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hr. J . Hear Moss Transfer. Vol. 27, No. 3, pp. 399407, 1984 0017-9310/84 $3.00+0.00Printed in Great Britain 0 1984Pergamon Press Ltd.

MODELING OF THE EFFECTIVE THERMALCONDUCTIVITY AND DIFFUSIVITY OF

A PACKED BED WITH STAGNANT FLUIDEMERSON F. JAGUARIBE

Centro de Tecnologia da UFPB, Departamento de Engenharia Meclnica,Campus Universitirio da UFPB, 58.000 Joso Pessoa, PB, Braziland

DONALD E. BEASLEY*University of Michigan, Department of Mechanical Engineering and Applied Mechanics, Ann Arbor, MI 48109, U.S.A.

(Recei ved 23 December 1982 and in evised form 22 June 1983)Abstract-Beginning with a model of radial energy transport in a fluid saturated porous medium formed by acubic array of cylinders with a stagnant fluid, a generalized method for determining overall effectiveconductivity, volumetric specific energy capacity, and effective thermal diffusivity is derived. The resultingmodel is not limited to any particular geometry, since it requires only the physical roperties of the fluid andsolid, and the void fraction as input. Comparisons demonstrate that the model agrees well with bothexperimental and theoretical values for the effective conductivity, while requiring no empirical or theoreticalmodel parameters. Results for effective thermal conductivity and effective thermal diffusivity as functions of

void fraction and as functions of the ratio of solid to fluid thermal conductivity are presented.

AacD4Gf l l

kL41' i jRRi jRl lRll

uu0

u,W

X

NOMENCLATUREarea [mm]tan 1specific heat [U kg- K-1parameter defined by equation (9)particle diameter [mm]sum of the angles which form the voidregion between consecutive radiithermal conductivity [W m- K- 1bed length [mm]heat flux [W m-1inner radius of a model void region [mm]cylinder, or particle radius [mm]outer radius of a model void regionradius, defined by equation (6) (see Fig. 4)Cmmlouter radius of the first continuous void(see Fig. 4) and defined by equations (10)and (11)total thermal resistance (electrical analogy)defined by equation (18) [K W - 1thermal resistance (electrical analogy)defined by equation (13) [K W - 1thermal resistance (electrical analogy)defined by equation (14) [K W - 1thermal resistance (electrical analogy)defined by equation (16) [K W-1thermal resistance (electrical analogy)defined by equation (17) [K W-l].

* Present address :Department ofMechanical Engineering,318 Riggs Hall, Clemson University, Clemson, SC 29631,U.S.A.

Greek symbols

;thermal diffusivity [m s- 1angle defined by equation (5) (see Fig. 2)

Y angle formed by the lines m and w (seeFig. 4) and defined by equation (12)E bulk mean voidageEl parameter defined by equation (15)e included angle of a defined void, given by

expressions (4), rd3, angle, represents the void fraction in the

volume 7c/4 (Rg - Rll)LP density [kg mm31* function of tan i [see equation (9)].

Subscriptsefi

j

effectivefluid phasedenotes the position of the void in eachsection, i = 1 indicates the first voidlocation from the center of the array,immediately above the axis ox (see Fig. 3)denotes the position of the section numbercontaining the particular void (see Fig. 3,the sections are represented by Romannumbers)limiting valuedenotes the particular radius in thecomposite cylinder R, = 1 = rl 2indicates the outer radius of the compositemediumpartial, referring to the shaded area inFig. 4solid

399

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400 EMERSON. JAGUARIBEnd DONALDE. BEASLEYT totalv void.

Superscriptsi denotes the position of the void in each

section, i = 1 indicates the first voidlocation from the center of the array,immediately below the axis oz (see Fig. 3)

V voidS solid.

1. INTRODUCTIONBECAUSEof their large interphase surface area to totalvolume ratio, and their capability ofgood fluid mixing,packed beds have been used in a wide variety ofprocesses which require an interaction between one ormore fluids and one or more solids. Catalytic reactors,pebble bed heaters, evaporators, absorbers, glassfurnaces, and thermal energy storage are examples ofapplications of packed beds.

Numerous studies have been dedicated to under-standing the various physical mechanisms of flow andheat transfer in packed beds, to improve their design.Recently the storage of thermal energy in packed bedshas prompted the study of transient temperaturedistributions in rock beds for thermal energy storage, aswould be employed in solar domestic heating. Duringan energy collection (charging) period, a temperaturedistribution, or thermocline, will be established in thepacked bed ; f there is not an immediate demand for thisstored energy, the thermocline will decay, with theassociated loss in available energy. For downward flowduring charging, an inherently unstable temperaturedistribution is established in the upper regions of thebed, since the outlet temperature from the collectordecreases after reaching a maximum near solar noon.Therefore, both natural convection and pure diffusionmay contribute to the thermocline degradation invarious regions of the bed. As an aid to understandingthe importance of this natural convection, in additionto the previously mentioned applications, a reasonablevalue for the effective thermal conductivity, andeffective thermal diffusivity in a stagnant packed bed isneeded. However, as Pai and Raghavan [l] report, theproblem of determining the thermal conductivity of aheterogeneous system has defied analytical solution fora considerable time.

The general problem of determining the overalleffective thermal conductivity ofa porous medium maybe approached by one of two basic ways :

(1) The overall value is taken as an appropriateaverage of the separate phase contributions.

(2) Transport among individual particles in aregular array is extended to the entire medium.In a medium composed ofa solid phase and a gas phase,the averages over the separate phases assume thatconduction through the solid acts over a volume 1 -Eand the conduction through the gas in the void spaces

over a fluid volume E. This approach has been used notonly to determine the effective thermal conductivity of aporous system, but also the electrical conductivity,magnetic permeability, coefficient of diffusion, etc. ofheterogeneous media (cf. Wyllie [2], Dulnev andNorikov [3,4], Beveridge and Haughey [S], etc.). Oftenthis method reduces to an analogous electricalnetwork. For random packings theeffectiveparametersare determined using unit cell elements of regularpackings, again with appropriate averages. Thesimplest models use the arithmetic mean of thecontributions from each phase ; others use harmonic orgeometric means. However, the more realistic modelsmake use of experimentally determined statisticalparameters, even for regular packings (cf. Tsao [6] andCheng and Vachon [7]).The method of extending transport among a fewparticles to the entire packing is exemplified by theworks of Kunii and Smith [S] and Yagi and Kunii [9].Parameters related to the angle bounding the heat flowarea for one contact point, the number of contactpoints, etc. are required in the work by Kunii andSmith, and the total area of perfect contact surfaces,effective thickness of the fluid film in the void spacerelative to the thermal conduction effects, etc. for themodel by Yagi and Kunii. Thus, these models requireempirical or theoretical parameters other than thephysical properties of the bed and packing materials ;still other parameters are needed to account forradiation effects, if the bed is operated at hightemperatures.

Krupiczka [lo] presents an interesting comprehen-sive analytical study of the thermal conductivity ofpacked beds, which expresses the thermal conductivityin terms of the properties of the bed. Assuming modelscomposed of both cylinders and spheres, solutions areobtained by non-orthogonal series methods. Sincethese results are difficult to utilize effectively,approximations with simpler functions are derived.Existing experimental data is then used to develop acorrelation in terms ofthe bulk mean voidage and k,/k,.The resulting expression is independent of the particlediameter and length of the bed. Thus, Krupiczkasmodel results in identical values of thermal conduc-tivity for beds with equal void fraction and solid andfluid, regardless of the particle diameter or geometricaldimensions for the bed.

The present work was directed toward finding theeffective thermal conductivity and effective thermaldiffusivity of a stagnant packed bed, at a low meantemperature where radiation effects are small, usingonly the void fraction, thermal properties of the solidand fluid, and the solid and fluid densities. Toaccomplish this, a specific geometry is assumed for theporous media, from which a generalized arrangementof solid and fluid regions is specified, the relativevolumes of which are governed by the bulk mean voidfraction, E. As such the final model is independent of thespecific geometry of the actual packing but rather is ageneralized model for diffusion in a porous media.

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Modeling of the effective thermal conductivity 4012. PROPOSED MODEL

It is desired to analyze the transport of thermalenergy in a fluid saturated porous medium composed ofcylinders, arranged as shown in Fig. 1, with the centralcylinder as the source of the energy. From symmetry,only the shaded region, with an included angle of 7~14must be considered in the analysis. Conductionthrough the two phases, solid and fluid, is assumed to bethe only mechanism for energy transport, implying thatthe fluid phase is stagnant and radiation effects may beneglected. Hereafter the fluid phase will be referred to asvoid regions. Conduction through the contact pointsbetween discrete solid particles is included in the modelas regions of continuity in the solid phase (see Fig. 3).2.1. General considerationsIn this approach, the existing arrangement of solidand void regions will be replaced by an equivalentarrangement of solid and void, whose geometry is moreamenable to calculating the overall effective conduc-tivity. The original voids are each replaced by a voidplaced within two concentric cylinders of radii rij andRij, and within the angle @(see Figs. 2 and 3). The modelgeometry is thus divided into sections, with the index jdenoting the section number containing the particularvoid. A section is defined as the region bounded by twoplanes perpendicular to the axis ox, each containing theaxis of one of the two neighboring cylinders, and theaxes ox and oz. (For example, the first section, Fig. 3, isthe triangular area OAA.) The index i denotes theposition of the void in each section; the subscript iindicates the first void location from the center of thearray, immediately above the axis ox. With thesedefinitions, rij may be determined from the expression

rij = [(2j-1)2+4(i-1)2]12R. (1)In the proposed arrangement of solid and fluid, the

volume ofthe model voids must be individually equal tothe volume of the voids they replace. To accomplishthis, R, is given by

Rij =

HEATED

(2)

FIG. 1. Diagram of the arrangement of cylinders forming theporous medium.

FIG. 2. Description of the angles 0; and /Ii.

when i z j, and by&R+r:, ,112R,, = (3)I

when i = j. The included angle for each new void isgiven by

ej = p;,Bj-i = pj-i-lsj, (4)

if j 2 2, where(5)

The radii rij and R,, defined above, are used tocharacterize the model voids, beginning with sectiontwo of the model geometry. The void associated withthe central cylinder will be considered as a special case.

Since the central cylinder is contacted by othercylinders only in line contacts, it is appropriate toconsider the central cylinder as being surrounded byvoid region of radius Rl 1.R, is defined by

R, = 6012R,n

With R, so defined, it is necessary to specify R 11, suchthat the volume bounded by R and Rl 1 is void andbetween Rll and R, both solid and void regions exist,with a common interface along the line which makes anangle 1 with ox. Then 41/7r represents the void fractionin the volume (7c/4)(Ri - Rll )L.

The total void fraction in the volume bounded by R,and l9 = 7r/4 is given by

2e~~ =~(R112--~~)+~(~:_-~~1~).Solving this equation for RI 1 yields

(7)

Rll = (16e+~c)R-AR; 12K-41 I . (8)

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402 EMERSONF. JAGUARIBEand DONALD E. BEASLEY

FIG. 3. Arrangement of void regions, described by the radii T,~,Rij

At this point the angle 1 is still an unknown the following relation is adopted when I > n/8, whichparameter, and must be specified in order to determine establishes the line E, Fig. 4, as the boundary for RllRll. It is assumed that the length of Rll will bedetermined by the following expression which relates sin yR1l =R1lLsin [rr-(L+y)] (11)the solid volume between R 11 and R, and the total voidvolume bounded by R and R,. The ratio of these R 1 I, is obtained from the equation (10) for 1 = rr/8 andvolumes is D, and for this work is assumed to be 0.05,yielding

where)k)-itO. t =I

y z g -tan- m sin n/8R,-m cos $3 (12)0.05, (9) It is clear that equations (10) and (11) are transcenden-tal, and must be solved by iteration.

With the geometry so defined, the diffusion ofa[2-(1-3aZ)Z] thermal energy may now be analyzed.IL= l+aZ

where a = tan I(,? < 7r/8), Ai is the shaded area in Fig. 4and A; is the total void area in the section. The value ofD was selected by first determining from the geometrythat the ratio was very small, of the order of 0.05 or less.A range of values were employed, to determine thesensitivity of the solution to changes in D. For D < 0.02problems existed in the solution to equation (lo), fromwhich A is determined. Computing times increasesignificantly for D < 0.05 ;results obtained for values ofD from 0.03 to 0.05 show no significant difference. ThusD = 0.05 is used for this study

i y( > = 0.2s-sin- ++2$. (10)

2.2 Analysis of energy transportUtilizing the proposed geometric model of a

generalized porous medium, the effective conductivityof the stagnant fluid case for the overall arrangementmay be determined. If we assume that a specified heatflux, ql, is flowing from the central cylinder, of radius R,limited by an angular range 0 < 0 < n/4, the proposedgeometry may be treated as a composite conductionmedium, with defined solid and fluid regions. Using anelectrical analogy, the system is represented by fourresistances in series

u 0 = ln @11/R)k,WW (13)Since for 1 > n/8 the values of Rl 1 increase faster than

for values of I < n/8, and when 1 = 7114,Rll = Ro,In (R,/Rll)

= [(1 -&I)ks+EIkf](n/4)L) (14)

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Modeling of the effective thermal conductivity 403

where

FIG. 4. Section I showing area A;, radius Rll, Rl l,, R, lines m and n, and angles 2 and y.

4/l&I =-) n

w = ln UWRo)k(~/W

(15)

(16)and

N-lx= 1 In UL + lR,)m = I Ck(44 G,) + @Znl~ (17)

G, represents the sum of the angles that form the voidregion between R, and R,, ,, where R,, 1 and R, aretwo consecutive radii of the composite cylindricalsolid/void model, beginning with ri2 and ending withthe external radius of the composite conductionmedium. This method of defining X, which determinesthe locations of the voids in the model geometry,contributes to the extended validity of the model forcases where the void (fluid phase) is randomlydistributed in a given system. Since equations (2) and(3) are written in terms of a bulk mean voidage, E, thestructural development of the model is not a limitingfactor, and the model is easily extended to othergeometric arrangements, with their associated vari-ations in E. Therefore, although the model is developedfor a particular geometry, the result is applicable toother geometries by allowing E to vary. It is alsoapparent that the model is equally valid for any twodistinct materials, fluid and solid, or just solids.

The overall resistance to the flow of energy, inanalogy to an electrical circuit, is given by

u = u,+u,+w+x, (18)

and the overall effective thermal conductivityk = ln (RN/R)

e (7r/4)UL (19)where R, = L.

Also, the thermal diffusivity ofa medium controls thediffusion of energy in a purely conductive mode. Assuch, the property of interest for the study ofthermocline degradation in a packed bed for thermalenergy storage is the effective thermal diffusivity

k,cIe = (1 - E)&C, + &P&f. (20)

The denominator of the function is linear in E, while k, isa complex function of several parameters. This impliesthat for a given packed bed, the effective thermaldiffusivitydependsstrongly on the voidfractionand thefluid and solid properties. To validate the model, it mustbe tested for a number of packings, including variationsin void and packing particle shape and size.

3. RESULTS AND DISCUSSIONThe proposed method of solution was programmed

for an IBM Personal Computer, with the requiredinput parameters of length of packing traversed by theenergy, L, the average spherical particle diameter, D,,the bulk mean voidage, E, and the thermalconductivities of the solid and fluid phases, k, and k,,respectively. For purposes of comparison, calculationsof effective conductivities were made for theexperimental cases reported in the summary by Ofuchiand Kunii [ 1 ] and one measured value by Beveridgeand Haughey [S]. The results of these calculations

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404 EMERSONF. JAGUARIBE and DONALD E. BEASLEY

Table 1

Case1 12.1 0.3762 8.7 0.3693 6.38 0.3374 3.69 0.3695 3.69 0.346 2.66 0.34I 3.69 0.3698 6.38 0.3589 8.70 0.369

10 3.69 0.36911 6.38 0.33712 8.70 0.36913 1.15 0.34014 3.69 0.36915 6.38 0.33716 6.38 0.35817 8.70 0.36918 3.09 0.39019 3.09 0.34220 10.9 0.40321 3.09 0.41322 6.32 0.39623 6.32 0.35224 6.32 0.35225 3.09 0.34226 3.09 0.3921 10.9 0.40328 8.18 0.47529 5.12 0.57230 5.12 0.603

4(mm)

- __-

k,lk0.900.900.900.900.900.903.73.13.7

3131312121212121

16501650165016501650165023702370

29470595959

Fluidkc,%Present

authors- SolidL

(mm) Exp.Ofuchi andKunii [l l]

Water Glass spheres 50 0.87 0.94 0.94Water Glass spheres 50 0.90 0.94 0.94Water Glass spheres 50 0.94 0.94 0.94Water Glass spheres 50 0.96 0.94 0.94Water Glass spheres 50 0.90 0.94 0.94Water Glass spheres 50 0.90 0.94 0.94He Glass spheres 50 2.46 2.40 2.35He Glass spheres 50 2.46 2.38 2.40He Glass spheres 50 2.49 2.43 2.32CO, Glass spheres 50 8.61 8.89 9.12CO* Glass spheres 50 8.24 9.25 9.40CO, Glass spheres 50 9.40 9.07 7.98Air Glass spheres 50 6.38 6.40 6.23Air Glass spheres 50 6.7 1 7.03 7.49Air Glass spheres 50 7.13 7.38 7.71Air Glass spheres 50 7.49 6.97 7.32Air Glass spheres 50 7.78 7.26 6.74Air Steel balls 50 10.1 16.6 16.61Air Steel balls 50 15.0 21.5 22.29Air Steel balls 50 18.6 15.5 11.20Air Steel balls 50 11.2 13.9 15.30Air Steel balls 50 10.4 17.2 14.32Air Steel balls 50 29.0 20.9 16.94CO, Steel balls 50 42.1 22.5 17.03CO* Steel balls 50 22.1 22.9 20.17He Steel balls 50 6.65 12.1 14.77

Water Steel balls 50 6.28 8.54 8.80Air Rashig rings 50 7.47 6.69 6.99Air Rashig rings 50 7.39 5.56 5.60Air Rashig rings 50 6.64 5.32 4.79

Case 4(mm) E kslk, Fluid Solidkc/k, klk,L Ofuchi and Present

(mm) Exp. Kunii [l l] authors31 8.18 0.475 87 co2 Rashig rings 50 10.032 5.12 0.603 87 co2 Rashig rings 50 7.9533 5.12 0.572 10.4 He Rashig rings 50 2.3134 8.18 0.475 2.53 Water Rashig rings 50 1.1135 5.12 0.603 2.53 Water Rashig rings 50 0.9836 7.31 0.366 34 Air Cement clinkers 50 8.9237 7.31 0.370 34 Air Cement clinkers 50 7.4038 7.31 0.370 50 CO, Cement clinkers 50 9.0239 7.3 1 0.366 5.9 He Cement clinkers 50 3.3440 7.31 0.370 1.44 Water Cement clinkers 50 1.2441 2.45 0.359 7.4 He River sand 50 3.1942 2.45 0.359 1.78 Water River sand 50 1.7743 2.45 0.359 42 Air River sand 50 8.2044 4.75 0.384 12.6 Air Nickel catalyst pellets 50 5.4645 4.75 0.384 18.5 CO, Nickel catalyst pellets 50 6.7346 4.75 0.384 2.22 He Nickel catalyst pellets 50 1.4547 4.75 0.384 1.82 H* Nickel catalyst pellets 50 1.20

andSmith [12]6.628 0.47 0.38 38.53 Air Glass sphere 12.7

Table 2

_______klk,

8.516.382.661.531.399.108.10

10.02.701.133.51.48.504.96.171.691.48Beveridgeand

Haughey [S]6.40

1.535.153.061.67I 468.758.659.983.231.253.621.4110.495.126.461.621.43

7.06

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Modeling of the effective thermal conductivity 405appear in Tables 1 and 2. The mean bed temperature inthe data of Ofuchi and Kunii was N 50C so thatradiative effects are relatively small. In general, theproposed stagnant fluid, conductive model is in goodagreement with both the experimentally determinedvalues and the predictions by Ofuchi and Kunii.However, there are specific cases for both models wherethe experimentally determined values are quitedifferent than the predicted values (i.e. Cases 18,22,23,24, 26,27, and 31). Error bounds for the experimentaldata could not be determined. However, the presentauthors could find no reasonable pattern ofdiscrepancy to indicate a particular physical expla-nation as the primary reason for the differences. In anycase, for physical systems as complex as these, thepredictions are quite reasonable, and follow the grossbehavior of the overall effective conductivity quite well,both for the present model and that of Ofuchi andKunii. It should be noted that the more complex modelof Ofuchi and Kunii shows significant deviation fromthe measured values in many of the same cases as thepresent authorsmodel. Also the present model appearsvalid for packings of irregular particles, producingreasonable values for the effective thermal conduc-tivity. The model ofKrupiczka [lo] was compared withthe values of k, found in Tables 1 and 2. Fork,/k, > 1650 this model predicted k, values consistentlyhigher than the experimental data and the model ofOfuchi and Kunii, and the present model. Fork,/k, < 294 the trend is reversed.

Figure 5 shows plots of a,/cc, and k,lk, for thepractical range of values of E for a packed bed. A solidline is used to characterize the range of E whereexperimental data are available. In these units, k,/k, and

16t 16

(5\ 1 Dp = 5mm

2414 \ 1 L = 50mm 22

7-6-5-

1::.2 0.3 0.4 0.5 0.6 07E

FIG. 5. Effective thermal conductivity and diffusivity as afunction of void fraction.

CI,/CL~re monotonic functions of c, with a limiting valuefor both curves as E approaches 0.8. From c: = 0.7 to 0.8the value of k,/k, continues to decrease while r,/c(rrepresented by a dashed line, exhibits a change in slope.Further experimental study is required to determinethe nature of the physical phenomena in this range ofvoid fraction. Figure 6 demonstrates the rapid increaseof k,/k, for 2 < In (k,/k,) < 5 with limiting valuesquickly reached outside this range.

4. CONCLUSIONSBeginning with a cubic arrangement of cylinders, a

general arrangement of solid and void regions has beendeveloped which allows for calculation of the overalleffective conductivity and the overall effective thermaldiffusivity of a packed bed. The model requires only thevoid fraction of the packing and the solid and fluidproperties. Comparisons with experimental data andpredictions for a range of void fraction, fluid and solidproperties, and packing particle shape and sizedistributions show good agreement. This model maythen be used to approximate the effective thermalconductivity and effective thermal diffusivity of apacked bed with a stagnant fluid phase. Also, knowingthe effective conductivity of a stagnant bed will allowthe effect of natural convective motion in the fluid to beassessed.

The model might be improved by starting from athree-dimensional spherical geometry, rather than acubic packing of cylinders. An additional improvementmay result from the introduction of radiative transfer inthe void spaces, in cases of high temperature operation.

The range of void fraction studied experimentally is

IE -17-i6-

14- L q Omm,3- E =0.3712-!I -to-

o. -2 -( 0 I 2 3 4 5 6 7 8h(k,/kt)

FIG. 6. Effective thermal conductivity as a function ofln (k,/k,).

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406 EMERSON . JAGUARIBE~~~DONALDE.BEASLEYsmall, and further comparisons would be useful,especially in the range of void below 0.3. Values of Edown to 0.0931 can be achieved with regular packingsof cylinders, so that additional comparisons arepossible.Acknowledgemen&-The first author was supported byConselho National de D~envolvimento Cientifico eTecnolbgico, CNPq, Brazil, while a Visiting Professor in theSolar Energy Laboratory, Department of MechanicalEngineering and Applied Mechanics, University of Michigan.The authors would like to express their thanks to Dr John A.Clark for his support and guidance, and to the Department ofMechanical Engineering and Applied Mechanics for use of thefacilities.

REFERENCES1. K. Pai and V. R. Raghavan, A thermal conductivity modelfor two phase media, L&t. Heat M ass Transfer 9, 21-27

(1982).2. M. R. J. Wyllie, An experimental investigation of the S.P.and resistivity phenomena in dirty sands, J. Petrol.

Tecknof. Trans. AIME I?@,4447 (1954).

3.

4.5.

6.7.

8.9.

10.11.

12.

G. N. Dulnev and V. V. Norikov, Conductivitydetermination for a filled heterogeneous system, J. EngngPhys. 31,657-661(1979).G. N. Dulnev and V. V. Norikov, Conductivity ofnonuniform systems, J. Engng Phys. 36,901-909 (1979).G. S. G. Beveridge and D. P. Haughey, Axial heat transferin packed beds, stagnant beds between 20 and 75OC, nt.J. Heat M ass Transfer 14, 1093-l 113 (1971).G. T. Tsao, Thermal conductivity of two phase materials,Ind. Engng Ckem. 53,395-399 (1961).S. C. Cheng and R. I. Vachon, Prediction of the thermalconducti~ty of two and three phase solid heterogeneousmixtures, fnt. J. Heat Mass Transfer 12,24%264 (1969).D. Kunii and J. M. Smith, Heat transfer characteristics ofporous rocks, d.I.Ck.E. Jf 6.71-77 (1960).S. Yagi and D. Kunii, Studies in effective thermalconductivities in packed beds, A.1.Ch.E. Jf 3, 373-381(1957).R. Krupiczka, Analysis of thermal conductivity ingranular materials, Int. Ckem. Engng 7, 122-144 (1967).K. Ofuchi and D. Kunii, Heat transfer characteristics ofpacked beds with stagnant fluids, lnt . J. Heat M assTransfer 8,749-757 (1965).S. Masamune and J. M. Smith, Thermal conductivity ofbeds of spherical particles, fnd. Engng Chem. Fundam 2,13&143 (1963).

MODELISATION DE LA CONDUCTIVITE THERMIQUE ET DE LA DIFFUSIVITEEFFECTIVES DUN LIT FIXE AVEC UN FLUIDE STAGNANTResume-A partir dun modele de transport radial denergie dans un milieu poreux sat& de fluide et formedun arrangement cubique de cylindres avec un fluide stagnant, on propose une mtthode getterale pourdeterminer la conductivite globale effective, la capacite thermique volumique et la diffusivite thermiqueeffective. Le modele nest pas limit.5 a une geombtrie particuliere puisquil requiert settlement la connaissancedes propri&s physiques du fluide et du solide et de la fraction de vide. Des comparaisons montrent que lemodele saccordent bien avec les valeurs exp&imentales et theoriques pour la conductivitb effective. Onpresente des resultats pour la fraction de vide et du rapport des conductivites du solide et du fluide.

MODELLBILDUNG DER EFFEKTIVEN WARMELEITFAHIGKEIT UNDTEMPERATURLEITFAHIGKEIT EINES FESTBETTES MIT RUHENDEM FLUID

Zusammenfassung-Zunachst wird ein Model1 entwickelt fiir den radialen Warmetransport in einem mitFluid gesittigten pordsen Medium, das aus einer kubischen Anordnung von Zylindern mit ruhendem Fluidbesteht. Daraus ergibt sich eine verallgemeinerte Methode, urn die effektive Warmeleitfahigkeit, dievolumetrische spezifische Warmekapazitiit und die effektive Temperaturleitfahigkeit zu ermitteln. Dasentstehende Model1 ist nicht aufeine besondereGeometrie beschrankt,daesnurdiephysikalischenStoffwerteder Fliissigkeit und des Feststoffs und die Porositiit als EingabegriiBen benbtigt. Vergleiche zeigen, daB dasModel1 gut mit expe~mentellen und theoretischen Werten fiir die effektive Leitfabigkeit ~bereinst~mt, ohneempirische oder theoretische Modellparameter zu benotigen. Ergebnisse fiir die effektive W~rmeleit~higkeitund die effektive Tem~raturleit~higkeit werden als Funktionen der Porositit und des Verh~Itni~es ausFeststoff- und Fluidw~~eleit~higkeit dargestellt.

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Modeling of the effective thermal conductivity 407

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