Transcript
FLUID FLOW THROUGH PACKED COLUMN S SABRI ERGU N Carnegie Institute of Technologyt, Pittsburgh, Pennsylvani a The equation has been examined from the point of view of its depend- ence upon flow rate, properties of the fluids, and fractional void volume, orientation, size, shape, and surface of the granular solids . Whenever possible, conditions were chosen so that the effect of one variable at atime could be considered . A transformation of the general equation indicates that the Blake-type friction factor has the following form : 1 -6 fa a 1 .75+ 150 ar A new concept of friction factor,/. representing the ratio of pressure drop to the viscous energy term is discussed . Experimental results ob- tained for the purpose of testing the validity of the equation are reported . Numerous other data taken from the literature have been included in the discussions The existing information an the flow of fluids through beds of granular Osborne flryadds (23) uas hest to formu- solids has been critically reviewed . It has been found that pressure late the resi ante offered by Iriceiat to the losses are caused by simultanstous kinetic and viscous energy losses, n'otiun of the flu,I as the sum of tw o ternts, of itioal respectively to der first and that the following comprehensive equation is applicable to all types power of t sr fluid velocity and to the o flow poduc o thedenstyo thefludwth APC - e) t PU 1- s GU„ secondpower of its velocity Lp =150a +1 75 .aDAPL=all +beV(1 h T HE pressure loss accompanying the flowof fluids through column s packed with granular material has beenthe subject of theoretical analysis andexperimental investigation . The pur- pose of the present paper is to smnmar . ice the existing information, to verifyfurther experimentally a theoretical de- retopment presented earlier, and to discuss practical applications of this newapproach . The experimental studies have been confined to gas flow throughcrushed porous solids . This case is the one usually encountered in practice, but is not identical with the case most thor-oughly studied by previous investiga- tors, viz., the flow of fluid through bells of nonporo(s solids, and more particu- larly . througi solids having uniform geometric shapes . Factors determining the energy loss(pressure drop) in the packed beds are numerous and some of than are not susceptible to complete and exact mathe- matical analysis . Various workers in the field have made simplifying assump- tions or analogies so that they could C o a l Resecrch Laboratory . Vol . 48, No . 2 NY 62592 $11 i f RITISH LIBRARY, BOSTON SPA LOAN/PHOTOCOPY REQUEST FOR M '_°p` coo No . 1979 Ossc,i ptio n . ã, .h„b aa a 41pii hinar, rn M,ry CHEM ENG . PROG . N .S . j `low through packe d aace of pybpUtion : ''rV 7 r a n o p p,s utilize some of the general equations ten These : factors ar e re important and will representing the forces exerted by the be discussed later, but they are irreic :ant fluids in motion (molecular, viscous, for the purpose of testing the linearit .ã ofkinetic, static, etc .) to arrive at a useful Equation (2) . As a typical plot . der . ob- expression correlating these factors . A , .rained for gas Sowthrough a be o f ff crushedporous solids are showninN ;tar e survey of the literature reveals various expressions derived from - different assumptions, correlating the particular experimental data obtained with or with- out sonic of the data published earlier . These correlations differ in many re- spects ; some are to be used only at low fluid flow rates . while others are ap- plicable only at higher rates . A separatesurvey of all these various correlations is not included here . As most authorities agree, the factorsto be considered are : (1) rate of fluid flow (2) viscosity and density of the fluid. (3) closeness and orientation of packing, and (4) site . shape, andsurface ai the particles . The first two variables concern the fluid, while the last two the solids ,1 . Rate of Fluid Flow, It is known that pressure drop through a granular bed is propor)ional to the fluid velocity atlow flow rates, and approximately to the square of the velocity at high tests Chemical Engineering Progres s mar k columns , Publ,snar : i Pam --- IS NlSNS N ar' '~Ild Yngwnl mere out a rot e here AP is the pressure on alont length L, a the density of the fluid, 11 its linear velocity, and a and b are factor s which are functions of the , system . A transformation of Equation (1) which yields a linear expression is : AP/LU= a+ bG(2 ) where 0,U has been replaced by G, the mass Sow rate . The above two-term preswre-drop equation has been found to be astir factory over the range of flow rates en- countered in pecked columns . Lindquist (19), Morcom (20) . and Ergun and Orning (7) have platted AP/LVag inst G and obtained straight lines as expectedfrom Equation (2) . The former two au- thors have included in their plots factor s ties of thã s a- the i kr t I The experimental results of the presen t investigation and those mentioned ax,ve ( :, 19, 20), as well as the data ott :inm d fromthe literature (3, 22). mlitate that . the two-term equation accurately cite tiar a the relation between flow rate and ptasure drop . 2 . Viscosity and Density ã of Fluid FromEquation (2) it is seen that as thevelocity a'sproaches zero as a lien t . the ratio of pressure drop to velocity ad, be- come constant : iAPC- e ms (3 U+ hi h i i fl s a coalit on for v scous nv, . Actcording to the Poiseuille equation andDar 'a law the factor a is propcr :ional to the viscosity of the fluid . The xher limiting condition is reached at hign flow rates when the constant a is negtigil It in comparison to bG . This is a condition forcompletely turbulent flowwhere k-aetic energy losses constitute the whole vain- tance . The effect of density is already contained in G . Equation (2) as be rewritten : API' -a a'pU+b9(P (4 ) r a S_--IT-q-11-13 a :nc,.a . . , t ,, .a I Source: http://industrydocuments.library.ucsf.edu/tobacco/docs/nnhp0209 . . i ., rr wi- Ihr t4 . o .it, of On 11 il Inl a i agor in'rlamfpMto th e sat :,bh, of I - -ã,Ikls ody, t y lir-t brut of 1?ryru .n,o t41 reprcv,ny riwnn vmãrgy I ... .. . aoal Ihr 'ao,MI Icon it . kiorlfc en . trill hã,ã .. . Thy Knaruy ,ppntiun (if )ã .( Ill, viwã.ON vu,rgy Term torpm-dot 11Mã prey-ore droop, while the ill and1lunuucr (3), sod I'hdoo and 6 .11,1u10 0) approarln nntdoysfor kneel i . nMray Iran a .nl n,ninoyatcsthe rd, .I „i .n ., n . . ids-rpc I,--,ã, wll, a r . .l , .,W, Uati .,,I i .u'U ã, 3 . Cleavers (Fractimul Void Vol- ume) and Orientation of Packing. Free . liã,n„I y . Md v .duunv has bete one of the na,-I :,oIr,nãrr .cd fads, . h . i 4'1 .'d 073- , .n,s . ulc lu,ã,rvlical Ircauucnls wiren,n a„r . ,,rod m-'al li,hinR the ticpend- air ãd )fir pr . .surc , .rq . niece iradmual vied v .dnnn It was sheet who first our *hllly ,rrtu,l Ile reecho by an ap- pr„orh ;mnh . ;wu lu I lull ' if Stanton and I'aun,ã11 I .5) In gwr„urt drop in circular pile,- . lliake .dnaiwsl Olt- foll .,wlpg dimes . ,and .ãs- grw1P.t _V' .I, It, .' Mt nhrrr t is the fractkmal void volume . p . ll ' ar,, vitatioINl ca .NanAand D . the dia .ndarr of the solid particles . The first of hcxw aruaps is recognised as the md(i- liy .l rfcl .an (actor aid the second as the n .,slifuvl Reynolds number . Blake sug- p,-a,l Il .at the inner of these groups be I11ã .it .s1 against the latter . Since both d - group rotafnthe fraciona , .id e,Aume . it can be deduced that pres- .ure drop is not a function of a single Group aluMs cThe failure of lux earlier attempts to arrive at a useful expression can be attn . hard to the want of recognition of the fact that pressure drop is caused by simultan- ew kinetic and viscous energy losses , C I 11nee r .ai I .t « shred„ o« One law .,4 Fig . 1. Typfiwl phsrs of the On . hewof pe tar .wrop .qull . . ro. cyncpochsda diltonm hoetlit of veld salver . ., 0g .'1 (2) aurae . . cowihrwak 1670 ere hgh1 .w pin .NNpencake . P nki domtpra1 046 g ./mCrowsnuik . .w .l oronof rob . 7 74 wtse . fall .1 724 ms Ms. and 21' C . Theoretical corsitkratwns of later workers (3, 7) indicate that dependency of each energy loss upon fractional void volume is different . Burke and Plummer proposedthe theory that the lowresistance of the packed bed can be treated as the stun of the separate resistances of the individual particles in it . Accordingly, viacoos energy loss was found to be wopor- uonal to (1-r)/, and kinetic loss to (1 -The authors, however, failed to recognise the additive nature of these losses and correlated the pressure drop bythe use of dimensionless groups similar to those of Blake . For viscous flow . Koarny- (14) arrived at an equation wdely usedlater (4 . 10, 11 . 13 . 1 .5, 261 by ssvunnng that the granular bed is equivalent to a group of simlar parallel darnels . The derival dependency up'Mn fractional void volume was (I-s=/e'. This factor is different by a (raaio . 0 - r)/q iron the factor derived by Burke for viscous flow Fair and Hatdt 410). Carman (4) . Inn and Surse (13), Fower and Hertel (11) . and others (6 . 13, 1 : 36) verified the Koteny factor experimentally, For a gen- eral correlation valid at all flow rotes, how- ever, Carman recommended the plot of thedimensionless groups of Blake . Recently . Leva (24) anal horse (22) also adopted Blake's procedure in presenting the pres- sure drop data in filed beds. Lena, et al . (18) stated that the pressure drop was pro- portiorw to (1 - .) / .' at lower dove rates and to (1 - s)/ .' at higher flowrates . Carman noted that at low fluid-flow rates the method of Blake leads to the Koreny eguatiou . hesxe to tux acto r APR . (I- )' p(' (5) On the other hand, at high flowrates Bake t tttethod iv ise t th e uati o o e q n sr fie . 2, la .ps.d .- ' is- we hkw * ansinl I .- - f oakr .ol said -I .-, aq . .ã oofBurke set Plummer for turbulenttint al .rod .(Q lasnapis .wet dopes .ro_ .bã . MN *d1 at fig . . I by--*Wo asks7 (6 Chemical Engineering Progress the garter needling tilt fractional vow ndunIe Icing (I - r)/e' . This range of tl .e plot at Blake tae generally 4mover . la .ke, L Basel no the theory of Reynolds for resfstanet to doid flowand the method of K .rsoy, a gateral olã .atirnl was developed by Ergun and Orniol; fur pressure drop thr .wgb fixed beds, In summary the fol- luwiou raxlusknra can he drawn from U .eir w .xk : 1 . Total cmrgy Irwses in hoot IMI> ran I .e erraloi us Its,, solo of viarr .ra AMkinetic energy tosses . 2 . Viacomclergy kenos art pr .pxglknual to') t -c)'/ .' ae .l tImkiotir energy I .- to (I - .1/0 . Since u a .Ml h of F .quatiat (4) represent the e .MTxkmls of viscous a,Ml kinetic energy losses . rcnprrdvtly . it is ,spoledtha ahepnpnebnal to(I - s)/ ' and h to O- .)/ ' in order for the theory to be valid . .\ItMmch the aboveauthor . have curr,lat .,l tirade data suctt s- fully single systems have nip been thor- oughly examined at various frarti .n .al v, .kl volumes . One of tow aims of IIMã present work bas, been to inveslieatc Owsinalr systems at various packing densities . A known amomn of solids was packed 6 t„ 20 different bulk densities each resulting ina different fractional void volume . For each packing the coefficients it and b of Equation (2) were determned frompres- sure drop and flowrate measurements (Fit. 1) . Firures 2 and 3 showtypical plowof a against(, andb .globe(I-t)/e' obtained fromFigure 1 . Saab plot, yield straight lines ach passing through the origin . The graphical repre- sentation is simple, yet most ective in tieinvestigation of the function of fractional void volume . A similar procedure has been adopted recently by Arthur, et at (1) it, testing the validity of the Soaeny vuatias and by ErFun (0) in camrctiun wth par-ticle density determnations for porous solids, It is of in :crust also to note that the two extreme ranges of the Blake plot lead to the tern of the general equation proposed by Ergun and Oreins- The pro . parliorralities an he expressed in the for- mulae : aneo (=-~r) (7) :I, = b 1 .~ (g ) wherea andb arefacors of proporti- O iE . s1 p-t)' i r rig . 7 . O .p . d . .r. at vista .,. .orgy f .w .. ã, . fc . .w .eel md wotw . . tq, . .riw . (7) . 0 . . .brok .od hr akregw4- through 7040 .lath, fags soh ., liamak deWry as 1 .27 0 ./oe . Croe'u . .ri o1 0- -0 It. %4 . w7 .24 pose . Ink Dos in 740 . . .u M* laid 23' C Februory, 1952 I I Source: http://industrydocuments.library.ucsf.edu/tobacco/docs/nnhp0209 I t alley . Their substitution into Equation (2) yields : Li e (9) A rearrangement of Equation (9) leads to : (10) Equation (10) makes it possible to group at data of Figure I on a single line by plotting APs LU(1 - .) against G/(I- .) This is demonstrated ht Figure 4 . Up to this point the aimhas been to formulate the effect of fractional void vol- ume in fixed beds, and the effect of orienta- tion was not included . The orientation of the randomly Packed beds is not susceptibleto exact mathematical formulation . This isespecially true it the particles have odd shapes and are not negligible in size con-pared wth the diameter of the container . Furnas (12) has treated the subject at length and introduced the concept of sar- nwl packing which was obtained by a slaunlard procedure . In tine present investi-gation, however, such a concept had to be abandoned, The problemwas to pack aknown amount of solids to various bulk densities . yet each packing had to be ant- forin and reproducible . This was accomplished by admitting gas belowthe supporting grid after the solidswere pound in . The gas rate was sufficient to keep the bed in an expanded state andthe use of a vibrator attached to she tube assured the uniformity of the packing . By varying the rate of upward gas flow the bulk density could he varied fromthe tightest possible to tie loosest stable pack . ing, For crushed material the most tightly packed bed having a height of 30 cm . couldeasily be expanded by 6 to 7 con . When thedesired pa,kiug density was at taincnl, the vibrator was ltxnlnulvcw1 anal the gas now rut off- The bed that was ready for pres . sure drop and flowrate measurements Highly reproducible packings can be ob- tained by this method, and more important . the particles are believed to be oriented bythe gas doming upward. This is evidenced by the existence of a theoretical relation- ship (7), verified experimentally, between the bed expansion and the flowrate . A further evidence for particle orientatio n was found in the fan that the most tightly ã packed beds have been obtained by slowlyreducing the rate of upward gas flow to an initially expanded bed while subjecting it to vibration . It wll be evident on inspection of theformof Equation (9) that the estimation rat fractional void volume is important, par- ticularly since it enters to second . and tlntrd-power terms aid is in many aces difficult sea measure directly . Whenever the particle density and the total weight of the granular material filling a given volume areknown . a may be readily alculated . But the particle density of crushed porous ma- terials is not readily known and its deter-mnation has presented a problemwhichwas much discussed Fractional void vol- umes were usually calculated by the use ofapparent specifu gravities which were de- termnes by variant procedures . Use of such values for a in the pressurcdrop equations masticated the introduction of correction factors . This often caused the workers to doubt the validity of the factors describing the dependence of pressure dro p ;rr ; t 7 ,,' ;?Vo 48, No . 2 upon. and to seek little correlations . How- ever, this was believed to be unwarranted (g) sitter the determnation of pressure drop through beds of porous panicles hinges upon the evaluation of the particle density . Therefore . a gas flow method wasdeveloped (8) for the determination of the particle density of porous granules . The method was ducked by the densities ob- tained for nonporous solids and the agree- wn was good . Use of the particle densi- ties of coke obtained by the method de- scribed, in the determination of fractional void volume sad hence in the promote drop equation, resulted in excellent agrexnwuu . 4 . Sits, Shape and Smrface of the ticles T h ffect ticl si t h r e e te e par e . - surface area, surface area, Ce and shape is best analysed in the light -of P e f di ti r theoretical implications of the Blake plot . The identity between the two extreme ranges of the Blake plot and the theoreticalequations developed respectively by Kozeny and Burke for viscous. and turbulent-flowranges has already been shown . Aso, is has been pointed out that these two expres- sions cotnsinnad the following general equation developed by Ergun and Qning (7) : APy ./L = 2 µS : U .(I --s)s/a ' +(p/8)GU .S (1 - .)/s (11) where a and p are statistical constants, g, is the gravitational constant, and S . is the specific surface of solids . i .e . . surfs of the solids per out volume, of the solids . Instead of specific surface . S., surface per unit packed volume . S . has been employedby some workers . Since the latter quantity' involves the fractional void volume, use of specific surface has been preferred in thepresent work . The relation between the two quantities is expressed b y Sea (1- .)S « Equation Ill) involves the concept of mean hydraulic radius in its theoreticaldevelopment (7) . Its validity has been tested wth spheres, cylinders, tablets, sotdoles, round sand and crushed materials (glass . coke, coal, etc .) and found to be sotisneunry . The experiments have not been extended to inctale solids having holes and other special shapes . Fewthos e B mtg . 4 . Agsaersl plat for ã single grins . petition to dgdarem #,*a* l odd 4eã . pats ol .ã~ ap . .» ~ arknj tqst wi p) ã 'a straigh t ln g Chemical Engineering Progress cases the concept of specific surface was believed to be not applicable by Burk: who suggested compensation by cmpirica fac- tors in connection with the use e f the Blake plot . Determination of specific surface in~clves the mcasurerne of the solid surface area as well as that of solid volume stint pre-sents no problemfor uniformgeo :nctricshapes . For irregular solids, especially fur porous materials, however, surfs area determination becomes involved . The sur- face of porous materials is necessari.y fullof holes and projections . Different surface arms are usually defined in connection with porous materials, viz ., total surface are a (including that of pores), external visibl e geometr ace, as s nct rom ex . s u ternal visible surface, may be visual.ted asthe surface of an impervious envelote sur- rounding the body in an aerodynamic sense . Irregularities and striae on the surface would not be taken into full accoui .t in a geometric surface area in contrast to ex- ternal surface area . Whether the value of the total, external or geometric surface areais .lesircd wll depend on the purpose for which it is to be used . Geometric surface armis believed (9) to be the relev .,nt one in connection with the pressure crop in parked columns . This is made evident by the close agreement between the nurface areas determined by gas-flaw methods and those by mcroscopic and light extinction meshetls . inasmuch as the surface rough . ness affects both the geometric surface area and the particle density, the deterntisation of its influence upon pressure drop ties in the evaluation of the effective values of then quantities . It has been customary to use a ch uracter-istic dimension to represent the part cle site in pressure-drop atculations . The charoc-terutie dimension generally used is thediameter of a sphere having the specific surface . S . . which is expressed by Snbatitutiat of Dr into Equation (11) yields : Ails . (I - )' AU . +k 1 - . GU . ~sok W 12) where k. en 72 a and k . =3/4 o, , Pinar torn of Equation (12) is : (13 ) N ., = D p The left-hand side of Equation (13) is the ratio of pressure drop to the viscau en- ergy termand will be designated by f .- APDã (1^ .) U (13a ) / = k.+k . -I V=~(136) According to Estwiot (13) a linear rela- tio udnip exists between I . and As./ 1- e . Data of the present investigation m those presented earlier have been treated accord- sngly, std the coefficients Jr . and Its havebeen determned by the method of least squares . The values obtained are lit o ISO and ter at 1.75 representing 64( experi- nicala . Data involved various- l spheres . sand, pulverized coke, and the ollowing gases : CO . N CH . and Hs . Otwe the constants Jr . and A . were obtained it wa s Page 91 I . C-D Ln 00 CD . CD CD 01 Source: http://industrydocuments.library.ucsf.edu/tobacco/docs/nnhp0209 N N . S . Asonars] yrapbKOl nrprewewias el pswwr drop opplieabb w bath risooar and w,b .&W Mw f . .ynu,o sonsidend . Solid nee, in oil 0, . . a . . , wo draws asssrdNe w FA- 14 . (13o) end sr . t . a an .rolwatk pasts . Th wdlnoh is sproo .nwd by 1fgoarlgn (13 . ) .IC ' I N I- a a h .nthso (liopldm npr0wModen of pwrwrg dreg I. poe#ed mkrwns, Salta 3bo r pnwals Qhossot Msvotipaflon Atosssa X hark . grad rl .w Cl .- sad Worwe II P, coos s,r igorr . 0 itf li r e ♦ r 00 r- er No s 0 N wwtnno dl plnwmo r f II 7 Iã, . I j I : 11111 s o a s a esin i ♦ eN0ae r w . o re r , i 1 ` I yaom~l .t ~ n ã arNO ã . Ne -E Ks . 7 . Gr pbksl mpeesewwdea at prwvra dreg I . toed hod . Cowof rigor 3anr ptatad . In all rhroo seas wild tees or . lde tbol sod ore dresr . seeardisg to rgnat]a . (]db) . The wdiass . Is rap's-tool by is, tgnatua, ll4 , l~ ii l0 I pe ar l II I H 4 i r C I i ~ ii .u r- a raes possible to construct the genera] equation . The results are shown on the top of Figure 5 . To be able to include a wder range ofdata, a - logarithmc scale has been usedwhich results in a curve for the straight line of Equation (13) . Data of Burke andPlummer and those of Morcomare also shown in Figure S . In all three cases the solid lines are identical and are drawn or . cording to the following equation : 10 . I . .—ISO+ 115 1 Data shown in Figure 5 and some addi- tional data obtained fromthe literature covering wder ranges of flowrate are in- cluded in Figure 6, together wilh the asymptotes of the resulting Curve on thelogarithmc scale . Again the solid line represents Equation (13x) . Adifferent formof Equation (121 is represented by : APg . D os :z k . 1a + k LGG . - rs N (14) The left-hand side of Equation (14) is the ratio of total energy losses to the terns repeetertling kinetic energy losses and willbe designated by / . ,_PEDso J . (14 .) E11- 150 1Ns . +1 .75 (1db IF February, 1952 hemical Engineering Progres s ogo 92 Source: http://industrydocuments.library.ucsf.edu/tobacco/docs/nnhp0209
