Ind. Eng. Chem. Res. 1989,28, 1684-1688
1684
GENERAL RESEARCH Liquid Film in Taylor Flow through a Capillary Said Irandoust and Bengt Andersson* Department of Chemical Reaction Engineering, Chalmers University of Technology, S-412 96 Gothenburg, Sweden
T h e thickness of the liquid film between the gas bubble and the tube wall in Taylor flow has been measured for air-water, air-ethanol, and air-glycerol. The filmthickness was fitted to the empirical formula 6/d, = 0.1811 - exp[-3.1(pU/u)O."]]. A complete solution of the Navier-Stokes and surface tension equations showed excellent agreement for air-ethanol, while for the air-water system the equations had to be corrected for the bubble rigidity due to a surface tension gradient. 1. Introduction The laminar gas-liquid two-phase flow in vertical tubes is of importance in many different fields. This area has been the subject of many detailed studies (Bendiksen, 1984, 1985; Bretherton, 1961; Brown, 1965; Brown and Govier, 1965; Collier and Hewitt, 1961; Collins et al., 1978; Goldsmith and Mason, 1963; Marchessault and Mason, 1960; Shen and Udell, 1985; Tung and Parlange, 1976; White and Beardmore, 1962; Zukoski, 1966). The flow pattern occurring in gas-liquid two-phase flows is an important governing parameter in their application. The flow patterns are dependent on the flow rates and properties of the liquid and the gas, besides the tube size. The common flow patterns observed are shown in Figure 1. In bubble flow the gas phase is dispersed in the liquid as bubbles much smaller than the tube diameter. Slug flow represents the case in which the gas moves in the form of large bubbles separated by liquid slugs which also contain small gas bubbles. Taylor flow is a special case of slug flow where the bullet-shaped bubbles (Taylor bubbles) are separated by liquid slugs with no gas entrained. In both slug flow and Taylor flow, there is a very thin liquid film between the gas bubble and the tube wall. Annular flow refers to the case in which the liquid flows as a wavy film at the tube wall with the bas phase moving in the core. Among several investigators, Taitel et al. (1980) give a qualitative discussion of the phenomena involved in the transition between the flow regimes. The flowing bubble systems are of great importance in connection with two-phase flow in several technical applications, such as two-phase flow through porous media, boiling in tubes, analytical devices, and monolithic catalyst reactors. In the monolithic reactor, the catalyst consists of a great number of straight porous channels in which the liquid and gas reactants flow cocurrently. Previous studies (Irandoust and Andersson, 1988, 1989) have shown that the slug flow gives the best mass-transfer properties in the monolith channels. In this flow pattern, Taylor bubbles give rise to a recirculation within the liquid slugs by preventing the development of any parabolic flow in the liquid slugs. This recirculation will enhance the radial mass transfer between the phases involved. In the slug flow, the thin liquid film surrounding Taylor bubbles affects the mass transfer and the axial dispersion. Irandoust and Andersson (1989) have recently shown that the recirculation within the liquid slugs, and the very 0888-5885/89/2628-1684$01.50/0
thin liquid film between the Taylor bubble and the tube wall, accounts for the superior mass-transfer characteristics in slug flow. The latter have made the study of film thickness of interest for many investigators (Fairbrother and Stubbs, 1935; Bretherton, 1961; Goldsmith and Meon, 1963; Marchessault and Mason, 1960; Taylor, 1961; Ozgu et al., 1973). These investigations deal either with special cases where both viscous and surface tension effects are negligible or where viscous effects dominate or with tubes of larger diameter than that common for monolithic catalysts. Many studies are mainly focused on the bubble rise velocity, pressure drop, and bubble stability. Little information, however, has been obtained about the liquid film thickness for systems where both surface tension, inertia, and viscous effects are of importance. Taylor (19611, Marchessault and Mason (1960), and Bretherton (1961) were among those who first developed empirical relationships for estimation of liquid-film thickness. Ozgu et al. (1973) have studied the variation of the liquid-film thickness for Taylor bubbles of different lengths. Brown (1965) and Brown and Govier (1965) have studied the liquid flow around an individual Taylor bubble, considering the effect of liquid viscosity. The authors concluded that the pressure drop, an important parameter in two-phase flow systems, is highly correlated to the thickness of the liquid film around the Taylor bubble. Recently Shen and Udell (1985) and Bendiksen (1984, 1985) studied the flow behavior of long bubbles in liquid. Numerical methods describing the motion of the bubbles were developed. In the present investigation, the liquid film thickness has been measured for various liquids in both upward and downward Taylor flow. The tube sizes used were those of interest in monolithic catalysts. Furthermore, the film thickness and the bubble shape have been calculated theoretically by numerical solution of the flow and the surface tension equations. Both surface tension and inertia and viscous effects have been taken into consideration. 2. Experimental Methods A schematic diagram of the experimental device is shown in Figure 2. All measurements were made with vertically mounted precision-bore glass tubing of diameters ranging from 1.0 to 2.0 mm. The test length of the tubes was 0.40 m. The gas-liquid flow through the glass tube was either upward or downwarn Taylor flow. The method of intro0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 1685
9
,.. .
*
. . . . e .
. .... ....... . . ..* .... ...
,,
*.
......... ...*.. .. ..:.. ... '.., . * . .,. . . *
*
*
.. . Q
T
Taylor flow
Slug flow
Annular flow
Figure 1. Flow patterns in vertical flow. Reprinted with permission from Irandoust and Andersson (1988). Copyright 1988, Marcel Dekker, Inc. gas out
A
photocell
Table I. Properties of Colored Solutions Used, Measured at 20 O C IO3&,kg/(m s) p , kg/m3 lo%, N/m 1 1000 71.2 water ethanol (95 wt 70) 1.17 804 21.5 142 1230 65.8 glycerol (88 wt W ) 336 1244 64.8 glycerol (94 w t W )
computer
of the fluids used. The physical properties of the liquids, with different concentrations of the color substance, were nearly constant. Other experimental methods for measuring liquid-film thickness are based on the relative velocity of the gas bubble, dispersion, direct observation through a microscope, and conductimetrical methods. 3. Theory
The problem concerns the estimation of the thickness of a liquid film surrounding Taylor bubbles. Such bubbles move under the influence of surface tension, inertia, gravitation, and viscous effects in a vertical cylindrical tube. These phenomena can partly be summarized as follows (Bird et al., 1960): au 1 a + - - (rv) = 0 az r ar
llquld out
1'
gas in
Figure 2. Experimental setup for film thickness measurements.
ducing the gas the the liquid needed careful design. The gas and liquid slugs were formed in the inlet cell (Figure 2). The slug lengths of gas and liquid were measured by use of a photocell enclosing the glass tube and could be varied between 4 and 200 mm independently of each other. The value of the average total linear velocity was between 0.04 and 0.66 m/s. The corresponding Reynolds number varied from 0.42 to 860. The flow systems used were water-air, ethanol-air, and glycerol-air. The film-thickness measurements were made spectrometrically with a photocell. The photocell was placed near the outlet of the tube in order to ensure fully developed laminar flow and steady value of the bubble velocity. The basic principle of the film-thickness measurements lies in comparing the extent of light absorption in the liquid slug with that of Taylor bubbles at a selected wavelength. Hence, the liquid phase was colored with a suitable color substance in such a way that the flow pattern was not disturbed. The fraction of light intensity detected by the photodiode was proportional to the thickness of the liquid layer and the concentration of the absorbing substance in the solution. In order to account for the loss of intensity due to the tube walls and light scattering at the interfaces, the measurements were carried out with two different concentrations of the color substance. Table I gives details
In the formulation of the equations of motion above, cylindrical coordinates with rotational symmetry were used. The fluid was a Newtonian one with constant viscosity and density. All symbols can be found in the Nomenclature section. The boundary conditions are attributed to the velocity components of the fluid and the stresses at the interface between the gas and the liquid. The velocity boundary conditions are simple due to the rigid impermeable tube wall and nonslip velocity at the w d . These conditions can be expressed as u=o
(4)
u=o
(5)
au/ar = o
(7)
av/ar = o
(8)
at the tube wall and
at the tube center due to the misymmetric flow conditions. The stress boundary conditions, on the other hand, are much more complicated due to the unknown physical properties of the interface layer. The motion at the interface is affected through the action of surface tension gradients caused by a variation in the surface contaminants along the surface. For a "clean" liquid-gas curved inter-
1686 Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989
;
-
Table 11. Variation Range of Dimensionless Groups lower limit upper limit pdtU2/a 0.07 9.3
dJla
&,2/
a
9.5 x 10-4
1.90
0.13
1.46
face, the liquid pressure, PL, at the interface is given by the Young and Laplace relation
where R1 and Rz are the principal radii of curvature of the interface and PG is the pressure inside the Taylor bubble. The principal radii of curvature, R1and R2,at a given point on the interface are expressed as 1 -1= (10) Rl r(1 + (ar/az)2)1/2
-1--
R2
(1
+(ar/a~)~)~/~
(11)
The differential equations of motion, eq 1-8, are solved by using a commercial “finite-difference” program, PHOENICS (CHAM Ltd.), which employs a “finite-domain” formulation of the problem considered. Since the solution of motion equations implies a correct bubble form and liquid-film thickness, an iterative method solving the differential equations (9)-(11) was used to adjust the bubble shape and hence the liquid film. When the surface tension gradients at the interface are considered, the bubble was assumed to be rigid over a certain distance (Clift et ai., 1978). More information about the solution method is given by Irandoust and Andersson (1989). Since the solution of hydrodynamic equations is quite complex, many investigators have treated their data empirically. The empirical relationships are generally based on dimensionless groups. The combination of different dimensionless groups has been discussed thoroughly by Bretherton (1961) and White and Beardmore (1962). For cylindrical bubbles rising through liquids, the magnitudes of inertial, viscous, and gravitational forces, relative to the surface tension force, depend upon a set of three dimensionless groups: (a) pdtU2/u,(b) p u l a , and (c) pgd,2/a, where p is the liquid viscosity, u is the surface tension, and p is the density of the liquid flowing in a tube of diameter dt with the total average linear velocity U. Groups a, b, and c are called the Weber number, capillary number, and Eotvos number, respectively. These equations are derived from the dimensional analysis of the differential equations and boundary conditions for the motion. These independent groups can also be combined to form other dimensionless groups. The Reynolds number, Re = pUdt/p, is the ratio of a to b, and the Froude number, Fr = U2/gdt, is the ratio of a to c. The magnitudes of the above groups reveal which forces act on the fluid. In capillaries of small diameters, the groups containing d, in the numerator are often not important. The equations of hydrodynamics are then determined by consideration of group b alone. 4. Results and Discussion
In order to measure the liquid-film thickness, 84 runs were performed. These experiments were carried out using different fluids, different flow rates, and various tube sizes for both upward and downward flow directions. The variation ranges of the dimensionless groups, a-c, are given in Table 11. All dimensionless groups were fitted to ex-
0.15
-
0.10
-
+
This work
- Calculated
0.2
0.0
0.4
0.6
0.8
1.0
1.2
1.4
using Eq 112
1.6
2D
1.8
LLL a
Figure 3. Dimensionless liquid-film thickness.
-
0 -
dt
-d2r/az2
0.2or
-I(
0.15-/, 0.00000
0.00015
0.00010
0.00005
0.000 2 0
0.20,
_.--
-
_ c-------,
0.10
-
.’
,,‘
- - - - - - 0.01
>J----
/’
Calculated using Eq.112)
,’ ,
I
From
-O.O(
Taylor. 1961
From Marchessault &Mason,-
1960
- O.O[ 0.05k.l” 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
V U U
Figure 4. Comparison of the experimental results. Table 111. Estimated Values of Parameters in Equation 12 B C A -(3.08 f 0.486) 0.54 f 0.032 0.18 f 0.007
perimental data by regression analysis by using empirical relationships. The results obtained indicated that the experimental data could all be explained by the capillary number pula. In the present work, the empirical formula
was fitted to the data with a nonlinear least-squares regression analysis. The values of the fitting parameters with an approximate 95% confidence interval are given in Table 111. Figure 3 shows the experimental results where the dimensionless film thickness is plotted against p u l a . The calculated values using eq 12 are also plotted. It is shown that eq 12 is in good agreement with observed experimental data. The residual analysis showed that the model explains well the dependence of liquid-film thickness on the capillary number. The residuals against dimensionless groups a-c showed no abnormality. Similar models of the type given by eq 12 have been used by many other investigators (Bretherton, 1961; Fairbrother and Stubbs, 1935; Marchessault and Mason, 1960). In Figure 4,the empirical model obtained in this work is compared to the data given by Taylor (1961) and the formula from Marchessault and Mason (1960), derived for p U / u < 2 x lo4. It is shown that the model given by eq 12 is generally in accordance with the data reported by Taylor (161) and Marchessault and Mason (1960). Fairbrother and Stubbs (1935) measured the liquid film thickness in horizontal
Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 1687
M f,lr
Table IV. Liquid-Film Thickness water-air ethanol-air
?:
35 70
31 61
btf?t(
ittttl
Ethanol - air
Figure 5. Domain under study.
tubes by direct measurement of the velocity difference between the gas bubble and the liquid, assuming that the film was at rest. They fitted their experimental data into the equation
--)
!- = 0.25( pu
lJ2
dt
(13)
examined for p U / a < 0.015. Bretherton (1961) has presented a theoretical treatment of Taylor flow in horizontal tubes valid for p U / a < 3 X The following equation, based on a constant film thickness at rest, was given:
!- = 0.6( 7) pu 2J3 dt
Water-air
Figure 6. Bubble shape in Taylor flow (U= 0.125 m/s, dt = 2 X m).
(14)
The film thickness values calculated from eq 13 and 14 are lower than the experimentally measured values in this work. The deviations might be due to errors inherent in the measurement methods. It should be noticed that eq 13 and 14 are derived for low values of the capillary number, where there is some doubt about the accuracy of the methods used. The measurements of a velocity difference as used by Fairbrother and Stubbs (1935) result in poor accuracy for the low-velocity range. Furthermore, eq 13 and 14 are derived for horizontal tubes. According to Bretherton (1961), the liquid film in a vertical tube will be thicker than that in a horizontal tube. Another fact which is relevant in comparison of different data is the type of film thickness reported. In the present work, the measured film thickness was an average value over the gas bubble apart from the ends. The results also showed that, for a given system and at a given flow rate, the liquid-film thickness was independent of the length of Taylor bubbles and liquid slugs. This has been reported by many authors (Goldsmith and Mason, 1963; Brown, 1965; White and Beardmore, 1962). The
above statement is true for bubbles of one tube diameter length and longer. It was also found that the direction of flow had a negligible effect on the measured values of the liquid-film thickness. In order to check the reliability of the measurements presented here, the flow equations and boundary conditions were solved for a fraction of the tube containing two Taylor bubbles and two liquid slugs (Figure 5). Because of the very long computer run time, only two cases, i.e., water-air and ethanol-air systems, were checked by numerical solutions of the equations. For a water-air system, the surface tension effects are much more pronounced than for an ethanol-air system. This is due to higher surface tension of the former, leading to larger surface tension gradient along the interface. Hence, in the present paper, the surface tension effects were considered for the water-air system by assuming the front and rear ends of the bubble to be rigid. The calculated values of the liquid-film thickness are compared to the experimentalvalues in Table IV. The final forms of Taylor bubbles are shown in Figure 6 for water-air and ethanol-air systems. It was found that the film thickness achieved a constant value first at the middle of the gas bubble. The bubbles are more elongated in the front and more compressed in the rear end where a node was detected. The theoretical calculations give evidence for the reliability of using eq 12. 5. Conclusions
The liquid-film thickness measured by the photometric technique developed in this paper confirms the type of the film thickness equations proposed by other investigators. The empirical model developed fits both present data and data reported by Taylor (1961) and Marchessault and Mason (1960) reasonably well. The theoretical solution for the f ' i i thickness developed from Navier-Stokes and surface tension equations is in excellent agreement with the ethanol-air system but
I n d . Eng. C h e m . Res. 1989,28, 1688-1693
1688
predicts a somewhat thinner film for the water-air system. This deviation is due to the surface tension gradients at the interface which makes the gas bubble more rigid. There is at the moment no theoretical model that can predict this phenomenon explicitly, but an empirical adjustment of the interface velocity gives a film thickness very close to the measured value. Acknowledgment The authors gratefully acknowledge the financial support of the Swedish National Board of Technical Development and the National Energy Administration of Sweden. Nomenclature A = constant defined by eq 12 B = constant defined by eq 12 c = constant defined by eq 12 d, = tube diameter, m Fr = Froude number g = acceleration due to gravity, m/sz p = pressure, Pa Pc = pressure inside the gas plug, Pa PL = pressure in the liquid at the interface, Pa r = radial coordinate, m R = tube radius, m R1 = principal radius of curvature, m R2 = principal radius of curvature, m Re = Reynolds number t = time, s u = velocity in the axial direction, m/s U = bubble velocity, m/s u = velocity in the radial direction, m / s z = axial coordinate, m
Greek Letters = viscosity, liquid, kg/m s p = density, liquid, kg/m3 u = surface tension, liquid-air, kg/s2 I.(
Literature Cited Bendiksen, K. H. Int. J. Multiphase Flow 1984, 10, 467-483. Bendiksen, K. H. Znt. J . Multiphase Flow 1985,II, 797-812. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960. Bretherton, F. P. J . Fluid Mech. 1961, I O , 166-168. Brown, R. A. S. Can. J. Chem. Eng. 1965, Oct, 217-223. Brown, R. A. S.; Govier, G. W. Can. J. Chem. Eng. 1965, Oct, 224-230. Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops, and Particles; Academic Press: New York, 1978. Collier, J. G.; Hewitt, G. F. Trans. Znst. Chem. Eng. 1961, 39, 127-136. Collins, R.; De Moraes, F. F.; Davidson, J. F.; Harrison, D. J. Fluid Mech. 1978,89, 497-514. Fairbrother, F.; Stubbs, A. E. J. Chem. SOC.1935, I , 527-529. Goldsmith, H. L.; Mason, S. G. J. Colloid Sci. 1963, 18, 237-261. Irandoust, S.; Andersson, B. Catal. Rev. Sci. Eng. 1988, 30(3), 341-392. Irandoust, S.; Andersson, B. Comput. Chem. Eng. 1989, 13(4/5), 519-526. Marchessault, R. N.; Mason, S. G. Znd. Eng. Chem. 1960,52,79-84. Ozgu, M. R.; Chen, J. C.; Stenning, A. H. Trans. ASME 1973,95, 425-427. Shen, E. I.; Udell, K. S. J. Appl. Mech. 1985, 52, 253-256. Taitel, Y.; Bornea, D.; Dukler, A. E. AZChE J. 1980, 26, 345-354. Taylor, G. I. J. Fluid Mech. 1961, 10, 161-165. Tung, K. W.; Parlange, J. Y. Acta Mechan. 1976, 24, 313-317. White, E. T.; Beardmore, R. H. Chem. Eng. Sci. 1962,17,351-361. Zukoski, E. E. J . Fluid Mech. 1966,25, 821-837.
Received for review March 21, 1989 Accepted July 31, 1989
Use of CI-MS for the Determination of Deuterium Content in Hydrocarbons. 2. Solutions for Systems Involving Multiple Ionization Processes Geoffrey L. Price* Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803
Enrique Iglesia Corporate Research Laboratories, Exxon Research and Engineering Corporation, Route 22 East, Annandale, N e w Jersey 08801
The application of chemical ionization mass spectrometry (CI-MS) to the analysis of hydrocarbons generally results in multiple ion reactions which produce analyte ions in the molecular ion region. Mathematical analysis of these processes for deuterium-containing species may be complex, but processes that involve mainly hydrogen addition or charge transfer may be modeled easily and the deuterium distribution in the analyte determined often without the complications of fragmentation or isotope effects. However, matrices that are used to model processes that result primarily from hydrogen abstraction with minor components of charge transfer and hydrogen addition have very large condition numbers, and direct solution is difficult and requires highly accurate spectra. The mathematics of these CI processes are also applicable to electron-impact (EI) spectra which produce fragmentation in the molecular ion region, and the mathematical ramifications on E1 are discussed. In a previous publication (Price and Iglesia, 1989b), we reported the application of chemical ionization mass spectrometry (CI-MS) to the analysis of deuterium distributions in hydrocarbons. However, our report was based
* To whom correspondence should be addressed. 0888-5885/89/2628-1688$01.50/0
solely upon the mathematical analysis of hydrogen abstraction spectra which result when the reagent gas abstracts a hydrogen (or deuterium) atom from the molecule:
R+ + A
-+
(A - H)+ + N
Two other reactions that may result in the production of 0 1989 American Chemical Society
