Numerical analysis of mass transfer in packed-bed reactors with irregular particle arrangements

Theodoros Atmakidis and Eugeny Y. Kenig*

Numerical Analysis of Residence Time Distributionin Packed Bed Reactors with Irregular ParticleArrangements

Abstract: The computational fluid dynamics approach isapplied to packed bed reactors with moderate tube-to-particle diameter ratios. In order to generate irregular parti-cle arrangements within the reactors, a modified ballisticdeposition method is used. While our previous studies werefocused on pressure drop and mass transfer characteristics,the present paper uses the same framework to investigatethe dispersion phenomena occurring in such reactors. Twodifferent methods, namely the tracer and the post-processing method, are applied. The first method imitatesthe experimental procedure in which a non-diffusive traceris injected into the packed bed. The residence time can beevaluated from theevolutionof the tracer concentration. Thesecond method is purely numerical and allows fast resi-dence time estimation. Simulation results are successfullyvalidated against experimental data from literature. Thesuggested numerical analysis is a valuable tool toward adeeper understanding of the fundamental transport phe-nomena in such reactors and, consequently, for theimprovement of the packed bed performance.

Keywords: packed beds, residence time distribution, CFD,channeling effect

DOI 10.1515/cppm-2014-0021

1 Introduction

Packed bed reactors represent one of the most importantreactor types widely used in the chemical industry,among others for large-scale manufacturing of basic che-micals and intermediates as well as for the removal ofharmful or toxic chemicals from gas or liquid streams [1].The main advantage of packed bed reactors that makes

them very attractive compared to other more sophisti-cated and novel reactor types is their low cost [2].

Pressure drop, residence time distribution (RTD),mass and heat transfer characteristics are crucial for theoptimal design and operation of packed bed reactors,and, hence, their accurate estimation is very important.In our previous works, we analyzed the influence of theconfining walls on pressure drop [3] and mass transferphenomena [4] at different tube-to-particle diameterratios and Reynolds numbers. In the present paper, weuse the same framework and focus on the dispersionphenomena that take place within packed beds.

The flow in packed bed reactors is rather complex. Onone hand, there are regions with low resistance to flow,allowing a major portion of the fluid to flow through thispathway. This phenomenon is called channeling and it isidentified both experimentally and numerically [3]. Suchpathways are mainly located near the walls, but they mayalso occur inside the packed bed, in regions with high localporosity. Molecules that follow these pathways do not haveenough time to react within the packed bed, thus reducingthe overall conversion efficiency of the reactor. On the otherhand, regions with high flow resistance also exist within apackedbed. These regions are characterizedby low local bedporosity and almost stagnant fluid flow. Molecules that fallin such regions remain longerwithin thepacked bed causingan additional reduction of the overall reactor efficiency. Acommonway to capture and to study such flow inhomogene-ities inside a packed bed reactor is given by the RTD curves.

Several experimental studies of the dispersion inpacked bed reactors with large tube-to-particle diameterratios can be found in the literature [5–8]. The treatmentof experimental data in these works differs significantlyresulting in different conclusions and correlations. Amongothers, the effect of the confining wall on the dispersionhas not been examined in detail. However, confining wallssignificantly influence the dispersion and other local phe-nomena in packed beds with low and moderate tube-to-particle diameter ratios. For instance, in the near-wallregion, solid particles appear as looser arrangements com-pared to other regions, thus resulting in higher local por-osities near the walls. These higher porosities enhance

*Corresponding author: Eugeny Y. Kenig, Chair of Fluid ProcessEngineering, University of Paderborn, Pohlweg 55, D-33098Paderborn, Germany, E-mail: [email protected] Atmakidis, Chair of Fluid Process Engineering, Universityof Paderborn, Pohlweg 55, D-33098 Paderborn, Germany,E-mail: [email protected]

Chem. Prod. Process Model. 2015; 10(1): 17–26

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channeling effects and increase flow inhomogeneitiesinside a packed bed reactor. Despite that packed bed reac-tors with low and moderate tube-to-particle diameter ratiosare most suitable for industrial processes dealing withhighly exothermic reactions, their studies published inliterature are very scarce [9].

Several correlations of the axial dispersion coeffi-cient in packed beds were developed for large tube-to-particle diameter ratios based on experimental data.Most often, these correlations are given in terms of thedimensionless axial dispersion Peclet number expressedas a function of the Reynolds number. Generally, Pecletnumber is found to vary between 0.3 and 0.8 forReynolds numbers in the range of 0.01–150 [10]. For awide range of Reynolds numbers, the Chung and Wen[11] correlation can be used:

Pe ¼ 1εð0:2þ 0:011Re0:48p Þ ð1Þ

The number of published computational fluid dynamics(CFD) studies on dispersion in packed beds is limited [12].However, some recent works appear promising.Dispersion in packed beds of spheres with the aid of thelattice Boltzmann method at low Reynolds numbers0:4<Re<0:77ð Þ and at 182<Pe< 350 has been studiedby Manz et al. [13]. Numerical results were compared withexperimental data obtained using nuclear magnetic reso-nance measurements, and a quantitative agreement wasobtained in all cases.

Dispersion inside packed bed reactors was also stu-died by Zeiser et al. [14] for tube-to-particle diameterratios of 5 to 10. In their investigations, the Monte Carlotechnique was employed to generate the random spherearrangement. To simulate the fluid flow inside the packedbed, the lattice Boltzmann automata method isemployed. From the simulation data, the following corre-lation for the dispersion is suggested:

Dax

Dm¼ 0:758þ Pe1:83m

39ð2Þ

In the work of Freund et al. [15], the random arrange-ments of spheres are generated by a two-step method. Asa first step, a random arrangement is created using a“raining processes”; as a second step, the stack ofspheres is compressed by rearranging the spheres withan increased probability into the direction of gravity. Thecomplex fluid field is resolved using the latticeBoltzmann method, and the axial and radial dispersioncoefficients are estimated and compared with experimen-tal data from literature. A good agreement is foundbetween the simulated and experimental results.

Using the lattice Boltzmann method, Maier et al. [16]investigated dispersion in packed beds with large tube-to-diameter ratios and at low Reynolds numbers. Theyfound that the presence of the confining walls have asubstantial impact on the dispersion enhancement forlarge tube-to-diameter ratios (10 to 50).

In most works reported in literature, the latticeBoltzmann method is employed to study dispersion inpacked beds. This method is found to be very efficient,especially for complex geometric structures [13, 14].However, nowadays, such complex structures can also besimulated using traditional, more rigorous finite volumemethods supplemented by advanced solution algorithms.The objective of our work is to apply these rigorous CFDmethods based on the finite volume discretization to studythe dispersion in packed beds with moderate tube-to-parti-cle diameter ratios. For the estimation of the RTD, twodifferent approaches are applied and compared, namely,the tracer method and the post-processing method. Thesemethods were preliminary tested in our earlier work [17].Here, we give a detailed description of both methods andthoroughly test them using two different chemical systems.

2 Systems studied

To accurately capture the local phenomena occurring inpacked bed reactors, possibly small but representativeelements should be identified. In our previous study, weshowed that packed beds consisting of regular particlearrangements could not serve as representative elements[3]. Therefore, it was necessary to apply a method gen-erating irregular packed bed configurations. The particleswere assumed spherical and these spheres were notallowed to overlap. The employed method is based onthe ballistic deposition method and modified using theMonte Carlo approach. This method allows the genera-tion of random configurations; it is simpler to code andfaster than the most rigorous ballistic deposition algo-rithms [18]. The geometry construction and its validationare described in detail by Atmakidis and Kenig [3].

Two different systems are considered, a gas one anda liquid one. Their basic geometrical parameters are pre-sented in Table 1. The tube-to-particle diameter ratios (λ)lie between 1 and 7. The values of the particle diametersin the gas system are taken equal to those studied byTsotsas and Schluender [19]. This system was used in ourearlier study on mass transfer [4].

In Figure 1, the representative particle arrangementsstudied in our work are shown. It is worth mentioning thatthe impact of the confining walls both on hydrodynamics

18 T. Atmakidis and E. Y. Kenig: Numerical Analysis of Residence Time Distribution



Table 1: Basic system parameters.

Liquid system Gas system

Fluid medium Water Air

Bed diameter (mm) 3 5 7 4 4 4Particle diameter (mm) 1 1 1 2.83 2 1.06Tube-to-particle diameter ratio (–) 3 5 7 1.41 2.00 3.77Bed length (mm) 20 10 5 60 40 10Active bed length (mm) 20 10 5 16.52 11.68 5.25Bed porosity (–) 0.52 0.48 0.46 0.64 0.61 0.50Specific surface area (m2/m3) 2,760.2 2,948.0 2,876.2 753.3 1,163.9 2,405.6

(a) (b) (c) (d)

(f)(e)

Figure 1: Simulated representative particle arrangements: λ¼ 1.41 (a), λ¼ 2.02 (b), λ¼ 3 (c), λ¼ 3.77 (d), λ¼ 5 (e), λ¼ 7 (f).

T. Atmakidis and E. Y. Kenig: Numerical Analysis of Residence Time Distribution 19



and on mass transfer is of crucial importance for such lowtube-to-particle diameter ratios. The application of empiricalcorrelations developed for larger ratios can lead to substan-tial inaccuracies here, because these correlations neglect theeffect of the confining walls. Thus, new correlations shouldbe developed in which this effect is directly included. Asalready mentioned, low and moderate tube-to-particle dia-meter ratio covers an important range of industrial interestfor highly exothermic heterogeneous reactions [20].

3 Method description

Flow inside packed bed reactors is very complex anddeviates from the ideal plug flow. Regions with eithervery high or zero local velocities are met in such reactors.These regions are undesirable, because they reduce theoverall reactor performance. Thus, it is important to ana-lyze and better understand the reasons that cause theselocal flow inhomogeneities. This can be done with thehelp of CFD-based simulations.

To describe the complex 3-D velocity and concentra-tion field between the particles of a packed bed reactor,classical equations of continuity, momentum, and spe-cies transfer are used. The solid particles keep fixedpositions, while the void between the particles remainsconstant. Flow is considered to be incompressible andsteady state. The simulations are performed for Reynoldsnumbers below 100 (laminar flow regime).

To study the dispersion in packed beds, first, the flowfield has to be resolved. At the inlet and outlet of the packedbed reactor, periodic boundary conditionswith respect to themain flow direction are imposed. In this way, one can simu-late a fully developed flow neglecting inlet and outlet effects.Consequently, the computational domain decreases result-ing in a substantial reduction of the required computationaltime and power. On all particle and confining wall surfaces,the standard no-slip boundary condition is imposed. Furtherinformation on the hydrodynamic simulation and its valida-tion can be found in Atmakidis and Kenig [3].

As the next step, dispersion within the packed bedcan be studied using the previously calculated flow field.The RTD in packed bed reactors can be determined withtwo different approaches, namely the tracer method andthe post-processing method. In our study, we appliedboth methods and compared them with regard to theiraccuracy, usability, and properties.

The tracer method just imitates the conventional experi-mental procedure for the RTD determination [21]. In theexperimental procedure, a non-diffusive tracer is injectedinto the packed bed reactor. For the injection, most often a

pulse or step input is used. In principle, it might also be aperiodic or a random input, but they are difficult to interpretand hence are rarely in use. The pulse injection (Dirac pulse)of a tracer is normally applied in practice, because it gives theRTD directly, requires less tracer, is simple, and providessufficient information. An injection is considered as instan-taneous, when its duration is less than 3%of thewholemeanresidence time within the system. The concentration of thetracer is recorded at the outlet of the packed bed. From thetracer concentration as a function of time the dispersioncoefficient can be estimated. Similar approach is used alsoin our CFD modeling. A non-diffusive tracer is injected intothe computational domain using a Dirac function. The con-centration of the tracer is calculated and followed until itcompletely exits the computational domain. Here, standardinlet/outlet boundary conditions are applied. Using thenumerically calculated tracer concentration at the outlet ofthe computational domain, the dispersion coefficient is esti-mated. It is worth noting that, due to the specifics of thesystem under study, e.g., low residence time in the studiedpacked beds and permanent flow disturbance due to parti-cles, the influence of molecular diffusion on the RTD char-acteristics is negligible.

The second method employed in our work to describethe flow inhomogeneities in packed bed reactors is thepost-processing method. Provided that the velocity fieldis determined and assuming no reactions in the packedbed, local residence time can be obtained numerically, bysolving the following equation [22]:

u � τ ¼ 1 ð3Þwhere u is local velocity and τ is local residence time.

4 Implementation

The simulations were performed using the commercialCFD code ANSYS CFX 11.0® by ANSYS Inc. For an appro-priate discretization of the computational domain, anunstructured tetrahedral mesh was applied. This choiceis justified by the geometric complexity and stochasticnature of the simulated packed bed. To avoid low-qualitymesh elements at the contact points particle/particle andparticle/confining wall, different techniques have beensuggested in the literature to modify the particle geome-try at these points (see, e.g., [23]). In our work, thediameter of all particles was reduced by 2% after thebed generation. More exactly, in the ballistic depositionmethod, particles of 1 mm diameter were used. Afterwardthese particles were scaled down and the meshing wasdone for particles of 0.98 mm diameter.




The proper choice of the spherical particles shrinkagedepends on the required simulation accuracy and avail-able CPU power and time. For low shrinkage, both parti-cle size and reactor void fraction are close to the reality.However, for an accurate calculation of the flow field indifficult regions (e.g., those close to the particle touchpoints and those between the particles and the confiningwall), a very fine mesh is necessary. The simulationsbecome very demanding in terms of computationalpower and time. On the other hand, using particles withsubstantial shrinkage can result in unrealistic increase ofthe void fraction and, hence, non-representative geome-try. The latter brings about significant errors in the flowfield and in the pressure drop estimation.

A thorough analysis of this issue was performed inAtmakidis and Kenig [3], whereby the void fraction (localand global) and the pressure drop were examined and agood agreement with experimental data from the literaturewas found. In addition, Atmakidis and Kenig [4] studiedmass transfer in similar packed beds and, again, success-fully validated their numerical results against experimentaldata. Taking into account that the bed geometry stronglyaffects flow and mass transport characteristics, the gener-ated particle configurations can be evaluated as represen-tative enough. The 2% shrinkage was finally found to be agood compromise providing a simplified yet realistic geo-metry and ensuring a reasonable evaluation of the localhydrodynamic characteristics in the packed bed reactor.This is in accordance with the results published byNijemeisland and Dixon [24] who demonstrate that the use

of a similar particle shrinkage yields satisfactory flow fieldand the heat transfer calculations.

The solution was obtained using a second-orderdiscretization scheme. Special attention was paid to ensurean accurate resolving of the boundary conditions near theparticle surface. This was achieved with the help of pris-matic elements, while five layers were found to be suffi-cient for the good fulfillment of the boundary conditions.A grid independence study was performed for all simu-lated cases in a similar way as in our previous work [3].Depending on the geometrical complexity and the totalnumber of particles in the computational domain, up to30 million tetrahedral and prismatic elements were used.These demanding calculations were performed using fourprocessors (AMD Opteron MP 852–2.6 GHz) with a totalmemory of 64 GBytes in parallel. In most demandingcases, about two days simulation time was required forthe solution. For all RTD simulations, a tolerance value of10−5 was chosen as a termination criterion.

5 Results

5.1 Hydrodynamics

As a first step, the hydrodynamics was studied, whilewater (liquid system) and air (gas system) at 20 °C wereused as fluid mediums. In Figure 2, streamline plots areshown for the air system, illustrating the velocity distri-bution within a packed bed with tube-to-particle diameter

Figure 2: Velocity field of the air inside a packed bed for λ¼ 3.77: Re¼ 14.1 (a), Re¼ 78.4 (b).




ratios of 3.77 for two different Reynolds numbers,Re¼ 14.1 and Re¼ 78.4.

For both Reynolds numbers, high velocities weremainly observed in the region near the confining wallwhere the void fraction was high. Regions with high velo-cities also appeared in the inner regions of the packing.These regions of small resistance to flow allow a majorportion of the fluid to pass through the packed bed inshort times, thus, reducing the overall residence time ofthe fluid. At the inner part of the packed bed, stagnantregions could be observed in regions with low void frac-tion. Due to the existence of such regions, the flow withina packed bed can be characterized as highly inhomoge-neous. Some further details on the hydrodynamic behaviorof similar systems can be found in our previous work [3].

5.2 RTD by the tracer method

In the tracer method, the experimental approach for thedetermination of the RTD is followed in the CFD simula-tions, while a non-diffusive tracer is injected into thecomputational domain using a Dirac function. In allsimulations, the injection time was 0.005 s (less than1% of the whole mean residence time). From the tracerconcentration averaged over the cross section at the exitof the computational domain, the RTD function can becalculated using the following equation [21]:

E tð Þ ¼ Cave tð ÞÐ10 Cave tð Þdt ð4Þ

In Figure 3, the radial-averaged RTD curves as functionsof time are shown for λ ¼ 3 and for Re¼ 19.5 at differentaxial positions. Near the inlet at plane 1, the RTD curveis steep and looks similar to the ideal plug flow curve.

When moving downward to the outlet, the curvebecomes smoother and a long tail appears. The smooth-ing of the RTD curve shows that the flow inside thepacked bed reactor deviates from the ideal plug flow.This mainly happens due to the stagnant regions, butalso due to the channeling effect observed in suchreactors.

In Figure 4, the calculated RTD curves at the outlet ofthe packed bed reactor with λ ¼ 3 as functions of time aredepicted for different Reynolds numbers. At low flowrates, the residence time is large and the RTD curve hasa long tail. At higher liquid flow rates, the RTD curvebecomes narrow with reduced tailing. This implies thatthe flow inside the packed bed tends to plug flow as theflow rate increases, because, at these flow conditions, theinfluence of stagnant zones on the overall flow is reduced.

From the cross-section-averaged tracer concentrations asfunctions of time, the axial dispersion coefficient can beestimated. As a first step, the mean residence time can beevaluated from the following equation:

�τ ¼Ð10 t Cave tð ÞdtÐ10 Cave tð Þdt ffi

Pti Cave;iΔtiPCave;iΔti

ð5Þ

The second central moment of the RTD curves gives thevariance σt, the degree of dispersion around the meanresidence time that can be calculated using the followingequation:

σt ¼Ð10 t � �τð Þ2Cave tð Þdt

Ð10 Cave tð Þdt

¼Ð10 t2Cave tð ÞdtÐ10 Cave tð Þdt � �τ2 ffi

Pt2i Cave;i ΔtiPCave;i Δti

� �τ2

ð6Þ0

1

2

3

4

5

6

0 0.5 1 1.5 2

Res

iden

ce ti

me

dist

ribu

tion

[s−1

]

Time [s]

plane 1

plane 2

plane 3

plane 4

plane 5

plane 6

plane 7

plane 8

plane 9

Figure 3: Radial-averaged RTD curves of a packed reactor atdifferent axial position as function of time: λ¼ 3, Re¼ 19.5.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4 5 6 7

Res

iden

ce ti

me

dist

ribu

tion

[s−1

]

Time [s]

Re = 6.1 (1)Re = 9.5 (2)Re = 19.5 (3)Re = 20.2 (4)Re = 24.6 (5)Re = 27.5 (6)

Figure 4: RTD at the outlet of the packed bed reactor with λ¼ 3 fordifferent Reynolds numbers.




The dimensionless variance of the distribution isequal to:

σ ¼ σt�τ2

ð7Þ

The one-dimensional convection–diffusion equation inits dimensionless form is applied to describe the tracermovement inside a packed reactor. This equation readsas follows:

@c@θ

¼ 1Pe

@2c@z2

� @c@z

ð8Þ

Assuming that a tracer molecule can enter the systemonly through the entrance boundary and can leave thesystem only through the exit boundary, eq. (8) can besolved analytically. Based on this solution, the RTD as afunction of time can be obtained and the second centralmoment (eq. (6)) can be estimated and expressed accord-ing to van Gelder and Westerterp [25]:

σ ¼ 2Pe

� 2Pe2

1� e�Pe� � ð9Þ

These results are now used for the post-processing of ourCFD simulations. The cross-section-averaged concentra-tions are implemented into eqs (5)–(6), and the dimen-sionless variance (eq. (7)) is estimated. Subsequently,using eq. (9) and knowing the dimensionless variance,the Peclet number is calculated and, consequently, theaxial dispersion coefficient. Applying this method, weestimated different dispersion coefficients for differentpacked bed reactors and at different Reynolds numbers.

In Figure 5, the calculated Peclet numbers as func-tions of the Reynolds number are shown together with

the reported values from literature [9, 11]. The latter aremeasured for high tube-to-particle ratios. The correlationby Chung and Wen [11] is well established and takesaccount of the packed bed porosity. In Figure 5, weplotted the Peclet numbers calculated by this correlationfor bed porosity of 0.46 and 0.64, the minimum andmaximum bed porosities studied by Chung and Wen[11]. These Peclet numbers are in a relatively good agree-ment with ours. However, the dependence of the bedporosity on the numerically estimated Peclet numbersseems to be weaker than that predicted by the correlationof Chung and Wen [11]. It is worth noting that the Pecletnumbers for the moderate tube-to-particle diameter ratiopacked beds estimated from our simulations are in thesame range as those from literature.

5.3 RTD by the post-processing method

The RTD evaluated with the post-processing method fordifferent Reynolds numbers is shown in Figure 6. In thismethod, one can estimate the local residence time of thefluid and identify both high and low flow regions. Thelow flow regions have higher local residence times. Theyappear in the rear part of the particles and in regionswith low local void fraction. Regions with low localresidence time are the regions where extensive channel-ing appears. Such regions mostly exist near the confin-ing walls, but they may also be met in the inner part ofthe packed bed.

Using the post-processing, all of these regions can beidentified. Once the velocity field is known, the estima-tion of the local residence time with this method isstraightforward and less demanding in terms of computa-tional power and time.

5.4 RTD method comparison

In Figure 7, a comparison of the average RTD at differentcross sections of the packed bed reactor obtained withboth methods is shown. Both methods yielded similarlocal averaged residence time values, with deviationsbeing smaller than 5%.

It is worth mentioning here that the main advantageof the post-processing method is that it requires signifi-cantly less computational time and power compared tothe tracer method, since it is just necessary to solve anadditional passive equation at steady state. Using thismethod, it is possible to identify stagnant or high-velocity(channeling) regions, and in this way to obtain a good

0.01

0.1

1

10

0.01 0.1 1 10 100

Pecl

et n

umbe

r [–

]

Reynolds number [–]

Tang et al. [9] – averaged curveChung and Wen [11] – ε = 0.46Chung and Wen [11] – ε = 0.64λ = 3.00λ = 5.00λ = 7.00λ = 1.44λ = 2.00λ = 3.77

Region of literature values

Figure 5: Calculated Peclet number as a function of the Reynoldsnumber for all examined packed bed geometries. Symbols – oursimulations performed with the tracer method, lines – data fromliterature.




first estimation of the flow inhomogeneities within thepacked bed reactor.

The tracer method requires significantly more com-putational time, because an additional transport equationhas to be solved in a transient mode. Simulation timesusing the tracer method were 10–20 times higher com-pared to the post-processing method. On the other hand,this method can provide better understanding of the flow

inhomogeneities and an accurate estimation of the axialdispersion coefficient.

6 Conclusions

In this work, dispersion in irregular beds of spheres wasinvestigated by means of CFD. Different packed bed con-figurations with different tube-to-particle diameter ratioswere studied.

For the generation of irregular packing configura-tions consisting of non-overlapping spherical particles,a ballistic deposition method was employed in combi-nation with the Monte Carlo method. This approachyields representative irregular particle configurations.

The numerical analysis of dispersion in packed bedswas performed using two different methods, namely, thetracer and the post-processing method. In the tracermethod, we imitated the experimental procedure usedfor the estimation of the RTD curves. At the inlet of thepacked bed, a non-diffusive tracer was injected. Using theCFD modeling, the evolution of the tracer concentrationwas followed until the tracer left the packed bed. Fromthe tracer concentration curves as functions of time,the axial dispersion coefficient was estimated. In thepost-processing method, an additional steady-state equa-tion was solved for the estimation of the local residencetime.

Figure 6: RTD (in seconds) obtained by the post-processing method for λ¼ 3.78: Re¼ 17.87 (a), Re¼45.08 (b).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Ave

rage

res

iden

ce ti

me

[s]

(pos

t-pr

oces

sing

met

hod)

Average residence time [s](tracer method)

z = 2 mm

z = 4 mm

z = 6 mm

z = 8 mm

z = 10 mm

z = 12 mm

z = 14 mm

z = 16 mm

z = 18 mm

Figure 7: Method comparison: Average residence time obtained forλ¼ 3 by the post-processing and the tracer method at differentpacked bed reactor heights.




Using the tracer method, RTD curves were deter-mined for different packed bed configurations and atdifferent Reynolds numbers. At low Reynolds numbers,the flow within a packed bed substantially deviates fromthe ideal plug flow. This deviation is mainly due to thestagnant regions. In these regions, tracer molecules areaccumulated and need more time to reach the packedbed reactor exit. For this reason, a long tail is observed atthe RTD curves at low Reynolds number. At higher flowrates, the effect of these stagnant regions decreases andthe flow approaches the ideal plug flow. From the RTDcurves, Peclet number was calculated and compared withexperimental data from literature. A good agreementbetween the calculated and the experimental Peclet num-bers could be found.

Using the post-processing method, the local resi-dence time can be calculated, and, consequently, flowinside the packed bed can be characterized. Additionally,stagnant or channeling zones can be identified.

A comparison between the two proposed methods forthe dispersion study in packed beds was performed. Bothmethods yielded almost identical local averaged residencetime values, with deviations smaller than 5%. The post-processing method requires less computational power andtime compared to the tracer method. This method can beused for a fast estimation of the flow field inhomogene-ities. On the other hand, the tracer method gives a betterunderstanding of the flow inhomogeneities within thepacked bed reactor. With this method, it is possible toderive RTD curves that can further be used to estimatedispersion coefficients with high accuracy and reliability,thus reducing the necessary number of experiments.Generally, the developed numerical approaches providethe knowledge that is often hard to obtain experimentallyand hence they can help to significantly improve design ofpacked bed units with low and moderate tube-to-particlediameter ratios.

Nomenclature

A Specific surface area, m2/m3

c ¼ CC0

Dimensionless tracer concentration, –C Local tracer concentration, mol/m3

Cave Cross-section-averaged tracer concentration, mol/m3

C0 Total injected tracer concentration, mol/m3

Dax Axial dispersion coefficient, m2/sDm Molecular dispersion coefficient, m2/sD Diffusion coefficient, m2/sE Residence time distribution, 1/sdp Particle diameter, m

L Packed bed length, m

Pe ¼ U0dpDax

Peclet number, –

Pem ¼ U0dpDm 1�εð Þ Molecular Peclet number, –

Re ¼ dpρU0

μ Reynolds number, –t Time, su Local velocity, m/sU0 Superficial velocity, m/sz ¼ Z

L Dimensionless vertical coordinate, –Z Vertical coordinate, m

Greek lettersε Porosity of the packed bed, –θ ¼ t

�τ Dimensionless time, –λ Tube-to-particle diameter ratio, –µ Dynamic viscosity, kg/m sρ Density, kg/m3

σt Second central moment of the RTD, s2

σ Dimensionless second central moment of the RTD,–

τ Local residence time, s�τ Mean residence time, s

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Numerical analysis of mass transfer in packed-bed reactors with irregular particle arrangements

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