Fluid Phase Equilibria 216 (2004) 201–217
Wax phase equilibria: developing a thermodynamic model using a systematic approach Hong-Yan Ji, Bahman Tohidi∗ , Ali Danesh, Adrian C. Todd Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh EH14 4AS, UK Accepted 28 May 2003
Abstract Reservoir hydrocarbon fluids contain heavy paraffins that may form solid phases of wax at low temperatures. Problems associated with wax formation and deposition are a major concern in production and transportation of hydrocarbon fluids. The industry has directed considerable efforts towards generating reliable experimental data and developing thermodynamic models for estimating the wax phase boundary. The cloud point temperature, i.e. the wax appearance temperature (WAT) is commonly measured in laboratories and traditionally used in developing and/or validating wax models. However, the WAT is not necessarily an equilibrium point, and its value can depend on experimental procedures. Furthermore, when determining the wax phase boundary at pipeline conditions, the common practice is to measure the wax phase boundary at atmospheric pressure, then apply the results to real pipeline pressure conditions. However, neglecting the effect of pressure and associated fluid thermophysical/compositional changes can lead to unreliable results. In this paper, a new thermodynamic model for wax is proposed and validated against wax disappearance temperature (WDT) data for a number of binary and multi-component systems. The required thermodynamic properties of pure n-paraffins are first estimated, and then a new approach for describing wax solids, based on the UNIQUAC equation, is described. Finally, the impact of pressure on wax phase equilibria is addressed. The newly developed model demonstrates good reliability for describing solids behaviour in hydrocarbon systems. Furthermore, the model is capable of predicting the amount of wax precipitated and its composition. The predictions compare well with independent experimental data, demonstrating the reliability of the thermodynamic approach. © 2003 Elsevier B.V. All rights reserved. Keywords: Wax; Solid-fluid equilibria; Equation of state; Model; Paraffin; Pressure
1. Introduction Petroleum fluids contain heavy paraffins that may form solid wax phases at low temperatures. Problems caused by wax precipitation, such as decreased production rates, increased power requirements, and failure of facilities, are a major concern in the production and transportation of hydrocarbon fluids. Techniques such as thermal treatment of pipelines, addition of chemical inhibitors, and/or pigging are commonly used to prevent wax accumulation. The costs associated with such measures could be reduced significantly if accurate means to predict the wax precipitation region were available. Therefore, it is crucial to develop reliable experimental techniques and/or predictive tools for determin∗ Corresponding author. Tel.: +44-131-451-3672; fax: +44-131-451-3127. E-mail address:
[email protected] (B. Tohidi).
0378-3812/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2003.05.011
ing wax equilibria. The industry has directed considerable efforts towards this goal over the past few decades. The cloud point temperature, or wax appearance temperature (WAT), where wax is first detected on cooling, is commonly measured in laboratories. However, WAT is not necessarily an equilibrium point, and its value depends on test procedures. Experimental studies conducted in this laboratory show that WAT is commonly a strong function of cooling rate; faster cooling rates often leading to a lower measured WAT. Furthermore, the measurement of WAT is significantly affected by the detection techniques employed. For example, WAT measured using visual microscopy can be 10–20 ◦ C higher than those determined using techniques such as differential scanning calorimetry, laser-based solids detection systems, and viscometry [1,2]. In contrast to WAT, the wax disappearance temperature (WDT) represents a true solid–liquid equilibrium (SLE) point. Accuracy of measured WDT is dependent on
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experimental techniques. By using a reliable experimental method, such as equilibrium step heating, the WDT measured by different laboratories should agree within a reasonable experimental error band. The difference between measured WDT and WAT can be very significant. For example, Ronningsen et al. [1], using microscopy, determined the WDT of a North Sea crude oil to be 28 ◦ C higher than the WAT measured using the same technique [1]. Therefore, WDT should be used instead of WAT for developing, tuning and validation of wax thermodynamic models. A thermodynamic model tuned and/or validated using WAT is subject to serious questions concerning reliability. Unfortunately, the majority of existing wax models are based on WAT, and as a result, errors in the order of 20 ◦ C between predicted and real wax phase boundaries are common [3]. The aim of this work is to develop and validate a new wax thermodynamic model that is based upon reliable experimental WDT data, generated both in this laboratory and reported in the literature. We first review some popular existing wax models, and then present our new wax model and its improvements. Predictions of this newly developed model are then validated against independent experimental data for wax phase boundaries, as well as the amount and composition of wax precipitated under different pressure conditions.
2. Review of existing wax thermodynamic models Several predictive thermodynamic models for wax have been presented in the literature. Some of the more popular models are detailed below. 2.1. Won [4,5] In 1986, Won presented early efforts to use a thermodynamic model for predicting wax phase boundaries [4]. The Soave–Redlich–Kwong (SRK) equation of state (EoS) was used for vapour–liquid equilibrium (VLE) calculations. A modified regular solution approach was employed for solid–liquid equilibrium (SLE) calculations, where activity coefficients were calculated using solubility parameters of individual components. The critical temperature, critical pressure and acentric factor were estimated using correlations suggested by Spencer and Daubert [6], Lydersen [7], and Lee–Kesler [8], respectively. The fusion temperature and heat of fusion were correlated to molecular weight using experimental data predominantly for pure n-paraffins with odd carbon numbers. In 1989, Won modified the model by using an approach that combined the modified regular solution with the equation of Flory–Huggins [9–12] for calculating activity coefficients in the liquid phase [5]. The wax model proposed by Won [4,5] was validated against cloud point temperatures measured for synthetic fuels, diesel fuels, and North Sea gas condensates. Many other
researchers adopted the model suggested by Won [4,5], sometimes without any modification, when developing their own model. However, there are several major shortcomings in the model proposed by Won [4,5] that limit its capability and reliability for predicting wax phase boundaries. Firstly, two different approaches are applied to the liquid phase for VLE and SLE; an EoS is used for VLE, while an activity coefficient model is applied to SLE. This leads to inconsistency in description of the liquid phase, and very often results in convergence issues. A further problem is that the modified regular solution approach used for describing wax solids does not vary greatly from the ideal solid solution approach, due to the similarity of the solubility parameters for n-paraffins. Both these approaches lead to overestimation of wax phase boundary temperatures. In addition, the model cannot provide reliable predictions of wax phase boundaries at high-pressure conditions, as the effect of pressure on wax equilibria is ignored. Finally the model is based on WAT data, which, as discussed, are not reliable for tuning and/or validating a model. 2.2. Hansen et al. [13] In 1988, Hansen et al. [13] presented a wax model that uses the SRK EoS for VLE calculations, with the ideal solid solution approach applied to the solid phase, and a polymer solution approach applied to the liquid phase for SLE. As parameters required in the polymer solution approach were determined by fitting to the measured cloud point temperatures (WAT) for 13 North Sea crude oils, it was not surprising that predicted WATs were in good agreement with measured WAT data for the same North Sea crude oils. The model proposed by Hansen et al. [13] has similar limitations to that of Won [4,5]. Furthermore, the polymer solution approach used by authors leads to activity coefficients in the liquid phase of the order of 10−10 [14,15], which does not correspond with reality [15]. 2.3. Pedersen et al. [14,15] In 1991, Pedersen et al. [14] presented a wax model based on modifications to the approach of Won [4]. A modified regular solution approach was applied to both the liquid and solid phases. Fusion properties and heat capacities for pure compounds were tuned to fit measured wax precipitation data for the North Sea oils. The model was validated using experimental WAT data for the North Sea oils. In 1995, Pedersen [15] further modified this model, employing a cubic equation of state for consistency in description of the liquid phase for VLE and SLE calculations. The ideal solid solution approach was applied to the solid phase. Fusion properties were calculated using correlations suggested by Won [4]. A problem with the model of Pedersen et al. [14] is that it uses unreliable values for fusion properties and heat
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capacity. The approaches used to describe wax solids in the models proposed in 1991 [14] and 1995 [15] (i.e. the approaches of regular solid solution and ideal solid solution), led to an overestimation of wax phase boundary temperatures. Again, models are flawed in that they are based upon non-equilibrium WAT data. 2.4. Erickson et al. [16] The model proposed by Erickson et al. in 1993 [16] was a modification of that of Won [4]. The ideal solution approach was applied to SLE calculations. Heat of fusion for pure compounds was tuned against experimental SLE data for binary mixtures. The proposed model was validated against experimental WAT data for crude oil and condensate samples. The model proposed by Erickson et al. [16] has similar limitations to that of Won [4]. 2.5. LiraGaleana et al. [17] In 1996, LiraGaleana et al. presented a wax thermodynamic model in which a multi-pure-solid approach was used for description of wax solids [17]. This approach assumed wax solids consisting of multiple solid phases, and each solid phase was a pure compound. The PR EoS was used for calculating fugacity in the liquid and vapour phases. Critical properties and the acentric factors were estimated using correlations suggested by Cavett [18]. The model was validated using experimental SLE data for binaries, and measured cloud point temperatures (WAT) for the North Sea crude oils. Studies on crystal structure in recent years reveal that the miscibility of n-paraffins in a solid state depends strongly on differences in molecular sizes (i.e. carbon number). An n-paraffin mixture with a significant carbon number difference (e.g. nC30 –nC36 ) appears to form eutectic solids [19], whereas an n-paraffin mixture with a consecutive carbon number distribution forms a single orthorhombic solid solution [20]. Thus assumptions of the multi-pure-solid approach are not consistent with real wax crystal behaviour. Therefore, the model proposed by LiraGaleana et al. [17] is of questionable reliability for systems consisting of compounds with similar molecular sizes. 2.6. Coutinho et al. [21–23] In 1995 and 1996, Coutinho et al. evaluated several approaches for calculating activity coefficients in SLE, including the Flory–Huggins, Universal Functional Group Activity Coefficient (UNIFAC), Flory free-volume, and entropic free-volume. In 1998, Coutinho [23] presented a wax thermodynamic model which used a combined UNIFAC and Flory free-volume approach to describe the liquid phase, with the universal quasi-chemical (UNIQUAC) equation being used to describe wax solids. In this UNIQUAC approach, the characteristic energy, uij , for calculating the
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adjustable binary parameter, τ ij , with Eq. (14), is expressed using λij and λii as below: uij =
z (λij − λii ) 2qi
where qi is the external surface area parameter of pure compounds. λii is calculated using enthalpy of sublimation for component i as follows: 2 λii = − (Hsubl,i − RT) z where z is the co-ordination number (set to 6 by the authors). λij is given by λkk as below, where k designates the smaller n-alkane of the pair ij: λij = λkk Using the above correlations, when k is i, uij is zero, and uji has a nonzero value. The model proposed by Coutinho [23] was validated using experimental data for the amount and composition of wax precipitated for mixtures. In 2000, Pauly et al. [24] modified the model of Coutinho [23] by using SRK EoS–GE for description of liquid and vapour phases. GE was obtained using a modified UNIFAC equation. Critical properties were estimated using correlations proposed by Twu [25]. The Poynting correction term was used to extend the model to high-pressure conditions. Partial molar volumes required for calculating the Poynting correction were estimated in accordance with crystallographic studies of n-paraffin solids. The model was validated using experimental WDT data for n-paraffin mixtures. Hydrocarbon solids present a positive deviation from the ideal solid solution, as shown later in this work. This positive deviation can be described using the UNIQUAC equation. The accuracy of this equation depends on parameters defined for the equation. Continho used thermodynamic properties to calculate binary parameters. Examining the model proposed by Coutinho [23] (as presented later in this work) shows that Coutinho’s UNIQUAC approach lacks reliability for mixtures containing molecules of similar sizes. 2.7. New wax model proposal Produced reservoir hydrocarbon fluids at pipeline conditions commonly consist of liquid and vapour phases. Vapour–liquid equilibrium is commonly calculated using a cubic equation of state. In order to ensure consistency in description of the liquid phase, the wax thermodynamic model must also use a cubic equation of state for calculating fugacity in the liquid phase for SLE. SRK EoS and PR EoS are popular for calculating fugacities in vapour–liquid systems. These EoS are therefore suitable choices for description of fluid phases in the wax model. When describing a compound using an equation of state, values for critical temperature, critical pressure, and acentric factor are required. Reliable experimental data for these parameters are available for n-paraffins up to C20 [26,27].
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However, it is almost impossible to measure directly critical properties for longer-chain n-paraffins due to thermal decomposition at high temperatures. As a result, different correlations have been suggested in the literature for estimation of critical properties and acentric factors for these compounds. However, estimated values using these correlations can differ considerably, potentially having a negative effect on the reliability of predicted wax equilibria. Considering the limitations of existing wax models as discussed, we propose here a new, more reliable model, Heriot-Watt WAX (HWWAX). Firstly, we present improved correlations for the estimation of fusion properties. Then, the most suitable correlations for calculating Tc and Pc are selected from those in the literature, while new correlations for calculating coefficients of temperature dependency functions for attraction terms in the SRK EoS and PR EoS are suggested for improving long chain n-paraffin fugacity calculations. Following this, a new approach for describing wax solids is developed. Finally, the effect of pressure on wax phase equilibria is addressed, and a method is proposed for extension of the developed wax model to high-pressure conditions based on measurements made at atmospheric pressure. 3. Thermodynamic modelling of wax phase equilibria To calculate solid–liquid equilibria, it is necessary to fulfil equality of fugacity for each component in both the solid and liquid phases. The fugacity of component i in the liquid phase is calculated using a cubic equation of state, expressed as follows: fiL = xi PϕiL
(1)
fiL
is the fugacity of component i in the liquid phase, where ϕiL is the fugacity coefficient, which can be calculated from an EoS, xi is the mole fraction of the component, and P is the system pressure. The fugacity (fiS ) of component i in the solid phase at system conditions is related to pure solid fugacity (fiOS ) at a reference pressure (PO ), based on solid solution theory, as expressed in Eq. (2): P v ¯ Si S S OS dP (2) fi = si γi f exp PO RT where si and γiS are the component mole fraction and activity coefficient in the solid phase, respectively. fiOS is calculated from pure liquid fugacity at the same temperature. As some n-paraffins exhibit solid–solid transitions before melting, the fugacity ratio of subcooled liquid versus solid can be calculated using Eq. (3): f OL Hf,i T T Htr,i ln iOS = + 1− 1− RT Ttr,i RT Tf,i fi T T CLS 1 1 p,i LS + Cp,i dT − dT (3) RT Tf,i R Tf,i T
where Ttr,i and Tf,i are the solid–solid transition temperature and fusion temperature of component i, respectively. Htr,i and Hf,i are latent heats of the solid–solid and solid–liquid LS is the heat capacity differtransitions, respectively. Cp,i ence between liquid and solid (heat capacity differences between the two solid forms are ignored). LS L S Cp,i = Cp,i − Cp,i
(4)
The pure liquid fugacity of compound i (fiOL in Eq. (3)) can be calculated using a cubic equation of state. 4. Fusion properties and heat capacity As shown in Eq. (3), fusion properties and heat capacity are required for calculating the fugacity of solids. Accuracy of values for these properties is vital for developing a reliable wax model. Experimental data show that fusion properties and heat capacities for n-paraffins are dependent not only on the carbon chain length, but also on whether carbon numbers are odd or even. This has been considered in the HWWAX model when developing correlations for fusion properties and heat capacity. In the HWWAX model, properties for both solid–solid (Ttr and Htr ) and solid–liquid (Tf and Hf ) transitions (at 0.1 MPa) have been regressed into correlations using available experimental data for pure n-paraffins up to C70 [28,29]. A third-order polynomial function has been developed to represent transition temperature as a function of carbon number, acknowledging the difference between odd and even carbon numbers. Latent heats for transitions have been correlated with the product of molecular weight and fusion temperature using a linear function. Several correlations have been developed to calculate heat capacity for solid n-paraffins of odd or even carbon numbers (as a function of temperature and carbon number) using available experimental data [30–32]. A single correlation as a function of temperature and carbon number has been developed for calculating the heat capacity of liquid n-paraffins. Correlations for both fusion properties and heat capacity are detailed in Appendix A of this paper. 5. Improving equations of state for calculating long chain paraffin fugacities Five sets of empirical correlations reported in the literature and widely applied to hydrocarbons for estimating Tc and Pc , have been evaluated. These include the correlations of Ambrose [33], Twu [25], Teja et al. [34], Constantinou and Gani [35], Riazi and Al-Sahhaf [36]. The critical volume is calculated according to: Zc RTc Vc = (5) Pc where the critical compressibility factor (Zc ) is a constant in both SRK EoS and PR EoS.
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Acentric factors (ω) for n-paraffins heavier than C20 are calculated using the correlation proposed by Lee–Kesler [8]. In the original SRK EoS and PR EoS, a second-order polynomial function based on the data of pure compounds up to C10 was used for correlating m with ω, which was required for calculating the attraction term in the equation of state. Direct extrapolation of m versus ω functions far beyond the acentric factor range in which those correlations were generated can impair the reliability of the equation of state. In this work, the m versus ω functions for SRK EoS and PR EoS, are extended to long chain n-paraffins using optimised m values in conjunction with calculated ω values. The objective function for the optimisation procedure is as follows: F=
whilst the SLE binaries consist of a low molecular weight n-paraffin (C5 –C12 ) and a long chain n-paraffin (C22 –C36 ) [41–46]. In the latter case, due to the large differences in the molecular sizes of the hydrocarbons, a pure solid may form. Therefore, the multi-pure-solid approach for describing wax solids can be used for the optimisation of m. Binary interaction parameters are then optimised by matching the experimental SLE data for binaries [41–46], using Tc and Pc calculated with the most suitable empirical correlations. When optimising binary interaction parameters, the multi-pure-solid approach for describing wax solids has been used for the same reason detailed in the above. Values of critical temperature, critical pressure and the new m versus ω functions suitable for paraffins from C1 up to at least C36 , as well as binary interaction parameters, have been determined for SRK and PR EoS, as presented below:
1 n1 + n 2 n n2 1 Pb,exp − Pb,cal WDTexp − WDTcal + × P WDT i=1
b,exp
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5.1. SRK EoS
exp
i=1
When using SRK EoS, correlations suggested by Riazi and Al-Sahhaf [36] are suitable for calculating critical temperature and pressure, leading to consistency between optimised m values for n-paraffins above C20 and below C20 , as shown in Fig. 1. Using the above data, a fourth order polynomial function for correlating m with ω in SRK EoS was developed, as presented in Eq. (7).
(6) where Pb is the bubble point pressure and WDT is the wax disappearance temperature for binaries. The subscript “exp” designates experimental data, and “cal” designates calculated data. n1 and n2 are the number for Pb data and WDT data, respectively. The m values for long chain n-paraffins have been optimised using estimated data of Tc and Pc with each set of empirical correlations. According to the consistency test of optimised m values, the most suitable empirical correlations for estimating Tc and Pc have been selected. Both VLE and SLE binary data were used for optimising m values, with binary interaction parameters set to zero. The VLE binaries consist of C2 or C3 and a n-paraffin heavier than C20 [37–40],
m = 0.4806 + 1.7137ω − 0.9207ω2 + 0.9620ω3 − 0.2595ω4
(7)
Comparison of the m versus ω plot using Eq. (7) with that using the original SRK second order polynomial function is also shown in Fig. 1. As shown in Fig. 2, binary interaction parameters optimised using SLE data for binaries are
5.0 Cn < C20: the original SRK 2rd order polynomial function Cn > C20: the original SRK 2rd order polynomial function Cn > C20: optimized m values with SRK EoS, this work C1 - C60: the 4th order polynomial function for SRK EoS, this work
4.5 4.0
m value
3.5 C36
3.0 C32 2.5
C22
C24
C20
2.0 1.5 1.0 0.5 0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
acentric factor Fig. 1. Relationship between m values and acentric factor (ω) in SRK EoS (Tc and Pc calculated using correlations suggested by Riazi and Al-Sahhaf [36] and ω calculated using the correlation suggested by Lee–Kesler [8] for n-paraffins above C20 ).
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0.00
Binary interaction parameter
-0.01 -0.02 -0.03 -0.04 -0.05
The heavier component: C22
-0.06
The heavier component: C24
-0.07
The heavier component: C28
-0.08
The heavier component: C32 The heavier component: C36
-0.09 -0.10 4
5
6
7
8
9
10
11
12
13
14
Cn of the lighter component in binary systems Fig. 2. Binary interaction parameter optimised using SRK EoS and SLE data in binaries (Cn : carbon number).
developing the following function for correlating m with ω:
almost constant and independent of the combination (pairing) of compounds when using SRK EoS. The value of binary interaction parameter is approximately −0.02 for all the binaries investigated.
m = 0.3748 + 1.5932ω − 0.5706ω2 + 0.3968ω3 − 0.092ω4
(8)
A comparison of the m versus ω plot using Eq. (8) with that using the original PR second-order polynomial function is shown in Fig. 3. As shown in Fig. 4, binary interaction parameters optimised using the SLE data for binaries are almost independent of compound combinations, with the average close to −0.024 when PR EoS is applied to the calculation of fugacity in the liquid phase.
5.2. PR EoS When using PR EoS, the correlations suggested by Twu [25] are suitable for calculating critical temperature and critical pressure for n-paraffins above C20 . As for SRK EoS, the m values optimised for n-paraffins above C20 and those for n-paraffins below C20 (shown in Fig. 3) were used in
5.0 Cn < C20: the original PR 2rd order polynomial function
4.5
Cn > C20: the original PR 2rd order polynomial function
4.0
Cn > C20: optimized m values with PR EoS, this work
m value
3.5
C1 - C60: the 4th order polynomial function for PR EoS, this work
3.0 2.5 2.0
C20
C22
C24
C32
C36
1.5 1.0 0.5 0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
acentric factor Fig. 3. Relationship between m values and acentric factor (ω) in PR EoS (Tc and Pc calculated using correlations suggested by Twu [25] and ω calculated using the correlation suggested by Lee–Kesler [8] for n-paraffins above C20 ).
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207
0.00
Binary interaction parameter
-0.01 -0.02 -0.03 -0.04 -0.05
The heavier component: C22
-0.06
The heavier component: C24
-0.07
The heavier component: C28 The heavier component: C32
-0.08
The heavier component: C36
-0.09 -0.10 4
5
6
7
8
9
10
11
12
13
14
Cn of the lighter component in binary systems Fig. 4. Binary interaction parameter optimised using PR EoS and SLE data for binaries (Cn : carbon number).
6. Describing wax solids The UNIQUAC equation with parameters determined in this work is used for describing wax solids, i.e. for calculating activity coefficients in the solid phase. The general UNIQUAC equation in terms of molar excess Gibbs energy is given as: n n gE ϑi z θi = si ln + qi si ln RT si 2 ϑi i=1 i=1 n n − qi si ln θj τji i=1
As shown in Eq. (9), for each pair of compounds, there are two adjustable parameters, τ ij and τ ji . These are given in terms of characteristic energies uij and uji . The general formulation for calculating τ ij is: uij τij = exp − (14) RT The characteristic energy, uij , is correlated with Cn,ij , the difference in carbon numbers for a pair of compounds: uij = a × Cn,ij
(9)
j=1
with
where a is a constant determined as 11 in this work using the experimental WDT data generated in this laboratory for C16 –C18 , C16 –C20 and C15 –C19 binaries. This work further considers that:
ri si ϑi = n j=1 rj sj
(10)
uji = uij
qi si θi = n j=1 qj sj
(11)
7. Modelling high pressure conditions
where gE is the molar excess Gibbs energy. ri and qi are molecular structure parameters of pure compounds, which depend on molecular sizes and external surface areas. z is the coordination number (6 ≤ z ≤ 12). The coordination number, z, is set to 10 here, according to the value suggested by Abrams and Prausnitz [47]. Based on the n-paraffin structure parameters provided in the literature [47,48], the following correlations have been developed for calculating ri and qi . ri = 0.675Cn,i + 0.4483
(12)
qi = 0.540Cn,i + 0.6200
(13)
where Cn,i is the carbon number for compound i.
(15)
(16)
The exponential term in Eq. (2), referred as the Poynting correction term, takes into account the effect of the difference between the operating pressure (P) and the reference pressure (PO ). In this work, the reference pressure is set to be the operating pressure, where the Poynting correcting term becomes unity. To calculate the pure solid fugacity at the reference pressure (fi OS ), fusion properties of pure compounds have to be those at the operating pressure conditions (i.e. the reference pressure in this work). Based on experimental measurements for pure n-paraffins [49,50], the following generalized correlation is proposed for calculating fusion temperatures at increased pressure conditions. Tf(P) = Tf(P=0.1 MPa) + 0.2 × (P − 0.1)
(17)
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where Tf(P=0.1 MPa) and Tf(P) are the fusion temperature of pure compounds at 0.1 MPa and the operating pressure (P) in MPa, respectively.
8. Results and discussion 8.1. Validating correlations of fusion properties and heat capacity Independent experimental WDT data for C6 –C16 and C6 –C17 binaries, generated in this laboratory, have been used for validating the suggested correlations for calculating fusion properties and heat capacity in this work. Critical properties and acentric factors for the compounds in these binaries are reliable and reported in the literature such as CRC [26]. Furthermore, pure solids may form in these binaries due to the significant differences in molecular sizes. Hence the accuracy of predicted WDT for these binaries largely depends on the reliability of values for fusion properties and heat capacity. As shown in Fig. 5, WDT predictions using the model developed in this work (HWWAX) are in good agreement with our experimental data, demonstrating the reliability of the fusion properties and heat capacity correlations. It may be inferred that the inclusion of a solid–solid transition in SLE calculations and acknowledging the differences between odd and even carbon numbers of n-paraffins, have improved the reliability of HWWAX predictions. 8.2. Comparing wax solid models Experimental data for binaries and ternaries, generated in this laboratory or reported in the literature, have been used for comparing several approaches for the description
of wax solids. These include the ideal solid solution, the multi-pure-solid, Coutinho’s UNIQUAC, and finally the HWWAX UNIQUAC approaches. Compounds in the binaries and ternaries are lighter than C20 . Their critical properties and acentric factors have been measured experimentally and reported in reference handbooks. Hence, the reliability of WDT predictions for these mixtures depends on the thermodynamic model used for describing wax solids. 8.2.1. Binaries Experimental WDT data have been generated for C16 – C18 , C16 –C20 and C15 –C19 binaries in this laboratory. As shown in Figs. 6–8, the predictions using the ideal solid solution approach generally overestimate WDTs for all the binaries investigated. This suggests that the solid solution for these binaries is non-ideal, and the deviation from ideal is positive. However, as shown in Figs. 7 and 8, the multi-puresolid approach is suitable for mixtures with components having significant differences in chain-length (e.g. C16 –C20 and C15 –C19 binaries), whereas, it is not satisfactory for predicting WDT for n-paraffins when molecular size differences are small (e.g. C16 –C18 binary). Coutinho’s UNIQUAC approach also shows limitations in the description of the nonideality of solid solutions formed by n-paraffins with similar carbon numbers (e.g. C16 –C18 binary). Experimental SLE data for C16 –C18 , C16 –C20 and C15 –C19 binaries have been used in the optimisation process for determining the constant in Eq. (15) when developing the HWWAX UNIQAUC approach for describing wax solids. Calculated wax phase boundaries using HWWAX are in good agreement with the experimental measurements, as shown in Figs. 6–8. Experimental WDT data for the binary C17 –C19 system have been reported in the literature [51]. As shown in Fig. 9, independent WDT predictions using HWWAX are in excellent agreement with the measurements, demonstrating
280 275 270
WDT/K
265 260 255 C6-C16: exp. data, this laboratory
250
C6-C16: HWWAX predictions 245
C6-C16: predictions without classifying odd/even Cn
240
C6-C17: exp. data, this laboratory C6-C17: HWWAX predictions
235
C6-C17: predictions without inclusion of S-S transition
230 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
C16 or C17 mole fraction Fig. 5. Experimental (this laboratory) and predicted WDTs for C6 –C16 and C6 –C17 binaries.
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Exp. data: this laboratory Calculations: HWWAX Predictions: the ideal solid solution Predictions: the multi-pure-solid approach Predictions: Countinho's UNIQUAC approach
305
300
WDT/K
209
295
290
285 0
0.2
0.4
0.6
0.8
1
C18 mole fraction Fig. 6. Comparison of experimental WDT data (this laboratory) for C16 –C18 binaries with model predictions using several approaches.
the reliability of the approach used for describing wax solids.
8.3. The wax phase boundary for multi-component mixtures: effect of pressure
8.2.2. Ternaries Experimental WDT data for C14 –C15 –C16 and C18 –C19 – C20 ternaries have been reported in the literature [52]. WDT data for C6 –C16 –C17 ternaries have been generated in this laboratory. As shown in Tables 1–3, HWWAX predictions show very good agreement with experimental data over a wide range of compositional distributions. A further observation is that the ideal solid solution approach generally overestimates WDTs, while the multi-pure-solid approach and Coutinho’s UNIQUAC approach shows significant deviations in some ternary systems.
Temperature is the major factor affecting wax precipitation. However, the impact of pressure on wax formation is also significant. Although wax may not form within operating temperatures at atmospheric pressure (the condition under which WATs and WDTs are often measured), wax precipitation may occur at higher pressures. Therefore, it is important for a wax model to have the ability to provide reliable prediction of wax phase equilibria at high-pressure conditions. Pressure affects wax precipitation in two opposing ways. Firstly, the solubility of gases in the liquid phase increases
312
307
WDT/K
302
297
292
Exp. data: this laboratory Calculations: HWWAX Predictions: the ideal solid solution Predictions: the mullti-pure-solid approach Predictions: Coutinho's UNIQUAC approach
287
282 0
0.2
0.4
0.6
0.8
1
C20 mole fraction Fig. 7. Comparison of experimental WDT data (this laboratory) for C16 –C20 binaries with model predictions using several approaches.
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H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
306 301
WDT/K
296 291 286 Exp. data: this laboratory 281
Calculations: HWWAX Predictions: the ideal solid solution
276
Predictions: the multi-pure-solid approach Predictions: Coutinho's UNIQUAC approach
271 0
0.2
0.4
0.6
0.8
1
C19 mole fraction Fig. 8. Comparison of experimental WDT data (this laboratory) for C15 –C19 binaries with model predictions using several approaches.
with an increase in the system pressure at conditions below the bubble point. This extra dissolved gas leads to a lower mole fraction of heavy compounds in the liquid phase, which may reduce the tendency for wax formation. On the other hand, as pressure increases, the solidification temperature of pure compounds increases, and the wax phase boundary may shift to a higher temperature at any given pressure. The impact of pressure on fugacity of condensed phases away from the critical point is not very significant in comparison with the above two effects. In a gas–liquid system, both factors mentioned above may affect the wax phase boundary. Under high-pressure conditions, depending on the dominant factor, the wax phase
boundary may be shifted with respect to the atmospheric pressure condition. Obviously, for a system above its bubble point, only the second factor will influence the wax phase boundary. In order to evaluate the reliability of HWWAX predictions, WDTs were measured for three multi-component mixtures at different pressures in this laboratory. Compositions for the mixtures are listed in Table 4. As shown in Fig. 10, the predicted wax phase boundaries using HWWAX are in good agreement with the independent experimental data measured in our laboratory. The pressure impact on WDT is obvious. As the pressure increases to 40 MPa, the WDT increases by approximately 8 K.
306
WDT/K
301
296
291 Exp: Robles et al. (1996) Predictions: HWWAX Predictions: the ideal solid solution approach Predictions: the multi-pure-solid approach Predictions: Coutinho's UNIQUAC approach
286
281 0
0.2
0.4
0.6
0.8
1
C19 mole fraction Fig. 9. Comparison of experimental WDT data [51] for C17 –C19 binaries with model predictions using several approaches.
H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
211
Table 1 Experimental WDT data [52] and model predictions for C14 –C15 –C16 ternary, at 0.1 MPa Experimental data
Predictions and deviations (Dev)
Mole fraction C14
C15
C16
0.06 0.14 0.17 0.24 0.21 0.27 0.37 0.32 0.43 0.57 0.73
0.57 0.23 0.06 0.33 0.56 0.66 0.05 0.24 0.33 0.17 0.14
0.37 0.63 0.77 0.43 0.23 0.07 0.58 0.44 0.24 0.26 0.13
Exp. WDT (K)
Ideal solid solution
Multi-pure-solid
Countinho’s UNIQUAC
HWWAX
WDT (K)
Dev (K)
WDT (K)
Dev (K)
WDT (K)
Dev (K)
WDT (K)
Dev (K)
283 285 286 282 281 280 283 282 279 278 276
287 288 289 286 285 283 287 286 284 284 282
4 3 3 4 4 3 4 4 5 6 6
277 284 287 279 272 275 283 280 272 273 274
−6 −1 1 −3 −10 −5 0 −2 −7 −5 −2
280 285 287 281 276 278 284 281 276 276 275
−3 0 1 −1 −5 −2 1 −1 −3 −2 −1
285 287 288 284 283 281 285 284 281 280 279
2 2 2 2 2 1 2 2 2 2 3
Table 2 Experimental WDT data [52] and model predictions for C18 –C19 –C20 ternary, at 0.1 MPa Experimental data
Predictions and deviations (Dev)
Mole fraction C18
C19
C20
0.02 0.05 0.05 0.1 0.1 0.1 0.14 0.15 0.2 0.2 0.26 0.33 0.4 0.43 0.48 0.6 0.79 0.9
0.02 0.05 0.9 0.1 0.4 0.55 0.73 0.15 0.2 0.6 0.26 0.33 0.1 0.43 0.15 0.2 0.11 0.05
0.96 0.9 0.05 0.8 0.5 0.35 0.13 0.7 0.6 0.2 0.48 0.34 0.5 0.14 0.37 0.2 0.1 0.05
Exp. WDT (K)
Ideal solid solution
Multi-pure-solid
Countinho’s UNIQUAC
HWWAX
WDT (K)
Dev (K)
WDT (K)
Dev (K)
WDT (K)
Dev (K)
WDT (K)
Dev (K)
309 309 305 308 306 306 304 307 306 305 306 304 305 303 304 302 301 301
310 309 305 309 308 307 306 308 308 306 307 306 307 304 306 304 303 302
1 0 0 1 2 1 2 1 2 1 1 2 2 1 2 2 2 1
309 309 303 307 302 298 300 306 304 296 301 298 302 292 299 295 298 299
0 −1 −2 −1 −4 −8 −4 −1 −2 −9 −5 −7 −3 −11 −6 −8 −3 −2
309 309 304 307 303 300 301 306 305 296 302 299 303 297 300 295 299 300
0 0 −1 −1 −3 −6 −3 −1 −1 −10 −4 −5 −2 −6 −4 −7 −2 −1
309 309 305 308 306 305 304 307 306 304 305 303 304 302 303 301 300 301
0 0 0 0 0 −1 0 0 0 −1 −1 −1 −1 −1 −1 −1 −1 −1
efficient and economical use of techniques to prevent wax accumulation. Experimental data for the composition and amount of wax precipitated for a multi-component mixture under several temperature and pressure conditions have been
8.4. Amount of wax precipitated and its composition The ability to predict the amount and composition of wax formed at given conditions could be extremely useful for
Table 3 Experimental WDT data (Heriot-Watt) and model predictions for C6 –C16 –C17 ternary, at 0.1 MPa Experimental data
Predictions and deviations (Dev)
Mole fraction C6
C16
C17
0.911 0.905 0.794
0.048 0.04 0.156
0.041 0.055 0.051
Exp. WDT (K)
Ideal solid solution
Multi-pure-solid
Countinho’s UNIQUAC
HWWAX
WDT (K)
Dev (K)
WDT (K)
Dev (K)
WDT (K)
Dev (K)
WDT (K)
Dev (K)
261 263 271
265 266 273
4 3 2
254 257 267
−7 −6 −4
258 260 268
−3 −3 −3
261 262 270
0 −1 −1
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H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
Table 4 Compositions (mol%) for mixtures A, B and C (Heriot-Watt)
reported in the literature [53]. The mixture consisted of consecutive normal alkanes from C6 to C36 , with decreasing concentration as a function of increasing in chain-length, representing a highly simplified crude oil system.
Components A
B
C
51.04 44.49 1.59 2.12 0.33 – – 0.17 – 0.19 – – 0.08
– 80.04 – – – 6.43 4.39 2.99 2.06 2.34 1.41 0.34 –
47.45 37.76 – 6.44 2.4 3.23 1.81 0.22 0.3 – 0.21 0.18 –
8.4.1. Amount of wax The amounts of wax (mass% of the feed) predicted using different models (i.e. HWWAX, Coutinho’s UNIQUAC approach, and the ideal solid solution approach) are compared against independent experimental data in Fig. 11. It is obvious that the ideal solid solution approach overestimates the solid amounts over the whole temperature range (i.e. 256–290 K). Predictions of the HWWAX model are in good agreement with the experimental data. As expected, the amount of wax deposition increases with a decrease in the system temperature.
50 45 40
P/MPa
35 30 25
A: exp. data, this laboratory
20
A: HWWAX predictions
15
B: exp. data, this laboratory B: HWWAX predictions
10
C: exp. data, this laboratory 5 0 285
C: HWWAX predictions 290
295
300
305
310
315
320
325
330
T/K Fig. 10. Measured (this laboratory) and predicted (using HWWAX) WDTs for mixtures A, B and C at different pressure conditions. 40 0.1MPa, Exp. data: Pauly et al. (2001) 0.1 MPa, Predictions: HWWAX 0.1 MPa, Predictions: Ideal solid solution approach 0.1 MPa, Predictions: Coutinho's UNIQUAC approach 50MPa, Exp. data: Pauly et al. (2001) 50MPa, Predictions: HWWAX
35
Solid depost/mass%
C7 C10 C13 C16 C18 C20 C21 C22 C23 C24 C28 C30 C36
30 25 20 15 10 5 0 250
260
270
280
290
300
T/K Fig. 11. Measured wax amounts (mass%) [53] and predictions using different wax models at 0.1 MPa.
H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
213
30
Solid composition/mass%
Exp. data: Pauly et al. (2001) 25
Predictions: HWWAX Predictions: Ideal solid solution approach
20
Predictions: Coutinho's UNIQUAC approach
15
10
5
0 5
10
15
20
25
30
35
40
Carbon numer Fig. 12. Measured [53] and predicted wax compositions using different wax models at 290.2 K and 0.1 MPa.
The reliability of HWWAX for prediction of the effect of pressure on wax deposition is also demonstrated in Fig. 11. In this case, as the pressure increases from atmospheric pressure to 50 MPa, the amount of wax precipitated increases from 4.6 to 9.3 mass% at 273.2 K.
deviation for the concentration of heavy hydrocarbons in the wax. Predictions of wax composition using HWWAX are in close agreement with experimental data. The predicted wax deposition at 290 K and 0.1 MPa, predominantly consists of paraffins heavier than C25 , which are in good agreement with the measured data. Fig. 13 shows the effect of temperature on the composition of wax precipitates (at 0.1 MPa). As temperature reduces, more light hydrocarbons take part in wax formation. The predictions of HWWAX are in good agreement with experimental data at different temperatures. Fig. 14 shows the impact of pressure on the composition of the wax phase. An increase in pressure has a similar impact as a reduction in temperature; both cause more light
8.4.2. Wax composition As shown in Fig. 12, the ideal solid solution approach over-estimates the amount of light components in the precipitated wax. Clearly, concentrations of n-paraffins below C28 in the wax are highly overestimated when using the ideal solid solution approach. Predictions based on the Coutinho’s UNIQUAC approach are in reasonable agreement with experimental data. However, this approach shows an obvious 30
Solid composition/mass%
290.2K, Exp. data: Pauly et al. (2001) 290.2K, Predictions: HWWAX
25
273.2K, Exp. data: Pauly et al. (2001) 273.2K, Predictions: HWWAX
20
256.2K, Exp.data: Pauly et al. (2001) 256.2K, Predictions: HWWAX
15
10
5
0 5
10
15
20
25
30
35
40
Carbon number Fig. 13. Measured [53] and predicted (using HWWAX) wax compositions at different temperature conditions, at 0.1 MPa.
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H.-Y. Ji et al. / Fluid Phase Equilibria 216 (2004) 201–217
25 0.1MPa, Exp.data: Pauly et al. (2001)
Solid composition/mass%
0.1MPa, Predictions: HWWAX 20 50MPa, Exp. data: Pauly et al. (2001) 50MPa, Predictions: HWWAX 15
10
5
0 5
10
15
20
25
30
35
40
Carbon number Fig. 14. Experimental [53] and predicted (using HWWAX) wax compositions at different pressure conditions (290.2 K).
hydrocarbons to take part in the wax phase. The predictions using HWWAX are in good agreement with experimental data at different pressures.
9. Conclusions A thermodynamically consistent phase behaviour model to predict the phase boundary, amount and composition of wax (HWWAX) has been developed in this work. The reliability of HWWAX has been verified by comparing its predictions with independent experimental data and those of other leading wax models. The reliability of the model is attributed to: (1) introduction of higher-accuracy values for fusion properties and heat capacities for n-paraffins, (2) optimisation of basic parameters for long chain n-paraffins for use in equations of state, (3) development of a new and more reliable approach for describing wax solids, and (4) extension of model capabilities to high-pressure conditions using the thermodynamic properties of pure compounds. List of symbols a constant Cn carbon number Cp heat capacity f fugacity g molar Gibbs energy H enthalpy P pressure q molecular external surface r molecular size R gas constant
s T u V x v z Z
solid mole fraction temperature characteristic energy volume liquid mole fraction molar volume coordination number compressibility factor
Greek γ ϕ ϑ θ
letters variation activity coefficient fugacity coefficient parameter related with molecular size parameter related with molecular external surface parameter related with characteristic energy acentric factor
τ ω
Superscripts E excess L liquid phase O pure component S solid phase Subscripts b bubble point c critical condition cal calculated data exp experimental data f fusion O reference condition subl sublimation tr solid–solid transition
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215
• Others
Acknowledgements This work was part of a Joint Industrial project funded by ABB Offshore Systems Ltd., the UK Department of Trade and Industry, Petrobras, Shell UK Exploration and Production, and TOTAL, whose support is gratefully acknowledged. Hongyan Ji wishes to thank James Watt Scholarship and the ORS Award Scheme for financial support. The authors also wish to thank Mr. Rod Burgass for his contributions to experimental work, and Mr. Ross Anderson for his assistance in the revision of manuscript.
Ttr (K) = Tf n-Paraffins with even carbon numbers: • For C22 ≤ Cn ≤ C42 Ttr (K) = 0.0032C3n − 0.3249C2n + 12.78Cn + 154.19 + ln(Cn ) • Others Ttr (K) = Tf
Appendix A A.2. Heat of fusion and heat of solid–solid transitions A.1. Fusion temperature and solid–solid transition temperature Correlations for calculating fusion temperatures and solid–solid transition temperatures have been developed in accordance with measured values reported in the literature [28,29]. Differentiation between odd or even carbon numbers for n-paraffins is applied to correlations in order to improve accuracy. A.1.1. Fusion temperature n-Paraffins with odd carbon numbers: • For Cn ≤ C9 Tf (K) = 0.3512C3n − 7.6438C2n + 72.898Cn − 73.9 • For C9 < Cn ≤ C43 Tf (K) = 0.0122C2n − 2.0861Cn − 775.598/Cn + 76.2189 ln(Cn ) + 156.9
n-Paraffins with even carbon numbers: Tf (K) = −0.0998C3n + 1.0812C2n + 18.602Cn + 49.216 • For C10 < Cn ≤ C42 − 0.3458C2n
+ 14.277Cn + 137.73
• For Cn > C42 Tf (K) =
Hsum (cal mol−1 ) = 0.119MW × Tf + 672.2 • For C9 < Cn ≤ C33 Hsum (cal mol−1 ) = 0.167MW × Tf + 432.47 • For Cn > C33 Hsum (cal mol−1 ) = 0.139MW × Tf + 3984.8
Hsum (cal mol−1 ) = 0.180MW × Tf + 522.7 • For Cn > C34
• For Cn ≤ C10
Tf (K) =
• For Cn ≤ C9
• For Cn ≤ C34
414.3(Cn − 1.5) Cn + 5.0
0.0031C3n
A.2.1. Sum of heats of fusion and heats of solid–solid transitions n-Paraffins with odd carbon numbers:
n-Paraffins with even carbon numbers:
• For Cn > C43 Tf (K) =
The sum of heats of fusion and heats of solid–solid transitions are considered to be dependent on the fusion temperature and the molecular weight. Correlations have been developed using data reported in the literature [28,29].
414.3(Cn − 1.5) Cn + 5.0
A.1.2. Solid–solid transition temperatures n-Paraffins with odd carbon numbers: • For C9 < Cn ≤ C43 Ttr (K) = 0.0039C3n −0.4239C2n +17.28Cn −ln(Cn )+95.4
Hsum (cal mol−1 ) = 0.139MW × Tf + 3984.8 A.2.2. Heat of fusion and heat of solid–solid transitions n-Paraffins with odd carbon number: • For Cn ≤ C9 Hf (cal mol−1 ) = 1.0 Hsum Htr (cal mol−1 ) = 0 • For C9 < Cn ≤ C43 Hf (cal mol−1 ) = 0.74 Hsum Htr (cal mol−1 ) = 0.26 Hsum
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• For Cn > C43
• For Cn > C20
Hf (cal mol−1 ) = 1.0 Hsum
aS = (1.6964 × Cn − 22.5000) × 10−6
Htr (cal mol−1 ) = 0
bS = −(1.1670 × Cn − 19.525) × 10−3
n-Paraffins with even carbon number:
d S = −(1.5093 × Cn − 31.209) × 10
• For Cn ≤ C20 Hf (cal mol−1 ) = 1.0 Hsum Htr (cal mol
−1
)=0
• For C20 < Cn ≤ C42 Hf (cal mol−1 ) = 0.64 Hsum Htr (cal mol−1 ) = 0.36 Hsum • For Cn > C42 Hf (cal mol−1 ) = 1.0 Hsum Htr (cal mol−1 ) = 0 A.3. Heat capacity Correlations have been developed for calculating heat capacity as a function of temperature and carbon number using measured data for n-paraffins [30–32]. Differentiation between odd or even carbon numbers for n-paraffins is applied to correlations when calculating the heat capacity of solids. A.3.1. Heat capacity for n-paraffin liquids CpL (cal mol−1 K−1 ) = aL T + bL with aL = 0.01 × Cn − 0.0138 bL = 4.529 × Cn + 3.8457 A.3.2. Heat capacity for n-paraffin solids CpS (cal mol−1 K−1 ) = aS T 3 + bS T 2 + cS T + d S with n-Paraffins with odd carbon number (based on data available up to C19 ): aS = (0.3571 × Cn + 2.1667) × 10−6 bS = −(0.2014 × Cn + 0.4300) × 10−3 cS = (0.4579 × Cn + 0.8105) × 10−1 d S = −(0.0678 × Cn − 0.0580) × 10 n-Paraffins with even carbon number: • For Cn ≤ C20 aS = (0.0929 × Cn + 4.9286) × 10−6 bS = −(0.0993 × Cn + 1.5929) × 10−3 cS = (0.3604 × Cn + 1.9115) × 10−1 d S = −(0.0459 × Cn + 0.2022) × 10
cS = (2.4703 × Cn − 39.848) × 10−1
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