Bubble growth dynamics in nucleate pool boiling with liquid subcooling effects

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Bubble Growth Dynamics in Nucleate

Pool Boiling with Liquid Subcooling

Effects

Muhad Rozi M at Nawi

Department of Mechanical and M anufacturing Engineering

Parsons Building

Trinity College

Dublin 2

Ireland

November 2014

I declare th a t this thesis has not been subm itted as an exercise fo r a degree at this or

any other university and it is entirely my own work.

I agree to deposit this thesis in the University's open access institutional repository or

allow the library to do so on my behalf, subject to Irish Copyright Legislation and

Trinity College Library conditions o f use and acknowledgement.

Muhad Rozi M at Nawi

November 2014

TRJNITY COLLEGE

ABSTRACT

Heat tra n s fe r in nucleate pool boiling has been characterized by very high

dissipated heat fluxes w h ils t requiring lo w drivin g te m p e ra tu re differences. The rate

o f bubble g ro w th and th e subsequent bubble m o tio n has a tre m e nd o us influ en ce on

th e heat tra n sfe r. In o rd e r to gain a deeper understanding o f th e m echanism s

responsible fo r this, basic know ledge o f bubble g ro w th dynam ics is required. To th is

end, single isolated bubble g ro w th dynam ics fro m an a rtificia l nucleation site in pool

boiling has been investigated e xp e rim e n ta lly in this study. An experim ental fa c ility has

been developed to p e rfo rm th e study. The experim ents have been conducted at

atm ospheric pressure w ith an e n viro n m e n ta l frie n d ly re frig e ra n t HFE-7000 as th e

w o rkin g flu id . A high speed video cam era w ith a co m b ina tio n o f p ow erfu l lens and a

tu b e extension has been used to capture th e bubble images during boiling. Image

processing in M a tla b has been used to process th e images and d ete rm ine relevant

param eters w hich characterize g ro w th and d eparture.

ACKNOWLEDGEMENTS

I would like to express my sincere thanks to my supervisor, Professor Anthony

Robinson fo r his trem endous guidance, patience, enthusiasm, understanding and

encouragement throughout my doctorate study. Also, thanks to Dr. Samuel Siedel fo r

his guidance, help and advice during this research work.

Apart from them , I would also like to express my appreciation to Gerry Byrne,

Mick Reilly, Gabriel Nicholson, Sean Doonan, Paul Normoyle and others in the

workshop fo r th e ir trem endous work, help, guidance and ideas. Thank you also to my

colleagues, David, Seamus, Gerrard and Rayhann fo r sharing ideas and support

thro ugho ut my tim e in college.

Thank you fo r the continuous support especially from my parent M at Nawi and

Mek Som, siblings and friends. To my lovely wife, Nor Hafeezah and my sons, Harith

and Hafeez, thank you fo r your continuing support and I am very grateful of your

presence in my life.

Special thanks to Hj Fauzi and Hjh Hamdah and families fo r th e ir great support

during my doctoral study.

I thank the Universiti Teknologi Mara (UiTM) fo r funding my doctoral studies

under the Young Lecturer Scheme. Last but not least, my sincere thanks, compliments

and regards to anyone who had helped and supported me in one way or another.

DEC LARATIO N... I

ABSTRACT... II

AC KN O W LED G E M E N TS...IV

TABLE OF C O N T E N T ... V

LIST OF FIGURES...IX

LIST OF T A B LE S ... XIV

N O M E N C LA T U R E ...XV

Latin Letters... XV

Greek Letters... XVII

Subscriptsan d Superscripts... XIX

Dimensionless Num bers...XX

CHAPTER 1 ...1

INTRO DUCTIO N... 1

1.1 B a c k g ro u n d...2

1 .2 M o t iv a tio n... 4

1 .3 O b je c tiv e s...5

1.4 Thesis O u tlin e s... 5

CHAPTER 2 ... 7

LITERATURE REVIEW ... 7

2.1.2 Bubble Departure...17

2.1.2.1 Bubble Departure Frequency... 26

2.1.3 Bubble Waiting T im e... 29

2.2 E x p e rim e n ta l Single B ub ble D y n a m ic s...31

2.3 S u b co o lin g Effects on B ub ble D y n a m ic s...32

2.3.1 Analytical Study...32

2.3.2 Experimental Investigations... 39

2.4 S u m m a ry...43

CH APTER S...45

EXPERIMENTAL DESCRIPTION AND DATA ANALYSIS... 45

3.1 E x p e rim e n ta l S e t-u p...46

3.1.1 Description of the Pool Boiling Facility... 46

3.1.2 Pool Boiler...47

3.1.3 Heating Element and Artificial Nucleation S ite ... 48

3.1.4 Measurement and Control E quipm ent... 50

3.2 E x p e rim e n ta l P rocedures...53

3.3 M e a s u re m e n t T e ch n iq u e s... 54

3.3.1 Heat Flux and Boiling Surface Temperature M easurem ent... 54

3.3.2 Image Processing... 56

3.3.3 Experimental Accuracy and U ncertainty...59

3.4 O p e ra tin g C o nd ition s...60

3 .5 D a ta A n a ly s is...61

3.5.4 Bubble growth curve...62

3.5.5 Non-dimensional parameter of shape and oscillation... 63

3.5.6 Forces acting on a grow/ing bubble... 63

3.5.6.1 Momentum variation... 65

3.5.6.2 Liquid inertia and added mass force... 66

3.5.6.3 Buoyancy force... 67

3.5.6.4 Triple line surface tension and adhesion forces... 68

3.5.7 Bubble Curvature... 69

CHAPTER 4 ... 71

RESULTS AND DISCUSSION...71

4.1 In tro d u c tio n...72

4.2 H eat Flux a t the H eated Surface... 75

4.3 W a itin g and G row th T im e s...77

4.3.1 Waiting tim e... 77

4.3.2 Grow/th tim e ...80

4.4 Bubble a t D e p a rtu re...82

4.5 V olum etric G ro w th...91

4.6 Energy Transfer a t Liquid-V apour In te rfa c e...95

4 .7 Bubble Shape and O scilla tio n s...98

4.7.1 Aspect ra tio ...98

4.7.3 Non dimensional description o f shape and oscillations... 105

4.7.4 Bubble tip and c u rv a tu re ... 107

4 .8 Contact Angle and Forces Analysis...1 09 4.8.1 Contact angle developm ent... 109

4.8.2 Forces acting on a growing b u b b le ...113

4 .9 Effects o f Liquid Subcooling...1 1 7 4.9.1 Isolated bubble: Inception, evolution and frequency of d e pa rtu re...118

4.9.2 Volum etric grovi/th and interface heat tra n s fe r... 124

4.9.3 Contact angle developm ent... 129

4.9.4 Forces acting on a growing b u b b le ...131

CHAPTERS...133

CO NCLUSIO N...133

Figure 1.1: Conceptual picture of pool boiling... 3

Figure 1.2: Typical boiling curve fo r w ater at 1 atm pressure [Faghri and Zhang (2006)]

... 4

Figure 2.1: Geometry of the growing bubble (Kiper, 1971)... 22

Figure 2.2: Relationship between dimensionless growth tim e and departure diam eter

(Yangetal., 2000)...26

Figure 2.3: Sketch fo r the growing bubble (Yang et al., 2000)... 26

Figure 2.4: One cycle of an individual bubble (Zhao and Tsuruta, 2 0 0 2 )...33

Figure 2.5: Dynamic microlayer (Zhao and Tsuruta, 2002)...34

Figure 2.6: Variation o f waiting and growing times w ith subcooling fo r bubbles form ing

at Site E [Ibrahim and Judd (1985)]...41

Figure 2.7: Superposition of the waiting tim e measurements fo r bubbles form ing at

site E fo r all levels o f heat flux investigated by Ibrahim and Judd (1 9 8 5 )... 41

Figure 3.1: Pool boiling fa c ility ... 47

Figure 3.2: Pool b o ile r... 48

Figure 3.3: Schematic of half slice of heating element and artificial nucleation site

(Unit: m m )... 49

Figure 3.4: Video camera s e tu p ... 52

Figure 3.5: Copper pipe coils fo r liquid subcooling...53

Figure 3.8: Sequence o f single bubble image processing fo r saturated b o ilin g ... 58

Figure 3.9: Sequence o f single bubble image processing fo r subcooled b o ilin g ...58

Figure 3.10: Estim ation o f te m p e ra tu re drop across th e layer o f A ra ld ite the rm al

cond u ctive adhesive... 60

Figure 3.11: Schematic o f c o n tro l v o lu m e ... 64

Figure 3.12: Schematic o f volum es, surfaces and lines in v o lv e d ... 64

Figure 3.13: Two principle radii at th re e selected points in a 3-dim ensional

re co nstru ctio n bubble[D i Bari and Robinson (2 0 1 3 )]... 70

Figure 4.1: Bubble g ro w th

a t A T w —

9.1 K w ith A t = 5 ms betw een the images...75

Figure 4.2: Heat flu x at various w all su pe rh e ats...76

Figure 4.3: N ucleate boiling versus natural convection correlated by Kobus and

W edekind (2001)at various w all s u p e rh e a ts ... 77

Figure 4.4: D e te rm in a tio n o f bubble w a itin g tim e f ^ f o r w all superheat,

A T w = 2.2

K,

w ith A t = 1 m s betw een th e im a g e s ... 78

Figure 4.5: W a itin g tim es,

o f six successive bubbles fo r th e range o f w all superheats

te s te d ... 78

Figure 4.6: W a itin g tim e , ti^ fo r th e average o f six successive bubbles at various w all

s u p e rh e a ts ...80

Figure 4.7: G row th tim es,

tg

o f six successive b u b b le s ... 81

Figure 4.8: G row th tim e , tg fo r the average of six successive bubbles at various wall

su p e rh e a ts...82

Figure 4.9: Bubble d ep a rture frequency at various w all superheats fo r the average o f

Figure 4. 11: D e p a r t u r e v o l u m e f or t h e a v e r a g e of six s uc c es s i v e b u b b l e s a t v ar io us

Vi/all s u p e r h e a t s ...86

Figure 4 . 12: Bubbl e d e p a r t u r e d i a m e t e r ,

Db

o f six s uc c e s s i v e b u b b l e s ... 8 8

Figure 4 . 13: D e p a r t u r e d i a m e t e r s for t h e a v e r a g e of six s u c c e s s i ve b u b b l e s a t v ar i o u s

wall s u p e r h e a t s ...8 9

Figure 4 . 14: R el at i on s hi p of e q u i v a l e n t b u b b l e d e p a r t u r e d i a m e t e r a n d b u b b l e

d e p a r t u r e f r e q u e n c y ... 9 0

Figure 4. 15: B ubbl e d e p a r t u r e f r e q u e n c y - d i a m e t e r a t v a r io u s wall s u p e r h e a t s ... 91

Figure 4. 16: G r o w t h c u r v e s o f six s u c c e s s i ve b u b b l e s a t A T w = 1 1 . 8 K ... 92

Figur e 4. 17: A v e r a g e of six b u b b l e s g r o w t h c u r v e s a t v a r i o u s wall s u p e r h e a t s ...94

Figur e 4. 18: A v e ra g e o f n o n - d i m e n s i o n a l six b u b b l e s g r o w t h c u r v e s a t v a r io u s wall

s u p e r h e a t s ...9 5

Figure 4. 19: Ra t e of b u b b l e v o l u m e c h a n g e ...97

Figur e 4. 2 0 : Bubbl e g r o w t h a s p e c t r a ti os

{ h /w )

a t v a r i o u s wall s u p e r h e a t s ... 99

Figur e 4. 2 1 : Early s t a g e of b u b b l e g r o w t h a f t e r d e p a r t i n g last b u b b l e f o r A7'w=2.2 K

w i t h A t = 1 m s ... 99

Figur e 4. 2 2 : Early s t a g e of b u b b l e g r o w t h a f t e r d e p a r t i n g last b u b b l e f o r

A T ^ - 1 1 . 8

K

w i t h A t = 1 m s ...100

Figure 4. 2 3 : Vertical c o a l e s c e n c e of t w o s u cc e s s i v e b u b b l e s f o r t h e c a s e

o f A T ^= 11.8

K

101

Figure 4.25: Bubble height o f th e centre o f gravity histories at non dim ensional tim e

fo r d iffe re n t w all s u p e rh e a ts ... 103

Figure 4.26: V elocity o f the centre o f gra vity

o f bubble g ro w th fo r d iffe re n t

wall s u p e rh e a ts ...105

Figure 4.27: A cceleration o f th e c e n te r o f gra vity

o f bubble g ro w th fo r

d iffe re n t wall su p e rh e a ts...105

Figure 4.28: Evolution o f non-dim ensional p aram eter

As

fo r various w all superheat 106

Figure 4.29: First derivative o f non-dim ensional param ete r

A ;

fo r various w all

superheats... 107

Figure 4.30: E volution o f bubble shape fo r low and high w all superheats... 108

Figure 4.31: Evolution o f radius at bubble tip fo r low, m o d erate and high wall

superheats... 109

Figure 4.32: D e fin ition o f co nta ct angle,

a

... 110

Figure 4.33: Calculation o f contact angle,

a

at the liq u id -va po u r in te rfa c e ...110

Figure 4.34: Contact angle histories fo r w all superheat,

[SJ^-2.2

K ...112

Figure 4.35: V ideo sequence o f bubble g ro w th fo r w all superheat,

hT^=2.2

K ...112

Figure 4.36: C ontact angle (corrected) histories fo r low, m o d erate and high w all

superheats...113

Figure 4.37: Various forces acting on a grow ing bubble at w all superheat AIiv=2.2 K:

(top) uncorrected contact angle, (b o tto m ) corrected co nta ct a n g le ... 115

Figure 4.38: Forces acting on a g ro w in g bubble at w all superheat, A7w=6.1 K ...116

ATsub

= 10 K w ith

A t — 1

ms between the im ages... 120

Figure 4.41: Evolution of a bubble in subcooled boiling w ith

ATsub

= 8 K fo r heat

flux,

q " =

36 k W /m 2 w ith

A t = 20

ms between the im ages... 121

Figure 4.42: Evolution of bubble shape fo r

ATsub

= 3 K and

ATsub

= 8 K ... 122

Figure 4.43: Comparative evolution of bubble shape fo r subcooled boiling

(ATsub

=

8 K) and saturated boiling

(ATw

= 2.2 K )... 122

Figure 4.44: Bubble departure frequency fo r various subcooling levels...123

Figure 4.45: Bubble volum etric growths fo r various subcooling levels...126

Figure 4.46: Rate of dimensional bubble volume chan ge...128

Figure 4.47: Rate of non dimensional bubble volume change... 129

Figure 4.48: Comparison of contact angle (corrected) histories of subcooled boiling

and saturated b o ilin g ...130

Figure 4.49: Vertical forces acting on a growing bubble in subcooled boiling fo r low

subcooling

{ATsu b-3

K)...132

LIST OF TABLES

Table 1: Properties of HFE-7000 at atmospheric pressure [3M (2014)]...61

Latin Letters

Symbol

N a m e

Units

rh

M a s s flow r a t e

kg.s’^

Ar e a

m^

A p a r a m e t e r in Eq. (2.6)

-a

A p a r a m e t e r in Eq. (2.49)

-B

A p a r a m e t e r in Eq. (2.10)

-b

A p a r a m e t e r in Eq. (2.49)

-C

A p a r a m e t e r in Eq. (2.13)

-c

A p a r a m e t e r in Eq. (2.49)

-Cp

Specific h e a t c a p a ci t y

J . k g ' . K '

D

A p a r a m e t e r in Eq. (2.20)

-D ,d

D i a m e t e r

m

Db

B u b b l e ' s d i a m e t e r a t d e p a r t u r e

m

Do

Base d i a m e t e r

Dt

T a t e ' s d i a m e t e r

m

F

Force

N

f

Body f o r c e

N.m'^

fd

B ub bl e d e p a r t u r e f r e q u e n c y

s'^

g

Gra vi ty a c c e l e r a t i o n

m.s-^

G(Q)

G eom etry o f bubble neck

h

Heat tra n s fe r c o e fficie n t

W.m'^.K'^

h

Enthalpy

J-kg'^

hcg

Height o f th e c e n te r o f gra vity

m

hfg

Latent heat o f vaporization

kJ.kg'^

h

Height o f th e bubble

m

I

Robinson and Judd crite rio n

k

Therm al c o n d u c tiv ity

W .m '\K '^

K

Empirical constant in Eq. (2.54)

L

Length

m

Lc

Capillary length

m

Lo

Characteristics length scale in Eq. (2.57)

M

M o le cu la r w e ig h t

g.m ol'^

m

Mass

kg

n

N um ber o f m olecules

mol

n

Em pirical constant in Eq. (2.54)

P

Pressure

Pa

P

Power

W

Q

Heat

W

q "

Heat flu x

W .m '^

R

Radius

m

r

Length in the radial d ire ctio n

m

R

Therm al resistance

K.W'^

*

T

K

t

tw W a it in g t i m e ms

G r o w t h t i m e ms

t *

-u

V V o lu m e m^

V * D im ensio nle ss v o lu m e

-X

y

z

G reek Letters

a

a

P

P

-P

y

-6

u D yn a m ic viscosity k g .m '\s '^

u

K in em atic viscosity

m^.s'^

ijj

A p a r a m e te r in Eq. (2.63)

(2.57)]_______________-00

bulk

b

related to the bubble

B.buoy

buoyancy

base

relative to the base o f the bubble

C

capillary

c

cavity, condensation

cq

center of gravity

CL

contact line

CP

contact pressure

Ctin

tip curvature

Cu

copper

d

detachm ent or departure

duy

unsteady growth

D

drag

e

evaporation

eq

equivalent

f

fluid

q

growth

1

inertia

£

liquid phase

LI

liquid inertia

mac

macrolayer

mic

microlayer

mom

m om entum

r

radial direction

s a t

s a tu r a ti o n

s ub

s ubcooling

V

v a p o u r p h a se

v f

viscous forces

vs

viscous s tr es s

w

h e a t e d wall or surface

X

x-direction

V

y-direction

z

z-direction

Dimensionless Numbers

Bond n u m b e r ,

B o

=

Grashof n u m b e r ,

Gy =

I I I , ,

PlCpliToo T^at^

Jakob n u m b e r , y a — —

---P v ^ f g

Nusselt n u m b e r ,

N u

= —

ilC r t

Prandtl n u m b er ,

P r

=

—

-ki

CHAPTER 1

INTRODUCTION

CHAPTER 1: INTRODUCTION

1.1

Background

Boiling is a complex process in which mass, m om entum , and energy are

transferred w ith in and between a solid wall, a liquid phase and vapour phase. When

liquid is boiled, a liquid-vapour phase change process occurs such th a t in some

situations/w all superheats, vapour bubbles are form ed either on a heated surface or

in a superheated liquid layer adjacent to the heated surface. This process is called

nucleate boiling and is known to be a highly effective mode of heat transfer and one

o f the most studied physical phenomena in therm al fluid science and engineering.

The form ation of bubbles w ithin superheated liquids is generally observed to

occur over a range o f tem peratures w ithin a metastable range o f superheats. Bubble

nucleation com pletely w ithin a superheated liquid is called homogeneous nucleation.

Meanwhile, bubble nucleation th a t occurs at the interface between a metastable

phase and another (usually solid) phase th a t it contacts is called heterogeneous

nucleation.

Liquid

Vapor

Heater

Figure 1.1: Conceptual picture o f pool boiling

CHAPTER 1: INTRODUCTION

10

10^

10

10

Region 1

1

Region II

'

Maximum

1

(critical) heat |

Region IV

/

'

Nucleate

9 ^

flux

Q *

1

" X 1

Film

/

1

boiling /

1

1

/

\ Transition

boiling /

Natural

1

/

1

\boiling

'

\

1

convection

boiling

1

/

1

^

y / 1 Slugs and j

\

1

D|

/ Minimum

--- heat flux,<?'^jj,

L /

1

columns '

Isolated

|

I bubbles!

|

1 1

1

Region III

1

1

1

1

30

100

320

A r= 7 ’, , - 7 ’.„„CC)

1000

Figure 1.2: Typical boiling curve fo r water at 1 atm pressure [Faghri and Zhang (2006)]

1.2

M otivation

Since the early to mid 1900s, theoretical studies of boiling have been

conducted attem pting to understand the boiling process and define theories and

m athematical models to predict its performance. Even still, global theories and robust

empirical form ulations seem to be elusive. This is due to a lack o f a com plete

understanding of the fundam ental physics of bubble dynamics, flo w and heat transfer

at these small tim e and length scales. This is complicated by the fact th a t bubble

dynamics are sensitive to a vast array of interdependent parameters which makes

exhaustive experim entation and exact numerical sim ulations difficu lt to achieve.

Although this type of heat transfer process has been utilized over the ages,

knowledge about the phenomenon and its physical mechanisms o f heat transfer is still

restricted, mainly because the technology required to measure the tim e and length

scales of the phenomenon has only been developed recently. This being the case, the

boiling phenomena w ithin the scope of isolated bubble dynamics from an artificial

nucleation site.

1.3

Objectives

The present study entitled

"Bubble Growth Dynamics in Nucleate Pool Boiling

w ith Liquid Subcooling Effects",

has the follow ing objectives:

• To design and construct an experimental test facility to allow the

visualization o f bubble dynamics fo r pool boiling from a single artificial

nucleation site at atmospheric pressure w ith the option of subcooling.

• To measure the bubble size more precisely using sophisticated modern

technique.

• To gain a deeper knowledge and understanding about the bubble

dynamics phenomenon in nucleate pool boiling under saturated

conditions.

• Evaluate

validity of classical

theories w ith

simple

controlled

experiments.

• To provide some initial understanding of the behaviour of single

isolated bubble growth during subcooled pool boiling.

1.4

Thesis Outlines

This thesis is divided into five chapters. It is organized as follows:

CHAPTER 1: INTRODUCTION

Chapter 2

- Analytical studies of bubble growth, bubble departure and

bubble waiting tim e are presented. It follow s by reviewing some o f the

experimental studies of single bubble dynamics. The effects of

subcooling on bubble grow th dynamics in both analytical and

experimental earlier studies will be presented in the last part in this

chapter.

Chapter 3

- The experim ent designed and b u ilt fo r the present study is

presented,

it

is

follow ed

by

the

experimental

procedures,

measurement techniques, operating conditions and data analysis.

Chapter 4

- Results from the experimental work are presented and

discussed fo r saturated conditions and subcooled conditions.

CHAPTER 2

LITERATURE REVIEW

Boiling heat transfer has received much attention fo r

about a century because o f the high heat transfer coefficients

associated w ith this mode of heat transfer. Numerous studies

have been carried out by many researchers attem pting to predict

boiling heat transfer rates since the first boiling curves produced

by Nukiyama (1934). This chapter is aimed at providing some

relevant inform ation regarding the research carried out pertaining

to the bubble dynamics and the effects of subcooling in nucleate

pool boiling from different researchers across the globe.

CHAPTER 2: LITERATURE REVIEW

2.1

Analytical Bubble Dynamics

The first im portant work on bubble dynamics was perform ed by Rayleigh

(1917). He form ulated the equation of m otion fo r spherical bubble expansion as a

problem of the dynamics of an incompressible and inviscid fluid, which was later to be

known as the inertia controlled growth regime. Integration of the momentum

equation in the liquid phase, neglecting surface tension influences, results in.

The surface tension term was later added by Plesset and Zwick (1954) by relating the

vapor and liquid pressures at the bubble interface through the Young-Laplace

pressure drop.

Eq. (2.2) is known as the extended or m odified Rayleigh equation. It relates the

pressure difference which drives growth to the inertial forces exerted by the liquid on

the bubble and surface tension forces at the interface.

2,1.1 Bubble Growth

Bubble growth in infinite superheated liquids was analytically investigated in the

1950s to the 1970s. Generally, the work has been divided into the follow ing tw o main

regions,

(i)

Inertia controlled growth

(ii)

Diffusion controlled growth

(

2.

1)

The region of bubble growth controlled by inertia forces was famously related to

the work of Rayleigh (1917). This region is restricted to the initial stages of rapid

grow th in which the bubble expansion rate is prim arily lim ited by its ability to 'push'

the surrounding liquid. During this stage, the rate of heat transfer to the interface is

assumed sufficiently high such that growth is not constrained by the resultant vapour

generation into the bubble. For the solution of this growth stage, the vapour pressure

is nearly constant and assumed to be near its maximum value of

~ f ’sat(T’oo)- Fof

bubbles large enough that the surface tension term is negligible [Eq. (2.1)], the

interface velocity can be calculated as.

By substituting the Clausius-Clapeyron equation to relate the vapour tem perature to

the saturation tem perature, the solution fo r inertial controlled growth is obtained

[Plesset and Zwick (1954), Mikic et al. (1970)],

From Eq. (2.4), it clearly shows that inertial controlled grow th is characterized by a

linear relationship between radius and tim e.

(2.3)

(2.4)

R(t) = A .t

(2.5)

w here

A

is.

CHAPTER 2: LITERATURE REVIEW

However, w ith no experim ental data existing at th a t tim e , th e a p p lica b ility o f Eq. (2.4)

was d iffic u lt to ascertain as it o nly becomes significant fo r very low system pressures.

Later, th e region o f bubble g ro w th co ntro lle d by th e rm a l diffu sio n was in tro du ce d ,

fo r exam ple in the works o f Forster and Zuber (1954) and Plesset and Zwick (1954). In

this case, th e bubble gro w th p re dictio n s was extended beyond th e in e rtia l co ntro lle d

g ro w th region by taking in to account th e fa ct th a t as th e bubble grows, th e la te n t heat

re q u ire m e n t o f evaporation depletes th e energy stored w ith in th e superheated layer

w hich has fo rm e d at the surface o f th e bubble [Plesset and Zwick (1954)]. As the

bubble grow s, its e qu ilibrium v a p o u r te m p e ra tu re decreases fro m

T^o

to its m inim um

value o f

Tsat (Poo)- As the in terfa cia l te m p e ra tu re and corresponding pressure drop,

bubble g ro w th becomes lim ite d by th e re la tive ly slo w e r diffu sio n o f heat to the

va p o u r-liq u id interface, causing th e g ro w th rate to c o n tin u a lly decrease.

Plesset and Zwick (1954) have obtained a so lu tio n fo r th e instantaneous

bubble radius prediction fo r th e case o f th e rm a l d iffu s io n co n tro lle d g ro w th by

supplying an approxim ate expression fo r th e liq u id te m p e ra tu re d is trib u tio n at the

interface. This approxim ate expression was fou n d u nd e r th e assum ption th a t th e drop

in te m p e ra tu re fro m Too to th e value o f

T at th e bubble boundary takes place in a

layer o f liquid adjacent to th e bubble w hich has a small thickness com pared w ith

bubble radius. This assum ption o f a 'th in th e rm a l b ou n da ry layer' resulted in an

a pp ro xim a te expression fo r th e liq u id te m p e ra tu re at th e m oving interface.

Then, by assum ing the rm al e q u ilib riu m betw een th e liq u id at th e in te rfa ce and the

va po u r and im p le m e n ting th e energy balance at th e in te rfa c e w hich relates th e rate o f

heat tra n s fe r to th e bubble to th e v a po u r mass balance, Plesset and Zwick d ete rm ine d

th a t.

/2

f t R \ d T / d r )

R(t) ~ 2 ^ / 3 J a ( ^ y ^

(2-8)

Here, Ja =

is the dimensionless superheat known as the Jakob number.

According to Prosperetti and Plesset (1978), Eq. (2.8) predicts th a t the radius w ill

increase asym ptotically w ith tim e if growth is diffusion controlled and is valid only fo r

tim es large enough th a t the grow th velocity is much smaller than the inertia

controlled velocity. From the Eq. (2.8), it clearly shows that the growth is asymptotic;

(2.9)

w here the Plesset and Zwick solution,

B is

12aja

n

Vz

(

2

Forster and Zuber (1954) perform ed a sim ilar analysis in which the interface

tem p e ra tu re was approxim ated by integrating the Green's function over the domain

o f the 'th in therm al boundary layer'. The Clausius-Clapeyron relation was once again

used in th e relationship between vapour pressure and tem perature to give the

asym ptotic expression,

CHAPTER 2: LITERATURE REVIEW

(

2.

12)

where fo r the Forster and Zuber solution, C is.

(2.13)

Meanwhile, w ith o u t the assumption o f a 'thin therm al boundary layer' as previously

used by Plesset and Zwick (1954) and Forster and Zuber (1954), Scriven (1959)

determined the exact solutions of the energy equation including the effects of radial

convection resulting from unequal phase densities to obtain an asym ptotic relation fo r

thermal diffusion controlled growth.

The constant

p

depends on the system pressure and the degree o f superheat. For the

case of moderate to high superheats or large Jakob numbers, Eq. (2.14) simplifies to,

When Eq. (2.15) is applied fo r common fluids and system condition in which

y

« 1, it

is identical to the solution given by Plesset and Zwick (1954) [Eq. (2.8)]. This implies

that fo r large enough Jakob numbers, the 'thin therm al boundary layer' assumption is

(2.14)

(2.15)

where.

valid [Robinson (2002)]. For th e case o f small Jakob num bers o r low superheats,

Scriven (1959) obtained,

(2.171

W hen applied to th e com m on flu id s and system c o n d itio n s (y « 1), it sim plifies to .

R ( t ) ^ y j 2 ] a . a t

(2.18)

It is also noticed fro m th e Eq. (2.18) th a t th e tim e take n to diffuse such a distance is

its e lf p ro p o rtio n a l to

R^,

/ ? « D . t V 2

(2.19)

w h e re the Scriven solu tion , D is,

D = yj2]a. a

(2.20)

From th e w orks o f Plesset and Zwick (1954), Forster and Zuber (1954) and

Scriven (1959) in th e case o f th e rm a l diffusion c o n tro lle d bubble expansion, it is

noticed th a t th e expressions given have th e same a s ym p to tic dependence on tim e

( ~ t ^ /

) b u t a d iffe re n t dependence on th e Jakob n u m b e r. This im plies th a t fo r small

Jakob num bers, th e 'th in th e rm a l b ou n da ry layer' assum ption may no longer be valid

[Robinson (2002)]. It suggests th e fo llo w in g dependence on th e Jakob num ber,

R ~ Ja

(large Ja)

CHAPTER 2; LITERATURE R E V IE W

Postulation can be made from the above efforts that the early stage of bubble

grow th is inertia controlled and the later stage is diffusion controlled. A general

relation fo r bubble growth rates in a uniform ly superheated liquid which is applicable

fo r the entire range of the bubble growth, including inertia controlled and diffusion

controlled growth, was derived by Mikic et al. (1970). They used the Clausius-

Clapeyron equation fo r the vapour pressure curve, assuming therm al equilibrium in

the vapour bubble so that the vapour pressure corresponds to the bubble wall

tem perature as the bubble grows. The relation was compared w ith the existing

experimental data of Lien and G riffith (1969) fo r w ater over a wide range of systems

pressures, including low pressure data w ith a significant inertia controlled region and

it was found to be in good agreement. The relation fo r the variation of bubble radius

w ith tim e which spans both regions is,

R* = ^ [ ( t + + 1 ) ^ / 2 - ( f ) V 2 - i j ( 2 . 2 2 )

where the scaled variables are given by.

R * =

B ^ / A ’

t*

(2.23)

A =

'^ s a tP l

B =

12

n

(2.24)

in which,

b = 2/3 fo r bubble growth in an infinite medium;

b =

/7

fo r bubble growth on a surface

describe grow th over th e en tire range of superheats. By assuming a linear variation o f

vapour pressure w ith te m p e ra tu re , th ey obtained th e expression,

3/ 3/

/ 2 / I \ / 2

-

1

(2.25)

w h e re th e scaled variables are expressed as.

\ ( 2 ( s a \ l ' ^ P i h f a ^ ^

_i,

I.(T T

^ (Pi[P vaoo)-P oc])

(2.26)

3 \ n / k ( T ^ - T s a t ) '

[ P v ( T o o ) - P c o ? ^ ^

2ap_

V

2

Analytical bubble grow th studies in th e region of th e rm a l diffusion controlled

g row th w ere later extended to consider non-uniform te m p e ra tu re fields which

approxim ate th e conditions of heterogeneous nucleate boiling at a heating surface;

fo r examples th e works of Zuber (1 9 6 1 ), Han and G riffith (1 9 6 5 ), Cole and Shulman

(1 9 6 6 ) and Mikic and Rohsenow (1969). Zuber (1 9 6 1 ) in his study o f non-uniform

te m p e ra tu re field effects took into account th e heat flux from th e heated surface to

th e liquid. For a spherical bubble, th e rate o f evaporation was given by:

d R _{'Pw T'sat}

y i n a t

(2 .2 7 )

CHAPTER 2: LITERATURE REVIEW

R = b —Ja^ na t

1 —

(2.28)

n

2k(T^ - T,at)

w ith a co nsta nt factor,

b

fo r th e e ffe c t o f sphe ricity w hich lies betw een

and

^f3

w ith

^ / 2

3*^ in te rm e d ia te value.

Han and G riffith (1965) to o k in to account th e th e rm a l boundary layer thickness

and a critical w all superheat re la tio n to th e cavity to o b ta in bubble g ro w th rates. The

th e rm a l layer thickness was o b ta in ed fro m th e co nsideration o f tra n s ie n t conduction

in to a layer o f liquid on th e surface. The Han and G riffith bubble g ro w th expression in

a n o n -u n ifo rm te m p e ra tu re fie ld is expressed as.

M ikic and Rohsenow (1969) considered bubble g ro w th in a n o n -u n ifo rm

te m p e ra tu re fie ld by using a one-dim ensional m odel corrected fo r sp h e ricity in ord er

to approach th e lim it fo r th e bub b le g ro w th in a u n ifo rm ly superheated liq u id when

th e w a itin g tim e ,

approaches in fin ity . They in tro d u ce d the assum ption th a t the

actual heat flu x can be expressed as.

^_(Ps(fc

k

\ 2 ( T „ - T s a t ) ^

/^

( T ^ - T ^ )

fAat

S \

(2.29)

where.

cu rva tu re factor,

w h e re

1 <

<

surface fa cto r,

cp^

=

4 n R ^

1 + cos 6

2

vo lu m e facto r,

q)^ =

1 ^

T'w-Tb

^

V

^ / { n a ( t + T ^

(2.30)

w here

A being area and c o rre ctio n fa c to r C was fo u n d to be V s . The second te rm in

Eq. (2.30) represents th e effects o f a n o n -u n ifo rm te m p e ra tu re fie ld . As th e g ro w th o f

th e v a po u r bubble is governed by th e heat tra n s fe r process, th e Eq. (2.30) can be

w ritte n as.

dR

p , h „ — = k ^

T ^ - T .

Tiu — Th

(2.31)

By in te g ra tin g Eq. (2.31) using /? = 0 at t = 0 gives an expression fo r bubble g ro w th .

/? = - V 3 7 a V ^ l

n

* s a t

Vz

n...^y2

(2.32)

2.1.2 Bubble D eparture

Perhaps th e firs t m odel fo r p redicting bub b le d e p a rtu re was in tro du ce d by

Fritz (1935). The Fritz e q u a tion , by utilizing th e co nta ct angle,

a and th e surface

tension,

a

gives th e bubble d ep a rtu re d ia m e te r as th e d ia m e te r th a t satisfies the

co n d itio n in w hich th e buoyancy force is balanced by th e capillary force,

tt

D

- ^ i P i - P v ) g = nDoa sin ao

(2.33)

CHAPTER 2: LITERATURE REVIEW

^ e q ~

6

^

7T

(2.34)

Here, it is supposed that the diam eter o f the bubble base is proportional to the

equivalent diameter of the bubble.

Do = CD,,

(2.35)

Then, by substituting Eq. (2.35) into Eq. (2.33) Fritz obtained.

- ^ ( P ; - P v ) 5 = CcTsinao (2-35 )

which gives the definition of bubble departure diam eter as.

a

a

where

C

is taken as 0.0208 and

is in degrees.

A similar method to predict the bubble departure diam eter has been proposed

by Chesters (1977). By using the same physical condition as Fritz (1935), he showed

that fo r small values of the shape factor defined as,

^

P.)sSj

,2.38)

where

R j

is the bubble radius at the apex, the bubble departure diam eter can be

(2.39)

The correction fa cto r

k < 1

was added to account fo r the over pressure inside the

bubble.

Kiper (1971) in his predictive study fo r determ ining the m inimum bubble

departure diam eter in saturated nucleate pool boiling, found that the m inimum

departure size varies w ith Jakob num ber only. In his work, the bubble is considered to

be o f spherical fo rm ending w ith a small neck which connects the bubble to the

heated surface as shown in Figure 2.1. He neglected both the drag force and the

vapour inertia force, though he considered the inertia of the fluid. Consistent w ith the

previous models o f bubble departure, individual bubbles w ill depart when the force

balance equation is satisfied, w ith the forces acting on the bubble defined as follows;

(i)

The buoyancy force

where

go

is a conversion constant. The contact area at the base of the bubble is small

w ith respect to the bubble size and the overpressure effect is negligible.

(ii)

The capillary force

Furthermore, if the geom etry o f the neck is known then the term

Dgcosa

can be

expressed as;

CHAPTER 2: LITERATURE REVIEW

Fq

— —a n G (6 )D

(2.42)

where G (0) is related to geom etry of the bubble neck.

(Hi)

The liquid inertia force

The inertial force of the apparent liquid mass surrounding the bubble was initially

derived by Keshock and Siegel (1964). They have included the affected mass of the

fluid which occupy the bubble volume by 1 1 /1 6 as suggested by Han and G riffith

(1962). The acceleration o f the fluid is approximated by the tim e rate of change of the

bubble growth velocity where the velocity is the change o f radius w ith tim e as

suggested by Clark and M erte (1963) and Adelberg (1963). Then, the inertia force

becomes,

d

— m u

(2.43)

which can be rew ritten in the final form.

(2.44)

By substituting the bubble growth law, i.e.,

D

=

Kiper (1971) obtained the

inertia force as.

(2.45)

where /? is the bubble growth parameter.

This force is due to th e specific pressure d is trib u tio n on th e bubble surface w hich

te n d s to fla tte n the bubble and hold it against th e w all. This fo rce is defined as

Fo =

-O.OlSlTT —

( 2. 46)

do

The force balance can be re w ritte n as;

Fs + F, = Fc + Fo

(2.47)

By using the foregoing fo rce expressions at th e m o m e n t o f bubble d e p a rtu re , th e

fo rc e balance equation can be w ritte n as,

a D l + bD^ + c = 0

(2.48)

w h e re ;

a = - g - —

b = - a G ( B ) a ,

c = 0.0105 —

( 2. 49)

6

go

9o

The equation has several solu tion s th o u g h ju s t one was fo u n d to p re d ict th e

e x pe rim en ta l result w hich was;

Dfc =

2.7Ja

(2.50)

w h e re

J a

is the Jakob num ber. For o th e rw ise constant flu id p ro pe rtie s, this expression

p re dicts th a t th e bubble d e p a rtu re d ia m e te r increases lin e arly w ith superheat. As

CHAPTER 2: LITERATURE REVIEW

B u b b !e _ _ ^ _ \ / ' n e c k I

inr:!i!niiin{i!jh^H )iii!iin!iT7

Figure 2.1: Geometry of the growing bubble (Kiper, 1971)

Zeng et al. (1993) proposed a somewhat d iffe re n t model considering th a t the

dom inant forces leading to bubble detachment would be the unsteady growth force

and the buoyancy force by neglecting the surface tension terms, which were

previously considered by many earlier researchers. Under these circumstances the

detachm ent condition occurs when,

The second term is the unsteady growth force in the vertical direction. The author

modelled this force by considering a hemispherical bubble expanding in an inviscid

liquid. An empirical constant, C was introduced to account fo r the presence of a wall;

^ b u o y ^ d u y

^ ^

(2.51)

where the first term is the buoyancy effect which is equal to.

Fb

y b ( p i - p v ) g

(2.52)

(2.53)

The evaluation of the growth stemmed from knowledge of the bubble growth rate

R( t ) .

In general this follows a power law;

where

K

and

n

were determined empirically. The unsteady growth force can then be

defined as;

By balancing of the buoyancy and unsteady growth forces, the value of the

detachment diam eter was obtained as;

From Eq. (2.56) it is possible to observe that for gravity tending to zero, the vapour

bubbles will not depart the heating surface unless there is some external mechanism

to induce an inertial force, such as system vibration.

Yang et al. (2000) have predicted the bubble departure diameter by correlating

the bubble departure diam eter only with the bubble growth tim e. They determined

this after plotting the relationship between dimensionless growth tim e and departure

diameter for the various different liquids and wide range of pressures experimented

by Cole (1967), Han and Griffith (1965), Stralen et al. (1975) and Keshock and Siegel

2/ 3

(1964) as shown in Figure 2.2. The data was correlated in the form

The dimensionless departure diam eter and growth tim e are as follows:

R(t )

=

K q

(2.54)

r3

PiuK^^^ —Pn^ + n ( n — l )

(2.55)

CHAPTER 2: LITERATURE REVIEW

D+ = — =

= - ( - ]

t

(2 57)

Lo

0 U j

^

where

A =

;

B = J a

^PiTs

12

—

cci

;

7T

0 =

1 /

7T \ 2 / 3

^

2

j

6] a

(2.58)

By using the correlation proposed by Rohsenow (1951);

RehPr,

- ^ = W

P r p

(2.59)

where

Nuf,

=

2 /(c ) Dfe/1

3c^

a j a

0

(2.60)

According to Mei et al. (1995),

/ ( c ) ~ 1 - ^ 1 - V l - c^j + ^ [ l - V l -

(2.61)

where the parameters c =

Rb/Rt,

w ith

R^

and

R^

defined in Figure 2.3. Connbining

3 / TT n' "

= (2 .6 2 )

w h e r e

ih = --- (2 .6 3 )

Predic tion results o f Eq. (2 .6 2 ) w e r e t h e n c o m p a re d w ith t h e m e a s u re d d a ta o f b o th

organics liquids and w a t e r fo r t h e e n t ir e range o f e x p e r im e n t a l con d itio n s and

p ro d u ce d e x p o n e n ts o f m = 1.4 and n - 0 .8 . In tro d ucin g t h e valu es of m and n and Eq.

( 2 .5 8 ) into Eq. ( 2 .6 2 ) gives;

D , = 8 .0 3 5 1 X (2 .6 4 )

P v

hfg

CHAPTER 2: LITERATURE REVIEW

1^1

3

2

1

0

■2

-3

*4

-2 -1 0 1 2 3 4 5 6

Figure 2.2: Relationship between dimensionless growth tim e and departure diameter

(Yang et al., 2000)

bubble

solid wa

Figure 2.3: Sketch fo r the growing bubble (Yang et al., 2000)

2.1.2.1B ubb le D e p a rtu re Frequency

Several models and correlations were found from the past analytical studies to

predict bubble departure frequency fo r pool boiling. In addition, bubble departure

frequency can be deemed as the reciprocal of the summation of bubble waiting tim e,

and bubble growth tim e,

tg,

fd =

(2-65)

Zuber (1959) proposed an equation describing the product of frequency and

departure diam eter in term s o f fluid properties. He suggested th a t it is related to the

velocity at which bubbles rise in a liquid, expressed as,

o g j P i - P v )

pf

V4

(

2

6 6

He then assumed that.

/ .

Du

a

(2.67)

and found an expression o f the form,

f . D ^ =

0.59

( ^a( Pi - Pv)

Pt

V4

(

2

6 8

However, Zuber's expression has been found to fit experimental data fo r only a

lim ited range of bubble boiling conditions. Im portantly, however, the Zuber

correlation predicts an inverse relation between bubble frequency and departure

diameter.

Later, McFadden and Grassmann (1962) assumed that the frequency-diam eter

product / .

Dfj

is a function o f Dj,,

a, pi

and Ap shown as follows.

CH A PT E R 2; LITERATURE R E V IE W

Eq. (2.69) describes the bubble frequency and bubble departure diameter in term s of

liquid and vapour properties. The le ft side of Eq. (2.69) represents the square root of

power o f the ratio of the buoyant force to the surface tension force. They found that

bubble departure frequency expression by McFadden and Grassmann is thus given as,

frequency w ith departure diameter. McFadden and Grassmann also considered the

case when bubble departure is acted upon by various oth e r forces, particularly at low

heat flux, where the bubble departure frequency can be expressed as.

Still, the functional relationship between the frequency and departure diameter

remains unchanged.

Soon after, Mikic and Rohsenow (1969) predicted a relationship between

bubble frequency and departure diam eter from th e ir bubble departure expression for

saturated boiling as follows.

inertial force to surface tension force ratio whereas the right side represents the

n

~

^/2 satisfactorily fitted most o f the available experim ental data at th a t tim e. The

(2.70)

This correlation predicts that /

a

which is an asym ptotic inverse relation of

(2.71)

4

y/SJa^natf.

1

+

(2.72)

V l/a V jr a

n

(2.73)

w here,

1

1

/ =

(2.74)

M ikic and Rohsenow stated th a t th e ra tio

fo r a given nucleation site is a fu n ctio n

o f pressure and w o u ld be d iffe re n t at d iffe re n t nucleus sites. From a w id e range of

In te re stin gly th e bubble fre q u e n cy is predicted to be a fu n c tio n o f

[(Tw-Tsat)/Dbf so

is a

fu n ctio n o f both superheat and d e p a rtu re diam eter.

2.1.3 Bubble W aiting Tim e

The bubble w a itin g tim e is th e interval b etw e e n w hen th e previous bubble

departs and th e next one nucleates. The w a itin g tim e ,

can be re la ted to the

p a rticu la r cavity size [M ikic and Rohsenow (1969) and Han and G riffith (1965)]. During

th e w a itin g tim e , when th e bubble is n o t grow ing, th e bubble te m p e ra tu re ,

has

been derived fro m the e quation o f a va po u r bubble in th e rm o d y n a m ic e q u ilib riu m .

/ th e y obta in ed th e sim p lified expression.

Di,./V2 ^

Qmja-Jna

(±10% )

(2.75)

CHAPTER 2: LITERATURE R E V IE W

which, after applying the Clasius-Clapeyron equation becomes,

r p _ r j . ,

^^T'sat

A hemispherical bubble of radius,

Rc

(located over a cavity o f the same radius)

w ill start to grow when the vapour tem p era ture is greater than

in Eq. (2.78). The

tim e required to achieve this tem perature is the w aiting tim e.

The expression fo r the tem perature distribution in the liquid was given by

Mikic and Rohsenow as follows:

T(y,

0 = T’oo + (7’w - 7’oo)erfc '

^

2yJ[a(t + t ^ ) ] j

(2.78)

By assuming that in a non-uniform tem perature field the tip of the considered bubble

(y =

Rc)

should be at 7^ fo r the beginning o f the bubble grow th, M ikic and Rohsenow

have an expression from the relations o f the Eqs. (2.77) and (2.78) as follows:

('

"

(2.79)

or:

erfc"

R r

In a sim ilar fashion, the earlier study of Han and G riffith (1965) have equated the

2.2

E xp e rim e n ta l Single Bubble Dynamics

Lee et al. (2003) perform ed an experiment o f nucleate pool boiling w ith

constant wall tem peratures to investigate single bubbles growing in saturated

conditions. They used R l l and R113 as w orking fluids. A microscale heater array was

used to maintain the constant wall tem perature. Each heater in the array had

dimensions of 0.27 x 0.27 mm, which is comparable w ith the diam eter of a typical

single bubble (0.25-0.7 mm). A high-speed CCD camera synchronized w ith the heat

flo w rate measurements was used to capture the bubble grow th images and the

geom etry of the bubble was obtained from those images. The captured images

showed a spheroidal-shaped bubble during growth. The images also showed an

asym ptotic bubble radius growth rate proportional to

which was much slower

than the previous analytical studies which show a

relationship. To analyse the data,

they used dimensionless parameters of tim e and bubble radius to characterize the

asym ptotic grow th behaviour irrespective o f the wall condition. The dimensionless

parameters were derived from the ratio of the corresponding latent heat transfer and

the conduction heat transfer rate through the bubble interface,

(q

-Quite recently, Siedel et al. (2008) have perform ed a nucleate pool boiling

bubble tem perature to the fluid tem p era ture at the distance of y =

^/2

Rc

from the

heating surface, consequently the w aiting tim e is obtained

times of the above

value in Eq. (2.80) as follows:

(Tw

CHAPTER 2: LITERATURE REVIEW

Pentane was used as a working fluid in this work. Bubble growth was recorded by a

high speed camera under various wall superheat conditions. They found that the

bubble volume at departure was independent of the wall superheat, whereas the

grow th tim e was dependant on the superheat. The bubble growth rate was found to

fo llo w non-dimensional growth laws;

V* =

fo r

t* >

0.2 and

V*

2 x

t* fo r

t* < 0.2. Furthermore, the bubbles growth tim es were found to be approxim ately

proportional to the wall superheat. However, the product ( / .

was found not to be

a constant, in contrast to many earlier models. This was due to the invariant bubble

departure diameter w ith wall superheat.

2.3

Subcooling Effects on Bubble Dynamics

2.3.1 Analytical Study

Based on the analytical study of waiting tim e

by Han and G riffith (1965) as

3

indicated in Eq. (2.81) fo r the consideration of y = -/?c, it clearly predicts that the

waiting tim e increases w ith the increasing liquid subcooling. In addition, the waiting

tim e also increases w ith increasing cavity radius,

The same trend fo r the waiting

tim e w ith changing liquid subcooling can be obtained from the proposed waiting tim e

theory of Mikic and Rohsenow (1969) as shown in Eq. (2.80) fo r the consideration of

y =

-Quite recently, relative to the earlier works, Zhao and Tsuruta (2002)

presented one of the more comprehensive models fo r bubble dynamics in subcooled

pool boiling. According to this study, one cycle of an individual bubble consists of tw o

parts; one is its lifetim e and the other is the waiting tim e. The lifetim e of the individual

bubble consists of three periods, i.e., an initial growth period (O < t <

tg),

a final

grow th period due to evaporation of a microlayer

{tg < t < tg +

tg ) and a

collapses as shown in Figure 2.4. Just after the individual bubble collapses and before

the next cycle begins, there is a waiting tim e,

tw-Waiting time

Activc nuclei

( d )

< e )

Figure 2.4; One cycle of an individual bubble (Zhao and Tsuruta, 2002)

(i)

Initial growth period

(O < t < t^)

During the initial growth of individual bubbles [Figure 2.4 (a)], semi-spherical bubbles

grow from an active site and a microlayer is form ed under the bubbles, as shown in

Figure 2.5 (a). The growth equation of individual bubbles was derived from the heat

balance between the latent heat of evaporation o f the liquid microlayer and the

conduction through the microlayer as follows,

- n R ^ p ^ h f g = 2 n k i f f

—

^ ^ r d t d r , 0 < t < t^

(2.82)

■ ' 0 ■ 'T g ^ m i c

CHAPTER 2: LITERATURE REVIEW

{ a )

The forming period ( b ) The evaporation perifxl

( c ) M icrolayer dryoui and niacrolayer

Figure 2.5: Dynamic m icrolaye r (Zhao and Tsuruta, 2002)

W hen th e w all su perheat is tow, th e fo llo w in g a p p ro xim a tio n was made,

^ r>3 u ' I r f f ' ^ s a t ) n ^ 4. ^ 4.

-nR^p^hfg

=

2nki

rdt dr ,

0 < t < t,

J(\ Jc\

<^>111>

(2.83)

By using th e in itia l thickness of th e m icrolaye r

at any p o in t (radius r ) by Cooper

and Lloyd (1969) w hich is,

^mic ~ 0 . 8 y f ^ = yjca. t, w ith c =

0.64 Pr

0 < t < tg

th e bubble radius expression then was obtained.

(2.84)

^ _ 2 k i ( T ^ T^at)

A — --- t ' 2

p ^ h f g ^ f c a

At the end o f the initial growth phase of an individual bubble ( t =

tg),

the bubble

diam eter

d

was given by,

or the initial grow th duration of the individual bubble is given as.

(ii)

Final growth period (^tg < t < tg + tg)

Meanwhile, fo r the period of final growth due to evaporation of the microlayer

(tg < t < tg +

tg), the microlayer was reported not to expand and the shape o f the

bubble changes from semi-spherical to a spherical segment geometry due to the

evaporation of the microlayer as shown in Figure 2.5. At the same tim e, a liquid layer

thicker than the microlayer is form ed under the bubble outside the microlayer area

which was called the macrolayer. The evaporation tim e of the microlayer

tg

was

determ ined by the conduction equation o f the liquid microlayer w ith the condition.

2

8 6