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A New Thermodynamic Model for Paraffin Precipitation in Highly Asymmetric Systems at High Pressure Conditions
Citation for published version: Ameri Mahabadian, M, Chapoy, A & Tohidi Kalorazi, B 2016, 'A New Thermodynamic Model for Paraffin Precipitation in Highly Asymmetric Systems at High Pressure Conditions', Industrial and Engineering Chemistry Research, vol. 55, no. 38, pp. 10208–10217. https://doi.org/10.1021/acs.iecr.6b02804
Digital Object Identifier (DOI): 10.1021/acs.iecr.6b02804
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Published In: Industrial and Engineering Chemistry Research
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Download date: 27. Sep. 2021 Subscriber access provided by Heriot-Watt | University Library Article A New Thermodynamic Model for Paraffin Precipitation in Highly Asymmetric Systems at High Pressure Conditions Mohammadreza Ameri Mahabadian, Antonin Chapoy, and Bahman Tohidi Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b02804 • Publication Date (Web): 06 Sep 2016 Downloaded from http://pubs.acs.org on September 7, 2016
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1 2 3 4 5 6 7 A New Thermodynamic Model for Paraffin 8 9 10 11 Precipitation in Highly Asymmetric Systems at 12 13 14 15 High Pressure Conditions 16 17 18 19 1 1, 2* 1 20 Mohammadreza Ameri Mahabadian , Antonin Chapoy , Bahman Tohidi 21
22 1 23 Hydrates, Flow Assurance & Phase Equilibria Research Group, Institute of Petroleum 24 25 Engineering, Heriot Watt University, Edinburgh, Scotland, UK 26 27 28 2Mines Paristech, CTP – Centre Thermodynamique des procédés, 35 rue St Honoré 77305 29 30 Fontainebleau, France 31 32 33 34 KEYWORDS 35 36 37 Solid fluid equilibrium, Paraffin wax, High pressure, Asymmetric systems, Clausius 38 39 Clapeyron equation, Thermophysical properties 40 41 42 ABSTRACT 43 44 The predictions of the crystallization temperature and the amount of precipitates of paraffin 45 46 47 waxes at high pressure conditions may be inaccurate using existing thermodynamic models. 48 49 This is mainly due to the lack of experimental data on the molar volume of solid paraffins at 50 51 high pressures. This inaccuracy is even more pronounced for mixtures of high asymmetry. 52 53 The present work provides a new accurate modelling approach for solid fluid equilibrium 54 55 56 (SFE) at high pressure conditions, more specifically, for highly asymmetric systems. In 57 58 1 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 2 of 38
1 2 3 contrast to the conventional methods for high pressure SFE modelling which define Poynting 4 5 molar volume correction term, to calculate the paraffin solid phase non ideality at high 6 7 pressures, the new method exploits the values of thermophysical properties of importance in 8 9 10 SFE modelling (temperatures and enthalpies of fusion and solid solid transition) evaluated at 11 12 the high pressure condition using a new insight to the well known Clausius Clapeyron 13 14 equation. These modified parameters are then used for evaluation of the fugacity in the solid 15 16 phase at higher pressure using the fugacity of pure liquid at the same pressure and applying 17 18 19 the well established formulation of the Gibbs energy change during melting. Therefore, the 20 21 devised approach does not require a Poynting correction term. The devised approach coupled 22 23 with the well tested UNIQAC activity coefficient model is used to describe the non ideality 24 25 of the solid phase. For the fluid phases, the fugacities are obtained with the SRK EoS with 26 27 binary interaction parameters calculated with a group contribution scheme. The model is 28 29 30 applied to highly asymmetric systems with SFE experimental data over a wide range of 31 32 pressures. It is first used to predict crystallization temperature in binary systems at high 33 34 pressures and then verified by applying it on multicomponent mixtures resembling 35 36 intermediate oil and natural gas condensates. 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 2 59 60 ACS Paragon Plus Environment Page 3 of 38 Industrial & Engineering Chemistry Research
1 2 3 1. Introduction 4 5 6 Formation of paraffinic solids is well documented to be able to impose considerable 7 8 operational costs due to decreasing flow efficiency and, in the worst case, pipeline blockage. 9 10 Due to high expenses of the remediation approaches for wax deposition problem (such as 11 12 13 chemical dissolution and pigging), prevention is always the best option which in turn calls for 14 15 accurate risk assessment of the wax formation problem, i.e. identifying the 16 17 temperature/pressure conditions under which the waxes form. Although not as important as 18 19 temperature, the pressure can have a significant effect on the wax phase boundary (see for 20 21 1 2 22 example the work of Pan et al. ). In fact, as outlined by Pauly et al. , in mixtures with 23 24 significant light end proportions, the pressure change can considerably affect the chance of 25 26 wax formation through retrograde condensation, depressurization and Joule Thomson effect. 27 28 Several thermodynamic models have been proposed in the literature for estimating wax 29 30 31 precipitation onset and the amount of wax formed inside the wax phase boundary. The 32 33 performance of existing models are (as will be shown later) good at low pressure conditions 34 35 as long as accurate thermodynamic models for the description of fluid and solid phases as 36 37 well as a precise correlation for calculating thermophysical properties of alkanes are utilised. 38 39 Paraffinic SFE calculations at high pressures using existing methodologies may show high 40 41 42 deviations compared to experimental data, more visibly in systems of high asymmetry with 43 44 high proportions of the light end which are the main subject of this study. The main motive 45 46 for studying such systems is their resemblance of volatile oils and gas condensates which 47 48 might form wax 1,3 . With similar intention, a handful of experimental studies in the literature, 49 50 51 mainly on binaries, have been focused on SFE in highly asymmetric systems. The purpose of 52 53 the current study is the development of a new thermodynamic model for the extension of one 54 55 of the accurate existing schemes for SFE modelling to high pressures. The next section 56 57 provides the background on the modelling wax precipitation at high pressures and the 58 3 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 4 of 38
1 2 3 complete formulation of the developed model. It also presents a modification of an existing 4 5 method. An extensive comparison of the devised methodology with the existing models is 6 7 then provided in the Results and Discussions Section. 8 9 10 11 2. Methodology 12 13 14 2.1. Background 15 16 The equilibrium calculations in the paraffin wax forming systems require evaluation of the 17 18 fugacity of precipitating components in the solid phase(s) which, consequently, calls for the 19 20 21 evaluation of fugacity of pure components in the solid state. The fugacity of pure paraffins in 22 23 the solid state, , are well established to be related to the pure components’ liquid 24 ∗ 25 fugacity, , by 4: 26 ∗ 27 28 29 f tr 30 ∗ f tr (1) 31 Δ Δ 32 ∗ = 1 − + 1 − 33 34 35 It is assumed here that the Gibbs free energy change due to thermal contributions during 36 37 phase changes (heat capacity effect) are negligible, as confirmed through sensitivity 38 39 analysis 5. In the equilibrium calculations, the pure components fugacities in the solid state 40 41 are then used to calculate the fugacity of components in the solid solution by: 42 43 44 S S S 45 (2) ∗ 46 = 47 48 Using Eq. 1 and Eq. 2 as well as an accurate thermodynamic model to describe fluid phases 49 50 and a robust activity coefficient model to calculate activity coefficient of components in the 51 52 solid solution, S, one can easily specify the solid fluid equilibrium state characteristics 53 54 55 applying a robust multiphase flash algorithm. The application of Eq. 1 requires accurate 56 57 values of thermophysical properties which are normally measured at components triple point 58 4 59 60 ACS Paragon Plus Environment Page 5 of 38 Industrial & Engineering Chemistry Research
1 2 3 pressure. Therefore, precise evaluation of wax phase boundary (or more accurately, wax 4 5 disappearance temperatures, WDT) at sufficiently low pressures near to the reference state 6 7 pressure (in this work 0.1 MPa) is an easy task, provided that a combination of strong 8 9 10 thermodynamic models are utilized. One such combination, as applied in the current work 11 12 consists of: 13 14 6 15 i. Thermodynamic model for fluid phases : Soave Redlich Kwong (SRK) EoS is used 16 17 to describe fluid phases and binary interaction parameters are calculated by Jaubert 18 19 and Mutelet 7 group contribution scheme (JMGC) as presented by Qian et al. 8. This 20 21 method was originally developed for the Peng and Robinson 9 (PR) EoS and then 22 23 extended to the SRK EoS as presented by Jaubert and Privat 10 is used. In the absence 24 25 26 of associating fluid, which is the case for the mixtures investigated here, the SRK 27 28 JMGC model has a proven capability to accurately describe fluid fluid equilibria 10 . 29 30 Jaubert and Mutelet 7 combined their group contribution method with a cubic EoS to 31 32 model VLE in highly asymmetric systems. They found that their group contribution 33 34 E 35 scheme coupled with a cubic EoS gives better results compared to EoS/G approaches 36 11 37 of LCVM (which is widely used for describing fluid in solid fluid equilibria of 38 39 waxy systems in several example works 3,12–16 ) and MHV217 . The JMGC method for 40 41 binary interaction parameters in modelling wax forming systems has been applied in 42 43 some publications 18,19 . In order to be consistent, in this work, the fugacity of pure 44 45 46 components in the liquid state are also calculated with the SRK EoS. 47 48 ii. Thermophysical properties estimation correlation: Experimental evidence has shown 49 50 that for pure odd alkanes, the dominant crystalline structures in multicomponent solid 51 52 solutions is orthorhombic 20–23 . This observation is also valid for cases where solid 53 54 24 55 solution consists only of even alkanes . In this regard, except for cases where the 56 57 solid solution is a pure even heavy paraffin, the thermophysical properties of 58 5 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 6 of 38
1 2 3 compounds, i.e., fusion temperature, f, solid solid transition temperature, tr , 4 5 enthalpy of fusion f and enthalpy of solid solid transition, tr , are evaluated 6 7 Δ 25 Δ 8 using the correlations of Coutinho and Daridon . In these set of correlations, the odd 9 10 paraffins properties are extended by extrapolation to the even alkane properties 26 . 11 12 Using these correlations, the values of thermophysical parameters are evaluated at 13 14 reference pressure, (assuming that the thermophysical properties of pure 15 16 24 17 components at the triple point pressure and the reference pressure are the same ). In 18 19 the cases of binary asymmetric systems, where the solid phase is a pure even paraffin, 20 21 the thermophysical properties of the paraffin used are those reported in the 22 23 comprehensive work of Dirand et al. 27 . 24 25 26 iii. Activity coefficient model for the solid solution : In this work the UNIQUAC activity 27 28 28 coefficient model in its original form as developed by Abrams and Prausnitz (later 29 30 utilized by Coutinho 29 for the non ideality of paraffinic solid phase(s)) is used to 31 32 evaluate paraffinic solid components activity coefficients in solid solution, S. Details 33 34 26 35 of this model and its formulation and parameterization can be found elsewhere . 36 37 Finally, accurate values for critical properties and acentric factor of components, especially 38 39 40 the heavy alkanes, are required. In this work, the critical properties and acentric factor pure 41 42 components are taken from the DIPPR database [35]. 43 44 2.2. Modeling wax phase boundary at high pressures 45 46 47 For high pressures, generally, two approaches can be utilized to evaluate fugacity of 48 49 components in the solid solutions in complex multicomponent waxy mixtures: 50 51 52 1 Poynting term models : In these models, the fugacities of the solid phase(s) evaluated at 53 54 the reference pressure (using Eq. 1 and Eq. 2) are translated to higher pressures using a 55 56 57 Poynting correction term i.e.: 58 6 59 60 ACS Paragon Plus Environment Page 7 of 38 Industrial & Engineering Chemistry Research
1 2 3 4 S S 5 (3) ̅ ∗ ̅ 6 , , = , , = 7 8 9 Here, is the molar volume of component in solid solution and is the reference 10 ̅ 11 pressure (0.1 MPa). Examples of this type are the works of Pauly et al. 12 , Morawski et 12 13 30 31,32 19 14 al. , Ghanaei et al. and Nasrifar et al. . Correct calculation of the Poynting 15 16 correction term requires an accurate model to evaluate the molar volume of components 17 18 in the solid solution. Due to scarcity of experimental data to develop such a model, 19 20 different authors have presented a variety of methods to estimate the Poynting term. 21 22 Pauly et al. 12 have assumed that the molar volume of components in solid solution is 23 24 25 equal to the pure component molar volume in the liquid state multiplied by a pressure 26 27 independent constant variable through: 28 29 30 31 ∗ (4) 32 ̅ 33 = = ∗ 34 35 36 Where and are the molar volume of the pure normal alkane in the solid and liquid 37 38 states, respectively. The assumption of a constant pressure independent contradicts the 39 40 41 fact that by increasing pressure, reduction in liquid the molar volume of a component is to 42 30 43 higher extents than in the solid state. Morawski et al. used Clausius Clapeyron equation 44 45 to modify , though again is considered to be pressure independent. They also 46 47 assumed that enthalpies of fusion and solid solid transition of normal alkanes are pressure 48 49 independent. Furthermore, Morawski et al. 30 model requires evaluation of a 50 51 52 composition dependant adjustable parameter and in this sense is not fully predictive. The 53 54 works of Nasrifar and Fani Kheshty 19 and Ghanaei et al. 31 are a modification of the 55 56 57 58 7 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 8 of 38
1 2 3 Morawski et al. 30 model, attempting to remove the adjustable parameter. Accordingly, 4 5 Nasrifar and Fani Kheshty 19 proposed the following formulation for the Poynting term: 6 7 8 f tr 9 10 (5) ̅ + 11 = − − 12 13 14 Here, is constant equal to 0.002 m 3/kmol obtained by fitting WDT of pure normal 15 16 31 17 paraffins. However, Ghanaei et al. , by assuming constant slopes for fusion and solid 18 19 solid transition temperatures of pure paraffins by increasing pressure, developed the 20 21 following formulation for the Poynting term: 22 23 24 25 f tr 26 (6) 27 ̅ − Δ Δ = ∗ + ∗ 28 29 30 Based on Ghanaei et al. 31 , with an accurate estimate and regardless of the carbon number 31 32 of the pure alkane the values of 4.5 MPa.K 1 and 3.5 MPa.K 1 can be assigned to 33 34
35 saturation pressure slope changes with temperature for fusion, ∗, and solid solid 36 37
38 transition, ∗ , for all heavy alkanes. To obtain this the authors have assessed a large 39 40 41 database of experimental fusion and solid solid transition temperatures of pure alkanes at 42 43 high pressure reported in the literature 27,33–39 . Based on the current study evaluations and 44 45 some work in the literature 30 this assumption is indeed precise (it will be shown later on, 46 47 utilized in a different scheme). This way they removed the need for parameter defined 48 49 19 50 in Nasrifar and Fani Kheshty work. However in both methods the same assumptions, as 51 52 that of Morawski et al. 31 hold. It should be noted that Ghanaei and co authors have also 53 54 presented another high pressure wax model 32 , again by devising a formulation for the 55 56 Poynting term, developed a few years prior to their latest approach described here. In our 57 58 8 59 60 ACS Paragon Plus Environment Page 9 of 38 Industrial & Engineering Chemistry Research
1 2 3 evaluations, only the performance of their recent model is assessed. Finally, there are 4 5 other works in the literature estimating the Poynting term by assuming the solid phase to 6 7 be incompressible and the liquid molar volumes are evaluated at average pressures. Due 8 9 10 to these questionable assumptions, especially in the cases studied here, such works are not 11 12 assessed here. 13 14 15 2 No-Poynting term models : In the second approach, the pure components solid fugacities 16 17 are calculated at high pressure using Eq. 1 with the thermophysical properties evaluated at 18 19 the same high pressure , i.e. no Poynting correction term is required. The method of Ji 20 21 40 f 22 et al. belongs to this group. In this method, a linear correlation is used to evaluate of 23 24 alkanes at higher pressure, with an accurate estimate that the slope of change of fusion 25 26 temperature by increasing pressure, ∗ is a constant value for heavy alkanes (as discussed 27 28 29 earlier). Therefore, one can write: 30 31 32 33 34 (7) − 35 = + ∗ 36 37 38 1 40 39 A constant of 5.0 MPa.K for ∗ is suggested by Ji et al. . In the original work of Ji et 40 41 40 42 al. the parameter is the only thermophysical property of pure heavy alkane for 43 44 which updated values are evaluated at higher pressures and the rest are held constant. 45 46 Based on our experience if is the only thermophysical properties modified at , the 47 48 model deviations from experimental behaviour can be significant at high pressures. The 49 50 19 51 same observation is made in the evaluations made by Nasrifar and Fani Kheshty . 52 53 Therefore, here, apart from the new model developed, first, a modified version of Ji et 54 55 56 57 58 9 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 10 of 38
1 2 3 al. 40 model is presented, in which not only the fusion temperatures but also the solid 4 5 solid transition temperatures of pure alkanes are updated at high pressure similarly by: 6 7 8 9 10 (8) 11 − = + ∗ 12 13 14 It will be shown later that, despite the simplicity of the approach, the modified Ji method 15 16 17 provides better results compared to that of “Poynting term” methods. In the modified Ji 18 19 31 model the value of slopes ∗ and ∗ are those suggested by Ghanaei et al. i.e. 4.5 20 21 22 MPa.K 1 and 3.5 MPa.K 1, respectively. As a final note to the methods in the second 23 24 category, it is assumed that the activity coefficient of components in the solution is fairly 25 26 27 constant with pressure. This is a reasonable assumption. In fact, differentiation of the
28 4 29 logarithm of activity coefficient with respect to pressure yields : 30 31 32 S 33 (9) ̅ 34 = 35 , 36 37 To see the effect of pressure on the activity coefficient the example case of binary n 38 39 pentane + n hexadecane is considered. For this highly asymmetric system, the absolute 40 41 value of excess molar volume is reported 41 to be as high as 1.1581 cm 3.mol 1 (for 42 43 0.7034:0.2966 molar ratio). Using this value in Eq. 9, at room temperature, a pressure 44 45 46 change of 100 MPa is translated into only about 4.5% change in activity coefficient. 47 48 Furthermore, the volume effect of mixing is decreasing by increasing pressure in 49 50 paraffinic systems (see for example 42 ) and, obviously, the excess molar volume of solid 51 52 solutions are smaller than that of liquid solutions, therefore one would expect even much 53 54 smaller changes in activity coefficient in the solid solution at high pressures and hence 55 56 57 the assumption of independency of activity coefficient from pressure is plausible. 58 10 59 60 ACS Paragon Plus Environment Page 11 of 38 Industrial & Engineering Chemistry Research
1 2 3 Based on several investigations, (and as will be shown for modified Ji, Pauly et al. 12 , Nasrifar 4 5 and Fani Kheshty 19 and Ghanaei et al. 31 models) the performance of the methods in both 6 7 categories are comparatively acceptable for mixtures of low asymmetry with overall 8 9 10 compositions having a low amount of light ends. The efficiency of the aforesaid methods, 11 12 however, is poor in mixtures of high asymmetry which have high proportions of light ends, as 13 14 will be presented later on. The deviations become even more as the pressure increases. This 15 16 issue is addressed in some works 2 and seemingly has prevented the authors accurately 17 18 19 modelling the experimental data. In this work, the aim is to tackle the problem of wax phase 20 21 boundary estimation at higher pressure, especially for highly asymmetric systems, by 22 23 developing a new model. Therefore, the work presents two new solid liquid equilibrium high 24 25 pressure models based on “No Poynting term” approach i.e. (i) the modified Ji model, 26 27 described earlier and (ii) a new accurate scheme described in the next section. The reason 28 29 30 why two new methods are presented here will be discussed in the results section. 31 32 33 2.3. New proposed method 34 35 In the current study, a new method based on the “No Poynting term” approach is developed 36 37 38 to model the non ideality of paraffinic solid phases at high pressures. The aim here is to have 39 40 accurate estimations of thermophysical properties of pure paraffins at high pressure, using 41 42 their values in the reference state and a proper formulation to modify them to account for 43 44 high pressure effect. Prior to discussing the development of the model, to have a better 45 46 47 understanding the of solid fluid equilibrium behaviour of highly asymmetric systems, 48 49 investigations are first carried out for simple binary systems of high asymmetry, for which 50 51 experimental solid fluid phase boundary data are available. According to Seiler et al. 43 the 52 53 combination of Eq. 1 and Eq. 2 can be extended to high pressure range if the pressure 54 55 dependence of both enthalpies of fusion and the solid solid transition is taken into 56 57 58 11 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 12 of 38
1 2 3 consideration. Here, the Clausius Clapeyron equation 44 is used to take the dependency of 4 5 to pressure into consideration. Using Clausius Clapeyron equation, for the fusion: 6 7 Δ 8 9 ∗ (10) 10 11 ∆ = − 12 13 As mentioned, for n alkanes, with an accurate estimate, the fusion temperature changes 14 15 linearly with pressure and the slope is a constant value independent of pressure and the 16 17 18 carbon number of the pure paraffin. Therefore, using Eq. 10, one can easily relate the 19 20 enthalpy of fusion at high pressure to the same property in reference pressure by: 21 22 23 24 (11) ∆ − 25 = 26 ∆ − 27 Here, as described, the fusion temperature of pure alkane at reference pressure ( is 28 29 25 30 calculated by the correlations proposed by Coutinho and Daridon except for the cases of 31 32 binary asymmetric systems where the solid solution is a pure even normal alkane in which 33 34 27 35 case for the heavy alkane , is taken from the work of Dirand et al. . Eq. 7 with ∗ 36 37 of 4.5 MPa.K 1 as suggested by Ghanaei et al. 31 is used to calculate fusion temperature of pure 38 39 40 alkane at high pressure, . 41 42 12 43 As mentioned, based on Pauly et al. one can relate the molar volume of heavy alkanes in the 44 45 liquid state to the same value in the solid state by multiplying it with constant value, i.e.: 46 47 48 (12) 49 50 = 51 Where, according to Pauly et al. 12 , for pure alkanes is equal to 0.86 and is assumed to be 52 53 pressure independent. For mixtures, due to excess volume effect they have suggested the 54 55 56 value of 0.9 for . A constant, pressure independent value, assigned for is questionable as 57 58 12 59 60 ACS Paragon Plus Environment Page 13 of 38 Industrial & Engineering Chemistry Research
1 2 3 obviously the effect of compaction due to high pressure is less in the solid state compared to 4 5 liquid state. Hence, one would expect that by increasing the pressure the value of should 6 7 increase. Accordingly, in this work, is defined to be pressure dependent, hereafter denoted 8 9 10 as for fusion, assuming to increase linearly with pressure (in the simplest possible way) 11 12 i.e.: 13 14 15 (13) 16 17 = + − 18 Here is a positive constant. Despite the simplicity of Eq. 13, as will be presented later on, 19 20 the formulation devised proves very accurate. Using the data reported by Schaerer et al. 45 an 21 22 23 average value of 0.895 is assigned to which is representing the ratio of pure alkane 24 25 liquid state to solid state molar volume at reference pressure (very similar to Pauly et al. 12 26 27 value of ). In this way Eq. 11 can be reduced to: 28 29 30 31 (14) 32 ∆ 1 − = 33 ∆ 1 − 34 35 Using the same approach, however with a different variable named to make a distinction, 36 37 for alkanes showing order disorder solid solid transitions, the following formula can be 38 39 written to update at high pressures: 40 41 ∆ 42 43 (15) 44 ∆ 1 − 45 = ∆ 1 − 46 47 Again by using Schaerer et al. 45 data the value of 0.958 is assigned to which 48 49 represents the ratio of pure alkane disordered to ordered solid state molar volume at reference 50 51 pressure. Similar to fusion, as mentioned, the same trend in change of of a pure alkane 52 53 1 54 by pressure is observed, i.e. increasing with constant ∗ of 3.5 MPa.K as suggested by 55 56 57 58 13 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 14 of 38
1 2 3 Ghanaei et al. 31 . Therefore, Eq. 8 is used to evaluate . With the same approach 4 5 applied for fusion as presented in Eq. 13, for solid solid transition it is proposed that: 6 7 8 (16) 9 10 = + − 11 In theory, the value of in Eq. 16 should be different from that of Eq. 13. However, the aim 12 13 14 here is to have one adjustable parameter, and as will be shown later on, a single value of for 15 16 both of the equations can accurately model highly asymmetric systems. Having adjustable 17 18 parameters to model solid fluid equilibrium at high pressures is acceptable (see for example 19 20 Morawski et al. 30 and Rodriguez Reartes et al. 46 models for paraffinic binary systems). 21 22 23 To sum up in the current approach the following steps should be taken to calculate non 24 25 26 ideality in the solid phase at high pressure : 27 28 29 1 Using proper correlations/database the thermophysical properties of pure components 30 31 i.e. f, tr , f, tr are evaluated at the reference pressure . 32 Δ Δ 33 1 1 2 With values of 4.5 MPa.K for ∗ and 3.5 MPa.K for ∗ , Eq. 7 and Eq. 8 are used 34 35 36 f tr 37 to evaluate and , respectively, at high pressure . 38 39 3 Adjusting the value of and using values of 0.895 for and 0.958 for , 40 41 Eq. 13 and Eq. 16 are used to evaluate and , respectively, at high pressure . 42 43 Guidelines for assigning a correct value for will be presented in the results section. 44 45 f tr 46 4 Using (i) the values of and at reference pressure and their values at high 47 48 pressure , calculated in Step 2, (ii) the values of and at reference pressure 49 50 (i.e. 0.895 and 0.958) and their values at high pressures (calculated in Step 3) 51 52 f tr 53 and (iii) and at reference pressure , (calculated in Step 1), Eq. 14 and Eq. 54 Δ Δ 55 15 are used to evaluate f and tr , respectively at high pressure 56 57 Δ Δ . 58 14 59 60 ACS Paragon Plus Environment Page 15 of 38 Industrial & Engineering Chemistry Research
1 2 3 5 Having calculated the value of all thermophysical properties at high pressure P, Eq. 1 4 5 and Eq. 2 are directly used to evaluate the fugacity of components in the solid solution 6 7 at high pressure. 8 9 10 As described earlier, for the modified Ji method, steps 3 and 4, are not required. In fact, the 11 12 13 modified Ji method is a special form of the, more general, new model proposed here for 14 15 which: 16 17 18 (17) 19 20 ∆ = 1 21 And: ∆ 22 23 24 (18) 25 ∆ = 1 26 As both modified Ji and the new model∆ are proposed here, to make the distinction, hereafter, 27 28 29 the term “this work model” refers to the newly developed method, not the modified Ji. 30 31 32 3. Results and Discussions 33 34 35 In this section, the performance of the developed model is compared with the modified Ji, 36 12 19 31 37 Pauly et al. , Nasrifar and Fani Kheshty and Ghanaei et al. models. To have fair 38 39 evaluations of the models, the SRK+JMGC model and UNIQUAC activity coefficient model 40 41 are used to describe the non idealities of fluid and solid phase, respectively, for all high 42 43 44 pressure methods. The SRK+JMGC model used here to describe fluids is fully predictive 45 46 and no tuning of model parameters to match saturation pressure data is made prior to wax 47 48 calculations. It will be shown that the performance of all these existing models in systems 49 50 with lower proportions of the light end in moderate pressure ranges is good, provided the 51 52 requirement discussed earlier in the Background Section. Therefore, the main focus here is 53 54 55 on highly asymmetric systems of high proportions of the light end at high pressures, 56 57 resembling gas condensates and volatile oils, for which experimental wax phase boundary 58 15 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 16 of 38
1 2 3 data are available. Data on such systems are scarce. The uncertainty in the experimental wax 4 5 phase boundary data used here to evaluate models is low as they are corresponding to wax 6 7 disappearance points. It is well established that wax disappearance temperature (WDT) data 8 9 10 are better representatives of true thermodynamic melting point compared to wax appearance 11 47,48 12 temperature (WAT) . The evaluations are first carried out for binary asymmetric systems, 13 14 then synthetic multicomponent asymmetric wax mixtures and are graphically represented for 15 16 selected systems with low, moderate and high proportions of methane. Furthermore, 17 18 19 reporting model errors from low to high pressure range using Average Relative Error (ARE) 20 21 percent, i.e. by: can be misleading as the high temperature ranges 22 23 ∑ × 100 24 would result is small AREs regardless of the model utilized. Therefore, instead, the Average 25 26 Absolute Error (AAE) is used for the comparisons which is defined by ( : number of points): 27 28 29 30 (19) 31 1 % = T − T × 100 32 33 34 3.1. Binary methane + heavy alkane mixtures 35 36 37 Here, the models results for binary asymmetric systems (with experimental wax phase 38 39 boundary data available) methane + n hexadecane 49 , methane + n heptadecane 35 , methane + 40 41 n eicosane 39 , methane + n docosane 50 , methane + n tetracosane 51 and methane + n 42 43 52 44 triacontane are presented. A total of 457 data points of WDT in binary asymmetric mixtures 45 46 are used for evaluations. In this study the WDT data of asymmetric systems for which the 47 48 data are not reported at high pressures of at least 50 MPa (e.g. 53–58 for which the solid fluid 49 50 phase boundary data are reported up to 12 MPa), or the systems for which the uncertainty in 51 52 the critical/physical properties of the heavy end component is high (e.g. 59 ) are not used in 53 54 55 evaluations. In all the systems evaluated wax phase boundary data in low to very high 56 57 proportions of methane with a variety of molar ratios were measured. The graphical 58 16 59 60 ACS Paragon Plus Environment Page 17 of 38 Industrial & Engineering Chemistry Research
1 2 3 comparisons are shown for selected low, moderate and high proportions of methane in 4 5 example binary mixtures methane + n heptadecane, methane + n eicosane and methane + n 6 7 triacontane in Figure 1 to Figure 3. As the deviations of existing models are very high in the 8 9 10 case of systems methane + n triacontane, depicting evaluation results in all proportions in a 11 12 single graph makes interpretations difficult, therefore it was decided to show them separately 13 14 in the way presented in Figure 3. As the first major observation in the model evaluations 15 16 (and as observed in Figure 1 to Figure 3), for the binary systems tested, adjusting a single 17 18 19 value of can accurately model the wax phase boundary, in a fixed binary system regardless 20 21 of the proportion of the methane in the mixture. As an example, as shown in Figure 3, the 22 23 equal to 1.38×10 4 MPa 1 gives an accurate match in low to high proportions of methane in 24 25 binary mixture methane + n triacontane. The adjusted value of for each case is presented 26 27 28 in Table 1. As shown in this table, except for the case of methane + n hexadecane system the 29 30 value of have almost the same order of magnitude in all the binary system. The models 31 32 deviations are presented in Table 2 in terms of AAE for all the data points. According to this 33 34 table, the results of “this work model” with a single for each case in very low to very high 35 36 37 proportions of light end (methane) are accurate. The superiority of the proposed model is 38 39 clearer in binary systems of higher asymmetry (see results for methane + n tetracosane and 40 41 methane + n triacontane in Table 2 where the deviation of existing models are very high at 42 43 high pressures. Interestingly, even if the average value of over the binary systems i.e. 44 45 1.23×10 4 MPa 1 is used for all the cases, still the performance of the proposed method is 46 47 48 much better than the alternative methods. 49 50 51 52 53 54 55 56 57 58 17 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 18 of 38
1 2 3 100 4 5 6 80 7 8 9 80.09 mol% C1 Experimental 10 60 60.14 mol% C1 Experimental 11 12 20.21 mol% C1 Experimental P(MPa) 13 40 This work model 14 15 Modified Ji model 16 20 Pauly et al. model 17 Ghanaei et al. model 18 19 Nasrifar and Fani Kheshty model 0 20 285 295 305 315 325 21 22 T(K) 23 24 25 Figure 1: Binary methane + n heptadecane solid fluid phase boundary. The results for “this 26 4 1 27 work model” are shown by adjusted of 1.18×10 MPa 28 29 30 100 31 32 33 80 34 35 9.9 mol% nC20 Experimental 36 36.3 mol% nC20 Experimental 37 60 38 84.8 mol% nC20 Experimental 39 Experimental SLVE data P(MPa) 40 40 This work model 41 Modified Ji model 42 Pauly et al. model 43 20 44 Ghanaei et al. model 45 Nasrifar and Fani Kheshty model 46 0 47 300 305 310 315 320 325 330 335 340 345 350 48 T(K) 49 50 51 52 Figure 2: Binary methane + n eicosane solid fluid phase boundary. The results for “this 53 4 1 54 work” model are shown by adjusted of 1.34×10 MPa 55 56 57 58 18 59 60 ACS Paragon Plus Environment Page 19 of 38 Industrial & Engineering Chemistry Research
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Figure 3: Binary methane + n triacontane solid fluid phase boundary. The results for “this 22 23 4 1 24 work model” are shown by adjusted of 1.35×10 MPa . The evaluation are made for 25 26 systems with (a) 89.7 (b) 50 and (c) 15 mol% methane in the binary mixture. : 27 28 Experimental SLVE WDT data ○: Experimental SLE WDT data; ─── This work model; · 29 30 31 · Modified Ji model; Pauly et al. model; Nasrifar and Fani Kheshty model; 32 33 34 Ghanaei et al. model 35 36 37 It is important to note that the value of AAEs reported in Table 2 may seem to be at odds with 38 39 those reported for similar systems in Pauly et al. 12 work for C1 nC24 and C1 nC22 systems. 40 41 However, in their evaluations for these mixtures Pauly et al. 12 ignored the data points of more 42 43 44 than 90% methane in these mixtures which correspond to region of high deviation with 45 46 existing models, whereas, here the models are evaluated with systems (binary and 47 48 multicomponent) having as high as 97 mol% light end. That is why for these two mixtures 49 50 they have obtained smaller values of AAEs. If the same data points are used here similar 51 52 53 AAEs will be obtained. This choice may be due to higher deviations in modelling VLE of
54 7 55 asymmetric systems of the higher light end as pointed out by Jaubert and Mutelet . 56 57 58 19 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 20 of 38
1 2 3 Table 1: The adjusted values of parameter (MPa 1) for the binary asymmetric systems 4 5 investigated 6 7 8 Binary system C1 nC16 C1 nC17 C1 nC20 C1 nC22 C1 nC24 C1 nC30 9 10 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 11 12 Average 1.23×10 4 13 14 15 16 17 Table 2: Average Absolute Error (AAE) of wax phase boundary calculated by different 18 19 models compared to experimental values for binary asymmetric systems 20 21 22 AAE (K) 23 24 )
25 1 26
el el MPa 31 19
4 4
27
28 12 29 30 =1.23×10 adjusted)
31 Binary Mixture This work mod ( This work model ( ModifiedJi (Proposed here) Pauly et al. Ghanaeiet al. Fani Kheshty Pressure range (MPa) 32 Number data of points Nasrifar and 33 C1 nC30 115 3.189 86.80 1.07 1.17 2.76 3.95 3.02 3.00 34 C1 nC24 136 1.820 95.62 0.67 1.18 4.34 3.97 4.38 4.06 35 36 C1 nC22 13 4.520 97.68 0.19 0.72 5.75 5.32 6.79 6.24 37 38 C1 nC20 81 95.66 160.19 0.44 0.47 1.12 1.57 1.64 2.26 39 40 C1 nC17 49 10.11 257.50 0.53 0.54 1.16 1.65 1.74 1.87 41 C1 nC16 63 6.540 193.10 0.25 0.43 0.34 0.51 0.84 1.43 42 43 Total 457 Average 0.53 0.75 2.58 2.83 3.07 3.14 44 45 46 47 48 3.2. Multicomponent Mixtures 49 50 51 Similar to binary systems, here the evaluations are made only for systems of high asymmetry, 52 53 where the light end is methane, as all of the existing models perform fairly well for low 54 55 asymmetric systems. For multicomponent systems again it is observed that a constant value 56 57 of is capable of accurately representing the wax phase boundary in a system with fixed 58 20 59 60 ACS Paragon Plus Environment Page 21 of 38 Industrial & Engineering Chemistry Research
1 2 3 light and heavy ends, regardless of their proportions. This is shown for selected systems of 4 5 different low to high proportion of methane combined with (i) a ternary mixture of nC16 to 6 7 nC18 (experimental data from 60 ) in Figure 4 and (ii) multicomponent wax of nC13 to nC22 8 9 2 10 (wax2 with experimental data taken from ) presented in Figure 5. For all the systems tested, 11 12 the complete results are presented in Table 3. A set of 403 data points for 5 highly 13 14 asymmetric mixtures are used for evaluations. As it is shown in this table, by assigning a 15 16 special for each system, compared to other models, very small values of AAE would be 17 18 4 19 achieved. For all the data points for multicomponent systems, a common value of 0.95×10 20 21 is assigned to . Using this value, again very small values of overall AAE are achieved as 22 23 shown in Table 3. Furthermore, according to the results, similar to what was observed for 24 25 binary systems, the modified Ji model performs much better than the other models. This 26 27 28 signifies the superiority of the “No Poynting term” models over “Poynting term” models, 29 30 when thermophysical properties of heavy alkanes (most significantly, fusion and solid solid 31 32 transition temperatures) are correctly evaluated at high pressures. Among the two “No 33 34 Poynting term” methods proposed, the new approach in which the parameter α is introduced 35 36 f tr f tr 37 is outperforming the modified Ji method as all parameters , , , are calculated at 38 T T ΔH ΔH 39 higher pressure while in the “modified Ji” method only f, tr are calculated at higher 40 T T 41 pressure. 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 21 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 22 of 38
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Figure 4: Multicomponent methane + (nC16 nC18) solid phase boundary. The results for
23 4 1 24 “this work model” are shown by adjusted of 1.71×10 MPa for (a) 89.48 (b) 59.97 and (c) 25 26 19.9 mol% methane in the mixture. ○: Experimental WDT data; ─── This work model; · 27 28 29 · Modified Ji model; Pauly et al. model; Nasrifar and Fani Kheshty model; 30 31 32 Ghanaei et al. model 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 22 59 60 ACS Paragon Plus Environment Page 23 of 38 Industrial & Engineering Chemistry Research
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Figure 5: Multicomponent methane + (nC13 nC22) solid phase boundary (“wax2” from the 22 23 2 24 work of Pauly et al. . The results for “this work model” are shown by adjusted of 0.55×10 25 26 4 MPa 1 (a) 89.76 (b) 79.73 (c) 69. 95 mol% methane in the mixture. ○: Experimental WDT 27 28 29 data; ─── This work model; · · Modified Ji model; Pauly et al. model; Nasrifar 30 31 and Fani Kheshty model; Ghanaei et al. model 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 23 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 24 of 38
1 2 3 Table 3: Average Absolute Error (AAE) of wax phase boundary calculated different models 4 5 compared to experimental values for multicomponent asymmetric systems 6 7 8 AAE (K) 9 10 11 )
12 1
31
13
Mixture 12
(MPa
14 15 19
16 work** 17 This This work* This Modified Ji (Proposed here) Pauly et al. Ghanaei et al. Kheshty Pressurerange (MPa) Adjusted 18 Number of pointsdata Nasrifar and Fani 4.22 4 19 C1+(nC16 nC18) 60 41 1.71×10 0.34 1.09 1.81 5.03 2.66 3.19 20 95.94 21 4 50 119 96.96 1.36×10 1.32 2.22 5.29 8.51 6.82 6.82 22 C1+(nC22+nC24) 171.4 23 24 4 C1+(nC24+nC30)38 105 103.1 1.17×10 1.16 1.99 6.22 11.24 7.69 6.22 25 193.1 26 99 0.1 0.55×10 4 27 C1+(nC13 nC22) 2 1.10 1.44 1.15 1.67 1.69 1.72 28 98.59 29 4 C1+(Multi paraffin) 61 39 0.1 45 0.32×10 0.38 0.85 0.49 0.93 1.03 1.03 30 31 0.1 Average 0.86 1.68 2.99 5.48 3.99 3.80 Total 403 32 193.1 33 34 *Adjusted for each case 35 36 **Common equal to 0.95×10 4 MPa 1 37 38 3.3. Notes on the adjustable parameter 39 40 41 Comparing the results presented in Table 2 and Table 3, it is observed that a smaller common 42 43 value can be assigned for parameter in binary systems compared to multicomponent 44 45 46 systems. This difference can be attributed to the excess volume effects in solid solutions in 47 48 comparison to pure solids, which results in having non unity values for activity coefficients. 49 50 Yet the common values of are in the same order. Furthermore there is insufficient 51 52 published experimental data at very high pressures (in orders of 100 MPa) and for a diverse 53 54 55 range of asymmetry in the systems. Also, the experimental data used here for model 56 57 evaluations and adjusting for each system are measured by different methods (for the data 58 24 59 60 ACS Paragon Plus Environment Page 25 of 38 Industrial & Engineering Chemistry Research
1 2 3 presented here, visual and calorimetric techniques), coupled with the general uncertainty with 4 5 wax phase boundary data 62 . Accordingly, it was not possible to check if more intricate 6 7 functions can model the behaviour of parameters and at high pressure or if can be 8 9 10 defined as a normal alkane dependant parameter or as a function of the heaviness and/or 11 12 degree of asymmetry of the system. In absence of wax phase boundary data for systems of 13 14 low to moderate asymmetry, the modified Ji model, is suggested. Overall as a general 15 16 guideline, based on the characteristics of a mixture it is advised to assign a value for within 17 18 4 1 4 1 19 in the lower and upper boundaries of 0.3×10 MPa to 1.4×10 MPa with a common value 20 4 1 21 of 0.95×10 MPa as discussed earlier. The performance of this value is shown for an 22 23 example system of high asymmetry with about 95.96 mol% methane (mixture “wax3” in the 24 25 work of Pauly et al. 2) in Figure 6. The grey region in this figure corresponds to the upper and 26 27 28 lower wax boundary change using the minimum and maximum values of . Using these 29 30 values, at pressures as high as 100 MPa an absolute error of ±2.5 K can be observed, 31 32 depending on the value of specified. For this system the best value for is 0.64×10 4 MPa 33 34 1. However, using both “this work model” with common value of 0.95×10 4 MPa 1 and 35 36 37 modified Ji method, very similar AAE compared to experimental data is achieved, though 38 39 “this work model” performs better than modified Ji at higher pressures as presented in Figure 40 41 6 compared to “Poynting term” methods which have almost similar performances. This 42 43 figure, again confirms that between methods without adjustable parameters the method of 44 45 modified Ji is the one offering the highest accuracy. 46 47 48 49 50 51 52 53 54 55 56 57 58 25 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 26 of 38
1 2 3 4 5 80 6 7 8 60 Experimental WDT data 9 10 This work model (α=0.64E 4) 11 40 This work model (α=0.95E 4) 12 P(MPa) Modified Ji model 13 Ghanaei et al. model 14 20 15 Pauly et al. model 16 Nasrifar et al. model 17 0 18 285 295 305 315 19 T(K) 20 21 22 Figure 6: Wax phase boundary modelled with different methods for the “wax3” system of 23 24 2 25 Pauly et al. with 95.96 mol% methane. The grey region corresponds to the upper and lower 26 4 1 27 wax boundary change using the minimum and maximum values of , i.e. 0.3 ×10 MPa and 28 29 1.4 ×10 4 MPa 1, respectively. For this system the best value of is 0.64×10 4 MPa 1 30 31 32 The performance of modified Ji model is not only comparably good at wax phase boundary 33 34 modelling but also in measuring the wax content and composition. Due to scarcity of such 35 36 37 data at high pressure, the comparisons are only made for a single system of nC10+(nC18 38 39 nC30), though with much lower degree of asymmetry compared to the rest of the systems 40 41 investigated here, for which the experimental wax content and composition are measured by 42 43 Daridon and Dauphin 63 at 293.15 K and at different pressures as high as 50 MPa (see Figure 44 45 4 1 46 7). For this system, the best value of to match wax composition data is 0.35×10 MPa .
47 4 1 48 This small value (compared to the common value of 0.95×10 MPa ) for this system of 49 50 lower asymmetry points to the possible correlation of with the heaviness of the light end 51 52 or, generally speaking, degree of asymmetry of system. Devising such a correlation requires 53 54 sufficient experimental solid liquid equilibrium data to be obtained for a vast range of 55 56 57 systems with different asymmetry and proportions of light end. 58 26 59 60 ACS Paragon Plus Environment Page 27 of 38 Industrial & Engineering Chemistry Research
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Figure 7: Composition of solid wax evaluated with different models for nC10+(nC18 nC30) 51 52 mixture of Daridon and Dauphin 63 . The results for “this work model” are shown by adjusted 53 54 4 1 55 of 0.35×10 MPa 56 57 58 27 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 28 of 38
1 2 3 Now, it is justified why here two models were proposed. The new model, by comparison 4 5 with the modified Ji and other models evaluated here, is capable of accurately modelling the 6 7 behaviour of solid liquid equilibria in highly asymmetric systems at high pressure, though it 8 9 10 does have an adjustable parameter. Using a common value for this parameter the new model 11 12 is still superior in a variety of cases. However, in the absence of high pressure WDT data and 13 14 when the uncertainty in choosing value of adjustable parameter is high or when it is not 15 16 desirable to use a model with an adjustable parameter the modified Ji model is suggested. As 17 18 19 presented earlier, the modified Ji prediction of wax phase boundary as well as wax 20 21 composition was more accurate than all of the “Poynting term” methods evaluated here. 22 23 24 4. Conclusions and Suggestions 25 26 27 A new model for the most problematic systems in the area of solid liquid equilibrium 28 29 modelling of the paraffinic waxy mixture, i.e. highly asymmetric mixtures at high pressure 30 31 conditions, is developed. The model is based on the “No Poynting term” approach and is 32 33 34 shown to be capable of accurately capturing the wax phase boundary behaviour of such 35 36 systems with a single adjustable parameter which can be fixed regardless of the proportion of 37 38 light and heavy ends in a mixture of variable ratios of fixed light end and heavy end. The 39 40 value of the adjustable parameter can vary from one mixture to another. Therefore, using a 41 42 total of 860 data points for evaluation, common values for the adjustable parameter are 43 44 45 assigned. Due to lack of experimental data, it is not possible at present to correlate the value 46 47 of the adjustable parameter with the degree of asymmetry of the mixture or even assign a 48 49 more complex form of function to the ratio of molar volume of n alkanes in the solid state to 50 51 the same value in the liquid state (here a linear function was used, in the simplest form, with 52 53 54 the slope to be the adjustable parameter). Therefore it is suggested to measure more 55 56 experimental solid fluid experimental data in systems with a wide range of asymmetry. 57 58 28 59 60 ACS Paragon Plus Environment Page 29 of 38 Industrial & Engineering Chemistry Research
1 2 3 Using common values for the adjustable parameter, the model works very well at high 4 5 pressures. Meanwhile, if it is not desirable to use a model with an adjustable parameter 6 7 (when uncertainty in its value is high) the modified Ji model, in the way revised in the current 8 9 10 study, is suggested as it was clearly shown that the “No Poynting term” methods performance 11 12 are superior to that of “Poynting term” methods for such problematic systems. 13 14 15 AUTHOR INFORMATION 16 17 18 Corresponding Author 19 20 * Tel: +44(0)1314513797 E mail: [email protected] (Antonin Chapoy) 21 22 23 Author Contributions 24 25 The manuscript was written through contributions of all authors. All authors have given 26 27 28 approval to the final version of the manuscript. 29 30 31 Funding Sources 32 33 34 This work was funded by the Ali Danesh scholarship of Heriot Watt University provided for 35 36 the first author. 37 38 39 ACKNOWLEDGMENT 40 41 42 The Ali Danesh scholarship of Heriot Watt University provided for the first author is greatly 43 44 appreciated. 45 46 47 ABBREVIATIONS 48 49 SFE, Solid fluid equilibrium; WDT, Wax disappearance temperature; SRK, Soave Redlich 50 51 52 Kwong equation of state; PR, Peng Robinson equation of state; JMGC, Jaubert Mutelet 53 54 group contribution method; VLE, Vapour liquid equilibrium; LCVM, Linear combination of 55 56 Vidal and Michelsen mixing rules; MHV2, Second order modified Huron Vidal mixing rule; 57 58 29 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 30 of 38
1 2 3 UNIQUAC, Universal quasi chemical activity coefficient model; WAT, Wax appearance 4 5 temperature; AAE, Average absolute error; ARE, Average relative error; SLE, Solid liquid 6 7 equilibrium; SLVE, Solid liquid vapour equilibrium. 8 9 10 REFERENCES 11 12 13 (1) Pan, H.; Firoozabadi, A.; Fotland, P. Pressure and Composition Effect on Wax 14 15 16 Precipitation: Experimental Data and Model Results. SPE Prod. Facil. 1997 , 12 (4), 17 18 250–258. 19 20 21 (2) Pauly, J.; Coutinho, J. A. P.; Daridon, J. L. High Pressure Phase Equilibria in 22 23 Methane+waxy Systems. 3. Methane+a Synthetic Distribution of Paraffin Ranging 24 25 from N C13 to N C22. Fluid Phase Equilib. 2012 , 313 , 32–37. 26 27 28 (3) Daridon, J. L.; Pauly, J.; Coutinho, J. A. P.; Montel, F. Solid Liquid Vapor Phase 29 30 31 Boundary of a North Sea Waxy Crude: Measurement and Modeling. Energy and Fuels 32 33 2001 , 15 (3), 730–735. 34 35 36 (4) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics 37 38 of Fluid-Phase Equilibria ; Pearson Education, 1998. 39 40 41 (5) Ghanaei, E.; Esmaeilzadeh, F.; Fathikaljahi, J. Wax Formation from Paraffinic 42 43 Mixtures: A Simplified Thermodynamic Model Based on Sensitivity Analysis 44 45 46 Together with a New Modified Predictive UNIQUAC. Fuel 2012 , 99 , 235–244. 47 48 49 (6) Soave, G. Equilibrium Constants from a Modified Redlich Kwong Equation of State. 50 51 Chem. Eng. Sci. 1972 , 27 (6), 1197–1203. 52 53 54 (7) Jaubert, J. N.; Mutelet, F. VLE Predictions with the Peng Robinson Equation of State 55 56 and Temperature Dependent Kij Calculated through a Group Contribution Method. 57 58 30 59 60 ACS Paragon Plus Environment Page 31 of 38 Industrial & Engineering Chemistry Research
1 2 3 Fluid Phase Equilib. 2004 , 224 (2), 285–304. 4 5 6 (8) Qian, J. W.; Jaubert, J. N.; Privat, R. Prediction of the Phase Behavior of Alkene 7 8 Containing Binary Systems with the PPR78 Model. Fluid Phase Equilib. 2013 , 354 , 9 10 212–235. 11 12 13 (9) Peng, D. Y.; Robinson, D. B. A New Two Constant Equation of State. Ind. Eng. 14 15 16 Chem. Fundam. 1976 , 15 (1), 59–64. 17 18 19 (10) Jaubert, J. N.; Privat, R. Relationship between the Binary Interaction Parameters (Kij) 20 21 of the Peng Robinson and Those of the Soave Redlich Kwong Equations of State: 22 23 Application to the Definition of the PR2SRK Model. Fluid Phase Equilib. 2010 , 295 24 25 (1), 26–37. 26 27 28 (11) Boukouvalas, C.; Magoulas, K. G.; Stamataki, S. K.; Tassios, D. P. Prediction of 29 30 31 Vapor −l iquid Equilibria with the LCVM Model: Systems Containing Light Gases with 32 33 Medium and High Molecular Weight Compounds. Ind. Eng. Chem. Res. 1997 , 36 (12), 34 35 5454–5460. 36 37 38 (12) Pauly, J.; Daridon, J. L.; Coutinho, J. A. P.; Lindeloff, N.; Andersen, S. I. Prediction of 39 40 Solid Fluid Phase Diagrams of Light Gases Heavy Paraffin Systems up to 200 MPa 41 42 43 Using an Equation of State GE Model. Fluid Phase Equilib. 2000 , 167 (2), 145–159. 44 45 46 (13) Pauly, J.; Daridon, J. L.; Coutinho, J. A. P. Measurement and Prediction of 47 48 Temperature and Pressure Effect on Wax Content in a Partially Frozen Paraffinic 49 50 System. Fluid Phase Equilib. 2001 , 187-188 , 71–82. 51 52 53 (14) Pauly, J.; Daridon, J. L.; Sansot, J. M.; Coutinho, J. A. P. The Pressure Effect on the 54 55 Wax Formation in Diesel Fuel. Fuel 2003 , 82 (5), 595–601. 56 57 58 31 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 32 of 38
1 2 3 (15) Pauly, J.; Daridon, J. L.; Coutinho, J. A. P. Solid Deposition as a Function of 4 5 Temperature in the nC10 + (nC24 nC25 nC26) System. Fluid Phase Equilib. 2004 , 6 7 224 (2), 237–244. 8 9 10 (16) Coutinho, J. A. P.; Pauly, J.; Daridon, J. L. Modelling Phase Equilibria in Systems 11 12 13 with Organic Solid Solutions. Comput. Aided Prop. Estim. Process Prod. Des. 2004 , 14 15 229–249. 16 17 18 (17) Dahl, S.; Fredenslund, A.; Rasmussen, P. The MHV2 Model: A UNIFAC Based 19 20 Equation of State Model for Prediction of Gas Solubility and Vapor Liquid Equilibria 21 22 at Low and High Pressures. Ind. Eng. Chem. Res. 1991 , 30 (8), 1936–1945. 23 24 25 (18) Nasrifar, K.; Moshfeghian, M. Multiphase Equilibria of Waxy Systems with Predictive 26 27 28 Equations of State and a Solid Solution Model. Fluid Phase Equilib. 2012 , 314 , 60–68. 29 30 31 (19) Nasrifar, K.; Fani Kheshty, M. Effect of Pressure on the Solid Liquid Equilibria of 32 33 Synthetic Paraffin Mixtures Using Predictive Methods. Fluid Phase Equilib. 2011 , 310 34 35 (1 2), 111–119. 36 37 38 (20) Craig, S. R.; Hastie, G. P.; Roberts, K. J.; Gerson, A. R.; Sherwood, J. N.; Tack, R. D. 39 40 Investigation into the Structures of Binary , Tertiary and Quinternary Mixtures of N 41 42 43 Alkanes and Real Diesel Waxes Using High Resolution Synchrotron X Ray Powder 44 45 Diffraction. J. Mater. Chem. 1998 , 8 (4), 859–869. 46 47 48 (21) Dirand, M.; Chevallier, V.; Provost, E.; Bouroukba, M.; Petitjean, D. Multicomponent 49 50 Paraffin Waxes and Petroleum Solid Deposits: Structural and Thermodynamic State. 51 52 Fuel 1998 , 77 (12), 1253–1260. 53 54 55 (22) Chevallier, V.; Provost, E.; Bourdet, J. B.; Bouroukba, M.; Petitjean, D.; Dirand, M. 56 57 58 32 59 60 ACS Paragon Plus Environment Page 33 of 38 Industrial & Engineering Chemistry Research
1 2 3 Mixtures of Numerous Different N Alkanes: 1. Structural Studies by X Ray 4 5 Diffraction at Room Temperature Correlation between the Crystallographic Long c 6 7 Parameter and the Average Composition of Multi Alkane Phases. Polymer 1999 , 40 8 9 10 (8), 2121–2128. 11 12 13 (23) Chevallier, V.; Petitjean, D.; Bouroukba, M.; Dirand, M. Mixtures of Numerous 14 15 Different N Alkanes: 2. Studies by X Ray Diffraction and Differential Thermal 16 17 Analyses with Increasing Temperature. Polymer 1999 , 40 (8), 2129–2137. 18 19 20 (24) Coutinho, J. A. P.; Stenby, E. H. Phase Equilibria in Petroleum Fluids: Multiphase 21 22 Regions and Wax Formation, PhD Thesis. Technical University of Denmark, 1995. 23 24 25 (25) Coutinho, J. A. P.; Daridon, J. L. Low Pressure Modeling of Wax Formation in Crude 26 27 28 Oils. Energy and Fuels 2001 , 15 (6), 1454–1460. 29 30 31 (26) Coutinho, J. A. P.; Mirante, F.; Pauly, J. A New Predictive UNIQUAC for Modeling 32 33 of Wax Formation in Hydrocarbon Fluids. Fluid Phase Equilib. 2006 , 247 (1 2), 8–17. 34 35 36 (27) Dirand, M.; Bouroukba, M.; Briard, A. J.; Chevallier, V.; Petitjean, D.; Corriou, J. P. 37 38 Temperatures and Enthalpies of (Solid + Solid) and (Solid + Liquid) Transitions of N 39 40 Alkanes. J. Chem. Thermodyn. 2002 , 34 (8), 1255–1277. 41 42 43 (28) Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures : A 44 45 46 New Expression for the Excess Gibbs Energy of Partly or Completely Miscible 47 48 Systems. AIChE J. 1975 , 21 (1), 116–128. 49 50 51 (29) Coutinho, J. A. P. Predictive UNIQUAC: A New Model for the Description of 52 53 Multiphase Solid Liquid Equilibria in Complex Hydrocarbon Mixtures. Ind. Eng. 54 55 Chem. Res. 1998 , 37 (12), 4870–4875. 56 57 58 33 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 34 of 38
1 2 3 (30) Morawski, P.; Coutinho, J. A. P.; Domańska, U. High Pressure (Solid + Liquid) 4 5 Equilibria of N Alkane Mixtures: Experimental Results, Correlation and Prediction. 6 7 Fluid Phase Equilib. 2005 , 230 (1 2), 72–80. 8 9 10 (31) Ghanaei, E.; Esmaeilzadeh, F.; Fathikaljahi, J. High Pressure Phase Equilibrium of 11 12 13 Wax: A New Thermodynamic Model. Fuel 2014 , 117 , 900–909. 14 15 16 (32) Ghanaei, E.; Esmaeilzadeh, F.; Fathikaljahi, J. A New Predictive Thermodynamic 17 18 Model in the Wax Formation Phenomena at High Pressure Condition. Fluid Phase 19 20 Equilib. 2007 , 254 (1 2), 126–137. 21 22 23 (33) Milhet, M.; Pauly, J.; Coutinho, J. A. P.; Dirand, M.; Daridon, J. L. Liquid–solid 24 25 Equilibria under High Pressure of Tetradecane+pentadecane and 26 27 28 Tetradecane+hexadecane Binary Systems. Fluid Phase Equilib. 2005 , 235 (2), 173– 29 30 181. 31 32 33 (34) Tanaka, Y.; Kawakami, M. Solid Liquid Phase Equilibria in Binary (Benzene, 34 35 Cyclohexane + N Tetradecane, N Hexadecane) Systems at Temperatures 230–323 K 36 37 and Pressures up to 120 MPa. Fluid Phase Equilib. 1996 , 125 (1 2), 103–114. 38 39 40 (35) Pauly, J.; Coutinho, J. A. P.; Daridon, J. L. High Pressure Phase Equilibria in Methane 41 42 43 + Waxy Systems. 1. Methane + Heptadecane. Fluid Phase Equilib. 2007 , 255 (2), 44 45 193–199. 46 47 48 (36) Machado, J. J. B.; de Loos, T. W.; Christian Ihmels, E.; Fischer, K.; Gmehling, J. High 49 50 Pressure Solid Solid and Solid Liquid Transition Data for Long Chain Alkanes. J. 51 52 Chem. Thermodyn. 2008 , 40 (12), 1632–1637. 53 54 55 (37) Machado, J. J. B.; de Loos, T. W.; Ihmels, E. C.; Fischer, K.; Gmehling, J. Erratum to 56 57 58 34 59 60 ACS Paragon Plus Environment Page 35 of 38 Industrial & Engineering Chemistry Research
1 2 3 “High Pressure Solid–solid and Solid–liquid Transition Data for Long Chain Alkanes 4 5 [J. Chem. Thermodyn. 40 (2008) 1632–1637].” J. Chem. Thermodyn. 2009 , 41 (5), 6 7 716. 8 9 10 (38) Machado, J. J. B.; de Loos, T. W. High Pressure Solid Fluid and Vapour Liquid 11 12 13 Equilibria in Model Hyperbaric Fluids: The System Methane + Tetracosane + 14 15 Triacontane. Fluid Phase Equilib. 2004 , 226 (1 2), 83–90. 16 17 18 (39) van der Kooi, H. J.; Flöter, E.; Loos, T. W. d. High Pressure Phase Equilibria of 19 20 {(1−x)CH4+xCH3(CH2)18CH3}. J. Chem. Thermodyn. 1995 , 27 (8), 847–861. 21 22 23 (40) Ji, H.; Tohidi, B.; Danesh, A.; Todd, A. C. Wax Phase Equilibria: Developing a 24 25 Thermodynamic Model Using a Systematic Approach. Fluid Phase Equilib. 2004 , 216 26 27 28 (2), 201–217. 29 30 31 (41) Aucejo, A.; Burguet, M. C.; Munoz, R.; Marques, J. L. Densities, Viscosities, and 32 33 Refractive Indices of Some N Alkane Binary Liquid Systems. J. Chem. Eng. Data 34 35 1995 , 40 (1), 141–147. 36 37 38 (42) Dzida, M.; Cempa, M. Thermodynamic and Acoustic Properties of 39 40 (Heptane+dodecane) Mixtures under Elevated Pressures. J. Chem. Thermodyn. 2008 , 41 42 43 40 (10), 1531–1541. 44 45 46 (43) Seiler, M.; Gross, J.; Bungert, B.; Sadowski, G.; Arlt, W. Modeling of Solid/fluid 47 48 Phase Equilibria in Multicomponent Systems at High Pressure. Chem. Eng. Technol. 49 50 2001 , 24 (6), 607–612. 51 52 53 (44) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering 54 55 Thermodynamics , 5th ed.; McGraw Hill: New York, 1996. 56 57 58 35 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 36 of 38
1 2 3 (45) Schaerer, A. A.; Busso, C. J.; Smith, a. E.; Skinner, L. B. Properties of Pure Normal 4 5 Alkanes in the C17 to C36 Range. J. Am. Chem. Soc. 1955 , 77 (7), 2017–2019. 6 7 8 (46) Rodriguez Reartes, S. B.; Cismondi, M.; Zabaloy, M. S. Modeling Approach for the 9 10 High Pressure Solid Fluid Equilibrium of Asymmetric Systems. Ind. Eng. Chem. Res. 11 12 13 2011 , 50 (5), 3049–3059. 14 15 16 (47) Bhat, N. V.; Mehrotra, A. K. Measurement and Prediction of the Phase Behavior of 17 18 Wax Solvent Mixtures : Significance of the Wax Dis Appearance Temperature. Ind. 19 20 Eng. Chem. Res. 2004 , 43 , 3451–3461. 21 22 23 (48) Hammami, A.; Ratulowski, J.; Coutinho, J. A. P. Cloud Points: Can We Measure or 24 25 Model Them? Pet. Sci. Technol. 2003 , 21 (3 4), 345–358. 26 27 28 (49) Glaser, M.; Peters, C. J.; van der Kooi, H. J.; Lichtenthaler, R. N. Phase Equilibria of ( 29 30 31 Methane +n Hexadecane) and (p,Vm,T) of N Hexadecane. J. Chem. Thermodyn. 1985 , 32 33 17 , 803–815. 34 35 36 (50) Flöter, E.; Hollanders, B.; de Loos, T. W.; de Swaan Arons, J. The Effect of the 37 38 Addition of Water, Propane, or Docosane on the Vapour–liquid and Solid–fluid 39 40 Equilibria in Asymmetric Binary N Alkane Mixtures. Fluid Phase Equilib. 1998 , 143 41 42 43 (1 2), 185–203. 44 45 46 (51) Flöter, E.; de Loos, T. W.; de Swaan Arons, J. High Pressure Solid Fluid and Vapour 47 48 Liquid Equilibria in the System (Methane + Tetracosane). Fluid Phase Equilib. 1997 , 49 50 127 (1 2), 129–146. 51 52 53 (52) Machado, J. J. B.; de Loos, T. W. Liquid Vapour and Solid Fluid Equilibria for the 54 55 System Methane + Triacontane at High Temperature and High Pressure. Fluid Phase 56 57 58 36 59 60 ACS Paragon Plus Environment Page 37 of 38 Industrial & Engineering Chemistry Research
1 2 3 Equilib. 2004 , 222-223 , 261–267. 4 5 6 (53) Rodrigues, A. B.; Kohn, J. P. Three Phase Equilibria in the Binary Systems Ethane N 7 8 Docosane and Ethane N Octacosane. J. Chem. Eng. Data 1967 , 12 (2), 191–193. 9 10 11 (54) Puri, S.; Kohn, J. P. Solid Liquid Vapor Equilibrium in the Methane N Eicosane and 12 13 Ethane N Eicosane Binary Systems. J. Chem. Eng. Data 1970 , 15 (3), 372–374. 14 15 16 17 (55) Estrera, S. S.; Luks, K. D. Liquid Liquid Vapor Equilibria Behavior of Certain 18 19 Ethane+ N Paraffin Mixtures. J. Chem. Eng. Data 1987 , 32 (2), 201–204. 20 21 22 (56) Peters, C. J.; Spiegelaar, J.; de Swaan Arons, J. Phase Equilibria in Binary Mixtures of 23 24 Ethane+ Docosane and Molar Volumes of Liquid Docosane. Fluid Phase Equilib. 25 26 1988 , 41 (3), 245–256. 27 28 29 (57) Peters, C. J.; De Roo, J. L.; Lichtenthaler, R. N. Measurements and Calculations of 30 31 Phase Equilibria in Binary Mixtures of Ethane+ Eicosane: Part 3. Three Phase 32 33 34 Equilibria. Fluid Phase Equilib. 1991 , 69 , 135–143. 35 36 37 (58) Peters, C. J.; De Roo, J. L.; de Swaan Arons, J. Measurements and Calculations of 38 39 Phase Equilibria in Binary Mixtures of Propane+ Tetratriacontane. Fluid Phase 40 41 Equilib. 1992 , 72 , 251–266. 42 43 44 (59) Gregorowicz, J. Phase Behaviour in the Vicinity of the Three Phase Solid Liquid 45 46 Vapour Line in Asymmetric Nonpolar Systems at High Pressures. Fluid Phase 47 48 49 Equilib. 2006 , 240 (1), 29–39. 50 51 52 (60) Pauly, J.; Coutinho, J. A. P.; Daridon, J. L. High Pressure Phase Equilibria in 53 54 Methane+waxy Systems. 2. Methane+waxy Ternary Mixture. Fluid Phase Equilib. 55 56 2010 , 297 (1), 149–153. 57 58 37 59 60 ACS Paragon Plus Environment Industrial & Engineering Chemistry Research Page 38 of 38
1 2 3 (61) Daridon, J. L.; Xans, P.; Montel, F. Phase Boundary Measurement on a Methane + 4 5 Decane + Multi Paraffins System. Fluid Phase Equilib. 1996 , 117 (1 2), 241–248. 6 7 8 (62) Coutinho, J. A. P.; Daridon, J. L. The Limitations of the Cloud Point Measurement 9 10 Techniques and the Influence of the Oil Composition on Its Detection. Pet. Sci. 11 12 13 Technol. 2005 , 23 (9 10), 1113–1128. 14 15 16 (63) Daridon, J. L.; Dauphin, C. Measurement of Pressure Effect on Wax Content in 17 18 Partially Frozen Paraffinic Systems. Meas. Sci. Technol. 1999 , 10 (12), 1309–1314. 19 20 21 TOC Graphic 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 38 59 60 ACS Paragon Plus Environment