Covolume Parameter Research Articles

Real gas effect on steady planar detonation and uncertainty quantification

The steady planar detonation in hydrogen/air mixture was simulated at elevated pressure. The real gas effect was incorporated by considering the Peng-Robinson equation of state (EoS), nonideal thermodynamic functions and reaction kinetic laws. The nonideal EoS and corresponding thermodynamic functions increase the Chapman-Jouguet (CJ) velocity but decrease the post-shock temperature compared to the ideal gas model, which induces a change of the induction time and distance. The effect of the nonideal reaction rate law depends on the amplitude of compressibility factor and fugacity coefficient and tends to shorten both the induction time and induction distance. These effects counterbalance each other in the complete real gas model which lead to induction time and distance close to the ideal model results. The total heat release was also found to be reduced for the complete model, but the key reactions and their relative importance remain the same as when using the ideal gas model. The nonideal portion of heat release is minor compared with its ideal portion as temperature increases and pressure drops in the reaction zone. The uncertainty of the real gas model, originating from the uncertainty of the parameters in the EoS, was quantified using a Monte Carlo sampling approach. The uncertainty increases linearly with initial pressure and is mainly determined by the species with higher uncertainty factor, larger mole fraction weighted molecular attraction and covolume parameters. Compared to the uncertainty caused by the chemical reaction model, real gas model uncertainty is negligible at low pressure, but becomes of the same order of magnitude at elevated pressure.

A Simple Method to Reduce the Optimized Parameters of Aqueous Amino Acid Solutions Using Association EOSs

Bringing up dissociation/ association equilibria, four new correlated equations are suggested for fixing the parameters of less investigated amino acid systems in a given value for two association models including PC-SAFT and CPA EOSs. Correlated parameters including co-volume parameter, association volumes, and the number of segments are dependent on the molecular weight of amino acids and have derived optimizing and fitting the parameters of the much-used aqueous amino acid solution. Moreover, to indicate the applicability of offered new equations, these formulas are utilized to reduce the number of optimized parameters from six to four for less investigated amino acid solutions. In the case of PC-SAFT EOS, the number of segments and association volume are fixed in a specified value and used to adjust four other parameters of PC-SAFT EOS including the segment of diameter, association energy, dispersion energy, and binary interaction parameter for less investigated amino acid solutions. In the case of CPA model, the correlated parameters are co-volume parameter and association volume, and these two parameters are applied to optimize four other parameters of CPA model such as temperature-dependent energy parameter containing a0 and c1, association energy, and binary interaction parameter. Also, the solubility of studied systems is predicted at different temperatures for aqueous binary solutions. Furthermore, osmotic coefficients and water activity of these less investigated systems are evaluated. In the case of PC-SAFT model, the calculated AADs for liquid density, activity coefficient, water activity, osmotic coefficient, and solubility are 0.0032, 0.0864, 0.0058, 0.051, and 4.46*10-4, respectively. For CPA model, the AADs of these thermodynamic properties are 0.0117, 0.075, 0.0023, 0.16, and 2.79*10-4, respectively. Both models can reproduce the literature data as well, and though the CPA model is a semi-empirical EOS, it doesn’t have irrelevant answers.

Robust and efficient determination of saturation pressure from constant mass expansion data

AbstractThis article presents an improved method for the determination of saturation pressure from pressure‐volume data of constant‐mass expansion (CME) for hydrocarbon mixtures. The conventional methods rely on the direct observation of an incipient phase and/or the change in total compressibility of the fluid sample near the saturation pressure in CME; however, they are unreliable for volatile oils, gas condensates, and near‐critical fluids. The method developed in this research is to capture the expansion behavior of the overall fluid through the attraction and covolume parameters of the Peng–Robinson equation of state. The reliability of the new method is demonstrated by comparing it with the conventional methods in the case studies using 59 different fluids. It was the only method that could reliably identify saturation pressures for five volatile oils, 11 gas condensates, and one near‐critical fluid among the datasets tested.

Accurate quantum-corrected cubic equations of state for helium, neon, hydrogen, deuterium and their mixtures

Cubic equations of state have thus far yielded poor predictions of the thermodynamic properties of quantum fluids such as hydrogen, helium and deuterium at low temperatures. Furthermore, the shape of the optimal α functions of helium and hydrogen have been shown to not decay monotonically as for other fluids. In this work, we derive temperature-dependent quantum corrections for the covolume parameter of cubic equations of state by mapping them onto the excluded volumes predicted by quantum-corrected Mie potentials. Subsequent regression of the Twu α function recovers a near classical behavior with a monotonic decay for most of the temperature range. The quantum corrections result in a significantly better accuracy, especially for caloric properties. While the average deviation of the isochoric heat capacity of liquid hydrogen at saturation exceeds 80% with the present state-of-the-art, the average deviation is 4% with quantum corrections. Average deviations for the saturation pressure are well below 1% for all four fluids. Using Péneloux volume shifts gives average errors in saturation densities that are below 2% for helium and about 1% for hydrogen, deuterium and neon. Parameters are presented for two cubic equations of state: Peng–Robinson and Soave–Redlich–Kwong. The quantum-corrected cubic equations of state are also able to reproduce the vapor-liquid equilibrium of binary mixtures of quantum fluids, and they are the first cubic equations of state that are able to accurately model the vapor-liquid equilibrium of the helium–neon mixture. Similar to the quantum-corrected Mie potentials that were used to develop the covolume corrections, an interaction parameter for the covolume is needed to represent the helium–hydrogen mixture to a high accuracy. The quantum-corrected cubic equation of state paves the way for technological applications of quantum fluids that require models with both high accuracy and computational speed.

Matching the critical point of associating fluids with the Cubic Plus Association equation of state

Simultaneous description of critical point, vapor pressure and liquid density with an equation of state has been a hot research topic. In this work, the possibilities and limitations of the Cubic Plus Association equation of state for describing these properties of associating fluids have been studied. First a procedure is developed to determine the energy and co-volume parameters by converting the system of three equations at the critical point to a single variable root-finding problem. The procedure is then used in parameter estimation to simultaneously match the critical temperature and pressure as well as fit the vapor pressure and liquid molar volume of typical associating fluids. The performance of the estimated parameters is subsequently compared with the parameters fitted only to vapor pressure and liquid molar volume. The possibility of introducing an extra adjustable parameter to improve the performance is further explored. The results show that, for the studied associating fluids, the Cubic Plus Association equation of state with the obtained parameters can match the critical temperature and pressure exactly, fit the vapor pressure within 2% and the liquid molar volume within 3–4% in the range of temperature up to 0.95 critical temperature, while the model overpredicts the critical volume up to 18%.

Modified SRK Equation of State for Modeling Asphaltene Precipitation

AbstractMany of oil reservoirs have dealt with operational problems due to probability of asphaltene deposition as a consequence of asphaltene precipitation during natural production and gas injection into the reservoir. So the prediction of asphaltene precipitation is very important and many equations of state (such as Ping Robinson (PR) and Soave–Redlich–Kwong (SRK)) are used for this reason. These common equations are suitable for non-polar components and a modification is necessary to use them for prediction of asphaltene precipitation because of the polar nature of asphaltene compounds. In this study, the SRK equation of state was modified by deriving a new equation for calculating b-parameter (co-volume parameter); and this modified SRK equation of state was used to model asphaltene precipitation. Finally asphaltene precipitation during natural depletion and first stage gas injection process (in different concentrations), was monitored at reservoir temperature and various pressures. The experimental results show a good match with the modified SRK equation of state.

Simplified local density model for gas adsorption in cylindrical carbon pores

Abstract Carbon materials, coal-bed methane and shale gas reservoirs contain various carbon pore geometries like slit and cylinder. Simplified local density (SLD) model is a beneficial and widely recognized method for gas adsorption in slit-like carbon pore, and hasn't been completely extended from slit-like carbon pore to cylindrical carbon pore, which restricts its actual application. Herein, a SLD-cylinder model is established by obtaining the key parameters (i.e. attractive parameter and co-volume parameter in adsorbed phase) with the introduction of dummy attractive potential and the validation with clean energy gas (i.e. H2 and CH4) computational adsorption results. Meanwhile, for this model, the effect of carbon layers and the sphere of application are fully discussed here. Furthermore, based on SLD-cylinder model, isobars and isotherms of H2 and CH4 at a temperature ranging from 213 to 413 K and a pressure ranging from 1 to 10 MPa are investigated in 2.7 nm and 1.35 nm cylindrical pores. We find that carbon layer number is an essential factor for gas adsorption calculation in a cylindrical pore. Also, the SLD-cylinder model is applicable for the pressure higher than ambient one and the pore size larger than 1 nm. Moreover, owing to the significant effect of bulk fluid density (BFD) on excess adsorption, for CH4 especially, the shapes of isobar and excess isotherm vary mightily with pressure and temperature, respectively. This understanding could be applied to the storage of clean energy gas and the recovering of coal-bed methane (CBM) and shale gas. Therefore, the new developed SLD-cylinder model is as useful as the original SLD-slit model in the applications of gas storage, unconventional gas reservoirs recovering and pore structure characterization.

Peng–Robinson Equation of State Model for Polycyclic Aromatic Hydrocarbons and Long-Chain Hydrocarbons Solubilities in Supercritical Fluids. Correlations Based on Solute Molecular Properties

The Peng–Robinson equation of state (EOS) with the classical van der Waals quadratic mixing rules is used in this study to calculate the solid solubilities of polycyclic aromatic and long chained hydrocarbons in supercritical carbon dioxide, ethane, and ethylene. The effect of the binary interaction parameters kij (for parameter a) and lij (for parameter b) in the van der Waals mixing rules, on the pressure dependence of the solute solubility in the supercritical fluid is examined. The addition of a third parameter, λ2, that modifies the pure component covolume parameter b of the solute, significantly enhanced the ability of the EOS to fit the experimental data. Reasonable optimally fitted binary interaction parameters kij and lij and pure component parameter λ2 were obtained. The EOS parameters were correlated with molecular properties and can be used as estimates for predicting solubilities when experimental data is missing. A phase stability test is proposed to guarantee that the stable root is always ...

A four parameter cubic equation of state with temperature dependent covolume parameter

Abstract A four-parameter, Ghoderao–Dalvi–Narayan 2 cubic equation of state (GDN2 CEOS), is presented which incorporates the following: 1. The experimental value of the critical compressibility factor has been used as a fixed input parameter for calculations; 2. All the parameters (a, b, c, d) of CEOS are temperature dependent functions in the subcritical region and are temperature independent functions in the supercritical region and; 3. A new α function is introduced with two compound specific parameters which are estimated by matching saturated vapor pressure at two fixed temperature points Tr = 0.5, 0.7. Our formalism enables us to cast three of the four parameters of the CEOS as a function of the remaining parameter. The proposed CEOS is used to predict properties of 334 pure compounds, including saturated vapor pressure and liquid density, compressed liquid density, heat capacities at the constant pressure and volume, enthalpy of vaporization, sound velocity. To calculate thermodynamic properties of a pure compound, the present CEOS require the critical temperature, the critical pressure, the Pitzer's acentric factor, the critical compressibility factor, and two parameters of the alpha function. The saturated liquid density predictions for pure fluids are very accurate when compared with GDN1 (Ghoderao–Dalvi–Narayan 1), MPR (Modified Peng–Robinson), and PT (Patel–Teja) equations of state. Unlike MPR EOS, the proposed temperature dependent covolume parameter b in the present work satisfies all the constraints mentioned in the literature to avoid thermodynamic inconsistencies at the extreme temperature and pressure. Using van der Waals one-fluid mixing rule, the present CEOS is further used to predict bubble pressure and the vapor mole fraction of binary mixtures.

Study of adsorption behavior in shale reservoirs under high pressure

Understanding adsorption behavior is of critical importance for the development of shale reservoirs. However, most extant adsorption models cannot properly describe isothermal adsorption when the pressure exceeds a specific value at which the Langmuir volume is obtained. In this study, a modified bi-adsorbent simplified local density/Peng-Robinson model (SLD-PR) was developed in which adsorption in both organic and inorganic components of shale was considered at the same time, and a new modification was imposed on the covolume parameter in equation of state. The model was demonstrated on high pressure methane isothermal adsorption tests of terrestrial shale samples and the high pressure methane adsorption data for marine shales available in the literature samples. It is found that the proposed model can fit isothermal adsorption well for both terrestrial and marine shales with different shapes of adsorption curves, especially in the high pressure range. With the application of the model, it was determined that the methane adsorption capacity of Yanchang Formation terrestrial shale exhibits a positive correlation with its TOC content. Other factors, such as specific surface area, clay content and temperature, also significantly affect adsorption capacity. Furthermore, in the calculation of in-situ gas storage in shale reservoirs, the proposed model produces more reliable results generally when compared with the original SLD-PR model and the Langmuir model.

Modeling phase equilibrium with a modified Wong-Sandler mixing rule for natural gas hydrates: Experimental validation

As natural gas hydrate forms at low temperature and high pressure (i.e., near critical region), vapor phase fugacity calculation becomes crucial in this region due to the intermolecular interactions. This work introduces the modified form of the Wong-Sandler mixing rule in the Peng-Robinson (PR) and Patel-Teja (PT) equation of state (EoS) models to revamp the fugacity calculation of vapor phase. This mixing rule which is based on the excess Gibbs free energy leads to precisely capture the polar and asymmetric properties of mixture components. The modified Wong-Sandler mixing rule is formulated by representing its correction factors for energy and co-volume parameters in terms of temperature and gas-phase composition. Here, the hydrate phase fugacity is determined by the Chen-Guo model that is coupled with an EoS model for hydrate equilibrium analysis. On the other hand, the liquid phase nonideality is estimated by using the Wilson activity-coefficient model. The proposed modified Wong-Sandler based phase equilibrium models are shown to be better than the existing phase rule based models with the three example natural gas hydrate systems (CH4+N2, CH4+C2H6, CH4+C2H4) using tetrahydrofuran asa thermodynamic promoter. Validating these models with real time data sets, it is further investigated that the PT based model shows a better performance compared to the PR equation of state. This apart, these two models show their superiority over the existing PT-van der Waals (PT-vdW) model, for which, a new parameter set is proposed for the said three systems for their improved performance.

Modeling Vapor Liquid Phase Equilibrium for CxHy + CxHyFz Using Peng–Robinson and Perturbed-Chain SAFT

Modeling the vapor liquid phase equilibrium of mixed refrigerants is the essential precondition for the calculation of refrigeration system. In this work, the study on 48 pure substances and 58 binary systems consisting of CxHy and CxHyFz was performed. Both Peng–Robinson (PR) and perturbed-chain SAFT (PCSAFT) were investigated. The relationship between the parameters in PR and PCSAFT was exhibited. Strong linear function of the volume parameter mσ3 in PCSAFT with covolume parameter b in PR were found, as well as the energy parameter me2 in PCSAFT with energy parameter a/b in PR. This correlated characteristic provides an idea for regressing pure substance parameters and a criterion whether the regressed parameters are reasonable or not. The interaction parameters in PR-VDW and PCSAFT show consistent tendency, and both are limited to a very narrow range. The simple predictive method was proposed by using the average value 0.146 for PR-VDW and 0.093 for PCSAFT. The internal correlated behaviors result that...

Development of a novel Peng–Robinson plus association equation of state for industrially important associating compounds

Cubic plus association (CPA) equations of state (EoSs) have found great interest in describing thermodynamic properties of associating fluids. In CPA EoSs, the association contribution proposed by Wertheim is added to cubic EoSs such as Soave–Redlich–Kwong (SRK) and Peng–Robinson (PR). In different developments of CPA EoSs, adjusting the pure component properties such as critical temperature and critical pressure in addition to the association parameters is proposed in some works in the literature. In this work, the PR EoS has been extended to water, phenol, and a number of alcohols (methanol up to dodecanol) by addition of the Wertheim association contribution. In contrast to other CPA EoSs, the experimental values of critical properties are used. The energy and co-volume parameters of PR EoS are modified by introducing a correction factor that is correlated as a function of reduced temperature. The results show that this model is capable of reproducing experimental saturated liquid density and vapor pressure data accurately.

Phase behavior of (CO2 + H2) and (CO2 + N2) at temperatures between (218.15 and 303.15) K at pressures up to 15 MPa

Vapor–liquid equilibrium data are reported for the binary systems (CO2+H2) and (CO2+N2) at temperatures between (218.15 and 303.15)K at pressures ranging from the vapor pressure of CO2 to approximately 15MPa. These data were measured in a new analytical apparatus which is described in detail. The results are supported by a rigorous assessment of uncertainties and careful validation measurements. The new data help to resolve discrepancies between previous studies, especially for the (CO2+H2) system. Experimental measurements of the three-phase solid–liquid–vapor locus are also reported for both binary systems.The vapor–liquid equilibrium data are modeled with the Peng–Robinson (PR) equation of state with two binary interaction parameters: one, a linear function of inverse temperature, applied to the unlike term in the PR attractive-energy parameter; and the other, taken to be constant, applied to the unlike term in the PR co-volume parameter. This model is able to fit the experimental data in a satisfactory way except in the critical region. We also report alternative binary parameter sets optimized for improved performance at either temperatures below 243K or temperatures above 273K. A simple predictive model for the three-phase locus is also presented and compared with the experimental data.

Direct perturbation of the Peng–Robinson attraction and covolume parameters for reservoir fluid characterization

Numerical simulation of enhanced oil recovery processes, such as gas and steam floods, requires that the reservoir fluid of interest is reliably modeled by use of an equation of state (EOS). Reservoir fluid characterization using an EOS can be challenging because of the uncertainties associated with non-identifiable heavy fractions, which are characterized as a few pseudo components.This paper presents a new method for reservoir fluid characterization that directly perturbs the attraction and covolume parameters of pseudo components from the n-alkanes’ values (direct perturbation from n-alkanes, or direct PnA). The direct PnA method systematically adjusts the attraction and covolume parameters of all pseudo components within a given reservoir fluid through a few adjustable parameters. As in our prior research, the ψ(=a/b2) parameter is used to ensure the proper interrelationship between the attraction (a) and covolume (b) parameters. A new approach to adjustment of the ψ parameters is developed with a linear relationship of ψ with molecular weight. The Peng–Robinson (PR) EOS is used throughout the research.The direct PnA method is applied to 84 different reservoir fluids, such as gas condensates, volatile oils, and heavy oils. Characterization of each fluid uses the saturation pressure and liquid density data at different pressures at the reservoir temperature. Other phase behavior data available are used to test the predictive capability of the fluid models developed using the PR EOS with the direct PnA method. Results show that the reservoir fluids tested are reliably characterized by the systematic control over phase behavior predictions in the direct PnA method. Unlike prior characterization methods, the direct PnA method does not require estimation of apparent critical parameters of non-identifiable heavy fractions.

Characterization of reservoir fluids using an EOS based on perturbation from n-alkanes

Characterization of reservoir fluids using a cubic equation of state (EOS) is conducted under unavoidable uncertainties in pressure–temperature–composition (P–T–x) space. The implicit, non-linear relationship between phase behavior predictions and adjustment parameters also makes fluid characterization non-unique and subjective. Although P–T–x space that phase behavior spans is continuous, different characterization methods have been proposed for different types of reservoir fluids.In our previous research, a method was developed for heavy-oil characterization using the Peng–Robinson (PR) EOS with the van der Waals mixing rules without volume shift. Uncertainty issues in heavy-oil characterization were addressed based on the concept of perturbation from n-alkanes (the PnA method). Pseudo components were initially assigned critical parameters that were optimized for n-alkanes in terms of liquid densities and vapor pressures using the PR EOS. The optimized reference values allowed for well-defined directions for perturbation of pseudo components’ critical parameters to match available experimental data. The robust regression algorithm required only three perturbation parameters.In this paper, we extend the PnA method to lighter fluids, such as gas condensates, volatile oils, and near-critical fluids. The main novelty of the new PnA method is that it considers proper interrelationship (ψ=a/b2) between the attraction (a) and covolume (b) parameters of pseudo components. The regression algorithm developed in this research controls the trend of the ψ parameter with respect to molecular weight using a fourth adjustment parameter γ. The ψ and γ parameters become more important for characterizing lighter fluids. For extra heavy oils, the new PnA method naturally reduces to the previous PnA method, where γ is zero.Case studies using 77 different reservoir fluids demonstrate the universal applicability, reliability, robustness, and efficiency of the new PnA method. The fluids used consist of 34 heavy and black oils, 12 volatile oils, and 31 gas condensates. Six fluids are near critical among them. The PnA method controls phase behavior predictions monotonically with parameter adjustments and systematically in P–T–x space. This is demonstrated by quantitative prediction of condensation/vaporization behavior of gas condensates and light oils and minimum miscibility pressures for various oil displacements. The PnA method requires no change in the thermodynamic model used, i.e., it can be readily implemented in existing software based on the PR EOS with the van der Waals mixing rules.We also explain how volume-shift parameters affect compositional phase behavior predictions when used as regression parameters in fluid characterization. The PnA method uses no volume shift, and properly couples volumetric and compositional phase behaviors.

Prediction of Henry's Constant in Polymer Solutions using the Peng-Robinson Equation of State

The peng-Robinson (PR), a cubic equation of state (EoS), is extended to polymers by using a single set of energy (A1, A2, A3) and co-volume (b) parameters per polymer fitted to experimental volume data. Excellent results for the volumetric behavior of the 11 polymer up to 2000 bar pressure are obtained. The EoS is applied to the correlation and prediction of Henry constants in polymer solutions comprising three polymer and many nonpolar and polar solvents, including supercritical gases. The correlation achieved with two adjustable parameter is satisfactory compared with the experimental data. As a result, the present work provides a simple and useful model for the prediction of Henry‘s constant for polymer containing systems including those containing polar, nonpolar and supercritical fluids. Keywords—Equation of state, Henry‘s constant, Peng-Robinson, Polymer solution.

Universal Mixing Rule for Cubic Equations of State Applicable to Symmetric and Asymmetric Systems: Results with the Peng−Robinson Equation of State

A mixing rule for cubic equations of state (CEoS) applicable to all types of system asymmetriesreferred to hereafter as the universal mixing rule (UMR)is proposed. For the cohesion parameter of the CEoS, the mixing rule involves the Staverman−Guggenheim part of the combinatorial term and the residual term of the original UNIFAC Gibbs free energy expression. For the covolume parameter of the CEoS, the quadratic concentration-dependent mixing rule is used with the combining rule for the cross parameter bij = [1/2(bi1/2 + bj1/2)]2. This UMR is applied to the volume-translated and modified version of the Peng−Robinson equation of state of Magoulas and Tassios (Fluid Phase Equilib. 1990, 56, 119), leading to what is referred to as the UMR−PR model. Very satisfactory results are obtained using the existing interaction parameters of the original UNIFAC model for vapor−liquid equilibrium predictions at low and high pressures for a wide range of system asymmetries including mixtures containing polymers. Satisfacto...

A new cubic equation of state for simple fluids: pure and mixture

A two-parameter cubic equation of state is developed. Both parameters are taken temperature dependent. Methods are also suggested to calculate the attraction parameter and the co-volume parameter of this new equation of state. For calculating the thermodynamic properties of a pure compound, this equation of state requires the critical temperature, the critical pressure and the Pitzer’s acentric factor of the component. Using this equation of state, the vapor pressure of pure compounds, especially near the critical point, and the bubble point pressure of binary mixtures are calculated accurately. The saturated liquid density of pure compounds and binary mixtures are also calculated quite accurately. The average of absolute deviations of the predicted vapor pressure, vapor volume and saturated liquid density of pure compounds are 1.18, 1.77 and 2.42%, respectively. Comparisons with other cubic equations of state for predicting some thermodynamic properties including second virial coefficients and thermal properties are given. Moreover, the capability of this equation of state for predicting the molar heat capacity of gases at constant pressure and the sound velocity in gases are also illustrated.

Vapor–liquid equilibrium in polymer–solvent systems with a cubic equation of state

A cubic equation of state (EoS), the Peng–Robinson (PR) one, is extended to polymers by using a single set of energy ( a) and co-volume ( b) parameters per polymer fitted to experimental volume data. Excellent results for the volumetric behavior of the polymer up to 2000 bar pressure are obtained. The EoS is applied to the correlation of low pressure vapor–liquid equilibrium (VLE) data for a variety of polymer solutions by employing only one interaction parameter in three mixing rules: the van der Waals one fluid (vdW1f) — using the Berthelot combining rule (B) for a ij , and the arithmetic mean (AM) and Lorentz (L) ones for b ij — the Zhong and Masuoka (ZM); and the modified Huron–Vidal 1st order (MHV1), by coupling the cubic EoS with the Flory–Huggins (FH) model. Best results are obtained with the ZM mixing rule and the same holds for the extrapolation with respect to temperature and polymer molecular weight (MW). Low errors in correlation and extrapolation are also obtained with the other mixing rules, but phase split is observed in certain cases. The performance of other methods for polymer parameters evaluation is also examined.