Ind. Eng. Chem. Res. 2006, 45, 1097-1104
1097
Local Mass Transfer in a Packed Bed: Experiments and Model Trong Dang-Vu,† Huu D. Doan,*,‡ and Ali Lohi‡ Department of Chemical Technology, Gdansk UniVersity of Technology, 11/12 Narutowicza, 80-952 Gdansk, Poland, and Department of Chemical Engineering, Ryerson UniVersity, 350 Victoria Street, Toronto, Ontario M5B 2K3, Canada
Liquid-to-packing local mass-transfer coefficients (LMTC) were measured in a 0.3 m diameter column with a bed height of 5.5 times the column diameter using the limiting-current technique. Several electrodes were placed at various radial and axial positions (packing heights) in the bed. Measurements were conducted at various liquid flow rates with two different liquid distributor designs: multipoint (MPLD) and single-point (SPLD) distributors. For shallow beds, mass-transfer variation with radial location and liquid flow rate using MPLD was less than that for SPLD. However, for larger bed depths, the local mass-transfer coefficient became less dependent on liquid flow rate with both liquid distributors. A mathematical model for the LMTC in a packed bed was also developed, taking into account the axial and radial positions in the bed. The model predicts the LMTC well with the root-mean-square errors (RMS) of predicted and measured values of 0.07 and 0.08 for the multipoint and single-point liquid distributors, respectively. Introduction Packed columns are widely used as contactors in the chemical process industry. Distillation, absorption/desorption, and extraction are typical applications of packed columns. In the design of a packed-bed contactor, the mass-transfer coefficient is usually assumed to be constant at all locations in the column. This approach relies on observations made at the macroscopic scale, i.e., measurements of flowrates and concentrations are made only at the inlet and outlet streams of the contacting device. However, the local mass-transfer coefficient actually varies with locations in the packed bed due to the complexity of the hydrodynamic behavior near the particle-fluid interface and the random nature of liquid spreading in the bed. Knowledge of the local mass-transfer coefficient in a packed bed is essential for packed-column design and scaling up. Although much research work1-5 has been done on liquid flow distribution or mass transfer in a packed bed, most studies neither present nor fully study spatial variation of the local masstransfer coefficient in the bed. Gostick et al.6,7 was the first group who measured the local fluid-particle mass-transfer coefficients in a packed bed of Pall rings using an electrochemical technique. They reported that the variation of the local mass-transfer coefficient with packing height decreased until it reached a minimum limiting value. They also pointed out that the axial profile of the mass-transfer coefficient for the outer section exhibited different trends from that for the inner or center section of the packed bed. However, a mathematical model for the local mass-transfer coefficient was not developed. Several mathematical models of mass transfer in a packed bed have been proposed.8-11 Generally, the mass-transfer coefficient is expressed in terms of the relevant dimensionless groups as follows,
Sh ) a‚Reb Scc
(1)
* Corresponding author. Tel.: (416) 979-5000 ext. 6341. Fax: (416) 979-5083. E-mail:
[email protected]. † Gdansk University of Technology. ‡ Ryerson University.
where a, b, and c are experimentally determined constants, Re is the particle Reynolds number, Sc is the Schmidt number, and Sh is the Sherwood number. The values of the constants a, b, and c found in the reported literature vary from one study to another. The liquid-solid mass-transfer coefficient is usually correlated with the Schmidt number to the 1/3 power.8-10,12,13 However, different values of the power of the Schmidt number ranging from 0.25 to 0.5 and higher can also be found in the literature.14-16 The coefficient, a, in eq 1 was also presented as a function of the packed-bed characteristics such as porosity and wetting area. Karabelas et al.17 studied mass transfer under different flow regimes and reported that, for very low Reynolds and Peclet numbers, the mass-transfer coefficient was constant. On the other hand, for high Peclet numbers, the mass-transfer coefficient was found to be proportional to the Reynolds number to the exponent of 0.33 and 0.5 for still higher Reynolds numbers. Pfeffer8 assumed the bed as an assemblage of isolated spheres, in which the distance between spheres was considered as a function of the bed porosity, . The author proposed a correlation for mass transfer of a single active sphere in an inert bed as below:
Sh ) Sc0.33
(
1.26
1 - (1 - )1.67
)
2 - 3(1 - )0.33 + 3(1 - )1.67 - 2(1 - )2
0.33
Re0.33 (2)
Taking into account the bed porosity, analogous equations for mass transfer in a packed bed were also proposed by Sirkar9 and Kawase and Ulbrecht.10 Guo and Thompson18 studied local mass transfer in a packed bed of benzoic acid spheres using the dissolution technique. Unfortunately, in this technique, the shape and the texture of the surface of packing particles may be altered during the dissolution process; hence, local mass transfer may vary significantly with time and space, and the transfer process would be under unsteady state throughout the experiment duration. Consequently, the measurements at the end of the experiment
10.1021/ie0505312 CCC: $33.50 © 2006 American Chemical Society Published on Web 12/27/2005
1098
Ind. Eng. Chem. Res., Vol. 45, No. 3, 2006
Figure 1. Schematic diagram of the experimental setup.
yield the time-averaged values over the whole experiment duration, from which the temporal or steady-state mass-transfer coefficients in the bed cannot be extracted. Several mathematical models have been presented in the literature. However, the local mass-transfer coefficients at various axial and radial locations were neither measured nor reported. Scale-up of pilot-plant data to the design of a largescale commercial column may be subject to significant deviation due to the lack of data and knowledge of local mass transfer in the column. In the present study, the limiting-current technique was used to measure local mass-transfer coefficients between the liquid and packing throughout the packed bed at varied axial, radial, and angular locations. Using the experimental data, a mathematical model was proposed, considering the variation of the local mass-transfer coefficient with the axial and radial positions in the bed. Experimental Section In the present study, the experimental apparatus consists of a 0.3 m diameter column filled with 24.5 mm stainless steel Pall rings (Figure 1). A bed height of 5.5× the column diameter was used to allow observation of the development of flow patterns in the bed. Local mass-transfer coefficients were measured at 54 locations over six axial levels in the bed. The axial levels were 30 cm apart, starting at 15 cm below the liquid distributor. These axial levels are equivalent to the ratios of the axial distance from the top of the bed to the bed diameter, x/D, of 0.5, 1.5, 2.5, 3.5, 4.5, and 5.5. At each axial level, to account for different angular and radial locations, nine electrodes were arranged as following: two interconnected electrodes at the center of the column cross section (r/R ) 0), where r is the radial position and R is the column radius; four at the inner radial positions (r/R ) 0.42); and the last four at the outer radial positions (r/R ) 0.83). The arrangement of electrodes at each axial level is shown in Figure 2. Note that two interconnected electrodes in the center are shown as one. The averaged value of two electrodes provides a more reliable measurement of the mass-transfer coefficient for the central position. To investigate the effect of initial liquid distribution on local mass transfer, two types of liquid distributors were used: singlepoint (SPLD) and multipoint (MPLD) distributors. The properties of these liquid distributors are shown in Table 1. Liquid
Figure 2. Schematic diagram of the electrode arrangement in the layer: (1) center electrode, (2) inner electrode, and (3) outer electrode. The distance between centers of electrodes is shown in units of r/R. Table 1. Characteristics of Liquid Distributors Used in the Present Study liquid distributor type
no. of nozzles
nozzle diam. (mm)
no. of nozzles per unit area (m-2)
single-point (SPLD) multipoint (MPLD)
1 16
23.8 4.76
14 219
flow rate to the column was varied from 9 350 to 56 120 kg‚m-2‚h-1, and the corresponding particle Reynolds numbers (Re) were between 95 and 560. The Schmidt number was held at ∼1 500. The electrochemical technique was used to measure the local mass-transfer coefficient. General aspects of the method are given by Selman and Tobias.19 The electrochemical reaction system used in the present study was the ferricyanide/ferrocyanide redox couple, as detailed below:
Reduction of ferricyanide to ferrocyanide at the cathode: Fe(CN)63- + e- f Fe(CN)64-
(3a)
Oxidation of ferrocyanide to ferricyanide at the anode: Fe(CN)64- f Fe(CN)63- + e-
(3b)
Because of the oxidation of ferrocyanide to ferricyanide at the anode, the bulk concentration of ferricyanide remains constant. This system exhibits fast electrode kinetics. In addition, it does not involve any solid reactants or products that would alter the packing surface, geometry, or shape. The concentrations of the ferricyanide and ferrocyanide used in the present study were 3.6 and 4.0 mol‚m-3, respectively. An excess amount (2 wt %) of NaOH as a support electrolyte was added to minimize the effect of ionic migration on mass transfer; hence, the transfer of ferricyanide from the bulk liquid to the cathode was predominantly diffusion-controlled. The test circuit consisted of nickel-coated Pall ring cathodes and stainless steel Pall ring anodes. Nonconductive polymer-coated Pall rings were used to separate the anode and cathode layers. The anode and cathode were connected to a DC power supply. A sufficient voltage was applied to the circuit to ensure the cathodic reduction masstransfer-controlled process. Under this condition, the limiting current obtained (iL) is proportional to the mass-transfer
Ind. Eng. Chem. Res., Vol. 45, No. 3, 2006 1099
Figure 3. Contour plots of mass-transfer distribution: (a) SPLD, Re ) 465; and (b) MPLD, Re ) 425. Index numbers 1, 2, and 3 show the level x/D ) 1.5, 3.5, and 5.5, respectively.
coefficient (kL) and can be expressed as
kL )
iL aczFC∞
(4)
where ac is the surface area of the cathode, C∞ is the bulk concentration of ferricyanide, F is the Faraday constant, and z
is the covalent number of the reaction. A more detailed description of the electrochemical experimental apparatus and procedures is given elsewhere.6,7 Results Local Mass-Transfer Distribution. Spatial mass-transfer distributions for single-point and multipoint liquid distributors
1100
Ind. Eng. Chem. Res., Vol. 45, No. 3, 2006
Figure 4. Mass-transfer coefficient at different axial positions (x/D) for SPLD: (a) center section and (b) outer section.
Figure 5. Mass-transfer coefficient at different axial positions (x/D) for MPLD: (a) center section and (b) outer section.
at different x/D levels (indicating the bed depth from the top of the packing) are presented in Figure 3. The numbers next to the contour lines are the values of Sh/Sc1/3. It can be seen that mass transfer is not uniformly distributed throughout the column because of the random nature of liquid spreading and the inherent liquid maldistribution in a packed bed, as observed by several other researchers. The ratio of the column diameter and the packing size is 12, which is not much higher than the minimum value of 8 (the recommended value is 15). Therefore, in this small system, the wall flow may also affect the masstransfer distribution. In addition, nonuniform liquid holdup due to local heterogeneity in the bed has also been reported in the literature.20-22 This may be another factor attributing to the nonuniform mass-transfer distribution in the bed. For SPLD at the higher axial level (smaller x/D, i.e., shallow bed), the highest local mass-transfer coefficient is observed at the center section. The difference between the maximal value of the mass-transfer coefficient in the center region and the minimal value in the outer region is more pronounced at higher axial levels. The smaller number of nozzles per unit crosssectional area of the column with SPLD tends to have a poor initial liquid distribution, resulting in higher mass-transfer coefficients in the center section. Sherwood and Halloway23 reported similar observations. At lower axial levels (larger bed depth or x/D), the variation of the local mass-transfer coefficient with radial location becomes more uniform. This indicates that
almost no liquid reaches the outer section of the bed at higher axial levels, while at lower axial levels, liquid distribution is uniform, resulting in less variation in the mass-transfer coefficient in the radial direction. For MPLD, a similar trend is also observed, as can be seen in Figure 3. However, the magnitude of the variation is lower due to more uniform initial liquid distribution provided by the MPLD. The effect of liquid flow rate on mass transfer was also investigated. The local mass-transfer coefficients, expressed in terms of Sh/Sc0.33, for the center and outer sections of the column at different x/D levels are plotted against the particle Reynolds number, Re, in Figures 4 and 5. Each point in the graph of the outer section represents the arithmetic average of all four electrodes at a given radial position. The value for the center section is the averaged data of two electrodes. To show the deviation of the mass-transfer coefficient of a given section at each x/D level from their average of all six levels, the arithmetic average with 95% confidence interval was shown in the graph as the “average” curve. The large value of standard deviation resulted from a nonuniform profile of the local mass-transfer coefficient in a packed bed. For both SPLD and MPLD, the mass-transfer coefficient was found to increase with liquid flow rate, as expected. Figure 4 illustrates the variation of the local mass-transfer coefficient for SPLD with Re. Along the packing height, the mass-transfer coefficient decreased in the center section but increased in the outer section due to liquid spreading
Ind. Eng. Chem. Res., Vol. 45, No. 3, 2006 1101
Figure 6. Comparison of the experimental results obtained and the literature data.
from the central region to the outer section and the development of wall flow. The maximal values of the mass-transfer coefficient were ∼110 and 45 for the center and outer sections, consecutively. The variation of the local mass-transfer coefficient with the axial and radial positions again indicates that the assumption of a constant mass-transfer coefficient throughout the packing bed is erroneous and may lead to large errors in designing and scaling up packed columns. The lines shown in Figure 4 are least-squares fitted lines for various x/D values. These lines show clearly that the masstransfer coefficient does not conform to a single relationship of Sh/Sc0.33 vs Re, which corresponds to the case with the assumption of constant mass-transfer coefficients. Instead, it is apparent that the spatial distribution of the local mass-transfer coefficient is significant. Figure 5 shows the variation of Sh/Sc0.33 with Re for MPLD. The variation of the mass-transfer coefficient with the bed height and the radial locations followed a similar trend to that for SPLD. However, the magnitude of the mass-transfer variation for MPLD was much less than that of SLPD. In general, the local mass-transfer coefficient decreased with the bed height, which is in agreement with the observations of Yoshida and Koyanagi.24 The difference between the maximal and minimal values of the mass-transfer coefficient for MPLD is smaller than that for SPLD. This indicates that liquid distribution was uniform and the development of wall flow was less with MPLD. It is also noted that, for both SPLD and MPLD (Figures 4 and 5), the increase of the local mass-transfer coefficient with Re is larger at the center section than at the outer section. For SPLD, the value of Sh/Sc1/3 varies from ∼80 for the center section to 50 for the outer section, while it is ∼55 at the center section and 30 at the outer section for MPLD. This indicates that local mass transfer with SPLD is more sensitive to liquid flow rate due to poor initial liquid distribution. The overall liquid distribution thus relies heavily on the capability of the packing in spreading out liquid in the column, which in turn is significantly dependent on liquid flow rate. In light of this observation, the importance of the initial liquid distribution on the overall performance of a packed column is illustrated. For comparison between the results obtained in the present study and the literature values, the arithmetic average of local values for all electrodes in the column is plotted against the Reynolds number in Figure 6 along with correlations given by Jolls and Hanratty25 and Karabelas et al.,26 which were obtained under comparable conditions. In both of those studies, the ferri/ ferrocyanide redox system was used with Re from 100 to 1100
Figure 7. Sample of the linear trend between ln(Sh/Sc0.33) and ln(Re) at the outer section with SPLD (r/R ) 0.83).
with the packing size of 25.4 mm (Jolls and Hanratty25) and from 12.7 to 76.2 mm (Karabelas et al.26) in a column of 0.3 m diameter. It can be seen that the results of the present study fall in the range of the literature data. The percent deviations are 21 and 18% with the results from Jolls and Hanratty25 and Karabelas et al.,26 respectively. The difference could be due to different packing types and sizes and liquid distributors used in each study. In those literature studies, the spherical packing was used. Mathematical Model of Local Mass-Transfer Coefficient. As shown above, the spatial distribution of mass transfer exists in the relationship of Sh/Sc0.33 vs Re. Therefore, the goal of this section is to find the coefficients a and b in eq 1 as a function of axial (x/D) and radial (r/R) positions, i.e., a ) f(x/ D,r/R) and b ) g(x/D,r/R). Equation 1 can be rewritten as
ln
( )
Sh ) ln(a) + b ln(Re) Sc0.33
(5)
Equation 5 is a linear equation of the following form,
Y ) AX + B
(6)
( ) Sh Sc0.33
(7)
X ) ln(Re)
(8)
where
Y ) ln
[ (Dx ,Rr )] x r A ) g( , ) DR
B ) ln f
(9) (10)
Linear regression was used to fit the data Y and X for each value of x/D and r/R. The values of the determination coefficient obtained are close to unity, indicating the high quality of the regression equations. Figure 7 is an example of the linear trend of Y versus X at three x/D levels of 1.5, 3.5, and 5.5 for the outer section with SPLD.
1102
Ind. Eng. Chem. Res., Vol. 45, No. 3, 2006
Table 2. Regression Results for the Mass-Transfer Model liquid distributor type SPLD MPLD
a
b
x r 5.468 - 0.09 - 5.683 D R x r 6.571 - 0.1 - 1.776 D R
x r 0.486 - 0.023 + 0.233 D R x r 0.442 - 0.0074 - 0.028 D R
Note: The difference between the RMS value calculated using the rounded-off constants shown in eqs 11 and 12 and that using the constants in Table 2 is within 5%.
calculated from the model and the experimental data is presented in Figure 8. The dotted lines show the boundary of (20% deviation from the model prediction. Almost all of the experimental values are encompassed in the envelope of the dotted lines. To evaluate the proposed models, the values of Ymod ) ln(Sh/Sc0.33)mod, which were calculated from eqs 11 and 12 for SPLD and MPLD, respectively, were compared to the experimental values Yexp ) ln(Sh/Sc0.33)exp using the root-mean-square error (RMS) defined as
The coefficients A and B obtained from linear regression of the relationship Y )AX + B of all positions (all x/D and r/R values) were used to establish the functions a ) f(x/D,r/R) and b ) g(x/D,r/R) by the least-squares method. The model for the local mass-transfer coefficient for SPLD can then be written as
( )
(
)
r x ) 5.5 - 0.1 - 5.7 Re0.5-0.02(x/D)+0.2(r/R) D R SPLD (11)
Sh Sc0.33
and the model for MPLD is
( ) Sh Sc0.33
(
)
r x ) 6.6 - 0.1 - 1.8 Re0.4-0.01(x/D)-0.03(r/R) D R MPLD (12)
From eqs 11 and 12, it can be seen that, for small x/D values, the effect of the Reynolds number on the local mass-transfer coefficient for MPLD is less than for the case with SPLD, as indicated by a lower value of the exponent of Re in the correlation for MPLD. This can be attributed to a better overall liquid distribution in the packed bed due to a good initial liquid distribution provided by MPLD. For higher x/D, i.e., deeper bed, the liquid spreads out more, leading to better liquid distribution and, hence, more even local mass transfer. Therefore, the local mass-transfer coefficient becomes less dependent on liquid flow rate. This is indicated by the negative sign of the x/D term that becomes significant when the x/D value becomes large, leading to decreases in the magnitude of the exponent of Re for both liquid distributor types. It is also noted that the r/R term in the exponent of Re for MPLD has a very small and negative coefficient (-0.03), indicating that the variation of the local mass-transfer coefficient with Re is almost independent of radial positions in the bed. On the other hand, for SPLD, the coefficient of r/R is positive (+0.20) and much larger than that of x/D (-0.02). This indicates that, for SPLD, the radial variation (r/R) has a stronger effect on the local masstransfer coefficient than the axial variation, especially for low x/D values. For SPLD, liquid tends to concentrate in the central region of the packing at low liquid rates, as expected. This was also observed in our separate study on liquid distribution in a packed bed of Pall rings. As a result, the mass transfer is enhanced significantly with increases in liquid flow rate, which helps with spreading liquid to the outer region of the packing. Therefore, the local mass-transfer coefficient increases more quickly with the liquid flow rate or Re, especially in the outer region (i.e., larger r/R). In addition, the coefficient of the Reynolds number in eq 11 becomes rather small with large values of r/R (near the column wall), and mass transfer becomes much more dependent on Re, i.e., liquid flow rate, which helps by spreading the liquid to the wall region. For the simplicity of the equation presentation, the constants in Table 2 were rounded off and shown in eqs 11 and 12. However, all calculations were based on the full expression of the constants as shown in Table 2. The comparison of the values
RMS )
x
n
∑1 (Yexp - Ymod )2 (13)
n
∑1 Yexp2
The RMS values obtained for SPLD and MPLD were 0.08 and 0.07, respectively. The models developed were also validated by data obtained by Zhu et al.27 The values of the RMS were 0.07 and 0.06 for the cross and ladder liquid distributors, respectively. The sensitivity of mass transfer with x/D and r/R was also analyzed, which was defined as the absolute values of ∂(Sh/ Sc0.33)/∂(x/D) and ∂(Sh/Sc0.33)/∂(r/R), respectively. From eqs 11 and 12, the sensitivity of mass transfer with x/D and r/R for both SPLD and MPLD can be expressed as follows:
∂
( )
Sh Sc0.33 SPLD ) x ∂ D
()
[{(5.5 - 0.1Dx - 5.7Rr )(-0.02 ln(Re))} ]
0.1 Re0.5-0.02(x/D)+0.2(r/R) (14) ∂
( ) Sh Sc0.33 r ∂ R
SPLD
)
()
[{(5.5 - 0.1Dx - 5.7Rr )(0.2 ln(Re))} ]
5.7 Re0.5-0.02(x/D)+0.2(r/R) (15) ∂
( )
Sh Sc0.33 MPLD ) x ∂ D
()
[{(6.6 - 0.1Dx - 1.8Rr )(-0.01 ln(Re))} ]
0.1 Re0.4-0.01(x/D)-0.03(r/R) (16) ∂
( )
Sh Sc0.33 MPLD ) r ∂ R
()
[{(6.6 - 0.1Dx - 1.8Rr )(-0.03 ln(Re))} ]
1.8 Re0.4-0.01(x/D)-0.03(r/R) (17) The results of the sensitivity analysis are presented in Figure 9. The solid lines represent the averaged values with all x/D values in Figure 9a and all r/R values in Figure 9b. The mass transfer for SPLD is more sensitive to both x/D and r/R than that for MPLD. For the values of x/D in the range of 0.5-5.5, the average sensitivity of mass transfer with x/D decreases ∼51%
Ind. Eng. Chem. Res., Vol. 45, No. 3, 2006 1103
Figure 8. Comparison between values calculated from the model and the experimental data: (a) SPLD and (b) MPLD.
Figure 9. Sensitivity of mass transfer (a) with x/D and (b) with r/R at Re ) 300. Solid lines represent the average values of the sensitivity (a) for all x/D and (b) for all r/R values.
and 29% for MPLD and SPLD, consecutively (Figure 9a). For the values of r/R in the range of 0.0-0.83, the average sensitivity of mass transfer with r/R increases ∼4.4× for SPLD and decreases much less (∼21%) for MPLD (Figure 9b). The high sensitivity of mass transfer with x/D and r/R reconfirms the variation of liquid distribution with radial and axial positions in the column for both types of liquid distributors. Consequently, the mass-transfer coefficient, particular for SPLD, varied drastically with radial locations in the bed, as indicated by its extremely high sensitivity with r/R.
in the bed. The model agrees well with the experimental data obtained from different studies from our research group. The values of the root-mean-square error smaller than 0.10 for both single-point and multipoint liquid distributors can be considered as satisfactory for the model prediction of local mass transfer in a packed bed, of which large deviation is often encountered in numerous reported studies. However, for model generalization, further measurements of local mass transfer for various packing types and sizes are needed, and a packing factor accounting for the packing geometry and size may then be introduced into the model.
Conclusion The experimental data obtained shows a significant spatial distribution of local mass-transfer coefficient in a packed bed. It was dependent on liquid flow rate as well as liquid-distributor design. For liquid flow rates used in the present study, the more uniform distribution of the mass-transfer coefficient in the bed was observed with the multipoint liquid distributor (MPLD), which provided better initial liquid distribution than that of the single-point liquid distributor (SPLD). For MPLD, the variation of the local mass-transfer coefficient was also less dependent on liquid flow rate and radial positions in the bed compared with SPLD. This can be attributed to a better overall liquid distribution in the packed bed with MPLD, even at low liquid flow rates, while liquid distribution with SPLD improved significantly with increases in liquid flow rate, causing higher liquid spreading from the center to the outer region of the packed bed. To our knowledge, the model proposed in the present study is the first model of local mass transfer in a packed bed. It allows the prediction of the local mass-transfer coefficient at a location
Acknowledgment Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) to this project is greatly appreciated. Nomenclature ac ) surface area of the cathode (m2) A ) coefficient of mass-transfer correlation C∞ ) bulk concentration of ferricyanide (mol‚m-3) D ) column diameter (m) de ) equivalent diameter of packing particle (m) F ) Faradays Constant (96 487 C/equiv) iL ) limiting current (A) kL ) liquid-phase mass-transfer coefficient (m‚s-1) r/R ) dimensionless radial distance from center of column R ) column radius x/D ) dimensionless axial distance from liquid distributor z ) number of electrons transferred in the electrode reaction
1104
Ind. Eng. Chem. Res., Vol. 45, No. 3, 2006
Greek Letters (m3‚m-3)
) porosity of packed bed υ ) kinematic viscosity of solution (m2‚s-1) Dimensionless Groups Re ) Reynolds number Sh ) Sherwood number (Sh ) kLde/D) Sc ) Schmidt number (Sc ) ν/D) Literature Cited (1) Hoek, P. J.; Wesselingh J. A.; Zuiderweg F. J. Small Scale and Large Scale Liquid Maldistribution in Packed Columns. Chem. Eng. Res. Des. 1986, 64, 431. (2) Al-Dahhan, M.; Highfill, W.; Tee Ong, B. Drawbacks of The Dissolution Methods for Measurement of The Liquid-Solid Mass-Transfer Coefficient in Two-Phase Flow Packed-Bed Reactors Operated at Low and High Pressures. Ind. Eng. Chem. Res. 2000, 39, 3102. (3) Wang, Y. F.; Mao, Z. S.; Chen, J. Scale and Variance of Radial Liquid Maldistribution in Trickle Beds. Chem. Eng. Sci. 1998, 53, 1153. (4) Sun, C. G.; Yin, F. H.; Afacan, A.; Nandakumar, K.; Chuang, K. T. Modeling and Simulation of Flow Maldistribution in Random Packed Columns with Gas-Liquid Countercurrent Flow. Chem. Eng. Res. Des. 2000, 78 (Part A), 378. (5) Edwards, D. P.; Krishnamurthy, K. R.; Potthoff, R. W. Development of An Improved Method to Quantify Maldistribution and Its Effect on Structured Packing Column Performance. Trans. Inst. Chem. Eng. 1999, 77 (Part A), 656. (6) Gostick, J.; Doan, H. D.; Lohi, A.; Pritzker, M. D. Investigation of Local Mass Transfer in A Packed Bed of Pall Rings using a Limiting Current Technique. Ind. Eng. Chem. Res. 2003, 42, 3626. (7) Gostick, J.; Pritzker, M.; Doan, H. D.; Lohi, A. Mass Transfer Variation within a Packed Bed and Its Relation to Liquid Distribution. Chem. Eng. J. 2004, 100, 33. (8) Pfeffer, R. Heat and Mass Transport in Multiparticle Systems. Ind. Eng. Chem. Fundam. 1964, 3, 380. (9) Sirkar, K. K. Creeping Flow Mass Transfer to A Single Active Sphere in A Random Spherical Inactive Particle Cloud at High Schmidt Numbers. Chem. Eng. Sci. 1974, 29, 863. (10) Kawase, Y.; Ulbrecht, J. A New Approach for Heat and Mass Transfer in Granular Beds based on The Capillary Model. Ind. Eng. Chem. Fundam. 1985, 24, 115. (11) Segiun, D.; Montillet, A.; Brunjail, D.; Commiti, J. Liquid-Solid Mass Transfer in Packed Beds of Variously Shaped Particles at Low Reynolds Numbers: Experiments and Model. Chem. Eng. J. 1996, 63, 1. (12) Kumar, S.; Upadhyay, S. N.; Mathur, V. K. Low Reynolds Number Mass Transfer in Packed Beds of Cylindrical Particles. Ind. Eng. Chem. Process. Des. DeV. 1977, 16, 1.
(13) Gaunand, A.; Coeuret, F. Influence of The Relative Electric Conductivity of the Two Phases on the Potential Distribution in FlowThrough Porous Electrodes under Limiting Current Conditions. Electrochim. Acta 1978, 23, 1197. (14) Olive, H.; Lacoste, G. Application of Volumetric Electrodes to The Recuperation of Metals in Industrial Effluents. I. Mass Transfer in Fixed Beds of Spherical Conductive Particles. Electrochim. Acta 1979, 24, 1109. (15) Williamson, J. E.; Bazaire, K. E.; Geankoplis, C. J. Liquid-Phase Mass Transfer at Low Reynolds Number. Ind. Eng. Chem. Fundam. 1963, 2, 126. (16) Wilson J. E.; Geankoplis C. J. Liquid-Phase Mass Transfer at Very Low Reynolds Numbers in Packed Beds. Ind. Eng. Chem. Fundam. 1966, 5, 9. (17) Karabelas, A. J.; Wegner, T. H.; Hanratty, T. J. Use of Asymptotic Relations to Correlate Mass Transfer Data in Packed Beds. Chem. Eng. Sci. 1971, 26, 1581. (18) Guo, G.; Thompson, K. E. Experimental Analysis of Local Mass Transfer in Packed Beds. Chem. Eng. Sci. 2001, 56, 121. (19) Selman, J. R.; Tobias, C. W. Mass Transfer Measurements by The Limiting-Current Technique. AdV. Chem. Eng. 1978, 10, 211. (20) Yin, F.; Afacan, A.; Nandakumar, K.; Chuang, K. T. Liquid Holdup Distribution in Packed Columns: Gamma Ray Tomography and CFD Simulation. Chem. Eng. Process. 2002, 41, 473. (21) Wen, X.; Afacan, A.; Nandakumar, K.; Chuang, K. T. Geometry Based Model for Predicting Mass Transfer in Packed Columns. Ind. Eng. Chem. Res. 2003, 42, 5373. (22) Li, M.; Bando, Y.; Tsuge, T.; Yasuda, K.; Nakamura, M. Analysis of Liquid Distribution in Non-Uniformly Packed Trickle Bed with Single Phase Flow. Chem. Eng. Sci. 2001, 56, 5969. (23) Sherwood, T. K.; Halloway, F. A. L. Performance of Packed Bed Towers Liquid Film Data for Several Packings. Trans. Inst. Chem. Eng. 1939, 36, 39. (24) Yoshida, F.; Koyanagi, T. Liquid-Phase Mass Transfer Rates and Effective Interfacial Area in Packed Absorption Columns. Ind. Eng. Chem. 1958, 50, 365. (25) Jolls, K. R.; Hanratty, T. J. Use of electrochemical techniques to study mass transfer rates and local skin friction to a sphere in a dumped bed. AIChE J. 1969, 15, 199. (26) Karabelas, A. J.; Wegner, T. H.; Hanratty, T. J. Use of asymptotic relations to correlate mass transfer data in packed beds. Chem. Eng. Sci. 1971, 26, 1581. (27) Zhu, Y.; Doan, H. D.; Lohi, A. Relation of Liquid Distribution and Local Mass Transfer Coefficient in A Packed Bed. In Proceedings of the 7th World Congress of Chemical Engineering, Glasgow, Scotland, July 10-14, 2005.
ReceiVed for reView May 5, 2005 ReVised manuscript receiVed November 4, 2005 Accepted November 30, 2005 IE0505312
