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Alkanes induced asphaltenes precipitation studies at high pressure and temperature in the presence of argon Daniela Hartmann, Humberto E. Lopes, Carmen S. Teixeira, Marcia Cristina Khalil de Oliveira, Gaspar González, Elizabete F. Lucas, and Luciana S Spinelli Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.5b02217 • Publication Date (Web): 03 Mar 2016 Downloaded from http://pubs.acs.org on March 3, 2016
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Alkanes induced asphaltenes precipitation studies at high pressure and temperature in the presence of argon D. Hartmann1, H. E. Lopes2, C. L. S. Teixeira2, M. C. K. de Oliveira2, G. Gonzalez1, E. F. Lucas1,3, L. S. Spinelli1 1
Federal University of Rio de Janeiro, Laboratory of Macromolecules and Colloids in Petroleum Industry/IMA, Rio de Janeiro, Brazil
2
Petrobras/CENPES, Av. Horácio Macedo, 950, Cidade Universitária, 21941915, Rio de Janeiro, Brazil
3
Federal University of Rio de Janeiro, Metallurgy and Materials Engineering Program/COPPE, Av. Horácio Macedo, 2030, block F, 21941598, RJ, Brazil,
[email protected] * Corresponding author:
[email protected] ABSTRACT. The continuous increase of petroleum production under adverse sub-sea conditions and the preeminent need to adequate operational condition and the efficient use additives to warrant flow assurance, makes interesting to set up experimental procedures to carry out n-alkane precipitation studies under high pressure (p) and high temperature (T) conditions. In this contribution, some preliminary experimental studies developed to characterize asphaltenes precipitation in model systems consisting of asphaltenes solutions in toluene or mixtures of hydrocarbons by the addition of propane, n-heptane or other alkanes at various pressures and temperatures, using a commercial compact equipment, are reported. In general terms, it was established that these tests follow the same pattern described at ambient p and T conditions and the one single study reported in the literature for a stock tank oil sample at 3000 psi and room T. Four crude oils of different characteristics were tested, using diluted or undiluted samples, and it was possible to detect the asphaltene precipitation onset. However, these results cannot be used to infer the stability of the crude oils because results correlating onset and stability at high p and T are not available yet. The effect of pressure at high pressures was not entirely resolved because the argon, used as an assumed inert gas to pressurize the system, dissolves in the hydrocarbons and displaces the precipitation onset towards lower values. The need to develop a compact equipment to assess the effect of solvents, inhibitors and other additives on the phase behavior of crude oil at high pressure and temperature, and in the presence of CO2 and other gases, although represents a valuable contribution to the petroleum industry in the area of flow assurance, still persists. Keywords: Asphaltenes precipitation; high-pressure conditions; translucence
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INTRODUCTION The stability of crude oil in relation to asphaltenes precipitation is one of the key factors for the developing of petroleum fields as well as the design of the basic production facilities required for flow assurance. Moreover, asphaltenes are a class of molecules responsible for crude oil emulsion stabilization.1-4 The thermodynamic conditions for asphaltenes precipitation are conveniently determined through depressurization tests of live oil samples at a pressures (p) and temperatures (T) consistent with the values prevailing for these parameters along the production process.5 Preliminary efforts to identify sensitive and precise experimental techniques for the measurement of organic solid deposition from live petroleum samples were presented by Ferworn et al. in 1995.6,7 The authors briefly review various techniques to detect live oil asphaltenes and wax precipitation including an apparatus consisting of a specially designed pressure cell equipped with a probe containing a fiber-optic near infrared light source and fiber optic detectors to measure the light transmission through oil sample. Solids formation were detected by an abrupt reduction in the transmittance caused by the appearance of particles, The apparatus worked well at pressures up to 10.000 psia and 350 oF for light oils but presented limitations for the heavy, dark crudes. In a later work,8 the authors presented a more detailed description of this experimental device; the Solid Detection System (SDS) reproduced in the article by courtesy of DB Robinson Ltd. (DBR). The system permitted, using a software package, to depressurize at a defined rate live oil samples contained in a visual PVT cell and to record in real time the pressure, temperature, solvent volume, time and the transmittance of the fluid using a fiber optic light transmission probe. As an example of the apparatus operation the pressure of a single phase fluid from the Gulf of Mexico was isothermally reduced from 9.600 to around 5.000 psia. The ACS Paragon Plus Environment
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graphs traces showed an initial increase of transmittance due to the reduction of the fluid density and then a sharp decrease at 5.700 psia due the formation of asphaltenes particles. Various articles were published subsequently during this period to describe the experimental procedure9 and to define the optimal experimental conditions Hammami et al.5,10 carried out laboratory depletion tests to determine the stability of four samples of live oils from the Gulf of Mexico. The authors identified the asphaltenes precipitation onset and concluded that, under-saturated crude oils presented a higher tendency to precipitate in agreement with de Boer criteria11 and that asphaltenes precipitation was reversible. Joshi et al.12 used optical scattering techniques and sedimentation rates to characterize asphaltenes precipitation at the precipitation onset and at further depletion ratios and concluded that near the onset the particles formed are small remain disperse whereas at lower pressures the flocs are observed to grow and sediment quite rapidly. Jamaluddin et al.13 compared the efficiency of gravimetric analysis, acoustic resonance, near infrared light scattering (NIR-LS) and filtration as alternative laboratory techniques to identify the asphaltenes precipitation onset of live oil samples. They concluded that NIR-LS and filtration determined the asphaltenes precipitation boundaries and the bubble points. However some subjectivity was detected in the definition of the onset from the transmittance against pressure graphs in the NIR-LS technique. Although filtration was considered slower than LS it presents the advantage that provides, in addition to the onset, the amount precipitate and, even considering that the amount of solids retained depended on the filter porosity and the asphaltenes particles size, it was considered that a combination of NIR-LS and filtration could represent a powerful tool to define asphaltenes phase behavior. Muhammad et al.14 used a combination of these two techniques (NIR-LS and high pressure, high temperature filtration) to determine the onset pressure and to estimate the particle size of the asphaltenes precipitated from reservoir fluid samples. For these purpose the onset for the oil samples was determined using LS and small ACS Paragon Plus Environment
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volume samples, which were collected isobarically and isothermally at different pressures during the depressurization for filtration tests using different pore size filters. The results basically confirm that particles formed near the precipitation pressure are smaller than those formed at lower pressures as reported by Joshi et al.12 The authors also concluded that filtration tests using 0.2 µm filters were more sensitive than LS to detect the asphaltenes onset pressure. In 2003, DBR incorporated a high-pressure microscope (HPM) and a particle size analysis software (PSA) that operated synchronized with the HPM to scan digital photomicrographs in real time to generate particle size distribution histograms. These new facilities permitted to detect solids formation by light transmittance, solids visual identification by HPM and particle size distribution monitoring by PSA as a function time and test conditions. The SDS equipment in its original form or including the HPM and PSA have been extensively used to identify asphaltenes precipitation onset pressure and characterize the particles size and growth continuously in isothermal pressure reduction experiments. Karan et al.15 presented results that clearly show the precipitation onset followed by the asphaltenes particles formation and flocculation. They also showed particles size histograms that confirm that particles size and number increase below the precipitation onset. A more recent publication present an updated diagram of the SDS-HPM-PSA set up.16 Jamaluddin et al.17 used thermal depressurization tests to estimate the effect of nitrogen on asphaltenes precipitation when this gas is used for pressure maintenance to improve oil recovery. It was found that the asphaltenes inject pressure and the bubble point pressure increased with the increase of the nitrogen injection. Gholoum et al.18 measured the asphaltenes onset pressure for a reservoir fluid from Kuwait by depressurization tests at various temperatures. The authors also investigated the effect of various short chain hydrocarbons and CO2 on asphaltenes stability. They observed that lowest precipitation onset was obtained using CO2 as precipitant and that the ACS Paragon Plus Environment
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onset increase as the precipitant carbon number increased. Negahbam et al.19 used the same equipment to identify the effect of rich gas and CO2 on the asphaltenes stability of live oil from a Abu Dhabi reservoir. The oil was stable at reservoir conditions but precipitated by hydrocarbon gas addition. For CO2 the precipitation was observed at lower temperatures only. HPM showed that the particle size was 0,5 µm at the onset and grew to 1.5 µm at the bubble point pressure. The equipment has also been used to confront the PC-SAFT asphaltenes precipitation predicting model with experimental results model systems and live crude oil.20 In subsequent publications21 the authors compare the PC-SAFT output with results reported by Jamaluddin for nitrogen17 and experimental results for methane22 and CO2.17,23
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In the
development of the Asphaltenes Instability Trend (ASIST),24 another asphaltenes precipitation predicting model that uses the solubility parameter at the precipitation onset and the molar volume of the precipitant to estimate the stability of stock tank oils, the asphaltenes onset pressure obtained by depressurization using the SDS was used to compare with the model results.25 Considering that the two techniques produced different estimates, the authors infered that using both procedures together it be could possible to identify a range of pressure over which asphaltenes would precipitate. The SDS, in its original design or as the devise including the HPM and PSA has also been used, although to a lower extent to, select asphaltenes inhibitors for field operations. Early results were reported by Allenson et al.26 The authors determined the critical CO2 concentration for asphaltenes precipitation high p and T for a reconditioned south American live crude oil by titration with CO2 using the SDS and developed a chemical that according to the authors, inhibited both asphaltenes precipitation and deposition and sticking. Y. R. Yin et al.27 developed and tested an inhibitor for continuous capillary injection to prevent asphaltenes flocculation. The product was evaluated with the SDS using downhole live oil by comparing the amount of ACS Paragon Plus Environment
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asphaltenes residual oil left after a depressurization experiment for both treated and untreated samples. Field tests confirmed the SDS results on the efficiency of the inhibitor. A similar study performed later for a crude sample from Alaska with asphaltenes precipitation problems.28 In subsequent developments the authors describe details of the experimental procedures and apply the tests to oil samples from production29 and refining30 operations. A more comprehensive study is reported by Karan et al.15 also for reservoir fluid from South America, using the SDS-HPNPSA system. The combination of these three techniques permits, in fact, an accurate examination of the asphaltenes particles formation that could be useful to discern inhibitors performance and mechanism of action. An experimental setup to study the effect of gases on asphaltenes stability at high pressures and temperatures was developed at the Complex Fluids Laboratory of the University of Pau. The equipment was based on isobaric filtration and consisted basically on a PVT mixing and titration cell, an injection unit and a high pressure filtration device. The equipment has been used mainly to estimate the effect of CO2 on the stability of crude oils. In a first study31 two asphaltenic dead crude oils: one from South America and the other one from the Middle East were used. Both samples presented similar SARA analysis and their asphaltenes content was 9 wt%. The authors measured the asphaltenes onset pressure and the bubble point for crude oil-CO2 mixtures for both oils and observed that asphaltenes are more soluble when temperature is decreased and when pressure is increased. The new high-pressure cell apparently produced reproducible results and according to the authors, its main advantages were the use of a small mass of oil (5 g), the injection of the precipitant under the actual investigation conditions and the continuous mixing of sample. In subsequent publications an optical endoscope was incorporated to the equipment for visual detection of the asphaltenes flocculation onset and the bubble point and it was used to estimate the effect of CO2 on the stability of a Brazilian live crude oil sample32 and to study the ACS Paragon Plus Environment
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asphaltenes phase behavior in the presence of CO2 at high pressures, as function of the temperature for a model oil consisting of asphaltenes from a Venezuelan crude oil dissolved in a toluene/n-heptane mixture.33 As shown in the brief preceding review, these two experimental techniques have been applied mainly to assess the asphaltenes onset for live crude oils in the case of SDS as it was clearly stated by the authors in the first paper on this subject6 or to study the effect of gases, (mainly CO2) in the case of the University of Pau setup. Except by some results reported by Gholoum,18 asphaltenes precipitation by different chain length alkanes from model asphaltenes solution systems or crude oil has not been studied at pressures and temperatures higher than the ambient conditions. This type of studies may furnish useful information on asphaltenes stability as it represents the basis of the ASSIST predictive model. The screening for efficient asphaltenes inhibitors, particularly in CO2 containing fluids, represents another subject in which the information available is limited and require high p and T asphaltenes dispersability tests rather than only the asphaltenes onset pressure. For these two applications, the techniques previously reviewed present limitations. Both maintain some degree of subjectivity in the estimation of the onset pressure or in the identification of the particles observed counted or dimensioned. Both are time-consuming and SDS is rather expensive. None of them permits to run spectra analysis which may be useful for other complementary results as reported in literature.12 SDS does not quantify the precipitate and the University of Pau equipment does not inform particle size. As an alternative procedure, in most cases, the n-heptane titration of dead oil has been adopted as a suitable procedure to preliminary assess the tendency of different crudes to precipitate asphaltenes,11,34-37 and it has been shown38-39 that there is a clear correlation between the asphaltenes separation induced by n-heptane and that occurring by the depressurization of the ACS Paragon Plus Environment
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live crude. Some discrepancies have recently been detected however, as for some crude oil samples that do not present a well-defined precipitation onset by titration with n-heptane but show evidence of particles formation in depressurization tests. These results and the pre-eminent need to identify and qualify asphaltenes inhibitors to be used under reservoir condition and, eventually, in the presence of CO240 makes interesting to set up experimental procedures to carry out n-alkane precipitation studies under high p and high T conditions. In this contribution, some experimental studies developed to achieve this objective are presented. Most of the tests were carried out using model system consisting of solutions of asphaltenes, separated form asphaltic residue by the conventional procedure, dissolved in toluene or mixtures of hydrocarbons and titrated with propane, n-heptane or other alkanes at 25 bar and 56 °C. As part of the work, a commercially available high pressure, high temperature compact flocculation device was used to identify the precipitation onset to assess the efficiency of the equipment. Tests at higher pressures and temperatures and studies extended to dead oil samples were also carried as the equipment is supposed to analyze crude oil samples at high p and high T without dilution. Such study is relevant since the reliability on a compact equipment to assess the effect of solvents, inhibitors and other additives on the phase behavior of crude oil at high pressure and temperature represents a valuable contribution to the petroleum industry in the area of flow assurance.
EXPERIMENTAL Materials The liquid hydrocarbons, n-pentane (C5), n-hexane (C6), n-heptane (C7), n-octane (C8), cyclohexane and toluene were commercial PA products obtained from VETEC Química Fina, Rio de Janeiro, Brazil, and were at least 99% pure. Benzene was obtained from Rio Lab, ndecane (C10) was obtained from Aldrich and n-dodecane (C12) was obtained from Merck and ACS Paragon Plus Environment
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were commercial PA products at least 99% pure. Propane (C3) was obtained from Air Liquide, Rio de Janeiro, Brazil, and n-butane (C4) was obtained from White Martins, Rio de Janeiro, Brazil, and were at least 99.5% pure. These gases were maintained at room temperature in 10 kg cylinders for propane and 5 kg cylinders for n-butane and liquefied volumes were transferred to a stainless steel bottle and pressurized to 40 bar. The argon gas was obtained from Air Liquide, Rio de Janeiro, Brazil, and was 99.999% pure. The asphaltic residue was obtained from a local refinery that operates an industrial propane de-asphaltation unit (Duque de Caxias Refinery, Rio de Janeiro, Brazil). The petroleum samples used were dead oil from Brazil fields, supplied by Petrobras, and main characteristics of these samples were presented in Table 1.41
Asphaltenes extraction The asphaltenes extraction was carried out using the following procedure modified from IP143 standard method.42 Approximately 20 g of asphaltic residue solids were mixed with 7 mL of toluene and left still overnight to dissolve. The solution was then diluted by the addition of 1000 mL of n-heptane and left to precipitate under moderate agitation for 24 hours. The precipitated solids were separated by filtration under vacuum using a qualitative filter paper and Soxhlet extracted with n-heptane up to complete removal of the n-heptane-soluble material present in the sample. Half liter of n-heptane was added to the system and held under reflux for 6 hours. Subsequently, the heptane insoluble fraction was dissolved in 350 mL of toluene and held under reflux for at least 4 hours. Finally, asphaltenes solution was dried in a rotary evaporator.
Asphaltenes precipitation onset at room p and T The asphaltenes precipitation onsets were measured by the n-heptane titration method using a Photonics 440 UV-VIS spectrometer. Titrations were carried out by the addition of n-heptane to ACS Paragon Plus Environment
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8.0 mL of asphaltenes-in-toluene solutions at 0.5 wt% and 1.0 wt%. The absorbance at 850 nm was measured using a probe with 5 mm path length directly immersed in the solution and one measurement per minute was normally recorded. The n-heptane flow rate was controlled by a HPLC pump at 0.2 mL/min. The asphaltenes precipitation onset was identified as the n-heptane volume corresponding to the minimum in absorbance and expressed in mL n-heptane/mL toluene solution. Onset measurements were carried out in duplicates. The use of UV-VIS spectrometer was a deliberate choice to check whether precipitation with propane at high T and p followed a similar pattern to that observed using C7 at ambient condition. There is a great amount of information on the asphaltenes precipitation onset for model systems and for crude oil samples using NIR spectroscopy but UV-VIS is closer to the translucence used in the TF5.
Asphaltenes precipitation onset at high p and T The asphaltenes precipitation onset at high pressures and temperatures were determined using the FT5 Flocculation Titrimeter; a high pressure, high temperature flocculation and titration system supplied by the firm PSL Systemtechnik GmbH, Clausthal-Zellerfeld, Germany. Figure 1 presents a scheme of the experimental setup used in the experiments. Titrations were carried out by the addition of a specific titrant to a sample of asphaltenes-in-toluene solutions at 0.25, 0.5, 1.0, 2.0 and 3.0 wt%. Also solutions in a mixture of solvents were prepared, containing aliphatic, aromatic and naphthenic solvent in different proportion as the solvent media. In the case of crude oil samples the procedure was basically the same except that for dark or viscous samples the oil was previously diluted with cyclohexane. The FT5 Flocculation Titrimeter consists basically of the following modules:
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A high grade stainless steel double jacket cylindrical cell clamped with a head containing ports for titration inlet, light probe, pressure sensor, and high pressure temperature sensor, integrated magnetic stirrer and outlet. A 200 mL glass titration vessel is fitted within this flocculation cell module. A high pressure (maximum pressure 700 bar) high precision liquid chromatographic pump used to pressurize the transfer and connection lines of titrant and the stainless steel bottle when pressures higher than 40 bar were required. A compressor used to pressurize the flocculation cell module with argon to the pressure required by the experiment. The gas compressor was used just for experiments where pressures higher than the argon cylinder pressure were required. The use of argon as a pressurization option was necessary because the apparatus does not incorporate a devise to mechanically pressurize the sample in the titration cell. As it will be further discussed in a subsequent section this represents an important limitation of the equipment. A heating/cooling circulation thermostat with a temperature range of -25 to 200⁰C used to heat the flocculation cell module and maintain a specified temperature. A insulation jacket can be used to maintain high temperatures. An optical solid detection system with light sensitivity up to 0.005 lx used in connection with a high p high T (up to 700 bar and 200⁰C) optical fiber probe used to detect and measure the change in light intensity over the probe 3 mm diameter optical gap and register it as translucence. A set including valves, pressure gauge, and lines to interconnect the flocculation cell module to the high pressure titrant bottles for the titration process. Another set including back-pressure and depressurization control to interconnect the flocculation cell module to the argon cylinder for the pressurization process.
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A WinFT software used to control and register the translucence of the liquid medium as well as other physical variables of the process such as volume of titrant injected, pressure and temperature as a function of time. In a typical experiment, 500 mL of propane were transferred from the gas supply line to a high-pressure titrant bottle and mechanically pressurized to 40 bar. This bottle was connected to the flocculation cell module of the FT5 Flocculation Titrimeter. In the case of tests with liquid hydrocarbons (C5 - C12), they were added to the flocculation cell directly from the graduated bottle by a chromatographic pump, after line pressurization to the test conditions. The glass titration vessel containing 30 mL of the test sample and a magnetic stirrer was fixed into flocculation cell that was then tightly clamped. The cell was subsequently heated up and when the temperature was close to 10 degrees Celsius below to the temperature required to the test argon was injected to the flocculation cell to reach the pressure specified for the experiment. The pressure was adjusted by the back-pressure valve and the system was kept for 15 to 30 minutes to stabilize the temperature. Before start the experiment alcohol volume was completed at the graduated bottle, line of titrant was pressurized at the test conditions and translucence was adjusted to 100% at WinFT program. Simultaneously, the HPLC pump, the titrant valve and the computer program were started. The titrant was injected using a JASCO model PU-2086 chromatographic pump at a flow rate of 2.0 mL/min. All experiments were carried out at least in duplicates, and the deviations were within 5%.
RESULTS Asphaltenes Precipitation Onset at Ambient p and T Figure 2 presents a typical graph of absorbance against dilution for the titration of 0.5 wt% and 1.0 wt% asphaltenes solutions in toluene using n-heptane as flocculant at ambient ACS Paragon Plus Environment
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temperature and pressure. The main features in Figure 2 are an initial reduction in the absorbance due to the dilution of the particles free sample by the addition of n-heptane followed by a decrease in the rate of absorbance reduction due to the appearance of asphaltenes particles that scatter part of the incident radiation. At a certain n-heptane/toluene solution ratio a minimum is reached and beyond this point the absorbance increases as the number of particles in the medium increase to subsequently level off or decay due to dilution or particles sedimentation. The asphaltenes precipitation onset is usually identified as the mL of titrant per mL of asphaltenes solution or per gram of oil at the point of minimum absorbance (or maximum transmittance)43 corresponding to 1.5 mL C7/mL asphaltenes solution for the two solutions shown in Figure 2. Other experimental methods detect the precipitation at and earlier stage and identify onset at slightly lower dilutions.44 Titration graphs as that included in Figure 2 may provide valuable information and the n-heptane titration of dead oil has been used as a method to investigate the tendency of a crude to precipitate asphaltenes in order to anticipate its stability under reservoir conditions,45 to select asphaltenes dispersants46 or to identify mixtures of solvents or petroleum fractions presenting a strong tendency to solubilize asphaltenes or asphaltenes-containing deposits.47-48 Another application derives from the extension of the regular solution theory to multicomponent systems for which the solubility parameter (δ) of the solvent medium has been defined as the volume-fraction (ϕ) average of the solubility parameter of all the component of the solution,49-50 −
δ = ∑ϕ i δ i i
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(1)
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Assuming that the asphaltenes may be represented as a single pseudo-component, the system represented in Figure 2 corresponds to a ternary mixture and its solubility parameter, for instance, at the precipitation onset is given by, ̅ = + +
(2)
where the sub-indexes t, a and h denote, respectively toluene, asphaltenes and the hydrocarbon used as flocculant, in the case of Figure 2, n-heptane. Considering the high molecular weight of the asphaltenes fraction and that in this type of experiments its concentration is normally rather low it may be assumed that its volume fraction at the precipitation onset always is negligible compared with the volume fractions of the toluene and the hydrocarbon. This approach has been extensively used to calculate the solubility parameter for polymers using binary solvent mixtures.51 Equation 2 may be then expressed solely in terms of the solvent, and the flocculant, ̅ = +
(3)
The application of Equation 3 to the results reported in Figure 2 using the values of 18.2 and 15.2 MPa1/2, respectively, for the solubility parameter of toluene and n-heptane conduce to a value of 16.4 MPa1/2 for the solubility parameter at the precipitation onset. It is worth to remember that this solubility parameter is related to inferior limit of the asphaltenes solubility parameters range and depends on the crude oil.52-53 Further details on the use of Equation 3 and its extension to assess and modeling compatibility in crude blends are given elsewhere.54-55
The Solubility Parameter of Various Solvents at High p and T In order to use Equation 3 to estimate the effect of composition on the stability and phase behavior of asphaltenes solution and crude oil samples at high p and T it is necessary to know the solubility parameter of some of the crude oil components and other hydrocarbons at these
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temperatures and pressures. This information was gathered using different equations and the following procedure. Values of Tc, Pc, ρc and ω, respectively, critical temperature (K), critical pressure (bar), critical density (Kg/m3) and acentric parameter for all hydrocarbons evaluated were obtained at NIST Chemistry WebBook,56 at National Institute of Standards and Technology homepage. This information at a specific pressure and temperature were used to calculate V, molar volume (m3/mol), by Peng-Robinson cubic equation, using a computational program for solving cubic equations of state from University of Cambridge, available at Patrick Barrie homepage of this University.57-58 The Equation 4 describe the Peng-Robinson model.49
= −
(4)
where, = 0.45724
! "! #"
*
$1 + 0.37464 + 1.54226( − 0.26992(* +1 − , -. ; / = 0.07780 "
" #"
The solubility parameter of a substance was calculated by the Equation 559 using the molar volume calculated like described previously. The ∆vH, molar enthalpy of vaporization (kJ/mol), also was a data available at NIST Chemistry WebBook. =1
∆3 4 6/*
5
(5)
where R is the ideal gas constant (J/K.mol), T is the experimental temperature (K) and δ is the solubility parameter (MPa1/2). In most of the calculations the equation used to calculate the molar volume was PengRobinson cubic equation. However Redlich-Kwong model was also used with the same computational program. The Equation 6 describe the Redlich-Kwong model.60
= − ACS Paragon Plus Environment
(6)
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where, ! !.8
" = 0.42748 # 9/! ; / = 0.08664 "
" #"
The solubility parameter δ was calculated by the Equation 5 like described to Peng-Robinson procedure. To compare solubility parameter data obtained from different models an equation suggested by Giddings et al.61 and modified by Marcus59 for supercritical fluids was also used, Equation 7. = 3.02 : 6/* ;
