Ind. Eng. Chem. Res. 2006, 45, 6325-6329
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GENERAL RESEARCH Onset of Rayleigh-Be´ nard-Marangoni Convection in Gas-Liquid Mass Transfer with Two-Phase Flow: Comparison of Measured Results with Theoretical Results Z. F. Sun* Physics Department, UniVersity of Otago, P.O. Box 56, Dunedin, New Zealand
Experimental results of the critical points of the Rayleigh and Marangoni instabilities in absorption and desorption of carbon dioxide into and from organic solvents have been compared with theoretical results obtained using a theoretical linear stability analysis in which the effect of nonlinear velocity profiles and nonlinear concentration profiles of fluid flows and the effect of the Gibbs adsorption layer have been considered. The comparison shows that, when the surface viscosity number of the Gibbs adsorption layer is ignored, the calculated values of the critical concentration difference are in satisfactory agreement with the measured data for the absorption processes but are about 7- to 9-fold less than the measured data for the desorption processes. Using reasonably estimated values of the surface viscosity and surface dilational viscosity of the Gibbs adsorption layer, this difficulty has been solved. The calculated values of the critical concentration difference under different operating conditions show good agreement with the measured data. Introduction Rayleigh and Marangoni instabilities in a solute transfer between a gas phase and a liquid phase which are in parallel, laminar, stratified flow between two horizontal plates have been investigated theoretically by Sun and Fahmy.1 In this linear stability analysis of the general Rayleigh-Be´nard-Marangoni problem, the effects of nonlinear velocity profiles and nonlinear concentration profiles of fluid flows on the critical Rayleigh number and the critical Marangoni number in gas-liquid mass transfer of practical importance have been considered. In addition, the effects of surface convection, surface diffusion, and surface viscosity of the surface-active solute in the Gibbs adsorption layer have been combined together. The linear stability analysis of the Rayleigh-Be´nardMarangoni problem has demonstrated that the critical Rayleigh number, the critical Marangoni number, and the critical wavenumber are dependent on the ratio of the gas viscosity to the liquid viscosity, µg/µl; the ratio of the gas diffusivity to the liquid diffusivity, Dg/Dl; the ratio of the gas velocity to the liquid velocity, ugm/ulm; the ratio of the thickness of the gas layer to that of the liquid layer, b/h; and the gas-liquid equilibrium property, m/RT. In particular, the critical parameters of the Rayleigh-Be´nard-Marangoni problem are the functions of the downstream location, x, and the Peclet number, Pe, of the liquid layer. The effects of these transport and equilibrium properties of fluids and the geometry of gas and liquid contactors on the onset of convective instability have been investigated.1 The linear stability analysis for the Rayleigh-Be´nardMarangoni problem has also demonstrated that the system stability is reinforced by the surface convection number A and the surface viscosity number Vi, when the operation line, defined as the ratio of the Marangoni number to the Rayleigh number, * Tel.: 64 3 479 7812. Fax: 64 3 479 0964. E-mail: zhifa@ physics.otago.ac.nz.
is in the first and second quadrants of the Ma-Ra plane.1 However, when the operation line is in a certain region of the fourth quadrant of the Ma-Ra plane, the critical threshold of Rayleigh convection and Marangoni convection is decreased by increasing the magnitudes of the surface convection number A or the surface viscosity number Vi.1 In this paper, experimental results of the critical points of Rayleigh and Marangoni instabilities in absorption and desorption of carbon dioxide into and from organic solvents2 are compared with theoretical results obtained using the theoretical stability linear analysis.1 Measurements Gas-liquid absorption and desorption experiments were conducted in a horizontal gas-liquid flow channel where the gas phase and liquid phase were in parallel, laminar, and stratified flow.2 The length, width, and height of the gas-liquid exposure section of the gas-liquid channel are 11, 11, and 4 cm, respectively. Thin liquid films on the lower horizontal plane of the gas-liquid channel were of a thickness of 3-5 mm. Using the horizontal gas-liquid flow channel, the influence of both the Rayleigh and Marangoni effect, which may enhance or inhibit each other, can be investigated. Seven tests of steady absorption and desorption experiments with cocurrent gas-liquid flows were conducted using the gasliquid channel.2 The gas used was CO2, and the absorbents were nonaqueous organic solvents, methanol and toluene, which were analytical reagents. All experiments were carried out at a temperature of 25 ( 0.5 °C and a pressure of 1 atm. Detailed information about the experimental apparatus and the experiments procedures can be found in ref 2. The operating conditions of the seven tests are listed in Table 1. For liquid-phase controlled mass transfer, the penetration model has been shown to be appropriate,3 and the average mass-
10.1021/ie060178f CCC: $33.50 © 2006 American Chemical Society Published on Web 07/29/2006
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Table 1. Operating Conditions for Tests 1-7 test
solvent
process
h (mm)
r ) σ0/gRFh2
A × 107
Pe
ugm/ulm
b/h
µl/µg
Dg/Dl
m/RT
1 2 3 4 5 6 7
methanol methanol methanol methanol methanol methanol toluene
absorption absorption absorption desorption desorption desorption desorption
3.0 3.5 4.0 3.0 3.0 3.0 3.0
-1.21 -0.89 -0.68 -1.21 -1.21 -1.21 -4.53
-5.77 -7.84 -5.84 2.72 2.74 2.74 5.65
4160 5546 6933 4160 5546 6933 3205
1.35 1.20 1.09 1.23 0.92 0.73 1.31
12.33 10.43 9.00 12.33 12.33 12.33 12.33
31.3 31.3 31.3 31.1 31.1 31.1 31.0
4477 4477 4477 4477 4477 4477 3634
0.26 0.26 0.26 0.26 0.26 0.26 0.43
Table 2. Physical Properties of the Solvents (25 °C)5 solvents
F × 10-3 (kg/m3)
methanol toluene
0.7853 0.8620
a
C* × 10-1
a
(mol/m3)
15.7 9.43
µl × 103 (Pa s)
Dl × 109
0.553 0.552
b
(m2/s)
σ0 × 106 (N m2/mol)
R × 106 (m3/mol)
0.955 2.121
-11.34 -6.15
3.75 4.62
Saturated solubility of CO2. b Diffusion coefficient of CO2.
transfer coefficients of the liquid phase over the gas-liquid contact distance, L, are expressed as
k0cl,ave
( )
uiDl )2 πL
Table 3. Physical Properties of Gaseous CO2 and N2 µCO2 × 105 (Pa s)a
µN2 × 105 (Pa s)b
DgP × 105 (m2 s-1 bar)c
1.4897
1.7796
1.69
1/2
(1)
Viscosity of CO2 at 298.15 K and 1 Viscosity of N2 at 298.15 K and 1 atm.14 c Diffusion coefficient of CO2-N2 at 298 K.6
Equation 1 is valid only for gas-liquid mass-transfer processes without the occurrence of interfacial turbulence. According to Brian et al.,4 the enhancement factor for the liquid-phase mass transfer, which is the ratio of the real mass-transfer coefficient to that predicted by the penetration theory, is defined as
the bulk liquid and the interface. When the value of the averaged concentration difference over the whole gas-liquid contactor is used, eqs 5 and 6 calculate the overall values of the Rayleigh number and the Marangoni number for the gas-liquid masstransfer process. When the value of the concentration difference at a downstream location x is used, eqs 5 and 6 calculate the local values of the Rayleigh number and the Marangoni number at the downstream location x. The physical properties of liquid methanol and toluene are listed in Table 2, where the values of the solute expansion coefficient, R, and the negative of the slope of the curve of the surface tension versus solute concentration, σ0, are estimated from the surface tension and liquid density differences between pure solvents and gas-saturated solvents.5 The physical properties of CO2 and N2 are listed in Table 3. The relation of Chapman and Enskog6 has been used to calculate the diffusion coefficient of CO2 and N2 at temperatures and pressures different from those for the diffusion coefficient listed in Table 3. The method of Wilke6 has been used to evaluate the viscosity of the gaseous mixture of CO2 and N2. Since σ0 is positive and R is negative for the CO2-methanol and CO2-toluene systems, the Rayleigh effect (Ra > 0) is inhibited by the Marangoni effect (Ma < 0) in absorption processes of carbon dioxide into methanol and toluene, while the Marangoni effect (Ma > 0) is inhibited by the Rayleigh effect (Ra < 0) in desorption processes of carbon dioxide from methanol and toluene. Figure 1 shows the variations of the measured overall liquidphase enhancement factors against the values of the concentration difference of carbon dioxide between the bulk liquid and the interface, for two typical tests: test 1 of the absorption of the CO2-methanol system and test 4 of the desorption of the CO2-methanol system, respectively.2 For brevity, figures corresponding to other tests, which are similar to Figure 1, have not been shown here. In Figure 1, the circle points (test 1) and the square points (test 4) represent the measured mass-transfer enhancement factors and the straight lines represent best-fitted curves for the enhancement factors. A best-fitted correlation between the overall liquid-phase mass-transfer enhancement factor and the concentration difference is obtained as follows,
Φ)
kcl,ave
(2)
k0cl,ave
Since both k0cl,ave and kcl,ave are the averaged parameters over the whole gas-liquid contact area, the enhancement factor defined by eq 2 is a property for the overall gas-liquid masstransfer process, instead of a local property. By comparison with the overall mass-transfer enhancement factor defined by eq 2, an expression for the local enhancement factor for the liquid-phase mass transfer can be defined as
Φx )
kcl,x
(3)
0 kcl,x
where kcl,x is the real local mass-transfer coefficient of the liquid 0 phase at a downstream location x and kcl,x is the local masstransfer coefficient of the liquid phase predicted by the penetration theory at the same downstream location as follows,
( ) uiDl πx
0 ) kcl,x
1/2
(4)
0 Because both kcl,x and kcl,x are the functions of the downstream location x, therefore, Φx is also a function of the downstream location x. The Rayleigh number and the Marangoni number are defined as follows,1,2
Ra )
gR∆Ch3 νlDl
(5)
σ0∆Ch µlDl
(6)
and
Ma )
where ∆C ()C0 - Ci) is the concentration difference between
a
atm.14 b
Φ ) K ln
( )
∆C + 1, (∆C g ∆Cc) ∆Cc
(7)
where ∆Cc is the critical concentration difference between the
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Figure 2. Gas-liquid mass transfer between two horizontal plates.1
mass-transfer rates enhanced by the Marangoni effect in gasliquid absorption. Figure 1. Enhancement factors and critical points for absorption of CO2 into methanol film in test 1, h ) 3 mm, Qg ) 1.15 L/min, and Ql ) 68.6 mL/min; and in test 4, h ) 3 mm, Qg ) 1.05 L/min, and Ql ) 68.6 mL/ min. Table 4. Critical Concentration Difference and the Value of the Parameter K
test
K
measured ∆Cc (mol/m3)
1 2 3 4 5 6 7
1.7262 1.9070 1.8519 1.3688 1.3874 1.6719 0.6661
-7.2930 -6.8772 -6.6165 13.7744 11.4239 11.6924 12.4105
a
calculated ∆Cc (mol/m3)
relative error (Vi ) 0) (%)
calculated ∆Cc (Vi * 0a) (mol/m3)
relative error (Vi * 0)
-8.0611 -6.6115 -5.1096 1.8580 1.5481 1.4424 1.2896
-10 4 29 641 638 711 862
-6.8416 -6.4342 -5.8450 14.7168 11.6590 10.5504 12.3352
7 7 13 -6 -2 11 1
Calculation Results The linear stability analysis for the general RayleighBe´nard-Marangoni problem in a solute transfer between a gas layer and a liquid layer which are in parallel, laminar, stratified flow between two horizontal plates (see Figure 2) has been discussed in detail in ref 1. In addition to the Rayleigh number and the Marangoni number defined in eqs 5 and 6, other relevant parameters, the Peclet number, Pe1; surface convection and surface diffusion numbers, A and S;1,8,9 and surface viscosity number, Vi;1,10-12 are defined as follows,
Estimated values of Vi have been used.
bulk liquid and the interface for the onset of Rayleigh-Be´nardMarangoni convection and K is a constant parameter. The critical concentration difference can be determined from Figure 1 and from the figures corresponding to other tests, as the values at which Φ approaches unity. It should be noted that the measured critical concentration difference by this way is the critical concentration difference for the overall gas-liquid mass-transfer process. The measured values of the critical concentration difference and the fitted parameter K in eq 7 are listed in Table 4. Equation 7 is a better correlation than the following correlation which was obtained previously by Sun et al.,2
Φ)
( ) ∆C ∆Cc
n
(8)
The values of the critical concentration differences appear to be underestimated by using eq 8 (see Figures 13 and 14 of ref 2). According to the Rayleigh number and the Marangoni number defined by eqs 5 and 6, we have
Ra Ma ∆C ) ) Rac Mac ∆Cc
(9)
Therefore, eq 7 can be rewritten as
( )
(10)
( )
(11)
Φ ) K ln
Ra + 1, (Ra g Rac) Rac
or
Φ ) K ln
Ma + 1, (Ma g Mac) Mac
Equation 11 is similar to that obtained by Lu et al.7 for the
ulmh
(23) D
Pe )
(12)
l
A)
δCi h∆C
(13)
S)
δDs hDl
(14)
µs + κs µlh
(15)
and
Vi )
Owing to the lack of measured data of the surface viscosity, µs, and the surface dilational viscosity, κs, for the CO2-methanol and CO2-toluene systems, µs and κs are assumed to be zero in the initial calculations described below, and accordingly, Vi ) 0. The effect of the surface diffusion number S may be neglected, as demonstrated by the theoretical linear analysis.1 Neglecting the slight change in the solute concentration at the interface along the flow direction, the surface convection number A has been calculated for tests 1-7. As shown in Table 1, the surface convection number A has the order of 10-7. Therefore, the effect of the surface convective number on the critical parameters can be neglected.1 Nevertheless, the surface convection number A has been included in the calculation. In the absorption processes of tests 1-3, the operation lines, expressed by the relation r ) Ma/Ra ) σ0/gRFh2,1 are in the fourth quadrant (Ra > 0, Ma < 0) of the Ma-Ra plane, while in the desorption processes of tests 4-7, the operation lines are in the second quadrant (Ra < 0, Ma > 0) of the Ma-Ra plane.1 Therefore, the slope of the operation lines is negative in all the tests, as listed in Table 1. The calculated values of the Peclet number for tests 1-7 are also listed in Table 1. Using the theoretical linear stability analysis,1 the local critical Rayleigh number and the critical Marangoni number along the flow direction in tests 1-7 have been calculated under the operating conditions listed in Table 1. The values of the local
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Figure 3. Variation of the critical concentration difference for absorption of CO2 into methanol film in test 1: h ) 3 mm, Qg ) 1.15 L/min, and Ql ) 68.6 mL/min.
Figure 4. Variation of the critical concentration difference for desorption of CO2 from methanol film in test 4: h ) 3 mm, Qg ) 1.05 L/min, and Ql ) 68.6 mL/min.
critical concentration difference ∆Cc have been obtained from the calculated critical values of the local Rayleigh number and the Marangoni number, using eqs 5 and 6. Figures 3 and 4 show the variations of the calculated critical concentration difference with the downstream distance of the methanol liquid flow in test 1 and test 4, respectively. To have cellular convection formed in the liquid layers, the operating concentration difference must be larger than the minimum value of the critical concentration difference. As indicated by Figures 3 and 4, with an increase in the operating concentration difference, cellular convection should occur first at the exit region (x ) 0.11 m) of the gas-liquid flow channel, where the absolute critical concentration difference is at a minimum. At the entrance of the gas-liquid channel in tests 1 and 4, the values of the critical concentration difference are very large, even larger than the saturation concentration of CO2 listed in Table 2. In fact, the minimum values of -∆C ) 8.0611 mol m-3 at x ) 0.11 m in test 1 and ∆C ) 1.8580 mol m-3 at x ) 0.11 m in test 4 are the calculated values of the critical concentration difference for the overall gas-liquid mass-transfer processes in tests 1 and 4, respectively, since the averaged liquid mass-transfer rate over the whole gas-liquid contact area is enhanced by the formation of cellular convection and the averaged liquid-phase mass-transfer coefficient starts to depart from that predicted by the penetration theory at the onset of cellular convection. The variations of the critical concentration difference in the absorption experiments of test 2 and test 3
and in the desorption experiments of tests 5-7, which are not shown here, are similar to those shown in Figures 3 and 4. Since it is difficult to obtain experimentally the local concentration difference, the local liquid-phase mass-transfer coefficients, and the local mass-transfer enhancement factors in the gas-liquid mass-transfer experiments, the values of the measured critical concentration difference of the overall gasliquid mass-transfer processes are compared with the values of the calculated critical concentration difference, which is the minimum value of the local critical concentration differences along the flow direction as described above. The calculated values of the critical concentration difference of the overall gas-liquid mass-transfer processes in tests 1-7 are compared with the measured results in Table 4. It is seen from Table 4 that, under the operating conditions listed in Table 1 together with Vi ) 0, for the CO2-methanol absorption processes (tests 1-3), the calculated values of the critical concentration difference are in satisfactory agreement with the measured values, with the largest relative error being 29%, as shown in Table 4. However, for the CO2-methanol and CO2toluene desorption processes (tests 4-7), the calculated values of the critical concentration difference are 7- to 9-fold less than the measured values, as shown in Table 4. It is suggested that the discrepancy between the calculated values and the measured values of the critical concentration difference is due to the fact that the effects of the surface viscosity and the surface dilational viscosity have been ignored in the initial calculations above. As discussed in the theoretical linear stability analysis of the general Rayleigh-Be´nardMarangoni problem,1 in addition to the surface convection number A, the surface viscosity number has a significant effect on the critical Rayleigh number and the critical Marangoni number. The surface viscosity µs of the Gibbs adsorption monolayer has the order of 10-7 to 10-5 kg s-1, and the surface dilational viscosity κs is often numerically larger than µs.13 Using the values of µl of methanol and toluene listed in Table 2 and assuming κs has the same order as µs, the values of (µs + κs)/µl for the CO2-methanol and CO2-toluene systems have been estimated to be in the order of 10-4 to 10-2 m. Assuming (µs + κs)/µl ) 5 × 10-3 m for the CO2-methanol system and using the liquid thickness listed in Table 1, the surface viscosity number Vi is estimated to be 1.7, 1.4, and 1.3 for tests 1, 2, and 3 and to be 1.7 for tests 4, 5, and 6, respectively. Under the operating conditions listed in Table 1 together with the estimated values of Vi above, the variations of the local critical concentration difference along the liquid flow direction in test 1 and test 4 have been recalculated and plotted in Figures 3 and 4, respectively. It is seen that, for the absorption process of CO2 into methanol in test 1 where the Rayleigh effect (Ra > 0) is inhibited by the Marangoni effect (Ma < 0), the calculated values of the local critical concentration difference slightly decrease from those calculated using Vi ) 0. However, for the desorption process of CO2 from methanol in test 4 where the Marangoni effect (Ma > 0) is inhibited by the Rayleigh effect (Ra < 0), the calculated values of the local critical concentration difference are largely increased from those calculated using Vi ) 0. The values of the critical concentration difference of the overall gas-liquid mass-transfer processes have been recalculated using the estimated values of Vi above for tests 1-6. The recalculated values are compared with the measured results in Table 4. As shown in Table 4, although a single estimated value of (µs + κs)/µl ) 5 × 10-3 has been used without adjusting from tests 1-6, the critical values calculated under the different
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operating conditions of both the CO2-methanol absorption processes and the CO2-methanol desorption processes in tests 1-6 are in good agreement with the measured data, with the largest relative error being 13% in test 3 (Table 4). This indicates that the estimated value of the surface viscosity property of the Gibbs adsorption layer in the CO2-methanol system is reasonable. Assuming (µs + κs)/µl ) 10-2 m for the CO2-toluene system, which is slightly larger than that of the CO2-methanol system, the value of the surface viscosity number Vi is estimated to be ∼3.3 for test 7. Under the operating condition for test 7 listed in Table 1 and Vi ) 3.3, the critical value of the concentration has been recalculated, which is in good agreement with the measured data, with a relative error of 1% (Table 4). Conclusions
x ) x-coordinate (m) z ) z-coordinate (m) Greek Symbols R ) solutal expansion coefficient of liquid phase (R ) -dF/ dCF0) (m3 mol-1) δ ) Gibbs adsorption depth (δ ) -dξ/dCRT) (m) Φ ) mass-transfer enhancement factor κs ) dilational viscosity of Gibbs adsorption layer (N m-1 s) µ ) viscosity (N m-2 s) µs ) viscosity of Gibbs adsorption layer (N m-1 s) F ) density (kg m-3) σ0 ) negative of the slope of the curve of surface tension versus solute concentration (σ0 ) - ∂ξ/∂C) (N m2 mol-1) ξ ) surface tension (N m-1) Subscripts
results2
The experimental of the critical points of the Rayleigh and Marangoni instabilities in absorption and desorption of carbon dioxide into and from organic solvents, methanol and toluene, have been compared with the theoretical results obtained using the theoretical linear stability analysis, in which the effect of nonlinear velocity profiles and nonlinear concentration profiles of fluid flows and the effect of the Gibbs adsorption layer have been considered.1 The calculated results indicate that the surface viscosity of the Gibbs adsorption layer plays an important role in the prediction of the critical parameters of the Rayleigh-Be´nard-Marangoni problem in gas-liquid mass transfer with thin liquid layers. When the effect of the surface viscosity of the Gibbs adsorption layer is ignored, the calculated values of the critical concentration difference are in satisfactory agreement with the measured data for the absorption processes of the CO2 into methanol, but they are about 7- to 9-fold less than the measured data for the desorption processes of CO2 from methanol and toluene. Using reasonably estimated values of the surface viscosity and the surface dilational viscosity of the Gibbs adsorption layers, the calculated values of the critical concentration are in good agreement with the measured data. Nomenclature A ) surface convection number b ) thickness of gas layer (m) C ) unperturbed solute concentration in liquid phase (mol m-3) D ) diffusivity for solute (m2 s-1) Ds ) surface diffusivity for solute in Gibbs adsorption layer (m2 s-1) g ) gravity acceleration (m s-2) h ) thickness of liquid layer (m) kcl ) liquid-phase mass-transfer coefficient (m s-1) K ) constant parameter m ) Henry’s law constant for solute (Pa m3 mol-1) Ma ) Marangoni number P ) pressure (Pa) Pe ) Peclet number Q ) volumetric flow rate (m3 s-1) R ) ideal gas law constant (N m mol-1 K-1) Ra ) Rayleigh number S ) surface diffusion number T ) temperature (K) u ) unperturbed velocity in the x-direction (m s-1) ui ) unperturbed surface velocity in the x-direction (m s-1) Vi ) surface viscosity number
ave ) average value over the gas-liquid contact distance c ) critical value g ) gas phase i ) gas-liquid interface l ) liquid phase m ) average value over the thickness of gas or liquid layer x ) local value at a downstream location x 0 ) bulk phase Literature Cited (1) Sun, Z. F.; Fahmy, M. Onset of Rayleigh-Be´nard-Marangoni Convection in Gas-Liquid Mass Transfer with Two-Phase Flow: Theory. Ind. Eng. Chem. Res. 2006, 45, 3293-3302. (2) Sun, Z. F.; Yu, K. T.; Wang, S. Y.; Miao, Y. Z. Absorption and Desorption of Carbon Dioxide into and from Organic Solvents: Effects of Rayleigh and Marangoni Instability. Ind. Eng. Chem. Res. 2002, 41, 1905. (3) Byers, C. H.; King, C. J. Gas-Liquid Mass Transfer with a Tangentially Moving Interface: Part I. Theory. AIChE J. 1967, 13, 628. (4) Brian, P. L. T.; Vivian, J. E.; Mayr, S. T. Cellular Convection in Desorbing Surface Tension-Lowering Solutes from Water. Ind. Eng. Chem. Fundam. 1971, 10, 75. (5) Hozawa, M.; Komatsu, N.; Imaishi, N.; Fujinawa, K. Interfacial Turbulence during the Physical Absorption of Carbon Dioxide into NonAqueous Solvents. J. Chem. Eng. Jpn. 1984, 17, 173. (6) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. (7) Lu, H. H.; Yang, Y. M.; Maa, J. R. On the Induction Criterion of the Marangoni Convection at the Gas/Liquid Interface. Ind. Eng. Chem. Res. 1997, 36, 474. (8) Brian, P. L. T. Effect of Gibbs Adsorption on Marangoni Instability. AIChE J. 1971, 17, 765. (9) Brian, P. L. T.; Ross, J. R. The Effect of Gibbs Adsorption on Marangoni Instability in Penetration Mass Transfer. AIChE J. 1972, 18, 582. (10) Scriven, L. E.; Sternling, C. V. On Cellular Convection Driven by Surface-Tension Gradients: Effects of Mean Surface Tension and Surface Viscosity. J. Fluid Mech. 1964, 19, 321. (11) Berg, J. C.; Acrivos, A. The Effect of Surface Active Agents on Convection Cells Induced by Surface Tension. Chem. Eng. Sci. 1965, 20, 737. (12) Palmer, H. J.; Berg, J. C. Hydrodynamic Stability of Surfactant Solutions Heated from Below. J. Fluid Mech. 1972, 51, 385. (13) Adamson, A. W.; Gast A. P. Physical Chemistry of Surfaces, 6th ed.; John Wiley & Sons: New York, 1997. (14) Touloukian, Y. S.; Saxena, S. C.; Hestermans, P. ViscosityThermophysical Properties of Matter. Plenum Press: New York, 1975; Vol. 11.
ReceiVed for reView February 13, 2006 ReVised manuscript receiVed June 13, 2006 Accepted June 30, 2006 IE060178F
