Mass Transfer into Laminar Gas Streams in Wetted...
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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