Gas Flow Distribution in Packed Columns 2002 Chemical Engineering and Processing Process Intensification

7/24/2019 Gas Flow Distribution in Packed Columns 2002 Chemical Engineering and Processing Process Intensification

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Chemical Engineering and Processing 41 (2002) 385393

Gas flow distribution in packed columnsRumen Darakchiev *, Chavdar Dodev

Institute of Chemical Engineering, Bulgarian Academy of Sciences, Acad. G. Bonche Street, Building 103, 1113Sofia, Bulgaria

Received 4 April 2001; received in revised form 6 July 2001; accepted 6 July 2001

Abstract

The distribution of gas flow over the cross section of packed columns depends on the manner of gas flow feeding, on packing

type, packing arrangement, and internal devices in the apparatus body. Various types of gas inlet devices are studied in order toestimate their ability for creation of uniform velocity profiles. The gas flow uniformity is characterized by the maldistribution

factor, Mf. Various methods for measurement of gas velocity and for calculation of this factor are presented in the paper. Data

for Mf from own experiments and from literature references are given along with detailed description of the particular

experimental conditions. It is shown that any packing has its own non-uniformity that cannot be improved. For a number of

packings, the limits of Mf are found. These values are used for determination of the penetration depth that ensures maximal

uniformity of the gas flow provided by the particular packing. 2002 Elsevier Science B.V. All rights reserved.

Keywords:Packed columns; Gas distributors; Gas flow distribution; Maldistribution factor; Depth of penetration

www.elsevier.com/locate/cep

1. Introduction

Many design methods for packed column appara-

tuses assume the existence of an ideal counter-current

flow of gas and liquid phases in the packing layer. In

reality, neither the liquid nor the gas follow the ideal

plug-flow pattern. This is the reason for process deteri-

oration and installation oversizing, which increases the

expenses for column manufacture and operation.

The liquid flow structure is improved by design of

new distributors that create flows with high initial

uniformity, by development of packings for redistribu-

tion of the liquid flow, and by internal devices for

elimination of the wall flow [1].Recently, the gas flow in packed columns is inten-

sively studied in order to improve its uniform distribu-

tion [213]. The research is focused mainly on the

influence of packing height or layer pressure drop on

the velocity profile [2,5 7,10 13]. The divergence of

flow from the ideal uniformity is characterized by the

maldistribution factor, Mf [2,5,6,8,11,13,14]. When in-

crease the packing height, this factor tends to a fixed

value. It is specific for each packing and depends on a

number of parameters, mainly on the packing type and

arrangement [2,9,12,13,15]. The bed height that ensures

maximal uniformity, possible to be attained with this

type of packing, is called penetration depth [8].

The uniformity of a gas flow depends on the manner

of its feeding in the apparatus and in the packing layer

[2,3,5,6]. The initial distribution is particularly impor-

tant in case of technological processes that do not need

beds higher than the penetration depth of the used

packing. It is valid especially for high efficient and low

pressure drop packings. Most of the modern packings

are of this type. The investigations are focused on the

rate of uniformity created by various types of gas inletsor specially designed distribution devices [2 5,13,16].

A number of papers have reported that the velocity

of gas flow does not affect significantly its distribution

over the cross-section [2,3,6,11,13,17]. However, inlets

with high initial irregularity can provide faster equaliza-

tion of the velocity profile at lower velocities.

2. Gas inlet devices

The gas flow in packed columns depends on the

manner of its feeding and on apparatus internals, in-

cluding the type of packing and its arrangement.

* Corresponding author. Tel.: +359-2-979-3227; fax: +359-2-

707523.E-mail address:[email protected] (R. Darakchiev).

0255-2701/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 2 5 5 - 2 7 0 1 ( 0 1 ) 0 0 1 5 1 - 9
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R. Darakchie, C. Dode/ Chemical Engineering and Processing41 (2002) 385393386

The manner of gas flow feeding is essential for the

initial irregularity that has to be improved in the ap-

paratus. The irregularity is due to the jet-type flow

before the packing layer. Larger difference between the

dynamic pressure in inlet pipe and in column body

creates more irregular flow. Of importance for flow

uniformity is the direction of feeding, as well as the

possibility to supply uniform circular feeding over the

cross-section ensured by local resistance or by otherappropriate methods of feeding towards solid wall or

water mirror.

The most common type of gas inlets are straight

inlets in lower part of the column below the supporting

grid that only cross the wall; and straight inlets that

penetrate at a certain distance L in the column body

towards the column axis (Fig. 1a). It is found that L

does not affect the initial gas distribution [2]. More

important is the distance between the inlet pipe and

packing layer. When it is smaller than (0.15/0.18)D,

back gas flows are formed in the packing near to the

inlet [7,18]. They increase the initial irregularity of the

gas flow.

Feeding devices of type slope inlet (Fig. 1b), which

are expected to direct the gas towards the liquid mirror,

do not represent a successful solution. The flow reflects

from the opposite column wall and creates similar

maldistribution as in above case. This behavior hasbeen registered in our study [6] as well as by other

authors [2].

When the inlet feeds the gas at the axis in downward

direction towards the lower column end or towards the

liquid mirror (bend inlet, Fig. 1c), significant improve-

ment of the initial irregularity is obtained [2,6].

The best initial uniformity is obtained with circular

inlets where the gas enters in radial or tangential direc-

tion through orifices of special shape [4,13]. A distribu-

Fig. 1. Various types of gas inlets (GI): (a) straight inlet (GI 1); (b) slope inlet (GI 2); (c) bend inlet (GI 3); (d) circular inlet (GI 4); 1 gas

distributing latticeVariant Ione lattice (GI 4-I, circular gas inlet with one lattice); Variant IItwo lattices (GI 4-II, circular gas inlet withtwo lattices); 2supporting grid; 3annular chamber; 4packed bed.


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R. Darakchie, C. Dode/ Chemical Engineering and Processing41 (2002) 385393 387

Fig. 2. Arrangement of measuring points.

methods [1,2,57,913]. Because of difficult measure-

ments in two-phase flow, the results are mainly in the

absence of liquid phase. However, it is known [9,11]

that under the loading point where packed columns

normally operate, the liquid phase is uniformly dis-

tributed and does not affect the gas flow distribution.

For this reason, single flow thermoanemometry results

are assumed to be reliable. When the velocity is over

the loading point, the friction forces between the liquidand gas flows increase, the liquid velocity profiles be-

come more irregular, liquid hold-up increases and the

free area of various packing zones changes. These fac-

tors influence the velocity profile and gas flow

distribution.

Thermoanemometry methods measure the velocity in

points over the packing layer. It can be supposed that

the registered profile does not coincide with the profile

in the layer [20]. Moreover, the last one or two packing

rows strongly influence over-the-layer profile [1720].

We have not found in the literature information about

the errors due to above factors. There are methods forvelocity measurements in the packing, but in this case

the gas flow is disturbed by the measuring probe, and

the results are not very correct. These methods are used

with random packings and are not applicable for struc-

tured packings. The velocity over the packing can also

be measured by pneumatic probes and LaserDoppler

anemometry. The last method is the most precise and

applicable in the presence of liquid flow, however the

necessary experimental equipment is very sophisticated.

A number of arrangements are used for local velocity

measurements. For example, the apparatus cross-sec-tion is divided in cells with equal surface, and the

velocity is measured in the cell center (Fig. 2a). Another

arrangement is division in coaxial ring sectors with

equal surface. The velocity is measured in different

points on the median of ring sectors (Fig. 2b). The use

of cells with equal surface leads to the most popular

expression for Mf [2]

Mf=1

nn

wiw0w0

2(3)

where n is the number of measuring points.

The gas velocity measurements are strongly depen-dent on the distance between the packing layer and

thermoanemometric sensor. The velocity profile be-

comes more uniform at larger distance and the corre-

sponding results supply rather qualitative picture,

different from the real situation. The place of measure-

ment over the cross-section is also of importance. The

walls of the packing elements divide the cross-section in

cells, and the velocity changes from zero over the wall

till certain maximal value inside the cell. However, in

usual practice the measuring points are arranged ac-

cording to a prefixed scheme not accounting for the

type and form of the layer beneath (Fig. 2a [12] and

tor of this type is studied in our paper [5]. The gas is fed

in an annular chamber (3) with distributing lattice (1) at

the inlet level (Fig. 1d). Two modifications of the gas

distributing lattice have been studied: (1) a single metal

plate with uniform perforation supplying free area of

25% (GI 4-I) and (2) the lattice is formed by two grids

as above placed at 100 mm distance (GI 4-II) (as shown

at the figure). The lattice resistance equalizes the flow

velocity and creates better flow distribution before its

entry in the packed bed.

3. Maldistribution factor

The gasflowing through the packing is not uniformly

distributed over the cross-section. This irregularity can

be estimated by the ratio of maximal and superficial

velocity wmax/w0. However, this is a point value, which

is not representative for the whole cross-section. It doesnot give a picture of the real distribution, especially

when the velocity is maximal over a small surface only.

The non-uniform velocity distribution over the cross-

section can be characterized by the Mf. It is a quantity

that integrates the surface and its corresponding veloc-

ity [8,14]

Mf=1

F0

F00

wiw0w0

2dF (1)

where F0 is total column cross-section; wi, the flow

velocity in point i; w0, mean velocity defined as

w0=1

F0

F00

widF (2)

In case of uniform distribution, the value of this

factor is zero.

There are other mathematical expressions for Mf that

result in determination of different values [13,19]. In all

cases, the expressions consider the flow distribution

over the entire cross-section.

The experimental determination of Mf is done by

measurements of local velocity in various points of the

cross-section. Most practiced are thermoanemometry


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Fig. 3. Variation of the Mf along the height of honeycomb packing.

Fig. 4. Sketch of the studied honeycomb packings: (a) ceramic block

packing with vertical walls and (b) ceramic block packing with

inclined walls.

Fig. 2b [2]). For this reason the corresponding results

are of accidental type.It is clear that there is no unified method for gas

velocity measurements. Additionally, different expres-

sions for determination of Mf are used and each pack-

ing is characterized by its own Mf. It can be concluded

that the value of Mf cannot be considered as a unique

criterion for comparison. The estimation of gas distri-

bution uniformity over the cross-section for each par-

ticular case has to take into account the type of

packing, height of the layer, the manner of gas feeding,

the method of velocity measurement, and the method

for calculation of Mf.

4. Uniformity limit

The uniformity limit is a certain sufficient rate of

uniform distribution that can be really attained by

particular packing. The determination of its value de-

pends upon the process, design efficiency, and cost/per-

formance penalties for the column [8]. In the best case,

this rate of flow regularity will be the same as the

internal irregularity created by the packing.

Inlet devices always create initial non-uniform distri-bution of the gas flow before its entrance in the packing

layer. This irregularity is reduced along the bed height

due to the pressure drop and to redistribution proper-

ties of the layer. Although this equalizing action, some

non-uniformity of the gas flow always exists. It is due

to the fact that the packing itself creates some irregular-

ity over the cross-section even in case of initial uniform

distribution of the gas flow. The rate of irregularity is

specific and depends mainly on the type of packing.

The packing height necessary to attain this minimal

irregularity depends on the initial distribution (type and

dimensions of inlet device), on column diameter (D), as

well as on packing pressure drop. In case of more

uniform initial distribution, lower height is necessary to

attain the internal irregularity specific for the particular

packing.

Some references [2,9,1113] report internal maldistri-

bution data for a number of packings used in practice.

Here we present our investigations in the field. Fig. 3

shows results for the Mf of two ceramic block packings

of honeycomb type with vertical walls. The main pack-

ing dimensions are given in Table 1. A sketch of the

packing is seen on Fig. 4a. The velocity profile has been

measured thermoanemometrically in the absence of liq-

uid phase.Fig. 3 represents the Mf along the packing height for

superficial velocity 2 and 2.5 m/s and for two types of

inlets. The D is 0.47 m. Curve 1 refers to honeycomb

Table 1

Main parameters of the ceramic honeycomb packing

Packing s1 (mm) () d (mm) 1 (% m3/m3)hp (mm) a (m

2/m3)

3.5210 134Honeycomb no. 1 66 0.72

96Honeycomb no. 2 0 6127 3.5 0.75

0.811273.54924 27Honeycombinclined walls


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packing no. 1 at both velocities and inlet 3 (GI 3). The

packing is 0.26 m high. The values of Mf (Eq. (3)) at

height levelH=0 m are obtained by measuring velocity

profile on two diameters (parallel and perpendicular to

the inlet axis). For determination of gas distribution

over the packing layer, maximal local velocities have

been measured immediately (25 mm) over the holes of

the packing elements (Fig. 4). The results for heights

0.26 and 1.2 m are obtained by measurements on twoperpendicular diameters, while for 0.8 and 1.6 m mea-

surements are taken at each hole of the cross-section.

The packed bed with height, H=1.6 m consists of two

layers1.2 m basic layer of honeycomb packing no. 1

and 0.4 m distribution layer for the liquid flow made of

honeycomb packing with inclined walls (Fig. 4b) [1]. A

tendency for approaching a value of Mf=0.17 is

clearly seen.

Gas distribution in honeycomb packing no. 2 is

studied using inlet GI 3 and two versions of inlet GI 4.

For inlet GI 4 the value of Mf tends to 0.11. With inlet

GI 4-II this value is already attained with 0.7 m pack-

ing. It is due to the higher pressure drop of two grids

that causes more regular flow distribution. The experi-

mental results for inlet GI 3 show higher values of Mf.In this case, the bed with height 1.6 m is composed

analogously to packing no. 1. Evidently, the specific

structure of redistribution layer increases flow non-uni-

formity. The results for height 1.2 m are obtained by

measurements on two perpendicular diameters only,

and some flow non-uniformity is possibly not regis-

tered. However, the tendency for approaching the char-

acteristic value of internal maldistribution for the

particular packing is clearly seen.

Fig. 5 shows the velocity profile with inlet GI 4-II for

gas velocity w=2 m/s and packing height 1.5 m.

The Mf in a rectangular industrial apparatus withcross-section 1.61 m2 is shown also in Fig. 3 (point +).

This apparatus is used for cooling offlue gases from a

steam boiler by direct contact with water. The inlet is of

straight type with cross-section 25% of apparatus cross-

section. The measurements are taken at mean flow

velocity 2.8 m/s with 1.2 m layer of honeycomb packing

no. 2 and redistribution layer of 0.4 m. The sample

points are arranged in rectangular grid pattern, and

maximal local velocity values are measured immediately

after the holes of the packing elements. There is a good

similarity with results for D=0.47 m. The correspond-ing velocity profile is illustrated in Fig. 6.

For packing no. 2, the value of Mf is about 0.10

0.11. The redistribution layer of packing no. 1 hinders

to determine the corresponding Mf value, expected to

be lower than 0.17.

Have been also investigated the influence of bed

pressure drop on Mf for packing made of expanded

metal sheets (Holpack) [7]. The sheets are arranged

horizontally in the column (Fig. 7). Each sheet is turned

at 90 clockwise with respect to the preceding sheet. A

sketch of this packing is given in Fig. 8. The main

dimensions and notifications are shown in Table 2. Thepacking with smaller size of holes and lower inter-sheet

distance, causing radial spreading, evidently creates

more uniform flow.

5. Maldistribution factor of various packings

Table 3 collects data for Mf of various packings

obtained by us and other authors.

As it was mentioned, the Mf of ceramic block honey-

comb packing is determined by measuring maximal

local velocity in each hole of the packing elements. We

Fig. 5. Three-dimensional picture of gas velocity profile over the bed

of honeycomb packing no. 2. Bed height 1.5 m, gas inlet GI 4-II,

superficial velocity w=2 m/s. Arrow represents the direction of

incoming gas flow.

Fig. 6. Three-dimensional picture of gas velocity profile over the

composed bed of honeycomb packing no. 2 with height 1.2 m and

redistribution layer with height 0.4 m. Arrow represents the directionof incoming gas flow. Industrial experiment.


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Fig. 7. Mf of Holpack (expanded sheet metal horizontal packing asdepending on packing) pressure drop.

w0 takes the mean value of the measured maximal local

velocities. The results for two honeycomb packings

with inlet GI 3 (Fig. 1c) are worse due to the presence

of redistribution layer.

The results for Holpack and for plastic Pall rings

[7,10] are obtained by velocity measurements on two

perpendicular diameters at 5080 mm over the bed.

They should not characterize the internal packing

maldistribution and should be used only for qualitativeestimation of flow non-uniformity because the sample

points cover too small part of the entire cross-section.

Stikkelman and Wesselingh [12] have reported the

values of Mf for Intallox saddles, Mellapak 250Y, and

plastic packing BX obtained by thermoanemometry in

presence of countercurrent liquid phase flow. The mea-

surements are taken over the liquid distributor in a

specially designed sampling device shown on Fig. 2a.

These values are not comparable to other data because

both the distributor and sampling device equalize addi-

tionally the gas velocity profile. Also, the Mf is calcu-

lated by the relation for standard deviation

Mf=1

nn

(wiw0)2 (4)

Kouri and Sohlo [11] have studied gas distribution in

beds of Pall rings and plastic packing BX with various

bed heights. They have reported results for single gas

phase and with liquid flow irrigated at various

flowrates. The gas velocity has been measured in a

special sampling device over the distributor. Beds of 1,

2, and 3 m height have been studied with gas flow

already uniformly distributed over the cross-sectionbefore its entry in the bed (inlet regular distribution

(RD)). The velocity has been measured following a

scheme similar to that of Fig. 2b, however the ring

sections do not have equal area and the expression for

Mf is

Mf=1

F0n

Fiwiw0

w0

2(5)

The above authors have not registered visible influ-

ence of gas velocity and packing height on the Mf.

However, it should be mentioned that in their experi-

ments the layers are thicker than the bed height neces-sary to equalize the velocity profile. Regarding Pall

rings of 50 mm, the value of Mf=0.085 is a mean value

for all three bed heights, various gas velocities and

irrigation densities. No influence of the liquid flow rate

on gas distribution is found.

Yuan and Li [2] have reported data for Mf of Mella-

pak 250Y obtained without liquid flow. Gas velocity is

measured by a scheme analogous to that on Fig. 2b.

The sample points are located on four diameters at the

median of ring sections with equal area. The distance

between the sensor and packing is not reported, thus

the influence of packing structure and eventual addi-

Fig. 8. Sketch of expanded metal sheet packing.

think this method gives a real picture of gas distribu-

tion in the packing layer, which is better than measure-

ments at regularly distributed points not following the

packing structure. Also, the thermoanemometric sensor

placed closer over the packing holes gives the real

velocity values in the upper part of the packing layer.

The Mf is calculated by the Eq. (3) where the velocity


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tional equalization could not be estimated. Addition-

ally, due to the low number of sample points (32 for a

column 1 m in diameter) it seems possible to omit

significant disturbances in the velocity profile.

6. Penetration depth

The penetration depth is defined as the packingheight at which the Mf value reaches the uniformity

limit [8].

Typical example for determination of penetration

depth is shown on Fig. 9, representing the influence of

packing height (Pall rings 16, 25, and 50 mm) on Mf

[13]. Here Mf values are calculated by a different

expression

Mf=

n

1

wi

w0

2(6)

For this reason they are rather different from the

other data given in this paper. Nevertheless, for a fixed

flow uniformity necessary for successful technological

operation of the apparatus, different penetration depths

are obtained, i.e., the desired gas flow uniformity is

provided by layers of different height depending on thepacking type.

The penetration depth of honeycomb packing no. 1

and 2 is about 1.6 m (Fig. 3). For Holpack packing no.

1 (Table 2) it is about 0.3 m.

It can be seen also from Fig. 3 that the penetration

depth is strongly affected by the inlet device. For

honeycomb packing no. 2 with inlet GI 4-II, the unifor-

Table 2

Main parameters of a packing element of expanded metal sheet

h (mm)Symbol (Fig. 8) s (mm)Packing l (mm) (m2/m2) (mm)h1 (mm)

7.0 22.2 3.0 0.8 0.87 Holpack 1 50

17.8 90.0 11.0 3.0Holpack 2 0.900

Holpack, expanded sheet metal horizontal packing.

Table 3

Mf for various types of packings

H (m) GI deviceSymbol (Fig. 3) Velocity (m/s)Packing Mf SourceD (m)

Own data0.1712.0GI 31.60.47Honeycomb 1ceramic

2.5GI 31.6 0.1670.47Honeycomb 1ceramic Own data

1.6 GI 3 2.0Honeycomb 2ceramic 0.136 Own data0.47

Honeycomb 2ceramic 0.47 1.6 GI 3 2.5 0.171 Own data

Honeycomb 2ceramic 0.47 1.5 GI 4-I 2.0 0.110 Own data

0.47 Own data0.103Honeycomb 2ceramic 2.0 GI 4-II1.5

Holpack 0.47 0.27 1.8GI 1 Darakchiev [7]0.180

Pall rings 50 mmplastic Dodev and0.1502.3GI 10.80.47

Darakchiev [10]

0.5Pall rings 50 mmplastic 1.0/3.0 RD 1.55/2.0 0.085a Kouri and Sohlo

[11]

Pall rings 50 mmplastic 0.5 3.0 RD 1.67 0.077 Kouri and Sohlo

[11]

Pall rings 25 mmplastic 0.058 Kouri and SohloRD 1.673.00.5

[11]

Stikkelman and0.5 0.035Intalox saddles 25 mmceramic 3.0GI 11.0Wesselingh [12]

0.032 Stikkelman and3.0GI 11.00.5Intalox saddles 25 mmplastic

Wesselingh [12]

3.0 0.048 Stikkelman andMellapak 250Ymetal 0.5 0.8 GI 1

Wesselingh [12]

0.8 GI 1 2.85 0.040 Yuan and Li [2]Mellapak 250Ymetal 1.0

3.0 RDPlastic BX 1.67 0.045 Kouri and Sohlo0.5

[11]

0.8 GI 1 3.0 0.014 Stikkelman andPlastic BX 0.5

Wesselingh [12]

Own data0.1562.8Honeycomb 2ceramic (industrial GI 11.61.441.12+

experiment)

a Mean value of Mf for various bed heights, gas, and liquid velocities.


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Fig. 9. Determination of penetration depth by the Mf for various Pall

rings.

In this study the Mf is determined for two ceramic

honeycomb packings, horizontal packing of expanded

metal sheets, and plastic Pall rings. The experiments

with honeycomb packing in industrial scale column

have confirmed the laboratory results. The results of

other authors for Intallox saddles, Pall rings, Mella-

pak 250 Y and Packing BX are presented along with

the experimental conditions and the manner of data

treatment. It gives the possibility for better data inter-pretation aiming at selection of most reliable results.

It is also shown that the minimal value of Mf can be

used as characteristic parameter for the internal pack-

ing irregularity.

The results obtained serve also for determination of

the penetration depth that ensures maximal gas flow

uniformity for the particular conditions.

Appendix A. Nomenclature

a specific surface of the packing (m2/m3)

diameter of the circumference inscribed in thed

packing hexagon hole (mm)

D column diameter (m)

F0 column cross-section (m2)

Fi cross-section of the ith measuring element

(m2)

packed bed height (m)H

h distance between expanded sheets (mm)

hP height of a packing element (mm)

h1 thickness of the expanded sheets (m)l length of holes in the expanded sheets (mm)

L length of insertion of the gas inlet pipe into

the column (mm)

Mf maldistribution factor

n number of measuring points

pressure drop (Pa)p

s width of holes in the expanded sheets (mm)

wall thickness of the packing element (mm)s1w gas flow superficial velocity (m/s)

wi local value of the gas flow velocity (m/s)

wmax maximal value of the gas flow velocity (m/s)

mean value of the velocity in a given cross-w0section (m/s)

local coordinates (mm)x, y

Greek symbols

angle of inclination of the wall of the packing

element ()

thickness of sheet metal (mm)

free cross-section of the expanded sheet metal

packing (m2/m2)

free volume of the Honeycomb packing (%1m3/m3)

regular distributionRD

mity is attained at penetration depth almost twice

lower than that using other inlets (GI 4-I and GI 3).

The results of Yuan and Li [2] are analogous. The

penetration depth of packing Mellapak 250 Y de-

pends strongly on the inlet. For inlet types GI 1/GI 3the uniformity is obtained at penetration depth 600

800 mm, while with circular inlet the penetration

depth is 300400 mm. If the particular technology

does not require thicker bed for heat and mass trans-

fer processes, apparatuses with significantly lower di-

mensions and capital costs can be designed.

7. Conclusion

The comparison of different inlet devices shows

that circular inlets create the most uniform distribu-tion of the gas flow before its feeding in the packing

layer. The use of such type of distributors ensures an

efficient process at reduced packing height and lower

capital costs.

It is shown that there is not a unified method for

experimental determination of the gas flow distribu-

tion over the cross-section of the packed columns.

The same is valid for the Mf that characterizes the

flow uniformity. For this reason, analyzing the avail-

able results, it is necessary to take into account the

particular experimental conditions and calculation ex-

pressions used.


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