Fluid Flow Through Packed Columns, Ergun

(1)

1 FLUID FLOW THROUGH PACKED

COLUMN S

SABRI ERGU N

Carnegie Institute of Technologyt, Pittsburgh, Pennsylvania

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 a

time 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

art

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

ternts, of itioal respectively to der first

o

and that the following comprehensive equation is applicable to all types power of t!sr fluid velocity and to the of flow, product of the density of the fluid with

AP Cl - e) t PU . 1 - s GU„ second power of its velocity :

L p, = 150 a + 1 .75 .a Da AP/L =all + beV (1)

h

T HE pressure loss accompanying the

flow of fluids through column s

packed with granular material has been

the subject of theoretical analysis and

experimental investigation . The

pur-pose of the present paper is to smnmar .

ice the existing information, to verify

further experimentally a theoretical

de-retopment presented earlier, and to

discuss practical applications of this new

approach . The experimental studies

have been confined to gas flow through

crushed 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.

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'_°p`

coo No .

₁₉₇₉

r

c

a n

o

w p

p,s

utilize some of the general equations ten These: factors are 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 .• of

kinetic, static, etc.) to arrive at a useful Equation (2) . As a typical plot. der.

ob-expression correlating these factors. A ,

_{ff crushed porous solids are shown in N. ;tar e}

.rained for gas Sow through a be of

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 separate

survey of all these various correlations

is not included here.

As most authorities agree, the factors

to 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, and

surface 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 at

low flow rates, and approximately to the

square of the velocity at high tests .

Chemical Engineering Progres s

mark

mere out

a rot

e

where AP is the pressure on alon

t

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/LV ag inst

G and obtained straight lines as expected

from Equation (2) . The former two

au-thors have included in their plots factor s

ties of th• s

a-i

the

hi k r

t t

I

. The experimental results of the present

investigation and those mentioned ax,ve ( :, 19, 20), as well as the data ott:inmd from the literature (3, 22) . militate that . the two-term equation accurately cite tiara the relation between flow rate and ptasure drop.

2. Viscosity and Density • of Fluid. From Equation (2) it is seen that as the velocity a'sproaches zero as a lien t. the ratio of pressure drop to velocity ad, be-come constant :

iAPC/-

₀

L m s (3

_{) U+}

e

hi h i i i fl w c s a coalit on for v

scous

nv, . Act cording to the Poiseuille equation and Dar '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 for completely turbulent flow where 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

a

I

(2)

. . i ., rr w i- Ihr t4 .. o.it, of On, 11"i'l Intl a i.agor in'rlamfpM to the

sat :,bh, of I - -•,Ikls ody, t y lir-t brut of 1?ryru .n,o t41 reprcv,ny riw,nn vm•rgy I . .. ... aoal Ihr 'ao,MI Icon it .. kiorlfc en. trill h•,•. .. Thy Knaruy ,ppntiun (if ) •.( Ill, viw•.ON vu,rgy Term to rpm-dot 11M• prey-ore droop, while the ill'

." and 1'lunuucr (3), sod I'hdoo and 6 .11,1u10 0) approarln nntdoys for kneel i. nMray Iran a .nl n,ninoyatcs the rd, .I „i .n ., n .. ids-rpc I,--,•, will, a r ..l , .,W, Uati.,,I i.u'U • ,

3. Cleavers (Fractimul Void Vol-ume) and Orientation of Packing. Free . li•,n„I y. Md v.duunv has bete one of the na,-I :,oIr,n•rr.cd fads,. h . i 4'1 .'d 073-, .n073-,s. ulc lu073-,•073-,rvlical Ircauucnls wire n,n a„r. ,,rod m -'al li,hinR the ticpend-air •d )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,•11 I .5) In gwr„urt drop in circular pile,- . lliake .dnaiwsl Olt- foll.,wlpg dimes . ,and.•s- grw1P.ti

_V'.I, It,

.' Mot ,

nhrrr t is the fractkmal void volume . p . ll.' ar,,.vitatioINl ca.NtanA and 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 I11• .it.s1 against the latter. Since both di - group, ro,tafn the fractiona l , .id e,Aume. it can be deduced that pres-.ure drop is not a function of a single Group aluMsc

The 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 1 1 need r .ai I .t«

shred„ o«, One law ., 4 Fig . 1 . Typfiwl phsrs of the On... hew of pe. .tar.wlrop .quill. . rot . cynic pochsd 'a diltonmt hoetlit of veld salver.., 0g.. .'1.. (2). aurae.. cow ihrwak 1670 ere, high 1.w pin .NN pen cake. P.nki domitp ra 1 .046 g./m Crowsnuik..w.l oron of 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 proposed the theory that the low resistance of the packed bed can be treated as the stun of the separate resistances of the individual particles in it. Accordingly, via coos 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 by the use of dimensionless groups similar to those of Blake . For viscous flow . Koarny-(14) arrived at an equation widely used later (4 . 10, 11 . 13. 1 .5, 261 by ssvunnng that the granular bed is equivalent to a group of similar 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), Fowler 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 the dimensionless 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-portiorwl to (1 - .) /.' at lower dove rates and to (1 - s)/.' at higher flow rates.

Carman noted that at low fluid-flow rates

the method of Blake leads to the Koreny

eguatiou. hesxe to tux actor

APR. (I- .)' p('

(5) On the other hand, at high flow rates Bake t tttethodgives rise to

th

e

e uatioq n fie. 2, la.ps.d.- .' is- wet hkw*

ansinl I.- - f .oakr .ol said -I.-, aq ...• oofBurke set Plummer for turbulent tint al .rod .(Q, lasnapis .wet dopes .ro_.b• .

M Nf *d 1. at fig .. I by --*,W of asks 7- (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 4m over . la .ke,L

Basel no the theory of Reynolds for resfstanet to doid flow and the method of K.rsoy, a gateral ol• .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 IM,I> ran I .e erraloi us Its,, solo of viarr.ra AM kinetic energy tosses.

2. Viacom clergy kenos art pr .pxglknual to') t -c)'/ .' ae.l tIm kiotir 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 ,spoiled that a he pnpn,etbnal to (I - s)'/.' and h to O- .)/.' in order for the theory to be valid. .\ItMmch the above author. 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 IIM• present work bas, been to inveslieatc Ow sinalr systems at various packing densities . A known amomn of solids was packed 6 t„ 20 different bulk densities each resulting in a different fractional void volume . For each packing the coefficients it and b of Equation (2) were determined from pres-sure drop and flow rate meapres-surements (Fit. 1) . Firures 2 and 3 show typical plow of a against(, and b .globe

(I-t)/e' obtained from Figure 1 . Saab plot, yield straight lines ach passing through the origin . The graphical repre-sentation is simple, yet most ective in tie investigation 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 S.oaeny vuatias and by ErFun (0) in camrctiun with par-ticle density determinations 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 )}

where a" and b' are factors of

proporti-O

iE .

s1

p-t)' ir

rig. 7. O.p..d. .r. at vista., ..orgy f.w .. •,. fc..w.eel mid wotw.. tq,..riw. (7). 0 ... .brok.od hr akregw 4- through 7040 .lath, fags soh., liamak deWry as 1 .27 0./oe. Croe'u ..ri.o1 0- -0 It . %4. w 7.24 pose . Ink Dos in 740 . ..u M* laid 23' C

Februory, 1952

I

t

(3)

I

t

alley . Their substitution into Equation (2) yields :

L i 1 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

AP s' LU (1 - .) "

against G/(I- .) . This is demonstrated ht Figure 4 .

Up to this point the aim has been to formulate the effect of fractional void vol-ume in fixed beds, and the effect of orienta-tion was not included . The orientaorienta-tion of the randomly Packed beds is not susceptible to exact mathematical formulation . This is especially true it the particles have odd shapes and are not negligible in size con-pared with 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 problem was to pack a known amount of solids to various bulk densities . yet each packing had to be ant-forin and reproducible.

This was accomplished by admitting gas below the supporting grid after the solids were pound in. The gas rate was sufficient to keep the bed in an expanded state and the 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 from the tightest possible to tie loosest stable pack . ing, For crushed material the most tightly packed bed having a height of 30 cm. could easily be expanded by 6 to 7 con . When the desired 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 flow rate measurements. Highly reproducible packings can be ob-tained by this method, and more important. the particles are believed to be oriented by the gas doming upward. This is evidenced by the existence of a theoretical relation-ship (7), verified experimentally, between the bed expansion and the flow rate . A further evidence for particle orientatio n was found in the fan that the most tightly • packed beds have been obtained by slowly reducing the rate of upward gas flow to an initially expanded bed while subjecting it to vibration .

It will be evident on inspection of the form of 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 are known . a may be readily alculated. But the particle density of crushed porous ma-terials is not readily known and its deter-mination has presented a problem which was much discussed Fractional void vol-umes were usually calculated by the use of apparent specifu gravities which were de-termines 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

upon . and to seek little correlations . How-ever, this was believed to be unwarranted (g) sitter the determination of pressure drop through beds of porous panicles hinges upon the evaluation of the particle density. Therefore. a gas flow method was developed (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-win 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

P

ar

e e

ffect

o

f th

e par

ticl si

e

te

. - surface area, surface area, Cie.

and shape is best analysed in the light -of . Pt e

f

A

i

r

f

di ti

theoretical implications of the Blake plot . The identity between the two extreme ranges of the Blake plot and the theoretical equations developed respectively by Kozeny and Burke for viscous . and turbulent-flow ranges has already been shown . Also, is has been pointed out that these two expres-sions cotnsinnad the following general equation developed by Ergun and Qrning (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 employed by some workers. Since the latter quantity' involves the fractional void volume, use of specific surface has been preferred in the present work. The relation between the two quantities is expressed by

Sea (1- .)S«

Equation Ill) involves the concept of "mean hydraulic radius" in its theoretical development (7) . Its validity has been tested with spheres, cylinders, tablets, sot doles, 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. Few those

B

mtg . 4. A gsaersl plat for • single grins. petition to dgdarem #,*a* l odd "4e•. pats ol.•~.ap..» ~ arknj tqst w i p) • 'a straighe t ln g

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 problem for uniform geo :nctric shapes . For irregular solids, especially fur porous materials, however, surfs" area determination becomes involved. The sur-face of porous materials is necessari .y full of holes and projections. Different surface arms are usually defined in connection with porous materials, viz., total surface area

(including that of pores), external visibl e

geometr

c su

ace, as

s nct

rom ex .

ternal visible surface, may be visual.ted as the 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 area is .lesircd will depend on the purpose for which it is to be used. Geometric surface arm is 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 microscopic 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 the diameter 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 .

~so k. W

(12)

where k. en 72 a and k. = 3/4 o, , Pinar

torn of Equation (12) is :

(13 )

N ., = Dp

The left-hand side of Equation (13) is the ratio of pressure drop to the viscau en-ergy term and will be designated by

f.-APD•

(1^ .)

L U .

(13a )

/. = k.+ k . -I V-=~ (136) According to Estwtiot (13) a linear rela-tio udnip exists between I. and A's ./ 1- e . Data of the present investigation mi those presented earlier have been treated accord-sngly, std the coefficients Jr. and Its have been determined 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

I

(4)

N

N . S. A sonars] yrapbKOl nrprewewtias 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 1. fgoarlgn (13..) . IC' I

N

I-at. a h

.nthso (liopldmi npr0wModen of pwrwrg dreg I. poe#ed mkrwns, Salta 3bo r

.pnwals

Q hossot Msvotipaflon Atosssa s X hark . grad rl .wr ❑ Cl.- sad Worwe.

II ' I P, coos s,r.igorr.0 it f 1 1 li

r

e ! . ♦ r 00 ! r- e r Neo ! s ! 0 N w w tnno dl plnwmo r

f 1

I I

I 7 I•, . I j I :

11111 1

s 'o ! a s a e sin ! i . ♦ e N0a e r w . o re r I I , i 1 ` I yaom~l .t ~ n ! ! • a r NO ! ! • ♦ . Nee ! ! !-E

Ks. 7. Gr.pbksl mpeesewwdea at prwvra dreg I. toed hod♦. Cow of rigor. 3 an r.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~l

i i

l0 1

I pear

l

I

_{I I}

I I H 4 i r

C

I i ~

i i

. 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 wider range of data, a - logarithmic scale has been used which results in a curve for the straight line of Equation (13) . Data of Burke and Plummer and those of Morcom are also shown in Figure S . In all three cases the solid lines are identical and are drawn or . cording to the following equation :

1'0. I. .— ISO+ 115

1 Data shown in Figure 5 and some addi-tional data obtained from the literature covering wider ranges of flow rate are in-cluded in Figure 6, together wilh the asymptotes of the resulting Curve on the logarithmic scale. Again the solid line represents Equation (13x) .

A different form of Equation (121 is represented by :

APg . D, os

:z k . 1a + k

L GG . - 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 will

be designated by / .

,_ PE D sot J

_{. (14.)}

E

11-150 1Ns. + 1

.75 (1db )

(5)

is

i

}

1. is similar to the friction factor more commonly used and is identical with the dimensionless group of Blake . It will be noted that Burke and Plummer plotted essentially ft vs, (►- .)/M. . which per, according to Equation (14), should yield a straight inc lon an arithmetic scale . The authors apparently failed to recognize this fact . The best curve drawn through the expert usI pants on an arithmetic scale does not differ markedly from the line representing Equation (14b) . The scatter to be seen an the plot of Burke and Plum-mer was largely due to the systems involv-ing mixtures and those for which the ratio of tube diameter to particle size was less than 10. 1 hew systems have been emitted in Figure 5, It has been customary, how-ever, to plot f. against N.,/(1 - e) instead of the inverse of the last variable. This type of plot is the one suggested by Blake and adopted by Carman, Morse and others. Figure 7 shows I. plotted vs. N . ./(I- .) for the data already presented in Figure 5 . Figure I is a more comprehensive presents-tion. The solid 1• act arc drawn according to &luation (lob) . A comparison of Figure 6 with 8 is analogous to that of 1. with ft. Both plots are capable of pre-senting the data. However, I. pus a big advantage over ft in that it is a linear functioc. of the modified Reynolds number , ,) . The curve of Figure 6 is a straight line on an arithmetic scale . On the other hard. I ., which has been used al-n,ost exclusively, is an inverse function. A comparison of various empirical represen-tations with Equation (I2) as to be seen in Figure 9.

The foregoing treatment so far has been confined to studying the factors in-volved in the pressure loss in packed beds and to analyzing experimentally the theoretical developments presented earlier . It is only proper that the equa-tions presented are also analyzed briefly from the standpoint of pure fluid dy-namics . Fortunately, the equations lend themselves for such analyses . By defi-nition :

and

1)

► to 6/S. (150 )

S. = S,/AL(1 - .) (15b)

where S, = total geometric surface

area of the solids and A =

cross-sec-tional area of the empty column . The

total iorep exerted by the fluid on the

solids = GPp,Ao . therefore the tractive

force per unit solid surface area, usually

referred to as the shear stress, e, is

expressed by :

-r o-r .>Ag.A ./S, (15e)

The ratio of the volume Occupied by

the fluid in the bed, AL., to the surface

area it sweeps, St, is the hydraulic

radius, rs,

rs or ALe/Sd (1Sd)

The actual average velocity of the fluid

in the bed is obtained from the ratio

of the superficial fluid velocity to th e

fractional voids,

a to f1/.

Substitution of Equations (lSarr) into

Equation (13a) gives

f. s. 36 r (16)

and into Equation (140) gives

!ass6 P- (17 )

Similarly proper substitution will yield

Na' as 6pnra (18)

Therefore, Equations (13) and (14)

respectively will become :

36 r=s •a 150 + 1 .75 6prs

# at

and

.0

6Puts Its

130 s +1.75

(19 )

(20)

It is seen that these transformations

employing the absolute values of shear

stress, fluid density, and velocity

elimi-nate the fractional void volume. The

terms involved in Equations (16.20)

are well known in the fields of

hydro-and aerodynamics. Other forms of

de-pendences upon . ascribed to a general

equation, as encountered in the

litera-ture, would not lead to complete

elimi-nation of the fractional void volume

upon transformation to these

fundamen-tat variables.

The theoretical significances of the

varied with the fractional void volume .

Whether or not kt is a constant is to

be decided on inspection of the lower

end of Figure 6 and the upper end of

Figure 8 where viscous energy losses

are dominant However, the inherent

inaccuracies involved in the

meusurc-ments of specific surface, fractional

void volume, eta, must be borne in

mind In the present work, moreover,

single systems were investigated at

dif-ferent fractional void volumes and no

evidence of variance of its with . was

found. This point is clearly supported

by the proportionality of a to (I- .)s/

es as to be seen from Figures 2 and 3,

and similar other graphical

representa-tions (1, 8, 9) . The factor ks(sa 3/4B)

is subject to treatment similar ta that

of kt (7, 8, 9),

Summary

The laws of fluid flow through

gran-ular beds have several aspects of

prac-tical consequence . They generally find

use in correlating the rate of mass and

heat transfer to and from moving fluids

(24) . The extension of such

relation-ships to packed columns will rtquire

formulation of the laws of fluid flow

through granular beds . Empirics,

cor-relations are generally useful for the

particular purpose for which tl..ey are

made, but may not shard light for a

different purpose. For the sake of

clarity in the application and use of the

constants let and lea have been omitted data obtained in packed columns, it

in the foregoing treatment The former-* seemed desirable to develop expressions

of these constants is discussed by Car- (Equation (12)) in a comprehensiv e

man and Lea and Nurse (15) in

con-nection with the Kozeny equation. As

a result of comparison of various

sys-tems involving different fractional void'

volumes, Lea and Nurse (16) concluded

that a(=let/72) was not a constant but

-'

her

CrN

a

4 3

ot1 ai t o

dla 4 3

form applicable to all typos of flow . I n

doing so the theoretical developments,

as well as the empirical approaches,

have been considered and the following

conclusions have been drawn :

1. Total energy loss in fixed beds can

]III I W ild

I

2 3 4 6

2

3 4 6 e 100

a

4 6 s1000 2 3 4

NR.

I-E

a a e.n

e.h.adv

4a at

nwr4 d in f d bed D

ta

d rf k bnd r1 . p

. s

p m0 un s,

a

d e.e or. ..p o . 15t

( ) This sep. at plus 4 idntnol with dear at sink. . a.ra atw is den rwu.rdt .9 sit aq ..t:o . (146) ,

(6)

I

i ,,fit l , a .n. .a •,IMS a a ...Yt ! 1

*

"Y . R

I

~

1

ra Nr111s .1 NYS la, Ns l l"t

;7. ii

a ,o

I

4 s • too

a

a 4 • a Io00 a

a

4 Na,

I-7

peg. 9. Coupwls.a at w eSa, .w*Uk•1 npr.mnta$ .ns rrta, e•w/fs . (12).

be treated as the sum of viscous and

kinetic energy losses.

2. Viscous energy losses per unit

length are expressed by the first term

of Equation (12) :

ISO 0(1-e)s ssU„

es per

and the kinetic energy losses by the second term :

.

3 . For any set of data the relative

amounts of viscous and kinetic energy

losses can be obtained from either

Equation (13) or (14) .

4 . A new form of friction factor, f,•

representing the ratio of pressure drop

to the viscous energy term has been

given (Equation I3c) and should have

advantages over the conventional type

of friction (actor.

5. A linear equation .too been shown

to represent the conventional type of

friction factor, vie ., the ratio of

pres-sure drop to energy term representing

kinetic losses (Equation 146) .

"

I

Acknowledgment

Tlae author acknowledges the

en-couragement and advice of H . H .

Lowry and J . C. Elgin, and the

assis-tance rendered by Curtis W. Dewalt.

Jr., in preparing this manuscript.

Notatfofl

n oo'd = coefficients in Equations (1),

k (4), and (7), respectively

A = cross-sectional area of the

empty column

6.6" coefficients in Equations (1)

and (8), respectively

Ds a effective diameter of particles

as defined by Equation

(ISa )

)'. = friction factor, which repro

.

9. sends the ratio of pressure

loss to viscous energy loss

and which is linear with

mass flow rate, defined by

Equation (13a )

friction factor . identical with

the dimensionless group of

Blake, defined by Equation

(1k)

gravitational constan t

G m mass-flow rate of fluid.

G=,v U

At n coefficient of the viscous ear

ergy term in Equation

(12) ; k, = 15 0

kg - coefficient of the kinetic

en-ergy term in Equation

(12) ; k2 - U S

L = height of bed

Nt,, = Reynolds number,

Na, = D,G/p: .

P s pressure loss, force units

ra = hydraulic radius of packed

bed, defined by Equation

(lSd )

S = surface of sagida per unit vol .

St =

time of the bed

total surface area of the

solids in the bed

S, = specific surface, surface of

solids per unit volume of

Solid s

is actual velocity of fluid in the

bed

U superficial fluid velocity based

on empty column croon

sec-tion

p.

Palo 94

Chemical Engineering Progress

Use _ ouptrticial fluid velocity nice

sacred at average pressure

• a coefficient of viscous energy

tern, in Equation (11 )

A = coefficient of kinetic energy

tern, is Equation (I1 )

• = fractional void volume in bed

p am absolute viscosity of fluid

p = density of flui d

= average shear stress, defined

by Equation (lSc )

Literature Cited

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Supplement, pp . 47 (1947) . 17 . Lewis, W. IC .• Gilliland, E. R ., and

Bauer, W . C.. led. Esp. Chen,., 41, 1104 (1949) .

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Fluid Flow Through Packed Columns, Ergun

1