Mass Transfer into Laminar Gas Streams in Wetted-Wall ... The ... Consider a gas and a liquid flowing cocurrently in a vertical wetted-wall column. It is assumed that the gas ...

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TitleMass Transfer into Laminar Gas Streams in Wetted-Wall Columns : Coc

urrent Gas-Liquid Flow with Circulation in Gas Phase

Author(s) Hikita, Haruo; Ishimi, Kosaku; Soda, Norifumi

Editor(s)

CitationBulletin of University of Osaka Prefecture. Series A, Engineering and nat

ural sciences. 1978, 27(1), p.79-89

Issue Date 1978-10-31

URL http://hdl.handle.net/10466/8304

Rights

79

Mass Transfer into Laminar Gas Streams in Wetted-Wall

Cocurrent Gas-Liquid Flow with Circulation in Gas

Columns

Phase

Haruo HiKrrA*, Kosaku IsHiMi* and Norifumi SoHDA**

(Received June 15, 1978)

The effect of the gas and liquid fiow rates on the mass transfer rate in laminar gas

streams in wetted-wal1 columns with cocurrent gas-liquid fiow was studied for the case

where a circulation flow of the gas exists. An approximate analytical solution was

obtained for the average gas-phase Sherwood number as a function of the gas-phase

Graetz number and the dimensionless interfacial gas velocity. Experiments were carried

out on the absorption of methanol vapor into water, using a 4.0 cm I. D. glass column of

80 cm long. The agreement between the experimental and the predicted effects ofbeth

gas and liquid flow rates on the gas-phase mass transfer rate was found to be fairly good.

1. Introduction

The influence of a moving interface upon gas-liquid mass transfer is a problem of

,both theoretical interest and practical importance. In a previous paperi), the results of

theoretical and experimental studies on the effect of the gas and liquid flow rates on the

mass transfer rate in larninar gas streams in wetted-wall columns with cocurrent gas-liquid

fiow have been reported. An analytical solution was obtained for the average gas-phase

Sherwood number ShG as a function of the gas-phase Graetz number GzG and the

dlmensionless interfacial gas velocity U. This analysis, however, is not applicable to the

case of U > 2 where the gas flow rate is very low and the circulation exists in the gas

phase. The present study was undertaken to present an approximate analytical solution

for the present case and to confirm this approximate solution experlmentally.

2. Theory

2.1 Velocity prornes

Consider a gas and a liquid flowing cocurrently in a vertical wetted-wall column. It is

assumed that the gas and liquid streams are laminar and fully developed. The velocity

proMes for this situation are shown in Fig. 1 (a).

The equations of motion fbr gas and liquid streams have been solved in a previous

paperZ and the following expression giving the velocity proMe for the gas has been

obtained:

* DepartmentofChemicalEngineering,CollegeofEngineering.** Graduate Student, Department of Chemical Engineering, College of Engineering.

80 Haruo HIKITA, Kosaku ISHIMI and Norifumi SOHDA

Liquid

fitm

lGasrlo

i1'l

z

Ua--pu!7uill,i

Fig. 1.

!1 k- k{ ,,//- l

(a)

Annulardownftow

i reglon

egggg

circutatien

ri

(b)

Oore

regionl

A! !

IS Ti

l

ulg'

Flow model and coordinate system.

u= u. (2-U)-2u. (1-U) (rlri)2 , (1)with

U= uilum, (2)where u is the gas velocity, u. and ui are the average gas velocity and the interfacial

gas velocity respectively, U is the demensionless interfacial gas velocity, r is the

distance in the radial direction, and ri is the radius of the gas passage. The values of u.,

ui and ri can be calculated from the followingequations2):

U = (GaG/16ReG)(p"/p")a2{2p*F(1 - a2)

+ (1 -p") (1 -a2 +a2 ln a2)], (3)

ReG = (GaG/16) [Fa2Ia2 +2u*(1 - a2)l +(pt*/p*)

× (1 -p") a2 (1 -or.2 +a2 ln a2) )], (4)

ReL = (GaL/32) [2p*F(1 -a2)2 +(1 -p*)

× l(1-a2)(1 -3a2)- a` ln a` i]. (5)

In these equations, ReG and ReL are the gas-phase and liquid-phase Reynoldsnumbers

defined by

ReG =4MllndptG, ReL =4P/ptL, (6)and GaG and GaL are the gas-phase and liquid-phase Galilei numbers defined by

GaG =pG2d3glptG2, GaL :pL2d3glptL2, (7)

Mbss T)ranstler into Laminar Gas Streams in Wetted- Wbll Cblumns 81

where Mi is the mass flow rate of the gas, r is the mass flow rate ofthe liquid per unit

perimeter, d is the column diarneter, g is the gravitational acceleration, "G and uL

are the viscosities ofthe gas and liquid, and pG and pL are the densities ofthe gas and

liquid. F is the dirnensionless shear stress defined by

F= ARf/2pGgZ, (8)where Z is the total height ofthe column and APf is the frictionalpressure drop for

the gas stream through the column. Further, a is the ratio of the gas passage diameter to

the column diarneter given by

a= 2ri/d, (9)and p* and pt" are the density and viscosity ratios defined by

P"=PG/PL, #" == ltGIS2L. (10) In cocurrent flow of the gas and liquid, the dimensionless interfacial gas velocity U is

always positive and increases with decreasing ReG and increasing ReL. When U is

equal to zero, the flow situation of the gas corresponds to the single phase flow in a circu-

lar tube, and when U is equal to unity, the gas flow corresponds to the plug flow with a

flat velocity proMe. Further, when U is equal to two, the gas velocity at the column

is zero, and when U is larger than two, the circulation exists in the gas phase, and the gas

near the liquid surface flows downward, while the gas in the core region of the column

flows upward, as shown in Fig. 1(a). Therefbre, in a wetted-wall column provided with

upper and lower calming sections, the upward gas flow in the central part of the column

and part of the downward gas flow adjacent to the upward gas flow would constitute a

circulation flow of the gas, and then the gas phase in the column may be divided into two

regions, the annular downflow region and the core circulation region, as shown in

Fig. 1(b). The radius r. ofthe boundary between the annular downflow region and the

core circulation region is given by

r.=riVi(i-=Z7S7fiTt75u) -/( ). (11)

2.2 Mass transfer analysis

Here, consider mass transfer from the gas phase to the liquid interface under the flow

situation for the case of U> 2, i.e. for the case ofcocurrent flow with circulation in the

gas phase. The existence of the core circulation region makes an exact solution to the

present problem much more difficult. In this paper, a simpler method of analysis similar

to that used in the previous work$ for the case of countercurrent flow is employed to

obtain an approximate solution.

Mass transfer in the annular downflow region is assumed to take place by convection

in the axial direction and by molecular diffusion in the radial direction. Mass transfer in

82 Haruo HIKITA, Kosaku ISHIMI and Norifumi SOHDA

the core circulation region, on the other hand, is assumed to take place only by molecular

diffusion in the radial direction, since the mean residence time of the circulating gas is

infinitely long and then the core circulation region can be treated as a stagnant gas film.

Thus, the diffusion equations for the present case can be written as:

rc f{: r Sl ri

DG(a2 Ch

OKrg rc

DG(

ar2

a2 cle

1 eCh ach+7 or )=" oz

ar2 1 OCZ+7 ar )=O,

'(12)

(13)

where Cl, and Cl, are the concentrations of the solute gas in the annular downflow

reglon and the core circulation region, respectively, DG is the gas-phase diffusivity of the

solute gas, and z is the distance from the inlet of the column. The boundary conditions

fbr Eqs. (12) and (13) may be written as:

z= O,

z> O,

z> O,

z> O,

rc S; r sgl ri ;

r= ri ;r= rc ;

r=O ;

Ch == Chi,

Ch = Ci,

Ch =q'ach/or = a( ,/or ,

oq/ar == o,

(14)

(1 5)

(16a)

(16b)

(17)

where Clf is the interfacial concentration and (hi is the inlet concentration in the

annular downflow region. The value of (,i is assumed to be uniform and can be

obtained from the following equation

Chi == Ci - ..2,i2 4e ru q, dr, (i s)

where Ci is the average inlet concentration at the top ofthe column.

After substituting Eq. (1) into Eq. (12) and solving the resulting equation and

Eq. (13) under the boundary conditions (14) to (17) and Eq. (18) by the method of

separatioh of variables, we obtain the solution for the concentration proMe of the solute

gas. The solution can be expressed conveniently in terms of the confluent hypergeome-

tric fUnctions9, M(a, b, x) and U(a, b, x) :

k-" qq =3001.,AnEn(i X.)exp(- e.G,Z,","2 ), (ig)

2, -- qc, = .jli,An4:(,',e,xn)exp(- D.G.Z,eX"2 ), (2o)

?Ifuss 77ansfer into Laminar GasStreams in Wetted-Ulall Cblumns 83

where 4i and A. are the eigenfuncitons and the expansion coefficients respectively,

and are defined by

Fh(t.,Xn) =[((t-4vX( EsU ) Xn)U(g-4th( U ) Xn,2,

pm( )x.(:f.)2)+Nbj(ri-:-t7J)Xn(t/)U(t-

4G(fSiU)Xn,1,pm( )Xn(t/)2)lM(-S--

4vft(iillt7iU ) Xn'i'pm( ) xn (t. )2)+ ((t-

4vit(¥7sU)X")M(-;--4vilfi(¥TiUu)Xn,2,pm( )Xn(t/)2)

-an( )Xn(-i/-)M(-S--4&(i¥EiiiU ) Xn,i,

pm( )Xn(-l}/')2)'tU(-ll--4th(U)Xn,1,

im( ) X. (tili.-)2)] exp [-Vr(i-=-05E) X. ((-Z.-)2 -(- b/ )21]/

1 2-U 3 2-U (-i--4pm( )Xn)[U(-iE--4pm( )Xn,2･

im( )Xn(t/)2)M(-lt-4<7ft(l¥vsU ) Xn,1,

pm( )Xn(-ii/fl)2)'U(-lt-4v i(i¥i}iU ) Xn'i'

pm( )Xn(-7/)2)M(-;--4vfi(ilSiU) Xn'2'

im( )Xn(-ll/-)2)]･ (21)and

An = -2/Xn (04i/aXn)r-ri' (22)

In these equations, X. are the eigenvalues and the roots of the following characteristic

equation:

(jFh )r-ri =O･ (23)The average concentra.tion C2 of the solute gas at the outlet of the column can be

obtained by multiplying Eqs. (19) and (20) by (2ut:lu.ri2) dr, integrating from r= r.

to r = ri and r = O to r= r. respectively, and adding the resulting two equations. The

final equation is given by

84 Haruo HIKITA, Kosaku lSHIMI and Norifumi SOHDA

giifi. -.;,Bn exp(- rr.X,"g ), (24)

where GzG is the gas-phase Graetz number defined by

GzG= W/pG DGZ, (25)and B. are the average expansion coefficients defined by

Bn = -2An(ri/Xn2)(OjF;i/Or),.,i ' (26)

The first six sets of values of X., A. and B. are given in Table 1 for four different

valuesof U.

Tabie 1. Valuesof Xn,An and Bn forvariousvaluesof U

n Xn An Bn

U= 2.5

U=3

U= 3.5

U=4

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

4.0257

10.058

16.314

22.639

28.994

35.365

4.5864

11.997

19.665

27.394

35.147

42.911

5.0906

13.68922.s4s

31.454

40.383

49.321

5.5511.

15.205

25.107

35.057

45.024

54.999

1.4445O

-O.71756

O.46723

-O.34396

O.27138

-O.22379

1.38924

-O.61721

O.38683

-O.28018

O.21925

-O.17996

1.36100

-O.56849

O.35113

-O.25300

O.19751

-O.16191

1.34383

-O.53955

O.33078

-O.23777

O.18542

-O.15192

O.726200

O.123542

O.047712

O.024931

O.O15246

O.OI0265

O.746306

O.116747

O.043998

O.022766

O.O13856

O.O09305

O.758820

O.112005

O.041698

O.021486

O.O13052

O.O08756

O.767292

O.108629

O.040153

O.020643

O.O12528

O.O08400

If the average gas-phase mass transfer coefficient kG ever the total height of the

wetted-wall column is defined in terms of the logaritlmic-mean concentration driving

force as

Mizss 1>nnstler into Laminar Gas Streams in Wetted- Wall Cblumns 85

- W(Ci - C2) kG-2vrripGz(Ac)i. ' (27)

with

(AC)i. = (Ci - C2)/ln [(Ci - CV )/(C2 - Ci)] , (28)

then the average Sherwood number may be expressed as

ShG = kG(2ri)/DG =: GzG(Ci - C2)1rr(AC)i. , (29)

and substitution of Eq. (24) into Eq. (29) gives

co Shc = -(GzG/T)ln[ Z B. exp (-fiX.2/GzG)] . (30) n=1Therefore, the average Sherwood number can be calculated from Eq. (30) with the use of

the values of X. and B. given in Table 1 asafunction of the Graetz number fbr four

valuesof U.

The computed results fbr the average Sherwood number Shc are shown in Fig. 2 as

a function of the gas-phase Graetz number GzG. The solid lines represent the approxi-

e=ut

100

60

40

20

10

6

4

2

U= 4

--- -2--

.--1

.--os."･ov

-3･5 -/.-3 /-:-J-!2･5 xt-J------1./., . ,. - ;-.".-- -f---- -f

- f..- -

-- -- f- -- .-' -- "t .- J-

46 10 20 40 60 100 200 400 GzG

Fig. 2. Approximate or exact analytical solution for ShG as a

functionof GzG forvariousvaluesof U.

mate analytical solution given by Eq. (30). The dashed lines show the analytical solution

presented in the previous paperi) for the case of cocurrent flow without circulation in the

gas phase, i.e. fbr the case of O < US 2, and the dash-dot line represents the analytical

solution for the case of U = O, i.e. for a single phase fiow through a circular tube. It can

be seen that also in the case of cocurrent flow of U > 2 the ShG value increases with

increasing value of U.

86 Haruo HIKITA, Kosaka ISHIMI and Norifumi SOHDA

3. Experimental

3.1 Apparatus and procedure

The wetted-wall column used was constructed of glass pipe, 4.0 cm in inside diameter

and 80 cm in length. The upper and lower gas calming sections of about1 m long were

provided.

The absorption of methanol vapor from air into water was studied. This system is

considered to be gas-phase controlled. The methanol vapor was taken from an electrically

heated evaporator and was mixed with air. The composition of the inlet gas stream was

approximately constant at a value of 3 vol% methanol. The methanol contents in the gas

and liquid streams were determined by the use of a gas chromatograph. To elminate the

rippling on the surface of the falling liquid fdm, O.05 vol% Scourol 100 (surface-active

agent) was added to the absorbent.

The liquid-phase Reynolds number ReL was kept at five constant values of 200,

300, 400, 600 and 800. The gas-phase Reynolds number ReG was varied from 3oo to

1OOO. The temperatures of gas and liquid were controlled and maintained at 200C.

3.2 Calculation of Sherwood and Graetz riumbers

[he average Sherwood number ShG was calculated from Eq. (29) with the

the following equation giving the logarithmic-mean driving force (AC) im :

use of

(AC)im "=(Ci - Ci") -(C2 - C2")

ln [(Ci - Ci")/(C2 - C2")] '(31)

where C* is the solute gas concentration in the gas phase in equMbrium with the bulk

liquid.

The equilibrium concentration of methanol vapor in the gas phase was calculated by

using the equation for the Henry's law constant presented by Fzljita$. In the calculation

of the Sherwood and Graetz numbers, the diffUsivity of methanol vapor in air at 200C

was taken as O.153 cm2/sec6). '

4. Results and Discussion

The experimental results are shown in Fig. 3, where the values of the average

Sherwood number ShG are plotted against the gas-phase Reynolds number ReG fbr

five values of the liquid-phase Reynolds number ReL. As can be seen in this figure, the

value of ShG increases with increasing ReL and decreases with increasing ReG until a

minimum value of S7iG is reached. The solid lines below the dashed line are the theore-

tical linesi) for the case of cocurrent flow without circulation in the gas phase, i.e. for the

case of O < U S; 2, and the dashed line represents the theoretical relationship between

ShG and ReG in the case of U :2. The solidlines above the dashed line, on the other

Mbss 7hander into Laminar (las Streams in Wetted- PVbll Cblumns 87

60

40

.e 2out

10

8

6

AA oU=2.-- - '

Kcy ReL

u eoOA 600A 400e 3ooo 200

-

2oo roO 600 8oo 1000 am ReG

Fig. 3. Effectsof ReG and ReL on ShG fbrabsorptionof methanol vapor into water at 200C.

hand, represent the approximate theoretical lines for the case of cocurrent flow with

circulation in the gas phase, i.e. fbr the case of U> 2, and are calculated from Eq. (30).

The measured values of Sh6 in the region where U> 2 are in good agreement with the

theoretical lines.

Figure 4 shows the values of Sh6 interpolated at three constant values of U from

eut

50

40

30

20

10

Key U a3 A 2.5 02

-･---o-----.-o--o--------

o

910 20 30 40 50 GzG

Fig. 4. Comparison between experimental and theoretical values of ShG.

the experimental data as a function of GzG. The solid lines are the approximate

theoretical lines representing Eq. (30). The agreement between the theory and the

experimental data is good, the average deviation of the data points from the theoretical

line being 1.9%.

88 Haruo HIKIEA, Kosaka ISHIMI and Norifumi SOHDA

5. Conclusion

'Ihe effect of the gas and liquid flow rates on the mass transfer rate in laminar gas

streams in wetted-wall columns with cocurrent gas-liquid flow has been studied theoreti-

cally and experimentally for the case where the circulation exists in the gas phase.

[he theoretical analysis indicates that the approximate analytical solution derived for

the average gas-phase Sherwood number ShG can be expressed in terms ofthe confluent

hypergeometric fUnctions as a fimction of the gas-phase Graetz number GzG and the

dimensionless interfacial gas velocity UL The ShG value increaseswithincreasing ReL

and decreases with increasing ReG.

The experimental results obtained with the metlianol vapor absorption runs show

that the experimental dependence of ShG on both ReG and ReL is in good agreement

with the theoretical predictions.

Notation

An

Bn

CCh･q

qC*

Cl,C2

Dcd

F

4GaG

GaL

GzG

gkG

M(a,b,x)

n ReG ReL

･r re

ri

:

:

:

:

:

:

:

:

:

:

:

:

:

:

:

:

:

:

:

:

:

:

expansion coefficients in Eqs. (19) and (20) and defined by Eq. (22)

average expansion coefTicients in Eq. (24) and defined by Eq. (26)

concentration of solute gas in gas phase, g-mol/cm3

values of C in annular downflow region and core circulation region,

g-mol/crn3

value of C at gas-liquid interface, g-mol/cm3

value of C in equilibrium with liquid, g-mollcm3

average values of C at inlet and outlet of column, g-mol/cm3

gas-phase diffusivity of solute gas, cm2 /sec

column diameter, cm

dlmensionless shear stress defined by Eq. (8)

eigenfunctions defined by Eq. (21)

gas-phase Galilei number defined as pG2d3g7(ptG2

liquid-phase Galilei number defined as pL2d3g)lpL2

gas-phase Graetz number defined as M,:lpGDGZ '

gravitational acceleration, cm/sec2

average gas-phase mass transfer coefficient, cm/sec

confluent hypergeometric function of argument x, parameters a and b

index of series

gas-phase Reynolds number defined as 4 "LITdptG

liquid-phase Reynolds number defined as 4P/"L

distance in radial direction cm 'radius of core circulation region, cm

radius of gas passage, cm

ShG

UU (a, b, x)

u

ui

um

wZz

Greek

a

r

(AC) im

MfXn

StC,StL

u*

P6,PLp*

:

:

:

･:

:

letters

:

:

:

:

Mbss 77ansti?r into Laminar Gas Streams in Wetted- Ulall Cblumns

average gas-phase Sherwood number defined as kG (2ri)IDG

dimensionless interfacial gas velocity defined as ui/u.

Ipgaritlmic type confluent hypergeometric function ofargument x,

parameters a and b

gas velocity, cm/sec

interfacial gas velocity, cm/sec

average gas velocity, cm/sec

mass flow rate of gas, g/sec

column height, cm

distance in flow direction, cm

ratio of gas passage diameter to column diameter defined as

mass flow rate of liquid per unit perimeter, g/cm sec

logarithmic-mean concentration driving force, g-mol/cm3

ftictional pressure drop for gas stream through column, g/cm sec2

nth eigenvalue

viscosities of gas and liquid, glcm sec

viscosity ratio defined as iiG/i2L

densities of gas and liquid, g/cm3

density ratio defined as pGlpL

89

2ri/d

1)2)3)4)

5)

6)

References

H. Hikita and K. Ishimi, J. Chem. Eng. Japan, 9, 362 (1976).

H. Hikita and K. Ishimi, ibid., 9, 357 (1976).

H. Hikita and K. Ishimi, Chem. Eng. Commun., 2, 181 (1978).

M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",

Standards, Washington D.C. (1964). 'S. Fiijita, Kagaku Kogaku, 27, 270 (1963).

"International Critical Tables", Vol. 5, p.62, McGraw-Hill, New York (1928).

National Bureau of