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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

*

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

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.

Anothersimpliﬁcationwidelyusedindroplet

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

ztZRdﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

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