ARTICLE pubs.acs.org/IECR
Population Balance Modeling of Pulsed (Packed and Sieve-Plate) Extraction Columns: Coupled Hydrodynamic and Mass Transfer Moutasem Jaradat,†,‡ Menwer Attarakih,§ and Hans-J€org Bart*,†,‡ †
Chair of Separation Science Technology, TU Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany Centre of Mathematical Computational Modelling, TU Kaiserslautern, 67653 Kaiserslautern, Germany § Chemical Engineering Department, Faculty of Engineering & Technology, University of Jordan, 11942 Amman, Jordan ‡
ABSTRACT: LiquidLiquid Extraction Column Module (LLECMOD) is a rigorous and comprehensive bivariate population balance framework for dynamic and steady-state modeling of liquidliquid extraction columns. Within LLECMOD, the user can simulate different types of extraction columns, including stirred and pulsed ones. The basis of LLECMOD depends on stable robust numerical algorithms based on an extended version of a fixed pivot technique (to take into account interphase solute transfer) and advanced computational fluid dynamics (CFD) numerical methods. In this work, mathematical models for pulsed packed and sieve tray extraction columns are developed. The models are programmed using visual digital FORTRAN and then integrated into the LLECMOD population balance model. As a case study, the steady-state performance of pulsed packed and sieve-plate columns, under different operating conditions, which include pulsation intensity and volumetric flow rates, are simulated. The effect of pulsation intensity is found to have a more profound effect on systems of high interfacial tension. On the other hand, the variation of volumetric flow rates has a substantial effect on the holdup, mean droplet diameter, and solute concentration profiles for chemical systems with low interfacial tension. Two chemical test systems recommended by the EFCE are used in the simulations. Model predictions are successfully validated against experimental data by adjusting the steady-state column hydrodynamics, using only droplet coalescence empirical parameters.
1. INTRODUCTION Liquidliquid extraction is an important separation process that is encountered in many chemical process industries,1,2 as well as biochemical and petroleum industries.3 Different types of liquidliquid columns are being used, which can be classified into two main categories: stirred (RDC, K€uhni, etc.) and pulsed (packed and sieve-plate) columns. The generally poor performance of spray columns can be improved by introducing column internals such as packing or sieve plates.46 Internal column geometry (packing/sieve plates) reduces axial mixing, increases droplet coalescence and breakage rates resulting in increased masstransfer rates, and affects the mean residence time of the dispersed phase, which allows the handling of large loads with small differences of interfacial tension and density,7 improving the hydrodynamic performance of the column and, subsequently, the masstransfer performance (extraction efficiency).3,814 In sieve-plate columns, the dispersed phase is redistributed over the column cross section at every tray. However, for columns with structured packing, the dispersed phase is redistributed over the column cross section at certain positions along the column using liquid distributers, if necessary. The performance of these columns can be enhanced by mechanical pulsation of one or both phases. This is a result of an increase in shear forces and consequent reduction in the size of dispersed droplets. This, in turn, increases the interfacial area and then the mass-transfer rate.15 Van Dijck16 devised the use of external energy in the form of pulsing in sieve-plate columns that had found wide applications in the processing of nuclear fuel. Since the pulsing unit can be remote from the pulsed column, the pulsed columns are recommended for processing the corrosive or r 2011 American Chemical Society
radioactive solutions3 and nuclear fuel reprocessing.1719 These columns have a clear advantage over other mechanical contactors when processing corrosive or radioactive solutions. Pulsed columns are one of the most important liquidliquid extraction equipment.14 Pulsed columns have distinct advantages: practically, the capability of high flow capacities (high throughputs),20,21 simple operation, extraction efficiencies, considerable flexibility,6,22,23 small footprint for multistage extraction systems,1 reduction in organic losses and organic inventory,24 and insensitivity toward contamination of the interface. Such types of extraction columns have been used successfully in a great variety of extraction processes,1,25,26 and pharmaceutical and hydrometallurgical industries, as well in the original Redox and Purex processes27 in atomic energy industry.28 However, pulsed columns are relatively trouble-free, have a reasonably small residence time, good extraction efficiency, better capacity-efficiency relation with minimum HTU occurring at close to flooding, a lesser power requirement, and suitability for heterogeneous phase contacting using small-sized particles with large holdup.29 Until now, the design of liquidliquid extraction columns has been based on a combination of pilot plant tests, the designer’s prior experience, and empirical correlations.30,31 A better understanding of the hydrodynamics and mass-transfer behavior of liquidliquid extraction columns can be used ingeniously to increase the reliability in the design of extraction columns.32,33 For this reason, there is Received: Revised: Received: Published: 14121
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an obvious need to include droplet interactions (breakage, coalescence, and droplet interface mass transfer), based on a population balance equation (PBE). The main objective of this paper is to develop a PBE-based model that is capable of describing the dynamic and steady-state behavior of pulsed (packed and sieve tray) extraction columns. The models of both columns are integrated into the existing program: LiquidLiquid Extraction Column Module (LLECMOD),33 which already can simulate agitated extraction columns (RDC and K€uhni). LLECMOD can simulate the steady-state and dynamic behavior of extraction columns, taking into account the effect of the dispersed phase inlet (the light or heavy phase is dispersed) and the direction of mass transfer (from continuous to dispersed phase and vice versa).34 The model predictions are extensively compared to the experimental data of Garthe35 with two EFCE test systems (wateracetonetoluene and wateracetonebutyl acetate) for extraction under varying operating conditions.
In this equation, f is the bivariate number concentration function, which takes into account the droplet diameter (d) and concentration (cy) as internal properties and space (z) and time (t) as external coordinates. Two velocity vectors are distinguished here: · velocity of droplets along internal coordinates (ζ = dζ/dt) and convective velocity along external coordinates (u). The diffusion operator (characterized by Dy) on the right-hand side is to take into account the nonideal flow behavior (deviation from plug flow), and the source term (S = SF + SB + SC) represents point volumetric source term due to droplet entry (SF), droplet breakage (SB), and droplet coalescence (SC). In the source term due to droplet entry (SF), the inflow of disperse phase is considered as a point source term at the point zd using the Dirac delta function δ. Thus, the equation for the source term SF is written as ! ! Q ind uf feed ð2Þ δðz zd Þ SF ¼ nF v̅ A
2. MATHEMATICAL MODELING The modeling of liquidliquid extraction columns aid to develop a computer simulation program that is capable of simulating the coupled hydrodynamic and mass-transfer behavior of the dispersed phase in extraction columns in order to avoid long and expensive pilot tests. Several attempts have been done to develop models for proper and reliable design of liquidliquid extraction columns and predict the influence of various operational and process parameters on the performance of such types of equipment without the need for a pilot-plant study.30,36 Blass and Zimmerman36 have developed a stage-wise model for the transient behavior of a sieve-plate extraction column, taking into account the backflow and assuming constant holdup. Hufnagl et al.37 evaluated a differential model for a K€uhni column. An empirical model for predicting the hydrodynamics in a pulsed sieve-plate column was proposed by Kumar and Hartland.38 Steiner et al.39 modeled a packed column using differential contact model without axial mixing. Flow models such as the dispersion or back-mixing model describe the nonideal flow, where one parameter accounts for all deviations from the ideal plug-flow behavior.31,40 These models are too simple to describe the real hydrodynamics, where one of the liquid phases is normally dispersed as droplets in the second continuous phase.41,42 Therefore, the influences of droplet movement, droplet interaction (breakage and coalescence), energy input (stirrer, pulsation), and mass transfer cannot be described satisfactorily. In order to tackle this problem, population balance models were proposed by various authors. Garg and Pratt43 developed a population balance model for a pulsed sieve-plate extraction column, taking into account experimentally determined values for droplet breakage and coalescence. Casamatta et al.44 proposed a population balance model, and Al Khani et al.45 applied this model for dynamic and steady-state simulations of a pulsed sieve-plate extraction column. Xiaojin et al.46 developed an improved dynamic combined model considering the influence of droplet size distribution. Recently, much work has been done in the population balance modeling in extraction columns.33,4751
In this equation, Q ind is the volumetric flow rate of the dispersed phase, A the cross-sectional area of the column, nF the normalized droplets number distribution density, f feed the feed distribution function, v the droplet volume, z the height of the column, and the inlet of the dispersed phase is zd. The source term SB describes droplet breakup events, which is given by
Multivariate Non-Equilibrium Population Balance Model. The coupled hydrodynamics and mass transfer in liquid extraction columns can be modeled by a nonhomogeneous bivariate population balance equation in one spatial domain:52
∂f þ ∇ 3 ½uf þ ∇ 3 ½ζ_f ¼ ∇ 3 ½Dy ∇f þ S ∂t
ð1Þ
SB ¼ ΓðvÞnðt, vÞ þ
Z v max v
Γðv, nÞβn ðv, v0 Þnðv, ξÞ δv ð3Þ
Here, Γ(ν) is the breakage frequency,and n(t, ν) is the droplet number density distribution; the daughter droplet distribution is βn(ν, ν0 ), and ν is the droplet volume. The source term SC describes droplet coalescence and is given by52 SC ¼ nðv, ξÞ þ
Z v v max vmin
ωðv, v v0 Þnðv v0 ; ξÞ δv0
1 Z vmax ωðv, v v0 Þnðv v0 Þnðv0 ; ξÞ δv0 2 v
ð4Þ
In this equation, ω(ν, νν0 ) is the coalescence frequency and νmax is the maximum droplet volume. By multiplying both sides of eq 1 by v(d) cy, and integrating with respect to cy (from zero to cy,max) and with respect to d (from dmin to dmax), one can get the mean solute concentration in the dispersed phase: ! Q iny c̅ yin ∂ðjy c̅ y Þ ∂ðjy c̅ y Þ ∂ ðuy jy c̅ y Þ Dy þ δðz zy Þ ¼ ∂z ∂t ∂z Ac þ
Z ∞Z c y, max 0
0
c_ y vðdÞfd, cy ðψÞ ∂d ∂cy
ð5Þ
In this equation, ϕy is the holdup of the dispersed phase, cy is the solute concentration in the dispersed phase. The first term on the right-hand side is the rate at which the droplets are entering the LLEC with volumetric flow rate Qin y (that is, perpendicular to the column cross-sectional area Ac, at a location zd with an initial solute concentration cin y , and is treated as a point source in space). The last term appearing in eq 5 is the total rate of solute transferred from the continuous to the dispersed phase, where the liquid droplets are treated as point sources. In this equation, the components of the vector ψ = [d cy z t] are those for the droplet 14122
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Table 1. Characteristic Droplet Diameters (dstab and d100) 1
a
column type
test system
af (cm s )
dstab (mm)
d100 (mm)
sieve-tray
w-a-t(d)
2.0
0.5
3.7
w-a-b(d)
2.0
0.3
3.0
w-a-t(d)
1.0
1.7
4.1
w-a-b(d)
1.0
2.0
3.8
w-a-t(d)
2.0
1.5
4.5
w-a-b(d)
2.0
0.8
3.1
w-a-t(d) w-a-b(d)
1.0 1.0
3.3 2.2
7.5 4.0
Table 2. Breakage Probability Constants (Ci) in Pulsed Columnsa column type
test system
packed
sieve-tray packed
a
a
Legend for systems used: w-a-t = wateracetonetoluene and w-a-b = wateracetonebutyl acetate.
internal coordinates (diameter and solute concentration), the external coordinate (column height) z, and the time t, where the velocity along the concentration coordinate (cy) is given as c_ y. Note that the dispersed phase velocity uy, relative to the walls of the column, is determined in terms of the slip velocity uo, with respect to the continuous phase.53 Conducting a component solute balance on the continuous phase leads to the solute concentration in the continuous phase cx equation:52 ∂ðjx cx Þ ∂ ∂ðjx cx Þ ðux jx cx Þ þ Dx ∂t ∂z ∂z ¼
Qxin cinx δðz zy Þ Ac
Z ∞Z c y, max 0
0
c_ y vðdÞfd, cy ðψÞ ∂d ∂cy ð6Þ
Note that the volume fraction of the continuous phase (ϕx) satisfies the following physical constraint: ϕx + ϕy = 1. The left-hand side of eq 6, as well as the first term on the right-hand side of the same equation, have the same interpretations as those given in eq 5, but with respect to the continuous phase. The last term appearing in eq 6 is the total rate of solute transferred from the continuous to the dispersed phase, where the liquid droplets are treated as point sources. Note that eq 1 is coupled to the solute balance in the continuous phase given by eqs 5 and 6 through the convective and source terms.
3. POPULATION BALANCE SUBMODELS The spatially distributed population balance equation (SDPBE) is general for any type of extraction column. However, what makes the equation specific is the internal geometry of the column, as reflected by the required correlations for hydrodynamics and mass transfer. Experimental correlations are used for the estimation of the turbulent energy dissipation and the slip velocities of the moving droplets along with interaction frequencies of breakage and coalescence. 3.1. Breakage frequency and daughter droplet distribution. For pulsed packed extraction column, the daughter droplet
distribution is assumed to follow the beta distribution function βn (d | d0 ), which is given by Bahmanyar and Slater:54 "
3 #ν 2 5 ! d d βn ðdjd Þ ¼ 3νðν 1Þ 1 0 d d0 6 0
ð7Þ
C1
C2
C3
C4
w-a-t(d)
1.98
0.08
1.39
0.80
w-a-b(d)
1.06
0.07
2.99
0.13
w-a-t(d) w-a-b(d)
3.81 2.49
0.61 0.27
1.11 0.95
3.47 1.77
Data taken from ref 35.
Table 3. Adjusted Coalescence Constant Values Used in the Simulation
w-a-b
w-a-t
Packed
Sieve-Plate
Column
Column
adjusted
af = 1
af = 2
af = 1
af = 2
constant, c
cm s1
cm s1
cm s1
cm s1
Q1
146.78
316.23
100.00
215.44
Q2
146.78
68.13
68.13
46.42
Q1
146.78
146.78
146.78
146.78
Q3
68.13
68.13
31.62
60.70
where d and d0 are characteristic droplet diameters, and ν is the average number of daughter droplets per breakage, which is given by 0 1:96 d ð8Þ 1 ν ¼ 2 þ 0:34 dstab Here, dstab is the stable droplet diameter, below which no more breakage takes place. For a pulsed sieve-plate extraction column, the experimental data of Henschke55 is used, where all parameters that influence the stable droplet diameter are considered. This includes breakage probability, number of daughter droplets per breakage, and daughter droplet distributions. According to Henschke,55 the mean number of daughter droplet is given by " # !ξ5 d 0 ξ4 af 1 þ 1 ð9Þ n̅ z ¼ 2:0 þ ξ3 ϕst υt dstab where af is the pulsation intensity, jst is the relative free crosssectional area of a sieve tray, υt is the terminal droplet velocity, and ξi are adjustable parameters. For a pulsed sieve-plate extraction column, the daughter droplet size distribution nx, after Henschke,55 is given by h pffiffiffi i 0:2 nz ¼ n̅ z 1 þ ð1 SÞ ð10Þ n̅ z 1:8 where S is a random number that is generated using the following distribution: " 3 #ðn̅ z 2Þ ð5=d0 Þ 3 d d d S ¼ 3n̅ z ðn̅ z 1Þ 1 0 d d0 d0 ð11Þ 14123
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Table 4. Velocity, Coalescence, and Mass-Transfer Model Parameters55 Velocity Model
a
Coalescence Model
Mass Transfer Model
test systema
dum (mm)
a15
a16
αum
αgr
ζ8
Dd (m2 s1)
CIP
b-w-a
2.97
2.24
1.82
8.0
8.0
1500
39.0 109
9687
t-w-a
7.10
1.52
4.50
8.0
8.0
2500
1.06 108
9445
Legend for test systems: b-w-a = butyl acetatewateracetone, and t-w-a = toluenewateracetone.
Table 5. Geometrical Data of Used Internals sieve-tray
Table 6. Operating Conditions for the WaterAcetone Toluene (w-a-t) Chemical Test System
Montz-Pak B1-350
diameter of sieve tray, diameter of holes,
volumetric surface area,
dh = 2 (mm)
aP = 350 m2 m3
rel. free cross-sectional area, 2
jst = 0.20 m m 2
height of compartment, hst = 100 mm
Cx,in
Cy,in
af
set
(L h )
(L h1)
(%)
(%)
(cm s1)
sieve-tray
Q1
40.0
48.0
5.44
0.36
1
Q3 Q1
61.3 40.0
74.1 48.0
5.52 5.35
0.15 0.40
1 2
Q3
61.3
74.1
5.45
0.41
2
Q1
40.0
48.0
5.73
0.00
1
Q3
61.3
74.5
5.69
0.00
1
Q1
40.0
48.0
5.84
0.61
2
Q3
61.3
74.5
5.89
0.60
2
diameter of a packing, DP = 79 mm
Dst = 79 (mm)
Q din
type
column
void fraction,
exp.
Qc 1
3
jP = 0.97 m m 3
height of a packing, hP = 100 mm
Here, dstab is the stable droplet diameter, below which no further droplet breakage takes place. The data for the stable droplet diazmeters for pulsed packed and sieve-tray columns under different operating conditions using the two standard test systems (wateracetonetoluene (w-a-t) and wateracetone butyl acetate (w-a-b)) are given in Table 1. The droplet breakage frequency and the daughter droplet distribution are correlated based on single droplet experiments. The droplet breakage frequency used in the simulation is given by Garthe:35 9 8 > ðd dstab Þ C3 > > > > > = < ðd100 dstab Þ C2 ð12Þ PB ðdÞ ¼ C1 πaf > ðd dstab Þ C3 > > > > > ; :C4 þ ðd100 dstab Þ where Ci are adjustable parameters for the breakage probability and d100 is the characteristic droplet diameter due to a breakage probability of 100%. The parameter πaf is a dimensionless number takes the influence of the pulsation intensity (pulse amplitude pulse frequency) on the breakage probability into account; it has been given by Garthe35 as !1=3 F2c πaf ¼ af ð13Þ μc ΔFg In this equation, Fc is the density of the continuous phase, μc the viscosity of the continuous phase, g the gravitational acceleration constant, and ΔF the density difference. This breakage frequency describes the breakage in a packed and sieve-tray column with only one set of constant parameters for a given liquidliquid system. The constants Ci in eq 12 are given in Table 2 for structured packing of type Montz 350 and for a sieveplate column. To predict the breakage probability in Montz packing and sieve tray compartments using eq 12, the characteristic droplet diameters d100 and dstab must be experimentally determined for each pulsation intensity.35
packed
3.2. The Coalescence Frequency. Coalescence rate analysis and modeling has been given greater attention. Many authors proposed phenomenological models based on the concept of collision frequency and coalescence efficiency, where the latter is assumed to be a function of the two coalesce droplets; an intensive review on the coalescence models has been provided by Liao and Lucas,56 whose report supports this assumption. Thus, the coalescence model after Henschke55 is modified to take into account the size of the two coalesce droplets. In this work, the modified coalescence model is used in the simulation of both pulsed (packed and sieve-plate) columns: 1=6
jy σ 1=3 Hcd ðΔFgÞ1=2 ωðd, d0 Þ ¼ c vðdÞn þ vðd0 Þn ξ8 μc d1=3
ð14Þ
In this modified coalescence frequency ω(d, d0 ), d, d0 represents the droplet diameter, υ is the droplet volume, ϕy is the dispersed-phase holdup, σ is the mixture interfacial tension, and Hcd is the Hamaker constant (the Hammaker constant value used in the simulation is 10 1020). The adjustable parameter ξ8 was fitted to experimental data for the two standard EFCE test systems: water acetonetoluene (w-a-t) and wateracetonebutyl acetate (w-a-b) and is given by Henschke.55 The values for this parameter are 2500 and 1500 for the first and second system, respectively. Adjusted constant c values that were used to fit the column hydrodynamics to the experimental data35 are listed in Table 3. 3.3. The Terminal Droplet Velocity. The hydrodynamic behavior of droplets during the movement in a surrounding continuous phase is dependent on the droplet diameter. Therefore, four different droplet velocity boundaries can be distinguished. The first of the droplet diameters identifies small droplets with rigid phase boundaries, which have no internal circulation. These droplets are moving as rigid spheres. For larger diameters, there are shear forces at the droplet surface, so an internal circulation begins to occur. Because of the circulation of the droplets, they move faster than rigid spheres. For droplets with even larger 14124
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Table 7. Operating Conditions for WaterAcetoneButyl Acetate (w-a-b) Chemical Test System Q din
Cx,in
Cy,in
af
type
set
(L h )
(L h1)
(%)
(%)
(cm s1)
sieve tray
Q1 Q2
40.0 60.0
48.0 72.0
5.17 5.46
0.16 0.00
1 1
Q1
40.0
48.0
5.35
0.00
2
Q2
60.0
72.0
5.15
0.00
2
Q1
40.0
48.0
5.22
0.00
1
Q2
60.0
72.0
5.1
0.00
1
Q1
40.0
48.0
5.61
0.00
2
Q2
60.0
72.0
5.21
0.00
2
column
packed
exp.
Qc 1
Figure 2. Simulated holdup profiles along the pulsed sieve-plate column height, as a function of pulsation intensity, compared to the experimental data,35 using the w-a-t test system: in the upper panel, the volumetric flow rate is Q1, whereas in the lower panel, the volumetric flow rate is Q3 (see Table 5).
Henschke55 proposed a terminal velocity model utilizing different droplet velocity models based on their droplet diameter. In his model, the spherical, oscillating, and deformed droplet velocity models are combined into a single model utilizing a crossover function, which is applicable over the entire diameter range. The resulting droplet velocity model is given by55 vt ¼
Figure 1. Simulated mean droplet diameter along the pulsed sieve-plate column height, as a function of pulsation intensity, compared to the experimental data,35 using the w-a-t test system. In the upper panel, the volumetric flow rate is Q1; in the lower panel, the volumetric flow rate is Q3 (see Table 5).
diameters, they lose their spherical shape; at the same time, the droplets begin to oscillate. With further increases in the droplet diameter, the droplet deforms and finally breaks up. The velocity of the droplets decreases as a result of the increasing flow resistance.55,57,58
voscillating or deformed vspherical a16 16 ðvoscillating or deformed þ vaspherical Þ1=a16
ð16Þ
where voscillating or deformed is a smooth transition from oscillating to deformed droplets and vspherical is the velocity model for spherical droplets.55 The exponents α15 and α16 describe the sharpness of the transition between submodels, where the parameters a15 and α16 are adjusted to measured values. The parameters in this velocity model can be fitted to the experimental data for certain test systems, based on single droplet experiments.55 For the description of the droplet motion in the pulsed structured packing column, a correlation for the slowing factor developed by Garthe35 is applied, which is given by 0:566 0:769 0:184 kv ¼ 0:077π0:138 πd πσ ð1 þ πaf Þ0:08 HPk π aPk
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ð17Þ
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Figure 3. Simulated mean droplet diameter along the pulsed sieve-plate column height, as a function of volumetric flow rate, compared to the experimental data,35 using the w-a-t test system. (In the upper panel, the pulsation intensity is 1; in the lower panel, it is 2.) (See Table 6.)
In the above equation πσ is the dimensionless interfacial tension (πσ = σ(F2c /μ4c ΔFg)1/3), πhp the dimensionless height of a packing (πhp = hp(FcΔFg/μ2c )1/3), πap the dimensionless volumetric surface area of a packing (πap = ap(μ2c /FcΔFg)1/3), and πd the dimensionless droplet diameter (πd = d(FcΔFg/μ2c )1/3). Based on the analysis of experimental results, Garthe35 developed a new correlation for the slowing factor in a pulsed sieveplate column: "
kv ¼ 1:406 ϕ0:145 π0:028 σ st
# 1:134 d 2:161 exp 0:129 ð1 ϕst Þ dh
ð18Þ where dh is the diameter of the holes in the sieve plates. 3.4. Mass Transfer. The mass transfer fluxes in the LLECMOD program are calculated based on the two-film theory. The two individual mass-transfer coefficients (kx and ky) are defined separately for the continuous and dispersed phases. The masstransfer model used in this work is taken from the work of Henschke.55 In this model, the author followed the idea of Handlos and Baron59 to develop a semiempirical model, which describes the mass transfer inside a droplet based on the diffusion and mass-transfer-induced turbulence. Their assumption used in this model is that mass transfer induces the interfacial vortices.
Figure 4. Simulated holdup profiles along the pulsed sieve-plate column height, as a function of volumetric flow rate, compared to the experimental data,35 using the w-a-t test system: in the upper panel, the pulsation intensity is 1 and 2 in the lower panel (see Table 6).
They define a transport coefficient, which is added to the molecular diffusion coefficient. It contains the so-called instability coefficient CIP as system-specific parameter and is based on the equation reported by Handlos and Baron.59 Table 4 depicts the parameters of Henschke’s models used in the simulations.55
4. NUMERICAL SOLUTION USING LLECMOD LLECMOD (LiquidLiquid Extraction Column Module) simulations can now be used successfully for different types of extraction columns, including agitated (RDC and K€uhni) and pulsed (sieve-plate and packed) columns. The design of LLECMOD is flexible in such a way that allows the user to define droplet terminal velocity, energy dissipation, axial dispersion, breakage and coalescence frequencies, and the internal column geometry. The correlation parameters that are obtained based on single droplet and droplet swarm experiments are considered in a modularized structure for the simulation program. The model equations are solved using an optimized and efficient numerical algorithm developed by Attarakih et al.,52 based on the generalized fixed-pivot technique60 and a first-order upwind scheme based on the finite-volume method. This is utilized successfully for the simulation of coupled hydrodynamics and mass transfer for general liquidliquid extraction columns with 14126
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Figure 5. Simulated mean droplet diameter along the pulsed packed column height, as a function of pulsation intensity, compared to the experimental data,35 using the w-a-b test system (in the upper panel, the volumetric flow rate is Q1 and in the lower panel, the volumetric flow rate isQ2) (see Table 6).
reasonable CPU time requirements. However, using the available commercial software, which based on the Galerkin method, the timeframe needed for hydrodynamics simulations is on the order of minutes, but that for mass-transfer simulations is on the order of hours. The complete mathematical model described above is programmed using Visual Digital FORTRAN and then integrated into LLECMOD. To facilitate the data input and output, a graphical user interface was designed. The graphical interface of the LLECMOD program contains the main input window and subwindows for parameters and correlations input. The basic feature of this program33 is to provide an easy tool for the simulation of coupled hydrodynamics and mass transfer in liquid liquid extraction columns, based on the population balance approach for both transient and steady-state conditions through an interactive windows input dialogue.
5. RESULTS AND DISCUSSION In this section, a sample problem is considered to illustrate the basic features of LLECMOD. For this purpose, a pilot-plant-scale pulsed column is considered. The dimensions of this column are as follows: column height, H = 4.40 m; inlet of the dispersed phase, zy = 0.85 m; inlet of the continuous phase, zx = 3.80 m,
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Figure 6. Simulated holdup profiles along the pulsed packed column height, as a function of pulsation intensity, compared to the experimental data35 using the w-a-b test system (in the upper panel, the volumetric flow rate is Q1, and in the lower panel, the volumetric flow rate is Q2) (see Table 6).
column diameter, dcol = 0.08 m. Two internal types are installed inside the active part of the column: sieve-tray and packing (Montz-Pak B1-350). The geometrical data of these internals are given in Table 5. The test systems published by the EFCE61 (wateracetone butyl acetate and wateracetonetoluene) are used in the simulations. These systems have a wide range of physical properties and, therefore, they permit the prediction of extraction columns for other systems. To completely specify the model, the inlet feed is normally distributed where the mean and standard deviation are dependent upon the test system used in the simulation and on the operating conditions (volumetric flow rate and pulsation intensity). The direction of mass transfer is from the continuous phase to the dispersed phase. The inlet solute concentrations in the continuous and dispersed phases are taken for the wateracetonetoluene test system and are shown in Table 6; those for the wateracetonebutyl acetate test system are shown in Table 7. The pulsation intensities and the volumetric flow rates used in the simulations are given in Tables 6 and 7. The performance of nonagitated counter-current extraction columns is often poor;62 a pulsation provides turbulence throughout the column and therefore improves the extraction efficiency. 14127
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Figure 7. Simulated mean droplet diameter along the pulsed packed column height, as a function of pulsation intensity, compared to the experimental data,35 using the w-a-t test system (in the upper panel, the volumetric flow rate is Q1, and in the lower panel, the volumetric flow rate is Q3) (see Table 5).
Consequently, the influence of pulsation intensity and volumetric flow rates on the performance of pulsed (sieve-plate and packed) extraction columns were investigated. 5.1. Steady-State Column Hydrodynamics: Pulsed SievePlate Column. Figure 1 shows the variation of the mean droplet diameter along the pulsed sieve-plate column height, compared to the experimental data for the chemical system (water acetonetoluene) using two different volumetric flow rates, as a function of pulsation intensity. In the upper panel, the set Q1 is used, and Q3 is used in the lower panel (see Table 6); from this figure, it is clear that the higher the pulsation intensity, the smaller is the droplet diameter. Good agreement between the experimental and simulated profiles is achieved for both cases. According to the definition of the dispersed phase holdup, when the dispersed phase flow rate and the number of dispersed phase droplets increase, the value of holdup will increase. By increasing the continuous phase flow rate, the drag force between the dispersed droplets and continuous phase increases, so the droplets movement will be limited and the residence time—and, consequently, the holdup—will increase. Figure 2 depicts the variation of the dispersed phase holdup along the pulsed sieve-plate column height and compared to the experimental data of Garthe35 for the wateracetonetoluene test system, where, by increasing the pulsation intensity, the holdup increased. In this figure, the effect
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Figure 8. Simulated holdup profiles along the pulsed packed column height, as a function of pulsation intensity, compared to the experimental data,35 using the w-a-t test system (in the upper panel, the volumetric flow rate is Q1, and, in the lower panel, the volumetric flow rate is Q3) (see Table 5).
of pulsation intensity on the holdup is investigated, where two different flow rates are used in the simulation: Q1 in the upper panel and Q3 in the lower panel. From this figure, the effect of pulsation intensity at higher volumetric flow (Q3) is less profound than at lower volumetric flow rates (Q1). The effect of the dispersed phase flow rate on the performance of the extraction column is found to be not appreciable, as shown in Figure 3 (the lower panel). Since the pulsation intensities are reasonably high, the effect of flow rate of the two phases on the performance of the extraction column had a minor contribution. A comparison between the simulated mean droplet diameter along the pulsed sieve-plate column height and the experimental data35 is shown in Figure 3. The test system used here is w-a-t, to show the effect of the volumetric flow rates of both phases under two different pulsation intensities: af = 1 cm s1 in the upper panel and af = 2 cm s1 in the lower panel. Again, good agreement is achieved. Figure 4 shows the effect of changing the flow rates on the holdup profiles along the pulsed sieve-plate column for the w-a-t test system; the simulation is conducting using two different pulsation intensities (1 and 2 cm s1 in the upper and lower panels, respectively). However, a comparison between the experimental and simulated results reveals good agreement. 14128
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Figure 9. Simulated mean droplet diameter along the pulsed packed column height, as a function of volumetric flow rate, compared to the experimental data,35 using different chemical test systems and the same pulsation intensity (see Tables 5 and 6).
5.2. Steady-State Column Hydrodynamics: Pulsed Packed Column. Figure 5 depicts the variation of the mean droplet dia-
meter along the pulsed packed column height to show the effect of pulsation intensity on the performance of the pulsed packed extraction columns, using two different volumetric flow rates. In this figure, the volumetric flow rate used in the upper panel is Q1, while a volumetric flow rate of Q2 is used in the simulation of the lower panel (see Table 7). The chemical test system used in the simulation is w-a-b; the effect of volumetric flow rates on the variation of mean droplet diameter is minor, especially at relative high pulsation intensities. It is clear that the higher pulsation intensity results in a smaller droplet diameter. As mentioned previously, according to the definition of the dispersed-phase holdup, when the dispersed-phase flow rate and the number of dispersed-phase droplets increase, the value of the holdup will increase. By increasing the continuous phase flow rate, the drag force between the dispersed droplets and the continuous phase increases, so the droplets movement will be limited and the residence time—and, consequently, the holdup—will increase. Figure 6 depicts the variation in the dispersed-phase holdup along the pulsed packed column height, compared to the experimental data35 for the w-a-b test system, where very good agreement is achieved. This figure shows the effect of pulsation intensity on the holdup; the results for two different flow rates are shown (Q1 in the upper panel and Q2 in the lower panel). From
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Figure 10. Simulated holdup profiles along the pulsed packed column height, as a function of volumetric flow rate, compared to the experimental data,35 using different chemical test systems and the same pulsation intensity (see Tables 5 and 6).
the two panels, as the pulsation intensity increased, the droplet diameter decreased and, therefore, the holdup increased. The performance of extraction columns is expected to vary with interfacial tension. To show the effect of interfacial tension on the performance of extraction columns, the test system with low interfacial tension (wateracetonebutyl acetate) is used in the simulations. It is known that, for higher interfacial tension test systems, the size of the droplets is larger than the droplet size in the lower interfacial tension test systems, which results in a decrease in their residence time in the column. Finally, the slip velocities increase and, consequently, the value of dispersed phase holdup will decrease and the column operates in a more-stable manner. According to Figures 5 and 7, it can be concluded that the mean droplet diameter of the w-a-b system (low interfacial tension) is smaller than that of the w-a-t (high interfacial tension) system, because of the interfacial tension effect. At low pulsation intensity, the breakup of droplets is controlled by the ratio of buoyancy and interfacial tension forces.19,63 As can be seen from Figures 5 and 6, compared to Figures 7 and 8, the smaller the droplet diameter, the higher the holdup. Figures 6 and 8 show the variation of the dispersed-phase holdup along the pulsed packed column height. Figure 6 shows the simulation for the w-a-b test system. Meanwhile, Figure 8 shows the simulation for the w-a-t test system. These figures show the effect of pulsation intensity on the holdup profiles. 14129
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Figure 11. Simulated solute concentration profiles in both phases along the pulsed packed column height, as a function of pulsation intensity, compared to the experimental data,35 using the w-a-b test system (in the upper panel, the volumetric flow rate is Q1, and in the lower panel, the volumetric flow rate is Q2) (see Table 6).
Two different test systems with different interfacial tensions are used to show the effect of the interfacial tension on the performance of a pulsed extraction column. Figure 5 shows the effect of pulsation intensity on the mean droplet diameter along the packed extraction column for the w-a-b test system, as a function of the volumetric flow rate, and to show the effect of interfacial tension. Figure 7 depicts the effect of pulsation intensity on mean droplet diameter along the packed column height for the w-a-t test system. A comparison of Figures 5 and 7 reveals that the pulsation intensity has a more-profound effect on the test system of higher interfacial tension. The effect of pulsation intensity on the Sauter mean diameter is shown in Figures 5 and 7. As can be seen, the Sauter mean diameter decreases with an increase in the pulsation intensity in the two chemical systems studied. As a conclusion, it is apparent from the simulation results plotted in Figures 5 and 7 that the droplet size decreases with pulsation intensity, as well as along the column height. Figure 9 depicts the variation of mean droplet diameter along the packed column height to show the effect of the volumetric flow rates on the performance of the pulsed extraction columns. In this figure, the test system used in the upper panel is water acetonebutyl acetate, while wateracetonetoluene is the test system used in the simulation in the lower panel. From this
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Figure 12. Simulated solute concentration profiles in both phases along the pulsed packed column height, as a function of pulsation intensity, compared to the experimental data,35 using the w-a-t test system (in the upper panel, the volumetric flow rate is Q1, and in the lower panel, the volumetric flow rate is Q3) (see Table 5).
figure, and for both test systems, the effect of the volumetric flow rate on the variation of mean droplet diameter is minor, especially at relative high pulsation intensities. In conclusion, it is apparent from the data plotted in Figures 3 and 9 that the changes in feed rate have less effect on the mean droplet diameter and, accordingly, on the performance of extraction columns when pulsation is used, rather than the usual column operation. Figure 10 shows the variation of holdup profiles along the pulsed packed column height. In the upper panel of this figure, the simulated and experimental holdup results are compared for the w-a-b test system. Meanwhile, the lower panel shows the comparison for the w-a-t test system. In both cases, an excellent agreement are achieved. A comparison between the two panels reveals that the variation of volumetric flow rates shows more profound effect on the test system with low interfacial tension, where the higher interfacial tension test system is more affected by the pulsation intensity. The pulsed (packed and sieve-plate) columns hydrodynamics (holdup and droplet diameter) is affected by the pulsation intensity and by changing in the flow rates as well. The hydrodynamics behavior of the dispersed phase against the pulsation intensity is shown in Figures 2,6, and 8 (holdup), and in Figures 1,5, and 7 (mean droplet diameter). According to these figures, it can be found that the droplet mean diameter decreased by increasing the pulsation intensity, and, hence, the holdup increased. The increase 14130
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Figure 13. Simulated solute concentration profiles in both phases along the pulsed packed column height, as a function of volumetric flow rates, compared to the experimental data,35 using different chemical test systems and the same pulsation intensity (see Tables 5 and 6).
in the pulsation intensity leads to higher shear stress and to intense droplet breaking (smaller droplets). It appears that the number of droplets in the column increases and, consequently, the values of holdup will increase. The droplet size and the degree of turbulence are dependent on the pulsation intensity; however, high pulsation intensities can increase axial mixing and reduce the extraction efficiency. Thus, the pulsation intensity can be used to control the droplet size, dispersed phase holdup, and, consequently, the performance of the extraction columns. The hydrodynamics behavior of the dispersed phase is varying with interfacial tension. Figures 6 and 8 show the effect of interfacial tension on the dispersed phase holdup; meanwhile, Figures 5 and 7 show the effect of interfacial tension on the droplet diameter. It is known that, for higher interfacial tension, the size of droplets is larger; thus, their residence time in the column will decrease. Finally, the slip velocities increasing and, consequently, the value of dispersed phase holdup will be decreased and the column operates in a more-stable manner. According to Figures 9 and 10, it can be concluded that the holdup of the w-a-t test system (large droplet size) is lower than the holdup of the w-a-b test system (small droplet size), because of interfacial tension. Figures 1, 3, 5, 7, and 9 show the mean droplet diameter profiles at different pulsation intensities; Figures 2, 4, 6, 8, and 10 show the holdup profiles of dispersed phase at different pulsation intensities. The free opening area and flow paths through the plates affect the
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dependency and magnitude of holdups on pulsation intensity. However, the minimum holdup exists at a certain pulsation intensity that is sometimes called the transition frequency at the corresponding amplitude, which is reached faster for perforated plates of small free opening area. Comparing Figures 1, 2, 5, 6, 7, and 8 with Figures 3, 4, 9, and 10 reveals that the effect of changing the flow rates of the two phases on hydrodynamics behavior (mean droplet diameter and holdup) is greater than that of changing the pulsation intensities. The hydrodynamics behavior, as a function of the volumetric flow rate, is shown in Figures 3, 4, 9, and 10. However, the effect of pulsation intensity on hydrodynamics behavior is given in Figures 1, 2, 5, 6, 7, and 8. In conclusion, it is apparent, from the above results, that the test systems with high interfacial tension (wateracetone toluene) presented the most significant changes in column performance with the variation of pulsation intensity. In contrast to this, the chemical test system wateracetonebutyl acetate presented the most significant changes in column performance with the variation of volumetric flow rate. 5.3. Steady-State Mass-Transfer Profiles: Pulsed (Packed and Sieve-Plate) Columns. The rate of mass transfer of a solute from one phase to the other in extraction columns is dependent mainly on the interfacial area between the two phases, the difference in the solute concentration in the two phases, the masstransfer coefficient, and the physical properties of the test system. A series of simulation runs were conducted to determine the best operating conditions and attainable efficiencies for a pulsed extraction column. All the simulation in this work was carried out with the mass-transfer direction being from the continuous phase to the dispersed phase. When the direction of mass transfer is from the continuous phase to the dispersed phase, the solute concentration in the wake of the droplet is larger than that at the top of the droplet; the consequence of the resulting interfacial tension gradient is that the interface moves opposite to the direction of the inner circulation generated inside the droplet. Therefore, it is expected that the interface deformations due to this mass-transfer direction enhance the droplet deformation, that is, the break-up process.25,64 Moreover, during solute transfer from the continuous phase to the dispersed phase, solute equilibrium is quickly established between the droplets and the continuous phase, whereas solute transfer continues over the rest of the droplet surface. This creates an interfacial tension gradient, which opposes the forces causing drainage in the film, thus reducing coalescence. Consequently, the droplet size becomes smaller than that in the case of no mass transfer and the values of holdup become higher and the column operates in a more unstable manner. Figures 11 and 12 depict the effect of pulsation intensity on the mass-transfer rate. The increase in pulsation intensity results in intense droplet breaking and, consequently, the Sauter mean droplet diameter will decrease. A decrease in relative velocity between the dispersed phase and the continuous phase results in an increase in the number of droplets in the column and, consequently, the values of holdup will also increase. Therefore, it can be concluded that the value of interfacial area (and, consequently, mass-transfer rate) increases with both effects.65 Both the simulation and experimental data from both systems (especially wateracetonebutyl acetate) shows that, at high pulsation intensity, increased entrainment and poor extraction efficiency is observed, with the production of fine dispersed droplets. Figures 11 and 12 show the simulated and experimental solute concentration profiles, as functions of the pulsed packed column height in both phases. The simulations are carried out at two different sets of volumetric flow rates. 14131
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operated near the flooding point to have a coalesced dispersedphase layer at each plate.
Figure 14. Simulated solute concentration profiles in both phases along the sieve-plate column height, as a function of volumetric flow rates, compared to the experimental data,35 using different chemical test systems and the same pulsation intensity (see Tables 5 and 6).
The effect of pulsation intensity on the mass-transfer rate of the w-a-t test system (high interfacial tension) is larger than that of the w-a-b test system (lower interfacial tension), because breakup of the dispersed-phase droplets into smaller ones is limited for the latter system, because of its lower interfacial tension. Figure 13 shows the simulated and experimental solute concentration profiles, as a function of the pulsed packed column height in both phases. This figure shows the effect of changing the volumetric flow rates on the mass-transfer profiles. The agreement between the simulation and experiment is excellent for both test systems. Figure 14 depicts the simulated and experimental solute concentration profiles in both phases along the pulsed sieve-plate column height. This figure shows the effect of changing the volumetric flow rates on the mass-transfer profiles. The agreement between the simulation and experiment is good for both test systems. In conclusion, there is an optimum pulsation intensity to obtain the best improvement in efficiency. These results lead to the conclusion that the best operation of pulsed (sieve-plate and packed) columns is obtained with a combination of high holdup and sufficient turbulence.66,67 Plates with smaller holes appear best for both effects. Sufficient pulsation intensity must be supplied for turbulence, and the feed rates must be maintained at a sufficiently high level for sufficient holdup. The column must be
’ CONCLUSIONS A comprehensive model for the dynamic and steady state simulation tool of extraction in pulsed (packed and sieve-plate) columns is developed. The model has been successfully validated against experimental data. This model is based on a nonequilibrium bivariate population balance model to describe the complex (steady-state and dynamic) coupled hydrodynamics and mass-transfer phenomena in extraction columns. In this work, the pulsed/unpulsed (packed and sieve-plate) are considered. Now, the present version of LiquidLiquid Extraction Column Module (LLECMOD) offers the simulation of pulsed/unpulsed (sieve-plate and packed) columns and the stirred (RDC and K€uhni) columns. The steady-state performance of the abovementioned columns is studied using a detailed population balance framework as an alternative to the commonly applied dispersion and back-mixing models. The model has been validated against the experimental data and good agreements were achieved. The model has been found to predict the real behavior of the agitated and nonagitated extraction columns accurately. Optimum operating conditions can be obtained by varying the pulsation intensity. Greater efficiency can be obtained with the proper pulsation intensity. The variation of the dispersed-phase holdup with height and the effect of operating conditions were studied experimentally in this work. The results were compared with similar studies carried out in other extraction columns. The purpose of this paper is to give a complete analysis of influence of operating conditions on the holdup profile of a pulsed (packed and sieve-plate) columns. The results show that, as the height increases, the holdup increases. As the flow rate of each phase increases, the holdup increases but the effect of the dispersed-phase flow rate is more than that of the continuous-phase flow rate. Pulse velocity has direct influence on holdup, but, at high pulse velocity, they have an inverse relationship to each other. In these studies, the systems with high interfacial tension (water acetonetoluene) presented the most significant changes in column performance with the variation of pulsation intensity, while the system with low interfacial tension presented the most significant changes in column performance with variation in the volumetric flow rates. ’ AUTHOR INFORMATION Corresponding Author
*Tel.: +(49) 631 205 2414. Fax: +(49) 631 205 2119. E-mail: bart@ mv.uni-kl.de.
’ ACKNOWLEDGMENT The authors acknowledge the financial support from DFG (Deutsche Forschungsgemeinschaft)Bonn, DAAD (Deutscher Akademischer Austauschdienst)Bonn, and the Federal State Research Centre of Mathematical and Computational Modelling (CM)2TU Kaiserslautern. ’ NOMENCLATURE a = pulsation amplitude (m) Ac = column cross-sectional area (m2) af = pulsation intensity (m s1) 14132
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Industrial & Engineering Chemistry Research a15, a16 = adjustable parameters aPK = volumetric surface area of a packing (m2 m3) Ci = constant parameter CIP = interface instability parameter cx, cy = solute concentration (continuous and dispersed phase) (kg m3) cx,in, cy,in = inlet solute concentration (continuous and dispersed phase) (kg m3) Dx, Dy = axial dispersion coefficient (continuous and dispersed phase) (m2 s1) dcol = column diameter (m) d, dm, d0 , d00 = droplet diameters (m) dmin, dmax = minimum and maximum droplet diameters (m) d100 = characteristic droplet diameter due to a breakage probability of 100% (m) dstab = stable droplet diameter (m) f = pulsation frequency (s1) fd,cy ∂d ∂cy = number of droplets having concentration cy and diameter d in the range [d,d + ∂d] [cy,cy + ∂cy] g(d, d0 ) = daughter droplet distribution based on droplet number (m1) h = coalescence frequency (s1) H = column and single compartment heights (m) Hcd = Hamaker coefficient (Nm) HPK = height of a packing (m) kv = slowing factor Koy = overall mass-transfer coefficient (m s1) K oy = average overall mass-transfer coefficient (m s1) kx, ky = individual mass transfer coefficients (continuous and dispersed phase) (m s1) Nz = number of daughter droplets N z = average number of daughter droplets Pr, PB = breakage probability PC = coalescence probability Qbot = total flow rate at bottom of the column (m3 s1) Qx,in, Qy,in = inlet flow rate (continuous and dispersed phase) (m3 s1) Qtop = dispersed phase flow rate at top of the column (m3 s1) t = time (s) ux, uy = relative velocity (continuous and dispersed phase) (m s1) ur = relative droplet (slip) velocity (m s1) ut = terminal droplet velocity (m s1) z = spatial coordinate (m) zd, zy = dispersed feed inlet (m) zc, zx = continuous phase inlet (m) Greek Symbols
βn = daughter droplet distribution based on droplet number (m1) Γ = droplet breakage frequency (s1) Δt = time interval (s) ε = energy dissipation (m2 s3) ζ = time and space vector λ = coalescence efficiency μx, μy = viscosity values (continuous and dispersed phase) (kg m1 s1) Fx, Fy = density values (continuous and dispersed phase) (kg m3) σ = interfacial tension (N m1) Σ = standard deviation (m) v, v0 = droplet volumes (m3) vmin, vmax = minimum and maximum droplet volumes (m3) ξi = parameter jx, jy = holdup (continuous and dispersed phase)
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ϑ(d0 ) = mean number of daughter droplets from mother droplet of diameter d0 γ = source term ϕst = relative free cross-sectional area of a sieve-tray plate (m2 m2) ω = coalescence frequency (m3 s1) Dimensionless Numbers
πaf = pulsation intensity πap = volumetric surface area of a packing πd = droplet diameter πhp = height of a packing πσ = interfacial tension
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