Fluid Dynamics in Countercurrent Packed Columnsssu.ac.ir/cms/fileadmin/user_upload/Daneshkadaha/...4...

4 Fluid dynamics in countercurrent packed columns

The flow patterns for the liquid and gas streams in packed separation columns dependon the geometry of the channels formed by the arrangement and size of the packing in thebed. Other relevant factors are the texture of the packing, the velocities of the counter-current streams, and the physical properties of the two phases under operating conditions.The very complexity of these parameters and their interrelationships renders an accuratetheoretical description of the flow patterns almost impossible. Consequently, simplifyingassumptions must be made, and their validity must be confirmed by experiment.

4.1 Fluid-dynamics model

It can be assumed that the liquid phase flows in the form of a film in channels of thesame free volume as those in the bed of packing under consideration. The geometry of thechannels can be described by the effective void fraction in the bed and by the ratio a of thesurface of the packing to the volume of the bed. From the aspect of the hydrodynamics, theeffective void fraction in a packed column is equivalent to the number of vertical flow chan-nels in which a liquid of density QL and dynamic viscosity r\L flows in the form of a film ofthickness so at an average velocity uL countercurrent to a stream of gas or vapour with anaverage velocity uv (cf. Fig. 4.1).

Packingsurface

Frictional force

Force of gravity

Interfacialarea

o

1

Packing Liquid holdup Vapour or gas

Fig. 4.1. Equilibrium of forces during two-phase flow within a channel of a packed column

Packed Towers in Processing and Environmental Technology. Reinhard BilletCopyright © 1995 VCH Verlagsgesellschaft mbH, WeinheimISBN: 3-527-28616-0

SOFTbank E-Book Center Tehran, Phone: 66403879,66493070 For Educational Use.

74 4 Fluid dynamics in counter cur rent packed columns

If steady-state conditions exist and inertia forces are neglected, it can be assumed thatthe force of gravity is in equilibrium with the shear forces at any point in a given coaxiallayer of thickness s within the liquid film and that the frictional force exerted by the vapourof density Qy acts at the surface of the film. Thus, in any given layer of thickness 5 in a liquidfilm of total thickness so, the equilibrium of forces can be described by

duL>s

If the downward liquid velocity uLs is expressed as a function of the density QL and thedynamic viscosity r\L, Eqn (4-1) can be integrated to give

uLtS = Ks-— (4-2)

Differentiation then yields the velocity gradient in the film, i.e.

ds S° r\L

Multiplication by i}L yields the shear stress in the liquid at the gas interface. In steady-state operation, the shear stress in the liquid is in equilibrium with that in the vapour of den-sity QV and velocity uv, i.e.

T\LK - QLgso = - | ~Y~ Qv (4-4)

The integration constant K can therefore be obtained. The equations thus derived arevalid for vertical flow in both phases and laminar flow in the liquid film. Examples of depar-tures from this model are oblique channels arising from the geometry of the packing, turbu-lent flow, and the texture of the packing. A practicable means of allowing for them is tointroduce a function for the flow or resistance factor §. Experimental studies to verify theequations in the model have confirmed that this procedure is permissible.

Rearranging Eqn (4-4) to give K and substituting this value in Eqn (4-2) yields the fol-lowing equation for the local liquid velocity uLs at any point in a layer of thickness s in theliquid film:

) s ( 4 . 5 )r\L 2 /

The theoretical average flow velocity uL for a liquid film of thickness so is defined by

uL = — I uLtS ds (4-6)so


4.1 Fluid-dynamics model 75

The solution to this equation is the mean local velocity uL of a liquid that descends coun-tercurrent to an ascending stream of vapour flowing at a mean local velocity uv within a bedof packing, i. e.

y g SOQL - ^ l u v Qvj (4-7)

If the liquid and vapour loads uL and uv are given, the mean velocities uL and uv dependon the packing geometry and the liquid holdup hL, which is defined by

where VL is the liquid volume and Vc is the column volume under operating conditions.

Experience has shown that hL is influenced by the following factors:- the texture of the packing,- the ratio a of the surface area of the packing to the volume of the bed, and- the void fraction 8.

The ratio a is defined by

where AP is the total area of the surface presented by the bed of packing and Vc is thecolumn volume.

The void fraction is given by

8 - Vc~Vp (4_io)

where VP is the volume occupied by the packing itself, i.e. without voids.

The following relationship exists between a and e:

s = l - a ^ - (4-11)AP

If the liquid holdup hL is known and the packing is uniformly wetted, the theoretical totalfilm thickness So is given by

so = — (4-12)a

The following relationship exists between the mean liquid velocity uL and the volumetricflow rate, which is normally given in column design problems:


76 4 Fluid dynamics in countercurrent packed columns

(4-13)so a

The corresponding relationship exists between the mean vapour velocity uv and theume flow rate of vapour uv, i.e.

pgvolume flow rate of vapour uv, i.e

Substituting the terms on the right-hand sides of Eqns (4-12), (4-13), and (4-14) for so,uL, and Uy respectively in Eqn (4-7) gives

a\LuL = V (f QL - } hL{;_htf «}*,) (4-15)

This equation describes flow in terms of the column load, the physical properties of theliquid-vapour system, and the packing characteristics. It is thus valid for all industrial pack-ing. The exponent n is exactly n — V3 for laminar flow in vertical channels.

The extensive experimental work required to verify the relationship is described in Chap-ter 3. The interrelationships between hL, uVy and uL, as described by Eqn (4-15), is shownqualitatively in Fig. 4.2, in which the subscript S refers to the loading point and the subscriptFl to the flood point. It can be seen from this diagram that, if the liquid load uL is kept con-stant, an increase in the gas velocity Uy does not significantly affect the liquid holdup hL untilthe loading point uv>s is reached at the surface of the film, i.e. at s = s0. In other words, ifthe liquid load uL is constant, the liquid holdup hL is practically equal to hLjS at loadsuv < uvs. At the point s = s0, the flow velocity is theoretically zero, and the liquid com-mences to hold up in the bed of packing. Afterwards, in the Uy > uvs range, the liquidholdup hL increases with the gas velocity until it attains a maximum hL>m at the flood point,when the gas velocity is uV}F[. The experiments demonstrated that it does not undergo anyfurther change once this point has been reached, even if uL > uLjF\ (cf. lower diagram in Fig.4.2) and if uv > UyFl. They also revealed that the vapour velocity at the loading point uv>s isabout 70 % of that at the flood point uFi.

Theoretically, these relationships are valid between uVjS, uL;S and hL}S', and between uv>Fh

uL>Fi and hL)Ft. They follow logically from the physical model, and the mathematical descrip-tion applies to the entire loading range.

4.2 Resistance to flow

The resistance factor § allows for the geometry of random packing or of the flow chan-nels in systematically stacked beds. It also allows for the transition from laminar to turbulentliquid flow and thus depends on the vapour and liquid loads.

The application of Eqn (4-15) to evaluate experimental results has revealed that § is ana-logous to the friction factor in pipe flow in that it is a function of Reynolds number Rey forthe gas phase. This is evident from Fig. 4.3, which shows the correlation between gas andliquid flow in both circular pipes and packed beds. It can be seen that the slope is practicallythe same in all cases.


4.2 Res is tan ce to flo w 11

ZDTD

ID

Gos or vapour load uv

-oao

Fig. 4.2. Qualitative relationship betweenthe liquid holdup and the phase loads in apacked column

Liquid holdup hL

The resistance factor § also depends on the liquid holdup, which is defined as the volumeof stationary liquid that exists in the form of a film on the surface of the packing or is pres-ent in voids, dead spots, and phase boundaries in the bed of packing within a two-phasecountercurrent column during the period in which the liquid phase descends at a constantrate onto the surface of the bed. There are two components of liquid holdup: a static hLst

and a dynamic hL>dyn. Thus

hL>dyn

(4-16)

The dynamic component flows downwards, and the static remains within the bed. As theliquid load increases, the difference between the two components becomes progressivelygreater, until the stage is reached when the static can be regarded as nonexistent. In mostindustrial mass transfer processes, the dynamic component preponderates by several ordersof magnitude.

An expression for hL that is valid over the entire loading range can be obtained by re-arranging Eqn (4-15). Thus



Gas Reynolds number ReV(Packing

&o3.

CO

ooCD

fe

00CD

Cd

G a s R e y n o l d s n u m b e r R e j

Fig. 4.3. Qualitative log-log relationship between the resistance factor and the gas Reynoldsnumber in the channels of a packed column with the liquid Reynolds number as parameter. Thecorresponding relationship for flow in tubes is presented as a comparison.

hL =g „

3 WiJ 4

a

/ IL) 2

(4-17)

The basic assumption in this case is that the flow of liquid over the packing is laminar. Itdiffers from flow in the vertical channels of the model in that it continually changes direc-tion and is interrupted at boundaries. Another point in which the theory departs from realityis that laminar flow in the film applies only at comparatively low liquid loads. The criticalReynolds number, above which flow becomes turbulent, has been derived from numerousmeasurements. According to definition,

uL ReL,cr = 10 (4-18)

For laminar liquid flow in the vertical channels of the model, the exponent n in Eqn(4-18) is exactly n = V3; and a figure of n = 2/3 has been determined for flow in system-atically arranged beds of packing. The value changes at higher loads, and the following rela-tionship for turbulent film flow was established by experiment on about 50 different types ofpacking dumped at random:

n = \ —, aN\\ lh

(4-19)

where h is the height and d the outer diameter of an individual element of packing, N is thenumber of elements per cubic metre, and a is the surface area per unit volume in m2/m3.


4.2 Resistance to flow 79

Although Eqn (4-17) can be solved only by iteration, its great advantage is that it is ana-logous to the equation for two-phase countercurrent flow in vertical tubes. The curves pre-sented in Fig. 4.4 convincingly verify this analogy. The first step in their compilation was torearrange the equation to solve for §, i.e.

uLhL(z-hLf\ 1

UyQv(4-20)

Values of | thus calculated for flow in a bed of metallic Bialecki ring packing were plot-ted against the corresponding Reynolds numbers with the liquid load uL as parameter. Thevalues for the Reynolds numbers were derived from the following relationship:

100

70

5040

30

20

10

S 5£ 4

Pipe flow Bialecki rings

COCD

ct:

\V\ \

V

u

_

\\ \

\ \^\ S

\ \

\

VV\

-A-

\ \^V \ \\ \ \\ \\ \\

L = 80m3/m2h"

- - -

41

\ \\

V

\\

\ \\>\\\\

V' \0 "10

V\ \\.

\

-—\

\

\\

XY2.8

\

\

yX\!_K\\\

\ \ V

\ s\V

\ \A s

\

w\v

- 2

A

N\y \_ ^ _ .

y \

\ ^\\

N Av V

\

5-mn

\

\ Ny. \

\ S

\ \

\>

\ y

NvNV\\

i Biale

- Sysl

\ \ \^ \ \\ \ \\\v\\\\ \ \s \< \\ \\ >s \

AS.\ Av > vy

:ki ri g tQlN =78323 1/m3

a =331 mz/m3

e =0.9381em: Air/Water

1 bar; 298 K

VV \\3^

\v \\\

= 80m3/m2h

L

\\\

\

1-

\\\

40

\\

r "

r-- - • 2 . 8

J

0.7

0.50.4

0.3

0.2

0 130 50 100 300 500 1000 3000 5000 10000

Gas Reynolds number Rev

Fig. 4.4. Resistance factor as a function of the gas Reynolds number for a packed column and atube with the liquid load as parameter. Valid for an air/water system at STP.



Uyid-lSp) QyKey — (4-21)

The corresponding curves for two-phase pipe flow, calculated from the standard equationscited in the literature, have been included in the diagram. The fact that the two sets ofcurves are largely parallel in the range of laminar gas flow provides striking evidence of theanalogy and confirms the validity of the model based on flow in vertical channels in packedcolumns. The relative positions of the two sets of curves show that the resistance factors arehigher in the bed of packing. However, this is only to be expected, because the pressuredrop for gas flow in beds of packing is considerably greater than that in vertical smoothpipes.

Evaluation of the results obtained on all the packings investigated led to the followingrelationship:

= C Rey2 ReL° (4-22)

in which C is a constant that is characteristic for individual types of packing and must bedetermined experimentally. Numerical values for C at the loading and flood points forvarious types of packing are presented in Table 4.1.

A relationship that is more useful in practice for the determination of § at the loadingand flood points in specific separation tasks is

C2 rMlL(4-23)

where r|L/ry is the viscosity ratio for the two phases and W is the well-known flow parameterdescribed by Eqn (1-3), i.e.

Table 4.1. Data for determining the loading and flood points in random beds of packing

Dumped Packing Material Sizemm

N1/m3

8

m3/m3CFI

Pall ring

Ralu ring

Metal

Plastic

Ceramic

Plastic

503525

503525

50

5050 hydr.

62421951753900

67651700052300

6215

57705720

112.6139.4223.5

111.1151.1225.0

116.5

95.295.2

0.9510.9650.954

0.9190.9060.887

0.783

0.9380.939

2.7252.6292.627

2.8162.6542.696

2.846

2.8432.843

1.5801.6792.083

1.7571.7422.064

1.913

1.8121.812


4.2 Resistance to flow 81

Table 4.1. (continued)

Dumped Packing

NOR PAC ring

Hiflow ring

Glitsch ring

Glitsch CMR ring

TOP-Pak ring

Raschig ring

VSP ring

ENVIPAC ring

Bialecki ring

Tellerette

Hackette

Raflux ring

DINPAC

Material

Plastic

Metal

Plastic

Ceramic

Metal

Metal

Alu

Ceramic

Metal

Plastic

Metal

Plastic

Plastic

Plastic

Plastic

Sizemm

5035256

2510

5025

5050 hydr.50 S25

503820

30 PmK30 P

1.5"1.5" T1.0"0.5"

50

5025

5025

806032

503525

25

45

15

7047

N1/m3

7330174505000048920

500040790

681568906050

46100

512013241

121314

2920031100

6074463547

158467560811

6871

599047700

784133434

20006800

53000

62781820048533

37037

12000

193522

976328168

am2 /m3

86.8141.8202.0197.9

92.3202.9

117.1118.482.0

194.5

89.7111.8286.2

180.5164.0

174.9188.0232.5356.0

105.5

95.0190.0

104.6199.6

60.098.4

138.9

121.0155.0210.0

190.0

139.5

307.9

110.7131.2

E

m3/m3

0.9470.9440.9530.920

0.9770.962

0.9250.9250.9420.918

0.8090.7880.758

0.9750.959

0.9740.9720.9710.952

0.956

0.8300.680

0.9800.975

0.9550.9610.936

0.9660.9670.956

0.930

0.928

0.894

0.9380.923

Cs

2.9593.1793.2772.865

2.7022.918

2.8942.8942.8662.841

2.8192.8402.875

2.6942.564

2.6972.7902.7032.644

2.528

2.4822.454

2.8062.755

2.8462.9872.944

2.9162.7532.521

2.913

2.832

2.825

2.9702.929

cFI

1.7862.2422.4722.083.

1.6262.177

1.8711.8711.7021.98?

1.6941.9302.410

1.9001.760

1.8411.8701.9962.178

1.579

1.5741.899

1.6891.970

1.5221.8642.012

1.8961.8851.856

2.132

1.966

2.400

1.9121.991

hydr. = made hydrophilic



( 4 - 2 4 )

The one and only constant C in Eqn (4-23) describes the shape and surface of thepacking. The viscosity m and load nL exponents are practically independent of the type ofpacking, and the form of Eqn (4-23) has been confirmed for all the types of packing investi-gated.

4.3 Loading conditions

If inertia forces are neglected and steady-state conditions exist in the main loading range,it is assumed for the purpose of drawing up the model that the shear forces and the force ofgravity are in equilibrium in a layer of any given thickness between 5 = 0 and s = s0 withinthe liquid film. It is also assumed that the frictional force exerted by vapour of density QV

and velocity uv acts at the surface of the liquid film (cf. Fig. 4.1).At a given liquid load wL, the liquid commences to hold up when its velocity at the inter-

face with the vapour becomes zero, i.e. when ML,S — 0 at s = s0'.

UL>S = 0 (ifs = so) (4-25)

The corresponding vapour velocity is the upper limit for the absolutely stable hydro-dynamic range defined by

V t V ^ V I F <4-26)The practical application of Eqn (4-26) necessitates a knowledge of the liquid holdup

hL = hLS at the loading point and the associated resistance factor (cf. Fig. 4.2).Substituting uv>s, as defined by Eqn (4-26), for uvin Eqn (4-15) yields the following theo-

retical relationship for the liquid holdup at the loading point:

Solving for hL>s and inserting in Eqn (4-26) yields the following equation for the vapourvelocity at the loading point:

—i/12

UL

8 QL

12 rwL n /

8 QL I V Qv

As has already been mentioned, the resistance factor §5 allows for the shape of the flowchannels and is therefore dependent on the packing geometry and on the vapour and liquidloads; and it also allows for the transition from laminar flow in the liquid phase. It can beexpressed as a function of the flow rate LIV, density QV/QL, and viscosity r\L/r\v ratios bycombining Eqns (4-23) and (1-3). Thus,


ciQv

QL

4.3 Loading conditions 83

- (4-29)

In this equation, the load point constant C5, which is specific for the packing, and theexponent ns must be determined empirically. The equation has been confirmed by experi-ment and is presented graphically in Fig. 4.5. It can be seen from this diagram that a discon-tinuity, which correlates with the phase inversion, divides the function into two linear sec-tions. The large majority of mass-transfer processes take place within the section representedby the lower values of flow parameter on the axis of abscissae. The values at the further end

CO

on

Phase inversion

Continuous phase :gas )Disperse phase : liquid S

0.4

Flow porameter I ffV I pL W

Fig. 4.5. Qualitative log-log relationship between the resistance parameter and the modified flowparameter

of the axis correspond to very high liquid loads, and the high slope of the linear relationshipin this range indicates an increase in the resistance factor 5s and a proportionally greaterpressure drop in the gas phase. It can thus be concluded that the intersect of the two straightlines, i.e. the discontinuity, is the point at which phase inversion occurs. In other words, thisis the point at which the gas becomes the disperse phase; and the liquid, the continuous. Itis given by

(4-30)

The associated values for the liquid load correspond to phase inversion at50-100 m3/m2h. Data on the higher loading range are not yet available for all the packingsinvestigated. An example of a known relationship is presented in Fig. 4.6.

Accordingly, the values for the constant Cs and the exponent ns in Eqn (4-29) depend onthe loading range. It has been found that the following empirically determined numericalvalues of ns apply in common for all types of packing in the loading ranges concerned:



£ W i*-> 0.4-*„, =-0.723

(4-31)

(4-32)

35-mm Bialecki ring, metal, System: Air/Water, 1 bar, 293 K

*-**

<s

0.1 0.2 0.3 0.4 0.50.6 0.8 1

Flow parameter | | / ^ ( A N

2 3 4 5 6

Fig. 4.6. Resistance parameter for a bed of metal packing

Numerical values of the constant Cs for specific types of packing are listed in Table 4.1(randomly dumped) and Table 4.2 (systematically stacked).

4.4 Flooding conditions

The flood point in a bed of packing can be described physically by empirical curves inwhich the liquid holdup hL has been plotted against the gas load uv with the liquid load uL

as parameter; or plotted against uL with the gas load uv as parameter. Curves of this natureare presented qualitatively in Fig. 4.2. In the upper diagram, the curves are vertical at theflood point, and can thus be described by the following equations:

—— = 0; uv^ uv>Fl (tor uv = uv>Fl)

-JJ^- = 0; uL -* uLtFl (for uL = uL>Fl)

(4-33)

(4-34)

Applying these equations to Eqn (4-15) for n = % leads to the following theoreticalflood point correlations:


Table 4.2. Data forpacking

4.4 Flooding conditions 85

determining the loading and flood points in geometrically arranged beds of

Arranged packing Material Sizemm

N1/m3

am2/m3

8

m3/m3CFI

Pall ring

Bialecki ring

Ralu pack

Mellapak

Gempak

Impulse packing

Montz packing

Euroform

for the gas load uV}Fi,

Ceramic

Metal

Metal

Metal

Metal

Metal

Ceramic

Metal

Plastic

Plastic

50 7502

35 20736

YC-250

250 Y

A2T-304

250

100

B1-200Bl-300

Cl-200C2-200

PN-110

i / T T (e-hL>Fl)2

155.2

176.6

250.0

250.0

202.0

250.0

91.4

200.0300.0

200.0200.0

110.0

\lhLZL

0.754

0.945

0.945

0.970

0.977

0.975

0.838

0.9790.930

0.9540.900

0.936

3.793

3.178

3.157

2.986

2.610

2.664

3.1163.098

2.653

3.075

3.024

2.558

2.464

2.099

1.996

1.655

2.3392.464

1.973

1.975

and for the liquid load,

1 Q L (4-36)

For a given liquid-gas mass flow ratio L/V, these equations are related as follows:

QvQvuL,Fl — Tv UV,Fl

QL V(4-37)

It follows that

I - 8) = ~y <T 8 UVtFig V QL QL

(4-38)

This is a purely theoretical relationship that shows how the liquid holdup at the floodpoint depends on the packing geometry, the physical properties of the system, and the gas or



vapour load. Only one of the solutions of this fourth-order equation is of physical signifi-cance: in the first place, the liquid holdup cannot possibly exceed the effective void fraction;and, in the second place, the right-hand side of the equation must be positive. This impliesthat the liquid holdup at the flood point must not be less than e/3 and must therefore lieuniquely within the following limits:

E/3 < hLtFl < e (4-39)

The holdup at the flood point hLFh as defined by Eqn (4-38), is theoretical, and itremains to be seen whether its existence can be verified in practice. If Eqn (4-38) is writtenin the following simplified form, the values determined for hLF[ will be of the same orderfor water (subscript W) as the reference liquid:

hLtFl = 0.3741 8 - ^ •SE] (for 0.1 <uL< 200 m3/m2h) (4-40)\ QL r\W /TI L >1-10" 4 kg/ms

Values determined by flood point measurements are usually lower, because the actualflood point is generally not attained owing to column instability and entrainment of packingelements in loose, dumped beds.

The resistance factor §F/ at the flood point, as defined by Eqn (4-35), is given by

(4-41)

Qv r\L

QL \ r\v

Except that the exponent for the viscosity ratio is 0.2 instead of 0.4, the value at theloading point, the form of this equation is identical to that of Eqn (4-29). The constant CFi

and the exponent nFh in common with Cs and ns in Eqn (4-9), must be determined empiri-cally. The values thus obtained for nFt apply to all types of packing. In analogy to the numer-ical values quoted for ns in Eqns (4-31) and (4-32), they depend on whether the loadingrange is above or below the phase inversion, i.e.

•y y — < 0.4 -> nFl = -0.194 (4-42)

L A Ov~77 \\ > °-4 -> nFi = -0.708 (4-43)V V QL

If the column load is higher than that at the phase inversion, the continuous gas phasebecomes the dispersed phase and bubbles upwards through the downward stream of liquid,which then becomes the continuous phase.

In analogy with Cs, Cm is a constant that depends on the shape and texture of the pack-ing and must be determined empirically. Values thus obtained in numerous loading tests withair/water absorption and rectification systems are listed in Tables 4.1 and 4.2. It can be seen


4.5 Relationship between boundary loads 87

that they lie between CFl = 1.5 and CFt = 3.0 in the range of liquid loads between uL = 5m3/m2h and uL = 80 m3/m2h. The theoretical limit is expressed by

(hL,Fi)th -> hLiFl = j e (for uL^ 0-^:^0) (4-44)

This relationship can be derived from the preceding equations. Thus, at extremely lowliquid loads, i.e. as uL or LIV tends to zero, the liquid holdup hLFi may account for aboutone-third of the effective void fraction 8 in the bed of packing. This fact leads to the follow-ing relationship for the theoretical maximum vapour load at the flood point uv>Fi,max'-

Uy,Fl,max = uVFl (for hLtFl = I-) = -J- \ -f- \ — \[^-3 ? y £>FI y a y Qv

\[ (4-45)a y Qv

This equation contains the well-known packing factor ale3 formerly resorted to in theliterature, e. g. by Sherwood, for flood point correlations.

Theoretically, the maximum liquid holdup at the flood point hLFi corresponds to the voidfraction e and would be reached at extremely high liquid velocities uL ^^> 0, in which casethe gas velocity would be negligibly small, i. e. uv —> 0 or LIV'—» a>. Thus,

(hL,Fi)th -» hL)Fi = e (for uL » 0; - y -> oo) (4-46)

Inserting this boundary condition in Eqn (4-38) yields the theoretical maximum liquidload uUmax\

UL,Fi,max = uL>Fi (for hL}Fi = E) = —g — — - (4-47)5 a r\L

4.5 Relationship between boundary loads

According to the flow model, all the liquid in the bed of packing must be thrust upwardsby the frictional force in the vapour if the following condition is satisfied {cf. Eqn (4-25)}

uLtS = 0 (ifs = 0) (4-48)

The associated vapour load at the flood point uv>Fi can be derived from Eqns (4-5),(4-12), (4-14), and (4-41), i.e.

\ I 2g A I hLFi * / QL (A ^ X(4-49)

Values of uV;Fi obtained from this equation are higher than those calculated from Eqn(4-35) by a factor /, which is given by


4 Fluid dynamics in countercurrent packed columns

f =1 -

hr(4-50)

The ratio of Eqn (4-26) to Eqn (4-49) is

Uy,s = _1 1 [%Fi e - /iL)5

and of Eqn (4-26) to Eqn (4-35),

MV^W ^ \ §S

- hL}S

L,S

hL,Fl

(4-51);

(4-52)

These ratios lead to the conclusion that the loading point can be anticipated at about70% of the flood point.

Flooding can also be expected if the boundary condition for Eqn (4-36) is uL —> 0, inwhich case Eqn (4-35) becomes

W^F/ = (4-53)

According to Eqn (4-44), the boundary condition uL —» 0 entails that hLFi = e/3. If thissubstitution is made in Eqn (4-53), Eqn (4-45) will again be obtained.

Comprehensive experimental studies have verified that the flow model presented here cangenerally be applied to all kinds of packed columns with countercurrent gas-liquid systems.All the measurements agreed very well with the theoretically derived relationships for theloading and flood points. The liquid load in the tests was varied between uL>Fi = 4.88 andUL,FI = 144 m3/m2h; the gas density, between QV = 0.3 and QV = 1.37 kg/m3; the liquid den-sity, between oL = 750 and QL = 1026 kg/m3; the dynamic viscosity of the gas, betweenr\v = 7.14 X 10~6 and r\v = 18.19 x 10~6 kg/ms; and the dynamic viscosity of the liquid,from T]L = 0.36 x 10"3 to y\L = 92.6 x 10"3 kg/ms.

4.6 Pressure drop in packed columns

All equations found in the literature for the determination of gas-side pressure drop inpacked absorption, desorption or rectification columns are empirical or semiempirical. Theyhave largely been derived from studies on traditional packing and cannot be applied unreserv-edly to modern types. Therefore, the aim of this chapter is to devise a theoretical relation-ship that will allow the pressure drop to be calculated for both conventional and moderntypes of packing in two-phase countercurrent columns. It is to be based on the fluid-dynam-ics model described in Section 4.1, in the compilation of which it was assumed that thebed of packing is equivalent to a multiplicity of flow channels through which the liquid flowsdownwards in the form of a film. If inertia forces are neglected, gravity and shear are held


4.6 Pressure drop in packed colums 89

in equilibrium within the film by the frictional forces corresponding to the shear stress TV

that acts in the gas or vapour at the surface of the film. xv is identical to the expression onthe right-hand side of Eqn (4-4), i. e.

TV= -^ - y - QV (4-54)

in which QV is the gas density, uv the average gas velocity, and % the gas-side resistancefactor.

The pressure drop Ap in a bed of height H, effective area ae and effective void fractionee = 8 - hL can be derived from Eqn (4-54). Thus,

Ap = - — H xv = - Qe H xv (4-55)Ee 8 — h,L

The effective area ae is given by

ae = fw a = au + aL (4-56)

where fw is a factor that allows for the change in resistance caused by wetting the surface of

packing with an area a,au is the area of the unwetted surface, andaL is the area of the surface of the liquid in the flow channels.

In other words, the effective area ae is the sum of the unwetted area au and the area ofthe liquid surface aL in the flow channels.

In an actual bed of packing, the hydraulic diameter of the flow channels differs from thetheoretical value to an extent that depends on the column diameter ds. The difference canbe allowed for by a wall factor fs, which can be obtained from the literature, i. e.

— = 1 + — (4-57)

Inserting the expressions given by Eqns (4-14), (4-54), (4-56), and (4-57) in Eqn (4-55)yields the following expression for the pressure drop per unit height:

Ap a Uy \

~W = %L (z-hLy ~QvJ1 ' (4-58)

in which the §L is the resistance factor for the wetted bed of packing and is given by

| t = i/W = S ^ ^ - (4-59)



The limiting case of hL = 0, i.e. fw = 1 and aL = 0, applies for the pressure drop perunit height Apo/H in a gas stream flowing through an unwetted bed of packing. Thus, inser-ting hL — 0 in Eqn (4-58) gives

Apo

1T Qvfs

(4-60)

where %0 is the resistance factor for the dry bed of packing.

The extent to which the pressure drop in the gas stream within the wetted bed exceedsthat within the dry bed corresponds to the ratio Ap/Ap0, which is given by

Ap0 z-hL

(4-61)

This equation states that the increase in pressure drop caused by wetting can be describedby the ratio of the resistance factors for two-phase and single-phase flow and a term that is afunction of the liquid holdup, viz.

e-hL(4-62)

Eqn (4-27) allows Eqns (4-61) and (4-62) to be solved graphically in terms of the liquidload UL. This is demonstrated in Fig. 4.7, which applies to 50-mm plastics Hiflow rings. Itcan be seen that, as uL decreases, the function f(hL) approaches a limiting value of unity,because the liquid holdup tends towards zero and thus loses its effect. The higher the liquidload, the greater the effect on the function f(hL) and the higher the pressure drop. The dia-gram also shows empirically determined values for the pressure drop ratio AplAp0 plottedagainst the liquid load. It can be seen that the i{hL) function fairly closely fits the charac-teristic shape of the curve for the experimental values. Hence, the two axes of ordinates inthe diagram can be equated, i.e.

50-mm Hiflow ring, PP. ds = 0.288 m, H = 1.37mAir/H2O. 1 bar, 293 K

0.8

\ i ITheoretical curve/ 1 1 ,d

=99—^htoi—^^^ii i i i

i —i

1

>>< » i H

i?

0.80.6 0.8 1 1.5 2 3 4 5 6 8 10 15 20

Liquid load uL 103 [m 3 /m 2 s ]

Fig. 4.7. Pressure drop ratio in a wetted and unwetted bed of packing as a function of the liquidload



(4-63)Apo " \e-hL

where W is an expression for the degree of wetting.

The resistance factor §# can be determined from the following equation, which has beentaken from the literature:

64 1.Rev

0.08 (4-64)

This relationship embraces the effect exerted on the gas stream by the gas Reynoldsnumber Rev, which is defined by

Uy dp Qy6V~ ( 1 - 8 ) %

where dp is the particle diameter and is given by

Js (4-65)

(4-66)

The constant Cp characterizes the geometry and the surface of the unwetted packing andis therefore specific for any given type. It has to be determined experimentally.

The relationship between the resistance factor %0 and the gas Reynolds number Rev isillustrated by the example given in Fig. 4.8, which is valid for an unwetted bed of 32-mmplastic Envipac rings. At low loads, the downward slope of the curve becomes more pro-nounced. In this range of Reynolds numbers (Rev ~ 2100 or less), flow is laminar, and thefirst summand in the expression on the right-hand side of Eqn (4-64) governs the shape ofthe curve. Above this, flow becomes turbulent, and the second summand becomes the moredecisive.

Fig. 4.8. Resistance factor for awetted bed of packing as a functionof the gas Reynolds number

g ™ 0.8• i s 0.6hu, Q 0.5

J % 0.45 0.3

32mm ENVIPAC,PP. ds=0.288m. H = 1.39mAir, 1 bar, 293 K

D — - i»

6 8 10J 2 3 4 5 6 8 104

Gas Reynolds number Rev

Equating Eqn (4-61) with Eqn (4-63) and substituting for §0 in Eqn (4-64) gives rise tothe following relationship for the resistance factor ^L in two-phase flow:



(3 -x)

(4-67)

As the liquid trickles through the bed, static holdup occurs at the points of contactbetween the individual elements of packing and in the intervening spaces, and the liquidforms a film on the surface of the packing. Hence, the surface structure differs from thatduring gas flow through a dry bed, and this fact is expressed by the difference between Eqn(4-67) and Eqn (4-64), i.e. Eqn (4-67) contains an additional term.

In Fig. 4.9, calculated values for the resistance factor §L have been plotted against theliquid load uL determined by measurements in a wetted bed of 32-mm plastic Envipac rings.

Eqns (4-64) and (4-67) have been checked against the experimental results obtained onmore than 50 types of packing. The check embraced the effects exerted by the various physi-cal properties of 24 different systems (cf. Table 3.3) including some intended for purelyhydraulic studies and mixtures intended for absorption, desorption, and rectification. Evalu-ation of the comprehensive data thus obtained revealed that the numerical value for theexponent x in Eqns (4-63) and (4-67) was x = 1.5 and that the expression for W can bereplaced by a function, viz.

W = exp 200/j \h(4-68)

32-mm ENVIPAC. PP. d s = 0.288 m, H = 1.39mAir/H2O, 1 bar, 293 K

CDOC

12"co

10.8

0.60.50.4

—A- A . 1 —

0.81 2 3 4 5 6 8 10Liquid load u L -10 3 [m 3 /m 2 s ]

20 Fig. 4.9. Resistance factor for awetted bed of packing as a functionof the liquid load

The Reynolds number ReL of the liquid in this case is given by

ReL =uL QL

a r\L

(4-69)

The second term in Eqn (4-68) becomes unity in the range below the loading point.Mathematical prediction of the theoretical liquid holdup hL by means of Eqn (4-27) is

restricted to the range below the loading point. The qualitative diagram presented in Fig.4.10 is intended for the mathematical description of hL above this limit. It shows the theo-retical relationship between the liquid holdup hL and the ratio of the gas velocity uy to thevalue at the flood point UV,FI- Up to the loading point, ht is practically independent of thegas velocity and assumes a constant value hLS- Above this point, the shear forces acting in



Fig. 4.10. Qualitative relationship between theliquid holdup and the relative gas load 0 Gas velocity ratio u v / u V F

the gas progressively support the liquid film, and the liquid holdup thus greatly increasesuntil it attains a maximum hLyFl {cf. Eqn (4-40)} at the flood point. An empirical equationthat satisfies this condition is

= hL>s + (hLtFi - hLjS)uv,m

(4-70)

In this equation, hL>s is the liquid holdup as defined by Eqn (4-47), and hLjFi is that at theflood point, as defined by Eqn (4-40) and confirmed for 0 < uL < 200 m3/m2h and r| >1 x 10=4 kg/ms. The empirically determined numerical value of the exponent n is n = 13.

Hence, all that has now to be known for the determination of pressure drop is the oneconstant Cp, which is specific for any given type of packing, regardless of whether or not thebed is wetted. Numerical values of Cp determined by experiment on various conventionaland modern types of packing are listed in Tables 4.3 (dumped at random) and 4.4 (stackedin regular arrangement).

Precise measurement of the pressure drop by experiment is rendered difficult by the un-steady gas and liquid flow above the loading point, and the range of scattering of the valuesthus determined is correspondingly greater.

Up to the loading point, the average deviation of the values calculated by means of Eqns(4-58), (4-67), (4-68), and (4-70) from the values determined by experiment amounts toabout 9 %. It is higher in the loading range extending up to about 90 % of the flood pointload uV;pi and attains the greatest value, viz. almost 11 %, in the actual vicinity of the floodpoint.

At particularly high loads, the liquid flows so densely in the channels within the bed ofpacking that it coalesces in the narrow parts of the bed. As a result, extensive zones of theeffective open cross-section become filled with liquid to form a continuous layer, throughwhich the gas phase rises in the form of bubbles. If this limiting load, which is referred to asphase inversion, is exceeded, the liquid holdup and the resistance factor or pressure drop inthe gas stream increase at disproportionately high rates. The points at which this stage isreached are defined by Eqns (4-42) and (4-43). At higher values of the parameters in theseequations, Eqn (4-67) no longer applies.

The pressure drop model derived here has been confirmed by measurements performedunder the conditions listed in Table 4.5.



Table 4.3. Data for determining the pressure drop in random beds of packing

Arranged packing

Pall ring

Ralu ring

Hilflow ring

Material

Metal

Plastic

Ceramic

Plastic

Metal

Plastic

Ceramic

Sizemm

5038352515

503525

50

5050, hydr.

5025

905050, hydr.25

75503520, 4 webs

N1/m3

6242157721951753900229225

67651700052300

6215

57705720

500040790

134068156890

46100

19045120

16840121314

am2/m3

112.6149.6139.4223.5368.4

111.1151.1225.0

116.5

95.294.3

92.3202.9

69.7117.1118.4194.5

54.189.7

108.3286.2

em3/m3

0.9510.9520.9650.9540.933

0.9190.9060.887

0.783

0.9380.939

0.9770.962

0.9680.9250.9250.918

0.8680.8090.8330.758

CP

0.7631.0030.9670.9570.990

0.6980.9270.865

0.662

0.4680.439

0.4210.689

0.2760.3270.3110.741

0.4350.5380.6210.628

Hiflow ring; Super

NOR-PAC ring

Plastic

Plastic

50 6050 82.0

Raflux ring Plastic 15 193522 307.9

0.942

0.894

0.414

503525,25,2215

TypeB10 webs

733017450478374434669274

193738

86.8141.8193.5179.4249.0311.4

0.9470.9440.9210.9270.9130.918

0.3500.3710.3970.3830.3970.365

0.595

VSP ring

ENVIPAC

Top-pak

Bialecki ring

Metal

Plastic

Aluminium

Metal

50, No.25, No.

80, No.60, No.32, No.

50

503525

21

321

784133434

20006800

53000

6947

62781930355000

104.6199.6

60.098.4

138.9

106.6

121.0164.4238.0

0.9800.975

0.9550.9610.936

0.956

0.9660.9650.940

0.7730.782

0.3580.3380.549

0.604

0.7191.0110.891




Arranged packing

Raschig ring

INTALOX

Hiflow saddle

Tellerette

Hackette

Material

Ceramic

Plastic

Ceramic

Plastic

Plastic

Plastic

Sizemm

25

50

50

50

25

45

N1/m3

48175

8656

8882

9939

35365

12252

am2/m3

185.4

122.1

114.6

86.4

182.0

133.4

em3/m3

0.662

0.908

0.761

0.938

0.900

0.931

CP

1.329

0.758

0.747

0.454

0.538

0.399


Table 4.4. Data for determining the pressure drop in geometrically arranged beds of packing

Arranged packing

Pall ring, stacked

Hilflow ring, stacked

Ralu pack

Impulse packing

Euroform

Material

Ceramic

Plastic

Metal

Ceramic

Metal

Plastic

Plastic

Sizemm

50

5050, hydr.

YC-250

100

B1-200B1-300

Cl-200C2-200

PN-110

N1/m3

7502

76408150

am2/m3

155.2

131.3140.1

250.0

96.7

200.0300.0

200.0200.0

110.0

E

m3/m3

0.754

0.9160.911

0.945

0.828

0.9790.930

0.9540.900

0.936

CP

0.233

0.1720.172

0.191

0.417

0.3550.295

0.4530.481

0.250


Table 4.5. Characteristic values required for fluid dynamic studies on random beds of packing

Gas capacity factorLiquid load

Column diameterPacked heightInterfacial areaVoid fraction

Number of packings investigatedNumber of measurements

F vuL-103

ds

HaE

m-l/2 s-l kgl/2

m3 / m2 s

mmm2/m3

m3/m3

543296

0.21-5.090.17-16.7

0.15-0.800.76-3.95

54-3800.66-0.98



4.7 Liquid holdup in packed beds

The relationships given in the literature for the liquid holdup in packed columns applymerely to a few conventional forms of packing and have often been derived solely from mea-surements on air/water systems. As was already mentioned in Chapter 3, the geometry ofmodern packing differs greatly in many respects from that of former types, with the resultthat the existing relationships can no longer yield reliable results for the liquid holdup. Themodel presented here for the prediction of liquid holdup has been based on the fundamentalrelationships enunciated in Sections 4.2-4.4. It is valid for both conventional and modernpacking in two-phase countercurrrent columns and allows reliable results to be obtained formultifarious two-phase systems. An assumption made in deriving the model is that the effec-tive void fraction in the bed of packing can be represented by a multiplicity of vertical flowchannels, along the walls of which the liquid trickles downwards in counterflow with theascending gas stream.

The actual flow behaviour can be visualized by allowing for the fact that the surfaces ofthe packing are not completely wetted at the liquid loads normally encountered in practice.In this case, the thickness s0 of the descending liquid film can be described in terms of theliquid holdup hL and the hydraulic area ^ of the packing surface, i.e.

so = — (4-71)ah

This equation differs from Eqn (4-12), which is valid for uniform and complete wetting ofthe packed surfaces. Inserting the expressions for uL, uv, and So given by Eqns (4-13),(4-14), and (4-71) in Eqn (4-7) yields the liquid holdup for the entire loading range in two-phase countercurrent flow, i.e.

hL =ah

-Qvuva

a"Vi (4-72)

This equation is analogous to Eqn (4-17), but includes the term (ah/a)2/3 to embrace theeffect of the hydraulic area ah. It allows for all the main parameters that affect hL up to theflood point in a two-phase countercurrent column: the phase densities QL and QV, the liquidviscosity \\L, the phase loads uL and uv, the acceleration due to gravity g, and the resistancefactor §L.

If the column is operated in the load range between the loading and flood points in two-phase countercurrent flow, the downward stream of liquid is no longer independent of thegas load, because it is held up by the shear forces in the gas stream. The boundary conditionthat applies in this case is defined by Eqn (4-25), which states that the velocity uL)S tends tozero at the surface of the liquid.

The results of the experiments described in Chapter 3 have shown that the liquid holdupin the uv < uv>s range is a function of uL only and that the gas stream exerts hardly anyeffect. Therefore, applying the boundary condition implied by Eqn (4-25) gives rise to thefollowing relationship for the liquid holdup in the uv < uvs range:


4.7 Liquid holdup in packed beds 97

<*h i , ( 4 _ 7 3 )

QLg I \ a

If this equation is compared with the theoretical relationship for the liquid holdup, asdescribed by Eqn (4-27), it will be seen that allowance for the actual flow conditions in apacked bed has been made by the term (ayjd)/z', a power of the ratio of the hydraulic to thegeometric area. The numerical value n 2/3 = for the index n has been generally confirmed byexperiments on both random and stacked beds of packing.

A relationship for the ratio aja has been determined in fluid dynamics studies on theliquid holdup and subsequently correlated to an empirical function. It was thus found that itdepends on the liquid Reynolds number ReL, as defined by Eqn (4-69), and the Froudenumber FrL, which is related to the liquid holdup by Eqn (4-27). The following relationshipsfor two ranges of Reynolds numbers were derived from the corresponding studies on the sys-tems listed in Table 3.3:

= ChRe£15Fr£1 (4-74)ReL<5

- ^ 1 = 0.85 Ch Rel25 FrlA (4-75)a lReL>5

Numerical values for the constant Cy, are listed together with characteristic geometric dataon the types of packing concerned in Tables 4.6 (dumped) and 4.7 (arranged). The datainclude the number N of packing elements per unit column volume and the void fraction e.On an average, the experimental results differ by only 6.7% from the figures for the liquidholdup calculated from Eqns (4-73), (4-74), and (4-75).

A relationship between the holdup at the loading point hLiS, as determined from Eqn(4-73), and that at the flood point hLFh was also derived from the results of the experi-ments. It is presented graphically in Fig. 4.11, in which the holdup ratio hLlhLS has beenplotted against the gas load ratio uv/uVjFi. The diagram confirms the theoretical consider-ations that were presented in Sect. 4.5 and led to the conclusion that the upper loading pointis to be anticipated at about 70 % of the load at the flood point. It can also be seen that theliquid holdup near the flood point is about 2.2 times greater than that in the capacity rangebelow the loading point, i.e.

hLiPl = 2.2 hLtS (4-76)

Inserting Eqn (4-76) in Eqn (4-70) leads to

hL = 1 + 1.2UVFI

(4-77)

This equation can be applied, together with Eqn (4-73), for the calculation of hL in theentire capacity range. The curve in Fig. 4.11 was plotted from these relationships.

The conditions for the tests in which the data for the holdup evaluations were acquiredare listed in Table 4.8.



Table 4.6. Data for determining the liquid holdup in random beds of packing

Dumped packing

Pall ring

DINPAC ring

Hiflow ring

Glitsch ring

NOR PAC ring

Raflux ring

VSP ring

ENVIPAC ring

Top-pak ring

Bialecki ring

Material

Metal

Plastic

Ceramic

Plastic

Metal

Plastic

Ceramic

Metal

Plastic

Plastic

Metal

Plastic

Aluminium

Metal

Sizemm

50352515

503525

50

7045

5025

50

20, 4 ribs20, 6 ribs

1.5" CMR1.5" CMR, T1.0" CMR0.5" CMR30 P30 PMK

50352515

15

50, No. 225, No. 1

80, No. 360, No. 232, No. 1

50

503525

N1/m3

62421951747500

229225

66641668252300

6455

976428518

500039917

6997

101444110688

6074463547

158467560811

3250928445

77101714744500

193738

193522

784133434

19626773

52681

6871

62781817752000

am2/m3

112.6139.4215.0368.4

102.0148.3225.0

121.0

110.7182.9

92.3197.5

120.2

239.3265.8

174.9188.0232.5356.0168.9180.2

95.1139.0180.0319.7

307.9

104.6199.6

58.8100.0138.0

105.5

121.0154.8225.0

8

m3/m3

0.9510.9650.9420.933

0.9260.9070.887

0.770

0.9380.922

0.9770.962

0.924

0.7970.776

0.9740.9720.9710.9550.9580.975

0.9490.9300.9270.918

0.894

0.9800.975

0.9590.9610.937

0.956

0.9660.9650.945

ch

0.7840.6440.7190.590

0.5930.7180.528

1.335

0.9911.173

0.8760.799

1.038

1.1670.958

0.9350.8701.0401.3380.8510.930

0.6510.5870.6010.343

0.491

1.1351.369

0.6410.7941.039

0.881

0.7980.7870.692


4.7 Liquid holdup in packed beds 99


Dumped packing

Raschig ring

Berl saddle

Tellerette

Hackette

Material

Metal

Ceramic

Carbon

Ceramic

Plastic

Plastic

Sizemm

15

2515106

25

2513

25

45

N1/m3

260778

48381220000847552

3022936

51913

63920529720

35000

12000

am2/m3

378.4

191.9310.2492.1771.9

205.7

205.4436.4

180.0

139.5

E

mVm3

0.917

0.7260.6900.5700.620

0.700

0.6950.660

0.900

0.928

ch

0.455

0.5770.6480.7911.094

0.623

0.6200.833

0.588

0.643

Table 4.7. Data for determining the liquid holdup in geometrically arranged beds of packing

Arranged packing

Pall ring

Bialecki ring

Gempak

Impulse packing

Montz packing

Mellapak

Euroform

Material

Ceramic

Metal

Metal

Metal

Ceramic

Metal

Plastic

Plastic

Sizemm

50

3525

A2 T-304

250

100

Bl-100Bl-200Bl-300

250 Y

PN-110

N1/m3

7510

2073663700

am2/m3

138.7

176.6275.0

202.0

250.0

102.0

100.0200.0300.0

250.0

110.0

£

m3/m3

0.670

0.9600.930

0.977

0.975

0.830

0.9870.9790.930

0.960

0.936

ch

1.066

0.6900.395

0.678

0.431

1.900

0.6260.5470.482

0.554

0.511



3

O

10.8

/"TH

/

^

odel

— &

correlcitio

W

n [in

\&

i -

e

o1

j£

i10.2 0.3 0.4 0.5 0.6

Gas velocity ratio UV /UV (FI

0.8 1

Fig. 4.11. Values of liquid holdup calculated from the model compared with those determined byexperiment

Table 4.8. Characteristic values required for fluid dynamic studies on geometrically arranged bedsof packing

Gas velocityLiquid load

Column diameterTotal area per unit volumeVoid fraction

Liquid densityLiquid viscositySurface tension

Number of systems investigatedNumber of packings investigatedNumber of measurements

uviiL-103

ds

a8

QL

TlL-103

GL • 103

m/sm3 / m2 s

mm2/m3

m3/m3

kg/m3

kg/mskg/s2

0.09-2.510.37-23.0

0.076-0.4459-7720.57-0.99

800-18100.78-185

22.6-86.3

1656

822

4.8 Liquid entrainment at high loads

If a bed of packing consists of large elements dumped at random or of elements stackedin a regular geometric pattern, the liquid flows through the channels largely in the form of afilm or lamellae. However, if the basic pattern in a stacked bed is a latticework, the possi-bility exists that the liquid film may become partly detached from the lowermost edges ofthe packing surfaces, particularly at high liquid loads. The droplets thus formed would thenfall through the void that lies between the tear-off edge and the underlying packing element.To an extent that depends on their size, the individual droplets may be held in suspension bya gas stream of the corresponding velocity. Alternatively, they may be entrained in theascending gas and conveyed to the packing elements immediately above them, on whichthey would be deposited and thus mixed with the local descending liquid film. Hence, withinthis loading range, the flood point would not be reached; and, as has been demonstrated byexperiment, the column efficiency could still be substantially improved (cf. Fig. 4.12).


4.8 Liquid entrainment at high loads 101

The onset of entrainment in cavities within beds of packing entails that the buoyancyforces acting on a liquid droplet of diameter d^r are in equilibrium with the force of gravity.In other words, the following condition must be satisfied:

le ddr — MKe Qv = — dl — g(QL~ Qv)O JO

(4-78)

Consequently, the capacity factor Fv>e, which governs entrainment, depends on the diame-ter d^ and stability of the droplets, i.e.

Fv,.= \/T -tddr(QL-Qv) (4-79)

Ethylbenzene/Styrene, L/V = 1

3

E

— 2

8000

7000

6000

5000

4000

3 =ALL

innlUU

^ ^

12%

/o

1

-

-—k>-78°/.

i

M P

-9

j

Pall ring 50 mm, metal

ds=500mm,Z=6001/m2

- 3

/

I

)ar|

y

1

"7i

/

/ /

\ Ir/

!/>/

|xe--23%

rALY

//

67 mbar

" • load at max.effy

- • maximum load —

4000 5000 6000 7000 8000 9000

Vapour load V [kg/m2h]

2.0 2.5 §5 35 Th 4 5Fv [m-V2s- lkg1/2](34mbar)

Fig. 4.12. Results of experiments on theentrainment of liquid droplets at the head ofthe column

1.5 2 l E 10 ?.5^s"1 kg'/2](67mbar)Fv



The lower diagram in Fig. 4.12 gives an idea of the amount of liquid AL that is entrainedby the gas stream at the head of a Pall-ring column for the separation of an ethyl-benzene/styrene system under vacuum and total reflux. It can be assumed that the volume ofliquid that may be entrained within the voids of a packed bed does not exceed AL. Thisvolume increases with the vapour capacity factor in the Fv > Fv>e range. Since the efficiencyattains a maximum in this loading range (cf. upper diagram in Fig. 4.12), it can be assumedthat the droplets are redeposited in the packing. As a result, the area of contact between thephases would be increased, and mass transfer would thus be intensified.

Actual flooding does not commence until the liquid load becomes even greater and anincreasing quantity of the liquid film or lamella is supported by the energy of the gasstream, as is expressed by a significant rise in the liquid holdup. The fluid dynamics modeldeveloped in Section 4.1 is thus confirmed, and it is physically unjustified to describe theonset of flooding in terms of descending liquid droplets when the equilibrium of forces hasbeen attained.

In actual fact, it can be deduced from Fig. 4.12 and the arguments presented above thatpacked columns are also eminently suitable for separating liquid droplets, and this fact haslong since been exploited in industrial practice.

4.9 Phase inversion in packed beds

Strictly speaking, the flow model presented in Section 4.1 is not valid unless the assump-tions made in its development apply. Experimental studies have revealed that the relation-ships for the loading and flood points derived from the model can be applied in the loadingrange described by a load parameter of i|> < 4 without reservation and to within an accuracythat is adequate for practical purposes. However, if this value of load parameter is exceeded,phase inversion occurs, i. e. the original disperse phase - the liquid - becomes the conti-nuous phase, and the previously continuous gas phase becomes the disperse phase. Thisstage sets in at the comparatively high liquid loads encountered in a number of absorptionprocesses and, particularly, in high-pressure rectification.

It ought to be possible to express phase inversion in physical terms by altering the expo-nent in the relationship for the resistance factor §, as is indicated by the example shown inFig. 4.6. This assumption has, in fact, been confirmed experimentally in numerous studies,some results of which have been collated in Figs. 4.13 to 4.15. The associated diagramsapply to packing of various geometries and textures produced from various materials andwith flow channels of various dimensions. It can be derived from them that the point atwhich phase inversion can be expected is at a flow parameter, as defined by Eqn (4-24), of

ip = xps = 0.4 (4-80)

The physical interpretation of these relationships is as follows (cf. Fig. 4.16). If thetransition point is reached at a given liquid load uL>i, the associated liquid holdup can beexpressed as a fraction / of the relative void fraction, i.e.

hL,i = {hLtS)t = is (4-81)


4.9 Phase inversion in packed beds 103

Nominal packing size 25 mmo Air/Water 1 bar 293 K; OToluene/n-Octane133mbar

• Chlorobenzene/Ethylbenzene 67 mbar

A Air/Ethylene glycol; v Air/Methanol.1 bar, 293 K

«. 0.8

1 0.8Z 6

= 3

2

0.8

•c

— *> ^ .

Raschig ring, ceramic

a> i< ^

Pall ring, plasticn

r- — .

Bialecki ring, metal

i

i s

s.k>c

Fig. 4.13. Resistance factor as afunction of the modified flowparameter

0.06 0.1 0.2 0.3 0.4 0.6 0.8 1 2 3 4 6 810

Modified flow parameter <\>s ( - p )

The corresponding vapour velocity at the loading point (uv?s)i can then be derived from Eqn(4-26). Thus

(4-82)KS)/ = (i-QVn/ir-A/v\/ -

and the corresponding theoretical liquid load, from Eqn (4-27), i.e.

.i = («u)i = 12(4-83)



Nominal packing size 50 mm

oAir/Water 1 bar 293 K;sEthanol/Wateri bar

0 Chlorobenzene/Ethylbenzene 67;133mbar

b

43iL

1

• ^

• Raschig

!

ring

- ^

, ceramic -

i

i

II

ai _aa.

"GO

o

a;So

pi

Dall

k

rin ramie -

1

p

iI

i

k

0.06 0.1 0.2 0.30.4 0.6 0.8 1 2 3

Modified flow parameter ^ s ( ^ pv Tlv>

0.4


Now, the following relationship exists between (uLS)i and (uv,s)i'-

L = (uL>s)j QL

V (uv,s)i Qv

Rearranging Eqn (4-24) gives

The relationship between §5>; and / then follows from Eqns (4-81) to (4-85), i. e.

(4-84)

(4-85)

(4-86)




a<_a

o

c

43

?

1

o0

6f.-)J

2

1

o Air/Water 1 bar 293 K

D,mChlorobenzene/Ethylbenzene 67-133 mbar

- NOR-PAC ring, plastic, 15 mm -

i i i i

1

1 U\—

iii

rTl

-Montz packing, metal, B1 - 300

1 1

iiii

K1i

i

Modified flow parameter 4M YTv0.4

0.06 0.1 02 0.30.4 0.60.8 1 2 3 4

The experimentally verified relationship expressed by Eqn (4-29) must apply below thephase inversion, with the consequence that the equation valid immediately before the phaseinversion is

(4-87)

Combining Eqns (4-86) and (4-87) gives rise to a relationship that allows the phase inver-sion coefficient i for a bed of packing to be determined, i.e.

(4-88)(i-02 QLIUL

where qs is a characteristic constant that applies for the packing under the conditions at theloading point and is given by

(4-89)



Phases/ V : disperse

ex

Gas load uv

Fig. 4.16. Qualitative description of phase inversion

It is evident from Eqn (4-88) that i depends on the viscosity ratio of the two phases andon the density of the liquid. The relationship between the phase inversion coefficient / andthe packing characteristic §5 is shown graphically in Fig. 4.17 for the standard air/water sys-tem. An idea of the effect exerted by the physical properties of the various systems (cf. Table3.3) can be obtained from Fig. 4.18.

Measurements in packed columns have shown that the pressure drop curves Ap/H =t(uv)uL = const run parallel below the loading point. Above the loading point, the slopeincreases with the liquid load. In other words, the difference between the load at the load-ing point and that at the flood point becomes progressively smaller and tends towards zero,i.e. uvs = uVyFh when complete phase inversion occurs (cf. Fig. 4.19). Hence, if hL>m = hLiS

— hLi, equating Eqn (4-26) with Eqn (4-35) would give

(4-90)

Equating the theoretical relationship derived from the model for the liquid holdup at theloading point {Eqn (4-27)} with that at the flood point {Eqn (4-38)} gives rise to



0.16

0.14

\Q 0.12oo

= 0.oCO

.£ 0.

/

f

/

(hUs)-,e

/ ^^ ^

*^ ^. ^ •

System: Air/Waterat STP

==—-=

0.06

0.040 50 100 150 200 250 300 350 400 450 500

Packing characteristic qs-10"6[m^2S"2]

Fig. 4.17. Relationship between the phase inversion coefficient and the packing characteristics foran air/water system at STP

chloride solution 1

Air /Calcium chloride solution 1

Air/Methanol

150 200 250 300 350 400 450 500

Packing characteristic qs-10~6[rrr2 s~z]

Fig. 4.18. Relationship between the phase inversion coefficient and the packing characteristics forvarious systems



- oa>ZDCOCOCD

Fig. 4.19. Qualitative pressure drop function

Gas or vapour load uv

(4-91)

The theoretical boundary condition is defined by

In this case, the liquid holdup is given by

(4-92)

(4-93)

In other words, the liquid holdup accounts for one-half of the void fraction. It thereforefollows that the resistance factors are all of the same magnitude, i. e.

%m = h,t = & (4-94)

As a consequence, Eqn (4-90) can be reduced to Eqn (4-93).Theoretically, the liquid holdup defined by Eqn (4-93) can be attained if the liquid load

assumes the limiting value given by Eqn (4-83) (for i = 0.5), i. e.

"v.s-uKFI % a2 ^ (4-95)



The corresponding vapour load is obtained from Eqn (4-82) (for / = 0.5), i. e.

The liquid holdup in the uL < uLi range is given by

hL,m = m (4-97)

where n < /2-The corresponding vapour load follows from the general theoretical relationship to de-

scribe the liquid holdup at the flood point, i. e. Eqn (4-38). Thus,

Since UL>FI > 0 and, consequently, (3n - 1) > 0, it follows that

%>n> % (4-99)

If Eqn (4-98) is applied to the conditions at the phase inversion, it can be equated to Eqn(4-83) to obtain the formal relationship between i and n, i. e.

i = n 3 V2(3n- l ) (4-100)

At the phase inversion, the relationship between them is given by (cf. Fig. 4.16)

mt = nli (4-101)

where m is the ratio of the liquid holdup at the flood point to that at the loading point, i.e.

m = ^ L (4-102)hus

Hence, if i is known from Eqn (4-88), n can be determined from Eqn (4-100); and m,from Eqn (4-101).

The relationships n = f(qs) and mf = f(qs) for various systems at standard temperatureand pressure are presented graphically in Fig. 4.20.

Rearranging Eqn (4-89) gives rise to

-Q-= (—)=— \ (4-103)qs \a I a a


110 4 Fluid dynamics in counter current packed columns

4.0in

vers

ion

CDCOO-c:d .

itio

rr

3.5

3.0

2.5

2.0

I 1.5

1.0

\-H\\\

/

*•

\\

V/

• .

^/

^ n -(I -

— Nitrogen/Heavi i

^ +

> ^ -^ - —

A-r...

, -i

— « ,

. —

— — -

— —

/ oil -

i

—— Air/Calcium

solution 1

——*A••.

— »ii

chloride

——-

- —

——

-Air/Water

i/i\

—

-——

—

= n 0.39

0.38

0.37

0.36

0.35

—I—I—J 0.34

QJOO

0.330 50 100 150 200 250 300 350 400 450 500

Packing characteristic q s 10"6 [m~2 s " 2 ]

Fig. 4.20. Characterization of the phase inversion point in various systems

Substituting for e3la2 in Eqn (4-83) then gives the liquid load at the phase inversion, i.e.

uL.i = ^T ^3 « — — (4-104)

This expression for the liquid load at the phase inversion is shown as a function of thearea per unit volume a with the void fraction e as parameter in Fig. 4.21.

Substituting ne for hL>m {Eqn (4-97)} and it for hL>s {Eqn (4-105)} in Eqn (4-102) gives

1(4-105)

Since the liquid holdup at the flood point is very difficult to determine by experiment,the values of m derived from it are not very accurate. The average is m — 2.2 for low-viscosity systems (cf. Section 4.7).

The following theoretical value of m is obtained by substituting hLyFllz for n in Eqn(4-105):

(4-106)

- 1

In analogy to Eqn (4-40), the holdup at liquid loads of uL < uL>i can be estimated from

(4-107)

where ri = 0.349.



120

£ 80

a

cr

•V e = 0.98

//

'""\?e--0.9(

•7

/£ =

]

Phase inversion

System:

at

W.I'.

Air/WaterSTP

Cs-

r

3.5

• — — .40

2050 100 150 200 250 300 350 400 450 500

Area per unit volume a [mz/m3]

Fig. 4.21. Liquid load at the phase inversion point as a function of packing characteristics

If r\L = r\w, inserting values of hLjFilz thus obtained in Eqn (4-106) would again yield avalue of about 2.2 for m. Solving Eqn (4-105) for n would then give rise to n = 0.349.

The flow pattern is altered once the flow parameter exceeds the value i - at which phaseinversion commences (cf. Figs. 4.13 to 4.15). If the set of equations in the model is to beformally retained, this alteration would be expressed as a change in the exponents andconstants in Eqns (4-29) and (4-41) for the resistance factors §5 / and I=F/)I- respectively.Numerical values for the exponents can be obtained from Eqns (4-32) and (4-43).

The following condition must be satisfied at the phase inversion:

55 - 55,i

Eqn (4-29) can therefore be converted into

Cs,i = Cs

Inserting the numerical values then gives

("s - ns,i)

(4-108)

(4-109)

CSi = 0.695 C5[ — (4-110)

Another condition that must be satisfied at the phase inversion is

(4-111)



Eqn (4-41) can therefore be converted to yield the following relationship for the constant:

\0.2 (nFl-nFl,i)

ZT) (4"112)

The simplified relationship in this case is thus

( ^ 0.1028

— (4-113)

Consequently, the vapour load at the loading and flood points can be formally deter-mined by Eqns (4-28) and (4-35), even in the range above the phase inversion. The valuesof ^s and §F/ required for the calculations and corresponding to n$ = nSj, Cs = Cs,i,HFI = nFiti, and CFl = CFiA can be obtained from Eqns (4-29) and (4-41) respectively.

4.10 Relationship between loading and flood points

The range between the loading and flood points is of utmost importance in the designand operation of packed columns. The liquid holdup and thus the pressure drop within thisrange increase at a higher rate than they do in the range below the loading point. The designload uVD, at which the column volume is a minimum, then lies between uV}s and uym, i- e.

uv,s ^ uV}D < uVyF{ (4-114)

Substituting for uvs from Eqn (4-26), f$ from Eqn (4-57), ^L from Eqn (4-59), and fw>s

for fw in Eqn (4-58) gives rise to the pressure drop per unit height (Ap/H)s in the uv = uV;S

loading range, i.e.

Likewise, the pressure drop per unit height (Ap/H)Fi at the flood point uv = uVtn can bederived from Eqns (4-35), (4-57), (4-58), and (4-59) and / w = fWiFl. Thus,

The pressure drop ratio is therefore

&PFI „ fw,Fi hLjm e - hLjS

= 2 — (4-117)fw,s h L i S e

Substituting m for hLiFilhL>s {cf. Eqn (4-102)} then gives


4.10 Relationship between loading and flood points

It follows from Eqns (4-97) and (4-102) that

113

(4-119)

Hence, the pressure drop ratio can be expressed by the following simple relationship:

ApFl n fw,Fl , x ,. 1 o m

— = 2 ^-r— (m - n) (4-120)Aps Jw,s

Alternatively, it can be expressed as follows by substituting the right-hand side of Eqn(4-105) for m:

fw,s \3V2(3n-l) - n

The factor n can be obtained by combining Eqns (4-97) and (4-107), i. e.

/ ^ \0.05

r«-uL < ut= 0.349

r\wlQw

(4-121)

(4-122)

The factors fw>s and fw,Fi m Eqn (4-121) are defined by Eqn (4-56) as the ratios of theeffective area ae to the area per unit volume a at the loading and flood points respectively.The effective area ae is that of the surface of the flow channels of hydraulic diameter dh inwhich the pressure drop would be the same as that encountered in the bed of packing underconsideration. The hydraulic diameter dh is defined by

(4-123)fwa

The pressure drop per unit height at the loading and flood points can be derived fromEqn (4-58). Thus,

H is

Ap

.? 5

Uy,S

- E 21

- hLyFi I dhiFtfs

(4-124 a)

(4-124 b)

Substituting ne for hL>m {cf. Eqn (4-97)} and zn/m for hLfS {cf. Eqn (4-119)} in theApFi/Aps ratio thus obtained gives rise to the following equation:

%Fl fw

fw,S

1 -

1-

n

m- n UV,S

(4-125)



The ratio of the vapour load at the flood point to that at the loading point can be derivedby equating Eqns (4-120) and (4-125). Thus,

(4-126)

(4-127)

uv,s

Substituting the right-hand side of Eqn (4-105) for m in this equation then gives

( I - " ) 2

uv,s- « V 2 ( 3 w - l )

where n is defined by Eqn (4-122) and the term h,s^Fi is the ratio of Eqn (4-29) to Eqn(4-41), i.e.

Im (nS - nFl)(4-128)

This ratio can also be estimated by applying Eqn (4-59). Thus,

fw,Fl

fw.S(4-129)

In this case, the ratioof this equation, i.e.

can be calculated from Eqn (4-67) or expressed in terms

lL,Fl

64hL,Fi \ ° ' 3 Rev,Fi

hL,s 1 64

• + •

- + •

1.8Reom

1.8- hLtFl \~

(4-130)

Rev,s Rev,s

The expression within the square brackets in this equation can be equated to(uViS/uViFif-08 = 0.97. Applying Eqns (4-97) and (4-119) then gives rise to

0.97

| t ,t,s

1 - n(4-131)

The fw,Filfw,s ratio is thus known from Eqn (4-59), and the pressure drop ratio can there-fore be obtained from Eqn (4-121).

Eqn (4-127) can also be derived by substituting Eqns (4-97) and (4-119) for hLiFt andhLS, respectively, in Eqn (4-52).

The flood point exponent n depends fundamentally on the system and the flow parameter\p and can be calculated from the model. The liquid holdup hL>s accounts for only a fractionof the relative void fraction 8, i. e. it differs from e by a factor/. Thus

hL.s = f e (4-132)


4.10 Relationship between loading and flood points 115

Hence, in analogy to the derivation of Eqn (4-88) for the determination of the phaseinversion coefficient /, the following relationship applies for the factor /:

f 144QL)\QLd-f)2' s 2 *

Equating Eqns (4-97), (4-102) and (4-132) gives

Substituting for /on the left-hand side of Eqn (4-105) then gives

/ 5 _ n5[2(3n-l)f3

(I-/)2 {l-n[2(3n-l)]V3}2

(4-133)

(4-134)

(4-135)

Three systems with different physical properties were taken as examples for evaluatingthe relationships discussed above: air/water, air/calcium chloride solution, and nitro-gen/heavy oil, each at STP in the loading range below the phase inversion, i. e. \p < 0.4.

Fig. 4.22 shows the relationship obtained from Eqn (4-131) between the flow parameteri|> and the modified resistance parameter ratio R, which is given by

R = (4-136)

« 2 8

O

S 2.4i_CD

1 2.0_a

s

•55 1.2

1 0.8TZJ

04

i

/ /

i

i .

y

/

*

«-—'

1

—

\\\0—-—— —. 1 —

1 bar, 293 K

—

— —

1 1

— —

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4

Flow parameter i|)

Fig. 4.22. Modified resistance parameter ratio as a function of the flow parameter


116 4 Fluid dynamics in counter current packed columns

Thus Fig. 4.23 would be obtained for packing with a CFilCs = 0.6 ratio.Fig. 4.24 was plotted from Eqns (4-133) and (4-135) and shows the coefficient n for the

liquid holdup under flooding conditions as a function of the flow parameter I|J. It also showsthe corresponding function for ra, which is the ratio of the liquid holdup at the flood pointto that at the loading point and was determined from the relationship between m and nexpressed by Eqn (4-105).

Finally, if the functions n = f(i|j) and §5/^7 are known, the following can also be deter-mined as functions of the flow parameter ip: the ratio of the resistance factors ^L,FI^L,S{Eqn (4-131)}; the vapour load ratio WV;F//"KS {Eqn (4-127)}; and the pressure drop ratioApFl/Aps {Eqn (4-120)}.

These relationships, as determined on some metal packings and an air/water system atSTP, are shown as an example in Figs. 4.25-4.27.

_, 1

<-n

O

"a

0.8

0.6

0.4

0.2

COCD

/

ArIf

\-1

F

C

1r"

. •

1 = 0 . 65

, ^-,——•• — "

Air Water

ii

-—-

— —

Dor, 293 K

^ —— —— —

.. • ••

— —

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4

Flow parameter ty

Fig. 4.23. Resistance parameter ratio as a function of the flow parameter


4.10 Relationship between loading and flood points 117

4.0

3.5

3.0

2.5

2.0

1.5

1.0

\

\

- q

\\

N

\\

\V

\

8 (

^ - — -

\

\

m s

——. • • •

n

Air/Water

S- A i r /Ca lc ium c

r2

\

i ••

* * —

—- —•——— — —

. . . .

1 bar

:hloricle solution 1 —

Nitrogen/Heavy oil

—.

— —

- * • —

m. —

— — . .

mi ——

——

I

, 2 9 3 K

" V\

*—-

....

\

~<—

m -—n .

0.39

0.38

0.37

0.36 §

0.35 I

0.34

0.33

0 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40

Flow parameter ty

Fig. 4.24. Ratio of the liquid holdup at the loading point to that at the flood point as a function ofthe flow parameter

Glitsch ring 30 PAir/Water, 1 bar, 293 K

™ 0.9

- 0.8

———.• •J.

— i — -^ — —

— — — ' _

fw,Fl/fw,S

Fig. 4.25. Characteristics defining theloading range between the loading andflood points

0.1 0.2 0.3

Flow parameter ty

0.4



QJCO

a

en _,

ao

1.5

1.3

7

5

1.1

™ 0.9

_ 0.8

CO

*" 0.4

0.34

0.33

Glitsch ring 30 Pmk

Air/Water. 1 bar. 293 K

— -— • • •

— — •

^ —

—*•

H 1

1

-4 ^- u-—«——

•W F l ' ' W S

.—- —— —- —

0.1 0.2 0.3

Flow parameter ^

0.4

Fig. 4.26. Characteristics defining theloading range between the loading andflood points

<

at _

a(—3

1.5

1.3

6

0.8

1

0.6

0.345

0.335

Glitsch ring 0.5 CMR 304

Air/Water. 1 bar. 293 K

s——• — —

V

^ ^

^ | — H —fw.Fl/fw,S

^- —

,

——— Fig. 4.27. Characteristics defining theloading range between the loading andflood points

0.1 0.2 0.3

Flow parameter <\>

0.4


Fluid Dynamics in Countercurrent Packed Columnsssu.ac.ir/cms/fileadmin/user_upload/Daneshkadaha/...4...

Documents

Transcript of Fluid Dynamics in Countercurrent Packed Columnsssu.ac.ir/cms/fileadmin/user_upload/Daneshkadaha/...4...