Chemical Engineering Science 66 (2011) 4382–4388

Contents lists available at ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

E-m

journal homepage: www.elsevier.com/locate/ces

Catalyst designs to enhance diffusivity and performance—I:Concepts and analysis

James Wei

Department of Chemical Engineering, Princeton University, Princeton, NJ 08544-5263, USA

a r t i c l e i n f o

Article history:

Received 30 October 2010

Accepted 9 February 2011Available online 23 February 2011

Keywords:

Catalysis

Design

Diffusion

Composite catalyst

Meso-structure connectivity

Macro-scale distribution

09/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ces.2011.02.010

ail addresses: jameswei@smtp.princeton.edu,

a b s t r a c t

A catalytic material with high reactivity but low diffusivity can achieve higher performance from blending

with an additive material with high diffusivity. The performance of such a composite catalyst depends both

on the meso-structure of the continuous connectivity of the additive, and on the macro-structure of the

gradient of additive in the catalyst pellet. Among the best meso-structures are the coated sphere and

the penetration structures. The composite catalyst with a uniform distribution of additive can be designed to

be many times more active than the starting catalytic material without additive. A gradient macro-structure,

with more additives on the surface than in the interior of a pellet, can be designed to be much more reactive

compared to the uniform structure with the same average loading of additives.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Consider an active catalytic material such as the ZSM-5 and thefaujasite with micropore diameters of 0.5–0.8 nm, with good activitybut poor diffusivity, so that its reaction rate would be diffusion-limited and less than its potential. The designer has a multi-scalechallenge to deal with pores, crystal, and pellet sizes, and ultimatelywith reactor sizes. Some typical values are given here.

ll rights reserv

jameswei@prin

Micro-scale, zeolite pores

Zeolite ZSM-5 0.5 nm Zeolite faujasite 0.7 nm MCM-41 4 nm

Meso-scale, zeolite crystals

Zeolite ZSM-5, Y 1 mm Macro-scale, catalyst pellets Fluid cracking 80 mm

Fixed bed

3000 mm

A catalyst pellet may have a high Thiele modulus j, leading toa low value for the effectiveness factor Z and poor performance.The design engineers can seek improvement in three scales:

(a)

A micro-scale method is to increase the zeolite pore size, suchas by ion exchange of the framework sodium with calcium orrare earth ions (Satterfield, 1980; Haag and Chen, 1987).

(b)

A meso-scale method is to mix two materials together that havecompensating properties, such as an active zeolite with ahigh diffusivity amorphous silica-alumina. When the zeolitecrystal also has a Thiele modulus that is too high to allow the

ed.

ceton.edu

easy access of bulky molecules, a method was patented togrow MCM-41 within zeolite crystals by chemical penetration(Ying and Garcia Martinez, 2009). Various methods to modifythe diffusivity of the matrix in important applications weretreated by Johannessen et al. (2007),Wang and Coppens (2010)and Wang et al. (2007).

(c)

A macro-scale method may be to reduce the catalyst particleradius R, and to engineer a non-uniform distribution of thecatalytic material with a higher concentration on the pelletsurface or interior (Becker and Wei, 1977; Morbidelli et al., 2001).

This paper is concerned with a composite catalyst of twocomponents: the catalytically active material A, and the highdiffusivity material B used at a volume fraction y. The blendingmust be engineered so that the material B is continuous andconnected, even if it is present only at a low volume fraction. Theaddition of B causes the value of the Thiele modulus j to decreaseand the value of the effectiveness factor Z to increase. However,as the volume fraction of active A is lowered to (1�y) in thecomposite, the reaction rate is actually (1�y)Z. So there is anoptimal level of y when the reaction rate per volume of pellet ismaximized. This paper is concerned with both the best meso-scale structure of the proportion and the method to mix the twomaterials in local regions, and the best macro-scale structure ofgradients in the catalyst pellet.

2. Diffusivity of composite materials: meso-scale

The meso-scale structure of the proportion and the physicalrelation between the active and the additive materials has a great

J. Wei / Chemical Engineering Science 66 (2011) 4382–4388 4383

influence on the resulting composite diffusivity. It would behighly desirable that the additive material forms a continuousnetwork, and there is no clumping of the active material so everyvolume element is not very far from the network. The diffusivityof non-homogeneous catalysts was considered in a series ofpapers (Theodorou and Wei, 1983), and is difficult to quantifyand generalize. There are many models of composite structures,and only some of them have predictable properties (Torquato,2002). Some of these structures approach the Hashin–Shtrikmanupper bound in composite diffusivity, and they include thelaminar and the coated sphere structures. Let us list some of thestructures that have constitutive equations of Dc ¼ f ðDA,DB,yÞ,where y is the volume fraction of the additive B.

(a)

Fig.(b) t

struc

The laminar structure can be manufactured by laminatingalternate thin layers of A and B together (Fig. 1(a)). Thisstructure is anisotropic, so that the composite diffusivity intwo directions parallel to the layers follows the arithmeticmean of Eq. (1), and both phases are continuous; much loweris the composite diffusivity in the direction perpendicular tothe layers, which follows the harmonic mean of Eq. (2)and both phases are discontinuous. The arithmetic mean is(Carslaw and Jaeger, 1959)

DC ¼ ð1�yÞDAþyDB

DC

DA¼ 1þða�1Þy where a¼ DB

DA41 ð1Þ

1. Six micro-structures of composite catalysts: (a) the laminar structure,

he penetration structure, (c) the coated sphere structure, (d) the random high

ture, (e) the random low structure, and (f) the shell structure.

and the harmonic mean is

1

DC¼

1�yDAþ

yDB

DC

DA¼

1

ð1�yÞþy=a ð2Þ

This laminar structure may be difficult to manufacture.

(b) The penetration structure, which can be manufactured by

driving rods or cones of B into a particle of A – in imitation ofthe kitchen method of driving aluminum rods into potato orturkey to shorten baking time. Another parallel is hydraulicfracture to enhance oil well production, by pumping fluidsat high pressure to fracture shale, to release oil and gasfrom tight rock formations (Economides and Nolte, 2000).A suggestion from nature is the fractal hierarchical treestructure seen in broccoli, lung, vascular bed, and river net-work where both phases are continuous from surface tointerior, Fig. 1(b) (Mandelbrot, 1977). A novel manufacturingmethod of this meso-scale structure in a crystal is chemicalpenetration of B into a particle of A (Ying and Garcia Martinez,2009). The composite diffusivity can approach the arithmeticmean of Eq. (1).

(c)

The coated sphere structure is built around a granularspherical core of A with a thin coating of B. Then the coatedspheres of microns in diameter are compressed and gluedtogether to form the catalyst pellets (Fig. 1(c)). This may bethe easiest structure to manufacture where the low volumehigh diffusivity phase is continuous, and the spheres need notbe of the same diameter. In this structure, B phase iscontinuous but A phase is not. The composite diffusivity fitsthe Coated Sphere Model (Torquato, 2002).

DC

DA¼ 1þða�1Þy�

ða�1Þ2yð1�yÞ3aþða�1Þy

ð3Þ

(d)

The random high matrix structure has a continuous phase ofhigh diffusivity B, embedded with a number of randomdisconnected spheres of A (Fig. 1(d)). This structure is usedfor the FCC catalyst, where the volume fraction of the zeolitemay be 10–30%. (Thomas, 1970) In a random mixing of twopowders, the high volume phase is likely to become thecontinuous phase. The composite diffusivity fits the Maxwellequation when the zeolite spheres are far apart, i.e. at highvalues of y (Carslaw and Jaeger, 1959).

DC

DA¼ a ð1þ2aÞþ2ð1�aÞð1�yÞ

ð1þ2aÞ�ð1�aÞð1�yÞ

� �ð4Þ

(e)

The random low matrix structure has a continuous phase oflow diffusivity A, with embedded high diffusivity B spheres.This structure is the most likely result when a small volume ofB is randomly introduced into a large volume of A, and shouldbe avoided for application in catalysis. The composite diffu-sivity fits the Maxwell equation when the spheres are farapart, i.e. at low value y.

DC

DA¼ð2þaÞþ2ða�1Þyð2þaÞ�ða�1Þy

� �ð5Þ

(f)

The concentric shell structure looks like an onion with manyalternating layers of A and B (Fig. 1(f)). Neither phase iscontinuous in this model in the radial direction from surfaceto interior. The composite diffusivity is the harmonic mean ofDA and DB, given in Eq. (2). This structure should also beavoided for application in catalysis.

Fig. 2 shows the diffusivities of the six composite materialsas functions of y the volume fraction of B, with the condition

Fig. 2. The diffusivity of the six meso-structure composite materials as functions

of the volume fraction of additive, where the diffusivity of the additive material is

10 times that of the catalytic material.

J. Wei / Chemical Engineering Science 66 (2011) 4382–43884384

a¼DB/DA¼10. They rank in the order of: laminar or penetra-tion4coated sphere4random high matrix4random lowmatrix4concentric shell. The structures that approach theHashin–Shtrikman upper bound share the characteristics thatthe high diffusivity component B is continuous. The structuresto maximize volumetric efficiency in catalysis require fairly lowvalues of y, such as yo0.3. The penetration and coated spherestructures are perhaps the most effective, as they have B phasecontinuous consistent with low y. The proven methods to formcatalyst granules of 3 mm diameter by mixing two powders ofmicrons in diameter include pelletizing, extrusion and pangranulation (Le Page, 1997).

3. Composite catalysts with uniform distribution

We follow the classic work of Thiele (1939) to design compo-site catalysts by distributing additives uniformly. It is convenientto use dimensionless parameters: x¼r/R, y¼C/Co. For the unmo-dified catalyst pure material A, let us designate

j2A ¼

R2kA

DA

The differential equation in spherical geometry becomes indimensionless form

d2y

dx2þ

2

x

dy

dx¼j2

Ay 0oxo1 ð6Þ

with the boundary conditions

yð1Þ ¼ 1

dy=dxð0Þ ¼ 0

The solution to this equation and boundary condition is

yðxÞ ¼sinhðjAxÞ

xsinhðjAÞand yð0Þ ¼jA=sinhðjAÞ

The effectiveness factor ZA is the ratio between the reactionrate for catalyst A in comparison with pulverized catalystA without diffusion effects.

ZA ¼3

j2A

jA

tanh jA

� ��1

!

We add material B to the catalyst pellet by replacing anddecreasing the concentration of A, and assume that we have amicrostructure that follows the arithmetic average of Eq. (1), suchas the laminar structure, so that

k

kA¼ 1�y ð7Þ

D

DA¼ 1þða�1Þy

j2u ¼

R2kA

DA�

1�y1þð�1Þy

¼j2A

1�y1þð�1Þy

or ju ¼jAb

The parameter b¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�y=1þða�1Þy

po1 is the factor to lower

the Thiele modulus jA of material A. The differential Eq. (6) hasnow been changed to

d2y

dx2þ

2

x

dy

dx¼j2

A

1�y1þða�1Þy

y¼ ðbjAÞ2y¼j2

uy ð8Þ

The effectiveness factor for this composite catalyst is

Zu ¼3

j2u

ju

tanhðjuÞ�1

� �

The reaction rate of this composite catalyst can be comparedwith the pulverized composite catalyst with the same value of y;the composite catalyst is more efficient in using material A, asZuZZA. However less catalytic material A is used in the compositeso that the dimensionless reaction rate in comparison to pulver-ized pure A catalyst is

ru ¼ ð1�yÞZu ð9Þ

One may regard Z as the efficiency of a catalyst particle basedon the weight of catalyst A actually used, and r as the efficiencybased on the volume of catalyst particle used.

A designer may have to accept values of R, kA, DA, DB, as given,and can compute the values of jA and a. The designer is free tomanipulate the microstructure of how to mix B with A, and thequantity of additive y used in the catalyst particle. Consider anexample here jA¼10, so that for a pure A catalyst the reactionrate compared to pulverized A is ZA¼0.2700. Then B with a¼10 isadded, and the result of Zu and r versus y are shown in Fig. 3. Theeffectiveness Zu in dotted line steadily increases with y; however,the reaction rate ru¼(1�y)Zu in solid line reaches a maximumvalue¼0.3720 when y¼0.30. The reaction rate of this compositeshould also be compared with pure A with ZA¼0.2700, so theratio is r¼1.378 or the reaction rate has been increased by 38%.

The optimal value of y is sensitive to the starting value of jA aswell as the relative diffusivity a of the additive B. Consider a range ofvalues of jA¼1–100, and a¼3, 10, 30, and 100. The results forspherical catalyst particles are shown in Fig. 4, which shows thereference reaction rate of catalyst A as a function of the initial jA.For each value of a, the resulting reaction rates r with the optimalamount of y are also shown. It is seen that when the starting valueof jA is less than 1, the efficiency ZA is above 0.9391 and there is notmuch advantage to adding B. But when jA¼10, the efficiency ZA islowered to 0.2700. For an additive with the value of a¼3, theimprovement in reaction rate is negligible; but when the additive

Fig. 3. The effects of additives on a diffusion-retarded catalytic material where the

initial Thiele modulus j¼10, and a¼100. The effectiveness factor Z in dotted line,

and the resulting reaction rate r in solid line are given as functions of y, the volume

fraction of additive.

Fig. 4. Reaction rate improvements in uniform catalysts by adding the optimal yto a range of initial j from 1 to 100, for the ratio of diffusivity a¼3, 10, 30,

and 100.

J. Wei / Chemical Engineering Science 66 (2011) 4382–4388 4385

has a¼10, then at the choice value of y¼0.30, the reaction rateshoots up to r¼0.3719, which is 38% higher than the startingmaterial A. The advantages of additives become even greater whenjA¼10 and a¼100, when the optimal y¼0.20 and r¼0.6492,which is to say that r¼0.6492/0.2700¼2.404 or the compositecatalyst has a reaction rate that is 2.4 times better than the originalA catalyst.

4. Composite catalyst with non-uniform distribution

Many industrial catalysts have non-uniform distribution of thecomponents, either by accident or by design. Haag and Chen(1987) pointed out that macroscopic gradients or compositionzoning can result from chemical treatments, leading to theenrichment of active sites towards the crystal surface that wouldbe equivalent to a reduction in crystal size. Some of the advan-tages of non-uniform distribution were analyzed by Becker andWei (1977) and by Morbidelli et al. (2001).

At the surface of the catalyst particle, there is a much higherrate of transport D(dC/dr) to feed reaction at the interior, sointuitively the loading of B expressed as y(r) should be lower inthe interior and higher at the surface. In the simple case of j¼0where all the catalyst volume elements consume reactant at thesame rate, then the ideal gradient to boost transport efficiencymight be y(x)px. At higher values of j, the internal concentrationof the reactant and consumption rate is lower in the centercompared to the surface, and the ideal profile of y(r) might beskewed even higher towards the surface, such as y(x)psinh(jx).This gain in transport efficiency is counter-balanced by thedisadvantage to move the active material to the interior, wherereactant concentration is lower.

Such a gradient in y(r) can be achieved by making a compositecatalyst with several types of coated spheres – some with thickercoating of B than others. The resulting catalyst pellet is con-structed as several concentric shells from a core, and the outsideshells use thicker coatings of B while the inside shells use thinnercoatings (Fig. 5). The most suitable manufacturing process may bepan granulation, by rolling moistened small seeds with low y in arolling pan with powder of higher y to increase diameter by asnow-ball phenomenon; then the granules move on to a seriesof pans with powders of increasing y. (Le Page, 1997) In a crystal,it might also be achievable by appropriate programming ofchemical penetration (Ying and Garcia Martinez, 2009) so thatthe outside shells would have higher value of y than interiorshells. To make sure that the tree structure is self-avoiding andthere are no isolated volume elements, one method is to use a fewtall trees to penetrate the interior, and many bushes to cover themiddle, see Fig. 5.

First we consider for illustrative purposes a rectangular cata-

lyst with two layers, where y1 the inside loading of B is smaller

than y2 the outside loading, to arrive at an average loading of y.There are now two differential equations that apply to region1 from 0oxo0.5, and to region 2 from 0.5oxo1. Let

j1 ¼ R

ffiffiffiffiffiffiffikA

DA

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�y1

1þða�1Þy1

s¼jA

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�y1

1þða�1Þy1

s

and

j2 ¼ R

ffiffiffiffiffiffiffikA

DA

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�y2

1þða�1Þy2

s¼jA

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�y2

1þða�1Þy2

s

So that the two differential equations are:

d2y1

dx2¼j2

1y1 0oxo0:5 ð10Þ

Fig. 5. Two macro-structures of the pellet, where loading y(r) is higher in zones

near the surface than in zones of the interior: (a) coated sphere structure and

(b) penetrating wedge structure.

Fig. 6. Improvement in reaction rates, using macro-structure linear gradient

catalysts by lowering interior loading I, for j¼10, a¼100, and average value of

y¼ 0:05, 0.1, 0.2, 0.25, and 0.33.

J. Wei / Chemical Engineering Science 66 (2011) 4382–43884386

d2y2

dx2¼j2

2y2 0:5oxo1:0

The boundary conditions are:

At x¼ 0 dy1

dx ¼ 0

At x¼ 0:5 y1 ¼ y2

At x¼ 0:5 D1dy1

dx ¼D2dy2

dx

At x¼ 1 y2 ¼ 1

The solutions can be given in the form of

y1 ¼ cacoshðj1 xÞ 0oxo0:5

y2 ¼ cbexpðj2 xÞþccexpð�j2 xÞ 0:5oxo1

When we apply the boundary conditions, we obtain the results

0 exp j2

� �exp �j2

� �cosh j1

2

� ��exp j2

2

� ��exp � j2

2

� �D1j1 sinh j1

2

� ��D2j2 exp j2

2

� �D2j2 exp � j2

2

� �2664

3775

ca

cb

cc

264

375¼

1

0

0

264

375

We can solve this equation by matrix inversion, and thenobtain the values of the three parameters ca, cb, and cc. Then weuse them to compute the concentration profile in the catalyst,

y(x), and also the boundary flux which is equal to the reaction rate

rG ¼1

j2A

1þða�1Þyð Þdy

dx

� �� �x ¼ 1

ð11Þ

The performance of the layered catalyst should be comparedwith the uniform composite catalyst with the same averageloading y of ru ¼ ð1�yÞZu, and also with the uniform pure Acatalyst with ru¼Zu.

Let us consider an example of reaction rates for a¼100, averagey¼0.33, and jA¼1, 3, and 10. The results show that when jA¼1,there is negligible difference between a two-layer and the referenceuniform catalyst; when jA rises to 3, the best two-layer catalystcan be 0.2% better than the uniform catalyst; but when jA rises to10, the best two-layer catalyst is better by 1.6%. This demonstratesthat the two-layer catalyst can be better than the best uniformcomposite catalyst with the same average loading of material B,although the improvement is only a few percent.

The advantages of a gradient of y(x) are greater with a lineargradient, y¼a+bx. This catalyst would have y¼a in the interior andy¼a+b at the surface, for an average y¼ aþb=2. We can quantifythe average inside loading in comparison with average loading asI¼(a+b/4)/(a+b/2). When b¼0, we have a uniform catalyst. Sincethe lowest possible value of a is zero, the index of inside loadingI must be higher than 0.5. Thus we have 0.5oIo1.0 where I¼1means a uniform catalyst, and I¼0.5 means the inside is loaded withB at half the average rate of loading.

The appropriate differential equation becomes

d2y

dx2þ

1

D�

dD

dx�

dy

dx¼j2

A

1�y1þða�1Þy

� �y ð12Þ

Consider an example when jA¼10, a¼100, for a range ofvalues y¼0.05, 0.10, 0.20, 0.25 and 0.33. Fig. 6 shows at I¼1, the

J. Wei / Chemical Engineering Science 66 (2011) 4382–4388 4387

values of r for the uniform catalyst steadily improve with theaddition of y, which reaches a local maximum value at r¼0.4218when y¼ 0:33. The values of r are very sensitive to the value ofI when y¼ 0:05, so that it rises from r¼0.2376 at I¼1 to the localoptimum of r¼0.2892 at I¼0.51. This represents an improvementof 21.7% of the gradient catalyst in comparison with the uniformcatalyst at the same loading. At higher values of y, the uniformcomposite has higher reaction rates, but it is less sensitive to thevalue of I as the distribution of B is shifted towards to the surface.At the value of y¼ 0:33, the rate of reaction becomes quiteinsensitive to the value of I. Note that the global optimum forlinear gradients is r¼0.4272 achieved at y¼ 0:25 and I¼0.66; thisrepresents an improvement of 4.27 times over the single-phasestarting catalyst where r¼0.1000.

When we program the quadratic loading y¼a+bx+cx2, it ispossible to lower the value of I almost to 0.25, and to increasefurther the rate of reaction. The best value of r¼0.3098 fory¼ 0:05 is attained at I¼0.26, which is a bigger local optimumof 30.4% improvement over the uniform catalyst. See the points inFig. 7. The global optimum has increased to r¼0.4299 wheny¼ 0:25 and I¼0.70.

A summary of the advantages of additives is given in Fig. 7,which shows a case of jA¼10, so the reaction rate of the startingmaterial A with no additive is r¼0.1000. When a uniform loadingwith a¼100 is employed, the reaction rate steadily increases withloading till it reaches the maximum at y¼ 0:33. The rate ofimprovements with the uniform addition of QUOTE is larger at thebeginning, and the march to the point of maximum rate at 33% is atdiminished return. A second line shows the additional advantage of

Fig. 7. Rate improvement for a catalyst with j¼10, by additive with a¼100, as

functions of the average y. The solid line is for adding B uniformly, the hollow line

is for adding B with the best linear gradients, and the dots at y¼ 0:05, 0.15, and

0.25 are for the best quadratic gradients.

formulating a linear gradient y¼a+bx for the additive, which is verysignificant at 0.05oyo0.25. Three points at y¼ 0:05, 0.15, and 0.25show the result of the quadratic gradient, y¼a+bx+cx2, which isbetter than the best linear gradient. Thus the global optimum ofr¼0.4299 is attained with a quadratic gradient when y¼ 0:25,which is 4.30 times better than the starting catalyst wherer¼0.1000. When the value for y is restricted to 0.05 or less, thelocal optimum of r¼0.3098 is attained also with a quadratic loading,which is 30.4% better than the uniform loading where r¼0.2376. Itis expected that a systematic exploration of arbitrary gradient of y(x)would result in even higher values of improvement.

5. Conclusions

The performance of a low diffusivity catalyst pellet can begreatly enhanced by judicious blending with a high diffusivityadditive material. The objective is to attain by controlling boththe meso-structure of the proportion and the continuity of theadditive phase, and the macro-structure of the gradient ofadditives in the catalyst pellet.

(1)

The diffusivity of the composite material is highly dependenton the meso-scale structure of the proportion and the mannerof mixing the two materials, and the best meso-structuredesign requires that the additive phase to be connected andcontinuous. The high diffusivity composite structures that areeasy to manufacture are the coated spheres and the penetra-tion structures.

(2)

For a composite catalyst pellet with uniform distribution, thevolume fraction of additive y, is the same throughout thecatalyst pellet. As a consequence of blending the Thielemodulus of the starting material j is reduced, and theeffectiveness Z; but since less active material is contained ina pellet, the volumetric reaction rate is reduced to r¼(1�y)Z,and the optimal y is the value when the rate reaches amaximum. With the diffusivity of the additive being muchhigher than that of the active material, it is possible to designa composite catalyst that is many times more active than thestarting active material.

(3)

A composite catalyst with a linear gradient in y(x) can bedesigned to have significantly higher reaction rates than theequivalent uniform catalyst. A catalyst with a quadraticgradient can outperform the best linear gradient catalyst.

Acknowledgment

The author would like to thank Christodoulos A. Floudas,Salvatore Torquato, and David Olson for excellent advice andextremely valuable information. This paper marks a friendshipbetween the author and Prof. Moosun Kwauk that spans overseven decades.

References

Becker, E.R., Wei, J., 1977. Nonuniform distribution of catalysts on supports.J. Catal. 46 (365–371), 372–381.

Carslaw, H.S., Jaeger, J.C., 1959. Conduction of Heat in Solids. Oxford UniversityPress, Oxford Chapter 16.

Economides, M.J., Nolte, K.G. (Eds.), 2000. Reservoir Stimulation. John Wiley &Sons, Ltd., New York.

Haag, W.O., Chen, N.Y., 1987. Catalyst design with zeolites. In: Hegedus, L.Louis(Ed.), Catalysis Design: Progress and Perspectives. Wiley-Interscience, NewYork Chapter 6.

Johannessen, E., Wang, G., Coppens, M.-O., 2007. Optimal distributor networks inporous catalyst pellets. I. Molecular diffusion. Ind. Eng. Chem. Res. 46, 4245.

J. Wei / Chemical Engineering Science 66 (2011) 4382–43884388

Le Page, J.F., 1997. Catalyst forming. In: Ertl, G., Knozinger, H., Weitkamp, J. (Eds.),Handbook of Heterogeneous Catalysis, vol. 1. Wiley-VCH, Weinheim Ger-many, pp. 412.

Mandelbrot, M.M., 1977. The Fractal Geometry of Nature. W.H. Freeman, New YorkChapter 16, 17.

Morbidelli, M., Gavriilides, A., Varma, A., 2001. Catalyst Design: Optimal Distribu-tion of Catalyst in Pellets, Reactors, and Membranes. Cambridge UniversityPress, Cambridge.

Satterfield, C.N., 1980. Heterogeneous Catalysis in Practice. McGraw-Hill,New York.

Theodorou, D., Wei, J., 1983. J. Catal. 83, 205–224.Thiele, E.W., 1939. Ind. Eng. Chem. 31, 916.

Thomas, Charles L., 1970. Catalytic Processes and Proven Catalysts. AcademicPress, New York.

Torquato, Salvatore, 2002. Random Heterogeneous Materials: Microstructure andMacroscopic Properties. Springer, New York.

Wang, G., Coppens, M.-O., 2010. Rational design of hierarchically structured porouscatalysts for autothermal reforming of methane. Chem. Eng. Sci. 65, 2344.

Wang, G., Johannessen, E., Kleijn, C.R., de Leeuw, S.W., Coppens, M.-O., 2007.Optimizing transport in nanostructured catalysts: a computational study.Chem. Eng. Sci. 62, 5110.

Ying, J., Garcia Martinez, J., 2009. Mesostructured zeolitic materials, and methodsof making and using them. US Patent US 7,589,041 B2, issued September 15,2009.