JOURNAL OF COLLOID AND INTERFACE SCIENCE 200, 95–103 (1998)ARTICLE NO. CS975377

The Drag Force Acting on a Small Evaporating Particle Exposedto a High-Temperature Diatomic Gas Flow

Xi Chen

Department of Engineering Mechanics, Tsinghua University, Beijing 100084, People’s Republic of China

Received July 17, 1997; accepted December 8, 1997

present analysis. A similar study but for the particle thermo-Kinetic-theory analytical results are presented concerning the phoresis was conducted in a previous paper (6) . As men-

effect of intense evaporation on the drag force acting on a spherical tioned before (6) , the research results presented here willparticle exposed to a high-temperature diatomic gas for the case be useful for many applications such as the synthesis of fineof a free-molecule regime. Molecule dissociation and atom recom-

powders by injecting solute droplets as the raw materialsbination are included in the analysis. It has been shown that evapo-into a high-temperature gas reactor and the electron depletionration may substantially enhance the drag force acting on a parti-technique by injecting liquid droplets into the plasma sheathcle, especially for the case of the particle materials with low-evapo-in spacecraft communication studies.ration latent heat and small molecular mass at high gas

The study of the drag force acting on a particle (or drop-temperatures. The values of the recombination fraction of atomsat the particle surface and of the fraction of the specularly reflected let) with intense evaporation represents a combined problemmolecules or atoms significantly affect the drag force acting on a of mass, heat, and momentum transfer as for the case ofparticle. Neglecting gas dissociation may appreciably overestimate thermophoresis study. Here the term ‘‘intense evaporation’’the drag force acting on a nonevaporating or evaporating particle means that the evaporation of the particle is controlled byat high gas temperatures. q 1998 Academic Press the heat transfer from the high-temperature gas to the parti-

Key Words: drag force; evaporating particle; kinetic-theory anal- cle and the surface temperature of the evaporating particleysis; high temperature gas.

is approximately equal to (or slightly lower than) the boilingtemperature of particle material. It is expected that an evapo-ration-added drag force will be caused by a nonuniformINTRODUCTIONevaporated-mass flux distribution around an intensely evap-orating particle, while the local evaporated-mass flux is de-The drag force acting on a small particle exposed to a gastermined through the local heat flux to the particle surfaceflow has been extensively studied (e.g., see (1–5)) due todivided by the latent heat of evaporation of the particleits importance in science and engineering. Especially, formaterial. Such a problem of simultaneous heat, mass, andfree molecule regime and ordinary temperature gases with-momentum transfer will be studied here for the case of aout dissociation or ionization, an analytical expression hasfree-molecule regime with great Knudsen numbers (Kn úbeen given in many textbooks for the drag force acting on10), where the Knudsen number (Kn) is defined as the ratioa nonevaporating spherical particle, while the effect of evap-of the mean free path length of gas particles to the diameteroration on the drag force was studied in (4) . Reference (5)of the immersed spherical particle.extended the analysis of Refs. (1–4) to the case of high-

temperature ionized gases (plasmas) but was restricted to amonatomic gas. The present paper intends to extend the ANALYSISprevious analysis (1–4) to the case of high-temperature di-atomic gases in which molecule dissociation in the bulk gas In the analysis, the following assumptions are employed:

(A) the mean free path length of gas particles (moleculesand atom recombination at the particle surface are involved,but the gas temperature is not too high so that gas ionization and atoms) is much greater than the diameter of spherical

particle ( free-molecule regime), (B) there exists a relativeis still negligible. Incomplete thermal accommodation and acombined specular and diffuse reflection of gas atoms or velocity between the bulk gas and the immersed particle,

but the temperature is everywhere uniform in the bulk gas,molecules at the particle surface will be considered in the

95 0021-9797/98 $25.00Copyright q 1998 by Academic Press

All rights of reproduction in any form reserved.

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96 XI CHEN

reflected specularly from the particle surface is the same asthat given in Eq. [1] , except that 0nz is used instead of nz

(4, 7) , while the velocity distribution function for the gasmolecules or atoms reflected diffusely from the particle sur-face is as follows. For the case of complete diffuse-reflec-tion, one has (4, 7, 8)

f /md1 Åcm

2p(kTW/mm) 2 expS0 n 2x / n 2

y / n 2z

2kTW/mmD [2]

for the first group of molecules which are incident upon theparticle surface and then reflected from the surface,

FIG. 1. The coordinate system used in this analysis.

f /ad Å(1 0 a)ca

2p(kTW/ma)2 expS0 n 2x / n 2

y / n 2z

2kTW/maD [3]

(C) equilibrium atom–molecule composition prevails in thebulk high-temperature diatomic gas, while atoms incidentto particle surface will partially recombine into molecules for the reflected atoms, whileat the cold particle surface, (D) the reflection of gas particles(molecules or atoms) from the particle surface is partiallyspecular and patially diffuse, (E) there is no interaction f /md 2 Å

(a /2)ca

2p(kTW/mm)2 expS0 n 2x / n 2

y / n 2z

2kTW/mmD [4]

between the incident gas particles and evaporated vapormolecules flowing out from the evaporating particle, and(F) the surface temperature of the immersed particle is con- for the second group of molecules which are formed in thestant (e.g., equal to the boiling temperature of the particle atom recombination process and reflected from the surface,material) . where TW is the surface temperature of the immersed particle

The coordinate system shown in Fig. 1 is employed in and a denotes the fraction of the atoms which recombinethis analysis. Namely, the x and y axes are tangential to the into molecules at the particle surface. cm and ca are thesphere surface, while the z axis is the inner normal of the molecule and atom fluxes incident upon the particle surfacesphere surface. The x axis is within the plane formed by the at point P in Fig. 1, respectively, and can be calculated byz axis and the oncoming velocity vector U` , while the yaxis is perpendicular to this plane. Hence, for any point Pat the particle surface with an angular position u, the x , y , cj Å *

`

0*

`

0`*

`

0`

nz f 0j dnxdny dnz . [5]and z components of the oncoming velocity are, respec-tively, U`sin u, 0, and U`cos u, in which U` is the magnitudeof the oncoming velocity vector. For the present case, the After inserting the expression for f 0j given in [1] into [5] ,velocity distribution function for gas molecules ( j Å m ) or one obtainsatoms ( j Å a ) incident to the particle surface can be ex-pressed as (4, 7, 8)

cj Å14

njnV j{exp(0S 2j cos2u)

f 0j Ånj

(2pkT /mj)3/2

/√pSj cos u[1 / erf ( Sj cos u) ]} [6]

1 expS0 (nx 0 U`sin u) 2 / n 2y / (nz 0 U`cos u)2

2kT /mjD ,

for atoms ( j Å a ) or molecules ( j Å m ) , where nV j Å√

8kT / (pmj) is the mean thermal speed, while Sj Å[1]U` /

√2kT /mj is the speed ratio related to the j th gas particles

(Sa Å Sm

√ma /mm) .where mj and nj are the particle mass and number density

of the j th gas species, k is the Boltzmann constant, and T By using Eq. [6] , the velocity distribution function in[4] of the molecules formed in the atom recombinationis the temperature of the bulk gas, while v is the velocity

vector of a gas particle and v Å nx i / ny j / nzk . process and diffusely reflected from the surface can be re-written asThe velocity distribution function for the j th gas particles

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97DRAG FORCE ON AN EVAPORATING PARTICLE

f /md 2 Å(a /2)cm

2p(kTW/mm)2 S na

nmD√

mm

ma

G(Sa)G(Sm) / (1 0 ea) *

0

0`*

`

0`*

`

0`

nz(manz) f /ad dnxdnydnz

/ (1 0 ea) *0

0`*

`

0`*

`

0`

nz(mmnz) f /md2dnxdnydnz1 expS0 n 2x / n 2

y / n 2z

2kTW/mmD , [4a]

Å nakTHF1 / ea√p

Sacos u / S (1 0 a) / a

2

√mm /maDin which G(Sj) is defined as

G(Sj) Å exp(0S 2j cos2u) 1 1 0 ea

2

√Tw /TGexp(0S 2

acos2u )

/√pSj cos u[1 / erf ( Sj cos u) ] . [7]

/ F (1 / ea )S12/ S 2

mcos2uDFor a nonevaporating particle, the surface pressure due

to the first group of molecules, i.e., the incident and then/ S (1 0 a) / a

2

√mm /maD (1 0 ea)reflected molecules, pm , can be calculated by (4, 7)

pm Å p0m / emp/ms / (1 0 em)p/md1 1√p

2

√Tw /TSacos uG [1 / er f ( Sacos u)]J , [9]

Å (1 / em)p0m / (1 0 em)p/md1 ,

where em or ea is the fraction of the specularly reflectedwhere p0m is the contribution to the molecule pressure com-molecules or atoms.

ponent pm of the incident molecules, p/ms is due to the specu-No shear stress appears if completely specular reflection

larly reflected molecules and p/ms Å p0m , while p/md1 is dueis concerned (4, 7) . On the other hand, no shear stress is

to the first group of the diffusely reflected molecules. Hence,caused by the diffusely reflected gas particles. Hence, thesurface shear stress component tj ( j Å a for atoms and j Åm for molecules) is calculated bypm Å (1 / em) *

`

0*

`

0`*

`

0`

nz(mmnz) f 0m dnxdnydnz

tj Å (tzx) j Å (1 0 ej) *`

0*

`

0`*

`

0`

nz(mjnx) f 0j dnxdnydnz/ (1 0 em) *0

0`*

`

0`*

`

0`

nz(mmnz) f /md1dnxdnydnz

Å (1 0 ej)njkTSj sin uH 1√p

exp(0S 2j cos2u)Å nmkTHF1 / em√

pSmcos u / 1 0 em

2

√Tw /TG

/ Sj cos u[1 / erf ( Sj cos u)]J . [10]1 exp(0S 2m cos2u) / [1 / erf ( Smcos u )]

1 F (1 / em )S12/ S 2

mcos2uD It is obvious that the drag force vector FD is in the directionof the oncoming velocity vector U` . The molecule and atomcomponents, FD j , of the drag force FD acting on a spherical/ (1 0 em)

√p

2

√Tw /TSmcos uGJ . [8]

nonevaporating particle can be calculated by

Considering that the atoms incident upon the particle sur- FD j Å 2pR 2 *p

0

(pj cos u / tj sin u)sin udu, [11]face recombine partially at the surface, the atom pressuredue to the incident and reflected atoms and the second group

in which R is the sphere radius. By using Eqs. [8] – [10],of molecules formed in the atom recombination process canone obtainsbe calculated by

pa Å p0a / eap/as / (1 0 ea)(p/ad / p/md2) FDm Å 4pR 2nmkTH2Sm / 1

4√pSm

exp(0S 2m) / 4S 4

m / 4S 2m 0 1

8S 2mÅ (1 / ea)p0a / (1 0 ea) (p/ad / p/md2)

1 erf ( Sm) / (1 0 em)√p

6Sm

√TW

T J , [12]Å (1 / ea) *`

0*

`

0`*

`

0`

nz (manz) f 0a dnx dnydnz

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98 XI CHEN

FDa Å 4pR 2nakTH2Sa / 1

4√pSa

exp(0S 2a) (qc)a Å (1 0 ea)HaaF*

`

0*

`

0`*

`

0`

nzSma

2n 2D

1 f 0a dnxdnydnz 0 *0

0`*

`

0`*

`

0`

(0nz)Sma

2n 2D/ 4S 4

a / 4S 2a 0 1

8S 2a

erf ( Sa) / F(1 0 a)

1 f /ad dnxdnydnz 0 *0

0`*

`

0`*

`

0`

(0nz)Smm

2n 2/ a

2

√mm

maG (1 0 ea)

√p

6Sa

√TW

T J . [13]

The drag force acting on a nonevaporating particle, FD Å / i

2kTwD f /md2dnxdnydnzG / acaErJ

FDm / FDa , can thus be readily calculated using Eqs. [12]and [13]. For the case of small speed ratio (e.g., for Sj õ

Å (1 0 ea)14

aananV akTHFS 2a / 2 / aEr

aakT0.2) , Eqs. [12] and [13] reduce to

FDm Å 4pR 2nmkTH 4

3√p/

√p

6(1 0 em)

√TW

T JSm , [14] 0 S2(1 0 a) / a

2 S2 / i

2DD TW

T Gexp(0S 2acos2u)

FDa Å 4pR 2nakTH 4

3√p/ F(1 0 a) /

√pSacos uFS 2

a /52/ aEr

aakT0 S2(1 0 a)

/ a

2

√mm

maG

√p

6(1 0 ea)

√TW

T JSa . [15] / a

2 S2 / i

2DD TW

T G[1 / erf ( Sacos u)]J , [17]

The drag force acting on a nonevaporating particle, FD Å where aa and am are the thermal-accommodation factors forFDm / FDa , for the case of small speed ratios can be calcu- energy transfer due to atoms and molecules, i is the numberlated using Eqs. [14] and [15]. of the internal freedom degree in a molecule, Er is the recom-

In order to calculate the evaporation-added drag force, one bination energy per recombined atom, and all the energymust calculate the local heat flux from the high-temperature released in the recombination reaction A / A r M is as-gas to the immersed particle, q Å qc 0 qr , where qc is the sumed to be transferred completely to the immersed particleheat flux due to the energy transport by the incident and for the case of diffuse reflection.reflected gas particles. qr is the radiative heat loss from the For an evaporating particle immersed in a gas flow, theparticle surface and qr Å csT 4

w , in which c and s are the involving speed ratio is usually small (e.g., Sm or Sa õ 0.2)surface emissivity and the Stefan–Boltzmann constant. Since For this case, [16] and [17] reduce tothere is no energy exchange between the gas and the im-mersed particle when the reflection of gas atoms or moleculesat the sphere surface is specular (4, 7) , the local heat flux (qc)m Å

1 0 em

4amnmnV mkTHF S2 / i

2DS1 0 TW

T DGcomponents due to gas molecules (qc)m and atoms (qc)a are

/√pSmcos uF5

2/ i

20 S2 / i

2D TW

T GJ , [18](qc)m Å (1 0 em)amF*`

0*

`

0`*

`

0`

nzSmm

2n 2 / i

2kTD

(qc)a Å1 0 ea

4aananV akTHF2 / aEr

aakT0 S2 0 a / i

4aD TW

T G1 f 0m dnxdnydnz 0 *0

0`*

`

0`*

`

0`

(0nz)

1 Smm

2n 2 / i

2kTWD f /md1dnxdnydnzG

/√pSacos uF5

2/ aEr

aakT0 S2 0 a / i

4aD TW

T GJ .

Å 1 0 em

4amnmnV mkTHFS 2

m / 2 / i

20 S2 / i

2D TW

T G [19]

It is noted that the local heat flux qc Å (qc)a / (qc)m or1 exp(0S 2mcos2u) /

√pSmcos uFS 2

m /52/ i

2 q Å qc 0 qr contains two parts. The first part does not dependon the u position, while the other part involving

√pSj cos u

is u-dependent. It is the second part that is critical for the0 S2 / i

2D TW

T G[1 / erf ( Smcos u)]J , [16]evaporation-added drag force. The local heat flux q assumes

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99DRAG FORCE ON AN EVAPORATING PARTICLE

its maximum at the frontal stagnation point (u Å 0) and its F*V Å 2pR 2 *p

0

(p *cos u / t*sin u)sin uduminimum at the rear stagnation point (u Å p) . Since thelocal evaporated-mass flux m

g

is equal to the local heat flux(q) divided by the evaporation latent heat of particle material Å 2pR 2 1

LV

√pkTW

2mV*

p

0

[(qc)m / (qc)a]cos u sin udu(LV ), i.e., m

g

Å q /LV, mg

will also assume a similar u-distribu-tion as q . As a result, the evaporation-added drag force dueto the reaction force of the evaporated-mass flux will also Å 4pR 2 p

12LV

√kTW

2mVH(1 0 em)amnmnV mkT

be in the direction of the relative velocity vector U` betweenthe bulk gas and the immersed particle. Namely, evaporation

1 F52/ i

20 S2 / i

2D TW

T GSm / (1 0 ea)aananV akTalways enhances the drag force acting on a particle.The vapor molecules issuing from the particle surface due

to evaporation are assumed to satisfy the Maxwellian veloc-1 F5

2/ aEr

aa

0 S2 0 a / i

4aD TW

T GSaJ ,ity distribution function (4, 5)

fV Å A exp[0B(V 2x / V 2

y / V 2z )] (for Vz ° 0), [20] or by using the relat ion n

V jS j Å√8 k T / ( p mj) 1

(U` /√2kT /mj ) Å 2U` /

√p ,

where A and B are two constants which are determined bythe following two physical conditions (4, 5) : ( I ) the aver-age kinetic energy of the vapor molecules leaving from the F*V Å 4pR 2

√p

6LV

√kTW

2mVHam ,effnmF5

2/ i

20 S2 / i

2D TW

T Gparticle surface corresponds to the surface temperature TW ,and (II ) the local flux of the outflowing vapor molecules

/ aa ,effnaF52/ aEr

aa

0 S2 0 a / i

4aD TW

T GJkTU` ,is equal to the local mass flux mg

divided by the vapormolecule mass mV . By using the two physical conditions,one obtains (4, 5) [24]

where the effective thermal accommodation factor for mole-A Å 1

2p(kTW/mV )2 S qc 0 qr

mVLVD , B Å 1

2kTW/mV

. [21]cules am ,eff Å am(1 0 em) and that for atoms aa ,eff Å aa(10 ea ) have been employed. The total drag force acting onan evaporating particle FDV can thus be calculated by FDVConsequently, the velocity distribution function fV in Eq.Å FD / F *V , where FD Å FDm / FDa ( the drag force for[20] concerning the evaporated vapor molecules takes thenonevaporating case) is calculated by using Eqs. [14] andform[15], while F *V ( the evaporation-added drag force) is ob-tained by Eq. [24].

fV Å1

2p(kTW/mV )2 S qc 0 csT 4W

mVLVD

RESULTS AND DISCUSSIONS

1 expS0 V 2x / V 2

y / V 2z

2kTW/mVD . [22] Equations [12], [13] (or Eqs. [14], [15]) , and [24] show

that the drag force acting on a nonevaporating or evaporatingparticle for free molecule regime (Kn @ 1) is directly pro-

It is easy to show that the additional shear stress due to portional to the square of particle radius. Hence, the dragthe evaporated vapor molecules t* is zero, while the addi- force reduced to per unit surface area of the particle, i.e.,tional pressure at the sphere surface due to those vapor FD / (4pR 2) or FDV/ (4pR 2) , or the drag force coefficient (2–molecules is 5) will be independent of the particle radius. The drag force

on the spherical particle is in the direction of the relativevelocity vector U` . Evaporation always enhances the dragp* Å *

0

0`*

`

0`*

`

0`

mVV 2z fVdVxdVydVz force acting on a particle, and the evaporation-added part of

the total drag force is inversely proportional to LV (evapora-tion latent heat) and

√mV (square root of the vapor-moleculeÅ qc 0 csT 4

W

mVLV

√pmV kTW

2. [23]

mass) . All these results are identical to those obtained beforefor the case of ordinary temperature gases without consider-ing molecule dissociation and atom recombination (4).The evaporation-added drag force acting on the evaporating

particle is thus Examination of Eqs. [12], [13], (or Eqs. [14], [15]) and

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100 XI CHEN

variations with the gas temperature of the ratio of the pre-dicted drag forces with evaporation to without evaporationare plotted in Figs. 2 and 3 for nitrogen and oxygen atatmospheric pressure, respectively. It can be seen that evapo-ration may significantly enhance the drag force, especiallyat high gas temperatures. For the case of aa Å am Å 1.0, aÅ 0.5, ea Å em Å 0, and T Å 7000 K, the calculated ratioof the drag forces with evaporation to without evaporation(FDV/FD) can be as great as 1.63 for nitrogen and 1.58 foroxygen. The values of the specular-reflection fraction foratoms and molecules (ea Å em) affect appreciably the ratioof the drag forces with evaporation to without evaporation,since the atomic and molecular components of the evapora-tion-added drag force are directly proportional to the effec-tive thermal accommodation factors aa ,eff or am ,eff , respec-tively. The difference between the calculated results pre-sented in Figs. 2 and 3 is due to the fact that an appreciabledissociation degree appears at a much lower temperature foroxygen (3500 K) than that for nitrogen (5500 K).

Figures 3 and 4 compare the effect of gas pressure on thedrag force ratio FDV/FD . It is expected that the equilibriumstate of the dissociation reaction O2 r O / O will shift toFIG. 2. Variation with the gas temperature of the ratio of the dragthe direction with greater mole fraction of atom species atforces with evaporation to without evaporation (FDV/FD) . Water droplet in

nitrogen: Er Å 4.88 eV, 105 Pa; TW Å 373 K; LV Å 2.26 1 106 J/kg; lower gas pressure according to LeChatelier’s principle. Inthermal accommodation factors aa Å am Å 1; atom recombination fraction addition, with the decrease of gas pressure, TW decreasesa Å 0.5. while LV increases. However, the difference between the

[24] derived for the case of high-temperature diatomic gasesand with a combined specular and diffuse reflection can alsoreveal the following new results: (A) the drag force actingon a nonevaporating particle depends on the fractions ofspecularly reflected molecules and atoms (ea and em) andthe atom recombination fraction a only through the termsinvolving the surface/gas temperature ratio (TW/T ) . Hence,it is expected that the effect of a, ea , and em on the dragforce will be small as small TW/T is concerned. (B) Theevaporation-added part of the drag force acting on an evapo-rating particle is dependent on the effective thermal-accom-modation factors aa ,eff and am ,eff , the atom recombinationenergy Er , and the atom recombination fraction a. (C) Theratio of the predicted drag forces with evaporation to withoutevaporation (FDV/FD) at small speed ratios does not dependon the relative velocity U` , since both FD and FDV are propor-tional to U` for this case. (D) The atom or molecule compo-nents of the drag force acting on a nonevaporating or evapo-rating particle are directly proportional to the atom and mole-cule number densities. Hence, the relative contribution ofatoms to the total drag force over that of molecules will bedependent on the dissociation degree of the diatomic gas oron the gas temperature level.

FIG. 3. Variation with the gas temperature of the ratio of the dragSome typical calculated results are shown in Figs. 2–8 forces with evaporation to without evaporation (FDV/FD) . Water droplet in

for a small water droplet immersed into a high temperature oxygen: Er Å 2.56 eV, 105 Pa; TW Å 373 K; LV Å 2.26 1 106 J/kg; thermalaccommodation factors aa Å am Å 1; atom recombination fraction a Å 0.5.diatomic gases within the free molecule regime. Calculated

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101DRAG FORCE ON AN EVAPORATING PARTICLE

calculated results for FDV/FD at 105 Pa and at 104 Pa shownin Figs. 3 and 4 is not too much.

Figure 5 plots the calculated variation with the recombina-tion fraction a of atoms at the particle surface of the dragforce on a nonevaporating or an evaporating particle for 3different oxygen temperatures. At 1000 K with negligiblegas dissociation, the atom number density is much smallerthan the molecule number density and thus a has no effecton the drag force. With the increase of temperature or oxygendissociation degree, the effect of a on the drag force becomesappreciable. In addition, with the increase of a, FDV/FDV0

increases while FD /FD0 decreases, where FDV0 and FD0 arethe drag forces for a Å 0. As seen from Fig. 5, the effectof a on the drag force on a nonevaporating particle is muchless than that on an evaporating particle.

Figure 6 compares the calculated variation with the gastemperature of the atomic and molecular fractions in thedrag force for a water droplet immersed in a nitrogen oroxygen. The calculated results for a nonevaporating particleare almost identical to those for an evaporating particle,especially for the droplet immersed into oxygen. As ex-pected, Fig. 6 shows that at low gas temperatures with negli- FIG. 5. Variation with the atom recombination fraction of the draggible molecule dissociation (e.g., less than 4000 K for nitro- force on a nonevaporating (lower) or evaporating (upper) particle. Water

droplet in oxygen: Er Å 2.56 eV, 105 Pa; TW Å 373 K; LV Å 2.26 1 106gen or less than 2000 K for oxygen), the values of theJ/kg; thermal accommodation factors aa Å am Å 1; ea Å em Å 0. FD0 andmolecular fraction are almost 1.0; i.e., the drag force is al-FDV0 are the values of FD and FDV for a Å 0. 1, 1000 K; 2, 3000 K; 3,most completely caused by the molecule species of the di-5000 K.

atomic gases. For this case, Eqs. [14] and [24] will reduce

to their counterparts obtained previously for ordinary tem-perature gases (2–4). On the other hand, at high gas temper-atures associated with appreciable gas dissociation degree(e.g., when the gas temperature is higher than 4000 K foroxygen or higher than 7000 K for nitrogen), the drag forcewill be caused mainly by the atom species. Including gasdissociation in the drag force calculation will be importantfor this case.

Figures 7 and 8 show the variation with gas temperatureof the calculated values FD ,WOD /FD and FDV,WOD /FDV, whereFD ,WOD and FDV,WOD are the drag forces on nonevaporatingand evaporating particles calculated without accounting forthe gas dissociation; i.e., all the gas particles are treated asmolecules. As seen from Figs. 7 and 8, when the gas temper-ature is lower than about 3000 K for oxygen, neglecting gasdissociation is permissible. However, at high gas tempera-tures with appreciable dissociation degree, ignoring gas dis-sociation will appreciably overestimate the particle dragforce.

CONCLUSIONSFIG. 4. Variation with the gas temperature of the ratio of the drag

forces with evaporation to without evaporation (FDV/FD) . Water droplet inFor the drag force acting on a nonevaporating or evaporat-oxygen: Er Å 2.56 eV, 104 Pa; TW Å 319 K; LV Å 2.39 1 106 J/kg; thermal

accommodation factors aa Å am Å 1; atom recombination fraction a Å 0.5. ing particle immersed in a diatomic gas, the present analysis

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102 XI CHEN

FIG. 8. Variation with the gas temperature of the ratio of the dragforces on an evaporating particle with gas dissociation to that withoutdissociation (FDV,WOD /FDV ). Water droplet in oxygen: Er Å 2.56 eV, 105FIG. 6. Variation with the gas temperature of the atomic (A) and molec-Pa; TW Å 373 K; LV Å 2.26 1 106 J/kg; thermal accommodation factorsular (M) drag force fractions. Water droplet in oxygen and nitrogen: 10 5

aa Å am Å 1; atom recombination fraction a Å 0.5.Pa; TW Å 373 K; LV Å 2.26 1 106 J/kg; thermal accommodation factorsaa Å am Å 1; ea Å em Å 0; a Å 0.5. Continuous lines ( —) withoutevaporation, dashed lines (---) with evaporation. for a free-molecule regime obtains the following conclu-

sions. As for the case of ordinary temperature gases, thedrag force is in the direction of the oncoming gas velocityand is proportional to the square of particle radius. Evapora-tion may significantly enhance the drag force, especially forthe case of the particle materials with low evaporation latentheat and small molecular mass. The ratio of the predicteddrag forces with evaporation to without evaporation doesnot depend on the relative velocity (U`) between the particleand the bulk gas at small speed ratios. The drag force actingon a nonevaporating particle depends on the specular-reflec-tion fractions (ea and em) and the atom recombination frac-tion (a) only through the terms involving the surface/gastemperature ratio TW/T , while the effective thermal-accom-modation factors and the atom recombination fraction (a)affect substantially the evaporation-added drag force. Therecombination fraction of atoms at the particle surface affectsthe drag force on an evaporating particle much more appreci-ably than that on a nonevaporating particle. At low gas tem-peratures with negligible gas dissociation, the drag force iscaused by the molecule species, while it can be attributedmainly to atom species at high gas temperatures. Ignoringgas dissociation will overestimate the particle drag force forthe latter case.FIG. 7. Variation with the gas temperature of the ratio of the drag

forces on a nonevaporating particle with gas dissociation to that withoutACKNOWLEDGMENTdissociation (FD ,WOD /FD) . Water droplet in oxygen: Er Å 2.56 eV, 105 Pa;

TW Å 373 K; LV Å 2.26 1 106 J/kg; thermal accommodation factors aa Å This work was supported by the National Natural Science Foundation ofChina through Grant 59676011.am Å 1; atom recombination fraction a Å 0.5.

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103DRAG FORCE ON AN EVAPORATING PARTICLE

4. Chen, Xi, ‘‘Kinetic Theory and Its Applications to Studies of HeatREFERENCESTransfer and Fluid Flow,’’ Chapt. 4. Tsinghua Univ. Press, Beijing,1996. (In Chinese)

1. Ashley, H., J. Aeronaut. Sci. 16, 95 (1949). 5. Chen, Xi, J. Phys. D 29, 995 (1996).2. Schaaf, S. A., and Chambre, P. L., ‘‘Flow of Rarefied Gases,’’ pp. 17– 6. Chen, Xi, J. Colloid Interface Sci. 191, 482 (1997).

21 and 35–52. Princeton Univ. Press, Princeton, NJ, 1961. 7. Bird, G. A., ‘‘Molecular Gas Dynamics,’’ pp. 73–75 and 91–98.3. Koshmarov, Yu. A., and Ruizov, Yu. A., ‘‘Applied Rarefied Gas Clarendon Press, Oxford, 1976.

Dynamics,’’ pp. 107–121. Mashnostroenue, Moscow, 1977. ( In 8. Chen, Xi, and He Ping, Plasma Chem. Plasma Process. 6, 313(1986) .Russian )

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