Ind. Eng. Chem. Res. 2007, 46, 2205-2211
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Mass Transfer in Bubble Column Reactors: Effect of Bubble Size Distribution Amol A. Kulkarni* Industrial Flow Modeling Group (iFMg), National Chemical Laboratory, Pune-411008, India
Mass transfer in a bubble column is analyzed in a different perspective. The experiments were performed in a bubble column reactor that was operated in a regime where the overall rate of absorption of gas into liquid was dependent on both the mass-transfer coefficient and the rate of reaction. Homogeneous catalytic oxidation of sodium sulfite was considered as a model reaction. The local instantaneous velocity data was measured using LDA in the presence of reaction and was used to yield the bubble size distribution at several locations in the column. The role of individual bubble size during mass transfer with chemical reaction was observed to vary. The findings suggest that a narrow bubble size distribution would help to achieve uniformity in the performance at bubble scale, so that an identical regime of mass transfer with reaction exists at the individual bubble level. 1. Introduction Interfacial mass transfer across the gas/liquid interface is an important phenomenon in bubble column reactors, and it is measured in terms of the overall mass-transfer coefficient kLa (in units of s-1), where kL is the true mass-transfer coefficient and a is the effective interfacial area. The second parameter can be estimated physically, either using the equilibrium bubble size and the average fractional gas holdup in the column or by taking into account the bubble size distribution, shape distribution, and average fractional gas holdup in the column. Recently, a few detailed analyses report the effect of bubble size on the true mass-transfer coefficient through some very elegant experiments1-4 and through numerical simulations for the analysis of physical mass transfer4-6 with and without surface contamination. The application of bubble column reactors and their effective utilization is dependent on the reaction system, the absorption regime, and the hydrodynamic behavior.7 In the heterogeneous regime of operation, bubble size distribution and eddy size distribution contribute differently to the interfacial mass transfer. The contribution of individual eddy size in the overall mass transfer is a function of eddy life, the bubble size distribution, and contact frequency between the individual eddy sizes and bubbles of different sizes (provided the liquid is far from the saturation limit). Among the two, the bubble size distribution is dependent on sparger specifications, gas-liquid properties, and the operating conditions (viz., gas velocity, temperature, pressure, etc.), whereas eddy size distribution is solely decided by the liquid viscosity and the superficial gas velocity. Hereafter, we focus on the bubble size distribution and its effects on the mass transfer in bubble column reactor. In a bubble column reactor that contains coalescing liquid, for bubbles >1 mm, the gas/liquid interface is flexible and the mass transfer across the interface is a dynamic process associated with the dynamics of the interface. In a multiphase system, kL is estimated either from the knowledge of the surface renewal rates or the contact time. The true mass-transfer coefficient is difficult to measure independently, whereas a can be estimated using various heuristics. The main reasons for difficulty in measuring the value of kL directly are either the inability to have an accurate estimation of different parameters proposed in various theories of interfacial mass transfer or an ambiguity * To whom correspondence should be addressed. Tel.: +91-2025902153. Fax: +91-20-25893260. E-mail address: aa.kulkarni@ ncl.res.in.
about the interfacial transport phenomena and the quantitative contribution of different governing parameters, such as eddy size distribution, age distribution, distance of eddies from the interface, eddy shapes, eddy motion at the interface, etc. In last several decades, many approaches have been developed to be more and more realistic toward capturing the interfacial mass transfer. A brief survey of these approaches is given in Table 1. In the absence of a clear idea of measuring specific parameters such as film thickness, age distribution function, eddy size distribution (as a function of the concentration gradient at the interface), penetration depth, surface renewal rate, etc., simple formulations that would follow a combination of two approaches are developed and used. In the present manuscript, due to its simplicity, we have used the penetration theory and its variants for our analysis. According to penetration theory,9 kL is given as
kL ) 2
x
DL πt
(1)
where DL is the diffusivity of gas in liquid and t is the contact time. In most cases, it is assumed that, during the contact of phases, the turbulence in the vicinity of the interface is isotropic (for the sake of mathematical convenience). Following Kolmogorov’s inertial range hypothesis based on the concept of isotropic turbulence, eq 1 can be given in an alternate as:
kL ) 2
x
DL
πx(ν/)
(2)
However, in reality, it is rare to find an isotropic flow24 and the aforementioned formulation can be used only with some correction factor to compensate for the intermittent variation in the energy dissipation rate (). One must note that this expression shows the dependence of kL on the energy dissipation rate,25 which has no easy ways of measurement. The kL values are known to be dependent on the interface mobility, and the mobility can be given in terms of time span for a size specific surface renewal. Following these arguments, Calderbank26 has derived the value of contact time in a bubbling system, in terms of the average bubble size (dB) and the average bubble slip velocity (VB):
x
kL ) 2
10.1021/ie061015u CCC: $37.00 © 2007 American Chemical Society Published on Web 03/06/2007
DLvB πdB
(3)
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Table 1. Survey of Models for Mass Transfer at Gas/Liquid Interfaces model two-film model8 penetration theory9 surface renewal model10
film penetration model11
description Steady-state mass transfer across two films of two fluids across the interface. Widely used to derive complex interfacial processes analytically. Mass transfer across the interface has been considered as an unsteady state process, whereas, for all eddies, equal age-distribution was considered.
equation for mass transfer DL kL ) δ kL ) 2
x
DL πt
Interfacial mass transfer is an unsteady state process. kL ) 2xDLs Random age distribution; eddy size distribution is not considered. The main disadvantage is that all the velocity fluctuation effects are lumped into a single-term surface renewal rate (s), which is difficult to measure. ∞ The resistance to the mass transfer in the phase considered is confined within DL 2L 2 L3 1 a film of thickness L at the interface. However, the fluid elements of the film kL ) 1exp(X) L 3DLθ πDLθ n)1 n2 or surface elements are constantly renewed as in the penetration theory or surface renewal theory. π2n2DLθ where x ) , L ) 151 dRe-0.875, L2 1 µ θ ) ) 13.7 × 10-5 Re1.75 S Fd2 L and S are taken from Brusset et al.12
(
∑
)
( )
random-eddy modification of penetration theory13
Modification of surface renewal theory by assuming that eddies arriving at random times come within random distances from the interface. Gamma distribution was approximated for the random quantities.
kL )
1 H 1 + DL 1.13
x
t DL
The concept of surface renewal theory could be expected when H/xD/S < 0.5, Distribution function for a random sequence of distance and time: where H is the average distance to where eddies approach the interface. 1 t φ(t) ) tR-1 exp β Γ(R)βRt
(
interfacial
turbulence14,15
Steady-state vertical component of the gas velocity, which should be sufficiently large for its penetration into liquid.
kL )
)
( )
D0.5F0.5ν0-1.5
σ0.5 (from Levich14)
Only flow in the viscous sublayer was taken into account.
kL ) 0.32
D0.5F0.5ν0-1.5
σ0.5 (from Davies15)
large-eddy model16
small-eddy model17
random-eddy surface renewal model18
The large eddy motion dominates the convective effect of turbulence on the flow field at the interface. A regular sequence of large cells is formed at the interface.
kL ) 1.46
x
DLxuj2 l
x
Steady state DL Highly mobile smaller eddies contribute the most to the mass transfer. kL ) 0.4 (ν/)0.5 The theory seems unreasonable for a few cases (with low surface tension fluids) where the surface renewal rates would be much less than the viscous forces. Extension of Harriot’s model.13 The distribution function for H and t The renewal procedure is formulated as a stochastic process. are given as φi(z) )
[
where β ) single-eddy model19
single-eddy model for random surface renewal20
combined penetrationsurface renewal model21
]
βR+1 zR exp(-βz)U(z) Γ(R + 1) R+1 〈zˆ〉
and U is a unit step function Steady-state motion at the interface. DL The contribution from every eddy that arrives at the interface is considered. kL ) 0.9 L/V The large eddies are assumed to follow a sinusoidal velocity pattern at the interface. The age distribution of eddies at the interface is measured experimentally Age distribution function: and is determined to have a gamma distribution. n t βk-1 k-1 Age distribution of fresh eddies is more important and is different from φ dt ) 1 t exp(-βt) 0 Danckwerts’ proposition. k)1 Γ(k) Modification of the penetration theory and extension of the single-eddy model. D Considers the eddy size distribution and age distribution. kL ) 2 Turbulent fluctuations at the interface are explained as eddy motions. πt n Each of the encountered eddies was considered to be fresh. t βk-1 k-1 t φ dx ) 1 exp(-βx) 0 k)1 Γ(k)
x
∫ ∫
∑
x
( )
∑
where x gives the eddy life and eddy size surface renewal stretch model22
The model is based on the equation of continuity and includes turbulence, as well as convective mass transfer at the fluid/fluid interface. Terminology of the stagnant film hypothesis is retained. model based on the vertical The model considers the gradient in the vertical velocity as the only velocity gradient at the criterion for correct estimation of the true mass-transfer coefficient. interface23 Experiments conducted over a wide range of conditions were observed to fit a general relationship that correlated interfacial hydrodynamics parameters with the mass-transfer velocity.
kL )
x
kL ) C
4DLs ) π
xβνSc
x x
where C ≈ 0.2
4DL π
VGg ν
Ind. Eng. Chem. Res., Vol. 46, No. 7, 2007 2207
which implies that, for low-viscosity liquids, such as water, as the bubble size increases, kL decreases or, very specifically, it will be largely dependent on the relationship between bubble size and its velocity. For stagnant liquids, the slip velocity is same as the bubble rise velocity. Equation 3 can also be obtained from the dimensionless numbers:
Sh ) 1.13Re0.5Sc0.5 Because eq 3 does not consider the complete flow field in the vicinity of the bubble, it will give some deviations from the actual values when the bubble size is larger and still the masstransfer coefficient would also increase. Taking into account the fact that not all bubbles are spherical, and, hence, the interface contact time cannot be given as dB/VB for all bubble sizes, Miller27 has introduced a correction factor for predicting the true mass-transfer coefficient (for stripping of carbon dioxide from aqueous solution with air):
k*L ) 683dB1.376
(4)
which can be multiplied by the kL from eq 3 to get the actual true mass transfer coefficient value. Such an approach that requires rigorous experiments has also been extended for gasoline and toluene systems.28 This approach combines the penetration theory of interfacial mass transfer and the wave theory for predicting bubble rise velocity. Thus, to estimate reliable kL, either bubble size distribution or energy dissipation rate is required and understanding its effects on the performance of a running bubble column is necessary. In this communication, we have used our experimental observations about the bubble sizes, and their rise velocity measured in a bubble column, and estimated specific reaction rates from the experiments in a stirred cell to determine the effect of bubble size and shape on the intrinsic mass-transfer-related features of a bubble column reactor. Here, we bring out an interesting observation that will illustrate the importance of knowledge about the bubble size and shape distribution in a bubble column reactor. 2. Experimental Procedure The experiments were conducted in a 150-mm inner diameter bubble column reactor with a clear liquid, height-to-diameter ratio of 6. An oil-free diaphragm-type compressor was used to introduce air through a sieve plate sparger with a hole diameter of 0.8 mm and a free area of 0.347%. Details about the experimental setup are given in the work by Kulkarni and coworkers.29,30 A precalibrated rotameter was used to measure the volumetric gas flow rate. Experiments were conducted at a superficial gas velocity (VG) of 24 mm/s. The axial component of velocity was measured using LDA at five different axial levels at several radial positions,29,31 while the mass transfer with chemical reaction of the homogeneous catalytic oxidation of sodium sulfite was taking place in the column. The use of the sulfite oxidation chemical method for mass transfer studies is discussed in detail in the work of Linek and Vacek32 and Kulkarni and Joshi.30 The details of the experimental conditions and the procedure used to evaluate the kinetic parameters are given elsewhere.30 For every experimental run, the value of pH and the ionic strength were maintained the same as those in the stirred cell, which was used to estimate kinetic parameters for the predefined operating conditions. Identical reactant (constant ionic strength by keeping [Csulfite + Csulfate] ≈ 0.8 M) and catalyst concentrations (10-6 < CCoSO4 < 2 × 10-5) were maintained in the bubble column to have the same specific reaction rate as
Figure 1. (A) Bubble rise velocity distribution and (B) estimated bubble size distribution in bubble column reactor at different axial measurement levels. Legends correspond to the distance from the sparger, in terms of column diameter D (L1 ) 2D, L2 ) 2.7D, L3 ) 3.4D, L4 ) 4.1D, and L5 ) 4.8D).
those in a stirred cell. The temperature of the system was maintained at 24 °C for all the experiments. The reduction in sulfite concentration was followed using iodometry. At the experimental superficial gas velocity, the Danckwerts’ plots were constructed for the estimation of kL and a separately. 3. Data Analysis The LDA data were mainly used to identify the passage of bubbles through the measurement volume29 and to estimate bubble size distribution in the bubble column. For that purpose, the slip velocity of bubbles was estimated by subtracting the local liquid velocity from the individual bubble rise velocity. The slip velocity data were transformed to terminal rise velocity data for individual bubbles by multiplying the slip velocity values by a correction term, 1/x∈L,33 where ∈L is the liquid holdup. The square-root dependence comes from the relationship
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Figure 2. Variation in true mass transfer coefficient with bubble size in pure liquids. Estimations are based on formulation by (i) Calderbank (1957): air-water and (ii) Miller (1974): stripping of carbon dioxide from aqueous solution with air. Data for Kawase and Moo Young35 and Aiba and Huang36 are also shown.
between the bubble shape and the drag coefficient. The bubble rise velocity (ub) information was then correlated to the bubble sizes, using the correlation by Nguyen34 for contaminated aqueous liquids. The value of ∈L was also obtained from the LDA data.29 The LDA data was acquired for a longer time period, to capture a sufficiently large number of bubbles (∼1200) at each location, which was later converted to a bubble size distribution31 (see Figure 1). The data from all the measurement locations in one measurement plane were subjected to further analysis. The estimated bubble sizes and the respective slip velocity data were used to estimate kL using eq 3, which is analogous to penetration theory (i.e., eq 1). Thus, the bubble size distribution and corresponding rise velocity information can be used to estimate the distribution of kL values at the respective measurement planes in the column. 4. Results and Discussion Estimated bubble sizes and the corresponding slip velocity values were subjected to eq 3 and the correction that was suggested by Miller27 (i.e., eq 4) to obtain the corresponding true mass-transfer coefficients. The resulting values of kL from eq 3 were observed to decrease continuously with bubble size dB and slowly level off (see Figure 2). The kL values based on the correction by Miller27 were observed to increase with dB, and the rate of increase in kL decreased gradually. The trend for the air-water system from eq 3 is consistent with the experimental data of Kawase and Moo Young,35 Vasconcelos et al.,4 and Aiba and Huang,36 whereas the majority of the experimental data published in the literature (for example, Figure 10 of Heijen and Van’t Reit37) follows the trends similar to those proposed by the combination of penetration theory and the wave theory. As mentioned previously, the chemical method of analysis (iodometry) was used to get Danckwerts’ plots for the estimation of kL. In the cases where the mass transfer is accompanied by a fast pseudo-mth-order reaction, the overall rate of oxygen absorption or the oxidation reaction (Ra) is given by
Figure 3. Distribution of true mass-transfer coefficient values estimated for a bubble size distribution in a bubble column at different axial measurement levels: (A) penetration theory alone and (B) penetration theory with correction for bubble shape variation. (See Figure 1 for an explanation of the symbols used.)
Ra ) a[A*]
{m +2 1D k
A mn[A*]
m-1
[B0]n + kL2
}
1/2
(5a)
l , i.e., where kmn ) k′CCoSO 4
Ra ) a{R2 + kL2[A*]2}1/2
(5b)
This equation was derived by Hikita and Asai,38 based on the two-film model. The first term on the right-hand side is the contribution to the overall rate of absorption due to reaction, whereas the second term corresponds to the overall mass-transfer rate. In such cases, the contribution of the specific rate of reaction (R) is comparable to the specific rate of mass transfer (kL[A*]). The intercept of the plots of (Ra)2 vs R2 yields the kLa value from the intercept and a is obtained from the slope. These two parameters can be used to estimate the value of kL for a given superficial gas velocity. The specific reaction rates R (measured using a model contactor: stirred cell) can be used
Ind. Eng. Chem. Res., Vol. 46, No. 7, 2007 2209
Figure 4. Distribution of xM values obtained in a bubble column at different specific reaction rates. The true mass-transfer coefficient is estimated from (A and B) penetration theory alone and (C and D) penetration theory with correction for bubble shape variation. For panels A and C, the bubble size distribution belongs to L2 in Figure 1, and for panels B and D, the bubble size distribution belongs to L1 in Figure 1.
to estimate the value of xM as
xM )
R [A*]kL
where [A*] is the concentration of oxygen at the interface. The value of xM indicates the ratio of rate of oxygen consumption in the film to that in the bulk. Following Doraiswamy and Sharma,39 for our system, for the range of catalyst concentrations under consideration, the value of xM < 3 confirmed that the system was in the regime between 2 and 3. The specific rate of reaction was varied by changing the catalyst concentration while all other variables (viz., initial concentration of the sodium sulfite, pH of the solution) were maintained constant. The instantaneous velocity data obtained using LDA at one measurement location during the oxidation at different catalyst concentrations did not show significant variation in the number density plots of the instantaneous velocity and also the bubble velocity ((3%). Hence, the velocity-time data obtained for the case of a catalyst concentra-
tion of 3.1 × 10-5 kmol/m3 were used for further analysis. The data were used to estimate the bubble size distribution at several axial measurement levels. When the kL values (given in terms of m/s), based on bubble size and slip velocity and the measured specific reaction rates, are used, the values of xM can be obtained at different measurement levels. Typical distributions of kL values obtained using eq 3 and using the correction for bubble shape are shown in Figure 3. Note that the range of abscissa for Figure 3A is wider than that for Figure 3B. Also, the range of kL on the abscissa has shifted toward the ordinate for the estimations that include a correction for the shape factor. Furthermore, the difference in the nature of distribution also gives an idea about the possible effects of bubble size distribution on individual hydrodynamic parameters such as kL, as shown here. These values are strongly dependent on the local bubble size distribution; therefore, they yield a distribution of xM values (Figure 4A-D) for different specific rates of reaction. Also, for the same bubble size distribution, xM is a strong function of the specific rate of reaction, and, hence, the range of distribution
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was observed to become narrow with reductions in R (see Figures 4A and 4B). The values from the formulation by Calderbank26 were smaller than those estimated on the basis of additional shape dependent correction factor. For a large number of (small) bubbles, kL values from the corrected formulation are less than the earlier case and, hence, the average xM values are relatively large. Figures 4A and 4C can be compared to realize the possible extent of variation in xM values for the estimations based on corrected penetration theory for higher specific rates. The averaged value of xM obtained (penetration theory with correction for shape effect) at each measurement level in the column lies in the range of 4.29-5.18 and indicates the mass transfer with reaction to be in the third regime (fast reaction), which is not true. The estimated xM values that use the kL values based on the penetration theory alone from different axial levels lie in the range of 2.17-2.28, which, when averaged, yields xM ) 2.23, which is comparable to the value of 2.276 obtained from the chemical method. This observation was seen to be consistent for all the catalyst concentrations that could achieve a value of xM < 3 from the chemical method. This observation supports the use of penetration theory for the estimation of kL. An important observation from Figures 4A and 4B is that, although the weighted average value of xM obtained from the data from all locations is 3 still exist, which diverts the system from being within the required regime completely. Thus, to achieve a desired performance from the system, it is required to maintain the bubble size in a narrow range, so that the extent of mass transfer with reaction would be identical in the entire column. This result also indicates that, in a bubble column under operation, one is required to analyze the validity of the xM values obtained using the chemical methods before estimating the important design parameters, such as the mass-transfer coefficient and the effective interfacial area, specifically when the mass transfer is accompanied by a fast pseudo-mth-order reaction everywhere in the column. It is necessary to realize that the observations of the distribution of xM values are equally valid for other multiphase reactors, viz., stirred tanks, fermentation systems, air-loop lift reactors, gas-induced systems, etc. In noncoalescing systems, the possibility of the occurrence of such an issue is not significant. On the other hand, for coalescing systems, it is necessary to identify the ways and means to avoid wide bubble size distribution, may be through proper sparger design and selecting suitable operating conditions. Although the observations from this analysis support the formulation by Calderbank,26 a thorough analysis is required to investigate the effect of the nonspherical shape of bubbles on the true mass transfer and, hence on the performance of the system. The bubble shape is a result of the internal pressure variation in the bubble. This might have an effect on the gas/ liquid interface at different regions on a bubble surface and, hence, possibly the film thickness variation over the bubble surface. In turbulent flows, the bubble motion has a nonlinear dependence on bubble size and the surrounding flow structures, which would cause totally different behavior of kL, relative to bubble size. Although these possibilities can shed some light on the relationship between bubble shape and interfacial mass transfer, more-detailed analysis is required to understand such intricate phenomena and their effects on the performance of a reactor.
5. Conclusions The mass transfer with chemical reaction is studied in a bubble column reactor with a different perspective. The estimated values of the true mass-transfer coefficient, based on Calderbank’s formulation, were more similar to the experimental results from the chemical method. Interestingly, although the weighted average value of xM obtained from the estimations from all locations is 3 exist, which indicates the extent of deviation of the system from being within the required regime completely. To achieve uniformity in the performance at bubble scale and below, it would be necessary to maintain the bubble size in a narrow range, so that the same regime of mass transfer with reaction exists at the individual bubble level. Nomenclature a ) effective interfacial area (1/m) [A*] ) concentration of oxygen at the interface (kmol/m3) C ) concentration (kmol/m3) C ) parameter in the model by Xu et al.23 DL ) diffusivity of gas in liquid (m2/s) H ) penetration depth (m) kL ) true mass-transfer coefficient (m/s) 27 k* L ) correction factor by Miller kLa ) overall mass transfer coefficient (s-1) l ) average eddy length scale (m) L ) eddy length scale (m) L ) penetration depth (m) xM ) ratio of rate of oxygen consumption in the film relative to that in the bulk n ) number of eddies R ) specific reaction rates Ra ) overall rate of absorption Re ) Reynolds number s ) surface renewal rate (1/s) Sc ) Schmidt number Sh ) Sherwood number t ) contact time (s) uB ) terminal rise velocity (m/s) uj ) average eddy velocity (m/s) VB ) bubble slip velocity (m/s) VG ) superficial gas velocity (m/s) Greek Symbols β ) parameter in the gamma distribution δ ) film thickness (m) ) energy dissipation rate (m2/s3) ∈L ) fractional liquid holdup in the bubble column φ ) distribution function ν ) kinematic viscosity (m2/s) θ ) penetration time (s) Acknowledgment The financial support during this work by Professor M. M. Sharma Endowment of Mumbai University is gratefully acknowledged. The author also thankfully acknowledges the discussions with Dr. S. S. Bhagwat of U.I.C.T., Mumbai. Literature Cited (1) Takemura, F.; Yabe, A. Rising Speed and Dissolution Rate of a Carbon Dioxide Bubble in Slightly Contaminated Water. J. Fluid Mech. 1999, 378, 319.
Ind. Eng. Chem. Res., Vol. 46, No. 7, 2007 2211 (2) Takemura, F.; Matsumoto, Y. Dissolution Rate of Spherical Carbon Dioxide Bubbles in Strong Alkaline Solutions. Chem. Eng. Sci. 2000, 55, 3907. (3) Botello-A Ä lvarez, J. E.; Navarrete-Bolan˜os, J. N.; Jime´nez-Islas, H.; Estrada-Baltazar, A.; Rico-Martı´nez, R. Improving Mass Transfer Coefficient Prediction in Bubbling Columns via Sphericity Measurements. Ind. Eng. Chem. Res. 2004, 43, 6527. (4) Vasconcelos, J. M. T.; Orvalho, S. P.; Alves, S. S. Gas-Liquid Mass Transfer to Single Bubbles: Effect of Surface Contamination. AIChE J. 2002, 48 (6), 1145. (5) Ponoth, S. S.; McLaughlin, J. B. Numerical Simulation of Mass Transfer for Bubbles in Water. Chem. Eng. Sci. 2000, 55 (7), 1237. (6) Haut, B.; Cartage, T. Mathematical Modeling of Gas-Liquid Mass Transfer Rate in Bubble Columns Operated in the Heterogeneous Regime. Chem. Eng. Sci. 2005, 60, 5937. (7) Schumpe, A.; Deckwer, W. D. Analysis of Chemical Methods for Determination of Interfacial Areas in Gas-in-Liquid Dispersions with NonUniform Bubble Sizes. Chem. Eng. Sci. 1980, 35 (10), 2221. (8) Lewis, W. K.; Whitman, W. G. Principles of gas absorption. Ind. Eng. Chem. 1924, 16, 1215. (9) Higbie, R. The Rate of Absorption of a Pure Gas Into a Still Liquid During Short Period of Exposure. Trans. Am. Inst. Chem. Eng. 1935, 31, 365-389. (10) Danckwerts, P. V. Significance of Liquid-Film Coefficients in Gas Absorption. Ind. Eng. Chem. 1951, 43, 1460. (11) Toor, H. L.; Marchello, I. M. Film-penetration Model for Mass and Heat Transfer. AIChE J. 1958, 4, 97. (12) Brusset, H.; Depeyre, D.; Quang, T. L. An Estimation of the FilmPenetration Model Parameters. Chem. Eng. Sci. 1973, 28, 31-39. (13) Harriot, P.; A Random Eddy Modification of the Penetration Theory. Chem. Eng. Sci. 1962, 17, 149-154. (14) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962; pp 689-692. (15) Davies, J. T. Turbulence Phenomena; Academic Press, New York, 1972. (16) Fortescue, G. E.; Pearson, J. R. A. On gas absorption into a turbulent liquid. Chem. Eng. Sci. 1967, 22, 1163-1176. (17) Lamont, J. C.; Scott, D. S. An Eddy Cell Model of Mass Transfer into the Surface of a Turbulent Liquid. AIChE J. 1970, 16, 513-519. (18) Bullin, J. A.; Dukler, A. E. Random Eddy Models for Surface Renewal: Formulation as a Stochastic Process. Chem. Eng. Sci. 1972, 27 (2), 439-442. (19) Luk, S.; Lee, Y. H. Mass Transfer in Eddies Close to Air-Water Interface. AIChE J. 1986, 32, 1546-1554. (20) Seo, Y. C.; Lee, W. K. Single-eddy Model for Random Surface Renewal. Chem. Eng. Sci. 1988, 43 (6), 1395-1402. (21) Balasubraniyam, J. Transport Phenomena at the Interface, M. Chem. Eng. Thesis, Mumbai University, India, 1996. (22) Jajuee, B.; Margaritis, A.; Karamanev, D.; Bergougnou, M. A. Application of Surface-Renewal-Stretch Model for Interface Mass Transfer. Chem. Eng. Sci. 2006, 61, 3917-3929. (23) Xu, Z. F.; Khoo, B. C.; Carpenter, K. Mass Transfer Across the Turbulent Gas-Water Interface. AIChE J. 2006, 52 (10), 3363.
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ReceiVed for reView August 3, 2006 ReVised manuscript receiVed December 21, 2006 Accepted January 18, 2007 IE061015U
