Effect of multiple particle interactions on burning droplets

C O M B U S T I O N A N D F L A M E 57:237-245 (1984)

237

Effect of Multiple Particle Interactions on Burning Droplets M. MARBERRY, A. K. RAY, and K. LEUNG Department of Chemical Engineering, University of Kentucky, Lexington, Kentucky 40506-0046

The interactions between burning droplets are analyzed on the basis of the quasi-steady assumption. A generalized treatment for burning rates of droplets in an array has been developed using a modified Laplace equation with point sources. This treatment is applied to two droplets of different sizes, as well as finite arrays containing up to eight symmetrically arranged monodisperse droplets. Particle interactions are shown to be a function of particle size, number density, and geometry of the array. Results are presented in terms of correction factors from which multiple particle burning rates can be calculated from single particle burning rates. The correction factors are shown to be in close agreement with results available in the literature.

INTRODUCTION It is well known that interactions among the constituent droplets in spray combustion reduce particle evaporation rates as compared with the single droplet case. Thus, particle interactions must be understood in order to predict rate processes for multiple particle systems. The importance of these droplet-droplet interactions has not gone unrecognized by experimental investigators. Rex et al. [ 1], have experimentally shown that the interaction effects between two burning fuel droplets depend on the distance between particles. Similarly, Sangiovanni and Labowsky [2] have found that the burning time of monodisperse droplets in a linear array gradually increases with decreasing droplet spacing. Recently, Miyasaka and Law [3] experimentally investigated linear arrays of two and three droplets and concluded that droplet vaporization rate and droplet heating are significantly retarded due to the interaction effect. Several theoretical studies have also dealt with the effects of particle interaction on combustion of fuel droplets. However, most of this work is limited to only two interacting particles [4-7]. A Copyright © 1984 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue, New York, NY 10017

more rigorous model based on the method of images was developed by Labowsky [8, 9] in order to examine nearest neighbor interactions on the evaporation rate of droplets. Although this method is applicable to interacting droplet combustion [10], it is difficult to use and involves lengthy numerical calculations. Ray and Davis [11, 12] have presented a generalized treatment of heat and mass transport between an assemblage of particles and a surrounding continuum. This analysis is based on the solution of unsteady state diffusive transport equations with unknown point sources located at the centers of the particles. Integral equations matching the boundary conditions at the surfaces of the particles are used to describe these point sources. This formalism is extended in the present paper to the case of interacting droplet combustion. The theory is applied to burning droplets of different sizes as well as to monodisperse droplets in multiparticle arays. ASSUMPTIONS The subsequent analysis is restricted to systems involving no convective transport. That is, the

0010-2180/84/$03.00

238

M. MARBERRY ET AL.

fuel droplets are considered to be burning in a quiescent oxidizing atmosphere. The quasisteady hypothesis is also applied. Thus, the moving interfacial boundary is ignored by assuming that the rate of change of particle size is small compared with the diffusive velocity of the oxidizing species. In addition, several simplifying assumptions are made which also appear in the classical single droplet studies: 1. The Lewis number (Le = p D C p / k ) is equal to unity for chemical species. 2. The spherical shape of each droplet is maintained throughout its combustion. 3. Natural convection and radiative energy transfer are neglected. 4. The temperature is uniform in each droplet and local thermodynamic equilibrium always exists at the surface of each droplet. 5. Physical properties such as thermal conductivity and gas density are constant and all species are assumed to have the same heat capacity and diffusion coefficient. PROBLEM FORMULATION The previous assumptions allow the governing transport equations to be expressed in the Schvab-Zeldovich form [ 13]:

(Ovwo - p D V wo) = 0,

(3b)

T= T~;

(3c)

Far From the Droplets: WF= 0,

(4a)

Wo = Wo.=,

(4b)

T= T~.

(4c)

For steady flow, the overall continuity equation reduces to v . ~ v ) = 0.

For a potential flow, the mass flux can be expressed as pv=

V "(PVt3M--PD V~M) = O;

(1)

V~.

mi =

(or) "h dAi

£'

V . (pVI3T -- p D V~T) = 0;

(2)

where the coupling functions are defined as

~T--

q~

M,,v,,

(8)

g,S(x-xi)6(y-yi)6(z-zi)=O,

(9)

i=l

•

Movo

For a system of N burning droplets the above equations are subject to the following boundary conditions: On The Surface Ofith Droplet (i = 1, N): wv = WF.~,

V T . h dAi.

N

V 2cb+ ~

wo +

S

In order to circumvent the problem of nonuniformities, the following modified Laplace equation is introduced:

Wo

Cp(T- Tr)

(7)

A,

miL = k

MI:•F

Vd)'hdAi

and

Energy Conservation:

WF

(6)

These differential equations are not valid in the space occupied by the droplets. Thus, a series ofdiscontinuites exists in the concentration, temperature, and flow fields. Moreover, the burning rate of each droplet is given by material and energy balances at the droplet surface:

=

Mass Conservation:

(5)

(3a)

where g/is a point source of unknown strength located at the center (xi, Yi, zi) of the ith droplet and 6(x) is the Dirac delta function. Equation (9) satisfies continuity in the region of interest, and the source strength will be determined from conditions given by Eq. (7) and (8). Since the mass flux vanishes far from the

BURNING DROP INTERACTIONS

239

droplets, the potential ~b assumes a constant value at infinity. The general solution of Eq. (9) can then be written as

the previous equations can be rearranged to yield

d~T [

L

qr Mouo

P D - ~ l i,s

~=0.+

-

(10)

,

= CT exp(~i,JpD),

i=1 ~i

where ¢0. is a constant and ri = [(X -- Xi) 2 +

( y __ Yi)

2 + (Z -- Zi) 2] 1/2

Solutions of Eqs. (1) and (2) for the coupling functions ~3r and t3M can be expressed in terms of the potential 0:

~M ~ AM + BM exp(O/pD), ~T =

w....

(11) (12)

A T + BT e x p ( O / p D ) ,

where A and B are arbitrary constants. Equation (12) can be rewritten as

(16)

where ~bi.~is the value of ~bat the surface of the ith droplet. However, the potential at the surface of each droplet, ~b~,is identical since all droplets are assumed to be at the same temperature. Thus, one obtains ~bi,~= ~bs

(constant).

(17)

Evaluation of the coupling function at the surface of each droplet,/3T.~, yields ~T.s -- / ~ T . . = CT[exp(~bJpD) -

II

CpiL--T.) qr

/3T--/3T,°" = CT [ e x p ( ~ D ) - 1 ] ,

(13) +

(w,,.~- Wo.,)

(18)

where The solution for ~ obtained from Eqs. (16) and (18) is given by

CT __.BT exp( ~b~*"), oD/

Cd T . - TO

(~T.'--

O~=O.-pD

wo..

(¢3T.~- ~3T,.)

+--

M,,uo

qr

• ln

The remaining unknowns in Eq. (13) are obtained by satisfying the boundary conditions at the droplet surfaces. Assuming that the oxidizer is not soluble in the droplet, then its mass flux at the surface is zero and the following condition must be satisfied: dwo I

l

= Wo

~.

(14)

Also, Eqs. (7) and (8) can be combined to give k

=L.

(15)

l,s

If the Lewis number is assumed to be unity, then

I

]

( L / q r + wo.SMou,O

(19) "

Equation (19) provides a relation for the surface potential 0s, which must be satisfied by each of the N droplets in the array. At the surface of the ith droplet the potential ~s, obtained using Eq. (10), is (i = 1, N),

(20)

j_ ~ rij

where ri.i = ai is the radius of the ith droplet. In developing Eq. (20), it is assumed that the average distance of the surface of the ith droplet from the center of thejth droplet can be approximated by the center-to-center distance given by rij=[(xi-xj)2 +(yi-yj)2+(zi-zj)2]

1/2.

(21)

240

M. MARBERRY ET AL.

SOLUTION PROCEDURE Equation (20) is applicable to each of the N droplets. In principle, one can then solve for N unknown source strengths (g~, g2 . . . . . gN) from N droplet equations. Once these values are obtained, the burning rate can be calculated from Eq. (7). Three cases of droplet combustion will be considered (i) burning rate of a single isolated droplet; (ii) interaction between two burning droplets of different sizes; and (iii) interactions between monodispersed burning droplets arranged in finite arrays of up to eight droplets. The findings of the present study will be compared with the results of Brzustowski et al. [5], Umemura et al. [6, 7], and Labowsky [10]. Case I. Single Droplets

For an isolated droplet burning in an oxdizing atmosphere, Eq. (20) gives gl = (~b~- 0~)a,

(22)

of arbitrary size using bispherical coordinates. Recently, Umemura et al. [6, 7] have examined the same problem for droplets of equal as well as different sizes and derived essentially the same solution as that of Brzustowski et al. [5]. In the present analysis, it is necessary to determine the strengths, gl and g2, of two point sources of unequal magnitude. Simultaneous solution of Eq. (10) and (20) results in l q~= ¢= + (~b~- ¢~) -~

( d - a2)

rl(d/aj-az/d)

d (a2--ala2/d)~

,

(25)

r2 ( d - ala:/d) ) where d is the center-to-center distance between droplets, while al and a2 represent the radii of the droplets. The solutions for burning rates m~ and mz are simplified considerably if the normal potential gradient at the droplet surface, V ¢ ' h , is replaced by the average potential gradient

where a is the droplet radius. The potential, ~b, can then be written as ~b= ~b~ + - -

a.

(23)

6

x i + ai,Yi,Z i

r

From Eqs. (7), (19), and (23), the burning rate of a single isolated droplet is obtained as a function of particle radius:

0~l + 3x~[ x t - ai,Yt,Z i

~

[ xi,Yi + ai,z i

miso(a)

( = 4~raoD In [.

[Cp(T~ - Ts)/qr + ((Wo,~ - Wo.s)/Movo)] ")

1+

(-~q~ + wo flM-~v~

For the assumption involved in this study, this solution is identical to that given by Williams [131.

(24)

3 "

+-Oy

+-xi,Yi - ai, Zi

OZ

xi,Yi,Zi + a i

Case II. Droplet Pairs

A simple case of interaction between burning droplets is that of a droplet pair. Brzustowski et al. [5] have examined this problem for droplets

"4--OZ

(26) xi,Yi,Z i - a i

where i = 1 or 2.

B U R N I N G DROP I N T E R A C T I O N S

241

By employing this approximation in conjunction with Eq. (7), the burning rates of the droplets are found to be

ml = mi~o(al)

t. al

6\al/

,

[

4]1

+ ( P l + 1)2 + ( p 2 + 1)3/2

mz=thiso(a2) f C 2 + l ~a 2

(C,) 6

~2

(P1

(27)

[

o-o

IITwo Particle Array

1) 2

,

Q-O

/

2 Triangular Array

I \''

1 (P2- l) 2 41Tetrahedral Array

1

- + (P2 + 1)2

+

41/

(P22 + 1)3/2

'

(28)

where

(P, -

1/P2)' (P2 - al/a2)

C 2= a2

(P2 -

51CubicalArray

Fig. 1. Geometrical arrangements of droplets in various arrays.

for all droplets in the array. Hence, point sources of the same strength, g, located at the centers of

(P1 - az/aO C1 = al

31Squanl Array

the droplets (x, y~, zi) may be used to satisfy the boundary conditions at the droplet surfaces. Under these conditions, the potential, ¢, can be written in the following form:

1/Pl)'

N

Pj= d/al,

P2 = d/a2.

By substituting al = a2 = a, the burning rate for two droplets of equal size can be obtained as a function of the burning rate of a isolated droplet, m~o(a), and the dimensionless droplet spacing,

1/r, (29)

~ 1/rl.i t

I

where

d/a. rl.i 2 = (xl - xi) 2 + (Yl - Yi) 2 + (Zl - Zi) 2, Case 111. Monodispersed Droplets in Finite Arrays

A more general situation will now be considered in which burning droplets are symmetrically located in a finite array. These droplets all have the same radius, a, and their arrangements in the arrays to be studied are shown in Fig. 1. The burning rates of droplets in similar arrays consisting of up to four droplets have been numerically calculated by Labowsky [8]. In symmetrical arrays, each droplet will " s e e " an equal number of droplets identically located with respect to its position. As a result, the effect of multiparticle interactions is the same

rl,l = a .

Subscript 1 represents an arbitrarily chosen reference particle and N refers to the number of droplets in the array. By applying the assumption of an average potential gradient, the following solution for burning rates of droplets in various arrays is obtained

ga

~ el,i

m = miso(a) 3(~bs- ~b=) i= 1 rj.i where e,.,= Ixl-x,I + [Yl-yi] + I z l - z i l

(30)

242

M. MARBERRY ET AL. R E S U L T S AND D I S C U S S I O N

TABLE 1

Expressions for the Strength of Point Sources in Various Arrays Type of Array

Several calculations using Eqs. (27), (28), and (30) have been performed in order to study the behavior of these solutions and elucidate the effect of droplet size, number density, and geometrical configuration of the array. It is convenient to plot the results in terms of a correction factor r/, defined by

Expression for g in Eq. (30)

a(~ - 6=) Triangular

.

Tetrahedral

.

.

.

.

.

1 + 2(a/d) --

-

I + 3(a/d)

Square

,7 = m / m . o

where rh~sois the burning rate of a single isolated droplet of the same size. This factor is applied to the burning rate of a single droplet in order to account for multiple droplet interactioq. The results for two interacting droplets using Eqs. (27) and (28) are shown in Fig. 2. This is a plot ofr/~ and r/2 versus the dimensionless droplet spacing d/(al + a2) for various values of droplet size ratio, a2/a~. The curve for az/a¿ = 1 represents the case of two equally sized droplets. The family of curves below this gives the correction factor r/2 for the smaller droplet, while the family of curves above this gives the correction factor rl~ for the larger droplet. Figure 2 shows that the interaction decreases the burning rate of both droplets and that the effect is more pronounced for the smaller droplet. Also the correction factors for all values ofa2/a~ decrease with decreas-

; a(~ - ¢=)

Cubical

i +(3 +3/-~-+l/-,~)(a/d)

and el,t= 3a. The source strength, g, for droplets arranged in various arrays is shown in Table 1. Equation (30) is the analytical equivalent of the numerical solution obtained by Labowsky. The analytical results of this study will be compared with the numerical calculations for various droplet arrays. 1.0

1

o,,~,-o 2._.__.~5

I

,

I

1

_

:

5

6 Spocinq

f

:

I

T

:

-

7

8

9

(

¢1 ) o1÷o z

0.50

0.9

0.8

PY/

0.7

0.6 2

3

4

Dimensionles=

I0

Fig. 2. The b u r n i n g rate correction factor f o r t w o interacting droplets as a f u n c t i o n o f

droplet spacing for various values of droplet size ratio.

B U R N I N G DROP INTERACTIONS

243 TABLE 3

ing separation distance between droplets. This reduction in burning rate is considerable when the droplets are closely arranged. The results of Fig. 2 are compared with those of Brzustowski et al. [5] or Umemura et al. [7] in Table 2. For the case of two idential droplets, the results are also compared with Labowsky's [10] numerical solutions in Table 3. The results of the present study are in close agreement with those of Brzustowski et al. [5] or Umemura et al. 17]--a maximum difference of 3% exists for a2/al = 0.75 and d~ (aj + a2) = 2. For two droplets of equal size, Table 3 shows that the results of all four studies are almost identical except for d / a = 3, where a difference of 5% is found between the results of Brzustowski et al. or Umemura et al. and the present study. For all droplet size ratios, the analytical results of this study predicts a slightly higher correction factor when the droplets are closely arranged. The results for finite arrays of symmetrically arranged monodispersed droplets are shown in Fig. 3. This figure indicates that deviations from single droplet behavior are significant for d / a < 20. These deviations increase as the number of

A Comparison of Results for Correction Factor for Two Identical Droplets Obtained from Brzustowski et al. 15] or Umemura et al. 17]. Labowsky [10]. and the Present Study

Dimensionless Spacing d/a

r/, Brzustowski et al. or Umemura et al.

rt. Labowsky

Present Stud}.

3 5 7 10 15 20 25 30

0.76 0.84 0.88 0.91 0.94 0.95 0.96 0.97

077 0.84 0.88 0.91 0.94 0.96 0.96 0.97

0,80 0,85 0,88 0.91 0,94 0.95 0.96 0.97

droplets in the array increases or as the spacing between droplets decreases. It is clear that the number density, which is directly proportional to the number of droplets and inversely proportional to ( d / a ) 3 largely affects the burning rate of an individual droplet in the array. This is so because competition for the oxidizer in the surrounding medium increases as the number den-

TABLE 2 A Comparison of Results for Correction Factor for Droplet Pair of Different Sizes Obtained from Brzustowski et al. [5] or Umemura et al. [7] and the Present Study

Brzustowski et al. or

Dimensionless Spacing d/(al + a2)

Umemura et al.

Present Study

rh

O:

rh

rt,

a,_/at = 0.25

2 3 4 5

0.937 0.952 0.963 0.969

0.629 0.746 0.810 0.846

0.957 0.958 0.961 0.969

0.640 0.753 0.812 0.848

a,./aL = 0.50

2 3 4 5

0.882 0.914 0.940 0.943

0.703 0.800 0.854 0.883

0.908 0.922 0.935 0.945

0.729 0.807 0.850 0.878

a:/a~ = 0.75

2 3 4 5

0.839 0.886 0.914 0.923

0.760 0.831 0.874 0.900

0.865 0.892 0.913 0.927

0.787 0.843 0.876 0.898

244

M. MARBERRY ET AL. 1.0

i

i

i

I

i

i

i

l

I

J

i

J

I

I

i

i

i

i

l

I

I

I

1

1

,

,

1

1

I

I

I

I

1

f 0.9 I

_~o.81 Z> u

g

:;0.6 '.~ 0.5 0.4 O.2

l

I

l

I0

20

30

40

Dime.lionllss Spoci. 0 (dlo) F i g . 3. T h e b u r n i n g

rate correction

factor for monodisperse

sity increases. Curves 3 and 4 in Fig. 3 show the effect of geometrical configuration on the burning rates of droplets in a four-membered array. The results indicate that, for d / a < 3, the droplets in the square array are subject to more interaction effects than the droplets in the tetrahedral array. The sitution reverses for d / a > 4. The arrays in curves 2 and 4 and Fig. 3 were also analyzed by Labowsky. A comparison of results is made in Table 4. This table shows that the numerical results of Labowsky's study and the analytical results of the present study are in excellent agreement. However, it should be pointed out that the method of images used by Labowsky requires extensive computations. His technique consists of constructing the solution by superimposing the fields for a series of point sources so that the boundary conditions are satisfied. For example, for an array containing eight droplets, his technique requires the use of 3200 sources to compute the first four generations of iterations and 456 sources for three generations. This becomes exceedingly tedious for arrrays containing a large number of droplets. CONCLUSIONS The generalized treatment of the effect of multiparticle interactions on the burning rate of fuel droplets is shown to be applicable to droplet pairs

droplets

in f i n i t e a r r a y s .

TABLE 4 A Comparison

between

the Correction

Factor

Obtained

by

Labowsky [10] and the Present Study

Type of Array

Dimensionless Spacing d/a

r/, 71, Present L a b o w s k y Study

Triangular

5 10 15 20 25 30

0.73 0.83 0.88 0.92 0.93 0.94

0.74 0.84 0.89 0,91 0.93 0.94

Tetrahedral

5 10 15 20 25 30

0.64 0.77 0.83 0.87 0.89 0.91

0.66 0.78 0.84 0.87 0.89 0.91

of different sizes as well as to finite arrays of symmetrically arranged monodispersed droplets. The analysis reduces to the classical solution for combustion of a single burning droplet. For droplet pairs, the results of the present study are in close agreement with those of Brzustowski et al. [5] and Umemura et al. [7], whose analysis is limited to two droplets only. For finite arrays of burning droplets, the results of Labowsky [ 10] are recovered without recourse to numerical so-

BURNING DROP INTERACTIONS lution. The corrections to single droplet burning rates are shown to depend strongly on the number density and to some extent on the geometrical configuration of the array. This analysis can also be applied to droplets burning at different temperatures. In a future study, the burning behavior of nonisothermal droplets will be investigated.

NOMENCLATURE Ai AM, BM AT, BT ai C1, C2 Cp

surface area of ith droplet arbitrary constants in Eq. (11) arbitrary constants in Eq. (12) radius of the ith droplet variables defined by Eq. (25) and (26) specific heat or constant pressure for the gaseous mixture CT arbitrary constant in Eq. (13) D binary diffusion coefficient between any two chemical species d center-to-center distance between two droplets el,i variable defined by Eq. (28) g strength of point source k thermal conductivity of the gaseous mixture L latent heat of vaporization per unit mass M, molecular weight of species i m~ burning rate of ith droplet miso(a) burning rate of an isolated droplet with radius a N number of droplets h unit vector orthogonal to droplet surface P~, P2 dimensionless variables defined by Eqs. (25) and (26) qr standard heat of formation per unit mass ri radial position from the ith droplet rj., center-to-center distance between the ith droplet and an arbitrary reference droplet T temperature v mass average velocity of the gaseous mixture wi mass fraction of species i xi, Yi, Zi coordinates for the center of the ith droplet

245

Greek Symbols tim fit r/ ¢ ,o 6(x) u, •~

coupling function in the mass conservation equation coupling function in the energy conservation equation dimensionless burning rate correction factor potential function mass density of the gaseous mixture Dirac delta function stoichiometric coefficient for species i variable defined by Eq. (13)

Subscripts F i

fuel ith droplet

m

mass

o

oxidizer standard reference condition droplet surface temperature T infinity

r s

T oo

REFERENCES 1. 2. 3.

4. 5. 6. 7. 8. 9. 10. I 1. 12. 13.

Rex, J., Fuhs, A., and Penner, S., Jet Propulsion 26:179 (1956). Sangiovanni, J. J., and Labowsky, M., Combust. Flame 47:15-30 (1982). Miyasaka, K., and Law, C. K.,. Eighteenth Symposium on Combustion, The Combustion Institute, Pittsburgh, 1981, pp. 283-291. Carstens, J. C., Williams, A., and Zung, J. T., J. A tmos. Sci. 27: 798-807 (1970). Brzustowski, T. A., Twardus, E. M., Wojciki, S., and Sobiesiak, A., A I A A J. 17:1234-1242 (1979). Umemura, A., Agawa, S., and Oshima, N., Combust. Flame 43:111-119 (1981). Umemura, A., Ogawa, S., and Oshima, N., Combust. Flame41:45-55 (1981). Labowsky, M., Chem. Engg. Sci. 31:803-813 (1976). Labowsky, M., Combust. Sci. Tech. 18:145-151 (1978). Labowsky, M., Combust. Sci. Tech. 22:217-226 (1980). Ray, A. K,, and Davis, E. J., Chem. Engg. Commun. 6:61-79 (1980). Ray, A. K., and Davis, E. J., Chem. Engg. Commun. 10:81-102 (1984). Williams, F. A., Combustion Theory, Addison-Wesley, Reading, MA, 1965).

Received 23 December 1982; revised 27 February 1984