AWARD NO. DE-FG22-92PC92527

Quarterly Report No. I2 Research Period: June 18, 1995 - September 17, 1995

HINDERED DIFFUSION OF COAL LIQUIDS

Theodore T. Tsotsis, Co-Principal Investigator Muhammad Sahimi, Co-Principal Investigator

University of Southern California Department of Chemical Engineering Los Angeles, California 90089-1211

Ian A. Webster, Co-Principal Investigator

UNOCAL Corporation 1201 West 5th Street

Los Angeles, California 90051

US/DOE Patent Clearance is not required prior to the publication of this document.

TABLE OF CONTENTS

INTRODUCTION PROJECT DESCRIPTION

A. EXPERIMENTAL TASKS A.l . Measuring Asphaltene Diffusivity in Model

Catalysts under Realistic Temperature and Pressure Conditions

A.2. Measuring Asphaltene Diffusivities

A.3.

B. B.l.

B.2.

B.3.

B.3.1.

B.3.2.

B.3.3.

B.3.4.

in Sol-Gel Ceramic Membranes Measuring Asphaltene Diffusivity in Model and Real Membranes under Reactive Conditions THEORETICAL TASKS Study of Hindered Transport in a Single Pore Transport and Reaction in Networks of Interconnected Pores Monte Carlo and MolecuIar Dynamics Simulations Dilute Simulations Low Density Diffusion with Adsorption/Desorption Role of Intramolecular, Interinolecular and Surface Forces-Accounting for Aggregation and Delamination Phenomena Molecular Dynamics Simulations RESEARCH COMPLETED THIS QUARTER LITERATURE REFERENCES APPENDIX

DISCLAIMER

This report was prepared as an acwunt of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

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HINDERED DIFFUSION OF COAL LIQUIDS

INTRODUCTION

The design of industrial catalysts requires that the diffusivity of the reacting species within the

catalyst be accurately known. Nowhere is this more important than in the area of coal liquefaction

and upgrading of coal liquids. In this area one is faced with the task of processing a number of

heavy oils, containing metals and other contaminants, in a variety of process dependent solvents.

I t is important, therefore, on the basis of predicting catalyst activity, selectivity, and optimizing

reactor performance, that the diffusivities of these oil species be accurately known.

Several studies exist [l], emphasizing the importance of diffusion in processes involving coal and

petroleum liquids. Spry and Sawyer [a] measured the demetallization rates of a crude oil using var-

ious catalysts and found these rates to depend on average pore size, in agreement with their simple

model, based on the theory of Anderson and Quinn [3]. Inoguchi [4] observed that optimum activity

for desulfurization of heavy crudes occurs with catalysts with pore sizes of lOOA and for vanadium

removal with pore sizes between 120 and 140A. Eigenson e t al. [5] found increased catalytic activity

for Moos on A1203, when the pore size increased from 70 to 150& Significant intraparticle diffu-

sion effects have been observed during asphaltene cracking and the desulfurization of asphaltenic

and nonasphaltenic fractions of a residuum by Philippopoulos and Papayannakos [6]. Ternan and

coworkers [7] have reported significant pore size effects during the catalytic processing of Athabasca

bitumen. Studies have also been published on the importance of intraparticle diffusion during coal

liquefaction and coal liquid upgrading, including those of Polinski and coworkers [8], Scooter and

Crynes [9], and Yen et al. [lo]. Workers at Amoco have found [ll] that micropore volume in a

critical pore size range, i.e. narrowly distributed about l20A diameter pores, was very important

in coal liquefaction catalyst activity. McCormick e t al. [12] have shown during their hydrotreating

studies that activities for both hydrogenation and dehydrogenation reactions are highly correlated

(> 99% confidence) with the pore volume in the preferred range (60-2OOA) in diameter. The im-

portance of catalyst average pore diameter and pore length during the hydrocracking of solvent

refined coals and lignites h a s been discussed by Berg and Kim [13]. Theoretical studies have also

been published, which incorporate hindered diffusion in order t o predict optimal pore sizes during

petroleum liquid upgrading [14] and catalytic coal liquefaction [15].

The molecules comprising coal liquids can range from less than 10 t o several hundred A in

diameter. Their size is, therefore, compara.ble to the average pore size of most hydroprocessing

catalysts. Thus, during processing, transport of these molecules in to the catalyst occurs mainly

by “configurational” or “hindered diffusion,” which is the result of two phenomena occurring in

1

the pores; the distribution of solute molecules in the pores is affected by the pores and the solute

molecules experience an increased hydrodynamic drag. The field of hindered diffusion has been

reviewed by Deen [16]. The earliest studies in the field were by Renkin et al. 1171. Based on the

work of Ferry [HI, they developed an expression to describe hindered diffusion D e = (1 - X ) 2 ( 1 - 2.1044A + 2.089A3 - 0.948X5) , Db

where De is the effective “hindered” diffusivity, Db the bulk diffusivity and X the ratio of solute to

pore radius. The first factor in this expression accounts for the partitioning effect, and the second

for increased hydrodynamic drag in the pore. Anderson and Quinn [3] and Brenner and Gaydos [19]

have- theoretically verified these results in the region of small As (their work is a generalization of

Taylor-Aris dispersion theory [20] for spherical molecules with sizes comparable to the diameter of

the cylindrical tube). A number of other theoretical studies have also been devoted to this problem

[21]. Since the work of Renkin et al. [17], there have been a number of additional experimental

studies of diffusion in restricted porous environments. Satterfield et aZ. [22] investigated hindered

diffusion of binary systems of paraffins and aromatic hydrocarbons in silica alumina catalysts. They

concluded tha t reduced diffusivity, in these systems, was only due to the hydrodynamic effects and

their results indicate a linear dependence of h(D,/Db) on A.

Prasher et al. [23] also carried out studies using hydrocarbon solutes in aromatic and hy-

droparaffinic solvents. Contrary to Satterfield’s results, they observed a solute partitioning effect

dependent on the nature of the diffusing chemical species, rather than i ts molecular size, as one

expects based on Renkin’s theory. Colton et aZ. [24] studied diffusion of proteins and polystyrene

in porous glass and found that protein diffusion follows Renkin’s theory, while polystyrene behaves

as a free draining molecule. Cannel and Rondelez [25] measured diffusion of polystyrene through a

Nuclepore membrane and found that polystyrene diffusion follows Renkin’s theory, if X is corrected

by a factor of 1.45. The above studies have been useful in showing that reduced diffusivities are

observed in restricted environments. The pore systems utilized, however, were not well-defined

and probably too complex t o allow for fundamental questions to be answered. In the sixties, well-

defined planar porous systems were developed. Price and Walker [26] demonstrated that uniform

parallel pores of molecular dimensions can be formed, by a track etching process) in a number of

alumino-silicate materials. Mica membranes prepared by this method were subsequently utilized

in diffusivity measurements of nonhydrocarbon systems [27].

Studies of diffusion of petroleum and coal liquids are recent and few in number. In one of

the earliest studies, Thrash and Pildes E281 studied the transport of asphaltenes derived from a

Middle East high sulfur vacuum resid through mica membranes. The mica membranes had a pore

diameter of N 1200A. Thrash a.nd Pildes claim that what they measure are bulk diffusivities from

2

which they calculate Stokes-Einstein diameters of 168L. For the asphaltene concentration range

of their experiments (50-500 ppm), they report no evidence of asphaltene molecular association.

Shimura et al. [29] studied asphaltene diffusion in toluene through porous catalysts by uptake-

type experiments. The bulk diffusivity of molecular weight (MW) fractionated petroleum pitches

in dilute chloroform soIutions was measured in a NorthrupMcBain diaphragm cell by Sakai and

coworkers [30]. Their studies led them to conclude that the pitch molecules examined are three-

dimensional structures with an oblate ellipsoidal shape, their equatorial diameter being 10-208, and

their axial ratio between 1 and 3. Jost et al. [31] also measured the bulk diffusivities of various

oil residues in a THF/methanol solution by Taylor dispersion techniques. Recently, Mieville et al.

[32] measured the diffusion of various asphaltenes, by uptake experiments, through catalysts with

both unimodal and bimodal pore size distributions. In their experiments, Hondo asphaltene was

found to have the highest diffusivity, which was attributed to its smaller average aromatic cluster.

Configurational diffusivity measurements of petroleum liquids, utilizing mica membranes, were

first made by Baltus and Anderson [33] using a Wicke-Kallenbach-type diffusion cell t o measure

the diffusivity, through mica membranes, of asphaltenes derived from Kuwaiti atmospheric bot- toms. They divided the overall MW range into five regions and assigned each to a “hypothetical”

asphaltene fraction. Using GPC, they measured the diffusivity of each fraction for pore diameters

ranging from 160 to 44008L. An experimental expression was found to relate the diffusivity De of

each fraction to its bulk diffusivity D,, i.e. De/D, = exp(-3.89X). Following the work of Baltus

and Anderson, two other research groups have, in the last five years or so, focused their attention

on the study of hindered transport of petroleum liquid macromolecules through restricted porous

environments. Baltus and coworkers [34] have measured the diffusivity through polycarbonate and

polyester membranes of asphaltenes isolated from Athabasca tar sand bitumen vacuum bottoms

and a vacuum resid from a blend of Canadian crudes. They have also coupled the diffusivity mea-

surements with intrinsic viscosity measurements of asphaltene solutions. Using models previously

developed by Garcia de la Torre and Bloomfield [35], which represent the asphaltene in terms of

assemblages of solid spheres, and by varying the size and number of such spheres, they were able

t o obtain a reasonable fit between their experimental results and theory.

a.

Our group h a s also studied the diffusivity of petroleum derived asphaltenes [36]. We have

investigated asphaltenes derived from a variety of California heavy crudes. Although our studies,

in some aspects, paralleled those of Baltus and coworkers, our main objective was distinctly different

from their objective. Voluminous literature exists on the detailed chemical structure of asphaltenes

and other petroleum and coal liquid fractions like resins and oils. For asphaltenes, in particular,

interest in their structure and reactive and transport properties h a s been steadily increasing among

those working in the field of petroleum and coal liquid upgrading.

The term asphaltenes has been historically used to collectively describe the chemical components

of a crude (or a coal liquid), which are generally more polar and larger in size. Since the earlier

stages of the field of heavy oil upgrading, asphaltenes were defined as a solubility class of various

components. Through the years, this definition attracted discussion and criticism (see, for example,

Bunger and Cogswell [37]). Since some asphaltene properties are dependent on their isolation

techniques, the relevant question here is whether or not the asphaltenes produced by such isolation

techniques bear any resemblance and similarity t o compounds found in the original crude, and

whether or not the asphaltenes themselves exist in the original crude rather than simply being

an artifact of their separation process. The debate over whether or not asphaltenes even exist,

and if they do, how does one properly isolate and characterize them, has continued for years and

will not be dealt with here. Comprehensive descriptions can be found in a number of papers [38].

The generally held view today is that asphaltenes do exist and are found in the original resid.

One, however, cannot uniquely characterize an asphaltene (and, for tha t matter, any other resid

fraction) based on a single property alone, such as solubility. For petroleum resids, two properties

are needed to differentiate one resid fraction from the other. Which two properties one should use,

however, still remains a matter of debate. In his pioneering paper, Long [39] first suggested that

the composition of petroleum liquids can be represented in terms of a plot of M W versus polarity.

Long and Speight [39], furthermore, opted to use the solubility parameter as a measure of polarity

and the 5% and 95% points of the MW distribution (as measured by GPC) as a measure of the

MW. Wiehe [38], on the other hand, uses the number average M W (as measured by VPO) as a

measure of M W and the average hydrogen content as a measure of the molecular attraction. For

coal liquids, a measure of molecular attraction involving oxygen functionality would be required in

addition to the average hydrogen content (in lieu of aromacity) and MW. This was the conclusion

of Snape and Bartle [40] who found that three independent variables, i.e. number average MW, the

proportion of internal aromatic carbon to total carbon and the percent acidic OH were needed to

distinguish between benzene insolubles, asphaltenes and n-pentane solubles for coal-derived liquids.

Generally, the heavier the liquid one deals with, the larger the percentage of asphaltenes one

finds in it. One of the obstacles in coal liquid (and resid) upgrading is the presence of heteroatoms,

such as sulfur, nitrogen, oxygen and metals, which poison the catalysts used for hydroprocessing.

In terms of composition, asphaltenes contain a greater percentage of heteroatoms than all other

fractions and, therefore, present the greatest challenge during resid upgrading. Furthermore, most

of the molecular components of an asphaltene are very large structures with sizes approaching that

of a typical catalyst pore. Hydroprocessing of asphaltenes involves a number of parallel and/or

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consecutive reactions like hydrodesulfurization, hydrodenitrogenization, hydrodemetallation and

cracking. Due t o the large size of most asphaltene entities, all these reactions, as they occur within

the catalyst pellets, are diffusion limited.

For many years, asphaltene and resid upgrading rector design has utilized the concept of as-

phaltenes being a single molecular entity, to which one can assign a unique effective diffusivity and

other effective transport properties. This was one of the first issues we addressed in our studies of

asphaltene diffusivity. Our experiments showed that the effective diffusivity of an asphaltene, as

measured in a standard Wicke-Kallenbach cell using model porous membranes, varies as a function

of experimental time by as much as an order of magnitude for the duration of an experimental run

[36]. The diffusivity, furthermore, is a strong function of temperature and pressure, and whether

the membrane is impregnated with a catalyst or not. One cannot, therefore, assign a unique exper-

imental diffusivity value to an asphaltene molecule. Our experimental findings, though new, were

surprising neither to us nor to those intimately involved in research on asphaltene chemical struc-

ture and properties. Asphaltenes are not uniquely defined simple molecular entities and generic

structures, but rather a complex mixture of colloidal entities (micelles) of various sizes and shapes,

consisting of assemblages of smaller particles, which in turn result from the clustering of lower MW components, all in a state of dynamic exchange strongly affected by the presence and polarity of

solvents, temperature, pressure and fluid mechanical conditions [36, 411.

Our studies subsequently focused on the issue of whether the picture of an asphaltene “molecule”

as described above, which emerges from the available chemical information, i.e., a dynamic system

in a constant state of interchange is consistent with the experimental findings of diffusivity and

transport of asphaltenes through model porous membranes. An experimental plan was devised,

as outlined in Fig. 1 t o test these concepts. The asphaltenes used were n-pentane insolubles

extracted from Hondo Crude, a California outer continental shelf oil, furnished by UNOCAL.

The diffusion cell was a stainless steel Wicke-Kallenbach-type apparatus consisting of two well-

mixed compartments (half cells). During experiments of asphaltene diffusion, the half-cells were

separated by a polycarbonate membrane containing straight, parallel, non-intersecting, cylindrical

pores (supplied by Nuclepore Corp., Pleasanton, California). The membrane pore size varied

between look and 4000A. (Further details about our experimental system and techniques can be

found elsewhere [36]. The first experiment was conducted with a lOOA membrane. We started

with a 5 weight percent solution of asphaltene in xylenes, which was loaded in one of the half-cells,

hereinafter referred t o as the high concentration side (HCS). Pure xylenes were loaded in the other

half-cell, the low concentration side (LCS). The asphaltenes were allowed t o diffuse through the

membrane for a predetermined period of time. During the experiment, samples were withdrawn

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5 WTW MEMBRANE PORE DIAMETER ASPHALTENES ANALYSIS

1) XRF (S. Ni. V)

100 A PURE XYLENE

ANALYSIS

150 A PURE XYLENE

300

ANALYSIS

4000 i\

PUREXYLENE

PURE XYLENE

L H L H L H

t EN0

t BEGIN

EXPERIMENT EXPERIMENT

Fig. 1 . Experimental plan

from the LCS and analyzed for sulfur, nickel and vanadium concentrations by X-ray Fluoresence

(XRF), and for MW distributions by Size Exclusion Chromatography (SEC), MW distribution of

the various metal containing components by SEC/ICP and for overall asphaltene content. After the

first experiment ended the second experiment was initiated with a l50A membrane. Again, pure

xylenes were loaded in the LCS. In this experiment, however, the HCS was filled with the asphaltene

solution remaining in the HCS a t the end of the first experiment with the l O O A membrane. Further

experiments were done with the rest of the membranes starting again with pure xylenes in the LCSs

and the HCSs containing the HCS solution carried over from the preceding experiment.

Some of the da t a collected from the diffusion experiments, depicting nickel and vanadium con-

centrations in the LCS as a function of run time are plotted in Figs. 2 and 3. Figs. 4 and 5 show

typical SEC analyses of the LCS samples at various experimental times for two different pore sizes.

One can draw the following conclusions from these figures (and the rest of the experimental data).

First note Figs. 4 and 5, which show SEC analyses of LCS samples withdrawn a t different run times.

The polydisperse nature of asphaltenes is clear from these figures. Asphaltenes are not simple and

uniquely defined molecules but rather consist of a whole range of molecular components of various

sizes. The components with smaller sizes diffuse faster than the components with larger sizes and

as a result the MW distribution of the asphaltene in the LCS a t smaller times is skewed towards

the lower MWs. At larger times, the MW distribution of the asphaltene in the LCS approaches

that of the asphaltenes initially loaded in the HCS. This, obviously, is an oversimplified picture of how asphaltenes diffuse through porous mem-

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I -

6 -

a 4

NICKEL . 4 o o o i

L m i . r s o i - o $00 A

/------

.. 0 40 80 120 160 200 240 2

T I M E (HOURS)

Fig.2. Luw s i d e concen t r a t ion

VANADIUM

I % 4

1

2

1

0 0 80 120 160 200 240 280 40

8( TIME (HOURS)

Fig.3. Low s i d e Concentrat ion.

I-INITIAL nicn.

TIME IHOURSI -

a 185.68

c 271.75 b 218.60

20 - 1006

PORE DIAMETER 0 -

2 3 . S

L O G MOLECULAR WEIGHT

Fig.4. Log Molecular weight

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L O G MOLECULAR WEIGHT

Fig.5. Log Molecular weight

-:- CHECK VALVE *- %UT OFF VALVE + + RUPTUREDISK BPA BACKPRESSURERECULATOR 111 LOIDlNG UANIFOLO

C w COOLING WATER 3 PRESSUUE GUAGE I W < . I)IVPI.I<E\TIII. i * n i : s i . n i : G ~ . A G I

JU

Fig.6. High temperature and pressure diffusion system.

b

Table 1: Activation energies in kcal/mole calculated from effective diffusivities.

branes. This mechanism of asphaltene transport is only consistent with the polydisperse nature of

asphaltenes. On the other hand, any polydisperse polymer would diffuse in a similar fashion, as

indicated in Figs. 4 and 5. To realize that asphaltenes diffuse in a uniquely different fashion, one

has t o examine Figs. 4 and 5 in conjunction with Figs. 2 and 3. Note, for example, Fig. 2. Clearly

there are components in the initial asphaltene that are larger than 150A. In fact, out of the total nickel diffused throughout the whole series of experiments, only approximately 20% permeates, in

the time allotted, through the 100 and 150A membranes. How could it then be possible that the

asphaltene at large times in Fig. 5 closely resembles the asphaltene initially loaded in the HCS? The only asphaltene structure capable of describing the data in Fig. 5 is the one we have previously

described, namely that “the asphaltenes are not simple, single generic species but rather complex mixtures of micelles of various sizes and shapes, of unformed micelles, of small particles and of

low MW components and attached alkyl chains, all in a state of dynamic exchange strongly ajJected

by the presence of solvents, temperature and pressure and fluid mechanical conditions.” Only a

dynamic association-disassociation process can explain the fact that the large components of the

asphaltenes, with nominal sizes larger than the diameter of the pore find their way from the HCS to the LCS.

The dynamic nature of the asphaltene molecule manifests itself in the effect of temperature

and concentration on its transport. Table 1 , for example, lists the activation energies calculated

from the diffusivity data from heptane-insoluble asphaltenes at various temperatures. Activation

energies for diffusion are generally of the order of 1 I<cal/mole. The values in Table 1 are more

typical of the association energies for individual asphaltene sheets reported by Dickie and Yen [41],

typically 14 to 20 Kcal/mole, ra.ther than diffusion activation energies. The trends in terms of

the pore size can be understood on the basis that asphaltene dissociation is expected to affect

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more strongly the transport through more constricted pores. The simplest model of an asphaltene

molecule, which accounts for both its polydisperse character and its dynamic nature will faithfully

reproduce the behavior in Table 1.

There is obviously a lot more information included in Figs. 2-5 and similar such figures. For

example, the mechanism of heteroatom transport emerging for these da ta is consistent with the

detailed chemical information as t o where these heteroatoms lie in the original asphaltene micelles

and particles. More detailed such discussions can be found in the original publications [36]. Of

more importance, however, are the implications of the transport mechanism for petroleum and coal

asphaltenes tha t is emerging on resid and coal liquid upgrading reactor design. In our opinion,

the implications are many and significant. Traditionally, upgrading reactor design has utilized

the concept of overall effective transport properties for asphaltenes and other oil fractions. In

view of the existing chemical information on the structure of asphaltenes, the practice has been

characterized as simplistic but of some engineering utility. Our thesis in this proposal is that

the practice is outright wrong and can lead to erroneous design calculations. For one thing, such

effective properties cannot be measured (one order of magnitude variation in effective diffusivity was

observed in some of our experiments). More elaborate techniques, furthermore, must be devised

to relate diffusivities measured in a Wicke-Kallenbach cell or by uptake experiments t o transport

properties of relevance under reactive conditions.

Is the previously stated concept of the structure of an asphaltene, i.e., of an asphaltene being

”a complex mixture in a state of dynamic exchange” significant for upgrading catalyst design? In

our opinion, it is, because it offers plausible explanations for what is happening during upgrading.

Look at Figs. 2 and 3. Most of the nickel, for example, found in a petroleum asphaltene should not

even be penetrating the structure of a typical upgrading catalyst, if it was not for the concept of

“dynamic exchange” between the various asphaltene components. The other plausible reason why

typical upgrading catalysts can process the heteroatoms found in resids or coal liquids is that the

structure of an asphaltene represented by the above model, although valid at low temperatures, will

not survive the higher upgrading temperature and pressure conditions, Le., the micelles and even

the particles will break apart. Superficially, this explanation is in contrast with experimental data

(like those, for example, by Wiehe [38] and our group [36]), which show tha t thermal treatment at

upgrading conditions of asphaltenes does not completely destroy the asphaltenes, but instead re-

sults in their incomplete conversion t o both lower MW fractions, like volatiles, saturates, aromatics

and resins but also t o higher MW components like coke. In our opinion, these data are signifi-

cant but not necessarily conclusive about the structure of asphaltenes at higher temperatures and

pressures in view of the asphaltenes being considered as “complex mixtures in a state of dynamic

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exchange.” Assuming, on the other hand, that the asphaltene structure does not survive thermal

treatment at upgrading conditions, what then is the purpose of studying such structures? In our

opinion, the issue here is one of rates. Even if one is to assume that at elevated temperatures and

pressures “micelles” and even “particles” break apart, the question t o be raised is at which rate

does this breakdown process occur? Contrary t o laboratory reactors, where most of the studies of

asphaltene’s chemical structure have taken place, most industrial reactors are continuous systems.

The state of the asphaltene “molecule” therefore does not only depend on the temperature, pressure

and polarity of the solvent but also on the reactor’s residence time. It is, therefore, very impor-

tant t o have a correct concept of the asphaltene’s structure and through careful experimentation,

one can then decide whether such a concept has any practical implications at realistic upgrading

conditions.

It is the purpose of the project described here to provide such a correct concept of coal as-

phaltenes by careful and detailed investigations of asphaltene transport through porous systems

under realistic process temperature and pressure conditions. The experimental studies will be

coupled with detailed, in-depth statistical and molecular dynamics models intended t o provide a

fundamental understanding of the overall transport mechanisms.

PROJECT DESCRIPTION

Throughout the experimental runs described below we will utilize a high pressure, high tem-

perature diffusion cell system, a schematic of which is shown in Fig. 6. This diffusion system has been tested through the measurement of the diffusivity of a number of model coal liquids.

The heart of the experimental system is a high pressure autoclave, which in its interior can

accommodate one or several ceramic membranes. One side of these membranes is exposed to the

contents of the autoclave, while the other side, through an independent flow system, is exposed to

flowing pure solvent. The pressure in the interior and exterior of the membranes can be indepen-

dently adjusted and controlled. This is also true with the flow rate of the solvent in the interior of

the membrane. The diffusion experiments are initiated by placing the coal liquid solution (model

liquids or asphaltenes) in the autoclave space exterior of the membrane, pressurizing the exterior

and interior membrane volumes and initiating the flow of the solvent. One has t h e option of running

the experiment in a batch (exterior)-continuous (interior) or batch-batch mode. The option also

exists for loading catalyst in the exterior volume either in a pellet or slurry form or using metal

impregnated membranes for simultaneously studying transport and reaction. Model membrane

preparation and characterization will be carried out both at USC and under the direction of one

of us (I. A. Webster) at the UNOCAL Science and Technology Division, of UNOCAL Corpora-

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tion (USTD). UNOCAL, in addition, will contribute technician and machine time on apparatuses,

such as Auger and XPS, preparative GPC, SEC, XRF, SEC/ICP, Low Angle Light Scattering

Photometer, Electron Microscope, Atomic Adsorption, Porosimeters and BET.

The project is of both experimental and theoretical nature and is divided into a number of

tasks, a brief description of which follows.

A. Experimental Tasks A. l . Measuring A s p h a l t e n e Diffusivity in Model Catalysts under Realistic Tempera-

ture and Pressure Condi t ions

The asphaltenes will be isolated from coal liquids provided by PETC using a continuous type

asphaltene isolation apparatus. This is a UNOCAL proprietary asphaltene isolation apparatus, a

replica of which has been constructed at USC. Both pentane and heptane insoluble asphaltenes will

be isolated.

The model membranes will be polymeric membranes, but also anodic aluminas, which have

straight, non-intersecting pores. The anodic aluminas are produced by direct electrochemical an-

odization of aluminum sheets [42]. The anodization technique allows [43] for control of pore size,

structure and morphology of the A1203 films formed, which depend on parameters such as voltage,

temperature, current density, pH and type of electrolyte used. Parallel pores of uniform size (15-

2000A) have been formed with pore densities as high as 10l1 pores/cm2. Our current anodization

cell is based on a design by Itaya et al. [43] and allows for in situ anodization and barrier film

removal (further details can be found in some of our recent publications [42]). During barrier film

dissolution the porous alumina film is protected by maintaining an immiscible fluid head on the

porous film side.

The experimental technique to be followed has already been described above. The asphaltene

solution will be loaded in the autoclave in the space exterior to the membrane and pressurized

under H2 or mixtures of HZ in Ar (see below). The interior and exterior membrane volumes will

be sampled periodically and analyzed by techniques such as XRF for heteroatom analysis, SEC for

M W distribution, SEC/ICP for heteroatom specific MW distribution, overall asphaltene content,

NMR and FTIR for overall functional group identification and by Gas Chromatography. Parame-

ters t o be investigated include (but are not limited to): (1) The effect of asphaltene concentration

on the overall transport mechanism; (2) The effect of temperature and overall pressure and pres-

sure gradient across the membrane (the pressure of the interior and exterior membrane volumes

can be independently adjusted and controlled); (3) The effect of solvent and type of asphaltene

(i.e., pentane or heptane insoluble, type of coal liquid it came from); and (4) The effect of pore

size (the anodic membranes have unimodal straight non-intersecting pores) and membrane surface

11

morphological properties, which can be varied by the choice of electrolyte and post-anodization

treatments.

When these experiments are completed a better picture, we hope, will emerge of the mechanism

of coal asphaltene transport under realistic temperature and pressure noindent.

A.2. Measuring Asphaltene Diffusivities in Sol-Gel Ceramic Membranes Anodic membranes have unimodal straight, non-intersecting pores. As such, they allow for

unique insight into the mechanism of the transport processes occurring since one does not have

t o worry about the complications resulting from the tortuous porous structure and to account

for an adjustable parameter, i.e., the tortuosity factor. On the other hand anodic membranes

are somewhat poor analogs of the true hydroprocessing catalysts, which have a complex tortuous

porous structure.

Recently developed Sol-Gel type ?-A1203 membranes offer a compromise and a middle ground

between the ideal structure of anodic aluminas and the complex structure of true upgrading cat-

alysts. These membranes also have an extremely narrow unimodal pore size distribution (like the

anodic aluminas) but have tortuous porous structures. Through their use we hope to gain an ad-

ditional insight into the complications in the transport mechanisms of asphaltenes arising from the

tortuosity of the porous medium.

The Sol-Gel membranes we will utilize will be MembraloxTM asymmetric alumina membranes,

which have been previously utilized by our group in catalytic membrane reactor investigations [44].

The minimum pore size of these membranes is 40A.

A.3. Measuring Asphaltene Diffusivity in Model and Real Membranes under Reactive Conditions

The issue of whether the intraparticle diffusivity measured under reaction conditions is the

same with tha t measured under non-reactive conditions is of fundamental interest and the subject

of ongoing debate [45-471. For example, Otani and Smith [45] reported the effective diffusivity under

reaction conditions t o be only about 20% of that measured in the absence of reaction. In contrast

Ryan et al. [46] have theoretically argued that the effective diffusivity should be independent of the

chemical reaction rate. We have also shown theoretically [47] that , as is also true with facilitated

transport through biological and polymeric membranes [48], in the presence of surface reactions

with fast and slow steps involving mobile surface species, diffusivities measured under non-reactive

conditions bear no similarity or relevance to diffusivities prevailing under reactive conditions. Under

such conditions, of course, even the strict applicability of classical transport reaction equations is in

doubt [4G-501. The asphaltene, due t o the action of solvents and its interaction with other asphaltene

molecules and the solid porous walls, is constantly fragmenting and agglomerating. In the absence

12

of a porous barrier one would expect an asphaltene solution to reach a state of pseudo-dynamic

equilibrium. When a porous barrier is present, however, as in the case with our experiments, this

dynamic equilibrium is shifted towards lower M W fragments, since these fragments preferentially

leak out of the membrane, i.e., one attains a form of facilitated fragmentation. In a real catalyst

particle the situation, of course, is very similar. One does not have a LCS, where the low MW fragments will diffuse t o but the low MW fragments are removed by the chemical reaction, i.e.,

again a chromatographic separation of the asphaltene occurs as i t moves from the bulk solution

into the particle’s center.

These are the types of questions and issues we hope to address in this task. Our experimental

system is very flexible and allows for the upgrading catalyst to be placed in the exterior membrane

volume in the form of pellets in a small rotating basket, to be slurried in the asphaltene solution

or in the solvent circulating in the interior membrane volume. The catalytically active component

to be utilized will be Mo promoted with either Co or Ni. Additional parameters to be investigated

(beyond those of Tasks I and 11) are the amount of metal present and the effect of H2 partial

pressure, by using mixtures of H2 and Ar.

B. THEORETICAL TASKS B. l . Study of Hindered Transport in a Single Pore

As mentioned in the Introduction, a number of theoretical studies have been devoted t o this

problem. Starting with the comprehensive works of Brenner and coworkers [19], Anderson and

Quinn [3] and Paine and Scherr [51]. The work of Brenner et al. and Paine and Scherr is a gen-

eralization of the Taylor-Aris [20] dispersion theory for spherical molecules with sizes comparable

t o the diameter of the cylindrical tube, in which hindered transport takes place. These and sub-

sequent studies on hindered diffusion have not considered the case of simultaneous transport and

reaction/adsorption in the pores.

As the first t a s k of our theoretical work, we intend to study hindered transport of non-spherical

molecules in cylindrical pores. The molecules comprising coal liquids have much more complex

structures and geometries than the molecules studied in the field of hindered diffusion, where even

the case of an ellipsoidal molecule in a cylindrical pore has not been completely analyzed (though

some progress has been made by Brenner and coworkers [19]). Furthermore, little is known on the

transport of large molecules in pores (cylindrical or not) in which a reaction is also taking place.

We intend to study such problems by generalizing the approach of Brenner and coworkers, which is

a statistical method, by which one calculates the probability density function for the center of the

molecule t o be at a position r at time t . From the second moment of this function, ( r 2 ) = 6D,t, the

effective diffusion coefficient is determined. The results of this part of our study will be essential to

13

the next stage of our investigation described below. Due t o the inherent complexity of the approach,

there is an upper limit on the degree of structural and geometric complexity of the molecules that

can be studied by this technique. More complex molecular shapes, conceivably more consistent

with our understanding of asphaltene molecules, will be employed in the Monte Carlo simulations

described below. The value of the techniques of Brenner and coworkers lies in their simplicity of

application, in tha t they do not require significant numerical computations.

B.2. Transport and Reaction in N e t w o r k s of In te rconnec ted Pores

As the second task of our work, we shall study hindered transport, reaction and adsorption

in three-dimensional networks of interconnected pores. Any porous system can be mapped onto

[50] a random network of bonds (pore throats) connecting t o each other at sites (pore bodies) of

the network. The resulting network usually has a random topology. The geometrical properties of

the porous medium can be incorporated into the structure of the network by assigning randomly-

selected effective radii to the bonds of the network. These can be chosen from a realistic pore size

distribution obtained from, e.g., porosimetry. The 'bonds can, in principle, take on any shape, e.g.,

cylindrical tubes, sinusoidal or converging-diverging channels, or constricted tubes. Our networks

will be constructed t o simulate our Sol-Gel ceramic membranes used in our experiments.

By modeling the porous medium as a network of interconnected pores, one can generalize

the work of Brenner and coworkers [19] to describe hindered transport and reaction/adsorption in complex porous systems. At the pore level, hindered transport of molecules is governed by a

convective-diffusion equation (CDE) or its generalization obtained in Task B.l. One then needs to

describe how transition of molecules from one pore t o the next takes place. A rigorous approach is

to write down a mass balance for a 2-coordinated pore body (node) i,

z

where S;j is the effective cross section area of the pore throat between pore bodies i and j and

Jij the flux in tha t pore. If this equation is written for every node of the network, one will obtain

a set of simultaneous equations for the concentration of the diffusing molecules throughout the

entire network. From the solution, one can then determine the effective transport properties of the

molecules. This approach necessitates the use of large scale Monte Carlo computer simulations t o

calculate the quantities of interest.

Since the description of transport at the pore level is rigorous (see Task B.l), our estimates of

the transport properties for the entire network will also be rigorous and exact, provided that the

network model closely resembles the actual porous medium. This t a sk h a s recently been initiated, and the preliminary results are encouraging [52].

14

B.3. Monte Carlo and Molecular Dynamics Simulations

Transport and reaction of molecules like asphaltenes in porous media is a complex process, in

which various interacting factors may all be significant. These include: (a) The structure of the

porous medium; (b) the molecular nature of the diffusing species, their shape, size and molecular

flexibility; (c) the intermolecular interactions between the diffusing molecules as well as their inter-

action with the porous media; (d) the adsorption and desorption of the molecular species on the

surface of the pores; and (e) the reaction on the pore surface, including the possible role of surface

diffusion.

Given the complexities of the problem, computer simulation is the ideal technique to conduct

such studies. In Tasks B.l and B.2 we described simulation methods for solving such problems.

These methods are restricted, however, to dilute systems, i.e., those in which only a few molecules

diffuse and react in the system, whereas in reality this may not be the case. If the concentration

of the diffusing species is high, then, intermolecular interactions, and also the interaction between

the molecules and the pore surface become very important. Moreover, the approaches described in

Tasks B.l and B.2 can only be implemented for relatively simple molecular shapes. Asphaltenes,

on the other hand, have extremely complex shapes. Since only limited detailed experimental da ta

on the chemical reactivity of asphaltenes exist in the literature at the present time, for the purposes

of this project, we will only concentrate our attention on understanding the physical characteristics

of, and the physical processes taking place, namely, the diffusion, adsorption and desorption of

molecular species in the catalyst particle and the dependence of these processes on the structure of

the pore space and the nature of the molecules. Our tools are Monte Carlo and Molecular Dynamics

simulations, which can naturally take into account the effect of every important factor. In what

follows we outline our research plan for a comprehensive, step by step, theoretical and computer

simulation studies of these phenomena.

B.3.1. Dilute Simulations In order t o understand the role that the porous medium’s structure and the intermolecular and

surface forces play, we will first consider the case of non-interacting molecules. This corresponds t o

dilute concentration experiments.

The appropriate structure, e.g., a single pore or a network of interconnected pores, will be

modelled depending on whether we simulate our experimental anodic or Sol-Gel membranes. After

generating the pore structure of the system t o initiate the computer simulations, N molecules will

be injected into the pore space; this is time t = 0. The molecular shapes used will be consistent

with our understanding of asphalthene molecules (see Introduction/Experimental section) since in

a simulation of the kind described here, any molecular shape and MW distribution can be used

15

without much difficulty. Each molecule will perform a random walk (Brownian motion) in the pore

space representing the diffusion process. At every step of the simulation, a molecule will select

one of the available directions, with an equal probability, t o make a transition to another part of

the pore space. The maximum length of the step is held constant determined by an acceptance rate of 50%. Each time a direction is selected, the molecule (i.e.? its center) is moved to the new

position. In the low density limit, a molecule does not meet another so there is no excluded volume

except for that associated with the pore surface. This type of simulation can be implemented in

two ways. One can inject into the system one molecule at a time, follow its motion and compile

the statistics of interest. Alternately, one can inject simultaneously N molecules into the system.

Each of the N molecules are then displaced in turn as described above for the sequential injection

method. For sufficiently large N , the results are independent of N . The simultaneous injection of

many molecules implementation is more efficient when parallel processing is available. While use

of random walks for simulating diffusion is not a new idea, to the best of our knowledge, aside from

our recent work [53], no one has used this technique to simulate diffusion of large and complex

molecules of various shapes in a restricted pore spaces such as those considered here.

For this series of initial simulations, we do not permit adsorption or reaction on the pore surface, to be accounted for in the next series of simulations (see below). The effective diffusivity of the

molecules can be evaluated by determining the time dependence of the mean-squared displacement

of the center of mass of the molecules

where ( e - ) denotes an average over all molecules and their initial positions in the system. In

the limit of long times and for large systems, the ratio r 2 ( t ) / t is proportional to the De of the

molecules. (The above calculations are, of course, repeated for a large number of time steps and

many randomly-selected initial positions of the molecules, and the averages of the quantities of

interest are computed, in order t o obtain representative values of effective diffusivities.) A series

of simulations for pore size distributions, and various molecular shapes are planned to probe the

effect of the pore structure on th i s very simple diffusion process. The key variable here will be the

relative sizes of the molecule and the pores, and the key question will be whether, consistent with

the experimental observations, De will eventually reach a constant value.

B.3.2. Low Density Diffusion w i t h Adsorp t ion/Desorp t ion

In this phase of the work, we will look at the effect of adsorption and desorption. We will first

consider the case of irreversible adsorption. In the simulation, when a molecule comes in contact

with a pore surface, it will be adsorbed with probability p which is proportional t o a Boltzman factor

16

exp(-cci), c being a constant and ~i the surface binding energy. p + 0 describes a system under

sorption kinetic control, whereas p + 1 represents a diffusion-controlled process. An adsorbed

molecule results in an excluded surface and volume in the pore space. The simulations will be

terminated when a large number of molecules have been adsorbed on the pore surfaces so that

there is no significant additional adsorption, or the adsorbed molecules, due to their finite volumes,

have effectively blocked the pore space and no further macroscopic molecular motion occurs. By this simulation procedure, termed the Dynamic Monte Carlo Method (DMCM), one can calculate

all the dynamic, and static properties of interest.

Next to be considered is the case of reversible adsorption. The problem is more complex since

now one must consider the kinetics of the adsorption/desorption process in addition to diffusion.

The injection rate of molecules must be controlled in a physically realistic manner. We believe

tha t it is physically plausible to assume that a state of equilibrium exists between the molecules

in the immediate surrounding of the porous medium and those on its outer periphery. This means

tha t a certain fraction, say f, of the layer of the pore space is occupied by diffusing molecules.

Throughout the simulation, f is maintained constant, which means tha t as soon as a molecule

leaves one of the peripheral positions, another molecule must be introduced into one of the vacant

peripheral cells, but not necessarily the one just vacated. The relative time scales of the diffu-

sion, adsorption and desorption processes can be varied by changing the adsorption probability p

and the ratio a = desorption rate/adsorption rate. The limit a + 0 corresponds to irreversible

adsorption. Setting a = 1 will require each molecule t o remain on the surface for a fixed period

of time (say rn time steps) before they are eligible for desorption. When interpreting the results

several factors must be considered. In case of the diffusion-limited regime, sorption will primarily

occur at the outside external perimeter of the catalyst particle and the phenomenon of poremouth

blocking will occur. The effect of molecular sorption on diffusivity is then a strong function of the

adsorption/desorption rates relative to the diffusive fluxes. When diffusion is not rate limiting, a

large number of molecules are adsorbed on t h e pore surfaces. Even then, however, pore plugging

can occur but the phenomenon is a percolation process. The DMCM technique to be developed in

this Task can be used t o systematically study the effect of all system parameters on the transport

process.

B.3.3. The Role of Intramolecular , Intermolecular and Surface Forces - Accounting

for Aggregation and Delaminat ion P h e n o m e n a

I

Intramolecular and intermolecular forces determine the size, shape and flexibility of the diffusing

molecules and the attraction/repulsion between molecular pairs. The number of possible parameters

can be very large, especially for polydisperse complex molecules like asphaltenes and cannot be

17

determined a priori in our simulations due t o the limited understanding of intermolecular and

surface forces. We will therefore s tar t by studying the diffusion and adsorption of molecules with

relatively simple shapes.

The simulations are started by generating a molecular analog of the asphaltene’s structure. One

then has to account for the intermolecular forces tha t play an important role in the diffusion of the

molecules. But what is the appropriate expression for the intermolecular forces? In most previous

MC simulations of diffusion processes, a Lennard-Jones (LJ) potential has been used, i.e.

where +;j is the potential energy experienced by the i-j pair of molecules e i j and a j j are the energy

and size parameter and r the intermolecular separation. The LJ potential, most likely is too simple

to account for the intermolecular forces between complex molecules. To resolve this problem, one

can treat c;j and u;j as adjustable parameters such that the simulation would reproduce one set

of data (e.g., at a given T and P) and, then, use the resulting values to study the system under

a different set of conditions. This procedure has recently been used by several authors with a

reasonable degree of success. At the next level of sophistication are site-site potentials and other

more complex-forms of intermolecular forces [54] that have been quite successful. These will also

be employed in our simulations. The molecule-wall interactions can similarly be modeled using the

following potential

(4) ST€ 2 u

+(z) = i3 (;yo - , where + is the potential energy due t o surface-molecule interactions, and z the distance between

the surface and the molecule’s center of mass. Of course, because of the complex structure of

asphaltene molecules, one can only use effective values of these parameters.

Different t: will be used to study the effect of the attraction of the pore surfaces. We start the

simulation by placing molecules in a manner dictated by our experiments. All molecules in the

system will then be given sequential random displacements in the normal Monte Carlo way, each

displacement accepted with probability:

Pdisp lacemeat = min[l, ezp(-cAu)] , ( 5 )

where Au is the potential energy difference resulting from the displacement, and c is a positive

constant which, for a macroscopic system, is related to its temperature. In the present case, it

is not necessary to exactly specify c. We would rather prefer to regard cAu as a dimensionless

potential energy difference. When the interaction between the molecule and the pore surfaces are

described using a potential equation (Eq. 4), we no longer need the parameter p defined above

1s

described previously, in order t o account for the adsorption/desorption rate relative to the random

diffusive flux, we have to introduce the parameter m, which is the number of units of time an

adsorbed molecule stays adsorbed before a desorption attempt is made. Since, in dealing with

asphaltene molecules one has to deal with aggregation-disaggregation phenomena, these must also

be accounted for in the simulation. Once a large molecule like an asphaltene delaminates and

fragments into several pieces, the motion of each fragment must then be followed, accounting for

the fact that the fragments can join together again (similar to aggregation of colloidal particles) at

a later time.

B.3.4. Molecular Dynamics Simulations

I The DMCM described above will give a satisfactory insight into the fundamental processes of

asphaltene diffusion. The MC method, however, has one major disadvantage. Real time is not

an explicit parameter of the simulation. M C methods are, therefore, strictly not applicable to

the study of highly non-equilibrium diffusion processes. Molecular dynamics (MD) calculations on

the other hand circumvent this difficulty. MD is a simulation technique in which the equations of

motion for a system are directly solved to determine the time evolution of the system. The solution

obtained in a MD study is

essentially exact to the extent tha t the potential model used accurately describes the intermolec-

ular forces. M D methods are now fairly well known and used for studying equilibrium properties

and one-particle dynamic properties such as self diffusion coefficients. They have been widely used

[55] to study simple fluids, nonpolar polyatomics, polar molecules, and bipolymers.

The feasibility of calculating accurate De with MD simulations was demonstrated by Jacucci

and McDonald [56], Jolly and Bearman [57] and Schoen and Hoheisel [58]. Stoker and Rowley [59]

used the techniques developed in these previous studies to simulate binary mixtures of polyatomic

molecules. Tirrell, Davis and co-workers [60] have used MD simulations to study diffusion of simple

molecules in a single pore. Either equilibrium or nonequilibrium MD can be used t o estimate

diffusion coefficients. In view of the structural complexities of the porous medium and the molecules

in our study, we shall only employ equilibrium MD simulations. One expects the qualitative trends

given by the two methods t o be similar. The long-time tail in equilibrium simulations can be

accounted for by appropriately long simulations. The purpose of the MD simulations here is two-

fold: first, to compare De evaluated using the mean-squared displacement methods [Eq. (a)] (from

both MC and MD) and those obtained using velocity autocorrelation functions,

(6) 1 0 0

a j j = 5 I ( 4 0 ) W ) d t 9

where w ( 0 ) and V ( T ) are velocities of a molecule at times zero and T respectively, and (.--) indicates

19

an average over sufficiently large number of molecules and for various choices of zero times. Secondly,

in the case of diffusion with adsorption, a comparison between MC and MD can be used t o establish

the significance of and, if possible, the appropriate quantitative magnitude of the parameter m.

The study by Schoen and Hoheisel [58] indicated that long runs are required to accurately

calculate the velocity correlation function. Of prime importance in improving the statistics of the

calculations are the number of time origins used in the correlation function determination, the

number of particles, and the length over which the integration is carried out. Despite its rigor, MD still remains a very promising but underdeveloped approach to the determination of transport

properties of real molecules in practical systems. To the best of our knowledge, a large scale MD calculation has never been attempted for a complex pore space and molecular structures, such as

those tha t in this study. The only drawback of MD is that for complex molecular structures such as

those of asphaltenes, the intramolecular and intermolecular potentials are not exactly known. One

way of attacking this problem is to use the most accurate and plausible potentials and t o adjust

the parameters potentials t o match the experimental data.

Before the full potential of MD simulation for these system can be realized, eventually several

problems must be overcome: (1) It is possible that for some systems the diffusional time scale and

tha t for adsorption/desorption are very different so that during the course of a normal simulation

of acceptable length, one of these processes cannot be adequately observed; (2) The type of MD

simulation described is not a typical MD simulation because the number of molecules in the system

( N ) is not constant. When more molecules are added to the system, they must be introduced with

the appropriate accelerations, velocities and intermolecular potential energies. These quantities

must be appropriately preselected so as not to introduce arbitrary variables or undesirable large

transients; and (3) The Green-Kubo expressions for the diffusivity [Eq. (6)] are derived for homo-

geneous systems. Their applicability t o porous media has not been established. Our studies will

hopefully provide definitive answers to some of these questions.

RESEARCH COMPLETED THIS QUARTER

The following were accomplished this quarter:

A paper was submitted during this quarter, a copy of which can be found in the Appendix.

In this paper, we use a lattice-gas method to simulate the slow flow of a fluid in systems

with complex and fractal surfaces and boundaries. Two such systems are studied. One is

flow in a single three-dimensional fracture with rough and fractal surfaces. The other is

flow over a three-dimensional diffusion-limited aggregate, which is relevant to suspension and

flow of colloidal aggregates, e.g., asphaltene molecules in a porous media such as membranes

20

and catalysts. In both cases, significant deviations from the classical results for non-fractal

systems are observed. Efficient use of the method allows us t o simulate very large systems,

and thus study the effect of the system’s size on the quantities of interest.

0 Our high pressure temperature transport experiments using mesoporous membranes with

the permselectiv layer facing the high concentration-side have been continued under reactive

conditions during thermal coal dissolution. In our BEM investigations, we have completed

the testing of the centerline approximation. A paper is in preparation which will be submitted

during the next quarter.

21

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26

Appendix A Papers

27

4 I

Lattice-gas siniulatioii of flow in systems wit 2 i complex surfaces and b o u ii d ar ie s

Xiaodong Zhang and Mark A. Knackstedt

Research School of Physical Sciences and Engineering, The Australian National University,

Canberra ACT 0200, Australia

Muhammad Sahimi*

Department of Chemical Engineering, University of Southern California, Los Angeles, CA

90089-1211, USA

We use a lattice-gas method to simulate the slow flow of a fluid in systems with complex

and fractal surfaces and boundaries. Two of such systems are studied. One is flow in a single

three-dimensional fracture with rough and fractal surfaces. The other is flow over a three-

dimensional diffusion-limited aggregate, which is relevant to suspension and flow of colloidal

aggregates e.g., asphaltene molecules in a porous medium such as a membrane. In both cases,

significant deviations from the classical results for non-fractal systems are observed. Efficient

use of the method allows us to simulate very large systems, and thus study the effect of the

system’s size on the quantities of interest.

*To whom all correspondence should be sent.

1

1. Introduct ion

Flow and transport i n systems with complex surfaces and boundaries are relevant to a

wide variety of scientific and industrial problems, and have been studied for a long time. For

example, natural porous media and rock contain a wide variety of pores and fractures with broad

distributions of sizes and shapes (for recent reviews see, for example, Sahimi [1,2]). There is

now experimental evidence [l,2] that, the internal surface of the pores and the fractures is very

rough, and the roughness may even obey fractal statistics. For example, if the fracture surface

were perfectly smooth, then flow in a single fracture could be modelled as flow between parallel

flat plates, for which volumetricflow rate Q is given by, Q = (20S3AP)/(l2qL), where w is the

width of the fracture and 6 its aperture, A P / L is the pressure gradient applied to the fracture,

and q is the viscosity of the fluid. Thus, according to this model Q depends on the third power

of 6, and the permeability I( of the fracture varies as 6'. However, there is ample experimental

evidence [3-51 for deviations from the cubic law, Q - S3, which has been attributed to the

surface roughness. For this reason, flow in a single fracture with a realistic shape cannot be

solved analytically, and only numerical solutions of the problem can be obtained, which is why

this problem has received considerable attention [1,2]. The numerical methods used so far have

been mostly based on discretizing the governing equations, the usual continuity and the Stokes'

equations, by a finite-difference or finite-element method and solving the resulting equations.

However, if the effect of surface roughness of a fracture is to be taken into account, the grid

has to be very refined near the surface and this requires prohibitive computations. Moreover,

because of the surface roughness, the fracture aperture varies from point to point. Thus, if the

aperture could be measured at a large number of points in the void space of the fracture, then

one could use the data with the discretized equations. However, this would also be very tedious

and computationally prohibitive.

As the second example, consider the problem of hydrodynamics of particles or aggregates

dispersed or suspended in a fluid, a phenomenon of considerable importance to a wide variety

of natural or man-made systems, such as colloids, polymers, granular and composite materials,

ceramics, and many more. For example, asphaltene aggregates are formed during catalytic coal

liquefaction and enhanced oil recovery by gas injection. They can be suspended in the fluid

within the porous medium, or precipitate on the pore surfaces. One of the most important

and unsolved problems is how the presence of the asphaltene aggregates affect the fluid flow.

1 c

The most important theoretical (and practical) problem is to predict the effective equilibrium

and transport properties of these materials, given their microstructural mechanics. The mi-

crostructural mechanics entails the distribution of various forces acting on the particles, such

as the hydrodynamic, Brownian, and interparticle forces, as well as the morphology or the

microstructure of the particles. From the knowledge of microstructural mechanics, given the

physical properties of the particles and the fluid that surrounds them, and the location and

motion of the boundaries of the system, one can, in principle, predict the macroscopic proper-

ties of the system with the given configuration, such as the diffusion coefficient and the thermal

conductivity. Although this seems to be a well-defined boundary value problem, solving it is

easier said than done, since one has to deal with a many-body problem which is extremely

difficult to solve. To tackle this problem, various schemes have been developed in the past

[6]. The problem is particularly difficult if the particles or aggregates have a complex shape.

For example, it is well-known [7-91 that many colloidal aggregates are fractal objects with very

complex structures, and moreover their properties scale with their linear size or mass, so that

their true fractal behavior does not manifest itself, unless the size of the aggregate becomes

large. For these reasons, calculating various properties of aggregates that are suspended in a

flowing fluid, such as the distribution of the hydrodynamic forces that act on the aggregates,

and the dependence of the friction factor on the Reynolds number, is a very difficult problem.

Even sophisticated approaches that have been developed in the past [6] cannot be used with

aggregates of more than a few hundred particles, unless one is willing to use a prohibitively

huge amount of computer time. Moreover, the distribution of hydrodynamic forces exerted by

the fluid on the particles of the aggregate becomes very broad, since (relatively) large forces

will be exerted on the most exposed parts of the aggregate, whereas the particles deep inside

the aggregate will feel small forces.

In this paper, we use an alternative method based on lattice-gas (LG) simulation, to study

flow in a three-dimensional fracture with rough and fractal surfaces, and the distribution of

the hydrodynamic forces that are exerted on a colloidal fractal aggregate that is suspended

in a slowly moving fluid. As the model of a colloidal aggregate, we use a three-dimensional

diffusion-limited aggregate (DLA), which has been shown to be relevant to a variety of im-

portant phenomena [9], and in particular, asphaltene aggregates. Lattice gas models are large

discrete structures where each site can be in one of several discrete states. Both time and

3

space are discrete, and usually connections between the sites are between nearest-neighbors

only, ideal conditions for high-speed simulations on vector or parallel computers. Moreover,

the simulation obeys certa.in rules (see below) such that, in the continuum limit, one recovers

the appropriate hydrodynamic equa.tions. Since such equations, even in their discretized form,

are not solved explicitly, the simulations axe carried out very efficiently, and thus large systems

can be studied.

The plan of this paper is as follows. In the next Section, we briefly discuss the lattice-gas

method that we use in this paper. In Section 3 we describe our simulation of flow in a fracture

with rough surfaces, while Section 4 presents our results for the hydrodynamics of flow over

DLAs. The paper is summarized in Section 5, where we also point out future directions in this

research field.

2. Lattice-Gas Simulation

As mentioned above, LG models are [1,2,10-121 large lattices where each site can be in one

of several discrete states. The lattice is populated by particles, and the variables describing the

state of the system are Boolean indicating the presence or absence of the particles in the bonds

of the lattice. The evolution of the system is governed by a set of collision rules that tell us

how the particles move in the lattice, and how they are scattered once they collide with each

other (see below). Both time and space are discrete, and usually connections between sites are

between nearest neighbors only, ideal conditions for high-speed simulations on vector or parallel

computers. The LG approximation of the Navier-Stokes equation in two dimensions is based on

particles of unit mass either resting on the site of a lattice or moving with unit velocity on one

of the six bonds emanating from each lattice site. Frisch, Hasslacher and Pomeau (FHP) [13]

showed that in order that the discrete equations reduce to the usual Navier-Stokes equations,

two-dimensional simulations have to be done on a triangular lattice.

On the triangular lattice the particles either have a unit velocity along a bond of the lattice,

or are at rest a t a site. Up to six particles may reside at any site on the triangular lattice.

However, multiple occupancy of a site is subject to the constraint that no two particles at a site

can have the same velocity. If particles hit each other, they are scattered according to the laws

of momentum conservation. For example, one particle hitting another one that is a t rest may be

scattered such that its direction chmges by GO", and that the previously resting particle moves

in a direction inclined at -60" with respect to the direction of the incoming particle. However,

I

this two-particle collision alone leads to anisotropy. One needs three-particle collisions in order

to remove the anisotropy. Thus two- and three-particle collisions constitute the minimum set of

collision rules necessary to obtain a n isotropic behavior for the LG models. Usually up to four

particle collisions are employed (5 and 6 particle collisions almost never happen). If a particle

hits a solid wall, it is reflected by 180" to simulate the no-slip boundary condition.

The extension of LG models to three dimensions is more complex, since no regular three-

dimensional lattice is isotropic, and thus in the continuum limit one has spurious terms, in

addition to those in the Navier-Stokes equation, which are caused by the anisotropy of the

lattice. However, there are now several methods of circumventing this difficulty [14,15]. For

example, one can use [14,15] the FHP model on a four-dimensional face-centered hypercubic

(FCHC) lattice. The nodes (x1,22,23, xq) of the lattice satisfy the condition that x1+z2+lc3+x4

is even, where xis are integer numbers. For this lattice, which has a coordination number of

24, all pairwise symmetric fourth-order tensors are isotropic, and therefore one can simulate

the Navier-Stokes equation on such a lattice. We may then make the observation that any

solution of the four-dimensional equation which does not depend on the fourth dimension is

a solution of the three-dimensional equation. This suggests the use of a FCHC lattice which

wraps around periodically in the fourth direction. One actually uses a lattice which is only

one lattice unit long in the fourth dimension, and therefore has an effectively three-dimensional

structure. This is the approach we use in this paper. The only disadvantage of this model is

that, although the fourth dimension is very thin, the discrete velocities still have components

in all directions; therefore the model is bit intensive (24 or 25 bits per site as compared with

6 in two dimensions). On the other hand, having a large number of discrete particle velocities

is not necessarily a disadvantage, because it allows more collisions between the particles,:and

therefore higher Reynolds number flows.

At the beginning of the simulation, one constructs a transition table that tells us how one of

the present states (determined by the velocity of the incoming particles) is transformed into the

next state, determined by the outgoing velocities. The table contains all of the possible states

(for example, in two dimensins there are 2' possible states). One then starts the simulation

using the many-body collisions at a site which conserve momentum and energy. The rules are

such that no new particle is created, and in one time step all particles on the lattice move first

to different lattice sites and then undergo transitions at the new sites. One iteration in the

5

c

simulation consists of updating all lattice sites according to the transition table. The number

of required iterations for reaching a steady-state solution depends on the microstructure of the

system. If the system has a complex configuration, then many iterations may be required. An

issue of particular importance is the mean free path of the particles and its relation to the

height of the roughness on the surface of the fracture. As Rothman 1161 showed, in order for

the LGA results to approach the continuum limit, the mean size of the void area in the lattice

must be at least twice the mean free path of the particles, which is about 9 lattice bonds. If this

condition is not obeyed in the simulations, the result may not represent a truely macroscopic

continuum.

The LG model that we described has been used by various authors to investigate flow phe-

nomena in porous media. Rothman [16] studied single-phase flow in porous media. Succi et

al. [17] studied the same problem in three dimensions. Ibackstedt e t a!. [18] (1993) used

the LG method to study dynamic (frequency-dependent) permeability of a porous medium, a

notoriously difficult problem, because one has to obtain the frequency-dependent permeability

by Fourier-transforming the numerical results. Normally, LG results are subject to relatively

large noise and fluctuations, and therefore their Fourier transform is very difficult to obtain.

Despite this, Knackstedt et al. [18] showed, by reproducing the exact results for the dynamic

permeability of channel-like systems, that if a large number of realizations are used, the LG re-

sults are reliable. This indicates the reliability of LGA for simulating flow problems. Brosa and

Stauffer [19], Kohring [20,21], and Sahimi and Stauffer [22] looked at flow in two-dimensional

porous media with various obstacle shapes and arrangements (random versus regular and peri-

odic), and paid particular attention to the efficiency of the simulation. Vollmar and Duarte [23]

studied flow through a porous membrane, and investigated the effect of various boundary con-

ditions. Chen, Doolen, and co-workers [24,25] used the LG and the related lattice-Boltzmann

methods to study a variety of flow problems.

3. Flow in a Single Frac ture wi th Fractal Surfaces

To simulate a given flow problem with the LG model, one first generates the configuration

of the system. One main advantage of using LG models is that any configuration of the system,

no matter how complex, can be used. Thus even the exact digital image of a natural system

can be used in the simulations. Once the desired configuration is generated, the simulation

can be started. For simulating flow in a single fracture, we first have to model the fracture

6

with rough surfaces. We model the fracture by a three-dimensional channel whose surface is

rough and obeys fractal statistics. The roughness of the surface was generated by a fractional

Brownian motion (fBm) [2G], which is a s ta t iomry stochastic process BH(r) wi th the following

proper ties

where r=(z, y, z ) and ro=(zo, yo, z o ) are two arbitrary points, and H is called the Hurst expo-

nent. A remarkable property of fl3m is that it generates correlations whose extent is in3nite.

For example, if we define a one-dimensional correlation function C ( r ) by

then one finds that C ( r ) = 22H-1 - 1, independent of T . Moreover, the type of correlations can

be tuned by varying H . If H > l /2, then C ( r ) > 0 and fl3m displays persistence, i.e., a trend

(for example, a high or low value) at T is likely to be followed by a similar trend at T- + Ar. If

H < 1/2, then C ( r ) < 0 and fl3m generates antipersistence, i.e., a trend at r is not likely to

be followed by a similar trend at r + Ar. For H = l / Z the trace of BH(r) is similar to that

of a random walk, while H = -1/2 represents the white noise limit. It has been shown by

Schmittbuhl et ab. [27] that, if we consider a two-dimensional roughness profile on a fracture

surface, then the average height h of the profile is related to its length L by

h N L H , (4)

where H is the roughness or Hurst exponent defined above. A value H c11 0.85 was found by

Schmittbuhl et al. for granitic faults, indicating strong positive and long-range correlations. It

is precisely the existence of such complex rough surfaces that make simulation of fluid flow in

a single fracture such a complex problem. In the present paper, we use a range of values for H

in order to assess its effect on the permeability of the fracture and its relation with the fracture

aperture. Figure 1 shows an example of a two-dimensional rough surface generated by fBm.

Once the roughness of the surface is generated, simulation of the flow problem are started.

We used an L, x L, x L, system, where z denotes the direction of flow, and in our simulations

L, = L, = 64 and L, = 128 (all distances are measured in units of lattice bonds). The

simulations start by introducing a flat velocity profile (piston flow) at the entrance to the

fracture. We then let the system to relax and reach a stea.dy state. Typically, this was achieved

after about 10,000 time steps, after which we measured the desired quantities for another

20,000 time steps. The pressure gradient A P / L , in the system was measured by calculating

the momentum transfer across imaginary planes perpendicular to the flow direction.

permeability K of the fracture was then calculated by using Darcy’s law:

The

IC A P ‘u =

r7 L z ’ (5)

where is the average fluid velocity. The simulations were carried out at low enough Reynolds

number to ensure that Darcy’s law is applicable. The results were then averaged over 20 different

realization of the fracture with a given H , the Hurst exponent. All of our computations were

carried out with a computer program that was optimized for the Connection Machine CM2.

Of prime interest is the dependence of the permeability of the fracture on its aperture. We

define the aperture as the distance between the two mean surfaces that are parallel to the flow

direction. We need to define a mean surface, since the fracture surfaces are all rough. Given

the two rough surfaces, we calculate their mean heights hl and hz, and thus the aperture is

simple 6 = L, - (h1 + hz). To change the aperture, the root mean square thickness -t of the

two surfaces constant, pushing the two surfaces together. The root mean square thickness -t

was also varied to probe the effect of . t /h on the results, where l / 6 can be interpreted as the

dimensionaless aperture.

Figure 2 shows the dependence of IC on the aperture for H = 0.8, a value close to what

Schmittbuhl et al. [27] obtained for rough fracture surfaces, and two values of l. As can be

seen, in both cases one has

IC - s p , (6)

with p N 2.67, except that for -t = 18 one has an apparent crossover to ,B N 2. The results

for -t = 10 differ from the classical result for fracture with smooth surfaces for which ,L? = 2.

Figure 3 shows the results for H = 0.3, and for both values of 4 we find p N 3.15, which is

again significantly different from the classical results. If we further decrease H , we find that ,L?

increases significantly. For example, for H = -1/2 we find ,8 > 4. Of course, such deviations

imply that the classical result for the volumetric flow rate Q N S3 does not hold.

Summarizing, our simulations show that for flow in a fracture with rough surfaces, one may

see significant deviations from the classical results for flow in a fracture with smooth surfaces.

More details of our results are given elsewhere [283.

8

4. Hydrodynamics of Flow over a Fractal Aggregate

We have also studied the problem of slow flow of a fluid over a fractal aggregate. This

study represents the first step towards our ultimate goal of studying suspension of colloidal

aggregates and calculating their dynamical properties, such as the viscosity of the suspension.

Since it is now well-established that many colloidal aggregates have a fractal structure, we

studied flow over a three-dimensional DLA aggregate, which has a fractal dimension D j N 2.5,

and has been studied extensively over the past many years [9]. A somewhat similar problem has

already been studied by Meakin [29,30], who used the Kirkwood-Riseman theory to investigate

hydrodynamics of a fractal aggregate moving with a constant velocity in a quiescent fluid.

However, due to computational difficulties, Meakin was able to use only very small aggregates

with at most 400 elementary particles in them (Meakin used diffusion-limited cluster-cluster

aggregates which have a fractal dimension Dj N 1.8 in three dimensions).

Using the LG method, we have studied the problem with aggregates whose masses M varied

between 427 and 4507 particles. We first generate the DLA cluster and then it place in the

lattice system. For clusters with M < 800 we used 32 x 32 x 64 lattices, while for the larger

aggregates we used 64 x 64 x 128 lattices. We also used periodic boundary conditions to minimize

the effect of finite lattice size. No-slip boundary condition was used at the interface between

the fluid and solid phases. The density of the fluid particles was about 3. The simulations

are similar to what we described for the fracture problem. At the entrance to the system, a

velocity profile is introduced, and the system (fluid plus the solid phase) is allowed to relax. In

this problem, the system reached equilibrium after about 7,000 time steps. We then measure

the quantities of interest in the next 3,000 time steps. The results are then averaged over more

100 realizations of the system.

An important property is the normalized force distribution defined by

where (F,;)i is the normalized force parallel to the flow direction, and (41); is the corresponding

unnormalized force. The logarithmic distributions are shown in Figure 4 for various aggregate

sizes, where N[ln(qi)] is the number of a.ggregate particles per unit interval of ln(4I) with

a force component 41. These distributions seem to be skewed and similar to a log-normal

distribution. Typical of fractd aggregates, these distributions can be collapsed onto a single

scaling curve. Shown in Figure 5 are In( N[ln( Fly)])/ln( M ) versus In( Fly). Although we used

large aggregates, due to the noise in the simulations the collapse is not complete. This figure

does indicate that there may be a single universal curve for the distribution of the hydrodynamic

forces for various aggregate sizes. If true, we expect to have

where g(x) is the scaling function that describes the collapsed curve in Figure 5. The force

distribution shown in Figure 4 is multifractal 1311, i.e., each of its moment scales with a different

power of M . From the knowledge of g(x) one can construct a spectrum of singularities f(a)

[31], describing the multifractal properties of the distribution, given by

Figure 6 shows the resulting spectrum.

Another important quantity is the friction factor Cj and its dependence on the Reynolds

number Re. Cf is defined by

where p is the density of the fluid particles, is the shear stress on the solid surface, and (v)

is the mean flow velocity. For a spherical particle in low Reynolds number fluid flow, one has

the classical result [32]

and we checked that our LG simulation does predict this equation correctly. Figure 7 shows our

results for the DLA aggregates with three different masses. For over two orders of magnitude

variations in Re in the range 0.1 <Re < 10 we have

a Rec ’ cj = -

where C N 0.85. We have no theoretical explanation for this value of C. More details of our

simulations and results are given elsewhere [33] .

5 . Summary and Conclusions

Flow in systems with rough surface is a complex phenomenon, and although several mod-

els have been developed for studying them, they were not completely satisfactory, and could

10

not provide quantitative, and in some cases even qualitative, predictions for the quantities of

interest. The most important factor that contributes to this complexity is the structure of

the surface of these systems which is very rough, and often obeys fractal statistics. In our

opinion, the systems that we studied in this paper represent definitive examples of the prob-

lems for which lattice-gas models are definitely more appropriate than the continuum, or even

the previous discrete models. This method is the ideal tools for vector computers and par-

allel computations, which are the way massive computations of the prersent are carried out.

As such models become more sophisticated and realistic, more traditional numerical methods

of simulating fluid flow in complex systems, such as the finite-difference methods, lose their

competitive edge, and will be phased out in the future.

Acknowledgments

11

This work was supported in part by the Petroleum Research Fund, administered by the

American Chemical Society, and by the US Department of Energy.

c

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