a new approach to fixed bed radial heat transfer modeling using velocity fields from computational...

8/13/2019 A New Approach to Fixed Bed Radial Heat Transfer Modeling Using Velocity Fields From Computational Fluid Dyna

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for convective transport. Methods to generate the packings andmesh development strategies are an essential part of this step. The

velocity and pressureelds then must be incorporated into a 3Denergy balance to give a 3D temperature eld. This is then to beaveraged to provide a 2D temperature eldT(r,z) which is the

basis for comparison to our new 2D pseudocontinuum model.For this new model, the 3D velocityelds from the CFD model

simulations have to be coarse-grained or averaged to extract theessential 2D informationto be supplied to the pseudocontinuummodel. Our early work5 suggested that we would need vz(r) andvr(r,z), to properly capture the radial convective transport of heatand the near-wall changes in transport. In addition, we use the

well-established ZehnerSchlunder formula6 for conduction heattransfer in a stagnant bed to represent the conductive contributionke

0(r) to the overall heat ux. These contributions are bothincluded in the pseudocontinuum model. The resulting 2D

temperature

elds are then compared against the 2D temperatureelds obtained from direct averaging of the 3D CFD results.Further details of each of these steps in Figure1 are provided inthe following sections.

2. BACKGROUND TO VELOCITY-BASED MODELS

The traditionalapproach to xed bed radial heat transfer for thelast 60 years7,8 has been the classical two-dimensional pseudo-continuum heat transfer model and its boundary conditions asembodied in the following equations:

=

c u

T

zk

r rr

T

r

1p 0 r

(1)

= | ==k

T

r h T T( )r Rr Rr w w

(2)

=

=

T

r0

r 0 (3)

| ==T T r( )z 0 in (4)

Several variations of this basic plug ow (PF) model exist,including those that incorporate axial dispersion or conductionterms. In this model the effective radial thermal conductivitykr

was used to represent all mechanisms for radial heat transferinside the bed, such as conduction, convective radial displace-ment ofuid, and particleparticle radiation. Despite evidence

thatkrvaried with tube radius, especially near the tube wall, itwas usually taken as constant to simplify both parameterestimation and solution of the model equations. The strongdecrease inkras the tube wall was approached was idealized to

be a heat transfer resistance located at the wall, and wasrepresented by the wall heat transfer coefficient, hw, along with atemperature jump at the wall. The parameters kr and hw eachreect the effects of several different heat transfer mechanisms, andhave proved difficult to determine over the years, especially at low

N, while reaction models based on them have been criticized asbeing oversimplied. Several papers and reviews have addressedthese concerns, many of which were summarized recently.2

One of the main perceived failings of the PF model has beenthe use of a radially uniform axial velocityu0to represent ow inthe tube. A comprehensive review ofuidow in packedtubes upto 1987 was given by Ziolkowska and Ziolkowski9 whichdemonstrated that the prevailing opinion was that the constantu0should be replaced byvz(r) and the wall heat transfer coefficienthw should not be used. Several research groups have developed

various approaches to obtain vz(r), including extended capillarymodels,10 the extendedBrinkmanDarcyForchheimerequationfrom either particle-based11,12 or porous-media based1315 methods,a combination of these two models,16 and variousmodels derivedfrom the volume-averaged NavierStokes equations.1720 Several ofthese ow models found it necessary to introduce an effective

viscosity into the equations,12,17 which introduces anotherparameter that requires estimation and to some degree negatesthe advantage of dispensing with the wall coefficient. In addition,the effective radial thermal conductivity is retained, but must now

be re-estimated for use in the altered model.21,22

In parallel with the problems associated with heat transfermodeling in packed tubes, several authors have expresseddissatisfaction with the standard dispersion model (SDM) whichuses effective diffusion to represent axial and radial dispersion23

and has drawbacks including innite speed of propagation andoverestimation of back-mixing. Some differentapproaches to this

problemhave included the cross-ow model,24 the alternatingowmodel,25 and the wave model, rst put forward by Stewart26 andmore recently strongly championed by Kronberg and hiscolleagues.27,28 The application of the wave model to xed bedheat transfer was demonstrated by Kronberg and Westerterp27

whose work showed that this model also results in parameters thatmust be determined from experimental data.

Kronberg and Westerterp27 in particular presented a strongargument for the need for a new approach to modeling transportinxed beds. The crux of their argument was that for many years

we have used effective diffusion and conduction models torepresent heat and mass transfer phenomena that are essentiallyuid mechanical in nature. The reason for this has largely beencomputational convenience, a constraint that is rapidly being eased

by the development of faster, larger computers and improvednumerical methods. It should be possible to move toward modelsthat more realistically represent the oweld in a xed bed. Someof these points of view have more recently been echoed bySchnitzlein29who pointed out that effective dispersion coefficientsare commonly used with gradients in concentration and that alarge contribution to dispersion isuid mechanical which is solelydriven by the packing structure and not by any concentrationgradient.

One alternative approach to radial heat transfer has been toconsider two- or three-dimensional ow elds, that is, toinclude velocity components transverse to the main direction ofow. Early attempts to obtain such ow elds were made by

Figure 1. Schematic of the relationships between the modelsinvestigated in this study.

Industrial & Engineering Chemistry Research Article

dx.doi.org/10.1021/ie4000568|Ind. Eng. Chem. Res. 2013, 52, 152441526115245


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Stanek and Szekely30 who substituted a two-dimensionalvoidage into the differential form of the vectorized Ergunequation. Their model could not satisfy the no-slip conditionon the tube wall, and also highlighted the need for a realistic

voidage or other suitable measure of bed structure to obtainow elds. This could be provided by packing networkmodels31,32 or cell models33 or even statistical models34 whichhave all had computational drawbacks. Several attempts have

been made to obtain two components of velocity from the two-dimensional NavierStokes equations, but most of themconcluded that radial velocity components were negligible infully developed ow in the packed bed and reported results foronly one-dimensional models.12,19,20

Radial heat transfer models that included two components of

velocity were investigated by several authors.3538

Theequations used took the general form

+

=

c G r z

T

zG r z

T

rk

r rr

T

r( , ) ( , )

1zp r r

(5)

= |

==k

T

rh T T( )

r Rr Rr w w

(6)

=

=

T

r0

r 0 (7)

| ==T T r( )z 0 in (8)

where some authors replaced eq6 with

| ==

T Tr R w (9)

and others included axial terms. Most neglectedthe radial massuxGr(r,z), for example Stanek and Vychodil

35 concluded thatradial ow termswere less than 1%ofthe velocity magnitude,

while Eigenberger36 and Froment37,38 and their co-workersstated that strong radial ow components were found only inthe rst particle layer or a short entrance region.

The conclusions of the previous paragraph seem tocontradict the well-established idea that the main contributorto radial heat transfer, at least as Re increases, is radialdisplacement of uid around the particles, that is, convectivedispersion. One explanation is that the uid mechanics modelsthat were used to obtain the velocity components were all

based on averaged measures of bed structure, usually voidage.Some used(r,z) and others only(r); however, all involved adegree of smoothing of the bed structure. We suggest here thatthe higher values of vr along the entire bed that would beneeded to account for the observed rates of radial heat transferare suppressed by the use of smoothed voidage elds in theNavierStokes equations. In fact, the void fraction at a point

can have only values of zero or one, and it changes abruptly atthe local level as the radial coordinate passes from particle touid and back many times. It is these abrupt changes in voidfraction that give rise to the redirection of ow, giving stronglocalvariations inp andvr, which in turn result in the observedradial heat transfer rates. These local variations can be averagedout in smoothed or global approaches to xed bed structureand uid ow.

To avoid uncertainties in the existing models ofow in xedbeds and the desire to avoid premature smoothing of thevelocity components at the local level, a different approach is touse simulations of velocity elds directly in heat transfermodels. For example, Dixon et al.5 put forward a model of

Figure 2.Simulated packed beds for CFD analysis: (a) N= 3.96, (b) N= 5.96, (c) N= 7.99.

Table 1. Comparison of Experimental to Computer-Generated Overall Voidage

N (expt) (computer model) % deviation

3.96 0.476 0.466 2.1

5.96 0.451 0.450 0.2

7.99 0.431 0.432 0.2




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xed-bed heat transfer that employed the uid and solidthermal conductivities of the phases and a two-dimensional

velocity eld composed of the components vz(r) and vr(r,z).The velocity components were calculated from a computer-generated packing, which was transformed into a networkmodel.31 Flows through the network branches were calculatedand then averaged to obtain the velocity components. Themodel was able to give heat transfer through the bed centerreasonably well but it was not as accurate in the wall region.One explanation was that the ow channels parallel to the wallcould not be included in the network and in addition during theaveraging process strong radial ows to and from the wall wereadded to give no net ow, although the net heat transfer wasprobably signicant. Problems with the network model near thetube wall prevented further development of the heat transfermodel at that time.

Ziolkowska and Ziolkowski, in their effective viscositymodel,17 derived a ow model that was macroscopically one-dimensional in the direction of the pressure drop, but whichcould have local radial components whichwere related to uidradial dispersion. In their later work18 a radial dispersioncoefficient was included directly into the equation of continuity

and obtained by analogy from friction factor correlations. Theequation of continuity was then integrated analytically to obtainthe interstitial component vr. More recently Schnitzlein

29

attempted to capture the uctuations in the local velocitiescaused by the local packing structure. His approach was to usecontinuum models dened in terms of a two-dimensionalspatially dependent porosity. From a computer-generatedsphere pack Schnitzlein obtained an angularly averaged voidage(r,z) which he used in the NavierStokes equations. Theasymptotic value for the radial Peclet number was found to bemore than twice as high as the experimental value ofPer() = 11.Using a three-dimensional network model which did not involveaveraging of the bed structure, better agreement for dispersion wasfound.39

Recent developments have been made in CFD which allowthe simulation ofow, heat, and mass transport in full beds ofspheres of several hundred particles.4043 Such computationscould allow the actual local values of the velocity componentsvzandvrto be obtained directly by simulation, with no need for amodel. Several of these studies have presented axial velocityvz(r) contours or proles, and more recent workhas begun togive axial proles of area-averaged radial velocity.44 The objectof the present work is to combine the model of Dixon et al.5

with CFD simulations to demonstrate the concept thatkrandhwcan be replaced by the velocity components vz(r) andvr(r,z),along with a model of stagnant bed conduction ke

0(r), to give amore physically realistic description of xed bed radial heattransfer.

3. DISCRETE PARTICLE (CFD) BED GENERATION

CFD simulations of full beds of spheres played a major role inour methodology. We have generated a range of tube-to-particle diameter ratios (3 N9.3) for sphere-packed beds.To do this, we adapted a published soft-sphere algorithm45

which produces sphere packs with lower voidage than the usualdrop-and-rollpacking algorithm. The algorithm rst places apredetermined number of spheres Np of given diameter dp atrandom positions inside a cylinder by allowing interpenetration

between the particles. The cylinder has diameter such as to givea speciedN, and a chosen initial voidage0sets the initial tubelength. The overlaps are then reduced in the absence of gravity

to a user-specied overlap tolerance by moving spheres in turnso as to expand the bed vertically. A gravitational force is thenapplied downward on each sphere to compact the bed, movingthe spheres so as to reduce the particle center of mass until astopping criterion is reached, while respecting the overlaptolerance. The results have been found to usually be in closeragreement with published experimental data than previous

algorithms for conned beds.We found that the random allocation of spheres in thiscollective rearrangement type of algorithm gave someunrealistic sphere packings at the bottom layers of the bed,especially for lowerN. For these simulations, it was decided to

build the packing from a base layer with spheres in a ringaround the wall. We therefore combined the original algorithm

with an initial position algorithm46 to more realistically locatethe wall layer of spheres at the tube bottom. With somemodication this solved the problem, and this combinedrandom-deterministic algorithm was used to generate a range ofsphere packs with nominal diameter dp = 0.0254 m.

Three values of N were chosen for detailed study in thiswork, N = 3.96, 5.96, and 7.99. These values were chosen

through consideration of the available experimental data, andalso to cover a reasonable range ofN. Side views of the threepackings are included in Figure2to give a sense of the type ofstructure that can be generated, along with the relativedimensions. For the N = 3.96 bed, 250 spheres were used,theN= 5.96 bed was generated from 400 spheres and for the

N= 7.99 bed there were 800 spheres in the model.For each packing, overall voidage was calculated from the

nominal tube and particle diameters, the packed bed length,and the number of particles.Experimental values were obtainedfrom the resultsofMueller47 as reported in the later paper bythe same author,48 and comparisons are presented in Table1.The overall voidage in the computer-generated beds is general-

ly lower, as would be expected from a soft-sphere algorithm.

Figure 3. Comparison of radial bed voidage proles for N = 3.96between experimental measurements and computer-generated spherepack calculations.




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The agreement is excellent for the three cases shown, withslightly higher error for lower N.

Further validation of the computer-generated structures wasconducted by comparison of the radial voidage prole toliterature data, again from Mueller.48 The validation compar-isons are shown in Figures35, as plots of voidage as a function of

the dimensionless distance from the tube wall, in multiples ofparticle diameter. The voidage proles were calculated from a list

of the sphere positions for the entire bed in each case using the

formulas developed by Mueller.49,50The voidage proles forN= 3.96 are shown in Figure3. The

prole shows two minima corresponding to the two layers ofspheres along the radial coordinate. Overall, the features andmagnitude of the experimental () are well-reproduced. Thehigh void fraction at the tube center ( 2) is caused by theholedown the tube center due to the packing structure. The

voidage prole forN= 5.96 is shown in Figure 4; some slightshift of the prole toward the tube wall may be attributed to thesoft-sphere algorithm which produces a more compactedpacking. This feature was also observed in similar algorithmspreviously.45 The downturn for values of 3 is due toanomalies at the center of the bed, where it is difficult to denesmall enough surfaces to obtain accurate values. Nevertheless,

this region is very small and of lesser importance comparedto the near-wall region where excellent agreement is found.The voidage prole of the N= 7.99 bed is shown in Figure 5and shows similar features and good agreement. Themagnitudes of the maxima and minima are especially accuratelyfound by the algorithm. The low voidage at 4 (tube center)for the computer-generated packing is not an anomaly; forthis particular packing the spheres lined up along thecenter-line.

The general good agreement shown in the graphsdemonstrates that the computer-generated sphere packreproduces the essential features of experimental measure-ments. The locations of maxima and minima are correctlyreproduced, as well as their magnitudes. This nding as well as

the results for overall voidage gives us condence in ourcomputer-generated models for the simulation of the velocityand temperatures in a xed bed.

4. 3D DISCRETE PARTICLE (CFD) SIMULATIONMODEL

The equations for the CFD simulation of uid ow and heattransfer in a single phase in this study are the equations ofconservation of mass, momentum, and energy. The con-servation of mass (continuity) equation is

+

=

t

u

xS

( )i

im

(10)

Figure 6.Verication of CFD solution for supercial axial velocityvzfor theN= 3.96 bed and Re = 240, using three mesh sizes.






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In the simulations described here, the source term Smwas equalto zero. The equation for conservation of momentum indirectioni is

+

=

+

+ +

u

t

u u

x

p

x xg F

( ) ( )i i j

j i

ij

ji i

(11)

In this equationp is the static pressure, ijis the stress tensor,andgiis the gravitational body force. For the present work theexternal body force termFiwas zero. The stress tensor ijfor aNewtonian uid is dened by

=

+

u

x

u

x

u

x

2

3ij

i

j

j

i

l

lij

(12)

Here is the molecular viscosity and the second term on theright-hand side of the equation is the effect of volume dilation.The energy equation is

+

=

+ +

+

h

t

u h

x

x

T

x

h J

x

Dp

Dt

u

xS

( ) ( )

( )

i

i

i i

j j j

iik

i

kh

(13)

In this equationh is the enthalpy and for the present study theuser-dened volumetric heat source termShwas zero. Radiation

was not included in the CFD simulation model due to therelatively low laboratory-level temperatures simulated. Theabove equations were solved in their original form for laminarows; for turbulent ows the Reynolds-averaged NavierStokes (RANS) models were used, with the original equations

being ensemble-averaged.With RANS models all turbulence length scales are modeled.

The solution variables are decomposed into mean, u i and

uctuating,ui components and integrated over an interval of

time that is large compared to the small-scale uctuations.When this is applied to the standard NavierStokes equations,the result is

+

=

+

+

+

u

t

u u

x

p

x x

u

x

u

x

u

x

u u

x

( )

2

3

( )

i i j

j i j

i

j

j

i

l

l

i j

j (14)

The velocities and other solution variables are now representedby Reynolds-averaged values, and the effects of turbulence are

represented by the Reynolds stresses, u u( )i j . To close the

system of equations the Reynolds stresses are put in terms ofthe averaged ow quantities. In the present work we used ak-two-equation model, which is a two-zone model designed to beintegrated all the way to the wall, provided that a sufficientlyne mesh is used there. Descriptions of the k- turbulencemodel are available in standard references and will not berepeated here. Laminar ow models were used for the threelowest ow rates simulated, and a turbulent ow model wasused at the highest ow rate, which corresponded toRe= 1900.

Boundary conditions for the momentum differentialequations were provided by taking the no-slip condition onall solid surfaces, both tube wall and particles. A uniform velocityprole was used at the tube inlet, and a pressure of 1 atm was

set at the tube outlet. For the energy balance, the tube walltemperature,Tw= 368.15 K and the temperature of the inletowTin = 298.15 K were specied. At the particle soliduidinterfaces continuity of temperature and heat ux was enforced.

CFD simulations were carried out to obtain velocity andtemperature elds in full beds of spheres for the three cases,

N = 3.96, N = 5.96, and N = 7.99. The nominal particle

diameter was 1 in. (0.0254 m) in all columns and the nominaltube diameters were 3.96 in. (0.1009 m), 5.96 in. (0.151384 m),and 7.99 in. (0.202946 m). The models had a length of 0.0254 mof empty tube before the bed inlet and a length of 0.0508 m ofempty tube after the bed to be able to place the inlet and outlet

boundary conditions away from the packing. The packedlengths of the columns were as given in Figure 2. Simulations

were run over a range of ow rates to give Re in the range802000.

The uid for the CFD simulations was taken as air withconstant properties corresponding to a bed average temper-ature of 333.15 K. These were density = 1.059545 kg/m3,

viscosity= 2.0291105 kg/ms, specic heatcp= 1800 J/kgK,and thermal conductivitykf= 0.0287 W/mK. The particles weretaken to be alumina with properties as densitys= 1947 kg/m3,specic heat cps = 1000 J/kgK, and thermal conductivity ks =1.0 W/mK.

The model geometries and the mesh were constructed usingthe commercial software GAMBIT 2.4.6, with the help of

journal les to carry out the repetitive creation and placementof the spheres. To obtain a ne enough near-wall mesh for thek- method we used boundary layer prism cells at outsideparticle surfaces and at the tube walls; tetrahedral cells wereused in the main uid volume and inside the particles. Theunstructured tetrahedral mesh cell size was 1.524 103 m(dp/16.7) and the boundary layer mesh thickness was 2.54 105 m (dp/1000) with three layers on the tube wall and a

single layer on the particle surfaces. The N= 3.96, 5.96, and

Figure 7.Verication ofnite element solution of 2D pseudocontin-uum model with constant coefficients against analytical solution atdifferent bed depths.




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7.99 total mesh sizes were 8.904 million, 15.579 million, and28.47 million cells, respectively.

To remove the problem of meshing around the contact pointsbetween the particles and between the tube wall and the particles,the technique of shrinking the diameters of the particles to 99% ofthe original diameter was used, so that the particles had an actualdiameter 0.025146 m. To provide the same size gaps for theparticlewall contact points the tube diameters were all increased

by 2.54 104 m also. This decision implied that heat transfer byparticleparticle or particle-wall area contacts was not representedin this model. Other approaches to the problem of meshingaround contact points have been developed, and these wererecently reviewed and compared.51 Although the use of gaps

between particles does affect heat transfer uxes and temperaturedistributions, in this study the same simplication was made inapplying the formula for the effective stagnant thermal

conductivity in the pseudocontinuum vzvr model, so that thecomparisons were made on the same basis.

To verify mesh independence, a mesh renement study wascarried out on the velocity proles in the N= 3.96 tube for Re =240. Three mesh sizes were compared, the base case size of 1.524 103 m (dp/16.7), and two ner meshes of 1.27 10

3 m (dp/20)and 1.016 103 m (dp/25). The three corresponding proles ofsupercial velocityvz(r) are presented in Figure6 where they areshown to coincide almost exactly except for a small region at the

bed center, where the velocity is higher due to the holein thepacking along the centerline which is typical for N= 4 beds. Thisshows that the base case mesh (dp/16.7) is acceptable for thepresent study of velocityelds.

The governing equations described above were solved usingthe nite volume commercial CFD code FLUENT 6.3.26. Thepressure-based segregated solver was used, with the SIMPLE

Figure 8.Contours of (a) axial velocity (m/s), (b) radial velocity (m/s), and (c) temperature (K), in the x = 0 plane of the N= 3.96 xed bed andforRe = 240. Dotted boxes indicate regions used for close-up velocity vectors shown in Figure 9.

Figure 9. Close-up analysis of boxed regions from Figure 8with N= 3.96 and Re = 240: (a) velocity vectors colored by axial velocity (m/s),(b) velocity vectors colored by radial velocity (m/s), and (c) temperature contours (K).




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scheme for pressure-velocity coupling. First-order upwindinterpolation was used for the convection terms; all diffusionterms used second-order discretization. Tests using second-order upwind interpolation for N= 3.96 andRe = 240 showedthat the velocity prole was unchanged for the most part,except for small differences very close to the tube wall and also

in the bed center. This result implies that rst-order upwindinterpolation is adequate for calculating velocity proles, but

will likely not be sufficient for calculations of pressure drop.Under-relaxation factors were left at the FLUENT defaultsettings, unless some instability was observed in the iterations,

when they were occasionally reduced. The convergence was

Figure 10.Cross-section ofN= 7.99 xed bed column showing radial surfaces and angular planes used for averaging and sampling of the CFDresults.

Figure 11.Averages over angular surfaces in theN= 3.96 bed,Re = 240: (a) void fraction, (b) temperature (K), (c) radial velocity (m/s), (d) axialvelocity (m/s).




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6.1. Temperature and Velocity Contour Analysis on aSingle Plane. The axial and radial velocity and temperaturecontours on a single plane at constant in the z- andr-directions for theN= 3.96 xed bed column andRe= 240 arepresented in Figure8 panels a, b, and c, respectively. In thesegures, only the central portion of the bed is presented, toavoid anomalies due to end effects. This corresponded to

0.1397 z 0.4191 m, approximatelyve particle layers fromthe inlet and three particle layers from the bed exit.

The axial velocity in Figure8a is high at the center of the bedwhere the bed voidage is close to unity and also near the tubewall at aboutr/R= 0.98. It then decreased to zero because ofthe wall boundary layer no-slip condition. In addition, the axial

velocity plot shows that the ow velocities near the particlesurfaces are zero, as dened by the no-slip condition on all thesolid surfaces in the geometry. Comparisons of the axial

velocityvzto the total velocity magnitude showed that the axialvelocity was usually the dominant component. When there wasa difference between vz and |v|, it meant that anothercomponent of ow played a signicant role, and in this caseit was the radial velocityvr.

The radial velocity in Figure 8b exhibited positive andnegative velocities extending over a range from 0.75 to 0.82m/s. Although most of the values clustered around zero, manysmall regions could be seen with positive and negative velocitiesup to 0.3 m/s in magnitude. The positive radial velocities meanthose velocity vectors which moved from center of the bed tothe tube wall, and negative radial velocities mean those

velocities which moved from tube wall to the center of thebed. Radial velocity had a signicant effect on the temperature

distribution at a local level in the bed; when radial velocity wasnegative and the ow direction was to the center of the bedthen high temperature uid owed from heated tube wall andpenetrated into the center of bed.

The temperature contours in Figure8c show that the radialtemperature prole did not develop smoothly from the inlet tothe outlet of the bed. The temperature contours haddevelopment and reduction in the radial position. This wasdue to the effect of the radial velocity that dominated the radial

Figure 13.Average of radial velocity contours in xed bed columns of (a)N= 3.96, (b) N= 5.96, and (c) N= 7.99 for Re = 80.

Figure 12. Supercial axial velocity proles for the three differentvalues ofN, Re = 240.




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heat transfer mechanism, supplemented by the uidparticleuid conduction path, rather than to the axial velocity. Thedotted-line box superimposed on the temperature contours inFigure8c highlights the uneven propagation of the temperatureinto the center of the bed. Corresponding dotted boxes werealso marked in Figure 8a and Figure8b for the velocity plots.

Velocity vector plots represent the uid velocity magnitude

and direction at each control volume. To study the localvelocities corresponding to the dotted line box in Figure8c, thevelocity vectors are presented in Figure 9a colored by axialvelocity and in Figure9b colored by radial velocity. Figure9cshows the temperature development and reduction moreclearly in the line box. The velocity vectors colored by axial

velocity were high at the center and close to the tube wall in thelow voidage area where the distance between particles or

between particles and the wall was larger. The velocity vectorscolored by radial velocity had negative and positive valuesdepending on the particle distributions. The radial velocity

vectors illustrate how the temperature distribution to the centerof the bed was changed locally in the radial position. When theradial velocities were negative and the velocity vectors left thetube wall to the center of the bed (between particles 4, 5 and 6),the temperature proles were more developed in the center of the

bed due to the enhanced transfer of high temperatureuid fromthe heated tube wall. However when the radial velocities werepositive and the velocity vector approached the tube wall thetemperature proles were reduced (between particles 1, 2, and 3)as the heat transfer from the tube wall was inhibited by the localmotion of the uid.

The analysis on this single plane emphasizes the importanceof the local radial velocity components, and the necessity forsampling and averaging methods to avoid cancellation effects

which would diminish the effect of the local velocityuctuations.

6.2. Coarse-Graining the 3D Velocity Fields. The voidfraction and axial velocityvzwere extracted from averaging of

diff

erent cylindrical planes in the radial direction inside thexed bed, as illustrated in Figure 10 at the left. These twoquantities were therefore averaged over both angular () andaxial (z) coordinates. In the z-direction, again only the centerparts of the beds were used, to avoid end effects, and thesecorresponded to 0.1397 z 0.4191 m for theN= 3.96 bed,0.127 z 0.296 m for theN= 5.96 bed, and 0.0762 z 0.3048 m for the N= 7.99 bed.

To determine radial velocity as a function of radial and axialposition, the straightforward extension of the averaging methodto radial cylindrical planes subdivided into small increments inthe axial direction was not used. Instead, we dened 32 angularplanes from the center of the bed to the tube wall at 11.25spacing. Figure 10 on the right shows the position of the

angular planes inside the

xed bed. All velocities were averagedat the samerand z position in all the angular planes together.But the cells in the different angular planes were not located atthe samerandzpositions because of the different unstructuredtetrahedral meshes in each plane. Therefore we usedinterpolation for all planes to have values of the radial velocitiesat the same rand z positions and then averaged them. In thiscase 200 (radial) 200 (axial) points were extracted for eachplane, and then velocity components at the corresponding





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points of all planes were averaged. The number of angularplanes used varied depending on the Nvalue and on Re, as it

was found that fewer planes could sometimes be used withgood results. This averaging procedure gave better resolution ofvr(r,z) than other methods that were tried, which was especiallyimportant to capture the variation of the radial velocity near thetube wall.

6.3. Local Features in the Averaged Fields. Figure11shows the angular averages of bed voidage, temperature, radialvelocity, and axial velocity forN= 3.96 atRe= 240, in the (r,z)plane. The gures show the entire tube length, including theunpacked inlet and outlet sections. As can be seen in Figure 11athe 2D voidage distribution in the bed represented con-rmation that assuming the voidage prole as a function of onlyradial direction was a reasonable assumption, since the high bed

voidage area was located at the center of the bed and close tothe tube wall for the entire axial direction. The axial velocityprole followed the voidage prole and it was not changedsignicantly in the axial direction so that it was also safelyassumed to be a function of radial position only (Figure 11d).The radial velocity, in contrast in Figure11c had a variety of

different values in both axial and radial directions, conrmingthat it had to be assumed as a function of both r and zdirections. The averaging damped out the more extreme local

values shown in Figure 8b for a single plane, but retainedsufficient variation to represent the radial heat transfer, asillustrated in the next section. The two-dimensional averagedtemperature distribution is also shown for comparison, inFigure11b. The region of lower temperature at the center of

the outlet area corresponds to an area of recirculating uidbehind the packing.

7. RESULTS OF COMPARISON OF MODELS

For the comparisons of vz() shown in Figure 12, only themiddle parts of the packed beds were used; that is, thecylindrical surfaces were clipped to the z-values listed insection

6.2. For all the two-dimensional comparisons of vr(r,z) andT(r,z) shown in Figures1320again only the middle parts ofthe packed beds were used, and the z-coordinate was reset tozero at the start of the sample region.

7.1. Axial Velocity Proles. The radially varying axialvelocity vz(r) was extracted from the averaging of differentcylindrical surfaces in the radial direction inside the xed bed.The averaged axial velocity corresponding to Re = 240 iscompared in Figure 12 for all three values of N. The axial

velocity followed closely the voidage prole, therefore slowvelocities were located in high void fraction regions but in thewall vicinity the axial velocity increased and then decreased dueto the boundary layer and the no-slip condition at the wall.

Overall, when plotted in terms of, the three proles ofvzare very similar, with maxima and minima in the same locationsand of comparable magnitude (note that the supercial velocity

was the same in each case). Some differences may be seenbetween theN= 3.96 prole and the other two proles in therst two velocity peaks from the wall, possibly due to thespecial structure at the lowest N. For each value ofN, there issome anomalous behavior at the bed center caused by the highor low void fractions there as discussed previously. In this plotthe values of within 0.1 from the bed center were discarded





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due to the difficulty in obtaining meaningful averages with verysmall radial surfaces.

7.2. Radial Velocity Fields. Radial velocities vr(r,z) were

obtained from the averaging of different angular planes atdifferent angular positions. The average radial velocity contoursfor the N = 3.96, 5.96, and 7.99 xed bed columns arepresented in Figure13, panels a, b, and c, respectively, for Re=80, and in the corresponding three parts of Figure14for Re =240, again for N = 3.96, 5.96, and 7.99. Similarly the radial

velocityelds for the threeNat the transitional valueRe= 950are given in Figure 15 and under turbulent ow conditions atRe= 1900 in Figure16.

Considering eachRe value separately, there does not appearto be a great deal of similarity between the three Nvalues foreach ow rate. BothN= 5.96 andN= 7.99 show higher valuesofvr near the bed center, and more extreme values than does

N= 3.96. This may be attributed to the stronger axial bypassing

both along the tube wall and down the bed center which ispeculiar to theN= 3.96 bed structure.

If the four different vr(r,z) elds corresponding to the fourdifferent Re are compared for each N individually, somepatterns start to emerge. The regions at high positive vrforN=5.96 andN= 7.99 are in the same positions near the centerline.They alternate axially with patches of strongly negative vr. Highmagnitudes ofvrappear to occur in bands in between the layersof particles arranged against the tube wall. Regions ofvrclose tozero correspond to averages at positions which are mostly insidethe layers of particles. It is possible to discern two, three, and fouraxial bands corresponding to the two, three, and four particlesalong the bed radius for N= 3.96, 5.96, and 7.99, respectively.

It would be expected thatvrwould alternate in sign in both theaxial and radial directions as the overall averages must come tozero, there being no net radial ow on the scale of the bed

radius. It appears that more analysis and more values ofNandRe will be needed to develop either empirical or mechanisticapproaches to predicting vr. For the purposes of the present

work, tables of values from the averaged CFD simulations aresufficient.

7.3. Radial and Axial Temperature Fields. Thecomparisons between the CFD 3D discrete particle and 2Dpseudocontinuum vzvr model temperatures are presented inFigure 17 for Re = 80. The CFD discrete particle modeltemperature contours in Figure 17 were obtained fromaveraging of the same 32 angular planes that were used forthe radial velocity. As can be seen, the pseudocontinuum modeltemperature contours had excellent quantitative agreement withthe averaged CFD temperature contours for all three Nvalues.

The model predicted the axial temperature distribution fairlywell; in addition the radial temperature distribution waspredicted very well. The temperature of the xed bed had arougher distribution in the axial direction due to the particleheat transfer by conduction and the radial velocities distribution

between particles. Both models showed development of thetemperature into the center of the bed at the same locations. Atthis lowest of the ow rates there is signicant thermalpenetration into the bed, across the entire radius for N= 3.96and 5.96, and across most of the radius for N = 7.99. Thetemperature elds for N= 3.96 have a slight difference at the

bed center where the higher void fraction and axial velocity givemore rapid temperature development in the CFD simulation.





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Figure 17.Comparison of CFD 3D discrete particle model temperature contours and 2D vzvr pseudocontinuum model temperature contours inxed bed column of (a)N= 3.96, (b)N= 5.96, and (c) N= 7.99 for Re = 80.





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ForN= 5.96 the two temperature elds are very similar, withthe CFD temperature level slightly lower at the outlet, butotherwise the comparison is good. For N = 7.99 the twotemperatureelds are in excellent agreement, with even smallerdetails the same. This demonstrates that the new model canpredict the convection and conduction heat transfer locally withthe pseudocontinuum energy equation very well in a 2D modelfor low Reynolds numbers for which conduction andconvection both play signicant roles.

The temperature contours for Re = 240 for allNare shown inFigure 18. It is seen that at this higher ow rate there is lesspenetration of the thermal front into the bed than for the lowerow rate. The temperature contours between CFD and the vzvrmodel were in generally good agreement. However, close to thecenter of the bed, the CFD temperature developed more than inthe vzvr model; this was especially so for N= 3.96 and somewhatthe case for N= 5.96. Since the center of the bed was taken as asymmetric boundary condition in the vzvr model the velocityprole close to the center had to satisfy this boundary conditionlimitation and could not develop in the same way as for the CFDmodel. For N= 7.99 the two temperature elds were again inexcellent agreement.

The temperature contours at the much higher ow rate Re =950 for all threeNare shown in Figure19.The simulations for thenear-turbulent regimeRe = 950 could be run as either laminar orturbulent models in the CFD; in this workRe= 950 was assumedas laminar ow and theow from CFD showed reasonable results.The results are more sensitive to the averaging method for higherReynolds number compared to the low Reynolds numbers sincethe radial velocities have lower negative and higher positive values,so it was necessary to average more angular surfaces to avoid lowor high radial velocities at different points, which would thencancel with the averaging method. Averaging of more angularsurfaces extracted better radial velocities to be used in thepseudocontinuum vzvr model. This may account for the strongerappearance of discrete temperature features in the N= 3.96 vzvr

model results, although generally the near-wall comparisons weregood. ForN= 5.96 the vzvr temperature is a little low, and also for

N= 7.99. The extent of the temperature contours and the generalshape of the developing contours are both good in all three cases.

The temperature contours for turbulent ow atRe= 1900 forallNare shown in Figure20. The CFD and pseudocontinuummodel temperature comparison showed some slight differencesin the results. This was due to the dimensional reduction of 3Dto 2D. There is a wider wall region of high temperature for the

vzvr model than in the CFD simulations, but comparisons showvery good results for all three N for the temperature levelsacross the tube radius, in terms of both extent and magnitude.

Overall, the 2D pseudocontinuum heat transfer model basedon the velocity elds from CFD produced very reasonable

results compared to the 3D CFD temperature simulationswithout the need to introduce any adjustable parameters suchaskr/kfand hwor an effective viscosity.

8. CONCLUSIONS

The main object of this work was to demonstrate the feasibilityof modeling radial temperature proles in xed beds of spheres

without using any adjustable parameters such as kr/kfand hw,and without using effective heat conduction approaches foruid mechanical phenomena. Instead, radial heat transfer wasto be predicted using local position-dependent components ofaxial and radial velocity to represent heat transfer by uidmotion and its decrease near the tube wall, and a local effective

medium cell model applied pointwise to account for heattransfer by thermal conduction and its dependence on local bed

voidage.Fluid owelds in validated xed beds of spheres of tube to

particle ratioN= 3.96, 5.96, and 7.99 were obtained by solvingthe 3D NavierStokes equations in a detailed CFD approach

which preserved the actual bed structure in the simulation. Amethodology was developed to obtain the axial velocity, v

z(r),

and radial velocity,vr(r,z) from the 3D discrete particle results.Stagnant effective thermal conductivity was calculated at eachradial position from the Zehner-Schlunder model as function oflocal bed porosity, uid thermal conductivity, and solid thermalconductivity.

Comparisons were made for Reynolds numbers in the range801900, for the three values ofN, under typical laboratory-scale conditions that would be used with a steam-heatedcolumn. The temperatures calculated by the new 2D velocity-

based heat transfer equation showed very good quantitative andqualitative agreement with the values given by the detailedCFD simulation. The trends withReandNwere captured well.

The results of this study suggest that the local radial velocitycomponents can account for the convective contributions to

radial heat transfer in a packed bed of spheres. They are notnegligible if computed from CFD simulations in modelgeometries that preserve the discrete bed structure instead ofreplacing it with a pseudocontinuum or effective medium. Asthe ultimate objective is a computationally tractable 2Dpseudocontinuum reaction engineering model, care needs to

be taken in averaging the information from the 3D discretesimulations for use in lower-dimensional effective models.Earlier approaches that began from effective medium models

with smoothed measures of bed structure substituted into thevolume-averaged NavierStokes equations or their equivalentto obtain velocity elds, all concluded that radial velocitycomponents were negligible as the smoothed structure led tothe result that (p/r) 0 and thus tovr 0, as the local radial

variations in pressure and velocity were averaged out. The axialand radial velocity proles obtained in this study suggested thatit may be possible to obtain generalized velocity componentsfor use in a predictive model.

AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected].

Present AddressDepartment of Chemical Engineering, Massachusetts Instituteof Technology (MIT), Cambridge, MA 02139, USA.

NotesThe authors declare no competing nancial interest.

ACKNOWLEDGMENTS

This material is based upon work supported by the NationalScience Foundation under Grant No. CTS-0625693.

NOMENCLATURE

cp= uid specic heat, J/(kgK)cps= solid specic heat, J/(kgK)

B = shape parameter for ZehnerSchlunder formulaBi= wall Biot number, hwR/krdp= particle diameter, mdt= tube diameter, mFi = external body force per unit volume, kg/(m

2s)


mailto:[email protected]:[email protected]


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gi = body force in i-direction per unit mass, m/s2

Gr= radial mass ux, kg/(m2s)

Gz = axial mass ux, kg/(m2s)

h= specic enthalpy, J/kghi= specic enthalpy of species i, J/kghw= apparent wall heat transfer coefficient, W/(m

2K)

Ji = mass diffusive ux of speciesi, kg/(m2s)

k= turbulent kinetic energy, J/kgke0 = stagnant effective thermal conductivity, W/(mK)kf= uid thermal conductivity, W/(mK)kp = thermal conductivity ratio, ks/kfkr = effective radial thermal conductivity, W/(mK)ks = solid thermal conductivity, W/(mK)

L= bed length, mN= tube-to-particle diameter ratio, dt/dpNM= parameter for ZehnerSchlunder formulaNp= number of particles in computer-generated packingp= static pressure, PaPer() = limiting value of Peclet number (Gzcpdp/kr) athigh Rer= radial coordinate, mR= tube radius, m

Re= Reynolds number based on particle diameter, dpvz/Sh= energy source term, J/(m

3s)

Sm = mass source term, kg/(m3s)

t= time, sT= temperature, KTin = inlet temperature, KTw= wall temperature, Kui= generic velocity component in direction i, m/su0= supercial plug-ow velocity, m/svr = radial velocity component, m/svz = axial velocity component, m/s

xi = coordinate direction i, mz = axial coordinate, m

Greek Letters

= bed voidage0= initial voidage for bed generation= effective thermal conductivity of the uid (molecular andturbulent), W/(mK)= angular coordinate, radians= uid viscosity, kg/(ms) = dimensionless distance from tube wall, (Rr)/dp= uid density, kg/m3

s= solid density, kg/m3

ij= viscous ux ofj-momentum in the i-direction, kg/ms2

= specic dissipation rate, s1

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