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Effect of surfactants on the growth and departure of bubbles from solid surfaces

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Effect of surfactants on the growth and departure of bubbles from solid surfaces

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Content EFFECT OF SURFACTANTS ON THE GROWTH AND DEPARTURE OF
BUBBLES FROM SOLID SURFACES

by

Leslie B. King

A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)

December 2009

Copyright 2009 Leslie B. King
ii
Acknowledgements
I thank God for making this endeavor possible. A special thanks to my wife
for her love and support. Thanks to my parents and brother who have supported
me all my life. I thank Prof. Satwindar Sadhal for sharing his considerable
experience and insight. My educational experience at USC was very positive in
large part because of his classroom and research instruction. Also, I would like
to thank the rest of my dissertation committee, Prof. Larry Redekopp,
Prof. Fokion Egolfopoulos, Prof. Bingen Yang, and Prof. Katherine Shing for their
helpful guidance. I would also like to thank The Aerospace Corporation for
funding my studies and Dr. Ejike Ndefo and Dr. Patrick Yee for their support,
guidance, and helping me manage the responsibility of work and school.
I dedicate my dissertation to my daughters, Nia and Nailah, with all my love.
I also dedicate my work to the memory of my grandmothers, Arronia Powell and
Theresa Craig, and cousins, Paul Tolden and Anthony Lucas.

iii
Table of Contents

Acknowledgements ii
List of Figures iv
List of Tables viii
Nomenclature ix
Abstract xi
Objectives xiii
Chapter 1: Introduction 1
Chapter 2: Theory 4
Chapter 3: Experiments 17
Chapter 4: Results 20
4.1: PIV Results 25
Chapter 5: CFD Analysis 54
Chapter 6: Model Validation 67
Chapter 7: CFD Results 71
Chapter 8: Conclusions 87
Chapter 9: Suggestions for Future Work 90
Bibliography 91
APPENDIX A: Gibbs adsorption isotherm derivation 95
APPENDIX B: Unit conversions 97
APPENDIX C: Surface tension gradient user defined function (UDF) 98
iv
List of Figures

Figure 1.1: Forces acting on interface. 1
Figure 2.1: Illustration of density discontinuity and molecular
force imbalance. 5
Figure 2.2: Principal radii of curvature. 7
Figure 2.3: Interfacial motion due to surface tension gradient 9
Figure 2.4: Surfactant distribution at an interface and the adjacent bulk
phases. 13

Figure 2.5: Surfactant effects on water density. 15
Figure 2.6: Effects of surfactant on the viscosity of water. 16
Figure 3.1: Surfactant interfacial distribution variation with concentration 19
Figure 3.2: Experimental setup. 19
Figure 4.1: High speed camera image of bubble before detachment 22
Figure 4.2a:Bubble growth from 1mm orifice. 22
Figure 4.2b:Bubble growth from 1.5mm orifice. 23
Figure 4.2c:Bubble growth from 2mm orifice. 23
Figure 4.3: Variation of heat transfer coefficient with concentration in
nucleate pool boiling with aqueous SDS solutions on a
22.2 mm diameter cylindrical heater. 24
Figure 4.4: Measurement quadrants during bubble growth. 26
Figure 4.5: Velocity components bubble flow with 1mm orifice in
deionized water. 32
Figure 4.6: Velocity components bubble flow with 1mm orifice in
291ppm SDS. 33
Figure 4.7: Velocity components bubble flow with 1mm orifice in
844ppm SDS. 34

v
Figure 4.8: Velocity components bubble flow with 1mm orifice in
1877ppm SDS. 35
Figure 4.9: Velocity components bubble flow with 1mm orifice in
3149ppm SDS. 36
Figure 4.10:Velocity components bubble flow with 1mm orifice in
4809ppm SDS. 37
Figure 4.11:Velocity components bubble flow with 1mm orifice
in ethanol. 38
Figure 4.12:Velocity components bubble flow with 1.5mm orifice
in water. 39
Figure 4.13:Velocity components bubble flow with 1.5mm orifice in
291ppm SDS. 40
Figure 4.14:Velocity components bubble flow with 1.5mm orifice in
844ppm SDS. 41
Figure 4.15:Velocity components bubble flow with 1.5mm orifice in
1877ppm SDS. 42
Figure 4.16:Velocity components bubble flow with 1.5mm orifice in
3149ppm SDS. 43
Figure 4.17:Velocity components bubble flow with 1.5mm orifice in
4809ppm SDS. 44
Figure 4.18:Velocity components bubble flow with 1.5mm orifice
in ethanol. 45
Figure 4.19:Velocity components bubble flow with 2mm orifice
in water. 46
Figure 4.20:Velocity components bubble flow with 2mm orifice in
291ppm SDS. 47
Figure 4.21:Velocity components bubble flow with 2mm orifice in
844ppm SDS. 48
Figure 4.22:Velocity components bubble flow with 2mm orifice in
1877ppm SDS. 49
Figure 4.23:Velocity components bubble flow with 2mm orifice in
3149ppm SDS. 50

vi
Figure 4.24:Velocity components bubble flow with 2mm orifice in
4809ppm SDS. 51
Figure 4.25:Velocity components bubble flow with 2mm orifice
in ethanol. 52
Figure 5.1: Grid of bubble growth simulation. 55
Figure 5.2a:Actual interface shape. 58
Figure 5.2b:Piecewise-linear scheme used by geo-reconstruct. 59
Figure 5.3: Surface tension of aqueous SDS solution
versus concentration. 63
Figure 5.4: Measuring the contact angle. 64
Figure 6.1: A schematic of a compound multiphase drop with a
spherical cap of an angle Φ. 68

Figure 6.2: Axisymmetric mesh. 69
Figure 6.3: Fluent solution comparison for Pe = 5. 69
Figure 6.4: Fluent solution comparison for Pe = 50. 70
Figure 7.1: Fluent geo-reconstruct bubble solution. 72
Figure 7.2: Fluent model bubble diameter comparison with
SDS concentration. 74
Figure 7.3a:Fluent model quadrant II u-velocity comparison with
SDS concentration. 75
Figure 7.3b:Fluent model quadrant II v-velocity comparison with
SDS concentration. 75
Figure 7.4a:Fluent model quadrant III u-velocity comparison with
SDS concentration. 76
Figure 7.4b:Fluent model quadrant III v-velocity comparison with
SDS concentration. 76
Figure 7.5: Velocity components for 2mm orifice in deionized water. 77
Figure 7.6: Velocity components for 2mm orifice in 291ppm
SDS mixture. 78

vii
Figure 7.7: Velocity components for 2mm orifice in 844ppm
SDS mixture. 79
Figure 7.8: Velocity components for 2mm orifice in 1877ppm
SDS mixture. 80
Figure 7.9: Velocity components for 2mm orifice in 3149ppm
SDS mixture. 81
Figure 7.10:Velocity components for 2mm orifice in 4809ppm
SDS mixture. 82
Figure 7.11:Surface tension at interface in quad II & III in
291ppm mixture. 83
Figure 7.12:Surface tension at interface in quad II & III in
844ppm mixture. 83
Figure 7.13:Surface tension at interface in quad II & III in
1877ppm mixture. 84
Figure 7.14:Surface tension at interface in quad II & III in
3149ppm mixture. 84
Figure 7.15:Surface tension at interface in quad II & III in
4809ppm mixture. 85
Figure 7.16:Pressure difference in neck region. 85
Figure 7.17:Fluent model general flow direction with surface
tension gradient. 86

viii
List of Tables
Table 4.1: Velocity ranges for bubbles formed with 1mm orifice 29
Table 4.2: Velocity ranges for bubbles formed with 1.5mm orifice 30
Table 4.3: Velocity ranges for bubbles formed with 2mm orifice 31
Table 5.1: Material properties 61
Table 5.2: Fluent model settings 61

ix
Nomenclature
A interface surface area
c surfactant concentration
C Courant number
D diameter; diffusion coefficient
Fr Froude number
Gr
m
Grashof number for mass transfer
G Gibbs function
g acceleration due to gravity
H total enthalpy
L length scale
n molar concentration
n ˆ unit vector normal to surface
P pressure
Pe Peclet number
Q total heat transfer
r radial coordinate in spherical coordinate system
R outer radius; principle radius of curvature; universal gas constant
Ra
m
Rayleigh number for mass transfer
Re Reynolds number
s bubble radius
S total entropy
Sc Schmidt number
t time
T temperature
u
ρ
velocity vector
x
U radial velocity; internal energy
V vertical velocity; volume
W total work done
x bubble length scale

Greek Symbols
β volumetric expansion coefficient
Г solute adsorption; surface concentration
К surface curvature
μ dynamic viscosity; chemical potential
ν kinematic viscosity
ρ density
ρ
0
reference density
σ surface tension
φ electrostatic potential
τ shear stress

Symbols
* dimensionless quantity

Subscripts
i radial direction
k tangential direction
xi
Abstract

Bubbles are an important mass and energy transport mechanism and are
present in many industrial systems. Boiling, fermentation, and aeration systems
are a few examples where bubbles are key to mass and energy transport. In
particular, there is scientific interest because of heat transfer fundamentals
associated with the boiling phenomenon in various systems. Typically, bubbles
form (nucleate) and grow on the crevices of solid surfaces. In a gravity field, they
detach due to the presence of buoyancy. The process begins with simple growth
after nucleation, followed by vertical elongation near the attachment point
(necking phenomenon), and ending with detachment. Bubble formation time and
departure size can influence the efficiency of these systems. It has been
observed that a surfactant added to an ebullient flow field reduces the departure
size of the bubble and the formation time. Although the surfactants being used
cause a reduction in surface tension, the effect on departure size is opposite to
that of a pure fluid with lower surface tension. To understand the reduction in
bubble size, the flow field around the bubble together with surfactant transport is
being studied. Bubble growth and departure models have been developed using
computational fluid dynamics (CFD) with the Fluent package. Experiments have
been conducted to capture the growth of the bubble with and without surfactants
in deionized water using injected air to create a bubble in an isothermal system
(instead of vapor bubbles as in boiling). This choice is made so as to isolate the
xii
effect of surfactant additives from thermal effects. Particle image velocimetry
(PIV) test were also carried out to quantify the flow around a forming bubble.
The results indicate that necking occurs more rapidly with a surfactant present.
A surface tension gradient creates a tangential stress at the surface and causes
it to move in the direction of the higher surface tension. This movement causes
the fluid in the neighborhood of the interface to move. This is called Marangoni
convection. The fluid motion, in the bulk, therefore has both radial and tangential
components. Final calculations have shown that the tangential velocity causes
an increase in total pressure on the neck of the bubble. The additional force
causes the neck to collapse more rapidly.

xiii
Objectives

Consider the growth of an isolated air bubble attached to a surface in a pool
of deionized water. The pool is in thermal equilibrium. The growth of the bubble
is based on a fundamental balance of forces. Buoyancy, gas momentum, and
pressure aid in bubble growth. Surface tension, liquid inertia, and drag force
retard bubble growth. The interplay of these forces dictates the volume of and
time the bubble stays at the surface before detachment. When a surfactant is
mixed with the deionized water, the surface tension of the solution is reduced.
As a result, one would expect that sharper curvature at the neck could be
afforded, and would cause larger bubbles to be sustained before pinch-off.
Instead, the bubble departure size is reduced in the dilute surfactant solution. In
the present investigation, it is shown that the presence of a surfactant causes the
necking and pinch-off to occur sooner than it would without a surfactant. The
surface tension gradient over the interface produces a resultant tangential force.
This tangential force leads to surface motion which, coupled with the bubble
growth as well as mass diffusion, causes redistribution of the surfactant in the
system. In deionized water, the flow field around the bubble and at the bubble
surface has primarily a radial velocity component. Additionally, in dilute
surfactant solution, the bubble flow field experiences an increase in the tangential
velocity component. The tangential component increases pressure at the neck of
the bubble causing pinch-off to occur sooner.
xiv
The investigation includes flow-visualization experiments that record the
bubble growth and detachment history. Surfactant concentration effects on
bubble growth and flow field have been studied. Additionally, the fluid flow and
mass transfer modeling have been carried out using the appropriate governing
equations. The system is kept isothermal and without phase change so as to
isolate the effect of surfactants. The purpose of the research is to acquire an
understanding of the physical phenomena that participate in the growth, necking,
and detachment processes.
1
CHAPTER 1
Introduction
Adiabatic single-bubble dynamics provides a fundamental basis for
understanding a variety of thermal and hydrodynamic transport processes
involving liquid-gas interfacial systems. These include: aeration, fermentation,
biochemical transformation, ebullient phase-change, and many others. The
liquid-gas interactions in bubble formation at an orifice in a liquid pool are
governed by a balance of aiding and restraining forces (see Figure 1.1).

Figure 1.1: Forces acting on interface [16,17].

Pressure
Buoyancy
Viscous
drag
Surface
tension
g
Pressure
Gas momentum
2
The gas momentum, pressure and buoyancy forces aid the growth of the gas
bubble. The inertia, viscous drag, and surface tension forces impede bubble
growth.
Surface tension has a force in the normal direction that drives fluid surfaces
toward a minimal energy state characterized by a configuration of minimum
surface area (spherical). Spatial variations in surface tension along the interface
cause the fluid to flow from regions of lower to higher surface tension (Marangoni
convection). In general, surface tension gradients along an interface can form
due to temperature gradients, concentration gradients, or electrostatic potential
gradients.
The purpose of this thesis is to carry out a thorough investigation of the
phenomena involved with bubble growth, necking, and detachment, with and
without the presence of surfactants. The numerical model entails the solution to
the Navier-Stokes equations for fluid flow, as well as convective and diffusive
mass transport of surfactant in the bulk and the liquid-gas interface. The
experiments involve flow visualization of air-injection bubbles with orifice sizes of
1 mm, 1.5 mm, and 2 mm and surfactant concentrations ranging from 0 ppm to
4809 ppm. The full set of experiments coupled with the numerical analysis
provide a clear understanding of the physical phenomena behind the early
bubble departure with the presence of surfactants, in contrast to the late
departure for pure liquids having lower surface tension. The results will be useful
for potential applications for enhancing boiling heat transfer, especially in low-
3
gravity environments where bubble departure is considerably delayed, adversely
affecting thermal transport.
4
CHAPTER 2
Theory

The flow field around a growing bubble determines many of its characteristics.
The small size of the bubble and its relatively low velocity results in flow fields in
the low Reynolds number (Re = O(1)) regime, indicating laminar flow conditions.
The inertia of the bubble system is weak, but not negligible, and it is therefore
considered in the numerical computation. Assuming axisymmetric,
incompressible, unsteady flow and considering gravity, the Navier-Stokes
equation can be written as [10,11,27]:
u P g
Dt
u D ρ ρ
ρ
2
∇ + ∇ − = μ ρ ρ (2.1)
where
Dt
D
is the substantive or material derivative,
∇ ⋅ +
∂
∂
= u
t Dt
D ρ

along with the continuity equation,
0 = ⋅ ∇ u
ρ
. (2.2)
Here u
ρ
is the velocity vector, g
ρ
is the gravity vector, ρ is the fluid density, and P
is the pressure. The length, velocity, and time parameters can be scaled using
[1]:
5
,
D
x
x =
∗
,
D
y
y =
∗
,
D
U
t t =
∗
,
U
u
u
ρ
ρ
=
∗
,
2
U
P
P
ρ
=
∗
∇ = ∇
∗
D , (2.3)
Here U and D are characteristic velocity and diameter. Substituting into
equations (1) and (2) gives the nondimensional forms.

,
Re
1 ˆ
0
2 ∗ ∗ ∗
∗
∗
∗ ∗
∇ + ∇ − =
= ⋅ ∇
u P
Fr
g
Dt
u D
u
ρ
ρ
ρ
(2.4)
where g ˆ is the unit vector in the direction of gravity. Reynolds number (Re) and
Froude number (Fr) are defined as:

υ
UD
= Re and
gD
U
Fr
2
= (2.5)
This completes the dynamic equations of motion. These equations of motion
are valid in a continuum and account for body forces, while surface forces are
implemented via boundary conditions.

Figure 2.1: Illustration of density discontinuity and molecular force imbalance.
Air molecules
Interface
Water molecules
6
Surface tension is present at the interface and it can be implemented as a
boundary condition. When two immiscible fluids are in contact, there is generally
a density discontinuity characterized as an interface. The interface is a few
molecular diameters thick, within which the density and composition change
rapidly from one side to the other as shown in Figure 2.1 [26,28]. The interface
behaves as if it were under tension because of intermolecular attractive forces.
This is called surface tension. Molecules in the bulk region are surrounded by
molecules of the same species and do not experience a net force on a time-
average basis. This is not the case of molecules in the vicinity of an interface
with a second fluid. This is the origin of surface or interfacial tension. Molecules
at the interface experience a pull toward the bulk phase (see Figure 2.1), as a
result the interface tends to minimize the area it occupies [22]. This is why small
gas bubbles, liquid drops, and soap bubbles are spherical in shape. Surface
tension, when it is constant, acts to minimize the surface energy and therefore
tends to keep a pure fluid particle spherical in shape. Surface tension can be
defined as the work required to form a new unit of interfacial area

dA
dW
= σ . (2.6)
Summing the forces acting on the motionless interface, which consist of the
pressure difference and the surface tension, results in the Young-Laplace
equation.
σκ = ΔP , (2.7)
7
where the curvature, is defined as:

2 1
1 1
R R
+ = κ (2.8)
with R
1
and R
2
as the principal radii of curvature as shown in figure 2.2
[2,6,7,16].

Figure 2.2: Principal radii of curvature.
If a surface tension gradient exists, then the interface will experience a
tangential force and mobility in the direction of the higher surface tension as
shown in figure 2.3. Viscous effects will cause the fluid at the interface to move
[3,4,5,18,22,23]. This is called Marangoni convection or, in the case of
exclusively thermally driven interfacial motion, it is referred to as thermocapillary
convection. Summing the normal and tangential forces and noting that the
8
surface tension will not be constant over the surface, the surface stress boundary
condition at the interface is given by:

i
k ik ik i
x
n n P P
∂
∂
+ − = + −
σ
τ τ σκ ˆ ) ( ˆ ) (
2 1 2 1
. (2.9)
Fluent is used to model the interfacial motion based on balancing this equation.
Surface tension gradients can form due to temperature gradients,
concentration gradients, or electrostatic potential gradients [17], i.e.,

t t
c
c t
T
T t ∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂ ϕ
ϕ
σ σ σ σ
(2.9)
where t is time, c is molar concentration, and ϕ is electrostatic potential. The
spatial gradient,
i
x ∂
∂σ
, is accounted for as a boundary condition. This will be
shown in the CFD section.
9

Figure 2.3: Interfacial motion due to surface tension gradient.

For this study, surfactant concentration gradients will be considered. The
surfactant concentration distribution is determined by Marangoni convection and
the coupled mass transfer processes of liquid-phase diffusion, adsorption, and
surface diffusion [12,13,14,15,24].
Surfactants are chemical substances that reduce surface tension of a liquid.
They are also known as surface active agents. Some examples are detergents
soaps, and sodium dodecyl sulfate (SDS). Surfactants reduce the surface
tension of water by adsorption at the liquid-gas interface. They also reduce the
interfacial tension between oil and water by adsorption at the liquid-liquid
interface.
10
A surfactant can be classified by the presence of charged groups. Anionic,
cationic, nonionic, and zwitterionic (dual charge) are the typical surfactant
classifications. An anionic surfactant has a negative charge, a cationic surfactant
has a positive charge, and a non-ionic surfactant has no charge. If a surfactant
has two oppositely charged groups, it is called zwitterionic. SDS is an anionic
surfactant. The effects of surfactant charge were not specifically considered in
this study, but in general the charge influences the thickness and molecular
makeup of the enveloping surfactant layer which can effect adsorption time,
which is on a microsecond time scale [15,41].
Surfactant mass diffusion is due to concentration gradients. Fick’s Law of
diffusion relates mass diffusion and concentration gradient. This process,
coupled with convective transport leads to the mass conservation equation,
c D c u
t
c
2
∇ = ∇ ⋅ +
∂
∂ ρ
(2.10)
where D is the diffusion coefficient (m
2
/s) and c is the surfactant concentration
(mol/m
3
) [1,5,9]. In addition, there is a high volumetric concentration of surfactant
in the thin interfacial region. It is convenient to describe this as a surface
distribution, Γ (mol/m
2
). Once adsorbed, the surfactant can also diffuse across
the interfacial area. The conservation within the interface is described by [43]:

s
s
s
s
s
Pe
c
n
Pe
u






∇ ⋅ = Γ ∇ − Γ ∇ ⋅
2
1 ρ
(2.11)
11
where
s
∇ is the surface gradient and Pe
s
is the Peclet number based on surface
diffusion.

s
s
D
UL
Pe = (2.12)
where D
s
is the surface diffusion coefficient. The term on the right-hand side of
eq. 2.11 accounts for surfactant adsorption or desorption. Generally, surface
diffusion is negligible because D > D
s
. Consequently, the time required for
surface diffusion is greater than bulk diffusion and bubble growth time (D ~ 10
-5

cm
2
/s and D
s
~ 10
-7
cm
2
/s).
To characterize the coupling of mass diffusion and flow field velocity, consider
the scaled molar concentration as defined in [1]:

∞
∗
−
−
=
c c
c c
c
o
o
(2.13)
Combining with scaled parameters from eq. (2.3), the mass conservation eq.
(2.10) can be written as:

∗ ∗ ∗ ∗
∗
∗
∇ = ∇ ⋅ +
∂
∂
c c u Pe
t
c
2
ρ
(2.14)
where Pe is the Peclet number,

D
UL
Pe = (2.15)
12
For the bubble growth process, the diffusive process is slow compared to the
convective transport, so the Pe >> 1.
Adsorption is the accumulation of solutes on the interface. Adsorbed atoms
change the specific surface free energy. An adsorbed molecule is bound to the
surface via a weak van der Waals type bond. This bond involves no charge
transfer from or to the substrate (interface). The attractive force is provided by
the instantaneous dipole moments of the adsorbed atom and its nearest-
neighboring surface atoms [28]. A molecular simulation would be ideal for
modeling adsorption, but for this analysis, a macroscopic approach based on
thermodynamics was applied. The surface and bulk concentrations along with
the surface tension are interdependent and described by thermodynamic
relations based on the Gibbs adsorption isotherm (see Appendix A for derivation)
.
ln
1






− = Γ
c d
d
RT
σ
(2.16)
For small deviation from equilibrium, equation (2.16) can be linearized as given
by Holbrook and LeVan [1, 3, 4]
(Γ
†
- Γ
∞
)=
dc
dΓ
(c
†
-c
∞
) (2.17)
where c
∞
and Γ
∞
are the equilibrium concentrations and c
†
and Γ
†
are the
interfacial concentrations (see Figure 2.4).
13

Figure 2.4: Surfactant distribution at an interface and the adjacent bulk
phases [3].

Since the concentration of surfactant on the interface can vary from point to
point, diffusion of molecules can occur. For this study, it is assumed the
distribution of surfactant in the tank is uniform at the inception of the bubble. So
as the bubble grows, its interfacial area increases and surfactant diffuses from
the continuous phase and is adsorbed. Marangoni convection and adsorption
are the mass transport mechanisms.
14
The density of the fluid is a function of temperature, pressure, and species
mass concentration. A variation of the fluid density with respect to these
properties can result in a noticeable buoyancy-driven motion of the bulk fluid.
Since the system in this study is isothermal, temperature does not affect density.
Similarly, pressure has no effect on density due to constant atmospheric
pressure and small hydrostatic pressure change. Density variation due to
species mass concentration variation is considered. However, the surfactant
concentrations applied in this study are on the order of a thousand parts per
million (ppm), which results in a negligible density difference [36, 37, 39]. The
Boussinesq approximation for species concentration only, can be written as:
, 1
0
0








Δ − =
ρ
ρ
β ρ ρ
i
mi
(2.18)

P T
i
mi
,
0
1
ρ
ρ
ρ
β
∂
∂
− = (2.19)
where
ρ
ρ
i
is the mass fraction,
mi
β is the volumetric species-concentration
expansion coefficient, which is a measure of the relative change in density with
respect to a change in concentration. It has dimensions of inverse of the mass
fraction of species i [37]. The subscript, 0 , denotes a reference state. Because
small surfactant concentrations were used, the mass fraction of surfactant is
small enough to neglect its effects on the density of the bulk fluid (see figure 2.5).
With nearly constant density, the mass transfer Grashof (Gr
m
<<1) number [37],
15

2
3
v
x g
Gr
i
mi
m
ρ
ρ
β Δ
= , (2.20)
for the dilute mixture is small. Thus the buoyancy effects due to surfactant
concentration are negligible.

Figure 2.5: Surfactant effects on water density [37, 39].
Fluid motion generated by buoyancy forces is influenced by viscous forces.
The viscosity of the surfactant-water mixture could vary with surfactant
concentration. Since SDS is a solid, it does not have a viscosity value.
However, its effect on the viscosity of water can be determined by its volume
fraction. The generalized mixture rule (GMR) is a phenomenological formula that
predicts the mechanical properties of isotropic multiphase materials in terms of
component properties, volume fraction, and microstructures. Considering the
16
surfactant-water mixture as a particle-liquid suspension, the viscosity can be
approximated by [40]
) 5 . 2 1 (
s w m
V + = μ μ . (2.21)
Where μ
m
is the mixture viscosity, μ
w
is the water viscosity, and V
s
is the volume
fraction of surfactant. A plot of this relation is shown in figure 2.6. The plot
shows that the effect of surfactant concentration on the viscosity of water is
negligible.

Figure 2.6: Effects of surfactant on the viscosity of water [40].
17
CHAPTER 3
Experiments

To better understand the flow field around a growing bubble, particle image
velocimetry (PIV) experiments were performed with and without surfactant. PIV
is a non-intrusive measurement technique for studying the velocity of some type
of flow. The advantage of this technique is that it does not require the placement
of any type of probe in the medium, which could affect the overall flow. In
addition, a probe can only measure the velocity at a single point, PIV can return
information about the entire flow field. PIV requires the fluid to be seeded with
tracer particles. The particles must be able to match the fluid properties
reasonably well. A high speed digital camera is needed to capture successive
images of the particles. Lasers are used to provide lighting at a frequency high
enough for high speed imaging. The images of the tracer particles are
processed using a tracking algorithm to get the flow field velocity vectors.
A 7759 cm
3
acrylic tank was used for the experiments. The tank was filled
with 3.785 L of deionized water. A plexiglass plate with a 1mm, 1.5mm, and
2mm orifices was used for the bubble surface. The plate was connected to a
compressed air tank with flex tubing. The air pressure was set to less than 5 psi.
The rate of bubble production was low enough such that the flow field was not
disturbed by the previous bubble. A high- speed digital camera, an argon laser,
and fluorescent microparticles were required for PIV measurements (see Figure
18
3.1). The argon laser operated at 2W and 514 nm. The argon laser and optics
were used to form a laser sheet, which dissects the bubbles. The microparticles
within the plane of the laser sheet are tracked. The microparticle diameter is 20
– 40 μm. When the laser comes in contact with the microparticles, they emit light
at 532 nm. A filter was used to block light below 530 nm. Light from the laser
was blocked but light from the particles was captured by the high speed camera.
The camera recorded the images at a rate of 682 Hz. The diameter of the
bubble was also measured.
Sodium dodecyl sulfate (SDS) was the surfactant used in the tests.
Measurements at surfactant concentrations of 291 ppm, 844 ppm, 1877 ppm,
3149 ppm, and 4809 ppm were performed. These values were chosen based on
the critical micelle concentration for SDS.
Surface tension measurements, for aqueous SDS solutions carried out by
Manglik [12], show a reduction with SDS concentration until approximately 2500
ppm. The surface tension varies little at concentrations above 2500 ppm. This
value is the critical micelle concentration. The critical micelle concentration
occurs when the interfacial area is completely covered by the surfactant, above
which micelles begin to form (see Figure 3.1). A micelle is an aggregation of a
large number of monomers [12]. The critical micelle concentration is an
important characteristic of a surfactant. Before reaching this critical point, the
surface tension changes strongly with the concentration of the surfactant and
after reaching this point, the surface tension stays more constant.
19

Figure 3.1: Surfactant interfacial distribution variation with concentration.
The image pairs were processed using the Very Swift Visualizer (VSV) code
and post-processing was done with Matlab. Bubble diameters were measured
using Photoshop. The average diameter and standard deviation was calculated
based on a sample of five or more bubbles. To compare the effects of surfactant
with a low surface tension pure liquid, ethanol was substituted for deionized
water.

Figure 3.2: Experimental setup.
Camera
Tank
Air tank
Regulator
Optics
Plate
micelle
monomer
20
CHAPTER 4

Results

The gas momentum, pressure, and buoyancy forces aid the growth of the air
bubble, which in turn is impeded by liquid inertia, viscous drag, and surface
tension forces. The interplay of these forces dictates the volume of and time the
bubble stays attached to the plate [19,20]. When the surfactant concentration of
the ebullient flow field was equal to or less than 1877 ppm, the formation time
and departure size of the bubble was reduced. However, when the surfactant
concentration was 3149 ppm and 4809 ppm, the formation time and departure
size of the bubble was comparable to the bubbles formed in deionized water.
This is due to the reduced surface tension gradient. When the surfactant
concentration becomes too high the surface tension gradient goes to zero. The
surface tension properties of the mixture imitate that of a pure fluid.
Manglik et al [14] observed a critical concentration level (~2500 ppm), beyond
which there was a reduction in the extent of heat transfer enhancement at high
heat fluxes for sodium dodecyl sulfate in water (see Figure 4.3). Heat transfer
enhancement depends on maximizing bubble departure frequency which
corresponds to maximum heat transfer coefficient. Manglik et al [14] also
observed the bubble departure diameter and growth time decreased with
surfactant concentration.
21
The graphs represent average values with standard deviations of up to 10%.
Three different sized orifices were tested (see Figures 4.2a, b, and c). The
diameter measurement is the maximum width of the bubble (see Figure 4.1). In
general, the pure fluid, without surfactant, produced larger bubbles over a
prolonged formation time. With reduced surface tension, an increase in
departure size would be expected. However, as discussed earlier, the effect on
departure size is opposite to that of a pure fluid. Consequently, the formation
time is reduced by 1 to 2 hundredths of a second. However, the formation time
difference can vary based on the air flow rate into the tank. The air flow was
maintained at a constant rate for the results shown in figures 4.2a, b, and c.
Considering the short time difference between the bubble growth in a pure fluid
and a fluid with surfactant, illustrates the short time required for surfactant
transport and adsorption.

22

Figure 4.1: High speed camera image of bubble before detachment.

Figure 4.2a: Bubble growth from 1mm orifice.
neck
diameter
0
1
2
3
4
5
6
0 0.01 0.02 0.03 0.04
time (s)
diameter (mm)
291 ppm
844 ppm
1877 ppm
3149 ppm
4809 ppm
water
ethanol
23

Figure 4.2b: Bubble growth from 1.5 mm orifice.

Figure 4.2c: Bubble growth from 2mm orifice.
0
1
2
3
4
5
6
7
0 0.01 0.02 0.03 0.04 0.05
time (s)
diameter (mm)
291 ppm
844 ppm
1877 ppm
3149 ppm
4809 ppm
water
ethanol
0
1
2
3
4
5
6
7
0 0.01 0.02 0.03 0.04 0.05 0.06
time (s)
d iam eter (m m )
291 ppm
844 ppm
1877 ppm
3149 ppm
4809 ppm
water
ethanol
24
0
5
10
15
20
25
0 1000 2000 3000 4000
C (ppm)
h (kW/m^2K)
60 kw/m^2
100 kw/m^2
160 kw/m^2

Figure 4.3: Variation of heat transfer coefficient with concentration in nucleate
pool boiling with aqueous SDS solutions on a 22.2 mm diameter
cylindrical heater [14].

25
4.1 PIV RESULTS
The PIV results are based on average velocities and the error bars represent
the standard deviation. To analyze the dynamic flow field around the bubble, it
was divided into four quadrants (see Figure 4.4). The results show the velocity
changes in each quadrant. There are three primary causes for the measurement
variation. One source of error is the non-uniform distribution of surfactant. How
the surfactant is distributed during the testing is unknown. The surfactant
distribution varied during testing because of diffusion and the motion of the
bubble during growth and departure. Consequently, the variation in surfactant
surface concentration, resulted in larger measurement variations for the samples
taken with surfactant versus the de-ionized water samples. Microparticle
distribution is another source of experimental error. During experimental runs,
the microparticles eventually settle on the bottom of the tank. To compensate for
the effects of gravity, the experiment had to be re-seeded every few minutes.
This is why the velocity at the initial time step is not zero for some test samples.
Another source of error is the measurement locations. Because the interface is
moving, the measurement points are not fixed. A considerable effort was made
to consistently take measurements in the same locations. For quadrants III and
IV, this was manageable because there is little interfacial movement. However
for quadrants I and II, the interfacial movement is substantial. As a result the
error bars in quadrants I and II are generally greater than quadrants III and IV.

26

Figure 4.4: Measurement quadrants during bubble growth.

For bubble growth in deionized water with 1mm, 1.5mm, and 2mm orifices,
the PIV results show fairly radial growth in time. Using the 1mm orifice, the u and
v velocity components in quadrants I and II show a range of 0.2 cm/s to 1.2 cm/s.
The u and v velocity components in quadrants III and IV show a range of 0.1
cm/s to 0.25 cm/s. The variations are small until the bubble separates from the
plate surface (see Table 4.1). Using the 1.5mm orifice, the u and v velocity
components in quadrants I and II show a range of 0.1 cm/s to 1.5 cm/s. The u
and v velocity components in quadrants III and IV show a range of 0.12 cm/s to
Quad I
Quad II
Quad III Quad IV
u
v
27
0.6 cm/s. This is more variation than the 1mm orifice case but overall the
variations are small (see Table 4.2). Using the 2mm orifice, the u and v velocity
components in quadrants I and II show a range of 0.4 cm/s to 2 cm/s. The u and
v velocity components in quadrants III and IV show a range of 0.05 cm/s to 0.8
cm/s (see Table 4.3). The velocity variation is greater than the 1mm and 1.5mm
orifices. There is very little fluid motion in these quadrants which is where the
bubble neck forms. However, the increases in velocity variation with orifice size
are due to the larger neck area formed with larger orifices. The larger neck area
requires more time to pinch-off from the plate which gives the bubble and the
flow around the bubble more time to develop. This resulted in stronger
recirculation in the neck region (see Figures 4.5, 12, and 19).
For the dilute solution of 291 ppm SDS, the u and v velocity components in all
four quadrants with all three orifices are similar to those observed in deionized
water. The surface tension gradients are weak because of the low surfactant
concentration (see Figures 4.6, 13, and 20). For the dilute solution of 844 ppm
SDS, the flow field around the bubble is noticeably changed. Using the 1mm
orifice, the u and v velocity components in quadrants I and II show a range of 0.1
cm/s to 1.6 cm/s. The u and v velocity components in quadrants III and IV show
a range of 0.01 cm/s to 0.7 cm/s, which is a considerable increase over the
deionized water and 291 ppm SDS samples (see Figures 4.7, 14 and 21). In the
necking regions, the wider ranges of velocities illustrate increasing circulation in
the region. Using the 1.5mm orifice, the u and v velocity components in
28
quadrants I and II show a range of 0.1 cm/s to 2 cm/s. The u and v velocity
components in quadrants III and IV show a range of 0.10 cm/s to 1.0 cm/s.
Using the 2mm orifice, the u and v velocity components in quadrants I and II
show a range of 0.2 cm/s to 3 cm/s. The u and v velocity components in
quadrants III and IV show a range of 0.22 cm/s to 1.2 cm/s. Flow field changes
continue for the dilute solution of 1877 ppm SDS. With the 1mm orifice, the u
and v velocity components in quadrants I and II show a range of 0.05 cm/s to 1.0
cm/s. The u and v velocity components in quadrants III and IV show a range of
0.1 cm/s to 0.6 cm/s. Measurements with the 1.5mm orifice show that the u and
v velocity components in quadrants I and II show a range of 0.1 cm/s to 1.6 cm/s.
The u and v velocity components in quadrants III and IV show a range of 0.03
cm/s to 0.7 cm/s. Next, using the 2mm orifice, the u and v velocity components
in quadrants I and II show a range of 0.5 cm/s to 3.5 cm/s. The u and v velocity
components in quadrants III and IV show a range of 0.06 cm/s to 1.0 cm/s (see
Figures 4.8, 15, and 22). For the dilute solution of 3149 ppm SDS, the velocity
ranges in all quadrants and orifice sizes begin to decrease. Using the 1mm
orifice, the u and v velocity components in quadrants I and II show a range of
0.02 cm/s to 0.2 cm/s. The u and v velocity components in quadrants III and IV
show a range of 0.08 cm/s to 0.16 cm/s. Using the 1.5mm orifice, the u and v
velocity components in quadrants I and II show a range of 0.05 cm/s to 0.5 cm/s.
The u and v velocity components in quadrants III and IV show a range of 0.03
cm/s to 0.2 cm/s. Using the 2mm orifice, the u and v velocity components in
quadrants I and II show a range of 0.1 cm/s to 1.6 cm/s. The u and v velocity
29
components in quadrants III and IV show a range of 0.05 cm/s to 0.5 cm/s (see
Figures 4.9, 16, and 23). For the dilute solution of 4809 ppm SDS, the velocity
ranges in all quadrants and orifice sizes continue to decrease. Using the 1mm
orifice, the u and v velocity components in quadrants I and II show a range of
0.05 cm/s to 0.5 cm/s. The u and v velocity components in quadrants III and IV
show a range of 0.02 cm/s to 0.35 cm/s. Using the 1.5mm orifice, the u and v
velocity components in quadrants I and II show a range of 0.1 cm/s to 1.4 cm/s.
The u and v velocity components in quadrants III and IV show a range of 0.02
cm/s to 0.7 cm/s. Using the 2mm orifice, the u and v velocity components in
quadrants I and II show a range of 0.1 cm/s to 0.8 cm/s. The u and v velocity
components in quadrants III and IV show a range of 0.06 cm/s to 0.16 cm/s (see
Figures 4.10, 17, and 24). The ethanol results are similar to deionized water
(see Figures 4.11, 18, and 25). The ethanol velocity range results, in all four
quadrants, are close to the deionized water velocity range results.
Table 4.1: Velocity ranges for bubbles formed with 1mm orifice
Orifice
1 mm
Quad I
(cm/s)
Quad II
(cm/s)
Quad III
(cm/s)
Quad IV
(cm/s)
Water u = 0 -
-
0.23
v = 0.05- 0.5
u = 0 -
-
0.88
v = 0.1 - 1.2
u = 0 -
-
0.15
v = 0.01-
-
0.25
u = 0.01-
-
0.01
v = -0.01-
-
0.1
291 ppm u = 0 -
-
0.2
v = 0.03-0.45
u =
-
0.1-
-
1.6
v = 0.2 - 1.4
u = 0.07 -
-
0.3
v = 0.08 -
-
0.4
u = 0.05-
-
0.04
v = 0.08-
-
0.06

30
Table 4.1: Continued
844 ppm u = 0.1-
-
0.7
v = 0.6 – 1.6
u = 0 -
-
1.5
v = 0.2 – 1.1
u = 0.01-
-
0.6
v =
-
0.3 - 0.7
u = 0.03 - 0.1
v = 0.03 - 0.13
1877 ppm u = 0.05-
-
0.22
v = 0.1 - 0.4
u =
-
0.2-
-
0.8
v = 0.1 - 0.6
u = 0.02-
-
0.35
v = 0.1 -
-
0.6
u = 0.1-
-
0.02
v = 0.1 -
-
0.1
3149 ppm u = 0.04 - 0.09
v = 0.1 - 0.45
u= 0.1-
-
0.6
v=0.25 - 0.7
u =
-
0.17- 0.15
v =
-
0.05 - 0.3
u = 0.07-
-
0.06
v =
-
0.04 -0.11
4809 ppm u = 0 -
-
0.15
v = 0.1 - 0.35
u=
-
0.05-
-
0.5
v= -0.1 - 0.4
u=
-
0.03-
-
0.26
v =
-
0.05 - 0.3
u = 0.02-
-
0.04
v = 0.12-
-
0.02
ethanol u= -0.05-
-
0.25
v = 0 - 0.68
u = 0 -
-
1
v = 0.1 - 1.5
u = 0.05-
-
0.45
v = 0 -
-
0.3
u= -0.01-
-
0.11
v =
-
0.1 - 0.15

Table 4.2: Velocity ranges for bubbles formed with 1.5mm orifice
Orifice
1.5 mm
Quad I
(cm/s)
Quad II
(cm/s)
Quad III
(cm/s)
Quad IV
(cm/s)
Water u = 0.2 - 0.95
v = 0.14 - 0.65
u = 0.2-
-
1.4
v = 0.1 - 1.3
u = 0.2 -
-
0.85
v = 0.1-
-
0.25
u = 0.1-
-
0.03
v = 0.25-
-
0.04
291 ppm u = 0 -
-
0.24
v = 0 - 0.61
u = 0 -
-
2.5
v = 1.5 -
-
1
u = 0.06-
-
0.12
v = 0.05 -
-
0.5
u = 0.14-
-
0.02
v = 0.15-
-
0.12
844 ppm u = 0.01- 0.5
v = 0.2 – 1.2
u = 0.1-
-
2.5
v = 0.2 – 1.8
u = 0.1-
-
0.6
v = 0.4 -
-
1
u = 0.12 - 0.28
v = 0.05-
-
0.25
1877 ppm u = 0.1-
-
0.65
v = 0.3 - 1.2
u = 0 -
-
1.8
v = 0.1-0.95
u = 0 -
-
0.5
v = 0.4 -
-
0.3
u = 0.04-
-
0.02
v = 0.02 - 0.13

31
Table 4.2: Continued
3149 ppm u = 0 -
-
0.3
v = 0.05 - 0.64
u= 0 -
-
0.2
v=0.15 -
-
0.1
u = 0.02-
-
0.05
v = 0.05-
-
0.17
u = 0.09-
-
0.02
v = 0.02 -
-
0.12
4809 ppm u = 0.1 -
-
0.45
v = 0.1 - 0.63
u= 0 -
-
0.8
v= 0.4 -
-
0.4
u= 0.05-
-
0.6
v = 0.2 -
-
0.25
u = 0.08-
-
0.04
v = 0.02- 0.15
ethanol u= 0.04-
-
0.23
v = 0.07-
-
0.04
u = 0 -
-
0.1
v=0.08-
-
0.07
u = 0.04-
-
0.1
v = 0.08-
-
0.03
u= 0.03-
-
0.01
v = 0.08-
-
0.03

Table 4.3: Velocity ranges for bubbles formed with 2mm orifice
Orifice
2 mm
Quad I
(cm/s)
Quad II
(cm/s)
Quad III
(cm/s)
Quad IV
(cm/s)
Water u = 0 -
-
1.5
v = 0.4- 1.8
u = 1 -
-
1.4
v = 0.5 - 1.5
u = 0 -
-
0.6
v = 0.2-
-
0.8
u = 0.05-
-
0.2
v = 0.3 -
-
0.15
291 ppm u = 0 -
-
0.5
v = 0.2 - 1.1
u = 0.2 -
-
2
v = 0.9 -1.6
u = 0.65 -
-
0.1
v =
-
0.4 -
-
1.6
u = 0.2-
-
0.27
v = 0.1 - 0.6
844 ppm u =
-
0.25-
-
1.1
v = 0.2 - 1.8
u = 2 -
-
3
v = 1.4 -
-
0.2
u = 0.3 -
-
1
v = 0.2 -
-
1
u =
-
0.22-
-
0.36
v =
-
0.27-
-
0.55
1877 ppm u =
-
0.2 -
-
1.6
v = 0.7 - 3
u = 2 -
-
2.5
v = 1.2 -
-
0.7
u = 0.3 -
-
0.9
v = 0.4 -
-
1
u =

0.04-
-
0.08
v = 0.15-
-
0.1
3149 ppm u = 0 -
-
0.55
v = 0.2 - 1.4
u= 0.3 -
-
1.6
v=0.15 - 0.8
u =
-
0.05-
-
0.47
v = 0.1 -
-
0.36
u =
-
0.03-
-
0.24
v = 0.08 - 0.24
4809 ppm u = 0.1 -
-
0.01
v = 0 - 0.25
u= 0.2-
-
0.7
v = 0.15-0.7
u= 0.05 - 0.15
v = 0.05-
-
0.12
u = 0.06-
-
0.01
v = 0 - 0.12
ethanol u=
-
0.02-
-
0.17
v = 0.1 -
-
0.05
u=
-
0.04-
-
0.25
v=0.04 -
-
0.13
u = 0.08-
-
0.1
v=-0.06 -
-
0.12
u= -0.01-
-
0.07
v = 0.1 -
-
0.05
32

Figure 4.5: Velocity components bubble flow with 1mm orifice in deionized
water.
33

Figure 4.6: Velocity components bubble flow with 1mm orifice in 291ppm SDS.
34

Figure 4.7: Velocity components bubble flow with 1mm orifice in 844ppm SDS.
35

Figure 4.8: Velocity components bubble flow with 1mm orifice in 1877ppm SDS.
36

Figure 4.9: Velocity components bubble flow with 1mm orifice in 3149ppm SDS.
37

Figure 4.10: Velocity components bubble flow with 1mm orifice in 4809ppm
SDS.
38

Figure 4.11: Velocity components bubble flow with 1mm orifice in ethanol.
39

Figure 4.12: Velocity components bubble flow with 1.5mm orifice in water.
40

Figure 4.13: Velocity components bubble flow with 1.5mm orifice in 291ppm
SDS.
41

Figure 4.14: Velocity components bubble flow with 1.5mm orifice in 844ppm
SDS.
42

Figure 4.15: Velocity components bubble flow with 1.5mm orifice in 1877ppm
SDS.
43

Figure 4.16: Velocity components bubble flow with 1.5mm orifice in 3149ppm
SDS.
44

Figure 4.17: Velocity components bubble flow with 1.5mm orifice in 4809ppm
SDS.
45

Figure 4.18: Velocity components bubble flow with 1.5mm orifice in ethanol.
46

Figure 4.19: Velocity components bubble flow with 2mm orifice in water.
47

Figure 4.20: Velocity components bubble flow with 2mm orifice in 291ppm SDS.
48

Figure 4.21: Velocity components bubble flow with 2mm orifice in 844ppm SDS.
49

Figure 4.22: Velocity components bubble flow with 2mm orifice in 1877ppm
SDS.
50

Figure 4.23: Velocity components bubble flow with 2mm orifice in 3149ppm
SDS.
51

Figure 4.24: Velocity components bubble flow with 2mm orifice in 4809ppm
SDS.
52

Figure 4.25: Velocity components bubble flow with 2mm orifice in ethanol.
53
The u and v velocity components in all quadrants show an increased range at
SDS concentrations of 844 ppm and 1877 ppm compared to deionized water.
More importantly, in quadrants III and IV the velocity components are no longer
small and constant. As a result of the surface tension gradient and subsequent
tangential bubble surface motion, the magnitude of the flow field in these
quadrants has increased. As a result of the increased flow velocity, the dynamic
pressure in this area is increased. The increase in dynamic pressure causes the
bubble necking to occur earlier than in solutions without surfactant. This will be
illustrated in the CFD section.
54
CHAPTER 5

CFD Analysis

The commercial computational fluid dynamics (CFD) package, FLUENT, was
used for the numerical computation of the bubble flow field. First, a refined mesh
was generated using the grid generation package, GAMBIT. The model of a tank
with air entering at the bottom was created. Several domains and meshes were
ran to find the optimum geometry. Experiments with uniform meshes with a
domain of 50 mm by 50 mm containing 40000 to 291600 cells were ran to test
the accuracy of the geo-reconstruct scheme. The results were good
approximations of the bubble geometry, but computation time was long. To
reduce the simulation run time and maintain accuracy, a non-uniform mesh was
created. In addition, the domain size was enlarged to speed up convergence.
The final grid has a domain of 200 mm by 90 mm and contains 41125 mesh
faces in a non-uniform distribution. More cells are located in the area where the
bubble forms and the flow field is dynamic. The grid becomes coarse in the
areas away from the bubble where there is little or no dynamic flow. In the areas
of no interest, there are fewer cells which results in substantially reduced
computation time. A single cell dimension in the bubble region is 0.1 mm by
0.1 mm (see Figure 5.1).

55

Figure 5.1: Grid for bubble growth simulation.

Since the pressure differences that drive the flow are small, the double precision
calculation scheme was used. The bubble geometry construction scheme is
more accurate using double precision. An unsteady, axisymmetric, and pressure
based model was used (see Table 5.2). Historically, the pressure-based solver
was developed for low-speed incompressible flows. In the pressure-based
approach, the pressure field is extracted by solving a pressure or pressure
inlet
axis
outlet
wall
wall
0.1 mm
0.1 mm
Z
r
56
correction equation which is obtained by manipulating continuity and momentum
equations [31]. In discrete form, these are:
0
1
= ⋅ ∇
+
∗
n
u
ρ
, (5.1)
and
( )
2
1
2
1
2
1
*
1
Re
1 +
∗ ∗
+
∗
+
∗
+
∗
∇ ⋅ −






∇ = ∇ +
Δ
− n
n
n
n
u u u P
t
u u ρ ρ
ρ ρ
. (5.2)
The dimensionless velocity vector is represented by
∗
u
ρ
. Equation (5.1)
represents continuity at time step n + 1. The convective and diffusion terms in
the momentum equation (5.2) are solved first at time step n+1/2. The velocities
are determined from the momentum equation (5.1), after the Poisson equation
2
1
2
2
1
2
Re
1
+
∗ ∗ ∗
∗
+
∗






∇ ⋅ − ∇ ⋅ ∇ +
Δ
⋅ ∇
= ∇
n
n
u u u
t
u
P
ρ ρ
ρ
(5.3)
has been solved for the pressure field, P*.
The Poisson equation is obtained by taking the divergence of the momentum
equation (5.2) subject to the continuity constraint. According to Tannehill, et al.
[31], the solution procedure consists of first computing
∗
u
ρ
from the momentum
equation while neglecting the pressure gradient terms. The pressure field, P*, is
determined from the solution of the Poisson equation, and the solution is used in
the momentum equation to determine the velocities. Then the pressure field in
57
the momentum equation becomes a pressure correction, which can be
determined from the solution of the Poisson equation.
The volume of fluid (VOF) multiphase model was enabled (see Table 5.2).
This model is used for analyzing immiscible fluids. The VOF model solves a
single set of momentum equations and tracks the volume fraction of each of the
fluids throughout the domain. In each control volume, the volume fractions of all
phases add to unity. The fields for all variables and properties are shared by the
phases and represent volume-averaged values, as long as the volume fraction of
each of the phases is known at each location. Thus, the variables and properties
in any given cell are either purely representative of one of the phases, or
representative of a mixture of the phases, depending upon the volume fraction
values. The tracking of the interface between the phases is accomplished by the
solution of a continuity equation for the volume fraction of one of the phases.
The discretization scheme used for VOF model is QUICK (see Table 5.2).
The geometric construction of the bubble was done using a piecewise linear
interpolation to the cells that lie on or near the interface (see Figures 5.2a and b).
The reconstruction based scheme used was geo-reconstruct. This scheme
assumes that the interface between two fluids has a linear slope within each cell
and uses this linear shape for calculation of the convection of fluid through the
cell faces. The position of the linear interface is calculated relative to the center
of each partially filled cell based on the volume fraction. The orientation of the
interface is based on the balance of forces acting on the interface. The normal
58
and tangential forces are computed in all the cells that make up the interface. In
each cell, normal forces are applied in the direction of the surface normal vector,
which is found from the gradient of the volume fraction. Tangential forces in the
cell are applied in the direction of the surface tension gradient. The amount of
fluid convected through each face is calculated using the computed linear
interface and information about the normal and tangential velocity distribution on
the interface. This distribution is based on balancing the normal and tangential
forces acting on the interface. The volume fraction calculation for the next time
step is based on values of the previous time step [33].

Figure 5.2a: Actual interface shape [33].
59

Figure 5.2b: Piecewise-linear scheme used by geo-reconstruct [33].
Water, air, and sodium dodecyl sulfate are the only materials considered.
The properties of water, air, and SDS are defined in Fluent (see Table 5.1).
Water or water-SDS mixture and air are defined as primary and secondary
phases. The greater quantity of fluid was defined as the primary phase.
For the surfactant transport, a diffusion model is coupled with the bubble
growth model. To enable the species diffusion model, the species transport and
reaction model was activated. Species transport enables the calculation of multi-
species transport, either non-reacting or reacting. No chemical reactions occur
for this study. The mass/species conservation equation (eq. 2.14) is solved for
each species. For this study, the only species is the surfactant. The surfactant,
SDS, was added to the material database. A mixture of liquid water and SDS
was created. The concentration of SDS is small such that there would be no
variation in density however the volume-weighted-mixing-law is used to calculate
60
the density. The volume-weighted-mixing-law for non-ideal gas mixture is
applied. Fluent computes the mixture density as:

∑
=
i
i
i
Y
ρ
ρ
1
(5.4)
where
i
Y is the mass fraction and
i
ρ is the density of species i.
The viscosity of the mixture generally depends on surfactant molecular weight
and geometry. However, because the surfactant concentration is low, there is no
significant change in the physical properties of water except for surface tension
[14]. The viscosity of the mixture is considered equal to the viscosity of water.

w
μ μ = (5.5)
The diffusion coefficient for SDS in water is 1e-9 m
2
/s [24].
An assumption of initial uniform surfactant distribution has been made. To
apply this to the computational domain, the surfactant is explicitly placed in the
domain by ‘patching’ the fluid. This allows the volume fraction of surfactant in
each cell to be specified and thus vary the surfactant concentration.
61
Table 5.1: Material properties
Air Water (l) SDS Water-SDS mixture
Density (kg/m
3
) 1.225 998.2 370 998.2
Viscosity
(kg/m*s)
1.7894e-5 0.001003 n/a 0.001003
Diffusion
Coefficient (m
2
/s)
n/a n/a n/a 1e-9

Table 5.2: Fluent model settings
Selection Setting
Pressure-
based solver
Compute velocity from momentum equations while
neglecting the pressure gradient terms. The pressure
Poisson equation is then solved for the pressure field, then
the velocities are computed.
Volume of
fluid (VOF)
solver
Multiphase model to compute momentum equation for each
fluid.
Materials Air density and viscosity are defined as 1.225 kg/m
3
and
1.7894e-5 kg/m*s. Water – SDS mixture density and
viscosity are computed using volume and mass weighted-
mixing-laws. D=1e-9 m
2
/s
Phase
interaction
Surface tension is specified as 72.8 dyn/cm for modeling
bubble growth in deionized water. The surface tension
functions used in the UDF is:
c c
2
ln 353 . 3 ln 516 . 6 49 . 68 − − = σ for ppm c 2500 ≤
39 25 . 0 + − = c σ for ppm c 2500 >
Operating
Conditions
Operating pressure of 1 atm is used. Gravity is -9.81 m/s
2
in
x-direction.
62
Table 5.2: Continued
Boundary
Conditions
Air inlet velocity set normal to boundary at 50 cm/s.
Outlet set to zero gauge pressure. The contact angle at
the plate surface is 60
o

Discretization Presto! scheme used for pressure, Quick used for
momentum, and Geo-Reconstruct used for volume
fraction, Quick used for water-SDS conservation
Adapt Region Patch surfactant concentration into fluid computation
domain
Initialization Volume fraction of water-SDS mixture is 1
Iteration Time step size set to 10
-6
sec. and 50000 time steps

Surface tension is defined in the phase interaction panel (see Table 5.2). The
surface tension can be assigned a constant value or given by a user-defined
function (UDF) (see Appendix C). The presence of surfactant is accounted for
using experimental relations from Mysels [25] (eq. 5.6),
c c
2
ln 353 . 3 ln 516 . 6 49 . 68 − − = σ (5.6)
Because surface tension varies with concentration until the critical micelle
concentration is reached and beyond that there is little surface tension variation,
two relations are needed to account for the range of concentrations considered in
this study (see Figure 5.3). Equation 5.6 is valid until the critical micelle
concentration is reached, which for SDS is approximately 2500 ppm. Above the
critical micelle concentration the surface tension varies linearly with concentration
[38]
63
39 25 . 0 + − = c σ . (5.7)

Figure 5.3: Surface tension of aqueous SDS solution versus concentration [38].
Surface tension variation with surfactant concentration is dependent on the
diffusion from the bulk fluid and Marangoni convection. Since the interfacial
position is constantly updated by Fluent, when the interface comes in contact
with the surfactant the concentration of surfactant is computed and the surface
tension at that cell is computed using equation (5.6). Because the attractive
force between the adsorbed atom and interfacial atom is instantaneous, when
surfactant contacts the interface it is assumed to be adsorbed. The Gibbs
c c
2
ln 353 . 3 ln 516 . 6 49 . 68 − − = σ
39 25 . 0 + − = c σ

c.m.c.
64
adsorption isotherm (eq. 2.16) is used to compute the adsorption, or surface
concentration Γ (mol/cm
2
), at the interface in the UDF.

Figure 5.4: Measuring the contact angle.
The operating conditions are defined by specifying the magnitude and
direction of the gravity acceleration and operating pressure (see Table 5.2). The
reference pressure location was set to a point where the fluid will always be
100% air. Boundary conditions are specified for the inlet, exit, and axis of
symmetry (see Figure 5.1). Air enters the domain at 50 cm/s. The inlet condition
is an approximation of the experimental inlet condition. The outlet allows only air
to exit (see Table 5.2). The contact angle, θ
w
, is defined as the angle between
the base plate and the tangent to the interface at the base plate (see Figure 5.4).
Based on experimental observations, the contact angle is 60
o
. The effect of
WATER
AIR
interface
wall
65
surface tension on the fluid interface is computed by Fluent. The surface stress
boundary condition at the interface is given by [29]:

i
k ik ik i
x
n n P P
∂
∂
+ − = + −
σ
τ τ σκ ˆ ) ( ˆ ) (
2 1 2 1
. (5.8)
Fluent resolves the spatial surface tension gradient and will simulate the resulting
interfacial motion.
Common solution parameters are set in the Solution Controls panel (see
Table 5.2). The Equations frame contains a list of the equations being solved for
the current model. Discretization schemes contain settings that control the
discretization of the convection terms in the solution equations. For the pressure
equation, the PRESTO! (Pressure Staggering Option) Scheme was used. This
scheme is generally used for high-Rayleigh number natural convection flow. We
have adapted the scheme whereby buoyant flow is generated by the very large
density difference between air and water. For solving the momentum equation,
the QUICK scheme was used because quadrilateral meshes were used to create
the grid.
For transient simulations, the governing equations must be discretized in both
space and time. The spatial discretization for the time-dependent equations is
identical to the steady-state case. Temporal discretization involves the
integration of every term in the differential equations over a time step. The
solution is iterated to convergence within each time step. The convergence
66
criteria is based on the value of the residuals and the Courant number. A
residual is defined as the number that results when the difference equation,
written in a form giving zero on the right-hand side, is evaluated for a solution
[31]. The residuals can be set and plotted during iteration to track convergence.
The Courant number is a dimensionless number that compares the time step in a
calculation to the characteristic time of transit of a fluid element across a control
volume (see eq. 5.9) [33]

fluid cell
v x
t
C
/ Δ
Δ
= . (5.9)
In general a Courant number ≤ 0.25 is desirable for stable computations. Lastly,
the size of the time step is set along with number of iterations (see Table 5.2).
The time step was chosen to be at most one-fourth the minimum transit time for
any cell near the interface.

67
CHAPTER 6
Model Validation
To validate the surface tension and mass transport model, an analytical case
for the motion of spherical bubbles in surfactant carried out by Oguz and Sadhal
[42] was considered. The fluid dynamics of moving drops or bubbles in the
presence of a soluble surfactant and an insoluble impurity was examined in
detail. The main purpose of this analysis was to establish a fairly general theory
that agrees with experimental measurements. Particular attention was paid to
situations involving a stagnant cap which arise when low-solubility surfactants
are present. The analysis was carried out semi-analytically using a matched
asymptotic analysis for weakly inertial flows (Re<<1).
A comparison study between the analytical results for interfacial velocity and
the Fluent simulation solution was performed. The results illustrated a similar
location of the stagnant cap around the rear stagnation point. The Fluent
simulation was able to capture the surfactant transport due to convection and
diffusion. Fluent also balanced the tangential forces acting on the interface. The
surface tension gradient was defined using the user defined function based on
results from Mysels [25].
68

Figure 6.1: A schematic of a compound multiphase drop with a spherical cap of
angle Ф [42].

A hemisphere was created in an axisymmetric grid (see Figure 6.2) to
simulate a moving spherical drop or bubble in a liquid with soluble impurity or
surfactant. The grid has 20,000 quadrilateral cells. The variable surface tension
acting on the interface was defined by the same UDF used in the bubble growth
model. The sphere diameter is 6cm and the diffusion coefficient is 3e-4 cm
2
/s.
Two runs with Pe = 5 and 50 were performed. As shown in Figures 6.3 and 6.4,
the results from Fluent compared well with analytical results from Oguz and
Sadhal [42]. Fluent is capable of computing the mass diffusion and convection
while accounting for the surface tension gradients acting on the interface.
69

Figure 6.2: Axisymmetric mesh.

Figure 6.3: Fluent solution comparison with results from Oguz and Sadhal [42]
for Pe = 5.
inlet
outlet
hemisphere
interface
70

Figure 6.4: Fluent solution comparison with results from Oguz and Sadhal [42]
for Pe = 50.
71
CHAPTER 7
CFD Results
Since the model is axisymmetric, Fluent model computations were conducted
for just Quadrants II and III. The geometric construction of the bubble was done
using the reconstruction scheme called geo-reconstruct. With the contact angle
boundary condition defined, the computed bubble geometry is similar to that
observed in the experiments, as shown in Figure 7.1. Fluent over-predicts
bubble growth early in the process but improves the prediction later in the growth
process. This is due to the fixed contact angle boundary condition. The contact
angle varies in the actual bubble growth process, but is treated as constant in the
Fluent model. In addition, the Fluent model results cannot approximate the
experimental results because the surfactant distribution in the experiment could
not be duplicated. Because of the assumption that the surfactant is uniformly
distributed in the domain, when the interface comes in contact with the surfactant
there is no surface tension gradient until the neck region forms. The surfactant
distribution becomes non-uniform when the bubble geometry starts to neck. The
fluid around the dome region has higher velocity than that around the neck region
causing the surface concentration to be higher on the dome. Surface tension
decreases with increased surface concentration until the critical micelle
concentration is reached. This results in the surface tension on the dome to be
less than the neck region, causing the interface to move from the dome to the
72
neck region. Viscous forces cause the fluid near the interface to be pulled in that
direction.

Figure 7.1: Fluent bubble reconstruction comparison with actual bubble
geometry at t = 0.012 s, 0.026s, and 0.038s respectively.
73
The Fluent model predictions exhibit similar trends of bubble diameter and
velocity variation with surfactant concentration as the experimental results. As
shown in Figures 7.5 – 10, the assumption of uniform surfactant distribution used
in the Fluent model causes differences from the experimental results with both
velocity components. The Fluent model cannot reasonably duplicate the
experimental results, but it can help explain the experimental findings. The
results of the Fluent model and the experiments are compared to illustrate how
the surfactant distribution affects the flow field. As shown in Figures 7.11 to 15,
the amount of surfactant concentration determines the surface tension gradient
and consequently, the velocity of the fluid at the interface. The fluid velocity at
the interface varies directly with the surface tension gradient. The increase in
velocity causes the pressure on the neck region to increase. As shown in Figure
7.16, the pressure difference between the inside and outside of the neck region
decreases with surfactant concentration. This resulted in shorter growth time
and slightly smaller bubbles with higher flow field velocities.
For bubble growth with constant surface tension, the detachment time is
longer. The detachment time and bubble diameter decrease with increasing
surfactant concentration until 3149ppm when it starts to increase (see Figure
7.2). The fluid velocity at the interface also increases with concentration until
3149ppm, after which, the velocity decreases. Surface tension gradient variation
with surfactant concentration causes the velocity variation with concentration.
Surface concentration of surfactant in Quad III gradually becomes less than
74
Quad II which results in the surface tension in Quad III to be higher than Quad II.
The surfactant concentration in Quad III, neck region, experiences little variation
due to the geometry of the bubble. The neck region surface area does not vary
much compared to the dome region as a result the velocity is lower. The
interface moves from Quad II to Quad III (see Figure 7.17). As the surface
tension difference increases the fluid velocity increases.
At the critical surfactant concentration, >2500ppm, a small surface tension
gradient exists. As the bubble grows the surfactant distribution remains mostly
uniform. At this concentration the bubble detachment time and diameter start to
increase and velocities start to decrease.

Figure 7.2: Fluent model bubble diameter comparison with SDS concentration.
75

Figure 7.3a: Fluent model quadrant II u-velocity comparison with SDS
concentration.

Figure 7.3b: Fluent model quadrant II v-velocity comparison with SDS
concentration.
76

Figure 7.4a: Fluent model quadrant III u-velocity comparison with SDS
concentration.

Figure 7.4b: Fluent model quadrant III v-velocity comparison with SDS
concentration.
77

Figure 7.5: Velocity components for 2mm orifice in deionized water.
78

Figure 7.6: Velocity components for 2mm orifice in 291ppm SDS mixture.
79

Figure 7.7: Velocity components for 2mm orifice in 844ppm SDS mixture.
80

Figure 7.8: Velocity components for 2mm orifice in 1877ppm SDS mixture.
81

Figure 7.9: Velocity components for 2mm orifice in 3149ppm SDS mixture.
82

Figure 7.10: Velocity components for 2mm orifice in 4809ppm SDS mixture.
83

Figure 7.11: Surface tension at interface in quad II & III in 291ppm mixture.

Figure 7.12: Surface tension at interface in quad II & III in 844ppm mixture.
84

Figure 7.13: Surface tension at interface in quad II & III in 1877ppm mixture.

Figure 7.14: Surface tension at interface in quad II & III in 3149ppm mixture.
85

Figure 7.15: Surface tension at interface in quad II & III in 4809ppm mixture.

Figure 7.16: Pressure difference in neck region.
86

Figure 7.17: Fluent model general flow direction with surface tension gradient.

Neck region Recirculation direction
87
CHAPTER 8
Conclusions
It has been observed that a surfactant added to an ebullient flow field reduces
the departure size of the bubble and the formation time. In general, the size and
formation time are reduced until the critical micelle concentration is reached. At
this critical point, the surface tension gradient is reduced to almost zero, causing
the bubble to have surface characteristics of one in a pure fluid. Although the
surfactants being used cause a reduction in surface tension, the effect on
departure size is opposite to that of a pure fluid with lower surface tension. Test
results using ethanol, which has a lower surface tension than water, show that
the bubbles formed generally have a greater formation time than those formed in
the SDS-water mixture with surfactant concentration below the critical micelle
concentration.
The presence of the surface tension gradient affected the flow field around the
bubble. The PIV results show that as the surfactant concentration increased the
fluid velocity increased. In general, the fluid regions around the bubble dome
(quadrants I and II) and neck (quadrants III and IV) increased with surfactant
concentration below the critical micelle.concentration. Flow field velocity
variation around the bubble is caused by the non-uniform surfactant distribution.
The surfactant distribution varies after bubble growth and departure. Steps were
88
taken to minimize this effect, however there was no way to reproduce the same
surfactant distribution for each bubble sample.
The Fluent model was created to assist in understanding how the mass
transport mechanisms effect the bubble growth and departure. Reproducing the
experimental results using the Fluent model was not possible. The surfactant
distribution in the experiments could not be duplicated in the Fluent model so a
neutral assumption of uniform distribution was used. The mass transport
mechanisms, diffusion, convection and adsorption, which determine the
surfactant distribution, are contained in the model. An equation for surface
tension variation with concentration was used in a user defined function.
Furthermore, the user defined function couples the mass transport mechanisms
with the bubble growth simulation. The model results qualitatively demonstrate
the effects of surfactant on bubble growth.
The Fluent model shows the surfactant concentration does not vary with the
bubble growth until the neck region starts to form. The change in bubble
geometry creates a surface tension gradient. The fluid at the bubble dome has
the higher velocity and consequently the highest concentration of surfactant.
High surfactant concentration produces lower surface tension. The fluid at the
neck region has the lower velocity and therefore a lower amount of surfactant is
transported. Lower surfactant concentration produces higher surface tension.
The dome region has a lower surface tension than the neck region so the
interface moves towards the neck region. Viscous forces cause the fluid at the
89
interface to move towards the neck region. The increase in velocity at the neck
region was observed in the PIV results and the Fluent model results. The
increase velocity caused a pressure increase on the neck interfacial area.
For pure fluids, the total pressure difference shows a positive pressure
gradient in the bubble neck area until the detachment time. The buoyancy force
causes the neck area to decrease and consequently the bubble will detach. But,
the total pressure difference shows an adverse pressure difference in the bubble
neck region earlier in the bubble development as the surfactant concentration
increases (see Figure 7.15). Because the pressure acting on the exterior of the
neck is greater than the interior pressure, the neck closes in on itself and pinch-
off occurs.
90
CHAPTER 9
Suggestions for Future Work

Additional PIV experiments with additional high speed cameras would be
beneficial. With the additional cameras more velocity planes could be measured
and a better understanding of the flow field could be attained. Applying a
technique to measure the surfactant distribution would be useful for future CFD
models. A 3-D fluent model could be created. A fully validated model could
simulate the effects in low-gravity environments and account for thermal effects.
Asymptotic analysis in the neck region could provide useful analytical information
about the dynamic character of the pinch-off process.
Attempts were made to compare the effects of surfactant in deionized water
with a low-surface tension pure liquid. Tests were conducted using ethanol.
However, properties such as viscosity and density vary from water. Thus it was
difficult to isolate the effects of reduced surface tension. If a few hydrophilic low-
surface tension pure liquid can be tested, it may be possible to isolate the effect
of varying surface tension and surface tension gradient.
91
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95
APPENDIX A: Gibbs adsorption isotherm derivation

Consider the First Law of Thermodynamics is applicable at the interface [30]
W Q dE δ δ + = , (A.1)
where Q represents heat transferred into the system and W is work done by the
system.
Incremental work can be defined as:
. dA PdV W σ δ + = (A.2)
Neglecting changes in potential and kinetic energy, equation (A.1) becomes:
. dA PdV Q W Q dU σ δ δ δ + + = + = (A.3)
Applying Maxwell relations to equation (A.3), can write:

i i
dn dA PdV TdS dU μ σ ∑ + + + = . (A.4)
Here μ
i
is the chemical potential and n
i
is molar concentration of species i.
Consider the Gibbs function, G
TS H G − = , (A.5)
leading to
. SdT Tds VdP PdV dU dG − − + + = (A.6)
Combining equations (A.4) and (A.6) gives
.
i i
dn dA VdP SdT dG μ σ ∑ + + + − = (A.7)
The process is isothermal and volume of the interface is negligible, so equation
(A.7) reduces to:
.
i i
dn dA dG μ σ ∑ + = (A.8)
Upon the application of Euler’s theorem, this becomes
.
i i
n A G μ σ ∑ + = (A.9)
Differentiation of equation (A.9) gives

i i i i
d n dn Ad dA dG μ μ σ σ ∑ + ∑ + + = (A.10)
96

Comparing equation (A.8) and (A.10), leads to
0 = ∑ +
i i
d n Ad μ σ , (A.11)
from where it follows that [8]
0 = ∑ +
i
i
d
A
n
d μ σ (A.12)
0 = Γ ∑ +
i i
d d μ σ , (A.13)
where Γ
i
=n
i
/A is the surface concentration of the surfactant.
For a single solute, equation (A.13) can be written as:
μ σ d d Γ − = , (A.14)
or









− = Γ
μ
σ
d
d
(A.15)
For an ideal solution,
c RT ln
0
+ = μ μ (A.16)
where μ
o
is the reference chemical potential.
c RTd d ln = μ (A.17)
Substituting equation (A.17) into (A.15), the final equation can be written as:
.
ln
1






− = Γ
c d
d
RT
σ
(A.18)
The energy content of a system can be changed by introducing more mass.
Chemical potential physically represents the energy changes of the system per
unit molar change of a species that are non-entropy and non-expansion work
associated. The change of energy is caused by the chemical energy contained
in the bonds of a species [35].
97
APPENDIX B: Unit conversions

1 ppm = 1 mg/l = 1 g/m
3
1 mol = 1000 mmol
1 dyn = 100000 N
1 dyn/cm = 1 mN/m
1 AMU = 1 g/mol

98
APPENDIX C: Surface tension gradient user defined function (UDF)

/*Surface Tension Gradient caused by nonuniform surfactant diffusion*/
#include "udf.h"
#define GC 8.3145 /*gas constant*/
#define Ta 293 /*Temperature K*/
#define GAMMA 2.45e-10 /*adsorption mol/cm^2*/
#define MW 288.38 /*SDS Molecular weight*/
#define phase_2_ID 2
#define CV 3.4676 /*convert concentration from kg/m^3 to mmol/l*/
#define CST 0.001 /*convert dyn/cm to N/m*/
#define rho 370 /*SDS density kg/m^3*/
#define cmcst -0.25 /*change in surface tension w/ SDS conc. above critical
micelle concentration (dyn/cm)*/
DEFINE_PROPERTY(surften,cell,ti)
{
Domain *d = Get_Domain(phase_2_ID); /*domain pointer*/
Thread *tp; /*thread from phase identified by phase_2_ID*/
real yi; /*mass fraction*/
tp = Lookup_Thread(d,THREAD_ID(ti));
/*rho = C_R(cell,tp);*/ /*concentration is yi*rho (kg/m^3) */
yi = C_YI(cell,tp,0);
if (yi <= 0.0055)
{ C_UDMI(cell,tp,0) = (68.49-
log(exp(yi*rho*1000/MW))*(6.516+3.353*log(exp(yi*rho*1000/MW))
))*CST;
C_UDMI(cell,tp,1) = (6.516+6.706*log(exp(yi*rho*1000/MW)))/(GC*MW);
99
return (68.49-
log(exp(yi*rho*1000/MW))*(6.516+3.353*log(exp(yi*rho*1000/MW))))*CST;
/*Experimental results from Mysels*/
return (6.516+6.706*log(exp(yi*rho*1000/MW)))/(Ta*GC/MW); /*adsorption
mol*10^-10cm^2*/
} else
{ C_UDMI(cell,tp,0) = (cmcst*yi*rho*1000/MW+39)*0.001;
C_UDMI(cell,tp,1) = 0.25; /*constant adsorption*/
return (cmcst*yi*rho*1000/MW+39)*0.001;
/*return 0;*/
}
}

Abstract (if available)

Abstract Bubbles are an important mass and energy transport mechanism and are present in many industrial systems. Boiling, fermentation, and aeration systems are a few examples where bubbles are key to mass and energy transport. In particular, there is scientific interest because of heat transfer fundamentals associated with the boiling phenomenon in various systems. Typically, bubbles form (nucleate) and grow on the crevices of solid surfaces. In a gravity field, they detach due to the presence of buoyancy. The process begins with simple growth after nucleation, followed by vertical elongation near the attachment point (necking phenomenon), and ending with detachment. Bubble formation time and departure size can influence the efficiency of these systems. It has been observed that a surfactant added to an ebullient flow field reduces the departure size of the bubble and the formation time. Although the surfactants being used cause a reduction in surface tension, the effect on departure size is opposite to that of a pure fluid with lower surface tension. To understand the reduction in bubble size, the flow field around the bubble together with surfactant transport is being studied. Bubble growth and departure models have been developed using computational fluid dynamics (CFD) with the Fluent package. Experiments have been conducted to capture the growth of the bubble with and without surfactants in deionized water using injected air to create a bubble in an isothermal system (instead of vapor bubbles as in boiling). This choice is made so as to isolate the effect of surfactant additives from thermal effects. Particle image velocimetry (PIV) test were also carried out to quantify the flow around a forming bubble. The results indicate that necking occurs more rapidly with a surfactant present. A surface tension gradient creates a tangential stress at the surface and causes it to move in the direction of the higher surface tension.

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Asset Metadata

Creator King, Leslie B. (author)

Core Title Effect of surfactants on the growth and departure of bubbles from solid surfaces

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 11/02/2009

Defense Date 09/24/2009

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag bubbles,hydrodynamic forces,mass transport,OAI-PMH Harvest,surface tension gradient,surfactant

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Sadhal, Satwindar S. (committee chair), Egolfopoulos, Fokion N. (committee member), Redekopp, Larry G. (committee member), Shing, Katherine S. (committee member), Yang, Bingen (committee member)

Creator Email leslie.b.king@aero.org,lking@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2707

Unique identifier UC1300337

Identifier etd-King-2488 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-270253 (legacy record id),usctheses-m2707 (legacy record id)

Legacy Identifier etd-King-2488.pdf

Dmrecord 270253

Document Type Dissertation

Rights King, Leslie B.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu