The prediction of transport properties of porous media using fractal models and NMR experimental techniques

$Page 1: The prediction of transport properties of porous media using fractal models and NMR experimental techniques$
Chemical Engineering Science 54 (1999) 3503}3512

The prediction of transport properties of porous media usingfractal models and NMR experimental techniques

Sean P. Rigby!,*, Lynn F. Gladden"

! Synetix, PO Box 1, Billingham, Cleveland, TS23 1LB, UK" University of Cambridge, Department of Chemical Engineering, Pembroke Street, Cambridge, CB2 3RA, UK

Abstract

A model has been constructed to determine the transport properties of porous media where the total tortuosity associated with thevoid space of a particular porous medium has independent contributions from the structural features of the material over variouslengthscales. CCA fractal model structures have been used to represent the structural characteristics of real pellets overmicro/mesoscopic (up to &10 lm) and macroscopic ('10 lm) lengthscales that are probed by NMR experimental techniquessensitive to these lengthscales. NMR relaxometry and mercury porosimetry have been used to study the micro/mesoscopic scale andMagnetic Resonance Imaging (MRI) has been used to study the macroscopic scale. In combination with simulations of moleculardi!usion using a "nite di!erence algorithm, a model structure consisting of a hierarchical combination of micro/mesoscopic andmacroscopic lengthscale models has been shown to give excellent predictions of the total tortuosity of real pellets determined by MRImethods. ( 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Porous media; Di!usion; NMR relaxometry; Porosimetry; MRI; Fractals

1. Introduction

Recent work (Hollewand and Gladden, 1993, 1995b)using NMR imaging techniques has shown that porouscatalyst support pellets display macroscopic heterogenei-ties in the spatial distribution of voidage and pore sizeover length scales from 0.01 to 1 mm. These heterogenei-ties have been found to signi"cantly in#uence bothsteady-state and transient di!usional processes (Hol-lewand and Gladden 1993, 1995b; Rigby et al., 1996) andthe deactivation of catalysts by coking (Cheah et al.,1994). Recently (Gladden et al., 1995), an image analysisalgorithm employing fractal concepts has been developedwhich, when applied to images of real pellets, suggeststhat over macroscopic dimensions these porous struc-tures possess a fractal character. Images of pellets fromthe same batch were found to be characterised by thesame value of the fractal parameter, whilst images ofpellets from di!erent batches have been found to becharacterised by di!erent values of the fractal parameter

*Corresponding author. Tel.: 01642 522246; fax: 01642 522606;e-mail: sean

}p}[email protected].

(Rigby and Gladden, 1996). It has also been found (Rigbyet al., 1996) that di!ering values of this fractal parameterfor two-dimensional slice NMR spin density images(which probe voidage) of axial and radial planes througha cylindrical alumina tablet correlate with the di!eringvalues of di!usivity for transient di!usion in the planesof the pellet being imaged. Results from mercury poro-simetry and nitrogen sorption experiments have shownthat the microscopic surface structure of the void space ofthe same types of porous materials also has a fractalcharacter (Farin and Avnir, 1989; Rigby and Gladden,1996). These two sets of "ndings have suggested that itwould be appropriate to employ a fractal model for thestructure of certain porous catalyst support pellets atboth the microscopic and the macroscopic scales.

It has recently (Rigby and Gladden, 1998) been pro-posed that the total tortuosity of a porous solid can beconsidered to have independent contributions from tor-tuosity derived from structural features of the void spaceover various lengthscales. In this work it was shown thatthe macroscopic contribution to the tortuosity may bepredicted from di!usion simulations on lattice-basedmodels derived directly from NMR spin density images

0009-2509/99/$} see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 5 1 7 - X

of thin slices through the porous medium of interest.These predictions were found to be in excellent agree-ment with experimental measurements of macroscopictortuosity obtained using a combination of MagneticResonance Imaging (MRI) and Pulsed-Gradient Spin-Echo (PGSE) NMR techniques. A di!erent approach willbe adopted here, which will not only predict the macro-scopic tortuosity but also the contribution to tortuosityfrom the void structure at the micro/mesoscopic scale.The methods described here will employ mercury po-rosimetry to characterise the pore structure at themicro/mesoscopic scale ((&10 lm) to obtain the overallaverage voidage fraction, overall average void surfacearea-to-volume ratio and modal pore size. MRI, togetherwith an image analysis algorithm based on fractal con-cepts, will be used to characterise macroscopic structuralheterogeneity in the form of a so-called image fractaldimension. Model structures used to represent the porestructure at these two di!erent lengthscales will be se-lected on the basis of agreement between the results ofsimulated characterisation experiments for the modelsand the respective experimental data for real pellets.NMR spin-lattice relaxation time data will be used tovalidate the micro/mesoscopic scale model. A hierar-chical combination of the macroscopic and micro/meso-scopic model representations of the porous structure willthen, in combination with di!usion simulations, be usedto predict transport properties. The predicted values oftotal tortuosity for the complete hierarchical modelstructure will be compared to experimentally determinedvalues using MRI techniques in order to validate thewhole modelling methodology.

A variety of di!erent experimental and modelling tech-niques will be used in this work. The following experi-mental and modelling sections will outline the methodsemployed, and where and how they are used. The "nalsection will show how the methods are combined toconstruct a hierarchical model structure from experi-mental characterisation data and validate themodel against further experimental data on transportproperties.

2. Experimental

Three types of commercial catalyst pellet were studiedin this work. The characterisations of the samples bymercury porosimetry are given in Table 1. The data frommercury porosimetry is used to construct the micro/mesoscopic scale model. Two di!erent structural para-meters were obtained using NMR methods. The experi-mental and subsequent analysis methods are describedbelow. The NMR spin-lattice relaxation time is an ex-perimental parameter that is sensitive to void surfacearea-to-volume ratio by which model structures of thepellet micro/mesoscopic structure will be validated. MRI

Table 1Characterisation of porous catalyst support pellets using mercuryporosimetry

Sample Form Mercury porosimetry results

Porosity Surface area Pore volume(m2 g~1) (cm3 g~1)

E2 Alumina tablet 0.65$0.03 250$20 0.67$0.05G1 Silica gel sphere 0.66$0.03 440$36 0.95$0.05G2 Silica gel sphere 0.69$0.03 127$10 0.94$0.05

experiments also provide spin-lattice relaxation time (¹1)

images from which a fractal parameter characterisingheterogeneities in the spatial distribution of void surfacearea-to-volume ratio is calculated. This characteristicparameter is used to constrain models of the macroscopicstructure of porous pellets. Finally, MRI is also usedto provide a value of tortuosity against which simula-tions of di!usion processes in the chosen models may bevalidated.

2.1. Determination of surface relaxation properties

If the void space of a porous material is completely"lled with a liquid then the measured NMR spin-latticerelaxation rate of the nuclei in the liquid is enhanced.This more rapid relaxation is attributed to interactionsbetween liquid molecules and the liquid}solid interface.According to Brownstein and Tarr (1977), if the surfaceinteraction is of short range and there is a fast di!usionalexchange of molecules between the surface region and theremaining (bulk) part of the liquid in the sample duringthe timescale of the experiment, then an induced nuclearmagnetisation will be expected to relax exponentiallywith time, with the relaxation rate (1/¹

1) given by the

weighted average between the surface and the bulk rates:

1

¹1

"C1!jS

<0D

1

¹1B

#

jS

<0

1

¹1S

, (1)

where ¹1S

and ¹1B

refer to the surface and bulk relax-ation times respectively; S and <

0represent the surface

area and pore volume, respectively, of the region probedby the di!using liquid molecule during the course of theexperiment; and j is a characteristic length representingthe extension of the surface interaction responsible forthe enhanced relaxation. In cases of variable saturation,where the void space is only partly occupied with liquid,and assuming ¹

1B<¹

1Sand that the liquid remains

wetting of the whole of the solid surface throughoutthe drying process, then the relaxation rate is given(D'Orazio et al., 1990), approximately, by

¹1"

<

<0

<0

jS C1

¹1S

!

1

¹1BD~1

, (2)

3504 S.P. Rigby, L.F. Gladden/Chemical Engineering Science 54 (1999) 3503}3512

Table 2Characterisation of porous catalyst support pellets using NMR techniques (the "gures in parentheses in the sample column correspond to the numberof samples of each pellet studied using MRI). Also shown is a comparison of micro/mesoscopic model predictions with the experimental results

Sample Mesoscopic model Mesoscopic Mesoscopic model Lengthscale of Ratio of image Averageinput model output validation mesoscopic model slice thickness image fractal

(lm) to pixel dimensionresolution (lm)

Size of 1 lattice ¹1S

(ms) Predicted Overall Sample Root mean (absoluteunit (nm) (obtained overall average ¹

1mean square, rms, dimensionless

(obtained from from drying average ¹1

(ms) overall molecular values are givenmercury expts.) (ms) (obtained average displacement in in brackets)porosimetry from ¹

1¹

1value time ¹

1(lm),

modal pore images) (ms) Sr2T1@2 (obtainedsize) using PGSE NMR)

E2 (3) 6.7 31 335$0.5 294 359$34 39.2 1000 : 40 ("25) 1.61$0.01407375

G1 (4) 5.0 39 317.5$0.4 261 312$20 34.6 1000 : 40 ("25) 1.66$0.03317310360

G2 (4) 18.7 22 712$4 516 610$50 55.3 700 : 40 ("17.5) 1.63$0.01692530700

where < is the volume of liquid in the pores. Thus (</<0)

represents the fractional saturation. The pore volume,<0, can be determined either gravimetrically or from

mercury porosimetry, and the surface area, S, can also bedetermined from mercury porosimetry, or nitrogen sorp-tion. These values, found using mercury porosimetry, aregiven in Table 1. Eq. (2) suggests that the observedrelaxation rate, ¹

1, will vary linearly with the saturation

(</<0). In the present work, the bulk relaxation rate is

assumed to be that of pure water (&3 s) and the thicknessof the surface layer was taken as 0.3 nm (D'Orazio et al.,1990). If the liquid is a wetting #uid, a change in pore"lling does not a!ect the value of S and the observedrelaxation time will be linear with liquid content. If thedrying was non-uniform and some regions dried outcompletely, whilst others remained wetted, this would beevident from a variation of the observed drying behav-iour from the linear form expected from Eq. (2). No suchdeviation was observed above a saturation level corre-sponding to the liquid volume su$cient for a completemonolayer coverage. NMR imaging studies (Koptyuget al., 1998) have shown that during the drying of hy-drophilic solids, such as alumina, capillary #ows com-pensate successfully for water losses at the periphery andmaintain a uniform water distribution across the sample.The spin-lattice relaxation time (¹

1) was determined

using the inversion recovery method (Fukushima andRoeder, 1981). The experiments described in this sectionwere performed on a Bruker Spectrospin MSL 200 spec-

trometer operating with a proton resonance frequency of200.13 MHz. Pellet samples, typically consisting of threepellets of a particular type, were prepared by impregnat-ing with deionised water at ambient conditions. Theoverall average relaxation time is then determined for thefully imbibed samples, and the pellets are allowed to dryunder ambient conditions. At di!erent stages in the dry-ing process the overall average value of ¹

1for the sample

is measured and the saturation level is determinedgravimetrically. The surface relaxation rate is determinedfrom the gradient of a best "t straight line of overallaverage relaxation time against variable saturation, inaccordance with Eq. (2). The surface relaxation rate char-acterises the liquid}solid interaction. From Eq. (1) themeasured relaxation time, ¹

1, is characteristic of the

surface area-to-volume ratio of the void space ofthe sample probed by a molecule over the timescale ofa typical ¹

1experiment. The values of ¹

1for the fully

saturated sample and ¹1S

for the three pellets consideredin this work are given in Table 2.

2.2. MRI

MRI is able to provide a spatially resolved map of thedistribution of a particular parameter, such as pelletvoidage or pore size. 1H NMR imaging experiments wereperformed using a Bruker DMX 200 NMR spectrometerwith a static "eld strength of 4.7 T yielding a protonresonance frequency of 199.859 MHz. Spin density and

S.P. Rigby, L.F. Gladden/Chemical Engineering Science 54 (1999) 3503}3512 3505

¹1

images were obtained using a spin-echo pulse se-quence employing 903 selective and 1803 non-selectivepulses. The imaging sequence was pre-conditioned usinga saturation recovery pulse sequence and an echo time of2.6 ms was used. All images acquired were of dimension128]128 pixels. The in-plane pixel resolution was 40 lm.More detail about the image acquisition is given else-where (Hollewand and Gladden, 1993, 1995b). A singleexponential saturation recovery "t was applied to the setof variable delay time data for each pixel to obtaina value of ¹

1for each pixel in the image plane.

Fractal physics provides a method for characterisingstrongly disordered systems (Mandelbrot, 1982). In thecase of the MRI images the fractal analysis is usedto classify the shape of homogeneous clusters of pixelintensity by estimating the fractal dimension of theboundary or perimeter of the clusters. The algorithmused has been described in more detail previously(Gladden et al., 1995). If the clusters obeyed planarEuclidean-type geometry then the perimeter, P, wouldbe expected to scale with the size of the cluster, R,as PJR and the area, A, to scale as AJR2. For afractal boundary the length of the perimeter willdepend on the resolution with which the perimeter ismeasured and will scale with the cluster size, R, as PJRd

where d is the fractal dimension of the boundary. Thefractal dimension then characterises the degree of dis-order and must satisfy d

501)d)d

emwhere d

501is the

topological dimension of the structure under analysis(d

501"1 for the boundaries in the images considered

here) and dem

is the embedding dimension, i.e. 2. For thesamples studied here, an area-perimeter relation for ho-mogeneous clusters of the form AJP2@d is obtained. Thefractal dimension is therefore estimated from the gradientof the log-area versus log-perimeter plot using a form ofthe boundary extraction method (Wang et al., 1991). Thefractal dimension obtained from this analysis will quan-tify the morphology of the heterogeneities in the spatialdistribution of voidage or pore size values from thelengthscale of the pixel resolution up to the size of theimage. The result expected for a completely randomarrangement of pixel intensities is 1.72}1.75 (Gladdenet al., 1995).

MRI is also able to probe transient transport processesin porous media and the results of these experiments areused to "nally validate the whole modelling methodo-logy. Transient di!usion processes are followed by ini-tially saturating a pellet with 1H

2O and then immersing

it in 2H2O. Since the MRI method is only sensitive to 1H,

the transient di!usion of 2H2O into the pellet can be

followed. The transient di!usion pro"les so-obtained canthen be "tted against solutions of the appropriate di!u-sion equation, assuming constant di!usivity; hence a tor-tuosity is evaluated (Hollewand and Gladden, 1995b).Since the characteristic lengthscale associated with thistype of experiment is that of the whole pellet the tortuo-

sity so-derived may be said to be representative of suchlengthscales.

3. Numerical modelling

The following section, "rstly, describes the principlesof construction for lattice-based model structures. Sec-ondly, the procedure by which the imaging experiment issimulated for model structures, in order to obtain overallaverage spin-lattice relaxation times and image fractaldimensions, is then described. The spin-lattice relaxationtimes are used to validate the micro/mesoscopic scalemodels and the image fractal dimension is the experi-mental constraint on the macroscopic scale models.Finally, the methods used for the simulation of di!usionprocesses within the models are then speci"ed. The pre-dictions of total tortuosity obtained from these simula-tions are used in the validation of the whole modellingapproach.

3.1. Construction of lattice-based model structures

Fractal objects are self-similar structures in which in-creasing magni"cations reveal similar features at di!er-ent lengthscales; they show a power-law relation betweenproperties such as mass, void volume or surface area andlengthscale, and would thus be termed mass, pore orsurface fractal, respectively. In this work three-dimen-sional (3D) Cluster}Cluster Aggregate (CCA) mass frac-tal model structures, grown in an analogous way to thosein 2D reported by Elias-Kohav et al. (1991), were used.The clusters were grown on a cubic lattice of a side lengthof 64 units; successively larger clusters are formed by thedi!usion and aggregation of smaller units, at "xed (i.e."nite) voidage fraction, starting from Random Cluster(RC) precursors. Models with voidage fractions in therange 0.50}0.75 were created. The RCs were created byoccupying lattice sites completely at random until therequired voidage fraction was reached. The procedure fordetermining the pore-size distribution for the simulated,3D fractal objects followed that of Ehrburger and Jullien(1988). The size of the lattice was chosen such that theface of one side of the lattice was of a size (in lattice units)similar to that of MR images of real pellets (in pixels).The mass fractal dimension was determined, using thebox counting method (Mandelbrot, 1982), for the wholecluster.

3.2. Construction of a simulated spin-lattice relaxationtime image

In a real image the signal associated with a pixel arisesfrom a volume in the pellet of size equal to the pixelresolution squared multiplied by the slice thickness. Inthe simulated images of model structures one face of the


lattice was selected to be the image surface. The e!ect ofvarying the resolution was simulated by taking arti"cialpixel sizes on the image surface corresponding to 1]1 or2]2 lattice units. Variable slice thickness could be ob-tained by taking di!erent heights for the square columnsof lattice units behind the image surface, whose long axesare perpendicular to that surface, which correspond tothe rest of a pixel volume in a real image. In order toobtain a simulated spin-lattice relaxation time image theprocedure below was followed:

(i) for each unoccupied lattice site, in a column corre-sponding to a simulated pixel volume, the number ofoccupied site neighbours is counted. The total for allthe unoccupied sites in the column is a measure ofthe surface area of the cluster, in lattice units, con-tained in the pixel volume;

(ii) the volume of the surface layer, mentioned above, isfound by multiplying the surface area found in (i) byan assumed value of j, the surface layer thickness,which is a small fraction of a lattice unit in size (seebelow);

(iii) the bulk liquid volume is represented by the di!er-ence in the values of the total voidage in asimulated pixel volume (found by counting up all theunoccupied lattice sites in a pixel volume) and thevolume of the surface layer found in (ii);

(iv) the ¹1

value for that pixel is then found by applyingthe equation:

1

¹1

"AfS

¹1SB#A

fB

¹1BB , (3)

where fSis the fraction of the pixel void volume occupied

by the surface layer and fB

is the fraction of the voidvolume considered as bulk liquid. ¹

1Sis the experi-

mentally measured surface relaxation time and ¹1B

istaken as the relaxation time for bulk water.

3.3. Simulations of diwusion

In the approach adopted here, similar to that used byElias-Kohav et al. (1991), the local tortuosity is approxi-mated by the number of sideways diversions that a mol-ecule needs to proceed in the void. If M is the locallyaveraged number of blocked pixels (occupied lattice sites)adjacent to an empty site, then the probability of a onesite diversion is M/6. After such a move there is a similarprobability of a further diversion and when M does notvary with every diversion, the local tortuosity aftern steps is

q"1#AM

6 B#AM

6 B2#2#A

M

6 BnP

1

1!(M/6), (4)

where the limit holds for large n. For a completely ran-dom arrangement, M/6"1!e and q"1/e.

The method then simulates di!usion occurring undera concentration gradient, typical of a Wicke}Kallenbachexperiment in the following manner. In this case the64]64]64 site lattice occupied by the model structure isdivided up into 512 small boxes of 8]8]8 sites, and thevoidage fraction and local tortuosity [using Eq. (4)above] is found in each box giving rise to a value ofdi!usivity in each box. The three dimensional (3D)steady-state di!usion equation:

LLX CD(X,>,Z)

LC

LXD#L

L> CD(X,>,Z)LC

L>D#

LLZ CD(X,>,Z)

LC

LZD"0 (5)

is then solved using a "nite di!erence algorithm. A valuefor the e!ective di!usivity for the whole structure isobtained from the #ux predicted by the simulation andthus a value for the tortuosity is deduced.

4. Results and discussion

It is generally the case that the overall tortuosity, q,for a macroscopic structure might be thought (Rigbyand Gladden, 1998) to consist of contributions from themicroscopic lengthscale (typically a few pore diameters)qi, mesoscopic lengthscale (up to the order of 10 lm)

qeand macroscopic lengthscales (&0.01}10 mm) q

asuch

that

q"qiqeqa. (6)

In the modelling approach described here the contri-butions from the micro- and mesoscopic lengthscales willbe considered together. The CCA structural modeldescribed above may be used to represent the real sampleat di!erent lengthscales depending on the parameterscharacterising it.

4.1. The selection and validation of model representationsof the micro/mesoscopic lengthscale

In the "rst case the model is used to represent thelengthscales of a pellet probed by water molecules duringthe course of an experiment to measure the non-spatiallyresolved value of overall average¹

1of the sample. At this

lengthscale the model is selected to have the same void-age fraction, as found by mercury porosimetry, as the realpellet. In this case the size scale of the model lattice is"xed by equating the modal pore size obtained for themodel structure (in lattice units) to the modal pore size ofthe real sample (in metres) from mercury porosimetry (seeTable 2). Only unimodal pore-size distributions will beconsidered here. For the CCA models used in this workthe modal pore sizes were all 1.5 lattice units. This pro-cedure will also "x the value of j in lattice units.


A simulated overall average value for ¹1

for the model isobtained from the average value of ¹

1from each of the

pixels. The overall average values of ¹1

found by experi-ment and the predictions for the model structures, alongwith the signi"cant model input parameters, are alsogiven in Table 2. Where there is agreement between theexperimental value and the model value of overall aver-age ¹

1for a particular type of pellet it is suggested that

the model structure accurately represents the averagegeometry of the real pellet over the characteristic len-gthscales probed by water molecules during the courseof a ¹

1experiment. A characteristic lengthscale can be

estimated by the use of the Einstein equation:

Sr2T"6DPGSE

¹1, (7)

where Sr2T is the mean square displacement of a molecu-le and D

PGSEis the average self-di!usivity of water found

from Pulsed-Gradient Spin-Echo (PGSE) NMR tech-niques (Hollewand and Gladden, 1995a). The values ofthis characteristic lengthscale for the pellets studied hereare given in Table 2 and may be compared with the sizeof one model lattice unit. It can be seen that the simulatedvalues for the value of overall average ¹

1obtained using

CCA structures are in good agreement with experimentsuggesting that these fractal structures are good modelsfor the structure of the real materials over the lengthscaleof the root mean squared displacement of the watermolecules. The lower characteristic lengthscale limit ofthe model structure is the size of a lattice unit. The factthat the experimental and simulated values of overallaverage ¹

1agree in the case of the CCA models con-

sidered here suggests that structural features of the pelletswith a characteristic size smaller than that correspondingto one lattice unit (such as possible smaller scale surfaceroughness) do not have a signi"cant e!ect on the di!u-sion/relaxation process. On lengthscales larger than thesidelength of the whole model lattice the mesoscopicmodel is considered to be homogeneous. On its own thismodel implies that the structure of a real pellet is homo-geneous over lengthscales greater than the lattice size.

4.2. The selection of a model representation of themacroscopic structure

The variations in the tortuosity of porous media ariseout of di!erences in the disposition in space of solidmatrix and void between di!erent samples. For this rea-son di!erent pellet types of the same overall voidagefraction may have very di!erent tortuosities. Di!erencesin the convolutions of the void structure will be re#ectedin variations in the spatial distribution of surface area-to-volume ratio of the void space. At the macroscopicscale, the CCA model is used to represent the heterogen-eous spatial distribution and irregular morphology ofregions of similar value of void surface area-to-volumeratio. The image fractal dimension characterises this par-

ticular feature of the pellet structure. The model CCAused to represent the porous structure of a particular realpellet over macroscopic scales is selected on the basis ofagreement between the image fractal dimensions cal-culated for the real pellet and simulated images, imagedwith the same value of the ratio of image slice thickness-to-pixel resolution.

The ratios of image slice thickness to in-plane pixelresolution for the MRI studies of the real pellets areshown in Table 2. The fractal dimensions for the spin-lattice relaxation time images, which probe the locallyaveraged pore surface-to-volume ratio, of the real pelletsbeing studied here are also given in Table 2. The fractalimage analysis algorithm described above was also ap-plied to the simulated ¹

1images of the model structures

discussed in the previous section and these results com-pared with those of the real pellets. Fig. 1 shows theimage fractal dimensions for simulated images of CCAand RC model structures of di!erent voidage fractions.Each data point is the sample mean value of 10}20 modelstructures of the same voidage grown with di!erent ran-dom number seeds. Several di!erent ratios of image slicethickness-to-pixel resolution have been considered toenable us to assess the importance of the so-called &pile-up' phenomenon in a comparison of simulated with ex-perimental data. The &pile-up' phenomenon is commonto tomography techniques where the value of a propertybeing imaged assigned to a particular in-image planepixel is the resultant average for the whole column vol-ume behind the image plane. The ratio of the image slicethickness-to-pixel resolution characterises the extent ofthis averaging. It is observed that, allowing for randomscatter, simulated images of a CCA using the range ofdimensionless ratios of slice thickness-to-pixel resolutionof 16}32 have the same fractal dimension. The absolutevalues of this ratio for the images of real pellets, given inTable 2, are all in this range 16}32. The other character-istic dimensionless ratio that is important in the case ofsimulations for the macroscopic scale features of pellets isthe ratio of pellet diameter (or radius) to pixel resolution.For the simulated images the equivalent ratio is the ratioof the sidelength of the image surface of the lattice to thepixel resolution. It is essential that the model for themacroscopic structure of the pellet represents su$cientvolume of the pellet to incorporate all the structuralfeatures arising at the macroscopic scale that in#uencetransport. As mentioned above, the models used here areof a size such that the image surface is of the same(equivalent) order in size as the real image. The dottedline in Fig. 1 shows the fractal dimension obtained, usingthe fractal image analysis algorithm, for model imagesconsisting of a square grid of size 64]64 sites, where theintensity of each pixel has been allocated completely atrandom from a rectangular distribution of intensities.A sample of 7 of these random images were found to havea fractal dimension of 1.756$0.003.


Fig. 1. Comparison of image fractal dimensions for simulated ¹1

im-ages of Random Cluster (RC) and Cluster-Cluster Aggregate (CCA)model structures. Numbers in parentheses indicate the ratio of theimage slice thickness to in-plane pixel resolution (the "gures are givenin lattice units). The dotted line indicates the result expected for a com-pletely random arrangement of pixel intensities. The random errorassociated with the simulated fractal dimensions is typically 2%.

The image fractal dimensions for all simulated imagesof CCA models and for real pellets (see Table 2) are lowerthan would be expected for a completely random ar-rangement of pixel intensities. The image fractal dimen-sions for RCs, given in Fig. 1, show a trend of increasingimage fractal dimension with increasing voidage fraction,whereas for the image fractal dimensions of simulatedimages of CCA structures there is a trend of decreasingfractal dimension with increasing voidage fraction. Thesetrends indicate that the degree of structural heterogeneityincreases with increased voidage fraction for CCAs, anddecreases with increased voidage fraction for RCs. Forsimulated images, with a given ratio of image slice thick-ness-to-pixel resolution, the image fractal dimensions forCCA structures are lower than the respective values forsimulated images of RC structures of the same voidagefraction, and thereby closer to the values for the realpellets, and so these results suggest the necessity for CCAfractal models of the macroscopic structure of porouspellets.

Fig. 2 indicates that the (mass) fractal dimension of the"nite concentration CCAs is close to a value of 3 (thevalue expected for a uniform structure) at low voidagefraction but does decrease as the voidage fraction of thelattice increases. Mass and pore fractals are also surfacefractals. From a comparison of Figs. 1 and 2, it can beseen, for CCAs of increasing voidage fraction, that as themass fractal dimension, as measured by the box-countingmethod, decreases, the image fractal dimension, as mea-sured by the image analysis algorithm, of simulated¹

1images (which probe surface area-to-volume ratio)

also decreases. This result suggests that the ¹1

imagingprocedure and the subsequent analysis of the resultantimages employing fractal concepts is sensitive to the

Fig. 2. Mass fractal dimensions for three-dimensional CCA modelsobtained using the box-counting method.

variation in the underlying fractality of the solid sur-face/pore space interface.

4.3. The use of a hierarchical model representation topredict total tortuosity of porous media

In previous studies mentioned above, it has been foundthat experimental methods of studying di!usion pro-cesses that probe the di!erent lengthscales that comprisethe hierarchical model give rise to di!erent values oftortuosity. This result suggested (Rigby and Gladden,1996) that a single CCA representation of the pore struc-ture, based on either the characterisation of the pelletmicro/mesoscopic structure or the macroscopic structurealone, is insu$cient for the real materials consideredhere. In the hierarchical modelling methodology thestructural characterisation data appropriate for each len-gthscale is used to abstract the structural features ofa porous material which in#uence di!usion over thatparticular lengthscale. The model representations consis-tent with that data are then used to assess the contribu-tion to the total tortuosity from that lengthscale. TheCCA has been shown to be a good model for the ge-ometry of the solid surface}pore volume interface at themeso/microscopic scale, as characterised by the ¹

1value,

in addition to the spatial distribution and morphology ofregions of similar pore surface area-to-volume ratio; thepresence of which imposes an additional tortuosity atlarger lengthscales. The CCA models used for the twodi!erent lengthscales are, however, not necessarily of thesame voidage fraction. This is because at the micro/me-soscopic scale the model is used to represent the typicalgeometry of the convolutions of the void}solid matrixinterface, whereas at the macroscopic scale it is used tomodel the spatial variation in the void}solid matrix inter-face geometry. The typical values of structural propertiessuch as void surface area-to-volume ratio and voidagefraction are determined by the structural features of the


Table 3A comparison of hierarchical model predictions for tortuosity with experimental results from MRI techniques. The quoted uncertainty in themodelling results represents the combined contributions from errors arising in the original experimental characterisation data and the random scatterin the simulation results

Sample Predicted Predicted Predicted Experimentallymicro/mesoscopic macroscopic total determined totalcontribution to contribution to tortuosity tortuositytotal tortuosity total tortuosity

E2 1.49$0.06 1.37$0.02 2.04$0.09 2.3$0.3G1 1.49$0.06 1.45$0.05 2.16$0.11 2.3$0.1G2 1.40$0.05 1.40$0.02 1.96$0.08 1.8$0.3

porous material over the lengthscales comprised by themicro/mesoscopic model.

Simulations of di!usion under an imposed concentra-tion gradient, such as is the case in the transient di!usionexperiments followed by MRI described above, may bemade using the Di!usion Flux method. It has been sug-gested (Rigby and Gladden, 1996) that the physical pro-cesses involved in the transient di!usion experiment aremore accuractely modelled by this simulation technique.Separate simulations on the model structures represent-ing the structural features at di!erent lengthscales can beused to assess and predict these individual factorial con-tributions to the total tortuosity. However, care must betaken with this approach, when comparing model predic-tions of tortuosity with experiment, to correctly match upthe characteristic lengthscales of each level in the modelhierarchy and any characteristic lengthscale over whichthe tortuosity is measured. Here the tortuosity over thesize of the whole pellet will be considered. For the sam-ples used here the CCA-type model may be used torepresent the void space of a pellet at the micro/meso-scopic and macroscopic scale. This is because the resultsof simulations of structural characterisation experimentsfor the two di!erent lengthscales on these models agreewell with the results obtained for the real materials theyare supposed to represent. For other samples, an obviousexample being bidisperse porous solids, the CCA maynot be appropriate at a particular lengthscale (e.g. be-cause it is unimodal) and these materials are the subjectof ongoing study. For each level in the hierarchy theparticular tortuosity associated with that lengthscale ofthe model is found from Di!usion Flux (DF) simulationscarried out using an appropriate CCA model of thecorrect voidage fraction to represent the pellet structureat that level. The results of these simulations are thencombined according to Eq. (6) to give a prediction of thetotal tortuosity of the porous solid. The factorial combi-nation of the individual tortuosities for each lengthscaleused in Eq. (6) gives a result which is equivalent to thecase of di!usion in a structure where the micro/me-soscopic scale model is nested in each of the void latticesites of the macroscopic scale model. The degree of

Fig. 3. A correlation of tortuosity, found by Di!usion Flux simula-tions, with image fractal dimension calculated for simulated images ofCCA model structures. The results are for CCA models imaged in therange of absolute values of ratios of image slice thickness-to-pixelresolution of 16}32. The dotted lines illustrate the process described inthe text which is followed to determine the macroscopic contribution tothe total tortuosity for pellet G2.

spatial variation in locally averaged (over lengthscalesof the size of an image pixel) void space surface area-to-volume ratio would be una!ected by this procedure.Although this concept is an abstract structural rep-resentation model it does mimic the essential propertyof the real material determining transport properties:contributions to the total tortuosity arise from thedegree of disorder in the structure over di!erent len-gthscales.

At the micro/mesoscopic scale the CCA model isselected on the basis of agreement between the values ofsample overall average ¹

1and voidage fraction, as mea-

sured by NMR relaxometry and mercury porosimetryrespectively, for the real pellets and the model structures.The correlation of tortuosity with voidage fraction hasbeen reported previously (Rigby and Gladden, 1996). Thetortuosities, found by the Di!usion Flux simulation, forthe micro/mesoscopic scale are shown in Table 3. Fig. 3shows the value of tortuosity, found by the di!usion #uxsimulations, for CCAs which, when imaged with a ratioof image slice thickness to pixel resolution in the range16}32, have a particular image fractal dimension. The


contribution to the overall tortuosity from the structuralfeatures of the pellet at the macroscopic scale is foundby reading o! the appropriate value of tortuosityfrom the correlation for a model with a particularimage fractal dimension which is the same as thatfor the real pellet. The dotted lines in Fig. 3 show anexample of this process for the case of the pellet sampleG2. The contributions to the total tortuosity of the pel-lets arising from the macroscopic structural heterogenei-ties, as obtained from Fig. 3, are also shown in Table 3.The predictions for the total tortuosity of the pelletobtained from the Di!usion Flux simulations and Eq. (6)are shown in Table 3 together with the experimentalresults for the total tortuosity of the pellet over len-gthscales of the order of the pellet size obtained by MRI.It can be seen that this hierarchical modelling approachyields excellent predictions of the total tortuosity of thereal pellets.

5. Conclusions

A hierarchical model for the structure of the void spaceof porous media has been presented. The particularmodel structure used to represent each of the levels in thishierarchy, characterised by di!erent lengthscales, is se-lected from a comparison of the model predictions for theresults of the characterisation experiments with experi-mentally determined data for the real materials. SimpleNMR relaxometry is used to probe the pore space overlengthscales of the order of 10 lm, which is characteristicof the rms displacement of water molecules during thecourse of such an experiment. MRI, in combination withfractal image analysis techniques, is used to characterisethe structure of pellets from the in-plane pixel resolutionupwards ('40 lm).

Hierarchical CCA fractal model structures have beenfound to be good models for the structure of the pelletsstudied here up to the lengthscales of the size of the wholepellets themselves (&mm). The overall average value of¹

1for a fully liquid-imbibed pellet sample is related to

the typical surface area-to-volume ratio of the solid/porespace interface. For model structures where the simula-tion of the relaxometry experiment gives rise to a value ofoverall average ¹

1similar to that for the real pellet it is

proposed that, that particular model structure is a goodrepresentation of the micro/mesoscopic scale structure ofthe porous pellet. MRI has shown that porous catalystsupport pellets possess macroscopic heterogeneities inthe spatial distribution of pore surface area-to-volumeratio. These heterogeneities also in#uence transport pro-cesses in those pellets. The imaging experiment has alsobeen simulated for model structures used to represent themacroscopic scale features of pellet structure. Wherethere is agreement between the image fractal dimensionobtained from image analysis of images of real pellets and

simulated images derived from model structures it isproposed that these models make good representationsof the real macroscopic structure. Di!usion simulationshave been carried out in each of these models. A factorialcombination of the contributions to total tortuosity fromthe structural features represented by each level in themodel hierarchy was used to predict the tortuosity, foundfrom MRI experiments, characteristic of the lengthscaleof the whole pellet. Good agreement has been foundbetween the predictions of this modelling approach andthe experimental results.

Notation

C concentration, dimensionlessD di!usivity, dimensionlessD

PGSEself-di!usivity of water found by PGSENMR, m2 s~1

fB

fraction of pore volume occupied by bulk-like phase, dimensionless

fS

fraction of pore volume occupied by sur-face-a!ected layer, dimensionless

M locally averaged number of blocked pixelsadjacent to an empty site, dimensionless

Sr2T mean square displacement, m2

S surface area, m2

¹1

spin-lattice relaxation time, ms¹

1Bbulk spin-lattice relaxation time, ms

¹1S

surface spin-lattice relaxation time, ms< pore volume occupied by liquid, m3

<0

total pore volume, m3

X cartesian co-ordinate, dimensionless> cartesian co-ordinate, dimensionlessZ cartesian co-ordinate, dimensionless

Greek letters

j thickness of surface-a!ected layer, mq total tortuosity, dimensionlessqa

macroscopic tortuosity, dimensionlessqe

mesoscopic tortuosity, dimensionlessqi

microscopic tortuosity, dimensionless

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The prediction of transport properties of porous media using fractal models and NMR experimental techniques