Geometry-Based Model for Predicting Mass Transfer in Packed Columns

Ind. Eng. Chem. Res. 2003, 42, 5373-5382

5373

Geometry-Based Model for Predicting Mass Transfer in Packed Columns X. Wen, A. Afacan, K. Nandakumar, and K. T. Chuang* Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6

We have extended our previous work on the modeling of geometry, liquid trickle flow, and pressure drop to study the mass-transfer process in a packed column. On the basis of the penetration theory and the detailed information of lateral and axial variations in packing geometry and fluid dynamics from our previous models, a predictive mass-transfer model has been developed on the scale much smaller than a packing particle in the case of random packings or a flow channel in the case of structured packings. Distillation experiments have been carried out with methanol/2-propanol and methanol/water for 16 mm metal Pall rings. The model has been validated with the experimental results and our previous data on a novel vertical-sheet structured packing. Simulations for uniform and uneven initial distributions have been carried out, which showed a strong influence of liquid-flow distribution on mass-transfer efficiency in a packed column. Introduction Packed columns are widely used in chemical process industries as liquid-gas (or vapor) contacting devices for mass transfer. Typical applications are packed distillation and absorption/desorption columns. In the modeling of mass transfer in packed columns, the concept of height of a transfer unit (HTU) is usually used in the analysis and design. The HTU concept is based on steady-state mass-balance equations with the assumption of plug flow. With this concept, the masstransfer process in a packed column is then modeled by correlations predicting mass-transfer coefficients and interfacial area. Many models of this kind are available in the literature.1-3 Some examples are correlations by Onda et al.4 and by Bravo and Fair5 for random packings, the University of Texas Separations Research Program (SRP) model and the Delft model for structured packings,6-8 and a correlation by Billet and Schultes9 for both random and structured packings. To account for axial dispersion of species, a dispersion model has also been used for modeling mass transfer in packed columns. The dispersion model describes the axial dispersion of a species by making an analogy to Fick’s law (see Sherwood et al.10). The usual assumptions for the model are that the solute concentrations in the two phases are uniform in the lateral direction and the velocities are constant throughout the column. Some recent work on the dispersion model includes the following: Benadda et al.11 determining dispersion coefficients, mass-transfer coefficients, and interfacial area with experiments at elevated pressures; Doan and Fayed12 evaluating the dispersion model for the gas phase with experiments and analyzing the sensitivity of the model to the model parameters; and Perrin et al.13 estimating the model parameters with transient-state techniques involving transferable and nontransferable tracers. The HTU concept is the plug-flow limit of the dispersion model. * To whom correspondence should be addressed. Tel.: 1-780-492-4676. Fax: 1-780-492-2881. E-mail: karlt.chuang@ ualberta.ca.

As the diameter of a column becomes large, heterogeneities in flow distribution develop naturally. The basic assumption of homogeneous conditions in the lateral direction of the above type of one-dimensional models become invalid. Because of this limitation of the models, the factors that influence the mass transfer in a packed column, such as the packing type and size, flow intensities of the liquid and gas, and flow distributions,1 are captured only indirectly through empirical or semiempirical correlations for model parameters: masstransfer coefficients, interfacial area, and dispersion coefficients. If the conditions under which the model parameters were experimentally obtained are not met, the models cannot be used reliably. It is the unreliability of these types of models that makes even rules of thumb for height equivalent to a theoretical plate (HETP) successfully compete with mass-transfer models in the design of packed columns.1 With the awareness of their influence on mass transfer, flow distributions have been introduced into masstransfer models. The zone/stage model14 splits fluid into adjacent zones after each stage. The zones are concentric rings of equal width that is larger than a packing particle. In the model, the characteristics of flow distribution are described by packing-dependent splitting factors and the inherent mass-transfer performance is characterized by “basic” HETP, a value under ideal flow conditions that can be inferred from mass-transfer data for small columns. The zone/stage model can predict the influence of the local liquid/vapor (L/V) ratio on the mass-transfer efficiency. Olujic´ et al.7,15 developed a discrete cell model, called LDESP, based on the liquidflow mechanisms in structured packings. The cell size is on the scale of the flow channel in the packings. With the predicted liquid and gas flow rates in the cells, the mass-transfer efficiency in each cell is estimated based on the local L/V ratio. Then the overall efficiency of the column is calculated. Basic HETP and experimentally determined splitting factors are also needed in the model. While the zone/stage model and the LDESP model use an equilibrium concept in a zone/cell, Higler et al.16

10.1021/ie030347l CCC: $25.00 © 2003 American Chemical Society Published on Web 09/09/2003

5374 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003

extended them into a nonequilibrium cell model. Because of the nonequilibrium characteristics of their model, the influence on the mass-transfer efficiency of not only local L/V ratio but also local hydrodynamics can be predicted by using empirical equations to calculate mass-transfer coefficients and interfacial area locally. In their examples, the Onda model was used for random packing and the SRP model for structured packing. The model works on a scale larger than that of a packing particle or a flow channel. With the advent of powerful computers and the development of sophisticated algorithms to solve partial differential equations, CFD (computational fluid dynamics) provides another approach for the analysis of mass-transfer processes in packed columns. Having the ability to capture lateral and axial variations in flow and mass transfer, CFD-based models can predict the separation efficiency based on detailed local conditions. Yin et al.17 has successfully analyzed randomly packed distillation columns with CFD simulation. Because of the complicated geometry of packings and the complex, dynamic nature of the gas-liquid interface, directly solving the basic Navier-Stokes equations is not practical for two-phase flow in packed columns. Therefore, a volume-averaged approach is used. To capture the information that is lost in the averaging process, closure models have to be provided, such as dispersion coefficients, mass-transfer coefficients, interfacial area, etc. For example, in the work of Yin et al.,17 the Onda model has been used as the closure model for interphase mass transfer. All of the above-mentioned models for predicting the influence of heterogeneous flow distribution on mass transfer are on a scale equal to or larger than that of a packing particle or a flow channel. Because the geometry of packing cannot be captured by the models on this scale, packing-dependent correlations or parameters for fluid dispersion and interphase mass transfer are required. If we work on a scale smaller than that of a particle size, in the case of random packings, or than that of a flow channel, in the case of structured packings, to some extent, the shape of a packing will disappear and local characteristics of fluid dynamics and mass transfer can be predicted based on fundamental theories and equations. Then packing-independent, more predictive models can be developed. In a series of papers,18-20 we have presented a threedimensional geometrical model for packed beds and geometry-based models for liquid trickle flow and pressure drop through a packed column on a subparticle scale. The present work is to extend the models into the mass-transfer process. Review of the Geometrical, Liquid-Flow, and Pressure-Drop Models In the geometrical model for packed beds,18 a threedimensional model of each packing element is first constructed by triangulating the complex surface of the element. Then for random packings, the packing process is simulated with a collision-detection algorithm and an optimization method to determine the position and orientation of each packing element within a bed such that any two packing elements merely touch each other. For structured packing, the position and orientation of each packing element (or brick) in a bed are predetermined, so the simulation of the packing process is unnecessary. The result from such a simulation is a

Figure 1. A layer of cells and the outward flows of a cell.

detailed three-dimensional model for the geometry of a packed bed, i.e., the shape, size, location, and orientation of each triangulated piece of the packing surfaces. The model is then used to extract useful geometrical properties such as spatial variation of the porosity and surface area. A geometry-based model to predict the single-phase trickle flow of liquid through a packed bed has been developed based on the three-dimensional geometrical model.19 In the liquid-flow model, a mesh system with layers of cuboid cells is set up for a packed bed (see Figure 1). The size of a cell is much less than the size of a packing element for random packings or the size of a flow channel for structured packings. The triangulated packing surfaces constructed by the geometrical model are clipped into each cell and then triangulated again to yield a group of subtriangles that are used to build up the geometrical properties within the cell. In each cell, the surface properties are assumed to be characterized by an average planar surface, with the surface area equal to the total surface area of all of the subtriangles within the cell and the gradient equal to the mean value of the gradients of all of the subtriangles weighted with their surface areas. On the basis of the mechanisms of liquid trickle flow in a packed bed, liquid flow at each point is decomposed into two possible directions: vertically down and horizontally in the direction of the negative gradient of the packing surface. In each cell, liquid may come from the cell above it and from some or all of the eight neighboring cells in the same layer (see Figure 1). The total volumetric flow rate in cell i of layer j is calculated by i8

j-1 Qlji ) Qlid +

δjli )

{

j δjliQllh ∑ l)i1

1 (flow from l to i) 0 (no flow from l to i)

(1)

(2)

δjli is determined by the local gradient of the surface in cell l of layer j. The outward liquid of cell i flows into one of the adjacent cells in the same layer along the direction of the negative gradient of the cell and/or into the cell under it in the next layer. The volumetric flow rates in the two directions are calculated by j j Qlid ) f di Qlji

(3)

j j ) (1 - f di )Qlji Qlih

(4)

j is assumed to The fraction of the downward flow f di equal the fraction of the free area (the area uncovered by packing surfaces) in the horizontal section of the cell.

Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5375

Unlike the zone/stage model,14 the LDESP model,7,15 and the nonequilibrium cell model,16 in which the splitting of liquid is calculated by empirical model j constants, here δjli and fdi are determined based on the detailed local geometrical properties of the packed bed. After the liquid-flow-rate distribution is obtained, the liquid holdup and the fraction of wetted surface area in each cell are also calculated based on fundamental hydrodynamic models for an inclined plate. Furthermore, with the mesh system kept the same, the average planar surface in each cell, the liquid-flowrate distribution, and the liquid-holdup distribution are used to build up the pressure-drop model together with fluid dynamic fundamentals for inclined plates.20 In the pressure-drop simulation, the gas-flow-rate distribution is also calculated by applying no lateral pressure gradient at each layer of the packed bed. Development of the Mass-Transfer Model In the development of the mass-transfer model, the same mesh system as that in the liquid-flow model and the pressure-drop model has been used. With the lateral and axial variations in the packing surface, liquid and gas flow rates, liquid holdup, and fraction of wetted surface predicted by the geometrical model, the liquidflow model, and the pressure-drop model, it is possible to predict locally the mass-transfer rates and then to predict the local and overall separation performances of a packed column. Assumptions. The following assumptions were made in the development of the model for mass transfer in a packed column with liquid-gas countercurrent flow: (1) There is no interaction between the gas phase and the liquid phase. Therefore, simulation results from the liquid-flow model, which is for single-phase trickle flow, and the results from the pressure-drop model, which also assumes no interaction between the two phases, can be used in modeling mass transfer. This assumption is valid in the operating region under the loading point. (2) The relative molar flow rates of the liquid and gas phases in each cell are identical with the corresponding relative volumetric flow rates, which include Rlji ) j j j j ) Qlid /QljT, Rlih ) Qlih /QljT, and Rgji ) Qgji/ Qlji/QljT, Rlid j QgT. (3) Mass transfer does not affect the relative flow rates of the two phases, although the overall flow rates of the two phases may change as a result of the transport of species. (4) As discussed in the pressure-drop model,20 gas spreads quickly in a packed bed and we assumed that, within each layer of the cells, the gas phase adjusts its distribution according to the variations of porosity, packing surface, and liquid holdup in the layer, so that there is no lateral pressure gradient across the section. Now we further assume that the gas phase is well mixed at the top of each layer of the cells; therefore, the solute concentration in the gas phase is uniform over the section. We also assume that spreading and mixing happen only at the interface of two adjacent layers; thus, the relative gas flow rate tunes to Rgji by lateral by spreading and the mole fraction is averaged to yj+1 k mixing, before the gas enters cell i in layer j from layer j + 1 (see Figure 2).

Figure 2. Schematic representation of a cell.

(5) The effective interfacial area for mass transfer is equal to the wetted surface area. Cell Equations. A schematic representation of a cell is given in Figure 2. For a contacting cell i (i ) 1, 2, ..., M) in layer j (j ) 1, 2, ..., N), there are the following: liquid coming from the cell above it with the relative j-1 and the component mole volumetric flow rate Rlid j-1 fraction xki (k ) 1, 2, ..., c); liquid flowing from some or all of the eight neighboring cells (l ) i1, i2, ..., i8) in the same layer (see Figure 1), each with the possible relative j and the possible component volumetric flow rate Rllh j mole fraction xkl; liquid leaving the cell into the cell below it and/or into one of the neighboring cells in the same layer, with the total relative volumetric flow rate j j + Rlih and the component mole fraction xjki; Rlji ) Rlid and gas flowing through the cell with the relative volumetric flow rate Rgji. Before entering layer j, the gas phase is well mixed and has a uniform mole fraction all over the section, yj+1 k . The mole fraction in the gas leaving the cell is yjki. Njki is the mass-transfer rate from the liquid phase into the gas phase. Lj is the total molar flow rate of the liquid phase in layer j and Gj the gas phase. The component mass balances can be made for cell i in layer j: i8

j-1 j-1 j-1 Rlid L xki +

j δjliRllh Ljxjkl - RljiLjxjki - Njki ) 0 ∑ l)i1

(5)

RgjiGj+1yj+1 - RgjiGjyjki + Njki ) 0 k

(6)

For a binary system, the individual mass-transfer coefficients are modeled by the penetration theory (see Knudsen et al.21)

kL′ ) 2xDL/πtL

(7)

kG′ ) 2xDG/πtG

(8)

with the contact time calculated by the fluid dynamics in each cell:

tL ) VhL/Ql

(9)

tG ) V( - hL)/Qg

(10)

The penetration model has been successfully used to calculate mass-transfer coefficients in packed beds and strings of spheres by Singh and Stockar.22 In the Billet and Schultes model, the penetration theory was also used to calculate mass-transfer coefficients in the liquid and gas phases.23


The mass-transfer rates are calculated based on the two-resistance theory:

N ) KOLSη(x - x*)

(11)

1 1 1 ) + KOL kL mkG

(12)

kL ) kL′FL′

(13)

kG ) kG′FG′

(14)

Equations 7-14 are all local equations for each cell; the sub- and superscripts i, j, and k are omitted for the convenience of writing. For a multicomponent system, the component masstransfer rates can be described by the Maxwell-Stefan equations. There are methods of calculating the masstransfer fluxes that could be used.24 Only c - 1 of eqs 5 and 6 are independent (k ) 1, 2, ..., c - 1). The mole fraction of the cth component is calculated from the mole-fraction summation equation for each phase. Layer Equations. For each layer of cells, the component mass-conservation equations for the liquid and gas phases can be written as

Ljxjk -

∑i Rlidj Ljxjki ) 0

(15)

Gjyjk -

∑i RgjiGjyjki ) 0

(16) Figure 3. Schematic diagram of the experimental setup.

or

xjk )

∑i Rlidj xjki

(17)

yjk )

∑i Rgji yjki

(18)

where xjk and yjk are the average mole fractions in the liquid and gas phases leaving layer j, respectively. Solving the Model Equations. Equations 5, 6, and 18 can be solved numerically with the detailed spatial distributions of the following properties predicted by the geometrical model, the liquid-flow model, and the pressure-drop model: porosity (ji), packing surface area (Sji), fraction of wetted area (ηji), relative flow rates of j j , Rlih , and Rgji), and liquid the two phases (Rlji, Rlid j holdup (hLi). The mass-transfer rates are calculated by eqs 7-14 and the inlet concentrations of the two phases are given as follows:

x0ki ) xki,in

(19)

) yk,in yN+1 k

(20)

Equations 19 and 20 are inlet conditions that liquid feeds at the top of the column and gas at the bottom, as in the cases of simple absorptions and total reflux distillations. Other feed conditions can be solved by sectioning the column. In the model, the total molar flow-rate profiles of the two phases (Lj and Gj) are assumed to be preknown. They can be obtained by solving additional energybalance equations or by assuming constant molar flow while adequate.

Determination of the Mass-Transfer Efficiency. Conventionally, there are two methods to characterize the mass-transfer efficiency of a packed column: HTU and HETP. In the present study, the HTU method is used to interpret simulation and experimental results. With the detailed concentration distributions from simulations, HTU for any height interval (H, H + ∆H) of the column can be calculated to give the local or overall efficiency of the column and to compare simulations with experiments:

NOG )

∫yy

dy y* - y

H+∆H

H

HOG ) ∆H/NOG

(21) (22)

Experimental Studies Mass-transfer experiments were carried out in a distillation column with an inner diameter (i.d.) of 0.15 m, packed with 16 mm metal Pall rings. The schematic diagram of the experimental setup is shown in Figure 3. The column was equipped with a total condenser and a partial reboiler. The packed height was 1.23 m. A liquid distributor with 12 irrigating points, which is equivalent to 658 points/m2, was employed at the top of the column for distributing reflux liquid. A total of six samplers were installed along the column to measure concentration profiles. The column was controlled by a computer with LabView 5.0 and an Opto-22 I/O system. Temperatures and flow rates were measured by thermocouples and orifice plates, respectively, and logged by the computer. The tests were conducted under total reflux and ambient pressure with two binary systems: methanol/2-propanol and methanol/water. Liquid samples


Figure 4. Comparison of predicted and measured HOG for Pall rings.

Figure 5. Comparison of liquid-phase concentration profiles, Pall rings, and the methanol/2-propanol system.

Table 1. Characteristics of Packings

packing

surface nominal area size (mm) porosity (m2/m3)

Pall rings 16 (5/8 in.) vertical-sheet structured packing

0.93 0.94

341 250

were taken from samplers after steady-state operation was reached. Samples were analyzed with a HP 5790A gas chromatograph with a thermal conductivity detector. For the methanol/2-propanol system, samples were taken from sampler nos. S1, S3, S4, and S6. For the methanol/water system, almost pure methanol and pure water can be obtained with the full packed height, so only samples from sampler nos. S3, S4, S5, and S6 were usually measured and samples from sampler no. S2 were taken in some cases. The packing characteristics and system properties are shown in Tables 1 and 2. Results and Discussion Random Packing. Simulations have been carried out for the experiments described above. In the simulations, each layer of the bed was divided into 48 × 48 cells in the diameter and the height of each layer was 0.0025 m. The resulting cell size was 0.0032 × 0.0032 × 0.0025 m3, which corresponded to a resolution of about 1/5 of the size of a packing particle. Heights of a transfer unit (HOG) were determined from the measured and predicted concentrations, respectively. Figure 4 shows a comparison of the overall HOG between model predictions and experimental results for the two systems. It can be seen from the figure that the predicted overall HOG results are in good agreement with the experimental data. While the overall HOG is a single measure of the column performance, more detailed comparisons between simulation results and experimental data are given in Figures 5 and 6 in terms of liquid-phase concentration profiles along the column. Figure 5 shows the concentration profiles of the methanol/2-propanol system for two sample operating conditions: F-factor ) 0.55 m/s (kg/m3)0.5 with a mole

Figure 6. Comparison of liquid-phase concentration profiles, Pall rings, and the methanol/water system.

fraction of 0.68 in the reflux and F-factor ) 0.81 m/s (kg/m3)0.5 with a mole fraction of 0.89 in the reflux. Figure 6 shows the concentration profiles of the methanol/water system also for two operating conditions: F-factor ) 0.54 m/s (kg/m3)0.5 with top x ) 0.94 and F-factor ) 0.76 m/s (kg/m3)0.5 with top x ) 0.93. In Figures 5 and 6, each predicted mole fraction in the liquid phase is an average value calculated by eq 17 based on the detailed distributions of flow and concentration over the column cross section at a given axial position. It is encouraging that the predicted concentration profiles also agree well with the experimental data. Structured Packing. To further test the predictability of the present model, mass-transfer data of a recently developed vertical-sheet structured packing25 have also been used to validate the model. Unlike the conventional structured packings with corrugated sheets, the novel structured packing is the assembly of vertical sheets. The liquid-flow model and the pressure-drop model have been validated against the vertical-sheet structured packing.20,25 The characteristics of the structured packing are listed in Table 1. Distillation experiments have been carried out in a column with an i.d. of 0.3 m.25 Seven elements of the

Table 2. Properties of the Test Systems system

FL (kg/m3)

FG (kg/m3)

µL × 104 (Pa‚s)

µG × 105 (Pa‚s)

σ × 103 (N/m)

DL × 109 (m2/s)

DG × 106 (m2/s)

R

methanol/2-propanol 12-91 (mol %) methanol/water 0.4-96 (mol %)

731-750 751-917

1.2-1.9 0.57-1.1

3.7-5.4 2.9-3.5

0.94-1.1 1.1-1.3

17-19 21-60

3.1-3.9 3.9-6.4

8.6-9.4 4.4-5.4

1.7-2.3 2.5-8.0


Figure 7. Computer-generated three-dimensional geometry of the vertical-sheet structured packing.

Figure 8. Comparison of predicted and measured HOG for structured packing.

structured packing were installed with the total packed height of 1.96 m. The liquid distributor for reflux was with 66 irrigating points, which is equivalent to 985 points/m2. The tests were conducted also under total reflux and ambient pressure with the methanol/2propanol and methanol/water systems. Only end concentrations were measured in the experiments. In the simulations of the structured packing, each layer of the bed was divided into 114 × 114 cells in the diameter, and the height of each layer was 0.0025 m. The resulting cell size was 0.0026 × 0.0026 × 0.0025 m3, which corresponded to a resolution of about 1/3 of the width of the flow channel of the packing (0.008 m). Figure 7 shows the geometry of two sheets of the packing generated by the three-dimensional modeling approach, on which the liquid-flow model, the pressuredrop model, and the mass-transfer model are built. The packing features the folded tabs, which act as spacers to control the spacing between two adjacent sheets and also as spreaders to spread liquid over the packing. Figure 8 shows a comparison of the overall HOG between model predictions and experiments. The result demonstrates reasonably good agreement between the predicted overall HOG and the experimental data for the

vertical-sheet structured packing in the 0.3 m i.d. column with the two tested systems. Discussion. In Figure 8, there are some discrepancies between predicted and measured HOG at low and high F-factors for the methanol/2-propanol system. At low F-factors, it was observed in the experiments that the distributor could not distribute liquid evenly over its drip points because of the low liquid flow rates and the high drip density (985 points/m2). There was no liquid flow at some of the irrigating points. However, in the simulation, the initial liquid flow was uniformly distributed over all of the drip points for any liquid load. The initial distributions used in the simulation, which were better than the actual conditions for low liquid loads, caused an overestimation of the mass-transfer efficiency for the methanol/2-propanol system at the low F-factor end in Figure 8. For the methanol/water system, there is a narrow pinch between the equilibrium curve and the operating line for total reflux at high methanol concentrations. This characteristic of the equilibrium relation makes the overall HOG, while calculated with eqs 21 and 22, more sensitive to the concentrations at the top of the column than to the concentrations at the bottom, within the mole-fraction range of the experiments. This shelters the model deviation at the low F-factor end in Figure 8, when the top concentrations are given as model inputs (eq 19) and the bottom concentrations are predicted by the model. All of the liquid-flow model, the pressure-drop model, and the mass-transfer model assume no interaction between the gas and liquid phases. This assumption is believed to be only valid in the preloading region. When the gas velocity and/or the liquid flow rate is increased to a certain point, liquid starts to accumulate or “load” the bed. Beyond the loading point, the interaction between the two phases gets stronger and stronger and cannot be omitted. The load of liquid above the loading point leads to extra increases of liquid holdup and interfacial area, which cannot be captured by the model. This caused an underestimation of the mass-transfer efficiency for the methanol/2-propanol system at the high F-factor end in Figure 8. The loading point was not reached for the methanol/water system. We can see from Figure 4 that for Pall rings the shapes of the HOG vs F-factor curves are almost flat and there are no obvious discrepancies between predicted and measured HOG at the low F-factor end. That is because the diameter of the column used for Pall rings is small, and therefore the initial liquid distribution has little effect on the mass-transfer efficiency. Figure 4 also shows the trend of the deviation of the predicted HOG for the methanol/2-propanol system when approaching the loading point. Noting that the investigations included two different systems and two different types of packings: a conventional random packing, Pall rings, and a recently developed structured packing with novel geometry as well as a wide range of operating conditions [F-factor from 0.35 to 3.9 m/s (kg/m3)0.5]; we can say that the reasonably good agreement between model predictions and experimental data has demonstrated the predictability and the packing-independent character of the present geometry-based mass-transfer model. The model itself, rather than by additional empirical parameters, is able to capture the geometrical characteristics of


Figure 9. Concentration distribution, structured packing, and the methanol/2-propanol system.

Figure 10. 100% initial distribution.

different packings and the fluid dynamic variations in a packed bed and to take into account the impacts of system properties. On the basis of the detailed information of lateral and axial variations in packing geometry and fluid dynamics, the mass-transfer model is capable of predicting spatial distributions of concentrations of species. Experimental data on such a detailed scale are not yet available. In Figure 9, we present a sample threedimensional visualization of the mole-fraction distribu-

tion of methanol in the liquid phase. The figure shows the result at the packed depth of 0.51 m from the top of the bed of a typical simulation, which is for the structured packing in the 0.3 m i.d. column with the methanol/2-propanol system. There is a strong variation of the concentration over the cross section of the column, while the average value is x ) 0.71. The present model can also be used to analyze the influence of liquid distribution on the mass-transfer efficiency. The 0.3 m i.d. column with the structured


Figure 11. 50% initial distribution.

based on the simulation resolution with the cell size of 0.0026 × 0.0026 m2.

Mf )

Figure 12. Maldistribution factors, structured packing, and the methanol/2-propanol system.

packing and the methanol/2-propanol system has been used to carry out the analysis by simulations. To compare, two liquid distributors were employed in the simulations. One was the liquid distributor with 66 irrigating points as used in the experiments. Another was a liquid distributor only covering half of the total cross section of the column, which was achieved by removal of half of the drip points from the 66-hole distributor. Those two initial distributions of liquid will be referred as the 100% initial distribution and the 50% initial distribution respectively hereafter. Figures 10 and 11 show the two initial distributions. Figure 12 shows the profiles of the simulated liquid maldistribution factor along the column for the two initial distributions at the reflux flow rate of 3.6 × 10-3 m3/m2/s. The maldistribution factor (Mf), as defined by eq 23, is a commonly used parameter for quantifying the flow maldistribution of a packed bed. The larger the Mf, the poorer the liquid distribution. The calculated Mf was

x ∑( ) 1

M

Mi)1

ui - u j u j

2

(23)

It can be seen from Figure 12 that, with the action of the packing, Mf values for both the 100% and 50% initial distributions reach the minimum values at the packed depth of 0.51 m (almost two layers of the packing that is 0.56 m), beyond that height, Mf increases because of the development of liquid wall flow. Because of severer initial maldistribution, Mf of the 50% initial distribution is much higher than that of the 100% initial distribution throughout the column. For interpretation of the effect of the liquid distribution, differential HOG has been calculated by eqs 21 and 22 with ∆H ) 0.1 m. Figure 13 shows the variations of differential HOG along the packed depth for the two initial distributions and at the same condition as that in Figure 12. We can see that, because of the poorer liquid distribution, local HOG are higher in the case of the 50% initial distribution. The calculated overall HOG for the 50% initial distribution, 0.80, is also higher than that for the 100% initial distribution that is 0.62. The curves in Figures 12 and 13 have similar shapes and find their minimums at the same packed depth. The higher values of Mf’s and HOG at the top and bottom of the column show that both the initial maldistribution and the development of liquid wall flow lead to poor mass-transfer efficiency. In both of the cases of the 100% and 50% initial distributions, HOG is a function of the packed height and there is a strong correspondence between the HOG variations and the Mf profiles along the column. The analysis has shown how strongly the mass-transfer efficiency depends on liquid distribution in a packed column.


Figure 13. Differential HOG, structured packing, and the methanol/ 2-propanol system.

Because the models can represent the detailed geometry of packings, this series of models, the threedimensional geometrical model and the geometry-based liquid-flow, pressure-drop, and mass-transfer models, are able to not only simulate existing packings but also predict the performances of nonexisting packings provided with the designed geometry. This is extremely useful in exploring new conceptual designs of packing and optimizing the geometry of the packing under development. Conclusions A geometry-based mass-transfer model has been developed for packed columns based on our previous geometrical model and geometry-based liquid-flow and pressure-drop models and the penetration theory. When a scale much smaller than that of a packing particle or a flow channel is used, the model can predict the influence of packing geometry, fluid dynamics, and system properties on mass transfer in packed columns. Distillation experiments have been carried out for 16 mm metal Pall rings. The experimental results together with our previous data on a novel vertical-sheet structured packing have been used to validate the model. Predicted and measured heights of a transfer unit (HOG) and concentration profiles have been compared for two binary systems: methanol/2-propanol and methanol/ water. The results have shown that the predictions of the present model are in agreement with the experimental data. On the basis of the detailed information of lateral and axial variations in packing geometry and fluid dynamics, the present model is capable of predicting detailed concentration distributions in a packed column and analyzing the influence of liquid distribution on the mass-transfer efficiency. Simulations for different initial liquid distributions have shown that HOG is a function of the axial position along a packed bed and also shown that the variation of HOG along a column is closely related to the profile of the liquid maldistribution factor, which demonstrated the strong effect of liquid-flow distribution on the mass-transfer efficiency. Nomenclature c ) total number of components D ) molecular diffusivity, m2/s fd ) fraction of liquid downward flow

F-factor ) uGxFG, m/s (kg/m3)0.5 G ) total gas flow rate, kmol/s H ) packed height, m hL ) liquid holdup HOG ) overall height of a transfer unit based on the gas phase, m k ) individual mass-transfer coefficient, kmol/m2/s k′ ) individual mass-transfer coefficient, m/s KOL ) overall liquid-phase mass-transfer coefficient, kmol/ m2/s L ) total liquid flow rate, kmol/s m ) local slope of the equilibrium line M ) total number of cells in a layer Mf ) maldistribution factor N ) total number of layers, or mass-transfer rate, kmol/s NOG ) overall number of transfer units based on the gas phase Qg ) gas flow rate in a cell, m3/s QgT ) ∑iQgi, total gas flow rate, m3/s Ql ) liquid flow rate in a cell, m3/s QlT ) ∑iQlid, total liquid flow rate, m3/s Rg ) relative volumetric flow rate of the gas phase Rl ) relative volumetric flow rate of the liquid phase S ) packing surface area in a cell, m2 t ) contact time, s u j ) average liquid superficial velocity, m/s uG ) gas superficial velocity, m/s ui ) liquid superficial velocity in cell i, m/s V ) volume of a cell, m3 x ) mole fraction in liquid phase y ) mole fraction in gas phase Greek Letters R ) relative volatility ) porosity η ) fraction of the wetted surface area µ ) dynamic viscosity, Pa.s F ) density, kg/m3 F′ ) molar density, kmol/m3 σ ) surface tension, N/m Indices * ) in equilibrium with the other phase d ) vertically downward flow G ) gas phase h ) horizontal flow i ) cell index in ) inlet j ) layer index, from the top down k ) component index l ) cell index L ) liquid phase

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Received for review April 22, 2003 Revised manuscript received July 18, 2003 Accepted August 4, 2003 IE030347L

Geometry-Based Model for Predicting Mass Transfer in Packed Columns

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