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.

( ⎥⎦

⎤⎢⎣

⎡−+⎟

⎠⎞

⎜⎝⎛ −−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ satm

m

m0

L2

L2 PP

RT∆v1

TT

RT∆HXln

φφ )

(1)

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=vL-vS, 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.

cuRT∆vP

TT

RT∆HX m

m

UoP

LP

LP

⎥⎦

⎤⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛ −−=⎟⎟

⎠

⎞⎜⎜⎝

⎛1ln

φφ

(2)

♣ To whom all correspondence should be made. E-mail: sunil_maity@che.iitkgp.ernet.in; Ph: 91-3222-281378

1

Presented at1st National Conference of Research Scholar and Young Scientists, Indian Institute of

Technology, Kharagpur, India, 2004.

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:

⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛ −−=⎟⎟

⎠

⎞⎜⎜⎝

⎛1

TT

RT∆H1

TT

RT∆HXln ss

ss

ssm

m

mo

L2

L2

φφ

(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.

Table 1: Table 1: PC-SAFT Parameters (Gross & Sadowski, 2001) of Organic Solutes and Solvents Hydrocarbon Segment no (m) Segment Diameter (σ, 0A) Energy Parameter (ε/κ, K)

Solutes

n-dodecane(C12) 5.3060 3.8959 249.21 n-hexadecane(C16) 6.6485 3.9592 254.70 n-octadecane(C18) 7.3271 3.9668 256.20 n-octacosane(C28) 10.3622 4.0217 252.0 n-dotriacontane(C32) 11.835 4.0217 252.0 biphenyl 3.8877 3.8151 327.42

Solvents n-hexane 3.0576 3.7983 236.77 n-heptane 3.4831 3.8049 238.40 n-decane 4.6627 3.8384 243.87 benzene 2.4653 3.6478 287.35 m-xylene 3.1861 3.7563 291.05 o-xylene 3.1362 3.7600 283.77 p-xylene 3.1723 3.7781 288.13

Table 2: Solid-Solid Transition and Melting Properties (Pan & Radosz ,1999; McLlaughlin & Zainal, 1959 )

Hydrocarbon Tss, K ∆Hss, J/mol Tm, K ∆Hm, J/mol n-dodecane (C12) 263.6 36977 n-hexadecane (C16) 291.2 53563 n-octadecane (C18) 301.1 59400 n-octacosane (C28) 331.2 35447 334.4 64658 n-dotriacontane (C32) 338.9 42700 342.1 76000 biphenyl 342.1 18732 Molar volume (cm3/mol) of n-octacosane and Temperature (K)

vL = 0.1238T +365.588; vs =0.11828T + 381.623

2

CRSYS, IIT Kharagpur, India, 2004.

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

Fig. 2 r

Fig. 2.3 EffeLee et ap-xylene

data taken from Hoerr & Harwood , 1999. These data could be predicted with PC-SAFT model with Kij = 0.0.

xperimbased prmodel p2.2 SoliMcLaugmodel-bthe mod

Solid-solid transition and melting properties are tabulated in Table 2.

0 .0 0 .2 0 .4 0 .6 0 .8 1 .02 4 0

2 5 0

2 6 0

2 7 0

2 8 0

2 9 0

3 0 0

3 1 0

3 2 0

3 3 0

3 4 0

3 5 0

h

e

T op G raph-C 3 2B o ttom G rap h- C 1 8

Tem

pera

ture

(K)

E x p e rim e n ta l (7 )

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 18,K ij= 0 .0 0 0 3 C 32,K ij= 0 .0 0 0 4

W e ig h t F ra c tio n o f S o lu te0.0 0.2 0.4 0.6 0.8 1.0

220

240

260

280

300Solid Symbol-Experimental Open Symbol-Model Predictions

C12, Kij=0.0C

16, Kij=0.0

Tem

pera

ture

(K)

Weight Fraction of Solutes

280

290

300

310

320

330

340

do

0.0

240

250

260

270T em

pera

ture

(K)

comparicorrelatefugacitycorrelatiassumedsolubilit 3. SENSA sensiunderstapressure

CRSYS, IIT Kharagpur, India, 2004.

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

epresents the solubility of n-octadecane and n-dotriacont

3: SLE for biphenyl in benzene at 1 bar

ct of Pressure on Solid-Liquid Equilibrium l, 1993 measured the saturation condition fo and n-dec

ental data were taken from Chang et al, 1983. The compedictions corresponding to Kij =0.0 and correlated results redictions could be used with only a small error. d-Liquid Equilibrium of Aromatic Compounds hlin & Zainal, 1959 studied the solubility of biphenyl inased predictions have been compared with thei experimel predictions give well agreement with the experimental d

10 mole% sobinary and tume change (

ure as desc-vapor pressure of the syste

all Kij.

SYST SLE model perature, adju.

r

0.2 0.4 0.6 0.8 1.0

Experimental Model Prediction(Kij=0.0)

3 0 6

0

5 0

SDS

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

Pre

ssur

e (B

ar)

Mole Fraction of Biphenyl

octa

r n-octacosanane solvents for approximately

sons of their experimental data for both d results are shown in Fig. 4 & 5. The vol coefficient of solute under the elevated pressons as shown in Table 2 and the saturated to be negligible in comparison to the pressuy data under elevated pressure with only sm

ITIVITY ANALYSIS FOR POLYETHYLENE tivity study was performed using the tunednd the effects of crystallinity, melting tem on solid–liquid equilibrium of polyethylene

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

ane in n-heptane solvent. The corresponding arisons of the experimental data with model-corresponding to the adjusted Kij showed that

benzene for higher mole fraction range. The ental data as shown in Fig. 3. It showed that ata.

3 0 9 3 1 2 3 1 5 3 1 8

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

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

T e m p e ra tu re (K )

Fig Effec ry SLE of n-cosane (C

. 4: t of pressure on bina28) in n-decane for different composition.

3

lid content and pressure up to 200 bar. The ernary systems with model predictions and ∆v) of n-octacosane required to calculate the ribed by equation 1, was calculated from the re of solute (Psat) in the same equation was m (P). The model correlates the experimental

EM as used for low molecular weight system to stable binary interaction parameter (Kij) and

e in n-decane and n-octacosane in mixture of

330

350

360

370

380

390

400

410

420

0.0 0.2 0.4 0.6 0.8

340

1.0 bar100 bar

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

Tem

para

ture

(K)

The PC-SAFT parameters of polyethylene (m/M= 0.0263mol/g, σ=4.02170A, ε/κ 49.5 K) used by Gross & Sadowski, 2002 for high-pressure phase equilibrium calculation was used for this study.

=2

Weight Fraction of Polyethylene(PE120K)

310 312 314 316

0

50

100

150

200

250

300

ExperimentalModel Predictions (Kij=0.0)

Pres

sure

(Bar

)

Correlated Results (Kij=-0.0004)

CRSYS, IIT Kharagpur, India, 2004.

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).

Temperature (K)

0.0 0.2 0.4 0.6 0.8 1.0

300

320

340

360

380

400

420

C=0.2C=0.4C=0.6

---------Kij=0.0,Tm=415K

Tem

pera

ture

(K)

Weight Fraction of Polyethylene(PE120K)

logging of pipelines due to crystallization of polyethyle

in bi

TUDY OF EXISTING EXPERIMEOLYETHYLENE WITH MODEL BASED PREDICTO

At fixed temperature solubility decreases with increase in preon solubility was found to be not substantial for only a smpressure will decrease solubility of polyethylene to a greatcpolyethylene decreases at a fixed temperature and pressure asolubility has been found to be substantial during this study. With increase in melting point, the solubility of polyethylenand crystallinity as shown in Fig. 8. With increasepolyethylene was decreased at a fixed temperature, pFrom the figures presented in this study, it was observed thmelting point of solute.

4. COMPARATIVE SPThe solubility data for polyethylene having molecular wcrystallinity 0.8 in m-xylene at atmospheric pressure is availHowever, tuning of PC-SAFT based predictor for real life general methodology proposed in this work was used to comwith experimental data. The PC-SAFT parameters of low d

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

34

35

36

37

38

39

40

41

42

43

44

Tem

pera

ture

(K)

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

ne. With increase in crystallinity, solubility of Fig. 7. The influence of crystallinity on

nary interaction parameter, Kij, the solubility of

NTAL SOLUBILITY DATA OF R

ssure as shown in Fig. 6. But the effect of pressure all change in pressure. But enormous change in

extent sufficient to cause industrial problem like

s shown in

e decreases at a fixed temperature, pressure, Kij,

ressure and crystallinity as shown in Fig. 9. at the solute is totally soluble in the solvent at the

eight 17000 g/mol, melting point 387.5K and able in existing literature, Pan and Radosz (1999). systems has rarely been studied in literature. The pare the model-based predictions and correlations ensity polyethylene (LDPE) reported by Gross &

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

0.0 0.2 0.4 0.6 0.8 1.00

0

0

0

0

0

0

0

0

0

0

Tm=415KTm=440K

-1 bar,Kij=0.0,C=0.4

Weight Fraction of Polyethylene(PE120K)

4

0.0 0.2 0.4 0.6 0.8 1.0330

340

350

360

370

380

390

400

410

420

kij=0.012 kij=0.01 kij=-0.0 kij=-0.008

-1bar,Tm=415K,C=0.4

Tem

pera

ture

(K)

Weight Fraction of Polyethylene(PE120K)

parameters. So the PC-SAFT parameters are

regressed using these solubility data of LDPE for beparameters for LDPE were determined to be m/M=

e model correlates well the solubility of LDPE.

NCLUSION computer based general methodology developed usine solubility of real solutes after tuning with the explubility of polyethylene. The model requires cry

olyethylene data to predict and correlate the solid-liqlvent systems and is useful to predict and the solubility

rganic solutes, the prediction of solubility using the lubility data. For polyethylene system this model could

arameter, binary interaction parameter (Kij).

ACKNOWLEFi 1 02

OMENCLATURE

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

390

380

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

0.0 0.2 0.4 0.6 0.8 1.0

340

350

360

370

Tem

pera

ture

(K)

Weight Fraction of Polyethylene

Experimental Kij m/M σ ε/κ

0.0 0.0263 4.0217 249.5 0.0 0.0243 4.50217 399.0.0015 0.0243 4.50217 399.0

comparison of predictions and correlations with experiwski, 2001,and newly evaluated parameters are shoSado

th COAthsopsoosop

DGEMENT nancial assistance from MHRD project No. F27- /20

NC = 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 associP

atin

u = number of ethyl units in backbone

sat = saturated pressure of solute at meltinR = gas constant SAFT = statistical associating fluid theorySLE = solid–liquid equilibrium SS = solid-solid T = temperature

CRSYS, IIT Kharagpur, India, 2004.

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

5

tter correlation and trend of results. The PC-SAFT 0.0243 mol/gm, σ = 4.50217 0A, ε/κ=399 K. The

n reported parameters of Gross & g the newly evaluated parameters,

g PC-SAFT equation of state, could be used to predict erimental data, was used to predict and correlate the stallinity, melting point, and molecular weight of uid equilibrium. The model can be used to multiple at elevated pressure as well. For low molecular weight same model gives well agreement with experimental be used to correlate the solubility by adjusting the one

T

[-] [kJ / mol]

[-] [-]

mental data based own in Fig. 10. Usin

S.V dated 19.3.2002 is thankfully acknowledged.

[kJ / mol] [J/K]

[g/mol] [bar]

g fluid theory [-] perature [bar] [J. Mole-1K-1]

[-] [-]

[-] [K] [-]

g tem

6

∆v = vL-vS, volume change X, x = mole fraction [-] <Greek Letters > ρ , ρc = density of completely amorphous and comp tely crystalline solute respectively [g/cm3]

[cm3/mol] [cm3/mol]

v = molar volume

lea

σ = segment diameter [ 0A ]

φ = fugacity coefficient [-] φ0 = fugacity coefficient of pure sub cooled liqu ute at given Temp and Press [-] ε = depth of pair potential <Subscripts> 2 = solute 02 = pure solute m = melting condition p = polymer ss = solid–solid phase transition <Superscripts>

hang S. S., Maurey J. R., and Pummer W. J., Solubilities of two n-alkanes in various solvents, J. Chem. Eng. -189.

ee H. G., Groves F. R., and Wolcott J. M., Effect of pressure on solid-liquid equilibrium for p-xylene + octacosane, and decane + p-xylene + phenanthrene mixtures, J. Chem.

ee H. G., Schenewerk P. A., and Wolcott J., and Groves F. R. Jr, Effect of pressure on solid-liquid equilibrium /n-decane/n-octacosane, Fluid Phase Equilibria, 128 (1997) 229-240.

Llaughlin E.a lub eh viour o mati art ΙΙ. Solubilitic n tetrachlori . M lin E. a ar ns in b , J

59, p. 3. hase Eq libria, Editio

Pa z ria in aphth ene, nosolu ons Fluid P

s, Inc., Ch ical and Engineering Th rmodyn

Ka

s lid id eq ilibria o tems containilac ne, J. C 90-149

id sol

[J]

L = liquid phase sat = property at saturation condition LITERATURE CITED CData, 28(1983)187Domanska 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. Ldecane+octacosane, decane +Eng. Data, 38(1993)257-259. Lfor the system carbon dioxideMc nd Zainal H.A., The so ility b a f aro c hydrocarbons. P es in arbo de, J. Chem. Soc., 1960, p.2485cLlaugh nd Zainal H.A., The solubility behaviour of aromatic hydroc bo enzene . Chem. Soc.,

19 86Prausnitz J. M., Molecular Thermodynamics of Fluid P iu 2nd n, 1986, p.419.

C., adosn R M., Modeling of solid–liquid equilib n al rmal-alkane and polyethylene 5 (1999) 57-73. ti , hase Equilibria,1 5

on , 3rdSandler S. I., John Wiles and S em e amics edition, 1999, p.575-585.

a Y.andTanak wakami M., Solid-liquid phase equilibria in binary (benzene, cyclohexane + n-tetradecane, n-u es 230-323K an pressu e up to hexadecane) systems at temperat r d r 120MPa, Fluid Phase Equilibria, 125

0 -114. (1996)1 3g R., Con J., Bina so -liqu s nWittin tantinescu D., and Gmehling ry u f organic sy g ε-

o (2001)1capr to hem. Eng. Data, 46 4 3.

CRSYS, IIT Kharagpur, India, 2004.