Droplet evaporation
• Combustion of liquid fuels requires their previous atomization, i.e., breaking the
liquid fuel into a spray of small drops (droplets). The droplets often vaporize
before combustion begins.
• A simplified model of droplet evaporation is described below. The interaction
between droplets is neglected, and the droplet surface temperature is assumed
to be close to the boiling temperature of the liquid, so that the evaporation rate is
controlled by heat transfer from the ambient to the droplet.
• These assumptions are reasonable if the spray is not dense and the temperature
of the medium is high.
• Heat transferred from the ambient supplies the energy necessary to vaporize the
liquid fuel, and the fuel vapor then diffuses from the droplet surface into the
ambient. The mass loss causes the droplet radius to shrink with time, until the
droplet is completed evaporated.
 fu , rs t  and t d
• We wish to determine m
Liquid fuel combustion
Combustion
Droplet evaporation
• Simplifying assumptions:
1. The droplet evaporates in a quiescent, infinite medium.
2. The evaporation process is quasi-steady, i.e., at any instant in time the
process can be described as if it were in steady state.
3. The fuel is a single-component liquid with zero solubility for gases.
4. The droplet temperature is uniform and equal to the boiling point of the fuel
(Ts = TBP).
5. Binary diffusion with Le=1.
6. All the thermophysics properties of the gaseous phase (lg, r e cp,g,) are
constant. Even though they may vary greatly from the droplet surface to
the ambient, a judicious choice of mean values allows reasonably accurate
predictions.
• Mass conservation equation in gaseous phase
d
(r r 2ur )  0
dr
Liquid fuel combustion
m  m fu  r ur 4π r 2  constant
Combustion
Droplet evaporation
• Energy conservation equation in gaseous phase
1 d
1 d  2 l g dh 
2
(
r
r
u
h
)

r
r
2
2

r dr
r dr  c p , g dr 
From mass conservation, and for constant
properties, this yields
m
dT
d  2 dT 
c p,g
 lg
r

4
dr
dr  dr 
with T = Ts for r = rs and T = T for r  .
The solution of this equation is
T  (T  Ts ) exp(  Z m r )  T exp(  Z m rs )
T (r )  s
1  exp(  Z m rs )
Liquid fuel combustion
with Z = cp,g/(4π lg)
Combustion
Droplet evaporation
• Energy balance at the droplet surface
4 rs2 l g
dT
dr
 m h fg
r  rs
Calculating the temperature derivative at the surface from the temperature profile
 , leads to
determined in previous slide, and solving for the evaporation rate, m
m 
4 l g rs
c p ,g
ln Bq  1
where the non-dimensional parameter Bq is known as transfer number or
Spalding number (subscript q indicates that it is only based on heat transfer)
given by
Bq 
c p, g (T  Ts )
h fg
Liquid fuel combustion
Combustion
Droplet evaporation
• Mass balance for the droplet
dmd
 m
dt
3
The mass of the droplet is given by md  rliq V  rliq D / 6 . Inserting the mass of
the droplet and the evaporation rate in the above equation
4 l g rs
d
3
( r liq D / 6 )  
ln Bq  1
dt
c p ,g
or, after some algebra,
8 lg
dD 2

ln ( Bq  1)
dt
rliq c p, g
K
8 lg
r liq c p , g
ln ( Bq  1)
Liquid fuel combustion
Combustion
Droplet evaporation
• Integration in time yield the D2 law:
D 2 (t )  Do2  K t
and the droplet lifetime
Do2
td 
K
The D2 law holds after an initial transient period associated with the heating of the
droplet to near boiling point.
The following approximations may be used to evaluate the gas phase properties:
c p, g  c p, fu T 
lg  0.4l fu T   0.6l T 
Liquid fuel combustion
with T 
Ts  T
2
Combustion
Droplet burning
• The previous model is now extended to include a spherically symmetric diffusion
flame that surrounds the droplet. The restriction that the droplet is at the boiling
point is removed.
• Simplifying assumptions
1. The burning droplet, surrounded by a spherically symmetric flame, exists in
a quiescent, infinite medium. There are no interactions with other droplets,
and the effects of convection are ignored.
2. The burning process is quasi-steady.
3. The fuel is a single-component liquid with zero solubility for gases. Phase
equilibrium prevails at the liquid-vapour interface.
4. The pressure is uniform and constant.
5. The gas phase consists only of fuel vapor, oxidizer and combustion
products, and is divided in two zones. The inner zone, between the droplet
surface and the flame, contains only fuel vapor and products, while the
outer zone consists of oxidizer and products. Binary diffusion prevails in
each zone.
Liquid fuel combustion
Combustion
Droplet burning
6. The fuel and the oxidizer react in stoichiometric proportions at the flame.
Chemical reaction is assumed to be infinitely fast, resulting in an infinitely
thin flame front.
7. The Lewis number is unity
8. Radiative heat transfer is negligible.
9. The thermophysical properties (lg, cp,g , rDM ) are constant.
10. The liquid fuel is the only condensed phase. No soot or liquid water is
present.
•
We may write five equations:
•
Mass species conservation in the inner and outer zones
•
Energy conservation in the inner and outer zones
•
Phase equilibrium at the liquid-vapour interface
These allow us to determine the five unknowns:
Liquid fuel combustion
m fu , r f , Ts , T f and y fu ,s
Combustion
Droplet burning
• Mass conservation still holds:
m  m fu  r ur 4π r 2  constant
• Fuel mass conservation in the inner zone
1 d
1 d  2 M dy fu 
2
 r r D

r
r
u
y

r fu
2
2
dr 
r dr
r dr 


These equations lead, after some algebra,
to
m  m fu   4 r
y fu  1 
2
r D M dy fu
1  y fu dr
exp(  Z fu m fu / r )
exp(  Z fu m fu / r f )
Liquid fuel combustion
Z fu 
1
4 r D M
Combustion
Droplet burning
• Oxidizer mass conservation in the outer zone:


1 d
1 d  2 M dyO2
2
r r ur yO2  2  r r D
2 dr
dr
r
r dr 




This equation and the global mass conservation lead, after some algebra, to
m  m fu  4 r
yO2
2
r D M dyO2
s  yO2 dr
s =(mO2/mfu)stoich
 exp(  Z fu m fu / r )

s
 1
 exp(  Z fu m fu / r f ) 
Noting that yO2  yO2,∞ when r  ∞, then
exp( Z fu m fu / r f ) 
yO2 ,
s
Liquid fuel combustion
1
Combustion
Droplet burning
• Energy conservation equation (the equation for droplet evaporation remains valid):
d  2 dT  m c p , g dT
r

dr  dr  4 l g dr
In the inner zone,
In the outer zone,
This yields
T (r ) 
T (rs )  Ts
T (r f )  T f
and
T (r f )  T f
and
T (r  )  T
 Z m

 Z m 
  T f exp   T fu   Ts exp   T fu

rs 
rf



 Z T m fu 
 Z T m fu 



exp  
 exp  


rs 
r f 


 Z T m fu
(Ts  T f ) exp  
r

 Z m
 Z T m fu 
  T exp   T fu
(T f  T ) exp  

r 
rf


T (r ) 
 Z T m fu 
 1
exp  

r f 

Liquid fuel combustion

  Tf






rs  r  r f
rf  r  
Combustion
Droplet burning
with ZT = cp,g/(4π lg)
• Energy balance at the droplet surface
q g i  m fu h fg  qi l
The heat conducted into the droplet interior
can be handled in several ways:
i) A common approach is to model the droplet as consisting of two zones: an
inner zone where the temperature is uniform and equal to its initial
temperature, To; and a thin surface layer at the surface temperature, Ts.
Hence,
qi l  m fu c p ,l ( Ts  To )
ii) Another approach is to assume that the droplet has a uniform temperature
with a transient heat-up period
qi l  md c p ,l
dTs
dt
Liquid fuel combustion
Combustion
Droplet burning
iii) The simplest approach is to assume that the droplet rapidly heats up to a
steady state temperature Ts, so that
qi l  0
• Expressing qg-i according to Fourier´s law and evaluating the temperature
gradient at the flame surface from the previously determined temperature
profile yields, after some algebra,
m fu c p , g (T f  Ts )
(qi l  m fu h fg )
exp( Z
exp(  Z T m fu / rs )
T
m fu / rs )  exp(  Z T m fu / r f )

1  0
• Energy balance at the flame sheet
 fu h fu  m
 O2 hO2  m
 prod hprod  q f i  q f 
m
(note that there is no flow of products inward
from the flame to the droplet surface due to
assumption 3)
Liquid fuel combustion
Combustion
Droplet burning

 fu h fu  s hO2  s  1hprod
m

 q f i  q f 
Noting that
h fu  hof , fu  c p, g T  Tref 
h prod  h of , prod  c p, g T  Tref 

hO2  hof ,O2  c p, g T  Tref

 
QP  hR Tref  hof , fu  s hof ,O2  1  s hof , prod
it follows that, for constant cp,g,
m fu Q p  q f i  q f 
using Fourier’s law and the temperature profiles formerly obtained, we then have
c p, g 
(Ts  T f ) exp(  ZT m fu / r f )
(T  T f ) exp(  ZT m fu / r f ) 


 1  0
Q p  exp(  ZT m fu / rs )  exp(  ZT m fu / r f )
1  exp(  ZT m fu / r f ) 

Liquid fuel combustion

Combustion
Droplet burning
• Liquid-vapor equilibrium at the surface of the droplet – Clausius-Clapeyron
equation
p fu,s  A exp(  B / Ts )
A and B are constants that depend on the fuel and which may be determined
from Clausius-Clapeyron equation.
Noting that
x fu,s 
p fu,s
and
p
y fu,s 
x fu,s M fu
x fu,s M fu  (1  x fu,s ) M prod
It follows that
y fu, s 
A exp(  B / Ts ) M fu
A exp(  B / Ts ) M fu   p  A exp(  B / Ts )M prod
Liquid fuel combustion
Combustion
Droplet burning
The model for droplet burning is constituted by the following five equations:
y fu ,s  1 
exp(  Z fu m fu / rs )
exp(  Z fu m fu / r f )
exp( Z fu m fu / r f ) 
m fu c p , g (T f  Ts )
(qi l  m fu h fg )
yO2 ,
s
exp( Z
1
exp(  Z T m fu / rs )


T m fu / rs )  exp(  Z T m fu / r f )
 1  0
c p, g 
(Ts  T f ) exp(  ZT m fu / r f )
(T  T f ) exp(  ZT m fu / r f ) 


 1  0
Q p  exp(  ZT m fu / rs )  exp(  ZT m fu / r f )
1  exp(  ZT m fu / r f ) 

y fu, s 

A exp(  B / Ts ) M fu
A exp(  B / Ts ) M fu   p  A exp(  B / Ts )M prod
Liquid fuel combustion
Combustion
Droplet burning
The solution of the governing equations may be obtained as follows:
i) Assuming Ts, equations (2), (3) and (4) may be solved to obtain, successively,
m fu 
4lg rs
c p,g
Bo,q
Tf 
ln( 1  Bo ,q )
m fu
 Q p yO2 ,

 
 c p, g (T  Ts ) 
s

 qi l  m fu h fg
 fu h fg
qi l  m
 fu c p, g ( yO2 ,  s)
m
r f  rs

where the transfer number Bo,q is defined as
ln 1  Bo,q
( sBo,q  yO2 , )  Ts

 y O ,  s 

ln  2

s


Liquid fuel combustion
Combustion
Droplet burning
ii) Then, yfu,s is calculated as follows from Eq. (1)
y fu ,s 
Bo,q  yO2 , / s
Bo,q  1
iii) Eq. (5) is solved to obtain a new value for Ts:
Ts 
B
 y fu, s pM prod


ln 

A
(
y
M

y
M

M
)

fu , s
fu
fu , s
prod
fu 

iv) Steps (i) to (iii) are repeated until the value of Ts guessed in step (i) is close
enough to that determined in step (iii)
Liquid fuel combustion
Combustion
Droplet lifetime
•
If it assumed that the droplet is at the boiling temperature, as in the simplified
model of droplet evaporation, the present droplet burning model greatly
simplifies.
In such a case, the previous equations allow the direct calculation of
m fu , T f and r f without iteration, and the equation for yfu,s becomes irrelevant,
since yfu,s = 1 in that case. This simplification is reasonable when the droplet
is burning vigorously after its initial heat-up transient.
•
Droplet lifetime
The equation for m fu in terms of the transfer number Bo,q is similar to that
derived for the evaporation of a droplet. Hence, a burning rate constant may
be defined as
K
8 lg
r liq c p , g
ln ( Bo ,q  1)
The burning rate constant is truly a constant only after a steady-state surface
temperature is reached, since only then Bo,q is a constant
Liquid fuel combustion
Combustion
Droplet lifetime
•
Assuming that the transient heat-up
period is small in comparison to the
droplet lifetime, the D2 law is
recovered for droplet burning:
D 2 (t )  Do2  K t
•
The droplet lifetime is found by
setting D to zero, yielding:
Do2
td 
K
•
The D2 law is a good representation
of experimental data after the heatup transient.
Liquid fuel combustion
Combustion
Burning rate
• The burning rate constant does not change much with the fuel. The droplet
lifetime is mainly dependent on its initial diameter, and therefore on the
atomization process.
• The burning rate constant is approximately equal to10-6 m2/s and 2x10-6 m2/s
for the burning of hydrocarbons in air and in oxygen, respectively, increasing
with the temperature of the ambient.
cetane
gasoil
kerosene
benzene
n-heptane
Liquid fuel combustion
Combustion
Droplet lifetime
• The thermophysical properties may be estimated as follows:
c p, g  c p, fu (T )
lg  0.4 l fu (T )  0.6 lox (T )
T 
Ts  T f
2
r liq  r liq (Ts )
• The spherical symmetry in the model arises from neglecting the relative
velocity between the droplet and the medium, as well as the buoyancy.
• The convective effects may be incorporated using the film theory, which
replaces the boundary conditions prescribed at r   by similar boundary
conditions prescribed at r = dM for the species and at r = dT for the
temperature.
Liquid fuel combustion
Combustion
Extension to convective environments
• The heat and mass transfer rates
at the surface of the droplet
increase due to convective effects.
• The radius dM and dT are related to
the
Nusselt
and
Sherwood
numbers as follows:
dT
rs
dM
rs


Nu
Nu  2
Sh
Sh  2
• In a medium at rest, Nu=2  dT  
• Assuming Le=1 for all species,
then Nu=Sh and dM =dT
Liquid fuel combustion
Combustion
Extension to convective environments
• The following correlation may be used to evaluate Nu for droplet burning in
forced convection:
Nu  2 
0.555 Re 1 / 2 Pr 1 / 3
1  1,232 /Re Pr 
4 / 3 1/ 2
where the Reynolds number is defined from the diameter of the droplet and the
relative velocity (physical properties may be determined at the average
temperature of the medium)
Convection influences the temperature and species profiles on the outer side of
the flame front. The 2nd and 4th equations of the system of equations are
modified as follows:

 yO2 ,  s  0
exp  Z fu m fu /rs Nu / Nu  2
exp( Z fu m fu / r f )
Liquid fuel combustion
s
Combustion
Extension to convective environments

c p, g 
(T f  Ts ) exp(  ZT m fu / r f )
(T f  T ) exp(  ZT m fu / r f )


 1  0
Q p  exp(  ZT m fu / r f )  exp(  ZT m fu / rs ) exp(  ZT m fu / r f )  exp  ZT m fu ( Nu  2) / rs Nu 


• The burning rate is now given by
m fu 
2lg rs Nu
c p, g
ln( 1  Bo,q )
with the transfer number still given by
Bo,q
m fu
 Q p yO2 ,




 c p, g (T  Ts ) 
s

 qi l  m fu h fg
The expression for the burning rate reduces to the original one for Nu=2, i.e.,when
the medium is quiescent.
Liquid fuel combustion
Combustion