notes lecture 12 - princeton university...of heat and mass transfer. the atomization of a 1cm...

Summer 2013

Moshe Matalon University of Illinois at Urbana-‐Champaign 1

Lecture 12

Droplet Combustion Spray Modeling

Moshe Matalon

Spray combustion:

Spray combustion involves many physical processes, including atomization, droplet

collision and agglomeration, vaporization, heat and mass transfer, droplet-air

and vapor-air mixing, ignition, turbulence, premixed and/or di↵usion flames,

pollutant production and flame extinction.

Many practical applications liquid fuel is injected into the combustion chamber

resulting in fuel spray.

The liquid fuel is atomized by the combined action of aerodynamical shear,

strain and surface tension producing a large number of small droplets which

increase the overall surface area exposed to the oxidizer and enhance the rates

of heat and mass transfer.

The atomization of a 1cm diameter droplet of liquid into droplets of 100 µmdiameter, for example, produces a million droplets and increases the overall

surface area by a factor of 10

4.

Moshe Matalon

Summer 2013


The droplets distribution range from few microns to around 500µm, primarily

because practical nozzles cannot produce sprays of uniform drop size at typical

operating conditions. Furthermore, many of the larger droplets produced in the

initial liquid disintegration undergo further breakup into smaller droplets.

Depending on the droplet distribution, the mode of operation and the state of

the ambience, di↵erent regimes of burning occur, including

• a di↵usion flame lies outside the cloud of droplets burning the ambient

oxygen with the fuel that originates from the vaporizing droplets

• droplet burning occurring all throughout the cloud or in a layer on the

edge of the cloud,

• internal group combustion with an internal flame that separates a group

of vaporizing droplets from a group of individually burning droplets

• individual droplet combustion where individual droplets burn with oxygen

di↵using across the resulting cloud of burning droplets

Moshe Matalon

Liquid jet

atomizer

liquid

oxidizer entrainment

flame

droplet burning

cloud burning

atomization

vaporization

S P R A Y

oxidizer entrainment

Moshe Matalon

Summer 2013


Droplet Vaporization and Combustion:

Fuel vapor

drop

Assumptions:

• droplet is a sphere

• spherical symmetry

• single component fuel

• uniform properties inside the droplet

• no fluid motion inside the droplet

• quiescent ambience

• no gravity

• quasi-steady approximation

• one-step overall chemical reaction (F + O ! Products)

Moshe Matalon

� d

dt

�43⇡⇢lr

3s

�= m

mass flow rate

mass per unit time leaving the droplet

m or ⇢gv can be determined from the steady equations in the gas phase, with

the droplet history determined a-posteriori from the relation above

�4⇡ r2s ⇢ldrsdt

= 4⇡r2s · ⇢g(v�drsdt

)

drsdt

⇠ �⇢g⇢l

v

a direct consequence of the jump relation [[ ⇢(v·n�VI) ]] = 0 across the interface

r = rs, namely �⇢lrs = ⇢g(v � rs) after using ⇢g/⇢l ⌧ 1.

quasi-steady approximation

Since ⇢g/⇢l ⌧ 1 (typically ⇠ 10

�2 � 10

�3) the droplet recedes very slowly

compared to the gas-phase processes, and

Overall mass conservation of the droplet

Moshe Matalon

Summer 2013


[[ ⇢Yi ((v+Vi) · n� VI) ]] = 0

⇢gYi(v � rs)� ⇢gDidYi

dr= �⇢lYi rs

⇢gvYF � ⇢gDFdYF

dr= ⇢gv

⇢gvYO � ⇢gDOdYO

dr= 0

Conditions for the mass fractions at the liquid-gas interface can be derived by

integrating the species equations across the moving liquid-gas interface, r = rs,or using the previously derived jump condition

where we have assumed that no reaction occurs along the interface

Using the approximation rs ⇠ (⇢g/⇢l)v

Moshe Matalon

⇢ghg(v � rs)� �gdT

dr

��a+

= �⇢lhl rs � �ldT

dr

��a�

[[ ⇢h(v·n�VI) + pVI + q·n ]] = 0

Similarly, integrating the energy (expressed in terms of enthalpy) across the

moving liquid-gas interface, r = rs, or using the previously derived jump con-

dition

where we have assumed that no reaction occurs along the interface and neglected

kinetic energy and viscous dissipation in the energy equation (the small Mach

number approximation). Then, using the approximation rs ⇠ (⇢g/⇢l)v

latent heat of

vaporization

⇢gv (hg � hl)| {z }Qv

��

✓dT

dr

��a+

� dT

dr

��a�

◆= 0

Moshe Matalon

Summer 2013


⇢gv Qv � �

✓dT

dr

��a+

� dT

dr

��a�

◆= 0

assuming all the heat conducted inwards

goes to heating the liquid fuel,

4⇡a2�

✓dT

dr

◆

a�⇡ 4

3⇡a3⇢lcpl

dT

dt

Alternatively it may be accounted for

by replacing Qv with an “e↵ective”

value that accounts for internal heating.

often neglected.

�dT

dr= ⇢v Qv

�dT

dr= ⇢vQv +

a

3⇢lcpl

dT

dt

��l| {z }

Moshe Matalon

! = B

✓⇢YF

WF

◆✓⇢YO

WO

◆e�E/RT

a

liquid

⇢T =P0

R/W

⇢vdYF

dr� ⇢DF

r2d

dr

✓r2

dYF

dr

◆= �⌫FWF !

⇢vdYO

dr� ⇢DO

r2d

dr

✓r2

dYO

dr

◆= �⌫OWO !

⇢vdT

dr� �

cp

1

r2d

dr

✓r2

dT

dr

◆=

Q

cp!

In the following, the gas density will be denoted by ⇢ (with no subscript) and

the droplet radius a.

d

dr(r2⇢v) = 0

The steady conservation equations are

Conditions far away, as r ! 1

T = T1, ⇢ = ⇢1 YF = 0 YO = YO1

Moshe Matalon

Summer 2013


If one assumes that the phase change at the interface occurs very fast, equilib-

rium conditions prevail and a definite relation exists between the vapor partial

pressure and the droplet surface temperature. This relation is of the form

pYF = k exp

⇢Qv

R

✓1

TB� 1

T

◆�

known as the Claussius Clapeyron relation. Instead we shall use for sim-

plicity that the surface temperature is constant, and at r = a

T = TB

Boundary conditions at r = a

⇢vYF � ⇢DFdYF

dr= ⇢v

⇢vYO � ⇢DOdYO

dr= 0

�dT

dr= ⇢vQv

T = TB

Moshe Matalon

M = r2⇢v ⇢T = 1at r = 1:

as r ! 1

M

r2dYF

dr� Le�1

F

r2d

dr

✓r2

dYF

dr

◆= �!

M

r2dYO

dr� Le�1

O

r2d

dr

✓r2

dYO

dr

◆= �⌫ !

M

r2dT

dr� 1

r2d

dr

✓r2

dT

dr

◆= q !

! = D ⇢2YFYOe��0/T

MYF � Le�1F

dYF

dr= M

MYO � Le�1O

dYO

dr= 0

dT

dr= MLv , T = Ts

T = ⇢ = 1, YO = YO1 YF = 0

We have a 6th-order system, with 7 BCs; the extra one for the determination of M.

We write dimensionless equations using a as a unit of distance, Dth/a as a unit

of velocity and normalize the density and temperature with respect to their

ambient values ⇢1 and T1.

We also use M =

m/4⇡

a(�/cp)to denote the dimensionless mass flux .

Moshe Matalon

Summer 2013


We note in, particular, the dependence on the Damkohler number on the droplet

radius D ⇠ a2 and on the ambient pressure D ⇠ P0.

We also assume unity Lewis numbers.

The parameters are

⌫ = ⌫OWO/⌫FWF �0 = E/RT1

q = Q/⌫FWF cpT1 D =⇢21a2⌫FB

(�/cp)WO

Lv = Qv/cpT1 Ts = TB/T1

Moshe Matalon

Pure vaporization: D = 0

YF = 1� e�M/r

YO = YO1e�M/r

T = Ts � Lv + LveM(1�1/r)

M = ln

✓1 +

1� Ts

Lv

◆

In dimensional form, the vaporization rate is

m =4⇡a�

cpln

✓1 +

cp(T1 � TB)

Qv

◆

Bv =

cp(T1 � TB)

Qv=

impetus for transfer

resistance to transfer

transfer number

Clearly Bv > 0, which implies that, for vaporization, T1 > TB.

Moshe Matalon

Summer 2013


T

TB

YF = 1

YF

T1

r1

Moshe Matalon

Combustion:

We will consider the Burke-Schumann limit D ! 1 and make use of the cou-

pling functions.

T + qYF = 1 + (Ts � 1 + q � Lv)(1� e�M/r)

T + q⌫�1YO = (Ts � Lv)(1� e�M/r) + (1 + q⌫�1YO1)e�M/r

from which we can also write

YF � ⌫�1YO = 1� (1 + ⌫�1YO1)e�M/r

The profiles are readily available

YF =

8<

:

1� (1 + ⌫�1YO1)e�M/r(r < rf )

0 (r > rf )

YO =

8<

:

0 (r < rf )

(⌫ + YO1)e�M/r � ⌫ (r > rf )

Moshe Matalon

Summer 2013


T =

8<

:

(Ts � Lv)(1� e�M/r) + (1 + q⌫�1YO1)e�M/r (r < rf )

1 + (Ts � 1 + q � Lv)(1� e�M/r) (r > rf )

transfer number

There are few conditions that remain to be satisfied; these are T = Ts at r = 1,

and continuity of all variables at r = rf . The first provides and expression for Mand the others determine the flame stando↵ distance and the flame temperature.

M = ln

✓1 +

1 + q⌫�1YO1 � Ts

Lv

◆

In dimensional form, the burning rate is

m =

4⇡a�

cpln

✓1 +

cp(T1 � TB) +QYO1/⌫OWO

Qv

◆

Bc =cp(T1 � TB) +QYO1/⌫OWO

Qv=

impetus for transfer

resistance to transfer

Moshe Matalon

rf =a ln(1 +Bc)

ln(1 + ⌫FWFYO1/⌫OWO)

Tf = T1 +(Q/⌫FWF �Qv)/cp � (T1 � TB)

1 + ⌫OWO/⌫FWFYO1

Note that rf > 1 implies that cp(T1�TB) > (Qv�Q/⌫FWF )⌫�1YO1 which

is not restrictive because typically Q > Qv.

The flame stando↵ distance

The flame temperature

Moshe Matalon

Summer 2013


T

TB

YF = 1

YF

T1YO

YO1

rrf1

Moshe Matalon

Bur

ning

rat

e

Damköhler number D D ~ a2, P0

Dext

BS limit Ta

Dign

Vaporization limit

Mc

Mv

Extinction may occur as the droplet size diminishes below a critical value; auto-ignition may occur if the ambient pressure is increased significantly.

If the activation energy parameter is considered large, the asymptotic analysis provides a more complete picture spanning the entire range of Damköhler number D.

Moshe Matalon

Summer 2013


Droplet history:

( use back the notation a = rs and ⇢ = ⇢g )

The vaporization and burning rates were found to have the form

m = 4⇡rs(�/cp) ln(1 +B) with the appropriate transfer number.

drsdt

= �⇢g⇢l

v =k

2rsk =

2�

cp

ln(1 +B)

⇢lwith

2rsdrsdt

= �k ) r2s = r2s(0)� kt

d2

t1/K

m = 4⇡r2s⇢gv ) ⇢gv = �cp

ln(1 +B)1

rs

d2 = d2(0)�Kt

K =8(�/cp)

⇢lln(1 +B)

d2-law

tlife =

✓⇢l⇢g

◆d20

8Dth ln(1 +B)

Moshe Matalon

Nuruzzaman et al. PCI, 1971

The d2-law prediction is verified experimentally for most of the droplet lifetime with the exception, perhaps, of a short initial period following ignition

Moshe Matalon

Summer 2013


Formulas that have been used are of the form

where K is the evaporation constant, K(0)the evaporation constant for a sta-

tionary burning droplet, Pr = µg/Dth⇢g is the Prandtl number.

K = K(0)⇣1 + C Re1/2Pr1/3

⌘C ⇡ 0.3

The behavior of a single droplet in a combustion chamber depends on the

Reynolds number Re = ⇢gd0|u � up|/µg, where u is a characteristic velocity

of the gas and up the particle velocity.

So far, the e↵ect of the relative velocity of the ambient gas on the droplet has

been neglected, practically taking Re = 0. The “spherical symmetry” assump-

tion cannot be justified when Re 6= 0. A general approach that has been often

used to address the problem of droplet combustion in a convective field has been

to correct a-posteriori the results derived for Re = 0 with empirical correlations.

Moshe Matalon

Similarly, burning rate corrections for free convection has been proposed, in the

form

K = K(0)�1 + C Gr0.52

�C ⇡ 0.5

where Gr is the Grashof number, representing the ratio of the buoyancy to

viscous forces.

Gr =

⇢2gd30 g

µ2g

✓�T

T

◆

Here �T may be taken as the di↵erence between the flame temperature and theambient temperature .

The assumption of spherical symmetry for suspended droplets can be justified

either in micro-gravity conditions (g ⌧ 1) or in reduced pressure environment

(P0 ⇠ ⇢g ⌧ 1)

Moshe Matalon

Summer 2013


Combustion of a solid particle:

Similar to a fuel droplet a solid particle may be surrounded by a di↵usion flame,

so that

rs

solid

For Le = 1, we have �/cp = ⇢DO.

m =4⇡rs�

cpln (1 +B)

if the flame is close enough to the surface, rf ⇡ 1

This is the di↵usion-controlled limit, where the di↵usion of oxidizer towards the

particle controls the burning.

ln (1 +B) = ln�1 + ⌫�1YO

�

m = 4⇡rs⇢gDO ln�1 + ⌫�1YO

�

Moshe Matalon

Using m = � d

dt

�43⇡⇢sr

3s

�= �4⇡⇢s r

2sdrsdt

namely a d2-law.

tlife =

✓⇢s⇢g

◆d20

8DO ln(1+⌫�1YO)

d2 = d20 �8⇢gDO

⇢sln (1 + ⌫�1YO) t

r2s = r2s(0)�2⇢gDO

⇢sln (1 + ⌫�1YO) t

m = 4⇡rs⇢gDO ln�1 + ⌫�1YO

�

we find in the di↵usion-controlled limit

Moshe Matalon

Summer 2013


solid

rs

kinetic-controlled

burning

If burning occurs on the surface (and possibly inside the pores) of the particle,

the burning rate is controlled by the chemical kinetics. For an Arrhenius-type

heterogenous surface chemical reaction rate, we have

This is the kinetic-controlled limit, where the chemical reaction rate controls

the burning.

m = 4⇡r2sks⇢YO

where ks = Bse�E/RTs

rs = rs(0)�ks⇢s

⇢YO t

Using m = � d

dt

�43⇡⇢sr

3s

�= �4⇡⇢s r

2sdrsdt

d = d0 � 2ks⇢g⇢s

YO t

tlife =

✓⇢s⇢g

◆d0

2BsYOe�E/RT

namely a d1-law (linear).

Moshe Matalon

t independent of T

tlife =

8>>>><

>>>>:

✓⇢s⇢g

◆d20

8DO ln(1+⌫�1YO)di↵usion-controlled

✓⇢s⇢g

◆d0

2BsYOe�E/RTskinetic-controlled

t ⇠ d20

t ⇠ d0

t ⇠ eE/RT

large particles, or higher ambient temperature ) Ds large

small particles, or low ambient temperature ) Ds small

) kinetic-controlled

) di↵usion-controlled

The controlling mechanism is determined by a surface Damkohler number

Ds =tdi↵usion

tkinetic

=d0

BsYOe�E/RTs

4DO ln(1+⌫�1YO)

Moshe Matalon

Summer 2013


Spray Modeling:

The accurate description of spray combustion, say by DNS, requires identifi-

cation of whether liquid or gas is present at every point in space, to employ

appropriate equations of state in each of the separate phases and to determine

the evolution of a large number of liquid-gas interfaces with boundary condi-

tions that involve phase transformation. The large number of droplets required

to simulate real dispersed flows in turbulent environments that are encountered

in practical applications, make such calculations prohibiitve.

There are a practically three approaches to spray combustion modeling.

• Continuum formulation

• Lagrangian formulation

• Probabilistic formulation

We will not be discussing any of these in detail, but we will try to get a general

idea of what each of these formulations involve.

Moshe Matalon

The spray contains N distinct chemical species and M di↵erent class of droplets

identified by fuel content, size and/or velocity. Let mi be the mass of species iand m(k)

the mass of droplets of class k.

The total volume of the mixture will be denoted by VT and will be given by

VT = Vg + Vl, where Vg and Vl are the volumes occupied by the gas and the

liquid phases, respectively.

Continuum Formulation. Properties of both the particles and the gas are

assumed to exist at every spatial point (the average value of a small parcel that

includes both phases) regardless of whether that point is actually in the liquid,

solid or gas at that instant.

1�' is the void fraction, or the ratio of the volume occupied by the condensed

phase from the total volume VT .

' = Vg/VT is the volume fraction of the gas, namely the ratio of the volume

occupied by the gas to the total volume of the mixture.

Moshe Matalon

Summer 2013


Similarly, the density of droplets of class k is ⇢(k), so that the bulk density of

droplets is

⇢ =MX

k=1

⇢(k) =1

VT

MX

k=1

m(k)

which di↵ers from the liquid density

⇢l =1

Vl

MX

k=1

m(k) =VT

Vl⇢ =

1

1� '⇢

The bulk density (mass per total volume VT ) of species i is ⇢i.

The bulk density of the gaseous mixture is

⇢ =NX

i=1

⇢i =1

VT

NX

i=1

mi

It di↵ers form the gas density

⇢g =1

Vg

NX

i=1

mi =VT

Vg⇢ =

1

'⇢ which is the mass per

unit volume that includes only gas.

For a dilute spray the condensed phase volume fraction is small (1� ' ⌧ 1) so

that ⇢ ⇠ ⇢g, but

⇢ = (1� ')⇢l ⇠1� '

⇢g/⇢l

⇢ = O(1) because ⇢g/⇢l ⌧ 1.

⇢ is often referred to as the spray density.

Moshe Matalon

@⇢

@t+r · ⇢v = M

@⇢

@t+r · ⇢v = �M

Statements of mass conservation of the gas and condensed phases lead to

Here M is the total mass (per unit volume, per unit time) added to the gasphase due to vaporization and v, v are the gas and particle velocities.

Using ⇢ = (1�')⇢l , we can recast the mass conservation of the droplets in the

following form

' is no longer conserved; the volume occupied by the droplets changes as a

result of (i)vaporization, and (ii) distance between droplets

@'

@t+r · (v') = M

⇢l

+r · v,

@

@t(⇢+ ⇢) +r · (⇢v + ⇢v) = 0

Note:

i.e., the total mass is conserved.

This can be determined, for example, from the expressions we derived for the

vaporization rate of isolated droplets.

Moshe Matalon

Summer 2013


Similarly, separate statements of momentum and energy conservation are writ-

ten for the gas and liquid phases with appropriate source/sink terms repre-

senting the total momentum and energy carried to the gas from the vaporizing

droplets, the total force exerted on the particles from their surroundings and

the internal heating of the droplets which, in general, could have a di↵erent

temperature than the bulk gas.

Moshe Matalon

f dr dx dv de... is the probable number of particles in the radius range dr about

r, in the spatial range dx about the position x, with velocities in the range dvabout v and internal energy in the range e about de, etc..., at time t.

the terms on the l.h.s. of the spray equation represent drop-carrier gas interac-

tions, the source term Q on the r.h.s. represents particle-particle interactions

(fragmentation, coalescence, etc..).

The time evolution of the distribution function f is given by

@f

@t+r

x

·(vf) +rv

·(af) + @

@r(rf) +

@

@e(ef) + · · · = Q,

and is known as the spray equation.

Probabilistic Formulation. In this approach the condensed phase is de-

scribed by a distribution function, or probability density function (pdf)

f(r,x,v, e, t)

Moshe Matalon

Summer 2013


dx

dt= v,

dv

dt= a,

dr

dt= r,

de

dt= e, etc....

The quantities in the spray equation are determined by the laws governing the

dynamics of individual droplets

The conservation laws for the gas-droplet mixture will include additional source

terms involving the integral of f . For example, the mass and momentum equa-

tions are

@⇢f

@t+r·(⇢fu) = �

Z4⇡r2⇢l rf drdvde,

⇢f

�@u@t

+ u·ru�= �rP+ ⇢fg �

Z ⇥43⇡r

3⇢la+ 4⇡r2⇢l r(v�u)⇤f drdvde,

where ⇢f is the fluid density, namely the mass of the gas mixture per unit volume

of space, which due to the presence of the particles di↵ers from the gas density

⇢g that corresponds to the mass per unit volume occupied by the gas.

For example, the dependence of r on the radius for liquid drops can be specified

by adopting the d

2-law

r =�/cp⇢l

ln (1 +B)

r,

The main di�culty in this approach is the high dimensionality of the distribution

function f(r,x,v, e, t) which, under the simplest assumptions, contains nine

variables.

Moshe Matalon

Lagrangian Discrete Particle Formulation. The droplets are considered as

individual point-particles and their trajectories are solved for, separately from

the flow field which exists only where there are no droplets. The main di�culty

is probably the inability to track a su�ciently large number of particles in order

to

Moshe Matalon

notes lecture 12 - princeton university...of heat and mass transfer. the atomization of a 1cm...

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