Fluid Flow Through Packed Columns, Ergun
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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
M
'_°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
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-
8/18/2019 Fluid Flow Through Packed Columns, Ergun
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
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-
8/18/2019 Fluid Flow Through Packed Columns, Ergun
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
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
: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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-
8/18/2019 Fluid Flow Through Packed Columns, Ergun
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l~
i i
l0
I
pe
ar
l
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 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
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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
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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
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
1 . Arthur, J
. R
., Lnnet, J. W Raynor,
E. J ., and Sington
. E
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2
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.
3
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., and Nurse, R. W., J
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19 . Lindquist, E., ?tvmier Ganges des
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V, pp
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99, Stockholm (1933)
.
20 . Marcum, A
. R
., Trans
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.
Engr,. (London), 24, 311 (1946) .
21 . Morse, R. D, lad. R .O. Chew
., 41 .
1117 (19 49 )
.
22. Oman . A
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3d
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. Pcaraleum New, 36, R79$
( 1 9 4 4 )
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21 Reynolds. O, "Papers on Mechanical
and Physical Subject
.,
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University Press (1900) ,
24, Sherwood
. T
. K, Absorption and
Extraction
.' McGraw-Hilt Book
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. 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.
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February, 1957
.,r
.
: r
.orp
T
nh r
R
that ;
9tJ~~
cc - 3
3nd1
to
ara.i
the
,nsc
mai
n
tar
V :
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