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8/11/2019 Mass Transfer in Packed Beds at Low Peclet Numbers
1/9
P e r g a m o n
( h , 'm~ , . I t im/m , ' . ' rm, / S: w e . Vol 52 . Nos . 21 22. pp . 3995 41~)3 . I t~97
1 '4 q 7 | q s c ~ . lc r , ~ w n c e l . l d A l l r i g h t s r e s c r ' . e d
P r i n t e d i n G r e a t B r i t a i n
Plh SO009-25A)9(97)OO242-X ~ ~ ~ s l ~ l ~ -~l~)
Mass t rans fer in p ac k ed b ed s a t l ow Pec l e t
n u m b e r s w r o n g e x p e r i m e n t s o r w r o n g
interpretat ions?
G. Rexwinkel ,* A. B. M.
Heesink and
W. P. M.
Van Swaaij
Department of Chemical Engineering, Twcnte Universi ty of Technology, PO Box 217. 7500 AE
Enschcdc, The Netherlands
(Accepted 7 July 1997)
Abstract Much research has been focused on mass transfer phenomena in packed beds. i-'or
Peclet numbers above 200, empirical relations have been derived that predict the valuc of the
mass transfer coefficient as a funct ion of the Reyno lds numb er and the Schmidt number . These
relations are more or less simila r to the well-known relation that Ranz and Marshall derived for
mass transfer aroun d a single sphere in an infinite medium
S h = ~ + f l R e S d .
For packed beds of spherical particles an :~-value of 3.89 can be calculated on basis of
fundame ntal considerations. However, Sherwood number s much lower than this minim um
value have been observed at Peclct numbers below 100. Several expl anat ions have been
proposed for this apparent discrepancy, such as misinterpr etation of the experimental results
due to unjustified neglection of axial dispersion or wall channeling. In this work, a model that
predicts the combined effects of axial dispersion and wall channel ing has been developed. With
this model, it is possible to explain the results obtained with undiluted beds in which all
particles are active in the process of mass transfer. However, such an explan atio n is not possible
for the results obta ined with dilu ted beds in which not all particles arc active. Therefore, in the
case of diluted beds other reasons for the apparen t drop in mass transfer rate must exist. In the
present investigation, it is demonstrat ed that the drop again originates from misinterpret ation
of the experimental results. It is shown, both experimentally and theoretically, that low
Sherwood number s can be obtained when large differences cxist between the local concent ra-
tion, experienced by an active particle and the mixed cup concentration of the whole bed
cross-section. (" 1997 Elsevier Science l,td
K e y w o r d s : Mass transfer; packed beds: low Peclet numbers; dilution.
I. INTRODUCTION
The rate of mass transfer between particles and fluid
in packed-bed contactors in often predicted with the
help of a generalized, dimensionless correlation
S h
= function
( R e , S c ,
Geomet ry . (11
Several correlation s have been published, e.g. by
Wakao and Funazkr i (1978) and Gnielinski (1978j.
Most of these were based on mass transfer measure-
ments performed at high Peclet numbers ( P e > 200),
and are modified versions of the well-known
Ranz- Mars hall equation, that was derived for a single
sphere in an infinite medium. Though these correla-
tions are accurate at high Peclet numbers, at low
*Corresponding aulhor. Te l. : 0031 534894338: filx:
0031534894774.
Peclet numbers ( P e < 100) Sherwood numbers have
been measured that are less than predicted. See Fig. 1.
in which the results of several studies are presented.
Sorcnscn and Stewart (1974) examined mass trans-
fer in packed beds on a fundamental basis. They
demonstrated that, at low Reynolds numbers
I R e < 10), the rate of mass transfer is only a function
of the Peclet number. They also showed that the
minimum value of the Sherwood number, which is
reached at
Pe = O,
amounts to 3.89 in the case of
ideally packed spherical particles. A similar result was
obta ined by Gu nn (1978). Up to now, this result could
not bc confirmed experimentally, not even when ap-
plying sh al lo t beds to minimize experimental inac-
curacies, that are inevitable due to two complications
that appear when operating at low Peeler numbers:
(i) the number of mass transfer units becomes quite
large, making it difficult to determine the drivi ng force
for mass transfer in an accurate way, and (i i) the
3995
8/11/2019 Mass Transfer in Packed Beds at Low Peclet Numbers
2/9
3996 G. Rexwinkel t a l
1 0 0 0 ~
100 ~ ~-
0
. . . . . . .
"
0.1
0.01
0.1 1 10 100 1000 10 00 0 100000
Williamson t al (1963)
Wilson nd Geankoplis 1966)
Appel nd Newman(1976)
Fedkiwand Newman(1982)
o Hsiung nd Thodos (1977)
13 Hobson nd Thodos (1951)
O Satterfield nd Resnick (1954)
Hsiung nd Thodos (1977)
Bar-Ilan nd Resnick(1957)
Wilson nd Geankop is (1966)
- - G n i e l i n s k i
Sc= 1000 (liquid)
....... GnielinskiSc=2.5 (gas)
1000000
P e [ ]
Fig. 1. Experimental Sherwood numbers from literature. Black symbols: undiluted bed--liquid; White
symbols; undiluted bed--gas; Gray symbols: diluted bed--gas (Bar-llan and Resnick, 1957; Hsiung and
Thodos, 1977) and diluted bed--liquid (Wilson and Geankoplis, 1966).
cont ribu tion of free convection to total mass transfer
may become significant. In order to overcome the first
complication some researchers applied diluted beds
in which only a part of the particles is active in the
mass transfer process, e.g . Bar -lla n and Resnick
(1957), Wilson and Ge ankopli s (1966)and Hsi ung and
Thodos (1977). By doing so they could lower the
concen tratio n in the effluent and calculate the drivi ng
force for mass transfer more precisely. Nevertheless,
their results are still not in accordance with expecta-
tions on basis of theory, as is illustrated by Fig. 1. The
correlation proposed by Gnielinski is assumed to de-
scribe the actual Sherwood number satisfactorily for
the whole Peclet range, since it predicts a limiting
Sherwood number of 3.8 at zero flow, which is in
agreement with the theoretical findings of Sorensen
and Stewart (1974), while at higher Peclet numbers
this correlation has also proven to be accurate.
In this paper, we will demonstrate that the discrep-
ancy between measured and predicted Sherwood
numbers, which is observed at low Peclet numbers, is
caused by wrong calculation of the (local) driving
force for mass transfer. We will do so both for un-
diluted beds, in which all particles are active, and for
diluted beds, in which only part of the particles in
active. However , emphasi s will be on diluted beds. It
will be shown that such beds need a comple tely differ-
ent approach than und iluted beds, and that the results
obtained with diluted beds are of no value to those
interested in mass transfer phenomena in undiluted
beds, at low Peclet numbers.
2. RE-EVALUATIONOF EXPERIMENTALDATA
Most Sherwood numbers, that were reported in
literature, where calculated while assuming plug flow
behavior of the fluid flowing through the bed, i.e.
while neglecting the effects of axial dispersion and
wall channeling. This may not always be justified,
especially not at small Peclet numbers. For example,
in the case of shallow beds, axial dispersion may have
a significant negative effect on the average driving
force for mass transfer, causing an apparent drop in
Sherwood number as was illustrated by Wakao and
Funazkri (1978). In the case of small diameter beds,
wall channeling may result in overestimation of the
average driving force in the bed, also resulting in an
apparent drop in Sherwood number (Martin, 1978).
As most reported data were obtained from
measurement s that were carried out with shallow beds
with a small diameter, it makes sense to re-evaluate
these data while correcting for the influences of axial
dispersion and wall channeling. We did so with the
help of the model developed by Martin (1978), which
was expanded in the following ways:
Mass transfer is assumed also to occur in the wall
zone. The mass t rans fer coefficients in the wall
zone and in the center part of the bed are as-
sumed to be equal.
The fluid concentrations in the effluents from
both zones are calculated on the basis of the
axially dispersed plug-flow model, in a similar
way as was done by Wakao and Funa zkri (1978).
The gas dispersion coefficient is calculated with
a correlation proposed by Gunn (1987), whereas
the liquid dispersion coefficient was calcula ted
using a correlation of Chung and Wen (1968).
These correlations have been proven to be accu-
rate also at low Peclet numbers.
The results of the re-evaluation are shown in Fig. 2. It
appears that, in the case of undiluted beds, the re-
evaluated Sherwood numbers reasonably agree with
8/11/2019 Mass Transfer in Packed Beds at Low Peclet Numbers
3/9
Mass transfer in packed beds 3997
1000
100
10
0.1
0.01
. . - ' ~ * " "
i,d J
[ ]
Ga - .11'
. t
O
0 . I 1 1 0
1 00 I (X X ) 1 0 0 0 0 1 0 0 ( O ) 1 0 0 00 0 0
P e [ - I
Williarnson et al (1%3)
W ilson and Geankopl is (1966)
A p p e l a n d Ne wma n ( 1 97 6 )
Fedkiw an d Newm an (1982)
H s i u n g a n d Th o d o s ( 1 97 7 )
Q H o b s o n a n d Th o d o s ( 1 95 1 )
o Sat ter f ie ld an d Resnick (1954)
# H s i u n g a n d Th o d o s ( 1 97 7 )
o Bar - l lan and Resnick (1957)
g W ilson an d Geankopl is (1966)
Gnielin ki Sc=1000 (l iquid)
. . . . . . . Gn i e li n sk i Sc = 2 .5 ( g a s )
Fig . 2. Re-evaluated S herwood num bers . Black symbols : und i lu ted bed - - l i qu id: Wh i te symbols : und i lu ted
b e d - - g a s , Gr a y s y mb o l s: d i l u te d b e d g a s iBa r - I l a n a n d R e s ni c k, 1 9 5 7 ; H s i u n g a n d Th o d o s , 1 9 7 7 ) a n d
di lu ted bed l iquid (Wilson and G eank opl is . 19661.
t h e o ry . A p p a r e n t l y , t h e d r o p i n S h e r w o o d n u m b e r .
t h a t i s o b s e r v e d i n u n d i l u t e d b e d s a t d e c r e a s i n g P e c l e t
n u m b e r , i s c a u s e d b y w r o n g i n t e r p r e t a t i o n o f t h e
e x p e r i m e n t a l r e s u lt s , i.e . b y u n j u s t i f i e d n e g l e c t i o n o f
a x i al d i s p e r s i o n a n d / o r w a l l c h a n n e l i n g .
R e - e v a l u a t i o n h a r d l y h a s a n e ffe ct o n t h e d a t a o b -
t a i n e d w i t h d i l u t e d b e d s . W e t h e r e f o r e c o n c l u d e t h a t ,
i n t h e c a s e o f d i l u te d b e d s , o t h e r p h e n o m e n a t h a n
a x i al d i s p e r s i o n o r w a l l c h a n n e l i n g c a u s e t h e d r o p i n
S h e r w o o d n u m b e r w h i c h i s o b s e r v e d a t d e c re a s i n g
P e c l e t n u m b e r . B e l o w , w e w i l l tr y t o i d e n t i f y th e s e
p h e n o m e n a . W e s t a r t w i t h a s t u d y o f a s i n g l e a c t i v e
p a r t i c l e s u r r o u n d e d b y i n a c t i v e p a r t i c l e s o n l y . T h e n ,
w e w i l l e x a m i n e b e d s c o n t a i n i n g m u l t i p le a c ti v e p a r -
t ic les .
3 . D I L l ? T E D B E D S : A S I N G L E A C T I V E P A R T I C I , E
3 .1 . Th em e t ica l
R a n z a n d M a r s h a l l ( 1 95 2 ) e x a m i n e d m a s s t r a n s fe r
a r o u n d a s i n g l e s p h e r e l o c a t e d i n a n i n f i n i t e m e d i u m ,
i n w h i c h n o o t h e r p a r t i c l e s a re p r e s e n t . T h e y s h o w e d
i n a s im p l e w a y t h a t t h e m i n i m u m v a l u e o f t h e S h e r -
w o o d n u m b e r (a t P e = 0 ) a m o u n t s t o 2 . I n t h e c a s e
t h a t t h e s p h e r e i s l o c a t e d i n a n i n f i n i t e b e d o f i n a c t i v e
p a r t ic l e s t h e v a lu e o f t h e m i n i m u m S h e r w o o d n u m b e r
i s b y d e f i n i t i o n o b t a i n e d b y m u l t i p l y i n g t h i s v a l u e o f
2 w i t h t h e p o r o s i t y ~: o f t h e b e d a n d b y d i v i d i n g i t b y
t h e t o r t u o s i t y r o f t h e b e d :
S h m i n
= - - 2 .
( 2 )
T
T h e t o r t u o s i t y o f a r a n d o m l y p a c k e d b e d c a n b e
c a l c u l a t e d a c c o r d i n g t o ( P u n c o c h a r a n d D r a h o s ,
1 9 9 3 ) :
r - (3)
y i e l d i n g , Sh,,~, = 2 t: 1 s . T h i s i s i n a g r e e m e n t w i t h t h e
r e s u l ts o b t a i n e d b y M i y a u c h i (1 97 1) . T h e m i n i m a l
v a l u e t h u s a m o u n t s t o a b o u t 0 . 5 w h i c h i s lo w e r t h a n
t h e m i n i m a l v a l u e p r e d i c t e d f o r u n d i l u t e d b e d s , i n
w h i c h t h e c o n c e n t r a t i o n p r o fi le a r o u n d a p a r ti c le d o e s
n o t r c a c h b e y o n d t h e n e i g h b o r i n g p a r t ic l e s .
3.2. Exp er imen ta l
S o l i d - g a s m a s s t r a n s f e r c o e f fi c i en t s o f a s i n g l e ac -
t iv e s p h e r e i n a n i n a c t i v e b e d h a v e b e e n d e t e r m i n e d
b y m e a s u r i n g t he e v a p o r a t i o n r a te o f a c a m p h o r
s p h e r e w i th a p u r i ty 9 6 % a n d a d i a m e t e r o f 1 c m i n
a n i t r o g e n g a s s t re a m . A c a m p h o r s p h e r e w a s w e i g h e d
b e fo r e t h e e x p e r i m e n t a n d s u b s e q u e n t l y p l a c e d i n s i d e
t h e c e n t e r o f a r e g u l a r l y p a c k e d b e d o f I c m d i a m e t e r
g l a s s s p h e r e s , h a v i n g a d i a m e t e r o f 1 0 . 5 c m a n d
a h e i g h t o f 1 2 c m [ F i g . 3 (a ) ] . B e f o re e n t e r i n g t h e
p a c k e d b e d , th e n i t r o g e n g a s w a s l e d t h r o u g h a d i s-
t r i b u t o r w i t h a h e i g h t o f 6 c m , c o n s i s t i n g o f g l a ss
b e a d s o f 1 m m , i n o r d e r t o e n s u r e a n e v e n l y d i s t r i b -
u t e d g a s f lo w . A f t e r a c e r t a i n t i m e o f o p e r a t i o n , d u r i n g
w h i c h t h e g a s f l o w w a s d i r e c t e d e i t h e r u p f l o w o r
d o w n f l o w t h r o u g h t h e b e d , t h e c a m p h o r s p h e r e w a s
w e i g h e d a g ai n . F r o m t h e c h a n g e i n m a s s t h e m a s s
t r a n s f e r c o e f f i c i e n t w a s c a l c u l a t e d :
k = m o - m . . . . ( 4 )
t ~ . ~ x d 2 ( C , - C b ) M
I n a f i r s t e x p e r i m e n t a c a m p h o r s p h e r e w a s w e i g h e d ,
p l a c e d i n s i d e t h e p a c k e d b e d a n d a f t e r a w h i l e , n o t
a p p l y i n g a n i t r o g e n g a s s t r e a m , t a k e n o u t o f t h e b e d
8/11/2019 Mass Transfer in Packed Beds at Low Peclet Numbers
4/9
3998 G. Rexwinkel t a l .
Sin~l ~
active
p r t i c l e
Vent ]
--L
12 cm
Distributor
10.5 cm
~ UV-meter J
Inert layer of
glass beads
Inert layer of
glass beads
5 c m
i
2 5m C m
).
10.5 cm
a) b)
Fig. 3. Schematicof the experimental setup for two different ypes of experimens: a) Single active sphere in
an inactive packed bed of glass spheres; (b) Multiple active spheres in a bed of inactive glass spheres which
is located between two layers of inactive glass spheres.
Table 1. Physical properties (at 25~C) ogether with design data and operating conditions
applied during the experiments
Single ac ti ve Multiple active
particle particles
Transferred species Camphor Methylbenzoate
Fluid Nitrogen gas Water
Temperature 20-25 C 20- 25:C
Molecular mass of transferred species 0.15224 kg/mol 0.13615 g/mol
Diffusion coefficient of transferred species 6.2
1 0 - 6 m / s *
9.3 10- lo m2/s
Solubility of transferred species -- 2.15 g/l:
Vapor pressure of transferred species 30.2 Pa~ ---
Particle diameter 10 mm 0.63-0.71 mm
Bed porosity 0.40 + 0.01 0.40 + 0.01
Percentage active particles -- 0.35%
Sc 2.54 1099
Re 0.11-141 0.001-- I
*Yaws (1995); *Wilke and Chang (1955); tGetzen e t
a l . (1992); Presser (1972)
and weighed again. It appeared that the reduction of
mass during this experiment could be neglected com-
pared to the reduction observed after a normal experi-
ment of several hours. During all experiments thc
decrease in particle diameter was negligible whereas
the increase of the camphor concentration in the ni-
trogen stream was small. Therefore, the bulk concen-
tration, Cb, could be considered equal to zero. The
saturation concentration, Cs, was determined from
the ideal gas law and the camphor vapor pressure,
P r e p , that was calculated for the applied temperature,
which varied between 20 and 25C but remained
constant during a single experiment. With these data
and the data from Table 1 available, the Sherwood
numb er was calculated according to
S h - R T ( m o - m r , , , ) (5)
t c x p n d p P v a p M D
In Fig. 4, the experimentally obtained Sherwood
numbers are plotted versus the Peclet number. The
8/11/2019 Mass Transfer in Packed Beds at Low Peclet Numbers
5/9
Mass transl~cr
solid line in this figure represents the correlation pro-
posed by Ranz and Marshall (1952). The upflow and
downflow experiments gave equal results indicating
that the influence of free convection on the mass
transfer rate can be neglected. Below a Peeler number
of 50 the presence of inert particles becomes notice-
able causing the measured Sherwood number of fall
below the value predicted by the Ran z Marshall
equation. At very low Peclet numbers the Sherwood
number indeed approaches the theoretical minimum
of approximately 0.5, which is lower than the minimal
value of 3.89 in undilu ted beds. We may thus conclude
that diluted beds cannot be used to examine mass
transfer phenomena in undiluted beds at low Peeler
numbers. We also conclude that the drop in Sher-
wood number to wdues below the limit of 0.5, as
observed with diluted beds. e.g. Bar-ilan and Rcsnick
(1953), must bc caused by interaction of the active
particles present in the diluted bed. Wc will examine
this possible interaction in the next section.
4. DILUTED BEt)S:
M I .
I .TIPI,E ACTIVE PARTICI.ES
4 1 Theoretical
As has been stated earlier, it is important to know
the local drivin g force for mass transfer in order to be
able to calculate the local mass transfer coefficient. In
1(~ )
I 0
- r , ~ f .
D
0.1
1
Single article u p f l o v .
O Smgteparticle downtlow
I~ Bar-Ilan and R~n ick (19571
Ranz-Mar~hall
e q u a t k m
.~x=-2.5
(~L~,)
0 . 0 1 L , , ,
O. I I 0 I ~ 100 0 1(X](30
Pe
[ I
Fig. 4. Experimental Sherwood numbers for a single active
particle together with the results of multiple acti~,e particles
of Bar-ilan and Resnick 11957).
in packed beds 3999
the case of a single active sphere in an infinite bed of
inert particles, the driving force for mass transfer
experienced by the active particle in exactly known
and trivial. Determination of the driving force experi-
enced by an individual active particle in a diluted
packed bed. in which more active particles are pres-
ent. is less trivial.
In Fig. 5, several hypothetical distributions of ac-
tive particles in a two-dimens iona l packed bed are
shown. The driving force experienced by the active
particles will be much different in each situation. In
situation 1 all active particles experience the same
driving force which has not been influenced by the
presence of any active particles upstream. In situation
It all active particles have been placed directly on top
of each other. The influence of particles upstream on
the driving force experienced by the particles dov
8/11/2019 Mass Transfer in Packed Beds at Low Peclet Numbers
6/9
4000
thoroughly mixed with spherical glass particles of the
same size. The mixture, of which 0.35% of all particles
was active, was brought into the bed resulting in an
active layer of 2 cm high. This layer was enclosed by
two 5 cm thick layers of inert glass beads. See Fig.
3(b). The methylbenzoate concentr ation at the exit of
the packed bed was continu ously measured using an
UV-spectrophotometer. The meth ylbenzoate concen-
tration in the effluent remained constant during the
first hou r of operation, indicatin g that pore limitat ion
in the porous Amberlite XAD-2 particles could be
neglected. The Sherwood numbers were calculated by
assuming plug-flow behavior
S h - d 2 ~ v p P In ( C s - C o u t )
6 D i n , \ C s - ~ (6)
The results are shown in Fig. 6 and are similar to
those of Bar-Ilan and Resnick (1957): Much smaller
values are obtained than with a single active particle.
In order to confirm that these smaller Sherwood
numbe rs are found because of intera ction between the
different active particles in the bed, extra experiments
were performed. These were done at a fixed Peclet
numb er of 2.89 using 0.5 g of impregnated Amberlite
XAD-2 particles. The number of mass transfer units
was thus kept constant. The height of the bed, includ-
ing the inert layers at the bottom and the top, always
was 12 cm. In orde r to simulate situati on I of Fig. 5 all
active particles were positioned in a horizontal layer
which was enlosed by two layers of inert glass beds
having the same diameter of 0.66 mm as the active
particles. In this way about 20% of the bed cross-
section was covered with active particles. Situat ion III
of Fig. 5 was also simulated. First all active particles
were mixed with equally sized inert glass beads. The
obtained mixture was then dumped into a tube with
a diameter of 1.35 cm a nd a height of 8 cm, which was
placed in the center of the bed, which was already
filled with a 2 cm thick layer ofiner t glass beads. Next,
the annular space in between the tube and the bed
wall was filled with inert glass beads and the tube was
slowly removed. Finally, a 2 cm thick layer of inert
G. Rexwinkelet al .
glass beads was added to complete the bed. Addi-
tional to these experiments, tests were performed us-
ing diluted layers of varying thickness [accord ing to
the set-up shown in Fig. 3(b)]. By appl ying a consta nt
amo unt of impregnated Amberlite XAD-2 particles of
0.5 g, the degree of dilution was varied. The results of
the interact ion experiments were evaluated on basis of
eq. (61. The ca lculated Sherwood numbers are shown
in Fig. 7.
If the as sumption of a radia lly well-mixed fluid
would hold in all cases, all experiments should have
given the same result, since equal amounts of active
particles have been applied. However, the results
show a significant influence of the particle distribu-
tion, which is in conflict with this assumption. The
calculated Sherwood numbers appear to increase
when the fraction of active particles is decreased,
especially at high degrees of dilution. This is in ac-
cordance with expectations, since the average axial
distance between the active particles strongly depends
on the degree of dilution at high dilution degrees,
whereas this dependence is relatively small at lower
degrees of dilut ion. So, the extent of radial mixing will
increase with the degree of dilution, in particular at
high degrees of dilution. The results obtai ned with the
particle distributions resembling situations I and III
of Fig. 5 also confirm that interaction between active
particles is an important item in diluted beds. This
interaction, which results from radial concentration
profiles, is not accounted for when assuming plug-
flow behavior. We will illustrate this with the model
discussed below.
4.3. T h e m o d e l
The importance of radial concentration profiles
and the resulting inter action between different active
particles in the bed can also be demonstrated in a the-
oretical way. To do so, the diluted bed is modelled by
a two-dimensional network of interconnected con-
tinuously stirred t ank reactors (CSTR); see Fig. 8.
Each CSTR contains exactly one particle which is
1 0 0
10
0 . 1
0 . 0 1
0.1
, Y
S m g t e p a r t i c l e . Th i s w o r k
I I I
0
M u l t i p l e p a r t i c l e s . T h i s w o r k
B a t - I l a n a n d R e sn i c k ( 1 9 5 7 )
1 3 t i s i u n g a n d T h o d o s ( 1 9 7 7 )
Ranz-Mar~,ha l l equa t i on Sc .=2 .5 (gas)
I I I
I 1 0 1 0 0 1 0 0 0 1 0 0 0 0
Pe
[-]
0 . 6 '
0.5
0.4
i
, ~ 0 . 3
0 . 2
0.I
S i n g l e m o n o l a y e r r e s e m b l i n g
s i t u a t i o n I o f F i g . 5.
o
T u b e n ' ~ u r e m e n t r e s em b l i ng
s i t u a t i o n I n o f F i g . 5 .
. . . . . . . . . . . . . : : ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
o d ~ l u t e d b e d m e a s u r e m e n t s
0 . 0 1 0 . 0 2 0 . 0 3
f rac t ion of partc i l es ac t ive
[-]
0 . 0 4
Fig. 6. Experimental Sherwood numbers for single particle Fig. 7. Experimental Sherwood numbers for several distri-
and multiple particles in a diluted bed. butions of active particle over the bed. P e = 2.89;S c = 1099.
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7/9
Mass transfer
Fig. 8. Schem atic representat ion of the interconnected
C STRs mod el where the grey dots represent act ive particles;
Flowdirect ion is downward,
n x
= number of part icles in
ho rizon tal direct ion and ny = number of particles in vert ical
direction.
e i t h e r a c t i v e o r i n a c t i v e . W h e t h e r a p a r t i c l e is a c t iv e
o r n o t i s r a n d o m l y d e t e r m i n e d . A s o m e w h a t s i m i l a r
a p p r o a c h w a s a p p l i e d b y v a n d e n B l e ek (1 96 9) , w h o
i n v e s t i g a t e d t h e e ff e c t o f d i l u t i o n o n t h e d e g r e e o f
c o n v e r s i o n i n f ix e d - b e d c a t a l y t i c r e a c t o r s .
M a s s t r a n s f e r b e t w e e n C S T R s t a k e s p l a c e in a x i a l
d i r e c t i o n t h r o u g h c o n v e c t i o n a n d a x i a l d i s p e r s i o n
a n d i n l a te r a l d i r e c t i o n b y d i sp e r s i o n . T h e a m o u n t o f
m a t e r i a l t r a n s f e rr e d b y d i s p e r s i o n b e tw e e n t w o n e i g h -
b o r i n g C S T R s i s c a l c u l a t e d a c c o r d i n g t o
N ~. j .~ . i ~ t = - D ~ C i . i+ t - C , . j ( 7a )
d ,
C i - l , j - C i . j
N c i ~ i - 1 .~ = D l~ ,
(7b/
t i p
A s t a t i o n a r y d i m e n s i o n l e s s m a s s b a l a n c e f o r C S T R i.
j y i e l d s
2 2 C 1
0 = ( - I B o , ~ , B - ~ , , x ) ~ J + (1 B-~a , )C~. j
1 1
+ zo ~ - c ' ' i ~ ~ + __Bol~--C~_ ~.j
1 . . r t S h
+ __Bo,~--C i + t
~
+ n,~ j ~: P e (A C i4) ' (8)
W i t h C ~ . j d e f i n e d a s C ~ . j / C . , , C ~ b e i n g t h e c o n c e n t r a -
t i o n a t t h e s u r f a c e o f a n a c t i v e p a r t i c l e w h i c h i s e i th e r
d i s s o l v i n g o r e v a p o r a t i n g . I n t h is e q u a t i o n n ~, c a n
a d a p t t h e v a l u e s 0 o r 1 a n d i n d i c a t e s w h e t h e r o r n o t
a n a c t iv e p a r t i c l e is p r e s e n t i n C S T R
i , j .
T h e C S T R s
in packe d beds 4001
t h a t a r e l o c a t e d a g a i n s t t h e p a c k e d - b e d w a l l a r e o n l y
c o n n e c t e d t o C S T R s i n t h e i n t e r i o r o f t h e b e d .
A n i m p o r t a n t t e r m i s A C ~ , ~ . w h i c h r e p r e s e n t s t h e
d r i v i n g f o r c e e x p e r i e n c e d b y a n a c t i v e p a r t i c l e in
C S T R i , j . I n t h e c a s e o f u n d i l u t e d b e d s t h i s d r i v i n g
f o rc e s h o u l d b y d e f i n i t io n b e b a s e d o n t h e m i x e d c u p
c o n c e n t r a t i o n o f th e w h o l e b e d c r o s s- s e c t io n . H o w -
e v e r , i t i s n o t t o b e e x p e c t e d t h a t t h e s a m e d e f i n i t i o n
h o l d s f o r d i l u te d b e d s b e c a u s e o f r a d i a l c o n c e n t r a t i o n
g r a d i e n t s. A s w e a r e i n t e r e st e d i n d e m o n s t r a t i n g t h e
e f f e c t s o f n e g l e c t i n g t h e s e r a d i a l c o n c e n t r a t i o n g r a d i -
e n t s , i n t h e p r e s e n t m o d e l t w o d e f i n i t i o n s o f d r i v i n g
f o r c e a r e a p p l i e d : ( ij m i x e d c u p d r i v i n g f o rc e , w h i c h is
c a l c u l a t e d f r om t h e m i x e d c u p c o n c e n t r a t i o n in
a h o r i z o n t a l r o w , a n d l i il C S T R d r i v i n g f o r c e , w h i c h i s
c a l c u l a t e d o n b a s i s o f t h e c o n c e n t r a t i o n w i t h i n t h e
C S T R i n w h i c h t h e a c t i v e p a r t i c l e is l o c a t e d . I n m a t h -
e m a t i c a l f o r m
A C , j - - I - ~ " '" ( 9 a )
t I t2 . (
A C cj - -- 1 - C~. j . (9b)
E q u a t i o n ( 8 ) h a s b e e n s o l v e d u s i n g b o t h e x p r e s s i o n s
f o r t h e d r i v i n g f o r c e b y t h e s u c c e s s i v e o v e r r e l a x a t i o n
m e t h o d . A s i t m a k e s n o s e n s e t o a p p l y e x p r e s s i o n s
t h a t w e r e d e r i v e d f or t h r e e - d i m e n s i o n a l b e d s a n d t h a t
p r e d i c t a f i n a l v a l u e o f S h e r w o o d a t P e = 0 . t h e S h e r -
w o o d n u m b e r w a s c a l c u l a t e d s o m e w h a t a r b i t r a r i l y
a c c o r d i n g t o
S h = 0.3 P e 1 3 . (10)
T h e v a l u e s o f B o t, , a n d B o ~ x w e r e c a l c u l a t e d f r o m
c o r r e l a t i o n s r e p o r t e d b y G u n n (1 98 7) . F i g u r e 9 s h o w s
t h e c a l c u l a t e d l a t e r a l c o n c e n t r a t i o n p r o f i l e s a t t h e e x i t
o f a p a c k e d b e d . c o n t a i n i n g 2 0 0 x 4 0 p a r t i c l e s w i t h
a d i a m e t e r o f 0. 5 m m o f w h i c h 3 0 0 a r e a c t i v e , u s i n g
t h e t w o d i f f e re n t d e f i n i t i o n s o f d r i v i n g f o r c e .
F i g u r e 9 c l e a r ly s h o w s t h a t l a r g e l a t e r a l c o n c e n t r a -
t io n g r a d i e n t s m a y e x i s t in d i l u t e d p a c k e d b e d s . F u r -
t h e r m o r e , i t s h o w s t h a t n e g l e c t i n g t h e s e l a t e r a l
c o n c e n t r a t i o n g r a d i e n t s , w h i c h i s d o n e w h e n c a l c u l a t -
i n g t h e l o c a l d r i v i n g f o r c e fo r m a s s t r a n s f e r a c c o r d i n g
t o e q . (9 a l , c a n r e s u l t in l o c a l d i m e n s i o n l e s s c o n c e n t r a -
t i o n s w h i c h a r e l a r g e r t h a n u n i t y , w h i c h b y d e f i n i t i o n
i s i m p o s s i b l e . T h e l o c a l d r i v i n g f o r c e f o r m a s s t r a n s f e r
i n a d i l u t e d b e d i s t h u s o v e r e s t i m a t e d w h e n t h e r a d i a l
c o n c e n t r a t i o n p r o t i l e in t h e b e d i s n e g l e c t e d , a s i s
d o n e w h e n t h e p l u g - f lo w m o d e l is a p p l i e d t o e v a l u a t e
t h e e x p e r i m e n t a l r e s u l t s . T h i s i s a l s o i l l u s t r a t e d b y
F i g . 1 0 i n w h i c h t h c p r c d i c t e d S h e r w o o d n u m b e r i s
s h o w n a s a f u n c t io n o f th e P e c l e t n u m b e r f o r b o t h
d e f i n i t i o n s o f t h e l o c a l d r i v i n g f o r c e A C i 4 . T h e p r e -
s e n t ed S h e r w o o d n u m b e r s w e r e c a l c u l a t e d w h i l e
a s s u m i n g p l u g - f l o w b e h a v i o r o f t h e f lu i d , w h i c h i s
c o m m o n p r a c t i c e w h e n e v a l u a t i n g m a s s t r a n s f e r
e x p e r i m e n t s
I1X c; ~ ~ i 1 i .n~ ,
S h - P e l n I . I11)
t / a t : i t , c 7 C ~ . ~
o r
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8/9
4O02
1 . 6
1 . 4
.2
1
0 . 6
0 . 4
0.2
0
G . R e x w i n k e l
et al .
I
M a J i m u m c o r v ~ m l rl U o n I ~ S S t bl
& ~ r u ~ o ~ l l l l ~ o n B I [ h e (
~ i t o f t l ~ I~ d
2 0 4 0 6 0 8 0 1 0 0
: : t , _ . . . . . . . o . _ , . , e
0.2
A l t l ' l g e Ol l t ' l l l l l l i o i l l [
~ t o t a , . I , ~
0
2 0 4 0 6 0 8 0 1 0 0
la teral positio n [ram] lateral pos ition [ram]
F ig . 9 . C a l c u l a t e d r a d i a l c o n c e n t r a t i o n p r o f i l e s a t t h e e x i t o f t h e p a c k e d b e d : ( a) M i x e d u p d r i v i n g f o rc e :
(b ) C STR dr iv i ng fo rce .
P e
= 10 : Sh = 1 .5 ; Bo i l , = 10 ; Bo~ , = 4 : d r = 0 .5 mm ; nx = 200 : n y = 40 :
z~
= 3 00: ~: = 0.48 = 1 - n:6.
10
s
,ll
0.1
O M xed cup driving orce
CSTRdriving orce
1 1 1
Pe 1-1
F ig . 1 0. S h e r w o o d n u m b e r s c a l c u l a t e d u s i n g e q . ( 1 1 ) f r o m
t h e m i x e d c u p c o n c e n t r a t i o n a t t h e e x i t o f t h e b e d a s c a l -
c u l a t e d b y t h e i n t e r c o n n e c t e d C S T R m o d e l u s i n g t h e t w o
d if fe ren t de f in i t ion s o f the d r iv ing fo rce . S o = 1000;
dp =
0.5 mm ; nx = 2(X): ny = 40; n~,,,, = 300: ~: = 0.48.
5 . CONCLUSIONS
T h e d e v i a t i o n b e t w e e n p r e d i c t ed a n d r e p o r t e d
S h e r w o o d n u m b e r s i n p a c ke d b e d s a t lo w P e c l e t n u m -
b e r s is c a u s e d b y t h e w r o n g i n t e r p r e t a t i o n o f t h e
e x p e r i m e n t a l d a t a . I n t h e c a s e o f u n d i l u t e d b e d s c o r -
r e c t i o n s s h o u l d b e m a d e f o r a x i a l d i sp e r s i o n a n d w a l l
c h a n n e l i n g . T h e S h e r w o o d n u m b e r s w h e n o b t a i n e d
r e a s o n a b l y a g r e e w i t h t h e o r y. S u c h c o r r e c t i o n s d o n o t
s u ff i ce in t h e c a s e o f d i l u t e d b e d s . T h e m i n i m a l S h e r -
w o o d n u m b e r i n d i l u t e d b e d s is l o w e r t h a n i n u n -
d i l u t e d b e d s. F u r t h e r m o r e . i n d i l u t e d b e d s t h e
e x i s t e n ce o f r a d i a l c o n c e n t r a t i o n p r o f il e s s h o u l d b e
c o n s i d e r e d , w h i c h i s n o t p o s s i b l e w h e n t h e d i s t r i b u -
t i o n o f t h e a c t i v e p a rt i c l es i s r a n d o m a n d t h e r e f o r e n o t
e x a c t l y k n o w n . N e g l e c t i o n o f th e s e r a d i a l c o n c e n t r a -
t i o n p r o f i le s b y a p p l y i n g o f th e p l u g - f l o w m o d e l r e -
s u it s i n u n d e r e s t i m a t e d S h e r w o o d n u m b e r s .
T h e r e f o r e , d i l u t e d b e d s s h o u l d n o t b e u s e d t o e x a m i n e
m a s s t r a n s f e r p h e n o m e n a i n u n d i l u t e d b e d s a t l o w
P e c l e t n u m b e r s .
T h e v a l u e s o b t a i n e d w h e n c a l c u l a t i n g t h e d r i v i n g
f o rc e o n b a s i s o f t h e m i x e d c u p c o n c e n t r a t i o n a r e
a l m o s t e q u a l t o t h o s e p r e d i c t e d b y e q . ( 1 0) . ( O n l y t h e
v a l u e a t
P e
= 1 i s s o m e w h a t l e s s d u e t o a x i a l d i s p e r -
s i o n w h i c h i s i n c l u d e d i n t h e m o d e l .) T h e v a l u e s o b -
t a i n e d w h e n c a l c u l a t i n g t h e d r i v i n g f o rc e o n b a si s o f C
t h e C S T R - c o n c e n t r a t i o n a r e m u c h l ow e r . W e , t he r e -
f o re , c o n c l u d e t h a t t h e d e v i a t i o n be t w e e n m e a s u r e d C
a n d p r e d ic t e d S h e r w o o d n u m b e r , w h i c h i s o b s e r v e d i n
d i l u t e d b e d s a t l o w P e c l e t n u m b e r s , is , a t l e a s t p a r t l y , A C ~ .j
c a u s e d b y t h e n e g l e c t i o n o f r a d i a l c o n c e n t r a t i o n p r o -
f i le s i n s i d e t h e b e d . d r
D u e t o i t s s i m p l i c i t y , t h e p r e s e n t m o d e l is n o t s u it e d D
t o q u a n t i f y t h e e f fe c t s o f r a d i a l c o n c e n t r a t i o n p r o f i l e s
o n t h e a v e r a g e r a t e o f m a s s t r a n s f e r i n s i d e a d i l u t e d D ~x
b e d . F o r t h is . C F D s i m u l a t i o n s a r e n e c e s s ar y . D la t
A c k n o w l e d g e m e n t s
T h i s i n v e s t i g a t i o n w a s s u p p o r t e d b y t h e D u t c h M i n i s t r y o f
E c o n o m i c A ff ai rs . T h e a u t h o r s a c k n o w l e d g e J . G 6 r t e n a n d
J . M . M e e r d i n k f o r t h e i r a s s i s t a n c e i n t h e e x p e r i m e n t a l
work
NOTATION
c o n c e n t r a t i o n o f t r a n s f e r r e d s p ec i e s in t h e
f l ui d , m o l / m 3
d i m e n s i o n l e s s c o n c e n t r a t i o n
C / C ~
d i m e n -
s i o n l e s s
d r i v i n g f o r ce e x p e r i e n c e d b y a n a c t i v e p a r -
t ic l e in C S T R
i . j
d i m e n s i o n l e s s
p a r t i c l e d i a m e t e r , m
d i f f u s i o n c o e f f i c i e n t o f t r a n s f e r r e d s p e c i e s ,
m 2 / s
a x i a l d i s p e r s i o n c o e f f i c i e n t , m Z / s
l a t e r a l d i s p e r s i o n c o e f f i c i e n t , m Z / s
8/11/2019 Mass Transfer in Packed Beds at Low Peclet Numbers
9/9
k
m
M
n ~ J
Ilactivc
n x
n y
N
Pv ap
R
tcxo
T
r
m a s s t r a n s f e r c o e f f i c i e n t, m / s
m a s s , k g
m o l e c u l a r m a s s , k g / m o l
r a n d o m f a c t o r (0 o r 1 ) o f e q . 8 , d i m e n s i o n l e s s
n u m b e r o f a c t iv e p a r t i c l e s, d im e n s i o n l e s s
n u m b e r o f p a r t ic l e s in h o r i z o n t a l d i r e c t io n ,
d i m e n s i o n l e s s
n u m b e r o f p a r t i c l e s i n v e r t i c a l d i r e c t i o n , d i -
m e n s i o n l e s s
f lu x , m o l / m 2 s
v a p o r p r e s s u re o f c a m p h o r , P a
g a s c o n s ta n t , J / m o l K
d u r a t i o n o f e x p e ri m e n t , s
t e m p e r a t u r e , K
i n t e r s t i t i a l f l u id v e l o c i t y , m / s
Greek letter.s
~: b e d p o r o s i t y , d i m e n s i o n l e s s
r / v i s c os i t y , Pa s
v fluid flOW, m3."s
p d e n s i t y , k g / m 3
r t o r t u o s i t y , d i m e n s i o n l e s s
Subscripts
m i n m i n i m a l
0 i n i t i a l l y
h i n t h e b u l k
f w i t h r e s pe c t t o f l u i d
i r a d i a l p o s i t i o n o f C S T R
i n a t t he i n l e t
j a x i a l p o s i t i o n o f C S T R
o u t a t t h e o u t l e t
p w i t h r e s p e c t t o p a r t i c l e ( s )
s s a t u r a t i o n
Dimensionless numbers
oax
BOla
Sh
P e
R e
Sc
a x i a l B o d e n s t e i n n u m b e r ( = vdp/D~d
l a t e r a l B o d e n s t e i n n u m b e r ( = vdp/D~,,)
S h e r w o o d n u m b e r ( =
kdp/D)
i n t e r s ti t ia l P e c l et n u m b e r ( = vdp/D)
R e y n o l d s n u m b e r ( = plvdp/qy)
S c h m i d t n u m b e r ( = q//pyD)
R E F E R E N C E S
A p p e l , P . W . a n d N e w m a n , J . ( 19 7 6) A p p l i c a t i o n o f
t h e l i m i ti n g c u r re n t m e t h o d t o m a s s t r a n s f e r i n
p a c k e d b e d s a t v e r y l o w R e y n o l d s n u m b e r s .
A. I .Ch.E.J . 22 , 979 -984 .
B a r - I l a n , M . a n d R e s n i c k , W . ( 19 5 7) G a s p h a s e m a s s
t r a n s f e r i n fi x e d b e d s a t l o w R e y n o l d s n u m b e r s . Ind.
Engng Chem. 4 9 , 3 1 3 - 3 2 0 .
B l e e k v a n d e n ,
C. M.,
W i e l e v a n d e r , K . a n d B e r g v a n
d e n , P . J . (1 9 69 ) T h e e f f e ct o f d i l u t i o n o n t h e d e g r e e
o f c o n v e r s i o n i n f i x e d b e d c a t a l y t i c r e a c t o r s . Chem.
Engnq Sci. 24 , 681 -694 .
Mass transfer in packed beds 4003
C h u n g , S . F . a n d W e n , C . Y . ( 19 6 8) L o n g i t u d i n a l
d i s p e r s i o n o f l i q u i d f l o w i n g t h r o u g h f i x e d a n d
f l u i d i z e d be ds . A. I .Ch.E.J . 14 , 857 -866 .
F e d k i w , P . S . a n d N e w m a n , J . ( 19 8 2) M a s s - t r a n s f e r
c o e f f ic i e n ts i n p a c k e d b e d s a t v e r y l o w R e y n o l d s
n u m b e r . Int. d. Heat Mass Transfer 25 , 935 -943 .
G e t z e n , F . , H e f t e r, G . a n d M a c z y n s k i , A . ( 19 9 2) Solu-
bility Data Series, V o l . 4 8 . P e r g a m o n P r e s s , O x f o r d .
G n i e l i n s k i , V . ( 19 7 8) G l e i c h u n g e n z u r B e r e c h n u n g d e s
W ~ i r m e - u n d S t o f f a u s t a u c h e s i n d u r c h s t r 6 m t e n
r u h e n d e n K u g e l s c h i . i t t u n g e n b e i m i t t l e r e n u n d
g r o s s e n P e c l e t z a h l e n . VT-verfahrenstechnik 12,
363--366.
G u n n , D . J . ( 19 7 8) T r a n s f e r o f h e a t o r m a s s t o p a r -
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