Background on Droplets and Sprays_1_2

8/10/2019 Background on Droplets and Sprays_1_2

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BACKGROUND ON DROPLETS AND SPRAYS

R. Bo rghi, COR IA-URA230/CNR..S

Facult des Sciences de Rouen

1. INTRODUCTION

The combustion of sprays tak:esplace in many industrial devices : Diesel and rocket engines,

turbojet combustion chambers, industrial burners. The propagation of a flame zone through a spay

,~.

dispersed into the air is also of interest for safety problems.

A spray, which contains droplets dispersed within a gas.mixture, is a random medium. This

randomness exist firstly due to the random position of the droplets in the spray, with different sizes

(random again),even when thespray is at rest or in laminar motion. The presence ofturbulence,

which is very general in practical cases, is an additional reason for the randomness. The computation

of the combustion within a spray cannot take into account the details of the small scale structure of the

spray, i.e. the positions and sizes of all the droplets and the gradients of velocity, temperature and

concentration, between the droplets. Such a kind of direct numerical simulaton is not possible for

practical purposes ; itcould be possible for the purpose of analysing

thephenomena

in very simple

cases; buthas not yet been attempted to my knowledge. Anyway, due to the randomness of the

spray, many of such calculations would be necessary in order to extract mean properties, the only

ones that are relevant for comparisons with experiments.

However, in the average, the spray can be considered as a continuous medium, similarly as

a gas containing many molecules, or as a turbulent flow containing many small seale eddies. Then, it

IS possible to write balance equations for the interesting quantities of.this mediurn- velocities, mass

fractions. temperatures ,- in a very classical manner. The first problemto be solvedfor thewriting

..

, ,-

.....

- ~ , . . . . .

of these equation is to chose the quantities for characterizing the spray. A first possibility is to

consider the spray as a single phase medium, with only asingle(mean) velocity, asingleImean)

temperature ; this is the framework of the so-called locally homogeneous spray (LHS) introduced

by G.Faeth and coworkers(l). But now a secondpossibility ismore often considered, with models

that distinguish the temperature and velocity of the gas phase and the ones of the liquid phase ; the

Eulerian-Eulerian approach defines only a single (mean) temperature or velocity for the liquid phase,

and the Lagrangian-Eulerian models are able to distinguish one temperature or velocity-for each

droplet. Anyway, the equations have to contain models in order to represent the small scale random

phenomena, and this constitutes the second modelling problem. The problem is not simply due to the

presence af two phases, with exchanges of mass, momentum and energy between them. The problem

is due to the fact that the quantities to be calculated by Eulerian equations are averaged quantities : the

average is taken statistically in theory, or spatially and temporally in practice.The situation hereis

.~

.

1

1

.

~

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similar to the one of turbulent flows, and combustion is known to produce additional difficulties due

to its non-linear character.

AlI the problems associated with this modelling are not yet solved. Actually, the modelIing of

the combustion of sprays is still in his infaney. However, important questions have been already

studied, and there is a sound basis for studying the problem. The purpose of this paper is to rewiew

this basic material in a synthetical manner, and to emphazises the main point to be studied further.

Beeause any modelling work has to be based on the analysis of the physical phenomena that are

oecuring, we shall present first our (assumed) knowledge on the small seale phenomena, at the scale

of the droplets or between the droplets. So, the first part of the paper considers the well known case

of a single droplet burning.or simply vaporizing. The seeondpart discuss the expectedstructureof '

- flames in sprays, either in the case of a premixed spray ,Le. a spray with droplets homogeneously

dispersed in a gaseous mixture on a very large scale, or for a spray jet-flame, when ajetof droplets is

dicharged into a gaseous medium. Then, the third section presents the basic equations, and discuss

the existing models. The open problems and some ideas towards their solutions are emphasized in lhe

.

.

conclusion.

,

2~ .

SINGLE DROPLET VAPORIZA TION AND COMBUSTION

We consider a single droplet of fuel in an oxidizing atmosphere, but th contrary could

equalIy be.considered. First, we describe in depth the case of a spherical droplet in a medium atrest,

without gravity, in the limit ofthe quasi-steady state. Discussions.concerning the departures from this

~. -

situation, and in particular thecaseofa movingdroplet, .willfollow.

2.1 The quasi-steady theory

The' simplest case is not the one of a single droplet burning, even without gravity and in a

mdium at rest, but the.one of a droplet continuously feeded at hiscenter and burning. So, for a given

mass flow rate fed, the radius of the droplet can be expected to remainconstant and all theprocesses

are in steadyjtate, The firstthebryf~{':this case has beendevelopped by D .B.S palding (2), and

Godsave (3); Numerous experimentshave beendone sofar, but with real conditions (see in

particular Kurnagai et aI. (4, in order to assess this assumption and the theory.

Let usassume that we have solved the problem and found the mass flow rate to be injected.

rn. Then, the radius rg of the droplet without feeding, if the problem can be assumed in quasi-steady

state all along the combustion period, would decrease following the law (PL is the density of the

iiquid) ,

.d . r ) = _ fi

dt g 47tr

P L

The solution of the problem is not very difficult if we use the simplifying assumptions

Low velocity and constant pressure around the droplet,

Uniform temperature within the droplet, equal to the surface temperature.

(lI


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- C

p

eonstant in the gas, equal diffusivity for fuel, heat, and oxidizer, sueh that pD=est,

-a single irreversible reaetion K + vOx ~ P takes place with infinite rate, i.e. fuel and oxidizer

eannot coexist,

a) Let us define now

Y

Ox

'

YK

T, the mass fractions of oxidiser and fuel, and temperature,

to be functions of the radius r around the droplet (see fig.1). vis the radial velocity,

p

the local

density in the gas, w the molar reaction rate, the M's the molar masses. The basic balance equations

are :

2)

Because we assume a single reaction, Z= v~x - ~~ do satisfy an equation withouj reaction

termo Because the reaction is very fast, with no coexistence of fuel and oxidizer, Z represents v~

if positive, and - ~~ if negative. The equation for Zis:

--(4m

2

pvZ) =--(41tpD r

2g

dr dr dr

i.e.iwith m

=

4m

2

pv;

r l.Z -

4m

2pD . d Z .

= Cte

.. dr

3)

Y

Equally, one see that Zr = ~/ .2..QL do satisfi' the same equation as Z :

q

c

p

vM

x

. rhZT -

4m

2

pD dZT = Cte

dr

( 3 )

b)

The boundary eonditions are needed now :

- At infinity from the droplet, we know Y

o

Y

K

T.

- At the droplet surface, the

conditions

for fuel and temperature do not deal with

T g

and

YK,g ,

but

with the mass and heat fluxes. Due to the infinitely fast reactions, Yox can be assurned zero at the

surface.

The heat flux vaporizes the liquid only, because the droplet is at uniform temperature ; so :

41tp 02 d t ) = + rilLv (4)

g dr g c

p


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Lv is the Iatent heat of vaporization, by unit of mass.

On the other hand, the global flux of mass for fuel is nothing but the total mass flux evaporated, and

also the total mass flux fed at the center, fi :

fiYKg - 41tp Dr~dd

YK

)

=

fi

/ r

g

That allows to fix the constants in (3) and (3') :

fiZ -

47tr

2

p D dZ - = _ fi

dr M

K

(5)

6)

.. T

. Z 4

2

D dZT .

g

fiLv

m

T-

nr P --=-m- ---

dr q/c

p

q

(6')

The integration of (6) and (6') gives Z and ~ as function of r:

r h

+

_ 1 _ ) =

4nr

pD

d Z

MK

dr

i.e.

dZ _

-liLd r.

. ( z +~ ) - 41tpD r

2

'

and

1 . ( Y O X . 1 ) - . . . )

=- - --

-+~

exp _ m ...

MK vMo~ MK. .

41tpDr

. (7)

EquaIIy, we get:

Zr

=

T

g

_

: :?+IT - T

g

- + Y o x .o o + Lv) exp (_ fi ) 7 )

.q/c

p

9 \

q/c

p

vM o

x

q 41tpDr

c) We ~a enow tocalculat ..:Y:Ki' fi,' arrd T

g

. The equilibrium forevaporation, a the

surface, allWsto write that thevapor partial pressure isa function of the temperature:

.... .. Y

~

Ps(Tg)=p Y ~~Y )

8)

--. S4.

K.g

MK M p

This additional boundary condition suffices for the complete solution. Indeed, it is possible to

express fi as function of Y

K.g

or as function of T

g-

In the first case, we use (7) at r

=

r

g :

(9)

=


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In the second case, we use (7') at the surface :

5

9 )

Then, it is possible to calculate Tg and YK,gindependently of rn by (8) and the result of equating (9)

and 9 ).If we define B

=

(c

p

(T_ - T

g )

+ : ~ : : )Lv ,called t~e Spalding's transfer.parameter ,

, . . . . . .4 0 : ~ .

we obtain:

YKg=JL

l+B

/

(10)

That gives a value smaller than unity (of course ), but only slight1y because B is usually of the order

of 10 to 20. Then, Tg' which is calculated from Y1(,gby (8), is only slight1y smaller than lhe boiling

temperature at the considered pressure.

d) From (7) and (7 ) one can plot YoX


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T T Lv Y q (LV q ) YOx,oo

T

-T Lv 00 g + C;+ Ox,oocpvM

ox

_ T

oo

+ T

g

- C; +c; ~

F - g -

C;+ . YOxooMK - YOx~K

(12)

1+

1+ 'v

x

vMox

The quasi-steady theory for the vaporizing droplet

The theory is easily transposed for the case of a vaporzing droplet, without combustion.

.

.

There isno need of the functions Z and ZT, and the solution for Z can be direct1y applied to YKand

T.

The results are given simply with a new transfer parameter

Bvap =

(c

p

(Too - T

g) )

Lv .

Then the formula (9') can be used again, but with Bvap instead of B. The fuel at the surface is given

with:

YKoo+Bvap

YKg 1 B

+

vap

(13)

2.3 The d

2

-law and itsdeficiencies

' ',. Finally, from

1)

and (9)or (9'), it is possible to get the law for the reduction of the droplet

r

radius, provided we know T

g :

dr . P O

g

= - m

= _ ~ JLog

1+ B)

dt 41t~PLPL r

g

Of course, this holds also for avaporizingdroplet with Bvap instead of B.

As rg appears in the denominator, one see that the derivative of the square of the radius is a

constant. Calling d

g

the droplet diameter, one get the so-called Id

2

aw :

, Pg

d~= d~(t::cO) - Kt, with K = 8Dg --. Log (1+B)

; ,. P L

1 4

Thfigures 3a ando, due to

S.

(jk~jima and S. Kumagai.show the results ofexperiments. '

The linear decrease of d

g

2

is not toofar fromr~~lity. Indeed, it is not satisfied

at

initial times because

the droplet has to be heated until its temperature is approximately uniform. During this first heating

period, the decrease of the radius is lower, of course. The law is notverified, equaly, at the end of

dr

the life of the droplet, hecause the quasi-steady approximation is not verified when d: is toa large.

Nevertheless, the agreement with the experiments is only qualitative. The numercal values

from

(10)-(12)

are not corresponding with the experimental ones, because pD and c

p

are not really

constant and do vary strongly around the droplet. Similarly, the ratio

rF/r

g is a constant in the

framework of the quasi-steady theory but do vary significantly in practice, as shown on the figure.

The influence of heating, of unsteadiness, changes of composition and temperature around

the droplet, have been studied, and approximate solutions for taking them into account have been

proposed. One can refer for that to the review of W.Sirignano (5).


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When the droplet is moving with respect to the surrounding air, the spherical symmetry is

broken and the theory is no more valido However, the theory can be improved provided the velocity

is low. The result is that the d2 law is always valid, but with a corrected coefficient :

d~ = d~(t = O) - K't

with K'

=

K

1 +

0;3 Re~12pr1l3)

(15)

Experiments have shown that this correction is useful until Reg of the order of 100,

unexpectedly. Reg is the droplet Reynolds number, and Pr the Prandtl number:

R

_ pgvgrg P _ IlgCpg

e

g

- r---. - ,

Ilg , ~g

- .

Nevertheless, even if the law is valid, it representsnly the integral of the vaporizing rate

along the surface of the droplet ; of course, the local vaporizing rate varies from the leading stagnation

point to the trailing point. Equally, it is clear that the flame is more or less elliptical around the

droplet, and there is the possibility of flame extinction, locally at the leading point or globally, ifthe

velocity is toa large. This occurs when the chemical reactions cannot be assumed infinitely fast,

probably when a Darnkhler nurnber, dg/v

g te

(where

te

is a chernical characteristic time scale) is toa

low. This situation has notrea11y been studied, to rny knowledge.

3 STRUCTURES IN SPRA Y COMBUSTION

3.1 Introduction

We will consider first the case of a very large, statistically hornogeneous spray where fuel

droplets are dispersed in an oxidizing gas. Thesizesof the droplets are notnecessarily identical, but

randomlydispersed around a mean value. The positions of the droplets are not regularly fixed, but

are randornly hornogeneously distributed.Wecall this case a premixed

spray ,

but this.premixing

appears only at large scale, of course.

The second interesting situation deals with the case of a flame around a jet of fuel droplets

that is discharged into a gaseous oxidant, atrest or coflowing. We call this case ali spray jet-flame ,

and we will consider sec.3.4.

3.2 Larninar, prernixed spray-Ilames.

We consider first the sirnplest case: the rnediurn is at rest, with no turbulent stirring, the

fuel droplets are dispersed randornly but in a statistically hornogeneous manner in an oxidant

atrnosphere, and a flarne zone, ignited sornewhere ona plane, propagates through this medium. What

is the fine scale structureof this flame zone ? This can be discussed by considering two non

dimensional pararneters : dg/eL, the ratio of the (mean) initial droplet diarneter to the characteristic

thickness of a laminar gaseous flame produced with the vaporized droplets, and n1l3 IF. where n is

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