Carman Fluid flow through granular beds.pdf

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IChemE 0263-8762/97/$10.00+0.00 Institution of Chemical EngineersFLUID FLOW THROUGH GRANULAR BEDS*

By P. C. CARMAN, Ph.D. (Graduate)INTRODUCTION

The laws of the flow of fluids through porousmedia have several aspects of practical importance.They are essential in the calculation of the movements of ground waters, of petroleum, and of naturalgas through sand and rock, in deciding the extent ofseepage through the subsoil of dams and of largebuildings, and in determining th e permeability ofconcrete and of other building materia ls . To thechemical engineer, such laws form the basis of designof packed towers and of converters containing granular catalysts, as in the contact process for sulphuricacid, or they allow the interpretation of data fromsmall models in the design of full-scale plant.lFurther, the fundamental laws of filtration, both forthe cake and for the filter medium, rest on the studyof flow through porous media. A brief review ofearlier work on this latter aspect has been made byUnderwood,2 and a more recent and comprehensivereview is that of Siegel,3 who covers the whole fieldof consolidated masses such as sandstones and porousearthenware, and of unconsolidated masses such assands, that is, the granular beds of the kind consideredin this paper.The following review deals wi th the simpler caseof unconsolidated grains and is mainly concernedwith the importance of the method of plottingby dimensionless groups introduced by Blake. 4 Afurther object is to suggest the application ofpermeability measurements to the determination ofthe surface of powders.TIlE D 'ARCY LAW AND ITS DERIVATIVES

The fundamental equation of permeability is thatof D'Arcy,5 an empirical equation based on measurements of the flow of water through sands and sandstones, and which may be represented as

6. Ptu = K . L (I )where K is the coefficient of permeability, or thepermeability, and is the rate of flow of water acrossa unit cube of the sand at unit pressure head. Thelaw is closely analogous to Poiseuille's law for theflow of a viscous fluid through a circular capillary,namely, d.2 L1p. gu = 32'1} . L (2)and much work has rested on the assumption that agranular bed is analogous to a group of capillaries Paper received May, 1937.t A Jist o f symbols is given at th e end of the paper.

1 Walker, Lewis and McAdams, Principlesof ChemicalEngineering, Second Edition, p. 116, McGraw-Hill BookCo., N.Y., 1927.2 A. J. V. Underwood, in Filtrationand Filters, by J. A.Pickard, page 87, Benn Bros., London, 1929.3 Siegel, in De r Chemie Ingenieur, Band I, Zweite Teil,p. 109, etc., Leipzig, 1933. Blake, Trans. Amer. Inst. Chem. Engrs., 1922, 14, 415. H. P. G. D'Arcy, LesFontaines Publiques de Ill. Villede Dijon, Victor Dalmont, Paris, 1856.

parallel to the direction of flow and of diameter, d. h e v i e ~ s on the natureandthe sizeofthese equivalentcapIllafles and whether they have any physical meaning have differed considerably.

The first extension of the simple D'Arcy law wasmade. by Dupuit,6 who realised that the apparen tvelOCIty, u, must be less than the actual velocity inthe pores. I f the pore-space in the bed be considered?,S e ~ e n ~ y d i s t r ~ b u t e d the porosity of a layer ofmfimtesImal thlCkness normal to the direction offlow will be equal to the porosity, , of the bed as awhole. As, for such a layer , the fractional free volumewill be equal to the fract ional free area, the truevelocity of flow must b e ~ Dupuit therefore gave:iPU = K 1 L . . . . . . . . . . . . . . (3)

The full importance of porosity was later realisedby Slichter,' whose treatment represents the firstreal attempt to derive expressions for the equivalent

c ~ a n n e l s from the general geometry ofa bedof equallySIzed s p h e r e ~ He assumed the average cross-sectionsof the e q U l v a l e ~ t channels to be triangular, anddeduced expresslOns for the cross sectional area andthe length of these channels in terms of particle sizeand of porosity. Then, applying a correcting factorto Poiseuille's law to allow for flow through a channelof t r i a ~ g u l a r cross section, he calculated the permeabIht,y of the bed. The resulting equation isd2 Mu=1O.2 K21}I; (4)where K 2 is a function of , varying from 843 for=026 to 12,8 for =046. SmithS has summarised

Slichter's treatment and revised it in a few detailsto give better agreement with experiment. 'A m o ~ i f i c a t i o n of Slichter's treatment was made byTerzaghl,9 who deduced a relationship betweenporosity and permeability in close agreementwith thatof Slich 'Cr, and, by introducing an empirical constant,h ~ o b t a m e ~ a formula.in moderately good agreementWIth experIment. ThIS formula is

( -0'13)2 d2 Mu = K a Vl- .;JY .. , (5)where K 3 is an empirical constantwith values between603 and 105 for all sands. A somewhat similar

t r e a t ~ e n t was attempted by Bou8sinesq,1 bu t hisequatIon has little interest as it contains no generalexpression for the effect of porosity.The main drawback to Slichter's treatment is thathe ?,ssumed a generalised mode of packing for spheres,whlCh Darapskyll has later shown to be impossible,

6 A. J. E. J. Dupuit, EtudesTheoretiques et Pratiquessu r Ie Mouvement des Eaux, 1863.1 Slichter, Nineteenth Ann. Rep. U.S . Geol. Surv. 1897-82,305. ' ,8 Smith, Physics, 1932, 3, 139. Tenaghi, Eng. New8 Ree., 1925,95, 832.10 Boussinesq, C. R. Acad. Sci., 1914,159,390 an d 519U Darapsky, Z. Math. Phys . , 1912,80, 170. .

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This table shows that th e mean hydraulic radiusdoes not affect correlation in the region of streamlineflow. On th e other hand, for th e shapes with most 10c. cit.

17 Kozeny, Ber. Wien Akad., 1927, 136a, 271.18 Davies, Engineering, 1929, 128, 69 and 98.19 Piercy, Hooper an d Winney, Phil . Mag., 1933 (7), 15, 647.20 Fair an d Hatch, Trans. Amer . Wat. W ks . Assn., 1933,25, 1551.

I U h d' . I RlO2 d puU e equa s - , t e ImenSlon ess groups, 2an -8 ' arelO pu 7J. . R=t;,.Py lO t;,.p.y. lO3 puobtamed, or, smce L . 8 L . pu28 and 7J8 areobtained . These are the groups recommended byBlake4 for plotting in the region of turbulent flow.For viscous flow, th e method of Blake gives riseto the following form of the D'Arcy equation,

t;,.p.y. lO3LU7J82 = k (6)lO3 t;,.p.yor, U=kYJ 8 2 -y;- (6a)

which is the equation later discussed by KozenyPHe derived thi s equat ion by assuming that thegranular bed is equivalent to a group of parallel,similar channels, such that th e total internal surfaceand the total internal volume are equal to the particlesurface and to the pore-volume, respectively, in thebed itself, that is, such that th e value of m for thesechannels He furthermore pointed ou t that,owing to th e tortuous character of th e flow through agranular bed, the length of th e equivalent channelsshould be L., where Le is greater than the depth, L,of the bed. The general law of streamline flow througha channel is m 2 t;,.p.yue=ko' YJ' ----y;;- (7)where ko depends upon the shape of the cross-sectionof the channel and has the following values for variousshapes.I8. 19, 20

TABLE 1.Values of ko for Streamline Flow in Various Gross-Sections.

an opinion confirmed in a recent criticism of Slichter'sassumptions by Graton and Fraser.12 Darapskymade a detailed study of flow through spheres in thetightest mode of packing, lO=026, and this was carriedfur ther by Burmester,13 but neither derived expressions of general applicability.In contrast to the analogy of equivalent channels,the work of Emersleben14 should be mentioned. Inhis paper, it is assumed that the grains in a bed ofsand are equivalent to a group of equally spaced,solid, circular cylinders, parallel to th e direction offlow. Thus, any section of the equivalen t bednormal t o the flow shows the whole of th e free spaceinter-connected, which is a much closer approximat ion to the actual system of pores in a granular bed.In his subsequent attempt to derive th e D'Arcy lawfrom fundamental hydrodynamical equations, however, Emersleben arrived at permeabilities whichare of the wrong order and for which th e variationwith porosity is much less than that found in experiment.Up to the present, therefore, the purely mathematical treatment of Emersleben14 has no t beensuccessful in providing a sound theoret ical basis forD'Arcy's law, nor has Slichter's7 geometrical treatmentof an idealised bed of spheres established th e analogybetween Poiseuille's law and D'Arcy's law. Greatersuccess has been obtained by semi-empirical methods,particularly the introduction by Blake4 of plottingby dimensionless groups.In the correlation of flow through smooth circularpipes, Stanton15 and his co-workers, following on thework of Osborne Reynolds, have shown that a unique

plot is obtained if th e dimensionless groups, ~u d and ~ are plotted against one another, whereR f r i ~ i o n a l force per unit area, and puede is called7Jthe Reynolds' number. For non-circular pipes,Schiller16 has shown that, in the turbulent region, the

I . 11 1 f R . I d . pU mcorre atlOn IS Stl C ose I --2 IS P otte agamst -'--PUe 7JThe f t _ cross-sectional area normal to flowac or m d f l 'dperimeter presente to Uland is termed th e mean hydraulic radius. Fo r acircular pipe, m = Since th e cross-section of thepipe is uniform, an alternative express ion forvolume of fluid in pipem is I f this expressionsurface presented to fluid'is applied to a granular bed, m = ~ and, acceptingDupuit's assumption that the interstitial velocity

410c. cit.7 loc. cit.

12 Graton an d Fraser, J. Geol., 1935, 43, 785; also Fraser'ibid, 1935, 43, 910.13 Burmester, Z. angew. Math. Meeh., 1924, 4, 33.H Emersleben, Phys. Z. , 1925,26,601.15 Stanton and Pannell, Collected Researches, ,NationalPhysical Laboratory, Vol. II , 1914.16 Schiller, Z. angew. Math. Meeh., 1923, 3, 2.

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Shape.I. Circle . .2. Ell ipses-(a) Major axis = twice minoraxis(b) Major axis = 10 by minoraxis.3. Rectangles-(a) Length = breadth, i.e.,square.(b) Length=2 by breadth . .(e) Length = 10 by breadth . .

(d) Length is infinite . .4. Equilateral Triangle . .5. Pipes with Coresl9 -(a) Core se t concentrically(b) Core se t eccentrically(c)

ko Remarks. 20 Poiseuille's law.21324517819426530167,

20-301,7-3,0 Eccentricity 07


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bearing on the present problem, ko ranges onlybetween th e l imits 18 and 25. I t is interest ing tonote that ko=20 does no t necessarily denote acircular cross section, nor even a shape resembling acircle. This probably explains much of th e successin applying Poiseuille's law for circular channels togranular beds.In substitution for U e and for Le in equation (7),Kozeny followed Dupuit's assumption that the inter-stitial velocity, U e, is equal to :-, bu t this should beEmodified stil l further. If , in any section of the bednormal to th e direction of flow, th e fractional freearea is , then the average velocity parallel to thedirection of flow must b e ~ . As, however, the actualEpath pursued by an element of th e fluid is sinuous,this represents only the component of velocity parallelto the direction of flow. Thus, th e time taken forsuch an element of fluid to pass over a sinuous trackof l e n g t h , L . , a t a v e l o c i t y , ~ . corresponds to thatt aken to pass over a distance, L, at a velocity, :-. Inshort, th e true value for u. is .t, and e q u t i o ~ (7),therefore, becomes

U = ~ ~ 2 . t . ~ . g . tY (8)and it is only necessary to substitute m ~ to obtainequation (6a), hereafter cal led Kozeny's equation.The only difference is that k is rep laced by theexpression ko ' ~ 2. According to th e experimentalwork reviewed in th e next section, th e val ue of k,and, therefore, of ko'(i) 2, is about 50.Bartell and Osterhof21 derived equation (8) byregarding the equivalent capillaries as circular, that is,ko=2 0, and with the aid of Hitchcock's22 assumptionthat fe This gives k=2 o ~ r =4,9, in goodagreement with experiment. As already noted, th evalue, ko=20 is reasonable, though it does notnecessarily denote a circular channel. On the o therhand, it is doubtful whether ~ can be as large as

and it is believed by t he writer that ~ V is amuch more probable value (seeAppendix I) . Itfollows

kthat kO=2 = 2,5, th e value for a nar row, rectangularchannel; these values are adopted in subsequentdiscussion.EXPERIMENTAL VERIFICATION OF KOZENY'S EQUATION.

In dealing with beds of spheres,6 ( I -E)S= d (9)21 Bartell and Osterhof, J. Phys. Chem., 1928, 32, 1553.22 Hitchcock, J. Gen. Physiol., 1926 9 755.

may be substituted in equation (6a), which thentakes th e formd2 E3 t.p.gU = kYJ 36 ( I -E)2 ' -:r- (10)

Fo r non-spherical part icles, a similar type of transformation may be made , in which

S= 6 ~ ~ E (9a)In this,


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TABLE III.Schriever's Data, for Small Glass Spheres.

Yiseosit.:v of gas-free oil (nujol) at. 99 C.=005 poise.Kinemat.ie viscosit.y=00598 (cm. 2)/sec.Average value of k=506.

26 Sehriever, Trans. Amer. Inst. Min. Metall. Engrs., Pet..Div., 1930, 86. 333.27 Muskat. an d Botset, Physics, IH31, 1. 27.

Very different conditions were employed by Muskatand Botset,27 who forced air through a bed of glassbeads at high pressure differences, so that duringits passage the ai r expanded to about thirty times

Diamet.er (d) Porosity k (Air). k (Water).em . (d00928 0391 5720370 58703675 59503635 574 60500709 0400 52003925 5350388 5450384 5370373 555 5'4300497 0376 5180373 5310363 5290361 521 5'1100319 03895 4880384 5'1103795 50203745 51803705 515 5360368 513 52903625 520 5320025 03H05 5290384 5390379 53H0373 5420370 5340366 549 5M

Average value of k=534.21 loco cit.26 loco cit.

its initial volume. In this case, equation (9) must beused in th e differential form,d2 3 pdP (/pu=G= kTJ . 3 6 1 - j 2 ~ (11)

the mass velocity G being preferred to the linearvelocity as it remains constant from cross-section tocross-section of th e bed. Since, for an ideal gas,:..=Po=a constant, equation (9a) is more usefullyP Poemployed in th e integrated form,

d2 3 D.(P2). {/ PoG=kTJ 36 ( 1 - ) 2 ' 2L . Po (12)Another form of equation (11) which is sometimesused is d2 3 pm . D.P . (/G=kTJ 36(1-)2' L (13)where pm is th e density of th e gas at th e arithmetic

(Po+P l Po pm dmean pressure 2 ,Slllce 2Po (PO+P I ) anD.(P2)=(po+P l ) (PO-PI) = (Po PI) . D.P.When equat ion (12) is applied to th e data ofMuskat and Botset27 for glass spheres of 00632 cm.diam., and =0 '338 , where they lie in th e region ofsteamline flow, it is found that k=465. When it isconsidered that, compared with Schriever's26 experiments, th e ratio of viscosit ies is about 23 : I, that has a considerably lower value, and that an oil isbeing compared with a gas undergoing a high degreeof expansion, th e agreement is remarkably good.TABLE IV .Green and Ampt s Data for Small Glass Spheres.Porosit.y k(.)

0387 50403777 51803653 51903533 52103889 50603779 49003689 50703603 50903958 4960384H 51203715 49503552 51403934 48403806 5060369 4H503597 516

that is,

0,02.';2

0'04i3

01025

Diamet.er (d)ems.

00528

viscosity of 06 to 11 poises (variation due to changesin temperature), and carried out experiments with-n in. to in. steel balls in a tower of 2 in. diameter.The sizes and porosities employed in his work are setou t in Table II , and, i n t he thi rd column, th e valuesk calculated from equation (6) are given. Thesevalues show a trend upwards as the size increases,but, as Coulson pointed out, this is probably dueto friction at th e wall, a term neglected in th ederivation of equation (6). The values of keGTT inth e last column have beencalculated by introducingth e

1f 8empulca actor S '1D.p. g 3 . ~ )keGT1 L 82 8 (6b)u TJ 1where 8 1=total surface area per unit of packed

volume = 8 + . The agreement in th e values ofkeGTT in the last column indicates that t hi s t ype ofcorrection is valid and that Coulson's experimentsgive approximately th e same value for k as do thoseo f Donat .Schriever26 passed a hot, gas-free oil, viscosity=0,05poise, through a column packed with small glassspheres, and varied and d. Neglecting th e empiricalequation which he advanced for his data, and usingth e data to calculate k, th e values in Table I II areobtained. Pract ical ly all the values are within 3%of the mean, k=506.

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Some earlier experiments by Green and Ampt28with small glass spheres cover approximately th esame range of values of d and of E as those ofSchriever,26 bu t the fluids employed were air andwater instead of oil (Table IV). Unlike the datapreviously discussed, however, these were notcorrected for the abnormal ly h igh resistance encountered in the first few layers of packing, a correction which, as Blake4 has pointed out, is importantin this type of experiment (see also Hatschek29 ).Consequently, while the values of k are fairly consistent for each size of grain, there are considerablevariations in the values for grains of different sizes.The average value, k=5 34, as would be expected,is somewhat above the value k=50 set by morerecent work, with proper corrections, bu t is sufficientlyclose to confirm Kozeny's equation. .Kozeny' s Theory Applied to Measurements ofCapillary Rise.-Equation (8) assumes that a granularbed can be s tudied as a group of equivalent channelsto which Poiseuille's law or its equivalent can beapplied. Kozeny's theory consists in the identificationof m, th e mean hydraulic radius of an equivalentchannel w i t h ~ th e mean hydraulic radius of th ebed. Interesting confirmation of this theory isobtained from measurement of capillary rise inpowders. In a circular capillary, th e capillary risefor a liquid making zero contact angle with thematerial of th e capillary is related to the diameter by

de U4= pgh (14)Schultze30 has investigated a wide variety of noncircular capillaries, and his data are in agreementwith the general equation,

Um= pgh (15)within an error of 10%. According to Kozeny'stheory, this equation should also be true for agranular bed, and the value of th e mean hydraulicradius, m, should whence,

E US pgh . . . . . . . . . . . . . . (16)or, substituting from equation (9a),

1>E =_u_:::6 ; ;1 ---E) pghd . . . . . . . . . . . . . . . . (17)Hacke tt and Strettan,31 using a bed of sphericalgrains, and four different liquids, found, for E=038,an average value of 96 for pghd, whereas th e6(I-E) Ucalculated value, , is 98.ESmith, Busang and Foote,32 tested a wide varietyof liquids in beds of sand containing nearly spherical

4 loco cit.26 loco cit.26 Green and Ampt, J. Agric. Sci., 1912-13, 5, No.1.2' Hatschek, J.S.O.l., 1908, 27, 538 .30 Schultze, Kolloidzschr., l 125, 36 , 65 ; 1925, 37, 10.31 Hackett, and Stl'ettan, J. A[lr'ic. Sci., 1928 18 671.32 Smith, Busang and Foote, Physics, 1931, 1, 18.

particles, and though their data are somewhatscattered, they near ly a ll lie within the l imits givenby equation (17) between 1>=0,75 and 1>=0'90,which are the probable limits of the surface factor, 1>,for the particular sands. The range of E was approximately E=O34 to E=O44.The only experiments in which both permeabilityand capillary rise have been measured for th e samepowder are those of Bartell and Osterhof.21 Theseshowed that the value of m calcula ted from

equation (8), assuming k= r ko = (i) 2. 2=5.0,was practically th e same as that calculated fromequation (15), as would be expected from the foregoingresults. As no measurement of the surface of thepowder was made, it is impossible to check that m

Ewas equal to S.Alternative Theories.-In opposition to Kozeny' streatment, many workers, e.g., Darapsky,ll Furnas,33Chilton and Colburn,34 do not accept th e Dupuit

assumption that the interstitial velocity is constantfrom cross-section to cross-section of the bed, andI u u Leequa to - , or to _.L ' I t is ev iden t, for instance,E Ethat for spheres packed in the most loose arrangement (cubic packing), th e porosity is =0476, whileth e fractional free area in a p lane of cent res for anylayer is only 0215. In the plane parallel to this anddis tant by hal f a diameter, the fractional free area isunity. This would mean a great contraction andexpansion in every distance, d, for a fluid flowingthrough the bed. On th e other hand, Graton andFraser12 have pointed ou t that such packings shouldoffer different permeabilities according to the directionof flow, whereas th e characteristic of natural, granular

beds is that they present random packings, and thatth e permeability is the same in all directions. In arandom packing, it may be assumed that the voidsare so evenly distributed throughout th e bed thatth e fractional free area at any cross-section is constantand equal t o the porosity, E.Thus, instead of considering that t he r at e of flowin a pore chan:qel is alternately increasing anddecreasing, it seems nearer t he t rue s ta te of randompacking to assume that it is constant. Within thebed there cannot be any isolated pore channel, sinceth e whole system of voids is inter-connected so thatwhere th e section of one void is decreasing in thedirection of flow the velocity does not increase, bu tth e excess of the fluid escapes to a neighbouring void,th e section of which is enlarging in the direction offlow. While this emphasises the constancy of th erate of flow at each point of the bed, it also makes th esinuous character of th e flow clear. I t is obvious thatevery flow-line of th e fluid, in th e continual divisionand rejunction with other flow-lines, must follow a

11 loco cit.1210c. cit.21 10c. cit. Furnas, Bull. U.S. Bur. Min., 1929, No. 307.'4 Chilton and Colburn, J. Ind. Eng. Ohern., 1931,23, 913 ;MacLarenWhite, Trans. Amer.1nBt.Ohern. Eng., 1935,31,390.

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CARMAN.-FLUID FLOW THROUGH GRANULAR BEDS.

TABLE VI.Values of ' calculated from Heywood's38 MicroscopicMeasurements.

T ABU V.Values of ' cnlculated from Permeability, using Kozeny sEquation.

I t would appear, therefore, that k=50 is independentof shape, so that equation (21) can be used forcalculating th e surface of any type of powder.TABLE VII.Pirie s o Results with Prisms and (:Ubp8.

Unfortunately, there is almost complete lack ofexperimental data for par ti cl es of regula'r, geometrical shape, and, therefore, of known specificsurface. Experiments of Uchida and Fujita 39 withsmall Lessing r ings indicated that, even thoughthese do not correlate with spherical particles in theturbulent region, agreement is rapidly approached as

O'S\O'SI065069073065038089055028

Material. Nature of 1> \\'orkers.Grain. -Average for various types 075 Fancher an d Lewis. 3 of sand.Flint sand Jagged. 065 Ottawa sand Nearly 095 spherical.Wilcox sand Jagged. 060 Chalmcrs, Taliaferroan d Bawlins.Sand . (Nature 0n5 Muskat and Botset .unknown.)Sand . Angular. 0'70-0'75 Grcen and Ampt. 'Sand . Roundcd. 083 Donat. 2 'Flint sand Jagged. 067 Flint sand Jagged flakes. 043

Tungsten powder . .Sand ( rounded grains)Crushed g la ss ( jagged g ra in s)Cork . .Pulver ised coalNatural coal dust (up to i in.)Fusain fibresFlue dust (fuscd, spherical)Flue dust (fused aggregates)Mica (flakes)

Substance an d Nature of Grain.

E k ltcorr.._Hexagonal prisms, 048 cm. 0377 502 471length by 047 em. d ia . 0426 428 402---Cubes 056 em. side . .. 0344 529 4n2

. .. 0397 471 437 . .. 0448 421 392

2. loc. (;i t.27 loe. cit.28 loco cit.3. Fancher an d Lewis, J. Ind. Enq. Chem., I n33, 25, 1I3H.51 Chalmers, Taliaferro an d Rawlins. Trans. Amer. Inst.1'.1in.Metall. Eng., Pet. Div., 1932, 98, 375 .38 loc. cit.39 Uchida and Fujita, J. Soc. Chern. Ind., Japan (suppl.binding), 1934, 37, 724B an d 791B.00 J . M. Pirie, private communication.35 Burke an d Plummer, J. Ind. Eng. Chern., 1928,20, 1197.3. Heywood, Proc. Inst. Mech. Eng., 1933, 125, 383.

very tortuous path. As far as th e length of th e pathis concerned, and it s influence on the interstitialvelocity, the correcting factor has been introduced. Later in this paper it will be necessary toinclude another important effect, that is, th e difference between flow in a curved channel and that in astraight channel.An interesting theory has been proposed by Burkeand Plummer,35 in which, instead of assuming th egranular bed to be equivalent to a group of parallelchannels, they regarded t he tot al resistance of th ebed to be made up from the sum of th e separateresistances of th e individual particles in it , asmeasured from th e rate of free fall. Fo r a bed ofspheres, however, th e resulting equation for streamlineflow is e2( I - e ) ::>P'g (18)U = k1 7J8 2 y; .. ~ ::>P.gthatrs, U=k 1 .-.;j36(I-e) L (19)where k1 has th e theoretical value k1=05. Since it3has been s ~ w that u is proportional to (l':'e)2 andnot to )' t hi s equat ion must be abandoned.( -e

Determination of the Surface of Powders.-Kozeny sassumption that ~ provides a convenient methodfor determining S, that is,

S = .... = Epgh (20)m aby th e capillary rise method, or

8= ' : '=e /e ::>Pgm ty 5uYJL (21)by the permeabil ity method. These equations,however, have been tested only for spherical particles,for which S is more easily calculated from d, that is,6(I-e)S= d To serve a useful purpose, they must beshown to hold for irregular particles. Unfortunately,there appear to be no published data suitable fortesting equation (20).In equation (21), assuming k=5 0, independentof particle shape, and substituting S = 6 ~ ~ e ,then 4> is given by f

6(I-e) 5u7JL4> = : :>P (22) E gThe values of 4> calculated from the data availablein the l ite ra tu re a re given in Table V. Theclassification, spherical, rounded, angular, jagged, hasbeen made on the basis of microphotographs presentedin th e various papers. The values of 4> appear to beconsistent and they are in agreement with valuesof 4> for typical powders, calculated from th e microscopic measurements of Heywood38 (Table VI).

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th e streamline region is entered. As, however, onlya few of their experiments extend to the streamlineregion, more defini te ev idence is requi red. Of muchgreater importance are a few figures recently obtainedby Pirie40, working entirely in the streamline region.These are summarised in Table VII.Apparently, while k is roughly of th e same orderas for spheres, it is not constant for changes inporosity, showing that a new factor has entered.Possjbly, flat-sided shapes behave differently, sincemany of the points of contact are really planes ofcontact. In spite of this, in the calculation of Sfrom permeability, the square root of k is taken, sothat th e maximum erro r f rom as suming k=5'Owould be only 10 .I t is suggested that, in the future, more attentionbe paid to these methods. The surface of powdersis a quantity required for many branches of work,and, at the present, it can be determined only fromrate of solution, rate of absorption methods, and bymicroscopic methods, al l of which are open to serious

error or are tedious.Mixtures of Sizes.-It is important to note that theprevious sections have dealt wholly with grains ofuniform sizes. When mixtures of sizes are studied,as has been done by Coulson25 , it is found thatKozeny's equation had certain important limitations.The value of S for spheres of mixed sizes is given by

S=6(1-) WI). dlwhere WI =fractional weight of part icles of size, dl land this, substituted in equation (6b), should givekeorr.=5O. The values obtained by Coulson are given

particles is the decrease in porosity. Normal ly, forspheres of one size, =038, independent of size,though for very small sizes, th e great ratio of surfaceto volume allows bridging t o t ake place more easily,so that tend s to increase. When, however, twosizes are mixed, and especially when the size-ratioexceeds four, th e mixture usually shows a lowerporosity than either of th e constituents. Theporosities in Table VIII emphasise the essentialpoints. For the ratio 125 : 1, no change in porosityis effected on mixing; for th e 2: 1 ratio, a slightminimum is reached; for th e 5 : 1 ratio, a porosityas low as =0286 is attained . Similar results havebeen obtained by Furnas33 and by Fraser12 In themixing of concrete, it has long been known from theresearches of Feret41 , that, if a low porosity is wanted,a ratio of at least 4: 1 must be maintained betweenone grade and the next.Furnas33 and Fraser12 each discussed this questionin some detai l, so that a fai rly clear picture can beformed. Assume a bed of large spheres to whichsmaller spheres are being added. Changes in porosity,, during th e addition of the first few spheres dependupon two opposing effects, (i) the small spheres fiJIthe voids between th e l arge ones and so tend todecrease , (ii) they tend to wedge apart the largerspheres and so to increase . As long as th e size.ratio is less than four, these two effects practicallybalance, bu t when it is greater than four, th e smallparticles can slip ,between th e larger without disturbing them, so that (i) rapidly becomes th e dominating effect. The l imit to this is when th e largevoids are a lmost filled, at which stage (ii) graduallydominates and passes through a minimum value.As more of th e small particles are added, another

TABLE VIII.(,oulson's25 Results for 1Ylixtures of Spherical Particles.

in. and in . Ratio 1 25 : 1. i\r in. and :& in . Ratio 2 : 1. .,s. in. an d 10 in. Ratio 5 : 1.% Smaller f keorr , % Smaller f k eorr . % Smaller f keorr .size. size. size.

106 0418 591 1l1 0397 533 1l7 0350 329403 0391 512 199 0386 53 190 0313 40620 0395 505 335 0383 5' 8 29'1 0313 36803 0394 52 500 0380 518 346 0286 5, )- - - 667 0383 5,)5 43,) 0293 503- - -- - 809 0390 517 533 0320 536-- - I - - - - 642 0334 548-- _ -- I - - - R21 0374 533I I I I

in Table VIII. With two exceptions, th e values ofkeorr. are reasonably close to 50 for th e size-ratios,125 : 1 and 2 : 1. When, however, the ratio of sizes is5: 1, k orr. varies widely between the extremes 33 and55. A better understanding of the reason for th ebreakdown of Kozeny's equation for such mixtures isobtained by an inquiry i nt o the packing of mixedsizes of spheres.The most noticeable effect from mixing two sizes of25 lac. cit.


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JUBILEE SUPPLEMENT-Trans IChemE, Vol 75, December 1997CARMAN.-FLUID FLOW THROUGH GRANULAR BEDS.

S39

th e packing of th e small particles in their vicinityand make it more open. The relative importance ofth e former volume effect to the latter surfaceeffect increases as th e size-ratio increases, so that,when the size-ratio exceeds 4: 1, the addition of largeparticles to a bed of small particles means a considerable decrease in E.The derivation of Kozeny's equation rests on th econception of a mean hydraulic depth for th e wholebed, which is, in effect, the average mean hydraul icdepth of a ll the pores of the bed. For it to hold trueth e pores must be reasonably even in size. This ist rue in beds of spheres of uniform size, and t he onlycase in which it would no t be likely to hold for mixedsizes occurs when th e size ratio exceeds 4: 1, andeven then, only when large spheres predominate, andthe large voids are incompletely filled by the smallerspheres. On first addition of small spheres to a bedof large ones, there are two widely different sizes ofvoid, namely, those between the larger spheres andthose between g roup s of smaller spheres nested inthe large voids. The large voids ini tial ly form aninterconnecting system, bu t as more small spheresare added, they gradually become filled, an d finally,only th e void-size characteristic of the smaller spheresremains. This is in agreement with th e general trendof Coulson's values, since it is only for the lowerpercentages of th e smaller spheres that k differs verymarkedly from 50.

That k should have low values when th e pore-spaceis divided between two different sizes of void is inaccordance with expectation, since it can be shown'that, for a group of parallel circular channels of givenvolume and given surface, the flow is greater whenthey are divided into two sizes than when they areall of th e same size, (see Appendix II).Deviations from D Arcy's Law.-Kozeny s equationis a derivative of D'Arcy's law, and is, therefore,subject to the same l imi tat ion s. The most important

limitation is that, when exceeds about 20, th epressure loss across th e bed r ises more quickly thanth e rate of flow. This is discussed in the next section.

An impor tant effect occurs at low values o fwhen th e rate of flow increases faster than th e dropin pressure. King 42 first pointed this out, andSiegeP has summarised th e more recent knowledge onth e subject. I t has, however, been subjected tolittle study and is not yet properly understood.King 42 postulated that an increase in the rate of flowincreases th e effective porosity of th e bed, th eeffective porosity, E', being less than the true porosity,E. Some confirmation of this was made by SvenErikson,43 who measured the actual velocity, U e , ina bed of sand by injecting a salt solution i nto waterflowing through th e bed and taking th e time for th e

3 loco cit.42 King, Nineteenth Ann. Rep. , U .S. Geol. Surv., 1897-8,2,59.43 Sven Erikson, J. GaBbeleucht., 1920, 63, 615.

TRANS INSTN CHEM ENGRS Vol. 15, 1937

sa lt to appea r at the out let . As suming that E' ~ Uewhere U is th e apparent velocity, he found that, atvery low rates of flow, E' increased steadily from 014and finally reached a steady value of about 047at ~ ~ = 0 0 0 6 th e porosity, 0,47, being the trueporosity of th e sand. He further found that, whenthe bed was drained and a ir was passed through, thefractional volume of water retained by th e wet sandwas 0,33, that is, 047 minus 014.

The hypotheses advanced to explain the phenomenonare (i) that a stationary film of liquid is retained onthe surface of th e particles, which decreases inthickness as th e rate of flow rises, and (ii) Darapsky'sllsuggestion that a small, stat ionary r ing of liquid isretained at each point of contact of the particles, thesize of th e ring being controlled by th e velocity offlow. The explanation most probably depends on thesurface forces between the liquid and the particlesof solid.Bozza and Secchi 44 showed that surface forces doenter into th e p robl em; they found that a bed ofvery fine quartz-sand gave permeabil it ies for waterand aqueous solutions about 13 times greater thanfor certain organic liquids. The surface tensionsof th e two groups were, respect ively about73-80 dynes./cm. and 24-30 dynes./cm. Furthermore, when a bed of finely ground galena was used, th eratio of permeabilities for t he two groups changed toabout 18. The range of va lues of covered by thiswork was mainly 1 X 10-3 to 1 X 10-6.

Sometimes an increase in permeability with riseof pressure can be explained on pur el y mechanicalgrounds. Fo r instance, Hatschek29 noticed that thisincrease was character is tic of fi lt er cloths , andUnderwood explained this by assuming that increaseof th e pressure difference across a cloth, supportedby a series of ridges, stretches th e fibres of the cloth,and thereby enlarges th e pores.One further exception to D'Arcy's law, importantin filtration, should be ment ioned . Thi s occurs whenth e particles forming the bed are easily deformed.Various workers 45 have dealt with thi s case, and haveshown that th e permeability decreases steadily asth e pressure drop across the cake is increased. Theflow is certainly no t turbulent, however, since, fora constant pressure difference, th e permeability ofth e bed is no t only inversely proportional to viscosity,bu t also to thickness of th e bed46, both of which arecharacteristic of streamline flow. What happens isthat an increase in the pressure difference furtherdeforms th e particles and thus, by decreasing th eporosity, reduces th e permeability.

11 loco cit.2 loc . ci t.. . Bozza an d Secchi, G. Chim. Ind., 1929, 11, 443 an d 487.4. Almy an d Lewis, J. Ind. Eng. Chem., 1912, 4, 528;A. J. V. Underwood, J.S.C.I., 1928, 47, 325T.46 P. C. Carman, J.S.C.I., 1933, 52, 280T; 1934, 53, 159Tan d 301T.


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S40 JUBILEE SUPPLEMENT-Trans IChemE, Vol 75, December 1997TRANSACTIONS.-INSTITUTION OF CHEMICAL ENGINEERS.

General Correlation for .flow through Granular Beds.The decrease of th e permeability as exceeds 20appears, at first sight, to be analogous to th e onset ofturbulence in a straight pipe. The change fromstreamline flow, however, takes place very gradually.

4

in which a, band n (1,6 < n < 2'0), are constantscharacteristic of the bed, and th e first and th e secondterms denote streamline flow and turbulent flow,respect ively. Chalmers, Tal iaferro and Rawlins37found th is type of equation accurately to fit theirexperimental data over wide ranges of u; it will be

I00I- ~ - _ - - - - - - - - - - - - _ I _ - - - - - - - - - _ - - - - - _ _ _ _ _ 1 - - - - - - -

I O I ~ _ _ _ i r _ _

~ l ~ , . D / - - - - - - - - - - - - - - - - - - ~ - - - - - = : . - . , . , ; : - _ - _ : - .... ID i l jII,

D I I ~ _ _ _ _ _

BO O I ~ - - - - - - - - - - - - _ _ _ - - - - _ - - - - _ - - - ~ - t _ - - - - - - = ~

D D D ~ n - - - - - n - r - - - - - _ ; ; _ - - - - ~ ; ; _ - - - - _ . _ h c _ _ _ _ _ - - - _ _ _ . h - - - - -001 01 10 10 100 1000pu .sFIG. 1.Correlationjor Beds oj Spherical Particles (jor key, see Table IX).

(AI and (B) Theoretical, based on relationshipfor straightpipes.Forchheimer47 has suggested that this is due tovariations of pore-size within th e bed, turbulenceoccurring first on the larger pores, and he suggests,therefore, th e following type of equation,

LV>=au +bun . ............ (23)., P. Forchheimer, Hydraulik, 3 Auti., p. 60, 1930,Leipzig an d Berlin.

shown later that a similar type of equation is bestsuited to fit th e curve in Fig. 1. Instead of adoptingForchheimer's hypothesis, however, these workersnoted that th e decrease in permeability correspondedless to the onset of turbulence than to th e increasein resistance exhibited by curved capillaries ascompared with B t r a ~ h t capillaries.37 IDe. cit.

TRANS. INSTN CHEM. ENGRS, Vol. 15,1937


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S41

I t is no t proposed to deal with the various empiricalequations proposed for turbulent flow, since nonehave any wide generality, and th e method of correlation by plotting dimensionless variables introduced byBlake4 has proved sufficiently successful over a widerange of data. I t has already been shown that th e

fluids include air, water and Schriever's nujol26,viscosity=0'05 poise. The ai r used by Muskat andBotset27 underwent a high degree of expansion init s passage th rough the bed, and a still greaterexpansion was attained in th e experiments withnatural gas. The whole range of the curve in Fig. ITABLE IX .

Data Used in Plotting Fig. 1.Curve or Range of e.oint on Workers. Nature of bed. Porosity f Nature of fluids.Fig. 1. IS

1 Schriever26 00252-0,1025 em. g la ss 0 3 53 -0 3 96 Hot., gas-free oil Approximately 0'15-0'7spheres. (1)=5'0 centipoises)2 Green and Amp t28 0025-0094 em . glass 0,361-0,391 Air; water. Range no t given. Probablyspheres. between 0,1-1,0.3 Muskat an d 00632 em. glass spheres 0317 Air (a t high degree of 14-20Botset. 27 expansion).4 Coulson2 0159-0794 em. s teel 0,392-0,417 Light oils, average vis- 0,01-1,0spheres. cosity about 09 poises.5 Burke an d 0'148-0,634 em. lead 0,303-0,421 Humid air. 02-250,0Plummer3 shot.6 Chalmers , 0127-0305 em. lead 0353 Natural gas, viscosity 100-4,000Taliaferro an d shot. about 0005centipoises.Rawlins (A t very high degreesof expansion)7 0305 em. lead shot 0347 500-9,0008 0406 em. lead shot 0346 600-13,0009 Arnould 48 25 em. wire spirals 090 Air 100-35010 Uchida an d 050-10 Lessing rings 0844 0'5-250X Fujita39 0184-0,441 lead shot 0384 06-250+ Furnas , Ray 0,138-0,905 em. lead 0,38-0,414 Range no t givenan d Kreisinger 49 shoto BerJO A MachO 1,0-5,0 em. porcelain 0,694-0,785 40-1,100saddlesdimensionless groups suggested by Blake, in thestreamline region, give rise t o t he Kozeny equation.. ::l.Pg3 puIn FIg. I, t hese groups, rp L 2S and -Spu 1 7Jhave been p lo tt ed on logarithmic scales, th e rangeof data exhibited on thi s p lo t being given in Table IX .For spherical particles, th e scale of sizes is fromd=0 025 em. to d=0905 em., and porosities rangefrom 10=030 to 10=0,42. The fluids vary fromCoulson's Oil25 , with a viscosi ty of about one poise toth e natural gas used by Chalmers, Taliaferro andRawlins37, the viscosity of which, according toRawlins52, was about 0005 centipoise; intermediate

4 lac. cit.' loc. cit.27 lac . c i t.28 lac. cit.33 lac . c i t.3. lac. cit.36 lac . c it ..7 lac. cit.'910c . cit.48 Arnauld, Chim. et Ind., 1929, 21, 478.49 Ray an d Kreisinger, Bul l. U .S . Bur . M in ., 1911, No. 21. 0 Berl, catalogue of Messrs. Ditt & Fries, , Viesbaden.01 Mach, Dechema Monographien, 1934, 6, 38.5. Rawlins, Trans. Amer. Inst. Min. Metall. Eng., Pet. Div.,1932, 98, 436.


has been extended back by Coulson25 (curve 4) to~ ~ = 0 0 1 in th e streamline region, and forward to~ ~ = 1 O 0 0 0 by Chalmers, Taliaferro and Rawlins inthe turbulent region (curves 6, 7 and 8). The onlyserious deviations are th e results with t,he largestsizes of lead shot used by Chalmers, Taliaferro andRawlins37 (curve 8), and with th e smallest sizesused by Furnas 33; th e results with th e small size dono t agree with th e data of Burke and Plummer35(curve 5) with th e same size of shot, and under thesame conditions. The deviation with th e large leadshot is more serious, and further experiments arerequired to decide whether th e Blake plot breaksdown for large values of ; ; .

Of even greater interest is th e extension of th ecorrelation to Arnould's wire spi ral s44 (curve 9),10=090, and to Berl saddles, 10=069--078, whichare no t only non-spherical (4)=02 and 4>=0'3,.5 lac . c it .26]OC. cit.2710c. ci t.33]OC. cit.3.lac. cit.37 lac . c i t.44]OC. cit.

L


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respectively), bu t present different orders of size andof porosity. On the other hand, while the ploteffects correlation for different sizes of Lessing rings,the curve for these rings lies well above the maincurve, though it appears to merge with the lat te rinside the region of streamline flow.The equation best fitting t he data in Fig. 1 is oneof the Forchheimer47 type in the form,(7)8) (7)8) 0.10/=5 pu + 04 pu (24)

in which the second term is negligible in the streamlineregion and the first in the turbulent region. Theonly region in which it fails is between - ~ ~ = 0 7 to

~ ~ = 7 0 where it gives results about 10% too high.All the data in Fig. 1, with the exception of thoseof Green and Ampt28, have been corrected for endeffect in the original papers . Another type ofcorrection is for the wal l effect. The simple type

of correction used for this is indicated in the followingsection. Owing to a considerable difference ofopinion as to t he na ture of the wall effect, it isdiscussed at some length.Correction for Wall Effect.-Photographs of crosssect ions of granular beds packed inside a tube havebeen given by Fumas33 and by Graton and Fraserl2 These show that the porosity of the bed is greateri n the layers next to the wall, so that , in this region, itwould appear that the permeability should be higherand that correction for this should be made incalculating the permeabil ity of the whole bed.Fumas33 approached the problem by imagining aconta iner being pressed into an infinite bed ofparticles, with simultaneous removal of par tic les

intercepted by the downward movement of the wallsof the container. The lengthy mathematica l argument which develops from this conception, however,is too uncertain, and, as Graton and Fraser12 havepointed out , particles are, in practice, fed into acontainer, so that the effect of the container wall isto give a different type of packing from the normalrandom packing. This may extend through severallayers of the mass, and the effect of the wall dependslargely upon i ts curvature and the sharpness of it sangles. While the proport ion of voids at the wallis greater than in the centre of the mass, these voidsare genera lly no wider, so that the mean hydraulicradius in this region is not necessarily greater.Exper iments upon the effect have not given

concordant results. Furnas found that a smallcontainer increased the permeabil ity of the bed,bu t Uchida and Fuj ita39 found, in one case, that itwas decreased, and, in two other cases, that the wallhad no effect.1210c. cit.2810c. cit.3310c cit.3910c. cit.n IDe. cit.

If , as seems probable, the only effect of the wallis to increase the average porosity of the bed, it isonly necessary to take care that the poros ity ismeasured in every run and that the correct value issubstituted in the dimensionless groups of Fig. 1.One factor, however, for which correction migh t bemade is the friction of the walls of the container. I thas already been pointed ou t that in the correlationof Coulson's25 results allowance for wall-friction hadto be made by introducing the lac to , ( 8.

1 s +1This is equivalent to writingJ /),P g . ,,3 ~ = f:,.p .g ,,3If. Lpu l8 8 1 Lpu l8 1

I t may be concluded, therefore, that no elaboratecorrection for wall effect is necessary, provided that represents the average porosity of the whole bed,including the region at the wall, and that SI is used inthe calculation of 0/.The following example i llus trates these points.Chilton and Colbum34 published data for 16 and 0.9 6spheres in a 36 container, in the region =1,000._According to Fig. 1, the value of 0/ should be about020 in thi s region. Values of were no t given, but,as shown in Fig. 2, drawn from experiments made by

7Or--------r------,-------,-----,

- 6 0 ~ _ / _ : . . - - ~ - - - - - - _ _ - - - _ _ _ _ f - - - - f

,,IIIIII

0 1\0 zo 3'0 100 15-0(DId)

FlO. 2.Relation between Por08ity of Bed and Size of Container.

x Experimental. Calculated.the writer in which lead shot was shaken down tothe closest possible packing in glass tubes, the valuesof are about 0485 and 0,47, respectively, thoughthe normal poros ity for spheres packed in a widetube is ,,=0,38. The geometrical reasoning forcalculations of voids is shown in Appendix III.6(1-,,)Since8 = d 8 =1215 cm.2/cm.aand 139cm.2/cm3/:,.p.g. ,,3for the two sizes, respectively, the group, L 'pUliS

' loc. cit.st IDe. cit.

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1

........ ........... t::::::--................. f - - I-..::=..4rr ; - - 13 -- 9 116 18- -- ---- I I

50

1

20

5

1.10 20 50

Key to Fig. 3.

100 2pulqS

FIG. 3.ReBultB Jar Rings.

500 1000 2000

Curve.12345678o1011

12131415161718III

Workers.Blake'

rnould 4Beri liO

MachaRose and Higby

Tower packing.0233 in. glass rings.0269 in .0394 in.0484 in.10 in. Raschig rings.10 in. Aluminium rings.30 cm. triangular rings.10 mm. porcelain rings.25mm.35mm.50mm .15mm.25mm. i in. stoneware rings.l in .i in.f in.1.0 in.11 in.

, loco cit.3< loco cit.n loco cit.ooloc. cit.ii i loco cit.

TRANS. INSTN CHEM. ENGRS, VoL IS, 1937


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equals 027 in each case, which is too high, bu t i f it ismultiplied by ,it is reduced to 019 for I spheresand to 0195 for 0'9 spheres. This is an extreme case,in which the conta iner is so small that it opens upthe packing of the spheres and i ts walls contr ibute alarge propor tion of the total friction. Normally,when th e ratio of the diameter of the container,D, to that of the particle, d, is greater than 10 : 1, thewall effect can be neglected, as it is less than experimental error.

Extension of the Blake Method to Non-SphericalParticles.-It has been shown that in the streamlineregion, permeability measurements give the samevalue for the surface factor, .p, as do Heywood's38microscopic measurements, and Fig. 1 shows thateven Lessing rings give concordant results in thisregion. Uch ida and Fujita's39 experiments on theserings, however, provide the sole example in which anon-spherical shape, for which .p is known, is tracedfrom the streamline region t o the turbulen t region.They provide an interesting case in that, whereasBerl saddles a,pd Arnould's spirals can be correlatedsatisfactorily with spheres in the turbulent region,Lessing rings show a steady deviation from the curvefor spheres between ~ ~ = 0 5 and ~ ~ = 2 0 and thereafter remain at about double the calculated values of{i. Probably the difference is that every element ofthe saddles and spirals is presented equally to flow,while th e interior of a ring is a source of eddies andof dead spaces at higher rates of flow. In thisrespect, the flow in granular beds would be analogousto flow in pipes and t o t he motion of particles throughfluids. Roughness of the pipe wal l only gives riseto increase of resis tance in the turbulent region.Wadell53, too, has shown that, in the correlation ofparticle motion through fluids by means of dimensionless groups , non-spherical par ti cles p resent muchgreater resistance to motion in the turbu lent regionthan do spheres, though, in th e streamline region,shape plays l it tle part.Blake found that his method of plotting did notgive correlation with small glass rings and I Raschigrings; the later data in Fig. 3 show lack of correlationfor rings. The later curves all lie between the limitsoriginally found by Blake4, while . the broken line,representing the data of Uchida and Fujita39(curve 10 from Fig. 1) is approximately midwaybetween these limits.I t seems probable that there are further unknownfactors with solid par ticles of more irregular shape.:For instance, Meldau and Stach54 have shown thatanthracite powders, which abound in acicular andlenticular shapes, tend, on the one hand, to form verysmall voids by juxtaposition of f la t faces or of convexwith concave faces, and, on the other hand, groups

loco cit.38 loco cit.39 loco cit. 3 Wadell. J. Franklin Inst., 1934, 217, 450. Mcldau aw l Stach (Trans.), J. Inst. Fuel, 1034, 7, 336.

of par ti cles lock together readily to form bridgesover large cavities. I t has already been pointed ou tin the section on mixed sizes, that the concept of mean hydrau lic radius for a granular bed uponwhich th e Blake plot rests cannot have a realmeaningunless the pores are of reasonably uniform size.Experimental data on these points, however, arelacking, since, though a large amount of publishedwork deals with beds of broken solids33.37 ,39. inno case can the surface of th e particle be calculated.Even for rings and saddles, only Uchida and Fujita39

have covered a wide range of values of Fo r acritical survey of the usefulness of the curves in Fig. 1,data are urgen tly requi red for par ticles of knownsurface area, including rings and particles with flatand with rounded surfaces, and extending from wellin the streamline region, say =0'1, into the turbulentpu Tjregion, sayTjS=500.

General Discussion of the Blake Method.-Theequations for flow through a straight pipe of lengthLe, and mean hydraulic radius, m, are;;=ko' (_TJ_) (for streamline flow) . . . (25)pu. puemand ~ = 0 0 2 8 (_TJ_) 0.25(for turbulent flow) . . (26)pu e puemsubstituting m=, and U e = ~ ( these equationsbecome applicable to a granular bed. For a fluidflowing across a unit cube of the bed, R is evaluatedas follows:Energy supplied per unit t ime=Energy expendedin overcoming friction per unit timethat is, l i ~ g .u =R .S .u.

l ipg (L)or, R= L S Le (27)and equations (25) and (26) take the forms

l ip g . 3 _ _ (Le) 2 (TJS) Lpu2S - {i-ko L pu .(2S)_. (Le)2.75 (TJS)0'25 ,and ifJ-O 028 L pu (29)

In an earlier section, th e most likely values of ko and( Le ) jof \ i were taken as 25 and V 2, respectively,whence 0/=5 (30)

( TJS) 025and {i=0'073 pu (31)These equations are plotted in Fig. 1, curves A and B.I t follows that the curve for granular beds bears aclose resemblance to curves for JIow in curved pipes,such as those investigated by White55 , both in (i) th e

33 loco cit.37 loco ci t .39 loco ci t . White, Proe .. Roy. ::Joe., 1 J2 J, A. 123 , 645.



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S45

gradual transitionfrom the streamline to the turbulentregion, and (ii) the low Reynold's number at whichdeparture from the law of streamline motion instraight pipes takes place. White expressed hisresults in terms of the rat io of the resistance of thecurved pipe to the resistance of the same length ofstraight pipe for the same Reynold's number. Thisratio, C, he plotted against pum . . where de is the7J ediameter of the pipe and de is the diameter of curva-ture, and thereby he was able to reduce all his resultsto a single curve. Full confirmation has been givenmore recently by Adler56 The resul ting curve isreproduced in Fig. 4 as curve 1; in the same figure,

Wilson, McAdams and SeltzerS? for standard elbows.The values of t for these are in th e region of 15.In view of the fact that this curve is an averagedrawn through widely scattered resu lt s (due tovariations in th e geometrical proportions of differentsizes of elbows), it follows th e trend of C for granularbeds very closely. A further important point is thatall three curves begin their upward trend at about thesame values of the abscissae.Though White's experiments did no t proceed farinto the turbulent region, they appeared to show that,down to ~ = 1 5 , the res istance for a curved pipe ine

/ :--- L.--- ____a__>l;L2/V V ------- .: 3a---

2U

10

5

1 20 50 10 20 50 100 zoo 1 2000 5000

(1) White's Curve(2) Elbows(3) Richter( ) Lorenz(5) Granular Bcds

FIG. 4.Comparison oj C Jor Curved Pipes and Jor Granular Bed8.

ABSCISSAE., ~ . A : .. de

the values of () for granular beds, as obtained fromequations (24), (30) and (31), are plotted as curve 5.Streamline region C= 1+0'08 0 9 (32). (Pu) 015Turbulent regiOn 6=5'5 7]S . (33)

(first term in equation (24) negligible).I t is apparent that, though i is taken as unity, thetwo curves are not in agreement. The lowest valuesof _ddr t es ted by White, however, was -dd=15, and it ise epossible that anomalous effects arise asi approachesunity. Confirmation for this is afforded by the dottedcurve (2) in Fig. 4, which is taken from that of

5 AllIer, Z. angcw. Muth. Mech., 1 J34, 14, 257.

fRANS. INSTN CHEM. ENGRS, Vol. 15, 1937

this region is only about 10 greater than that fora straight pipe. Richter58 carried ou t experimentswith smooth, curved, copper tubes over th e rangee =400-18,000. Down to = 15 his results veri.7] efied those of White in the turbulent region; theymay be represented fairly closely by the equation( pum) 0032C =0,92 (34)

Below this value of ie, th e relationship between ~e d and C was more uncertain, but, as i. decreased below5,0, C increased rapidly until, at = 1,7, the lowest

.7 Wilson, McAdams and Seltzer, J. Ind. Eng. Chem., 1922,14.105.os Richter, Forschv,ngsurbeiten, V.D.l., 1930, No. 338.


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S46 JUBILEE SUPPLEMENT-Trans IChemE, Vol 75, December 1997'l'RANSACTIONS.-INSTITUTlON OP' CHEMICAL ENGINEERS.

value employed in his work, the equation for CbecameG = 9 0 ~ m r 0 3 2 (35)

Lorenz 59 has prov ided a theoret ical equation forturbulent flow in curved pipes, which may be transformed to th e form given in equation (36).( pum)025 (d) 20=1 +57 - : ;- . i (36)

The relationship between 0 and i as given by thisequation is in general agreement with Richter 'sresults in that the term involvingPs very small ifde d 15 d 'd l eb ' t ded excee s , an rlses rapl y a ove um y as -d decreases below 5.0. In Fig. 4, equations (34), (35)and (36)-the last for ~ = 1 7 - a r e plotted as curves3a, 3b and 4, as also a re the average results ofWilson,McAdams and Seltzer57 for standard elbows. Inview of th e uncertainties arising from scantiness ofexper imental data, both for curved pipes and forgranular beds in the turbulent region, agreement isreasonably good for the two cases.

I t appears, therefore, that both in streamline andin turbulent flow, th e flow through granular beds isequivalent to flow in a helically wound pipe, suchthat th e diameter of the helix is approximate ly15 t imes th e diameter of the pipe. This does no tprovide an exact picture of th e pat h ta ken by anelement of fluid traversing a granular bed. On th ebasis of th e excellent correlation in Fig. 1, however,it does suggest that the tortuous nature of the pathsis not so chaotic as often imagined. The true pictureis that of an element of fluid winding i ts way along apath through the bed, changing direction at randomfrom point to point, bu t only in such manner thatth e curvature of the pa th and its mean hydraulicradius, bear a relationship to one another which ispractically constant at all points in t he bed and fora ll beds .

Design of Packed Towers.-Equation (24) may beapplied directly to the calculation of pressure dropfor packed towers where th e packing consists ofsolid spheres or saddles. For practical purposes,since flow is usually well in the turbulent region,the equations may be simplified tol:1P '(1.3 ( 18)'1t/s=L 28 =0,4 - (24a)pu 1 pu

With ring packings, it is no t possible to design witha high degree of accuracy, but safe values are givenby using 25 times the values in equation (24a), that is,l:1P(I3 1 8 )'1~ ~ = = 1 -Lpu28 1 pu

57 loe. cit.S. H. Lorenz, Phys. Z. , 1 J2 l, 30, 228.

SUMMARY.In the foregoing paper, it has been shown that th edimensionless groups originally used by B lake ' forflow. of fluid through granular beds provide anexcellent correlation for beds of spherical grains, andthat thi s extends over th e very wide range of

experimental data available, ~ ~ = O O I - l O , O O O ) .In the streamline region, where D'Arcy's law holds,Kozeny17 has provided a theoretical basis for Blake'smethod of correlation, and the form of D'Arcy'slaw which results has been termed Kozeny's equation.Certain deviations from D'Arcy's law at very low

values of have been discussed briefly. I t has alsobeen pointed out that Kozeny's equation does notextend to mixtures of two sizes of spherical particleswhen the size-ratio exceeds 4 : 1, and the proportionof smaller spheres i n the mixture is less than 40 .Satisfactory data for non-spherical particles arescanty, bu t it appears that, in the streamline region,Kozeny's equa.tion isvalidwithin an error of 10 -20for all shapes of par ticle. In tJl.e turbulent region,th e Blake plot correlates spheres and curved shapessuch as saddle tower-packings, bu t is not satisfactoryfor rings.The theoretical implications of the Blake plot havebeen discussed in detail, and it has been shown thatflow in granular beds bears a close analogy to thatin curved pipes with the same mean hydraulicradius.An interesting development is that, for a powder,Kozeny's conception of a meanhydraulic radiusfor a granular bed is applicable to the correlation ofmeasurements of cap il la ry rise and permeability.As a corol lary to this, since both correlations involveth e value of the specific surface of the powder,measurements of capillary rise and of permeability,respectively, offer two new and independent methodsfor determining the specific surfaces of powders.Acknowledgment.-The writer wishes to expresshis thanks to Mr. M. B. Donald and to Dr. P. Sillittofor helpful suggestions and criticism in the preparationof this paper. -

APPENDIX 1.Experiment to determine value of 7.

In an attempt to determine the path followed byan element of fluid in its way through a. granularbed, th e writer has introduced colour-bands in waterflowing ver tica lly downwards through a bed. ofi glass spheres in a 1 glass tube. Streamline flowwas maintained. By pareful adjustment, itfound possible to observe the colour-band throughthree or four layers of par ticles and sometimes evenfurther. The most' notable characteristic was thatt he path made an almost constant angle of 45 withthe axis of the tube, that is, with the direction offlow. Deviationswere observed, even to the extremesof 0 (vertical flow) and of 90 (horizontal flow), but loc . cit.

17 loe. cit.



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JUBILEE SUPPLEMENT-Trans IChemE, Vol 75, December 1997 S47CARMAN.-FLUID FLOW THROUGH GRANULAR BEDS.

cI

FIG. 5

FIG. 6

8

E

Now,

voJ. of solid enclosed by cylinder= vol. of 1 sphere ~ d6

7T D2 h 7TFree space = -4-- fjd3

P ~ i t y ~ 1 - i t l= 1 : ~ l 2 ~

h2=OA2_(OD-AC)2=cP-(D-d)2, ~ [ - ~ . ~ r V ~ - I J

which gives as a function of or ofThis calculation only applies as long as B C doe,;,no t intersect the lower sphere, th e l imiting case

these were remarkably few, and seemed to be evenlydistributed about the angle of 45, which could thusbe taken as the mean value. In one or two casesthe 45 track was almost a perfect helix for two orthree layers of th e bed.On the basis of these observations, th e writer hasassumed that the track followed by an element offluid corresponds to an average inclination of 45,and is therefore \1'2 times the depth of the bed, that is,L. ./_L =y2 . ApPENDIX II .Effect of Non-uniformity of Pore-Size on Rate of Flow.In th e following, a proof is given for the statementthat flow is greater through parallel channels unequalin size than through channels of even size, with th esame internal volume and internal surface, that is,with th e same average mean hydraulic radius.(a) Consider one large circular pipe, diameter, d,and n smaller pipes, diameter, rd, with r


17/17


LIST OF SYMBOLS .(In consistent c.g.s. units).

~ P p r e s s u r e difference in gm./cm.'~ ? ) = P O - P 1 ) in (gm./cm. ') ', where Po and P 1 are th eabsolute pressures, on t he two s ides of t he b ed .

g=acoeleration due to gravity,= 981 cm./(sec.2).R=frictional force pe r unit area of par ticle surface , indynes/(cm.2).h=height of capillary rise, in cm.L=depth or thi ckness o f bed , in cm.Le=actual length of path taken by fluid in traversingdepth, L, of the bed, in cm.m=mean hydraul ic radius, in em.d=mean spherical diameter of a pRJtide, in em.

de=diameter of equivalent channel, in em. I f channelnon-circular de=4m.dc=diameter of curvature for a curved channel, in cm.D=diameter of conta iner , in cm.S=area of par ti cl e sur face pe r unit volume of packedspace, in (cm.)2/(cm.)3, that i s, ( cm . '1).Sl=total surface area/unit vol ., inclusive of wall surface,

that is, Sl= S + ~ .u=apparent velocity, in cm./sec., that is velocity calcu.lated on empty container.u.=actual velocity in th e pore-channels, in cm./sec.p=density of fluid, in gm./(cm.3).

pm= density at (Po+P 1l, when fluid is a perfect gas, ingm./(cm. ).G=pll, that is appe,rent mass velocity in gm./sec./(cm.')of cross-section of empty container.7J= viscosity of fluid, in poises.p = = kinematic viscosity, in (cm.2)/sec.pf ' volume of pore.space pe r unit volume of the bed, thatis the porosity.

[{ = rate of flow of water at 100 C. through unit c ube o fgranular bed at unit difference of pressure, as givenby D'Arcy 's law.K 1, a, b, n, etc., various constants applicable to special cases.ko=constant in general law of streamline motion throughchann ,18 of uniform, but noncircular, cross.section, ~ P g m kO'7J mthat 13 = or, u. = - . , LeP'''''' pUem ko L.Fo r circular sections, ko=20.k=5'0=constant i n g ener al law of streamline motionthrough granular beds (Kozeny's equation),

e3 _ f3tha.t is, U = k7J/:j2 L - 57JS2 -y;-k 1=0'5=constant in Burke and Plummer's general lawof streamline motion through granular beds,

u= f2(1-.) . gkl7JS2 L

=surface factor, such that, for any shape of par ti cl e,S=6(1-f)fV;d. Fo r spherical particles, < >= 1.0, and,for al l other shapes, < > is less than unity.'f31J;= Lpll Sl

C = ~ a t i o of resistance in curvilinear flow to resistancein rectilinear flow fo r a given Reynold's number.

an View 2dFIG. 7.

Elevation 2FIG. 8 .

The spheres will lie in pairs, the lineeach pair lying perpendicular to the

DAt l=20.of centres for

line of centres for adjacent pairs. Consider twoadjacent pairs. The centres will form the comers ofa regular tetrahedron, with length of each side=d.For such a tetrahedron as in Figs. 7 and 8, th edistance, h, between opposite sides, isgiven by h= ~ 2

Now take the volume enclosed between the planes:. 'TTD2 'TTD2dVol. of cyhnde r= -h= - -_4 4V/2Vol. occupied by spheres=four herni-spheres=' ..d33

'TTD2d TTFree vol. =4 V2 -3 d3= l : ~ ; i d 3 l _ 4\/2 ' ( :.)2-'TTD2d 1 3- D4V2

= (1 - ~ =0'528


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