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On the effect of volatility on thermophoresis of aerosols
f1)
J. Aero'oI Sci.. Vol. 27. 5uppl. I. pp. 5279-5280. 1996
Pergamon
Copyright () 1996Elseyier Science Ltd Printed in Great Britain. All rights reserved 0021.8502/96 $15.00+0.00
PH: 80021-8502(96)00212-1
ON THE EFFECT OF VOLATILITY ON THERMOPHORESIS OF AEROSOLS. S.P.BAKANOV Russian Academy of Sciences Institute of Physical Chemistry 31 Leninsky Prospect,117915,Moscow.Russia. KEYWORDS Thermophoresis; Volatility
possible reasons for the deviation of the observed values of thermophoretic velocity of a particle from those calculated by Epstein's formula were analyzed by the author with coworkers. (We refer to the fact that the observed values of thermophoretic velocityexceed 30 to 50 times the computed ones). Since this analysis has not been directly confirmed by experiment, investigating alternative explanations is of interest as well. Up to now, when analyzing the thermophoresis of particles, we disregarded phase transitions at their surface, i.e. "non-volatile" particles were treated. Formally, this means that in the zero approximation with respect to Knudsen number Kn = the normal component of gas velocity is assumed to be vanishing at the particle surface (the surface is impermeable to gas molecules).Here we intend to replace this impermeability condition by a more general one. For instance, we can allow for phase transitions of one of the components of the particle substance at its surface. Consider the thermophoresis of volatile spherical particle with radius R. Temperature inside (subscript i )and outside the particle, and concentration of the volatile component are governed by the Laplace equations. In the zero approximation with respect to Knudsen number the boundary conditions for gas velocity, temperature and concentration of volatile component at the surface of the particle are as follows:
%
Vcr
( Ve
p; --;;:-= v,-v
PI VI' -p=O,
- VI ) (J
Vir
(
)
2 r'
la:'
vlaT
= k TS T R iJ() + Dk DS R iJ()'
sr a
iff
-K-+KI-'-Lpv =0
a-
er'
T= 1;, c= C.
Let us assume that at infinite distance from the particle gradl' =Const and C=
ni =
Canst and the particle non-isothermitcity
or, = 1;(R)- r;,
to be sufficiently small. Then the concentration of saturated (subscript s) vapor of the volatile componentin the neighborhood of the particle surface can be expressed as follows C.(T) =C.( -ra)+ ~ 67;.
The mutual diffusion gas component flux has a form I JK OinT) ( ) l'1-l'2 r =- C(l-C) l.JI....a- +kT~ . 5279
5280
Abstracts of the 1996 European Aerosol Conference
We introduce the notation
L C (r- ) Dpl = N. I 1 CI We present the Laplace equation solutions in the form 1; = To +(A ·r),
where
oc = CI(~)-C""
Sf= To - T.... From the boundary conditions after simple manipulations we obtain
N 1+ KL k T
3 A=-·
iCl) gradT
2 1+--+1 Ki N (k +T2 K KT r or
The last 6 equations determine the distributions of temperature inside and outside the particle, and of concentration of the volatile component outside the particle. As can be readily seen, these distributions naturally generalize the known results for non-volatile particles. The "volatility criteria", which have never been formulated before for such kind of problems, also emerge naturally. Indeed, if
I»
;(kr+T~)
then we obtain the temperature and concentration distributions in the limiting case of nonvolatile particles. The inequality N -kT»1 KJ' corresponds to the case of "highly volatile" particles, The velocity field around the sphere obeys the Navier-Stokes equations.Non-symmetrical component of v contributes to ~e thermophoretic force. The steady particle motion with the velocity v17'(thermophoretlc velocity) corresponds to the condition that the pressure force of the gas onto the particle equal to zero. As a result we have Vr p
2{
=-3"
v
k rs T+ DkDS
N (ie kr ) ] N k gradT or +L or +T" +"LT PI PI
K
I
l
r
For particles with high thermal conductivity the effect of volatility is governed by the last term of the equation. It does not depend on thermal conductivities of gas as well as of particle and can be essentially described as the formation around the particle of a gas mixture affecting the thermophoresis by providing an uncompensated momentum due to thermal diffusion. We computed the thermophoretic velocity for mercury and NaCI aerosols in the air in presence of water vapor. The latter can condense and evaporate from particle surfaces. It can be seen that water volatility at room temperature results in increasing the thennophoretic velocity for the aerosols considered only 2-3 times. Even if the temperature is of the order of 50 C the velocity can increase only 10 times compared to Epstein's value. Thus the experimentally observed values of thermophoretic velocity cannot be accounted for solely by the effect of volatility.
J. Aero'oI Sci.. Vol. 27. 5uppl. I. pp. 5279-5280. 1996
Pergamon
Copyright () 1996Elseyier Science Ltd Printed in Great Britain. All rights reserved 0021.8502/96 $15.00+0.00
PH: 80021-8502(96)00212-1
ON THE EFFECT OF VOLATILITY ON THERMOPHORESIS OF AEROSOLS. S.P.BAKANOV Russian Academy of Sciences Institute of Physical Chemistry 31 Leninsky Prospect,117915,Moscow.Russia. KEYWORDS Thermophoresis; Volatility
possible reasons for the deviation of the observed values of thermophoretic velocity of a particle from those calculated by Epstein's formula were analyzed by the author with coworkers. (We refer to the fact that the observed values of thermophoretic velocityexceed 30 to 50 times the computed ones). Since this analysis has not been directly confirmed by experiment, investigating alternative explanations is of interest as well. Up to now, when analyzing the thermophoresis of particles, we disregarded phase transitions at their surface, i.e. "non-volatile" particles were treated. Formally, this means that in the zero approximation with respect to Knudsen number Kn = the normal component of gas velocity is assumed to be vanishing at the particle surface (the surface is impermeable to gas molecules).Here we intend to replace this impermeability condition by a more general one. For instance, we can allow for phase transitions of one of the components of the particle substance at its surface. Consider the thermophoresis of volatile spherical particle with radius R. Temperature inside (subscript i )and outside the particle, and concentration of the volatile component are governed by the Laplace equations. In the zero approximation with respect to Knudsen number the boundary conditions for gas velocity, temperature and concentration of volatile component at the surface of the particle are as follows:
%
Vcr
( Ve
p; --;;:-= v,-v
PI VI' -p=O,
- VI ) (J
Vir
(
)
2 r'
la:'
vlaT
= k TS T R iJ() + Dk DS R iJ()'
sr a
iff
-K-+KI-'-Lpv =0
a-
er'
T= 1;, c= C.
Let us assume that at infinite distance from the particle gradl' =Const and C=
ni =
Canst and the particle non-isothermitcity
or, = 1;(R)- r;,
to be sufficiently small. Then the concentration of saturated (subscript s) vapor of the volatile componentin the neighborhood of the particle surface can be expressed as follows C.(T) =C.( -ra)+ ~ 67;.
The mutual diffusion gas component flux has a form I JK OinT) ( ) l'1-l'2 r =- C(l-C) l.JI....a- +kT~ . 5279
5280
Abstracts of the 1996 European Aerosol Conference
We introduce the notation
L C (r- ) Dpl = N. I 1 CI We present the Laplace equation solutions in the form 1; = To +(A ·r),
where
oc = CI(~)-C""
Sf= To - T.... From the boundary conditions after simple manipulations we obtain
N 1+ KL k T
3 A=-·
iCl) gradT
2 1+--+1 Ki N (k +T2 K KT r or
The last 6 equations determine the distributions of temperature inside and outside the particle, and of concentration of the volatile component outside the particle. As can be readily seen, these distributions naturally generalize the known results for non-volatile particles. The "volatility criteria", which have never been formulated before for such kind of problems, also emerge naturally. Indeed, if
I»
;(kr+T~)
then we obtain the temperature and concentration distributions in the limiting case of nonvolatile particles. The inequality N -kT»1 KJ' corresponds to the case of "highly volatile" particles, The velocity field around the sphere obeys the Navier-Stokes equations.Non-symmetrical component of v contributes to ~e thermophoretic force. The steady particle motion with the velocity v17'(thermophoretlc velocity) corresponds to the condition that the pressure force of the gas onto the particle equal to zero. As a result we have Vr p
2{
=-3"
v
k rs T+ DkDS
N (ie kr ) ] N k gradT or +L or +T" +"LT PI PI
K
I
l
r
For particles with high thermal conductivity the effect of volatility is governed by the last term of the equation. It does not depend on thermal conductivities of gas as well as of particle and can be essentially described as the formation around the particle of a gas mixture affecting the thermophoresis by providing an uncompensated momentum due to thermal diffusion. We computed the thermophoretic velocity for mercury and NaCI aerosols in the air in presence of water vapor. The latter can condense and evaporate from particle surfaces. It can be seen that water volatility at room temperature results in increasing the thennophoretic velocity for the aerosols considered only 2-3 times. Even if the temperature is of the order of 50 C the velocity can increase only 10 times compared to Epstein's value. Thus the experimentally observed values of thermophoretic velocity cannot be accounted for solely by the effect of volatility.