ARTICLE pubs.acs.org/IECR
Gas-Liquid Mass Transfer in Taylor Flow through Circular Capillaries Dingsheng Liu†,‡,§ and Shudong Wang*,† † ‡
Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, P.R. China Graduate School of the Chinese Academy of Sciences, Beijing 100039, P.R. China ABSTRACT: Based on computational fluid dynamics (CFD), gas-liquid mass transfer in upward Taylor flow through vertical circular capillaries was studied. To save computational resources and time, the numerical simulations were carried out in a moving frame of reference attached to Taylor bubbles. Three consecutive Taylor bubbles were used to mimic the behavior of Taylor flow in an infinitely long capillary. Steady-state solutions of concentration fields were obtained to describe gas transfer from Taylor bubbles to the liquid phase. The liquid-phase volumetric mass-transfer coefficient, KLa, was investigated as a function of various parameters, including the liquid-film length, liquid-slug length, liquid-film thickness, bubble rise velocity, liquid-phase diffusivity, capillary diameter, and gravity. One fitted equation, expressed with three dimensionless numbers, was developed to quantify the relationship between KLa and the above parameters. The examples show that the equation could predict KLa well. The contributions of the cylindrical bodies and hemispherical caps of Taylor bubbles on the overall mass transfer were studied separately.
1. INTRODUCTION Monolith reactors are attracting increasing attention and are considered as promising alternatives to conventional gas-liquidsolid reactors, such as slurry and trickle-bed reactors.1-3 In general, a monolith block consists of an array of straight, parallel, and uniform channels with square or circular geometries, typically having hydraulic diameters in the range of 1-5 mm. In principle, the information obtained from a single channel can be used to scale up commercial reactors if the gas and liquid phases are uniformly distributed in the channels of monolith reactors.4 A pioneer systematic study on gas-liquid flow in monolith reactors was performed by Satterfield and Ozel.5 They showed that the twophase flow in a single channel of monolith reactors could be characterized by that in a capillary.5 Taylor flow in a single capillary or microchannel has been extensively investigated because the flow pattern is very common and important in gas-liquid-solid monolith reactors. Taylor flow consists of trains of long bubbles separated by liquid slugs.1-3,6,7 The axial dispersion of the solute is reduced significantly because of the axial segregation of the bulk liquids.8,9 At the same time, recirculation flow patterns are induced in liquid slugs trapped between consecutive bubbles, which enhances the radial mass transfer greatly and makes solution concentration more uniform.10,11 The gas-to-wall mass transfer is increased markedly through the existence of thin liquid films between the bubbles and capillary walls. For chemical reactions with high intrinsic rates, the external mass transfer becomes rate-limiting.12 Kawakami et al. studied gas-liquid mass transfer in monolith reactors with different channel sizes13 and demonstrated the superior mass-transfer properties of a monolith reactor compared with those of a packed-bed reactor. Irandoust et al. investigated the gas-liquid mass transfer in Taylor flow through a capillary by absorption of oxygen into three different liquids: water, ethanol, and ethylene glycol.14 One semitheoretical formula calculating gas-liquid volumetric mass-transfer coefficient, KLa, was developed by means of penetration theory. The r 2010 American Chemical Society
theoretical results agreed well with the experimental data with one adjustable curve-fitting parameter.14 Bercic and Pintar measured gas-liquid mass transfer for Taylor flow through capillaries by physical absorption of methane into water.15 The role of the gas bubble and liquid-slug lengths on gas-liquid mass transfer was studied carefully.15 They presented an empirical correlation for estimating KLa KL a ¼ 0:111Ub 1:19 =½ð1 - εG ÞLUC 0:57
ð1Þ
Applying Higbie penetration theory16 to explain gas transfer across the cylindrical bodies and hemispherical caps of Taylor bubbles, Baten and Krishna17 put forward a set of formulas for calculating KLa
KL, film
KL a ¼ KL, cap acap þ KL, film afilm pffiffiffi KL, cap ¼ 2ð 2=πÞðDUb =dc Þ0:5
ð2Þ
acap ¼ 4=LUC pffiffiffi ¼ ð2= πÞðDUb =Lfilm Þ0:5 ,
ð4Þ
afilm ¼ 4Lfilm =ðdc LUC Þ
Fo < 0:1
ð3Þ
ð5Þ ð6Þ
where the Fourier number, Fo, was defined as Fo = Dtfilm/δ2. Recently, Vandu et al. gained KLa by means of experimental oxygen absorption dynamics.18 The experimental results were in good agreement with the theoretical values given by eqs 2-6, associated with the assumption that the dominant mass transfer occurred in the liquid film surrounding the Taylor bubble. Vandu et al. also investigated the gas-liquid mass-transfer characteristics of monolith reactors, with upward flow of the gas and liquid Received: December 27, 2009 Accepted: December 14, 2010 Revised: November 8, 2010 Published: December 30, 2010 2323
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Figure 1. Schematic diagram of Taylor bubbles flowing in an infinitely long capillary.
phases through channels.19 They showed that KLa is larger in monolith reactors than in airlift and bubble columns. This improvement is due to the superior mass-transfer characteristics of Taylor flow in narrow channels. In the present work, the mass transfer from Taylor bubbles to the liquid phase in circular capillaries was studied systematically based on computational fluid dynamics (CFD) simulations. The influences of various parameters on the volumetric mass-transfer coefficient were investigated. The numerical results were compared with the available published data.
2. THEORETICAL AND NUMERICAL METHODOLOGY Figure 1 shows Taylor bubbles flowing in an infinitely long capillary. Here, one combination of a Taylor bubble and two neighboring half-liquid slugs is called a “unit cell”. Mass transfer from the gas phase to the liquid phase occurs along with the flow of Taylor bubbles. It is well-known that the mass-transfer coefficient is defined as the ratio of the mass flux to the driving force. The driving force is usually expressed as the difference between solution concentrations. Generally, mass-transfer coefficients are influenced by the state of flow and physical properties of the fluids, but not the solution concentration. Therefore, if the end effects of the capillaries are eliminated, the number and position of unit cells used to estimate the mass-transfer coefficient should not affect the results, as was also confirmed in the present simulations. To evaluate KLa, the following assumptions were made: (1) the liquid phase is in plug flow and (2) the gas concentration at the gas-liquid interfaces along the capillary is constant at its equilibrium value.15 For the plug-flow model, taking into account the geometrical characteristics of Taylor flow, the volumetric mass-transfer coefficient is given by UTP C - Cinlet ln KL a ¼ ð7Þ Lc C - Coutlet 2.1. Governing Equations. For simplicity, the shapes of the Taylor bubbles were assumed to be known in advance. Each Taylor bubble consisted of a cylindrical body and two hemispherical caps. In the computational domains from which void gas bubbles were subtracted, only liquid-phase regions were considered. In addition, constant liquid density and laminar flow were assumed. The mass and momentum conservation equations for the liquid phase throughout the domains are given by
r3 U BL ¼ 0
ð8Þ
DðFL U BL Þ þ r 3 ðFL U BL U BL Þ Dt BL þ r ¼ - rp þ r 3 ½μL ðr U
T U BL Þ
Figure 2. Computational domains and boundary conditions used in the simulations.
2.2. Model Geometries and Boundary Conditions. Three unit cells were used to simulate gas transfer from Taylor bubbles to the liquid phase in infinitely long capillaries. The model geometries and boundary conditions are illustrated in Figure 2. Gas and liquid flowed from the bottom to the top in vertical capillaries. A two-dimensional coordinate system with axial symmetry about the capillary centerline was employed. Simulations were carried out in a moving frame of reference attached to the rising Taylor bubbles. Thus, the bubbles are stationary, whereas the capillary walls move downward with the rise velocity of the Taylor bubbles. No-slip boundary conditions were provided at the capillary walls, and zero-shear-stress boundary conditions were provided at the surfaces of the Taylor bubbles. With user-defined functions (UDFs), fully developed liquid velocity profiles were employed at the inlet and outlet of the computational domain in Figure 2b that were used to obtain stable flow fields. The velocity fields near bubbles are illustrated in one case in Figure 3a. Thulasidas et al.20 showed that the gas-bubble and liquid-slug velocities are related by
Ab Ub ¼ Ac Uslug þ Qfilm þ FL gB
ð9Þ
and the equation predicting the local mass fraction (regarded as the solution concentration in this work) takes the form D ðF CL Þ þ r 3 ðFL U BL CL Þ ¼ r 3 ðFL DrCL Þ ð10Þ Dt L
ð11Þ
where Qfilm is the liquid-film volumetric flow rate ! πFL gRc 4 n Qfilm ¼ 1 þ 4ðRb =Rc Þ4 3=4 - lnðRb =Rc Þ 8μL o - ðRb =Rc Þ - 2 ð12Þ 2324
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Figure 4. Effects of the number of mesh grids (three bubbles used in the simulations) on the volumetric mass-transfer coefficient. Conditions: Lfilm = 6 mm, dc = 1 mm, δ = 0.026 mm, Ub = 0.5 m/s, Luc = 16 mm, D = 2.0 10-9 m2/s. Figure 3. (a) Liquid velocity fields obtained from the simulations and (b) grid cells near bubbles. Conditions: Lfilm = 6 mm, dc = 1 mm, δ = 0.026 mm, Ub = 0.5 m/s, Luc = 16 mm, D = 2.0 10-9 m2/s.
For given values of Rb, Rc, and Ub, the liquid-slug velocity, Uslug, can be calculated from eqs 11 and 12. Therefore, the fully developed liquid-slug laminar flow profile relative to moving Taylor bubbles is Uinlet ¼ Uoutlet ¼ 2Uslug ½1 - ðr=Rc Þ2 - Ub
ð13Þ
The mesh grids used were structured quadrilateral cells generated by the preprocessor Gambit software. The grids were carefully designed and used near bubble surfaces and in liquid films. Figure 3b shows mesh grids near bubbles in detail. Confidence in the grid independence of the results was gained by increasing the number of grid cells for a few representative simulations. For example, the effect of the number of grid cells on KLa is illustrated in Figure 4. 2.3. Simulation Scheme. To evaluate the volumetric masstransfer coefficient, the overall simulation process was divided into two successive campaigns. First, steady flow fields were obtained by solving eqs 8 and 9. Second, based on the steady flow fields, mass transfer from bubbles to the liquid phase was studied by solving eq 10, and then steady concentration fields were realized. In the second campaign, the tracer concentration throughout the computational domain was set to zero [arbitrary units (au)] as the initial condition. At the surface of the bubbles, the tracer concentration was set to 1 au, which was used as the boundary condition. The solutions of the momentum and species conservation equations were approximated by the second-order upwind differencing scheme. The standard pressure interpolation scheme was employed, and SIMPLEC was used for pressure-velocity coupling. The simulations were realized using the Fluent software package.21
3. RESULTS AND DISCUSSION 3.1. Effect of the Number and Position of Unit Cells on KLa. As discussed previously, the volumetric mass-transfer coef-
ficient calculated from any unit cell should be the same if the end effects of the capillaries have been eliminated. In Figure 5, three
Figure 5. Volumetric mass-transfer coefficient calculated from every unit cell. Conditions: Lfilm = 6 mm, dc = 1 mm, δ = 0.026 mm, Ub = 0.5 m/s, Luc = 16 mm, D = 2.0 10-9 m2/s.
simulations were carried out under the same initial and boundary conditions for three, five, and eight unit cells. The results show that KLa is nearly a constant value for every unit cell (except for the unit cells at each end of the capillaries) because of the presence of the same flow environment. The end effects of capillaries on KLa were significant because the end bubbles were connected to only one other bubble, whereas internal bubbles were connected to two bubbles, one at either side. Figure 5 also shows that KLa evaluated from the internal unit cell in the simulation employing three unit cells was very close to that from the internal ones in the simulation using eight unit cells, and the error was within 5%. To save computational resources and time, three unit cells were used during the remaining simulations. 3.2. Effects of Various Parameters on KLa. Here, we investigated the dependence of KLa on the liquid-film length (Lfilm), liquid-slug length (Lslug), bubble rise velocity (Ub), liquid-phase diffusivity (D), liquid-film thickness (δ), and capillary diameter (dc). The details of the variations of the parameters 2325
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Table 1. Details of the Parameters Employed in the Simulations
a
dc (mm)
Lfilm (mm)
Lslug (mm)
δ (μm)
Ub (m/s)
D (10-9 m2/s)
1
4, 6, 8, 10, 12, 14
20, 18, 16, 14, 12, 10a
21
0.5
2
2
4, 6, 8, 10, 12, 14
20, 18, 16, 14, 12, 10a
42
0.5
2
1
4
8, 12, 16, 20, 24, 28
21
0.5
2
2
4
8, 12, 16, 20, 24, 28
42
0.5
2
1
6
10
10, 30, 50, 70, 90, 110, 120
0.5
2
2
6
10
10, 30, 50, 70, 90, 110, 120
0.5
2
1
6
10
26
0.1, 0.3, 0.5, 0.7, 0.9, 1.2
2
2 1
6 6
10 10
52 26
0.1, 0.3, 0.5, 0.7, 0.9, 1.2 0.5
2 0.1, 0.5, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
2
6
10
52
0.5
0.1, 0.5, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
1, 1.5, 2, 2.5, 3
6
10
26
0.5
2
1, 1.5, 2, 2.5, 3
6
10
52
0.5
2
Data for Lslug and Lfilm are counterparts in sequence in these rows.
employed in the simulation are given in Table 1. We carefully considered quantitatively the relationship between KLa and the parameters Lfilm, Lslug, Ub, D, δ, and dc. For generality and simplicity of practical application, the three dimensionless numbers Si, Li, and Pe were introduced; they are defined as Si = (KLadc2)/D, Li = (Lfilm/Lslug)(dc/δ), and Pe = (Ubdc)/D. Based on 78 sets of data given in Table 1 and their counterpart KLa values obtained in the simulations, we presented the fitted equation Si ¼ 0:12Li0:44 Pe0:54
ð14Þ
Figure 6 shows that the relationship between the equation data (calculated from eq 14) and the simulation data was good. Most errors were within (30%. Note that the term (dc/δ) in Li is a function of the capillary number and can be obtained by means of existing formulas.22,23 The other parameters in Li and Pe are easy to obtain based on experimental measurements or existing theoretical methods. Therefore, eq 14 is very convenient for evaluating KLa in practice. Further analyzing eq 14, it is shown that KLa increased with increasing Lfilm, Ub, and D, whereas it decreased with increasing Lslug, δ, and dc. The dimensionless number Fo was less than or close to 0.1 under the present simulation conditions. Consequently, the liquid film could be considered as unsaturated when gas bubbles passed by. To some extent, the penetration theory is suitable to describe gas-liquid mass transfer under these conditions. According to eqs 5 and 6, a longer liquid film has a smaller KL, film value, but a larger specific surface area, afilm. The overall results show that KLa increased with lengthening liquid film in eq 14. When the liquid-film length was kept constant, acap and afilm decreased with lengthening of the liquid slug (referenced to eqs 4 and 6), which resulted in KLa becoming smaller at larger Lslug values. The simulations show that KLa increased with increasing Ub and D. The reason for this behavior is that a larger Ub made the contact time shorter, which increased KL,cap and KL,film (see eqs 3 and 5). Similarly to Ub, a larger D value also made KL,cap and KL,film larger. Larger diffusivity made gas-phase transfer easier in the liquid phase. Liquid-film thickness, δ, is another important parameter determining KLa. Baten et al. showed that KLa is inversely proportional to film thickness for Fo > 1,17 whereas for Fo < 0.1, they considered KLa to be independent of film thickness, even though their results showed that KLa decreased with
Figure 6. Relationship between formulation and simulation volumetric mass-transfer coefficients.
increasing liquid-film thickness. Their reasons were that mass transfer from cylindrical bodies to liquid films occurred in the penetration regime and that the dependence of KLa on δ is perhaps due to grid convergence differences between thin and thick films in the simulations. In our study, the relation between δ and KLa was carefully checked. The results showed that KLa decreases with increasing liquid-film thickness. The reasons for this behavior are as follows: First, acap and afilm became smaller for thicker films, even though they are not shown in eqs 4 and 6. Equations 4 and 6 are approximate expressions evaluating specific interface area and are suited for thinner liquid films, but not for thicker liquid films. Second, thicker liquid films mean that more liquids flow from the front of the bubbles to the back of the bubbles, which changes the velocity and concentration fields around the bubbles. The dependence of KLa on capillary diameter is given in eq 14, which shows that KLa decreases with increasing capillary diameter. The change trend can also be explained by penetration theory. As shown in eqs 3 and 6, KL,cap and afilm become smaller when the capillary diameter becomes larger. The effect of gravity on KLa is shown in Figure 7. When the Taylor bubbles moved upward in the vertical capillaries, they were surrounded by liquid films that flowed downward due to 2326
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Figure 7. Effect of gravity on volumetric mass-transfer coefficient. Conditions: Luc = 16 mm, Lfilm =6 mm, D = 2.0 10-9 m2/s.
gravity. At low liquid-film flow rates, the flow along the wall was laminar, and a velocity profile was quickly established. For circular capillaries, the velocity profile in the liquid film was given by Reineit24 as Ur ¼
FL gRc 2 ½1 - ðr=Rc Þ2 þ 2ðRb =Rc Þ2 lnðr=Rc Þ 4μL
ð15Þ
From eq 15, the downward liquid velocity approaches its largest value at the surface of liquid film, r = Rb. The surface velocities of liquid films is in the range 4.3 10-3 m/s < Usur < 6.8 10-2 m/s [here, FL=998 kg/m3, μL=0.001 kg/(m 3 s)], obtained from eq 15, at 0.03 mm < δ < 0.12 mm for 1- and 2-mm capillaries in Figure 7. Usur increased with increasing liquid-film thickness. From the penetration theory, a larger Usur value makes the contact time between gas and liquid smaller and thus increases KL,film,17 although this is not shown in eq 5. The increment is more apparent at lower bubble velocity. Equation 5 ignores the effect of Usur on KL,film. Compared with the bubble velocity Ub = 0.5 m/s, Usur is so small here that it can be neglected. Thus, the effect of gravity on KLa in Figure 7 is unapparent for Ub = 0.5 m/s. In contrast, for Ub = 0.05 m/s, it was found that the gravity effect existed and the effect increased slowly with increasing liquid-film thickness. 3.3. Gas Transfer Across Various Parts of Taylor Bubbles. To evaluate the gas-liquid mass transfer across the cylindrical bodies and hemispherical caps of Taylor bubbles, two ideal models were employed. One model assumed that gas transfer occurred only across the cylindrical bodies of the bubbles. In this case, the solute (gas) concentration at the surface of the cylindrical bodies was set to 1.0 au, and the mass-transfer flux across caps was fixed to be zero; these values were used as the boundary conditions during the simulations. The other model presumed that gas transfer occurred only across the caps of bubbles. The boundary conditions were just the opposite to those in the first model: the solute concentration at the surface of caps was set to 1.0 au, and the masstransfer flux across the cylindrical bodies was fixed at zero. KL,filmafilm and KL,capacap obtained from the numerical simulations based on the above two models are shown in Figure 8. It is well-known that, in practice, gas transfer occurs simultaneously across the cylindrical bodies and caps of Taylor bubbles. Overall volumetric mass-transfer coefficient KLa, based on the practical model given in section 2.3, is also given in Figure 8. To reveal the
Figure 8. Volumetric mass-transfer coefficients obtained from various parts of Taylor bubbles. Conditions: Ub = 0.5 m/s, Luc = 24 mm, δ/dc = 0.021, D = 2.0 10-9 m2/s.
relations between KL,filmafilm, KL,capacap, and KLa, the algebraic sum of KL,filmafilm and KL,capacap, namely, (KL,filmafilm þ KL,capacap), is also shown in Figure 8. Under the conditions given in Figure 8a,b, (KL,capacap)simu kept a constant value, whereas (KL,filmafilm)simu increased with lengthening liquid film. (KL,filmafilmþKL,capacap)simu and (KLa)simu both followed the change trend of (KL,filmafilm)simu . One point worth noticing is that (KLa)simu was very close to (KL,filmafilm)simu but markedly smaller than (KL,filmafilmþKL,capacap)simu. Similar result were also confirmed experimentally by Vandu et al.19 To some extent, KL,filmafilm was found to approximately equal KLa, which was consistent with the experimental results.18,19 The simulation values are also compared with the values calculated based on eqs 2-6 in Figure 8. As a whole, the simulation values were consistent with the calculation ones, especially for KL,capacap . It is well-known that the volumetric mass-transfer coefficient, KLa, is influenced by the gas-liquid specific surface area, a. Excluding a, the mass-transfer coefficient, KL reveals the ability of mass transfer occurring in a unit area. KL,cap, KL,film, and KL obtained from the present simulations and the equations developed by Baten et al.17 are displayed in Figure 9. Note that (KL)formu was calculated as follows ðKL Þformu ¼ ðKL, cap acap þ KL, film afilm Þformu =ðacap þ afilm Þformu ð16Þ 2327
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Figure 9. Mass-transfer coefficient obtained from various parts of Taylor bubbles. Conditions: Ub = 0.5 m/s, Luc= 24 mm, δ/dc = 0.021, D = 2.0 10-9 m2/s.
According to the figure, (KL,cap)simu apparently does not vary, whereas (KL,film)simu decreases with increasing liquid-film length. (KL)simu was close to (KL,film)simu and followed the change trend of (KL,film)simu. (KL,cap)simu was larger than (KL,film)simu, which means that the ability for gas transfer across the caps is larger than that across the cylindrical bodies in units of mol/(s 3 m2). This finding contradicts the results of Irandoust et al.,14 who concluded that the two parts make the same contribution to the gas transfer in terms of mol/(s 3 m2). This might be due to the interaction of mass transfer across the cylindrical bodies and caps of bubbles, as (KL)simu is less than (KL,cap)simu and (KL,film)simu. It is clear that the formula values are consistent with the simulation ones for both KL,cap and KL,film, especially for KL,cap. Similarly to (KL)simu, (KL)formu is also close to (KL,film)formu and follows the change trend of (KL,film)formu. However, (KL)formu is larger than (KL,film)formu, which might be caused by the mathematical formula used to evaluate (KL)formu, eq 16. 3.4. Comparison with the Existing Empirical Correlation and Theoretical Formula. The present simulations are compared with the empirical correlation, eq 1, and theoretical formulas, eqs 2-6, in this section. Considering eq 1, one point worth noticing is that the capillary diameter and physical characteristics of the fluid do not appear in it, which reduces its general applicability. The term (1 - εG)Luc in the denominator
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Figure 10. Volumetric mass-transfer coefficient comparison of the present simulation data with the values calculated from eq 1. Conditions: (a) δ/dc = 0.021, Ub = 0.5 m/s, Luc = 24 mm, D = 2.0 10-9 m2/s; (b) Lfilm = 6 mm, δ/dc = 0.026, Ub = 0.5 m/s, Luc = 16 mm.
practically equals the liquid-slug lengths. Therefore, eq 1 shows that KLa depends on the liquid-slug length, not the gas-bubble length. In fact, the data of Bercic and Pintar15 were obtained from long bubbles (or liquid films) and long liquid slugs. From the penetration theory, the contact time between the liquid films and bubbles increased with increasing liquid-film length, which means that the liquid film easily approaches saturation and that its contribution to increasing the overall mass-transfer ability decreases. KLa is plotted against the liquid-film length for 1- and 2-mm-diameter capillaries in Figure 10a. In the experiment performed by Bercic and Pintar,15 the diffusivity of methane in water was estimated to be 2.0 10-9 m2/s, which is the same value as was employed in the present simulations in Figure 10a. Because the effects of capillary diameters on KLa were not considered, eq 1 present only one function curve. The curve follows the change trend of the simulation values, but does not agree well with them, especially for the 2-mm-diameter capillary. Equation 1 was originally proposed based on the physical absorption of methane in water. Some researchers considered that, if a different gas-liquid system were applied, eq 1 could be scaled to account for the different diffusivity by ðKL aÞA ¼ ðKL aÞB ðDA =DB Þn 2328
ð17Þ
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some regions. In fact, eq 2 is an approximate formula for calculating KLa, and it ignored all possible overlap and interaction between different mass-transfer processes at the cylindrical body and cap of a unit cell. In the simulations performed by Baten et al.,17 only one unit cell was considered, and periodic boundary conditions were employed at two ends of the unit cell, which meant that the solute concentration was the same at the two ends of the unit cell and did not correctly reveal the real environment around the Taylor bubbles. The errors introduced by the model geometry and periodic boundary conditions were not considered in their work.
Figure 11. Volumetric mass-transfer coefficient comparison of the values calculated from eq 14 with the data obtained by Baten et al.17 Conditions: (a) D = 1.0 10-9 m2/s and Ub = 0.45 m/s, with (1) dc = 1.5 mm, δ = 0.024 mm, Lfilm = 6.289 mm; (2) dc = 3 mm, δ = 0.048 mm, Lfilm = 5.321 mm. (b) D = 1.0 10-9 m2/s and Luc = 40 mm, with (1) dc = 1.5 mm, δ = 0.024 mm, Lfilm = 6.289 mm; (2) dc = 3 mm, δ = 0.048 mm, Lfilm = 5.321 mm.
where the scaling factor n was 1.0 for film theory and 0.5 for penetration theory.12 Based on eqs 1 and 17, KLa was calculated and is plotted against D in Figure 10b. Note that, here, the parameter n in eq 17 equaled 0.5. Figure 10b shows that KLa obtained from eqs 1 and 17 is consistent with the simulation values in terms of its trend. Similarly to Figure 10a, the calculated values agreed better with the simulation values for the 1-mm capillary than for the 2-mm capillary. The cause is not yet clear and might result from the present simulation conditions and the expression of eq 1. As reviewed in the Introduction, Baten et al.17 presented eqs 2-6, to predict KLa based on the penetration theory. They also investigated gas transfer from Taylor bubbles to the liquid phase by means of CFD. Our simulation results have already been compared with the values obtained from eqs 2-6 in Figures 8 and 9. The agreement between our values and Baten et al.'s17 values is good. To check the validity of the present formula, eq 14 was used to calculate KLa, and the results are compared with the values obtained by Baten et al.17 in Figure 11a,b. As shown in the figure, the curves given by eq 14 were consistent with Baten et al.'s data in the trend, although deviations exist in
4. CONCLUSIONS Gas-liquid mass transfer in Taylor flow through circular capillaries was studied numerically. The effects of various parameters on the volumetric mass-transfer coefficient were carefully checked. Based on the present numerical simulations, the following conclusions were drawn: (1) KLa increases with increasing liquid-film length, liquidphase diffusivity, and bubble velocity, whereas it decreases with increasing the liquid-slug length, capillary diameter, and liquid-film thickness. One important formula, Si = 0.12Li0.44Pe0.54, was developed that can predict KLa well. Because of the simplicity and generality of this formula, it is very convenient for evaluating KLa in practice. (2) The effect of gravity on KLa is very slight for Taylor flow through capillaries, especially for higher bubble velocities and thinner liquid films. For lower bubble velocities and thicker liquid films, KLa with gravity is slightly larger than that without gravity. (3) The gas-transfer behaviors across the cylindrical bodies and caps of Taylor bubbles were studied separately. To some extent, KL,filmafilm approximately equals KLa. In terms of mol/(s 3 m2), the ability of gas transfer across the caps of Taylor bubbles is larger than that across the cylindrical bodies. However, because the surface area of the cylindrical body is usually much larger than that of the caps, gas transfer across the cylindrical bodies is more important. (4) For Fo < 0.1, both Baten et al.'s method,17eqs 2-6, and the present formula, eq 14, can approximately describe mass transfer from bubbles to liquids. The two results follow the same trend. The variance between eqs 2-6 and eq 14 in Figure 11a,b, partially results from the fact that the overlap and interactions between the different masstransfer paths were all ignored in eq 2. The comprehensive understanding of gas-liquid mass transfer in Taylor flow will be helpful to optimize the design and operation of gas-liquid-solid monolith reactors and microreactors. ’ AUTHOR INFORMATION Corresponding Author
*Tel.: þ86-411-84662365. Fax:þ86-411-84662365. E-mail:
[email protected]. Present Addresses §
Department of Chemistry, University of Cambrigdge, Lensfield Road, Cambridge, UK CB2 1EW.
’ ACKNOWLEDGMENT The authors are grateful for the financial support for this work from the National Natural Science Foundation of China 2329
dx.doi.org/10.1021/ie902055p |Ind. Eng. Chem. Res. 2011, 50, 2323–2330
Industrial & Engineering Chemistry Research (Nos. 20590365 and 20776132); National High Technology Research and Development Program of China (No. 2007AA05Z148); and Innovation Fund of Dalian Institute of Chemical Physics, Chinese Academy of Sciences (No. K20060275). D.L. also thanks Professor Sankaran Sundaresan of Princeton University for his helpful comments, discussions, and encouragement.
’ NOMENCLATURE a = specific interfacial area, m-1 Ab = bubble cross-sectional area, m2 Ac = capillary cross-sectional area, m2 au = arbitrary units C = concentration (local mass fraction in eq 10), au C* = gas equilibrium concentration at the gas-liquid interface, au Cinlet = inlet concentration, au Coutlet = outlet concentration, au D = liquid-phase diffusivity, m2/s dc = capillary diameter, m Fo = Fourier number = Dtfilm/δ2 g = acceleration due to gravity, m/s B KL = liquid mass-transfer coefficient, m/s KLa = liquid volumetric mass-transfer coefficient, 1/s Lc = capillary length, m Li = (Lfilm/Lslug)(dc/δ) Luc = unit cell length, m Lfilm = liquid-film length, m Lslug = liquid-slug length, m n = parameter in eq 17 p = pressure, Pa Pe = Peclet number = (Ubdc)/D Qfilm = liquid-film volumetric flow rate, m3/s r = radial coordinate, m Rb = bubble radius, m Rc = capillary radius, m Si = (KLadc2)/D t = time, s UTP = two-phase average velocity, m/s Uinlet = inlet velocity, m/s Uoutlet = outlet velocity, m/s Uslug = liquid-slug velocity, m/s U BL = liquid velocity vector, m/s Ub = bubble rise velocity, m/s Ur = axial velocity at the radial coordinate r, m/s Usur = liquid velocity at the surface of a bubble, m/s Uwall = capillary wall velocity, m/s Greek Letters
δ = liquid-film thickness, m εG = volume fraction of gas phase in a unit cell μL = liquid molecular viscosity, kg/(ms) FL = liquid density, kg/m3 Subscripts
cap = hemispherical cap film = liquid film G = gas phase inlet = inlet of the computational domain or capillary L = liquid phase outlet = outlet of the computational domain or capillary simu = simulation values slug = liquid slug
ARTICLE
formu = formula values uc = unit cell
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