Mark A. Trebble

Development of a new four-parameter cubic equation of state

Abstract A new four-parameter cubic equation of state is presented which gives improved predictions of phase behaviour for both polar and non-polar fluids. The effect of the number of parameters contained in the equation is discussed in detail. New temperature functions are presented for the “a” and “b” parameters which are thermodynamically consistent and yet simple in form. Discussion of the optimization and generalization procedures used for parameter evaluation is given. Overall comparisons of PVT predictions with ten other recently developed cubic equations of state are made.

Extension of the Trebble-Bishnoi equation of state to fluid mixtures

Abstract Mixing rules are established for the Trebble-Bishnoi equation of state which are quadratic in form. Vapour—liquid data for thirty-three binary systems were used to evaluate the extension of the equation to fluid mixtures. As many as four interaction parameters were used to fit the binary data, however good correlation was often obtained with only one interaction parameter. Several isotherms were included for each binary system and many chemical types were included in the list of components evaluated. Extensive volumetric data for the binary system of carbon-dioxide and propane were compared to specific volumes calculated from the Trebble-Bishnoi equation in order to evaluate the accuracy of mixture density predictions. A ternary system of acetone—methanol—water was predicted very accurately using interaction parameters regressed from binary vapour—liquid equilibrium data alone. All of the results compared very favourably to results from both the Peng-Robinson and the Soave-Redlich-Kwong equations of state.

A modified Trebble—Bishnoi equation of state: thermodynamic consistency revisited

Abstract An inconsistency in thermodynamic property prediction using the Trebble—Bishnoi equation of state (TB EOS) was recently identified by scientists at the Shell research center in Amsterdam. Specifically, it was found that the TB EOS predicted negative values of isochoric heat capacity and imaginary values for speed of sound at extremely low temperatures. Subsequent investigation by the authors has led to the conclusion that any temperature dependence whatsoever in the co-volume term of van der Waals type equations will result in anomalies in the predicted thermodynamic properties of pure fluids at extreme conditions. This work presents a discussion of the implications of temperature dependence in the TB EOS parameters and describes a modification to the equation in which temperature dependence is completely removed from the co-volume term. By altering the temperature dependence in the attractive parameter we are able to correlate phase behavior with the modified equation more accurately than with the original equation, and the inconsistency has been corrected.

Accuracy and consistency comparisons of ten cubic equations of state for polar and non-polar compounds

Abstract A diverse component library containing extensive PVT data for 75 pure components has been used to critically evaluate ten recently published cubic equations of state. Several of the equations tested showed specific regions of very poor volume predictions and some failures to predict physically meaningful values for volume were even encountered. Temperature dependence in the “b” parameter was included by two of the researchers: Fuller (1976) and Hayen (1981). Investigation revealed that this dependence led directly to the prediction of negative heat capacity in the single phase region. Restraints upon the allowable forms of temperature dependence in the “b” parameter are therefore implied and are discussed in this work.

Estimation of multiple binary interaction parameters in equations of state using VLE data. application to the Trebble-Bishnoi equation of state☆

Abstract A systematic approach for the estimation of binary interaction parameters for equations of state is presented. A least-squares procedure which is computationally very efficient is advocated for the calculation of the binary interaction parameters. Subsequently, if the calculated phase behavior represents the experimental data without a gross bias, the statistically best parameters can be obtained by maximum likelihood (ML) estimation. For these cases, the use is advocated of an implicit ML estimation procedure which is computationally significantly more efficient than the “error in variables” method. The proposed approach is particularly suitable for equations of state which have more than one interaction parameter. In such cases, the best parameter set is chosen from among several combinations of interaction parameters present in the equation of state.

On the applicability of the Sandler-Wong mixing rules for the calculation of thermodynamic excess properties—VE, HE, SE, CpE

Abstract Investigation of derived thermodynamic properties using Sandler-Wong mixing rules shows that predicted excess properties can be unreliable since they are a complex function of the original Gibbs free energy model and the a and b terms. Sometimes the mixing rule will predict a different qualitative behaviour for the heat of mixing when compared to the one predicted by the original Gibbs excess model. Moreover the mixing rules can produce thermodynamic inconsistency at elavated pressures and lead to the prediction of negative heat capacities.

Thermodynamic property predictions with the trebble-bishnoi equation of state

Abstract Predictions of thermodynamic properties including second virial coefficients, enthalpies, isobaric heat capacities, speed of sound, and Joule-Thomson coefficients are presented for the Trebble-Bishnoi equation of state. These predictions are compared both to experimental data and to predictions from the Peng-Robinson equation of state. Both equations of state give reasonable and consistent values for all of the above properties.

Modelling of solid phases in thermodynamic calculations via translation of a cubic equation of state at the triple point

Abstract Salim P.H. and Trebble M.A., 1994. Modelling of solid phases in thermodynamic calculations via translation of a cubic equation of state at the triple point. Fluid Phase Equilibria 93; 75-99 A simple methodology is demonstrated by which a cubic equation of state (EOS) can be modified to allow calculation of equilibria involving solid phases as well as thermodynamic properties of the solid phase itself. Volume translation of the EOS at the triple point is performed using experimental values of solid density. Temperature-dependent parameters of the translated equation (TEOS) are then regressed to vapour pressure data at temperatures below the triple point if data are available. A generalized procedure for calculation of temperature-dependent parameters in the translated equation is also described (for instances where data are unavailable) which requires the heat of sublimation at the triple point and the isobaric heat capacity of the solid. The translated equation can then be utilized to describe solid phases at temperature above the triple point temperature by extrapolation, with the result that the entire pressure-temperature space for a pure component can be accurately reproduced. The methodology is successfully applied to the Trebble-Bishnoi-Salim EOS and is used to calculate both pure component phase behaviour and binary solid-liquid and solid-vapour equilibria. Criteria are also discussed to ensure that the translated equation does not violate thermodynamic consistency.

Correlation of VLE data for binary mixtures of 1-alkanols and normal hexane with the Trebble-Bishnoi equation of state

Abstract Vapour-liquid equilibrium data for six binary systems of primary alcohols (methanol-hexanol) and normal hexane were correlated with both the Trebble-Bishnoi equation of state and the Peng-Robinson EOS. Both equations yielded reasonable results for all of the systems except methanol-hexane where the prediction of liquid-liquid equilibria precluded an accurate fit of the VLE data. For the other five binary systems the average absolute deviation in vapour mole fractions for 98 data points was 0.0197 for the TB EOS and 0.0391 for the PR EOS. Bubble point pressures (60 points) and bubble point temperatures (38 points) were fit with deviations of 5.46% and 1.92 K respectively for the TB EOS while the PR EOS gave deviations of 10.82% and 2.10 K respectively. The binary systems were fit with two generalized interaction parameters for the TB EOS and with one binary-specific interaction parameter for the PR EOS. Vapour pressure and volumetric data for the primary alcohols up to 1-decanol were used to generate component dependent parameters for the TB EOS which were then generalized against molecular weight. Overall errors of 3.72% in vapour pressure were obtained for 237 data points while the PR EOS gave an average of 13.31%. Errors in liquid volumes for 110 data points were 3.15% for the TB EOS and 11.25% for the PR EOS. Vapour volume errors for 85 data points were 7.80% from the TB EOS and 6.48% from the PR EOS.

A generalized approach to prediction of two and three phase CO2-hydrocarbon equilibria

Abstract A generalized form of the Peng-Robinson equation of state is used to calculate two and three phase equilibria in systems containing dense CO 2 contact with both well denned hydrocarbons and with reservoir oils. The thermodynamic model combines completely generalized interaction parameters and critical property correlations with a truncated form of a gamma distribution model for describing the hexane plus fraction of a reservoir oil in a semi-continuous manner. Predictions of phase behaviour in well defined systems are used to demonstrate the accuracy of the method and are compared to regressed fits of the data. Characterization of the C + 6 fraction of the reservoir oil requires specification of the nature of the oil (paraffinic, olefinic, naphthenic, or aromatic), the average molecular weight of the C + 6 fraction, and a single parameter describing the skew of the gamma distribution model. Sensitivity of phase behaviour predictions to the above parameters is found to be reasonably limited and surprisingly good fits of the data are obtained over a wide range of parameter values.