droplet interactions during combustion of unsupported
TRANSCRIPT
Droplet Interactions during Combustion of Unsupported Droplet Clusters
In Microgravity:
Numerical Study of Droplet Interactions at Low Reynolds Number
A Thesis
Submitted to the Faculty
of
Drexel University
by
Irina N. Ciobanescu Husanu
in partial fulfillment of the
requirements for the degree
of
Doctor of Philosophy
December 2005
ii
DEDICATIONS
In the memory of my father,
Dedicated professor Dr. Mihail-Serban Voiculescu, whose life and career were my role models
for each moment of my life. His unconditional love and his continuing support and care were, are,
and will ever be my guiding light, even if now is watching over me from Heaven. You will be
forever alive in my heart.
To my mother,
Dr. Doina Voiculescu, a wonderful doctor and professor and loving and devoted mother, for
always encouraging me to pursue academic success. You always have been the shoulder to lean
on or to cry on, and your courage, determination and creativity motivated me to keep going
forward.
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ACKNOWLEDGEMENTS
I would like to express my deep gratitude to my advisors Dr. Gary A. Ruff and
Dr. Mun Y. Choi for their guidance and assistance throughout the duration of this
research. Their encouragement and support were critical to the success of this project. I
would like to extend my special thanks to Dr. Ruff, who always was there for me when I
needed and whose valuable advice and suggestions helped me performing the research.
I would like to thank all of my committee members, including Dr. Gary A.Ruff,
Dr. Mun Y. Choi, Dr. Nicholas Cernansky, Dr.Howard Pearlman, Dr. Alan Lau, and Dr.
Alexander Dolgopolsky for encouraging me and offering precious suggestions and
feedback.
My special appreciation is extended to NIST team for publicly releasing the Fire
Dynamic Simulator code that was of tremendous help in developing the present
investigation, along with their continuous support every time I needed.
Special thanks go to NASA Drop Tower and Microgravity Combustion Research
Laboratory teams from NASA Glenn Research Center, for their help and support in
design and fabrication of the equipment as well as for their important suggestions and
advice.
I am exceptionally grateful to my family, my husband, my daughters and my
stepson for their remarkable patience, encouragements and for always believing in me.
Special appreciation to my daughters Ana and Diana for their contribution to data
collection for my research, and for cooking for our entire family while I was researching.
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The support of the NASA Office of Biological and Physical Research,
Microgravity Combustion Research Program (Grant Number NCC3-847) is greatly
appreciated.
All my thanks to those who helped, encouraged and supported me during these
past years and whose names are not mentioned here, to all my friends and relatives.
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TABLE OF CONTENTS
Page
LIST OF TABLES........................................................................................................................viii
LIST OF FIGURES ......................................................................................................................... x
ABSTRACT................................................................................................................................... xv
CHAPTER 1: INTRODUCTION .................................................................................................... 1
CHAPTER 2: BACKGROUND ...................................................................................................... 4
2.1 Studies of Single and Multi-Component Isolated Droplets....................................... 4
2.2 Studies of Arrays of Droplets and Streams ............................................................. 11
2.2.1 Theoretical Analysis ..................................................................................................... 11
2.2.2 Experiments Involving Arrays of Drops....................................................................... 17
2.2.3 Radiative Heat Loss Studies on Fuel Droplet Combustion........................................... 19
2.3 Summary ................................................................................................................. 20
CHAPTER 3: DROPLET INTERACTIONS AT LOW REYNOLDS NUMBERS ..................... 23
3.1 Point Source Method............................................................................................... 23
3.1.1 Description of the Method ............................................................................................ 23
3.1.2 Advantages and Disadvantages of the Method ............................................................. 26
3.2. Extension of the PSM for Non-symmetric Arrays ................................................. 29
3.2.1 Method Description ...................................................................................................... 30
3.2.2 Results of the PSM for Non-symmetric Arrays ............................................................ 31
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CHAPTER 4: FORMULATION OF THE NUMERICAL SIMULATION.................................. 37
4.1 Overview ................................................................................................................. 37
4.2 Assumptions, Justifications and Implications ......................................................... 38
4.2.1 General Assumptions.................................................................................................... 38
4.2.2 Water Absorption.......................................................................................................... 40
4.2.3 Effect of Reynolds number on Interacting Droplets: Near Zero Re
Number .................................................................................................................................. 41
4.2.4 Internal Circulation within the Droplet......................................................................... 41
4.3 Numerical Solution ................................................................................................. 42
4.3.1 General Equations......................................................................................................... 44
4.3.2 Combustion Model ....................................................................................................... 47
4.3.3 Radiation Model in the Gas-phase................................................................................ 49
4.3.4 Numerical Algorithm.................................................................................................... 51
4.3.5 Problem Geometry........................................................................................................ 53
4.4 Burning Rate Calculation ........................................................................................ 54
4.5 Ignition Characteristics ........................................................................................... 57
4.6 Limitations of the Model and Error Margins .......................................................... 59
CHAPTER 5: RESULTS OF THE DIRECT NUMERICAL SIMULATION .............................. 62
5.1 Overview ................................................................................................................. 62
5.2 Isolated Droplet Combustion .................................................................................. 63
5.2.1 Grid Characteristics and Sensitivity Simulations.......................................................... 64
5.2.2 Validation Tests ............................................................................................................ 70
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5.3 Combustion of Droplet Arrays................................................................................ 83
5.3.1 Two Droplet Arrays: Modified Version of FDS_v3..................................................... 84
5.3.2 Two Droplet Arrays: FDS_v4....................................................................................... 87
5.3.3 Three Droplet Arrays: FDS_v4................................................................................... 100
CHAPTER 6: EXPERIMENTAL INVESTIGATION................................................................ 108
6.1 Operating Parameters ............................................................................................ 111
6.2 Description of the Test Hardware ......................................................................... 112
6.2.1 Microgravity Environment ......................................................................................... 112
6.2.2. Mechanical Design .................................................................................................... 112
6.3. Test Procedures .................................................................................................... 121
CHAPTER 7: SUMMARY AND RECOMMENDATIONS ...................................................... 123
LIST OF REFERENCES............................................................................................................. 132
APPENDIX A: Droplet Mass Burning Rate And Burning Rate Constant .................................. 138
APPENDIX B: Sample Input Files For FDS............................................................................... 139
APPENDIX C: Burning Rate Calculation Samples..................................................................... 149
APPENDIX D: Experimental Set-Up.......................................................................................... 152
APPENDIX E: Nomenclature...................................................................................................... 155
CURRICULUM VITAE.............................................................................................................. 159
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LIST OF TABLES
Table 1: Curve fitting coefficients for methanol (for temperatures in centigrade scale) ............................................................................................................................ 39
Table 2: Reaction rate parameters for methanol ............................................................................ 48
Table 3: Properties for fuel and products of combustion for methanol air reaction........................................................................................................................................... 48
Table 4 Minimum ignition energy for various pure methanol droplet diameters ........................................................................................................................................ 58
Table 5: Grid sensitivity analysis as a function of temperature for a 2 mm droplet burning in air and a domain size of 64mm/side................................................................. 66
Table 6 Domain size dependence for a 2mm droplet burning in air, using a grid cell size of 1mm................................................................................................................... 67
Table 7 Grid sensitivity analysis for a 2mm droplet burning in air ............................................... 69
Table 8 Validation analysis for a 2.2 mm burning droplet in air at 10%, 15%, 21%, 35%, 50% and 75% oxygen ........................................................................................ 71
Table 9 Variation of burning rates with time, case with radiation................................................. 77
Table 10 Burning rates for a 2mm droplet with and without radiation.......................................... 79
Table 11 Correction factors table: Comparison between Point Source Method and numerical data for two-droplet symmetric arrays. ..................................................... 85
Table 12 Correction factors table: Comparison between Point Source Method and numerical data for two-droplet asymmetric arrays, where the droplet diameter ratio is 2. ............................................................................................................. 85
Table 13 Correction factors and burning rates for symmetric two-droplet arrays; the isolated droplet mass burning rate is 2.815E-07 [kg/s] ................................................ 88
Table 14 Correction factors and burning rates for asymmetric two droplet arrays having droplet diameters’ ratio of 2 .................................................................................... 95
Table 15 Correction factors for three methanol droplet arrays of identical or different droplet sizes .............................................................................................................. 101
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Table 16 Burning rate calculations for a single 2 mm droplet burning in air ................................................................................................................................................. 149
Table 17 Mechanical Equipment List .......................................................................................... 152
Table 18 Experiment Controls Timeline...................................................................................... 153
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LIST OF FIGURES
Figure 1 PSM Droplet Coordinate System .................................................................................... 24
Figure 2 Comparison of MOI and PSM results for three-drop linear array (Annamalai and Ryan, 1993) ......................................................................................................... 27
Figure 3 Comparison of MOI and PSM results for a five-drop array (Annamalai and Ryan, 1993) ......................................................................................................... 28
Figure 4: Comparison of MOI and PSM results for a seven-drop array (Annamalai and Ryan, 1993) ......................................................................................................... 28
Figure 5 Ratio of the non-dimensional droplet vaporization rate in a stream to an isolated droplet – numerical solution (Leiroz and Rangel, 1994) .............................................................................................................................................. 29
Figure 6 Comparison between PSM and experimental data obtained in normal gravity, for a three droplet array performed by Liu. (2003); fuel: methanol, T=20C, p=1 atm. ........................................................................................................... 31
Figure 7 Calculated correction factor for a two-droplet array with different droplet sizes (a1/a2=1.5, a1/a2=5) ..................................................................................... 33
Figure 8 Correction factor calculated for a three-droplet asymmetric array with two larger droplets of similar diameter sizes and a much smaller droplet in the wake of larger drops.................................................................................... 33
Figure 9 Correction factor calculated for a three-droplet asymmetric array with two larger droplets of near similar diameter sizes and a much smaller droplet in the wake of larger drops.................................................................................... 34
Figure 10 Correction factor calculated for a three-droplet asymmetric array with two smaller droplets in the wake of larger drop ........................................................... 34
Figure 11 Correction factor calculated for a three-droplet asymmetric array with near similar droplet sizes .............................................................................................. 35
Figure 12 Slice planes around the droplet...................................................................................... 55
Figure 13 Slice position ................................................................................................................. 56
Figure 14 Visualization of the slice positions in a computational domain for two droplet array ...................................................................................................................... 57
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Figure 15 Temperature distribution for a 203mm computational domain for a single droplet burning in atmospheric pressure (case 4), positioned at the center of the domain............................................................................................................. 67
Figure 16 Temperature diagram for a 403mm computational domain for a single droplet burning in atmospheric pressure (case 6), positioned at the center of the domain....................................................................................................................... 68
Figure 17 Numerical estimated burning rates constants for a 2.2mm droplet burning in different oxygen/nitrogen concentration .......................................................... 73
Figure 18 Experimental and numerically predicted data for initially pure methanol droplets burning in various nitrogen/oxygen environments at 1 atmosphere (Marchese and Dryer, 1999). ...................................................................................... 73
Figure 19 Flame position for a 1.2 mm droplet burning in 15% oxygen, at 0.8 s of burning time without igniters. ....................................................................................... 74
Figure 20 Flame position for a 1.2 mm droplet burning in 15% oxygen, at 1.1s of burning time without igniters. ........................................................................................ 74
Figure 21 Flame position at 3.0s of simulation for a 1.2 mm droplet burning in 50% oxygen.................................................................................................................. 75
Figure 22 Flame position at 4.0s of simulation for a 1.2 mm droplet burning in 50% oxygen.................................................................................................................. 76
Figure 23 Flame position at 5.0s of simulation for a 1.2 mm droplet burning in 50% oxygen.................................................................................................................. 76
Figure 24 Comparison of temperature profiles as a function of droplet radii and burning times, with non-luminous radiation considered. Initial conditions: n-heptane, drop diameter, 3.0 mm; temperature, 298 K; atmosphere, air at 1atm pressure (Marchese et al., 1999).............................................................. 78
Figure 25 Numerical estimated burning rate constants for a 2 mm methanol droplet burning in air at 1atm, with and without considering non-luminous radiation. ................................................................................................................. 80
Figure 26 Measured and calculated diameter squared for 5mm methanol/water droplets (Marchese et al., 1999. ........................................................................... 80
Figure 27 Droplet combustion predictions (with and without non-luminous radiation considered) compared with the numerical results of King (1996) and the experimental results of Kumagai (1971). Initial conditions: drop diameter, 0.98 mm; temperature, 298 K, air at 1atm pressure Marchese et al. (1999) ..................................................................................................... 81
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Figure 28 Flame position approximated by the position of maximum heat release for a 1 mm droplet burning in air after 3.1s of simulation with non-luminous radiation included (0.5s of independent burning) ................................................... 81
Figure 29 Flame position approximated by the position of maximum heat release for a 1 mm droplet burning in air after 3.6s of simulation (1.1s of independent burning) with non-luminous radiation included ........................................................ 82
Figure 30 Flame position approximated by the position of maximum heat release for a 1 mm droplet burning in air after 4.0s of simulation (1.5s after ignition is off) with non-luminous radiation included. The slice position is 1 mm behind the droplet............................................................................................... 82
Figure 31 Comparison between PSM and Numerical Simulation Data (FDS_v3 modified) for a two-drop symmetric and asymmetric arrays ......................................... 85
Figure 32 Histories of droplet diameter squared for different spacing at atmospheric pressure, investigation performed by Okai et al. (2000) ........................................... 88
Figure 33 Comparison between PSM (Annamalai and Ryan (1993), Leiroz et al. (1997) and numerical solution for a symmetric two droplet array ............................................................................................................................................... 89
Figure 34 Experimental burning times as a function of separation parameter for a two droplet array of n-heptane burning in air at atmospheric pressure Mikami et al. (1994) ................................................................................... 91
Figure 35 Flow field around the droplets for two methanol droplet array having identical diameters, initial diameter 2mm. Velocity vectors are perpendicular to slice plane. Each cell is 1mm.............................................................................. 92
Figure 36 Velocity field around a two methanol droplet asymmetric array (l/a=4), a1/a2=2. Velocity vectors are perpendicular to slice plane. Each cell is 0.5mm.................................................................................................................................. 93
Figure 37 Velocity field around a two droplet asymmetric array (l/a=8); a1/a2=2. Velocity vectors are perpendicular to slice plane. Each cell is 0.5mm. ........................................................................................................................................... 93
Figure 38 Velocity field around a two droplet asymmetric array (l/a=16); a1/a2=2. Velocity vectors are perpendicular to slice plane. Each cell is 0.5mm. ........................................................................................................................................... 94
Figure 39 Comparison between PSM and numerical solution for a two droplet asymmetric array, a1/a2=2; fuel: methanol, burning in air at atmospheric pressure and g=0........................................................................................................ 95
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Figure 40 Flame position for a two droplet symmetric array burning in air. .................................................................................................................................................. 98
Figure 41 Flame contours for a two droplet asymmetric array (a1/a2=2, l/a=4).............................................................................................................................................. 98
Figure 42 Flame contours for a two droplet asymmetric array (l/a=16), based on heat release per unit volume. .......................................................................................... 99
Figure 43 Temperature profile for a two droplet array of different diameters (l/a1=16, l/a2=32). .......................................................................................................... 99
Figure 44 Correction factor for a three methanol droplet arrays of equal and different droplet sizes, mounted in the apices of a triangle compared against PSM results and Liu (2003) experimental data ............................................................... 102
Figure 45 Measurement of K/K0 ratio for a three droplet array where K is the burning rate of a center drop in a linear array and K0 is the isolated droplet burning rate (Dietrich et al., 1997) .................................................................................. 103
Figure 46 Flame two-dimensional contours for a three droplet symmetric array (front view) at 4.0s of simulation........................................................................................ 104
Figure 47 Flame two-dimensional contours for a three droplet symmetric array (top view) at 4.0s of simulation .......................................................................................... 105
Figure 48 Three-dimensional iso-contours for 3850kW/m3 heat release per unit volume of (flame approximate position) for a symmetric three droplet array burning in air after 4.0s of simulation .................................................................... 105
Figure 49 Flame contours for a three droplet asymmetric array, having different droplet sizes (a1/a2=1.33, a1/a3=2, l/a1~7) at 4.0s of simulation .................................... 106
Figure 50 Three-dimensional iso-contours for 6300kW/m3 heat release per unit volume of (flame position) for an asymmetric three droplet array burning in air after 4.0s of simulation.......................................................................................... 106
Figure 51 Flame shape history as a function of separation distance for seven droplet two dimensionally arranged clusters of droplets (L varies from 10mm to 30 mm) (Nagata et al., 2002) ............................................................................... 107
Figure 52 Bench-top apparatus (acoustic levitator) to study evaporation of levitated fuel droplets (Liu, 2003) ........................................................................................... 108
Figure 53 Drop Tower Rig System to study evaporation and combustion of unsupported fuel clusters of droplets under microgravity ....................................................... 110
Figure 54 Mechanical lay-out of the Droplet Cluster Rig (top view) .......................................... 116
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Figure 55 Mechanical lay-out of the Droplet Cluster Rig (side view)......................................... 117
Figure 56 Schematic of the igniter assembly inside the enclosure .............................................. 118
Figure 57 Igniter elevator assembly............................................................................................. 119
Figure 58 Loop igniter assembly ................................................................................................. 120
Figure 59 Experiment operation timeline .................................................................................... 122
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Droplet Interactions during Combustion of Unsupported Droplet Clusters in Microgravity: Numerical Study of Droplet Interactions at Low Reynolds Number
Irina N. Ciobanescu Husanu
Academic Advisor: Dr. Mun Y. Choi Research Advisor: Dr. Gary A. Ruff
Abstract
The present work developed a numerical model to study the combustion of well-
characterized drop clusters in microgravity environment using direct numerical simulation by the
means of Fire Dynamic Simulator – a CFD model of fire-driven fluid flow. The computational
research investigated the combustion of clusters of droplets of different sized and asymmetric
three-dimensional configurations in zero gravity environments for zero relative Reynolds
numbers. One of the aspects studied is droplet interaction during evaporation and combustion
over the lifetime of the droplet. The model developed accounts for variable gas-phase thermo-
physical properties, unity Lewis number, Stefan velocities and includes the gas-phase radiative
transfer (solved by a finite volume method) for finite rate reaction. Mass burning rates are
calculated for each droplet in an array and compared to mass burning rate of similar single
droplet, the ratio of these two being a correction factor η. Single droplet combustion has been
studied to evaluate and validate the model output. It was found that single droplet combustion
does follow the d2-law, mass burning rates being in excellent qualitative agreement with current
theories and experimental data. Direct numerical results of multiple droplet combustion were
obtained and compared with a point source method as well as with experimental and numerical
models developed in the past. Data obtained with proposed method provided results consistent
with and in qualitative agreement with multiple droplets combustion theories and experimental
xvi
investigations. Quantitatively, the numerical model results were in the range of 85% to 95% of
the results provided by the investigations found in the literature for droplet array combustion
models and in the range of 85% to 90% when compared with single droplet combustion models.
The numerical simulation along with the future proposed experiment described in the
project is a unique combination of investigative methods that will provide support for future
investigations and for understanding of droplet interaction phenomena.
1
CHAPTER 1: INTRODUCTION
The study of the evaporation of droplets is of fundamental importance in the context of
sprays and spray combustion. There are extensive bodies of work of both theoretical and
experimental studies of droplets evaporation, all of which have attempted to gain insight into
different aspects of the phenomena. Evaporation, ignition, and combustion of isolated droplets
have been the subject of many of these experimental and theoretical studies. However, the
behavior of a spray cannot be anticipated by a model of an isolated droplet that, by nature, does
not take into account the complex phenomena of droplet interactions (the coupling of heat and
mass transfer with the neighboring droplets), and the impact of the environment upon the droplet
cluster. While the combustion of small drops (1 - 100 µm diameter) is not greatly affected by
buoyancy, these drops are difficult to observe. Microgravity environments are required to allow
larger drops to be studied while minimizing or eliminating the confounding effects of buoyancy.
Even with the large number of isolated droplet, droplet array, and spray studies that have been
conducted in recent years, the extrapolation of the results from droplet array studies to spray
flames is difficult. The problem arises because even the simplest spray systems introduce
complexities of multi-disperse drop sizes and drop-drop interactions, coupled with more
complicated fluid dynamics. That’s why, recently, researchers have performed numerical
simulations and experiments that specifically examine drop-drop interactions to bridge the gap
between the isolated droplet studies and complete sprays processes.
The objective of this project is to study the combustion of well-characterized drop
clusters in microgravity environment using direct numerical simulation. The computational
research will investigate combustion of clusters of droplets of different sizes and asymmetric
three-dimensional configurations in zero gravity environments for low relative Reynolds
2
numbers. One of the aspects studied is droplets interaction during evaporation and combustion
over the lifetime of the droplet.
The numerical simulation uses Fire Dynamic Simulator – a CFD model of fire-driven
fluid flow. The numerical simulation accounts for variable thermo-physical properties, includes
the gas-phase radiative transfer (solved by a finite volume method) for finite rate reaction and
simulates the variation of the fuel mass flow rate with the radius of the droplets in the cluster.
Mass burning rates are calculated for each droplet in an array and compared to the mass burning
rate of similar single droplets, the ratio of these two being a correction factor η that depends on
droplet diameters and droplets interspacing in a cluster. The data obtained will be compared with
single droplet and arrays of droplets combustion theories and experimental data previously
developed.
These simulations will provide comparisons to support a future microgravity experiment
in which the formation of the clusters can be precisely controlled using an acoustic levitation
system. Using this system, dilute and dense clusters can be created and stabilized before
combustion is begun allowing the spectrum of droplet interactions during combustion to be
observed and quantified. Normal gravity experiments have been previously conducted on isolated
droplets and two-dimensional arrays containing between 2 and 13 droplets. These tests verified
the effect of droplet interactions when the average normalized droplet spacing, l/d, is less than 10
and the group combustion number, G, is around 1.0. Numerical simulations are underway to
evaluate these experimental results.
The future low-gravity experiments are to be conducted in a drop tower facility and will
focus on (1) the effect of droplet size, cluster size and number of drops on the combustion
3
process, (2) the effect of the type and composition of fuel on group combustion, and (3) the
ability of the group combustion number to scale the observed group combustion regimes.
4
CHAPTER 2: BACKGROUND
2.1 Studies of Single and Multi-Component Isolated Droplets
An extremely large number of theoretical and experimental studies have investigated the
behavior of a single fuel droplet in varying environments. Evaporation and combustion of drops
has been studied in quiescent conditions, with natural and forced convective flows (Matlosz et al.,
1972; Harpole, 1980), at low and high pressure (Hiroyasu and Kadota et al., 1974; Curtis and
Farrell, 1992), and at high and low Re numbers (0 < Re < 300 and up) (Sirignano and Raju, 1990;
Dandy and Dwyer, 1990). In experimental studies, isolated drops are generally suspended on thin
fibers to be easily examined in normal gravity (Kumagai and Isoda, 1957; Matlosz et al., 1972;
Chérif, 1994; Chesneau, 1994). Later, researchers tried to eliminate the effects induced by the
supporting fiber by using different methods such as free flying drops and microgravity
environments (Hartfield and Farrell, 1993; Ristau et al., 1993; Nomura et al., 1996) from
atmospheric to supercritical pressure. Most of the relevant studies that have dealt with the
evaporation of isolated droplets have been summarized in reviews by Faeth (1977), Law (1982,
1986), Sirignano (1983), and Givler and Abraham (1996). Fortunately, it is not necessary to
examine all of the work contained in these reviews in this document. Instead, this section will
examine computational and experimental methods that have been applied to isolated droplet
evaporation from the point of view of their relevance to the study of drop arrays and inter-droplet
interactions.
Early theories of isolated droplet evaporation were based on simplified models, mainly
due to the complex phenomena associated with the fluid mechanics and heat transfer that occurs
during evaporation. The classic assumption is that evaporation is a quasi-steady process so that
the droplet can be assumed to be of a fixed size. Essentially, this models the droplet as a porous
5
sphere into which fuel is fed at a rate equal to the mass evaporation rate. Solution of spherically-
symmetric fuel species continuity and energy equations yields the d2 evaporation law. The
objective of many computational and experimental studies has been to investigate departures
from the d2 law under varying conditions. These studies will be discussed shortly.
Of particular note is a recent investigation reported by Kozyrev and Sitnikov (2001) who
studied the slow evaporation of a single liquid droplet placed in a chemically inert gas of
moderate pressure that contains the vapor of the droplet. Instead of making a quasi-steady
assumption, they treated droplet evaporation by solving equations for diffusion and molecular
flux from a spherical liquid surface. Their approach uses Maxwell’s classical theory but under the
assumption that the vapor near the droplet surface is not saturated. Treatment in this manner
allows certain refinements in the theory to be made. Specifically, these are:
• The existence of the free energy of the surface, because it is this energy that determines
the pressure of saturated vapor above the curved interface
• The coefficient of condensation, defined as the probability of a vapor molecule incident
on the surface of the condensed phase not being reflected, considerably affecting the
rate of evaporation
• A kinetic description of the molecular flow (the formation of the diffusive flux of vapor
molecules)
• A consistent description of the process of energy exchange between gas molecules and
the condensed phase at the interface
The temperature of the evaporating droplet, the vapor tension near the droplet surface and
the droplet evaporation time were calculated using this analysis. This approach demonstrated that
a simple refinement of the typical quasi-steady theory complicates considerably the relevant
equations. The significant feature of this theory is that it addresses the variety of physical
6
processes associated with the evaporation of a droplet at temperatures much lower than their
boiling point. Their model predicted evaporation rates for water and mercury droplets well
because of the relatively low evaporation rates. Evaporation times for fuel droplets, n-hexane for
example, were overestimated due to the difficulties in modeling the vapor tension in the
environment. Even though this model is one of the most comprehensive applied to droplet
evaporation and included most of the complex kinetic, fluid mechanics and heat transfer
phenomena, it produced essentially the same results as quasi-steady droplet evaporation models.
This theory would give the correct estimate of evaporation for arrays of droplets only under the
conditions that their evolution in time doesn’t affect the temperature and vapor concentration of
the environment.
Upon having a theory for droplet evaporation rates, it is logical to conduct experiments to
verify the predictions. Unfortunately, the requirement of maintaining a droplet motionless while it
evaporates requires that the droplet be supported in some way; typically, it is suspended on a thin
fiber. Experimental measurements made it obvious that the fiber was altering the droplet
evaporation rate sufficiently that true comparisons could not be made. Therefore, theoretical
evaluations began to include the fiber in the droplet evaporation model. A general characteristic
of all these studies is that the geometry was assumed to be symmetric, while geometric
configurations of clusters of droplets in a spray may or may not be symmetrical, the symmetry
being considered a constraint in modeling the spray behavior using an array of fiber supported
droplets.
One of the first papers that addressed the effect of the fiber was the work of Kadota and
Hiroyasu (1976) who studied high-pressure droplet vaporization under natural convection. In this
study, they considered conduction through the fiber using a one-dimensional steady state analysis
and the radiative absorption was assumed to occur on the droplet surface. While the results were
7
somewhat qualitative, they calculated a significant enhancement of the vaporization rate because
of the presence of the fiber (enhanced heat transfer to droplet from fiber). Megaridis and
Sirignano (1993) calculated the behavior of a slurry droplet containing a spherical particle, and
vaporizing in a high temperature convective environment. They found that the relative motion of
the solid particle and the liquid carrier fluid is very significant during the early stages of the
simulation, and that the fluid mechanics dominate the heat and mass transport phenomena.
Treating the internal, spherical particle as a bead on a supporting wire, Shih and Megaridis (1995)
extended the model to evaluate the evaporation characteristics of a fuel droplet, suspended
concentrically from a spherical bead held at the tip of a thin filament. The drop is exposed to a
hot, laminar gaseous environment with Re~50. The model solves the Navier-Stokes equations in
conjunction with the heat conduction within the suspending filament. Results show that the size
of the quartz suspension fiber does not influence considerably the droplet surface regression rates.
However, the corresponding mass evaporation rates are considerably different for fiber-supported
drops in comparison to the free traveling droplets. The suspended drop configuration
overestimates the evaporation rates and it under predicts the droplet lifetime compared with a free
traveling droplets. This difference was found to be larger with increasing ambient temperature.
Another problem that occurred while conducting experiments was that buoyant
convective flow produced by either thermal and/or concentration gradients changed the symmetry
of the problem. Accordingly, droplet evaporation and combustion experiments have been
performed in microgravity environments to eliminate the effect of buoyancy (buoyancy it
becomes significant for droplets of 100µm or larger). However, as in normal gravity, a fiber was
generally used to support the droplet. Avedisian and Jackson (2000) observed the effect of the
fiber on soot particle production for droplets burning in a stagnant atmosphere in microgravity.
They found that the configuration of the soot shell inside the flame is non-symmetric due to the
8
non-symmetric distribution of thermophoretic and drag forces around the droplet. They also
observed the non-linearity of the d2-law due to the influence of the fiber. Several theoretical and
experimental studies that used microgravity environments were conducted to investigate various
aspects of droplet evaporation and combustion characteristics including vaporization and
combustion of bi-component droplets (Shaw, 1999), water dissolution in alcohol droplets during
evaporation and combustion and the radiative extinction of fiber-supported drops (Williams,
1997, 1999), single methanol droplet gasification at sub- and super-critical conditions (Chauveau
et al, 1995, 1999), and extinction of droplet pairs (Dietrich et al., 1999). Discrepancies were
observed between the theoretical models and the experimental results at normal and micro-
gravity. Recently, Yang and Wong (2001) conducted a study aimed at resolving the discrepancies
between the theory and experiments for droplet evaporation in microgravity. Radiative absorption
and fiber conduction enhance the evaporation rate significantly (Yang and Wong, 2001). (The
study by Megaridis and Shih also emphasized this phenomenon but they considered these effects
in a more simplified manner.) Yang and Wong (2001) used a comprehensive droplet evaporation
model that includes fiber conduction, liquid-phase radiative absorption, real-gas thermophysical
properties and the variation of the enthalpy of vaporization at elevated pressures. Their theoretical
method relies on direct calculation of governing equation and numerical simulation. To simplify
the problem, a one-dimensional, spherically symmetric system at steady-state conditions was
assumed. Their configuration was an n-heptane droplet having an initial diameter of 0.6-0.8 mm
suspended at the tip of a horizontal quartz fiber. They observed the droplet while it was
evaporating in a hot furnace filled with nitrogen pressurized to 20atm. Comparisons to the
experimental data obtained by Nomura et al. (1996) and Risteau et al. (1993) prove that the
method qualitatively agrees with the experimental data. The main issue remains at large
pressures, where both theoretical and experimental results were unreliable. The method is
difficult to extend to arrays of drops because it would require at least a two-dimensional model.
9
Also, for droplet arrays or cluster simulations, it is not possible to introduce a time-dependent
coordinate transformation as it was done for single droplet simulation, due to the fact that
different droplets vaporize at different rates (Dwyer et al., 2000). Chauveau, Chesneau and
Göklap (1995) have observed experimentally high-pressure vaporization and combustion of a
methanol droplet in reduced gravity (obtained during the parabolic flights of the CNES
Caravelle). The researchers used the combination of high pressure and reduced gravity to
minimize the pressure-enhanced natural convection effects and to extend the applicability of the
fiber suspended droplet technique. This paper studied the effect of ambient pressure on the
droplet evaporation rate. Low temperature vaporization experiments were conducted only at
normal gravity (1-g) in dry air up to P=100 bar. Droplet burning experiments were performed up
to P=80 bar under 1-g and up to P=50 bar under reduced gravity (approx. 0.01g). For all
experiments presented, the methanol droplet diameter was 1.5 mm and was suspended on a fiber
in an ambient gas having a temperature of 300K. For burning experiments, the investigated
regimes range from sub-critical to trans-critical. The conclusions of this research were that the
average evaporation rate decreases with decreasing pressure for low temperature conditions,
while at high-pressure, the burning rates will increase. It is also emphasized the strong effect
buoyancy has on the vaporization and burning rates. Deviations from the d2-law were noticed and
a corrected d2- law that takes into account the effect of buoyancy was applied. It represents a
useful area for further studies of spray combustion.
Another research area that has produced results relevant to the study of droplet
interactions is the evaporation of multi-component droplets (Labowsky, 1978, 1980; Law, 1976,
1978; Law and Law, 1982; and Xiong et al., 1984). Studies conducted by Annamalai and Ryan
(1992, 1993) regarding isolated multi-component drops produced approximate solutions for the
evaporation rate. The physical system generally modeled was a drop of arbitrary composition
10
placed in a quiescent atmosphere. Quasi-steady conditions with constant thermophysical
properties of the liquid-gas phase were assumed. The method determines the evaporation rate
through simple explicit solutions. The results obtained here are extended and used to study the
evaporation characteristics of multi-component drop arrays. Daïf, Bouaziz, Chesneau and Chérif
(1999) studied vaporization of multi-component isolated drops and droplet arrays, theoretically
and experimentally. These researchers examined the behavior of an isolated droplet and the
leading droplet in a system of several droplets. Their model is based on “film theory” that
assumes that mass and heat transfer between the droplet surface and the external gas phase take
place inside a thin gaseous film surrounding the droplet. The model is a generalization of that
used by Abramzon and Sirignano (1989) to study a droplet vaporization model for spray
combustion calculations, extended to the vaporization of a multi-component fuel droplet subject
to natural and forced convection. The generalization of the Abramzon and Sirignano (1989)
model, for natural convection, consists in considering the average Nusselt and Sherwood numbers
as a function of Grashof number (where 103 < Gr < 8x104) while, for forced convection, the
average Nu and Sh numbers equations provided by Renksizbulut et al. (1991), as a function of
Reynolds number (10<Re<300). This model is only valid for the leading drop in an array because
it does not account for the effect of droplet wakes on following droplets.
The calculations have been verified by the experiments conducted using an apparatus
placed at the end of a thermal wind tunnel, which is fitted with homogenization grids to obtain a
uniform flow in the channel. The airflow velocity varies from 0 to 10 m/s and the maximum
constant temperature is 150oC. A 0.4-0.8 mm diameter droplet was suspended to a thin capillary
and placed in a natural and a forced convective environment. The droplet was suspended at the
center of the test section with the supporting capillary tube positioned perpendicular to the flow.
Droplet evaporation was determined by recording the droplet diameter variation as a function of
11
time using a video system. A thermographic infrared system was synchronized with the video to
simultaneously record thermal images. As fuels, the researchers used pure heptane, pure decane
and a heptane-decane mixture. The authors considered that the results of the experiments are in
satisfactory agreement with the calculation model. However, their theoretical model doesn’t take
into consideration any effects of the fiber used to suspend the drop when they predicted the
evaporation rate.
2.2 Studies of Arrays of Droplets and Streams
2.2.1 Theoretical Analysis
In spite of the research conducted on isolated droplets, a model of an isolated droplet
cannot predict the spray behavior because the effect of inter-droplet interactions and the effect of
the gaseous environment on the cluster are not considered. Recent numerical and experimental
studies have evaluated the interaction of two or more droplets (Raju and Sirignano, 1990; Chiang
and Sirignano, 1993; and Daïf, Bouaziz, Chesneau, Chérif and Bresson, 1997, Imaoka R.T. and
Sirignano W.A., 2005 for example). Several researchers, such as Dunn-Rankin, Sirignano, Rangel
and Orme (1994) have performed a tremendous amount of work in the area droplet interactions.
Arrays and streams of droplets have been studied using various computational, theoretical and
experimental methods. Many of these that have been conducted in the last decade are included in
a comprehensive review by Dunn-Rankin, et al. (1994). The objective for much of this research
was to explain the effect of neighboring droplets on a droplet in an array and the field behavior
for liquid and gas properties in the arrays and streams. The approach that is generally used
considers three levels of interactions between droplets depending on the spacing between the
droplets. These are: (1) far apart one from other, so the drops in an array can be considered as
isolated drops and the study of this type of array can be reduced to that of single droplets; (2)
12
close enough to modify the ambient conditions and to be affected the lift and the drag
coefficients, Sh and Nu number (this case implies the study of droplet interactions and cannot be
treated, as the drops are isolated); and (3) the droplets are close enough to collide and
coalescence. Of particular interest for this review is the second level of interaction, for both
reactive and non-reactive situations, when the distance between droplets is up to one-drop
diameter but they do not collide or coalesce.
Theoretical studies taking into consideration two or three evaporating droplets of equal
diameters without forced or natural convection (performed by several researchers such as
Twardus and Brzustowski, 1977; Patnaik et al., 1986; Raju and Sirignano, 1990, etc.) concluded
that:
• The evaporation rate decreases with the inter-drop spacing;
• The proximity of the neighboring droplet inhibits the exchange of mass and energy
between the droplet and the surrounding gas (Twardus and Brzustowski, 1977;
Labowsky, 1976, 1980);
• Diffusion analyses must include natural and forced convection and variable
thermophysical properties to avoid the over prediction of the effect of droplet
interaction (Xiong et al., 1985);
• There is a critical ratio of the two initial droplet diameters below which droplet
collision does not occur (Raju and Sirignano, 1990)
• For more than two droplets in tandem, a particular droplet (generally the center drop)
is more affected by the nearest droplet that by the others (Chiang and Sirignano,
1993);
The studies were performed for a wide range of Reynolds numbers, droplet radii and
spacing, considering both constant and variable thermophysical properties. All of these results are
13
in qualitative agreement. The theoretical and computational results of two and three tandem
vaporizing droplets agree with the experimental investigations (Sirignano, Rankin, Rangel and
Orme, 1994). Other theoretical studies involved three-dimensional numerical analysis of two or
three spheres moving in parallel (Dandy and Dwyer, 1990; Tomboulides et al., 1991; Kim et al.,
1992, 1993), at Reynolds numbers between 50 and 150, showed how local aerodynamic
modifications can significantly affect droplet trajectories. Labowsky (1978, 1980) studied the
droplet arrays vaporization using the Method of Images, assuming a slow evaporation and a non-
convective environment. Then, the method was extended to rapidly evaporating arrays of
droplets.
Studies using a continuous droplet stream have offered an alternative method to get from
droplet arrays to full sprays studies. Numerical solutions using fine computational grids were
applied (Raju and Sirignano, 1990; Chiang and Sirignano, 1993) and their results highlighted the
effect of droplets interactions on heat transfer and lift and drag coefficients. The majority of the
numerical solutions and modeling are limited to high and intermediate Reynolds numbers. Leiroz
and Rangel (1994) developed a theoretical methodology based on direct numerical simulation to
investigate low and zero Reynolds number for vaporization of a droplet stream. The method
consists of applying potential flow theory to a long stream of vaporizing droplets, assuming that
the superposition of several droplets is valid for zero Reynolds. The vaporization rate is
approximated by a power function G representing the ratio of the non-dimensional droplet
vaporization rate in a stream to that of an isolated droplet. The theoretical approach is in good
qualitative agreement with other theories developed previously and also with experimental results
for high velocity streams of drops. Unfortunately, experimental data were not available for
comparison for low and zero Reynolds numbers. Of particular interest is the fact that this method
is quite similar to the theoretical approach developed by Annamalai and Ryan (1993), which will
14
be presented later. Reacting droplet streams have also been studied both in steady and unsteady
situations. Delplanque and Sirignano (1993, 1994) conducted a theoretical study of transcritical
vaporization of an array of liquid oxygen droplets. They found that the combined effect of the
high temperature from the reaction zone (combustion) and the reduced droplets relative velocities
cause the droplet surface temperature to reach the critical mixing conditions, which represents a
significant departure from the behavior predicted for a vaporizing isolated drop.
Three-dimensional effects of interacting droplets have been investigated to some extent,
both for steady and unsteady vaporization (effects of interacting droplets on droplet lift and drag
forces and on droplet torque). However, the 3-D effects of interacting droplets on heat and mass
transfer remain to be determined. The effect of modifications in lift and drag coefficients and Nu
and Sh on droplets in a stream must be included in droplet stream computational analyses
(Sirignano, Rankin, Rangel and Orme, 1994). Dwyer, Stapf and Maly (2000) have carried out an
interesting three-dimensional direct numerical approach for unsteady vaporization and ignition of
a stationary array of droplets, at low, intermediate and high relative Re numbers (relative Re were
obtained using the relative velocity of the free stream with respect to the droplet array). The main
purpose of the study was to quantify the droplet interactions in the array, to investigate the effect
of droplet interactions upon the flow field and chemistry, and to study the influence of Re
numbers on droplets interaction. A non-symmetric array of six identical heptane droplets at
intermediate Reynolds numbers was considered. The model considered variable thermo-physical
properties, one-dimensional heat conduction (for each droplet), and neglected gravity and thermal
diffusion effects. It was also assumed that the gas phase obeys the ideal gas law, which is
considered a good assumption for the simulation conditions (the errors due to real gas effects are
less than 5%). The method consisted of performing direct numerical simulations to investigate the
array behavior in terms of rate of vaporization, mass and heat transfer and lift and drag
15
coefficients. Droplet interactions are quantified by calculating the droplet drag coefficient for a
single drop and an array, and considered that any difference between the drag coefficients of the
droplets is due to droplet interactions. Simulation time is very short in the droplet lifetime – up to
0.902 ms (at t=0.902 ms there is a significant burning in the droplet wake), due to the reaction
zone moving into the droplet array and the lack of grid resolution where chemical reaction occurs
(Dwyer et al., 2000). For accurate simulations, a very dense main mesh or an adaptive mesh is
needed that was not possible at the time due to the lack of computational resources. Their results
infer that the influence of Re number is quite strong on the interactions between droplets,
especially for low Re. Also, the rate of vaporization in a droplet array is dependent on the
geometry of the array. Even for the small number of droplets studied, there can be a factor of 2
difference in the mass loss of the droplets. At high Re, droplets in the array behave like individual
isolated droplets, and at low Re the array behaves like a single entity. The authors inferred that
this technique could be efficient for investigating complex problems of droplet interaction, even
for large number of droplets. However, simulations of complete sprays were not feasible “due to
the billions of spray particles and the complex fluid physics that occurs in practical systems
(collisions, turbulence and secondary breakup of spray droplets)” (Dwyer, Stapf and Maly, 2000).
Future investigations were planned to include moving droplets and periodic arrays that model a
“slice” of a spray and also DNS for larger number of drops (>100). The authors did think that the
main challenge would be to design numerical simulations that are simple and yet contain the
essence of the physical processes of spray behavior. One important problem is that this method
was unable to simulate and investigate arrays of droplets having diameters larger than 1 mm due
to a need for a more sophisticated meshing. Continuing his previous research in droplet array
combustion area, Sirignano and Imaoka (2005) developed a three dimensional model and
numerical simulation of asymmetric fuel droplet arrays burning in quiescent conditions with
uniform and non-uniform spacing and variation of droplet size, considering Stefan convection,
16
diffusion and infinitely fast chemical kinetics. According to the authors, the model is based on
Method of Images developed by Labowsky (1976, 1978, and 1981) and later modified to account
for the effect of neighboring droplets by Marbery et al. (1984). Data obtained through this
investigation led to the conclusion that vaporization rates correlates well with existing data for
symmetric, uniform dispersed arrays of droplets of identical size. Although this model is a step
forward in quantification of droplet interactions, a drawback of the method is the fact that a
numerical code has to be developed for each array configuration and also that the model is
amenable only for atmospheric pressure, at near stoichiometric conditions and no thermal
radiation effect is included. For large arrays, the method will require a large number of sinks,
extending even more the computational time.
Another interesting theoretical approach is that presented by Annamalai and Ryan (1993).
They studied the evaporation of both single and multi-component arrays of droplets using the
Point Source Method (PSM). The method determines the mass loss rate of interacting drops by
treating each droplet as a point mass source and heat sink, and evaluates the steady-state mass
loss of arrays of interacting drops in a quiescent atmosphere with Le unity. This method is used to
obtain analytic expressions for the evaporation rate of an isolated droplet and arrays of single and
multi-component droplets. To determine the mass loss rate of interacting droplets, a correction
factor is defined as the ratio of mass loss rate of a droplet into an array to the isolated droplet
mass loss rate. Calculations were performed for symmetric arrays and the authors infer that the
method can be used for arrays 1000 drops or less (computational time being the limiting factor),
under the condition that the interparticle spacing, l/a, is much greater than 2. For arrays up to 5
drops, the results from the PSM are in excellent agreement with the results obtained through the
exact methods developed Brzustowski et al. (1979), Labowsky (1978,1980) and Annamalai and
Ramalingam (1987). However, for arrays with 7 and more drops, it was observed that the
17
correction factor for the center drop decreases dramatically and the average correction factor
could become negative (the worst scenario). By definition, the correction factor is positive and
any negative value has no physical meaning. As a result, the inter-drop spacing was set to avoid
this problem. This error increases with the number of drops in the array. Therefore, it was
concluded that PSM is limited in predicting the correction factor of primary drop (center drop)
especially for arrays containing more than 7 drops. The authors inferred that better results for
larger arrays (more than 7 drops) could be obtained by setting evaporation rate to be zero for the
center drop (Annamalai and Ryan, 1993). Physically, this could mean that the center drop doesn’t
evaporate due to vapor pressure created by the neighboring evaporating droplets around central
drop. One important issue is that this method wasn’t verified experimentally for asymmetric
configuration and for larger number of drops. For arrays of multi-component drops, the method
was adapted to match the experimental conditions of Xiong et al. (1984) and, therefore, is in good
qualitatively agreement with the experiment.
2.2.2 Experiments Involving Arrays of Drops
Compared to the number of investigations of isolated droplets, streams of droplets, and
full liquid sprays, experimental investigations of well-controlled droplet arrays are less common.
The main characteristic is that this type of investigation allows researchers to “isolate the effect of
neighboring droplets on drop aerodynamics, drop vaporization and drop combustion” (Dunn-
Rankin, Sirignano, Rangel and Orme, 1994). The most common experiments involve either two-
dimensional arrays of fiber-supported drops or parallel streams of droplets subjected to a hot
environment and ignited (Sangiovanni and Labowsky, 1982; Queiroz and Yao, 1990). This
configuration is fixed and symmetrical, and therefore, amenable to analysis. The main result of
these types of experiments is that the vaporization rate is affected by the neighboring droplets and
by buoyancy. The principal issue is that experimental studies that use the fiber-supported arrays
18
of drops cannot predict the effects of neighboring droplets on aerodynamic drag if convection is
present.
Nguyen and Rangel (1991) and Nguyen and Dunn-Rankin (1992) used freely flying
drops to evaluate the effect of neighboring droplets on aerodynamic drag. These experiments are
in agreement with the numerical models presented by Chiang and Sirignano (1993). The primary
conclusion was that the first droplet behaves essentially as an isolated drop (in terms of lift and
drag coefficients) with the trailing drops were perturbed by the droplet nearest to them. Another
experimental study conducted by Silverman and Dunn-Rankin (1994) focused on reacting droplet
streams. A self-sustained droplet stream was considered and the effect of droplet size and spacing
on the burning rate, flame size, and ignition delay was evaluated. Anti-Stokes Raman scattering
was applied to measure the thermal field near the flame surrounding a rectilinear droplet stream.
This method gave qualitative results on the concentration of fuel vapors between droplets but
could not predict the relationship between the concentration of fuel vapors and the decrease of the
vaporization rate. The authors concluded that a simplified “spray” configuration could predict the
behavior of actual sprays even if they don’t have all the characteristics of real sprays.
The evaporation and combustion of multiple drops have also been studied in microgravity
environments to determine the effects of drop-drop interaction. Studies in reduced gravity
emphasized various aspects of droplet evaporation and combustion, focusing on high pressure
burning of droplet arrays (Chauveau et al., 1999) combustion of unsupported droplets in a
convection-free environment (Jackson and Avedisian, 1998), combustion of two-dimensionally
arranged fuel samples (Nagata et al., 1999), combustion of mono-dispersed and mono-sized fuel
droplet clouds (Nomura et al., 1999), and exploration of the thermal structure of an array of
burning droplet streams (Queiroz and Yao, 1990). Chen and Gomez (1995, 1997) evaluated how
well the results from droplet arrays can be extrapolated to spray flames. Dietrich et al. (1999,
19
2001) examined combustion of interacting two dimensional droplet arrays in microgravity
environment. The experiment uses the classical fiber-supported droplet combustion technique and
examines droplet interactions under conditions where flame extinction occurs at a finite droplet
diameter. The authors found that the droplet lifetime or average burning rate varies by less than
10% for drop interspacing greater than six diameters. They also investigated arrays up to three
droplets using multidirectional viewing to observe transient drop size and flame position. The
authors showed that inter-droplet spacing played an important role in the extinction of the droplet
array.
2.2.3 Radiative Heat Loss Studies on Fuel Droplet Combustion
Thermal radiative effects on droplet combustion had been neglected in most of the works
developed in the past because of the mathematical and physical complexity of research on
radiative transfer (Faeth 1983, Law 1982 and Viskanta et al., 1987). However, the last 10 – 15
years of research has considered thermal radiative effects in their droplet combustion models,
many of them considering soot-related radiation effects. Saitoh et al. (1993) included radiative
transfer in their droplet combustion model by treating the gas phase as a participating medium
while assuming the droplet to be an opaque material. Their numerical investigation showed that
when thermal radiation is considered for the case of n-heptane, the maximum flame temperature
was reduced by at least 25% compared to that without considering thermal radiation. Thus, they
concluded that thermal radiation should not be ignored in modeling droplet combustion. Also,
Marchese et al. (1999) developed experimental and numerical studies of a burning n-heptane
droplet and compared the model predictions for cases with and without non-luminous radiation
considered with experimental data provided by Kumagai et al, 1971 (Choi and Dryer (2001). The
numerical model developed by Bergeron and Hallett (1989) included radiation to extract reaction
rate constants from the measured data using the suspended droplet technique. Lage and Rangel
20
(1993) investigated droplet vaporization by including thermal radiation absorption. The model
used assumes that the incident radiation is spherically symmetric and there is a blackbody spectral
intensity distribution. However, the gas phase is assumed to be a non-participating medium.
Simulations using decane droplets with a radius of 25-100 µm, tested with ambient temperatures
from 500 to 1800 K, concluded that under normal spray combustion conditions, there is not
enough radiative energy to induce explosive vaporization of mono-component hydrocarbon
droplets, and only the total absorptance values are needed for vaporization studies. Flame
radiation is classified as being non-luminous or luminous. In non-luminous flames, carbon
dioxide and water vapor are the most prominent constituents at temperatures up to 3000 K. When
soot is present, however, the flame becomes luminous. Siegel and Howell (4th Ed.) indicated that
soot, usually produced in the fuel-rich region of the hydrocarbon flames, can often double or
triple the radiant energy emanated by the gaseous products alone. However, the purpose of this
research is to consider only non-luminous radiative heat loss due to the non-sooting characteristic
of the fuel used (methanol) and for the simplicity of the calculations. The model employed is
designed to capture the general characteristics of droplet interactions, and considering soot for
this model would not have a significant impact upon the output, however will have an adverse
effect upon the computational resources available at this time. Therefore, soot will not be
considered as reaction product.
2.3 Summary
The previous sections have reviewed a broad variety of theoretical and experimental
work performed in the area of droplet evaporation and droplet interaction. The main goal of most
of this research was to obtain results that would be useful to understand or predict spray behavior
and characteristics. They used configurations of varying complexity and applied assumptions that
21
tried to model, as accurately as possible, the phenomena at a real scale. In general, they found that
many aspects of the sprays behavior could be studied using these simplified models.
The study of isolated droplets is an important step in understanding the processes of
evaporation of droplets, but the extrapolation of these results to the more complex configurations
that occurs in sprays requires the knowledge of how droplet interactions modify the isolated drop
results. Many of the studies concluded that a good approximation of a real spray behavior could
be achieved through the investigation of the arrays of droplets. Studies of isolated droplets did
reveal some interesting aspects that have helped the development of investigations using droplet
arrays. These include the following:
• For configurations of fiber-supported isolated droplets, the heat conduction induced by
the fiber and the radiative absorption dramatically affects the droplets evaporation
characteristics. There is an increasing influence with the temperature;
• The environment temperature has a strong influence on evaporation rates;
• The evaporation and combustion rates are greatly affected by buoyancy and by
ambient pressure (for low temperatures)
• Reynolds, Sherwood and Nusselt numbers are important in describing the droplets
evaporation process;
• Although thermal radiation is not all that important for small isolated droplets (up to
1.5 mm diameter) burning in quiescent environment and temperatures up to 3000K
(Choi and Dryer, 2001 and Kadota and Hiroyasu, 1976), droplet interactions are
affected by thermal radiation even for smaller droplet sizes.
The interactions between droplets are very important in the vaporization and combustion
process. Several models and methods to study droplet interactions during the vaporization and
burning process have been developed with many of these theories treating symmetric arrays.
22
However, the complexity of the calculation increases rapidly with the number of drops so many
were suitable for arrays not larger than 2-3 drops. Numerical simulations for larger arrays of
drops have been developed but these were limited to very small droplets (less than 50 µm
diameter) with one exception, namely the most recent study of Imaoka and Sirignano (2003,
2004). One important conclusion is that the proximity of a neighboring droplet affects the mass
and energy transfer between the droplet and the surrounding gas, the lift and drag coefficients,
and the Nu and Sh numbers. Droplet interactions and vaporization rates are strongly affected by
Reynolds number, and they are very dependent on the geometry of the array. Most of the
experimental studies are based on fiber-supported droplet arrays, but additional unknowns are
induced because of the fiber. While the effects of the fiber have been studied for isolated drops,
similar analyses have not been done for arrays of fiber-supported droplets because the required
models have been sufficiently complex that they would require a large investment of time and
computational resources. Approximate methods to investigate multiple droplets, such as the Point
Source Method (PSM) have been developed and are in good agreement with other exact or direct
numerical solutions except when droplets are within one diameter of each other. These methods
are applicable for large arrays, but they have not been verified by experimental data. The main
conclusion is that investigations of droplet arrays can yield results that are relevant solution to
actual spray systems. However, there is a substantial work to be performed in the arrays of drops
and sprays domain.
23
CHAPTER 3: DROPLET INTERACTIONS AT LOW REYNOLDS NUMBERS 3.1 Point Source Method
One important issue is to extend a current theory that studied droplet interactions so that
it can match an experimental configuration that can actually be produced. To do so will require
modifications to the theory and adaptation of the experiment. The best way to achieve this is to
start with current theoretical (approximate) solutions that have been shown to provide quite
accurate results for arrays of droplets larger than 20. The Point Source Method is simple and
sufficiently comprehensive to take into account variable thermophysical properties and, as has
been presented above, is in a very good agreement with DNS results and other theoretical
investigations (see Figure 2, Figure 3, Figure 4, and Figure 5 in the next section). The model used
by Annamalai and Ryan (1993) for single-component arrays of drops could be adapted for non-
symmetric arrays, as presented in next section. This method will yield a set of equations for
variable droplet diameters and droplets interspacing, based on the equation given for Point Source
Method. With this formulation, the theory could be compared directly to an experiment in which
the droplets might not be exactly symmetrical. In any event, the Point Source Method can be used
to demonstrate the anticipated effects of non-symmetric droplet arrays (variable spacing and
droplet sizes). However, before discussing these results, the method itself will be discussed and
developed in the next section.
3.1.1 Description of the Method
Annamalai and Ryan (1992, 1993) developed the Point Source Method to investigate the
effects of droplet interactions for arrays of droplets, for both single component and multi-
component drops.
24
The purpose of the method is to evaluate the mass loss rate of interacting drops,
quantifying this through a correction factor η defined as:
isom
m&
&=η (1)
Figure 1 PSM Droplet Coordinate System
where m& is the evaporation rate of a droplet in array and
)1ln(2 BDaShmiso += πρ& (2)
is the evaporation rate of an isolated drop. The transfer number, B, is given by
fgwp hTTcB /)( −= ∞ (3)
ρ is the gas phase density, a is the drop radius and D is the diffusivity.
25
In this method, an array of N droplets is assumed, evaporating under quiescent
atmosphere (Sh = 2), with the drops located at radial positions jrr rr= , where j=1…N, as shown in
Figure 1. Assuming slow evaporation (negligible Stefan flow), the governing equation is the
Poisson equation written as:
( )∑=
−=∇N
jjj rrDmrY
1
2 4/)( rr& πρ (4)
where jrr rr− is the distance from the center of the j-th drop and D is diffusion coefficient. The
system is solved using a superposition method obtained by treating the evaporation of each drop
as though it is the only drop in the array (Labowsky, 1976, 1978). The solution will be given by
assuming each drop acts as a point source, the strength of a single drop is concentrated at the
center of the drop, and the “j” drop evaporates with a source strength jm& . Using the generalized
species conservation equation for a point source at an arbitrary location jrr rr= under the
assumptions shown above and using the superimposing of the solutions, will yield that the mass
fraction at “r” is given as:
( ) ( )( )∑=
∞ −=−N
jjj rrDmYrY
14/ rr
&r πρ (5)
The point source method assumes jii rra rr−<< , where “a” is the droplet radius in the array (see
Figure 1), so the mass fraction of vapors at the drop surface is:
( ) ( ) ( )∑≠=
∞
−
+=−+
N
ijj ji
j
j
j
i
iiiw rr
aDa
mDa
mYarY
,1 44 rr&&rr
πρπρ (6)
26
It also can be shown from the interface mass and energy conservation that the temperature of all
drops in the array is the same under steady state conditions (Labowsky, 1976) and the expression
of surface mass fraction is same as the relation for an isolated drop. This implies that the vapor
mass fraction at the surface of the drops is the same throughout the cloud and the value is the
same as that for an isolated drop.
Using these assumptions and the above equation, the correction factors can be determined
by solving a set of N linear equations given by:
( ) NirraN
ijjjijji ....1,1/
,1==−+ ∑
≠=
rrηη (7)
3.1.2 Advantages and Disadvantages of the Method
This method could be applied to arrays of droplets up to 1000 drops, under the quasi-
steady assumption and a quiescent atmosphere. The PSM yields simple linear algebraic equations
for the correction factor of any array of a given configuration. It agrees very well with other exact
methods, but comparisons with experimental data were available only for arrays up to 7 drops and
generally for symmetric configurations. However, the PSM has been verified for a two-droplet
configuration having non-equal droplet diameters. In this particular case, the results agree very
well with the bi-spherical coordinate method developed by Brzustowski et al. (1979). The PSM
results were also compared with the exact solutions provided by Method of Images (MOI)
(Labowsky, 1976, 1978, 1980) as can be seen in Figures 2 - 4 that show results for three-droplet,
five-droplet and seven-droplet configurations, respectively. In these figures, the primary droplet is
the center drop. The correction factors for both the primary drop and the average for all drops are
shown in these figures. For all droplets in these configurations, the PSM and MOI differ for l/a
less than 6. The agreement at larger droplet spacings is very good. The results obtained using
27
Navier-Stokes equations (Raju and Sirignano, 1990; Sirignano, 1993) are also in a good
qualitatively agreement with those obtained through Point Source Method. PSM also agrees very
well with the results obtained through numerical simulation developed by Leiroz and Rangel
(1994) for droplet streams at zero and low Reynolds numbers, as is shown by Figure 5. The plot
obtained by Leiroz and Rangel for the ratio of the non-dimensional droplet vaporization rate in a
stream to an isolated droplet is quite similar to that determined by PSM and MOI for the average
correction factor for a linear three-drop array. There are no comparisons with experimental data.
Using the point source method, the average correction factor for a large array can be
reasonably predicted, but the approximation of the point source is valid only if l/a>>2. One of the
problems of this method is that, for arrays larger than 7 drops, the correction factor of the center
drop is very severe, in some cases even being negative. The error for the center drop increases
with the number of drops in the array. Setting the evaporation rate for the center drop to be zero,
knowing that, for large arrays, the center drop contribution to the average correction factor is very
small, could solve the problem.
Figure 2 Comparison of MOI and PSM results for three-drop linear array (Annamalai and Ryan, 1993)
28
Figure 3 : Comparison of MOI and PSM results for a five-drop array (Annamalai and Ryan, 1993)
Figure 4: Comparison of MOI and PSM results for a seven-drop array (Annamalai and Ryan, 1993)
29
Figure 5: Ratio of the non-dimensional droplet vaporization rate in a stream to an isolated droplet – numerical solution (Leiroz and Rangel, 1994) 3.2. Extension of the PSM for Non-symmetric Arrays
As presented above the point source method was evaluated only for symmetric arrays and
only against other theoretical results. However, it has been demonstrated by Liu and Ruff (2001)
that while free-floating droplet arrays can be generated, the droplets are seldom of the same size
or spaced evenly. For non-symmetrical arrays, under the same assumptions, Eq. (7) can be used
to calculate a correction factor that can then be compared to experimental data. For two droplet
array, the correction factors for droplet 1 and 2, if droplet sizes are different, are given by:
( ) ( )1221
212111 aaalaaalal −−=η
( ) ( )2122
221122 aaalaaalal −−=η
30
For a three droplet configuration, where droplet spacings as droplet diameters are
variable and droplets are mounted in the apices of a scalar triangle, the correction factors are
given by the relations below, obtained from equation (7) using Cramer’s Rule.
[ ] [ ]
22,1
212
3,2
3223,1
31
3,23,12,1
321
3,132,123,23,12,13,2321 2
1
1111
laa
laa
laa
lllaaa
lalallllaa
−−−+
−−+−+=η
[ ] [ ]
22,1
212
3,2
3223,1
31
3,23,12,1
321
3,232,113,12,13,23,1312 2
1
1111
laa
laa
laa
lllaaa
lalallllaa
−−−+
−−+−+=η
[ ] [ ]
22,1
212
3,2
3223,1
31
3,23,12,1
321
3,223,112,13,23,12,1213 2
1
1111
laa
laa
laa
lllaaa
lalallllaa
−−−+
−−+−+=η
3.2.1 Method Description
First, to demonstrate the types of solutions that can be obtained for these configurations,
the PSM will be applied for a two-drop configuration of non-equal diameters, progressing to a
three-droplet arbitrary configuration, with the drops mounted on the apices of a scalar triangle.
For this three-droplet array, a system of three equations has been developed. Based on this
algorithm, the correction factor can be determined for larger arrays of arbitrary configuration in
terms of droplet diameters and drop interspacing. While has not been yet developed, from the
results obtained we could infer that the same equation (7) can be used to generate a system of
algebraic equations for larger arrays of droplets, varying in size and locations. The solution of
such system is the similarity parameter described above, that accounts for quantifying droplet
interactions.
31
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50
L/a
η
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50
L/a
η PSM
0.8840.8750.822
isomm && /=ηDiameter (mm) Expt.
0.512 0.8600.476 0.7510.293 0.694
PSMPSM
0.8840.8840.8750.8750.8220.822
isomm && /=ηDiameter (mm) Expt.
0.512 0.8600.476 0.7510.293 0.694
isomm && /=ηDiameter (mm) Expt.
0.512 0.8600.476 0.7510.293 0.694
isomm && /=ηDiameter (mm)Diameter (mm) Expt.Expt.
0.5120.512 0.8600.8600.4760.476 0.7510.7510.2930.293 0.6940.694
Figure 6: Comparison between PSM and experimental data obtained in normal gravity, for a three droplet array performed by Liu. (2003); fuel: methanol, T=20C, p=1 atm.
Liu (2003) proved that experimental data obtained are in good qualitative agreement with
PSM and his findings are presented in Figure 6.
3.2.2 Results of the PSM for Non-symmetric Arrays
Calculations were performed for two-drop arrays of non-equal diameters and for three-
drop non-symmetrical arrays. The three-drop configuration consists of three drops of non-equal
diameter mounted in the apices of an equilateral triangle.
For a two-drop configuration, droplet 1 is larger than droplet 2 and the inter-drop spacing
cannot be less than the sum of the radii of the two drops (the drops do not collide and do not
coalesce). It has been observed that as the ratio of droplet radii increases, the smaller drop has
less of an affect on the larger drop and the correction factor of the larger drop is almost 1,
indicating that it behaves like an isolated droplet. However, the correction factor of the smaller
drop decreases dramatically to values significantly below unity, indicating that the evaporation
rate of the smaller drop will be considerably affected by the presence of the bigger drop. A
32
smaller drop that is in the wake of a larger drop will be affected even at large droplet inter-
spacing (l/a ~50). These results indicate that as the ratio of droplet radii increases, the droplet
interactions are present and strong even for large values of l/a, as shown in Figure 7.
When the droplets have comparable diameters, both are equally affected when they are
up to 15 or 20 radii apart. As the spacing increases, the droplet interactions are weaker and the
droplets behave like isolated drops. The diagram presented in Figure 7 illustrates this behavior.
Calculations were also performed for a three-drop configuration; the first drop is the
largest with the diameters decreasing for drops 2 and 3. As a condition of non-colliding and non-
coalescing, the spacing between drops cannot be less then the sum of the first 2 drops. Four
different three-droplet configurations were evaluated. In the first configuration, two droplets have
similar or nearly similar sizes and the third one has a much smaller diameter. As shown in Figure
8 and, the larger two drops behave like a two-drop array as we have seen in the previous analysis,
while the third, smaller drop is strongly affected by the other two. At closer distances between
droplets, the vaporization rate could be as small as 40% of the vaporization rate of an isolated
drop in similar conditions. The droplet interactions are predicted to be significant up to l/a~40.
The next three-droplet configuration we will consider is a single large drop with the other
two being considerably smaller but having similar diameters (). While the large drop behaves like
an isolated drop, the evaporation rate of the two smaller drops in the array decreases by up to
30% compared to an isolated drop. The behavior is similar to the previous case only for l/a>20.
When the droplets are closer than l/a=20, the effect on the smaller droplets is even stronger.
33
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30 35 40 45 50
l/a
η
η1, a1/a2=1.5
η2, a1/a2=1.5
η1, a1/a2=5
η2, a1/a2=5
Figure 7 Calculated correction factor for a two-droplet array with different droplet sizes (a1/a2=1.5, a1/a2=5)
a1/a2=1.1, a1/a3=5
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30 35 40 45 50
l/a
ηη1η2η3
Figure 8 Correction factor calculated for a three-droplet asymmetric array with two larger droplets of similar diameter sizes and a much smaller droplet in the wake of larger drops
34
a1/a2=1.3, a1/a3=7
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30 35 40 45 50
l/a
ηη1η2η3
Figure 9 Correction factor calculated for a three-droplet asymmetric array with two larger droplets of near similar diameter sizes and a much smaller droplet in the wake of larger drops
a1/a2=3, a1/a3=5
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30 35 40 45 50
l/a
η η1 η2η3
Figure 10 Correction factor calculated for a three-droplet asymmetric array with two smaller droplets in the wake of larger drop
35
a1/a2=1.3, a1/a3=2.5
0
0.2
0.4
0.6
0.8
1
1.2
0 10 20 30 40 50 60
l/a
ηη1η2η3
Figure 11 Correction factor calculated for a three-droplet asymmetric array with near similar droplet sizes
These examples could be considered as extremes. Usually, the droplets in a cluster have
similar diameters with ratios of droplet radii varying between 1.1 and 3 (Figure 11). In this case,
the array behaves like a single entity with all droplets being affected by the presence of the others.
The droplet interactions are significantly strong in the range of l/a from 2 to 20 and, as it can be
seen in Figure 11, the effect on all three droplets is similar.
Up to now, experimental data for comparison with these theoretical plots are not
available. The experimental method that will be developed at NASA Glenn Research Center will
be used to obtain data necessary for these comparisons. This data and the resulting comparisons
will be the topic of future research and presentations.
36
While waiting for experimental investigation, a numerical model has been developed that
will be based on estimation of the same similarity parameter as in PSM and the data will be
validated against PSM and other theoretical and available experimental data from past
investigations present in the literature. Also, the numerical simulation results will be used to
support our future experimental investigation still in progress.
37
CHAPTER 4: FORMULATION OF THE NUMERICAL SIMULATION
In a real spray, the distance between droplets is small enough to allow droplets to
interact. One important conclusion inferred by several researchers was that for real sprays, a drop
is affected by its nearest neighboring droplets. Therefore, isolated drop studies are difficult to
extend to descriptions of complete spray systems. The most comprehensive theoretical models of
isolated drops are generally fairly complex methods and not amenable to the analysis of droplet
arrays or clusters. Some of them (Imaoka and Sirignano, 2004, 2005 and Yang and Wong, 2002,
for example) are expandable to and/or treat array of drops, under certain assumptions, but will
require prohibitively long computational times or do not predict individual burning rates but
rather an array burning or vaporization rate. As presented in the previous review, theoretical and
experimental studies have been performed that examined arrays of drops and streams and many
of the aspects of droplets evaporation and droplets interaction have been revealed. These
investigations have pointed the way towards new areas of research one of them being the
developed in the present research.
4.1 Overview
The effect of two- and three-dimensional configurations on these phenomena could be
important and should be quantified. However, three-dimensional interactive effects on heat and
mass transfer remain to be determined. Studies on these topics will undoubtedly start with arrays
having only a few drops. Nevertheless, the studies should be extended to arrays of a greater
number of drops for more direct application to practical sprays. As presented in recent studies,
methods developed up to now have several drawbacks, as indicated in Imaoka and Sirignano
(2005) whose model itself being still in progress and does not include thermal radiation effects.
Another difficulty is that the majority of them have not been compared with experimental data
38
taken in similar conditions as the model described by the theory. Through the review of this
literature and the development of an experimental method, it was found that many of the
theoretical or numerical studies are not sufficiently flexible to mimic an experiment as it would
have to be performed.
Therefore, a three dimensional model of symmetric and asymmetric arrays of fuel droplets
burning under microgravity conditions have been developed. The numerical simulation along
with the future proposed experiment described further in the project is a unique combination of
investigative methods that will provide support for future investigations and for understanding of
droplet interaction phenomena.
4.2 Assumptions, Justifications and Implications
4.2.1 General Assumptions
As previously stated, the model will mimic (as close as possible) the experimental
investigation, supporting in this way the experiments performed by Liu (2003) under normal
gravity and future experimental data obtained through a microgravity experiment in which the
formation of dilute and dense clusters can be created and stabilized before combustion is begun
allowing the spectrum of droplet interactions during combustion to be observed and quantified. In
this numerical simulation, each drop in the array will vaporize and burn according to the
conditions created by the environment around the droplet and also the array can be considered as
an entity. Therefore, the reaction will happen only if the temperature and fuel and oxidizer
concentrations are high enough to sustain the combustion. Furthermore, if all the fuel in a droplet
is consumed, it will be removed completely from the array. This approach is closer to what
happens in reality in the evaporation and combustion of groups of droplets. However, the fixed
position of the droplets during simulation will not allow quantifying the effect of droplet motion
39
due to the Stefan velocities. This assumption is valid as long as we refer to an array of droplets
surrounded by other arrays of droplets (as in the middle of a spray cloud) where the movement of
a droplet relative to other droplets is negligible. Also, this assumption will be valid on the time
constraints imposed during an experiment, i.e., the data are collected after the igniters are
removed and burning of the droplets is self-sustaining.
The model developed in this study accounts for radiative heat transfer using a Finite Volume
Method, Stefan convection, diffusion and finite-rate chemical kinetics, and unity Lewis number
but does not account for forced convection and for internal circulation within the droplet. The
assumptions made for the model are similar to those used in previous studies, i.e., a quiescent
environment, near zero Reynolds number, and gas-phase variable thermo-physical properties. The
temperature inside the droplet is considered to be uniform and constant throughout the droplet at
a frozen moment in time. The specific heat and thermal conductivity of the liquid fuel is assumed
to be variable in time according to gas-phase temperature near the droplet. The liquid-fuel
specific heat will be obtained using the curve fitting coefficients from tables (Perry’s Chemical
Engineers Handbook) (Table 1), with the curve approximated as function of temperatures ranging
from 293K to 337K (vaporization temperature) using Eqn. (8).
1000/)( 45
34
2321 TCTCTCTCCc p ++++= [J/mol-K] (8)
Table 1: Curve fitting coefficients for methanol (for temperatures in centigrade scale)
C1 C2 C3 C4 C5 Coefficients
1.0580E+05 -3.6223E+02 9.3790E-01 0 0
40
Similarly, the variation of thermal conductivity is simulated using two linear functions of
temperature, one ranging from 298K to 323K, having the limit values of 0.202W/m-K to
0.195W/m-K and the second from 323K to 348K, whose values at the margins are 0.195W/m-K,
respectively 0.189w/m-K. The values are from tables found in Handbook of Chemistry and
Physics. Only gas-phase phenomena are considered to drive the droplet combustion around the
droplet. The liquid-gas phenomena are included. Due to finite rate fast chemical reaction, the
oxidizer is consumed at the flame sheet and the flame sheet itself is very thin, being the limit
between fuel and oxidizer (McGrattan (2004)). There is no soot yield considered although the
simulation is able to account for soot production.
4.2.2 Water Absorption
The model does not consider water absorption of methanol droplets, investigating just pure
methanol droplet combustion in dry air. Water absorption is an important phenomenon that is still
under investigation, being proved that methanol droplets can and do absorb water during drop
burning lifetime. The burning rates would be retarded with water being absorbed and then
vaporized. However, the amount of water absorbed is not significant at initial stages of
combustion (Ross, 2001), and including thermal radiation effects, the gasification rates are lower
by about 10% (Marchese et al.) when water/methanol droplets are considered. The conclusion
inferred by Choi and Dryer (2001) in their comprehensive review (Ross, 2001) is that the water
diffusivity is not that important in the calculation of burning rates, one of the important output
parameters of the current work. Therefore, water absorption into methanol droplets during
combustion is neglected for the purpose of this study.
41
4.2.3 Effect of Reynolds number on Interacting Droplets: Near Zero Re Number
An important issue in the evaluation of droplet evaporation and interactions is that the
relative Reynolds number, e.g., the Reynolds number based on the local droplet conditions and
the relative velocity between the droplet and the ambient. Many researchers concluded that for
low relative Re numbers (0 to ~30), the droplet interactions are very strong, much stronger than
for higher Re (>50). The lower Re number regimes (0-30) are more relevant to practical sprays
flows, after the droplets have been decelerated from the spray injection conditions. In fact, if the
drop is moving with the surrounding flow, the Reynolds number could be assumed to go to zero.
Most of the studies were performed mainly for high and intermediate Re numbers limits for a
vaporizing droplet array or stream. The zero and low Reynolds number limit for a vaporizing
droplet stream is the logical next step beyond the classical isolated droplet theory resulting in the
d2-law. This issue has been investigated using direct numerical simulation and approximate
methods, but experimental studies for this case are not available yet. The investigation of
unsupported arrays of drops for zero relative Re number could offer a better understanding of the
droplets interaction phenomena. A combined theoretical and experimental approach for
unsupported arrays of drops for low Reynolds numbers would be a significant research
contribution that could provide relevant insight concerning droplet interactions and the
characteristics of droplet evaporation for arrays of fuel drops.
4.2.4 Internal Circulation within the Droplet
The presence of the internal liquid circulation due to droplet generation and deployment
techniques and heterogeneity of the surface temperature (Marchese and Dryer, 1996) has been
demonstrated and such internal circulation inside a pure fuel droplet enhances mass and heat
transfer inside the droplet. Evaluating the effect of this contribution is quite complex and would
42
require further computational efforts. Formulating and solving a comprehensive solution of the
complete problem is quite difficult for current computers to handle. Marchese et al. (1999),
approximated this effect using an “effective” liquid-phase conductivity (“enhanced mass and
thermal diffusivities” (Marchese et al., 1996), but this introduces a new parameter that must be
determined. In his solution, he found the droplet heat-up transient becomes slower, while the
burning rate or extinction phenomena are not greatly affected. Generally, good agreement with
experimental measurements is observed, even when the presence of internal circulation in the
liquid-phase is ignored.
4.3 Numerical Solution
Fire Dynamic Simulator computer code (FDS), publicly released by National Institute for
Standards and Technology, is a computational fluid dynamic numerical model of thermally-
driven fluid flow that solves Navier-Stokes equations for low-speed flow. It is easy to use and
adapt and has the main advantage of having the source files available for download from a public
website, unlike the majority of commercial computational fluid dynamics codes. Since its first
release in 2000, the code has been reviewed, improved and validated through numerous
experimental and theoretical investigations in the past years as presented in the Technical
Reference Guide (McGrattan et al., 2005), making it a valuable tool for fire research. Another
advantage of FDS is Smokeview, a software tool that works concerted with FDS and was
designed to visualize in two or three dimensions numerical simulations generated by FDS.
Therefore, taking into account all the advantages presented by FDS it was easier, less time
consuming and more reliable to use rather than writing a personalized code for this investigations
or using a commercial CFD code.
43
Initially, a modified version of Fire Dynamic Simulator computer code (McGrattan et al.
2005) – version 3.0 (FDS) was been used to simulate ignition and combustion of isolated droplets
and droplet arrays. The code was modified to account for thermal radiation for finite-rate reaction
using direct numerical simulation. The radiation subroutine contained in the code was originally
developed for a mixture-fraction combustion model but was adapted to calculate radiation heat
loss through boundaries using the finite rate combustion modeling based on CO2 and H2O
absorption coefficients under ideal, non-sooting gray gas assumption. The modifications were
made on version 3 of FDS and the radiation model was based on “RadCal” (McGrattan et al.
2005, Grosshandler, 1993). The modified radiation solver used a finite volume solution for the
thermal radiation equation using point-wise values of the mass fraction, temperatures and partial
pressures to determine the absorption coefficient throughout the computational domain. The finite
volume solver is used to determine also the radiation heat flux from the gas-phase back to the
vaporizing/burning droplet. To reduce the computational time involved, the radiation solver
skipped computation of variable specific heats for the species in the gas-phase. The assumption is
valid considering that there is not much of variation of temperature during self-sustained
combustion.
The initial data obtained using the modified version were in qualitative agreement with
other theoretical results for isolated drop combustion and with the combustion of droplet arrays as
presented in our previous work (Ciobanescu and Ruff, 2004) and these results will be also
presented later in this paper in Chapter 5. One difficulty encountered was a slight over-prediction
of the burning rates caused by over prediction of the temperatures. This behavior was
acknowledged by FDS authors and the code was further improved (McGrattan et al. 2005). The
fourth version of FDS offers the possibility of including radiation in calculations that involves
one-step chemical reaction. Also, the data obtained using the new version recently released
44
(2005) gave similar results with the modified version developed previously to model droplet
combustion. One other problem of the modified version was the large amount of CPU time
required to run a simulation. The new version (FDS_v4) offers improved CPU times for similar
cases, requiring 20% to 30% less CPU time than the modified version based on FDS_v3,
depending on grid resolution and complexity of the droplet configuration.
Considering all the above and the advantages of a “debugged” and improved code that
eliminated much of the errors of previous version used to create the modified version, we decided
to use FDS_v4 in the remainder of this investigation. FDS_v4 does include a finite volume solver
for finite-rate reactions that is used with either the gray gas or band model, similar to that
included in modified FDS. Another advantage of FDS_v4 is the possibility of varying the fuel
properties as a function of temperature, namely the specific heat and thermal conductivity of the
liquid fuel, giving the opportunity of more accurate predictions. However, because of the number
of simulations performed using FDS_v3 and the fact that the modified version 3 and FDS_v4
provided identical simulations, results obtained using the modified version of FDS_v3 will be
presented in the following sections.
4.3.1 General Equations
For the gas-phase, the equations used were the equation of state and the conservation
equations for total mass, momentum, and chemical species. The energy equation is not directly
solved, but its source terms are included in the equation for flow divergence. The partial
derivatives of conservation equations of mass, momentum and energy are approximated as finite
differences, and the solution is built on a three-dimensional, rectilinear grid and it has time as
fourth coordinate. The divergence constraint contains the influence of conduction, mass diffusion,
enthalpy diffusion, thermal radiation, and chemical heat release on the velocity divergence. A
45
Poisson equation for the hydrodynamic pressure was solved using a fast direct solver. Second
order temporal and spatial discretization schemes were used (McGrattan et al. 2005). Most of the
theoretical basis for the model is described in the Technical Reference Guide of FDS (version 4)
(McGrattan et al. 2005). Highlights of hydrodynamic, combustion and radiation model and also
details of the numerical method used by this research will be given below from the above cited
source:
Mass Conservation:
0=⋅∇+∂∂ uρρ
t (9)
Species Conservation:
( ) lllll mYDYYt
′′′+∇⋅∇=⋅∇+∂∂
&ρρρ u (10)
Momentum Conservation
( ) τρρ ⋅∇++=∇+
∇⋅+
∂∂ fguuu p
t (11)
Equation of Energy Conservation
( ) ∑ ∇⋅∇+∇⋅∇+⋅∇−=⋅∇+∂∂
llllr YDhTk
DtDphh
tρρρ qu (12)
The f term is the external force of the fluid. The term ptp
DtDp
∇⋅+∂∂
= u is a material derivative.
Equation of State
( ) MTMYTp il ℜ=ℜ= ∑ ρρ0 , (13)
where 0p is the background pressure.
This equation is valid only for low Mach number where the time step in the numerical
algorithm is influenced only by the flow speed, and the modified state equation leads to a
46
reduction in the number of dependent variables in the system of equations by one. The divergence
of the fluid flow u⋅∇ is given from the material derivative of the equation of state and
substituting the terms of energy and mass conservation equations. The specific heat of the
mixture is considered constant and is the summation of the product between temperature
dependent specific heats of the species and mass fraction of the species in the mixture. The
enthalpy is defined similarly.
The approximate form of divergence used in the calculations is:
dtdp
pTcqYDdTcTk
Tc plrlllp
p
0
0,
111
−+
′′′+⋅∇−∇⋅∇+∇⋅∇=⋅∇ ∑∫ ρρ
ρ&qu (14)
To account for the pressure rise in sealed enclosures, the pressure equation is given by the
integral of the divergence over the computational domain.
A simplified form of the momentum equation is also used in this formulation. Following
McGrattan et al. (2005), we start from Eqn. (11), substitute g∞+∇=∇ ρpp ~ , apply the vector
identity for second term in the right had side of momentum equation and decompose the pressure
term using Eqn. (14)
ppp ~11~~∇
−+
∇=
∇
∞∞ ρρρρ (15)
This results in a simplified form of Eqn. (11):
( )[ ]τρρρρρρ
ω ∇++−=∇
−+
++×−
∂∂
∞∞∞
fgu
uu 1~11~
2
2
ppt
(16)
The pressure equation is obtained by taking the divergence of Eqn. (16). The pressure term in the
pressure equation is decomposed and approximated (Eqn. 17) using the average density and the
values of the pressure from the previous time step.
47
ppp ~11~~1∇
−⋅∇+
∇⋅∇=∇∇
ρρρρ (17)
Whether the extra pressure term neglected or approximated (to simplify the numerical solution
for the pressure equation) is determined by its contribution to the creation of vorticity. In the
model investigated here, restoring baroclinic vorticity is automatically invoked and is described
in the FDS Technical Reference Manual (McGrattan et al. 2005).
The binary diffusion coefficient of a species diffusing into other species is given by:
Ω
×=
−
sm
M
TTD
Dlmlm
lm
2
221
23371066.2
σ (18)
where ( ) 1112 −+= mllm MMM , the Lennard-Jones coefficient is the average of the
corresponding species coefficients and DΩ is the diffusion collision integral. It is assumed that
nitrogen is the background species, therefore:
( )2,, NlDNSl DD ρρ = . (19)
4.3.2 Combustion Model
According to the FDS Technical Guide, the one-step reaction considers the general reaction of
oxygen and a hydrocarbon fuel approximated by Arrhenius equation, in our case, methanol,
where the reaction rate is given by:
[ ] [ ] [ ] RTEba eOOHCHBdt
OHCHd −−= 233 (20)
The values for B, a, b, and E have been taken from Stiech (2003) and Westbrook and Dryer
(1981) (Table 2.):
48
Table 2: Reaction rate parameters for methanol Property Value Units
B = Pre-exponential factor
(“BOF”)
3.12E12
smolcm
⋅
3
E = Activation energy 125,604
kmolkJ
a = exponent for oxygen
concentration (“XNO”)
0.25 -
b = exponent for fuel
concentration (“XNF”)
1.5 -
Table 3: Properties for fuel and products of combustion for methanol air reaction
Lennard-Jones
potential parameters
Species Diffusion
coefficient
s
m 2
Viscosity
⋅ smkg
Thermal
conductivity
⋅ KmW
Molecular
weight
mol
g
σ [Å] k
ε [K]
CH3OH 1.32 E-5 96.27E-7 0.01565 32 - -
O2 - - - 32 3.467 106.7
H2O - - - 18 2.641 809.1
CO2 - - - 44 3.941 195.2
The reaction between fuel and air takes place at a finite rate given by Arrhenius equation
(Eqn. 20) and the reaction zone is an infinitely thin sheet where the fuel-oxygen concentration is
high enough to sustain the combustion (McGrattan, 2005). Species properties as well as the
49
stoichiometric coefficients must be specified. In any cell where the reaction is “on” the chemical
reaction time is much shorter that any convective or diffusive transport time scale. Nitrogen is a
background species and does not participate in the reaction except as a diluent. The properties of
fuel and products of reaction are tabulated below (Table 3.) and are according to Perry’s
Chemical Engineers Handbook and JANNAF tables:
For methanol, the effective heat of combustion is 19,937kJ/kg, the heat of vaporization is
1100kJ/kg, and the thickness of the liquid is the droplet diameter. Although the thickness is very
small, the fuel will be considered thermally thick that allows the thermal conductivity of the
liquid to be specified (see the input files in the Appendix B). Also the ignition temperature is
specified, for methanol being the boiling point (64.7oC). The model does not consider the
convection within the liquid droplet.
4.3.3 Radiation Model in the Gas-phase
FDS solver uses the Radiative Transport Equation for an absorbing, emitting and
scattering medium. A summary of the general model used by FDS and pertinent to our
investigation will be described below. This model is included in the FDS_v4 Technical Reference
Guide (McGrattan et al., 2005). The radiative transport equation used in FDS is
( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )∫ Ω′′′Φ+++−=∇⋅π
λλ πλσλλσλκ
4
,,4
,,,,,, dIBII ss sxssxxsxxxsxs (21)
where “ ( )sx,λI ” is the radiation intensity at wavelength λ, “s” is the direction of the intensity
vector, ( )λκ ,x and ( )λσ ,x are the local absorption and scattering coefficients, and B(x, λ) is
the emission source term.
50
FDS can operate in two modes when solving the RTE, specifically a gray gas model and
a wide-band model. Due to the lack of soot in the gas-phase, the lumped gray gas assumption
may produce significant over-predictions of the emitted radiation (McGrattan et al., 2005).
The wide-band model implies that for each band, the RTE can be derived and the source
term can be written as a fraction of the blackbody radiation. The total intensity is the summation
of all the intensities integrated over each band, and determined using the mean absorption
coefficient inside the band. The radiative heat loss term is the divergence of the radiation flux in
the gas. The gray gas model assumes that the spectral dependence is lumped into one absorption
coefficient and the source term is given by the blackbody radiation.
To select the right model to use in this application, the wide-band model has been tested,
using six spectral bands (CH4 as fuel) versus the gray gas model, for single droplet combustion.
The source term in the RTE is based only on the gas (CO2 and H2O mixture) temperature and
composition. Upon analysis, it was concluded that the slight increase in mass burning rates and
temperatures using wide-band model (less than 7% for mass burning rates and less than 3% in
temperatures) does not overcome the dramatic increase in the computational time. Therefore, for
the purpose of this research, the gray gas model for radiation solver has been chosen.
The RTE is solved using a finite volume solution similar to that used by Raithby and
Chiu (1990) and is based on “RadCal” developed by Grosshandler (1993).More details about the
model can be found on Technical Reference Guide (McGrattan et al., 2005). To increase the
accuracy of the radiation model, the number of solid angles may be increased and the time and
angle increments could be reduced. However, this technique will considerably increase the
computational time.
The modified version of FDS_v3 used the gray gas approximation of the mixture-fraction
model. The modifications consisted of creating a subroutine that will invoke the radiation solver
51
even if the absorption coefficient is not specified (the mixture-fraction combustion model) and if
the finite-rate reaction is present. The radiation solver would compute the mean absorption
coefficient based on gas temperature using one band. Considering that the size of the droplet is
relatively small and the radiative heat loss is lower than convective heat loss, it was assumed that
the over-prediction of the emitted radiation will not greatly affect the overall heat release.
Alternate methods of solving the RTE using FDS_v3 would also significantly increase the
computational time. Therefore, this approach seemed to be the reasonable choice. FDS_v4,
however, offers a more accurate modeling of gas-phase radiation at reasonable computational
times and has been used in many of the simulations. The data for radiative heat loss obtained
from the modified FDS version and FDS_v4 were compared and the results are in qualitative
agreement. The main improvement was in the computational efficiency of the code as well as in
code stability. Simulations of more complex cases were made possible using FDS_v4.
4.3.4 Numerical Algorithm
The numerical method will be very briefly described here; it is fully described in the
FDS_v4 Technical Reference Guide (McGrattan et al., 2005). For all scenarios, all spatial
derivatives are approximated by second order finite differences on rectilinear grid and the flow is
updated with a second order predictor-corrector time step. The scalar quantities are approximated
at the center of the cell and all vector quantities are defined at center of the cells faces, normal to
the face. For a DNS simulation, the diffusion coefficients, dynamic viscosity, and thermal
conductivity are defined at the center of the cells. For the time discretization, the pressure, density
and mass fractions are estimated using an Euler step from the initial condition quantities,
considered known. The divergence is calculated from these quantities and then the Poisson
equation is solved with a direct solver. The corrector step will adjust the thermodynamic
quantities at the next time step. The pressure is computed from estimated quantities and after that
52
the velocity vector is corrected. The mass and energy equation are combined using the divergence
of the flow field which is discretized on both predictor and corrector steps. The convective terms
are written as upwind-biased differences in the predictor step and downwind biased differences in
the corrector step (McGrattan et al., 2005), while the material and thermal diffusion terms are
central differences (same for corrector and predictor steps) and the temperature is calculated from
equation of state using the density.
For finite-rate reactions, the heat release is averaged over a time step for a given cell (ijk),
using the density of the gas in a cell, the mass fraction of the fuel and the heat of combustion
(specified in the input file). Because the specifics of the one-step reaction, the chemical time of
the reaction is much shorter than the time step of the solver. Therefore, all the processes will be
considered “frozen” at the beginning of the time step. The mass fractions of the species are
updated before the convection and diffusion update. The mass fractions of the fuel and oxygen
are calculated using a Runge-Kutta second order method, for each grid cell using the reaction rate
parameters as given data. The RTE is integrated over discrete solid angles and over the rectilinear
grid (Howell and Siegel, 2002). The detailed scheme is presented in reference (McGrattan et al.,
2005) and it has no particularity for the finite-rate reaction.
A peculiarity of the DNS calculations for one-step chemical reaction is the evaluation of
the convective flux to the wall (a wall being any “solid” surface defined as either boundaries of
the computational domain or user-defined obstructions, in our case the droplet surface).
For the thermally thick wall (or solid), a one dimensional heat transfer calculation is
performed at each boundary cell of the solid (the droplet, in the current study). The solid is
partitioned in a number of cells. The temperatures are updated using a Crank-Nicholson scheme
and the boundary condition
53
( )44 TdqqxTk rc εσ−′′+′′=
∂∂
− && (22)
is discretized, the radiative term being linearized.
The technical guide has further detailed descriptions of the discretization of momentum
and pressure equations. Of importance is to notice that due to very small time steps involved in
the simulations performed and fine grids used, the main constraint for the time step is the von
Neumann criterion. The pressure equation is obtained from the divergence of the momentum
equation as a Poisson equation which is solved directly.
4.3.5 Problem Geometry
The general configuration that will be solved using the numerical method presented in the
previous sections consists of a computational domain spatially discretized using rectilinear grid.
Generally, a coarse grid was used for the entire computational domain and a finer grid embedded
into the coarse grid was used around the droplet cluster. The coarser grid has its cell dimension
equal to the biggest droplet radius in the array and the finer grid has the cell dimension equal to
the smallest droplet radius in the array. The droplet cluster is positioned at the center of the
computational domain. Droplets are represented by cubic obstructions created using four or more
cells. A special feature of the code allows defining the edged as curves by “cutting” the sharp
corners and avoiding the creation of steep temperature gradients at cube corners. When visualized
using Smokeview, the droplets have a quasi-spherical shape. The geometric configuration of each
case studied will be presented in the Chapter 5 that will follow.
Each cluster is ignited using four igniters, equally positioned respective to the center of
the cluster. The igniters are defined as parallelepipeds and have the cell walls facing the droplets
54
heated to a specified temperature (3000ºC). The igniters are at 20ºC at the initial time (t=0) and
the temperature is increased using a defined linear function of temperature versus time for the
0.25s, time when the temperature reaches the maximum. The maximum temperature is
maintained for another 2.25s. After 2.5s from the initial time the igniters are removed from the
computational domain.
There is no temperature gradient normal to the surfaces, the solid is considered thermally
thick and the boundary layer is solved using DNS. Gas phase temperatures are defined at cell
centers, while the velocities and other vector parameters are defined at the cell face’ centers.
The planes that form the boundary walls of the computational domain are considered
open, denoting a passive opening to the outside.
4.4 Burning Rate Calculation
Although the mass burning rate is one of the FDS outputs, the results produced are just a
verification of the total heat release to the boundaries and should be carefully evaluated. To
estimate the droplet mass burning rate, a separate algorithm was developed, based on the
temperature, velocity, species mass fractions and density for each node in a plane of the
computational domain. In Figure 12 below, four of the six planes used to estimate mass burning
rates are shown. Each plane represents a slice through computational domain at a specified
coordinate (xmin., xmax., ymin, ymax zmin, or zmax). This slice is divided into rectangular cells
(following the user specified spatial grid) and, at each node, the specified combustion parameters
values are calculated stored by the code into an ASCII slice file. The file provides values for the
entire slice but the data used to determine the burning rate are only those that are inside the faces
of the cube defined by the intersection of the defined planes. For example, for a slice defined at
55
xmin, the face of the cube will be defined by the intersection of the slice with x-ymax, x-ymin, z-ymin
and respectively z-ymax planes.
Figure 12: Slice planes around the droplet
Given this geometry and knowing all computational parameters on the six faces of the
volume surrounding the droplet, the burning rate is obtain by numerical integration of Eqn (23)
( ) dAm ijij uρ=& (23)
over all six planes surrounding a droplet. Data used to calculate the burning rate are collected
after the igniters are removed and the mass fraction of the fuel, temperature, Stefan flow
velocities and density for each node are averaged for a time increment and for each drop in a
cluster. The slice position has been carefully chosen to be inside the flame front to account for the
parameters of the investigated evaporating droplet Figure 13.
56
Figure 13: Slice position
The code outputs instantaneous values averaged on each cell node using in the slice files
and data are extracted using an ASCII code. First, the summation of the product between average
density, average velocity and cell face area, has been calculated and then the results were
summed, yielding the mass burning rate around each droplet in the array.
∑∑=
=6
1
2
i navgavgb lm uρ& (24)
where l is the length of a cell, i (i=1 to n) is the number of the cube faces and n is the
number of the cells on a side face of the cube defined by the intersection of the slice planes.
To account for the spherical shape of the flame, this summation is multiplied by the ratio
between the averaged sphere surface and the cube surface area: The average surface of the
equivalent sphere for the cube is obtained by the arithmetic average between the surface of the
sphere inscribed in the cube and the surface of the sphere circumscribe to the cube. This
correction was used only to overcome the under-prediction of the burning rates due to insufficient
grid resolution inside the flame zone. A similarity parameter or correction factor η was then
computed for each drop in an array.
isomm&
&=η (25)
57
The correction factors were calculated imposing the no collision or coalescence condition
generated for each configuration. Figure 14 shows an example of the format of the slices in the
computational domain, obtained using Smokeview_v4.0, a visualization code provided by NIST
team along with FDS.
Figure 14: Visualization of the slice positions in a computational domain for two droplet array
For multiple droplets configurations, several slices will be defined in the input files for
FDS, around each droplet being defined four slices providing necessary data.
4.5 Ignition Characteristics
For the finite-rate model, the heat loss controls the combustion phenomena. The
minimum ignition energy was analytically calculated using models presented by Kuo (1986) and
Turns (2000) as being the energy necessary to raise the temperature of a spherical volume of air-
fuel mixture to the stoichiometric flame temperature.
58
3,min 6 qstoichairairp dTcE πρ ∆= (26)
( )
5.0
1ln
+Φ
=B
dair
fq ρ
ρ (27)
Where dq is the air-fuel mixture diameter, Φ is the equivalent ratio and B is the Spalding transfer
number. Following this model, the following Minimum Ignition Energy (MIE) values have been
found for methanol.
Table 4 Minimum ignition energy for various pure methanol droplet diameters Drop diameter
[mm]
0.3 0.5 1.0 2.0
MIE [J] 0.41 1.91 15.3 122
In our model, the ignition is performed by an external source and heat is supplied to the
environment around the droplet. The external source is defined as four parallelepiped
obstructions, whose walls are hot in the proximity of the droplets (the walls facing the droplets).
Their size is similar to the size of the droplet. It was observed that the reaction begins to occur
after at least 0.4s when four igniters having 3000ºC wall temperature are applied. These findings
are in concordance with the experimental and numerical data obtained by Marchese and Dryer
(1999) where they observed the beginning of the reaction after 0.45s during their computational
investigation for an n-heptane 1.325mm droplet and surrounding air temperature of 1208K. Also,
based on previous experiments performed by several researchers (Faeth and Olson, 1968, Dryer
and Marchese, 1999), for methanol droplets the ignition time (the time required to get to flame
59
temperature) varies from 0.5s to 1.5s, depending on droplet diameter and surrounding
temperature.
To determine the ignition time, several cases were simulated, varying the number of
igniters from one to four, igniters’ wall temperature in the range of 500ºC to 6000ºC and the
ignition time from 0.5s to 2.5s.
For the heat release to be high enough to sustain the combustion reaction, it was
determined that four igniters were necessary and that they had to be energized for at least 1.5s.
Due to steep temperature gradients created during the sudden ignition, a ramp function of time
has been applied for the first 0.25s and the total ignition time was increased to 2.5s. During this
time, the temperature will linearly increase from 20ºC to 3000ºC.
A sensitivity study was performed to determine if the igniter temperature impacted the
burning rate of the droplets. Igniter temperatures between 3000ºC and 6000ºC were evaluated
and no influence on the combustion parameters during or after ignition was observed.
For multiple droplet combustion, a shorter ignition time was applied (2.0s) due to the
increased mass flux into the reaction zone.
4.6 Limitations of the Model and Error Margins
According to FDS_v4 Technical Reference Guide (McGrattan et al., 2005), the general
limitations applicable to our model are related to the rectilinear geometry, fire growth and spread,
combustion model and radiation. The FDS_v4 code is limited to all cases involving low speed
flow, ruling out any scenario that involves flow speeds approaching speed of sound.
The only limitation of our model due to the use of the rectilinear grid is in some
situations where certain geometric configurations do not conform to the rectangular grid. To
60
offset the effect of sharp edges, there is a technique in FDS_v4 to lessen the effect of “saw-tooth”
obstructions used to represent nonrectangular objects, and it was used to reduce steep temperature
gradients at the corners of the cubes representing the droplets. Caution has to be involved when
using the “saw-tooth” feature because the volume is reduced by this approximation of the curved
boundaries, mostly on a coarse grid, and the corresponding (by volume) droplet diameter could be
significantly smaller.
While the code allows using multiple meshes that “leak” one onto the other, meaning that
the coarser mesh will uses the data from the finer grid at its boundaries, when a finer grid
embedded into a coarser grid, both grids may behave independently and no data will “leak” into
the coarser grid. In some cases the overlapping of the grids may cause unsteadiness, leading to a
disruptive burning phenomenon. This can be overcome by using the grid meshes that share a
boundary instead of overlap.
For models where the heat release rate is predicted, the uncertainty of the model is
generally higher, the reasons for this being that the physical processes of combustion, radiation
and solid phase heat transfer are more complicated than their mathematical representations in
FDS and that the results of calculations are sensitive to both the numerical and physical
parameters (McGrattan, 2005). However, local velocities calculated using FDS_v4 are in the
range of 5% to 20% of experimental measurements and do not introduce significant deviations in
the predicted quantities greater than experimental investigations.
The combustion model involves the finite-rate assumption and this global approach for
combustion is still an area of research. Generally, most of numerical and analytical approaches
used both mixture fraction or finite-rate reaction but both of them were based on one step
chemical kinetics, their analyses producing reliable results while predicting combustion
61
parameters. However, if the level oxygen concentration at infinity is below 10%, the mixture
might not burn unless certain modifications are made to the input files.
Radiative heat transfer is included in the model via the solution of the radiation transport
equation for a non-scattering gray gas. The equation is solved using a technique similar to finite
volume methods for convective transport. One of the sources of error is the radiative heat flux
from the fire to the fuel surface. The error is due to a combination of insufficient grid resolution
in the boundary layer, and uncertainty in the absorption coefficient and flame temperature
calculated by the numerical code. As a result, the heat flux to the fuel surface is often over-
predicted (as per validation results presented in Technical Reference Guide, McGrattan et al.
(2005)). The radiative flux in the near-field, where coverage by the default number of angles is
much better, is better predicted.
The accuracy in calculation of flow velocities is very good, knowing that a corrector
predictor scheme is used to solve the divergence. The errors in calculating the normal velocities at
one cell face are several orders of magnitude less than the characteristic flow velocity. Using a
direct fast solver of the Poisson (pressure) equation had led to relative errors between the
computed and the exact solution of the discretized Poisson equation smaller than 10-12
(McGrattan, 2005).
One of the most important errors is related to the discretization of governing equations.
FDS_v4 uses a second-order accurate spatial and temporal numerical schemes, meaning that the
discretization is directly affected by the grid resolution. Halving the grid for example will reduce
the discretization error by a factor of 4, but that does not have to have a similar reflect for the
output results. Sensitivity tests have to be performed to quantify the errors on the output
quantities (sensitivity analysis and validation will be presented in the following chapters).
62
CHAPTER 5: RESULTS OF THE DIRECT NUMERICAL SIMULATION
5.1 Overview
Several factors have been studied before starting to investigate the combustion of
multiple droplets. For example, the sensitivity of the temperature and burning rates to domain size
and grid size (number of cells into a domain and cell size), ignition time, temperature of the
igniters and number of igniters used must be determined. As it has been shown in previous
sections, a large diversity of experimental, numerical and analytical studies of isolated droplets
have been developed in the past decades, providing data with which to compare the numerical
simulations of single droplet combustion.
The assumptions made in the beginning of Chapter IV are valid for all the cases studied
and presented in the next sections. The droplets combustion occurs in a quasi-quiescent
environment, considering unity Lewis number, Stefan convection, diffusion, gas-phase variable
thermo-physical properties, radiative heat transfer, and finite-rate chemical kinetics. There is not
considered water absorption by the methanol droplets, nor internal circulation or forced
convection considered.
The droplets are considered fixed in space and they do not move due to Stefan velocities.
However, the velocities due to thermal convection are considered when calculating burning rates.
The Reynolds number for all the investigated cases is considered to be near zero. If for single
droplet combustion, as shown in the next subchapter, the Reynolds number varies from 0.2 to 0.5,
following similar calculations, for two and three droplet arrays, the Reynolds number varies from
0.3 to 0.05.
63
To investigate grid and domain sensitivity as well as to validate the code against
theoretical and experimental results provided by studies found in literature, we developed single
droplet simulations that will be presented in the following sections. Therefore, the next step was
to investigate multiple droplets arrays, starting with two-droplet arrays and continuing with three-
droplet arrays.
5.2 Isolated Droplet Combustion
General assumptions stated previously in the above sections apply to all of the isolated
droplet combustion cases studied. In addition to that, the assumption of near zero Reynolds
number was based on average Re number calculated using average density and velocity values at
L=1mm distance from the droplet. The viscosity value used is from Table 3.
µρuL
=Re (28)
where the average density is in the range of
−= 328.025.0mkgρ , the average velocity is in
the range of
−=
smu 017.00083.0 , and viscosity
⋅−=
smkgE 0727.96µ .
The average Reynolds number obtained varies from 0.23 to 0.45 as function of droplet
diameter (droplet diameter varies from 2mm to 1mm). Therefore, the assumption of near zero
Reynolds number holds.
64
5.2.1 Grid Characteristics and Sensitivity Simulations
The grid characteristic dimension or the optimal grid size was initially established by
estimating the characteristic flame diameter using
52
=
∞∞
∗
gTcQD
pρ
& (29)
This first estimation is stating value. The time step is very important in establishing the
grid size. Therefore, the estimated value of the cell size will be adjusted using the time step
increment. This adjustment is made iteratively, by running several cases, using the as initial case
the estimated value given by Eq. 29. For the next cases, the grid will be gradually increased as
number of cells. A factor that has to be monitored is the time step size from the output data file. If
the time step starts to decrease dramatically, the cell size has to be increased for the next case.
The size of the time step is normally set automatically by dividing the size of a grid cell
by the characteristic velocity of the flow (McGrattan et al., 2005). During the calculation, the
time step is adjusted so that the CFL condition is satisfied. The default value of the time step is
( ) gHzyx 31
5 δδδ where δx, δy, and δz are the dimensions of the smallest grid cell, H is the
height of the computational domain, and g is the acceleration of gravity. Another condition that
has to be satisfied when using small fine grids (cell less than 5mm) is
1111,,max2 222 <
++
zyxt
ckD
p δδδδ
ρυ , (von Neumann criterion) (30)
65
where viscosity, material diffusivity and thermal conductivity were used. If the velocities
developed during combustion are too high compared to grid cell dimension, the time steps
become very small, causing the code to enter an infinite loop.
Another parameter considered was the concentration of methanol vapor around the
droplet. The fuel concentration has to be high enough for the reaction to take place. However,
since the mass fraction of fuel is averaged over a cell, the value will generally be under-predicted
and, if the grid is too coarse, ignition times will have to be extremely long or no ignition will
occur. The use of finer grids can solve this problem and it was found that a cell dimension of one
half of the smallest droplet diameter used in the array or smaller is sufficient for a droplet to burn.
Due to the complex nature of the factors that had to be considered, including the total CPU time
for each case (reducing cells size by half will incurs a 16-fold increase in total CPU time); each
case had to be evaluated to determine the optimal grid. There was no general grid dimension set
for all cases.
Grid sensitivity tests (Table 5) have been performed in addition to those already
performed by the authors of FDS and described in the technical guide. To study grid sensitivity
for the current investigation, several cases were created using different grid sizes for a single 2
mm droplet. Droplets were ignited using four igniters, at 3000ºC. Igniters were removed after
2.5s for the cases studied.
The maximum temperature is averaged in a cell and because the flame sheet is very thin,
and the temperature is averaged at the center of each cell, the adiabatic flame temperature will not
be achieved (Note: for methanol, the adiabatic flame temperature is 1904 oC). It was observed
that using smaller domains, for a 4 mm droplet FDS_v3 and the modified version of FDS_v3
over-predicted the flame temperatures. As we reduce the cell size the temperatures will slightly
66
increase for FDS_v4. A smaller domain was used for sensitivity analysis under the same
conditions as the previous analysis and the same conclusions were reached. The maximum
temperature will increase by about 20% when the number of the cells in the grid increases by 23
but with a dramatic cost in computational time (almost 10 to 15 times longer). Also the radiation
subroutine increases the CPU time by about 25%. The data were collected at 1.5s after the
igniters were removed from the computational domain and averaged for a sampling time of
0.002s. Mass burning rates were slightly affected by the grid size, increasing the grid size by a
factor of 2 for each axis caused in increase in the burning rate of about 8.5%. The calculated mass
burning rates were 2% to 14% higher than the bulk mass burning rate predicted by the code using
the total heat rate due to combustion.
Table 5: Grid sensitivity analysis as a function of temperature for a 2 mm droplet burning in air and a domain size of 64mm/side
Max. Temperature [oC] Case Number of cells
FDS_v3
modified
FDS_v4
no Rad.
FDS_v4
Rad.
Case 1 64x64x64 1210 1051.8 1008
Case 2 80x80x80 1140 1108 1096
Case 3 96x96x96 1190 1220 1153
The mass burning rates are governed by the Arrhenius equation and therefore affected by
the variations of the temperature due to the asymptotic relation between mass burning rates and
temperature. Considering this aspect, the computational domain size has to be large enough to not
influence the temperature. As the domain increases in size, the combustion parameters become
67
independent of the domain size as shown in Table 6 below. For single droplet, burning rates
become domain size independent for dimensions larger than 40 mm x 40 mm x 40mm (403 mm).
Table 6 Domain size dependence for a 2mm droplet burning in air, using a grid cell size of 1mm
Case Domain size
[mm]
Max. Temperature
[oC]
Burning rate
[1.0E-07*kg/s]
Case 4 203 1155.5 2.475
Case 5 303 1084 2.678
Case 6 403 1010.2 2.753
Case 1 643 1008 2.757
Temperature Distribution after 4.0s of Simulation (Case 4)
0
200
400
600
800
1000
1200
1400
0.E+00 2.E-03 4.E-03 6.E-03 8.E-03 1.E-02 1.E-02 1.E-02 2.E-02 2.E-02 2.E-02
x [m]
T [ºC]
Figure 15 Temperature distribution for a 203mm computational domain for a single droplet burning in atmospheric pressure (case 4), positioned at the center of the domain.
68
Temperature Distribution after 4.0s of Simulation (Case 6)
0
200
400
600
800
1000
1200
0.E+00 5.E-03 1.E-02 2.E-02 2.E-02 3.E-02 3.E-02 4.E-02 4.E-02
x [m]
T [ºC]
Figure 16 Temperature diagram for a 403mm computational domain for a single droplet burning in atmospheric pressure (case 6), positioned at the center of the domain
Another analysis performed concerned the variation of the temperature and burning rates
with grid size (cell size). Cases were investigated employed cells of 1mm and 0.5mm, for various
domain sizes.
Doubling the number of cells of the grid for each axis (the number of cells increase by a
factor of 8), for same domain size, the burning rates increase by less than 8%. The mass burning
rates increase by about 11% as the domain size is increased from 203mm to 403mm for grid cell
size of 1mm and 5.5% for a cell size of 0.5mm. For the finer grids, the burning rates are not as
sensitive to the domain size.
69
Table 7 Grid sensitivity analysis for a 2mm droplet burning in air
Case Domain size
[mm]
Number of cells /
cell size [mm]
Max. Temperature
[oC]
Burning rate
[1.0E-07*kg/s]
Case 4 203 203
1mm
1155.5 2.475
Case 9 203 403
0.5mm
1197.8 2.67
Case 5 303 303
1mm
1084 2.678
Case 12 303 603
0.5mm
1074.5 2.704
Case 6 403 403
1mm
1010.2 2.753
Case 13 403 803
0.5mm
1016.5 2.815
Increasing the number of radiation angles from 100 to 200 and also decreasing the angle
increment from 5 to 3 time steps does increase the maximum calculated temperature by 9% to
10% due to a more accurate calculation of radiation loss (decreasing the heat loss due to
radiation) but there was less than a 5% increase on predicted mass burning rates compared with
the 203 grid size case). However, the computational costs are dramatic (the CPU time doubled
compared to using the wide band model).
Nevertheless, a combination of fine grid resolution and a fine tuning of the RTE solver
will provide a more realistic simulation. Due to the extensively long computational time, this type
of case will not be investigated at this time, but the results from previous tests provided us with
enough elements to conclude that this solution is possible.
70
The calculated burning rates are in excellent qualitative agreement with the d2-law and
the values obtained are in excellent agreement with analytical solutions obtained by direct
calculation using theoretical droplet combustion model presented in Turns (2000). The mass
burning rate calculated using the ideal droplet combustion yielded a value of 3.83E-07 kg/s for a
2 mm droplet diameter and the calculations are presented in Appendix A. However, studies
performed by Faeth and Olson (1968), Marchese et al. (1996, 1999), Dryer et al. (1996), Dietrich
et al. (1995, 1997, 1999) concluded that the experimental values of mass burning rates to be
smaller that the values calculated using d2-law.
The burning rates were found to be affected by the grid size, and less affected by the
domain size (slightly more affected for small computational domains); therefore a grid size that
satisfies the ignition condition, i.e., cell size to be at least half of the smallest droplet in the array,
will be used for the simulation of clusters of droplets. Combustion parameters for two and three
droplet arrays become domain independent for a domain size larger than 403mm. If the distance
between droplets increases, the domain size has to be increased accordingly.
5.2.2 Validation Tests
For validation purposes, several test cases were created to investigate droplet combustion
using a single droplet. Comparisons for multiple droplets and arrays will be presented in
following sections) For this case, the droplet diameter was defined as a 2 mm solid cube filled
with liquid methanol that vaporizes with smooth edges to simulate a spherical droplet and reduce
steep gradients at the droplet surface. The grid size is specified to be 64x64x64 (1mm cell
characteristic dimension). Ignition occurs by four igniters that rapidly ramp up to 3000K at the
start of the simulation. The igniters are removed from the domain after 2.5 sec. All data are
collected 1.5s after igniters are removed and the duration of simulation varies from 1.5s to 20s.
71
Burning rates and temperature variations have been evaluated through these simulations. The
oxygen concentration has been varied from 10% O2 by volume to 75% O2 by volume and the
results are presented in Table 8.
Burning rates constants K were estimated using the d2-law and the ideal combustion
model for an isolated droplet (Turns, 2000).
sl
f
r
mK
ρπ2
&= (31)
where, lρ is methanol density, sr is droplet initial radius and fm& is the mass burning rate.
Table 8 Validation analysis for a 2.2 mm burning droplet in air at 10%, 15%, 21%, 35%, 50% and 75% oxygen
Case: Oxygen
Concentration
Max.
Temperature
[oC]
Burning rate
[kg/s]
Equivalent
K [mm2/s]
K [mm2/s]
Marchese et
al. (1999)
Case 15 10% 375 no burning - -
Case 16 15% 1060 2.73E-07 3.56E-01 0.4E-01 18% O2
Case 17 21% 1010.6 3.29E-07 4.29E-01 0.35 – 0.48
Case 18 35% 1070 3.54E-07 4.62E-01 6.0E-01 30% O2
Case 19 50% 1126.5 4.07E-07 5.31E-01 -
Case 20 75% 1121.8 3.83E-07 5.00E-01 -
The low mass burning rate value obtained for the 75%O2 case is caused by the flame
being too close to the droplet. The slice is “cutting” through the flame and the instantaneous
densities and velocities are not adequately resolved using this grid resolution (cell size is 1 mm).
72
For a more accurate prediction of the burning rate, a finer grid would be needed for this particular
case, although the error is less than 10% of the equivalent ideal burning rate calculated using d2-
law, and may be considered acceptable. However, the data from this particular simulation are
excluded form the final analysis.
The graph from Figure 17 was obtained by using the same slope-intercept 1.0 as in Figure
18 and estimated burning rates constants K as slopes. The burning rate constant K was estimated
using Eqn. (28), considering ideal combustion. The K values are less than 10% lower than those
obtained by Marchese and Dryer (1999) through numerical and experimental predictions, which
is a very good quantitative agreement (within the quoted experimental uncertainty). The lower
values for K could be also attributed to slight over-prediction of thermal radiation by FDS_v4
code and also by the fact that their predicted K values are obtained over the first 1.5s of burning
history, while the estimated K values tabulated above from are obtained from our numerical data
collected after 3.0s of burning time (considering that the incipient reaction starts after 0.5s of
simulation). The simulation results for low and high oxygen concentrations are in good
qualitative agreement with numerical and experimental data obtained by Marchese and Dryer
(1999) following a similar trends as can be noted from the figures below.
For the low oxygen concentrations, the flame diameter will constantly increase and the
flame will move further away from the droplet, as concluded by Choi and Dryer (2001).
The flame around the burning 2 mm diameter droplet simulated by the heat release rate in
our numerical investigation is increasing in diameter moving further away from the droplet
surface (see Figure 19 and Figure 20 below).
73
0.00
0.20
0.40
0.60
0.80
1.00
0 0.5 1 1.5 2
t/D02 [s-mm-2]
D2/D02
15% O221% O235% O250% O2
Figure 17 Numerical estimated burning rates constants for a 2.2mm droplet burning in different oxygen/nitrogen concentration
Figure 18 Experimental and numerically predicted data for initially pure methanol droplets burning in various nitrogen/oxygen environments at 1 atmosphere (Marchese and Dryer, 1999).
74
Figure 19: Flame position for a 1.2 mm droplet burning in 15% oxygen, at 0.8 s of burning time without igniters.
Figure 20: Flame position for a 1.2 mm droplet burning in 15% oxygen, at 1.1s of burning time without igniters.
75
The flame position is considered to be at the location of maximum heat release rate. As
can be observed from the figures below, as the time increases, the flame reduces in diameter and
reaches a quasi-steady diameter in the proximity of the droplet. Also the maximum temperature
and mass burning rate estimated at 3.6s (~1.0s after the igniters are removed), are higher than for
a droplet burning in 21% oxygen. For the case of a droplet burning with 75% oxygen, the flame
position is steady near the droplet at earlier times than for droplet burning with 50% oxygen.
For higher oxygen concentrations, the normalized square of the droplet diameter as a
function of normalized time has a steeper slope, corresponding to a larger mass burning rate,
whereas for the low oxygen concentrations, the slope is less steep, corresponding to a reduced
mass burning rate (Choi and Dryer 2001). As observed, the simulation predicted a gradual
increase in the flame diameter as the oxygen concentration decreased, in good qualitative
agreement with the data presented by Choi and Dryer (2001).
Figure 21 Flame position at 3.0s of simulation for a 1.2 mm droplet burning in 50% oxygen
76
Figure 22: Flame position at 4.0s of simulation for a 1.2 mm droplet burning in 50% oxygen
Figure 23: Flame position at 5.0s of simulation for a 1.2 mm droplet burning in 50% oxygen
77
Another validation test was the variation of the burning rates with and without radiation.
For small computational domains and/or coarse grids, the burning rates are under-predicted,
mainly due to higher temperatures developed when radiation solver was turned off. Larger
enclosures and finer grids were necessary to ensure adequate domain size and grid independence
for combustion parameters and to predict burning rates in agreement with theoretical results. It
was observed that for numerical simulations without radiation, the mass burning rate would
stabilize after 6 sec of simulation, while for those with radiation, the stabilization occurs much
sooner (after 4s of simulation).
Table 9 Variation of burning rates with time, case with radiation
Time [s]
2.6s 2.8s 3.0s 3.2s 3.4s 3.6 3.8 4.0 4.5 5.0
T [ºC]
Burning rate
[1E-07*kg/s]
1428
2.84
1315
2.69
1261
2.61
1228
2.58
1201
2.56
1184
2.5
1170
2.488
1156
2.475
1554
2.474
1150
2.474
The simulation performed does not allow observing the regression of the droplet while
burning, therefore, is not possible to have a similar plot as those obtained by Marchese et al.
(1999). However, a similar temperature evolution is noticed from our simulations (see Table 9).
During ignition the temperature will abruptly increase, peak and then decrease with time.
78
Figure 24 Comparison of temperature profiles as a function of droplet radii and burning times, with non-luminous radiation considered. Initial conditions: n-heptane, drop diameter, 3.0 mm; temperature, 298 K; atmosphere, air at 1atm pressure (Marchese et al., 1999).
In cases without radiation and smaller computational domains (see Table 10), the
temperatures increases rapidly and accordingly the burning rates are higher. In these cases, the
oxygen concentration in the domain decreases rapidly, thereby slowing the combustion reaction
rate. No burning rates could be calculated from these simulations. As the computational domain is
increased the combustion process becomes steady and mass burning rates and temperatures are in
the range of the experimental and theoretical data described in literature (Marchese et al. (1999),
Kumagai (1971) and King (1996)).
The numerical model predictions are also in good qualitative agreement with the model
developed by Marchese et al. (1999) for a 5mm methanol/water droplet for the case where only
pure methanol was considered and also with the numerical and experimental results of Kumagai
(1971) and King (1996) for a 0.98mm n-heptane burning droplet. Their data shows a slight
reduction in burning rates when radiation is considered. Comparisons are made only with the 0%
79
water concentration case from Figure 26. The figure below shows this trend, using an equivalent
burning constant estimated using Eqn. 31. Also, as the domain size increases, the burning rates
are more accurately predicted, as they are when the number of the cells in the grid is increased.
As previously stated, the burning rates without non-luminous radiation will become grid and
domain independent for larger domains and finer grids that those cases in which non-luminous
radiation is considered.
Table 10 Burning rates for a 2mm droplet with and without radiation
Case Domain size
[mm]
Number of cells /
cell size [mm]
Max. Temperature
[oC]
Burning rate
[1.0E-07*kg/s]
K
[mm2/s]
Case 4 203 203
1mm
1155.5 2.475
Case 8 (no Rad.)
203 203
1mm
1755 -
Case 9 203 403
0.5mm
1197.8 2.67
Case 14 (no Rad)
203 403
0.5mm
1998 -
Case 6 403 403
1mm
1010.2 2.753 3.591E-01
Case 10 (no Rad)
403 403
1mm
1074.2 2.80 3.64E-01
Case 13 403 803
0.5mm
1016.5 2.815 3.673E-01
Case 11 (no Rad)
403 803
0.5mm
1102 2.955 3.725E-01
80
Numerical estimated K for a 2 mm droplet burning in air
0.00
0.20
0.40
0.60
0.80
1.00
0 0.5 1 1.5 2 2.5
t/D02 [s-mm-2]
D2/D02 80^3 (Radiation)
80^3 (no Radiation)no Radiation
Radiation
Figure 25 Numerical estimated burning rate constants for a 2 mm methanol droplet burning in air at 1atm, with and without considering non-luminous radiation.
Figure 26 Measured and calculated diameter squared for 5mm methanol/water droplets (Marchese et al., 1999.
81
Figure 27 Droplet combustion predictions (with and without non-luminous radiation considered) compared with the numerical results of King (1996) and the experimental results of Kumagai (1971). Initial conditions: drop diameter, 0.98 mm; temperature, 298 K, air at 1atm pressure Marchese et al. (1999)
Figure 28 Flame position approximated by the position of maximum heat release for a 1 mm droplet burning in air after 3.1s of simulation with non-luminous radiation included (0.5s of independent burning)
82
Figure 29 Flame position approximated by the position of maximum heat release for a 1 mm droplet burning in air after 3.6s of simulation (1.1s of independent burning) with non-luminous radiation included
Figure 30 Flame position approximated by the position of maximum heat release for a 1 mm droplet burning in air after 4.0s of simulation (1.5s after ignition is off) with non-luminous radiation included. The slice position is 1 mm behind the droplet.
83
Comparing the model predictions for a 1mm methanol droplet burning in air at
atmospheric pressure (Figure 30) with experimental and numerical data of Kumagai (1971) and
King (1996), a good quantitative agreement can be observed for the flame position. In both
numerical and experimental data extracted from Figure 27, the flame diameter is about 6 mm for
the n-heptane droplet, which is similar with the numerical prediction of our model for a 1 mm
pure methanol droplet, whose flame diameter is about 7.5mm as observed in the above figures. It
also has a similar quasi-constant flame diameter as in the Kumagai experiment and King
numerical model. The initial increase of flame diameter cannot be predicted by our model due to
the presence of the ignition stage at the beginning of simulation.
After evaluating the sensitivity and validation tests, we may conclude that the combustion
parameters predicted by this numerical model for single droplet are in very good qualitative
agreement with theoretical and experimental results presented in the literature, quantitatively
being lower by 10% to 15% than experimental and numerical models of Marchese et al. (1999),
Kumagai (1971) and King (1996). The model is amenable for analysis and prediction of mass
burning rates with a reasonable margin of error (less than 10%) for an adequately grid resolved
domain. Single droplet combustion predicted by the code follows the d2-law as shown in the
Table 9 where the mass burning rates remain constant throughout the combustion process,
meaning that a constant burning rate K can be calculated from the predicted data. The data
provides confidence that calculated parameters adequately characterize the combustion process of
single and multiple droplets.
5.3 Combustion of Droplet Arrays
Isolated droplet studies cannot fully describe the complex nature of spray behavior;
therefore, the study of droplet arrays is an important step forward towards achieving more in-
84
depth understanding of combustion processes in a spray. One important task on this path is to
investigate droplet interactions in a cluster of droplets. In the previous sections a thorough
analysis of single droplet combustion obtained using FDS_v4 yielded reliable numerical model to
predict mass burning rates for isolated droplets of different sizes as well as for droplets in an
array.
The next sections will describe the analysis of multiple droplet arrays and droplet
interactions, beginning with two droplet symmetric and asymmetric arrays and continuing with
three droplet asymmetric arrays. The numerical data are compared with PSM results and with
other available data from literature.
5.3.1 Two Droplet Arrays: Modified Version of FDS_v3
During the earlier stages of this investigation, simulations were performed using a
modified version of FDS_v3. Because of the extensively long CPU times needed to run each case
for droplet diameters of 2 mm and smaller, the maximum size of the droplets was increased to 4
mm. Thus, for symmetric two-droplet arrays, the droplet size is 4 mm and for the asymmetric
case, the larger droplet is 4 mm and the smaller droplet is 2 mm. The grid resolution is 323 for a
643mm computational domain. Although as observed in the figure below (Figure 31), the
correction factor predicted for various l/a values follow the same behavior predicted by the PSM
developed by Annamalai and Ryan (1993), the actual burning rates predicted by the code were
40% to 50% higher. In the figure and tables below, the index 1 is attributed to the larger droplet,
and index 2 to the smaller drop. For the asymmetric case, the radius of the larger droplet is used
to calculate the ratio of the droplet spacing-to-radius ratio.
85
Table 11 Correction factors table: Comparison between Point Source Method and numerical data for two-droplet symmetric arrays.
l/a PSM η numerical η
8 0.89 0.62
14 0.93 0.66
24 0.96 0.73
34 0.97 0.74
Table 12 Correction factors table: Comparison between Point Source Method and numerical data for two-droplet asymmetric arrays, where the droplet diameter ratio is 2.
l/a PSM η1 PSM η2 numerical η1 numerical η2
16 0.97 0.94 0.8 0.76
28 0.98 0.96 0.89 0.85
48 0.99 0.98 0.94 0.9
68 0.99 0.98 0.96 0.92
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60 70
l/a
η
PSM η1,a1/a2=2
PSM η2,a1/a2=2
PSM η,a1/a2=1
numerical η,a1/a2=1
numerical η1,a1/a2=2
numerical η2,a1/a2=2
Figure 31: Comparison between PSM and Numerical Simulation Data (FDS_v3 modified) for a two-drop symmetric and asymmetric arrays
86
The over-prediction of mass burning rates is caused by the insufficient grid resolution
that led to higher combustion temperatures (up to 2500 oC) and an inadequate computational
domain size. A finer grid and a larger computational domain could not be used when these
simulations were performed due to lack of computational resources. However, almost the same
percentage error was propagating to combustion of both single droplet and arrays of droplets. The
correction factor is based on the ratio of mass burning rate of an individual droplet in an array to
burning rate of isolated droplet and when correction factor is evaluated, most of this error will be
cancelled. The asymmetric droplet size model is better predicted than the symmetric
configuration, the correction factors being closer to PSM results for same ‘l/a” and droplets
diameter ratios.
The high temperatures developed inside the computational domain led to stronger droplet
interactions for a symmetric two-droplet array for l/a as large as 50 or more. The high rates of
oxygen consumption become the dominant process during combustion at elevated temperatures
and the burning rates were dramatically affected by the droplet interactions, causing flame
extinction for the smaller droplet spacings, before the droplets were completely burned. The
results from these simulations led to following conclusions:
• The code allows several droplets to burn simultaneously, the slices can be defined
around each droplet and, using data provided by the code for each slice, burning rate
can be calculated;
• Although preliminary results provided by FDS_v3 code are grid dependent, a finer
resolution as well as adaptive meshes around droplets and flame sheet can make the
parameters grid independent;
• Preliminary numerical results are in good qualitatively agreement with PSM
predictions
87
• Improved results are expected for simulations using a fine grid and smaller droplets.
More accurate predictions were performed using FDS_v4 and the results are presented below.
5.3.2 Two Droplet Arrays: FDS_v4
In this section, results will be presented for simulations of two-droplet arrays using
FDS_v4. These will be compared with PSM results for both symmetric and asymmetric
arrays. All the simulations employed a 643mm domain and two meshes, a coarse mesh (1mm
cell size) which covers the entire domain and a finer mesh around the cluster that
encompasses the flame zone. The burning rates are calculated using the same techniques as
for the isolated droplet cases, but without accounting for the spherical shape of the flame. For
the clusters considered in this work, the flame has an ellipsoidal shape rather than a spherical
one and is closer approximated by a cube, having “smoothed” corners to avoid sharp
gradients and to simulate a spherical drop, and determined by the intersections of the slice
planes.
Droplet interactions are quantified by the correction factor η, defined by Eqns. (1)
(PSM) and (23). The inter-drop spacing has been gradually increased and, for the asymmetric
model, the droplet diameters have been varied along with spacings between droplets. Flame
position is represented by the maximum heat release around the cluster at a given point in
time and will be visualized using Smokeview_v4.
For inter-droplet spacing l/a greater than 4, the mass burning rates are increasing
gradually as the distance between droplets increases, which is consistent with experimental
findings of Okai et al. (2000) for a pair of pure methanol droplets burning at atmospheric
pressure under microgravity conditions (Table 13 and Figure 32). For inter-drop spacing
88
smaller than 3, a higher correction factor has been found and the explanation is given below,
along with the analysis of Figure 33.
Table 13 Correction factors and burning rates for symmetric two-droplet arrays; the isolated droplet mass burning rate is 2.815E-07 [kg/s] Normalized
inter-drop
spacing
l/a
Burning rate
droplet in array
[kg/s]
Correction
factor η
PSM
Correction
factor η
numerical
simulation
3 2.31 0.75 0.82
4 2.079 0.80 0.738
8 2.166 0.889 0.769
16 2.339 0.941 0.831
30 2.502 0.968 0.889
Figure 32 Histories of droplet diameter squared for different spacing at atmospheric pressure, investigation performed by Okai et al. (2000)
89
The correction factors for a two-droplet array calculated from the data predicted are
in the range of 10% of PSM and, for droplet spacings larger than 10, the numerical results are
in the range of 5% to 10% of data obtained by Leiroz and Rangel (1997) for a droplet into a
stream. The numerical method developed by Leiroz and Rangel (1997) assumes that the
droplet is in the middle of a stream surrounded by other droplets of similar diameter. A
similar droplet would be present in the center of an array of 25 droplets or larger. Our
numerical model for a two droplet symmetric array consists only of the two droplets into the
computational domain. Therefore, the corresponding mass burning rate ratios for small
droplet interspacing are dramatically lower than those predicted by our numerical solution
(see Figure 33). In the Leiroz and Rangel case, this is consistent with stronger droplet
interactions caused by the interference of the other droplets surrounding the central droplet.
Correction factor for a symmetric two dropet array
0
0.2
0.4
0.6
0.8
1
1.2
0 10 20 30 40 50
l/a1
ηPSM ηnumerical η @4.0sLeiroz et al. η
Figure 33 Comparison between PSM (Annamalai and Ryan (1993), Leiroz et al. (1997) and numerical solution for a symmetric two droplet array
90
The comparison analysis could be divided in two regions: one for l/a >8 and one for
l/a >8. For droplet small spacings, up to l/a=4, the heating of the droplet due to radiative heat
transfer is predominant and the correction factors are closer to those predicted by PSM. As
previously stated, PSM is valid for l/a>>1, therefore a comparison with PSM is not valid.
The increase in mass burning rates as the droplet spacing is further reduced to values as small
as 3 is consistent with findings of Mikami et al. (1994) for a two droplet heptane and
heptane/hexadecane array burning under microgravity environment, where for a two droplet
array, there is a minimum of burning times (consistent with higher burning rates) as a
function of separation parameter, for droplet spacings between 4 and 8 (Figure 34 and Table
13). Okai et al. (2000) concluded following their experimental investigation that for pairs of
methanol of 0.9 mm diameter, this minimum is not present due to the absence of strong
radiative effects as methanol is a non-sooting fuel. Our numerical model is for 2 mm diameter
droplets and, for droplets of this size, the radiation effects are stronger than for droplet of
smaller sizes (less than 1mm), leading to droplet heating and mass burning rates
enhancement. As the droplet spacing increases, up to 50 ~ 60, the oxygen starvation
phenomenon is predominant over droplet heating in the reaction zone.
Correction factors also do follow a similar trend as PSM and Leiroz and Rangel
similarity parameters for l/a >8. Although the numerical study of Leiroz and Rangel implied
a stream of droplets, their assumption being that the droplet studied behaves like the central
droplet in a 25 droplet linear array, while the configuration used by our numerical simulation
is a two droplet array; these two results are very close quantitatively. As the droplets spacings
are larger, the influence from the droplets downstream or upstream is not as strong; therefore
the proximity of the results is expected.
91
As it can be seen from Figure 33, the droplet interactions for a model including
radiation are stronger even for larger droplet spacing (larger than l/a=50). This conclusion
can be inferred by extending the trend-line given by the correction factors yielded from the
studied cases (Figure 33). At inter-drop spacings from l/a=8 to 30 the mass burning rates
could decrease to ~15 to 30% of the isolated droplet burning rate, while for PSM the decrease
is only to 5 to 20% of single droplet burning rate. The numerical solution yielded results
similar to those of Leiroz and Rangel (1997) for droplet spacings between l/a=10 to 30.
Figure 34 Experimental burning times as a function of separation parameter for a two droplet array of n-heptane burning in air at atmospheric pressure Mikami et al. (1994)
Figure 35 shows the flow field around a two droplet array visualizing the Stefan flow
velocities due to thermal convection developed during combustion. Both droplets have a similar
velocity vector field, velocities having similar magnitudes, equally affecting each other.
92
Figure 35 Flow field around the droplets for two methanol droplet array having identical diameters, initial diameter 2mm. Velocity vectors are perpendicular to slice plane. Each cell is 1mm.
Figures below (Figure 36, Figure 37, and Figure 38) present the velocity field
generated by the Stefan flow around a two droplet asymmetric array for various droplet non-
dimensional spacings. Even for larger spacings between droplets, as large as l/a=16, the
droplets affect each other, between them being created a region where the velocities are very
low, the opposing velocity vectors canceling each other. For smaller inter-drop distances, up
to l/a= 8, the larger droplet is also affected by the smaller droplet, this influence being
stronger than for larger spacings.
93
Figure 36 Velocity field around a two methanol droplet asymmetric array (l/a=4), a1/a2=2. Velocity vectors are perpendicular to slice plane. Each cell is 0.5mm.
Figure 37 Velocity field around a two droplet asymmetric array (l/a=8); a1/a2=2. Velocity vectors are perpendicular to slice plane. Each cell is 0.5mm.
94
Figure 38 Velocity field around a two droplet asymmetric array (l/a=16); a1/a2=2. Velocity vectors are perpendicular to slice plane. Each cell is 0.5mm.
The interactions are strong even for distances as large as l/a=40. Figure 39 and Table
14 show that the smaller droplet in the wake of the larger droplet is strongly influenced even
for spacings as large as 30 (60 when spacing is normalized to smaller droplet radius). For the
larger droplet, the correction factors for l/a between 4 and 16 are lower by 8% to 16% than
the PSM results, mainly because the radiative heat transfer lowers the temperature and,
consequently, the associated burning rates. For the smaller droplet, the effect is even stronger
for inter-drop spacings up to 50 or even larger as shown in Figure 39.
For reduced distances between droplets (l/a=4 and 8), the array behaves like the
droplets have similar diameter, almost equally influencing each other. It is the same
maximum in mass burning rates found by Mikami et al. (1994) and presented above for the
symmetric droplet array case.
95
Table 14 Correction factors and burning rates for asymmetric two droplet arrays having droplet diameters’ ratio of 2 Distance l/a1
Burning rate isolated droplet
[1E-07*kg/s]
Burning rate droplet in array
[1E-07*kg/s]
Correction factor η
PSM
Correction factor η
numerical simulation
a1 2.815 2.306 0.903 0.819 4
a2 0.669 0.554 0.774 0.828
a1 2.815 2.27 0.945 0.806 8
a2 0.669 0.545 0.882 0.815
a1 2.815 2.483 0.962 0.882 12
a2 0.669 0.551 0.920 0.824
a1 2.815 2.56 0.971 0.909 16
a2 0.669 0.556 0.939 0.831
a1 2.815 2.698 0.984 0.958 30
a2 0.669 0.601 0.967 0.898
Correction factor for an asymmetric two droplet array
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
0 10 20 30 40 50
l/a1
η
PSM η1numerical η1PSM η2numerical η2
Figure 39 Comparison between PSM and numerical solution for a two droplet asymmetric array, a1/a2=2; fuel: methanol, burning in air at atmospheric pressure and g=0
96
This phenomenon of a minimum in correction factor trend could be explained by the
effects of radiation and thermal convection: on the larger drop and on the smaller drop.
Radiation is absorbed by both droplets and does lower burning rates of droplets. However,
the smaller droplet being subjected to thermal convection generated by combustion of the
larger droplet (smaller droplet is in the wake of the larger droplet) will be exposed to two
phenomena: one is the enhancing of the gasification rate due to convection and the other is
the oxygen starvation due to combustion of the larger droplet in its vicinity. At certain
spacings between the two droplets, the enhancing of the gasification rate is predominant upon
the oxygen starvation, leading to a higher than expected burning rate for the smaller droplet.
Therefore the correction factor of the smaller droplet is higher than that predicted by PSM.
Same phenomena occur to the larger droplet, being affected by the thermal convection
generate by combustion of the smaller droplet, enhancing also the burning rate, but at a
reduced scale. As it can be observed, the correction factor for the larger droplet has also an
increased value for inter-drop spacings of 4. However, the smaller droplet is affected by the
bigger droplet, its mass burning rate being lower than the similar isolated droplet burning in
air. This phenomenon is more significant when the droplets are closer. As the droplets move
further apart, the influence of smaller droplet upon the larger one becomes less significant
and their behavior becomes similar to that obtained using PSM.
The mass burning rates for the larger drop at inter-drop distances up to 10 are
approximately 80% of the isolated drop burning rates while PSM data shows a 10% reduction
in mass burning rate for same droplet when compared to single droplet burning rate. As the
distance between droplets is increased, the smaller droplet burning rate is in the range of 80%
to 90% of the similar single droplet burning rate, while the larger drop burning rate is from
87% to 96% of isolated droplet burning rate. Compared to PSM, the numerical simulation
97
predicts more significant droplet interactions, even for very large droplet spacings. This is in
very good agreement with theoretical model of Leiroz and Rangel (1997) that predicts strong
droplet interactions for a center droplet in a large array. This result also agrees with the
experimental data of Okai et.al (2000) for a pair of methanol droplets burning in air at
atmospheric pressure under microgravity (diameters of 0.9mm ±0.15mm). Their experimental
data predicted values for K/K0 in the range of ~0.8 to ~0.5 for inter-drop spacings from 13.6
to 5.4.
For the symmetric case, the flame around the array has a cylindrical shape with
ellipsoidal ends (Figure 40), consistent with conclusions of Okai et al., 2000. For the asymmetric
case, the flame will be, for reduced distances apart, of the shape of an asymmetric ellipsoid
(Figure 41) and for larger spacings, the flame become more of a surface created by the
intersection of two pear shaped flames, posed tip to tip (Figure 42 and Figure 43). At a droplet
radii ratio of 2, the droplet pair burns as an array and not as individual droplets even for droplet
spacings up to l/a=20.
The heat release and temperature profiles for an asymmetric methanol droplet pair
shown in Figure 42 and Figure 43, indicate that even for spacings as large as 16 (l/a2=32),
both droplets interact strongly with each other with the smaller droplet being more affected.
98
Figure 40 Flame position for a two droplet symmetric array burning in air.
Figure 41 Flame contours for a two droplet asymmetric array (a1/a2=2, l/a=4)
99
Figure 42 Flame contours for a two droplet asymmetric array (l/a=16), based on heat release per unit volume.
Figure 43 Temperature profile for a two droplet array of different diameters (l/a1=16, l/a2=32).
100
5.3.3 Three Droplet Arrays: FDS_v4
We next modeled arrays containing three methanol droplets having varying droplet
diameters. To simplify the calculations of the PSM correction factors, the droplets were mounted
in the apices of a quasi-equilateral triangle. Burning rates were calculated using the same method
as for a single droplet, described in a previous section. Both symmetric and asymmetric
configurations assume that the droplets radii ratios are a1/a2=1.33 and a1/a3=2. For a symmetric
configuration, droplet sizes were 2 mm and the normalized spacings between droplets were ~6
and ~16, respectively. For asymmetric cases l/a1=~7.2, and 19.6, respectively for the same 2 mm
size droplet for the larger droplet. In descending size order, the droplets will be a1 (largest), a2 and
a3 (smallest).
The isolated droplet mass burning rates used to calculate correction factors (Table 15) are
as follows: for 2 mm diameter droplet 2.815E-7[kg/s], for 1.5 mm diameter droplet 1.41[kg/s]
and, for 1 mm droplet diameter 0.669E-07[kg/s].
For both configurations, the burning rates of the droplets in the array are decreasing by
25% for l/a=6 and by 10% to 12% for l/a = 16 or larger. Note that the normalized distance
between droplets is the ratio of the spacing between two neighboring droplets to the radius of
largest droplet. For the symmetric configuration, the numerical correction factor is in excellent
quantitative agreement with PSM.
101
Table 15 Correction factors for three methanol droplet arrays of identical or different droplet sizes
Configuration l/a Burning rate
1E-07*[kg/s]
PSM
η
Numerical
η
Experimental
η
(Liu, 2003)
6 2.181 0.75 0.774 - Symmetric
16 2.525 0.889 0.897 -
a1 2.264 0.837 0.804 -
a2 1.106 0.797 0.783 6
a3 0.498 0.761 0.745
a1 - 0.908 - 0.86
a2 - 0.888 - 0.751 12
a3 - 0.868 - 0.694
a1 2.564 0.942 0.911 -
a2 1.248 0.93 0.883 -
Asymmetric
20
a3 0.574 0.918 0.859 -
Analyzing the asymmetric configuration, the correction factors are 5% to 8% lower than
those predicted by PSM, showing stronger droplet interactions at the same droplet spacings. This
behavior is similar to that predicted by the two droplet array simulations. As shown in the Figure
44, the burning rates for the smallest droplets decrease by 12% at l/a larger than 20, and by 25%
when l/a is less than 6. Extrapolating the results for larger droplet separation distances, as the
droplet size is decreasing the effect of droplet interactions is stronger, even for droplet normalized
spacings larger than 20 (l/a = 40 when normalized with smallest droplet radius). The main
conclusion here is that no matter how large the inter-drop non-dimensional spacing is, the effect
of the larger drops upon the smallest droplet is strong. These results are in very good agreement
with those predicted by the PSM.
102
a1/a2=1.33, a1/a3=2
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50l/a1
η
PSM ηsnumerical ηsPSM η1numerical η1expt. η1 (Liu, 2003) PSM η2numerical η2expt. η2 (Liu, 2003) PSM η3numerical η3expt. η3 (Liu, 2003)
Figure 44 Correction factor for a three methanol droplet arrays of equal and different droplet sizes, mounted in the apices of a triangle compared against PSM results and Liu (2003) experimental data
The experimental investigation performed by Dietrich et al (1997), shown in Figure 45,
indicates a decrease in the burning rate of the center methanol drop of only about 8% when l/a is
close to 8 (l/d=4, where “d” is droplet diameter whereas, “a” is the droplet radius). These results
also show a maximum in the burning rate at a specific value of l/a. This phenomenon was also
captured by the numerical simulation for the two droplet array as well as the experimental data of
Mikami et al (1994). As of now, there is not enough data from our numerical model for three
droplet arrays to conclude that this maximum in burning rates would occur. Further developments
of our model would have to be performed before evaluating these observations. Nonetheless, the
numerical results are following the same general trend as the data of Dietrich et al. (1997), being
in a range of 15% or less of their results.
Liu (2003) found that vaporization rates for droplets in a cluster are decrease by 15% to
30% of the single droplet values for droplet normalized spacings of 10 to 12. These results are 8%
103
to 15% lower than those predicted by PSM and 10% lower than current numerical solution, for
similar values of l/a (Figure 44). The experimental investigation of Liu (2003) studied droplet
interactions of vaporizing droplets under normal gravity, interactions being generated by the fuel
vapor concentration around the droplets while the present numerical solution investigates droplet
interactions during combustion. The explanation of the difference between the results of these
two studies could be that while one is based on vapor accumulation effects, the other is a balance
between the enhanced burning rates due to heat release and the lowering of burning rates due to
radiation and oxygen starvation effects.
Figure 45 Measurement of K/K0 ratio for a three droplet array where K is the burning rate of a center drop in a linear array and K0 is the isolated droplet burning rate (Dietrich et al., 1997)
Another aspect that was investigated was the flame shape around the droplet cluster. As
illustrated in Figure 46 - 46, the droplets are burning as a group even when the distance between
them increases to distances as large as 20 mm. The same phenomenon was observed by Nagata et
al.(2002) for arrays of seven hexanol and butanol droplets (2 mm to 2.5mm diameter) suspended
on glass fiber. They show a clear trend from external group combustion to single droplet
104
combustion with an increase in sample spacing, as shown in Figure 51. Similar to their
conclusions, as the distance increases, the flame surrounding the array tends to approach the edge
of the array with increasing droplets’ spacing (Figure 42 and Figure 43 for a two droplet
asymmetric array). When the inter-drop spacing is 6 mm, a single envelope flame encloses the
cluster throughout the combustion process. For symmetric configuration and relatively small
droplet spacings (up to 15mm), the flame has an ellipsoidal shape. As the distance is further
increased, the flame shape tends to become more of a triangular shape (from a two dimensional
point of view). For the asymmetric case, as the distance between droplets increases to 16mm, the
flame envelope shows more independent burning of the droplets, even if the droplet interactions
are still present and there is still a common flame surrounding the cluster.
The current numerical solution proved to be in very good qualitative agreement with
experimental results of Nagata et al (2002) and with the group combustion theory.
Figure 46 Flame two-dimensional contours for a three droplet symmetric array (front view) at 4.0s of simulation
105
Figure 47 Flame two-dimensional contours for a three droplet symmetric array (top view) at 4.0s of simulation
Figure 48 Three-dimensional iso-contours for 3850kW/m3 heat release per unit volume of (flame approximate position) for a symmetric three droplet array burning in air after 4.0s of simulation
106
Figure 49 Flame contours for a three droplet asymmetric array, having different droplet sizes (a1/a2=1.33, a1/a3=2, l/a1~7) at 4.0s of simulation
Figure 50 Three-dimensional iso-contours for 6300kW/m3 heat release per unit volume of (flame position) for an asymmetric three droplet array burning in air after 4.0s of simulation
107
Figure 51 Flame shape history as a function of separation distance for seven droplet two dimensionally arranged clusters of droplets (L varies from 10mm to 30 mm) (Nagata et al., 2002)
Evaluating the results of numerical simulations for two and three droplet arrays, it can be
concluded that the predictions are in excellent agreement with theoretical and experimental data
from existing literature and also that this model could be extended for droplets arrays having
more than 3 droplets. To investigate arrays with larger numbers of droplets, the configuration
should include multiple meshes around groups of droplets positioned abut to limit the number of
parameters calculated for each mesh and for time increment and thus decrease the computational
time and the likelihood of stalled calculations. The major limitation is the actual computational
resources available. Another issue is the limited number of meshes (five) that could be employed
with FDS. Under these circumstances, the model could be extended to study up to 15 droplets
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CHAPTER 6: EXPERIMENTAL INVESTIGATION
The previous sections identified several areas of droplet interactions that could be
investigated. Numerical simulation is an approach that, while complex and time-consuming, has
been accomplished. However, interpretation of the theoretical results would benefit greatly by
experimental data obtained using similar configurations. Most previous experimental studies have
used fiber-supported droplet arrays, but the vaporization rates are generally overestimated
because of the effect of the fiber. While the effect of a supporting fiber has been analyzed for an
isolated droplet, such an analysis has not been performed for a droplet array. Given the close
proximity of the fibers in an array of fiber-supported drops, it is not reasonable to assume that the
extrapolation of the results to un-supported droplets will be the same. The presence of the fibers
has also complicated the interpretation of data from previous experiments and limited
investigations to rather small arrays.
RF Power Amplifier
Microphone Amplifier
Oscilloscope
Droplet Generator
Reflector
Positioner
Horn
Transducer
Signal Generator
Computer
Recirculating Cooler
Temperature Reading
Impedance Matching Circuit
Figure 52 Bench-top apparatus (acoustic levitator) to study evaporation of levitated fuel droplets (Liu, 2003)
To make progress in this area, an experimental configuration or technique must be developed
to reduce the effect of the droplet suspension mechanism or eliminate it altogether. To eliminate
109
the fiber, another means must be found to support the droplets. In their attempt to eliminate the
suspension fiber, some researchers used in their experiments arrays of free flying drops in a
convective stream or free falling setups to simulate the reduced gravity. The complexity of
induced by buoyancy and fibers effects should be also eliminated. A microgravity environment
would be ideal to help eliminate the effect of buoyancy.
An apparatus has been built at Drexel University’s Frederic O. Hess Engineering
Research Laboratory to study the unsupported fuel droplet evaporation and combustion (see
Figure 52 by Liu, 2003). This acoustic levitator can stabilize a two-dimensional unsupported
cluster of droplets. Using this apparatus, several sets of data were collected and it was calculated
the rate of evaporation for isolated drops, arrays of 2 drops and arrays of 3 drops through a linear
interpolation. There were obtained sets of evaporation constants K for all three configurations
mentioned above. The experimental conditions were adjusted to match the theoretical model
proposed. Also, the isolated drop experiment was performed in the similar conditions (ambient
and drop temperature and pressure, calibration, fuel used) as the 2-drop and 3-drop experiments
(Liu and Ruff, 2001).
Because of the unique nature of the apparatus to be used in this experiment, a
considerable amount of development and testing in normal gravity was required. A normal-
gravity test facility was constructed and completed at Drexel University. Specifically, two
essential development issues were overcome: (1) to design and fabricate an acoustic levitator to
stabilize a two-dimensional droplet cluster prior to combustion and (2) to develop a method to
introduce a specified number and size of droplets into the acoustic field. This facility has been
used to conduct tests of the evaporation of interacting droplets (Liu and Ruff, 2001).
110
An important step forward is to perform experiments in microgravity environments to
eliminate the effects of gravity and to reduce the influence of the acoustic field, knowing that in
microgravity the droplets will remain suspended without an acoustic field. To achieve this
objective it is necessary to build a system that will incorporate the components of the acoustic
levitator and that will meet the requirements to perform experiments in a drop-tower facility (see
Figure 53). This rig is useful to investigate many of the complex fluid mechanic and chemical
kinetic issues surrounding the combustion of clusters of drops. One of the tasks will be to study
the evaporation of drops in an array. Mirroring the previous 1-g experiments, similar sets of data
can be collected, having the advantage of more accurate data provided.
Figure 53 Drop Tower Rig System to study evaporation and combustion of unsupported fuel clusters of droplets under microgravity
111
In this experiment, the formation of the clusters can be precisely controlled using an
acoustic levitation system so that dilute and dense clusters can be created and stabilized before
combustion is begun. Therefore, this experiment will allow the spectrum of droplet interactions
during combustion to be observed and quantified. This data will provide a realistic and logical
intermediate step in the progression of experiments from individual droplet combustion to spray
flames. It will reduce the extrapolation required to extend results of droplet array studies to those
obtained in spray combustion systems. The data will also provide the needed experimental
verification of group combustion models currently being applied to complex flow situations.
6.1 Operating Parameters
The primary effects to be investigated in this proposed experiment include (1) the effect
of droplet size, cluster size and number of drops on the combustion process, (2) the effect of the
type and composition of fuel on group combustion, and (3) the ability of the group combustion
number to scale the observed group combustion regimes. Initially, only a small subset of these
will be investigated and the tests will be conducted at one atmosphere pressure. Currently, tests
are planned using clusters containing 1, 5, 10, 15, and 20 drops. These increments are selected to
compare with various droplet array studies reported in the literature. For each of these clusters,
tests will first be performed with methanol. We will first develop the test procedures using a
single levitated drop and then expand the test matrix to increasing numbers of droplets. Once
successful atmospheric tests have been achieved and the test procedures clearly defined,
modifications to the base apparatus to enhance its capabilities can be considered.
112
6.2 Description of the Test Hardware
6.2.1 Microgravity Environment
The droplets cannot be ignited in normal gravity environment because the presence of the
igniter disrupts the acoustic field. Therefore, the challenges that remained in the development of
the reduced gravity facility were to (1) develop a method to ignite the droplet cluster and (2)
assemble all of these components into a controllable experiment in a drop tower rig. The
mechanical and electrical design described in the Final Design Review document at NASA will
also briefly be presented in the following sections, and all the tables containing technical
specifications will be presented in the Appendix D.
6.2.2. Mechanical Design
A mechanical layout for the components required for this experiment has been
developed. Figure 54 and Figure 55 show top and side views of the components placed in a
standard A-frame drop rig. A list of components and the manufacturer is shown in the Appendix
D.
Acoustic Levitator
This system includes the piezoelectric transducers, the acoustic reflector, front and rear
transmitter blocks, and the reflector traverse. The droplet cluster is formed between the acoustic
driver and reflector.
113
Signal Generation
This system produces and controls the signal that operates the acoustic levitator. It
includes a programmable function generator, a signal amplifier, a step-up transformer, and a
signal generator control circuit.
Droplet Insertion
This system produces the droplet cluster in the levitator. It consists of a syringe pump,
a needle positioner motor, and a high voltage power supply. The syringe pump and needle
positioner introduce a droplet into the acoustic field while the high voltage power supply is used
to break up the ‘parent’ droplet and form the cluster. Methanol and ethanol will be used for the
initial tests.
Igniter Assembly
This system moves and controls the igniter and igniter motion. It consists both of the
igniter traverse, hot wire, and an elevator that drops the igniter out of the plane of the droplet
cluster.
Diagnostics
Two color CCD cameras form the primary diagnostics for this experiment. The
droplets are illuminated from the side by an LED. One camera looks at a backlit image of the
cluster while the other looks down on the cluster through the reflector. Thermocouples are used to
monitor the ambient temperature inside the enclosure and of the acoustic transmitter.
114
General
The levitator and needle positioner are located inside an enclosure. This is a non-sealed
volume intended to acoustically isolate the acoustic levitator and provide a still ambient
environment during the drop. The purge bottle is present so that the contents of the enclosure can
be purged to the drop tower vent system after each drop.
As shown in Figure 54 and Figure 55, the batteries are placed at the bottom of the rig
along with the base for the acoustic levitator. A second level from the bottom holds the
components for the signal generation system, i.e., the power amplifier, signal generator and
transformer, and box for the control circuit. The fuel syringe and motor, stepper motor control
box and high voltage power supply are mounted on a second shelf at the level of the droplet
array. The side-view camera and the diagnostic control box are also mounted on this shelf. The
droplet array is viewed from the side and the top through a window in the center of the reflector.
The power distribution module, computer control system, and relay box and DC-AC converter
are shown placed on the top of the rig.
While Figure 54 and Figure 55 show the general layout of the rig, the detail of the
components inside the enclosure is not well represented. The components contained inside the
enclosure are:
• igniter and elevator assembly
• acoustic reflector
• reflector traverse motor and assembly
• thermocouples (air, acoustic levitator)
• needle traverse
115
A schematic of the layout of the components inside the enclosure is shown in Figure 56. The
igniter assembly has been designed and is shown in Figure 57. Figure 58 shows detail of the loop
igniter assembly. This component has been fabricated at NASA Glenn Research Center and is
currently undergoing testing. The acoustic reflector, reflector traverse, and needle traverse will be
mounted onto the igniter and elevator assembly.
116
Signal Generator
Ther
moc
oupl
eIn
terfa
ce B
ox
0 5 10 15 20 25 30 35 38
0
5
10
15
Fiber OpticTransmitter 2
24 VDC Battery BoxIgniterAssembly 24 VDC Battery Box
Signal Amplifier
Fiber OpticTransmitter 1
Signal GenControl Box
Stepper MotorControl Box
High VoltagePower Supply
Power Distribution
Module
TT8-DDACSTransformerCircuit BoxCameraLens
PurgeBottle Fuel Syringe and Motor
LED Control Box
Relay Box
Enclosure
Signal Generator
Ther
moc
oupl
eIn
terfa
ce B
ox
0 5 10 15 20 25 30 35 380 5 10 15 20 25 30 35 38
0
5
10
15
0
5
10
15
Fiber OpticTransmitter 2Fiber Optic
Transmitter 2
24 VDC Battery BoxIgniterAssembly 24 VDC Battery Box
Signal Amplifier
Fiber OpticTransmitter 1Fiber Optic
Transmitter 1Signal GenControl Box
Stepper MotorControl Box
High VoltagePower Supply
Power Distribution
Module
Power Distribution
Module
TT8-DDACSTransformerCircuit BoxCameraLens CameraCameraLensLens
PurgeBottle Fuel Syringe and Motor
LED Control Box
Relay Box
EnclosureEnclosure
Figure 54 Mechanical lay-out of the Droplet Cluster Rig (top view)
117
0
5
10
15
20
25
30
33
0 5 10 15 20 25 30 35 38
IgniterAssembly
24 VDC Battery Box
Cam
era
Lens
Power Distribution Module
Fiber OpticTransmitter 1
24 VDC Battery Box
Fiber OpticTransmitter 2
TT8-DDACS
ThermocoupleInterface Box
Signal Generator
Stepper MotorControl Box
Signal GenControl Box
TransformerCircuit Box
High VoltagePower Supply
Fuel Syringe and Motor
CameraLens
PurgeBottle
Signal Amplifier
Relay Box
DC-AC InverterLED Control Box
0
5
10
15
20
25
30
33
0
5
10
15
20
25
30
33
0 5 10 15 20 25 30 35 38
IgniterAssembly
24 VDC Battery Box
Cam
era
Lens
Cam
era
Cam
era
Lens
Lens
Power Distribution ModulePower Distribution Module
Fiber OpticTransmitter 1Fiber Optic
Transmitter 1
24 VDC Battery Box
Fiber OpticTransmitter 2Fiber Optic
Transmitter 2
TT8-DDACS
ThermocoupleInterface BoxThermocoupleInterface Box
Signal Generator
Stepper MotorControl Box
Signal GenControl Box
TransformerCircuit Box
High VoltagePower Supply
Fuel Syringe and Motor
CameraLens CameraCameraLensLens
PurgeBottlePurgeBottle
Signal Amplifier
Relay Box
DC-AC InverterLED Control Box
Figure 55 Mechanical lay-out of the Droplet Cluster Rig (side view)
118
SignalGenerator
Amplifier
Transformer
1
Signal ControlCircuit
High VoltagePower Supply
2
3
4
5
11
6
10
13
7
8
9
15
12
14
16
T1 T1
1718
19
SignalGenerator
Amplifier
Transformer
11
Signal ControlCircuit
High VoltagePower Supply
22
3
44
55
11
6
10
13
7
8
99
15
12
14
16
T1T1 T1T1
1718
19
Figure 56 Schematic of the igniter assembly inside the enclosure
Valve19
Valve18
Purge bottle17
Thermocouples16
LED15
High Voltage Power Supply14
Transformer13
Acoustic Signal Amplifier12
Signal Generator11
Acoustic Signal Control Circuit10
Enclosure9
CCD Camera8
Syringe Pump7
Needle Insertion Drive6
Droplet Insertion Needle5
Igniter Elevator4
Reflector Drive3
Reflector2
Acoustic Transmitter1
ItemItem No
Valve19
Valve18
Purge bottle17
Thermocouples16
LED15
High Voltage Power Supply14
Transformer13
Acoustic Signal Amplifier12
Signal Generator11
Acoustic Signal Control Circuit10
Enclosure9
CCD Camera8
Syringe Pump7
Needle Insertion Drive6
Droplet Insertion Needle5
Igniter Elevator4
Reflector Drive3
Reflector2
Acoustic Transmitter1
ItemItem No
121
6.3. Test Procedures
The entire experimental investigation will be conducted at the 2.2 Second Drop Tower at
NASA Glenn Research Center. The operation of this experiment is best explained using the
experiment timeline shown in Figure 59. The drop tower will be prepared for a drop with the
experiment package enclosed in the drag shield and positioned at the top of the tower. At this
time, electrical connections will still be in place so an external operator can control the formation
of the droplet cluster.
First, the acoustic field will be initiated by turning on the power amplifier. The operator
will then dispense a specified amount of fuel to the tip of the hypodermic needle and the reflector
traversed to form the droplet cluster in the acoustic field. The droplet generator will be retracted.
After the operator verifies that a stable cluster has been formed, he will initiate automated control
of the experiment. (The formation of the cluster will be verified visually using the on-board CCD
camera.) If an external connection has been used for initial experiment control, it will be detached
at this time. The facility operators will then be notified that the rig is prepared to drop.
The acoustic field will be turned off when the computer receives the drop initiation
signal. It is unclear whether it will be best to terminate the field instantly or to more gradually
decrease the levitating force. The igniters will also be energized and, after a specified time,
retracted so they don’t interfere with the cluster during the drop. In any event, the timing for the
removal of the acoustic field and energizing of the igniters will be controlled by the computer and
are variables that will be worked out during the first checkout drops. The experiment will proceed
and data collected until the end of the drop, at which time all systems will be de-energized.
This timeline is also described in Table 18 from Appendix D in terms of the experiment
control actions.
122
Figure 59 Experiment operation timeline
-15 0 2.2 -90
Release signal
Initiation signal
Turn off acoustic field
Time (sec)
-50
Retract drop generator
End of experiment
Begin data logging
Ignition
Begin cluster formation
Initiate acoustic field
123
CHAPTER 7: SUMMARY AND RECOMMENDATIONS
The present work developed a numerical model to study the combustion of well-
characterized drop clusters in microgravity environment using direct numerical simulation. The
computational research investigated the combustion of clusters of droplets of different sized and
asymmetric three-dimensional configurations in zero gravity environments for zero relative
Reynolds numbers. One of the aspects studied is droplet interaction during evaporation and
combustion over the lifetime of the droplet. The model proved to be able to provide reliable data
that will support future microgravity experimental investigations.
Background and Motivation
In general, isolated droplet investigations are not amenable to the analysis of arrays of
drops or clusters because of the complexity of the methods used and the effect of the gaseous
environment and drop-drop interactions. The study of isolated droplets has revealed aspects of
droplet behavior that have aided the development of investigations using arrays of drops. These
findings include that the evaporation rates are strongly affected by (1) the heat conduction
induced by a supporting fiber, (2) radiative absorption, (3) the ambient temperature and pressure,
(4) buoyancy, and (5) the Re, Sh and Nu numbers.
In an array or cluster of drops, a drop is affected by its nearest neighboring droplets. In
the analysis of an array of droplets, droplet interactions are considered. The proximity of a
neighboring droplet affects the mass and energy transfer between the droplet and the surrounding
gas, the lift and drag coefficients and the Nusselt and Sherwood numbers. Droplet interactions are
strongly affected by the Reynolds number and they are dependent on the geometry of the array.
124
The effect of the fiber on droplet interaction and droplet vaporization and combustion
phenomena has not been studied for arrays of fiber-supported drops because the cumulative effect
of the fibers would eventually mask the details of droplet behavior. Furthermore, this type of
analysis would require a large amount of time and computational resources. Droplet interactions
for non-symmetric and larger arrays of drops can be large and should also be investigated. The
effect of the Reynolds number on interacting droplets is an important issue that has been
investigated both through analytic and numerical methods. Unfortunately, for low Reynolds
numbers, experimental studies are not available yet. The investigation of unsupported arrays of
drops for near zero Reynolds numbers, both theoretical and experimental, could offer a better
understanding of the droplet interaction phenomena.
Point Source Method
The Point Source Method investigates the effect of droplet interactions for arrays up to
1000 droplets, arranged in symmetric configurations, evaluating the mass loss rate of interacting
drops, under quasi-steady and quiescent atmosphere. The method accounts for variable thermo-
physical properties. PSM is valid only if l/a>>1. The Point Source Method has been verified
against data provided by current theories but there are no comparisons with experimental data. An
extension of a current theory (Point Source Method) has been investigated for non-symmetric
two- and three-drop arrays. Droplet interactions have been evaluated using the ratio of the
vaporization rate in an array to an isolated drop that defined the correction factor.
The PSM results led to the conclusion that droplet interactions are strong for non-
dimensional inter-drop spacings between 2 and 20 (for configuration using drops with similar or
near similar diameters). As the spacing between droplets increases, droplet interactions are
weaker. For small droplets in the wake of a larger drop, droplet interactions are present and strong
125
even for large inter-spacing (up to l/a~50). The evaporation rate of the smaller drops could
decrease by up to 30-40% compared to an isolated drop.
Model Description and Problem Formulation for Numerical Solution
A three dimensional model of symmetric and asymmetric arrays of fuel droplets burning
under microgravity conditions have been developed, that will mimic (as close as possible) the
experimental model allowing the spectrum of droplet interactions during combustion to be
observed and quantified. The particularity of this numerical simulation is that each drop in the
array vaporizes and burn according to the conditions created by the environment around the
droplet and also the array can be considered as an entity.
The model study accounts for radiative heat transfer using a Finite Volume Method,
Stefan convection, diffusion and finite-rate chemical kinetics, a quiescent environment, near zero
Reynolds number, and gas-phase variable thermo-physical properties and unity Lewis number but
does not account for forced convection and for internal circulation within the droplet.
The numerical solution is based on Fire Dynamic Simulator, computational fluid dynamic
numeric model of thermally-driven fluid flow that solves Navier-Stokes equations for low-speed
flow. The combustion model assumes one-step chemical reaction. Spatial and temporal
discretization is based on rectilinear three dimensional grids and a time step increment adjusted
according to von Neumann criterion.
Initially, a modified version of FDS_v3, which included a radiation solver for finite-rate
reaction combustion, was used and preliminary results were in good qualitative agreement with
PSM and other theoretical data for combustion of single drop and arrays of drops. Later during
the project development the simulations were performed using FDS_v4, based on the advantages
126
offered by new version of FDS, including but not limited to a finite volume fraction solution of
radiation equation for finite-rate reaction model, a faster solution and reduced CPU time.
Mass burning rates are calculated using a numerical algorithm using the output data
provided by FDS_v4 and a non-dimensional factor η (defined similar to PSM) was calculated to
account for droplet interactions. Several cases were investigated to determine the ignition times
and the necessary ignition energy to sustain the combustion. As a conclusion, it was determined
that the ignition time required for methanol to reach flame temperature was 2s to 2.5s, using four
igniters positioned at one cell apart from the droplet and having a wall temperature of at least
3000ºC.
Validation of the Code Using Isolated Droplet Combustion
Both sensitivity and validation has been performed using a single droplet configuration,
the numerical results being compared to the theoretical and experimental data found in literature.
Several factors have been studied for single droplet combustion before investigating the
combustion of multiple droplets, such as the sensitivity of temperature and burning rates to
domain size and grid size (number of cells into a domain and cell size). For single droplet
combustion, as the domain increases in size, the combustion parameters become independent of
the domain size and burning rates for dimensions larger than 403mm. If a finer grid is used, the
burning rates are not as sensitive to the domain size.
The burning rates were found to be affected by the grid size, and less by the domain size
(slightly more affected for small computational domains); therefore a grid size that satisfies the
ignition condition has been employed for the simulation of clusters of droplets. Combustion
parameters for two and three droplet arrays become domain independent for a domain size larger
127
than 403mm. If the distance between droplets increases, the domain size has to be increased
accordingly.
Varying the oxygen concentration in the ambient surroundings from 10% to 75% by
volume the numerical results (burning rates and flame positions during combustion) are in good
qualitative agreement with numerical and experimental data obtained by Marchese and Dryer
(1999), being in the range of 10% of their result-and data plotted following similar trends.
Turning on and off the radiation solver, the numerical model predictions shows a
reduction in mass burning rates when radiation is considered being in good qualitative agreement
with the model developed by Marchese et al. (1999) for a 5 mm isolated pure methanol droplet
and also with the numerical and experimental results of Kumagai (1971) and King (1996) for a
0.98mm n-heptane burning droplet. Burning rates without non-luminous radiation will become
grid and domain independent for larger domains and finer grids that those cases with non-
luminous radiation considered. The flame diameter for a 1 mm droplet burning in air with and
without non-luminous radiation considered is in good quantitative and qualitative agreement with
experimental and numerical data of Kumagai (1971) and King (1996).
After evaluating the sensitivity and validation tests, we may conclude that the predictions
of combustion parameters by the numerical model for single droplet are in very good qualitative
agreement with theoretical and experimental results presented in the literature. The model is
amenable for analysis and prediction of mass burning rates in a reasonable margin of error (less
than 10%) for an adequately grid resolved domain. Single droplet combustion predicted by the
code follows the d2-law. The predicted data confers confidence that combustion parameters
resulted adequately characterize the combustion process of single and multiple droplets.
128
Multiple Droplet Combustion
The next step was evaluation of combustion of multiple droplet arrays under zero-gravity
conditions. Two and three droplet arrays of identical or different sizes were investigated,
increasing gradually the distance between droplets in the array.
For two droplet array configuration, the correction factors determined from the numerical
model are lower by about 10% than those predicted by PSM but 5% to 10% higher than those
predicted by the numerical model of Leiroz and Rangel (1997) for non-dimensional distances
between droplets greater than 10. For droplet spacings of l/a greater than 4, the mass burning
rates increase gradually as l/a increases, which is consistent with experimental findings of Okai et
al. (2000) for a pair of pure methanol droplets burning at atmospheric pressure under
microgravity conditions. As the normalized spacing between droplets is further reduced, burning
rates tend to slightly increase, consistent with experimental data of Mikami et al. (1994) for a n-
heptane droplet array, but contradicted by experimental findings of Okai et al. (2000) for a 0.9
mm methanol droplet. In the latter work, a minimum burning time (maximum in burning rates)
was not observed and justified by the absence of the soot in methanol combustion. Additional
investigation will be needed to clarify this issue.
An explanation of the enhancement of the burning rates for very small distances between
droplets could relay on the larger size of the droplets employed by our numerical simulations that
yields on a stronger radiative effects.
For the asymmetric case, droplet interactions are even stronger than those predicted by
PSM, the smaller droplet being more affected by the bigger droplet. Droplet interactions are
present and strong even for normalized distances between droplets greater than 50.
129
The flame envelope has a quasi-cylindrical shape, consistent with Okai et al. (2000)
conclusions for a two droplet array.
The next configuration studied was a three droplet array, with similar or different droplet
sized, and mounted in the apices of a triangle. Analyzing the asymmetric configuration, the
correction factors are 5% to 8% lower than those predicted by PSM, showing stronger droplet
interactions at same droplet spacings, being similar to that predicted by the two droplet array
simulations. The burning rates for the smallest droplets decreases by as much as 12% for l/a
larger than 20, and by as much as 25% compared to isolated droplet burning rate when l/a is
decreasing up to 6. The general behavior is following the trend of PSM results, the current
numerical solution being in very good agreement with PSM. Mass burning rates are generally
15% lower than those predicted by the experimental data of Dietrich et al. (1997). However, the
numerical data are in good qualitative agreement with experimental results of Liu (2003) and
Dietrich et al. (1997).
Varying the distance between droplets, the numerical solution predicted a flame envelope
variation consistent with experimental results of Nagata et al (2002).
The predictions of the numerical simulations are in excellent agreement with theoretical
and experimental data from literature and also that this model could be extended for droplets
arrays having more than 3 droplets. Due to limitation the actual computational resources
available, the current model could be extended to study up to 15 droplets. Further developments
to our model as well as to FDS have to be made to increase even more the number of droplets in
the cluster.
130
The model was proven to be able to investigate different array configurations to predict
the behavior of cluster of droplets. Therefore the model could mimic future experimental
configurations and provide numerical data to support experimental results.
Recommendations
As described previously in Chapter V and above in the summary section, as the droplets
in an array are closer one to each other, a maximum in burning rates has been noticed for both
two and three droplet arrays, being consistent with Mikami (1994) findings for a two droplet
array and with Dietrich (1997) experimental results for a three droplet linear array. However,
these results are contradicted by experimental findings of Okai et al. (2000) for 0.9 mm methanol
droplets, where a maximum in burning rates was not observed. Stronger radiative effects due to
larger droplets employed could explain this phenomenon. To further explore this problem, several
simulations will be needed for non-dimensional spacings slightly smaller and larger than the
inter-drop spacing for which the maximum has been found, as well as different spatial
configurations of cluster of droplets having diameters varying from 0.5mm to 4 mm should be
studied to clarify for what type of configurations this maximum in burning rates occurs. Also, an
experimental investigation of similar configurations as in the numerical study will be necessary to
validate the numerical data.
Imposing a zero Reynolds number for the numerical solution was justified by the relative
motion of the droplets in the cluster and also to match a potential experimental investigation
described previously. For intermediate and high Reynolds numbers the theoretical study of
particle interaction becomes more complex due to the necessity of taking into account the
influence of the non-linear terms of the Navier–Stokes equations on the hydrodynamic interaction
forces. Numerical approaches were used for describing the particle interactions in the non-zero
131
Reynolds numbers regime: Tal et al. (1983), Kim et al. (1993), Raju and Sirignano (1990), Tsai
and Sterling (1990), Chiang and Sirignano (1993), Liang et al. (1996), among others.
It still remains to be determined that gas phase radiative losses have a significant impact
when droplets are exposed to a slow convective flow field in microgravity. There has been little
work showing the effects that a slow convective flow field will have on the radiative heat loss and
implicit on droplet interactions for a droplet cluster burning in microgravity. An interesting
approach will be to perform similar investigation when Reynolds numbers are low, but not zero.
This could be done using the same computational code and geometrical configurations as in the
zero Reynolds case, imposing in the computational domain a convective environment: a flow of
air through one of the boundary walls that has a specified velocity.
An experimental apparatus developed at Drexel University and currently being installed
in a drop tower experiment at NASA Glenn Research Center can provide data on the behavior of
droplet clusters without the effects of fibers or buoyancy.
An important step forward is to perform similar experimental investigations in
microgravity environment to those performed in one-g, to eliminate the effects of gravity and to
reduce the influence of the acoustic field.
The numerical simulation along with the future proposed experiment described in the
project is a unique combination of investigative methods that will provide support for future
investigations and for understanding of droplet interaction phenomena.
132
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138
Appendix A: Droplet Mass Burning Rate And Burning Rate Constant Droplet mass burning rateIdeal case (d2-law) after S. Turns
CH3OH+1.5(O2+3.76N2)=>CO2+2H2O+1.5*3.76N2
ath=1.5
(A/F)= 6.455 kg air/kg fuel
MW air= 28.97 kg/kmol (S. Turns)MW fuel= 32.042 kg/kmol Gas Engineers Handbook, 1965, Industrial Press
kg 0.4kF(T)+0.6kair(T) Law and Williams, Kinetics and Convection in the combustion of alkane droplets, 1972
Tboil 337.65 K Gas Engineers Handbook, 1965, Industrial PressTsurr 293 KTad (100% th. air) 2177 K Gas Engineers Handbook, 1965, Industrial Press
T=(Tad+Tboil)/2 1275.3 K
kF 0.060885021 W/m-KkF(550)*(T/550)^0.5
kF(550K)= 0.039983944 W/m-K S. Turns: Appendix B3
kair(T) 8.06E-02 W/m-K S. Turns: Appendix C1
kg 7.27E-02 W/m-K
cp,F= 3.1067 kJ/kg-K Perry's Chemical Engineers' Handbook (7th Edition)cp,air= 1.187 kJ/kg-K Perry's Chemical Engineers' Handbook (7th Edition)
hfg(Tboil) 1100.81 kJ/kg Perry's Chemical Engineers' Handbook (7th Edition)(deltaHv)=5.2390*((1-(T/337.6))^(0.3682+(0*T/337.6)+(0*((T/337.6)^2))))*10(latent heat of vaporization)
∆hc= 19918.85 kJ/kg JANAF Thermochemical Tables(Enthalpy of Combustion)
B0,q= 2.6770
mass fuel burning rate 3.82951E-07 kg/s S. Turns page 390, eqn. 10.68a
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Appendix B: Sample Input Files For FDS
Single Droplet &HEAD CHID='Case6',TITLE='Drop Comb, 4 ign sources, 40x40x40 grid' / &GRID IBAR=40,JBAR=40,KBAR=40 / &PDIM XBAR0=0.0,XBAR=0.04,YBAR0=0,YBAR=0.04,ZBAR0=0,ZBAR=0.04 / &TIME TWFIN=5. / &MISC DNS=.TRUE.,DTCORE=0.25, BACKGROUND_SPECIES='NITROGEN',TMPA=20.,GVEC=0.0,0.0,0.0 / &SPEC ID='METHANOL',MASS_FRACTION_0=0.00,NU=-1,MW=32.042, VISCOSITY=96.27E-7,THERMAL_CONDUCTIVITY=0.01565, DIFFUSION_COEFFICIENT=1.32E-5 / &SPEC ID='OXYGEN',MASS_FRACTION_0=0.21,NU=-1.5, SIGMALJ=3.467,EPSILONKLJ=106.7 / &SPEC ID='WATER VAPOR',MASS_FRACTION_0=0.0,NU=2, SIGMALJ=2.641,EPSILONKLJ=809.1 / &SPEC ID='CARBON DIOXIDE',MASS_FRACTION_0=0.0,NU=1, SIGMALJ=3.941,EPSILONKLJ=195.2 / &OBST XB=0.019,0.021,0.019,0.021,0.019,0.021, SURF_ID='METHANOL' / droplet &SURF ID='METHANOL', RGB = 0.40,0.40,0.40, HEAT_OF_VAPORIZATION=1101., PHASE='LIQUID', DELTA=0.002, KS=0.20, ALPHA=8.85E-8, BURNING_RATE_MAX=1.00E-02, DENSITY=787., C_P=2.5 TMPIGN=65. BURN_AWAY=.TRUE./ &REAC FUEL='METHANOL', FYI='Methyl Alcohol, C H_4 O', MW_FUEL=32.042 , NU_O2=1.5 , NU_H2O=2., NU_CO2=1., EPUMO2=13290., SOOT_YIELD=0.0,
140
RADIATIVE_FRACTION=0.0, Y_O2_INFTY=0.233, BOF=3.2E12,E=125604,XNF=0.25,XNO=1.5,DELTAH=19937./ ! ! -- ignition from hot spot ! &SURF ID='IGNITION',TMPWAL=3000.,RAMP_Q='IGN RAMP' / &RAMP ID='IGN RAMP',T=0.0,F=0.0 / &RAMP ID='IGN RAMP',T=0.25,F=1.0 / &RAMP ID='IGN RAMP',T=2.5,F=1.0 / &OBST SURF_ID6='IGNITION','INERT','INERT','INERT','INERT','INERT', XB=0.022,0.023,0.019,0.021,0.019,0.021,T_REMOVE=2.5 / igniter right &OBST SURF_ID6='INERT','IGNITION','INERT','INERT','INERT','INERT', XB=0.017,0.018,0.019,0.021,0.019,0.021,T_REMOVE=2.5 / igniter left &OBST SURF_ID6='INERT','INERT','IGNITION','INERT','INERT','INERT', XB=0.019,0.021,0.022,0.023,0.019,0.021,T_REMOVE=2.5 / igniter back &OBST SURF_ID6='INERT','INERT','INERT','IGNITION','INERT','INERT', XB=0.019,0.021,0.017,0.018,0.019,0.021,T_REMOVE=2.5 / igniter front ! ! &VENT CB='XBAR' ,SURF_ID='OPEN' / &VENT CB='XBAR0',SURF_ID='OPEN' / &VENT CB='YBAR' ,SURF_ID='OPEN' / &VENT CB='YBAR0',SURF_ID='OPEN' / &VENT CB='ZBAR' ,SURF_ID='OPEN' / &VENT CB='ZBAR0',SURF_ID='OPEN' / &SLCF PBY=0.022,QUANTITY='TEMPERATURE',VECTOR=.TRUE. / &SLCF PBY=0.022,QUANTITY='METHANOL' / &SLCF PBY=0.022,QUANTITY='HRRPUV' / &SLCF PBY=0.022,QUANTITY='DENSITY'/ &TAIL /
141
Two Droplet Array &HEAD CHID='2drops_16a',TITLE='Burning rate for 2 Drops with 4 thick ignition sources, l/a1=16, a1/a2=2' / &GRID IBAR=80,JBAR=30,KBAR=30 / &PDIM XBAR0=0.0,XBAR=4.0E-2,YBAR0=1.25E-2,YBAR=2.75E-2,ZBAR0=1.25E-2,ZBAR=2.75E-2 / &GRID IBAR=40,JBAR=40,KBAR=40 / &PDIM XBAR0=0.0,XBAR=4.0E-2,YBAR0=0,YBAR=4.0E-2,ZBAR0=0.0,ZBAR=4.0E-2 / &TIME TWFIN=4.0 / &MISC DNS=.TRUE.,DTCORE=0.25,SUPPRESSION=.FALSE.,RESTART=.TRUE., BACKGROUND_SPECIES='NITROGEN',TMPA=20.,GVEC=0.0,0.0,0.0 / &MISC GVEC=0.0,0.0,0.0 / &SPEC ID='METHANOL',MASS_FRACTION_0=0.00,NU=-1.0,MW=32., VISCOSITY=96.27E-7,THERMAL_CONDUCTIVITY=0.01565, DIFFUSION_COEFFICIENT=1.32E-5 / &SPEC ID='OXYGEN',MASS_FRACTION_0=0.21,NU=-1.5, SIGMALJ=3.467,EPSILONKLJ=106.7 / &SPEC ID='WATER VAPOR',MASS_FRACTION_0=0.0,NU=2, SIGMALJ=2.641,EPSILONKLJ=809.1 / &SPEC ID='CARBON DIOXIDE',MASS_FRACTION_0=0.0,NU=1, SIGMALJ=3.941,EPSILONKLJ=195.2 / &OBST XB=0.011,0.013,0.019,0.021,0.019,0.021, SURF_ID='METHANOL',SAWTOOTH=.FALSE. / droplet1 &SURF ID='METHANOL', RGB = 0.40,0.40,0.40, HEAT_OF_VAPORIZATION=1101., PHASE='LIQUID', DELTA=0.002, ALPHA=8.85E-8, BURNING_RATE_MAX=1.00E-02, DENSITY=787., C_P=2.5 TMPIGN=65. BURN_AWAY=.TRUE./ &OBST XB=0.0275,0.0285,0.0195,0.0205,0.0195,0.0205,SURF_ID='METHANOL',SAWTOOTH=.FALSE. / droplet2 &SURF ID='METHANOL', RGB = 0.40,0.40,0.40, HEAT_OF_VAPORIZATION=1101., PHASE='LIQUID',
142
DELTA=0.002, ALPHA=8.85E-8, BURNING_RATE_MAX=1.00E-02, DENSITY=787., C_P=2.5 TMPIGN=65. BURN_AWAY=.TRUE./ &REAC FUEL='METHANOL', FYI='Methyl Alcohol, C H_4 O', MW_FUEL=32. , NU_O2=1.5 , NU_H2O=2., NU_CO2=1., SOOT_YIELD=0.0, RADIATIVE_FRACTION=0.0, BOF=3.2E12,E=125604,XNF=0.25,XNO=1.5,DELTAH=19937./ ! ! ! -- ignition from hot spot ! &SURF ID='IGNITION',TMPWAL=3000.,RAMP_Q='IGN RAMP' / &RAMP ID='IGN RAMP',T=0.0,F=0.0 / &RAMP ID='IGN RAMP',T=0.2,F=1.0 / &RAMP ID='IGN RAMP',T=2.0,F=1.0 / &OBST SURF_ID6='INERT','IGNITION','INERT','INERT','INERT','INERT',XB=0.009,0.010,0.019,0.021,0.019,0.021,T_REMOVE=2.0 / igniter left &OBST SURF_ID6='IGNITION','INERT','INERT','INERT','INERT','INERT', XB=0.0295,0.0305,0.019,0.021,0.019,0.021,T_REMOVE=2.0 / igniter right &OBST SURF_ID6='INERT','INERT','INERT','IGNITION','INERT','INERT',XB=0.019,0.021,0.017,0.018,0.019,0.021,T_REMOVE=2.0 / igniter front &OBST SURF_ID6='INERT','INERT','IGNITION','INERT','INERT','INERT', XB=0.019,0.021,0.022,0.023,0.019,0.021,T_REMOVE=2.0 / igniter back ! ! &VENT CB='XBAR' ,SURF_ID='OPEN' / &VENT CB='XBAR0',SURF_ID='OPEN' / &VENT CB='YBAR' ,SURF_ID='OPEN' / &VENT CB='YBAR0',SURF_ID='OPEN' / &VENT CB='ZBAR' ,SURF_ID='OPEN' / &VENT CB='ZBAR0',SURF_ID='OPEN' / &SLCF PBX=0.0105,QUANTITY='TEMPERATURE',VECTOR=.TRUE. / &SLCF PBX=0.0105,QUANTITY='METHANOL' / &SLCF PBX=0.0105,QUANTITY='DENSITY'/
143
&SLCF PBX=0.0135,QUANTITY='TEMPERATURE',VECTOR=.TRUE./ &SLCF PBX=0.0135,QUANTITY='METHANOL' / &SLCF PBX=0.0135,QUANTITY='DENSITY'/ &SLCF PBX=0.027,QUANTITY='TEMPERATURE',VECTOR=.TRUE. / &SLCF PBX=0.027,QUANTITY='METHANOL'/ &SLCF PBX=0.027,QUANTITY='DENSITY'/ &SLCF PBX=0.029,QUANTITY='TEMPERATURE',VECTOR=.TRUE. / &SLCF PBX=0.029,QUANTITY='METHANOL'/ &SLCF PBX=0.029,QUANTITY='DENSITY'/ &SLCF PBY=0.0185,QUANTITY='TEMPERATURE',VECTOR=.TRUE. / &SLCF PBY=0.0185,QUANTITY='METHANOL' / &SLCF PBY=0.0185,QUANTITY='DENSITY'/ &SLCF PBY=0.0215,QUANTITY='TEMPERATURE',VECTOR=.TRUE./ &SLCF PBY=0.0215,QUANTITY='METHANOL'/ &SLCF PBY=0.0215,QUANTITY='DENSITY'/ &SLCF PBY=0.019,QUANTITY='TEMPERATURE',VECTOR=.TRUE./ &SLCF PBY=0.019,QUANTITY='METHANOL' / &SLCF PBY=0.019,QUANTITY='DENSITY'/ &SLCF PBY=0.021,QUANTITY='TEMPERATURE',VECTOR=.TRUE./ &SLCF PBY=0.021,QUANTITY='METHANOL' / &SLCF PBY=0.021,QUANTITY='DENSITY'/ &SLCF PBZ=0.0185,QUANTITY='TEMPERATURE',VECTOR=.TRUE. / &SLCF PBZ=0.0185,QUANTITY='METHANOL' / &SLCF PBZ=0.0185,QUANTITY='DENSITY'/ &SLCF PBZ=0.0215,QUANTITY='TEMPERATURE' ,VECTOR=.TRUE./ &SLCF PBZ=0.0215,QUANTITY='METHANOL' / &SLCF PBZ=0.0215,QUANTITY='DENSITY'/ &SLCF PBZ=0.019,QUANTITY='TEMPERATURE',VECTOR=.TRUE./ &SLCF PBZ=0.019,QUANTITY='METHANOL' / &SLCF PBZ=0.019,QUANTITY='DENSITY'/ &SLCF PBZ=0.021,QUANTITY='TEMPERATURE',VECTOR=.TRUE./ &SLCF PBZ=0.021,QUANTITY='METHANOL' / &SLCF PBZ=0.021,QUANTITY='DENSITY'/ &TAIL/
144
Three Droplet Asymmetric Array &HEAD CHID='3SF_6a',TITLE='Burning rate for 2 Drops with 4 thick ignition sources, d=2a=2mm, 1.5mm and 1mm' / &GRID IBAR=64,JBAR=64,KBAR=40 / &PDIM XBAR0=1.6E-2,XBAR=4.8E-2,YBAR0=1.6E-2,YBAR=4.8E-2,ZBAR0=2.2E-2,ZBAR=4.2E-2 / &GRID IBAR=44,JBAR=44,KBAR=44 / &PDIM XBAR0=1.0E-2,XBAR=5.4E-2,YBAR0=1.0E-2,YBAR=5.4E-2,ZBAR0=1.0E-2,ZBAR=5.4E-2 / &TIME TWFIN=5. / &MISC DNS=.TRUE.,DTCORE=0.25,SUPPRESSION=.FALSE., BACKGROUND_SPECIES='NITROGEN',TMPA=20.,GVEC=0.0,0.0,0.0 / &SPEC ID='METHANOL',MASS_FRACTION_0=0.00,NU=-1.0,MW=32.042, SIGMALJ=3.626, EPSILONKLJ=481.8 / &SPEC ID='METHANOL1',MASS_FRACTION_0=0.00,NU=-1.0,MW=32., VISCOSITY=96.27E-7,THERMAL_CONDUCTIVITY=0.01565, DIFFUSION_COEFFICIENT=1.32E-5 / &SPEC ID='OXYGEN',MASS_FRACTION_0=0.21,NU=-1.5, SIGMALJ=3.467,EPSILONKLJ=106.7 / &SPEC ID='WATER VAPOR',MASS_FRACTION_0=0.0,NU=2, SIGMALJ=2.641,EPSILONKLJ=809.1 / &SPEC ID='CARBON DIOXIDE',MASS_FRACTION_0=0.0,NU=1, SIGMALJ=3.941,EPSILONKLJ=195.2 / ! !Symmetric configuration, droplets size a1=1mm, a2=0.75mm, a3=0.5mm ! &OBST XB=0.029,0.031,0.03,0.032,0.032,0.034, SURF_ID='METHANOL',SAWTOOTH=.FALSE. / droplet1 &SURF ID='METHANOL', RGB = 0.40,0.40,0.40, HEAT_OF_VAPORIZATION=1101., PHASE='LIQUID', DELTA=0.002, KS=0.20, ALPHA=8.85E-8, BURNING_RATE_MAX=1.00E-02, DENSITY=787., C_P=2.5 TMPIGN=65. BURN_AWAY=.TRUE./
145
&OBST XB=0.032,0.0335,0.037,0.0385,0.032,0.0335, SURF_ID='METHANOL',SAWTOOTH=.FALSE. / droplet2 &SURF ID='METHANOL', RGB = 0.40,0.40,0.40, HEAT_OF_VAPORIZATION=1101., PHASE='LIQUID', DELTA=0.002, KS=0.20, ALPHA=8.85E-8, BURNING_RATE_MAX=1.00E-02, DENSITY=787., C_P=2.5 TMPIGN=65. BURN_AWAY=.TRUE. &OBST XB=0.036,0.037,0.03,0.031,0.032,0.033, SURF_ID='METHANOL',SAWTOOTH=.FALSE. / droplet3 &SURF ID='METHANOL', RGB = 0.40,0.40,0.40, HEAT_OF_VAPORIZATION=1101., PHASE='LIQUID', DELTA=0.002, KS=0.20, ALPHA=8.85E-8, BURNING_RATE_MAX=1.00E-02, DENSITY=787., C_P=2.5 TMPIGN=65. BURN_AWAY=.TRUE./ &REAC FUEL='METHANOL', FYI='Methyl Alcohol, C H_4 O', MW_FUEL=32. , NU_O2=1.5 , NU_H2O=2., NU_CO2=1., EPUMO2=13290., SOOT_YIELD=0.0, RADIATIVE_FRACTION=0.0, BOF=3.2E12,E=125604,XNF=0.25,XNO=1.5,DELTAH=19937./ ! ! ! -- ignition from hot spot ! &SURF ID='IGNITION',TMPWAL=3000.,RAMP_Q='IGN RAMP' / &RAMP ID='IGN RAMP',T=0.0,F=0.0 / &RAMP ID='IGN RAMP',T=0.25,F=1.0 / &RAMP ID='IGN RAMP',T=2.5,F=1.0 /
146
&OBST SURF_ID6='IGNITION','INERT','INERT','INERT','INERT','INERT', XB=0.038,0.039,0.03,0.032,0.032,0.034,T_REMOVE=2.5 / igniter right &OBST SURF_ID6='INERT','IGNITION','INERT','INERT','INERT','INERT', XB=0.027,0.028,0.03,0.032,0.032,0.034,T_REMOVE=2.5 / igniter left &OBST SURF_ID6='INERT','INERT','IGNITION','INERT','INERT','INERT', XB=0.032,0.034,0.040,0.041,0.032,0.034,T_REMOVE=2.5 / igniter back &OBST SURF_ID6='INERT','INERT','INERT','IGNITION','INERT','INERT', XB=0.032,0.034,0.027,0.028,0.032,0.034,T_REMOVE=2.5 / igniter front ! ! &VENT CB='XBAR' ,SURF_ID='OPEN' / &VENT CB='XBAR0',SURF_ID='OPEN' / &VENT CB='YBAR' ,SURF_ID='OPEN' / &VENT CB='YBAR0',SURF_ID='OPEN' / &VENT CB='ZBAR' ,SURF_ID='OPEN' / &VENT CB='ZBAR0',SURF_ID='OPEN' / &SLCF PBX=0.0285,QUANTITY='TEMPERATURE',VECTOR=.TRUE. / &SLCF PBX=0.0285,QUANTITY='METHANOL' / &SLCF PBX=0.0285,QUANTITY='CARBON DIOXIDE' / &SLCF PBX=0.0285,QUANTITY='WATER VAPOR' / &SLCF PBX=0.0285,QUANTITY='DENSITY'/ &SLCF PBX=0.0315,QUANTITY='TEMPERATURE',VECTOR=.TRUE./ &SLCF PBX=0.0315,QUANTITY='METHANOL' / &SLCF PBX=0.0315,QUANTITY='CARBON DIOXIDE' / &SLCF PBX=0.0315,QUANTITY='WATER VAPOR' / &SLCF PBX=0.0315,QUANTITY='DENSITY'/ &SLCF PBX=0.034,QUANTITY='TEMPERATURE',VECTOR=.TRUE. / &SLCF PBX=0.034,QUANTITY='METHANOL'/ &SLCF PBX=0.034,QUANTITY='CARBON DIOXIDE' / &SLCF PBX=0.034,QUANTITY='WATER VAPOR' / &SLCF PBX=0.034,QUANTITY='DENSITY'/ &SLCF PBX=0.0355,QUANTITY='TEMPERATURE',VECTOR=.TRUE. / &SLCF PBX=0.0355,QUANTITY='METHANOL'/ &SLCF PBX=0.0355,QUANTITY='CARBON DIOXIDE' / &SLCF PBX=0.0355,QUANTITY='WATER VAPOR' / &SLCF PBX=0.0355,QUANTITY='DENSITY'/ &SLCF PBX=0.0375,QUANTITY='TEMPERATURE',VECTOR=.TRUE. / &SLCF PBX=0.0375,QUANTITY='METHANOL'/ &SLCF PBX=0.0375,QUANTITY='CARBON DIOXIDE' / &SLCF PBX=0.0375,QUANTITY='WATER VAPOR' / &SLCF PBX=0.0375,QUANTITY='DENSITY'/
147
&SLCF PBY=0.0295,QUANTITY='TEMPERATURE',VECTOR=.TRUE. / &SLCF PBY=0.0295,QUANTITY='METHANOL' / &SLCF PBY=0.0295,QUANTITY='CARBON DIOXIDE' / &SLCF PBY=0.0295,QUANTITY='WATER VAPOR' / &SLCF PBY=0.0295,QUANTITY='DENSITY'/ &SLCF PBY=0.0325,QUANTITY='TEMPERATURE',VECTOR=.TRUE./ &SLCF PBY=0.0325,QUANTITY='METHANOL' / &SLCF PBY=0.0325,QUANTITY='CARBON DIOXIDE' / &SLCF PBY=0.0325,QUANTITY='WATER VAPOR' / &SLCF PBY=0.0325,QUANTITY='DENSITY'/ &SLCF PBY=0.0365,QUANTITY='TEMPERATURE',VECTOR=.TRUE./ &SLCF PBY=0.0365,QUANTITY='METHANOL' / &SLCF PBY=0.0365,QUANTITY='CARBON DIOXIDE' / &SLCF PBY=0.0365,QUANTITY='WATER VAPOR' / &SLCF PBY=0.0365,QUANTITY='DENSITY'/ &SLCF PBY=0.039,QUANTITY='TEMPERATURE',VECTOR=.TRUE./ &SLCF PBY=0.039,QUANTITY='METHANOL' / &SLCF PBY=0.039,QUANTITY='CARBON DIOXIDE' / &SLCF PBY=0.039,QUANTITY='WATER VAPOR' / &SLCF PBY=0.039,QUANTITY='DENSITY'/ &SLCF PBY=0.0315,QUANTITY='TEMPERATURE',VECTOR=.TRUE./ &SLCF PBY=0.0315,QUANTITY='METHANOL' / &SLCF PBY=0.0315,QUANTITY='CARBON DIOXIDE' / &SLCF PBY=0.0315,QUANTITY='WATER VAPOR' / &SLCF PBY=0.0315,QUANTITY='DENSITY'/ &SLCF PBZ=0.0315,QUANTITY='TEMPERATURE',VECTOR=.TRUE. / &SLCF PBZ=0.0315,QUANTITY='METHANOL' / &SLCF PBZ=0.0315,QUANTITY='CARBON DIOXIDE' / &SLCF PBZ=0.0315,QUANTITY='WATER VAPOR' / &SLCF PBZ=0.0315,QUANTITY='DENSITY'/ &SLCF PBZ=0.0345,QUANTITY='TEMPERATURE',VECTOR=.TRUE./ &SLCF PBZ=0.0345,QUANTITY='METHANOL' / &SLCF PBZ=0.0345,QUANTITY='CARBON DIOXIDE' / &SLCF PBZ=0.0345,QUANTITY='WATER VAPOR' / &SLCF PBZ=0.0345,QUANTITY='DENSITY'/ &SLCF PBZ=0.0340,QUANTITY='TEMPERATURE',VECTOR=.TRUE./ &SLCF PBZ=0.0340,QUANTITY='METHANOL' / &SLCF PBZ=0.0340,QUANTITY='CARBON DIOXIDE' / &SLCF PBZ=0.0340,QUANTITY='WATER VAPOR' / &SLCF PBZ=0.0340,QUANTITY='DENSITY'/ &SLCF PBZ=0.0335,QUANTITY='TEMPERATURE',VECTOR=.TRUE./
148
&SLCF PBZ=0.0335,QUANTITY='METHANOL' / &SLCF PBZ=0.0335,QUANTITY='CARBON DIOXIDE' / &SLCF PBZ=0.0335,QUANTITY='WATER VAPOR' / &SLCF PBZ=0.0335,QUANTITY='DENSITY'/ &TAIL/
149
Appendix C: Burning Rate Calculation Samples Table 16 Burning rate calculations for a single 2 mm droplet burning in air X Z V-VELOCITY average velocity DENSITY average density Burning rate m m m/s kg/m3 0.018 0.018 0.002833 0.004146 0.28954 0.282415 2.95E-10
0.0185 0.018 0.003775 0.005464 0.28314 0.27895 3.86E-100.019 0.018 0.004764 0.006745 0.28053 0.278253 4.78E-10
0.0195 0.018 0.005594 0.007597 0.27997 0.278438 5.41E-100.02 0.018 0.005972 0.007673 0.27993 0.278378 5.46E-10
0.0205 0.018 0.005741 0.006975 0.27987 0.278095 4.94E-100.021 0.018 0.005061 0.005853 0.28038 0.27875 4.13E-10
0.0215 0.018 0.004223 0.004685 0.28301 0.282248 3.33E-100.022 0.018 0.00342 0.28951
0.018 0.0185 0.004199 0.005923 0.28065 0.275935 4.14E-10
0.0185 0.0185 0.005777 0.008063 0.27633 0.27413 5.65E-100.019 0.0185 0.007541 0.010235 0.2758 0.274698 7.26E-10
0.0195 0.0185 0.009079 0.011687 0.27671 0.2755 8.36E-100.02 0.0185 0.009745 0.011766 0.27714 0.275423 8.41E-10
0.0205 0.0185 0.009232 0.010482 0.27657 0.274483 7.42E-100.021 0.0185 0.007863 0.008486 0.27556 0.273818 5.94E-10
0.0215 0.0185 0.006265 0.006509 0.27605 0.275603 4.54E-100.022 0.0185 0.004834 0.28042
0.018 0.019 0.005673 0.007596 0.27482 0.271833 5.26E-10
0.0185 0.019 0.008044 0.010582 0.27194 0.270828 7.39E-100.019 0.019 0.01089 0.013694 0.27245 0.271743 9.72E-10
0.0195 0.019 0.013432 0.015776 0.27383 0.272575 1.13E-090.02 0.019 0.014491 0.015862 0.27432 0.27249 1.14E-09
0.0205 0.019 0.013595 0.013962 0.27366 0.271495 9.9E-100.021 0.019 0.011239 0.011043 0.27214 0.27044 7.7E-10
0.0215 0.019 0.008578 0.00823 0.27152 0.271385 5.69E-100.022 0.019 0.006362 0.27442
0.018 0.0195 0.006827 0.008612 0.27143 0.26977 5.94E-10
0.0185 0.0195 0.00984 0.012099 0.26914 0.268978 8.44E-100.019 0.0195 0.013554 0.01576 0.26978 0.269833 1.12E-09
0.0195 0.0195 0.0169 0.018217 0.27091 0.27052 1.31E-090.02 0.0195 0.018282 0.018309 0.27124 0.270435 1.31E-09
150
0.0205 0.0195 0.017078 0.016048 0.27074 0.269575 1.14E-090.021 0.0195 0.013935 0.012593 0.26944 0.26856 8.76E-10
0.0215 0.0195 0.01042 0.009282 0.26866 0.26927 6.38E-100.022 0.0195 0.00756 0.27094
0.018 0.02 0.007272 0.00864 0.27033 0.269755 5.95E-10
0.0185 0.02 0.01051 0.012127 0.26818 0.268958 8.46E-100.019 0.02 0.014493 0.015789 0.26881 0.26981 1.12E-09
0.0195 0.02 0.018092 0.018249 0.26983 0.270495 1.31E-090.02 0.02 0.019592 0.018344 0.2701 0.27041 1.31E-09
0.0205 0.02 0.018282 0.016087 0.26966 0.269553 1.14E-090.021 0.02 0.014898 0.012634 0.26846 0.268543 8.79E-10
0.0215 0.02 0.011119 0.009322 0.26768 0.269258 6.41E-100.022 0.02 0.008028 0.2698
0.018 0.0205 0.006881 0.007676 0.27141 0.271798 5.31E-10
0.0185 0.0205 0.009896 0.010663 0.2691 0.270775 7.45E-100.019 0.0205 0.013611 0.013781 0.26974 0.27168 9.78E-10
0.0195 0.0205 0.016961 0.015873 0.27086 0.272503 1.14E-090.02 0.0205 0.01835 0.015969 0.27119 0.27242 1.14E-09
0.0205 0.0205 0.017152 0.014079 0.27069 0.271435 9.98E-100.021 0.0205 0.014017 0.011165 0.2694 0.27039 7.78E-10
0.0215 0.0205 0.010503 0.008349 0.26863 0.271353 5.77E-100.022 0.0205 0.007638 0.27092
0.018 0.021 0.005779 0.006045 0.2748 0.275905 4.22E-10
0.0185 0.021 0.008148 0.00818 0.27188 0.27407 5.73E-100.019 0.021 0.010999 0.010371 0.27238 0.274613 7.35E-10
0.0195 0.021 0.013554 0.011847 0.27374 0.275398 8.47E-100.02 0.021 0.014628 0.011941 0.27422 0.275323 8.53E-10
0.0205 0.021 0.013746 0.010667 0.27358 0.274398 7.55E-100.021 0.021 0.011399 0.008677 0.27207 0.273753 6.07E-10
0.0215 0.021 0.00874 0.006701 0.27146 0.275573 4.68E-100.022 0.021 0.006516 0.2744
0.018 0.0215 0.004352 0.00431 0.28065 0.282425 3.07E-10
0.0185 0.0215 0.0059 0.005619 0.27629 0.278918 3.97E-100.019 0.0215 0.007674 0.006929 0.27573 0.278178 4.91E-10
0.0195 0.0215 0.009259 0.007818 0.2766 0.27834 5.57E-100.02 0.0215 0.009949 0.007907 0.27703 0.278283 5.63E-10
0.0205 0.0215 0.009442 0.007215 0.27646 0.27802 5.11E-100.021 0.0215 0.00808 0.006103 0.27548 0.278713 4.31E-10
0.0215 0.0215 0.006491 0.00494 0.276 0.282268 3.51E-10
151
0.022 0.0215 0.005058 0.28043 0.018 0.022 0.003037 0.2896
0.0185 0.022 0.003951 0.28316 0.019 0.022 0.004952 0.28049
0.0195 0.022 0.005833 0.27989 0.02 0.022 0.006232 0.27984
0.0205 0.022 0.006006 0.2798 0.021 0.022 0.005334 0.28034
0.0215 0.022 0.004506 0.28303 0.022 0.022 0.003704 0.28961
2.82E-07
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Appendix D: Experimental Set-Up
Table 17 Mechanical Equipment List
System Component Manufacturer Model Status
Piezoelectric transducers Channel Industries C-0509-620 Operational at NASA GRC
Front/rear transmitter blocks Drexel University Custom Operational at NASA GRC
Acoustic Horn SonicEase Custom Operational at NASA GRC
Acoustic Reflector Drexel University Custom On hand at NASA GRC
Reflector Traverse Haydon Switch and Instrument, Inc. (HSI) 20542-12-035 On hand at NASA GRC
Stepper Motor Controller Haydon Switch and Instrument, Inc. L/R 39105 On hand at NASA GRC
Signal Amplifier Optimus XL-200 Operational at NASA GRC
Frequency Generator Agilent 33120A Operational at NASA GRC
Step-up Transformer Circuit Drexel University Custom Breadboard operational at NASA GRC
Control Circuit NASA GRC Custon Breadboard operational at NASA GRC
Syringe Pump NASA GRC Custom Use as-is from HPDCE
Needle positioner Haydon Switch and Instrument, Inc. 20842-12 On-hand at NASA GRC; bracket
must be designed.
Stepper Motor Controller Haydon Switch and Instrument, Inc. L/R 39105 On-hand at NASA GRC
High Voltage Power Supply American High Voltage EC Series Use circuit as-is from HPDCE
Ignitor Igniter Assembly NASA GRC Custom Hot wire, igniter movement, and elevator in testing at NASA GRC
LED and control NASA GRC Custom Use as-is from HPDCE
Camera 1 mount (side view) NASA GRC Custom Need to design and fab mount
Camera 2 mount (top down) NASA GRC Custom Need to design and fab mount
Lens supports NASA GRC Custom Need to design and fab
Power Distribution Module NASA GRC Custom to be obtained
TT8-DDACS NASA GRC Custom to be obtained
Relay box NASA GRC Custom Use as-is from HPDCE
DC-AC Inverter Sinergex PureWatts 300 On-hand at NASA GRC
Purge bottle Swagelok TBD must be sized and ordered
Enclosure NASA GRC Custom 8020 structure, top and two sides Lexan, other sides aluminum
General
Acoustic Levitator
Signal Generation
Droplet Insertion
Diagnostics
153
Table 18 Experiment Controls Timeline Time (sec) Action Control Requirement
-90 Initiate acoustic field Turn on waveform generator (Manually)
-70 Turn on LED Close Relay 4
-60 Turn on cameras digital output from D/O - 3: (ch. 15 and 18)
-60 Turn on VCR Manual Control
-50 Move droplet
insertion needle into
the acoustic field
Activate Needle Control Motor FWD (digital output from
D/O - 3 (ch. 16))
-40 Turn on HV power
supply
Close Relay 7
-40 Disperse liquid
droplet to end of
needle
Move Fuel Supply Motor FWD (Close Relay 3 FWD;
open upon moving specified amount)
-20 Jog reflector to
release droplets
digital output from D/O - 3 (ch. 17)
-15 Turn off HV power
supply
Open Relay 7
-15 Withdraw droplet
insertion needle
Activate Needle Control Motor REV (digital output from
D/O - 3 (ch. 16))
-10 Move igniter elevator
to HIGH position
Move Igniter Elevator Motor FWD (Relay 5 FWD)
154
-5 Move igniter arm into
position
Move Igniter Arm Motor FWD (Relay 6 FWD)
-5 Initiate data logging Thermocouples
-2 Turn on hot wire Activate Relay 8
0 Initiate acoustic field
ramp profile
Initiate ramp profile on the drop indication (digital output
from D/O - 4 (ch. 21&22))
0.5 Turn off hot wire De-active Relay 8
0.5 Retract igniter arm Move Igniter Arm Motor REV (Relay 6 REV)
0.5 Move igniter elevator
to LOW position
Move Igniter Elevator Motor REV (Relay 5 REV)
2.2 End of Drop
2.5 Turn off cameras digital output from D/O - 3: (ch. 15 & 18)
3 Terminate data
logging
3 Turn off LED,
acoustic field ramp,
waveform generator,
etc.
Remove power from all relays
5 Turn off VCR Manual Control
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Appendix E: Nomenclature
a fuel droplet radius
ai, aj fuel droplet radius for the “i” or “j” droplet in the array
b* non-dimensional half inter-droplet spacing
B pre-exponential factor for Arrhenius reaction; Spalding transfer number
cp constant pressure specific heat
dq air-fuel mixture diameter
D diffusion coefficient
D* characteristic fire diameter
E activation energy
f external force vector (excluding gravity)
g acceleration of gravity
H total pressure divided by the density
h enthalpy; heat transfer coefficient
hi enthalpy of ith species
h0i heat of formation of ith species
hfg enthalpy of formation
I radiation intensity
Ib radiation blackbody intensity
Ja Jacob number, ( ) fgsatp hTTc /−∞
k thermal conductivity; suppression decay factor
156
l distance between droplets in an array, measured from droplet’s center
l/a non-dimensional spacing between droplets in an array, as the ration of distance between droplets and droplet radius
M molecular weight of the gas mixture
Mi molecular weight of ith gas species
m& mass evaporation rate of a droplet in array
isom& mass evaporation rate of an isolated droplet
fm ′′& fuel mass flux
im ′′′& mass production rate of ith species per unit volume
Om ′′& oxygen consumption rate per unit area
N number of droplets in a cluster (array)
Nu Nusselt number
Pr Prandtl number
p pressure
p0 background pressure
p~ pressure perturbation
qr radiative heat flux vector
q& ′′′ heat release rate per unit volume
rq ′′& radiative flux to a solid surface
cq ′′& convective flux to a solid surface
Q& total heat release rate
157
*Q characteristic fire size
rr radial position vector
ℜ universal gas constant
Re Reynolds number
s unit vector in direction of radiation intensity
Sc Schmidt number
Sh Sherwood number
T temperature
∞T ambient temperature
wT droplet surface temperature
t time
u = (u,v,w) velocity vector
x = (x,y, z) position vector
Xi volume fraction of ith species
Yi mass fraction of ith species
∞OY mass fraction of oxygen in the ambient
IFY mass fraction of fuel in the fuel stream
H∆ heat of combustion
OH∆ energy released per unit mass oxygen consumed
ε emission coefficient
δ wall thickness
η correction factor (similarity parameter)
158
λ wavelength
kε Lennard-Jones potential parameter ([K])
κ absorption coefficient
µ dynamic viscosity
iν stoichiometric coefficient, species i
Φ dissipation function; equivalence ration
DΩ diffusion collision integral
ρ gas-phase density
τ viscous stress tensor
σ Stefan-Boltzmann constant, Lennard-Jones coefficient ([Å])
lmσ Lennard-Jones coefficient ([Å])
sσ scattering coefficient
159
Curriculum Vitae Irina N. Ciobanescu Husanu
EDUCATION
Drexel University, Philadelphia, PA Ph.D. in Mechanical Engineering 2005 Dissertation: “Droplet Interactions during Combustion of Unsupported Droplet Clusters in Microgravity: Numerical Study of Droplet Interactions at Low Reynolds Number”
Polytechnic University of Bucharest, Romania B.S. /M.S. in Aeronautical Engineering 1990
Thesis: “Calculus and Design of a Turboprop Aircraft Engine. The Influence of the Profiling Law on Engine Performances for Rotor Blades of the Compressor”
TEACHING AND RESEARCH EXPERIENCE Delaware County Community College, Media, PA Adjunct Instructor – MATH 100 “Intermediate Algebra” 2006 Developed syllabus and overall course structure, and administered all grades. Developed teaching strategies for individual instruction.
Drexel University, Philadelphia, PA Teaching Assistant –in “Thermodynamics”, “Thermodynamics Analysis”, “Heat Transfer”, “Fluid Mechanics”, “Internal Combustion Engines”, “HVAC Controls”, “Dynamics”, “Aerospace Structures”
2003-2005
Collaborated on curriculum, projects and exam development, held lectures and all recitation classes, met with students on regular basis, and graded all written work, including final exam papers.
Research Assistant (Research Fellow) 2000-2004
Study of Ice Accretion using LEWICE Code - Computational Evaluation of Icing Scaling Methods and development of Ice Accretion Model Droplet Interactions at Low Reynolds Numbers - Analytical investigation of droplet interactions during vaporization and combustion of asymmetric droplet clusters in microgravity (spatial asymmetric geometry of the array as well as different droplets sizes for a given fuel)
RELATED EXPERIENCE
Ministry of Foreign Affairs, Romania Diplomat 1997-2000
Ministry of Research and Development, Romania Senior Governmental Expert 1994- 1997
Company for Turbo-machinery and Engineering, Bucharest, Romania Researcher 1990-1997
University “Titu Maiorescu” Bucharest, Romania
Part Time Mathematics Instructor 1992-2000
PUBLICATIONS AND PAPERS
“Droplet Interactions during Combustion of Unsupported Droplet Clusters in Microgravity: Numerical Study of Droplet Interactions at Low Reynolds Number”, Bell & Howell, 2006 “Combustion of Unsupported Droplet Clusters in Microgravity”, Poster Session of 30th International Symposium of Combustion Institute, 2004 “Combustion of Unsupported Clusters of Droplets in Microgravity”, G.A. Ruff, S. Liu, I. Ciobanescu, KAIST Workshop, Drexel University, 2003
160
“Droplet Vaporization in a Levitating Acoustic Field”, G.A. Ruff, S. Liu, I. Ciobanescu, Proceeding of 7th International Workshop on Microgravity Combustion and Chemically Reacting Systems, 2003 “Teleworking, Concept and Definitions”, A. Toia, S. Dragomirescu, I. Ciobanescu, G. Macri, 1996 “Tele-Health”, A. Toia, I. Ciobanescu, 1997 “Social and Economic Impact of Teleworking”, A. Toia, S. Dragomirescu, I. Ciobanescu, G. Macri, 1997
LANGUAGES English – speak fluently and read/write with high proficiency
Romanian – native language French – speak fluently and read/write with high proficiency
MEMBERSHIPS Member of Student Chapter of Combustion Institute HISTORY Born in Bucharest, Romania in 1966
Married to Cristian Husanu and has three children: Catalin, Diana and Ana