Ind. Eng. Chem. Res. 2002, 41, 3429-3435
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Axial Mixing Effects in Packed Gas-Liquid Contactors Ricardo Macias-Salinas* and James R. Fair Separations Research Program, The University of Texas at Austin, Austin, Texas 78712
The usual assumption of plug flow of gas and liquid in packed columns may not be correct in many instances. If there is departure from this ideal flow case, then the driving force for mass transfer is diminished, and allowance must be made for additional packed height if the specified separation is to be made. On the basis of our earlier experimental work, we present here the methodology for making the nonideal flow correction and show examples of how the correction can have a significant impact on the separating capability of the contactor. In particular, newer high-efficiency packings are included in the study, specifically “through-flow” rings and strucured packings of both the gauze and sheet-metal types. Introduction When designing a packed column, for example, a simple absorber or stripper, the chemical engineer usually develops the equivalent of a y-x diagram and locates on the diagram the equilibrium and operating lines. The lines may not be straight, and the engineer is well equipped to allow for curvature. However, the assumption is invariably made that the phases are in plug flow and the lines are the loci of actual and equilibrium concentration profiles. If a computer program is being used, the program also makes this simple assumption. However, packing geometry and fluid mechanics dictate the need for correcting the profiles in many cases. The needed correction is attributed to “axial mixing”, also called “longitudinal dispersion” or “backmixing”. Axial mixing will be used here, with the recognition that radial mixing can also occur, but is thought to be minimal in larger, commercial-type contactors. Axial mixing results from viscous effects, molecular diffusion as in laminar flow, and eddy diffusion for turbulent flow. This suggests that dispersion parameters are needed if the corrections are to be made. Background It has been known for many years that certain types of contactors are prone to backmixing problems. Early studies of spray contactors showed the number of achievable transfer units or stages to be severely limited by the observed backmixing in the continuous phase for both gas-liquid and liquid-liquid systems.1,2 The same situation was found for gas-liquid bubble contactors, holding especially for the continuous (liquid) phase. However, it was thought that the presence of packing, especially small, random-type elements, would ensure plug flow. Awareness of nonideal flow behavior in flowtype chemical reactors was stimulated by the text of Levenspiel,3 where differences between plug-flow and well-mixed reactors were shown to influence strongly the conversion and yield attainable in the reaction step. For fluid-fluid contactors, the work of Miyauchi and Vermeulen4 is considered seminal and provided correction factor charts for dealing with departure from ideal * To whom correspondence should be addressed. Present address: Instituto Polite´cnico Nacional, ESIQIEUPALM, Zacatenco, DF 07738, Mexico. E-mail: rms@ ipn.mx.
countercurrent or cocurrent flow conditions. More recently, Von Stockar and Lu5 discussed the effect of axial mixing on countercurrent columns but did not provide data on parameters useful for quantification. Because reported packed column efficiencies have invariably been based on the assumed plug flow of the phases, possible effects of axial mixing on concentration driving forces have not been considered. This leads to questions regarding the accuracy of generalized correlations based on such reported efficiencies. One could take this point further and suggest that reported generalized correlations of packed column efficiency should be reexamined in light of the axial mixing methods given in the present paper. The present authors have reported earlier on studies of axial mixing in a 0.43 m column with air and water flowing countercurrently.6,7 A dynamic measurement technique was used to detect mixing effects; pulses of tracers were added to each phase, and dispersion of the pulses was related mathematically to mixing phenomena; the resulting dispersion parameters could be correlated with flow and geometry variables. A representative plot of dispersion in the gas phase is shown in Figure 1. Here, the liquid rate has been held constant and the gas rate has been increased up to 78% of the maximum (flooding) velocity. As the flow rate increases, the curves broaden with corresponding lower heights. While the curves shown are for 25.4 mm Raschig rings, similar curves are available for other packings.8 Mixing Model The experimental response curves of both phases (gas and liquid) were interpreted in terms of the diffusiontype model, which is derived from a material balance for the tracer over an element πr2 dz of the packed bed:
De
∂2C ∂C ∂C ) -v 2 ∂z ∂t ∂z
(1)
where C is the tracer concentration, v is the interstitial velocity, and De is an axial mixing coefficient. No radial concentration gradients are assumed to be present. For designing or rating a packed gas-liquid contactor, values of the mixing coefficient are needed, as influenced by fluid properties and flow rates and the geometry of the packing and the bed. In this work, De is incorporated into a dimensionless parameter, the
10.1021/ie010538t CCC: $22.00 © 2002 American Chemical Society Published on Web 06/17/2002
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Figure 1. Representative dispersion data for the gas phase with a constant liquid rate. Points are experimental values, and curves are based on model predictions. From ref 6.
Hog ) Koga/vs
Bodenstein number:
Bo ) Lv/De
(2)
where L is the height of the packed section under study and v is the interstitial velocity of the flowing phase. The Bodenstein number is similar to the Peclet number, except that the bed length is used instead of the particle diameter as the characteristic length. Correlations of Bo presented by the authors,6,7 based on experimental work in a larger diameter column, will be used in the present work. Design Approach A large amount of mass-transfer data have been reported on packed absorbers. The majority of the data, usually expressed in terms of transfer units, have been determined under the implicit assumption of plug-flow conditions for both phases, for which the following equation of Colburn9 applies:
Nog )
[
(
) ]
y1 - mx2 1 -λ ln (1 - λ) 1-λ y2 - mx2
which is little affected by departure from plug flow. Thus, the total packed height (for the column or a section of the column) is taken as
h ) NogHog
(6)
Axial Mixing Correlations The following correlations have been developed in terms of dimensionless quantities to enable prediction of axial mixing in both phases under two-phase flow conditions:
Random packings Bog ) 0.0878Reg-0.8915 × 10-0.00075Rel(dpap)8.231 Bol ) 24.461Rel0.5544Ga-1/3(dpap)2.1127
(7) (8)
valid operating ranges: 340 < Reg < 4066 (3)
where y1 and x2 are the mole fractions of the entering gas and liquid, y2 is the mole fraction of the solute in the exit gas, and λ is the stripping factor, defined as
GmMl λ)m LmMg
(5)
(4)
The height of an overall transfer unit, on a gas concentration basis, is
90 < Rel < 237 1.79 × 108 < Ga < 1.93 × 108 4.83 < dpap < 521 Structured packings Bog ) 4.2468 × 108Reg-0.896 × 10-0.00208Rel(deqap)-7.792 (9)
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Bol ) 8.154Fi0.7082
(
3 sin2 φ
)
1.159
(deqap)6.337
(10)
(3) plug flow in the x phase and axial mixing in the y phase (see the appendix). The values of A and B serve to obtain Nox* as follows:
valid operating ranges: 381 < Reg < 3516 Nox* )
25 < Rel < 122
3.16 < deqap < 3.53
1 1 1 1 ) + Nox Nox NoxP Nox*
45° < φ < 60° The above correlations were developed from experimental data for 25.4 mm ceramic Raschig rings and 25.4 mm metal Pall rings (random packings) and Sulzer BX and Flexipac 2 (structured packings). They have been generalized by means of packing diameter (dp, deq), specific surface area ap, and corrugation angle θ. Dimensions of the packings may be found in standard handbooks. Axial Mixing Effect on Interphase Mass Transfer To evaluate the impact of axial mixing on masstransfer efficiency, mixing numbers Bo can be used to predict true values of the numbers of overall transfer units Nox that take axial mixing into account. The suitability of this approach, of course, depends on how valid the tracer measurements are under mass-transfer conditions. The model used here, based on the previous experimental work of the authors6-8 as well as that of many others, is valid when relatively small amounts of solute are transferred. When there is significant mass transfer, as in distillation, the dynamic method of Linek et al.10 may be more appropriate. Hartland and Mecklenburgh11 presented a rigorous analytical solution of the differential model for the case of countercurrent extraction with straight equilibrium lines. In the present work that solution was applied to the case of absorption in packed columns with gas as the raffinate phase and liquid as the extract phase. The rigorous solution involves an iterative procedure to evaluate the axial mixing correction of the apparent number of transfer units. This represents a simulation case in which the height of the packing h is known. The calculation method is outlined below. Algorithm for Correcting Nox Values for Axial Mixing. 1. The Bodenstein numbers in each phase are first expressed in terms of the known height of the packing h used to conduct the mass-transfer experiments as follows:
(11)
2. The initial estimate of the true number of transfer units Nox is set equal to the reported number of transfer units under plug-flow conditions NoxP:
Nox ) NoxP
( )
(13)
4. A new estimate of Nox is calculated via the following equation derived from Newton’s method:
1.53 × 10-6 < Fi < 1.59 × 10-5
h Box,y ) Bog,l L
1 B ln λ-1 A
(12)
3. The coefficients A and B in the solution given by Hartland and Mecklenburgh are then evaluated for three cases: (1) axial mixing in both phases, (2) axial mixing in the x phase and plug flow in the y phase, and
(14)
5. The new value of Nox is checked for convergence, and steps 3-5 are repeated until convergence is achieved. Alternatively, the true height of overall transfer units can be computed from
Hox ) h/Nox
(15)
The above converging method has been implemented in a computer program named NTUTRUE. The listing of the program may be obtained from the authors. Case Studies To illustrate the importance of axial mixing on the performance of countercurrent gas-liquid flow contactors, the above procedure was used to correct selected mass-transfer data reported for the cases of gas absorption, stripping, water cooling, and distillation, using the same types of packings for which the degree of axial mixing was measured by the authors. Most of the masstransfer data were reported in terms of the apparent values of the overall mass-transfer coefficients KoxPa or the height of the overall transfer units HoxP prevailing in the test column. The conversion to NoxP values can be readily attained through the following equations:
NoxP ) KoxPah/vsx
(16)
NoxP ) h/HoxP
(17)
Absorption of Ammonia in Water (Fellinger).12 In an extensive study of absorption of ammonia in water from air, Fellinger presented Koga values for various random packings. His data for 25.4 mm ceramic Raschig rings were selected to assess the importance of axial mixing. This system represents the case in which both gas- and liquid-phase resistances are important. The corrected number of transfer units should therefore include the effect of axial mixing in both phases. Accordingly, the mixing correlations for 25.4 mm ceramic Raschig rings were used to compute the gas- and liquid-phase Bodenstein numbers:
Bog ) 61.66Gm-0.8788 × 10-0.0223Lm
(18)
Bol ) 6.786Lm0.6125
(19)
where the above mixing numbers were then converted to Box and Boy under mass-transfer conditions (eq 11) using the experimental packing height of 0.65 m. The stripping factor λ was estimated using m ) 0.80. Figure 2 gives the conditions of the selected Fellinger runs along with the results corrected for axial mixing. (More detailed information for the case studies, given in
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Figure 2. Corrected mass-transfer data of Fellinger12 due to axial mixing. Absorption of ammonia in water from air at 20 °C using 25.4 mm ceramic Raschig rings.
tabular form, may be obtained from ref 8 or directly from the authors.) The higher true values of Nox indicate that more transfer units are required to achieve the same separation compared with the plug-flow assumption. Neglecting axial mixing might therefore lead to an overestimation of the mass-transfer driving forces, yielding higher Hox values. The number of transfer units under plug-flow conditions NoxP and those corrected for axial mixing Nox are shown in Figure 2 as functions of the gas flow rate. The curves show expected trends. In the preloading region, Nog decreases consistently with the gas flow rate until reaching a minimum near the loading point. Beyond this point, the increased turbulence and change of flow pattern explain the rapid increase of Nog with the gas rate. Also, the increase of Nog (or the decrease of Hog) with the liquid rate can be explained by the increase of the effective interfacial area for mass transfer. As evidenced by Figure 2, there is a small difference between NoxP and Nox below the loading point with deviations varying from 5 to 11% for the two liquid rates. This indicates that axial mixing has only a slight effect on the performance of the absorber at lower loadings. As the flooding point is approached, the effect of axial mixing is more pronounced with a maximum of 18% difference between NoxP and Nox. The increased difference is mainly ascribed to the small gas-phase Bodenstein numbers obtained at high gas rates. This agrees with the general recognition that for this type of application most of the mass-transfer resistance is in the gas phase. Desorption of Oxygen from Water (Sherwood and Holloway).13 These researchers reported Hol values for the desorption of oxygen from water into air at 25 °C, using several random packings. For this case mass-transfer resistance lies entirely in the liquid phase (very low solubility of oxygen, m ≈ 4.4 × 104). Thus, the stripping factor λ is essentially zero. Also, because the liquid phase controls, axial mixing in the gas phase is irrelevant. Only one set of experiments deals with 25.4 mm ceramic Raschig rings, and that set has been corrected for axial mixing. Liquid-phase Bodenstein numbers were first computed and then converted to Box. A plot of NoxP and Nox versus liquid rate is given in Figure 3. The difference between apparent and true Nox values decreases as the
Figure 3. Corrected mass-transfer data of Sherwood and Holloway13 due to axial mixing. Desorption of oxygen from water into air at 25 °C using 25.4 mm ceramic Raschig rings.
Figure 4. Corrected mass-transfer data of Whitney and Vivian14 due to axial mixing. Absorption of sulfur dioxide in water from air at 21.1 °C using 25.4 mm ceramic Raschig rings.
liquid flow increases, i.e., as plug-flow conditions are approached. At the lowest liquid rate, however, the large deviation from plug flow (about 25%) clearly anticipates a strong effect of axial mixing on efficiency. This case demonstrates the importance of properly assessing the impact of axial mixing effects in order to be able to establish the range of operating conditions over which these effects can be safely neglected. Sulfur Dioxide Absorption by Water (Whitney and Vivian).14 These investigators studied the performance of 25.4 mm ceramic Raschig rings for the absorption of sulfur dioxide from air into water at 21 °C. This is a system in which both gas and liquid resistances are expected to be important. Accordingly, axial mixing in both phases will contribute to the true Hol values. Two sets of runs were considered for the mixing correction. As in the preceding application, Box and Boy were determined. The stripping factor λ is based on m ) 24, obtained from SO2-water solubility data at 20 °C.15 Figure 4 shows the results in terms of NoxP and Nox as functions of gas rate. The differences between the apparent and true Nol values exhibit consistent trends with respect to the gas rate; i.e., the deviation from plug flow increases slightly as the flood point is approached. A decrease of the liquid rate, however, has a more pronounced effect on the true Nol value, despite
Ind. Eng. Chem. Res., Vol. 41, No. 14, 2002 3433
n-butane system at total reflux and at 689, 1138, 2068, and 2758 kPa pressures. In this case, the mixing characteristics of both phases are equally important because both provide mass-transfer resistance. Thus, the so-called “point” approach of Linek et al.10 could be more appropriate, as mentioned earlier. The present approach is still useful for indicating mixing problems. Because Mellapak 250Y and Flexipac 2 are essentially identical geometrically, the present correlations for the latter were used to predict the mixing behavior in the vapor and liquid:
Figure 5. Corrected mass-transfer data of McNulty and Hsieh16 due to axial mixing. Water cooling with air using 0.61 m of Flexipac 2.
the fact that axial mixing in the gas phase is more significant at high liquid loadings. This leads to the conclusion that nearly all of the resistance to mass transfer lies in the liquid phase for this particular absorption. Water Cooling with Air (McNulty and Hsieh).16 The water-air system was used to study the performance of various structured packings of the corrugated type. For water cooling (air humidification), the investigators obtained Hog values for Flexipac 2 for a packed height of 0.61 m. For this system all of the mass-transfer resistance is in the gas phase; hence, m ≈ 0 and λ ) 0. Axial mixing in the liquid phase is thus unimportant. The required mixing data in the gas phase were obtained from the correlations given earlier:
Boy ) 137.08Gm-0.8835 × 10-0.0276Lm
(20)
Two sets of runs for a given liquid rate were examined to verify the impact of axial mixing. Figure 5 shows the difference between the plug-flow and true Nog values as a function of the gas flow rate. As in the preceding examples, there is an appreciable difference between the apparent and true Nog values for the two liquid rates. As expected, this difference is larger at higher liquid loadings because of the low values of Bodenstein numbers in the gas phase. The largest deviation from plug flow was found to be approximately 16%, indicating that axial mixing has only a moderatly detrimental effect on the column performance. Importantly, near the flood point the difference between NoxP and Nox tends to decrease, probably because of the rapid decrease of efficiency caused by factors other than gas-phase mixing. High-Pressure Distillation, iC4/nC4 System (Fitz et al.).17 The poor performance exhibited by structured packings under high-pressure distillation conditions has received considerable attention.17-19 The factors causing this anomalous behavior are not yet well identified, but severe axial mixing in the vapor phase may be a key factor because of the high liquid and vapor flow rates typically encountered in high-pressure distillations. In an attempt to assess the contribution of axial mixing to the poor performance at high pressures, at least approximately, the present correction procedure was applied to the efficiency data of Fitz et al., for Mellapak 250Y structured packing with the i-butane/
Box ) 4.688Lm0.2586
(21)
Boy ) 137.08Vm-0.8825 × 10-0.0276Lm
(22)
Correlations containing dimensionless groups were discarded because they predicted extremely low values of the mixing number in the vapor phase as a result of the high Reynolds numbers of both phases at high pressures. One should recognize, however, that the mixing correlations of the present work have been confirmed only for air-water at atmospheric pressure. Hydrocarbon systems at high pressures may not fit the models because of a lack of experimental validation as well as the experimental methodology as discussed earlier. Although the suitability of the present approach may be questionable, it can at least provide a means of estimating roughly the mixing behavior occurring in structured packings in high-pressure distillation. Two sets of runs at 1138 and 2758 kPa were chosen and corrected for axial mixing. Bodenstein numbers Box and Boy were obtained using the total packed height (3.78 m). The stripping factor λ is close to unity (total reflux), indicating that both vaporand liquid-film resistances should be considered. Masstransfer results were reported in terms of HETP (height equivalent to a theoretical plate) values as a function of the Fs factor ()vsvFp1/2). The conversion to the above Vm and NovP values was possible via the following equations:
Vm ) FsFv1/2 HogP ) HETP
λ-1 ln λ
(23) (24)
The variation of NoxP (plug flow) and Nox (axial mixing) with vapor mass velocity is shown in Figure 6 for the two pressures. A clear loss of efficiency is indicated as the pressure increases. Also, the difference between the plug-flow and true Nov values is significant at both pressures with deviations varying from 45 to 200% at 1138 kPa and from 25 to 113% at 2758 kPa. This indicates that axial mixing in both phases can have an overwhelming effect on the column performance, almost destroying the countercurrent contacting scheme. However, this does not necessarily imply that axial mixing is mainly responsible for the substantial decrease of efficiency at higher pressures. In fact, the average deviation from plug flow at 2758 kPa turns out to be smaller than that obtained at 1138 kPa, with 51% and 108% average, respectively. Based on these findings, other factors that may cause low efficiency in structured packings should be taken into account, such as severe frothiness due to low surface tensions experienced at high pressures, as suggested by Kister.19
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high-porosity packing with regular flow channels may reduce mixing in the gas phase. For the liquid phase, packings having good wetting properties may promote low levels of axial mixing. For structured packings, based on the present results, a corrugation angle greater than 45° from the horizontal may have advantages. Finally, an experimental investigation should be conducted to confirm or invalidate the potentially significant effects of axial dispersion on the performance of distillation columns operating at high pressure. In this respect, use of a liquid-liquid flow system is recommended because the small phase density differences, typically found in high-pressure distillation, can be conveniently evaluated. It is possible that this density difference can cause severe mixing in the continuous (vapor) phase. Figure 6. Corrected mass-transfer data of Fitz et al.17 due to axial mixing. High-pressure distillation of the i-butane/n-butane system at total reflux using Mellapak 250Y structured packing.
As mentioned earlier, the present results represent an attempt to determine whether axial mixing can be an important factor in high-pressure distillation if mixing correlations developed for air-water at atmospheric pressure are used. Because no justification was given to the above approach, the interpretation of the results may not represent the real physical behavior of axial mixing in high-pressure distillation. The evidence seems clear, however, that axial mixing can play an important role in the performance of packings operated at higher pressures. Further experimental work would be necessary to confirm the mixing effect.
Acknowledgment The authors thank the Separations Research Program at The University of Texas at Austin and the National Council of Science and Technology of Mexico (CONACyT) for providing financial support for this research. Appendix. Evaluation of Coefficients A and B in the Hartland-Mecklenburgh Differential Model (i) Axial Mixing in Both Phases.
A ) µ1g2g3eµ1-µ2(µ2 - µ3) + µ2g3gl(µ3 - µ1) + µ3g1g2eµ3-µ2(µ1 - µ2) B ) µ1g1eµ3(µ2 - µ3) + µ2g2eµ1+µ3-µ2(µ3 - µ1) + µ3g3eµ1(µ1 - µ2)
Conclusions A procedure has been proposed to account for axial mixing effects. Application of the procedure shows that such mixing has only a moderately detrimental influence on the column performance in the range of operating conditions reported for gas absorption, stripping, and water cooling. The assumption of plug-flow conditions may be valid in these cases. The use of the present correction procedure is recommended for establishing the range of operating conditions in which axial mixing effects can be neglected. Neglecting this nonideal flow behavior in countercurrent contactors can result in underestimation of the overall number of transfer units, thus leading to an unsafe design. In systems where both gas- and liquid-side resistances are important, the performance of the column can be affected by the extent of axial mixing in the two phases. However, there are cases in which mixing in one of the phases may be seriously detrimental to the performance even though the other phase mostly contributes to the mass-transfer resistance. In such cases, the correction procedure presented here can be an essential diagnostic tool. For the i-butane/n-butane distillation system at high pressures, the application of the present correction procedure predicts a large effect of vapor- and liquidphase axial mixing on the column efficiency. This finding, however, requires further experimental verification because the above approach was based on correlations of air-water mixing data. To minimize the adverse effects of axial mixing, better packing designs should be considered. For example, a
where the three roots µ consist of the principal one µ1, which is small and close to zero, a large positive root µ2, and a large negative root µ3 obtained from
µ3 - µ2(Box - Boy) - µ(λNoxBoy + BoxNox + BoxBoy) + (λ - 1)NoxBoxBoy ) 0 and g1, g2, and g3 are given by
gi ) (1 - µi/Box)(1 + µi/Boy)-1
for i ) 1-3
(ii) No Axial Mixing in the y Phase.
A ) µ2(1 - µ1/Box)e-µ1 - µ1(1 - µ2/Box)e-µ2 B ) µ1 - µ2 where µ1 is the smaller and µ2 the larger positive root of
µ2 - µ(Box + λNox) + (λ - 1)BoxNox ) 0 (iii) No Axial Mixing in the x Phase.
A ) µ2 - µ1 B ) µ1(1 + µ2/Boy)eµ2 - µ2(1 + µ1/Boy)eµ2 where µ1 is the smaller and µ2 the larger negative root of
Ind. Eng. Chem. Res., Vol. 41, No. 14, 2002 3435
µ2 + µ(Boy + Nox) - (λ - 1)BoyNox ) 0 Notation a ) interfacial area, m2/m3 ap ) packing surface area, m2/m3 A, B ) coefficients in the Hartland-Mecklenburgh model (eq 13) Bo ) Bodenstein number, vL/De C ) normalized tracer concentration deq ) equivalent diameter (structured packing), m dp ) nominal diameter (random packing), m De ) axial mixing coefficient, m2/s Fi ) film number, ηvs/deq2Fg g ) acceleration due to gravity, m/s2 Ga ) Galileo number, dp3F2g/η2 Gm ) gas flow rate, kg/m2‚s h ) packing height, m HETP ) height equivalent to a theoretical plate, m Ho ) height of the overall transfer units, m Ko ) overall mass-transfer coefficient, m/s L ) test section length, m Lm ) liquid flow rate, kg/m2‚s m ) slope of the equilibrium curve M ) molecular weight, kg/kgmol No ) number of overall transfer units No* ) value of No used in the iterations (eq 13) Re ) Reynolds number, vsL*F/η t ) time, s v ) interstitial velocity, m/s vs ) superficial velocity, m/s Vm ) vapor flow rate, kg/m2‚s x ) liquid-phase concentration (mole fraction) y ) vapor-phase concentration (mole fraction) Greek Letters η ) viscosity, kg‚m/s θ ) corrugation angle of structured packing, from horizontal, deg λ ) stripping factor (eq 4) p ) density, kg/m3 Subscripts g ) gas phase l ) liquid phase o ) overall p ) packing P ) plug-flow basis v ) vapor phase x, y ) phase designations
Literature Cited (1) Pigford, R. L.; Pyle, C. Performance characteristics of spraytype absorption equipment. Ind. Eng. Chem. 1951, 43, 1649.
(2) Von Stockar, U.; Cevey, P. F. Influence of the Physical Properties of the Liquid on Axial Dispersion in Packed Columns. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 717. (3) Levenspiel, O. Chemical Reaction Engineering, 1st ed.; John Wiley: New York, 1962. (4) Miyauchi, T.; Vermeulen, T. Longitudinal Dispersion in Two-Phase Continuous-Flow Operations. Ind. Eng. Chem. Fundam. 1963, 2, 113. (5) Von Stockar, U.; Lu, X. Simple and Accurate Shortcut Procedure to Account for Axial Dispersion in Countercurrent Separation Columns. Ind. Eng. Chem. Res. 1991, 30, 1248-1257. (6) Macias-Salinas, R.; Fair, J. R. Axial Mixing in Modem Packings, Gas and Liquid Flow. I. Single-Phase Flow. AIChE J. 1999, 45, 222. (7) Macias-Salinas, R.; Fair, J. R. Axial Mixing in Modem Packings, Gas and Liquid Phases. 2. Two-Phase Flow. AIChE J. 2000, 46, 79. (8) Macias-Salinas, R. Gas- and Liquid-Phase Axial Dispersion through Random and Structured Packing. Ph.D. Dissertation, The University of Texas at Austin, Austin, TX, 1995. (9) Colburn, A. P. The Simplified Calculation of Diffusion Processes. General Consideration of Two-Film Resistances. Trans. AIChE 1939, 35, 211. (10) Linek, V.; Benesˇ, P.; Sinkule, J.; Ksˇlvsky´, Z. Simultaneous Determination of Mass Transfer Coefficient and of Gas and Liquid Axial Dispersions and Holdups in a Packed Absorption Column by Dynamic Response Method. Ind. Eng. Chem. Fundam. 1978, 17, 298. (11) Hartland, S.; Mecklenburgh, J. C. A Comparison of Differential and Stagewise Countercurrent Extraction with Backmixing. Chem. Eng. Sci. 1966, 21, 1209. (12) Fellinger, L. L. Absorption of Ammonia by Water and Acid in Various Standard Packings. Sc.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1941. (13) Sherwood, T. K.; Holloway, F. A. L. Performance.of Packed Towers. Liquid Film Data for Several Packings. Trans. AIChE 1940, 36, 39. (14) Whitney, R. P.; Vivian, J. E. Absorption of Sulfur Dioxide in Water. Chem. Eng. Prog. 1949, 45 (5), 323. (15) In Perry’s Chemical Engineer’s Handbook, 6th ed.; Perry, R. H., Green, D., Eds.; McGraw-Hill: New York. (16) McNulty, K. J.; Hsieh, C. Hydraulic Performance and Efficiency of Koch Flexipac Structured Packing. Paper presented at the 1982 AIChE National Meeting. (17) Fitz, C. W.; Kunesh, J. G.; Shariat, A. Performance of Structured Packing in a Commercial Scale Column at Pressures of 0.02-27.6 bar. Ind. Eng. Chem. Res. 1999, 38, 512. (18) Kurtz, D. P.; McNulty, K. J.; Morgan, R. D. Stretch the Capacity of High-Pressure Distillation Columns. Chem. Eng. Prog. 1991, 87 (2), 43. (19) Kister, H. Z. Distillation Design; McGraw-Hill: New York, 1992.
Received for review June 21, 2001 Accepted April 8, 2002 IE010538T
