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Engineering

Procedía

ELSEVIER

Procedía Engineering 42 (2012) 1574-1591

www.elsevier.com/locate/procedia

20th International Congress of Chemical and Process Engineering CHISA 2012 25 - 29 August 2012, Prague, Czech Republic

PPBLAB: A New Multivariate Population Balance Environment

for Particulate System Modelling and Simulation

M. Attarakihab a*, S. Al-Zyoda, M. Abu-Khaderb, H. J. Bartcd

aThe University ofJordan, Faculty of Engineering & Technology, Department of Chemical Engineering, 11942Amman, Jordan bAl-Balqa Applied University, Faculty ofEng. Tech. Chem. Eng. Dept., POB 15008, 11134Amman, Jordan cChair of Separation Science and Technology, TU Kaiserslautern P.O. Box 3049 - 67653 Kaiserslautern, Germany dCentre of Mathematical and Computational Modelling, TU Kaiserslautern, Germany

This work presents a new windows-based MATLAB program, which is called PPBLAB (Particulate Population Balance Laboratory) for modelling and numerical simulation of particulate systems. As a first step, liquid-liquid extraction columns are modelled using the population balance equation as a mathematical framework. Up to date population balance models and solvers are incorporated. The discretization of the spatial domain is based on the finite volume method with flux vector splitting. A strongly stable semi-implicit first order time integration scheme is used to resolve such a large and stiff ODE system. The MATLAB GUI is used to make PPBLAB a user friendly program, which allows the user to define and simulate liquid extraction columns. A thermodynamics package TEA-COCO, which obtained from CAPE-OPEN, is linked to PPBLAB. Therefore, a special interface is designed for the purpose of data exchange between PPBLAB and CAPE-OPEN TEA tool. The solute distribution coefficient in ternary systems is predicted using the UNIQUAQ model with a special optimization tool to estimate the binary interaction parameters based on infinite conditions. A pilot plant Kuhni extraction column is simulated and tested. Full analysis and performance of the PPBLAB are carried out and validated against experimental data.

© 2012 Published by Elsevier Ltd. Selection under responsibility of the Congress Scientific Committee (Petr Kluson)

Keywords: PPBLAB; liquid extraction; CAPE-OPEN; population balance; Kuhni column

1. Introduction

a* Corresponding author. Tel.: +962-6-5355000; fax: +962-6-5355522. E-mail address: attarakih@yahoo.com, m.attarakih@ju.edu.jo.

Abstract

1877-7058 © 2012 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.07.538

Liquid-liquid extraction is considered as an important unit operation where it has various applications in the chemical industries [1,2].

Nomenclature

Greek symbols

UNIQAUQ binary interaction parameters

Breakage and coalescence birth source terms repressively. Solute concentration, kg/m3

Empirical parameters in the droplet coalescence and breakage probability models

Axial dispersion coefficient, m2/s

Droplet diameter, m

Sauter mean droplet diameter, m

Number concentration function, 1/m/m3

Acceleration of gravity, m/s2

Overall and individual mass transfer coefficients, m/s

Reynolds number

Schmidt number

Sherwood number

Time, s

Velocity vector, m/s

Particle volume, m3

Modified Weber number

Inlet of continuous or dispersed phases

Phase holdup

Daughter particle distribution, 1/m Particle breakage frequency, 1/s Energy dissipation, m2/s3

Velocity along particle space property, (property unit)/time

Mean number of daughter particles (droplets)

Viscosity ration ( y/ x)

Viscosity, Pa.s

Density, kg/m3

Interfacial tension, N/m

Coalescence frequency, m3/s

Subscripts

A Axial dispersion

D Drag coefficient

In Inlet

R Relative

X Heavy phase

Y Light phase

0 Terminal or slip

In counter-current extraction columns the state of the art is the one dimensional dispersion or back-mixing models, where the dispersed phase is assumed to be pseudo-homogeneous and is characterised by the mean Sauter droplet diameter, which is invariant along the column height [1,2].

An alternative to this is to combine a particle population balance model (PPBM) to take into account the physical behaviour of the droplet swarm such as droplet rise velocity (forward mixing) and back-mixing, droplet coalescence and breakage and solute interphase transport [3, 4]. Recent advances in population balance modeling make it possible to take these interacting phenomena using one bivariate PPBM [3-5]. This PPBM takes into account not only the discreet character of the dispersed phase but also it describes the complex interaction between hydrodynamics and mass transfer. Recent research shows that there is a strong coupling between column hydrodynamics, mass transfer and the type of the dispersed phase (whether light or heavy phase) [6,7].For a comprehensive review of mathematical modelling of liquid-liquid extraction columns, their advantages and disadvantages, the interested reader can refer to Mohanty and Attarakih et al. [4,5]. In an up to date work, Kislik [8] highlighted the importance of design and operation of liquid extraction equipment. He presented a review and analyses of different types of the phase dispersion and the effect of droplet coalescence on the design of agitated extractors (mixer-settler), pulsing mixers, and centrifugal mixer-settlers. In his review paper, Grinbaum [9] discussed the present approach in the design of liquid extraction equipment, where the design is still based on experimental pilot plant data with the help of steady state flowsheet simulators. The design of extraction equipment based on the pilot plant approach has proven to be efficient, but time consuming and expensive. In the long term powerful PPBM models may enable the solution of many extraction equipment problems with reliability that does not require experimental verification [9, 10]. In order to be widely useful, these models have to be transformed into user friendly simulators that require only easily available data as input. In this regard, Attarakih et al. [11], presented the first version of a Windows-based computer program, which is called Liquid-Liquid Extraction Column MODule (LLECMOD) to simulate only the hydrodynamics of RDC extraction column based on the differential PPBM. LLECMOD comes to fill the gap in detailed liquid extraction process simulation, and t o allow the description of the local and integral dispersed phase properties of the droplet swarm and its interaction with the surrounding continuous phase. In 2008, Attarakih et. al, [12] further developed the software to describe the steady state and the dynamic behavior of the coupled hydrodynamics and mass transfer. In the 2008 version of LLECMOD, two agitated contactors were modeled: RDC and Kuhni extraction columns. The basic feature of LLECMOD program is to provide a simulation tool for both transient and steady state hydrodynamics and mass transfer using an interactive Windows input dialogs. These include dialogs to input functions for droplet terminal velocity taking into account swarm effect, slowing factor due to column internal geometry, breakage and coalescence frequencies, daughter droplet distribution and the axial dispersion coefficient. In a recent version of LLECMOD, two pulsed columns (sieve plate and packed column) were modeled using the bivariate PPBM. These are sieve tray and packed columns [13]. Other development in this direction is the ReDrop software, which simulates extraction columns based on Monte Carlo approach [14]. Monte Carlo models are known for their high memory usage and CPU time requirements, with the ability to handle multivariate population balance models [4].

In the present work, a new comprehensive windows-based MATLAB environment, which is called PPBLAB software (Particulate Population Balance LABoratory) is developed. PPBLAB uses the bivariate PPBM with respect to particle internal properties: Concentration and size. A mixed fixed-pivot and QMOM (Quadrature Method Of Moments) discrete PPBM is developed, which makes use of the recent advances in solving the PBE [15]. On the hierarchical level, the bivariate PPBM consists of detailed and reduced models, which are combined in an efficient way to overcome the computation complexity. In the new PPBLAB software, one is able to model and simulate liquid-liquid extraction columns using the PPBM as a mathematical framework. The MATLAB GUI (Graphical User Interface)

is used to facilitate model implementation, which makes the program as an effective simulation tool. Recent advances in column design and performance using new population balance models and solvers are utilized. These include the One Primary and One Secondary Particle Model (OPOSPM) and the bivariate fixed-pivot technique [10, 11]. For space solvers, the program offers one dimensional stagewise models or grid refinement discretization using the finite volume method with flux vector splitting. The resulting semi-discrete model is a large DAE system, where explicit, semi-implicit first order Euler methods or standard MATLAB ODE solvers are implemented.

On the modelling level, PPBLAB offers population balance models for agitated and non-agitated extraction columns, which includes: RDC, Kuhni, pulsed sieve tray and packed columns. The main objectives of the present work are to highlight the main features of the PPBLAB software in modelling both agitated and non-agitated extraction columns, and to illustrate the numerical performance and experimental validation of the PPBLAB. A pilot plant Kuhni extraction column of 0.08 m diameter and 4.4 m height is simulated. The experimental data on dispersed phase holdup, mean Sauter diameter and solute concentration profiles were validated against the experimental data of Garhte [16].

2. PPBLAB-CAPE OPEN Link

In a novel software development step, a new feature of PPBLAB is the TEA (Thermodynamics for Engineering Applications)thermodynamics package, which is obtained from CAPE OPEN as a part of COCOC (Cape-Open to Cape-Open) steady state flow sheet simulator [17]. TEA has standard CAPE OPEN interface to communicate with MATLAB. For coupling TEA with PPBLAB, there is a need for interface design to exchange data between TEA and PPBLAB. In this version ofPPBLAB, an interface is designed using MATLAB. The flow of information between TEA and PPBLAB is shown in Fig. 1.

PPBLAB

Fig. 1. Data exchange between PPBLAB and TEA through a MATLAB designed interface.

The CAPE-OPEN thermodynamics package TEA is linked to PPBLAB program. The package contains more than 190 commonly chemical components and more than 100 property calculation methods. Special interface is designed using MATLAB for the purpose of data exchange between PPBLAB and CAPE-OPEN TEA. The chemical components interface of PPBLAB is shown in Fig. 2, where the user can define solute, dispersed and continuous phase chemical components. An important

advantage of CAPE-OPEN TEA is being an open source package, where the user can easily add and define new chemical components.

Fig. 2. PPBLAB-CAPE OPEN component data base interface where a link between PPBLAB and TEA thermodynamics package is provided.

3. PPBLAB-UNIQUAQ-TEA package

Another new feature of PPBLAB is the prediction of solute distribution coefficient in ternary systems using UNIQUAQ thermodynamic model. The binary interaction parameters were optimized based on infinite dilution distribution coefficients. A special binary parameter estimation tool is developed, which imports the structural parameters for binary systems from TEA thermodynamics package through a designed interface using MATLAB. As a result of this, PPBLAB contains fitted interaction parameters for more than one hundred popular ternary systems, which are collected from the open published literature on ternary liquid-liquid equilibria. The UNIQUAQ thermodynamic model, with its new interaction parameters, is used successfully to predict the solute equilibrium concentration over practical ranges encountered in liquid-liquid extraction columns (up to 10 percent).

4. Mathematical model

The population of particulate phase (here liquid droplets) is adequately described by a bivariate number concentration function with two internal particle properties: particle size (d) and solute concentration cy. Accordingly, a number continuity equation, which describes the evolution particulate phase and its interaction with the surrounding continuous phase, can be written as [3]:

N ^ V- &

D^W) = s

In this equation, the bivariate number concentration function is f(d,cy,r,t), where r is the spatial coordinate system and t is time, the first term inside brackets represents the movement of droplet clouds due to the droplets rising velocity uy and its transport by axial dispersion is expressed in terms of axial dispersion coefficient Day. This axial dispersion is correlated experimentally in one dimensional models to take into account the back mixing of the dispersed phase [7,9]. The summation inside the brackets represents the convection of the particulate phase along internal particle coordinates. In this model, particle growth due to solute transfer (^) is represented by the rate of change of particle size with respect to time, while the rate of change of solute concentration inside the particle is ^2 . For small solute concentrations particle growth rate (Cll) can be neglected, while (^2) can be related to the convective solute transfer from the continuous phase. Assuming the particle is well mixed (vigorous internal circulation), the rate of change of solute concentration in individual droplet is given by [3,7]:

Where c* is the droplet equilibrium solute concentration and the overall mass transfer coefficient (Koy) is estimated using the two film resistance theory in terms of the individual mass transfer coefficients in both continuous and dispersed phases. PPBLAB provides many phenomenological and experimentally correlated mass transfer models to estimate the individual mass transfer coefficients. A sample of these is shown in Tables 1 and 2.

4.1 Droplet terminal -velocity

The droplet velocity, Uy, relative to the walls of the column is determined in terms of the slip velocity (u0) with respect to the continuous phase as a result of momentum balance on the rising droplet of diameter (d):

In the above equation the function ks takes into account the slowing of droplet motion and is function of droplet diameter and column internal geometry. The function f(ay) is used to account for the hindering effect due to the presence of other droplets in the dispersion. The simplest form of this function is: f( y) = (1- y)n where the exponent (n) is function of the droplet Reynolds number and is introduced to compensate for the increase of the drag coefficient[16]. The slip velocity (u0) is a function of the droplet Reynolds number, mean droplet diameter and the physical properties of the two phases. In this work we propose the use of the following form of the slip velocity:

Table 1. Individual mass transfer correlations outside droplets.

Reference Treybal [18]

Heertjes [19]

Kronig and Brink [20]

Garner and Tayeban [21]

Kummar and Hartland [22]

Correlation

k = ^. ( 0.725 Re0'57 .Sc0A2

x d 1 ^ x

k, = 0.83«' x r

k = . (o.ôjRe Sc

* A \ V P

Comments

Circulating droplet in swarm

Oscillating droplet Circulating droplet Rigid droplet

k = . 12 + 0.67./Re .Sc

x d \ V p

Sherwood number is given by correlations which Circulating and oscillating

depends on the column type (RDC or Kuhni droplets

column)

Table 2. Mass Transfer Correlations inside droplets.

Reference

Kummar and Hartland [22]

Handlos and Baron [23]

Kronig and Brink [20]

Laddha and Degaleesan [24]

Correlation

Sherwood number is given by correlations which depends on the column type (RDC or Kuhni column)

ky = 0.00375,-

k = 0.079. -

%/ni r

17.66,—

k = 0.023. ——

* Sc.05

Comments

Circulating and oscillating droplets

Oscillating droplets, internal circulation

Diffusion and internal circulation

Circulating droplets, empirical based on single file of droplets, derived from penetration theory

Pilhofer and Mewes [25]

Slater [26]

ky = 0.002,-

I A? F

Empirical

Stagnant cap model, effective

Î)8 — ^y"cxp I~ n j diffusion, one empirical

X" . SZ Or 1 & 1 2Z ± E

„2 fF

=1« n

pij contamination factor

d.a .u.

2048(1 + k)

a given in [26]

where the drag coefficient (Co) is due to Schiller and Nauman [27]. The second form of the slip velocity is due to Wesselingh, and Bollen [28] where the viscous and inertial regions of the particle motion are covered in one equation, and thus eliminating the implicit nature of Eq.(4) above. The droplet relaxation time ( p) is given by:

t = g 24 6= r_zUL ^

p 18 nj §CD Re r gDr

It is worthwhile to note that the relaxation time is only explicit (independent of droplet velocity) in the viscous region where Stokes's law applies. In [13] it was shown that the Wesselingh and Bollen approximation is in good agreement with droplet terminal velocity as given by Eq.(4). Moreover, Eq.(3) is reduced to the algebraic slip velocity model at steady state [3,16]. The particle relaxation time is found to be in the order of 1 second for the test chemical system water-butyl acetate, where the latter is the dispersed phase. Therefore, steady state approximation of Eq.(3) is used in PPBLAB. Other slip velocity correlations for liquid droplets are also available in PPBLAB based on the algorithm developed by Godfrey and Slater [29].

4.2 Particle breakage and coalescence

The source term on the right hand side is a functional of the number concentration function, which represents net number of particle concentration as a result of particle breakage and coalescence. The bivariate source term, which is shown in Table 3 is complicated by many double integrals to account for the conservation of total droplet number and solute concentration [3, 12]. Particle breakage in agitated liquid extraction columns is a result of turbulent energy dissipation of the continuous phase and mechanical forces resulting from shearing of the liquid droplets by rotor blades [7, 16]. Schmidt et al. [7] and Garthe [16] observed experimentally that shear forces are dominant above certain critical rotor speed, which is function of column internal geometry and physical properties. Based on this, breakage of droplets can be adequately correlated using single droplet experiments in laboratory scale devices.

Table 3. The source term in Eq.(l), which is split into birth and death terms for droplet breakage and coalescence respectively [3].

Bb n n K c; d 0

Db - G(d:^)/(d~c;i,r)

Bc 1 d c;!m" Qd cf - n n ^ ' h> ay ' Cr<>z CP> r)iw *- ^ )x . Cy % I" g if1. = min [C„ № )/nK))) c;;mln = max ( O.C^ll - n[d)/n[d (1 - C/C^ ^ )

Dc (dmax - d3 Sm„ nd,Cy [dt,r) n "r\ ^d,day y{dt,r)^^ d 0

The breakage frequency ( ) is found to be proportional to droplet breakage probability, which is function of Weber number, and inversely proportional to the particle residence time in a given column compartment [7,16]. On the other hand, droplet coalescence is complicated by not only interactions of droplets with the surrounding turbulent continuous phase, but also by those between particles themselves once they are brought together by the external flow or by body forces. Many theories are proposed for

droplet coalescence process; however, the most popular theory is the film drainage model of Shinnar and Church [30]. The proposed theory postulates that two colliding liquid droplets may cohere together but are prevented from coalescence by a thin film of trapped liquid between them. If the attractive forces between these droplets are enough to drain this thin film, then droplet coalescence will follow. In this way, droplet coalescence is considered to be successful due to the prevailing fluctuations and coalescence only if the interaction time is sufficient for the intervening film to drain out down to the critical rupture thickness [31]. One of the well-known and accepted models for droplet coalescence frequency ( ), which is based on film drainage theory is that of Coulaloglou and Tavlarides [32]:

■/3 " .2/3 ,2/3,1/28- LJ_ c2njrxe | (dd') QLS

Hd,da ) = %——(d +d f(d2n +d '2/3)1/2 V |xpi - J ¡J (6)

Based on the kinetic theory of gases, the first term in this model describes the collision between two liquid droplets in the same way as that two gas molecules. This follows from the Kolmogoroff's local isotropic turbulent, where coalescence between liquid droplets is induced by turbulent random collisions. The second term is based on the film drainage theory and describes the efficiency of collision between two droplets of sizes d and d . This collision efficiency depends on system physical properties (viscosity , density and interfacial tension ), dispersed phase holdup ( y) and energy dissipation ( ). A recent and critical review on particle coalescence models is presented by Liao and Lucas [31]. The solute concentration in the continuous phase, cx, is predicted by making a solute balance in the same phase at constant density; this is because the solute quantity which is transferred from the continuous phase to the dispersed phase is very small. The solute balance in the continuous phase is given by [3,7]:

^ - N 1]t

Dax N^

„in in Cx Ux

n n Cyu\ d )f(d> Cy' f>

5. Case study: Simulation of pilot plant Kuhni extraction column

In this section, PPBLAB is validated against steady state experimental data published in Garthe thesis [16]. The 0.08 DN Kuhni extraction column is shown in Fig. 3 with an overall height of 4.40 m. In this figure the column geometrical details are shown along with operating conditions and the specified inlet distribution used in the simulation. The details of these measuring points are found in [16]. The extracted experimental data are divided into two sets: the hydrodynamic data (inlet size distribution, dispersed phase holdup and the Sauter (d32) mean droplet diameter) and the mass transfer data consisting of the solute concentration profiles along the column height. Both set of data are taken for the EFCE (European Federation of Chemical Engineering) chemical system water-acetone-toluene [33]. The dispersed phase is toluene which is introduced at the bottom of the column at a flow rate of 48 L/h, while the continuous phase is water at a flow rate of 40 L/h and the rotor speed is 150 rpm. The first data set (the dispersed phase holdup and the Sauter mean droplet diameter) is used in estimating the coalescence model parameters: Ci and C2 appearing in Coulaloglou and Tavlarides [32] model given by Eq.(6). These parameters are fitted against the experimental data of Garthe [16].

Column dimensions

Column diameter (m) 0.080

Column height (m) 4.4

Stator diameter (m) 0.045

Blade diameter (m) 0.050

Compartment height(m) 0.05

Inlet and outlet compartment numbers

Dispersed phase inlet 17 phase

Continuous inlet

Total number compartments

Column operating conditions

Continuous phase flow rate (L/h) Dispersed phase flow rate (L/h) Continuous phase inlet concentration (Kg/m3) Dispersed phase inlet concentration (Kg/m3) Rotor speed (rpm)

Inlet distribution parameters

Number of pivots Population mean Population variance

2.7 0.5

Fig. 3. Schematic diagram, dimensions, operating conditions and inlet feed distribution specifications for a pilot plant Kuhni extraction column [16].

5.1 Column hydrodynamics

In PPBLAB, the column hydrodynamics are specified and defined using the user input dialogue as shown in Fig. 4. The left part of this input dialogue is used to specify the inlet feed distribution, which can be normal Gaussian, lognormal or weibul distribution. In Fig. 4, the number of pivots (droplet diameter grid) is taken 10, with minimum and maximum droplet diameters as 0.01 and 6 mm respectively. On the right hand side of Fig. 4, the user can define the column hydrodynamics. This includes choosing the active particle interactions inside the column, where breakage and coalescence, only breakage, only coalescence or even no breakage and no coalescence can be chosen. The particle breakage frequency is taken after Schmidt et al. [7], which is given by:

= c„

+ C 4.md.\

Where Wem0d is the modified Weber number with respect to the critical rotor speed. It was shown experimentally that below this critical rotor speed negligible particle breakage takes place [7, 16]. The

constants c3, c4and c5 are fitted to experimental data for both Kuhni and RDC columns and given in [16]. To complete the breakage model, the daughter particle distribution is found to follow a modified beta distribution and is given by [7,16]:

b ( d I d | = 3 JI J

In the above equation ( 2) is the mean number of daughter droplets which results from break up of mother particle of size d . The above equation should satisfy total number and mass conservation as well [5,7]. Based on the test chemical system (water-acetone-toluene) the terminal droplet velocity is chosen as Vignes law [34]. The last element in the user input dialogue ofFig.4 is the slowing factor appearing in Eq.(3). PPBLAB offers a constant value or prompt the user to choose one of the available correlations for the Kuhni column. In this case study, a mean constant value of Garhte [16] experimental data is used, which is 0.33.

Fig. 4. PPBLAB user input dialogue to define and specify extraction column hydrodynamics.

5.2 Mass transfer model

The choice of the individual mass transfer coefficients is a difficult task due to the presence of many phenomenological models and correlations (a sample is presented in Tables 1 and 2) with limited ranges of applicability. This is because the dispersed phase individual mass transfer coefficient is found dependent on the behavior of single droplet in the sense whether it is stagnant, circulating or oscillating [26, 29, 35]. The models shown in Tables 1 and 2 can be used to estimate the individual mass transfer coefficient when the particle state is known. Unfortunately, there is no general acceptable model or correlation, which can be used to cover all the particles internal state. The choice becomes even more

complicated when dealing with contaminated industrial systems [26, 29]. In the present work, the correlation of Garner and Tayeban[21] is used for the continuous phase and Laddha [24] correlation for the dispersed phase, which is guided by their excellent prediction of the solute concentration profile along the column height. However, the criterion based on droplet Reynolds number as suggested by Zhang et al. [35] may be used as a guide for selecting the proper mass transfer model.

5. 3 Distribution coefficient

To predict correctly the solute distribution coefficient, the PPBLAB-UNIQUAQ-TEA parameter estimation tool is used to estimate the binary interaction parameters for the chemical system: water-acetone-toluene at 20°C with infinite dilution distribution coefficient, which equals to 0.71 based on mass concentration units. The optimized UNIQUAQ binary interaction parameters are shown in Table 4.

Table 4. Optimized UNIQUAQ binary interaction parameters at 20 C° using PPBLAB-UNIQUAQ-TEA

Atoluene-acetone Aacetone-toluene Aacetone-water Awater-acetone

74.172 -115.130 97.992 -109.172

Fig. 5. Comparison between the experimental equilibrium solute concentration [33] and the prediction of the UNIQUAQ model based on the fitted binary interaction parameters shown in Table 4.

In the calculation of the distribution coefficient based on the UNIQUAQ model, it is assumed in PPBLAB that the light and heavy phases (toluene and water respectively) are immiscible. Therefore, only the solubility of acetone in both phases is considered. Despite this simplifying assumption, which is required to minimize the size of the DAE system dictated by the population balance modelling approach, the model predicts fairly well the experimental data [33] as shown in Fig. 5.

5. 4 PPBLAB numerical solvers

Based on the defined input in the previous sections, PPBLAB starts up the Kuhni column simulation by introducing the specified dispersed phase at the bottom of the extraction column. This allows the column simulation using two predefined modes: The first one is the detailed population balance simulation, where the solute in each droplet group is calculated using a bivariate mixed QMOM and fixed pivot technique [3]. The second mode, which is called the reduced model, is designed to simulate the mean solute concentration in all droplet groups, and hence the size of the DAE(Differential Algebraic Equations) system is drastically reduced from (2NP+1)(N) to (NP+2)(N). Here, Np is the number of groups (pivots) and N is the total number of stages or numerical spatial cells. Note that the degree of system complexity is represented by the size of system of ODE being solved. This system of ODEs is classified as a stiff system due to the presence of different time scales. First, the time scale of the diffusional mass transfer is much longer than that of droplet hydrodynamics (droplet convection). Second, each droplet size has its own time scale, with small droplets characterized by small time constants. The diversity in the process time scales calls for special ordinary differential equation solver to ensure numerical stability under different user inputs. Therefore, PPBLAB provides strongly stable semi-implicit first-order time integrators [3,11,12]. The MATLAB ODE solver: odel5s is known to handle very difficult and stiff problems. Hence, odel5s is made available inthe PPBLAB inadditionto otherMATLAB ODE solvers.

mmtmmdtmm

Column Height (m )

-PPBLAB Results-D etaile d Model

• PPBLAB R esults-R educ ed Model

• E xpe rim e ntal Data

Fig. 6. Comparisons between the simulated and experimental hold-up profiles [16] using water-acetone-toluene system and 10 droplet groups (pivots).

5. 5 Simulation results

In this section, we present a sample of PPBLAB results for the Kuhni extraction described in Fig. 3. The number of particle groups (pivots) is chosen 10 and the final simulation time is 9000 s with time step equals to 20 s.

-P P BLAB R esulls-D elailed Model

4 P P B LA B R esulls-R edited M odel % E ipeiii enlal D ala

Fig. 7. Comparisons between the simulated and experimental Sauter (d32) mean droplet diameter profiles [ 16] using water-acetone-toluene system and 10 droplet groups (pivots).

The relatively long final simulation time is chosen to ensure steady state mass transfer profiles along the column height. On the other hand, the hydrodynamics profiles are found to reach steady state at a relatively short time (in the order of 100 to 500 s). Figs. 6 and 7 show the steady state simulated holdup and Sauter mean diameter profiles along the column height. These two hydrodynamic integral properties are fitted to the experimental data by adjusting the two fitting constants in Eq.(6). The values of these constants are found to be : Ci = 0.005 and C2 = 1.33xl0um"2, which are in the same order of magnitude for water-acetone-toluene chemical system used in previous published work [3, 7, 11]. From the holdup and Sauter profiles along the Kuhni column height, it is clear that droplet breakage is dominant. This result is expected, since droplet breakage in Kuhni extraction columns is higher than that in other agitated extraction columns such as the RDC column [16]. PPBLAB can produce not only the integral population properties, but also the local properties such as droplet volume distribution as shown in Fig. 8. Again, it is clear that the 3D representation of the particle size concentration shows the dominance of droplet breakage along column height. Once the hydrodynamics of the column is correctly predicted, the parameter adjustment procedure can be viewed as model training to predict mass transfer profiles in a later independent step. To test this hypothesis, the solute concentration profiles are simulated under the same conditions described above. Fig. 9 shows the excellent prediction of these solute concentration profiles using the correlation of Garner and Tayeban [21] for the continuous phase and Laddha [24] correlation for the dispersed phase. It is important to mention here that the equilibrium solute concentration is predicted using the UNIQUAQ model with the optimized binary interaction parameters as shown in Table 4.

Droplet Diam eter (m m )

Column H e ig h t (m )

Fig. 8. 3D droplets volume concentration in a Kühni extraction column.

C o lu m n Height (m )

Detailed Model Dispersed Phase Concentration Detailed Continuous Phase Concentration

— a- — Reduced Model Dispersed Phase Concentration

— -«-— Reduced Model Dispersed Phase Concentration % Continuous C o n c e n tra tio n -E x p e rie m n t

V Dispersed Phase C oncentration-E xperim ent

Fig. 9. Comparison between simulated and experimental solute concentration profiles[16] in a Kuhni extraction column.

From the authors experience using the previous versions of LLECMOD [12, 13], the prediction accuracy of the steady state solute concentration profiles are remarkable. Moreover, Fig. 9 presents a comparison between the detailed and reduced population balance models, which are described in section 5.4. It is clear that the reduced model is sufficiently accurate to predict the solute concentration profiles with a considerable reduction in computational time.

5.6 Numerical convergence analysis

The performance of PPBLAB depends on the number of droplet groups (pivots) and the number of compartments (spatial numerical cells). Two mean proprieties of the droplet population are studied as function of number of droplet groups. These are the holdup and the Sauter mean diameter. The convergence measure used here is the L2-norm of the deviation between experimental and numerical profiles along the column height at steady state. The final simulation time is fixed at 9000 s and the time step is taken as 20 s. Table 5 summarizes the convergence analysis results and the dependence of CPU time on the number of droplet groups. First, the two integral population properties converge as function of number of droplet groups, where better sampling of the distribution is achieved. The cost of this accuracy is evident by the increase of CPU time; however, it is clear that above 10 droplet groups, there is no significant reduction in the error in both integral properties.

Table 5. CPU time dependence on the number of pivots using the reduced model with final simulation time 9000 s and t = 20s»

Number of droplet groups a exp a sim ^32 exp d32sim CPU time (s)

5 0.0240 0.3149 137.734

10 0.0236 0.2452 196.920

15 0.0237 0.2334 255.749

20 0.0236 0.2298 315.763

♦All simulations are run on a laptop of2.40 GHz and 4 GB RAM using MATLAB 2011a.

6. Summary and conclusions

In this work, PPBLAB which is a newly developed windows-based software is introduced. PPBLAB is intended to be a user-friendly environment for modelling particulate systems using population balances. As a first step, liquid-liquid extraction columns, which include RDC, Kuhni, sieve plate and pulsed packed extractions models are now available. Another advanced step is the coupling of PPBLAB with CAPE-OPEN tools such as the thermodynamics engine TEA. Concerning phase liquid-liquid equilibria, PPBLAB provides a coupled UNIQUAQ-TEA parameter estimation tool to estimate binary solute interaction parameters using only infinite dilution distribution coefficients. The results show that the equilibrium solute concentration is predicted with a good accuracy when compared to the experimental data. Modelling and numerical analyses show that the detailed model complexity is proportional to 2NPN where Np is the number of particle groups and N is the number of compartments. On the other hand the reduced model offered by PPBLAB has a complexity, which is only proportional to NPN.

In the simulation of pilot plant RDC Kuhni column, the size of the ODE system is found to be in the order of 3000 for pilot plant extraction column and is even more for industrial ones. So, stiff numerical ODE solvers are needed to resolve such a large system in a reasonable computational time. PPBLAB uses strongly implicit first-order ODE solvers, which proves to be fast and robust when compared to MATLAB ODE solvers (e.g. odel5s). The numerical solution is found to converge (in terms of the dispersed phase integral properties) by increasing the number of droplet groups. On the experimental level, the model flexibility to predict the experimental performance is demonstrated in the prediction of dispersed phase holdup, Sauter mean diameter and solute concentration profiles. The only adjusted and sensitive parameter is found to be the pre-exponential parameter in the droplet coalescence model of Coulaloglou and Tavlarides [32].

Acknowledgment

The authors are very grateful to the German Research Association (DFG) for the financial support.

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