Microsoft Word - A_04_Borsos_R.doc HUNGARIAN JOURNAL OF INDUSTRIAL CHEMISTRY VESZPRÉM Vol. 39(1) pp. 1-6 (2011) CRYSTALLIZATION OF PARA-XYLENE IN A BATCH COOLING CRYSTALLIZER A. BORSOS , B. G. LAKATOS University of Pannonia, Department of Process Engineering H-8200 Egyetem Street 10., HUNGARY E-mail: borsosa@fmt.uni-pannon.hu Crystallization of para-xylene in a batch cooling crystallizer was investigated by means of computer simulation. The mathematical model used in this modelling study consists of the population balance equation for crystals completed with the differential equations governing the mass balance of solute and the heat balance of the crystalline suspension. The population balance equation was reduced to a finite set of ordinary differential equations for moments of the crystal size variable using the standard moment method. The resulted finite system of ordinary differential equations was solved in MATLAB environment. The simulation results revealed that both the dynamic trajectories of the crystallizer and the final properties of the crystalline product depend strongly on the cooling profiles. Using the numerical model presented the batch crystallization system can be optimized by choosing the best temperature profile for producing a crystalline product exhibiting the required properties. Keywords: Para-xylene, Cooling crystallization, Population balance model, Moment method, Optimization, Simulation Introduction Xylenes are widely used materials in industry since the three xylene isomers are important components of diverse paints and varnishes and also of the gasoline and fuel of airplanes. The leather industry consumes xylenes in great amounts too. Xylene as a mixture of three isomers, i.e. orto-xylene, meta-xylene and para-xylene is produced mostly from petroleum. These isomers have the same chemical formula C6H4(CH3)2 but different structural configuration. Among the xylene isomers para-xylene plays important role since it is the principal precursor to terephthalic acid and dimethyl terephthalate. Both monomers are used in the production of polyethylene terephthalate (PET) which is a widely used raw material of polyesters. Some modern technologies use PET-based materials to produce vehicle parts as engineering resins often in combination with glass fibre [1]. After the first steps in crude oil separation technology the isomers and other components are in the same phase and that needs a further separation step for producing pure para-xylene. The three xylene isomers exhibit rather close boiling points but taking advantage of differences in the melting points cooling crystallization is a well suited separation technique for producing pure para- xylene product [2]. Crystallization is an important, often used unit operation of chemical industry as a separation and purification technique. During the course of process solid particles are formulated from solution. The driving force for crystallization is supersaturation which is, in essence, the difference between the actual and the equilibrium saturation concentration of solute depending on temperature. In crystallization, the crystalline product exhibits size distribution (CSD) even under ideal process conditions. This is an important quality parameter in processing. CSD depends on the degree of supersaturation that influences the nucleation and growth of crystals significantly. Investigation by properly constructed mathematical models is a feasible tool for process design of crystallization. The balance equations which contain process and kinetic parameters as constitutive expressions are coming into view by development of computer science and technology. In the case of crystallization the population balance equation describes the behaviour of the crystal population by means of which the size distribution of crystals and dynamics of crystallizer can be investigated. The population balance equation is often solved by applying the moment method [3-9]. The moment equations of low order moments describe some important properties of crystals such as the number of crystalline particles, the total crystal volume in the solution, the size distribution and the mean size of the crystalline product. A well designed moment equation model beside the properties of crystalline population involves information also on the mass and heat balance and dynamic behaviour of crystallizer [10, 11]. The aim of this paper is to investigate the para-xylene crystallization process by using a controlled cooling 2 profile in a batch crystallizer. The process is described by a suitable population balance model, and the population balance equation is solved by means of the moment method. The properties of the crystalline product and dynamic behaviour of the crystallizer obtained by numerical experimentation in MATLAB environment are presented and analysed. Population balance model Consider a stirred batch crystallizer in which supersaturation is generated by cooling as it is illustrated in Fig. 1. In this case, the crystallizer is considered as a two phase operational unit having liquid and solid phases. Assuming that: - Crystals are characterized by a linear dimension L; - All new crystals are formed at a nominal size Ln ≥ 0 so that we take Ln ≈ 0; - Crystal breakage and agglomeration are negligible; - No growth rate fluctuations occur, then the set of process equations, termed population balance model consists of the following balance equations. Population balance equation: ( ) ( ) ( ) 0 ,,,, = ∂ ∂ + ∂ ∂ L tLnTwwG t tLn sc (1) subject to the initial and boundary conditions ( ) ( )LntLn 00, == (2) ( ) ( ) ( )TwwBtLnTwwG scscL ,,,,,lim0 =→ (3) ( ) 0,lim = ∞→ tLn L (4) Figure 1: Schematics of a cooling crystallizer cooled by means of a jacket In Eq.(1), B denotes the nucleation rate given by the power law expression ( ) b s scb bsc w ww RT E kTwwB ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= exp,, 0 (5) where kb0 is the constant kinetic coefficient, Eb is the energy of nucleation, T is the temperature of crystalline suspension, wc is the concentration of solute and ws denotes the equilibrium saturation concentration. The growth rate of crystals is calculated by the following equation ( ) g s scg gsc w ww RT E kTwwG ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −= exp,, 0 (6) where kg0 denotes the constant kinetic coefficient. The equilibrium saturation concentration ws, i.e. the solubility of para-xylene is given by the second order polynomial 2210)( TbTbbTys ++= (7) obtained by fitting to the measurement data published by de Geode [13]. The results of fitting of Eq.(7) to the experimental data of equilibrium saturation concentration of para-xylene dissolved in a mixture of xylene isomers is presented in Fig. 2. Further thermodynamic properties of xylene isomers are listed in Table 1. Figure 2: Equilibrium saturation concentration of para-xylene in solution of xylenes Table 1: Thermodynamic properties of xylene isomers Melting point (°C) Boiling point (°C) para-xylene 13.3 138.5 ortho-xylene -25.2 144.0 meta-xylene -47.9 139.3 The mass and energy balance equations are also parts of the mathematical model. These equations are required to complete the population balance equation to obtain a full crystallizer model for computing the cooling profile, crystallization process and the properties of the product. The solute mass balance is given as 23 μρ GMkdt dw M svcV c sv −= (8) where Msv denotes the total mass of solvent in the crystallizer, kV is the volumetric shape factor, ρc is the density of crystals and µ2 is the second order moment from the raw moments of the crystal size variable L weighted by the population density function defined as qh, Th,in c, Msv, T, n qh, Th 3 .3,2,1,0,),()( max 0 == ∫ kdLtLnLt L k kμ (9) Naturally, here the mass Msv of solvent is constant. The heat balance equation for the crystalline suspension is expressed as ( ) ( ) ( )TTMUa TTVUa dt dT MmCC control sv sv V controlsvVsvccsv −= −=+ ρ (10) where Cc and Csv are the heat capacities of crystals and solvent, Tcontrol denotes the controlled temperature of fluid medium in the jacket, U is the overall heat transfer coefficient and aV is the heat transfer surface in a unit volume of suspension. In Eq.(10), mc denotes the total mass of solute and crystals in a unit volume of suspension 3μρcVcc kwm += (11) assuming that the heat capacities of crystals and solute are the same value. Moment equation model The population balance model (1)-(11) describes the properties of crystalline particles and the behaviour of crystallizer but the numerical solution of the population balance equation (1) is a complex time consuming procedure. The moment method provides an alternative way of handling the problem. This is a widely used method in studying the disperse systems which often provides exact results. The mass and heat balances are formulated by means of the third order moment which can be determined from the set of equations for the first four leading moments in the moment equation hierarchy: ( ) ( )TwwB dt td sc ,, 0 = μ (12) ( ) ( ) ( )tTwwG dt td sc 0 1 ,, μ μ = (13) ( ) ( ) ( )tTwwG dt td sc 1 2 ,,2 μ μ = (14) ( ) ( ) ( )tTwwG dt td sc 2 3 ,,3 μ μ = (15) As a consequence, the ordinary differential equations for moments μ0, μ1, μ2 and μ3 together with the mass and heat balance equations (8) and (10) provide a closed model consisting of six ordinary differential equations, as well as of the kinetic equations (5) and (6) and the constitutive expression (7) describing the solubility of para-xylene in a xylene solution. Simulation results and discussion Numerical solution of the set of differential equations (8), (10) and (12)-(15) were solved using the ODE solvers in MATLAB environment. In numerical experiments the temporal temperature profile was simulated by means of the following function ( ) startendstart bn endc bn c control TTT t t T +− ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + − = 11 , 1 (16) where fixing the start and end values, the shape of the profile was controlled by changing the parameter bn. Fig. 3 presents some possible temperature profiles depending on the value of parameter bn. The thin lines present the evolution of temperature in the suspension while the thick lines show the temperature changes in the jacket. The differences reflect, in principle, the dynamics of the crystalline suspension coupled with the wall of the crystallizer. The process and kinetic parameters used in numerical experimentation are listed in Tables 2 and 3. The kinetic parameters were chosen as published by Mohameed et al. (2006) Table 2: Process parameters used in simulation V=10 m3 Vh=3 m 3 T0=5 °C Uav=1,1e3 qh=1 m 3 s-1 Tend=-18 °C b0=0.6943 b1=0.02 b2=0.0002 Table 3: Kinetic parameters used in simulation kg0=28.5 ms -1 g=3.95 kv=0.52 b=0.89 kb0=812.6 #m -3s-1 ρh=1000 kg m -3 ρc=1100 kg m -3 ρsv=1000 kg m -3 Eg=785. J mol -1 Eb=-56.37 J mol -1 Figure 3: Temperature profiles by Eq.(16) 4 A number of numerical experiments have been carried out to choose the so called optimal temperature profile which provide the maximum final mean size of crystals 32L defined as ( ) ( ) ( )end end end t t tL 2 3 32 μ μ = (17) where tend denotes the final moment of time corresponding to the final temperature Tend. Fig. 4 shows the temporal profiles of solute concentration and the equilibrium saturation concentration corresponding to the temperature profiles in Fig. 3. The evolutions of the zero order moment μ0 are presented in Fig. 5 for the cooling methods shown in Fig. 3. This moment provides, in principle, the total number of crystals in a unit volume of suspension. Differences are observed in the final values of the number of crystals as a function of the cooling profiles. The optimal cooling profile produced the lowest value of crystal number. Figure 4: Solute and saturation concentration Figure 5: Evolution of the zero order moment μ0 The histories of the third order moment μ3 as a function of time are shown in Fig. 6. This moment is related to the total volume of crystals in a unit volume of suspension from which the mass of the crystalline product can be obtained by multiplying that with the density of crystals. There are significant differences in the time profiles but the final volumes of the crystalline product becomes of nearly the same value at the end of the process. Temporal evolutions of the mean size of crystals 23L , defined by Eq.(15) are presented in Fig. 7. The graphs in Fig. 7 illustrate well that variation of the mean crystal size exhibit significant differences during the course of the batch process. The optimal cooling profile gives the largest mean crystal size in accordance with the results shown in Figs 5 and 6. The other two cooling methods produced nearly the same mean sizes. Figure 6: Evolution of the third order moment μ3 relating to the total volume of crystals in a unit volume of suspension for different cooling methods Figure 7: Temporal evolutions of the mean crystal size 32L depending on the cooling methods of Fig. 3 5 It should be noted that the temporal evolution of the mean crystal size passes through maximum in each case as it is shown in Fig. 7. It is a consequence of that although all the moments increase monotonously in time the rates of their increase differ leading to such an unusual phenomenon. Similar phenomena can be observed in Fig. 8 comparing the temporal evolutions of the mean crystal size 32L with that of 10L defined as ( ) ( ) ( )t t tL 0 1 10 μ μ = (18) Both graphs presented in Fig. 8 were computed applying the linear temperature profile. The corresponding evolutions of the first order moment are shown in Fig. 9 for the sake of illustration that these moments also increase monotonously in time. Figure 8: Temporal evolution of the mean sizes 32L and 10L Figure 9: Evolution of the first order moment μ1 It seems to be interesting to note that the mean crystal size 32L is greater than 10L at each moment of time during the course of the process leading to greater value even at the final moment. Mohameed et al. (2006) published kinetic parameters differing to some extent, obtained by fitting their model to experimental data produced by using diverse cooling profiles. A second set of their kinetic parameters is shown in Table 4. The differences between the processes of crystal mean sizes obtained by Table 4: Kinetic parameters used in simulation kg0=16.25 ms -1 g=4.74 kv=0.52 b=0.71 kb0=345.59 #m -3s-1 ρh=1000 kg m -3 ρc=1100 kg m -3 ρsv=1000 kg m -3 Eg=370.23 J mol -1 Eb=-63.37 J mol -1 using the original parameters listed in Table 3 and those in Table 4 are shown in Fig. 10. Fig. 10 illustrates well that the time histories of the mean crystal sizes differ from each other significantly but the final mean sizes of the crystalline product exhibit just about 10% difference. This means that the parameters obtained from different experiments reflect different dynamic behaviour of the batch crystallizer but rather similar final product properties. Figure 10: Evolution of the mean size 32L by using the kinetic parameters in Tables 3 and 4 Conclusions A population balance model was developed and applied for simulation of batch cooling crystallization of para- xylene. In order to apply the moment method a moment equation hierarchy was generated from the population balance equation. The set of equations for the first four moments coupled with the mass and energy balance equations proved suitable to investigate the dynamic 6 behaviour of the crystallizer and to predict the product properties of the cooling crystallization process. The applied model allowed simulating the heat transfer through the wall of crystallizer by using a controlled jacket cooling method. In this case, there was negligible difference in the product values from the perfectly controlled crystallizer and that when the controlled jacket temperature model was applied. The results of numerical experimentations revealed that different cooling profiles generate significant differences in the characteristic values of the crystalline product. Analysis of the simulation results allows concluding that the product properties can be computed and designed successfully by using a mathematical model and numerical experimentation. By means of the numerical model presented the batch crystallization system can be optimized by choosing the best temperature profile for producing a crystalline product exhibiting the required properties. ACKNOWLEDGEMENTS This work was supported by the Hungarian Scientific Research Fund under Grant K 77955. Also, this work was supported by the European Union and co-financed by the European Social Fund in the frame of the TAMOP-4.2.1/B-09/1/KONV-2010-0003 and TAMOP- 4.2.2/B-10/1-2010-0025 projects. SYMBOLS aV - heat transfer surface in a unit volume, m 2 m-3 B - nucleation rate, # m-3 s-1 b - kinetic parameter bn - optimization variable, Eq.(14) C - heat capacity, J kg-1 K-1 E - Energy, J mol-1 G - growth rate, m s-1 k - kinetic coefficient L - size variable, m n - population density function, # m-4 T - temperature, K t - time, s U - heat transfer coefficient, J m-2 s-1 K-1 V - volume, m3 w - concentration, kg_solute kg_solvent-1 GREEK SYMBOLS ε - volumetric ratio of solution μk - k th order moment ρ - density, kg m-3 INDICES b - nucleation c - crystal g - growth in - inlet s - equilibrium saturation concentration sv - solvent REFERENCES 1. H. A. MOHAMEED, B. A. JDAYIL, K. TAKROURI: Separation of para-xylene mixture via crystallization, Chem. Eng. Proc., 46, (2006), 25–36 2. J. H. PERRY: Chemical engineers Handbook (In Hungarian); Technical Press, Budapest, Hungary (1968) 3. T. BICKLE, B. LAKATOS, CS. MIHÁLYKÓ: Mathematical models of particulate system. (in Hungarian); The New Results in Chemistry 89, Academic Press, Budapest (2001) 4. B. G. LAKATOS: Population balance modelling of crystallization processes, Hungarian Journal of Industrial Chemistry 35, (2007), 7–17 5. B. G. LAKATOS, CS. MIHÁLYKÓ, T. BICKLE: Modelling of interactive populations of disperse systems, Chemical Engineering Science 61, (2006), 54–62 6. D. RAMKRISHNA: Population Balances, Theory and Applications to Particulate Systems in Engineering, Academic Press, San Diego (2000) 7. A. D. RANDOLPH, M. A. LARSON: Theory of Particulate Processes, Academic Press, New York (1988) 8. N. S. TAVARE: Industrial Crystallization. Process Simulation, Analysis and Design. Plenum Press, New York (1995) 9. ZS. ULBERT: Investigation and modelling of dynamic processes of crystallizers (in Hungarian). PhD Thesis, University of Veszprém, Veszprém (2002) 10. Á. BORSOS, B. G. LAKATOS: Modelling of crystal size distribution in batch cooling crystallizer (in Hungarian); Proceedings of Conference Chemical Engineering Days, Veszprém (2011) 11. Á. BORSOS: Modelling and optimizing cooling crystallizers (in Hungarian), MSc Thesis, University of Pannonia, Veszprém (2010) 12. Á. BORSOS, B. G. LAKATOS: Investigation of dynamic processes of cooling crystallizers in case of needle-shape crystals (in Hungarian). Acta Agraria Kaposváriensis (in press) (2010) 13. R. DE GEODE: Crystallization of para-xylene with scraped surface heat exchangers; Ph.D. Thesis, Delft University of Technology, Netherlands (1988) << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. 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