UNIVERSITY OF CINCINNATI
January 16
04
_____________
, 20 _____
JUNTAO ZHANG
I,______________________________________________,
hereby submit this as part of the requirements for the
degree of:
DOCTOR OF PHILOSOPHY (Ph.D.)
________________________________________________
in:
Department of Mechanical, Industrial and Nuclear Engineering
________________________________________________
It is entitled:
EXPERIMENTAL AND COMPUTATIONAL STUDY OF
________________________________________________
NUCLEATE POOL BOILING HEAT TRANSFER IN AQUEOUS
________________________________________________
SURFACTANT AND POLYMER SOLUTIONS
________________________________________________
________________________________________________
Approved by:
Raj M. Manglik
________________________
Milind A. Jog
________________________
Michael Kazmierczak
________________________
Rupak K. Banerjee
________________________
________________________
EXPERIMENTAL AND COMPUTATIONAL STUDY OF
NUCLEATE POOL BOILING HEAT TRANSFER IN AQUEOUS
SURFACTANT AND POLYMER SOLUTIONS
A Dissertation Submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of
DOCTOR OF PHILOSOPHY (Ph.D.)
in the Department of Mechanical, Industrial and Nuclear Engineering
of the College of Engineering
2004
by
Juntao Zhang
B.S., University of Science and Technology Beijing, China, 1992
M.S., University of Science and Technology Beijing, China, 1995
Committee Chair: Professor Raj M. Manglik
ABSTRACT
Saturated, nucleate pool boiling on a horizontal, cylindrical heater and the
associated bubble dynamics in aqueous surfactant and polymer solutions are
experimentally and computationally investigated. Boiling curves ( qw" vs. ∆Tsat data) for
different additive concentrations and photographic records of the salient features of the
ebullient behavior are presented, along with a characterization of the interfacial properties
of the aqueous solutions. The surfactant additive significantly alters the nucleate boiling
in water and enhances the heat transfer. The enhancement increases with concentration,
with an optimum obtained in solutions at or near the critical micelle concentration (CMC)
of the surfactant.
On the other hand, boiling in aqueous polymer solutions shows
contrary results. The enhancement is seen to increase with polymer concentration in the
aqueous surface-active HEC solutions, while boiling in shear-thinning Carbopol 934
solutions is seen to decrease continuously with increasing concentrations.
In order to develop a theoretical model and understand the associated convective
mechanisms, the dynamics of a single growing and departing bubble during nucleate
boiling from a horizontal heated surface has been numerically simulated. The results
highlight the role of the microlayer in nucleate boiling, as well as the effects of altered
surface tension and viscosity, apparent contact angle, and wall superheat on the bubble
dynamics and boiling behavior in aqueous surfactant and polymer solutions.
In multiphase heat transfer, the nature and dynamics of surface contact often plays
a dominant role. In fact, almost all the transport processes in nucleate boiling, from the
inception of an embryonic bubble to the subsequent phase change, are intricately
connected with this interface. To better characterize the molecular dynamics of the
additives at the interfaces, an extensive literature review is presented that delineates the
surfactant adsorption process and its electrokinetic effects (zeta potential), and the
concomitant surface wetting behavior. This fundamental understanding is supported by
the measured interfacial properties of the solutions reported in this study. And as shown
by the boiling data, the heat transfer performance is changed dramatically when the
interfacial transport process is altered.
Finally, as documented by the pool boiling experiments with aqueous surfactant
solutions, a markedly different ebullient behavior than not only that of water is observed,
but between pre- and post-CMC solutions as well. The characteristic bubble dynamics is
found to correlate well with the measured surface wettability and the adsorption isotherm,
which in turn correlates with the zeta potential. In essence, the unique structure of a
surfactant - a hydrophilic head with a hydrophobic tail - provides the means to control of
nucleate boiling by altering surface wettability. The advantage of this very effective
method is that there is no need to make any changes to any existing boiling equipment,
except for the addition of desired amount of surfactant into the solution. The potential
application impact extends to any process that involves phase-change (large-scale as well
as micro-scale heat exchange devices). Furthermore, the surfactant physisorption and its
electrokinetics are not only basic to a fundamental understanding of nucleate boiling
control, but also can provide insights into related natural phenomena in other applications
as well. There include chemical and biological sensors, microgravity applications, and
microfluidic (or lab-on-chip) devices, among others.
Also, some nanofluids or
nanoparticles with special surface properties based on the different structures of
surfactant micelles can be developed to provide even greater interfacial control in these
broad spectrum of emerging applications.
ACKNOWLEDGEMENTS
First, I would like to thank Professor Raj M. Manglik for his excellent guidance,
support, encouragement, and inspiration that make this dissertation possible. I am also
thankful to the committee members Drs. Milind A. Jog, Michael Kazmierczak and Rupak
K. Banerjee for their valuable suggestions, and Mr. Bo Westheider of the Instrumentation
Laboratory, MINE department, and Mr. Doug Hurd of the machine shop for their help in
the construction of the experimental facility. Also, the valuable communications with
Professor Vijay K. Dhir, UCLA, on the microlayer and phase-change modeling, the
generous help of Dr. Osamu Tatebe, National Institute of Advanced Industrial Science
and Technology, Japan, on the MPCG implementation, and instructive suggestions of
Professor Douglas W. Fuerstenau, University of California – Berkeley, on the surfactant
adsorption characteristics, are gratefully acknowledged.
I specially thank the love and support of my wife, Jia Li, which made my work
possible and worthwhile. Also, I would like to thank my parents for their constant
encouragement and support through my studies.
The financial support provided by National Science Foundation, the University
Research Council, and Ohio Board of Reagents is gratefully acknowledged. My fellow
graduate students: Manish Bahl, Satish Vishnubhatla, and S. Sethu Raghavan provided
much needed assistance in acquiring some rheology and interfacial data.
Finally, I would like to thank all the members in the TFTPL laboratory and all my
friends in UC for enriching my everyday life.
TABLE OF CONTENTS
Page
ABSTRACT
ACKNOWLEDGEMENTS
TABLE OF CONTENTS
i
LIST OF TABLES
v
LIST OF FIGURES
vi
NOMENCLATURE
xi
1. INTRODUCTION
1
1.1 Surfactants, Colloid Systems, and Interfacial Phenomena
2
1.1.1 Surfactants
5
1.1.2 Colloid systems and interfacial phenomena
5
1.1.3 Surface tension and micelles
7
1.1.4 Electrokinetic effects and zeta potential
10
1.1.5 Surfactant physisorption at the solid-liquid interface
14
1.2 Surface Wettability and Nucleate Boiling
18
1.2.1 Contact angle and surface wettability
18
1.2.2 Surface wetting effects on nucleate boiling
21
1.3 Nucleate Pool Boiling with Surfactants
22
1.4 Nucleate Pool Boiling with Polymers
26
1.5 Computational Fluid Dynamics with Moving Boundaries
30
1.6 Scope of Study
32
i
2 INTERFACIAL PROPERTIES AND RHEOLOGY MEASUREMENTS
2.1 Surface Tension Measurements
34
34
2.1.1 Introduction
34
2.1.2 Viscosity effects on surface tension measurements
38
2.1.3 Results and discussions
39
2.1.3.1 Aqueous surfactant solutions
39
2.1.3.2 Aqueous polymeric solutions
46
2.2 Contact Angle Measurements
52
2.3 Rheology Measurements
55
2.3.1 Aqueous surfactant solutions
56
2.3.2 Aqueous polymer solutions
56
3. POOL BOILING HEAT TRANSFER
61
3.1 Experimental Setup
61
3.2 Nucleate Pool Boiling in Aqueous Surfactant Solutions
65
3.2.1 Pool boiling in aqueous cationic surfactant solutions
65
3.2.2 Optimum heat transfer and critical micelle concentration
71
3.3 Nucleate Pool Boiling in Aqueous Polymer Solutions
72
3.3.1 Pool boiling in aqueous polymer solutions
74
3.3.2 Surface-active and rheological effects
77
4. VISUALIZATION AND CHARACTERIZATION OF NUCLEATE POOL
BOILING IN AQUEOUS SUFACTANT SOLUTIONS
81
4.1 Introduction
81
4.2 Zeta Potential and Contact Angle
82
ii
4.3 Ethoxylation Effect
85
4.4 Dynamic Surface Tension – Molecular Weight Effect
86
4.5 Surface Wettability Effect on Nucleate Boiling Heat Transfer
92
4.6 Ebullient Dynamics Visualization
94
4.6.1 Visualization in aqueous surfactant solutions
94
4.6.2 Visualization in aqueous polymer solutions
99
4.7 Characterization of Nucleate Pool Boiling in Aqueous
Surfactant Solutions
102
5. SIMULATION OF A SINGLE BUBBLE
112
5.1 Introduction
112
5.2 Mathematical Formulation
114
5.3 The Numerical Method
120
5.4 Solution Validation
122
5.4.1 Efficacy of the MPCG method
122
5.4.2 Efficacy of the level-set method
125
5.4.3 Verification of phase change modeling
127
5.5 Results and Discussion
130
5.5.1 Microlayer
130
5.5.2 Surface tension effect
131
5.5.3 Viscosity effect
138
5.5.4 Temperature, velocity, and pressure fields
141
5.5.5 Apparent contact angle and superheat effect
144
5.6 Significance and Limitations of Nucleate Boiling Simulations
iii
146
6. CONCLUSIONS AND RECOMMENDATIONS
149
6.1 Conclusions
149
6.2 Recommendations for Future Research
153
BIBLIOGRAPHY
161
APPENDIX A. SURFACE TENSION σ (mN/m) DATA
177
A.1 Aqueous Surfactant Solutions
177
A.2 Aqueous Polymer Solutions
181
A.3 Dynamic Surface Tension with Time
182
A.4 Surface Tension with Temperature
186
APPENDIX B. CONTACT ANGLE DATA
187
APPENDIX C. POOL BOILING DATA
188
C.1 Aqueous Surfactant Solutions
188
C.2 Aqueous Polymer Solutions
197
APPENDIX D. TEMPORAL AND SPATIAL DISCRETIZATION
iv
200
LIST OF TABLES
Page
1.1
Typical Colloidal Systems (Birdi, 2003)
1.2
Chronological Listing of Nucleate Pool Boiling Studies
of Aqueous Surfactant Solutions
23
Chronological Listing of Nucleate Pool Boiling Studies
of Aqueous Polymeric Solutions
27
2.1
Physico-Chemical Properties of Surfactants
37
2.2
Physico-Chemical Properties of Polymers
38
2.3
Increase in Viscosity of Aqueous Polymer Solutions with respect to Water
at 23°C as a Function of Concentration at a Shear Rate of 500 s-1
60
1.3
5.1
Combination of Non-Dimensional Parameters Studied (D = 2mm)
v
6
138
LIST OF FIGURES
Page
1.1
1.2
Typical boiling curve for controlled wall heat flux and schematic
representation of the different heat transfer regimes
2
The conjugate problem in modeling nucleate pool boiling with
or without additives
4
1.3
Schematic illustration of the primary structure of a surfactant molecule
5
1.4
Different possible micellar structures (Evans and Wennerström, 1999)
9
1.5
Schematic diagram of a typical electrokinetic boundary layer
10
1.6
Four basic electrokinetic phenomena and the relationship between them
13
1.7
Schematic representation of a typical adsorption isotherm and aggregate
states of a surfactant on a solid surface in aqueous solutions
15
1.8
Different wetting conditions of a liquid drop on a solid surface
19
1.9
Surface forces involved in spreading of a liquid
20
2.1
Schematic of surface tensiometer and data acquisition system
36
2.2
Surface tension measurements at equilibrium and higher bubble frequency
(surface age of 50 ms) for aqueous surfactant solutions at 23°C
40
Surface tension measurements at equilibrium and higher bubble frequency
(surface age of 50 ms) for aqueous surfactant solutions at 80°C
42
Dynamic surface tension relaxation for aqueous solutions
of CTAB and Ethoquad 18/28
44
2.5
Equilibrium surface tension measurements as a function of temperature
45
2.6
Dynamic surface tension measurements for aqueous HEC solutions at 23°C
47
2.7
Surface tension measurements at equilibrium and higher bubble frequency
(surface age of 50 ms) for aqueous polymer solutions at 23°C
50
Surface tension measurements at equilibrium and higher bubble frequency
(surface age of 50 ms) for aqueous polymer solutions at 80°C
51
2.3
2.4
2.8
vi
2.9
(a) Measured contact angle for aqueous CTAB, Ethoquad 18/25, SDS,
Triton X-100, and Triton X-305 solutions; (b) corresponding ionic surfactant
adsorption surface state; and (c) EO group effect on surface wettability
53
(a) Contact angle and adsorption isotherms for nonionic surfactants
Triton X-100 and Triton X-305 in aqueous solutions; and
(b) non-ionic surfactant adsorption.
54
2.11
Relative viscosity changes of aqueous CTAB and Ethoquad 18/25 solutions
58
2.12
Variation of apparent viscosity with shear rate for aqueous
HEC QP-300 solutions
59
Variation of apparent viscosity with shear rate for aqueous Carbopol 934
solutions
60
Schematic of experimental facility: (a) pool boiling apparatus, and
(b) cross-sectional view of cylindrical heater assembly
63
3.2
Optical microscope images of the roughness characteristics of heater surface
64
3.3
Nucleate pool boiling data for aqueous solutions of DTAC; all data are for
decreasing heat flux except as otherwise indicated
67
Nucleate pool boiling data for aqueous solutions of CTAB; all data are for
decreasing heat flux except as otherwise indicated
68
Nucleate pool boiling data for aqueous solutions of Ethoquad O/12 PG;
all data are for decreasing heat flux except as otherwise indicated
69
Nucleate pool boiling data for aqueous solutions of Ethoquad 18/25;
all data are for decreasing heat flux except as otherwise indicated
70
Variation of the relative heat transfer performance of aqueous cationic
surfactant solutions with heat flux and additive concentration
(decreasing qw" )
73
3.8
Nucleate pool boiling data for aqueous solutions of HEC-QP300
75
3.9
Nucleate pool boiling data for aqueous solutions of Carbopol 934
76
3.10
Variation of the enhanced boiling heat transfer performance of HEC-QP300
solutions with heat flux and additive concentration
78
Effect of dynamic surface tension on the boiling heat transfer coefficient
80
2.10
2.13
3.1
3.4
3.5
3.6
3.7
3.11
vii
4.1
Measured streaming zeta potential and contact angle for the adsorption of
SDS and CTAB in their aqueous solutions
84
4.2
Measured contact angle for aqueous CTAB and Ethoquad 8/25 solutions
85
4.3
Dynamic surface tension relaxation for aqueous anionic SDS and
SLES solutions
87
Dynamic surface tension relaxation for aqueous nonionic
Triton X-100 and Triton X-305 solutions
88
Effect of surfactant molecular weight and its ethoxylation on the heat
transfer coefficient enhancement
90
Surfactant molecular weight dependence of the maximum enhancement in
heat transfer coefficient enhancement (Wasekar and Manglik, 2001)
91
(a) Effect surface wettability (or contact angle θ) on nucleate pool boiling
(Liaw and Dhir, 1989)
(b) Nucleate pool boiling data for Refrigerant of R113 on a copper tube
(Jung and Bergles, 1989)
93
Ebullient behavior in nucleate boiling of distilled water, and aqueous
cationic CTAB and Ethoquad 18/25 solutions of different concentrations
(C/CCMC = 0.5, 1, and 2) at qw" = 20 kW/m2 and 50 kW/m2
97
Ebullient behavior in nucleate boiling of distilled water, and aqueous
SDS (anionic) and Triton X-305 (nonionic) solutions of different
concentrations (C/CCMC = 0.5, 1, and 2) at qw" = 20 kW/m2 and 50 kW/m2
98
4.4
4.5
4.6
4.7
4.8
4.9
4.10
Ebullient behavior in nucleate boiling of distilled water, and aqueous
HEC-QP300 and Carbopol 934 solutions of different concentrations at
different heat fluxes ( qw" = 20 kW/m2, 50 kW/m2, and 100 KW/m2)
101
Schematic of interfacial phenomena in aqueous surfactant solutions
(not to scale)
104
4.12
Surfactant effects on nucleate boiling heat transfer in its aqueous solutions
106
4.13
(a) Schematic of surfactant transport process during a bubble formation and
departure (not to scale); (b) Dynamic surface tension effect on bubble
dynamics (evolution of pre-departure shape and size).
108
Physical domain of a boiling bubble decomposed into a macro region
and a microlayer
116
4.11
5.1
viii
5.2
MAC-staggered grid to show where the variables u, p, T, and φ are located
117
5.3
(a) Poisson problem with jump diffusion coefficients (T-shape); and
(b) test computational results for u(x,y) for the MPCG method (f = 100)
123
(a) Poisson problem with jump diffusion coefficients (arc); and
(b) test computational results for u(x,y) for the MPCG method (f = 100)
124
Contours of level-set function φ(x) with the solid line circle
representing φ(x) = 0
125
Rising bubble interfaces plotted at different times for U = 0, V = 1
using the mass-preserved level-set method
126
5.7
Schematic of a growing bubble in an extensive superheated liquid pool
127
5.8
Bubble growth in an extensive superheated liquid pool: (a) bubble interfaces
plotted at different times; (b) bubble growth with time
129
5.9
Microlayer shape and vapor-liquid interface temperature distribution for a
nucleated bubble
131
Temporal evolution of bubble shapes for various Weber numbers
(Re = 278)
133
5.11
Change in bubble rising velocity with time for various Weber numbers
133
5.12
Bubble growth and its departure in nucleate pool boiling for ∆T = 10K, ϕ = 45°.
(a) σ = 58.86 mN/m (water), (b) σ = 47.0 mN/m (SDS, C = 1000 wppm),
135
(c) σ = 37.5 mN/m (SDS, C = 2500 wppm )
5.13
Bubble departure diameter vs. surface tension (∆T = 10°C, ϕ = 45°)
137
5.14
Temporal evolution of bubble shapes for various Morton numbers
(We = 0.538)
139
5.15
Change in bubble rising velocity with time for various Morton numbers
139
5.16
Comparison of predicted result with the experiment for water:
(a) present simulation; (b) experimental result of Bhaga and Weber (1981)
140
Temperature isotherms during different bubble growth stages
142
5.4
5.5
5.6
5.10
5.17
ix
5.18
(a) Velocity field; (b) filled pressure contours; and (c) line pressure contours,
in and around a detached isolated bubble for ∆T = 10K, ϕ = 45°,
and σ = 37.5 mN/m, t = 0.06s
143
5.19
(a) Bubble shape at departure for different contact angles; and (b) bubble
departure diameter vs. apparent contact angle (∆T = 10°C, σ = 58.86 mN/m) 145
5.20
Effect of wall superheat ∆T on bubble growth (ϕ = 45°, σ = 58.86 mN/m):
(a) bubble shape at departure; (b) bubble departure time; and
(c) bubble departure diameter vs. wall superheat
147
5.21
Adsorption-desorption controlled surfactant interfacial transport process
148
6.1
Schematic representation and AFM detection images of adsorbed layer
structures consisting of (A) spherical micelles, (B) cylindrical micelles,
and (C) a bilayer (Schulz et al, 2001)
155
Conceptualization of possible surfactant adsorbate layers
at the solid-liquid interface
156
Proposed approach to correlation of nucleate boiling in aqueous
surfactant solutions
160
Proposed investigations to correlation of nucleate boiling in aqueous
surfactant solutions
160
6.2
6.3
6.4
x
NOMENCLATURE
A
Hamaker constant [J]
heater surface area (= 2πroL) [m2]
C
concentration [dimensionless]
Ca
Capillary number [dimensionless]
D
bubble diameter [m]
f
source term
bubble frequency [1/s]
Fr
Froude number [dimensionless]
g
gravity vector [m/s2]
Gr
Grashof number [dimensionless]
H
Heaviside function
h
grid spacing for macro region [m]
h
boiling heat transfer coefficient [kW/m2 K]
hev
evaporative heat transfer coefficient [W/m2 K]
hfg
latent heat of evaporation [kJ/kg]
I
unit vector
I
current [A]
Ja
Jacob number [dimensionless]
k
thermal conductivity [W/m2]
interface curvature in Eq. (5.7)
diffusion coefficient in Eq. (5.24)
xi
L
length of heated cylinder [m]
Γ
surface concentration [mol/m2]
L
characteristic length [m]
M
molecular weight [kg/kmol]
Mo
Morton number [dimensionless]
Na
active nucleation site density [m-2]
P
pressure [Pa]
Pe
Peclet number [dimensionless]
q
heat flux [W/m2]
q w′′
wall heat flux [W/m2, or kW/m2]
r
radial coordinate
R
radius of wall thermocouple location [m]
ro
cylindrical heater radius [m]
R
universal gas constant [J/mol K]
R
radial location of the interface at y=h/2
Re
Reynolds number [dimensionless]
S
sign function
T
temperature [K]
∆Tw
wall superheat [K]
t
time [s]
u
characteristic velocity vector [m/s]
U
x-direction velocity [m/s]
V
y-direction velocity [m/s]
xii
Vmicro
rate of vapor volume production from the microlayer [m3/s]
∆Vmicro
control volume near the micro region [m3]
V
voltage [V]
We
Weber number [dimensionless]
x
horizontal coordinate
Y
height of computational domain [m]
y
vertical coordinate
Greek
α
thermal diffusivity [m2/s]
β
Coefficient of thermal expansion [1/K]
D
diffusion term in Eq. (5.11)
δ
liquid film thickness [m]
φ
level set function [m]
γ
shear rate [s-1]
η
apparent viscosity [Pa⋅s]
ϕ
apparent contact angle [deg]
σ
surface tension [mN/m]
θ
dimensionless temperature
static contact angle [deg]
ρ
density [kg/m3]
µ
dynamic viscosity [N s/m2]
xiii
ν
kinematic viscosity [m2/s]
τ
surface age [ms]
Ω
computational domain in Eq. (5.24)
Subscripts
′
dimensionless
0
Initial
cap
Capillary
int
Interface
l,v
liquid, vapor
poly
pertaining to aqueous polymer solution
o
outer surface
r
radial location
sat
Saturation
surf
pertaining to aqueous surfactant solution
s
property at the surface
w
Wall
water
pertaining to pure water
∞
bulk, far field condition
50 ms
pertaining to surface age of 50 ms
xiv
CHAPTER 1
INTRODUCTION
Nucleate boiling is an important and efficient thermal management process with a
broad spectrum of applications, because relatively small temperature differences can
sustain very high heat transfer rates. Extensive research on numerous facets of boiling
heat transfer has been reported in the literature (Carey, 1992; Collier and Thome, 1996;
Dhir, 1998; Kandlikar et al. 1999). Pool boiling essentially occurs at a heated surface in
quiescent liquid, where its motion near the surface is driven by natural convection and
ebullient (bubble inception, growth, and departure) conditions; whereas in forced flow
boiling, liquid flow over the heater surface is imposed by external applied pressure
gradients. Figure 1.1 shows a typical pool-boiling curve, a plot of wall heat flux qw" vs.
the wall superheat ∆T (= Tw – Tsat), for the entire set of regimes that are encountered in
the heat transfer process.
The complete boiling curve is characterized by the following transport
characteristics. As the heat-input rate to the surface is increased, the first heat transfer
mode to appear is natural or free convection. Subsequently, at a certain value of the
superheat (Point A), vapor bubbles appear on the heater surface, and this is referred to as
the onset of boiling (ONB). In liquids that wet the surface well, the onset of nucleation
may be delayed. For these liquids, a sudden activation of large number of cavities at an
increased superheat causes a reduction in the surface temperature while the heat flux
remains constant, and this feature is often referred to as “Temperature overshoot”. This
behavior is not observed when the boiling curve is obtained by reducing the heat flux,
1
Log qw″
C
E
B
D
A
Hysteresis
Log ∆T
Fig. 1.1 Typical boiling curve for controlled wall heat flux and schematic
representation of the different heat transfer regimes
and, thus, a temperature hysteresis results. After inception, a dramatic increase in the
slope of the boiling curve is observed. In partial nucleate boiling (Region II, A-B),
discrete bubbles are released from randomly located active nucleation sites on the heater
surface. The transition from isolated bubbles to fully developed nucleate boiling (Region
III, B-C) occurs when bubbles at a given site begin to merge in the vertical direction in
the form of jets. The maximum or critical heat flux (CHF) sets the upper limit of the
2
fully developed nucleate boiling. After CHF, most of the surface is very rapidly covered
with vapor; the surface temperature rises quickly. When the heat input rate is controlled,
the heater surface will rapidly pass through Regions IV (C-D) and V (D-E), and stabilize
at Point E in the film boiling regime. If the temperature at E exceeds the melting point of
the heater material, the heater will be burnt out. Further details about nature of boiling
and boiling behaviors in different regimes can be found in Dhir (1998) and Carey (1992),
among others.
In recent years, enhancement of nucleate boiling heat transfer has received much
attention, and different active and passive techniques have been documented in several
reviews (Thome, 1990; Bergles, 1997; Manglik, 2003). The use of liquid additives in
particular, which include surfactants or surface-active substances that significantly alter
the surface tension of the boiling liquid even at very low concentrations, has been the
focus of some current research. The reviews by Wasekar and Manglik (1999, 2001)
provide an extended discussion of several issues associated with enhanced boiling heat
transfer in surfactant and polymeric solutions.
Several studies have investigated
enhanced pool boiling in aqueous surfactant and polymeric solutions under atmospheric
conditions, and a variety of different predictive parameters and mechanisms have been
proposed to describe the complex phase-change process. The primary determinants of
the general boiling problem, however, can essentially be classified under three broad
categories: heater, fluid, and heater-fluid interface (Nelson et al., 1996; Nelson, 2001).
For nucleate boiling with additives in aqueous solutions, the associated potential
mechanisms that may be involved are depicted as a conjugate problem in Fig. 1.2.
3
HEATER (smooth or structured)
Physical
properties
Geometry
Heat flux (wall
superheat)
Shape: Plate,
Orientation
cylinder, Thickness,
wire, etc. diameter
Surface characteristics
(fractal dimension)
Cavity density
Roughness
Constant wall temperature or
Constant heat flux
HEATER-FLUID
INTERFACE (with
or without additives)
Contact angle Additive physi(wettability) sorption characteristics
(zeta potential)
Active site density
Interaction or potential interactions
FLUID (with or without additives)
Physics properties
Thermal
conductivity
Additive
Viscosity
surface tension σ
Near-surface features Far-surface features
Meniscus Vapor stems
Vapor
columns
Liquid in microlayer Discrete Vapor
and macrolayer
bubbles mushrooms
Liquid-vapor interface Foaming
Physico-chemical properties
Marangoni convection
(adsorption/desorption)
Dynamic σ Ionic nature Ethoxylation Molecular weight Apparent viscosity
Fig. 1.2 The conjugate problem in modeling nucleate pool boiling with or without additives
4
1.1
Surfactants and Interfacial Phenomena
1.1.1
Surfactants
Surfactant is a generic term for a surface-active agent, which literally means
active at a surface. It is fundamentally characterized by its tendency to adsorb at surfaces
and interfaces when added in low concentrations to an aqueous system. Surfactants have
a unique long-chain molecular structure that is composed of a hydrophilic head and a
hydrophobic tail as illustrated in Fig. 1.3. They have a natural tendency to adsorb at the
liquid-vapor interface with their polar head oriented towards the aqueous solution and the
hydrocarbon tail directed towards the vapor. Based on the nature of the hydrophilic part
of the molecule, which is ionizable, polar, and polarizable, surfactants are generally
categorized as anionics, nonionics, cationics, and zwitterionics (Holmberg et al., 2003).
Hydrophilic
(polar) head
Hydrophobic (non polar) tail
Fig. 1.3 Schematic illustration of the primary structure of a surfactant molecule
1.1.2
Colloid systems and interfacial phenomena
There are three states of matter – gas, liquid, and solid, and when one of these
states is finely dispersed in another, then a colloidal system is obtained. This can be in
the form of aerosols, emulsions, inverse emulsions, sols or colloidal suspensions, gels,
biocolloids, or association colloids, based on the different dispersion state (Hunter, 2001).
5
Colloidal systems are of great practical importance, and Table 1.1 lists various types of
colloidal systems and their common examples.
Table 1.1 Typical Colloidal Systems (Birdi, 2003)
Dispersed
Phases
Continuous
System type
Liquid
Gas
Aerosol fog, spray
Gas
Liquid
Foam, fire extinguisher foam
Liquid
Liquid
Emulsion (milk)
Solid
Liquid
Sols, suspension waste water, cement
Corpuscles
Serum
Blood
Hydroxyapatite
Collagen
Bone
Liquid
Solid
Solid emulsion (toothpaste)
Solid
Gas
Solid aerosol (dust)
Gas
Solid
Solid foam – insulating foam
Solid
Solid
Solid suspension/solids in plastics
Biocolloids
Colloidal systems often exhibit rather unusual phenomena at their phase
boundaries (interfaces), relative to the expected bulk phase interactions, such that the
behavior of the entire system is controlled by interfacial processes (Rosen, 1989; Evans
and Wennerström, 1999; Hunter, 2001; Birdi, 2003; Holmberg, 2003; Chen, 2003).
Interfacial phenomena are important in almost every industrial process, from
heterogeneous catalysis to the manufacturing of composite materials, from medical
technology to detergency, as well as in numerous processes such as thermal treating,
coating, nanofluids, dispersion, flotation, solubilization of chemicals, oil exploitation,
crystallization, fabrication of compound systems (reinforced materials and coated
6
materials), and boiling. With nucleate phase-change and ebullience in aqueous surfactant
solutions, which are association colloid systems, where molecules of surface-active
substances (e.g. surfactants) are associated together to form small aggregates (micelles)
in water, the aggregates formed may often adopt an ordered structure. The consequent
interfacial changes significantly affect boiling. Thus, it is critical to understand the
causes and results of interfacial behavior and variables that affect it in order to predict
and control the properties of not only boiling in surfactant solutions but all colloidal
systems in general.
Altogether five different interfaces can exist: gas-liquid, gas-solid, solid-liquid,
liquid-liquid, and solid-solid. In the case of surfactant solutions, the additive may adsorb
at all of the five types of interfaces. For nucleate pool boiling in aqueous surfactant
solutions, however, there are two primary interfaces that have a dominating influence: (1)
vapor-liquid interface, at which the surface tension reduces because of the surfactant
adsorption-desorption process, and (2) solid-liquid (or heater-liquid) interface, where the
surfactant physisorption occurs and the surface wetting behavior changes.
1.1.3
Surface tension and micelles
Surfactant additives in aqueous solutions naturally tend to diffuse towards the
vapor-liquid interface and subsequently get adsorbed on it.
Depending upon their
chemistry (ionic and molecular structure) and orientation at the interface, some
desorption may also occur. The primary effect of the surfactant adsorption-desorption
process at the vapor-liquid interface is to reduce the surface tension of the solution. This
entire process is time-dependent and it manifests in a dynamic surface tension behavior at
7
an evolving vapor-liquid interface (as in ebullience), which eventually reduces to an
equilibrium value after a long time span.
Surface tension reduction of an aqueous solution decreases continually with
increasing concentrations till the critical micelle concentration (CMC) is reached, at
which point the surfactant molecules cluster together to form micelles. All surfactants in
their solutions show significant changes in adsorption behavior at or around their
respective CMC. The CMC is characterized by micelle formation, or micellization,
which is the property of surface-active solutes that leads to the formation of colloid-sized
clusters, i.e., at a particular concentration, additives form aggregates in the bulk phase or
a surfactant cluster in solution that are termed micelles (Edwards et al., 1991). Different
shapes and sizes of micelles exist depending upon the surfactant type and its packing,
concentration, solution temperature, presence of other ions, and water-soluble organic
compounds in the solution, and typical micelles are shown in Fig. 1.4. However, as
pointed out by Porter (1994), the micelle is a dynamic entity and its structure and shape
can change with time. Further aggregation above a certain temperature referred to as the
cloud point, leads to the separation of the liquid phase that gives the solution a turbid
appearance.
The presence of ethoxy or ethylene oxide (EO) group in surfactants changes the
critical packing parameter (CPP)1, which in turn is a key determinant of their micellar
structure and CMC in their aqueous solutions (Rosen, 1989; Holmberg et al., 2003). The
micelle formation in highly ethoxylated surfactants consistently yields vesicular
mesophases, and their formation results in a reduction in CMC at elevated temperatures
1
CPP is the ratio between the cross-sectional area of the hydrocarbon tail part and that of the polar head
group of the surfactant molecule.
8
(Holmberg et al., 2003). On the other hand, molecules of CTAB (a non-ethoxylated
cationic) typically cannot pack into a cone truncated by surfaces of high and opposite
curvature as needed to direct the vesicular mesostructure (Lu, et al., 1999); instead,
lamellar mesophases are commonly formed in bulk and thin-film samples.
(a) Spherical micelle
(d) Reversed micelle
(b) Cylindrical micelle
(e) Bicontinuous micelle
(c) Lamellar phase
(d) Vesicle
Fig. 1.4 Different possible micellar structures (Evans and Wennerström, 1999)
9
1.1.4
Electrokinetic effects and zeta potential
When two phases come in contact, they generally develop a potential difference
between them. With the presence of ions, or excess electrons, or ionogenic groups in one
or both phases, there is a tendency for the electric charges to distribute themselves in a
particular direction at the interface (Hunter, 1981; Lyklema, 1991; Evans and
Wennerström, 1999; Hunter, 2001). An electrokinetic boundary layer develops when a
solid surface containing immobilized electrical charges comes in contact with an aqueous
solution of mobile ions. Also referred to as an electric double layer (EDL) as shown in
Fig. 1.5 (Hunter, 2001), it generally consists of the following two layers: (1) an immobile
or stern layer of ions opposite in sign to that of the surface, and (2) a diffuse layer or a
cloud of hydrated ions, which transition from the significant excess of counterion at the
SOLID SURFCAE
fixed charge layer to a balance of cations and anions in the bulk solution.
Stern Layer
+
+
_ +
_
+
+
_
+ _
+
_
+
_ + _
+
+
+
_ + +
+ _ _ _ +
_
+
Diffuse
Layer
_
_
+
_
DISTANCE
ζ POTENTIAL
0
Fig. 1.5 Schematic diagram of a typical electrokinetic boundary layer
10
A variety of phenomena or “electrokinetic effects” are observed when one of
these phases is caused to move tangentially past the second phase. The consequent forces
on the solid or liquid interface can be characterized in terms of either the charge or
electrostatic potential (Lyklema, 1991; Evans and Wennerström, 1999; Hunter, 2001). In
the later case, the average potential in the surface of shear or zeta potential ζ is the
fundamental quantifying parameter, and it provides the basis for explaining a variety of
natural phenomena in colloid chemistry and electrochemistry. These include electrode
kinetics, electrocatalysis, corrosion, adsorption, and crystal growth, among others, and
the concomitant flow behavior and colloid stability cannot be treated without knowledge
of charge distribution in the interfacial region (Lyklema, 1991; Evans and Wennerström,
1999; Hunter, 2001; Birdi, 2003). Many of the important properties of colloidal systems
are determined directly or indirectly by the potential at the interfaces. Adsorption of ions
and molecules, for instance, is determined by the charge and potential distribution, which
itself determines the interaction energy between the molecules or particles.
There are four distinct electrokinetic or zeta potential effects depending on the
way in which motion is induced: electrophoresis, electroosmosis, streaming potential, and
sedimentation potential (Hunter, 1981; Delgado, 2002), and Fig. 1.6 shows the
relationship between them. For the surfactant adsorption at the solid-liquid interface, the
streaming zeta potential will arise when the solution is in contact with the stationary
surface. The measurement of streaming potential is relatively straightforward, as has
been described by Hunter (1981), Gu and Li (2000), and Gusev and Horváth (2002), and
others. In principle, the double layer ions are carried downstream by the flow through a
capillary, and their accumulation there generates a field that causes a back conduction.
11
When the forward and the back currents are equal, then the potential difference across the
capillary is the streaming potential. Streaming potential measurements are primarily
dependent upon the following: (i) the applied pressure, (ii) the conductivity of the liquid
in the capillary or plug, and (iii) the streaming potential developed.
The fundamental importance of the zeta potential change and physisorption
process in characterizing nucleate boiling in aqueous surfactant solutions cannot be
overstated. In these association colloids, molecules of the surface-active additive form
small aggregates or micelles in water, which tend to adopt an ordered structure. The
adsorption of ionic surfactants, in particular, at the solid-liquid interface alters the
behavior of the solid surface considerably (Hunter, 2001; Fuerstenau, 2002). Based on
this adsorption process, the electrokinetics and wettability behaviors of the solid surface
can be explained from the ions exchange in the EDL, and are directly reflected in the
change in ζ.
12
Di
sp
es y
v
o ar S
Li erse
E
M
n
LE qui Ph
o I
aseStati RES
CT d P ase
h
P e O
ROhas St
e
rs has PH
e
- O e M a t io
sp id P RO
i
SM ov nar
D qu T
OSes y
C
i
Applied
Field
L LE
Causes Movement
IS
E
ZETA
POTENTIAL
ST
L
R
IA
T
Di EA
Applied Force Results
N n
in Potential
LiqspersMIN
TEotio y
e
ui d P G
PO M nar
Ph has POT
N
in io
ase e S E
IO ase Stat
N
t
T
a
in tio T
h
Mo na IA
TAse P hase
L
N
tio r y
E per d P
n
i
IMDis qu
D
Li
SE
Fig. 1.6 Four basic electrokinetic phenomena and the relationship between them
13
1.1.5 Surfactant physisorption at the solid-liquid interface
The adsorption of ionic surfactants at a solid-liquid interface in physically
adsorbing systems, i.e., in systems where chemical interactions are absent and adsorption
at low concentrations occurs through electrostatic interaction between the surface and
surfactant ions charged oppositely to the surface, has been extensively investigated in the
literature. Various theories about the structure of the adsorbate layer have also been
presented (Hunter, 2001; Dobiáš and Rybinski, 1999). Of these, the hemimicelle or
Fuerstenau model has been broadly adopted in colloid science, and the pioneering work
on the nature of surfactant adsorption that lead to this model was reported by
Somasundaran and Fuerstenau (1966).
The typical isotherm in Fig. 1.7 shows four distinct regions of surfactant
adsorption that are associated with the aggregation mode of adsorbed ions at the solidliquid interface (Somasundaran and Fuerstenau, 1966; Scamehorn et al. 1981; Dobiáš and
Rybinski, 1999; Fuerstenau, 2002). These can be summarized as follows:
Region I: At low concentrations, surfactant adsorption takes places as individual ions in
the diffuse part of the double layer and in the stern plane, and obeys Henry’s
law. There is no chain-chain association, and it is independent of the
hydrocarbon chain length. Under these conditions, the zeta potential is almost
constant and surface wettability remains relatively unchanged.
Region II: There is sharp increase in the slope of the adsorption isotherm due to selfassociation of adsorbed surfactant ions, and the formation of hemimicelles
(Bisio, 1980; Fuerstenau, 2002). Adsorption takes place primarily in the stern
plane, and the polar heads of the surfactant ions are oriented toward the
14
CMC (critical micelle concentration)
Adsorption Density
Region IV
Plateau adsorption region
(Double layer)
Region III
Reverse Hemimicelles
PZR (point of zeta potential reversal )
Region II
Hemimicelles
Region I
Individual Ions start to aggregate
Individual Ions
Concentration
Hydrophilic head
Hydrophobic tail
Hydrophobic
Hydrophilic
Fig. 1.7 Schematic representation of a typical adsorption isotherm and aggregate
states of a surfactant on a solid surface in aqueous solutions
15
surface. The surface is hydrophobic in this region for most surfactants except
for high molecular weight ones, whose bulky polar head would occupy a
larger surface area.
Region III: Is characterized by a decrease in the slope of the adsorption isotherm due to
the reversal of the ζ potential, which becomes increasingly negative for
anionic surfactants and increasing positive for cationic surfactants because of
the opposite charges they carry.
The surfactant ions adsorb as reverse
hemimicelles (Fuerstenau, 2002), with their polar heads oriented both toward
the surface and liquid, and the surface becomes increasing hydrophilic.
Region IV: As the CMC is approached, the adsorption isotherm becomes independent of
the surfactant concentration in solution, forming a bilayer or equivalent
(Manne and Gaub, 1995; Schulz et al., 2001; Fuerstenau, 2002; Richard et al.,
2003). The ζ potential tends to remain constant and the surface becomes
strongly hydrophilic.
In this characterization, it should be noted that there are two important transition points in
the overall adsorption process. The first is at the end of Region II, or the intersection
between Regions II and III, where the ζ potential is zero and all of the adsorbed ions must
be in the Stern plane. This point is variously referred to in the literature as the isoelectric
point (IEP)2 (zero electrokinetic potential), or point of zero charge (PZC), or point of zeta
potential reversal (PZR). In colloidal systems, detecting the IEP poses considerable
difficulties, and ζ potential seems a more reliable indictor of specific or inner layer
adsorption (Hunter, 2001; Birdi, 2003). Under these conditions, one part of the double
2
The concentration of the potential determining ion at which the zeta potential is zero is defined as the
isoelectric point (IEP).
16
layer reflects the surface charge on the solid surface and the other part constitutes the
oppositely charged surfactant ions. The second transition point is at the end of region III,
when CMC is reached. The bilayer is formed at this transition and the contact angle
tends to remain constant even when C ≥ CMC.
Several variations of this adsorption model have been postulated in the literature.
Bisio et al. (1980), Scameborn et al. (1981), and Chandar et al. (1987) modified the
Fuerstenau model by postulating the formation of surfactant double layers (bilayers) as
well as surfactant monolayers in the second adsorption stage. They called them lamellar
layers, where on top of the first layer (with head groups are oriented toward the surface) a
second layer (with head groups are oriented toward the bulk) is formed by means of
hydrophobic forces. In their “admicelle” concept, Harwell et al. (1985) and Bitting and
Harwell (1985) took into account the surface inhomogeneities, and indicated that the
monolayer aggregates are never formed, which contrasts with the “bilayer” model. They
regard these bilayer sections on the surface as a pseudophase and call them “admicelles”
(adsorbed micelles) to differentiate them from Fuerstenau’s hemimicelles. Koopal and
Ralston (1986) further provide a quantitative analysis of chain-chain interactions and the
influence of chain conformation without reference to the notion of hemimicelle
formation.
These models are different, however, in almost all the systems that have been
investigated, generally there is increasing hydrophilization of the surface with increasing
surface coverage when IEP has been exceeded, which is typically expressed by an
improvement in dispersibility of the solid particles in the aqueous solutions (Dobiáš et al.,
1999). Some exceptions are charged or polarized surfaces, which are going to alter the
17
orientations of the adsorbed ionic surfactants because of the electro-repulsions. Also,
with some types of surfactants that may form trilayers or reversed micelles when CMC is
approached, different adsorbate layers and surface wetting behaviors may be exhibited
(Evans and Wennerström, 1999).
1.2
Nucleate Boiling and Surface Wettability
1.2.1
Contact angle and surface wettability
Surface wettability is critical to many applications that are as diverse as power
generation, chemical processing, and biological systems. It generally refers to the
manifestation of the molecular interactions between liquids and solids in direct contact at
the interface (Blake, 1993), and is generally reflected in the contact angles (Lyklema,
1991; Kwok and Neumann, 1999). In multiphase heat transfer, the nature and dynamics
of surface liquid-solid contact often plays a dominant role (Chen, 2003).
The liquid-solid system can be either completely wetting (θ = 0°), or have
different degrees of wetting (0 < θ < 180°), or be complete non-wetting (θ = 180°) as
schematically illustrated in Fig. 1.8. Knowledge of the intermolecular interactions, both
within the liquids, and across the liquid-vapor (or liquid-gas) and liquid-solid interfaces,
is an important part of characterizing surface wettability or contact angle. When a drop
of liquid is placed on a solid substrate, it may spread so as to increase the liquid-solid and
liquid-gas interfacial areas. Simultaneously, the solid-gas interfacial area decreases and
the contact angle θ between the drop and solid is reduced. The value of θ can be seen as
a measure of the balance between the tendency of the drop to spread so as to cover the
solid surface and to contract in order to minimize its own surface area (Decker et al.,
18
1999; Holmberg et al., 2003); spreading would typically continue until the system
reaches equilibrium.
θ = 0º
θ = acute
complete wetting
(hydrophilic)
θ
different degrees
of
θ = obtuse
θ
wettability
θ = 180º
complete non-wetting
(hydrophobic)
Fig. 1.8 Different wetting conditions of a liquid drop on a solid surface
19
The degree of spreading is governed by the surface tension of the liquid σLG, the
surface tension of the solid σSG (usually referred to as the surface free energy), and the
interfacial tension of the liquid and the solid σSL. These forces that essentially represent
the liquid-gas, solid-gas, and solid-liquid interfacial tensions, and their interactions are
depicted in Fig. 1.9.
σLG
Vapor or gas
Liquid
θ
σSG
Solid
σSL
Fig. 1.9 Surface forces involved in spreading of a liquid
The surface free energy of the solid σSG, tends to spread the drop, i.e., to shift the threephase point forward or along the surface. Thus, spreading is generally favored on highenergy surfaces. The interfacial tension σSL and the horizontal component of the surface
tension force σLGcosθ act in the opposite direction. At equilibrium state, the resultant
force is thus zero and
σ SG = σ SL + σ LG cosθ
(1.1)
This expression is very well known as Young’s equation and has become the basis for
understanding the phenomenon of contact angle or surface wetting on solid surfaces.
20
Contact angle measurements are typically made using either sessile drops or
adhering bubbles, and are usually automated and computerized, which enables values of
the contact angles to be determined with a high degree of reproducibility. Also, the
contact angle can be influenced by the surface roughness, surface chemical heterogeneity,
and impurities, among some other factors (Kwok and Neumann, 1999).
1.2.2
Surface Wetting Effects on Nucleate Boiling
The phenomenological modeling of mechanisms for heat and fluid transport in
nucleate boiling is still not completely developed, and is the subject of much study. One
reason for this is the complexity of interfacial contact at the surface, which is affected by
intermolecular forces. Given that the primary heat transfer is by evaporation and its
efficiency is directly related to nucleation site density and bubble dynamics, surface
wetting becomes an important predictor. To characterize the inter-relationship, insights
are often obtained from a visual observation of the ebullience.
As a critical determinant of nucleate boiling, active nucleation site density is
found to be a function of wettability and heat flux, and it directly accounts for the energy
transfer by ebullience at the heater surface (Dhir, 1998; Barthau, 1992). Wang and Dhir
(1993), Dhir (1998), and Basu et al. (2002) have systematically studied the effect of pure
liquid wettability on the active nucleation site density and onset of nucleate boiling
(ONB) in pool boiling. They have correlated their data for active site density as a
function of the wall superheat and contact angle, and pointed out that the fraction of the
nucleated cavities decreases as the wettability of the surface increases. The wettability of
the surface in their collective work was changed by controlling the degree of oxidation of
21
the heater surface. Hibiki and Ishii (2003) have also presented results that map active
nucleation site density as a function of contact angle and wall superheat, as well as the
critical cavity size. Similarly, as reviewed by Dhir (1998) and Kenning (1999), increased
surface wettability, which results in fewer nucleation sites, typically produces larger
bubbles with lower departure frequencies.
1.3
Nucleate Pool Boiling with Surfactants
Small amounts of surfactant additives in water tend to change and enhance the
boiling heat transfer in water by essentially modifying nucleation and the concomitant
bubble dynamics. The importance of surfactant-enhanced boiling heat transfer has been
widely recognized, and many studies have investigated the pool boiling behavior in
aqueous surfactant solutions under atmospheric conditions. A recent review of this body
of work was provided by Wasekar and Manglik (1999), and Table 1.2 gives a more
comprehensive chronological listing of the available literature on experimental
investigations.
Boiling with surfactant additives is a very complex process as shown earlier in
Fig. 1.2. Besides the effects of heater geometry, its surface characteristics and wall heat
flux level, the bulk concentration of additive, surfactant chemistry (ionic nature and
molecular weight), dynamic surface tension of the solution, surface wetting and
nucleation cavity distribution, marangoni convection, surfactant adsorption and
desorption, and foaming are some of the factors that appear to have a significant
influence (Wu, et al. 1998; Hetsroni, et al., 2001; Wasekar and Manglik, 2001, 2002).
Also, the bubble dynamics (inception and gestation → growth → departure) has been
22
found to be considerably altered with reduced departure diameters, increased frequencies,
and decreased coalescence (Wu et al. 1995; Wasekar and Manglik, 2000; Hetsroni, et al.,
2001). A direct correlation of the heat transfer with suitable descriptive parameters for all
of these effects, however, remains elusive because of the complicated nature of the
problem.
Table 1.2 Chronological Listing of Nucleate Pool Boiling Studies
of Aqueous Surfactant Solutions
Author(s)
Heater
Geometry
Surfactants
Stroebe et al. (1939)
Morgan et al. (1949)
Jontz and Myers (1960)
Roll and Myers (1964)
Cylinder
Cylinder
Plate
Plate
Duponol
Drene; SDS
Tergitol; Aerosol-22
Aerosols: OT, AY, IB, and MA;
Hyonics: PE-200
Frost and Kippenhan
(1967)
Huplik and Raithby
(1972)
Shah and Darby (1973)
Shibayama et al. (1980)
Cylinder
Ultra Wet 60L
Podsushnyy et al. (1980)
Cylinder
Filippov and Saltonov
(1982)
Yang and Maa (1983)
Saltanov et al. (1986)
Chang et al. (1987)
Tzan and Yang (1990)
Cylinder
Octadecylamine
Plate
Cylinder
Cylinder
Cylinder
SLS and SLBS
Octadecylamine
SDS
SDS
Plate
FC-176
Plate
Plate
Joy
Sodium oleate; Rapisool B80;
Puluronic: F98, F88, F208
PVS-6 polyvinyl alcohol, NP-3
sulfonol, and SV1017 wetting agent
23
Table 1.2 (continued)
Liu et al. (1990)
Plate
Chou and Yang (1991)
Wu and Yang (1992)
Plate
Cylinder
BA-1, BA-2, BA-3, BA-4, DPE-1,
DPE-3, Gelatine, Oleic acid,
Trimethyl octadecyl ammonia
chloride, trialkyl methyl ammonia
chloride, and polyvinyl alcohol
SDS
SDS
Wang and Hartnett (1992)
Wu et al. (1993)
Tan and Wang (1994)
Lin et al. (1994)
Wu et al. (1994)
Wang and Hartnett (1994)
Wire
Tube
Cylinder
Sphere
Sphere
Wire
SDS
SDS
WY
SDS
SDS
SDS and Tween-80
Wu et al. (1995)
Cylinder
SDS, Tergitol, Aerosol-22,DTMAC,
Tween-20, 40, 80, n-Octanol, and
Triton X-100
SDS
Ammerman and You
(1996)
Qiao and Chandra (1997)
Manglik (1998)
Wu et al. (1998a)
Wu et al. (1998b)
Wu et al. (1999)
Wire
Plate
Cylinder
Cylinder
Cylinder
Cylinder
Yang et al. (2000)
Cylinder
Wasekar and Manglik
(2000)
Hetsroni et al. (2001)
Yang et al. (2002)
Wen and Wang (2002)
Wasekar and Manglik
(2002)
Cylinder
SDS
AGS
SDS, and Triton X-100
n-octanol in water and LiBr
SDS; DTMAC; Triton X-100,
Aerosol
CPC, SDS, DTMAC, DTMADS,
CPDS
SDS
Plate
Cylinder
Plate
Cylinder
Habon G
Triton SP-190, SP-175
SDS, Triton-X-100, Octadecylamine
SDS, SLES, Triton X-100, X-305
24
A variety of different predictive parameters and mechanisms have been proposed
to describe the complex phase-change process, but much of the focus has been on the
effect of reduced interfacial tension.
Because of the highly dynamic nature of the
nucleate boiling, typically in the range of 0-100 ms (Prosperetti and Plesset, 1978), the
dynamic surface tension of the aqueous surfactant solution instead of the equilibrium
surface tension has been proposed as a predicator of the nucleate boiling process
(Wasekar and Manglik, 2002). Furthermore, the addition of a small amount of surfactant
to water, not only changes the liquid-vapor interfacial behavior, but, more importantly, it
also alters the solid-liquid interfacial characteristics. All the factors related to surface
wettability thus get affected, including one of the most important ones in nucleate boiling
- active nucleation density.
The wetting of aqueous surfactant solutions is further found to be influenced by the
additive’s chemical structure. The presence of the ethoxy or ethylene oxide (EO) group
in its molecular-chain, in particular, increases the overall size of the polar head and makes
the surfactant more hydrophilic (Barry and Wilson, 1978; Evans and Wennerström, 1999;
Holmberg et al., 2003). This increases surface wettability due to the adsorption of
surfactant molecules on the solid surface (Ashayer et al., 2000).
The concomitant
influence on the active nucleation site density and dynamic contact angle should therefore
be taken into consideration in characterizing nucleate boiling of aqueous surfactant
solutions.
25
1.4
Nucleate Pool Boiling with Polymers
Thermal processing of fluid media to produce biochemical, pharmaceutical,
personal care, and hygiene products is a very complex heat transfer problem. It typically
entails heating and drying of aqueous polymeric solutions by boiling, in order to thicken
them and make pastes. Rather anomalous phase-change behaviors have been observed in
this process (Kotchaphakdee and Williams, 1970; Wang et al., 1992; Shul’man et al.,
1993), and the consequent lack of close thermal control often leads to product loss and
quality degradation. A variety of factors play a role, and they include the type of
polymer, its molecular weight and concentration, solution rheology and interfacial
properties (surface tension and wettability), heated surface geometry, and heat flux levels,
among others.
Several studies have investigated nucleate pool boiling characteristics of
polymeric solutions under atmospheric conditions, and Table 1.3 gives a chronological
listing of most of the available literature. In one of the earliest studies on the effects of
polymer additives on boiling of water on a plate heater, Kotchaphakdee and Williams
(1970) found the heat transfer to be enhanced in HEC-H and PA-10 solutions. While
both additives make the solution more viscous with a shear-thinning flow behavior, HECH has surface-active properties as well (i.e., it reduces surface tension of the solution
appreciably) and a lower molecular weight (M = 2×105); for PA-10, M = 106. The
combined effects of nucleation site density and solution concentration on boiling are
reported by Ulicny (1984). Appreciable enhancement was observed in aqueous HEC
solution on a 600 grit roughened surface, whereas no apparent enhancement was
observed for the same solution on a much smoother surface. The experimental data of
26
Table 1.3 Chronological Listing of Nucleate Pool Boiling Studies
of Aqueous Polymeric Solutions
Author(s)
Kotchaphakdee and
Heater
Geometry
Polymers
Plate
Acrylamide, PA-10, PA-20, HEC-
Williams (1970)
Miaw (1978)
Yang and Maa (1982)
Paul and Abdel-Khalik
L, HEC-H, HEC-M
Plate
HEC-H, PA-10, PA-30
Plate and wire HEC-250HR, 300HR, 250GR
Wire
Separan AP-30, NP-10P, MGL;
(1983)
PEO; HEC 250MR, 250HR, 250
HHR
Ulicny (1984)
Plate
PA-10
Hu (1989)
Wire
Separan AP-30, HEC 250HHR
Wang and Hartnett
Wire
SLS and Separan AP-30
Shul’man et al. (1993)
Plate
PAA, HEC-H, PEO
Shul’man and Levitskiy
Plate
PAA, HEC-H, PEO
Plate
PAA, HEC-H, PEO
(1992)
(1996)
Levitskiy et al. (1996)
Bang, et al. (1997)
Sphere
PEO
Wang and Hartnett (1992), Paul and Abdel-Khalik (1983), and Hu (1989), however,
indicate a deterioration in boiling heat transfer for very dilute aqueous polymeric
solutions when compared to that of pure water. All of these studies (Paul and AbdelKhalik, 1983; Hu, 1989; Wang and Hartnett, 1992) used platinum wire heaters instead of
a plate heater (Kotchaphakdee and Williams, 1970; Ulicny, 1984) and geometry effects
may be present. The results of Yang and Maa (1982) are even more contrary, and the
same boiling heat transfer performance for dilute aqueous HEC solutions with both a
27
plate and platinum wire heater has been reported. More recently, Shul’man et al. (1993),
and Levitskiy et al. (1996) have shown that even with the same kind of polymers of the
same molecular mass, the boiling performance can substantially change with
concentration, temperature, and external conditions. They report enhanced boiling heat
transfer in dilute solutions (C = 15-500 wppm), but a decreased boiling heat transfer in
highly concentrated aqueous solutions (C = 1%) of HEC-H on a plate heater, which has a
characteristic size that is much larger than the mean size of the bubbles.
The changed boiling heat transfer in polymer solutions is also displayed in a
markedly different bubbling behavior (shape and size of bubbles, their growth rate,
foaming, and nucleation frequency, etc.) compared to that of pure water (Shul’man et al.,
1993; Hu, 1989; Levitskiy et al., 1996). Bubbles of smaller sizes and regular shapes are
released from the heater with higher frequencies than seen in pure water, and they rise in
a more orderly fashion. Adjacent bubbles tend to coalesce less due to the effects of
normal stresses and longitudinal viscosity in thin films formed between them. Levitskiy
et al. (1996) suggest that the change in the wetting angle along with a reduction in surface
tension for HEC-H solutions account for the observed decrease in bubble sizes. The
changes in the interfacial characteristic are perhaps a direct consequence of the molecular
adsorption dynamics of the additive.
Polymers are typically large molecules, macromolecules, or agglomerates of
smaller chemical units called monomers, and are broadly classified as biological or nonbiological macromolecules. Their addition in water primarily increases the solution
viscosity, which tends to increase with concentration as well as the molecular weight of
the polymer, and often display a shear-rate dependent shear-thinning rheology (Chhabra
28
and Richardson, 1999; Carreau et al., 1997). With the exception of some surface-active
polymers (or polymeric surfactants) such as hydroxyethyl cellulose (HEC) and
polyethylene oxide (PEO), most polymeric solutions do not show any significant change
in surface tension σ (Manglik et al., 2001; Hu et al., 1991). The viscosity of the polymer
solution, however, can influence σ measurements considerably, especially at higher
viscosity and bubble frequency (Manglik et al., 2001; Fainermann et al, 1993; Janule,
1998). The reduced surface tension in solution of surface-active agents is largely brought
about by the molecular adsorption of the additives to the vapor-liquid interface
(Holmberg et al., 2003). The time scales of this process vary from order of seconds to
minutes depending upon the polymer chemistry and its concentration in solution, which
is possibly due to the slow processes of diffusion transport of polymer molecules to the
interface and their subsequent reorientation (Persson et al., 1996).
This dynamic
adsorption process, along with time scales of 10-100 ms for boiling bubble dynamics in
water (Prosperetti and Plesset, 1978), thus results in a rather complex interfacial
behavior, that significantly alters the nucleate pool boiling in polymeric solutions.
29
1.5
Computational Fluid Dynamics with Moving Boundaries
In order to develop theoretical constructs for the complex ebullient phase-change
behavior, computational modeling provides an attractive tool.
However, this poses
considerable difficulties, where, for example, a direct calculation of the nucleate boiling
flows without empirical correlations essentially requires accurate tracking of the twophase interfaces. These flows are characterized by the discontinuity of many variables
across the phase interface. These discontinuities pose several computational challenges
requiring special treatment. In addition, the location of the interface is not known a priori
and must be found as part of the solution.
Under the broad categories of Langrangian and Eulerian methods, several
numerical techniques have been developed so far by researchers in the area of moving
boundary problems. In the class of finite-difference methods, such as the marker-and-cell
(Harlow and Welch, 1965) and Volume-of-Fluid (VOF) (Hirt and Nichols, 1981)
methods, the moving interface is traced with marker particles or functions that are
advected through the finite difference mesh. In these methods, the physical quantity at a
computing cell implying an interface is calculated by volumetric averaging of vapor and
liquid phases; the discontinuous interface is therefore likely to be smoothed out. The
numerical schemes such as the Arbitrary-Lagrangian-Eulerian (ALE) method (Hirt et al.,
1974), Boundary-Element Method (BEM) (Brebbia et al., 1984), Boundary-FittedCoordinates (BFC) method (Ryskin and Leal, 1984), and Finite-Volume-Method (FVM)
(Rhie and Chow, 1983), have an advantage in that their mesh can be generated to fit the
interface shape. The generation of interface-fitted mesh, however, is troublesome when
the interface merges or has large deformation. In a combined Langrangian-Eulerian
30
method or the Front-Tracking Method (FTM) (Unverdi and Tryggvason, 1992) have
employed a moving unstructured boundary-confirming grid in combination with a
stationary Cartesian grid to track the motion of bubbles undergoing severe deformation,
which is in essence the front tacking schemes applied by Glimm et al. (1987). The
Unified Coordinate method proposed by Hui et al. (1999) is an innovative scheme to
conduct Langrangian and Eulerian calculations on a single unified coordinate by a simple
mathematical transformation, in which the Eulerian or Langrangian coordinate is just one
of the special cases.
More recently, newer approaches of interface evolution problems have been
developed. Phase-field model (Kobayashi, 1993; Wheeler et al., 1993), Phase-field
Fourier-Spectral Method (Liu and Shen, 2003), Ghost Fluid Method (GFM) (Fedkiw et
al. 1999), Level-Set Method (LSM) (Osher and Sethian, 1988; Sethian, 1990), and
Segment Projection Method (Front-Tracking + Level Set) (Tornberg, 2000), have made
significant improvements on tracking the liquid-vapor interfaces. Among these, the
Level-Set Hamilton-Jacobi formulation has been widely used for capturing interface
evolution especially when the interface undergoes extreme topological changes, e.g.,
merging or pinching off. It is also attractive because it admits a convenient description of
topologically complex interfaces and is quite simple to implement. Even though the
level-set method does not have the same conservation properties as VOF or front-tracking
methods, the strengths of the level set method lie in its ability to accurately compute
flows with surface tension and changes in topology.
31
1.6
Scope of Study
As stated previously boiling of aqueous surfactant or polymer solutions is not
only important in thermal processing of biochemical, pharmaceutical, personal care, and
hygiene products, but the addition of desired amounts of surfactants to water is also an
innovative and promising technique to control the phase change process itself. The
presence of a small amount of surfactant or polymer additive in water changes both the
liquid-vapor (bulk chemistry) and the solid-liquid (surface chemistry) interfacial
characteristics. This, however, along with the fluid rheology change and the complex
nature of nucleate boiling, results in a complex two-phase problem, which is investigated
in this dissertation. The primary scope of this study in addressing the many facets of the
problem is summarized in the following:
1. Characterization of the interfacial phenomena and the associated transport
processes in aqueous surfactant and polymeric solutions at both the liquid-vapor
(dynamic and equilibrium surface tensions) and the solid-liquid (surface
wettability or contact angle) interfaces.
2. Conducting pool boiling experiments to quantify the effects of surfactant
concentration, its ethoxylation, solution rheology, surface wettability, dynamic
and equilibrium surface tensions, and additive molecular weight on the nucleate
boiling performance of water on a horizontal cylindrical heater.
3. Rheology measurement and characterization of the effects of polymerization
degree and surface-active properties on nucleate pool boiling heat transfer in
aqueous polymer solutions.
32
4. Surface wetting behavior qualification with the delineation of electrokinetic
effects or zeta potential due to the physisorption at the solid-liquid interface,
which in turn correlates with the corresponding adsorption isotherm.
5. Visualization and characterization of nucleate pool boiling in aqueous surfactant
solutions in order to qualitatively relate the heat transfer performance to the
nucleation process and ebullience. This facilitates the phenomenological
understanding of the rather complex and elusive transport process associated with
boiling heat transfer.
6. Computational simulation of the dynamics of a single bubble in pool boiling of
water, which includes the microlayer modeling and its role in nucleate boiling,
and the effects of altered contact angle, wall superheat, surface tension and
viscosity on the bubble dynamics.
33
CHAPTER 2
INTERFACIAL PROPERTIES AND RHEOLOGY MEASUREMENTS
2.1
Surface Tension Measurements
2.1.1
Introduction
For boiling in aqueous surfactant solutions, the detailed knowledge of dynamic σ
and corresponding adsorption behavior is very crucial. The dynamic surface tension of
liquids can be measured by several different techniques, and these are either direct
methods (maximum bubble pressure, oscillating jets, and Langmuir through methods) or
indirect methods (surface wave, oscillating bubble, and pulsed drop methods). The
equilibrium σ, on the other hand, can be measured by either standard static methods (duNüoy, sessile drop, and Wilhelmy plate) or as long time asymptotes of the dynamic
surface tension measurements. Joos (1999) and Dukhin et al. (1995) documented several
different equilibrium and dynamic surface tension measurement methods along with their
respective characteristic time and temperature ranges of operation, and suitability of
application.
Surface tension measurements in this study were made by the maximum bubble
pressure method using a twin orifice computerized surface tensiometer (SensaDyne
QC6000, CSC Scientific Company). Figure 2.1 schematically gives the details of the
surface tensiometer and the instrumentation employed. Dry air at 3.4 bar is slowly
bubbled through a parallel set of small and large glass orifice probes of 0.5 and 4.0 mm
diameter, respectively, which are immersed in the test fluid pool in a small beaker to
produce a differential pressure signal proportional to the fluid surface tension. The
temperature of the test fluid is measured using a well-calibrated thermistor (±0.1°C
precision, 0 - 150°C) attached to the orifice probes. The aqueous solution container is
34
immersed in a constant temperature bath in order to control and maintain its desired
temperature. The time interval between the newly formed interface and the point of
bubble break-off is referred to as “surface age,” and it gives the measure of bubble
growth time that corresponds to the dynamic surface tension value at a given operating
bubble frequency. Thus, by altering the air-bubble frequencies through the probes, both
static or equilibrium and dynamic surface tension can be measured. Detailed descriptions
of the solution preparation, instrument calibration, and validation procedures, along with
measurement uncertainties can be found in Manglik et al. (2001).
The maximum
uncertainties in the measurement of concentration, temperature, and surface tension were
found to be ±0.4% for powder form additives, ±5% for additives in liquid form, and ±
0.5% and ±0.7%, respectively.
Several different water-soluble surfactants that are representative of a wide
spectrum of commonly used additives are employed: dodecyltrimethylammonium
chloride (DTAC, cationic), cetyltrimethylammonium bromide (CTAB, cationic),
oleylmethylbis[2-hydroxyethyl]ammonium chloride (Ethoquad O/12 PG, cationic),
octadecylmethyl[15-polyoxyethylene]ammonium chloride (Ethoquad 18/25, cationic),
sodium dodecyl sulfate (SDS, anionic), sodium lauryl ether sulfate (SLES, anionic),
octylphenol ethoxylate (Triton X-100, nonionic), and octylphenoxypolyethoxyethanol
(Triton X-305, nonionic). They have different molecular weights, ionic nature, and
number of EO groups or degree of ethoxylation.
Their chemical composition and
relevant physico-chemical properties are listed in Table 2.1. Two polymers with different
degree of polymerization and surface-active properties: hydroxyethyl cellulose (HECQP300), a nonionic compound with surface-active properties, and Carbopol 934, a
35
cationic shear-thinning polymer without any surface-active properties, are also employed.
Their chemical composition and relevant physico-chemical properties are listed in Table
2.2.
Sensor
Package
Differential
Pressure
Gas
Gas
Transducer
Flow
Flow
Flow Controller
Metering
Valves
Small
Orifice
Probe
Large
Orifice
Probe
Pressure Controller
P
Interface
Card
Temperature Probe
Test Fluid
Computer
Constant-temperature bath
Air Supply
Tank
Fig. 2.1 Schematic of surface tensiometer and data acquisition system
36
Table 2.1 Physico-Chemical Properties of Surfactants
(Octadecylmethyl
[15-polyoxyethylene]
ammonium chloride)
(Sodium
dodecyl
sulfate)
SLES
(Sodium
lauryl ether
sulfate)
RN(CH3)(CH2CH2O
H)2Cl, R=oleyl
RN(+)(CH3)[(CH2CH
2O)mH][(CH2CH2O)n
H]Cl(-), R=C18H37
C12H25SO4
Na
Cationic
2
Yellow viscous
liquid
Cationic
15
Yellow viscous
liquid
364.5
403
Sigma-Aldrich
DTAC
CTAB
Ethoquad O/12 PG
(Chemical
Name)
(Dodecyltrimethyl
ammonium
chloride)
(Cetyltrimethyl
ammonium
bromide)
(Oleylmethylbis[2hydroxyethyl]
ammonium chloride)
Chemical
formula
C15H34ClN
C19H42BrN
Ionic nature
a
EO group
Cationic
0
Cationic
0
Appearance
White powder
White powder
Molecular
weight
263.9
Manufacturer
Sigma-Aldrich
Surfactant
Purity
≥ 99%
Melting point
> 246°C
Solubility
50 mg/mL
(20°C)
Specific
Gravity
Viscosity
b
(cp)
(pure liquid)
Surface
Tension
(mN/m) 25°C)
a
Ethoxy or ethylene oxide group
b
Brookfield viscometer
Ethoquad 18/25
SDS
Triton X-100
Triton X-305
(Octylphenol
ethoxylate)
(Octylphenoxy
plyethoxy
ethanol)
C12H25(OCH2
CH2)3 SO4Na
C14H21(OCH2
CH2)9-10OH
C14H21(OCH2CH
2)30OH
Anionic
0
White
powder
Anionic
3
Nonionic
9-10
Nonionic
30
Slightly yellow
viscous liquid
Clear liquid
Clear liquid
994
288.3
422
624(average)
1526 (average)
AkzoNobel
AkzoNobel
Fisher
Henkel
≈ 99%
> 230°C
≥ 99%
-
≥ 99%
-
≥ 99%
> 206°C
≥ 99%
-
Union
carbide
-
10 % (w/v)
> 25% (w/v)
-
150mg/mL
-
-
-
-
0.986 (25°C)
1.058 (25°C)
0.4
1.03
1.065
1.095
-
-
1750(23°C),
110(90°C)
-
500 (25°C)
240 (25°C)
470 (25°C)
-
40.3 (0.1%),
40.7 (1.0%)
50 (0.1 %)
-
-
-
-
37
Union carbide
-
Table 2.2 Physico-Chemical Properties of Polymers
Polymer additive
(Trade name)
Chemical name
Ionic nature
Appearance
Molecular weight
Manufacturer
Purity
Specific gravity
Viscositya, cps (25°C)
a
Brookfield viscometer
HEC
(NATROSOL QP-300)
Hydroxyethyl cellulose
Nonionic
White to light tan dry powder
~ 4-6 x 105
Amerchol
~ 99%
1.033
300-400 (2% solution)
Carbopol
(CARBOPOL 934)
Polyacrylic acid
Cationic
White dry powder
~ 3 x 106
BF Goodrich
~ 99%
1.41
~ 8-9 x 104 (2% solution)
2.1.2 Viscosity effects on surface tension measurements
Under dynamic conditions, the higher σ is primarily obtained due to the viscous
resistance offered by the fluid against the growing bubble interface, and it has been found
to be dependent upon fluid viscosity, capillary radius, and surface age (Fainermann et al.,
1993; Janule, 1998), and to predict the apparent increase in σ, Fainermann et al. (1993)
give the following correlation based on the Stokes’ flow approximation:
∆σ = 1.5 ( µ rcap τ )
(2.1)
Alternatively, for measurements made by the SensaDyne QC6000 surface tensiometer,
the viscosity effect can be corrected by adjusting the bubble frequency (or the surface
age) of each orifice by employing the following inverse relationship with their respective
radius (Janule, 1998):
br r g = bτ τ g
1
2
1
2
(2.2)
Here r1, r2 and τ1, τ2 are the respective radius and surface age of the small and large
orifice.
38
2.1.3
Results and Discussion
2.1.3.1 Aqueous surfactant solutions
The surface tension variations with concentration at both the equilibrium and
higher bubble frequency f the four cationic surfactant solutions at 23°C are graphed in
Fig. 2.2. While σ at higher f (represented by a surface age of 50 ms) is always larger than
the corresponding equilibrium value (surface age 17-59 s), both are seen to decrease with
increasing surfactant concentration to asymptotically attain a constant value beyond the
critical micelle concentration. The CMC for the four surfactants at 23°C (obtained from
the asymptotic intersection point of the equilibrium adsorption isotherm) are ~ 6000
wppm for DTAC, ~ 400 wppm for CTAB, ~ 600 wppm for Ethoquad O/12 PG, and ~
2300 wppm for Ethoquad 18/25. For aqueous CTAB solutions, the σ - C data compare
quite well with the Razafindralambo et al. (1995) results at 20°C, and so does the CMC
with the 392.5 wppm value at 25°C reported by Holmberg et al. (2003).
The σ values at equilibrium and higher f for the different surfactant solutions at an
increased bulk temperature of 80°C are presented in Fig. 2.3.
The higher bubble
frequency measurements are, once again, for a bubble surface age of 50 ms, which is
representative of bubble frequencies typically encountered in nucleate boiling of water.
The surfactant adsorption at the bubble vapor-liquid interface is a time-dependent process
that gives rise to the dynamic surface tension behavior; this, however, eventually reduces
to the equilibrium condition after a long time period (Bahl et al., 2003; Manglik et al.
2001; Iliev and Dushkin, 1992). The variation of σ with surface age in Fig. 2.3 clearly
illustrates this. Also, a lower molecular weight surfactant diffuses faster than its higher
39
90
80
Equilibrium
Surafce age (50 ms)
DTAC
CTAB
Ethoquad O/12
Ethoquad 18/25
σ [mN/m]
70
60
50
40
T = 23°C
Razafindralambo et al. (1995), CTAB
30
100
101
102
103
104
C [wppm]
Fig. 2.2 Surface tension measurements at equilibrium and higher bubble frequency
(surface age of 50 ms) for aqueous surfactant solutions at 23°C
40
molecular weight counterpart, and this is seen in the faster σ relaxation of CTAB in
comparison with that for Ethoquad 18/25 in Fig. 2.4. The presence of EO groups in
Ethoquad 18/25 makes its polar head more bulky and lowers its mobility. A similar trend
is obtained at elevated temperature (Bahl, 2003). As such, the dynamic σ isotherms at
80°C are atypical of the additive adsorption-desorption kinetics during atmospheric
pressure boiling of aqueous surfactant solutions3. The equilibrium σ at CMC, on the
other hand, represents the maximum possible surface tension reduction for the solutions
at 80°C. In general, it is observed that the process of micelle formation takes place over a
range of concentrations (Rosen, 1989; Adamson, 1976), and in the present measurements,
they are found to be ~ 4500 wppm for DTAC, ~ 500 wppm for CTAB, ~ 850 wppm for
Ethoquad O/12 PG, and ~ 500 wppm for Ethoquad 18/25.
The results in Fig. 2.3, when compared with the respective values at 23°C in Fig. 2.2,
indicate overall reductions in σ at the higher temperature (80°C), which are due to
increased surfactant diffusivity with increased temperature (Holmberg, et al. 2003;
Rosen, 1989). These reductions, however, are not uniform over both the dynamic and
equilibrium conditions. The degree of variation depends on the surfactant ionic nature
and molecular structure among some other factors (Bahl et al., 2003; Manglik et al, 2001;
Holmberg, 2003), and it reflects completely different adsorption-diffusion kinetics at
elevated temperature during short and long transients. Another salient feature is that
surfactants with EO groups in their hydrocarbon chain show larger reductions in σ with
increasing temperature (Ethoquads versus DTAC and CTAB), which is clearly evident in
3
While 80°C is the upper limit for QC-6000 surface tensiometer, the surface tension data at real boiling
temperature can be obtained by extrapolation of surface tension data with temperature, which typically
has a linear relationship.
41
65
T = 80°C
60
55
σ [mN/m]
50
45
40
35
30
25
100
Equilibrium
Surface age (50 ms)
DTAC
CTAB
Ethoquad O/12
Ethoquad 18/25
101
102
103
104
C [wppm]
Fig. 2.3 Surface tension measurements at equilibrium and higher bubble frequency
(surface age of 50 ms) for aqueous surfactant solutions at 80°C
42
Fig. 2.5, where σ - T variations for Ethoquad 18/25 and CTAB are graphed along with the
results for pure water, and its shows larger gradients for Ethoquad 18/25 at all
concentrations. The surfactant’s molecular-chain geometry and packing essentially
determine the aggregate/micelle structure, and it is well known (Holmberg et al., 2003)
that the polyoxyethylene chain compresses as the temperature increases. This leads to an
increased CPP (critical packing parameter) value, which lowers the CMC as well as the
surface tension (Holmberg et al., 2003). The temperature effect on CMC is even stronger
for surfactants with larger number of EO groups, as the polar head size increases with
increasing number of ethylene oxide units, and because they tend to form vesicle micelles
instead of spherical or lamellar micelles (Lu et al., 1999), thereby exhibiting a totally
different temperature dependence (Bahl et al, 2003; Manglik et al., 2001; Partearroyo et
al. 1996). However, the σ - T variation generally tends to be linear for a surfactant at a
given concentration as shown in Fig. 2.5.
43
70
Ethoquad 18/25
CTAB
C/Cc.m.c. = 0.5
C/Cc.m.c. = 1
C/Cc.m.c. = 2.0
σ [mN/m]
60
50
40
Typical bubble frequencies
in nucleate boiling of water
30
10-2
10-1
100
Surface Age [s]
Fig. 2.4 Dynamic surface tension relaxation for aqueous cationic CTAB
and Ethoquad 18/28 solutions
44
101
80
Ethoquad 18/25
CTAB
100 wppm
300 wppm
800 wppm
200 wppm
400 wppm
1000 wppm
70
Water
σ [mN/m]
60
50
40
30
10
20
30
40
50
60
70
80
90
T [°C ]
Fig. 2.5 Equilibrium surface tension measurements as a function of temperature
45
2.1.3.2 Aqueous polymeric solutions
The measured surface tension values at different bubble frequencies and polymer
concentrations for HEC QP-300 solutions at 23°C are graphed in Fig. 2.6. Similar to the
observations made earlier for surfactants, the surface tension is found to increase with
increasing bubble frequency and decreasing concentration. A critical polymer
concentration (CPC) akin to CMC in surfactants is observed, such that the σ relaxation
attains a saturation value at or around CPC. The value of CPC ascertained from the
equilibrium σ - C isotherm (lowest bubble frequency of 0.017 Hz) is estimated to be ~
600 wppm.
The experimental data given by Hu et al. (1991), as well as the
manufacturer’s (Union Carbide, 1998) reported σ value for 0.1% (concentration by
weight) solution are also graphed in Fig. 2.6. Except for Hu et al. (1991) data, the present
measurements for equilibrium σ agree well with the other results. The Hu et al. (1991)
data appear to fall in the dynamic conditions represented approximately by the bubble
frequency of 0.33 Hz, which corresponds to a surface age of 3 seconds4. As noted by
Persson et al. (1996) for polymers belonging to the class of nonionic cellulose derivates
that include HEC, time required for the complete relaxation of σ to an equilibrium value
is of the order of minutes, possibly due to the slow processes of diffusion transport of
polymer molecules to the interface and their subsequent reorientation. In concurrence
with this, the present results indicate typical relaxation times for HEC to be around 1-2
minutes. The surface tension measurements for Carbopol 934 solutions displayed a
similar behavior with varying bubble frequency and hence are not discussed separately.
4
It may be noted that though the data by Hu et al. (1991) were obtained by the maximum bubble pressure
method, the bubbling frequency (as surface age) has not been mentioned in the paper.
46
74
Aqueous HEC-QP300 solutions at 23°C
73
72
σ [mN/m]
71
70
69
Bubble frequency: 10 Hz
Bubble frequency: 2Hz
Bubble frequency: 0.33Hz
Bubble frequency: 0.017Hz
Hu et al. (1991) at 25°C
Manufacturer's Data
(Union Carbide, 1998), 20°C
68
67
66
100
101
102
103
104
C [wppm]
Fig. 2.6 Dynamic surface tension measurements for aqueous HEC
solutions at 23°C
47
The higher bubble frequency and equilibrium σ values for HEC QP-300 and
Carbopol 934 at room temperature are shown in Fig. 2.7. At equilibrium conditions, σ is
seen to reduce with increasing concentration for both polymers.
In comparison to
Carbopol solutions, however, HEC solutions show significantly higher σ relaxation both
at dynamic and equilibrium conditions, with a rather sharp change in slope or surface
tension gradient near CPC. It should be noted that HEC is a surface-active polymer, and
thus its adsorption behavior would tend to be similar to those of surfactant solutions. On
the other hand, a reversed trend of increasing σ with concentration is observed in
dynamic measurements of aqueous Carbopol solutions. This apparent increase in σ,
especially under dynamic conditions, is due to the viscous resistance offered by the fluid
against the growing bubble interface.
Such an increase in σ measurements by the
maximum bubble pressure method has been typically observed in highly viscous liquids
like aqueous solutions of polymers and glycerol (Hirt et al, 1990; Fainermann, 1993).
The dynamic surface tension measured according to Janule’s procedure (1998) is
graphed in Fig. 2.7 along with the corrected values of σ using Eq. (2.1). For both
polymer solutions, there is an excellent agreement between the corrected experimental σ
values obtained from the modified measurement procedure (Eq. (2.2)) and those
computed using Eq. (2.1). More notably, there is an appreciable difference between the
viscosity compensated and non-compensated values for Carbopol solutions, with a
characteristic decrease in σ with concentration (similar to HEC) even under dynamic
conditions. HEC-QP300 being a sparingly viscous polymer, contrastingly shows smaller
change in σ after viscosity correction. The surface tension data reported by Hu et al.
(1991) and Ishiguro and Hartnett (1992) for Carbopol solutions are also graphed in Fig.
48
2.7. The discrepancies between these and the present data set are relatively large and
may perhaps be attributed to the differences in methods of measurement, varying sources
of polymer samples, and possible non-accounting of viscosity effects.
Figure 2.8 depicts the variations in σ at both the equilibrium and higher f for the
two polymers at an elevated solution temperature of 80°C. An overall depression in σ is
observed with a maximum reduction in equilibrium σ of around 4% and 8%,
respectively, for 2000 wppm Carbopol and HEC solutions. The viscosity effect for
Carbopol solutions under dynamic conditions is found to be reduced, presumably due to
the decrease in the solution’s apparent viscosity from room temperature to 80°C. Also,
the dynamic σ values are close to the surface tension of water, and they do not show a
significant variation with concentration. For HEC solutions, the viscosity compensation
results in a negligible change in dynamic σ, which is quite similar to that observed in the
behavior at room temperature.
The polymer adsorption process at the bubble vapor-liquid interface is also time
dependent, which manifests in a dynamic surface tension behavior that eventually
reduces to the equilibrium condition after a long time period. This σ relaxation behavior
is essentially the outcome of the molecular kinetics of the additive in water. In solutions
with lower than overlap concentrations or CPC, the polymer molecules remain as isolated
macromolecules with little intermolecular interactions.
At overlap or “semi-dilute”
concentrations, the polymer molecules “touch” each other, and with increasing
concentration the frequency of collisions between the polymer coils eventually causes
overlapping and entanglement of their chains.
49
74
72
70
σ [mN/m]
68
EXPERIMENTAL DATA (polymer solutions at 23°C)
66
64
Carbopol 934
Equilibrium
50ms, w/o viscosity correction
50ms, with viscosity correction by Eq. (2.1)
50ms, with viscosity correction by Eq. (2.2)
Hu et al. at 25°C (1991)
Ishiguro and Hartnett at 25°C (1992)
HEC-QP300
62
60
101
Equilibrium
50 ms, w/o viscosity correction
50ms, with viscosity correction by Eq. (2.1)
50ms, with viscosity correction by Eq. (2.2)
102
103
C [wppm]
Fig. 2.7 Surface tension measurements at equilibrium and higher bubble
frequency (surface age of 50 ms) for aqueous polymer solutions at 23°C
50
65
σ [mN/m]
60
EXPERIMENTAL DATA (polymer solutions at 80°C)
Carbopol 934
55
Equilibrium
50ms, w/o viscosity correction
50ms, with viscosity correction by Eq. (2.1)
50ms, with viscosity correction by Eq. (2.2)
HEC-QP300
Equilibrium
50ms, w/o viscosity correction
50ms, with viscosity correction by Eq. (2.1)
50ms, with viscosity correction by Eq. (2.2)
50
101
102
103
C [wppm]
Fig. 2.8 Surface tension measurements at equilibrium and higher bubble
frequency (surface age of 50 ms) for aqueous polymer solutions at 80°C
51
2.2
Contact Angle Measurements
The liquid-solid contact angle was measured by the sessile drop method, using a
Kernco GI Contact Angle Meter / Wettability Analyzer. The measurement uncertainty in
this case is estimated to be a max of ±1.4% for powder form additives and ±5% for
additives in liquid form. The change in surface wettability (measured by the contact
angle) with concentration in ionic surfactants (SDS-anionic, CTAB and Ethoquad 18/25cationic) and nonionic surfactants (Triton X-100 and X-305) are graphed in Fig. 2.9(a).
Ionic surfactants undergo a different adsorption process than that for nonionic surfactants
due to the latter’s lack of charge. The adsorption isotherms for ionics (SDS, CTAB, and
Ethoquad 18/25) correlate well with the physisorption characterization schematically
illustrated in Fig. 2.9(b). The contact angle reaches a lower plateau around the CMC
where bilayers start to form on the surface. Wettability of nonionic surfactants in aqueous
solutions, on the other hand, shows that the contact angle data attains a constant value
much below CMC.
Direct interactions of their polar chain are generally weak in
nonionics, and it is possible for them to build and rebuild adsorption layers below CMC
(Levitz, 2002).
The reduced contact angle trough at lower concentrations (C < CMC)
can also be attributed to the absence of any electrical repulsion that could oppose
molecular aggregation unlike that associated with ionic surfactants (Miller and Neogi,
1985). Furthermore, the continuous decrease in contact angle for Triton solutions prior to
reaching a constant value is brought about by the presence of larger EO groups in the
surfactant molecular chain. The number of EO groups increases the overall size of the
polar head, and controls the hydrophilic/hydrophobic balance on the surfactant molecule
(Holmberg et al., 2003).
52
90
CTAB (Cationic)
Ethoquad 18/25 (Cationic)
SDS (Anionic)
Triton X-100 (Nonionic)
Triton X-305 (Nonionic)
Contact angle [deg]
80
70
CMC
60
50
CMC
CMC
CMC
CMC
40
30
101
102
103
104
Concentration [wppm]
(a)
Individual
Ions
Reversed
Hemimicelles
Hemimicelles
Hydrophobic tail
Bilayer
Hydrophilic head
Hydrophobic
Hydrophilic
(b)
Increasing EO groups
(c)
Fig. 2.9 (a) Measured contact angle for aqueous CTAB, Ethoquad 18/25, SDS,
Triton X-100, and Triton X-305 solutions; (b) corresponding ionic surfactant
adsorption surface state; and (c) EO group effect on surface wettability
53
The surfactant wetting behavior can be directly related to its adsorption isotherm,
figure 2.10(a) shows that measured contact angle correlates well with the adsorption
density. For nonionics, their molecules may just get adsorbed randomly at the liquidsolid interface due to lack of charge as illustrated in Fig. 2.10(b). If the molecule is
adsorbing on a substrate like a steel sample that exhibits a contact angle, the more EO
groups on the molecule should reduce the adsorption because that molecule would then
rather be in the aqueous phase for nonionic surfactants (Ottewill, 1967).
80
60
4
3
CMC
50
2
CMC
40
30
101
Adsorption density [10-6mole/m2]
Contact angle [deg]
70
5
Adsorption density
Contact angle
(Levitz, 2002)
Triton X-100
Triton X-305
1
102
103
0
Concentration [wppm]
(a)
(b)
Fig. 2.10 (a) Contact angle and adsorption isotherms for nonionic surfactants Triton
X-100 and Triton X-305 in aqueous solutions; and (b) non-ionic surfactant adsorption.
54
2.3
Rheology Measurements
Viscosity measurements were carried out using a rotating cylinder rheometer
(AR-2000; TA Instruments) that can function in both a controlled-stress and controlledshear-rate environment. It contains an electronically controlled induction motor with an
air bearing support for all rotating parts. The drive motor has a detachable draw rod
arrangement to which the measuring geometry (rotating cylinder, cup and cylinder,
parallel plate, and cone and plate) can be attached. The angular displacement is measured
by an optical encoder device, which can detect very small movements down to 40 nRad.
The encoder consists of a non-contacting light source and a photocell arranged on either
side of a disc-shaped diffraction grating attached to the drive shaft. A stationary segment
of a similar disc is positioned between the light source and the encoder disc. The
interaction of these discs results in diffraction patterns that are detected by the photocell.
When the liquid sample strains under stress, the encoder disc moves, and the diffraction
patterns change, and the associated digital signals are directly related to the angular
deflection, and, therefore, to the strain of the sample.
Sample weights of additives in power form were measured using a precision
electronic weighing machine of ±0.1 mg accuracy. For the additives available in liquid
form, different ranges of precision syringes were used for the measurement of sample
volumes, and the corresponding weight calculated by the known value of specific gravity
of each additive. Then samples were dissolved in distilled deionized water (at slightly
elevated temperatures to aid solubility for polymers) to obtain the test samples. The
instrument calibration was validated using a standard oil sample, and pure water. The
test liquid viscosity was then measured using a DIN geometry for shear rates less than
55
100 s-1, and a double concentric cylinder geometry system for higher shear rates.
Temperature control was attained via a Peltier system that allows for rapid and accurate
heating and cooling of the liquid sample. The data reproducibility was further checked
using multiple runs for a few fixed-concentration samples. Again, the maximum singlesample, error propagation uncertainty in viscosity and temperature were ±1.4% for
powder form additives and ±5% for additives in liquid form, and ±0.5%, respectively.
2.3.1 Aqueous surfactant solutions
The viscosity-shear rate data for CTAB and Ethoquad 18/25 are presented in Fig.
2.11, where the relative changes in the apparent viscosity from that of water are graphed
for three different concentration (C/CCMC= 0.5, 1, 2). Taylor-Couette instability will take
effect after 150/s to show an untrue shear thickening behavior, therefore, the data after ~
150/s were cut off in Fig. 2.11. It clearly shows that the viscosity of CTAB and Ethoquad
18/25 are close to that of the water in dilute aqueous surfactant solutions, the date scatter
is well within ±5%. That dilute solutions of ionic and nonionic surfactants usually
behave as Newtonian liquids, and the viscosity of these solutions is always close to that
of solvent was also found in other studies (Wang and Hartnett, 1994; Hoffmann and
Rehage, 1986). The measured viscosities for both SDS and Tween-80 aqueous solutions
at room temperature were also found to be constant and independent of shear rate over a
concentration range of 125 to 500 wppm (Wang and Hartnett, 1994).
2.3.2 Aqueous polymer solutions
The viscosity-shear rate data for aqueous HEC QP-300 and Carbopol 934
solutions are graphed in Figs. 2.12 and 2.13, respectively. These were obtained in a
56
controlled-rate mode, where the shear rate was ramped and allowed to equilibrate to a
steady-state value before the next successive increase. It is seen in Fig. 2.12 that HEC
solutions are significantly more viscous than water, and that viscosity increases with
concentration. Also, at low concentrations the solutions virtually behave as Newtonian
fluids. The Hu et al. (1991) data are also included in Fig. 2.12 for comparison, though
these are for a different grade of the HEC family of polymers (HEC 250HHR) that has a
higher molecular weight (M = 1.3 x 106) as well as a higher degree of polymerization
(NATROSOL-Hercules, 1999), and their shear-thinning behavior in higher concentration
solutions is evident. A similar rheological behavior has also been observed by Maestro et
al. (2002). Their data for the lower molecular weight HEC9 (M = 9 x 104) show near
Newtonian characteristics even at very high concentration (10% by weight), and a nonNewtonian shear-thinning behavior in a 0.75% HEC130 (M = 1.3 x 106) solution. These
results are clearly indicative of the role of molecular weight and degree of polymerization
in the rheological behavior of polymers. The data for Carbopol 934 solutions in Fig.
2.13, when compared with the respective values for HEC QP-300 in Fig. 2.12, indicate
higher viscosity to reflect the increased degree of polymerization (M = 4-6 x 105 and 3 x
106, for HEC and Carbopol, respectively).
Also, the shear-thinning behavior for
Carbopol 934 solutions is more obvious at higher concentrations. At a shear rate of 500
s-1, atypical of the bubble-fluid motion in nucleate boiling, the relative change in the
apparent viscosity of different concentration solutions from that of water for both HEC
QP-300 and Carbopol 934 are given in Table 2.3.
57
C/CCMC = 0.5
C/CCMC = 1.0
C/CCMC = 2.0
1.4
1.2
1.0
CTAB
η s /η w
0.8
T=23°C
0.6
1.4
C/CCMC = 0.5
C/CCMC = 1.0
C/CCMC = 2.0
1.2
1.0
0.8
Ethoquad 18/25
T=23°C
0.6
101
γ [s-1]
102
Fig. 2.11 Relative viscosity changes of aqueous CTAB and Ethoquad
18/25 solutions
58
10-1
Hu et al. (1991)
(HEC 250HHR @22°C)
500 wppm
1000 wppm
2000 wppm
η [Pa.s]
10-2
10-3
Present Data (HEC-QP300) @ 23°C
300 wppm
600 wppm
3000 wppm
water
10-4
101
102
103
γ [s-1]
Fig. 2.12 Variation of apparent viscosity with shear rate for aqueous
HEC QP-300 solutions
59
10-1
Present Data (Carbopol 934) @ 23°C
100 wppm
500 wppm
1000 wppm
3000 wppm
water
η [Pa.s]
10-2
10-3
10-4
101
102
103
γ [s-1]
Fig. 2.13 Variation of apparent viscosity with shear rate for aqueous
Carbopol 934 solutions
Table 2.3 Increase in Viscosity of Aqueous Polymer Solutions with respect to Water
at 23°C as a Function of Concentration at a Shear Rate of 500 s-1
HEC
Concentration
η x 103
[wppm]
[Pa.s]
Water
0.935
300
1.08
600
1.25
3000
3.65
Carbopol
Concentration
η x 103
[Pa.s]
[wppm]
100
1.03
500
1.43
1000
1.77
3500
4.39
%
increase
15.50
33.68
290.37
60
%
increase
10.16
52.94
89.30
369.52
CHAPTER 3
POOL BOILING HEAT TRANSFER
Nucleate pool boiling experiments and the measured heat transfer performance of
aqueous solutions of the four cationic surfactant and two polymer solutions are described
in this chapter. The results for different concentration solutions are presented, and the
optimum enhancement in heat transfer is identified.
3.1
Experimental Setup
The experimental setup used for the pool boiling studies is shown schematically
in Fig. 3.1(a). The inner glass tank, which contains the surfactant or polymer solution
pool and the cylindrical heater, is encased in an outer glass tank that has circulating
mineral oil fed from a constant-temperature recirculating bath (not shown in figure) to
maintain the test pool at its saturation temperature. A water-cooled reflux condenser,
along with a second coiled-tube water-cooled condenser, helps condense the generated
vapor and maintain an atmospheric-pressure pool.
A pressure gage (±0.0025 bar
precision) is mounted on top of the boiling vessel to monitor the pressure in the pool
throughout the experiments. The heating test section (shown in Fig. 3.1b) consists of a
horizontal, gold plated, hollow copper cylinder of 22.2 mm outer diameter; the 0.0127
mm thick gold plating mitigates any surface degradation and oxidation from chemicals in
the test fluids.
A 240 V, 1500 W cartridge heater, with insulated lead wires, is press-
fitted in the hollow cylinder with conductive grease to fill any remaining air gaps and
provide good heat transfer contact with the inside of the tube. The cartridge heater is
centrally located inside the copper tube, and the gaps at each end are filled with silicone
61
rubber to prevent water contact. Also, because the heater surface condition significantly
influences the boiling behavior, it was examined using an optical microscope as well as
an atomic force microscope (AFM). The optical microscope images of the heated surface
in Fig. 3.2 show a random distribution of pits, cavities, and machining grooves of varying
shapes and sizes along the heater surface. The overall r.m.s. roughness from AFM scans,
measured at four different locations, range from 0.076 µm to 0.347 µm.
The heater-wall and pool-bulk temperature measurements were recorded using
copper-constantan precision (±0.5ºC) thermocouples, interfaced with a computerized data
acquisition system with an in-built ice junction and calibration curve.
A variac-
controlled AC power supply, a current shunt (0.15 Ω with 1% accuracy), and two highprecision digital multimeters (for current and voltage measurements) provided the
controls and measurements of the input electric power. At each incremental value of
power input or heat load, the dissipated wall heat flux qw′′ and the wall superheat ∆Tw
were computed from the measured values of V, I, the four wall thermocouple readings
(Ti,r), and saturation temperature of the pool from the following set of equations:
qw′′ = (VI A)
(3.1)
 4

Tw =  ∑ Ti ,r − ( qw′′ ro k ) ln ( ro r ) 4 
 i =1

(3.2)
∆Tw = (Tw − Tsat )
(3.3)
The maximum experimental uncertainties in qw′′ and ∆Tw, based on a propagation of error
analysis (Moffatt, 1998), were 1.44% and 0.5% respectively. Details of the experimental
procedure, uncertainty analysis, and the extended validation of test measurements with
boiling data for water are given by Wasekar and Manglik (2001).
62
Leads for
cartridge
heater and
thermocouple
Cooling
water
Condenser
P
To circulating bath
Leads for
Thermocouples
Cylindrical
heater
Inner
tank
Test fluid pool
Mineral
oil bath
Outer
tank
Thermocouples
for pool
temperature
(a)
(b)
Fig. 3.1 Schematic of experimental facility: (a) pool boiling apparatus, and (b) crosssectional view of cylindrical heater assembly
63
100X
200X
Fig. 3.2 Optical microscope images of the roughness characteristics of heater surface
64
3.2
Nucleate Pool Boiling in Aqueous Surfactant Solutions
Aqueous solutions of DTAC, CTAB, Ethoquad O/12 PG, and Ethoquad 18/25
with different concentrations were prepared by dissolving weighted samples in distilled
water. The sample-measurement procedure and corresponding measurement uncertainties
are described in detail by Manglik et al. (2001). The boiling of curve for pure distilled
water was first established over a period of four months to verify its repeatability and the
effects of heater surface aging, and it provides the baseline reference for the surfactant
solutions results as well as validates the experimental reliability of the apparatus
(Wasekar and Manglik, 2001).
3.2.1
Pool boiling in aqueous cationic surfactant solutions
The pool boiling data for different concentration aqueous solutions of DTAC,
CTAB, Ethoquad O/12 PG, and Ethoquad 18/25, are presented in Figs. 3.3, 3.4, 3.5 and
3.6, respectively. In general, with the addition of surfactant to water, the nucleate boiling
curve shifts to the left indicating enhancement in heat transfer. CTAB and Ethoquad
18/25, typically representative of the four cationics, were analyzed below in details.
While CTAB is a higher molecular weight cationic surfactant without EO groups,
Ethoquad 18/25 has an even higher molecular weight but with a relatively high
ethoxylation of 15 EO groups (Table 2.1). The impact of their different chemistry is
clearly seen in the respective nucleate boiling curves for their aqueous solutions. All data
graphed in Figs. 3.3 - 3.6 are for decreasing heat flux unless indicated otherwise.
The data for CTAB (Fig. 3.4) show considerable heat transfer enhancement with
increasing concentration, as represented by the characteristic leftward shift in the boiling
65
curve relative to that for distilled water. Also, there was early incipience or onset of
nucleate boiling (ONB) (observed visually with onset of bubbling activity): for C = 400
wppm, ONB was seen at qw′′ ≅ 8.5 kW/m2 or ∆Tw ≅ 3.7 K, as compared to that for
distilled water at qw′′ ≅ 12.83 kW/m2 or ∆Tw ≅ 5.13 K. The optimum heat transfer
enhancement is seen to be obtained with 400-500 wppm solutions (~ CMC for CTAB at
80°C). But with C > CMC, the enhancement decreases and the heat transfer even
deteriorates below that in distilled water in high concentration (≥ 800 wppm) solutions,
particularly at low heat fluxes. A similar dependence on C with a different cationic
surfactant (Habon G, M = 500) is seen in the Hetsroni et al. (2001) data, as well as those
for anionic and nonionic surfactant solutions reported in other studies (Wu, et al. 1998a,
Wasekar and Manglik, 2000; Wasekar and Manglik, 2002).
The boiling curves for aqueous Ethoquad 18/25 solutions in Fig. 3.6 display a
somewhat different behavior, with considerably less enhancement. In fact significant
enhancement is seen only at higher heat fluxes, and, once again, the peak performance is
with C ~ CMC (~ 500 wppm). With higher concentrations (C > CMC), there is a
rightward shift in the boiling curve, and substantially lower heat transfer coefficients than
those for distilled water are obtained when C ≥ 3000 wppm. This is also accompanied
with delayed incipience and thermal hysteresis or temperature overshoot, as seen in the
increasing and decreasing qw′′ - ∆Tw data for 5000 wppm solution; such hysteresis was not
seen in lower (C < CMC) concentration solutions. This boiling behavior is akin to that
normally observed in highly wetting liquids (Kandlikar et al. 1999; Kenning, 1999; BarCohen, 1992; Bergles, 1988).
66
103
9
8
7
6
5
4
3
q"w (kW/m2)
2
500 w ppm, q"w ⇑
500 w ppm, q"w ⇓
2000 w ppm
3000 w ppm
4000 w ppm
5000 w ppm
10000 w ppm, q"w ⇑
10000 w ppm, q"w ⇓
Distilled Water
DTAC
Tsat=100°C
102
9
8
7
6
5
4
3
2
Nucleate boiling
101
hysteresis
9
100
4
7
∆Tw (K)
101
2
Fig. 3.3 Nucleate pool boiling data for aqueous solutions of DTAC; all data are
for decreasing heat flux except as otherwise indicated
67
q"w (kW/m2)
103
100 wppm, q"w ⇑
100 wppm, q"w ⇓
200 wppm
300 wppm
400 wppm
500 wppm
700 wppm
800 wppm
1000 wppm, q"w ⇑
1000 wppm, q"w ⇓
CTAB
Distilled Water
102
Tsat = 100°C
101
100
4
7
∆T w (K)
101
2
Fig. 3.4 Nucleate pool boiling data for aqueous solutions of CTAB; all data are
for decreasing heat flux except as otherwise indicated
68
1039
8
7
6
5
4
3
2
200 wppm
400 wppm
600 wppm
800 wppm
1000 wppm
1500 wppm
3000 wppm, q"w ⇑
3000 wppm, q"w ⇓
Ethoquad O/12 PG
q"w (kW/m2)
Distilled Water
Tsat = 100°C
1029
8
7
6
5
4
3
2
1019
100
4
7
∆T w (K)
101
2
Fig. 3.5 Nucleate pool boiling data for aqueous solutions of Ethoquad O/12
PG; all data are for decreasing heat flux except as otherwise indicated
69
103
200 wppm
500 wppm
700 wppm
1500 wppm
3000 wppm
5000 wppm, q"w ⇑
5000 wppm, q"w ⇓
Ethoquad 18/25
q"w (kW/m2)
Distilled Water
Tsat = 100°C
102
101
100
4
7
101
2
∆T w (K)
Fig. 3.6 Nucleate pool boiling data for aqueous solutions of Ethoquad 18/25;
all data are for decreasing heat flux except as otherwise indicated
70
3.2.2
Optimum heat transfer and critical micelle concentration
The surfactant additive significantly alters the nucleate boiling in water and
enhances the heat transfer. A closer inspection of Figs. 3.3 through 3.6 reveals that the
enhancement increases with concentration, with an optimum obtained in solutions at or
near the critical micelle concentration or CMC of the surfactant. Such an optimum has
also been observed by Wasekar and Manglik (2000,2002), Hetsroni et al. (2001), and Wu
et al. (1998a). As discussed earlier, the process of micelle formation characterizes this
range of concentration (Manglik, et al., 2001; Rosen, 1989; and Tsujii, 1998).
The effects of heat flux and surfactant concentration on the nucleate boiling heat
transfer are further highlighted in Fig. 3.7, where the relative increase in the heat transfer
coefficient from that of water for all four cationic surfactants are graphed. The relative
heat transfer coefficient defined as:
(h − hwater )
hwater
 (q ′′ ∆Tw ) − (q w′′ ∆Tw )water 
= w

(q w′′ ∆Tw )water


(3.4)
Besides depicting the improved heat transfer in solutions with 0 < C ≤ CMC, it clearly
shows the decrease in the enhancement in high concentration (C > CMC) solutions. In
fact, at low heat fluxes there is even a degradation in heat transfer compared to that for
water in all surfactant solutions except in those with DTAC.
With a maximum
enhancement of 63% in 4000 wppm aqueous DTAC solution, the performance is seen to
be dependent upon the wall heat flux, concentration, and surfactant molecular weight and
EO group content. The enhancement is significantly greater in aqueous solutions of
DTAC and CTAB (non-ethoxylated cationics) as compared to that in Ethoquad O/12 PG
71
and Ethoquad 18/25 (ethoxylated cationics) solutions.
As pointed out previously
(Wasekar and Manglik, 2001; Wasekar and Manglik, 2002), the process of micelle
formation and the molecular dynamics in a concentration sublayer at the vapor-liquid
interface characterizes the resultant optimum heat transfer enhancement in surfactant
solutions with C ~ CMC.
3.3
Nucleate Pool Boiling in Aqueous Polymer Solutions
In order to determine how nucleate pool boiling heat transfer of water is affected
by the addition of polymers that have different degrees of polymerization and surfaceactive properties, hydroxyethyl cellulose (HEC-QP300), a nonionic polymer, and
Carbopol 934, a cationic polymer, are employed. While both produce viscous aqueous
solutions, the former displays significant surface-active properties and the latter renders a
shear-thinning rheology in the shear-rate range of interest (10-1000 s-1). Their chemical
composition and relevant physico-chemical properties are listed in Table 2.2. Variations
in their shear-rate dependent viscosity, along with temperature-dependent equilibrium
and dynamic surface tension are recorded and presented in chapter 2, in order to
characterize the rheological and interfacial behaviors of the polymeric solutions. Pool
boiling curves ( qw" vs. ∆Tsat) for the incipience to fully developed nucleate boiling
regimes under atmospheric pressure saturated conditions are presented, which highlight
the effects of polymer concentration, dynamic surface tension or surface active, and wall
heat flux on boiling and the associated heat transfer coefficients.
72
0.9
0.8
C/Cc.m.c.= 0.5 C/Cc.m.c.= 1
C/Cc.m.c.= 2
DTAC
CTAB
Ethoquad O/12 PG
Ethoquad 18/25
0.7
(hsurf - hwater) / hwater
0.6
Tsat=100°C
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
101
q"w (kW/m 2)
102
Fig. 3.7 Variation of the relative heat transfer performance of aqueous cationic
surfactant solutions with heat flux and additive concentration (decreasing qw" )
73
3.3.1
Pool boiling in aqueous polymer solutions
The experimental pool boiling data for surface-active HEC solutions of different
concentrations as well as that for water are presented in Fig. 3.8. The heat transfer
enhancement with increasing surfactant concentration is evident from the leftward shift in
the boiling curve relative to that for pure water. This boiling process was further visually
observed to have an early incipience or onset of nucleate boiling (ONB). However, the
enhanced heat transfer is seen to “peak” with a 600 wppm concentration (~ CPC for
HEC-QP300) solution, and then decrease with higher concentrations. The results for
3000 wppm solutions even show a degradation in performance relative to water at lower
heat fluxes. Boiling curves for higher concentrations that exhibit a rightward shift were
also seen to have delayed incipience as well. A similar performance with HEC-H has
been reported by Shul’man et al. (1993), and their results show that the heat transfer
coefficient reaches its maximum at C ~ 500 wppm on a plate heater with a size that is
much larger than the mean size of the boiling bubbles. This 500 wppm concentration is
probably the CPC for HEC-H, which is a different grade of the HEC family of polymers
and has lower molecular weight than HEC-QP300.
The boiling data for aqueous
Carbopol 934 solutions in Fig. 3.9 display a somewhat different behavior, and the heat
transfer is seen to continuously continue to decrease with increasing concentrations
compared to that for water, with the rightward shift in the boiling curve. Also, delayed
incipience, low bubble departure frequency, and some vapor explosions due to higher
viscous resistance were observed. Deterioration of boiling heat transfer has also been
reported by Paul and Abdel-Khalik (1983) in drag-reducing polyacrylamide (Separan AP30) solutions.
74
103
100 wppm
300 wppm
500 wppm
600 wppm
700 wppm
1000 wppm
3000 wppm
HEC-QP300
q"w (kW/m2)
Distilled Water
Tsat = 100°C
102
101
3
4
5
6
7
∆T w (K)
8
9
101
2
Fig. 3.8 Nucleate pool boiling data for aqueous solutions of HEC-QP300
75
103
100 wppm
300 wppm
500 wppm
1000 wppm
1500 wppm
3000 wppm
Carbopol 934
q"w (kW/m2)
Distilled Water
Tsat = 100°C
102
101
3
4
5
6
7
8
9
∆T w (K)
101
2
Fig. 3.9 Nucleate pool boiling data for aqueous solutions of Carbopol 934
76
The effects of heat flux and surfactant concentration on the nucleate boiling heat
transfer in HEC-QP300 solutions are further highlighted in Fig. 3.10, where the relative
increases in heat transfer coefficients from that of water are graphed for different
concentrations. A maximum enhancement of 22.9% in a 600 wppm aqueous solution is
seen, and the improved performance tends to be somewhat weakly dependent upon the
wall heat flux. Enhanced heat transfer in nucleate boiling of dilute (C < CPC ~ 500
wppm) aqueous HEC-H solutions on a plate heater is also evident from the
Kotchaphakdee and Williams (1970) data. Furthermore, Fig. 3.10 clearly shows the
decrease in the boiling heat transfer enhancement in HEC solutions with C > CPC (700
wppm, 1000 wppm, and 3000 wppm). In the very high concentration (3000 wppm)
solution, up to 7.5% degradation in the heat transfer coefficient when compared to that
for pure water is evident for qw′′ < 70 kW/m2; the degradation also tends to be strongly
dependent upon wall heat flux.
3.3.2 Surface-active and rheological effects
In general, the factors that affect the nucleate boiling performance of polymeric
compounds include, among others, the changes in surface tension of the liquid,
adsorption of macromolecules on the heating surface, nucleate site density, heater
geometry and its surface characteristics, influence of macromolecules on diffusion heat
transfer in solvent evaporation, hydrodynamics of convective flows in the boiling boundary layer and bubble motion (micro-convection), thermodynamics features of the
polymer-solvent solutions, and rheological effects. At high heat fluxes and a sufficiently
large time duration of nucleation boiling, the possibility of macromolecular
77
0.3
HEC-QP300
Tsat = 100°C
(hpoly - hwater) / hwater
0.2
0.1
0.0
Present Data
-0.1
(Cylindrical heater)
Kotchaphakdee & Williams (1970)
100 wppm
300 wppm
500 wppm
600 wppm
700 wppm
1000 wppm
3000 wppm
(HEC-H, Plate Heater)
-0.2
-0.3
62.5 wppm
125 wppm
250 wppm
101
102
q"w [kW/m2]
Fig. 3.10 Variation of the enhanced boiling heat transfer performance of HECQP300 solutions with heat flux and additive concentration
78
thermodestruction (degradation) should also be taken into account (Levitskiy et al.,
1996). For surface-active HEC solutions, the reduction in dynamic surface tension σ
(which decreases the required superheat for the onset of boiling), and the macromolecular
adsorption on the heating surface (which could contribute to the formation of new
nucleation sites and increased bubble frequency) are perhaps the two main factors for the
boiling heat transfer enhancement in lower concentration (C < CPC) HEC solutions. On
the other hand, the decreases in the nucleate boiling heat transfer coefficients in HEC
solutions with higher concentrations (C > CPC) and pure shear-thinning Carbopol 934
solutions are possibly associated with the substantial increase in the liquid viscosity that
tends to suppress the micro-convection in the bubble boundary layer as well as retard the
growth of vapor bubbles.
Finally, Fig. 3.11 provides further insights on the role of dynamic surface tension
or surface-active effects on the heat transfer performance. The normalized pool boiling
heat transfer coefficient data for HEC (300 and 600 wppm) and Carbopol (100 and 300
wppm) solutions are graphed. While their respective concentrations are different, the
apparent viscosity of their dilute solutions is comparable (ηHEC,
100wppm, and
300wppm
vs. ηCarbopol,
ηHEC, 600wppm vs. ηCarbopol, 300wppm). In the measured range of heat fluxes in the
nucleate boiling regime, heat transfer enhancement is seen in HEC solutions, while,
contrastingly, there is only heat transfer deterioration in Carbopol solutions. Considering
that the only drastic physical property change in these four concentration solutions is the
dynamic surface tension relaxation5 (dσ/dτ is 0.9 for the 600wppm HEC solution, which
5
The dynamic surface tension gradient (dσ/dτ) used in this study is the value obtained at a surface age of
50 ms, which is representative of bubble frequencies typically encountered in nucleate boiling of water.
79
shows a much larger dynamic surface tension reduction compared to the values for other
three solutions), and that the measured surface wettability (represented by the solid-liquid
interface contact angle6) for both HEC and Carbopol are close to that of water (77°),
these results clearly suggest that the dynamic surface tension is perhaps one of the more
significant prediction parameters.
0.6
0.5
(hpoly - hwater) / hwater
0.4
HEC
HEC
Carbopol
Carbopol
300 wppm
600 wppm
100 wppm
300 wppm
dσ/dτ
η
C
1.08e-3
1.25e-3
1.03e-3
1.29e-3
(x 10-2)
θ
0.3
0.9
0.1
0.1
76
75
76
77
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
Cylindrical heater
101
Tsat = 100°C
102
q"w [kW/m2]
Fig. 3.11 Effect of dynamic surface tension on the boiling heat transfer coefficient
6
The solid-liquid contact angle for HEC QP-300 and Carbopol 934 measured in this study by the sessile
drop method for different concentration solutions, though not presented here, showed insignificant
change from those for water (Manglik et al., 2003).
80
CHAPTER 4
VISUALIZATION AND CHARACTERIZATION OF NUCLEATE POOL
BOILING IN AQUEOUS SURFACTANT SOLUTIONS
4.1
Introduction
Nucleate boiling in aqueous surfactant or polymer solutions is a complex
conjugate process as illustrated earlier in Fig. 1.2. It depends on surfactant concentration,
wall heat flux level, surfactant chemistry, dynamic surface tension, surface wettability
and nucleation cavity distribution, and marangoni convection, heater geometry and its
surface characteristics, and rheological properties of the solutions, etc.
The nucleation is a process in which finite size clusters of molecules encompassing
properties of the second phase appear in the host liquid (Brennen, 1995), and generally
subdivided into two categories: homogeneous nucleation and heterogeneous nucleation.
This study is only related to heterogeneous nucleation – a process in which bubbles form
discretely at pits, scratches, and grooves on a heated surface submerged in a pool of
liquid. The onset of nucleate boiling on a heater submerged in a pool of liquid is
characterized by the appearance of vapor bubbles at discrete locations on the heater
surface (Kenning, 1999; Dhir, 1998).
Surface finish, surface wettability, heater
geometry, surface contamination, system pressure, liquid subcooling, gravity, heat flux
modes (steady state or transient) are considered to have a significant influence. However,
the primary heat transfer is by evaporation and its efficiency is directly related to
nucleation site density and bubble dynamics, and phenomenological insights can be
obtained from a visual observation of the ebullience. A knowledge of nucleate site
density as a function of wettability and wall superheat is critical in order to develop a
81
credible model for predication of nucleate boiling. The latter includes the processes of
bubble growth, bubble departure, and bubble waiting time (reformation of the thermal
layer).
For phenomenological study of nucleate boiling in aqueous surfactant or polymer
solutions, visualization is an important approach to investigate how the active nucleation
density changes due to surfactant adsorption at the solid-liquid interface, and how the
reduced dynamic surface tension affects the bubble dynamics. The boiling heat transfer
performance can be quantitatively related to its ebullience (nucleation, inception and
gestation → growth → departure). The growth of nucleating vapor bubbles and their
motion near the cylindrical heater surface were recorded by a PULNiX TMC-7 highspeed color CCD camera with shutter speeds of up to 0.1 micro-second. The CCD
camera is interfaced with a PC through a FLASHBUS MV Pro image capture kit that has
high-speed PCI-based bus-mastering capabilities (up to 132 Mbytes/s). It delivers
consecutive frames of video in real time into the system memory while keeping the CPU
free to operate on other applications. Furthermore a FUJI 12.5-75mm micro lens was
used on the CCD camera to facilitate high quality close-up photography.
4.2
Zeta Potential and Contact Angle
The surface wettability can be characterized by zeta potential ζ (an electrokinetic
control parameter for the stability of hydrophobic colloids), which shows distinct regions
of change along the adsorption isotherm that are associated with the aggregation mode of
adsorbed ions at the solid-water interface. The higher wettability tends to suppress
nucleation and bubble growth, thereby weakening the boiling process.
82
Zeta potential and contact angle measurements complement each other and allow
the understanding of electrokinetic characteristics and the surface wetting behavior due to
the physisorption of surfactant molecules at the solid-liquid interface. The change in zeta
potential and measured surface wettability (represented by the contact angle) with
concentration in aqueous SDS (anionic) and CTAB (cationic) are graphed in Fig. 4.1,
depicting the distinct regions of change in adsorption and the corresponding wettability
variation. The change in wettability with the adsorption of ionic surfactants with
concentration, and its direct correlation with zeta potential is clearly evident. SDS shows
a stronger adsorption than CTAB at the liquid-solid surface, which is reflected in the
magnitude of zeta potential and the larger changes in surface wettability. After the point
of zeta potential reversal (PZR) (Lyklema et al., 1991; Fuerstenau, 2002), or isoelectric
point (IEP) (Hunter, 1981), the slope of the ζ potential curve becomes negative for the
anionic surfactant SDS or positive for the cationic surfactant CTAB because of the
opposite charges they carry, and suggests that some of the adsorption may start taking
place in reverse orientation to form reverse hemimicelles (Fuerstenau, 2002), where the
surface becomes increasing hydrophilic, which is evident from the contact angle
measurement as shown in Fig. 4.1. As CMC is approached, a bilayer is formed, the
contact angle tends to be constant, and surface becomes highly hydrophilic.
83
101
90
102
103
80
-30
70
60
-10
PZR
50
Zeta Potential [mv]
Contact angle [deg]
-20
0
40
30
20
101
Contact angle, CTAB
Contact angle, SDS
Zeta potential (Vanjara and Dixit, 1996), CTAB
Zeta potential (Sakagami et al. 2002), SDS
102
103
10
20
104
Concentration [wppm]
Fig. 4.1 Measured streaming zeta potential and contact angle for the adsorption of
SDS and CTAB in their aqueous solutions
84
4.3
Ethoxylation Effect
The presence of ethoxy or ethylene oxide (EO) group in their hydrocarbon chain
(ethoxylation) increases the overall size of the polar head and makes the surfactant more
hydrophilic.
Consequently, it will alter the solution’s surface wettability drastically
(Barry and Wilson, 1978; Ashayer et al. 2000), and this is clearly seen from the contact
angle measurements for CTAB and Ethoquad 18/257 graphed in Fig. 4.2. The later has
significantly lower contact angles with increasing C, particularly when C ≥ CMC. This is
a direct consequence of the surfactant chemistry and its physisorption dynamics
(Fuerstenau, 2002), and the altered surface wettability probably accounts for the boiling
deterioration in Ethoquad O/12 PG and Ethoquad 18/25 sloutions at C > CMC
concentrations as shown in Fig. 3.7.
Contact angle [deg]
101
80
102
103
70
c.m.c.
60
CTAB
Ethoquad 18/25
50
101
102
c.m.c.
103
104
Concentration [wppm]
Fig. 4.2 Measured contact angle for aqueous CTAB and Ethoquad 8/25 solutions
7
Both CTAB and Ethoquad 18/25 are cationic and comparable. While CTAB has a higher molecular
weight, Ethoquad 18/25 has an even higher M but with a relatively high ethoxylation of 15 EO groups.
Surfactant with ethoxylation doesn’t necessarily mean it would have higher wettability. Physisorption at
the solid-liquid interface is a complex process, which depends on the surfactant structure, chemical and
physical properties, and surface characteristics etc. Electrokinetics (zeta potential) and adsorption
isotherm are more direct indictors for the surfactant adsorption process.
85
4.4
Dynamic Surface Tension – Molecular Weight Effect
When a new surface in a surfactant solution is created, a finite time is required to
reach equilibrium state between the surface concentration and the bulk concentration.
This non-equilibrium surface tension is called dynamic σ. The surface tension relaxation
is a diffusion-rate dependent process, the time required to reach equilibrium varies from a
few milliseconds to a few hours, and is typically found to depend on the type of
surfactants, its diffusion-adsorption kinetics, micellar dynamics, ethoxylation, and bulk
concentration levels (Manglik et al., 2001). The time scale for complete surface tension
relaxation tends to be smallest for lower molecular weight compounds (Iliev et al., 1992).
The effectiveness of an additive is often judged by its ability to lower a liquid's static
surface tension at the lowest possible concentration. Equally important, the additive
should not cause undesirable side effects, such as interference with solid interface
adhesion, or increased tendency to foam. For boiling applications with small surface age
interface (10-100 ms, Prosperetti and Plesset, 1978), however, the dynamic surface
tension relaxation process rather than the equilibrium surface tension is perhaps the more
critical determinant (Wasekar and Manglik, 2002).
A lower molecular weight surfactant diffuses faster than its higher molecular
counterpart, and this is seen in the faster σ relaxation of CTAB (cationic) in comparison
with that for Ethoquad (cationic) 18/25 in Fig. 2.4. Anionics (SDS and SLES) and
nonionics (Triton X-100 and Triton X-305) show the same dynamic surface tension
behavior with molecular weight as present in Figs. 4.3 and 4.4. It should be noted that
the molecular weight effect might not necessarily be the same for different ionic
categories of surfactants.
In most surfactant solutions, the time scale to reach the
86
equilibrium value (total relaxation) at a newly-created interface is of the same order as
that of bubble formation and departure in nucleate boiling (0-100 ms).
70
SLES
SDS
C/Cc.m.c. = 0.5
C/Cc.m.c. = 1
C/Cc.m.c. = 2.0
σ [mN/m]
60
50
40
Typical bubble frequencies
in nucleate boiling of water
30
10-2
10-1
100
101
Surface Age [s]
Fig. 4.3 Dynamic surface tension relaxation for aqueous anionic SDS and SLES solutions
87
70
Trotion X-305
Triton X-100
C/Cc.m.c. = 0.5
C/Cc.m.c. = 1
C/Cc.m.c. = 2.0
σ [mN/m]
60
50
40
30
20
10-2
Typical bubble frequencies
in nucleate boiling of water
10-1
100
101
Surface Age [s]
Fig. 4.4 Dynamic surface tension relaxation for aqueous nonionic Triton X-100
and Triton X-305 solutions
Figure 4.5 provides insights on the role of surfactant molecular weight,
ethoxylation, and dynamic surface tension of their solutions on the heat transfer
enhancement. The normalized pool boiling heat transfer coefficient data for cationic
DTAC (4000 wppm), CTAB (400 wppm), Ethoquad O/12 PG (600 wppm), and Ethoquad
18/25 (700 wppm) solutions are graphed.
While their respective concentration is
different, the dynamic surface tension value (nominally representative of nucleate boiling
bubble frequencies) of their aqueous solutions at 80ºC is the same (~ 47 mN/m) in each
case. In the measured range of heat fluxes, the heat transfer enhancement is seen to be in
88
the order of DTAC > CTAB > Ethoquad O/12 PG > Ethoquad 18/25, which is in the
reverse order of their respective molecular weights and number of EO groups. It also
shows a boiling deterioration at low heat flux ( qw′′ < 50 kW/m2) for Ethoquad 18/25.
Within the typical time transients for bubble growth in nucleate boiling of surfactant
solutions, the faster diffusion of lower molecular weight surfactants (higher mobility)
leads to a larger number of surfactant molecules approaching the growing bubble
interface. They, therefore, reduce the surface tension faster (dσ/dτ8) and increase the
bubble growth and departure frequencies to yield better heat transfer performance. Also,
in this dynamic ebullient and additive adsorption process, a measure of the dynamic
surface tension is the more appropriate scaling property instead of a static value at a fixed
bubble frequency.
The normalized pool boiling heat transfer coefficient data for anionic (SDS and
SLES) and nonionic (Triton X-100 and Triton X-305) solutions (Wasekar and Manglik,
2002) are also presented here in Fig. 4.6. The lower qw′′ range, which represents the
partially developed nucleate boiling regime, is characterized by the thermal hysterisis and
wetting behavior of the surfactant solution. At higher heat fluxes, typically in the fully
developed nucleate boiling regime ( qw′′ > 150 kW/m2), however, the maximum heat
transfer enhancement is in the order of SDS > SLES > Triton X-1009 > Triton X-305.
This is also, in the reverse order of their respective molecular weights and number of EO
groups. These results therefore suggest that the surfactant molecular weight, and dynamic
surface tension relaxation instead of the static value are critical parameters in predicting
8
9
The dynamic surface tension gradients (dσ/dτ) used here were also obtained at a surface age of 50 ms.
Triton X-100 is highly surface active and depresses σ very quickly; however, it also has 9-10 EO groups.
It is the combination of dynamic surface tension and surface wettablity effects that determine the heat
transfer performance.
89
their enhanced nucleate pool boiling heat transfer performance under the given operating
conditions.
1.0
C
EO σ50ms,80°C dσ/d-2τ
(x 10 )
0
47.1
3.6
0
46.5
3.1
2
47.0
2.5
15
46.7
1.6
M
DTAC
4000 263.9
CTAB
400 364.5
Ethoquad O/12 600 403
Ethoquad 18/25 700 994
0.8
(hsurf - hwater) / hwater
0.6
0.4
0.2
0.0
Fully developed
nucleate boiling
-0.2
0
50
100
150
"
200
250
2
q w (kW/m )
Fig. 4.5 Effect of surfactant molecular weight and its ethoxylation on the heat
transfer coefficient enhancement
90
1.0
C
0.9
EO σ50ms,80°C dσ/d-2τ
(x 10 )
SDS
2500 288.3 0
SLES
1750 422
3
Triton X-100 200 624
9
Triton X-305 750 1526 30
0.8
(hsurf - hwater) / hwater
M
42.8
43.1
45.1
47.5
3.5
2.1
2.9
2.1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
50
100
150
200
250
q"w (kW/m2)
Fig. 4.6 Surfactant molecular weight dependence of the maximum enhancement in heat
transfer coefficient enhancement (Wasekar and Manglik, 2001)
91
4.5
Surface Wettability Effect on Nucleate Boiling Heat Transfer
With an increase in surface wettability, the nucleate boiling curves tend to shift to
the right. This is seen in Fig. 4.7(a) where the saturated nucleate boiling data of Liaw and
Dhir (1989), obtained on vertical surfaces having different static contact angles, are
graphed. These data are for the fully developed nucleate boiling regime, and the data of
Nishikawa et al. (1984) for a vertical surface are also included. It is noted that although
the boiling curves shift to the right with an increase in the wettability (decreasing contact
angle), the qw" − ∆T gradient remains nearly unchanged. Another interesting point is that
critical heat flux (CHF) increases as the surface wettability improves. For liquids that
wet the surface well (contact angle → 0), a temperature overshoot or hysteresis is
observed in the ONB region of the boiling curve as shown in Fig. 4.7(b) for nucleate pool
boiling of a highly-wetting liquid R113. For these liquids, the convective heat removal
process continues to persist up to a higher superheat; upon incepience, a large reduction
in superheat is observed, and it only occurs with increasing heat flux (Dhir, 1998,
Kenning, 1999).
A closer inspection of Figs. 3.3 through 3.6 shows that boiling in aqueous
surfactant solutions when C ≥ CMC exhibits a similar behavior to that in Fig. 4.7. The
temperature overshoot or hysteresis was also observed when C ≥ CMC, especially in the
highly wetting aqueous Ethoquad 18/25 solutions. From this observation, one can deduce
that there is indeed a wettability change considering that the viscosity and thermal
conductivity remains almost unchanged in dilute surfactant solutions. This is further
clearly established by the contact angle measurements presented in Fig. 2.9.
92
105
100
Water
CHF
q"w ⇓
q"w ⇑
q"w (W/m2)
qw" (W/cm2)
104
50
θ = 90°
3.3
θ = 69°
103
1
θ = 27°
θ = 38°
θ = 14°
Nishikawa et al. (1984), θ = 90°
20
10
15
20
30
102
10-1
40
∆Τ (Κ)
100
∆T (K)
101
Fig. 4.7 (a) Effect surface wettability (or contact angle θ) on nucleate pool boiling (Liaw and Dhir, 1989)
(b) Nucleate pool boiling data for Refrigerant of R113 on a copper tube (Jung and Bergles, 1989)
93
102
4.6
Ebullient Dynamics Visualization
The enhanced boiling performance can be related to the ebullient characteristics.
The photographic records for different concentration aqueous solutions of CTAB and
Ethoquad 18/25, SDS, Triton X-305 (typically representative of the four cationics,
anionics, and nonionics), HEC-QP300 (surface-active polymer), and Carbopol 934
(shear-thinning polymer), are presented in Figs. 4.8, 4.9 and 4.10, respectively.
4.6.1
Visualization in aqueous surfactant solutions
Figure 4.8 represents the boiling behavior in water, and CTAB and Ethoquad
18/25 solutions (non-ethoxylated and ethoxylated cationic surfactants, with number of
EO groups = 0 and 15, respectively) of three different concentrations (C/CCMC = 0.5, 1,
and 2) at two different heat flux levels ( qw′′ = 20, and 50 kW/m2). In comparison to that in
water, boiling in CTAB solutions is more vigorous and is characterized by clusters of
smaller-sized, more regularly shaped bubbles that originate at the underside of the
cylindrical heater. These bubbles then slide along the heater surface at departure, thereby
knocking off much smaller bubbles growing on the top surface of the heater. This
process was observed to increase with heat flux and the consequent higher bubble
departure frequency.
Also, because of the considerable reduction in σ for CTAB
solutions, much smaller-sized bubbles are nucleated in a cluster of active nucleation sites,
especially at lower heat fluxes.
They have a significantly higher bubble departure
frequency, with virtually no coalescence of either the neighboring or sliding bubbles that
come in contact with others around the heater’s periphery when C < CMC. However,
when C ≥ CMC, some foaming patches begin to occur, the liquid only coverage of the
94
heater surface increases, and slightly larger bubbles are formed, all of which indicate a
surface wetting condition change. A foam layer, whose thickness increases with heat
flux, is also seen to form at the free surface of the pool. This boiling history is very
similar to that also seen in anionic and nonionic surfactant solutions (Wasekar and
Manglik, 2002; Zhang and Manglik, 2003).
Boiling in Ethoquad 18/25, on the other hand, shows much smaller-sized bubbles
in pre-CMC solutions, and considerably fewer and larger-sized bubbles are formed with
increasing concentrations. This presents a contrastingly different behavior from not only
that of water but also that of CTAB, and is perhaps due to the surfactant’s high degree of
ethoxylation. The presence of large number of EO groups in their hydrocarbon chains
totally changes the surface wettability at the solid-water interface. This is quite evident
from the measured contact angle data presented in Fig. 4.2.
The observed ebullience and boiling data cannot be explained by the reduced
dynamic surface tension alone. If this were so, then smallest-sized bubbles would be
seen in C ≥ CMC solutions, where the surface tension reaches the lowest possible value.
Instead, because of the adsorption of surfactants and their different orientations in the
adsorption layer (Fig. 2.9b), the heater surface wettability increases with increasing
concentration. Fewer bubbles are thus nucleated that have relatively larger departure
diameters.
As pointed out by Fuerstenau (2002), the adsorption isotherm and
corresponding surface state can be divided into four regimes that are associated with the
aggregation mode of adsorbed ions at the solid-water interface: (1) at low concentrations,
95
surfactant adsorption takes place as individual ions10; (2) there is a sharp increase in the
adsorption density due to self-association of adsorbed surfactant ions and the formation
of hemimicelles; (3) the surfactant ions adsorb as reverse hemimicelles, with their polar
heads oriented both toward the surface and liquid, and the surface becomes increasingly
hydrophilic; and (4) as the CMC is approached, the adsorption becomes independent of
the bulk concentration in solution, and the surfactant molecules form a bilayer on the
surface to make it strongly hydrophilic. The measured contact angle data in Fig. 2.9(a)
correlate well with this characterization schematically illustrated in Fig. 2.9(b). The
reason that Ethoquad 18/28 shows more hydrophilic behavior that CTAB at lower
concentrations is because of its bulky polar head that occupies larger portion of the heater
surface, and the variation in ebullience at the heater surface (Fig. 4.8) also reflects this
wettability change.
The ebullience of SDS and Triton X-305 (non-ethoxylated anionic and
ethoxylated nonionic surfactants, with number of EO groups = 0 and 30, respectively) as
presented in Fig. 4.9 displayed a similar behavior with degree of ethoxylation and
varying surfactant concentration and hence are not discussed separately.
10
Adsorption takes place with polar heads of the surfactant ions oriented toward the surface that yields a
hydrophobic surface with most surfactants except for high molecular weight ones, whose bulky polar
head would occupy a larger surface area.
96
Water
Water
22.2 mm
CTAB
CTAB
Ethoquad 18/25
Ethoquad 18/25
C/CCMC = 0.5
C/CCMC = 0.5
C/CCMC = 1
C/CCMC = 1
C/CCMC = 2
C/CCMC = 2
kW/m2
50 kW/m2
20
Fig. 4.8 Ebullient behavior in nucleate boiling of distilled water, and aqueous cationic CTAB
and Ethoquad 18/25 solutions of different concentrations (C/CCMC = 0.5, 1, and 2) at qw" = 20
kW/m2 and 50 kW/m2
97
Water
Water
22.2 mm
SDS
SDS
TRITON X-305
TRITON X-305
C/CCMC = 0.5
C/CCMC = 0.5
C/CCMC = 1
C/CCMC = 1
C/CCMC = 2
C/CCMC = 2
kW/m2
50 kW/m2
20
Fig. 4.9 Ebullient behavior in nucleate boiling of distilled water, and aqueous SDS (anionic)
and Triton X-305 (nonionic) solutions of different concentrations (C/CCMC = 0.5, 1, and 2) at
qw" = 20 kW/m2 and 50 kW/m2
98
4.6.2
Visualization in aqueous polymer solutions
The nucleate boiling performance of HEC and Carbopol solutions can also be
qualitatively related to the ebullient characteristics, and typical photographic records are
presented in Fig. 4.10. The photographs depict the boiling history with increasing heat
flux ( qw′′ = 20, 50, and 100 kW/m2) for aqueous solutions of two different concentrations
each (300 wppm and 1000 wppm for HEC, 100 wppm and 300 wppm for Carbopol11),
along with that for deionized distilled water. Boiling and the attendant bubbling process
in HEC solutions are seen to be distinctly different from that in pure water. It is more
vigorous, and is characterized by smaller-sized and more regularly shaped bubbles that
have a reduced tendency for coalescence when C < CPC. It is also noted that the HEC
additive leads to an early inception of bubbles with a faster covering of the heating
surface and a higher bubble departure frequency. This is essentially the outcome of
reduced surface tension at the liquid-vapor interface. Also, molecular adsorption on the
heating surface may contribute to the formation of new sites (Levitskiy et al., 1996),
which in turn would explain the increase in number of bubbles as there was no change in
the surface wettability (measured by the contact angle) (Manglik et al., 2003). However,
at very large concentration (C > CPC), the bubbles that originate at the underside of the
cylindrical heater tend to coalescence and form bigger bubbles as they slide along the
cylindrical periphery of the heater surface at departure. At the same time, there are some
small patches that are covered by liquid, and no bubbles formed underneath these
patches. This phenomenon is totally different from that in boiling of surfactant solutions,
in which surface wetting condition significantly changes or foaming begins to occur
11
In Carbopol 934 solutions, the boiling pool became significantly more cloudy with increasing
concentration and it was very difficult to get clear photographs when C > 300 wppm. Hence, snapshots
for a higher concentration solution are not presented.
99
when C > CMC. Perhaps the significantly increased viscosity of higher concentration
polymer solutions tends to suppress the bubble nucleation process and growth of vapor
bubbles. As a result, some nuclei do not get activated at all, and this also then leads to the
deterioration of boiling performance in aqueous solutions with C > CPC.
Boiling in Carbopol 934 solutions, on the other hand, shows a totally different
ebullient behavior than that of water, as well as that of surface-active HEC solutions.
Considerable bubble suppression is observed in Carbopol 934 solutions, as well as
dispersed vapor explosions (bright white spots captured in the picture; Fig. 4.10) in some
regions of the heater surface. Same kind of bubbling activity was also observed by Bang
et al. (1997) in dilute polyethylene oxide (PEO) solutions. Furthermore, the Carbopol
additive in water lead to a delayed inception of bubbles, or ONB, and the sparsely formed
bubbles have a slower departure frequency. This is essentially the outcome of increased
viscosity in Carbopol solutions, which gives a higher viscous resistance for the bubbles to
nucleate and depart. Again, contrasting the boiling ebullience with that of pure water, as
well as between HEC and Carbopol solutions, there appear to be more nucleation sites
activated in aqueous HEC QP-300 solutions than that with Carbopol 934. The higher
active nucleation site density would also explain the increased heat transfer performance
in surface-active HEC solutions.
100
Water
HEC
300 wppm
HEC
1000 wppm
Carbopol
100 wppm
Carbopol
300 wppm
20 kW/m2
50 kW/m2
100 kW/m2
Fig. 4.10 Ebullient behavior in nucleate boiling of distilled water, and aqueous HECQP300 and Carbopol 934 solutions of different concentrations at different heat fluxes
( qw" = 20 kW/m2, 50 kW/m2, and 100 KW/m2)
101
4.7
Characterization of Nucleate Pool Boiling in Aqueous Surfactant Solutions
A successful correlation always depends on the recognition of the problem
(Mordecai, 1930). For boiling in pure liquids, a variety of different parameters and
mechanisms have been proposed to describe the complex behavior (Dhir, 1998; Nelson et
al., 1996). However, the potential mechanisms that affect nucleate boiling in aqueous
surfactant solutions are not clearly established; particularly, the adsorption processes at
liquid-vapor and solid-liquid interfaces perhaps require an in-depth scaling of the various
observed mechanism. A direct correlation of the heat transfer with suitable descriptive
parameters for all effects is difficult because of the complex nature of the problem. A few
recent attempts have been made (Wen and Wang, 2002; Sher and Hetsroni, 2002;
Wasekar and Manglik, 2002) to correlate the nucleate boiling heat transfer for aqueous
surfactant solutions, but the adopted methodologies have significant limitations, and their
general applicability is not established.
Wen and Wang (2002) modified the pure-fluid Mikic-Rohsenow (1969)
correlation with the available contact angle and surface tension data to represent their
own boiling data. However, equilibrium surface tension at room temperature, rather than
the dynamic effects, and that too without stipulating the level of concentration has been
used in their analysis. Because a single concentration value seems to be the implicit
guiding factor in its applicability, a generalization of the correlation is circumscribed.
Sher and Hetsroni (2002) considered a surfactant diffusion mechanism for both the
liquid-vapor and solid-liquid surface tensions, and proposed a model based on the
Rohsenow (1952) correlation. It gave an acceptable agreement with their own
experimental data for pre-CMC solutions, and that too for only one surfactant (Habon G).
102
No data for post-CMC solutions were presented, and it was indicated that the diffusion
and adsorption mechanisms might be altered in such a case. It may be noted that at low
concentrations the adsorption process is generally in the monolayer region, which will
not show drastic wetting changes on the heater surface (Fuerstenau, 2002). Wettability is
essentially governed by the surfactant concentration and its adsorption process at the
solid-liquid interface. Also, nucleate boiling heat transfer is influenced by variations in
the surfactant chemistry. Wasekar and Manglik (2002) considered the effect of surfactant
ionic nature and molecular weight (exponent of -0.5 for the ratio of molecular weights for
anionics and 0 for the nonionics when C < CMC at fully developed boiling regime) on
the pool boiling performance of water, and pointed out that dynamic surface tension is
perhaps a critical predictor of the ebullient phase-change behavior. However, rather
simplistically, the value of σ at a fixed bubble frequency or surface age τ was adopted to
represent the dynamic surface tension effects. This study explores the role of surfactant
adsorption and interfacial phenomena as illustrated in Fig. 4.11, manifest in the dynamic
surface tension relaxation due to the dynamic adsorption-desorption process at the liquidvapor interface, and the surface wettability changes due to surfactant physisorption at the
solid-liquid interface, on nucleate boiling heat transfer.
The nucleate boiling heat transfer depends on the nucleation, bubble growth rates,
bubble departure frequencies and sizes, and surface wettability. With the addition of
small amounts of surfactants, the saturated nucleate pool boiling of water on a cylindrical
heater is altered significantly. The heat transfer is enhanced in solutions with C ≤ CMC,
but decreases when C > CMC. In general, besides the heat flux (or wall superheat) levels,
the relative extent of performance change is seen to be governed by the surfactant
103
C
Liquid-vapor
interface
Liquid-solid
interface
q"
(a) Schematic of a nucleated bubble
Individual
Ions
Reversed
Hemimicelles
Hemimicelles
Hydrophobic tail
Bilayer
Hydrophilic head
Hydrophobic
Hydrophilic
C
(b) Adsorption at the solid-liquid interface
C
(c) Adsorption at the liquid-vapor interface
Fig. 4.11 Schematic of interfacial phenomena in aqueous surfactant solutions (not
to scale)
104
interfacial phenomena at both the liquid-vapor and solid-liquid interfaces. These in turn
are determined by several additive physico-chemistry-based factors, and their rather
complex inter-relationships are characterized in Fig. 4.12. Some key factors are analyzed
in details as following:
1. Equilibrium and dynamic surface tension
Surface tension is an important variable for nucleate boiling. The nucleate pool
boiling heat transfer coefficient h is related to the equilibrium surface tension σ of the
pure liquids by the following (Rohsenow, 1952)
h ∝ 1/σ 3
(4.1)
and the bubble departure diameter was proposed by Fritz (1935) as
D ∝ σ 1/2
(4.2)
However, equilibrium surface tension could be deceiving when applied to nucleate
boiling heat transfer in aqueous surfactant solutions.
When a new surface is created in a surfactant solution, a finite time is required for
the reagent adsorption to reach an equilibrium state between the surface concentration
and bulk concentration. This time-dependent surfactant adsorption at the vapor-liquid
interface of a bubble gives rise to the dynamic surface tension (DST) behavior, which,
however, eventually reduces to the equilibrium condition after a long time period
(Holmberg et al., 2003; Chang and Franses, 1995; Rosen, 1989). For boiling applications,
the formation and departure of bubbles is also a highly dynamic process where the vaporliquid interface has a small surface age; typically in the range of 0-100 ms (Prosperetti
and Plesset, 1978). The dynamic surface tension relaxation rather than the equilibrium or
static surface tension, therefore, becomes the more critical determinant.
105
Bubble growth rate and departure
Marangoni convection
Bubble Size
Nucleation
Liquid-vapor interface
adsorption/desportion
Physico-Chemical
Properties
Dynamic Surface Tension*
•Ionic Nature
(Liquid-vapor, bulk chemistry)
•Molecular structure
•Ethoxylation
Contact Angle/Wettability
(Equilibrium and dynamic)
Microlayer thickness
Heat flux removal rate
Wetting/spreading
Surfactant
•Molecular Weight
•Micelle
•Packing
Zeta Potential (Physisorption)*
(Solid-liquid, surface chemistry)
Other factors •Heater size and its characteristics
Active nucleation site density
•Heat flux level (superheat)
Bubble nucleation rate and size
•Fluid transport properties
Bubble collapse and merge
•Far surface features and foaming
Boiling hysteresis
•Physical properties
Fig. 4.12 Surfactant effects on nucleate boiling heat transfer in its aqueous solutions
106
When a bubble starts to form at a nucleating site, the initial surface tension is that of
water and it then reduces continually with time τ as more and more surfactant molecules
move to the interface during the adsorption process. At the same time, desorption may
also occur till an equilibrium condition is reached and surface tension becomes constant.
This time-dependent adsorption/desorption process is schematically illustrated in Fig.
4.13(a), and is reflected in the measured dynamic surface tension behavior seen in Figs.
4.3 and 4.4. The effects of dynamic surface tension on bubble formation and departure
are more clearly demonstrated in Fig. 4.13(b), where results of a controlled adiabatic
single-bubble experiment are presented. Photographic images of a near-departure air
bubble, captured by a high-speed (2000 frames/sec) camera under identical operating
conditions, are presented. Constant-pressure bubbling activity in the following three
liquids was recorded: water (pure liquid, σeq = 72.3), 2500 wppm SDS solution
(surfactant solution, σeq = 37.5 at CMC), and N,N-Dimenthyl Formamide or DMF (pure
liquid, σeq = 37.1). The bubble surface age was controlled at ~100ms, which is of the
same order as the time scales for the dynamic adsorption/desorption process in aqueous
surfactant solutions and ebullience in nucleate boiling. Dynamic σ effects are selfevident in Fig. 4.13(b), where a larger bubble is seen in aqueous SDS solution when
compared to that in a pure liquid (DMF) that has the same equilibrium σ value (~ 37
mN/m). This is essentially due to the time-dependent surfactant adsorption/desorption
process at the liquid-vapor interface.
107
τw
τg
(a)
Water (pure-liquid)
(σ = 72.3)
SDS solution
(σ = 37.5)
DMF (pure-liquid)
(σ = 37.1)
(b)
Fig. 4.13 (a) Schematic of surfactant transport process during a bubble formation
and departure (not to scale); (b) Dynamic surface tension effect on bubble
dynamics (evolution of pre-departure shape and size).
108
2. Surface wettability (contact angle)
One critical determinant of nucleate boiling is the active nucleation density, which
is a function of wettability and directly accounts for the energy transfer by ebullience at
the heater surface.
Surface wettability is usually measured by the solid-liquid
equilibrium contact angle. On a typical large heater surface, the fraction of nucleated
cavities decreases as the surface wettability increases (Dhir, 1998). Also, the volume of
trapped air or vapor in a cavity, incipient superheat, boiling incipience, hysteresis, critical
bubble radius, and departure bubble size are strongly influenced by surface wettability.
In the case of aqueous surfactant solutions, the reagent will undergo a certain adsorption
pattern (different orientation of the surfactant molecule head or tail) with changing
concentration at the solid-liquid interface. Consequently, the heater surface will show
different wetting behavior at different adsorption stages because of the unique structure
of a surfactant – a hydrophilic head with a hydrophobic tail.
3. Microlayer
There is increasing evidence with a growing body of literature to support the
hypothesis that a thin layer of liquid (the microlayer) forms beneath a vapor bubble, and
that it accounts for most of the wall heat transfer during saturated nucleate boiling
process (Cooper and Lloyd, 1968; Lee and Nydahl, 1989; Lay and Dhir, 1995; Zhao et
al., 2002). With surfactant additives in water, the liquid-vapor and solid-liquid surface
tension are altered, which changes the energy balance at the solid-liquid-vapor threephase contact line. Therefore, the dynamic (advancing and receding) contact angle and
microlayer thickness will be different in aqueous surfactant solutions compared to that of
pure liquid. The consequent heat flux removal mechanism and the role of the microlayer
109
under these conditions are yet to be developed.
4. Marangoni effect
Variation of the temperature distribution at a vapor-liquid interface results in local
surface tension gradients along the interface. This in turn produces displacements in the
interface in the direction of increasing surface tension, and the phenomenon is commonly
referred to as Marangoni effect (Named after Italian physicist Carlo Marangoni) (Scriven
and Sternling, 1960). Similarly in aqueous surfactant solutions, local variations in the
reagent concentration produce a similar convective effect. Marangoni convection is not
only important to the bubble incipience and waiting periods, but also plays a role in the
reduced bubble coalescence of boiling in aqueous surfactant solutions (Yang, 1990). The
effects of surfactant concentration on the initial short-time-scale Marangoni convection
around boiling nuclei in aqueous solutions have been computationally investigated by
Wasekar and Manglik (2003). Their model consists of an adiabatic, rigid, hemispherical
bubble on a downward facing constant temperature heated wall, in a fluid pool with an
initial uniform temperature gradient. With a surfactant present in solution, a surface
concentration gradient develops at the bubble interface that will promote diffusocapillary
flows along with the temperature-gradient induced thermocapillary flows. However, the
Wasekar and Manglik (2003) model is rather constrained and simplistic. The complex
interactions of both thermo- and diffuso- capillary Marangoni convection, the excess
surfactant concentration due to the physisorption at the solid-liquid interface, mass
transfer at the liquid-vapor interface and its development, adjacent bubble effect, and the
convection in the bulk solution caused by the bubble growth and departure, are yet to be
understood and modeled.
110
5. Nuclei formation
Trace amounts of foreign particles in liquids known to affect nucleation process.
Though there is no convincing evidence, the additives may agglomerate in solutions and
form particle large enough to serve as boiling nuclei. This speculation may be especially
valid for long chain polymeric surfactants or polymers in aqueous solutions.
It is desirable that the results of all surfactant effects on the ebullient heat transfer
can be analyzed and formulated into one convenient equation form for practical design
applications. A successful correlation of nucleate boiling data for aqueous surfactant
solutions, however, should include the complete interfacial phenomena. That is not only
the adsorption/desorption process at the liquid-vapor interface, but also the physisorption
at the solid-liquid contact. Excluding one or the other would be an oversimplification of
this complex process. Clearly further systematic research is necessary to establish these
and other surfactant effects on pool boiling heat transfer before a generalized model can
be developed.
111
CHAPETR 5
SIMULATION OF A SINGLE BUBBLE DYNAMICS
5.1
Introduction
Several mechanistic and semi-mechanistic models have been developed to
facilitate the understanding of boiling behavior and provide performance prediction tools.
A critical element of this work is the simulation of the single-boiling-bubble dynamics, in
order to extend the fundamental understanding of the phase-change process. However,
because of the complexity involved in modeling the continuously evolving vapor-liquid
interfaces in pure liquids, greatly simplifying assumptions are often made in developing
various models (Shyy et al. 1996). This is further complicated by the need to include the
adsorption-desorption controlled interfacial surfactant transport for ebullience in
surfactant solutions. Over the past decade, some studies have expanded the complexity
of nucleating bubble-dynamics models, by implementing a direct numerical tracking of
the changing liquid-vapor interface. The finite-volume method (FVM) (Welch, 1998),
volume of fluid (VOF) method (Welch and Wilson, 2000), level-set method (LSM) (Son
et al., 1999), and arbitrary Lagrangian-Eulerian (ALE) method (Yoon et al., 2001) have
been typically used. Some recent work (Takada et al., 2000; Palmer and Rector, 2000)
has also considered the Lattice-Boltzmann method (LBM).
Plesset and Zwick (1954) were probably the first to present a one-dimensional
analytical model for the motion of a spherical bubble. More recently, Lee and Nydahl
(1989), and Patil (1991) have numerically simulated the bubble growth in nucleate
boiling with an expanded model by including the microlayer, for whose thickness the
formulation of Cooper and Lloyd (1969) was employed. Although the heterogeneous
112
temperature field and the hydrodynamics were accounted correctly by solving the
momentum and energy equations in the liquid, it was assumed that the bubble remained
hemispherical during its growth, and, as such, the bubble departure process could not be
demonstrated. For a direct calculation of the moving interface in nucleate pool boiling,
Welch (1998) has applied the finite-volume method on a moving unstructured mesh.
However, the calculation was limited to a small deformation of the two-phase interface
due to numerical instabilities caused by the severe distortion of the computational grid.
The VOF method (Welch and Wilson, 2000) has been widely used because of its nice
property of preserving the volumes of the two phases in flows, though a very fine grid is
needed to capture a smooth interface. Also because it is difficult to compute accurate
local curvatures from volume fractions, modeling surface tension forces becomes a
problem. A notably newer approach to the solution of interface evolution problems is the
level-set Hamilton-Jacobi formulation (Osher and Sethian, 1988; Sethian, 1990), which
has been applied to incompressible two-phase flows by Sussman et al. (1994). A
modified version that accommodates the liquid-vapor phase change process in nucleate
boiling has been employed by Son et al. (1999) to capture the dynamics of a single
bubble growing on a horizontal heated surface. It is noted, however, that this application
of the level-set method does not have a liquid-vapor volume-preservation property. A
fluid particle tracking mesh-free method (MPS-MAFL) has been extended by Yoon et al.
(2001) for an ALE simulation of this problem. In this calculation, the particle number
density is not constant and particles are allowed to be concentrated locally for higher
resolution; the microlayer beneath the bubble, however, has not been modeled.
113
This study presents a complete numerical simulation of the ebullient dynamics of
a single bubble in pool boiling of water with effects of apparent contact angle, wall
superheat, and altered surface tension and viscosity. The nucleating bubble is
decomposed into a microlayer and a macro region in order to account for the peripheral
heat transfer. The fluid motion, pressure drop, and heat transfer in the microlayer are
modeled by a fourth-order differential equation, similar to that developed by Lay and
Dhir (1995). The governing mass, momentum, and energy conservation equations in the
macro-vapor and -liquid regions are solved numerically, with the surface tension modeled
by a continuum method introduced by Brackbill et al. (1992). The latter eliminates the
need for interface reconstruction, thereby simplifying its calculation and fully integrating
it into the momentum equation.
The vapor-liquid interface in the macro region is
captured by a PDE-based fast local level set method (Peng et al, 1999), which easily
handles the breaking and merging of interfaces. To overcome the disadvantage posed by
the non-conservation form of the level-set equation, an interface-preserving level-set
redistancing algorithm developed by Sussman and Fatemi (1999) is used to preserve the
mass in both the vapor and liquid phases.
5.2
Mathematical Formulation
The coordinate system and the physical domain for the computational model,
indicating the microlayer and macro region, similar to the scheme used by Son et al.
(1999), are shown in Fig. 5.1. The microlayer envelops the thin film that forms
underneath of the bubble, whereas the macro region consists of the bubble and the liquid
surrounding it. The flow is assumed to be axisymmetric and laminar, with all liquid
114
physical properties considered to be constant at 100°C (saturated atmospheric condition).
In the computational domain, as illustrated in Fig. 5.2, a MAC-staggered grid is used for
the discretization of the differential equations. The discrete velocity field uin, j ,
temperature field Ti ,nj , and the level set function φin, j are located at cell centers, but the
pressure pin+−1/1/2,2 j +1/ 2 is located at cell corners.
Microlayer
The mass, momentum, and energy conservation equations governing the
microlayer, based on the Lay and Dhir (1995) treatment, can be respectively stated as
∂δ
= vl − q / ρ l h fg
∂t
(5.1)
∂pl
∂ 2ul
= µl 2
∂x
∂y
(5.2)
q = kl (Tw − Tint ) / δ
(5.3)
vl = −
1 ∂ δ
xul dy
x ∂x ∫0
(5.4)
By employing a modified Clausis-Clayperon equation (Dasgupta et al. 1993), the
evaporative heat flux can be written as
q = hev [Tint − Tv + ( Pl − Pv )Tv / ρ l h fg ]
(5.5)
hev = 2( M / 2π RTv )0.5 ρ v h 2fg / Tv ; Tv = Tsat ( Pv )
(5.6)
Thus, noting that the pressure in the vapor and liquid phases are related as
pv = pl + σ ⋅ k − ρ l g (δ − δ e ) +
115
A
δ3
−
q2
ρ v h 2fg
(5.7)
Macro Region
Liquid
g
T=Tsat
Vapor
y
ϕ
x=X
Heater Surface
y
δ0
Microlayer
r=0
h/2
r=R
r
Fig. 5.1 Physical domain of a boiling bubble decomposed into
a macro region and a microlayer
116
Physical Boundary
Liquid
T,u,
T,u,
T,u,
φi-1,j+1
φ i,j+1
φi+1,j+1
Pi-3/2,j+1/2
Pi-1/2,j+1/2
Pi+1/2,j+1/2
T,u,
T,u,
T,u,
φi-1,j
φi,j
φi+1,j
Pi-3/2,j-1/2
Pi-1/2,j-1/2
Pi+1/2,j-1/2
T,u,
T,u,
T,u,
φi-1,j-1
φi,j-1
φi+1,j-1
Pi-3/2,j-3/2
Pi-1/2,j-3/2
Pi+1/2,j-3/2
Fig. 5.2 MAC-Staggered grid to show where the variables u, p, T,
and φ are located
117
Equations (5.1)-(5.7) can be combined to yield the following final governing equation for
the microlayer:
δ '''' = f (δ , δ ', δ '', δ ''')
(5.8a)
which is subject to the following boundary conductions:
r = 0 : δ = 1, δ ' = 0, δ ''' = 0
h
r = R :δ =
, δ '' = 0
2δ 0
(5.8b)
Where curvature of the interface k and apparent contact angle ϕ are defined as
k=
1 ∂
∂δ
∂δ
(x
/ 1 + ( )2 )
∂x
x ∂x ∂x
tan ϕ =
h
( X1 − X 0 )
2
(5.9)
(5.10)
Macro Region
The dimensionless conservation equations for momentum, energy, and mass in
the macro region are, respectively
∂u
∇p y GrL
+ ui∇u = −
+
+
θ
∂t
ρ Fr Re 2
1 1
1
+ ( ∇i(2 µD ) −
k ∇H )
We
ρ Re
(5.11)
∂θ
1
+ u ⋅ ∇θ =
∇ ⋅ ∇θ
Pe L
∂t
(5.12)
∇⋅u = −
=
.
1 ∂ρ
( + u ⋅ ∇ρ ) + V micro
ρ ∂t
m
ρ2
.
⋅ ∇ρ + V micro
118
(5.13a)
where
m = ρ ( ui − u) = k ∇T / h fg
i
V micro = ∫
X1
X0
kl (Tw − Ti )
rdr
ρ v h fgδ∆Vmicro
(5.13b)
Also, the level-set function to capture the evolving vapor-liquid interface is advanced as
φt + uint ⋅ ∇φ ( x (t ), t ) = 0
(5.14)
And φ is initialized to be the signed normal distance from the interface
if x ∈ the liquid
 > 0,

φ ( x, t ) =  = 0, if x ∈ Γ{x | φ ( x, t ) = 0}
 < 0,
if x ∈the vapor

(5.15)
Equations (5.11)-(5.15) are bounded by the following set of conditions:
y = 0:
u = v = 0, T = Tw ,φ = − x cos ϕ
∂v ∂T ∂φ
=
=
=0
∂x ∂x ∂x
∂u ∂v ∂φ
=
=
= 0, T = Tsat
∂y ∂y ∂y
r = 0, r = R : u =
y =Y :
(5.16)
The various dimensionless variables and groups in Eqs. (5.11)-(5.13) are defined as
L = σ /[ g ( ρl − ρ v )], U = gL , x = Lx ',
u = Uu ', t = ( L / U )t ', P = P ' ρ lU 2 ,
ρ = ρl ρ ', µ = µl µ ', k = kl k ',
θ = (T − Tsat ) /(Tw − Tsat )
U2
ρ l LU 2
LU
Fr =
, We =
, Re =
σ
νl
gL
UL
g β (Tw − Tsat ) L3
Pe L =
, GrL =
ν l2
a
119
(5.17a)
(5.17b)
5.3
The Numerical Method
The fourth-order differential equation governing the microlayer was integrated
using an adaptive, step-size control, fifth-order Runge-Kutta method. In order to obtain
the governing equation for pressure that achieves mass conservation, the accurate
projection method or fractional-step is used (Brown et al., 2001), which is an improved
fully second-order accurate projection algorithm over the method used by Bell and
Colella (1989). The diffusion terms are treated by a fully implicit scheme, and the
convection terms in level-set function are discretized by the third-order essentially nonoscillatory (ENO) schemes given by Fedkiw et al (2002). For other convection terms, a
predictor-corrector method (Puckett et al., 1997; Sussman et al., 1999) is adopted. The
time-stepping procedure is based on the third-order total time diminishing (TVD) RungeKutta method as
φ n +1 = φ n + ∆tL(φ n )
∆t
 L(φ n ) + L(φ n +1 ) 
4 
∆t
φ n +1 = φ n +  L(φ n ) + 4 L(φ n +1/ 2 ) + L(φ n +1 ) 
6
φ n +1/ 2 = φ n +
(5.18)
To prevent numerical instability arising from discontinuous physical properties, the
smooth Heaviside function in Eq. (5.11) is defined as
if φ < −ε
0

1  φ 1

H ε (φ ) =  1 + + sin (πφ / ε )  if φ ≤ ε

2  ε π
1
if φ > ε
(5.19)
where ε = α∆x , and which will not exhibit a jagged or sharp irregular shape when α > 1 .
120
The fluid density, viscosity, and thermal conductivity are defined in terms of H
ρ (φ ) = ρ g + ( ρ l − ρ g ) H (φ )
µ (φ ) = µ g + ( µl − µ g ) H (φ )
(5.20)
k (φ ) = kl H (φ )
The key to success of the level-set method is to maintain level set φ a distance
function; φ can become irregular after some period of time, which will cause steep
gradients in the distance function, and Eq. (5.14) can be reinitialized as follows:
d τ + S ( d 0 )( ∆d − 1) = 0
d ( x,0) = d 0 ( x ) = φ ( x, t )
(5.21)
Where
Sε (φ 0 ) =
φ0
φ 20 + ε 2
(5.22)
To overcome the disadvantage posed by the level-set equation not being in a conservation
form, the following constraint (Sussman et al., 1999) is used to improve the accuracy of
Eq. (5.14):
φn +1 = φn +1 + ∆tλi , j H ∆' x (φ0 ) ∇φ0
− ∫ Ωi , j H ∆' x (φ0 )
φn +1 − φ0
∆t
λi , j =
2
'
∫ Ωi , j  H ∆x (φ 0 ) ∇φ0
(5.23)
In implementing the level-set method, a PDE-based fast local level-set method
developed by Peng et al. (1999) is used to track the water-vapor interface, which reduces
the computational effort by one order of magnitude when compared to the method used
by Sussman et al. (1994). In order to compute the projection of the momentum equation,
the multigrid-preconditioned conjugate gradient method (MPCG) developed by Tatebe
(1993), which allows the computation of cases that can not be executed at the proper
121
density ratio (~1000:1) by using the standard Gauss-Seidel iterative method or multigrid
methods, was implemented. The preconditioner to the conjugate gradient is a multigrid
V-cycle, and symmetric multicolor Gauss-Seidel relaxation is used as the smoother at
each level of the V-cycle. Details of temporal and spatial discretization are given in
Appendix D.
5.4
Solution Validation
5.4.1
Efficacy of the MPCG method
In order to ascertain the efficacy of the MPCG method, the results for the Poisson
equation with jump diffusion coefficients for different shapes (T-shape and Arc) are
graphed in Figs. 5.3(b) and 5.4(b), and the physical problems for these results are
depicted in Figs. 5.3(a) and 5.4(a), respectively. This model can be expressed as follows:
−∇ ( k ∇u ) = f
Ω = (0,1) x (0,1)
(5.24)
The MPCG method has robust and efficient convergence properties compared to other
methods, and it was found to take much less time to converge than using the incomplete
Cholesky (IC) as a preconditioner. Moreover, the MPCG method can be efficiently
implemented in parallel computations.
122
1
y
k=1
0.5
k=1000
0
0
0.5
1
x
(a)
y
1
0.5
0
0
0.5
1
x
(b)
Fig. 5.3 (a) Poisson problem with jump diffusion coefficients (T-shape); and
(b) test computational results for u(x,y) for the MPCG method (f = 100)
123
1
y
k=1
0.5
k=1000
0
0
0.5
1
x
(a)
y
1
0.5
0
0
0.5
1
x
(b)
Fig. 5.4 (a) Poisson problem with jump diffusion coefficients (arc); and
(b) test computational results for u(x,y) for the MPCG method (f = 100)
124
5.4.2
Efficacy of the level set method
In the level-set method, an interface is represented as a zero level set of a
continuous function, designed to have a positive sign on one side of the curve and a
negative sign on the other site of the curve. In the present application, the level set φ(x)
is defined as a signed distance function φ (x) that yields the closest distance to the
interface as illustrated in Fig. 5.5. Here the level set is the distance to a circle centered at
the origin (0.5,0.5) with a positive sign outside of the circle (r = 0.15). The full level-set
function φ(x) is updated with the velocity field determined by the governing equations.
To demonstrate the applicability of the level-set method, the axisymmetric rise of
a sphere with a constant velocity v(x,y) = {0, 1} was simulated, and these results are
shown in Fig. 5.6 as a test problem. In this case, the interface is accurately tracked while
the mass is conserved with reinitialization, Eq. (5.21), and the constraint of Eq. (5.23) is
applied.
y
1
0.5
0
0
0.5
x
1
Fig. 5.5 Contours of level-set function φ(x) with the solid line circle representing for φ(x) = 0
125
y
1
0.5
0
0
0.5
x
1
Fig. 5.6 Rising bubble interfaces plotted at different times for U = 0, V = 1
using the mass-preserved level-set method
126
5.4.3
Verification of phase change modeling
The liquid-vapor phase change process in nucleate boiling has been employed by
Son et al. (1999) to accommodate the level-set function to capture the dynamics of a
single bubble growing on a horizontal heated surface. A simpler case of bubble growth
in an extensive uniformly superheated liquid, as shown in Fig. 5.7, is tested before
simulating of the bubble growth during nucleate boiling on a heated surface.
Tl
q
uint
Tsat
Tl > Tsat
Fig. 5.7 Schematic of a growing bubble in an extensive superheated liquid pool
There are two limiting cases of the bubble growth process that are generally
acknowledged in the literature (Carey, 1992):
(a) Inertial-controlled growth. Heat transfer to the interface is very fast and is not a
limiting factor to growth.
The growth rate is therefore governed by the
momentum interaction between the bubble and surrounding liquid.
These
conditions usually exist during the initial stages of bubble growth, just after the
embryonic bubble forms and begins to grow.
(b) Heat-transfer-controlled growth.
In this regime, growth is limited by the
relatively slower transport of heat to the interface. These conditions generally
127
correspond to the later life stages of bubble growth when it is larger and the liquid
superheat near the interface has been significantly depleted.
This study is only related to the heat-transfer-controlled bubble growth period.
The governing equations for the transport of heat in the liquid surrounding the bubble are
the following:
∂T
∂T  α l  ∂  2 ∂T 
+u
=   r

∂t
∂r  r 2  ∂r  ∂r 
u=
kl
dR  R 
 
dt  r 
(5.25)
2
(5.26)
∂T
dR
( R, t ) = ρv hlv
dt
∂r
(5.27)
The boundary and initial conditions on these conservation equations are
T ( r ,0 ) = T∞ ; T ( R, t ) = Tsat ( Pv ) ; T ( ∞, t ) = T∞
(5.28)
Plesset and Zwick (1954) have given an analytical solution as
R(t ) = 2CR α l t
(5.29)
where
CR =Ja 3/π , Ja=
C p ρ l (Ts − Tsat )
ρ v h fg
(5.30)
and which provides the reference for testing the computational modeling.
Two cases with different Jacob numbers were tested using the level-set method
with phase-change modeling. The results are presented in Fig. 5.8, and it is clearly seen
that the numerical simulations agree well the analytical solutions.
128
12
12
Numerical
11
Numerical
11
Analytical
9
9
8
8
7
7
y[mm]
10
y[mm]
10
6
5
6
5
4
4
3
3
2
2
1
0
1
Ja = 1.873e-3
0
1
2
3
4
Analytical
5
0
6
Ja = 3.745e-3
0
1
2
r[mm]
3
4
5
6
r[mm]
(a)
6
Numerical
Analytical
5
Ja
r [mm]
4
3
3
5e.74
3
=
-3
.873e
Ja = 1
2
1
0
0
10
t [ms]
20
30
(b)
Fig. 5.8 Bubble growth in an extensive superheated liquid pool: (a) bubble
interfaces plotted at different times; (b) bubble growth with time
129
5.5
Results and Discussion
5.5.1 Microlayer
Cooper and Lloyd (1969) were probably the first to present an analytical model
and experimental results for the formation of a microlayer, and treated it as a liquid
wedge underneath the growing bubble. More recently, Wayner and his co-workers
(Wayner, 1992; Dasgupta et al. 1993, 1994) have experimentally and theoretically
addressed the problem of the shape of the microlayer. Lay and Dhir (1995) subsequently
considered a more detailed theoretically analysis of the pressure distribution in the
microlayer to predict the shape of the microlayer and the heat flux.
The present
investigation is based on the Lay and Dhir analytical model; however, different boundary
conditions are applied to accommodate the heat transfer and growth of a nucleated
boiling bubble.
Figure 5.9 depicts the computed liquid-vapor interface shape and the microlayer
thickness, as well as the temperature distribution for a bubble of apparent contact angle of
30°. It can be seen that the slope of the interface in the microlayer decreases to zero
rapidly near the nonevaporating region (r = 0, δ = δ0). As the thickness of the liquid
microlayer increases, the interface temperature decreases from the wall temperature to the
liquid saturation temperature, which suggests that the more significant evaporative heat
transfer occurs in the microlayer region. Cooper and Lloyd (1969) have suggested that
bubble growth rates are of the same order as the evaporation rates from the microlayers.
In Lee and Nydahl (1989), it has been shown that microlayer evaporation accounts for
nearly 90 percent of the wall heat transfer during saturated boiling of water at 1 atm and
130
8.5 K wall superheat. Also, the total energy removal from the microlayer is coupled with
the heat and mass transport from the macro region.
5e-005
384
4e-005
380
Interface Temperature
378
3e-005
Microlayer Shape
2e-005
376
1e-005
374
372
Microlayer Thickness δ (m)
Interface Temperature (Κ)
382
0
1e-005
2e-005
3e-005
0e+000
r (m)
Fig. 5.9 Microlayer shape and vapor-liquid interface temperature
distribution for a nucleated bubble
5.5.2
Surface tension effect
The shape and dynamics of a bubble rising in water is governed essentially by the
following set of physical quantities: liquid/vapor viscosity, gravity, liquid/vapor density,
bubble mass, bubble initial diameter, surface tension, pressure, and velocity of the bubble
relative to the surrounding fluid. A bubble immersed in a fluid maintains its shape
mainly due to the surface tension forces that help the bubble surface adapt to variations in
131
the external stresses. Surface tension is associated to a surface energy that it tends to
minimize.
The variations in shape of an adiabatic spherical rising gas bubble in quiescent
water, as computed by the level-set method, are depicted in Fig. 5.10. The results are for
different surface tension values while keeping the remaining parameters constant, and
which are reflected in the changes of Weber number:
We =
ρU 2 L
σ
(5.31)
With time the spherical shape is seen to change to an oblate ellipsoid that has a flattening
or convex underside. The elliptical stretching increases with We, i.e., the aspect ratio
(minor axis vs. major axis) is smaller for larger Weber number (smaller surface tension).
However, the rising velocity is larger when compared to that for smaller Weber number,
and these computations are graphed in Fig. 5.11 for Re = 278.
The bubble terminal velocity is reached after a certain time, which is the result of
an equilibrium of the following forces: (1) The drag force, which represents the viscous
effects that tend to slow the bubble motion as penetrates the fluid; (2) pressure and
surface tension forces, where in general, the main component is the hydrostatic pressure
gradient under static conditions, but under dynamic conditions, the dynamic pressure
gradient balances out the surface tension force, and they are the primary determinants of
the changing bubble shape; (3) Gravity and buoyancy forces, where gravitational force is
usually negligible on a bubble, and the buoyancy force is generated due to density
variation - also referred to as the Archimedes force; (4) The virtual mass force; the fluid
in its immediate surrounding flows at the same speed as that of the bubble, and in fact,
the bubble is a gas-liquid coherent object. Therefore, bubbles demonstrate a much higher
132
8
8
7
8
7
7
t = 3.5
6
6
6
5
5
5
4
4
t=2
3
3
2
2
2
t=0
0
-2
-1
0
t=0
1
1
2
0
-2
(a) We=0.538
-1
t=2
4
t=2
3
1
t = 3.5
t = 3.5
t=0
1
0
1
0
-2
2
-1
(b) We=0.782
0
1
(c) We=1.03
Fig. 5.10 Temporal evolution of bubble shapes for various Weber numbers (Re = 278)
1.0
0.9
u [dimensionless]
0.8
0.7
0.6
0.5
0.4
Re=278, We=0.538
Re=278, We=0.782
Re=278, We=1.03
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
t [dimensionless]
Fig. 5.11 Change bubble rising velocity with time for various Weber numbers
133
2
inertia than that of their own mass, and the effect of this added mass is to dampen or
retard the bubble acceleration; (5) The lift force, by which when a bubble experiences a
vertical shear fluid flow, it develops a movement orthogonal to the main flow with the
bubble deformation, especially when a surfactant or other reagents are present; (6) The
basset history force, which is the resulting effect of the virtual added mass that must
adjust itself to Lagrangian acceleration and take into account the recent past of the bubble
evolution; (7) wall force that accounts for the wall effect; (8) Marangoni forces; with a
surfactant in solution, a surface concentration gradient develops at the bubble interface
that promotes diffusocapillary flows, and so also thermocapillary flows are generated due
to the temperature-gradient induced density variations.
Figure 5.12 shows the time evolution of bubble growth and departure in saturated
nucleate pool boiling of both pure water and aqueous solutions of different surface
tensions for a wall superheat of ∆T = 10 K and ϕ = 45°. Compared to the behavior in
pure water, the ebullience in solutions with reduced surface tension (σ = 47.0 mN/m for
SDS at C = 1000 wppm,) is seen to be characterized by a smaller departure diameter, and
higher bubble detachment and departure frequency. The reduced surface tension due to
the interfacial adsorption-desorption of the additive causes the bubble to become more
spherical with time, and this results in a small increase in speed of the detached bubble,
before steady state is reached. Focusing on the bubble detachment process in particular, a
neck is seen to form near the wall and thereafter the bubble breaks off, and an accelerated
necking, break-off, and subsequent nucleation is seen with reduced surface tension. The
small portion left after the bubble departure serves as a nucleus for the next bubble. The
134
nucleation of smaller-sized bubbles and their removal at high frequency essentially
promote enhanced heat transfer in aqueous surfactant solutions.
t=0.03s
t=0.03s
t=0.03s
t=0.04s
t=0.04s
t=0.04s
t=0.08s
t=0.08s
t=0.08s
(a)
(b)
(c)
Fig. 5.12 Bubble growth and its departure in nucleate pool boiling for ∆T = 10
K, ϕ = 45°. (a) σ = 58.86 mN/m (water), (b) σ = 47.0 mN/m (SDS, C = 1000
wppm), (c) σ = 37.5 mN/m (SDS, C = 2500 wppm)
135
The diameter to which a bubble grows before departing is dictated by the balance
of various forces acting on the bubble, namely, inertia of the liquid and vapor, liquid drag
on the bubble, buoyancy, and surface tension.
Fritz (1935) correlated the bubble
departure diameter by balancing buoyancy and surface tension as follows:
σ
Dd = 0.0208ϕ
g ( ρl − ρ g )
(5.32)
Subsequently, Cole and Rohsenow (1969) also proposed a correlation of bubble departure
diameter as
Dd = 1.5 × 10−4 ϕ
σ
g ( ρl − ρ g )
Ja *5 / 4
(5.33)
where
Ja* =
ρl c plTsat
ρ v hlg
(5.34)
While the wall superheat and apparent contact angle effects on the bubble departure
diameter were not considered in Eqs. (5.32) and (5.33), they do provide a reasonably
good correct length scale for the boiling process. The calculated departure diameters for
the fluids with different surface tension values but an apparent contact angle of 45° (~
representative of the nominal contact angle in pure water (Han and Griffith, 1965)) and a
wall superheat of 10°C are graphed in Fig. 5.13. Also included are the results from Eqs.
(5.32) and (5.33) for comparison, and a fair agreement is seen where the two results
enveloping the computational values.
136
101
D [mm]
Cole and Rohsenow's correlation (1969)
100
Fritz's correlation (1935)
Present simulation
10-1
100
101
102
σ [mN/m]
Fig. 5.13 Bubble departure diameter vs. surface tension (∆T = 10°C, ϕ = 45°)
137
5.5.3 Viscosity effect
The effect of viscosity (represented by Morton number) on the motion of the
bubble is illustrated in Fig. 5.14. The Morton number is defined as
Mo =
g µ 4 ( ρl − ρ v )
(5.35)
ρl2σ 3
As seen from the simulated shapes graphed in Fig. 5.14, the bubble is more spherical or
the aspect ratio (minor axis vs. major axis) is larger for a larger Morton number (higher
viscosity). Meanwhile, the rising velocity is smaller for a larger Morton number, as
shown in Fig. 5.15 due to the higher viscous force. That, also, in the case of water, the
bubble reaches a terminal velocity after a relatively longer time compared to that in
surfactant solutions. Furthermore, the predicated shape using the modified level-set
method in this study is in good agreement with the experimental results of Bhaga and
Weber (1981) for water (Mo =2.59e-11, We = 0.538) as seen in Fig. 5.16. All the cases
studied are summarized in Table 5.1.
Table 5.1 Combination of Non-Dimensional Parameters Studied (D = 2mm)
Re = 278 (water at 23°C)
Case No.
1
2
3
Case No.
1
2
3
σ
We
[mN/m]
5.38e-1
72.3
7.82e-1
48.9
1.03
37.5
We = 5.38e-1 (water at 23°C)
η x 103
Mo
[Pa.s]
2.59e-11
0.935
2.49e-10
1.77
9.47e-09
4.39
138
C (SDS)
[wppm]
0
1000
2500
C (Carbopol 934)
[wppm]
0
1000
3500
8
7
8
8
7
7
t = 3.5
t = 3.5
6
6
6
5
5
5
4
4
t=2
4
t=2
3
3
3
2
2
2
t=0
1
0
-2
-1
0
t=0
1
1
0
-2
2
-1
0
t=2
t=0
1
1
0
-2
2
(b) Mo=2.49e-10
(a) Mo=2.59e-11
t = 3.5
-1
0
1
2
(c) Mo=9.47e-09
Fig. 5.14 Temporal evolution of bubble shapes for various Morton numbers (We = 0.538)
1.0
0.9
u [dimensionless]
0.8
0.7
0.6
0.5
0.4
Mo=2.59e-11, We=0.538
Mo=2.49e-10, We=0.538
Mo=9.47e-09, We=0.538
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
t [dimensionless]
Fig. 5.15 Change in bubble rising velocity with time for various Morton numbers
139
(a)
(b)
Fig. 5.16 Comparison of predicted result with the experiment for water: (a) present
simulation; (b) experimental result of Bhaga and Weber (1981)
140
5.5.4
Temperature, velocity, and pressure fields
Figure 5.17 provides an indication of the changes in thermal field around the
bubble during growth.
Because the bubble represents an isothermal sink12 in a
nonuniform temperature field, the isotherms near the bubble do not uniformly surround
the bubble in a boundary-layer-type manner (Lee and Nydahl, 1989). The overall energy
for bubble growth comes from both the bubble cap and the microlayer, and the folding of
inflexions in the isotherms in a thin layer near the bubble seen in the computations are
somewhat difficult to resolve and identify experimentally.
The crowding of the
isotherms at the bubble base is reflective of the very high heat flux in that region.
Initially when the bubble is located inside the thermal boundary layer in Fig. 5.17(a), then
it growth is driven by the evaporation all around the vapor-liquid interface and the bubble
quickly grows out of the thermal boundary layer. During a major portion of this time
span, however, there is no wrapping of the bubble cap and the energy required for
evaporation is restricted to a small portion around the bubble base as shown in Fig.
5.17(b).
The flow field and pressure contours in and around a detached bubble in pure
water under conditions where σ = 58.86 mN/m and ∆T = 10 K are shown in Fig. 5.18.
During the early period of the upward motion of this bubble, the liquid around it is seen
to be pushed out. A circulatory flow pattern inside the bubble as well as in the liquid
outside is also clearly seen for the freely rising detached bubble. The vapor velocity
vectors in the bubble are reflective of its bulk movement in the upward direction and the
changes in the bubble shape as it rises in the pool. In the present illustrative calculations,
12
Uniform temperature and pressure are assumed for the vapor phase in the simulation. In reality, the
bubble may be non-isothermal (Beer, 1979), and further investigation is needed to establish this
condition.
141
the apparent contact angle is assumed to be constant (ϕ = 45°). The pressure distribution
around the bubble and the velocity field are coupled together, and the crowding of the
isotherms around the bubble as shown in Fig. 5.18(b) is reflective of the very high-
2.0
2.0
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
y/L
y/L
pressure gradients that are balanced out by the surface tension forces.
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0
x/L
x/L
(a) early growth
(b) later growth
Fig. 5.17 Temperature isotherms during different bubble growth stages
142
2
y/L
1.5
1
0.5
0
-1
-0.5
0
0.5
1
x/L
2
2
1.5
1.5
y/L
y/L
(a)
1
0.5
0
-1
1
0.5
-0.5
0
0.5
1
x/L
0
-1
-0.5
0
0.5
x/L
(b)
(c)
Figure 5.18 (a) Velocity field; (b) filled pressure contours; and (c) line pressure
contours, in and around a detached isolated bubble for ∆T = 10K, ϕ = 45° and
σ = 37.5 mN/m, t = 0.06s
143
1
5.5.5 Apparent contact angle and superheat effect
The dependence of bubble growth on apparent contact angle is depicted in Fig.
5.19. Three different contact angles (ϕ = 35°, 45°, and 55°) were selected for the
computations with a fixed wall superheat (∆T = 10°C) and surface tension (σ = 58.86
mN/m). It is seen from Fig. 5.19 that the bubble departure diameter becomes larger and
the vapor volume required for bubble departure increases as the contact angle increases.
This is probably caused by the surface tension forces at the three phase contact line. The
increase of bubble departure diameter with contact angle is found to be generally in
agreement with the Fritz correlation (1935) as shown in Fig. 5.19(b). In the present
calculations, all apparent contact angles are assumed to be constant. In reality, however,
the dynamic contact angles (advancing and receding contact angles) may differ from the
static contact angle. The consequent apparent contact angle hysteresis on a real surface
during nucleate boiling may have some effect on the shape of the liquid-vapor interface at
the heater surface during bubble departure.
Gorenflo et al. (1986) proposed the following correlation for bubble diameter at
departure for the heat flux controlled bubble growth, which includes the wall superheat
effect and is a function of the thermal diffusivity α and the Jacob number Ja:
1/ 3
 Ja 4 ⋅ α l2 
Dd = C1 ⋅ 

 g 
1/ 2
 
2π  
⋅ 1 +  1 +
 
  3 ⋅ Ja  
4/3
(5.36)
where
Ja =
ρl c pl ∆T
k
,αl = l
ρ v hlv
ρ l c pl
144
(5.37)
4
3
y[mm]
ϕ = 55
ϕ = 45
2
ϕ = 35
1
0
-2
-1
0
1
2
r[mm]
(a)
3.5
D [mm]
3.0
Present simulation
2.5
2.0
Fritz correlation (1935)
1.5
25
35
45
55
65
ϕ [deg]
(b)
Fig. 5.19 (a) Bubble shape at departure for different contact angles; and (b) bubble
departure diameter vs. apparent contact angle (∆T = 10°C, σ = 58.86 mN/m)
145
Different values of the proportionality constant were suggested for different boiling
liquids, and for water, C1 = 2.63. Figure 5.20 illustrates the effect of wall superheat on the
bubble growth, and its departure time, along with a comparison between simulation
results and predictions from the correlation proposed by Gorenflo et al. (1986) for a fixed
apparent contact angle (ϕ = 45°) and surface tension (σ = 58.86 mN/m). The bubble
departure diameter is seen to increase with the wall superheat. However, the bubble
departure time or growth period decreases when ∆T increases, and it indicates a higher
heat transfer rate and vapor production rate as well.
5.6
Significance and Limitations of Nucleate Boiling Simulations
The computational results presented in this chapter are insightful in making a
good assessment of the extent of the effects of surface tension, viscosity, wall superheat,
apparent contact angle, and microlayer when boiling in aqueous surfactant solutions.
Nevertheless, there are certain limitations of the present simulations, as a more
meaningful and complete model requires the inclusion of the adsorption-desorption
controlled interfacial surfactant transport process as illustrated in Fig. 5.21 into the
governing equations.
When a surfactant is present in an aqueous pool, its adsorption at the interface
results in an equilibrium surface concentration Γeq and surface tension σeq, whereby the
process solely results in a surface tension reduction. However, at the dynamic interface
of a nucleate boiling bubble, the surface concentration rarely remains at its equilibrium
value. Surface convection creates concentration gradients that alter the local surface
tension, thereby producing Marangoni stresses on the interface, with larger deformations
146
4
3.5
3.5
3.0
y [mm]
2.5
∆ T = 13K
2.5
∆ T = 10K
2.0
D [mm]
3
2
∆ T = 7K
1.5
1.5
1.0
1
0.5
0.5
-1
0
1
0.0
0.00
2
0.01
0.02
0.03
r [mm]
t [s]
(a)
(b)
0.04
101
9
8
7
6
Present simulation
5
4
D [mm]
0
-2
13K
10K
7K
3
2
Correlation by Gorenflo et al. (1986)
100
9
8
5
6
7
8
101
9
∆Τ [K]
(c)
Fig. 5.20 Effect of wall superheat ∆T on bubble growth (ϕ = 45°, σ = 58.86
mN/m): (a) bubble shape at departure; (b) bubble departure time; and (c)
bubble departure diameter vs. wall superheat
147
0.05
when compared to the surfactant-free case (Cuenot et al, 1997; Eggleton and Stebe, 1998;
Wasekar and Manglik, 2003). In fact, physico-chemical processes such as bulk mass
transfer, and additive adsorption/desorption have a strong influence on the flow around
the bubble, which in turn affects the surfactant transport process. A few recent studies
have attempted to model to investigate the complex interactions between surfactant
transport process and single bubble dynamics (Wang, 1999; Liao and McLaughlin, 2000;
Palaparthi, R., 2001) with encouraging results. However, all these studies have treated
the bubble to be adiabatic, and free-slip or non-slip boundary condition was applied at the
liquid-vapor interface; this, unavoidably, will alter the transport process at the interface.
Further in depth simulation modeling and computational methods are yet to be
developed.
U
Diffusion From
Bulk to Surface
Kinetic Adsorption
Surface Convection
to Trailing End
Kinetic Desorption
Diffusion From
Surface to Bulk
Fig. 5.21 Adsorption-desorption controlled surfactant interfacial transport process
148
CHAPETR 6
CONCLUSIONS AND RECOMMENDATIONS
6.1
Conclusions
With the addition of small amounts of surfactants or polymers, the saturated pool
boiling of water on a cylindrical heater is found to be altered significantly. In general,
besides the heat flux (or wall superheat) levels, the relative extent of change in boiling
performance is seen to be governed by the interfacial phenomena at the surface contact
(solid-liquid and vapor-liquid), which in turn are determined by several additive physicochemistry-based factors. The major accomplishments, findings, and salient features of
this study are summarized as follows:
1. An extensive literature review is presented that assess the available information
on pool boiling heat transfer of aqueous surfactant and polymer solutions,
interfacial phenomena and electrokinetic effects, especially the physisorption
process at the solid-liquid interface, and its effects on surface wettability and
active nucleation density.
2. Extended measurements and data for interfacial properties (dynamic and
equilibrium surface tensions, and wettability) are presented, in order to
characterize the adsorption behaviors of additives in their aqueous solutions at
both the solid-liquid and liquid-vapor interfaces. It was generally found that the
surfactant physisorption process tends to follow a specific adsorption isotherm,
which in turn is determined by the electrokinetics (zeta potential) at the solidliquid interface that lead to corresponding changes in the surface wetting
conditions.
The measured contact angle correlates well with this adsorption
149
isotherm and zeta potential variations with concentration. Also, additives have
surface-active properties that lower the surface tension of water considerably.
The surfactant adsorption-desorption process, however, is time-dependent and it
manifests in a dynamic surface tension behavior, which eventually reduces to an
equilibrium value after a long time span. For boiling applications with small
surface-age interface, it is the dynamic surface tension relaxation process rather
than the equilibrium or static surface tension at a fixed bubble frequency that is
perhaps the more critical determinant of the phase-change performance.
3. In the case of polymer additives, the measured shear-rate- and temperaturedependent viscosity for dilute surfactant solutions showed insignificant change
from that for water.
However, the aqueous polymer solutions become
significantly more viscous, and those with a much higher degree of
polymerization or molecular weight (Carbopol 934 vs. HEC QP-300) exhibit a
distinct non-Newtonian shear-thinning rheology.
4. The heat transfer in saturated nucleate boiling of aqueous cationic surfactant
solutions is found to be enhanced considerably. The performance is seen to
depend upon the dynamic σ reduction of the solution, micellar structure of the
surfactant, its degree of ethoxylation (which influences both surface tension and
wettability), and its molecular weight and ionic nature (which influence coverage,
inter-molecular repulsion or lack thereof at the vapor-liquid and solid-liquid
interfaces, and σ relaxation time). The heat transfer generally increases with qw′′
and additive concentration up to a C ≤ CMC. Depending on C and qw′′ , the heat
transfer coefficient is found to increase by as much as 63% over that for pure
150
water for DTAC (a low molecular weight cationic) solutions. With C > CMC, the
enhancement decreases and the heat transfer can even deteriorate below that for
water depending upon qw′′ and the surfactant chemistry.
High concentration
solutions of the ethoxylated cationic Ethoquad 18/25 (15 EO groups), for
example, show considerable heat transfer deterioration as well as incipience
thermal hysteresis that is typically found in highly wetting fluids. The boiling
process in non-ethoxylated surfactant solutions was observed to be characterized
by an early incipience of regularly shaped smaller-sized bubbles, with a reduced
tendency for coalescence and relatively higher bubble departure frequencies. The
presence of EO groups in the molecular chain of the surfactant, which changes the
surface wettability and alters the active nucleation site density and their
distribution, tends to promote the inception of smaller-diameter bubbles in premicellar concentrations and suppress the nucleation process in post-micellar
solutions.
The different boiling mechanisms between non-ethoxylated and
ethoxylated surfactants can essentially be related to the different physisorption of
surfactant molecules at the solid-water interface, which is characterized by the
surface wettability variations as a function of surfactant concentration.
5. Reflecting its surface-active nature and molecular adsorption at the vapor-liquid
interface, the polymeric HEC solutions show much greater relaxation of both the
dynamic and equilibrium surface tension in comparison with Carbopol. As a
consequence, nucleate boiling in polymeric HEC solutions (C < CPC or the
critical polymer concentration) is observed to be characterized by significantly
larger number of considerably smaller bubbles that have much higher departure
151
frequencies than that in pure water. The reduced surface tension, along with the
molecular adsorption on the heating surface (liquid-solid interface) perhaps also
contributes to the formation of new nuclei. The combined mechanisms result in
considerable enhancement, with up to 22.9% higher heat transfer coefficients,
relative to water, in the near CPC or overlap concentration solutions of HECQP300.
In post-CPC solutions, however, a decrease in the heat transfer
coefficient from the maximum values obtained at CPC is observed. This is
perhaps due to the retardation of vapor bubble growth and suppression of microconvection in the boundary layer because of the high viscosity of high
concentration solutions. Boiling in aqueous Carbopol 934 solutions, on the other
hand, shows a continuous deterioration in heat transfer, relative to that in pure
water, at all concentrations because of the viscous suppression of ebullience
activity. Some vapor explosions are also observed on the heater surface, akin to
the boiling behavior normally found in highly viscous liquids. These results also
suggest that the dynamic surface tension and apparent viscosity are the dominant
performance predictors, and should perhaps be accounted for in developing any
predictive model for nucleate boiling heat transfer in aqueous polymer solutions.
6. An in-depth systematic characterization of nucleate pool boiling in surfactant
solutions is delineated, which is based on the extensive deductive analysis boiling
heat transfer experiments, coupled with photographic visualization and interfacial
property and fluid rheology measurements.
The effects of dynamic surface
tension and surface wettability (represented by contact angle) on nucleate boiling
heat transfer are analyzed.
The roles of molecular weight, ionic nature,
152
ethoxylation, and zeta potential are discussed. Because of the highly dynamic
nature of nucleate boiling in surfactant solutions, the measure of dynamic surface
tension is seen to be an effective scaling property for the heat transfer data. The
faster diffusion of lower molecular weight surfactants tends to reduce the surface
tension faster in a short period of time, which is reflected in the better heat
transfer performance of their solutions. Besides the dynamic surface tension
relaxation, the additive physico-chemical properties, which alter the surface
wetting of aqueous solutions due to the interfacial physisorption of surfactant
molecules, are shown to be critical parameters in predicting their enhanced
nucleate pool boiling heat transfer performance.
7. Finally, in an effort to understand the ebullience behavior, the single-bubble
dynamics is computationally modeled. To this end, first, a general review of the
computational fluid dynamics methods with moving boundaries is presented.
Next, a computational model is developed for the complex single bubble
dynamics that addresses the effects of surface tension, viscosity, microlayer, wall
superheat, and apparent contact angle on the bubble dynamics. The liquid-vapor
interface was tracked by a modified fast local level-set method that
accommodates the liquid-vapor phase change process, and which also has mass
preservation property.
6.2
Recommendations for Future Research
Nucleate boiling in aqueous surfactant or polymer solutions is a complex
conjugate problem. This, along with the interfacial phenomena at both solid-liquid and
153
liquid-vapor interfaces, results in a rather complicated transport process. Even though an
elaborate and controlled investigation has been conducted experimentally and
computationally modeled in this dissertation, many aspects of their boiling heat transfer
quantification are yet to be resolved before a generalized design correlation can be
developed or the nucleate boiling process can be effectively controlled. Therefore, the
following areas are recommended for future research:
1. In-depth interfacial phenomena characterization - zeta potential, surface
wettability, and adsorption isotherm complement each other and allow a better
understanding of the eletrokinetic characteristics, surface wetting, and the
concomitant flow behavior. Extensive investigations are necessary for a better
understanding of the surfactant physisorption process and its electrokinetic effects
at the solid-liquid interface. Furthermore, different adsorption structures may
exist (Richard et al., 2003; Schulz et al., 2001; Manne and Gaub, 1995) besides
the well-known double-layer model, which is evident from the images in Fig. 6.1
taken by Schulz et. al. (2001) using Atomic Force Microscopy (AFM).
AFM,
which can detect and characterize surface aggregates only a few nanometers in
diameter, and Neutron Reflectometry (NR), which can be used to measure the
thickness and concentration profile of an adsorbed surfactant at an interface, are
two of the most important techniques to visualize and measure the adsorption
layer. Such measurements are critical for characterizing the surfactant adsorption
process at the solid-liquid interface, and the different possible adsorption layers
based on different micelle structures are illustrated in Fig. 6.2.
154
Fig. 6.1 Schematic representation and AFM detection images of adsorbed layer
structures consisting of (A) spherical micelles, (B) cylindrical micelles, and
(C) a bilayer (Schulz et al, 2001)
155
Surfactant molecule
Hydrophobic
tail
Hydrophilic
head
C < HMC
HMC < C < CMC
Hydrophilic
Hydrophobic
Hydrophobic
Bilayer
Monolayer
Spherical
Vesicle
Slightly
Hydrophobic
Trilayer
C > CMC
Reversed
Fig. 6.2 Conceptualization of possible surfactant adsorbate layers at the solid-liquid interface
156
2. The surfactant physisorption and its electrokinetics are not only basic to a
fundamental understanding of nucleate boiling control, but also can provide
insights into related natural phenomena in other applications. For example, using
the changed bubble dynamics to detect certain substances (surfactants, DNA, or
proteins) leads to quick, simple and inexpensive chemical or bio sensing, and
microfluidic devices (Thomas et al., 2003; Cornell et al, 1997; Huber et al., 2003).
Another interesting phenomenon is to make water flow uphill (Chaudhury and
Whitesides, 1992) on a surface with a wettability gradient produced by coating of
self-assembled monolayers (SAMS). All these topics are related to the interfacial
phenomena. The idea to alter the transport process from the interfacial contact to
improve the heat transfer performance drastically would lead to some interesting
research directions. For example, some nanofluids or nanoparticles (Wasan and
Nikolov, 2003; Chaudhury, 2003, Hentze et al., 2002; Das et al., 2000) with
special surface properties based on the different structures of surfactant or
micelles can be developed and the potential impact applies to any process that
involves phase change or surface contact. Extensions of the present study with
such consideration will be able to reach out to new frontiers of research.
3. Experimental investigation of bubble dynamics using controlled single bubble
experiments to visualize and quantify the altered transport process with presence
of additives. This will make a significant contribution to the fundamental
understanding of the altered bubble dynamics with surface-active and other
additives in aqueous solutions.
157
4. Computational modeling of a single boiling-bubble dynamics (inception and
nucleation→ growth → departure) to investigate the effect of surfactant additives
on the nucleation process, bubble growth, Marangoni convection, and surfactant
transport process during the bubble lifetime. Effective and innovative numerical
methodologies are probably needed to treat the microscale transport process,
surfactant adsorption/desorption, and the free interface properly.
5. The unique structure of a surfactant – a hydrophilic head with a hydrophobic tail –
provides an effective means to control the surface wettability, which is critical to
control of many applications, including nucleate boiling. Surfactant and polymer
additives in water can be used to shift the water-boiling curve to the left or right,
CHF and the boiling regimes will change accordingly. The ability to control
phase-change process is of great importance to thermal processing, which include
not only the traditional phase change devices that work in the nucleate boiling
regime, but also the transport processes under extreme high heat flux, for
instance, quenching. Additional boiling experiments are required to extend the
nucleate boiling curve for aqueous surfactant and polymer solutions on different
heaters (shape and size) up to CHF and film boiling regime.
6. Phase-change enhancement and control experiments under microgravity
conditions. The method shown in this study to alter the transport process at the
surface contact is determined by intermolecular electrokinetics, which could be
free of gravity effect, and it would provide an effective solution for cooling
devices used in the space program applications.
158
7. Correlation and qualification of the heat transfer performance of aqueous
surfactant solutions. Nucleate boiling of water with additives is a complex
problem as illustrated in Fig. 1.2. There are some encouraging attempts to
correlate this process from one or two aspects. However, due to the rather
intricate and interdependent nature of the problem, these models are either
oversimplified, or only applied to a small concentration range, or focused on a
specific surfactant. A thorough understanding of the associated phenomenon and
mechanisms is still required, and the development of generalized predictive
correlations is yet to be resolved. Using a systematic and controlled methodology
is critical to reach out to reliable and robust models when correlating the boiling
heat transfer of water with additives. As for boiling in aqueous solutions, the
most suitable and fundamental approach to this conjugate problem could extend
the work from the well-established basic mechanistic models for pure liquids. For
example, the well established Mikic and Rohsenow model (Mikic and Rohsenow,
1969; Judd and Hwang 1976) for nucleate boiling in water could form a good
starting reference:
i
K12
π (λρ c p )l f Dd2 N a ∆T
2
K2
π


+  1 − 1 N aπ Dd2  α nc ∆T + α ev ∆TN a Dd2
2
4


q=
(6.1)
By combining all the factors affected by surfactant additives as shown in Fig. 6.3,
the final correlation can then be established eventually. Figure 6.4 proposes the
possible future investigations based on the current recognition of this transport
process.
159
Basic Model (Pure Liquid)
(well-established)
Active
density
Bubble departure
diameter
Bubble departure
frequency
Marangoni
convection
Physical
properties
Microlayer
evaporation
…
Surfactant (Additives)
Fig. 6.3 Proposed approach to correlation of nucleate boiling in aqueous surfactant solutions
Second stage correlation
Physico-Chemical
Properties
•Ionic Nature
First stage correlation
?
Dynamic Surface
Tension
?
?
Zeta Potential/
Contact Angle
•Molecular structure
Bubble departure
diameter
Bubble release
frequency
dσ/dt or τD*
Single
bubble
experiment
•Ethoxylation
•Molecular Weight
•Micelle
•Packing
•Others
* Diffusion time
Active site density
Or correlation from literature (θ vs. Na)
Dynamic
(advancing)
Contact Angle
High speed
video (actual
heater)
Microlayer thickness
(flux removal rate)
Fig. 6.4 Proposed investigations to correlation of nucleate boiling in aqueous surfactant solutions
160
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APPENDIX A
SURFACE TENSION σ (mN/m) DATA
A.1 AQUEOUS SURFACTANT SOLUTIONS
AQUEOUS CTAB SOLUTIONS
T = 23°C
T = 80°C
σ (mN/m)
σ (mN/m)
(50ms)
(Equilibrium)
59.2
58.7
C (wppm)
σ (mN/m)
50
(50ms)
69.1
σ (mN/m)
(Equilibrium)
67.5
100
67.7
60.1
55.2
52.1
200
65.7
49.9
50.4
43.8
300
62.9
43.1
48.4
40.1
400
60.7
39.2
46.5
37.9
500
57.9
38.4
45.2
36.5
600
-
38.4
-
35.9
700
54.6
38.4
42.6
35.9
800
53.4
38.3
-
-
900
-
38.3
40.5
35.8
1100
52.6
38.2
39.1
35.7
1300
52.4
38.2
38.7
35.7
AQUEOUS DTAC SOLUTIONS
50
71.9
71.9
-
-
100
-
70.9
-
58.4
200
71.4
69.9
-
-
300
71.3
68.9
-
-
177
400
-
68.2
-
-
500
-
67.5
59.6
53.4
600
-
66.4
-
-
700
-
65.6
-
-
900
-
63.1
-
-
1000
66.5
-
57.8
-
1100
-
60.5
-
-
1300
-
59.2
-
-
1500
-
56.8
55.6
47.2
2000
59.9
54.5
-
-
2500
-
52.5
-
42.2
3000
55.1
50.2
50.4
40.2
3500
-
48.1
-
-
4000
50.1
-
-
38.2
4500
-
44.7
-
-
5000
-
-
44.7
37.1
5500
-
42.5
-
-
6000
-
-
-
37.0
6500
43.2
41.7
41.9
-
7000
-
-
-
37.0
7500
-
41.6
-
-
8000
42.4
-
-
-
8500
-
41.6
39.6
-
178
AQUEOUS Ethoquad O/12 PG SOLUTIONS
T = 23°C
σ
T = 80°C
C
C
σ
(wppm) (Equilibrium) (wppm)
70
32.87
31.41
(50ms)
58.48
67.6
98.6
52.5
65.7
53.8
69.2
96.87
62
164.3
51.3
98.6
51.8
262.9
68.9
131.58
59
262.9
49.4
197.2
50.3
460.13
68.4
219.3
52.3
460.1
48.4
295.8
48.2
657.33
66.1
282.96
49.9
558.7
47.5
394.4
45
953.13
64
379.59
46.3
657.3
47
424.4
43.3
1248.9
62.4
471.6
44.7
953.1
44.3
591.6
40.8
1544.7
61.1
526.31
43.3
1248.9
41.7
788.8
38.6
1840.5
59.9
660.24
42.8
1544.7
39.6
1320.49
37.8
2136.3
58.7
848.88
42.3
1906.27
39.5
1037.53
41.8
1162.5
41.6
1320.49
41.6
1603.45
41.6
C
(wppm)
32.87
(50ms)
98.6
69.5
164.3
71.1
σ
55.9
C
σ
(wppm) (Equilibrium)
32.87
56.1
AQUEOUS Ethoquad 18/25 PG SOLUTIONS
35.27
71.3
35.27
63.2
35.27
54.1
35.27
47.9
105.8
70
70.53
60
105.8
52.1
70.53
45.8
179
35.27
71.3
35.27
63.2
35.27
54.1
35.27
47.9
105.8
70
70.53
60
105.8
52.1
70.53
45.8
317.4
66.9
105.8
58.6
211.6
50.6
105.8
44.6
634.81
62.1
211.6
56
317.4
49
211.6
43.3
952.21
60.1
317.4
55
634.8
47.1
423.21
42.6
1058.01
59.6
423.2
54.2
952.2
46
740.61
42.3
1163.82
59.2
634.8
53
1375.4
43.9
1058.01
42.3
1375.42
58.3
846.4
52.5
2080.76
42.7
1728.09
57.4
1058.0
51.9
3138.78
42.6
2080.76
56.8
1269.62
51.5
6665.49
42.6
2186.56
56.5
1622.29
51.1
2503.97
56
2327.63
50.3
3561.98
55
2458.8
50.2
5325.34
53.8
3032.97
50.2
8852.05
52.4
4090.99
50.2
180
A.2 AQUEOUS POLYMER SOLUTIONS
AQUEOUS HEC QP-300 SOLUTIONS
T = 23°C
σ (mN/m)
T = 80°C
σ (mN/m)
σ (mN/m)
(50ms)
(Equilibrium)
62.5
61.5
C (wppm)
σ (mN/m)
50
(50ms)
72.4
(Equilibrium)
72.3
100
72.3
72
62.5
61.3
200
71.8
69.8
62.1
60.4
300
71.2
68.6
-
59.9
400
70.5
68
60.7
59.1
500
70.1
67.5
60.3
58.7
600
-
67.3
-
-
800
69.1
67.2
59.3
57.5
1000
69
67.1
58.7
57.3
1200
68.9
67
58.5
57.2
1600
68.8
66.9
58.3
57.2
2000
68.8
66.8
58.4
57.1
AQUEOUS Carbopol 934 SOLUTIONS
100
72.4
71.7
62.6
62.3
200
-
-
-
61.8
250
72.5
71.1
62.5
-
181
300
-
-
-
61.1
400
-
70.7
-
61
500
72.5
70.5
62.5
60.8
700
-
70.2
-
60.5
800
-
70.1
-
-
900
-
70
-
60.1
1000
72.5
69.9
62.4
-
1200
-
69.7
-
-
1300
-
69.5
-
59.9
1500
72.7
69.4
62.5
-
1800
-
69.3
-
-
2000
72.9
69.2
62.4
59.9
A.3 DYNAMIC SURFACE TENSION WITH TIME
Aqueous SDS Solutions
C = 1250wppm
Surface Age σ (mN/m)
(s)
0.026
69.3
C = 2500wppm
Surface Age
σ (mN/m)
(s)
0.028
50.7
C = 5000wppm
Surface Age
σ (mN/m)
(s)
1.2
38
0.031
62.6
0.037
45.2
0.32
38.3
0.042
58.3
0.041
43.9
0.183
38.6
182
0.026
69.3
0.028
50.7
1.2
38
0.031
62.6
0.037
45.2
0.32
38.3
0.042
58.3
0.041
43.9
0.183
38.6
0.057
55.6
0.058
42.1
0.09
39.3
0.092
54
0.088
41.1
0.047
40.5
0.166
52.6
0.330
40.0
0.037
41.9
0.214
52.3
1.350
39.8
0.027
45
0.354
51.9
1.653
51.7
Aqueous CTAB Solutions
C = 200wppm
1.3
48.4
C = 400wppm
1.29
39
C = 800wppm
1.10
38.3
0.324
48.6
0.339
39.6
0.356
38.3
0.25
49.1
0.266
40
0.240
38.7
0.186
50.4
0.200
40.9
0.181
39.4
0.108
53.3
0.135
43.5
0.147
40
0.056
57.7
0.097
45.7
0.104
41.4
0.040
61.4
0.065
48.9
0.070
43.5
0.040
53.6
0.040
48.0
Aqueous TRITON X-305 Solutions
C = 500wppm
Surface Age σ (mN/m)
(s)
C = 1000wppm
Surface Age
σ (mN/m)
(s)
183
C = 2000wppm
Surface Age
σ (mN/m)
(s)
1.5
53
1.5
49.4
1.5
47.9
0.9
53
0.863
49.4
0.821
48
0.5
53.2
0.5
49.5
0.5
48.2
0.348
53.4
0.357
49.8
0.33
48.5
0.277
54.1
0.27
50.5
0.26
48.9
0.2
55.9
0.21
51.5
0.21
49.5
0.153
57.5
0.15
53.2
0.151
50.7
0.117
59.2
0.113
54.8
0.114
52.4
0.07
62
0.07
57.9
0.07
54.7
0.04
64.8
0.04
61.1
0.04
58.1
Aqueous TRITON X-100 Solutions
C = 100wppm
1.4
35.7
C = 200wppm
1.4
34.1
C = 400wppm
1.4
33.1
0.6
35.9
0.6
34.3
0.6
33.2
0.45
36.1
0.45
34.4
0.45
33.4
0.35
36.8
0.335
34.9
0.33
33.9
0.27
37.6
0.25
35.4
0.25
34.5
0.18
39.2
0.18
36.6
0.18
35.3
0.12
41.4
0.12
38.3
0.13
36.4
0.097
42.7
0.097
39.2
0.09
37.9
0.07
44.6
0.07
41.5
0.06
40.3
0.04
48.3
0.04
45.5
0.04
42.6
184
Aqueous Ethoquad 18/25 Solutions
C = 500wppm
σ
C = 1000wppm
(mN/m)
52.5
Surface
Age (s)
3.2
1.60
52.5
0.80
C = 2000wppm
σ
(mN/m)
51.1
Surface
Age (s)
3.0
1.6
51.1
52.8
0.8
0.585
53
0.392
σ
C = 4000wppm
σ
(mN/m)
49.3
Surface
Age (s)
3.1
(mN/m)
48.8
1.5
49.5
1.53
48.8
51.3
0.80
49.6
0.8
48.9
0.558
51.6
0.562
49.9
0.55
49.2
54
0.375
52.3
0.368
50.5
0.369
49.8
0.25
55.5
0.25
53.7
0.25
51.9
0.270
50.6
0.19
56.9
0.19
55.2
0.192
52.9
0.200
51.5
0.128
59.2
0.14
56.6
0.13
54.6
0.12
53.8
0.099
60.8
0.094
59.0
0.090
56.8
0.09
55.2
0.07
63.3
0.070
61.1
0.060
59.2
0.06
57.4
0.040
67.6
0.036
65.9
0.040
62.1
0.04
59.5
Surface
Age (s)
3
185
A.4 SURFACE TENSION WITH TEMPERATURE
Aqueous CTAB Solutions
T (°C)
C = 100wppm
C = 300wppm
C = 1000wppm
22.7
60.1
43.1
38.3
40
57.4
42.1
37.7
60
55.1
40.8
36.5
70
53.7
-
-
80
52.1
40.1
35.8
Aqueous Ethoquad 18/25 Solutions
T (°C)
C = 200wppm
C = 400wppm
C = 1000wppm
23
58.6
54.1
51.9
40
55.0
50.1
49.3
60
49.1
47
45.2
80
44.6
42.9
42.3
186
APPENDIX B
CONTACT ANGLE DATA
SDS
CTAB
Ethoquad 18/25
TRITON
X-100
C
θ
TRITON
X-305
C
θ
C
θ
C
θ
C
θ
(wppm)
deg)
(wppm)
(deg)
(wppm)
(deg)
(wppm)
(deg)
(wppm)
(deg)
10
77
10
77
10
77
10
70
10
70
30
76
20
77
20
77
20
66
20
66
50
75
30
77
30
77
30
64
30
63
100
75
50
77
50
75
50
61
50
60
200
72
70
75
100
70
70
59
100
56
500
66
100
73
200
66
90
57
200
55
700
60
150
69
500
61
100
56
500
55
1000
55
200
65
700
57
130
48
700
55
1300
53
290
63
1000
55
140
45
1000
55
1500
50.5
380
59.8
2000
52
150
41
2000
55
1700
45
500
60
2500
51
180
41
3000
55
2000
44
700
60
4000
51
300
41
2300
37.8
5000
51
500
41
2400
35.3
3300
34.5
3700
35
4500
34.4
4700
34.8
5000
35.2
187
APPENDIX C
POOL BOILING DATA
C.1 AQUEOUS SURFACTANT SOLUTIONS
PURE WATER
Increasing qw" ⇑
Decreasing qw" ⇓
∆Tw
qw"
∆Tw
qw"
3.289
2.473
3.289
2.513
5.443
9.562
5.443
9.2
6.795
21.819
6.795
21.766
7.786
38.212
7.786
38.317
8.539
59.634
8.539
59.630
9.283
85.685
9.283
84.850
10.024
116.122
10.024
117.816
10.853
154.279
10.853
154.992
11.467
193.169
11.467
197.569
12.3677
238.305
12.368
244.298
AQUEOUS DTAC SOLUTIONS
C = 100 wppm
C = 500 wppm
C = 500 wppm
C = 1000 wppm
∆Tw
qw" ⇓
∆Tw
qw" ⇑
∆Tw
qw" ⇓
∆Tw
qw" ⇓
3.101
2.318
3.077
2.453
2.977
2.453
3.996
2.503
5.065
9.718
5.000
8.767
4.990
9.467
4.650
7.583
188
6.310
21.458
6.1086
20.470
6.213
22.470
5.744
17.964
7.292
37.944
7.005
37.817
6.805
39.817
6.676
35.357
7.957
60.213
7.660
59.090
8.060
69.490
7.487
58.765
8.854
85.576
8.295
83.057
8.595
89.957
7.920
85.133
9.341
117.326
8.795
117.854
9.095
127.854
8.502
116.804
9.960
154.612
9.424
144.939
9.724
168.939
9.060
153.847
10.792
194.836
10.121
187.562
10.391
197.872
9.515
198.140
11.312
240.125
10.497
231.178
10.392
241.178
10.181
243.079
C = 2000 wppm
C = 3000 wppm
C = 4000 wppm
C = 5000 wppm
∆Tw
qw" ⇓
∆Tw
qw" ⇓
∆Tw
qw" ⇓
∆Tw
qw" ⇓
2.766
2.333
2.730
2.393
2.541
2.326
2.172
2.740
4.191
7.692
4.009
8.740
4.138
9.709
4.393
9.661
5.744
21.417
4.9502
21.051
4.806
22.009
5.103
21.394
6.194
39.370
5.515
38.682
5.297
38.645
5.637
39.078
7.0886
60.272
6.100
60.255
5.793
59.174
6.289
61.777
7.627
86.996
6.576
87.707
6.285
85.463
6.689
86.241
8.101
116.31
7.293
112.192
6.586
117.385
7.117
118.369
8.609
156.323
7.685
156.47
6.911
156.206
7.392
159.661
9.107
197.173
7.862
196.619
7.497
196.757
7.988
197.281
9.286
237.989
8.0284
239.852
7.696
238.016
8.442
243.322
189
C = 7000 wppm
C = 10000 wppm
C = 10000 wppm
∆Tw
qw" ⇓
∆Tw
qw" ⇑
∆Tw
qw" ⇓
3.391
2.342
3.772
2.525
3.701
2.393
5.213
8.790
6.127
9.615
5.1556
7.739
5.762
18.640
7.115
21.790
6.507
18.051
6.938
38.991
8.072
39.114
7.838
41.682
7.554
61.032
8.177
59.533
8.324
63.255
7.946
86.419
8.800
85.750
9.334
93.707
8.573
118.355
9.348
117.505
9.647
126.192
9.406
157.038
10.366
152.850
10.302
166.470
9.939
197.738
11.135
195.711
11.029
196.619
10.673
240.382
11.2521
220.582
11.410
244.852
AQUEOUS CTAB SOLUTIONS
C = 50 wppm
C = 100 wppm
C = 100 wppm
C = 200 wppm
∆Tw
qw" ⇓
∆Tw
qw" ⇑
∆Tw
qw" ⇓
∆Tw
qw" ⇓
2.990
2.307
2.995
2.514
2.965
2.414
3.0379
2.788
5.0743
9.987
5.0222
9.256
5.018
8.958
4.976
9.938
6.167
23.953
6.162
21.947
5.957
20.134
6.141
23.938
7.113
39.822
7.010
38.538
7.020
36.135
6.221
37.413
7.785
61.094
7.592
60.159
7.577
58.199
6.897
60.109
190
7.999
87.449
7.692
85.436
7.667
80.567
7.165
85.654
8.115
119.338
7.962
118.739
7.962
108.345
7.551
117.437
8.397
157.693
8.196
154.145
8.190
143.321
7.708
154.735
8.768
199.952
8.580
196.676
8.611
189.458
7.983
199.052
9.011
246.856
8.808
240.408
8.873
233.804
8.395
241.134
C = 300 wppm
C = 400 wppm
C = 500 wppm
C = 600 wppm
∆Tw
qw" ⇓
∆Tw
qw" ⇓
∆Tw
qw" ⇓
∆Tw
qw" ⇓
2.988
2.491
2.413
2.191
2.514
2.473
3.286
2.130
3.966
7.259
4.216
8.991
4.422
9.994
5.0188
9.102
5.455
21.290
5.191
21.749
5.261
23.953
6.318
22.025
6.536
41.327
5.778
37.644
5.872
41.427
6.995
38.642
6.477
58.738
6.186
58.894
6.282
63.647
7.159
56.714
6.793
86.172
6.381
85.260
6.533
89.166
7.853
85.625
7.043
118.126
6.7845
117.002
6.879
123.991
7.9642
111.376
7.205
155.390
6.979
152.259
6.979
143.781
8.275
141.557
7.597
194.557
7.183
196.050
7.228
188.188
8.613
184.609
8.092
235.465
7.602
238.469
7.722
247.439
9.131
233.527
C = 800 wppm
C = 1000 wppm
C = 1000 wppm
∆Tw
qw" ⇓
∆Tw
qw" ⇑
∆Tw
qw" ⇓
4.170
2.116
4.318
2.820
4.112
2.457
191
6.102
9.592
6.253
9.733
5.587
8.712
6.770
20.926
7.213
22.031
6.810
17.899
7.694
38.325
7.721
38.665
7.422
34.981
7.849
56.897
8.109
59.116
8.179
52.567
7.991
81.339
8.834
85.939
8.679
83.107
8.405
111.005
9.314
117.629
9.325
111.547
8.980
154.055
10.270
154.701
10.199
150.108
9.404
194.616
10.954
198.896
10.921
187.236
10.102
239.443
11.833
238.603
11.577
228.615
AQUEOUS Ethoquad O/12 PG SOLUTIONS
C = 200 wppm
C = 200 wppm
C = 400 wppm
C = 400 wppm
∆Tw
qw" ⇑
∆Tw
qw" ⇓
∆Tw
qw" ⇑
∆Tw
qw" ⇓
2.317
2.333
2.464
2.513
3.397
2.795
2.403
2.522
5.189
9.142
5.363
9.582
5.141
9.837
4.441
9.911
6.538
21.397
6.441
21.766
6.322
21.844
6.2361
20.725
7.235
39.556
7.193
38.317
6.914
33.522
6.9191
36.498
7.640
58.441
7.453
59.630
7.292
51.173
7.663
59.748
8.099
86.587
7.982
84.850
7.916
79.479
7.981
85.631
8.389
111.064
8.397
117.816
8.034
110.983
8.295
120.128
8.573
153.230
8.568
154.992
8.227
147.686
8.195
157.901
8.873
204.064
8.947
197.569
8.576
189.749
8.655
201.673
192
9.153
238.262
C = 600 wppm
9.378
243.298
C = 600 wppm
8.702
237.45
C = 800 wppm
8.793
241.518
C = 800 wppm
∆Tw
qw" ⇑
∆Tw
qw" ⇓
∆Tw
qw" ⇑
∆Tw
qw" ⇓
2.633
2.620
3.098
2.556
3.189
3.952
3.244
2.782
5.028
10.293
5.042
9.525
5.843
11.144
5.109
9.933
6.000
21.323
6.410
22.079
6.323
17.796
6.277
21.535
6.7302
37.674
6.971
38.113
7.162
36.410
7.008
38.304
7.1664
57.971
7.347
58.865
7.805
59.338
7.515
59.135
7.620
85.419
7.605
84.729
8.132
91.016
7.778
84.997
7.844
117.48
8.027
117.759
8.542
119.323
8.077
115.441
8.051
157.106
8.213
151.856
8.506
163.774
8.2
150.72
8.351
203.378
8.437
195.853
8.813
212.883
8.532
190.697
8.341
246.079
8.905
241.583
8.923
249.528
9.184
246.66
C = 1000 wppm
C = 1000 wppm
C = 1200 wppm
C = 1200 wppm
∆Tw
qw" ⇑
∆Tw
qw" ⇓
∆Tw
qw" ⇑
∆Tw
qw" ⇓
2.775
3.041
4.228
2.733
3.407
2.616
3.420
2.521
4.990
10.340
6.818
9.626
7.115
11.142
6.610
9.996
6.281
21.456
7.522
22.132
7.212
24.973
7.0953
22.328
8.481
37.998
8.230
36.098
7.856
41.1607
7.954
36.939
9.597
59.483
8.578
59.373
9.055
61.576
8.756
59.407
9.869
86.114
9.086
85.238
9.315
86.526
9.114
87.746
8.961
121.256
9.432
118.942
9.379
119.954
9.366
116.024
193
8.834
161.144
9.559
152.078
9.505
166.254
9.542
155.099
8.349
206.648
9.960
195.831
9.926
203.959
10.092
196.28
8.740
251.480
10.674
230.956
10.690
239.923
11.038
250.617
C = 1500 wppm
C = 1500 wppm
C = 3000 wppm
C = 3000 wppm
∆Tw
qw" ⇑
∆Tw
qw" ⇓
∆Tw
qw" ⇑
∆Tw
qw" ⇓
4.295
2.865
5.086
2.394
6.389
2.735
8.389
2.733
7.862
10.357
7.662
9.996
8.197
10.457
9.775
10.857
8.050
21.479
8.517
21.056
9.782
22.118
10.97232
21.118
9.060
41.765
9.499
35.453
11.402
37.697
12.407
35.697
10.828
61.543
10.785
59.596
12.869
59.429
12.999
56.429
11.574
87.706
11.822
86.492
14.318
87.491
14.213
81.491
12.471
118.505
12.203
113.264
14.954
115.720
14.916
110.206
12.479
159.852
12.248
145.424
15.007
161.479
15.019
152.479
12.059
200.774
11.689
192.078
15.267
201.170
15.271
211.1699
12.575
239.036
12.037
236.478
15.118
237.866
15.213
246.866
AQUEOUS Ethoquad 18/25 SOLUTIONS
C = 200 wppm
C = 200 wppm
C = 500 wppm
C = 500 wppm
∆Tw
qw" ⇑
∆Tw
qw" ⇓
∆Tw
qw" ⇑
∆Tw
qw" ⇓
2.478
2.604
3.0945
2.356
3.488
2.837
3.303
2.551
4.779
10.261
5.266
9.113
5.841
10.299
5.308
9.811
7.317
22.279
6.630
20.902
6.513
21.809
6.598
22.541
194
8.155
38.112
7.729
38.710
7.799
38.799
7.381
37.036
8.945
59.542
8.477
63.431
8.649
58.806
8.0231
59.565
9.222
86.186
9.114
87.561
8.709
89.923
8.497
82.911
9.790
121.214
9.439
121.086
8.931
121.653
8.922
115.732
9.832
158.663
9.682
155.828
9.191
160.872
9.197
158.256
9.774
201.926
9.968
206.479
9.444
205.921
9.488
199.771
10.380
243.156
10.485
245.777
10.271
245.068
10.037
242.113
C = 700 wppm
C = 1000 wppm
C = 1000 wppm
C = 1500 wppm
∆Tw
qw" ⇓
∆Tw
qw" ⇑
∆Tw
qw" ⇓
∆Tw
qw" ⇑
3.120
2.351
2.185
2.703
3.228
2.367
2.575
2.776
5.665
8.755
4.515
9.939
6.317
9.880
6.143
11.134
7.198
20.856
7.648
22.8249
7.195
18.854
7.403
22.426
7.906
37.460
8.349
37.611
8.157
32.899
8.817
39.289
8.359
58.837
9.0963
62.051
8.599
55.085
9.652
59.899
8.559
78.847
9.419
84.564
9.058
81.783
9.969
89.547
9.126
105.989
9.711
120.431
9.654
117.421
10.553
116.809
9.816
145.469
9.900
160.360
10.169
160.379
10.759
160.360
10.142
183.634
10.579
207.225
10.548
196.738
11.204
204.136
10.607
231.967
10.999
242.619
11.106
237.097
11.021
237.854
195
C = 2000 wppm
C = 2000 wppm
C = 3000 wppm
C = 3000 wppm
∆Tw
qw" ⇑
∆Tw
qw" ⇓
∆Tw
qw" ⇑
∆Tw
qw" ⇓
3.363
2.605
3.446
2.196
4.155
2.621
4.5022
3.008
6.796
10.697
5.422
8.847
6.107
9.933
6.946
8.459
8.921
22.954
7.108
21.4739
7.515
22.589
8.6759
21.065
11.101
38.382
8.876
37.424
8.997
37.971
10.575
37.701
11.653
60.021
9.616
59.162
9.543
59.703
12.412
57.384
12.287
86.718
10.300
84.915
10.755
91.270
12.804
80.366
12.543
123.132
10.830
115.305
11.84
124.507
13.536
110.575
12.520
163.174
11.251
149.605
12.464
165.501
13.734
146.783
12.429
203.866
11.871
187.255
12.751
208.22
14.120
187.567
12.884
244.473
12.469
235.527
13.103
251.025
14.640
233.913
C = 5000 wppm
C = 5000 wppm
C = 5000 wppm
C = 5000 wppm
∆Tw
qw" ⇑
∆Tw
qw" ⇓
∆Tw
qw" ⇑
∆Tw
qw" ⇓
8.218
16.727
7.259
2.624
16.727
80.782
17.375
84.608
11.959
18.185
9.276
9.676
18.184
111.393
18.568
115.599
13.1098
18.993
12.001
23.544
18.994
152.120
19.439
159.109
14.780
20.239
14.578
38.750
20.239
191.914
20.536
199.4
16.231
21.344
16.131
62.849
21.344
247.135
21.439
237.806
196
C.2 AQUEOUS POLYMER SOLUTIONS
AQUEOUS HEC-QP300 SOLUTIONS
C = 100 wppm
C = 100 wppm
C = 300 wppm
C = 500 wppm
∆Tw
qw" ⇑
∆Tw
qw" ⇓
∆Tw
qw" ⇑
∆Tw
qw" ⇑
3.527
3.2423
3.1734
2.561
3.132
2.749
2.6576
2.776
5.370
9.742
5.483
9.502
5.272
10.147
4.903
8.732
7.466
21.609
7.1880
20.870
7.195
21.903
6.638
20.105
8.822
36.450
8.864
39.068
8.393
37.026
7.610
33.416
10.005
58.689
9.874
55.465
9.443
59.442
8.849
51.833
10.819
83.373
11.083
88.182
10.260
89.928
9.707
79.335
11.621
118.335
12.012
126.059
11.079
119.690
10.517
109.844
12.405
159.421
12.838
169.028
11.850
169.207
11.110
147.576
13.340
202.736
13.684
211.758
12.671
214.09
12.075
188.616
14.166
247.398
14.453
254.611
13.278
259.741
12.442
232.257
C = 600 wppm
C = 700 wppm
C = 1000 wppm
C = 3000 wppm
∆Tw
qw" ⇑
∆Tw
qw" ⇑
∆Tw
qw" ⇑
∆Tw
qw" ⇑
2.547
2.723
3.105
2.818
3.463
2.794
3.480
2.603
4.691
9.099
5.518
10.714
5.955
9.794
6.253
9.290
6.358
21.335
7.188
22.764
7.681
22.992
8.199
21.146
7.497
37.673
8.1994
34.724
8.758
41.730
9.595
37.479
8.523
57.724
9.159
58.268
9.893
63.717
10.479
59.655
9.378
86.562
10.071
89.823
10.312
85.098
10.895
80.556
197
10.264
121.388
10.456
123.075
10.817
118.782
11.203
127.479
10.836
159.376
11.096
165.723
11.1617
159.495
11.758
156.263
11.736
203.888
12.196
215.480
12.1467
202.343
12.610
201.237
11.984
244.399
12.388
251.148
12.626
246.124
13.404
248.535
AQUEOUS Carbopol 934 SOLUTIONS
C = 100 wppm
C = 300 wppm
C = 500 wppm
C = 1000 wppm
∆Tw
qw" ⇑
∆Tw
qw" ⇑
∆Tw
qw" ⇑
∆Tw
qw" ⇑
2.810
2.687
2.718
2.889
3.265
2.949
2.493
2.1859
5.018
9.711
5.095
8.975
5.764
10.201
5.935
9.113
7.035
20.724
7.157
20.276
7.461
20.740
8.009
20.8
8.207
34.310
8.608
35.904
8.798
35.711
9.588
37.228
9.647
54.819
9.944
55.627
10.634
56.688
10.902
56.227
10.895
79.457
11.345
80.435
11.923
81.568
12.661
83.449
12.067
110.303
12.453
107.476
13.382
110.683
14.214
109.745
13.065
144.867
13.972
143.528
14.810
147.121
15.387
146.162
13.918
182.583
14.870
178.229
15.817
180.565
16.872
190.318
14.645
220.928
16.098
221.016
17.101
221.880
18.111
226.122
C = 1500 wppm
C = 3000 wppm
C = 3000 wppm
∆Tw
qw" ⇑
∆Tw
qw" ⇑
∆Tw
qw" ⇓
3.852
3.157
2.304
1.715
3.962
3.010
6.535
8.763
7.195
9.954
7.073
9.976
8.737
19.816
9.186
20.820
9.424
20.539
198
10.092
36.011
10.803
35.906
10.714
37.766
11.729
56.230
12.194
56.071
12.569
56.694
13.516
82.704
14.331
82.149
14.358
83.232
15.114
112.918
16.639
111.222
16.051
113.539
16.571
146.963
17.806
148.242
17.509
146.817
17.940
187.080
19.660
190.946
18.860
183.752
18.877
224.166
21.176
225.901
20.266
228.451
199
APPENDIX D
TEMPORAL AND SPATIAL DISCRETIZATION
The whole simulation is solved in the domain
Ω = {( x , y )|0 ≤ x ≤ X ,0 ≤ y ≤ Y}
D1. Projection Methodology (Fractional Step Method or Time-Split Method With
Lagged Pressure)
The accurate projection method or fractional-step (Brown et al., 2001) is used,
which is an improved fully second-order accurate projection algorithm over the
method used by Bell and Colella (1989).
The first step of the projection method: semi-implicit viscous solver for the
intermediate velocity u*
U* −Un
y GrL n Gp n −1/ 2 L* + Ln Mn +1/ 2
θ − n +1/ 2 + n +1/ 2 − n +1/ 2
= −[(U i∇)U ]n +1/ 2 +
+
ρ
2ρ
ρ
Fr Re 2
∆t
For the intermediate field u* on boundary
u* = ubn +1
Next, un+1 is recovered from the projection of u* by solving
∆t∇i
1
ρ
∇q n +1 = ∇i u* in Ω
n +1/ 2
ni∇q n +1 = 0 on ∂Ω
And setting
un +1 = u* −
∆t∇q n +1
ρ n +1/ 2
The new pressure is computed by utilizing the correct pressure update
200
∇p n +1/ 2 = ∇p n −1/ 2 + ∇q n +1 −
∆t∇i
1
ρ
n +1/ 2
∆t
1
∇µ n +1/ 2∇i n +1/ 2 ∇q n +1
ρ
2
∇q n +1 = ∇i u*
1
∇p n +1/ 2 = ∇p n −1/ 2 + ∇q n +1 − ∇µ n +1/ 2∇i u*
2
1
p n +1/ 2 = p n −1/ 2 + q n +1 − µ n +1/ 2∇i u*
2
This formula is different from Bell et al. (1989)
∇p n +1/ 2 = ∇p n −1/ 2 + ∇q n +1
Bell’s formula is not consistent with a second-order discretization of the NavierStokes equations, because the normal component of this equation imply that
ni∇p n +1/ 2 = ni∇p n −1/ 2
For all n, which cannot be correct in general.
Discretization of the projection
∆t∇i
1
ρ
n +1/ 2
∇q n +1 = ∇i u* in Ω
ni∇q n +1 = 0 on ∂Ω
u +u
−u −u
v +v
−v −v
( DU )i +1/ 2, j +1/ 2 = i +1, j i +1, j +1 i , j i , j +1 + i , j +1 i +1, j +1 i , j i +1, j
2 ∆x
2∆y
 1

 D ρ Gφ 

i +1/ 2, j +1/ 2
=
1  1
( 2φi−1/ 2, j −1/ 2 + φi+1/ 2, j−1/ 2 + φi−1/ 2, j+1/ 2 − 4φi+1/ 2, j+1/ 2 )

6h 2  ρi , j
+
+
+
1
ρi , j +1
1
ρi +1, j
( 2φ
i −1/ 2, j + 3/ 2
( 2φ
1
ρi +1, j +1
i + 3/ 2, j −1/ 2
( 2φ
+ φi +1/ 2, j +3/ 2 + φi −1/ 2, j +1/ 2 − 4φi +1/ 2, j +1/ 2 )
+ φi +1/ 2, j −1/ 2 + φi +3/ 2, j +1/ 2 − 4φi +1/ 2, j +1/ 2 )
i + 3/ 2, j + 3/ 2

+ φi +1/ 2, j +3/ 2 + φi +3/ 2, j +1/ 2 − 4φi +1/ 2, j +1/ 2 ) 


201
MPCG matrix
Ax = b
A is matrix of ( n * n ), i, j = 0......n
A = [......]
 x (1 + 1/ 2,1 + 1/ 2)  b(1 + 1/ 2,1 + 1/ 2) 
 x (2 + 1/ 2,1 + 1/ 2)  b(2 + 1/ 2,1 + 1/ 2) 


 
 x (3 + 1/ 2,1 + 1/ 2)  b(3 + 1/ 2,1 + 1/ 2) 


 

.
 .

.
 .


 

.
 .



* x (i + 1/ 2, j + 1/ 2) = b(i + 1/ 2, j + 1/ 2) 


 

.
 .

.
 .


 

.
 .
 x ( n − 5 / 2, n − 1/ 2)  b( n − 5 / 2, n − 1/ 2) 


 
 x ( n − 3/ 2, n − 1/ 2)  b( n − 3/ 2, n − 1/ 2) 
 x ( n − 1/ 2, n − 1/ 2)  b( n − 1/ 2, n − 1/ 2) 


 
b(2 / 3, j + 1/ 2), b(i + 1/ 2, 2 / 3), b( n − 1/ 2, j + 1/ 2), b(i + 1/ 2, n − 1/ 2)
are different from others, boundary nodes are included , corner has two boundary nodes
D2. Temporal Discretization
When discretizing the governing equations temporally, the diffusion terms are
treated by a fully implicit scheme and the convection and source terms by explicit
method. Therefore:
∇ ⋅ u n +1 = (
ρc pl
m n +1
ρ2
.
⋅∇ρ + V micro )
T n +1 − T n
= − ρc pl [ u ⋅ ∇T ]n + [∇ ⋅ k∇T ]n +1
∆t
φ n +1 = φ n + ∆tL(φ n )
∆t
 L(φ n ) + L(φ n +1 ) 
4
∆t
φ n +1 = φ n +  L(φ n ) + 4 L(φ n +1/ 2 ) + L(φ n +1 ) 
6
φ n +1/ 2 = φ n +
202
| ∇φ n +1 | = 1 for φ ≠ 0
1
2
φ n +1/ 2 = (φ n + φ n +1 )
ρ n +1/ 2 = ρ (φ n +1/ 2 )
µ n +1/ 2 = µ (φ n +1/ 2 )
D3. The surface Tension term (1/ W )k (φ )∇H (φ )
( M )ij =
1
( DN )ij (GH node )ij
W
n = ∇φ / | φ |
k = ∇in = ∇i(∇φ / | φ |) |φ =0
ni +1/ 2, j +1/ 2 =
(Gφ )i +1/ 2, j +1/ 2
| (Gφ )i +1/ 2, j +1/ 2 |
(Gφ )i +1/ 2, j +1/ 2
 φi +1, j +1 + φi +1, j − φi , j +1 − φi , j 


2 ∆x

=
 φi +1, j +1 − φi +1, j + φi , j +1 − φi , j 


2 ∆y


ni1+1/ 2, j +1/ 2 + ni1+1/ 2, j −1/ 2 − ni1−1/ 2, j +1/ 2 − ni1−1/ 2, j −1/ 2
( DN )i , j =
2 ∆x
1
1
ni +1/ 2, j +1/ 2 − ni +1/ 2, j −1/ 2 + ni1−1/ 2, j +1/ 2 − ni1−1/ 2, j −1/ 2
+
2∆y
D4. Discretization of Pressure Gradient
(Gp )i , j
 pi +1/ 2, j +1/ 2 + pi +1/ 2, j −1/ 2 − pi −1/ 2, j +1/ 2 − pi −1/ 2, j −1/ 2 


2∆x

=
 pi +1/ 2, j +1/ 2 − pi +1/ 2, j −1/ 2 + pi −1/ 2, j +1/ 2 − pi −1/ 2, j −1/ 2 


2∆y


D5. ENO Scheme
A third-order ENO method was used for the approximation of the convective terms
in level set function. For u divergence free:
203
(u ⋅∇)φ = (uφ ) x + (vφ ) y
(u ⋅∇)u = f x + g y
Where
2
f = ( uuv ), g = ( uv
)
v2
(uφ ) x + (vφ ) y
≈ (ui +1/ 2 , j + ui −1/ 2 , j )(φ i +1/ 2 , j − φ i −1/ 2 , j ) / (2h)
+ (vi , j +1/ 2 + vi , j −1/ 2 )(φ i , j +1/ 2 − φ i , j −1/ 2 ) / (2h)
(uT ) x + (vT ) y
≈ (ui +1/ 2 , j + ui −1/ 2 , j )(Ti +1/ 2 , j − Ti −1/ 2 , j ) / (2h)
+ (vi , j +1/ 2 + vi , j −1/ 2 )(Ti , j +1/ 2 − Ti , j −1/ 2 ) / (2h)
( f 1 ) x + ( g1 ) y
≈ (ui +1/ 2 , j + ui −1/ 2 , j )(ui +1/ 2 , j − ui −1/ 2 , j ) / (2h)
+ (vi , j +1/ 2 + vi , j −1/ 2 )(ui , j +1/ 2 − ui , j −1/ 2 ) / (2h)
( f 2 ) x + ( g2 ) y
≈ (ui +1/ 2 , j + ui −1/ 2 , j )(vi +1/ 2 , j − vi −1/ 2 , j ) / (2h)
+ (vi , j +1/ 2 + vi , j −1/ 2 )(vi , j +1/ 2 − vi , j −1/ 2 ) / (2h)
For computing ui +1/ 2 , j , similarly for ui , j +1/ 2 ,φ i +1/ 2 , j ,Ti +1/ 2 , j , ... , The ENO scheme is
defined as
i.
Upwind
i
u
≥0
+1/ 2 , j
k l = {i +1 iotherwise
ii.
First order
f i +(11)/ 2 , j = f kl , j
iii. Second order
204
a=
b=
f kl , j − f kl −1, j
∆x
f kl +1, j − f kl , j
∆x
if |a |≤ |b |
otherwise
c={
a
b
if |a |≤ |b |
k 2 = {kkll −1otherwise
f i +( 21)/ 2 , j = f i +(11)/ 2 , j +
∆x
c(1 − 2( k l − i ))
2
iv. Third order
a=
b=
f k2 −1, j − 2 f k2 , j + f k2 +1, j
f k2 , j
( ∆x ) 2
c = {ba
f
( ∆x ) 2
− 2 f k2 +1, j + f k2 + 2 , j
( 3)
i +1/ 2 , j
if |a |≤ |b |
otherwise
= f
(2)
i +1/ 2 , j
( ∆x ) 2
+
c(3(( k 2 − i ) 2 − 1)
3
D6. Approximation of Advection Terms  (U i∇ ) U 
n+1/ 2
This discretization of the advection terms in this algorithm is based on the method
used by Puckett et al. (1997) and Sussman et al. (1999). It is a predictor-corrector
method evolved from unsplit Godunov method introduced by Colella (1990).
For face (i+1/2,j):
U
n +1/ 2, L
i +1/ 2, j
n
∆x ui , j ∆t n
∆t
1 ∆t n
=U +( −
)U x ,ij − ( vU y )ij + n
(Lij − Gpijn −1/ 2 − Mijn + Fijn )
2
2
2
ρij 2
n
ij
Extrapolated from cell (i,j)
n
∆x ui +1, j ∆t n
∆t
n +1/ 2, R
n
Ui +1/ 2, j = Ui +1, j + ( −
)U x ,i +1, j − ( vU y )i +1, j +
2
2
2
1 ∆t n
(L − Gpin+−1,1/j 2 − Mi +n1, j + Fi +n1, j )
ρin+1, j 2 i +1, j
Analogous formulae are used to predict values at each of the other faces of the cell:
205
Uin, +j +1/1 2,F / B , φin++1,1/j 2, L / R , φin, +j +1/1 2, F / B
U xn and φ xn are evaluated using a monotonicity-limited fourth-order slope
approximation (Colella,1985):
2
1
( q j +1 − q j −1 ) − ( q j +2 − q j −2 )
12
= 3
2
1
∆rj
( rj +1 − rj −1 ) − ( rj + 2 − rj −2 )
3
12
δ qj
The transverse derivative terms ( vU y ) are evaluated as below, by first extrapolating
from above and below to construct edge states, using normal derivatives only, and
then choosing between these states using the upwinding procedure defined below.
B
 ∆y vij ∆t  n
−
U i , j +1/ 2 = Uijn + 
U y ,ij
2 
 2
T
 ∆y vi , j +1∆t  n
+
U i , j +1/ 2 = Uin, j +1 − 
U y ,i , j +1
2 
 2
Where U y are limited slopes in the y direction. In this upwinding procedure we first
define the normal velocity on the edge:
 v B if v B > 0, v B + v T > 0

B
T
B
T

= 0 if v ≤ 0, v ≥ 0 or v + v = 0
 T
T
B
T
 v if v < 0, v + v < 0

adv
v i , j +1/ 2
adv
Now upwind U based on v i , j +1/ 2 :
U i , j +1/ 2
U B

U B + U T
=
2

T
U

adv
if v i , j +1/ 2 > 0
adv
if v i , j +1/ 2 = 0
adv
if v i , j +1/ 2 < 0
After constructing U i , j −1/ 2 in a similar manner, these upwind values can be used to
form the transverse derivative:
( vU )
y
ij
=
(
)(
adv
adv
1
v i , j +1/ 2 + v i , j +1/ 2 U i , j +1/ 2 − U i , j −1/ 2
2 ∆y
206
)
A similar upwinding procedure is used to choose the appropriate states Ui +1/ 2, j
given the left and right states:
uin++1/1/2,2 j
u L

= 0
u R

if u L > 0, u L + u R > 0
if u L ≤ 0, u R ≥ 0 or u L + u R = 0
if u R < 0, u L + u R < 0
Following a similar procedure to construct Ui −1/ 2, j , U i , j +1/ 2 , Ui , j −1/ 2 .
In general, the normal velocities at the edges are not divergence-free; in order to
make these velocities divergence-free, the MAC projection is applied before
construction of the convective derivatives. The equation
 1

D MAC  n G MAC ∅  = D MACU n +1/ 2
ρ

is solved for ∅ , where
 u n +1/ 2 − uin−+1/1/2,2 j uin, +j +1/1/22 − uin, +j −1/1/22 
,
D MACU n +1/ 2 =  i +1/ 2, j

∆x
∆y


and
∅
−∅
x
(G MAC∅ )i+1/ 2, j = i+1,∆j x i , j
∅
−∅
y
(G MAC∅ )i+1/ 2, j = i , j+∆1 y i , j
The face-based advection velocities at t n +1/ 2 are then defined by
n +1/ 2
uiADV
+1/ 2, j = ui +1/ 2, j −
n +1/ 2
viADV
, j +1/ 2 = vi , j +1/ 2 −
ρin+1/ 2, j =
1
ρin+1/ 2, j
1
ρ
n
i , j +1/ 2
(G
MAC
(G
MAC
∅)
∅)
x
i +1/ 2, j
y
i , j +1/ 2
1 n
ρij + ρin+1, j )
(
2
207
The next step, after construction the advective velocities
Uin++1/1/2,2 j
U L
if u ADV > 0

1
=  (U L + U R ) if u ADV = 0
2
U R
if u ADV < 0
The advection terms can now be defined by
n +1/ 2
 (U i∇ ) U  i , j
1 ui +1/ 2, j + ui −1/ 2, j
=
(Ui+1/ 2, j − Ui−1/ 2, j )
2
∆x
ADV
ADV
1 vi , j +1/ 2 + vi , j −1/ 2
+
(Ui, j+1/ 2 − Ui , j−1/ 2 )
2
∆y
ADV
ADV
D7. Discretization of the divergence of the stress tensor ∇ ⋅ (2 µD)
The components of the viscous stress tensor D is estimated using central
differencing:
∇ ⋅ (2 µ D ) = 2(
( µux ) x +( µ
( µ v y ) y +( µ
u y +vx
2
u y + vx
2
)y
)x
)
The first component of the viscous term ∇ ⋅ 2 µ (φ )D is discretized as
2 µi +1/ 2, j (ui +1, j − ui , j ) − 2 µi −1/ 2, j (ui , j − ui −1, j )
+
+
∆x 2
µi , j +1/ 2 (ui , j +1 − ui , j ) − µi , j −1/ 2 (ui , j − ui , j −1 )
∆y 2
µi , j +1/ 2 ( vi +1, j +1 − vi −1, j +1 + vi +1, j − vi −1, j ) − µi , j −1/ 2 (vi +1, j − vi −1, j + vi +1, j −1 − vi −1, j −1 )
4 ∆x∆y
Where
1
2
1
2
µi +1/ 2, j = ( µ (φi , j ) + µ (φi +1, j )), µi , j +1/ 2 = ( µ (φi , j ) + µ (φi , j +1 ))
The second component of the viscous term is discretized in a similar manner.
208
D8. Discretization of the Laplace L *
2 µi +1/ 2, j iui +1, j + 2 µi −1/ 2, j iui −1, j
∆x
+
2
+
µi , j +1/ 2 iui , j +1 + µi , j −1/ 2 iui , j −1
∆y 2
µi , j +1/ 2 ( vi +1, j +1 − vi −1, j +1 + vi +1, j − vi −1, j ) − µi , j −1/ 2 (vi +1, j − vi −1, j + vi +1, j −1 − vi −1, j −1 )
4∆x∆y
 2( µi +1/ 2, j + µi −1/ 2, j ) ( µi , j +1/ 2 + µi , j −1/ 2 ) 
−
+
 ui , j
∆y 2
∆x 2


A 9-point stencil approximating the Laplacian
 1

 D ρ Gφ 

i +1/ 2, j +1/ 2
=
1  1
( 2φi−1/ 2, j−1/ 2 + φi +1/ 2, j−1/ 2 + φi−1/ 2, j+1/ 2 − 4φi+1/ 2, j+1/ 2 )

6h 2  ρi , j
+
+
+
1
ρi , j +1
1
ρi +1, j
( 2φ
i −1/ 2, j + 3/ 2
+ φi +1/ 2, j + 3/ 2 + φi −1/ 2, j +1/ 2 − 4φi +1/ 2, j +1/ 2 )
( 2φ
i + 3/ 2, j −1/ 2
+ φi +1/ 2, j −1/ 2 + φi +3/ 2, j +1/ 2 − 4φi +1/ 2, j +1/ 2 )
1
ρi +1, j +1
( 2φ
i + 3/ 2, j + 3/ 2

+ φi +1/ 2, j +3/ 2 + φi +3/ 2, j +1/ 2 − 4φi +1/ 2, j +1/ 2 ) 


A 5-point stencil approximating the Laplacian
 1

 D ρ Gφ 

i +1/ 2, j +1/ 2
=
2 
1
(φi+3/ 2, j+1/ 2 − φi+1/ 2, j+1/ 2 )
2 

h  ρi +1, j + ρi +1, j +1
+
1
−φ
(φ
)
ρi , j + ρi , j +1 i −1/ 2, j +1/ 2 i +1/ 2, j +1/ 2
+
1
−φ
(φ
)
ρi , j +1 + ρi +1, j +1 i +1/ 2, j +3/ 2 i +1/ 2, j +1/ 2
+

1
φi +1/ 2, j −1/ 2 − φi +1/ 2, j +1/ 2 ) 
(
ρi , j + ρi +1, j

209