Mathematical modeling of gas distribution in packed columns
Tatiana Petrova *, Krum Semkov, Chavdar Dodev
Institute of Chemical Engineering, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Building 103, Sofia 1113, Bulgaria
Received 15 May 2002; received in revised form 9 November 2002; accepted 9 November 2002
Abstract
A dispersion model is proposed for the description of gas distribution in a packed column in the absence of liquid flow. The radial
gas velocity profile is simulated for three different packing heights and compared with appropriate experimental data, obtained in a
column with ‘honey-comb’ structured packing and gas distribution device (GDD). By separating the effect of the distributor from
that of the packing layer, the additivity of packing redistribution capability and GDD is proved. The model parameters (distribution
ability) are identified by means of non-linear optimization, minimizing the residual variance between the experimental and
theoretical gas velocity. The model and experiment adequacy is checked. An analytical formula for the maldistribution factor is
proposed, based on the model distribution parameter and initial condition only, and compared with the measured experimental
maldistribution factor. Considering the studied packing, gas distribution coefficient is found based on the additivity of the obtained
model parameters.
# 2003 Elsevier Science B.V. All rights reserved.
Keywords: Gas distribution; Packed columns; Maldistribution factor; Structured packing; Identification of the parameters
1. Introduction
Modeling of gas or liquid flow distribution in a
packing layer is thoroughly discussed in the literature
[1�/11]. Most popular are: Diffusion model*/Porter and
Jones [1], Cihla and Schmidt [8], Stanek and Kolar [9],
the Natural Flow Model*/Albraight, Hoek et al.,
Stikkelman [2�/4], and Zone-Stage model of Zuiderweg
et al. [5]. All above models require preliminary state-
ment of distribution coefficients or splitting coefficient
of fluid flow. In most cases, the results obtained for
modeled distribution often show discrepancy from the
experimental data, especially pronounced in the wall
zone. The discrepancies are also due to inlet flow non-
uniformity and packing structure in the wall zone, i.e. to
the assumed initial and boundary conditions.
Many authors use the term maldistribution factor Mf,
characterizing the deviation of gas flow from uniform
distribution. Most often Mf is defined as a mean
quadratic difference between the local and mean velo-
city. The maldistribution factor depends on packing
type, packing arrangement inside the column, etc. It is
disclosed by Darakchiev and Dodev [10,11] that increas-
ing packing height, the maldistribution factor of gas
phase tends to a certain minimal value, specific for the
packing type and depending on its local structure. The
height, after which Mf remains unchanged is called
‘penetration depth’. It measures packing ability toequalize the gas flow velocity profile. The theoretical
foreseeing of penetration depth is important for scaling-
up and design of industrial apparatuses.
The present paper aims at checking the possibility to
use the dispersion model for the description of gas flow
in a packed column by separating the effect of the
distributor device and from that of the packing layer.
The model parameters are also identified and thedistributor and packing gas redistribution capabilities
are compared.
2. Experimental
Part of the experimental data of Darakchiev and
Dodev [10,11], in the absence of liquid phase, have been
used. The experimental gas velocity profiles are obtained
* Corresponding author. Tel.: �/359-2-979-3281; fax: �/359-2-707-
523.
E-mail address: [email protected] (T. Petrova).
Chemical Engineering and Processing 42 (2003) 931�/937
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PII: S 0 2 5 5 - 2 7 0 1 ( 0 2 ) 0 0 1 8 4 - 8
for specific gas distribution device (GDD) [10] without
packing, and for the same distribution device with
structured ‘honey-comb’ packing at two different pack-
ing heights: 0.7 and 1.5 m [11]. The experimental set-up
is shown at Fig. 1, while packing characteristics are
given in Table 1. The adjacent packing layers are
ordered such the each layer is displaced with respect to
previous one by half a hexagonal ‘honey-comb’ cell; in
the same manner, gas flow attains radial distribution
due to gas stream split, as shown in Fig. 2.
The diameter of column (6) is 470 mm, that of the
outer cylinder (2) is 670 mm. The gas (air) is fed laterally
in the outer cylinder by inlet (1) with radius 0.1175 m.
The distribution lattices (3), made of perforated metal
plates, serve to equalize the flow velocity. The support-
ing grid (5) has large free area*/more than 90%.
Gas (air) flow velocity profile is measured by thermo-
anemometer (9) (Kurz 440 USA). It is directly connected
to computer (10) for data acquisition and processing.
High precision differential manometer ZAKLADY-
CYNOWICE-KATOVICE is used to measure pressure
drop Dp .
The experiments are carried out for three different
combinations of GDD and packing height, namely:
Configuration number 1: empty column (without
packing), h�/0, gas velocity, 2 m/s, related to entire
column cross-section and GDD with two identical
lattices with 25% free area, placed at 100 mm
distance.
Configuration number 2: same conditions, with layer
of ‘honey-comb’ packing h�/0.7 m high.Configuration number 3: same conditions, packing
layer h�/1.5 m high.
The measurements in an empty column (Configura-
tion number 1) have been done with thermoanemometry
probe placed at about 10 mm over the supporting grid in
order to eliminate the effect of air jets passing through
its orifices. Each value of gas velocity is averaged over
500 measurements. The number of points in this case is
N�/97.
In the runs with ‘honey-comb’ packing (Configura-tions number 2 and 3), the probe has been placed at the
upper end of the layer just over the orifices of each
packing element (see Fig. 1). The maximal local velocity
for each orifice has been registered. The number of
measuring points for Configurations 2 and 3 is N�/184.
The maldistribution factor Mf of gas flow is deter-
mined by the relation:
Mf �1
Wexp0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
N � 1
XN
i�1
(W expi �W
exp0 )2
vuut ; (1)
where N is the number of points at which dimensionless
velocity Wexp is calculated. The value W0exp is the
arithmetic mean over the local values Wiexp.
The maldistribution factor of Configuration number
1 (without packing) presents the ability of the distribut-
ing device GDD to equalize the gas flow. In cases
number 2 and 3, Mf summarize the equalizing ability of
GDD and packing, expressing also the effect of the layer
height. The values of measured pressure and maldistri-
bution factor for all configurations, determined byrelation Eq. (1), are given in Table 2.
3. Mathematical model
Liquid spreading in a packed column is described by a
dispersion model, proposed by Cihla and Schmidt [8].
The same model applied to gas flow, regarding cylind-
rical symmetry, takes the form:
@W
@Z�
1
r
@W
@r�
@2W
@r2: (2)
Assuming full reflection at the column wall (r�/1), the
boundary condition reads:
@W (1; Z)
@r�0: (3)
The initial condition is approximately simulated by gas
feeding from a disk with radius r1 equal to the inlet
Fig. 1. Scheme of the experimental set-up. (1) Gas inlet, (2) external
cylinder, (3) gas distribution lattice, (4) free space under the supporting
grid, (5) supporting grid, (6) column, (7) packing, (8) liquid surface, (9)
hot-wire anemometer, (10) computer.
T. Petrova et al. / Chemical Engineering and Processing 42 (2003) 931�/937932
radius of pipe 1 (Fig. 1):
g(r)�W (r; 0)�1
r21
for 05r5r1;
g(r)�W (r; 0)�0 for r1Br51: (4)
The dimensionless height Z�/Dh /R2 corresponds to the
Fourier number of the theory of heat conduction and
includes packing layer height h , as well as the coefficient
of gas radial spreading D . At a sufficiently large height,
the model reduces to:
W (r; Z)�1 for Z 0 �
which means that the velocity profile becomes uniform.It is easy to see that the model value of maldistribution
factor tends to zero at Z 0/�.
The analytical solution of Eqs. (2)�/(4) for a dimen-
sionless gas velocity W is:
W (r; Z)�1�X�n�1
AnJ0(qnr)exp(�q2nZ); (5)
where qn are roots of the characteristic equation,
J?0(qn)��J1(qn)�0; n�1; 2; . . . : (6)
The coefficients An are found from the initial conditions
by using the expression:
An�2
J20 (qn) g
1
0
g(r)J0(qnr)r dr: (7)
Let us present the considered column apparatus
consisting of GDD and packing as two separate
consecutive units, the first one containing GDD only,
and the second one*/packing only (Fig. 3a). Eachapparatus has some spreading capability, expressed by
the model parameter Z . According to the scheme on
Fig. 3, Z1 represents the redistribution ability of GDD
and Z2 that of the packing.
Gas distribution after the GDD, starting from a disk
with diameter equal to that of the inlet gas pipe, will be
for Z�/Z1:
f (r; Z1)�1�X�n�1
Adiskn J0(qnr)exp(�q2
nZ1);
where Adiskn �
2J1(qnr1)
r1qnJ20 (qn)
; A0�1;
(8)
This distribution stands for an initial distribution for the
packing layer, and the solution after this layer is given
by Eq. (5), but with new coefficients:
Table 1
Packing characterization
Packing d (mm) H (mm) s (mm) a (m2/m3) o (m3/m3)
‘Honey-comb’ 27 61 3.5 96 0.75
Fig. 2. Scheme of ‘honey-comb’ packing and layer arrangement.
Table 2
Experimental values of pressure drop and maldistribution factor
Configuration number Dp1 (Pa) Dp2 (Pa) Dp3 (Pa) Mf (%)
1 84.3 94.6 10.3 59.75
2 82.4 108.4 25.9 10.71
3 82.9 128.9 46.1 10.30
Fig. 3. Scheme of fictitious division of column to GDD and packing.
T. Petrova et al. / Chemical Engineering and Processing 42 (2003) 931�/937 933
Anewn �
2
J20 (qn)
� limd00
[ g1�d
0
J0(qnr)r dr�Adiskn exp(�q2
nZ1)
� g1�d
0
J20 (qnr)r dr�
X�m"n
Adiskn Adisk
m exp(�q2mZ1)
� g1�d
0
J0(qnr)J0(qmr)r dr] (9)
Taking into account the ortogonality of the Bessel
functions, as well as the characteristic Eq. (6),
g1
0
Jt(kr)Jt(lr)r dr
�1
2
�r2�
t2
k2
�J2
t (kr)�1
2r2[J?t(kr)]2; if k� l
0; if k" l
;
8<:
the formula Eq. (9) reduces to:
Anewn �Adisk
n exp(�q2nZ1): (10)
Then, the solution after the packing layer will be:
W (r; Z2)�1�X�n�1
Anewn J0(qnr)exp(�q2
nZ2)
�1�X�n�1
Adiskn J0(qnr)exp[�q2
n(Z1�Z2)] (11)
On the other hand, considering the GDD with the
packing as being one apparatus (Fig. 3b), the outlet gas
distribution can be calculated as follows:
W (r; Z3)�1�X�n�1
Adiskn J0(qnr)exp(�q2
nZ3): (12)
Consequently, it follows from Eqs. (11) and (12) that:
Z3�Z1�Z2 (13)
i.e. the redistribution capabilities of the GDD and the
packing can be considered as being independent and
additive.
The solution Eq. (5) of the dispersion model gives thepossibility to derive a theoretical expression of the
maldistribution factor. According to the discrete defini-
tion Eq. (1) of Mf, and using integral interpretation, a
following expression is obtained:
This formula is valid for all initial conditions. The
maldistribution factor does not depend on r , but on the
initial conditions and model parameter Z , only.
4. Results and discussion
The optimal values of model parameters Z for all
configurations are identified by non-linear optimization,
minimizing the residual variance s2 between theoretical
and experimental velocity:
s2�1
N � 1
XN
i�1
(W modeli �W
expi )2 (15)
For Configuration number 1 the optimal value found
is Z1�/0.09 and for Configuration number 2 the optimal
value found is Z2�/0.265. For Configuration number 3
it is found that after a certain value of Z , the residual
difference s2 does not further depend on this parameter.The limit value of Z in this case is Z3�/0.5 and
optimum is not found. Results of optimization for all
cases is given in Fig. 4.
Mf �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
F0gF0
0
�Gi � G0
G0
�2
dF
vuuut �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig1
0
2r[W (r; Z)�1]2dr
vuuut
�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX�n�m
A2nexp(�2q2
nZ) g1
0
2rJ21 (qnr)dr�
X�n
X�m"n
AnAmexp[�Z(q2n�q2
m)] g1
0
2rJ1(qnr)J1(qmr)dr
vuuut
�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX�n�1
A2nJ2
0 (qn)exp(�2q2nZ)
vuut (14)
T. Petrova et al. / Chemical Engineering and Processing 42 (2003) 931�/937934
For the optimal Z values at configuration 1 and 2 the
adequacy of the model has been proved using the Fisher
criteria.For Configuration number 2 the relative error be-
tween experimental and calculated velocity profiles is
shown in Fig. 5. The arithmetical mean relative error is
8.10% and mean quadratic deviation is 13.17%.
Fig. 6 illustrates experimental and theoretical maldis-
tribution factor. The theoretical curve Eq. (14) tends to
zero when increasing Z . The experimental minimal
value is determined by the term called ‘uniformity limit’,related to that discrete packing structure. In case of
‘honey-comb’ packing, its discreteness can be expressed
by the ratio:
s
d � s�0:12;
which obviously coincides with ‘uniformity limit’ from
Fig. 6.
The penetration depth of honey-comb packing with
the same GDD has been found experimentally to be
about 1.6 m [11]. As seen in Table 2 and in Fig. 6,
however, Mf, decreases insignificantly between valuesh�/0.7 and 1.5. Although weakly expressed, dispersion
minimum is still observed at h�/0.7 (Fig. 4) while in
Configuration 3 (h�/1.5) there is already no such
minimum, i.e. the penetration depth lies somewhere
between the two points.
Based on the results of Configurations 2 and 3 and
using the additivity*/Eq. (13), one can specify the
redistribution capability of GDD and the packing(Fig. 3):
�/ from Configuration 1 for GDD Z1�/Z1�/0.09;
�/ from Configuration 2 for packing Z2�/Z2�/Z1�/
0.175.
Hence, the packing redistribution coefficient will be:
D�(Z2)R2
h2
�0:0138 m:
The separation of the effect of GDD from that of the
packing enables one to assess the redistribution cap-
abilities of both GDD and packing. Gas redistribution
takes place at the expense of ‘pressure drop’ that
Fig. 5. Contour plot for relative error between model and measured
velocity profile for configuration number 2.
Fig. 6. Comparison between theoretical and experimental maldistri-
bution factor Mf for model parameter Z .
Fig. 4. Nonlinear optimization of parameter Z by means of compar-
ison between model and experiment at min (s2). Optimal Z1�/0.09;
Z2�/0.265; asymptotic Z3�/0.5.
T. Petrova et al. / Chemical Engineering and Processing 42 (2003) 931�/937 935
requires energy introduction. Hence, an appropriate
comparison parameter can be the ratio between the
redistribution capability, (model parameter Z ), and the
pressure drop, by means of which that redistributiontakes place.
Then, using the values Dp1, Dp2 and Dp3 for Config-
urations 1 and 2, given in Table 2, one gets (see Fig. 1),
that the mean value of pressure drop of GDD is
DpGDD�/93.5 Pa. For the packing a pressure drop is
Dppack�/25.9�/10.3�/15.6 Pa, respectively. Thus, the
ratio for GDD will be:
Z1
DpGDD
�0:09
93:5�9:63�10�4;
and that for the packing:
Z2 � Z1
Dppack
�0:175
15:6�1:12�10�2:
Hence, the packing has 11.6 times larger effectiveness of
gas redistribution, or it can provide the same effective-
ness but for values of Dp being 11.6 times smaller.
Another basis of comparison can be the so-called
‘equivalent height’ of GDD-heq, or packing with height
that provides the same distribution capability as that of
GDD. We get in the specific case that:
heq�Z1
R2
D�0:360 m:
Such a packing layer would have the following resis-
tance:
Dp�15:6
0:70:360�8 Pa:
while GDD would provide the same redistribution for
Dp�/93.5 Pa.
5. Conclusions
The results of this study show the possibility to use the
dispersion model in order to describe gas distribution in
packed columns. Separate estimation of the influence of
inlet device and packing layer on gas flow irregularity isalso done. It helps to a more expedient design of the
inlet devices.
In the presence of more layers of different packing,
one can formally divide the column into several sections.
In each case the redistribution capability of each section
is characterized by a single parameter Z , only. The link
between Mf and Z is expressed by Eq. (14), and it is one
and the same for the whole apparatus. The systematicdeviation between experiment and theory is observed at
attaining the ‘uniformity limit’ or the ‘penetration
depth’, and it is due to the packing discrete structure.
Note that in the presence of packing, one can trans-
form the parameter Z into ‘redistribution coefficient’
D�/ZR2/h , being dependent on the packing type. That
coefficient, however, has nothing in common with thesimilar spreading coefficient involved in the dispersion
model of a liquid phase, due to the totally different
mechanisms of redistribution of gas and liquid. The
liquid is a disperse phase, which occupies a very small
portion of the column volume. The coefficient D
characterizes the spreading of the liquid phase on the
surface of the packing*/as films, drops and jets with
different direction and length. In contrast to the liquid,gas is a continuous phase, which passes along the whole
free column volume through parallel channels with
different length and direction. The redistribution driving
force is in fact the difference between the local pressures
that tend to equalize.
In that case the dispersion model is formally applied,
since it does not involve pressure. We make analogy to a
diffusion process where the redistribution properties areexpressed by the coefficient D , only.
The additive contribution of inlet device and packing
layer, the determination of model parameters and
‘equivalent packing height’ of inlet device makes possi-
ble to optimize column design for minimal pressure
drop. It will be an object of future study.
Appendix A: Nomenclature
A0, An coefficients in Eq. (5)
a specific surface of packing (m2/m3)
D coefficient of fluid (gas or liquid) radial
spreading (m)
d inner circumference diameter of a packing
element (Fig. 2) (mm)F , F0 current surface element and total column
cross-section (m2)
Gi , G0 local and mean gas velocity (m/s)
J0, J1 zero and first order Bessel function of first
kind
H height of a packing element (Fig. 2) (mm)
h height of a packing layer (Fig. 1) (m)
Mf maldistribution factorN number of measuring and calculated
points
Dp1,2,3 pressure drop for GDD, GDD�/packing
and for packing only (Pa)
qn roots of Eq. (6)
R column radius (m)
r�/r ?/R dimensionless radial coordinate
r1 dimensionless radius of gas feeding diskr ? radial coordinate (m)
s wall thickness of a packing element (Fig.
2) (mm)
T. Petrova et al. / Chemical Engineering and Processing 42 (2003) 931�/937936
Wimodel,
Wiexp
calculated and measured dimensionless
velocity, i�/1, . . ., N
W�/G /G0 local dimensionless gas velocity
W0 mean dimensionless gas velocityZ�/Dh /R2 dimensionless axial coordinate
Z1, Z2, Z3 current values of Z in Fig. 3
Z1, Z2, Z3 optimal values of Z from nonlinear
optimization
Greek symbols
g function of initial condition
d limit in Eq. (9)o packing free volume (m3/m3)
s residual variance for comparison of the
model and experimental values
Subscripts
i , j , k , l , m ,
n
summation index
t order of Bessel function
Superscripts
disk coefficients for initial disk condition
model calculated model values
new coefficients of initial condition-existing
model solution
exp experimentally measured values
References
[1] K.E. Porter, M.C. Jones, A theoretical prediction of liquid
distribution in a packed column with wall effect, Trans. IchemE
41 (1963) 240�/247.
[2] M.A. Albraight, Packed tower distributors tested, Hydrocarbon
Processings 9 (1984) 173�/177.
[3] P.J. Hoek, J.A. Wesselingh, F.J. Zuiderweg, Small scale and large
scale liquid maldistribution in packed columns, Chem. Eng. Res.
Des. 64 (1986) 431�/449.
[4] R.M. Stikkelman, Gas and liquid maldistribution in packed
columns, Ph.D. thesis, Delft University, The Netherlands, 1989.
[5] F.J. Zuiderweg, J.G. Kunesh, D.W. King, A model for the
calculation of the effect on the efficiency of a packed column,
Trans. IchemE 71 (1993) 38�/44.
[6] M.F. Song, H. Yin, K.T. Chuang, K. Nandakumar, A stochastic
model for the simulation of the natural flow in random packed
columns, Can. J. Chem. Eng. 76 (1998) 183�/189.
[7] R.J. Kouri, J. Sohlo, Liquid and gas flow patterns in random
packings, Chem. Eng. J. 61 (1996) 95�/105.
[8] Z. Cihla, O. Schmidt, A study of the flow of liquid when freely
trickling over the packing in a cylindrical tower, Collect. Czech.
Chem. Commun. 22 (1957) 896�/907.
[9] V. Stanek, V. Kolar, Distribution of liquid over a random
packing. VII, Collect. Czech. Chem. Commun. 38 (1973) 1012�/
1026.
[10] R. Darakchiev, Ch. Dodev, Gas flow distribution in packed
columns, Chem. Eng. Proc. 41 (2002) 385�/393.
[11] Ch. Dodev, N. Kolev, R. Darakchiev, Gas flow distributor
for packed bed columns, Bulg. Chem. Commun. 31 (1999) 414�/
423.
T. Petrova et al. / Chemical Engineering and Processing 42 (2003) 931�/937 937