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Asphaltene flocculation in crude oil systems D. Browarzik ) , H. Laux, I. Rahimian Institute of Physical Chemistry, Martin Luther UniÕersity Halle-Wittenberg, Geusaer Straße, 06217 Merseburg, Germany Technical UniÕersity Clausthal, Institut fur Erdgasforschung, Walther-Nernst-Str. 7, 38678 Clausthal-Zellerfeld, ¨ Erdol-und ¨ Germany Received 22 October 1997; accepted 30 September 1998

Abstract Flocculation points of complex systems of the type crude oil q solventq precipitant were experimentally and theoretically studied. In this work 6 crude oils are combined with the solvents toluene and cyclohexane and the precipitants n-pentane, n-hexane, n-heptane and i-octane. Experimentally, the oil solution was titrated by the precipitant to determine the precipitant volume necessary for flocculation. Detecting the intensity of a light beam which passes the sample, the flocculation point is assumed to correspond to the maximum of the titration curve. The modeling of the flocculation points is based on the simple Scatchard–Hildebrand solubility theory in the framework of continuous thermodynamics. The crude oil is considered to consist of maltenes and asphaltenes. The composition of these two subsets is described by separate continuous distribution functions. The oil species are identified by their solubility parameters reflecting their degree of aromaticity and their content of heteroatoms. Both maltenes and asphaltenes are assumed to obey Gaussian distribution functions with respect to the solubility parameter. In this way only the polydispersity with respect to the solubility parameter is taken into account. The molar mass polydispersity, here, is much less important and, therefore, it will be neglected. To calculate the flocculation points only a single equation has to be numerically solved. The only fit parameter is the number average of the molar mass of the asphaltenes, the experimental values of which are too inaccurate. The comparison of the calculated and the experimental results are surprisingly reasonable considering the simplicity of the model and the inaccuracies of some input data. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Experimental method; Method of calculation; Continuous thermodynamics; Flocculation; Crude oils; Asphaltenes

1. Introduction Crude oils are complex mixtures of different hydrocarbons involving sulfur, nitrogen and oxygen, too. Even organic metal compounds occur predominantly as porphyrins. There are large differences )

Corresponding author. Tel.: q49-3461-46-2133; fax: q49-3461-46-2129; e-mail; uni-halle.de

0378-3812r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 8 . 0 0 4 3 4 - 8

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D. Browarzik et al.r Fluid Phase Equilibria 154 (1999) 285–300

both in the molar masses and in the structure of the oil species. For example, the asphaltenes consist of relatively high molecular weight species which are polyaromatic hydrocarbons with saturated side chains, which also contain heteroatoms. Therefore, between the asphaltene species and the other species, particularly the aliphatic species, strong interactions occur. For this reason the asphaltenes are soluble in toluene but insoluble in n-alkanes as n-pentane or n-heptane. Usually, this solubility behaviour is used to distinguish the asphaltenes from the rest of the oil. Some of the asphaltene species are even insoluble in crude oil. These are spherical crystallites consisting of usually not more than five condensed aromatic layers including porphyrins. The size of the crystallites varies in the range of 1.5–2 nm. Larger crystallites occur seldomly because of the presence of saturated side chains on the aromatic rings. The p–p bonds of the crystallites result in a relatively high thermodynamic stability. The solvated asphaltene crystallites are the primary particles of a colloidal dispersed phase. These particles can associate and aggregate forming larger particles dependent on temperature, pressure and concentration. Flocculi and precipitates are formed in instable colloidal systems. During the production and processing of crude oil, the colloidal dispersed phase is important because of its influence on the physical and chemical behaviour of the mixture. For example, instabilities of this phase are the reason for the formation of deposits and coke. To investigate the factors influencing the stability of the colloidal dispersed phase one has to consider the following points: - There is no possibility to distinguish exactly the asphaltenes from the resins and the resins from the lighter oils. - Practically, the asphaltenes are defined by their solubility in toluene and insolubility in n-alkanes. - Experimentally, the amount and the composition of the asphaltenes, the resins and the lighter oils cannot directly be determined. The only source to get such information is the precipitation experiment. Therefore the experimental determination of flocculation point is used to study the factors on which the stability of colloidal dispersed crude oil systems depends. Furthermore, a simple thermodynamic model is developed to describe the experimental flocculation point data.

2. Experimental 2.1. Flocculation point determination The flocculation points were determined by a titration method. A sample of the crude oil was solved in an organic solvent Ž toluene, cyclohexane. and, five days later, titrated with a precipitant Ž n-pentane, n-hexane, n-heptane, i-octane. in a thermostated mixing apparatus. The titration is monitored by means of a light beam passing the sample. The measured light intensity is plotted vs. the added precipitant volume. First, the light intensity increases because of the dilution of the solution. However, at the beginning of the flocculation the light intensity decreases. When the first flocculi are formed the flocculation point is reached. This point is assumed to correspond to the maximum of the titration curve discussed above. All measurements relate to the following standard conditions: initial solution volume V L 20 ml

D. Browarzik et al.r Fluid Phase Equilibria 154 (1999) 285–300

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Table 1 Characterization of the crude oils: number averaged molar mass M EM of the maltenes, density r E , mean solubility parameter d E , standard deviation s E and, mass fraction y EA of the asphaltenes of the crude oil

(

(

Crude oil

M EM rg moly1

r E rg cmy3

d E r MJrm3

s E r MJrm3

y EA

Arabian Heavy BCF Black Minnel California Laguna Maya

399 413 501 413 496 495

0.8801 0.9120 0.9396 0.9855 0.9201 0.9200

17.86 17.80 18.22 18.35 18.45 18.10

0.96 0.90 0.91 0.86 0.81 0.98

0.060 0.062 0.106 0.117 0.118 0.127

initial oil concentration m E 50 g oilr1 solvent temperature T 298.15 K titration rate 1 mlrmin stirring speed 200 rpm The experimental precipitant volumes V F needed for flocculation for 48 systems of the type crude oil q solventq precipitant Ž6 crude oils were combined with 8 solvent–precipitant pairs. are listed in Tables 2 and 3. 2.2. Colloid precipitation The colloids Žasphaltenes and resins. were precipitated by Neumann’s method w1x using ethyl acetate and, subsequently, extracting with soxhlet apparatus the filtered precipitate by either n-pentane to obtain the petroleum resins or toluene to obtain the asphaltenes. After removing ethyl acetate from the filtrate, the oily phase Ž dispersion medium. remains. The total of all oil components, except the asphaltenes, is called maltenes. 2.3. Molar mass determination For determination of the molar mass of the crude oil or, respectively, of its fractions, vapor pressure osmosis at 378C was applied. Here, the solvent was toluene and, the mass fraction of the oil, was 0.02 in most cases. The molar masses of the maltenes of the crude oils which were experimentally studied are presented in Table 1.

3. Characterization of the crude oil and its fractions The modeling of phase equilibria of complex mixtures needs lumping methods for describing the composition of the phases. Either pseudocomponents may be introduced or a continuum treatment is necessary w2x. Both methods are applied to the precipitation of asphaltenes and the simple Scatchard– Hildebrand solubility theory is often used as the thermodynamical model w3x. The results show that the description of the asphaltenes by a molar mass distribution is more successful than considering them as a single pseudocomponent. However, association and aggregation

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of the asphaltenes complicate the exact determination of molar masses. The molar mass values experimentally obtained depend strongly on the experimental conditions Ž e.g., solvent, temperature. . Therefore, the molar mass is not suitable for characterizing the crude oils and, especially, the asphaltenes. Further, the oil species are not only different in their molar masses but, more important, there are differences in the degree of aromaticity and in the content of heteroatoms w4x. The solubility parameter of the Scatchard–Hildebrand theory reflects the intermolecular interactions originating from the aromaticity and the heteroatoms in some way. Therefore this solubility parameter seems to be a more convenient identification variable for the oil species than the molar mass. A further reason for the oil characterization by the solubility parameter is the possibility to correlate it with experimentally available data. So, in this paper the characterization of crude oils and asphaltenes is based on the solubility parameter. The solubility parameters of crude oil and its fractions including asphaltenes were calculated by w4x

ž

d s 16.55 q 0.00464 q

3.2285 nC

/

Ž Z Ra y 2 . q D d He

Ž1.

According to Eq. Ž 1. the solubility parameter is a function of the number of carbon atoms n C and, of the hydrogen deficit Z Ra compared with the number of hydrogen atoms in a n-alkane of the same number of carbon atoms. The quantity D d He accounts for the influence of the heteroatoms on the solubility parameter. This quantity is given by yi D d He s r Ý Fi Ž2. Ai i

r is the density of the sample, Fi is the increment value, y i is the mass fraction and A i is the molar mass of heteroatoms of the type i. Eqs. Ž 1. and Ž 2. were tested for numerous substances for which the experimental values of the solubility parameter are available. It was found that the agreement between the calculated and the experimental values are within the accuracy of the experimental data. Applying Eqs. Ž1. and Ž2. the calculation of the solubility parameter of an oil needs only experimental values of the molar mass average and the content of carbon, hydrogen and heteroatoms. These data are well available. For crude oils the solubility parameter varies approximately in the range of 16 up to 21 MJrm3 . For asphaltenes this range is approximately 19–21 MJrm3 . Experimental investigations w5,6x of the solubility of asphaltenes in organic solvents proved the asphaltenes to have very similar values of the solubility parameter independent of the kind of crude oil the asphaltenes originated from. After fractionating crude oils or asphaltenes and calculating the solubility parameters of the fractions the plot of the amount of the fractions vs. the corresponding solubility parameters results in Gaussian distributions w4,7x. The relation between the Gaussian distributions of a crude oil and of its asphaltenes is discussed in Section 5. The parameters of the Gaussian distributions Ž mean value and standard deviation. of the crude oils which were studied experimentally are listed in Table 1. Further experimental data concerning these oils may be found in the thesis of Hatke w8x. Knowing mean solubility parameter and standard deviation of the crude oil and of the asphaltenes as well as the total mass fraction y EA ŽTable 1. of the asphaltenes the crude oil is sufficiently characterized. Studying the solubility of crude oils in polar solvent–precipitant pairs the solubility parameter of the simple Scatcard–Hildebrand theory is not useful. For these cases, the application of the Hansen

(

(

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289

three-component solubility parameter is recommended w9x. This generalized solubility parameter concept takes into account interactions between dipoles and specific interactions. However, in this paper flocculation points for non-polar solvent–precipitant pairs are considered. Therefore, the solubility parameter of the simple Scatchard–Hildebrand theory w3,10x is sufficient to characterize the oil samples and to calculate the flocculation points.

4. Model To describe the flocculation of asphaltenes the phase equilibrium of the complex system crude oil ŽE. q solvent ŽS. q precipitant ŽP. is considered. The treatment is based on continuous thermodynamics and is similar to that of liquid–liquid equilibria of a polymer solution w2,11x. However, unlike a polymer solution the polydispersity with respect to the molar mass is much less important than the polydispersity with respect to the intermolecular interactions. Therefore, in the following, instead of the molar mass as identification variable, the solubility parameter d of the Scatchard–Hildebrand theory w3x is chosen. According to experimental results w4,7x the crude oil may be characterized by the Gaussian distribution WE Ž d . s

1

'2p s E exp

y

Ž d yd E.

2

s E2

Ž3.

the parameters of which are the mean value d E and the standard deviation s E . WEŽ d .d d gives the volume fraction of all species with d-values between d and d q d d . The distribution function is normalized by the condition q`

Hy` W

E

Ž d . d d s 1.

Ž4.

The calculation of phase equilibria of polymer solutions is usually based on the segment-molar excess Gibbs energy G E w11x. This quantity originates from the excess Gibbs energy of the mixture reduced by the Flory–Huggins contribution and related to one mole of segments. Here, the molecules are imagined to be divided into segments of the same size. Corresponding to the chosen identification variable d the quantity G E is calculated on the basis of the Scatchard–Hildebrand theory w3x. GE s V ) Ý i

Ý wi wj Ž di y dj .

2

Ž5.

j

Here, w i and w j are the volume fractions of the components i and j and V ) is the volume of one mole of segments. Assuming the segment number of a molecule to be proportional to its volume, G E is related to the segment-molar activity coefficients g i by G E s RT Ý w i lng i

Ž6.

i

where R is the universal gas constant and T is the temperature. With the aid of Eq. Ž5. one can show lng k s

V)

ž d yd / RT k

2

;

d s Ý wi di . i

Ž7.

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290

For the solvent ŽS. and the precipitant ŽP. of the considered system these equations read lng S s

V)

ž d yd / RT S

2

; lng P s

V)

ž d yd / RT P

2

.

Ž 8a.

For the oil species, applying the notation of continuous thermodynamics, the segment-molar activity coefficients are given by lng E Ž d . s

V) RT

Ždyd .

2

Ž 8b.

where the mean value d of the mixture may be expressed by q`

d s Ž1 y w E y w P . d S q w P d P q w E d E ;

dEs

Hy` d W

E

Ž d .d d .

Ž 8c.

To calculate the phase equilibrium the segment-molar chemical potentials are needed. These expressions read w11x

m S s m S ) q RT m P s m P ) q RT

1 rs

ln w S q

1 rP

ln w P q

m E Ž r . s m E ,0 ) Ž r . q RT

1 r

1

1 y

rS

r

1

1 y

rP

r

q RT lng S

Ž 9a .

q RT lng P

Ž 9b.

ln w EW E Ž d . q

1

1 y

r

r

q RT lng E Ž d .

Ž 9c .

The first term in each equation is a function of temperature and pressure only w11x. In the case of Eqs. Ž9a. and Ž9b. this is a pure-component term w11x. The second term in each case Ž Flory–Huggins contribution. describes the effects arising from the size differences between the molecules. r S , r P and r are the segment numbers of the solvent molecules, of the precipitant molecules and, of the oil species. r is the number average of the segment number of the mixture. Now, the equilibrium between the feed phase I and an infinite amount of a new formed phase II Žflocculated phase. is considered, following the same procedure as in the calculation of the cloud point curve of a polymer solution w11x. At constant temperature and pressure the equilibrium conditions read

m SI s m SII ;

mIP s mIIP ;

mIE Ž d . s mIIE Ž d . .

Ž 10.

The third condition applies to all values of d even if there is no real species taking this value. In a similar way as shown for polymer solutions w11x with the aid of Eqs. Ž 9a. , Ž 9b. and Ž 9c. the equality of the segment-molar chemical potentials results in 1 y w EII y w PII s Ž 1 y w EI y w PI . exp Ž r S r S ) .

Ž 11a.

w PII s w PI exp Ž r P r P ) .

Ž 11b.

w EII WEII Ž d . s w EI WEI Ž d . exp r E r E ) Ž d .

Ž 11c.

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where the auxiliary quantities r S ), r P ), r E )Ž d . are defined by

rS) s rP)s

1

1

II

y

r 1

r II

y

rI 1

y lng SII q lng SI

Ž 12a.

y lng PII q lng PI

Ž 12b.

rI

1

1

y lng EII Ž d . q lng EI Ž d . . Ž 12c. r rI Here, r a is the number average of the segment number of the mixture in the phase a with a s I,II. This quantity is given by

r E )Ž d . s

1 r

s a

II

y

1 y w Ea y w Pa

q

rS

w Pa

q

rP

w Ea rE

.

Ž 13 .

In Eqs. Ž11a., Ž11b., Ž11c., Ž12a., Ž12b., Ž12c. and Ž13. the polydispersity with respect to molar mass is neglected. All oil species are assumed to be of the same segment number Ž r E . but different in their values of the solubility parameter. Whereas the polydispersity of polymers is mainly one with respect to molar mass the polydispersity of crude oils consisting of many different classes of compounds is predominantly one with respect to the solubility parameter d . Using Eqs. Ž 12a., Ž12b., Ž12c. , Ž8a., Ž 8b. and Ž8c. one finds

r P ) s rS) q 2

V) RT

Ž d II y d I . Ž d P y d S .

Ž 14a.

V)

Ž d II y d I . Ž d y d S . . RT Combination of Eqs. Ž11a., Ž 11b. and Ž 14a. results in r E )Ž d . s r S ) q 2

1 rP

ln

w PII

ž / w

I P

1 y rS

ln

1 y w EII y w PII

ž

1yw

I Ey

w

I P

/

y2

V) RT

Ž d II y d I . Ž d P y d S . s 0.

Ž 14b.

Ž 15 .

Combination of Eqs. Ž11a. and Ž 12a. using the first of Eq. Ž8a. one finds 1 rS

ln

ž

1 y w EII y w PII 1 y w EI y w PI

/

1 q r

y I

1

V)

q II

RT

r

Ž d II y d I . Ž d II y d I . q 2 ž d I y d S / s 0

Ž 16.

Eq. Ž14b. gives with Eq. Ž 11a. r E r E ) Ž d . s A q Bd

Ž 17.

where As

rE rS

ln

B s 2 rE

ž

1 y w EII y w PII 1yw

V) RT

I Ey

w

I P

Ž d II y d I . .

/

y 2 rE

V) RT

d S Ž d II y d I .

Ž 18a. Ž 18b.

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292

Considering the normalization condition Ž 4. or, respectively, the second of Eq. Ž 8c. integration of Eq. Ž11c. results in q`

w EII s w EI exp Ž A .

Hy` W

I E

Ž d . exp Ž Bd . d d

Ž 19a.

q`

w EII d EII s w EI exp Ž A .

Hy` d W

I E

Ž d . exp Ž Bd . d d .

Ž 19b.

If the feed phase I obeys a Gaussian distribution function as given by Eq. Ž 3. the integrals of Eqs. Ž19a. and Ž 19b. possess analytical solutions and one obtains

w EII s w EI exp A q Bd EI q

1 2

Ž Bs EI .

2

Ž 20a.

2

w EII d EII s w EI d EI q B Ž s EI . exp A q Bd EI q

1 2

Ž Bs EI .

2

.

Ž 20b.

The quotient of the last both equations results in 2

d EII s d EI q B Ž s EI . .

Ž 21.

With the aid of Eqs. Ž 3. , Ž 11c. , Ž 17. and Ž 20a. one can show that the oil distribution function of the flocculated phase II is also of the Gaussian type Ž Eq. Ž 3.. with the mean value d EII , given by Eq. Ž 21. and, the standard deviation s EII s s EI . In the following the standard deviation is simply denoted by s E . With the aid of Eqs. Ž8c., Ž18a., Ž18b. and Ž21. one obtains WEII Ž d .

II

I

d yd s

Ž w PII y w PI . Ž d P y d S . q Ž w EII y w EI .Ž d EI y d S . 1 y 2 rE

V) RT

2 E

s w

.

Ž 22 .

II E

Now, there are the three phase equilibrium conditions Ž 15. , Ž 16. and, by using Eqs. Ž 18a. , Ž 18b. and Ž20a., the relation 1 rE

ln

w EII

ž / w EI

1 y rS

ln

ž

1 y w EII y w PII 1 y w EI y w PI

/

y2

V) RT

Ž d II y d I . d EI y d S q r E

V) RT

s E2 Ž d II y d I . s 0.

Ž 23. Here, the difference d II y d I is given by Eq. Ž22.. The difference 1rr I –1rr II occurring in Eq. Ž16. may be calculated with the aid of Eq. Ž 13.. The quantity d I of Eq. Ž 16. is obtained using Eq. Ž 8c.. Based on Eqs. Ž15., Ž 16. and Ž 23. three scalar unknowns, e.g., w EII , w PII and the precipitation volume V F may be calculated. However, an essential simplification is possible, taking into account the flocculated phase II to be practically free of the solvent and the precipitant. Then w EII s 1 and w PII s 0 is nearly fulfilled. Thus, instead of three phase equilibrium conditions only a single one is needed. Forming the difference of Eqs. Ž 16. and Ž23. this one condition reads 1 rE

ln w EI q

1 r

y II

1 r

y I

V) RT

Ž d II y d I . Ž d II y d I . 1 y 2 rE

ž

V) RT

s E2 q 2 d I y d EI s 0.

/

ž

/

Ž 24.

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293

Applying Eq. Ž8c. to the phase I and using Eq. Ž22. the rearrangement of Eq. Ž24. gives

ž

1y2

V) RT

r E s E2

1

/

rE

ln w EI q

1 r

1

II

y r

V) q

I

RT

Ž 1 y w EI .Ž d EI y d S . y w P Ž d P y d S .

2

s0

Ž 25. where 1 r

1

II

y r

s Ž 1 y w EI .

I

1

ž

1 y

rE

rS

/ ž y w PI

1

1 y

rP

rS

/

Ž 26.

is obtained using Eq. Ž13. . Practically, phase II consists of asphaltenes only. Therefore a refinement of the model dividing the crude oil into maltenes and asphaltenes seems to be useful. Both the maltenes and the asphaltenes are assumed to obey Gaussian distributions with the parameters d EM , s EM for the maltenes and, respectively, d EA , s EA for the asphaltenes. Then, in analogy to Eq. Ž 26. one finds the condition

ž

1y2

V) RT

V) q RT

2 r EA sAE

1

/

r EA

I ln w EA q

1 r

II

1 y

rI 2

I I I I yw EM y d S / q Ž 1 y w EA y dS / y w PŽ d P y dS . s 0 . ž d EA ž d EM

Ž 27.

with 1 r

y II

1 r

I s yw EM I

ž

1

1 y

r EM

rS

/

I q Ž 1 y w EA .

ž

1

1 y

r EM

rS

/ ž y w PI

1

1 y

rP

rS

/

.

Ž 28 .

In the experiments, the precipitant is added to a solution volume VL Ž20 ml.. The initial concentration of the oil solution is given by m E Ž g crude oilrl solvent.. Therefore, the initial volume fraction of the crude oil is m ErŽm E q r E .. After adding a precipitant volume V F to the initial solution the volume fractions w EI w PI w SI of the total crude oil, the precipitant and, of the solvent are

w EI s

mE

VL

m E q r E VL q VF

;

w PI s

VF VL q VF

;

w SI s 1 y w EI y w PI .

Ž 29 .

I Denoting the mass fraction of the asphaltenes within the crude oil by y EA the volume fraction w EA of I the asphaltenes and the volume fraction w EM of the maltenes can be calculated from the volume fraction w EI of the total crude oil by I w EA s w EI

y EA r EM y EA r EM q Ž 1 y y EA . r EA

;

I I w EM s 1 y w EI y w EA .

Ž 30.

Here, r EA and r EM are the densities of the asphaltenes and of the maltenes. With the aid of Eqs. Ž 29. I I and Ž30. the calculation of the three volume fractions w EM , w EA , w PI requires only the knowledge of

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294

the precipitant volume VF . Thus, the only unknown is V F and altogether with Eqs. Ž28. – Ž30. the only equation to solve numerically is Eq. Ž 27..

5. Application of the model and comparison with experimental data Based on Eq. Ž27. by using Eqs. Ž28. – Ž30. the volume VF of the precipitant needed for flocculation can be calculated and compared with experimental data. For this the segment numbers have to be known. The segment number of the solvent molecules is arbitrarily assumed to be r S s 1. The other segment numbers are considered to be volume ratios and are given by rP s

MP r S MS r P

;

r EM s

M EM r S MS r EM

;

r EA s

M EA r S

Ž 31.

MS r EA

where MS , M P , M EM , M EA are the molar masses Žor the corresponding number average molar masses. of the solvent, the precipitant, the maltenes and the asphaltenes, respectively. The density of the maltenes may be calculated from experimental density data of the crude oil and of the asphaltenes using the relation

r EM s

Ž 1 y y EA . r EA r EArr E y y EA

.

Ž 32.

Flocculation points for systems of the type crude oil q solventq precipitant were calculated. The solvents are toluene Ž d s 18.2 MJrm3 ., cyclohexane Ž d s 16.8 MJrm3 .. The precipitants are n-pentane Ž d s 14.5 MJrm3 ., n-hexane Ž d s 14.9 MJrm3 ., n-heptane Ž d s 15.1 MJrm3 ., i-octane Ž d s 14.1 MJrm3 .. The solubility-parameter values of these substances were found in the handbook of Barton w10x. The crude oils considered and the input data of them are presented in Table 1. The number of average molar mass of the maltenes was measured by vapor-pressure osmosis w8x, neglecting the small influence of the resins. The average solubility parameter of the crude oil is calculated by means of Eq. Ž1. using experimental data of the amounts of the elements and of the average of the molar mass. After fractionating of the crude oil, the amounts of the fractions in dependence on the corresponding average solubility parameters proved to fulfill the Gaussian distribution function. The standard deviation is given in Table 1. The quantities d EA and, particularly, s EA are difficult to determine experimentally. Therefore, the following approximation is introduced. The fractionation of crude oils and of asphaltenes show that both the crude oils and the asphaltenes can be represented approximately by Gaussian distributions. Thus, WEŽ d . is given by Eq. Ž3. and the analogous distribution function WEAŽ d . with the parameters d EA and s EA is of the same type. Now one can introduce a distribution function WEA )Ž d . by rE WEA ) Ž d . s y EA W Žd . Ž 33. r EA EA

(

(

(

(

(

(

which describes the contribution of the asphaltenes within the distribution function of the crude oil. Integration of Eq. Ž 33. for the total domain of definition of d gives the volume fraction of the asphaltenes within the crude oil. If y EA is given, the parameters of both distribution functions are not

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295

independent of each other. However, the requirement that both distributions are to be of the Gaussian type cannot be exactly fulfilled. A reasonable approximation may be found by the requirements dWE

ž /

WE Ž x . s WEA ) Ž x . ;

dd

s dsx

ž

) dW EA

dd

/

Ž 34. d sx

meaning that there is a value d s x of the solubility parameter related to both distribution functions that possess the same value and slope. Applying Eqs. Ž3., Ž33. and Ž34. one obtains 2

xs

ln

ž

d EAŽ s Ers EA . y d E

Ž 35a.

2 Ž s Ers EA . y 1

y EA r E s E

r EA s EA

/

q

1 d EA y d E

ž

2

/

2 2 s E2 y s EA

s 0.

Ž 35b.

The parameters of the crude oil and the mass fraction of the asphaltenes are known. The unknowns are the parameters of the asphaltenes which, in accordance to the experimental experience, are assumed to have the same values independent of the type of the crude oil. Fitting the asphaltene Table 2 Comparison of the calculated precipitant volumes Ž V F .calc and the experimental precipitant volumes Ž V F .exp for the solvent toluene Precipitantqcrude oil n-Pentane

n-hexane

n-heptane

i-octane

qArabian Heavy qBCF qBlack Minnel qCalifornia qLaguna qMaya qArabian Heavy qBCF qBlack Minnel qCalifornia qLaguna qMaya qArabian Heavy qBCF qBlack Minnel qCalifornia qLaguna qMaya qArabian Heavy qBCF qBlack Minnel qCalifornia qLaguna qMaya

Ž V F .calc rml

Ž VF .exp rml

45.5 64.4 41.7 43.3 53.0 31.2 58.3 90.9 52.3 54.6 69.8 37.5 63.5 103.0 56.4 58.9 76.6 39.9 25.2 31.0 23.8 24.4 27.6 19.4

43.5 45.7 39.5 43.2 48.8 32.7 62.0 90.8 52.9 54.5 66.2 36.1 64.0 97.7 79.2 80.5 77.0 40.1 46.0 63.1 40.3 45.3 55.1 32.1

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parameters, only Eq. Ž 35b. is needed. For given values of the asphaltene parameters with the aid of Eq. Ž35b. the quantity y EA and the quadratic deviation from the corresponding experimental value is calculated for all crude oils of Table 1. Matching the asphaltene parameters according to the method of least squares it is found

(

(

d EA s 19.58 MJrm3 ; s EA s 0.55 MJrm3 . Ž 36. The most difficult quantity to determine experimentally is the average of the molar mass M EA of the asphaltenes. On the one side, measurement by vapor-pressure osmosis results in unrealistic high values, probably originating from association effects. On the other side M EA influences considerably the calculated V F-values. Therefore this quantity was fitted to the experimental V F-values of the flocculation points. For each crude oil one parameter Ž M EA . was fitted to the experimental V F-values ŽTables 2 and 3. of all eight solvent–precipitant pairs, using the method of least squares. In this, values between 1780 grmol and 2200 grmol, which seems reasonable, were found. More in detail, the molar mass averages M EA of the asphaltenes for the crude oils considered are: 1980 grmol ŽARABIAN HEAVY., 1780 grmol Ž BCF., 2000 grmol ŽBLACK MINNEL. , 1970 grmol Ž CALIFORNIA. , 1840 grmol ŽLAGUNA., 2200 grmol ŽMAYA.. Using these values VF-values were calculated for 48 systems of the type crude oil q solventq precipitant and compared with experimentals data. The results are listed in Tables 2 and 3. One can Table 3 Comparison of the calculated precipitant volumes Ž V F .calc and the experimental precipitant volumes Ž V F .exp for the solvent cyclohexane Precipitantqcrude oil n-Pentane

n-hexane

n-heptane

i-octane

qArabian Heavy qBCF qBlack Minnel qCalifornia qLaguna qMaya qArabian Heavy qBCF qBlack Minnel qCalifornia qLaguna qMaya qArabian Heavy qBCF qBlack Minnel qCalifornia qLaguna qMaya qArabian Heavy qBCF qBlack Minnel qCalifornia qLaguna qMaya

Ž V F .calc rml

Ž VF .exp rml

20.5 31.1 18.3 19.2 24.8 12.0 25.9 42.7 22.7 23.9 32.1 14.4 28.1 47.7 24.3 25.7 35.0 15.2 11.5 15.4 10.6 11.0 13.2 7.5

18.2 29.7 16.6 16.6 23.5 8.6 25.5 46.5 20.3 20.3 34.6 14.3 29.2 53.4 27.9 33.3 38.9 20.4 20.6 34.0 15.3 16.6 25.8 9.2

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Fig. 1. The precipitant Ž n-hexane. volume V F in dependence on the asphaltene mass fraction y EA of the crude oil Maya for the solvents toluene Župper curve. and cyclohexane Žlower curve..

see a reasonable agreement between the calculated and the experimental values, particularly, for the n-hexane systems. Only the results for systems with i-octane as precipitant are somewhat worse. In this case the values are always too small. Taking into account the simplicity of the model and the use of only one fit parameter fitted, these results are even surprisingly good. Furthermore, one has to consider the wide range of experimental VF-values. For example, the combination of the good solvent toluene and of the bad precipitant n-heptane results in very high VF-values Ž until nearly 100 ml. , and, the combination of the bad solvent cyclohexane and of the good precipitant n-pentane results in relatively small VF-values Žpartially less than 10 ml..

Fig. 2. The precipitant Ž n-hexane. volume V F in dependence on the initial crude oil concentration m E for the solvent cyclohexane and the crude oils Maya Žthick line., Arabian Heavy Žthin line., Laguna Ždashed line., BCF Ždotted line..

298

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Fig. 3. The precipitant Ž n-hexane. volume V F in dependence on the initial crude oil concentration m E for the solvent toluene and the crude oils Maya Žthick line., Arabian Heavy Žthin line., Laguna Ždashed line., BCF Ždotted line..

In Fig. 1 the precipitant volume VF is plotted vs. the asphaltene mass fraction y EA of the crude oil Maya. Both curves relate to the precipitant n-hexane. The upper curve relates to the good solvent toluene and the lower curve relates to the less good solvent cyclohexane. Although there are no experimental values for comparison, the curves seem to be qualitatively correct considering experimental data of polar solvent–precipitant pairs w8x. An increasing asphaltene content results in a decrease of the precipitant volume needed, particularly, in the range of relatively small asphaltene contents. In Fig. 2 the precipitant volume is plotted vs. the initial crude oil concentration m E for the solvent–precipitant pair cyclohexanern-hexane and for some crude oils. Not surprisingly, with increasing m E-values the precipitant volume needed for flocculation decreases slightly. Such behaviour for polar solvent–precipitant pairs is well known. Fig. 3 shows similar results for the better solvent toluene. Here, the VF-values are essentially higher than in Fig. 2 but else the curves are similar to those of Fig. 2. The number of measured flocculation points w11x is much larger than presented here. However, these data relate to polar solvent–precipitant pairs being outside of the range of validity of the model. Further progress seems to be possible by generalization of the model introducing the Hansen three-component solubility parameter w9x.

6. Conclusions Asphaltene flocculation of complex systems of the type crude oil q solventq precipitant may be modeled by the simple Scatchard–Hildebrand solubility theory applying continuous thermodynamics. In this, the crude oil is assumed to consist of maltenes and asphaltenes describing the polydispersity of both by Gaussian distributions. Unlike other applications of continuous thermodynamics these

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distributions are considered to depend on the solubility parameter of the oil species. Doing so, the different aromaticity of the oil species and their different contents of heteroatoms, which are essential for flocculation, are taken into account. To model only one fit parameter is used. Nevertheless, compared with the measured flocculation data the calculated results are reasonable. Unfortunately, because of the use of the Scatchard–Hildebrand solubility parameter the model is restricted to non-polar solvents and precipitants. However, in the future the generalized solubility parameter concept of Hansen w9x could help to overcome this disadvantage.

7. List of symbols GE Mi Mi mE R r r ri ri T VF VL V) Wi WEA ) Y EA

segment-molar excess Gibbs energy molar mass of a discrete component i number average of the molar mass of a continuous ensemble i of oil species initial concentration of the crude oil solution Žg oilrl solvent. universal gas constant segment number number average of the segment number of the mixture segment number of a discrete component i number average of the segment number of a continuous ensemble i of oil species temperature precipitant volume needed for flocculation initial volume of the oil solution in the flocculation experiment Ž VL s 20 ml. volume of one mole of segments distribution function of a continuous ensemble of oil species contribution of the asphaltenes to the distribution function WEŽ d . of the crude oil total mass fraction of the asphaltenes within the crude oil

Greek letters gi segment-molar activity coefficient of a discrete component i Ž . gi d segment-molar activity coefficient of an oil species identified by the solubility parameter d of the continuous ensemble i d solubility parameter d number average of the solubility parameter of the mixture di solubility parameter of a discrete component i di number average of the solubility parameter of a continuous ensemble of oil species mi segment-molar chemical potential of a discrete component m iŽ d . segment-molar chemical potential of an oil species identified by the solubility parameter d of the continuous ensemble i ri density of a discrete component or of a continuous ensemble i of oil species ri ) auxiliary quantities introduced by Eqs. Ž12a., Ž12b. and Ž 12c. si standard deviation of the continuous distribution function of an oil ensemble i wi volume fraction of a discrete component i or of a continuous ensemble i of oil species

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Subscripts E EA EM P S

crude oil asphaltenes of the crude oil maltenes of the crude oil precipitant solvent

Superscripts I, II phases I, II

Acknowledgements The authors are grateful for the financial support of the Deutsche Forschungsgemeinschaft in accomplishing this work.

References w1x H.J. Neumann, J. Wilkens, Bitumen, Teere, Asphalte, Peche 6 Ž1974. 246–247. w2x G. Astarita, S.I. Sandler ŽEds.., Kinetic and Thermodynamic Lumping of Multicomponent Mixtures, Elsevier, Amsterdam Ž1991.. w3x M.R. Islam, in: T.F. Yen, G.V. Chilingarian ŽEds.., Asphaltenes and Asphalt, Vol. 1, Elesevier, Amsterdam Ž1994., pp. 249–298. w4x H. Laux, Erdol-Erdgas-Kohle 108 Ž1992. 227–232. ¨ w5x G. Zenke, Dissertation, TU Clausthal Ž1989.. w6x H. Lian, J.R. Lin, T.F. Yen, Fuel 73 Ž1994. 423–428. w7x H. Laux, I. Rahimian, Erdol ¨ und Kohle 47 Ž1994. 430–435. w8x A. Hatke, Grundlagen der Asphaltenausfallung bei der Erdolgewinnung und Verarbeitung: Ein Beitrag zur Dis¨ ¨ pergierung und Stabilisierung von Asphaltenen, Thesis, TU Clausthal Ž1992.. w9x C.M. Hansen, Ind. Eng. Chem. Proc. Res. Dev. 8 Ž1969. 2–11. w10x A.F.M. Barton, Handbook of Solubility Parameters and Other Cohesion Parameters, CRC Press, Boca Raton, Florida Ž1985.. w11x M.T. Ratzsch, H. Kehlen, Prog. Polym. Sci. 14 Ž1989. 1–46. ¨