- Email: [email protected]
Transport properties of gas in silica aerogel
] O U R N A L OF
I~(]~I.CI~I,I,II~~LI~ ELSEVIER
Journal of Non-Crystalline Solids 186 (1995) 264-270
Transport properties of gas in silica aerogel S.Q. Zeng a,b,*, A. Hunt a, R. Greif b Lawrence Berkeley Laboratory, UniL~ersityof California, Berkeley, CA 94720, USA b Department of Mechanical Engineering, UniL,ersity of California, Berkeley, CA 94720, USA
Abstract The motion of gas molecules in silica aerogel is restricted by the solid silica matrix. As a result, the mean free path, velocity distribution function, diffusivity, viscosity and thermal conductivity are changed. In this work, relations for these quantities are derived for a gas in silica aerogel using an approach similar to that used for a gas in free space. Results for the mean free path predict that, for p > 10 bar, the mean free path o f the gas molecules in aerogel will be almost the same as in free space. However, as the pressure is reduced, the mean free path reaches a constant finite value instead o f increasing as in free space. The thermal conductivity o f a gas in aerogel starts to decrease at p = 10 bar and is almost negligible at 0.01 bar, while the thermal conductivity o f a gas b e t w e e n parallel walls 1 cm apart starts to decrease at p = 10 - 4 bar and is almost negligible at p = 10 - 7 bar. The predicted thermal conductivity o f a gas in aerogel is in good agreement with experimental results.
1. I n t r o d u c t i o n
A striking feature of silica aerogel is its fine solid matrix with extremely fine pore size. Base-catalyzed tetraethoxysilane (TEOS) aerogel has a mean pore size of about 20 nm. Such a small pore size greatly restricts the motion of gas molecules in the aerogel. The mean free path of gas molecules in free space obeys the equation /m,,-
1 kuT v/~ngwd~ -- v~wdgp'
(1)
where ng is the number density of gas molecules and dg is the diameter of a gas molecule. From Eq. (1), one finds that at 3.7 bar pressure at 296 K the mean
free path of gas molecules is about 20 nm, which indicates that at pressures much above one atmosphere the motion of gas molecules in aerogel is obstructed by solid matrix. As the gas pressure in the aerogel is reduced, the effects of the solid aerogel matrix restricting the motion of the gas molecules become more pronounced. As a result of the retarded motion of the gas molecules in aerogel, the thermal conductivity of the gas in aerogel is much smaller (about 1/2) than that of the gas in free space. A widely used formula [1,2] for the calculation of the thermal conductivity of the gas in aerogel was derived by Kaganer [3]: o Ag o = ,~.g/( 1 + 213 Kn)
(2)
where 13 is a constant and Kn is the Knudsen number defined as * Corresponding author. Tel: + 1-510 486 4292. Telefax: + 1510 486 4260. E-mail: [email protected].
Kn = Zm/~;
0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 0 2 2 - 3 0 9 3 ( 9 5 ) 0 0 0 5 2 - 6
(3)
S.Q. Zeng et al. /Journal of Non-Crystalline Solids 186 (1995) 264-270
6, the characteristic system size, is the distance between the heat-exchanging surfaces. Eq. (2) is applicable for heat transfer between parallel surfaces. Its validity for the gas in aerogel is questionable, because temperature slip conditions are used on the solid surfaces. For gas molecules in aerogel, the solid surfaces are a silica matrix distributed over the entire space. Accordingly, temperature slip conditions cannot be simply applied. In addition, it is very difficult to use Eq. (2) because the characteristic system size or the Knudsen number is often difficult to define. It is obvious that the distance between the surfaces is not a suitable characteristic system size because the mean free path of gas molecules in aerogel is very small compared with the distance between the surfaces. Gas molecules in aerogel, except for those which are near the surfaces, cannot reach the surfaces no matter how low the pressure is, because of the presence of the silica matrix. In this study, transport processes in aerogel are investigated. From the kinetic theory of gases, results for the mean free path of gas molecules in free space and for the corresponding transport properties are obtained, but these results cannot be used for the transport processes of gas molecules in aerogel because the solid matrix of silica obstructs the motion of gas molecules. Accordingly, those formulae must be modified for transport processes in aerogel.
265
composed of the scattering cross-section of the solid particles of aerogel and gas molecules, i.e., l m = 1 / S C S of (gas molecules + solid particles) per unit volume. This is expressed as [4] 1
lm
1
=
~
,
(4)
v/-2ngwd2 + ~ fo ~r2U~(r~) drs where Ns(r~)dr s is the number density of the solid particles of radius between r S and r~ + d r s and H is the porosity defined as the volume of the pores divided by the total volume of the matrix. For spherical particles, Eq. (4) becomes 1
lm =
v/2ngwd 2 + S~ Ppor/H'
(5)
where S~ is the specific surface area of the aerogel defined as surface area per unit mass and Ppor is the aerogel density.
2.2. Velocity distribution function Assume that we are dealing with a physical situation close to equilibrium conditions. One then expects that the velocity distribution function, f(r, c, t), does not differ greatly from the Maxwell distribution, f(°)(r, c, t), which describes local equilibrium conditions, i.e.,
2. Analysis
2.1. Mean free path of gas molecules in aerogel
f(o) =
3/2rn ng (2~kBT)
expt[ - m (
c - co)2/2kBT], (6)
The mean free path of gas molecules in free space is expressed in Eq. (1). The denominator in Eq. (1), 2 rr dgng, is the scattering cross-section (SCS) of gas molecules per unit volume, i.e., one can write Eq. (1) approximately as 1
lm" = SCS of gas molecules per unit volume"
where the number density, rig, temperature, T, and mean molecular velocity, c o , may be slowly varying functions of position, r, and time, t, but are independent of molecular velocity, e. In a gas transport process under no external forces, the velocity distribution function obeys the Boltzmann equation 0f --
Gas molecules in aerogel collide with each other and also with the solid particles which constitute the solid silica matrix. Hence, for the calculation of the mean free path of gas molecules in aerogel, the total scattering cross-section should be used which is
Ot
+c"
Of Or
0cf -
-
-
Ot
(7)
Assume that the effect of collisions is always to restore a local equilibrium situation described by the distribution function, f(0). Further assume that, if the molecular distribution is disturbed from the local
266
S.Q. Zeng et al. /Journal of Non-Crystalline Solids 186 (1995) 264-270
equilibrium so that the actual distribution f ( r , c, t) differs from f(0), then the effect of the collisions is to restore f to the local equilibrium value, f(0), exponentially with a relaxation time, ~-, which is of the order of the time between molecular collisions, i.e., Ocf/Ot = - ( f - f ( ° ) ) / , r . In addition, we assume that the gradients, i.e., concentration gradient for mass transfer, velocity gradient for momentum transfer and temperature gradient for energy transfer, are small, and hence Of/Or is small and f differs from f(o) by only a small amount. Hence, we replace f on the left-hand side of Eq. (7) by f(0). Eq. (7) then becomes
2.3. Transport properties of a gas in a porous medium To show the difference between transport processes in free space and those in porous media, we first present an elementary deduction of the diffusivity for particle diffusion in free space [5]. We consider three transport processes: mass diffusion, momentum transport and energy transport; the corresponding transport properties are diffusivity, viscosity and thermal conductivity. Fig. l(a) exemplifies molecular diffusion in free space. Let ly ° be the y component of the mean free path. Across the plane at y there is a molecular flux density in the positive y direction equal to 7Wng G, 1and in the negative y direction equal to 5Wng ly° where N is the molecular velocity in the y direction and the number density, ng, of gas particles takes values at y +_ lyo, respectively. The molecules in a volume having a base of unit area and a height equal to N (Fig. l(a)) can cross the plane at y in unit time; this is not true for porous media! The net flux density, JY, averaged over all directions on a hemisphere is -
of.,) -
-
8t
+ c"
oft(,}
f_ f(0)
-
-
-
Or
-
(8a)
,
~-
where ~- is given by ~"= l m / C .
(8b)
Rearranging Eq. (8a) yields Of(0) Of(o) f = f ( o ) _ 7" + cr" -8t Or
(9)
1 JnY=(2 [ngly-l~°-ngly+ly°]~)=(-
Since the relaxation time, r, for gas molecules in aerogel differs from that for gas molecules in free space, the velocity distribution function will be different for the two conditions.
° ,
,, Base ~ 1 J
o
W
o
o
o
o
o
o
o
o o
°o
o
o
o
o
o
o
o
o °o
y+l y Y
oo o
o
o
o o
W
o
o o
o
o O
°
Silica particles
Gas molecule~
y+l Yo
W
1 dng_ dng 3 ~ y clm~'= - D o d---~'
o
¢-ly
W
o o o
o
o
o
y-ly°
o
° Base = 1 o
o
(a)
(b) Fig. 1. (a) Gas in free space; (b) gas in aerogel.
d ng Fly,, ) dy (lO)
267
S.Q. Zeng et al. /Journal of Non-Crystalline Solids 186 (1995) 264-270
where D0=½~lm .
(11)
The thermal conductivity is related to the diffusivity according to
where ( w f H dc) is the number of particles with speed in the range from c to c + d c crossing a plane at y per unit area, per unit time in aerogel. Substituting Eq. (19) into Eq. (20) yields (
Ag O =
D0~v = ~Cv 1 ^ C/mo,
(12)
Of(°)}
J Y = f wf(°'H dc + r f w 2 - -
On
dc
and the viscosity is given by For diffusion in a porous medium, only some of the molecules in the volume having a base of unit area and a height equal to ~ (Fig. l(b)) can cross the plane at y because of the blockage from the existing solid particles, which constitute the solid matrix of a porous medium. In addition, the number density of gas particles will take values at y +_ ly where ly, the y component of the mean free path, is different from lyo. Hence, the net flux density, Eq. (10), becomes
dng = ( - dy wlyM) -
af(o) D = r /" w 2 dc. (22) J On For a gas of uniform density and temperature, moving parallel to the x direction with a mass velocity, u, which is a function of y alone, Eq. (9) becomes
(14)
~'xy =
(15)
Ag = D ~ , = ~1 C^v c l m n ,
(16)
1.£ = D o = l p ~ l m H .
(17)
From Eqs. (11)-(13) and (15)-(17), the ratios of the transport properties in a porous medium to those in free space are given by D Ag /~ lm --
-
Ago
U.
~0
(18)
lm o
It is possible to express the transport properties in terms of the velocity distribution function. For a gas at rest, uniform in pressure and temperature but having a concentration gradient in the y direction, Eq. (9) becomes f = f(0) _ w~"
dn Of (°) dy
On
f
m ( u - Uo)Wfll dc = - t x ( a U o / O y ) . (24)
Substituting Eq. (23) into Eq. (24) yields
= f
mw2f(°)II dc = r p F l ,
(19)
(20)
(25)
where p ( = f m w Z f (°) dc) is the hydrodynamic pressure. Using Eq. (8b) and the energy equipartition principle in Eq. (25) yields the same result for /z as Eq. (17). Similarly for a gas at rest, uniform in pressure but with a temperature which is a function of y, Eq. (9) becomes f = f ( 0 ) _ w7
dT Of(°) dy
(26)
0T
The energy flux in the y direction is q = f e w f l I dc,
The molecular flux in the y direction is Jny = f w f H dc,
(23)
The viscous stress in the x direction on a plane at y is
Eqs. (11)-(13) become accordingly ID = ~cl m I~,
Do
.
The first term on the right-hand side is an odd function of w and therefore vanishes upon integration. From the definition of diffusivity, one obtains
Ou Of (°) f = f(0)_ w'r-- -0y Ou
dng dng ½ d y ? l m I I = - D d y 17.
dy
(21) (13)
tx o = D o p = ½P~lmo.
an
(27)
where e is the energy of a molecule. Substituting Eq. (26) into Eq. (27) yields [6] or( ° ) f ew2--II Ag = 7 Y 0T
dc =
~Cv,
(28)
268
S.Q. Zeng et al./Journal of Non-Crystalline Solids 186 (1995) 264-270
where ~"= 2.5 for a simple gas with smooth spherically symmetric molecules. For diatomic and polyatomic gases, ~" is given by the Eucken formula
i.e., the measured value at low pressure, from the total conductivity.
~"= ¼ ( 9 T - 5),
4. Results
where 3' is the ratio of specific heats. An air molecule has a diameter 3.53 × 10 lo m and a mass 4.648 x 10 -26 kg [9], which yields from Eqs. (1), (5), (2) and (28)
3. Experimental The specific surface area, S~, which is used to determine the mean free path of gas molecules in aerogel, is obtained from nitrogen adsorption-desorption measurements (Brunner, Emmett and Teller, BET). The aerogel sample used was opacified with porosity of 94% and density 0.11 g cm -3. A description of the measurement is given in Ref. [4]. The hot wire technique [7] was used to measure the thermal conductivity of the opacified aerogel [8]. The hot wire technique yields the total conductivity which includes contributions from the gas and the solid matrix as well as radiative transfer. When the gas pressure is lower than 10 -4 bar, the apparent gas conductivity is very small and the total conductivity essentially results from the solid matrix and radiative contributions which are independent of the gas pressure. The apparent gas conductivity is then obtained by subtracting the solid and radiative conductivity,
10 -4
I
i
l m = 0-25Ss Ppor/~- 1 + 4.01 X 109 pT- ~ 0.026 10 -9 Tp-16 -1
Ag,, = 1 + 0 . 8 0 8
Ag
(31)
X
=
1 + 4.01 X 109 p T - 1 ,
0"25Ss Ppor//
I
i
i lllltl
J
i
~ illlll
~
t
i Iiii
J
i
Eq. (29), gas between parallel walls 1 cm apart Eq. (30), gas in aerogel
[]
10 -6
L. 0
o
Q
el.
0[3
[]
[]
[]
[]
D
[3
O0
[]
eQ D O
10-8 m
10-1o
i
10-3
i
i
iiiii
I
10 -2
f
i
~
~111
(32)
with p (bar), T (K), S s ( m e / k g ) and Ppor (kg/m3)" The specific surface area, S S, from a BET measurement is 797 mZ/g. The overall uncertainty in the determination of S s is 4%. The mean free path of the gas molecules at a temperature 296 K is calculated from Eq. (30) and the mean free path in free space is calculated from Eq. (29). The results are presented in Fig. 2.
D o
o
(30)
60.22 X 105 pT-°SII
~ illlll
•
(29)
4.01 X 109 pT-1 1
u
o
1
=
lm"
I
10 -1
i
~
i
illll
I
100
i
,
i
ii~ll
I
101
Pressure, bar Fig. 2. Pressure dependence of the mean free path of gas molecules.
i
Fill
102
S.Q. Zeng et al. /Journal of Non-Crystalline Solids 186 (1995) 264-270 0.04 •
~:
0.03
E -~. '{
0.02
i
0.01
measured -- Eq. (31), gas between
parallel
walls 1 cm apart -* - Eq. (32), gas in aerogel
/
¢..--*'
/
6. Conclusions ,,,,,d , ,,,,,,,I j H,,,d , ,,,,,,,I , ,,,,,,d , ,,,,,d , ,,,,,,d L,H,J
-0.01
e.g., at 10 7 bar, can also be achieved by using only a moderate vacuum in aerogel, e.g. 10 2 bar; (2) the calculated apparent thermal conductivity of a gas in aerogel from Eq. (32) is in good agreement with the experimental results.
,/
0 r~
269
10-7
10-5
10-3 10-1 Pressure, bar
, ,,,,,,
101
Fig. 3. Pressure dependence of gas thermal conductivity.
The results for the thermal conductivity of the gas calculated from Eqs. (31) (for 6 = 1 cm) and (32) are shown in Fig. 3. The results from the hot wire experiment are also shown in Fig. 3. The overall uncertainty in the determination of Ag is 4% [8].
5. Discussion
From Fig. 2 it is seen that, for p > 10 bar, the mean free path of the gas molecules in aerogel is almost the same as in free space. When the pressure is reduced, the mean free path in the aerogel deviates from that in free space and approaches a constant finite value instead of increasing as in a free space. From Fig. 3 it is seen that: (1) when the pressure is reduced to 0.1 bar, the thermal conductivity of the gas in the aerogel decreases while the thermal conductivity of a gas in free space is not affected by this decrease in pressure. Note that the small heat transfer that results from a very high vacuum in free space,
The differences between transport processes in a free space and those in aerogel are summarized in Table 1. The solid network of aerogel partitions the space into fine pores which restrict the motion of the gas molecules. Thus, at low pressures the mean free path in aerogel is greatly reduced. The ratio of the transport properties in aerogel to those in a free space is obtained, which depends both on the ratio of the mean free path of the gas molecules in aerogel to that in a free space and on the porosity of aerogel. The derived equation for the thermal conductivity of a gas in aerogel is in good agreement with the experimental results. This equation depends on the specific surface area, density and porosity of the aerogel. The authors wish to express their appreciation to Dr W. Cao and Dr M. Ayers for their help with the experiments. This work was supported by the Assistant Secretary for Conservation and Renewable Energy, Advanced Industrial Concepts (AIC) Materials Program of the Advanced Industrial Concepts Division, Office of Industrial Technologies of the US Department of Energy under Contract No. DEAC03-76F00098.
Table 1 Comparison of transport processes of gas molecule in free space with those in aerogel In free space In aerogel Mean free path: Relaxation time: Velocity distribution: Diffusivity: Viscosity: Thermal conductivity:
l m ~ ]/[ff2-ng,rr
~o l~
d g2 -}- S s P p o r / 1 - / ]
I~0 = pSlmo/3
= lm/~ f = f(o) + cr(OfO/ar) D o = ~lm 11/3 tz = p~l m [1/3
Ag. = ~t~ v ~1mo/3
Ag= ~'C~~lmI I / 3
=
/ C~
fo = f(o~ + c z o ( a f . / a r ) D o = ~lm,,/3
270
S.Q. Zeng et aL /Journal of Non-Crystalline Solids 186 (1995) 264-270
References [1] X. Lu, M.C. Arduini-Schuster, J. Kuhn, O. Nilsson, J. Fricke and R.W. Pekala, Science 255 (1992) 971. [2] L.W. Hrubesh and R.W. Rekala, J. Mater. Res. 9 (1994) 731. [3] M.G. Kaganer, Thermal Insulation in Cryogenic Engineering (Israel Program of Scientific Translation, Jerusalem, 1969) p. 6. [4] S.Q. Zeng, A.J. Hunt and R. Greif, ASME J. Heat Transfer, in press.
[5] C. Kittel and H. Kroemer, Thermal Physics, 2nd Ed. (Freeman, New York, 1980) p. 399. [6] S. Chapman and T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd Ed. (Cambridge University, Cambridge, 1990) p. 105. [7] G.D. Morrow, Ceram. Bull. 58 (1979) 687. [8] S.Q. Zeng, A.J. Hunt, W. Cao and R. Greif, ASME J. Heat Transfer 116 (1994) 756. [9] L.B. Loeb, The Kinetic Theory of Gases (McGraw-Hill, New York, 1934) p. 216.
I~(]~I.CI~I,I,II~~LI~ ELSEVIER
Journal of Non-Crystalline Solids 186 (1995) 264-270
Transport properties of gas in silica aerogel S.Q. Zeng a,b,*, A. Hunt a, R. Greif b Lawrence Berkeley Laboratory, UniL~ersityof California, Berkeley, CA 94720, USA b Department of Mechanical Engineering, UniL,ersity of California, Berkeley, CA 94720, USA
Abstract The motion of gas molecules in silica aerogel is restricted by the solid silica matrix. As a result, the mean free path, velocity distribution function, diffusivity, viscosity and thermal conductivity are changed. In this work, relations for these quantities are derived for a gas in silica aerogel using an approach similar to that used for a gas in free space. Results for the mean free path predict that, for p > 10 bar, the mean free path o f the gas molecules in aerogel will be almost the same as in free space. However, as the pressure is reduced, the mean free path reaches a constant finite value instead o f increasing as in free space. The thermal conductivity o f a gas in aerogel starts to decrease at p = 10 bar and is almost negligible at 0.01 bar, while the thermal conductivity o f a gas b e t w e e n parallel walls 1 cm apart starts to decrease at p = 10 - 4 bar and is almost negligible at p = 10 - 7 bar. The predicted thermal conductivity o f a gas in aerogel is in good agreement with experimental results.
1. I n t r o d u c t i o n
A striking feature of silica aerogel is its fine solid matrix with extremely fine pore size. Base-catalyzed tetraethoxysilane (TEOS) aerogel has a mean pore size of about 20 nm. Such a small pore size greatly restricts the motion of gas molecules in the aerogel. The mean free path of gas molecules in free space obeys the equation /m,,-
1 kuT v/~ngwd~ -- v~wdgp'
(1)
where ng is the number density of gas molecules and dg is the diameter of a gas molecule. From Eq. (1), one finds that at 3.7 bar pressure at 296 K the mean
free path of gas molecules is about 20 nm, which indicates that at pressures much above one atmosphere the motion of gas molecules in aerogel is obstructed by solid matrix. As the gas pressure in the aerogel is reduced, the effects of the solid aerogel matrix restricting the motion of the gas molecules become more pronounced. As a result of the retarded motion of the gas molecules in aerogel, the thermal conductivity of the gas in aerogel is much smaller (about 1/2) than that of the gas in free space. A widely used formula [1,2] for the calculation of the thermal conductivity of the gas in aerogel was derived by Kaganer [3]: o Ag o = ,~.g/( 1 + 213 Kn)
(2)
where 13 is a constant and Kn is the Knudsen number defined as * Corresponding author. Tel: + 1-510 486 4292. Telefax: + 1510 486 4260. E-mail: [email protected].
Kn = Zm/~;
0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 0 2 2 - 3 0 9 3 ( 9 5 ) 0 0 0 5 2 - 6
(3)
S.Q. Zeng et al. /Journal of Non-Crystalline Solids 186 (1995) 264-270
6, the characteristic system size, is the distance between the heat-exchanging surfaces. Eq. (2) is applicable for heat transfer between parallel surfaces. Its validity for the gas in aerogel is questionable, because temperature slip conditions are used on the solid surfaces. For gas molecules in aerogel, the solid surfaces are a silica matrix distributed over the entire space. Accordingly, temperature slip conditions cannot be simply applied. In addition, it is very difficult to use Eq. (2) because the characteristic system size or the Knudsen number is often difficult to define. It is obvious that the distance between the surfaces is not a suitable characteristic system size because the mean free path of gas molecules in aerogel is very small compared with the distance between the surfaces. Gas molecules in aerogel, except for those which are near the surfaces, cannot reach the surfaces no matter how low the pressure is, because of the presence of the silica matrix. In this study, transport processes in aerogel are investigated. From the kinetic theory of gases, results for the mean free path of gas molecules in free space and for the corresponding transport properties are obtained, but these results cannot be used for the transport processes of gas molecules in aerogel because the solid matrix of silica obstructs the motion of gas molecules. Accordingly, those formulae must be modified for transport processes in aerogel.
265
composed of the scattering cross-section of the solid particles of aerogel and gas molecules, i.e., l m = 1 / S C S of (gas molecules + solid particles) per unit volume. This is expressed as [4] 1
lm
1
=
~
,
(4)
v/-2ngwd2 + ~ fo ~r2U~(r~) drs where Ns(r~)dr s is the number density of the solid particles of radius between r S and r~ + d r s and H is the porosity defined as the volume of the pores divided by the total volume of the matrix. For spherical particles, Eq. (4) becomes 1
lm =
v/2ngwd 2 + S~ Ppor/H'
(5)
where S~ is the specific surface area of the aerogel defined as surface area per unit mass and Ppor is the aerogel density.
2.2. Velocity distribution function Assume that we are dealing with a physical situation close to equilibrium conditions. One then expects that the velocity distribution function, f(r, c, t), does not differ greatly from the Maxwell distribution, f(°)(r, c, t), which describes local equilibrium conditions, i.e.,
2. Analysis
2.1. Mean free path of gas molecules in aerogel
f(o) =
3/2rn ng (2~kBT)
expt[ - m (
c - co)2/2kBT], (6)
The mean free path of gas molecules in free space is expressed in Eq. (1). The denominator in Eq. (1), 2 rr dgng, is the scattering cross-section (SCS) of gas molecules per unit volume, i.e., one can write Eq. (1) approximately as 1
lm" = SCS of gas molecules per unit volume"
where the number density, rig, temperature, T, and mean molecular velocity, c o , may be slowly varying functions of position, r, and time, t, but are independent of molecular velocity, e. In a gas transport process under no external forces, the velocity distribution function obeys the Boltzmann equation 0f --
Gas molecules in aerogel collide with each other and also with the solid particles which constitute the solid silica matrix. Hence, for the calculation of the mean free path of gas molecules in aerogel, the total scattering cross-section should be used which is
Ot
+c"
Of Or
0cf -
-
-
Ot
(7)
Assume that the effect of collisions is always to restore a local equilibrium situation described by the distribution function, f(0). Further assume that, if the molecular distribution is disturbed from the local
266
S.Q. Zeng et al. /Journal of Non-Crystalline Solids 186 (1995) 264-270
equilibrium so that the actual distribution f ( r , c, t) differs from f(0), then the effect of the collisions is to restore f to the local equilibrium value, f(0), exponentially with a relaxation time, ~-, which is of the order of the time between molecular collisions, i.e., Ocf/Ot = - ( f - f ( ° ) ) / , r . In addition, we assume that the gradients, i.e., concentration gradient for mass transfer, velocity gradient for momentum transfer and temperature gradient for energy transfer, are small, and hence Of/Or is small and f differs from f(o) by only a small amount. Hence, we replace f on the left-hand side of Eq. (7) by f(0). Eq. (7) then becomes
2.3. Transport properties of a gas in a porous medium To show the difference between transport processes in free space and those in porous media, we first present an elementary deduction of the diffusivity for particle diffusion in free space [5]. We consider three transport processes: mass diffusion, momentum transport and energy transport; the corresponding transport properties are diffusivity, viscosity and thermal conductivity. Fig. l(a) exemplifies molecular diffusion in free space. Let ly ° be the y component of the mean free path. Across the plane at y there is a molecular flux density in the positive y direction equal to 7Wng G, 1and in the negative y direction equal to 5Wng ly° where N is the molecular velocity in the y direction and the number density, ng, of gas particles takes values at y +_ lyo, respectively. The molecules in a volume having a base of unit area and a height equal to N (Fig. l(a)) can cross the plane at y in unit time; this is not true for porous media! The net flux density, JY, averaged over all directions on a hemisphere is -
of.,) -
-
8t
+ c"
oft(,}
f_ f(0)
-
-
-
Or
-
(8a)
,
~-
where ~- is given by ~"= l m / C .
(8b)
Rearranging Eq. (8a) yields Of(0) Of(o) f = f ( o ) _ 7" + cr" -8t Or
(9)
1 JnY=(2 [ngly-l~°-ngly+ly°]~)=(-
Since the relaxation time, r, for gas molecules in aerogel differs from that for gas molecules in free space, the velocity distribution function will be different for the two conditions.
° ,
,, Base ~ 1 J
o
W
o
o
o
o
o
o
o
o o
°o
o
o
o
o
o
o
o
o °o
y+l y Y
oo o
o
o
o o
W
o
o o
o
o O
°
Silica particles
Gas molecule~
y+l Yo
W
1 dng_ dng 3 ~ y clm~'= - D o d---~'
o
¢-ly
W
o o o
o
o
o
y-ly°
o
° Base = 1 o
o
(a)
(b) Fig. 1. (a) Gas in free space; (b) gas in aerogel.
d ng Fly,, ) dy (lO)
267
S.Q. Zeng et al. /Journal of Non-Crystalline Solids 186 (1995) 264-270
where D0=½~lm .
(11)
The thermal conductivity is related to the diffusivity according to
where ( w f H dc) is the number of particles with speed in the range from c to c + d c crossing a plane at y per unit area, per unit time in aerogel. Substituting Eq. (19) into Eq. (20) yields (
Ag O =
D0~v = ~Cv 1 ^ C/mo,
(12)
Of(°)}
J Y = f wf(°'H dc + r f w 2 - -
On
dc
and the viscosity is given by For diffusion in a porous medium, only some of the molecules in the volume having a base of unit area and a height equal to ~ (Fig. l(b)) can cross the plane at y because of the blockage from the existing solid particles, which constitute the solid matrix of a porous medium. In addition, the number density of gas particles will take values at y +_ ly where ly, the y component of the mean free path, is different from lyo. Hence, the net flux density, Eq. (10), becomes
dng = ( - dy wlyM) -
af(o) D = r /" w 2 dc. (22) J On For a gas of uniform density and temperature, moving parallel to the x direction with a mass velocity, u, which is a function of y alone, Eq. (9) becomes
(14)
~'xy =
(15)
Ag = D ~ , = ~1 C^v c l m n ,
(16)
1.£ = D o = l p ~ l m H .
(17)
From Eqs. (11)-(13) and (15)-(17), the ratios of the transport properties in a porous medium to those in free space are given by D Ag /~ lm --
-
Ago
U.
~0
(18)
lm o
It is possible to express the transport properties in terms of the velocity distribution function. For a gas at rest, uniform in pressure and temperature but having a concentration gradient in the y direction, Eq. (9) becomes f = f(0) _ w~"
dn Of (°) dy
On
f
m ( u - Uo)Wfll dc = - t x ( a U o / O y ) . (24)
Substituting Eq. (23) into Eq. (24) yields
= f
mw2f(°)II dc = r p F l ,
(19)
(20)
(25)
where p ( = f m w Z f (°) dc) is the hydrodynamic pressure. Using Eq. (8b) and the energy equipartition principle in Eq. (25) yields the same result for /z as Eq. (17). Similarly for a gas at rest, uniform in pressure but with a temperature which is a function of y, Eq. (9) becomes f = f ( 0 ) _ w7
dT Of(°) dy
(26)
0T
The energy flux in the y direction is q = f e w f l I dc,
The molecular flux in the y direction is Jny = f w f H dc,
(23)
The viscous stress in the x direction on a plane at y is
Eqs. (11)-(13) become accordingly ID = ~cl m I~,
Do
.
The first term on the right-hand side is an odd function of w and therefore vanishes upon integration. From the definition of diffusivity, one obtains
Ou Of (°) f = f(0)_ w'r-- -0y Ou
dng dng ½ d y ? l m I I = - D d y 17.
dy
(21) (13)
tx o = D o p = ½P~lmo.
an
(27)
where e is the energy of a molecule. Substituting Eq. (26) into Eq. (27) yields [6] or( ° ) f ew2--II Ag = 7 Y 0T
dc =
~Cv,
(28)
268
S.Q. Zeng et al./Journal of Non-Crystalline Solids 186 (1995) 264-270
where ~"= 2.5 for a simple gas with smooth spherically symmetric molecules. For diatomic and polyatomic gases, ~" is given by the Eucken formula
i.e., the measured value at low pressure, from the total conductivity.
~"= ¼ ( 9 T - 5),
4. Results
where 3' is the ratio of specific heats. An air molecule has a diameter 3.53 × 10 lo m and a mass 4.648 x 10 -26 kg [9], which yields from Eqs. (1), (5), (2) and (28)
3. Experimental The specific surface area, S~, which is used to determine the mean free path of gas molecules in aerogel, is obtained from nitrogen adsorption-desorption measurements (Brunner, Emmett and Teller, BET). The aerogel sample used was opacified with porosity of 94% and density 0.11 g cm -3. A description of the measurement is given in Ref. [4]. The hot wire technique [7] was used to measure the thermal conductivity of the opacified aerogel [8]. The hot wire technique yields the total conductivity which includes contributions from the gas and the solid matrix as well as radiative transfer. When the gas pressure is lower than 10 -4 bar, the apparent gas conductivity is very small and the total conductivity essentially results from the solid matrix and radiative contributions which are independent of the gas pressure. The apparent gas conductivity is then obtained by subtracting the solid and radiative conductivity,
10 -4
I
i
l m = 0-25Ss Ppor/~- 1 + 4.01 X 109 pT- ~ 0.026 10 -9 Tp-16 -1
Ag,, = 1 + 0 . 8 0 8
Ag
(31)
X
=
1 + 4.01 X 109 p T - 1 ,
0"25Ss Ppor//
I
i
i lllltl
J
i
~ illlll
~
t
i Iiii
J
i
Eq. (29), gas between parallel walls 1 cm apart Eq. (30), gas in aerogel
[]
10 -6
L. 0
o
Q
el.
0[3
[]
[]
[]
[]
D
[3
O0
[]
eQ D O
10-8 m
10-1o
i
10-3
i
i
iiiii
I
10 -2
f
i
~
~111
(32)
with p (bar), T (K), S s ( m e / k g ) and Ppor (kg/m3)" The specific surface area, S S, from a BET measurement is 797 mZ/g. The overall uncertainty in the determination of S s is 4%. The mean free path of the gas molecules at a temperature 296 K is calculated from Eq. (30) and the mean free path in free space is calculated from Eq. (29). The results are presented in Fig. 2.
D o
o
(30)
60.22 X 105 pT-°SII
~ illlll
•
(29)
4.01 X 109 pT-1 1
u
o
1
=
lm"
I
10 -1
i
~
i
illll
I
100
i
,
i
ii~ll
I
101
Pressure, bar Fig. 2. Pressure dependence of the mean free path of gas molecules.
i
Fill
102
S.Q. Zeng et al. /Journal of Non-Crystalline Solids 186 (1995) 264-270 0.04 •
~:
0.03
E -~. '{
0.02
i
0.01
measured -- Eq. (31), gas between
parallel
walls 1 cm apart -* - Eq. (32), gas in aerogel
/
¢..--*'
/
6. Conclusions ,,,,,d , ,,,,,,,I j H,,,d , ,,,,,,,I , ,,,,,,d , ,,,,,d , ,,,,,,d L,H,J
-0.01
e.g., at 10 7 bar, can also be achieved by using only a moderate vacuum in aerogel, e.g. 10 2 bar; (2) the calculated apparent thermal conductivity of a gas in aerogel from Eq. (32) is in good agreement with the experimental results.
,/
0 r~
269
10-7
10-5
10-3 10-1 Pressure, bar
, ,,,,,,
101
Fig. 3. Pressure dependence of gas thermal conductivity.
The results for the thermal conductivity of the gas calculated from Eqs. (31) (for 6 = 1 cm) and (32) are shown in Fig. 3. The results from the hot wire experiment are also shown in Fig. 3. The overall uncertainty in the determination of Ag is 4% [8].
5. Discussion
From Fig. 2 it is seen that, for p > 10 bar, the mean free path of the gas molecules in aerogel is almost the same as in free space. When the pressure is reduced, the mean free path in the aerogel deviates from that in free space and approaches a constant finite value instead of increasing as in a free space. From Fig. 3 it is seen that: (1) when the pressure is reduced to 0.1 bar, the thermal conductivity of the gas in the aerogel decreases while the thermal conductivity of a gas in free space is not affected by this decrease in pressure. Note that the small heat transfer that results from a very high vacuum in free space,
The differences between transport processes in a free space and those in aerogel are summarized in Table 1. The solid network of aerogel partitions the space into fine pores which restrict the motion of the gas molecules. Thus, at low pressures the mean free path in aerogel is greatly reduced. The ratio of the transport properties in aerogel to those in a free space is obtained, which depends both on the ratio of the mean free path of the gas molecules in aerogel to that in a free space and on the porosity of aerogel. The derived equation for the thermal conductivity of a gas in aerogel is in good agreement with the experimental results. This equation depends on the specific surface area, density and porosity of the aerogel. The authors wish to express their appreciation to Dr W. Cao and Dr M. Ayers for their help with the experiments. This work was supported by the Assistant Secretary for Conservation and Renewable Energy, Advanced Industrial Concepts (AIC) Materials Program of the Advanced Industrial Concepts Division, Office of Industrial Technologies of the US Department of Energy under Contract No. DEAC03-76F00098.
Table 1 Comparison of transport processes of gas molecule in free space with those in aerogel In free space In aerogel Mean free path: Relaxation time: Velocity distribution: Diffusivity: Viscosity: Thermal conductivity:
l m ~ ]/[ff2-ng,rr
~o l~
d g2 -}- S s P p o r / 1 - / ]
I~0 = pSlmo/3
= lm/~ f = f(o) + cr(OfO/ar) D o = ~lm 11/3 tz = p~l m [1/3
Ag. = ~t~ v ~1mo/3
Ag= ~'C~~lmI I / 3
=
/ C~
fo = f(o~ + c z o ( a f . / a r ) D o = ~lm,,/3
270
S.Q. Zeng et aL /Journal of Non-Crystalline Solids 186 (1995) 264-270
References [1] X. Lu, M.C. Arduini-Schuster, J. Kuhn, O. Nilsson, J. Fricke and R.W. Pekala, Science 255 (1992) 971. [2] L.W. Hrubesh and R.W. Rekala, J. Mater. Res. 9 (1994) 731. [3] M.G. Kaganer, Thermal Insulation in Cryogenic Engineering (Israel Program of Scientific Translation, Jerusalem, 1969) p. 6. [4] S.Q. Zeng, A.J. Hunt and R. Greif, ASME J. Heat Transfer, in press.
[5] C. Kittel and H. Kroemer, Thermal Physics, 2nd Ed. (Freeman, New York, 1980) p. 399. [6] S. Chapman and T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd Ed. (Cambridge University, Cambridge, 1990) p. 105. [7] G.D. Morrow, Ceram. Bull. 58 (1979) 687. [8] S.Q. Zeng, A.J. Hunt, W. Cao and R. Greif, ASME J. Heat Transfer 116 (1994) 756. [9] L.B. Loeb, The Kinetic Theory of Gases (McGraw-Hill, New York, 1934) p. 216.