RATIONAL CONSTRUCTION OF AN AUGMENTED HARD CORE EQUATION OF STATE FOR PURE COMPOUNDS AND STUDY OF ITS APPLICATION TO MIXTURES

The Carnahan-Starling hard-sphere equation of state perturbed by attractive forces expressed in a virial expansion has been studied in the vicinity of the critical point. The three van der Waals conditions for the critical isotherm at the critical point were used to determine the minimum number of terms required in the perturbation series. P-v-T data at other temperatures were used for the determination of the temperature dependence of the virial terms. The equation has been tested for mixtures using the mixing rules of Arai et at.     

Critical isotherms from virial series using asymptotically consistent approximants

The low‐density equation of state of a fluid along its critical isotherm is considered. An asymptotically consistent approximant is formed having the correct leading‐order scaling behavior near the vapor‐liquid critical point, while retaining the correct low‐density behavior as expressed by the virial equation of state. The formulation is demonstrated for the Lennard–Jones fluid, and models for helium, water, and n‐alkanes. The ability of the approximant to augment virial series predictions of critical properties is explored, both in conjunction with and in the absence of critical‐property data obtained by other means. Given estimates of the critical point from molecular simulation or experiment, the approximant can refine the critical pressure or density by ensuring that the critical isotherm remains well‐behaved from low density to the critical region. Alternatively, when applied in the absence of other data, the approximant remedies a consistent underestimation of the critical density when computed from the virial series alone. © 2014 American Institute of Chemical Engineers AIChE J, 60: 3336–3349, 2014     

Correlation of Zeno (Z = 1) Line for Supercritical Fluids with Vapor-Liquid Rectilinear Diameters

For a wide range of substances, extending well beyond the regime of corresponding states behavior, the contour in the temperature-density plane along which the compressibility factor Z = P/ρkT is the same as for an ideal gas is nearly linear. This Z = 1 contour, termed the Zeno line, begins deep in the liquid region and ascends as the density decreases to the Boyle point of the supercritical fluid, specified by the temperature T_B for which (dZ/dρ)_T = 0 as ρ → 0; equivalently, at T_B the second virial coefficient vanishes. The slope of the Z = 1 line is — B₃/(dB₂/dT), in terms of the third virial coefficient and the derivative of the second, evaluated at T_B. Previous work has examined the Zeno line as a means to extend corresponding states and to enhance other practical approximations. Here we call attention to another striking aspect, a strong correlation with the line of rectilinear diameters defined by the average of the subcritical vapor and liquid densities. This correlation is obeyed well by empirical data for many substances and computer simulations for a Lennard-Jones potential; the ratios of the intercepts and slopes for the Zeno and rectilinear diameter lines are remarkably close to those predicted by the van der Waals equation, 8/9 and 16/9, respectively. Properties of the slightly imperfect fluid far above the critical point thus implicitly determine the diameter of the vapor-liquid coexistence curve below the critical point.     

Equation of state for gaseous ethane determined from isotropic model potentials

A four-term virial equation of state was combined with isotropic model potentials to predict accurate volumetric and caloric thermodynamic properties of ethane in the gas phase. The parameters of the model potentials were determined from a fit to speed-of-sound data alone; no other data were used. The approximation used for the fourth virial coefficient included all interactions that contain up to two triplet potentials. Predicted ordinary second and third virial coefficients are in agreement with the data of Funke et al. [8]; we believe that predicted fourth virial coefficients are reliable and accurate. In the subcritical temperature region, the equation of state predicted compressibility factors that deviate by less than 0.04 percent at densities of up to 2.7 mol/dm³ (≈ 0.4ρ _c ). At supercritical temperatures, compressibility factors deviate by less than 0.02 percent at densities of up to 2.6 mol/dm³; also, in this region predicted isobaric heat capacity agrees with available data to within uncertainties of 0.4 percent at densities above 3 mol/dm³. We demonstrated that the four-term virial equation is more accurate than the three-term analogue.     

Equation of state for gaseous nitrogen determined from isotropic model potentials

A four‐term virial equation of state was combined with isotropic potential models to predict accurate volumetric and caloric thermodynamic properties of nitrogen in the gas phase. The parameters of the model potentials were determined from a fit to acoustic data alone; no other data was used. For nitrogen, it was only necessary to approximate the fourth virial coefficient at the level of interactions that contained no more than one triplet potential; higher order approximations offered no further advantage. It was shown that the four‐term virial equation was more accurate than the three‐term analogue. It was found that predicted virial coefficients became consistent with experimental values when temperature was >150 ± 30 K; conversely, below this range virial coefficients predicted by the model did not agree with experiment. It was believed that predicted fourth virial coefficient was reliable and accurate only above about 150 K. Predicted compressibility factors deviated by <0.05% at pressures of up to 10–12 MPa, or densities up to 4 mol/dm³ (≈0.4ρ_c), only when temperature was >220 K. Values of enthalpy predicted from the equation of state showed good agreement with experimental data.     

ON THE PREDICTION OF HEATS OF VAPORIZATION OF PURE COMPOUNDS

An expression to predict the values of the latent heat of vaporization of pure compounds at their normal boiling point has been developed. A knowledge of the critical temperature, critical pressure, normal boiling point and acentric factor of the compound coupled with the use of generalized correlations for second virial coefficient and liquid molar volume allows to obtain good estimates, particularly for alcohols and ketones. Results for 71 various substances are compared with experimental values and with the results of eight other equations available in the literature. The possibility of using the calculated heats of vaporization at the normal boiling point to estimate values at other temperatures is discussed.     

ON THE PREDICTION OF HEATS OF VAPORIZATION OF PURE COMPOUNDS

An expression to predict the values of the latent heat of vaporization of pure compounds at their normal boiling point has been developed. A knowledge of the critical temperature, critical pressure, normal boiling point and acentric factor of the compound coupled with the use of generalized correlations for second virial coefficient and liquid molar volume allows to obtain good estimates, particularly for alcohols and ketones. Results for 71 various substances are compared with experimental values and with the results of eight other equations available in the literature. The possibility of using the calculated heats of vaporization at the normal boiling point to estimate values at other temperatures is discussed.     

EVALUATION OF PENG ROBINSON EQUATION OF STATE IN CALCULATING THERMOPHYSICAL PROPERTIES OF COMBUSTION GASES

A set of simple equations of the thermodynamic and transport properties of the combustion gases of a gas turbine have been derived based upon the critically evaluated data and two equations of state: The virial equation of state and Peng-Robinson (PR) equation of state.

The properties which have been considered were, density, specific heat at constant pressure, enthalpy, entropy, viscosity and thermal conductivity.

The temperature range was (200-2600 K) theoretically while the pressure range was (0.3-1.2 MPa).

A computer program, to evaluate the departure of thermophysical properties using virial and PR equations of state, was used.

The Peng Robinson (PR) equation of state gave better estimated accuracy than the virial equation of state especially in evaluating the departure of thermodynamic properties.     

Automated batch characterization of polymer solutions by static light scattering and viscometry

Using a programmable mixing pump, light scattering flow chamber, refractive index detector, and single capillary viscometer, the batch (unfractionated) characterization of polymers in solution has been automated. Three different schemes to produce polymer concentration gradients were used, and values for weight average mass M_w, root mean square radius of gyration 〈S²〉^1/2, second virial coefficient A₂, and intrinsic viscosity [η] were determined for a broad distribution sample of poly(vinyl pyrrolidone) (PVP) and a narrow fraction of poly(ethylene oxide) (PEO). High concentration experiments on the PVP also allowed determination of the third virial coefficient A₃. The method has several advantages over traditional manual methods in terms of accuracy, sample preparation, and amount of labor required. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 73: 2359–2368, 1999     

AN EMPIRICAL VIRIAL-LIKE CUBIC EQUATION OF STATE FOR PHASE EQUILIBRIUM CALCULATIONS

An empirical cubic equation of state(EOS) was obtained by truncating the virial expansion in reciprocal of molar volume after the third term. The constants of the EOS was generalized in terms of critical temperature, critical pressure and Pitzer's acentric factor.

In pure component applications the EOS exhibited a performance comparable to Peng-Robinson (1976) EOS in the reduced temperature range of 0.5 to 1. The present EOS tends to predict better saturation liquid volumes at reduced temperatures below 0.8, and better estimations for second virial coefficient at high reduced temperatures.

The EOS was successfully employed for vapor liquid equilibrium calculations for some mixtures of normal or slightly polar fluids with traditional one binary parameter mixing rule at moderately high pressures. At low reduced temperatures, where conventional one adjustable parameter applications of the cubic equations compare unfavorably with dual methods based on excess Gibbs energy functions for the liquid phase, a new two constant mixing rule introduced by Stryjek and Vera (1986) was employed for the present equation of state.     

Temperature dependence of unperturbed dimension and interaction parameters of polyester resin in solvents

The flow behaviour of a polyester in various solvents was studied at temperatures ranging from 10 to 80°C. The practical data obtained from the temperature dependence of limiting viscosity number [η] were used to calculate unperturbed dimensions and interaction parameters of the polyester resin in poor, moderate and good solvents. The data provided information regarding conformational transitions in the polymer chains in terms of exothermic or endothermic local ordering of solvents on resin segments and their fixation on polymer coils. The temperature dependence of unperturbed dimensions K_e, Flory–Huggins interaction parameter χ₁₂, the second virial coefficient A₂, entropy parameter U₁, enthalpy parameter K₁, and viscosity expansion factor α_n, has been used to estimate the solvent quality for the resin.     

Solubility parameters of hydrocarbons

Information available in the literature on vapor pressures, saturated vapor and liquid densities, and critical constants, for different hydrocarbons, has permitted the calculation of δ, the solubility parameter advanced by Hildebrand, at temperatures up to and including the critical point. For these hydrocarbons, the residual quantity, δ-δ_c was found to depend on 1-T_R according to the relationship, where k appears to be a constant within different classes of hydrocarbons. Values of δ calculated with this equation were compared with the corresponding values used to develop it, and produced an average deviation of 0.85% for 153 values considered which represented 17 hydrocarbons.     

Critical temperatures and pressures and high temperature vapour pressures of seven alcohols

Critical temperatures and pressures have been determined experimentally for the four lowest n-alcohols, 2-propanol, 2-butanol, and 2-methyl-1-propanol, and are given in the penultimate line of Table 3. The last line of Table 3 lists the weighted mean critical pressures, with the corresponding critical temperatures obtained from he vapour pressure equation. Comparison with the corresponding data from standard reference books reveals deviations from the experimental data in some [2, 4] and incorrect critical pressure for ethanol in all five reference books quoted in Table 3. In addition, vapour pressure data were obtained for the same alcohols in the temperature range approaching the critical point. The measurements are reproduced by the simplified Clausius-Clapeyron equation with the constants and standard deviation given in Table 2.     

一些半经验立方型流体状态方程的近似推导

将virial方程高次项的贡献近似地整合到三项截断式中，然后利用第2 virial系数B的近似式对virial方程的二项及三项截断式进行不同的形式变换和扩展，近似地得到了Martin、Abott的总包性立方型方程以及一些半经验立方型方程或其类似形式，如van der Waals、Clausius、Redlich-Kwong、Usdin-McAuliffe (Fuller)、Martin、Peng-Robinson、Patel-Teja、Harmens-Knapp、Tong-Liu、LHSS、MKS、Du-Guo、CCOR、SIRK，Lee-Edmister方程等，这表明上述方程具有一定的统计力学基础.另用同样方法得到了9个新的三次方程，其中有些方程在形式和参数方面具有较大的灵活性，因而可能具有较好的应用前景.     

Flame Temperature Calculations at High Temperature and Pressure

In the past, real gas effects on the flame temperature were evaluated using the virial equation of state. Usually, the virial expansion was truncated after the third term. In this work the equation of state for dense gases proposed by Haar and Shenker is considered. The implementation developed for H₂O, CO, CO₂, H₂, and N₂ by Powell, Wilmot, Haar and Klein is used. The contribution of all minor species is assumed to be approximated by a Lennard-Jones gas with ϵ/k = 100 K and σ = 3.0 Å. It is found that the more conventional approach is valid up to a loading density of 0.2 g/cm³. As density increases real gas effects cause the calculated flame temperature to decrease and the calculated pressure to increase. A computer program to perform the calculations has been devised for a personal computer.     

On the derivation and application of a new vapor pressure equation

Starting from the Clapeyron equation and making use of the equivalent expression for the reference state for latent heat of vaporization of pure liquids, namely at 0°K., a new vapor pressure equation is derived as follows: _χ(T_r) has been tabulated for T_r values in the range 0.50 < T_r <0.99 after programming on an IBM 1620 computer. The new vapor pressure equation has been tested with a variety of liquids both polar and non-polar, and found to predict vapor pressure data in the orthobaric range with an average error of ± 1.90% and maximum error of ± 3.20%. A significant outcome of this investigation is the identification of Riedel's factor α_o and Pitzer's acentric factor ω These have been shown to be functions of critical properties and reduced boiling point.     

Virial coefficients of normal fluids

The force constants of the square-well potential for normal fluids were correlated in terms of the critical properties P_c, T_c, and the acentric factor ω, by means of a non-linear regression technique. The second virial coefficients, a total fo 100 points chosen uniformly with respect to P_r and ω from the compilation of Dymond, together with the compressibility factors, taken from the generalized tables of Pitzer. were employed in the correlation. The valid ranges of the correlation are 0.5 ?T_r ? 2·0 and 0 ? ρ_r ? 1·0. The calculated results for the second virial coefficients compare favourably with the generalized correlation of Pitzer and Curl. The calculated third virial and cross third virial coefficients, and the compressibility factors also agree well with the literature values.     

The new simple extended corresponding-states principle: vapor pressure and second virial coefficient

A new simple extended corresponding-states principle has been developed to represent and predict the thermophysical properties of fluids. The extended corresponding-states principle only requires the substance-dependent critical parameters and acentric factor which enhances the corresponding-states principle of Pitzer et al. to include the behavior of substances whose force fields deviate strongly from spherical symmetry. The additional corresponding-states parameter defined in terms of the deviation of the critical compression factor of a real molecule from that of spherical molecules is independent of experimental data for any specific property. The new simple extended corresponding-states principle presented here remarkably improves the representation of the vapor pressure from the triple point to the critical point and the second virial coefficient from the triple point to the highest temperatures over which experimental data exist. Accurate results for these two well-understood properties are given for simple, normal, polar, hydro-bonding and associating compounds. The results also show that the new simple extended corresponding-states principle is more reliable and accurately predicts the vapor pressure and second virial coefficient of a strongly nonspherical fluid than any other existing methods.     

COMPARISON OF A PAIR OF THREE PARAMETER EQUATIONS OF STATE

The lattice fluid (LF) equation of state derived by Sanchez and Lacombe from a lattice model is compared to the empirical Peng-Robinson (PR) equation for normal alkane fluids ranging from methane to heptadecane in molecular weight. With respect to vapor pressure predictions, the equations are both good. The LF equation is superior, especially for higher molecular weight fluids, to the Peng-Robinson equation in predicting saturated liquid densities. For carbon numbers less than 6, the PR equation predicts heats of vaporization more accurately, whereas for carbon numbers greater than 9 the LF equation is more accurate than the PR one for temperatures lower than about 95% of critical.     

Dew‐point measurements for water in compressed carbon dioxide

When transporting CO₂ for sequestration, it is important to know the water dew point in order to avoid condensation that can lead to corrosion. A flow apparatus to measure the water content at saturation in a compressed gas has been constructed. A saturator humidifies the flowing gas by equilibrating it with liquid water. Then, a gravimetric hygrometer measures the water mole fraction of the humid gas. Dew‐point data for H₂O in CO₂ on six isotherms between 10 and 80 °C at pressures from 0.5 to 5 MPa are reported. The uncertainties in water content at the dew point (expanded uncertainty with coverage factor k = 2) are on average 0.3%, significantly smaller than in any previous work. The data have been analyzed to extract the interaction second virial coefficient; the values are consistent with the theoretical estimates of Wheatley and Harvey but have a much smaller uncertainty. Published 2015 American Institute of Chemical Engineers AIChE J, 2015 © 2015 American Institute of Chemical Engineers AIChE J, 61: 2913–2925, 2015