A Coupled DEM and CFD Simulation of Flow Field and Pressure Drop in FixedBed Reactor with Randomly Packed Catalyst Particles

Hua Bai,* Jorg Theuerkauf, and Paul A. Gillis

The Dow Chemical Company, 2301 N. Brazosport BlVd., Freeport, Texas 77541

Paul M. Witt

The Dow Chemical Company, Midland, Michigan 48674

Packed bed unit operations are required for many commercial chemical processes. The ability to a prioripredict void fraction and pressure drop in a packed bed would significantly improve reactor design as wellas allow for optimization around catalyst performance, catalyst design, and the resulting process pressuredrop. Traditionally, the packed bed reactor designs are based on a homogeneous model with averaged empiricalcorrelations. These correlations are often inapplicable for low tube-to-particle diameter ratios (D/d < 4) inwhich tube wall and local phenomena dominate. In this work, the discrete element method (DEM) andcomputational fluid dynamics (CFD) are coupled to model a fixed bed reactor with low tube-to-particle diameterratios (D/d < 4). DEM is used to generate a realistic random packing structure for the packed bed withspherical or cylindrical particles, which is then imported into the CFD preprocessor (Gambit) to generate themesh for the CFD simulation. Two types of experiments were conducted: the laboratory-scale experimentswith up to ∼150 particles to allow simulation of entire packed beds in CFD, including random packing andstructured packing, and the plant-scale experiments conducted with up to ∼1500 randomly packed particles.The concept of a “porosity correction factor” was introduced to compensate for the effect of porosity deviationbetween actual packing and the CFD model, which can be accumulated during DEM simulation, and particleshrinkage for purposes of grid generation in the CFD model. The predicted pressure drops match well withthe experimental measurements with errors less than the desired limit (10%) for industrial design of packedbed reactors. The pressure drops calculated by the empirical correlations confirmed the inconsistency andunreliability of the empirical correlations for the packed beds with low tube-to-particle diameter ratios (D/d< 4) as well as the advantage of the DEM/CFD approach.

1. Introduction

Traditionally, packed bed reactors are designed by a trial-and-error process. The packing void fraction and the pressuredrop across the packed bed are two critical variables for thedesign, and they are usually predicted using empirical correla-tions, such as the Ergun equation for pressure drop1 and theLeva2 or Dixon3 equation for void fraction (packing porosity).A small error in the predicted void fraction can translate to asignificant error in pressure drop prediction. Because of thisinherent error in the pressure drop prediction, many packed bedsare oversized with a capacity safety factor. An alternate methodto handle the expected error is to conduct specific pilot planttests with representative particles. Given the packing materialand the correct tube diameter, the Ergun equation can be fittedto the test data by varying the Ergun equation constants as wellas the effective particle diameter and void fraction. If the actualtube-to-particle diameter (D/d) combination is used in the testrig, the predicted pressure drop from the modified Ergunequation can be sufficient. However, these tests are expensiveand require significant prework to investigate. Typical catalystsupports are alumina or silica which can break and degrade afterpouring. To minimize the effects of broken particles, the testsneed to be completed with fresh particles, which may requiredrums of catalyst to be tested. Because of this, there is asignificant limitation on the variations of catalyst shapes andsizes that can be reviewed for optimal performance.

The empirical correlations are usually based on homogeneousassumptions with averaged characteristics; thus they often donot apply to the packed beds with low D/d ratios where thetube wall local phenomena dominate. The local changes inporosity can lead to large variations in the predicted velocityprofile and therefore nonuniform head loss along the packedbed.4 Accurate prediction of local voidage is also important forpredicting heat transfer in packed beds, which is critical forstability analysis5,6 and reactor control. In order to reflect thelocal effects, a spatially resolving three-dimensional (3D) flowsimulation is needed. A first requirement for such a detailedsimulation is an appropriate 3D representation of the geometricstructure of the packing.

The advent in high power computing technology has resultedin significant advancements in fundamental modeling whichprovide an alternative approach, by coupling DEM (discreteelement method) and CFD (computational fluid dynamics)technologies to address this issue. DEM is used to generate arealistic packing structure for the packed bed. The simulatedpacking structure is then imported into the CFD preprocessorto generate a mesh for the CFD simulation. The flow throughthe space between particles in the packed bed is obtained bysolving the transport equations. This yields a very detailedsolution containing local values of all relevant variables suchas pressure, velocity, shear stress, turbulence properties, andtemperature. Such detailed information is of great importancein understanding the phenomena occurring in the packed bed.

The ability to a priori predict void fraction and pressure dropin a packed bed would significantly improve reactor design as

* To whom correspondence should be addressed. Tel.: (979) 238-7621. Fax: (979) 238-7463. E-mail: hbai@dow.com.

Ind. Eng. Chem. Res. 2009, 48, 4060–40744060

10.1021/ie801548h CCC: $40.75 2009 American Chemical SocietyPublished on Web 03/12/2009

well as allow for optimization around catalyst performance,catalyst design, and the resulting process pressure drop. In thiswork, the packing structure in cylindrical tubes randomly packedwith spherical or cylindrical particles and the flow through thepacking were modeled by coupling DEM and CFD. Two typesof experiments were conducted: laboratory-scale experimentswith less than 160 particles packed to allow modeling of entirepacked beds, including random packing and structured packing,and plant-scale experiments conducted with up to ∼ 1500randomly packed particles. The goal is to predict the packedbed pressure drop with an error of less than 10%. The workwas focused on the packed beds with low tube-to-particlediameter ratio (D/d < 4) where empirical correlations oftenbecome unreliable.

2. DEM and CFD

DEM7 is an explicit numerical scheme which simulates thedynamic and static behavior of assemblies of particles basedon contact mechanics. A soft particle model is used in the DEMto simulate the particle contact behavior with springs, dashpots,and frictional sliders. The motion of each particle is tracked,and interaction with other particles or boundaries is considered.Forces are calculated based on interaction of particles and thephysical properties of the entities, including the hardness ofparticles, expressed with a spring, and the particle energydissipation, expressed with a dampener or dashpot. The hardnessof the particle is proportional to the Young’s modulus, whilethe dashpot is related to the coefficient of restitution. The frictionbetween entities is defined with a Coulombic type of frictionand implemented with a friction factor. Based on the physicalproperties the particle forces are calculated, which leads to newparticle positions.7,8 The commercial DEM software packagePFC3D by ITASCA9 was used in this work.

Once the packing structure of the packed bed is predictedwith DEM, it is then incorporated into CFD to simulate theflow through the bed. Using CFD to model the detailed flow inpacked beds has gained increasing attention in the past few

years.10-15 In these published works, usually only a smallnumber (a few to tens) of particles were modeled to reduce therequirements for computational resources, and the particles werepacked in special patterns such as regular or periodic packing(also called “structured packing”) so that the packing structurescan be directly reconstructed in CFD without DEM or otherpacking predictions. These works provided some insights intohow the packing structure influences the transport characteristics.However, they are not applicable directly to industrial applica-tions, where hundreds and thousands of particles are randomlypacked typically. The lattice Boltzmann method has also beenused to compute the flow through a packed bed of spheres.16-18

It has a potentially high efficiency which allows somewhat largerpacking to be simulated, but it is still in its early stage and hassome limitations such as difficult to handle energy balance.11

3. Experiments

Two types of experiments were conducted. One is thelaboratory-scale experiment in which less than 160 particleswere packed to allow modeling of the entire packed beds inCFD, including random packing and structured packing. Withthe structured packing simulation, particles are positioned inspecific patterns and the CFD geometry can be directly builtup without DEM simulation. This experiment is mainly for thepurpose of validating the CFD model. The other type is themore realistic plant-scale experiment in which up to thousandsof particles were randomly packed but only a segment of a entirepacked bed could be included in the CFD simulations due tothe limit in available computational resources. The plant-scaleexperiment provided data for investigation of pressure dropscale-up from a segment to the corresponding entire packed bed.

3.1. Experimental Setup. Figure 1 shows a photograph (a)and sketch (b) of the laboratory setup for the laboratory-scaleexperiment. The tubes for these experiments were made ofacrylic so that the structure of the packing could be visualized.The air flow, directed downward through the packing, wascontrolled by a manual valve and measured by an Aquamatic

Figure 1. Photograph (a) and schematic (b) of the laboratory experiment setup.

Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 4061

AT21052 flow cell rotometer. A water manometer was used tomeasure pressure drop across the packing. The manometermeasuring locations were more than 2 times the tube diameterfrom the packing to ensure stable manometer readings. The backpressure (or the pressure at the packing top) was also measuredalong with the pressure drop measurement. Two kinds ofparticles were used in the laboratory-scale experiment: 10 mmdiameter steel balls for the packing with spherical particles andacrylic cylinders with a diameter of 10 mm and a length of 10mm for the packing with cylindrical particles. The packing wassupported by three pins that passed through the cross sectionof the column. This design was chosen to minimize pressuredrop from the packing support. The plant-scale experiment hada setup similar to that in Figure 1 except that pressurized air(1.72 bar) was used and the packing tubes were constructed ofsteel; therefore, the packing structures were not visible.

3.2. Experimental Cases and Data. Table 1 summarizes thekey parameters and measurements of both laboratory-scale andplant-scale experiments, including the tube diameter (D), particlediameter (d), and length of the cylindrical particles (h), the totalnumber of particles in packing (N), the packing height (H), theflow rate (Q), and the measured pressure drop (∆P). Each caseor setup was tested under three or four different flow rates. Foreach flow rate, the test was repeated at least three times toquantify experimental precision. The pressure drop shown inTable 1 is the average of the repeats. All experiments wereconducted at ambient temperature (∼25 °C).

The first three cases (LS-1, LS-2, and LS-3) in Table 1 arethe laboratory-scale experiments using structured packing ofspherical particles. The square duct, with internal dimensionsof 20 mm by 20 mm, allows four 10 mm steel balls to bepositioned in a simple cubic lattice structure, giving a tube-to-particle dimension ratio of 2.0. The three cases have 16, 32,and 64 particles, corresponding to 4, 8, and 16 packing layers.All layers have the same four-ball packing. This structuredpacking can be built in the CFD model without DEM simulationsince all particles’ positions are known. These three structuredpacking cases were used to verify the CFD model since the

errors due to randomness of packing, DEM simulation, and useof a segment of entire packed bed were eliminated.

The next case in Table 1 (LU-1) is a packed bed with 153steel balls randomly packed in a 26.67 mm diameter tube, givinga tube-to-particle diameter ratio of 2.67. The relatively smallnumber (153) of particles allowed the entire packed bed to besimulated in CFD so that the effect of using only a segment ofentire bed could be eliminated. The last case (LU-2) of thelaboratory-scale experiments is a packed bed with 82 cylindricalparticles randomly packed in a tube with an inner diameter of20.51 mm. Each cylindrical particle has 10 mm diameter and10 mm length, and its effective particle diameter, dp, definedas the diameter of a sphere with the same volume as the particle,is 11.45 mm, giving a tube-to-particle diameter ratio of 1.79.The entire packed bed (LU-2) was included in the CFDsimulation.

The last three cases (PU-1, PU-2, and PU-3) in Table 1 are theplant-scale experiment with hundreds of (up to 1545) steel balls(10 mm diameter) randomly packed in three different steel tubesof diameters of 21.41, 26.12, and 32.28 mm, giving tube-to-particlediameter ratios of 2.14, 2.61, and 3.23, respectively. The largenumbers (799, 916, and 1545) of particles, more realisticallyreflecting the plant operation conditions, were beyond the capabilitylimit of the available computational resources. Therefore, only asegment of the entire packed bed with 125 particles was includedin the corresponding CFD simulation.

3.3. Packed Bed and Packing Porosity. Figure 2 shows afew examples of packed beds investigated in this paper. Figure2a shows a packed bed with structured packing of sphericalparticles. Figure 2b shows two packed beds with randompacking of spherical particles,19,20 and Figure 2c shows two fixedbeds packed with random packing of cylindrical particles.21

Marked in Figure 2 are the tube inner diameter (D), the heightof the packed bed (H), the particle diameter (d), and the heightof cylindrical particles (h). The packed bed porosity, ε, is definedas the fraction of the void volume within the packing height.For the packed beds with spherical particles, the packingporosity can be expressed as

Table 1. Experimental Cases and Measurements

case labelpacked bed tubediam (D), mm D/d

no. particlespacked (N)

packing height(H), mm

flow rate (Q),kg/h

measd press.drop (∆P), Pa particles/packing/scale

LS-1 square duct, 20 × 20 2.00 16 40 10.41 862 steel balls/structured/lab scale17.69 2 48220.81 3 51626.43 5 723

LS-2 square duct, 20 × 20 32 80 10.41 1 986 steel balls/structured/lab scale17.69 4 96420.81 6 89526.43 11 238

LS-3 square duct, 20 × 20 64 160 10.41 3 123 steel balls/structured/lab scale17.69 9 51520.81 14 54826.43 24 270

LU-1 26.67 2.67 153 312.7 10.41 343 steel balls/random/lab scale17.69 97720.81 1 39826.43 2 285

LU-2 20.51 1.79 82 444.7 10.41 5 594 cylinders/random/lab scale16.65 16 13820.81 26 242

PU-1 21.41 2.14 799 1722.1 3.30 1 686 steel balls/random/plant scale6.59 6 4619.89 14 178

PU-2 26.12 2.61 916 2319.0 3.65 884 steel balls/random/plant scale7.31 3 491

10.96 7 593PU-3 32.28 3.23 1545 1742.4 5.75 1 581 steel balls/random/plant scale

11.49 5 96517.24 12 945

4062 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009

For the packed beds with cylindrical particles, the packingporosity can be expressed as

where N is the total number of particles in the packing.

4. Models and Methodology

The general procedure to model the flow field and pressuredrop in a packed bed is briefly described below. First, thepacking structure in a packed bed is simulated using PFC3D, acommercial DEM package.9 Then, the DEM-simulated packingis imported into Gambit,22 a commercial CFD preprocessor tocreate the geometry and mesh for CFD simulation. If a CFDsimulation of the entire packed bed becomes unmanageable orbeyond the limit of available computational resources (commonfor commercial-scale packed beds), a representative segmentfrom the entire packed bed is selected for the mesh generation.Next, the flow field through the packing is solved usingFLUENT,23 a commercial CFD package. Finally, the pressuredrop across the packing is retrieved from the CFD simulationresult. If only a segment of the entire packed bed is modeled,the CFD-predicted pressured drop needs to be scaled up for theentire packed bed.

4.1. Packed Bed Simulation with DEM. For the DEMsimulation with spherical particles, the packing is generated bydropping spheres from a specified height into a tube. The launchposition of the spheres at the specified height is randomlyassigned in the tube to mimic the real filling procedure of thetube with particles. Figure 3a illustrates the filling process, whilecross sections are shown in Figure 3b for D/d ) 2.6. Localdifferences in the packing can be observed. Further, the DEMpacking (Figure 3c) is compared with the steel balls in aPlexiglas pipe (Figure 3d). Another example of particle packingis depicted for a D/d ) 2.2 in Figure 3e,f. The DEM result isa view from the top, while the experimental result is a sideview. The packing structure is in very good agreement.

For simulation with cylindrical particles, it is more compli-cated since the DEM code (PFC3D) allows only spheres as thebasic particle geometry. If a shape other than a sphere isrequired, it has to be built by assembling spheres. With thespheres the cylinders are generated by only placing particleson the surface of the cylinder which is defined by the diameterand number of particles on the surface. The number of particleson the surface has an impact on the roughness. Since the spheresare staggered on the surface, “valleys” and “hills” can be found.However, the smoothness of the surface can be adjusted by therelative positions of the spheres and their diameters. A smoothersurface can be achieved by more spheres with smaller diametersfor the composition of one cylinder. This adds to the numericalcomplexity of the packing simulation. During the packingprocess of the cylinders in the tube, the surface structure canbe important if the resolution of the particles is not high; e.g.,

Figure 2. Examples of packed beds with low tube-to-particle diameter ratios (D/d): (a) structured packing with spherical particles, (b) random packing withspherical particles, and (c) random packing with cylindrical particles.

ε ) 1 - 2Nd3

3D2H(1)

ε ) 1 - Nd2h

D2H(2)

Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 4063

few spheres lead to a rough surface. If two cylinders are incontact and they interlock due to surface resolution, frictionaleffects will be masked. Further, the CFD mesh generation forsuch intersecting cylinders is difficult. A sensitivity study wascarried out to determine how many single spheres are requiredto avoid influence on the packing. The number of particles usedfor each cylinder depends on the size of the particle that is tobe represented in DEM without introducing numerical effectsthat influence the packing. For the cylinder packing case (LU-2) in this work, each cylinder was created with 1000 spheres,which is computationally manageable but fine enough to notsignificantly influence the packing. A typical DEM cylindricalparticle is shown in Figure 4a. By default the PFC3D code is asingle phase simulation tool; thus it does not use drag orbuoyancy forces on particles falling though the air. To minimizethe computational time to fill a tube, cylindrical particles werenot dropped from the very top of the tube but were generatedat defined heights in the tube. Based on the vertical location ofa cylinder, the velocity of the cylinder was calculated based onpotential and kinetic energies. It was ensured that the terminalsettling velocity of the particles was not exceeded. Thisprocedure ensures a correct starting velocity which affects theinteraction of the cylinder at impact on the packing. Dependingon the packing, the impact of falling particles can affect thestructure of the packing. To mimic the effect that the particlesrotate while they are falling in the tube, the cylinders are rotatedrandomly before they fall in the tube. The initial rotation of thecylinders prevents a preferential orientation and packing struc-ture of the cylinders. Figure 4b shows the filling of the tube.Once the cylinders fill the pipe, the next cylinder’s initial releaselocation is moved upward. Figure 4c is a close-up of a sectionof the DEM-simulated packing. It can be seen that local packingstructure varies based on the vertical position. The structure ofthe packing in the tube changes locally based on geometry andthe way the cylinders interact during the filling process.

In addition to the geometric effects, the friction parametersalso influence the packing. For the DEM simulations theparticle-particle friction as well as the particle-wall frictionwere considered. This effect was studied with an experimentaldesign where the geometries of the pipe and cylinder were kept

constant but the friction factors were varied between 0 and 1.The results reveal that the wall friction has a higher impact onthe porosity than the ball friction. Based on the observationsthat the selection of the simulation parameters affects thepacking structure, here expressed with the overall packingporosity, experiments are required to calibrate the simulationparameters.24

4.2. Geometry and Mesh Generation. A DEM-simulatedpacking geometry is exported into a text file of multiple columnscontaining the information regarding dimensions and positionsof all particles in the packing. Current computational powerallows DEM simulation of a entire packed bed containingthousands of particles. However, the number of particles thatcan be modeled in CFD is significantly less and is limited byavailable computational resources. For a typical commercial-scale packed bed such as the plant-scale experiment conductedin this work, only a small section of the entire packed bed isincluded in the CFD model to fit the available computationalresources. The chosen section should have a packing porositythat is the closest to that in the corresponding entire packedbed.

Particles contact each other, and some also contact the tubewall. At these points of contact, mesh cells become highlyskewed, which results in a poor-quality mesh and often causesa convergence problem in CFD simulation. To avoid thisproblem, each particle is slightly shrunk in size but the particleposition remains unchanged during the geometry buildup.The particle shrinkage used in this work was 0.5% for sphericalparticles and 1% for cylindrical particles. Correspondingly, thespherical particle diameter used in the CFD model is 99.5% ofthe actual ball diameter and the cylindrical particle diameterused in the CFD model is 99% of the actual particle diameter.The length of the cylindrical particle is also shrunk by 1%.Shrinking particles in the CFD model is a common approachadopted in published works10-15 using body-fitted mesh for thepacked bed. Typically, 1% shrinkage was used in thosepublications. The particle shrinkage directly affects the packingporosity (voidage) and therefore the pressure drop, and this effectis further discussed later in this paper.

Figure 3. Fixed beds with low tube-to-particle diameter ratios, randomly packed with spherical particles (DEM simulation and photos of experimentalpacking)

4064 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009

Tetrahedral elements were used to generate unstructured meshfor the packed bed, and the mesh size can be characterized bythe edge length of a tetrahedral element. Small mesh size isoften required not only for accommodating complex packinggeometry but also for avoiding grid generation problems,especially when the shrinkage is very small. For example, with1% particle shrinkage, the mesh size can be as large as 1 mm(10% particle diameter) for 10 mm particles. However, if theparticle shrinkage is reduced to 0.5%, the mesh size has to be0.7 mm or less in order to avoid failure in the grid generation.

Empty sections are added to each end of the packed bed, asshown in Figure 2. This is necessary in order to minimize theeffect of boundary conditions specified the inlet and outlet ofthe tube in CFD. The mass flow rate specified at the inlet (topof the modeling domain) is internally converted into a uniformvelocity at the inlet boundary. The flow velocities start to varysignificantly within the packing. The pressure-outlet boundaryspecifies a uniform pressure at the outlet (bottom of the modelingdomain). The empty sections allow flow to naturally developas approaching or leaving the packing. Parametric study showedthat the length of the empty sections should be at least one tubediameter (D). It was also found that there was no noticeabledifference in the pressure drop predictions if longer emptysections were used.

Scripts have been developed to automate the tedious processof generating the packing geometry and mesh described abovefor spherical particles and cylindrical particles separately. The

script reads in the data file exported from DEM simulation andconverts the packing data into a Gambit journal file. Runningthe journal file in Gambit generates a mesh for CFD simulation.Mesh size and particle shrinkage can be adjusted either in thescript or in the Gambit journal file if needed.

4.3. Flow Field Simulation with CFD. The three-dimen-sional steady-state flow field through the packed bed is simulatedby solving the Reynolds averaged Navier-Stokes (RANS)equations and the mass conservation equation. The commercialCFD package FLUENT 6.1 was used in this work. For theinterested gas flow range, flow in the packed bed channel isturbulent with the tube Reynolds number ranging from 2000 to20 000. The renormalization group (RNG) k-ε model25 ischosen for the turbulence closure. Different turbulence modelswere examined to understand their effects on the accuracy ofpressure drop prediction, including the standard k-ε model, therealizable k-ε model, RNG k-ε model, the Spalart-Allmaras model, the standard and SST k-ω model, and theReynolds stress model (RSM). These models are available andbuilt in FLUENT 6.1. Detailed descriptions of each model canbe found in the FLUENT 6.1 User’s Manual;23 thus they arenot repeated here. The turbulence model study was conductedwith the same packed bed and flow conditions so that theturbulence model was the only difference. The study shows thatthere is no significant difference among various k-ε and k-ωmodels with variations of pressure drop of less than 4%. The

Figure 4. DEM simulation of a fixed bed with randomly packed cylindrical particles: (a) a cylindrical DEM particle assembled with balls, (b) packing ofthe tube at different stages in time, and (c) close-up of a section of the DEM-simulated packing.

Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 4065

Spalart-Allmaras model predicted the lowest pressure drop andthe largest discrepancy from the measurement. The RSM modelpredicted the largest pressure drop that was the closest to themeasurement. However, the simulation took significantly (2-3times) more CPU time and the predicted pressure drop was lessthan 3% larger than the selected RNG k-ε models. The RNGk-ε model25 was developed for the flows featuring strongstream-line curvature, separation, vortices, and boundary layersunder strong adverse pressure gradients; thus it was considereda proper choice for the packed bed applications.

The turbulent flow field through a packed bed is dominatedby fluid-surface interactions. These surfaces include both theparticles and the tube walls which are prevalent throughout thebed. The k-ε models are primarily valid for turbulent core flows(i.e., the flow in the regions somewhat far from walls). Veryclose to the wall, viscous damping reduces the tangentialvelocity fluctuations, while kinematic blocking reduces thenormal fluctuations. Toward the outer part of the near-wallregion (boundary layer), however, the turbulence is rapidlyaugmented by the production of turbulence kinetic energy dueto the large gradients in mean velocity. Directly resolving thenear-wall velocity profile requires very high mesh resolutionfor the boundary layer with y+ close to 1 (y+ is the dimensionlessnormal distance from the wall-adjacent cell to the wall). Thishigh resolution requirement can significantly increase thenumber of mesh elements for the CFD model to the level beyondthe limit of available computational resources. Otherwise, thenumber of particles included in CFD has to be significantlyreduced, such as less than a dozen,26,27 which would be farinsufficient to represent the packing characteristics of a randomlypacked bed for a pressure drop study. In this work, the standardwall functions28 were used for the near-wall modeling, whichallows relatively larger near-wall mesh elements; thus sufficientparticles were included in the CFD simulation. With the standardwall functions, the viscous sublayer and buffer layer are notresolved. Instead, semiempirical formulas called “wall functions”

are used to bridge the viscosity-affected region between the walland the fully turbulent region. During the CFD simulations, y+

on the particles and tube wall were checked to make sure thaty+ is within the applicable range: 30 < y+ < 250. Local gridrefinements were conducted to refine boundary layer mesh forthe surface region with y+ > 250.

The second-order UPWIND discretization scheme23 was usedfor all equations. The air density is calculated by the ideal gaslaw at room temperature (25 °C). Details of model formulationscan be found in the FLUENT 6 User’s Manual;23 thus they arenot repeated here. Figure 5 shows a typical CFD simulationresult for a segment of packed bed, including the flow field,represented by the flow path lines (Figure 5a), the pressurecontours on particles as well as on the tube wall (Figure 5b),the velocity magnitudes on the central plane across the packedbed (Figure 5c), and the pressure profile on the same centralplane (Figure 5d). Each point on the pressure profile plotcorresponds to one spatial position in the packing. The twoalmost flat lines at both ends correspond to the empty sectionsof the flow channel, indicating little pressure change in the non-packing zones (the empty sections). These local pressurevariations correspond to the local velocity variations throughthe packing structure.

4.4. Pressure Drop across Packed Bed. The pressure dropis simply the difference of average pressures at the inlet, P1,and outlet, P2, as illustrated in Figure 5d. This is the pressuredrop across the packed bed if the entire bed is simulated inCFD. However, if only a segment of the entire packed bed isincluded in CFD, the predicted pressure drop must be scaledfor the entire packed bed. Two linear scaling approaches wereexamined: scaling by packed bed height (H) and scaling by thenumber of particles (N). Both approaches gave similar results,but the latter was more robust and consistent; it was thus usedin this paper. Scaling by the particle number was also foundmore accurate when the number of packing layers (H/d) in a

Figure 5. Typical CFD simulation results: (a) flow pattern shown as flow path lines, (b) pressure contours on particles and tube wall, (c) velocity on centralplane, and (d) pressure profile across the packed bed central plane.

4066 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009

CFD simulation was small. The pressure drop for the entirepacked bed, ∆P, can be expressed as follows:

where N is the number of particles in the entire packed bed andNS is the total number of particles in the segment included inthe CFD simulation. In chemical plants, an accurate number ofparticles in a packed bed is often hard to obtain but the packingheight information can be available. In this case, scaling canbe done using packed bed height. The fundamental reason forthe linear scaling of pressure drop is explained from the typicalpressure profile shown in Figure 5d. The average pressure dropsalmost linearly along the packing height even though localpressure can be noticeably higher or lower than the average.

4.5. Porosity Deviation between CFD Model and PackedBed. Porosity for a packed bed can be calculated from eq 1 or2 for an actual packing (experiment) or a DEM-simulatedpacking. In CFD, the packing porosity can be easily obtainedfrom the volume of fluid zone and the tube volume. Since thepressure drop is very sensitive to packing porosity, the porositydeviation between the packing in the CFD model and actualpacking can greatly affect the accuracy of model prediction.The sources of the deviation are (1) DEM simulation, (2) onlya segment of the entire packed bed included in CFD, and (3)particle shrinkage in the CFD model.

4.5.1. DEM Simulation. The nature of packing randomnessnaturally reflects as the difference between an actual orexperimental packing and corresponding DEM simulation. Inaddition, the parameters in the DEM simulation, such as theparticle dropping position, the initial dropping velocity, and thefriction factor, can also affect the simulation result.24 Overall,it has been found that DEM is capable of simulating the packedbeds of spherical or cylindrical particles with porosity deviationless than 3%. DEM-simulated packing may have a larger orsmaller porosity than the actual packing. For all experimentalcases listed in Table 1, the DEM simulations were done forwhole packed beds.

4.5.2. Selected Segment of Entire Packed Bed. In mostindustrial applications where thousands of particles are packed,only a segment of the entire packed bed can be afforded to beincluded in CFD due to the limit of available computational

resources. The packing porosity of a selected segment can belarger or smaller than the corresponding entire bed, dependingon where the segment is selected and how many particles areincluded in the selected segment. Figure 6 illustrates how theporosity of the selected segments deviates from the entire packedbed when the numbers of particles in the segments vary. Thesedata are from the plant-scale experiments with spherical particlesand the corresponding DEM simulations. Generally speaking,the deviations decrease as the number of particles included inthe selected segments increases. There are also small-scale (upand down) periodic variations in the porosity deviation as thenumber of particles changes, which can be more clearly seenin the zoomed-in plot in Figure 6b. These periodic variationscorrespond to the layer changes in the packing. The porositydeviation suffers a small jump every time an additional particleconstitutes a new layer of packing, and then decreases as moreparticles are added to the same layer. The segments with zeroporosity deviation from the entire packed bed have minimum159, 800, and 1182 particles for three different D/d ratios 2.14,2.61, and 3.23, respectively. As the particle number furtherincreases, the segment porosity can be either larger or smallerthan the entire bed, but the deviation is relatively small (lessthan 0.5%). The larger the D/d ratio is, the more particles areneeded to reach this low level deviation of packing porosity.This is because sufficient layers are required to represent theentire packed bed and a packing with a larger D/d ratio containsmore particles per layer.

Obviously, the packing porosity of a segment also varieswithin the selected segment. A recommended practice is toinclude as many particles as possible in the segment and selectthe segment position to have minimum porosity deviation fromthe entire packed bed. In this paper, for the three plant-scaleexperimental packed beds where up to 1500+ spherical particleswere packed, only 125 particles were included in selectedsegments in the CFD simulations. Compared to the minimumnumber of particles required to have close-to-zero deviation fromthe entire packed bed (Figure 6), this number (125) is close forthe bed with the lowest D/d ratio (2.14) but insufficient for theother two beds with larger D/d ratios (2.61 and 3.23).

4.5.3. Particle Shrinkage. As discussed earlier, particleshrinkage is necessary to avoid a highly skewed CFD meshwhich can cause failure in mesh generation or a serious

Figure 6. Effect of number of particles included in selected segment on packing porosity deviation between the segment and the corresponding entire packedbed.

∆P ) ∆PSNNS

(3)

Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 4067

convergence problem in the CFD simulation. Because theparticle volume is proportional to its diameter cubed, a slightshrinkage in the particle diameter results in significant changesin the packing voidage. Figure 7 shows the packing porositydeviation versus the particle shrinkage for the three plant-scaleexperimental cases, based on eq 1. The particle shrinkage, η, isdefined as the relative change between the particle diameter usedin the CFD model (dCFD) and the actual particle diameter, d:

The packing porosity deviation percentage is linearly pro-portional to the particle shrinkage for each case. With 1%particle shrinkage, the packing porosity increases about 3%.With 2% particle shrinkage, the packing porosity can increaseby more than 6%. Even for 0.5% particle shrinkage, the packingporosity deviation is about 1.6%.

The pressure drop across the packed bed is known to be verysensitive to the packing porosity. Figure 8 shows how theporosity deviation may affect the pressure drop deviation, basedon the Ergun equation1 for pressure drops and other correlationsfor packing porosity as discussed later in this section. A 10%porosity deviation can transfer into 30% error in pressure drop.In order to control the error for the pressure drop of less than10% (required for packed bed reactor design), the porositydeviation should be less than 3%, which would require theparticle shrinkage to be less than 1% even without additionalporosity deviation due to the DEM simulation or to use asegment of the entire packed bed. On the other hand, the smallerthe particle shrinkage is, the smaller mesh size is required togenerate the CFD mesh, which would lead to a significantincrease in the total number of CFD mesh elements. Forexample, the CFD mesh with 0.5% shrinkage generated (usingmesh size 0.7 mm) generally has 2-3 times more elements thanthose with 1% shrinkage using mesh size 1 mm. The better and

more accurate representation of the packing in the CFD modelcomes with increased cost of computational resources.

4.6. Compensation for Porosity Deviation. The packingporosity deviation between CFD and experiment is accumulatedduring the processes of DEM simulation, use of a selectedsegment of entire packed bed in CFD model, and particleshrinkage. Each of these processes is an inherent part of theDEM/CFD methodology, and the porosity deviation cannotbe avoided in the DEM simulation and particle shrinkage. Aparticle shrinkage of 1% alone would cause 3% porositydeviation which may lead to ∼10% error in the pressure dropprediction. Therefore, it is necessary to compensate the effectof the porosity deviation in order to meet the prediction accuracygoal (error <10%).

Calculating the porosity deviation is straightforward, asdefined in the following equation:

where ε is the porosity of an actual packing using eq 1 or 2.εCFD is the packing porosity in the CFD model, which can beeasily calculated within CFD from the volume integrals of allfluid cells and the packing height in the CFD model.

From the Ergun equation, the pressure drop can be related tothe porosity deviation, as shown in Figure 8. The concept of a“porosity correction factor” was introduced to use this informa-tion to compensate the effect of the porosity deviation. Thecorrected pressure drop, ∆PC, is related to the CFD-predictedpressure drop, ∆P, by

where k is the porosity correction factor, which is calculatedby

where y is the relative pressure change responding to the packingporosity deviation σ (in %). The correlation between y and σ isderived by curve-fitting the plots in Figure 8. When there is noporosity deviation between the CFD model and the actualpacking (σ ) 0%), the porosity correction factor k ) 1, or therewill be no correction for the pressure drop prediction. If thepacking in CFD is 3% larger than the actual (σ ) 3%), thenk ) 1.113, or the porosity correction factor would correct thepredicted pressure drop by increasing it by 11.3%.

In the design phase for a packed bed reactor, the pressuredrop may need to be predicted but no actual packing is availableto be used as the basis to quantify the packing porosity deviation.In this case, DEM simulations of the entire packed bed can beused as the basis instead.

The porosity correction factor in eq 7 is derived from thewidely used Ergun-based correlations (details in the nextsection), and its accuracy can be improved with improved Erguncorrelations. It is widely accepted that the Ergun equationprovides a proper platform to correlate pressure drops across apacked bed with packing void fraction and flow condition. Theefforts to improve predictions have been mostly focused onmodifying the void fraction correlations and on recalibrationof the Ergun constants (a and b in the Ergun equation) forspecific applications. On the other hand, DEM and CFD canalso be used to improve the Ergun equation especially for apacked bed with a low tube-to-particle diameter ratio. Forexample, the void fraction of a packed bed can be obtained

Figure 7. Packing porosity deviation varying with particle shrinkage.

Figure 8. Effect of packing porosity deviation on pressure drop deviation.

η )d - dCFD

d(4)

σ )εCFD - ε

ε(5)

∆PC ) k(∆P) (6)

k ) 11 - y

) 1

1 - 3.385σ + 0.045σ2(7)

4068 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009

directly from a DEM simulation or new void fraction correla-tions can be derived from the data generated from massive DEMsimulations for interested applications. The pressure dropsobtained from the DEM/CFD approach can be used to recali-brate the Ergun constants. The “improved Ergun equation” canthen be used to improve the porosity correction factor. Thisprocess can be iterated until the desired accuracy is achieved.In this paper, the porosity correction factor was obtained simplyfrom the existing empirical correlations detailed in the nextsection, with which the predicted pressure drops were foundwithin the desired error limit (10%).

4.7. Empirical Correlations. For comparison purposes, theempirical correlations for predicting the pressure drop and voidfraction of a packed bed are presented below. The most familiarcorrelation for pressure drop through a packed bed is the Ergunequation.1

The Ergun equation consists of two terms, corresponding tothe Ergun constants a and b. The first term is nominallyconsidered the turbulent term, representing the contribution topressure drop from expansion and contractions. The second term,loosely called the laminar term, represents contributions fromform drag. According to Ergun, the advantage of this equationwas that void fraction sufficiently characterized tube packing.This allows the parameters a and b to be universal, dimension-less constants (a ) 1.75, b ) 150).1 Ergun further argued thatthe parameters a and b were independent of catalyst particlegeometry and tube size. However, all of Ergun’s data1 wereobtained with large tube-to-particle diameter ratios, D/d > 10.Much research has been conducted to identify more accurateErgun constants a and b for different particle geometries andparticle-to-tube diameter ratios. A widely used set of Ergunconstants is from Handley and Heggs:29

For the void fraction of a packed bed, the most widely usedcorrelations are probably Dixon’s correlations: for sphericalparticles (eqs 11a-11c)

and for cylindrical particles (eqs 12a-12c):

In eqs 12a-12c, dp, the effective particle diameter, is definedas the diameter of a sphere with the same volume as the particle,that is

The correlations in eqs 8-13 are used to calculate the pressuredrops for the cases in Table 1 except the first three cases, forwhich the square duct tubes were used and the correlations onlyapply to cylindrical tubes. The calculation results are comparedwith the predictions using the DEM/CFD simulations as wellas the measurements.

5. Results and Discussion

5.1. Structured Packing and CFD Model Validation. Theexperimentally measured pressure drops in Table 1 are plottedtogether with the corresponding CFD predictions for a directcomparison. Figure 9 shows the results for the three laboratory-scale cases with structured packing of spherical particles (LS-1, LS-2, and LS-3). The entire packed beds were directlymodeled in the CFD without DEM for the structured packingseen in Figure 9. The only difference between the CFD packingand the experimental packing is that the particle diameter inCFD was shrunk by 0.5%. The predicted pressure drops matchwell the experimental measurements, although the CFD predic-tions are consistently lower than the measurements. The averageerror (3.1%) is within the desired range (10%). This goodagreement between the predictions and measurements validatesthe CFD model. Also shown in Figure 9 are the CFD predictionswith the porosity correction factor k to compensate the porositydeviation due to the particle shrinkage (marked as “CFD-k”).The packing porosity for the experimental packing with 10 mmspheres is 0.476. For the CFD packing with the shrunken particlediameter d ) 9.95 mm, the porosity becomes 0.484, which is1.68% larger than the actual packing. It is this slightly largerporosity that caused the consistently lower predicted pressuredrops than the measurements. By including the porosity cor-rection factor, the model prediction errors are reduced to lessthan 2%.

5.2. Random Packing of Spherical Particles (Labora-tory Scale). Figure 10 shows the results for the laboratory-scale case with random packing of 153 spherical particles (LU-1). The entire packed bed, simulated by DEM, was included inthe CFD simulation. The particle diameter in the CFD simulationwas shrunk by 0.5% to 9.95 mm. The predicted pressure drops,as shown in Figure 10, are noticeably lower than the measure-ments with average error of 13.7%, which is beyond the desirederror limit (10%). This discrepancy is mainly caused by twofactors: the DEM simulation and the particle shrinkage. TheDEM-predicted packed bed height is about 1.8% taller than themeasured packing height, which makes the DEM-simulatedpacking looser than the actual packing. Combined with theparticle shrinkage, the packing porosity deviation between theexperimental packing and the CFD model is 3.52%. Applyingthe porosity correction factor, the model predictions match theexperimental data with an average error less than 4%. Alsoshown in Figure 10 are the pressure drops calculated using theempirical correlations (eqs 8-11c), which significantly over-predicted the pressure drops with an average error of 61.5%.

5.3. Random Packing of Cylindrical Particles (Labora-tory Scale). For the laboratory-scale case with 82 randomlypacked cylindrical particles (LU-2), the entire packed bedsimulated by DEM was included in the CFD simulations. Figure11 shows the flow field in the packed bed corresponding to themedian flow condition (16.65 kg/h). The flow field is illustratedby the flow path lines (Figure 11a) and the velocity vectors onthe central plane (Figure 11b). The flow path lines are the tra-jectories of tracers released at the top inlet and colored by initial

∆P ) Po - (Po2 - 2000HG2RT(1 - ε)

Mwdpgε3 (a + b1 - ε

Re ))0.5

(8)

a ) 1.248 and b ) 368 for spherical particles (9)

a ) 1.28 and b ) 458 for cylindrical particles(10)

ε ) 0.4 + 0.05(d/D) + 0.412(d/D)2 if d/D e 0.5 (11a)

ε ) 0.528 + 2.464(d/D - 0.5) if 0.5 e d/D e 0.536(11b)

ε ) 1 - 0.667(d/D)3(2d/D - 1)-0.5 if d/D g 0.536 (11c)

ε ) 0.36 + 0.1(dp/D) + 0.7(dp/D)2 if d/D e 0.6 (12a)

ε ) 0.677 - 9(dp/D - 0.625)2 if 0.6 e d/D e 0.7 (12b)

ε ) 1 - 0.763(dp/D)2 if d/D g 0.7 (12c)

dp ) √3 (3/2)d2h for cylindrical particles (13)

Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 4069

locations, thus providing a three-dimensional overview of thegas flow through the packed bed. The velocity vectors can revealthe flow structures, such as eddies behind/between the particles,and local channeling. The local flow velocity in the packed bedcan be a few times higher than the superficial flow velocity inthe packed bed. Figure 11c shows the pressure contours on theparticles and tube wall, depicting local pressures and gradualvariations through the packed bed.

The CFD-predicted pressure drops across the packed bed areplotted in Figure 12 to compare with the measurements as wellas the calculations with the empirical correlations (eqs 8-13).The CFD predictions (marked as “CFD” in Figure 12) areconsistently lower than the measurements. The average deviationis 8.3%. Also shown in Figure 12 are the pressure dropscorrected by the porosity correction factor (marked as “CFD-k”). The DEM-simulated packing, calculated using eq 2, has aporosity of 0.5573 that is 1.3% less than the packed bed in theexperiment, which has a porosity of 0.5646, calculated usingeq 2 also. This means that the DEM-simulated packing is morecompact than the actual packing. The porosity of the packed

bed used in the CFD simulations (εCFD) is 0.5704, which is1.05% larger than the porosity of the packed bed in theexperiment (0.5646). The difference between CFD and DEMis caused by the particle shrinkage (1%) and rounded edges(r ) 1 mm) of the particles during the CFD grid generation,which turns the slightly more “compact” than actual packinginto a slightly less “compact” than actual packing. Plugging inσ ) 1.05% (the packing porosity deviation between the CFDand the experiments) into eq 7 gives the porosity correctionfactor k ) 1.04. After the k-correction, the average error (fromthe measured pressure drops) now drops from 8.3% to 6.1%.The CFD predictions before and after the k-correction are bothwithin the acceptable error range (10%). On the other hand,the empirical correlations significantly underpredicted the pres-sure drops with an average error of 40.1%.

5.4. Random Packing of Spherical Particles (Plant Scale).Figure 13 shows the results for the three plant-scale cases withrandomly packed 10 mm spherical particles (PU-1, PU-2, andPU-3). The packed beds in the experiment contain 799, 916,

Figure 9. Comparison of predicted and measured pressure drops in structured-packed beds.

Figure 10. Comparison of predicted and measured pressure drops in laboratory-scale packed bed with random packing of spherical particles (LU-1).

4070 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009

and 1545 particles for tube-to-particle diameter ratios of 2.14,2.61, and 3.23, respectively. The entire packed beds weresimulated by DEM. However, limited by the available compu-tational resource, only 125 particles (taken from the bottomsection of each entire packed bed, shown in Figure 13a) were

included in the CFD simulations. The particle diameter wasshrunk by 0.5% in the CFD model. As seen in Figure 13b, theagreement between the CFD-predicted pressure drops (withoutthe porosity correction factor) and the measurements variessignificantly, from relatively good (for PU-1, average error4.9%) to slightly over the limit (for PU-2, average error 10.4%),and then to very poor (for PU-3, average error 26%). Comparingthe packing porosity in the CFD model with the experimentalpacking, it is clear that the porosity deviations are the maincontributor to the mismatch. For the three plant-scale experimentcases (PU-1, PU-2, and PU-3), the packing porosities in theCFD model are 1.36%, 3.09%, and 9.25% larger than theexperiments, respectively, resulting from the combination effectof the DEM simulation and the use of only 125 particles in theCFD model and the particle shrinkage. For the case withthe largest tube-to-particle diameter ratio (PU-3, D/d ) 3.23),the selected segment is more loosely packed than the experi-mental bed. In addition to the position of the selected segment,an insufficient number (125) of particles included in the CFDsimulation is the main cause for this large deviation of packingporosity, as suggested in Figure 6 that more that 1000 particles

Figure 11. Sample CFD simulation results for packed bed with random packing of cylindrical particles: (a) flow pattern shown as flow path lines, (b)velocity vector on central plane, and (c) pressure contours on particles and tube wall.

Figure 12. Comparing model predictions with measured pressure dropsfor packed bed with random packing of cylindrical particles.

Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 4071

are required in order to have the segment porosity close to theentire packed bed. After compensation by the porosity correctionfactor, the match between the CFD predictions and the measure-ments is significantly improved as shown in Figure 13, and theaverage error is reduced to less than 4%.

The empirical correlations (eqs 8-13) overpredicted thepressure drops in all cases. The deviations from the measuredpressure drops vary dramatically, from pretty accurate (averageerror 3.8% for PU-1) to acceptable (average error 9.3% for PU-2) to considerable (average error 93.4% for PU-2).

Comparing the measurements and the calculations shown inFigures 10, 12, and 13 confirms that the empirical correlationsare unreliable in predicting pressure drop across a packed bedwith a low tube-to-particle diameter ratio (D/d < 4). Theirdeviations are inconsistent and can be significantly beyond theacceptable error limit. In contrast, the predictions using the

DEM/CFD approach consistently matched well with the mea-sured pressure drops.

6. Conclusions

The ability to a priori predict void fraction and pressure dropin a packed bed would significantly improve reactor design aswell as allow for optimization around catalyst performance,catalyst design, and the resulting process pressure drop.Traditionally, the packed bed reactor designs are based on ahomogeneous assumption with averaged empirical correlationsalong with a sophisticated process of trial and error. Thosecorrelations often become invalid or unreliable for low tube-to-particle diameter ratios (D/d < 4), where the tube wall effectand local phenomena dominate. In this paper, the discreteelement method (DEM) and computational fluid dynamics

Figure 13. Comparison of predicted and measured pressure drops in plant-scale randomly packed beds.

4072 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009

(CFD) are coupled to model the flow field in packed beds. DEMis used to generate realistic random packing for a packed bedof spherical or cylindrical particles, which is then convertedinto a packing geometry to generate mesh for CFD simulation.Scripts have been developed to automate the tedious process.Only a segment of entire packed bed would be included in CFDsimulation if modeling the entire packed bed becomes unman-ageable or beyond the limit of available computational resources.Pressure drop across the bed is part of the CFD simulationresults.

Two types of experiments were conducted: laboratory-scaleexperiments with up to 153 particles packed to allow modelingof entire packed beds, including random packing and structuredpacking, and plant-scale experiments conducted with up to 1545randomly packed particles. A concept of the “porosity correctionfactor” was introduced to compensate the effect of porositydeviation between actual packing and the CFD model sincepressure drop is very sensitive to packing porosity. The porositydeviation can be caused by DEM simulation, use of a segmentof entire packed bed, and particle shrinkage in CFD. They areinherently part of the DEM/CFD methodology and cannot beeliminated. The random nature of packing naturally reflects asthe difference between DEM-simulated packing and actualpacking. The number of particles included in CFD simulationsis limited by available computational resources. Shrinkingparticles is necessary to prevent highly skewed mesh at thosesingle-point contacts between particles or between a particleand the reactor wall. The CFD model was validated by goodagreement between the CFD predictions and the measurementsfor the structured packing cases in which the error due to theDEM simulation and use of a segment of packed bed wereeliminated. For the packed beds with randomly packed particles,the agreement between the CFD-predicted pressure drops andthe measurements varied, depending on the porosity deviation.After compensated by the porosity correction factor, the matchbetween the CFD predictions and the experimental data isimproved and the average errors are reduced to within thedesired limit for industrial design (10%). The pressure dropscalculated by the empirical correlations confirmed the advantageof the DEM/CFD approach as well as the inconsistency andunreliability of the empirical correlations for the packed bedswith low tube-to-particle diameter ratios (D/d < 4).

The DEM/CFD approach is being extended to study fixedbeds packed with more complicated designs of catalyst particlessuch as ring or penta-ring as well as with more complicatedphysical models such as heat transfer, mass transfer, andchemical reactions. The DEM/CFD approach can be used tohelp design packed bed reactors such as optimizing the tube-to-particle diameter ratio, optimizing the tube diameter, andoptimizing the size and shape of catalyst particles. Furthermore,the DEM/CFD simulations provide detailed information aboutlocal voidage, local head loss, and local heat transfer rate throughthe packed bed, which can be critical for stability analysis andreactor control.

Acknowledgment

Dan Friedhoff, Billy Smith and Mike Cloeter in The DowChemical Company are thanked for their contributions inconducting the experiments.

Nomenclature

a ) Ergun constant in Ergun equation for pressure drop (dimen-sionless)

b ) Ergun constant in Ergun equation for pressure drop (dimen-sionless)

D ) tube inner diameter (mm)d ) spherical particle diameter; cylindrical particle diameter (mm)dCFD ) particle diameter used in CFD (mm)D/d ) tube-to-particle diameter ratio (dimensionless)dp ) effective particle diameter (mm)g ) gravitational constant (9.81 m/s2)G ) mass flux (kg/m2 · s)h ) height of cylindrical particle (mm)H ) height of packed bed (mm)k ) porosity correction factor (dimensionless)Mw ) molecular weight of gas through packed bed (g/mol)N ) total number of particles in a packed bedNS ) total number of particles in a segment of the packed bedPo ) inlet pressure to a packed bed (Pa)R ) gas constant (8.314 m3 Pa/mol ·K)Re ) Reynolds number (dimensionless)∆P ) pressure drop across packed bed (Pa)∆PS ) pressure drop across a segment of the packed bed (Pa)∆PC ) pressure drop prediction corrected by porosity correction

factor (Pa)T ) temperature (K)ε ) void fraction of packed bed, packing porosity; packed bed

porosity (dimensionless)εCFD ) void fraction of packed bed in CFD (dimensionless)σ ) deviation of packing porosity between actual and CFD model

(%)η ) particle shrinkage (%)

Literature Cited

(1) Ergun, S. Fluid Flow through Packed Columns. Chem. Eng. Prog.1952, 48, 89.

(2) Leva, M.; Grummer, M. Pressure Drop through Packed Tubes: PartIII Prediction of Voids in Packed Tubes. Chem. Eng. Prog. 1947, 43, 713–718.

(3) Dixon, A. G. Correlations for the wall and particle shape effects onfixed bed voidage. Can. Chem. Eng. 1988, 66, 705–708.

(4) Freund, H.; Zeiser, T. Numerical simulations of single phase reactingflows in randomly packed fixed-bed reactors and experimental validation.Chem. Eng. Sci. 2003, 58, 903–910.

(5) Tstotsas, E.; Schlunder, E.-U. Heat Transfer in Packed Beds withFluid Flow: Remarks on the Meaning and the Calculation of a Heat TransferCoefficient at the Wall. Chem. Eng. Sci. 1990, 45, 819–837.

(6) Li, C.; Finlayson, B. A. Heat Transfer in Packed BedssA Reevalu-ation. Chem. Eng. Sci. 1977, 32, 1055–1066.

(7) Cundall, P.; Strack, O. D. L. A discrete element model for granularassemblies. Geotechnique 1982, 29, 47.

(8) Cundall, P. A. Distinct element models of rock and soil structure.Analytical and computational methods in engineering and rock mechanics;Brown, I. T., Ed.; Allen & Unwin: London, Chapter 4, 1987; pp 129-163.

(9) PFC3D User’s Guide, 2nd ed.; 2003.(10) Dixon, A. G.; Nijemeisland, M. CFD as a Design Tool for Fixed-

Bed Reactors. Ind. Eng. Chem. Res. 2001, 40, 5246–5254.(11) Nijemeisland, M.; Dixon, A. G. CFD Study of Fluid Flow and Wall

Heat Transfer in a Fixed Bed of Spheres. AIChE J. 2004, 50, 906–921.(12) Logtenberg, S. A.; Nijemeisland, M.; Dixon, A. G. Computational

fluid dynamics simulation of fluid flow and heat transfer at the wall-particlecontact points in a fixed-bed reactor. Chem. Eng. Sci. 1999, 54, 2433–2439.

(13) Calis, H. P. A.; Nijenhuis, J.; Paikert, B. C.; Dautzenberg, F. M.;van den Bleek, C. M. CFD modeling and experimental validation of pressuredrop and flow profile in a novel structured catalytic reactor packing. Chem.Eng. Sci. 2001, 56, 1713–1720.

(14) Romkes, S. J. P.; Dautzenberg, F. M.; van den Bleek, C. M.; Calis,H. P. A. CFD modelling and experimental validation of particle-to-fluidmass and heat transfer in a packed bed at very low channel to particlediameter ratio. Chem. Eng. J. 2003, 96, 3–13.

(15) Petre, C. F.; Larachi, F.; Iliuta, I.; Grandjean, B. P. A. PressureDrop through Structured Packings: Breakdown into the ContributingMechanisms by CFD Modeling. Chem. Eng. Sci. 2003, 58, 163.

Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 4073

(16) Freund, H.; Zeiser, T.; Huber, F.; Klemm, E.; Brenner, G.; Durst,F.; Emig, G. Numerical simulations of single phase reacting flows inrandomly packed fixed-bed reactors and experimental validation. Chem.Eng. Sci. 2003, 58, 902–910.

(17) Zeiser, T.; Steven, M.; Freund, H.; Lammers, P.; Brenner, G.; Durst,F.; Bernsdorf, J. Analysis of the flow field and pressure drop in fixed-bedreactors with the help of lattice Boltzmann simulations. Philos. Trans., Ser.A: Math., Phys., Eng. Sci. 2002, 360, 507–520.

(18) Mantle, M. D.; Sederman, A. J.; Gladden, L. F. Single- and two-phase flow in fixed-bed reactors: MRI flow visualisation and lattice-Boltzmann simulations. Chem. Eng. Sci. 2001, 56, 523–529.

(19) Bai, H.; Theuerkauf, J.; Witt, P.; Gillis, P. Modeling Fluid flowand Pressure Drop in Packed Beds with Coupling DEM and CFD. Presentedat the AIChE 2004 Fall Annual Meeting, Nov 7-9, 2004, Austin, TX.

(20) Theuerkauf, J.; Witt, P.; Bai, H. Using Discrete Element Methodto Generate Correct Packing Structures in Packed Beds of Spheres andCylinders for use in CFD Modeling. Presented at the AIChE 2004 FallAnnual Meeting, Nov 7-9, 2004, Austin, TX.

(21) Bai, H.; Theuerkauf, J.; Witt, P. Predicting Fluid Flow and PressureDrop in Randomly Packed Beds of Cylindrical Particles with Coupling DEMand CFD. Presented at the AIChE Annual Meeting, Oct 30-Nov 4, 2005,Cincinnati, OH.

(22) FLUENT Inc. Gambit 2.1 User’s Manual; 2004.

(23) FLUENT Inc. FLUENT 6.1 User’s Manual; 2004.(24) Theuerkauf, J.; Witt, P.; Schwesig, D. Analysis of Particle Porosity

Distribution in Fixed Beds Using the Discrete Element Method. PowderTechnol. 2006, 165, 90–97.

(25) Yakhot, V.; Orszag, S. A. Renormalization Group Analysis ofTurbulence. I. Basic Theory. J. Sci. Comput. 1986, 1 (1), 3.

(26) Nijemeisland, M.; Dixon, A. G.; Stitt, E. H. Catalyst design byCFD for heat transfer and reaction in steam reforming. Chem. Eng. Sci.2004, 59, 5185–5191.

(27) Dixon, A. G.; Nijemeisland, M.; Stitt, E. H. Packed Tubular ReactorModeling and Catalyst Design Using Computational Fluid Dynamics. AdV.Chem. Eng. 2006, 31, 307–389.

(28) Kim, S.-E.; Choudhury, D. A Near-Wall Treatment Using WallFunctions Sensitized to Pressure Gradient. ASME FED, Separated andComplex Flows; 1995; Vol. 217.

(29) Handley, D.; Heggs, P. J. Momentum and Heat Transfer Mechanismin Regular Shaped Packing. Trans. Inst. Chem. Eng. 1968, 46, 251–264.

ReceiVed for reView October 14, 2008ReVised manuscript receiVed January 9, 2009

Accepted January 15, 2009

IE801548H

4074 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009

本文献由“学霸图书馆-文献云下载”收集自网络，仅供学习交流使用。

学霸图书馆（www.xuebalib.com）是一个“整合众多图书馆数据库资源，

提供一站式文献检索和下载服务”的24 小时在线不限IP

图书馆。

图书馆致力于便利、促进学习与科研，提供最强文献下载服务。

图书馆导航：

图书馆首页 文献云下载 图书馆入口 外文数据库大全 疑难文献辅助工具