liquid-vapor mixture at this time, and phase
changes occur during the subsequent bubble oscillations. Unfortunately,
the light passing through the bubble is strongly scattered due to the
enormous difference of host and vapor indices of refraction and no
information is available on this question. However, the pressure
produced by the oscillating bubble strongly suggests that its contents
are predominantly gaseous and this is the point of view taken through-
out this discussion.

Examples of explosions near the end of the evaporative stage
are shown in Figure 3.5; in prints (a), (b), (d) and (e), a small
amount of butane liquid can still be seen in the drop. The char-

acteristic cap and annular waves are visible in all these prints;

also distributed over the surface is a spectrum of more irregular
disturbances originating on the roughened evaporating interface. All
of these surface features appear ''softer'’ than at earlier stages, in-
dicating that the smoothing effect of surface-wave propagation has
taken effect.

From nucleation to the final consumption of the liquid butane,
the evaporation process is summarized in Figure 3.15. Outlines of
the vapor bubble within the drop are shown (taken from actual photo-
graphs) at the various stages of the explosion. Evaporation begins
at (a) with the creation of a nucleus near the drop surface; by the
time 8 psec has elapsed the nucleous has grown to a bubble of ~ 200
fim diameter in (b). Interacting with the drop surface, the growing
bubble displaces the surrounding fluid and develops its distinctive
structure in (c), (d) and (e). Simultaneously, the characteristic
wrinkled structure of the evaporating interface develops and persists
for the remainder of the evaporation. Ultimately all the liquid butane

is vaporized and only a gas bubble remains in (f).

3.3.5 Gas Bubble Oscillations and Breakup

Immediately following the cessation of evaporation, the gas
bubble is at a net positive overpressure, a consequence of the dy-
namic (nonequilibrium) nature of the previous event, occurring in the
evaporation process. Evaporation at the superheat limit occurs suf-
ficiently rapidly that the increased pressure in the bubble can only
be partially alleviated by volume expansion and acoustic radiation
before the droplet is completely vaporized. The overpressure remain-
ing in the bubble represents stored energy, which is then released

during the subsequent oscillations of the gas bubble. A portion of

this energy is radiated away into the environment (the "ping" heard
upon evaporation), the rest is dissipated through the creation of new
bubble surface (Taylor instability), the viscous dissipation associated
with the gas and host liquid motion and the process of thermally
equilibrating the vapor with the host.

The fate of the oscillating bubble is determined by the coupling
between volume oscillations and surface oscillations (capillary waves
on the bubble surface), the spherical analog of the Taylor instability
(Taylor 1950) for plane surfaces. This coupling results in
amplification of small surface disturbances into drastic deformations,
ultimately causing the breakup of the bubble. Some results of the
linear stability theory of this coupling and the application to the
freely oscillating bubbles, such as occur in the present work, are
presented in Appendix C. In summary, growth is stabilizing, col-
lapse destabilizing and the most unstable region is near the minimum
radius (Taylor instability). Furthermore, in the spherical case the
possibility arises of algebraic growth, a behavior which does not
occur in the plane case.

Initial surface distortions are damped out as the bubble expands
(Figure 3.6), at the maximum radius (print (e)) only an evenly dis-
tributed small amplitude modulation (''orange peel'') remains. As the
bubble collapses (Figure 3.7) toward the minimum radius, the first
distortions seen (print (a)) are already in the nonlinear stage. At
the minimum radius, the tremendous outwardly directed acceleration
(on the order of 10*g! C.f, 50 g in the original experiments on the
Taylor instability (Lewis 1950)) on the interface produces the con-

voluted surface seen in print (b). Rebounding from the minimum,

(print (c)), the bubble stretches its surface, the scale of the defor-
mations correspondingly increases and the number of nodes decreases.
Away from the minimum, expansion of the bubble has a stabilizing
effect on small surface disturbances. However, the deformations
have developed to such an extent (prints (d), (e) and (f)) that this has
little influence and the smooth surface of the first maximum is never
recovered,

Although the bubble is quite deformed after passing through the
first minimum, it continues to oscillate in volume. These volume
oscillations are confirmed by the extinction meter measurements
described in Appendix A and the bubble pulses radiated at the minima
(in comparison, the surface deformations contribute very little to the
acoustic emission). The selection of prints in Figure 3.8 reveals
the highly complex and nonlinear shapes the bubble surface assumes.
After several cycles of oscillation, distinct smaller bubbles can be
seen (prints (e) and (f)) pinching off from the original bubble. The
destructiveness of the surface instability is quite evident in Figure
3.16. These five pictures and the pressure trace are representative
of smaller initial drop sizes than seen in Figure 3.8. Deformations
appear to be even more drastic, and in two cases (prints (d) and
(e)), the bubble has disintegrated altogether. Finally, a completely
different appreciation of the deformed bubbles is obtained from the
image of the scattered light. In Figure 3.17, print sets (b), (c) and
(d) show simultaneous views of the transmitted (white background) and
scattered light (dark background).

Bubble oscillations and surface instabilities have been exten-

Sively studied in recent years (e.g. Plesset and Prosperetti 1977)

and the phenomena seen in the present experiment contribute nothing
fundamentally new to this field. In particular, the instability of an
oscillating gas bubble has been previously observed in underwater
explosions (Cole 1948), cavitation research (Benjamin and Ellis

1966, Lauterborn 1974) and acoustically driven bubbles (Kornfeld and
Suvorov 1944, Hullin 1977).

However, the oscillating gas bubbles produced by droplet vapor
explosions are unique. The bubble is filled with an inert gas, unlike
the vacuous bubbles of cavitation or the high temperature bubbles of
underwater explosions, and the radial oscillation amplitude is large
so that the results are dramatic, but the minimum bubble radius is
not so small that the interesting events which occur there cannot be
observed (a problem typical of cavitation bubbles). This combination
of ideal circumstances in the present experiment has resulted in
particularly good photographs of the instability which show much more
detail than previous work.

There has been one previously published photographic investiga-
tion of the oscillating bubbles resulting from droplet vapor explosions.
Apfel and Harbison (1975) took movies at a framing rate of 3500/sec
of the superheat limit explosion of an ether droplet suspended in
glycerin. The actual evaporation process was not resolved, since
the framing rate of the camera was too low, but the resulting bubble
oscillations and instability upon bubble collapse were observed.
Overall, these observations are consistent with those of the present
work, but it is difficult to judge from their published photographs

whether the details of the instability are the same.

3.4 Pressure Signals

Each explosion emits a distinctive and unique pressure signal.
No two are exactly the same in detail, but there is an obvious
similarity of form; yet systematic differences occur between individual
explosions. Orientation of the bubble, its location within the drop and
the unsteady nature of bubble growth are uncontrollable factors which
produce unsystematic differences during the early stages of explosion.
During the later stages, the unsteady motion associated with the
instability of the oscillating gas bubble produces even more unsystematic
variations (this effect is also pointed out by Cole (1948)). Variations
in drop size and drop-transducer orientation produce systematic dif-
ferences in pressure signals. Despite these individual differences,
all pressure signals show the initial rise due to evaporation and the
succeeding pulses from gas bubble oscillations.

There are several characteristic time scales on which there is
significant structure in the pressure signal. The smallest of these
are the first 10-20 psec associated with the development of the vapor
bubble and its evaporating surface. Intermediate time scales are the
duration, 40-160 psec, of the evaporative stage and the interval, 0.3-
1.2 msec, between the subsequent bubble pulses produced by the
volume oscillations. The longest time scale present in the signal,
3-10 msec, characterizes the decay in bubble-pulse peak amplitude
that is due to the damping of the volume oscillations. Examples of
pressure signals, measured ~ 2.5 cm from the explosion with trans-
ducer configuration C, Figure 2.5, are shown in Figures 3.18, 3.19,
3,21 and 3,23, Each of these figures illustrates the effect of initial
drop size on the pressure signal for one of the characteristic time

intervals listed above.

Figure 3.18 compares pressure signals measured during the
initial 50 psec of 3 different explosions. In all 3 cases, the signal
begins with a characteristic two-step structure followed by a generally
monotonic increase, on which may be superimposed a nonrepeatable
oscillation.

Initial steps or peaks in the first 10-15 usec are a feature of
almost every pressure signal measured in the present study and seem
to be reasonably independent of the initial drop size. The universality
of this intriguing characteristic signal suggests a fundamental and re-
peatable unsteadiness in the nucleation and bubble development taking
place at this time. A quantitative investigation of this possibility is
undertaken in Chapter 4.

In Figure 3.19, pressure signals from 4 different explosions
are compared over the entire evaporative stage, including in some
cases the end of evaporation and the resulting gas bubble expansion.
The size of the initial drop, decreasing from trace (a) to trace (d),
affects both the scale and details of the pressure signal produced
during evaporation. The most obvious effect is that larger drops
require a longer time to completely evaporate and produce a higher
ultimate pressure. A quantitative expression of this correlation be-
tween evaporation time and ultimate pressure is shown in Figure 3.20
for 46 different explosions. Fundamentally, both of these quantities
must depend on the initial drop diameter. However, this dependence
was not determined in the present investigation. More subtle is the
effect on the characteristic initial structure, which is generally more
prominent for smaller drops (e.g. trace (d)) than larger ones (e. g.

trace (a)). Possibly, this is due to the reflection, at the interface

of the glycol and remaining liquid butane, of the pressure signal
produced by the growing bubble. Such an effect, discussed further
jn Section 4.2, would only be important for the earliest stages of
bubble growth and would affect a greater portion of the initial signal
as the initial drop size was increased.

Figure 3.21 compares pressure signals measured over the first
2 msec of 3 different explosions. Both the initial pressure increase
due to evaporation and the succeeding bubble pulses due to volume
oscillation can be seen in all traces. Spikes on the signals (also
seen in Figure 3.23) at 1.05 msec are the record of the electro-
magnetic noise produced by the spark gap used in the photography.
Initial drop size decreases from trace (a) to trace (c). Larger drops
produce larger gas bubbles and the bubble oscillation period increases
with bubble volume. Therefore, the time interval between bubble
pulses is an increasing function of initial drop size. This correlation
is quantitatively expressed in Figure 3.22 by plotting the interval
between the end of evaporation and the peak of the first bubble pulse
vs. the length of the evaporative stage. Bubble-pulse peak amplitude
is of the same order as the ultimate pressure produced in the evap-
orative stage. This is in sharp contrast to the case of underwater
explosions produced by detonations; the initial pulse (shock wave) due
to the detonation process has an amplitude many orders of magnitude
larger than the following bubble-pulse peak amplitudes. Finally, the
pressure pulses produced by the oscillating gas bubbles are not as
clean and simple as the idealized theory would predict (e.g. trace (c) of

Figure 3.2), Many of the smaller peaks, seen between the main pulses, are

artifacts from secondary waves generated by reflection inside the

test section. Also, as mentioned previously, the instability of the
oscillating bubble surface contributes to the irregular appearance of
the signal.

The longest interval over which pressures were recorded was
10 msec. Examples of signals over this interval are shown in Fig-
ure 3.23. Smaller explosions (e.g., trace (c)) show a regular damp-
ing of the signal followed by a nearly constant amplitude oscillation.
Larger explosions (traces (a) and (b)) show the same general damping
but a more complex oscillatory signal, which suggests a superimposed
oscillation in the mean test section pressure. Such an oscillation in
the mean pressure could be due to the net displacement effect of the

explosion exciting a low frequency response of the test section.

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sions of Superheated Droplets", J. Acous. Soc. Am. 57, 1371.

Benjamin, T.B. and Ellis, A.T. 1966 ''The Collapse of Cavitation
Bubbles and the Pressures Thereby Produced Against Solid
Boundaries", Phil. Trans. Roy. Soc. A260, 221.

Cole, R.H. 1948 Underwater Explosions, Princeton University Press,
Princeton, New Jersey.

Dergarabedian, P. 1953 ''The Rate of Growth of Vapor Bubbles in
Superheated Water", J. App. Mech. 20, 537.

Dergarabedian, P. 1960 "Observations on Bubble Growth in Various
Superheated Liquids", J. Fluid Mech. 9, 40.

Ellis, A.T. 1965 “Parameters Affecting Cavitation and Some New
Methods for Their Study'', California Institute of Technology,
Hydrodynamics Laboratory, Report No. E-115.1.

Faneuff, C.E., McLean, E.A. and Scherrer, V.E. 1958 ‘''Some
Aspects of Surface Boiling’, J. App. Phys. 29, 80.

Florshuetz, L.W., Henry, C.L. and Khan, A. Rashid 1969 "Growth
Rates of Free Vapor Bubbles in Liquid at Uniform Superheats
Under Normal and Zero Gravity Conditions", Int. J. Heat Mass
Transfer 12, 1465.

Hewitt, H.C. and Parker, J.D. 1968 '"'Bubble Growth and Collapse
in Liquid Nitrogen", J. Heat Transfer 90, 22.

Hickman, K. 1972 ''Torpid Phenomena and Pump Oils", J. Vac.
Sci. Tech. 9, 960.

Hooper, F.C., Eidlitz, A. and Faucher, G. 1970 'Bubble Growth
and Pressure Relationships in the Flashing of Superheated
Water", Vols. 1-3, Technical Publication 6904, University of
Toronto, Department of Mechanical Engineering.

Hullin, C. 1977 "Stabilitats grenze pulsierender Luftblasen in
Wasser'', Acustica 37, 64.

Kosky, P.G. 1968 ‘Bubble Growth Measurements in Uniformly
Superheated Liquids", Chem. Eng. Sci. 23, 695.

Jarvis, T.J., Donohue, M.D. and Katz, J.L. 1975 ''Bubble
Nucleation Mechanisms of Liquid Droplets Superheated in Other
Liquids", Journal of Colloid and Interface Science 50, 359.

Kornfeld, M. and Suvorov, L. 1944 ''On the Destructive Action of
Cavitation’, J. App. Phys. 15, 495.

REFERENCES (Continued)

Lauterborn, W. 1974 ''General and Basic Aspects of Cavitation",
in Finite-Amplitude Wave Effects in Fluids, (ed. L. Bjérng¢),
IPC Science and Technology Press Inc., Surrey, England, 195.

Lewis, D.J. 1950 ''The Instability of Liquid Surfaces when Acceler-
ated in a Direction Perpendicular to Their Planes. II", Proc.
Roy. Soc. A202, 81.

Miller, C.A. 1973 "Stability of Moving Surfaces in Fluid Systems
with Heat and Mass Transport - Il. Combined Effects of
Transport and Density Difference Between Phases'', AIChE
Journal 19, 909.

Moore, G.R. 1959 '"Vaporization of Superheated Drops in Liquids",
AIChE Journal 5, 458.

Niino, M., Toda, S. and Egusa, T. 1973 ''Experimental Investiga-
tion of Nucleation and Growth of a Single Bubble Using Laser
Beam Heating'', Heat Transfer Japanese Research 2, 26.

Palmer, J. 1976 ''The Hydrodynamic Stability of Rapidly Evaporat-
ing Liquids at Reduced Pressure’, J. Fluid Mech. 75, 487.

Plesset, M.S. and Prosperetti, A. 1977 ''Bubble Dynamics and
Cavitation'’ in Am. Rev. Fluid Mech. (ed. M. van Dyke, et al),
Annual Reviews Inc., Palo Alto, California, 145.

Simpson, H.C., Beggs, G.C. and Nazir, M. 1974 ''Evaporation of
Butane Drops in Brine", Desalination 15, 11.

Taylor, G.I. 1950 ''The Instability of Liquid Surfaces when Acceler-
ated in a Direction Perpendicular to Their Planes. I", Proc.
Roy. Soc. AZOl, 192.

Tochitani, Y., Mori, Y.H. and Komotori, K. 1977 ‘'Vaporization
of Single Liquid Drops in an Immiscible Liquid Part I: Forms
and Motions of Vaporizing Drops", Warme-und Stoffubertragung
10, 51.

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IV. BUBBLE GROWTH AND EVAPORATION RATE

4.1 Introduction

It is possible using the data obtained in these experiments to
obtain quantitative information about the early stages of explosive
evaporation. In particular, a first estimate can be made of the
evaporative mass flux during the explosion. The pressure signal
emitted by the explosion can be used to calculate an equivalent
acoustic source, whose strength is the time derivative of the fluid
volume displaced by the growing bubble. The net volume displaced
by the bubble is determined by the balance between fluid flowing into
the bubble (evaporative mass flux) and fluid pushed ahead of the
growing bubble. Digital analysis of selected photographs has been
used to determine the bubble volume and mean evaporating interface
area as a function of time. Thus, knowing the net fluid volume
displaced by the bubble (from the pressure signal) and the physical
volume of the bubble (from the photographs), the total mass flux into
the bubble can be determined as a function of time. Further, know-
ing the mean evaporating surface area, the mean evaporative mass
flux can be calculated.

In this chapter, the physical basis of the calculation outlined
above is derived, the methods of data reduction are described and
results using data from the first 30 usec of the explosions observed
in the present study are presented. Unfortunately, due to the sparse-
ness of the bubble volume data, a rigorous and detailed comparison
to the predictions of the standard model of bubble growth cannot be
made. However, a general comparison is possible; there is clear

indication that the mass flux is much larger than predicted by

available theories, as might be expected in view of our observations

of highly wrinkled evaporative interfaces.

4.2 Principles of the Mass Flux Calculation

The calculation of the equivalent acoustic source for the initial
phase of the explosion is greatly simplified for two reasons, First,

the growing bubble can be considered a compact source; that is, the

wavelengths } of the radiated sound are all much larger than the
characteristic size of the bubble. For example, an oscillation with
period 3 psec would have a 4.5 mm wavelength (in glycol), which is
typically 10 bubble diameters at times of order 15 wsec. Second,
the pressure transducer is in the far field of the bubble; that is, the
characteristic size of the bubble R is much smaller than the
bubble-to-transducer distance r, r/R >> 1. For example, the ex-
ploding drops (therefore bubbles) were nominally 25 mm from the
transducer for these measurements; this distance is ~ 50 bubble radii
at 15 psec.

Under the simplifications given above, the first approximation
to the acoustic potential © is a simple source of strength Q,
located at the center of the bubble (Landau and Lifshitz 1959,

Lighthill 1978) .
o(r, t) = - Serle)

4nr

Source strength Q(t) is defined to be volume of fluid displaced per
unit time by the source. The simplest quantity to compute is the

total amount of fluid displaced up to a time t

J Q(t" )dt! . (4-1)

This is just the difference between the physical volume of the bubble

(the displaced volume if there is no mass flux)

V(t) = J dv (4-2)
bubble
and the volume formerly occupied (while in the liquid state) by the

vapor presently in the bubble

— J o. dv. (4-3)
Py, v
bubble

The liquid density (butane) has been assumed uniform throughout

PL
the drop. Recognizing that (4-3) can be simply expressed in terms of
the total amount of mass M(t), in the drop at time t, the overall

displaced volume is

J Q(t)dt! = V(t) - aan (4-4)
0 L

Differentiating and rearranging, the key equation is obtained
dM dv QO. (4-5)

dt ~ °L dt ~ PL
The final step in this calculation is to relate the measured
pressure signal to the effective source strength Q. If the region
surrounding the bubble was filled only with host fluid of density P oo?
from acoustics the perturbation pressure p' due to the source

would simply be

pi =- Poot . (4-6)

In fact, during the evaporative stage, the bubble is surrounded by

both liquid butane remaining from the original drop and the host

liquid. Due to the differing densities and acoustic impedances of
the liquid glycol and butane, the butane-glycol interface at the
drop surface may alter the radiation from the bubble, making
(4-6) invalid. However, it is possible to show that the interface's
effect is negligible except possibly for a short period of time just
after the initiation of bubble growth. The basis of this approxima-
tion is that, under the compact source conditions assumed earlier,
true acoustic waves exist only in the far field of the source and

in the near field the fluid moves as if it were incompressible

(Landau and Lifshitz 1959).

This fundamental property of compact sources implies that
actual acoustic wave reflection from the drop surface only occurs at
the very earliest times of bubble growth, when the drop surface can
be considered in the far field of the bubble radiation. At later
times, but still small compared to the total test time, the liquid butane
and butane-glycol interface are within the near field of the bubble
radiation. Since the dynamics of the incompressible motion in the
near field do not appreciably deform the liquid butane-glycol interface (as
shown in Figures 3.2 and 3.3), the kinematics of the motion will
be the same as if the liquid butane and glycol densities were equal. In
conclusion, for times greater than 3-5 usec, the volume displacement
of the bubble will be transmitted unaltered to the far field of the
host, where the pressure can be calculated by (4-6), substituting the
host density for Por
The actual pressure detected by the transducer at point r is

denoted Ap(t) and can depend on p' in a complex way (as dis-

cussed in Section 2.4.1). However, before the first secondary waves

arrive (t < 30 psec), the pressure Ap, on the transducer face, is
just ap', where a is the coefficient of reflection for waves
normally incident on the transducer-mount combination (a = 1. 83 for
an aluminum-glycol interface). Integrating equation 4-6, the relation

between measured pressure and source strength is obtained

An t
Q(t) = —= f Ap(t'+r/c)dt! . (4-7)

The far-field and compact-source assumptions imply that a
detailed description of the source is not required and cannot be
obtained from this analysis. Only the integral properties, V, M
and Q, of the source are involved and any dependence of the emitted
pressure on the source (bubble) orientation is ignored. The reason-
ableness of such a treatment is borne out by the experimental ob-
servations of similar pressure signals from bubbles at very different

orientations to the transducer.

4.3 Data Reduction

Ideally, the pressure signal and bubble volume should be re-
corded simultaneously throughout each explosion in order to properly
calculate the mass flux. However, in the present experiment, the
bubble volume was only measured (at most) at one time in each ex-
plosion. In the spirit of ensemble averaging (and with a belief that
the initial bubble growth is similar for all explosions), volume data
obtained at different times in separate explosions was combined to
define au effective bubble volume history. This volume, together
with an ensemble-averaged pressure, was used in equations 4-5 and

4-7 to calculate the average mass flux.

The method used to calculate bubble volumes (and mean
evaporating interface area) from photographs and the resulting
volume history is presented in Section 4.3.1. The processing of
the pressure signals and the ensemble average signal is presented

in Section 4.3.2.

4.3.1 Photographs

Bubble volume and mean evaporating surface area were nu-
merically calculated, approximating the bubble as a body of revolu-
tion, from the coordinates describing the outline of the bubble on the
photograph. The calculation was simplified by analyzing only those
bubbles whose axis of symmetry was in or near the plane of the
photograph, i.e., the bubble must appear as in cross-section to the
viewer.

A total of 22 different bubbles, representing stages of the ex-
plosion from 8.3 to 91 psec, were analyzed. Using a graphics
digitizer (HP9872), the perpendicular distance to the bubble outline,
on both sides of the estimated symmetry axis Z, was recorded at
evenly spaced points. The averages of the distances at a given
location z on the symmetry axis was taken to be the curve R(z)
which described the profile of the approximating body of revolution.
Typically, 30 to 50 points were recorded for a single bubble, smaller
increments being used for smaller bubbles. The measured magnifi-
cation of the overall reproduction process was used to convert dis-
tances on photographs to the actual bubble dimensions.

Effective bubble volume V and mean evaporating surface area
A were both calculated from these data using the trapezoidal rule

approximation to the standard integrals for a body of revolution

Vv =| mR®(z)dz ,

A =| 2nR(z) (1 + (G=)) dz

The derivative = ,» in the area integral, was approximated by the
first difference. The limits of integration for the area integral

were chosen so that only the area of the bubble surface estimated

to be in contact with the liquid butane was calculated. No attempt
was made to correct for optical distortion (lensing effect of the drop)
or out-of-plane symmetry axes; these errors are estimated to be of
the same order as the intrinsic error of this technique due to the

natural asymmetry of the bubbles.

The results for the bubble volume V, the mean evaporative

1/3
interface area A and the mean bubble radius R = cae are
given in Table 4.1. The limits of the evaporating interface were

not well defined for some of the larger drops, so values of the

area were not calculated. One remarkable feature of these results
is the nearly linear growth of the mean radius R_ with time over
the entire interval (Figure 4.1), with a mean velocity of 14.3 m/sec.
For the purposes of calculating the mass flux, a restricted set of
the data (times < 30 psec) was fit to a simple power law

(x) = pt, (4-8)

Coefficients a and b were determined by a least squares fit to

the logarithm of equation 4-8. The results are:

2. 649 em?

V = 6.443 x 10° ¢

; (4-9)

1.362 2
t cm

A 1.472 x 10 4

(4-10)

the units of t are psec. Equations 4-9 and 4-10 are plotted,
together with the individual data points, in Figures 4.2 and 4.3,
respectively.

a , which is needed in order

to calculate the mass flux. In those calculations, the crude estimate

Unfortunately, it is not V, but

obtained by analytically differentiating equation 4-9 was used. This
procedure produces a definite (but unknown) bias in the result, the

actual a is certainly less smooth than this estimate.

4.3.2 Pressure Signals

Pressures at a distance of ~2.5 cm from the drops were

measured by a PCB transducer in configuration C, Figure 2.3.
The transducer was mounted in an aluminum plug 2.3'' in diameter;
diffracted waves from the plug's edges limited the test time to the
initial 30 psec of the explosion. During this time, it was assumed
the pressure on the transducer face was the pressure produced by
normal reflection of the incident wave.

The transducer output was digitized and stored in the Nicolet
oscilloscope at a sampling rate of 20 MHz. Overall frequency re-
sponse was limited by the transducer (which has a natural resonance
at 500 KHz) to a bandwidth of 1 MHz. The initial zero level of the
signal was subtracted, an overall scaling factor (a product of detector
sensitivity and amplifier gain) computed and the raw data converted
to psi on the transducer face. Each signal was inspected to deter-
mine the exact location (within + 0.1 usec) of the beginning of the
first pressure increase due to the explosion. This location was used
as the absolute origin of time ("time zero'') in the computation and

averaging of the signals.

A total of 7 pressure signals were processed by this scheme
and averaged together (all signals were aligned so that every "time
zero'' coincided) to yield an ensemble average pressure signal, shown
in Figure 4.4. This is the average signal which is used to compute
the average source strength from equation 4-7. Comparing the
average signal with the individual signals in Figure 3.19, some
common features stand out. In particular, the characteristic steps
at 5 and 10 usec appear distinctly in the average signal and can be
seen to various extents in all of the individual signals. However,
the oscillatory structure which is superposed on the general pressure
rise after 10 sec in many individual signals is not universal and is

washed out in the averaging process.

4.4 Results

The stage of the explosion considered in these calculations is
displayed in Figure 4.5. In this figure, a series of photographs
(some of which were used in the volume calculations) and a repre-
sentative pressure trace illustrate bubble growth from 16 to 40 msec.
Actually, the most intriguing features of the pressure trace are the
steps or bumps occurring at the very earliest times, during the first
15 psecs. From the discussion in Section 4.2, it is clear that these
rapid changes in radiated pressure are due principally to rapid
changes in the volume growth rate or evaporative mass flux. It is
tempting to relate this unsteadiness in the pressure with evapora-
tive instabilities which could cause large and rapid changes in the
mass flux. The possibility of such an instability has already been
suggested in connection with the interface wrinkling (Section 3.3.3).

However, since no photographs have been obtained before 8 psec,

very little volume data is available to support any conclusions about
the mass flux at these early times.

Lacking a better alternative, the analytical fits to the data,
equations 4-9 and 4-10, were used to calculate the mass flux from
t = 0 onward. This is, in effect, a backward extrapolation of the
smoothed volume data. Therefore, any rapid changes of pressure
appear to arise only from rapid changes of mass flux, whereas, in
reality, both mass flux and volume contribute to the pressure signal.

Results have been calculated up to 50 usec, despite the injunc-
tions against using the pressure data past 30 psec. This was done
to show that effects of secondary waves are gradual and no discon-
tinuity occurs when they first appear. The volume derivative calcu-
lated from equation 4-9 is shown in Figure 4.6. This is to be com-
pared with the average source strength, shown in Figure 4.7. The
difference between these two terms is the volume of liquid butane
that has been evaporated. Using these data in equations 4-5 and 4-6,
the total mass M_ in the bubble and its derivative oe have been
computed and are plotted in Figure 4.8. Notice that the bumps in
the pressure produce only a slight waviness in os and have no
visible effect on M.

Mean evaporative mass flux m was calculated by

=i aM
~ A dt ;

(4-11)
where A is given by equation 4-10. As shown in Figure 4.9, the
mass flux rises very rapidly (most of the increase occurs in the

first microsecond) and after several wiggles levels off at about 400

gm/cm* sec. The nature of the wiggles and their possible relation

to bubble growth is considered in the next section.

An evaporative mass flux of 400 gm/cm* sec is large under
any circumstances: Mass fluxes typical of low superheat bubble
growth experiments (e.g. Table 3.2) are of the order 107+ to 10°73
gm/cm* sec. Other common examples of evaporation (tea kettles,
power plant steam generators, etc.) have similarly low mass fluxes.
Only in extreme situations, such as vacuum distillation (Hickman
1972) or laser ablation of metals (Ready 1965), does the mass flux
reach ~ 10° gm/cm?® sec.

There is a simple argument which confirms that this large
value of the mass flux nevertheless is not unreasonable. Suppose
the liquid to be evaporated at constant rate through the surface of a
sphere growing at a constant radial velocity R. The net rate of
consumption of liquid mass would be

<> = mA = m4nR?t? (4-12)

Integrating, the time t to evaporate a mass of liquid originally

contained in a sphere of radius R, is found to be

t= ; Aa 173 (4-13)
(= = R*)
PL,

Substituting the values mh = 400 gm/cm” sec, = 0.43 gm/cm®

PL
and R= 1.4 x 10° cm/sec, a 0.5 mm radius drop is found to
evaporate completely in the quite reasonable time of 50 sec. In
reality, only a portion of bubble surface is evaporating and the

estimate given here is a lower bound to the actual time required

for evaporation.

100

4.5 Comparison with the Classical Model

In this section, the predictions of the ‘'classical model" of
vapor bubble growth (introduced in Section 1.2.3) are compared with
the experimentally measured and inferred results. Strictly speaking,
this comparison is possible only for that small interval of time
(~ 10 psec) where it has been conjectured (Section 3.3.1) that the
bubble is completely surrounded by the drop liquid. As the bubble
grows and interacts with the host fluid, a smaller fraction of the
bubble surface is actually evaporating in the experimental case than
in the idealized bubbles of the theory. Therefore, the predictions
of the theory should overestimate bubble growth for t > 10 psec; in
fact, it is found that the growth is vastly underestimated at these
later times. The extent to which this occurs is demonstrated by
comparing bubble volumes, then mass fluxes and finally pressure
signals over the first 30 psec of growth. It should be kept in mind
that experimental results for the effective bubble volume, in the
first 10 usec, are an extrapolation of the actual data, which were all
obtained at later times.

From this point on, the ''classical'' theory of bubble growth
will mean the latest and most complete version of that theory as
presented by Prosperetti and Plesset (1978). This theory describes
the growth of a smooth, spherical vapor bubble from a critical
nucleus in a uniformly superheated liquid. Bubble growth is con-
veniently divided into three stages: first, a surface-tension-controlled
stage in which the bubble grows from a critical nucleus, radius R),;
second, an inertia-controlled stage in which the bubble surface grows

with a characteristic velocity R, (determined by the vapor pressure

101

and density of the superheated fluid); third, an asymptotic stage in
which bubble growth is dominated by heat transfer and is character-
ized by a Rw~ pb /2 dependence. Once the bubble has grown out of
the surface-tension-dominated stage, the growth is approximately
described (under the assumptions of the theory) by a universal equa-
tion in scaled variables. The approximations involved in deriving
this universal equation are: a) the surface tension is negligible;

b) the vapor pressure p(T) is a linear function of temperature T;
c) both latent heat and vapor density are constants, independent of
the evaporating surface temperature. The scale length and time are
the radius of the bubble R* and the time t* at which the cross-
over from inertia to asymptotic stages occurs.

Two additional parameters pw and 17, characterizing bubble

growth, are defined;

1/2
p= (8)

rake eR (4-14)
R- Ro

Parameter wp determines the relative importance of the inertial
stage of growth. A decreasing function of superheat, wu = 1 for low
superheats (predominantly heat-transfer-controlled growth) and de-
creases to ~ 10° at superheat-limit conditions (predominantly inertia-
controlled growth). Parameter 7 is the characteristic time required
for the bubble to escape the surface-tension-controlled stage and is
also a decreasing function of superheat. Parameter values and scale
lengths are given in Table 4.2; these have all been calculated from
the equilibrium properties of saturated n-butane by the prescription
given in Prosperetti and Plesset. Since the properties of the

saturated liquid and vapor play such an important role in this theory,

102

selected properties of saturated n-butane are given in Table 4.3.
These properties were calculated from the following thermodynamic
correlations and empirical formulas: vapor pressure, Riedel equation
(Reid et al 1977); latent heat, Watson relation (Reid et al 1977); sur-
face tension, corresponding-states correlation (Reid et al 1977); vapor
density, Thompson correlation (Thompson and Sullivan 1979); quid
density, Rackett equation (Rackett 1970).

In this theory, the vapor in the bubble is assumed to be uniform
and in equilibrium with the liquid surface; thus, there are only two
independent variables, bubble radius R and surface temperature
TS The approximations in the theory determine all other quantities
as a function of R, T and their derivatives. For convenience,
approximate analytical expressions for R and T were used in
this comparison, instead of exactly computing the Prosperetti-Plesset
model for this particular case. These analytic expressions are
equations (8) and (13) of Mikic et al (1970) with the modified evalua-
tion of scaling constants suggested in Prosperetti and Plesset, who
show that these expressions are in reasonable agreement with the
solutions to the universal bubble growth equation. The thermodynamic
properties of saturated butane used in this comparison were obtained
from the analytic formulas mentioned above in connection with Table
4.3.

Figures 4.10 and 4.11 compare the predicted bubble volume
and volume growth with the fit to the experimental data over the
first 30 psec. Predicted V and =~ are initially larger and then
drop far beneath the experimental values. As pointed out at the

beginning of this section, the large discrepancy at later times is due

103

to the inadequacy of the classical model rather than any invalidity of
the comparison. An alternative comparison is shown in Figure 4.1
where both the bubble radius computed by the classical model and
the limiting inertial growth rate (Ry = 40 m/sec) are shown together
with the experimental effective radius data. Note that the predicted
growth very rapidly deviates from the inertial limit and only the
asymptotic diffusion-limited stage is apparent on the time scale of
Figure 4.1.

The predicted evaporative mass flux m is approximated in

the theory as

YT IS , (4-15)
{a term involving mal has been neglected here; including this term
can only decrease the predicted mass flux) where p(T.) is the
saturated vapor density at temperature TS: Figure 4.12 compares

this prediction with the experimental results for this first 10 psec
of bubble growth. It is in the mass flux comparison that the
differences between theory and experiment are most clearly demon-
strated. Predicted and inferred mass fluxes at 10 usec differ by
10°. This situation only worsens with increasing time, since the
theory predicts that R and T. decrease monotonically; hence,
by equation 4-15, so must m. Further, this predicted behavior of
m rules out any oscillatory behavior with time. This is not neces-
sarily the case with the emitted pressure wave, discussed next.
Within the framework of the theory, the pressure pulse pro-
duced by the bubble can be calculated from incompressible bubble
dynamics. While this appears inconsistent with the acoustic treat-

ment of the experimental pressure, it can be shown that, in the far

104

field of a compact source, the pressure calculated from acoustics
agrees with that calculated from potential flow (except for the
retarded time in the acoustic result, which has already been com-
pensated for in the data processing). The far field pressure at a
distance r from the bubble center can be written

p= 2 wyT) - pts eR) (4-16)

The combination of an increasing bubble radius and a decreasing
temperature and velocity suggests that the pressure can have a max-
imum. The pressure calculated by equation 4-16 (divided by a_ to
simulate reflection at the transducer) is compared to the ensemble
average pressure in Figure 4.13. A broad maximum occurs at

3 psec; in view of the approximations used in this comparison and in
view of the foregoing comparisons of experiment with theory the co-
incidence of the peak value with the measured signal is strictly
fortuitous. Under the assumptions of the theory only one maximum
can occur in the predicted pressure so the subsequent bumps and
much greater rate of increase measured experimentally are unex-

plained by the theory.

105

REFERENCES

Hickman, K. 1972 ''Torpid Phenomena and Pump Oils", J. Vac.
Sci. Tech. 9, 960.

Landau, L.D. and Lifshitz, E.M. 1959 Fluid Mechanics, Pergamon,
New York.

Lighthill, J. 1978 Waves in Fluids, Cambridge University Press,
Cambridge.

Mikic, B.B., Rohsenow, W.M. and Griffith, P. 1970 ''On Bubble
Growth Rates", Int. J. Heat Mass Transfer 13, 657.

Prosperetti, A. and Plesset, M.S. 1978 "Vapor-Bubble Growth in
a Superheated Liquid", J. Fluid Mech. 85, 349.

Rackett, H.G. 1970 "Equation of State for Saturated Liquids",
J. of Chem, Eng. Data 15, 514.

Ready, J.F. 1965 "Effects Due to Absorption of Laser Radiation",
J. App. Phys. 36, 462.

Reid, R.C., Prausnitz, J.M. and Sherwood, T.K. 1977 The
Properties of Liquids and Gases, McGraw-Hill Book Co.,
New York.

Thompson, P.A. and Sullivan, D.A. 1979 "A Simple Formula for
Saturated-Vapor Volume", Ind. Eng. Chem. Fund. 18, 1.

BUBBLE VOLUME, EQUIVALENT RADIUS AND

106

TABLE 4.1

EVAPORATING SURFACE AREA

Experiment Time
(Date) (Run #) _(us)
3/22 2 8.3
3/14 3 9.0
3/14 4 11.0
3/14 13 11.8
4/29 31 12.9
4/29 27 16.4
4/28 12 20.2
4/28 20 24.0
4/28 13 25.2
4/29 5 27.3
4/28 17 28.5
4/29 10 30.4
4/29 21 32.5
4/29 17 34.1
4/29 20 39.6
3/23 51
3/23 2 52
3/23 1 55°
3/7 12 65
3/30 18 69.3
3/23 19 88
3/23 ll 91

.54 x 10°
~257
292
335
- 455
. 423
- 589
-571
. 754
. 02
. 82
. 09
18
.97
5. 42
10.1
11.3

myn NY &—-& &F-& CO CO CTC Coo oO Oo (Dm Ef NB Ne

mee FY CO OO 900 CO ODO OOClUlUDUlUlUcUOUlUCcUNUNULUCUCUOULUCUCOUNULCONUCUCUCOUOUCUCUNUCUCOOUCUCOULUDOD

CN a cn oo a OE OD ED EE > EE a> EE > EE > EP)

107

TABLE 4.2

SCALE LENGTHS AND PARAMETERS IN THE PLESSET-PROSPERETTI

THEORY EVALUATED FOR BUTANE AT THE SUPERHEAT LIMIT

Parameter

Ry
Re

Description

Radius of critical nucleous

Characteristic radial velocity

of interface in inertia stage

Scaling length,
at crossover from inertia to

radius of bubble

asymptotic stage

Scaling time,

to reach crossover from inertial

time required

to asymptotic stage

The time required for the bubble
to crossover from surface tension
to inertia stage (= a7?

Prosperetti and Plesset)

defined by

Value

107 cm

10° cm/sec

107% cm

10°° sec

107?° sec

10°

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122

V. LIQUID-VAPOR INTERFACE INSTABILITY

5.1 Introduction

The wrinkling of the evaporative surface observed in the present
investigation presents a challenging problem: What is the underlying
physical mechanism responsible for this phenomenon? In this chapter
one possible mechanism, Landau instability, is proposed and the im-
plications for the classical model of bubble growth are examined.
Only the possible inception of surface distortion by this mechanism
is considered and the more difficult nonlinear problem of the structure
and dynamics of the persistent roughening is not addressed.

All bubble surfaces observed in the present study were wrinkled
to some extent, including those bubbles seen at the earliest times.
Unfortunately, since no bubbles younger than 8 psec could be observed
in these experiments, it was impossible to determine definitely
whether the bubble surface was ever smooth. However, the regular
appearance and small amplitude of the disturbances at the earliest
times suggests that the surface may once have been smooth, Fur-
ther, the classical theory of nucleation implies that the bubble is
smooth at the very instant it is created from a fluctuation. On this
basis, it will be assumed that the surface is initially smooth, and
the stability of this smooth surface to perturbations in shape will be
investigated.

This chapter is organized as follows: Results of the previous
investigations of the linear stability of a plane evaporating surface
are summarized in Section 5.2. Stability criteria and the physical
basis of Landau instability are discussed in Section 5.3. Accounting

for the effects of sphericity in an ad hoc way, the possibility of

123

instability during vapor bubble growth (described by the classical
model) is investigated and the relation to the experimental observa-
tions discussed in Section 5.4. Digital image enhancement of the
photographs of bubble surfaces at the earliest times is described

and an example is presented in Section 5.5. Finally, the results

of the stability analysis are summarized and some concluding remarks

about the validity of the calculation are made in Section 5.6.

5.2 Review of Possible Instabilities

The linear stability of an idealized, steadily evaporating, plane
liquid surface has been investigated by both Palmer (1976) and Miller
(1973). Palmer identifies four distinct types of instability which

explicitly are caused by evaporative flux:

a) Differential vapor recoil. Variations in surface tempera-

ture produce variations in the evaporation rate. The consequent
variation in the recoil force due to the evaporative flux produces
surface deformations, which, in turn, promote convection and so
sustain the variations in surface temperature.

b) Inertial or Landau instability. Distortions of the evaporat-

ing surface produce vorticity in the vapor, which is convected down-
stream. The vorticity field feeds back to the interface to amplify the
initial distortions of the surface, producing more vorticity.

c) Viscous dissipation. Distortions of the evaporating surface

produce a variation in dissipation rate along the surface. The heat-
ing caused by dissipation promotes increased evaporation, which in
turn sustains the surface deformation.

d) Moving boundary instability. Distortions of the evaporating

surface are accompanied by changes in thickness of the adjacent

124

boundary layer, which in turn causes local variations in the heat
flux into the surface. These heat flux variations result in evapora-
tion rate changes which positively feedback to increase surface dis-
tortions.

All of these mechanisms can be stabilized to some extent by
the action of surface tension, stabilizing accelerations, viscous dis-
sipation and thermal diffusivity. Therefore, in each case the driving
force, the evaporative mass flux, must exceed a threshold level
before the instability occurs. These instabilities are not peculiar to
evaporating liquid surfaces. Most have been proposed previously in
various other contexts such as combustion, chemical reaction and
solidification, i.e. any discontinuous change of phase in which fluid
dynamics and/or heat release is important.

Conditions under which individual mechanisms dominate are
extensively discussed by Palmer. However, the particular case of
the initial stages of bubble growth at the superheat limit was not
discussed, but, as outlined below, it can be inferred that the most
likely instability is the inertial or Landau instability.

The inertial mechanism is the simplest possibility; it does not
depend on a specific model of evaporation or require an intrinsic
variation in surface temperature to support it. Neither viscosity nor
thermal conductivity are essential; only a difference in density between
liquid and vapor phases is required to produce vorticity. Further,
the particular conditions of evaporation under consideration are not
very favorable to the functioning of the other mechanisms. For the
liquid-vapor density ratios found at the superheat limit, vapor recoil

instability exists only in a very narrow range of conditions. The

125

moving boundary instability is manifested on very long wavelengths,
which are effectively stabilized by the deceleration of the bubble
surface. Viscosity is a very inefficient mechanism for producing
instability and has a much higher threshold than the other mechan-
isms.

In addition to instabilities which are driven only by evaporation,
there exist several other well known instabilities which are always
possible at liquid-vapor interfaces. These include: e) Rayleigh-

Taylor (driven by imposed accelerations); f) Kelvin-Helmholtz (due

to a shear or discontinuity in the velocity parallel to the interface);
g) Marangoni (due to convection induced by the temperature depen-
dence of surface tension), The initial state of the unperturbed inter-
face precludes possibility f) and the accelerations present in bubble
growth are stabilizing, eliminating e). Surface tension gradients do
not initially exist, but could arise as the result of other instabilities:
In general, Marangoni type effects could be coupled to inertial (or
any other) instabilities, but are not expected to be a primary source
of instability at large evaporation rates,

Both Miller and Palmer suggest the possibility of an inertial
type instability occurring for rapid evaporation. Indeed, Miller con-
cludes that "the density difference effect could be of importance
during rapid growth of cavitation bubbles under low pressure or
tension.'' The conclusion of this chapter is to support that statement
and generalize it to the possibility of instability in bubble growth

from highly superheated liquids, especially at the superheat limit.

5.3 Physics of the Landau Instability

The inertial or Landau instability was first discussed by Landau

(1944) in the context of laminar flames. An analysis (first made in

126

‘ “that paper) of the stability of a flame over a liquid will be applied
pere, without modification, to the stability of evaporating interfaces.
At the level of approximation in the problem, the only difference
petween combustion and pure evaporation would be in the vapor
density, which can be arbitrarily chosen anyway.

Heuristically, instability can be traced back to the generation
of vorticity, in the initially irrotational flow, by the deformations of
the interface. | This vorticity is convected downstream and is of
such sign to induce velocities at the interface which amplify the
surface distortions, further increasing the vorticity production.

The situation is schematically indicated in Figure 5.1 for a two-
dimensional flow. As indicated, spatial gradients in the perturbed
(time dependent) pressure interact with density gradients (actually,
the density jump across the interface) to produce the vorticity at the
4nterface (this is just the baroclinic torquc, Yp X Vp, mechanism of
producing vorticity). The vorticity field alternates in sign, with

the period of the surface deformation, in the direction transverse

to the flow and undergoes an exponential decay as it is convected
downstream.

In the absence of any other forces on the interface, the surface
is. found to be unstable to deformations of any wavelength, smaller
wavelengths having larger amplification rates. The effect of surface
tension is to stabilize the interface to disturbances of short enough

wavelength. The magnitude of the cutoff wavelength Ae can be

] .

The author is indebted to Professor Frank Marble for the vorticity
explanation presented here and the original suggestion that Landau
instability was important,

127

estimated by equating the characteristic force/unit area on the inter-

face due to surface tension o

{eo}
Ls

to the characteristic force developed by the instability

. ] ]
m°(— - —) (5-2)
P., Pe
The ratio of these two forces can be used to define a dimensionless

parameter which will be referred to as the Inertia number N,;

Tv
m 1 i
N,=6—(—-—) , (5-3)
I GRY Py

where 6 is some scale length (which we will take for convenience

to be 1 cm). In terms of the dimensionless wavenumber

K = =~ ; (5-4)

the precise criterion is: the interface is unstable when K < Ny ;

stable when K > Ny and neutrally stable for K = N

The effect of stable accelerations (accelerations of the liquid

toward the vapor) is to damp out deformations of a long enough wavelength.
The ratio of the accelerative forces to the surface tension forces

is characterized by a dimensionless parameter referred to as the

Weber number WN

WwW? ,
Aps
Nw = 8 5 ? (5-5)
where g = -R, the deceleration of the bubble surface (N

WwW

is often also known as the Bond or Edtvos number). With the

inclusion of accelerations the stability criterion is modified to:

128

WwW .
Ke t KON, » Stable ;
Nw
= +K-= N, , Neutrally Stable ;
vw

It follows directly from the first inequality that for N, < 2 Nw:

the interface is absolutely stable to disturbances of all wavenumbers.

In the linearized analysis, unstable disturbances grow expon-
entially in time, i.e. all amplitudes <= eft The growth rate Q
can be written in dimensionless form w = TQ, where the character-
istic time T = J (pt py). From the analysis of Landau, which is
repeated in the more accessible Landau and Lifshitz (1959), the dis-
persion relation w = w(K, Ni. Nyy Nos where Ny = Py/P.» can be

written out directly

1-N* N,/K + K®
w? + 2wK + xv? (K® - — oo) =O. (5-6)
p I

The stability criteria mentioned above, the most unstable wavenumber

K" and the corresponding amplification rate w can all be derived

from this relation. For sufficiently unstable flows (Ny >> 24/ Nw).
K’ « Ny, where the constant of proportionality varies between 5 and
= as Ny varies between 1 and o; corresponding amplification rates

are ww NUN - 1). Curves of constant growth rate, the neutral

stability boundary and the most unstable wavenumber K are plotted

on the K vs. Nr plane in Figure 5.2 (N,, and Nw are held con-

stant).

There is an even more compact, but not necessarily simpler,

129

choice of dimensionless parameters. If K/j Nw and N,/J Nw
are used as the new variables, then the parameters can be combined

and the dispersion relation written,
w= WK Ny, Ny// Nye No) . (5-7)
No further consideration or use of (5-7) will be made.

5.4 Application to the Classical Model of Bubble Growth

The theory developed by Landau is based on an initially plane
interface in a steady flow. Strictly speaking, application of the
results to nonsteady spherical interfaces could only be justified if
the disturbance wavelength is much smaller than the bubble radius
and the disturbance growth rate is much faster than the rate charac-
terizing bubble growth. It is not known in advance whether these
conditions will always be met and the success of the application will
be judged afterwards. Intuitively, the effect of sphericity is to
suppress all disturbances with wavelengths of the order of or larger
than the bubble dimensions. To account for this effect an arbitrary
cutoff wavenumber kp, corresponding to a wavelength of one drop
radius, will be introduced. The possible unstable modes will be
calculated as though the interface were plane, but only disturbances
of wavenumber k > Kp will be considered as actual unstable modes
of the spherical interface. In particular, if all possible unstable
modes have wavenumbers k < kp, then the spherical interface will
be considered stable.

The cutoff wavenumber method of correcting the planar insta-

bility theory for the effects of spherical geometry can be tested for

the special case of nonevaporating bubbles, for which the spherical

130

case (described in Appendix C) has been worked out exactly. The
exact theory does indeed predict the usual Taylor instability (R> 0)
will be suppressed if the bubble radius is small enough. As assumed
in the evaporating case, the exact criterion for damping in the non-
evaporating case can be stated in terms of a cutoff wavenumber based
on the radius; more precisely, the cutoff wavelength is ~ 1.7 R.
For this particular example, the cutoff wavenumber idea does account
for the major effect of the spherical geometry, although more subtle
effects such as algebraic growth of instabilities associated with the
radial motion of the surface cannot be treated by this method.
Neglecting the effect of accelerations, the Inertia number N

and the normalized cutoff wavenumber Kp have been calculated as
a function of time during bubble growth. The approximate formulas
of universal bubble growth (Mikic et al, 1970) described in Section
4.5 were used in this calculation. The path of the system is shown
on the K - Nr plot, Figure 5.3; the region of possible instability
lies in the shaded area below K = Ny and above Kp: This calcu-
lation does indicate that in the interval ~ 10 ° sec to 2 x 10° sec
bubble growth would be unstable. Outside of this range the bubble
radius is smaller than the shortest possible instability wavelength,
Kp > Ny- If accelerations are included, the main effect is to de-
crease the length of the interval over which the motion is unstable
to 10°° - 2 x 10°° sec. The accelerations referred to here are
actually decelerations of the interface (R < 0) as the bubble transitions
from the inertial to asymptotic stage; the order of magnitude of R

is 108 cm/sec” (10° g). In Table 5.1, the Inertia number, Weber

number, cutoff wavenumbers, growth rate and most unstable

131
wavenumbers are tabulated as a function of time during bubble growth.
The growth rates and most unstable wavenumbers have been calculated
for the purely surface-tension-stabilized interface.

Also shown on Figure 5.3 is the region corresponding to the con-
ditions at the interface of the regularly wrinkled bubbles observed experi-
mentally at the earliest times. The range in K of this region corres-
ponds to the range of disturbance scale sizes observed in the earliest photo-
graphs (this point is considered in more detail in the next section); the

range of values of N, correspond to the inferred mass flux of 400 gin/em®

sec and the possible range of surface tensions in metastable butane (3-16)
dynes/cm) and vapor densities (0.05-0.1 gm/cm*) that may occur in the
experiments.

A possible interpretation of the experimental data can be made by
recognizing that the linear instability must have occurred well before
8-10 psec. All the experimental data refer then to the fully developed,
nonlinear stage of the instability. This is consistent with the results of
Chapter 4 (e.g. Figure 4.9), which suggests large and rapid changes in
the mass flux occur in the first 1-2 usec. Assuming that the most un-
stable wavelength chosen by the linear amplification process is "frozen in"
when the growth reaches the nonlinear stage, the range of wavelengths
observed at 8 psec and the ad hoc model of instability (excluding accelera-
tions) imply that linear instability could have occurred about 1,0 Usec.
However, the effect of accelerations is to terminate the instability at
0.3 psec. Corresponding instability wavenumbers at this time are 2-10
times larger than those observed at 8-10 usec, implying that a large de-
crease in wavenumber must have occurred during the nonlinear stage if
the present explanation is, in fact, correct. Further work is in progress

on the exact linear instability theory in an effort to clarify this situation.

132

Thus, the bubble could have grown according to the classical model
until about 0.3 psec, at which time the linear instability occurred and
consequently, the mass flux increased. Following the onset of instability,
the mass flux and bubble growth rate rapidly increased until the instability
growth saturated at a mass flux 10° larger than in the absence of instability.
This rapid increase of mass flux corresponds to a sharp veer to the right
of the (Ni, Kp) trajectory (on Figure 5.3 from the classical path at about

0.3 psec to the region experimentally observed at 8-10 psec.

5.5 Image Processing and Experimental Evidence of Instability

As proposed above, the experimental observations of a roughened
evaporating surface and an abnormally large evaporative mass flux
could both be explained by a fundamental instability of the evaporating
interface. Photographs of the bubble at the earliest times (e.g.
Figures 3.2 and 3.10) are the most direct experimental evidence that
this instability developed from an initially smooth surface, Bubble
surfaces in these photographs appear covered by a fairly regular |
pattern, that could be produced by surface deflections or waves.

Such waves may have originally been formed in the first few micro-
seconds of bubble growth by the linear instability process described
in the preceeding sections. This idea is the motiviation for compar-
‘ing (in Figure 5.3) the length scale of the intensity variations making
up the pattern on the bubble surface to the most unstable wavelength
of the linear instability theory. An order of magnitude of the length
scale can be obtained simply by estimating the average distance be-
tween dark (or light) patches on the pattern. For example, the
bubble seen in print (a), Figure 3.10 has a 150 fam radius and the

length scale of the pattern is 20-40 pm.

133

To obtain a clearer idea of the organization (if any) present in the
pattern and a better estimate for the length scale, several bubble images
taken at the earliest times were digitized and computer processed to en-
hance the pattern. 2 An example of the results can be seen in Figure 5. 4;
print (2) is a contrast enhanced (stretched) but otherwise unprocessed image

of the bubble seen in print (a), Figure 3.10; print (b) is the enhanced ver~

sion. The processing used to enhance the image was: a) the original
gray level data, contained in a 64 X 64 array of pixels, was expanded to a
512 x 512 array using linear interpolation; b) a 3 x 3 median filter (each
pixel gray level is replaced by the median level of the surrounding 3 X 3
square of pixels) was applied; c) a high pass filter, with a cutoff length of
13 pixels, was applied; d) finally, the contrast scale was stretched (linearly)
to accentuate the difference between light and dark arcas of the pattern.

The enhanced image shows that the pattern covers the entire bubble,
and in particular that it occurs at the periphery, where the unprocessed
version appears uniform. The regularity of the pattern is much more
striking in the enhanced image, but there is substantial asymmetry and
randomness. This is consistent with the possibility that linear instability
occurred at about 0.3 usec whereas the experimental observations are
made at & psec, well into the nonlinear regime. The possibility exists
that a more rigorously defined length scale could be computed from digital
data representing the picture, e.g. a characteristic length could be deter-
mined from the 2-D autocorrelation of the image. However, the complexity
of the image (i.e. the presence of the circular boundary of the bubble and

the nonuniform background due to the lensing effect of the bubble) prevents

The author would like to thank Daryl Madura and the staff of the Medical}
Imaging Analysis Facility of the Jet Propulsion Laboratory for perform
ing this work.

134

a simple interpretation of the autocorrelation function and the best estimate

of scale size remains that determined above by human judgement.

5.6 Summary

The purpose of this chapter was to investigate the possibility
that the evaporating bubble surface is fundamentally unstable at the
conditions of the superheat limit found in the present study. Out of 7
possible instabilities, the Landau or inertial instability was chosen as
being the most likely to occur. Landau's original analysis of a flame over
a plane liquid surface was supplemented by an ad hoc sphericity condition
and the resulting theory applied to the classical model of bubble growth.
Taking into account the stabilizing effects of both surface tension and de- |
celeration, the bubble surface was found to be unstable for the portion
8x 10°? to3 x 10” sec of the initial growth.

Experimentally, the observations made in the present siudy at
8 psec show regular disturbances with a scale length (20-40 um) on
the order of the unstable wavelengths predicted by the linear theory
to occur at about 0.3 psec. An explanation consistent with all the
available evidence and the theory is that the observed instability at
8 psec is far into the nonlinear stage and the linear stage of the in-
stability did occur in the first 1-2 usec of bubble growth. Observed
length scales are 2-10 times those predicted by the linear instability
theory; this may be due to the nonlinear growth process and is a
point requiring further investigation.

Of course, the calculation carried out in Section 5.4 does not

rigorously prove that Landau instability will occur. Indeed, the ad

135

hoc method of including sphericity suggests that only modes repre-

sented by a small band of wavenumbers (K, < K < Ny) can be un-

stable at any time. Further, the growth rates of these unstable
modes are such that interaction between the bubble growth and in-
stability development must be important. Inspection of Table 5.1
reveals that the product of instability growth rate and time is nearly
a constant over the unstable region. Algebraic growth, rather than
exponential growth, would characterize such a process. However,
both the narrow band of unstable modes and the algebraic instability
are plausible effects that are known to occur in the related insta-
bility of nonevaporating spherical surfaces discussed earlier and in
Appendix C,

In addition to the difficulties associated with the approximate
nature of the calculation, there is the possibility that the bubble
growth could be unstable from the very outset. The surface-tension-

controlled stage of bubble growth, which is not included in the

universal growth model used in this calculation, is characterized by

positive, Taylor unstable, accelerations (R > 0). Since the duration
(~ tT = 107° sec) of this stage is short, the accelerations can be

enormous, R~ 10!2 cm/sec®. The possibility of instability at the
earliest stage due to these accelerations alone can be investigated
using the exact linear stability theory (reviewed in Appendix C)
developed for the surfaces of nonevaporating, but unsteady, bubbles.
In fact, under the conditions at the superheat limit of butane given
by the conventional theory of nucleation, this exact theory predicts
no instability will occur. This is due to the cutoff effect of the

bubble radius; the most unstable wavelength for a plane interface

136

under the same conditions is 10° larger than the critical nucleus

radius Ry, = 4 x 107 cm,

137

REFERENCES

Landau, L.D. 1944 "On the Theory of Slow Combustion", Acta
Physiochimica U.R.S.S. 19, 77.

Landau, L.D. and Lifshitz, E.M. 1959 Fluid Mechanics, Pergamon
Press, New York, 479.

Miller, C.A. 1973 "Stability of Moving Surfaces in Fluid Systems
with Heat and Mass Transport - II. Combined Effects of
Transport and Density Difference Between Phases", AIChE
Journal 19, 909.

Palmer, H.J. 1976 ''The Hydrodynamic Stability of Rapidly Evap-
orating Liquids at Reduced Pressure", J. Fluid Mech. 75,
487.

138

TABLE 5.1

LANDAU INSTABILITY PARAMETERS FOR BUBBLE GROWTH

(usec) SR ‘w “I 7 (see? )
0 1.7 x 107 00 2.1x 10° 1.4.x 10° -
0.02 7.9x10* 6.7 x 10° 1.1x108 7.3 x 104 1.6 x 108
0.2 8.6 x 10° 1.4x10® 4.5x10* 3.0x10* 2.7.x 10’
2.0 1.1 x 10° 1.6x 107 4.9x10% 3.3 x 10° 2.5 x 10°

20.0 1.4x 107 8.4x10° 1.2 x10 8.0 x 10: -

200.0 5.4 x 10° 2.9x10* 1.2x10 8.0 -

139

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141

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(Ny, Kp) For Bubble |
Growth Described By |
1io® - The Classical Theory —F
Stable K>Ny
5 Lb -
v 10 0.02
~ Region Of Instability For ‘ Sec
= The Spherical Interface
Z jOF + +0.2 Sec
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- ; pSec Experimentally
Zio“ - . At 8-lOpwSec 4

je Sec VA Most Unstable Mode
Te) L. { ai it {
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Inertia Number, N;

FIG.5.3 REGION OF INSTABILITY FOR BUBBLE GROWTH
DESCRIBED BY THE CLASSICAL MODEL AND

COMPARISON TO THE EXPERIMENTAL
OBSERVATIONS AT 8-10 # SEC

142

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143

VI. A ONE-DIMENSIONAL ANALOG OF THE CLASSICAL MODEL
AND SOME LIMITATIONS AT THE SUPERHEAT LIMIT

6.1 Introduction

Throughout the previous chapters, frequent reference has been
made to the "classical model'' of bubble growth and its failure to
adequately describe the experimental observations made in the present
work. One possible limitation of the classical model was considered
in Chapter 5; namely, that the evaporating bubble surface is in-
trinsically unstable at large evaporation rates. Aside from this
fundamental limitation, there are others due to the approximate
character of the theory and the possible violation of key assumptions
at the extreme conditions encountered at the superheat limit. Key
assumptions which may not be valid under conditions of rapid evapora-
tion include: equilibrium of liquid and vapor at the evaporating
surface; neglect of the dynamics of the vapor inside the bubble; and
the neglect of the flux of liquid toward the interface in the calculation
of heat transfer in the liquid.

No attempt will be made here to comprehensively discuss any
of these issues or develop a less restrictive theory. However, to
illustrate the issues we will set up and solve a simplified one-
dimensional version of the classical model. This simplified model uses com-
pressible dynamics (acoustics) instead of the usual incompressible
bubble dynamics, but otherwise uses the same kind of approximations,
and the solution has the same character, as the classical spherical
bubble growth. In the context of this geometrically simplified model,
the limitations of one particular approximation, namely the neglect of
differential liquid-interface motion in the heat transfer problem, will

be examined.

144

Under the conditions of very rapid evaporation, the flux of
liquid into the evaporating surface also represents a substantial con-
vection of thermal energy into the thermal boundary layer adjacent
to the surface. This convective heat transfer has a fundamentally
different character than the usually assumed diffusive heat transfer.

In particular, such convection can support a steady-state evaporation

process under the appropriate conditions. Clearly, this implies the
possibility of a completely different picture of bubble growth in highly
superheated liquids than previously obtained from the classical, dif-

fusion-limited theory.

6.2 One-Dimensional Version of the Classical Model

In this section the one-dimensional analog of the classical model
of bubble growth is developed and, under certain approximations,
solved analytically. By one-dimensional analog is meant the growth
of a region of vapor ("planar bubble'') bounded by two parallel evap-
orating surfaces adjoined by infinite regions of superheated liquid.
Obviously, the problem is symmetric so that only half of the bubble
and one evaporating surface has to be considered. All motion, heat
transfer and variation of physical properties occurs only in the direc-
tion normal to the evaporating surface.

The coordinate system and basic notation is shown in Figure
6.1; vapor fills the region between the center of the bubble x = 0,
and the right hand evaporating surface x = X(t). To the right of
the evaporating surface, X(t), is superheated liquid moving with
velocity u(x,t). At the bubble surface, the velocity in the liquid

will be denoted us,

u(t) = u(X(t), t)

145

The initial undisturbed conditions in the liquid are p = Po? T= TO
and u(x,0) = 0. The interface starts at the origin x = 0 at time
t = 0 and subsequently moves to the right with a velocity X(t),
pushing the liquid ahead of the interface and radiating pressure waves.
At the interface, there is a net evaporative mass flux m from the
liquid into the vapor. Therefore, the liquid velocity at the interface,
u., will be less than the interface velocity x by an amount deter-
mined by the conservation of mass,

m =p Fe -ul) ,

where is the liquid density. The ''differential liquid-interface

Py
motion'' referred to in the introduction of this chapter is merely this
difference between X and ua:

The liquid is assumed to be slightly compressible so that dis-
turbances generated at the evaporating interface are propagated out
into the undisturbed liquid with the speed of sound c. Other than
this, the approximations of the classical model as set forth by
Prosperetti and Plesset (1978) are used to describe the vapor dy-

namics, evaporation process and heat transfer, The most important

of these are given below for completeness:

a) Conditions in the vapor are uniform and are taken to be
those of the saturated vapor at the temperature of the bubble surface

i,e, throughout the bubble the vapor density is p(T.) and the
vapor pressure is p(T). The dynamics of the vapor inside the
bubble and at the interface are neglected.

b) Boundary conditions at the interface are simplified by
assuming: the pressure difference due to the momentum flux is

negligible, p(T.) = Po the interface-liquid velocity difference due to

146

mass flux is negligible, Us = xX; and the enthalpy difference between
liquid and vapor is the same as for the equilibrium liquid-vapor sur-
face, i.e. the kinetic energy of the flow is assumed to be negligible
and the heat flux q into the interface is determined only by the
mass flux m and the latent heat L,

c) The heat transfer problem is simplified by assuming that

only the conduction in the liquid is important

. _ 5 _
G=- kp (6-2)

and convection is negligible, i.e., there is no differential interface-
liquid motion. Further, the heat transfer is all assumed to take
place in a thin thermal boundary layer of thickness 6 adjacent to
the evaporating surface.

Later, additional approximations will be made in order to
obtain an analytical solution to the problem. These approximations
are the same as those used by Prosperetti and Plesset (1978) in
deriving the universal model of bubble growth mentioned earlier in
Chapter 4.

Before going into the details of the model a simplified physical
picture of bubble growth (from the classical point of view) will be
given. This picture is valid for both spherical and planar cases
except that a stage of spherical growth dominated by surface tension
has been omitted. The motion of the bubble is determined by two
processes: first, the rate at which fluid can be pushed out ahead of
the growing bubble, and, second, the rate at which vapor can be

evaporated into the bubble. The first rate is determined by the

147

inertia of the liquid (i.e. for spherical bubbles, potential flow) and
the driving force exerted on the liquid (i..e., the vapor pressure in
the bubble). The second rate is determined by heat transfer at the
bubble surface and the requirement that the vapor always be in
equilibrium with the liquid. The crucial coupling between these two
rates occurs at the bubble surface, whose temperature determines
the magnitude of the driving force (vapor pressure) and in turn, is
determined by the heat transfer accompanying the evaporative mass
flux.

At early times in the bubble growth, little heat transfer has
taken place and the bubble surface temperature is just slightly lower
than ambient. This initial period is referred to as the "inertial stage!’
since the vapor pressure is roughly constant and pushes the bubble
surface out as fast as liquid dynamics (inertia) will allow. As bubble
growth proceeds, the cumulative effects of heat transfer reduce the
interface temperature; hence, the vapor pressure falls and the bubble
growth rate decreases. Asymptotically, as t—o, the bubble pressure
tends toward Po and the growth is strictly diffusion-limited, then
the bubble 'radius'' R(t) (or, in the one-dimensional case, X(t))
~ ti 2

The details of the model are logically separated into two parts:
first, the heat transfer will be discussed in Section 6.2.1; second,
the liquid dynamics (acoustics) will be discussed in Section 6.2.2.
The model equations will be summarized and the approximate uni-

versal equation solved in Section 6.2.3.

6.2.1 Heat Transfer

The heat transfer problem is to determine the interface

148

temperature T as a function of the heat flux into the interface.
Under the assumptions of this model only the diffusive transport is
important, and then only within a thin boundary layer adjacent to the
interface where the liquid velocity is constant. In interface-fixed
coordinates the boundary layer problem is identical to the one-
dimensional transport of heat in an initially uniform semi-infinite
solid, with the flux prescribed as a function of time on the boundary.
It is a standard problem whose solution can be found in Carslaw and

Jaeger (1959, p.76), and, in terms of the temperature gradient

dT
= at the interface,
dx
1/2 dT.
Ky t dx §7)
T =T -(€&) J Ss art (6-3)
s 00 T -
0 vj t-T

This solution is identical to the limit R - o of the solution to the
spherical case (Plesset and Zwick 1952), a solution universally used
in the conventional models of bubble growth.

in order to connect this solution to the bubble growth, equa-
tions 6-1 and 6-2 must be used to determine the temperature gradient

at the interface as a function of the mass flux. Mass flux can then

be related to the bubble size X(t) by the overall mass balance of

the bubble

mh = = (Xp(T)) (6-4)

6.2.2 Liquid Dynamics

The acoustic formulation of the problem is used in the one-

dimensional case since incompressible dynamics of a semi-infinite

149

one-dimensional fluid are trivial. Only waves propagating away
from the interface into the liquid are considered. From acoustics,
the perturbation pressure p'’ is related to the fluid velocity per-

turbation 6u by

p' = peéu_ , (6-5)

where pc_ is the acoustic impedance of the liquid. Under the
assumptions (b) of this model, in the liquid adjacent to the interface
the perturbation pressure is p' = p(T.) - Py and the velocity is
Su = X. Altogether, at the interface

PT.) - P= pe X , (6-6)
which, together with equations 6-1 through 6-4 and the saturated vapor

thermodynamics, p(T.) and p(T), forms the complete set of

equations for this model.

6.2.3 Summary of the Model and Its Approximate Solution

Combining the previous equations and simplifying, the following
pair of equations for Ts and X_ result

M2 ¢ uty & ((T,)x)

s ee) T k

) (6-7)

pulT,) - P= pe X

For general Py» Py and L these equations must be sim-
ultaneously solved by numerical integration. However, as shown by
Prosperetti and Plesset (1978) for the analogous spherical model,
further approximation yields a single, more tractable equation with
solutions similar in form to those of (6-7). The necessary approxi-

mations are that the vapor pressure depends linearly on temperature

150

p(T.) - p(T) = A(TS-T ) (6-8)

and the vapor density and latent heat are constants independent of

interface temperature. Similar to the spherical version of the
problem, a scaling velocity Xo and characteristic time T can
be defined
o PAT )- Pp
_ Vv 00 _
X= 5c (6-9)
and
T pe.” Ky °

In terms of the dimensionless variables

t= t/t and XY = X/Xpg
the approximate equation is

x(#) = 1 -

o- cre

2G ae (6-11)
t-§

A straightforward application of the Laplace transform technique

yields the universal solution:

X = e. ; erfc Jat . (6-12)
Integration determines the bubble surface position X(t) and equations
6-6 and 6-8 determine the interface temperature from X. All of the
properties ot the model solution that were mentioned in the discussion
at the end of Section 6.2 can be verified from this solution and its
expansions at early times,

yrl- 2y%-1/2

and late times,

1 ~-1/2
X~7t

151

6.3 Limitations of the Approximate Heat Transfer Solution

In this section, the heat transfer problem will be considered
from a rather more abstract point of view as the one-dimensional
transfer from a plane sink or source (representing the interface)
to the fluid flowing into it from the right half plane. Fixing the
location of the interface at x = 0, the governing equation in inter-

face-fixed coordinates is

8 o*
- w(x, t) ox 7 K ;
Ox

re)
( )T = 0 (6-13)
where

w(x,t) = X(t) - u(x+X(t), t)

In the lab frame X and wu are respectively, the interface and

liquid velocities.

For incompressible one-dimensional flow,

u = u(t) (6-14)

only. If we assume a slightly compressible flow, then u_ repre-
sents an acoustic disturbance and is, in general, a function of
position also. To simplify the problem in this case, we suppose
that the temperature variation is contained within a thermal boundary
layer, within which the velocity u is taken to be uniform and equal

to the velocity Us at the interface, which is a function of time

only, us = u(t). In this case, the general equation is
f) f) 9
Be - wl) ag 7 KTS) T= 0 (6-15)
where

Boundary conditions are that the temperature is uniform initially,

there is no disturbance at infinity and the gradient is specified at

152

the source;

T(x,0) = T) ,
T(o,t)= T , (6-16)
T (0, t) = g(t)

The problem then is to determine the temperature field or, more
specifically, the temperature of the interface as a function of the
gradient T (0, t) at the interface. Solutions can be found for the
simplified cases of constant or vanishing velocity w by Laplace
transform and Green's function methods.

An order-of-magnitude analysis of the terms in equation 6-15
reveals the general properties of its solutions. The equation has
the form of a transport equation with the basic property that the
disturbance at the origin will be contained within a boundary layer
of thickness 6(t), The proper form of 6 and its dependence on
time will be considered next in this section, Outside of this boundary
layer diffusion is negligible and material is being transported to the
origin without a change in temperature. In the special case treated
here of uniform initial conditions (T = constant everywhere), the
solution outside the boundary layer is trivial and the equations and
poundary layer solution are the same as for the full problem.

The thickness 6(t) and the interface temperature T,(t) display
either a quasi-steady or nonsteady behavior depending on the relative
importance of the convective transport or diffusive (nonsteady) trans-
port. At the interface, the following order-of- magnitudes estimates
will be made for the various terms in equation (6-15)

oT . ds ,
at tO”

153

oT UT
—nw 2 .
W Ox é ,
3 (6-17)
k oT _ KTp ;
ax" &°
where U is a constant velocity characterizing w(t). Inside the
boundary layer the diffusion term 5 is always important and
ox

must be balanced by one or both of the other terms to obtain non-
trivial solutions.

In the case where nonsteadiness dominates, the previous re-
marks imply that

ct |

(6-18)

The result 6 ~ /Kt is typical of a diffusion-controlled process and

as long as ww t"" where n2 5, there will always exist a stage
near t= 0 where this process dominates. The approximate equa-
tion governing this stage is

oT eT _

ow kK BE =O (6-19)
It has the solution

1/2 tT (0,t) -—-—
T(x,t) = TT. - (S) —x 4K (t-T) dt (6-20)
oe] nis
0 t-T

Note that for x = 0, this is simply equation 6-3 which appeared in
our one-dimensional analog of the classical model of bubble growth.
In the case where convection dominates, the relations between

the scaling parameters must be

U K 1

154

The thickness is now 6 ~ x and depends on time only if w(t) is

a strong function of time. This is a result typical of steady-state
heat transfer such as in forced convection or a steady flame front.

If the boundary condition TO, t) and velocity w(t) are sufficiently
well behaved functions of time, there will always be a stage

where inequality (6-21) is satisfied for sufficiently large time t.

The approximate equation governing the asymptotic stage is

= 0 (6-22)

which has the solution

KT (0,t) - MH,
x K

T(0,t) = - e (6-23)

w(t)

For the case in which the velocity w(t) is constant and
T (0, t) is reasonably well behaved, there must be a crossover be-
tween the diffusive behavior at the early times and the asymptotic
quasi-steady state. The time t at which this occurs is, say,

when the thicknesses 6(t) for both cases are equal

= /«t-

Cla

or, t = K/U?. The analytical solution for the case w(t) = w=
constant substantiates these order-of-magnitude estimates, and
expansion of that solution at early and late time yields solutions
6-20 and 6-23 respectively. The existence of the characteristic
time t" has previously not been pointed out in the context of heat
transfer for the bubble growth problem.

At the low superheats typical of all previous bubble-growth
experiments, the velocities w(t) induced by the mass flux through

Re
the liquid-vapor interface were very small, thus the times t are

155

enormous compared to the duration of the experiment. For example,
for water at 3.5°C superheat, t = 10©° sec! Therefore, the commonly
used diffusive approximation for the heat transfer (e.g. Plesset and
Zwick 1952) with w= 0 agrees quite well with the low superheat
experimental data. However, the time t in the present experi-
ments with butane at the superheat limit is estimated to be on the
order of 10°° sec! This time is much smaller than the time

t= 2 x 10°° sec characterizing the length of the inertial stage, in-
dicating that convective transport must play an important role in
bubble growth. The neglect of this effect in the theory could account
for some of the discrepancy with experimental observations. However,
the presence of the highly roughened evaporative surface indicates

that there are other, equally serious, problems with the model. In
conclusion, the results of this section indicate the need for a more
careful treatment of bubble growth at large superheats. Particularly,
the role of convective heat transport should be reconsidered and the

possibility of ''steady-state'' solutions investigated.

156

REFERENCES

Carslaw, H. and Jaeger, J. 1959 Conduction of Heat in Solids
2nd ed., Clarendon Press, Oxford.

Plesset, M. and Zwick, S. 1952 ''A Nonsteady Heat Diffusion
Problem with Spherical Symmetry", J. App. Phys. 23, 95.

Prosperetti, A. and Plesset, M. 1978 "Vapor-Bubble Growth in
a Superheated Liquid", J. Fluid Mech. 85, 349.

157

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158

VII. CONCLUSIONS

The vapor explosion of a single droplet (~ 1 mm dia.) of liquid
butane at the superheat limit has been experimentally investigated,
The purpose of this experiment was exploratory, and was motivated
by the difficult and unknown dynamics of highly nonequilibrium evapora~
tion which occurs at the limit of thermodynamic stability. High speed
photography and pressure measurements have been used together for
the first time to obtain a description of the complete explosion pro-
cess within a drop superheated in the bubble-column apparatus.
Emphasis was placed on the early (microsecond time scale) evapora-
tive stage. Despite the apparent simplicity of the vapor explosion of
a single superheated droplet, the present experiment revealed a wide
range of phenomena of varying complexity occurring at different
stages of the explosion,

The explosion is initiated by a single nucleus spontaneously
forming in the drop, close to, but not on the drop surface. The
nucleus grows by evaporation of the surrounding superheated butane
to form a butane vapor bubble asymmetrically located within the
drop. Bubble growth involves several new and unique features which

are remarkably repeatable for different explosions:

a) Photographs of the evaporative surface show a highly
roughened and disturbed interface for most of the evaporative stage.
At the earliest times in the explosion for which photographs have been
obtained (~ 8 psec), the roughening appears to begin as a regular
pattern on an otherwise spherical surface, suggestive of a fundamental
instability due to evaporative mass flux.

b) Due to the asymmetric location of nucleation within the

159

drop, a portion of the evaporating bubble surface is observed to contact
the host fluid first and become nonevaporating. A unique, axisymmetric
structure of circumferential surface waves culminating in a spherical
cap appears on this protruding nonevaporating surface as the bubble grows.
Both the surface waves and the protrusion of the nonevaporating surface
are conjectured to be driven by the "jet'’ of evaporated vapor coming
from the opposing evaporating surface and impinging on the host liquid.

c) Nucleation and initial development of the bubble in the first
10 psec is almost universally accompanied by an intriguing pulsating
pressure signal, characterized by two steps, which suggests that a
fundamental and repeatable unsteadiness, perhaps connected with the
above-mentioned instability, is taking place at this stage. Unfortunately,
no volume data were obtained in the present experiment prior to 8 usec
so that a definite source for this signal could not be determined.

Overall, bubble growth is characterized by a constant effective
radial velocity of ~ 14.3 m/sec, resulting in the complete vaporiza-
tion of the original drop liquid within 40-160 psec (depending on the
initial drop size) of the beginning of the explosion. Pressure signals
measured ~ 2.5 cm from the explosion during the evaporative stage
(but after the occurrence of the two-step structure which is always ob-
served) show a roughly linear increase of pressure with time, upon which
nonrepeatable oscillations may be superposed. The maximum pressure
reached at the end of evaporation ranges from 1 to 7 psi, larger drops
producing larger maximum pressures.

Immediately following the completion of evaporation, the bubble
is at a net positive overpressure, due to the dynamic, nonequilibrium
nature of the foregoing evaporation process. This stored energy is

released through subsequent volume oscillations of the gas bubble.

160

These oscillations produce characteristic pressure waves known as
"bubble pulses'', which are the dominating feature of all pressure
signals measured in the later stages of the explosion. At the
minimum in volume of the first oscillation cycle, the surface of the
bubble experiences large adverse accelerations, resulting in Taylor
instability, and catastrophic deformation of the bubble surface results.
Continuing volume oscillation of the deformed bubble is accompanied
by further Taylor instability and increasing deformations until the
original bubble is broken up into a large number of smaller bubbles;
in all cases, this is the final state visible to the human observer.

All of these features of bubble dynamics observed at later
times have been previously reported by many investigators in other
contexts; the merit of the present observations is the clarity of the
photographs of the instability process and the ideal nature of droplet
vapor explosions for producing large amplitude oscillations of an
inert gas bubble.

By far the most important discovery of the present work is the
instability of the evaporative surface and its implication for greatly
increased evaporative mass transfer. A simple acoustic model of the
evaporating bubble was developed to relate evaporative mass flux to
the volume growth rate and emitted pressure signal. Using photo-
graphically determined bubble volumes and pressure signals measured
in the first 30 psec of the explosion, a preliminary estimate of the
evaporative mass flux has been made with this model. After a very
rapid increase in the first 1-2 psec of bubble growth, the mass flux
remains approximately constant at ~ 400 gm/cm”* sec thereafter.

This mass flux is much larger than encountered in typical evaporation

161

experiments and is 10° larger than would be predicted on the basis

of classical, diffusion-limited bubble growth. - Further, comparison
with measured volume growth rates and emitted pressure signals shows
that the classical theory vastly underestimates both quantities at later
times. It is not possible to determine from this preliminary inve sti-
gation whether the abnormally large mass flux is solely produced by
the increase of evaporative surface area due to the roughening or
whether more subtle effects associated with the highly nonequilibrium
state of the liquid are also involved. In any case, the present results
clearly indicate that the assumptions of the previous, near-equilibrium
theories of evaporation need reconsideration at the extreme conditions
of the superheat limit.

It is proposed in the present work that the roughening of the
evaporative surface originates in a fundamental instability of the
liquid-vapor interface driven by the evaporative mass flux. We have
investigated this for the simplest type of instability mechanism, the
Landau or inertial instability, which depends only on the existence of
a sufficiently large mass flux and a density difference at the inter-
face. Landau's original analysis for a plane surface was supplemented
by an ad hoc correction for sphericity, and the resulting theory was
applied to the classical model of bubble growth. Taking into account
surface tension and deceleration, the bubble surface was found to be
unstable for the portion 8 x 10° to 3 x 10°7 sec of the initial growth,

The earliest observations in the present study at 8 sec showa
regular pattern with a scale length of 20-40 um on the surface of the bubble;
a maximum length scale of 2-10 times smaller than that is predicted by
by the ad hoc theory to occur at times of order 0.3 psec. An explana-

' tion consistent with the available evidence and the theory is that the

162

observed instability at 8 sec has developed well into the nonlinear
stage and the actual instability did occur in the first 1-2 psec of
bubble growth. Bubble images enhanced by digital processing show
substantial asymmetry and randomness in the pattern observed at the
earliest times, supporting the above conclusion.

A one-dimensional analogue of the classical model of bubble
growth has been developed and, under certain assumptions, analytically
solved. In the context of this one-dimensional model, we have reex-
amined the assumption used in the classical theory that convective
heat transfer is neglibible. It is concluded that under the conditions
found at the superheat limit, this assumption is incorrect. Entirely
different types of solutions may be possible; in particular, steady-
state solutions can exist.

In conclusion, the present observations are fundamentally dif-
ferent from what would be expected by the mere extrapolation of
previous near-equilibrium theories and results. Preliminary efforts
have been made to calculate the evaporative flux and stability prop-
erties of rapidly evaporating surfaces, but only a superficial treat-
ment has been possible. The extent of the generality of the present
experimental results needs to be verified in detail, but clearly, the
present observations indicate evaporation at the superheat limit can

be much more complex than previously envisioned.

163

APPENDIX A, PRELIMINARY MEASUREMENTS

A.1l Shadowgraphy

Initial visualization of the explosions was made with a simple
shadowgraph system (Figure A.1) which consisted of a point source
spark gap, a collimating mirror and a Polaroid film holder. The
spark gap was triggered from the pressure signal by the same tech-
nique as described in Section 2.4.2 and the images were recorded
on Polaroid type 667 film (ASA 399%). Due to focusing effects
mentioned in Section 2.4.2, the drops and bubbles appeared as
relatively featureless opaque blobs; only the relatively large surface
perturbations (Taylor instability) at late times were obvious.

From these shadowgraphs and the pressure signals the existence
of bubble oscillations and the possibility of instability at late times
were inferred. The evaporation process remained unrevealed and

clearly better diagnostics were called for.

A.2 Extinction Meter

Shadowgraph observations of the nearly circular shadows cast
by the exploding droplets suggested that a continuous measurement of
the extinction of incident light could yield the mean diameter as a
function of time. A system was assembled to do this as shown in
Figure A.2. The principle of operation is that the optical cross-
section of such a scatterer as the exploding droplet is proportional
to the geometric cross-section. Asymptotically in the far field of
the scattered radiation, the constant of proportionality is 2; directly
reflected and refracted rays account for half this, the other half is
contributed by the light diffracted from the edge of the drop, see

Brillouin (1949). Generally, the far field approximation is not

164

achieved experimentally (Sinclair 1947) and a calibration scale has
to be established for individual instruments.

The illumination for the instrument shown in Figure A.2 was
provided by a 5 mW He-Ne laser whose output was spatially filtered
and collimated. The collimated beam passed through a beam
splitter where a portion of it was focused onto a phototransistor
(reference detector); the remainder of the beam proceeded through
the test section intercepting the exploding droplet. After exiting the
test section the beam passed through an aperture and was focused
onto a second phototransistor (signal detector). The outputs from
the reference and signal detectors were recorded by the Nicolet
oscilloscope for later processing by the computer. Given the initial
drop diameter and the instrument calibration, the projected area
as a function of time could be determined from the ratio (used to
eliminate any overall drift in intensity) of signal detector output to
reference detector output. An example of the signal detector output
is shown in Figure A.3, the bubble oscillations and mean volume
increase can be seen quite clearly.

These measurements confirmed the existence of oscillating
bubbles, at late times, and indicated that an overall expansion of
the drop occurred during the evaporation stage. However, there
was no chance of revealing the details of the evaporation process

by this method and it was not further pursued.

165

REFERENCES

Brillouin, L. 1949 "The Scattering Cross Section of Spheres for
Electromagnetic Waves", J. App. Phys. 20, 1110.

Sinclair, D. 1947 "Light Scattering by Spherical Particles", J.
Opt. Soc. Am. 37, 475..

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169

APPENDIX B, START UP OF AN AIR JET INTO WATER

This appendix describes some exploratory flow visualization of
the start up of an air jet submerged in water. The purpose of this
experiment was to determine if there was any similarity of the air-
water interface to the butane vapor-glycol interface described in
Section 3.3.2. The primary interest was in the form of the instabil-
jties that must certainly be present on the jet surface. No effort
was made to achieve true dynamic similarity between the two flows.

The apparatus (Figure B.1) consisted of an aquarium filled
with water, an air injection system (jet air supply) and a photographic
set up similar to that described in Section 2.4.2. The jet issued
downward from a flanged 1/8'' pipe placed just beneath the free
surface of the water, about 8'' from the bottom of the aquarium.

A solenoid valve connected the jet pipe with the air reservoir, a
surplus oxygen bottle (~ 340 cm® volume) pressurized with house air
(30 psig). The triggered spark gap-diffuser described in Section 2. 4.2
was used to illuminate the jet at a variable delay time from the solenoid
opening. The image was recorded at a magnification of 2 on Polaroid type
47 film, using the same Graphex camera body as in the principal
investigation. By carefully maintaining constant initial conditions
for every shot (i.e. the same reservoir pressure and the same size
initial air bubble protruding from the jet orifice), single photographs
taken at different delay times on separate shots could be used to
construct the course of jet development.

Several different combinations of jet orientation (upwards or
downwards) and orifice configurations (flanged and unflanged) were

experimented with. All of these combinations showed the development

170

of instabilities on the jet surface; the most striking of these were
obtained with the particular arrangement of Figure B.1l. The
development of the jet is shown in Figure B.2; the times indicated
under each print are the number of milliseconds that have elapsed
since the current was first applied to the solenoid. Prior to 4 msec,
there was no detectable motion; this corresponds to the opening time
of the solenoid.

The jet is initially just a growing half bubble of air, originating
from the small half bubble naturally present on the orifice when there
is no flow. Mean half bubble shape develops from an initially oblate
to a prolate axisymmetric surface; bubble volume increases approxi-
mately linearly in time (flow through the solenoid valve orifice is
choked). The surface appears very smooth until ~ 10 msec, when
an annular bulge can be seen developing near the base of the bubble.
Further development of surface deformations can be seen in the sub-
sequent prints at 11, 19 and 26 msec. These surface deformations
are very strikingly axisymmetric and exhibit the sharp corners
(particularly the 26 msec print) and regularity seen in the vapor
explosions. In the 19 msec print, the surface is observed to have

a re-entrant shape following the bulge nearest to the flange.

171

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173

APPENDIX C, STABILITY CRITERIA FOR OSCILLATING BUBBLES

Linear stability of the surface of an oscillating bubble was
first considered in connection with the observed instability of col-
lapsing cavitation bubbles (Plesset 1954) and bubbles resulting from
underwater explosions (Cole 1948). General criteria for stability
were established by Birkhoff (1956), who found that, in addition to
the usual Taylor instability associated with acceleration, there could
exist an algebraically growing instability associated with a decrease
in radius. Although the stability for freely oscillating bubbles was
investigated numerically by Strube (1971), the role of accelerations
was not emphasized and the simple results which can be obtained
from Birkhoff's criteria were not stated. The purpose of this
appendix is to explain some general features of the instability and
to give the results of Birkhoff's criteria for stability of the freely
oscillating bubbles observed in the present experiments.

A standard linear stability analysis assuming inviscid, incom-
pressible flow (for example see Plesset and Prosperetti (1977)) yields
the Rayleigh equation

RR +> R? = pF _ 20 (C-1)

Px Poo

for the mean bubble radius R anda linear, second order equation

for the surface perturbation amplitude an

* R, _
a, t3 Ra, - Aa, = 0 (C-2)
where
A = (n-1) cy - even) (C-3)
p_ R
[oe]

and the vapor density has been neglected in comparison to the liquid

174
density in the last expression. The angular variation of the surface

perturbation is assumed to be given by a spherical harmonic of

order n, Yn (8,2); note that the azimuthal index m _ does not

appear in equation C-2.

Birkhoff's stability criterion for equation C-3 is:

ae
dt

wave amplitudes oscillate in sign and grow in time.

a) A 0; oscillatory instability, surface
b) A> O, “7 (R&A) SS 0; unstable, surface wave amplitudes
grow monotonically in time.
c) A amplitudes decrease monotonically in time.
Generally, instability b) grows much faster than a), exponentially vs.
algebraically, so b) tends to be more important and will be discussed
first. From equation C-3, A > 0 whenever R>O and sufficiently
large; thus instability b) can be expected whenever the surface ac~
celerations are large and positive. This is simply the spherical
analog of the Taylor instability (Taylor 1950) for an accelerated plane
interface. Both the plane and spherical cases share the common
feature that if the wavelength } of the disturbance is small enough
(i.e. mode number n large enough) i < Le then surface tension
will damp out any possible instability. For large mode number n,
the spherical interface appears locally plane to the disturbance and

an effective wavelength } can be defined,
nk = 2nR . (C-4)

The effective cutoff wavelength Le can then be determined by
setting A = 0, approximating (ntl)(n+2) = n® and using equation

c-4,

175

ho = 2m / Rp, . (C-5)

This expression has exactly the same form as the corresponding
result for a plane interface. -Another common feature is that there
is a most unstable mode in both spherical and planar cases. The
form of equation C-3 and its dependence on n assures that A
will have a maximum for n between n = 2 (the lowest order mode
which actually represents a surface deformation) and the cutoff mode
number, n= 2nR/h..

The major difference between the spherical and planar Taylor
instabilities is that the longest wavelength possible in the spherical
case is finite (corresponding to n= 2), while in the planar case
modes of infinitely large wavelength can occur. This difference can
be interpreted as the existence in the spherical case of another cut-

off wavelength i approximately equal to the bubble radius, which

R?
has the effect of damping out all modes with \ > Ap: An important
consequence is that the spherical interface can be absolutely stable
if AG > Ap: irrespective of the magnitude of the accelerations.
Physically, this case occurs whenever the bubble radius is so small
that surface tension forces associated with the mean curvature of the
interface damp out any possible disturbances. The criterion for the
absolute stability under the conditions where R> 0 can be found

by requiring A to be negative for the lowest mode (insuring that

it will be negative for all modes), n = 2. This implies that

(R) < 120
R?

max
12.6)

(C-6)

176

Instability a) is specific to spherical surfaces and occurs under
conditions (A < 0) considered stabilizing or at worst, neutrally stable
for a plane interface. General circumstances under which this in-
stability occurs are difficult to specify, but there are some simple
particular cases. One standard example is the collapse of a spherical
void suddenly created in a fluid; this is Rayleigh's original cavitation
bubble collapse problem, discussed in Plesset and Prosperetti (1977).
In that case, the accelerations are stabilizing but the decreasing
"bubble'! radius causes any initial surface disturbances to grow, the
amplitude being proportional to R14 and oscillating in sign as the
"bubble" collapses. Another simple case is the freely oscillating
bubble, which will be discussed next.

An idealized model for the oscillating gas bubbles observed in

the present study is the Rayleigh equation with the gas pressure

given by the adiabatic relation
p(R)~ ROY (c-7)

With appropriate boundary conditions, this equation yields periodic,
nonlinear oscillations of R (schematically indicated in Figure C. 1)
which have been used extensively to represent the motion of freely
oscillating bubbles, e.g. Cole (1948). Using the stability criterion
given above the unstable portions of an oscillation cycle can be de-
termined as a function of the amplitude and frequency of the oscilla-
tion,

As expected, the motion is stable for all frequencies of oscilla-
tion if the amplitude is small enough. With increasing amplitude,
the first possibility of instability is parametric excitation for par-

ticular volume oscillation frequencies corresponding to subharmonics

177

of the linear surface wave oscillation frequencies. This mechanism
causes surface deformations to grow slowly over many volume
oscillation periods (Hullin 1971; Francescutto and Nabergoj 1978),
and is obviously irrelevant to the catastrophic instability observed in
the present experiments.

The catastrophic type of instability seen in the present work is
associated primarily with the large, positive radial accelerations that
occur at the minimum in volume during the oscillation cycle. For
large oscillation amplitudes, the surface tension can be neglected in
the Rayleigh equation and the accelerations (R) at the volume

max

minimum can be written
° Ap(R vin!
(R) oo . (C-8)

max p R_.
oo min

Using the above expression for (R) in inequality (C-6),

max

we find that it is possible that A > 0 (for n = 2) only if

l2o0

R_.
min

Ap(R (C-9)

min
Whether inequality (C-9) is satisfied depends on the amplitude of
oscillation; the larger the amplitude, the better it is satisfied.
Therefore, if the amplitude of the oscillation is sufficiently large,
then A > 0 for some mode numbers n_ when the bubble radius is
a minimum. Taylor instability then, is only possible in that portion
of the oscillation cycle near the minimum radius (labeled (b) in Figure
6.1); whether the instability will actually occur depends on both the

amplitude of the oscillation i.e. (R) and the magnitude of the
max

surface tension.

178

Away from the region near the minimum radius, R< 0 and
therefore A <0 always. Then, the only type of behavior possible

is case (a) or (c). Using the Rayleigh equation, the deriviative
dt
absolute stability; if R< 0 case (a) holds, with the possibility of

(R®°A) can be calculated and we find that if R > 0 case (c) holds,

algebraic instability. The regions of the oscillation cycle where

these two cases occur is shown on Figure 6.1. In summary:

a) R< 0, R< 0; unstable, surface waves grow algebraically
in time;

b) RS 0, R> 0; unstable, surface waves grow exponentially
in time;

c) R> 0, R <0; absolute stability, surface waves are

damped (exponentially) in time.

The three regimes where the different instabilities, (a) - (c), are
possible are indicated on a typical cycle of oscillation in Figure C.1
(the corresponding pressure pulse Ap _ that would be produced in the
far field of this oscillation is also shown). Away from a region near
the minimum radius, growth of the bubble damps the surface waves
until the maximum radius occurs. As the bubble collapses from the
maximum radius, surface waves can first grow algebraically in
amplitude, then exponentially if R becomes sufficiently positive near
the minimum radius.

Experimentally, onset of instability and the importance of non-
linear effects, at the first minimum in bubble volume, is shown in
Figure C.2. The pressure trace in this figure was obtained in the

far field (configuration C of Figure 2.3) so that a comparison to

179

Figure C.1 is possible. From print (c), the characteristic wave-
length of the deformations can be estimated to be 0.3 mm. _ Using
equation C-8, the radial acceleration can be estimated from the peak
amplitude of the bubble pulse and the minimum bubble radius (2 mm)
to be 10* g. The most amplified wavelength, * = AMS predicted
by the plane Taylor instability theory from this acceleration (i.e.
equation C-5) is within a factor of 2 of the observed characteristic
wavelength. The even more complex process which occurs at the

second minimum is illustrated in Figure C.3.

180

REFERENCES

Birkhoff, G. 1956 ''Stability of Spherical Bubbles", Q. Appl. Math.
13, 451.

Cole, R.H. 1948 Underwater Explosions, Princeton University Press,
Princeton, New Jersey.

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181

A (a) (b) (c)

Rmax

Bubble Radius, R(t)

Rmin

Time

Far. Field Pressure, Ap

Time
FIG. C.!1 A NONLINEAR BUBBLE OSCILLATION CYCLE
SHOWING STABILITY REGIMES AND THE
FAR FIELD BUBBLE PULSE

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