Film Thickness, Flow Regimes, and Flooding in Countercurrent

water films. With increasing liquid flow rate, the falling film exhibits the following flow regimes: wavy film, troughs/crests on the film, drop forma...
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Film Thickness, Flow Regimes, and Flooding in Countercurrent Annular Flow of a Falling Film at High Pressures Ron Stockfleth*,† and Gerd Brunner‡ Degussa AG, Abteilung VT-F, Rodenbacher Chaussee 4, 63457 Hanau-Wolfgang, Germany, and TU Hamburg-Harburg, Arbeitsbereich Thermische Verfahrenstechnik, Eissendorfer Strasse 38, 21071 Hamburg, Germany

The film thickness, flow regimes, and flooding points for the countercurrent annular flow of corn oil (Mazola) and carbon dioxide at 338 K and pressures between 7.6 and 20.6 MPa are examined experimentally. The film flows down a stainless steel rod with a diameter of 10 mm and a length of 1000 mm that is inserted in a glass tube with an internal diameter of 15 mm, yielding an annulus of 2.5 mm. The film thickness adheres to Nusselt’s equation for smooth water films. With increasing liquid flow rate, the falling film exhibits the following flow regimes: wavy film, troughs/crests on the film, drop formation, and flooding. The countercurrent gas flow affects the flow regime of the liquid film. The flooding points for the falling film column can be correlated with the same equation as the flooding points for packed columns with various packings and fluids at different states (high and ambient pressure). Introduction The flow regime of a falling film is an important aspect of the design of gas-liquid contacting devices, because it determines the interfacial area, the rate of its renewal, and the mixing and backmixing of the liquid phase. In countercurrent annular flow, the liquid flows downward in different flow regimes: smooth, waves, crests, or droplets. The gas flows upward and exerts a shear force on the liquid. The magnitude of this force depends on the momentum of the gas flow, its physical properties, and the topology of the interfacial area. The two phases interact, and this can lead to flooding if the annular area is too small to allow the countercurrent flow of both phases. For the design of countercurrent contacting devices, knowledge about the flooding point is required to determine the minimum annular area needed for the countercurrent throughput of the required amounts of both phases without flooding of the device, and knowledge about the topology of the interfacial area is needed to specify the mass transfer coefficient. There have been numerous investigations1-4 of falling films for normal pressures (P ≈ 0.1 MPa). In the following, only the literature dealing with falling films at high pressures is reviewed briefly; an extensive general review can be found elsewhere.5 Beyer6 examined phase equilibria for the liquid phases pelargonic acid, oleic acid, linoleic acid, squalane, and oleic acid monoglycerides with the gaseous or supercritical phases carbon dioxide and ethane at temperatures of 313, 333, and 353 K and pressures between 2 and 12 MPa. He also investigated the mass transfer coefficients with a falling film cell. Above a certain pressure, he observed that the falling film disintegrated and became a veil of tiny droplets. The residence time * Author to whom correspondence should be addressed. E-mail: [email protected]. Phone: +49-6181-2833. Fax: +49-6181-4697. † Degussa AG. ‡ TU Hamburg-Harburg.

of the gaseous phase in the falling film cell and the volumetric flow rate of the liquid phase were constant in all experiments. Because the fluid dynamic parameters were not changed, it was assumed that physical properties are the reason for the film disintegration. In diagrams where the liquid-film or KF number was shown as a function of the pressure, the film disintegration always occurred at the maximum of the KF number

KF )

FLσ3 gµL4

(1)

where FL is the liquid density, σ is the interfacial tension, g is the gravitational acceleration, and µL is the dynamic viscosity of the liquid. Blaha-Schnabel7 expanded Beyer’s6 work. He measured the mass transfer coefficients of pelargonic acid with carbon dioxide or ethane and of stearinic acid with carbon dioxide or ethane at 333 and 353 K and pressures between 1 and 14 MPa in a falling film cell with inclinations between 0° and 90°. Again, the residence time of the gaseous phase in the falling film cell and the volumetric flow rate of the liquid phase were constant for all experiments. Film disintegration was observed and attributed to the activity of the gaseous phase. Both authors6,7 concluded that the main reason for the disintegration of the falling film is the drastic change of the physical properties at high pressures. However, the scope of the experiments conducted does not allow this conclusion, because the fluid dynamic parameters (i.e., gas and liquid velocities) were not varied. Moser8and Moser and Trepp9 challenged and falsified the claim that the main reason for the disintegration of the falling film is the drastic change of the physical properties at high pressures with falling film experiments at temperatures between 313 and 353 K and pressures between 8 and 35 MPa in which the residence time of the gaseous or supercritical phase was constant, but the velocity of the liquid phase varied. He employed R-tocopherol and squalane as liquid phases

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and carbon dioxide as gaseous phase and found that the cause for the film disintegration is a combination of fluid dynamic (liquid velocity) and physicochemical (property) effects. Kerst and co-workers10,11 investigated the mass transfer coefficient, the film flow regimes, and the jet flow regimes of the binary system methyl myristate with carbon dioxide at 313 and 323 K and pressures between 0.1 and 8 MPa. He supplied some excellent photographs of falling films at high pressures. In his apparatus, the liquid flow rate varied, and the gas remained motionless. At high flow rates the film disintegrated. In accordance with Moser and Trepp8,9 he attributed the disintegration to physicochemical and fluid dynamic effects. Both sets of authors8-11 use a ReL-KF diagram dating back to Grimley12 to show the borders between the different flow regimes

KF ) f(ReL) ) C1ReLC2 with ReL )

mL µLU

(2)

where mL denotes the mass flow rate of the liquid and U represents the circumference of the rod. C1 and C2 are dimensionless constants. The objective of this work is to examine the effects of varying velocities of the gaseous or supercritical phase on the flow regimes of the falling film and to investigate the flooding of falling films at high pressures. It was achieved by observing the flow regime of a falling film at different flow rates of gas and liquid in a highpressure column equipped with several windows. In addition, the liquid film thickness and the pressure drop that the gas flow experiences when flowing upward countercurrently to the liquid were measured. The falling film was produced on a stainless steel rod with a diameter of 10 mm. The rod was placed in a glass tube with an internal diameter of 15 mm, leaving an annulus of 2.5 mm through which the two phases must pass countercurrently. Corn oil (Mazola) was employed as the liquid phase, and the gaseous phase was supercritical carbon dioxide. The experiments were conducted at 338 K and pressures between 7.6 and 20.6 MPa. Experimental results for the change of flow regimes, film thickness and flooding are presented, and compared with data from other authors and predictions of existing theories. Experimental Apparatus The apparatus shown in Figure 1 can be operated at pressures between 8 and 30 MPa and temperatures between 313 and 373 K. It consists of two main parts: a 1.89-m-high column with four long windows and an equilibrium autoclave equipped with a stirrer. The bottom segment of the column is also equipped with a long window to allow continuous monitoring of the liquid level. The coexisting phases are brought into equilibrium by circulation and stirring. This is necessary to observe the fluid dynamic phenomena independently of the mass transfer phenomena. Two high-pressure gear pumps (one for the supercritical phase and one for the liquid phase) convey the two phases from the autoclave to the column and back into the autoclave after they have passed through the column. The pressure drop is measured with a differential pressure transducer that contains two membranes with resistive strain gauges on them. It measures differential

Figure 1. Flowsheet of the experimental apparatus: 1, column; 2, autoclave; 3, differential pressure transducer; 4, gear pumps; 5, flow meters; 6, steel rod with wavy liquid film; 7, halogen headlight; 8, CCD camera; full line, liquid cycle; dashed line, supercritical fluid cycle.

pressures between 0 and 2 kPa at absolute pressures up to 40 MPa and temperatures between 293 and 343 K with an accuracy of 10 Pa (143-DP-A2SS1EP2EA4HA, Foxboro-Eckhardt, Stuttgart, Germany). The entire apparatus is placed within two modified heating ovens to control the temperature (FED 720 and FED 400, WTB Binder, Tuttlingen, Germany). The pressure in the apparatus is generated by an airdriven pump (Maximator, Schmidt, Kranz & Co., Zorge/ Su¨dharz, Germany), which supplies liquid carbon dioxide at a maximum discharge pressure of 45 MPa. The densities of the coexisting phases are measured with an oscillating U-tube densimeter, which can be operated at a maximum temperature of 423 K and a maximum pressure of 40 MPa (DPR 2000, Anton Paar, Graz, Austria). The mass flow rates of the gas and liquid phases are measured with vibrating-tube mass flow meters, which can be operated at a maximum temperature of 393 K and a maximum pressure of 32.5 MPa (RHM 01 and RHM 03, Rheonik, Maisach, Germany). A CCD camera with a shutter, a 50-mm lens, and several intermediate rings was used to record the visual observations on a SVHS videotape. One 50-W halogen headlight was used as a back light on the opposite side of the steel rod and another one as a front light on the same side of the rod as the camera. The falling film was observed in the lowest of the four long windows, which was 0.4 m downstream of the liquid inlet. This entire apparatus is described in more detail elsewhere.13-15 The film carrier is a 1000-mm-long stainless steel rod with a diameter of 10 mm inserted into a glass tube with an inner diameter of 15 mm. The glass tube does not hold the pressure; it just provides a circular flow channel through which gas and liquid must pass.14 At the inlet, the liquid has to flow over a radial weir and through a narrow annulus of 0.2 mm between the rod and a restriction cap. This ensures that the liquid wets the entire circumference of the rod. The lower section of the rod is slightly conical with an angle of 1°. The diameter of the rod is only 9 mm at the bottom where a collector gathers the liquid, which then flows to the bottom segment of the high-pressure column through a bore with a diameter of 7 mm. The diameters of all bores in the liquid outlet were overdesigned to make sure that the resistance to the flow was so small that flooding

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would occur not in the outlet but in the narrow annulus between the rod and the glass tube. Materials The carbon dioxide had a purity of 99.95% and was supplied by Kohlensa¨urewerke Deutschland, Germany. The Mazola corn oil is available in most grocery stores. It was employed as the liquid phase, because its physical properties (density, interfacial tension, dynamic viscosity) at high pressures have already been investigated extensively.16,17 Experimental Procedure The heating ovens are used to bring the apparatus to the desired temperature. Next, the plant is filled with liquid from a storage tank. The carbon dioxide is then conveyed to the column by an air-driven pump until the desired pressure is reached. The two phases are circulated and stirred until they are in equilibrium. Then, the experimental runs start. The phases are circulated at constant liquid-phase level in the bottom, the pressure drop is measured, and 1 min of visual observation of the falling film is recorded on videotape. Then, the bottom valve is closed, and the gear pumps are switched off. The film thickness is measured by the drainage method. When the liquid level in the bottom is no longer changing, the previous liquid level (before the gear pumps are shut off) is subtracted from the final liquid level after drainage to yield the volume of the liquid holdup from which the film thickness can be calculated. The next experiment is conducted at the same flow rate of the supercritical phase but with increased liquid flow rate. This procedure is repeated until the flooding point or the capacity limit of the gear pumps is reached. Then, the mass flow rate of the supercritical phase is increased, and the procedure is repeated. Visual Observations Four different flow regimes were observed and are displayed in Figure 2. “Wavy flow” consists of sinusoidal waves with amplitudes below 0.5 mm. “Trough or crest flow” is characterized by nonsinusoidal irregularities with amplitudes up to 2.3 mm. The irregularities look like troughs or crests protruding from the base of the film. The irregularities differ in their amplitude and expansion: the small irregularities are like droplets imbedded in the film, whereas the large ones form troughs across the entire circumference of the rod. Most irregularities have an intermediate expansion, i.e., they form troughs across a fraction of the circumference of the rod. In “drop formation”, single droplets tear off and splash against the inner wall of the glass tube. This regime is difficult to observe and hard to distinguish from crests or troughs that have an amplitude of 2.5 mm (the dimension of the annulus) or more, because both effects yield the same result: a liquid splash on the inner wall of the glass tube. Finally, with regard to “flooding”, the column always floods from the bottom to the top as already described elsewhere.14 The liquid accumulates in the annulus between the rod and glass tube and is suspended by the momentum of the gas flow. The liquid column grows until it finally reaches the long windows where it can be observed visually. The gas flow no longer passes along the liquid film but rather passes through the liquid column accumulated in the annulus,

whereby it entrains liquid droplets of which some, depending on their size and settling velocity, are carried upward. These entrained droplets can be observed through the windows before the front of the accumulated liquid reaches the observable zone. Entrained droplets, along with a continually increasing pressure drop, are definitive indicators for flooding. At normal pressure conditions “smooth liquid films” were observed by other authors,2,3 but in this work, in accordance with Moser and Trepp’s8,9 observation, even at the lowest reasonable liquid velocity, the liquid film was still wavy. With increasing liquid flow rate the flow regime changes in the following order: wavy, troughs/crests, drops. The flow regime flooding is a special case. This regime can only be reached if the flow rate of the gas is high enough to dam up the liquid. The transition between the different regimes is fluent. The following definitions were used to distinguish the different regimes. For the wavy regime, waves with small amplitudes visible to the naked eye appear. For troughs or crests, five or more irregularities with significant amplitude pass one window in 10 s. For drops, two or more liquid drops splash against the inner wall of the glass tube in 10 s. For flooding, more liquid enters the column at the inlet than leaves the column at the outlet. Drops are entrained by the gas flow, and the pressure drop rises continually until the entire annulus is filled with liquid. Results and Discussion Film Thickness. The average film thickness without countercurrent gas flow was measured with the drainage method. According to Brauer,2 the liquid friction factor ψL for the film flow is similar to a dimensionless film thickness

ψL )

2δ3gFL2 (mL/U)2

(3)

Figure 3 depicts the dimensionless film thickness as a function of the Reynolds number of the liquid. The straight line shows Nusselt’s1 equation for the film thickness of laminar smooth films.

ψL )

6 ReL

(4)

Nusselt’s equation describes the measured film thickness with acceptable accuracy. Equation 4 was developed for smooth water films flowing down vertical surfaces. Two of the assumptions necessary for the derivation of eq 4 are not valid for high-pressure falling films. First and most obvious, the film is not smooth. Second, the buoyant force exerted on the liquid film by the surrounding gas cannot be neglected at high pressures. The buoyant force inhibits the flow of the liquid and should result in a film thickness greater than that calculated with eq 4. The waves, troughs, and crests on the real film travel downward with a higher velocity than the base or substrate film,18 yielding a smaller average film thickness than predicted by eq 4. These two effects not taken into account by eq 4 seem to cancel each other. Transition between Different Flow Regimes. The borderlines between the flow regimes waves-troughs/

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Figure 2. Flow regimes: (a) waves, (b) crests, (c) drop formation, (d) flooding. T ) 338 K, P ) 20.6 MPa.

crests and troughs/crests-drops are shown in a ReLKF diagram, as suggested by Grimley.12 Figure 4 shows the wave-crest transition and the crest-drop transition with and without countercurrent gas flow. The flow rate of the gas is not further quantified in Figure 4; the empty symbols denote data points obtained at different flow rates of the gaseous phase. The data for the crestdrop transition can be described with sufficient accuracy using Moser and Trepp’s8,9 correlation

ReL ) 0.84KF0.18

(5)

From Figure 4, it is obvious that the gas flow has a significant impact on the flow regime of the liquid film. This effect is quantified as follows. The gas flow exerts a shear force on the liquid film, which affects the shape of the interface, i.e., the flow regime

Fshear ≈ τHdH

(6)

Figure 3. Film thickness at different Reynolds numbers: diamonds, experimental data; line, Nusselt’s1 theory.

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Figure 4. Flow regimes, diagram suggested by Grimley:12 solid square, drop formation without gas flow; empty squares, drop formation with gas flow, solid triangles, crest formation without gas flow; empty triangles, crest formation with gas flow; line, Moser and Trepp’s8,9 correlation.

Figure 5. Flow regimes, new diagram accounting for shear stress: solid squares, drop formation without gas flow; empty squares, drop formation with gas flow, solid triangles, crest formation without gas flow; empty triangles, crest formation with gas flow; line: Moser and Trepp’s8,9 correlation.

where τ is the shear stress; H is the height of the film; and dH is its hydraulic diameter, which, in this case, is the difference between the rod diameter and the inner diameter of the glass tube (5 mm). The gas flow experiences a pressure drop and must therefore exert the following force on the liquid surface and the inner wall of the glass tube

j/G ) f(j/L) with j/G )

Fgas ≈ ∆PdH2

(7)

where ∆P is the pressure drop. If the shear force exerted by the gas on the inner wall of the glass tube is neglected, a force balance yields

dH H

Fgas ) Fshear f τ ≈ ∆P

(8)

The shear stress on the liquid surface is proportional to the pressure drop. Rating the pressure drop to the impact pressure of the gas flow gives the dimensionless gas resistance factor ψG

ψG )

∆PdH FGuG2H

(9)

where uG is the superficial velocity of the gaseous phase. The influence of the gas flow on the flow regime is taken into account by using the property ReL(1 + ψG)n instead of ReL as the ordinate in Figure 5. For experiments without gas flow, ψG was set to zero, resulting in the same ordinate as Figure 4. The modified diagram is displayed in Figure 5. For the exponent, the value of n ) 1/3 yielded the best coincidence between transition points with and without countercurrent gas flow. The newly introduced ordinate appropriately accounts for the impact of the gas flow on the transition of the flow regimes. Flooding. For the correlation of the flooding points, an approach dating back to Wallis4 was employed

x

uG 

FG

gdH(FL - FG)

j/L )

x

uL 

and FL

(10)

gdH(FL - FG)

with uL being the superficial liquid velocity and  the fractional void volume, which is unity for a falling film column but smaller than unity for packed columns. j/G and j/L are modified Froude numbers rating the respective impact pressure to the difference between liquid head and buoyancy. It is assumed that these two dimensionless parameters are sufficient to describe the flooding phenomenon in empty and packed columns. For packed columns, slightly different parameters than these two are commonly used. According to the laws of dimensional analysis,20 these new dimensionless parameters can be obtained from the old ones with the use of a nonsingular transformation matrix

[] [

][ ]

x

uL j/G j/G 1 0 / T / f jG and Φ ) - 1/2 1/2 jL uG Φ

FL FG (11)

where the matrix contains the exponents that will yield the new parameters from the old ones. Φ is the flow parameter. It rates the kinetic energy fluxes of the two phases. The flooding diagram using these two parameters differs only marginally from the diagram originally suggested by Sherwood et al.21 in 1938. Figure 6 shows that flooding diagram where j/G is a function of Φ. Figure 6 displays various flooding data for packed columns for a variety of packings (Sulzer CY, Sulzer EX, Sulzer Mellapak, 5 × 5 × 0.5 mm Raschig rings, and 4-mm Berl saddles) and substances (water, air, carbon dioxide, olive oil deodorizer distillate, soybean oil deodorizer distillate, fatty acid methyl esters, and tocopherols) and the flooding data for the falling film column. For the packed columns, the hydraulic diameter for use in eq 10 was calculated with the following

Ind. Eng. Chem. Res., Vol. 40, No. 25, 2001 6019 Ci ) constant parameter dH ) hydraulic diameter, m g ) gravitational acceleration, m/s2 Fgas ) force exerted on the gas flow, N Fshear ) shear force exerted on the liquid surface, N H ) height of column, m j/G ) Wallis parameter for the gas phase defined in eq 10 j/L ) Wallis parameter for the liquid phase defined in eq 10 KF ) KF number defined in eq 1 Ki ) constant parameter mL ) mass flow rate of liquid phase, kg/s n ) constant parameter ∆P ) pressure drop, Pa ReL ) Reynolds number of the liquid phase defined in eq 2 U ) circumference, m uG ) superficial velocity of gaseous phase, m/s uL ) superficial velocity of liquid phase, m/s Figure 6. Flooding diagram: thick line, correlation; dashed lines, ( 30% interval; empty triangles, structured and random packings at high pressures;14,15 circles, structured packings at high pressures;13 solid diamonds, Mellapak at normal pressure; 19 solid triangles, falling film flooding at high pressures.

equation

dH )

4 a

(12)

where  is the fractional void volume of the packing and a is its specific surface area. A flooding point correlation suggested by Wallis4 was employed

K

xj/G + K1xj/L ) xK2 T j/G ) (1 + K2xΦ)2

(13)

1

where K1 and K2 are dimensionless parameters. For the correlation of the data displayed in Figure 6, the values K1 ) 0.4222 and K2 ) 1.1457 resulted in a relative standard deviation of 19%. This is an acceptable value, especially considering that only one set of parameters for eq 13 correlates the flooding points for a wide variety of geometries (structured packings, random packings, and falling film columns), substances, and states (temperature and pressure). Conclusions The effect of the flow rate of countercurrent gas flow on the flow regime of a falling liquid film at high pressures is both demonstrated and quantified. The thickness of the falling film can be described successfully with Nusselt’s1 equation (eq 4). This is surprising, because two of the simplifying assumptions on which the derivation of this equation is based are not valid for high-pressure falling films. The flooding points for falling films at high pressures are correlated effectively with the equation suggested by Wallis4 (eq 13). That same equation also correlates the flooding points of packed columns with various packings and various fluids at different states. List of Symbols Latin Symbols a ) specific surface area of the packing, 1/m

Greek Symbols δ ) film thickness, m  ) fractional void volume of the packing Φ ) flow parameter µG ) dynamic viscosity of the supercritical phase, Pa s µL ) dynamic viscosity of the liquid phase, Pa s FG ) density of the supercritical phase, kg/m3 FL ) density of the liquid phase, kg/m3 σ ) interfacial tension, N/m τ ) interfacial shear stress, N/m2 ψG ) resistance factor for the gaseous phase defined in eq 9 ψL ) resistance factor for the liquid phase defined in eq 3

Literature Cited (1) Nusselt, W. Die Oberfla¨chenkondensation des Wasserdampfes. Z. Ver. Dtsch. Ing. 1916, 60, No. 27, 151. (2) Brauer, H. Stro¨ mung und Wa¨ rmeu¨ bergang bei Rieselfilmen; VDI-Verlag: Du¨sseldorf, Germany, 1956. (3) Feind, K. Stro¨ mungsuntersuchungen bei Gegenstrom von Rieselfilmen und Gas in lotrechten Rohren; VDI-Verlag: Du¨sseldorf, Germany, 1960. (4) Wallis, G. B. One-Dimensional Two-Phase Flow; McGrawHill: New York, 1969. (5) Guedes de Carvalho, J. R. F.; Talalia, M. A. R. Interfacial shear stress as a criterion for flooding in counter current film flow along vertical surfaces. Chem. Eng. Sci. 1998, 53, No. 11, 2041. (6) Beyer, A. Stoffu¨bergang bei der Auflo¨sung hochmolekularer Stoffe in dichten Gasen. Ph.D. Thesis, University of ErlangenNu¨rnberg, Erlangen, Germany, 1990. (7) Blaha-Schnabel, A. Stoffu¨bergang und Stabilita¨t von Flu¨ssigkeitsfilmen bei der Auflo¨sung von Lipiden in dichten Gasen. Ph.D. Thesis, University of Erlangen-Nu¨rnberg, Erlangen, Germany, 1992. (8) Moser, M. Fallfilmstabilita¨t und Grenzfla¨chenspannungen in Systemen mit einer u¨berkritischen Komponente. Ph.D. Thesis, University of Zu¨rich, Zu¨rich, Switzerland, 1996. (9) Moser, M.; Trepp, C. Investigating the Stability of Falling Films at Round Vertical Film Carriers under High Pressure. Chem. Eng. Technol. 1997, 20, 612. (10) Kerst, A. W. Fluiddynamik und flu¨ ssigkeitsseitiger Stofftransport bei hohen Dru¨ cken; VDI-Verlag: Du¨sseldorf, Germany, 1998. (11) Kerst, A. W.; Judat, B.; Schlu¨nder, E.-U. Flow regimes of free jets and falling films at high ambient pressure. Chem. Eng. Sci. 2000, 55, 4189. (12) Grimley, S. S. Liquid flow conditions in packed towers. Trans. Inst. Chem. Eng. 1945, 23, 228. (13) Meyer, J.-T. Druckverlust und Flutpunkte in Hochdruckgegenstromkolonnen betrieben mit u¨berkritischem Kohlendioxid.

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Ph.D. Thesis, Technical University of Hamburg-Harburg, Hamburg, Germany, 1998. (14) Stockfleth, R.; Brunner, G. Hydrodynamics of a Packed Countercurrent Column for the Gas Extraction. Ind. Eng. Chem. Res. 1999, 38, 4000. (15) Stockfleth, R.; Brunner, G. Holdup, Pressure Drop, and Flooding in Packed Countercurrent Columns for the Gas Extraction. Ind. Eng. Chem. Res. 2001, 40, 347. (16) Jaeger, P. T. Grenzfla¨ chen und Stofftransport in verfahrenstechnischen Prozessen am Beispiel der Hochdruck-Gegenstromfraktionierung mit u¨ berkritischem Kohlendioxid; Shaker Verlag: Aachen, Germany, 1998. (17) Jaeger, P. T.; Schnitzler, J. v.; Eggers, R. Interfacial Tension of Fluid Systems Considering the Nonstationary Case with Respect to Mass Transfer. Chem. Eng. Technol. 1996, 19, 197.

(18) Moalem Maron, D.; Brauner, N.; Dukler, A. E. Interfacial structure of thin falling films: Piecewise modeling of the waves. PCH, PhysicoChem. Hydrodyn. 1985, 6, 87. (19) Sulzer Chemtech Strukturierte Packungen fu¨ r Destillation und Absorption; Sulzer Chemtech: Winterthur, Germany, 1997. (20) Stichlmair, J. Kennzahlen und A ¨ hnlichkeitsgesetze im Ingenieurwesen; Altos Verlag: Essen, Germany, 1990. (21) Sherwood, T. K.; Shipley, G. H.; Holloway, F. A. L. Flooding Velocities in Packed Columns. Ind. Eng. Chem. 1938, 7, 765.

Received for review January 30, 2001 Revised manuscript received September 13, 2001 Accepted September 19, 2001 IE0100885