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    Porosity and surface area of a granular bed from measurements of the flow of air through the

    bed and measurements of the granular shape factors

    This article has been downloaded from IOPscience. Please scroll down to see the full text article.

    1961 Br. J. Appl. Phys. 12 172

    (http://iopscience.iop.org/0508-3443/12/4/310)

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    SIGNIFICANCE O F T H E PROPERTIES O F M A T E R I A L S I N T H E PACKING O F R E A L SPHERICAL PARTICLESMACRAE,. C., FINLAYSON,. C., and GRAY,W. A., 1957,MEWAU,R., and STACH, ., 1934, J. Inst. Fuel, 7, 336.Nuss, W. F., and WHITIPI'G, . L. , 1947, Bull, Amer. Ass.ROSE,H. E., 1945,Proc. Instn Mech. Engrs, 153, 141.SMITH,W. O., FOOTE,. D., and BUSANG,. F., 1929,Phys.

    STEWART, D. A., 1951, High Quality Concrete (London:VERMAN,. C., and BANERJEE,., 1946, Nature (London),WESTMANN,. E. R., and HUGILL,H. R., 1930, J. Amer.WHITE,H. E., and WALTON,. F., 1937, J. Amer. Ceram.

    Nature (London), 179, 1365. Spon).157, 584.Ceram. Soc., 13 , 767.Soc., 20, 155.

    Petrol. Geol., 31? 2044.

    Rev., 34, 1272.

    Porosity and surface area of a granular bed frommeasurements of the flow of air through the bedand measurements of the granular shape factorsby W. M. JONES,D.Phil., F.lnst.P., and D. MILLS, B.Sc., Grad.Inst.P., Department of Physics, UniversityCollege of Wales, AberystwythMS. received 21st December 1959, in revised orm 13th October 1960

    AbstractFlow experimentsgive an equation relating E to S in termsof measurable quantities, where E is the inter-granularporosity of the bed, and S is the envelope surface area ofthe granules in 1 em 3 of bed. Heywo od's shape factor sfor the granules give a second equation relating E to S.Using the tw o equations E and S can be determineduniquely. If the tortuosity factor be ta ken to be 7712 inthe equationsgiven previously (Jones 1956), connecting thefriction factor h and the Reynolds number (Re), thenh = 6/( Re) or granular beds, as for long straight tubes.

    HE flow of a fluid through a granular bed is related toe3 / S and other measurable quantities, where E is theinter-granular porosity of the bed, and S s the envelopesurface area of the granules in 1cm3 of bed. E and S cannotbe determined uniquely from flow alone and when thegranules are themselves porous diffculties exist in determiningeither E or S separately by other means; for example, if E bedetermined by a liquid displacement method then doubtexists as to the extent of the penetration of liquid into thepores of the granules and if attempts are made to seal thepores before immersion in liquid doubts exist as to theefficiency of the sealing. S cannot be determined by anadsorption method because then the internal area of thepores would contribute to the area which is measured.However, Heywood's (1947) shape factors can be used togive a set of equations connecting E with S and these can beused in conjunction with flow experiments, to determine Eand S uniquely.

    TheoreticalThe principle of the method is the application of theresult of the previous paper (Jones 1956), namely, that if theflow resistance of a bed is represented on a graph showinglog h as a function of log (Re),where h is the friction factorand (Re) the Reynolds number, then the graphs will be thesame for beds of irregular granules as they are for beds ofregular granules. To represent the flow resistance on such agraph use is made of the equationsh = 1 / 2 A p ~ ~ g l / p ~ ~ ( S(410))and (Re)=21/2up/~{S+ (4/D)} (1)

    where A p is the flow resistance of the bed (cm w.g.), E is theporosity of the bed, D its diameter, and 1 its length, S is theenvelope surface area of the granules in 1cm3 of bed, U isthe velocity of flow of the fluid of density p and viscosityinto, or out of, the bed. For beds of regular solid particles(e.g. lead shot) all the quantities in Eqn (1) can be determinedand the position of the graph established. The values of Eand S can then be chosen for the beds of irregular particlesso that the flow results lie on the same graph.TO chose a value for E assume a value for the envelopedensity d of the granules (e.g. 0 - 7 g ~ m - ~or charcoalgranules) and hence knowing the weight of the granules perunit volume of bed, E can be determined. Once d is fixedthe value of S s also fixed since d and S are interconnectedthrough the following sequence of equations. (The termsused in the equations are defined after the complete sequencehas been written down.)k (Heywood's volume constant) = l /dNdp3. (2 )

    (3)f (Heywood's surface coefficient) =So(the specific surface determined optically) = N f d p 2 . (4)

    Once So is found S follows from the weight of unit volumeof the granular bed.In Eqns (2) to (4),N s the number of granules per gramme,n is the elongation index being the ratio LIB where L is theaverage length of the granules and B their average breadth(Heywood 1947), C is a constant put equal to 3.1, dp 'themean projected diameter' of a granule is defined bydp = 4Ai.rN, where A is the specific projected area of thegranules settled in their most stable position.If an incorrect value is chosen for d i n the first place thenE and S will be wrong and in the streamline region of flow thegraph representing log h as a function of log (Re) for thegranules will be parallel to the graph established using leadshot. E and S must now be altered to make the graphscoincide in the streamline region of flow. Consider thechoice of d to be wrong by a factor x then k will be wrongby a factor l/x, f will be wrong by a factor y , wherey = 1.57 + Ck413(n+ l ) / ~ ~ / ~ n ' / ~ > / ( 1 . 5 7Ck413(n+ )/n113}and So and S will both be wrong by a factor y. E will bewrong by a factor z, where z = (x - 1 + E ) / X E . Thus,

    1.57 + ~ k 4 / 3 { ( ~ )in1/3}

    VOL. 12, APRIL 1961RITISH JOURNAL OF APPLIED PHYSICS 172

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    P O R O S I T Y A N D S U R F A C E A R E A O F A G R A N U L A R B E Dinstead of log h and log (Re) the wrong values log h j og 2 3- ogy and log (Re)- ogy have been calculated; n practice,z3 > y , hence the displacement of the graph representingresults for the granules is approximately logz3 ; thus z isfound and, subsequently, the correct values of E and S.

    ExperimentalThe quantities in the equations (1) to (4) which are measur-able are AP, , N , dp and n. These were measured withapparatus very similar to that described in the previouspaper (Jones 1956), using the same experimental technique;indeed, the only constructional alteration in the apparatuswas in the flow apparatus in that alcohol manometers afterthe design of KovaEiE (1953) were used for measuring pressuredifferences ess than 2 cm w.g., instead of the aniline/sodium-carbonate manometer used previously. Two alcohol mano-meters were used; one measuring a pressure difference of2 mm w.g. full scale, the other measuring 2 cm w.g. full scale.The other quantities occurring in Eqns (1) to (4) areknown from the known properties of air and the knowndimensions of the charcoal beds, except of course, E and Swhich are being determined. Having determined E and Sthe quantities V (envelope volume per gramme) and Socouldbe calculated. V andSo should be constant for each charcoalindependent of any variation in E and S from bed to bed.The dimensions of the beds used in the experiments weredetermined by the dimensions of the canisters in which thebeds were packed. Twelve canisters in all were used,comprising four canisters of each of the diameters 2, 3 and4cm. The four canisters of each diameter had lengths of1, 2, 3 and 4cm, respectively. More than one bed of anygiven dimension was made up during the course of theexperiments. The beds were not used for the sole purpose ofdetermining E and S, ut were used in uptake experiments;on the other hand, the flow characteristics of each wasmeasured and the results can be used for the purpose offinding E and S. For the largest granules flow experimentswere also made with a bed 120 cm long and 2 cm in diameter.In taking samples of charcoal for measurement of L and B

    care was taken to avoid choosing the larger particles becausethey are more evident in a number of granules spread on awhite background; therefore, in sampling the charcoal, aspoonful was taken from the bottle and all the granules inthe spoonful examined. Another spoonful was then takenand so on until at least 50 granules had been measured. Thesame procedure was followed in counting granules for thedetermination of N except that here the sample counted wasabout 0. 5 gm. (In the case of the -52 $60 charcoal thesample was dispersed in a large known volume of glycerineand the number of granules in a known small part of thiscounted, after filtration.) To avoid dust adding to the valueof A the granules were washed in alcohol before evaluatingA.Five sieve fractions separated from one stock of charcoalwere studied in the experiments. The charcoals are describedin the Table.

    Results obtained with flows of air from 0.03 I/& to501/min through beds of lead shot are shown in Fig. I(u).The straight portion of the curve corresponds to streamlineflow through the beds and the point of departure of the curvefrom the straight line to the onset of turbulence; this occursat a value of Reynolds number of about 3.0.Several errors of observation can occur. At the smallestflow by far the largest are the errors in determining thepressure difference across the bed and in determining theflow itself; the other errors are negligible. In plotting thepoints on Fig, l ( a ) at low flow (small values of Reynolds

    D m f leadshotI \ T mm.

    0 0 4 2 5 2 .8 00 0 . 4 2 2 2.80A 0408 2 8 0X 0408 2 4 10 0 - 4 0 3 2 .4 1

    Size 01bed (DxC)Cm

    2U 42 x23x 32 x 43 x 3

    Fig. 1. Curves showing the relationship between the frictionfactor and Reynolds number for beds of (U ) lead shot, (b)char-coal granules.number), the most unfavourable case is represented, that is,where the two errors work in the same direction to producea larger error. At high flows a number of observationalerrors of about the same magnitude occur; it is unreasonableto assume they can all act in the same direction so that theroot mean square error is given for these.The curve established in the streamline region of flowusing lead shot beds is shown again in Fig. I(b)by the brokenline. The experimental points obtained using three differentbeds of charcoal from the BAS. range -8 -F 10 are shownon the figure. To calculate E and S the superficial densitywas taken to be 0.715 g cmP3; the best straight line throughpoints a t values of Reynolds number less than 3 .O is shownby the full line. This line represents streamline flow throughthe charcoal beds. The full line can be made to coincidewith the broken line by taking the density of the charcoal tobe 0 . 695g~m-~ .Results for other grades of charcoals,together with shape factors relating to the charcoals aregiven in the Table. The symbols used in the Table are defmedby equations (2) to (4).The experimental errors in Fig. l(b) are determined asfor those in Fig. l(u). It is important to note that care mustbe taken in measuring the diameter of the beds, since itultimately occurs as the fourth power in and squared in (Re) .Discussion

    It is known from other experiments that values determinedfor the surface area of granular particles depend on theexperimental method used and, consequently, the surfacearea should be defined in relation to the use to which thevalues of the surface area are put. The work which has beendescribed was carried out as part of a series of experimentsto measure the uptake of benzene by charcoal beds as afunction of the physical variables of the beds, the benzeneVOL. 2, Apm 1961 173 BRITISH JOURNAL OF APPLIED PHYSICS

    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  • 8/2/2019 Diamtre de Heywood

    4/4

    P O R O S I T Y A N D S U R F A C E A R E A O F A G R A N U L A R B E DTable. Description o f the charcoals

    Directly measurableD , N A dp fcm)B A S . range ( cm ) g -1 (cmzg-1) i to 1%

    - 8 +1 0 0.187 318 5 1 6.4 i 0 . 4 0.256-10$12 0.154 560 6 19.7 i .3 0.212-144-16 0.110 1578 9 27.3 ? 0.4 0 .148--18+22 0.078 4740 & 25 41.8 i . 2 0.108-52+60 0,027 1 .2 2 0.1 X lo 5 131 2 2 0.0375being carried in an air stream passing through the beds(Mills 1959), and it was thought that flow porosity andenvelope surface area (as defined by flow) would be relevanrvariables to consider and therefore measure. But values of Ean d S as defined by flow would be of value in many otherproblems where the flow of fluids around granular particlesoccm, for example, not only in consolidated beds, but alsoin fluidized beds and under conditions where particles faUthrough fluids (Institute of Physics Conference 1951). Th eusefulness of determining E an d S (and hence V an d So)by aflow method is therefore clear. The question arising is howfar th e method described here does measure the flow porosityand surface area of the granular beds. This question was ineffect answered by the work of the previous paper (Jones1956) when it was shown that Eqn (1) did apply to regularand irregular particles, the value of S fo r the irregular particleshaving been determined from the Eqns (2) to (4) and theenvelope volume (and hence E) from liquid displacementan d a relationship between E and DJD. In this work theenvelope volume was not found from liquid displacement,bu t V an d Sowere found by trial and error by using thesequence of Eqns (1) t o (4). This paper is, in fact, anaccount of the method of trial and error. As a check onthis present work the volume of fluid displaced statically bythe charcoal was found at the Chemical Defence Experi-mental Establishment, Porton, by immersing the unsievedcharcoal in mercury (under a pressure of 1 atm) and alsoby immersing the charcoal in water, after treating the char-coal with silicone fluid. Th e purpose of the silicone reatmentwas to make the charcoal water-repellent so that water didnot penetrate the interior of the granules. The spe cs cvolume V was found to b e 1 e38 cm3 g-' from immersion inmercury and also from immersion in water. On the otherhand, the average value of V taken from the Table is1-4 3cm3 g-' which is 39 % greater than the value found byliquid displacement. Since the value of V is the same byboth mercury displacement and water displacement, thevalue 1.38 must be right since if there were much penetrationinto the interior it would be unlikely that it would be thesame for the different liquids. The slightly greater volumefound by air flow could be accounted for by supposing astagnant layer of air of 10-4cm thickness to cling to theparticles during the flow experiments; considering the rough-ness of the particles such a stagnant layer would not besurprising. In any event it is felt the agreement between theresults of the various methods of finding V,which have beendescribed is sufficiently close to justify the use of air flowtogether with Heywood's shape factors to iind porositiesand surface areas for use in problem s where fluid flow occurs.Con sider now the abs olute position of the cu rve of Fig. l(a)with respect to the coordinate axes. In the previous papercomparison was made between the results given there andthose summarized by Rose and Mott (Institute of PhysicsConference 1951), for regular particles, allowance beingmade for different definitions of h and ( Re ) , and it was seenthat the results were in mutual agreement over those parts

    Deduced f ro m best f i r inpowexoerimenrs

    0.250 0.189 1.32 1.37 1.45 0.229 68-10.179 0.139 1.29 1.35 1.41 0.276 94.80.144 0.100 1.44 1.38 1.43 0.253 140.00.046 0.033 1.41 1.37 1.41 0.221 416of the curves whkh were common to all investigations(values of log ( Re )> 0.6). The results given in Fig. l(a)are similarly in agreement with previous results. Apartfrom the results of the previous paper and those given hereno results are known for values of log ( Re )< 0 .6 ; closeexamination of the results given here and those given in theprevious paper will show they deviate from one anotherwhen log (Re)< 0.0. Several precautions were taken incalibrating instruments for the determination of the absoluteposition of the line on Fig. l(a)and its position is consideredto be reliable; the linear part of the curve satisfies he equationh = 19*S/(Re). In streamline flow through straight tubesh = 16/(Re). The numerical constant 19.5 depends ondefinitions used in deriving expressions for and (Re) (cf.Rose (Institute of Physics Conference 1951),whose num ericalconstants are different from ours by a factor of loo), andalso on the numbers substituted for quantities such as thetortuosity factor l e / l ; if in deriving Eqn (1) lc /I were putequal to 7~12nstead of 2/2 then h = 16/(Re) for beds as fortubes. The tortuosity factor was equated to 4 2 n Eqn (1)after Carman (1956, p. 45), but there is considerable justi-fication for taking 712 as the tortuosity factor in that it isthe ratio between the semi-circumferenceand the diam eter ofa circle and thus represents the longer path taken round asphere rather than through it (Carman 1956). To be able torepresent flow through granular beds on the same curve asflow through tubes is a satisfying result and a pleasing appli-cation of the principles of similarity. To be able to do thisdirectly Eqn (1) should be modiiied by putting le/l = 4 2instead of equal to 4 2 ; Eqn (1) then becomes

    In conc lusion, the absolu te position of the curve of Fig. l(u)does no t affect the values found fo r E an d S for the charcoalbeds. These values are chosen so that the curve representingthe flow through th e ch arcoal beds coincides with th at repre-senting the flow through the lead shot beds of known E and S.Hence, constants in Eqn (1) and constants of calibration offlow apparatus do not alter the values found for E an d S ofthe charcoal beds.The work was done in consultation with the ChemicalDefence Experimental Establishment, Porton. We areparticularly grateful to Dr. F. A. P. Maggs for his interest

    in the work. ReferencesCARMAN,. C., 1956, Flow of Gases through Porous MediaHEYWOOD, H., 1947, Symposium on Particle Size Analysis,INSTITUTEF Pmsics, 1951, Some Aspects of Fluid FlowJONES,W. M., 1956, Brit . J. Appl. Phys., 7 , 370.K O V A ~ I E ,., 1953, J. Sci. Instrum., 30, 304.MILLS,D. , 1959, Ph.D. Dissertation, University of Wales.

    (London: Butterworths Scientific Publications).Supplement to Trans Instn Chem. Eng., 5,18.(London: Edw ard Arnold).

    BRITISH JOURNAL OF APPLIED PHYSICS 174 VOL. 12, APRIL1961

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