Cellular Convection in Desorbing Surface TensionLowering Solutes from Water P. 1. 1. Brian, J. E. Vivian, and S. T. Mayr IlepartitLeirt oj’ Chemiccil I:’ngineering, .If nssacliusetts Institute of Teclinology, Cnmbritlge, .Ilriss. 02139
The effect of cellular convection, driven b y surface tension gradients, upon the gas-liquid. mass transfer rate was studied b y desorbing four surface tension-lowering solutes from aqueous solution and monitoring the transfer process in each phase b y the simultaneous transfer of tracer components. kL was enhanced as much as 3.6-fold b y the cellular convection when the Marangoni number was increased above its critical value. kf, was unaffected b y the cellular convection, in agreement with predictions based on estimation of the linear and velocity scales of the cellular convection. Critical Marangoni numbers showed the expected dependence on Hk,,*/k,* and the surface tension equilibration time but were very much larger than theoretically predicted values.
Results and Discussion
M a s s Transfer in Absence of Marangoni Instability. Figure 2 Iire>riit> reiiilt> f o r tlie dcwirjitioii of ~ i ~ ’ ~ i ~ i ~ ~ I ~ ~ i i c ~ from water, ivhich is coiitlolletl by t i i t s liquitl lih:i>c, 1,e-i.t,aiice aiid >lion.. i i o 3Iaraiigntii c,ffect.-. T h e \.ariablc> ;ire grouiieti a > > h o \ v i t i t i order t o rorrcct (Gi1lil:iiid cd d . ,1958) for variation.: i t i tile coluniti Iieiglit, tlic, liquitl ~ i l t n i editI)rwluct for the liquid fli..’ 211.’ it-, :itid tlie visco>it!.-drit.ity i ) l i a b e > hut the-e quatitities did i i c i t v:ir>. aIi1ircci:il)lj~. tialiy>therefore, Figure 2 i> :I plot of t h e prop~.IciieiIc>o coefficient 2s. tlie liquid flow rkite i n the hliort wettml-\r:iIl column. The upper liiie reprebents :I tlieoreticnl ec1u:itioii (Gillila~idet al.? 1958; Tiviati aiid Peaceniait, 1956) 1 x 1 4 on l)elirtratioii tlieory and uitaccelerated ,-tre:imliiie flo\v theor).. The data poitits define a liiie 6 tu 97c below tile lieiietrtitioii theory liiie, iii good ayreenieiit witli tlic rrsult‘ of other iiivestigatiow of liquid I~li:isr-coiitr~illtdniasi tr:ius:fer iii a hart wetted-n.al1 column. Ilata :ire 4iowii iii Figure 2 for propylene desorption into dry nitrogen :iiid iiitrogeii liresnturated with n-ater vapor. The rewltb agree, cotifirmiiig calculatioiis (lI:iyr, 1970) Irhicli shon- tli:it cooling of the interface due to water vaiiorizatioii into the dry nitroyeti was negligibly small. Figure 3 1iresent.i re.;nlt:, for water valiorizatioil into nitrogen gas; which i- coiitrolled hy the gas phase re short wetted-wall colunin. The effert of liquid flon- rate upoii tlie gas pliase dlierwmd nriinber is sliowii for >everid gas phase Reynold. nuinhers and therefore Graetz numbers. The Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
75
I
I
l3 I
0'
02
0
WT
Figure 1.
04
08
06
I
I
1
0 Calculated Value 8nclud,ng
Ef'ect
NatLral C o n v e c t on
-
1.0
S O L U T E IN W A T E R
Surface tensions of dilute aqueous solutions a t
25°C Figure 3.
Evaporation of water into nitrogen
N
Effect of liquid flow and gas flow rates
I
t
I
I
I.' c ;
k
_I:k
-1
s-1
I
I 11
3
m
Dry Nitrogen rn Wet N i t r o g e n 0
1
] 1
X
I
Y
u p 100
10
r / E(g/crn
-.
P
min
I
Figure 2. Desorption of propylene from water in dry and wet nitrogen in absence of Marangoni instability
I 31 10
I
I
I
20
30
43
r/Jr ( s i c m m r
Iirokeii liorizoiital liiie represents the Graetz solutioii for each Graetz iiuniher, and tlie 1)oiiits sliowi a t zero liquid flow rate represent the reaiilt of llartinelli aiid 13oelter (1942), which corrects tlie Graetz solution for natural convection. The (lata points show n nioderatc effect of liquid flow rate upon the gas phase i n a s traiisfer coefficieiit, iiidicatiiig that the liquid surface velocity is affecting the g a s pliase fluid iiiecliaiiics, probnbly creating circulatioii in tlie g ~ i splinse. Desorption of Ether. Figure 4 preseiits the over-all iuass traiider coefficient for the desorption of ether from water iiito iiitrogeii gas. ('urves are showii for various ether concentrations in the liquid feed, Ji-hicli reprebent the mass transfer driving force hecause the iiitrogeii feed gas coiitained 110 ether. The bottom curve, for ether coiicentratioris of 0.009 aiid 0.012 weight yo,is in agreement n.ith the overall rnass transfer coefficieiit computed by adding the liquid phase and tlie gas phase resistances as obtained from Figures 2 and 3, suitably corrected for changes in the diffusivities. At higher ether concentrations, howeyer, the over-all coefficient for ether desorption is higher than that predicted by adding the liquid aiid gas phase resistances as given in Figures 2 and 3, and indeed tlie ether desorption coefficient shuns a substantial variation with the inass traiisfer driving force. For the highest driving force studied, 0.94 weight % ether in the liquid feed, K L is P pproxiniately twice the value obtained ivith very IOK inlet conceiitrations of ether. This 76
Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1971
50
, I 1 60 70 80
)
Figure 4. Desorption of ethyl ether at various ethyl ether concentration levels
variation of the ether desorption coefficient with the driving force suggests the presence of JIarangoiii iiistability causing cellular convection a t the higher ether coiiceiitrntioiis, but it is not possible froni thc,3e results to deterniine whether thc convection i-. enhancing the liquid plinse iii coefficient, the gas phase coefficient, or both. The propylene and water vapor tracer studies supply this additional informa tion. Figure 5 presents the propylene desorptioii coefficient measured while propylene desorption was occurring siniultaiieously with the desorption of ether. The bottom line represents the desorption of propyleiie in the absence of simultaneous ether desorption, as given in Figure 2. Es~~erirnental results with simultaneous et'her desorption agreed with this bottom line when the ether concentration in the liquid feed was 0.012 weight % or less. .%t higher concentrations of ether in the liquid feed, the propylene desorption coefficient was a sensitive function of the ether concentration but independent of the propylene concentration. TT%h 0.94 weight % ether in the liquid feed, the propylene desorption coefficient was enhanced 3.6-fold. This enhancement is attributed to cellular convection in the liquid phase driven bl- surface
W t % Ether in Liquid Feed
Sroichiometr'c Removal of
.-
r
22
-0 -0 -0
-
/
~
1
94
I
42 24
I 3
-011
*f ~ ~ 2 -0077
004 4 00 2 9
N s e =~ 500
,
3
33
23
8 13
a s coo
63
30
r/it; ( g i c r n m i n i
EThE;?
Figure 5. Desorption of propylene with simultaneous desorption of ethyl ether
01
0 0 0 5 001
150
CONC
10
IN L QUID
=EED , A t
Figure 6. Enhancement of propylene desorption due to simultaneous desorption of ethyl ether Arrows indicate two runs for which no enhancement was observed
77--7--I_ c,
N R ~ G = ~ ' ~
''
(
1
Zero E t h e r Conc
\
. -E" L? 0
N G =~ 1 4 4
m
,Groetz
Solutlon
~
0 X
c
1 VI
2
7-
I
Wt
Oi.
Erher
0 0012 0
5 c
-
0016
4L 0
-
-
10
/
23
-
I
-.-C
4 U
30
I
40
31
50
10
-
/
5-
1
0029 a 0.077 0 011 0
./
/
Ether Ccnc
- 0 94
N R ~ G
:500
L
.
Wt
Oi0
i
1L - L L J
20 30 4 0 50 r / , F ( g i c m m n,
70 90
Figure 7. Evaporation of water in presence and absence of ethyl ether desorption
Figure 8. sorption
tension yrudieiits produced by the ether desorption. Defining Q L :I> the ratio of tlie propylene de5orption coefficient, :it a given ether concentration to that a t zero ether coiiceiitriitioii, a t the Lame liquid flow rate, it i i seen from Figure 5 that QI,\-ark> little ivitli the liquid flow rnte but varies subktatitialij. nitli tlie ether de-orptioii driving force. Choo-ing a liquid flow rate iii the middle of the range studied, the results in Figure 5 are crosqplotted in Figure 6 to sliow the variation of Q L witli ether coiicentratioii. The propylene desorption coefficient>in Figure 5 are for a gas phase Reynolds number of 500, but t1ie.e results ivere essentially independent of gas pha-e f l o ~rate over a r m g e of Reynolds nuinllers from 350 to 900, as were the coefficients in Figure 2 . Figure 7 pre>eiit. tlie Sherwood number for 11-ater vaporization simultaneous1~- with the desorption of ether. For reference, the appropriate curve from Figure 3 is ,shown for water vaporization in the absence of ether desorption. Data points are 5Iion.ii for ether conceiitratioiis in the feed liquid
froin 0.012 to 0.11 weight 5;; at higher ether coiiceiitrations, the water vn1)orizatioll rate could iiot be tleterniiiietl :iccurately because the ether iiiteiieretl wit11 the oijeration of the electrolytic n-ater vapor analyzer. Over tlie raiige of ether coiicentratioiis ii1j to 0.11 weight yc,Fiqures 5 a l i i 1 6 41ow U ~ I to a twofold enliaiicemeiit iii tiie liquid ph:me inas> tr:m>fer coefficient, h i t Figure i sliow no euliaiicenieiit in tlie gas pliase mass transfer coefficient as iuonitored hy the rate of water vaporizntioii. Figure 8 shon-s :i test of the additivity of tlie liquid and gas phase rehtaiices for ether desorption iii tlle pre-eiice of tlie cellular convection produced hj- the 3Iarangoiii effect . The upper curve reprebents the liquid phase ni efficient for ether desorption, obtained from the propylene de>orption coefficients sliorrn in Figure 5 niter correction by tiie >quare root of the ratio of the diffusivities for ether and for prol)yleiie. 'This curve therefore represents a value of k~ which is enhanced :rpproximately 3.6-fold by the cellular
Test of additivity equation for ethyl ether de-
Ind. Eng. Chem. Fundarn., Vol. 10,
No. 1, 1971 77
10
9L
I
1
cu
: - 6k
Line From F i g u r e 2
U
cn . E
U
a,
. a, 1
-E
I
I
n
I
'I
\
7 c
U
U
i
1
0021 0 0 2 3
A3:6 0318
3L 10
I 20
r/F
1
V 0.10
00.20
20
30
t0.41
1
f33C
I
1
I
30
40
53
1
( S i c m min)
10
r/p ( g i c m min )
40
50
Figure 9. Desorption of propylene with simultaneous desorption of methyl chloride
Figure 10. Liquid phase coefficients calculated from methyl chloride data
convection. The second curve from the top in Figure 8 represents tlie product of the Henry's law coilstant for ether and the gas phase niass transfer coefficient, corrected to the diffusioii coefficieiit for ether but unaffected hy the cellular coiivection a- indicated in Figure 7. The next two curves show the experimental over-all coefficients for ether desorption, a s reported in Figure 4> and the over-all coefficients calculated by reciprocal addition of the liquid and gas phase coefficieiits .51ion.11 iii the upper two curves. The agreement of the experimental and calculated over-all coefficieiits for ether desorption is excellent. For reference, the bottom curve iu Fig\irc 8 ~liowsthe over-all coefficient for ether desorption which rvould have been expected if the liquid phase coefficient had not been enlianced by tlie cellular convection. Tllis curve call be compared Ivitli the bottom curve in Figure 4, where it is been to be ill good agreenieiit with tlie experimentally iiieasured over-all coefficient for very low liquid phase ether concentratioiis, where the llarangoiii effect was negligible. Figurc 8 slions very good agreement with the additivity of resistances a t the Iiiglie.st ether concentration studied; equally good ayreenieiit was foiiiid (Ala>-r)1970) a t all other ether concentrations. Desorption of Methyl Chloride. Figure 9 presents results for propylene desorption with the simultaneous desorptioii of nirthyl chloride. T h e solid line represents propylene demrption from water and is taken from Figure 2 ; the broken line.: \ha\\- the 9570 confidence limits on the propylene desorption coefficient froni pure water. The data point? for nietliyl cliloride conceiitratioiis uli t o 0.41 weight yo show that the niethyl chloride desorptioii process did iiot affect tlie propylene desorption coefficient, indicating that 110 ceIlular convection was preseiit in the liquid phase. Figure 10 presents the correspoiiding methyl chloride desorption coefficient. The value lxesented, k ~ represents , the liquid phase mass transfer coefficient and was obtained hy correcting the over-all methyl chloride desorption coefficient for gas phase resistance. The correction amounted to only 5% and could therefore be made with confidence. The methyl chloride liquid phase desorption coefficients thus computed are divided by the square root of the diffusivity of methyl chloride in the ordinate of Figure 10 and are therefore comparable to any liquid phase mass transfer coefficient in the short wet'ted\Tall column. The data in Figure 10 are compared with the line for propylene desorption from Figure 2 together with the 95% confidence limits. Excellent agreement is seen between these methyl chloride desorption coefficients and the line for propylene, again indicating that methyl chloride desorption
is not accompanied by any cellular convection, K a t e r vaporization meai;urements sirnultaneous v i r h the methyl chloride desorption experiments agreed with the data reported in Figure 3 for the vaporization of pure water. These results indicate that the gas phase mass transfer coefficient behaved normally, and therefore that llarangoni effects were iiot irnportant in the gas phase during methyl cliloride desorption. Desorption of Acetone. Figure 1 1 presents results for propyleiie desorption during t h e simultaneous desorption of acetone. As in t h e case of ether, high concentrations of acetone in the liquid feed resulted in propylene desorption coefficient3 substantially greater than those measured in the abseiice of acetone. .It tlie highest acetone concentration of 0.86 neight 7c,~ I determined L from the prop\-lene desorption coefficients slioivii in Figure 11 varied from approximately 2.1 a t the low liquid flow rates to approximately 1.7 at the highest liquid flow rate,$. The experiments with 0.34 weight' % acetone eniployed a nitrogen stream presaturated with water vapor, while the other experiments employed dry iiitrogeii. The agreement among tlie two sets of results indicates tlint interfacial cooliiig due to water vaporization was negligible iu these experiments also. Although the cellular convection liad a subatantial influence upon tlie liquid phase mass traiisfer coefficient! simultaneous water valmrizatioii nieasiirements agreed with those presented in Figure 3, indicating that the cellular coiivectioii had a negligible effect upon the gas phase i n coefficient. The acetone desorption process was controlled by the gas pliase niass transfer resistance to the extent of 90 t o 957c, and therefore the substantial eiihancenient in k~ slionn in Figure 11 did iiot appreciably affect the over-all coefficients for acetone desorption. Indeed, only with the propylene tracer could the cellular convection produced by acetone desorption be detected. Gai phase coefficients calculated from the acetone over-all coefficients by subtracting the liquid phase resistance agreed with the values of kc obtained from water vaporization measurements when corrected for the difference in the gas phase Sclimitt numbers. Desorption of Triethylamine. Figure 12 presents results for propylene desorptioii during the simultaneous desorption of triethylamine. .Is n-ith acetone and ether, t h e triet h yla mine desorption resulted i i i a substantial enhancement in t h e propylene desorption coefficieiit, +L varying from approximately 2 a t low liquid flow rates to approsirnately 1.7 a t high liquid flow rates a t the highest triethylamine concentration studied (Figure 12). Scale of Convection Cells. Cellular convection enhanced
78
Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1971
Wt. Ol0 A c e t o n e in Liquid F e e d
Propylene
c1 c
v
2
N ~ ~ ~ = 5 1 5
* Gas 38 1 1 8 910 Figure 1 1. acetone
20I
I
Phase S a t u r o t e d witi- h a t e r
I
I
20
30
I
I L
I
5 0 60
40
80
Desorption of propylene with simultaneous desorption of
1
____ I
S t0 1c h iomet r ic of Propylene
7
1
Re mova I -
N, h
U
% . E U
v
NReG
3'
4
!
5
6
7
,
!
lJ
I
I
8910
513
20
30
40
50
r / E ( g i c m min) Figure 1 2. Desorption of propylene with simultaneous desorption of triethylamine (TEA)
the liquid phase mass transfer coefficients severalfold, a t t h e same time having a negligible effect upon t h e gas phase mass transfer coefficient,. T h e most likely esplaiiatioii for t h e failure of t h e convection t o influence X.0 \vhile substantially increasing k~ appears to be that the linear dimension of the convection cells was too small and the velocity of the conrectioii too low to affect k ~ even , though X.L was increased substantially. Based on the work of Bakker et al. (1966), it is reasonable to suppose that the depth of the convection cells in the liquid layer should be approximately equal to the mass transfer penetration depth. The mass transfer penetration depth in the liquid phase is approximately 0.001 ern (Table I), and a best guess at the depth of the convection cells would be this value. Surely the cell depth could be no greater than the liquid
layer thickness on the wetted-wall coluiiiii, 0.025 cin. T h e linear diinension of the convection cells in tlie plane of the gas-liquid interface is expected t o he coinparable to the depth of tlie cells in the liquid phase, and the linear dimension of tlie convection cells in the gas phase would have to be app-osiniately the same, for the convection cells to circulate synchronously in the two phases. In view of the mass transfer penetration depth in the gas phase of approximately 0.4 cni (Table I), it is not surprising that convection cells of such small linear dinieiision have relatively little influence upon the gas phase mass transfer coefficient because they probably penetrated no niore than 0.3y0into t,lie gas phase bouiidary layer and surely could have penetrated no niore than 7%. Even when the convection cells 1)enetrate t'lie entire inass transfer boundary layer, the convection would have a n irnInd. Eng. Chern. Fundarn., Vol. 10, No. 1, 1971
79
4.0
I
I
l / / / / / ~
I
I
I
I I I l l
I
l 1 l 1 1 1 1
I
1
1
I ETHYL ETHER
o
-
r/fi= 3 0 g / c m rnin.
r / Q= 5 0 g / c r n rnin
Rz 1.1
I I
OL
2 .o
/
-.
ii+ MARANGONI
Figure 13. number
~-Phase
~
Liquid
Gas
30 0 026 0 01 0 001
30 1 3 0 6 0 4
portant effect upoii tlie inass traiisfer coefficient only if the velocity scale of the convection cells were comparable to the transfer coefficient expressed in velocity units. Defining a convection velocity, V , as the t o t d quantity of fluid exchanged between the bulk and the interface per unit time per unit of total area of the gas-liquid interface, the flus of the transferring solute due to the cellular convection would he given by
xc! =
V(C,
- Cj)
(1)
The diff usive flux, on the other hand, in the absence of l l a r a n goni effects is given by N u = k(Ce
- Cj)
(2)
Since the cellular convection was observed to enhance the liquid phase mass transfer coefficient severalfold, it is inferred that the velocity of the convection cells, V , mas about equal to the liquid phase mass t'ransfer coefficient in the absence of Narangoni instability, R-hich was approximately 0.01 cm per second. The gas phase mass transfer coefficient in velocity units, on the other hand, was approximately 0.6 cm per second in the short wetted-wall column, and convection cells with a velociby scale of the order of 0.01 cm per second would be expected to have negligible influence on the gas phase mass transfer coefficient even if those cells were large enough to penetrate through the gas phase boundary layer. These considerations suggest that the convection cells 80 Ind.
Eng. Chem. Fundom., Vol.
1 1
I
I
1
NUMBER,NM~
Liquid phase enhancement factors as a function of Marangoni
Table 1. Approximate Mass Transfer Parameters for Two Phases
Flow velocit) of pha\e, cm/sec Depth of pha>e, cm Mas5 traii4er coefficient, k , cm/sec Penetr:ttioii depth. z , cm
I
10, No. 1, 1971
probably had a linear dimension of approximately 0.001 cm and a velocity scale of approximately 0.01 cm per second, the cells being too small and the velocity being too low to affect the gas phase mass transfer coefficient appreciably. These roiisiderations also suggest a time scale for the cellular convection which is approximately 0.1 second, the same as the liquid phase contact time. Correlation of Results. Figure 13 prc>eiits a correlation of t h e liquid phase enhancement factors rs. t h e JIarangoni number. T h e definition of t h e JIarangoiii number employed here is h a m i oii t,he use of the liquid phase mass transfer penetratioii depth as t h e linear dimension, and the penetration depth was approximately 7% of the liquid depth on the column wall. Figure 13 shows that the curve of + L us. llarangoni number is different for the different surface active solutes. These differences are believed to be due largely to variations in R , the ratio of the liquid phase resistance t o the gas phase resistance to mass transfer for the solute in question. Table I1 shows some of the physical constants for the various solutes employed and includes the percentage of the mass transfer resistance in the gas phase a t average liquid and gas phase flow rates and in the absence of Marangoni effect's. Methyl chloride desorption is largely liquid phasecontrolled, acetone desorption is largely gas phase-controlled, and ether and triethylamine desorption are intermediate. 4 system would be expected to be more stable to Xarangoni effects for higher values of R , and indeed, with a n infinite gas phase mass transfer coefficient, concentration gradients in the surface could not exist, and surface tension-driven convection would not be possible. This is in agreement with linearized stability analyses (Pearson, 1958; Scriven and Sternling, 1964; Smith, 1966) which show the critical Marangoni number to increase with increasing values of R . The results in Figure 13 do not, however, line u p in order of increasing R; the order of ether and triethylamine is reversed. This is also seen in Figure 14, where the critical hlarangoni number is plotted against R. The critical values of the Marangoni number were determined from Figure 13 as the values at which $ L approaches unity. Also shown in
Figure 14 are the values of the Marangoni numbers corresponding to the maximum concentrations employed for methyl chloride and propylene desorption, although for these solutes no enhancement in the liquid phase mass transfer coefficient was observed. The broken line in Figure 14 is drawn through the data for acetone and ether. The methyl chloride data fall only slightly below the line, indicating that perhaps the methyl chloride system was not far below the critical llarangoni number a t the maximum concentration employed. The propylene Narangoni number, on the other hand, is almost two orders of magnitude below its critical value, as estimated by the estrapolation of the broken line, in agreement with the experimental observat,ion that the propylene concentration had no influence upon k ~ either : for desorption of propylene alone or for simultaneous desorption of propylene and one of the other solutes (Mayr, 1970). It would appear in Figure 14 that either the critical llarangoni number for triethylamine is abnormally high or that for ether is low, indicating the influence of an additional dimensionless group upon the attempted correlation in Figure 14. It is believed that the triethylamine critical Marangoni number is abnormally high. I n Figure 15 the ether and acetone data from Figure 13 have been replotted after normalizing the llarangoni number with respect to its critical value as given in Figure 14. T h e slope of the curve for acetone is approximately 0.25, and the slopes of the curves for the ether data are in the vicinity of 0.5. Thus 4~ for t,he ether system is more sensitive to increases in ATMaabove its critical value than is $ L for acetone. There is no reason to believe t h a t the slope of such a curve should not, be a function of R, but, neither is it known how the slope would be expected to vary with R. Therefore more data will be required to determine whether or not R alone is responsible for the higher slope for ether than that for acetone. Abnormal Behavior of Triethylamine. 'The triethylamine d a t a show stability to t h e Jlaraiigoni effect greater t h a n t h a t which \vould be expected if t h e Maraiigoiii number and R were t h e only parameters to affect, $ L (Figures 13 aiid 14), probabll- becauhe a freshly formed surface of a solution containing a surface tension-lowering solute displays a dynamic surface tension higher than the static value until diffusion from within the liquid establishes the equilibrium excess surface concentration as given by the Gibbs equat,ion. The time required for establishment of the equilibrium surface t,ension depression depends not only upon the diffusivity of the solute but also upon the magnitude of the excess surface concentration, which is proportional to the derivative of surface tension with respect to concentration, 6
I
1
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10'
1o6
I o5
DATA OF CLARK AND KING TRlE THYLAMINE
,/
1
' METHYL CHLORIDE
PROPYLENE /
,
"'
I
iI
/
I o3
,
/
,
I
2
I 0"
/
/
H
,'ACE TON E
1o2
,0
IO'
C r i t i c a l Maiangoni NumbQrs M a x i m u m M a r a n g o n i N u m b e r s in AbsQnce O f lnstab l l t y
1
L U lo2
16'
1
IO'
lo2
lo3
R = Hk,*/k: Figure 14. Influence of distribution of mass transfer resistance on critical Marangoni number
Table II. Properties of Solutes at 25"C, 1 A t m
% Solute
l o h X DL"
Do"
H C
Mass Tranfer Resistance in G a s Phased
Propylene 1 . 4 4 0.122 i.32 0.3 Methyl chloride 1 . 6 0 0.130 0,385 5 Ethyl ether 0 . 9 6 0.088 0.0367 36 Triet,hylamine 0,82 0,074 0,0065 i i .1cet o ne 1.27 0.104 0.0012 95 Water .., 0.231 -10-5 -100 D L from iitidersoii ct U L . ~ 1938; Byers a i d King, 1067; Iteid and Sherwood, 1966; Viviaii and King, 1964. * DC from Reid and Sherwood, 1966. e H from Glen and lIoelw-yii-Hughes, 1!153, "Interiiatioiial Critical Tables," 1928; Lattey, 1907; arid Llayr, 1970. ,4t r di = 30, *YI~Ec; = ,500, in absence of Llaraiigotii effects.
--
(1
according to the Gibbs equation. Batkker et al. (1966) have analyzed theoretically the time required for the establishment of the equilibrium surface tension. Employing their method, calculations ( X a y r , 1970) show that the time required for the surface tension depression to reach 90% of its equilibrium value is substantially less than a millisecond for
I
I
I
I
I +Data
5-
of Clark and Kin
NMa/ N M a ( C c )
Figure 15. Effect of Marangoni instability on liquid phase mass transfer enhancement factor Ind. Eng. Chem. Fundam., Val. 10, No. 1, 1971
81
-
Critical Marangoni Absence
.-3l
of
Numbers
Instability
/ /
C L A R K anb KING
I
1
I
I
1
!
Figure 16. Comparison of experimental and predicted critical Marangoni numbers
the solutes employed in this study, with the exception of triethylaniine a t low conceiitrations, for which the calculated equilibration time was 11 milliseconds. The time required for the convective flow across the cells was estimated to be approxiinately 100 milliseconds, based on the inferred h e a r aiid velocity scales, and it is possible that the equilibration time for trietliylamiiie a t low concentrations was too loiig for that solute to exert its equilibrium surface tensioii. thus stabilizing the system against the Marangoiii instability. Generality of Correlation. A general correlatioii of c $ ~vs. .Yyz aiitl R , of the type attempted in Figure 13, would be useful. The experimental results ol' this study indicate t h a t a t lenst one additional pnranieter is iiivol\-ed, aiid it has been speculated here that this might involve the surface teiision equilibration time. Even for solutes with suitably short surface tensioii equilibrat'ion times, experimental coiifirmatioii would be desirable before much confidence could be placed in the assumption that $ L is uniquely determined by lYMaand R. Furthermore, it mould be reckless to assume that a correlntioii for a short wet'ted-mall column such as the one employed here would be quantitatively identical with the correspoiidiiig correlatioii for a sieve trap contactor, packed column, or eome other fluid mechanical system. Finally, there must be some concern about the possible influence of noiivolatile surface active impurities in the distilled water employed in this investigation. Berg aiid Acrivos (1965) discussed the stabilizing influence of such nonvolatile surface active agents against Marangoiii disturbances, aiid variations in water purity might be expected to destroy the hoped for agreement between the results of different iiivestigatioiis of the type reported here. I t is perhaps significant that the data of Clark and King (1970), who desorbed several solutes from n-tridecaiie, show reasonable agreement with the results of this study in Figures 14 and 15. The organic solvent employed by those investigators would be expected to be less susceptible to stabilization by trace quantities of surface active impurities than the distilled water employed here, but the agreement might be fortuitous. 82
Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
The absorber used by Clark and King was different fluidmechanically from that employed here, and because their liquid layer was horizontal it appears that a destabilizing density gradient developed which probably augmented the surface tension-driven cellular convection. It would seem wise, therefore, to await further experimental results before placing too much confidence in the generality of these correlations. Comparison with Theory. T h e critical 1Iaraiigoni numbers of Figure 14 are compared iii Figure 16 with results of linearized hydrodynamic stability theory. T h e upper curve is Pearsoii's (1958) theoretical result for t h e case in which t h e coiiceiitratioii gradient exteiidb throughout the liquid layer, e = 1. Vidal and Acrivo:: (1968) obtained the solution for E < 1 but only a t R = 0. The present authors have extended their results to R > 0, producing the loner curve in Figure 16 for E = 0.07, correspondiiig to the present experiments. The esperimeiital critical Narangoiii numbers lie several orders of magnitude above the theoretical predictioiis. Of course, a t the theoretical critical values the convection niight just be too weak to be detected, but it appears more likely that the disagreement really represents a contradiction. This might be the result of surface rigidity due to lionvolatile surface active impurities, as discussed earlier. ,hiother 110ssible explanation is that' all of the results were affected by finite surface tension equilibration times, not just the triethylamine results. The theoretical analyses did not account for the finite time required for the establishmelit of the surface tension depression. The discrepancy between theory aiid esperinieiit seen in Figure 16 caiiiiot he attributed to n .short liquid pha.se residence time wliicli is insufficient to allow growth of the disturbance>, as implied in similar contexts (Daiickwerts and Tavares Da Silva, 1967; Thomas and Sicholl, 1967, 1969). The theoretical analyses of Pearson and of Tidal niid -1crivos were made for stagnant liquid layers, but an esamiiiation of the differential equations of the linearized perturbation theory s h o w that an niialysis for a flowiiig liquid film would yield the same critical 1Iarangoiii number for the disturbance mode which takes the form of stationary roll cells lvith ases aligned in the flow direction. This mode of disturbance is unaffected by the mean flow velocity and cannot propagate doivnstream. Liquid phase residence time is, therefore, of no significance, and the theoretical results in Figure 16 are upper limits for flowing liquid films as well as stagnant films, The resolutioii of the discrepancy between theory and experiment shown in Figure 16 \vi11 require more study, perhaps along both theoretical and experimental lines. I t is hoped that the speculation above will aid in that study. Conclusions r ,
1he use of two tracers has proved to be ai1 effective nieaiis of studying 1Iarangoiii effects accompaiiyiiig the desorption of surface teii3ioii-lowering solutes from water, providing quaiititatire measures of the eiihaiicenieiit in kL and in kc due to the cellular convection. The luck of aiiy observed effect on kG in coli eiit with the inferred linear dinieiisioii and velocity scale the coiivectioii cells, 0.001 c m and 0.01 cm per second, respectively. The experimental critical values of the Xu-aiigoiii number, show the expected dependence 011 R, escept that the results for triethylamine appear incongruous. -111 of these values are several orders of magnitude higher than theoretical predictions based on linearized hydrodynamic stability analyses,
and it is speculated that, this may be due to surface impuritieh or to the fiiiite time required for equilibration of the aurfacr tension depression.
SUPERSCRIPT
*
=
in nb:,eiice of cellular convection
Literature Cited Nomenclature
I1
bulk fluid coiiceiit,ration, g moles solutejcni3 = interfacial concentration, g mole solute/cni3 = coluiiin diameter, ciii = diff usivity , c tii2jsec = column height, cni
II
=
1: I
ec-cin? gas pha-e Graetz numher, QlhDG &I-\!, = 1I:iraiigoni number, ( y , - y B ) / p k ~ * .YM>= gap phase Reynold. nuniher, 4 Q , ' 7 d ~ .YW, = gas phaie Shernood number, k ~ ( d- 26)l Dr, c l = volumetric gab floiv rate, cni3//iec R = ratio of liquid ph:ise resistance to pas ph in ali>eiice of 1Inraiigoiii effect:,, Mk = cellular convection velocity, c i n 'sec z = i)enetrntion denth. or conceiitratioii lioundnrv laver
SI>
= i i i i i + s trniisfer
Andermi. I). K., Ilall. J. I?., Babb, h.L., J . Phris. Chrtn. 62. 401 f 19.58). Bakker,~C..4.P., vaii Biiytenen, P. AI., Beek, IV.J., Chcm. Eny. Sei. 21, 1039 (1966). Bakker, C. A . P., vaii Vlih;.iirgen, F. 11. F., Beek, W. J., Chott. Eng. Sei.22, 1349 (1967). Benard, H., Rtzl. Gen. Sei. Pirrc A p p / . 11, 1261 (1900). Berg, J . C., ilcrivo-, -I., Chctn. Eny. Sei. 20, 737 (196.5). Brian. P. I,. T., Viviail, J. IC., lIatiato3, I). C., A.I.Ch.E. ,I. 13, 28 (1967). Bvers, C. I € . , King, c'. J.. d . I . C h . E . J . 13, 628 (1967). Ci:rrk, 11,IV,, King, C. J., d.1.Ch.E. J . 16, 69 il970,. I)anckwert,~,P. V., Tavare. I)a Silva, A,, Chon. Eny. Sci. 22, 1,513 (1967). Gillilmrd, 1'. l t . , Baddoiu,, I:. F., Brimi, P. L.T., -4.I.CR.E. .J. 4,
=
t
Y
r
=
6 cc
=
Y
t
= =
QL
=
=
pure n-iter) liquid flow rate per unit of columii perimeter, g,' niin-cni depth of liquid layer 011 column n-all, c m liauid x.is(lositv. dvnes-aec;cni2 " _ kinematic viscosity of gas, cm2, sec product of liquid viacosit,y and density divided by same product for water at' 25°C liquid pha$e mass transfer enhancenieiit factor, k ~FL* ,
SUR~CRIPT~ CR = critical value G = gas phase L = liquid phase
lupported by the Sational Science Foundation. S . T. l l a y r received financial .upport from Rand lIines, Ltd., the South African Industrial Cellulose Corp. (SAICCOR), and the South African Council for Scientific and Industrial Research (CSIR).
Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
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