Chemical
Enyineering
Science,
1974, Vol. 29, pp. 1227-I
DIFFUSION
236.
Pergamon
Press.
Printed
in Great
IN POROUS MEDIA STRUCTURE
Britain
OF A RANDOM
L. M. PISMEN Institute of Electrochemistry, (Received
Moscow, U.S.S.R.
23 July 1973)
Abstract-The present paper deals with diffusion in a porous solid, which is considered as a discrete random medium composed of random structural elements (pores) chaotically connected with each other. Each structural element is characterized by the passage time distribution function calculated with taking into account the correlations of flows passing through the neighboring pores. The analysis of a nonstationary diffusion process, after transition to the limit of times much exceeding the mean passage time of a single pore, enables to obtain general expressions connecting the effective diffusivity in a porous medium with the statistical parameters of structural elements. At the same time the values are calculated which characterize the deviation of the concentration distribution in a porous medium from the gaussian one obtained as a solution of the quasi-homogeneous diffusion equation. To make the general expressions more concrete it is necessary to introduce a model of structural elements (pores) which determines the passage time distribution. The models of straight and corrugated pores considered yield expressions for the effective diffusion coefficient in the Fick and Knudsen regions. A similar analysis of the nonstationary diffusion accompanied by the first order reaction allows to get the expression for the effective rate constant of the catalytic reaction in a porous medium.
An investigator dealing with the problem of the theoretical analysis of porous media properties faces a rather paradoxical situation. On the one hand, the complexity of the real porous media structure rules out the application of simple models; on the other hand, the low accuracy, and, most commonly, the discrepancy of experimental data make it senseless at first sight to predict the properties of porous media on using complex calculational methods. The settling of this paradox lies in the fact that the phenomenological characteristics of porous media depend on numerous concealed structure parameters which cannot be measured and checked in a normal experiment. The consideration of these parameters is necessary for the theory that should describe the processes in real media, even though, for lack of information on the detailed structure of a porous medium, the accuracy of the description will be unattainable. Extensive studies are devoted to experimental and theoretical determination of effective diffusivities in a porous medium; a detailed review is given in the Satterfield’s book[l]. The basic theoretical result obtained was the calculation of two factors - the tortuosity factor [2-61, and the shape factor[7-91, - which take account of the diffusion deceleration in a porous medium under the influence of tortuosity and corrugation of pores.
These results are based on the simple models of the porous media structure, and partially contradict one another. This model approach predominated in the theory of diffusion in porous catalyst pellets. As an alternative, other methods [ 10,l l] were suggested and adopted from the dielectric permeability theory of nonhomogeneous media [12], in which mathematically similar problems have been set as early as the time of Maxwell. The second approach is of interest, since it abandons primitive, though visual, model representations, such as Wheeler’s “bee model”. However, it was not successful, in so far as the material structure of a porous medium was not taken into account. The common shortcomings of both opposite approaches lies in the fact that they fail to combine the allowance for the porous medium structure with its statistical description as a random medium. Both structural and statistical descriptions of the porous medium can be achieved by separating the pae space into discrete random structural elements chaotically connected to each other. In this case the substance transfer in a porous medium is divided into numerous elementary transfer processes inside and among structural elements. Such methods have been employed in the theories of capillary equilibrium [6,13- 151, permeability and flow dispersion [ 16-241 in porous media.
1227
L. M.
1228
The present paper deals with the calculation of the effective diffusivity and the effective rate constant of a chemical reaction in a porous catalyst considered as a discrete random medium. This method is in general similar to that which the author has used earlier in the study of the flow dispersion in granular catalyst beds [2 1,223. The advantage of this method is that general formulas (Sections 3, 4) are derived without any model representations on the nature of the structure elements; the parameters of specific models are introduced only at the last stage. Apart from the calculation of the basic macroscopic characteristics of the transfer processes, this method enables one to indicate the applicability limits of the quasihomogeneous description of a porous medium. An essential improvement of the previous method consists in taking account of short-range correlations, which are important for the diffusion process and lead to the increase of flux through the pores with high permeability. The parameters of structure elements are regarded as random variables with a prescribed distribution; the order of the element combination is also considered to be random. A particle leaving the structure element is assumed to lose its history completely; therefore, only the correlations between the fluxes through the pores with a common joint are taken into account. The parameters of the neighbouring structural elements are assumed to be statisticaliy independent. Therefore, the processes in spatially non-homogeneous porous media are not studied in the present paper. 2. STRUCTURAL
ELEMENTS
Consider the pore space to consist of discrete structural elements (“pores”) with random parameters. Let each pore be characterized by the distribution function of its passage time (p(~/s), which will be referred below as microdistribution. By a passage we call the transfer of a diffusing particle from one end of the pore (“inlet”) to the opposite end (“outlet”) resulting in the displacement of the particle by a certain distance. In case the particle that has entered into a pore does not succeed in getting out of it, the passage is considered not to occur, and the particle coordinate to coincide with the inlet coordinate (on describing the substance transfer by distances which much exceed the length of a pore, this assumption causes only a negligibly small error). The value (p(r/s)dr is, thus, equal to the probability that the diffusing particle appearing at the inlet of the pore will reach the outlet after the period of time from T to T+ dr. The microdistribuction (P(T/s) depends
PISMEN
on the random vector s, whose components are the parameters defining the size, shape and orientation of the pore. The vector s distribution is characterized by the probability density f(s), being called below as parametric distribution. The porous medium is supposed to be statistically homogeneous, since the parametric distribution does not depend on the coordinate x. 3. MACRODISTRIBUTION
Let us introduce the macrodistribution function $(t, x) that has a sense of the probability density of the diffusing particle displacement during the time t from the origin to the point with the coordinate x. Apparently, this function defines completely the process of the substance transfer in a porous medium. The statistical characteristics of the transfer process we are interested in may be obtained from the Fourier-Laplace transform of the macrodistribution function (moment generating functionMGF) G(p, o) = i eeP’dt f eioX$ (t,x)d%. (1) -cc 0 Consider the fixed trajectory passing through n pores with some values +(j = 1,2,. . ., n) of the random vector s. Let &‘(t 1{Sj}) be the probability of the event that the diffusing particle during the time t will pass n-l pores and will stay in the nth pore, whereas G,O@I {sj}) being its Laplace transform. The passage time of the trajectory is the sum of passage times of single pores tj, which are independent random variables. Therefore, the MGF G,” equals to the product of MGF’s of elementary events. The MGF of the single pore passage time is the Laplace transform g(p)s,) of the microdistribution function cp(t( sj). Since the probability of retaining in the nth pore during the time t, is equal to
tn
I-j-dW 0
the MGF connected event the non-passage
p-‘[I-_g(hJl.
with the last elementary through the nth pore is
Thus,
Y
j=l
The displacement x on passing the trajectory is equal to the sum of displacements xj on passing single pores. These values are considered to be statistically independent; therefore, the FourierLaplace transform (MGF) G,(p, Wl{Sj Xi}) of the
Diffusion in porous media of a random structure
probability +, (t, XI {si, Xj} ) of passing the same n-unit trajectory during the time t and arriving at the point x is
G,(P, 4~
xjl) =
1229 a+im
=A
P.(1)
I
eDtG(P) dp
(8)
and examine l-g(ptsn) n~[g(pls,i)e~?“]. p (3)
Summing over the trajectories involving any number of units, an averaging over the distribution of pore parameters, yields:
G(P,~=(C Gn) 7l=1
Here the angular brackets denote the averaging over the parametric distribution (the displacement along the pore xj is also considered to be involved into the parameter vector sj ). Assuming the parameters of the neighbouring pores to be independent random values, we obtain:
the asymptotic behaviour of the statistic moments at t + ~0. So far as the averaged microdistribution function (cp(rlsj)) should be normalized by unity, (g(0)) = 1, and function (7) has the second order pole at the point p = 0. From the normalization condition of function (cp( 7)) it also follows that in the right semiplane, including the imaginary axis, k&(p)) < 1 and function p2(p) has no singularities there. Ail the singularities of function F*(P) other than p = 0 lie, thus, in the left semiplane. Since the microdistribution function results from the solution of the equation of a single pore diffusion, all singularities should be simple poles with the only point of accumulation at infinity. Consequently, the asymptotic behaviour of the macrodistribution dispersion is defined by residue of the intergrand in Eq. (8) at the point p = 0: p2(f)
=
@%Mpl~j)) 1AdP[p l-(g(plsj)) I)
(9)
p = 0.
Using the expansion of function (g (~1s~))
l-
= p[l-
k(Pbj)) (g(p]sj) e(03cj)]’
4. STATISTICAL
(g(plsj))
(5)
= i-k(akGlawk)w=o.
1+c (-l)k@$pPk
(10)
k=l
(ok (sj) ) being the averaged moments of the microdistribution function, we obtain:
MOMENTS
Differentiating Eq. (5) with respect to o and putting o = 0, we obtain the Laplace transforms of the macrodistribution moments cc,(p)
=
lx,*)
/+2(r) =&jjy+
w (%(G)) [ 2(Ql(Sj))’
Cxps,,,) 1
J
I.
(11) (6)
In the case of an isotropic medium it is obviously sufficient to study the moments characterizing the displacement along one of the coordinates x only. For the zeroth moment we have pO(p) = G (p, 0) = p-l, that corresponds to the normalization of the macrodistribution function $(t, X) by unity. The first moment is proportional to the average displacement along the pore and vanishes in an isotropic medium. The second moment (dispersion) is of greatest interest: its Laplace transform according to Eqs. (5) and (6) has the form:
Use the formula of the Laplace reverse transform
The exact formula for pZ( t) should contain additional exponentially decaying terms. The relaxation time characterizing the approach to the asymptotic formula (11) is determined by the position of the pole of the function (8) in the left semiplane nearest to the imaginary axis, and is of the same order of magnitude as the characteristic time of passing through a single pore. Deviations from the asymptotic formula (11) at the macroscopic times under consideration should, apparently, be negligible. In case of describing the transfer processes in a porous medium by the quasi-homogeneous diffusion equation
(12)
1230
L. M. PISMEN
the variation of the macrodistribution dispersion with the time can be given by the formula /.&(I) = 2(D*le)t
(13)
where D * is the effective diffusivity and E the porosity. Comparing Eqs. (11) and (13) we define the effective diffusivity in a porous medium as D”
_
$f’;. ffl
As it should be expected, the effective diffusivity appears to be proportional to the ratio of the average square displacement on passing the pore (x,‘) to the average passage time of a single pore (LY~),both values being averaged over the parametric distribution. It is essential that Eq. (11) not only allows to determine the value of the effective diffusivity, but also, it indicates the limits of the quasi-homogeneons diffusion equation applicability for the description of the substance transfer in a porous medium. First of all, quasi-homogeneous description of a porous medium is not rightful at small distances which are comparable with the length of a single pore, when the asymptotic formula (11) is not valid without the addition of exponentially decaying terms. Eq. (I 1) contains another condition of the diffusion equation applicability: the constant term in square brackets must be small in comparison with the first one proportional to the time. The additional condition of the diffusion equation applicability is that the higher cumulants of the macrodistribution are small (they are exactly zero for the Gauss distribution, which is a solution of the quasi-homogeneous diffusion equation). Only even macrodistribution moments are likely to be different from zero in the isotropic medium. By differentiating Eq. (5) four times with respect to o, we get the Laplace transform of the fourth moment
cL4(P)=
c%%) +6 P(l-k))
((xj%>)2.
P 1-k)
The coefficient of excess tends to zero a bit slower than the relation of the term in square brackets in Eq. (11) to the term proportional to t in the same equation. 5. CALCULATION OF FUNCTION (e(p)) For further investigation, it is necessary to make the porous medium model more elaborate in order to calculate the microdistribution moments in Eqs. (1 l), (14) and (16). The most simple way of obtaining the microdistribution function would be to solve the diffusion equation for a single pore with the initial condition having a form of the deltafunction, and with the reflecting and absorbing boundary conditions at the inlet and outlet of the pore, respectively. The microdistribution function (P(T) is then determined as the substance flux at the pore outlet at the moment 7. Such a procedure has an obvious shortcoming; the fact that the substance may return back to the inlet of the pore is not taken into account. Besides, in this case the flux of the substance passing through narrow and long pores with high resistance is exaggerated, and the flux through the pores with high permeability is underestimated, respectively. A more sensible approach involves the consideration of correlations of the fluxes through the neighbouring pores. Consider N pores converging in a common joint (Fig. 1). For the time being we characterize each pore by two parameters: the cross-sectional area S, and the length 1: a pore may be, thus, represented as a straight cylinder with the constant cross-section. The substance transfer in each ith pore is described by the diffusion equation
D
a2ci
=
ac, at
a.$i
(17)
where D is the molecular diffusion coefficient. Putting the origin of the porewise coordinates & at the joint (their common inlet), we set the initial condition as follows
(15) The boundary conditions in the joint are
The asymptotics of function p4(f) is obtained by the same method exactly that has been described in the foregoing. The main term in the asymptotic expression for the coefficient of excess Z%(t) = /14/hZ - 3 is: (al)].
(16)
Ci=idem;
i i=l
St
(19)
So far as by definition the passage through the pore is considered to occur when the diffusing particle reaches the pore outlet (whose coordinate is equal to Zi)>the absorbing boundary conditions
Diffusion in porous media of a random structure
0
/
k .
-c
1231
(ids) = 1,2/2D; (01~) = 3 l,“/D’.
The result happens to be independent of both the distribution of the pore cross-sectional areas and the number of pores converging in a joint. TO determine the values (Xj”), (Xj’ CX~) etc. it is necessary to expand the following function in the same way:
Ii
(Xj”g) = (2 Sili’ cosech (p/D)‘Yi
Fig. 1. A pore joint.
X coth (p/D) “‘&
should be determined
>
at the pore outlets Cj(li)
(24)
= 0.
Performing the Laplace transformation over t, from the solution of Eqs. (17)-(20) we obtain the Laplace transform of the distribution function of the passage time through thejth pore
I
2 &
(COS’8).
(25)
Here the averaging over the angle 13between the pore axis and the axis x, is separated. The value K = (cos20)-’ is usually called as the tortuosity factor. Expanding (25) with respect top we get:
($9 =
5(gg); (~lXl9 = & (($$
+2((Ey))~
The value (xf) involved into Eq. (16) is obtained by expanding function (xj4g (p) ) :
= NSJsinh Note that the microdistribution function obtained in this way is not normalized by unity, and the total flux through the pore g(O(q) appears to be proportional to its permeability Sl/lj. However, function (g(p)) averaged over the parametric distribution will be normalized, since
Rearranging function (g(p) ) (g(P))
= (Z:
Si
cosech ((p/D)“‘&)
x coth ( (p/D)Ti))
I
C Si (22)
(X$)
=
( COS48)
(z
Si&‘,/C
~(~);
(a2) =
j&((Z)
+4(($$-)2)). (23) In the particular case for the pores of similar lengths 1, we have:
(27)
S,iii).
Inserting Eqs. (23), (26) into Eq. (14) we obtain: D* =
ED/K.
(28)
Equation (28) is the usual expression of the effective diffusivity involving the porosity E and the tortuosity factor K. In this case the effective diffusivity turns out to be independent of the distribution of the pore lengths and cross-sections. The formula for the coefficient of excess appears to be rather cumbersome, but it may be simplified by assuming the procedure of summing over the pores outgoing from the joint to be approximately equivalent to that of averaging:
and expanding it into a series (10); we obtain averaged microdistribution moments:
(a,) =
(26)
1 .
(29)
It can be seen from here, that provided the scatter of pore lengths is not so wide the coefficient of excess is always small at the times much exceeding the average passage time of a single pore (12)/D. 6. TORTUOSITY
FACTOR
In an isotropic medium with all pore orientations
1232
L. M.
PISMEN
equally probable the tortuosity factor is
KC
-1= cos% sin t3de -I = 3. II2 ) 0
(30)
Different values of the tortuosity factor have been derived theoretically in Refs: K = fi[2], K = 2[6], K = fi[3], K = 3 [4,5]. Though the methods leading to the values K = tiand K = 2 are vulnerable from the theoretical point of view, these results are most widely known since they better agree with the experimental data. The value K = 3 exceeds the tortuosity factor values which are usually observed experimentally. This contradiction is, however, only seeming. The thing is; that the obtained result is valid only in the limit of infinitely thin pores. In a pore of finite width the main orientation of the substance transfer does not coincide with the orientation of the pore axis, but forms an angle with it (Fig. 2). Taking account of this clear geometric fact, one may obtain a smooth transition with the increasing porosity from the porous medium, characterized by the tortuosity factor exceeding unity, to the free space with K = 1. A simple formula with necessary properties can be derived when diffusion in a pore is assumed to proceed along the direction which is the nearest to the chosen axis x of all those allowed by the pore geometry. Let the maximum angle between the diffusion direction and the pore axis be equal to @r with sin O1= d/l (d is the pore diameter, see Fig. 2). At 0 < B,, or n - 0 < &, the diffusion will proceed along the axis x, whereas at 0, < 0 < r - 0, at an angle 87 (?I (the sign minus refers to the angles which are less than 7r/2 the sign plus, to those which are larger). It should be noted, that upon the pore widening the porosity changes proportionally to d2; putting d/l = l, we get w K-1 = I 0 71-h +;
I rrie
7r/2 sin 0 do+;
I 81
06 o-5
l g
0.4
o-3
02
0. I
o-2
0.3
0.4
0.5 0.6
0.6
I 0
Fig. 3. The plot of Eq. (3 1) superimposed on Cunie’s data [9] representing effective difisivity vs porosity for various unconsolidated porous media.
literature[9]. Some divergence from the experimental data at large porosities can be naturally explained by the fact that such highly porous medium as steel wool etc. cannot have a purely random structure. In this region the experimental data are also the least precise. 7. THE EFFECTIVE DIFFUSIVITY REGION
IN THE KNUDSEN
Upon thinning-down the pores the dependence of the diffusion coefficient in pores and also, that of the effective diffusivity on the pore diameter is manifested. Let us write the improved expressions for the microdistribution moments, taking account of unequal diffusion coefficients in different pores:
cos2(f3- 0,) sin 8 de
cos?(e+e~)sinede=
Fig. 2. The flux orientation in a pore of finite width.
l-~(l+~)(l-e)~~~ (31)
As it may be seen from Fig. 3 this formula well agrees with the experiment at small and intermediate porosities. The solid line in the figure corresponds to Eq. (3 l), K being equal to the ratio of abscissa to ordinate. The experimental data for unconsolidated porous media are taken from the
(4 Using
=~($g)+f((&J’).
Eq. (14) the expression
for the effective
Diffusion in porous media of a random structure diffusivity
is obtained:
D” = f(B)/(&).
(33)
Unlike the case of the constant D above, Eq. (33) the dependence of D* on the
1233
discrepancy is small for not so wide spread distributions. By a lucky chance, it turned out, as though Eq. (36) takes account in some way of the pore competition, despite of the essentially arbitrary choice of the mean pore diameter of all possible expressions of(d”)/(d”-I) type. 8. CORRUGATION OF PORES AND SHAPE FACTOR Consider now the case of corrugated pores with varying diameter. The diffusion deceleration due to variation in the cross-section of pores is characterized by the shape factor. The effect of different types of corrugation has been investigated in papers[7-91. The approach used here allows to find the general solution of the problem without introducing model representations up until the last stage of calculation. The diffusion in a pore with a varying cross-section area S(&) is described by the equation
0:
= f (12)l(lzID).
(34)
If D = ud (in the Knudsen region) and the diameter and the length of the pore are considered to be uncorrelated, then Dj+ = (m/K) (l/d)-‘. Now consider the opposite case of the extremely high competition. It is realized at a great number of pores in a joint, when the operation of summing over the pores outgoing from the joint is approximately equivalent to that of averaging. In this case Eq. (33) yields: D
ZIG= m
1. _ WD) K
=-_
(sl)
EU(d3) K (d’)’
(35)
The boundary conditions are given, as usual, by Eqs. (1 Q-(20). The microdistribution moments can be found by multiplying Eq. (37) by tk, integrating over t, and by solving consecutively the resulting equations for functions gik(5;) = JtkCi(ti,t) ((ak)
=
((SiD
dgik/d5i),i=l,)):
&(sOi)
$$)
D* _ 4.5’~ _ II KU
lu K
(d*)
=~gik-‘.
(38)
The boundary conditions are: gi”(&) = 0;
It can be easily seen that, since D,*lD: = (d3)(d-l) (dZ)-’ > 1 the competition of pores leads to the increase of the effective diffusivities; the wider the pore diameter distribution, the higher the diffusivity. Usually the effective diffusivity in the Knudsen region is expressed through the pore hydraulic diameter 4~/o = (d2)/(d) (u is the surface area in the unit grain volume):
dt
0
gik(0) = idem;
x (S*$r,=o
=;akO.
(39)
i
The solution of the equation Of gi” yields: ‘i
(36)
(d)’
deviation of the relation D,*/D$ = (d3)(d) ( d2)-l from unity may be of either sign, and this
The
CES VOL.
29 NO. 5-L
Inserting
function
gio(ti)
into the equation
for
L. M. PISMEN
1234 Ri’(Si),
Another expression with F < F,,, is obtained in case the procedure of summing over the pores outgoing from the joint is assumed to be equivalent to that of averaging:
we get:
F = (Ji S(<)d+/(r,“/; 0
S-‘(5)dg)
.
(44)
0
The more the pore cross-sectional area distribution is smeared, the more the limiting value of the shape factor exceeds unity. 9. THE EFFECTIVE RATE CONSTANT OF THE REACTION ON A POROUS CATALYST
The last equality is written with allowance for the symmetry of the pore inlet and outlet. The value (Xj”) = (12)/~ can be also determined from Eq. (40):
. Thus, the expression for the effective Eq. ( 14) can be given by
(42)
diffusivity
Dj:=DE. KF'
The approach under consideration enables to solve one more problem, viz. the definition of the effective rate constant of the reaction in a porous medium, k*. With all attention given to the definition of the effective diffusivity D”, the problem of obtaining another important value k* has been neglected until now. Meanwhile, the validity of a usual definition of k* as the product of the rate constant per unit of the active surface x, by the surface per unit volume of the porous medium u, is not a priori evident. On the contrary, it might be expected that the effective rate constant should decrease due to a limited reagent access into narrow pores, which have the largest surface to volume ratio. For the definition of k”, it is convenient to consider the following system. Let a steady state unit flux of the substance diffusing into an infinite porous medium and reacting within pores be delivered to the origin. Provided the substance transfer is described by a quasi-homogeneous diffusion equation D*V2C = k*C
where F is the corrugation factor, or the shape factor. Fixing model functions S ([) or random distributions of value S, one may obtain from Eq. (43) different specific expressions for the form factor. At S = const, F = 1, naturally. The maximum value F is achieved in the limiting case of a rapid change of the pore cross-section along its length, when the procedure of integrating along the length of a single pore becomes equivalent to that of averaging over the distribution ‘of the pore crosssectional area. In this case Eq. (43) is reduced to the following: F max
=
(S) (S-1) = (S) (d-2).
(444
(45)
the total amount of the unreacted substance in the porous medium is equal to e/k”. Therefore, if Gr( p, w) is the Fourier-Laplace transform of the macrodistribution function in the medium with a chemical reaction, then, k” = c/G’(O, 0). The function G’ may be determined in the same way as the above function G (p,w). The probability of passing through the pore during the time r without the reaction is e-“(p(r), the reaction rate constant for the given pore kj being proportional to the ratio of the pore surface to its volume. This function corresponds to the passage time distribution for the same pore (P(T). Function Gr(p,~) may be, thus, written as in Eq. (4) with substitution of p by p+ k,+ Performing the sum-
Diffusion in porous media of a random structure
ming procedure we obtain the following equation instead of Eq. (5): Gr(p
“)
(P+kj)-‘)
=([l-g(P+kjlsj)l (1 -g(p+kJsj)
(46)
eiwXj)
and, consequently, k* T=~=([l-g(k,,s,),k,-‘)’
(I-g(kjIsj)) (47)
Using Eq. (22) for the microdistribution function corrected by taking into account a possible difference of the diffusion coefficients in different pores
cosech (%)I’2
S/x
SiD:/2 (p + ki)r(2
coth r+)1’2/1)
(48)
yields:
k*=(g$/($gJ E
(49)
Let us put down simplified expressions for k*, assuming, as in the above case, the operation of summing over the pores outgoing from the joint to be approximately equivalent to the operation of averaging:
(50) This expression turns out to be independent of the diffusion coefficient D. The usual expression for k* may be represented as k*=p=qa=e$$
E
(51)
It must be underlined that the coincidence of Eqs. (50) and (51) follows from the assumptions of sharp competition between different pores and statistical independence of pore lengths and crosssections. The traditional expression for k* is valid only within this approximation. 10.
CONCLUSIONS
This paper presents a general approach to the problem of diffusion in catalyst pores based on the
1235
representation of a porous body as a discrete random medium. It allows to calculate the effective diffusivities at arbitrary suppositions about the pore length and pore radii distribution, the number of pores in joints, the type of pore corrugation, and so on. Unfortunately, the information required for such a calculation much exceeds that the investigator usually has: moreover, in general it is much more difficult to obtain such information than to measure directly the effective diffusivity for a certain porous substance. The author believes that the principal value of this method, as well as of all theoretical papers on the porous medium structure and properties, is that it gives a possibility to estimate qualitatively the role of various structural factors, and also, it enables to find out the conditions under which they may be essential. It has long been realized by investigators, see for instance[7], that such simple characteristics as porosity and the mean pore diameter are far from being sufficient for the unequivocal definition of the effective diffusivity, and the experimental data scatter obtained for media of equal porosity but prepared in different manners, or of different substances is not incidental, and cannot be explained by measuring errors only. Even such a property as the distribution function of pore radii appears to be insufficient for an unequivocal prediction of the effective diffusivity, since it by no means involves such factors as the pore space structure (the number of pores converging in a joint), and the intensity of corrugation. At the same time, under certain conditions various complicating factors may be of little importance. For instance, in the Fick region the effective diffusivity appears to be independent (in the absence of some noticeable corrugation) of the distribution of the random pore parameters. Accordingly, the least experimental data scatter can be observed in unconsolidated wide-porous media, in which the effect of random structural factors is shown least of all. In consolidated media in which the role of corrugation increases, greater values of DID”’ different for various substances, may be observed, that is naturally explained by the unpredictable increase of the shape factor. A wide scatter of experimental values of effective diffusivities is typical for the Knudsen region, in which the effect of random structural factors is also essential. It should be noted, that those structural factors, which effect the effective diffusivitity, also define the properties of a porous medium, such as capillary equilibrium, permeability, etc. Therefore, a simultaneous experimental study of different
L. M. PISMEN
1236
properties of one and the same porous medium, combined with a detailed study of its geometric structure would be mostly useful for the subof calculational stantiation and verification methods and porous medium models. NOTATION
c
concentration D molecular diffusivity D* effective diffusivity d pore diameter F shape factor G macro, MGF macro, MGF g k rate constant per unit volume k” effective rate constant 1 pore length N number of pores in a joint Laplace
; s t
time
X
coordinate
pore
cross
parameter
variable section vector
Greek symbols
microdistribution moments porosity azimuthal angle tortuosity factor macrodistribution moments distance along the pore surface area per unit volume passage time macrodistribution function microdistribution function rate constant per unit area Fourier variable REFERENCES
[l]
SATTERFIELD C. N., Mass Transfer in Heterogeneous Catalysis MIT Press, Cambridge Mass. (1970).
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