- Email: [email protected]
Modelling the Mixing and Dissolution Kinetics of Partially Miscible Liquids
MODELLING THE MIXING AND DISSOLUTION KINETICS OF PARTIALLY MISCIBLE LIQUIDS S. Ibemere and S. Kresta Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Canada.
Abstract: The problems of liquid–liquid extraction and solids dissolution in stirred tanks are classic mass transfer problems which have been studied extensively in the literature. An analogous problem, that of dissolution of pure drops close to their solubility limit, has been almost completely neglected. This problem has practical application in the dissolution and dispersion of small amounts of surface active additives, particularly in the water treatment industry and in oil field applications where the cost of the chemical additive is one of the major process costs. The question is also of interest for some chemical reactions, such as the third Bourne reaction (diethylmalonate in water) and the hydrolysis of acetic anhydride to acetic acid. In this paper, the drop size distribution and solute concentration in the bulk are measured throughout the dissolution period and the dissolution process is successfully modelled using an Eulerian –Lagrangian approach. The validated model is used to identify the key variables driving the dissolution rate. The approach to saturation, the impeller rotational speed, and the continuous phase viscosity all play an important role. When the solute is injected close to the impeller, the role played by surface tension is surprisingly small. Keywords: liquid dissolution; multi-mechanism model; mixing; liquid– liquid system; mass transfer.
INTRODUCTION
Correspondence to: Professor S. Kresta, Department of Chemical and Materials Engineering, University of Alberta, Edmonton, T6G 2G6, Canada. E-mail: [email protected]
DOI: 10.1205/cherd06185 0263–8762/07/ $30.00 þ 0.00 Chemical Engineering Research and Design Trans IChemE, Part A, May 2007 # 2007 Institution of Chemical Engineers
extremes, there is a fixed solubility limit and two liquids are partially miscible. In this work, we are interested in liquid –liquid systems which must operate close to their solubility limit, and particularly in systems where this solubility limit occurs at fairly low concentrations of the minor component. In these systems, the minor phase is first dispersed as drops, and then completely dissolves over time. The drop size distribution and the concentration of solute in the continuous phase both change over time. Early models treated turbulent conditions in stirred tanks as homogenous (Park and Blair, 1975; Hsia and Tavlarides, 1980, 1983; Skelland and Kanel, 1992); however, it is well known that the turbulence intensity is typically 100 higher close to the impeller than in the bulk of the tank. Consequently, models composed of several compartments (zones) have evolved. Some authors have modelled the impeller region and the rest of the tank separately using a two zone model (Coulaloglou and Tavlarides, 1977; Tsouris and Tavlarides, 1994; Maggioris et al., 2000) while others have used pre-selected volumes with dissipation energy and volumetric flow rates determined using computational fluid dynamics (CFD) analysis and/or experimental data (Baldyga et al., 1995; Alopaeus et al., 1999; Kresta et al., 2005).
The problem of dissolution kinetics in liquid– liquid systems has long been recognized by static mixer manufacturers and by surface chemists. In the first case, field engineers report ‘stranding’ of a completely soluble liquid as it passes through a static mixer; and in the second case, surface chemists insist on a mixing period of 24 h in very small shaker flasks when determining solubility limits of long chain polymer surfactants. Recent studies on partially miscible liquid– liquid systems (Hemsing, 2001; Kennedy, 2003) have shown dissolution times that are 50 –75 times longer than the blend time. Chemical engineers have generally focused on the limits of blend time for completely miscible liquids (Grenville, 1992), on the mass transfer of a solute between phases in liquid –liquid extraction, and on drop size distributions in completely immiscible liquids. In order to approach the problem of dissolution kinetics of a pure liquid, we first define the class of liquid–liquid systems of interest. Liquids which are mutually soluble at all concentrations have no solubility limit. They are completely miscible. Conversely, immiscible liquids (e.g., the two carrier phases in liquid –liquid extraction) approach zero mutual solubility. Between these two 710
Vol 85 (A5) 710–720
MODELLING THE MIXING AND DISSOLUTION KINETICS OF PARTIALLY MISCIBLE LIQUIDS The dispersed drop population is commonly modeled using either population balance models or simulation techniques such as the Monte Carlo method (Spielman and Levenspiel, 1965), or the quiescence interval method (Shah et al., 1977). Due to the difficulty of accurately modelling the simultaneous drop–drop interactions (coalescence), drop –eddy interaction (breakup) and solute mass transfer, most models eliminate one of these mechanisms. Models developed to predict the drop size distribution in liquid –liquid dispersions typically neglect the effects of mass transfer (Alopaeus et al., 1999; Valentas et al., 1966; Valentas and Amundson, 1966; Ramkrishna, 1974; Bajpai et al., 1976). Other models incorporating solute transfer (Curl, 1963; Bayens and Laurence, 1969; Jeon and Lee, 1986; Skelland and Kanel, 1992) eliminate coalescence. Full reviews of population balances can be found in Attarakih et al. (2004) and Ramkrishna (2000). In this work, coalescence is neglected, but the the effects of both breakup and mass transfer on the drop size distribution are modelled directly. The model provides a way to identify the variables which have the biggest impact on the dissolution rate, and to estimate the range of dissolution times that might be expected for various solute/solvent pairs.
711
the tank volume is divided into five mixing zones: the impeller swept volume; three zones in the impeller discharge region; and the bulk of the tank, as shown in Figure 1(a). The distribution of the energy dissipation over each of these regions is very sensitive to the location of the mixing zones. Data reported in the literature shows a range of results, primarily because the measurements use different specifications for the control volumes and somewhat different ways of calculating the dissipation. Jaworski and Fort (1991) measured axial velocity and pressure profiles using a three-hole Pitot tube and reported that the power was distributed as follows: 32% in the impeller region; 54% in the region below the impeller; and 14% in the remaining volume of the tank. Macro-scale energy balances reported by Zhou and Kresta (1997) using LDA gave: 52% in the impeller region; 23.1% in the impeller discharge region; and 24.9% in the remaining volume of the tank. In this model, the dimensions of the zones are refined to give better resolution and the percent of the power dissipated in each zone is determined using the local dissipation measurements shown in Figure 1(b) (Zhou, 1997). An overall energy balance in the tank is satisfied. The zone equations are presented below and summarized in Table 1.
MODEL Accurate prediction of dissolution time for slightly miscible systems is difficult (if not impossible) because of the multimechanism processes taking place simultaneously, coupled with the turbulence inhomogeneity in the stirred tank. Prediction of the dissolution time requires an accurate description of the hydrodynamics of the stirred tank, breakup, coalescence and solute mass transfer, and an accurate and stable numerical scheme to resolve the systems of equations for each drop size class in the polydispersed system. Discretization of the mixing tank into the tiny cells (grids) required for CFD increases the computational time and imposes a precision that far exceeds the accuracy of the breakup and dissolution equations. The modelling approach used here is the zone model, which uses zones to represent regions where significant variations in local hydrodynamic properties exist within the mixing tank. Each zone has a residence time and energy dissipation rate, e , with the shortest residence times and the most intense turbulence in the four zones which define the impeller region. The three zones just below the impeller have widely varying residence times and energy dissipation rates and each drop travels through only one of these three zones on each pass through the impeller, thus building up a distribution of drop experiences over time. Reducing the number of discretized regions just to the extent that steep variations in turbulence and a distribution of drop experiences over time can be accurately represented provides dramatic savings on the computational time and resources. In each mixing zone, the turbulence and flow properties are defined, the drop breakup and mass transfer are modelled using published correlations; and the resulting transient drop size population balances are solved.
Mixing Zones The mixing zone model is defined for a D ¼ T/3 PBTD geometry, identical to that used in the experiments. Based on the varying levels of turbulence shown in Figure 1(b),
Zone 1: Impeller swept volume The volumetric flow rate through the impeller is Q1 ¼ NQND 3 and the turbulent kinetic energy dissipation rate (e ) is set to 22% of the total power (P) injected, divided by the mass of fluid in the impeller swept volume (rcV1). The impeller volume is V1 ¼ (pD 3 cos(458))/20 and the residence time (t1) for the fluid in this region is the ratio of the impeller swept volume to the volumetric flow rate.
Zones 2, 3 and 4: Impeller discharge region The velocity and local energy dissipation profiles at the lower edge of the impeller reveal a maximum close to the tip of the blade (Hockey and Nouri, 1996; Jaworski et al., 1996). Close to the hub, the energy dissipation and velocity drop close to the bulk values. In order to provide an orderly transition from one extreme to the other, three zones are used to define the impeller discharge region. The volume of each zone is annular; Vk ¼ p(z2,k 2 2 2 z1,k)(r2,k 2 r1,k ), where r1 and r2 define the radial limits and (z2 2 z1) is the height of the discharge zone, 0.2 D. The volumetric flow rate for each zone was obtained by integrating the axial velocity profile over the radial limits: rð 2:k
Qk ¼ 2 p
Urdr
(1)
rl:k
The width of the trailing vortex is 0.15 D from the tip of the impeller blade, and this is represented by zone 2, where 38% of the power is dissipated. Zone 3 is next to the trailing vortices and has an annulus width of 0.1 D; 10% of the power is dissipated in zone 3. Zone 4 is a disk extending from the centre of the tank to 0.25 D, with 2% of the total energy dissipation. A total of 50% of the energy is dissipated in the impeller discharge stream.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710–720
712
IBEMERE and KRESTA
Figure 1. (a) Half-plane showing the dimensions of the mixing zones and (b) profiles of the dissipation rate under the impeller, showing the limits of the mixing zones.
Zone 5: bulk The volume of the bulk is obtained by subtracting the volumes of zones 1, 2, 3 and 4 from the total tank volume. The volumetric flow rate in the bulk is equal to the volumetric flow rate into the impeller (zone 1) by
continuity, since all the fluid from the impeller discharge region is unfolded into the bulk and re-entrained into the impeller suction. The power dissipated in the bulk must be 28% of the total power input in order to ensure an overall energy balance.
Table 1. Defining equations for each zone in the mixing field. zone (1) Impeller swept volume (2) Discharge stream: trailing vortex (3) Discharge stream: transition (4) Discharge stream: hub region (5) Bulk
Dimensions (m) z1 ¼ D z2 ¼ 1.15 D r1 ¼ 0.0 r2 ¼ 0.5 D z1 ¼ 0.8 D z2 ¼ D r1 ¼ 0.35 D r2 ¼ 0.5 D z1 ¼ 0.8 D z2 ¼ D r1 ¼ 0.25 D r2 ¼ 0.35 D z1 ¼ 0.8 D z2 ¼ D r1 ¼ 0.0 r2 ¼ 0.25 D Everywhere else
Volume (m3) Vi ¼ p(z2 2 z1)(r22 2 r12)
Volumetric flow rate (m3 s21)Qi
Turbulent energy dissipation rate (W kg21)1i
Residence time (s) t ¼ Vi/Qi
0.11 D 3
NqND 3 ¼ 0.79 ND 3 0.22 NpN 3D 5/V1 ¼ 2.368 N 3D 2
V1/NqND 3 ¼ 0.15/N
0.08 D 3
0.17 ND 3
0.38 NpN 3D 5/V2 ¼ 6.033 N 3D 2
0.47/N
0.04 D 3
0.41 ND 3
0.1 NpN 3D 5/V3 ¼ 3.34 N 3D 2
0.093/N
0.04 D 3
0.21 ND 3
0.02 NpN 3D 5/V4 ¼ 0.651 N 3D 2
0.186/N
20.93 D 3
0.79 ND 3
0.28 NpN 3D 5/V5 ¼ 0.017 N 3D 2
26.49/N
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710– 720
MODELLING THE MIXING AND DISSOLUTION KINETICS OF PARTIALLY MISCIBLE LIQUIDS Population Balance
Drop Breakup
The classical population balance equation is applied to each mixing zone, k, to give @Ni @Nm,i @Nb,i ¼ þ rz (UNi ) @t @t @t k
(2)
where Nm is the change in the number of drops of size i due to mass transfer, Nb is the change due to breakup, and the third term is the change in zone k due to convection. Although ¯ Ni) vanishes when integration is the convective term rz . (U done over the batch stirred vessel, it is retained in each mixing zone because it reflects the redistribution of the drops. The drop population must be discretized such that the simulation is both accurate and numerically stable. Drop breakup and dissolution both lead to a reduction in drop size, and most of the change occurs at small drop sizes, so a geometrically increasing discretization was selected. The geometric length scale requires three parameters to fully describe the discretization: the minimum drop size, ‘0, maximum drop size, ‘p, and the number of drop size classes, p. The range of drop sizes in the model was set to match the range of the experimental measurements (‘0 ¼ 5 mm, ‘p ¼ 450 mm). The number of size classes required to accurately match the numerical solution to an analytical solution was p ¼ 80 (Ibemere, 2005), giving 1=p ‘p rg ¼ ¼ 1:05786 ‘0
(3)
‘i ¼ rg ‘i1
(4)
rg þ 1 ‘i þ ‘i1 Li ¼ ¼ ‘i ¼ 0:9727‘i 2 2rg rg 1 DLi ¼ ‘1 ‘i1 ¼ ‘i ¼ 0:0547‘i rg
(5) (6)
where p is the total number of drop size classes, rg is the geometric factor, ‘0 is the minimum drop size, ‘p is the maximum drop size, ‘i is the upper limit of size class i, Li is the drop size in class i, and DLi is the geometric spacing for the ith class size. As the drops dissolve, they are removed from interval i by reducing the number of drops in interval i by an equivalent volume, rather than by changing the drop diameter continuously. The experimental drop size distribution acquired after 20 s was fit to a log-normal distribution and used to initialize the simulation. The total number density in the tank was then determined using a volume balance on the undissolved material: N(Li , t) ¼ P(Li )
Vd p P i¼0
P(Li )
¼
pL3i =6
Ni NT V d p P NT pNi L3i =6
713
(7)
i¼0
where Ni is the number of drops in interval i, P(Li) is the log normal probability function, Vd is the total volume of the undissolved solute in the stirred tank at t, and ‘iþ1 represent the interval limits for the ith drop class size. The drops were divided into the five mixing zones according to the volume of each zone. Subsequent experimental drop size distributions were compared with the population in zone 5, the bulk.
The general expression for the drop breakup rate is @Nb (L0 ) ¼ g(L0 )N(L0 , t) @t ð1 m(L0 )f (L, L0 )g(L0 )N(L0 , t)dL þ
(8)
L¼L0
The first term on the right hand side of the breakup equation accounts for the rate of breakup of drops of diameter size L0, while the second term accounts for the rate of formation of drops of size L0 from the breakup of drops of sizes larger than L0. The function g(L0) is the drop breakup frequency, f (L, L0) is the size probability density function of the fragments, m(L0) is the number of fragments resulting from the breakup of a particle of diameter L0, and @Nb (L0 )=@t gives the net change in the number of drops of size L0 that has occurred during the time step. For binary breakup, m(L0) ¼ 2 for all drop sizes. The drop breakup frequency model proposed by Luo and Svendsen (1996) was selected for this work because it requires no experimentally determined or fitted parameters. Alopaeus et al. (1999) provide a numerically explicit formulation of this model using incomplete gamma functions: !1=3 2:4738(1 f) 1 g(Li , Lj ) ¼ b8=11 L2j 8 9 > < (G(8=11, tm ) G(8=11,b))þ > = 2b3=11 (G(5=11, tm ) G(8=11, b)) > > : 6=11 ; þb (G(2=11, tm ) G(2=11, b)) 5:8634Cf s b¼ rc 12=3 L5=3 j 2 2=3 L3 Li Cf ¼ þ 1 i 1 Lj Lj 11=3 h tm ¼ b Lj 3 1=4 n h¼ 1
(9)
(10)
(11) (12) (13)
The local dissipation (e ) in each zone is used to calculate the corresponding Kolmogorov microscale (h). Equation (9) is used to find g(Lo) for a finite length interval as g(Li ) ¼
i1 X g(Li , Lj ) j¼1
2
DLj
(14)
The probability distribution of daughter drops (Konno et al., 1980) can be approximated by a combination of gamma functions (Lasheras et al., 2002): f (Li , Lj ) ¼ p X
8 G(12) Li Li 2 1 G(9)G(3)Lj Lj Lj
f (Li , Lj )DLj ¼ 1:0
(15) (16)
j¼0
The change in drop size distribution due to breakup for a
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710–720
714
IBEMERE and KRESTA solving for @L=@t gives
specific drop size class Li can now be obtained @Nb (Li ) ¼ g(Li )N(Li , t) @t þ
p X
dLi 2kL,i (Cs Cbuilk ) ¼ dt rd
2f (Li , Lj )g(Lj )N(Lj , t)
(17)
j¼iþ1
where p is bin number for the largest drop size. The first and second terms on the right hand side denote the rates of breakup and formation of drops of size Li. The functions f (Li, Lj) and g(Li) are calculated once, at the beginning of a run, for all size classes. The breakup rate is then calculated at each time step for each size class, Li. This breakage function is valid only for drops larger than the Kolmogorov microscale, and assumes that the inertial range of turbulence (25/3 slope of the energy spectrum) is fully developed in the flow.
Mass Transfer Correlations for mass transfer coefficients depend on the phases involved, the stirred tank geometry, and the dispersed phase fraction. Only mass transfer correlations for pure dissolving solids or liquid –liquid extractions were found in the literature, and neither is perfectly suited to the study of pure dissolving drops. Although solid dissolution and liquid – liquid extraction both result in the transfer of solute from particles into the continuous phase, they both differ considerably from the dissolution of pure dissolving drops. Pure dissolving drops shrink until they disappear completely, like solid particles, while in liquid–liquid extraction the drop size is nearly constant with time. Pure dissolving drops have no mass transfer resistance on the drop side, again like dissolving solids, while liquid–liquid extraction systems present mass transfer resistances from both immiscible liquids. Conversely, drops are subject to deformation, breakup, internal circulation and oscillation, unlike solid particles, and surface motion on drops increases the slip velocity relative to a solid sphere. Reviews of mass transfer correlations for liquid–liquid extraction in stirred tanks are given in Perry’s (1997) Handbook (Tables 5–21 to 5–28) and Kumar and Hartland (1999), while the topic of solids dissolution is discussed by Pangarkar et al. (2002) and Atiemo-Obeng et al. (2004). The selection of an appropriate correlation for the mass transfer coefficient was based on the similarity of geometries, phases present, and performance in the model. Table 2 gives a selection of the correlations considered. The dependence of the mass transfer coefficient on particle size varies, with exponents ranging from 20.7 to 0.33 for both solid–liquid and liquid–liquid extraction systems. In view of the uncertainty on the exponent for particle size, several correlations were compared to experimental data for this system (Ibemere, 2005). The correlation by Glen (1965) gave the best results, as shown in Figure 2, and is restated here in terms of the length scale L: kL ¼
DAB Dr 1=2 L 1=3 ReSc1=3 D D rd
(18)
The mass transfer rate for a single drop of size L is dm @Vd @Vd @L ¼ kL (pL2 )(Cs Cbulk ) ¼ rd ¼ rd dt @t @L @t
(19)
(20)
and the number of drops, dni,out that will shrink into the (i-1)th class size from the i th class size in time dt is given by dNi dLi Ni ¼ (21) dt out dt DLi Similarly, a number of drops will shrink into the i th class size from the (i þ 1)th class size according to dNi dLiþ1 Niþ1 ¼ (22) dt in dt DLiþ1 and the overall change in the ith size class due to dissolution is dNi 2(Cs Cbulk ) Niþ1 Ni ¼ kL,iþ1 kL,i (23) dt rd DLiþ1 DLi
Material Balance Mass conservation of the dispersed phase concentration in the stirred tank is ensured by closing the material balance between the continuous and dispersed phases. The stirred tank is assumed to be well mixed, so the solute concentration in the continuous phase can be obtained by difference: Ni L3i k¼1 i¼1 ¼ p 5 P P VTANK p=6 Ni L3i md,0 rd p=6
Cbulk
p 5 P P
(24)
k¼1 i¼1
where Cbulk is the solute concentration in the bulk at any time, md,0 is the initial mass of dispersed phase added, and VTANK is the total fluid volume of the stirred tank (dispersed phase volume inclusive). Recall that k is the mixing zone index.
Numerical Solution The limiting time scale Dt based on a Courant number of 0.5 is given by Dt ¼
DL1 Vk or Dt ¼ 2@L1 =@t 2Qk
(25)
where the first timescale is based on the rate of mass transfer from the smallest drops, and the second timescale is based on the convection time scales through each of the five mixing zones. The smallest of the six calculated timesteps is used for the integration. A modified Hounslow discretization with forward differencing (Hounslow et al., 1988; Hounslow, 1990; Ibemere, 2005) was implemented. This scheme predicted the first four moments of the distribution without any error and eliminates oscillations which occur at the small end of the drop size distribution when using either the backward or central differencing schemes.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710– 720
MODELLING THE MIXING AND DISSOLUTION KINETICS OF PARTIALLY MISCIBLE LIQUIDS
715
Table 2. Particle size dependence for different mass transfer coefficient correlations. S-L denotes solid–liquid dissolution; L-L denotes liquid–liquid extraction; and G-L denotes gas–liquid mass transfer. Researcher
System studied
Tank configuration
Correlation for kL 0
Asai et al. (1988)
S-L
Baffled cylindrical flat bottomed stirred tank with turbine impeller
0 15:8 11=5:8 !0:58 11=2 dp4=3 n 0:33 A C DAB B 5:8 @ @2 þ 0:61 A DAB n dp
Exponent on dp
0
0 1 1 !1=3 0:5 1dp4 n DAB B A C @2 þ 0:49@ 3 A DAB n dp
Kuboi (1974)
S-L
Baffled stirred tank with turbine impeller
Levins and Glastonbury (1972)
S-L
Baffled tank with flat, if Dr is not significant; curved, and 0 ! 4=3 1=3 0:62 0:17 pitched blade D n 0:36 DAB @2 þ 0:44 dp 1 turbine impellers T DAB n dp
if Dr is significant;
Miller (1971)
S-L
Baffled stirred tank with turbine impeller
Sano et al. (1974)
S-L
Baffled stirred tank with turbine impeller
Boyadzhiev and Elenkov L-L (1966)
Baffled stirred tank with turbine impeller
Glen (1965)
L-L
Six blade disk turbine in baffled stirred tank
Skelland and Moeti (1990)
L-L
Six flat blade turbine in baffled stirred glass vessel
20.23
0
1=3 4=3 @2 þ 0:222 1 dp n
0
4=3 1=3 @2 þ 0:4 dp 1 n
! dp m 0:5 n 0:38 DAB 2 þ 0:47 DAB n dp !4=3
!0:62
n DAB
n DAB
1 0:33 A DAB dp
0:33
1
Af DAB dp
20.33
20.18
2 0.5
20.7
20.2
Lamont and Scott (1970)
Calderbank and Moo-young (1961)
G-L studies Aerated mixing compared vessels and with L-L data columns
DAB m 0:5 n 1=6 0:65 DAB dp
20.5
2 0:33 5=4 D N dp Dr n 0:33 DAB n rd DAB D D
0.33
0:67 2 0:5 2 5=12 D2 N D dp DN n dp T g !5=4 rdp2 g n 0:33 0:5 DAB f DAB s dp
1:23 105
! 0:25 P n 0:67 P (gDr)4=3 m1=3 n where ¼ A 0:13 V DAB V r2=3
EXPERIMENTAL For phases that are not in equilibrium, the rate of mass transfer and/or chemical reaction is greatly influenced by the available interfacial area between the phases and the deviation from equilibrium (i.e., the chemical potential between the two phases). Investigations into the mass transfer kinetics coupled with drop breakup and shrinkage require simultaneous measurement of the concentration of the solute and the interfacial area available for mass transfer. The drop
0.0
0.0
size distribution defines the available interfacial area for mass transfer while the solute concentration defines the extent of deviation from equilibrium. Accurate measurement of these parameters allows a full model validation and direct determination of the overall mass transfer coefficient, kL. In the experiments, a flat bottomed cylindrical glass tank is mounted inside a flat bottomed square glass tank and the space between the tanks is filled with water to minimize refraction. The tank diameter equals the liquid height, H ¼ T ¼ 145 mm. Four equally spaced baffles (W ¼ T/10)
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710–720
716
IBEMERE and KRESTA the impeller at 2r/D ¼ 0.8 to give the maximum data rate. The position of the probe does not affect the measured DSD (Zhou and Kresta, 1998). The gas chromatograph (GC) and phase Doppler particle analyser (PDPA) were used in their standard configurations. 2-Butanone was used as the internal standard for the GC, giving a linear calibration curve with a coefficient of determination of 0.99 (diethyl malonate concentration versus area ratio). The PDPA is an extension of laser Doppler velocimetry (LDV) and allows simultaneous measurement of drop diameter and velocity. The PDPA used in this work (Zhou and Kresta, 1998) has a focal length of 500 mm and a drop diameter range of 1.1 mm , d , 338 mm at a collection angle of 22.58. The PDPA imposes some constraints on the selection of continuous and dispersed phases. The liquid has to be transparent, and the ratio of refractive indices between the continuous phase and the dispersed phase must be moderate. It is also desireable to have a small density difference, and low viscosities. The continuous phase is deionized ultra-filtered (DIUF) water, which is completely free of particulate. The dispersed phase is diethyl malonate. Diethyl malonate has a specific gravity of 1.055, a viscosity of 2.15 mPa s, a surface tension of 31.83 mN m21, a refractive index of 1.41, and a solubility limit of 22.6 kg m23 in water at 208C. For each experiment, 25 ml of diethyl malonate was pumped into the mixing tank over a period of 20 s. The feed time was selected to ensure that all drops experienced turbulence and breakup at the impeller before being convected into the bulk of the tank. Drop size data was collected continuously using the PDPA, and concentration measurements were taken every 20 s. The GC samples were collected using a 2 mm pipette placed 2–3 cm above the impeller blade. The mouth of the pipette was significantly larger than the drops of solute. The samples were allowed to settle for about 10 min at a 458 angle. This angle of inclination allows fast separation of the drops, and gave more repeatable results than fast centrifugation. Because of the small settling velocity, the mass transfer during settling can be neglected. Using a micropipette, 1.2 ml of the continuous phase was withdrawn from the supernatent, combined with 0.02 ml of 2-butanone, and injected into the GC.
RESULTS AND DISCUSSION Figure 2. Comparison of model and experimental results for the cumulative drop size distribution as it evolves over time. (a) The correlation by Glen (1965) gives the best fit to the experimental data for cumulative number density at 650 rpm; (b) the fit at 550 rpm is poor and (c) the prediction at 750 rpm is quite good. This figure is available in colour online via www.icheme.org/cherd.
are placed around the periphery of the mixing tank and the tank is agitated using a four-bladed, down-pumping, stainless steel 458 pitched blade turbine (PBT) impeller, with a diameter (D ¼ T/3) and blade width (w ¼ D/5) placed at an off-bottom clearance equal to the impeller diameter (C/D ¼ 1.0). The dissolution kinetics were measured at three impeller speeds (550, 650 and 750 rpm); all higher than the impeller speed required for fully turbulent flow (510 rpm). A 3 mm internal diameter feed tube, placed at an offset of 20 mm from the impeller shaft and 4 mm above the impeller blades is used to feed the dispersed phase. The drop size distribution (DSD) was measured just below
Other than the selection of a mass transfer correlation, there are no fitting parameters in the model. The model results are evaluated in terms of the concentration versus time, the evolution of the Sauter mean diameter over time, and the evoluton of the drop size distribution over time. It will be shown that the model results are more accurate for higher rotational speeds, so further analysis is done at 750 rpm or higher. The final objective is to determine how the rate of mass transfer for various liquid –liquid systems compares with the blend time for completely miscible systems. We emphasize that this comparison is meant to illustrate the large differences in the time scales of these two processes. It will be evident to the reader from the results that it is not a physical dimensionless group, since the mass transfer variables dominate the dissolution time. The first thing that was determined is the relative importance of drop breakup and mass transfer. The liquid was slowly injected directly into the most turbulent part of the tank, and the first measurements were started at 20 s. The
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710– 720
MODELLING THE MIXING AND DISSOLUTION KINETICS OF PARTIALLY MISCIBLE LIQUIDS
717
initial drop size distributions at all three rotational speeds were completely composed of drops smaller than the limiting drop size for breakup in zone 2. At 550 rpm, the breakup rate is negligible for drops smaller than 300 mm. At 650 rpm, the breakup rate is neglible for drops smaller than 280 mm; and at 750 rpm, the breakup rate is neglible for drops smaller than 240 mm. From the initial cumulative drop size distributions shown in Figure 2, it is clear that that there is no drop breakup in the tank after the first 20 s at any rotational speed. This means that mass transfer will be the dominant mechanism in the reduction of the drop size distribution. Changes in surface tension of a factor of 1000 gave the same simulation results for both the dissolution time and the drop size distribution, confirming this observation. Of course, the initial drop breakup is critical to the generation of surface area for mass transfer, but because of the nature of the liquids and the experimental design, this part of the model could not be directly validated. Figure 2 compares the predicted and experimental normalized drop size distributions. The normalized number density is given by ni ¼ Ni/NT DL where N is the absolute number of drops in a size class, NT is the total number of drops in the distribution and DL is the size class interval. The correlation by Glen [Figure 2(a)] gave the best fit to the experimental data at 650 rpm and was used for all subsequent simulations. At 550 rpm [Figure 2(b)], the model fails to accurately predict the drop size distribution and the flow is just at the lower limit of turbulent conditions (Re ¼ 21 700), but at 750 rpm [Figure 2(c)], the model agreement with experiment is very good. As the rotational speed is increased from 550 to 750 rpm, the dissolution time drops by a factor of two. This is a very significant effect, which should also be apparent in the mean drop size and bulk concentration results. Figure 3 shows the model predictions and experimental results for mean drop size. The agreement is better than the detailed results shown in the drop size distribution data, because the differences across the distribution are averaged out. The model tracks the mean diameter equally well at all three rotational speeds, across significant changes in the rate of mass transfer. As the impeller rotational speed increases, fewer data points are measured due to reduced dissolution time, but the reader will recall that the mean diameter does not tell us anything about how many drops are
present in the tank, so it cannot be used to accurately determine the dissolution time. The bulk concentration, shown in Figure 4, is used to determine the dissolution time. The concentration in the bulk is fixed in both the model and the experiment by a mass balance. The initial measured concentration corresponding to the initial drop size distribution 20 s after injection of diethyl malonate in the tank is about 10.5 kg m23. This dissolved concentration represents about 85% of the injected diethyl malonate indicating very fast dissolution rates in the early seconds of the process. Once again, the model predictions are much better at higher rotational speeds. The model results at 550 rpm cannot be considered reliable. In order to determine the range of dissolution times which might be observed with various partially miscible liquid – liquid systems, three additional liquid were selected for simulations: O-xylene, toluene and amyl acetate. Their properties and the simulation results are given in Table 3. The saturation concentrations range from 22.6 kg m23 to 0.1 kg m23 and all of the simulations were performed at 50% of the saturation concentration. While the raw dissolution time is given in Table 3, it is more useful to consider the ratio of dissolution time to blend time. This gives an indication of the relative magnitude of the two processes, and gives a clearer picture of the impact of N on the results. The ratio of dissolution time to blend time ranges from 91 to 190, as shown in Figure 5. While the results at 750 rpm are linear, extrapolation of this trend is not supported by the last point for diethyl malonate at 11.3 kg m23 and a dissolution/blend time ratio of 91. While increasing the experimental rotational speed from 550 to 750 rpm halved the dissolution time; further increasing the rotational speed by a factor of 4 reduces the dissolution time by less than 30%. Operating at this very high speed (3350 rpm) will increase the power consumption by a factor of 64, and at the experimental scale is not physically achievable. Clearly, when dissolution times are to be minimized, the best operating conditions will be achieved by using high, but not excessive, rotational speeds. These results define some interesting bounds for the problem of liquid dissolution in stirred tanks. We suggest that further studies revisit the question of a physically meaningful local mass transfer coefficient, and variables such as the percent approach to saturation or the absolute driving force for
Figure 3. Sauter mean diameter decay with time at three rotational speeds.
Figure 4. Dissolved dispersed phase concentration profile.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710–720
718
IBEMERE and KRESTA
Table 3. Model predictions for sample systems of liquids dispersed in water (m ¼ 0.894 mPa s) at 50% of their saturation concentration. Dispersed phase
Saturation concentration (Kg m23)
Initial concentration difference (kg m23)
Interfacial tension (mN m21)
O-Xylene
0.175
0.08875
30.31
Toluene
0.5
0.25
28.50
Amyl acetate
1.0
0.5
25.67
22.6
11.3
31.83
Diethyl malonate
N(rpm)
Dissolution time (s)
Dissolution to blend time ratio
750 3350 750 3350 750 3350 750 1500
700 155 645 125 560 100 340 130
190 190 175 150 150 120 91 70
Simulation studies done using the model show that the dissolution times are of the order of 100 –200 times the blend time, and that the dissolution time to blend time ratio decreases with increased agitation. The dominant variables affecting the mass transfer rate are the approach to the solubility limit, expressed as the absolute concentration difference, and the rotational speed of the impeller. An optimum balance between rotational speed and power consumption was observed. The main outcomes of the work are the problem definition (dissolution rates for partially miscible fluids are much longer than the blend time for completely miscible fluids) and identification of the key parameters which determine the dissolution time. Further research is warranted, particularly with respect to the effect of less optimum feed strategies and systems in which drop breakup kinetics would have a more significant effect. Figure 5. Effect of the initial concentration difference on the ratio of dissolution time to blend time for selected process fluids (see Table 3 for details).
mass transfer rather than a simple scaling with the blend time.
CONCLUSIONS An experimental procedure for the simultaneous measurement of transient drop size distribution and the corresponding solute concentration of a slightly miscible liquid–liquid system (diethyl malonate dispersed in deionized ultra filtered water) was developed. A mechanistic model which captures the inhomogeneity in the stirred tank using 5-mixing zones with local dissipation energy rates estimated from experimental data tracks the transient drop size distribution in the stirred tank using population balances and published correlations for drop breakup and mass transfer, predicts the solute concentration in the bulk of the stirred tank and the dissolution time. The model was validated by comparing the model predictions with experimental results for the solute (diethyl malonate) concentration in the tank; Sauter mean diameter, drop size distribution and cumulative number distribution of the dispersed drops. The model gave good agreement with experimental results, although the subtleties of the experimental conditions meant that the effects of drop breakup could not be directly validated. In fact, it was found that differences in interfacial tension are quite small for many process fluids, so this limitation is not as important as was initially expected.
NOMENCLATURE b C Cbulk Cf CS d dp D DAB f g g H i, j k kL L Li ‘0 ‘i DLi md m N ni Ni NP NQ NT P p P Q rg r Re Sc
constant off-bottom impeller clearance, m bulk concentration in stirred tank, mol L21 or kg m23 constant solute concentration at equilibrium, mol L21 or kg m23 drop diameter, m particle diameter, m impeller diameter, m molecular diffusion coefficient, m2 s21 probability density function of fragments drop breakup frequency, s21 gravitational constant, ms22 initial fluid height, m subscripts, drop interval index subscript, mixing zone index mass transfer coefficient, m s21 drop size, m drop size in interval i, m smallest drop size, m upper limit of drop size interval i, m spacing of i th size interval, m mass of dispersed drops, kg number of daughter drops formed impeller rotational speed, rps or rpm number of drops in interval i normalized with DLi, m21 number of drops in interval i impeller power number impeller flow number total number of drops impeller power requirement, W total number of drop size classes log normal probability function volumetric flow rate, m3 s21 geometric size interval factor radial distance, m Reynolds number Schmidt number
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710– 720
MODELLING THE MIXING AND DISSOLUTION KINETICS OF PARTIALLY MISCIBLE LIQUIDS t tm Dt T U Vd Vk VTANK w W z
time, s constant time step, s tank diameter, m axial velocity, m s21 total volume of drops, m3 volume of mixing zone k, m3 total volume of stirred tank, m3 blade width, m baffle width, m axial distance, m
Greek symbols e n rc rd Dr s f h m
rate of kinetic energy dissipation, W kg21 kinematic viscosity, m2 s21 continuous phase density, kg m23 dispersed phase density, kg m23 density difference between phases, kg m23 interfacial tension, N m21 volume fraction dispersed phase Kolmogorov length scale, m dynamic viscosity, kg m21 s21
REFERENCES Alopaeus, V., Koskinen, J. and Keskinen, K.I., 1999, Simulation of the population balances for liquid-liquid systems in a non-ideal stirred tank. part 1: description and quantitative validation of the model, Chem Eng Sci, 54(24): 5887– 5899. Asai, S., Konishi, Y. and Sasaki, Y., 1988, Mass-transfer between fine particles and liquids in agitated vessels, J Chem Eng Japan, 21(2): 107–112. Atiemo-Obeng, V.A., Penney, W.R. and Armenante, P., 2004, Solidliquid mixing, in Paul, E.L. et al. (eds). The Handbook of Industrial Mixing. (Wiley, New York, USA). Attarakih, M.M., Bart, H.J. and Faqir, N.M., 2004, Numerical solution of the spatially distributed population balance equation describing the hydrodynamics of interacting liquid-liquid dispersions, Chem Eng Sci, 59(12): 2567–2592. Bajpai, R.K., Ramkrishna, D. and Prokop, A., 1976, Coalescence redispersion model for drop size distributions in an agitated vessel, Chem Eng Sci, 31(10): 913– 920. Baldyga, J., Podgorska, W. and Pohorecki, R., 1995, Mixing-precipitation model with application to double feed semi batch precipitation, Chem Eng Sci, 50(8): 1281–1300. Bayens, C.A. and Laurence, R.L., 1969, A model for mass transfer in a coalescing dispersion, Ind & Eng Chem Fund, 8(1): 71. Boyadzhiev, L. and Elenkov, D., 1966, On the mechanism of liquidliquid mass transfer in a turbulent flow field, Chem Eng Sci, 21(11): 955–959. Calderbank, P.H. and Moo-young, M.B., 1961, The continuous phase heat and mass-transfer properties of dispersions, Chem Eng Sci, 16(1–2): 39– 54. Coulaloglou, C.A. and Tavlarides, L.L., 1977, Description of interaction processes in agitated liquid-liquid dispersions, Chem Eng Sci, 32(11): 1289– 1297. Curl, R.L., 1963, Dispersed phase mixing I: theory and effects in simple reactors, AIChE J, 9(2): 175– 181. Glen, J.B., 1965, Thesis, cited from: Skelland, A.H.P. and Kanel, J.S., Simulation of mass transfer in a batch agitated liquid-liquid dispersion, Ind Eng Chem Res, 1993, 31(3): 908– 920. Grenville, R.K., 1992, Blending of viscous Newtonian and pseudoplastic fluids, PhD dissertation, Cranfield Institute of Technology, Cranfield, Bedfordshire, UK. Hemsing, J.L., 2001, Kinetics of blending for soluble polymer additives, MSc dissertation, Department of Chemical & Materials Engineering, University of Alberta, Canada. Hockey, R.M. and Nouri, J.M., 1996, Turbulent flow in a baffled vessel stirred by a 608 pitched blade impeller, Chem Eng Sci, 51(19): 4405–4421. Hounslow, M.J., Ryall, R.L. and Marshall, V.R., 1988, A discretized population balance for nucleation, growth and aggregation, AIChE J, 34(11): 1821– 1832. Hounslow, M.J., 1990, A discretized population balance for continuous systems at steady state, AIChE J, 36(1): 106– 116.
719
Hsia, M.A. and Tavlarides, L.L., 1980, A simulation model for homogeneous dispersions in stirred tanks, Chem. Eng. J. & BioChem. Eng. J., 20(3): 225 –236. Hsia M.A. and Tavlarides, L.L., 1983, Simuation analysis of drop breakage, coalescence and micromixing in liquid-liquid stirred tanks, Chem. Eng. J. & Biochem. Eng. J., 26(3): 189–199. Ibemere, S., 2005, Dissolution kinetics of liquid-liquid dispersions using local rates of turbulent dissipation, MSc dissertation, Department of Chemical & Materials Engineering, University of Alberta, Canada. Jaworski, Z. and Fort, I., 1991, Studies on mixing .79.: Energy dissipation rate in a baffled vessel with pitched blade turbine impeller, Collection of Czechoslovak Chem Comm, 56(9): 1856–1867. Jaworski, Z., Nienow, A.W. and Dyster, K.N., 1996, An LDA study of the turbulent flow field in a baffled vessel agitated by an axial, down-pumping hydrofoil impeller, Can J Chem Eng, 74(1): 3– 15. Jeon, Y.M. and Lee, W.K., 1986, A drop population balance model for mass transfer in liquid-liquid dispersion I: Simulation and its results, Ind & Eng Chem Fund, 25(2): 293– 300. Kennedy, C., 2003, Mixing, precipitation and coagulation with alum, MSc dissertation, Department of Chemical & Materials Engineering, University of Alberta, Canada. Konno, M., Matsunaga, Y., Arai, K. and Saito, S., 1980, Simulation model for breakup process in an agitated tanks, J Chem Eng Jpn, 13(1): 67–73. Kresta, S., Anthieren, G. and Parsiegla, K., 2005, Model reduction for prediction of silver halide precipitation, Chem Eng Sci, 60(8/9): 2135–2153. Kuboi, R., Komasawa, I., Otake, T. and Iwasa, M., 1974, Fluid and particle motion in turbulent dispersion .3. Particle-liquid hydrodynamics and mass-transfer in turbulent dispersion, Chem Eng Sci, 29(3): 659 –668. Kumar, A. and Hartland, S., 1999, Correlations for prediction of mass transfer coefficients in single drop systems and liquid-liquid extraction columns, Chem Eng Res & Des, 77(A5): 372–384. Lamont, J.C. and Scott, D.S., 1970, An eddy cell model of mass transfer into the surface of a turbulent liquid, AIChE J, 16(4): 513– 519. Lasheras, J.C., Eastwood, C., Martinez-Bazan, C. and Montanes, J.L., 2002, A review of statistical models for the break-up of an immiscible fluid immersed into a fully developed turbulent flow, International Journal of Multiphase Flow, 28: 247–278. Levins, D.M. and Glastonbury, J.R., 1972, Particle-liquid hydrodynamics and mass transfer in a stirred vessel II: mass transfer, Trans Inst Chem Eng, 50(2): 132. Luo, H. and Svendsen, F., 1996, Theoretical model for drop and bubble breakup in turbulent dispersions, AIChE J, 42(5): 1225– 1233. Maggioris, D., Goulas, A., Alexopoulos, A.H., Chatzi, E.G. and Kiparissides, C., 2000, Prediction of particle size distribution in suspension polymerization reactors: effect of turbulence nonhomogeneity, Chem Eng Sci, 55(20): 4611– 4627. Miller, D.N., 1971, Scale-up of agitated vessels – mass transfer from suspended solute particles, I&EC Proc Des Dev, 10(3): 365. Pangarkar, V.G., Yawalkar, A.A., Sharma, M.M. and Beenackers, A.A.C.M., 2002, Particle-liquid mass transfer coefficient in two-/ three-phase stirred tank reactors, Ind Eng Chem Res, 41(17): 4141–4167. Park, J.Y. and Blair, L.M., 1975, The effect of coalescence on drop size distribution in an agitated liquid– liquid dispersion, Chem Eng Sci, 30 (9): 1057–1064. Perry, R.H. and Green, D.W. (eds). Perry’s Chemical Engineers’ Handbook, 1997, 7th Edition (McGraw-Hill, New York, USA). Ramkrishna, D., 1974, Drop-breakage in agitated liquid-liquid dispersions, Chem Eng Sci, 29(4): 987–992. Ramkrishna, D., 2000, Population Balances: Theory and Applications to Particulate Systems in Engineering (Academic Press, San Diego, USA). Sano, Y., Yamaguchi, N. and Adachi, T., 1974, Mass transfer coefficients for suspended particles in agitated vessels and bubble columns, J Chem Eng Japan, 7: 255–261. Shah, B.H., Ramkrishna, D. and Borwanker, J.D., 1977, Simulation of particulate systems using concepts of interval of quiescence, AIChE J, 23(6): 897– 904. Skelland, A.H.P. and Kanel, J.S., 1992, Simulation of mass transfer in a batch agitated liquid-liquid dispersion, Ind Eng Chem Res, 31(3): 908– 920.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710–720
720
IBEMERE and KRESTA
Skelland, A.H.P. and Moeti, L.T., 1990, Mechanism of continuousphase mass-transfer in agitated liquid-liquid systems, I&EC Res, 29(11): 2258– 2267. Spielman, L.A. and Levenspiel, O., 1965, A Monte Carlo treatment for reacting and coalescing dispersed phase systems, Chem Eng Sci, 20(3): 247. Tsouris, C. and Tavlarides, L.L., 1994, Breakage and coalescence models for drops in turbulent dispersions, AIChE J, 40(3): 395– 406. Valentas, S.K. and Amundson, N.R., 1966, Breakage and coalescence in dispersed phase systems, Ind & Eng Chem Fund, 5(4): 533. Valentas, S.K., Bilous, O. and Amundson, N.R., 1966, Analysis of breakage in dispersed phase systems, Ind & Eng Chem Fund, 5(2): 271.
Zhou, G. and Kresta, S.M., 1998, Evolution of drop size distribution in liquid-liquid dispersions for various impellers, Chem Eng Sci, 53(11): 2101– 2105. Zhou, G. and Suzanne M. Kresta, 1996, Distribution of energy between convective and turbulent flow for three frequently used impellers, Chem Eng Res Des, 74A: 379–389. Zhou, G., 1997, Characteristics of turbulence energy dissipation and liquid-liquid dispersions in an agitated tank, PhD dissertation, Department of Chemical & Materials Engineering, University of Alberta, Canada. The manuscript was received 3 October 2006 and accepted for publication after revision 2 January 2007.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710– 720
Abstract: The problems of liquid–liquid extraction and solids dissolution in stirred tanks are classic mass transfer problems which have been studied extensively in the literature. An analogous problem, that of dissolution of pure drops close to their solubility limit, has been almost completely neglected. This problem has practical application in the dissolution and dispersion of small amounts of surface active additives, particularly in the water treatment industry and in oil field applications where the cost of the chemical additive is one of the major process costs. The question is also of interest for some chemical reactions, such as the third Bourne reaction (diethylmalonate in water) and the hydrolysis of acetic anhydride to acetic acid. In this paper, the drop size distribution and solute concentration in the bulk are measured throughout the dissolution period and the dissolution process is successfully modelled using an Eulerian –Lagrangian approach. The validated model is used to identify the key variables driving the dissolution rate. The approach to saturation, the impeller rotational speed, and the continuous phase viscosity all play an important role. When the solute is injected close to the impeller, the role played by surface tension is surprisingly small. Keywords: liquid dissolution; multi-mechanism model; mixing; liquid– liquid system; mass transfer.
INTRODUCTION
Correspondence to: Professor S. Kresta, Department of Chemical and Materials Engineering, University of Alberta, Edmonton, T6G 2G6, Canada. E-mail: [email protected]
DOI: 10.1205/cherd06185 0263–8762/07/ $30.00 þ 0.00 Chemical Engineering Research and Design Trans IChemE, Part A, May 2007 # 2007 Institution of Chemical Engineers
extremes, there is a fixed solubility limit and two liquids are partially miscible. In this work, we are interested in liquid –liquid systems which must operate close to their solubility limit, and particularly in systems where this solubility limit occurs at fairly low concentrations of the minor component. In these systems, the minor phase is first dispersed as drops, and then completely dissolves over time. The drop size distribution and the concentration of solute in the continuous phase both change over time. Early models treated turbulent conditions in stirred tanks as homogenous (Park and Blair, 1975; Hsia and Tavlarides, 1980, 1983; Skelland and Kanel, 1992); however, it is well known that the turbulence intensity is typically 100 higher close to the impeller than in the bulk of the tank. Consequently, models composed of several compartments (zones) have evolved. Some authors have modelled the impeller region and the rest of the tank separately using a two zone model (Coulaloglou and Tavlarides, 1977; Tsouris and Tavlarides, 1994; Maggioris et al., 2000) while others have used pre-selected volumes with dissipation energy and volumetric flow rates determined using computational fluid dynamics (CFD) analysis and/or experimental data (Baldyga et al., 1995; Alopaeus et al., 1999; Kresta et al., 2005).
The problem of dissolution kinetics in liquid– liquid systems has long been recognized by static mixer manufacturers and by surface chemists. In the first case, field engineers report ‘stranding’ of a completely soluble liquid as it passes through a static mixer; and in the second case, surface chemists insist on a mixing period of 24 h in very small shaker flasks when determining solubility limits of long chain polymer surfactants. Recent studies on partially miscible liquid– liquid systems (Hemsing, 2001; Kennedy, 2003) have shown dissolution times that are 50 –75 times longer than the blend time. Chemical engineers have generally focused on the limits of blend time for completely miscible liquids (Grenville, 1992), on the mass transfer of a solute between phases in liquid –liquid extraction, and on drop size distributions in completely immiscible liquids. In order to approach the problem of dissolution kinetics of a pure liquid, we first define the class of liquid–liquid systems of interest. Liquids which are mutually soluble at all concentrations have no solubility limit. They are completely miscible. Conversely, immiscible liquids (e.g., the two carrier phases in liquid –liquid extraction) approach zero mutual solubility. Between these two 710
Vol 85 (A5) 710–720
MODELLING THE MIXING AND DISSOLUTION KINETICS OF PARTIALLY MISCIBLE LIQUIDS The dispersed drop population is commonly modeled using either population balance models or simulation techniques such as the Monte Carlo method (Spielman and Levenspiel, 1965), or the quiescence interval method (Shah et al., 1977). Due to the difficulty of accurately modelling the simultaneous drop–drop interactions (coalescence), drop –eddy interaction (breakup) and solute mass transfer, most models eliminate one of these mechanisms. Models developed to predict the drop size distribution in liquid –liquid dispersions typically neglect the effects of mass transfer (Alopaeus et al., 1999; Valentas et al., 1966; Valentas and Amundson, 1966; Ramkrishna, 1974; Bajpai et al., 1976). Other models incorporating solute transfer (Curl, 1963; Bayens and Laurence, 1969; Jeon and Lee, 1986; Skelland and Kanel, 1992) eliminate coalescence. Full reviews of population balances can be found in Attarakih et al. (2004) and Ramkrishna (2000). In this work, coalescence is neglected, but the the effects of both breakup and mass transfer on the drop size distribution are modelled directly. The model provides a way to identify the variables which have the biggest impact on the dissolution rate, and to estimate the range of dissolution times that might be expected for various solute/solvent pairs.
711
the tank volume is divided into five mixing zones: the impeller swept volume; three zones in the impeller discharge region; and the bulk of the tank, as shown in Figure 1(a). The distribution of the energy dissipation over each of these regions is very sensitive to the location of the mixing zones. Data reported in the literature shows a range of results, primarily because the measurements use different specifications for the control volumes and somewhat different ways of calculating the dissipation. Jaworski and Fort (1991) measured axial velocity and pressure profiles using a three-hole Pitot tube and reported that the power was distributed as follows: 32% in the impeller region; 54% in the region below the impeller; and 14% in the remaining volume of the tank. Macro-scale energy balances reported by Zhou and Kresta (1997) using LDA gave: 52% in the impeller region; 23.1% in the impeller discharge region; and 24.9% in the remaining volume of the tank. In this model, the dimensions of the zones are refined to give better resolution and the percent of the power dissipated in each zone is determined using the local dissipation measurements shown in Figure 1(b) (Zhou, 1997). An overall energy balance in the tank is satisfied. The zone equations are presented below and summarized in Table 1.
MODEL Accurate prediction of dissolution time for slightly miscible systems is difficult (if not impossible) because of the multimechanism processes taking place simultaneously, coupled with the turbulence inhomogeneity in the stirred tank. Prediction of the dissolution time requires an accurate description of the hydrodynamics of the stirred tank, breakup, coalescence and solute mass transfer, and an accurate and stable numerical scheme to resolve the systems of equations for each drop size class in the polydispersed system. Discretization of the mixing tank into the tiny cells (grids) required for CFD increases the computational time and imposes a precision that far exceeds the accuracy of the breakup and dissolution equations. The modelling approach used here is the zone model, which uses zones to represent regions where significant variations in local hydrodynamic properties exist within the mixing tank. Each zone has a residence time and energy dissipation rate, e , with the shortest residence times and the most intense turbulence in the four zones which define the impeller region. The three zones just below the impeller have widely varying residence times and energy dissipation rates and each drop travels through only one of these three zones on each pass through the impeller, thus building up a distribution of drop experiences over time. Reducing the number of discretized regions just to the extent that steep variations in turbulence and a distribution of drop experiences over time can be accurately represented provides dramatic savings on the computational time and resources. In each mixing zone, the turbulence and flow properties are defined, the drop breakup and mass transfer are modelled using published correlations; and the resulting transient drop size population balances are solved.
Mixing Zones The mixing zone model is defined for a D ¼ T/3 PBTD geometry, identical to that used in the experiments. Based on the varying levels of turbulence shown in Figure 1(b),
Zone 1: Impeller swept volume The volumetric flow rate through the impeller is Q1 ¼ NQND 3 and the turbulent kinetic energy dissipation rate (e ) is set to 22% of the total power (P) injected, divided by the mass of fluid in the impeller swept volume (rcV1). The impeller volume is V1 ¼ (pD 3 cos(458))/20 and the residence time (t1) for the fluid in this region is the ratio of the impeller swept volume to the volumetric flow rate.
Zones 2, 3 and 4: Impeller discharge region The velocity and local energy dissipation profiles at the lower edge of the impeller reveal a maximum close to the tip of the blade (Hockey and Nouri, 1996; Jaworski et al., 1996). Close to the hub, the energy dissipation and velocity drop close to the bulk values. In order to provide an orderly transition from one extreme to the other, three zones are used to define the impeller discharge region. The volume of each zone is annular; Vk ¼ p(z2,k 2 2 2 z1,k)(r2,k 2 r1,k ), where r1 and r2 define the radial limits and (z2 2 z1) is the height of the discharge zone, 0.2 D. The volumetric flow rate for each zone was obtained by integrating the axial velocity profile over the radial limits: rð 2:k
Qk ¼ 2 p
Urdr
(1)
rl:k
The width of the trailing vortex is 0.15 D from the tip of the impeller blade, and this is represented by zone 2, where 38% of the power is dissipated. Zone 3 is next to the trailing vortices and has an annulus width of 0.1 D; 10% of the power is dissipated in zone 3. Zone 4 is a disk extending from the centre of the tank to 0.25 D, with 2% of the total energy dissipation. A total of 50% of the energy is dissipated in the impeller discharge stream.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710–720
712
IBEMERE and KRESTA
Figure 1. (a) Half-plane showing the dimensions of the mixing zones and (b) profiles of the dissipation rate under the impeller, showing the limits of the mixing zones.
Zone 5: bulk The volume of the bulk is obtained by subtracting the volumes of zones 1, 2, 3 and 4 from the total tank volume. The volumetric flow rate in the bulk is equal to the volumetric flow rate into the impeller (zone 1) by
continuity, since all the fluid from the impeller discharge region is unfolded into the bulk and re-entrained into the impeller suction. The power dissipated in the bulk must be 28% of the total power input in order to ensure an overall energy balance.
Table 1. Defining equations for each zone in the mixing field. zone (1) Impeller swept volume (2) Discharge stream: trailing vortex (3) Discharge stream: transition (4) Discharge stream: hub region (5) Bulk
Dimensions (m) z1 ¼ D z2 ¼ 1.15 D r1 ¼ 0.0 r2 ¼ 0.5 D z1 ¼ 0.8 D z2 ¼ D r1 ¼ 0.35 D r2 ¼ 0.5 D z1 ¼ 0.8 D z2 ¼ D r1 ¼ 0.25 D r2 ¼ 0.35 D z1 ¼ 0.8 D z2 ¼ D r1 ¼ 0.0 r2 ¼ 0.25 D Everywhere else
Volume (m3) Vi ¼ p(z2 2 z1)(r22 2 r12)
Volumetric flow rate (m3 s21)Qi
Turbulent energy dissipation rate (W kg21)1i
Residence time (s) t ¼ Vi/Qi
0.11 D 3
NqND 3 ¼ 0.79 ND 3 0.22 NpN 3D 5/V1 ¼ 2.368 N 3D 2
V1/NqND 3 ¼ 0.15/N
0.08 D 3
0.17 ND 3
0.38 NpN 3D 5/V2 ¼ 6.033 N 3D 2
0.47/N
0.04 D 3
0.41 ND 3
0.1 NpN 3D 5/V3 ¼ 3.34 N 3D 2
0.093/N
0.04 D 3
0.21 ND 3
0.02 NpN 3D 5/V4 ¼ 0.651 N 3D 2
0.186/N
20.93 D 3
0.79 ND 3
0.28 NpN 3D 5/V5 ¼ 0.017 N 3D 2
26.49/N
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710– 720
MODELLING THE MIXING AND DISSOLUTION KINETICS OF PARTIALLY MISCIBLE LIQUIDS Population Balance
Drop Breakup
The classical population balance equation is applied to each mixing zone, k, to give @Ni @Nm,i @Nb,i ¼ þ rz (UNi ) @t @t @t k
(2)
where Nm is the change in the number of drops of size i due to mass transfer, Nb is the change due to breakup, and the third term is the change in zone k due to convection. Although ¯ Ni) vanishes when integration is the convective term rz . (U done over the batch stirred vessel, it is retained in each mixing zone because it reflects the redistribution of the drops. The drop population must be discretized such that the simulation is both accurate and numerically stable. Drop breakup and dissolution both lead to a reduction in drop size, and most of the change occurs at small drop sizes, so a geometrically increasing discretization was selected. The geometric length scale requires three parameters to fully describe the discretization: the minimum drop size, ‘0, maximum drop size, ‘p, and the number of drop size classes, p. The range of drop sizes in the model was set to match the range of the experimental measurements (‘0 ¼ 5 mm, ‘p ¼ 450 mm). The number of size classes required to accurately match the numerical solution to an analytical solution was p ¼ 80 (Ibemere, 2005), giving 1=p ‘p rg ¼ ¼ 1:05786 ‘0
(3)
‘i ¼ rg ‘i1
(4)
rg þ 1 ‘i þ ‘i1 Li ¼ ¼ ‘i ¼ 0:9727‘i 2 2rg rg 1 DLi ¼ ‘1 ‘i1 ¼ ‘i ¼ 0:0547‘i rg
(5) (6)
where p is the total number of drop size classes, rg is the geometric factor, ‘0 is the minimum drop size, ‘p is the maximum drop size, ‘i is the upper limit of size class i, Li is the drop size in class i, and DLi is the geometric spacing for the ith class size. As the drops dissolve, they are removed from interval i by reducing the number of drops in interval i by an equivalent volume, rather than by changing the drop diameter continuously. The experimental drop size distribution acquired after 20 s was fit to a log-normal distribution and used to initialize the simulation. The total number density in the tank was then determined using a volume balance on the undissolved material: N(Li , t) ¼ P(Li )
Vd p P i¼0
P(Li )
¼
pL3i =6
Ni NT V d p P NT pNi L3i =6
713
(7)
i¼0
where Ni is the number of drops in interval i, P(Li) is the log normal probability function, Vd is the total volume of the undissolved solute in the stirred tank at t, and ‘iþ1 represent the interval limits for the ith drop class size. The drops were divided into the five mixing zones according to the volume of each zone. Subsequent experimental drop size distributions were compared with the population in zone 5, the bulk.
The general expression for the drop breakup rate is @Nb (L0 ) ¼ g(L0 )N(L0 , t) @t ð1 m(L0 )f (L, L0 )g(L0 )N(L0 , t)dL þ
(8)
L¼L0
The first term on the right hand side of the breakup equation accounts for the rate of breakup of drops of diameter size L0, while the second term accounts for the rate of formation of drops of size L0 from the breakup of drops of sizes larger than L0. The function g(L0) is the drop breakup frequency, f (L, L0) is the size probability density function of the fragments, m(L0) is the number of fragments resulting from the breakup of a particle of diameter L0, and @Nb (L0 )=@t gives the net change in the number of drops of size L0 that has occurred during the time step. For binary breakup, m(L0) ¼ 2 for all drop sizes. The drop breakup frequency model proposed by Luo and Svendsen (1996) was selected for this work because it requires no experimentally determined or fitted parameters. Alopaeus et al. (1999) provide a numerically explicit formulation of this model using incomplete gamma functions: !1=3 2:4738(1 f) 1 g(Li , Lj ) ¼ b8=11 L2j 8 9 > < (G(8=11, tm ) G(8=11,b))þ > = 2b3=11 (G(5=11, tm ) G(8=11, b)) > > : 6=11 ; þb (G(2=11, tm ) G(2=11, b)) 5:8634Cf s b¼ rc 12=3 L5=3 j 2 2=3 L3 Li Cf ¼ þ 1 i 1 Lj Lj 11=3 h tm ¼ b Lj 3 1=4 n h¼ 1
(9)
(10)
(11) (12) (13)
The local dissipation (e ) in each zone is used to calculate the corresponding Kolmogorov microscale (h). Equation (9) is used to find g(Lo) for a finite length interval as g(Li ) ¼
i1 X g(Li , Lj ) j¼1
2
DLj
(14)
The probability distribution of daughter drops (Konno et al., 1980) can be approximated by a combination of gamma functions (Lasheras et al., 2002): f (Li , Lj ) ¼ p X
8 G(12) Li Li 2 1 G(9)G(3)Lj Lj Lj
f (Li , Lj )DLj ¼ 1:0
(15) (16)
j¼0
The change in drop size distribution due to breakup for a
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710–720
714
IBEMERE and KRESTA solving for @L=@t gives
specific drop size class Li can now be obtained @Nb (Li ) ¼ g(Li )N(Li , t) @t þ
p X
dLi 2kL,i (Cs Cbuilk ) ¼ dt rd
2f (Li , Lj )g(Lj )N(Lj , t)
(17)
j¼iþ1
where p is bin number for the largest drop size. The first and second terms on the right hand side denote the rates of breakup and formation of drops of size Li. The functions f (Li, Lj) and g(Li) are calculated once, at the beginning of a run, for all size classes. The breakup rate is then calculated at each time step for each size class, Li. This breakage function is valid only for drops larger than the Kolmogorov microscale, and assumes that the inertial range of turbulence (25/3 slope of the energy spectrum) is fully developed in the flow.
Mass Transfer Correlations for mass transfer coefficients depend on the phases involved, the stirred tank geometry, and the dispersed phase fraction. Only mass transfer correlations for pure dissolving solids or liquid –liquid extractions were found in the literature, and neither is perfectly suited to the study of pure dissolving drops. Although solid dissolution and liquid – liquid extraction both result in the transfer of solute from particles into the continuous phase, they both differ considerably from the dissolution of pure dissolving drops. Pure dissolving drops shrink until they disappear completely, like solid particles, while in liquid–liquid extraction the drop size is nearly constant with time. Pure dissolving drops have no mass transfer resistance on the drop side, again like dissolving solids, while liquid–liquid extraction systems present mass transfer resistances from both immiscible liquids. Conversely, drops are subject to deformation, breakup, internal circulation and oscillation, unlike solid particles, and surface motion on drops increases the slip velocity relative to a solid sphere. Reviews of mass transfer correlations for liquid–liquid extraction in stirred tanks are given in Perry’s (1997) Handbook (Tables 5–21 to 5–28) and Kumar and Hartland (1999), while the topic of solids dissolution is discussed by Pangarkar et al. (2002) and Atiemo-Obeng et al. (2004). The selection of an appropriate correlation for the mass transfer coefficient was based on the similarity of geometries, phases present, and performance in the model. Table 2 gives a selection of the correlations considered. The dependence of the mass transfer coefficient on particle size varies, with exponents ranging from 20.7 to 0.33 for both solid–liquid and liquid–liquid extraction systems. In view of the uncertainty on the exponent for particle size, several correlations were compared to experimental data for this system (Ibemere, 2005). The correlation by Glen (1965) gave the best results, as shown in Figure 2, and is restated here in terms of the length scale L: kL ¼
DAB Dr 1=2 L 1=3 ReSc1=3 D D rd
(18)
The mass transfer rate for a single drop of size L is dm @Vd @Vd @L ¼ kL (pL2 )(Cs Cbulk ) ¼ rd ¼ rd dt @t @L @t
(19)
(20)
and the number of drops, dni,out that will shrink into the (i-1)th class size from the i th class size in time dt is given by dNi dLi Ni ¼ (21) dt out dt DLi Similarly, a number of drops will shrink into the i th class size from the (i þ 1)th class size according to dNi dLiþ1 Niþ1 ¼ (22) dt in dt DLiþ1 and the overall change in the ith size class due to dissolution is dNi 2(Cs Cbulk ) Niþ1 Ni ¼ kL,iþ1 kL,i (23) dt rd DLiþ1 DLi
Material Balance Mass conservation of the dispersed phase concentration in the stirred tank is ensured by closing the material balance between the continuous and dispersed phases. The stirred tank is assumed to be well mixed, so the solute concentration in the continuous phase can be obtained by difference: Ni L3i k¼1 i¼1 ¼ p 5 P P VTANK p=6 Ni L3i md,0 rd p=6
Cbulk
p 5 P P
(24)
k¼1 i¼1
where Cbulk is the solute concentration in the bulk at any time, md,0 is the initial mass of dispersed phase added, and VTANK is the total fluid volume of the stirred tank (dispersed phase volume inclusive). Recall that k is the mixing zone index.
Numerical Solution The limiting time scale Dt based on a Courant number of 0.5 is given by Dt ¼
DL1 Vk or Dt ¼ 2@L1 =@t 2Qk
(25)
where the first timescale is based on the rate of mass transfer from the smallest drops, and the second timescale is based on the convection time scales through each of the five mixing zones. The smallest of the six calculated timesteps is used for the integration. A modified Hounslow discretization with forward differencing (Hounslow et al., 1988; Hounslow, 1990; Ibemere, 2005) was implemented. This scheme predicted the first four moments of the distribution without any error and eliminates oscillations which occur at the small end of the drop size distribution when using either the backward or central differencing schemes.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710– 720
MODELLING THE MIXING AND DISSOLUTION KINETICS OF PARTIALLY MISCIBLE LIQUIDS
715
Table 2. Particle size dependence for different mass transfer coefficient correlations. S-L denotes solid–liquid dissolution; L-L denotes liquid–liquid extraction; and G-L denotes gas–liquid mass transfer. Researcher
System studied
Tank configuration
Correlation for kL 0
Asai et al. (1988)
S-L
Baffled cylindrical flat bottomed stirred tank with turbine impeller
0 15:8 11=5:8 !0:58 11=2 dp4=3 n 0:33 A C DAB B 5:8 @ @2 þ 0:61 A DAB n dp
Exponent on dp
0
0 1 1 !1=3 0:5 1dp4 n DAB B A C @2 þ 0:49@ 3 A DAB n dp
Kuboi (1974)
S-L
Baffled stirred tank with turbine impeller
Levins and Glastonbury (1972)
S-L
Baffled tank with flat, if Dr is not significant; curved, and 0 ! 4=3 1=3 0:62 0:17 pitched blade D n 0:36 DAB @2 þ 0:44 dp 1 turbine impellers T DAB n dp
if Dr is significant;
Miller (1971)
S-L
Baffled stirred tank with turbine impeller
Sano et al. (1974)
S-L
Baffled stirred tank with turbine impeller
Boyadzhiev and Elenkov L-L (1966)
Baffled stirred tank with turbine impeller
Glen (1965)
L-L
Six blade disk turbine in baffled stirred tank
Skelland and Moeti (1990)
L-L
Six flat blade turbine in baffled stirred glass vessel
20.23
0
1=3 4=3 @2 þ 0:222 1 dp n
0
4=3 1=3 @2 þ 0:4 dp 1 n
! dp m 0:5 n 0:38 DAB 2 þ 0:47 DAB n dp !4=3
!0:62
n DAB
n DAB
1 0:33 A DAB dp
0:33
1
Af DAB dp
20.33
20.18
2 0.5
20.7
20.2
Lamont and Scott (1970)
Calderbank and Moo-young (1961)
G-L studies Aerated mixing compared vessels and with L-L data columns
DAB m 0:5 n 1=6 0:65 DAB dp
20.5
2 0:33 5=4 D N dp Dr n 0:33 DAB n rd DAB D D
0.33
0:67 2 0:5 2 5=12 D2 N D dp DN n dp T g !5=4 rdp2 g n 0:33 0:5 DAB f DAB s dp
1:23 105
! 0:25 P n 0:67 P (gDr)4=3 m1=3 n where ¼ A 0:13 V DAB V r2=3
EXPERIMENTAL For phases that are not in equilibrium, the rate of mass transfer and/or chemical reaction is greatly influenced by the available interfacial area between the phases and the deviation from equilibrium (i.e., the chemical potential between the two phases). Investigations into the mass transfer kinetics coupled with drop breakup and shrinkage require simultaneous measurement of the concentration of the solute and the interfacial area available for mass transfer. The drop
0.0
0.0
size distribution defines the available interfacial area for mass transfer while the solute concentration defines the extent of deviation from equilibrium. Accurate measurement of these parameters allows a full model validation and direct determination of the overall mass transfer coefficient, kL. In the experiments, a flat bottomed cylindrical glass tank is mounted inside a flat bottomed square glass tank and the space between the tanks is filled with water to minimize refraction. The tank diameter equals the liquid height, H ¼ T ¼ 145 mm. Four equally spaced baffles (W ¼ T/10)
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710–720
716
IBEMERE and KRESTA the impeller at 2r/D ¼ 0.8 to give the maximum data rate. The position of the probe does not affect the measured DSD (Zhou and Kresta, 1998). The gas chromatograph (GC) and phase Doppler particle analyser (PDPA) were used in their standard configurations. 2-Butanone was used as the internal standard for the GC, giving a linear calibration curve with a coefficient of determination of 0.99 (diethyl malonate concentration versus area ratio). The PDPA is an extension of laser Doppler velocimetry (LDV) and allows simultaneous measurement of drop diameter and velocity. The PDPA used in this work (Zhou and Kresta, 1998) has a focal length of 500 mm and a drop diameter range of 1.1 mm , d , 338 mm at a collection angle of 22.58. The PDPA imposes some constraints on the selection of continuous and dispersed phases. The liquid has to be transparent, and the ratio of refractive indices between the continuous phase and the dispersed phase must be moderate. It is also desireable to have a small density difference, and low viscosities. The continuous phase is deionized ultra-filtered (DIUF) water, which is completely free of particulate. The dispersed phase is diethyl malonate. Diethyl malonate has a specific gravity of 1.055, a viscosity of 2.15 mPa s, a surface tension of 31.83 mN m21, a refractive index of 1.41, and a solubility limit of 22.6 kg m23 in water at 208C. For each experiment, 25 ml of diethyl malonate was pumped into the mixing tank over a period of 20 s. The feed time was selected to ensure that all drops experienced turbulence and breakup at the impeller before being convected into the bulk of the tank. Drop size data was collected continuously using the PDPA, and concentration measurements were taken every 20 s. The GC samples were collected using a 2 mm pipette placed 2–3 cm above the impeller blade. The mouth of the pipette was significantly larger than the drops of solute. The samples were allowed to settle for about 10 min at a 458 angle. This angle of inclination allows fast separation of the drops, and gave more repeatable results than fast centrifugation. Because of the small settling velocity, the mass transfer during settling can be neglected. Using a micropipette, 1.2 ml of the continuous phase was withdrawn from the supernatent, combined with 0.02 ml of 2-butanone, and injected into the GC.
RESULTS AND DISCUSSION Figure 2. Comparison of model and experimental results for the cumulative drop size distribution as it evolves over time. (a) The correlation by Glen (1965) gives the best fit to the experimental data for cumulative number density at 650 rpm; (b) the fit at 550 rpm is poor and (c) the prediction at 750 rpm is quite good. This figure is available in colour online via www.icheme.org/cherd.
are placed around the periphery of the mixing tank and the tank is agitated using a four-bladed, down-pumping, stainless steel 458 pitched blade turbine (PBT) impeller, with a diameter (D ¼ T/3) and blade width (w ¼ D/5) placed at an off-bottom clearance equal to the impeller diameter (C/D ¼ 1.0). The dissolution kinetics were measured at three impeller speeds (550, 650 and 750 rpm); all higher than the impeller speed required for fully turbulent flow (510 rpm). A 3 mm internal diameter feed tube, placed at an offset of 20 mm from the impeller shaft and 4 mm above the impeller blades is used to feed the dispersed phase. The drop size distribution (DSD) was measured just below
Other than the selection of a mass transfer correlation, there are no fitting parameters in the model. The model results are evaluated in terms of the concentration versus time, the evolution of the Sauter mean diameter over time, and the evoluton of the drop size distribution over time. It will be shown that the model results are more accurate for higher rotational speeds, so further analysis is done at 750 rpm or higher. The final objective is to determine how the rate of mass transfer for various liquid –liquid systems compares with the blend time for completely miscible systems. We emphasize that this comparison is meant to illustrate the large differences in the time scales of these two processes. It will be evident to the reader from the results that it is not a physical dimensionless group, since the mass transfer variables dominate the dissolution time. The first thing that was determined is the relative importance of drop breakup and mass transfer. The liquid was slowly injected directly into the most turbulent part of the tank, and the first measurements were started at 20 s. The
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710– 720
MODELLING THE MIXING AND DISSOLUTION KINETICS OF PARTIALLY MISCIBLE LIQUIDS
717
initial drop size distributions at all three rotational speeds were completely composed of drops smaller than the limiting drop size for breakup in zone 2. At 550 rpm, the breakup rate is negligible for drops smaller than 300 mm. At 650 rpm, the breakup rate is neglible for drops smaller than 280 mm; and at 750 rpm, the breakup rate is neglible for drops smaller than 240 mm. From the initial cumulative drop size distributions shown in Figure 2, it is clear that that there is no drop breakup in the tank after the first 20 s at any rotational speed. This means that mass transfer will be the dominant mechanism in the reduction of the drop size distribution. Changes in surface tension of a factor of 1000 gave the same simulation results for both the dissolution time and the drop size distribution, confirming this observation. Of course, the initial drop breakup is critical to the generation of surface area for mass transfer, but because of the nature of the liquids and the experimental design, this part of the model could not be directly validated. Figure 2 compares the predicted and experimental normalized drop size distributions. The normalized number density is given by ni ¼ Ni/NT DL where N is the absolute number of drops in a size class, NT is the total number of drops in the distribution and DL is the size class interval. The correlation by Glen [Figure 2(a)] gave the best fit to the experimental data at 650 rpm and was used for all subsequent simulations. At 550 rpm [Figure 2(b)], the model fails to accurately predict the drop size distribution and the flow is just at the lower limit of turbulent conditions (Re ¼ 21 700), but at 750 rpm [Figure 2(c)], the model agreement with experiment is very good. As the rotational speed is increased from 550 to 750 rpm, the dissolution time drops by a factor of two. This is a very significant effect, which should also be apparent in the mean drop size and bulk concentration results. Figure 3 shows the model predictions and experimental results for mean drop size. The agreement is better than the detailed results shown in the drop size distribution data, because the differences across the distribution are averaged out. The model tracks the mean diameter equally well at all three rotational speeds, across significant changes in the rate of mass transfer. As the impeller rotational speed increases, fewer data points are measured due to reduced dissolution time, but the reader will recall that the mean diameter does not tell us anything about how many drops are
present in the tank, so it cannot be used to accurately determine the dissolution time. The bulk concentration, shown in Figure 4, is used to determine the dissolution time. The concentration in the bulk is fixed in both the model and the experiment by a mass balance. The initial measured concentration corresponding to the initial drop size distribution 20 s after injection of diethyl malonate in the tank is about 10.5 kg m23. This dissolved concentration represents about 85% of the injected diethyl malonate indicating very fast dissolution rates in the early seconds of the process. Once again, the model predictions are much better at higher rotational speeds. The model results at 550 rpm cannot be considered reliable. In order to determine the range of dissolution times which might be observed with various partially miscible liquid – liquid systems, three additional liquid were selected for simulations: O-xylene, toluene and amyl acetate. Their properties and the simulation results are given in Table 3. The saturation concentrations range from 22.6 kg m23 to 0.1 kg m23 and all of the simulations were performed at 50% of the saturation concentration. While the raw dissolution time is given in Table 3, it is more useful to consider the ratio of dissolution time to blend time. This gives an indication of the relative magnitude of the two processes, and gives a clearer picture of the impact of N on the results. The ratio of dissolution time to blend time ranges from 91 to 190, as shown in Figure 5. While the results at 750 rpm are linear, extrapolation of this trend is not supported by the last point for diethyl malonate at 11.3 kg m23 and a dissolution/blend time ratio of 91. While increasing the experimental rotational speed from 550 to 750 rpm halved the dissolution time; further increasing the rotational speed by a factor of 4 reduces the dissolution time by less than 30%. Operating at this very high speed (3350 rpm) will increase the power consumption by a factor of 64, and at the experimental scale is not physically achievable. Clearly, when dissolution times are to be minimized, the best operating conditions will be achieved by using high, but not excessive, rotational speeds. These results define some interesting bounds for the problem of liquid dissolution in stirred tanks. We suggest that further studies revisit the question of a physically meaningful local mass transfer coefficient, and variables such as the percent approach to saturation or the absolute driving force for
Figure 3. Sauter mean diameter decay with time at three rotational speeds.
Figure 4. Dissolved dispersed phase concentration profile.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710–720
718
IBEMERE and KRESTA
Table 3. Model predictions for sample systems of liquids dispersed in water (m ¼ 0.894 mPa s) at 50% of their saturation concentration. Dispersed phase
Saturation concentration (Kg m23)
Initial concentration difference (kg m23)
Interfacial tension (mN m21)
O-Xylene
0.175
0.08875
30.31
Toluene
0.5
0.25
28.50
Amyl acetate
1.0
0.5
25.67
22.6
11.3
31.83
Diethyl malonate
N(rpm)
Dissolution time (s)
Dissolution to blend time ratio
750 3350 750 3350 750 3350 750 1500
700 155 645 125 560 100 340 130
190 190 175 150 150 120 91 70
Simulation studies done using the model show that the dissolution times are of the order of 100 –200 times the blend time, and that the dissolution time to blend time ratio decreases with increased agitation. The dominant variables affecting the mass transfer rate are the approach to the solubility limit, expressed as the absolute concentration difference, and the rotational speed of the impeller. An optimum balance between rotational speed and power consumption was observed. The main outcomes of the work are the problem definition (dissolution rates for partially miscible fluids are much longer than the blend time for completely miscible fluids) and identification of the key parameters which determine the dissolution time. Further research is warranted, particularly with respect to the effect of less optimum feed strategies and systems in which drop breakup kinetics would have a more significant effect. Figure 5. Effect of the initial concentration difference on the ratio of dissolution time to blend time for selected process fluids (see Table 3 for details).
mass transfer rather than a simple scaling with the blend time.
CONCLUSIONS An experimental procedure for the simultaneous measurement of transient drop size distribution and the corresponding solute concentration of a slightly miscible liquid–liquid system (diethyl malonate dispersed in deionized ultra filtered water) was developed. A mechanistic model which captures the inhomogeneity in the stirred tank using 5-mixing zones with local dissipation energy rates estimated from experimental data tracks the transient drop size distribution in the stirred tank using population balances and published correlations for drop breakup and mass transfer, predicts the solute concentration in the bulk of the stirred tank and the dissolution time. The model was validated by comparing the model predictions with experimental results for the solute (diethyl malonate) concentration in the tank; Sauter mean diameter, drop size distribution and cumulative number distribution of the dispersed drops. The model gave good agreement with experimental results, although the subtleties of the experimental conditions meant that the effects of drop breakup could not be directly validated. In fact, it was found that differences in interfacial tension are quite small for many process fluids, so this limitation is not as important as was initially expected.
NOMENCLATURE b C Cbulk Cf CS d dp D DAB f g g H i, j k kL L Li ‘0 ‘i DLi md m N ni Ni NP NQ NT P p P Q rg r Re Sc
constant off-bottom impeller clearance, m bulk concentration in stirred tank, mol L21 or kg m23 constant solute concentration at equilibrium, mol L21 or kg m23 drop diameter, m particle diameter, m impeller diameter, m molecular diffusion coefficient, m2 s21 probability density function of fragments drop breakup frequency, s21 gravitational constant, ms22 initial fluid height, m subscripts, drop interval index subscript, mixing zone index mass transfer coefficient, m s21 drop size, m drop size in interval i, m smallest drop size, m upper limit of drop size interval i, m spacing of i th size interval, m mass of dispersed drops, kg number of daughter drops formed impeller rotational speed, rps or rpm number of drops in interval i normalized with DLi, m21 number of drops in interval i impeller power number impeller flow number total number of drops impeller power requirement, W total number of drop size classes log normal probability function volumetric flow rate, m3 s21 geometric size interval factor radial distance, m Reynolds number Schmidt number
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710– 720
MODELLING THE MIXING AND DISSOLUTION KINETICS OF PARTIALLY MISCIBLE LIQUIDS t tm Dt T U Vd Vk VTANK w W z
time, s constant time step, s tank diameter, m axial velocity, m s21 total volume of drops, m3 volume of mixing zone k, m3 total volume of stirred tank, m3 blade width, m baffle width, m axial distance, m
Greek symbols e n rc rd Dr s f h m
rate of kinetic energy dissipation, W kg21 kinematic viscosity, m2 s21 continuous phase density, kg m23 dispersed phase density, kg m23 density difference between phases, kg m23 interfacial tension, N m21 volume fraction dispersed phase Kolmogorov length scale, m dynamic viscosity, kg m21 s21
REFERENCES Alopaeus, V., Koskinen, J. and Keskinen, K.I., 1999, Simulation of the population balances for liquid-liquid systems in a non-ideal stirred tank. part 1: description and quantitative validation of the model, Chem Eng Sci, 54(24): 5887– 5899. Asai, S., Konishi, Y. and Sasaki, Y., 1988, Mass-transfer between fine particles and liquids in agitated vessels, J Chem Eng Japan, 21(2): 107–112. Atiemo-Obeng, V.A., Penney, W.R. and Armenante, P., 2004, Solidliquid mixing, in Paul, E.L. et al. (eds). The Handbook of Industrial Mixing. (Wiley, New York, USA). Attarakih, M.M., Bart, H.J. and Faqir, N.M., 2004, Numerical solution of the spatially distributed population balance equation describing the hydrodynamics of interacting liquid-liquid dispersions, Chem Eng Sci, 59(12): 2567–2592. Bajpai, R.K., Ramkrishna, D. and Prokop, A., 1976, Coalescence redispersion model for drop size distributions in an agitated vessel, Chem Eng Sci, 31(10): 913– 920. Baldyga, J., Podgorska, W. and Pohorecki, R., 1995, Mixing-precipitation model with application to double feed semi batch precipitation, Chem Eng Sci, 50(8): 1281–1300. Bayens, C.A. and Laurence, R.L., 1969, A model for mass transfer in a coalescing dispersion, Ind & Eng Chem Fund, 8(1): 71. Boyadzhiev, L. and Elenkov, D., 1966, On the mechanism of liquidliquid mass transfer in a turbulent flow field, Chem Eng Sci, 21(11): 955–959. Calderbank, P.H. and Moo-young, M.B., 1961, The continuous phase heat and mass-transfer properties of dispersions, Chem Eng Sci, 16(1–2): 39– 54. Coulaloglou, C.A. and Tavlarides, L.L., 1977, Description of interaction processes in agitated liquid-liquid dispersions, Chem Eng Sci, 32(11): 1289– 1297. Curl, R.L., 1963, Dispersed phase mixing I: theory and effects in simple reactors, AIChE J, 9(2): 175– 181. Glen, J.B., 1965, Thesis, cited from: Skelland, A.H.P. and Kanel, J.S., Simulation of mass transfer in a batch agitated liquid-liquid dispersion, Ind Eng Chem Res, 1993, 31(3): 908– 920. Grenville, R.K., 1992, Blending of viscous Newtonian and pseudoplastic fluids, PhD dissertation, Cranfield Institute of Technology, Cranfield, Bedfordshire, UK. Hemsing, J.L., 2001, Kinetics of blending for soluble polymer additives, MSc dissertation, Department of Chemical & Materials Engineering, University of Alberta, Canada. Hockey, R.M. and Nouri, J.M., 1996, Turbulent flow in a baffled vessel stirred by a 608 pitched blade impeller, Chem Eng Sci, 51(19): 4405–4421. Hounslow, M.J., Ryall, R.L. and Marshall, V.R., 1988, A discretized population balance for nucleation, growth and aggregation, AIChE J, 34(11): 1821– 1832. Hounslow, M.J., 1990, A discretized population balance for continuous systems at steady state, AIChE J, 36(1): 106– 116.
719
Hsia, M.A. and Tavlarides, L.L., 1980, A simulation model for homogeneous dispersions in stirred tanks, Chem. Eng. J. & BioChem. Eng. J., 20(3): 225 –236. Hsia M.A. and Tavlarides, L.L., 1983, Simuation analysis of drop breakage, coalescence and micromixing in liquid-liquid stirred tanks, Chem. Eng. J. & Biochem. Eng. J., 26(3): 189–199. Ibemere, S., 2005, Dissolution kinetics of liquid-liquid dispersions using local rates of turbulent dissipation, MSc dissertation, Department of Chemical & Materials Engineering, University of Alberta, Canada. Jaworski, Z. and Fort, I., 1991, Studies on mixing .79.: Energy dissipation rate in a baffled vessel with pitched blade turbine impeller, Collection of Czechoslovak Chem Comm, 56(9): 1856–1867. Jaworski, Z., Nienow, A.W. and Dyster, K.N., 1996, An LDA study of the turbulent flow field in a baffled vessel agitated by an axial, down-pumping hydrofoil impeller, Can J Chem Eng, 74(1): 3– 15. Jeon, Y.M. and Lee, W.K., 1986, A drop population balance model for mass transfer in liquid-liquid dispersion I: Simulation and its results, Ind & Eng Chem Fund, 25(2): 293– 300. Kennedy, C., 2003, Mixing, precipitation and coagulation with alum, MSc dissertation, Department of Chemical & Materials Engineering, University of Alberta, Canada. Konno, M., Matsunaga, Y., Arai, K. and Saito, S., 1980, Simulation model for breakup process in an agitated tanks, J Chem Eng Jpn, 13(1): 67–73. Kresta, S., Anthieren, G. and Parsiegla, K., 2005, Model reduction for prediction of silver halide precipitation, Chem Eng Sci, 60(8/9): 2135–2153. Kuboi, R., Komasawa, I., Otake, T. and Iwasa, M., 1974, Fluid and particle motion in turbulent dispersion .3. Particle-liquid hydrodynamics and mass-transfer in turbulent dispersion, Chem Eng Sci, 29(3): 659 –668. Kumar, A. and Hartland, S., 1999, Correlations for prediction of mass transfer coefficients in single drop systems and liquid-liquid extraction columns, Chem Eng Res & Des, 77(A5): 372–384. Lamont, J.C. and Scott, D.S., 1970, An eddy cell model of mass transfer into the surface of a turbulent liquid, AIChE J, 16(4): 513– 519. Lasheras, J.C., Eastwood, C., Martinez-Bazan, C. and Montanes, J.L., 2002, A review of statistical models for the break-up of an immiscible fluid immersed into a fully developed turbulent flow, International Journal of Multiphase Flow, 28: 247–278. Levins, D.M. and Glastonbury, J.R., 1972, Particle-liquid hydrodynamics and mass transfer in a stirred vessel II: mass transfer, Trans Inst Chem Eng, 50(2): 132. Luo, H. and Svendsen, F., 1996, Theoretical model for drop and bubble breakup in turbulent dispersions, AIChE J, 42(5): 1225– 1233. Maggioris, D., Goulas, A., Alexopoulos, A.H., Chatzi, E.G. and Kiparissides, C., 2000, Prediction of particle size distribution in suspension polymerization reactors: effect of turbulence nonhomogeneity, Chem Eng Sci, 55(20): 4611– 4627. Miller, D.N., 1971, Scale-up of agitated vessels – mass transfer from suspended solute particles, I&EC Proc Des Dev, 10(3): 365. Pangarkar, V.G., Yawalkar, A.A., Sharma, M.M. and Beenackers, A.A.C.M., 2002, Particle-liquid mass transfer coefficient in two-/ three-phase stirred tank reactors, Ind Eng Chem Res, 41(17): 4141–4167. Park, J.Y. and Blair, L.M., 1975, The effect of coalescence on drop size distribution in an agitated liquid– liquid dispersion, Chem Eng Sci, 30 (9): 1057–1064. Perry, R.H. and Green, D.W. (eds). Perry’s Chemical Engineers’ Handbook, 1997, 7th Edition (McGraw-Hill, New York, USA). Ramkrishna, D., 1974, Drop-breakage in agitated liquid-liquid dispersions, Chem Eng Sci, 29(4): 987–992. Ramkrishna, D., 2000, Population Balances: Theory and Applications to Particulate Systems in Engineering (Academic Press, San Diego, USA). Sano, Y., Yamaguchi, N. and Adachi, T., 1974, Mass transfer coefficients for suspended particles in agitated vessels and bubble columns, J Chem Eng Japan, 7: 255–261. Shah, B.H., Ramkrishna, D. and Borwanker, J.D., 1977, Simulation of particulate systems using concepts of interval of quiescence, AIChE J, 23(6): 897– 904. Skelland, A.H.P. and Kanel, J.S., 1992, Simulation of mass transfer in a batch agitated liquid-liquid dispersion, Ind Eng Chem Res, 31(3): 908– 920.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710–720
720
IBEMERE and KRESTA
Skelland, A.H.P. and Moeti, L.T., 1990, Mechanism of continuousphase mass-transfer in agitated liquid-liquid systems, I&EC Res, 29(11): 2258– 2267. Spielman, L.A. and Levenspiel, O., 1965, A Monte Carlo treatment for reacting and coalescing dispersed phase systems, Chem Eng Sci, 20(3): 247. Tsouris, C. and Tavlarides, L.L., 1994, Breakage and coalescence models for drops in turbulent dispersions, AIChE J, 40(3): 395– 406. Valentas, S.K. and Amundson, N.R., 1966, Breakage and coalescence in dispersed phase systems, Ind & Eng Chem Fund, 5(4): 533. Valentas, S.K., Bilous, O. and Amundson, N.R., 1966, Analysis of breakage in dispersed phase systems, Ind & Eng Chem Fund, 5(2): 271.
Zhou, G. and Kresta, S.M., 1998, Evolution of drop size distribution in liquid-liquid dispersions for various impellers, Chem Eng Sci, 53(11): 2101– 2105. Zhou, G. and Suzanne M. Kresta, 1996, Distribution of energy between convective and turbulent flow for three frequently used impellers, Chem Eng Res Des, 74A: 379–389. Zhou, G., 1997, Characteristics of turbulence energy dissipation and liquid-liquid dispersions in an agitated tank, PhD dissertation, Department of Chemical & Materials Engineering, University of Alberta, Canada. The manuscript was received 3 October 2006 and accepted for publication after revision 2 January 2007.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A5): 710– 720