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A New Thermodynamic Model for Paraffin Precipitation in Highly Asymmetric Systems at HighPressure ConditionsAmeri Mahabadian, Mohammadreza; Chapoy, Antonin; Tohidi Kalorazi, Bahman

Published in:Industrial and Engineering Chemistry Research

DOI:10.1021/acs.iecr.6b02804

Publication date:2016

Document VersionPeer reviewed version

Link to publication in Heriot-Watt University Research Portal

Citation for published version (APA):Ameri Mahabadian, M., Chapoy, A., & Tohidi Kalorazi, B. (2016). A New Thermodynamic Model for ParaffinPrecipitation in Highly Asymmetric Systems at High Pressure Conditions. Industrial and Engineering ChemistryResearch, 55(38), 10208–10217. DOI: 10.1021/acs.iecr.6b02804

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Article

A New Thermodynamic Model for Paraffin Precipitation inHighly Asymmetric Systems at High Pressure Conditions

Mohammadreza Ameri Mahabadian, Antonin Chapoy, and Bahman TohidiInd. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b02804 • Publication Date (Web): 06 Sep 2016

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1

A New Thermodynamic Model for Paraffin

Precipitation in Highly Asymmetric Systems at

High Pressure Conditions

Mohammadreza Ameri Mahabadian1, Antonin Chapoy

1, 2*, Bahman Tohidi

1

1Hydrates, Flow Assurance & Phase Equilibria Research Group, Institute of Petroleum

Engineering, Heriot-Watt University, Edinburgh, Scotland, UK

2Mines Paristech, CTP – Centre Thermodynamique des procédés, 35 rue St Honoré 77305

Fontainebleau, France

KEYWORDS

Solid-fluid equilibrium, Paraffin wax, High pressure, Asymmetric systems, Clausius-

Clapeyron equation, Thermophysical properties

ABSTRACT

The predictions of the crystallization temperature and the amount of precipitates of paraffin

waxes at high pressure conditions may be inaccurate using existing thermodynamic models.

This is mainly due to the lack of experimental data on the molar volume of solid paraffins at

high pressures. This inaccuracy is even more pronounced for mixtures of high asymmetry.

The present work provides a new accurate modelling approach for solid-fluid equilibrium

(SFE) at high pressure conditions, more specifically, for highly asymmetric systems. In

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contrast to the conventional methods for high pressure SFE modelling which define Poynting

molar volume correction term, to calculate the paraffin solid phase non-ideality at high

pressures, the new method exploits the values of thermophysical properties of importance in

SFE modelling (temperatures and enthalpies of fusion and solid-solid transition) evaluated at

the high pressure condition using a new insight to the well-known Clausius-Clapeyron

equation. These modified parameters are then used for evaluation of the fugacity in the solid

phase at higher pressure using the fugacity of pure liquid at the same pressure and applying

the well-established formulation of the Gibbs energy change during melting. Therefore, the

devised approach does not require a Poynting correction term. The devised approach coupled

with the well-tested UNIQAC activity coefficient model is used to describe the non-ideality

of the solid phase. For the fluid phases, the fugacities are obtained with the SRK EoS with

binary interaction parameters calculated with a group contribution scheme. The model is

applied to highly asymmetric systems with SFE experimental data over a wide range of

pressures. It is first used to predict crystallization temperature in binary systems at high

pressures and then verified by applying it on multicomponent mixtures resembling

intermediate oil and natural gas condensates.

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1. Introduction

Formation of paraffinic solids is well documented to be able to impose considerable

operational costs due to decreasing flow efficiency and, in the worst case, pipeline blockage.

Due to high expenses of the remediation approaches for wax deposition problem (such as

chemical dissolution and pigging), prevention is always the best option which in turn calls for

accurate risk assessment of the wax formation problem, i.e. identifying the

temperature/pressure conditions under which the waxes form. Although not as important as

temperature, the pressure can have a significant effect on the wax phase boundary (see for

example the work of Pan et al.1). In fact, as outlined by Pauly et al.

2, in mixtures with

significant light end proportions, the pressure change can considerably affect the chance of

wax formation through retrograde condensation, depressurization and Joule-Thomson effect.

Several thermodynamic models have been proposed in the literature for estimating wax

precipitation onset and the amount of wax formed inside the wax phase boundary. The

performance of existing models are (as will be shown later) good at low pressure conditions

as long as accurate thermodynamic models for the description of fluid and solid phases as

well as a precise correlation for calculating thermophysical properties of alkanes are utilised.

Paraffinic SFE calculations at high pressures using existing methodologies may show high

deviations compared to experimental data, more visibly in systems of high asymmetry with

high proportions of the light end which are the main subject of this study. The main motive

for studying such systems is their resemblance of volatile oils and gas condensates which

might form wax1,3

. With similar intention, a handful of experimental studies in the literature,

mainly on binaries, have been focused on SFE in highly asymmetric systems. The purpose of

the current study is the development of a new thermodynamic model for the extension of one

of the accurate existing schemes for SFE modelling to high pressures. The next section

provides the background on the modelling wax precipitation at high pressures and the

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complete formulation of the developed model. It also presents a modification of an existing

method. An extensive comparison of the devised methodology with the existing models is

then provided in the Results and Discussions Section.

2. Methodology

2.1. Background

The equilibrium calculations in the paraffin wax forming systems require evaluation of the

fugacity of precipitating components in the solid phase(s) which, consequently, calls for the

evaluation of fugacity of pure components in the solid state. The fugacity of pure paraffins in

the solid state, ��∗�, are well-established to be related to the pure components’ liquid

fugacity, ��∗�, by4:

��∗��∗� = ��Δ �f�� 1 − ��f� + Δ �tr�� 1 − ��tr�� (1)

It is assumed here that the Gibbs free energy change due to thermal contributions during

phase changes (heat capacity effect) are negligible, as confirmed through sensitivity

analysis5. In the equilibrium calculations, the pure components fugacities in the solid state

are then used to calculate the fugacity of components in the solid solution by:

��S = �S��∗��S (2)

Using Eq. 1 and Eq. 2 as well as an accurate thermodynamic model to describe fluid phases

and a robust activity coefficient model to calculate activity coefficient of components in the

solid solution, ��S, one can easily specify the solid-fluid equilibrium state characteristics

applying a robust multiphase flash algorithm. The application of Eq. 1 requires accurate

values of thermophysical properties which are normally measured at components triple point

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pressure. Therefore, precise evaluation of wax phase boundary (or more accurately, wax

disappearance temperatures, WDT) at sufficiently low pressures near to the reference state

pressure (in this work 0.1 MPa) is an easy task, provided that a combination of strong

thermodynamic models are utilized. One such combination, as applied in the current work

consists of:

i. Thermodynamic model for fluid phases: Soave-Redlich-Kwong (SRK) EoS6 is used

to describe fluid phases and binary interaction parameters are calculated by Jaubert

and Mutelet7 group contribution scheme (JMGC) as presented by Qian et al.

8. This

method was originally developed for the Peng and Robinson9 (PR) EoS and then

extended to the SRK EoS as presented by Jaubert and Privat10

is used. In the absence

of associating fluid, which is the case for the mixtures investigated here, the SRK-

JMGC model has a proven capability to accurately describe fluid-fluid equilibria10

.

Jaubert and Mutelet7 combined their group contribution method with a cubic EoS to

model VLE in highly asymmetric systems. They found that their group contribution

scheme coupled with a cubic EoS gives better results compared to EoS/GE approaches

of LCVM11

(which is widely used for describing fluid in solid-fluid equilibria of

waxy systems in several example works3,12–16

) and MHV217

. The JMGC method for

binary interaction parameters in modelling wax forming systems has been applied in

some publications18,19

. In order to be consistent, in this work, the fugacity of pure

components in the liquid state are also calculated with the SRK EoS.

ii. Thermophysical properties estimation correlation: Experimental evidence has shown

that for pure odd alkanes, the dominant crystalline structures in multicomponent solid

solutions is orthorhombic20–23

. This observation is also valid for cases where solid

solution consists only of even alkanes24

. In this regard, except for cases where the

solid solution is a pure even heavy paraffin, the thermophysical properties of

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compounds, i.e., fusion temperature, ��f, solid-solid transition temperature, ��tr, enthalpy of fusion Δ �f and enthalpy of solid-solid transition, Δ �tr, are evaluated

using the correlations of Coutinho and Daridon25

. In these set of correlations, the odd

paraffins properties are extended by extrapolation to the even alkane properties26

.

Using these correlations, the values of thermophysical parameters are evaluated at

reference pressure,�� (assuming that the thermophysical properties of pure

components at the triple point pressure and the reference pressure are the same24

). In

the cases of binary asymmetric systems, where the solid phase is a pure even paraffin,

the thermophysical properties of the paraffin used are those reported in the

comprehensive work of Dirand et al.27

.

iii. Activity coefficient model for the solid solution: In this work the UNIQUAC activity

coefficient model in its original form as developed by Abrams and Prausnitz28

(later

utilized by Coutinho29

for the non-ideality of paraffinic solid phase(s)) is used to

evaluate paraffinic solid components activity coefficients in solid solution, ��S. Details

of this model and its formulation and parameterization can be found elsewhere26

.

Finally, accurate values for critical properties and acentric factor of components, especially

the heavy alkanes, are required. In this work, the critical properties and acentric factor pure

components are taken from the DIPPR database [35].

2.2. Modeling wax phase boundary at high pressures

For high pressures, generally, two approaches can be utilized to evaluate fugacity of

components in the solid solutions in complex multicomponent waxy mixtures:

1- Poynting term models: In these models, the fugacities of the solid phase(s) evaluated at

the reference pressure (using Eq. 1 and Eq. 2) are translated to higher pressures using a

Poynting correction term i.e.:

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��, �, �� = ��, ��, �� ̅��

� !�� = �S��∗��S�� ̅��

��

!�� (3)

Here, �̅�� is the molar volume of component " in solid solution and ��is the reference

pressure (0.1 MPa). Examples of this type are the works of Pauly et al.12

, Morawski et

al.30

, Ghanaei et al.31,32

and Nasrifar et al.19

. Correct calculation of the Poynting

correction term requires an accurate model to evaluate the molar volume of components

in the solid solution. Due to scarcity of experimental data to develop such a model,

different authors have presented a variety of methods to estimate the Poynting term.

Pauly et al.12

have assumed that the molar volume of components in solid solution is

equal to the pure component molar volume in the liquid state multiplied by a pressure

independent constant variable# through:

� �̅��

� !� = � #��

��

!� = # $%��∗��$%��∗�� (4)

Where �� and �� are the molar volume of the pure normal alkane " in the solid and liquid

states, respectively. The assumption of a constant pressure-independent # contradicts the

fact that by increasing pressure, reduction in liquid the molar volume of a component is to

higher extents than in the solid state. Morawski et al.30

used Clausius-Clapeyron equation

to modify#, though again # is considered to be pressure independent. They also

assumed that enthalpies of fusion and solid-solid transition of normal alkanes are pressure

independent. Furthermore, Morawski et al. 30

model requires evaluation of a

composition-dependant adjustable parameter and in this sense is not fully predictive. The

works of Nasrifar and Fani-Kheshty19

and Ghanaei et al.31

are a modification of the

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Morawski et al.30

model, attempting to remove the adjustable parameter. Accordingly,

Nasrifar and Fani-Kheshty19

proposed the following formulation for the Poynting term:

� �̅��

� !� = −& '( �f + ( �tr�)�) * �� − �� (5)

Here, & is constant equal to 0.002 m3/kmol obtained by fitting WDT of pure normal

paraffins. However, Ghanaei et al.31

, by assuming constant slopes for fusion and solid-

solid transition temperatures of pure paraffins by increasing pressure, developed the

following formulation for the Poynting term:

� �̅��

� !� = � − ��) + Δ �f,!�∗!�-. +

Δ �tr/!�∗!�0123 (6)

Based on Ghanaei et al.31

, with an accurate estimate and regardless of the carbon number

of the pure alkane the values of 4.5 MPa.K-1

and 3.5 MPa.K-1

can be assigned to

saturation pressure slope changes with temperature for fusion, 4�∗456, and solid-solid

transition, 4�∗4578, for all heavy alkanes. To obtain this the authors have assessed a large

database of experimental fusion and solid-solid transition temperatures of pure alkanes at

high pressure reported in the literature 27,33–39

. Based on the current study evaluations and

some work in the literature30

this assumption is indeed precise (it will be shown later on,

utilized in a different scheme). This way they removed the need for parameter & defined

in Nasrifar and Fani-Kheshty19

work. However in both methods the same assumptions, as

that of Morawski et al.31

hold. It should be noted that Ghanaei and co-authors have also

presented another high pressure wax model32

, again by devising a formulation for the

Poynting term, developed a few years prior to their latest approach described here. In our

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evaluations, only the performance of their recent model is assessed. Finally, there are

other works in the literature estimating the Poynting term by assuming the solid phase to

be incompressible and the liquid molar volumes are evaluated at average pressures. Due

to these questionable assumptions, especially in the cases studied here, such works are not

assessed here.

2- No-Poynting term models: In the second approach, the pure components solid fugacities

are calculated at high pressure using Eq. 1 with the thermophysical properties evaluated at

the same high pressure�, i.e. no Poynting correction term is required. The method of Ji

et al.40

belongs to this group. In this method, a linear correlation is used to evaluate ��f of

alkanes at higher pressure, with an accurate estimate that the slope of change of fusion

temperature by increasing pressure,4�∗456 is a constant value for heavy alkanes (as discussed

earlier). Therefore, one can write:

�-�� = �-�� + +� − ��,!�∗!�-.3 (7)

A constant of 5.0 MPa.K-1

for4�∗456 is suggested by Ji et al. 40

. In the original work of Ji et

al. 40

the parameter �- is the only thermophysical property of pure heavy alkane for

which updated values are evaluated at higher pressures and the rest are held constant.

Based on our experience if �- is the only thermophysical properties modified at�, the

model deviations from experimental behaviour can be significant at high pressures. The

same observation is made in the evaluations made by Nasrifar and Fani-Kheshty19

.

Therefore, here, apart from the new model developed, first, a modified version of Ji et

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al.40

model is presented, in which not only the fusion temperatures but also the solid-

solid transition temperatures of pure alkanes are updated at high pressure similarly by:

�01�� = �01�� + +� − ��/!�∗!�0123 (8)

It will be shown later that, despite the simplicity of the approach, the modified Ji method

provides better results compared to that of “Poynting term” methods. In the modified Ji

model the value of slopes 4�∗456 and

4�∗4578 are those suggested by Ghanaei et al.31

i.e. 4.5

MPa.K-1

and 3.5 MPa.K-1

, respectively. As a final note to the methods in the second

category, it is assumed that the activity coefficient of components in the solution is fairly

constant with pressure. This is a reasonable assumption. In fact, differentiation of the

logarithm of activity coefficient with respect to pressure yields4:

�9�$%:��S;�9� �5,< = ��̅�=�� (9)

To see the effect of pressure on the activity coefficient the example case of binary n-

pentane + n-hexadecane is considered. For this highly asymmetric system, the absolute

value of excess molar volume is reported41

to be as high as -1.1581 cm3.mol

-1 (for

0.7034:0.2966 molar ratio). Using this value in Eq. 9, at room temperature, a pressure

change of 100 MPa is translated into only about 4.5% change in activity coefficient.

Furthermore, the volume effect of mixing is decreasing by increasing pressure in

paraffinic systems (see for example42

) and, obviously, the excess molar volume of solid

solutions are smaller than that of liquid solutions, therefore one would expect even much

smaller changes in activity coefficient in the solid solution at high pressures and hence

the assumption of independency of activity coefficient from pressure is plausible.

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Based on several investigations, (and as will be shown for modified Ji, Pauly et al.12

, Nasrifar

and Fani-Kheshty19

and Ghanaei et al.31

models) the performance of the methods in both

categories are comparatively acceptable for mixtures of low asymmetry with overall

compositions having a low amount of light ends. The efficiency of the aforesaid methods,

however, is poor in mixtures of high asymmetry which have high proportions of light ends, as

will be presented later on. The deviations become even more as the pressure increases. This

issue is addressed in some works 2 and seemingly has prevented the authors accurately

modelling the experimental data. In this work, the aim is to tackle the problem of wax phase

boundary estimation at higher pressure, especially for highly asymmetric systems, by

developing a new model. Therefore, the work presents two new solid-liquid equilibrium high

pressure models based on “No-Poynting term” approach i.e. (i) the modified Ji model,

described earlier and (ii) a new accurate scheme described in the next section. The reason

why two new methods are presented here will be discussed in the results section.

2.3. New proposed method

In the current study, a new method based on the “No-Poynting term” approach is developed

to model the non-ideality of paraffinic solid phases at high pressures. The aim here is to have

accurate estimations of thermophysical properties of pure paraffins at high pressure, using

their values in the reference state and a proper formulation to modify them to account for

high pressure effect. Prior to discussing the development of the model, to have a better

understanding the of solid-fluid equilibrium behaviour of highly asymmetric systems,

investigations are first carried out for simple binary systems of high asymmetry, for which

experimental solid-fluid phase boundary data are available. According to Seiler et al.43

the

combination of Eq. 1 and Eq. 2 can be extended to high pressure range if the pressure

dependence of both enthalpies of fusion and the solid-solid transition is taken into

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consideration. Here, the Clausius-Clapeyron equation44

is used to take the dependency of

Δ - to pressure into consideration. Using Clausius-Clapeyron equation, for the fusion:

∆ -�� = �� − ��-�� !�∗!�-�� (10)

As mentioned, for n-alkanes, with an accurate estimate, the fusion temperature�-changes

linearly with pressure and the slope is a constant value independent of pressure and the

carbon number of the pure paraffin. Therefore, using Eq. 10, one can easily relate the

enthalpy of fusion at high pressure � to the same property in reference pressure �� by:

∆ -��∆ -�� = � �� − �� − �� -��-�� (11)

Here, as described, the fusion temperature of pure alkane at reference pressure (�-�� is calculated by the correlations proposed by Coutinho and Daridon

25 except for the cases of

binary asymmetric systems where the solid solution is a pure even normal alkane in which

case for the heavy alkane ", �-�� is taken from the work of Dirand et al.27

. Eq. 7 with 4�∗456

of 4.5 MPa.K-1

as suggested by Ghanaei et al.31

is used to calculate fusion temperature of pure

alkane "at high pressure,�-��. As mentioned, based on Pauly et al.

12 one can relate the molar volume of heavy alkanes in the

liquid state to the same value in the solid state by multiplying it with constant value, i.e.:

�� = #�� (12)

Where, according to Pauly et al. 12

, for pure alkanes # is equal to 0.86 and is assumed to be

pressure independent. For mixtures, due to excess volume effect they have suggested the

value of 0.9 for#. A constant, pressure independent value, assigned for # is questionable as

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obviously the effect of compaction due to high pressure is less in the solid state compared to

liquid state. Hence, one would expect that by increasing the pressure the value of #should

increase. Accordingly, in this work, # is defined to be pressure dependent, hereafter denoted

as#- for fusion, assuming to increase linearly with pressure (in the simplest possible way)

i.e.:

#-�� = #-�� + ?�� − �� (13)

Here ? is a positive constant. Despite the simplicity of Eq. 13, as will be presented later on,

the formulation devised proves very accurate. Using the data reported by Schaerer et al.45

an

average value of 0.895 is assigned to#-�� which is representing the ratio of pure alkane

liquid state to solid state molar volume at reference pressure (very similar to Pauly et al.12

value of#). In this way Eq. 11 can be reduced to:

∆ -��∆ -�� = �1 − #-��1 − #-�� -��-�� (14)

Using the same approach, however with a different variable named #01 to make a distinction,

for alkanes showing order-disorder solid-solid transitions, the following formula can be

written to update ∆ 01at high pressures:

∆ 01��∆ 01�� = �1 − #01��1 − #01�� 01��01�� (15)

Again by using Schaerer et al.45

data the value of 0.958 is assigned to #01�� which

represents the ratio of pure alkane disordered to ordered solid state molar volume at reference

pressure. Similar to fusion, as mentioned, the same trend in change of �01 of a pure alkane

by pressure is observed, i.e. increasing with constant 4�∗4578 of 3.5 MPa.K

-1 as suggested by

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Ghanaei et al.31

. Therefore, Eq. 8 is used to evaluate�01��. With the same approach

applied for fusion as presented in Eq. 13, for solid-solid transition it is proposed that:

#01�� = #01�� + ?�� − �� (16)

In theory, the value of ? in Eq. 16 should be different from that of Eq. 13. However, the aim

here is to have one adjustable parameter, and as will be shown later on, a single value of ? for

both of the equations can accurately model highly asymmetric systems. Having adjustable

parameters to model solid-fluid equilibrium at high pressures is acceptable (see for example

Morawski et al. 30

and Rodriguez-Reartes et al. 46

models for paraffinic binary systems).

To sum-up in the current approach the following steps should be taken to calculate non-

ideality in the solid phase at high pressure�:

1- Using proper correlations/database the thermophysical properties of pure components

i.e. ��f, ��tr, Δ �f, Δ �tr are evaluated at the reference pressure��.

2- With values of 4.5 MPa.K-1

for 4�∗456 and 3.5 MPa.K

-1 for4�∗4578, Eq. 7 and Eq. 8 are used

to evaluate ��f and ��tr , respectively, at high pressure�.

3- Adjusting the value of? and using values of 0.895 for #-�� and 0.958 for#01�� , Eq. 13 and Eq. 16 are used to evaluate#- and#01, respectively, at high pressure�.

Guidelines for assigning a correct value for ?will be presented in the results section.

4- Using (i) the values of ��f and ��trat reference pressure ��and their values at high

pressure �, calculated in Step 2, (ii) the values of #- and#01 at reference pressure

��(i.e. 0.895 and 0.958) and their values at high pressures � (calculated in Step 3)

and (iii) Δ �f and Δ �tr at reference pressure ��, (calculated in Step 1), Eq. 14 and Eq.

15 are used to evaluate Δ �f and Δ �tr, respectively at high pressure �.

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5- Having calculated the value of all thermophysical properties at high pressure P, Eq. 1

and Eq. 2 are directly used to evaluate the fugacity of components in the solid solution

at high pressure.

As described earlier, for the modified Ji method, steps 3 and 4, are not required. In fact, the

modified Ji method is a special form of the, more general, new model proposed here for

which:

∆ -��∆ -�� = 1 (17)

And:

∆ 01��∆ 01�� = 1 (18)

As both modified Ji and the new model are proposed here, to make the distinction, hereafter,

the term “this work model” refers to the newly developed method, not the modified Ji.

3. Results and Discussions

In this section, the performance of the developed model is compared with the modified Ji,

Pauly et al. 12

, Nasrifar and Fani-Kheshty19

and Ghanaei et al. 31

models. To have fair

evaluations of the models, the SRK+JMGC model and UNIQUAC activity coefficient model

are used to describe the non-idealities of fluid and solid phase, respectively, for all high-

pressure methods. The SRK+JMGC model used here to describe fluids is fully predictive

and no tuning of model parameters to match saturation pressure data is made prior to wax

calculations. It will be shown that the performance of all these existing models in systems

with lower proportions of the light end in moderate pressure ranges is good, provided the

requirement discussed earlier in the Background Section. Therefore, the main focus here is

on highly asymmetric systems of high proportions of the light end at high pressures,

resembling gas condensates and volatile oils, for which experimental wax phase boundary

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16

data are available. Data on such systems are scarce. The uncertainty in the experimental wax

phase boundary data used here to evaluate models is low as they are corresponding to wax

disappearance points. It is well established that wax disappearance temperature (WDT) data

are better representatives of true thermodynamic melting point compared to wax appearance

temperature (WAT)47,48

. The evaluations are first carried out for binary asymmetric systems,

then synthetic multicomponent asymmetric wax mixtures and are graphically represented for

selected systems with low, moderate and high proportions of methane. Furthermore,

reporting model errors from low to high pressure range using Average Relative Error (ARE)

percent, i.e. by: AB∑ DEF�GHI�JEF�KLM�DEF�GHI�B� × 100 can be misleading as the high temperature ranges

would result is small AREs regardless of the model utilized. Therefore, instead, the Average

Absolute Error (AAE) is used for the comparisons which is defined by (P: number of points):

QQR% = 1PT DT��VWX� − T��YZ[�DB� × 100 (19)

3.1. Binary methane + heavy alkane mixtures

Here, the models results for binary asymmetric systems (with experimental wax phase

boundary data available) methane + n-hexadecane49

, methane + n-heptadecane35

, methane +

n-eicosane39

, methane + n-docosane50

, methane + n-tetracosane51

and methane + n-

triacontane52

are presented. A total of 457 data points of WDT in binary asymmetric mixtures

are used for evaluations. In this study the WDT data of asymmetric systems for which the

data are not reported at high pressures of at least 50 MPa (e.g. 53–58

for which the solid-fluid

phase boundary data are reported up to 12 MPa), or the systems for which the uncertainty in

the critical/physical properties of the heavy end component is high (e.g. 59

) are not used in

evaluations. In all the systems evaluated wax phase boundary data in low to very high

proportions of methane with a variety of molar ratios were measured. The graphical

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comparisons are shown for selected low, moderate and high proportions of methane in

example binary mixtures methane + n-heptadecane, methane + n-eicosane and methane + n-

triacontane in Figure 1 to Figure 3. As the deviations of existing models are very high in the

case of systems methane + n-triacontane, depicting evaluation results in all proportions in a

single graph makes interpretations difficult, therefore it was decided to show them separately

in the way presented in Figure 3. As the first major observation in the model evaluations

(and as observed in Figure 1 to Figure 3), for the binary systems tested, adjusting a single

value of ? can accurately model the wax phase boundary, in a fixed binary system regardless

of the proportion of the methane in the mixture. As an example, as shown in Figure 3, the ?

equal to 1.38×10-4

MPa-1

gives an accurate match in low to high proportions of methane in

binary mixture methane + n-triacontane. The adjusted value of ? for each case is presented

in Table 1. As shown in this table, except for the case of methane + n-hexadecane system the

value of ? have almost the same order of magnitude in all the binary system. The models

deviations are presented in Table 2 in terms of AAE for all the data points. According to this

table, the results of “this work model” with a single ? for each case in very low to very high

proportions of light end (methane) are accurate. The superiority of the proposed model is

clearer in binary systems of higher asymmetry (see results for methane + n-tetracosane and

methane + n-triacontane in Table 2 where the deviation of existing models are very high at

high pressures. Interestingly, even if the average value of ? over the binary systems i.e.

1.23×10-4

MPa-1

is used for all the cases, still the performance of the proposed method is

much better than the alternative methods.

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Figure 1: Binary methane + n-heptadecane solid-fluid phase boundary. The results for “this

work model” are shown by adjusted ? of 1.18×10-4

MPa-1

Figure 2: Binary methane + n-eicosane solid-fluid phase boundary. The results for “this

work” model are shown by adjusted ? of 1.34×10-4

MPa-1

0

20

40

60

80

100

285 295 305 315 325

P (

MP

a)

T(K)

80.09 mol% C1-Experimental



This work model

Modified Ji model

Pauly et al. model

Ghanaei et al. model

Nasrifar and Fani-Kheshty model

0

20

40

60

80

100

300 305 310 315 320 325 330 335 340 345 350

P(M

Pa)

T(K)

9.9 mol% nC20-Experimental



Experimental SLVE data

This work model

Modified Ji model

Pauly et al. model


Nasrifar and Fani-Kheshty model

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19

Figure 3: Binary methane + n-triacontane solid-fluid phase boundary. The results for “this

work model” are shown by adjusted ? of 1.35×10-4

MPa-1

. The evaluation are made for

systems with (a) 89.7 (b) 50 and (c) 15 mol% methane in the binary mixture. �:

Experimental SLVE WDT data ○: Experimental SLE WDT data; ─── This work model; -·-

·-Modified Ji model; ----- Pauly et al. model; ········ Nasrifar and Fani-Kheshty model; - - -


It is important to note that the value of AAEs reported in Table 2 may seem to be at odds with

those reported for similar systems in Pauly et al.12

work for C1-nC24 and C1-nC22 systems.

However, in their evaluations for these mixtures Pauly et al.12

ignored the data points of more

than 90% methane in these mixtures which correspond to region of high deviation with

existing models, whereas, here the models are evaluated with systems (binary and

multicomponent) having as high as 97 mol% light end. That is why for these two mixtures

they have obtained smaller values of AAEs. If the same data points are used here similar

AAEs will be obtained. This choice may be due to higher deviations in modelling VLE of

asymmetric systems of the higher light end as pointed out by Jaubert and Mutelet7.

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Table 1: The adjusted values of parameter ? (MPa-1

) for the binary asymmetric systems

investigated

Binary system C1-nC16 C1-nC17 C1-nC20 C1-nC22 C1-nC24 C1-nC30

Adjusted ? 0.74×10-4

1.18×10-4

1.34×10-4

1.34×10-4

1.43×10-4

1.35×10-4

Average 1.23×10-4

Table 2: Average Absolute Error (AAE) of wax phase boundary calculated by different

models compared to experimental values for binary asymmetric systems

AAE (K)

Bin

ary

Mix

ture

Num

ber

of

dat

a p

oin

ts

Pre

ssure

ran

ge

(MP

a)

Th

is w

ork

mod

el

(? adju

sted

)

Th

is w

ork

mod

el

(?= 1.2

3×

10

-4 M

Pa-1

)

Mod

ifie

d J

i

(Pro

po

sed h

ere)

Pau

ly e

t al

. 12

Ghanae

i et

al.

31

Nas

rifa

r an

d

Fan

i-K

hes

hty

19

C1-nC30 115 3.189-86.80 1.07 1.17 2.76 3.95 3.02 3.00

C1-nC24 136 1.820-95.62 0.67 1.18 4.34 3.97 4.38 4.06

C1-nC22 13 4.520-97.68 0.19 0.72 5.75 5.32 6.79 6.24

C1-nC20 81 95.66-160.19 0.44 0.47 1.12 1.57 1.64 2.26

C1-nC17 49 10.11-257.50 0.53 0.54 1.16 1.65 1.74 1.87

C1-nC16 63 6.540-193.10 0.25 0.43 0.34 0.51 0.84 1.43

Total 457 Average 0.53 0.75 2.58 2.83 3.07 3.14

3.2. Multicomponent Mixtures

Similar to binary systems, here the evaluations are made only for systems of high asymmetry,

where the light end is methane, as all of the existing models perform fairly well for low

asymmetric systems. For multicomponent systems again it is observed that a constant value

of ? is capable of accurately representing the wax phase boundary in a system with fixed

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21

light and heavy ends, regardless of their proportions. This is shown for selected systems of

different low to high proportion of methane combined with (i) a ternary mixture of nC16 to

nC18 (experimental data from60

) in Figure 4 and (ii) multicomponent wax of nC13 to nC22

(wax2 with experimental data taken from2) presented in Figure 5. For all the systems tested,

the complete results are presented in Table 3. A set of 403 data points for 5 highly

asymmetric mixtures are used for evaluations. As it is shown in this table, by assigning a

special ? for each system, compared to other models, very small values of AAE would be

achieved. For all the data points for multicomponent systems, a common value of 0.95×10-4

is assigned to?. Using this value, again very small values of overall AAE are achieved as

shown in Table 3. Furthermore, according to the results, similar to what was observed for

binary systems, the modified Ji model performs much better than the other models. This

signifies the superiority of the “No-Poynting term” models over “Poynting term” models,

when thermophysical properties of heavy alkanes (most significantly, fusion and solid-solid

transition temperatures) are correctly evaluated at high pressures. Among the two “No-

Poynting-term” methods proposed, the new approach in which the parameter α is introduced

is outperforming the modified Ji method as all parameters T\f, T\tr, ΔH\f, ΔH\tr are calculated at

higher pressure while in the “modified Ji” method only T\f, T\tr are calculated at higher

pressure.

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Figure 4: Multicomponent methane + (nC16-nC18) solid phase boundary. The results for

“this work model” are shown by adjusted ? of 1.71×10-4

MPa-1

for (a) 89.48 (b) 59.97 and (c)

19.9 mol% methane in the mixture. ○: Experimental WDT data; ─── This work model; -·-

·-Modified Ji model; ----- Pauly et al. model; ········ Nasrifar and Fani-Kheshty model; - - -


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Figure 5: Multicomponent methane + (nC13-nC22) solid phase boundary (“wax2” from the

work of Pauly et al.2. The results for “this work model” are shown by adjusted ? of 0.55×10

-

4 MPa

-1 (a) 89.76 (b) 79.73 (c) 69. 95 mol% methane in the mixture. ○: Experimental WDT

data; ─── This work model; -·-·-Modified Ji model; ----- Pauly et al. model; ········ Nasrifar

and Fani-Kheshty model; - - - Ghanaei et al. model

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Table 3: Average Absolute Error (AAE) of wax phase boundary calculated different models

compared to experimental values for multicomponent asymmetric systems

AAE (K)

Mixture

Nu

mber

of

dat

a p

oin

ts

Pre

ssu

re r

ang

e (M

Pa)

Ad

just

ed

? (MP

a-1)

This

work

*

This

work

**

Mo

dif

ied Ji

(P

rop

ose

d

her

e)

Pau

ly e

t al

. 12

Gh

anae

i et

al.

31

Nas

rifa

r an

d

Fan

i-

Kh

esh

ty 1

9

C1+(nC16-nC18)60 41 4.22-

95.94

1.71×10-4

0.34 1.09 1.81 5.03 2.66 3.19

C1+(nC22+nC24)50 119 96.96-

171.4

1.36×10-4

1.32 2.22 5.29 8.51 6.82 6.82

C1+(nC24+nC30)38 105 103.1-

193.1

1.17×10-4 1.16 1.99 6.22 11.24 7.69 6.22

C1+(nC13-nC22)2 99 0.1-

98.59

0.55×10-4

1.10 1.44 1.15 1.67 1.69 1.72

C1+(Multi-paraffin)61 39 0.1-45 0.32×10-4

0.38 0.85 0.49 0.93 1.03 1.03

Total 403 0.1-

193.1

Average 0.86 1.68 2.99 5.48 3.99 3.80

*Adjusted ? for each case

**Common?equal to 0.95×10-4

MPa-1

3.3. Notes on the adjustable parameter ^ Comparing the results presented in Table 2 and Table 3, it is observed that a smaller common

value can be assigned for parameter ? in binary systems compared to multicomponent

systems. This difference can be attributed to the excess volume effects in solid solutions in

comparison to pure solids, which results in having non-unity values for activity coefficients.

Yet the common values of ? are in the same order. Furthermore there is insufficient

published experimental data at very high pressures (in orders of 100 MPa) and for a diverse

range of asymmetry in the systems. Also, the experimental data used here for model

evaluations and adjusting ? for each system are measured by different methods (for the data

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presented here, visual and calorimetric techniques), coupled with the general uncertainty with

wax phase boundary data62

. Accordingly, it was not possible to check if more intricate

functions can model the behaviour of parameters #- and #01at high pressure or if ? can be

defined as a normal alkane dependant parameter or as a function of the heaviness and/or

degree of asymmetry of the system. In absence of wax phase boundary data for systems of

low to moderate asymmetry, the modified Ji model, is suggested. Overall as a general

guideline, based on the characteristics of a mixture it is advised to assign a value for ? within

in the lower and upper boundaries of 0.3×10-4

MPa-1

to 1.4×10-4

MPa-1

with a common value

of 0.95×10-4

MPa-1

as discussed earlier. The performance of this value is shown for an

example system of high asymmetry with about 95.96 mol% methane (mixture “wax3” in the

work of Pauly et al.2) in Figure 6. The grey region in this figure corresponds to the upper and

lower wax boundary change using the minimum and maximum values of?. Using these

values, at pressures as high as 100 MPa an absolute error of ±2.5 K can be observed,

depending on the value of? specified. For this system the best value for ? is 0.64×10-4

MPa-

1. However, using both “this work model” with common value of 0.95×10

-4 MPa

-1 and

modified Ji method, very similar AAE compared to experimental data is achieved, though

“this work model” performs better than modified Ji at higher pressures as presented in Figure

6 compared to “Poynting term” methods which have almost similar performances. This

figure, again confirms that between methods without adjustable parameters the method of

modified Ji is the one offering the highest accuracy.

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Figure 6: Wax phase boundary modelled with different methods for the “wax3” system of

Pauly et al.2 with 95.96 mol% methane. The grey region corresponds to the upper and lower

wax boundary change using the minimum and maximum values of?, i.e. 0.3×10-4

MPa-1

and

1.4×10-4

MPa-1

, respectively. For this system the best value of ? is 0.64×10-4

MPa-1

The performance of modified Ji model is not only comparably good at wax phase boundary

modelling but also in measuring the wax content and composition. Due to scarcity of such

data at high pressure, the comparisons are only made for a single system of nC10+(nC18-

nC30), though with much lower degree of asymmetry compared to the rest of the systems

investigated here, for which the experimental wax content and composition are measured by

Daridon and Dauphin63

at 293.15 K and at different pressures as high as 50 MPa (see Figure

7). For this system, the best value of ? to match wax composition data is 0.35×10-4

MPa-1

.

This small value (compared to the common value of 0.95×10-4

MPa-1

) for this system of

lower asymmetry points to the possible correlation of ? with the heaviness of the light-end

or, generally speaking, degree of asymmetry of system. Devising such a correlation requires

sufficient experimental solid-liquid equilibrium data to be obtained for a vast range of

systems with different asymmetry and proportions of light end.

0

20

40

60

80

285 295 305 315

P(M

Pa)

T(K)

Experimental WDT data

This work model (α=0.64E-4)

This work model (α=0.95E-4)

Modified Ji model


Pauly et al. model

Nasrifar et al. model

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Figure 7: Composition of solid wax evaluated with different models for nC10+(nC18-nC30)

mixture of Daridon and Dauphin63

. The results for “this work model” are shown by adjusted

? of 0.35×10-4

MPa-1

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Now, it is justified why here two models were proposed. The new model, by comparison

with the modified Ji and other models evaluated here, is capable of accurately modelling the

behaviour of solid-liquid equilibria in highly asymmetric systems at high pressure, though it

does have an adjustable parameter. Using a common value for this parameter the new model

is still superior in a variety of cases. However, in the absence of high pressure WDT data and

when the uncertainty in choosing value of adjustable parameter is high or when it is not

desirable to use a model with an adjustable parameter the modified Ji model is suggested. As

presented earlier, the modified Ji prediction of wax phase boundary as well as wax

composition was more accurate than all of the “Poynting term” methods evaluated here.

4. Conclusions and Suggestions

A new model for the most problematic systems in the area of solid-liquid equilibrium

modelling of the paraffinic waxy mixture, i.e. highly asymmetric mixtures at high pressure

conditions, is developed. The model is based on the “No Poynting term” approach and is

shown to be capable of accurately capturing the wax phase boundary behaviour of such

systems with a single adjustable parameter which can be fixed regardless of the proportion of

light and heavy ends in a mixture of variable ratios of fixed light end and heavy end. The

value of the adjustable parameter can vary from one mixture to another. Therefore, using a

total of 860 data points for evaluation, common values for the adjustable parameter are

assigned. Due to lack of experimental data, it is not possible at present to correlate the value

of the adjustable parameter with the degree of asymmetry of the mixture or even assign a

more complex form of function to the ratio of molar volume of n-alkanes in the solid state to

the same value in the liquid state (here a linear function was used, in the simplest form, with

the slope to be the adjustable parameter). Therefore it is suggested to measure more

experimental solid-fluid experimental data in systems with a wide range of asymmetry.

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Using common values for the adjustable parameter, the model works very well at high

pressures. Meanwhile, if it is not desirable to use a model with an adjustable parameter

(when uncertainty in its value is high) the modified Ji model, in the way revised in the current

study, is suggested as it was clearly shown that the “No Poynting term” methods performance

are superior to that of “Poynting term” methods for such problematic systems.

AUTHOR INFORMATION

Corresponding Author

* Tel: +44(0)1314513797 E-mail: [email protected] (Antonin Chapoy)

Author Contributions

The manuscript was written through contributions of all authors. All authors have given

approval to the final version of the manuscript.

Funding Sources

This work was funded by the Ali Danesh scholarship of Heriot-Watt University provided for

the first author.

ACKNOWLEDGMENT

The Ali Danesh scholarship of Heriot-Watt University provided for the first author is greatly

appreciated.

ABBREVIATIONS

SFE, Solid-fluid equilibrium; WDT, Wax disappearance temperature; SRK, Soave-Redlich-

Kwong equation of state; PR, Peng-Robinson equation of state; JMGC, Jaubert-Mutelet

group contribution method; VLE, Vapour-liquid equilibrium; LCVM, Linear combination of

Vidal and Michelsen mixing rules; MHV2, Second-order modified Huron-Vidal mixing rule;

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UNIQUAC, Universal quasi-chemical activity coefficient model; WAT, Wax appearance

temperature; AAE, Average absolute error; ARE, Average relative error; SLE, Solid-liquid

equilibrium; SLVE, Solid-liquid-vapour equilibrium.

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