ABSTRACT
Title of thesis: ANALYSIS AND MEASUREMENT OF CANDLE
FLAME SHAPES
Gregory A. Tabaka, Master of Science, 2008
Thesis directed by: Assistant Professor Peter B. Sunderland
Department of Fire Protection Engineering
A combined analytical and experimental study was performed to produce and
validate a method by which the lengths and widths of flames burning on wax candles
can be predicted. Two analogies are drawn: first, that heat transfer from the
wick can be modeled as a slender vertical rod undergoing a natural convection
process surrounded by a stagnant film; and second, that fuel pyrolyzed from the wick
surface that remains unburned until above the wick can be modeled as flow from
a circular port burner. Empirical correlations applied to the theoretical equations
have produced reasonable agreement with measured flame lengths and widths for
paraffin candles.
ANALYSIS AND MEASUREMENT OF CANDLE FLAME
SHAPES
by
Gregory A. Tabaka
Thesis submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Master of Science
2008
Advisory Committee:
Professor Peter B. Sunderland, Chair
Professor James G. Quintiere, Co-Chair
Professor Arnaud Trouv?e
Copyright c? Gregory A. Tabaka 2008
Dedication
To Thomas Jefferson, patron saint of freedom and architect of American liberty.
ii
Acknowledgements
As most any graduate student laments, the academic process of writing a thesis
takes longer than anticipated and requires the touch of other people to become
successful. Few of us have the capacity to make discoveries and change the world
in isolation like Isaac Newton. I am no exception and feel the need to express my
gratitude towards those who have contributed to completing this paper.
First, I would like to thank my thesis advisors at the University of Maryland,
College Park, professors Peter B. Sunderland and James G. Quintiere. Their will-
ingness to advise a student without outside funding bespeaks their devotion to the
pursuit of knowledge and discovery. Their tutelage was central to pouring the the-
oretical foundation of this work and their subsequent efforts allowed it to grow into
something substantial and contributory to the body of combustion literature, yet
they provided the broad measure of independence and freedom I required to make
this thesis the fruits of my own labor.
The experimental data were graciously provided by Danjun Lian while in the
United States during a brief junket away from life in China. Her observation and
digitization of candle flame shapes allowed me to not only validate the theory pre-
sented herein but also to demonstrate trends that I did not anticipate. If one good
experiment is worth the opinion of a thousand experts, then her thirty good exper-
iments are worth far more than the solitary opinion of this student.
I owe a large debt of gratitude to Dr. Javier Trelles, who graciously donated his
time to help my understanding of axisymmetric flows, combustion within annular
iii
structures, the numerical mathematical techniques that ultimately did not make it
into this thesis and?most of all?for academic direction when I was wandering.
Also, I wish to thank Dr. Arnaud Trouv?e of the University of Maryland, College
Park particularly for serving on my thesis defense committee. I also thank him
for lecturing the graduate Diffusion Flames and Burning Rates Theory class, which
served as excellent preparation for the subject matter of this thesis.
In a similar vein, I thank the professors at the University whom I have not
yet mentioned for their instruction throughout the years and for preparing me for
the profession of fire protection engineering: Andr?e W. Marshall, Marino di Marzo,
James A. Milke and Frederick W. Mowrer. In fact, if Dr. di Marzo had not spoken to
my class about the Graduate School, I would not have made the decision to pursue
a Master?s degree.
Also, I wish to thank my friends and family for their encouragement, patience
and ability to preserve my sanity. This thesis was researched and written largely
while on long business trips away from home that endured for months at a time. The
ability to come home and rejoin my friends and family has made all the difference.
Greg Tabaka
November 9, 2007
Airborne somewhere between Tampa and Baltimore
iv
Table of Contents
List of Tables vi
List of Figures vii
Nomenclature viii
1 Introduction 1
2 Experimental 6
2.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Flame Shape Measurements . . . . . . . . . . . . . . . . . . . . . . . 7
3 Candle Flame Shape Theory 10
3.1 Boundary Layer Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Fuel Vaporization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Top Disc Mass Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Total Boundary Layer Mass Flow . . . . . . . . . . . . . . . . . . . . 25
3.5 Boundary Layer Thickness and Flame Standoff Distance . . . . . . . 28
3.6 Overwick Flame Height . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Results 38
4.1 Fuel Vaporization Rates . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Flame Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Lateral Burning Rate and Boundary Layer Thickness . . . . . . . . . 50
4.4 Total Burning Rate and Flame Length . . . . . . . . . . . . . . . . . 53
5 Conclusion 60
5.1 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . 61
A Summarized Measurements 63
B Candle Flame Photographs 64
C Constants and Physical Values 67
Bibliography 68
v
List of Tables
3.1 Names and properties of unbranched alkanes . . . . . . . . . . . . . . 37
A.1 Summary of data from candle flame experiments . . . . . . . . . . . . 63
vi
List of Figures
3.1 Candle and flame schematic . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Boundary layer profiles for a cooling hot rod and flame analogy . . . 14
3.3 Boundary layer velocity profile curve, u(y,z)/C0?z . . . . . . . . . . 27
4.1 Wick top fuel vaporization rate ?mF,Dcp/kD . . . . . . . . . . . . . . . 39
4.2 Wick top fuel vaporization rate ?mF,Dcp/kL . . . . . . . . . . . . . . . 40
4.3 Wick lateral fuel vaporization rate ?mF,Lcp/kL and consumption rate . 41
4.4 Sensitivity of w/D to B . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.5 Sensitivity of w/D to NuD,L . . . . . . . . . . . . . . . . . . . . . . . 46
4.6 Observed and predicted w/D using an approximation for NuD,L . . . 48
4.7 Relative position of the flame front,yf/y?, using the shell approximation 51
4.8 Flame height above the wick top, Lf/L. . . . . . . . . . . . . . . . . 56
vii
Nomenclature
Latin symbols
B Spalding B number (unitless)
cp Specific heat of gas at constant pressure at Tm (J/kg?K)
C0 Axial velocity modulus (m1/2/s)
C1 Axial velocity scaling factor (unitless)
D Diameter of the wick at the vertical midpoint (m)
D Diffusion coefficient (m2/s)
emp Empirical fit
f Mixture fraction (unitless)
g Gravitational acceleration (m/s2)
Grx Grashof number = g??Tx3/?2 (unitless)
h Heat transfer coefficient (W/m2?K) or enthalpy (J/kg)
H Enthalpy of gasification (J/kg?K)
k Thermal conductivity at Tm (W/m?K)
L Length of wick from the molten pool to the wick tip (m)
m Mass (kg)
M Molecular weight (kg/mol)
n Empirical data fit parameter (unitless)
Nux Nusselt number = hx/k (unitless)
Pr Prandtl number = ?/? (unitless)
r Radial coordinate (m)
R Radius of the wick (m)
Rax Rayleigh number = ?m(Tf ?T?)x3/Pr?2m (unitless)
s Stoichiometric fuel-oxygen mass ratio (unitless)
S Stoichiometric fuel-air molar ratio (unitless)
T Temperature (K)
w Maximum diameter of the flame (m)
u Axial component of velocity (m/s)
V Volume (m3)
v Radial component of velocity (m/s)
X Mole fraction (unitless)
Y Mass fraction (unitless)
y Standoff distance from wick surface (m)
z Axial coordinate (m)
viii
Greek and other symbols
? Thermal diffusivity (m2/s)
? Similarity variable (J/g)
? = R+y?; Velocity boundary layer position (m)
?x Difference variable (for x = T: K, for x = { F, Ox, N, P }: unitless)
? Stretched coordinate (see [1])
? Dimensionless mass flow per unit height = ?mprimeF/? (unitless)
? Thermal volumetric expansion coefficient (1/K)
? Kinematic viscosity (m2/s)
? Mass density (kg/m3)
? = 2pik/cp = 2pi?D
?hc Enthalpy of combustion (J/kg)
Subscripts
0 Reference state of formation (superscript)
b Relating to the apparent circular port burner
B Burned
D Wick diameter
E Entrained
F Fuel
f Flame sheet
g Gas
l Liquid
L Wick length
m Fuel/products region between wick surface and flame sheet
N Inert diluent
Ox Oxygen
P Product of combustion
v Vaporization of liquid fuel
? Ambient
? Boundary layer
Superscripts
?x x per unit time (x/s)
xprime x per unit wick length (x/m)
xprimeprime x per unit surface area (x/m2)
xprimeprimeprime x per unit surface volume (x/m3)
x? x is nondimensional (?)
ix
Chapter 1
Introduction
The candle flame has formed a fundamental basis in the study of fire and has en-
lightened researchers about the nature of steady laminar flames, soot production,
burning rates, and combustion in microgravity amongst other topics. An improved
understanding of the candle flame itself may further the state of the art and lead
to other advances in combustion and fire science. For instance, simplified ana-
lytical methods such as that posed within can be used to validate more complex
code or provide valuable estimation or design techniques. This analysis can also
be extended to other geometries or gravitational conditions where data exist about
free-convective heat transfer. Candle research can lead to reduced frequency of fires
and soot emissions in homes. It can also be used by manufacturers to select wax
and wick properties to obtain desired flame widths, lengths and luminosities.
One of the earliest studies of candle flame dynamics was presented in 1860 by
Michael Faraday [2]. More recent scholarship on wicked candles began circa 1990
and has continued since. These studies were primarily simplified numerical studies,
themselves representing the solution of a simplified set of equations of motion. This
facilitated solution by computer, which is why the timeframe coincides with the
rising popularity of the personal computer.
However, if we generalize our study to flames on vertical surfaces (with a con-
1
centration on vertical cylinders), the body of research swells.
Multiple studies exist for burning under forced convection. Emmons? [3] seminal
paper studied combustion of burning droplets with forced convection and obtained
a similarity solution, assuming a stagnant boundary layer. The approach taken by
Emmons would be influential in subsequent analyses. Also, the coupling of convec-
tive effects with stagnant films used by Spalding [4] and Emmons was also influential
in this field of study. Abdel-Khalik et al. [5, 6] studied horizontal cylinders of porous
material saturated with n-heptane under forced convection as an approximation to
droplet burning. Tao and Kaviany [7] and Tao et al. [8] studied the burning rate of
fuel supplied to a porous slab under forced convection.
Several studies exist for free convection as well, for a variety of liquid and solid
fuels. Kaviany and Tao [9] studied vaporization effects on a liquid-soaked block of
porous material in free convection. Raju and T?ien [10] experimentally studied heat
and mass transport within porous planar wicks. Kim et al. [11] studied laminar
burning on vertical, horizontal and inclined geometries saturated with liquid fuel
and obtained an approximate solution for burning rate based on the Grashof and
Spalding B numbers, albeit employing theory that neglects wick curvature effects.
Kosdon et al. [12] studied burning of alpha-cellulose vertical cylinders under laminar,
free-convective conditions and obtained a scaling law for flame standoff distance as
a function of vertical position. Ahmad and Faeth [13] analyzed liquid burning on a
vertical planar surface, but also studied the flame shape in a region where no fuel
is being added, which is similar to the region above a candle wick. Malary and
Sibulkin [14] examined laminar burning of methyl methacrylate (a solid material
2
with favorable chemical decomposition behavior) of vertical flat plates and found
thatauniqueburningratedidnotexist. GroffandFaeth[15]exploredaliquid-fueled
wick fed by reservoir otherwise geometrically similar to a candle but they worked in
rectilinear coordinates. Maxworthy [16] examined the instability of propane flames
using a burner whose form is very similar to the wick of Groff and Faeth [15].
Pagni [17] presented nine ?classic? diffusion flames with theoretical solutions, in-
cluding a cylindrical stagnant layer solution similar to the analysis presented within
this thesis. Pagni was able to relate the width of the flame to the mass burning rate
but did not express his solution in terms of practical dimensionless groups. He was
able to obtain an expression for flame height by treating fuel unburned until above
the fuel source to be flow from a Burke-Schumann burner but presented his results
as a curve fit in thermophysical and chemical quantities.
A number of studies in microgravity were performed as well. Linan [18] studied
burning of solid fuel cylinders in microgravity. Dietrich et al. [19, 20] studied candle
flames in microgravity and non-buoyant atmospheres, in which molecular diffusion
is shown to be the governing phenomenon. Du et al. [21] performed experiments to
measure the temperatures of candle flames in microgravity.
In the past two decades, several computational fluid dynamics analyses have
been made. Lyu and Chen [22] performed a numerical study of planar and cylin-
drical liquid-fueled porous wicks that demonstrated that the fuel vaporization rate
on cylinders deviated from the z?1/4 dependence for flat plates, where z is distance
in the vertical direction. Wu and Chen [23] applied a mixture fraction approach
to study the shape and oscillation of near-laminar flames on cylindrical wicks by
3
numerical solution of a system of partial differential equations (PDE). Riley [24]
created a PDE model using the flame sheet model for steady burning on candle
wicks much longer than the flame itself, though he predicted only the steady state
of the system. Hamins et al. [25] used Fire Dynamics Simulator (FDS) from the
National Institute of Standards and Technology (NIST) to model a candle flame,
including the bent geometry of a wick. Alsairafi et al. [26, 27] created a detailed
computational fluid dynamics model of a candle flame with chemical kinetics, as-
suming that the wick surface was saturated with liquid fuel and studied the effect
of varying gravitational accelerations. Raju [28] incorporated a model of heat and
mass flow within a candle wick into Alsairafi?s computational model to remove the
saturated surface assumption and to numerically study heat and mass transport
within the wick, thereby adding a direct coupling mechanism between flame and
fuel supply.
Literature pertaining to heat transfer or gaseous combustion has been omitted
in this history. This thesis will base burning behavior on heat transfer literature
and extend the process to combustion. Specifically, the processes of natural con-
vection along vertical cylinders and burning from circular gaseous port burners are
employed.
In particular, this thesis proposes a methodology for predicting the width, length
and burning rate of a paraffin candle flame as the geometry of the wick?namely,
length and diameter?changes. The effect of variation in fuel is not experimentally
explored, though the method does allow for chemical effects to be included through
the concepts of stoichiometry, B number and transport rates.
4
Thus motivated, the objectives of this study are as follows:
1. Predict the widths of candle flames;
2. Predict the lengths of candle flames; and
3. Compare predicted quantities against experimental observations.
This thesis will begin by describing the experimental data gathering scheme
that was implemented by Dan-jun Lian, which led to the validation set for the
analytical method that follows. This analytical approach can best be described
as an approximate treatment based on the heat transfer literature with a classical
application of fluid mechanics and combustion theory. The analysis extends the
burning rate and flame sheet treatments presented in Quintiere [29] to cylindrical
coordinates to produce scaling laws. A further chapter is devoted to discussing the
accuracy and limitations of the formulae created, as well as possible extrapolation
to conditions different than those tested. Finally, measurements are presented for
substantiation and for later researchers to utilize.
5
Chapter 2
Experimental
Laboratory experiments were conducted in which five different diameters of wick
were molded into paraffin candles with various lengths of wick exposed above the
plane of wax that forms the top of the candle. The candle was then set on a bench,
ignited and observed. When steady state burning was achieved (as evidenced by
the formation of a liquid pool of 25 mm in diameter), a high resolution photograph
was taken for later measurement on a personal computer. No measurements of mass
loss, temperature or velocity were obtained as the purpose was only to determine the
extent of the luminous region of flame under the assumption that the wick geometry
would be the governing influence.
2.1 Procedure
The experimental procedure that was followed is described below and is similar to
that used previously in this laboratory to measure candle smoke points by Allan et
al. [30].
1. Prepare the candle.
(a) Cut wick to desired length.
(b) Place paraffin beads into a beaker that serves as the mold for the candle.
6
(c) Melt paraffin beads and allow to solidify.
(d) Remove from beaker.
(e) Drill hole on axis.
(f) Stiffen wick by adding liquid wax.
(g) Insert wick into hole.
2. Position camera to obtain a clear photograph.
3. Light the candle and place it inside a Plexiglas draft shield.
4. Wait for the candle to achieve a laminar, stable burning regime.
5. Snap the photograph with the Nikon D100 digital camera.
6. Record observation notes and photograph serial number.
2.2 Flame Shape Measurements
Once the catalog of photographs has been uploaded to a personal computer, they
were analyzed to obtain flame measurements. Once the image in question is on
screen, the measurement of interest (e.g., flame width) is taken in pixels. Because
the scaling factor between pixels and physical distance is known, the actual dis-
tance represented on screen can be measured. Using this technique, all 33 flame
photographs were measured. The following measurements were recorded:
1. Height of the flame attachment point above the wax pool
2. Length of the wick above the molten wax pool, L
7
3. Diameter of the wick at the vertical midpoint, D
4. Diameter of the wick at the top
5. Dry wick diameter using a caliper with minimal force applied
6. Length from the molten wax pool to the flame tip
7. Maximum diameter of the flame, w
Mass loss rates were not recorded.
The described technique offers several benefits. First, all dimensions of the flame
are available at one particular instant in time. That is, the width and length are
presented as a couple, rather than having been measured sequentially, introducing
error. Second, measuring the width of a flame directly is difficult and may affect
gas flow or introduce cooling effects, which could influence the measurement. Third,
an electronic record is made that can be checked in the future. Finally, the high
resolution camera produces images that can be measured very precisely. A con-
version factor of one millimeter per 33.95 pixels was employed, so measurements
should have an accuracy of one half-pixel (? 0.059 mm); however, the actual error
in measurement is significantly higher.
The source of this error is threefold: first, the base of the candle flame as mea-
sured has a faint blue corona that does not appear clearly in digital photographs and
makes discerning the flame attachment point particularly difficult. Second, despite
attempts to produce steady flames, flicker and drafts still do occasionally coincide
with the photograph. Third, the physical endpoints to measurements are some-
8
times difficult to discern, especially in flame regions that are not sharply defined.
One cause of the nonexistence of ideal measurement points can be attributed to the
fact that the wick could not always be aligned perfectly to gravity, nor do candle
wicks have the uniform cylindrical shapes with constant diameters and plane ends
that theorists apply to simplify the mathematics.
The raw data was compiled to a spreadsheet application. This electronic format
provided enough flexibility to be able to store, transmit, manipulate, calculate and
plot results to gain qualitative and quantitative understandings of candle flame
behaviors.
9
Chapter 3
Candle Flame Shape Theory
Previous analyses of flame structure have been attempted on a broad spectrum
of theory: some are simplistic one-dimensional analyses with planar flames; some
have attempted direct numerical simulation using detailed information on elaborate
chemical reaction kinetics and flow phenomena; still others utilize a blend of science
rooted in fundamentals and a pragmatic empiricism to obtain solutions. This thesis
can comfortably claim to be in the lattermost category.
The most fundamental approach encountered within the literature is to concep-
tualize a saturated, porous fuel source with a flame attached and attempt to solve
the boundary layer equations in rectilinear coordinates. This approach was followed
by Kim et al. [11], Ahmad and Faeth [13], and Groff and Faeth [15]. This thesis will
depart from their strategy by referencing the boundary layer equations mainly for
the purpose of calculating upward mass flow but will retain the porous fuel source
component. Instead of solving the boundary layer equations or computing the flow
field such as Lyu and Chen [22], Wu and Chen [23], Riley [24], Alsairafi [26, 27],
and Raju [28] have done, this paper will employ a heat transfer correlation and sto-
ichiometry to quantify upward fuel flow. This semi-empirical approach is employed
because?unlike Emmons [3]?a similarity solution for the problem of a flame within
stagnant film does not exist for a vertical cylinder, thereby limiting the methods of
10
solution available. This approach was deemed to be more amenable to producing
scaling law relationships than CFD studies and therefore was chosen. Also, the
flame height is calculated via correlation rather than as a result of a flow field com-
putation. This flame height calculation is reminiscent of Pagni?s [17] approach in
which he assumed fuel not burned lateral to the wick entered a Burke-Schumann [31]
port burner above.
The analytical theory presented here begins by making an analogy between a
candle flame and a slender vertical rod undergoing natural convective cooling for
the purposes of calculating the mass flux of fuel through the wick surface along the
vertical face. More precisely, the mass transfer rate due to vaporization is coupled
to heat transfer. Next, stoichiometry and stagnant laminar boundary layer theory
are employed to approximate the mass flow rate of oxidizer joining the flame front.
Knowing the amount of fuel vaporized and the amount of oxidizer available, it is
possible to calculate the amount of fuel that remains unburned in the lateral region
and is therefore transported to the overwick region for eventual combustion. It
is hypothesized from experimental observation that the maximum flame diameter
resides at this interface. The vertical flow of fuel there resembles a circular port
gaseousfuelburnerandsoflameheightiscalculatedusingliteratureavailableonthat
topic. The candle, flame and geometry described above are shown schematically in
Fig. 3.1. The theory is then adjusted empirically to improve the agreement between
theory and observed fact.
This methodology presented relies on several key assumptions:
11
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mi
no
us
 zo
ne
Fla
me
 at
tac
hm
en
t p
oin
t
Mo
lte
n w
ax 
po
ol
So
lid 
wa
x r
es
erv
oir
L f L
w
D
yyyy
rrrr
zzzz
Figure 3.1: Candle and flame schematic
1. The heat transfer from the flame to the wick can be modeled as a vertical
cylinder at flame temperature cooling under the influence of natural convection
without a surrounding flame front;
2. Pyrolyzate that has not burned laterally to the wick can be modeled as fuel
flowing vertically from a circular port burner;
3. Compressibility effects can be ignored;
4. The flame and near flow field remain steady, laminar and axisymmetric;
12
5. The wick is shaped as a cylinder and remains uniformly saturated with liquid
fuel at the surface; and
6. The fuel is burned in stoichiometric proportions with oxidizer.
3.1 Boundary Layer Flow
Central to this paper is the hypothesis that a candle flame can be modeled as
residing within a stagnant laminar boundary layer where the fuel vaporization rate
is coupled with heat conduction. In this idealized model, the candle wick is perfectly
cylindrical and aligned parallel to the gravity vector and is therefore axisymmetric.
In Fig. 3.2, the dashed curve schematically represents the boundary layer known to
classical heat transfer as the stagnant boundary layer and the solid curve represents
the candle flame sheet. Specifically, the stagnant boundary layer has a thickness
proportional to the quarter power of the vertical coordinate [32] (i.e., y? ?z1/4) and
the flame lies at some fraction of this thickness, as will be shown.
The natural point of departure for this analysis is a consideration of the flow
within the boundary layer. We can express the mass flow rate in the z-direction by
abstractly integrating the upward mass flux, ?mprimeprime, as
?m(z) =
integraldisplay ?
R
?mprimeprime(r,z)dA, (3.1)
13
y/y
?
0
1
z / L 01
Fla
me
Bo
un
da
ry 
La
yer
Figure 3.2: Qualitative stagnant boundary layer profiles for a slender vertical rod
undergoing natural convective cooling (dashed) and for a candle flame (solid) below
the top of the wick.
or, more concretely, while emphasizing the cylindrical nature of the problem as
?m(z) = 2pi
integraldisplay R+y?
R
?mprimeprime(r,z)rdr. (3.2)
For tractability, we assume a constant density of ?m, which represents the average
density of the region between the wick surface and flame sheet at a given axial
coordinate, z. This value for density has been chosen because this region occupies
the majority of the space between the wick and the thermal boundary layer (as will
14
be shown later). By substituting ?mprimeprime = ?mu(r,z), we can approximate the mass flow
rate as
?m(z) = 2pi?m
integraldisplay R+y?
R
u(r,z)rdr. (3.3)
To obtain an estimate of the proportions of fuel and oxidizer that comprise
?m(z), we realize that ?m(z) reflects the sum of mass flow from fuel vaporized from
the wick surface and the air entrained to combust that fuel (recall that the boundary
layer extends beyond the flame sheet, so the presence of air between the limits of
integration is still consistent with the flame sheet model). The total vertical mass
flow within the boundary layer at the top of the wick can be expressed as
?m(L) =
integraldisplay L
0
parenleftbig?mprime
F,z + ?m
prime
E,z
parenrightbigdz = ?m
F,L + ?mE,L. (3.4)
For candle flames that extend beyond the top of the wick, we can conclude that
some fuel vaporized from the wick remains unburned in the lateral region and it
therefore becomes interesting to calculate this quantity. To do so, we assume fast
chemistry to achieve stoichiometric burning and a uniform mass fraction of oxygen in
air. Further, we assume that all air entrained in the lateral region is consumed there
and is not transported to the overwick region. Thus, the stoichiometric burning rate
of fuel in the lateral region is given by
?mF,B = ?mOx,E,Ls (3.5)
15
and the mass flow rate of oxidizer is given by
?mOx,E,L = YOx,? ?mE,L. (3.6)
By combining Eqs. 3.5 and 3.6, we can quantify the amount of fuel transported to
the overwick region to be
?mF,? = ?mF,L? ?mF,B = ?mF,L? YOx,? ?mE,Ls . (3.7)
By Eq. 3.4, we can rewrite this as
?mF,? = ?mF,L
parenleftbigg
1+ YOx,?s
parenrightbigg
? YOx,?s ?m(L). (3.8)
This equation is still incomplete and requires an expression for ?mF and some repre-
sentation of the axial velocity profile for ?m(L) included through Eq. 3.3.
3.2 Fuel Vaporization
To produce an expression for the sidewall fuel vaporization rate, ?mF, we begin with
a simplified set of conservation equations expressed in cylindrical coordinates for
axisymmetric flow [33]:
?
summationdisplay
h0f,j ?mprimeprimeprimej = 1r ??z
parenleftbigg
r?u
integraldisplay T
T0
cp(T)dT
parenrightbigg
+ 1r ??r
parenleftbigg
r?v
integraldisplay T
T0
cpdT
parenrightbigg
? 1r ??r
parenleftbigg
r?D ??r
integraldisplay T
T0
cpdT
parenrightbigg
(3.9)
16
?mprimeprimeprimej = 1r ??z (r?uYj)+ 1r ??r (r?vYj)? 1r ??r
parenleftbigg
r?D?Yj?r
parenrightbigg
(3.10)
in which ?mprimeprimeprimej is defined as the rate of production of species j and a Lewis number of
unity was assumed, so ?D = k/cp.
Forthepurposeofcalculatingthemassflowoffuelvaporizingfromawick, wewill
apply additional simplifications. First, we approximate integraltextTT0 cp(T)dT = cp(T ?T0)
with cp held constant to remove the integral in Eq. 3.9. Next, we assume that the
first terms in Eqs. 3.9 and 3.10 are negligible in comparison to the others on the
basis that temperature, mass fraction and velocity gradients in the axial direction
are small in comparison to those in the radial direction. Indeed, the temperature and
mass fraction of the wick surface are assumed to be uniform and at the boiling point
and saturation concentration of the fuel and is assumed to have an axial velocity near
zero. The large effect of this assumption is to remove any dependence on derivatives
in the z-direction, resulting in a system of ordinary differential equations. Also, we
mandate that the enthalpies of formation are defined for the ambient temperature,
so T? = T0. Generally, these enthalpies are available at 298 K, which is typical
of room temperature. Furthermore, we dispense with the summation notation and
allow
summationdisplay
h0f,j ?mprimeprimeprimej = ?hc ?mprimeprimeprimeF. (3.11)
From this point on, we parallel an analysis from Quintiere [29]; however, the
reference analysis was developed for a one-dimensional vaporization process in rec-
tilinear coordinates reminiscent of the Stefan problem [34]. For more concise nota-
17
tion, we also introduce ?T = T ?T? and ?F = YF ?YF,?. By incorporation of the
assumptions and notation above, we may now write the conservation equations as
?r?hc ?mprimeprimeprimeF = ddr (r?vcp?T)? ddr
parenleftbigg
r?D ddr (cp?T)
parenrightbigg
(3.12)
r ?mprimeprimeprimeF = ddr (r?v?F)? ddr
parenleftbigg
r?Dd?Fdr
parenrightbigg
(3.13)
where the species conservation equation has been written for the ?fuel species.?
Noting the obvious similarity between Eqs. 3.12 and 3.13, we multiply Eq. 3.13 by
?hc and add the product to Eq. 3.12 to obtain the homogenous ordinary differential
equation,
d
dr (r?v(cp?T +?hc?F))?
d
dr
parenleftbigg
r?D ddr (cp?T +?hc?F)
parenrightbigg
= 0. (3.14)
Next, we will specify the boundary conditions for this ODE by using a stagnant
film model. Far from the wick surface (more precisely, at the boundary layer inter-
face), we have ambient conditions, namely, YF = YF,? and T = T?, so ?T = ?F = 0.
A mass balance at the surface of the wick demands that the amount of mass passing
through the wick in the condensed phase is equal to the sum of the amount leaving
due to diffusion and bulk velocity transport as a vapor. This can be expressed as
?mprimecondensed = ?mprimevapor + ?mprimediffusion. (3.15)
The bulk mass flow rate through the surface of the wick is equal on the inside and
18
outside infinitesimal volumes and will be denoted ?mprimer. Thus, we can state that
?mprimecondensed = ?mprimerYF,l where YF,l is the mass fraction of fuel in the liquid (taken as
unity) and ?mprimevapor = ?mprimerYF(R). Additionally, we can express the mass flow due to
diffusion as ?mprimediffusion = ?2pirDdYFdr evaluated at r = R. Therefore, the boundary
condition imposed at the wick surface is
?mprimeFYF,l = ?mprimeFYF(R)?2piR?DdYFdr
vextendsinglevextendsingle
vextendsingle
r=R
, (3.16)
where ?mprimer = ?mprimeF because YF = 1 within the wick (the term YF(R) is retained for
clarity in subsequent analysis).
A similar argument is made to determine the heat transfer boundary condition
at the wick surface. We know that the rate of energy loss due to fluid transport
through the wick surface must balance that of conduction inwards from the fluid
because no increase in temperature is possible with the liquid at its boiling point.
This enables us to balance heat as
0 = ?m
prime
FH
cp ?2piR
k
cp
dT
dr
vextendsinglevextendsingle
vextendsingle
r=R
. (3.17)
Above, the quantity H represents the heat of gasification adjusted to T?, which
includes the enthalpy of solid-solid phase changes, enthalpy of fusion (solid-liquid),
heating from melting temperature to vaporization temperature, and enthalpy of
vaporization (liquid-vapor). By again invoking the unity Lewis number assumption
19
and multiplying the equation above by cp,
0 = ?mprimeFH?2piRcp?DdTdr
vextendsinglevextendsingle
vextendsingle
r=R
. (3.18)
Again noting a similarity, we multiply Eq. 3.16 by ?hc and add the product to
Eq. 3.18 to obtain
0 = ?mprimeF (H +?hc(YF(R)?YF,l))?2piR?D
parenleftbigg
cpdTdr +?hcdYFdr
parenrightbigg
. (3.19)
At this point, it is expedient to introduce ? and ?,
? = cp?T +?hc?F ? = 2pi?D = 2pik/cp (3.20)
so that the governing equation, Eq. 3.14, and its boundary conditions can be rewrit-
ten in the succinct form
?mprimeFd?dr ?? ddr
parenleftbigg
rd?dr
parenrightbigg
= 0 (3.21)
r = R : 0 = ?mprimeF (H +?hc(YF(R)?YF,l))?R?d?dr
r = ? : ? = 0
(3.22)
where use has been made of the fact 2pir?v(r) = YF,l ?mprimer = ?mprimeF is independent of
r for a nonreacting flow. Although these equations are more compact than their
predecessors, we are able to move a step further by nondimensionalizing ? as
?? = ?H +?h
c(YF(R)?YF,l)
. (3.23)
20
The resulting equation and boundary conditions are as follows:
?mprimeFd?
?
dr ??
d
dr
parenleftbigg
rd?
?
dr
parenrightbigg
= 0 (3.24)
r = R : d??dr = 1R ?mprimeF?
r = ? : ?? = 0.
(3.25)
The problem specified by Eqs. 3.24 and 3.25 is well-posed and has the analytical
solution
??(r) =parenleftbigr????parenrightbigR?? (3.26)
where ? = ?mprimeF/?. In Eq. 3.26, ?? is the dependent variable, r is the independent
variable, and both R and ? are constants presumed to be known. The dimensionless
vaporization rate, ?, however, is an eigenvalue of this equation. By solving for ??
at r = R, we obtain
??(R) = 1?
parenleftbigg?
R
parenrightbigg?
, (3.27)
which can be manipulated to produce the vaporization rate,
?mprimeF
? =
ln(1???(R))
ln(?/R) . (3.28)
This equation can be rewritten in a more familiar form by starting with a different
form of the conserved scalar, ??. This is accomplished by considering Eq. 3.10 from
the perspective of the oxidizer species rather than the fuel species and following a
21
similar analysis. This results in
??Ox = cp(T ?T?)+(YOx ?YOx,?)(?hc/s)H , (3.29)
which can be substituted into Eq. 3.28 as
???(R) = YOx,? (?hc/s)?cp(Tv ?T?)H ?B. (3.30)
where YOx(R) = 0 and H can be found in Table 3.1. Equation 3.30 defines the
Spalding B number. The numerator can be viewed as the net energy released per
unit mass of fuel consumed whereas the denominator is a measure of the amount
of energy needed to vaporize that unit mass of fuel. Finally, we rewrite Eq. 3.28 in
terms of k/cp as
?mprimeF(z)cp
k = 2pi
ln(1+B)
ln(?/R) , (3.31)
which is equivalent to the result obtained by Pagni [17].
To proceed, we approximate the boundary layer as a conducting annular shell
(the ?shell approximation?) surrounding the wick. It can be shown [35] that the
relationship between boundary layer thickness from a heated, vertical cylindrical
surface to the Nusselt number of the flow may be expressed as NuD,z = 2/ln(?/R).
By substitution into Eq. 3.31, we can relate the mass transfer process to convection
as
?mprimeF(z)cp
k = piln(1+B)NuD,z(z) (3.32)
22
A correlation exists that approximates the local Nusselt number [36] along the wick
in the vertical direction,
NuD,z
Gr1/4z = 0.24
z
R
bracketleftbiggGr
z
4
bracketrightbigg?1/4
+0.40Pr3/10, (3.33)
which can be expressed as
NuD,z = 0.68+0.40Pr1/20 (RaDD/z)1/4. (3.34)
This correlation is accurate within 5% for RaDD/L = 210 Pr/?4 for 0 ? ? ? 10, or
RaDD/L ? 0.074 [36]. Also, the heat transfer over the full length of the wick can
be integrated using Eq. 3.34 to obtain the average Nusselt number for wick length
as
NuD,L = 0.68+0.53Pr1/20 (RaDD/L)1/4. (3.35)
Similarly, it is possible to integrate Eq. 3.32 to obtain an expression for the total
mass flow from the lateral region of the wick,
?mF,Lcp
kL = piln(1+B)NuD,L. (3.36)
Equations 3.34 and 3.35 both suggest a limiting Nusselt number of 0.68 as the
wick becomes infinitely slender. Also, for gases, the term Pr1/20 is negligible as it
holds a value of approximately 0.98.
23
3.3 Top Disc Mass Flow
One of the assumptions of this analysis was that the entire wick surface remains
bathed in liquid fuel. One consequence of this assumption is that self-trimming wicks
are excluded from this application for two reasons: first, such wicks are designed to
desiccate at a desired length so that they may be trimmed by the flame; and second,
such wicks also are made to burn in a bent shape, in contrast to the wicks that were
intentionally kept as vertical as possible in these experiments. Another consequence
of the bathed wick assumption is that the idealized disc surface atop the wick also
releases fuel.
The mass flux from this surface is modeled as being driven by natural convection
exclusively. However, this is a false condition because viscosity effects from the
upward flow of fuel-products mixture from the wick sidewall will affect mass flux in
the circular region. Nonetheless, this combined forced/free convection problem is
treated as pure free convection.
As will be shown later, the range of RaD examined is approximately 10?3 to 106
and experimental data gathered for candles represents a range of approximately 0.1
to 50. One expression for the Nusselt number of a flow over a finite, horizontal hot
plate in air for this range of RaD can be fitted to curves in Gebhart [37],
NuD,D = 0.43+0.6Ra1/4D . (3.37)
By making the approximation that the slope of the gas temperature curve at
24
the wick surface is approximately equal to the ratio of temperature difference to an
abstract ?boundary layer thickness??t atop the disc and assuming that the saturated
liquid fuel is present, one can write hc ?k/?t and obtain the approximate mass flux
at the disc [29]
?mprimeprimeF ? hcc
p
ln(1+B). (3.38)
We can then multiply through by piD2/4 and simplify to obtain the mass flow from
the disc,
?mF,D ? pi4 kc
p
Dln(1+B)NuD,D, (3.39)
or, in dimensionless form as
?mF,Dcp
kD ?
pi
4 ln(1+B)NuD,D. (3.40)
3.4 Total Boundary Layer Mass Flow
In Eq. 3.3, we left ambiguous the profile of the axial velocity curve. To complete
this analysis, however, we must specify one. We have selected to model the axial
velocity as the zeroth-order expansion from LeFevre and Ede [32],
u(y,z) = C0?zyy
?
parenleftbigg
1? yy
?
parenrightbigg2
, (3.41)
where
C0 = 80?mB2
0
and B40 = 8021(20+21Pr)?
2
m
g?m?1 . (3.42)
25
In these expressions, material properties have again been calculated for the fuel-
products mixture. The expression for B0 reflects a correction made to the original
source. Furthermore, while the term ?1 reflects the temperature difference between
the cylinder?s surface and ambient air in a pure heat transfer model, the situation
is more tenuous due to the temperature profile caused by the flame. Because the
thrust of the idea is to model the buoyancy effects, we choose ?1 = Tf ?T?. Thus,
we can simplify to
C0 = 4
radicalbigg
105g?m(Tf ?T?)
20+21Pr . (3.43)
Variable C0 has dimensions of L1/2 t?1.
The parameterization in Eq. 3.41 has some correct qualitative behaviors, as
shown in Fig. 3.3. For instance, the velocity becomes zero at the wick surface
(no-slip condition) and also at the boundary layer edge, which is close to the 1%
of ambient commonly taken as defining the boundary layer. Also, the curve rises
quickly to a peak velocity before tapering off.
Equation 3.3 calls for the integral of ru(r,z) with respect to r. To calculate this
integral, we apply the transformation r = y+R to Eq. 3.41 and evaluate at z = L
to obtain
?m(L) = pi15C0?m?LD2y?D
parenleftbigg5
4 +
y?
D
parenrightbigg
, (3.44)
whereanextraD/Dfactorhasbeenintroduced. Equation3.44nowhastheexpected
?L scaling factor. By multiplying through by c
p/k (again evaluated at Tm), we can
26
y/y
?
0.0
0.2
0.4
0.6
0.8
1.0
u ( y , z )  /  C 0  z
1 / 2
0.0
0
0.0
2
0.0
4
0.0
6
0.0
8
0.1
0
0.1
2
0.1
4
0.1
6
Figure 3.3: Boundary layer velocity profile curve for a slender vertical rod in natural
convection [32].
express the total mass flow in nondimensional terms as
?m(L)cp
kL =
pi
15C0
?mcp
k
?
L3
bracketleftbiggD
L
bracketrightbigg2 y
?
D
parenleftbigg5
4 +
y?
D
parenrightbigg
, (3.45)
where an extra?L2/Lfactor has been introduced. Next, we substitute?m = k/?mcp
and C0 from Eq. 3.43. After rearranging terms, we have
?m(L)cp
kL =
pi
15
?Prbracketleftbigg 42 105
20+21Pr
bracketrightbigg1/2bracketleftbiggg?
m(Tf ?T?)D3
?2mPr
D
L
bracketrightbigg1/2 y
?
D
parenleftbigg5
4 +
y?
D
parenrightbigg
.
(3.46)
27
Finally, we replace the first bracketed term in this equation with C1 and the second
with radicalbigRaDD/L to produce
?m(L)cp
kL =
pi
15C1
?PrradicalbigRa
DD/L
y?
D
parenleftbigg5
4 +
y?
D
parenrightbigg
, (3.47)
where C1 is a velocity scaling factor and is calculated to have a value of 6.9 from
an analytical approach. This factor is not known precisely and its interpretation is
complicated by the presence of a flame sheet, which does not appear in the analogous
pure convection scenario.
3.5 Boundary Layer Thickness and Flame Standoff Distance
The thickness of the boundary layer can be calculated by modeling the boundary
layer as an annular shell around the wick, which transfers heat via conduction [35].
This thickness can be calculated from the Nusselt number by again invoking the
shell approximation,
NuD,z = 2ln(1+y
?(z)/R)
, (3.48)
and solving for y? to obtain the thermal boundary layer thickness,
y?(z) = R(exp(2/NuD,z)?1). (3.49)
The position of the flame relative to the boundary layer can be calculated by
extending the conserved scalar approach of Section 3.2, which has been taken from
Quintiere [29]. We begin by realizing that because the original fuel atoms can neither
28
be created nor destroyed, we can define a mixture fraction f of fuel atoms that must
obey the sourceless conservation equation
?mprimeFdfdr ?? ddr
parenleftbigg
rdfdr
parenrightbigg
= 0, (3.50)
with the boundary conditions
r = R : ?mprimeFYF,l = ?mprimeFf(R)?R?dfdr
vextendsinglevextendsingle
vextendsingle
r=R
,
r = ? : f = 0.
(3.51)
The mixture fraction at the surface is denoted byf(R) and has a boundary condition
identical to Eq. 3.16, the boundary condition for the chemical fuel species. The
mixture fraction at r = ? equals zero because fuel is assumed to come from the
wick only (pure diffusion flame) and be consumed entirely in the flame sheet. As
with ?, we normalize f in accordance with its boundary conditions to obtain f? =
f/(f(R)?YF,l). In so doing, the governing equation, Eq. 3.50, once rewritten for
f?, becomes identical to Eq. 3.21 and the boundary conditions become identical
to Eq. 3.25. Because the governing equations and the boundary conditions are
identical, uniqueness dictates that one solution exists and is common to both ??
and f?. Thus,
?? = f? = ff(R)?Y
F,l
. (3.52)
Equation 3.52 allows us to calculate the mixture fraction at the surface by eval-
29
uating the equation at r = R to obtain
f(R) = ?
?(R)YF,l
??(R)?1 =
BYF,l
1+B. (3.53)
By choosing to express ?? in terms of the oxidizer and fuel species mass fractions,
it can be shown [29] that the stoichiometric value of mixture fraction, ff, is
ff = (YOx,?/s)
slashBigparenleftbiggYOx,?
sYF,l +1
parenrightbigg
. (3.54)
This value of mixture fraction defines the position of the flame sheet, or ff = f(rf).
Equation 3.26 can be solved for rf using ?? = f? to obtain r?f = ?? +R?f?f. Next,
we can substitute Eq. 3.54 into Eq. 3.52 to obtain the stoichiometric value of f?f to
produce the position of the flame sheet in radial coordinates,
r?f = ???R? YF,lYOx,?(Y
F,l ?f(R))(sYF,l +YOx,?)
. (3.55)
The negative sign comes from inverting the order of terms in the denominator of
Eq. 3.52. In a sense, Eq. 3.55 states that position of the flame front is a deduction
from the position of the boundary layer, which is based on the radius of the wick
and chemistry conditions.
Although the expression for rf given by Eq. 3.55 expresses the quantity desired,
it does so in terms of the unknowns ? and ?. To eliminate these variables, we divide
30
Eq. 3.55 by R? and simplify to obtain
rf
R =
bracketleftBiggparenleftbigg
?
R
parenrightbigg?
? YF,lYOx,?(Y
F,l ?f(R))(sYF,l +YOx,?)
bracketrightBigg1/?
. (3.56)
We continue by rewriting the fractional term (equal to ???f) in more familiar terms
by utilizing Eqs. 3.52?3.54 to obtain
??f = ? 1+B1+sY
F,l/YOx,?
. (3.57)
By expressing Eq. 3.31 in terms of ? as ? = ln(1+B)/ln(?/R), we can rewrite
Eq. 3.56 as
rf
R =
bracketleftbigg
1+B? 1+B1+sY
F,l/YOx,?
bracketrightbigg1/?
. (3.58)
Equation 3.58 can be simplified further into the compact form
rf
R =
bracketleftbigg 1+B
1+YOx,?/sYF,l
bracketrightbigg1/?
. (3.59)
Next, weinvoketheshellapproximationfromEq.3.48tosubstitutefor?, thereby
coupling mass and heat transfer via the dimensionless group,
? = 12NuD,z ln(1+B). (3.60)
31
This allows us to write
rf
R =
w
D =
bracketleftbigg 1+B
1+YOx,?/sYF,l
bracketrightbigg2/NuD,L ln(1+B)
, (3.61)
where Eq. 3.61 has been expressed at z = L. Equation 3.61 can be written in terms
of the stoichiometric mixture fraction via Eq. 3.54 as
rf
R =
w
D =
bracketleftbigg 1+B
YOx,?/sff
bracketrightbigg2/NuD,L ln(1+B)
. (3.62)
It is possible to separate the chemistry terms from the fluid mechanical terms
by writing Eq. 3.61 as
rf
R =
w
D =
bracketleftBiggparenleftbigg
1+B
1+YOx,?/sYF,l
parenrightbigg1/ln(1+B)bracketrightBigg2/NuD,L
, (3.63)
where the chemistry-driven terms are denoted by parentheses and the fluid mechan-
ical terms by the square brackets.
Yet another form of Eq. 3.63 is possible: we can express yf/y?, or the ratio
between flame position and boundary layer thickness. By substituting rf = R+yf
and employing Eq. 3.49 for the boundary layer thickness, ? = R+y?, the resulting
relationship is
yf
y? =
bracketleftBig
1+B
1+YOx,?/sYF,l
bracketrightBig2/NuD,L ln(1+B)
?1
exp(2/NuD,L)?1 . (3.64)
32
3.6 Overwick Flame Height
The final task to calculate the gross dimensions of a candle flame is to determine
the height of the flame above the wick. As previously mentioned, all experimental
cases produced flame heights that extended beyond the wick tip, even those with
narrow diameters. This allows us to conclude that the fuel combusted over the
wick largely originates from the wick sides. Furthermore, it was observed that the
maximum flame diameter coincides with the tip of the wick; therefore, we consider
the horizontal circular regions formed by the flame. This region will be modeled as
a circular port burner with the flow rate of fuel prescribed by Eq. 3.8.
A well-respected body of literature on the subject of gaseous port burners exists
due to Roper [1, 38]. In particular, these sources provide expressions of the length of
the flame in the overwick region (where the volumetric flow rate has been adjusted
to ambient temperature) based on theory as
Lf = 14pi
?VF(T?)
ln(1+1/S)
(T?/Tf)2/3
D? (3.65)
and one adjusted based on experimental observations with gaseous fuels as
Lf =
parenleftBig
1330 sm2
parenrightBig ?VF(T?)
ln(1+1/S), (3.66)
in which S is the molar stoichiometric air-fuel ratio.
The measurements given by Eq. 3.66 and those experimentally measured in a
similar fashion generally pertain to flames exhibiting yellow luminosity from soot.
33
In contrast, this analysis considers the width and length of stoichiometric regions.
For yellow flames, the stoichiometric and luminous widths are nearly the same.
However, the luminous lengths are typically twice as long as stoichiometric, as shown
in Sunderland et al. [39].
Application of Eqs. 3.65 and 3.66 is problematic because they are expressed for
gaseous fuels with a uniform fluid temperature at the burner mouth. A secondary
issue is that Roper?s [1] formulation considers the volumetric flow rate of fuel whereas
analysis of candle flames is more naturally expressed in mass flow rate; therefore,
an approximate average fuel density is required to marry the approaches.
In the candle flame, the ?apparent port burner? coincides with the plane of the
flame at its widest diameter and actually has a temperature profile where the tem-
perature is equal to vaporization temperature, Tv, for 0 ?r?R and then increases
to Tf at r = rf. However, in keeping with the original model, we shall model the
gaseous fuel at its vaporization temperature, Tv. Using a constant temperature, an
estimate of the mass density of the fuel can be identified. To express Roper?s [1]
theory from the perspective of mass flow rate rather than volumetric, we begin with
his equation,
? = 4DfzTvw2uT
f
, (3.67)
where the diffusivity has been taken at the flame temperature because the narrow
zone near the flame sheet in which concentration gradients are steep dominates the
diffusion process. Similarly, Eq. 3.65 adjusts D to the flame temperature with the
34
approximation D(T) = Dref (T/Tref)5/3. Furthermore, Roper shows that
1?exp(?1/4?) = (1+S)?1 (3.68)
when z = Lf. If we assume a uniform axial velocity, u, and ?mF = 14piw2u1?v, then
Eq. 3.67 can be substituted into Eq. 3.68 and rearranged to obtain
Lf = TfT
v
?mF
?vDf
1
4piln(1+1/S). (3.69)
Recalling that k/cp = ?D is to be evaluated at Tm, we must adjust both the density
and diffusivity of the above equation to this temperature. Because the molecular
weight of the vaporized fuel is greatly different than the surrounding air, we must
consider its effect via the ideal gas law,
?m
?v ?
M?Tv
MFTm. (3.70)
After substituting Eq. 3.70 into Eq. 3.69 and adjusting the diffusivity to Dm, we
may divide through by L to obtain
Lf
L =
?mFcp
kL
M?
MF
bracketleftbiggT
m
Tf
bracketrightbigg2/3 1
4piln(1+1/S). (3.71)
Finally, we apply the definition Tm = (Tf +Tv)/2 and simplify to obtain
Lf
L =
?mFcp
kL
M?
MF
bracketleftbigg
1+ TvT
f
bracketrightbigg2/3 1
44/3piln(1+1/S). (3.72)
35
It can be shown [33] for a hydrocarbon fuel with a composition CxHy that S is
S = x+y/4X
O2
, (3.73)
where XO2 is the mole fraction of oxygen in air and has a value of approximately
21%. Paraffin is composed primarily of high molecular weight alkanes, which have
the general form CnH2n+2 where 22 ? n ? 27 and participates in the general
combustion reaction (in air)
CnH2n+2 +
parenleftbigg3
2n+
1
2
parenrightbiggparenleftbigg
O2 + 1?XO2X
O2
N2
parenrightbigg
?
nCO2 +(n+1)H2O+ 1?XO2X
O2
parenleftbigg3
2n+
1
2
parenrightbigg
N2. (3.74)
We can rewrite S in terms of n or s as
S = 3n+12X
O2
= MFM
?
s
YOx,?. (3.75)
Calculated values of S and s are listed in Table 3.1 for straight-chain alkanes of
weights typical to paraffin waxes. For this analysis, we arbitrarily select n = 24
(tetracosane) to represent the wax used in our candle experiments.
36
n Chemical name MF (g/mol) S (mol/mol) s (kg/kg) H (kJ/kg)
22 n-Docosane 310.60 159.52 3.45 620.7
23 n-Tricosane 324.63 166.67 3.45 603.5
24 n-Tetracosane 338.65 173.81 3.45 621.0
25 n-Pentacosane 352.68 180.95 3.45 592.6
26 n-Hexacosane 366.71 188.10 3.45 634.0
27 n-Heptacosane 380.73 195.24 3.45 615.9
Table 3.1: Names and properties of unbranched alkanes. H assumes an initial tem-
perature of 298.15 K and is based on Chickos and Wilson [40]. (XO2 = 0.21,YO2 =
0.233,M? = 28.85 g/mol)
37
Chapter 4
Results
In this chapter, the model of Chapter 3 is compared with flame length and width
measurements for paraffin candles.
4.1 Fuel Vaporization Rates
The candle flame model presented offers two distinct regions of fuel supply: the
lateral and horizontal (top) areas of the idealized cylindrical wick. For the top,
Eq. 3.40 predicts the fuel vaporization rate and is plotted in Fig. 4.1. It is possible
to rewrite Eq. 3.40 to include the dimensionless group RaDD/L and slenderness
ratio D/L as
?mF,Dcp
kL =
pi
4 ln(1+B)
parenleftBigg
0.43DL +0.6
bracketleftbigg
RaDDL
bracketrightbigg1/4bracketleftbiggD
L
bracketrightbigg3/4parenrightBigg
. (4.1)
The wick top fuel supply rate is plotted along these axes in Fig. 4.2.
For the sides of the wick, the fuel vaporization rate is presented in Eq. 3.36 and
is plotted in Fig. 4.3.
38
Ra
D
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
m F , D c p / k D
10
-1
10
0
10
1
10
2
B =
 1 
B =
 5 
B =
 10
 
Figure 4.1: Wick top fuel vaporization rate ?mF,Dcp/kD.
39
Ra
D 
D/L
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
m F , D c p / k L
10
-2
10
-1
10
0
10
1
10
2
B =
 1 
B =
 5
B =
 10
D/L
 =
 0.
1
D/L
 =
 1
Figure 4.2: Wick top fuel vaporization rate expressed as ?mF,Dcp/kL for comparison
with the lateral fuel vaporization rate.
40
Ra
D 
D/L
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
m F , L c p / k L
10
-1
10
0
10
1
10
2
B =
 1 
B =
 5 
B =
 10
 
La
ter
al 
Fu
el 
Co
ns
um
pti
on
C 1
 =
 6.
9
C 1
 =
 13
.8
C 1
 =
 20
.0
Figure 4.3: Wick lateral fuel vaporization rate ?mF,Lcp/kL and lateral fuel consump-
tion rate. (For constants, see Appendix C).
41
4.2 Flame Width
By modeling a candle flame as residing within the boundary layer of a stagnant,
free-convective flow, we were able to obtain an expression (Eq. 3.63) for the position
of the flame front at the interface between the lateral and overwick regions,
rf
R =
w
D =
bracketleftbigg 1+B
1+YOx,?/sYF,l
bracketrightbigg2/NuD,L ln(1+B)
.
This expression implicitly defines the position of the maximum flame standoff
distance to be at the interface mentioned above. This is necessary for the model
because the region above the sidewall is modeled as a gaseous port burner with
an unknown flame profile. However, this approximation is also a reasonable one
because, in general, the majority of fuel supplied to the flame is supplied through
the wick sides and therefore a near-maximum quantity of fuel has already been
contributed by this position. Further, experimental evidence shows that the flame
does indeed bulge only slightly above this location.
Equation 3.63 shows that there are several phenomena in play.
First, the position of the flame is normalized by the radius of the wick, rather
than its length. The lack of a similar term on the right hand side of the equation
is significant because this reveals that R (or D) is the dominant geometric factor
in determining the position of the flame front. However, in this application, it is
incorrect to state that wick length plays no role because NuD,L, which is evaluated
at z = L, is dependent upon L via correlation. This dependence manifests itself
42
physically as representing the length of the surface that provides the majority of
fuel. Equation 3.63 predicts larger wick radii produce larger flames, as is expected.
Second, the Spalding B number plays a prominent role as the numerator of the
exponent term and as part of the exponent itself. Recalling that larger values of B
represent a lower resistance to vaporization, the flame expands with increasing B.
This can be best understood by considering the inverse situation: a lower B number
requires that the flame resides closer to the wick so that more heat can be conducted
into the wick surface to vaporize the more resistant fuel. A larger B number is then
more tolerant of greater standoff distances (i.e., shallower temperature gradients);
combined with the propensity of fuel to seek oxidizer, the flame extends to the
more plentiful oxygen at farther distances. That is, for a flow penetrating a circular
surface in the normal direction at a given speed, a larger radius results in a higher
volumetric flow rate, which provides more oxygen. The actual quantitative effect
of changes in B are small, as can be seen in Fig. 4.4. This figure also shows that
B takes on greater significance for smaller Nusselt numbers due to diminished heat
flow into the liquid phase.
Third, stoichiometry and fuel composition play a role in the fractionYOx,?/sYF,l.
For tetracosane in air, this has a value of 0.0676. This fraction has a practical range
of about 0.025?1.0. Here, the lower end of the range represents fuels requiring high
amounts of oxidizer where the ambient oxygen mass fraction is on the threshold of
quenching, or YOx,? ? 0.1 and s ? 4. The high end represents fuels requiring low
amounts of oxidizer in an oxygen-rich environment, orYOx,? = 1 ands? 1. In both
cases, it was assumed that pure fuels (YF,l = 1) were being burned. The net effect is
43
B
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
w / D 1
00101102
Nu
D,L
 =
 2
Nu
D,L
 =
 10
Nu
D,L
 =
 1
Nu
D,L
 =
 0.
68
Nu
D,L
 =
 8
Nu
D,L
 =
 4
Figure 4.4: Sensitivity to w/D to changes in B. The range of NuD,L selected
bounds the possible values calculated via Eq. 3.35 for 10?3 < RaDD/L < 105.
(YOx,?/sYF,l = 0.0676).
44
that fuels under ?easier? burning conditions (i.e., with higher values of the fraction)
serve to decrease w/D. Numerically, this occurs because the inner denominator of
Eq. 3.63 grows, causing the parenthesized term to decrease. Physically, this effect
can be interpreted as similar to requiring less oxidizer, enabling the flame to dwell
closer to the fuel surface.
Fourth, the greatest effect is given by the Nusselt number, which is a measure
of the relative importance of convection to conduction heat transfer. There are two
significant effects. First, larger values of NuD,L vaporize more fuel from the wick
sides (though that effect is not visible in Eq. 3.63) and second, larger values of
NuD,L result in higher flow velocities, which have the effect of sweeping the flame
into a thinner profile. In this model, the second factor has a dramatic effect upon
the flame width as can be seen in Fig. 4.5, which shows that for weakly convective
scenarios (NuD < 1), flames are expected to have wide breadths. Again, this is the
manifestation of a conduction- (or mass diffusion-) dominated flame; in this sense,
flames tend toward their behavior in microgravity [19]. Figure 4.5 also demonstrates
the insensitivity of w/D to changes in B.
Comparison between observations and predictions require an intermediate cal-
culation because NuD,L is not known a priori. To calculate NuD,L, we can employ
Eq. 3.34. Calculating NuD,L in this way, the flame width predicted is plotted as
the solid curve in Fig. 4.6, where the abscissa has been replaced with the more
basic variable, RaDD/L. This figure shows that, on average, the observed flame
width is about 52% as large as the predicted value. This is in line with Kosdon et
al. [12], whose experimental flame standoff distance, yf, was about half as large as
45
Ra
D 
D/L
 
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
w / D 1
00101102
Nu
D,L
10
0
10
1
B =
 10
B =
 1
Lim
itin
g N
u D
,L 
= 
0.6
8
Figure 4.5: Sensitivity to w/D to changes in NuD,L. (YOx,?/sYF,l = 0.0676)
46
that predicted by their theory. Also, the CFD results of Alsairafi [26], Raju [28]
and Riley [24] are plotted for comparison. For consistency, material properties for
tetracosane were substituted for any specified in the original reference to calculate
the Rayleigh number as RaD/D3 = 3.50?108 m?3. In this way, all data points are
directly comparable with results from this thesis.
Figure 4.6 shows qualitative agreement but quantitative disagreement. That is,
the flame widths descend at the rate predicted with RaDD/L but the width itself is
too high. As has been previously shown, w/D is not sensitive to choices of B and
YOx,?/sYF,l is constant for the possible hydrocarbon fuels; therefore, we must look
elsewhere for the source of the error.
The most likely source of this discrepancy is that the shell approximation,
Eq. 3.49, poorly approximates reality. This is excusable as the presence of the flame
likely influences the position of the boundary layer. Furthermore, the existence of
two colliding streams (fuel and oxidizer) creates radial velocities that are different
from a cooling vertical cylinder undergoing natural convection. Unfortunately, mul-
tiplication of the right hand side of Eq. 3.60, which constitutes the reciprocal of the
exponent in Eq. 3.61, does not result in the fit desired, suggesting a more complex
relationship is required or that another source of error is involved.
OnemightsuspectthatexperimentalvaluesofRaDD/Lcouldbeunderpredicted,
sliding the data points to the left in the plot. Classically, the Rayleigh number is
definedbythetemperaturedifferencebetweenthecylindersurfaceandambientfluid.
This is not representative of a cylinder surrounded by a flame, however, so instead
the characteristic temperature difference was defined as that between the maximum
47
RaD D/L
10-310-210-1100101102103104105
w / D
10-1
100
101
102
Theroretical
Empirical fit (R2 = 0.96)
D = 0.76, 2.39 < L < 6.74
D = 1.33, 2.52 < L < 7.47
D = 3.29, 3.40 < L < 11.92
D = 2.97, 3.68 < L < 9.99
D = 5.03, 3.21 < L < 8.07
Alsairafi [25]
Raju [27]
Riley [23]
  CFD Results
Figur
e4
.6:
Obs
erv
ed
and
predic
ted
w/
D
using
an
appro
ximatio
n
for
Nu
D
,L
.O
bs
erv
ed
values
of
length,
diameter
and
fla
me
width
were
obtained
via
pho
tomet
ry.
Dia
meter
sho
wn
refl
ects
the
me
an
diameter
for
the
samples
tested
and
differs
substa
ntially
from
the
un
burn
twic
k.
Unit
s:
mm.
(Y
Ox,
?
/sY
F,l
=
0.06
76
,B
=
4.29
;fo
rother
consta
nts,
see
App
endix
C).
48
flame temperature and the ambient gas. Any other choice of temperature difference
would reduce RaD. The Rayleigh number can also be manipulated by alternate
selections of ?, Pr or ?m. Of these, we know that the Prandtl number will not vary
greatly. The expansion coefficient?m is defined as 1/Tm; defining it as 1/Tf or 1/T?
would not change raise or lower RaD by more than a factor of 3, which is insufficient
to shift the experimental data points to the right by the factor of 100 as is needed.
Because Pr is approximately constant over a wide temperature range for most gases,
? is defined as ?Pr, giving heavy weight to the choice of ?. To translate the data
points to the right as necessary for agreement with the curve, ? must decrease by
a factor of ten (because ?2 appears in the denominator of Ra). Such a departure
from the nitrogen properties used is unlikely because the fuel-products mixture has
a comparable molecular weight to nitrogen, the heavy fuel being among the minor
species. Furthermore, an error in estimating thermal diffusivity would promulgate
to the fuel vaporization rate due to the inclusion of k/cp, the major component of
the fuel mass flow nondimensionalizing group, within its definition.
Empirically, Eq. 3.63 can be adjusted by a constant factor of 0.52 to achieve a
fit to data,
wemp
D = 0.52
bracketleftbigg 1+B
1+YOx,?/sYF,l
bracketrightbigg2/NuD,L ln(1+B)
. (4.2)
This is plotted as the dotted line in Fig. 4.6.
49
4.3 Lateral Burning Rate and Boundary Layer Thickness
It is possible to express ?mprimeF(z), the local burning rate per unit length of wick, in
terms of the lateral vaporization rate by dividing Eq. 3.32 and 3.36 to obtain
L ?mprimeF(z)
?mF,L =
NuD,z
NuD,L. (4.3)
As is typical with boundary layer analyses, there is a singularity at z = 0 in the
calculation of NuD,z, which implies that the equation above fails near the leading
edge of the boundary layer. If we limit our analysis to z = L, we can utilize flame
width measurements that were taken nearby to approximate the lateral vaporization
rate. This is necessary because burning rates were not measured and so direct com-
parison is not possible. However, it is possible to calculate the lateral vaporization
rate from the flame width using Eq. 3.31 if a relationship between ? and w can be
found. Previously, we employed the shell approximation to produce Eq. 3.64, which
is an expression for yf/y?. Fig. 4.7 shows that, for practical fuels, the flame lies near
the edge of the boundary layer. That the flame lies close to the oxidizer is no sur-
prise; this is reflected also in the small stoichiometric value of mixture fraction. In
fact, in the limit of infinite air-fuel ratio, s, the ?flame? coincides with the boundary
layer.
Equation 3.32 suggests that ?mprimeF ? NuD,z, where NuD,z grows with (RaDD/z)n.
The parameter in parentheses is proportional to D4nz?n; and therefore to ?mprimeF ?
D4nz?n. Temporarily setting aside the fact that n is a affixed a value of 0.25, this
50
Ra
D 
D/L
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
y f / y?
0.7
5
0.8
0
0.8
5
0.9
0
0.9
5
1.0
0
B =
 10
B =
 1
B =
 2B
 =
 3
B =
 4
B =
 5
B =
 10
Figure 4.7: Relative position of the flame front, yf/y?, using the shell approximation
for NuD,L. (YOx,?/sYF,l = 0.0676)
51
scaling law has some interesting properties.
First, it states that the fuel mass flux per unit length (g/s-m) decreases as one
moves upwards along the wick; logically, this must be so because we prescribed an
isothermal wick surface. Stated differently, a constant temperature differential be-
tween wick surface and ambient must have a decreasing mass flux because the flame
grows in the radial direction, reducing the heat flux by decreasing the temperature
gradient between the flame and wick surface. By contrast, the opposing case of a
constant fuel mass flux boundary condition would require varying temperature or
constant flame position. The latter case does not apply to the evaporative burning
regime. This behavior is known in classical heat transfer literature.
Second, as expected, the mass flow increases with D. However, if we express
the mass flux proper (i.e., mass flow per unit area) by dividing by piD, we find that
?mprimeprimeF ? z?n1D4n1?1. The significance of this scaling law is the transition to a flat-
plate burning regime, i.e., for cylinders with large D/L ratios, flat plate behavior
is anticipated. Indeed, the correlation employed to calculate NuD,z is based on the
Nusselt number of a flat plate with a correction for curvature applied. Further,
Nagendra [41] provides an exponent of n = 0.25 for ?short cylinders/flat plates.?
Using the value of n = 0.25, we find that the dependence of ?mprimeprimeF on D vanishes.
Additionally, the proportionality to z?0.25 is also expected for a flat plate. Both of
these facts are in agreement with flat plate theory.
The values for n have been discretized by correlation by Nagendra et al. [41] to
include 0.05 for ?wires? and 0.16 for ?cylinders.? However, physically, a continuous
range is expected. Lyu and Chen [22] performed a numerical study of evaporation on
52
vertical porous cylinders in which they determined that the mass flux ?mprimeprimeF decreases
with z and D. Although they do not supply exponents to fit into the Nagendra cor-
relation, their results demonstrate that scaling law above has the correct qualitative
behavior.
4.4 Total Burning Rate and Flame Length
The total mass flow rate of fuel being vaporized from the wick is the sum of the
lateral and top component flow rates given by Eqs. 3.36 and 3.40,
?mFcp
kL = piln(1+B)
parenleftbigg1
4NuD,D
D
L +NuD,L
parenrightbigg
, (4.4)
where the dimensionless group Nuln(1+B) appears twice. This group contains a
both a chemical term, ln(1+B), and convective-geometric term, Nu, which together
give a measure of the fuel vaporization rate due to the presence of a flame in the
surrounding stagnant layer.
The final property of the flame that we measure and model is the length of the
flame. Both are based on straight wicks, whereas many consumer candles are of the
self-trimming type in which the flame consumes the combustible wick by coming
into contact with it. However, this description is slightly inaccurate because these
typically slender wicks will curve somewhat and so the tip of the wick actually
extends beyond the high point of the wick.
Nevertheless, the region of the flame above the wick is modeled as the circular
port of a gaseous burner using the theoretical relationship from Roper [1] as the
53
basis and is presented as Eq. 3.72. Following from Eqs. 3.8, 3.47 and 4.4, we can
consolidate an expression for the amount of fuel flowing from this imaginary burner
as
?mF,bcp
kL = piln(1+B)
parenleftbigg1
4NuD,D
D
L +
parenleftbigg
1+ YOx,?s
parenrightbigg
NuD,L
parenrightbigg
?
YOx,?
s
pi
15C1
?PrradicalbigRa
DD/L
y?
D
parenleftbigg5
4 +
y?
D
parenrightbigg
, (4.5)
where the second line of the equation is the lateral fuel consumption rate. Equation
4.5 is valid only for flames that extend above the wick. We rewrite Eq. 3.49 to
include D rather than R and substitute for y?/D to produce
?mF,bcp
kL = piln(1+B)
parenleftbigg1
4NuD,D
D
L +
parenleftbigg
1+ YOx,?s
parenrightbigg
NuD,L
parenrightbigg
?
YOx,?
s
pi
60C1
?PrradicalbigRa
DD/L
parenleftbige2/Nu
D,L ?1
parenrightbigparenleftbigge2/Nu
D,L + 3
2
parenrightbigg
. (4.6)
Equations 3.34, 3.35 and 3.37 can be substituted for NuD,L, NuD,L and NuD,D,
respectively, to obtain the apparent circular port mass burning rate in terms of
RaDD/L. Finally, Eq. 4.6 can be substituted into Eq. 3.72 to calculate the dimen-
sionless flame height above the wick,
Lf
L =
?mF,bcp
kL
M?
MF
bracketleftbigg
1+ TvT
f
bracketrightbigg2/3 1
44/3piln(1+1/S). (4.7)
This equation is plotted in Fig. 4.8 against flame measurements. The plot demon-
strates qualitative agreement between theory and data. Also, the CFD results of
54
Alsairafi [26], Raju [28] and Riley [24] are plotted for comparison. For consistency,
material properties for tetracosane were substituted for any specified in the original
reference to calculate the Rayleigh number as RaD/D3 = 3.50?108 m?3. In this
way, all data points are directly comparable with results from this thesis.
Candle flame lengths grow with increasing values of RaDD/L, as expected, and
at approximately the same rate as the data. Additional fuel is being vaporized
commensurate with increased free-convection conditions. However, the quantitative
disagreement of a factor of 2.7 too high on average implies that either the predicted
fuel vaporization rate is too high, flame heights for a given amount of fuel are too
high, the lateral fuel consumption rate is too low, or a combination of the above.
Following the pattern of Eq. 3.66 (based on Roper [38]), an empirically-adjusted
flame height equation can be expressed as
Lf,emp.
L = 0.023
?mF,bcp
kL
M?
MF
1
ln(1+1/S), (4.8)
in which a correction factor of 2.7 was applied, temperature terms have been folded
into the numerical constant but the chemical terms have been retained explicitly.
Also, the 44/3pi term has been included within the constant. Inclusion of the tem-
perature terms within the constant is believed to be appropriate as the value of
1 +Tv/Tf is not highly sensitive to the choice of fuel for paraffins. Equation 4.8 is
plotted in Fig. 4.8 with the label ?Fit.?
The largest component of the discrepancy between Eq. 4.7 and data is likely
due to the overprediction of fuel vaporization rates. In the absence of mass loss
55
RaD D/L
10-310-210-1100101102103104105
L f / L
10-1
100
101
102
D/L = 0.1  
D/L = 1.0 
0.12 b D/L b 0.19 
0.20 b D/L b 0.29 
0.36 b D/L b 0.46
0.52 b D/L b 0.56
0.75 b D/L b 0.85
0.97 b D/L b 1.60
Alsairafi [25]
Raju [27]
Riley [23]
C1 = 6.9
Fit (C1 = 6.9)
  CFD Results
C1 = 20.0
Figur
e4
.8:
Flame
heigh
tab
ov
ethe
wic
kto
p,
Lf
/L
.(F
or
consta
nts,
see
App
endix
C).
56
measurements, we refer to the existing work of Hamins et al. [25] for corroboration
with respect to mass loss rates. Steady state conditions for a self-trimming wick of
diameter 1 mm and exposed length of approximately 12 mm embedded in a paraffin
candle resulted in a mass burning rate of 0.105 g/min and overwick flame height of
approximately 34 mm. In the dimensionless system used, these experimental data
correspond to RaDD/L = 2.92?10?2, ?mFcp/kL = 2.36 andLf/L = 2.83. To check,
Eq. 4.4 predicts a total burning rate of ?mFcp/kL = 4.80, which is composed of the
top fueling rate ?mF,Dcp/kL = 9.72?10?2 and lateral fueling rate ?mF,Lcp/kL = 4.71.
Also, the predicted flame height isLf/L = 4.38.The calculated dominance of lateral
fueling rate over top fueling rate is not unexpected given the 12/1 D/L ratio of the
wick. The results, although coarse, demonstrate that Eq. 4.4 calculates burning
rates and flame lengths that are order-of-magnitude correct but too high by 103%
and 55%, respectively, on this single data point.
A minor component of the disagreement can be attributed to the low lateral fuel
consumption rate, plotted in Fig. 4.2. That is, the lateral fuel consumption rate as
predicted with C1 = 6.9 following the method of LeFevre and Ede [32] is also too
low by perhaps a factor of two. The constant C1 can be adjusted if velocity data are
available or a sensitivity analysis approach can be applied as in Fig. 4.3. The three
lateral fuel consumption curves correspond to the original value of C1, 2C1 and an
upper limit of 20 above which the lateral burning rate exceeds the lateral supply
rate for B = 1. It can be seen that each increase of C1 increases the importance
of burning lateral to the wick relative to burning above the apparent port burner.
Selecting C1 = 0 is nonphysical because such a choice implies that no oxygen is
57
present to combust fuel at the wick lateral region, yet a flame is present to vaporize
fuel through the first term of Eq. 4.6.
The value assigned to C1 is important because it effectively scales the axial
velocity profile and therefore directly scales the upward mass flow term represented
in Eq. 3.47. Consequentially, more fuel is predicted to be burned in the lateral
region of the wick owing to increased air entrainment. Also, this corresponds to a
reduction in the amount of fuel being transported to the apparent port burner and
so flame heights will be shorter.
Even nearly triplingC1 to a value of 20 has only a minor effect upon the apparent
port burner flame height as shown in Fig. 4.8. From this, we conclude that that the
majority of fuel burned in a candle does so in the overwick region. Intuitively, this
can be understood as the result of increased air entrainment due to higher upward
gas velocities produced by experiencing buoyant acceleration within the flame for a
longer period. That is, fuel in the overwick region is moving faster because it has
been accelerating over a longer period compared with the slow-moving fuel lower in
the flame; these high velocities entrain more air and so have a higher burning rate.
One point of qualitative agreement between the flame length theory and experi-
mental data is the absence of a strong trend due to the slenderness ratio, D/L. The
theory predicts that the slenderness ratio plays a small role in determining the pro-
portion of lateral- and top-vaporized fuel and that higher values ofD/Lshould have
taller flames, symbolizing that more fuel has been released from the top. Figure 4.8
shows that, overall, higher values of D/L do predict taller flames, but this effect is
difficult to distinguish. Also, the wicks with the narrow range of 0.75 ?D/L? 0.85
58
exhibit flame lengths over nearly the entire range of measurements for other slen-
derness ratios. The theoretical curves for D/L = 0.1 and 1 should bound 30 of the
33 experimental data points, but they do not. It is likely that any effect from D/L
is small as theory predicts, but is obscured in the experimental data because this
effect is contained within measurement error.
More plausible explanations for the quantitative error lie within this thesis. One
possibility is that the decision to use the material properties (k/cp) of dry nitrogen
introduces error, which could account for part of the fuel rate discrepancy as well.
The unity Lewis number assumption for the mixture of hydrocarbons, products of
combustion and air could contribute error. A nonunity Lewis number would affect
the nature of the lateral burning rate math problem as well. Finally, the resultant
equivalent diffusion rates may not be representative of the fuel in question, providing
oxygen to the flame at a rate faster than expected, resulting in shorter flames. That
is, axial diffusion of fuel towards oxidizer may be a non-negligible factor and one
that causes more fuel to be burned at lower elevations than is predicted. Even with
these discrepancies, however, we may assume from Roper?s theoretical approach that
overwick flame heights are proportional to the fuel rate supplied.
59
Chapter 5
Conclusion
A stagnant film model for characterizing the mass burning rate, length and width of
the luminous region for candle flames was proposed and developed. In this analysis,
radiation and sooting were ignored. The mass burning rate has been calculated by a
heat transfer analogy in which empirical correlations based on the Rayleigh number
were employed. The Rayleigh number is defined in terms of the temperature dif-
ference between peak flame temperature and the ambient, with material properties
evaluated at the average temperature between the wick surface and flame sheet. The
length of flames was estimated by modeling a circular port gaseous burner located
immediately above the wick. The effects of curvature of the wick has been included
by using a heat transfer correlation for cylinders.
The following assumptions were made:
1. The heat transfer from the flame to the wick can be modeled as a vertical
cylinder at flame temperature cooling under the influence of natural convection
without a surrounding flame front;
2. Pyrolyzate that has not burned laterally to the wick can be modeled as fuel
flowing vertically from a circular port burner;
3. Compressibility effects can be ignored;
60
4. The flame and near flow field remain steady, laminar and axisymmetric;
5. The wick is shaped as a cylinder and remains uniformly saturated with liquid
fuel at the surface; and
6. The fuel is burned in stoichiometric proportions with oxidizer.
Experimental flame lengths and widths were obtained for paraffin candles and
compared with the model. Flame heights were overpredicted by approximately
170%. Flame widths were also overpredicted by approximately 100%. Both flame
length and width followed the scaling trends predicted. However, the expected
influenceoftheratioofwickdiametertowicklengthwasnotseenintheexperimental
data; this may be phenomenological or this may be due to shortcomings in the
photometric method of measuring candle flames.
5.1 Recommendations for Future Work
The following subsequent research is suggested.
1. A comparison of predicted and observed burning rates is warranted. There-
fore, an inexpensive study in which candle mass loss is measured could be
implemented to validate the mass burning rate model.
2. Comparison with the Alsairafi [26] or Raju [28] CFD models may provide
mutual model validation.
3. National Institute of Standards and Technology?s Fire Dynamics Simulator is
61
in the midst of releasing a new version. A study of candle flames using the
liquid burning submodel may again provide mutual validation.
4. The experimental data bear out the fact that the unburnt and burning wick
dimensions differ significantly. The change in wick dimensions can be studied
by a photometric method. Wick porosity may play a role in the swelling or
shrinking of the wick.
5. Photometric methods to measure flame and wick dimensions can be improved.
In particular, several flame dimensions are difficult to discern in the visible
spectrum. Alternate methods may lead to more precise measurements of can-
dle flames.
6. This thesis assumed that the entire length of the candle wick was releasing fuel
in a consistent way. However, the photographs show that there exists a region
of wick near the wax pool that is below the flame. Here, the heat flux is lower
than within the flame and consequently the vaporization rate is reduced. A
study of the actual vaporization rate of this region or an analytical method of
predicting it can be studied.
7. In contrast to the saturated conditions prescribed in this thesis, wicks may
dry out near their tips, resulting in lower burning rates than predicted. This
drying could be studied experimentally to determine the effect upon mass
burning rate. Raju?s [28] CFD model presents a computational approach to
the question.
62
Appendix A
Summarized Measurements
Relative to Pool Diameters
No. Flame Wick Attachment Flame Wick Wick Dry Wick
Height Length (at top) (at Midpoint) (via Caliper)
1. 5.81 2.39 0.85 3.92 0.92 0.68 0.81
2. 8.23 3.48 1.08 4.21 0.98 0.71 0.81
3. 9.40 5.66 1.50 4.57 1.09 0.74 0.81
4. 9.82 4.40 1.38 4.48 1.18 0.77 0.81
5. 10.81 6.74 1.60 4.68 0.63 0.82 0.81
6. 10.08 6.12 1.40 4.65 0.65 0.83 0.81
7. 13.23 7.47 1.65 5.19 0.94 1.13 1.10
8. 17.47 6.84 1.94 5.24 1.47 1.18 1.10
9. 14.24 5.67 1.96 5.07 1.58 1.24 1.10
10. 10.78 3.72 1.65 4.63 1.60 1.40 1.10
11. 8.31 3.07 1.15 4.33 1.60 1.40 1.10
12. 7.10 2.52 1.23 4.12 1.56 1.48 1.10
13. 7.16 2.70 1.33 4.13 1.57 1.51 1.10
14. 50.76 11.92 3.74 7.90 3.73 2.54 2.08
15. 51.90 9.61 3.24 7.84 3.86 2.65 2.08
16. 43.93 8.51 3.21 7.75 3.86 3.13 2.08
17. 26.07 5.67 2.26 7.25 4.52 3.62 2.08
18. 19.61 4.89 1.92 7.04 4.46 3.68 2.08
19. 36.35 6.99 2.56 7.54 4.25 3.69 2.08
20. 9.02 3.40 1.77 6.07 4.37 3.74 2.08
21. 20.24 4.66 2.06 6.36 3.30 2.86 2.58
22. 30.58 6.72 2.39 6.87 3.27 2.89 2.58
23. 50.55 9.99 2.33 7.69 3.30 2.89 2.58
24. 10.20 3.80 2.38 5.48 3.27 3.09 2.58
25. 7.99 3.68 2.36 5.21 3.19 3.12 2.58
26. 59.95 8.07 3.51 9.14 5.29 4.89 3.34
27. 44.22 6.13 2.65 8.66 5.31 4.91 3.34
28. 54.56 7.49 3.54 8.99 5.37 4.92 3.34
29. 14.30 3.21 2.10 7.34 5.40 5.04 3.34
30. 42.04 6.39 2.92 8.66 5.31 5.05 3.34
31. 34.71 6.13 3.19 8.63 5.32 5.07 3.34
32. 29.69 5.24 2.68 8.16 5.39 5.13 3.34
33. 21.46 4.10 2.10 8.01 5.38 5.22 3.34
Table A.1: Summary of data from candle flame experiments. Unit: mm.
63
Appendix B
Candle Flame Photographs
64
No.
1
cm
No.
2
No.
3
cm
No.
4
No.
5
cm
No.
6
No.
7
cm
No.
8
No.
9
cm
No.
10
No.
11
cm
No.
12
No.
13
cm
No.
14
No.
15
cm
No.
16
No.
17
cm
No.
18
65
No.
19
cm
No.
20
No.
21
cm
No.
22
No.
23
cm
No.
24
No.
25
cm
No.
26
No.
27
cm
No.
28
No.
29
cm
No.
30
No.
31
cm
No.
32
No.
33
cm
66
Appendix C
Constants and Physical Values
Symbol Value Units Comment Source
B 4.29 - Tetracosane at 298.15 K, see Eqn. 3.30 -
cp (1) 1.053 kJ/kg?K Nitrogen at 481 K (for B) [42]
cp (2) 1.199 kJ/kg?K Nitrogen at 1169 K [42]
C1 6.9 - 4radicalbig105/(20+21Pr) -
k 0.074 W/m?K Nitrogen at 1169 K [42]
g 9.81 m/s2 - -
?hc/s 13,100 kJ/kg - -
H - - See Table 3.1 for tetracosane -
MF - - See Table 3.1 for tetracosane -
M? 28.85 g/mol - -
Pr 0.72 - - -
Rax/x3 3.50?108 m?3 Ideal gas, g?m(Tf ?T?)/Pr?2m -
s - - See Table 3.1 for tetracosane -
S - - See Table 3.1 for tetracosane -
Tf 1673 K Given as 1400 ?C [25]
Tv 664.5 K Tetracosane [43]
T? 298.15 K - -
XOx,? 0.21 - -
YOx,? 0.233 - -
? 2.14?10?4 m2/s Nitrogen at 1169 K [42]
?m 8.56?10?4 1/K Ideal gas, 1/Tm = 2/(Tf +Tv) -
Notes:
1. 481 K is the average temperature between Tv and T?
2. 1169 K is the film temperature between Tf and Tv
67
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