Plasma Chemistry and Plasma Processing, Vol. 12, No. 4, 1992

Enhancement of Heat Transfer from a Rarefied

Plasma Flow to Thermoemitting Particle

A. G. Gnedovets 1 and A. A. Uglov 1

Received June 4, 1991; revised December 21, 1991

The methods of molecular gas dynamics are applied to the description o f heat transfer from a rarefied plasma flow to a spherical metallic or nonmetallic particle. It has been shown that under conditions of strong thermoemission the rate of heat transfer is considerably enhanced as a result o f the contribution o f hot plasma electrons due to the decrease in particle potential by its absolute value.

KEY WORDS: Rarefied plasma flow; spherical metallic or nonmetallic particle; particle charging; thermal electron emission; heat transfer.

1. I N T R O D U C T I O N

One of the most important effects in the plasma processing of fine particles of materials is associated with gas ionization and presence of charges--negative electrons and positive ions together with neutral molecules in thermal plasma flow. The rate of heat transfer from the ionized gas is determined not only by collisions of the particle surface with molecules but also with electrons and ions. The laws of plasma-particle interaction are determined by the accumulation of charges on a particle and by the establishment of a local electric field affecting the motion of electrons and ions. The effect of particle charging on heat and momentum transfer from a rarefied plasma has been studied in refs. 1-11 on the basis of the free molecular flow approach. C~2-~4)

Due to the different mean thermal velocities of plasma electrons and ions [ t~e /~ - (m~Te/meTi )~ /2>> 1], a particle in a quasi-neutral plasma has a negative potential ~oy < 0, at which charge fluxes carried by electrons and ions balance each other. At high temperatures of the material, thermal

JA. A. Baikov Institute of Metallurgy, Russian Academy of Sciences, Leninsky Prospekt 49, Moscow 117911, Russia.

371 0272-4324/92/1200-0371506.50/0 © 1992 Plenum Publishing Corporation

3 7 2 Gnedovets and Uglov

electron emission from the particle surface may lead to a decrease in the absolute value or a change in the sign of the equilibrium (floating) potential ~r. This variation of the particle potential results in corresponding changes in the heat fluxes transferred by electrons and ions.

In the present work, kinetic theory analysis is applied to the problem of heat exchange between a rarefied (Kn = I /R >> 1) thermal plasma flow and a thermoemitt ing metallic or nonmetallic spherical particle with an extremely thin plasma sheath ( rD/R << 1 ). The diffuse scattering law is chosen for molecules and neutralized ions reflecting from the particle. The intensity of electron emission is considered to be limited, so that the particle floating potential remains negative. The above limitations make it possible to use in the calculations the already known half-range Maxwellian velocity distri- bution functions for each kind of plasma species (molecules, electrons, and ions) incident on the particle or reflected from its surface, and for emitted thermal electrons. The assumptions made above are realistic for many practical cases in the plasma processing of powder materials, ~-~'t~ and the results of the present analysis concern the micron-sized particles in low- pressure thermal plasma flows.

2. ANALYTICAL D E S C R I P T I O N

The following mechanisms of energy transport processes must be intro- duced in the power balance at the surface of thermoemitt ing particles immersed in a rarefied plasma flow:

1. Transfer of kinetic energy from each kind of plasma species (molecules, electrons, and ions) to the particle surface, including contributions of electron speed reduction and ion acceleration in the local electric field of the particle;

2. Transfer of potential energies of charge states as a result of electron recombination and ion neutralization at the surface;

3. Heat losses due to diffuse reflection of plasma molecules and neutral- ized ions from the particle;

4. Energy losses due to electron emission, including both kinetic and potential energies.

In some special situations, additional terms accounting for evapor- ation (41 and radiation ~ 17~ losses must be introduced in the heat balance given above.

The free molecular flow approach describes the dimensionless local fluxes of the number of j th species j , = J~/J}' and their kinetic energy

± ± 0 ej = E j / E j transferred to ( - ) and from (+) the surface at a given point A

Heat Transfer in Rarefied Plasmas 373

de te rmined by the center angle O measured f rom the frontal s tagnat ion point o f the sphere in the fol lowing way:

j f = exp[-(cj , , , - sin) 2] + zrl/2sjn erfc(cjm - sin) . + . +

J h = - - J h , J~ = - - jT~

e l = [ 1 1 2 +~(C~,. + Cj,,,sj. + s~)] e x p [ - ( c j ~ - sj.) ~] (1)

+ ½zr'/2sj.(~+ s2) erfc(cj,,, - sj.) + ½zij;'r~jy r + +

eh = --jh r~, ee = --jTeT~/ Ze

where y f= -e~of/kTe, re = Te/Th, r~ = T~/Th, r~j= Te/Tj, sj= V / ( 2 k T j / m j ) t/~, s~. =sj cos 0, c ~ =c. , , = 0 , ce,. =y)/2, z~ = 0 , z~ = 1, z~ = - 1 , j o = Nj(kTj/2~rrn~)~/2, and E ° = NjkTj(2kTJzrmj) ~/2. Here V is p lasma flow velocity, ~ is t empera ture , Nj is n u m b e r density, subscripts h = a, i denote heavy p lasma species (molecules a and ions i), e are electrons, Te are the rmoemiss ion electrons, and s is particle surface. The coordinate system is shown in Fig. 1.

The d imensionless local electron emission current jr~ = Jre/J ° is deter- mined by the part icle surface t empera tu re T~ according to the R ichardson- D u s h m a n fo rmula t~8~

JTe = (AR /e ) T~ e x p ( - ~ e / k T ~ ) (2)

The d imensionless local heat flux densities qj = Q j / E ° caused by the j t h kind of species at a given point on the surface of thermoemi t t ing particle

Fig. l . Coordinate system.

V

374 Gnedovets and Uglov

can be written as

qh = ef, + j-h (½w,, - "(~) (3)

qe = e~, +½j;we " t - - J T e ( ~ W e " ~ - ' r J T e )

where wj = W j / k ~ , Wa =0, We =~, , , W~ = L - t h e , ~,, is work function, and L is ionization energy.

The dimensionless total heat fluxes (q j) = ( 1 / S p E °) j Qj dSp transferred by the j th kind of species to the whole particle surface Sp are calculated as

(q,,) = (e~) + (j~)(½w. - %) (4)

( q,) = ( e ;) + ½(j ;) We -- (jrc)(½ We + r~ )

where ( j r ) = ( 1 / S i l °) J J f dSp and (e j ) = ( 1 / S p E °) ~ E f dSp. Evaluation of the total fluxes transferred to the particle can be perfor-

med by numerical integration. Charge accumulation on the particle goes on until the charge fluxes of

incident negative electrons, positive ions, and emitted thermoelectrons bal- ance each other. The charging process is very rapid as compared with thermal and hydrodynamical phenomena. (1'3-s) Hence, the description of plasma-part icle interaction can be carried out in the quasi-steady approach with an equilibrium (floating) potential of the particle ~pp = q~r. Since the surface of a metallic (conducting) particle is an equipotential surface, the value of the floating potential q~j is determined by equating the total charge flux of electrons to that of ions. For nonmetallic (nonconducting) particle, the local value of the floating potential q~: is determined by equating the local charge flux densities of electrons and ions at every point of the particle surface. These conditions are written in dimensionless form as

( j ; ) - (jre) = (IZe/7",.)'/2(j,) (5)

for metallic particle (y: = const), and

j~, - j r ~ = (I.t~/ ~',,)t/2j[ (6)

for nonmetallic particle [y f= y:( O)]. The mass ratio parameter /zj = m j / m s is introduced here. In the case of plasma flow with low speed ratio s = Sh < 1, the formulas

describing heat transfer to thermoemitting particle can be simplified. Both for metallic and nonmetallic particles, Eq. (1) for local fluxes reduces to

j~- = exp( -y / ) , ee = exp ( -y r )

jh = 1 + rrl/2s. + s~, (7) - - 5 1/2 - - 1 2 - - 3 2 1 .

eh= l - r aTr s, n-~_s ~-~_Sn'+-~ZhJh reJ{t.

where s, = sh, and the condition Se << Sh = S < 1 is taken into account.

Heat Transfer in Rarefied Plasmas 3 7 5

For metallic particles, the expressions for total fluxes with an accuracy up to second-order corrections - s 2 are written as

(jh) = 1 + ~s z

I "~ (j~) = (1 +_~o~s') exp( -y~) (8)

2 I 0 1 0 (el,) = 1 + s +~z , , -c , , [y t+~(yc-a)s 2]

I ~' (e~,) = (1 + go~s-) exp(-y.~)

Here

yf = y O _ ~as 2, y~=ln[a(7"e/i , ze)'/2], a = l l [ l + ( ' r , , / p , e ) ' / ~ ' r ~ ]

Averaging over the surface o f nonmetal l ic particles leads to the same expressions for (ji,), (jS), and (e~). The equation for the dimensionless total flux of kinetic energy transferred by heavy plasma species can be written in this case as

2, f o 5 J f (e~,)= l + s +~z , , ' r , ,~y f+? (9)

3. RESULTS AND D I S C U S S I O N

The influence o f thermal electron current jr~, emitted from a metallic or nonmetal l ic particle and the speed ratio s = sh o f rarefied argon plasma flow on the distributions o f dimensionless particle floating potential . v r = - e ~ , . / k T e , local heat flux densities q * = Q J E * = ( e ~ / ~ ; ) ' / 2 q j , and total heat fluxes ( q ) ~ ) = ( 1 / S p E * ) ~ q j d S p = ( e ~ / m ) ' / 2 ( q j ) (here E * = NjkT~,(2kT,,/'n'm;,) I/2, ej = Tj/ T~,, txj = mi/m;,) was studied with the help of numerical modeling. The calculations were performed for the following values o f the dimensionless parameters: d~e/kTe=4.5 , L / k T e = 15.8, T~ = T~/Tj,<< 1, r,. = T,,/Tj, = 1 (Figs. 2-4), and r, ,= T,,/T~, = 10 (Fig. 5).

In a stat ionary (s = 0) plasma, no matter whether the particle is metallic or nonmetallic, it has a constant floating potential and uniform distribution o f local flux densities o f each kind of plasma species all over its surface. If the thermal electron emission is insignificant (jr~, = 0), the dimensionless floating potential and heat fluxes transferred from a one-temperature (re = Te/T;, = 1) argon plasma give the following values: yr = 5.60, q* = (q*)= 1, q* = (q*) = 3.25, q* = (q*) = 9.35. The total heat fluxes are enhanced by the presence of plasma flow relative velocity.

Figures 2 and 3 compare variations o f the dimensionless local floating potential Yr and the local heat flux densities q* with the center angle 0 on the surface o f metallic and nonmetall ic spherical particles for different

376 Gnedovets and Uglov

10

8

6

4

2 -

J J

1 / i J

f /

/ /

/

2

06 4-'5 90 1 35 180 @ (Degree)

Fig. 2. Surface distributions of dimensionless local floating potential y/ for metallic (solid lines) and nonmetallic (dashed lines) spherical particle in Ar plasma flow (s = 1, % = 1) for ditterent thermal electron emission currents Jr~ = 0 (1) and 0.1 (2).

intensities of thermal electron emission jre = 0 and 0.1 at fixed argon plasma flow speed ratio s = 1 and electron-heavy species temperature ratio ze-- Te/Th ---- 1. A metallic particle always has a constant floating potential over its surface. In the absence of electron emission, a nonmetallic particle has a noticeably nonuniform surface distribution of floating potential, with a minimum at the frontal stagnation point of the sphere. Substantially nonuni- form distributions are observed for the heat flux densities of heavy plasma species, molecules, and ions, with higher values at the upstream surface and lower values at the downstream surface of both the metallic and the nonmetallic particles. The electron heat flux density has a nonuniform distribution over the nonmetallic particles and a comparatively uniform

Heat Transfer in Rarefied Plasmas 377

40

30

1o

0

- -2 .....

\\\

\\ 5

\\

5\\\

45 90 ]55 180 0 (Degree)

40

30

20

~0

b \

\\\

. \ 5 \ \ .

4'5 90 (S) (Degree)

Fig. 3. Surface distributions of dimensionless local heat flux densities q* of molecules (1), electrons (2), and ions (3) for metallic (a) and nonmetallic (b) particles in Ar plasma flow ( s= 1, r e = 1) for different thermal electron emission currents ire =0 (solid lines) and 0.1 (dashed lines).

d i s t r ibut ion over the m e t a l l i c part ic les d u e to the d i f f erence b e t w e e n the d i s t r ibut ions o f the f loat ing po tent ia l in these t w o cases . T h e r m a l e lectron

e m i s s i o n leads to the decrease o f the f loat ing potent ia l by its a b s o l u t e value . T h e surface d i s t r ibut ion o f the loca l f loat ing po tent ia l o f n o n m e t a l l i c parti- c les b e c o m e s s m o o t h e r , so that the d i f f erences b e t w e e n m e t a l l i c and non- m e t a l l i c part ic les d i sappear . E v e n at l o w va lues o f e lec tron e m i s s i o n current JTe (< 1, the in tens i ty o f the e lec tron heat f lux dens i ty q~* rises c o n s i d e r a b l y , w h i l e the ion heat f lux dens i ty q,* decreases s l ight ly .

CI

~v

1 0 -~ 1 0 -2 1 0 - '

10 b

10 . . . . . /

JTe -JTe

Fig. 4. Influence of thermal electron emission current Jre on (a) floating potential yj and (b) total heat fluxes (q~') of molecules (1), electrons (2), and ions (3) for metallic particle in Ar plasma flow (re = 1). Solid lines, s=0; dashed lines, s = 1.

378 Gnedovets and Ugiov

5

\ \

\ \

JTe

l O

l o

1

o

01

/ / .// '

S k 5 5

- . . . . . . 5

1

J'r~

Fig. 5. Influence of thermal electron emission current Jr,. on (a) floating potential kit and (b) total heat fluxes (q*) of molecules (1), electrons (2), and ions (3) for metallic particle in stationary (s =0) Ar plasma. Solid lines, % = 1; dashed lines, r,, = 10.

The effect o f the thermal electron emission current JT,. = (JT,') on the floating potential Yl and the total heat fluxes (q*) for metallic particles in a stationary (s = 0 ) and a flowing (s = 1) rarefied one- temperature (r,. = 1) argon plasma is illustrated in Fig. 4. With the growth of JT,', the particle floating potential Yr decreases, and the total heat flux (q*) remains the same for molecules, decreases a little for ions, and increases strongly for electrons. At fixed jT~., the lower values of Yr and higher values of (q*) correspond to flowing plasma as compared with stat ionary plasma. Under the condit ions of strong electron emission (jT,. ~ 1, Yr ~ 0), the total heat flux o f electrons (q*) rises two orders of magni tude in compar ison with the case of nonemit- ting (jT,. = 0) particle. In spite of the differences in the local floating potential distributions, nearly the same results are obtained for the total heat fluxes transferred to the nonmetall ic particles.

The temperatures of electrons T,. and heavy species Th could be different for low-pressure plasma conditions. The distinctions between charge and heat transfer to a thermoemit t ing spherical particle from a one- temperature (re = T,.I Th = 1) and a two-temperature (re = Te/Th = 10) stationary (s = 0) argon plasma at fixed electron temperature T~ are shown in Fig. 5. In the case of a nonemit t ing (JT,. = 0) particle, the higher values o f the dimension- less floating potential Yr and, hence, the lower values o f the electron heat flux (q*), correspond to the two-temperature plasma. At strong electron emission (jTe ~ 1), the particle floating potential Yr drops to zero, and the total electron heat fluxes (q*) coincide for both the one- and the two- temperature plasma. Therefore, the enhancement of heat transfer due to thermal electron emission rises with the increase in the e lec t ron-heavy species temperature ratio ~'~ for a two-temperature plasma.

Heat Transfer in Rarefied Plasmas 379

The reasons for the enhancement of the intensity of heat transfer from plasma to thermoemitting particle are the following. If the emission from the surface is negligible, the flux of incident plasma electrons is limited by the high negative potential of the particle. Thermal electron emission leads to a decrease in the height of the potential barrier and an increase in the collision frequency between particle surface and "hot" plasma electrons, with the energy exceeding the energy removed from the particle by "cold" emission electrons. It is important that in the case of cold nonemitting particles, the intensities of electron and ion heat fluxes depend on the contribution of the potential energies of their charge states, and in the case of thermoemitting particles, the main mechanism of heat transfer is associated with the flux of kinetic energy of plasma electrons. For example, in a stationary plasma, Eq. (3) for the electron heat flux taking into account Eqs. (5) and (7) can be rewritten as ( q * ) = q * = l+(re/~e)~/~jTe(1 --r.,)+(1/2)We. AtjTe >> (/z,./~'e) ~/-~, r,<< 1, one gets (q*)= (Te/l, Ze)l/2J'Te. Hence, the contribution of the kinetic energy of plasma elec- trons to the total heat balance at the particle surface is in direct proportion to the intensity of thermal electron emission. It should also be noted that under plasma conditions (q*)+(q*)>> (q*)= 1, which means that even at low degrees of gas ionization the contribution of electrons and ions to particle heating remains significant.

The results of the numerical modelling of heating, melting, and evapor- ation of a thermally thin (Bi<< 1, Tp = T~) molybdenum particle (R = 1/~m) in an argon plasma (Tg = 14,000 K, P~ =0.1 atm) are presented in Fig. 6, which illustrates the relative importance of various mechanisms involved in plasma-particle heat transfer. These calculations were performed on the

7 7

6

~ 5

o 4

2

]

/ /

f _ _ _ _ m f

J

J /

O 0.'1 0.'2 O.L3 O J4 t ( m s )

6

5

3

2

1

O

Fig. 6. Heating, melting, and evaporation of Mo particle (R = 1 ~m) in Ar plasma (Tg = 14,000 K, Pg =0.1 atm). (I) With account for ionization and emission; (2) with account for ionization but without account for emission; (3) without account for ionization and emission.

380 Gnedovets and Uglov

basis of three different models: (1) with account for gas ionization and thermal electron emission from the particle; (2) with account for ionization but without account for emission; (3) without account for ionization and emission. Heat losses due to particle melting, evaporation, and radiation were also included in these models. The results of calculations show that participation of electrons and ions in heat transfer leads to considerable growth in the particle heating rate in the plasma as compared with a hot molecular gas. At high particle temperatures corresponding to the regime of thermal electron emission, the increase in heat transfer to particles from the plasma takes place as a result of the contribution of plasma electrons. Due to the exponential dependence of the free-molecular vapor mass flow rate on metal temperature, the three chosen models give significantly different calculated times of particle evaporation: ~0.2 ms for emitting particles in a plasma; ~ 1.2 ms for nonemitting particles in a plasma; and ~8.4 ms for nonemitting particles in a hot gas. Therefore, ionization at high gas temperatures and thermal electron emission at high particle material temperatures ensure considerable increase in plasma processing efficiency. Similar calculations carried out for particles of a number of metals in an argon plasma within the pressure range Px =0.001-1 atm at temperature T~ -~ 104 K show that thermal electron emission enhances the heating rate of particles of refractory metals (W, Mo, Zr). This effect may play an important role for heat transfer from a low-pressure thermal plasma to particles of materials with low values of electron work function (oxides, interoxides, or metal compounds).

4. C O N C L U S I O N

The main results of this study may be summarized as follows:

1. Plasma-particle heat exchange is greatly influenced by the processes of charge transfer and particle charging. The high efficiency of plasma action on a particle is due to the participation of electrons and ions in heat transfer process.

2. Heat transfer from plasma to particle is enhanced by thermal electron emission from its surface as a result of the contribution of the kinetic energy of hot plasma electrons due to the decrease in particle floating potential by its absolute value.

3. The enh'ancement of heat transfer due to thermal electron emission increases with increase in the electron-heavy species temperature ratio for a two-temperature plasma.

4. The effect of thermal electron emission on heat transfer from a low-pressure thermal plasma is of great importance for particles of refractory metals and materials with low values of electron work function.

Heat Transfer in Rarefied Plasmas 381

NOMENCLATURE

A R

e ±

E j li

± J j k I m~

oj rD

R s

s. rj

V 0

Richardson constant electron charge flux density of kinetic energy ionization energy flux density of number of plasma species Boltzmann constant mean free path plasma species mass number density of plasma species pressure heat flux density Debye length particle radius speed ratio particle surface temperature thermal motion speed of plasma species plasma flow velocity relatively to the particle center angle particle floating potential work function

Subscripts a

e

g h i P s

Te +(-)

molecules electrons plasma (gas) heavy plasma species (molecules and ions) ions particle surface emitted thermal electrons reflected from the particle (incident) plasma species

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382 Gnedovets and Uglov

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