- Email: [email protected]
18 P 32 Thermophoresis of aerosol particles—a kinetic analysis
J Aem,.~ol Sci., Vol 24. Suppl. I. pp. S I 8 3 - S I 8 4 , 1993 Printed in Great Britain
0021 8502/93 $6.00 + 0 . 0 0 Pergamon Press Ltd
18 P 33 ~/~FB~SIS
C~ A~K)SOL PARTIC[~/~ - A KINETIC ANALI"SIS
S. Beresnev
Ural State University, Department of Molecular Physics, Lenin Str. 51, 620083 Ekatherinburg, Russia
KEYkDRDS thermophoresis,
Knudsen
number,
acccmmodation
coefficients,
gas-kinetic
equation, integral-moment method
METHODS A kinetic theory for the thermophoretic force and
velocity
of
a
spherical
aerosol particle is presented. The analysis is carried out on the basis of the linearized BGK and S model (Shakhov, 1968) kinetic equations. The S model ( a third order model kinetic equation) yields 13 correct moments of the velocity distribution function and gives the correct
Prandtl
number
Pr
=
2/3
for
monatomic gas. The integral-moment method of solution for arbitrary values of the Enudsen number is employed.
The
possibility
of
arbitrary
energy
and
tangential momentum acccmr~x~ation of gas molecules on the particle surface is taken
into
account
in
the
boundary
condition.
The
particle-gas
conductivity ratio A is assumed to be arbitrary. NumericaI results
for
heat the
thermophoretic force and velocity for the whole range of Knudsen numbers have been obtained.
RESULTS In the case of complete accommodation the dimensionless thermophoretic force Fth for the whole range
of
/62 numbers
is
determined
by
the
following
analitical expression: Fth = ( ~ + ~
A)/[ 1 + (I + 2.5 A A ~ ) ~ ]
,
(i)
I/2
where
* v " 32[ ~m Fth = Fth / Fth , Fth : - I--5 8kT o
1
The functions ~,,z,a are dependent only on the A~ table I. S183
z
R o %g (v T)~, number
and
A = are
%e/Xg, given
in
SI84
~, BER~,S~[V
1000 800 600 400 200 100 80 60 40 20 I0 8
-1.324(-5) -2.067(-5) -3.673(-5) -8.256(-5) -3.291(-4) -1.307(-3) -2.036(-3) -3.598(-3) -8.004(-3) -3.115(-2) -1.109(-1) -1.836(-1)
1.366(-9) 2.677(-9) 6.379(-9) 2.175(-8) 1.791(-7) 1.502(-6) 2.987(-6) 7.225(-6) 2.439(-5) 1.066(-4) -1.852(-3) -6.174(-3)
-1.004 -1.006 -1.007 -1.011 -1.022 -1.044 -1.055 -1.074 -i. II0 -1.218 -1.415 -1.504
6 4 2 1 0.8 0.6 0.4 0.2 0.I 0.08 0.06 0.04
-0.2626 -0.4780 -1.059 -1.749 -1.952 -2.181 -2.439 -2.717 -2.857 -2.884 -2.910 -2.935
-0.0224 -0.1013 -0.7320 -3.078 -4.501 -7.034 -12.37 -28.89 -~2.20 -78.85 -106.6 -182.1
i.' The • ~, ~2, ~9 dependence on the inverse 102 number R = ~ di)~nsionless ~ h o r e t i e following form: Uth = U t ~ U ~ n =
Fth/F
velocity Uth in the same 8 % U th=
U,
5(8+~)
T
case
(v T)~ ,
-1.635 -1.847 -2.226 -2.533 -2.609 -2.691 -2.781 -2.881 -2.937 -2.949 -2.961 -2.973 I/2
/(210~)
has
the
(2)
O
where FU
is the dimensionless isothermal drag (Beresnev et al., 1990).
In flg 1,2 c ~ a r i s o n of th~oretical(for S model)and experimental data the thermophoretic force and velocity is presented. As may be seen, results for the S model agree satisfactorily with the experiment over range of /62 numbers studied at quite real tangential momentum and energy com~x~tion coefficients values. O. S
"~'"
~
"Ft h
~~%'~""~ •.Z/
$/
Oi ¢
-,
~
0. t
:L.O
~0.0
Mr*udmer,
i o
0.0
0.£ number
/" ~--~-~ / , e~/ /v // //
o
I
o. o ~
" "/ " ~:/" ! I
I i*u~r
o. ~
I I llJlJ
~. o
l
for the all ac-
The thermophoretic forcegFLh: /'2.. ~ . " a - A=I0 , b _/./~ .. theory and ~ i m e ~ t . .//*/" A=I0,1- o~=~_=I.0, 2- ~ =0.8, 3- ~ =0.6 4- Jacobsen ~nd Brock, 1369, NaCI-A~, A~240,5 ~ ek =~ =i.0, 6- ~ =0.8, m-Schadt and C ~ l e , ~ 9 6 t , NaCl-air ~, A~240, o-TC~air, A~8, A- f~-alr, A~440, x - Rosenblatt and LaMer,1948, TCP-air, A~8, • Schmitt, 1959, silicon oil- At, A~II. t.0 Mr,
----~-_-~-'_-~~ The thermophoretic velocity Ut~ for NaCl-particles in air(A~240) :theory and experiment. I- ~ = ~ = I 0,2- ~=0.8, I I IIIIII
~o. o
I
3- a_=0.6,4- Lipmtov and Chex~ova, 1989, • - ~redi et ai.,1979, x - Derjaguin et ~oo. o al., 1966. I IIIII
Bere~ev, S.A. ,V.G.Chavnyak, G.A.Fo~gagin( 1990), J.Fluid Mech. ,Vol. 219,405-421. Llpmtov,G.N. ,E.A.Chernova (1989), J.Aeroaol 8ci., Vol.20, 931"933. Hhakhov,E.M. (1988), Fluid Dyn., Vol. 3, 95,99.
0021 8502/93 $6.00 + 0 . 0 0 Pergamon Press Ltd
18 P 33 ~/~FB~SIS
C~ A~K)SOL PARTIC[~/~ - A KINETIC ANALI"SIS
S. Beresnev
Ural State University, Department of Molecular Physics, Lenin Str. 51, 620083 Ekatherinburg, Russia
KEYkDRDS thermophoresis,
Knudsen
number,
acccmmodation
coefficients,
gas-kinetic
equation, integral-moment method
METHODS A kinetic theory for the thermophoretic force and
velocity
of
a
spherical
aerosol particle is presented. The analysis is carried out on the basis of the linearized BGK and S model (Shakhov, 1968) kinetic equations. The S model ( a third order model kinetic equation) yields 13 correct moments of the velocity distribution function and gives the correct
Prandtl
number
Pr
=
2/3
for
monatomic gas. The integral-moment method of solution for arbitrary values of the Enudsen number is employed.
The
possibility
of
arbitrary
energy
and
tangential momentum acccmr~x~ation of gas molecules on the particle surface is taken
into
account
in
the
boundary
condition.
The
particle-gas
conductivity ratio A is assumed to be arbitrary. NumericaI results
for
heat the
thermophoretic force and velocity for the whole range of Knudsen numbers have been obtained.
RESULTS In the case of complete accommodation the dimensionless thermophoretic force Fth for the whole range
of
/62 numbers
is
determined
by
the
following
analitical expression: Fth = ( ~ + ~
A)/[ 1 + (I + 2.5 A A ~ ) ~ ]
,
(i)
I/2
where
* v " 32[ ~m Fth = Fth / Fth , Fth : - I--5 8kT o
1
The functions ~,,z,a are dependent only on the A~ table I. S183
z
R o %g (v T)~, number
and
A = are
%e/Xg, given
in
SI84
~, BER~,S~[V
1000 800 600 400 200 100 80 60 40 20 I0 8
-1.324(-5) -2.067(-5) -3.673(-5) -8.256(-5) -3.291(-4) -1.307(-3) -2.036(-3) -3.598(-3) -8.004(-3) -3.115(-2) -1.109(-1) -1.836(-1)
1.366(-9) 2.677(-9) 6.379(-9) 2.175(-8) 1.791(-7) 1.502(-6) 2.987(-6) 7.225(-6) 2.439(-5) 1.066(-4) -1.852(-3) -6.174(-3)
-1.004 -1.006 -1.007 -1.011 -1.022 -1.044 -1.055 -1.074 -i. II0 -1.218 -1.415 -1.504
6 4 2 1 0.8 0.6 0.4 0.2 0.I 0.08 0.06 0.04
-0.2626 -0.4780 -1.059 -1.749 -1.952 -2.181 -2.439 -2.717 -2.857 -2.884 -2.910 -2.935
-0.0224 -0.1013 -0.7320 -3.078 -4.501 -7.034 -12.37 -28.89 -~2.20 -78.85 -106.6 -182.1
i.' The • ~, ~2, ~9 dependence on the inverse 102 number R = ~ di)~nsionless ~ h o r e t i e following form: Uth = U t ~ U ~ n =
Fth/F
velocity Uth in the same 8 % U th=
U,
5(8+~)
T
case
(v T)~ ,
-1.635 -1.847 -2.226 -2.533 -2.609 -2.691 -2.781 -2.881 -2.937 -2.949 -2.961 -2.973 I/2
/(210~)
has
the
(2)
O
where FU
is the dimensionless isothermal drag (Beresnev et al., 1990).
In flg 1,2 c ~ a r i s o n of th~oretical(for S model)and experimental data the thermophoretic force and velocity is presented. As may be seen, results for the S model agree satisfactorily with the experiment over range of /62 numbers studied at quite real tangential momentum and energy com~x~tion coefficients values. O. S
"~'"
~
"Ft h
~~%'~""~ •.Z/
$/
Oi ¢
-,
~
0. t
:L.O
~0.0
Mr*udmer,
i o
0.0
0.£ number
/" ~--~-~ / , e~/ /v // //
o
I
o. o ~
" "/ " ~:/" ! I
I i*u~r
o. ~
I I llJlJ
~. o
l
for the all ac-
The thermophoretic forcegFLh: /'2.. ~ . " a - A=I0 , b _/./~ .. theory and ~ i m e ~ t . .//*/" A=I0,1- o~=~_=I.0, 2- ~ =0.8, 3- ~ =0.6 4- Jacobsen ~nd Brock, 1369, NaCI-A~, A~240,5 ~ ek =~ =i.0, 6- ~ =0.8, m-Schadt and C ~ l e , ~ 9 6 t , NaCl-air ~, A~240, o-TC~air, A~8, A- f~-alr, A~440, x - Rosenblatt and LaMer,1948, TCP-air, A~8, • Schmitt, 1959, silicon oil- At, A~II. t.0 Mr,
----~-_-~-'_-~~ The thermophoretic velocity Ut~ for NaCl-particles in air(A~240) :theory and experiment. I- ~ = ~ = I 0,2- ~=0.8, I I IIIIII
~o. o
I
3- a_=0.6,4- Lipmtov and Chex~ova, 1989, • - ~redi et ai.,1979, x - Derjaguin et ~oo. o al., 1966. I IIIII
Bere~ev, S.A. ,V.G.Chavnyak, G.A.Fo~gagin( 1990), J.Fluid Mech. ,Vol. 219,405-421. Llpmtov,G.N. ,E.A.Chernova (1989), J.Aeroaol 8ci., Vol.20, 931"933. Hhakhov,E.M. (1988), Fluid Dyn., Vol. 3, 95,99.