Sheath combustion of sprays

Twentieth Symposium (International) on Combustion/The Combustion Institute, 1984/pp. 1789-1798

SHEATH COMBUSTION OF SPRAYS M. SICHEL AND S. PALANISWAMY

Department of Aerospace Engineering The University of Michigan Ann Arbor, Michigan 48109-2140 Most fuel sprays do not burn or evaporate as individual droplets but as a group. Based on previous studies it is shown that the transition from single droplet to group combustion depends on a single parameter 9 In the limiting case 9 < < 1, which corresponds to most spray parameters of practical interest, evaporation of the fuel occurs in a thin sheath or vaporization wave at the edge of the cloud and combustion occurs at a diffusion flame outside the cloud. In this case 9 is the ratio of vaporization wave thickness to cloud radius. For this regime, known as sheath combustion, it is shown that the cloud lifetime and cloud ignition and extinction are simply related to analogous results for individual droplet burning.

Introduction

ing occurs when a group combustion number

It is by now well established that most fuel sprays do not burn, or evaporate as individual droplets but as a group 1'2'3 or in a mode commonly known as "Group Combustion." In the limiting case evaporation of the fuel occurs in a thin sheath or vaporization wave at the edge of the cloud and combustion then occurs at a diffusion flame outside the cloud, so that the cloud behaves very much like a large single fuel droplet, however, with modified boundary conditions. This mode of combustion, which Chiu and Liu 1 have called sheath combustion, forms the subject of this paper. The important problems are to establish a criterion for the onset of group combustion, to determine the total cloud burning time, and to find the external conditions for cloud ignition and extinction. It is shown below that previous analyses of several authors lead to similar criteria for the onset of group combustion and that most sprays of practical interest fall in the sheath combustion regime. The most significant result of sheath combustion analysis is that cloud life and ignition and extinction criteria can be simply related to analogous results for single droplet burning.

G = (1 + 0.276 Rel/2Sc I/3) 4~r h d l~c2Le < < 1 (1) while group combustion is characterized by G > > 1. Symbols used throughout are defined in the Nomenclature. Labowsky and Rosner 2 indicate that group combustion will occur when the Thiele Modulus

tb = (3dp)l/2(Rc/a) = (4~r fi h)l/2Rc > ~J*

where ~* is a critical value found to have a value of 0.26. At the other extreme, Correa and Sichel3 show that sheath, and hence group combustion, will always occur when a parameter e = (47r h a l~cZ)-1 < < 1

(3)

These three parameters, although derived differently, are closely related. The parameter e, which is governing in the case of sheath combustion arises when the conservation equations are expressed in suitable dimensionless form as described in detail by Correa and Sichel. 3'4 The procedure can be illustrated using the vapor mass conservation equation within the cloud:

O0

Range of Applicability

(2)

1 0

+ ~ r r (~0~) dt

There are several regions of group combustion-one of the most important of which is that of incipient group combustion in which the transition from individual droplet to group combustion occurs. Several criteria for establishing the onset of group combustion have been proposed. Chiu and coworkers 1'5 suggest that individual droplet bum-

=h

(~v In

1+

~

(4)

The generation of vapor by the individual droplets has been replaced by a continuous mass source distribution based on the single droplet evaporation

1789

1790

SPRAY COMBUSTION

law. The equations are made dimensionless using I~, a, the cloud thermal diffusion time Rc2/k,. and the convective velocity kr/Rc. The dimensionless equation is then

0t

~ ~r (~1513) = a In (1 + 0)

(5)

In the asymptotic limit 9 --->0 conditions in the cloud interior are uniform without vaporization. Evaporation occurs in a thin sheath or vaporization wave at the edge of the cloud and this is precisely the sheath limit. Suuki and Chiu 5 started from a series solution of the same equations, and found that the radius of the diffusion flame, which lies outside the cloud during group combustion, decreases with decreasing G until at the critical value Gc = 0.152, the flame just touches the cloud surface. This condition is taken as the division between external group combustion and single droplet burning, at least at the edge of the cloud. Labowsky and Rosner2 used both continuum and discrete particle analysis. The continuum approach is based on a solution of the steady state diffusion equation for the oxidizer with the effect of droplet burning replaced by a continuous mass sink. With increasing values of the Thiele modulus r the oxygen concentration at the center of the cloud decreases until at a critical value t~ = $* it drops to zero. This condition is then taken to signal incipient group combustion. In the second approach the droplet flame locations in a particle array of finite size are calculated by reducing the Shwab-Zeldovitch form of the mass conservation equations to the Laplace Equation and then using the method of superposition. For an array with a given total number of particles incipient group combustion was taken to correspond to the particle center to center distance for which the flames at the center of the array just touch. The center to center distance for which the particles at the corners of the array just touch their neighbors is called the total group combustion point and is similar to what Chiu and Liu 1 term the onset of external group combustion. At incipient group combustion, these calculations lead to the relation

and using K = 1.707 • 10-3 as determined by Labowsky/Rosner2 $* ~ 0.26 which, remarkably, is identical to the results from the continuum theory. Both the group combustion number G and the Thiele Modulus t~ can be expressed in terms of 9 so that with Le = 1, and Re = 0 G = 9

iI/ _ 9

(S)

The parameters derived from these analyses of group combustion are thus equivalent. At incipient group combustion for G = G~, 9 = 6.6 while the criterion t~* = 0.26 leads to 9 = 14.5. Sheath combustion requires 9 < < 1. These results are summarized in Table I below. The parameter 9 can be assigned several closely related physical interpretations. Thus, Correa and Sichel3 show that 9 = (~/~)2 where/~ is the thickness of the vaporization wave at the edge of the cloud. Chiu and Liu I indicate that 9 = G -1 represents the ratio of the heat transfer between the gaseous and liquid phase to the heat transfer in the gas phase, while Labowsky and Rosner~ interpret 9 = r as the ratio of a characteristic reaction time to a characteristic diffusion time or a Damkohler Number. The relevance of these results to practical sprays becomes apparent when 9 is expressed in terms of the ratio C of droplet separation distance to droplet radius for then, 9 = (1/4~) c 3 (~/~<)2

(9)

Figure 1 shows the variation of 9 with l~c/d for constant values of C. Values of 9 at incipient group combustion are also indicated. For a typical fuel cloud with C = 10, ~ = 100 I~ and Rc = 10 cm, it follows that e < 10 -4 and even for a more dilute cloud with C = 100, 9 < < 1. The most significant conclusion from Fig. 1 is that most clouds of practical interest fall into the sheath combustion regime with 9 < < 1. Clouds falling in the incipient group combustion regime are extremely dilute so that a droplet array analysis like that of Labowsky and Rosner 2 may be more appropriate than the continuum analysis of Befs. 1 and 3. Cloud Burning Time

N

4 -n71"-Ptc 3 = K

C3/2

3

(6)

between the total number of droplets N in the cloud and the ratio c = L/a of particle separation to cloud radius with K a constant. In terms of the Thiele Modulus (6) leads to the relation I1/* = (614 V~) 1/3

(7)

Most fuel droplet clouds of practical interest will, as shown above, fall into the sheath combustion regime. The asymptotic analysis of Correa and Sichel3'4 then shows that the expression for cloud burning time is identical to that for single droplets if the droplet radius ~ is replaced by the cloud radius Rc and the liquid density 15e is replaced by Of, the liquid mass per unit spray volume. The cloud thus satisfies a D 2 law given by

SHEATH COMBUSTION OF SPRAYS

1791

TABLE I Group combustion parameters

Reference

Labowsky and Rosner [2]

Chiu, et al [1],[5],[14]

Correa and Sichel [3],[4]

Parameter Name

Thiele Modulus

Group Comb. No.

Sheath Comb. Parameter

Symbol

G

Definition

(34,)'/~/~oa

4~rahl~2Le,

(4~ral~2)- l

In terms of e

1/V"e

I/e**

e

Values at Inc. Group Combustion

0.26

0.152

Corresp. e

14.5

6.6

*For Re = 0 **For Le = 1.0

(fVf~D ~ =

(1 -

7/'~c)

with the cloud life time "?c given by

(11)

~c = (flcla) 2 (•df><) ~d

in terms of the lifetime § of a single fuel droplet. Equation (11) can also be expressed in the form

which shows that Zrc/;:
Labowsky- Rosner

10-

1,0

10 - 1

to-<

10 -3,

110

i lO3

ll0Z R

e

, 1~

/ a

FIG. 1. Group combustion parameters.

'2 10

'3 10

'4 10 R

c

/

J 10 5

a

Fie. 2. Relation between #c/ga, R~/?i, e, and C.

SPRAY COMBUSTION

1792

variation of %ffra with f~c/?~ for different values of the droplet separation ratio C. The regression velocity of the vaporization wave at the cloud surface is another important parameter associated with sheath combustion and its initial minimum value lJ = I3,, corresponding to lq = 1~ can be expressed in the form (~l.~)

= [~333

-1 = 3 Re -

(13)

t~ where 15a = (~/2~'a) is the minimum surface regression velocity of a single burning droplet. Equation (13) is represented graphically in Fig. 3.

Cloud Ignition and Extinction Given that a droplet is suddenly exposed to a high temperature environment, the basic problem is to establish the minimum temperature or ignition temperature at which the droplet will ignite. The converse, i.e., determining the ambient temperature at which a burning droplet is extinguished, constitutes the extinction problem. A closely related although somewhat different problem is establishing the delay time between exposure to a hot environment and ignition. Single drop ignition and extinction have been discussed in the recent reviews by Williams, 13a5 and Law1~ while a recent experimental and theoretical study of single drop ignition delay has been presented by Sangiovanni and Kesten. 17 Comparable studies for sprays are limited although the effect of adjacent droplets on ignition delay has been considered by Sangiovanni

104 .

103 .

\

,/

,

d/dr(pvr z) = 0

//

i

(14)

and

ov dr (r

- 7

= -DpZ~'rYo exp (-Ta/T)

i = 1,2,3

(15)

with ~b1 = 'Yf, ~b2 = Yo and ~3 = T. The temperature has been non-dimensionalized using hHc/gp where ~'-gc is the heat of combustion per unit mass of fuel. The Damkohler number D appearing on the right hand side is the ratio of a characteristic diffusion time based on initial cloud radius Rc to a characteristic chemical time

D = (Rc2/k~)/(@o/gaf),v)

:/\/

(16)

where 13 is the frequency factor. It has been assumed that the Lewis number (I3/I~r) is unity and that ~I3 is constant. Correa and Sichel 3'4 have formulated the first order sheath combustion boundary conditions at the cloud edge r = 1~/I~ = ~, in terms of the cloud edge temperature 0e0 or the dimensionless mass flux fit as follows

1~"~ 10Z

10

and Kesten is while Ballal and Lefebvre 19 have studied minimum ignition energies and quenching distances for sprays. Again, this latter problem differs from the basic ignition problem above. It is shown below that the ignition and extinction temperatures of a monodisperse, quiescent spray cloud can in the sheath combustion mode, be related to the corresponding analysis for single droplets. The cloud is assumed to be in a quiescent atmosphere so that there are no convective effects and the pressure is assumed to be uniform. Only the continuity, species, and energy conservation equations therefore need to be considered. Correa and Sichel3'4 have shown that the quasi steady analysis is valid in the sheath combustion limit, since the characteristic time based on the evaporation wave velocity is then much larger than the characteristic gas phase diffusion time. The chemical reaction is assumed to be single step, irreversible, first order in both fuel F and oxidizer O so that aF + bO --~ pP, and governed by Arrhenius kinetics. The governing equations for mass species, and energy conservation in the Shvab-Zeldovich form are then:

I

l~

1; 2

1'o3

1'o'

,'oS

Re/ a

FIG. 3. Relation between lJ~/15a, R~/~, C and e.

d-~

= -- - - ~ 0e0 (1 --

fit

(17) r=~

1793

SHEATH COMBUSTION OF SPRAYS

d~o r=~ "~F

=--~0"o~o~==~111Yo r=~

(18)

and the mass fractions of fuel and oxidizer at the cloud edge r = ~ will be r

d~ r=~

V~

(20)

The Clausium-Clapeyron relation between the surface fuel concentration and temperature is replaced by the condition T = T~

(21)

although this relation is used in determining the cloud equilibrium temperature T~. Far from the droplet cloud as r ~ oo T = W| r

= 0, Yo = Yo,|

(22)

From Eq. (15), it is now readily shown that ~F+T=T|162 exp (rh/~j)[exp(-rh/r) - 1] (23) r + T = }'o,| + T= + [I? + 'i'or] exp (fn/~)[exp(-in/r) - 1 ]

~[or

(19)

since

pVl'21r=~ = /h = (~/'5/3) 06e~2

= {1 - exp (-rho/~)}

(24)

where ~'f~ and Yo~ are the fuel and modified oxidizer mass fractions at the cloud edge r = ~. This completes the basic formulation. Two distinct reaction regimes associated with the energy equation must now be considered in detail.

=

Yo,~ exp (-rho/~)

(27)

The fuel and oxidizer mass fractions within the cloud remain constant at g{fr and Yo~. For large activation energy, Ta > > 1, following Lifian, 6 it is possible to define a small parameter el = T2JTa < < 1, and the energy equation (15) is rewritten in terms of a modified Damkohler No. D~ as r dT

r2dr

1 d (rzdT)

r2 dr

~rr = D~pZYFY~

(2s) where D| = D exp (-Ta/T~). In the large activation energy limit, 1/Ta -'* 0, it is obvious from (28) that for r - 0(1), near the cloud, where T < T| chemical reactions will be frozen to all orders. This is no longer true for large r, when T--* T~. A uniform perturbation of the energy equation about the frozen flow solution, therefore, fails and the singular perturbation method developed by Fendell; and Law s has been used. FollOwing Lifian9 a perturbed mass flow is defined as: ~t = ~l0 4- E.lrh1 4- (21 rh2 4- ...

(29)

In the inner region, r - 0(1), Tin, = T ~ - r + e

1. Near Ignition Regime

exp [rho + liffhi)(i/~ - l/r)]

When the Damkohler number D < < 1, the source term in the energy equation can, in the first approximation, be neglected corresponding to pure evaporation, so that there is then a diffusive-convective balance in the gas phase surrounding the cloud. The energy equation (15) is then satisfied by T = T~ - ~ + (T~ - Tr + e)e-e"o/r

(25)

(30)

And in the outer region r ~ o% where the reaction term is of O(e) Tou t = W~ + lil201('q) 4- E.21 n2(~) + . . .

(31)

where the outer variable -q = elr. Following Matkowsky and Reiss,10 it is necessary to introduce the variable A and expand it as

In the case of pure evaporation rh and 0o~ are given by

A = Ao + elA1 + ...

(32)

where rho = / j in T |

e e

h = D~p~2Yo.~E1-2

(26)

so that the first order outer energy equation will be:

SPRAY COMBUSTION

1794

1 d f.qzdOl~ [~ d~q J = Ao

~q2 d'q I

]

+ 01 exp (01)

This equation and the boundary conditions obtained by matching Tinn and Tout are to order ~I i, where 0 < i < 1, identical to those of Lifian. 9 Matching to the next order, O(~1) yields the first correction to the mass flow rate due to chemical reaction in the far field r ---> ~. It was found by Lifian9 that for a < 0, multiple solutions satisfying the above equation and boundary conditions existed, for all values of the reduced Damkohler number Ao less than a critical value Aoi, while there were no solutions for Ao > Aoi. This result implies branching of solutions at Ao = Aoi and, thus, defines an ignition Damkohler number. For a -> 0, unique solutions were found for all Ao. As shown later, a < 0, implies that the far field temperature T= is less than the equilibrium flame temperature Tye. Lifian9 found that for a < 0, Aoi could be related to a and Ix by AOI1/2 = (1.531 - a)/Ixa z = N/lxa 2

Further, for ~ -< r <

(37)

rfe

T = Tr - ~ + (T. - Tr + e + Yo,-) exp

(-rh/r)

(38)

and the mass fraction of fuel at the cloud edge will be given by r

= [1 + to.,) exp (-rh/~)]

Outside the flame front for r > T=(T~-e+

(39)

rf

1) + ( T ~ - Tr + e -

1) e x p ( - r h / r )

(40)

The flame temperature and radius are obtained by equating (38) and (40) at r = rfe, so that

Tie= T. +

(34)

and for large negative values Of r Aol1/2 = (1.531 - 0.998a)/Ixa 2 = N/Ixa 2

Ooe = ln {T" - Tr + ~ + t~

(33)

~o=(1

- T= + T~ -

2)

(1 + to,=)

(41)

rs = rh/ln (1 + to,:)

(42)

and (35)

Equations (26), (34) and (35) together with the definition of Ix reveal that the ignition Damkohler number is proportional to 1/~ ~ = (I~/I~) -2 so that the ignition Damkohler number increases with decreasing cloud radius.

In order to study the structure of this flame and thereby the extinction characteristics, it is convenient to us." the inner variables = 8o n

r

- rfe) 2~2rZye

(43)

2. Near Equilibrium Regime In the equilibrium limit, l i D --> 0, there will be a thin diffusion flame away from the fuel cloud with no leakage of reactants past the flame front. When the Damkohler number D is not so large, it is to be expected that the flame front will be thicker with leakage of one or both the reactants on either side of the flame. To study the extinction characteristics of the fuel cloud, it is necessary to perturb the temperature distribution about the equilibrium limit. Following Lifian6 ~2 = Tfe21Ta < < 1 will be chosen as the small parameter. In the equilibrium limit, with a flame front located at r = rye to zeroth order in ~2, there will be no oxidizer for ~ -< r < rfe and no fuel for rye -< r < ~. From equations (23) and (24), the mass flux will then be rh= ~ln{T=-Tr

+"~ ~ + Yoa}

(36)

In the flame region the temperature is then expanded as: Tinn = T f e - 80m(~2131 + E2~"q + ~22~2 + ...)

With m = - n = 1/3 and with 8o a reduced Damkohler number expanded a s 8 = 80 + E281 + , . .

= 4D exp

(-Ta/Tfe)pfe2~zarfe4/m 2 (45)

and ~/= - 1 - [2(T= - Tr + e - 1)/(1 + to=)]

(46)

The first order inner energy equation can then be written as

dZf31 d .qz

and from Eq. (20) the cloud edge temperature is

(44)

= (131 + n)(131 -

n) exp [-8o-1/3(13i + ~/'q)] (47)

SHEATH COMBUSTION OF SPRAYS The boundary conditions are obtained by matching with the outer expansions r<

rye

T = Tr

--

e + (T= - - T~ + f + ~'o,=)exp (,m/r) - [exp(-rh/r)- exp(-rh/~)] 9 (e,zB1 + eeZBz + ...)

(48)

r > rfe

T = Tr + e + 1 + (T= - T~ + e - 1) exp -

(-rh/r)

[1 - exp (-rh/r)](e2A1 + e22A2 + ...)

(49)

The inner variable "q, Tinn, ~, and the outer expansions were chosen to reduce the energy equation and boundarv conditions to those obtained by Lifian. 6 Here (1 ~- "y) is the ratio of twice the heat lost to the oxidizer side of the flame to the total heat lost from the flame. Lifian6 found that the reduced equation had multiple solutions satisfying the boundary conditions for 50 > 50E a critical value and no solutions for 50 < 50E, when I~/I < 1, thus defining an extinction Damkohler number 50z. When > 1, the flame begins to receive heat from either the far field or the fuel side and therefore no extinction characteristics could be found. As found by Lifian, 6 and also numerically verified here, for ]~/] <1. 5oE ~ e[(1 -I~1) - (1 - I 1) 2 + 0.26(1 -I~1) 3 + 0.055(1 - 1~1)']

(50)

3. Results

1795

The ignition and extinction Damkohler numbers are proportional to 1/~ 2, where ~ = R/Re, and therefore increase as the cloud shrinks. The cloud Damkohler Number D (Eq. (16)) is fixed for a given cloud radius and pre-exponential factor I], and since the conditions for ignition and extinction are D > Dig, and D < Dext it is possible that a burning cloud may extinguish as Dext increases with decreasing cloud radius. Such behavior is suggested by the unsteady asymptotic analysis of single drop burning by Waldman ll and the numerical analysis of Saitoh and Nagano 12 who observed that the ratio of flame to droplet radius ~'re/abecame unbounded as t ~ 0% implying flame extinction with decreasing droplet radius. It is important to note again that the analysis would be identical to that for a single fuel droplet if the reference length scale were changed from the cloud radius Ptc to the droplet radius a. Sample calculations were carried out for an octane (CsHls) fuel cloud, with properties given in Table II. Figure 5 gives the ignition and extinction Damkohler numbers as a function of the ambient temperature T= for different values of ~. There is no experimental data for the type of fuel cloud considered here; however, with the close analogy between single droplet and cloud ignition demonstrated here, it is possible to estimate cloud ignition temperatures from single droplet ignition data. Thus Williams 15 shows a measured ignition temperature of 1023 K for No. 2 fuel droplets with a = 100 I~m. Considering octanes as a typical hydrocarbon Fig. 5 then suggests a single droplet Damkohler No. D - 102. For the fuel cloud with l~c = 0.1 m, the Damkohler No. will be 106 times the droplet value,

The above analysis has made it possible to predict the ignition and extinction of a fuel cloud. From Eqs. (32) and (33), it follows that the Damkohler number for ignition is given by: R/R =1.0 Dig R/R =0.5

N2 exp (Ta/T~)ez2 o~4(T~- Tr + e)2~2 In {T=

-T,+e} s ~ .-A

(51) The Damkohter number for extinction is given by DExt --

5oE exp

(TalTfe)[In(1 +

4e23pf~2ffe4~2IlnlT=-

r

T~ + ~? +

4 o,=

e

~400.00

(52) where 5OE is given by Eq. (50).

i 500.00

i 600,00

i 700.00

9 i 800.00

I 900.00

ANBIENT TEMPERATURE T (K)

FIG. 4. Variation of ignition and extinction Damkohler numbers with ambient temperature.

SPRAY COMBUSTION

1796

TABLE II Data for ignition and extinction studies Fuel Liquid Density Drop Number Density Initial Cloud Radius Initial Drop Radius Boiling Temperature Fuel Molecular Weight Oxidizer Molecular Weight Specific Heat Thermal Conductivity Vaporization Latent Heat Heat of Combustion Ambient Oxidizer MassFraction Ambient Pressure Stoichiometric Coefficient Fuel Stoichiometric Coefficient Oxidizer Activation Temperature

Octane Pe

707

kg/m 3

n

1 • 109

/m 3

~

0.1

m

~

1 •

10 -4

m

Tb

398.7

~

~Tv" s

114.2

kg/mol

V~/o (~p

32 0.5

kg/mol kcal/kg-k

h

0.5

kcal/m-s-k

~

71.7

kcal/kg

AH,

1.07 • 104

kcal/kg

Yo.~

0.23

~

1.0

a

1.0

b

12.5

"i~o

12,200

atm

OK

i.e., D - 10s. Figure 5 then indicates a cloud ignition temperature of 460 K. A fuel cloud therefore ignites at a much lower ambient temperature than one of the individual droplets, a result which is consistent with the observation 1~ that single droplet ignition temperature decreases with increasing droplet diameter.

4. Discussion It has been shown that the Thiele modulus, ~, group combustion number G, and sheath combustion parameter, ~, which have been used to characterize spray combustions are all equivalent and expressible in terms of any one of them, e.g., e. When t < < 1 combustion or vaporization occurs in the sheath combustion mode and then it is shown

that the cloud burning or evaporation time can be simply calculated once the single droplet burning characteristics are known. Similarly, it is shown that the analysis of cloud ignition and extinction is the same as that for single droplet ignition if the droplet radius is replaced by the cloud radius as the characteristic length. Single droplet ignition data therefore can be used to estimate cloud ignition temperatures. The ignition and extinction analysis is based on steady state burning rather than on a transient analysis of cloud heat up during an induction period in which the ignition temperature would correspond to that for which the ignition delay time t~ --o ~. In the present analysis, which parallels that used for single droplets, the quasi-steady assumption reduces the problem to a system of coupled ordinary differential equations with multiple solutions. The analysis is nevertheless unsteady in the sense that the stability properties of the governing equations are used in discarding the unstable solutions and in locating the ignition and extinction Damkohler numbers. A stationary monodisperse cloud is not really a fuel spray, but the results may, nevertheless, prove useful for the analysis of real sprays if these are treated as a collection of smaller cloudlets. The effect of local cloudlet conditions could then be introduced through the expression for single droplet burning times. The results presented here should be treated with caution since many important effects have been neglected. Some of the more important ones are the effect of convection on motion outside and within the cloud, the effect of the actual particle size distribution vs. a monodisperse cloud, the effect of nonuniform fuel distribution within the cloud, and the effect of turbulence to name just a few. So far, there is no direct correlation between the theoretical results presented here and experiment.

Nomenclature

a b C C_n D D F G Le r~ h

Stoichiometric coefficient of fuel Droplet radius Stoichiometric coefficient of oxidizer Frequency factor in Arrhenius kinetics Separation--radius ratio Constant pressure specific heat Mass diffusion coefficient Damkohler number Fuel Group combustion number Gas phase thermal diffusivity I~ = h/15(~p Latent heat vaporization ~ = ~/cA--Hc Lewis number Mass flux rh = r~/(47rOrKrRc) Droplet number density

SHEATH COMBUSTION OF SPRAYS O P p_ R

Oxidizer Products Number of moles of products Instantaneous cloud radius Initial cloud radius r ~/1~ T Temperature T = "i'cp/CA--Hc 0 Vaporization wave velocity V Velocity V = V / ( I ( r / ~ ) 9r Fuel molecular weight Vro Oxidizer molecular weight r Fuel mass fraction ~o Oxidizer mass fraction Yo = ~[o/v a (T| - Tr + e - 1)/T| - T r + e) A0I Reduced ignition Damkohler number ~0E Reduced extinction Damkohler number ]~'-Hc Heat of combustion per unit mass of fuel 9 (4"r n a ~)tc2)-1 91 T| 92 Tfe2/Ta 00e Cloud edge temperature )~ Gas phase thermal conductivity ~1, (T| - Tr "~- e ) ~ v bWo/a~VF p % ~'d d~ *

Density O/(Jr Cloud burning time Drop burning time Volume fraction Thiele modulus

Subscripts fe

Flame property Liquid property r Cloud reference condition 0o Ambient condition 0,1, 2 Solution order REFERENCES 1. Cmv, H. H. AND LIU, T. M.: Comb. Sci. and Tech., 17, 127-142, (1977)

1797

2. LABOWSKY,M. AND ROSNER, D. E., Advances in Chemistry Series, No. 166, Amer. Chem. Soc. (1978). 3. COItaEA, S. M. AND SICHEL, M: Nineteenth Symp. (International) on Combustion, pp. 981-991, The Combustion Institute, 1982. 4. COaaEA, S. M. AND SICHEL, M: Comb. Sci. Teeh., 28, 121-130(1982). 5. SUZUKI, J. AND CHIU, H. H.: Proc. of Ninth (International) Symposium on Space and Tech. and Sci., 145-154 (1971). 6. LI/qAN, A.: Astronautica Acta, 1, 1007-1039 (1974). 7. FENDELL, F.E.: Astronautica Acta, 11, 418-422

(1965). 8. LAW, C. K.: Combust. Flame, 24, 89-98 (1975). 9. LIlqAN, A.: Acta Astronautica, 2, 1009-1029 (1975). 10. MATKOWSKY,B. J. AND Rmss, R. L.: SIAM Journal of Applied Mathematics, 33, 230-255, (1977). 11. WALDMAN, C. H.: Fifteenth (International) Symposium on Combustion, The Combustion Institute, p. 429, 1975. 12. SAITOH, T. AND NAGANO, O.: Comb. Sci. and Tech.. 22, 227-234 (1980). 13. WILLIAMS,A.: Oxidation Comb. Rev. 3, 1-45, (1968). 14. Cmv, H. H., KIM, H. Y., AND CaOKE, E. J.: Nineteenth Symposium (International) on Combustion, pp. 971-980, The Combustion Institute, 1982. 15. WILLIAMS,A.: Comb. Flame, 21, 1-31, (1973). 16. LAW, C. K.: Prog. in Energy and Comb. Sci., 8, No. 3, 169-199, (1982). 17. SANCIOVANNI,J. j., AND KESTEN, A. S.: Comb. Sci. and Tech. 16, 59-70, (1977). 18. SANCIOVANNI,J. J. AND KESTEN, A. S.: Sixteenth Symposium (International) on Combustion, pp. 577-592, The Combustion Institute, (1977). 19. BALLAL,D. R., AND LEFEBVEE, A. A.: Proc. R. Soc. London, A. 364, 277-294, (1978).

COMMENTS S. Aggarwal, Carnegie Mellon University, USA. Your steady-state analysis for a droplet group appears to be exactly similar to that for a single droplet; only difference being due to the different sizes involved. Another very important difference between the cloud and a single droplet would be the number density effect. The droplet group results should be plotted to show the influence of number density explicitly.

Authors" reply. The results presented here have been plotted with the separation ratio C as a parameter, since this seems to provide a good characterization of cloud density. By using the relation C = n -1/3 ~-: these results can also be expressed with n and ~ as the significant parameters. Thus Eq. (9) for e becomes

1798

SPRAY COMBUSTION = (rtola)14

~

n ~3

so that t varies inversely with the total n u m b e r of droplets in the cloud. The ratio of cloud to droplet life time "~c/ia is very strongly influenced by n since it follows from Eq. (12) that

lets. We thus agree that the interaction effects for single droplets could be applied to interacting clouds. In the case of turbulent clouds the size of the cloud packets might be related to a suitable turbulent length scale.

Ll~rd = (4~r/d) fi a 1~2 It is, however, of interest that neither the flame radius (Ref. 3 in the paper) or the ignition and extinction Damkohler Nos. as given by Eqs. (51) and (52) depend explicitly on the n u m b e r density n.

L. Kennedy, Ohio State Univ., USA. Your results were presented for one droplet size and you indicate that they behave essentially similar to individual droplets. If one had spatially separated cloud packets of different size droplets, do you believe that the interaction effects known for single droplets could also be extended to the interacting clouds described by your model? Authors" Reply. As I stated in the paper, while a spherical monodisperse cloud is not a spray, the results of the analysis may be applicable to real sprays if these are treated as a collection of cloud-

R. Priem, Amer. Gas Assoc. Labs, USA. W h e n you have a cloud of drops that is within the sheath or cloud limits for combustion, what is the stoichiometry of the cloud (including the liquid)? It would appear that they would be very rich, i.e. greater than 100 times stoiehiometric. Authors" Reply. D r o p l e t clouds in the s h e a t h combustion range, and probably in most practical sprays are indeed very rich. The octane cloud chosen as an example in our paper has an equivalence ratio of about 36 which is of the order of magnitude you suggested in your comment. When viewed in this way it is not surprising that individual droplet burning is probably restricted only to very dilute clouds, and that, as indicated in the paper, for most clouds the rate of oxidizer diffusion toward the cloud center is insufficient to support individual droplet burning. The analysis by Labowsky and Rosner (Ref. 27 in the paper) for droplet arrays provides a vivid illustration of this phenomenon.