Modeling and Simulation of Solid-Liquid Equilibrium: Model Validation Using Solubility Data and Sensitivity Study for Polyethylene System. Sunil Kumar Maity♣, Kalyan Gayen, Sirshendu De and Saibal Ganguly Department of Chemical Engineering, Indian Institute of Technology, Kharagpur-721302, India

ABSTRACT In this work, PC-SAFT equation of state was used to model solid-liquid equilibrium (SLE). With the experimental SLE data available in literature for low molecular weight n-alkanes and aromatic compounds both at atmospheric and elevated pressure, the suitability of the developed SLE model based on PC-SAFT equation of state was tested. Subsequently a sensitivity study was performed to understand the effects of different parameters that affect the solubility of polyethylene. Keywords: Solid-Liquid Equilibrium, Polyethylene, Model based Predictor INTRODUCTION The study of solubility of polyethylene is of great technical interest for developing and designing separation processes, such as crystallization and fractionation and solving industrial problems. In the polyethylene industry, crystallization of polyethylene on the reactors surfaces, heat exchangers, and flush drums and clogging of pipelines are some of the frequently encountered industrial problems. These industrial problems can be investigated and rectified with the proper knowledge of solid-liquid equilibrium and availability of a validated computer based model. In the present work, PC-SAFT equation of state of Gross & Sadowski, 2001 was used to model solid-liquid equilibrium since it has wide applicability starting from low molecular weight organic compounds to highly non-ideal macro-molecular weight system such as polymers. This equation of state requires three pure component parameters: segment number (m), segment diameter (σ), and energy parameter (ε/κ). Additionally PC-SAFT has adjustable solvent-solute binary interaction parameter (Kij). This model was tested from totally crystalline to partially crystalline solutes such as polymer. Using the experimental solubility data available in literature for low molecular weight solutes and polyethylene the suitability of the developed model was first tested. Sensitivity analysis for polyethylene system was performed using the tested model to understand the effects of pressure, crystallinity, melting temperature, and binary interaction parameter (Kij) on solid–liquid equilibrium. The binary interaction parameter acts as tuning parameter for the model based predictor. This work presents a general methodology for development of model based predictors for industrial usage and its tuning with experimental data for any general polymer and solvent system. 1. MODELING OF SOLID-LIQUID EQUILIBRIUM The fundamental principle of any phase equilibrium calculation is that the fugacity of any component is equal in all the phases under equilibrium condition. With this fundamental principle, Cheng & Radosz, 1999 developed the solid-liquid equilibrium model that is applicable to totally crystalline solutes. ⎡ ∆H m ⎛ T m ⎤ ⎛ φ LX L ⎞ ∆v ⎞ ( − 1⎟ + ln ⎜⎜ 2 0 2 ⎟⎟ = − ⎢ P − P sat )⎥ ⎜ (1) RT ⎠ ⎝ φ ⎠ ⎣ RT m ⎝ T ⎦ where ϕ0 is the fugacity coefficient of pure sub-cooled liquid solute at constant T and P. ϕ2L is the fugacity coefficient of solute in solution, χ2L is the equilibrium solubility of solute, Psat is the saturated-vapor pressure of solute at its melting temperature (Tm). ∆v is the volume difference of liquid and solid solute defined as∆v=vLvS, and ∆Hm is the enthalpy of fusion of solute. Subsequently the author extended this equation for partially crystalline solutes such as polymer with the hypothesis of Harismiadis & Tassios, 1996, who assume that the logarithm of the ratio of fugacities is proportional to crystallinity (C). The crystallinity is the fraction of crystalline substances present in the polymer. The amorphous polymers will have zero crystallinity.

⎛φ L X ln ⎜⎜ P o ⎝ φP ♣

L P

⎡ ∆H ⎞ ⎟⎟ = − ⎢ ⎠ ⎣ RT

U m

∆vP ⎛ Tm ⎞ − 1⎟ + ⎜ RT ⎝ T ⎠

⎤ ⎥ cu ⎦

(2)

To whom all correspondence should be made. E-mail: [email protected]; Ph: 91-3222-281378

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Where ∆Hu is the enthalpy of melting per mole of crystal unit. For polyethylene, ∆Hu =8.22 kJ/mol, as reported by Van Krevelen, 1990. The polymer-volume change, ∆v, is determined from the densities of an amorphous polymer, ρa, and a crystalline polymer, ρc: ∆v=1/ρa-1/ρc. For polyethylene, ρa=0.853 g/cm3 and ρc=1.04 g/cm3. Solid crystalline normal alkanes such as n-octacosane and n-dotriacontane exhibit a solid-solid (ss) phase transition a few degrees below its melting point. Since the two solid phases are in a state of thermodynamic equilibrium, the effect of the solid-solid phase transition is included as follows:

⎛ φ LX L ln ⎜⎜ 2 o 2 ⎝ φ

⎞ ∆H ss ⎛ Tss ∆H m ⎛ Tm ⎞ ⎞ ⎟⎟ = − − 1⎟ − − 1⎟ ⎜ ⎜ RT T RT T ⎠ ⎠ ⎠ m ⎝ ss ⎝

(3)

where Tss is the ss transition temperature and ∆Hss is the enthalpy of the ss transition. The fugacity coefficients of solute appeared in the above three equations are calculated by PC-SAFT equation of state. For the correlation and prediction of phase equilibrium in macromolecular systems, the equations of state for chain molecules have been successfully used for more than two decades. Sadowski et al, 2001 developed perturbed-chain SAFT equation of state based on perturbation theory. According to their model, molecules are considered as chains composed of spherical segments. According to the perturbation theory, the potential function can be divided into repulsion and attraction components. Attraction may be due to dispersion or association. In the present work, only dispersion based attraction is taken into consideration. Special types of forces like hydrogen bonding and dipole-dipole interaction have not been considered. The model is applicable to real chain molecules of any length, from spheres to polymers and can be used to calculate density, vapor pressure, and caloric properties. 2. TUNING OF THE SLE MODEL USING EXISTING SOLUBILITY DATA The suitability of the developed composite model was tested by using the experimental solubility data for low molecular weight organic compounds collected from existing literature. PC-SAFT parameters of these solvents and solutes are listed in Table 1. (Gross & Sadowski, 2001) Table 1: Table 1: PC-SAFT Parameters of Organic Solutes and Solvents Hydrocarbon Segment no (m) Segment Diameter (σ, 0A) Energy Parameter (ε/κ, K) Solutes n-dodecane(C12) n-hexadecane(C16) n-octadecane(C18) n-octacosane(C28) n-dotriacontane(C32) biphenyl

5.3060 6.6485 7.3271 10.3622 11.835 3.8877

n-hexane n-heptane n-decane benzene m-xylene o-xylene p-xylene

3.0576 3.4831 4.6627 2.4653 3.1861 3.1362 3.1723

3.8959 3.9592 3.9668 4.0217 4.0217 3.8151

249.21 254.70 256.20 252.0 252.0 327.42

3.7983 3.8049 3.8384 3.6478 3.7563 3.7600 3.7781

236.77 238.40 243.87 287.35 291.05 283.77 288.13

Solvents

Table 2: Solid-Solid Transition and Melting Properties

(Pan & Radosz ,1999; McLlaughlin & Zainal, 1959 )

∆Hss, J/mol Tm, K Hydrocarbon Tss, K n-dodecane (C12) 263.6 n-hexadecane (C16) 291.2 n-octadecane (C18) 301.1 n-octacosane (C28) 331.2 35447 334.4 n-dotriacontane (C32) 338.9 42700 342.1 biphenyl 342.1 Molar volume (cm3/mol) of n-octacosane and Temperature (K) vL = 0.1238T +365.588; vs =0.11828T + 381.623

∆Hm, J/mol 36977 53563 59400 64658 76000 18732

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2.1 Solid-Liquid Equilibrium of n-Alkanes Fig. 1 shows the solubility of n-dodecane and n-hexadecane in n-hexane solvent, on the basis of experimental data taken from Hoerr & Harwood , 1999. These data could be predicted with PC-SAFT model with Kij = 0.0. Solid-solid transition and melting properties are tabulated in Table 2. 350

300

Solid Symbol-Experimental Open Symbol-Model Predictions

330

280

320 310

Temperature (K)

Temperature (K)

T o p G rap h -C 3 2 B o tto m G rap h - C 1 8

340

260

240

300 290 280

(7 )

E x p e rim e n ta l M o d e l P re d ic tio n s (K ij= 0 .0 ) C o rre la te d R e s u lt C 1 8 ,K ij= 0 .0 0 0 3 C 3 2 ,K ij= 0 .0 0 0 4

270 260

C12, Kij=0.0 C16, Kij=0.0

220

0.0

0.2

0.4

0.6

0.8

250 240 0 .0

1.0

0 .2

Weight Fraction of Solutes

0 .4

0 .6

0 .8

1 .0

W e ig h t F ra c tio n o f S o lu te

Fig. 1: SLE for n-dodecane and nhexadecane in n-hexane solvent at 1 bar

Fig. 2: SLE for n-octadecane and ndotriacontane in n-heptane solvent at 1 bar

Fig. 2 represents the solubility of n-octadecane and n-dotriacontane in n-heptane solvent. The corresponding experimental data were taken from Chang et al, 1983. The comparisons of the experimental data with modelbased predictions corresponding to Kij =0.0 and correlated results corresponding to the adjusted Kij showed that model predictions could be used with only a small error. 2.2 Solid-Liquid Equilibrium of Aromatic Compounds McLaughlin & Zainal, 1959 studied the solubility of biphenyl in benzene for higher mole fraction range. The model-based predictions have been compared with their experimental data as shown in Fig. 3. It showed that the model predictions give well agreement with the experimental data. 300

340 330

S o lid S y m b o l-E x p e rim e n ta l D o tte d L in e -M o d e l P re d ic tio n s (K ij = 0 .0 ) S o lid L in e -C o rre la te d R e s u lts (K ij = 0 .0 0 0 6 )

250

Pressure (Bar)

320

Temperature(K)

310 300 290 280 270

200

150

100

50

X 2 = 0 .0 6 0 1 3 X 2 = 0 .0 8 1 9 8 X 2 = 0 .1 0 7 4

260

Experimental Model Prediction(Kij=0.0)

250

0 306

240 0.0

0.2

0.4

0.6

Mole Fraction of Biphenyl

0.8

1.0

309

312

315

318

T e m p e r a tu r e ( K )

Fig. 4: Effect of pressure on binary SLE of noctacosane (C28) in n-decane for different composition. 2.3 Effect of Pressure on Solid-Liquid Equilibrium Lee et al, 1993 measured the saturation condition for n-octacosane in n-decane and n-octacosane in mixture of p-xylene and n-decane solvents for approximately 10 mole% solid content and pressure up to 200 bar. The comparisons of their experimental data for both binary and ternary systems with model predictions and correlated results are shown in Fig. 4 & 5. The volume change (∆v) of n-octacosane required to calculate the fugacity coefficient of solute under the elevated pressure as described by equation 1, was calculated from the correlations as shown in Table 2 and the saturated-vapor pressure of solute (Psat) in the same equation was assumed to be negligible in comparison to the pressure of the system (P). The model correlates the experimental solubility data under elevated pressure with only small Kij. Fig. 3: SLE for biphenyl in benzene at 1 bar

3. SENSITIVITY ANALYSIS FOR POLYETHYLENE SYSTEM A sensitivity study was performed using the tuned SLE model as used for low molecular weight system to understand the effects of crystallinity, melting temperature, adjustable binary interaction parameter (Kij) and pressure on solid–liquid equilibrium of polyethylene.

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The PC-SAFT parameters of polyethylene (m/M= 0.0263mol/g, σ=4.02170A, ε/κ=249.5 K) used by Gross & Sadowski, 2002 for high-pressure phase equilibrium calculation was used for this study. 300

420 410

250

400 390

Temparature(K)

Pressure (Bar)

200

150

100

380 370 360

1.0 bar 100 bar --------Kij=0.0,Tm=415K, C=0.4

350

50

Experimental M odel Predictions (Kij=0.0) Correlated Results (Kij=-0.0004)

0 310

312

314

340 330 0.0

316

Temperature (K)

0.2

0.4

0.6

Weight Fraction of Polyethylene(PE120K)

0.8

Fig. 6: Effect of pressure on solubility of polyethylene in m-xylene.

Fig. 5: Effect of pressure on SLE of n-octacosane(3) in mixture of n-decane (1) and p-xylene (2) (x1/x2=2,X3=0.09803).

At fixed temperature solubility decreases with increase in pressure as shown in Fig. 6. But the effect of pressure on solubility was found to be not substantial for only a small change in pressure. But enormous change in pressure will decrease solubility of polyethylene to a great extent sufficient to cause industrial problem like clogging of pipelines due to crystallization of polyethylene. With increase in crystallinity, solubility of polyethylene decreases at a fixed temperature and pressure as shown in Fig. 7. The influence of crystallinity on solubility has been found to be substantial during this study. With increase in melting point, the solubility of polyethylene decreases at a fixed temperature, pressure, Kij, and crystallinity as shown in Fig. 8. With increase in binary interaction parameter, Kij, the solubility of polyethylene was decreased at a fixed temperature, pressure and crystallinity as shown in Fig. 9. From the figures presented in this study, it was observed that the solute is totally soluble in the solvent at the melting point of solute. 440

420

430 400

Temperature(K)

360 340

C=0.2 C=0.4 C=0.6 ---------Kij=0.0,Tm=415K

320 300 0.0

0.2

0.4

0.6

0.8

Weight Fraction of Polyethylene(PE120K)

Fig. 7: Effect of crystallinity on solubility of polyethylene in m-xylene at 1 bar.

1.0

Temperature(K)

420

380

410 400 390 380 370

Tm=415K Tm=440K -1 bar,Kij=0.0,C=0.4

360 350 340 0.0

0.2

0.4

0.6

0.8

Weight Fraction of Polyethylene(PE120K)

1.0

Fig. 8: Effect of melting point (Tm) on solubility of polyethylene in m-xylene.

4. COMPARATIVE STUDY OF EXISTING EXPERIMENTAL SOLUBILITY DATA OF POLYETHYLENE WITH MODEL BASED PREDICTOR The solubility data for polyethylene having molecular weight 17000 g/mol, melting point 387.5K and crystallinity 0.8 in m-xylene at atmospheric pressure is available in existing literature, Pan and Radosz (1999). However, tuning of PC-SAFT based predictor for real life systems has rarely been studied in literature. The general methodology proposed in this work was used to compare the model-based predictions and correlations with experimental data. The PC-SAFT parameters of low density polyethylene (LDPE) reported by Gross &

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Sadowski, 2001, were determined solely by regressing the liquid density. The unsatisfactory results were found to predict and correlate the solubility of LDPE by using these parameters. So the PC-SAFT parameters are 420 390

410 400

Temperature(K)

380 370

kij=0.012 kij=0.01 kij=-0.0 kij=-0.008 -1bar,Tm=415K,C=0.4

360 350 340 330 0.0

0.2

0.4

0.6

0.8

Weight Fraction of Polyethylene(PE120K)

Fig. 9: Effect of Kij on solubility of polyethylene in m-xylene.

Temperature (K)

380

390

1.0

370

360

Experimental σ ε/κ Kij m/M 0.0 0.0263 4.0217 249.5 0.0 0.0243 4.50217 399.0 .0015 0.0243 4.50217 399.0

350

340 0.0

0.2

0.4

0.6

0.8

1.0

Weight Fraction of Polyethylene

Fig. 10: Solubility of polyethylene in m-xylene at 1 bar pressure and prediction by PC-SAFT model.

regressed using these solubility data of LDPE for better correlation and trend of results. The PC-SAFT parameters for LDPE were determined to be m/M=0.0243 mol/gm, σ = 4.50217 0A, ε/κ=399 K. The comparison of predictions and correlations with experimental data based on reported parameters of Gross & Sadowski, 2001,and newly evaluated parameters are shown in Fig. 10. Using the newly evaluated parameters, the model correlates well the solubility of LDPE. CONCLUSION A computer based general methodology developed using PC-SAFT equation of state, could be used to predict the solubility of real solutes after tuning with the experimental data, was used to predict and correlate the solubility of polyethylene. The model requires crystallinity, melting point, and molecular weight of polyethylene data to predict and correlate the solid-liquid equilibrium. The model can be used to multiple solvent systems and is useful to predict and the solubility at elevated pressure as well. For low molecular weight organic solutes, the prediction of solubility using the same model gives well agreement with experimental solubility data. For polyethylene system this model could be used to correlate the solubility by adjusting the one parameter, binary interaction parameter (Kij). ACKNOWLEDGEMENT Financial assistance from MHRD project No. F27-1/2002 TS.V dated 19.3.2002 is thankfully acknowledged. NOMENCLATURE C = crystallinity ∆H = molar enthalpy change ∆HU = enthalpy of melting per crystal unit κ = Boltzmann constant Kij = binary interaction parameter m = number of segments per chain M = molar mass P = pressure PC-SAFT = perturbed-chain statistical associating fluid theory P sat = saturated pressure of solute at melting temperature R = gas constant SAFT = statistical associating fluid theory SLE = solid–liquid equilibrium SS = solid-solid T = temperature u = number of ethyl units in backbone

[-] [kJ / mol] [kJ / mol] [J/K] [-] [-] [g/mol] [bar] [-] [bar] [J. Mole-1K-1] [-] [-] [-] [K] [-]

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v = molar volume [cm3/mol] L S ∆v = v -v , volume change [cm3/mol] X, x = mole fraction [-] = density of completely amorphous and completely crystalline solute respectively [g/cm3] ρ a, ρ c 0 = segment diameter [ A] σ φ = fugacity coefficient [-] 0 = fugacity coefficient of pure sub cooled liquid solute at given Temp and Press [-] φ ε = depth of pair potential [J] 2 = solute 02 = pure solute m = melting condition p = polymer ss = solid–solid phase transition L = liquid phase sat = property at saturation condition LITERATURE CITED Chang S. S., Maurey J. R., and Pummer W. J., Solubilities of two n-alkanes in various solvents, J. Chem. Eng. Data, 28(1983)187-189. Domanska U., Groves F. R. Jr., and McLlaughlin E., Solid-liquid phase equilibria of binary and ternary mixtures of benzene and polynuclear aromatic compounds, J. Chem. Eng. Data, 38(1993)88-94. Gross J.and Sadowski G., Modeling polymer systems using the perturbed-chain statistical associating fluid theory equation of state,Ind. Engg. Chem. Res., 41(2002)1084-1093. Gross J. and Sadowski G., Perturbed-chain SAFT: An equation of state based on a perturbation theory for chain molecules, Ind. Eng. Chem. Res., 40(2001)1244-1260. Harismiadis V.I., Tassios D.P., Solid-liquid-liquid equilibria in polymer solution, Ind. Eng. Chem. Res., 35 (1996)4667–4681. Lee H. G., Groves F. R., and Wolcott J. M., Effect of pressure on solid-liquid equilibrium for decane+octacosane, decane + p-xylene + octacosane, and decane + p-xylene + phenanthrene mixtures, J. Chem. Eng. Data, 38(1993)257-259. Lee H. G., Schenewerk P. A., and Wolcott J., and Groves F. R. Jr, Effect of pressure on solid-liquid equilibrium for the system carbon dioxide/n-decane/n-octacosane, Fluid Phase Equilibria, 128 (1997) 229-240. McLlaughlin E.and Zainal H.A., The solubility behaviour of aromatic hydrocarbons. Part ΙΙ. Solubilities in carbon tetrachloride, J. Chem. Soc., 1960, p.2485. McLlaughlin E. and Zainal H.A., The solubility behaviour of aromatic hydrocarbons in benzene, J. Chem. Soc., 1959, p.863. Prausnitz J. M., Molecular Thermodynamics of Fluid Phase Eqiulibria, 2nd Edition, 1986, p.419. Pan C., Radosz M., Modeling of solid–liquid equilibria in naphthalene, normal-alkane and polyethylene solutions, Fluid Phase Equilibria,155(1999) 57-73. Sandler S. I., John Wiles and Sons, Inc., Chemical and Engineering Thermodynamics, 3rd edition, 1999, p.575585. Tanaka Y.and Kawakami M., Solid-liquid phase equilibria in binary (benzene, cyclohexane + n-tetradecane, nhexadecane) systems at temperatures 230-323K and pressure up to 120MPa, Fluid Phase Equilibria, 125 (1996)103-114. Witting R., Constantinescu D., and Gmehling J., Binary solid-liquid equilibria of organic systems containing εcaprolactone, J. Chem. Eng. Data, 46(2001)1490-1493.

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