Figure 3 is a similar plot for metal deposition on a rotating disk. Results for I L / I D for growing mercury drops also apply to unsteady diffusion into an infinite, stagnant fluid. Except for the hydrogen ion discharge, the results are not very diflerent between the disk and the drop, and the ratio I J I D probably could be approximately applicable to other electrochemical problems which are more difficult to analyze.
U
Y zi
= = =
GREEKLETTERS constant in rate of growth of mercury drops, cm./ sec.113 0.89298 thickness of Nernst diffusion layer, cm. dimensionless independent variable (see Equations 7, 13, 16, 18, and 20) kinematic viscosity, sq. cm./sec. electrostatic potential, volts rotation speed of disk, radians/sec.
Nomenclature a
e-
= = = =
F
=
Z
= = = = = = = = = = = = =
6,
D,
I M, n
.YE 7 7
R R St
t T Ui
0.51023 concentration of species z, mole/cc. diffusion coefficient of species z, sq. cm./sec. symbol for electron Faraday’s constant, coulomb/equiv. current densiity, amp./sq. cm. dimensionless current density symbol for species z number of electrons transferred in electrode reaction flux of species z, mole/sq. cm.-sec. radial distance in spherical coordinates, cm. concentration ratio radius of mercury drop, cm. gas constant, joule/mole-” K. stoichiometric coefficient in electrode reaction time, sec. temperature, O K. mobility of species z, sq. cm.-mole/joule-sec.
fluid velocity, cm./sec. distance from electrode surface, cm. charge number of species i
literature Cited (1) Cochran, \Y. G., Proc. Cambridge Phzl. SOC.30, 365-75 (1934). (2) Conway, B. E., “Electrochemical Data,” Elsevier, Amsterdam, 1952. (3) Eucken, Arnold, Z. Physik. Chem. 59,72-117 (1907). (4) IlkoviE, D., Collection Czech. Chem. Commun. 6, 498-513 (1934). (5) IlkoviE, D., J . Chim. Phys. 35, 129-35 (1938). (6) Levich, B., Acta Physicochim. URSS 17, 257-307 (1942). (7) MacGillavry, D., Rideal, E. K.. Rec. Trau. Chim. 56, 1013-21 (1937).
(8) Slendyk, I., Collection Czech. Chem. Commun. 3, 385-95 (1931). RECEIVED for review February 14, 1966 .ACCEPTED May 27, 1966 Work supported by the U. S. Atomic Energy Commission.
DIFFUSION OF GASES ACROSS POROUS MEDIA K. S. S P I E G L E R College of Engineering, University of California, Berkeley, Calif.
Basic laws of gaseous diffusion in porous media are derived directly from an extension of the “friction model,” without recourse to detailed kinetic theory.
HEN two gases held in two separate and closed vessels and W o r i g i n a l ly a t equal pressure interdiffuse across a porous diaphragm, the transport phenomenon depends on the characteristic pore dimension (“pore size”). If the pore size is very large compared to the free path (“normal diffusion” range), the gases diffuse a t almost equal rates in opposite directions. I n other words, the fluxes are completely coupled and characterized by a single interdiffusion coefficient, and only a very small (if any) pressure gradient develops between the vessels. O n the other hand, if the pore size is smaller than the free path (“Knudsen diffusion” range), each gas diffuses a t the beginning a t its own independent rate, characterized by individual diffusion coefficients, D A K and D B K , respectively, and a n appreciable transient pressure gradient develops. Different diffusion rates are also observed in the steady-state interdiffusion of gases across a porous plug, whose terminal faces are flushed with the two gases a t equal pressure.
I n self-diffusion experiments on gas A the equal diffusion coefficients, DA,of the interdiffusing species are given by
1
1
DA
DAK
--_ -_
1 +T--
DAA
where D A K is the Knudsen diffusion coefficient of A through the porous medium which is independent of the pressure (as long as the free path is made larger than the pore size), and D)lAis a coefficient related to the self-diffusion coefficient of gas A in free space and inversdy proportional to the pressure ( 3 ) . T h e physical model corresponding to Equation 1 is that of two diffusion “resistances” in series; a plot of l/DA us. pressure is linear with a positive intercept (77). Equation 1 was derived by Bosanquet (73) by elementary arguments based on the kinetic theory of gases. I t is pertinent to ask what minimum assumptions are necessary to VOL. 5
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NOVEMBER 1 9 6 6
529
derive the very basic laws of gas interdiffusion across porous media, such as Equation 1, or the fact (3, 76) that the steadystate interdiffusion flux ratio for binary mixtures (at absolutely uniform pressure) is always equal to the ratio of the Knudsen diffusion coefficients: This is true a t all pressures, not just in the Knudsen region. I n this paper, it is shown that many useful relationships in the field of gas interdiffusion can be obtained by elementary formulation in terms of the flux equations of irreversible thermodynamics, combined with a simple “friction” model which translates the phenomenological coefficients into friction coefficients. The friction model is a n extension of the model of Stefan (79), Rayleigh (75),and Einstein (2). This concept is now being applied frequently to transport in condensed phases-e.g., ionic membranes (5, 9, 78), molten salts (6, 7), and electrolyte solutions (8, 7 7 ) . In all of the following considerations, maintenance of uniform temperature is assumed. Adsorption and chemical interaction phenomena are neglected. The porous diaphragm is taken as the stationary frame of reference for the diffusion fluxes. Basic Friction Model
Consider two ideal gases, A and B, interdiffusing across a porous plug or diaphragm a t uniform total pressure and temperature. The generalized driving force, FA, acting on 1 mole of A is the gradient of the chemical potential of A, p A :
FB
where X A B and XAc are proportionality constants termed “friction coefficients” (watt sec.2 ern.-? mole-1). Since the friction between the opposing fluxes of A and B is due to collisions between A and B, and since the number of such collisions which 1 mole of A experiences per unit time is proportional to F B , we can set the friction factor, XAB,proportional to ?B
where p” (watt set.* cm. mole-?) is a constant independent of the concentrations of A and B, but dependent on the geometry of the porous medium. From Equations 3 and 4, we have
FA = P”?B(uA -
UBI
and the analogous equation for FB is 530
l&EC FUNDAMENTALS
+ XACUA
(5)
XBCUB
(6)
We rewrite Equations 5 and 6 in terms of the molar fluxes
JA =
=
uACA, JB
uBEB:
FA = [@“(FBB/FA) f X A C / C A ] J A FB =
-P”JA f [ P ” ( ~ A / ~ Bf)
-~“JB
(XBC/CB)]JB
(7) (8)
I n these equations, the reciprocity relations are satisfiedLe., if we write them in the short form
FA
=
FB
=
f RABJB
~ A A J A
+
RBAJA RBBJB
(9) (10)
we see that R A B = R E A . Thus the friction model, which is compatible with Newton’s third law, does not violate the reciprocity relation of nonequilibrium thermodynamics. (In the case o$ binary “normal” interdiffusion, there is only one independent flux with respect to our frame of reference. Therefore the reciprocity expressed by the pair qf Equations 9 and 10 is purely formal in that case, as further discussed under “Normal Interdiffusion.”) Solving Equations 7 and 8 in terms of JA and J B , which is frequently a more useful way to present the result, yields J A
= (FA/d”)(fl”CA
+
XBc)FA
f (P”CAFB/ld”)FB
J B = (p”CA?B/d”)FA f (FB/~”)(P”CB $. XA,)FB d”
where
where FA is the gross concentration of A, in mole per cubic centimeter total volume. (Gross concentrations FA and FB are equal to the respective gas concentrations in free space times the porosity, E . ) Forces F, concentration gradients dc/dz, as well as fluxes J and superficial velocities u in later equations are vectors, the sign depending on the direction. O p e may consider this force to be counterbalanced by a “friction” force which consists of two terms: one due to momentum excbange in collisions between gas molecules diffusing in opposite directions, the other in collisiow with the wall. T h e first friction term is proportional to the difference in the superficial velocities of the gases uA - uB (centimeters per second) similar to the assumption for the viscous drag of ions and molecules in solution ( 7 7 , 78). The other is proportional to the difference of the velocities of gas A and the matrix. The latter difference is simply uA because the matrix is stationary. uA and uB are taken as vectors qnd have therefore opposite signs in interdiffusion:
+
P ” ~ A ( u B- u A )
=
5
+
+ XACXEC
~ ” ( C A X A C EBXBC)
(11)
(12)
(13)
The reciprocity relation also holds for the pair of Equations 11 and 12, which is not surprising because it is derived simply from the pair of Equations 7 and 8 which obey the reciprocity relation. Using the expressions for the forces FA and FB from Equation 2, we can rewrite Equations 11 and 12 in terms of the concentration gradients : JA
=
(RT/~”)(P”FA
+ XB,)(-d?A,/dZ) + ( R Tfl”F~/d”) (- d F s / d z )
(14)
+
JB = (RTp”FB/d”)(-ldC~/dt)
( R T / d ” ) ( p ” C B f X A c ) ( - d F ~ / d z ) (15) Knvdsen Diffusion
When the gas-wall collisions greatly outnuqber the gas-gas collisions (low pressure, small characteristic pore dimension), we can neglect the A-B friction terms, characterized by p”, in Equations 14 and 15 in comparison with the gas-wall collision terms. If we designate the fluxes under these conditions (“Knudsen fluxes”) by J A K and J B K , respectively, Equations 14 and 15, respectively, yield (with e denoting the porosity of the plug) : JAK
=
(RT/XAc)(-d?A/dz)
(RTe/XAc)(-dcA/dz)
JBK
=
(RT/XBc)(-dC,/dz)
(16)
= ( R T E / X B C ) ( - ~ C B / ~ Z ) (17)
These expressions are of the type J = D ( - d c / d z ) , and hence the Knudsen diffusion coefficients D A K and D B K , respectively, are DAK = ERT/XAC DBK
=
eRT/XBcf
(18)
(19)
Inasmuch as XBc and X A c can be different for different gases, the Knudsen interdiffusion coefficients can be different too, which is not surprisisg, since the two gases diffpse in-
dependently of each other. diffusion coefficients is DAs/’DBs
T h e ratio of the Knudsen inter=
xBC/XAC
(20)
[Elementary kinetic theory teaches that this ratio equals the square root of the molecular mass ratio m B / f ? i A (7).] Individual Diffusion Coefficients,
fi,
and
Consider a gas interd.iffusion experiment in which the two faces of a porous plug o f thickness dz are being flushed with mixtures of the two ideal gases A and B a t exactly the same pressure (3>76). T h e total concentration of the gases is F, and therefore -dC.< = a’?, (assuming no chemical reaction). T h e fluxes of A and B, respectively, are obtained from Equations 14 and 15: J A
=
+ .X,c)(-ddCA/d~) -
(RT/d”)(P”?A
(RTP”FA/d”) (-d?.t/’dz) J B
=
= t ( RTXnc/d”) ( -d ~ , / d ~ )
(21)
( R T ~ ~ A c i d ” ) ( - d F , / d z )= (ERTX~c/d”)(--c,/dz) (22)
These equations demonstrate proportionality betLveen flux and respective concentration gradient, although the individual diffusion coefficients of A and B, respectively, are not necessarily equal : D A :=
eRTXscld”
I)B = eRTX,,/d”
(23) (24)
The ratio of the individual diffusion coefficients is, from Equations 23, 24, and 20: I)A/I)B =
x B r / x A c
=
D.iK/DBK
(25)
Thus the flux ratio and the ratio of the individual diffusion coefficients are expected to be numerically equal to the ratio of the Knudsen diffusion coefficients? D A x / D B s ,a t all pressures (not just in the Knudsen region). This was found in experiments under carefully controlled conditions (3>76).
porous plug (72, 77). At the start of the experiment the forces F and Concentration gradients for the two gases are equal, but this equality will prevail throughout the diffusion process only if the opposing diffusion fluxes remain numerically equal. Otherwise, a transient pressiire gradient w7ill develop and progressively decay as the gas composition in the two vessels tends to equalize. If XAcand X B c are finite, we see from Equation 25 that in general the fluxes a t uniform pressure (start of the experiment) are by no means numerically equal, except when the Knudsen diffusion coefficients of the two gases are equal. The latter is the case, for instance, in the interdiffusion of very similar gases, such as para- and orthohydrogen, and, to a lesser degree, in the interdiffusion of isotopic species, provided the percentage difference of their molecular masses is small. I n such cases, called self-diffusion, the transient pressure differences are very small, both the concentration gradients and the fluxes of the two gases are numerically equal and of opposite sign throughout the experiment, and the interdiffusion process then follows a logarithmic decay law, characterized by a single interdiffusion coefficient (72, 77). But froin Equations 27 and 28 there is yet another situation in which both the concentration gradients and the diffusion fluxes betwecn the closed vessels are numerically equal and opposed. This is the case when X,, and X,, are negligible, and even when the gases are not of similar nature. Physically, this means that the number of gas-wall collisions is negligible compared to the number of gas-gas collisions-i.e., the plug is highly permeabl. This phenomenon is the opposite of “Knudsen diffusion” and may be called “normal diffusion” (3)or “normal interdiffusion.” This condition is realized when the characteristic pore dimension is much larger than the mean free path (experiments a t relatively high pressures, \\.it11 relatively highly permeable plugs). In this case the total pressure remains constant: with respect to time, and uniform, and we obtain from Equations 27 and 2 J;I
=
-JB
= FAFA/(p”t) =
[RT/(p”C)](--dF,/dz)
“Normal” Interdiffusion
[RT / (@”c) ] ( -d~A/’&)
I n the type of experiment discussed above, the concentration gradients are equal in magnitude and opposed. This is achieved by flushing the tWo terminal faces of the porous plug or diaphragm with ideal gases of constant composition and equal pressure. Numerical equality of forces also obtains when the plug is placed between two large gas reservoirs of equal volume which are connected by a n external duct (70). I n another type of experiment, the fluxes, rather than the , forces, are equal in magnitude and of opposite sign : JA
+
J B
=
0
(26)
To demonstrate the effect of this condition, we introduce it into the basic force Equations 7 and 8, respectively, and multiply them by and - EB, respectively.
+
=
where C = E ~ 4 EB. !Ye see that, in general, the condition of numerically equal and opposed concentration gradients CAFA/RT= -FBFB/RT (Equation 2) is incompatible with the condition of numerically equal and opposed fluxes J.i = - J B , except when X A r = X B r . Consider, for instance, the type of gas interdiffusion experiment in which t\vo pure gases originally a t equal pressure are held in two closed vessels of equal volume, separated by a
(29)
‘This expression is of the type of Fick‘s law with a single interdiffusion coefficient, BAB; D.dB
=
RT,’(p”c) = eRT;‘(p‘‘F)
(30)
As a limiting case, Equations 29 and 30 even describe interdiffusion in free gas space. I n this case one can consider the walls of the container in which the interdiffusion experiment is carried out as a “porous medium” consisting of a single pore. Here the gas-\vall friction Coefficients are entirely negligible. Equation 30 then applies to any pair of gases a t any pressure, Since, for ideal gases, 6’’is independent of the composition (albeit not independent of the temperature and pore geometry), we see that for ideal gases there is only one interdiffusion coefficient which varies inversely with the concentration or total pressure. These are well knoivn experimental facts ( 4 ) . Pressure Dependence of Self-Diff usion
Bosanquet Equation. \Ye consider interdiffusion through a porous plug a t uniform prewire and total gas concentrations F = Fi Substituting for d“ from Equation 13, we obtain from Equations 2 3 and 24, resprLtively,
+
VOL. 5
NO. 4
NOVEMBER 1 9 6 6
531
=
CB
(32) These are equations of the Bosanquet type (Equation 1). The first terms on the right are the reciprocals of the Knudsen diffusion coefficients, DA, and DBK,respectively (Equations 18 and 19), and hence independent of the pressure, while the second terms depend on the pressure and are not equal. For the interdiffusion of two gases of equal Knudsen diffusion Coefficients, DAK, we obtain from Equations 18,30, 31, and 32
~ / D A= ~ / D B=
+ ( ~ / ’ D A B ) ( ~ / D A K+) ( ~ / D A A )
(~/DAK)
=
(33) because for the case of self-diffusion, D A B is designated as B A A , by definition. Thus the application of the friction model to self-diffusion leads directly to the Bosanquet equation.
Comparison of Different Descriptions of Interdiffusion through Porous Plugs
For a given gas pair, interdiffusion through a specific porous plug is characterized by a set of three parameters: the experimental ones DA, D, (measured a t -dc,/dz = d C B / d Z ) , and DAB (measured under conditions of “normal” interdiffusion, when JA = - J B ) ; or the friction parameters X A c ,XBc, and p”, which can be calculated from the three experimental coefficients by use of Equations 18, 19, and 30, respectively; or the phenomenological coefficients Lu, L I Z and , L22of the general transport equations of nonequilibrium thermodynamics (74).
+ LIPB
JA
=
LuFA
JB
=
L ~ ~ 4F ALZVFB
(34) (35)
The description in terms of friction coefficients is useful because it leads to a number of important observed facts with a minimum of assumptions, and yet does not seem to contradict any observed findings. At least for ideal gases, this description also permits the calculation of interdiffusion a t any desired pressure from data obtained a t other pressures, which cannot be done directly by use of the L coefficients in the formal treatment of irreversible thermodynamics (Equations 34 and 35). The friction model treatment is not meant to supersede more sophisticated traditional treatments in terms of the kinetic theory of gases which also contain the basic assumption about numerical equality of applied force and friction force (Newton’s third law) expressed by Equation 3, and, for ideal gases, are based on the three parameters m A , mB (molecular masses), and the concentration of the “dust” representing the solid ( 3 ) . However, even the very simple assumptions of the friction model lead directly to many pertinent facts about gas interdiffusion which have been observed in experiments.
The suggested units are identical or consistent with those of Spiegler (78).
532
=
concentrations of gases A and B, respectively, in porous medium, mole per cc. (gross volume)
I&EC FUNDAMENTALS
respective concentrations in free space
= total gas concentration in porous medium,
mole per cc. an expression defined in Equation 13, wattZ sec4 cm.-4 mole+ = individual diffusion coefficients of A and B, dA, D B respectively, a t uniform pressure, sq. cm. per sec. = Knudsen diffusion coefficients of A and B, reqpectively, in porous medium, sq. cm. per sec. = self-diffusion coefficient of A in medium, under “normal” interdiffusion conditions, sq. cm. per sec. = interdiffusion coefficient of A and B in DAB porous medium (when J A = - JB),sq. cm. per sec. = generalized forces acting on A and B, FA, FB respectively, watt sec. cm.-1 mole-’ = phenomenological coefficients as defined in L11, Ll?, L22 flux Equations 34 and 35, mole2 c m - 1 watt-’ sec.-Z = molar fluxes of A and B, respectively, mole cm. --z sec. -1 = molar fluxes of A and B, respectively, under Knudsen diffusion conditions, mole c m . + set.-' R = universal gas constant, u a t t sec. mole-‘ io K,)-I RAA,R I B , RBB = phenomenological coefficients as defined in flux Equations 9 and 10, watt sec.z cm. mole+ T = absolute temperature, O K . UA, UB = average linear (superficial) velocities of gases A and B, respectively, cm. per sec. XAC, XBC = friction coefficient of A and B, respectively, with pore \valls, watt sec.z cm.-2 mole-1 z = length coordinate in direction of diffusion, an. B = friction coefficient between A and B, as defined in Equation 4, watt sec.2 cm. mole -2 = chemical potential of -4and B, respectively, W A ! C(B watt sec. mole-‘ e = porosity (fraction of free volume) of plug d I’
=
literature Cited (1) Carman, P. C., “Flow of Gases through Porous Media,”
Academic Press, New York, 1956. (2) Einstein, A , , Ann. Physik 17 (4), 549 (1905). (3) Evans, R. B., 111, Watson, G. M., Mason, E. A., J . Chem. Phys. 35,2076 (1961). (4) Jost, W., “Diffusion,” Academic Press, New York, 1960. (5) Kedem, O., Katchalsky, A., J . Gen. Physiol. 45, 143 (1961). (6) Klemm, A., 2. Naturjorsch. 15a, 173 (1960). (7) Laity, R. W., DiscussionsFaraday Soc. 32, 172 (1961). (8) Lamm, O., ArkivKemiMineral. Geol. 18B, N o . 5 (1944). FUNDAMEXTALS (9) Lightfoot, E. N., Wills, G. B., IKD.ENG.CHEM. 5,114(1966). (10) McCarty, K. P., Mason, E. Phys. Fluid.r3, 908 (1960). (11) Newman, J., Bennion, D., Tobias, C. W., Ber. Bunsenges. 69, 608 (1965). (12) Ney, E. P.,Arristead,R. C., PhyJ.Rev. 71, 14(1947). (13) .Pollard, M’.G., Present, R . U., Ibid.,73,762(1948); quoting British T A Rept. BR-507 (1944) (classified, not available). (14) Prigogine, I., “Thermodynamics of Irreversible Processes,” Interscience. New . York. 1961. ( 1 5 ) Rayleigh; Lord, Proc: Math. Soc. London4, 363 (1873). (16) Scott,D. S., Dullien, F. A. L., A.I.Ch.E. J . 8 , 113 (1962). (17) Spjegler,K, S., J,Electrochem.Soc. 113, 161 (1966). (18) Spiegler, K. S., Trans. Faraday Sod. 54,1408 (1958). (19) Stefan, J., Wien.Sitzber 63 (2), 63 (1871); 65 ( 2 ) , 323 (1872). ~~~~
Nomenclature
E A , EB
F
~~~
RECEIVE,D for review February 4, 1966 ACCEPTED hfay 5, 1966
