Departure of vapor- and gas-bubbles in a wide pressure range

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 Hogeschool

Eindhoven. https://doi.org/10.6100/IR33763

DOI:

10.6100/IR33763

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Published: 01/01/1984

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DEPARTURE OF VAPOR- AND GAS-BUBBLES

IN A WIDE PRESSURE RANGE

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:

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. 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

REFERENCES I l l

SUMMARY 115

SAMENVATTING 117

LEVENS LOOP 120

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/s1

1

2 ]

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: F

8 = pgV [N] surface tension force (adhesion force) [N]

resultant surface tension force, cf. eq. (2. 14) [N] correction force, cf. eq. (2.8) [N]

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 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/s2l 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

p2 (p 2 -po\n (p 2 -p o) dyn p (z) 1_hydr 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•2735jK1•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) [-]

R

R*

R*

1 R' R _top R _top 1

R

_top 1 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) 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

₁

2_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

0_in 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•05m]

velocity of the bubble centre of mass [m/s]' liquid velocity (AI) [m/s]

bubble volume [m3 ] ib., at departure [m3 ] V /a3 [ - ] 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 /a2 [-] 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

<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"

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.

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

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.

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).

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].

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).

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 • (R_{1 /c)2} ~ p2 (water: 2 bar< p.< 40 bar).

Semeria (1962, 1963), however, observed a sharp decrease of R_{1 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).

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.

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.

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

p (z) can be written as: l p (z) l where: - p (z) 1_hydr - pext p (z) + pext + p 1_hydr 1_dyn

(the hydrostatic pressure)

p (H) + p g(H-z) 1_hydr 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 1_dyn

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

p 2 ~

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•

FcJJ

p (z)e.ndA+Jf pext~z·E_dA+Jf p .ndA+ 1_{hydr ~ -} 1_dyn 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) 1_hydr .n dA +

JJ

p ex t ~-e .n dA-A +B 0 0

- JJ

P (z) e .n dA-

JJ

1_{hydr z} -B 0 B 0

JJ

p _{2 - z -}e .n dA - F _cr B 0 (2.5) (2.6)

The first term of (2.6) represents the upward directed Archimedes-or buoyancy fArchimedes-orce FB

=

p

1gV.

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)

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 _op ) when Rt _op ~

the "capillary length", defined by a cap Finally, (2.8) becomes (see fig. 2.2):

with: FB F corr p (2o/R - p 1gH)'ITrB 2 top 2'ITrBosin<jl 0 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'ITorB 2_/R

_I

(directed downwards) o top (2. 14)

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 _corr , which is incorrect since 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, i f 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)

where: def B

def

!

(1+cos~*) and A Now we assume (for ~ ₀ < _~ 45°)

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).

top 1 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 ])

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}

Hence (assuming again: p >> p ):

1 2

p - {p (z1_{) + 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 _top' · ~ = x/R _top and S = p _{1 top} gR2

/cr.

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.

The angle of inclination a (see fig. 2.3) of the profile at the (sharp) edge of the cavity changes with Rt _op (see fig.

2.5).

b. For spreading bubbles that subset of profiles is selected for

which a "cut off" with a

=

cp

=

const. is possible. The

corres-o

ponding bubble foot radius changes with R _top (see fig. 2.6).

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

1

gR_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 _top is passed through, belonging to the above 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 _{. op} is of the order of 1 mm at departure.

Maximal volume:

It appears (see fig. 2.5) that there exists a maximal volume V max

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 < v 1 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

-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°.

- I f the subscript(,,,) indicates the value at departure, then, 1 for S < 0.1 l -r = Rt

{~

3

S + 0(132)} 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 _top

..

lj.a 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

1

2) (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:

(2.40)

- Before departure, rB diminishes from

= 3 9 212.a sin 2_{¢ {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 z1_-coordinate

1

of the inflexion point I, is:

H 1 z I I 2Rtop {1 +

~

sin 2 ¢ 0

~n(16/sin

2

_¢

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

top₁ 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)

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. _corr

=

1rrB crsin<P ,

1 o l we obtain from (2.16): p gV + 1rrB crsin<P - 21rrB crsin<P 0

=

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 _top 1 1

1 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 <jl2•

()

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 (!)

(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 _D

=

0 _. 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 ga3 _{we have with}

1 cap R.*

=

R*/a cap fR.*3 - hR.*

=

0 (2.16£) where:

- 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:

-*

_{R =}

_1-f

h

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

sin

2

~*

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

= a

1 ~* 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:

-*

_{R = >'-·~} J3 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.)

As mentioned in sect. 2.8.2, this expression is accurate for ~ ₀ < _~ ~/2(!)

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 _op can

n0

longer be used to 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 ga3 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.