Liquid–solid equilibria

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Ž . Fluid Phase Equil ibria 149 1998 191–207 Liquid–solid equilibria in a decane q multi-paraffins system J. Pauly, C. Dauphin, J.L. Daridon )  Laboratoire Haute Pression, Centre Uni Õersitaire de Recherche Scientifique, UniÕersite de Pau, AÕenue de l’UniÕersite, ´ ´ 64000 Pau, France Received 18 December 1997; accepted 13 May 1998 Abstract Measurements, performed at atmospheric pressure on mixtures made up of decane plus various distributions of heavy normal paraffins from octadecane to triacontane, were carried out to provide information on wax Ž . cont ent amount and composit ion as functi on of temper at ur e. At vari ous temper at ur es below the wax app eara nce temper atu re, liq uid and sol id pha ses in partia lly frozen mix tur es wer e separated by iso the rma l filtration and analyzed by gas chromatography. Amount and composition of both phases were then deduced fr om mass balance af te r corre ction of the entr apped li quid in the soli d resi due. Furt hermor e, the L–S equilibrium data obtained are compared with the values predicted by means of several models. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Experimental method; Data; Solid–fluid equilibria; Heavy hydrocarbons 1. Introduction Certain petroleum fluids, such as paraffinic crudes, contain heavy hydrocarbons which, when they are present in significant proportions, can precipitate during production or during transport through a pip eli ne tra ver sin g col d reg ion s. Acc umu lat ion of these sol id dep osits, whi ch clo g up fil ters and obstruct pipelines if the phenomenon is not treated, represents a major risk of deterioration for this ki nd of equi pment. In or der to pr event this pr ocess, which is li nked to changi ng pr essure and temper ature con dit ion s and flu id compos iti on cau sed by bri ngi ng the wel l int o pro duc tio n, it is essential to be able to predict the phase behavior of the reservoir fluid by means of thermodynamic models. To develop models or to test the validity of existing models, it is useful to have experimental liquid–solid phase equilibrium data for systems whose composition is precisely identified. With this ) Corresponding author. Tel.: q33-5-59-92-3 0-54; Fax: q33-5-59-80- 83-82; E-mail: jean-luc.dar idon@univ-pa u.fr 0378-3812r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. Ž . PII: S0378-3812 98 00366-5

Transcript of Liquid–solid equilibria

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Ž .Fluid Phase Equilibria 149 1998 191–207

Liquid–solid equilibria in a decane q multi-paraffins system

J. Pauly, C. Dauphin, J.L. Daridon )

 Laboratoire Haute Pression, Centre UniÕersitaire de Recherche Scientifique, UniÕersite de Pau, AÕenue de l’UniÕersite,´ ´64000 Pau, France

Received 18 December 1997; accepted 13 May 1998

Abstract

Measurements, performed at atmospheric pressure on mixtures made up of decane plus various distributions

of heavy normal paraffins from octadecane to triacontane, were carried out to provide information on waxŽ .content amount and composition as function of temperature. At various temperatures below the wax

appearance temperature, liquid and solid phases in partially frozen mixtures were separated by isothermal

filtration and analyzed by gas chromatography. Amount and composition of both phases were then deduced

from mass balance after correction of the entrapped liquid in the solid residue. Furthermore, the L–S

equilibrium data obtained are compared with the values predicted by means of several models. q 1998 Elsevier

Science B.V. All rights reserved.

Keywords: Experimental method; Data; Solid–fluid equilibria; Heavy hydrocarbons

1. Introduction

Certain petroleum fluids, such as paraffinic crudes, contain heavy hydrocarbons which, when theyare present in significant proportions, can precipitate during production or during transport through apipeline traversing cold regions. Accumulation of these solid deposits, which clog up filters andobstruct pipelines if the phenomenon is not treated, represents a major risk of deterioration for thiskind of equipment. In order to prevent this process, which is linked to changing pressure andtemperature conditions and fluid composition caused by bringing the well into production, it is

essential to be able to predict the phase behavior of the reservoir fluid by means of thermodynamicmodels. To develop models or to test the validity of existing models, it is useful to have experimentalliquid–solid phase equilibrium data for systems whose composition is precisely identified. With this

)

Corresponding author. Tel.: q33-5-59-92-30-54; Fax: q33-5-59-80-83-82; E-mail: [email protected]

0378-3812r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .P I I : S 0 3 7 8 - 3 8 1 2 9 8 0 0 3 6 6 - 5

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w xaim in mind, we performed 1 liquid–vapor and liquid–solid phase transition measurements underpressure on synthetic mixtures whose composition was close to those encountered in natural fluids.Thus, the phase envelope and the temperatures of appearance of solid paraffins with differentmixtures made up of a methane q decane solvent in equimolar distribution and a heavy fraction madeup of a series of alkanes between C and C were determined experimentally up to 50 MPa.18 30

Unfortunately, these measurements give no data about the composition or the quantity of the

emerging solid phase. In order to make up this lack of information, we characterized experimentallyin this paper the solid deposits created by determining the composition and the quantity deposited vs.temperature at atmospheric pressure. To ensure consistency with previously studied mixtures, wemaintained the same heavy fractions; however, because the mixed solvent was completely degassed atatmospheric pressure, it was replaced by pure decane.

The equilibrium data were then compare with the values calculated by various predictive models.w xThe first model tested is the regular solution model of Won 2 in which both liquid and solid phases

w xare assumed regular. The second model, developed by Hansen et al. 3 , is based on the generalized

Ž . Ž . Ž .Fig. 1. Scheme of the transparent cell for liquid–solid separation. 1 Piston. 2 Transparent cylinder. 3 Heat conductingŽ . Ž . Ž . Ž . Ž .liquid. 4 Two-phase system. 5 Transparent cell. 6 Filter. 7 Valve. 8 Septum.

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w xpolymer solution theory of Flory 4 for the description of the liquid state and the solid phase isw xregarded as ideal. Whereas in the third model proposed by Coutinho et al. 5,6 , the solid state is

described by a local composition model. Finally, the experimental data have been used to test theprediction capacities of the approach which rest on the hypothesis that each component of the heavy

w xfraction can crystallize pure 7 , thus leading to several solid phases.

2. Experimental technique

The method generally used to characterize equilibria between fluid phases vs. temperature consistsof taking and then analyzing a micro-sample of each of the phases present in equilibrium at all thetemperatures investigated. In the case of liquid–solid equilibrium, where the solid phase is made up of an amalgam of crystals which is not necessarily homogeneous, sampling cannot be consideredrepresentative of the composition of the solid phase. In this context, to determine accurately thecomposition of the solid phase, it has to be completely isolated from the rest of the system and thenanalyzed as a whole. To do this, the mixture of heavy compounds is initially prepared in substantial

quantity by weighing and then diluted in the solvent until the desired overall composition is reached.The system obtained is then heated until complete dissolution of the paraffins and then homogenized.At this stage, the mixture is broken down into several samples, of an average capacity of 3 to 4 cm3,to allow for investigation at different temperatures.

Table 1Ž .Feed composition mass % of the C qdistribution of paraffins systems10

Mixture A Mixture B Mixture C References Purity

Feed composition

% of  n-C 64.73 47.76 65.02 Aldrich )99.010% of heavies 35.27 52.24 34.98

( ) HeaÕ  y fraction mass %

% of  n-C – 13.72 10.15 Fluka )99.018

% of  n-C – 12.27 10.15 Aldrich 99.019

% of  n-C 29.21 10.98 10.17 Aldrich 99.020

% of  n-C 20.97 9.87 10.15 Fluka )98.021

% of  n-C 15.01 8.86 10.09 Aldrich 99.022

% of  n-C 10.74 7.96 10.03 Aldrich 99.023

% of  n-C 7.66 7.12 9.96 Fluka )99.024

% of  n-C 5.46 6.35 9.86 Fluka )98.025

% of  n-C 3.88 5.69 9.76 Aldrich )99.026

% of  n-C 2.76 5.06 9.68 Fluka )98.027

% of  n-C 1.95 4.58 – Fluka )98.028

% of  n-C 1.38 3.99 – Fluka )99.529

% of  n-C 0.98 3.54 – Aldrich )99.030

a 0.684 0.858 0.951Ž .MW grmol 310.51 310.17 311.83

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The phase separation, which is obtained by compression of the two-phase liquid–solid systemŽ .through a filter, is carried out in a transparent cell Fig. 1 , thermoregulated at the desired temperature

Ž .with an accuracy of 0.02 K .When the filtration is complete, the two phases recovered are weighed and then analyzed on a

Hewlett-Packard 6890 gas chromatograph. Based on precise knowledge of the total mass of thesample and the quantity of matter present in the two phases, we were able to estimate by means of a

mass balance that the quantity of the chemical retained in the filtration system, valve or on the cellwalls was less than 1% for all the tests performed.At this stage, only the composition of the liquid phaseŽL. is determined accurately. The solid

phaseŽS., which is made up of a myriad of micro-crystals, tends to keep part of the liquid, coating thecrystals during compression filtering. So, the filtration residueŽSR. corresponds in fact to the superim-position of the solid phaseŽS. and of the trapped liquidŽLT.. Given the nature of the systems studies,

Table 2

Amount of solid deposit and composition of liquid and solid phases as a function of temperature for system A

Ž .T  K290.85 288.15 283.15 278.25 273.25 268.25

( ) Amount of solid deposit mass %

2.46 6.80 14.86 22.16 23.71 28.33

( )Composition of the solid phase mass %

C 4.09 7.26 13.33 18.04 21.18 25.0520

C 6.9 11.04 16.59 19.38 20.47 21.4121

C 10.2 14.00 16.8 17.03 16.57 15.9122

C 13.1 15.19 14.84 13.61 12.56 11.5523

C 14.36 14.07 11.74 10.02 9.13 8.2424

C 13.56 11.63 8.7 7.19 6.51 5.8225

C 12.11 9.3 6.48 5.3 4.8 4.2626C 9.54 6.78 4.55 3.72 3.4 3.0127

C 7.31 4.91 3.22 2.65 2.43 2.1628

C 5.3 3.44 2.23 1.87 1.75 1.5429

C 3.52 2.38 1.52 1.28 1.2 1.0630

( )Composition of the liquid phase mass %

C 67.14 70.48 77.13 83.46 88.04 93.2610

C 10.93 10.93 10.33 8.53 6.57 4.2720

C 7.37 7.01 5.64 3.87 2.62 1.2321

C 4.98 4.39 2.93 1.73 1.14 0.4922

C 3.36 2.72 1.55 0.88 0.61 0.2823

C 2.21 1.65 0.86 0.51 0.37 0.1824

C 1.43 1.01 0.52 0.33 0.23 0.1125

C 0.97 0.67 0.35 0.23 0.16 0.0726

C 0.64 0.44 0.25 0.17 0.10 0.0527

C 0.44 0.31 0.19 0.13 0.07 0.0328

C 0.31 0.23 0.15 0.10 0.05 0.0229

C 0.21 0.17 0.12 0.08 0.04 0.0130

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Table 3

Amount of solid deposit and composition of liquid and solid phases as a function of temperature for system B

Ž .T  K

293.15 290.85 288.05 283.15 277.55 268.95

( ) Amount of solid deposit mass %

4.57 6.80 8.15 12.92 21.58 28.03

( )Composition of the solid phase mass %C 0.00 0.00 0.01 0.10 0.25 4.3418

C 0.00 0.00 0.18 0.41 2.13 8.8519

C 0.00 0.00 0.40 1.25 5.28 11.0120

C 0.02 0.32 1.00 2.93 8.33 10.5021

C 0.48 1.29 2.79 5.60 10.52 9.7522

C 1.74 3.55 6.05 9.11 11.55 9.1523

C 4.16 7.07 9.84 11.55 11.04 8.3724

C 7.57 10.78 12.55 12.27 10.11 7.3925

C 11.86 14.34 14.27 12.80 9.56 6.9626

C 16.17 16.65 15.08 12.78 9.27 6.8027

C 18.76 16.44 13.80 11.80 8.27 6.3028

C 19.44 16.01 12.47 10.28 7.22 5.5329

C 19.80 13.54 11.58 9.13 6.47 5.0630

( )Composition of the liquid phase mass %

C 64.63 66.96 70.22 72.60 77.99 90.7410

C 5.00 5.25 5.09 5.61 6.56 4.8418

C 4.72 4.90 4.76 5.16 5.59 2.4419

C 4.43 4.50 4.40 4.56 4.14 0.9520

C 3.93 3.93 3.79 3.66 2.46 0.4121

C 3.62 3.52 3.31 3.23 1.34 0.2522

C 3.36 3.16 2.82 2.06 0.75 0.1823

C 2.80 2.48 2.04 1.25 0.42 0.1124

C 2.21 1.76 1.32 0.70 0.25 0.0825

C 1.72 1.24 0.85 0.41 0.16 0.0026C 1.34 0.87 0.56 0.29 0.13 0.0027

C 0.97 0.60 0.37 0.19 0.08 0.0028

C 0.72 0.47 0.27 0.16 0.07 0.0029

C 0.56 0.35 0.19 0.11 0.06 0.0030

which are all made up of a pure solvent much lighter than the heavy fraction, the proportion of liquidtrapped in the solid phase can easily be determined from the quantity of solvent measured in the solidresidue. Because the difference in length of chain between decane and the first distribution paraffin is

Ž .substantial eight carbons , there cannot be any partial miscibility between heavy paraffins and decane

w x8 . Now, as the melting temperature of the solvent is 243 K, it cannot be crystallized in our range of investigation temperatures. So its presence in the solid phase is solely due to the existence of thetrapped liquid:

mSR s mLT 1Ž .C C10 10

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As we know the exact composition in the liquid phase, it is possible to calculate the total amount of entrapped liquid by:

mLT s mSR r X L 2Ž .C C10 10

which leads to the fraction Y LT of liquid trapped in the solid residue of filtration:

 X SRC10

LTY  s 3Ž .L X C10

in which X SR and X L correspond to the mass fraction of solvent analyzed in the two partsC C10 10

Ž .recovered after separation. The quantity of solid crystallized expressed in mass fraction can thus becorrected by:

F S s F SR 1 y Y LT 4Ž . Ž .

where F SR and F S represent the mass fractions of the solid residue and the solid which has actually

Table 4

Amount of solid deposit and composition of liquid and solid phases as a function of temperature for system C

Ž .T  K

293.05 288.15 283.05 278.15 272.95 267.85

( ) Amount of solid deposit mass %

5.86 11.93 14.97 20.33 22.80 27.03

( )Composition of the solid phase mass %

C 0.00 0.03 0.07 0.10 0.58 1.2318

C 0.00 0.12 0.28 0.37 1.92 3.4619

C 0.29 0.56 0.98 1.23 4.65 6.8620

C 1.08 1.78 2.81 3.23 8.02 9.6521

C 3.21 4.70 6.43 6.84 11.16 11.4622C 7.57 9.72 11.62 11.72 13.66 12.9023

C 13.33 15.03 15.86 15.43 14.40 13.2024

C 19.38 19.05 18.29 17.86 14.49 13.0625

C 24.86 22.77 20.64 20.26 15.00 13.5826

C 30.29 26.25 23.03 22.96 16.12 14.5927

( )Composition of the liquid phase mass %

C 60.08 69.49 72.94 82.72 82.71 85.9310

C 3.94 4.15 4.43 3.64 4.62 4.7418

C 4.12 4.31 4.58 3.60 4.32 3.9419

C 4.25 4.38 4.57 3.31 3.47 2.6020

C 4.15 4.13 4.09 2.58 2.20 1.3021

C 4.16 3.76 3.34 1.75 1.21 0.6222

C 4.26 3.27 2.48 1.06 0.64 0.3423

C 4.01 2.42 1.53 0.58 0.34 0.2024

C 3.64 1.70 0.91 0.33 0.21 0.1325

C 3.64 1.30 0.63 0.23 0.16 0.1026

C 3.75 1.10 0.50 0.19 0.13 0.1027

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crystallized in the whole system. Finally, the real composition of the solid phase X S can be obtainedi

using:

 X SR y Y LT X LŽ .i iS X  s 5Ž .i LT1 y Y Ž .

for any given i.

3. Measurements

Measurements were made on three mixtures of different compositions of the decaneq heavyfraction system on which the liquid–vapor and liquid–solid transition measurements had been

w xpreviously performed by Daridon et al. 1 in the presence of methane. The molar proportion of Ž . Ž .decane 80% and heavy fraction 20% are the same in the three mixtures. Only the composition of 

Ž .the compounds present in the heavy fraction differs from one sample to the other Table 1 . Thesesubstances, which all belong to the n-alkanes, are distributed within the heavy fraction according to adecreasing distribution characterized by the recurrence relationship:

 x sa x 6Ž .c cnq1 n

corresponding to a simplified representation of the heavy fractions found in paraffinic crudes. Thecoefficient a , which is different for the three mixtures studied, was defined so as to maintain the

Fig. 2. Temperature dependence of the amount of solid deposit for system C.

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Ž .Fig. 3. Composition mass % of the solid phase vs. temperature for system B.

mean molar mass at 310.6 grmol which corresponds to the molar mass of docosane. However, thenumber of components must be different from one mixture to the other in order to keep the sameaverage molar mass with different values of  a coefficient. Thus, mixture A was prepared in order thatthe composition of the heavy fraction is continually decreasing from eicosane to triacontane while theheavy part of mixture B contains paraffins from octadecane to triacontane and mixture C is limited toheptacosane.

Measurements were carried out every 5 K from the phase change temperature which is estimated at293.35 and 299.35 K for systems A and B and 297.65 K for mixture C. The amount of solid deposit

Table 5

Percentages of paraffin crystallized vs. temperature for system A

290.85 K 288.15 K 283.15 K 278.25 K 275.25 K 268.25 K

Overall percentage of paraffins crystallized 7.13 19.81 43.02 63.25 72.21 85.37

Percentage of C crystallized 0.94 4.62 18.23 37.60 50.05 69.8220

Percentage of C crystallized 2.31 10.29 36.69 58.80 70.80 87.2821

Percentage of C crystallized 4.91 18.86 49.72 73.66 81.90 92.8022

Percentage of C crystallized 8.95 28.96 62.20 81.33 86.47 94.0723

Percentage of C crystallized 14.07 38.42 70.22 84.85 88.60 94.6124

Percentage of C crystallized 19.30 45.69 74.36 86.20 89.71 95.2225Percentage of C crystallized 24.01 50.47 76.15 86.66 90.44 95.7426

Percentage of C crystallized 27.33 52.90 76.18 86.29 90.98 95.9127

Percentage of C crystallized 29.51 53.50 74.87 85.79 91.53 96.7628

Percentage of C crystallized 30.17 52.30 72.66 84.85 91.36 97.1029

Percentage of C crystallized 29.45 50.42 69.32 82.77 90.71 97.0230

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Table 6

Percentages of paraffin crystallized vs. temperature for system B

293.15 K 290.85 K 288.05 K 283.15 K 277.55 K 268.95 K

Overall percentage of paraffins crystallized 11.92 18.11 22.97 35.13 55.55 80.79

Percentage of C crystallized 0.00 0.00 0.01 0.28 1.04 25.8918

Percentage of C crystallized 0.00 0.00 0.34 1.16 9.48 58.5519

Percentage of C crystallized 0.00 0.00 0.79 3.91 25.99 81.8720

Percentage of C crystallized 0.024 0.59 2.29 10.62 48.22 90.8921

Percentage of C crystallized 0.63 2.61 6.96 20.44 68.36 93.8322

Percentage of C crystallized 2.42 7.59 16.00 39.61 80.90 95.1923

Percentage of C crystallized 6.63 17.23 29.97 57.81 87.85 96.7424

Percentage of C crystallized 14.10 30.90 45.76 72.23 91.75 97.3025

Percentage of C crystallized 24.82 45.78 59.83 82.24 94.27 100.0026

Percentage of C crystallized 36.62 58.29 70.50 86.74 95.15 100.0027

Percentage of C crystallized 48.08 66.67 76.80 90.21 96.60 100.0028

Percentage of C crystallized 56.38 71.31 80.39 90.50 96.59 100.0029

Percentage of C crystallized 62.87 73.85 84.39 92.49 96.74 100.0030

vs. temperature is reported in Tables 2–4 and plotted as example in Fig. 2 for mixture C. Similarly,the compositions analyzed in the two phases are reported in Tables 2 –4, and the distribution of paraffins within the solid phase is represented schematically in the histogram in Fig. 3 for mixture B.From the results obtained on the quantities and compositions of the phases present in equilibrium, it ispossible to express the percentage of crystallized paraffins. These data are given in Tables 5–7 and inFig. 4 for mixture B which display the percentages corresponding to the individual crystallizedparaffins as a function of temperature.

In Fig. 3 within which the paraffin content in solid phase of system B is plotted as a function of temperature and number of carbon atoms, it can be noticed that at the highest temperature, the solidphase is not composed of all the paraffins present in the mixture. The lightest, from C to C , only18 20

precipitate 5 to 10 K below the wax appearance temperature according to the system studied. It can

Table 7

Percentages of paraffin crystallized vs. temperature for system C

293.05 K 288.15 K 283.05 K 278.15 K 272.95 K 267.85 K

Overall percentage of paraffins crystallized 13.48 30.74 39.41 59.64 63.07 72.48

Percentage of C crystallized 0.00 0.08 0.28 0.71 3.56 8.7918

Percentage of C crystallized 0.00 0.38 1.06 2.59 11.60 24.5519

Percentage of C crystallized 0.42 1.71 3.63 8.65 28.33 49.4720

Percentage of C crystallized 1.60 5.52 10.78 24.20 51.83 73.3621

Percentage of C crystallized 4.58 14.47 25.30 49.92 73.21 87.2922

Percentage of C crystallized 9.96 28.72 45.20 73.85 86.30 93.3423

Percentage of C crystallized 17.14 45.69 64.60 87.18 92.61 96.1124

Percentage of C crystallized 24.88 60.29 77.93 93.24 95.39 97.3725

Percentage of C crystallized 29.82 70.37 85.19 95.67 96.56 98.0126

Percentage of C crystallized 33.44 76.37 89.09 96.92 97.25 98.2227

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Fig. 4. Percentage of paraffins crystallized as a function of temperature for system B.

also be noted in Fig. 4, which shows the percentage of crystallized paraffin as a function of temperature for each component, that the slope of precipitation is especially large at the beginning of the crystallization for heavy paraffins.

4. Comparison of models

Liquid–solid phase equilibria are reflected by equations of equality of fugacity which lead to thefollowing expression of the distribution coefficient K  for each component present in both phases:i

L SS L L o x g  f  Õ yÕPi i i i i

K  s s exp d P 7Ž .Hi L S S ož / ž / x g  f RT Pi i i o

In this expression, the equilibrium ratio is defined by the sum of three terms. The first one, whichcharacterises the nonideality of liquid and solid solutions, requires further hypotheses on the nature of the liquid and solid solutions to be evaluated. The second term, which takes into account the effects of 

w xtemperature, can be evaluated 9 directly from determination of the change in Gibbs free energybetween the pure solid and the pure subcooled liquid at pressure P . The last term, which representso

the pressure dependence, is generally little different from unity at low pressure and can be neglectedas the volume change is small. Thus, when solid–solid transition ŽSS. occurs between temperature T 

and melting temperature T m, the equilibrium ratio can be written:

L m SS m m mg  D H T  D H T  DC T T i i i P i i

K  s exp 1 y q 1 y q 1 y q ln 8Ž .i S m SSž /g  RT T RT T R T T  i i i

In this formulation, the phase equilibrium ratio K  is connected only to the melting and solid–solidi

transition properties of pure components and to the ratio of activity coefficients. The properties of 

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pure components can be correlated to molecular weight, while evaluation of the activity coefficientsratio requires models which can describe the properties of mixture in both states. In order to identifythe most appropriate activity models combination for predicting the liquid–solid equilibria of thiskind of mixtures, and in particular the amount and the composition of the solid phase as a function of temperature, a comparative study of the predictive abilities of various models has been performedwith reference to the data measured. The models selected for these tests are listed below.

4.1. The model of Won

w x Ž .Won 2 has introduced two simplifications in Eq. 8 . Firstly, the variation of heat capacity isneglected as it provides only a slight contribution in the exponential function. Secondly, the meltingtemperatures and solid–solid transition temperatures are considered as equal since they are generallyvery close, which leads to the following expression:

L mqSSg  D H T i i

K  s exp 1 y 9Ž .i S mž /g  RT T i i

w xin which pure n-paraffin properties are estimated from correlation functions. Moreover, Won 2 hasproposed to use the regular solution theory of Scatchard–Hildebrand in order to model the nonidealityof liquid and solid phases.

4.2. The model of Pedersen et al.

w xPedersen et al. 10 have also proposed to use a regular solution model. However, two modifica-tions were introduced in Won’s model: the heat capacity difference was taken into account in theequilibrium ratio and correlation functions were given to express the solubility parameters as functionof carbon number of components.

4.3. The model of Hansen et al.

w x Ž .The basis of the model of Hansen et al. 3 is identical to Won’s model since both use Eq. 9 tow x w xexpress the equilibrium ratio K  ; however, unlike Won 2 , Hansen et al. 3 have assumed the solidi

phase as homogeneous ideal. Whereas nonideality in the liquid phase is described by the polymerw x w xsolution theory of Flory 4 , Hansen et al. 3 have also proposed a new correlation for estimating the

melting temperature of heavy components of crude oils. However, as the mixtures studied containw xonly n-paraffins, this correlation was replaced by that of Won 2 for these tests.

4.4. The model of Coutinho et al.

In order to take into account the significant difference in size between molecules in liquid state,w xCoutinho et al. 5 have proposed to use a free-volume model to describe the nonideality of the liquid

w xphase. Moreover, for describing the nonideality of the solid phase, Coutinho et al. 6 have developedw xa predictive version of the local composition model of Wilson 11 in which interaction energies are

related to the coordination number Z  and to the enthalpy of sublimation of pure alkanes.

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4.5. The model of Ungerer et al.

w xUngerer et al. 7 have proposed another approach which rests on the assumption that each heavycomponent crystallizes pure, thus leading to several solid phases. In this method, the condition of equilibrium is given for each component which may precipitate, by the following expression:

 f L s f So 10Ž .i i

where fugacity of the pure component i in the solid state is determined from the fugacity of the samesubstance in subcooled liquid state. When the melting and solid–solid transition temperatures areassumed equal and the change of heat capacity is neglected, the equilibrium condition can beexpressed by the following relationship:

mqSSD H T iL L o f  s f  exp y 1 y 11Ž .i i mž /  RT T  i

In this expression, the fugacity of the component i in the liquid solution and the pure liquid fugacityw xare calculated from the cubic Peng–Robinson 12 equation of state using original mixing rules.

4.6. The ideal solution model

Moreover, the simple model in which both phases are assumed ideal has also been tested.Assuming, furthermore, the heat capacity change as negligible and the melting and solid–solidtransition temperatures as equal, the equilibrium ratio is expressed by the following relation:

mqSSD H T i

K  s exp 1 y 12Ž .i mž /  RT T  i

in which it is connected only to the pure component properties.

To appreciate the respective abilities of these models to predict the L– S transition temperatures, theŽdeviations between the calculated and the experimental values which correspond to the temperature

.of disappearance of the last solid part were listed in Table 8. On the other hand, as the error on the

Table 8

Deviation DT  between experimental and calculated wax appearance temperatures

Ž .DT  K Mixture A Mixture B Mixture C

Ž .T  Kexp

293.35 299.35 297.35

Ideal model q7.0 q4.1 q4.0w xWon 2 q7.2 q4.3 q4.1

w xHansen et al. 3 q5.0 q2.1 q2.0w xPedersen et al. 10 q5.1 q2.2 q2.6w xCoutinho et al. 5,6 q1.0 q0.6 q0.8

w xUngerer et al. 7 y3.1 y0.1 y0.6

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Fig. 5. Amount of paraffins crystallized calculated with various models for system A.

temperature of phase change introduces a shift of the solid deposits curves along temperature axis, theŽ .deviation between calculated and experimental quantities amount and composition cannot be used to

test the capacities of these models to represent the behavior of solid deposits. It was because of thisŽ .difficulty that graphical representation Figs. 5–9 was used to compare the models.

The deviations listed in Table 8 show that all activity models overestimate the solid appearancetemperatures; nevertheless, this overestimation keeps slight, the deviation never exceeding 4 K. In

Fig. 6. Percentage of hexacosane crystallized calculated as a function of temperature for system C.

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Fig. 7. Percentage of tricosane crystallized calculated as a function of temperature for system C.

w xparticular, the deviations observed with the model proposed by Coutinho et al. 5,6 are for the threesystems of the order of 1 K. This performance is all the more remarkable, as the model is used in itsoriginal form, in other words, without any adjustment of parameters. On the other hand, the pureparaffin crystallisation model underestimates the liquid–solid transition temperatures. Still there, thevalues predicted by the model are not so far from experimental values.

Fig. 8. Percentage of eicosane crystallized calculated as a function of temperature for system C.

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Fig. 9. Mass fraction of eicosane calculated as a function of temperature for system A.

Study of solid deposit curves shows that Won’s model and ideal solution model lead to similarprediction. This result comes from values of regular activity coefficients which are nearly similar inboth phases. Thus, as for ideal solution, the g  ratios are close to unity and the equilibrium ratiosi

become only function of pure components parameters. On Fig. 5, which represents the amount of solid deposits as a function of temperature for mixture A, it can be observed that the improvements of 

w xPedersen et al. 10 solely bring about a translation of curves calculated by Won’s model. These threew xmodels, as well as Hansen et al. 3 , overestimate the crystallisation rate at the wax appearance

Žtemperature. This is particularly true for mixtures B and C which contain more light components C ,18

.C . Actually, all these models, which are based on the assumption that heavy components19

Ž .precipitate in one solid solution, predict that all components heaviest as well as lightest of the heavyŽ .fraction are present in the first solid bulk, whereas measurements Tables 2– 4 show that the lightest

do not precipitate at the wax appearance temperature. This means, as it can be seen on Figs. 6 and 7,that the models can predict the heaviest component content in solid phase but give a poor

Ž .representation of composition of lightest components Fig. 8 .w xUnlike these models, the approach of Ungerer et al. 7 always underestimates the slope of solid

deposit vs. temperature. Actually, as the model assumes a pure component crystallisation, it predictsthat only the heaviest component precipitates at the solid appearance temperature, and thus minimisesthe amount of solid deposit at the beginning. Consequently, the percentage corresponding to theindividual paraffins crystallized as a function of temperature is better represented for heaviestcomponent than for lightest. However, despite the assumption made, representation is not aberrant andthe deviations observed by reference to experimental data are not higher than those observed withsolid solution models.

Finally, as it can be observed on Fig. 5, the curves predicted by the procedure of Coutinho et al.w x5,6 show a good fit with experimental points. In particular, the model predicts an inflection pointwhich is also experimentally observed on curves corresponding to the individual paraffins crystallized

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Ž .as a function of temperature Figs. 6–8 . Comparison of calculated and experimental solid phaseŽ .composition Fig. 9 is more convincing of the quality of the model. So, a correct description of solid

solutions is a necessary condition for a satisfactory representation of solid–liquid equilibrium of manycomponent systems. Nevertheless, as all other models, this model gives a better representation of 

Ž . Ž .heaviest Fig. 6 than lightest components Fig. 8 .

5. Conclusion

Experimental equipment of solid–liquid filtration and measurement procedure have been developedin order to study the solid precipitation in synthetic mixtures. The apparatus was used to perform newexperimental measurements on mixtures made up of decane plus various distributions of heavynormal paraffins from octadecane to triacontane. These measures, which concern the amount and thecomposition of solid precipitate as a function of temperature on synthetic complex systems, provide adetailed characterization of the solid precipitation in synthetic systems.

The numerical verifications performed on database demonstrate that the procedure proposed byw xCoutinho et al. 5,6 leads to a satisfactory representation of the liquid–solid equilibria of these

multi-component systems.

6. List of symbols

C  Heat capacityP

 f  fugacityF  j mass fraction of phase j in the whole system H  enthalpyK  equilibrium ratiom mass

P pressure R ideal gas constantT  temperatureV  volume x mole fraction of  ii

 X  mass fraction of  ii

Y j mass fraction of phase j in the solid residue of filtration

Greek letters

a distribution coefficient

g  activity coefficient

Superscripts

L liquidLT trapped liquidm melting

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S solidSR solid residueSS solid–solid transition

Subscripts

i componentn carbon numbero pure component at reference pressure

Acknowledgements

The authors wish to thank Doctor J. Coutinho for helpful discussions.

References

w x Ž .1 J.L. Daridon, P. Xans, F. Montel, Fluid Phase Equilibria 117 1996 241.w x Ž .2 K.W. Won, Fluid Phase Equilibria 30 1986 265.w x Ž .3 J.H. Hansen, A. Fredenslund, K.S. Pedersen, H.P. Ronningsen, AIChE J. 34 1988 1937.w x4 P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, New York, 1953.w x Ž .5 J.A.P. Coutinho, S.I. Andersen, E.H. Stenby, Fluid Phase Equilibria 103 1995 23.w x Ž .6 J.A.P. Coutinho, K. Knudsen, S.I. Andersen, E.H. Stenby, Chem. Eng. Sci. 51 1996 3273.w x Ž .7 P. Ungerer, B. Faissat, C. Leibovici, H. Zhou, E. Behar, Fluid Phase Equilibria 111 1995 287.w x Ž .8 V. Kravchenko, Acta Physicochim. 21 1946 335.w x9 J.M. Prausnitz, Molecular Thermodynamics of Fluid Phase Equilibria, Prentice-Hall, Englewood Cliffs, NJ, 1969.

w x Ž .10 K.S. Pedersen, P. Skovborg, H.P. Rønningsen, Energy and Fuels 5 1991 924.w x Ž .11 G.M. Wilson, J. Am. Chem. Soc. 86 1964 127.w x Ž .12 D.Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fundam. 15 1976 59.