Chemical Engineering Science 60 (2005) 6947–6957

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Flow through packed bed reactors: 1. Single-phase flow

Damjan Nemeca,1, Janez Levecb,∗aLaboratory for Catalysis and Chemical Reaction Engineering, National Institute of Chemistry, SI-1000 Ljubljana, Slovenia

bDepartment of Chemical Engineering, University of Ljubljana, SI-1001 Ljubljana, Slovenia

Received 26 November 2004; received in revised form 31 March 2005; accepted 15 May 2005Available online 3 August 2005

Abstract

Single-phase pressure drop was studied in a region of flow rates that is of particular interest to trickle bed reactors(10< Re∗� <500).

Bed packings were made of uniformly sized spherical and non-spherical particles (cylinders, rings, trilobes, and quadralobes). Particleswere packed by means of two methods: random close or dense packing (RCP) and random loose packing (RLP) obtaining bed porositiesin the range of 0.37–0.52. It is shown that wall effects on pressure drop are negligible as long as the column-to-particle diameter ratio isabove 10. Furthermore, the capillary model approach such as the Ergun equation is proven to be a sufficient approximation for typicalvalues of bed porosities encountered in packed bed reactors. However, it is demonstrated that the original Ergun equation is only able toaccurately predict the pressure drop of single-phase flow over spherical particles, whereas it systematically under predicts the pressuredrop of single-phase flow over non-spherical particles. Special features of differently shaped non-spherical particles have been taken intoaccount through phenomenological and empirical analyses in order to correct/upgrade the original Ergun equation. With the proposedupgraded Ergun equation one is able to predict single-phase pressure drop in a packed bed of arbitrary shaped particles to within±10%on average. This approach has been shown to be far superior to any other available at this time.� 2005 Elsevier Ltd. All rights reserved.

Keywords:Packed bed; Momentum transfer; Pressure drop; Single-phase flow; Ergun equation; Non-spherical particle

1. Introduction

The application of heterogeneous catalysis gives rise to avariety of reactor types. Of those, the packed bed reactorsbelong to the most widely applied reactors, their popularityoriginating from their effectiveness in terms of performanceas well as low capital and operating costs. The reactantsflowing through a packed bed reactor can be both in theform of gas or liquid. In numerous applications both phasesare present. However, a study of single-phase flow is ofparticular interest to this work since it is not only essential forsingle-phase applications, but also constitutes the basis for

∗ Corresponding author. Tel.: +386611760280; fax: +386611259244.E-mail address:janez.levec@ki.si(J. Levec).

1 Current address: Akzo Nobel Chemicals b.v., Arnhem, The Nether-lands.

0009-2509/$ - see front matter� 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2005.05.068

studying two-phase flow through packed beds as discussedin part 2 of this work (Nemec and Levec, 2005).The shape and size of catalyst particles that make up the

bed are determined by the characteristics of the processin question. As a general rule, the particle size and shapeshould aim at high effectiveness, so as to utilize the catalystmaterials and reactor volume and therefore increase the bedactivity (Worstell, 1992). For practical reaction rates as ap-ply to most processes and typical diffusivities in gas-filledpores of catalysts, diffusion limitation will generally occurwith particles having a diameter of a few millimeters (Sieand Krishna, 1998). In catalytic processes where liquid ispresent, the catalyst pores are likely to be filled with theliquid and low diffusivity in the liquid phase may even in-creasing the likelihood of diffusion limitation. In the case ofporous solid catalysts, by far the largest portion of catalyti-cally active surface area consists of pore walls. For a givenconversion rate, the external surface determines the flux

6948 D. Nemec, J. Levec / Chemical Engineering Science 60 (2005) 6947–6957

density for diffusion of reactants to the catalytic surfaceinside the volume of catalyst particle. Therefore, the spe-cific surface (ratio between the external surface area andthe volume of the particles) of the catalytic bed has to be ashigh as possible so that, as a consequence, a smaller reactorvolume is then required. By decreasing the size of particlesthe specific surface area of the bed can be increased, how-ever, there exists a limit to the minimum size of the catalystparticle determined by the acceptable overall pressure dropthrough the bed. Another option to enlarge the surface tovolume ratio can be done by introducing different shapes ofparticles (cylinders, rings, etc.) since by doing so the effec-tive particle size remains relatively unchanged. Especiallyfor liquid phase processes extrudates with a clover leafcross-section (polylobed), namely trilobes and quadrolobes,are often used since they offer a greater surface to volumeratio than cylindrical extrudates of the same maximal out-side diameter and also retain their advantage in liquid phaseoperation (Sie, 1993).However, it has to be kept in mind that differently shaped

particles also pack with different degrees of bed porosity,which results in different pressure drops as well as differentoverall bed activities. As a rough rule it is said that the bedporosity increases the more the shape of particles deviatesfrom the spherical shape. This problem can be met to somedegree with the use of loading techniques, which give higherbed densities (Wooten, 1998). Obviously, the comparison ofefficiency of different particles is not straightforward as thechoice of the appropriate shape and size of the catalyst par-ticles as well as the loading technique will be determined bythe specific transport and kinetic characteristics of a givenprocess.Cooper et al. (1986), for example, provided a clearpicture of the interplay in choosing the optimum catalystshape, size and packing procedure for a specific system,considering two possible hydroprocessing conditions. Twoadditional criteria, namely the strength of the catalyst parti-cles and their manufacturing cost also need to be taken intoaccount in determining the appropriate particle shape (Sieand Krishna, 1998).The considerations regarding the optimum structure of

packed bed reactors are complex, therefore a systematicstudy is needed if we are to find a model or correlationfor predicting single-phase flow pressure drop of any prac-tical value as no shape and size of particles constituting thepacked bed as well as loading technique can be ruled out.The goal of the present work was therefore to study the ef-fect of particle shape and size, and bed loading techniqueon the single-phase pressure drop in packed beds, find anappropriate model, and test it with a wide variety of avail-able experimental data as well as by comparing it to otherexisting approaches.

2. Modeling single-phase flow through packed beds

The modeling of flow through porous media is one ofthe oldest subjects of interest to engineering in general and

still continues to attract the interest of engineers and re-searchers alike due the complexity of the modeling involved(Liu and Masliyah, 1996). However, the modeling can beconsiderably simplified if one is to consider a homogeneousporous medium where the possible porosity does not varya lot and a uniform flow distribution within the bed can beassumed. This in general is the case encountered in (com-mercial) packed bed reactors, which are made up of roughlyuniform particles in terms of both shape and size, wherethe possible porosity span encountered is relatively narrow(0.35< �<0.55), and the wall effect is negligible.Basically flow through packed beds can be modeled by

analogy with flow in pipes when the bed porosity is uniformthroughout the bed and below 0.6 (Dullien, 1992; Liu andMasliyah, 1996; Punˇcocháˇr and Drahoš, 2000). The pressuredrop through packed beds is the result of frictional lossescharacterized by the linear dependence upon the flow ve-locity and inertia characterized by the quadratic dependenceupon the flow velocity (Forchheimer effect). Adding thesetwo contributions results in the well-known Ergun equation(Ergun, 1952), which can also be written in dimensionlessform in terms of modified Reynolds and Galileo numbers asfollows (Niven, 2002)

�P/L

��g= �� = A

Re∗�

Ga∗�

+ BRe∗2

�

Ga∗�. (1)

Ergun (1952)had shown that the above equation fitted datafor spheres, cylinders, and crushed solids over a wide rangeof flow rates within acceptable engineering accuracy. In or-der to check the functional dependency upon bed porosityhe also varied the packing density for some materials to ver-ify the (1 − �)2/�3 term for the viscous loss part and the(1− �)/�3 term for the kinetic energy part. Note that a smallchange in porosity has a large effect on the pressure drop,which makes it difficult to predict the latter accurately and toreproduce experimental values after a bed is repacked. De-spite all the problems, Ergun determined the constant for theviscous term (often referred to asBlake–Kozeny–Carmanconstant) to be 150 and the constant for the inertial term(Burke–Plummerconstant) to be 1.75.It is now generally accepted that satisfactory predictions

of pressure drop in packed beds can be done with the useof simple semi-empirical models like the Ergun equation.However, this is only true for infinitely extended packed bedscomposed of particles that do not differ much in shape fromthat of spheres. Should this not be the case, corrections to theErgun equations should be applied and this can usually bedone by modifying the constants of the viscous and inertialterms as discussed in the following section.

3. Evaluating Ergun constants

Universal values of the Ergun constants have been asubject of considerable debate since 1952. Probably themost known work regarding this question is thatMacdonald

D. Nemec, J. Levec / Chemical Engineering Science 60 (2005) 6947–6957 6949

et al. (1979)who have proposed the use of modified val-ues of 180 and 1.80 as more acceptable for particles ofarbitrary shape. However, even their results indicate, al-though not conclusively, that the original Ergun constantsin fact do an adequate job in predicating the pressure dropfor beds of spherical particles, whereas for beds made upof non-spherical particles the values should be even 15%or more higher than 180 and 1.80. Thus the later univer-sal constants are no more than a compromise of limitedvalue.It seems now that most researchers have satisfied them-

selves with the fact the values of the Ergun constants oughtto be determined empirically for each bed. This philosophyarises from the belief that the values are not only depen-dent on the particle geometry but in addition they can varyfrom macroscopic bed to bed (made of the same particles)due to different structures of the packing within the bed af-ter repacking (Jiang et al., 2000). Nevertheless, we believethat there do exist some principles for the values. For one,inherent inaccuracies involved in the measurements of spe-cific surface and bed porosity must be borne in mind whenevaluating the constants. Small differences especially in thevalues of mean porosity can give rise to big differences in theconstants of the general Ergun equation and in some caseswall effect has not been considered. Secondly, it is agreedthat there exist no universal values of constants but ratherthat general values should be sought for groups (families) ofparticles of similar geometry. Therefore, the effect of differ-ent parameters or bed characteristics on the Ergun constantshave been studied with the help of existing findings and dataavailable in the literature as well as new experimental dataobtained during the course of this work.

3.1. Experimental setup and methods

A high-pressure experimental setup had been designedfor studying two-phase flow hydrodynamics in packed beds(Nemec et al., 2001), which was also useful for precisesingle-phase flow studies. A stainless steel column had a di-ameter of 41mm and height of 70 cm. The gas flow rate wasmonitored and measured by a mass flow controller, whereasthe pressure drop was measured with the use of a precisedifferential pressure transducer. Measurements were mostlyconducted with nitrogen at 10bar, although measurements at5 and 20bar were also performed as well as with other gases(H2, O2, CO2, He, Ar) to check for the effect of gas density.Different types of packings have been employed, the

properties of which are summarized inTable 1. Most of thepackings employed are those typically encountered incommercial reactors. The particles have been packed intwo different manners as first proposed byScott (1960):random close or dense packing (RCP)—by slowly pour-ing small amounts of particles and tapping the columnin between packing stages—and random loose pack-ing (RLP)—by tipping the filled container horizontally,

slowly rotating it about its axis and gradually returningit to the vertical position. Great care was taken in deter-mining the accurate values of bed porosities due to thetremendous effect it has on the hydrodynamic phenomena.This was done with the weighing method: dried parti-cles (free of moisture) were packed into the bed and theporosity was then determined form the mass of particlesand density of a representative particle (determined byHg-porosimetry). More precise information about the ex-perimental setup and procedures can be found elsewhere(Nemec, 2003).

3.2. Effect of wall

There exists an annular wall zone where the averageporosity is greater than in the core of the bed since spatialdistribution of particles must conform to the shape of thewall. The influence of the wall upon flow via channelingbecomes more significant asD/ds decreases (Fand andThinakaran, 1990). On the other hand, the viscous friction atthewall, which increases the pressure drop,may not be negli-gible in comparison to that caused by particles due to the factthat friction surface of the wall increases relative to the totalbed surface corresponding to particles asD/ds decreases.Winterburg and Tsotsas (2000)have quantified the two

effects with numerical simulations, and in a separate workEisfeld and Schnitzlein (2001)have come up with an em-pirical correction to the Ergun equation which also correctlyreflects the counteracting effects of flow maldistribution andwall friction as well as the fact that the magnitude of the ef-fects is Reynolds number dependent. Very recentlyDi Feliceand Gibilaro (2004)have also presented their analysis of thephenomena.Without going into too much detail the messageis clear. The failure to recognize the need of incorporatingsuch corrections into the Ergun equation can result in devi-ations of the constants with tube-to-particle ratio diameterwith experimental analyses. However, the general conclu-sion of all of the above works is that the Ergun equation(with average values of porosity and superficial velocity)from a practical point of view is applicable down to quitelow tube-to-particle-diameter ratios(D/ds �10). It shouldbe emphasized that the beds tested in this work had the tube-to-particle-diameter ratio higher than 10 so the experimen-tally determined Ergun constants should not be affected bythe reactor column wall. This is proven by the experimentalresults presented inFig. 1 for four different sizes of spherespacked in two different manners as all data points fall onthe same line.

3.3. Effect of bed porosity

As mentioned before, the constants can vary from macro-scopic bed to bed even if repacked with the same batch ofparticles. If the repacking of the bed changes the values ofthe Ergun constants this could mean that the porosity is not

6950 D. Nemec, J. Levec / Chemical Engineering Science 60 (2005) 6947–6957

Table 1Properties of the packing utilized

Shape �s (dimensionless) dp × hp × di (mm) ds (mm) de (mm) D/ds (dimensionless) Material � (dimensionless)

Spheres 1 1.66 1.66 1.66 24.7 �-Al2O3 0.382–0.431Spheres 1 2.10 2.10 2.10 19.5 Glass 0.385–0.431Spheres 1 2.57 2.57 2.57 16.0 �-Al2O3 0.396–0.429Spheres 1 3.50 3.50 3.50 11.7 �-Al2O3 0.400–0.443Cylinders 0.866 3.2× 4.2 4.03 3.49 10.2 �-Al2O3 0.368–0.420Cylinders 0.782 1.6× 4.7 2.62 2.05 15.7 �-Al2O3 0.437Cylinders 0.672 1.3× 7.5 2.67 1.80 15.4 �-Al2O3 0.484–0.526Rings 0.590 3.2× 5.0× 1.3 4.25a 2.35 9.7 �-Al2O3 0.474Trilobes 0.630 1.27× 5.5 2.54 1.41 16.1 Ni-Mo 0.466–0.511Quadralobes 0.593 1.35× 5.2 2.13 1.26 19.3 Ni-Mo 0.471–0.502

aInner void not taken into account.

0 50 100 150 200 250 300

103

104

105Data: This work5 < Reγ < 300MRD = 4.0%

(104 data points)

Reγ*

Spheres: D/ds

1.66 mm RCP 0.382 25 1.66 mm RLP 0.431

2.10 mm RCP 0.385 20 2.10 mm RLP 0.431

2.57 mm RCP 0.396 16 2.57 mm RLP 0.429

3.50 mm RCP 0.400 12 3.50 mm RLP 0.443

Ergun (1952)

ε

Ψγ G

a γ*

*

Fig. 1. The effect of bed porosity and wall effect(D/ds) on pressuredrop for beds made of spheres.

adequately taken into account by the capillary model.Rumpfand Gupte (1971)have analyzed the effect of various distri-butions of spherical particles over a relatively wide range ofporosities(0.35< �<0.70) and proposed a different depen-dence upon porosity. For the region of packed bed reactorrelevance(0.35< �<0.55) the proposed porosity functionof Rumpf and Gupte (1971)does not differ very much fromthat of the capillary model, considering an average differ-ence of only about 10%. Other porosity functions like theone determined byLiu et al. (1994)in general yield val-ues between those of the capillary model and the empiricalmodel proposed byRumpf and Gupte (1971).Furthermore, it has to be pointed out that the results of

Rumpf and Gupte (1971)have been obtained from mediacreated with higher porosities than normally encounteredin beds composed of spherical particles and could there-fore lead to non-uniformly packed beds giving us the wrongimpression. Thus, it was deemed necessary to recheck theporosity effect on pressure drop with more natural parti-cle distributions. To obtain a wide variety of bed porosi-ties two different packing procedures were employed in thepresent work; RCP and RLP. Spheres were used, as they, byvirtue of their unique shape, are incapable of influencing the

structure of the bed by their orientation. Some additionaldifferences between the porosities of beds, despite the samepacking procedures, were due to wall effect, which as men-tioned before did not affect the overall pressure drop. Theseresults are also shown inFig. 1. One can conclude that theporosity dependence seems to be well described by the cap-illary model, reflected by the fact that all the data lay on asingle curve for all packed beds. This is in agreement with anumber of works for the viscous regime reviewed byCarman(1937)as well as a more recent one ofEndo et al. (2002).With regards to the porosity dependence within the inertialregime,Hill et al. (2001b)reported, on the basis of theoret-ical simulations of flow through random arrays of spheres,that the porosity function is also well taken into accountas long as the porosity is around 0.4 as is indeed the casefor packed bed reactors when made up of spheres.Ergun(1952)also made an interesting point that if a transforma-tion of his equation is made employing the fundamental ex-pressions for the shear stress, hydraulic radius and intersti-tial velocity, this leads to complete elimination of porosity,and the resulting equations are in accordance to those usedin the field of aerodynamics. Therefore, the porosity func-tion of the capillary model can be assumed as an accurateone within the region of interest(0.35< �<0.55) as the ar-guments for overweight those against.

3.4. Effect of particle shape

It is hard to expect that the simple assumptions of thecapillary model can describe the flow through packed bedsmade up of differently shaped particles with universalconstants. However, if we brake up the problem down tosmaller parts that involve groups (or families) of particles ofsimilar geometry some general principles regarding Ergunconstants may be found. Particle shapes studied involvethose most often encountered in packed bed reactors, i.e.,spheres, cylinders, rings (or hollow cylinders) and polylobedextrudates.Spheres: As can be concluded fromFig. 1, the orig-

inal Ergun equation with values of viscous and inertial

D. Nemec, J. Levec / Chemical Engineering Science 60 (2005) 6947–6957 6951

constants of 150 and 1.75, respectively, fits the pressure dropdata for beds of spheres extremely well. This is in fact inagreement with a number of works found in the open litera-ture (Macdonald et al., 1979; Burghardt et al., 1999; Lakotaet al., 2002; Dullien, 1992), which claim that the originalErgun constants are able to predict the pressure drop inbeds composed of spherical particles to within 10%, whichis certainly within acceptable limits. As a matter of fact,even “more normal” bed assemblies ofRumpf and Gupte(1971) (porosities of 0.366 and 0.409) agree with suchconclusions (Macdonald et al., 1979), whereas for otherbed assemblies with higher porosities the original Ergunconstants would over predict the pressure drop. Again, it isthought that the most probable reason for this is due to thenon-uniformities in construction of such beds rather thaninadequate representation of the porosity function.Cylinders: A similar study as that for flow through bed of

spheres has been done for bed of cylindrical shaped particlesand the results exhibited a greater scatter of data comparedto the case of spheres. However, no trend was observed,which would indicate that an inadequate porosity functioncould be blamed for the scatter of data. Such scatter ofdata has also been observed byPahl (1975)who has madean extensive study of flow through packed beds involvingcylindrical particles of a wide range of aspect ratios(hp/dp)

as well as considered the effects of different ways of packingthe particles. Therefore, it is thought that the scatter is dueto different orientation of particles when packing the bedin a different manner. This is substantiated by the fact thateven reproducibility in pressure drops after repacking a bedof cylinders in the same manner is not as good as in thecase of spheres. What this basically means is that cylinders(and other non-spherical particles) can orientate themselvesat various angles with respect to the axis of the bed, as wellas to the horizontal base which leads to different pressuredrops even in the case of equal bed porosities. Spheres, byvirtue of their unique shape, are incapable of influencingthe structure of bed by their orientation. The fact that theorientation of particles or the bed structure in general has adefinite effect on the permeability is not new (Scheidegger,1974; Tukaˇc and Hanika, 1992), although results cannot berepresented by a simple correlation.The above discussion could also serve as a way of ex-

plaining why cylinders (or other non-spherical particles forthat matter) have higher fitted Ergun constants than in thecase of spherical particles. The mean porosity is simply notsufficient to characterize the structure of the bed thus addi-tional parameters would be required to explain the pressuredrop characteristics of beds packed with different shapes ofparticles. Tortuosity is an obvious candidate and it may wellbe that beds of cylindrical particles have a more tortuousstructure than those of spherical particles (Foumeny et al.,1996). Other parameters like the dynamic specific surface,in other words wetted surface (Comiti and Renaud, 1991) orform drag (Dolejš and Machaˇc, 1995) have also been pro-posed. However, all these additional structure parameters are

rather vague physical concepts and are very hard to measureexperimentally. Usually they act as fitted parameters frompressure drop data. It seems, however, if there is no wayaround using fitted values then we are better off using theErgun constants for families of non-spherical particles. Theadditional structure parameters can only serve for qualitativedescription of the phenomenon.Table 2provides the average values of Ergun constants for

cylindrical particles of different aspect ratio as fitted fromexperimental data obtained from different sources.As can beseen the more the aspect ratio(hp/dp) of a cylindrical par-ticle differs from a value of 1 (in other words, the more non-spherical in shape) the higher theBlake–Kozeny–Carmanas well as theBurke–Plummerconstant. This provides theidea to correlate the fitted Ergun constants to the spheric-ity of a particular cylindrical particle. Eqs. (2) and (3)have been found to give reasonable fits as also shown inFig. 2

A = 150/�3/2s , (2)

B = 1.75/�4/3s . (3)

Admittedly Eqs. (2) and (3) are purely empirical by nature,however, they are based on observations for a wide varietyof data from several sources. The equations provide a rea-sonable guess for the Ergun constants in case of an a prioriprediction of pressure drop over a bed made of cylinder par-ticles withhp ≈ dp or higher. In no case are these empiricalcorrelations applicable to tablet type particles(hp/dp <0.5)since these particles are characterized by marked stratifica-tion and partial overlapping which can considerably increasethe already tortuous path of the flowing fluid (Comiti andRenaud, 1991).Rings(hollow cylinders): Rings were among the first types

of packings that have indicated that shape of particles mayhave an additional effect to that already been taken into ac-count by the specific surface (equivalent particle diameter)since much higher pressure drops have been measured thananticipated from calculations based on spherical particles.Carman (1937)had proposed using values of (Ergun) con-stants for ring packings that are about 2.5 times higher thanthose for spheres, and attempted to explain by claiming thatthe interior of rings could be a source of eddies and deadspaces at higher rates of flow.LaterSonntag (1960)derived a correction for the possible

dead space within a ring particle. After fitting the correctedpressure drop correlation, the results suggested that onlyabout 20% of the interior of a particular ring particle wasavailable for flow. Therefore, the lower “effective” porosityof the bed as seen by the flowing fluid, although counterbal-anced by the lower specific surface of the bed to some extent,is responsible for the higher-pressure drops than originallyanticipated by the Ergun equation. By applying Sonntag’scorrections in deriving the Ergun equation the appropriate

6952 D. Nemec, J. Levec / Chemical Engineering Science 60 (2005) 6947–6957

Table 2Ergun constants (rounded values) for cylindrical particles of various aspect ratios

hp/dp �s � A B Source

0.37 0.757 0.418–0.500 280 4.60 Pahl (1975)0.72 0.862 0.323–0.490 190 2.70 Pahl (1975)0.91 0.872 0.336–0.588 200 2.50 Pahl (1975)1.00 0.874 0.363 200 2.00 Reichelt (1972)1.00 0.874 0.350 180 2.00 England and Gunn (1970)1.33 0.866 0.368–0.420 210 1.90 This work1.91 0.835 0.334–0.682 210 2.50 Pahl (1975)2.94 0.782 0.437 240 2.40 This work3.81 0.722 0.402–0.492 230 2.50 Pahl (1975)5.77 0.672 0.484–0.526 250 2.50 This work

0.5 0.6 0.7 0.8 0.9 1.0

150

200

250

300

350 0.37 0.72 0.91 1.00 1.00

Eq. (2)

A

�s

0.5 0.6 0.7 0.8 0.9 1.0�s

Symbol hp/dp

Symbol hp/dp

1.311.912.943.815.77

Eq. (2)

0

1

2

3

4

5

Eq. (3)B

Eq. (3)

Symbol hp/dp

1.311.912.943.815.77

0.37 0.72 0.91 1.00 1.00

Symbol hp/dp

(a) (b)

Fig. 2. Dependency of fitted Blake–Kozeny–Carman (a) and Burke Plummer (b) constants on sphericity for cylindrical particles of various aspect ratios.(see for symbols inTable 2).

values for the Ergun constants would read (Nemec, 2003)

A = 150

[�3

(1− (1− �)(Vf c − m Vi)/Vp)3

]

×(

(Sf c + m Si)

Vp

de

6

), (4)

B = 1.75

[�3

(1− (1− �)(Vf c − m Vi)/Vp)3

]

×(

(Sf c + m Si)

Vp

de

6

)2

, (5)

whereVf c andSf c are the volume and surface of a hypo-thetical full cylinder with the same outer diameter(dp) andheight(hp) as the ring andVi andSi are the inner volumeand surface of the ring with an inner diameterdi . The frac-tion of the interior of ring available for flow(m) can betaken as 0.2 as proposed by Sonntag.Fig. 3demonstrates theusefulness of the Sonntag’s correction for hollow cylinder

0 100 200 300 400 5000.0

2.0x105

4.0x105

6.0x105

8.0x105

1.0x106

1.2x106

ΨγG

a γ *

Reγ *

Hollow cylinders - This work [de = 2.35 mm (3.2x 1.3 x 5); ε = 0.474] Raschig rings - Lakota et al. (2002) [de = 3.98 mm (8.5x 5 x 5.5); ε = 0.583]

Ergun (1952) corrected Ergun constantsEqs. (4) & (5)

Fig. 3. Measured and predicted pressure drop values for hollow cylindersand Raschig rings.

particles (from this work) and Raschig rings (Lakota et al.,2002). The agreement with Sonntag’s correction is simplyremarkable.

D. Nemec, J. Levec / Chemical Engineering Science 60 (2005) 6947–6957 6953

Table 3Fitted Ergun constants for polylobed particles

Shape hp/dp �s � A B Source

Trilobe 4.33 0.630 0.466 295 4.71 This work‘’ 4.33 0.630 0.511 263 4.99 This workQuadralobe 3.85 0.593 0.471 292 3.93 This work‘’ 3.85 0.593 0.502 294 4.19 This work

0 20 40 60 80 100 120 140 160 180 2000.0

2.0x104

4.0x104

6.0x104

8.0x104

1.0x105

1.2x105

1.4x105

1.6x105

1.8x105

2.0x105

32 data pointsMRD = 9.1%

*Reγ

Data: This workShape: Trilobesde = 1.41 mm (1.27 x 5.5)

ε = 0.466 (RCP) ε = 0.511 (RLP)

Ergun (1952) corrected Ergun constantsEqs. (6) & (7) Ψ

γGa γ

*

Fig. 4. Measured and predicted pressure drop values for beds consistingof trilobes packed in two different manners.

Polylobes: There is very little information in the open lit-erature about packed beds made up of polylobed particles(trilobes, quadralobes and pentalobes), which is surprisingsince such shapes are often encountered in the refinery in-dustry as they provide better accessibility to the inner surfaceof the catalyst (and thus improve diffusion characteristics)due to their shape (Cooper et al., 1986). Due to the lack ofdata we are forced to make assumptions on a limited amountof our own data. The fitted Ergun constants seem to follow asimilar trend to that of cylinders when it comes to the depen-dency upon sphericity as seen fromTable 3. However, theBlake–Kozeny–Carmanconstant seems to be slightly lessdependent on sphericity whereas theBurke–Plummercon-stant is much more dependent on the sphericity of particlesthan for the case of cylinders. The following correlations forpredicting the Ergun constants for beds made up of poly-lobed particles can be employed

A = 150/�6/5s , (6)

B = 1.75/�2s , (7)

which can predict single-phase pressure drop to about10% accuracy as shown inFigs. 4 and 5 for the caseof trilobes and quadralobes, respectively. Note also thatthe particles have been packed in two different manners,which had no significant influence on the fitted Ergunconstants.

0 20 40 60 80 100 120 140 160 1800.0

2.0x104

4.0x104

6.0x104

8.0x104

1.0x105

1.2x105

1.4x105

1.6x105

34 data pointsMRD = 8.5%

Data: ThisworkShape: Quadralobesde = 1.26 mm (1.35x 5.2)

ε = 0.471 ε = 0.502 Ergun (1952) corrected Ergun constants Eqs. (6) & (7)Ψ

γGa γ

*Reγ

*

Fig. 5. Measured and predicted pressure drop values for quadralobe-shapedpackings (data from this work) packed in two different manners.

3.5. Other effects

The operational region of flow rates(10< Re∗� <500)

was that of particular interest to trickle-bed reactors (Nemecand Levec, 2005) where the viscous and inertial term ofthe Ergun equation have more or less even or at least com-parable contributions to the overall pressure drop. How-ever, in pure creeping flow the situation is different fromthat predicted by the general Ergun equation and the evalu-ated constants as proposed so far. One might argue that theBlake–Kozeny–Carmanconstant for beds of spheres as de-termined by Ergun(A=150) and substantiated by the resultsof this work is not in accordance with the results reportedby Carman (1937)(A=180). This should necessarily resultin approximately 17% under-prediction of drag force (pres-sure drop) within viscous regime(Re∗

� <1). The differencearises presumably in order to achieve a better fit of the Er-gun’s equation to experimental results at moderate Reynoldsnumbers(1< Re∗

� <1000) which are of interest to commer-cial packed bed systems. Flow through porous media is farmore complex than what we might be led to believe by thesimple Ergun equation (seeHill et al., 2001a, for more onthis).On the other hand there is considerable evidence that the

Ergun equation overpredicts the pressure drop at values ofReynolds numbers above 300 (Dolejš and Machaˇc, 1995;Hicks, 1970; Salatino and Massimilla, 1990; Tallmadge,1970), with the difference becoming significant within theturbulent flow regime (Reynolds numbers above 900).

6954 D. Nemec, J. Levec / Chemical Engineering Science 60 (2005) 6947–6957

Both of the above-mentioned phenomena are hard to spotespecially if one is not aware of their existence or lookingfor them in the first place. Simple fitting of constants to theErgun equation cannot and will not yield the same values asfor instance to the one found byCarman (1937)for creepingflow andFand et al. (1987)for turbulent flow, meaning thatwe should differ between the several types of fluid flows,which include creepingmotion, laminar-inertial flow and tur-bulent flow. If we employ the Ergun equation, the functionaldependency of pressure drop upon different parameters willin essence stay the same; however the proportionality (Er-gun) constants will differ from case to case owing to smallrelation inaccuracies. Again, the results given in this workapply to the laminar-inertial flow(1< Re∗

� <1000), whichis of particular interest to packed bed reactors. Furthermore,it should be noted that the Ergun equation is inapplicablefor packed beds with broad size distributions. The reasonfor this is that the average value of the permeability of par-ticles having different sizes is not necessarily equal to thatof particles having the mean diameter (Endo et al., 2002;MacDonald et al., 1991).

4. Comparison of pressure drop predictions

The approach of employing the Ergun equation with con-stants for the viscous and inertial terms corresponding todifferent types of particle shapes has been evaluated in termsof its ability to predict available experimental data and bycomparing it to other existing approaches. The comparisonsare made by means ofMean Relative Deviation(MRD) val-ues as statistical indicators

MRD(%) = 1

N

N∑i=1

|�i,calc− �i,exp|�i,exp

100. (8)

4.1. Upgraded Ergun equation

The constants proposed for different shapes of parti-cles are given by Eqs. (1)–(7). Since the original valuesof the Ergun constants (150 and 1.75) have been retainedand the quality of the predictions have been improvedthrough empirical and phenomenological corrections to theBlake–Kozeny–CarmanandBurke–Plummerconstants, wecan state that the original Ergun equation has been upgradedrather than corrected.The accuracy of predictions with the proposed upgraded

Ergun equation has been tested on a large data base avail-able to us that included all the important particle shapes:spheres, cylinders, rings and polylobed extrudates. The re-sults are summarized inTable 4. As can be seen, the pre-dictions are very reliable with the MRD rarely exceeding10% regardless of particle shape. The somewhat worse re-sult for the prediction of theEngland and Gunn (1970)datafor rings does, however, stand out. This could be due to the

complexity of the system since it is hard to expect that inall cases of rings of different dimensions only 20% of theinner void volume is available to flow as was assumed in thiswork for simplification of the general problem. Therefore,the overall error of only about 11% for the whole databasefor ring-shaped particles should be viewed as respectable.Certainly more data would need to be tested to determinethe reliability of the correlations for rings and polylobes.

4.2. Universal Ergun constants

One approach often cited in the literature is that ofMacdonald et al. (1979), whereby the authors suggest em-ploying the Ergun equation and using universal values of180 and 1.8 for the viscous and inertial constant, respec-tively. Although there is no doubt that the work is of greatgeneral value, the results should, however, be applied withreservation. Of the number of data sources employed onlythe data ofRumpf and Gupte (1971)for spheres andPahl(1975) for cylinders were comprehensive and precise. Fur-thermore, there existed some evidence that better agreementwith experimental data would result by using different pa-rameter values for spherical particles than for other shapes(about 15% lower than 180 and 1.8 for spheres and about15% higher for other shapes) which is in accordance to somedegree with the results obtained in this work.Macdonaldet al. (1979)have also argued that the roughness affects thepressure drop in a similar manner to that in pipes, thereforethey set the viscous constant to 180 and allowed the iner-tial constant to vary from 1.8 for smooth particles to 4.0for roughest particles. However, the applicability of such acorrelation is limited since the effect of roughness is not de-pendent on a parameter indicating the degree of roughnessand thus basically serves as an open or fitted parameter.Furthermore, it was later shown byCrawford and Plumb(1986) for artificially roughened media andJordi et al.(1990)for particles of natural or random roughness that thepressure drop is substantially increased by the presence ofsurface roughness over the entire range of Reynolds num-bers (both viscous and inertial contributions are affected).Such behavior is quite different from the one proposed byMacdonald et al. (1979)by drawing analogy with flow inrough pipes and can be explained by taking into accountthe increased surface area due to surface roughness.The results on the predictions of the approach are shown in

Table 4. Certainly, fixing the values of constants to a certainvalue reduces the flexibility of the model to successfullypredict experimental data for a variety of particle shapes. Ascan be seen the predictions for particles of simpler shapeslike spheres and cylinders are reasonable if not good (datafor spheres is over predicted, whereas the data for cylindersis generally under predicted). However, prediction resultsbecome much worse for more complicated shapes like ringsand polylobed extrudates where the data are considerablyunder predicted.

D. Nemec, J. Levec / Chemical Engineering Science 60 (2005) 6947–6957 6955

Table 4Comparison of model predictions with available experimental data in terms of MRD as statistical parameter

Shape ds (mm) � (dimensionless) Source No. of data MRD (%)

Upg. EE Uni. EC NN

Spheres 1.66–3.50 0.382–0.443 This work 104 4.0 11.1 42.2Spheres 3.0–6.0 0.382–0.391 Lakota et al. (2002) 75 8.1 16.8 19.9Cylinders 2.62–4.03 0.368–0.526 This work 162 9.0 16.4 20.6Cylinders 2.66–3.66 0.323–0.682 Pahl (1975) 407 10.7 18.0 59.9Cylinders 5.72 0.363 Reichelt (1972) 9 4.9 6.8 97.3Cylinders 7.27 0.35 England and Gunn (1970) 29 5.3 6.2 104Hollow cylinders 4.25a 0.474 This work 16 5.6 21.6 78.5Rings 8.42 0.583 Lakota et al. (2002) 26 2.4 58.2 88.0Rings 7.27a 0.53 England and Gunn (1970) 28 20.7 18.7 82.0Trilobes 2.54 0.466–0.511 This work 32 9.1 48.0 55.7Quadralobes 2.13 0.471–0.502 This work 34 8.5 46.5 59.9

Upg. EE, Upgraded Ergun Equation (This work—Table 4); Uni. EC, Universal Ergun constants (Macdonald et al., 1979); NN, Neural network approach(Iliuta et al., 1998).

aInner void not taken into account.

4.3. Neural networks approach

Another approach to predicting single-phase pressuredrop through packed beds has been presented more re-cently by Iliuta et al. (1998). As the Ergun constants areusually claimed to be complex lumps accounting for thebed geometry the authors have cross-correlated several bedparameters to the two Ergun constants by using weightsobtained from feed-forward neural network fitting of avail-able experimental data. The parameters taken into accountare particle and column diameter, bed porosity, and particleshape or sphericity. The range of tested experimental datafrom over 100 sets generally encompasses the whole rangerelevant to packed bed reactors.As can be seen fromTable 4the neural network approach

of Iliuta et al. (1998)has proven to be extremely unreliableas the predictions are much worse than those of the Up-graded Ergun Equation approach proposed by this work oreven the Universal Ergun constants approach ofMacdonaldet al. (1979)for that matter. There is no apparent order inthe predictions as groups of the available experimental datacoming from a single source and covering a single particleshape were noticed to be sometimes considerably under pre-dicted as well as over predicted. This considerable scatter ofpredictions for a certain shape of particles is most likely tobe the consequence of the extreme sensitivity of the corre-lations for the Ergun constants on absolute particle size andbed porosities. Recall that the effect of these two bed param-eters has been found to be either non-existent or marginalduring the course of this work.

5. Conclusions

The capillary model based on the hydraulic radius con-cept, and the Ergun equation, as the most recognizable math-ematical form of the model, have undergone a considerable

degree of criticism over the last several decades when ap-plied to flow in porous media. This is not surprising sincethe rather simple-minded model of porous medium as a setof parallel identical channels is a crude simplification of thevery complex structure of porous media. It would be unre-alistic to expect that such a model could cover all types ofporous media like consolidated and unconsolidated mediaas well as fibrous beds.The analysis of single-phase flow through packed beds

and the subsequent discussion put forward by this worksuggests that the Ergun equation is well suited for the de-scription of flow through packed bed reactors. In such casesthe porosity range is rather narrow(0.35< �<0.55), thebed is made up of similar sized particles, which followsthe suitability of the hydraulic radius concept, and the flowrates are moderate(1< Re∗

� <1000). Nevertheless, theErgun equation has to be applied in its generalized form(Eq. (1)) with the Ergun constants suited for particulartype of particles. The particles have been divided into fourgroups of the most popular shapes encountered in packedbed reactors and empirical corrections for the constantshave been provided to take into account the special fea-tures of different shapes of particles (these are given byEqs. (2)–(7)). When predicting single-phase pressure dropthrough packed beds this approach has been demonstratedto be superior to other approaches encountered in the litera-ture with average expected relative deviations of about 10%regardless of particle shape or packing procedure. Morework on polylobed packings, however, is recommended,as there is very little information available in the openliterature.

Notation

A Blake–Kozeny–Carmanconstant—viscous term ofthe Ergun equation, dimensionless

6956 D. Nemec, J. Levec / Chemical Engineering Science 60 (2005) 6947–6957

B Burke–Plummerconstant—inertial term of the Er-gun equation, dimensionless

de equivalent particle diameter[6Vp/Sp], mdi inner diameter or ring/hollow cylinder, mdp particle nominal diameter, mds equivalent volume sphere diameter[(6Vp/�)1/3],

mD diameter of column, mg gravitational acceleration(9.81m/s2), m/s2

Ga∗� modified Galileo number for �-phase,

[(��/��)2g(de�/(1− �))3], dimensionless

hp height (length) of particle, mL length of bed, mm fraction of the interior of ring available to flow,

dimensionlessRe∗

� modified Reynolds number for �-phase,[��v�de/��(1− �)], dimensionless

Sf c surface area of a hypothetical full cylinder, m2

Si inner surface of ring or hollow cylinder, m2

v� superficial velocity of�-phase, m/sVf c volume of a hypothetical full cylinder, m3

Vi inner volume of ring or hollowcylinder, m3

Vp volume of particle, m3

Greek letters

�P total pressure drop—including gravitational contri-bution, Pa

� bed porosity, dimensionless�� dynamic viscosity of�-phase, Pa s�� density of�-phase, kg/m3

�s spherictiy of particle[(36�V 2p /S3p)1/3], dimension-

less�� dimensionless pressure drop for�-phase

[(�P/L)/��g], dimensionless

Subscripts

� gas or liquid phase� gas phase

Acknowledgements

The authors would like to thank the Slovenian Ministryof Education and Science for the financial support underGrant no. P0-0521-0104. We are extremely thankful to Nor-ton Chemical Process Products Corporation, for supplyingus with a large number of samples of catalyst carriers with-out which this work would not be possible. The help of othercompanies (Condea Chemie GmbH, Nikki Universal Co.,Engelhard de Meeren, Akzo Nobel Catalysts, Catalyst Trad-ing Company, Ltd.) by providing smaller samples is alsogratefully acknowledged.

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