cfd as a design tool for fixed-bed reactors

CFD as a Design Tool for Fixed-Bed Reactors

Anthony G. Dixon* and Michiel Nijemeisland

Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts 01609

Computational fluid dynamics (CFD) is a tool that is becoming more realistic for use in thedescription of the detailed flow fields within fixed beds of low tube-to-particle diameter ratio(N). The motivation for the use of CFD is presented by reviewing the current state of fixed-bedreactor modeling, with an emphasis on the treatment of the description of fluid flow within thebed. Challenges in the use of CFD for fixed beds of particles are treated here, selected resultsare presented for N ) 2 and N ) 4, and potential uses of the simulation information in designmodels for fixed-bed reactors are discussed.

Introduction

Computational fluid dynamics is a well-establishedtool for several areas of reaction engineering. However,for gas-solid reactors such as fixed beds, geometriccomplexity has so far prevented detailed modeling oftheir hydrodynamics. The usual approach to modelingfixed-bed reactors assumes plug flow and effectivetransport mechanisms. Recent work suggests that im-proved predictions of reactor performance can be ob-tained for slim tubes if the radial variation of the axialflow component is included, but effective parametersmust still be retained, and good predictions of measuredvelocity profiles can only be obtained if an effectiveviscosity is also introduced.

Modern CFD codes and the exponential growth ofcomputer power are bringing realistic fixed-bed flowsimulations into the realm of possibility. It is nowfeasible to obtain detailed flow fields in fixed beds oflow tube-to-particle diameter ratio (N). The flow fieldis especially interesting in the near-wall region, whereflow features can differ from those in the bed center.This information can be used directly in detailed three-dimensional reactor simulations or to provide informa-tion for inclusion in process design models.

This contribution presents the motivation and frame-work for the use of CFD calculations in derivingimproved fixed-bed models for low N. Challenges infixed-bed CFD are discussed, including the generationof representative bed geometries and formulation of themodel for laminar and turbulent flows. Some examplesof detailed calculations are presented that can be usedto develop simplified models of fluid flow, heat transfer,and chemical reaction. Future needs are identified,including the extraction of useful knowledge from thelarge data sets that result from simulations.

Background

Modeling and simulation are essential tools in theanalysis and scale-up of reactors, and as demands onreactor performance increase, model performance needsto be able to give the spatial distribution of reactants,catalysts, inerts, and products in detail at all times.1Current heterogeneous reactor models have been basedon fairly radical simplifying assumptions: homogeneity,

effective transport parameters, pellet effectiveness fac-tors, etc. These simplifications have been driven by a(fast-disappearing) need for computational simplicityand by the difficulties of the complex structure ofrandom packed tubes. These idealized models have led,however, to problems. Even the most advanced modelstoday cannot quantitatively represent reactor behaviorif independently determined kinetics and transportparameters are used.2,3

Nowhere have the simplifications in reactor modelingbeen so sweeping as in representing the fluid flowthrough the reactor. Despite the realization that theglobal behavior of a flow or transport system dependsdirectly on local flow structures,4 in most cases, thehydrodynamic modeling of fixed-bed reactors is stillbased on plug flow. The most recent developments haveonly gone as far as extending simple uniform one-dimensional plug flow with a single constant velocitycomponent to a single velocity component with variationperpendicular to flow.5

The simplified models of the past have been mainlyintuitive or empirical responses to the need to decidewhich factors are crucial in a reactor model and whichare of minor importance. The phenomena that must beaccurately represented are those that have strong effectson rates of reaction or rates of heat and mass transfer.The traditional approach to this problem of modelingtransport rates in fixed-bed reactors has been to intro-duce radial, and axial when needed, effective thermalconductivities and an apparent wall heat-transfercoefficient.6-10 The parameters lump together all of thephysical phenomena contributing to the heat-transferpicture. These parameters are then estimated fromexperimental data by regression analysis of appropriatemodels,11 which usually assume the fluid flowing throughthe tube to be in unidirectional axial “plug” flow. Despitenearly 50 years of such efforts, a large degree ofdisagreement exists between the data from variousworkers,12-14 especially for the dimensionless heat-transfer coefficient or wall Nusselt number. In addition,this approach yields no insight into the physical mech-anisms at work or possible methods of improving theheat-transfer rates.

The main limitation in our understanding is the lackof resolution of the detailed flow picture in these beds.15

In reality, thermal energy is transported by strongradial convective flows as fluid is displaced around thepacking elements. Regions of stagnant and reverse flow

* Corresponding author. Tel.: (508) 831-5350. Fax: (508)831-5853. E-mail: [email protected] (A. G. Dixon).

5246 Ind. Eng. Chem. Res. 2001, 40, 5246-5254

10.1021/ie001035a CCC: $20.00 © 2001 American Chemical SocietyPublished on Web 07/28/2001

have been identified by nuclear magnetic resonance(NMR) imaging experiments,16-19 which are limited toliquid flows at very low flow rates. These flow featuresare thought to be strongly connected to poor heat-transfer performance near the wall. To understandthem, and to quantify them for gas flows at the largeflow rates of industrial practice, realistic three-dimen-sional simulation of the fluid flow in a fixed bed isnecessary.

The design approach discussed here is the develop-ment of gas-solid reactor models based on the system-atic analysis of the flow fields within them. Computa-tional fluid dynamics (CFD) can provide details of flowpatterns in the interstices of a fixed-bed reactor, iden-tifying regions that could lead to enhanced or poortransport. The connection to process- or macroscopic-level models for reactor design and analysis is the mainchallenge.

Fixed-Bed Transport Models

Classical Approaches. The earliest approaches6,7 tomodeling transport in low-N fixed beds assumed thefluid to be in uniform, one-dimensional axial flow (plugflow), lumped all mechanisms for radial transport intoan effective thermal conductivity kr or diffusivity Dr, andrepresented the observed increase in resistance to heattransfer near the containing wall by an apparent wallheat-transfer coefficient hw. These parameters wereincorporated into models that viewed the bed as asingle-phase continuum (pseudohomogeneous models).Axial transport was usually regarded as negligibleunder the high flow rates of industrial conditions.

The 50 years of research that have followed the initialintroduction of effective transport parameters have seenthe parameters correlated against fluid flow rates, fluidproperties, bed properties, and catalyst particle shapesand sizes.8,20,21 Theories for the prediction of the pa-rameters have been developed, on the basis of morefundamental mechanisms that could be isolated andmeasured, at least in theory.9,22 Much discussion in theliterature has taken place over how to measure tem-perature data in fixed beds,23 how to eliminate un-wanted effects such as bed length,24 how to obtainunbiased estimates of the effective parameters,25 whetherthe parameters depend on reaction rates in the catalystpellets,26 and many more issues. Continuum modelshave been developed that discard the pseudohomoge-neous assumption27 and regard the bed as being com-posed of two phases, each with its own set of effectiveparameters as well as interphase coefficients. Ap-proaches from the literature of porous media have beenused28-30 to try to put the derivation of the continuumequations and the parameters in them on a morerigorous basis.

The result of all of this effort is a plethora ofcorrelations in the literature, which are usually in verygood agreement with the data of the authors whodeveloped them and in less good agreement with thedata of other workers. There is a well-documentedinability to combine independently measured kineticswith heat and mass transport to model reactor perfor-mance without further adjustment of the parameters.There is a flourishing debate as to what the effectiveparameters really represent9,22,26 and, in some cases, asto whether they should be used at all.31 The inescapableconclusion is that a designer of a fixed-bed multitubular

reactor would not be able to predict a priori with anycertainty the behavior of the reactor.

Recent Approaches. The deficiencies in our abilityto model fixed-bed reactors might stem from the valuesof the transport parameters, the limited ranges of themodels used, the inadequacy of the kinetics, or acombination of all three and maybe other sources.Several recent attempts have been made to radicallychange the approach to fixed-bed reactor modeling. Arevival of the cell model approach has been tried,32,33

which is likely to meet the same problems as theoriginal, as it must rely on idealized pictures of mixingin the interstices of the packing, and as it is extendedto accommodate both heat and mass transfer only withdifficulty. The approach to the dispersion of mass andheat through Fick’s and Fourier’s laws has been chal-lenged, and a wave model first developed over 30 yearsago34 is now being extended and revised.35

A data-based approach has been proposed36 thatdispenses with the need to measure and correlateeffective parameters. Microscopically valid transportequations are constructed for a segment of the bed. Theyallow the theoretical use of a Green’s function to writethe solution at one space location and time (x, t) in termsof a known solution at another (x′, t′). The inlet feed,reaction, and wall heat-transfer terms then becomeknown sources in the equations.

Although a large amount of work has been done onbed structure and velocity profiles, this work has focusedon the usually empirical relation of the radial variationof the axial velocity component, vz(r), to the radialvariation of the bed porosity, ε(r), and has been exten-sively reviewed.37 Recently, there has been considerablework by the groups of Vortmeyer38 and Eigenberger39

using the extended Brinkman-Forchheimer-Darcyequation to obtain vz(r). This equation is already well-known in the literature of flow in porous media,28,29

where it has been derived through the volume-averagingmethod and gives vz(r) in terms of ε(r)

where

The solution of this equation gives vz(r), which is thenincorporated into the energy balance and mass balanceequations for the reactor, along with reaction terms.Either homogeneous or heterogeneous bed-scale modelscould be used. For a homogeneous model, for example,the energy balance gives

with a boundary condition at r ) R that idealizes theextra resistance near the wall to a resistance at the wallby introducing the wall heat-transfer coefficient via theequation

0 ) - dPdz

+ µeff(∂2vz

∂r2+ 1

r∂vz

∂r ) -µf

Kvz - FF

xKvz

2 (1)

K )ε

3(r)dp2

150(1 - ε(r))2; F ) 1.75

x150ε3(r)

(2)

Fcpvz(r) ∂T∂z

) keff(∂2T∂r2

+ 1r

∂T∂r ) (3)

-keffdTdz

) hw(T(R) - Tw) (4)

Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5247

The authors38,39 have claimed improved predictionsof reactor behavior at low N, confirming that the nextgeneration of models needs to include more informationabout the flow field in the reactor. One drawback to thisparticular approach is that the effective radial conduc-tivity, keff, and apparent wall heat-transfer coefficient,hw, are retained and, indeed, represent different phe-nomena than they did in the previous plug-flow models,so that new correlations must be developed for them.40

Because the use of vz(r) is again an approximation,although a better one than plug flow, we can expect thatthe disagreement and confusion regarding the lumpedparameters is likely to be repeated. In addition, neithergroup could obtain reasonable results for the velocityprofile without introducing an effective viscosity, µeff,into the Brinkman term in the momentum equationabove. This is a new lumped parameter that must becorrelated. These approaches do not bring us any closerto understanding and representing the phenomena infixed beds at a fundamental level.

What is needed are experimental and computationaltechniques that allow us to understand and model fixed-bed phenomena on the particle or subparticle level.Experimental techniques to do this have improvedmarkedly in recent years, such as laser-Doppler veloci-metry (LDV),41 nuclear magnetic resonance (NMR)imaging,42 and particle tracking methods.43,44 Thesemethods all allow observations of flow inside the fixedbed without disturbing the bed structure and can thusfurther our understanding, but each is subject to severelimitations. LDV requires windows for optical accessand is restricted to beds of very low N where such voidsoccur naturally. It also requires the fluid to be refrac-tive-index-matched with the transparent material of thecolumn. NMR methods are also restricted to liquids, andusually, the flow rate must be very low for the signalstrength to be high enough for detection. Particletracking methods require observation and counting ofthe markers, and problems with choice of fluid similarto those with LDV are found.

Computational techniques for fluid flow at the porescale in fixed beds have so far been few. Network modelshave been used by several investigators.45,46 The latticeBoltzmann method (LBM) has also enjoyed some promi-nence recently.47-49 Solution of the full Navier-Stokesequations to obtain the microscopic flow field aroundparticles in a fixed bed, for realistic flow rates, hasusually been considered to be unachievable as a resultof computational limitations and geometrical complexi-ties, which so far have been beyond computational fluiddynamics (CFD) codes.50 Early computations were madefor creeping flow in a cubic array of spheres51 and forsimplified geometries such as two spheres near a wall.52

Flow around single particles53,54 and several particlesin a row55 has also been studied, in which the axisym-metric nature of the flow allowed a two-dimensionalsimulation. Full three-dimensional simulations of bedsof small numbers of spheres, including wall effects, havebeen started by our group.56-59 Most recently, a com-plete bed of 44 spheres with N ) 2 has been solved andvalidated by comparison to experimental data.60

Both experimental and computational methods forfixed beds at the pore (interstice) scale have revealed avery complex picture, especially near the containingwall, of strong radial flow components, regions of reverseflow (back flow), stagnant eddies, and so on. Thesehydrodynamic features can be expected to play impor-

tant roles in heat transfer, dispersion, and chemicalreaction. To date, no studies have investigated thisconnection at the level of the phenomena occurring inthe interstices between particles.

Computational Fluid Dynamics Modeling ofFixed Beds

In CFD simulations, the Navier-Stokes mass andmomentum conservation balances are solved for aquantity of mesh volumes; additional balances for heattransfer and turbulence modeling parameters can beadded. The basic equations and background of thesebalances are stated in standard references.61 In thissection, we present a brief summary of some of the mainpoints regarding the use of CFD for fixed-bed modeling.We also present some results from two studies of fixedbeds with N ) 2 and N ) 4 to show the kind ofinformation that can be obtained from CFD modeling.The study with N ) 2 was conducted as a benchmarkingcase to validate the use of CFD in fixed-bed modeling.60

Only a summary of the main points is presented here.The first step in CFD simulation is the creation of a

representative geometric model of the desired flowsituation. After a geometric model has been created, avolume mesh for the numerical simulation is needed.The mesh must differentiate between the solid and fluidparts within the geometry and have a proper controlvolume density so that it shows all flow features withoutincreasing computational intensity unnecessarily. Es-pecially for turbulent conditions, the mesh needs to befine in constricted flow areas, i.e., near sphere-spherecontact points and sphere-wall contact points.

Another point for the numerical simulation of flow isthat all mesh elements need a finite dimension on alledges, which does not allow actual contact pointsbetween solid parts in the geometry. Thus, a gap mustbe introduced between solid surfaces in the model. Asensitivity study to the introduced gap was performed60

to allow for an appropriate flow solution.When a sufficiently accurate mesh has been gener-

ated, the numerical simulation can be performed.Boundary and initial conditions need to be specified.Boundary conditions include wall temperatures and flowinlet and outlet conditions; initial conditions define thefirst step in the numerical simulation and the iterationprocess and concern the fluid flow field and the initialtemperatures of both the fluid and the bed internals.

The geometric model of the flow situation needs tobe identical to the experimental setup to perform avalidation, which necessitates an accurate modelingtechnique. For the direct validation study, a structuredpacked bed with N ) 2 was chosen; this specific ratioallows for a very structured bed that can be modeledwith mathematical accuracy. Beds with low tube-to-particle diameter ratios show structured packings,which is an advantage in creating a simulation geom-etry. The N ) 2 bed packs regularly in the experimentalsetup, and an identical geometric model was created (seeFigure 1).

The mesh used in this validation model was anunstructured tetrahedral mesh, also shown in Figure1. The complexity of the packed-bed geometry neces-sitated an unstructured mesh. The minimal density ofthe mesh was determined after a sensitivity study wasperformed on a single-sphere model. Specific care wastaken in meshing the areas in the flow-constricted areasnear sphere-sphere and sphere-wall contact areas; the

5248 Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001

mesh was slightly refined in these areas to allow forless distorted mesh volumes. The resulting mesh con-tained 430 000 control volumes and required approxi-mately 11 h of CPU time on a 500-MHz DEC Alphaworkstation for a turbulent solution and approximately5.5 h for a laminar solution.

Numerical simulations were performed under condi-tions similar to those under which the experiments wereperformed. In a range of Reynolds numbers (373-1922),radial temperature profiles above the bed were estab-lished, and these values were directly compared withexperimentally obtained radial temperature profiles.Both simulation and experimental temperature profileswere acquired under steady-state conditions. A com-parison of the temperature profiles showed that CFDcould qualitatively and quantitatively predict the ex-perimental results accurately.

A selection of the flow fields and temperature contoursavailable from the CFD simulations is shown in Figures2-5. In Figure 2, the flow field shows strong axiallydirected bypass flow down the near-wall voids presentin the N ) 2 packing (refer to Figure 1) and weakerflow components passing the particle contact points. Thearrows representing the velocity vectors point in thedirection of the local flow, and their lengths are pro-portional to the magnitude of the velocity at each

location. Open spaces correspond to solid volumes thathave been removed from the picture for clarity.

The velocity field shown in Figure 3 corresponds to abed section passing through the particle centers in they-z plane. This section clearly shows the circulatory

Figure 1. Geometry and mesh for fixed bed with N ) 2.

Figure 2. Velocity vector plot for N ) 2 for a four-layer sectionover the entire bed diameter in the y ) x plane at Re ) 1922;legend shows velocity magnitude in meters per second.

Figure 3. Velocity vector plot for N ) 2 for a four-layer sectionover the entire bed diameter in the x ) 0 plane at Re ) 1922;legend shows velocity magnitude in meters per second.


flow typical near particle-particle contact points, thestrong radially directed flow components immediatelybefore and immediately following a particle, and regionsof reverse flow and near-stagnant flow next to the walls.Thus, in some parts of the bed, flow is rapidly bypassingthe packing in the wall regions, whereas in other partsdownstream of wall-particle contact points, the flow isslow, and even backflow is taking place. Accelerated flowcan be seen in the narrow interstices between particles.A closer view of these features is presented in Figure4. The region of near-stagnant flow corresponds to thecontact point of two of the particles that have beenremoved for clarity. The last picture for N ) 2 showstemperature contour plots for the entire tube of 44spheres for three Re values in Figure 5. A comparisonof the three tubes shows the development of the tem-perature profile occurring further downstream withincreasing Re, as would be expected. The regions of lowtemperature at the bottoms of the figures correspondto the unheated flow-calming section of the apparatus.The higher-temperature contours within the outlines ofthe particles demonstrate the effects of the higherthermal conductivity of the solids. The development ofan interrupted boundary layer near the wall is alsoevident, with temperature intrusions into the bedthrough the solid. These pictures of near-wall temper-ature fields demonstrate the complexity of the flow andheat-transfer phenomena near the wall of a fixed bed.

Following the validation study, a geometric model ofa larger N ) 4 packed bed was developed. When thetube-to-particle diameter ratio is increased the numberof particles in the packing increases substantially.Because the mesh density needs to be similar to thedensity used for the N ) 2 bed, the mesh size increasesdramatically with the increase in N. To keep the meshsize limited and the numerical simulation reasonable,

translational periodic boundaries on the flow inlet andoutlet are necessary. Periodic boundaries can be en-forced on a packing by defining particle positions in aplane and forcing identical particle positions on the topand bottom of the packing, thus creating identicalboundaries.

During close examination of the N ) 4 packing, it wasfound that several packing structures developed, fromthe imposed planar particle organization at the bottomthrough a transition region to a repeating 3D structure.The repeating structure could be isolated and modeledwith translational periodic boundary conditions as arepresentative part of a continuing structure in aregular N ) 4 bed.

In low-N beds, the wall-induced packing structuredominates. The N ) 4 geometry showed axially repeat-ing alternating layers of nine spheres along the wall.In the central void created by the wall layer, weobserved a spiraling three-sphere arrangement. Thespiraling occurred to obtain identical layer spacing inthe central and wall structures. The wall layer sup-ported the central spiral and imposed its layer spacingthereupon, resulting in an axially repetitive structure.

The N ) 4 geometry was created with a coarse meshfor a laminar flow prestudy, which is shown in Figure6. The mesh for this model was too coarse to find anaccurate flow field, but it can give an indication of thelarge-scale flow structures, such as bypassing, reverse

Figure 4. Velocity vector field for N ) 2 at a sphere-spherecontact point in the x ) 0 plane at Re ) 1922; legend shows velocitymagnitude in meters per second.

Figure 5. Temperature contour plot for N ) 2 for section of entirebed in the x ) 0 plane for (a) Re ) 373, (b) Re ) 986, (c) Re )1922; legend shows temperature in Kelvin.


flow, and radial flow components. These are shown inFigure 7 for the flow field in a bed section parallel tothe overall axial flow direction for Re ) 180. Thetortuous nature of the flow as it moves past particle-particle contact points is clearly seen, as well as the

bypassing features near the wall and regions of backflowdownstream of the particles. These features are furtherreinforced in Figure 8, in which pathlines colored byaxial velocity components are shown; thus negativevalues can be seen, showing regions of backflow.

Discussion

The CFD results of the previous section show that thisapproach can rapidly generate a great deal of informa-tion. Even a steady-state CFD analysis without heattransfer yields a database of results that comprises thespatial coordinates (r, θ, and z) and the associatedvelocity components vr, vθ, and vz, for up to 500 000control volumes in the fluid. The addition of conductionthrough particles and temperature for each of thevolumes complicates the analysis of the data stillfurther. As computer speed and memory increase andthe size of CFD models increases along with them, theproblem of obtaining useful knowledge from a largedatabase of information will only get worse.

For fixed-bed modeling and design, we can considerdifferent uses of the data.

(i) The velocity components could be used directly ina three-dimensional heterogeneous model of the fixedbed. This would be specific to the particular packingused in the CFD simulation and would be of little helpin new reactor designs or in assessing variability fromone reactor tube to another. It would also result in anextremely computationally complex reactor model, inwhich the level of detail of the transport was incompat-ible with the level of knowledge of the reaction kinetics.

(ii) The minimum use of the CFD information wouldbe to average the axial velocity component over asuitable representative averaging volume (or area) toobtain vz(r) for insertion into eq 3. This use of the datawould have the advantage of providing information that

Figure 6. Geometry and mesh for fixed bed with N ) 4.

Figure 7. Velocity vector plot for N ) 4 for a six-layer sectionover the entire bed diameter in the x ) 0 plane at Re ) 180; legendshows velocity magnitude in meters per second.

Figure 8. Path lines for N ) 4 for a six-layer section over theentire bed diameter in the x ) 0 plane at Re ) 180; legend showsaxial velocity component in meters per second.


could be directly used in reactor simulations and thatwould be compatible with existing approaches. The vz(r)expression in eqs 1 and 2 could be used to estimate µeff,i.e., the CFD calculations could be used to provide aconstitutive relation for an averaged tube-scale descrip-tion. This use of the data disregards a great deal of thefine structure of the flow field, as two of the threevelocity components are discarded as being averagedout. This would not be our preferred approach, as webelieve that heat transfer and reactor performance arestrongly affected by the particle-scale features of theflow field.

(iii) An attractive approach to fixed-bed modeling isto move forward, away from the use of effective param-eters, but still retaining a general, structure-basedapproach to transport that would be useful for reactordesign. For example, Dixon et al.62 postulated thatradial transport of heat takes place by solid conductionand the radial displacement of flow around particles.They constructed a model of fixed-bed heat transfer thatemployed the true fluid and solid thermal conductivitiesof the phases and a two-dimensional velocity fieldcomposed of the components vz(r) and vr(r,z). Theoriginal work used a computer-generated packing,which was then tesselated and transformed into anetwork model. A pressure gradient was applied, andflows through the network branches were calculated.These flows were then averaged to obtain the velocitycomponents. The resulting model represented heattransfer through the bed center quite well but was notaccurate near the wall. This result was explained bythe fact that the flow channels parallel to the wall couldnot be included in the network and also by the fact that,during the averaging process, a “cancellation effect” wasobserved, in which strong radial flows to and from thewall were added to give no net flow, although the netheat transfer could have been significant. Nonetheless,the development of a model completely without effectiveparameters was a significant step toward more physi-cally based fixed-bed models.

CFD simulations could be used in conjunction withthe approach of Dixon et al.62 to replace the need for anetwork model and computed branch flows. One couldanticipate improved modeling of flows near the wall,using CFD. However, the choice of which velocitycomponents to retain and determination of their depen-dence on spatial coordinates was purely intuitive. Amore systematic approach is desired, in which impor-tant features of the flow could be identified and linkedto structural features of the bed, such as particle size,shape, and N. Searching through the myriad of flowpictures that could be generated by CFD postprocessingis becoming beyond human capability as models oflarger-N beds are created, especially as more featuresof the flows are internal to the bed. The efficientperformance of such searches will require tools from therealm of information sciences, such as pattern recogni-tion, classification, and feature extraction. The incor-poration of prior knowledge will be necessary to guidethese steps. The coupling of CFD to information tech-nology tools has the potential of making a strong impacton reactor design.

Conclusions

An approach to fluid flows in low tube-to-particlediameter fixed beds is needed that is general enough tobe applicable for reactor design purposes, but that

retains the detailed flow features that contribute totransport of heat and mass and to strong local gradientsthat could influence reaction kinetics. This approachshould provide a link to bed structure, so that, if apacking or packing characteristics are known, flowfeatures, transport rates, and reactor performance canbe rapidly assessed. Computational fluid dynamicssimulations of flow through packed-bed structures canprovide highly detailed and reliable information aboutthe temperature and flow fields. The challenge for thefuture is to use the information that will be availableto gain knowledge and understanding that will allowus to develop reduced models that are detailed enoughfor design purposes, but still intuitively understandableand computationally tractable. Some of the recent toolsof information technology will be of use in this endeavor.

Acknowledgment

The authors thank the DuPont educational fund fortheir financial support. Also, Fluent, Inc., is acknowl-edged for the university license of their CFD software.

Notation

cp ) fluid heat capacity [J/(kg K)]dp ) particle diameter (m)F, K ) constants defined in eq 2hw ) apparent wall heat-transfer coefficient [W/(m2 K)]keff ) effective radial thermal conductivity [W/(m K)]N ) tube-to-particle diameter ratio (dt/dp)P ) pressure (Pa)r ) radial coordinate (m)R ) tube radius (m)Re ) Reynolds number (Fvzdp/µ)T, Tw ) temperature and wall temperature, respectively

(K)vz ) axial gas velocity (m/s)V, Vf ) averaging volume and volume in fluid phase,

respectively (m3)x, y, z ) coordinates (m)

Greek Symbols

ε ) bed voidageF ) fluid density (kg/m3)µf ) fluid viscosity (N s/m2)µeff ) effective bed viscosity (N s/m2)θ ) angular coordinate

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Received for review December 7, 2000Revised manuscript received June 6, 2001

Accepted June 6, 2001

IE001035A


cfd as a design tool for fixed-bed reactors

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