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The thermal force in the low density limit
The Thermal Force in the Low Density Limit N. T. T O N G AND G. A. B I R D Department of Aeronautical Engineering, University of Sydney, N. S. W. Australia
Received June 9, 1970; accepted October 7, 1970 The Waldmann result for the thermal force on a suspended spherical particle in the free molecule regime is critically examined. It is shown that the assumptions implied by this result can lead to significant error and a more accurate collisionless result is derived and discussed. IN TRODUCTION Studies of the thermal force (1-3) on suspended spherical particles almost invariably adopt the W~ldmann solution (4) for the theoretical free molecule limit. This limit is usually defined by K , = X/r~ --~ oo where K~ is the Knudsen number, X is the molecular mean free path, and r~ is the radius of the sphere. Since the mean free path is very large compared with the dimensions of the sphere, the reflected molecules do not suffer collisions until they have traveled a very large distance from the sphere. There is then a negligible probability that products of these collisions collide with the sphere, and the incident molecules are adequately described by tile undisturbed ambient gas distribution function. Some eollisional effects must exist at any finite Knudsen number but, for most flows, they are negligible at Knudsen numbers above ten. The free molecule analysis is therefore applied at large but finite Knudsen numbers, to give what may be called the "collisionless" result to distinguish it from the infinite Knudsen number limit. However, in the present problem, there are finite Knudsen number effects that are quite unrelated to the importance or otherwise of intermoleeular collisions. The purpose of this paper is to present a "collisionless" theory which ineludes these effects. It also includes a higher
approximation to the distribution function and provides a small correction to the Waldmann result even in the Kn --~ oo limit. Since the theory does not include the effects of intermoleeular collisions, the inclusion of the other finite Knudsen effects may appear to be inconsistent. However, no exact analysis exists for the eollisional effects and, at the present stage of development of rarefied gas dynamics, there appears to be no prospect of one being carried out. Approximate analyses and numerical studies of the eollisional effects will continue to be made and, in order for these to be properly assessed, it is desirable to have a eollisionless theory which includes all effects that are amenable to exact analysis. Moreover, it appears that, at I~:nudsen numbers of the order of 5, the magnitude of these effects is rather greater than the probable magnitude of the eollisional effects. Since the highest Knudsen numbers in existing experiments are in this range, the extended theory should aid the interpretation of the experimental results. The Waldmann solution assumes that the ambient distribution function is given by the first approximation to the Chapman-Enskog solution for a monatomie gas with temperature gradient. The surface temperature of the sphere is assumed to be uniform and equal to that of the ambient gas at the 1OCaJournal of Colloid a.nd Interface Science, Vol.35, No. 3, March1971
403
404
T O N G AND B I R D
tion of the sphere. A mixture of diffuse and specular reflection is assumed for the interaction of the incident molecules with the sphere. The result for the net force F . on the sphere is F~ = - a ~ 5 ( r ~ k J ~ )
VT.
[1]
Here r~ is the radius of the sphere, k~ is the coefficient of thermal conductivity of the gas, ~ is the mean molecular speed, and T is the temperature of the ambient gas. A survey (3) of experimental results indicates that the high Knudsen number limit agrees with Eq. [1] to approximately 5 %. At low Knudsen numbers, intermolecular collisions become important. The free molecule regime gives way to the transition regime, which is much less amenable to exact analysis. However, the experiments indicate the following empirical form (2, 3) for the force Ft in the transition regime.
effects can modify the Waldmann result to a significant degree and a more accurate eollisionless result for monatomic gases is derived and discussed. ANALYSIS
1. Waldmann Result. The assumptions implied by the use of the Waldmann equation are best illustrated by an outline of its derivation. The first approximation to the ChapmanEnskog distribution function for the gas is (~) f=f0
1--~ul O
[31
"@~ujuj - 5) pRTk° dT;
Here, f0 = (~a/Trm) exp (-~2uiuj) is the equilibrium distribution function, R is the F~ = F. exp (-~-rJX), [2] gas constant, ~ = 1/(2RT) 1/2is the reciprocal of the most probable molecular speed, p where r is an empirical factor. Values of r is the density, and ui(i = 1, 2, 3) are the have been calculated (3) from an approxi- molecular velocity components in the xl mate theory and a comparison with the directions. Repetition of subscript within a experimental values has been used as a test term indicates summation. of the theory. If this procedure is to be For the calculation of the thermal force, meaningful, it is essential that the collision- take the origin at the center of the sphere and less limit be given by the Waldmann solution choose the axis X in the direction of the to a high degree of accuracy. temperature gradient as shown in Fig. 1. In order for the Chapman-Enskog solu- Axes x ~, yt, z~ are then taken with x t normal tion to be accurate, the ratio of the mean free to and directed towards the surface element path to the temperature scale length must be of the sphere at P and y' along the intersecsmall compared with unity. Even when this tion of the planes defined by OPX and the is satisfied, higher approximations to the surface element. The x', y', z' directions are Chapman-Enskog distribution may be calcu- then taken as the xl directions in Eq. [3] and lated and this is the first effect examined in the flux of any quantity Q to the element of the analysis. The assumption of uniform surface is surface temperature implies that the material of the sphere has infinite thermal conductiv7r 8/2 ~ ¢~ ity and the effect of finite thermal conductivity is next examined. The Waldmann •exp [ - - ~ ( u '~ -t- v'~ + w')~] result also assumes a single value for the [4] ambient gas temperature. However, this --5 + w') 2J temperature may vary significantly over the diameter of the sphere and this is the third d') , , , kg (u' cos 0 -t- v' sin 0) du dv dw. effect to be analyzed. It is shown that these pR ~"
57
Journal of Colloid and Interface Science, Vol. 35, No. 3, March 1971
T H E R M A L F O R C E IN LOW D E N S I T Y L I M I T
405
molecules, i.e., 16~- 1/~
Fw
1
T
Fig. 1. Coordinate system for analysis
Here n is the number density and p = n m , where m is the mass of a molecule. B y putting Q = m u ~ and Q = m J we obtain, respectively, the pressure p~ and shear stress r~ on the element, i.e.,
o
2fllq dT 57rln cos 0 dX
[5]
and flkg dT rl = 5~v?2 sin 0 ~ .
[6]
A term such as the first one in pi that is uniform around the sphere obviously makes no contribution to the force on the sphere. A property of the Chapman-Enskog distribution is that the number flux is isotropic so that, if the surface temperature is uniform, the force exerted by the diffusely reflected molecules is also uniform and makes no contribution. The pressure due to the speeularly reflected molecules is equal to p~. The shear stress due to the diffusely reflected molecules is zero and that due to the speeularly refleeted molecules is equal to r~. The net force is therefore given by F~o-
¢~2k~ d T 57rll~ dX •
flr2kg ddX T'
[7]
and, since ~ = 2/(flrY2), this is equivalent to Eq. [1]. Note that the disappearance of e is due to the spherical geometry and would not occur for other shapes. 2. Second Approximation to the Distribution Function• When the second terms in the Sonine polynomials (6) are taken into aeeounb, the second approximation to f for inverse power law molecules is
X
Pi - 4fl2
15
{2(2 -- e) cos 2 0 -}- e sin ~ 0} •27rr:2 sin 0 dO,
where + is the fraction of diffusely reflected
f = - fo 1 - 1 6 flui~ 5 45c~~ -- 16~ -}- 16
Is]
65 2
0)1 p R T
2a + ~ -
'
where a is the power law in the molecular potential and k~ is now the second approximation to the coefficient of thermal conduction. Note that this reduces to the first approximation for the special case of Maxwell molecules (a = 4). When this is used in place of the first approximation the thermal force becomes F46c~2 -- 20o~ + 161
F=F.L4~j
16~+ig
"
[9]
2. Effect of Finite Sphere Conductivity. The Waldmann result assumes that the temperature of the sphere is uniform; this requires the thermal conductivity lc~ of the sphere to be infinite• The energy flux Ei from the incident molecules to an element of the sphere may be obtained from Eq. [4] with Q = m ( u '2 + v '2 @ w'~)/2, i.e.,
1 h~ cos 0 d T Diffuse reflection may be regarded as effusion Journal of Colloid and Interface Science, Vol. 35, No. 3, March 1971
TONG AND BIRD
406
from a fictitious gas of density pz in equilibrium at the surface temperature T~. The energy flux from the element due to the reflected molecules is therefore E~ = epff(2~r~mfl,3) -~- (1 - e)E~.
The heat input to the element must be equal to the heat conducted away from it so that, equating Eqs. [15] and [17], we obtain
[11]
The fictitious gas density of is obtained from the condition that the number fluxes to and from the surface must be equal, i.e.,
c=~k~32
p~ = ~psRir~ + (1 - ~)pi -
p (1 ~e #~
or
IrlJ2~3 dT ko cos 0 - p dX"
[131
Therefore, for finite conductivity, it seems reasonable to assume a temperature distribution
T~/T = 1 -- c cos 0,
pr--~
1--~eos0
[14]
cos0 [15] 1
-3
/c~ cos 0 d T j"~
32"
The general solution (7) for the temperature T~ at radius r in the sphere is
T~ = ~_, a.~r~P~ (cos 0),
+
where Pn(COS 0) are Legendre polynomials. EvMuating this at the surface and comparing with Eq. [14] we find a0 = T, at = - c T / r , , an = 0.
The heat conducted from the surface of the sphere is
-/~° . - 3 T . .... [17]
n~0 =
The mean free path X is conventionally defined through g -- 0.5 paX, where g is the coefficient of viscosity which may be related to kg through the Prandtl number P~. The net force on the sphere can then be written
F=F~o+F~=F~
1+ 5 [19] • l + - - - g
= --l~, ~_, na~ r2-~Pn(cos 0)
( 1 - - e)p~.
[16]
n=0
and a2 = a3 . . . . . .
)+
This may be integrated over the surface to give the force due to the reflected molecules
where e is a constant. The heat input to the element from the gas is then ~[ ~ c
cos 0)m
c
+ (1 - ~)p~
=1
Er=
to/"
Now the pressure due to the reflected molecules is
p f f ~ = p/~. [12] If the sphere is a perfect insulator (k~ = 0), than E¢ must be equal to E~, and Eqs. [10] to [12] give
E~-
~ +
T
--k~e-COS0. rs
Journal of Colloid and Interface Science, Vol. 35, No. 3, March 1971
2e P~
.
~. Effect of Finite Sphere Size. The temperature T in the above analysis is the gas temperature at the location of the sphere. At low Knudsen numbers, the temperature change over the diameter of the sphere m a y be significant. The ambient pressure is uniform so the density is inversely proportional to the temperature. The equilibrium component of the pressure due to the incident gas is proportional to pT and is therefore unaffected. The perturbation of the nonequilibrium component of p~ and of ri is of
THERMAL FORCE IN LOW DENSITY LIMIT the second order in d T / d X . Since the Chapman-Enskog solution (including the second approximation in Section 2 above) is accurate only to the first order in dT/dX, this need not be taken into account. The first-order perturbation to the energy flux disappears when it is integrated over the surface so the temperature of a perfectly conducting sphere is unaffected. The number flux is also affected to the first order in d f / d X and since it is proportional to pT 1/2, the increased flux is from the cooler gas. The resulting change in the pressure due to the diffusely reflected molecules therefore opposes the net force. The net force is readily shown to be
K~
......
F
1.5
4=07
i
0.00 g
1.4
[3
1.2
I.I O.5
1.0
co
05 . . . . . . . 3
' I0
J
|
30
,
,
, , ,,I I00
300 Kn
F = F~
1
~re 8
[20]
5. Combined Effect. For clarity, the three effects have been dealt with separately. An analysis including all the effects yields the final result F F~
46a s - 20a + 16 45a 2 -- 16a -k 16
+
57re/32 5K~k~ 1+ - - 2 Pr d%
[21]
l + - - -2- P~ - &~ . 5Kn t~ DISCUSSION The molecular model affects only the first term in Eq. [21]. This is the term resulting from the second approximation to the Chapman-Enskog distribution function. The correction is zero for Maxwell molecules (a = 4) and increases to a few per cent for hard sphere molecules (a = ~ ) . The appropriate value of a for a monatomic gas can be estimated from the temperature dependence of the viscosity coefficient since, for inverse power law molecules (6), u ~ T I/2+2/~.
[22]
For most gases, a is of the order of 10 and
Fig. 2. Modification of WMdinann result for = 10 ande = 1. under normal conditions, the molecular reflection is diffuse (e = 1). Figure 2 then shows the ratio of the thermal force to the Waldmann result as a function of Xnudsen number for a number of values of the spheregas thermal conductivity ratio. Most experiments have been carried out with liquid aerosols having a thermal conductivity approximately ten times that of the gas. Figure 2 shows that the error in the Waldmann result is approximately 2 % at Kn = 10 and 8% at Kn = 3. Since this is within the range of agreement quoted (2) for theory and experiment, no discrepancy has been introduced by the fuller analysis. However, the magnitude of the correction is of the same order as the eollisional (or transition regime) effects at Knudsen numbers of about 3 to 5 and must be taken into account when assessing these effects. It is the finite size effect that is most significant in practical eases. The effect of finite sphere conductivity is in the opposite direction to the finite size effect and increases the thermal force, but is insignificant when the sphere-gas thermal conductivity ratio is high. A typical liquid droplet in hydrogen or helium would have Journal of Colloid and Interface ~cience, V o l . 35, N o . 3, l~[arch 1971
408
TONG AND BIRD
a conductivity ratio near unity and the effects of finite size and conductivity would very nearly cancel. The conductivity effect would dominate if the thermal conductivity of the sphere was small compared vdth that of the gas. However, this condition could be met only for the most extreme material combinations such as particles of solid insulating materials in hydrogen or helium. Such systems would probably have the interesting property that the thermal force would increase above the Waldmann result near K= = 1, reach a maximum around K~ = 3 to 5, and then decline.
Journal of Colloid and Interface Science, Vol. 35, No. 3, March 1971
REFERENCES 1. BROCK, J. R., Or. Colloid and Interface Sci. 25,
564 (1967). 2. BRocI(, J. R., J. Colloid and Interface Sci. 25, 392 (1967). 3. BROCK,J. R., Y. Colloid and Interface Sci. 23, 448 (1967). 4. WALD~ANN, L. Z., Z. Naturforsch, A14, 589 (1959). 5. VINOENTI, W. G. AND KRrdGER, C. I-I., Introduction to Physical Gas Dynamics, Wiley, 1965. 6. C~A~MAN, S., AND COWLING, T. G., "The Mathematical Theory of Non-Uniform Gases," 2nd ed. Cambridge, 1958. 7. Am'~ci, V. S., "Conduction Heat Transfer," Addison-Wesley, 1966, p. 247.
Received June 9, 1970; accepted October 7, 1970 The Waldmann result for the thermal force on a suspended spherical particle in the free molecule regime is critically examined. It is shown that the assumptions implied by this result can lead to significant error and a more accurate collisionless result is derived and discussed. IN TRODUCTION Studies of the thermal force (1-3) on suspended spherical particles almost invariably adopt the W~ldmann solution (4) for the theoretical free molecule limit. This limit is usually defined by K , = X/r~ --~ oo where K~ is the Knudsen number, X is the molecular mean free path, and r~ is the radius of the sphere. Since the mean free path is very large compared with the dimensions of the sphere, the reflected molecules do not suffer collisions until they have traveled a very large distance from the sphere. There is then a negligible probability that products of these collisions collide with the sphere, and the incident molecules are adequately described by tile undisturbed ambient gas distribution function. Some eollisional effects must exist at any finite Knudsen number but, for most flows, they are negligible at Knudsen numbers above ten. The free molecule analysis is therefore applied at large but finite Knudsen numbers, to give what may be called the "collisionless" result to distinguish it from the infinite Knudsen number limit. However, in the present problem, there are finite Knudsen number effects that are quite unrelated to the importance or otherwise of intermoleeular collisions. The purpose of this paper is to present a "collisionless" theory which ineludes these effects. It also includes a higher
approximation to the distribution function and provides a small correction to the Waldmann result even in the Kn --~ oo limit. Since the theory does not include the effects of intermoleeular collisions, the inclusion of the other finite Knudsen effects may appear to be inconsistent. However, no exact analysis exists for the eollisional effects and, at the present stage of development of rarefied gas dynamics, there appears to be no prospect of one being carried out. Approximate analyses and numerical studies of the eollisional effects will continue to be made and, in order for these to be properly assessed, it is desirable to have a eollisionless theory which includes all effects that are amenable to exact analysis. Moreover, it appears that, at I~:nudsen numbers of the order of 5, the magnitude of these effects is rather greater than the probable magnitude of the eollisional effects. Since the highest Knudsen numbers in existing experiments are in this range, the extended theory should aid the interpretation of the experimental results. The Waldmann solution assumes that the ambient distribution function is given by the first approximation to the Chapman-Enskog solution for a monatomie gas with temperature gradient. The surface temperature of the sphere is assumed to be uniform and equal to that of the ambient gas at the 1OCaJournal of Colloid a.nd Interface Science, Vol.35, No. 3, March1971
403
404
T O N G AND B I R D
tion of the sphere. A mixture of diffuse and specular reflection is assumed for the interaction of the incident molecules with the sphere. The result for the net force F . on the sphere is F~ = - a ~ 5 ( r ~ k J ~ )
VT.
[1]
Here r~ is the radius of the sphere, k~ is the coefficient of thermal conductivity of the gas, ~ is the mean molecular speed, and T is the temperature of the ambient gas. A survey (3) of experimental results indicates that the high Knudsen number limit agrees with Eq. [1] to approximately 5 %. At low Knudsen numbers, intermolecular collisions become important. The free molecule regime gives way to the transition regime, which is much less amenable to exact analysis. However, the experiments indicate the following empirical form (2, 3) for the force Ft in the transition regime.
effects can modify the Waldmann result to a significant degree and a more accurate eollisionless result for monatomic gases is derived and discussed. ANALYSIS
1. Waldmann Result. The assumptions implied by the use of the Waldmann equation are best illustrated by an outline of its derivation. The first approximation to the ChapmanEnskog distribution function for the gas is (~) f=f0
1--~ul O
[31
"@~ujuj - 5) pRTk° dT;
Here, f0 = (~a/Trm) exp (-~2uiuj) is the equilibrium distribution function, R is the F~ = F. exp (-~-rJX), [2] gas constant, ~ = 1/(2RT) 1/2is the reciprocal of the most probable molecular speed, p where r is an empirical factor. Values of r is the density, and ui(i = 1, 2, 3) are the have been calculated (3) from an approxi- molecular velocity components in the xl mate theory and a comparison with the directions. Repetition of subscript within a experimental values has been used as a test term indicates summation. of the theory. If this procedure is to be For the calculation of the thermal force, meaningful, it is essential that the collision- take the origin at the center of the sphere and less limit be given by the Waldmann solution choose the axis X in the direction of the to a high degree of accuracy. temperature gradient as shown in Fig. 1. In order for the Chapman-Enskog solu- Axes x ~, yt, z~ are then taken with x t normal tion to be accurate, the ratio of the mean free to and directed towards the surface element path to the temperature scale length must be of the sphere at P and y' along the intersecsmall compared with unity. Even when this tion of the planes defined by OPX and the is satisfied, higher approximations to the surface element. The x', y', z' directions are Chapman-Enskog distribution may be calcu- then taken as the xl directions in Eq. [3] and lated and this is the first effect examined in the flux of any quantity Q to the element of the analysis. The assumption of uniform surface is surface temperature implies that the material of the sphere has infinite thermal conductiv7r 8/2 ~ ¢~ ity and the effect of finite thermal conductivity is next examined. The Waldmann •exp [ - - ~ ( u '~ -t- v'~ + w')~] result also assumes a single value for the [4] ambient gas temperature. However, this --5 + w') 2J temperature may vary significantly over the diameter of the sphere and this is the third d') , , , kg (u' cos 0 -t- v' sin 0) du dv dw. effect to be analyzed. It is shown that these pR ~"
57
Journal of Colloid and Interface Science, Vol. 35, No. 3, March 1971
T H E R M A L F O R C E IN LOW D E N S I T Y L I M I T
405
molecules, i.e., 16~- 1/~
Fw
1
T
Fig. 1. Coordinate system for analysis
Here n is the number density and p = n m , where m is the mass of a molecule. B y putting Q = m u ~ and Q = m J we obtain, respectively, the pressure p~ and shear stress r~ on the element, i.e.,
o
2fllq dT 57rln cos 0 dX
[5]
and flkg dT rl = 5~v?2 sin 0 ~ .
[6]
A term such as the first one in pi that is uniform around the sphere obviously makes no contribution to the force on the sphere. A property of the Chapman-Enskog distribution is that the number flux is isotropic so that, if the surface temperature is uniform, the force exerted by the diffusely reflected molecules is also uniform and makes no contribution. The pressure due to the speeularly reflected molecules is equal to p~. The shear stress due to the diffusely reflected molecules is zero and that due to the speeularly refleeted molecules is equal to r~. The net force is therefore given by F~o-
¢~2k~ d T 57rll~ dX •
flr2kg ddX T'
[7]
and, since ~ = 2/(flrY2), this is equivalent to Eq. [1]. Note that the disappearance of e is due to the spherical geometry and would not occur for other shapes. 2. Second Approximation to the Distribution Function• When the second terms in the Sonine polynomials (6) are taken into aeeounb, the second approximation to f for inverse power law molecules is
X
Pi - 4fl2
15
{2(2 -- e) cos 2 0 -}- e sin ~ 0} •27rr:2 sin 0 dO,
where + is the fraction of diffusely reflected
f = - fo 1 - 1 6 flui~ 5 45c~~ -- 16~ -}- 16
Is]
65 2
0)1 p R T
2a + ~ -
'
where a is the power law in the molecular potential and k~ is now the second approximation to the coefficient of thermal conduction. Note that this reduces to the first approximation for the special case of Maxwell molecules (a = 4). When this is used in place of the first approximation the thermal force becomes F46c~2 -- 20o~ + 161
F=F.L4~j
16~+ig
"
[9]
2. Effect of Finite Sphere Conductivity. The Waldmann result assumes that the temperature of the sphere is uniform; this requires the thermal conductivity lc~ of the sphere to be infinite• The energy flux Ei from the incident molecules to an element of the sphere may be obtained from Eq. [4] with Q = m ( u '2 + v '2 @ w'~)/2, i.e.,
1 h~ cos 0 d T Diffuse reflection may be regarded as effusion Journal of Colloid and Interface Science, Vol. 35, No. 3, March 1971
TONG AND BIRD
406
from a fictitious gas of density pz in equilibrium at the surface temperature T~. The energy flux from the element due to the reflected molecules is therefore E~ = epff(2~r~mfl,3) -~- (1 - e)E~.
The heat input to the element must be equal to the heat conducted away from it so that, equating Eqs. [15] and [17], we obtain
[11]
The fictitious gas density of is obtained from the condition that the number fluxes to and from the surface must be equal, i.e.,
c=~k~32
p~ = ~psRir~ + (1 - ~)pi -
p (1 ~e #~
or
IrlJ2~3 dT ko cos 0 - p dX"
[131
Therefore, for finite conductivity, it seems reasonable to assume a temperature distribution
T~/T = 1 -- c cos 0,
pr--~
1--~eos0
[14]
cos0 [15] 1
-3
/c~ cos 0 d T j"~
32"
The general solution (7) for the temperature T~ at radius r in the sphere is
T~ = ~_, a.~r~P~ (cos 0),
+
where Pn(COS 0) are Legendre polynomials. EvMuating this at the surface and comparing with Eq. [14] we find a0 = T, at = - c T / r , , an = 0.
The heat conducted from the surface of the sphere is
-/~° . - 3 T . .... [17]
n~0 =
The mean free path X is conventionally defined through g -- 0.5 paX, where g is the coefficient of viscosity which may be related to kg through the Prandtl number P~. The net force on the sphere can then be written
F=F~o+F~=F~
1+ 5 [19] • l + - - - g
= --l~, ~_, na~ r2-~Pn(cos 0)
( 1 - - e)p~.
[16]
n=0
and a2 = a3 . . . . . .
)+
This may be integrated over the surface to give the force due to the reflected molecules
where e is a constant. The heat input to the element from the gas is then ~[ ~ c
cos 0)m
c
+ (1 - ~)p~
=1
Er=
to/"
Now the pressure due to the reflected molecules is
p f f ~ = p/~. [12] If the sphere is a perfect insulator (k~ = 0), than E¢ must be equal to E~, and Eqs. [10] to [12] give
E~-
~ +
T
--k~e-COS0. rs
Journal of Colloid and Interface Science, Vol. 35, No. 3, March 1971
2e P~
.
~. Effect of Finite Sphere Size. The temperature T in the above analysis is the gas temperature at the location of the sphere. At low Knudsen numbers, the temperature change over the diameter of the sphere m a y be significant. The ambient pressure is uniform so the density is inversely proportional to the temperature. The equilibrium component of the pressure due to the incident gas is proportional to pT and is therefore unaffected. The perturbation of the nonequilibrium component of p~ and of ri is of
THERMAL FORCE IN LOW DENSITY LIMIT the second order in d T / d X . Since the Chapman-Enskog solution (including the second approximation in Section 2 above) is accurate only to the first order in dT/dX, this need not be taken into account. The first-order perturbation to the energy flux disappears when it is integrated over the surface so the temperature of a perfectly conducting sphere is unaffected. The number flux is also affected to the first order in d f / d X and since it is proportional to pT 1/2, the increased flux is from the cooler gas. The resulting change in the pressure due to the diffusely reflected molecules therefore opposes the net force. The net force is readily shown to be
K~
......
F
1.5
4=07
i
0.00 g
1.4
[3
1.2
I.I O.5
1.0
co
05 . . . . . . . 3
' I0
J
|
30
,
,
, , ,,I I00
300 Kn
F = F~
1
~re 8
[20]
5. Combined Effect. For clarity, the three effects have been dealt with separately. An analysis including all the effects yields the final result F F~
46a s - 20a + 16 45a 2 -- 16a -k 16
+
57re/32 5K~k~ 1+ - - 2 Pr d%
[21]
l + - - -2- P~ - &~ . 5Kn t~ DISCUSSION The molecular model affects only the first term in Eq. [21]. This is the term resulting from the second approximation to the Chapman-Enskog distribution function. The correction is zero for Maxwell molecules (a = 4) and increases to a few per cent for hard sphere molecules (a = ~ ) . The appropriate value of a for a monatomic gas can be estimated from the temperature dependence of the viscosity coefficient since, for inverse power law molecules (6), u ~ T I/2+2/~.
[22]
For most gases, a is of the order of 10 and
Fig. 2. Modification of WMdinann result for = 10 ande = 1. under normal conditions, the molecular reflection is diffuse (e = 1). Figure 2 then shows the ratio of the thermal force to the Waldmann result as a function of Xnudsen number for a number of values of the spheregas thermal conductivity ratio. Most experiments have been carried out with liquid aerosols having a thermal conductivity approximately ten times that of the gas. Figure 2 shows that the error in the Waldmann result is approximately 2 % at Kn = 10 and 8% at Kn = 3. Since this is within the range of agreement quoted (2) for theory and experiment, no discrepancy has been introduced by the fuller analysis. However, the magnitude of the correction is of the same order as the eollisional (or transition regime) effects at Knudsen numbers of about 3 to 5 and must be taken into account when assessing these effects. It is the finite size effect that is most significant in practical eases. The effect of finite sphere conductivity is in the opposite direction to the finite size effect and increases the thermal force, but is insignificant when the sphere-gas thermal conductivity ratio is high. A typical liquid droplet in hydrogen or helium would have Journal of Colloid and Interface ~cience, V o l . 35, N o . 3, l~[arch 1971
408
TONG AND BIRD
a conductivity ratio near unity and the effects of finite size and conductivity would very nearly cancel. The conductivity effect would dominate if the thermal conductivity of the sphere was small compared vdth that of the gas. However, this condition could be met only for the most extreme material combinations such as particles of solid insulating materials in hydrogen or helium. Such systems would probably have the interesting property that the thermal force would increase above the Waldmann result near K= = 1, reach a maximum around K~ = 3 to 5, and then decline.
Journal of Colloid and Interface Science, Vol. 35, No. 3, March 1971
REFERENCES 1. BROCK, J. R., Or. Colloid and Interface Sci. 25,
564 (1967). 2. BRocI(, J. R., J. Colloid and Interface Sci. 25, 392 (1967). 3. BROCK,J. R., Y. Colloid and Interface Sci. 23, 448 (1967). 4. WALD~ANN, L. Z., Z. Naturforsch, A14, 589 (1959). 5. VINOENTI, W. G. AND KRrdGER, C. I-I., Introduction to Physical Gas Dynamics, Wiley, 1965. 6. C~A~MAN, S., AND COWLING, T. G., "The Mathematical Theory of Non-Uniform Gases," 2nd ed. Cambridge, 1958. 7. Am'~ci, V. S., "Conduction Heat Transfer," Addison-Wesley, 1966, p. 247.