SIMULATION OF THE OIL-WATER
INVERSION PROCESS
By
JUSTIN JUSWANDI
Bachelor of Science
Oklahoma State University
Stillwater, Oklahoma
1993
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
May, 1995
SIMULATION OF THE OIL-WATER
INVERSION PROCESS
Thesis Approved:
Thesis Advisor
ean of the Graduate College
ii
ACKNOWLEDGMENTS
I wish to express my sincere appreciation to my
advisor, Dr. Alan D. Tree for his guidance, motivation, and
inspiration throughout this study. I also appreciate his
patience in going through my thesis corrections. I am also
thankful to my other committee members, Dr. Martin High and
Dr. Robert L. Robinson, Jr. for their support and
encouragement.
I wish to express gratitude to those who provided
suggestions and assistance in this study: Dr. Guohai Liu,
Mr. Sivakumar Sambasivam, Mr. Xiaofeng Guan, and Mr. Lim
Khian Thong.
I would also like to give my special appreciation to my
mom, brothers, sister, and all my relatives who have always
provided me with their everlasting support and
encouragement.
Finally, I would like to thank all the industrial
sponsors and School of Chemical Engineering for their
financial support during this study.
iii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION.................................... 1
I I. LITERATURE REVIEW..... . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Downhole Corrosion.... . . . . . . . . . . . . . . . . . . . . . . 4
Ernul s ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Stability of Emulsions...................... 9
Viscosity of Emulsions...................... 10
Phase Inversion............................. 11
Droplet Coalescence and Breakup....... ...... 12
Drop Size Distribution in Emulsions......... 15
The Monte Carlo Method...................... 16
Accuracy of the Monte Carlo Method..... ..... 19
III. MODEL DEVELOPMENT.............................. 21
Annular Flow in Gas Wells.. ....... ..... ..... 21
As sumpt ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Scheme of the Simulation.................... 24
Start of the Simulation.................... 26
Movement of the Droplets................... 26
Droplet Coalescence............... . . . . . . . . . 27
Probability of Droplet Coalescence......... 30
Probability of Droplet Breakup.. ...... ..... 30
1. Droplet Size and Flow Conditions..... 31
2. Viscosity Ratio............. . . . . . . . . . 32
Accepting and Rejecting Droplet Movement... 33
IV. RESULTS AND DISCUSSION......................... 35
Prediction of Drop Size Distribution... .... 35
Evolution of the System Energy............. 37
Comparison of Simulated Drop Size
Distributions with Experimental Data.... 42
Conservation of Mass in the Simulation..... 45
Prediction of the Phase Inversion.......... 48
Effect of the Viscosity Ratio. .......... ... 50
Effect of Turbulence....................... 50
iv
Chapter Page
Comparison of the Simulation Results on
Phase Inversion with Experimental Data.. 52
V. CONCLUSIONS AND RECOMMENDATIONS................ 55
Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Recommendations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
RE FERENCE S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 7
APPENDICES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
A. Nomenclature................................ 62
B. Procedure for Running the Computer Code..... 65
C. Computer Code to Predict Phase Inversion.... 67
D. Sample Calculations of Droplet Volume
and Surface Area........................ 82
v
LIST OF TABLES
Table Page
I. Truncation Error in the Simulations.. .......... 48
vi
LIST OF FIGURES
Figure Page
1. Drop Size Distribution of Viscous Paraffin
in Na oleate Solution....................... 17
2. Drop Size Distribution of Water in
Schoonebeck Crude Oil....................... 17
3. Drop Size Distribution of Shellsolv in Water... 18
4. Schematic Diagram of an Annular Flow.... ....... 22
5. Algorithm for Predicting Phase Inversion....... 25
6. Schematic Diagram of Unacceptable
Droplet Movement..... . . . . . . . . . . . . . . . . . . . . . . . 28
7. Schematic Diagram of the Critical Distance..... 29
8. Drop Size Distribution for Case I
(probability of Droplet Breakup = 0). ....... 36
9. Drop Size Distribution for Case II
(Probability of Droplet Breakup = 1) ........ 38
10. Energy of Droplets in Case I as a Function
of the Number of Droplet Moves..... ...... ... 39
11. Energy of Droplets in Case II as a Function
of the Number of Droplet Moves...... ..... ... 40
12 Comparison of Experimentally Obtained and
Simulated Drop Size Distribution of Water
in Crude Oil..... . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
13 Drop Size Distribution of Viscous Paraffin
in 1% Na Oleate (Sibree, 1933).............. 46
Vll
Figure
14 Simulated Drop Size Distribution of
Viscous Paraffin in 1% Na oleate Solution ..
15 Energy of Droplets in Water-in-Oil and Oilin-
Water Emulsion as a Function of the
Page
47
Volume Fraction of Water..... ....... .... ... 49
16
17
Dependency of Phase Inversion on the Viscosity
Ratio of Water to Oil ..................... .
Dependency of Phase Inversion on the Viscosity
Ratio with Reynolds Number as a Parameter ...
viii
51
53
~.
CHAPTER I
INTRODUCTION
Corrosion is a serious problem in gas and oil
production. Corrosion causes an increase in production cost
due to the additional spending on corrosion controls. Loss
of production can also occur in a severely corroded well due
to downtime. The expense associated with corrosion has led
to continuous research to understand the nature of corrosion
in gas and oil wells and to find more effective ways to
prevent the corrosion.
Corrosion in oil and gas wells is generally controlled
by the use of corrosion resistant metals, protective
coatings, and inhibitors. The type of corrosion control
used in a particular well depends on the conditions in the
well. Sometimes, two or more corrosion control methods are
implemented in a single well in order to provide better
protection.
The most reliable method to control corrosion is to use
corrosion resistant metals. However, the cost of the
corrosion resistant metals is very high and is generally too
expensive.
A more affordable but less reliable method is to use
protective coatings. The coatings can be plastic,
1
inorganic, metallic, and non-metallic materials. The
drawback of using protective coatings is that they have to
be free of any defect since a small defect may spread
quickly and cause a rapid failure.
Inhibitors are also used to protect gas and oil wells.
Inhibitors are generally organic chemicals which adhere to
the surface of the metal and promote the formation of an oil
film which protects the metal. However, in order for an
inhibitor to be effective in protecting the metal, it has to
cover all the metal surfaces, which is difficult to
accomplish in practice.
To reduce the cost of corrosion controls and to protect
gas and oil wells more effectively, the ability to predict
the location where corrosion begins is valuable. The
ability to make such a prediction has the potential of
saving capital and operating costs because the use of
corrosion resistant metals, protective coatings, or
inhibitors can be greatly reduced. In this project, a model
to predict the conditions under which corrosion begins has
been developed.
The presence of water in contact with the metal on the
tube wall is necessary for the corrosion to occur. The
water may contain dissolved C02 or H2S which is acidic and
corrosive to most metals.
Predicting the location where water first wets the
metal in a gas well is the goal in this project. The flow
of gas and liquid inside a gas well is very turbulent and
2
chaotic. An annular flow typically exists in a gas well.
The gas flows in the core and the liquid film flows on the
tube wall. The liquid film consists of oil and water which
are present as an emulsion. Due to the immicibility of oil
and water, one phase is dispersed in the other. Near the
bottom of the well, liquid water is usually present in a
small amount, therefore, water is initially dispersed in the
oil. The amount of water condensate in the film increases
in the upper part of the well due to the temperature drop.
At a certain point in the well, the volume fraction of water
reaches a critical value and the water inverts to become the
continuous phase and wets the metal. This process is called
phase inversion.
No experimental work has been done to allow the
prediction of phase inversion in the annular flow.
Conducting an experiment in a laboratory to mimic the
condition in gas wells will be very difficult. In this
project, a computer simulation is used to study the phase
inversion in gas wells.
3
CHAPTER II
LITERATURE REVIEW
All topics relevant and important to the development of
the model will be reviewed here. Specifically, a review of
downhole corrosion, emulsions, droplet coalescence, droplet
breakup, drop size distribution, and computer simulation
will be presented in this chapter.
Downhole Corrosion
Many factors have been found to affect the corrosion
rate in gas and oil wells. Bacon and Brown (1943) found
that a highly turbulent two-phase flow downstream of various
fittings and orifice plates caused corrosion due to
corrosion-erosion effects. Other factors that affect the
corrosion rate include the partial pressure of C02 and H2S
present in gas phase, temperature, properties of the
corrosion product film, fluid velocity, the type of flow
regime, concentration of various inorganic ions in the
formation water, and gas and water production rate.
Several researchers (Shock and Sudbury, 1951, Tuttle,
1987) indicated that the partial pressure of C02 and H2S had
a strong effect on corrosion by affecting the pH of water
found in oil and gas field. As a rule of thumb, a well was
4
-."-'-
classified as corrosive if the partial pressure of the
corrosive gases was above 15 psi. If the partial pressure
of the corrosive gases was between 7 and 15 psi, the well
was classified as probably corrosive. When the partial
pressure of the corrosive gases was below 7 psi, the well
was most likely non-corrosive.
Temperature affects the corrosion rate in gas and oil
wells by changing the pH, the solubility of C02 in water,
the electrochemical anodic and cathodic reaction rates, and
by the formation of corrosion product layer. DeWaard et al.
(1975) found that corrosion rate depended on temperature in
an exponential manner, much like the exponential
relationship in the Arrhenius equation. However, at higher
temperature and higher partial pressure of C02, the
corrosion rate did not depend on the temperature as strongly
as predicted by DeWaard et al.
The deviation from the exponential relationship was due
to the formation of corrosion product layer which partially
protected the well. Ikeda (1984) found that at temperatures
below 60°C, the corrosion product layer that formed on the
surface of the metal was soft and non-adhesive. At
temperatures near 100°C, the corrosion product layer was
thick and loose. The corrosion rate was a maximum in this
temperature range. At temperatures above 150 °C, the
corrosion rate was found to be a minimum due to the
formation of a fine, tight, adhesive film on the surface of
the metal which prevented the metal from corroding further.
5
----------------------------------------------.------------------ --
Another important factor that affects corrosion rate is
the flow velocity. Choi, Cepulis, and Lee (1989) found that
as the fluid velocity increased in water-in-oil (w/o) or
oil-in-water (o/w) emulsion, the corrosion rate also
increased. They argued that in the oil-continuous phase,
the fluid velocity increased both the local turbulence and
the contact time of the water phase with the metal. In the
water-continuous phase, the increase in fluid velocity was
claimed to have washed off the protective corrosion product
and to have increased the mass transfer through the pores of
the film.
The flow regime also affects the corrosion rate.
Johnson et al. (1991) found that in slug flow the corrosion
rate could be as high as seven times that in annular flow.
Chemical species in the formation water had also been
found to affect the corrosion rate. Chloride ion with the
presence of oxygen had been shown to greatly increase the
localized corrosion. However, in the absence of oxygen,
chloride ion actually reduced the uniform corrosion by
surface inhibition. Chemical species that increased the
alkalinity of the formation water were found to reduce the
corrosion rate.
Water and gas production rate has a great influence on
the corrosion rate as well. Bradburn (1977) observed that
the water production rate was a better indicator of the
corrosiveness of gas wells than the partial pressure of C02.
The effect of gas production rate was later considered by
6
Gatzke and Hausler (1984) who found that the corrosion rate
increased when the water or gas production increased.
Robertson and Erbar (1988) assumed that corrosion is
most likely to occur in the water condensation zone. They
developed a model to predict the water condensation zone in
gas wells. The model took into account the two-phase flow
regime and non-linear pressure drop in gas wells.
Liu and Erbar (1990) first developed a model which
includes fluid mechanics, mass transfer, and surface
reactions to predict uniform corrosion rates without
protective films. In their model, the key corrosive species
was hydrogen ion in the aqueous medium. The model also
assumed that corrosion began at the location where
condensation first occurred. Liu (1991) later modified the
model to include the calculation of corrosion rate in the
presence of protective films.
Sambasivam (1992) developed a computer model to predict
the conditions under which corrosion was most likely to
occur in gas wells. The model assumed that corrosion began
when water actually wet the metal as opposed to when water
condensation occurred. Sambasivam's model used a stochastic
simulation (Monte Carlo method). Given the flow conditions,
the model generated drop size distributions for both waterin-
oil (wjo) and oil-in-water (ojw) emulsions. The energy
of both type of emulsions was then calculated and compared.
The emulsion with the lowest energy was taken as the stable
emulsion. The volume fraction of water in the emulsion was
7
then increased and simulation was repeated until phase
inversion occurred. Some problems were encountered in his
model. The problems are listed below:
1. The simulation could only be performed with the ratio of
maximum to minimum diameter of two;
2. Phase inversion always occurred at 0.5 volume fraction
of water; and
3. The phase inversion did not depend on the physical
properties of the system.
The project reported here continued Sambasivam's work and
corrected these problems.
Emulsions
An emulsion is a dispersion of oil droplets in water
(o/w) or water droplets in oil (w/o). In this document, any
highly polar, hydrophilic liquid is categorized as water,
and any non polar, hydrophobic liquid is categorized as oil.
Lissant (1974) argued that the behavior of emulsions
depended more on their physical and topological
configuration than on the chemical properties of their
constituents. He divided emulsions into three categories.
The first category was the emulsions with less than 30%
volume of dispersed phase. The droplets in this type of
emulsions did not have a close interaction with each other
because of the large space between them. The property of
the emulsion was determined mainly by the property of the
continuous phase. Emulsions with 30% to about 74% volume of
8
dispersed phase fell into the second category. The droplets
in this category had more collisions or interactions with
each other. The close interactions of the droplets caused
an increase in the viscosity of the emulsion. The third
category was emulsions with more than 74% volume of
dispersed phase. The emulsions in this category were
usually unstable to shear unless special emulsifiers were
used.
Stability of Emulsions
Emulsions are formed as a result of two competing
processes: droplet coalescence and droplet breakup. Droplet
coalescence is a natural process because it reduces the
surface area, and therefore lowers the energy of the system.
On the other hand, droplet breakup requires energy.
Therefore, emulsions are thermodynamically unstable.
Emulsions can breakdown in several ways. One way is by
the separation of the dispersed and continuous phases into
two layers. Another way is by "creaming". Creaming is
characterized by the formation of two different emulsion
systems: an oil-rich system and an oil-poor system.
To form a stable emulsion, droplet coalescence has to
be prevented. Emulsifiers are generally used to prevent
droplet coalescence. Lissant (1974) stated that emulsions
with small volume fraction of the dispersed phase were
stabilized by using ionic emulsifiers and by producing very
small droplets. For emulsions with higher volume fractions
9
of the dispersed phase, emulsifiers which formed thin film
around the droplets were found to be more effective in
stabilizing the emulsion.
Viscosity of Emulsions
Viscosity of emulsions is a very important physical
property because it affects the stability of emulsions. The
viscosity of an emulsion depends on the viscosity of the
continuous phase, the volume fraction of the dispersed
phase, and the droplet size distribution, as well as the
temperature.
High viscosity in the continuous phase is found to
stabilize an emulsion. Droplets in this system move slower
because they experience greater resistance in the continuous
phase. Slower movement of the droplets results in less
droplet collision. Therefore, less droplet coalescence
occurs and the emulsion is stable.
An increase in temperature reduces the viscosity of
emulsion. The decrease in the viscosity results in an
increase of droplet mobility. Consequently, the rate of
droplet coalescence increases and the emulsion becomes less
stable.
An increase in the volume fraction of the dispersed
phase is found to increase the viscosity of the emulsion.
However, the increase in the viscosity of emulsion due to
the increase in the volume fraction of dispersed phase
destabilizes the emulsion instead of stabilizing it.
10
Droplets which are close together in this type of system are
more likely to collide and coalesce.
Droplet size has less effect on the viscosity of the
emulsion than the other factors discussed above. Emulsions
with smaller droplet sizes are found to be more viscous than
the ones with larger droplet sizes. The stability of the
emulsion is also increased by using smaller droplets with
uniform size.
Phase Inversion
Emulsions can be inverted from one type to the other.
In phase inversion, the dispersed phase inverts to become
the continuous phase and vice versa.
Many factors can influence phase inversion. Lissant
(1974) stated that phase inversion occurs when the volume
fraction of the dispersed phase reached a critical value.
Shinoda and Kunieda (1983) observed that phase inversion
occurred at a certain temperature range which they called
phase inversion temperature (PIT). Emulsions with
temperature above the PIT were water-in-oil type and those
with temperature below PIT were oil-in-water type. Smith,
Covatch, and Lim (1991) observed that at a certain range of
concentration, emulsions were always water-in-oil type and
at another range they were always oil-in-water type. This
observation contradicted the report of Shinoda and Kunieda.
Bhatnagar (1920) found that emulsifiers also affected
phase inversion. He observed that trivalent electrolytes
11
were more effective than bivalent electrolytes in causing
phase inversion.
Chemical species can also affect phase inversion.
Simon and Poynter (1968) were able to invert highly viscous
wlo emulsion to olw by chemical means.
Mao and Marsden (1977) found that the inversion could
also be achieved by varying oil concentration, temperature,
and shear stress. Increasing the oil concentration and the
temperature of an emulsion favored the formation of wlo
emulsion. Increasing shear stress also favored the
formation of wlo emulsion but its effect was negligible at
high temperature.
Brooks and Richmond (1994) examined the effect of oilphase
viscosity and stirrer speed on phase inversion. They
found that as the oil viscosity increased, the volume
fraction of water required for the phase inversion
decreased. The turbulence in the liquid was found to have
less effect on phase inversion. They found that the volume
fraction of water required for the inversion increased only
slightly as the stirrer speed was increased.
Droplet Coalescence and Breakup
Droplets in an emulsion have different sizes. The drop
size distribution exists because of the coalescence and
breakup of droplets in the emulsion. The processes and
mechanisms of droplet coalescence and breakup are still not
fully understood.
12
---- ....... ---.-.. ............ ., ....
Taylor (1934) studied droplet deformation and breakup
under shear and extensional flows. He derived an equation
to calculate the maximum drop diameter that could exist in a
given flow condition:
(II-1)
where ~ is the maximum drop diameter, a is the maximum
velocity gradient in the flow field, a is the interfacial
tension, and ~c and ~d are the viscosity of the continuous
and dispersed phase, respectively.
Clay (1940) obtained drop size distribution data
produced in a turbulent pipe flow. He used his data to
propose mechanisms for droplet coalescence and breakup. He
observed that droplets coalesced on collision or after they
clung to each other for some time. He suggested that a
droplet brokeup because of a velocity gradient or a pressure
difference on the surface of the droplet.
Kolmogoroff (1949) and Hinze (1955) studied droplet
breakup based on the balance between two forces: the
external forces which deformed the droplet and the
interfacial tension forces which counteracted the
deformation. Hinze defined a Weber number (We) as the ratio
of the external forces to the interfacial tension forces:
We= 'f
(a/d)
(II-2)
13
where t is the turbulent stresses, a is the interfacial
tension, and d is the diameter of the droplet. Kolmogoroff
and Hinze also postulated that for a given flow condition a
critical Weber number (Wecrit ) existed. If We of the droplet
was greater than Wecrit the droplet brokeup. Therefore, Wecrit
was defined as follows:
't'
We rit = ( I I - 3 )
C (J / dfD1JX
where ~ax is the maximum stable diameter. Hinze showed that
Wecrit varied for different types of flow and deformation.
Using Clay's data, he found that Wecrit for droplets produced
in turbulent pipe flow was about 1.
Kolmogoroff and Hinze suggested that in turbulent flow
the spectrum of eddies which could break the droplet should
have the size in the same order as the droplet diameter.
Eddies with size much greater than the drop diameter could
only translate the droplet. Eddies with size smaller than
the drop diameter caused only small deformation which could
not result in droplet breakup. Based on the above notion
and equation (11-3), Hinze derived an equation to calculate
the maximum stable diameter:
(11-4)
where e is the local energy dissipation per unit mass.
Levich (1962) derived an equation to calculate dm=
based on the balance of the internal pressure of the drop
14
_ .. "'-
with the capillary pressure of the deformed drop. For flow
in the tube, the equation is given by:
drmx = (0' / kjpy2)O.6 A/·6 (11-5)
where k f is a numerical constant, Y is the kinematic
viscosity, Aa is the scale of eddy at which the Reynolds
number is unity, p is the density of continuous phase.
Levich postulated that for pipe flow, droplets with the
minimum diameter were found near the wall because it was the
region where rapid changes in velocity occurred. The
equation for the minimum diameter derived by Levich is given
below:
dmin = (cw/25pVo3) (11-6)
where Va is the characteristic eddy velocity.
Drop Size Distribution in Emulsions
Studies on emulsification processes and properties
require the knowledge of drop size distribution.
Experimental data of drop size distribution had shown that
there was no general drop size distribution for all
emulsions. The data suggested that the drop size
distribution depended on how the emulsions were made.
Schwarz and Bezemer (1956) proposed a drop size
distribution which was derived statistically. The equation
which contains two parameters was found to agree with
experimental data of emulsions which were prepared
mechanically, but not with those prepared by phase
15
- --.. --
inversion, vapor condensation, or electrical disintegration.
Schwarz and Bezemer found that drop size distributions of
viscous paraffin in a Na Oleate solution and water in
Schoonebeck crude oil were log-normally distributed. The
drop size distributions are shown in Figure 1 and 2. The
maximums of the distributions in Figure 1 and 2 are skewed
to the left.
Collins and Knudsen (1970) experimentally obtained the
drop size distribution of oil-in-water in a well defined
turbulent pipe flow. The drop size distribution is given in
Figure 3. The drop size distribution did not follow any
kind of known distributions, such as log-normal, upper-limit
log normal, etc. They argued that the drop size
distribution measured was actually a superposition of two
distributions, one was initially present and the other
produced by turbulence in the flow field.
Karabelas (1978) measured the drop size distribution of
water in two liquid hydrocarbons of viscosity approximately
2 and 20 mNs/m2 in pipe flow. He found the distribution
could be fitted to an upper-limit log normal function.
The Monte Carlo Method
Computer simulation has been used to solve various
problems in engineering. Data from experiments can be
compared to the results from the computer simulation. If
the comparison is good, the computer simulation can be used
to provide insight into the experiment or used to predict
16
dn 100 AND dv 100
dx N dx V
50
40
30
20
10
2
HISTOGRAMS: OBSERVATIONS BY SIBREE
CURVES. • • EOUATION (24)
0------0 EOUATION (23)
0=8.71'
,X = 19 I'
12 14 16
Figure 1:
18
x,p'
20
Drop Size Distribution of Viscous Paraffin in Na Oleate
Solution (Taken from Kolloid Zeitschrift 1956, Vol. 146,
p. 142)
dn 100 d'/ 10C
--AND- -
dx N dx V
10
60
50
40
30
. ?O
10
2
HISTOGRAMS: OWN OBSERVATIONS
CURVES' • • EOUATION (24)
0------0 EOUATION (23)
0= 5.61'
X = /491'
6 8 10
X,I'
Figure 2:
Drop Size Distribution of Water in Schoonebeck Crude Oil
(Taken from Kolloid Zeitschrift 1956, Vol. 146, p. 143)
17
OBSERVED
120
~
1.3°m SHELLSOLV
z 576 DIAMETERS FROM INJECTION
1&.1
~ y/R .. 0.1 " 80
1&.1 a:: I&. U = -16 FT/SEC
a::
1&.1 40
en
2
~ z
100 200 300 400
DIAMETER - _MICRONS
Figure 3:
Drop Size Distribution of Shellsolv in Water (Taken from
AIChE Journal 1970, Vol. 16, p. 1079)
the result of the experiment under different conditions.
Computer simulation has also been used as a substitute for
experiments of extreme conditions which are impossible or
dangerous to be carried out in a laboratory.
One of the methods of computer simulation is the Monte
Carlo method which is also called the method of statistical
trials. The method solves a problem by constructing a
random process whose parameters are equal to those in the
original problem. The variable of interest is then solved
by the observing the random process.
The earliest example of Monte Carlo computation is the
description of the calculation of the quantity n by Buffon
(1777) using the "needle-tossing" experiment. Volser in
18
1850 performed the "needle-tossing" experiment and found the
value of n to be 3.1596.
Sometimes the accurate modeling of a random physical
process is difficult. A simplified artificial process which
approximates the original process and which can be modeled
by a computer may be used instead. This simplification is
often necessary because of two reasons: the complete
knowledge of the original process is unavailable and the
computer is unable to perform complex process calculations
in a reasonable time.
The Monte Carlo method has been used successfully in
solving problems which are random in nature. Examples of
some of these problems are found in neutron physics and
detection of signals on a phone with random noise. The
Monte Carlo method has also been used successfully to solve
deterministic problems such as boundary-value problems and
linear algebraic equations.
Accuracy of the Monte Carlo Method
The error of the Monte Carlo method, 0, can be
calculated as follows:
(II-7)
where p is the probability that event A occurs, N is the
number of trials, and L is the number of trials in which
event A occurs. The error is found to be of the order
19
(11-8)
Equation (11-8) shows that a large increase in the number of
tests is required in order to significantly reduce the
error. In a practical problem, the error in the Monte Carlo
method is of order 0.01 to 0.001 (Shreider, 1966).
20
Chapter III
MODEL DEVELOPMENT
The review of Sambasivam's stochastic model (1992) and
the revision to his model are given in this chapter. This
chapter also describes the physical system of the annular
flow in gas wells and the assumptions used in the model.
The chapter then describes the development of the
probability function of droplet breakup in detail.
Annular Flow in Gas Wells
The flow regime under consideration in this model is
the annular flow which often exists in gas wells. Other
flow regimes such as slug or bubbly flow are less often
encountered in gas wells, and in general these types of flow
regimes are more difficult to study. Therefore, they have
not been considered in this model.
A schematic diagram of annular flow in a gas well is
shown in Figure 4. The gas phase flows upward in the core
of the tube. The liquid film flows upward on the tube wall.
The liquid film consists of oil and water condensate. Due
to the immicibility of oil and water and the turbulence in
the liquid film, oil and water in the liquid film are
21
!!!!!!!!!!!!!Oo
~:~:~:~:~:~:~ 0
1:1 °0
!!!!!!!!!!!!!Oo
!!!!!!!!!!!!!OoO
Gas Flow
Figure 4:
Schematic Diagram of an Annular Flow
22
present as an emulsion. The emulsion can be either waterin-
oil or oil-in-water.
At some point in a gas well, phase inversion may occur.
Near the bottom of the gas well, water is usually present in
a small amount in the liquid film, therefore, water is
usually dispersed in oil. Due to temperature drop, the
amount of water condensate in the film increases in the
upper part of the well. At some point in the gas well, the
amount of water in the liquid film reaches a critical value
and water inverts to become the continuous phase and wets
the tube wall. Corrosion is assumed to begin at the
location where the phase inversion occurs.
Assumptions
Several assumptions were made about the physical system
of the annular flow:
1. The liquid film on the tube wall is thin compared to the
diameter of the tube. For gas wells, this assumption is
justified because the liquid condensate production is
usually very small compared to the gas production. For thin
liquid film, all droplets in the emulsion are approximately
at the same distance from the tube wall. Consequently, the
breakup of droplets in the liquid film is independent of
their distances from the wall.
2. The droplets in the liquid film are spherical. In
reality, the droplets in the liquid film may be nonspherical.
23
3. Two droplets coalesce when the distance between their
centers is less than O.3(dl + d2), where dl and d2 are the
diameters of the droplets.
4. No emulsifier is present in the liquid film.
5. The following properties are known:
- temperature and pressure,
viscosity and density of oil and water in the
liquid film,
- velocity and thickness of the liquid film, and
- volume fraction of water in the liquid film.
Scheme of the Simulation
The droplet coalescence and breakup in the liquid film
are modeled as stochastic processes. The use of a
stochastic method to model droplet coalescence and breakup
in the liquid film is appropriate due to the inherent
randomness of these processes in the turbulent and chaotic
flow which exists in gas wells.
Figure 5 shows the algorithm used to predict phase
inversion. For given flow conditions in a gas well, the
stochastic process simulation predicts the stable emulsion
type of the liquid film. First, the simulation produces the
equilibrium drop-size distribution for both w/o and o/w
emulsion. Then from the drop size distributions, the energy
of w/o and o/w emulsions is calculated and compared. The
emulsion with a lower energy is taken as the stable and
favored emulsion. The simulation begins with a small volume
24
Increase
vol.frac. ~------~
of water
Volume fraction of
water = vwat
Vol. frac. of disp.
phase = vwat
Generate drop size
dist. of wlo emulsion
of
Generate drop size
dist. of olw emulsion
Figure 5:
Algorithm for Predicting Phase Inversion
25
fraction of water. The volume fraction of water is then
increased and the simulation is repeated until the stability
of the emulsion shifts from w/o to o/w. The simulation
predicts the volume fraction of water at which the phase
inversion occurs.
Start of the Simulation
In the beginning of the simulation 6084 water or oil
droplets are placed in a lattice. The droplets are arranged
in a face cubic center configuration. All the droplets have
the same initial diameter which is equal to 30% of the
maximum diameter.
The volume of the lattice depends on the volume
fraction of the phases in the emulsion. The volume of the
lattice is calculated as follows:
V. _ 6084 x 1/ 6n din/
la1tiCt! - i/J
(111-1)
where dini is the initial diameter of the droplets in the
lattice and i/J is the volume fraction of the dispersed phase.
Movement of the Droplets
One droplet is chosen at random. The droplet is then
moved in a random direction within the lattice.
position of the droplet is calculated as follows:
Xnew = Xo1d +(2.0£1 -lO)S
Ynew = Yo1d +(2.0£2 -lO)S
Znew = Zo1d + (2.0£3 -lO)S
26
The new
(111-2)
(111-3)
(111-4)
- -_._ ....
where Xnew ' Ynew ' and Znew are the new x, y, and z coordinate
of the droplet. Xo1d ' Yo1d ' and Zold are the initial x, y,
and z coordinate of the droplet. el , e2 , and e3 are random
numbers between zero and one. 8 is the maximum allowable
displacement of the droplet inside the lattice. In our
model, 8 is set to 10% of the maximum diameter. The terms
in the brackets in the above equations allows droplets to
move in the positive and negative direction. If the new
position of the droplet is outside the lattice, the droplet
is moved back into the lattice in a manner shown in Figure
6.
Droplet Coalescence
After the droplet is randomly moved, the possibility of
the droplet coalescence is checked. Two droplets are
assumed to coalesce if the distance between the centers of
the droplets is less than a critical distance (dcrit). In
our model, the critical distance is set to 0.3 (d1 + d2 ) •
The schematic diagram of the critical distance between two
droplets is given in Figure 7.
Two droplets that coalesce will form a single droplet
with diameter equal to:
d = (d 3 + d 3 )1/3
combine 1 2 (111-5)
where d1 and d2 are the diameters of the coalescing
droplets.
27
a
o o Inilio. poslUon
Figure 6:
a
r-
\
I
position outside
the lattice [not
accepted]
Schematic Diagram of Unacceptable Droplet Movement
28
_._ ....
dcrit = O.3[dl +d2)
10( ?ol
Droplet 1 Droplet 2
Figure 7:
Schematic Diagram of the Critical Distance
29
probability of Droplet Coalescence
The probability of droplet coalescence depends on the
volume fraction of the dispersed phase. A droplet is more
likely to coalesce with another droplet if the volume
fraction of the dispersed phase is high. On the other hand,
when the volume fraction of the dispersed phase is low, a
large space exists between droplets and thus reduces the
probability of coalescence.
Probability of Droplet Breakup
The droplet breakup in the model can be described as
follows. After the random droplet movement, another droplet
is chosen randomly and its probability of droplet breakup is
calculated and compared with a random number. If the
probability of breakup is higher than the random number, the
droplet is allowed to breakup into two equal droplets.
Otherwise, the droplet does not breakup.
The probability of droplet breakup in turbulent flow
depends on several factors. The computer model developed in
this project accounts for the effects of droplet size, flow
conditions, and viscosities. The probability of breakup is
given as follows:
P=O~~+A) (111-6)
where P1 accounts for the effect of droplet size and flow
conditions and P2 accounts for the effect of viscosity. 0.5
is a normalizing factor.
30
1. Droplet Size and Flow Conditions
The size of a droplet affects the probability of the
droplet breakup. Hinze (1955) and Sleicher (1962) used the
term maximum diameter (dm=) to define the largest stable
drop size that can exist in a given flow. The closer the
diameter of the droplet to ~, the greater the probability
for the droplet to breakup. In our model, the term minimum
diameter (dmin ) is used to define the smallest drop size that
can exist in a given flow. The w/o and o/w emulsions in the
gas well are assumed to have a drop size distribution
between dmin and dmax • Dmin can be estimated by using equation
11-6. Dm= can be estimated by using equation 11-4 or 11-5.
A linear relationship of droplet size to the
probability of droplet breakup is used in our model. The
relationship is similar to the model used by Collins and
Knudsen (1977). However, a constant, k, is present in our
model to take into account the flow conditions in the gas
well. The equation is given as follows:
_ k (d -dmiD.)
PI - (dmax -dmiD.)
(111-7)
The greater the turbulence in the liquid film, the
larger the value of k is. The constant, k, can have a value
from zero to one. The turbulence in the liquid film is
indicated by the Reynolds number. In this model the
relationship of the constant k and the Reynolds number is
proposed as follows:
31
k = exp(-lOOOJ
Rei (111-8)
where Rei is the Reynolds number of the liquid film. The
constant, 1000, is to give k a value of 0.62 when Rei is
2100. The exponential form of equation 111-8 limits the
value of k from 0 to 1.
The Reynolds number in equation 111-8 is defined as
follows:
Rei = 2p,V8
J.1, (111-9)
where PI is the density of the liquid film. V is the
average velocity of the liquid film. B is the thickness of
the liquid film. ~I is the viscosity of the liquid film.
2. Viscosity Ratio
Viscosity ratio of the dispersed to the continuous
phase is a very important factor in droplet breakup.
Viscosity is a measure of resistance to deformation. A
material with high viscosity requires a greater energy to
deform. Stone (1994) argued that the viscosity ratio of the
dispersed phase to the continuous phase (~d/~C) was the most
important variable in determining droplet breakup. If ~d/~c
was of order of magnitude greater than one, the internal
flow processes of droplets in an emulsion were damped,
resulting in less frequent droplet breakup. The condition
resulted in a drop size distribution with a small number of
uniformly size drops. When ~d/~c is low (i.e. 0.01), Stone
32
observed that droplets broke readily, resulting in a drop
size distribution with many small droplets.
Sambasivam (1992) in the study of phase inversion also
concluded that viscosity played an important factor in
droplet breakup. When the viscosity of the dispersed and
the continuous phase was not taken into account in droplet
breakup, Sambasivam's model always predicted phase inversion
at 50% volume fraction of the dispersed phase.
All studies in droplet breakup conclude that an
increase in droplet viscosity results in an increase of the
energy needed to break the droplet. The reason is that an
additional energy is needed to overcome the internal viscous
dissipation. Based on the above observations, the following
relationship of the viscosity ratio to the droplet breakup
is proposed:
(111-10)
The probability of droplet breakup approaches one as ~d/~c
approaches zero. As ~d/~c becomes large, the probability
approaches zero.
Accepting and Rejecting Droplet Movement
In the simulation, droplet movement which causes the
system energy to decrease or remain unchanged is always
accepted. When a droplet coalesces with another droplet,
they form a single droplet with a lower surface area and
thus lowers the system energy. When droplet movement does
33
not result in coalescence, the system energy remains
constant.
Droplet movement which causes the system energy to
increase is only accepted on certain conditions. When a
droplet breaks into two equal droplets the surface area and
thus system energy increases. The droplet is allowed to
break only if its probability of breakup is higher than a
random number. Otherwise, the droplet breakup is rejected.
34
Chapter IV
RESULTS AND DISCUSSION
Prediction of Drop Size Distribution
Since the drop size distribution of an emulsion
determines its surface energy, a good prediction of drop
size distribution is required in the model in order to
predict the stability of the emulsion. The ability of the
model to predict the drop size distribution was tested
against two limiting cases. In both cases, the simulations
were started with the initial diameter of 5.463 micron.
Case I was a hypothetical case in which all droplets in
the emulsion had a zero probability of breakup. The result
of the simulation is shown in Figure 8. The droplets in
Case I were found to have a uniform size close to the
maximum diameter.
The result of the simulation for Case I agrees with our
expectation. If droplets in an emulsion can only undergo
coalescence, eventually all the droplets will coalesce to
form the droplets with the maximum allowable size.
The second case tested was a hypothetical case in which
all droplets had a probability of breakup equal to one. The
35
100
90
80
C/)
~ 70
c... a Ci 60
4- a
Q) 50
0> ..r.o..
~ 40
u
L-.
~ 30'
W
m
20
10
0
0
Probability of Droplet Breakup = 0
Maximum Diameter = 18.21 micron
Minimum Diameter = 0.1 micron
Volume Fraction of Water = 0.30
3 6 9 12
Diameter of Droplets, Micron
Figure 8:
15
Drop Size Distribution for Case I
(Probability of Droplet Breakup = 0)
17
result of the simulation is shown in Figure 9. The majority
of the droplets in Case II were found to have drop sizes
close to the minimum diameter. The tail of the distribution
was the result of droplet coalescence that existed in the
system. Even though the tail of the distribution accounted
for only 20% of the total number of droplets, it accounted
for about 90% of the total volume in the system. The
droplets in Case II were log-normally distributed. The
maximum of the distribution was skewed to the minimum
diameter.
Again, the result of the simulation was as expected.
In Case II, droplet breakup was a more dominant process than
droplet coalescence. Since each of the droplets in the
emulsion underwent droplet breakup more often than
coalescence, eventually they formed droplets with smaller
diameters.
Evolution of the System Energy
The evolution of the system energy for Case I and II as
a function of the number of Monte Carlo moves are given in
Figures 10 and 11. For Case I where no droplet breakup
existed, all the droplets coalesced to reduce their surface
areas and formed a more stable system. The system reached
equilibrium at about 100,000 moves. At the equilibrium, the
droplets stopped coalescing because the large space in
between them made coalescence difficult. The evidence is
37
90
80 L
70
..(.J..).
~ 60
a L...
0
'+- 50
a
Q)
0> 2 40
c
Q) u ID 30
a.. w
co 20
10 I--
0
0
Probability of Droplet Breakup = 1
Maximum Diameter = 18.21 micron
Minimum Diameter = 0.1 micron
Volume Fraction of Water = 0.30
t---l.
3 6 9 12
Diameter of Droplets, micron
Figure 9:
Drop Size Distribution for Case II
(Probability of Droplet Breakup = 1)
15 17
w
~
E 10
()
I
<D
C> - 9
"0
co
t
.0. - 8
X
(If ...... \ Probability of Droplet Breakup = 0
<D 7 a.
0
10-
0
<D 6 .....c..:.
~
0
6). 5
10-
<D
C
W
<D 4 ()
~
::J
(f) 3
0 50000 100000 150000 200000
Number of Moves
Figure 10:
Energy of Droplets in Case I as a Function of the Number of Droplet
Moves
~
0
10.5
E
()
I 0>10 c
>.
-0
~ b9.5
..--
><
..e..n. 0> 9
a..
0
L-
0
0>8.5
..r.:.:.:..
'+- 0
~ 8
L-
0> c
UJ
~7.5
~
::::J en 7
0
Probability of Droplet Breakup = 1
50000 100000
Number of Moves
Figure 11:
150000 200000
Energy of Droplets in Case II as a Function of the Number of Droplet
Moves
from the number of droplets that had been greatly reduced
from 6084 to 234 at the equilibrium.
For Case II, droplet breakup was more dominant than
droplet coalescence. More surface areas were created at the
beginning; hence a small rise in the energy of the system
occurred. As the simulation progressed, the frequency of
droplet coalescence increased. More and more droplets with
diameters greater than the initial diameter were formed.
The energy of the system started to decrease and eventually
reached an equilibrium after 200,000 moves.
The evolution of energy in Case II at first seemed to
contradict with ones expectation. Since the simulation
produced many droplets (about 95%) with diameter less than
the initial diameter, one would expect the surface areas and
thus the energy of the system to increase instead of
decreasing. However, the effect of droplets with diameter
greater than the initial diameter could not be
underestimated.
Eventhough the droplets with diameters greater than the
initial diameter (5.463 micron) accounted for only 5% of the
total droplet population, they actually accounted for about
50% of the total droplet volume. These droplets reduced the
surface areas greatly and counteracted the effect of the
smaller droplets. In Case II, the droplets with diameters
greater than the initial diameter were found to have a
greater effect in reducing the surface areas than the
smaller droplets in increasing the surface areas.
41
Therefore, the energy of the system decreased as shown in
Figure 11.
The model predicted drop size distribution as expected
for the hypothetical Case I and II. In Case I and II, the
probability of droplet breakup was set to be zero and one,
respectively, for all the droplets in the system. For a
general case, the droplet breakup in the model is dependent
on the size of the droplet, the viscosity ratio of the
dispersed and continuous phase, and the turbulence in the
liquid phase.
Comparison of Simulated Drop Size Distributions with
Experimental Data
For general cases, comparisons of simulated drop size
distributions with experimental data are desirable.
Unfortunately, no drop size distribution data in a verticalannular
flow has been reported in the literature.
Therefore, comparisons were made with systems other than the
vertical-annular flow.
A comparison was made (Case III) with the drop size
distribution obtained by Schwarz and Bezemer (1956) which is
shown in Figure 12. The emulsion was prepared from water
and crude oil. Other data such as the viscosities and
Reynolds number were not specified. Therefore, the input
data for the simulation had to be estimated. Several
simulations with different input variables were performed.
The simulation produced a drop size distribution in good
42
-(J) (1)
Q.
0
L..
0
'+-
0
(1)
0> -ro c
(1) u
"-
(1)
,j:>. a..
w
70
60
50
40
30
20
10
0
1 4
• Schwarz and Bezemer, 1956
D Simulation
6 8 10
Diameter of the Droplets, micron
Figure 12:
12 14
Comparison of Experimentally Obtained and Simulated Drop Size
Distribution of Water in Crude Oil
agreement with the experimental data when the following
input data were used: the viscosities of the water and oil
equal to 1.0 cP, the Reynolds number equal to 2000, the
volume fraction of the water equal to 0.30.
The simulated drop size distribution which is also
given in Figure 12 compared favorably with the data obtained
by Schwarz and Bezemer (1956). Both drop size distributions
show that about 70% of the droplets have diameters less than
2 micron. Both distributions also show that only a small
fraction of droplets have diameter in the range of 5 to 15
micron.
Another comparison was made (Case IV) with the drop
size distribution obtained by Sibree (1933) using a Hurrell
mill. The Hurrell mill consists of a casing in which a
rotor composed of two discs shaped in section like a
truncated cone revolves in close proximity to a similarly
shaped stator ring. The emulsion was made from a viscous
paraffin dispersed in 1% sodium oleate solution. The sodium
oleate acted as an emulsifier. The volume fraction of the
dispersed phase was 0.50. The viscosity of the dispersed
and continuous phase, as well as the Reynolds number of the
liquid, was not specified in the data. Thus, the input data
to the simulation had to be estimated. Again, several
simulations with different input variables were performed.
The simulation produced a drop size distribution in closest
agreement with the experimental data when the viscosity
ratio of the dispersed to the continuous phase was set to
44
one, the Reynolds number was set to 10000, and the maximum
and the minimum diameters of the droplets were set to 18.21
and 1.0 micron, respectively.
The simulated drop size distribution did not compare
favorably with Sibree's data. The difference of the drop
size distribution produced by the simulation and that
obtained from the experiment could be accounted for. The
drop size distribution from Sibree's data is given in Figure
13. The simulated drop size distribution from the model is
shown in Figure 14. The difference in the two distributions
was that droplets with diameter greater than 6 micron was
present only in a very small fraction in the distribution
obtained by Sibree. The difference could be explained as
follows. Since the emulsion used by Sibree contained an
emulsifier, droplet coalescence was very small or nonexistent
in the emulsion. Therefore, the formation of
bigger droplets in the emulsion was not favored. In
contrast, our model which produced drop size distribution in
Figure 14 did not account for the presence of the
emulsifier. Therefore, droplet coalescence which formed
larger droplets were allowed in our model.
Conservation of Mass in the Simulation
The mass of the system was always conserved in the
simulations. The mass was calculated at the beginning and
at the end of the simulations. Only small truncation errors
45
~
0"1
30
r--
25
Volume Fraction of Dispersed Phase = 50%
..(./.).
Q) 20 a.
0 - L..
0
'0
Q) 15
,....-
Cro ) - ..... c
Q)
~ 10 Q) a.. r--
- 5
r--
o - Ih
o 3 6 9 12 15 17
Diameter of Droplets, micron
Figure 13:
Drop Size Distribution of Viscous Paraffin in 1% Na Oleate Solution
(Sibree, 1933)
01:>0
-.J
14 r-
12
.(..f.) 10 Q)
0.. e o 8
'0
Q)
C> .r..o. c
()} e Q) a.
6
4
2
ol:::::
o
I--
I--
Vol. Frac. of Dispersed Phase = 50%
Vis. of Water and Oil = 1.0 cP
Maximum Diameter = 18.21 micron
Minimum Diameter = 1.0 micron
Reynold Number = 10000
r--
I--
r-
__ I-I--
r- r-I~ r-
I-- I--
I--
3 6 9 12 15
Diameter of Droplets, Micron
Figure 14:
1--1--
[
17
Simulated Drop Size Distribution of Viscous Paraffin in 1% Na Oleate
Solution
were observed in all cases. The truncation errors for Case
I, II, III, and IV are given in Table I.
Table I
Truncation Error in the Simulations
Case No. # of Moves % Error
I 200,000 0.089
II 200,000 0.015
III 100,000 0.0009
IV 200,000 0.010
Prediction of the Phase Inversion
To predict the volume fraction of water at which the
wlo emulsion inverts to olw emulsion, simulations were
performed for both types of emulsions at different volume
fractions of water. Simulations are usually started with a
lower volume fraction of water. If wlo emulsion is more
stable than olw emulsion for a given volume fraction of
water, the volume fraction of water is increased and
simulation is repeated. Phase inversion occurs when olw
emulsion has a lower surface energy than wlo emulsion.
A typical run is shown in Figure 15. The viscosity of
water and oil were 1.0 and 2.5 cP, respectively. The
Reynolds number of the liquid film was 10,000. The
simulations were performed with a volume fraction of water
48
,j:::.
\0
8.5
Water-in-Oil
Vis. of Water = 1.0 cP
N
8t
E Vis. of Oil =2 .5 cP -u "- Re = 10000 Q)
c:
>.
,"0. t 7.5
0 ...-
><
Q)
E 7 :J
0 <: >.
0>
L...
~ 6.5 ......... " w Oil-in-Water
6 I I I I I I I I 1 I
0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60
Volume Fraction of Water
Figure 15:
Energy of Droplets in Water-in-Oil and Oil-in-Water Emulsion as a
Function of the Volume Fraction of Water
ranging from 0.38 to 0.56 with an increment of 0.03. At
volume fraction of water about 0.46, the olw emulsion became
more stable than that of w/o. Therefore, the model
predicted that phase inversion occurred at volume fraction
of water of 0.46 for the given flow conditions.
Effect of the Viscosity Ratio
To investigate the effect of the viscosity ratio on
phase inversion, simulations were performed at different
viscosity ratios. In these simulations, the Reynolds number
of the liquid film was constant. The result of these
simulations with Reynolds number of 100 is given in Figure
16. Viscosity ratios used were 0.01, 0.05, 0.4, 1.0, 2.5,
20, and 100. The result shows that as the viscosity ratio
decreased, the volume fraction of water at which phase
inversion occurs also decreased. Figure 16 shows that the
effect of the viscosity ratio vanished when the ratio was
greater than 100 or smaller than 0.01. Figure 16 also shows
that as the viscosity of the oil phase increased, the oil
phase became more likely to be dispersed which agrees with
the observation by Clarke and Sawistowski and Selker and
Sleicher (1965).
Effect of Turbulence
To investigate the effect of turbulence of the liquid
film on phase inversion, simulations were performed at
different Reynolds number. The results of the simulations
50
c
0
(/)
L...
Q)
>c
Q)
(/)
CO
.c a.......
CO
L...
..Q..).
-~ 0
c
0
I.J1 :;:
I-' () CO
L...
LL
Q)
E
:::J
0 >
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.01
Oil in Water
•
0.1
Re = 100 •
•
•
• Water in Oil
1 10
Viscosity Ratio of Water to Oil
Figure 16:
Dependency of Phase Inversion on Viscosity Ratio of Water to Oil
•
100
for Reynolds number of 100, 1000, and 10000 are shown in
Figure 17.
As the Reynolds number of liquid increased, the phase
inversion occurred at a higher volume fraction of water if
the viscosity ratio was less than one. The results in
Figure 17 agree with those obtained from stirred tank
experiments. Quinn and Sigloh (1963) and Selker and
Sleicher (1965) found that the volume fraction of the
dispersed phase at phase inversion increased as the stirrer
speed (Reynolds number in the liquid phase) increased.
Figure 17 also shows that when the viscosity ratio was
greater than one, the phase inversion occurred at a higher
volume fraction of water as the Reynolds number decreased.
Comparjson of the Simulation Results on Phase Inversion with
Experimental Data
One of the latest studies on phase inversion was done
by Brooks and Richmond (1993). They studied phase inversion
of water-in-oil emulsions to oil-in-water emulsions in a
stirred tank. Oils with viscosity ranging from 0.7 to 200
cP were used to investigate the effect of viscosity on phase
inversion. The stirrer speed was varied from 400 to 800 rpm
in order to investigate the effect of turbulence. The
results from their experiments were found to agree with the
results obtained from our model.
The results of the study by Brooks and Richmond (1993)
and by our model showed that when the oil viscosity
52
1Jl w
0.65
c • Re = 10000
. Q I • U) 0.6 • Re = 1000 L- a> >c • Re = 100
a> III
U) .c.co 0.55 •• a....... • co
L- a>
0.5 + Oil in Water ..... co ;: "
'0
c I 1 • 0.45 • Water in Oil
A
a>
E 0.4 ::J
~ • •
0.35
0.01 0.1 1 10 100
Viscosity Ratio of Water to Oil
Figure 17:
Dependency of Phase Inversion on the Viscosity Ratio with Reynolds
Number as a Parameter
increased, the volume fraction of water at phase inversion
decreases. By increasing the oil viscosity from 0.7 cP to
200 cP, Brooks and Richmond found that the volume fraction
of water decreased and reached a constant value of 0.15 when
the oil viscosity was above 200 cPo The simulation results
showed the same trend. The simulation predicted the volume
fraction of water to remain constant at 0.38 when the oil
viscosity was above 100 cPo The trend observed in both
studies showed that a minimum amount of water needed to be
present in wlo emulsion in order for phase inversion to
occur.
Both studies also showed that when the turbulence
increased, the volume fraction of water at phase inversion
also increased. By increasing the stirrer speed from 400 to
800 rpm, Brooks and Richmond (1993) found that the volume
fraction of water at phase inversion increased but not
significantly. Simulation results showed that the effect of
turbulence was only significant at low Reynolds number and
high oil viscosity. At high Reynolds number, the effect of
turbulence became insignificant.
54
Conclusions
CHAPTER V
CONCLUSIONS AND RECOMMENDATIONS
A unique model of phase inversion prediction has been
developed. Given the conditions in a gas well, the model
predicts the emulsion type of the liquid film on the tube
wall. The model predicts the water-wet zone in gas wells
and therefore predicts the location where corrosion is most
likely to occur.
The phase inversion prediction from our model agrees
qualitatively with the experimental results obtained by
Brooks and Richmond (1993). The model and the experimental
results of Brooks and Richmond (1993) agrees on the
following major points:
• As the oil viscosity is increased, the volume fraction of
water at phase inversion decreases and reaches a critical
value when the oil viscosity reaches a certain value.
• As the turbulence in the liquid phase is increased, the
volume fraction of water at phase inversion also
increases.
• Turbulence in the liquid phase has a less significant
effect on phase inversion than viscosity.
55
Recommendations
To gain more confidence in our model, a comparison with
the experimental data of phase inversion in the annular flow
is needed. Currently, the results from our model have not
been compared with the experimental data obtained in the
annular flow because no such study has been reported in the
literature.
A correlation can be proposed from the simulation
results and incorporated into the DREAM software. The DREAM
software is a computer software developed at Oklahoma State
University. The software is used to predict the corrosion
rate in gas wells. The present work can be incorporated
into DREAM software as follows. For any zone in a gas well,
the DREAM software can be used to provide all the necessary
properties needed as the input to the phase inversion
simulation. The physical properties needed are the
viscosity of water and oil, and the Reynolds number of the
liquid phase. The correlation can then be used to determine
whether a particular zone in a gas well is water-wet or oilwet.
56
REFERENCES
Allen, M. P., Tildesley, D. J., Computer Simulation of
Liquids; Oxford University Press: New York, 1987.
Bhatnagar, S. S. The Reversal of Phases by Electrolytes
and the Effects of Free Fatty Acids and Alkalis on
Emulsion Equilibrium. J. Chem. Soc. 1920, 119, 61.
Bradburn, J. B.; Water Production - An Index to Corrosion.
South Central NACE Meeting, Houston, Texas, 1977.
Brooks, B. W.; Richmond, H. N. Phase Inversion in Non Ionic
Surfactant-Oil-Water System-III. The Effect of the OilPhase
Viscosity on Catastrophic Inversion and the
Relationship Between the Drop Sizes Present Before and
After Catastrophic Inversion. Chem. Eng. Sci. 1994,
49(11), 1843-1853.
Choi, H. J.; Cepulis, R. L.; Lee, J. B. Carbon Dioxide
Corrosion of L-80 Grade Turbular in Flowing Oil-Brine Two
Phase Environments. Corrosion 1989, 45, 943-950.
Clarke, S. I.; Sawitowski, H. Phase Inversion of Stirred
Liquid-Liquid Dispersions Under Mass Transfer
Conditions. Trans. Instn. Chem. Engrs. 1978, 56, 50-55.
Clay, P. H. Mechanism of Emulsion Formation in Turbulent
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T-1-3, Eds.; National Association of Corrosion Engineers;
Houston, 1984, 11-58.
Collins, S. B.; Knudsen, J. G. Drop Size Distributions
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AIChe J. 1970, 16, 1072-1080.
57
Crolet, J. L.; Bonis, M. R. An Optimized Procedure for
Corrosion Testing Under CO2 and H2S Gas Pressure.
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DeWaard, C.; Milliams, D. E. Carbonic Acid Corrosion of
Steel. Corrosion, ~9'S, 31, ~ll-~S~.
Gatzke, L. K.; Hausler, R. H. A Novel Correlation of Tubing
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Hinze, J. 0. Fundamentals of the Hydrodynamic Mechanism.
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Ikeda, A.; Ueda, M.; Mukai, S. CO2 Behavior of Carbon and Cr
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Johnson, B. V.; Choi, H. J.; Green, A. S. Effects of Liquid
Wall Shear Stress on CO2 Corrosion of X-52 C-Steel in
Simulated Oilfield Production Environments. Corrosion
1991, Paper No. 573, Cincinnati, Ohio.
Karabelas, A. J. Drop Size Spectra Generated in Turbulent
Pipe Flow of Dilute Liquid/Liquid Dispersion. AIChE J.
1978, 24(1), 170-180.
Kolmogoroff, A. N. About Breaking of Drops in Turbulent
Flow. Dokl. Akad. Nauk. SSSR 1949, 66, 825.
Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall:
Eaglewood Cliffs, 1962, Chapter 8.
Lissant, K. J., Emulsions and Emulsion Technology; Marcel
Dekker: New York, 1974; Vol 6, Part 1, Chapter 1.
Liu, G. A Mathematical Model for Prediction of Downhole Gas
Well Uniform Corrosion in CO2 and H2S Containing Brines;
Ph.D. Thesis, Oklahoma State University, Dec. 1991.
Mao, M. L.; Marsden, S. S. Stability of Concentrated Crude
Oil-In-Water Emulsions as a Function of Shear Rate,
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Technol. 1977, 16, 54-59.
58
Quinn, J. A.; Sigloh, D. B. Phase Inversion in the Mixing
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18.
Robertson, C. A. Down*Hole Phase I: A Computer Model for
Predicting the Water Phase Corrosion Zone in Gas and
Condensate Wells; Master Thesis, Oklahoma State
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Sambasivam, S. A Model to Predict Oil/Water Emulsion Phase
Inversion in the Downhole Environment; Master Thesis,
Oklahoma State University, 1994.
Schwarz, N.; Bezemer, C. A New Equation for the Size
Distribution of Emulsion Particles. C. Kolloid
Zeitschrift 1956, 146, 139-144.
Selker, A. H.; Sleicher, C. A. Factors Affecting Which
Phase Will Disperse When Immicible Liquids Are Stirred
Together. Can. J. Chem. Engng. 1965, 43, 298-301.
Shinoda, K.; Kunieda, H. Encyclopedia of Emulsion
Technology; Marcel Dekker: New York, 1983; p 337
Shock, D. A.; Sudbury, J. D. Prediction of Corrosion in Oil
and Gas Wells. World Oil 1951, 133, 180-192.
Sibree, J. O. The Viscosity of Emulsions. Part II. Trans.
Faraday Soc. 1931, 27, 161-175.
Simon, R.; Poynter, W. G. Down-Hole Emulsification for
Improving Viscous Crude Production. J. Pet. Tech. 1968,
20, 1349-1353.
Smith, D. H.; Covatch, G. C.; Lim, K. H. Temperature
Dependence of Emulsion Morphologies and the Dispersion
Morphology Diagram. J. Phys. Chem. 1991, 95(3), 1463-
1466.
Taylor, G. I. Viscosity of a Fluid Containing Small Drops of
Another Fluid. Proceedings of the Royal Society 1932, 201
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Tuttle, R. N. Corrosion in Oil and Gas Production. J. Pet.
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59
Tuttle, R. N.; Hamby, T. W. Deep Wells-A Corrosion
Engineering Challenge. Material Performance 1977, 16(10),
9-12.
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60
APPENDICES
61
a
d
dcombine
k
L
N
p
v
APPENDIX A
NOMENCLATURE
maximum velocity gradient in flow field
droplet diameter
diameter of the droplet resulting from
coalescence of two droplets
initial droplet diameter
maximum diameter
minimum diameter
constant in equation 111-7
a numerical constant in equatin 11-5
number of trials in which event A occur
number of trials
probability that event A occurs (eqn. 11-7)
probability of droplet breakup (eqn. 111-6)
probability of droplet breakup which accounts
for the effect of droplet size and flow
conditions
probability of droplet breakup which accounts
for the effect of viscosity
Reynolds number of the liquid
average velocity of the liquid
62
volume of the lattice used in simulation
We Weber number
critical Weber number
new x-coordinate of droplet
initial x-coordinate of droplet
new y-coordinate of droplet
initial y-coordinate of droplet
new z-coordinate of droplet
initial z-coordinate of droplet
Greek Symbols
E
v
p
interfacial tension
turbulent stresses
local energy dissipation per unit mass
kinematic viscosity
density of liquid (eqn. II-S)
density of continuous phase
density of the liquid film
error of the Monte Carlo method (eqn. II-7)
maximum allowable droplet displacement inside
the lattice (eqn. III-2,3,4)
thickness of the liquid film (eqn. III-B)
volume fraction of the dispersed phase
63
Ilc viscosity of the continuous phase
Ild viscosity of the dispersed phase
III viscosity of the liquid film
1..0 scale of eddy at which Reynolds number is
unity
'\)0 characteristic eddy velocity
Ell E21 E3 random numbers
64
APPENDIX B
PROCEDURE FOR RUNNING THE COMPUTER CODE
The computer program for phase inversion prediction is
written in FORTRAN (Appendix C). The output from the
program is written into three files: DROP.OUT, ENERGY.OUT,
and COORD.OUT. DROP.OUT contains the equilibrium drop size
distribution. ENERGY.OUT contains the energy of the
droplets in a 100-move increment. COORD.OUT contains the
coordinates of all droplets in the lattice.
The input variables are given below:
- dmax:
dmin:
- pwat:
- visoil:
- viswat:
- reynolds:
iter:
maximum droplet diameter
minimum droplet diameter
volume fraction of water
viscosity of oil
viscosity of water
reynolds number of liquid
number of iteration (100,000 to 200,000) is
recommended.
To compile the program using RS6000 machines (located
in Engineering North 301 and 516), type the following
command:
xlf -0 executable file source file
65
where executable file is the name of the executable file
chosen by the user and source file is the name of the source
code file.
To run the program in the background (recommended),
type the following command:
nice nohup executable file &
66
C***
C
C
C
C
C
C
C
C
C
C
C
C***
APPENDIX C
COMPUTER CODE TO PREDICT PHASE INVERSION
THIS PROGRAM IS USED TO CALCULATE THE ENERGY LEVEL OF WATER-IN-OIL
AND OIL-IN-WATER EMULSIONS. THE EMULSION WITH A LOWER ENERGY IS
THE STABLE AND FAVORED EMULSION. SIMULATION STARTS WITH WATER-INOIL
EMULSION, THEN FOLLOWS BY OIL-IN-WATER EMULSION.
THE INPUT VARIABLES IN THIS PROGRAM ARE:
DMAX MAXIMUM DIAMETER IN THE EMULSION
DMIN = MINIMUM DIAMETER IN THE EMULSION
PWAT = VOLUME FRACTION OF WATER
VISOIL = VISCOSITY OF OIL
VISWAT = VISCOSITY OF WATER
ITER = NUMBER OF ITERATION
REYNOLDS = REYNOLDS NUMBER OF THE LIQUID PHASE
DOUBLEPRECISION DMAX, DMIN, SIDE, ENERGY
DOUBLEPRECISION XNEW,YNEW,ZNEW,DMEAN,FINAL_ENG1,FINAL_ENG2
DOUBLEPRECISION DIA(50000), X(50000), Y(50000), Z(50000)
DOUBLEPRECISION PHI, MASS, INI_MASS, DIST, VTOT
INTEGER N, NMOV, COUNT, FCOALESCE, FBREAK
INTEGER ITER,EMUL_TYPE
INTEGER TOT_ITER, SEED,SEED1,SEED2,PRINT_RESULT
REAL PWAT,PDISP,VISDISP,VISCONT,VISWAT,VISOIL,REYNOLDS
INTEGER INC_ENERGY, PRINT_ENERGY
C*** THERE ARE 6084 DROPLETS IN THE CUBE INITIALLY.
3
FCOALESCE = 0
FBREAK = 0
PHI = 3.141592654
PRINT ENERGY = 0
OPEN (UNIT=8, FILE
OPEN (UNIT=9, FILE
OPEN (UNIT=10,FILE
PWAT 0.62
DMAX
DMIN
VI SWAT
18.21
0.1
1.0
VISOIL 0.01
REYNOLDS = 1000.0
'DROP.OUT', STATUS = 'NEW')
'ENERGY.OUT', STATUS = 'NEW')
'COORD.OUT', STATUS = 'NEW')
67
WRITE(8,180)PWAT
180 FORMAT('VOLUME % OF THE WATER IS ',F5.3)
TOT ITER = 200000
SEED = 869696
SEEDl= 386754
SEED2= 985872
EMUL TYPE = 1
IF (EMUL_TYPE.EQ.1) THEN
ENDIF
PDISP = PWAT
VISDISP VI SWAT
VISCONT = VISOIL
2 IF (EMUL_TYPE.EQ.2) THEN
ENDIF
PDISP = 1.0 - PWAT
VISDISP VISOIL
VISCONT = VISWAT
N = 6084
INC ENERGY = 1
DMEAN = 0.3*DMAX
CALL INITIALIZE_ARRAY(DIA,X,Y,Z)
CALL CUBE_SIZE(PDISP,DMEAN,SIDE,VTOT)
CALL PLACE_DROPLET(SIDE,DMEAN,DIA,X,Y,Z)
CALL CALC_MASS(N,DIA,MASS)
INI MASS = MASS
WRITE(8,*) '&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&'
WRITE(8,l10) EMUL_TYPE
WRITE(8,120) INI_MASS
110 FORMAT('EMULSION TYPE IS ',12)
120 FORMAT('INITIAL MASS = ',E15.7)
CALL ENERGY_CONFIGURATION(DIA,ENERGY)
WRITE(8,156) ENERGY/VTOT
156 FORMAT('INITIAL ENERGY OF THE SYSTEM IS ',E15.5)
68
1
155
C
C
C
C
C
C
C
C
970
980
ITER 0
ITER ITER + 1
CALL MOVE_DROPLET(N,DIA,SIDE,X,y,Z,XNEW,YNEW,ZNEW,NMOV,SEED,
& DMAX)
CALL COALESCENCE (X,Y,Z,DIA,XNEW,YNEW,ZNEW,N,NMOV,DMAX,
& SEED1,FCOALESCE)
CALL BREAKUP(N,DIA,DMAX,DMIN,X,Y,Z,SEED2,SEED1,FBREAK,SIDE,
& FCOALESCE, DMEAN,VISDISP,VISCONT, REYNOLDS)
IF (ITER.EQ. (100*INC_ENERGY)) PRINT ENERGY=l
IF (PRINT_ENERGY.EQ.1) THEN
CALL ENERGY_CONFIGURATION (DIA, ENERGY)
WRITE(9,155) ITER, ENERGY/VTOT
FORMAT(I8,5X,E14.7)
INC ENERGY = INC ENERGY+1
PRINT ENERGY = 0
ENDIF
IF
IF
IF
IF
IF
IF
IF
IF
IF
IF
IF
IF
(ITER. EQ. 500) PRINT RESULT = 1
(ITER.EQ.1000) PRINT RESULT = 1
(ITER.EQ.5000) PRINT RESULT = 1
(ITER.EQ.10000) PRINT RESULT 1
(ITER.EQ.50000) PRINT RESULT = 1
(ITER. EQ. 70000) PRINT RESULT = 1
(ITER.EQ.100000) PRINT RESULT = 1
(ITER.EQ.150000) PRINT RESULT = 1
(ITER.EQ.200000) PRINT RESULT = 1
(ITER. EQ. 300000) PRINT RESULT = 1
(ITER.EQ.TOT_ITER) PRINT RESULT 1
(PRINT_RESULT.EQ.1) THEN
CALL CALC_MASS (N,DIA,MASS)
WRITE(8,970) MASS
FORMAT ('MASS=' , E15. 7)
DIST = 0.71
CALL SORT_DROPLET(DIA,DIST,DMIN,DMAX,ITER)
PRINT RESULT = 0
WRITE(8,980) FCOALESCE, FBREAK
FORMAT('FCOALESCE=' ,I8,2X, 'FBREAK=' ,18)
ENDIF
IF(ITER.LT.TOT ITER) THEN
69
GOTO 1
ENDIF
DO 333 COUNT = 1,50000
WRITE(10,334) X(COUNT),Y(COUNT) ,Z(COUNT)
333 CONTINUE
334 FORMAT(FI0.5,5X,FI0.5,5X,FI0.5)
130
140
IF (EMUL_TYPE.EQ.l) THEN
ENDIF
CALL ENERGY_CONFIGURATION(DIA,ENERGY)
FINAL_ENGI = ENERGY/VTOT
WRITE(8,130) FINAL_ENGI
EMUL TYPE = 2
GOTO 2
IF (EMUL_TYPE.EQ.2) THEN
ENDIF
CALL ENERGY_CONFIGURATION(DIA,ENERGY)
FINAL ENG2 = ENERGY/VTOT
WRITE(8,140) FINAL ENG2
FORMAT('ENERGY AT EQUILIBRIUM FOR WATER IN OIL
FORMAT('ENERGY AT EQUILIBRIUM FOR OIL IN WATER
IF (PWAT.LE.0.56) STOP
IF (FINAL_ENG2.LT.FINAL_ENGl) THEN
ELSE
ENDIF
STOP
END
WRITE(8,*) 'PHASE INVERSION OCCURRED'
PWAT PWAT - 0.02
GOTO 3
WRITE(8,*) 'NO PHASE INVERSION'
PWAT PWAT - 0.02
GOTO 3
, ,E15.8)
',E15.8)
C***************************************************** *****************
SUBROUTINE INITIALIZE_ARRAY(DIA,X,y,Z)
DOUBLEPRECISION DIA(50000), X(50000), Y(50000), Z(50000)
70
INTEGER COUNT
C*** THIS SUBROUTINE IS USED TO INITIALIZE ALL ARRAYS TO ZERO.
DO 10 COUNT = 1, 50000
DIA(COUNT) = 0.0
X (COUNT) 0.0
Y(COUNT) 0.0
Z(COUNT) 0.0
10 CONTINUE
RETURN
END
C *********************************************************************
SUBROUTINE CUBE_SIZE(PDISP,DMEAN,SIDE,VTOT)
DOUBLEPRECISION DMEAN,SIDE,PHI,VOL_DROP,VTOT
REAL PDISP
INTEGER TOT DROP
C*** THIS SUBROUTINE CALCULATE THE SIZE OF THE CUBE WITH 6084 DROPLETS
C*** INSIDE FOR THE GIVEN VOLUME FRACTION OF WATER.
C*** SIDE = THE SIDE OF THE CUBE, CM
TOT DROP = 6084
PHI = 3.141592654
VOL_DROP = TOT_DROP/6.0*PHI*DMEAN**3
VTOT VOL_DROP/PDISP
SIDE = VTOT**(1.0/3.0)
RETURN
END
C**********************************************************************
SUBROUTINE PLACE_DROPLET(SIDE,DMEAN,DIA,X,Y,Z)
DOUBLE PRECISION DIA(50000) ,X(50000) ,Y(50000) ,Z(50000)
DOUBLE PRECISION SIDE,DMEAN,CONST
INTEGER COUNT, NDR, I, J, K
C*** THIS SUBROUTINE PLACES DROPLETS IN THE LATTICE IN A FACE CUBIC
C CENTER CONFIGURATION.
C*** NDR = NUMBER OF DROPLET ON THE SIDE OF THE CUBE.
C*** A TOTAL OF 6084 DROPLETS ARE PLACED IN THE CUBE INITIALLY WITH
C NDR 12.
NDR 12
COUNT 1
CONST SIDE/(2.0*(NDR-1.0»
DO 10 I 1, NDR
DO 20 J = 1, NDR
DO 30 K = 1, NDR
X (COUNT) (REAL{I)-1.0)*SIDE/{NDR-1)
(REAL(J)-1.0)*SIDE/{NDR-1)
(REAL(K)-1.0)*SIDE/{NDR-1)
Y(COUNT) =
Z(COUNT) =
DIA(COUNT) = DMEAN
71
COUNT COUNT + 1
30 CONTINUE
20 CONTINUE
10 CONTINUE
DO 40 I
DO 50 J
DO 60
1, NDR-1
1, NDR-1
K 1, NDR
X (COUNT)
Y(COUNT) =
Z(COUNT) =
DIA(COUNT)
REAL(I)*SIDE/(NDR-1)-CONST
REAL(J)*SIDE/(NDR-1)-CONST
(REAL(K)-1.0)*SIDE/(NDR-1)
= DMEAN
COUNT = COUNT + 1
60 CONTINUE
50 CONTINUE
40 CONTINUE
DO 70 I 1, NDR-1
DO 80 J 1, NDR
DO 90 K 1, NDR-1
X (COUNT) REAL(I)*SIDE/(NDR-1)-CONST
Y(COUNT) = (REAL(J)-1.0)*SIDE/(NDR-1)
Z(COUNT) = REAL(K)*SIDE/(NDR-1)-CONST
DIA(COUNT) = DMEAN
COUNT = COUNT + 1
90 CONTINUE
80 CONTINUE
70 CONTINUE
DO 100 I 1, NDR
DO 110 J 1, NDR-1
DO 120 K 1, NDR-1
X (COUNT) (REAL(I)-1.0)*SIDE/(NDR-1)
Y(COUNT) = REAL(J)*SIDE/(NDR-1)-CONST
Z(COUNT) = REAL(K)*SIDE/(NDR-1)-CONST
DIA(COUNT) = DMEAN
COUNT = COUNT + 1
120 CONTINUE
110 CONTINUE
100 CONTINUE
RETURN
END
C**********************************************************************
SUBROUTINE ENERGY_CONFIGURATION (DIA,ENERGY)
DOUBLEPRECISION DIA(50000), PHI, ENERGY, SUM
INTEGER COUNT
REAL SURF TENS
C*** THIS SUBROUTINE CALCULATES THE ENERGY OF THE DROPLETS IN THE
C LATTICE.
SUM 0.0
PHI 3.141592654
72
SURF TENS = 30
DO 10 COUNT = 1,50000
SUM = SUM + DIA(COUNT) **2
10 CONTINUE
ENERGY
RETURN
END
PHI*SUM*SURF TENS
C**********************************************************************
SUBROUTINE MOVE_DROPLET(N,DIA,SIDE,X,Y,Z,XNEW,YNEW,ZNEW,NMOV,
& SEED,DMAX)
DOUBLEPRECISION DIA(50000) ,SIDE, X(50000),Y(50000) ,Z(50000)
DOUBLEPRECISION XNEW,YNEW,ZNEW,DRMAX,DMAX
REAL J J, RANDX
INTEGER NMOV,SEED,N
C*** THIS SUBROUTINE MOVES A DROPLET RANDOMLY INSIDE THE LATTICE.
C IF THE DROPLET MOVES OUT FROM THE LATTICE, IT IS MOVED BACK INSIDE
C THE LATTICE.
10 NMOV = 0
JJ = 0.0
CALL RANDOM(SEED,RANDX)
JJ = N*RANDX
NMOV = ANINT(JJ)
IF (NMOV.EQ.O) GOTO 10
IF (DIA(NMOV) .EQ.O.O) GOTO 10
C*** MOVE DROPLET NMOV RANDOMLY.
DRMAX = O.l*DMAX
CALL RANDOM(SEED,RANDX)
XNEW = X(NMOV) + (2.0*RANDX-1.0)*DRMAX
CALL RANDOM(SEED,RANDX)
YNEW = Y(NMOV) + (2.0*RANDX-1.0)*DRMAX
CALL RANDOM(SEED,RANDX)
ZNEW = Z(NMOV) + (2.0*RANDX-1.0)*DRMAX
IF (XNEW.LT.O) XNEW = -XNEW
IF (XNEW.GT.SIDE) XNEW = 2*SIDE-XNEW
IF (YNEW.LT.O) YNEW = -YNEW
IF (YNEW.GT.SIDE) YNEW = 2*SIDE-YNEW
IF (ZNEW.LT.O) ZNEW = -ZNEW
IF (ZNEW.GT.SIDE) ZNEW = 2*SIDE-ZNEW
RETURN
END
C**********************************************************************
SUBROUTINE COALESCENCE (X,Y,Z,DIA,XNEW,YNEW,ZNEW,N,
& NMOV,DMAX,SEED1, FCOALESCE)
DOUBLE PRECISION X(50000), Y(50000) , Z(50000), DIA(50000)
DOUBLEPRECISION XNEW, YNEW, ZNEW, RCRIT, D2, DMAX
DOUBLE PRECISION DBREAK
73
INTEGER N, NMOV, JJ
INTEGER NEIGH, SEED1,FCOALESCE,COA
REAL RANDX1
C*** THIS SUBROUTINE CHECKS IF A DROPLET COALESCE WITH ITS NEIGHBOUR.
C IF THE DROPLET COALESCES WITH ANOTHER, THE DIAMETER OF THE NEW
C DROPLET IS CALCULATED.
NEIGH 0
C*** CHECK IF DROPLET NMOV WILL COALESCE WITH ITS NEIGHBOUR.
DO 10 JJ = 1,N
IF (JJ.EQ.NMOV) GOTO 10
IF (DIA(JJ) .EQ.O.O) GOTO 10
RCRIT = 0.3*(DIA(NMOV)+DIA(JJ»
IF (ABS(XNEW-X(JJ» .GT.RCRIT) GOTO 10
IF (ABS(YNEW-Y(JJ» .GT.RCRIT) GOTO 10
IF (ABS(ZNEW-Z(JJ» .GT.RCRIT) GOTO 10
NEIGH = JJ
GOTO 20
10 CONTINUE
C*** DROPLET NMOV DID NOT COALESCE WITH ITS NEIGHBOUR, ENERGY REMAIN THE
C SAME. PLACE THE DROPLET TO ITS NEW POSITION.
x (NMOV) XNEW
Y (NMOV) YNEW
Z(NMOV) ZNEW
GOTO 30
C*** DROPLET NMOV COALESCED WITH ITS NEIGHBOUR, CHECK IF THE RESULTING
C*** SIZE EXCEEDS THE MAXIMUM DIAMETER.
20 D2 (DIA(NMOV)**3+DIA(NEIGH)**3)**(1.0/3.0)
IF (D2.LE.DMAX) THEN
DIA(NMOV) = 0.0
X (NMOV) 0.0
Y(NMOV) = 0.0
Z(NMOV) = 0.0
DIA(NEIGH) = D2
C*** REPLACE THE DROPLET DELETED WITH THE LAST DROPLET FROM THE ARRAY.
IF (NMOV.NE.N) THEN
DIA(NMOV) = DIA(N)
X (NMOV) X (N)
Y (NMOV) Y (N)
Z (NMOV) Z (N)
C*** DELETE THE LAST DROPLET FROM THE ARRAY.
74
ENDIF
N = N-l
FCOALESCE
ENDIF
DIA(N) = 0.0
x (N) 0.0
Y(N) 0.0
Z(N) 0.0
FCOALESCE+l
C*** IF THE DIAMETER RESULTING FROM COALESCENCE IS GREATER THAN THE
C*** MAXIMUM DIAMETER, THE DROPLET BREAKS INTO TWO, ONE OF THEM WITH
C*** DIAMETER EQUAL TO MAXIMUM DIAMETER.
IF (D2.GT.DMAX) THEN
DIA(NEIGH)
DIA(NMOV)
DMAX
(D2**3-DMAX**3)**(l.0/3.0)
ENDIF
30 RETURN
END
C**********************************************************************
SUBROUTINE BREAKUP(N,DIA,DMAX,DMIN,X,Y,Z,SEED2,
& SEED1, FBREAK,SIDE, FCOALESCE, DMEAN,VISDISP,VISCONT, REYNOLD S)
INTEGER N,MM,NBRE
INTEGER SEED1,SEED2,FBREAK,FCOALESCE,BRE,NNEW
REAL KK,RANDX2,VISDISP,VISCONT,CONSTANT,REYNOLDS
DOUBLEPRECISION DIA(50000),X(50000),Y(50000) ,Z(50000)
DOUBLEPRECISION DB, DMAX, DMIN
DOUBLEPRECISION PROB, PROB1, PROB2, SIDE
DOUBLEPRECISION DMEAN,RRCRIT,XXNEW,YYNEW,ZZNEW
C*** THIS SUBROUTINE CALCULATES THE PROBABILITY OF DROPLET BREAKUP.
C IF THE PROBABILITY OF BREAKUP IS HIGHER THAN A RANDOM NUMBER, THE
C DROPLET IS ALLOWED TO BREAK.
120 KK = 0.0
NBRE = 0
CALL RANDOM2(SEED2,RANDX2)
KK = N*RANDX2
NBRE = ANINT(KK)
IF (NBRE.EQ.O) GOTO 120
IF (DIA(NBRE) .EQ.O.O) GOTO 120
CONSTANT EXP(-lOOO.O/REYNOLDS)
PROBl = EXP(-(VISDISP/VISCONT»
75
PROB2
PROB
CONSTANT *((DIA(NBRE)-DMIN)/(DMAX-DMIN))
0.5* (PROB1+PROB2)
DB = DIA(NBRE)*0.5**(1.0/3.0)
CALL RANDOM2(SEED2,RANDX2)
BRE = 0
C*** IF THE SIZE OF THE DROPLET RESULTING FROM THE BREAKUP IS LESS THAN
C*** THE MINIMUM DIAMETER, DROPLET BREAKUP IS REJECTED.
IF(RANDX2.LT.PROB) THEN
IF (DB.LT.DMIN) THEN
BRE 0
ELSE
BRE 1
ENDIF
ELSE
BRE 0
ENDIF
IF (BRE.EQ.l) THEN
DIA(NBRE) = DB
C*** THE NEW DROPLET FORMED FROM THE BREAKUP IS PLACED AT THE
C*** END OF THE ARRAY.
NNEW = N+l
20 DIA(NNEW) = DB
C*** THE DROPLET CREATED FROM BREAKUP IS PLACED RANDOMLY INSIDE THE
C*** LATTICE.
C***
C
CALL RANDOM2(SEED2,RANDX2)
X (NNEW)= RANDX2*SIDE
XXNEW = X(NNEW)
CALL RANDOM2(SEED2,RANDX2)
Y(NNEW)= RANDX2*SIDE
YYNEW = Y(NNEW)
CALL RANDOM2(SEED2,RANDX2)
Z(NNEW)= RANDX2*SIDE
ZZNEW = Z(NNEW)
FBREAK = FBREAK+l
N = N + 1
THE DROPLET RESULTING FROM THE BREAKUP IS CHECKED FOR THE
POSSIBILITY OF COALESCENCE WHEN PLACED RANDOMLY INSIDE THE
76
C***
&
ENDIF
RETURN
END
LATTICE
CALL COALESCENCE(X,Y,Z,DIA,XXNEW,YYNEW,ZZNEW,N,NNEW,
DMAX,SEED1,FCOALESCE)
C**********************************************************************
SUBROUTINE CALC_MASS (N,DIA,MASS)
DOUBLEPRECISION DIA(50000), MASS, SUM, PHI
INTEGER N, COUNT
REAL DENSITY
C*** THIS SUBROUTINE CALCULATES THE TOTAL MASS OF DROPLETS IN THE
C EMULSION.
PHI = 3.141592654
DENSITY = 1.0
SUM = 0
MASS = 0.0
DO 100 COUNT = 1,N
SUM = SUM + DIA(COUNT)**3
100 CONTINUE
MASS = DENSITY*SUM*PHI/6.0
RETURN
END
C**********************************************************************
C***
SUBROUTINE SORT_DROPLET(DIA,DIST,DMIN,DMAX,ITER)
DOUBLEPRECISION DIA(50000) ,DIST,DMIN,DMAX
INTEGER RANGE1,RANGE2,RANGE3,RANGE4,RANGE5,RANGE6,RANGE7
INTEGER RANGE8,RANGE9,RANG10, RANGEMIN,RANGEMAX, TOTDROP,ITER
INTEGER RANG11,RANG12,RANG13,RANG14,RANG15,RANG16,RANG17,RANG18
INTEGER RANG19,RANG20,RANG21,RANG22,RANG23,RANG24,RANG25,RANG26
INTEGER RANG27,RANG28,RANG29,RANG30,RANG31,RANG32,RANG33,RANG34
INTEGER RANG35,RANG36,RANG37,RANG38,RANG39,RANG40
INTEGER COUNT
THIS SUBROUTINE SORTS THE DROPLETS IN A CERTAIN SIZE RANGE.
RANGEl 0
RANGE2 0
RANGE 3 0
RANGE 4 0
RANGE 5 0
RANGE 6 0
77
RANGE 7 0
RANGE 8 0
RANGE 9 0
RANG 1 0 0
RANG 11 0
RANG12 0
RANG 13 0
RANG14 0
RANG15 0
RANG16 0
RANG17 0
RANG18 0
RANG19 0
RANG20 0
RANG21 0
RANG22 0
RANG23 0
RANG24 0
RANG25 0
RANG26 0
RANG27 0
RANG28 0
RANG29 0
RANG 3 0 0
RANG31 0
RANG32 0
RANG33 0
RANG34 0
RANG35 0
RANG36 0
RANG37 0
RANG38 0
RANG39 0
RANG40 0
RANGEMIN 0
RANGEMAX 0
TOTDROP = 0
DO 10 COUNT = 1, 50000
IF (DIA(COUNT) .NE.O.O) THEN
IF (DIA(COUNT) .NE.O) TOTDROP = TOTDROP+1
IF (DIA(COUNT) .LT.DMIN) RANGEMIN= RANGEMIN+1
IF (DIA(COUNT) .LT. (l.*DIST» RANGE1= RANGE1+1
IF (DIA(COUNT) .LT. (2.*DIST» RANGE2= RANGE2+1
IF (DIA(COUNT) .LT. (3.*DIST» RANGE3= RANGE3+1
IF (DIA (COUNT) . LT. (4 . *DIST) ) RANGE 4 = RANGE 4 +1
IF (DIA (COUNT) . LT. (5. *DIST) ) RANGE 5 = RANGE5+1
IF (DIA (COUNT) . LT. (6 . *DIST) ) RANGE 6 = RANGE 6 +1
IF (DIA (COUNT) . LT. (7. *DIST) ) RANGE 7 = RANGE7+1
IF (DIA (COUNT) . LT. (8 . *DIST) ) RANGE 8 = RANGE 8 +1
IF (DIA (COUNT) . LT. (9. *DIST) ) RANGE9= RANGE9+1
IF (DIA(COUNT) .LT. (10.*DIST»RANG10= RANG10+1
IF (DIA(COUNT) .LT. (11.*DIST»RANG11= RANGl1+1
78
ENDIF
IF (DIA(COUNT) .LT. (12.*DIST»RANG12= RANG12+1
IF (DIA(COUNT) .LT. (13.*DIST»RANG13= RANG13+1
IF (DIA(COUNT) .LT. (14.*DIST»RANG14= RANG14+1
IF (DIA(COUNT) .LT. (15.*DIST»RANG15= RANG15+1
IF (DIA(COUNT) .LT. (16.*DIST»RANG16= RANG16+1
IF (DIA(COUNT) .LT. (17.*DIST»RANG17= RANG17+1
IF (DIA(COUNT) .LT. (18.*DIST»RANG18= RANG18+1
IF (DIA(COUNT) .LT. (19.*DIST»RANG19= RANG19+1
IF (DIA(COUNT) .LT. (20.*DIST»RANG20= RANG20+1
IF (DIA(COUNT) .LT. (21.*DIST»RANG21= RANG21+1
IF (DIA(COUNT) .LT. (22.*DIST»RANG22= RANG22+1
IF (DIA(COUNT) .LT. (23.*DIST»RANG23= RANG23+1
IF (DIA(COUNT) .LT. (24.*DIST»RANG24= RANG24+1
IF (DIA(COUNT) .LT. (25.*DIST»RANG25= RANG25+1
IF (DIA(COUNT) .LT. (26.*DIST»RANG26= RANG26+1
IF (DIA(COUNT) .LT. (27.*DIST»RANG27= RANG27+1
IF (DIA(COUNT) .LT. (28.*DIST»RANG28= RANG28+1
IF (DIA(COUNT) .LT. (29.*DIST»RANG29= RANG29+1
IF (DIA(COUNT) .LT. (30.*DIST»RANG30= RANG30+1
IF (DIA(COUNT) .LT. (31.*DIST»RANG31= RANG31+1
IF (DIA(COUNT) .LT. (32.*DIST»RANG32= RANG32+1
IF (DIA(COUNT) .LT. (33.*DIST»RANG33= RANG33+1
IF (DIA(COUNT) .LT. (34.*DIST»RANG34= RANG34+1
IF (DIA(COUNT) .LT. (35.*DIST»RANG35= RANG35+1
IF (DIA(COUNT) .LT. (36.*DIST»RANG36= RANG36+1
IF (DIA(COUNT) .LT. (37.*DIST»RANG37= RANG37+1
IF (DIA(COUNT) .LT. (38.*DIST»RANG38= RANG38+1
IF (DIA(COUNT) .LT. (39.*DIST»RANG39= RANG39+1
IF (DIA(COUNT) .LT. (40.*DIST»RANG40= RANG40+1
IF (DIA(COUNT) .GT.DMAX) RANGEMAX = RANGEMAX+1
10 CONTINUE
WRITE(8,*) '**************************************************'
WRITE(8,18) ITER
WRITE(8,19) DMIN,RANGEMIN
WRITE (8,199) (1. *DIST), RANGEl
WRITE(8,20) (1.*DIST), (2.*DIST), (RANGE2-RANGE1)
WRITE(8,20) (2.*DIST), (3.*DIST), (RANGE3-RANGE2)
WRITE(8,20) (3.*DIST), (4.*DIST), (RANGE4-RANGE3)
WRITE(8,20) (4.*DIST), (5.*DIST), (RANGE5-RANGE4)
WRITE(8,20) (5.*DIST), (6.*DIST), (RANGE6-RANGE5)
WRITE(8,20) (6.*DIST), (7.*DIST), (RANGE7-RANGE6)
WRITE(8,20) (7.*DIST), (8.*DIST), (RANGE8-RANGE7)
WRITE(8,20) (8.*DIST), (9.*DIST), (RANGE9-RANGE8)
WRITE(8,20) (9.*DIST), (10.*DIST), (RANGIO-RANGE9)
WRITE (8,20) (10. *DIST) , (11. *DIST) , (RANG11-RANG10)
WRITE(8,20) (11.*DIST) I (12.*DIST), (RANG12-RANG11)
WRITE (8,20) (12. *DIST) I (13. *DIST) , (RANG13-RANG12)
WRITE(8,20) (13.*DIST), (14.*DIST) I (RANG14-RANG13)
WRITE(8,20) (14.*DIST) I (15.*DIST), (RANG15-RANG14)
WRITE(8,20) (15.*DIST) I (16.*DIST), (RANG16-RANG15)
WRITE(8,20) (16.*DIST), (17.*DIST), (RANG17-RANG16)
79
WRITE(8,20) (17.*DIST), (18.*DIST), (RANG18-RANG17)
WRITE(8,20) (18.*DIST), (19.*DIST), (RANG19-RANG18)
WRITE(8,20) (19.*DIST), (20.*DIST), (RANG20-RANG19)
WRITE(8,20) (20.*DIST), (21.*DIST), (RANG21-RANG20)
WRITE(8,20) (21.*DIST), (22.*DIST), (RANG22-RANG21)
WRITE(8,20) (22.*DIST), (23.*DIST), (RANG23-RANG22)
WRITE(8,20) (23.*DIST), (24.*DIST), (RANG24-RANG23)
WRITE(8,20) (24.*DIST), (25.*DIST), (RANG25-RANG24)
WRITE(8,20) (25.*DIST), (26.*DIST), (RANG26-RANG25)
WRITE(8,20) (26.*DIST), (27.*DIST), (RANG27-RANG26)
WRITE(8,20) (27.*DIST), (28.*DIST), (RANG28-RANG27)
WRITE(8,20) (28.*DIST), (29.*DIST), (RANG29-RANG28)
WRITE(8,20) (29.*DIST), (30.*DIST), (RANG30-RANG29)
WRITE(8,20) (30.*DIST), (31.*DIST), (RANG31-RANG30)
WRITE(8,20) (31.*DIST), (32.*DIST), (RANG32-RANG31)
WRITE(8,20) (32.*DIST), (33.*DIST), (RANG33-RANG32)
WRITE(8,20) (33.*DIST), (34.*DIST), (RANG34-RANG33)
WRITE(8,20) (34.*DIST), (35.*DIST), (RANG35-RANG34)
WRITE(8,20) (35.*DIST), (36.*DIST), (RANG36-RANG35)
WRITE(8,20) (36.*DIST), (37.*DIST), (RANG37-RANG36)
WRITE(8,20) (37.*DIST), (38.*DIST), (RANG38-RANG37)
WRITE(8,20) (38.*DIST), (39.*DIST), (RANG39-RANG38)
WRITE(8,20) (39.*DIST), (40.*DIST), (RANG40-RANG39)
WRITE(8,31) DMAX,RANGEMAX
WRITE(8,30) TOTDROP
18 FORMAT('ITERATION=' ,I8)
19 FORMAT(14X,'< ',F15.5,'=',I8)
199 FORMAT('O',14X,'-',F15.5,'=',I8)
20 FORMAT(F15.5,'-',F15.5,'=',I8)
30 FORMAT('TOTAL DROPLET LEFT = " I8)
31F FORMAT (14X, '> ',F15.5,'=',I8)
RETURN
END
C**********************************************************************
SUBROUTINE RANDOM (SEED, RANDX)
INTEGER SEED
REAL RANDX
SEED = 2045*SEED + 1
SEED = SEED - (SEED/1048576)*1048576
RANDX = REAL(SEED + 1) / 1048577.0
RETURN
END
C**********************************************************************
SUBROUTINE RANDOM 1 (SEED1, RANDX1)
INTEGER SEED1
REAL RANDX1
80
SEED1 = 2045*SEED1 + 1
SEED1 = SEED1 - (SEED1/1048576)*1048576
RANDX1 = REAL(SEED1 + 1) / 1048577.0
RETURN
END
C***************************************************** *****************
SUBROUTINE RANDOM2(SEED2, RANDX2)
INTEGER SEED2
REAL RANDX2
SEED2 = 2045*SEED2 + 1
SEED2 = SEED2 - (SEED2/1048576)*1048576
RANDX2 = REAL(SEED2 + 1) / 1048577.0
RETURN
END
81
APPENDIX D
Sample Calculations of Droplet Volume and Surface Area
The drop size distribution for Case II after 50,000 and
100,000 moves are given below:
Dia. (micron) Ave.Dia Move = Move =
(micron) 50,000 100,000
0-0.71 0.355 4926 22414
0.71-1.42 1. 065 4430 6720
1.42-2.13 1. 775 2207 2215
2.13-2.84 2.485 1644 1277
2.84-3.55 3.195 832 708
3.55-4.26 3.905 571 474
4.26-4.97 4.615 436 320
4.97-5.68 5.325 348 271
5.68-6.39 6.035 280 196
6.39-7.10 6.745 208 170
7.10-7.81 7.455 162 136
7.81-8.52 8.165 122 133
8.52-9.23 8.875 96 105
9.23-9.94 9.585 75 77
9.94-10.65 10.295 63 46
10.65-11.36 11.005 37 42
11.36-12.07 11.715 35 33
12.07-12.78 12.425 28 23
12.78-13.49 13.135 18 22
13.49-14.20 13.845 13 15
14.20-14.91 14.555 6 19
14.91-15.62 15.265 6 8
15.62-16.33 15.975 3 3
16.33-17.04 16.685 4 5
17.04-17.75 17.395 2 2
17-75-18.46 18.105 3 7
Total 16555 35441
Droplets
82
The simulations were started with 6084 droplets which had
initial diameter of 5.463 micron. So, the initial diameter
was in the range of 4.97 - 5.68.
The initial volume of droplets
= 6084 x n/6 x (5.463}3 = 519,376
The initial surface area of droplets
= 6084 x n x (5.463}2 = 570,429
Note: The average diameter was used in the following
approximater calculations.
At 50,000 Moves:
1. Number of droplets with diameter less than 5.68 micron =
15,394/16,555 x 100% = 93%
Number of droplets with diameter greater than 5.68
micron = 7%
2. Volume of droplets with diameter less than 5.68
micron = n/6 (4926xO.3553 + 4430x1.0653 +2207x1.7753 +
1644x2.4853 +832x3.1953 +571x3.9053 +436x4.6153
+348x5.3253)
= 199,680
= 199,680/519,376 x 100% = 38%
Volume of droplets with diameter greater than 5.68
micron = 62%
3. Surface area of droplets with diameter less than 5.68
micron = n ( 4926xO.3552 + 4430x1.0652 +2207x1.7752
+1644x2.4852 +832x3.1952 + 571x3.9052 +436x46152
+348x5 .3252 )
= 185,684
Surface area of droplets with diameter greater than 5.68
micron
= n ( 280x6. 0352 +208x6. 7452 +162+7.4552
+122x8.1652 +96x8.8752 +75x9.5852 +63x10.2952
+37x11.0052 + 35x11.7152 +28x12.4252 +18x13.1352
+13x13.8452 +6x14.5552 +6x15.2652 +3x15.9752
+4x16.6852 +2x17.3752 + 3x18.1052 )
= 248,795
83
Total surface area at 50,000 moves
= 185,684+248,795 = 434,479 < 570,429
At 100,000 moves
1. Number of droplets with diameter less than 5.68 micron =
34,399/35,441 x 100% = 97%
Number of droplets with diameter greater than 5.68
micron = 3%
2. Volume of droplets with diameter less than 5.68 micron
n/6 (22414xO.3553 +6720x1.0653 + 2215x1.7753 +
1277x2.4853 +708x3.1953 +474x3.9053 +320x4.6153
+ 271x5.3253 ) = 164,792
= 164,792/519,376 x 100% = 32%
Volume of droplets with diameter greater than 5.68
micron = 68%
3. Surface area of droplets with diameter less than 5.68
micron = n (22,414xO.3552 + 6720x1.0652 +2215x1.775 +
1277x2.4852 + 708x3.1952 + 474x3.9052 +
320x4.6152 + 271x5.3252 ) = 154,884
Surface area of droplets with diameter greater than 5.68
micron = n (196x6.0352 + 170x6.7452 + 136x7.4552 +
133x8.1652 + 105x8.8752 + 77x9.5852 + 46x10.2952
+ 42x11.0052 + 33x11.7152 + 23x12.4252 +
22x13.1352 + 15x13.8452 + 19x14.5552 + 8x15.2652
+ 3x15.9752 + 5x16.6852 + 2x17.3952 + 7x18.1052 )
258,558
Total surface area at 100,000 moves
= 154,884+258,558 = 413,442 < 570,429
84
Thesis:
Major Field:
Biographical:
VITA
Justin Juswandi
Candidate for the Degree of
Master of Science
SIMULATION OF THE OIL-WATER INVERSION
PROCESSES
Chemical Engineering
Personal Data: Born in Medan, Indonesia, on September
2, 1970, the son of Jasin and Betty.
Education: Graduated from Sutomo High School, Medan,
Indonesia, in May 1989; received Bachelor of
Science degree in Chemical Engineering from
Oklahoma State University, Stillwater, Oklahoma in
May 1993. Completed the requirements for the
Master of Science degree in Chemical Engineering
at Oklahoma State University in May 1995.
Experience: Undergraduate and graduate research
assistant, School of Chemical Engineering,
Oklahoma State University, January, 1993 to May,
1995. Undergraduate research assistant,
Department of Chemistry, Oklahoma State
University, January ,1991 to May, 1991.
Professional Membership: American Institute of
Chemical Engineers.