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3007

This is to certify that the
thesis entitled

DROPLET EVAPORATION IN A QUIESCENT, MICRO-
GRAVITY ATMOSPHERE

presented by

Paul Raphael Cole

has been accepted towards fulﬁllment
of the requirements for the

MS. degree in Manical Engineering

 

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DATE DUE DATE DUE ’ DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2/05 p:/C|RC/DateDue.indd-p.1

DROPLET EVAPORATION IN A QUIESCENT, MICRO—GRAVITY ATMOSPHERE
By

Paul Raphael Cole

A THESIS

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Mechanical Engineering

2006

ABSTRACT
DROPLET EVAPORATION IN A QUIESCENT, MICRO—GRAVITY ATMOSPHERE
By
Paul Raphael Cole

Droplet evaporation is important in many applications and spans numerous
research topics ranging from rocket fuel to crystal growth. In this thesis, the author gives
a detailed analysis of the sensitivity of the droplet surface temperature and the droplet
radius—squared to various physical parameters of common hydrocarbon fuels.
Furthermore, the author has provided a brief optimization analysis in which the physical
parameters are optimized in such a way that the time required for the droplet to
completely evaporate is minimized. Due to the nonlinearity of the governing equations, a

numerical algorithm was developed so that these analyses might be conducted.

Ad Matris Sapientia quod est nonfactum

iii

ACKNOWLEDGEMENTS

First, I wish to thank the Mechanical Engineering department for its ﬁnancial and
academic support. Second, I wish to thank Mr. DeVon Washington for his many useful
comments and suggestions. Third, I wish to thank Dr. Ronald Averill for introducing me

to the power of the HEEDS© software.

And last but by no means least, I wish to thank Dr. Indrek S. Wichman, without

whose knowledge and guidance this thesis would not have been possible.

iv

TABLE OF CONTENTS

LIST OF TABLES ................................................................................ viii
LIST OF FIGURES ................................................................................ ix
Chapter 1: Introduction ....................................................................................................... 1
1 .1 . Background ..................................................................................................... 1
1 .2. Motivation ....................................................................................................... 1
1 .3. Methodology ................................................................................................... 2
Chapter 2: Literature Survey ............................................................................................... 4
2.1. Overview ......................................................................................................... 4
2.2. Survey ............................................................................................................. 4
Chapter 3: The Governing Equations .................................................................................. 9
3.1. Overview ......................................................................................................... 9
3.2. The Ideal Gas Law ........................................................................................ 11
3.3. The Clausius—Clapeyron Relationship .......................................................... 12
3.4. General Governing Equations ....................................................................... 14
3.5. Derivation of the Dimensionless Governing Equations ................................ 21

3.6. Solutions for a Propane Droplet .................................................................... 23
Chapter 4: The Sensitivity Analysis .................................................................................. 25
4.1. Overview ....................................................................................................... 25
4.2. Computational Methodology ........................................................................ 26
4.3. Results and Discussion .................................................................................. 27
4.3.1. Sensitivity to the Gas Constant ..................................................................... 27
4.3.2. Sensitivity to the Latent Heat of Vaporization .............................................. 31
4.3.3. Sensitivity to the Boiling Temperature ......................................................... 36
4.3.4. Sensitivity to the Ambient Temperature ....................................................... 40
4.3.5. Sensitivity to the Initial Temperature ............................................................ 44
4.3.6. Sensitivity to the Speciﬁc Heat of the Droplet Vapor ........ . .......................... 47
4.3.7. Sensitivity to the Speciﬁc Heat of the Liquid Comprising the Droplet ........ 51
4.3.8. Sensitivity to the Lewis Number of the Fuel ................................................ 55
4.3.9. Sensitivity to the Ambient Mass Fraction ..................................................... 59
4.3.10. Sensitivity to the Mass Fraction above the Evaporated Surface ................... 63
4.3.13. Final Conclusions for the Sensitivity Analysis ............................................. 67

vi

Chapter 5: The Optimization Study .................................................................................. 71

5.1. Overview ....................................................................................................... 71
5.2. Computational Methodology ........................................................................ 72
5.3. Results and Discussion .................................................................................. 76

5.3.1. Optimization of the Gas Constant, the Latent Heat of Vaporization, the
Boiling Temperature, the Vapor Speciﬁc heat, the Droplet Speciﬁc Heat, the Vapor
Thermal Conductivity, and the Droplet Density ............................................................ 76

5.3.2. Optimization of the Gas Constant, the Latent Heat of Vaporization and the

Boiling Temperature ...................................................................................................... 79
5.4. Final Conclusions .......................................................................................... 81
Chapter 6: The Asymptotic Analysis ................................................................................ 83
Chapter 7: Notes concerning the Computational Algorithms ........................................... 91
Bibliography ...................................................................................................................... 93

vii

LIST OF TABLES

Table 1. Parameter ranges for those that are varied ............................................. 77
Table 2. Top — ranking designs ................................................................... 77
Table 3. Parameter ranges for those that are varied ............................................ 79
Table 4. Top — ranking designs ................................................................... 80

viii

LIST OF FIGURES

Figure 1. The solution for the surface temperature and for the radius—squared for a

propane droplet ..................................................................................... 24
Figure 2. The sensitivity of the surface temperature to the gas constant .................... 29
Figure 3. The surface temperature solution ...................................................... 29
Figure 4. The sensitivity of the radius—squared to the gas constant ......................... 30
Figure 5. The radius—squared solution ........................................................... 30

Figure 6. The sensitivity of the surface temperature to the latent heat of
vaporization .......................................................................................... 33

Figure 7. The surface temperature solution ..................................................... 33

Figure 8. The sensitivity of the radius—squared to the latent heat of
vaporization .......................................................................................... 34

Figure 9. The radius—squared solution ........................................................... 34

Figure 10. Variation of the surface temperature sensitivity with the latent

heat at select instances of time ...................................................................... 35

Figure 11. Variation of the radius—squared sensitivity with the latent

heat at select instances of time .................................................................... 35
Figure 12. The sensitivity of the surface temperature to the boiling temperature. .......... 38
Figure 13. The surface temperature solution ................................................... 38

ix

Figure 14.

Figure 15.

Figure 16.

Figure 17.

Figure 18.

Figure 19.

Figure 20.
Figure 21.

Figure 22.

Figure 23.

Figure 24.
vapor ......

Figure 25.

Figure 26.

vapor ......

Figure 27.

Figure 28.

Figure 29.

The sensitivity of the radius—squared to the boiling temperature ............... 39
The radius—squared solution ......................................................... 39
The sensitivity of the surface temperature to the ambient temperature. . . . . ...42
The surface temperature solution ................................................... 42
The sensitivity of the radius—squared to the ambient temperature .............. 43
The radius—squared solution ......................................................... 43
The sensitivity of the surface temperature to the initial surface temperature..45
The surface temperature solution ................................................... 45
The sensitivity of the radius—squared to the initial surface temperature..... . ..46
The surface temperature solution ................................................... 46
The sensitivity of the surface temperature to the speciﬁc heat of the droplet
............................................................................................ 49
The surface temperature solution ................................................... 49
The sensitivity of the radius—squared to the speciﬁc heat of the droplet
............................................................................................ 50
The radius—squared solution ......................................................... 50

The sensitivity of the surface temperature to the droplet speciﬁc heat. . . .53

The surface temperature solution ................................................... 53

Figure 30. The sensitivity of the radius—squared to the droplet speciﬁc heat .............. 54

Figure 31. The radius—squared solution ......................................................... 54
Figure 32. The sensitivity of the surface temperature to the fuel Lewis number. . . . . . ..57
Figure 33. The surface temperature solution ................................................... 57
Figure 34. The sensitivity of the radius—squared to the fuel Lewis number ................ 58
Figure 35. The radius—squared solution ......................................................... 58

Figure 36. The sensitivity of the surface temperature to the ambient mass fraction ...... 61

Figure 37. The surface temperature solution .................................................... 61
Figure 38. The sensitivity of the radius—squared to the ambient mass fraction. . . . . . ...62
Figure 39. The radius—squared solution .......................................................... 62

Figure 40. The sensitivity of the surface temperature to the evaporated mass fraction....65

Figure 41. The surface temperature solution ................................................... 65
Figure 42. The sensitivity of the radius—squared to the evaporated mass fraction... . . . . .66
Figure 43. The radius—squared .................................................................... 66
Figure 44. A plot of eqn. (11.i) for contours of constant latent heat ........................ 70
Figure 45. A close—up of Figure 44 in the vicinity of the contour intersections ........... 70

xi

Figure 46. Burn—out time per design evaluation. The horizontal line represents the bum—
out time for a propane droplet, i.e., the baseline design ........................................ 77

Figure 47. Bum—out time per design evaluation. The horizontal line represents the bum—
out time for a propane droplet, i.e., the baseline design ........................................ 80

Figure 48. Solutions for the surface temperature, the radius—squared and the z—parameter
for a large latent heat. Note I is 10 times larger for illustration purposes .................. 84

Figure 49. Asymptotic solutions for the interval of increasing surface temperature and
radius—squared ....................................................................................... 86

Figure 50. Asymptotic solution for the surface temperature in the vicinity of the
maximum radius—squared .......................................................................... 89

Figure 51. Asymptotic solution for the radius—squared in the vicinity of the maximum
radius—squared ....................................................................................... 89

Figure 52. Asymptotic solution for the z—parameter in the vicinity of the maximum
radius—squared ....................................................................................... 90

xii

Chapter 1: Introduction

1.1. Background

In the study of a physical system described by mathematical relationships it is
essential to pare the system down to its most important elements, while at the same time
retaining that system’s essential nature. Then its behavior can be described in terms of
those elements. In theoretical analysis the elements are the dimensionless parameters that
appear in the equations describing the problem. The number of parameters may range
from only one (e. g. the Reynolds number in laminar, incompressible ﬂuid mechanics), to
dozens (as in combined ﬂuid mechanics/combustion/heat transfer studies). A theoretical
or mathematical model is commonly understood to be optimized for the description of a
particular process when the number of parameters is minimal. How this minimization is
to be accomplished has been a subject of discussion since the dawn of the scientiﬁc age.
On a historical note, Antoine de Saint-Exupery (French writer and aviator, 1900—1944)
said “A designer knows he has achieved perfection not when there is nothing left to add,

but when there is nothing left to take away.”

_ 1.2. Motivation

This thesis examines an evaporating droplet in a quiescent atmosphere. This
problem was chosen for two reasons. First, as a pseudo—combustion problem it has

relevance to the huge and growing discipline of research into the combustion of fuel

droplets in oxidizing gas ﬂows. This is a mainstay topic of combustion research on which
many lengthy articles and reviews have been written. Furthermore, evaporation is not
limited to fuels but is also important in other sciences such as cloud physics and

biological crystal growth [10].

The second reason is that this problem can be simpliﬁed to eliminate all
combustion and ﬂow quantities while still retaining a fairly large number of
dimensionless parameters in the governing equations. Hence, the relatively simple
physical problem of an evaporating quiescent droplet can be extremely difﬁcult to
examine from the standpoint of running sufﬁcient numbers of cases to encompass the
phenomenon or even to make reliable generalizations and predictions. It is diﬁicult to run
a one—parameter ﬂow system within a speciﬁed range of operating Reynolds numbers;
one can imagine the difﬁculty of operating a system with approximately ten parameters

that must be speciﬁed and controlled to remain within speciﬁed intervals.

1.3. Methodology

The ﬁrst goal of this thesis is to examine the inﬂuence of a large number of
parameters on droplet evaporation. The inﬂuence must be examined in a careful, rational
way even for this relatively simple example. The procedure outlined here exempliﬁes the
need for even more careful considerations of complicated problems that may, in addition,
involve chemical reactions, non—symmetric ﬂows and other difﬁculties which were

removed at the outset from our model. If it is difﬁcult to sort out the importance of

parameters when there are few, it will be correspondingly more difﬁcult to accomplish

this sorting when there are many.

Sensitivity analyses related to this thesis have been systematized for large systems
of chemical reaction equations. These studies beneﬁt from the qualitatively identical
mathematical structure of the chemical kinetic equations. Their response to temperature
is functionally the same, and the sensitivity analysis reduces to one of sorting out areas in
the temperature spectrum where certain groups of reactions dominate other groups of
reactions. In the problem considered here, by contrast, there is a set of “physical”
processes that lead to a set of equations, each of which describes a different physical
process. The problem is then not one of sorting out the dominant members of a certain
class (e. g. among a large set of reactions) but rather one of sorting out the important and
dominant physical terms in the equations themselves before any approximations were

made. This is an altogether different approach with an altogether different goal.

The second goal of this thesis is to study the results of parameter optimization so
that droplet lifetime is minimized. This is something of a novel approach to studies in
combustion although the importance of evaporation cannot be disregarded: the
evaporation process can be the slowest process, which consequently determines the

overall burning rate [19].

Chapter 2: Literature Survey

2.1. Overview

There are relatively few studies devoted to pure evaporation; rather, the majority
of archival research on droplets seems to be in the actual combustion process and/or
ﬂame behavior. This is somewhat surprising in light of the fact that evaporation would
seem to constitute the simpler “model” problem which should be solved ﬁrst. However,
the papers that are described in the following section seem to be relevant to this study in
that they offer commensurate computational and experimental results. The following

section seeks to highlight the important aspects of the corresponding works.

2.2. Survey

I. Hegseth et a1. [10] experimentally demonstrated to show that when a droplet

evaporates sufﬁciently fast, it exhibits a vigorous ﬂow in the interior; a ﬂow that is driven

2
by surface tension gradients. The D —law theory formulated by Maxwell assumed that

the evaporation process is controlled by a mass diffusion process exterior to the droplet.
If there is no convection, then the liquid interior of the evaporating droplet must supply
the heat to the surface for vaporization. In this thesis, we have assumed that the motion

inside the droplet is negligible and that Marangoni effects are negligible.

Vai-meng Ha et al. [9] intended to investigate the onset and physical mechanisms
of the Marangoni instability of an evaporating droplet. Generally, evaporation at the
surface of a liquid is an unsteady process involving heat and mass transport.
Additionally, the process of evaporation also leads to a local reduction in the temperature
of the bulk liquid near the surface. The Marangoni instability may then be induced by
surface temperature reduction and the consequent variation in surface tension. With the
quasi—steady approximation (i.e., the regression of the droplet is negligible during
evaporation), the surrounding gas motion is asymptotically steady and the temperature
distribution of the droplet is temporarily frozen at each speciﬁed instant. Thus, the
authors derived a lower limit of the onset of Marangoni instability via a linear stability
analysis. Although the evaporation of a droplet is never a steady process since the radius,
surface temperature, and hence the evaporation rate decrease during evaporation, the time
scales of the gas phase are two orders of magnitude smaller than the time scale of heat
diffusion in the liquid phase. Therefore, when the interest is mainly on the liquid phase,
the gas phase can be treated as asymptotically steady since its response time is so much
faster than that of the liquid phase. For this thesis we use the assumption of a negligible

regression rate to simplify the boundary conditions.

T. Elperin et al. [7] numerically investigated the evaporation of a spherical droplet
immersed in a stagnant, sub—critical, radiative gaseous mixture. Here, heat ﬂux from the
high temperature surrounding conducts to the droplet causing the droplet to evaporate.
The authors showed that, in the case of sub—critical evaporation, neglecting the liquid
volumetric expansion causes underestimation of the evaporation rate at the initial stage

and overestimation of the evaporation rate at the ﬁnal stage of the evaporation process.

Furthermore, the authors demonstrated the signiﬁcance of liquid thermal expansion
during the evaporation process. Some of the oversimplifying assumptions used in this
work were: (1) the quasi—steady behavior for the gaseous phase heat and mass transfer,
(2) the therrno-physical properties were evaluated at some reference condition, (3) the
“1/3” was rule used for averaging the thermo—physical properties, (4) liquid thermal

expansion and radiation effects were negligible. However, the quasi—steady assumption

for droplet evaporation is valid when thermal (t7) and diffusion relaxation (t0) times are

much smaller than the drOplet lifetime. The authors demonstrated that evaporation of
highly volatile droplets in a hot gaseous medium cannot be assumed quasi—steady.
Additionally, liquid volumetric expansion is signiﬁcant. For this thesis, the temperature
of the ambient, i.e., the gaseous medium, is taken as room temperature thereby allowing

the quasi-steady assumption.

R. Digilov et al. [5] showed that Trouton’s empirical rule for the entropy of
vaporization follows from the similarity of the potential function of molecular interaction,
thus illustrating that the rule is associated with the law of corresponding states. Frederick
Thomas Trouton (1863-1922) developed an empirical rule that makes the following
statement: for most liquids at atmospheric pressure, the ratio of latent heat of
vaporization to normal boiling point has a value between 75 and 80 kJ/kmol—K for a large
number of different pure substances. If one wishes to obtain a dimensionless result,

Trouton’s rule can be formulated to read:

 

LV
~010
RT ()

Water and alcohols do not follow this rule exactly -— in fact, only for elemental liquids
whose molecules are quasi-spherical and non-polar does this rule hold reasonably well.
Furthermore, the authors showed that Trouton’s empirical rule should be reformulated to
read “constant within a family of similar substances alone, and possibly differing for
substances from different groups.”1 For this thesis, we have restricted possible fuels to

hydrocarbons, thus appreciably maintaining Trouton’s rule.

Kozyrev et al. [15] examined a diffusion model for droplet evaporation which
considered the reduction in the droplet temperature and vapor pressure near its surface.
The evaporation of a single spherical droplet can be regarded as classical; in its simplest
form, it was solved by Maxwell in 1877. However, Maxwell assumed the vapor above
the droplet surface was always saturated, which is generally not true. Furthermore, the
concentration of saturated vapor at the surface is determined by the droplet temperature
rather than the ambient temperature. In their study of the vapor and heat ﬂuxes, the
authors conﬁned themselves to the simplest assumptions, viz., the linear diffusion of
molecules and the supply of energy to the evaporating droplet by heat conduction.
Overall, the model allowed estimation of the condensation coefﬁcient of the vapor
molecules from the experimental form of this dependence. For this thesis, we have not
assumed that the vapor above the droplet is a function of the ambient temperature; on the

contrary, the droplet temperature is one of the variables sought.

Zhu et al. [27] investigated the evaporation of fuel droplets (n—heptane) in high-

pressure environments where the thermodynamic conditions exceed the critical state of

 

‘ Pg. 20

the fuel. Droplet gasiﬁcation in such surroundings is important for diesel engines and
liquid rockets. To model such conditions is no simple task since the conventional “low—
pressure” models generally break down in high—pressure situations; in high-pressure
surroundings, mass transport within the droplet interior cannot be neglected. To make
matters more complex, as the droplet surface approaches the critical state, the latent heat
of vaporization, as well as the surface tension, goes to zero. For this thesis, it has been

assumed that the droplet evaporates under atmospheric conditions.

Ghassemi et a1. [8] experimentally investigated the evaporation characteristics of
single and multi—component droplet which were suspended from the tip of a quartz ﬁber
in normal gravity conditions. In this study, the authors varied the environment pressure

from atmospheric to 25 times atmospheric; the temperature, 400—700 °C. For single

. 2 . .
droplets, the authors observed the classrcal D —Iaw 1n the last stages of the evaporation

process. They also observed that an increase in ambient temperature would act to
decrease the total droplet lifetime. From their experiments, they found that when the
ambient temperature is not too high (relative to the boiling temperature), the interior

temperature is not uniform. Under this condition, the evaporation process does not

2
exactly follow the D —law. For this thesis, there is a large enough difference between the

ambient and the boiling temperature to allow a linear evaporation process.

Chapter 3: The Governing Equations

3.1. Overview

The equations and boundary conditions for an evaporating droplet in a quiescent
atmosphere are derived in the following sections. These include the conservation

equations for the liquid phase and the gas phase. For the liquid phase, the domain is 0 < r
< rs(t); for the gas phase (i.e., the vapor above the surface of the droplet, which is
assumed to be an ideal gas), the domain of interest is rs(t) < r < 00. The surface
temperature and square of the droplet radius (i.e., the surface area) are both functions of
time. The physical parameters are as follows: R (the gas constant), Lv (the latent heat of

vaporization), T b (the boiling temperature), T 00 (the ambient temperature), T 0 (the initial

droplet temperature), Cp (the constant—pressure speciﬁc heat of the vapor), CL (the

constant—pressure speciﬁc heat of the liquid), Lep (the fuel Lewis number), Y Foo (the
ambient fuel mass fraction), YFb (the fuel mass fraction that exists at the boiling
temperature), kg (the thermal conductivity of the droplet vapor), and pL (the density of

the droplet).

A relevant quantity in this analysis is a, which is a ratio of gas and liquid

densities. This parameter is deﬁned in equation (1).

-pg
8— 1L (1)

Here the subscript “g” refers to the vapor above the droplet surface (assumed to obey the

ideal gas law) and the subscript “L” refers to the liquid comprising the droplet. Under

. . . -3 . . .
atmospheric conditions, £~ 0(10 ). At hrgh pressure, 8 increases and achieves a value

of unity at the critical point. The parameter 6 relates to the derivation of appropriate

boundary conditions and also serves as an indication of suitable “disperse” conditions.

A major assumption in this study is that the gas-phase analysis is quasi—steady,
i.e., at any instant in time, the process can be treated as if the gas phase were at steady

state. It can be shown that the characteristic response time of the gas is much shorter than

the vaporization time, i.e., tg << tv where

0V
t ~ g.0
V a
and
2
3r
tg~ 0
aT,goo

Clearly, as a diminishes, tv becomes large. This means that evaporation is a faster

process relative to the thermal equilibration of the gas. Hence, it is not unreasonable to

expect a nearly uniform droplet temperature throughout the vaporization process.

10

3.2. The Ideal Gas Law

The “gas” under consideration is the vapor around the droplet surface as well as
the surrounding gas phase. The ideal gas law (IGL) is typically written in the following

form:

pV=mRT
where m = MN, i.e., (mass [kg]) = (molar mass [kg/kmolj) x (number of moles [kmol]), p
is the pressure in kPa, V is the volume in m3, R is the particular gas constant for the gas

under consideration in kJ/kg—K and T is the temperature in K. Note that the IGL can also

be written in the following form:

p=pRT
. . . 3
where p rs the densrty rn kg/m of the gas at the pressure p and temperature T. It has been

assumed that atmospheric pressure exists at the surface of the droplet, i.e., p = Patm =

101.325 kPa. So, at the droplet surface toward the gas side, we can write the following

1050): pat%Ts (t)

For the purposes of this study, we write the following:

I7
_ atm
pg— ﬁTb

expression:

11

3
Now, we shall test equation (1); for ethane, pg : 1.986 kg/m -) 52: 0.0036 thus equation

(1) seems to be satisﬁed for this hydrocarbon. In terms of the IGL, we can describe the

zpatm
pL RTb

3.3. The Clausius—Clapeyron Relationship

a—terrn in the following way:

In this section, a method for expressing the Clausius—Clapeyron relationship in
terms of mass fraction rather than pressure is given. Such a form will allow this

relationship to be used as a boundary condition.

A detailed derivation of this relationship is usually found in most thermodynamics

textbooks. The conventional form is written as:

The subscripts “l” and “2” refer to the states under consideration. Speciﬁcally, the
derivation considers two thermodynamic states, 1 and 2, for the same chemical species.
One of the assumptions associated with this relationship is that the change in volume
(induced by evaporation) is equal to the volume of the vapor produced. Another
assumption is that the latent heat of vaporization is constant over the temperature range

from state 1 to state 2.

12

The mass fraction of species j is deﬁned as the ratio of the mass of species j to the

total mass, i.e.,

This means the IGL can be written as ij = ijT. For an ideal gas mixture at constant

volume, we can deﬁne the gas phase mole fraction as

y.=”j/
J P

where pj is the partial pressure of species j and p is the total pressure. Alternatively, we

can write

where N] is the number of moles of species j and N is the number of moles of the mixture.

Since m = MN, we can write the relationship between yj and Y] as

M1
Y.= .—
1 y] M
Finally, we can say
P2=y_2=M1Y_2=_Y_2_
P1 y1 M2 Y1 1

Note that M 1 = M 2 since there is just one specie under consideration, viz., the fuel vapor.

Hence, the Clausius—Clapeyron relationship can be written as follows:

L
Y2=Y1exp _v i—J—
R T1 T2

For this study, state 2 exists at the droplet surface; state 1 exits at the boiling condition.
In [9], the authors used a similar form of this relationship for a boundary condition at the

liquid/vapor interface.

3.4. General Governing Equations

The spherically symmetric conservation equations of mass, species, and energy
are given by equation (2). The initial conditions are given by equation (3) and the

boundary conditions are given by equation (4).

6
pg

at

 

1 6 2) .
+.__.__ V :0 2.1
r2 6r(pg gr ( )

1 6 61’. (Z'ii)
=—— p D.r2——'—
2 6r

l4

I 6 2 j
— C T +—-—- V C T
alps p g] zarlpg g pr g
6T (2.iii)

=IapCLeDr2

767g!) F ar

 

 

 

rs (t =0)=r0 (3-i)
T(r,t =0)=T0 (3.ii)
T(r—)oo,t)=Too (4.i)
T(r= rs ,t)=Ts (t) (4.11)
L 1 1
YF s (r 2 rs ,t): YFb exp[7:— {E— T5 (0“ (4.111)
pg (Vg -f5 )=m"=— p, is (4.iv)
aYF
pg(Vg —fS)YF—p DF-a—rzn'r" (4.v)
6 Y0 _
pg(Vg —r's)Y0 -pg 00x arx =0 (4.v1)
Pg lVg ‘r's )Lv + 41' =4; (4.vii)

Equation (4.iii) is the Clausius-Clapeyron equation which is discussed in section
3.3. Equation (4.v) and (4.vi) are mass—conservation equations for the fuel and oxidizer,
respectively [19]; they are also the classic boundary conditions in mass diffusion
problems [25]. Equation (4.vii) is an energy balance at the liquid-vapor interface, i.e.,

this equation equates the difference in conductive ﬂuxes to the energy required for

15

vaporizing the liquid at the surface [19]. Note that the vapor and the liquid heat ﬂuxes

can be expressed according to equation (5).

6T

 

.,,_ __1_ .

ql—kl 6r (5.1)
aTg

"' =k 5."

qg g 6r ( 11)

Note that equation (5.i) describes the heat ﬂux entering the liquid interior, and equation

(5.ii) describes the heat entering the vapor above the droplet surface.

The droplet regression rate is considered negligible insofar as the gas—phase

response is concerned [9]. This can be exploited when we examine the relative velocity

term, i.e. (Vg 4's) . If we make the following deﬁnitions:

V
g g,0 s 0 v

then we can obtain an estimate for the relative velocity, viz.,

( \
V*_ 8’0 drS

g S g.0 g xv d,*

 

 

Recalling the magnitude of the vaporization response time, we can write the following

expression:

16

 

S
:1:

dt

 

Clearly, as the value of a diminishes, then the term 8 becomes negligible.

This means that the regression rate of the droplet can be neglected in the boundary

conditions in comparison with the vapor velocity. Similarly, since t1 << tv, the droplet

temperature during most of the burning process is essentially uniform, as already

mentioned.

The heat ﬂux entering the liquid interior (multiplied by the surface area) must be
equal to the volumetric heat capacity of the liquid droplet (multiplied by the transient rate

of temperature rise), i.e.,

(3T
.. 2 _ if 3 __1
(qIX4ﬂ'rs j—( 3 rs ’01 C11 6t]

6 T d T
where—a-tl- =—d-ti since the droplet temperature is spatially uniform [19].

Now equations (2.i—iii) need to be integrated under the boundary conditions
(equations (4.i—vii)) and the quasi-steady assumption for the gas phase i.e., the partial
derivatives with respect to time are zero. On integrating equation (2.i), we obtain an

expression of the mass ﬂow rate.

p V r =———=M (6.i)

l7

Using this result in conjunction with a negligible regression rate, we can re—cast

equations (4.v-vi) as

 

. 2 an. . ..
MYF—pgDFr 6r =M (6.11)
, 61/

MY r2 —0—x—=0 (6.iii)

Orr—pgDOx 5,-

Equation (2.iii) will be evaluated via integration, substitution of the M term, then
utilizing equations (4.ii) and (5.ii) to evaluate the integration constant. For convenience,

equation (5.ii) will be expressed in terms of the Lewis number and mass diffusivity term.

6T

 

 

 

2 2 g
V C r T = C LeD r +C
(pg g p g) pg 1) F ar 0
- 2aTg
MC T = C LeD r +C
p g pg .0 F ar 0
Tglr=rs,t)=TS(t)
ZaTg 2
C LD = '”
pg 10 e F r 6r rs qg
r,T
S S
C=MC T-r2q"
0 ps 3 3

Thus, we arrive at the following equation for the gas—phase energy equation:

This equation must be integrated again. We say that T g(t, r —> oo) — T 5(t) = T 00, which

 

 

 

yields equation (7).
r32 q; r32 4' M C
T (r t)—Ts(t)+MCp = T00 — s( )+MCP exp — kg r (7)

Now we examine equation (6.ii), which can be re-arranged to read as follows:

M(YF-1)=pg DF r2 (:27,

 

On integration under the limits Y ﬁr = rs) = YFs and Y F(r —> oo) = Y Foo, we obtain the

following solution (for kg/Cp = pg DF LeF):

 

MC I—Y
p Foo
Lerln 1 Y
grs — Fs

Later on, it will prove convenient to deﬁne the following surface parameter given in

 

 

equation (8):
MCp 1 I-YF00
I: k =L 1" 1 Y (8)
g rs eF - Fs

If now we evaluate equation (7) at the droplet surface, i.e., at r = rs, we can make the

following statement:

19

 

 

 

 

n = _ S —1_ ( _ ) -1 "' —l
q r2 TOO Ts(t)+MC r2 T00 TS(t)e +q e
S S
01'
.. _MCP (TOO-T30” e-Z
qg r2 I—e-Z

 

8 r

S eZ—I

,, k,(T..-r,(:))[ , ]

On recalling equation (4.vii) and the expression for the heat ﬂux entering the

liquid, we can use the above result to obtain the following expression:

kg [Too—T50” z . 1 :2
61—1

 

r
S

This expression can be re—arranged to yield equation (9.1).

(It r52 81—] CP
£7T

3k
where 677 = % L CL is a particular and convenient deﬁnition of a combined

(9.i)

dTS Z [(Too_Ts(t)) Lv]

gas/liquid thermal diffusivity.

From equation (4.iv) we can extract an equation for the radius—squared of the

droplet which is given in equation (9.ii).

20

 

5 :45 x (9.ii)

In this equation,l=2C% C , i.e., xi is a ratio of liquid/gas speciﬁc heats. Equations
P

(9.i—ii) constitute the governing equations of the droplet evaporation process.

3.5. Derivation of the Dimensionless Governing Equations

The governing equations are rendered dimensionless since such equations are
typically easier to program. The dimensionless variables and parameters are given in

equations (lO.i-vii). This method of non—dimensionalization is adapted from [7].

T: [58.9] (10.1)

 

Too
62 4520) 2 (lO.ii)
’0
to {TS i: 0)] (lO.iii)
Tb 26%] (lO.iv)

s :%0 (lO.v)
r2 ,
t0 = ET (10.V1)

2]

L ..
Iv 2 ﬂy?» (10.v11)

(1 O.viii)

 

Note that 7 is the ratio of speciﬁc heats which is usually tabulated in thermodynamics

tables and that r0 7% 0. From these deﬁnitions the governing equations take the following

 

form:
dr .
——=—”—(#—0) (11.1)
613 4:2

2
d: =—/lz (11.ii)
ds

Here, we have used

YFS =YFb exp(lv {% — ZR (12.i)

 

 

 

I—Y

a: F°° (12.ii)
I—YFS

z: ("(9) (121a)
LeF

,u= 1" (12.iv)
ex —1

The initial conditions for the coupled, non—linear equations (11.i—ii) are

r(0)=2'0 (13.i)

22

.52 (0)=1 (13.ii)

An interesting quantity is the transfer number B [24]. Qualitatively, this transfer
number represents the ratio of an impetus for inter—phase transfer to a resistance
opposing the transfer. For a vaporizing droplet, the gas—phase diffusion of the fuel is the
primary impetus for vaporization, as opposed to the heat conduction. Thus the transfer

number takes the following form:

3.6. Solutions for a Propane Droplet

Figure 1 describes a representative solution for the evaporation process of a
propane droplet. The plot will describe the dimensionless surface temperature solution
and the dimensionless radius—squared solution, i.e., equations (11.i—ii). The

representative hydrocarbon is propane, so the parameters assume the following values:

R = 0.1885 kJ/kg—K; Lv = 42 7.8 kJ/kg,‘ T}, = 231.05 K; Too = 300.15 K; To = 100.15 K;
(3,, = 1.6794 kJ/kg—K; CL =2.25 kJ/kg—K; LeF = 1.0; YFOO = 0.010;

-3 3
YFb = 0.9; kg = 0.0151 x 10 kW/m—K; pL = 581kg/m
Hence, the dimensionless variables (equations (lO.iii—viii)) are as follows:

to =0.3337; :3 =0. 7698; IV =7.5612; .0 =0.8487;

23

7: 1212.1; ET = 1212.1; 2 = 12121.0; t0 =7.2144 seconds;

For a propane droplet under these conditions, the bum—out time is Sb = 4.304158,

i.e., tb = 0.64202 seconds. Qualitatively, the solutions shown in Figure 1 agree very

closely with the results from several sources, viz., [3], [7], [8], [9], and [26]. However,
the actual numerical quantities differ mainly because of differences in fuels, initial

conditions, and non—dimensionalization techniques.

 

 

 

 

 

 

 

 

1.1 T . , , r
s a a 5 +114
1E=:= """ '=_=‘. ----- i ————————— J'- ———————— Jr —————— +§2[__] —1

0.9 ————————— 4 ......... ______ 4 ........ .3 ......... L ........ 1

0.8~—————————: --------- --------- : --------- : ----------

0.7 --------- f ------- -—i --------- ----- J --------- --------- _

0.6 --------- ' --------- --------- :--— ————— :- ---------

as -------- e --------- a --------- + -------- 1 --------- 1 -------- 1

0.4—4 ------ 4 --------- : --------- + --------- : --------- :— ----- 5 --1
i I I I i a
i l i l l
o 05 1 1.5 2 25 3
51—]

Figure l. The solution for the surface temperature and for the radius—squared for a propane droplet.

24

Chapter 4: The Sensitivity Analysis

4.1. Overview

The approach to the sensitivity analysis for this study is to induce small changes
in the vicinity of a set of representative values in the relevant physical parameters. This
seems to be a very intuitive way of conducting a sensitivity analysis since such an
approach will aid in determining the inﬂuence of the physical parameters on the droplet

vaporization rate and the droplet surface temperature.

For this thesis, when a parameter is varied, the other parameters are maintained at
their representative values for propane. Also, in each section, there are a total of four
plots; two plots for the surface temperature and two plots for the radius—squared. One
plot is the actual sensitivity, which contains ﬁve contours to show how the value of the
parameter itself affects the sensitivity. The other plot is the solution, which also contains
ﬁve contours in order to show how the value of the parameter itself affects the solution.
An interesting characteristic of the solution for surface temperature is that the surface
temperature is always less then unity as well as less than the boiling temperature. In fact,

the surface temperature converges to a wet—bulb temperature [25].

It has been found that the derivatives of the surface temperature and the radius—
squared (equations (ll.i—ii)) play an important role in the sensitivity behavior. The
derivatives under consideration are those taken with respect to time. In equations (11.i—

ii), the derivatives are taken with respect to s, which is a dimensionless time.

25

4.2. Computational Methodology

The majority of sensitivity work was conducted in MATLAB®. The

methodology for inspecting the sensitivity of r (the dimensionless surface temperature)

and :2 (the dimensionless radius—squared) to small changes in the physical parameters is

as follows:

1. Deﬁne an offset value. For example, if the sensitivity of r and 4’2 to some arbitrary

parameter P is under consideration, the notation for this offset value is AP; the

method of deﬁnition will be given later.

2. Deﬁne P1 and P2 where P] is ﬁxed and P2 = P1 + AP. Subsequently, P3, P5, P7, and

P9 will be deﬁned. Note that the (0)0461 terms are referred to as primary values, and
the (ﬁeven terms are the secondary values; thus the general method for computing

the Peven values is Peven = Podd + AP. As mentioned earlier, the reason for deﬁning

ﬁve primary values was to see how the value of the physical parameter itself

inﬂuenced the sensitivity.
3. The formula for the sensitivity of an arbitrary solution X to an arbitrary parameter P is

X=P

_6_)£ where 11’ X
P avg

’ven as S”
g] 8P P

is interpreted to read “the sensitivity of X to

 

small changes in P.” For this thesis, the arbitrary solution X can be either ror .52; the

arbitrary parameter can be any of the physical parameters examined in this thesis.

26

Since the solution was obtained numerically, the sensitivity was also evaluated

numerically, viz.:

X —X
”"11? =Pang mi); PI

 

4. The offset value (AP) is taken as the average of the absolute difference of the Pm,g

value and the primary values, i.e.,

Pill

AP = angPavg —

where i = 1, 3, 5, 9, and Pm,g is the average of the primary parameter values.

4.3. Results and Discussion

4.3.1. Sensitivity to the Gas Constant

Recall that the normalized gas constant, R, is deﬁned by the expression R W = R“

where Ru is the universal gas constant (8.31434 kJ/kmol—K) and W is the molar mass of

the substance (kg/kmol). Hence, if a substance has a large gas constant, then the
substance must have a smaller molar mass. The following plots illustrate the sensitivity
of the surface temperature and the radius—squared to the gas constant for s e [0, 3. 0]

where AR = 0.044 7 kJ/kg—K.

27

The sensitivity of the surface temperature and the solution for the surface
temperature are shown in Figure 2 and Figure 3, respectively. From Figure 2, there
exists a very brief interval in which surface temperature has negligible sensitivity to the
gas constant; this is the same interval in which the droplet experiences condensation (see
Figure 5). Also, there is a reversal in the surface temperature sensitivity contours; at ﬁrst
a large value for gas constant induces a low sensitivity; then it induces a high sensitivity.
This reversal occurs because the heat-up rate decreases, i.e., a larger value of the gas
constant causes the surface temperature to reach its constant value faster. It is interesting
to note that a large gas constant always induces a smaller temperature (Figure 2).
Additionally, a large gas constant always corresponds to a smaller surface temperature

derivative, i.e., a slower rate of change.

The sensitivity of the radius—squared and the solution for the radius—squared are
shown in Figure 4 and Figure 5, respectively. Similar to the behavior of the surface
temperature, the radius—squared initially experiences negligible sensitivity to the gas
constant. Then there is a steady rise in sensitivity throughout the remainder of the
evaporation process. However, there is a reversal, i.e., an instance where the value of the
gas constant has an opposite effect on the sensitivity (see Figure 5). This reversal occurs
because larger gas constant values allow the radius—squared to experience faster decay.
However, the radius—squared grows progressively more sensitive to the gas constant for
larger values of the gas constant. Additionally, a larger gas constant allows a smaller

degree of condensation as well as faster evaporation.

28

 

IIIIII

I'Ill

 

. ill-
..

 

1

__.._L_-_--..__L.____-_--t._--..---..t_--_-__.._.1

—

 

—*— R: 0.1064
—e-— R: 0.1482

 

—*-— R: 0.1976 b T
—B—- R: 0.2765

 

 

"I'ul 'IIII

IIIII'|1

_
L
_
_
_
_
_
_
_
_
L 11111 r. 11111
_
_
_
_
_
.
_
.
L
_
.
_
_
.
_

 

 

0.08

0.04—-------«L---

.1.
P11:
9-

 

2.5

1.5
‘I'Ime, s {—1

Figure 2. The sensitivity of the surface temperature to the gas constant.

 

 

1|" Iaal I

 

—-————————d

.——___.____.—1

all!

 

 

—6—- R: 0.1482 - ~

 

 

I'llllll'll

llllllllll

 

 

0.55 ~- -

o.5~--—-

f—W
J.
P

2.5

1.5
Time, s {—1

0.5

Figure 3. The surface temperature solution.

29

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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0 1. 1 1. 1. 0 0. 1 1 0 0 0 0 0 0 0 0
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2.5

1.5

Time, s {—1
30

0.5
Figure 5. The radius—squared solution.

4.3.2. Sensitivity to the Latent Heat of Vaporization

Recall that the latent heat of vaporization is the amount of energy a substance
needs to absorb in order to change from the liquid phase to the gas phase. The following

plots illustrate the sensitivity of the surface temperature and the radius—squared to the

latent heat of vaporization for s e [0, 3.0] where ALV = 27.4 kJ/kg.

The sensitivity of the surface temperature and the solution for the surface
temperature are shown in Figure 6 and Figure 7, respectively. It is apparent from Figure
6 that, overall, a large value of latent heat always induces a relatively low sensitivity in
the surface temperature. Additionally, from Figure 7, a large latent heat allows the
surface temperature to achieve a higher value. This stands to reason because if a
substance has a large latent heat then that substance will require more energy in order to
change phase. An interesting characteristic of the surface temperature sensitivity is the
initial, steady increase in sensitivity, up to a peak; each contour has its own peak, and
these peaks do not occur at the same instant. Speciﬁcally, these peaks occur when the
contours of the derivative of surface temperature (equation (1 l .i)) reverse. Note that the
derivative contours do not intersect at exactly the same instant, i.e., there is no “ﬁxed
point.” Speciﬁcally, on the LHS of the peaks, a large latent heat allows a faster rate of
change in surface temperature, whereas on the RHS of the peaks, a large latent heat

allows a slower rate of change.

The sensitivity of the radius—squared and the solution for the radius—squared are
shown in Figure 8 and Figure 9, respectively. From Figure 8 and Figure 9, it can be seen

that during the small condensation period, the radius—squared has negligible sensitivity to

31

latent heat. However, once the droplet starts to evaporate, the radius—squared becomes
progressively more sensitive to changes in the latent heat, although a large value of latent
heat induces less sensitivity. Additionally, a large latent heat always corresponds to a
larger radius—squared, i.e., a large latent heat induces more condensation as well as a

longer evaporation time.

Figure 10 and Figure 11 illustrate the variation of the sensitivity of the surface
temperature and the radius—squared with the value of the latent heat for select instances
of time. From these ﬁgures, it is apparent that when the latent heat has a lower value,
both the surface temperature and the radius—squared experience lower levels of
sensitivity for all time. It is also apparent that the radius—squared is more sensitive to the

latent heat than is the surface temperature.

32

 

 

 

 

 

 

 

 

 

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Figure 6. The sensitivity of the surface temperature to the latent heat of vaporization.

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Figure 7. The surface temperature solution.

33

 

 

 

____——————.1

2.5

 

 

 

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Figure 8. The sensitivity of the radius—squared to the latent heat of vaporization.

11
o.9--------—

1.1

0.7

2.5

1.5

Time, s {—1
34

0.5
Figure 9. The radius—squared solution.

 

 

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Figure 10. Variation of the surface temperature sensitivity with the latent heat at select instances of time.

 

 

 

 

 

 

 

 

 

 

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_ _ h _ _ _ _ m _

_ _ ~ _ _ _ _ - _
11111 T1111TIP11111117111171111.1111111111111 1%

_ . u _ _ . _ - _

_ _~ _ . . . w _

_ _~ _ _ _ _ _

_ .N _ . _ _ u _

_ _ _ . _ _ _
11111 111131.1111.11111.11111_11111_1111w1l1lJ111-lm

. _~ _ . _ . n. _

. _ _ . _ o. _

_ ~N _ . _ . 0.. _

_ _ . . _ _ _ _
lllll _I I I I ~.I I I I I.I l I I I.I I I I 1.11 I I I l_I I I ImI.I I I I I_I I I II. E
_ ~1. _ _ . _ on. _ 4

. ~_ _ . . _ -_ _

_ ~_ _ . _ _ I. _

_ ~_ . . . _ -_ _
_ Q _ _ _ _ _ m. _ 0
11111 T1511_11111.11111_11111.11111_111o1.11111111111
_ ~ _ _ . _ _ -_ _ 4

_ h _ _ . _ _ m. _

_~ _ . . _ _ o. _

_~ _ _ _ . _ - _ _

_~ . _ _ _ _ a _ _

11111 .1T111_11111_11111_11111.11111_11$11.11111.11111

_~ _ _ _ _ _ u _ .

_ _ . _ _ o _ _

N _ _ _ . _ u _ _

_ _ . _ _ _ _ .
_ p h _ a p _ L m
5 4 “w 3. 5 2 5 1. 5 03

4. 0 . 0 2 O 1. 0 O.

O o 0 0 0

3
TL w 9

LV [kJ/kg]

Figure 11. Variation of the radius—squared sensitivity with the latent heat at select instances of time.

35

 

_‘ ~

 

4.3.3. Sensitivity to the Boiling Temperature

The boiling temperature of a liquid is that temperature at which the liquid changes
to a gas at a solid interface. At atmospheric pressure, the boiling temperature of propane
is 231.05 Kelvin, i.e., -42.1 0C. The following plots illustrate the sensitivity of the

surface temperature and the radius—squared to the boiling temperature for s e [0, 3. 0]

where ATb = 36K.

The sensitivity of the surface temperature and the solution for the surface
temperature are shown in Figure 12 and Figure 13, respectively. There is a brief interval
in which Surface temperature has negligible sensitivity to the boiling temperature; this is
the interval in which the droplet experiences a small amount of condensation —- the longer
the condensation process, the longer is the interval of negligible sensitivity. Furthermore,
a large boiling temperature induces a lowered sensitivity in the surface temperature. It
also allows the surface temperature to achieve a higher value. This characteristic stands
to reason because a lower boiling temperature means that a substance does not require so

much energy to boil.

The sensitivity of the radius—squared and the solution for the radius—squared are
shown in Figure 14 and Figure 15 , respectively. During the brief condensation phase, the
sensitivity of the radius—squared is very similar to the sensitivity of the surface
temperature in that during the condensation process, the radius—squared has negligible
sensitivity to boiling temperature. However, once the droplet starts to evaporate, the
radius—squared experiences a progressively larger degree of sensitivity until the droplet is

completely evaporated. Additionally, a higher boiling temperature allows more

36

condensation to occur, thus a longer evaporation process, while at the same time allowing

the radius—squared to be less sensitive to small changes in boiling temperature.

37

 

 

   

 

 

 

 

 

   

 

 

0. 2 «2.3 2 H 2 2 2 2 2
2 _ 2 2 2 2 2
a 2 9+ _ . 2 2 2.. 2 2
_ 2 _ 2 2 2 _ 2 2
A :
2 2 H2 2 2 2 2 .. 2 2
2 2 2 2 2 2 2
2 2 .222 n 2 2 .. 2 2
2, __
111 .. 1.. 1 111111111 2. 11111 2 111111 k 11111 2 111111
2.2 .22.. 2 _ _ .. _ _
A 2 6 2 . _ 2 : . _
2 2 2 2 2 2 2
.m 2 wH. 2 2 _ = 2 2
_ 2 __ 2 2 _ 2 2 2
a _ .2 _ 2 2 2 2. 2 2
F _ _ 2 2 _ 2 2
2 2 _ _ 2 =2 2
2 _ _ 2 2 _
III‘II—II IIIII IIIII ._1 IIIII .IIII 5” IIIII _ IIIIII
_ 2
2. n u u .. _
a _ 2 2 2.. _
A _ _ 2 2 2
a _ 2 2 2: _
. _ 2 2 2 _
m _ _ 2 2 z 2
2 2 2 _ _ 2
11122.... 11111111111111111111 2 11111 +1151: 11111 L
2 2 . 2 2
_ _ _ 2 : 2
2 _ 2 2 2
2 2 _ 2 = 2
_ _ _ _ _
_ _ 2 :2
_ _ _ 2 _
_ . _ _ =—
IIIII —I"IIIIII‘IIIII-|"D .IIIII—I'II I—I"I.u—IIIII1
_ _
2 2 2 2n 2 =2
5 5 5 9 5 _ 2 _
5 7. 0. 8. 2 H _. H .._
5 1. 3 ,
M 2 3 Mm % n u ..
1 2 2 2 2 2 2 a/ 1..
VIII ..b ..b l0b ..b ..b II— IIIII - 'IIIIwIIIA "III'Il
T T T T T 2 _ 2 .22
2 _ 2. 22
2 2 2 .
2 2 2
2 2 2
2 _ 2
_ 2 2 _ _ 2
Ir L 2 h 2 F I
7. 6. 5 4. 3 2 1. O 1.
0 0 0 0 0 0 O nw

2.5

1.5
Tune, s {—1

0.5

Figure 12. The sensitivity of the surface temperature to the boiling temperature.

 

IIIIII

 

H—r-m-r—«r— «I

+H—4»—+ -+-—+—+——+—

DwIII

IIIII.:I III
I.

—.a

 

 

__2_ Tb: 184.55 .2.

 

 

—L—A——A— 45444244.» a .A —a~

, -—H~-

IIIIII

IIIIII

IIIIIIIIIIIIIIIIIA

_—_—_—.?—.———_._—__

IIIIII

 

 

0.4 —

2.5

1.5
Trme, s [—-]

0.5

Figure 13. The surface temperature solution.

38

 

I|"Ilv

I'll-

r'-ltl

lllll

 

 

 

 

 

266.89 1

Tb:

 

 

 

 

 

 

 

18
1.6--

 

Tlme, s {—1

Figure 14. The sensitivity of the radius—squared to the boiling temperature.

 

 

   

 

---—.-——-~——J

 

 

 

 
 

 

 

 

 

 

5 5

o. 2.

..I 3

3 5

2 3

.U ...D

T T

2 2

2 2 o
4. 2 o
0 0

Tune, s {—1

Figure 15. The radius—squared solution.

39

4.3.4. Sensitivity to the Ambient Temperature

The ambient temperature is the temperature of the medium in which the droplet is
suspended. For this thesis, it was set at a nominal value of 300.15 Kelvin, i.e., 27 °C.

The following plots illustrate the sensitivity of the surface temperature and the radius—

squared to the ambient temperature for s e [0, 3.0] where AT 00 = 12 K.

The sensitivity of the surface temperature and the solution for the surface
temperature are shown in Figure 16 and Figure 17, respectively. The ﬁrst noticeable
characteristic is that the sensitivity decreases, with a larger ambient temperature inducing
a larger decrease (see Figure 16). These relative minimae do not occur at the exact same
instant; rather they occur in the interval in which the contours of the derivative of the
surface temperature intersect. Again, the derivative contours do not intersect at the exact
same instant. Speciﬁcally, on the LHS of the sensitivity peaks, a larger ambient
temperature induces a larger derivative; on the RHS, this behavior reverses. Although the
contours of the surface temperature derivatives and the contours of the surface
temperature sensitivity intersect, the contours of the surface temperature solution do not;
in fact, a lower ambient temperature allows surface temperature to achieve a higher value

for all time.

The sensitivity of the radius—squared and the solution for the radius—squared are
shown in Figure 18 and Figure 19, respectively. During the brief condensation phase, the
radius—squared has negligible sensitivity to changes in the ambient temperature (unlike
the sensitivity of the surface temperature during this interval). However as soon as

evaporation begins, the radius—squared grows progressively more sensitive until the

40

droplet has evaporated. Furthermore, the radius—squared has greater sensitivity to the
ambient temperature when the ambient temperature is large; however, the radius—squared
converges to the same amount of sensitivity regardless of the actual value of the ambient
temperature. Additionally, a larger value for the ambient temperature allows less

condensation, thus, a faster evaporation process.

4]

      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 2 2
~ _ _ _ _ _ _ _ .
2v 2 2 2 2 2 2 2 e n u
_ _ _ _ _ _ — _ m
2 2 2 2 20 0 0
.1 a m m m
2 2 2 2 2m 2 2 m ..
_ 2 _ 2 2 .. .. .. .. ..
w w m w m
2- ..... 2 ..... 2..1 r T r T ---5. m -- -. - ........... T
2 2 2 2 _ 2 m
_ _ _ _ _ .1
2 u 2 u 2 .m
_ 2 2 2 2
- _ _ _ _ e <
2 2 _ . 2 2 2 .
_ 2 2 _ _ _ _ . m H
2- ..... 2 ..... 2 ..... 2 ..... 2 ..... 2 m -- 1 - ..................................
2 2 2 2 2 2 2 2 m
_ 2 _ _ _ _ _ _
_ _ 2 _ 2 _ _ _ m
2 22. n2 2 2 2 2 2 ] p
_ _ _ . _
H h _ _ _ _ _ F- m
..... 5 s 2... 7 --
2 n2 2 2 2 2 2 1 e. C
2 ..2 2 2 2 2 2 m a
2 2 ...2 2 2 _ 2 2 .H M
.2 2 2 2 2
" 2111.1 :2. 2 2 2 u 2 H
2 211.. 2 2 2 2 2 m
..... 2 ..... 2:11.. :2 ..... 2 ..... 2 ..... 2 ..... 1 ..m ..................................
2 2 2 _,11./1..1 . 2 2 2 ..v.. 2
2 2 2 2 /.uI.I u _ 2 W 2
2 2 2 2 2,”, a. 2 2 .U 2
2 2 2 _ 2 lu/ n; 2 .l 2
2 2 2 2 2 2 l/u/ﬁ/ 2 m 2
n n u u n _ n _ m _
2 2 2 2 2 _ _ .u 5 .1111 11111111 1 .. . 111111111111111
..... 2 ..... 2 ..... 2 ..... .._...1....-o. 2m ._ _ .
_ _ _ _I
2 2 _ 2 _ _ _ .2 . . _ 2 2 _ I
2 2 2 2 2 _ _ . 6 _ _ 2 2 ’1
2 _ 2 _ 2 2 H _ _ 2 _ _
2 2 2 2 2 _ 1
_ _ _ . _ _ m 2 2 2 2 2 _ ..
2 2 2 2 2 . u. 2 g _ _ 2 2 _ _ 21
2 2 2 2 2 .- n 2 r“ 2 2 2 2 2 2 2
2 2 _ t 1 c. 2 0 8 5 7 5 6 5 5 5 4
6 5 5 5 4 5 3 5 2 7 . 6 . 5 4 «w
0. 5. 0. 4. 0. 3 0. 2 0 0 0. O 0. 0 0 0 0. 0 0
o o 0 0

0.65

2.5

1.5

Tlme, s [—]
42

0.5
Figure 17. The surface temperature solution.

 

 

 

 

 

 

 

 

 

~ 2 2 a 2 2 J 2i
2 2 2 2 2 2 2 2
2 2 2 2 2 2 _ 2
2 2 2 2
0 0 o
2 2 2 2
. _ _ L. .1 a m m
2 2 2 2 2n]2 2 2
2 2 2 2 2 w m
2111112 11111111 2 IIIII _ 11111 2 T T T T T 12.3.
2 2 2 2 2
2 2 2 2 2
2 2 2 2 2
2 2 2 2 2
2 2 2 2
2 2 2 2
2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2111112 11111 2 11111 2.. 1.2 11111 2 11111 2 11111 2 11111 2 11111 2
2 2 2 2 2 2 2
2 2 2 2 2 2 2
2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2
2 2 2 2 2 2 2
211.1112 11111 2 IIIII 2 !!!!! 2 1111111 2111112111112 11111 r3.
2 2 2 2 _ 2 2 2 1
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2
2 2 2 2 2 2 2
2 2 2 2 2 2 2
2 2 2 2 2 2 2
2 2 2 2 2 2 2
2 2 2 2 2 2 2
lllll 2.I II I I2I I I I I-I I II '12.! IIII2I II I.I2I I .I II2III II. 1
2 2 2 2 2 2 , 2
2 2 2 2 2 2 2
2 2 2 2 2 2 2
2 2 2 2 2 _ 2
2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 _ 2 2 2 2 2
2 2 2 2 2 2 2 2 5
11111 21111121111121111121111121111121111121111121 .
2 2 2 2 2 2 2 2 0
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
p 2 r r 2 r L 2 l0
8. 6. 4. 2. 1 8. 6. 4. 92 0
4| 1 4| 1 0 0 0 O
8

Time, s {—1

Figure 18. The sensitivity of the radius—squared to the ambient temperature.

 

 

 

 

 

 

 

 

 

2 2 2 a 12 1.1 r 2L.»
_ 2 2 2 2 e
2 2 2 2 2. 2 e.
2 2 2 2 2.. 2 2
2 2 2 2 . 2 2
2 2 2 2 .2 2
2 2 2 2 2 2 2
2 2 2 2. 2 2‘ 2
2 2 2 . 2
21.111.1112 11111 F 11111 2 111111 2.1.1.. 11 11111111 2 111111
2 2 2 2 2 2 2
2 _ _ 2. 2 e 2 2
2 2 2 2.. , 2 2
_ 2 2 a 2 2
2 2 2 ..2 2 2 2
2 2 2 2 ‘ 2 2
2 2 2 e 2 2
2 2 2, 2 2 2
T 2 2 2 . 2 2 1 21 1.11
1I1112 11111 2 11111111 1 11111 2111.111... 11121 1
2 2 .2 2 2 2
2 2 .. 2 2 2
2 2 ..2 ..2 2 2 2
2 2 2 2 2 2
2 2 . 2 2 2 2
2 2 2 2 2 2 2
2 2 2 2 2 _
TIIIIJ !!!!! l IIIII 4. 11111 2 IIIII 4 IIIII 2 111111
2 I 2 2 _ 2
2 2.. .2 2 2 2 2
2 .. _ 2 _ 2 2
2 .2 2 2 2 2 2
2 .. 2 2 2 2 2
2 .. .. 2 2 2 2 2
2 ... .. 2 2 2 2 2
2 ..k. 2 2 2 2 2
2 .. 2 111111 21.111112- 11111 211.111... 11111 _ 111111
IIIIII. ”the. 2 2 2 2 2
2a 2 2 2 2 2 2
_“».‘§ _ _ 2 P 2 »
n.»‘\§
.2.... 2 _ _ 0 0 O O
i... 2 2 6 7 8 w
a... 2 2 2 2 2 2
L - — — C. C. I. I. ..
. _ _ _ m w 22 22
1111111111 111I1J111114111 111.
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2 2 2 2
2 2 2 2 2 2
I 2 2 i2 2 2 2
1 4| 9 8 7 6 5 4
1 0 0 0 0 O 0

2.5

1.5
Tume, s [—]

0.5

Figure 19. The radius—squared solution.

43

4.3.5. Sensitivity to the Initial Temperature

The initial temperature is the temperature of the droplet surface at time zero. For
this thesis, it was set to a nominal value of 100.15 Kelvin, i.e., -173 °C. The following

plots illustrate the sensitivity of the surface temperature and the radius—squared to the

initial temperature for s e [0, 3. 0] where AT 0 = 30 K.

The sensitivity of the surface temperature is given in Figure 20 and the solution
for the surface temperature is given in Figure 21. At ﬁrst, the surface temperature has
relatively high sensitivity to its initial temperature, but this sensitivity quickly diminishes
to zero. Additionally, a larger value of the initial temperature induces a quicker decay in
the sensitivity. Additionally, the surface temperature becomes independent of initial
temperature (unlike the behavior of the other parameters). Furthermore, the sooner the
surface temperature reaches its constant—value behavior, the faster the sensitivity decays

to zero.

The sensitivity of the radius—squared and the solution for the radius—squared are
shown in Figure 22 and Figure 23. Between these ﬁgures, the radius—squared exhibits
remarkable and unique behavior toward this parameter. During the condensation phase,
there is a quick rise in the sensitivity of radius—squared. As the radius—squared
approaches its linear decay regime, the sensitivity starts to level off to a constant value.
This behavior is unique to this parameter in that such behavior does not show up in any
of the other parameters. Furthermore, the sooner the radius—squared starts to decay, the

more sensitive it is to the initial temperature.

44

 

 

 

 

 

 

       

 

 

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_ . . _ _ . _ . .
_ . _ _ _ _ _ _ _
. _ . _ _
m 5 w 5 O _ . . _ .
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1 1 4| 1 2 . _ . _ _
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0 0 o 0 0 _ _ _ _ .
r--T T T: T T .-uruuiu.----ruuuuuunarunu 5.
_ _ _ _ _ 2
_ . _ _ _
_ _ _ _ _
. _ _ _ .
. . _ _ .
_ _ _ _ .
a . q 4 _ . _ _ _
. . _ _ . _ _ _ _
_ . . _ . _ _ _ _
IIIIAIIII.IIII._.IItI_IIIIﬂIIIJIIIIﬁIIIJIIIIWIII 2
_ _ _ _ _ _ _ _ _ ,
. _ . _ _ _ _ _ _
_ _ . _ _ . . _ _
_ . _ _ _ . . _ _
. _ _ _ _ _ . . .
_ _ . _ _ _ . _ .
_ _ . _ _ . _ _ .
_ _ _ _ _ _ . _ _ 5
ritualillvtlu+II..I_I|II+IIII_IIIITIIILIIIITIII .
. _ _ . _ _ _ _ _ 1
_ . _ _ . _ . _ _
. _ _ _ _ _ _ _ .
_ . . _ . _ _ _ _
_ _ _ _ . . _ _ _
_ . _ . _ _ _ _ _ .
_ _ _ _ _ _ _ _ . ..
_ _ _ _ _ _ . _ . ..m.
_ . _ . _ _ . _ _ s
lulu41:i|_uiuiﬂ|it...|||...||lll.|ll|.ll|..I...|..i_tll3.9.1
_ _ _ _ _ _ _ _ _ ...w
_ . . . _ . _ . _ w
_ _ _ _ _ _ _ _ .,
_ . _ _ _ . . _ m,
_ _ _ _ _ _ _ _ .. M“
_ . . _ . _ _ _ aw
_ _ . _ _ _ _ .
_ _ . _ _ .. _ =5
IIIIJIIIJI11141IIJIIII u 11: l. .
_ _ _ _ u _ . : O
_ _ _ ,. . . _ a
_ . _ u _ _ _
_ _ .. it. _ _ _ i u
_ _ i _ . _ t
_ , . . _ o .
_\\. _ _ _ I. _
\X. _ u _ _ . .

\ _ _ _ _ _ L 0
5. 5 4. 5 3. 5 2 5 1. 5 0
0 4. 0 3. 0 2 0 1. 0 0.

0 O 0 O 0

Time, s [—]

Figure 20. The sensitivity of the surface temperature to the initial surface temperature.

 

 

  

 

 

 

 

 

 

 

ﬂ n _ a a a n _
. _ _ _ _ _ _
_ _ _ . _ . _
_ . _ _
_ _ _ . 0 5 w 5 0
_ _ _ _ 0 2 7 0
. _ _ _ 1 1 1 1 2
. _ _ . an n. so a. on
0 o O 0 0
. _ _ .
. _ . _
. _ _ _
. _ _ .
. _ . _
_ _ . _
_ _ _ _ i a a
_ _ _ _ _ _ _
_ _ . _ _ _ _
Tl l4 lllll .l IIIII _ IIIII A. IIIII . IIIII A IIIII _ llllll
_ _ _ _ _ _ _
_ _ _ _ _ _ _
. _ . _ _ _ _
. _ _ _ _ _ _
_ _ . _ _ _ .
_ _ _ _ _ _ .
. _ _ . _ _ _
_ . _ _ _ _ _
It 1.... IIIII T ..III. nnnnn + nnnnn _ IIIII + IIIII . llllll
_ _ . _ _ _ _
. . _ _ . _ .
. . . _ _ . _
. _ . _ . _ .
. _ _ _ _ _ _
_ _ _ . _ . _
. _ _ _ . _ _
_ _ _ _ _ _ _
_ _ . _ _ _ _
II II. lllll .I lllll _|||..|_t lllll .iitllt. uuuuu _ llllll
.._, _ _ . _ _ _ .
..r . _ _ . _ . .
_ . _ _ _ _ _ .
n... .._ _ _ _ . _ _
w . _ _ _ _ . .
a .. _ . _ _ . _
..w a _ . _ _ _ _
.t .r ._ _ _ _ _ _
II: 4| ulu1.|||J IIIII 4 lllll _ lllll 4 lllll . uuuuuu
3 _ _ _ _ _ _
_ a . _ _ . _
_ _ _ a _ _
. .. _ . _
_ _. . _ _
_ _ _ i o . . _
_ . _. - _ ._ _
_ _ . i .. _ _
9. p _ 4 p. <_ A
0 6. 0 5. 0 A 0 3.
0 O 0 O

2.5

1.5
Time, s [—]

0.5

Figure 21. The surface temperature solution.

45

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a . F \I b 3 DU. I \l D I

. q <44 . . a . m _ . . . _ . . . .e
B _ W . _ . . . _ . . . . . . .. . .
n. m _ . _ . . . m . . . _ . _ . .. ..a
a _ W . . C . . . . _ _ .. e.

G _ . _ . . 0 5 w 5 0 p _ . _ . . . . .
B . H. W . _ 0 2 7 o m . _ . . . . _ . .._
m . H. . . 1 4| 4| 4| 2 w . . _ . . . ....
_ _ _ u. D. an no no . _ _ - ‘5 —
B . _ W _ _ To To To To To 5 W . U u . n u . _: .
éllll llll .ill.|l_| llIFll lllz a TlllPIllnLllllrlllLlIllrlllL I l I lLV-lll. lllll
_ . . _ . . _ . . a. _
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_ . . . . . a .
m _ .M _ IQ” " u u n . a u _
W. . . . .m . . _ . . . _ e . _
t é
a 4 u M .4 u _ _ _ .m n n n u .w _ . u
W H... . O . _ . . _ .C . _ _ . . . ~a . _
. . . . _ . ._ . .
...4: TL.--”- ..... _ ..... . ..... _ ...... 2 m -- .....
B _ H G _ . _ 0 . _ . _ . _‘ . . _
W j W _ _ _ t . _ _ . 5. .VA _ . .
O +. . . . d . . . . . e. . . _
a 4. W H n u 1 e u u n .w .. . . . u
m 4. M n . . u _ w . u . _ . u u n n
W m M 0 . . _ . S m- _ . _ .._. .... . . . .
IIII . lllll . llllll IIII+I|IITIIIT lat. u 11..aiiliinTunuallilTnnll
D H H M . . . 999 w . _ . s . _ _ _
B 0 ., . _ m m . _ . a. _ . _ .
m H... O . _ II a . . . . .._ . . . .
a G . _ T f. _ . c . . . . _
a. . e _ u § _ _
W h .. H . .m u . . .a u u u . _
r , .. , . _ f _ . ..: _ . . _ _
W H. .... . _ 0 _ . c. . . . _ .

. . _ . . . . . . . .
gill-1+. . .I IIIIIIII I II In IIIIIIIIIIIIIIIIIIIIIIIIIIIII
m .. . u ...w u . u u u u .

W . ..U . . .e . . . . p .
_ .1 _ - _ _ _
a 0 5 O 5 0
_ . .u _ _
a _ m __ _ _ _ m n w n m
. S .: . e. . _
_ C . .. . . _ ..0 ..0 ..0 .0 ...U
_ m ... _ .. _ . _ T T T T T
IIIII .illui lull... |_I.I|1IIIJIIII1II nil
. ... .._. . . _
2 ,. : . . .
2 . .._ _
e .s .a. H H .
m a. . . . _
. . _ _
.Wb m... . . . . _ _ _ _
F .._ _ . . _ . . _ _
l. . r L L h r p .
1. 1 9. 8. 7 6. 5. 4. 3. 2 1.
1 0 0 O 0 O 0 0 0 O

2.5

1.5

Time, s {—1
46

0.5

 

Figure 23. The radius—squared solution.

4.3.6. Sensitivity to the Specific Heat of the Droplet Vapor

The speciﬁc heat of the droplet vapor is the constant—pressure speciﬁc heat. At
atmospheric pressure, the constant—pressure speciﬁc heat of propane is 1.6794 kJ/kg—K.

The following plots illustrate the sensitivity of the surface temperature and the radius——

squared to the vapor speciﬁc heat for s e [0, 3. 0] where ACp = 0.2263 kJ/kg—K.

The sensitivity of the surface temperature is given in Figure 24 and the solution
for the surface temperature is given in Figure 25. From Figure 24, it is apparent that the

surface temperature has very small sensitivity to the vapor speciﬁc heat, i.e., the

. . . . -3
sensrtiwty lS always of the order of 10 . However, there are some noteworthy

characteristics. The ﬁrst is that a higher vapor speciﬁc heat always induces a smaller
sensitivity in the surface temperature. Also, there exist “peaks” in the sensitivity. These
contour peaks occur when the contours of the surface temperature derivative reverse;
speciﬁcally, on the LHS of the peak, a larger vapor speciﬁc heat induces a smaller
derivative. On the RHS of the peak, the reverse is true. At ﬁrst, a large vapor speciﬁc
heat induces a lower temperature (on the LHS of the peak); later, on the RHS of the peak,
this characteristic has reversed. In fact, this reversal in the solution occurs at the
instances the sensitivity contours have returned to zero. In the steady—state regime, a
high vapor speciﬁc heat allows smaller sensitivity, although a higher value is allowed for
the surface temperature. It must be noted that the derivative contours do not intersect at
the exact same instant, thus the sensitivity contours do not peak at the exact same instant;

this means that there is no ﬁxed point in the behavior of the surface temperature.

47

The sensitivity of the radius—squared and the solution for the radius—squared are
shown in Figure 26 and Figure 27 respectively. From Figure 26, it is apparent that the
radius—squared is substantially more sensitive to changes in vapor speciﬁc heat than the
surface temperature. Similar to the surface temperature sensitivity, a large vapor speciﬁc
heat allows the radius—squared to be less sensitive to the vapor speciﬁc heat. However,
the sensitivity contours also experience peaks. These peaks occur at the instances the
contours of the derivative of radius—squared intersect. These derivative—intersections do
not occur at the exact same instant, thus the sensitivity peaks do not occur at the exact
same instant. The sensitivity contours all return to zero at the instances the solution
contours intersect. Prior to this intersection, a large vapor speciﬁc heat allows less
condensation to occur; after this intersection, a vapor speciﬁc heat corresponds to a

longer evaporation time.

48

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
     

 

 

 

 

 

 

 

m.
m.
V
C
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H. 4. 4 .4. . . m m e Z ..4 . . . m e
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5...... . .ummev. .m 4.1.. _mwme7
...4.................. m .4....................
{-._-1.4..- .. aw. C... hwy GP CP 12%. m Tram, - .. OF C? CP CP CP ---N
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_ _ _ ﬂ. . . e _ .
t. .4 .4. . . w .2: . . .
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r- ...42 o :f-fn--._-:-.----..::.:--_.-:-_-:-_-:--2
....4... . . . . .. u 4...... . . . . ..
.m J H. . . . . . . 1 m HM 4W . . . . . . .
_ _ . . . . . . . h .- . . . . . . . . .
_ . a. . .. . _. _ _ 5.s W - ..5
. . . . m m . . . . . .
. . . . T .m . . . . . .
. m m m m _ . . . . .
_ _ . _ PM” _ _ n _ _
_ O .
. Wu; H
. _ Hm _
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_ .. . .. m n
. . . . T .
m m m m . u. ,
. _ . . . e
. . . . _ . m.
m m. m a. m m m m u. m m n
0 0 0 0 0 0 0 0 0

Time, s {—1
49

Figure 25. The surface temperature solution.

 

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2.5

1.5
Time, s {—1

0.5

Figure 26. The sensitivity of the radius—squared to the speciﬁc heat of the droplet vapor.

 

 

 

 

 

 

 

 

 

 

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_ _ . _ _ . a
_ . . _ _ _ _
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s u u u 1 1 1 1 1
1-- . ..... u ..... if- .P P P P P -1
w . _ _ _ C C C C C
_ . _
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I _ _ F u u u
1 9 8. 7. 6. 5. 4.
0 0 0 0 0 0

2.5

1.5
Time, s [—-]

0.5

Figure 27. The radius—squared solution.

50

4.3.7. Sensitivity to the Speciﬁc Heat of the Liquid
Comprising the Droplet

The speciﬁc heat of the liquid comprising the droplet is the constant—pressure
speciﬁc heat. At atmospheric pressure, the constant—pressure speciﬁc heat of liquid

propane is 2.25 kJ/kg—K. The following plots illustrate the sensitivity of the surface

temperature and the radius—squared to the droplet speciﬁc heat for s c [0, 3. 0] where AC L
= 0.12 kJ/kg—K.

The sensitivity of the surface temperature and solution for the surface temperature
are shown in Figure 28 and Figure 29, respectively. Overall, the surface temperature is

essentially insensitive to the droplet speciﬁc heat, since, from Figure 28, it is apparent

. . . . -4 .
that the sen31t1v1ty lS always of the order 10 . However, even at this scale, there are

some interesting characteristics. The ﬁrst is that the surface temperature becomes
insensitive to the droplet speciﬁc heat quite quickly. Also, there are two peaks in
sensitivity, both of which occur when the derivative contours of the surface temperature
intersect, although the intersections do not occur at the exact same instant. Thus, the
contour peaks do not occur at exactly the same instant. The instant where the surface
temperature has zero sensitivity to the droplet speciﬁc heat occurs at the instants where
the droplet returns to its original dimensions. However, relative to the sensitivity of the
radius—squared, the surface temperature has negligible sensitivity to the droplet speciﬁc

heat.

The sensitivity of the radius—squared and the solution for the radius—squared is

shown in Figure 30 and Figure 31, respectively. Similar to the behavior of the surface

51

temperature sensitivity, the sensitivity contours of the radius—squared return to zero when
the droplet returns to its original dimensions. Although a large droplet speciﬁc heat
induces more condensation, it also induces quicker evaporation. During the period of
condensation, a high droplet speciﬁc heat allows the radius-squared to be more sensitive
to the droplet speciﬁc heat, whereas when the droplet is evaporating, the radius—squared

is less sensitive to changes in the droplet speciﬁc heat.

52

x104

 

 

 

 

 

 

 

 

I3

_ _ _ _ _
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t _ _ . _ .
. _ _ _ _ _
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i _ _ . _ _
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_ _ . . _ _
F IIIIII L IIIIIII r llllll L IIIIIII _ ttttttt r1 lull—.0.
. _ _ . . 1
_ _ . _ _ _
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llllll llllllllTllllll4| Ill|_lllllll1l|lllll -
. _ _ _ _ _ O
f _ . _ _ _
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_ . _ _ . _
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_ _ . _ .
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_ . _ . _ _

_ C _ h C 0

5 4 3 2 4| 0

._
0
Tal s?

Time, s [—]
Figure 28. The sensitivity of the surface temperature to the droplet speciﬁc heat.

 

 

 
 

 

 

 

 

 

 

 

T_ p

.H. _ a s _ 2 _ _
u n u u 2 m n 5 3
. . _ . 7. 1. 1. 2 w
u u u u 1 2 2 2 2
. _ u u n ..L .r :L L .r
--.H_----.- ..... C C C C C ---
w m m m m A + A + +
H u n u ”
:Iuw ----- _. ----- u ----- .. ----- n ..... ._ ----- .- -----
- ----- _. ..... _ ----- ----- n ----- ._ ----- TI;
_ . _ . _
7 .._. .._. .._. ...s ...s -
0 6. O. 5. 0. A
0 O 0

2.5

1.5
Time, s [—]

0.5

Figure 29. The surface temperature solution.

53

 

llllll

lull'l'l'llsll'

  

I... lllu'l'l'llinl"|l.|1

 

 

   

 

 

I lllll

 

levlulllll«IllaIllvll'llll|'lIII-IIIII¢DI.|'III'

—-—.-—_—...—I

l'lo'llvlllull‘lllll

2.5

————-—————-—————————1

r---—--—--I——--——----
I
I
l
I
I
1
2

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1.5
TIme, s [—]

   
 

 

0.5

 

o.2------—--
0.1——---—-——
0

Figure 30. Sensitivity of the radius—squared to the droplet speciﬁc heat.

 

 

 

 

 

 

 

 

_ _ ﬂ
. . _
_ . _
_ . _
_ _ _
_ _ _
. _ .
_ _ _
IIIIII _IIaI-.IIIIII.
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1.5
TIme, s [—]

0.5

Figure 31. The radius-squared solution.

54

4.3.8. Sensitivity to the Lewis Number of the Fuel

The Lewis number is the ratio of the thermal diffusivity of the fuel to the mass
diffusivity of the fuel. In most combustion—type problems, the Lewis number is of the
order unity, which means that the mass is diffusing away from the droplet surface about
as fast as heat is conducting toward the droplet surface. If thermal diffusivity is constant,
then the Lewis number can only vary if the mass diffusivity varies. The following plots

illustrate the sensitivity of the surface temperature and the radius—squared to the fuel

Lewis number for s e [0, 3. 0] where ALeF = 0.1 .

The sensitivity of the surface temperature and the solution for the surface
temperature is given in Figure 32 and Figure 33, respectively. There is a small peak in
the sensitivity at the instant the contours of the surface temperature derivatives intersect;
speciﬁcally, on the LHS of the peak, a lower Lewis number corresponds to a lower
derivative (i.e., slower rate of change). On the RHS of the peak, this behavior reverses.
Furthermore, a low Lewis number always allows the surface temperature to be more
sensitive to changes in the Lewis number. The sensitivity returns to zero when the
contours of the surface temperature solution contours intersect again. This intersection is
signiﬁcant since prior to the intersection, a low Lewis number corresponds to a higher

surface temperature — a characteristic which reverses after the intersections occured.

The sensitivity of the radius—squared and the solution for the radius—squared
contours are given in Figure 34 and Figure 35, respectively. Some interesting behavior is
that the solution contours intersect a few instances before the droplet returns to its

original dimensions for a given Lewis number. This intersection is signiﬁcant since a

55

low Lewis number corresponds to a larger amount of condensation as well as a faster
evaporation. This phenomenon appears to correspond to the instances that the sensitivity
returns to zero. Additionally, the sensitivity peaks occur slightly before the droplet
attains is maximum size for a given Lewis number; note that radius—squared attains its
maximum value when the derivative of radius—squared is zero. Speciﬁcally, the
sensitivity peaks occur when the contours of the derivatives of the radius—squared
intersect. It must be noted that the radius—squared solution contours do not intersect at
the exact same instant, nor do the radius—squared derivative contours intersect at the
exact same instant, nor do the radius—squared sensitivity contours intersect at the exact

same instant; this means no “ﬁxed point” exists in the radius—squared behavior.

Finally, a low Lewis number always allows the radius—squared to be more
sensitive to changes in the Lewis number, and also allows a larger amount of
condensation (as well as a faster evaporation). Similarly, a low Lewis number always

allows the surface temperature to be more sensitive to changes in the Lewis number.

56

 

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a
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1.5

TIme, s [—]

Figure 32. The sensitivity of the surface temperature to the fuel Lewis number.

 

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2.5

1.5
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0.5

Figure 33. The surface temperature solution.

57

 

 

 

 

  
    

 

 

 

 

 

 

 

 

 

 

 

 

 

a T 2
_ . _ _ _ _ . . _
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TIme, s [—-]
58

0.5
Figure 35. The radius—squared solution.

4.3.9. Sensitivity to the Ambient Mass Fraction

The ambient mass fraction of the ﬁle] represents the portion of the fuel which is in
the ambient; as such, it usually has a very small value. For this thesis, ambient mass
fraction was set to a nominal value of 0.01. The following plots illustrate the sensitivity

of the surface temperature and the radius—squared to the ambient mass fraction for s e [0,

3.0] where AYFOO = 0.012.

The sensitivity of surface temperature and solution of surface temperature is given
in Figure 36 and Figure 37, respectively. Each sensitivity contour experiences a steady
rise to a peak; then there is a steady decrease to constant—value behavior. However,

before each sensitivity contour attains its peak, there is a reversal (Figure 36).

Speciﬁcally, on the LHS of the reversal, a large YFoo—value induces greater sensitivity in

the surface temperature; also, a large Y FOO—value corresponds to a higher surface

temperature, as well as a higher value for the derivative of the surface temperature. On

the RHS of the reversal, a large Y FOO—value induces a low sensitivity in the surface

temperature which is maintained for all time. The sensitivity contours attain their
respective peaks in the same interval in which the surface temperature derivative contours
intersect one another, although the sensitivity contours peaks do not occur at the exact

same instant nor do the derivative contours intersect at the exact same instant. For the

entire evaporation process, a higher Y Foo—value allows surface temperature to achieve a

higher value.

59

The sensitivity of the radius—squared and the solution for the radius—squared are
given in Figure 38 and Figure 38, respectively. From Figure 38, the radius—squared
sensitivity has similar behavior to the surface temperature sensitivity. The sensitivity
contours intersect one another just after the solution contours attain their respective peaks

(i.e., before condensation reaches a maximum) and just before the solution contours

return to unity. Prior to the intersection of the sensitivity contours, a high Y Foo—value

induces a high sensitivity in the radius—squared; afterward, a high Y Foo—value induces a

low sensitivity in the radius—squared which is maintained for the remainder of the
evaporation. The ﬁrst peaks (i.e., relative maxima) of the sensitivity contours occur in
the same interval in which the respective radius contours return to unity. This is also the

same interval in which the contours of the derivatives of the radius—squared intersect

each other. On the LHS of the ﬁrst sensitivity peaks, a high Y Foo—value corresponds to

higher derivative; on the RHS, a lower derivative. After the sensitivity contours have
attained their respective peaks, the sensitivity contours of the radius—squared
monotonically decrease until they reach their respective relative minima. These relative
minimae occur in the same interval in which the contours of the derivatives of the radius-

squared intersect each other a second time.

60

 

 

 

 

 

 

 

 

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Figure 36. The sensitivity of the surface temperature to the ambient mass fraction.

  
          

 

 

 

 

 

 

 

 

 

 

 

 

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Time, S [—-]

0.5

Figure 37. The surface temperature solution.

61

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. _ _ _ . _ _ _ _
_ _ _ . _ . _ _
. _ _ _ _ . _ _
_ _ _ _ . _ _ _
_ _ _ _ . _ _ _ _
. _ . . . _ _ .
. _ _ _ . . . .
_ Li L _ L _ _ _ _
8 6 4 2 1 8 6 4 2 0
1 1 1 1 0 0 0 0 0
o. o. o. o. o. o. o. o. o.
0 0 O 0 0 O 0 0

B
u.

J.

r—w
l

I—l
LU‘

N

2.5

1.5
Time, s {—1

0.5

Figure 38. The sensitivity of the radius—squared to the ambient mass fraction.

 

 

 

 

—+——YF.° 004
‘ —e—YFOOIO.05 ‘

 

 

 

lllllll'l'lllllulll

l’lll illI-IJ

 

 

0.7L‘---~--—4———_---_-

|
i—u—l

   

0.5

Time, s [—]

Figure 39. The radius—squared solution.

62

4.3.10. Sensitivity to the Mass Fraction above the
Evaporated Surface

The mass fraction of the fuel above the droplet surface represents the portion of
fuel that exists immediately above the droplet surface. As such, it is always of the order
unity. For this thesis, the mass fraction above the droplet surface was assigned a nominal
value of 0.9. The following plots illustrate the sensitivity of the surface temperature and

the radius—squared to the mass fraction above the evaporated surface for s e [0, 3. 0]

where AYFb = 0.06.

The sensitivity of the surface temperature and the solution for the surface

temperature are given in Figure 40 and Figure 41, respectively. At ﬁrst, the surface

temperature has negligible sensitivity to Y Fb — this is the interval of condensation. Then

there is a steady increase in sensitivity up to a steady state value. For all time, a large

YFb—value induces lower sensitivity in the surface temperature. Additionally, a large

Y Fb—value allows the surface temperature to achieve a lower value.

The sensitivity of the radius—squared and the solution for the radius—squared are
given in Figure 42 and Figure 43 respectively. Similar to the surface temperature

sensitivity, there is an interval in which the radius—squared has negligible sensitivity to

Y Fb- Then the radius—squared becomes progressively more sensitive for the remainder of

the evaporation process. Also, a high Y Fb—value induces lower sensitivity in the radius—

63

squared, as well as allows the radius—squared to achieve smaller values, i.e., a high YFb-

value corresponds to a smaller condensation, thus a shorter evaporation time.

64

 

 

 

___—..—____‘

 

 

 

'l'llll

IuIIIIII 'I'l'l

‘l'l.II'IIIIII||II'IIIIIII.III'I

 

 

0

 

2.5

.5

1

Time,

0.5

S {—1

Figure 40. The sensitivity of the surface temperature to the evaporated mass fraction.

 

 

 
     

VIII!

——--———-——4

F———————-

 

__.-___--—4

 

 

 

 

 

 
 

 

 

0.6~—-—-——--

O.5~--—— ---4___-_____

TL 9

0.4”

2.5

1.5
Time, s [--]

0.5

Figure 41. The surface temperature solution.

65

 

 

 

 

 

 

 

 

0.5

 

0.16

0.14~----——--

0.02~--------

Time, s [—]

Figure 42. The sensitivity of the radius—squared to the evaporated mass fraction.

 

 

 

 

 

 

 

 

 

Time, s {—1

Figure 43. The radius—squared solution.

66

 

 

 

 

 

 

 

 

 

 

Time, s [—]

vll'-|

 

Figure 42. The sensitivity of the radius—squared to the evaporated mass fraction.

 

 

r
I
l
I
o

.1

I'lll'lil

     

lll'lnl

 

 

 

1.1

 

2.5

1.5
Time, s [—]

0.5

Figure 43. The radius—squared solution.

66

4.3.13. Final Conclusions for the Sensitivity Analysis

Essentially, both the sensitivity of the surface temperature and the sensitivity of
the radius—squared follow the same pattern. For example, if the surface temperature
sensitivity contours intersect, the radius—squared sensitivity contours also intersect. The
ﬁrst major difference is that the surface temperature sensitivity contours level off to
constant values, whereas the radius-squared sensitivity contours progressively increase

throughout the evaporation process.

The only parameter that deﬁed this generality was the initial temperature. The
surface temperature quickly became totally insensitive to the initial temperature of the
droplet, whereas the radius—squared developed a constant sensitivity to the initial
temperature. The second major difference was in the magnitude of the sensitivity —
usually the radius—squared was substantially more sensitive to small changes in the
parameters than surface temperature. The exceptions were the boiling temperature, the
ambient temperature, the ambient mass fraction, and the evaporated—surface mass
fraction. The surface temperature and the radius—squared had approximately equal

sensitivity to these parameters.

Overall, the surface temperature experienced some interesting behavior with
respect to certain parameters. The sensitivity of the surface temperature experienced

reversals for the following parameters: the gas constant, the ambient temperature, and the

ambient mass fraction. Speciﬁcally, large R and large Y Foo at ﬁrst induced more

sensitivity in the surface temperature, whereas large T 00 induced lower sensitivity (Figure

67

2, Figure 16, and Figure 36, respectively). The reversal was maintained for the gas
constant and the ambient mass fraction for the remainder of the evaporation process.
However, the ambient temperature induced another reversal in the surface temperature

sensitivity (for a total of two reversals).

Although these parameters reversed their inﬂuence on the sensitivity of the

surface temperature, they did not appear to reverse their inﬂuence on the surface
temperature itself, i.e., large R and large T 00 allowed the surface temperature to achieve
smaller values (Figure 3 and Figure 17, respectively) for the entire evaporation process,
and large Y Foo—values allowed the surface temperature to achieve higher values (Figure
35).

The radius—squared tended to be relatively insensitive to all of the parameters
during the condensation process. The exceptions to this were the initial temperature
(Figure 22) and the ambient mass ﬁ'action (Figure 38), both of which saw an immediate
increase in the sensitivity. After the condensation process ended and the evaporation

started, the radius—squared grew progressively more sensitive to the parameters for the

remainder of the evaporation process.

Concerning an overall “sensitivity magnitude,” the surface temperature is

sensitive to the following parameters (from lowest sensitivity to highest): T 0, C L Y Foo

Lv, C , LeF, Y Fb, R, T 00, T b- The radius—squared is sensitive to the following parameters

(from lowest sensitivity to highest): YFoo. LeF, Y Fb» R, Cp, T 0, C L» Lv, T b, T 00.

68

Another interesting characteristic was the presence of “peaks” in the sensitivity
contours; these contour peaks occurred as either relative maxima or relative minima.
Speciﬁcally, these contour peaks occurred for the following parameters: the gas constant,
the latent heat of vaporization, the ambient temperature, the vapor speciﬁc heat, the
droplet speciﬁc heat, the fuel Lewis number, and the ambient mass fraction. The contour
peaks always occurred in the same interval in which the surface temperature derivative
contours or the radius—squared derivative contours intersected — the derivative contours
never intersected at the exact same instant, thus the sensitivity contours never peaked at
the exact same instant. Figure 44 is a plot of equation (11.i) for contours of constant
latent heat of vaporization; Figure 45 is a close—up of Figure 44 in the vicinity of the
contour intersections. The purpose of these ﬁgures is to illustrate that there is no “ﬁxed

point” in the governing equations.

69

 

 

 

 

 

 

 
     
 

 

 

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7. 6. 5. 4. 3. 2 1 o 1
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TIHmEPU

2.5

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Time, s [—]

0.5

Figure 44. A plot of eqn. (1 Li) for contours of constant latent heat.

 

 

 

 

 

 

 

 

a _ a a J J A ~ _ _ _
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0.822 0.824 0.826 0.828 0.83 0.832

Tune, 3 [—]

0.82

0.812 0.814 0.816 0.818

Figure 45. A close—up of Figure 44 in the vicinity of the contour intersections.

70

Chapter 5: The Optimization Study

5.1. Overview

This part of this thesis is concerned with parameter optimization. For this portion
of the study, the HEEDS®2 software package was utilized. Since this software was
developed to ﬁnd optimal designs based on certain constraints, the parameter
optimization study can be considered as ﬁnding a hydrocarbon fuel that possesses these
optimized parameters. In a HEEDS® parameter optimization study, optimized designs
are sought by iteratively changing the values of certain variables, executing the deﬁned
analyses, and evaluating the responses of each design candidate. New design candidates
are generated based on the various mathematical search algorithms in HEEDS®.
Primarily, the SHERPA3 search algorithm was utilized since it is a particularly effective
method for most parameter optimization problems. During a single search, this method
uses multiple search algorithms simultaneously as opposed to sequentially. Furthermore,
the SHERPA method is capable of learning about the design space and adaptively

navigating many types of design spaces.

 

2 Acronym for Hierarchical Evolutionary Engineering Design System

3 Acronym for Systematic Hybrid Exploration that is Robust, Progressive, and Adaptive.

71

The parameters tp be optimized are the relevant physical constants (e.g. latent

heat, boiling temperature, etc.); they are optimized in order to minimize 51,, where S], is

the dimensionless instant of time that the droplet disappears. At this condition, we have

5261,) = 0. For this study, all values for these parameters are restricted to those of

hydrocarbon fuels. In other words, the range of the parameter space of the governing
parameters is restricted to that which corresponds to hydrocarbon fuels. For all of the
parameter optimization runs, the following items will be shown: an agent response plot, a

table containing the best designs, and a table containing the parameter ranges.

For this study, a good design is one in which the bum—out time is less than the
baseline as well as one in which the parameters lie within the speciﬁed range. The

baseline hydrocarbon is propane; the baseline parameter values are given in section 3.6.

5.2. Computational Methodology

This optimization study is done in two parts. The ﬁrst part is the necessary
background analysis work, viz., solving the governing equations based on the given
values for the physical parameters; this part is done in the C ++ programming language.
The governing equations are solved with the classical 4th order Runge—Kutta method, the

form of which is given in equation (14).
x(t+h)=x(t)+-;-(K, +2K, +2K, +K,) (14.i)

K 2h f(t,x) (14.ii)

72

K2 =hf£t+%,x+K%] (14.iii)

K3 =hf[t+%,x+K%] . (14.iv)
K4 =hf(t+h,x+K3) (14.v)

Note that the function f(t, x) refers to the ordinary differential equation or system thereof.
Since we have a system of two ordinary differential equations, we can write them in

vector form as follows:

$40.30))

This algorithm was validated by comparing its results to those acquired by MATLAB©;

there is agreement between the two usually to the second or the third decimal place.

The second part is the optimization process, which is conducted in HEEDS®. It
is necessary to explain some of the terms that are found in a HEEDS® optimization
study. All HEEDS® projects, regardless of type, have the following elements: at least
one process which contains at least one analysis; a set of project variables; a set of
project responses; and at least one agent [28]. In this thesis, the analysis solves the
governing equations in order to determine the instant at which the droplet is completely
evaporated. A process is the analysis (or set of analyses) that are performed by an agent
to fully evaluate a particular design. Some evaluations may require a single analysis,
while other evaluations may require several analyses and use different analysis tools.

The project variables are the quantities that are varied — here, the variables are the

73

physical parameters. The project response is simply the value of sb, i.e., the

dimensionless instant of time that total evaporation has occurred. Finally, an agent
executes and controls the design exploration. In this application, a design evaluation is
simply the bum—out time. In other words, the computation of a new value for the bum—

out time, given new parameter values, constitutes a design evaluation.

For this particular study, HEEDS® requires three items from the user: an input
ﬁle, an executable ﬁle, and an output ﬁle. The input ﬁle contains a list of values for the
physical parameters, or the baseline design, which is used by HEEDS® as a starting point
for the optimization process. As mentioned earlier, propane was chosen as the baseline
design. The executable ﬁle is created when the C ++ source code is compiled. Finally,

the output ﬁle is created when the executable ﬁle is run (or executed).
Finding the instant that the radius—squared goes to zero makes this is a root—

ﬁnding problem. The derivative of 1:2 is known from the governing equations (equation

(11.ii)), so 52 itself is needed in order to utilize the Newton—Raphson method, the general

form of which is the following:

 

tn+1 :tn '—

A subtle point about all root-ﬁnding methods is the choice of an initial guess for the root.
This means that if the initial guess is “close” to the actual root, then the method will
converge to that root. For the Newton—Raphson method, a quadratic convergence rate

exists for an initial guess which is sufficiently close to the root.

74

In this particular study it is known a posteriori that If] develops a linear decay.

Also, the Newton—Raphson method converges in exactly one iteration if F (t) is linear,

i.e., if F (t) has the form F (t) = a + bt. Thus, for this study, the bum—out time, sb, is

calculated via equation (15).

52
Sb :Sb,initial — ——(_ 42’) (15)
The algorithm is as follows:
1. Compute 6 until Bis about constant; if 6 > 1, evaporation is present so the algorithm

can proceed.

Check for condensation. If 42 > 1 at s = h, i.e., the next time step, then condensation
is present. Thus a critical time (sc) can be calculated (sc E instant where .52 returns to
unity). The current code will return sc — h. This is acceptable since sc is ONLY used

for computing an initial guess for sb, which is sb, initial = 2.0*sc. It is known a
posteriori that the factor of 2.0 moves the evaluation far enough into the linear region

of :2 such that the single-iteration N-R method returns reasonable results.

. If condensation is not present, then the slope of 52 must be considered in order to

determine an initial guess for sb. The slope is ﬁrst computed at s = h/IO; then the

slope is computed a second time at s = h (note that h has been ﬁxed at 0.001). Then

75

I? I'l‘f'!‘ nu imrﬂmrwmrcrmyw

these results are passed to a pretest loop where subsequent slopes are computed.

-8 _
When two consecutive slopes are within 1.0 x 10 of each other, the loop stops srnce

the slope of {2 is sufﬁciently linear. The initial guess to S], is taken as the s — value

where the most recent slope was computed.

5.3. Results and Discussion

5.3.1. Optimization of the Gas Constant, the Latent Heat
of Vaporization, the Boiling Temperature, the Vapor Specific
heat, the Droplet Speciﬁc Heat, the Vapor Thermal
Conductivity, and the Droplet Density

For this optimization run, the following parameters are optimized in order
to minimize the burn—out time: the gas constant, the latent heat of vaporization, the
boiling temperature, the vapor speciﬁc heat, the droplet speciﬁc heat, the thermal
conductivity, and the droplet density. For this study the parameter domain is given by
Table 1. These ranges were selected since they are representative of common
hydrocarbon fuels such as ethane and benzene. The computed bum—out time per
evaluation (agent response plot) is given in Figure 46. The top-ranking designs (from
worst down to best) are given in Table 2. The units of the parameters are given in section

3.6.

76

Table 1. Parameter ranges for those that are varied.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

R L v Tb CP CL kg pl.
Minimum 0.072783 300 180 1.2 1.7 0.0136 546.49
Baseline 0.1885 427.8 231.05 1.6794 2.25 0.0151 581
Maximum 0.2765 500 400 1.823 3.11 0.148 680.5
Table 2. Top — ranking designs.

Sb R Ly Tb Cp CL kg(10-4) pl.
2.50135 0.1441 492 186.6 1.5987 2.024 1.22 671.1193
1.91127 0.0769 304 224 1.4617 2.433 0.19 638.9569
1.63547 0.0728 500 180 1.2000 3.110 1.48 680.5
1.19067 0.0728 300 180 1.2000 3.110 1.48 680.5
1.18933 0.1054 300 180 1.2000 3.110 1.48 680.5
1.18578 0.1013 300 180 1.2000 3.110 1.48 680.5

10 T I I l I
9 ---------- 5 ---------- 5 ---------- f --------- J: --------- i -------- 1
8 ---------- s —————————— s --------- a --------- e --------- I -------- —
7 a a s s a -
I 6 __________ : __________ : _________ .' _________ 1 ......... 1 ........ -
“5’ - ' . -. : : : : :
s 5":- """ .;5 ““““““ 3 """"" T‘.“"”“* “““““ T ““““ 7
<7 _-i - . . 3 z 1 1
E 4 __________ . .......... . __________ - ________ -
a I . t I l I
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l I - l l l
3------.----:+ ------- : --------- 1 --------- 1 --------- t -------- ..
I
I' ' ' *- ' i ' . i - ! .'.
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i I I _ .1. I . .d i . . i
1 __________ :___-_____-_--:_-_':'e.-_-':1-_Ir_e!-- _!--.-I_-egt-__-r._-_.-_~.-___-
0 I 1 1 1 :
0 20 40 60 80 100 120

Design Evaluation

Figure 46. Burn—out time per design evaluation. The horizontal line represents the
bum—out time for a propane droplet, i.e., the baseline design.

77

{fr M'Emwnrmzmv WWW?"

From Figure 46, it is apparent that in about the ﬁrst twenty evaluations, HEEDS®
has found about as many “bad” designs as good designs; this is because the SHERPA

method is randomly sampling the design space in order to acquire an understanding of the

, th . .
desrgn space. However, by the 30 evaluation, HEEDS® has essentially converged.

The presence of the outliers is due to the fact that the SHERPA method never stops

searching the design space for an even better design.

From the Table l and Table 2, we see that the optimal gas constant is between the
minimum allowable value and the baseline, albeit closer to the minimum; the optimal
latent heat is consistent at its minimum allowable value as is the optimal boiling
temperature. The vapor speciﬁc assumed its minimum allowable value whereas the
liquid speciﬁc heat assumed its maximum allowable value. The thermal conductivity and

liquid density both assumed their maximum allowable values.

Intuitively, a fuel droplet will experience the quickest evaporation if the droplet
has a low latent heat of vaporization and a low boiling temperature (thus the droplet must
absorb a smaller amount of energy in order to change phase). Also, vapor with a low

heat capacity will serve to accelerate evaporation.

A minimal gas constant indicates a large molar mass (recall section 3.2 and 4.3.1).
A larger molar mass would aid in evaporation since it would allow the substance to have
a higher heat capacity. Thus it follows that a droplet with a large heat capacity would
serve to accelerate evaporation since a large speciﬁc heat essentially means that the

droplet “holds” a large quantity of heat that aids in the evaporation of near—surface

molecules.

78

To optimize kg and pL may not yield reliable results since it is known ﬁ'om the

sensitivity analysis that both the surface temperature and the radius—squared are

completely insensitive to these parameters.

5.3.2. Optimization of the Gas Constant, the Latent Heat
of Vaporization and the Boiling Temperature

For this Optimization run, the following parameters are optimized in order to
minimize the bum—out time: the gas constant, the latent heat of vaporization, and the
boiling temperature. For this study the parameter domain is given by Table 3. These
ranges were selected since they are representative of common hydrocarbon fuels such as
ethane and benzene. The computed burn—out time per evaluation (agent response plot) is
given in Figure 47. The top-ranking designs (from worst down to best) are given in

Table 4. The units of the parameters are given in section 3.6.

Table 3. Parameter ranges for those that are varied.

 

 

 

 

 

 

 

 

R L , T b
Minimum 0.072783 300 180
Baseline 0.1885 427 .8 231.05
Maximum 0.2765 500 400

 

 

79

1‘" ”WWW?“ 1!"! Y‘ﬁ‘l‘éttw

Burn-out time [——]

Table 4. Top — ranking designs.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sb R Lv Tb
33.06541 0.1278 462 380.2
14.97371 0.0789 396 318.6
7.828994 0.2419 396 303.2
3.184831 0.1033 378 191
3.131561 0.1033 404 182.2
2.561621 0.0728 300 180
2.556044 0.0769 300 180
2.553503 0.0789 300 180
2.54645 0.0932 300 180
2.545874 0.0911 300 180
2.545723 0.0893 300 180
10 l l T l l - l l
9 ------- e ------ ; ------ e ------ ------- s ------- I ----- J

8.. ....... s ....... s ....... a .......

7 ----- ----- i ------ +2 ------ «1 ------- s ------- ------

e ------- ....... a ..... s ....... a ....... a ———————

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0 i : 1 I 1 i l
0 10 20 30 40 50 60 70

 

 

 

 

Design Evaluation

Figure 47. Bum—out time per design evaluation. The horizontal line represents the
burn—out time for a propane droplet, i.e., the baseline design.

80

From Figure 47, it is apparent that in about the ﬁrst twenty evaluations, HEEDS® found

th
about as many “bad” designs as good designs, but by the 30 design, HEEDS® has

converged. However, the minimal burn—out time for this run is not as small as for the

previous run.

Between Table 3 and Table 4, the optimal gas constant is close to the minimum
allowable value; the optimal latent heat is equal to the minimum allowable value; the
optimal boiling temperature is also equal to the minimum allowable value. These results

seem physically plausible for the reasons given in the previous section.

From the sensitivity analysis, it is known that the radius-squared exhibits a high
level of sensitivity to the boiling temperature and a smaller level of sensitivity to the gas
constant. Such a characteristic might explain why the gas constant required more
evaluations to converge to its optimal value, whereas the boiling temperature converged

to its optimal value relatively quickly.

5.4. Final Conclusions

Unfortunately, there was only sufﬁcient time to thoroughly investigate two

optimization runs. However, the results from the two runs seem to suggest a certain

pattern which supports the sensitivity analysis.

Between the two optimization runs, the radius—squared was relatively insensitive
to some parameters and sensitive to other parameters. The order of the radius—squared
sensitivity is given in section 4.3.13. For the insensitive parameters (i.e., the parameters

to which the radius—squared had relatively no sensitivity), HEEDS® seemed to converge

81

_. 1"" 9"?‘1‘1'6W1rrq-9W'

 

to their optimal values rather quickly. For the sensitive parameters (i.e., the parameters to
which the radius—squared was sensitive), HEEDS® seemed to require slightly more
evaluations to converge to their optimal values (if indeed optimal values exist). This
stands to reason because if the radius—squared is insensitive to a given parameter, then
that parameter would have minimal effect on the bum—out time; thus, that parameter
could be assigned almost any value. Of course, this is only suggested by the current

studies; more optimization studies would be necessary to verify this pattern.

82

1". MMiﬁtthia-mrrrvnrm

Chapter 6: The Asymptotic Analysis

Due to the non—linearity of the governing equations, they must be solved

numerically. However, if we consider a hypothetical fuel that possesses a high activation

energy, i.e., 1,, >> 0(1), then it is possible to conduct an asymptotic analysis that might be

analytically tractable. The purpose of the asymptotic analysis is to remove all but the
most essential elements of a system so that the system is simpliﬁed but still illustrates the

relevant physics.
When lv >> 0(1) (say that 1,, ~ 100) and while the surface temperature (7) is still

increasing, it can be shown that the Z parameter is about constant. This would constitute
the ﬁrst interval in which the asymptotic analysis is conducted. As the surface
temperature approaches the transition to its constant—value behavior, 1 is still about
constant; this is the second interval for the asymptotic analysis. These intervals are

quantiﬁed in Figure 48. In this ﬁgure are plots of r (the dimensionless surface

temperature), 52 (the dimensionless radius—squared) and the z—parameter for a large lv—

value. For the remainder of this section, the dimensionless parameters take the following

values:

To =0.3337; 2], =0. 7698; 1,, =97.21; 0:10.91; 21 = 0.893; 7: 1.126.

C: 0.009; K23 0.01;]: 1.005; B 2 111.4; m z 112; Uz—0.306.

83

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1.015

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1.005~-—--—

0.985»-——-

1.6 1.8

1.4

1.2

0.8

0.6

0.4

0.2

S {—1

Figure 48. Solutions for the surface temperature, the radius—squared and the z—parameter for

a large latent heat. Note 1 is 10 times larger for illustration purposes.

84

The ﬁrst interval of the asymptotic analysis extends from s = 0 to just before é]
attains its maximum value. The second interval extends from there to just before rattains
its constant—value behavior.

In the ﬁrst interval of this limiting case, we have 2’ 2 —0.0101. If we
deﬁne Z = —K in this particular interval, and substitute this deﬁnition into equation (11.ii),

then the solution for the dimensionless radius—squared is:
52 :1 + Cs

On substitution of this expression into equation (11.i), the ordinary differential equation

(which is now linear) is:

 

 

 

d1 _ ds
KB—Jr I+Cs
I K .
Here, C=21K , B: K +0, J = . It can be shown that the solution for the
1—e_ 1—e_

dimensionless surface temperature is:

1:5-‘1—3-{1—(1+Cs)—2m}+ 70(1+Cs)—2m

Here,m=%. These two solutions suggest that the dimensionless time variable, s, be

. -1 . . .
scaled wrth C , re, 77 = Cs; also, that the dimensronless surface temperature be scaled,

i.e.,¢=(J%( B)“ Thus, the radius—squared and the surface temperature can be

expressed in the following ways:

85

f'mnrthﬂﬁrmnmrmy

1+7}

52

¢=1+U§'2m

(J T% B)_1 . Figure 49 illustrates these two asymptotic solutions. From

Here, U

inspection, it is clear that as s —) 00, (D —+ 1. From Figure 46, the ﬁrst interval extends to

abouts = 0. 6. From the deﬁnition of 17, the corresponding n—value is about 0.0054.

 

---..-.1_____

1.-----

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ll"l llulll

11"" 111-1.

T"--"‘l"‘"--_l-’-_-

Illll 'II'l'

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llllllllll

 

 

 

VIIIII

 

 

—_--—_I—--——+J

'll'lll

'l'Il

 

 

 

0.9~--——

 

0.018 0.02

0.014 0.016

0.012

0.004 0.006 0.008

0.002

11 l—l

Figure 49. Asymptotic solutions for the interval of increasing surface temperature

and radius—squared.

86

As the dimensionless surface temperature approaches its constant—value behavior,

this ﬁrst asymptotic analysis breaks down. Thus we start the second analysis in the next

interval which begins at the instant 1:2 attains its maximum value; at this instant, I is still

about equal to —K. In this particular analysis, 62 (so :0. 75 )2 5,3,“ z [.007 . We make the

following deﬁnition:

In the above expression, a=oo+ 0% + 0212 +... Also, we expand s in the
v
V

neighborhood of sc, viz., h = Iv(s - 5.). Thus, with the z = —K approximation, and
assuming IV is large, we can integrate equation (11.ii) from {2(sd to .52 (s); then we
substitute the deﬁnition for h. The solution for If? is now the following:

2 2 2K
5 “gmaxz'l—h
v

If we deﬁne g(h) = h, we can extract the following differential equations:

d0’_ 2’ 10' .+_Q
dh 5771ax [vb-911

dg

h

 

>=|>~e

87

11"‘FW'Whﬁ3M '1"!- r‘n. "-nvmv

One advantage that the above system offers over the original system (equations (11.i—ii))
is that the a—equation has no explicit dependence on the g—function. Additionally, it is
clear from the deﬁnition of g(h) that g(0) = 0; and that this occurs at the instant the

radius—squared attain its maximum value. In order to acquire an initial condition for the
o—equation, we note that 0': 1,,(1 — 7). Thus, 0(h = 0) = 1,, (I — zfs = sd). Consistent
with the numerical values of the other parameters, 0(0) 2 25. If the asymptotic

deﬁnition of r is substituted into equations (12.i—iii), we can approximate the z—

parameter with the following expression:

2’_ I In 1"YFoo
LeF I—YFbe—ﬂ

242-110-211

The asymptotic solutions are shown in Figure 50, Figure 51, and Figure 52. From

 

 

Figure 48, this analysis only applies from s = 0.75 to about s = 0.85; from the deﬁnition

of the h—variable, the corresponding h—value is about 10.

88

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4 4 4 4 4 4 4 .
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30

h {—1

Figure 51. Asymptotic solution for the radius—squared in the vicinity of the maximum radius—squared.
89

 

 

 

 

 

 

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_ . . . _ _
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. . _ . _ _
_ _ . _ _ .
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. _ _ _ _ _
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5 3 5 2 5 1 5 2
3 2 2 2 1 2 0 0
2 0. 2 O. 2 0. 2 0

0. O O. 0 O. O 0.

O 0 0 O

x

h {—1

Figure 52. Asymptotic solution for the z—parameter in the vicinity of the maximum radius—squared.

9O

Chapter 7: Notes concerning the
Computational Algorithms

The ﬁrst issue that was problematic to this study was not realizing the importance
of the (9 parameter in the evaporation process. Essentially, if 0 becomes greater than
unity at some point in time, then evaporation is present. Speciﬁcally, it is the constant

portion of 6that has the greatest relevance to the evaporation process.

The second issue was setting a ﬁnal time, Sﬁnal, in order to provide a time interval

in which the governing equations would be solved. At ﬁrst, Sﬁnal was set to a value of 6

since the surface temperature reached its constant—value behavior around this time in

many simulations. Then it seemed to be a simple matter to check if aSﬁnal) > 1; if so,

then evaporation was present. However, it was discovered that for certain values of the

physical parameters, total evaporation occurred before Sﬁnal = 6! This created havoc

with the algorithm. This was resolved by setting an s-window from 0 to 400. Then, 6

was computed until it reached its constant—value behavior.

The last noteworthy issue was that Bhad two constant—value regions under certain
conditions. Typically, the ﬁrst constant—value region was less than unity. As the then—
current algorithm was written, this ﬁrst constant—value region of 6 indicated to the rest of
the algorithm that no evaporation was present when in fact evaporation was present!

Thus, the algorithm was modiﬁed to read:

91

1. Compute 61 = 6(3 = 0).

2. Compute 62 = 6(5 = h).

3. Compute A6 = 192 — 61.

4. If A65 A9 max: then there is “ﬂat behavior” toward the start of 6.

4.a. 62 = 61; continue until A6’> A6 max-

4.b. At the instant A6> A6max, 62 = 61.

4.c.Continue on in computing 6] and 62 until A6> A6? tolerance-

5. If A6> A19 max: then there is no ﬂat spot, so go to step 4.c.

92

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[3]

[9]

[10]

Bibliography

Cengel, Y.A. & Boles, MB. (2002) Thermodynamics: An Engineering Approach
4th ed. McGraw Hill Book Co.

Cheney, W. & Kincaid, D. (2004) Numerical Mathematics & Computirrg 5111 ed.
Thompson-Brooks/Cole

Dai, M., Perot, J.B., & Scmidt, DP. (2002, September) “Heat Transfer within
Deforrning Droplets” Proceedings of ASME: Internal Combustion Engine
Division, New Orleans

Dash, S. K., Sengupta, S. P., & Som, S. K. (1991) “Transport Processes &
Associated Irreversibilities in Droplet Evaporation” J. Thermophvsics & Heat
Trans. Vol. 5, No. 3, pp. 366—373

Digilov, R., & Reiner, M. (2004) “Trouton’s rule for the law of corresponding
states” Eur. J. Phys. Vol. 25, pp. 15—22

Ebbing, DD. (1996) General Chemistry 5tlrl ed. Houghton Miﬁlin Co.

Elperin, T. & Krasovitov B. (2006) “Transient Analysis for Sub-critical
Evaporation of Fuel Droplet in Non-isothermal Stagnant Gaseous Mixtures:
Effects of Radiation & Thermal Expansion” J. Heat and Mass Transfer Vol. 42,
pp.427—436

Ghassemi, H., Back, 8., & Khan, Q. (2006) “Experimental Study on Binary
Droplet Evaporation at Elevated Pressures and Temperatures” Combust. Sci. and
Tech. Vol. 178, pp. 1031—1053

 

Ha, V. & Lai, C. (2000) “The Onset of Stationary Marangoni Instability of an
Evaporating Droplet” Proceedings of the Royal Society of London, pp. 885—909

Hegseth, J. J ., Rashidnia, N., & Chai, A. (1996) “Natural Convection in Droplet
Evaporation” msical Review B, Vol. 54, Issue 2, pp. 1640—1644

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Hiwase, S.D., Datta, A., & Som, SK. (1998) “Entropy Balance & Exergy
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Hombeck, R.W. (1975) Numerical Methods Quantum Publishers, Inc.

 

Incropera, F.P. & DeWitt, DP. (2002) Introduction to Heat Transfer 4th ed. John
Wiley & Sons

 

Kays, W. M. (1966) Convective Heat & Mass Transfer McGraw Hill Book Co.

Kozyrev, A.V. & Sitnikov, A.G. (2001) “Evaporation of a Spherical Droplet in a
Moderate-pressure Gas” Uspekhi Fizicheskikh Naul_(_., Vol. 44, pp. 725—733

Poinsot, T. & Veynante, D. (2001) Theoretical and Numerical Combustion R.T.
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Potter, M.C. & Wiggert, DC. (2002) Mechanics of Fluids 3Id ed. Thompson-
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Sherwood, T.K., Pigford, R.L., & Wilke, CR. (1975) Mass Transfer McGraw Hill
Book Co.

Sirignano, W.A. (1999) Fluid Dynamics & Transport of Droplets & Sprays
Cambridge University Press

Som, S. K., & Dash S. K. (1993) “Thermodynamics of Spray Evaporation” _J_.
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Turns, SR. (2000) An Introduction to Combustion: Concepts and Applications
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Wichman, 1.8. Notes on Droplet Combustion Dept. of Mechanical Engineering,
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Williams, F.A. (1985) Combustion Theory 2nd ed.The Benjamin/Cummings
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Tseng, C. C. & Viskanta, R. (2004) “Effect of Radiation Absorption on Fuel
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Zhu, G. S. & Aggarwal, SK. (2002) “Fuel Droplet Evaporation in a Supercritical
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HEEDS User Manual
(MSU UNIX path: /opt/soﬁ/heeds-4.1/manuals/usersman.pdf)

Red Cedar Technology, 4572 S. Hagadom Rd., Suite 1- E, E. Lansing, MI 48823,
www.redcedartech.com

Gunnar Norman’s homepage http://www4.tsl.uu.se/~norrnan
The Svedberg Laboratory, Box 533, S-75121 Uppsala, Sweden

http://www.princeton.edu/~ﬂdryer/nasa.dir/backgrnd.htm
“General Background for Spherically Symmetric Isolated Droplet Combustion”

Dr. Frederick L. Dryer, Dept. of Mechanical and Aerospace Engineering, D-316
Engineering Quadrangle, Princeton University, Princeton, NJ 08544

95

IL ”2m.mﬁlﬁ135§3 ‘II‘P"

IIIIIIIIIIIIIIIIIIIIIIIIIIIII

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