An Inverse Problem Method for RDC Simulation

Abstract

- The hydrodynamic and mass transfer behavior of a diameter, holdup and concentration profiles for a pilot plant Rotating Disk Contactor (RDC) extraction column is investigated scale column. for two different liquid-liquid systems recommended by the European Federation of Chemical Engineering (EFCE). An II. THE MATHEMATICAL MODEL ipnavrearmseetperrosbilnemasRmDeCtheoxdtriascatipopnliceodlutomnes.tiSminagtlee-tdhreopcoleatlesstcuednicees The general spatially distributed population balance in a small lab scale RDC is used to evaluate the coalescence equation (SDPBE) for describing the coupled hydrodynamics parameters necessary for column simulations, which were and mass transfer in liquid-liquid extraction columns (LLECs) obtained by an inverse solution of the population balance model in a one spatial domain could be written as [2]: using the generalized fixed-pivot technique for the discretization roefstuhlteeddrforpomlet sionltveirnngatlhceoionrvdeirnsaetep.rTobhleemcoaarleescdeenpceendpeanrta monettehres ∂fd,cy (ψ ) + ∂ uy fd,cy (ψ ) + ∂ cɺy fd,cy (ψ ) = chemical test system used for the simulation. Then the resulted ∂t ∂z ∂cy tcRhoDeaClhesycedexrntorcdaeycpntiaaormnamicceoatleunrmds nvm.aalTussehsetrwasenimrsefueulrasetbededhinaSvLaiouLrtEeorCfMmpOielaoDnt tpodlarsontuptldoeytf ∂∂z Dy fd,c∂yz(ψ )  + QAyicn f yin (d , cy ;t)δ (z - zy +) ϒ {ψ } diameter, hold-up values and concentration profiles for organic and aqueous phase were found to be well predicted compared to the experimental data.

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I. INTRODUCTION

Liquid-liquid extraction is a powerful separation technique after distillation and it is widely used in many industrial fields such as petrochemical and biochemical industries. Liquidliquid extraction columns are classified into three main categories: stirred columns (Rotating Disc Contactor (RDC) and Kühni columns), non-agitated columns and pulsed columns (sieve plate and packed columns).

Liquid–liquid extraction column (LLEC) simulation based on population balance modeling has difficulties due to contiguous change in the droplet properties such as Sauter mean droplet diameter, hold-up values and concentration profiles for organic and aqueous phase. The most common mechanisms effects on the population balance model are breakage, coalescence and mass transfer [ 1 ].

Inverse problem is an ill-posed in general and need some stabilization techniques to get reliable optimized parameters. Solving it is a necessary part for reliable modelling strategy. However, this is complicated due to the increase in the size of the differential-algebraic system when dealing with inverse problems using population balance models.

The objective of this paper is to evaluate the coalescence parameter in a small lab scale RDC for two EFCE systems as is butylacetate-water and toluene-water. We then use these estimated constants to predict the Sauter mean droplet In this equation the components of the vector = [ ] are those for the droplet internal coordinates (diameter and solute concentration), the external coordinate is (z) and for time is (t). The velocity vector along the internal coordinates is given by (cy) is ( ) and (Υ) represent the source term. (ζ) represent the net number of droplets produced by breakage and coalescence per unit volume and unit time in the coordinates range [ζ, ζ+∂ζ]. The dispersed phase velocity (uy) relative to the walls of the column is determined in terms of the slip velocity (us) with respect to the continuous phase velocity (ux) with respect to the walls of the column [ 3 ], [ 4 ].

The source term is given by the following equation: ϒ {ψ } = Bb (d , cy ; t, z) - Db (d , cy ; t, z) Source Term

Birth by Breakage

Death by Breakage + Bc (d , cy ; t, z) - Dc (d , cy ; t, z)

Birth by Coalescence Death by Coalescence

The bivariate source term for the four rates of droplets birth and death due to droplet breakage and coalescence are quite complicated they are expressed in details [ 2 ].

Simulating liquid-liquid extraction columns is a challenging task due to the discrete character of the dispersed phase. In technical geometries the population balance model has no general analytical solution and hence a special numerical technique is required to solve the PBE [ 4 ]. Several numerical approach are proposed to solve the PBE which are classified into two categories namely classes method Kumar and Ramkrishna [ 5 ] and method of moments [ 6 ]. Attarakih has invented a new technique for the solution of the population balance equation using the sectional quadrature method of moments (SQMOM) which combines the advantages of classes method and method of moments [ 7 ].

III. INVERSE PROBLEM APPROACH

Inverse problems for population balances have attracted many researchers in the recent years [ 3 ], [ 8 ]–[ 10 ]. Inverse problem is considered an ill-posed in general and need some stabilization techniques to get reliable optimized parameters. In more general cases a mathematical programming procedure has to be applied in order to find the best fit of the experimental data. Since the inverse problems are highly sensitive to errors in the experimental data, the experimental data has been fitted to a normal distribution to give a smother curve and remove some noise, by regularization.

By solving the population balance equations the optimization algorithm derives the unknown coalescence parameters by fitting the simulated outlet distribution to the experimental one using an objective function.

In this work an example of the inverse problem program has been used to obtain the coalescence parameters of a lab scale RDC for two cases of two EFCE systems as is butylacetate-water and toluene-water.

IV. ESTIMATION PARAMETERS

Despite the rich literature on population balance equation modeling, there are few studies on optimization packages for coalescence models that helps to choose the best coalescence model for the experimental data. These optimization packages must fully describe the system by solving the population balance problem inversely to get the best values of constants in the coalescence model. In addition the objective function should be wisely chosen to minimize the error and to have a unique solution for a given data. Since droplet coalescence is highly sensitive to the hydrodynamics and physic-chemical properties, interfacial dynamics and mass transfer [ 3 ].

Coulaloglou & Tavlarides (1977) developed a model for stirred vessel, which is based on the kinetic theory of gases and drainage film theory; and calculate the coalescence frequency from the collision rate (frequency) (h) and the coalescence efficiency (λ) [ 11 ].

ω (d , d ',ϕd ) = h(d , d ',ϕd ) λ(d , d ', ϕd )

The Coulaloglou & Tavlarides coalescence model is given by the following equation [ 12 ]:

 ε 1/3  ω(d, d ',ϕd ) = c1 1+ϕ (d + d ')2(d2/3 + d '2/3)1/2 ×    c2ηxρxε  (dd ') 4  exp -     σ 2(1+ϕ)3  (d + d ')  

An alternative to this model is developed by Sovova (1981) introducing a different formula for the efficiency of collisions, the mechanism of coalescence of drops by film drainage is replaced by a mechanism based on the effect of collision impact [ 13 ]:

The coalescence frequency model proposed by Laso (1986) is expressed by the following equation [ 14 ]:

We- 0.51ϕ 0.9O h- 0.05 d

A semi-empirical model proposed by Casamatta and Vogelpohl in (1985) in which the first term at the right hand of the equation is the coalescence frequency caused by random motion, which is assumed to be proportional to the drop volumes. The second term represents the coalescence induced by different rise velocities are expressed by the following equation [ 11 ], [ 15 ]: ω (d, d ',ϕd ) = c5 π362 (d × d ')3  + c6 π62  d +2d ' 2 d 3 - d '3  The inverse problem program can estimate the constants c1……c7 of the previous coalescence models along with the ability to choose one of the breakage models available to run the simulation or with no breakage mode. Additional results will be discussed in further publications.

V. LLECMOD PROGRAM

The mathematical model described above was programmed using Visual Digital FORTRAN and to facilitate the data input and output a LLECMOD program was designed. A graphical user interface of LLECMOD program (LiquidLiquid Extraction Column MODule) simulates liquid-liquid extraction columns based on the population balance model to predict the column performance at steady state and transient conditions. Furthermore the design of LLECMOD is flexible in such a way it enables the user to define the input operating condition, the coalescence and breakage parameters and other different options according to the simulated case studied. Also LLECMOD enables to input the droplet terminal velocity, energy dissipation, axial dispersion, breakage and coalescence frequencies and the other internal geometrical details of the column.

LLECMOD is considered a user-friendly and flexible windows-based program that contains the main input window and other sub-windows for parameters and correlations input. Using LLECMOD simulations can now be carried out successfully for different types of extraction columns including agitated (Rotating Disc Contactor (RDC) and Kühni columns) and non-agitated or pulsed columns (sieve plate and packed columns) with different internal geometry.

VI. RESULTS AND DISCUSSION

In this study we simulate two cases of two phase liquidliquid system for butylacetate-water (B-W) and toluene-water (T-W) in a lab scale RDC [ 16 ]. The RDC extraction column has five compartments, its internal diameter is 150 mm, column height 150 mm, compartment height is 30 mm, internal stator diameter is 105 mm, rotator diameter 90 mm and diameter of the rotating shaft is 54 mm. The angular velocity of the shafts is constant 300 rpm and volumetric flow rate for continuous and dispersed phase is Qc = Qd = 100 l/hr for both systems.

The values of coalescence parameters for Coulaloglou and Tavlarides coalescence model (c1 and c2) were estimated using the inverse problem for both systems shown in Table 1.

The inlet and outlet cumulative volume distribution are used from the work of Simon [ 16 ]. The unknown coalescence parameters values were obtained by minimizing the square sum of errors according to the following equation: = ( , −

#,$%) ! where; Q3 is the cumulative volume distribution.

) *+, = ' ( ( ) . where; d: droplet diameter [mm]. Fig. 1 shows the resulted curve form simulating the cumulative volume distribution at the outlet to the experimental data, where a very good agreement is achieved.

Then the estimated constants were used to simulate a pilot plant RDC extraction column at steady state using the LLECMOD program; for the column height 4.4 m, column diameter 0.08 m, inlet of the dispersed and continuous phase are 0.85 m and 3.8 m, respectively. The two EFCE test systems butylacetate-acetone-water (B-A-W) and tolueneacetone-water (T-A-W) are used with inlet feed that is normally distributed with mean value equal to 2.1 mm and standard deviation of 0.6 mm.

2 3

Column Height (m)

The pilot plant RDC extraction column geometric internal data: its compartment height is 50 mm, internal stator diameter is 50 mm, rotator diameter 45 mm and diameter of the rotating shaft is 10 mm. The angular velocity of the shafts is constant 200 rpm and volumetric flow rate for continuous and dispersed phase is 40 l/hr and 48 l/hr, respectively for both systems. The direction of mass transfer is from the continuous to the dispersed phase. 0 1 2

3 Column Height (m)

B-A-W Sim T-A-W Sim B-A-W Exp 4 5 B-A-W Sim T-A-W Sim B-A-W Exp T-A-W Exp 4 5 0 1

2 3

Column Height (m)

A comparison between the simulated Sauter mean droplet diameter along the column height and the experimental data taken from the work of Garthe (2006) [ 17 ] is shown in Fig. 2 where a good agreement is achieved for both tested chemical systems.

Fig. 3 shows the simulated holdup along the column height compared to the experimental data for both chemical systems; where a very good agreement is achieved for both tested systems.

B-A-W Sim T-A-W Sim B-A-W Exp T-A-W Exp 4 5 0 1

2 3

Column Height (m)

In this paper the inverse problem method was used for parameter estimation for different droplet coalescence models for lab scale RDC extraction column. The resulted parameters derived are valid for pilot plant extraction column but it is dependent on the chemical system used. The user friendly windows-based program LLECMOD was used for the hydrodynamics and mass transfer simulation of RDC extraction columns for steady state simulation of pilot plant RDC using the derived values of coalescence parameters resulted from the inverse problem program. The resulted simulation for the Sauter mean droplet diameter, hold-up and concentration organic and aqueous phase profiles showed a good agreement to the experimental data.

ACKNOWLEDGMENT

The authors wish to thank the State Research Centre of Mathematical and Computational Modelling.

NOMENCLATURE 2 Ac: column cross-sectional area [m ] C: solute concentration [kg/m3] B, D: birth and death source terms [1/m3/s] c1……c7: coalescence parameters d: droplet diameter [m] d′: mother droplet diameter [m] DR: column diameter[m] f: Flux vector Oh: Ohnesorge no. = ℎ = 01

(2)3 1)4.6 Q: volumetric flow rate [m3/s] Q3: cumulative volume distribution u: velocity [m/s] t: time [s] We: Weber no. =we = z: space coordinate [m] 9:;/=>6/= ?

GREEK SYMBOLS ε: energy dissipation [m2/sec3] η: dynamic viscosity [kg/m sec] λ: coalescence efficiency

3 ρ: Density [kg/m ] σ: surface tension [N/m] Υ: source term that represents the net number of droplet produced by breakage and coalescence [1/s] φ: volume fraction φd: hold up ψ: internal and external coordinates vector ([ d cy z t] ) ω: coalescence rate [m3/sec] x, y: continuous and dispersed phases respectively s: slip velocity

SUBSCRIPTS

SUPERSCRIPTS

●: derivative with respect to time B: breakage C: coalescence exp: experimental in: inlet max: maximum out: outlet sim: simulated

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