departure of vapor- and gas-bubbles in a wide pressure range · height difference between two...
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Departure of vapor- and gas-bubbles in a wide pressure range
Citation for published version (APA):Slooten, P. C. (1984). Departure of vapor- and gas-bubbles in a wide pressure range. Technische HogeschoolEindhoven. https://doi.org/10.6100/IR33763
DOI:10.6100/IR33763
Document status and date:Published: 01/01/1984
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DEPARTURE OF VAPOR- AND GAS-BUBBLES
IN A WIDE PRESSURE RANGE
P.C. SLOOTEN
DEPARTURE OF VAPOR- AND GAS-BUBBLES IN A WIDE PRESSURE RANGE
Druk: Dissertatie Drukkerij Wibro, Helmond, Telefoon 04920-23981.
DEPARTlTRE OF VAPOR- AND GAS-BUBBLES IN A WIDE PRESSURE RANGE
PROEFSCHRIFf
TER VERKRIJGING VAN DE GRAAD VAN DOCfOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. S. T. M. ACKERMANS, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP
VRIJDAG 22 JUNI 1984 TE 16.00 UUR
DOOR
PIETER CORNELIS SLOOTEN
GEBOREN TE ZAANDAM
Dit proefschrift is goedgekeurd
door de promotoren:
prof.dr. D.A. de Vries
prof.dr.ir. G. Vossers
co--promotor:
dr. S.J.D. van Stralen
CONTENTS
LIST OF SYMBOLS
INTRODUCTION
1. BUBBLE DEPARTURE AND BOILING
1. 1 Boiling
1.1.1 Boiling research
1.1.2
1.1.3
1. 1. 4
1. 1.5
1.1.6
1. 1. 7
1.1.8
Simplifications; bubble types
1.1.2-1 Static situation
1.1.2-2 Dynamic situation
Growth types
Binary systems
Waiting time. Bubble cycle
Departure models
Pressure dependence
Relation between bubble frequency and departure
radius
1.1.9 Vertical walls
1.1.10 Viscosity
1. 2 Gas bubbles
2. DEPARTURE OF SPREADING BUBBLES FROM A HORIZONTAL WALL
2.1 Restrictions
2.2 The resultant' force on a bubble, growing at a wall
2.3 Equation of motion
2.4 Classification with respect to growth rate
2.5 Alternative description using the quantities of the
equivalent spherical segment
2.6 Bubble shape; Methods to obtain departure formulae
2.6.1 Static situation
2.6.2 Transition and dynamic situation
2. 7 Intermezzo: The "equilibrium bubble shape theory"
2.7.1 Numerical results
2.7.1.a Cavity bubble evolution;
Criterion for the bubble mode
2.7.1.b Spreading bubble evolution
5
I I
13
13
14
15
16
16
17
17
18
18
20
21
21
21
22
23
23
24
28
29
29
30
31
31
31
33
35
36
2
2. 7.2 Analytic results 37
2.7.2.a Analytic results on cavity bubble departure 37
2.7.2.b Analytic results on spreading bubble
departure
2.8 Static departure (of spreading bubbles)
2.8.1 Use of the force equation (2.16)
2.8.2 Use of the quantities of the spherical segment
2.8.3 Static departure: derivation of (2.37b),using the
force equation
2.9 Transition departure (of spreading bubbles)
2.10 Dynamic departure (of spreading bubbles)
38
39
39
41
42
44
48
2.11 Microlayer bubbles 49
2. 12 The departure time t 50 l
3. DEPARTURE OF CAVITY BUBBLES FROM A HORIZONTAL WALL 52
3.1 Static departure (of cavity bubbles) 52
3. 1. 1 Use of the force equation (2. 16) 52
3.1.2 Use of the quantities of the equivalent spherical
segment
3.2 Transition departure (of cavity bubbles)
3.3
3.4
3.5
3.6
3.2.1 Comparison with spreading bubble results
Dynamic departure (of cavity bubbles)
Microlayer bubbles
The departure time t l
Numerical example
4. DYNAMIC DEFORMATION; MICROLAYER FORMATION
4.1 Dynamic deformation
4.2 Microlayer formation
4.3 Numerical example 1
4.3.1 Superheat dependence
4.3.2 Pressure dependence
4.4 Numerical example 2
4.5 Conclusions of Chapter 4
54
55
57
57
58
58
59
61
61
67
70
70
71
73
74
5. EXPERIMENTS
5.1 Purpose of the experiments
5. 1. 1
5 .1. 2
5.1. 3
Elevated pressures
Low pressures
Theoretical pressure dependence of R , t 1 1
75
75
75
75
76
5.2 Description of the experiments 76
5.3 Experimental results 79
5.4 Comparison of the experiments with the (static) theory 84
5.4.1 Pressure dependence of R 1
5.4.1.1 Experiments using a heated wall.
Comparison with the relaxation microlayer
theory
5.4.2 Cavity size dependence of R 1
84
86
86
5.4.3 Pressure dependence of t 88 1
5.5 Comparison of low pressure data from literature with the
dynamic theory
5.6 Conclusions; suggestions for further experiments
5.6.1 Elevated pressures
5.6.2 Low pressures
5.6.3 Minimal t 1
5.6.4 Suggestions for further experiments
APPENDICES
A1. Bubble growth
A2. Comparison of the exact numerical solutions for the bubble
shape with the analytical approximation
88
91
91
91
91
92
93
98
A3. Nucleation 100
A4. Viscous force,Fv; force due to surface tension gradients, F~cr 102
AS. Physical properties 104
A6. Bubble mass 105
A7. Derivation of the liquid inertia force FD for hemi-spherical
bubble growth
A8. Extension to gas bubbles
A9. Extension to binary systems
A10. Vertical walls
106
107
108
109
3
REFERENCES Ill
SUMMARY 115
SAMENVATTING 117
LEVENS LOOP 120
NAWOORD 120
4
LIST OF SYMBOLS AND SI-UNITS. =~==~=~~=====================
a l
a cap
A
A 0
B
B 0
c 1
llc
c
c
D
e -z
f
F
F cr,r
F corr
Quantity
liquid thermal diffusivity [m2 /s]
capillary length, defined: a l(a/p g). For water cap 1
at I bar, 100° C: a = 2.48 10- 3 m [m] cap
coefficient, defined by eq. (2.23) [-]
surface area of a bubble [m2]
coefficient, defined by eq. (2.24) [-]
area of the bubble base [m2]
specific heat of the liquid [J/kgK]
supersaturation of dissolved gas [kg/m3]
growth constant, cf. eq. (2.1) [m/s 1 12]
di~ensionless growth constant, defined: C C/(gas )1/li [-] cap
drag coefficient (A4) [-]
rrdcrolayer thickness [m]
discriminant of cubic equation, cf. sect. 2.9 [-];
diffusion coefficient of dissolved gas [m2 /s]
unit-vector in z-direction, cf. fig. 2.2 [-]
coefficient, cf. sect. 2.8.3 [-); bubble frequency,
inCh. I [s- 1 ]
total force, applied on the bubble, cf. eq. (2.4) [N]
Archimedes- or buoyancy force: F8 = pgV [N]
surface tension force (adhesion force) [N]
resultant surface tension force, cf. eq. (2. 14) [N]
correction force, cf. eq. (2.8) [N]
dynamic force or liquid inertia force, cf. eq. (2.5) [N]
5
F \)
Ffla
F(R*) 1
g
G
h
flh
H
ii
H*
ii*
I
j
k
k 1
K 0
Jl,
m, m'
m 2
n, n'
.!!.
p
p1
P (r, 1
t)
Po
6
viscous force, cf. A4 [Nl
force by surface tension gradients, cf. A4 [Nl
function of R*; cf. fig. 2.8 [-l 1
gravitational acceleration [m/s 2 l
vaporized mass diffusion fraction [-l
coefficient, cf. sect. 2.8.3 [-l
height difference between two points at the bubble
surface, cf. sect. 4.2 [ml
bubble height [ml
H/acap [-l
height of the equivalent spherical segment [ml
H* /a [-l cap
inflexion point of the bubble profile [- l
term of eq. (2. t6g), representing the liquid inertia
force: j = FD/2np ga [-l 1 cap
thermal conductivity [J/(sKm)l
ib. of the liquid [J/(sKm)l
curvature of the bubble profile at the bubble base [m- 1]
latent heat of evaporation [J/kg]
exponents, cf. sect. 5.t.3 [-]
bubble mass [kg]
exponents, cf. sect. 5. t.3 [-]
inside directed unit vector, normal to the bubble
surface [-]
ambient pressure or pressure of boiling vessel (Ch. 5) [Pal
liquid pressure [Pa]
liquid pressure field (At) [Pa]
p (oo, t) (At) [Pal 1
p2
(p 2 -po\n
(p 2 -p o) dyn
p (z) 1hydr
Pext
r
r
q
Q
R
R(t)
R
R 1
R(*)
pressure of the vapor inside the bubble [Pa]
initial pressure, needed for bubble growth [Pa]
(dynamic) bubble overpressure due to liquid inertia [Pa]
vapor pressure [Pa]
hydrostatic liquid pressure [Pa]
external pressure of the system [Pa]
dynamic liquid pressure [Pa]
dynamic pressure difference, cf. eq. (4.1.2) [Pa]
surface tension pressure difference, cf. sect. (4.1) [Pa]
hydrostatic pressure difference, cf. sect. (4. I) [Pa]
pressure difference across the interface, at the bubble
base [Pa]
cavity mouth radius, cf. fig. 2.1 [m]; radial coorlinate
in AI [n]
radial vector in Al [-]
radius of the bubble base, cf. fig. 2.1 [m]
heat flux [J/m2 s]
number, defined by eq. (3.21) [Pa m0•
2735 jK1•
09"]
radius of the equivalent sphere, cf. eq. (2.1) [m];
radius of hemisphere in A7 [m]
radius of free bubble in AI [m]
dR/dt [m/s]
radius of equivalent sphere, at departure [m]
radius of equivalent sphere in A7 [m]
principal radii of curvature of the bubble surface [m]
radius of the equivalent spherical segment, cf. fig. 2.1 [m]
ib., at departure [m]; or used fori* in sect. (2.9), 1 ... ' (3.3) [-]
7
R
R*
R* 1
R'
R top
R top1
R top1
R 0
it. ~n
Rdiff
Re
s
t
t 1
t'
t w
T
T(!_, t)
T sat
T 1
T 0
T 100
TR(t)
8
R/a [-] cap
R*l [ ] acap -
ib., at departure; in sect. (2.9), ••• , (3.3) written
as R* [-] 1
value of R, chracterising the transition from inertia
controlled to diffusion controlled bubble growth [m]
radius of curvature of the bubble top [m]
ib., at departure [m]
Rtop /acap [-] 1
radius of the equilibrium vapor cluster or nucleus of
homogeneous boiling [v. Stralen/Cole, 1979, Ch. 3] [m]
dR/dt in the inertia controlled growth regime [m/s]
dR/dt in the diffusional growth regime [m/s]
Reynold number, defined by eq. (A4.3) [-]
number, defined in eq. (4.3.4) [m/ (s 1 12 K)]
time since the bubble initiation [s]
departure time: t at departure [s]
t at the transition from inertia controlled to diffusion
controlled bubble growth [s)
waiting time between bubble departure and the initiation
of the next bubble [s]
temperature [K]
temperature field [K)
saturation temperature [K]
liquid temperature [K]
T(oo, t) (Al) [K)
lb., in the u~disturbed liquid [K]
bubble wall temperature (Al) [K]
"' ~ 2
T (R) !
L'.T
u
v
v(E, t)
v
v !
v !
v max
VI
w
X
X
z
z'
z
a
f3
0
E
e 0
e 0in
8 0 ,w
\)
p
vapor temperature [K]
T at the bubble wall for a spherical bubble [K) 1
reduction of bubble wall superheat in binary mixtures [K]
number, defined by eq. (3.20) [Pa0 • 05 m]
velocity of the bubble centre of mass [m/s]'
liquid velocity (AI) [m/s]
bubble volume [m3 ]
ib., at departure [m3]
V /a 3 [-]
1 cap
maximal V [m3]
V, when inflexion point I is at the bubble base [m3 ]
wall growth reduction factor; 0 < W ~I, cf. Ch. 4 [-]
normal distance to the axis of rotational symmetry [m]
x/Rtop [-]
normal distance to the wall [m]
H - z [m]
z/R [-) top
angle of contact of a cavity bubble [-]; circular
parameter (AIO) [-]
shape factor; f3 = p gR2 /a = R2 /a 2 [-] 1 top top cap
thermal boundary layer thickness [m)
I - P /p [-] 2 !
superheat, e T - T [K]; wall superheat in sects. o 1 sat (4.3), (5.3), (5.5), (A3) [K); initial superheat in AI [K]
initial superheat, needed for bubble growth [K]
(local) wall superheat [K]
kinematic viscosity [~2 /s]
density difference between liquid and vapor; p
[kg/m3 ]
p - p ! 2
9
10
<Po
<P * ( t)
liquid density [kg/m3]
vapor density [kg/m3 ]
surface tension [N/m]
contact angle of a spreading bubble [-]
static value of <P [-]
angle of contact of the equivalent spherical segment,
cf. fig. 2.1 [-]
Subscripts; superscripts.
( ... ) l
( ... ) 1
( ... ) 2
( ... ) max.
( • ··)min
( ... ) *
"at departure", when used in combination with R, R*,
-* * R , rB, V, t, B, A, ¢> , f, h, j, ...
"of the liquid phase", when used in combination with T,
p, p, k, c,a
"of the gas phase", when used in combination with T, p,
p, m
"maximal"
"minimal"
"belonging to the description with the equivalent spherical
segment"
"dimensionless"
"belonging to the inflexion point I of the profile"
INTRODUCTION
1. Subject matter of the thesis.
In this thesis a theoretical and experimental study of the departure
of bubbles from a horizontal wall is presented. The phenomenon of
bubble departure is of importance in various fields of technology. We
mention in particular heat transfer from a wall. to a boiling liquid
and gas production by electrolysis. In both cases the departure of
bubbles is one of the dominant factors that influence the efficiency
of the process.
Boiling heat transfer plays an important role in such areas as power
generation, jet propulsion and space exploration. Nowadays much
attention is paid to water electrolysis used for the production of
hydrogen, which may become an important energy carrier.
Although much work has been done on the departure of bubbles (from a
horizontal wall), no satisfactory theory has yet been developed, which
applies to the various situations, occurring in practice. For example,
low pressure boiling bubbles behave quite differently from high
pressure bubbles; much confusion exists about the validity ranges of
the departure formulae that are available.
In this work we have attempted to bring more order in this matter by
distinguishing different types of bubbles and domains of their growth
rate, while considering the factors that govern departure in each
case. Special attention is paid to the effect of pressure on departure
in the range of 40 bar to subatmospheric pressures.
For a common type of bubble, i.e. a bubble with its foot attached to
the mouth of the cavity at which it originated (so-called
cavity bubbles), experiments have been performed with the aim of
elucidating the pressure influence on the departure mechanism in the
case of an artificial cavity of known geometry.
II
12
2. Structure and contents of the thesis.
In Chapter l a number of concepts are introduced that are found in the
literature on the subject, whilst an introduction to this literature is
given.
Chapter 2 deals with the theory of departure of so-called "spreading
bubbles". These are bubbles with a foot spreading beyond the mouth of the
originating cavity. The behaviour of these bubbles is considered at various
rates of growth.
Chapter 3 contains the departure theory of cavity bubbles at various growth
rates.
In Chapter 4 criteria are derived for the formation of a microlayer, i.e. a
thin layer of liquid between the bubble and the wall. Also the phenomenon
of dynamic deformation is considered. Dynamic deformation occurs when the
bubble shape is influenced to a large extent by liquid inertia.
In Chapter 5 the specific experiments (mentioned at the end of section 1)
are described and their results_ are compared with the theory for cavity
bubbles. Conclusions are drawn and some suggestions for further work are
given.
A number of detailed developments and some extensions are presented in the
Appendixes.
A separate list of symbols is added to facilitate the interpretation of the
formulae.
CHAPTER 1 ==========
BUBBLE DEPARTURE AND BOILING.
1. Bubble departure.
Bubble departure from a wall occurs in boiling, in electrolysis, but
also, for instance, in a glass of carbonated water. Although most of
the results of this thesis are applicable to the latter two examples
(cf. AS), it will be the boiling process that is considered in more
detail.
1 .1 Boiling.
Few scientific papers about boiling were reported before the 1930s. In
the 1930s Jakob (1933) first observed single bubbles by a camera (with
a speed of 500 frames/s). Nukiyama (1934) observed the existence of
several modes of boiling on a heated wire.
It has now been well established that there are three basic modes of
boiling: nucleate boiling (considered in this thesis), transition
~~~~and filmboiling. In the so-called boiling-curve of fig. 1.1
(for water) these three modes are represented by the regions AB, BC
and CD respectively [v. Stralen/Cole, 1979].
Water
1 1.25 Convection =~Transition Frlmboiling
, (radiation}
1
, reg1on ' I tOO I !
I
B I " I
'E I ' ;: ' ' ::!' 0.75 ' \
0"
i 0.50
0.25 c
0 I 5 10 2 s 102 2 103
e •. K
Ug. 1 . 1 H eirt 6£ux q Vf.> • J.. upeJLheiU e : boJ.Ling euJtv e. 6 oJt wiU:eJL 0
13
14
Each of the modes occurs over a specific range of the_superheat
(temperature exceeding the saturation temperature) measured at the
surface. Nucleate boiling is the most commonly observed mode. It is
characterised by the periodical formation of bubbles from sites on the
surface, called nuclei or cavities.
Beyond point B in fig. 1.1 the wall becomes more and more
"vapor-blanketed". reducing the heat transfer. The heat flux at point
B is called the critical heat flux or "burn-out" heat flux. Beyond C,
radiation makes q increase again, provided that the wall is able to
resist the increase of temperature. If not, the wall is damaged and
"burn-out" may occur. Usually boiling in a vessel is called pool
boiling. In presence of a forced convection along the surface we have
flow boiling (two-.phase flow heat exchange).
1.1.1 Boiling research.
In research on boiling, generally, two ways are followed: 1. The
microscopic method, in which one tries to determine size, shape,
hydro- and thermodynamics of single bubbles. Thereafter superpositions
~an be made. 2. The macroscopic method, in which one tries to find
(semi-empirical) correlations which predict for instance the heat flux
of a given surface at which boiling occurs.
From the 1930s to the early 1950s most work was done on pool boiling
in a microscopic way. Fritz (1935) found an expression for the maximal
volume of slowly growing "spreading bubbles", which was experimentally
confirmed [Tong, 1965]. Plesset/Zwick (1954) and Scriven (1959)
described the expansion of a spherical bubble in an initially uniformly
superheated liquid. Experimental confirmation was found by
Dergarabedian (1953), Darby R. (1964).
Under influence of the rapid advances (since the 1950s) in engineering
technology, related to jet propulsion, electric power generation and
nuclear energy in particular, the scientific interest in boiling became
more and more directed at macroscopic flow boiling research.
Particularly subcooled flow boiling (bulk temperature below the
saturation temperature) at high pressures appeared to be advantageous
in cooling technology. The total flow boiling heat flux may be
thought of as the sum of several heat transport phenomena: 1. free
convection; 2. forced convection along the surface; 3. transport of
latent heat by the bubbles (large at high pressures); 4. transport of
a thermal boundary layer by the bubbles; 5. a bubble induced micro
convection (a major effect, bringing cooler liquid to the hot surface.
[Tong (1965): p.21]). See v. Stralen/Cole (1979, Ch.6).
1.1.2 Simplifications; bubble types.
Pool boiling is far too complicated to be described theoretically in
detail.
Even the size and shape of single bubbles is influenced by so many
factors (gravity, surface tension, contact angle, liquid- and vapor
inertia, thermal properties, temperature-gradient, geometry of the
heating surface, etc.), that only simplification will lead to insight
that may be extended to more complex situations.
In this thesis, among other things, the size and shape of single
bubbles at a horizontal wall are considered, in particular at their
departure.
We assume in all cases an active cavity to be present. The question
whether a cavity (at given conditions) is active or not is, unitil
now, difficult to answer ([v. Stralen/Cole, 1979, Ch.S], [Hsu/Graham,
1976]). It is almost certain at the moment, that, apart from the
cavity-mouth radius, the inner geometry of the cavity is of importance
as well (cf.sect.5.3). In our experiments the difficulties with the
construction of a cavity which is active at low superheats pointed in
that direction too.
A major distinction that must be made is that between the (quasi-)
"static" (slowly growing bubbles; "static" departure in
which the liquid interia force does not play a role) and the "dynamic"
situation (rapidly growing bubbles; "dynamic" departure in which the
liquid intertia force does play a role).
15
16
1.1.2-1 In the static situation the bubble shape is now fairly well
understood ([Bashforth/Adams, 1883], [Hartland/Hartley, 1976],
[Cheaters, 1977]). The theory of this will be referred to as the
"equilibrium bubble shape theory" (cf. sect. 2.7).
Depending on contact angle, surface roughness and growth rate (cf.
Ch.2), two static bubble types appear to exist:
a. "cavity bubbles" (or mode A bubbles, [Chesters, 1978]) for which
the bubble foot remains attached to the cavity mouth during the
bubble evolution (see fig. 2.5).
b. "spreading bubbles" (or mode B bubbles, [Chesters, 1978]) for which
the bubble foot spreads out freely on the surface in the bubble
evolution (see fig. 2.6).
These two static bubble types behave quite differently and it
certainly is unfortunate that only a few authors, until now,
incorporated this difference in their considerations. For example, the
most widely used boiling heat flux correlation [Rohsenow, 1952] is
based on Fritz' departure formula, which only applies to spreading
bubbles, despite the fact that; in practic~, the bubble foot usually
does not spread out [Chesters, 1978].
1.1.2-2 In the dynamic situation the bubble shape is less understood.
Experimentally it is known (Johnson et al. 1966] that a flattening
occurs. Many authors reported the existence of a microlayer beneath a
very rapidly growing bubble ([Moore/Mesler, 1961], [Sharp, 1964]). The
question at which growth rates "microlayer bubbles" are formed is
treated in Ch.4 of this thesis. Microlayers have been treated
theoretically a.o. by v. Ouwerkerk (1970) and Zijl (1978). Many
experiments with these large bubbles (for instance in water at low
pressures) have been carried out ([Zijl, 1978], [v. Stralen/Cole,
1979], [Cole/Shulman, 1966], [Cooper, 1978]).
The influence of liquid inertia is quite complex. In the present thesis
(Ch.4) we distinguish: 1. The influence of the inertial force on bubble
departure (cf. Ch.2, Ch.3); 2. the occurrence of important dynamic
deformation of the bubble (cf. Ch.4); 3. the occurrence of microlayer
bubbles (cf. Ch.4); 4. the occurrence of the inertia controlled mode of
growth during a major part of the departure time (cf. Al).
It should be noted that these aspects of liquid inertia become
important at different (critical) values of the growth rate (cf. Ch.4).
1.1.3 Growth types.
If we do not consider highly viscous liquids, two types of bubble
growth may be distinguished, as is pointed out in A1 {[Plesset/Zwick,
1954], [Plesset/Prosperetti, 1977]).
a. Inertia controlled growth (also called Rayleigh growth or ~~~~
growth). Here the liquid inertia governs the bubble expansion.
b. Heat-diffusion controlled growth.
The bubble wall temperature has decreased to the saturation
temperature at the bubble pressure. Heat diffusion due to the
temperature gradient at the bubble wall is now the "driving force" of
the bubble expansion.
The problem (a Stefan problem with a moving boundary) is not purely
diffusional. Also radial convection must be taken into account. For a
spherically symmetric expansion in an initially uniformly superheated
liquid this problem was solved by Scriven (1959). The bubble radius is
proportional to the square root of the growth time: R cr t\.
As is shown in Al for boiling water the contribution of the initial
growth type to the bubble size at departure is negligible unless at
extreme superheats and/or at sub-atmospheric pressures. In this thesis
we limit ourselves to diffusion controlled growth, where R rr t~ ([Sernas/Hooper, 1969], [Cooper, 1978], [Labuntsov, 1975]).
1.1.4 Binary systems.
Although in this thesis only pure liquids are considered, an extension
can be made to binary systems, i.e. liquids with a dissolved second
liquid component. See A9 [v. Stralen/Cole, 1979].
17
18
1.1.5 Waiting time. Bubble cycle.
From an active nucleus bubbles are formed periodically with bubble
frequency f. We denote the growth time (till departure) by t 1 and the
time ~rom departure till the initiation of the next bubble by tw
(waiting time}. Then f • 1/(tl + tw>•
In many practical situations, ~ is reported to be small compared to
t 1 , except at very low superheats ([Perkins/Westwater, 1956],
[Donald/Haslan, 1958], [Johnson et al., 1966], this thesis Ch.S}. A
theoretical description of tw is given by Hsu/Graham (1976}.
1.1.6 Departure models.
Using the tables of Bashforth, Adams (1883) for the bubble shape,
Fritz (1935) obtained an expression for the departure radius of a
spreading bubble on a horizontal wall. Fritz derived that for the
equilibrium liquid-vapor interface of a bubble (controlled by surface
tension and hydrostatic pressure) with constant contact angle, a
maximum volume exists (see also sect. 2.7.lb, 2.7.2b). Larger volumes
can not be stable and departure will occur. Though the contact angle
is difficult to measure, Fritz' formula is widely accepted, but often,
abusively, used for cavity bubbles or dynamic bubble shapes.
As shown in Ch.2, it is possible to derive Fritz' formula from a force
equation, containing the upward buoyancy force, the downward surface
tension force and an upward net wall reaction force (or correction
force Fcorr•whicb is zero when the bubble is closed). The latter,
being of the same order of magnitude as the other two forces (cf.
sect. 2.8), has been ignored in many papers.
However, it should be noted that equating the sum of all forces to
(approximately) zero is not a departure criterion just by itself.
Since the bubble mass is reiatively small far below the critical point
the righthand side of Newtons second law (eq. (2.15)) is approximately
zero for all stages of growth (cf. sect. 2.3; A6). Hence a geometric
departure criterion is needed (cf. sect. 2.6.1). This fact has been
taken into account by only a few authors ([Saddy/Jameson, 1971],
[Chesters, 1978]). As mentioned, cavity bubbles behave quite dif
ferently from spreading bubbles. Yet, this fact is not sufficient to
explain the differences between the static theory and some experimen
tal results regarding large bubbles. Many authors included the dyna
mic effects of liquid inertia to explain for example the large bubbles
observed at low pressures.
Cole/Shulman (1966) propose some correlations for the departure radius
R1 , including low pressure results (at one cavity) and high-pressure
results of Semeria (1962, 1963) (at undefined cavities). The latter
results do include the effect of pressure on the cavity size: an
increase of pressure activates smaller cavities [Hsu/Graham, 1976],
which, in turn, yield smaller bubbles. Cooper et al. (1978) propose a
correlation for R1 at low pressures. Not distinguishing static and
dynamic bubble types they suggest that bubble departure is quickened
(!)as a result of surface tension, while the contact angle is assumed
to be of minor influence.
v. Stralen [v. Stralen/Cole, 1979] defines a superheated "relaxation
microlayer" around a bubble growing as a spherical segment with a
constant contact angle. This layer is considered to act as a buffer of
superheat enthalphy consumed by bubble growth. It is assumed in this
theory that the bubble departs when the excess enthalpy of the layer
is depleted. The thickness of the layer determines the departure time
and is assumed to be half of the thickness of the thermal boundary
layer at the wall. The theory, using a thermal departure criterion,
predicts a sharp decrease of R1 at increasing pressure, due to the
reduction of the thermal boundary layer thickness. This prediction
contradicts the experimental results described in Ch. 5 and the theory
of Chs. 2, 3 using a force model combined with a geometric departure
criterion.
Keshock, Siegel (1964), and Saini et al. (1975) introduce a dynamic
force, using the "virtual mass" of a bubble in an unbounded liquid
(neglecting the asymmetry of the problem, introduced by the wall).
19
20
Another method to introduce dynamic effects is sometimes referred to
as Macceleration controlled" departure. Saddy/Jameson (1971),
Kirichenko (1974) and Zijl (1978) estimate the value of the
acceleration of a growing and rising bubble. It is supposed that the
bubble departs when its centre of mass has a distance from the wall of
1.5 times (1.0 according to Zijl) the equivalent bubble radius.
An excellent fundamental description of bubble departure has been
given by Cheaters (1977, 1978), for the first time distinguishing
spreading bubbles and cavity bubbles and (for a special case)
introducing dynamic effects.
1.1.7 Pressure dependence.
Ihe static departure theories of Fritz (1935), Cheaters (1978) and of
this thesis (sect. 2.8; 3.1) predict a slow decrease of the departure
radius R1 at increasing pressure. For water, from 2 tot 40 bars, this
means a reduction of about 16% for cavity bubbles or 24% for spreading
bubbles (cf. Cbs. 2,3; AS). Since the growth constant C is roughly
inversely proportional to the pressure p (eq. 4.3.6), we have (using
the diffusional relationship (2.1)) for the departure time t 1:
t 1 • (R1/c)2 ~ p2 (water: 2 bar< p.< 40 bar).
Semeria (1962, 1963), however, observed a sharp decrease of R1 and a
decrease of t 1 at increasing pressure (for water and elevated
pressures, associated with static departure,. cf. sect. 3.6).
However, Semeria did~ observe cavities of~ size, so his results
will include the considerable reduction of the size of active cavities
with increasing pressure [Hsu/Graham, 197.6].
One of the purposes of the experiments presented in the present thesis
is a verification (or otherwise) of the predicted pressure dependence
of the static departure mechanism, with elimination of the effect of
pressure on nucleation.
The dynamic departure theory (cf. [Zijl, 1978] and sect. 2.10, 3.3)
predicts (for low pressures and/or high superheats):
Rl ~ c4/3 ~ l/p4/3,
hence t 1 ~ 1/p213 (roughly)
Low pressure data ([Cole/Shulman, 1966], [Cooper, 1978]) are in
agreement with these qualitative results.
1.1.8 Relation between bubble frequency and departure radius.
Many authors expected a simple relationship between bubble frequency f
and departure radius R1• Ivey (1967) gives a literature survey on this
subject. He concludes that a (hydro-)dynamic, a transition- and a
static region must be distinguished. Even then, only an approximate
fit of experimental data is possible.
1.1.9 Vertical walls.
A fundamental description of bubble departure at vertical walls has,
until now, .shown to be too complicated. By the lack of rotational
symmetry the exact shape of the bubble has not been described.
Adopting some crude assumptions, however, an upper limit for the
static departure volume of a cavity bubble at a vertical wall is
derived in AlO.
1.1.10 Viscosity.
For water, the force on a bubble due to viscosity (Fv), may usually be
neglected in the departure mechanism (A4). For viscous liquids,
however, Fv will be important. In that case, an expression for Fv may
be incorporated in a force equation like eq. (2.16).
21
22
1.2 Gas bubbles.
For gas bubbles growing in a (locally) supersaturated liquid
(electrolytically evolved bubbles for example) much of the boiling
theory applies (cf. AS). Bubble growth, in this case, is controlled by
mass diffusion. Since the latter is a much slower process than heat
diffusion, we now usually have the simpler "static" situation of sect.
1.1.2-1 ([Plesset/Prosperetti, 1977], [v. Stralen/Cole, 1979}).
CHAPTER 2 =========
DEPARTURE OF SPREADING BUBBLES FROJ:.f A HORIZONTAL WALL.
2.1. Restrictions.
In this chapter the departure of so called spreading bubbles (from
a horizontal wall) is considered.
[Note: The results of sect. (2.2), .•• , (2.5), however, also
apply to cavity bubbles, if we replace rB and ~ 0 by rand a respectively;
compare figs. 2.5 and 2.6].
It is .assumed that the contact angle $ ~ 0 , where ~ 0 is the static
value for smooth surface [Dussan, 1979].
We restrict ourselves to heat diffusion controlled bubble growth,
for which (cf. A1):
(2 .1)
with:
- R ~ the equivalent bubble radius , i.e. the radius of the sphere
with equal volume as the real bubble;
- C the growth constant ;
- t the time since the bubble initiation.
Heat diffusion controlled growth is the common situation in boiling
(cf. A1). Eq. (2.1) is experimentally and theoretically founded for
a free bubble, expanding in an initially uniformly superheated liquid
([Plesset/Zwick, 1954], [Scriven, 19581, A1). For bubble growth on a
wall in presence of a thermal boundary layer, (2.1) is supported by
experimental evidence ([Sernas/Hooper, 1969], [Cooper, 1978],
[Labuntsov, 1975]). Cis defined then as given in sect. 4.3.
The growth rate is assumed to be limited ~n such a >vay that no
"microlayer bubbles" are formed and that no serious dyna.rcic deformation
occurs at departure. Criteria for both phenomena have been derived in
Ch. 4.
23
24
We further neglect those forces on a departing bubble which are
due to viscosity and surface tension gradients (cf. A4). Since we
consider situations far below the critical point, we have
p /p << 1, where p , p are the specific masses of liquid and 2 1 1 a
vapor respectively.
2.2.
z
H bubble - - equivalent
spherical segment
Fig. 2.1 Sp!te.acUn.g bubble a.nd m equiva.len:t .ophe!tic.a.£. .6egmen:t.
Fig. 2.1 shows a spreading bubble and its equivalent spherical
segment of radius R*, i.e. the spherical segment with equal volume
and the same base-area (B). 0
The height of the bubble is H and of the spherical segment
H*. The radius of curvature at the bubble top is Rt op (rotational symmetry about the z-axis). rB denotes the
radius of the circle of contact, while r is the cavity
radius. The surface area of the bubble is A • The time-o
dependent angle of contact between the equivalent bubble
segment and the wall is denoted by ~*(t).
Inside the bubble the pressure is p • Since we neglect hydrostatic 2
pressure differences inside the bubble, p is not z-dependent. 2
Outside the bubble, in the liquid, the pressure is denoted by p (z), 1
which depends on z.
p (z) can be written as: l
p (z) l
where:
- p (z) 1hydr
- pext
p (z) + pext + p 1hydr 1 dyn
(the hydrostatic pressure)
p (H) + p g(H-z) 1hydr 1
(the external pressure of the system)
(the dynamic pressure in the liquid).
(2. 2)
(2.3)
In absence of forced convection p is due to the bubble expansion 1dyn
only. We then have rotational symmetry about the z-axis. Therefore,
the resultant force on the bubble can only have a component in the
z-direction, which we call F.
The force F on the vapor mass of the bubble is given by:
F
where:
.n dA + JJ p2 ~2 .n dA +
B 0
circle of contact
(-a s in<j> ) dt 0
(2.4)
~z is the unit vector in the z-direction, ~ is the unit vector normal
to the bubble boundary pointing to the inside of the bubble (see
. 2. 2).
The first term of (2.4) represents the force by the liquid on the
bubble; the second term represents the reaction force of the wall;
the third term is the line-integral of the z-component of the
surface tension along the circle of contact with radius rB. This
downward directed surface tension force Fa can be written as:
F 0
= 2rrrB O'sin<j> 0
•
Using (2.2), we have from (2.4):
25
26
FcJJ p (z)e.ndA+Jf pext~z·E_dA+Jf p .ndA+ 1hydr ~ - 1dyn
A A A 0 0 0
p e .n dA - F z -z- a B
0
The third .term on the right hand side of (2.5) represents the
dynamic force or liquid inertia force FD' directed downwards.
From (2.5) it follows:
F= If A +B
0 0
p (z) 1hydr
.n dA + JJ p t e .n dAex ~-A +B
0 0
- JJ P (z) e .n dA- JJ 1hydr -z-
B 0
B 0
JJ p e .n dA - F 2-z- cr B
0
(2.5)
(2.6)
The first term of (2.6) represents the upward directed Archimedes
or buoyancy force FB = p1gV.
Using Jf pext ~·E. dA = O, we obtain from (2.6):
or:
A +B 0 0
F = FB + fJ B
0
{p -p (z) - p t} e .n dA - FD - F 2 1hydr ex -z - a
F = FB + ~rB2 {p -p (0) - p t} - F - F 2 1hydr ex D cr
Using Laplace's law:
p - {p (H) + p t} = 2cr/Rt , 2 lhydr ex op
and (2.3), we obtain from (2.7):
(2. 7)
(2.8)
The second term on the right hand side of (2.8) is an upward directed
correction force F , which becomes zero when the bubble is closed corr (rB=O), or (with H ~ 2Rt ) when Rt ~ op op the "capillary length", defined by a cap
Finally, (2.8) becomes (see fig. 2.2):
with:
FB
F corr
p
(2o/R - p1
gH)'ITrB2
top
2'ITrBosin<jl0
a , where a cap cap /(a/p g).
1
denotes
(2.9)
(2.10)
(2.11)
(2. 12)
The liquid inertia force FD is not known a priori. However, assuming
potential flow, Witze et al. (1968) found the analytical solution
of the exterior flow pattern around an expanding sphere adhering
to a horizontal wall. Assuming in addition that growth law (2.1)
is valid, they found:
(2.13)
Eq. (2.13) can be considered as a good approximation for FD' provided
that the effect of viscosity can be neglected and that no serious
deviation of the spherical shape occurs due to dynamic or hydrostatic
pressure differences (cf. Ch. 4) (Witze et al. were the first to solve
this asymmetric flow problem).
Using (2.11) and (2.12) we define the --~~~--~~~~~~~~~~~
F as: o,r
F o,r IF - 2'ITorB2 /R I (directed downwards) o top (2. 14)
27
28
z
Fig. 2.2 Foltc.IU on a. .6p!tea.cii.ng bubble.
Fig. 2.2 gives a schematic picture of the forces acting on a
spreading bubble.
Some earlier attempts to derive a spreading bubble departure law
from a force consideration neglect F , which is incorrect since corr F is of the same order of magnitude as- Fcr (cf. sect. 1. 1. 6; corr sect. 2.8).
2.3. Equation of motion.
Considering the bubble as a moving body of mass m , we have for the 2
resultant force:
(2. 15)
Here v denotes the z-velocity of the centre of mass of the bubble.
Since p /p << 1, the right hand side of eq. (2.15) may be neglected 2 1
(cf. A6), giving:
(2.16)
In general, eq. (2.16) is a dynamic equation containing the effects
of liquid inertia.
But, if FD = 0 (i.e. the "static" case), eq. (2.16) is an actual
force balance. Then, eq. (2.16) has to be satisfied exactly at any
time t between the initiation and the departure of the bubble.
In case FD f 0 and Witze's time independent solution is used,
eq. (2.16) has to be satisfied approximately in the period under
consideration.
Note: As mentioned in sect. 2.1, eq. (2.16) also applies to
cavity bubbles, if we replace rB and ~ 0 by r and a respectively.
2.4.
1. "Slow growth" resulting finally in "static departure" (cf. sect.
2.8; 3.1) will occur for those values of C for which the resul-
2.
tant surface tension force F is dominant as compared to the o,r
liquid inertia force FD.
resulting finally in "transition departure"
(cf. sect. 2.9; 3.2) will occur for those values of C for which
FD and F are of the same order of magnitude. o,r 3. "Rapid growth" resulting finally in "dynamic departure" (cf.
sect. 2.10; 3.3) will occur for those C-values for which
~ FD dominates F o,r b the growth rate limitation of section (2.1) is satisfied.
Note: The values of C, corresponding with the above conditions
will be derived in section 2.9 and section 3.2.
2.5. Alternative description using the quantities of the equivalent
spherical segment.
With (2.10), (2.11), (2.12) and (2.13) force equation (2.16) becomes:
(2. 16a)
In (2.16a),V, Rtop' Hand rB are time dependent. If R*, u* and
~*denote the radius, height and angle of contact of the equiva-
lent spherical segment (see 2 .1), then:
rB R*sin~ *
u* 2BR*
v = ~TIAR*3 3
(2.20)
(2.21)
(2.22)
29
30
where: def
B def
! (1+cos~*) and A
Now we assume (for ~ < 45°) 0 ~
R l';j R* top
B2 (3-2B) (2.23),(2.24)
(2.25) ,(2.26)
(In case¢ = 45°, the relative inaccuracies of these assumptions 0
are 10% and 5% respectively; cf. A2.)
With (2.20), ••• , (2.26), equation (2.16a) becomes:
(2.16b)
Reorganizing the terms, we have:
(2.16c)
2.6. Bubble shape; Methods to obtain departure formulae.
In absence of forced convection we have rotational z-symmetry. Then
the bubble profile (fig. 2.1) determines the bubble shape completely.
In case of slow growth (with static departure), the bubble profile
is known:
1. Numerically ([Hartland/Hartley, 1976], [Bashforth/ Adams, 1883 ])
(cf. sect. 2.7.1);
2. Analytically [Chesters, 1977l<cf. sect. 2.7.2).
When more rapid growth is considered this "equilibrium bubble
shape theory" can only be applied with some accuracy if no great
dynamic deformation occurs (cf. Ch. 4).
2.6.1. Static situation.
To derive an expression for the static departure radius, Rt , for op instance, two ways can be followed. 1
1. Direct application of the "equilibrium bubble shape theory"
([Hartland/Hartley, 1976], [Chesters, 1977]). For cavity bubbles
as well as for spreading bubbles it can be shown (see sect. 2.7)
that there exists a maximal bubble volume V max
As will be treated in sect. 2.7, this maximal volume
is reached when the inflexion point of the bubble profile
appears at the bubble base (for spreading bubbles) or
when a 90° (for cavity bubbles). These criteria are
(for this static case) the ~~~~~~~~~~~~~~~·
mentioned in sect. 1.1.6.
When bubble growth continues, the bubble shape becomes
unstable and the bubble will detach itself from the
surface.
2. Substitution of a geometric departure criterion in force equation
(2.16a) yielding an expression for R (cf. sect. 2.8). top1
Both methods, of course, n.eed to be cons is tent.
2.6.2. Transition and dynamic situation.
followed.
Since the "equilibrium bubble shape theory" is not widely known,
its fundaments and some results will be treated briefly in the next
section.
2. 7. Intermezzo: The "equilibrium bubble shape theory".
([Hartland/Hartley, 1976 ], [Chesters, 1977 ])
[Whenconsideringjust bubble departure, the most important parts
31
32
of this section are: 1) Fig. 2.4 (possible bubble shapes);
2) Figs. 2.5, 2.6 (bubble evolution); 3) Fig. 2.7 (the
existence of a maximal volume);4) Eqs. (2.3.4), (2.3.6),
(2.3.7) (departure formulae) ].
Supposing again rotational symmetry about the z-axis and assuming
that the dynamic pressure differences in the fluid are neglible,
the bubble profile (z' = f(x), see fig. 2.3) is determined by
Fig. 2. 3 Bubble pMfJ.l.u a.eeo!tding :to e.q. (2 .29).
the surface tension (tending to curve the profile) and by the
hydros.tatic pressure differences in the liquid (tending to reduce
the curvature at increasing z').
Then we have the following local interface condition (Laplace's
law):
(2.27)
where:
p2 -
face
{p (z') + p t} is the pressure difference across the inter-lhydr ex
and Ra• ~ are the principal radii of curvature of the bubble
surface.
The left hand side of (2.27) can be evaluated at the bubble top,
giving:
p - {p (z'=O) + Pext} 2 1hydr
2o/Rt op
Hence (assuming again: p >> p ): 1 2
p - {p (z 1 ) + p } = 2a/R - p gz'. 2 1hydr ext top 1
Therefore, (2.27) becomes:
2a/Rt - p gz' op 1 (2.28)
Substitution of the appropriate expressions for Ra and ~ yields the
ordinary differential equation for the bubble profile (or for the
profile of a pendant drop, cf. Chesters (1977 ):
-------- + --------- 2 R top
Using dimensionless variables, (2.29) becomes:
d~/d~ + --------
{1+(d~/dx)2}3/2 x{1+(d~/dx) 2 } 1 / 2 2 - Sz
p gz' 1
a
Here~= z'/R · ~ = x/R and S = p gR2 /cr. top' top 1 top
From (2.30) we see that the profile is determined by
the dimensionless parameterS (the shape factor), while
R determines the scale. top
2.7.1. Numerical results.
(2.29)
(2.30)
Numerical solution of (2.30) [Hartland/Hartley, 1976] results in a
set of possible equilibrium bubble profiles, labeled by S (see fig.
2 .4).
Note: It is important to realize that the bubble mode (spreading
bubbles or cavity bubbles) enters in the theory as a boundary
condition:
a. For cavity bubbles, that subset of profiles of the above set is
selected for which a "cut off" at x = r = const. is possible.
33
34
The angle of inclination a (see fig. 2.3) of the profile at the
(sharp) edge of the cavity changes with Rt (see fig. 2.5). op
b. For spreading bubbles that subset of profiles is selected for
which a "cut off" with a = cp = const. is possible. The correso
ponding bubble foot radius changes with R (see fig. 2.6). top
Flg. 2.4
2 11- x/ Rtop
Nu.me!U.cal. ,oo.e.u..ti..oru, ot1 ( 2. 30} 601t cU..66eJr.ent valuu
o6 :the. dime.ru,-ionl.U-6 paJUtmeteJL 13 = p 1gR_t
0/cr.
For a vapor bubble growing in a superheated liquid V is monotonously
increasing. This means that a continuous set of profiles with (in
general) increasing R is passed through, belonging to the above top defined subsets for case a. or case b.
2.7.1.a Cavity bubble evolution. Criterion for the bubble mode.
() ... ct ' > '*' m:tn a at inflexion
departure
v ~ v max
for u '"" 90°
hypothetical
profile with
u > 90°;
F-Lg. 2.5 Evofu:Uon a c.a vJ.;ty bu.b bte.
The evolution of a cavity bubble has been
2.5.
v < v max
schematically in
Of course, a has its minimal value amin when the location of attach
ment is at the inflexion point I of the profile.
For a perfectly smooth with no dynamic hysteresis of the
contact angle, no spreading of the bubble foot on the wall will
occur, if amin > ¢0
, ¢0
being the static contact angle. On the
other hand, this spreading does occur and a change to the spreading
bubble shape takes place, if a . < ¢ ([Chesters, 1978], [Dussan, mi.n o
1979]).
In most practical situations:
r << R top 1
(2.31)
shortly after nucleation. For boiling water at I bar for instance,
r varies from 5 to 100 ~m, while Rt is of the order of 1 mm at . op departure.
Maximal volume:
It appears (see fig. 2.5) that there exists a maximal volume V max
35
36
in the bubble evolution. Larger bubbles can not exist at a cavity with
the same radius. A criterion for the occurrence of this maximal volume
appears to be: a 90°. At the same time Rt (and S) are maximal. op These statements can be proven analytically [Chesters, 1977].
2.7.l.b Spreading bubble evolution.
" ... • spreadit).g
begins.
Beginning of
typical
spreading
bobble shapta
v < v1
1\ < R top topmax
a < Bmax
when inf lex.ion
point I is
reached. Then:
R • R top topmax
B • 6
'"""
Fig. 2.6 Evotution aS a ~p4eading bubbte.
departure;
v • vmax Rl v1 Rtop and 5 are
somewhat below
their maxima
Fig. 2.6. gives a schematic representation of the growth of a
spreading bubble. Some results of the numerical approach are:
-For q, < 90° the bubble has a "neck". This corresponds with: 0 s < 0.58.
-At increasing B (Rtop) their exists for every q,0
a maximal volume
vmax' See fig. 2.7.
-We see (fig. 2.6, 2.7) that S (Rt ) reaches its maximum op B (Rt ) when max opmax
the location of attachment is just at the
inflexion point of the profile. The angle of inclination a is
then: a= a1
= q,0
• In that case the value of the bubble volume
VI is only little less than the maximal value Vmax; when
ai(=q,) < 60° (corresponding to S < 0.30), then V ~VI within 0 rv 1*-.1 max
an accuracy of about 7%, cf. A2.
-From fig. 2.7 it appears that in the first stage of the bubble
evolution S(Rt ) and V both increase. This goes on until just op before departure. Then B (R ) and V are reached.
max topmax I
4 V/a~ap ! Vmax
Fig. Z. 7 Sc.he.ma.tic. pic..tuJte. o6 .the. s-de.pe.nde.nc.e o6 VI a~ap' ;.,!towing .the.
e.)(w.teJ1c.e. afi a maximum voiame. V max. The. a.Joww;., indic.a..te. .the.
bubble. gJtow.th. The. bJtake.n line. Jte.6e.Jtf., .to the. at.tac.hme.n.t a..t .the.
irt6ieexio n point. The. ha..tc.he.d Mea. tte.fi eM .to .the. domain ~ < 9 0 °. During the final stage of volume growth S(Rtop) decreases a little.
Any further volume growth causes necessarily a discontinuous change
to a (more or less) spherical shape (departure).
2.7.2. Analytic results.
[Chesters, 1977]
Eq. (2.30) can be solved analytically for small S, i.e.:
s ;_, 0.1 (2.32)
This condition is satisfied in many boiling situations. For boiling
water at I bar it means for instance: Rt < 0.8 mm op ~
2.7.2.a Analytic results on cavity bubble departure.
(2. 33)
S(R ) has its maximal value at departure. A criterion for this top is: a= 90°.
0
37
38
-If the subscript(,,,) indicates the value at departure, then, 1
for S < 0.1 l -
r = Rt {~3 S + 0(13 2)}
op 1 1
Therefore:
R top 1
l
(lra2 ) 1 / 3 {I + 0(13 )} 2 cap 1
(2.34}
- It has been shown that the volume, V, of the bubble above any
horizontal plane cut:ting the profile in the neck region is:
(2.35)
Therefore, with (2.34):
V = 2Tira2 {I + 0(13 )} 1 cap 1
(2.36)
2.7.2.b Analytic results on spreading bubble departure.
- Condition (2.32) is now equivalent to: ~ ~ TI/6 rad, as follows 0
from the numerical solution (sect. 2.7.1)
- Departure occurs when the inflexion point of the profile is reached.
R .. lj.a top 1
cap
= l~.a cap
sin¢ {I + 0(~ 2 )} 0 0
1P {l + 0(¢2)} • 0 0 with· ¢
0 in radians
(2. 37)
(2. 37a)
This result was first obtained by Fritz (1935) who found 0.596
instead of lj = 0.613 as conRtant. 813 1/2
Or, from (2.37): sin + = (-1 ) +0(13 3 12 ) (2.38) 0 3 1
Again, we see that 13 < 0.1 corresponds to A < TI/6 rad. 1 I"V . '+'o ~
+ 2_ ¢2 + O(.p")} 8 0 0
- The bubble base radius at departure,rB , which equals the 1
x-coordinate of the inflexion point I, appears to be:
(2.39)
(2.40)
- Before departure, rB diminishes from
= 39212.a sin2 ¢ {1 + 0(¢ 2
)} cap o o
(2.41)
to rB . Except in case r > rB. Then the bubble ends its life as 1 1
a cavity bubble.
-At departure, the bubble height H , which equals the z 1-coordinate 1
of the inflexion point I, is:
H 1
z I I 2Rtop {1 + ~ sin2 ¢
0 ~n(16/sin2 ¢ 0 ) + 0(¢~)}
1 (2.42)
Note: From the comparison of the above approximations for
R and V with the numerical results (cf. A2) it follows top 1 1
that the inflexion point as departure criterion and the
departure formulae (2.37), (2.39) happen to be good approxi
mations not just for 0 < ¢0
< TI/6, but even up to ¢0
TI/4
or more. The deviation in (2.37a) for instance is only 5% if
¢0
= TI/3. This fact extends its applicability considerably.
2.8. Static departure (of spreading bubbles).
We consider low growth rates (for C-values as given in (2.63)) with
static departure for which the results of the "equilibrium bubble
shape theory" may be applied. In section 2.7.2 the direct application
of this theory led to expressions for R and V . top 1
1
2.8.1. Use of the force equation (2.16).
As mentioned in 2.6.1 expressions for R and V can also be ob-top 1
tained from force equation (2.16), with 1 FD = 0, and taken at
departure.
Two assumptions are needed, which are valid for
¢ < TI/3 rad(S ~ 0.3) and 9 < TI/6 rad(S < 0.1) 0 ~ 1 ·- 0 ~ 1 ~
39
40
respectively (cf. A2):
a) Departure (V = V = V ) occurs when the inflexion point of the 1 max
bubble profile appears at the bubble base.
b) The departure volume can be represented by:
Application of Laplace's law at the bubble base gives:
Hence, with FD. = 0 and F. = 1rrB crsin<P , · corr 1 o l
we obtain from (2.16):
p gV + 1rrB crsin<P - 21rrB crsin<P0
= 0,-and l 1 1 0 l
Using Laplace's law at the bubble top and (2.3) we have:
2cr p2 -{p (z=O) + p } = -R-- - P gH
lhydr ext 1 1 top1
With (2.43) and (2.45) we obtain:
R top --
1 sin<P rB o
l
= 2 -
H R 1 top
1
a2 cap
(2. 35a)
(2.43)
(2.44)
(2.45)
(2.46)
It can be shown (using the results of this section) that the left
hand side of (2.46) is of order I while the second term of the
right hand side is of order <jl 2 • ()
Therefore:
(2.47)
Using (2.47) and (2.35a), we obtain from (2.44):
R top 1
(2. 37)
which again is Fritz' expression, now derived from a force balance.
Also, with (2.44), (2.47) and (2.37):
(2.39)
and
(2.40)
which are the required expressions.
2.8.2. Use of the quantities of the spherical segment.
If we choose to use the description with the quantities of the
equivalent spherical segment, we can show that the analytic
results of sect. 2.7.2 lead to:
H* H 1
R* R l
<t>*(t ) 1
Also then:
B 1
A 1
+ 0 (<P 3) 0
+ 0(1>3) top 0
1
H + 0(<1>3) 0 0
I 2 - 16 4>0
I 4 + 768 <I> 0 +
3 4 + 0(<1>6) - 256 <Po 0
O(q,s) 0
(2. 48)
(2.49)
(2.50)
(2.51)
(2.52)
From (2.37a) and (2.49) we directly obtain Fritz' equation for the
quantity R*: l
, 4>0
in rad (2.37b)
A comparison of the first order expression of (2.37b) and the exact
numerical value of R*, derived from the results of Hartland/Hartley 1
shows that (2.37b) is accurate within a few percent for <f> < "/2 (!) 0 ~
41
42
(cf. A2). This fact makes (2.37b) to be the most useful analytic ex
pression, given in this thesis, for the static departure radius of
spreading bubbles.
Another way to obtain (2.37b) is substituting (2.50), (2.51) and
(2.52) in the force equation (2.16c). Since this method is well
suited for a generalization to higher C-values (sect. 2.9) we shall
now carry out briefly all steps of this derivation, with mention of
the underlying assumptions.
2.8.3. Static departure: derivation of (2.37b), using the force
equation.
Assumptions
- p << p 2 1
- no viscosity;
smooth surface;
no forced
--
convection;
F = 0 D .
F 0
JJ {p (z)+p }e .ndA+~r~ p -F =0 A 1hydr ext -z - 2 a
()
(2.16)
(2.16d)
(Here, only the second term is positive, i.e.
the wall reaction force is directed upward).
FB- rB2{p (O)+p t}+~Bz.p -F =0 1hydr ex 2 a
or:
(2. I 6e)
where ~P I stands 21 0 for {p -p (0)-p t} ,i.e.
2 1hydr ex the pressure difference across the interface, at
the bubble base.
4 AR* 3 *2 • 2 * ( 2a ) 2 * . * . O p 8J ~ ~R s1.n .p -R-- - p gH - ~aR s1.n4> SI.n4>0=
l top 1
After dividing by 2~p ga 3 we have with 1 cap
R.* = R*/a cap
fR.* 3 - hR.* = 0 (2.16£)
where:
h * a* * sine!~ {sine!~ - (--) sin,P } o Rtop
- small ~ 0
- small ~ 0
- geometric
departure
criterion
(accurate for
~ < rr/3, cf.A2) 0 ~
- small ~ 0
so, at any time in bubble evolution:
R* = /-f (exact expression)
At departure we write:
-* h R = 1-f
1
(Both h and f become negative 1 1
~* > 54° (~ > 770)' cf. A2). 1 0
Analytic approximation:
R top
1
Hence:
(2.48);
f =~A - B sin2 ~* 3 1 1 .
h sin~* 1
A r::; 1
Hence:
- 2 f1 - 3
1
'
(sin~ -0
(2 .52);
. * s1n~ ) 1
• 2 * B SHJ,, ~ 1 1
h = sin~* (sin~* - sin~*) 1 1 0 1
<<
for
2 3
(2.53)
(2.54)
(2. 49)
The bubble leaves the surface when the inflexion
point of the profile is reached: ~ 0 = a1
~* r::; !~ (2.50); 1 0
sin~ r::; ~ (~ in rad). 0 0 0
=·> f 2
1 -3 (2.55)
h I ~2
1 =4 0
(2.56)
Therefore:
-* J3 R = >'-·~ 1 8 0
(2.37c)
-* * for R = R /a . The overbar 1 1 cap
be omitted in the continuation of this
(Fritz' expression -* on R will
1 Chapter.)
43
44
As mentioned in sect. 2.8.2, this expression is
accurate for ~ < ~/2(!) 0 ~
2.9. Transition departure (of spreading bubbles).
We next turn our attention to higher growth rates (for C-values as
given by (2.6.4)) associated with transition departure. For this
case the "equilibr:i.um bubble shape theory" does not apply.
According to section 2.6.2 we therefore can only make use of the
force equation (2.16) (now with FD ~ 0) to.obtain departure formulae.
In the C-domain under consideration, only small dynamic deformation
occurs at departure, and no microlayer bubbles are formed, as is
shown inCh, 4. Therefore we may assume Witze's expression (2.13)
for FD to be valid.
Since, as explained in Ch. 4, dynamic deformation first appears as
a flattening of th.e bubble top, Rt can n0 longer be used to op determine the scale of the bubble. It is therefore advantageous to
consider here only the quantities of the equivalent spherical· seg
ment.
The procedure of sect. 2.8.3 will now be generalized to higher growth
rates, for which FD ~ 0.
Instead of (2.16f) we then obtain the following dimensionless equation
(taken at departure)
or,
f R_*a 1 1
with j
f R* 3
l l
l
-* - h R -FD
1 1 2~p ga 3 1 cap
def FD/21Tp ga 3
1 cap
h R* -j = 0 1 1 1
0,
R.* and dropping the overbar of again: 1
(2. 16g)
The more "dynamic" the departure is, the larger the ~*-values (and ' . 1
~-values), at which f or h becomes negative, will be (cf. A2). 0 1 1
For our purpose, we therefore safely can take: f > 0, h > 0. 1 1
We consider only small dynamic deformation (particularly at the
bubble base it will be srmll). Therefore we again adopt the assump
tions which led us to (2.55) and (2.56), which means physically that
we again introduce the geometric (inflexion point) departure criterion
in the force equation
with:
If (for
R* I
2 f1 = 3
h =.!. <1>2 1 4 0
O.I45C 4 /ga 3
cap
the present) we assume
. * c +J /R y/2 1 1 1
f 1
(2.16h)
(2.55)
(2.56)
(2.57)
j to be small, than from (2.16h): 1
(2. 58)
It is clear then that the dynamic force, represented by j1
, enlarges
the departure radius.
Graphically, we can represent
of F(R*) R* 3 - !:J._ R* -.h. 1 1 fl 1 f1
given (for different values of
the solutions of (2.16h) as the zero's
F(R*) and its zero's are schematically l
the discriminant
def h j I (.....1..) 3 I (.....1..) 2) • f. 2 8 - 2] f + 4 f 1n 1g, • • D
1 1
From fig. 2.8 we see that only one positive root of eq. (2.16h)
exists, being the desired departure radius.
Tile positive analytical solutions of (2.16h) (for different values
of D) are listed in table 2.1.
Static departure, of course, appears as the limit for j1
+ 0
(FD negligble).
45
46
Ei..g. Z. 8 G!taphi..eal. !te.p!te& e.n:ta:tio n o 6 F ( R*) • 1
With dynamic departure we here mean the limit for h + 0 1 -
(F negligible). We introduce a dimensionless growth constant C, cr,r
given by: !
C = C/(ga3 )
4
cap
More precisely, the values of the dimensionless growth constant C, which determine the boundaries of the transition departure domain,
are derived in the following way.
We say that departure is static if the values of R* given by eqs. 1
(2.37c) and (2.59) (see table 2.1) differ by less than 10%,
On the other hand we call departure dynamic if the values of R* 1
given by eqs. ( 2. 61) and (2. 62) differ by less than 10%.
With these definitions it is easily derived that we have:
Table 2.1
Type of D =
I ( 1 3 departure - 2]y-) + 1
Static
0
Transition = 0
Dynamic
C=C/(ga 3 )
1/
4
cap
(using (2 .55),
(2.56), (2.57))
0.80¢ l/ 4
0
< 0
= 0.80~ 3/
4
0
»0
R:(dimensionless)
h I 1 t'f
1
* R1
(dimensionless),
(using (2.55),
(2.56) and (2.57))
, (2.37c)
(2.61)
.6oc'l 3 , (2.62)
48
- static departure, if C < 1.01 (> y/s = 0.70tj>0
3 1'+, (2.63) 1'
(h )3/8 (h )3/B -transition departure, if 1.01 ~ ~ C ~ 2.29 ~ , 1' ' 1
or: 0.704> 3 1~ < C < 1.594> 3 1~, 0 - - 0
(2.64)
-dynamic departure, if C > 2.29 ~:)318 = 1.59tj> 0 3 /~ (2.65)
The analytical
in fig. 2.9. results of table 2.1 for the above regions are given
R1* lf/Jo
2.0
static departure
1.5
1.0
fig. 2.9
transition departure
2.10. Dynamic departure (of spreading bubbles).
For C > 1.594> 3 14, (2.65), we obtained:
0
/' /
/
7 /
if' ~
/dynamic departure
(2.62)
This result only applies under the following conditions:
-heat diffusion controlled growth(cf. AI);
-no microlayer bubbles (cf. Ch. 4);
-no serious dynamic deformation (cf. Ch. 4) and
FD = 0.29~p 1 C4 ,(2.13);
- FD >> F · cr,r' - f = 2/3 (from A • z * ~ 1 and B s1n ¢ << 2/3, cf. sect. 2.8.3).
1 1 l 1
The latter two assumptions mean
original force equation (2.8):
that, regarding the
FB- ~r~{p 2 - p (0) -1hydr
} - o,
both F and the second term (F : the net wall reaction force) cr corr
are neglected. The ignorance of the latter is not a priori expected,
since p can be large compared to (p (O) + P t) during bubble 2 ex
growth.
However, at departure, Tir~{p 2 - p (0) -1hydr
will be small for
two reasons:
a) rB tends to stay relatively small (on rough surfaces) at high
growth rates;
b) the curvature K of the bubble profile at the bubble base will 0 .
be negative for this type of bubble just before detachment
(see fig. 2.10 and 4.3), Using Laplace's law:
~r~{p 2 - p (0) - p } = cr(K + --1--)
1hydr ext o rB 1
we see that the pressure difference must be reduced.
Fig. 2.10 Vyn.am.Lc.bubbte.
2.11. Microlayers bubbles.
If C > 2.96 (4.2.12) weshallhave microlayer bubbles (cf. sect. 4.2).
Since no direct contact between the bubble and the wall exists
[Zijl, 1978], no longer a distinction can be made then between
cavity bubbles and spreading bubbles.
49
50
The bubble shape tends to be hemi-spherical (see fig. 4.3, or fig. 2.11).
For an expanding hemisphere at a wall, the dynamic force has been
derived in A7: FD = 0.31~p 1 C4 (2.66}
If we, as in sect. 2.10.1, neglect again the second and third terms
in eq. (2.8), then we find (with A1
= 1):
(2.67)
F..ig. z. 11 M.iCJLolafjeJL bubble
As mentioned earlier, only forslow growth with static departure,
the bubble shape is exactly known. Therefore, no estimate can be
made of. the accuracy of eqs. (2.62) and (2.67). A comparison of
these departure radii with experimental low pressure data is given
in Ch. 5.
2.12. The departure time
Expressions fot the departure time can be derived from (2.1)
together with the expressions for R* given in the last column 1
of table 2. I •
Using dimensionless quantities it follows from (2.1) that we have
(g/a ) 1 /2t cap 1
(2.68)
For static departure we find from (2.37c):
(2.69)
and for dynamic departure from (2.62):
(2.70)
Hence, for a given ¢ , t decreases with increasing C in the static 0 I
departure region, whilst it increases with increasing C in the
dynamic departure region.
In the transition region t passes through a minimum. Numerical 1
calculations show that this minimum occurs for D > 0 at a C-value
given by
c I. 39 ¢ 3 I 4 0
From (2.61) it then follows that
and
R* = 1.065 ¢ 1 0
0.587 ¢ l/Z 0
Using C/¢ 3/
4 as an independent variable (cf. 0
(2.69) and (2.70) can be rewritten as:
0.36 (C/$ 3/4)2/3 0
(2.71)
( 2. 7 2)
(2. 73)
• 2.9) equations
(2. 69a)
(2. 70a)
51
52
CHAPTER 3 •========
DEPARTURE OF CAVITY BUBBLES FROM A HORIZONTAL WALL.
Parts of the theory developed in the previous chapter also apply to
cavity bubbles. We again assume the restrictions of sect. 2.1 to
be valid. Then the results of sects. (2.2), ••• , (2.5) can be applied,
if we replace rB and ~ 0 by rand a respectively.
3.1. Static departure (of cavity bubbles).
We first consider low growth rates (corresponding to C-values given
by (3. 13)), for which the "equilibrium bubble shape theory" (sect.
2.7) may be applied.
In section 2. 7. 2 the direct application of this theory led to ex-
pressions for Rt (2.34) and V (2.36). op l l
We can also derive these from a force equation. This will turn
out to be much easier than in the case of spreading bubbles, since
Fcorr' the second term in (2.16) is small now as compared to FB.
3.1.1. Use of the force equation (2.16).
For (static) cavity bubbles, (2.16) becomes:
p gV + (R 20
- p gH) n 2 - 21rras ina = 0 l top l
Adopting again a= 90° as a departure criterion (cf. sect. 2.7.1.a;
2.7.2.a), we obtain:
p gV + (~- p gH )u2 - 21Tr0' = 0 1 1 Rtop 1 1
1
(3.l)
giving:
v 1
2ura2 {I -~(I - 6 H /2R )} cap top 1 1 top
1 1
(2.36a)
Since r/R and 6 are generally small for cavity bubbles (see top1
1
below), we now obtained an expression for V , consistent with and more l
accurate than (2.36).
Using (2.35a) again (for 8 ~ 0.1, (2.32)), we have from (2.36a): 1
(2.34)
From (2.34) we derive:
3 r {1 + 0(8 )} Z -R-- 1 top
l
or:
_r_ = ~ f3 {I + 0(8 ) } Rtop . 3 1 l
1
(3. 2)
Since r/R is small for cavity bubbles (cf. sect. 2.7.l.a), top 1
this is true for 6 as well. 1
The condition: S < 0.1 (2.32) 1 ~
corresponds to:
r -R-- < 0.07
top (2.3la)
1
Now (2.36a) can also be written as:
(2.36b)
53
54
3.1.2. Use of the quantities of the equivalent spherical segment.
Since r/Rt << I, .p* will be small, i.e. the spherical segment op 1 I
very closely approximates a sphere.
Using definitions, (2.35a) and (3.2),we can easily show that:
A 1
- 2._ .p*4 + O(.p*"> 16 1 1
R* + .;...,* * = R {I + 0(¢ 2)} 1 top
1 2 l 1
r ¢*{I + .;...,* + 0(¢*2>} --= R l 2 l 1 top 1
B 1
= 1 ¢*{1 2 1
+ 0(¢*)} l
From (2.34) and (3.4) we directly obtain:
(3.3)
(3.4)
(3.5)
(3.6)
(2. 34a)
Another way to obtain (2.34a) is the substitution of (3.3) in force
equation (2.16b), with F0
= 0 and taken at departure.
However, as we see from (3.1), Fcorr is one order of r/Rtop smaller
than FB. Therefore, we simply have: 1
With (3.5) we find:
(3. 7)
which gives (2.34a) again.
The advantage of using R* instead of Rt was already explained 1 op 1
in sect. 2.9.
3.2. Transition departure (of cavity bubbles).
We next consider higher growth rates (for C-values according to
(3.14)), associated with transition departure.
Analogous to sect. 2.9 we again assume only small dynamic deformation
to occur and Witze's expression (2.13) for FD to be valid.
If, as a departure criterion, we assume a = 90° to hold also for
this case, then force equation (3.7) becomes:
(3 .8)
Easier than in the analogous case of spreading bubbles (sect. 2.9)
we now directly obtain:
{~(-r- + j )}1/3a 2 a 1 cap cap
(3. 9)
where:
j1
= FD/2rrp ga 3 = 0.145C 4
1 cap (2. 57)
The dimensionless departure radius R* = R*/a then becomes (if we -* 1 1 cap
drop the overbar on R1
again for the rest of this chapter):
(3. 10)
It is clear that the dynamic force, represented by j , enlarges the 1
departure radius.
Static departure, of course, appears as the limit for C + 0.
From (3.10):
(~ _r_) 1/3 2 a cap
(3 .II)
With dynamic departure, we here mean the limiting case for r/a + 0. cap
From (3.10):
55
(3. I 2)
More precisely, the values of the dimensionless growth constant C,
which determine the boundaries of the transition departure domain,
are found in the same way as was done for spreading bubbles:
We consider departure to be static if the relative difference between
the R*-values according to (3. I 0) and (3. I I) is less than I 0%. 1
On the other hand we consider departure to be dynamic if the relative
difference between the R*-values according to (3.10) and (3.12) is 1
less than 10%.
With these definitions we easily find:
- static departure, for:
C< 1.23 (-r-) 1/" a cap
- transition departure, for:
1.23 (-r-)l/'+ < c'< 2.14 a - -cap
- dynamic departure, for:
C > 2.14 (-r-) 1 /" a cap
(-r-)1/'+ a cap
(3. 13)
(3. I 4)
(3.15)
For these three regions the C_dependence of R* is given in fig. 3.1. 1
-* 11/3 R1 /(r/acap
2.5
static de arture 2.0
1.0 I j1.23 1.~ : 1.62
/ /
- )1/4 zp 12.14_~2.,_5 __ ___.lo ___ ct .... [r/acap
3.2.1. Comparison with spreading bubble results.
I) Contrary to the case of spreading bubbles, the static and
transition departure radius now depend on the cavity mouth
radius r. Hence, nucleation directly affects R*. For boiling l
water it is known that at high pressures more and smaller cavities
become active [Hsu/Graham, 1976]. On technical surfaces, cavity
type bubbles are most common [Chesters, 1978]. Observations of
Semeria (1962) and Tolubinsky et al. (1966) in the static de
parture regime (cf. sect. 3.6) show a drastic reduction of R* at
increasing pressure.
This can be explained by two additional phenomena:
a. smaller cavities are into play;
b. the departure mechanism itself is pressure dependent (a cap decreases if p increases, see AS).
As is shown by our experiments (Ch. 5) the second effect indeed
plays a role but is by far insufficient to explain the earlier
* observed reduction of R • l
2) Spreading bubbles are often somewhat larger than cavity bubbles,
at departure. For example, in boiling water (I bar), common static
values are:
a. cavity bubbles: IO~m ~ r ~ 100 ~m,
0.45mm < R* < 0.97rnm ~ l ~
with:
b. spreading bubbles: n/6 < ~ < ~/3, corresponding with: ~ 0 ~*
0.79mm < R < 1.59mm. ~ 1 ~
3) Spreading bubble departure is independent of r; in a way, the
bubble "forgets" its origination.
3.3. Dynamic departure (of cavity bubbles).
We now consider high growth rates, for which FD dominates F0
•
Departure is then called "dynamic". It appeared in the last section
as an extreme case of transition departure.
For C > 2.14 1 /4 ,(3.15), we had:
57
58
(3 .I 2)
This result was derived under the following conditions:
-Heat diffusion controlled bubble growth (cf. AI);
-No microlayer bubbles (cf. Ch. 4);
- No serious dynamic deformation (cf. Ch. 4) and
FD = 0,29wp1C4 (2.13);
- F >> F >> F • D a corr'
- r is small compared to the scale of the bubble at departure.
It is noticed here that if any dynamic deformation (flattening) of
the bubble takes place, this effect will be minimal at t = t .(See 1
Ch. 4, fig. 4.2: ~Pd decreases, ~P increases: the bubble tends yn a to "round off" by the effect of surface tension).
Therefore the inaccuracy of Witze's spherical approximation in the
calculation of FD will also be minimal at t = t1
•
3.4. Microlayer bubbles.
If C > 2.96 (4. 2, 12) we shall have microlayer bubbles (cf. sect. 4.2).
No longer a distinction can be made between cavity bubbles and
spreading bubbles, as was already mentioned in sect. 2.10.1.
3.5. The departure time t1
_
The departure time t has its minimal value in the transition del
parture region.
Using. (2.1) and (3.10) we obtain:
t • {l· __ r_ + 0.22(H2/3 tcap) 1/2 1 2 a -. g cape•
t is minimal for dt /dC = 0, leading to: 1 1
C • 2.13(r/a ) 1 / 4 cap
Hence:
(3.16)
(3.17)
t = 0.73/a /g (r/a ) 1 /6 1min cap cap
while:
1.82(r/a ) 113 cap
See, for example, fig. 5.4.
3.6. Numerical example.
(3 .18)
(3. 19)
On technical surfaces most of the bubbles are cavity bubbles
[Chesters, 1978]. In the important high pressure boiling technology
static cavity bubble departure is a common phenomenon, as will be
shown in the following example for water.
According to (3.13) we expect static departure, if:
C < 1.23 (gra2 ) 1 / 4
cap (3 .13)
Table 3.1 shows the numerical values of both sides of (3.13), for
different values of p , r and e . Use has been made of the empirical v 0
pv-dependence of for water (cf. AS) and of eq. (4.3.6) with
W=l.
Table 3.1
c (mm/sl/2) 1.23(gra2 ) 1 /~ (mm/sl/2) pv(bar) a cap (mm)
cap '
e =5K r=50)lm r=l00)lm 0
2.48 11.60 7.65 9.11 10.83
10 2.21 1.33 2.67 5.34 5.75 7.23 8.59 10.22
20 2.07 0.70 I. 39 2.79 5.56 6.99 8.32 9.89
30 1.93 0.48 0.95 1.90 5.37 6.76 8.03 9.55
40 1. 85 0.36 0. 73 1.45 5.26 6.61 7.87 9.35
59
60
We see in this example that departure is static for p > 10 bar. --- v~
For I0 5Pa < p < 1.5 106 Pa we can represent a by the formula: ~ v ~ cap
a = U/p O•OS with U = 4.41 10- 3 Pa0 • 05m cap v • (3.20)
Then, from (3.13) and (3.20) we obtain, for water, that departure
is static if:
(3.21)
CHAPTER 4 =========
DYNAMIC DEFORMATION; MICROLAYER FORMATION.
In this chapter the influence of liquid inertia will be examined
on two phenomena:
I) The existence of a dynamic deformation or flattening before
bubble departure (sect. 4.1).
2) The formation of a thin "microlayer" underneath an approximately
hemispherical bubble (sect. 4.2).
In a numerical example (sect. 4.3) for water the criteria for both
phenomena will be expressed in terms of superheat and pressure.
A third phenomenon involving liquid inertia: dynamic departure has
been treated in Ch. 2 and Ch. 3.
A fourth aspect of liquid inertia, the inertia controlled growth
(or Rayleigh growth) will be treated briefly in Appendix I.
4.1. deformation.
According to the equilibrium bubble shape theory (sect. 2.7) the
curvature of the bubble profile is determined by surface tension
and by hydrostatic pressure differences along the bubble profile.
For rapidly growing bubbles, the static bubble shape is deformed
by a dynamic pressure as a result of liquid inertia.
In this section we shall derive a growth rate criterion for the
occurrence of this dynamic deformation.
For the present purpose (viz. the comparison of dynamic and static
effects on the bubble shape) we shall neglect the hydrostatic
pressure differences and assume the bubble foot to be very small.
The static bubble shape will then approximately be a sphere (see
fig. 4.1).
61
62
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;.,pheJLe. i.e.. the. ;.,pheJLe. w.i.:th e.quctf. volume. a.6 the. a.c..tu.a1
bubble.; 1. he.mi-6phrvU£!.ai.. ;.,hape. a.6 de.te!Lmin.e.d by Uquid in.en;ti.a.
on.!y; 2. ac:tua.e ;.,hape. i6 v-ioeo~>dtj -£6 not ne.gUgible.;
miCJtol.atjeJL6 oJtmatio n
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the. ~>uJt6aee. te.n;.,ion. onl:g.
From Witze's (1968) analysis it is known that dynamic pressure is
maximal at the bubble top. Dynamic deformation will therefore start
as a flattening of the bubble top.
The question to be considered here is at which growth rate the
dynamic pressure ~Pd between the points P and Q (at the bubble yn top and base, see fig. 4.1) becomes important as compared to the
pressure difference ~p (=2a/R) across the wall of the undeformed q
bubble.
For this growth rate we have:
~Pd > ~P = 2a/R yn ~ a (4.1.1)
We shall consider this as a criterion for the occurrence of dynamic
deformation [Chesters, 1978].
From the exact solution of the liquid pressure distribution of an
expanding sphere adhering to a wall (neglecting viscosity) it
follows that [Witze, 1968]:
t,p dyn r:.s 2P 1 R_z (4.1.2)
With (4.1.2), (4.1.1) becomes:
2p R2 >2a/R 1 ~.
(4. 1.3)
Since, for the purpose of this section, we are not interested in the
initial stage of bubble growth (cf. AI), we may assume diffusion
controlled bubble growth and use (2.1) to obtain from (4.1.3):
(4 .I .4}
As a numerical example both sides of (4.1.4} and (for completeness)
the hydrostatic pressure difference between P and Q (fig. 4.1)
2p gR) are represented as a function of R in fig. 4.2 (for 1
boiling water at 1 bar).
From • 4.2 we see t,pd decreasing more rapidly than ~p at in-yn a creasing R. Physically, this means that t,pd tends to loose its yn influence when the bubble. grows; the bubble shape will become more
"static" at increasing R.
Another fact that is illustrated by fig. 4.2 is the relative unim
portance of t,ph for R < 0.3 mm, while ~ph = ~p0 for R acap = 2.48 mm.
In • 4.2, the maximal R is chosen to be the dynamic departure radius
(for C = 35.0 mm/s 1 12).
In dimensionless variables, (4.1.4) becomes:
(4.1.5)
Hence, dynamic deformation occurs, if
c ~ (4R) 1 1~ (4.1 .6)
63
64
"' 0..
QJ ... ::J
"' "' QJ c.. c.
t!Pdyn ( (, 35.0 mm Is 1121 lr-~--- 27.8
lr-"<--T--- 20.0 1'\r----->,,--::.,.---->.,---- 1 0. 0
·---''\---'~-'x- s.o
F~g. 4.2 The R-dependenQe o6
l'!.pdyn p1C4 /2R 2
; l'!.pcr = 2cr/R; !'!.ph= 2p1gR 6o!t WILteJt.,
bo~ng a.t 1 b~; p = 958 kg/m 3; cr 0,0587 N/m; 1
a.c.a.p = 2.48 mm.
Considering the situation at departure we distinguish five cases.
Use is made of results obtained in Ch. 2 and Ch. 3 (in general for
small .p , r). 0
4.1.1.1. Static departure (cavity bubbles).
R.* 1
[C < 1.23 (-r-) 1 1" (4.1.7)] a '
c2 _r_)l/3 2 a cap
cap
4.1.1.2. Static departure (spreading bubbles).
ii* 1
[c < o.7o.p 3 14, (4.1.9)1
0
3 1s·<Po (accurate for .p ~ ~/2 rad:) 0
4.1.2.1. Transition departure (cavity bubbles).
ii* 1
[1.23(~) 1 / 4 :::, c:::, 2.14(~) 1 1", (4.1.11)] cap cap
{2 c-r-) + o.2z'c"} 1 13 2 a cap
4.1.2.2. Transition departure (spreading bubbles).
1. [o.7o.p 3 14 < c < o.so.p 3 / 4 , (4.1.13)1 0 0
ii* = ih.p0 cos{~arccos 2 •46C
4
} 1 cp 3
0
2. [0.80cp 314 < c < 1.59cp 314 , (4.1.15)] 0 0
4.1.3. Dynamic departure.
Cc > 2.14 (~) 1 1" (cavity bubbles), (4.1.17), cap
(4.1.8)
(4.1.10)
(4.1.12)
(4.1.14)
(4.1.16)
65
66
or:
C > 1.59~ 3 14 (spreading bubbles), (4.1.18)] 0
Ad 4.1.1.1.
(4.1.19)
Substitution of (4.1.8) in (4.1.6), setting R = R* and comparison 1
with (4.1.7) leads to inconsistency unless:
r/a > 2.94 cap (4. 1.20)
In general, (4.1.20) can not be satisfied (e.g. for boiling water
at I bar, a = 2.48 mm; with r of the order of 10- 5 m). Hence we cap ' do not have dynamic deformation in this case.
Substitution of (4.1.10) in (4.1.6), setting R = R* and comparison - 1
with (4.1.9) leads to inconsistency unless:
<jl 0
> 3.19 rad (4.1.21)
Eq. (4.1.21) is both unrealistic and beyond the validity range of
the theory (cf. Ch. 2).
Ad 4.1.3.
Substitution of (4.1.19) in (4.1.6) and setting R R* leads to: 1
(4. 1. 22)
a) Cavity bubbles: Eq. (4.1.22) is more stringent than eq. (4.1.17)
if r/a < 0.18, a situation that can be realized. For water, cap boiling at I bar, in most cases: r/a < 0.18. Then (4.1.22) cap is a meaningful condition for having dynamic deformation at
departure.
b) Spreading bubbles: Eq. (4.1.22) is more stringent than eq.
(4.1.18) if~ < 0.84 rad, a situation that can be realized. 0
For water on most surfaces: ~ < 0.84 rad. Then (4.1.22) is 0
a meaningful condition for having dynamic deformation at de-
parture.
Hence, in many situations (4.1.22) is a suitable condition for
having dynamic deformation at departure. With (4.1.19), this
condition can also be written as:
R 1
-* (R:J R ) > 0 • 9 3 (4. I. 22 I) 1
For every individual situation with given a , r (cavity bubbles) cap or with given a , ~ (spreading bubbles) one can determine (from cap 0 eqs. (4.1.13), (4.1.14), (4.1.15), (4.1.16) and (4.1.6)) whether
or not dynamic deformation can occur in a part of the transition de
parture region.
4.2. Microlayer formation.
It is known that very rapidly growing bubbles have a flattened
shape and a thin liquid "microlayer" between them and the wall
([Moore/Mesler, 1961], [v. Stralen/Cole, 1979]).
For very high growth rates, the dynamic deformation (sect. 4.1)
of the bubble is extremely large. The bubble shape ~Jill tend to
be hemi-spherical (see fig. 4.1, fig. 4.3). However, as are
sult of viscosity, a thin liquid boundary layer ("microlayer")
remains at the wall, below the rapidly expanding bubble. A de
tailed theoretical treatment of this phenomenon was presented by
Zijl (1978).
In this section we consider the question how rapid growth must
be in order to give rise to the extreme dynamic deformation which
is a necessary condition for the occurrence of a microlayer.
From sect. 4.1 we obtain that extreme deformation occurs if:
2p R » 2cr/R l
(4.2.1)
67
68
In the diffusion controlled growth regime, (4.2.I) becomes, with
(2. I):
(4.2.2)
Or, using dimensionless variables:
(4.2.3)
Contrary to the case of sect. 4.I, we are, for the moment, not only
interested in the diffusion controlled growth regime, but also in
the inertia controlled growth regime (initial growth, cf. AI). It
is not known a priori at which moment in the growth period the
greatest chance exists for satisfying (4.2.I). However, as is shown
in Appendix I, this moment appears to be the transition timet'
[corresponding to the (dimensionless) transition radius R'] between
both growth regimes.
Therefore, it is sufficient to investigate condition (4.2.3) for
R = R':
(4.2.4)
Eq. (4.2.4) represents the inequality ~Pd >> ~p , evaluated at yn o the minimal value of R in the diffusion controlled growth regime.
From fig. 4.2 we see that, indeed, ~pd has the greatest chance yn being dominant if R is minimal in this growth regime.
From (4.2.4) we obtain:
(4.2.5)
The >>-sign, expressing for instance a ratio of about 10 in (4.2.4),
now expresses a ratio of I0 1 1~ ~ I.8 in (4.2.5). We therefore re
place (4.2.5) by:
c > (40R') 1 /~ (4.2.6)
In Appendix I (eq. (AJ.IS)) R' is approximated by its value for
a spherically symmetrical expansion of a free bubble:
(4.2. 7)
where R is the dimensionless radius of the metastable equilibrium 0
vapor nucleus [v. Stralen/Cole, 1979: Ch. 3].
With (4.2.7) we obtain from (4.2.6):
C>4.16R 1/
4
0
Replacing by the (dimensionless) cavity mouth radius r/a
(cf. A3), we obtain:
C > 4.16(r/a ) 1 / 4
cap
(4.2.8)
cap
(4.2.9)
Eq. (4.2.9) is the condition that has to be satisfied to have micro
layer formation at the very beginning of the diffusion controlled
growth period; referred to as "initial microlayer formation" in the
remainder of this chapter. But, as we already saw in fig. 4.2,
~pdyn tends to loose its dominance when the bubble grows. Hence,
the microlayer may disappear.
We now define a "microlayer bubble" to be a bubble for which a micro
layer exists till the bubble radius has reached 75% of its value at
departure. This criterion is chosen because in the final phase until
departure the bubble detaches itself from the surface and a micro
layer will no longer exist at the moment of departure (cf. fig. 4.3).
For having "microlayer bubbles" now (4.2.3) has to be satisfied for
an R. < o. 7sii . 1
A sufficient condition for this is (4.2.3), being satisfied for
R R , resulting in: 1
(4.2.10)
Or:
(4. 2. l 1)
69
70
It is clear that (4.2.11) is a more stringent condition then (4.2.9).
Substituting the dynamic departure radius (4.1.19) in (4.2.11) we
finally obtain the following criterion for the occurrence of micro
layer bubbles:
c > 2.96 (4. 2.12)
4.3. Numerical example 1.
For water, boiling at I bar the criteria, derived in sect. 4.1 and
sect. 4.2 are evaluated in this section, using acap = 2.48 mm;
(ga 3 ) 1 1~ 0.0196 m/s 112 (AS), hence C ~ 20C mm/s 1 12 • cap
a) For a cavity mouth radius r = 25 ~m initial microlayer formation
will occur for: C > 25.8 mm/s 1 / 2 (4.3.1)
b) We shall have "microlayer bubbles" for:C > 58.1 mm/s 1 12 (4.3.2)
We shall now express (4.3.1) and (4.3.2) in terms of the superheat
6 , and the pressure p respectively. [For heat diffusion controlled 0
growth, the ambient pressure p approximately equals the saturation
vapor pressure pv at the bubble wall temperature (cf. AI).]
4.3.1. Superheat dependence.
Let us adopt a modified Plesset/Zwick relation [v. Stralen/Cole,
1979]: r-----------------------~
(4.3.3)
Here W is a wall growth reduction factor. W ~ 1 for an almost
spherical bubble growing in a very thick thermal boundary layer
as compared to the bubble diameter.
W can decrease to values below 0.5 if the thermal boundary layer
thickness is much smaller than the bubble diameter, during mdst
of the bubble's life.
Substituting the numerical values for p
we have:
1 bar (cf. AS) in (4.3.3)
c = w s e 0
with S = 2.40 10- 3 m/(s 1 12 K)
(4.3.4)
Assuming W = I and, substituting (4.3.4) in (4.3.1) and (4.3.2) res
pectively we obtain:
a) From (4.3.1): initial microlayer formation
for e > 10.7 K 0
b) From (4.3.2): "microlayer bubbles"
for e > 24.2 K 0
4.3.2. Pressure dependence.
(4.3.11
)
(4.3.2')
We now consider boiling at various pressures for a given value of the
superheat e • 0
The pressure dependence of C is mainly caused by the quantity p , 2
occurring in the denominator of eq. (4.3.3).
[For example, for boiling water, raising the pressure from 0.1 bar
to 10 bar, p increases by a factor 71; (p k c ) 1 / 2 /~ just by a . 2 1 1 1
factor 1.07 (cf. A5).] Therefore we neglect the p-dependence of
(p k c ) 1 / 2 /~ and evaluate it at I bar and 373 K, giving: l 1 l
• (4.3.4)
For boiling water in the pressure range from 0.1 bar to 30 bar the
numerical values of pv(Pa) and p2
(kg/m 3) are related (within an
accuracy of 3%, cf. A5) by
p (4.3.5) 2
Combining (4.3.41
), (4.3.5) and (4.3.3) we obtain the following re
lation between the numerical values (using SI-units) of C, e and p : 0 v
or:
c = 115 we /p 0 ' 939
0 v
p = (115 we /C) 1' 065
v 0
(4.3.6)
(4.3. 7)
71
72
Assuming W = I and choosing, for example, 6
( 4. 3 . 6) , ( 4. 3 . I ) and ( 4 . 3 . 2) :
SK, we find with
a) From (4.3.1): initial microlayer formation
for (4.3.8)
b) From (4.3.2): "microlayer bubbles"
for (4.3.9)
Resuming, the criteria for the occurrence of initial microlayer form
ation and of a "microlayer bubble" in the present example are given
in table 3.1.
Table 3.1.
-'
c, rmn/sl/2 e (p =I 0 v
bar) ,K p (6 =SK),b v 0
ini ti"al microlayer formation > 25.8 > 10.7 < 0.43
"micro layer bubbles" > 58.1 > 24.2 < 0.18
An illustration of the occurrence of microlayer bubbles is given in
fig. 4.3 [v. Stralen/Cole, 1979].
F-i.g . 4 • 3. Vepa!l..ti.n.g miCJtol£t!jeJt bubble p v b R* 0.7.6 ~, ~ 7.9 mm.
4.4. Numerical example 2.
In this example we shall (for a few cases) give a schematic repre
sentation (fig. 4.4) of the growth rate criteria, derived in this
chapter. These will be compared to the growth rate criteria for
static and dynamic departure, derived in Cbs. 2 and 3.
In fig. 4.4 three cases are considered for boiling water (physical
properties from AS):
a) Cavity bubbles r 25 llm;
b) Spreading bubbles: <P 'If /6; 0
c) Spreading bubbles: <P 'ff/4. 0
From fig. 4.4 we see that dynamic deformation in this example is
not compatible with static departure, but only with dynamic departure
(the shaded area in fig. 4.4).
g cavity bubbles (r=25pm)
I static
jdep. I 0 7.6
Q_ static
13.3 19.6
dyn. departure
spreading 1 dep. 1
bUb~6 i 0~--~8~.2---~~~~=-~~~LLLL~~L_
(<Po =TC/6) dyn. departure
eq.(4.1.22l // dyn. deformation
....--,--... microlayer bubbles
58.1 C(mmts112J
I micro layer
1 bubbles
58.1 C(;m/s1/2l
micro taxer bubbles
58.1 1/2 C(mm/s l
F ..i..g. 4. 4. Schema.:ti..c. Jte.pJt.e..o e.n:ta.t.i..an 6oJt :the. oc.c.uMe.nc.e. in wa.:teJt o 6 (1) ~ta:tic. and dynamic. de.pa!ttu!te, (?) dynamic. de6o!tma.:ti..on
a:t de.pa!ttu!te. (.&haded Mea.), ( 3) rtU.CJtota.yeJt bu.bb.te..o.
The. dotted Une..o incLLc.a:te. :the. hypo:the.tic.a.t c.au o6 dynamic.
de.6o!tma.:ti..o n ( e.q . ( 4. 1 • 6 l ) a:t J..:ta.:ti..e de.pa!ttu!te..
73
74
4.5. Conclusions of Ch. 4.
I) (Ad. 4.1) Dynamic deformation at departure does not occur in the
static departure region.
2) (Ad. 4.1) Dynamic deformation at departure may occur in the tran
sition departure region. In most cases, however, it will only
occur in the dynamic departure region for
C ;:,_ I .39
3) (Ad. 4.2) Microlayer bubbles occur for
c > 2.96
4) Dynamic departure occurs if (cf. Chs. 2 and 3):
for cavity bubbles C > 2.14 (r/a ) 114
cap for spreading bubbles: C > 1.59 ~ 3 / 4
0
(4 .I • 22)
(4.2.12)
(4.1.17)
(4.1.18)
5) A clear distinction must be made between dynamic departure,
dynamic deformation and microlayer formation.
a, In many cases (as for instance for water at I bar and the
(r, $ )-values of the example of fig. 4.4) there exists a 0 -
range of C-values for which liquid inertia is dominant in
the departure mechanism (i.e. dynamic departure, or
FD/F >> I) while liquid inertia is not dominant with re-a,r gard to the bubble shape at departure (i.e. no serious dyna-
mic deformation, or ~pd /~p < 1). yn a
b. Generally, a wide range of C-values exists (see the example
of fig. 4.4) for which the liquid inertia is dominant in the
departure mechanism, while it is by far not as important as
to give rise to microlayer bubbles (for which the criterion
used is ~pd /~p » 1). yn a
CHAPTER 5 =========
EXPERIMENTS.
5.1. Purpose of the experiments.
5.1.1. Elevated pressures.
The first purpose of this chapter is the presentation of experiments
at elevated pressures, with cavity bubbles generated at single arti
ficial cavities and a comparison of their results with the theory
presented in Ch. 3.
In particular, the pressure dependence of R and t is of interest. 1 1
As is already mentioned in sects. 1.1.6, 1.1.7 the experiments need
to be performed at single cavities in order to eliminate the effect
of pressure on nucleation. Semeria 1 s (1962, 1963) experiments
~bowing a sharp decrease of R and a decrease of t at increasing p) l l
therefore can not be used for testing the departure theory developed
in Ch. 3.
Since for the high pressure range and single cavities no experimental
departure data are available, while the departure mechanism of cavity
bubbles at these pressures is of great practical interest, we have
carried out a series of water boiling experiments using artificial
cavities of known geometry and size in the pressure range from 1 bar
to 40 bar.
5.1.2. Low pressures.
As mentioned in the Introduction of this thesis, much confusion exists
about the validity ranges of several departure formulae.
Cole/Shulman (1966) fit both their experimental low pressure data and
Semeria's high pressure results with one straight line (in a double
logarithmic plot) finding as an empirical relationship:
R a: 1/p 1
(0.01 bar ::;, p ::;.. 130 bar)
75
76
v. Stralen compares his theoretical predictions: R « 1/p2 ; t ~ 1/p2 , ' 1 1
based on a thermal departure criterion (cf. sect. 1. 1.6) with (dynamic)
low pressure data [v. Stralen/Cole, 1979, p. 477, 480, 481].
It will be the second purpose of this chapter to make a comparison
between the experimental low pressure data, obtained by v. Stralen
(1979) and Cole/Shulman(l966), and the theory presented inCh. 2 and
Ch. 3.
5.1.3. Theoretical pressure dependence of
According to Cbs. 2, 3 (and using the empirical relationships acap(p)
(AS) and C(p) (eq. 4.3.6 for water) we obtain the following
pressure dependences:
a) Static departure
- Cavity bubbles R a: 1/pn l
with 0.03 < n < 0.05 for 1 bar ~P < 40 bar.
Using (2.1): t a: pm 1
with 1. 78 ~ m ~ 1.82 for 1 bar ~ p ~ 40 bar.
- Spreading bubbles
with 0.05 ~ n' ~ 0.08 for 1 bar ~ p ~ 40 bar.
m' Using (2.1): t a: p l
with 1.72 < m' ~ 1.78 for 1 bar ~p ~40 bar.
b) Dynamic departure
Using (2.1): t a: 1/p 0 • 63
1
5.2. Description of the experiment's.
(5 .1)
(5. 2)
(5.3)
(5.4)
(5.5)
(5.6)
Fig. 5.1 shows a schematic top view of the experimental set up.
A boiling vessel (stainless steel) made for pressures up to 50 bars,
was filled with degassed, demineralized water. The windows (thickness
10 mm, diameter 30mm) were made to resist 50 bars overpressure and a
temperature gradient of 25 K/mm.
10
Fig. 5.1 A -~Jc.he.ma;tLc. .top view o6 the expe!Umen.tal. -!JU-up.
1 J Baiting veM el
2) Electtic. heating
6
3) Smail'- ho!Uzon:to.l pla.te c.ontaA.nin.g :the aJ!..t.i6ieict.t cavU..iu
4] Window 1
5) Window 2
6) Rec.oJu:Ung o6 p and T
7J High -6peed came~ta
8) "Poin.t .6oWtc.e" Hg-Xe lamp
9) Vapo~ leak c.o~ol
1 0 l Vac.tJ.Wl'l pwnp
77
78
A small horizontal plate behind window 1 contains the artificial
cavities at which the bubbles were formed when the superheat e 0
was sufficiently large.
The light beam from a "point source" Hg-Xe lamp, passing window 2,
was focused on the cavities. A high speed camera was used to observe
the bubbles. In our experiments the maximum speed used was 1000 frames/s.
To obtain a stable superheat in a wide pressure range the following
procedure was followed.
Before each series of observations the water was electrically heated
while the vessel was closed. When p exceeded 45 bar, the heater was
disconnected and, after the free convectional flow had come to rest
a vapor leak was introduced.
As a result of this leak p slowly decreased, creating an (approximately
constant) superheat e • For different values of p cinematographic 0
shots were taken of growing and departing bubbles (10-30 bubble cycles
in each shot, depending on p and e ). From the seperate frames the 0
R(t) dependence was obtained for the life-cycle of a single bubble
and the corresponding C-value (which was approximately constant
during growth) was calculated from (2.1) by means of a computer
program. During one shot, the values of R and reproduced well 1
(variance 5% and 10%).
Two different plates, made out of stainless steel, were used: one
containing a single cavity with a mouth radius r = 25 ~m and one
containing two single cavities with mouth radii of 25 ~m and 50 ~m.
All cavities us~d were of the reservoir type (see 5.2)
Cv. Stralen/Cole, 1979, p. 123].
liquid r=25)Jm (50)Jm)
F-ig. 5.2 RueJr..vo-Ur. type ca.vli.y.
The reservoir was made by drilling a hole (0: 0.35 mm) in the lower
side of the plate and filling it up (almost completely) by a tightly
fitting plug, thus creating a closed reservoir.
Next the cavity hole (0: 25 ~m or 50 ~m; length 0.3 rum) was drilled
from the upper side to the reservoir.
5.3. Experimental results.
The experimental results are in Table 5.1. Each line
represents the average results obtained in one shot at similar con
ditions of r, p and a • T t is the corresponding saturation temperao sa ture at pressure p; e is derived from the cinematographically ob-
0
tained value of C by means of eq. (4.3.6) with W = 1.
No observations below p ~ 1 bar (r = 50 ~m) or p ~ 2 bar (r = 25 ~m)
could be made as a result of an inactivation of the artificial cavities
(preceded by the occurrence of considerable waiting times). This
inactivation was probably caused by the increase of the minimal value
of a , required for the activation of a cavity, at decreasing pressure. 0
This minimal value is often related to r by:
e = (2uT t)/(p ~r) omin sa 2 ·
[Hsu/Graham, 1976].
Some of our (similarly constructed)artificial cavities could not be
activated at all in our experiments having an approximately uniform,
low superheat. In additional experiments using a heated wall any
inactivation of the artificial cavities could be compensated by inten
sifying the wall heating.
Apparently, besides the r-value , more factors (for instance small pits
in the inner surface of the cavity; small amounts of gas)play a role.
Another phenomenon observed was the inactivation of cavities after one
or two series of experiments. An explanation for this could be a re
placement of all the gas and vapor in the reservoir by liquid
[v. Stralen/Cole, 1979, p. 123].
In fig. 5.3, for two cavity sizes, the observed departure radii R l
are represented as a function of p. The curves (1) and (2) represent
79
80
series no. 1 (r = 25 ~m)
p
(bar)
43.0 38.3 23.7 22.5 17.7 15.2 13.4 12.7 9.64 7.53 5.85 4.87 4.02 3.23 2. 77 2.09 I. 76
T sat (K)
527.7 520.8 494.2 491 .5 .479 .2 472.0 466.0 463.6 451 .4 440.9 430.8 423.9 416.7 409.0 403.9 394.6 389.1
0.34 0.28 0.42 0.40 0.46 0.52 0.65 0.60 0.78 0.84 0.96 1.08 1.06 1.15 1.20 1.33 1.37
e 0
(K)
5.0 3.7 3.5 3.2 2.9 2.9 3.2 2.8 2.8 2.4 2.2 2. 1 1.7 1.5 1.3 1.1 1 .o
Rl
( 10-1 mm)
4.98 4.06 4.11 4.30 4.46 5. 19 5.29 5.47 4.94 5.72 5.73 5.43 5.03 4.91 5.08 5.07 5.10
t I
(10- 1 s)
21.5 21.0 9.78
11.6 9.40 9.96 6.62 8.31 4.01 4.64 3.56 2.52 2.25 1.82 I. 79 1.45 1.38
series no. 2
32.4 25.6 19.7 12.5 7.5
Table 5 .I
I r=25 ~m r=25 ~m r=25 ~m 1 r=SO ~m r=25 11m
511 .2 498.2 484.7 463.0 440.7
0.34 0.37 0.61 0.74 1.10
3.8 3.4 4.3 3.1 3. 1
4.22 4.29 4.89 5.12 5.35
I 6.o2 I 6.43
6.66 I 6. 79 I 7.16
E.xperimental values of R and t for different I I
pressures. Two series are shown. Each line represents
15.4 13.4 6.42 4.78 2.36
a shot at approximately constant p, C and e . R and t 0 I I
are average values obtained from one shot (variance 5%
and 10% respectively); C has been derived from eq. (2.1),
e from eq. (4.3.6). Some of these results are also 0
represented in figs. 5.3 and 5.4.
p I Tsat (bar) (K)
a) 23.6 493.4 b) 18.3 480.9 b) 14.0 468.0 c) 13. 1 465.0 b) 12.5 462.9 b) 8. 17 444.4 c) 7.46 440.5
I b) 3.98 416.4
Table 5.2
c e R t 0 1 1
(nnn/sl/2) (K) ( 10-1nnn) (10- 1 s)
r=25].lm : 25].lm I I
50l!m 50 ].1m 25].lm 50 )1m 25].lm 1 50].lm
0.39 I 0.38 3.4 I 3.2 4.54 I 6.66 12.5 : 30.5 0.62 I 0.62 4. 1 4. 1 5.32 I 6.78 7.28 I I 12 .o o. 72 I 0.72 3.7
I 3.7 5.52 I 6.76 5.81 I 8.73
1.06 1.07 5. 1 5.2 5.61 I 6.93 2.75 4. 15 0. 73
I 0.76 3.4 I 3.5 I 5.41 6.91 5.44 I 8. 15 I
0.97 I 0.92 3.0 I 2.9 5.42 6.99 3.08 I 5.78 I
1.48 I 1.28 4.2 .56 7.10 1.42 I 3.22 I
1.50 I 1. 31 2.4 .40 I
7.27 1.30 I 3.06 I i I
Results of the additional experiments, described in sect.
5.4.1.1, using a heated wall. Each line represents a shot
at approximately constant p, C and e . R and t are the 0 1 1
average values obtained from one shot (variance 5% and
10% respectively). C has been derived from eq. (2.1),
e from eq. (4.3.6). 0
In situation a) the wall heating was disconnected. In the
situations b) the current through the wall heating system
was SA (resistance approximately 0.01 ohm); in situationB
c): 10A.
It is seen from this table that changes in the wall heating
(e ) do not influence R • 0 1
These results are also represented in figs. 5.3 and 5.4.
81
I
82
R1lmml
0.8 t ~~
---=+~o--~*~o~--~0~~---------
0.
" .... \ \ . .131
\ ..... _;_~;··- .. ·~ ..... . - p(barl
0 10 20 so
Fig. 5. 3. Pl!.e.&-6Wte depen.den.c.e o6 the depaM:Wte Jta.cLiu.-6 R • 1
CWtve ( 7) : :the.oll.e.ti.c.al value.& 601!. 1!. = 50 llm;
CWtve. (2): ib. fioll. 1!. = 25 llm;
o expeJL.i.me.J'I;tai value.& 601!. Jr. = 50 llm (-6etie.& 2);
6 ib. 6oll. 1!. = 25 ]lm {-6etie-6 2), (efi. table. 5.1);
CWtve. ( 3): Semluua'-6 ( 1962) obMJr.va.ti.on.-6;
CWtve. (4): v. Stltale.M ( 7979, p. 480) pll.e.dic.:tion..
The. ll.Uuli:-6 o6 :the a.ddi.tion.al e.x.pell.ime.nZ-6, ducJtibed in
-6ec.t. 5.4.1.1, Min.g a. hea-ted wall, Me ll.e.pJr.uente.d bif
x(Jr. = 25 J.lm) a.n.d +(!!.=50 ]lm) (c.fi. table. 5.2).
the theoretical values of R according to eq. (2.34a), sect. 3.1.2, 1
using the relation a (p), given in AS. cap
Curves (3) and (4), representing Semeria's (1962, 1963) observations
and v. Stralens (1979, p. 480) prediction respectively are drawn in
fig. 5.3 for comparison.
Fig. 5.4 represents, for the two cavity sizes, the theoretical and
observed values of the departure time t as a function of the growth 1
constant C.
The lines in the high p-region of fig. 5.4 represent the theoretical
predictions according to eqs. (2.34a) (sect. 3.1.2) and (2.1).
The minimal value oft , shown in fig. 5.4,was derived in sect. 3.5. l
For completeness, also the low pressure data of v. Stralen
[v. Stralen/Cole, 1979, p. 480] are given in fig. 5.4, while the line,
drawn in this pressure domain, represents the prediction of our theory
for dynamic departure:
(3. 12)
The choice of C as independent variable is not arbitrary. Since
t = R2 /C 2 (2.1) and we already saw that (in the high p-region) no l l
drastic variation of R with p occurs, the major p-dependence of t 1 1
is expected to be due to the variation of C. This is also to be expected
in the low p-region, where R (theoretically) depends on C (3.12). 1
For water, the empirical p-dependence of C is given by:
c 115 we /po·939 0
(4.3.6)
For the high p-region the C-values are obtained cinematographically.
It should be noted that the C-values represented by curve (3) are
obtained from (4.3.6) (with W 1) and the experimental values of
6 and p. 0
33
84
5.4. Comparison of the experiments with the (static) theory.
5.4.1. Pressure dependence of
From fig. 5.3 it appears that the observed p-dependence of R is in 1
good agreement with the theoretically predicted one. No resemblance
with the drastic reduction of R at increasing pressure, observed by 1
Semeria (1962, 1963) is found •
This result confirms the presumption that the pressure dependence of
nucleation was responsible for the very small bubbles, observed by
Semeria.
Quantitatively, the experimental R -values are smaller than the l
predicted ones. Possibly the effective cavity mouth radius was
reduced, due to impurities or roughness elements in the tiny cavity
mouth. The theory was derived for an ideal sharp-edged cavity mouth _
without roughness.
Also a contribution of the bubble induced liquid flow in
the departure process is to be expected, resulting in a smaller average
departure radius. Although the observed bubbles were only slightly
agitated by this flow, a small force contribution in the departure
process seems likely in our experiments.
radius was found, in agreement with the This (static)
result contradicts v. Stralens prediction [v. Stralen/Cole, 1979,
p. 303, 313], derived for a heated wall.
00 \J1
1000
100
10
F..tg. 5.4
high p-region atmospheric p-region
r-50J.Jm r=25)Jm ~
10
GJtow.th M.te dependenc.e o 6 .the dep<Vltulte fue .t 1
•
(3)
100
low p-region
--+- C(mmts112]
1000
1 n .the !Ugh p-!teg-ion .the .tlteoltetic.<tl'. .Unu r 11. 12 I Me ob.t<Uned 6Jtom eq~. r 2. 34al I Hct. 3. 1. 21 and 12. 1 I.
w-i.th It = 50 ~m and It = 25 "m ltel>pec.Uvel.y. OWl expelt-irnen.tltt JtUu.Ui. Me ltepltuen.ted by oilt = 25 ~m. ~e!t-iu 11,
•Ill = 25 "m, ~e!t-iu 21 k6 . .table 5.11. In .the tow p-!teg-ion .the .theo!tetic.al.Une (31 ~obta-ined 61tom eq~.
(3.121 and (2.11. The expelt-imen.tltt value/> o6 v. S.tllalen ( 19791 Me lteplluen.ted by 6. The .tlr.aMWon depCVt.tWle
-in.teJtvat hM been -ind-ic.a.ted 601t IL = 25 "m and It = 50 "m. The Jtuu.Ui. o6 .the addWonal expelt-irne>!U ~-ing a
he.a.ted waU (u.ct. 5.4.1.11 aM !tep!te.&enterl btj x(!t • 25 1•ml and +(ll • 50 pml k6 . .tabte. 5.21.
86
5.4.1.1. Experiments using a heated wall. Comparison with the
relaxation microlayer theory.
Some additional experiments have been performed to investigate the
influence of wall heating. The same cavities were used but this time
the wall was heated electrically. The heating caused an increase of
C as compared to experiments using the non-heated wall under the same
circumstances. At a given pressure in the range of 1 bar to 40 bar no
influence of the increase of C nor of the non-uniformity of the tem
perature field on R was observed (compare Table 5.2 and figs. 5.3 1
and 5.4).
It is now possible to compare the results of the experiments described
in this chapter with the "relaxation microlayer theory" of v. Stralen
(cf. sect. 1.1.6) in which a thermal criterion is used for bubble de
parture from a heated wall. According to this theory the departure
radius increases with increasing thermal boundary layer thickness, a. The latter decreases with increasing pressure. However, it appears
from our experiments both for heated and non-heated walls that neither the
existence of a thermal boundary layer nor the reduction of a at
increasing p influences R in the pressure range under consideration. . 1
At least for static departure it therefore must be concluded that the
thermal departure criterion used by v. Stralen can not explain the
observed trends. This confirms the basic idea that bubble departure
is a matter of forces.
Full results of the experiments using a heated wall will be published
elsewhere.
5.4.2. Cavity size dependence of R • l
From eq. (2.34a) (sect. 3.1.2) we see that, for a given pressure, the
ratio of the departure radii at the cavities with r = 50 ~m and r = 25 ~m
must be 2113 ~ 1.26.
Our observations (fig. 5.3) show a mean value of this ratio of 1.35,
which is in good agreement with the theory. See also fig. 5.5.
I.
2.
3.
Fig. 5. 5 Th!tee ~ubHquen.:t pic:twt~ o6 ~ta-Uc. bubb.te depaJL:tv.Jte a-t two
di66eJten.t a.!!il6ic.io..l c.avi:ti..~ (.te6.t: Jt = 50 11m; Jtigh.t: Jt = 25 11ml;
p = 79,7 baJt; 6oft .th~e pic:twt~: R\(50 11ml = 0.647 mm, R
1(25 11ml = 0.494 mm and .the Jta.tio o6 .th~e Jtadii ~ 1.31
(.theoJte.tic.aUy 7.26); .the c.ame.Jta ~peed WM 152 6~tam~ /~.
87
88
5.4.3. Pressure dependence of t • l
From fig. 5.4 we see that the experimental growth rate dependence of
t is in good agreement with the theoretically predicted one. l
None of the experiments shows a reduction of t at increasing p in 1
the high pressure region (compare sect. 5.1.1 and sect. 5.4.1.1).
Quantitatively the t -values are below the predicted ones l •
for the same reasons as were mentioned in case of R (sect. l
probably
5.4.1).
5.5. Comparison of low pressure data from literature with the
dynamic theory.
Fig. 5.6 represents the low pressure R -data of v. Stralen/Cole l
(1979, p. 481) and Cole/Shulman (1966). The theoretical curves of the
static theory in the high p-region are- added for completeness
and comparison.
The line in the low pressure region of fig. 5.6 shows the theoretically
predicted R (C)-dependence, according to eqs.(3.12) or (2.67). l
The C-values used for the experimental low pressure data were obtained
(as in fig. 5.4) from (4.3.6) and the experimental values of 8 and p. 0
It is pointed out that, if W< 1,the experimental R -data of fig.S.6 corres-1
pond to lower C-values. For the extremely low value W = 0.3 these data
points are indicated in fig. 5.6 by (&) [v. Stralen/Cole, 1979] and
(o) [Cole/Shulman, 1966]. In reality, for these large low pressure
bubbles, W will decrease from 1 to possibly 0.3 during the bubble
growth, because the bubble emerges from the thermal boundary layer,
coming into contact with the cooler bulk liquid present in these
experiments.
Comparing experiments and theory in the low p-region of fig. 5.6 we
see (regardless of the value of U) in both cases a sharp decrease of
R at increasing p (or decreasing C). l
For these low pressure bubbles, it is difficult to compare the data
and the theory quantitatively because:
f
10 (D) (t.)
___ j1J.. _____ _ - __ __ !]) _____ _
--------------
high p- region atmospheric p-region
(3
(A)
(D)
(D)( I Q
low p-region
0.1
.__,_ ______ . ______ .. ______ .££#4:[=50}Jm - Clmm/s112j
I
100 10-' 10
F-<.g. 5. 6 Gltowth IULte depe~denee o 6 the. depaJLtwr.e. >uuUu.l R
1 •
In the. h-igh p:h'l.eg-ion tke theoJtetioa! .Unet> I 11, ( Zl CVte obtJ:Uned 6ttom eq. ( Z. 34a) !Hc.t. 3. 1. Z), w.Uh Jt • 50 um
and fL • 25 um itet>pec.t<.vely. (Owr. <Oxpe.Mme.l!tat itUu.l..t¢ Me g-iven .in 6,ig. 5.3}. Tn the low p-!Leg.ion the theoJte.tic.a.t.
Une. !31 .U. obt<Uned 61t0m eq. f.L 121. The. expe.Mme.n:ta.e vttfM.! o6 v. St.IULfen/Cote !1979} aM. Upitet>ented by
~rw • 11 OIL (61!111 • 0.31; thoH o0 Cote/Shulman (19661 b!l a(W • 11 011. (a} (W 0.31. The. tM~Ltion depaJLt:Wte
L>!teJtva.l. luu. been LndLcated 6oJt fL • 25 um and fL • 50 ""'·
1000
90
- On the average the experimental R -values for W = 1 are below the 1
theoretical line and the R -values for W = 0.3 are above this line. 1
This suggests: 0.3 < W < 1.
- In our analysis departure coincided with the occurrence of a closed
shape. For large hemi-spherical bubbles this can not easily be ob
served. In that case departure is often defined as the moment at
which a certain distance to the wall is reached. Longer departure
times .are to be expected, using the latter definition (compare
fig. 5.4).
- For extremely low pressures the contribution of inertia controlled
bubble growth (cf. Appendix I) must be taken into account. This
effect also complicates the comparison of the theory and the low
pressure data referred to in this Chapter.
Comparing the experimental t -values and the theoretically predicted l
ones in the low pressure region of fig. 5.4 we notice an overall
agreement of the experimental growth rate dependence of t with the 1
theoretically predicted trend. Quantitatively, the comparison is com-
plicated by the reasons mentioned above.
5.6. Conclusions; suggestions for further experiments.
5.6.1. Elevated pressures.
1) Only a small reduction of R was observed at increasing p, being 1
in agreement with the static theory, presented in Ch. 3.
2) An important increase of t 1
was observed at increasing p, being
in good agreement with the theoretical prediction.
3) For a given p, no influence of the growth rate on R was observed, 1
in agreement with the static theory (Ch. 3).
4) Due to an increase of the cavity mouth radius by a factor 2, the
experimental R -value increased on the average by a factor 1.35; 1
the theoretical prediction of this factor is 1.26.
5) A thermal departure criterion, as used by v. Stralen, can not
explain the observed trends, mentioned above (cf. sect. 5.4.1.1),
which confirms the basic idea that departure is a matter of forces.
5.6.2. Low pressures.
Qualitatively , the p-dependence of experimental low pressure R -data l
and t -data, taken from literature, is roughly in agreement with the l
theoretical prediction of the dynamic theory. However, the quantitative
agreement is poor for this pressure range.
5.6.3. Minimal t • l
Regarding the whole pressure range of 5.4 it may be concluded
that theoretical and experimental evidence exists for the existence
of a minimum value of the departure time t , located in the
atmospheric pressure range. 1min
91
92
5.6.4. Suggestions for further experiments.
Further experiments using single artificial cavities are suggested
to verify the theoretically predicted growth rate criterion for the
occurrence oft (eq. 3.17), not only for water but also for other 1 •
liduids. mLn
Low pressure experiments in an approximately uniformly superheated
liquid and at single cavities are suggested for a quantitative test
of the theoretical predictions of the dynamic theory.
Concerning the important field of nucleation it is suggested to in
vestigate the p-dependence of nucleation both theoretically and
experimentally, using a set of artificial cavities and a wall heating.
APPENDIX I
Bubble growth.
a. The bubble growth of a spherical bubble in an initially uniformly
superheated and unbounded liquid has been described by Plesset/
Zwick (1954), Forster/Zuber (1954), Scriven (1959), Plesset/
Prosperetti(1977). A short exposition of the theory is given in
this Appendix.
The basic equations governing the radial expansion of the bubble
under consideration are:
1) Conservation of momentum:
av + v av at or
(A 1.1)
2) Conservation of mass:
r 2v sR2 :R (A 1. 2)
with €: 1 - p /p 2 1
3) Conservation of energy:
<lT + ()T + 2 <lT) v- = a at ar 1 ar 2 r <lr (A 1. 3)
Here v:JT/<lr denotes the radial convection term mentioned in sect.
1.1.3. The initial conditions (t=O) are:
v(!_, 0) o, 0) T 0
The boundary conditions for r
p(oo, t) p , T(oo, t) 0
T 0
p (r, 0) l-
"'are:
const.
R • 0
(A1 .4)
93
94
At the bubble wall (r~R) one has:
(A 1. 5)
Here the liquid heat capacity has been neglected as compared to
the heat of evaporation, orca /~ << 1. l 0
Combining (A1.1) and (A1.2) and integrating between r=R and r=oo
yields the so-called Rayleigh-equation:
12 R2 + RR = {p (R)- p }/~p (A1.6) l 0 1
Using Laplace's law: p - p (R) ~ 2o/R, we obtain from (A1.6): 2 l
(A 1. 7)
Physically, (A1.7) says that the bubble_overpressure equals the
sum of the liquid inertial pressure and the Laplace overpressure.
T
To···········------h<at t----_,.
F.i.g. A1.1
0 R
T e.mpeJUI.tuJr.e. cJ.i6:tJU.bu.t£on. t\OIL
a,& ymp.tot.i.e gJLow:th
r
It is further assumed that the temperature inside the bubble (T ) 2
is uniform. As pointed out by Scriven (1959), the bubble wall
temperature TR(t) (~ T2
) diminishes rapidly after the bubble
initiation from the initial value T to T T t(p ) (cf. fig. o sa o
A1.1). The solution of the growth problem, using T T is called 0
initial growth or inertia controlled growth).
Tsat is =L:::.E...::.::..=.::::.::_.sz:::...::..:.:,= or diffusion controlled growth (see fig. A1.2).
Fig. A.J. 2
R
inertia controlled growth !roughly: R«tl
Gltow.th Jtegim~ •
The transition point R', t 1 between both growth regimes is defined
by (cf. [v. Stralen/Cole, 1979]):
(A1 .8)
For the entire initial period of growth it is assumed that the
bubble overpressure equals the one corresponding with the vapor
nucleus:
p - p ~ 2o/R 2 0 0
(A1.9)
Using R /R << and E = 1, this approximation gives with (A1.7) 0
(Plesset/Prosperetti, 1977):
3 • 2 pz-po~ZplR
Hence, R = {4o/(3p R )} 1 12 t l 0
(A1.10)
(A 1.11)
95
96
~ Asymptotic growth.
Assuming equilibrium conditions and eR linearized Clapeyron relation gives:
p - p = p t9R/ET t ~ p t9R/T t 2 o 2 sa 2 sa
T t(p ) - T t(p ) the sa 2 sa o
(A1.12)
(where; since p << p , Kelvins correction for the surface cur-2 l
vature has also been neglected). Since eR << Tsat we obtain:
p - p ~ 0 (no overpressure, only heat diffusion governs bubble 2 0
growth).
Scriven (1959), using (on dimensional grounds) r/(a t) 112 as a ' 1
similarity variable, obtained from {A1.3, 4, 5):
with:
c
or:
c
(2k /p t)l/2 e 1/2 1 2 0 •
for p c e /p t << 2~ l 1 0 2
for p c e /p t >> 2~. l l 0 2
(A1.13)
(A1.14)
(A1.15)
Eq., (A1.14) was derived earlier by Plesset/Zwick (1954). It is
experimentally confirmed (Dergarabedian, 1953).
An expression for the case that p c e /p t is of the order of 1 1 0 2
magnitude of 2~ has been given by Zijl (1978).
b. The transition point is defined by (A1.8). With (A1.9), (A1.10)
and (A1.13) we obtain:
(4o/3p R ) 1 / 2
l 0 (A1. 16)
Expressing p - p (eq. (A1.9)) in terms of the initial superheat 6 2 0 0
(using the Clapeyron equation) we obtain:
(2p ~e /3p T t) 1/
2 = C2 /2R' (A1.17) 2 o 1 sa
For water, boiling at 1 bar, using (A1.15) and choosing e lOK 0
we obtain from (A1.17):
R' 57.9 ~m, corresponding to
t 1 (R') 2 /C 2 = 6.22 ~s.
Obviously, the contribution of inertia controlled growth is neg
ligible for water boiling at atmospheric pressures.
Considering water, boiling at 0.2 bar, it follows from (A1.17):
R' 1.4 mm, while the dynamic departure radius (3.12) is:
R ~ 14 mm. 1
For that case we have an inertial contribution of 10%.
c. In sect. 4.2 it was mentioned that the greatest chance to satisfy
(4.2.1) is for R = R1 • In the diffusional growth regime this was
shown already (below eq. (4.2.4)). In the inertia controlled growth
regime we obtain from (A1.10):
p R2 4cr/3R 1 D
Substituting this in (4.2.1) we obtain:
R » l_ R 4 0
(A 1. 18)
From (A1.18) we see that the greatest chance satisfying (4.2.1) is,
indeed, for R R', being the maximal R in this growth regime.
97
98
APPENDIX 2.
Comparison of the exact numerical solutions for the bubble shape
with the analytical approximation.
Table A2.1 represents the exact numerical values (directly or indirectly
obtained from Hartland/Hartley (1976) of various quantities associated
with the bubble shape, evaluated at departure. It is assumed that the
departure occurs when the inflexion point of the bubble profile appears
at the bubble base (hence a1
= ~ 0 , cf. fig. 2.6). The latter assumption
is accurate within a few percent for ~ < TI/3 rad. For a comparison 0 ~
several departure formulae, obtained in Ch. 2, are also listed. The
numbers referring to these formulae are mentioned in the column heads.
All quantities have been made dimensionless by means of a • The last cap columns of table A2.1 show the high accuracy of eq. (2.37b)- for a wide
range of ~ -values. 0
Being beyond the validity ranges of the theory, not all values for
~ 0 > 88° have been given. For ~: 90° the spherical segment is a hemi
sphere, hence H* R*. 1 1
I Exact numerical solution
description using t:quivalent spherical segment
R ii I v I
* i\* -· f h /(h /f) 'i 'i H
' top l
l l l 1 ' l 1 1 1 (deg) (deg)
19 0.199 0.413 0.0346 10 0.1971 0.3891 0.642 0.026 0.201
l-t5 0.436 0.981 0. 429 25 0.4609 0.8787 0.483 0.099 0.453
63 0.575 I .328 I. 152 39 0.6567 1.167 0.24 t 0.109 0.672
75 0.661 1. 466 1.869 51.5 0.1888 1.280 0.034 0.027 0.891
'11 (), 7UM 1.49J 2.JO! 59 U.Bo:>:r l. J 15 -0.01:' -0.050 0~84~
88 0. 759 1.488 2.730 67 0.9448 1.314
I -0.151 -0.131 0.934
94 0.81) 1.455 3.110 75 1.0274 1. 293
100 0.871 1.401, 3 ./;17 83 I. 1125 1.2M5
106 0.933 1.345 3.642 90 1. 2033 1.2033 -0.233 -0.337 1. 203 I
A 1alyt ic approximation ,.
¢ Rtop i\ v ;2 ~ R*(eq. (2. 37b)) 0 top \ 8 ' l
(deg) \ l !{•(exact)
eq.(2.37) eq. (2.37al eq.(2.39) eq,(2.37b) 1
19 0.199 0.2031 0.035 0.2031 1.03
45 0.433 0.4809 0.466 0.4809 1.02
bJ 0.546 0.6733 1.279 0.6733 1.02
75 0.591 0.8016 2. 157 0.8016 1.02
81 0.605 0.8660 2. 718 0.8660 1.00
SH 0.612 0.9401 3 .t.85 0.9401 0.99
:)4 O.Yti
100 0.96
106 0.94
A2 1
99
100
APPENDIX 3.
Nucleation.
In the theory on homogeneous nucleation (i.e. nucleation in the bulk
mass of a liquid) the occurrence of a very small metastable equilibrium
nucleus is described [v. Stralen/Cole, 1979]. This nucleus, with radius
R, has an overpressure of (p - p ). = 2cr/R. For bubble growth it is 0 2 0 1n 0
necessary that p - p exceeds this value. Any further growth then is 2 0
facilitated by the reduction of 2cr/R.
In the literature on heterogeneous nucleation [Hsu/Graham, 1976] this
concept has been used explaining bubble nucleation at a wall. The
initial pressure that has to be exceeded for bubble growth is thought
to be (cf. fig. A3.1):
2cr/r (A3. 1)
Ei.9. A3.1 Model. eofL nude.a.tf.on a.t a. wa.U-c.avUy.
Assuming equilibrium conditions and p << p the application of the 2 1
linearized Clapeyron equation yields;
(A3.2)
Considering the above theory two remarks should be made:
1. The application of the homogeneous nucleation theory to the situation
of nucleation at a wall (i.e. locating the nucleus at the cavity
mouth) has a speculative character. Although no better model is
available, it could fairly well be that other factors are decisive
in heterogeneous nucleation.
For instance, the inner geometry of the wall cavity (inner rough
ness) or gas residues.
2. The interpretation of (A3.2) as a functional relation between the
superheat 6 and the cavity radius r has been given by many authors. Q
However, only 60 . is related tor by (A3.2). In fact (A3.1) and ~n
(A3.2) express minimal values of (p - p ) and 6 respectively, 2 Q 0
needed for bubble growth. At a heated wall the values of (p p ) 2 0
and 6 may be many times larger. 0
101
102
APPENDIX 4.
Viscous force, Fv; force due to surface tension gradients, F~v·
1. Suppose, for simplicity, the bubble is a sphere, adhering to the
wall. Then, the bubble velocity v = R. The viscous force may then
be written as:
1 • 2 F = -irp C (RR)
v 2 1 D (A4.1)
where CD denotes Batchelors (1967) expression for the drag coefficient
of a free rising bubble:
CD = 48/Re (A4.2)
For our case:
Re (A4.3)
Substituting (A4.2), (A4.3) in (A4.1) and using (2.1) we obtain:
F v
(A4.4)
For high growth rates this must be compared to FD (eq. (2.13)):
(A4.5)
Regarding water at 373 K, 1 bar, this ratio is smaller than 0.1
for
C > 7. 7 mm/ s 1 /Z (A4.6)
Since we consider high growth rates (dynamic departure region),
(A4.6) will generally be satisfied (compare fig. 4.4).
For low growth rates (associated with static departure), Fv must
be compared to F , the order of which may be estimated by its a,r value for cavity bubbles at departure:
F ~ 2nrcr (A4.7) a,r
Therefore:
(A4.8)
For water at 373 K, 1 bar, this ratio is smaller than 0.1 for
(A4.9)
For cavity sizes in the range 10 ~m-100 pm, (A4.9) is satisfied at
the low growth rates under consideration.
Resuming, we may conclude that for water, Fv generally may be
neglected.
2. As mentioned by Chesters [v. Stralen/Cole, 1979, p. 894], surface
tension gradients, resulting from temperature differences along the
bubble surface, result in a net viscous force on the bubble (di
rected downward). This force F60 may be estimated by
F60 ~ R!Ja (Chesters). Comparing this force to FB we
find for water at 373 K and a !Ja corresponding to 1 K, that
FA /FB < 0.02 for R 0.5 mm, indicating that, for our purpose, ua ~ ~
F!Ja safely may be neglected.
103
104
APPENDIX 5.
Physical properties.
In Table A5.1 various physical properties are given for water at
saturation. In the last column the
Plesset/Zwick (1954) relation C/6 values of C/6 , according to
(~)l/2(p koc )1/2/p ~are the
0 'If l 1 l 2 given.
By graphical means one can easily verify eqs. (3.20) and (4.35).
T p P, p~ I c, ... k ., a, " "C bal k&Jm"l kJjkgK rr.w/t<,., mm 2/s m'J• IO-lN/m
0.01 G.0061 D 1999.8 0.~850 4.217 1.864 S69 1.7~ O,llS 75,60 10 0,012211 999,7 0,009397 4.193 I.HbH 587 1.30 0.140 74.24 20 0.023368 998.3 0.01129 U82 1.874 603 1,00 0,144 72,78 30 o.04Z417I 995.7 0.03037 4.179 1.883 618 0.800 0.148 71.23 40 0,073149 I 992.3 0,05116 4.179 1.894 632 0,656 0,153 69,61 ~Q 0,12B4 988.0 0,08300 4.181 1.907 643 O.~SI 0.156 67,93 (,!,) 0.19919 983.2 0.1302 4.185 1,924 !>54 0,471 0,159 66.19 70 0,31161 977.7 0,1981 4,190 1,944 662 0.409 0,162 64,4() 8G G,47359 971.6 0,293~ 4.197 1.969 670 (),361 0,164 62.57 90 I 0,70108 965.2 0.4233 4.205 1.999 676 0.322 (),166 60.69
100 1,0132 958.1 0.597~ 4.216 2,034 681 0,291 0.168 58,78 110 1.4326 950.1 0.8260 4.229 2.G75 1684 0,265 0.110 56.83 120 1,9854 942.9 1.121 4.245 ~~124 i 687 0,244 o.1n 54,85 130 2.7012· 934,6 1.496 4,263 uso 1688 0.226 0,173 52.83 I4G 3.6136 925.8 1,966 4.285 2,245 I 688 0.211 0,174 5G,79 150 4,7597 916.8 2,S47 4,310 2,320 1687 0,197 0.174 48,70 160 1>,1804 907.3 3.259 4,339 2,406 1684 0,186 0,114 46.59 170 1,9202 897.3 4.122 4,31! 2,504 '681 0,117 0,174 44,44 180 10.003 886,9 5,160 4,408 2.615 1677 0,168 0.173 42.26 190 12.552 876.0 6.398 4,449 2~741 611 0.161 (),172 40.05 200 15,551 864.7 7.865 4,497 2,883 665 0,!55 0.171 37.81 210 19,080 852,8 9,596 4,551 3,043 657 0,149 0.169 35,53 220 23,201 840.3 11,63 4,6!4 3,222 648 G.l45 0,167 33.13 230 27,979 827.3 14,00 4,686 3.426 639 0,140 04164 30,90 240 33,48() 813.6 16,77 4,77() 3,6S6 628 0,136 0,161 28.56 250 39,776 799,2 19,99 4.869 3.918 618 0,134 (),159 26.19 260 46.940 783.9 23.74 4,986 4.221 6()3 0,131 G.l54 23.82 270 SS.051 767,8 28.11 5,126 4,574 590 0.129 O,ISO 21,44 28G 64,191 150.5 33,21 S.296 4,996 515 0,128 0,145 19.07 290 74,448 732.1 39,2() 5,501 5,501 558 0,121 0,139 16,71 300 85,917 712.2 46,25 5,773 6,144 541 0,127 0.133 14.39 310 98,697 690.6 54,64 6,120 6,962 523 0,125 0,125 12,11 320 l12,9G 666.9 64,75 6.586 8.053 508 0,124 0.116 9.89 330 128,65 640.5 77,1S 7,248 9,589 482 0,124 0,104 7,75 340 146,08 610.3 92,76 8.270 11,92 460 G.l24 0.091 5,71 350 165,37 514,5 1U,4 10,08 IS,9S 437 0.123 oms 3.19 36G 186,74 528,3 143,S ,14,99 26.79 399 I 0.124
O,GS2 2.03 310 :210.S3 448,3 201,7 IS);!l 112.9 )48 0,125 0,017 0.47 374,1$ 1221.20 3U,S 315,5 .. 238 0,143 (l 0
Table A5.1 Physical properties of water at saturation
(from V.D.I., Warmeatlas, 1974).
a""*' f.. mm kl/kg
2,777 2501.0 l.Hl 24 77,4 2,7!7 2453.9 2,701 2430,3 2.675 2406,5 2.648 2382.6 2.620 2358.4 2.592 2333,8 2.563 2308,8 2.533 :!283.4 2.501 2257,3 2.470 2230.5 2,437 2202.9 2.403 2174,4 2.368 2144,9 2.331 2114,2 2.292 2082.2 2.252 2048,8 2.211 2014,0 2,167 1977,4 2,121 1939,0 2.073 1898,7 2.02l 1856,2 1.968 1811,4 1,912 1764,0 1,851 I 713.7 1,788 166G.l 1.719 !603.0 1,646 154l.b 1,568 1475.2 1.484 1403.1 1,393 1324,1 1,294 1236,5 1.184 1138,1 1,061 1025,6 0,915 893.1 0,733 722.6 G.44l 44,0
0 0
C/e~ mm/sl/2~
249.6 131.7 73.09 42.46 2~. 70 16.11 10.44 b.961 4.772 3 .l4'J 2.404 I. 759 1.312 0.9947
o. 7657 0.5980
0.4726 0.3782 0.3060 0.2637 0.2061 0.1713
. 0.1435 0.1213 0.1031 0.08847 0.07613 0.0660~
0.0.5'171 0.05069 0.04492. 0.04016 0.03646 o.o3:no \).03130 0.03069 0.03350 0.6387 ...
APPENDIX 6.
Bubble mass.
The right hand side of eq. (2.15) can be written as:
41Tp dR + .!!_ 1TO R3 dv dt 3 2 dt
(A6.1)
If we suppose, for simplicity, that the bubble is a sphere, adhering
to the wall, then: v = dR/dt, dv/dt d 2R/dt 2 •
Using this and (2.1) we obtain with (A6.1):
(A6.2)
For high growth rates, (A6.2) must be compared with FD (eq. (2.13)):
d (m v))/FD = 2.3 p /p , independent of C. 2 2 1
Far below the critical point, we have:
(A6.3)
For low growth rates, FD « F , hence from (A6. 3) we have in that CJ,r case:
(ddt (m v))/F << 1 2 o,r (A6.4)
Resuming, we may conclude that the momentum change of the bubble mass
(A6.1) is negligible as compared to the dominant forces.
105
106
APPENDIX 7.
Derivation of the liquid inertia force FD for hemi-spherical bubble growth.
From (A1.7) we have for the inertial pressure of a spherical bubble ex
pansion in an unbounded, non-viscous liquid:
3 •z (p - p ) ep 1 (-2
R + RR) 2 o dyn
(A7. 1)
. 7
F,ig.A7.1
By the symmetry of the problem·we can use this result for the hemi
spherical expansion of a microlayer bubble. Then, the (downward) liquid
inertia force may be represented by:
(A7.2)
Assuming diffusional growth and s 1, (A7. 1), (A7·.2) and (2 .1) yield:
(A7.3)
When using the radius R (~ of the equivalent sphere with volume 2 3
3 ~R we have:
For that case, a combination of (A7.1) and (A7.2) yields:
(2.66)
This is the expression referred to in sects. (2.11) and (3.4); it differs
little from Witze's expression (2.13) for a spherical expansion!
APPENDIX 8.
Extension to gas bubbles.
The "static" departure formulae derived in the present thesis and
their validity ranges, may be applied to the case of gas bubbles
growing in a locally supersaturated liquid (for example: electrolyti
cally evolved gas bubbles).
Since the mass diffusion process governing bubble growth in this case
is a much slower process than heat diffusion, only the "static" results
apply.
For this case, C contains the mass diffusion parameters D (diffusion
coefficient) and ~c (the supersaturation of the dissolved gas).
v. Stralen/Cole (1979, p. 413) give an application of Scrivens (1959)
theory to the case of electrolytically evolved gas bubbles:
for ~c/p < rr /6 2
C ~ (2! D) 1 12 ~c/p ,for ~c/p > rr/6 1T 2 2
Sillen (1983) gives a detailed description of the behaviour of electro
lytically evolved gas bubbles.
107
108
APPENDIX 9.
Extension to binary systems.
For a binary system, i.e. a liquid with a dissolved second liquid
component, the departure theory of the present thesis applies, if
we take the change in o and other physical properties into account.
Also the relation (4.3.3) for the growth constant C has to be trans
formed to include the effect of a combined heat and mass diffusion
to the bubble.
When considering the situation in which the second liquid component
is more volatile than the pure liquid a result of the combined heat
and mass transport is the reduction of the effective superheat,
governing the bubble growth. An extensive description of these pheno
mena has been presented by v. Stralen [v. Stralen/Cole, 1979, p. 222].
It is shown by v. Stralen that, for binary mixtures with a more
volatile second component, C may be derived from its value for a pure
liquid by multiplication by a factor
Here 6T denotes the reduction of 6 by the combined heat-mass diffusion. 0
G is called the vaporized mass diffusion fraction.
APPENDIX 10.
walls.
As mentioned in sect. 1.1.9 a fundamental description of the asymmetric
shape and the departure mechanism has until now shown to be too com
plicated. Nevertheless, for cavity bubbles on a vertical wall, an upper
limit of the static departure volume V can be derived. 1
The force equation which has to be satisfied at any moment of attach
ment is:
where FB p gV (upward buoyancy force) and F is the downward com-1 a
ponent of the adhesion force. During bubble growth, FB increases. As
long as the bubble is attached to the cavity, Fa equals FB. Hence
increases during bubble growth.
However, there is a maximum value of F0
• This maximum is reached in
the hypothetical case in which the tangent surface at any point of the
line of contact (at the bubble base) is parallel to the wall (see
A10.1). If we also assume the bubble foot to be circular, Fa can
be expressed by:
'lf
2o J r sinCLdo. 4ro, CL=O
where CL is the circular parameter of the circle of contact.
Now we easily obtain:
4r a 2
cap
This volume is a factor 4/2n smaller than the departure volume of a
cavity bubble on a horizontal wall (eq. (2.36b), sect. 3.1.1).
109
Fig. A10.1 Sl<.ei:.dt o6 a c.avliy bubb.te. at a veJLti..c.a.t wail..
110
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Chesters, A.K., "An analytical solution for the and volume
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Chesters, A.K., ·~odes of bubble growth in the slow-formation regime
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Cole, R., Shulman, H.L., "Bubble departure diameters at subatmospheric
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Cooper, M.G., Judd, A.M., R.A., "Shape and departure of
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Darby, R., "The dynamics of vapour bubbles in nucleate boiling", Chem.
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Donald, M.B., Haslan, F., "The mechanism of the transition from
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(1958).
Dussan, E.B., "On the spreading of liquids on solid surfaces; static
and dynamic contact lines", Ann. Rev. Fluid Mech., Vol. 11,
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Ill
112
Fritz, W., "Berechnung des Maximalvolumens von Dampfblasen", Phys.
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Hartland, S., Hartley, R.W., "Axisymmetric fluid-liquid interfaces",
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Hsu, Y.Y., Graham, R.W., "Transport processes in boiling and two
phase systems", Hemisphere Publ. Corp. {1976).
Ivey, H.J., "Relationships between bubble frequency, departure diameter
and rise velocity in nucleate boiling", Int. J. Heat Mass Transfer,
Vol. 10, pp. 1023-1040 (1967).
Jakob, M., Linke, W., "Heat transfer from a horizontal plate", Forch.
Gebiete Ingenieur W., Vol. 4, no. 2, pp. 75-81 (1933).
Johnson, M.A., de la Peiia, J., Mesler, R.B., "Bubble shapes in
nucleate boiling", A. I. Ch. E. Journal, Vol. 12, no. 2, pp. 344-
348 (1966).
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113
114
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In this thesis formulae are given for the departure radius R and the 1
departure time t of vapor and gas bubbles growing on a horizontal l
wall. The theoretical predictions are compared with experimental re-
sults, obtained by the author in the pressure range of 1 bar to
40 bar and by other investigators at subatmospheric pressures.
Starting point is a consideration of the forces acting on a growing
bubble. A criterion, deduced from the theory describing the bubble
shape, determines the departure conditions. Using this criterion and
assuming diffusion controlled growth (i.e. the bubble radius proportional
to the square root of time) expressions for R and t have been derived 1 l
from a force equation (Chs. 2, 3).
The departure mechanism appears to be fundamentally different for slow
growth with "static" departure (for which the liquid inertia force FD
is negligible as compared to the surface tension force F0
) and for
rapid growth with "dynamic" departure (for which dominates F). (J
Dependent on contact angle, surface roughness and growth rate, two
"statically" departing bubble types must be considerd, behaving quite
differently: (1) the (common) type of "cavity bubbles" (for which the
bubble foot is attached to the cavity which originated the bubble),
(2) the "spreading bubble" type (for which the bubble foot spreads out
freely on the surface, having a constant contact angle). The well-known
Fritz-formula (1935) for R , derived from a force equation in Ch. 2, l
only applies to the "spreading bubble" type.
At very high growth rates (for instance for water boiling at subat
mospheric pressures) "microlayer bubbles" may occur, i.e. bubbles
with an approximately hemi-spherical shape growing on a thin liquid
boundary layer ("microlayer"). Critical values of the growth constant,
C = R/t 1l 2, have been derived for the occurrence of dynamic deformation
of the bubble (due to liquid inertia) and, as an extreme case, the
occurrence of "microlayer bubbles" (Ch. 4). It is shown that a clear
distinction must be made between dynamic departure, dynamic deformation
and microlayer formation (cf. sect. 4.5).
115
116
Since for the high pressure range (with "static" departure) and single
cavities no experimental departure data are available, while the de
parture mechanism of cavity bubbles is of great practical interest,
a series of water boiling experiments has been carried out, using ar
tificial cavities of known geometry and size,in a pressure range from
1 bar to 40 bar. When using single cavities, the effect of pressure
on nucleation is eliminated, contrary to earlier experiments
(Semeria, 1962, 1963). By means of high speed cinematography the
pressure dependence of R and t has been observed. The pressure de-l l
pendences of both appeared to be in good agreement with the theoretical
predictions derived in Ch. 3: only a small reduction of R with increas-1
ing pressure was observed, while t increased considerably (contrary l
to Semeria's (1962, 1963) observations).
Two artificial cavities, differing a factor 2 in mouth-radius were
simultaneously observed. The observed R -values differ by a factor 1.35 l
on the average,as compared to 21 / 3 <~ 1.26) theoretically (Ch. 3).
At given pressures no influence of the growth rate (hence of the super
heat) on R was observed at elevated pressures. l
Combination of our high pressure results with low pressure results from
literature confirmed the existence of the theoretically predicted mini
mum value of t in the atmospheric pressure range. 1
Additional experiments under similar conditions, using an electrical
wall heating, showed that the wall heating did not affect the departure
radius R , while the p-dependences of R and t showed equal trends as 1 1 1
observed in the experiments without a waH heating.
The "relaxation microlayer" theory [v. Stralen/Cole, 1979] does not
predict the dependence of R on the cavity radius. Further it pre-1
diets R 1
to be proportional to the growth rate C for boiling at a
given pressure. Therefore, it is concluded that this theory does
not properly describe static bubble departure in the pressure range
of I bar to 40 bar.
SAMENVATTING =====·=========
In dit proefschrift worden uitdrukkingen afgeleid voor de loslaat
straal R en de loslaattijd t van damp- en gasbellen die groeien 1 1
op een horizontale wand. De theoretische voorspellingen worden ver-
geleken met experimentele resultaten van de auteur (voor het druk
gebied van 1 bar tot 40 bar) en van andere onderzoekers (voor sub
atmosferische drukken).
Uitgangspunt is een beschouwing van de krachten die op een groeiende
bel werken. Een criterium, afgeleid van de theorie voor de belvorm,
bepaalt de loslaatcondities. Door gebruik te maken van dit criterium
en de aanname van diffusiegroei (d.w.z. de belstraal met
de wortel uit de tijd) zijn uit een krachtenvergelijking uitdruk
kingen voor R en t afgeleid (Hoofdst. 2, 3). 1 1
Ret loslaatmechanisme blijkt fundamenteel verschillend te zijn voor
langzame met "statisch" loslaten (waarvoor de vloeistoftraag-
heidskracht F0 verwaarloosbaar is ten opzichte van de oppervlakte
spanningskracht F0
) en voor snelle groei met "dynamisch" loslaten
(waarvoor FD veel groter is dan F0).
Afhankelijk van de contacthoek, de oppervlakte-ruwheid en de groei
snelbeid moeten we twee "statisch" loslatende beltypes onderscbeiden
die zich verschillend gedragen: ( 1) de (veel voorkomende) "caviteits
bellen" (waarvan de belvoet vastzit aan de kern waaruit de bel is
ontstaan), (2) de "spreidende bellen" (waarvan de belvoet zich vrij
uitspreidt over bet oppervlak, waarbij de contacthoek constant
blijft). De bekende Fritz-formule (1935) voor R is aileen van toe-1
passing op "spreidende bellen". Deze formule wordt uit een kracbten-
vergelijking afgeleid in Hoofdstuk 2.
Bij zeer boge groeisnelbeden (bijvoorbeeld optredend voor kokend
water bij subatmosferische drukken) kunnen "microlaag-bellen" ge
vormd worden. Dit zijn bij,benadering, halfbolvormige bellen die
op een dunne vloeistof-grenslaag ("microlaag"). Kritieke
waarden voor de groeiconstante, C = R/t 1 12, zijn afgeleid voor
bet optreden van dynamische vervorming van de bel (door vloeistof
traagheid) en, als extreem geval, het optreden van "microlaag-bellen"
(Hoofdstuk 4).
117
118
Er blijkt een duidelijk onderscheid te moeten worden gemaakt tussen
dynamisch loslaten, dynamische vervorming en microlaag-vorming
(zie sect. 4.5).
Omdat er voor hoge drukken (met "statisch" loslaten) en afzonderlijke
kernen geen experimentele loslaatgegevens beschikbaar zijn, terwijl
bet loslaatmechanisme van caviteits-bellen van groot praktisch belang
is, is een· serie experimenten uitgevoerd met kokend water en kunst
matige kernen van bekende afmetingen en vorm. De druk werd gevarieerd
van 1 bar tot 40 bar. Door afzonderlijke kernen te gebruiken werd de
invloed van de druk op kernvorming geelimineerd, in tegenstelling tot
eerdere experimenten door Semeria (1962, 1963). Met een "high-speed"
filmcamera is de drukafhankelijkheid van R en t geobserveerd. De l l
drukafhankelijkheid van beide bleek goed overeen te stemmen met de
theoretische voorspellingen, afgeleid in Hoofdstuk 3: slechts een
geringe afname van R bij toenemende druk werd waargenomen, terwijl l
t aanzienlijk toenam (in tegenstelling tot Semeria's (1962, 1963) 1
waarnemingen) •
Twee kunstmatige kernen die een factor 2 verschilden in openingsdia
meter, werden gelijktijdig geobserveerd. De waargenomen R -waarden 1
verschillen gemiddeld een factor 1.35, vergeleken met
2 1 / 3 (~ 1.26) theoretisch (Hoofdstuk 3).
Bij gegeven druk werd geen invloed van de groeisnelheid (dus de
oververhitting) op R waargenomen bij verhoogde drukken. 1
Door onze hoge druk-resultaten te combineren met lage druk-resultaten
uit de literatuur werd het bestaan van de theoretisch voorspelde
minimumwaarde van t (in het atmosferische drukgebied) bevestigd. l
Toegevoegde experimenten, uitgevoerd onder dezelfde omstandigheden,
maar nu met een elektrisch verhitte wand, toonden aan dat de wand
verhitting geen invloed had op de loslaatstraal R , terwijl de druk-1
afhankelijkheid van R en t dezelfde "trends" vertoonden als werd l l
waargenomen in de experimenten zonder wandverhitting.
De "relaxatie microlaag"-theorie [v. Stralen/Cole, 1979] doet geen
voorspelling over de invloed van de kernstraal op R • Verder voor-1
spelt deze theorie een evenredig verband tussen R en de groei-1
constante C, bij gegeven druk. Daarom kan worden geconcludeerd dat
deze theorie geen juiste beschrijving geeft van het statisch los
laatmechanisme van bellen in het drukgebied van I bar tot 40 bar.
119
120
LEVENSLOOP. =============
P.C. Slooten werd geboren in 1949 te Zaandam. Doorliep de Rijks-HBS
te Purmerend en deed bet HBS-B examen in 1966. Studeerde bedrijfskunde
en later natuurkunde aan de TH-Eindhoven.
Verrichtte zijn afstudeeronderzoek in de vakgroep Theoretische Natuur
kunde onder leiding van dr.ir. H.J. van Ouwerkerk en prof.dr. L.J.F. Broe
Behaalde bet ingenieursdiploma in maart 1979.
Bleef eerst in genoemde vakgroep werkzaam, werd later als wetenschappelij:
assistent benoemd in de werkeenheid Kookverschijnselen en Tweefasenstromi1
van de vakgroep Transportfysica (afdeling Technische Natuurkunde} van de
TH-Eindhoven.
In het kader van deze' laatste aanstelling werd bet theoretische en later
ook experimentele onderzoek van dit proefschrift verricht.
Veel dank ben ik verschuldigd aan D.A. de Vries, die zich op diepgaande
wijze heeft ingezet voor de totstandkoming van dit proefschrift. Ook
dank ik S.J.D. van Stralen voor de discussies die ik met hem had en de
adviezen die daaruit voortvloeiden.
Ik ben G. Vossers en A.K. Chesters erkentelijk voor hun steun op be
langrijke momenten. Voor het experimentele onderzoek ben ik veel dank
verschuldigd aan C.A. Copray. Kees, het welslagen van de experimenten
is voor het belangrijkste deel jouw werk. De discussies met jou over
de opzet van de experimenten waren van beslissend belang.
Ik dank ook d~ leden van de vakgroep Transportfysica voor hun onmisken
baar positieve installing en de prima sfeer.
Pim Sluijter, Jan Niessen en Bernadette Bongenaar dank ik voor hun hulp
en discussie.
Het maken van een kern, actief bij Jage oververhittingen was niet een
voudig. Jan van Asten, Henk Heller en }mrius Bogers dank ik daarom
voor hun creativiteit en flexibele planning.
Ruth Gruyters ben ik erkentelijk voor de prachtige bellen-plaatjes.
Je remercie ~rjon Dahlmans qui a soigne la dactylographie, pour sa
perfection et !'engagement avec laquelle elle a acheve ce travail.
Stellingen
behorende bij het proefschrift
van P.C. Slooten
22 juni 1984
I
De vaak gemaakte veronderstelling dat de formule van Fritz voor de los
laatstraal van een damp- of gasbel kan worden afgeleid uit een balans
van opwaartse kracht en adhesiekracht is niet juist.
Hsu, Y.Y., Graham, R.W., Transport processes in boiling and two-phase
systems, Hemisphere Publ. Corp. (1976).
Cole, R., Shulman, H.L., Int. J. Heat Mass Transfer, Vol. 9, pp. 1377-
1390 ( 1966) .
Fritz, W., Phys. Zeitschr., Vol. II, pp. 379-384 (1935).
Dit proefschrift, hoofdstuk 2.
II
Het idee dat een bel van een horizontale wand loslaat wanneer de resul
tante van de omhooggerichte krachten op de bel ongeveer gelijk en tegen
steld gericht is aan de resultante van de omlaaggerichte kra,.hten, zoals
dit onder anderen gebruikt wordt door Beer (1969) en Keshock/Siegel (1964),
is, voor drukken ver beneden de kritieke druk, onjuist.
Beer, H., Progress in Heat and Mass Transfer, Pergamon Press, Vol. 2,
p. 364 (!969).
Keshock, E.G., Siegel, R., NASA Report TND-2299 (1964).
Dit proefschrift, hoofdstuk 2.
a 1
a cap
A
A 0
B
B 0
c 1
/1c
c
D
e -z
f
F
F o,r
F corr
Quantity
liquid thermal diffusivity [m2 /s]
capillary length, defined: a l(cr/p g). For water cap 1
at 1 bar, 100° C: a = 2.48 J0- 3 m [m] cap
coefficient, defined by eq. (2. 23) [-]
surface area of a bubble [m2]
coefficient, defined by eq. (2.24) [-]
area of the bubble base [m2 ]
specific heat of the liquid [J/kgK]
supersaturation of dissolved gas [kg/m3]
growth constant, cf. eq. (2.1) [m/s 1 12 ]
di~nsionless growth constant, defined: C
drag coefficient (A4) [-]
microlayer thickness [m]
discriminant of cubic equation, cf. sect. 2.9 [-];
diffusion coefficient of dissolved gas [m2 /s]
unit-vector in z-direction, cf. fig. 2.2 [-]
coefficient, cf. sect. 2.8.3 [-]; bubble frequency,
inCh. 1 [s- 1 ]
total force, applied on the bubble, cf. eq. (2.4) [N]
Archimedes- or buoyancy force: FB = pgV [N]
surface tension force (adhesion force) [N]
resultant surface tension force, cf. eq. (2.14) [N]
correction force, cf. eq. (2.8) [N]
dynamic force or liquid inertia force, cf. eq. (2.5) [N]
p 1
(J
<Po
<P * ( t)
liquid density [kg/m3 ]
vapor density [kg/m3 ]
surface tension [N/m]
contact angle of a spreading bubble [-]
static value of tjl {-]
angle of contact of the equivalent spherical segment,
cf. fig. 2.1 [-]
Subscripts; superscripts.
( ... ) 1
( ... ) 1
( ... ) 2
( ... ) max
( ... )min
( ... )*
"at departure", when used in combination with R, R*,
-* * R , rB' V, t, B, A, tjl , f, h, j , ...
"of the liquid phase", when used in combination with T,
p, p, k, c , a
"of the gas phase", when used in combination with T, p,
p, m
"maximal"
"minimal"
"belonging to the description with the equivalent spherical
segment"
"dimensionless"
"belonging to the inflexion point I of the profile"