fluid flow through packed columns, ergun

8/18/2019 Fluid Flow Through Packed Columns, Ergun

1/7

FLUID FLOW THROUGH PACKED

COLUMN S

S A B R I E R G U

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

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 fow poduc o hedensyo hefudwh

APC - e) t PU

1- s GU„ secondpower of its velocity

Lp =150a +175

aDAPL=all +beV(1

h

T HE pressure loss accompanying the

flowof 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 .

Vol . 48, No. 2

NY 6 2592

$11 if

R I T I S H L I B R A R Y , B O S T ON S P A

LOAN/PHOT OC OPY R EQUEST FOR

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'_°p` c o o N o .

1979

Ossc,iptio

n

. • ,

.h„b aa

a

41pii hinar, rn

M,ry

CHEM ENG

. PROG

.

N

.S

.

j `low through packe

d

aace of pybpUtion

:

''rV

7

ra

n

o

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 ,

.rained for gas Sowthrough a be of

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 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

columns ,

Publ,snar

:

i Pam

--- IS NlSNS

N

a

r' '~Ild Yngwnl

mere out

a rot

e

here 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/LVag 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-

theikr t

I The experimental results of the present

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 • of Fluid

FromEquation (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-

ems (3

hi

h i

i

fl

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 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


2/7

.

. i ., rr wi- Ihr t4.

o.it, of On 11 il Inl

a iagor in'rlamfpMto the

sat

:,bh, of I - -•,Ikls ody, t y lir-t brut

of 1?ryru .n,o t41 reprcv,ny riwnn 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 and1lunuucr (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,--,•, wll, 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-

,

.n,s. ulc lu,•,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

.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

I 1 1 •

.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

G r o u p a l u M s

c

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

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

. Pnki domtpra1046

g./mCrowsnuik

.

.w.l oronof rob. 774

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 lowresistance 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 wdely used

later (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 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-

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

A PR. (I-)' p('

(5)

On the other hand, at high flowrates

Bake t tttethod iv

ise t

th

e

uatioo e q n

s r

fie

. 2, la

.ps.d

.-' is- we hkw

*

ansinl I.- - foakr

.ol said -I.-, aq

.

.• oofBurke set Plummer for turbulent

tint al .rod .(Q lasnapis

.wet dopes

.ro_.b• .

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 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 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 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 Owsinalr

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 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 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 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

..

•,

. 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

Fe br uo r y , 19 52

I

I



3/7

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 fract ional void vol-

ume in fixed beds, and the effect of orienta-

tion was not included

. The orientation 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 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 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

belowthe 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 fromthe

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 flowrate 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 flowrate

. 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 wll be evident on inspection of the

formof 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-

mnation has presented a problemwhich

was much discussed Fractional void vol-

umes were usually calculated by the use of

apparent 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

; r r

;

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 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-

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 thr e e tee par

e

. - surface area, surface area, Ce

and shape is best analysed in the light -of

P e

fdi tir

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 . 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 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 wth 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

. Fewthos

e

B

m tg

. 4. Agsaersl plat for • single grins.

petition to dgdarem #,*a* l odd 4e•. pats

ol.•~

ap

.

.»

~ arknj tqst wi p) • 'a straigh

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 problemfor uniformgeo

: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 ace, as s nc t rom ex

.

su

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 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 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.

~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

.-

APD•

(1̂

.)

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 have

been 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

g a s e s

: 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



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itf

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Is rap's-tool by is, tgnatua, ll4

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l~

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pe

ar

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r

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r- a

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possible to construct the genera] equation

.

The results are shown on the top of Figure

5

. To be able to include a wder range of

data, a - logarithmc scale has been used

which results in a curve for the straight

line of Equation (13)

. Data of Burke and

Plummer 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 D a t a s h o w n i n F i g u r e 5 a n d s o m e a d d i -

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 the

logarithmc scale

. Again the solid line

represents Equation (13x) .

Adifferent formof Equation (121 is

represented by :

A Pg

. 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 will

be designated by /

.

,_PEDso J

. (14

.)

E11-

150 1Ns

. +1

.75 (1db

IF

February, 1952hemical Engineering Progres s

ogo 92



5/7

is

i

}

1. is simlar to the friction factor more

commonly used and is identical wth 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 in

c 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 fromthe 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). Acomparison 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

fromthe 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 or

. >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) give

s

f. s

. 36r (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

:

3 6 r =s • a 150 + 1 .75 6prs

a

t

and

.0

6Puts Its 130 s +1.7 5

( 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). T he 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 bu

t

-'

her

CrN

a

4

3

ot1 ai t

o

da

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

] I I I I W

i l d

2 34 6

2

3 4 6 e 100 a

4 6 s1000 2 3

4

NR.

I-E

a a e.n

eh

adv

4a at nwr4 d in f d bed d rfta bn d

p

. s

p

un s,

e

.e

0

a or. ..p o ..

15t

) This sep

. at plus 4 idntnol with dear at sink.

. a

ra atwis den rwu

rdt

.9 sit aq ..t:o

. (146) ,

.

: 9 5 2 Vo l. 48, No

. 2

Chemical Engineering progress

pogo 93



6/7

I

i

,,fit

l a .n

.

.a•,IMS

a a

.

.

Yt

*

"

. R

I

~

r

Nr111s

.1

NYS la, Ns

l

l"t

;7 i

i

a ,o

I I

4

s •

a

a 4 • a Io00 a

oo a 4

Na,

I-7

peg

. 9

. Coupwls.a at w eSa,

.w Uk•1 npr

.mnta$

.ns r r t a, 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 pe

and the kinetic energy losses by the

second term:

.

3

. For any set of data the relativ

e

amounts of viscous and kinetic energy

losses can be obtained fromeither

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) respectivel

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

Gmmass-flowrate of fluid

G=vU

At n coefficient of the viscous ear

ergy termin Equation

(12) ; k, = 15 0

kg - coefficient of the kinetic en-

ergy termin Equation

(12) ; k2 - US

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 be

d

S, = specific surface, surface of

solids per unit volume of

Solid

s

is actual velocity of fluid in the

be

d

U superficial fluid velocity based

on empty column croon sec-

tion

p.

Palo 94


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

1 . Arthur, J

. R

., Lnnet, J. W Raynor,

E. J ., and Sington

. E

. P

. E, Trans .

Faraday Sat. . 46, 270 (1950) .

2

. Blake, F

. E

. Tram

. Am

. Inst. Cheap .

Sages . 14, 415 (1922)

.

3

. Burke, S. P ., and Plummer, W

. B

., l,sd

Env. Chen ,., 20. 1196 (1928)

.

4

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. last

Chsa,

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Engrs

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lad

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8 . Ergun

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Chem, 23, 151

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9. Ergun

. S

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10

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., and Hatch, L P .

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It

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Hatch, L

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Clem . I n d . , 5 8 , 2 7 7 ( 1 9 3 9 )

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16. Lea, F. It ., and Nurse, R . W

., To-s

less. Chem

. Loge

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Supplement, pp . 47 (1947)

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17

. Lewis, W

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Bauer, W. C. led. Esp. Chen,., 41,

1104 (1949) .

18

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., and Grammer, H

., Chre,

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Eng

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; 43, 549, 633, 713

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19 . Lindquist, E., ?tvmier Ganges des

Grander Barrages

. Vol

V, pp

. 81-

99, Stockholm (1933)

.

20 . Marcum, A

. R

., Trans

. lest. Chum

.

Engr,. (London), 24, 311 (1946) .

21 . Morse, R. D, lad. R .O. Chew

., 41 .

1117 (19 49 )

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22. Oman . A

. O, and Watson K

3d

. .

NaN

. Pcaraleum New, 36, R79$

( 1 9 4 4 )

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21 Reynolds. O, "Papers on Mechanical

and Physical Subject

.,

." Cambridge -_

University Press (1900) ,

24, Sherwood

. T

. K, Absorption and

Extraction

.' McGraw-Hilt Book

Ca, New York N. Y. (1937) .

25. Stanton, T. E, and Pannell

. J

. R»

19

9 T r

a n s

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R 9

)o y . So. (Leaden), A214

.

26. Traxlee

. R . N

., and Baum. LA. K.

Physic, 7, 9 (1936)

.

27. Wilhelm, R. H, and Kwauk, al

.

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Chew

. Exp. Progress, 44. 201

( 1 9 4 8 )

.

February, 1957

.,r

.

: r

.orp

T

nh r

R

that ;

9tJ~~

cc - 3

3nd1

to

ara.i

the

,nsc

mai

n

tar

V :



7/7

fluid flow through packed columns, ergun

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