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: tanchotp@netscape.net (T. Petrova).

Chemical Engineering and Processing 42 (2003) 931�/937

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0255-2701/02/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved.

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

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