A corrected tortuosity factor for the network calculation ... - Science Direct

Shorter Communications Greek letters ratio of pore radii at entrance and constrictions volume fraction of water in macrostructure '£rn total volume fraction of water in the matrix gt 6 factor accounting for the pores of varying crosssection r tortuosity p density

REFERENCES Atto, A. T., Marsden, E. P. and Nursten, H. E., 1968, Kinetics of leather dyeing--Part I. J. Am. Leather Chem. Assoc. 63, 315-325, Barter, R. M., 1953, A new approach to gas flow in capillary systems. J. Phys. Chem. 57, 35-40. Bear, R. S., 1952, The structure of collagen fibrills. Adv. Protein Chem. 7, 68-160. Bhatia, S. K., 1985, Directional autocorrelation and the diffusional tortuosity of capillary porous media. J. Catalysis 93, 192-196. Bienkiewicz, K., 1983, Physical Chemistry of Leather Making. Krieger, Huntington. Dinh, S. M., Luo, C. W, and Berner, B., 1993, Upper and lower limits of human skin electrical resistance in iontophoresis. A.I.Ch.E.J. 39, 2011-2018. Dunlop, P. J., 1965, Diffusion and fractional coefficients for two concentrated compositions of the system, watermannitol-sodium chloride at 25°C: test of the Onsager reciprocal relation. J. Phys. Chem. 69, 4276-4283. Elden, H. R., 1971, Biophysical Properties of the Skin. Wiley, New York. Fatt, I., 1962, Modified diffusion time lag. d. Phys. Chem. 66, 760-762. Grigera, R. J. and Acosta, A. A., 1974, Determination of the equivalent pore radius in leather. J. Am. Leather Chem. Assoc. 69, 373-375. Heidemann, E., 1982, Newer developments in the chemistry and structure of collagenous connective tissues and their impact on leather manufacture. J. Soc. Leather Traders Chem. 66, 21-29. Heidemann, E. and Keller, H., 1970, X-ray Studies of tanned collagen. J. Am. Leather Chem. Assoc. 65, 512-536. Johnson, M. F. L. and Stewart, W. E., 1965, Pore structure and gaseous diffusion in solid catalysts. J. Catalysis 4, 248-252. Kannagy, J. R., 1964, Application of Archimedes' principle for determination of the apparent volume of leather, J. Am. Leather Chem. Assoc. 59, 636-649.

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Miglyachenko, A. F., 1972, Determination of the porous structure parameters of leather. Kozh. Obuvn. Prom. St. 2, 31-32. Mitton, R. G., 1946, The air permeabilities of light leathers and their specific surfaces. J. Int. Soc. Leather Traders Chemists 29, 255. O'Brien, J. A., 1983, Mathematical model for unsteady state salt diffusion from brine cured cattle hides. J. Am. Leather Chem. Assoc. 78, 286-299. Payatakes, A. C., Ng, K. M. and Flumerfelt, R. W., 1980, Oil Ganglion dynamics during immiscible displacement: model formulation. A.I.Ch.E.d. 26, 429-442. Petersen, E. E., 1958, Diffusion in pore of varying cross section. A.I.Ch.E.J. 4, 343-345. Ramasamy, D. A., 1970, Study of the chrome tanning complexes. TR UR-A7(60), U.S. Dept. of Agriculture. Samatha, G., 1991, Coupled diffusion in solvent assisted leather dyeing. M. Tech. thesis, Department of Chemical Engineering, A.C. College of Technology, Madras, India. Sanjeevi, R., Ramanthan, N. and Viswanthan, B., 1954, Pore size distribution in collagen fiber using water vapor adsorption studies. J. Colloid. Sci. 57, 207-21 I. Schmit, F. O. and Gross, J., 1948, Further progress in the electron microscopy of collagen. J. Am. Leather. Chem. Assoc. 43, 658-675. Silverston, P. L., 1986, Effective diffusivity and structure of porous catalyst Chap. 7, in Multiphase Chemical Reactors. (Edited by A. Gianetto and P. L. Silverston). Springer, Berlin. Stromberg, R. R. and Swerdlow, M., 1957, Pores in collagen and leather, J. Am. Leather Chem. Assoc. 47, 336-354. Thorstensen, T. C., 1976, Practical Leather Technology. 2nd Edition. Krieger, Huntington. Vogel, A. L., 1978, Text Book of Quantitative Inorganic Analysis, 4th Edition. Longman Group, London. Zakharenko, V. A. and Pavlin, A. V., 1973, Macro porous structure of natural leather. Kozh. Oburn. Prom. St. 15, 46-48. Zapotocky, J. A., 1966, Precipitation, complex formation and oxidation-reduction methods, in Pharmaceutical Chemistry, Vol. I. (Edited by L. G. Chatten}. Marcel Dekker, New York. Zettlemoyer, A. C., Schweitzer, E. D, and Walker, W. C., 1946, The internal surface area of hide. J. Am. Leather. Chem. Assoc. 41,253-264.

Pergamon

Chemical Engineering Science, Vol. 50, No. 5, Pp. 897 900, 1995 Copyright ~) 1995 Elsevier Science Ltd Printed in Great Britain. All fights reserved 0009 2509/95 $9.50 + 0.00

0009-2509(94)00220-7

A corrected tortuosity factor for the network calculation of diffusion coefficients (Received 27 April 1994; accepted in revised form 30 June 1994)

Intersecting capillary models, in which pores of constant cross section (often cylinders) are distributed along the bonds of a regular or irregular lattice, are very widely used in the modelling of diffusion (with or without reaction) in porous solids. I-Applications of these models have been reviewed by Sahimi et al. (1990).] The diffusion and reaction behaviour of

the pore network depends strongly on the coordination number, Z, which is the mean number of pores attached to a node of the network, and on the pore size distribution; other aspects of the lattice topology are of less significance. One attraction of intersecting capillary models lies in their tractability. If it is assumed that the pores are much longer

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than they are wide, diffusion and reaction is effectively one dimensional in individual pores and the solution of the network diffusion problem reduces to satisfying a mass balance at each node. An additional advantage of this type of model is that their parameters can often be estimated experimentally. Despite their popularity, intersecting capillary models suffer from a defect which becomes apparent when their predictions are compared with experimental diffusion results for granular porous media with high porosities. The effective diffusion coefficient in a porous medium, De, may be related to the molecular diffusion coefficient, Do, by De = Do-. r

(1)

E is the porosity of the medium and T is the tortuosity factor. For any regular lattice consisting of identical pores z = 3 (Sahimi et al., 1990). This result is also obtained for a network of randomly oriented cylindrical pores (Johnson and Stewart, 1965). If the pore size is distributed, z > 3. On the other hand, measurements of the tortuosity factor in packs of spheres with a narrow particle size distribution (giving a narrow pore size distribution) give r < 3 [see e.g. Kim et al. (1987) and references contained therein; a more complete list of references is given by Friedman (1993)]. This discrepancy stems, in part, from the assumption that the pores can be regarded as infinitesimally wide, in comparison to their length, and therefore that diffusion in each pore is unidimensional in a direction parallel to the pore axis. In this communication we propose an approximate correction to take into account the finite aspect ratio of the pores. The correction is derived for networks of cylindrical pores but the approach outlined here could be adapted to pores of other cross sections. The porosity of a medium composed of cylindrical pores is related to the coordination number, Z, the pore lengths, I, and the pore diameters, d, of the network by conservation of volume (Burganos and Sotirchos, 1987): =

n(d2)Z

(2)

8(l 2) •

The brackets indicate an arithmetic mean over all the pores in the network. In writing this equation, we have assumed that d and l are uncorrelated. (Other assumptions are possible; this relationship is in any case difficult to obtain experimentally.) We focus now on networks of identical pores, for which the aspect ratio is 6 = ~- =

VC

.

(3)

If we take values typical of real porous solids, we obtain relatively large values of 6; e.g. for e = 0.5 and Z = 6, 6 = 0.46. The reason that the assumption 6 ~ 0 causes an error for large aspect ratios is that in a real solid the macroscopic concentration gradient is not in general aligned along the pore axis. If a sufficiently large element of the isotropic solid is considered, the pores will sample all orientations equally. Thus, in a real solid, diffusion is not restricted to be parallel to the pore axis and the rate of diffusion along each pore is underestimated by a one-dimensional calculation, with the error increasing with 6. [It is worth pointing out that 6 ~ 0, which, from eq. (3), implies e ~ 0, is unlikely to be relevant for any real, amorphous porous solid as the percolation threshold, at which the pore network becomes disconnected and transport ceases, will be reached first.] For a pore of finite aspect ratio, we assume that the diffusing molecules always take the shortest possible path from one end of the pore to the other. The calculation is illustrated using Fig. 1. The pore is oriented in a direction forming an angle co with the macroscopic concentration gradient, VC. For co < ~ and co > n - ~ (where tan ~ = 6), we assume that diffusion takes place in the direction of VC; for ~ < co < n - ~, we assume that diffusion takes place at

Fig. 1. A cylindrical pore oriented at an angle ~o to the macroscopic concentration gradient.

an angle ~k from the direction of the macroscopic concentration gradient, that is, as close as possible to the macroscopic gradient, within the cylindrical envelope. In making this calculation it is necessary to invoke the smooth field approximation (SFA) (Burganos and Sotirchos, 1987; Sahimi et al., 1990), which states that the local concentration gradient experienced by the diffusing molecules in any pore is the projection of the macroscopic concentration gradient, VC, along the pore axis. Johnson and Stewart (1965) calculated the tortuosity factor for randomly oriented pores (of zero aspect ratio) by integrating over all possible directions in the range 0 < co < n and obtained z-1 _

?0

1 COS2~oSin C°do ) 2 = 3'

~4)

The factor sin co/2 is the probability density function that the pore has orientation w. For a pore of finite aspect ratio, making use of our assumptions, eq. (4) becomes ~ sin Z l l = Jo COS2I ~ - - ~ d c ° = f ] cos20 -sinco - ~ - d m + f./2 cos 2 (co - ~b)sin COde° ~ 2 +

-~

sin co

f~ cos~(~o ~)--~-d~ +

/2

+f~

COS2n sin °gdto. 2

(5)

J,,_¢,

From the pore geometry, sin ¢

d d2x/.~_ ~,

I cos ¢

~

12

(6)

Substituting eq. (6) in eq. (5) gives ri -1 = 1

2 (6 2 + 1) 1/2 -- 6 3 62 + 1

(7)

As the streamlines would, in fact, be curved rather than straight as we have assumed here, eq. (7) is strictly a lower bound on r (or an upper bound on r-1). The reciprocal of the corrected tortuosity factor, ri-1, is plotted against the aspect ratio, 6, in Fig. 2. For infinitesimally wide pores the assumption that diffusion occurs parallel to the pore axis is valid and zi- 1 ~ 31. As the aspect ratio increases, ri- 1 diverges from r - 1 until it approaches the physically correct value of unity as 6 --, pp. Pismen (1974) proposed a correction with a similar physical basis, but resulting in an incorrect mathematical formulation. He set 6 = sin 4~, rather than tan q~. In addition, Pismen assumed the aspect ratio to be related to the porosity through 6 = e 1/2 which, from eq. (3), implies a fixed value of

Shorter Communications I.O ! t

~r ~

- - ~

I

'

I

' ~

~ '

899

on the contrary, gives values which are smaller than unity and decrease with increasing Z (Fig. 4). This discrepancy is a consequence of the fact that neither free space nor a medium of very high porosity can be constructed from cylindrical elements. We introduce an empirical correction factor, f, to give a modified, corrected tortuosity factor, rz, that has the correct limiting behaviour as e --* 1:

I

06 04

"C; 1 = f T 1 1

t

0.2 {

i

O , O [ ~ . ~ _ ~ _ ~ ,

0.0

0.2

0.4

~_,

0.6

0.8

,

~-1.0

f must be unity at a = O, where eq, (7) is exact, and give r~- ~ = 1 at ~ = 1. For the sake of simplicity we assume that this factor increases linearly with the porosity.

f(Z,e,)

8 - diameter to length ratio Fig. 2. The corrected reciprocal tortuosity factor vs the aspect ratio of the pore.

IS)

= [ r l ( Z , e = 1) -- 1]e + 1.

(9)

Combining the aspect ratio for e = 1, 3 = 8 x / ~ Z [eq. [3)] with eqs (7) and (9) gives

23[0 +

f ( z , ~) =

8/gZ

+

8 / n z ) '/2 -

1 - - 23[(1 +

(8/nz) ''2]

8/7cZ) 1,2 - (8/7~Z) 1/2 ]

e+l. (10)

1.0

Substituting eqs (3), (7) and (10) in eq. (8) yields

.o (1.8

I

23[(l+8/ItZ)I/2-(8/nZ)I/2] 8/nZ + 1

OO

-23[(1 +

q

8 / n Z ) 1/2 - ( 8 / n Z p / 2 j

1J ~+

06

X [1 -- 23(8~/gZ + 1) 1/2 -- (88/~Z) 1/2] -8~G-z + i j

0.4 ,

02

°(g.o~

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

Z - coordination number Fig. 3. The dependence of the pore aspect ratio on t h e coordination number of the lattice, with porosity as a parameter.

O.S

~-

--1/'C

/ - ~ / / -

0.4

-- -- 4

{--

,11)

r ; 1 is plotted as a function o f t with Z as a parameter in Fig. 4. Figure 5 compares experimental values of the tortuosity factor, obtained from measurements of gas diffusion, for 48 sphere packs with narrow particle size distributions (and therefore narrow pore size distributions). These measurements were reported in thirteen publications: Kim el al. (1987) and 12 others listed by Friedman (1993). Although the scatter in the data is high, it is clear that the measured tortuosities are m u c h better predicted by eq. (11) than by 1; = 3, using a physically reasonable range of coordination numbers. In the limit 6--, 0, the effective diffusion coefficient of a network having a distribution of pore sizes can be calculated by directly solving the set of linear equations, representing conservation of mass at all the nodes of the network, or by using approximate theories such as the effective medium approximation (EMA) (Kirkpatrick, 1973), the renorrealized E M A (Sahimi et al., 1983) or the Monte Carlo renormalized E M A (Zhang and Seaton, 1992). The effective diffusion coefficient is related to the pore size distribution by 1

l/-r = 1/3

D~ = ~-¢Do .5

( d2)at

(12)



O.2

(1.2

0.4

0.6

018 '

1.0

C - porosity Fig. 4. r f ~and r~ 1 vs porosity, with coordination number as a parameter.

0.8 0.6

: ~

/

0.4 the coordination number, Z = 8/~ = 2.55, which is unrealistically low for most porous solids. Figure 3 shows the dependence of the aspect ratio on the coordination n u m b e r Z, with porosity, e, as a parameter, calculated using eq. (3). Substituting eq. (3) in eq. (7) yields the dependence of the corrected tortuosity factor on e and Z (Fig. 4). The corrected tortuosity factor is significantly different from r = 3 except at what, for most porous materials of practical interest, are unrealistically low porosities. For free space (i.e. e = 1), zi -~ should be unity. Equation (7),

~ "

n..... red m

l~r = 1/3

t

1

0.2

0.0

t

0.2

0.4

0.6

0.8

1.0

c - porosity Fig. 5. z~-1 and measured reciprocal tortuosity factors for packings of glass beads with narrow particle size distributions.

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Shorter Communications

where (d2)e, is the "effective" value of d 2, the diffusional conductance of an individual pore. (Here we have made the more restrictive assumption that the pore length is constant; as the length distribution is unlikely to be known in practice, this is probably as good an approximation as any. Equation (12) is easily modified to reflect an assumed length distribution.) To properly take into account the finite aspect ratios of the pores, eq. (12) should be modified as follows: D, = eD0

(d zTi- 1)©ff (d25

(13)

Here we have simply introduced the corrected tortuosity factor into the diffusional conductance of an individual pore. [For 6 ---,0, z[ 1 ...}~ and eq. (12) is recovered.] As, in any case, a real solid will correspond only approximately to our model of intersecting capillaries of constant cross section, the detailed calculation of eq. (13) is probably not justified in practice. We suggest the following approximate factorisation:

De = z[ l(~)eDo (d2)~rr

(14)

C De Do d

f

I Z

NOTATION concentration of the diffusing molecules, M/L 3 effective diffusion coefficient in porous medium, L2/T molecular diffusion coefficient in a single phase, L2/T pore diameter, L correction factor pore length, L mean coordination number of the network

Greek letters pore aspect ratio (d/l) 8 porosity T tortuosity factor corrected tortuosity factor TI modified, corrected tortuosity factor T2 angle defined by the aspect ratio of the pore angle between VC and the assumed diffusion di0 rection O) angle between the pore orientation and VC

( d 2)

where ~ = (d2)~/2/1, so that zi- 1(3) is the reciprocal tortuosity factor corresponding to a pore of the mean diffusional conductance. It is known that increasing the width of the distribution of diffusional conductances about its mean value reduces the diffusion coefficient [see, e.g. Zhang and Seaton (1992)]. The effect of introducing the correction for finite aspect ratio at the level of individual pores, in eq. (13), is to widen the conductance distribution (as the tortuosity decreases with increasing pore width) compared with the conductance distribution of eq. (14) where the tortuosity factor corresponding to the mean pore width is used. Thus, the value of De calculated using eq. (14) is an upper bound to the detailed calculation of eq. (13). As discussed above, z i ~ is itself an upper bound to the true value for a single pore, Thus eq. (14) is, in principle, an upper bound on the diffusivity of a network of pores of distributed size. Note that in order to reproduce the limit T- ~ --* 1 as e --, 1, an empirical correction would need to be made, along the lines of the correction to the tortuosity factor for a single pore expressed by eq. (8).

Acknowledqement--SPF thanks the Leo Baeck (London) Lodge for its financial support and the Department of Chemical Engineering of the University of Cambridge for its hospitality. S. P. FRIEDMAN t

N. A. SEATON*

Department of Chemical Engineering University of Cambridge Pembroke Street Cambridge CB2 3RA, U.K.

*Author to whom correspondence should be addressed. ~Present address: Institute of Soils and Water, The Volcani Center, P.O. Box 6, Bet-Dagan, 50250, Israel.

REFERENCES

Burganos, V. N. and Sotirchos, S. V., 1987, Diffusion in pore networks: effective medium theory and smooth field approximation. A.1.Ch.E.J. 37, 1678-1689. Friedman, S. P, 1993, Transport of microcapsules and slow release in saturated and unsaturated soil. Ph.D. thesis, Hebrew University of Jerusalem. Johnson, F. L. and Stewart, W. E., 1965, Pore structure and gaseous diffusion in solid catalysts. J. Catalysis 4, 248-252. Kim, J.-H, Ochoa, J. A. and Whitaker, S., t987, Diffusion in anisotropic porous media. Transport in Porous Media 2, 327-356. Kirkpatrick, S., 1973, Percolation and conduction. Rev. Mod. Phys. 45, 574-588. Pismen, L. M., 1974, Diffusion in porous media of a random structure. Chem. Enong Sci. 29, 1227-1236. Sahimi, M., Gavalas, G. R. and Tsotsis, T. T., 1990, Statistical and continuum models of fluid-solid factions in porous media. Chem. Engng Sci. 45, 1443-1502. Sahimi, S., Hughes, B. D., Scriven, L. E. and Davis, H. T., 1983, Real-space renormalization and effective-medium approximation to the percolation conduction problem. Phys. Rev. B. 28, 307-311. Zhang, L. and Seaton, N. A., 1992, Prediction of the effective diffusivity in pore networks close to a percolation threshold. A.LCh.E.J. 38, 1816-1824.