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Advanced models of fuel droplet heating and evaporation

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Recent developments in modelling the heating and evaporation of fuel droplets are reviewed, and unsolved problems are identified. It is noted that modelling transient droplet heating using steady-state correlations for the convective heat transfer coefficient can be misleading. At the initial stage of heating stationary droplets, the well known steady-state result Nu=2 leads to under prediction of the rate of heating, while at the final stage the same result leads to over prediction. The numerical analysis of droplet heating using the effective thermal conductivity model can be based on the analytical solution of the heat conduction equation inside the droplet. This approach was shown to have clear advantages compared with the approach based on the numerical solution of the same equation both from the point of view of accuracy and computer efficiency. When highly accurate calculations are not required, but CPU time economy is essential then the effect of finite thermal conductivity and re-circulation in droplets can be taken into account using the so called parabolic model. For practical applications in computation fluid dynamics (CFD) codes the simplified model for radiative heating, describing the average droplet absorption efficiency factor, appears to be the most useful both from the point of view of accuracy and CPU efficiency. Models describing the effects of multi-component droplets need to be considered when modelling realistic fuel droplet heating and evaporation. However, most of these models are still rather complicated, which limits their wide application in CFD codes. The Distillation Curve Model for multi-component droplets seems to be a reasonable compromise between accuracy and CPU efficiency. The systems of equations describing droplet heating and evaporation and autoignition of fuel vapour/air mixture in individual computational cells are stiff. Establishing hierarchy between these equations, and separate analysis of the equations for fast and slow variables may be a constructive way forward in analysing these systems.
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Advanced models of fuel droplet heating and evaporation
Sergei S. Sazhin
*
School of Engineering, Faculty of Science and Engineering, The University of Brighton, Cockroft Building,
Lewes Road, Brighton BN2 4GJ, UK
Received 18 April 2005; accepted 2 November 2005
Available online 6 January 2006
Abstract
Recent developments in modelling the heating and evaporation offuel droplets are reviewed, and unsolved problems are identified.
It is noted that modelling transient droplet heating using steady-state correlations for the convective heat transfer coefficient can be
misleading. At the initial stage of heating stationary droplets, the well known steady-state result NuZ2 leads to under prediction of
the rate of heating, while atthe final stage the same result leads to over prediction. The numerical analysis of droplet heating using the
effective thermal conductivity model can be based on the analytical solution of the heat conduction equation inside the droplet. This
approach was shown to have clear advantages compared with the approach based on the numerical solution of the same equation both
from the point of view of accuracy and computer efficiency. When highly accurate calculations are not required, but CPU time
economy is essential then the effect of finite thermal conductivity and re-circulation in droplets can be taken into account using the so
called parabolic model. For practical applications in computation fluid dynamics (CFD) codes the simplified model for radiative
heating, describing the average droplet absorption efficiency factor, appears to be the most useful both from the point of view of
accuracy and CPU efficiency. Models describing the effects of multi-component droplets need to be considered when modelling
realistic fuel droplet heating and evaporation. However, most of these models are still rather complicated, which limits their wide
application in CFD codes. The Distillation Curve Model for multi-component droplets seems to be a reasonable compromise between
accuracy and CPU efficiency. The systems of equations describing droplet heating and evaporation and autoignition offuel vapour/air
mixture in individual computational cells are stiff. Establishing hierarchy between these equations, and separate analysis of the
equations for fast and slow variables may be a constructive way forward in analysing these systems.
q2005 Elsevier Ltd. All rights reserved.
Keywords: Droplets; Fuel; Heating; Evaporation; Convection; Radiation
Contents
1. Introduction . . . ....................................................................... 163
2. Heating of non-evaporating droplets ........................................................ 166
2.1. Convective heating ................................................................ 167
2.1.1. Stagnant droplets .......................................................... 167
2.1.2. Moving droplets ........................................................... 172
2.2. Radiative heating ................................................................. 178
2.2.1. Basic equations and approximations ............................................. 178
2.2.2. Mie theory ............................................................... 179
Progress in Energy and Combustion Science 32 (2006) 162–214
www.elsevier.com/locate/pecs
0360-1285/$ - see front matter q2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.pecs.2005.11.001
*
Tel.: C44 1273 642300; fax: C44 1273 642301.
E-mail address: s.sazhin@brighton.ac.uk.
2.2.3. Integral absorption of radiation in droplets ........................................ 181
2.2.4. Geometric optics analysis ..................................................... 182
3. Droplet evaporation . .................................................................... 184
3.1. Empirical correlations .............................................................. 184
3.2. Hydrodynamic models ............................................................. 185
3.2.1. Classical model ............................................................ 185
3.2.2. Abramzon and Sirignano model ................................................ 188
3.2.3. Yao, Abdel-Khalik and Ghiaasiaan model ......................................... 190
3.3. Multi-component droplets ........................................................... 191
3.4. Kinetic models ................................................................... 193
3.5. Molecular dynamics simulations ...................................................... 197
3.6. Evaporation and autoignition ......................................................... 198
3.7. Coupled solutions ................................................................. 200
4. Concluding remarks .................................................................... 202
Acknowledgements . .................................................................... 203
Appendix A. Physical properties of fuels ..................................................... 203
Appendix B. Physical properties of tetradecane ................................................ 203
Appendix C. Physical properties of n-heptane ................................................. 204
Appendix D. Physical properties of n-dodecane ................................................ 204
Appendix E. Physical properties of diesel fuel ................................................. 204
References ........................................................................... 205
1. Introduction
The problem of modelling droplet heating and
evaporation is not a new one. A discussion of the
models developed prior to the early fifties is provided in
[1,2]. A number of widely known monographs and
review papers have been published since then,
including [3–17]. Various aspects of this problem
have been covered in numerous review articles,
including those published in this journal [18–21].In
all these monographs and review articles, however, this
problem was discussed as an integral part of a wider
problem of droplet and spray dynamics. This inevitably
limited the depth and the breadth of coverage of the
subject. Also, most of the relevant monographs and
reviews were published more than 5 years ago, and thus
do not include the most recent developments in this
area.
In contrast to the articles referenced above, this
review will focus on the relatively narrow problem of
droplet heating and evaporation. Although the
application of the models will be mainly illustrated
through examples referring to fuel droplets, most of
them could be easily generalised to any liquid droplets
if required. Only subcritical heating and evaporation
will be considered. Near-critical and supercritical
droplet heating and evaporation was covered in the
relatively recent reviews published in this journal [22,
23], and in [24]. Analysis of the interaction between
droplets, collisions, coalescence, atomization,
oscillations (including instabilities of evaporating
droplets) and size distribution will also be beyond
the scope of this review, although all these processes
indirectly influence the processes considered (see [25–
40]). Neither will the problem of heating and
evaporation of droplets on heated surfaces be
considered (see [37,41]). Although the phenomena
considered in this review can be an integral part of the
more general process of spray combustion, the
detailed analysis of the latter will also be beyond
the scope of this work (see [42–45]). Although the
problem of radiative heating of droplets is closely
linked with the problem of scattering of radiation, the
formal modelling of the two processes can be
separated. The models of the latter process were
reviewed in [46] (see also [47]), and their analysis
will be beyond the scope of this paper.
Soret and Dufour effects will be ignored. Soret effect
describes the flow of matter caused by a temperature
gradient (thermal diffusion), while Dufour effect
describes the flow of heat caused by concentration
gradients. The two effects occur simultaneously. Both
effects are believed to be small in most cases although
sometimes their contribution may be significant (see
[48–52]).
In most models of droplet evaporation it is
assumed that the ambient gas is ideal. This
assumption becomes questionable when the pressures
are high enough, as observed in internal combustion
engines. The main approaches to taking into account
S.S. Sazhin / Progress in Energy and Combustion Science 32 (2006) 162–214 163
Nomenclature
acoefficient introduced in Eq. (73) (m
Kb
)
a
l
liquid fuel absorption coefficient (1/m)
a
w
,b
w
,c
w
constants introduced in Eq. (34)
Apre-exponential factor (1/s)
A
v
,B
v
functions introduced in Eqs. (40) and (41)
a
0,1,2
coefficients introduced in Eq. (74) (m
Kb
,1/
(K m
b
), 1/(K
2
m
b
))
bcoefficient introduced in Eq. (73)
b
0,1,2
coefficients introduced in Eq. (74) (1, 1/K,
1/K
2
)
Bbranching agent
B
f
parameter introduced in Eq. (87)
B
M
Spalding mass number
B
T
Spalding heat transfer number
B
l
Planck function (W/(m
2
mm))
cspecific heat capacity (J/(kg K))
c
k
,d
k
functions introduced in Eqs. (62)–(64)
C
f
fuel vapour molar concentration (kmol/m
3
)
C
1,2
coefficients in the Planck function
(W mm
3
/m
2
,mmK)
C
g1
,
g2
coefficients introduced in Eq. (79)
d
f
diameter of fuel molecules (m)
Dbinary diffusion coefficient (m
2
/s)
Eactivation energy (J)
E
r,q,f
components of wave electric field (N/C)
fmolecular distribution function
f
c
function introduced in Eq. (35)
f
m
relative contribution of components (see
Eq. (137))
Fforce (N)
F
T,M
correction factors: d
T
/d
T0
;d
M
/d
M0
Fo Fourier number: tkg=R2
d
g
0
(R) function defined by Eq. (78) (W/(m
2
mm))
hconvection heat transfer coefficient
(W/(m
2
K))
h
m
mass transfer coefficient (m/s)
h
0
(hR
d
/k
l
)K1
H
e
enthalpy (J)
Iproperty of the component (see Eq. (137))
I
l
spectral intensity of thermal radiation in a
given direction (W/(m
2
mm))
I0
lspectral intensity of thermal radiation
integrated over all angles (W/(m
2
mm))
Iext
lspectral intensity of external radiation in a
given direction (W/(m
2
mm))
I0ðextÞ
lspectral intensity of external radiation
integrated over all angles (W/(m
2
mm))
jmass flux (kg/(m
2
))
kthermal conductivity (W/(m K))
k
B
Boltzmann constant (J/K)
Kn Knudsen number
l
coll
characteristic mean free path of molecules
(m)
l
K
thickness of the Knudsen layer (m)
Lspecific heat of evaporation (J/kg)
Le Lewis number: k
g
/(c
pg
r
total
)
mmass (kg)
m
i
mass of individual molecules (kg)
m
l
complex index of refraction: n
l
Kik
l
Mmolar mass (kg/kmol)
nindex of refraction (does not depend on l)
n
l
index of refraction (depends on l)
n
0
1.46
N
A
Avogadro number (1/kmol)
Nu Nusselt number
_
qheat flux (W/m
2
)
Qintermediate agent
Q
a
efficiency factor of absorption
Q
f
specific combustion energy (J/kg)
p(R) radiative power density (see Eq. (77)) (W/
m
3
)
p
n
coefficients introduced in formula (58) (K/s)
p
l
(R) spectral distribution of radiative power
density (W/(m
3
mm))
~
pvp
v
/p
amb
P(R) radiative term in Eq. (57) (K/s)
P
ch
chemical power per unit volume released in
the gas phase (W/m
3
)
P
total
total amount of radiation absorbed in a
droplet (K/s)
P1
kassociated Legendre polynomials
Pe Peclet number
Pr Prandtl number
Rdistance from the droplet centre (m)
R
cut
parameter introduced in Eq. (151) (m)
R
g
gas constant (J/(kg K))
R
i
positions of individual molecules (m)
R
ij
distance between molecules (m)
R
m
the value of R
ij
when VZK3
ij
(m)
R
ref
reflection coefficient
R
u
universal gas constant (J/(kmol K))
R* radical
R
*
R
d
/n(m)
RH hydrocarbon fuel
Re Reynolds number
Sfunction introduced in Eq. (65)
Sc Schmidt number
Sh Sherwood number
S.S. Sazhin / Progress in Energy and Combustion Science 32 (2006) 162–214164
ttime (s)
Ttemperature (K)
~
T0ðRÞparameter introduced in Eq. (21) (K)
ufluid velocity (m/s)
Uvalue of the net velocity of the mixture
(m/s)
vmolecular velocities (m/s)
kv
n
(r)kparameter introduced in Eq. (21)
VLenard–Jones 12-6 potential (J)
V
shift
parameter introduced in Eq. (151) (J)
w
l
normalised absorbed spectral power density
of radiation
We Weber number
xposition in space (m)
x
l
size parameter: 2pR
d
/l
X,Yn- and m-dimensional vectors
Yrelative concentration
zparameter introduced in Eq. (104)
Greek symbols
a,b,gparameters introduced in Eq. (138)
a
v
(Re) parameters introduced in Eq. (49)
b
c
coefficients introduced in Eq. (33)
b
m
evaporation or condensation coefficient
b
v
parameter introduced in Eq. (53)
g
c
parameter introduced in Eq. (6)
gparameter introduced in Eq. (85)
d
T,M
film thickness (m)
dttime step used for calculation of droplet
parameters (s)
Dtglobal time step (s)
eemissivity
3small positive parameter
3
i
,3
j
parameters introduced in Eq. (149) (J)
3
ij
minimal energy of interaction between
molecules (J)
3
m
species evaporation rate
zparameter introduced in Eq. (24)
z
k
Riccati–Bessel functions
qangle relative to the velocity of unperturbed
flow or the angle of wave propagation
q
R
radiative temperature (K)
QHeaviside unit step function
kthermal diffusivity (m
2
/s)
k
R
kl=ðclrlR2
dÞ(1/s)
k
l
index of absorption
lwavelength (m or mm)
l
m
3.4 mm
l
n
eigen values obtained from the solution of
Eq. (22)
l
v
m
l
/m
g
l
1,2
spectral range of absorbed radiation (mm)
L
0
function introduced in Eq. (73)
mdynamic viscosity (kg/(m s))
m
c
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1Kð1=n2Þ
p
m
0
(t)(hT
g
(t)R
d
/k
l
)(K)
m
q
cos q
m0
qparameter introduced in Eq. (76)
m
*
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1KðR=RÞ2
p
nkinematic viscosity (m
2
/s)
xparameter introduced in formula (86)
p
k
function introduced in Eqs. (62)–(63)
rdensity (kg/m
3
)
r
b
modified density defined by Eq. (147) (kg/
m
3
)
r
l
2pR/l
(rc)
12
r
g
c
pg/
(r
l
c
l
)
sStefan–Boltzmann constant (W/(m
2
K
4
))
s
i
,s
j
parameters introduced in Eq. (150) (m)
s
ij
zero energy separation between molecules
(m)
s
s
interfacial surface tension (N/m)
t
del
time delay before the start of autoignition
(s)
t
k
function introduced in Eqs. (62)–(64)
t
l
a
l
R
t
0
a
l
R
d
fasymuthal angle measured from the plane of
electric field oscillations
F(u) function introduced in Eqs. (5) and (6)
ck
eff
/k
l
(see Eq. (54))
c
l,m
molar fraction of the mth species in the
liquid
c
t
parameter defined by Eq. (17)
j
k
Riccati–Bessel functions
U
Y
parameter introduced in Eq. (118)
Subscripts
b boiling
abs absorbed
amb ambient
c centre or convection
coll collision
cr critical
d droplet
diff diffusive
dr drift
eff effective
ext external
f film surrounding droplets or fuel
g gas
ins incident
S.S. Sazhin / Progress in Energy and Combustion Science 32 (2006) 162–214 165
‘real gas’ effects have been discussed in [53–57]. The
analysis of these effects, however, will be beyond the
scope of this review.
This review is intended to be both an introduction to
the problem and a comprehensive description of its
current status. Most of the review is planned to be a
self-sufficient text. On some occasions, however, the
reader will be referred to the original papers, without
detailed description of the models. Experimental results
will be discussed only when they are essential for
understanding or validation of the models.
The focus will be on the models suitable or
potentially suitable for implementation in computational
fluid dynamics (CFD) codes. These are the public
domain (e.g. KIVA) or commercial (e.g. PHOENICS,
FLUENT, VECTIS, STAR CD) codes. The structures of
these codes can vary substantially. However, basic
approaches to droplet and spray modelling used in them
are rather similar. This will allow us to link the models,
described in this review, with any of these codes, without
making any specific references.
Following [13] the models of droplet heating can be
subdivided into the following groups in order of
ascending complexity:
(1) models based on the assumption that the droplet
surface temperature is uniform and does not change
with time;
(2) models based on the assumption that there is no
temperature gradient inside droplets (infinite
thermal conductivity of liquid);
(3) models taking into account finite liquid thermal
conductivity, but not the re-circulation inside
droplets (conduction limit);
(4) models taking into account both finite liquid
thermal conductivity and the re-circulation inside
droplets via the introduction of a correction factor
to the liquid thermal conductivity (effective
conductivity models);
(5) models describing the re-circulation inside droplets
in terms of vortex dynamics (vortex models);
(6) models based on the full solution of the Navier–
Stokes equation.
The first group allows the reduction of the dimension
of the system via the complete elimination of the
equation for droplet temperature. This appears to be
particularly attractive for the analytical studies of
droplet evaporation and thermal ignition of fuel
vapour/air mixture (see e.g. [58–62]). This group of
models, however, appears to be too simplistic for
application in most CFD codes. The groups (5) and (6)
have not been used and are not expected to be used in
these codes in the foreseeable future due to their
complexity. These models are widely used for
validation of more basic models of droplet heating, or
for in-depth understanding of the underlying physical
processes (see, e.g. [13,63–66]). The main focus of this
review will be on models (2)–(4), as these are the ones
whichareactuallyusedinCFDcodes,ortheir
incorporation in them is feasible.
The review consists of two main parts. First,
models of non-evaporating droplets will be reviewed
(Section 2). Then evaporation models will be discussed
(Section 3). The main results of the review will be
summarised in Section 4.
2. Heating of non-evaporating droplets
Someone wishing to model heating of non-evapor-
ating droplets would be required to take into account a
number of important processes. These include the
deformation of droplets in the air stream and the
inhomogeneity of the temperature distribution at the
droplet surface. Rigorous solution of this general
problem, however, would have been not only difficult,
but also would have rather limited practical appli-
cations. Indeed, in realistic engineering applications
l liquid
lg from liquid to gas
mtype of species in the liquid phase
p constant pressure
r radial component
R radiation
R
d
outside the Knudsen layer
s surface
sv saturated fuel vapour
t time dependent
v fuel vapour
qcomponent in qdirection
fcomponent in fdirection
K0 inner side of the droplet surface
C0 outer side of the droplet surface
Ninfinitely far from the droplet surface
Superscripts
— average
00 absolute value per unit area of the droplet
surface
S.S. Sazhin / Progress in Energy and Combustion Science 32 (2006) 162–214166
modelling of simultaneous heating of a large number of
droplets would be required. Moreover, this modelling
would have to be performed alongside gas dynamics,
turbulence and chemical modelling. This leads to the
situation where the parameters of gas around droplets
may be estimated with substantial errors, which are
often difficult to control. This is why the focus has to be
on finding a reasonable compromise between accuracy
and computer efficiency of the models, rather than on
the accuracy of the models alone.
The most commonly used assumption is that the
droplet retains its spherical form even in the process
of its movement. This assumption will be made in this
review as well. The generalisations of the models to
droplets of arbitrary shapes were discussed in a number
of books and articles (see [2,65,67]). These generalis-
ations could be applied to simple models of droplet
heating, but it is not obvious how they can be applied to
the more sophisticated models discussed below.
Anothersimplicationwidelyusedindroplet
heating models is the assumption that the temperature
over the whole droplet surface is the same (although it
can vary with time). This assumption effectively allows
the separation of the analysis of heat transfer in gaseous
and liquid phases. It is expected to be a good
approximation in the case of a stationary or very fast
moving droplet, when the isotherms almost coincide
with streamlines [63]. The errors introduced by this
assumption in intermediate conditions are generally
assumed to be acceptable.
In what follows the models for convective and
radiative heating of droplets will be considered
separately (Sections 2.1 and 2.2).
2.1. Convective heating
2.1.1. Stagnant droplets
In the case of stagnant droplets, there is no bulk
motion of gas relative to the droplets, and the problem
of their heating by the ambient gas reduces to a
conduction problem. The heat conduction equation
can be solved separately in the droplet and the gas, and
the solutions are matched at the droplet surface.
Assuming the spherical symmetry of the problem, its
mathematical formulation is based on the solution of
the equation
vT
vtZkv2T
vR2C2
R
vT
vR

;(1)
where
kZ
klZkl=ðclrlÞwhen R%Rd
kgZkg=ðcpgrgÞwhen Rd!R%N;
((2)
k
l(g)
,k
l(g)
,c
l(pg)
, and r
l(g)
are the liquid (gas) thermal
diffusivity, thermal conductivity, specific heat capacity,
and density, respectively, Ris the distance from the
centre of the sphere, tis time, subscripts l and g refer to
liquid and gas, respectively.
This equation needs to be solved subject to the
following initial and boundary conditions:
TjtZ0Z
Td0ðRÞwhen R%Rd
Tg0ðRÞwhen Rd!R%N;
((3)
TjRZRd
K0ZTjRZRdC0;
kl
vT
vRRZRd
K0Zkg
vT
vRRZRdC0
;TjRZNZTgN:
(4)
Assuming that T
d0
(R)ZT
d0
Zconst
1
,andT
g0
(R)Z
T
gN
Zconst
2
, Cooper [68] has solved this problem
analytically, using the Laplace transform method. His
solution can be presented as
TðR%RdÞZTgNC2kl
pkgffiffiffiffiffi
kg
kl
rðTgNKTd0ÞRd
R
!ð
N
0
duFðuÞexp Ku2Fo kl
kg

sin uR
Rd

;
(5)
TðRRRdÞZTgNC2kl
pkgffiffiffiffiffi
kg
kl
rðTgNKTd0ÞRd
R
!ð
N
0
du
uFðuÞexp Ku2Fo kl
kg

!cosðgcuÞsin uCffiffiffiffiffi
kg
kl
rsin gc
!kl
kgðucos uKsin uÞCsin u

;ð6Þ
where
FðuÞZðucos uKsin uÞ
u2sin2uCkg
kl
kl
kgðucos uKsin uÞCsin u
hi
2;
Fo Ztkg=R2
dðFourier numberÞ
S.S. Sazhin / Progress in Energy and Combustion Science 32 (2006) 162–214 167
gcZffiffiffiffiffi
kl
kg
rRKRd
R

u:
As expected, this solution predicts diffusion of heat
from gas to droplets, if T
gN
OT
d0
. As a result, droplet
and gas temperatures approach each other in any finite
domain. Initially, the heat flux from gas to droplets
predicted by the equation
_
qZKkg
vT
vRRZRdC0
(7)
is infinitely large. It approaches zero at t/N.
This solution was originally applied to the problem
of heating a stationary liquid sodium sphere in UO
2
atmosphere [68]. However, the applicability of this
solution to more general problems, involving the time
variations of gas temperature due to external factors
and evaporation effects, is questionable. The only
practical approach to solve this general problem is
currently based on the application of computational
fluid dynamics (CFD) codes. In this case one would
need to take into account the distribution of temperature
inside droplets at the beginning of each time step and
the finite size of computational cells. These effects were
taken into account in the solution suggested in [69,70].
However, these papers did not discuss how practical
it is to implement the solution into a CFD code.
Practically all the available and, known to me, CFD
codes are based on separate solutions for gas and liquid
phases, followed by their coupling [19,28,44,71].
Hence, some kind of separation of the solutions for
gas and liquid phases would be essential to make them
compatible with these codes.
The required separation between the solutions could
be achieved based on the comparison between the
thermal diffusivities of gas and liquid. Let us consider
typical values of parameters for diesel fuel spray
droplets and assume that these droplets have initial
temperature 300 K and are injected into a gas at
temperature 800 K and pressure 30 atm [61]:
rlZ600 kg=m3;klZ0:145 W=ðmKÞ;
clZ2830 J=ðkg KÞ;rgZ23:8kg=m3;
kgZ0:061 W=ðmKÞ;cpg Z1120 J=ðkg KÞ:
A more detailed discussion of fuel properties is given in
Appendix A. For these values of parameters we obtain
k
l
Z8.53!10
K8
m
2
/s and k
g
Z2.28!10
K6
m
2
/s. This
allows us to assume:
kl/kg:(8)
This condition tells us that gas responds much more
quickly to changes in the thermal environment than
liquid. As a zeroth approximation we can ignore the
changes in liquid temperature altogether, and assume
that the droplet surface temperature remains constant in
time. This immediately allows us to decouple the
solution of Eq. (1) from the trivial solution of this
equation for the liquid phase (T(R%R
d
)Zconst). The
former solution can be presented as
TðRORdÞZTgNCRd
RðTd0 KTgNÞ1Kerf RKRd
2ffiffiffiffiffiffi
kgt
p

;
(9)
where
erfðxÞZ2
ffiffiffi
p
pð
x
0
expðKt2Þdt:
In the limit RZR
d
, Eq. (9) gives TZT
d0
. In the limit
t/0, but RsR
d
, this equation gives TZT
gN
.
Having substituted (9) into (7) we obtain the
following equation for the heat flux from gas to
droplets [72]:
_
qZkgðTd0 KTgNÞ
Rd
1CRd
ffiffiffiffiffiffiffiffiffi
pkgt
p

:(10)
The same expression follows from the analysis reported
later in [73,74], who were apparently not aware of the
original paper [72]. Moreover, this expression might
have been derived even earlier, as in 1971 it was
referred to in [75] as the ‘well known conduction
solution’ without giving any references.
For t[tdhR2
d=ðpkgÞEq. (10) can be further
simplified and rewritten as:
j_
qjZhðTgNKTd0Þ;(11)
where his the convection heat transfer coefficient
defined as
hZkg
Rd
:(12)
Remembering that the convection heat transfer is
commonly described by the Nusselt number NuZ2R
d
h/
k
g
, Eq. (12) is equivalent to the statement that NuZ2.
Solution (11) could be obtained directly from Eq. (1) if
the time derivative of temperature is ignored (steady-
state solution). It gives us the well known Newton’s law
for heating of stationary droplets so long as the
boundary layer around droplets has had enough time
to develop. Note that for the values of parameters
mentioned above t
d
Z3.5 ms. That means that except at
S.S. Sazhin / Progress in Energy and Combustion Science 32 (2006) 162–214168
the very start of droplet heating, this process can be
based on Eqs. (11) and (12). These equations are widely
used in CFD codes.
Comparing Eqs. (10)–(12), it can be seen that
Newton’s law (Eqs. (11) and (12)) can be used to
describe the transient process discussed above, if the gas
thermal conductivity k
g
is replaced by the ‘time
dependent’ gas thermal conductivity k
g(t)
defined as
[72,74]
kgðtÞZkgð1CztÞ;(13)
where
ztZRdffiffiffiffiffiffiffiffiffiffi
cpgrg
pkgt
r:(14)
This is applicable only at the very start of droplet
heating (at the start of calculations when a droplet is
injected into the gas). Unless abrupt changes in gas
temperature occur, one may assume that the boundary
layer around the droplet has had enough time to adjust to
varying gas temperature. This would justify the
application of Newton’s law in its original formulation
(Eqs. (11) and (12)).
Note that in the limit t/Nsolution (9) is simplified
to
DThTKTgNZRd
RðTd0 KTgNÞ;(15)
where DTindicates the local changes in gas
temperature after the boundary layer around the
droplet has been formed. The change of gas enthalpy,
due to the presence of the droplet, in this case can be
obtained as:
DHeZð
N
Rd
rgcpgDT4pR2