G. f. Carrier, f. e. Fendell, s. f. Fink IV, And c. n. Folley, Particle Transport in a Counter-flow

Particle Transport in a Counter-flow

G. F. CARRIER, F. E. FENDELL,* S. F. FINK IV, and C. N. FOLLEYSpace & Technology Division, TRW Space & Electronics, Redondo Beach, CA 90278 USA

We consider properties of a steady two-dimensional isothermal low Mach-number counter-flow, into which adilute loading of small spherical particles is introduced at the local gas velocity, at a finite axial distance fromthe stagnation plane for the axial velocity component of the gas. The particles are introduced on one side of thatstagnation plane only, and the consequences of any subsequent velocity slip of the particles with respect to thelocal gas are examined. The self-similarity (planar symmetry in the axial coordinate, for most of the keydependent variables), familiar from particle-free counter-flow, also holds for the two-phase flow under theseconditions. Results, obtained by Lagrangian tracking of the motion of a single particle, distinguish: thenonoscillatory trajectory of that particle for relatively small strain-rate, large drag-rate conditions (the particledoes not cross the stagnation plane for the axial velocity component); and the oscillatory trajectory of thatparticle under relatively large strain-rate, small drag-rate conditions (the particle may cross the stagnation planerepeatedly). However, for the multi-particle scenario for self-similar two-phase flow, the results for bothconditions have commonalities. A single, densely particle-loaded, very thin slab region arises: one planar sideof the slab interfaces with a particle-free, purely gaseous counter-flow, and constitutes the axial stagnation planefor that flow; the other planar side of the slab interfaces with a dilutely particle-laden region, in which theparticle behavior is unaltered by the presence of the close-packed thin slab. In fact, the thin slab effectively isthe stagnation plane for the axial velocity component of the gas, the value of the strain-rate/drag-rate ratiocharacterizing whether the plane is displaced from its pure gas counter-flow position (and, if so, to whatdifferent axial position). 2001 by The Combustion Institute

NOMENCLATURE

a spherical particle radiusDW particle dragL magnitude of x-coordinate location at

which the particle-free stream entersthe counter-flow domain

l magnitude of the x-coordinate locationof a particle at time t 5 0

m mass of a spherical particlet time since a particle being tracked

entered the counter-flow domain acrossx 5 2l

t* time at which the velocity component uof a particle being tracked is stagnated

t1 time at which a particle being trackedenters the dense stream

u fluid velocity component in the xdirection

V y-directed velocity of the dense streamv fluid velocity component in the y

directionwW fluid velocity vectorX x-coordinate thickness of the dense

stream

x Cartesian similarity coordinate for asteady two-dimensional counter-flow

x(t) x-coordinate location at time t of aparticle being tracked

x(t) x-coordinate speed at time t of aparticle being tracked

Y y-coordinate location at time t 5 0 of aparticle being tacked

y Cartesian coordinate perpendicular to xy(t) y-coordinate location at time t of a

particle being trackedy(t) y-coordinate speed at time t of a

particle being tracked

Greek Letters

a1 [1 2 4(V/b)]1/2

a2 [1 1 4(V/b)]1/2

a3 [4(V/b) 2 1]1/2

b Stokes drag rate, 6pma/mh(t) multiple to which a line segment of

simultaneously entering particles isstretched in the y direction at time t

h(t) time rate of change of h(t)m dynamic viscosity of fluidro mass of particles per volume of space,

at the entry plane* Corresponding author. [email protected]

COMBUSTION AND FLAME 126:16301639 (2001)0010-2180/01/$see front matter 2001 by The Combustion InstitutePII S0010-2180(01)00279-6 Published by Elsevier Science Inc.

rs characteristic value of r(t) under closepacking

r(t) mass of particles per volume of space attime t

t dimensionless time, bt/2t* bt*/2V strain rate of a steady two-dimensional

counter-flow

INTRODUCTION

Counter-flow configurations have played an im-portant role in both the theoretical understand-ing and the gathering of experimental data for avariety of combustion-related phenomena. Weanalyze here the transport of initially sparsedistributions of particles in such flows. We dealwith steady counter-flows, and the phrase ini-tially sparse distributions refers to the numberdensity of particles at their sites of introductioninto the flow; wherever there is no large varia-tion from the initial sparseness, the conse-quences of particle collisions are negligible.

Although our analysis is broadly applicable,our work is particularly motivated by phenom-ena such as soot formation in hydrocarbon/aircounter-flow diffusion flames [1] and metal ox-ide particle formation (as feedstock for ceram-ics) in counter-flow diffusion flames [25].Wooldridge [4] cites the counter-flow-diffusion-flame apparatus as one of six basic experimentsespecially suitable for examining the gas-phasecombustion synthesis of particles. Scenarios inwhich fuel-droplet sprays are evaporated toform a combustible vapor, such that the drop-lets do not persist to the vicinity of the gas axialvelocity component stagnation plane, have beendiscussed [68], and do not constitute the majormotivation for our work.

A particular relevant context concerns a pro-posed modification in the schedule (duty cycle)for transverse injection, into the hydrogen-richflow in the single nozzle of Stage 3 of theMinuteman III missile, of a helium-pressurizedaqueous solution of the oxidizer strontium per-chlorate Sr(ClO4)2. The liquid-injection trans-verse vector control (LITVC) system, in gener-ating side force for attitude control (and somemodest thrust augmentation as well), may incur

an unacceptably large local augmentation of theheat load on the portion of the char-forming,ablative nozzle liner in the vicinity of utilizedLITVC ports. We expect that oxygen from thedecomposition of the injected strontium per-chlorate reacts with the nozzle-flow hydrogen togenerate steam and release heat. At 500 to 600K, we expect the decomposition of strontiumperchlorate also to generate both hydrogenchloride and, in the presence of steam, thecondensed-phase product strontium oxide SrO.As a first step to unraveling the two-flamephenomena, we isolate the issue of the fate ofthe inert condensed-phase product generatedby the decomposition. Of course, the con-densed-phase product aluminum oxide Al2O3 isalso present in the nozzle flow from the com-bustion of the aluminized composite grain in thechamber of the rocket stage.

We adopt the simplification used by others [9,10]: we examine a spherical particle in theabsence of phase change in a constant-densitygas flow. For convenience, we also track inLagrangian fashion the trajectory of a singleparticle in the steady counter-flow, described inconventional Eulerian fashion. However, ourinterest lies in a steady counter-flow, which isinitially sparsely particle laden in the vicinity ofthe particle-entry plane, and in which the famil-iar planar similarity holds, so that the keydependent variables remain functions of theaxial coordinate only. The conditions underwhich this planar similarity holds with particlespresent require, inter alia, that: 1) the adoptionof the Stokes drag (here for spherical particles)is appropriate; and 2) the particles are intro-duced without slip (i.e., introduced with thelocal velocity of the gas flow) at finite distancefrom the stagnation plane for the axial velocitycomponent [1113, 14]. Of course, if the needfor modification of the Stokes-drag hypothesis(which pertains to small particles with modestvelocity slip with respect to the motion of theco-located gas) were to become evident in thecourse of solution, we would adopt that modi-fication. Nevertheless, we reiterate that we arenot concerned with tracking the fate of one orjust a few particles introduced spasmodically ina counter-flow for diagnostic purposes; our in-terest lies in a steady counter-flow in which a

1631PARTICLE TRANSPORT IN A COUNTER-FLOW

reasonable anticipation is that planar similarityholds extensively (if not everywhere), and atransversely invariant, temporally unchangingflux of particles enters the counter-flow.

Finally, we take the particles to be introducedin the counter-flow exclusively on one side ofthe stagnation plane for the axial velocity com-ponent; of course, in the presence of slip, theparticles introduced on one side of the stagna-tion plane may or may not remain confined tothat side of the stagnation plane. Any crossingof the stagnation plane is here not related tothermophoresis, which is not present in ourisothermal flow; neither is any other diffusivetransport of particles taken to enter. Of partic-ular interest is the possible existence, and loca-tion if it exists, of a dense (typically thin) layerof particles separating particle-free regions ofthe counterflow from only dilutely particle-laden regions of the counter-flow; such a denselayer may arise as a consequence of axial-flowstagnation and finite interphase velocity slip.Suggestions from theory of the existence ofsingularly large particle number densities verynear the axial gas velocity stagnation plane forsmaller values of the Stokes number (the Stokesnumber being here defined as the ratio of thestrain rate of the gaseous flow to a rate thatincreases with the drag on particles) have notbeen verified experimentally [9, 10]; furtherstudy seems warranted to resolve the uncer-tainty [9, 10]. Furthermore, for larger values ofthe Stokes number, single particle-trackingmodels and experiments yield trajectories withmultiple reversals of direction [9, 10, 1517];these results have been interpreted, for a mul-tiple particle context, to imply the coexistence ofoppositely directed particles, so that analyticcomplications arise [12]. Clarification againseems warranted [17].

SINGLE-PARTICLE DYNAMICS IN ACOUNTER-FLOW

It is very convenient to begin with one particlein a steady two-dimensional gaseous counter-flow at very small Mach number. The gas-phasevelocity field wW (x, y) 5 u(x) x 1 v(y) y where:

u 5 2V x, v 5 V y in x . 2l, . y . 2

(1)

where x,y denote Cartesian coordinates: V de-notes a constant strain rate; and l denotes apositive number (Fig. 1). We denote the posi-tion of a single particle by x(t),y(t), where tdenotes time since the particle entered the flowfield defined by Eq. (1). We adopt the initialparticle position and initial particle velocity sothat initially there is no slip of the particle withrespect to the gas:

x~0! 5 2l, y~0! 5 Y, x~0! 5 Vl, y~0! 5 VY

(2)

We take the particle drag DW to be that appro-priate for a small spherical particle (m denotesthe dynamic viscosity of the gas and a, the radiusof the particle):

Fig. 1. Sketch of the streamlines of a steady two-dimen-sional low-speed isothermal gaseous counterflow (solidcurves), with a stagnation plane (for the axial velocitycomponent u) located at axial-coordinate position x 5 0.Small spherical particles are continuously introduced withthe local gas speed at the flow-domain boundary x 5 2l. Thetrajectories of the particles introduced at the end points ofa differential line segment dY at x 5 2l are shown (dashedcurves), with the y-direction separation increasing to h dYwhen the particles subsequently lie at an (algebraically)larger value of the axial coordinate x. No particles areintroduced in x . 0, and no particles enter that region forsmall-strain-rate, large-drag-rate conditions.

1632 G. F. CARRIER ET AL.

DW 5 26pma @~ x 1 Vx! x 1 ~ y 2 Vy! y# (3)

We reiterate that this Stokes-drag hypothesis isto be modified if and when that need becomesevident. Thus, in 2l , x, uyu , ,

x 1 b ~ x 1 Vx! 5 0, y 1 b ~ y 2 Vy! 5 0 (4)

where b [ 6pma/m, with units of inverse time,and m denotes the mass of spherical particle ofradius a. Solution of Eq. (4) subject to Eq. (3)introduces the ratio b/V, which is termed theStokes number, but we choose not to introducefurther notation. For what follows, we note thatm ; a3, so b ; a22; if the particle size isincreased while all other parameters are heldfixed, then the ratio b/V is decreased in value.We prefer henceforth to deal mainly with theinverse, V/b.

For (V/b) , (1/4), referred to here as thehigh-drag low strain-rate case, the solution,familiar from oscillation theory, is [15, 9, 10]:

xl

5 [exp (2t)] F2cosh(a1t)2

1 2 2(V/b)a1

sinh(a1t)G for (5)2l , x , x~Vt1!,

xlV

5 @exp (2t)]

Fcosh(a1t) 1 1a1 sinh(a1t)G (6)h

yY

5 [exp(2t)] Fcosh(a2t)1

1 1 2 ~V/b!a2

sinh(a2t)G (7)h

V5

y/YV

5 Fexp(2t)] Fcosh~a2t! 1 1a2 sinh~a2t!G(8)

where time t1 is discussed below, and

a1 @1 2 4~V/b!#1/ 2, a2 @1 1 4~V/b!#

1/ 2,

t 5 ~bt/ 2! 5 @Vt/ 2#/~V/b! (9)

For (V/b) . (1/4), referred to here as thelow-drag, high strain-rate case, Eqs. (5) and (6)are taken in the form:

xl

5 @exp(2t)# F2cos~a3t!2

1 2 2~V/b!a3

sin~a3t!G (10)x

lV5 @exp(2t)# Fcos~a3t! 1 1a3 sin~a3t!G

(11)

where:

a3 @4~V/b! 2 1#1/ 2 (12)

For the transitional case (V/b) 5 (1/4):

xl

5 2 @exp(22Vt!] ~1 1 Vt! (13)

xlV

5 2 @exp(22Vt!] ~1 1 2Vt! (14)

NUMBER DENSITY DISTRIBUTION

We now consider the steady flow of a sparsehomogeneous (monodisperse) collection of par-ticles, each of which enters the flow at x 5 2l(where the condensed-phase density is denotedro, the product of the (true) density of theparticles times the number of particles pervolume of space) and follows a trajectory de-scribed by Eqs. (59) if (V/b) , (1/4), and Eqs.(79) and (1012) if (V/b) . (1/4).

The layer of particles that crosses x 5 2lthrough the segment dY at time t 5 0 will crossthe segment of length (dY)h(t) and axial posi-tion x(t) with axial velocity component x(t) attime t. (We regard Y and t as parametricindependent variables.) The requirement thatvolume occupied by the particles is preservedthen implies that:

roVl~dY! 5 r~t! x~t! ~dY! h~t!, (15)

where r(t) denotes the mass of particles pervolume of space. Under Eq. (15) the density ofparticles anywhere in the domain at issue wouldbe independent of y and be given by:


r~t!ro

5Vl

x~t! h~t!(16)

were it not for the fact that r(t)/ro, as given byEq. (16), becomes, for large t, larger than rs/ro,where rs is the close-packed density of theparticles. [The close-packed density is on theorder of half the (true) particle density, themass of particulate per volume occupied byparticulate.] Of course, before r becomes rs, rbecomes so large that the drag law that we haveused is not descriptive.

Accordingly, we hypothesize that there existsa layer of thickness X(y) in which the x-directedvelocity component is zero, the particle densityis rs, and the y-directed velocity component V issuch that both particle mass and particle mo-mentum are conserved. Furthermore, the ab-sence of any y dependence in any of the statevariables at issue (other than the y-directedvelocity) suggests that X would also be indepen-dent of y. We denote by x(t1), to be found, theposition of the interface between the denselayer and the dilutely particle-populated region.We anticipate that the particles are so sparselydistributed in the region between the entryplane x(0) 5 2l and the dense-layer interfacex(t1) that the foregoing recipes for the particlebehavior in the region require no modificationin any of that region [9].

Mass and momentum conservation, usingEqs. (59) and 16 [or Eqs. (712) and (16), asappropriate], require that:

~rsVX!Y 5 roVl, (17)

~rsV2X!Y 5 roVlY h~t1!, (18)

where V(Y) is the y-directed velocity of thedense stream at y 5 Y h(t1). Integration of Eqs.(17) and (18) gives, since V 5 0 at Y 5 0:

rsVX 5 roVlY, (19)

rsV2X 5 roVlh~t1! Y

2/ 2. (20)

From Eqs. (19) and (20):

V 5 h~t1! Y/ 2; (21)

X 5rors

VlYV

5rors

2Vlh~t1!

, (22a)

or, alternatively:

rors

5Xh~t1!

2Vl. (22b)

The above integral-like statements concern-ing conservation of mass and momentum for thedense layer would not be significantly altered ifthere were fewer but larger particles in thedense layer. The similarity of the flow in thedilutely particle-populated region is what setsthe spatial rate of incrementation of the denselayer, and the dilutely particle-populated regionis not likely to be altered by some aggregationamong particles already in the dense layer.

LOW STRAIN-RATE, HIGH DRAG-RATECASE

For the high drag-rate, low strain-rate case(V/b) , (1/4), with the aid of Eqs. (59), weanticipate that the dense layer lies contiguous tothe stagnation plane for the axial velocitycomponent, x 5 0; i.e., the dense layer lies inx(t1) , x , 0, where: x(t1) , 0;

X/l 5 ux~t1!u/l; (23)

and, to reiterate, x(t1) is to be found. The gasflow in the particle-free region x . 0 is unal-tered from that holding for the particle-freeflow.

Equations (58) and (16) for t 5 t1, (21),(22a), and (23) constitute eight coupled alge-braic equations for the eight dimensionless un-knowns:

Xl

,ux~Vt1!u

l,

x~Vt1!lV

, h~Vt1!,

h~Vt1!V

,V

YV,

r~Vt1!rs

, and Vt1, (24)

as functions of the two specified dimensionlessratios ro/rs and V/b. Operationally, it is conve-nient to interchange the roles of one unknownquantity, Vt1, and one specified ratio, ro/rs. Byadopting values for Vt1 and inferring ro/rs fromEq. (22b), which replaces Eq. (22a), we conve-niently avoid seeking the roots of any transcen-dental equations. However, only those valuesadopted for Vt1 which give 0 , (ro/rs) , 0.01(say), produce results of physical interest.

Results, obtained by assigning a succession of


values to (V/b) . (1/4), for each of which arange of values of Vt1 are adopted, are given inFigs. 2 to 5. These plots indicate that the denselayer is so thin (perhaps a few particle widths)that the layer is challenging to detect experi-mentally; inability to detect the layer in thelaboratory [9] is owing to limitation in diagnos-tic resolution, not to inadequacy of the model.Because the amount of displacement of thedilute particle-density solution to account forthe presence of the close-packed particlesboundary layer is vanishingly small, then, overeffectively all the domain in which particlesexist, the simple dilute-population modelingsuffices, without the need for any modification.

LARGE STRAIN-RATE, SMALL DRAG-RATE CASE

For (V/b) . (1/4), the inertia of a largerindividual particle in a higher-speed flow is such

that the particle may cross the stagnation planefor the axial velocity component of the gas, andmay proceed to the other side of that gas-flowstagnation plane from the side on which the

Fig. 2. For several values of the dimensionaless ratio b/V,where b denotes a Stokes-drag-related rate and V, the strainrate of the gaseous counterflow [with (b/V) . 4 character-izing small-strain-rate, large-drag-rate conditions], the di-mensionless time Vt1 at which a particle, introduced withoutslip with respect to the gas at x 5 2l at time t 5 0, reachesthe interface between the dilutely-particle-loaded domainand the very thin densely-particle-laden layer (situatedcontinuous to the stagnation plane x 5 0). The abscissa isthe ratio of the gas density ro to the mass per volume ofspace of closely packed particles rs.

Fig. 3. The normalized position of the interface x(Vt1)/lbetween the relatively broad dilutely-particle-laden domainand the very narrow densely-particle-laden domain, pre-sented as a function of the density ratio ro/rs, for severalvalues of the reciprocal of the Stokes number, b/V. Thequantitly | x(Vt1)|/l gives the normalized dense-layer thick-ness X for (b/V) . 4.

Fig. 4. The normalized axial velocity of a paticle at thedilute-particle-domain, dense-particle-layer interface,x(Vt1)/(Vl ), vs. the density ratio ro/rs, for several values ofthe reciprocal of th Stokes numbr, b/V. The axial motion ofparticles is virtually stagnated as they join the dense layer.


particle was introduced into the counter-flow.Such a particle encounters an opposed gas flowupon crossing the stagnation plane, eventuallyhas its axial velocity stagnated and nominallyreversed, but in the steady state for a sparsehomogeneous collection of particles, immedi-ately encounters oncoming particles that subse-quently entered the counter-flow and are ontheir way to the same fate. The upshot is theformation of a dense layer of particles of thick-ness X, with particle density rs, and havingy-directed velocity component V; again, X and Vare to be found. However, for (V/b) . (1/4), thedense layer lies in x(t1) , x , x(t*), where:

x~t*! 5 x~t1! 1 X, with t* . t1, X . 0 (25)

so x(t*) . x(t1); since we may anticipate, on thebasis of previous results, that, for physicallyrealistic conditions, X is small (i.e., the denselayer is exceedingly thin) for parameter valuesof physical interest, in general we expect x(t1) .0, and, therefore, x(t*) . 0. We do not attemptto track the trajectory of individual particlesafter their axial speed is first stagnated byencountering the dense layer, but instead ac-count for the mass and momentum balance of

the close-packed particles in the dense layer(just as was done for the small strain-rate, largedrag-rate case).

We identify t* [from which, by substitution inEq. (10), we identify the position x(t*)] as thesmallest positive value t at which x(t) 5 0, wherex(t) is given by Eq. (11), and the relationbetween t* and t* is given by Eq. (9). Accord-ingly,

tan a3 t* 5 2a3; (26)

the positive quantity a3 was defined in Eq. (12),and is regarded as specified. This is the nominalvalue at which x(t) would become zero if adense layer did not intervene. If

a3 t* p 2 h* (27)

then:

tan h* 5 a3, h* 5 arc tan a3 (28)

so (Fig. 6):

t* 5 ~p 2 arc tan a3!/a3 (29)

Substitution of Eq. (29) into (10) gives (Fig. 7):

x~t*!l

5 SVbD1/ 2 exp(2t*) (30)

The quantity t*( bt*/2) is seen to be afunction of (V/b) only. Because X is antici-

Fig. 5. The bulk density at the dilute-region/dense-layerinterface, r(Vt1), normalized by the bulk density at entry,ro, vs. (ro/rs), for several values of the reciprocal of theStokes number, b/V(. 4). The density of particles, whileappreciably increased at the interface above the entry value,remains well below half that of a close-packed layer.

Fig. 6. The normalized time t*([bt*/2) to the first stagna-tion of axial motion of a single particle vs. the Stokesnumber (V/b) for the large-strain-rate, small-drag-ratecircumstances [(V/b) . (1/4)]. A particle that soonertravels into the region of stronger opposed flow is stalledearlier.


pated to be very small for very small values of(ro/rs), the dimensionless time t1 is onlyslightly smaller than the dimensionless timet*; hence, t1 is large for values of (V/b) onlyslightly greater than (1/4), for fixed smallvalues of (ro/rs), and there is continuous

behavior of the solution on the ratio (V/b) inthe vicinity of the dimensionless ratio (V/b) 5(1/4).

With t* in hand from Eq. (29) for a givenvalue of (V/b) in the range of interest, we mayobtain the eight unknowns:

Xl ,

x~Vt1!l ,

x~Vt1!lV , h~Vt1!

y~Vt1!Y ,

h~Vt1!V

y~Vt1!VY ,

r~Vt1!rs

,V

VY, and Vt1 (31)

from the eight relations Eqs. (7, 8, 10, 11, 16, 21,22a, 25). As previously, for convenience, weoperationally regard (ro/rs) to be deduced andVt1 as specified, and replace Eq. (22a) with itsequivalent, Eq. (22b).

Results, again obtained by assigning a succes-sion of values to (V/b) . (1/4), for each ofwhich a range of values for Vt1 is adopted, aregiven in Figs. 8 to 10. As is to be expected, muchof the behavior of the dependent variables isseen to be about the same as in the previouslytreated case for small strain rate and high dragrate. In fact, the key difference is that thegas-flow stagnation plane (for the axial velocitycomponent) is now at [x(Vt1) 1 X], not at x 5 0as before (Fig. 11). This translation of axialcoordinate is to be accounted for in writingexpressions for the purely gaseous flow in thedomain x . [x(Vt1) 1 X]. Because the dilutelyparticle-laden domain extends over the expanse2l , x , x(Vt1), which interfaces with a thin

dense layer, the scenario holding for the multi-particle counter-flow for (V/b) . (1/4) is, inmany respects, an axially displaced version ofthe scenario holding for the multi-particlecounter-flow for (V/b) , (1/4).

For an experimental study of the multi-parti-cle counter-flow described in this manuscript, itis noted, concerning the x-momentum flux law,that the momentum of gas and particles crossingx 5 2l is equal to momentum of gas crossingx 5 L{.[x(Vt1) 1 X]}. However, any momen-tum balance-motivated adjustment of pressure,to account for the dilute particle loading of onlyone of the two opposed streams constituting thecounter-flow, probably would be of modestmagnitude.

Fig. 7. The normalized position of the interface x(Vt1)/lbetween the relatively broad dilutely-particle-laden domainand the very narrow densely-particle-laden domain, pre-sented as a function of the density ratio ro/rs, for severalvalue of the Stokes number, (V/b) . (1/4). Figure 3 givescorresponding results for (V/b) , (1/4).

Fig. 8. Normalized thickness of the dense layer, X/l, vs. theratio of entry bulk density to close-packed bulk density,ro/rs, for (V/b) . (1/4). Corresponding results for (V/b) ,(1/4) are given in Fig. 3. Uniformly, the dense layer is verynarrow.


CONCLUSION

We have considered particle transport in asteady two-dimensional low Mach-numbercounter-flow. More specifically, we have ad-dressed scenarios in which the planar symmetry(similarity) familiar from purely gas-phase con-texts is extended to a two-phase context, suchthat particles persist to the stagnation plane ofthe axial velocity component. Examination ofsingle particle trajectories in a counter-flowprovides a convenient, informative introductionto the phenomena; such studies distinguish: 1)small strain-rate, large drag-rate circumstances(in which no reversal of direction of a relativelysmall particle is observed), from 2) large strain-rate, small drag-rate circumstances (in whichtypically multiple reversal of direction of arelatively large particle are observed). However,description of the transport of multiple particlesin a self-similar counter-flow goes beyond La-grangian tracking, and identifies the existence ofa densely particle-laden layer(s) even in acounter-flow in which particles are introducedwith dilute number density at the local gasvelocity holding at a flow-domain boundary.

We have considered nonvaporizing monodis-perse spherical particles introduced at modestnumber density and with no slip with respect to

the local constant-density gas flow, in a planeparallel to (and at a finite distance from) thestagnation plane for the axial velocity compo-nent of the gas. On the other side of thestagnation plane, no particles are introduced tothe gaseous influx. We find that, for physicallyinteresting conditions for which interphaseforce is described by Stokes drag, a single, verythin, slab-like, densely particle-laden layer aris-es; to the particle-introduction side of thisclose-packed particles layer lies a dilutely parti-cle-laden region, in which the particle behavioris well described by single particle analysis; tothe other side of this close-packed particleslayer lies a particle-free gas flow. The normal-ized position of the dense layer varies with thevalue of the Stokes number (the ratio of theflow rate to the drag rate) only. The dense layeris the stagnation plane for the axial velocitycomponent of the gas flow. Aside from a trans-lation of the dense-layer position from the gasaxial velocity component stagnation-plane posi-tion for larger Stokes numbers, the scenario forparticle transport in a counter-flow is essentiallythe same for all finite Stokes numbers.

We have not examined whether instabilityphenomena arise in connection with the iner-tially produced particle-concentration buildup

Fig. 10. The bulk density at the dilute-region, dense-layerinterface, r(Vt1), normalized by the bulk density at entry,ro, vs. (ro/rs), for (V/b) . (1/4). Corresponding results for(V/b) , (1/4) are given in Fig. 5. Uniformly, the densifica-tion at the interface is much less than that in the dense layer.

Fig. 9. The normalized axial velocity of a particle at thedilute-particle-domain, dense-particle-layer interface, x(Vt1)/(Vl), vs. the density ratio ro/rs for (V/b) . (1/4). Correspond-ing results for (V/b) , (1/4) are given in Fig. 4. Uniformly,the normalized axial velocity at the interface is small.


in a laminar counter-flow. However, we reiter-ate our belief that the reason that the denselayer is challenging to detect seems relatedprimarily to its thinness, rather than to othercause, such as instability phenomena. If thedense layer of particles were unstable in thecounter-flow, the resulting thicker (if somewhatless dense) layer would be expected to be morereadily detectible. The fact that the dense layerhas not been detected is consistent with thelayer being thin and remaining stable. Also, thetime of residence of a laterally flowing particlesin the dense layer (within a counter-flow appa-ratus of feasible dimension in the presence ofEarth gravity) is to be compared with the timefor any instability to grow to finite amplitude,and thus be of practical consequence. Hence,even were an instability to arise, the experimen-tal evidence suggests that the instability is of no

practical consequence because it has insufficienttime to grow significantly within the laboratoryflow domain (or within the span of time that aquasisteady counterflow scenario describes thelocal strain-rate field in a turbulent flow, afterthe local organized translation and rotationhave been subtracted from that flow).

REFERENCES

1. Zhang, C., Atreya, A., and Lee, K., Twenty-FourthInternational Symposium on Combustion, The Combus-tion Institute, Pittsburgh, 1992, p. 1049.

2. Zachariah, M. R., Chin, D., Semerjian, H. G., andKatz, J. L., Combust. Flame 78:287 (1989).

3. Chagger, H. K., Hainsworth, D., Patterson, P. M.,Pourkashanian, M., and Williams, A. Twenty-SixthInternational Symposium on Combustion, The Combus-tion Institute, Pittsburgh, 1996, p. 1859.

4. Wooldridge, M. S., Prog. Energy Combust. Sci. 24:63 (1998).5. Pratsinis, S. E., Prog. Energy Combust. Sci. 24:197 (1998).6. Li, S. C., Libby, P. A., and Williams, F. A. Twenty-

Fourth International Symposium on Combustion, TheCombustion Institute, Pittsburgh, 1992, p. 1503.

7. Darabiha, N., Lacas, F., Rolon, J. C., and Candel, S.,Combust. Flame 95:261 (1993).

8. Gao, L. P., DAngelo, Y., Silverman, I., Gomez, A.,and Smooke, M. D. Twenty-Sixth International Sympo-sium on Combustion, The Combustion Institute, Pitts-burgh, 1996, p. 1739.

9. Li, S., C., Libby, P. A., and Williams, F. A., Combust.Flame 94:161 (1993).

10. Li, S. C., Prog. Energy Combust. Sci. 23:303 (1997).11. Continillo, G., and Sirignano, W. R., Combust. Flame

81:325 (1990).12. Gutheil, E., and Sirignano, W. A., Combust. Flame

113:92 (1998).13. Massot, M., Kumar, M., Smooke, M. D., and Gomez, A.

Twenty-Seventh International Symposium on Combustion,The Combustion Institute, Pittsburgh, 1998, p. 1975.

14. Graves, D. G., and Wendt, J. O. L. Nineteenth Inter-national Symposium on Combustion, The CombustionInstitute, Pittsburgh, 1982, p. 1189.

15. Wendt, J. O. L., Kram, B. M., Masteller, M. M., andMcCaslin, B. D. Twenty-First International Symposiumon Combustion, The Combustion Institute, Pittsburgh,1986, p. 419.

16. Puri, I. K., and Libby, P. A., Combust. Sci. Tech 66:267(1989).

17. Chen, N.-H., Rogg, B., and Bray, K. N. C. Twenty-Fourth International Symposium on Combustion, TheCombustion Institute, 1992, p. 1513.

Received 28 November 2000; revised 7 May 2001; accepted 27May 2001

Fig. 11. Schematic of the gas-phase streamlines (solidcurves), the particle trajectories (dashed curves), and thedense layer of thickness X, for large-strain-rate, small-drag-rate conditions, for which the stagnation plane for the axialvelocity component is situated at x(t1) 1 X. In Fig. 1(holding for small-strain-rate, large-drag-rate conditions),the stagnation plane lies at x 5 0. The thickness of thedense layer X is greatly exaggerated for depiction. Theposition x(t1) is the interface between the relatively broaddilutely-particle-laden region and the relatively narrowdensely-particle-laden layer.


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