Wilson Equation Research Articles

Prediction of azeotropic behavior using physicochemical properties for Sn-based binary liquid alloys in vacuum distillation

Crude tin must undergo separation from impurities to enhance its value and expand its applications. The phase diagrams and physical properties in liquid state of Sn-based alloys are the theoretical basis for vacuum impurity removal from crude tin. The formation of azeotrope affects the effect of impurity removal in vacuum distillation. This study focuses on the correlation between the azeotropic region and the parameters of the Wilson equation of 13 Sn-based binary systems under 10 Pa. The results reveal that the Al–Sn, Cu–Sn, and Ti–Sn systems can form azeotropes that are challenging to separate through straightforward vacuum distillation. In contrast, the other Sn-based binary alloys could not form azeotropes at the experimental conditions, thereby facilitating their removal through vacuum distillation. This method provides a reliable support for the separation and purification of crude tin by vacuum distillation.

Soliton structures of fractional coupled Drinfel’d–Sokolov–Wilson equation arising in water wave mechanics

This article delves into the dynamic constructions of distinctive traveling wave solutions for wave circulation in shallow water mechanics, specifically addressing the time-fractional couple Drinfel’d–Sokolov–Wilson (DSW) equation. Introducing the previously untapped exp(-ϕ(ξ))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$exp(-\\phi (\\xi ))$$\\end{document}-expansion method, we successfully generate a diverse set of analytic solutions expressed in hyperbolic, trigonometric, and rational functions, each with permitted parameters. Visualization through three-dimensional (3D) as well two-dimensional (2D) plots, including contour plots, reveals inherent wave phenomena in the DSW equation. These newly obtained wave solutions serve as a catalyst for refining theories in applied science, offering a means to validate mathematical simulations for the proliferation of waves in shallow water as well as other nonlinear scenarios. Obtained wave solutions demonstrate the bright soliton, periodic, multiple soliton, and kink soliton shape. The simplicity and efficacy of the implemented methods are demonstrated, providing a valuable tool for approximating the considered equation. All figures are devoted to demonstrate the complete wave futures of the attained solutions to the studied equation with the collaboration of specific selections of the chosen parameters. Moreover, it may have summarized that the attained wave solutions and their physical phenomena might be useful to comprehend the various kind of wave propagation in mathematical physics and engineering.

Three-phase crystallization for purification of acrylic acid from acrylic/propionic acid mixture

A new separation technology, three-phase crystallization (TPC), was utilized to purify acrylic acid (AA) from a binary liquid mixture comprising AA and propionic acid (PA). The solid–liquid equilibrium data of AA/PA mixture reported in the literature were adopted to recover the activity coefficients of the liquid mixture; and, subsequently, the vapor–liquid equilibrium conditions of AA/PA mixture were predicted using Raoult’s law and the Wilson equations for the recovered activity coefficients. The reported solid–liquid equilibrium data and the predicted vapor–liquid equilibrium conditions were coupled to determine the three-phase transformation conditions of AA/PA mixture, which were employed to conduct the batch TPC experiments. Upon conclusion of the TPC, the yield and purity were measured to estimate the amount of the AA crystals and volume of the liquid mixture retained in the overall final product, respectively.

Experimental determination and theoretical modeling of isobaric vapor–liquid equilibria, liquid mass density, surface tension and dynamic viscosity for the methyl butyrate and tert-butanol binary mixture

The purpose of this work is to provide consistent experimental measurements on the vapor–liquid equilibria (VLE) at the isobaric conditions of 50.00, 75.00 and 94.00kPa, as well as experimental determinations of the liquid mass densities, the surface tensions and the dynamic viscosities at 298.15K and 101.3kPa of the methyl butyrate + tert-butanol binary system within the whole composition range. For these goals, a Gillespie-type cell is used to carry out the VLE measurements, whereas an oscillating densimeter, a maximum differential bubble pressure tensiometer, and a rotating viscometer are used to quantify the remaining physical properties. The thermodynamical quality of the reported VLE data is validated by Fredenslund’s consistency test. Based on the VLE results, this zeotropic binary mixture exhibits a positive deviation from Raoult’s law. Furthermore, the validated data are correlated with three activity coefficient models (i.e., Wilson, NRTL, and UNIQUAC), with the Wilson equation being the most accurate one. Excess volumes, surface tensions, and liquid viscosities data are correlated by Redlich–Kister-type expansions, and the Grunberg–Nissan equation is also applied to viscosity modeling. The calculated excess volumes display a positive deviation from the ideal solution model. Regarding surface tensions, this system showcases both negative and positive deviation, althouhg slight, from the linear behavior while exhibiting the linear trend at a molar fraction of methyl butyrate of 0.758. In addition, this bimodal deviation is adequately predicted by the Chunxi model with the Wilson parameters determined from the VLE data. Finally, the dynamic viscosities data obey a strictly monotonic mole dependence predicted by Eyring’s theory, although without high accuracy. These results also reflect some interesting phenomena related to competitive association effects, which are discussed.

Exploring Novel Soliton Solutions to the Time-Fractional Coupled Drinfel’d–Sokolov–Wilson Equation in Industrial Engineering Using Two Efficient Techniques

Exploring Novel Soliton Solutions to the Time-Fractional Coupled Drinfel’d–Sokolov–Wilson Equation in Industrial Engineering Using Two Efficient Techniques

Activity Calculation and Vacuum Separation Theoretical Research concerning Ag–Cu, Ag–Sb and Cu–Sb Binary Alloys

The Ag–Cu–Sb system is a key component of lead anode slime and boasts an exceptionally high economic recovery value. In this work, six models, including the Molecular Interaction Volume Model (MIVM), Modified Molecular Interaction Volume Model (M-MIVM), Wilson equation, Miedema model, Regular Solution Model (RSE) and Sub-Regular Solution Model (SRSE), are used to calculate the predicted values of the activity and its deviations with experimental data for binary alloys in the Ag–Cu–Sb system for the first time. The result reveals that the overall means of the average relative deviation and average standard deviation of the M-MIVM are 0.01501 and 3.97278%, respectively, which are about two to six times smaller than those of the other five models, indicating the stability and reliability of the M-MIVM. In the meantime, the predicted data of the Cu–Ag binary alloy at 1423 K, Sb–Ag binary alloy at 1250 K and Sb–Cu binary alloy at 1375 K calculated from the M-MIVM are more reliable and pass the Herington test. Then, the separation coefficient–composition (β–x), temperature–composition (T–x–y) and pressure–composition (P–x–y) of the Cu–Ag, Sb–Ag and Sb–Cu binary alloys are plotted based on the M-MIVM and vacuum theories, showing that the Cu–Ag binary alloy is relatively difficult to separate and that high temperatures or high copper contents are detrimental to obtaining high-purity silver. Meanwhile, theoretical data of the T–x–y diagram are consistent with the available experimental data. These results can guide vacuum separation experiments and industrial production concerning Ag–Cu, Ag–Sb and Cu–Sb binary alloys.

Exploration of soliton solution of coupled Drinfel’d–Sokolov–Wilson equation under conformable differential operator

The aim of this paper is to study the nonlinear couple Drinfel’d–Sokolov–Wilson (DSW) equation under conformable differential operator (CDO) that plays an important role in describing the shallow water in fluid mechanics, plasma physics, nonlinear optics and Bose–Einstein condensate (soliton formation). The Tanh–Coth and energy balance techniques are implemented in this work to explore some travelling wave solutions. These analytical methods yield various solutions for the considered DSW equation. The solutions obtained from both method are expressed in the form of hyperbolic and trigonometric function. Furthermore, the dynamic behaviour of the obtained solutions are illustrated, via 2D and 3D curves. Additionally, the effect of the conformable differential operator on dynamics of wave solutions is studied via 2D graphs.

Propylene and DME solubility in PAG oil: Experimental investigations and simplified modeling

This work experimentally investigated the solubility of Propylene and Dimethyl Ether (DME) in polyalkylene glycol (PAG) oil. The solubility experimental data were used to fit different local composition models from the literature, the Wilson and Heil equations, and the Non-Random Two-Liquid (NRTL) model. Two new empirical correlations were proposed to decrease the deviation between experiments and the models. The first correlation used six fitting coefficients and resulted in an average deviation of 1.2 % for Propylene and 2.8 % for DME. The second empirical correlation was proposed with a single solubility coefficient (SSC) and a compact form, which allows solubility to be determined explicitly. The correlation resulted in average deviations from experimental data of 1.6 % for Propylene and 2.5 % for DME. Moreover, the SSC correlation showed performance robustness against the decrease of available data points. The correlation was used to compare the solubility of Propylene and DME with the solubility of Propane. At the same conditions, Propane showed the lowest solubility while DME resulted in the most soluble refrigerant.

Research on triamterene equilibrium solubility in mono-solvents and aqueous solutions according to solvent effect, models, thermodynamic analysis and quantum chemical calculation

The equilibrium solubilities of triamterene in pure solvents {isopropanol, methanol, n-propanol, water, 1,2-dichloroethane, 2-methoxyethanol, ethyl acetate, acetonitrile, N-methylformamide (NMF), n-butanol, isobutanol and dimethylsulfoxide (DMSO)}, as well as solvent mixtures of methanol/DMSO + water, were determined by dint of the saturation shake-flask technique. Triamterene solubility (105x) rose as the determination temperature rose and fell in the order at 298.15 K: 2-methoxyethanol (453.9) > DMSO (281.3) > NMF (142.6) > methanol (8.424) > n-propanol (1.432) > n-butanol (1.152) > ethyl acetate (0.9441) > isobutanol (0.8738) > acetonitrile (0.7991) > isopropanol (0.7419) > 1,2-dichloroethane (0.5232) > water (0.1795). No crystal transition or solvation in the trial process was shown via X -ray power diffraction patterns. The solvent effects, in which the interactions of solvent–solvent and solvent–solute were considered, were studied via linear solvation energy relationship. Using equations such as Wilson, Apelblat, λh, and NRTL for mono-solvents and Jouyban-Acree as well as modified van't Hoff-Jouyban-Acree for solvent mixtures, the magnitudes of equilibrium solubility were associated. The greatest relative average deviation of 5.787 × 10−2 and the maximum deviation value of root mean square of 219.1 × 10−4 were computed for neat solvents. According to the solubility information in pure solvents, the Wilson equation was used to compute the mixing solution parameters. At 298.15 K, the inverse Kirkwood-Buff integral was utilized here to examine the preferred solvation of triamterene. In blends of DMSO/methanol + water with middle and rich DMSO/methanol compositions, the parameters of preferred solvation for DMSO/methanol were positive, indicating the preferential solvation of triamterene by DMSO/methanol. In addition, the electrostatic acidity-basicity properties were demonstrated using the lowest negative electrostatic potential in company with the minimum m local ionization energy of molecular surface. The N in ring of the triamterene is the preferential target for electrophilic assault; and –NH2 of the triamterene, for nucleophilic assault. The weak triamterene-solvent interactions were graphically displayed by the use of the independent gradient approach on the basis of Hirshfeld partition analysis.

The secret of the Wilson equation

60 years ago, Grant Wilson proposed the first successful activity coefficient model based on the local composition concept he introduced. The model has received wide acceptance and extensive applicability as it could represent accurately the vapor-liquid equilibria of complex mixtures. It could be readily extended to multicomponent systems without additional parameters and with good results. It has not dominated the chemical engineering practice compared to models proposed a few years after (NRTL and UNIQUAC) possibly due to one important shortcoming. The Wilson equation could never represent liquid-liquid equilibria no matter the values of the model's parameters. While this is well-established, the “physical” reasons behind this deficiency had not been fully explored. This work presents and explores the secret of the Wilson equation related to its ability to predict partial miscibility. It will be established that the capabilities of the Wilson model are possibly broader compared to what was originally thought. The presentation is also inspired by several discussions with professor Michael L. Michelsen over the years which are also cited in the manuscript.

A boundary integration method for calculating diffraction intensity of crystalline materials with planar deformation

Defects and strain states in crystals can be investigated using X-ray and neutron diffraction, two types of penetrating, non-destructive detection techniques. Various approximations are frequently employed to minimize the amount of computation in the practical calculations because the present Fourier transform approach, established by Warren, Averbach et al. based on the Kinematic diffraction theory, has a huge computational cost. This paper develops a numerical method to precisely calculate diffraction intensity by integrating kinematic diffraction amplitude over the boundary, which is applicable to crystals with planar deformation. The precision of this method is assessed by contrasting it with Wilson's analytical equations for the X-ray diffraction results of a single screw dislocation. As an example, the difference in diffraction spots on a single edge dislocation irradiated by different incident X-ray widths are calculated and analyzed. When compared to the X-ray diffraction spots of an ideal material with a dislocation core dug out, the results indicate that if lattice information close to the dislocation core is to be retrieved by X-ray diffraction, the X-ray irradiation area should be restricted.

Solubility measurements, correlation, and thermodynamic characterization of 4-guanidinobenzoic acid hydrochloride in pure and binary solvents from 294.55 K to 333.65 K

The solubility of 4-guanidinobenzoic acid hydrochloride (4-GABCL) in pure solvents (H2O, methanol, ethanol, n-propanol, isopropanol, 1-butanol, and 1-pentanol) and two binary solvents (H2O + ethanol/isopropanol) were determined by the dynamic saturation method in the temperature range of 294.55 K–333.65 K under atmospheric pressure. From the measurement results, it could be observed that the solubility of 4-GABCL in both pure and binary solvents all increased with increasing temperature. The solubility order of 4-GABCL in pure solvents was ranked as follows: methanol > H2O > n-propanol > ethanol > n-butanol > isopropanol > n-pentanol. The solubility of 4-GABCL in binary solvents tended to increase and then decrease with increasing molar fractions of ethanol and isopropanol, and the solubility of the solute in the binary solvents was greater than that of each in the monosolvents. In addition, Ideal model, Apelblat model, λh model, Wilson model, and NRTL model had been used to correlate the solubility of 4-GABCL in pure and binary solvents. Correlated effects show that the Wilson model gives the best correlation. The influence of solvents on the solubility of 4-GABCL was further studied by KAT-LSER model. The results showed that the solubility was mainly affected by the polarizability and the self-viscosity of the solvent.The mixing and dissolution thermodynamic properties of 4-GABCL in all experimental solvents had been calculated and analyzed by using the Wilson equation and solvent data. It was determined that the mixing and dissolution of 4-GABCL were spontaneous, endothermic and entropy-driven in the study systems. Besides, the solubility order of the pure solvent is consistent with the dissolution entropy at a temperature of 318.15 k.

Can salinity variability drive the colonization dynamics of periphytic protozoan fauna in marine environments?

To investigate effects of salinity variability on colonization dynamics of periphytic protozoan fauna, a 21-day study was conducted in temperature-controlled circulation systems (TCCSs). Periphytic protozoan communities were incubated using glass slides as artificial substrata in five TCCS aquaria with a large-scale salinity gradient of 9, 19, 29 (control), 39, and 49 PSU, respectively. The colonization dynamics were observed on days 3, 5, 7, 10, 14, and 21. The colonization dynamics were well fitted to the MacArthur–Wilson and logistic model equations in colonization and growth curves in all five treatments, respectively. However, the maximum species richness and abundance were reduced, and the colonization patterns were significantly shifted in four treatments with salinity changed by 20 PSU compared to the control (29 PSU). Thus, it is suggested that the large-scale salinity variability may reduce the species richness significantly and affect colonization dynamics of periphytic protozoan fauna in marine environments.

Computational approach and dynamical analysis of multiple solitary wave solutions for nonlinear coupled Drinfeld–Sokolov–Wilson equation

In this article, the coupled Drinfeld–Sokolov–Wilson equation under investigation base on computational approach named extension of modified rational expansion approach. By this computational approach secured various kinds of solitons named kink solitons, dark solitons, anti kink solitons, periodic solitons, bright solitons, kink dark solitons, kink bright solitons, anti kink dark solitons, anti kink bright solitons for coupled DSW system. The calculated results are interesting, precise, novel, and different which have not been secured in the previous research. We represent the graphical behaviour of determined solutions by 2-D, 3-D and contour through computational software. The determined results may helpful and play significant role in the study of nonlinear behaviour in various branches of physical sciences such as laser optics, communication system, fluid dynamics, transfer of heat, shallow water waves, acoustics, optics and many others. The constructed results prove that proposed computational approach is concise, powerful, and straightforward for securing the solitons results of other nonlinear partial differential equations.

Analysis of the exact solutions of nonlinear coupled Drinfeld–Sokolov–Wilson equation through [formula omitted]-model expansion method

The coupled Drinfeld–Sokolov–Wilson (DSW) equation is a system of nonlinear partial differential equations (NLPDEs) in (1+1)-dimensions that play a fundamental role in soliton theory and integrable systems. Due to its ability to accurately describe wave phenomena in dispersive water waves, coupled DSW equation has wide-ranging applications in fluid dynamics, plasma physics, and mathematical physics. This study investigates the solitary wave solution of coupled DSW equation by using ϕ6-model expansion. This technique reveals the diverse types of solutions, including bright, dark, singular, and periodic singular solitons. The 3D, contour, and 2D plots are also illustrated to demonstrate the physical behavior of the obtained solutions. The findings of this study can contribute to the development of new analytical and numerical tools for solving other nonlinear equations in the future. The inclusion of constraint conditions in the ϕ6-model expansion technique enhances its applicability and reliability, rendering it a valuable approach for investigating nonlinear systems in various scientific domains.

Thermodynamic Behavior of (2-Propanol + 1,8-Cineole) Mixtures: Isothermal Vapor-Liquid Equilibria, Densities, Enthalpies of Mixing, and Modeling.

Vapor pressures and other thermodynamic properties of liquids, such as density and enthalpy of mixtures, are the key parameters in chemical engineering for designing new process units, and are also essential for understanding the physical chemistry, macroscopic and molecular behavior of fluid systems. In this work, vapor pressures between 278.15 and 323.15 K, densities and enthalpies of mixtures between 288.15 and 318.15 K for the binary mixture (2-propanol + 1,8-cineole) have been measured. From the vapor pressure data, activity coefficients and excess Gibbs energies were calculated via the Barker's method and the Wilson equation. Excess molar volumes and excess molar enthalpies were also obtained from the density and calorimetric measurements. Thermodynamic consistency test between excess molar Gibbs energies and excess molar enthalpies has been carried out using the Gibbs-Helmholtz equation. Robinson-Mathias, and Peng-Robinson-Stryjek-Vera together with volume translation of Peneloux equations of state (EoS) are considered, as well as the statistical associating fluid theory that offers a molecular vision quite suitable for systems having highly non-spherical or associated molecules. Of these three models, the first two fit the experimental vapor pressure results quite adequately; in contrast, only the last one approaches the volumetric behavior of the system. A brief comparison of the thermodynamic excess molar functions for binary mixtures of short-chain alcohol + 1,8-cineole (cyclic ether), or +di-n-propylether (lineal ether) is also included.

Investigation on solid-liquid equilibrium for binary mixtures of carbon dioxide (R744) and alkanes: Propane (R290) and isobutane (R600a)

The solid-liquid equilibrium (SLE) of binary systems containing carbon dioxide (R744) and alkanes propane (R290) and isobutane (R600a) was experimentally investigated. A novel experimental setup using the cooling curve method was designed, built and verified, enabling SLE measurements at temperatures down to 173 K. The temperature and pressure of the triple point of carbon dioxide (CO2) were measured. The results for R744 and the binary mixture of carbon dioxide and propane were in agreement with the available literature data, demonstrating the accuracy of the experimental setup. Activity coefficients were determined for the binary mixtures studied. The Schröder and Wilson equations were used to calculate solubility over various temperatures and compositions. The Wilson model produced a very accurate correlation for the studied non-ideal binary mixtures, while the Schröder equation did not provide satisfactory results. Determined activity coefficients and SLE curves should be useful for the design of low-temperature refrigeration systems operating on carbon dioxide mixtures with low global warming potential.

Chaotic behavior and solitary wave solutions of stochastic-fractional Drinfel’d–Sokolov–Wilson equations with Brownian motion

The stochastic-fractional Drinfel’d–Sokolov–Wilson equation is a presently improved nonlinear partial differential equation in electromagnetism and fluid mechanics. Using the theory of planar dynamical systems, the authors first analyzed chaotic behavior, including sensitivity, Poincaré section and Lyapunov exponent, and then constructed solitary waves and periodic solutions. Secondly, the author analyzed the effects of random noise and fractional derivatives on the solution. Finally, a comparative study was conducted to demonstrate the innovation of this study. The methods and results of this paper can be applied to the study of a wider range of nonlinear wave phenomena.

Solubility and thermodynamic parameters of nicotinic acid in different solvents

Nicotinic acid is an important compound needed for the body to keep proper functioning and stay healthy. The determination of solubility parameters in different solvents can be useful for operations such as crystallization or encapsulation of nicotinic acid. In this study, we critically evaluated the solubility of nicotinic acid in nine solvents: water, methanol, ethanol, propanol, propan-2ol, butanol, acetone, ethyl acetone and dimethyl sulfoxide (DMSO). Solubility data are provided as mole fractions of solvent versus temperature. Results showed that DMSO afforded the highest nicotinic acid solubility, greater than the ideal solubility values. Thermodynamic parameters such as the activity coefficient, melting temperature and enthalpy of fusion were determined by differential scanning calorimetry. Non-predictive activity coefficient models were obtained to represent the experimental data through Margules, van Laar, Wilson, NRTL, and UNIQUAC equations. The van Laar, Wilson, and NRTL models showed broadly similar performance. The UNIQUAC model showed an optimum performance for six of the nine solvents.

Clean and efficient recovery of In–Sn and In–Sn–Bi lead-free brazing materials by vacuum volatilization and thermodynamic modeling

In this study, a novel method was proposed for the clean and efficient recovery of high-value-added In–Sn and In–Sn–Bi alloys. Vacuum volatilization of the In–Sn binary alloy was conducted in a temperature range of 1373–1573 K with a system pressure of 5 Pa. The results showed that In and Sn were effectively separated. The Sn content in the residue was 99.9908%, and the In content in the volatile fraction reached 99.66%. Vacuum volatilization of the In–Sn–Bi ternary alloy was conducted out at 1073–1373 K and 5 Pa. The results showed that Bi could be separated from In and Sn, the Bi content in the residue decreased from 32.5% to 3.88%, and the Bi content in the volatile fraction reached 95.67%. The residual In–Sn alloy could be effectively separated by vacuum volatilization. The MIVM, M-MIVM, Wilson equation and NRTL were used to calculate and analyze the activity and activity coefficients of In–Sn, In–Bi, Sn–Bi and In–Sn–Bi alloy elements, respectively, and the results showed that M-MIVM had the highest prediction accuracy. The vapor-liquid equilibrium (VLE) data of the In–Sn, In–Bi, Sn–Bi, and In–Sn–Bi alloys were predicted, and the corresponding VLE phase diagrams were plotted based on the M-MIVM and VLE theory. The predicted VLE values were in good agreement with the corresponding experimental data determined in this study, indicating that the M-MIVM was reliable for indium-based alloys. The results of this work provided reliable experimental data and a thermodynamic model for the VLE study of In-based alloys, opening a new technique for the clean and efficient recycling of binary and multicomponent In-based alloys.