PR Equation Of State Research Articles

Correlation of Viscosity of Aqueous Solutions of Alkanolamine Mixtures Based on the Eyring's Theory and Wong-Sandler Mixing Rule

A viscosity model, based on Eyring's absolute rate theory combined with a cubic PR equation of state and Wong-Sandler mixing rule, has been proposed in order to correlate viscosities of aqueous solutions of alkanolamine mixtures at atmospheric pressure and different temperatures. In the proposed method, the energy and size parameters in studied Equation of State (EoS) have been obtained using the Wong - Sandler (WS) mixing rule combined with the NRTL and Wilson Gibbs equations. The NRTL and Wilson parameters for aqueous solutions of alkanolamine mixtures have been correlated using measured viscosity data at atmospheric pressure and different temperatures. The overall average deviation between the experimental and calculated viscosities of studied aqueous solutions of alkanolamine mixtures using Wilson model is 0.92%.

A modified differential-model for interaction parameters in PR EoS with vdW mixing rules for mixtures containing HFCs and HCs

In a previous work, a predictive model for the interaction parameter kij was proposed as kij=ki−kj. The assumed mixing factor ki for the pure components can be obtained by fitting to the correlated kij of mixtures which have available experimental data. In this work, up to 62 binary mixtures containing HFCs and HCs were involved and their reported experimental VLE data were correlated using the PR equation of state with the vdW mixing rules. According to the correlation results, a modified differential-model was proposed for the prediction of the interaction parameter kij with the introduction of the acentric factors and the critical properties of the components. The values of mixing factors ki in the modified model were obtained by the least square fitting of the correlated 62 kij data for 15 HFCs and HCs, namely HFC23, HFC32, HFC125, HFC143a, HFC134a, HFC134, HFC152a, HFC161, HFC227ea, HFC236fa, HFC236ea, HFC245fa, propane, isobutane, and n-butane. 105 predicted kij for mixtures containing these HFCs and HCs were given. The predicted results by using the modified model show a good agreement compared to the correlation results. The overall average absolute deviations of pressures and vapor phase compositions of the predicted results are 1.77% and 0.0086, respectively.

Modeling high-pressure phase equilibria of coalbed gases/water mixtures with the Peng–Robinson equation of state

Reservoir simulations for enhanced coalbed methane recovery and CO2 sequestration require an equation of state (EOS) model that can provide accurate predictions of vapor–liquid equilibria (VLE) of mixtures of coalbed gases (methane, nitrogen and carbon dioxide) with water at high-pressures and near-critical temperatures. An EOS should be capable of providing rapid and accurate estimates of phase equilibrium for gas+water mixtures under the conditions encountered in coalbed reservoir work. Although more complex equations of state may be used, the computational efficiency of cubic equations of state offers a distinct advantage for reservoir simulation purposes. Therefore, in this work, we utilized the Peng–Robinson equation of state (PR EOS) to model the high-pressure phase equilibria of coalbed gas+water mixtures, then developed expressions for the binary interaction parameters as functions of temperature for each of the important binary systems methane+water, nitrogen+water and carbon dioxide+water. For this purpose, an expanded database was assembled that contains vapor–liquid equilibrium measurements for the three binary mixtures. Several case studies were conducted to investigate both the correlative and predictive capabilities of the PR EOS. The PR EOS was found to be capable of describing the phase equilibria of these systems over the range of conditions encountered in coalbed reservoirs; however, the level of precision varies with the degree of complexity used in representing the interaction parameters in the mixing rules. Overall, an absolute average percentage deviation (%AAD) ranging from 0.3 to 1.7 was obtained for the liquid-phase compositions of the gas for the three binary systems when two system-specific interaction parameters were used. Expressions were developed to account for the temperature dependence of the binary interaction parameters. Using these expressions, the liquid-phase compositions for the three systems can be predicted within about 3–6%AAD, which is within three times the experimental errors in the measurements used in the analysis. The predictions for the vapor-phase composition of water, however, were less accurate (in terms of %AAD) due to the low values of the mole fraction of water in the vapor phase, where small errors in composition could lead to large percentage errors.

Experimental determination and prediction of the compressibility factor of high CO 2 content natural gas with and without water vapor

In order to study the effect of different CO 2 contents on gas compressibility factor (Z-factor), the JEFRI-PVT apparatus has been used to measure the Z-factor of dry natural gas with CO 2 content range from 10.74 to 70.42 mol% at the temperature range from 301.2 to 407.3 K and pressure range from 7 to 44 MPa. The results show that Z-factor decreases with increasing CO 2 content in natural gas at constant temperature and increases with increasing temperature for natural gas with the same CO 2 content. In addition, the Z-factor of water-saturated natural gas with high CO 2 content has been measured. A comparison of the Z-factor between natural gas with and without saturated water vapor indicates that the former shows a higher Z-factor than the latter. Furthermore, Peng-Robinson, Hall-Yarborough, and Soave-Benedict-Webb-Rubin equations of state (EoS) are used for the calculation of Z-factor of high CO 2 content natural gas with and without water vapor. The optimal binary interaction parameters (BIP) for PR EoS are presented. The measured Z-factor is compared with the calculated Z-factor based on three models, which shows that PR EoS combined with van der Waals mixing rule for gas without water and Huron-Vidal mixing rule for water-saturated gas, are in good agreement with the experimental data.

Calculation of Carbon Dioxide Solubility in Liquefied Natural Gas

It is important to evaluate the solubility of solid carbon dioxide in liquefied natural gas for natural gas liquefaction at relatively high temperature. The regular solution method and the equations-of-state (EOS) are used to calculate the solubility of carbon dioxide in saturated liquid methane in this paper. The calculation results are compared with the experiment data, and it certifies that the EOS method can be recommended for this kind of solubility calculation. In addition, nitrogen and ethane are common components in natural gas. In this paper, PR EOS is selected to calculate the solubility of carbon dioxide in CH4+N2 and CH4+C2H6 mixtures. Results show that the solubility of carbon dioxide in liquid CH4+N2 mixtures increases with the addition of nitrogen content in the relatively low temperature region (lower than 155K). With the temperature increases, the solubility of carbon dioxide decreases with the increase of nitrogen content. While in liquid CH4+C2H6 mixtures, it increases with the increase of ethane content. liquid nitrogen, liquid oxygen or LNG. In 1940, Fedorova calculated the solubility of carbon dioxide in liquid oxygen and in liquid nitrogen according to ideal solution theory. At the same time, he did some experiments and found that the theoretical calculations are more than 100 times larger than the experimental values(7). In 1962, Davis et al performed a series of experiments on the methane-carbon dioxide system and got the solubility of carbon dioxide in methane at different temperatures(8). Most of these researchers are experts in the field of chemistry, who were focus on a variety of experimental methods of solubility determination. Li from Zhejiang University used the regular solution method and modified Scatchard-Hildebrand relation in her PhD thesis to calculate the solubility of carbon dioxide in liquid nitrogen and liquid oxygen, and obtained good results(9). As liquid methane is a cryogenic non-polar liquid similar with liquid nitrogen and liquid oxygen, similar method has been imitated in the calculation of the solubility of carbon dioxide in the saturated liquid methane in this paper. Additionally, simple cubic equations-of-state has been widely used in non-polar fluid phase equilibria calculations. In 2006, ZareNezhad and Eggeman(10) used PR EOS to predict CO2 freezing points of hydrocarbon liquid and vapor mixtures at cryogenic conditions of gas plants. The overall average absolute relative deviation between the experimental and predicted CO2 freezing temperatures for this binary system is 0.26%. So EOS method is selected for the solid-liquid phase equilibria calculation in this paper.

Prediction of Critical Endpoints Based on the PR Equation of State

In the design of chemical processes, such as catalytic cracking of bitumen and heavy oil, the knowledge of phase behavior at the critical endpoint is essential. Based on the PR equation of state, the algorithm developed by Heidemann and Khalil for calculating critical properties was used to compute critical points. An algorithm for determining the equilibrium phase of the critical point using the tangent plane criterion was developed, and was used to calculate the critical endpoints of different mixtures, including non-polar, polar and associating systems. The critical endpoint, representing the type of the phase behavior, was employed to fit the interaction parameter of mixtures in critical state at high pressure. Lines of critical endpoints for ternary mixtures were also determined with the algorithm.

Study on elution ability of salicylic acid on ion exchange resins in supercritical carbon dioxide

The elution ability of salicylic acid on ion exchange resins in supercritical carbon dioxide has been studied. Some factors influencing elution recovery, including entrainer, temperature, pressure and the flow rate of supercritical fluid CO2 are discussed in this work. The addition of a small amount of entrainer, such as ethanol, triethanolamine and their mixture to supercritical CO2 can cause dramatic effects on the elution ability. The results show that the salicylic acid can be only slightly eluted from the resin with supercritical CO2 alone with temperatures ranging from 307.15 to 323.15 K and pressures ranging from 10 to 30 MPa. Meanwhile, with the same T, P conditions, 40.58% and 73.08% salicylic acid can be eluted from the ion exchange resin with ethanol and ethanol + triethanolamine as the entrainer, respectively. An improved PR equation of state with VDW1 mixing rules is used to calculate the elution recovery of salicylic acid in supercritical CO2 and the results agree well with the experimental data.

Predicting the phase equilibria of CO 2 + hydrocarbon systems with the PPR78 model (PR EOS and kij calculated through a group contribution method)

In 2004, we started to develop a group contribution method aimed at estimating the temperature dependent binary interaction parameters ( k ij ( T)) for the widely used Peng–Robinson equation of state (EOS). Because our model relies on the Peng–Robinson EOS as published by Peng and Robinson in 1978 and because the addition of a group contribution method to estimate the k ij makes it predictive, this model was called PPR78 (predictive 1978, Peng Robinson EOS). In our previous papers eleven groups were defined: CH 3, CH 2, CH, C, CH 4 (methane), C 2H 6 (ethane), CH aro, C aro, C fused aromatic rings, CH 2,cyclic and CH cyclic C cyclic. It was thus possible to estimate the k ij for any mixture containing alkanes, aromatics and naphthenes at any temperature. In this study, the PPR78 model is extended to systems containing carbon dioxide. To do so, the group CO 2 was added. The results obtained in this study are in many cases accurate.

Application of the simulated annealing technique to the calculation of critical points of multicomponent mixtures with cubic equations of state

In this work we have applied a global optimization method for the calculation of critical points of multicomponent systems. This method, based on the simulated annealing algorithm that does not depend on initial guesses and gives preference to global optima, allows locating multiple critical points of a given mixture without any convergence difficulty. Here, it is showed that using appropriate cooling schedule parameters in the simulated annealing algorithm, the computing time for solution of a critical point is reduced drastically so that it can be used for practical purposes that require extensive critical point calculation in multicomponent mixtures. The criticality conditions, based on the tangent plane distance in terms of the Helmholtz energy, were evaluated using temperature and volume as independent variables. The procedure was tested on binary and multicomponent mixtures using the SRK and PR equations of state. The total computing time for forty multicomponent mixtures containing three to twelve components was 4.11 s ranging from 0.06 s for three components to 0.23 s for twelve components, whereas for a multicomponent mixture containing twenty-five components the average computing time was 0.57 s. The results of the critical points calculated in this work illustrate the capability of the simulated annealing algorithm to locate multiple critical points of binary and multicomponent mixtures.

Modified Cubic Equation of State for Prediction VLE Phase Behavior of Fluids: Pure and Mixture

In this paper, a modified form of DPTG (Dashtizadeh‐Pazuki‐Taghikhani‐Ghotbi) cubic equation of state is proposed. To this end parameters of the original DPTG equation (a, b) are also corrected. In order to calculate the saturated vapor pressure, the liquid density, and the vapor volume of 25 pure fluids modified DPTG equation of state is applied and the final results are compared with the SRK and PR equations of state. Finally, the above‐mentioned DPTG EOS is used for calculation of the second virial coefficient, enthalpy and entropy of vaporization, surface tension of pure substances, and the density of normal hydrocarbons and Z. On the other hand, the modified DPTG equation of state is applied to calculate bubble pressure, binary systems mole fraction of the vapor phase, and the density of normal hydrocarbons mixture. The results show that the modified DPTG equation of state could be used for calculation of pure and mixtures of fluids phase behavior with good accuracy.

Application of a new cubic equation of state to computation of phase behavior of fluids and asphaltene precipitation in crude oil

In this study, a new equation of state has been presented based on the perturbation theory. The vapor phase volume, liquid density, saturation pressure enthalpy and entropy of 20 pure components have been calculated by this new EOS. The average absolute deviations of the predicted saturation pressure, liquid density and vapor phase volume are 0.64, 2.69 and 2.390%, respectively. Also, liquid densities of n-alkanes (C 1–C 20) have been estimated by the new EOS whose average absolute deviation is 4.3%. The obtained results have been compared to PR and M–PR equations of state and the accuracy of the new EOS has been approved. Then, the Flory–Huggins model has been modified for prediction of the amount of asphaltene precipitation in crude oil. The estimated amounts of asphaltene precipitation by this model and the new EOS have been compared to the amounts obtained by the Flory–Huggins model and the experimental data. This comparison shows that the modified Flory–Huggins model is more accurate for prediction of asphaltene precipitation in crude oil.

Stereoisomeric effects on volumetric properties under pressure for the system cis- + trans-decalin

In order to study the influence of stereoisomeric effects on the volumetric properties, an extensive experimental study of the density of the binary system composed of the two stereoisomeric molecular forms of decalin – cis and trans – have been measured (294 points) for 7 different compositions including the pure compounds at 3 temperatures (303.15, 323.15 and 343.15 K) and 14 isobars up to 65 MPa with a vibrating-tube densimeter. The experimental relative uncertainty is estimated to be 0.05%. The isothermal compressibility and the isobaric thermal expansivity have been derived and discussed with respect to their behavior due to stereoisomerism. These results have been interpreted as due to the molecular structure as well as changes in the molecular distance and free volume. Three different types of equations of state (EoS) are studied in order to investigate how they take the stereoisomeric influence and effect into account through their required model parameters, revealing that some model parameters are influenced by stereoisomerism, whereas other parameters are not. In all cases for the density, the models are able to distinguish between the two stereoisomeric decalin compounds. The considered models are the PR EoS, the PC-SAFT approach and a molecular dynamics approach. Further, based on the analysis of the model parameters performed on the pure compounds, it has been found that the use of simple mixing rules without introducing any binary interaction parameters is sufficient in order to predict the density of cis- + trans-decalin mixtures with the same accuracy in comparison with the experimental values as obtained for the pure compounds.

High-Pressure Phase Behavior and Equilibria for Chinese Petroleum Residua and Light Hydrocarbon Systems. Part II

The influence of temperature, pressure, and solvent to oil ratio on phase compositions has been determined in a Hg-free high-pressure PVT instrument by using a whole phase sampling technique. Supercritical fluid extraction and fractionation (SFEF) technology, in combination with SARA analysis, have resolved a key problem, the characterization of heavy oil in the modeling of the phase equilibria of the complex heavy oil and light hydrocarbon systems. Phase equilibria of the heavy oil system can be adequately modeled by the SRK and PR equations of state using a new estimation method for critical properties. This information has been used to establish operational envelopes for conventional solvent deasphalting process (SDA) and a new supercritical solvent-selective separation (SSELS) process.

Gas solubilities in molten polymers based on an equation of state

The molecular thermodynamic model developed previously for fluids of chain-like molecules has been extended to correlate the solubilities of gases such as N 2, CO 2, CH 4 and C 2H 4 in molten polymers. Classical mixing rules with one adjustable binary parameter was used to adjust the interaction energy between segments of mixtures. For 21 binaries including 52 systems tested, the average relative deviation of pressure is 19.3%. The accuracy of the correlation can be notably improved by using two parameters κ ij and l ij , the latter was used to adjust the collision diameter between segments of mixtures. The average relative deviation of pressure is lowered to 5.6%. For the systems of CO 2-molten polymer, the model parameters of CO 2 should be re-determined from the experimental solubility in one polymer, the accuracy of the correlation can be notably improved with the use of only one parameter. The comparisons among the calculated results of this work, SAFT and PR equation of state have been made.

Phase diagram studies of methanol–CHF 3 and methanol–H 2O–CHF 3 mixtures

Comprehensive phase diagrams of methanol/CHF 3 and methanol/H 2O/CHF 3 mixtures over the temperature range of 25–100°C and pressure range of atmospheric to 340 atm are reported. Fluoroform is expected to be useful as an alternative to CO 2 for enhancing the fluidity of liquid mixtures due to its high polarity and low viscosity. Therefore, these mixtures will be studied as mobile phases for enhanced-fluidity liquid chromatography and extraction. The phase behavior of methanol/CHF 3 and methanol/H 2O/CHF 3 was compared to that of methanol/CO 2 and methanol/H 2O/CHF 3. Fluoroform is markedly more miscible with methanol or methanol/H 2O than is CO 2. Data for methanol/CHF 3 binary mixtures were also modeled by the Peng–Robinson equation of state. The correlation results showed that the PR equation of state with two temperature-independent binary parameters was capable of representing the experimental data over the entire temperature range with an average relative deviation within 6%.

Special Algebraic Properties of Two-Parameter Equations of State: Homogeneous Azeotropy

Two-parameter equations of state (EOSs), such as the Peng-Robinson and Redlich-Kwong EOSs, or their variants, are widely used for the simulation of industrial processes. In this work, some algebraic properties of two-parameter cubic and noncubic equations of state and of their translated forms are investigated. For saturated pure compounds, translated and untranslated forms describe the same universal curves for pairs of certain dimensionless variables. In the present work we provide general parameters, not previously given in the literature, representing the universal relations set by the van der Waals (VdW), Redlich-Kwong (RK), and modified Carnahan-Starling-van der Waals (MCSV) equations of state. Direct calculation procedures, which make use of the pure compound universal relations, are proposed for the calculation of properties at infinite dilution, leading to the determination of regions of azeotropy. For mixing rules which do not depend on density, it has been found that the universal relations which are valid for saturated pure compounds are also valid for homogeneous azeotropes of any number of components. This finding can be exploited within any existing algorithm designed to find azeotropic points, in order to get faster and more reliable outputs. On the basis of the identity between pure compound and azeotropic universal relations, noniterative methods of calculation are proposed to establish the existence of simple or multiple azeotropy and of pseudocritical points. Simplified calculation procedures are also proposed to obtain values of binary parameters from experimental azeotropic information. All numerical examples have been done using the Peng-Robinson EOS (PR EOS).

Modeling of gas solubilities in polymers with cubic equation of state

The PR equation of state (EOS), as a representative of cubic EOSs, is applied to correlate gas solubilities in molten polymers in this work. Two existing mixing rules, the conventional van der Waals one-fluid mixing rule with one adjustable parameter (vdW1) and that with two adjustable parameters (vdW2), are used to test the capability of the PR EOS. The results show that the vdW1 is not enough, while the vdW2 can give acceptable correlations. A new mixing rule proposed in our previous work, which has been proved to work well for correlating VLE of polymer solutions using cubic EOSs, is further modified here. The new proposed mixing rule with one adjustable parameter shows greatly improved correlations than the vdW1, and is comparable with the vdW2 which requires two adjustable parameters. For 17 systems tested, the total average absolute deviation of pressure for the new mixing rule is 10.0% against 47.8% for the vdW1, and 12.1% for the vdW2.

High-Pressure Vapor−Liquid Equilibria of the Ternary System Carbon Dioxide + Vinyl Acetate + 2-Butanol. Experimental and Modeling Results

Bubble-point and dew-point pressures and vapor−liquid critical points of the ternary system (x1)carbon dioxide + (x2)vinyl acetate + (x3)2-butanol were measured in the temperature range 278−510 K up to 14 MPa in a high-pressure capillary glass tube apparatus, using the synthetic method. The system has been investigated for varying carbon dioxide compositions at fixed ratios of vinyl acetate and 2-butanol (x2/x3 = 1/3 and 3/1) and at fixed 2-butanol composition (x3 = 0.15). Modeling of the vapor−liquid equilibria of the ternary system has been performed using two equation of state (EOS) methods: (1) the Stryjek−Vera modification of the Peng−Robinson EOS with the Wong−Sandler mixing rules (PRSV/WS) implementing the NRTL-GE model; (2) the LCVM model, which is the translated modified PR EOS with the linear combination of Vidal and Michelsen mixing rules implementing the UNIFAC-GE model.

Compositional simulation of reservoir performance by a reduced thermodynamic model

In compositional reservoir simulations, it is usually assumed that a local thermodynamic equilibrium between all phases exists throughout the reservoir within each time step. Phase behavior calculations are conducted based on an equation of state (EOS) in each block. This computation normally takes a very significant percentage of the total CPU time. In this work, an extension of the reduced thermodynamic model (Wang and Stenby, Computers chem. Engng 16, 5449-5456, 1992) to the processes in which both pressure and feed composition vary is demonstrated. This modified model has been tested against the experimental data of gas condensates and the PVT data computed from the EOS. The results indicate that the new model can well represent the influences of changes in the feed composition and the pressure on the K-values. The new model has been implemented into a compositional reservoir simulator, UTCOMP (Chang Ph.D. Thesis, Univ. of Texas, Austin, 1990). Simulations of natural depletion for three well-defined mixtures and of a dry gas recycle process for a gas condensate in the North Sea are performed. The results simulated with the new method are comparable to those computed by the PR EOS. But the CPU time is reduced by approximately 50% by means of the new method compared to the one required by the PR EOS. In addition, this work shows that the CPU time can be further decreased by using the newly suggested procedure (Leibovici and Neoschil, Fluid Phase Equil. 74, 303-308, 1992) for solving the Rachford-Rice equation for phase split.

Phase equilibrium calculation in compositional reservoir simulation

In a reservoir simulation, it is usually assumed that a instantaneous local thermodynamic equilibrium between all phases is reached within each time step. An equation of state (EOS) is used to calculate the amount and composition of each phase in each block. This approach is computationally slow because the solution is obtained by iteration. In this work, use of local thermodynamic model instead of the EOS to reduce the time of computation in compositional reservoir simulation problems is demonstrated. This method has been implemented in a compositional simulator UTCOMP ( Chang, 1990). The natural depletion simulations of three well-defined mixtures are carried out with this new method over a period of 1500 days. The results simulated by the new method are in agreement with those calculated by the Peng-Robinson EOS. But around 40% of the CPU time can be saved by using the new method compared to that required by using the PR EOS for the simulations performed in this work.