Adaptation of van der waals equation of state to the gaseous and liquid behavior of argon: Its application to simple substances

Utilizing the fact that the co-volume parameter, b = v_c/3, van der Waals equation of state has been reduced to the form where β = a/P_cv²_c and z = P_cv_c/RT_c. PVT data for argon, extending into the liquid region, have been utilized to establish the cohesive pressure parameter a as a function of temperature and density. This dependence is expressed in equation form and has been used to predict the PVT behavior of argon in its gaseous and liquid states, with an average deviation of 1.35% (250 points). The deviations resulting from this relationship for substances having similar z_c-values (z_c ≈? 0.291) were found to be of this order of magnitude.     

Saturation and metastable properties of the van der waals fluid

Modern applications require accurate thermodynamic equations of state that portray stable and metastable states of fluids. Earlier work has shown that the famous van der Waals equation, contrary to widely held views, finds its natural niche in the rank of real substances ordered according to their values of the acentric factor. In this note, the saturated and metastable state properties of a van der Waals fluid are presented in the form of a temperature table and a pressure table. A comparison of these results with the available data for mercury is made and it is established that mercury is approximated rather well as a van der Waals fluid.     

Ternary liquid—liquid equilibria: the van laar equation

The capabilities of the three suffix van Laar equations in predicting the more complex types of ternary liquid—liquid phase diagrams have been examined. While the limitations enforced by thermodynamic consistency on the binary parameters seriously affect the quantitative accuracy with which correlations can be achieved, the van Laar equations are capable of describing qualitatively a variety of complex phenomena. In the system water, n-hexane, phenol, the eruption of a three liquid phase region inside a binodal curve as temperature is decreased is accounted for by the van Laar equation. The similar eruption of a three liquid phase region inside a band in the system glycol, lauryl alcohol, nitroethane is also described. A variety of other types of computed phase diagrams, all of which (save one) have experimental counterparts, are presented. It is also shown that the van Laar equation is capable of quantitatively correct correlation of data for some selected simple systems.     

Modified van Laar’s Equation and Its Application to VLE ofPolymer Solutions

The original van Laar‘s theory has been modified. The internal pressures of components and mixture are expressed by Prank‘s relation and the excess entropy for mixing of components is also considered. A new activity coefficient equation, which can be satisfactorily applied to polymer solutions, is obtained. The calculated results for the VLE of 179 polymer solutions show that the accuracy of fit is evidently superior to UNIQUAC equation.     

A calculation for the viscosity of fluids by using van der Waals equation of state

A new equation for the viscosity of fluid is presented by considering that the viscosity is equal to the product of the shear pressure and the shear relaxation time. The shear pressure and the shear relaxation time are calculated thermodynamically by applying the van der Waals model for fluids. The calculated viscosities for various simple substances are in good agreements with those of the observed values through liquid-critical point-gas region.     

Reduced van der Waals equation of state and critical properties of rubbers

A reduced equation of state describing the deformation mode as simple elongation is derived for van der Waals networks, thus manifesting general relationships for molecular networks also embracing their stability characteristics and the occurrence of a phase transition.     

Critical behavior of pure confined fluids from an extension of the van der Waals equation of state

The critical behavior of pure fluids confined in porous solids is investigated using an extension of the van der Waals equation of state. The effects of pore size and of the interaction between fluid molecules and pore walls are evaluated. Fluid molecules were assumed spherical, interacting with each other and with the walls of cylindrical pores through distinct square-well potentials. It was found that our model may predict either one or two mechanically stable critical points for the confined fluid, depending on its specifications. When two critical points are predicted, one is attributed to a major contribution of molecule-molecule interactions and the other to a major contribution of molecule-wall interactions. The confined fluid critical point(s) presented a complex dependence on the pore size, due to the interplay of molecule-molecule and molecule-wall interactions. It is shown that the prediction of two critical points for confined fluids is useful to describe adsorption isotherms with two phase transitions.     

A molecular interpretation of the parameters of the van der Waals equation of state for real networks

Summary A molecular interpretation of the phenomenological van der Waals parameters of the equation of state for real molecular networks is presented. The theoretical treatment is eludicating the special role of the crosslinks.Herrn Prof. Dr. G.V. Schulz zu seinem 75. Geburtstag gewidmet     

Some properties of the NRTL equation in correlating liquid—liquid equilibrium data

The NRTL equation proposed by Renon and Prausnitz has been discovered to have some properties that may complicate its use. Specifically, there is more than one set of parameters, τ₁₂ and τ₂₁, which fit given solubility data with a fixed value of the nonrandomness parameter. Also, some sets of parameters can predict two different miscibility gaps at a fixed temperature and pressure in a binary system. It is shown that it is possible to predict more than one miscibility gap even in symmetric systems and that this occurs for at least some values of the nonrandomness parameter greater than 0·426, which was the limit for phase separation reported by Renon and Prausnitz.The value of the NRTL equation in correlating and extrapolating equilibrium data of various kinds has been amply demonstrated in the literature. However, some NRTL parameter sets arrived at simply by correlating data may predict the complex physical behavior discussed here.     

Modified Redlich—Kwong equation of state for saturated vapour—liquid equilibrium

Defining the dimensionless temperature T* as: T* = ($\text{T}{}_{\mathrm{c}}\text{T}$?1)/($\text{T}{}_{\mathrm{c}}\text{T}{}_{\mathrm{NB}}$?1) a new temperature dependence of the attractive force part of the Redlich—Kwong equation of state is presented. The new modified Redlich—Kwong equation predicts the saturated liquid—vapour equilibrium states accurately over the entire liquid range from the triple point up to the critical point. A comparison is made with experimental data for a series of selected organic, inorganic and quantal fluids (34 compounds, 740 data points pairs) and results obtaine by the use of two modifications by Soave of the Redlich—Kwong equation of state.     

On the modified Stefan-Maxwell equation for isothermal multicomponent gaseous diffusion

Understanding multicomponent gaseous diffusion in porous media is crucial to describing the transport of fuel and reaction products in the anode of a solid oxide fuel cell, for which this work was originally pursued. The Stefan-Maxwell approach provides a general theoretical framework so that measured or predicted binary diffusion coefficients may be utilized for multicomponent diffusion (for which Fick's law is invalid). This approach has since been extended to account for a porous solid structure resulting in what is usually referred to as the “modified Stefan-Maxwell equation”, which is the subject of the present work. Using a virtual experiment involving ternary diffusion and the modified Stefan-Maxwell equation, it is shown that multicomponent diffusion in the Knudsen regime (in which wall drag is significant) produces a gradient in total pressure, which then drives the diffusion of gaseous components for which there are no mole fraction gradients. To the author's knowledge, this peculiar phenomenon has not been verified by a real experiment. The analysis also shows that bulk diffusion in the present virtual experiment is equimolar, which contradicts the common assertion that Graham's relation is valid even in conditions where bulk diffusion is dominant. Finally, the present work shows the importance of the term involving the total pressure gradient in the modified Stefan-Maxwell equation. In the literature, the gradient in total pressure is often mistakenly associated only with permeation. This paper demonstrates that it is an essential part of the driving force for diffusion and its omission leads to an erroneous prediction in the present virtual experiment. A detailed derivation of the modified Stefan-Maxwell equation is also provided, underscoring the relevance of the total pressure gradient term.     

The enskog modulus for p-hydrogen in the gaseous and liquid states

The recent PVT measurements of p-hydrogen reported by the National Bureau of Standards^(1,2,3) have been utilized to establish the Enskog modulus, bpx, for pressures up to 350 atm and temperatures up to 100°K. These conditions define both the gaseous and liquid regions and also include the saturation envelope from the triple point to the critical point of this substance. This modulus is expressed graphically as a function of reduced pressure and reduced temperature. The best available PVT measurements coupled with a precise calculational procedure conducted on an IBM 709 computer have produced a bpx correlation which readily permits the calculation of the pressure dependence of the transport properties of p-hydrogen according to the Enskog theory for the dense gaseous and liquid regions.     

Modified parameters for the Redlich-Kwong equation of state

Two new parameters are proposed for the Redlich-Kwong equation of state. They are developed for pure components at T>T_c with emphasis placed on the application of the Redlich-Kwong equation to vapor-liquid equilibrium studies. The proposed parameters reflect the departure from the experimental T_c and P_c values, and are related to the conventional parameters. Calculated critical isotherms of pure propane and hydrogen by means of the proposed method and two methods available in the literature are compared and discussed. A set of mixing rules is proposed for correlating vapor-liquid equilibrium data. Critical volumes and acentric factors of pure components are not required in the application of these mixing rules. Applicability of the proposed new parameters and mixing rules is demonstrated by the correlation of vapor-liquid equilibrium data for five binary systems at thirty-two isothermal conditions and by the prediction of the triple-values dew-point curve for the methane-n-butane mixtures. Based on a paper presented at the Research Seminar on “Vapor-Liquid Equilibria in Multicomponent Mixtures” November 2–6, 1975 at Jablonna, Poland.     

Effects of pressure and inlet temperature on coaxial gaseous methane/liquid oxygen turbulent jet flame under transcritical conditions

This study has investigated numerically turbulent flames of cryogenic oxygen and methane under supercritical pressures relevant to liquid propellant rocket engines. A real-fluid version of the flamelet equations is employed to accommodate simultaneously non-equilibrium chemistry of hydrocarbon fuel and non-ideal thermodynamics in local flame structures while the effect of turbulent fluctuations is accounted for via a presumed probability density functions. The present model reproduced qualitatively well the experimentally observed unique feature of a transcritical flame of coaxial gaseous methane/liquid oxygen injector, which is characterized by sudden flame expansion, abruptly terminated flame tip, and expansion induced flow recirculation. Numerical results reveal that pseudo-boiling phenomena occurred in the transcritical mixing layer between the cryogenic oxygen core and the surrounding hot gas play a crucial role in mixing and combustion processes. It is also found that the transcritical flame structure is drastically affected by elevating the chamber pressure or increasing the oxygen inlet temperature in terms of flame length, sudden expansion angle, and reverse flow strength. Detailed discussions are made for effects of the real-fluid behaviors on the turbulent flame field as well as on the local flame structure in mixture fraction space.     

A single parameter equation for the prediction of multicomponent vapor-liquid data from binary isobaric data

A single parameter Wilson equation that is suitable for use with isobaric systems is presented. Parameters for both the original and modified equations are obtained from isobaric binary data and from infinite dilution activity coefficients and then used to predict multicomponent vapor-liquid data for systems up through a quinary. Satisfactory results were obtained from the use of infinite dilution activity coefficients and the sensitivity of the final result to the accuracy of these end points is discussed.