Simplified Perturbed Hard Chain Theory Research Articles

A practical equation of state for non‐spherical and asymmetric systems for application at high pressures. Part 1: Development of the pure component model

AbstractA practical, mathematically and computationally simple, equation of state (EOS) has been developed to accurately describe pure component phase behaviour of spherical and chain‐like molecules. The EOS consists of a newly developed hard sphere model and a perturbation term based on the Barker and Henderson approach using the Chen and Kreglewski intermolecular potential model and a double constrained summation as a mathematical expression thereof. The perturbed hard chain theory (PHCT) approach is used to extend the EOS to non‐spherical molecules. The EOS compares well with other more complex models such as the simplified perturbed hard chain theory (SPHCT) and statistically associating fluid theory (SAFT) models and will be extended to describe mixtures in Part 2 of this series. © 2011 Canadian Society for Chemical Engineering

Soft Repulsion and the Behavior of Equations of State at High Pressures

The so-called characteristic curves of Brown—the Amagat (Joule inversion), Boyle, and Charles (Joule–Thomson inversion) curves—of hydrogen are calculated with several equations of state. This work demonstrates that not all equations can generate physically reasonable Amagat curves. After inclusion of corrections for soft repulsion (based on the Weeks–Chandler–Andersen perturbation theory) and quantum effects into the simplified perturbed-hard-chain theory (SPHCT) equation of state, this equation is able to not only generate an Amagat curve, but also predict pVT data, residual Gibbs energies, and heat capacities of several gases at and above 100 MPa reasonably well.

Prediction of critical transitions of ternary mixtures containing ammonia and n-alkanes

The analysis of the critical transitions that occur in ternary mixtures is important to describe their physical behavior. It also enables the phase behavior of multi-component mixtures at high temperatures to be inferred. The objective of this work was to identify the critical transitions that occur in ternary mixtures containing ammonia and n-alkane. The mixture’s critical loci were obtained and tested for stability using thermodynamic criteria expressed in terms of the Helmholtz free energy. Two equations of state were used to represent the Helmholtz free energy: the Carnahan–Starling–Redlich–Kwong (CSRK) and the Simplified Perturbed Hard Chain Theory (SPHCT). In order to identify the existing critical transitions, profiles of the critical loci were calculated along constant compositional ratios χ = x 1 / x 2 . Some of the curves depict higher order critical transitions between liquid–liquid and gas–liquid critical point regions, or two different liquid–liquid critical regions. One of the critical transitions found could be considered as a new sub-class within existing classifications for ternary mixtures proposed by Sadus [R.J. Sadus, J. Phys. Chem. 96 (1992) 5197–5202].

THE MODIFIED PGR EQUATION OF STATE: ASYMMETRIC MIXTURE VLE REPRESENTATIONS AND PREDICTIONS

A set of mixing rules was proposed for the modified Park-Gasem-Robinson (PGR) equation of state (EOS) to extend its predictions to mixtures. The phase behavior predictive capability of this segment-segment interaction model was evaluated for selected binary asymmetric mixtures involving ethane, carbon dioxide, and hydrogen in normal paraffins. The predicted bubble point pressures for the ethane + n-paraffin and carbon dioxide + n-paraffin binaries were compared to those of the Peng-Robinson (PR), simplified perturbed hard-chain theory (SPHCT), and original PGR equations. The a priori predictive capability of the modified PGR EOS is significantly better than that of the PR, SPHCT, and original PGR equations of state for ethane binaries with absolute-average percent deviation (%AAD) of 5%. However, this EOS produces comparable representations for ethane binaries (%AAD of 1.9%) and carbon dioxide binaries (%AAD of 2.0). For hydrogen binaries, the modified PGR EOS showed much better representations (%AAD of 1.7) than the original PGR equation and was comparable to the PR equation.

Modeling of critical lines and regions for binary and ternary mixtures using non-cubic and cubic equations of state

In this work, two non-cubic equations of state, Perturbed Chain Statistical Association Fluid Theory (PC-SAFT) and Simplified Perturbed Hard Chain Theory (SPHCT) were used to predict critical lines and regions for several binary and ternary systems, by fitting one binary interaction parameter ( κ ij ) for each binary system. Results were compared with experimental critical data, which included inorganic gases, hydrocarbons (light and heavy paraffins, non-saturated olefins and aromatics), alcohols, carbon monoxide and carbon dioxide. These experimental data covered wide ranges of temperatures and pressures which are commonly used in industry. For comparison, the data were also modeled using two cubic equations of state: the well-known Peng–Robinson equation (PR) and the Patel–Teja–Valderrama equation (PTV). PC-SAFT and SPHCT pure-component parameters were obtained by regression, adjusting pure-component data such as vapor pressure and saturated liquid molar volume (taken from DIPPR 1.2.0). Excellent results were obtained with the PC-SAFT EoS, while the SPHCT, PR and PTV EoS did not have a good performance. The Heidemann and Khalil algorithm, with two nested single-variable iteration loops (an internal loop for volume and an external loop for temperature), was used for locating a single liquid–liquid or vapor–liquid critical point.

THE MODIFIED PGR EQUATION OF STATE: PURE-FLUID PREDICTIONS

ABSTRACT The Park-Gasem-Robinson (PGR) equation of state (EOS) has been modified to improve its volumetric and equilibrium predictions. Specifically, the attractive term of the PGR equation was modified to enhance the flexibility of the model, and a new expression was developed for the temperature dependence of the attractive term in this segment-segment interaction model. The predictive capability of the modified PGR EOS for vapor pressure and saturated liquid and vapor densities was evaluated for selected normal paraffins, normal alkenes, cyclo-paraffins, light aromatics, argon, carbon dioxide, and water. The generalized EOS constants and substance-specific characteristic parameters in the modified PGR EOS were obtained from pure component vapor pressures and saturated liquid and vapor molar volumes. The calculated phase properties were compared with those of the Peng-Robinson (PR), the simplified-perturbed-hard-chain theory (SPHCT), and the original PGR equations. Generally, the performance of the proposed EOS (%AAD of 1.3, 2.8, and 3.7 for vapor pressure, saturated liquid, and vapor densities, respectively) was better than the SPHCT and original PGR equations in predicting the pure fluid properties.

Rescaling of three-parameter equations of state: PC-SAFT and SPHCT

Two-parameter equations of state like PR or SRK predict a universal critical compressibility factor for pure compounds, being intrinsically unable to reasonably describe the PVT properties of different fluids and their asymmetric mixtures, which require at least three component specific parameters in the density dependence. In this paper, we have explored the capabilities of the Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) and the Simplified Perturbed Hard Chain Theory (SPHCT) equations of state to represent pure compound properties. Original parameters published by their respective authors overestimate critical temperatures, which is detrimental for the study and design of supercritical separation processes. Therefore, we have investigated here the performance of rescaled parameters for each model, i.e. those matching the experimental critical temperature and pressure of pure compounds. The rescaling of the size and energetic parameters as implemented in recent works in the literature was found to provide poor results, and instead a simple procedure to obtain optimized rescaled parameters from only T c, P c and the acentric factor is proposed. These parameters show a regular behaviour for n-alkanes with PC-SAFT and their performance is tested in this work also for carbon dioxide, using the recent equations by Span and Wagner as reference data. The proposed parameters reproduce vapour pressures with the same accuracy of PC-SAFT original parameters but assure an exact representation of the experimental critical pressure and temperature. However, this is achieved at the expense of underestimated liquid densities, this limitation being common to different models and becoming more pronounced for larger molecules.

The evolution of multicomponent systems at high pressures Part III. Effects of particle shape upon the Alder–Wainwright transition

The thermodynamic stability of pure, aspherical hard-body fluids has been examined throughout the range of pressure 1–100000 atm, and over a range of temperature extending from 10 to 10000 K, using the equation of state developed from scaled particle theory (SPT) by Boublik for convex, aspherical, hard-body systems (T. Boublik, J. Chem. Phys., 1975, 63, 4084). The scaled particle theory description of the spherical hard-body system is an exact solution in statistical mechanics, as is also its extension to aspherical hard bodies. Therefore, the Boublik solutions share the absolute property of scaled particle theory. Accordingly, the constraints of the third law of thermodynamics have been applied to this analysis of the thermodynamic stability of aspherical hard-body fluids. Aspherical hard-body fluids are shown to undergo the Alder–Wainwright transition at monotonically lower pressures and reduced volumes with increasing degree of asphericity, in consistent agreement with data from computer simulation experiments. The results developed by the Boublik equations are compared with those of the simplified perturbed hard chain theory (SPHCT); and the SPHCT formalism is shown to give acceptably accurate results for the density and Gibbs free enthalpy at pressures less than approximately 50000 atm, and modest degrees of asphericity.

Heat Capacities at Constant Volume of Pure Water in the Temperature Range 412−693 K at Densities from 250 to 925 kg·m-3

The heat capacity at constant volume CV of pure water has been measured in the temperature range from 412 K to 693 K at 13 isochores between 250 kg·m-3 and 925 kg·m-3. Measurements cover the critical region and coexistence curve. Measurements have been made in both the one- and two-phase regions near the phase transition points. The measurements were made in a high-temperature, high-pressure adiabatic nearly constant-volume calorimeter. The uncertainty of the heat capacity measurements is estimated to be within ±2.5%. Liquid and vapor one- ( ) and two-phase ( ) heat capacities, temperatures (TS), and densities (ρS) at saturation were obtained by the method of quasi-static thermograms. The parameters (c, T*,V*) of the simplified-perturbed-hard-chain-theory (SPHCT) equation of state have been optimized to allow calculations of heat capacities for water in the vapor and liquid phases. The relative average deviations for H2O were within about ±4.5%, except in the critical region where differences reached 15−2...

Prediction of the Condensation Behavior of Natural Gas: A Comparative Study of the Peng−Robinson and the Simplified-Perturbed-Hard-Chain Theory Equations of State

For the prediction of the condensation behavior of natural gas, one has to select an equation of state (EoS) which will be accurate in the temperature and pressure range of interest (10 < P/bar < 70 bar and 250 < T/K < 310). Another requirement of the selected EoS is that it easily can be adapted to a characterization procedure for the heavy-end fraction. For that purpose, two equations were tested: the Peng−Robinson (PR), which is one of the most applied cubic EoS, and the simplified-perturbed-hard-chain theory (SPHCT) equation, which is one of the simplest EoS based on sound statistical mechanical principles. In the underlying study, their predictive capabilities for the prediction of saturated vapor pressures of pure compounds and vapor/liquid equilibrium pressures for binary mixtures are compared. Only components present in natural gas are considered. In addition, new pure-component parameters for the SPHCT EoS for n-alkanes are evaluated. Also a method to find the characteristic energy for non-n-alk...

Predicting calorimetric properties using the original and the modified simplified-perturbed-hard-chain theory equations of state

Previous evaluations of the simplified-perturbed-hard-chain theory (SPHCT) equation of state (EOS) have documented its ability to predict the equilibrium and volumetric properties of many pure fluids and mixtures. Shaver et al. (Shaver, R. D., Robinson, R. L., Jr., and Gasem, K. A. M., 1995. Modified SPHCT EOS for improved predictions of equilibrium and volumetric properties of pure fluids, Fluid Phase Equilibria, 112: 223–248) have offered modifications to the SPHCT model, which have resulted in improved pure-fluid vapor pressure and phase density predictions. In this complementary work, we evaluate the predictive abilities of the SPHCT EOS for calorimetric properties. Specifically, the accuracy of enthalpy and entropy predictions using the SPHCT and its modifications are compared with those of the widely used Peng-Robinson (PR) EOS. Our evaluations were conducted for six pure fluids of varying chemical structure and covering the two-phase and single-phase regions. The results indicate that the abilities of the PR EOS, the original SPHCT EOS and the modified model to predict calorimetric properties are similar to their comparative abilities to predict volumetric properties. The absolute average percentage deviations obtained for liquid density, enthalpy, and entropy, respectively, are: 7.6%, 4.2% and 3.3% for PR; 7.3%, 9.1% and 5.6% for the original SPHCT; and 10.9%, 4.2% and 2.5% for the modified model. Further, the modified SPHCT EOS of Shaver et al. (1995) appears to be more accurate at predicting calorimetric properties than the original SPHCT model.

High-Pressure PVT Behavior of Natural Fats and Oils, Trilaurin, Triolein, and n-Tridecane from 303 K to 353 K from Atmospheric Pressure to 150 MPa

High-pressure PVT data are lacking for many natural oil systems that are processed with supercritical fluids. In this work, we report on the PVT behavior of beef shank, beef tallow, coconut, palm, and palm kernel fats and castor, menhaden, linseed, olive, perilla, safflower, sesame, and soybean oils, each of which was characterized by a fatty acid distribution, iodine value, and saponification value. In addition, we report on the PVT behavior of pure components trilaurin, triolein, and tridecane. Well over 800 PVT data were measured with a static-type bellows apparatus. PVT behavior of the fats showed a marked change in crystallinity with increasing temperature and pressure. PVT behavior for the fats and oils above their melting point was similar, as expected, but could be explained in terms of molecular weight, iodine value, and to some extent, fatty acid composition. At a given temperature and pressure, the specific volume of the oils decreased with increasing iodine value. The data were correlated with the Tait, Peng-Robinson, Hederer-Peter-Wenzel, lattice, Flory, and simplified perturbed hard chain theory (SPHCT). As expected from their functional form, all cubic equations performed poorly in the compressed liquid region. Only the Tait, Flory, and SPHCT equations provided satisfactory correlation. SPHCT parameters were determined for all fats and oils. These parameters were generalized in terms of measurable oil properties, saponification value, and iodine value. The parameter, c, which is related to the number of degrees of freedom, was found to be slightly temperature dependent. With the correlated parameters, PVT behavior of the oils could be described to within 4.9% average error in pressure.

Solution algorithms and parameter sensitivity analysis for the SPHCT equation of state

The simplified perturbed hard chain theory (SPHCT) equation of state (EOS) possesses several attractive features. We have been exploring possible modifications to the equation to improve its performance for both equilibrium and volumetric property calculations. (In a companion study, we have outlined our strategies for modifying the SPHCT EOS.) As a precursor to our study of modifications to the SPHCT EOS, we (a) developed a robust solution algorithm for the SPHCT, (b) established a novel approach to solving the critical-point constraint equations, and (c) performed a parameter sensitivity analysis study for the equation, each of which is described in the present work. These results provided valuable guidance to our efforts in modifying the SPHCT EOS, which are presented in a companion article. The robust algorithm developed for solution of the SPHCT EOS employs a solution equation written in terms of the compressibility factor. This algorithm exhibits better behavior near both the liquid and vapor roots than previous solution equations. However, this robust behavior requires increased computation time during parameter regressions. The SPHCT parameter sensitivity analysis shows that the characteristic temperature ( T ∗ ) and the maximum coordination number ( Z M) have very strong influences on calculated vapor pressures and phase densities. Futher, application of the critical constraints yields more stable parameterization than is obtained by utilizing the SPHCT equation in its original form. Simple correlations are presented for solving the critical point constraints. The correlations (a) significantly reduce computational time and complexity and (b) facilitate application of the critical point constraints without the need to embed complicated numerical routines within existing EOS computer codes.

Modified SPHCT equation of state for improved predictions of equilibrium and volumetric properties of pure fluids

Segment-segment interaction models, such as the simplified-perturbed-hard-chain theory (SPHCT) equation of state (EOS), offer the potential of improved volumetric and equilibrium property predictions. Our previous evaluations of the SPHCT EOS and similar evaluations in the literature lead us to believe that proper modifications can make the SPHCT more accurate. In the present study, we report our results for pure-fluid property predictions. The SPHCT EOS is reduced from a three-parameter to a one-parameter equation of state by using classical critical point constraints. The temperature dependence of the attractive portion of the equation is modified to provide improved vapor pressure predictions. In addition, volumetric predictions are enhanced by incorporating a phenomenological volume translation strategy. The modified SPHCT equation of state is compared with the orginal version and with the Peng-Robinson (PR) equation using experimental data for a variety of chemical species. The modified SPHCT equation yields better vapor pressure predictions than either the original SPHCT or PR equations (1.2% average absolute percent deviation for the modified SPHCT, 3.4% for the original SPHCT, and 4.2% for PR). Improved results are also obtained for liquid and vapor phase densities (2.7% and 1.9% for the modified SPHCT, 6.8% and 6.2% for the original SPHCT, and 6.6% and 3.1% for PR for liquid and vapor phases respectively).

Prediction of one-component vapour-liquid equilibria from the triple point to the critical point using a simplified perturbed hard-chain theory equation of state

A simplified perturbed hard-chain theory (SPHCT) equation of state is used to predict the vapour-liquid equilibria of 69 one-component fluids including n-alkanes, n-alkenes, n-alkynes, n-alkanols, polyatomic polar molecules and noble gases. A procedure is reported for obtaining the equation of state parameters from critical properties and the acentric factor. The predicted vapour pressures are compared with experimental data over a wide range of temperature, which in many cases, extends from the triple point to the critical point of the fluid. The calculations are also compared with results obtained using the Christoforakos-Franck equation of state. It is concluded that the hard-chain theory methodology represents vapour-liquid equilibria of non-spherical molecules more accurately than hard-sphere approaches. The average absolute deviation in the vapour pressure predicted by the SPHCT equation of state is typically within the range of 5%–15%.

Global phase behavior based on the simplified-perturbed hard-chain equation of state

The equation of state that results from the simplified-perturbed hard-chain theory (SPHCT) has been used to calculate phase diagrams for binary fluid mixtures and to classify these phase diagrams in accordance with the system of van Konynenburg and Scott. For molecules with equal or similar sizes, the global phase diagrams are similar to the ones obtained with the van der Waals, Redlich–Kwong, and Carnahan–Starling–Redlich–Kwong equation of state. In addition to the types I–V, one can calculate also types VI, VII, and VIII with the SPHCT equation. For molecules with large size differences two new, main types of phase behavior have been discovered. We propose to call them type IX and X.

Critical point calculations for oil reservoir fluid systems using the SPHCT equation of state

Garcia-Sánchez, F., Ruiz-Cortina, J.L., Lira-Galeana, C. and Ponce-Ramirez, L., 1992. Critical point calculations for oil reservoir fluid systems using the SPHCT equation of state. Fluid Phase Equilibria, 81: 39-84. The ability for predicting the critical points of reservoir fluids using the Simplified Perturbed Hard-Chain Theory (SPHCT) equation of state proposed by Kim et al. is analyzed. The computational procedure developed by Heidemann and Khalil was used for locating the critical points in any multicomponent mixture described by this equation of state. Experimental binary vapor-liquid equilibrium data of interest in the petroleum industry have been used to evaluate the interaction coefficients in this equation for 164 binary mixtures. The performance of the SPHCT EOS to predict critical points is demonstrated on four oil reservoir fluid systems containing up to forty-eight components. Convergence to all the critical points existing in these large systems was obtained in a few iterations without any difficulty.

Application of the simplified-perturbed-hard-chain theory for pure components near the critical point

Abstract van Pelt, A., Peters, C. and de Swaan Arons, J. 1992. Application of the Simplified-Perturbed-Hard-Chain Theory for pure components near the critical point. Fluid Phase Equilibria, 74: 67-83. This investigation is part of a study to classify the fluid phase behaviour of binary mixtures, using the Simplified-Perturbed-Hard-Chain Theory (SPHCT). The SPHCT equation of state has been evaluated with new parameter values for c, V★ and T★. They allow the saturated vapour pressures of non-polar substances to be predicted near their critical point. A procedure is presented that allows the fitting of saturated vapour pressure data while fixing pc and Tc at the experimental values. Moreover, a very simple relation is proposed to obtain values of the parameters ϵ k and σ for the normal alkanes. These parameters are necessary for mixture calculations. The maximum deviation in the saturated vapour pressure curve is less than 5%, and the maximum average absolute deviation is about 3% within a temperature range of 0.75

Effect of hard-sphere structure on pure-component equation of state calculations

Hard-sphere fluid structure is shown to be important in describing the thermodynamic properties of the square-well fluid. A simple local composition model using the exponential approximation (EXP) of Anderson and Chandler (1972) for the pair correlation function is developed. The hard-sphere pair correlation function is determined using the analytic solution to the Percus-Yevick theory (Wertheim, 1964). The model is shown to predict the properties of the square-well fluid that are in good agreement with molecular dynamics data. The equation is used to predict the properties of argon and methane. The equation is also incorporated into the Simplified-Perturbed-Hard-Chain Theory (SPHCT), and the new equation improves calculations of vapor pressures and saturated densities for non-polar and polar fluids.

Measurements and calculations of phase equilibria in binary mixtures of propane + tetratriacontane

Abstract Peters C.J., de Roo J.L. and de Swaan Arons J., 1992. Measurements and calculations of phase equilibria in binary mixtures of propane + tetratriacontane. Fluid Phase Equilibria , 72: 251-266. This paper reports on the phase behaviour of binary mixtures of propane and tetratriacontane ( n -C 34 H 70 ) in the near-critical region of propane. The measurements cover a temperature region from about 350 K up to 420 K. Pressures up to 10 MPa were applied. It has been found that in the binary mixture propane + tetratriacontane partial miscibility occurs in the liquid phase, i.e. in this binary a three-phase equilibrium liquid + liquid + vapour (1 1 1 2 g) with a lower and upper critical endpoint exists. As well as the measurement of this three-phase equilibrium, various types of two-phase boundaries and the three-phase equilibrium solid tetratriacontane + liquid + vapour (s B 1g) have been determined experimentally. In addition simplified-perturbed-hard-chain theory (SPHCT) was applied to model the vapour + liquid equilibrium data. The results obtained are compared with similar calculations, using the Peng-Robinson (PR) equation of state. The calculations established that when modelling vapour + liquid equilibria of binaries with molecules differing largely in chain-length, the SPHCT equation of state should be preferred over the PR equation.