Two-parameter Equation Research Articles

Prediction of pure and mixture thermodynamic properties and phase equilibria using an optimized equation of state – part 1: Parameter estimation

This study presents a novel three-parameter equation of state (EOS) wherein attraction parameter polynomial coefficients are optimized to enhance predictive capabilities. The performance of the proposed EOS is systematically compared with established models including Peng-Robinson, Patel-Teja, and Twu-Coon-Cunningham equations. Through comprehensive evaluation, it is demonstrated that the proposed EOS exhibits superior accuracy in vapor pressure and latent enthalpy predictions, while maintaining comparable precision in liquid density estimations. Notably, the fugacity expression of the proposed EOS closely resembles that of a two-parameter equation, resulting in significantly reduced computational overhead. Additionally, a comprehensive table of equation of state parameters for all four equations is available in an online repository, facilitating easy implementation and comparison. Furthermore, a reliable generalized polynomial correlation is provided for the proposed EOS parameters against the true compressibility factor and acentric factor, leveraging data accessibility and enhancing its applicability and versatility. These findings underscore the potential of the optimized attraction parameter polynomial coefficients approach in advancing the accuracy and efficiency of EOS modeling, thereby offering promising avenues for diverse applications in thermodynamics and process engineering.

Data transformation and model selection in bivariate allometry.

Students of biological allometry have used the logarithmic transformation for over a century to linearize bivariate distributions that are curvilinear on the arithmetic scale. When the distribution is linear, the equation for a straight line fitted to the distribution can be back-transformed to form a two-parameter power function for describing the original observations. However, many of the data in contemporary studies of allometry fail to meet the requirement for log-linearity, thereby precluding the use of the aforementioned protocol. Even when data are linear in logarithmic form, the two-parameter power equation estimated by back-transformation may yield a misleading or erroneous perception of pattern in the original distribution. A better approach to bivariate allometry would be to forego transformation altogether and to fit multiple models to untransformed observations by nonlinear regression, thereby creating a pool of candidate models with different functional form and different assumptions regarding random error. The best model in the pool of candidate models could then be identified by a selection procedure based on maximum likelihood. Two examples are presented to illustrate the power and versatility of newer methods for studying allometric variation. It always is better to examine the original data when it is possible to do so.

Assessing the Role of Patient Generation Techniques in Virtual Clinical Trial Outcomes.

Virtual clinical trials (VCTs) are growing in popularity as a tool for quantitatively predicting heterogeneous treatment responses across a population. In the context of a VCT, a plausible patient is an instance of a mathematical model with parameter (or attribute) values chosen to reflect features of the disease and response to treatment for that particular patient. A number of techniques have been introduced to determine the set of model parametrizations to include in a virtual patient cohort. These methodologies generally start with a prior distribution for each model parameter and utilize some criteria to determine whether a parameter set sampled from the priors should be included or excluded from the plausible population. No standard technique exists, however, for generating these prior distributions and choosing the inclusion/exclusion criteria. In this work, we rigorously quantify the impact that VCT design choices have on VCT predictions. Rather than use real data and a complex mathematical model, a spatial model of radiotherapy is used to generate simulated patient data and the mathematical model used to describe the patient data is a two-parameter ordinary differential equations model. This controlled setup allows us to isolate the impact of both the prior distribution and the inclusion/exclusion criteria on both the heterogeneity of plausible populations and on predicted treatment response. We find that the prior distribution, rather than the inclusion/exclusion criteria, has a larger impact on the heterogeneity of the plausible population. Yet, the percent of treatment responders in the plausible population was more sensitive to the inclusion/exclusion criteria utilized. This foundational understanding of the role of virtual clinical trial design should help inform the development of future VCTs that use more complex models and real data.

The determination of the solubility, enthalpy, and entropy of solutions of methyl Furan-2,5-Dimethylenedicarbamate and its hydrides in different solvents

The solubilities of methyl furan-2,5-dimethylenedicarbamate (FDC) and methyl tetrahydrofuran-2,5-dimethylenedicarbamate (THFDC) in dimethyl carbonate (DMC), anhydrous ethanol (EtOH) and benzene were studied using the static equilibrium method combined with high-performance liquid chromatography under atmospheric pressure at temperatures ranging from 278.15 K to 318.15 K. The experimental results demonstrate that the solubility of FDC and THFDC exhibits an upward trend as temperature increases. Among the six correlation models used, all of them (modified Apelblat equation, Buchowski-ksiazczak (λh) model, simplified two-parameter equation, polynomial equation, NRTL model and Wilson model) can effectively correlate the solubility data. Notably, the Wilson model demonstrates superior correlation performance compared to others in FDC system and the NRTL model demonstrates superior correlation performance compared to others in THFDC system. Using the modified Van’t Hoff equation and Gibbs equation, we calculated the apparent standard dissolution enthalpy (ΔHsol), the standard dissolution entropy change (ΔSsol), and the standard dissolution Gibbs energy (ΔGsol) for FDC and THFDC in different solvents. The research resulted indicate that all these dissolution processes are non-spontaneous, entropy-increasing endothermic processes. This study provides valuable reference data for crystallization and purification in industrial production.

A uniformly convergent method for two-parameter singularly perturbed parabolic partial differential equations with a large shift

A class of singularly perturbed two-parameter parabolic partial differential equations with a large shift is studied. These equations include time-dependent variables and exhibit sensitivity to small perturbations in the system parameters. Moreover, the shift captures the influence of neighbouring states or events on the current evolution of the system. It adds memory-like behaviour to the problem, making their analysis and numerical solution more intricate. The solution of these equations exhibits a multiscale character. The interior and boundary layers exist across which the solution has a steep gradient. A higher-order accurate numerical method is proposed on a non-uniform mesh to solve the problem. The method combines an upwind difference scheme in space and a Crank-Nicolson scheme in time. The method is unconditionally stable, and parameters uniformly convergent with second-order accuracy in time and first-order accuracy in the space variable. Besides, the paper presents numerical results and illustrations to support theoretical estimates.

A semi-empirical equation modeling long-range contributions to the activity coefficients of individual ions at high ionic strengths

Molecular thermodynamics is well developed with models like UNIQUAC, UNIFAC and SAFT. These models include robust codes and extensive databases for chemical and petroleum industries through ASPEN, DORTMUND, and others. However, important systems in the pharmaceutical and biotech industries now exist where these codes and their databases are not always sufficient. Cell culture media used to grow mammalian cells are multicomponent aqueous solutions with 50, 100, or more compounds including amino acids, salts, acids or bases, as well as sugars, fatty acids, etc. These components can exhibit both complexation and incomplete dissociation. The thermodynamics of these systems have proven challenging to model, as current solubility descriptions frequently assume complete dissociation in solutions containing a single solute in water at relatively low concentrations. However, not all compounds that are generally considered to be strong electrolytes are completely dissociated even at low concentrations. Further, mixtures of such components can show dramatic changes in dissociation and complexation with concentration, pH, and temperature. In this paper, we present a semi-empirical model for individual ions in aqueous solutions. We use data from 317 compounds to isolate the ionic contribution to a compound’s activity coefficient in aqueous systems from confounding short-range effects. We compare eight robust regression M−estimators with a least-squares estimate and provide a two-parameter equation relating ionic strength, valence, and the activity coefficient not requiring arduous characterization of the solute of interest. We demonstrate that the Fair M−estimator generates an accurate model that can serve as an appropriate reference state for use with a model based on perturbation theory up to 29 molal.

Determination of the solubility, dissolution enthalpy and entropy of 2,4-diaminotoluene, methyl 3-amino-4-methyl-N-phenyl carbamate and dimethyl toluene-2,4-dicarbamate in different solvents

The solubilities of 2,4-diaminotoluene (2,4-TDA), methyl 3-amino-4-methyl-N-phenyl carbamate (4-TMC) and dimethyl toluene-2,4-dicarbamate (2,4-TDC) in dimethyl carbonate (DMC) and di(2-ethylhexyl) phthalate (DOP) were determined by dynamic method at 298.15 ∼ 398.15 K under atmospheric pressure. The solubility data were correlated by the modified Apelblat equation, polynomial equation and simplified two-parameter equation. The modified Apelblat equation was more accurate than the others. In addition, the thermodynamic properties of the solution process, including the Gibbs energy, enthalpy and entropy were calculated by the modified Van't Hoff equation. The dissolution process of 2,4-TDC in DOP solvent is a non-spontaneous entropy reduction endothermic process. The dissolution processes of 2,4-TDA, 4-TMC and 2,4-TDC in DMC solvent and 2,4-TDA and 4-TMC in DOP solvent are all non-spontaneous entropy-increasing endothermic processes.

Russian per capita consumption of fish products in raw mass or marketable product weight: how to recalculate back and forth

Aim. Find a simple way to transfer the average per capita consumption of fish products (kg/person/year) from the raw mass to the mass of marketable products and vice versa.Methods. Statistical processing of official data on the consumption of fish products in the USSR and the Russian Federation, published by Rosstat (Russian Federal State Statistics Service) and FAO (Food and Agriculture Organization of the United Nations) for 58 years, was made using the least squares method.Results. Two-parameter equations are found that best describe the relationship between the average per capita consumption of fish products in the mass of raw fish and in the mass of commercial products ready for consumption, as well as coefficients for simplified approximate quick calculations of the mass of consumed raw fish by the mass of commercial products and the mass of consumed commercial products by mass of raw fish.Novelty. Based on the results of the original analysis of the material, a new method for recalculating the Russian per capita consumption of fish products in the mass of raw fish by the mass of consumed commercial products and vice versa was obtained for the first time.Practical significance. Restoration of long-term series of consumption and targets for food security, the efficiency of the functioning of the fishery complex, and the social policy of the state. The article may be useful to economists, marketers, sociologists, medicals, businessmen, officials and politicians.

What is complex allometry?

Complex allometry describes a smooth, curvilinear relationship between logarithmic transformations of a biological variable and a corresponding measure for body size when the observations are displayed on a bivariate graph with linear scaling. The curvature in such a display is commonly captured by fitting a quadratic equation to the distribution; and the quadratic term is typically interpreted, in turn, to mean that the mathematically equivalent equation for describing the arithmetic distribution is a two-parameter power equation with an exponent that changes with body size. A power equation with an exponent that is itself a function of body size is virtually uninterpretable, yet numerous attempts have been made in recent years to incorporate such an exponent into theoretical models for the evolution of form and function in both plants and animals. However, the curvature that is described by a quadratic equation fitted to logarithms usually means that an explicit, non-zero intercept is required in the power equation describing the untransformed distribution - not that the exponent in the power equation varies with body size. Misperceptions that commonly accompany reports of complex allometry can be avoided by using nonlinear regression to examine untransformed data.

Comprehensive study on deep eutectic solvent density based on various EoSs: SRK, PT, VTSRK, sPC-SAFT

In this communication, the capability of a two-parameter cubic equation of state (EoS) i.e. Soave-Redlich-Kwong (SRK) EoS to predict the density behavior of deep eutectic solvents (DESs) in the wide range of temperature conditions and at atmospheric pressure, was improved using the volume-translated concept. At first, the c-parameter of 139 DESs (1793 experimental data points) was obtained referring to volume-translated SRK (VTSRK) EoS. The obtained results were compared with the original SRK and Patel-Teja (PT) EoSs and sPC-SAFT (simplified perturbed chain statistical associating fluid theory) EoS. Afterward, three c-parameter correlations were recommended based on molecular weight and acentric factor (Case 1), acentric factor (Case 2), and critical compressibility factor (Case 3). The capability of recommended c-parameter correlations was investigated by density modeling of 67 new DESs systems (419 experimental data). The AARD% (average absolute relative deviation) of Case 1, Case 2, Case 3 and sPC-SAFT EoS was 4.24%, 3.24%,4.19% and 0.5677%, respectively.

Physical origin of adsorption heat and its significance in the isotherm equation

Adsorption heat is a cornerstone concept underpinning almost all the existing theoretical isotherms for porous media. Yet, it remains elusive what this concept stands for and what are the physical sources for it. Here, a rigorous thermodynamic formulation is derived for the adsorption process with the aid of the law of mass action, leading to an expression of adsorption heat as the external chemical potential difference between two adjacent adsorbate layers. The interaction between adsorbent and adsorbates like van der Waals is theoretically identified as the intermolecular source for the change of adsorption heat. Thereby, adsorption heat is formulated as a spatially varied function of the distance to adsorbent surfaces. This adsorption heat function facilitates the establishment of a two-parameter sorption isotherm equation. The equation can be theoretically converted to two widely used adsorption isotherm equations, namely Brunauer–Emmet–Teller (BET) and Guggenheim-Anderson-de Boer (GAB) equations, and shows excellent performance in capturing experimental isotherm data of a wide array of materials. The comparative analysis demonstrates that the derived two-parameter isotherm equation outperforms the two-parameter BET equation and the three-parameter GAB equation in most cases. Such performance verifies the generality of the derived isotherm equation, thus further confirming the validity of the derived formulation of adsorption heat.

The Dual-Parameter Control of Synchronization in Steel Box Girder Incremental Launching Construction

When a steel box girder is constructed using the jacking method, the contact area between the jack and the bottom of the girder is subjected to complex forces, and it is very critical to ensure the local stability of the girder. When the phenomenon of unsynchronized jacking occurs, it will lead to changes in the contact area and affect the structural safety. In order to solve the above problems, this paper takes the background of the incremental launching construction of the main bridge across the Yellow River on Jiao Ping Expressway, adopts the Midas FEA NX 2021 finite element software to establish a finite element hybrid unit model under the maximum cantilever condition for the first time, and analyzes the local stresses in this state. The results show that the local maximum equivalent stress of the steel box girder is 198.301 MPa, which meets the requirements. The effect of jacking asynchrony on the structural forces is analyzed by simulating jacking asynchrony in the local model. The results show that both vertical jacking asynchrony and lateral deflection will lead to an increase in local stresses in the steel box girder and even steel yielding. On the basis of the above single-parameter study, a two-parameter correlation analysis is carried out to obtain the two-parameter control equation of jacking, the control threshold of the vertical jacking height difference is formulated to be 15 mm, and the dynamic control of lateral deflection is realized according to the control equation. Through comparison, it is found that the two-parameter control threshold of jacking synchronization is reduced, which can supplement the unfavorable state missed during single-parameter control and is a safer and more effective means of control.

Potential reinforcing properties of silane-modified rapeseed straw fibers in friction composites: Fractal dimension and circularity characterise powder flowability

The powder flowability of natural reinforcing fibers significantly affects their reinforcement. Cellulose reinforced fibers (RSF) were extracted from rapeseed straw waste and treated with silane-ethanol solution modification. Fractal dimension and three major shape factors (circularity, elongation and compactness) were used to characterise the flowability of RSF. The silane-modified RSF was characterised by fractal features and fractal dimension was the main factor affecting its flowability. k-mean clustering and box-plot analyses indicated that circularity had the most significant effect on flowability among all shape factors. Multiple stepwise regression analysis further yielded a two-parameter equation characterising the quantitative relationship between flowability and the combination of fractal dimension and circularity. Moreover, the incorporation of RSF resulted in an important transformation in the wear mechanism of the friction composites towards single abrasive wear, with a 44.1% increase in wear resistance.

Регрессионные модели смешанных эффектов зависимости высоты от диаметра ствола в сосновых древостоях европейской части России

Evaluation of the relationship between tree heights and diameters in the course of logging sites can be done through the use of tables of height categories or regression models. Models of mixed effects make it possible to improve the accuracy of forest inventory to a large extent. The purpose of the study is to develop a mixed effects model for describing the dependence of heights on the diameters of pine trees in the European part of Russia. In the course of study we used the data from measurements of 3 571 model pine trees growing on 201 trial plots from 13 regions of the European part of Russia from Karelia to the Samara region and from the Tver region to the Komi Republic. The paper analyzes 28 simple regression models of the dependence of height on diameter, selected according to literary sources. The selection of the best models was based on quality metrics accepted in statistics, such as the square root of the root mean square error (RMSE), the average percentage of absolute error (MAPE), the average absolute error (MAE), bias (Bias), coefficient of determination (R2), information criteria Akaike (AIC) and Bayesian (BIC). The two-parameter Neslund equation is recognized as the most simple and universal among the fixed effects models. To increase the predictive power of the Neslund equation, a separate trial plot was added as a random effect, which made it possible to significantly improve the quality metrics. It has been established that the height curves predicted by the obtained model are flexible and have a significant response to the initial ratios of the height and diameter of individual trees. The resulting model of mixed effects is an alternative to the tables of height categories used in the practice of forest accounting, which show only conditional relationships between the heights and diameters of trees. The introduction of industry standards developed on the basis of the model will improve the efficiency of accounting for wood resources in pine forests in the European part of Russia.

Effects of Nano-SiO2 grafting on improving the interfacial and mechanical properties of concrete with rice straw fibers

As a green and clean product, plant fiber has become a hot spot in the research of concrete. However, the weak interfacial properties of plant fibers with cement matrix restricted the development of plant fiber reinforced concrete. This study intends to prepare rice straw fiber reinforced concrete (RSFRC) grafting by Nano-SiO2 on the surface of rice straw fibers. The interface properties of silica grafted on straw fibers with cement were evaluated by the pull-out test of a single fiber from the cementitious matrix. The mechanical properties and fatigue resistance of RSFRC were investigated and a two-parameter Weibull probability fatigue equation of RSFRC was established. The enhancement mechanism of ITZ was revealed by Scanning Electron Microscopy (SEM) and X-ray Diffraction (XRD) tests. Results show that straw fibers grafted with Nano-SiO2 can momentously improve the mechanical and fatigue properties of RSFRC. The compressive strength, flexural strength, and abrasion resistance of RSFRC were increased by 14.6%, 22.3%, and 87.1%, compared with ordinary RSFRC after 28 days of curing. The specimens with 3% Nano-SiO2 concentration and the addition of 2% rice straw fiber to the cement weight exhibited the best flexural fatigue performance. From the microstructure analysis, Nano-SiO2 can enhance the interfacial properties of straw fibers in cement matrix, decrease the presence of calcium hydroxide and facilitate the formation of calcium silicate hydrate(C-S-H) gel. Beyond that, this research has a positive environmental impact on lowering the carbon footprint in concrete from using agriculture-based waste materials to extend the service life of infrastructures.

Potential of Oil-free-Cake Dry Seed Kernels of Desert Date (Balanites aegyptiacus (L.) Delile) for simultaneous Removal of Pb (II), Cu (II) and Cd (II) ions from Battery Industry Effluent

This study was carried out to investigate the potential of oil-free-cake dry seed kernels of desert date (Balanites aegyptiacus (L.) Delile) for simultaneous removal of Pb (II), Cu (II) and Cd (II) ions from Battery Industry Effluent using standard methods in a batch sorption process.. The battery effluent was mimicked by dissolving known concentration of metal ions in distilled water. The optimum conditions for the simultaneous removal of Pb (II), Cu (II) and Cd (II) ions were: pH (6), contact time (105 min) and adsorbent dosage (6.5 g). The results obtained showed that the biosorbent was effective for the simultaneous removal of Pb (II), Cu (II) and Cd (II) ions with removal efficiencies of up to 87.7%, 78.6% and 61.6% respectively. The consistency of the model-predicted optimum conditions were confirmed by conducting experiments under those conditions. It was found that the experimental removal efficiencies under optimum conditions were very close (less than a 5% error) to the model-predicted value. The three parameter isotherm equations (Redlich-Peterson and Sips) provide better fitting than the two-parameter isotherm equations (Freundlich and Langmuir), which can be explained by the fact that these have three adjustable parameters. The best fitted was achieved with the Redlich–Peterson equation. The results of the sorption kinetic study showed that the data fitted well with the pseudo-second order model for Pb (II), Cu (II) and Cd (II) ions. In conclusion, desert date oil-free cake exhibits great potential as an efficient sorbent for removal of metal ions.

Commentary: Allometric analyses of data with outlying observations: The ontogenetic shift in metabolic allometry of American eels (Anguilla rostrata)

Authors of a recent report concluded that different patterns of metabolic allometry characterize juvenile and subadult stages in the life cycle of American eels (Anguilla rostrata). This conclusion was based on a comparison of straight lines fitted to logarithmic transformations of the original observations for metabolic rate and body mass, with the line fitted to transformations for 30 juveniles having a substantially lower slope than the line describing observations for 30 subadults. However, the authors failed to account for an influential outlier in the sample of juvenile eels, and this one outlier was determinative for the outcome of the analysis. When the outlier is removed from the combined data set for juveniles and subadults, the resulting sample of 59 observations is well described by a single straight line, which implies, in turn, that untransformed observations can be described by a two-parameter power equation with lognormal error. This supposition is confirmed by a graph of the two-parameter equation against the backdrop of the untransformed data. Thus, no change in the pattern of metabolic allometry occurs during the ontogeny of American eels: the same pattern of allometric variation characterizes both juvenile and subadult animals.

Two-parameters single equation stem taper models of Pinus sibirica in Siberia, Russia

A large number of studies have been devoted to the cedar forests in Russia, but studies of the conicity of trunks remain limited. The main goal of this study was to analyze two-parameter single equation stem tapper models for Siberian cedar trees, which will serve as guidelines for future research. The study tested six two-parameter models of one equation for the conicity of the trunk of Siberian cedar trees. For this, data from 1805 measurements of diameters in 193 felled trees were used. The data were collected in natural pure and mixed Siberian pine stands in Irkutsk region (Southern Siberia) and Tyumen region (Western Siberia), Russia. Our analysis showed that for our data set, the Reed and Green model turned out to be the best model. The model is characterized by the smallest average absolute error percentage, the square root of the root mean square error, and the largest coefficient of determination. The residuals of this model are independent and characterized by constant variance. The analysis performed showed that two-parameter single equation models have a good fit in the central part of the trunk, therefore, forestry management in cedar forests should be used with caution. For the purposes of forestry management in cedar forests, it is recommended to develop and put into practice more complex models.

General Adsorption Model to Describe Sigmoidal Surface Tension Isotherms of Binary Liquid Mixtures

Surface tension (σ) isotherms of liquid mixtures can be divided into Langmuir-type (L-type, including LI- and LII-type) and sigmoid-type (S-type, including SI- and SII-type). Many models have been developed to describe the σ-isotherms. However, the existing models can well describe the L-type isotherms, but not the S-type ones. In the current work, a thermodynamic model, called the general adsorption model, was developed based on the assumption of surface aggregation occurring in the surface layers, to relate the surface composition with the bulk one. By coupling the general adsorption model with the modified Eberhart model, a two-parameter equation was developed to relate the σ with the bulk composition. Its rationality was examined using the σ data of 10 binary mixtures. The results indicate that the new model can accurately describe the S- and L-type isotherms of binary liquid mixtures, showing a good universality. One advantage of the model is that its two parameters, i.e., the adsorption equilibrium constant (K) and the average aggregation number (n), can be estimated by linear fitting experimental σ data, thereby obtaining unique values. This model suggests that the S- and LII-type isotherms arise from the surface aggregation (n ≠ 1). In addition, the standard molar Gibbs free energy of surface adsorption (ΔG̃ad0) and the apparent surface layer thickness (τ) were analyzed for 10 binary mixtures. The ΔG̃ad0 data suggest that the order of adsorption tendency is LI-type ≫ SI-type ≈ SII-type > LII-type, and the strong adsorption usually corresponds to large τ. This work provides a feasible model for describing the S-type isotherms and a better understanding of the surface properties of liquid mixtures.

Patel-Teja cubic equation of state – A review of modifications and applications till 2022

The Patel-Teja equation of state (PT EOS) is extensively researched, modified, and applied three-parameter cubic equation of state for the prediction of volumetric and thermodynamic properties by both academia and industry. As a result, we have presented a review article on PT EOS covering its history and progress up to 2022. We attempted to compile all the modifications along with applications to PT EOS for both pure fluids and mixtures. The prediction of a similar critical compressibility factor for all fluids, followed by unreliable liquid density estimation using two-parameter equations of state, prompts researchers to seek a better alternative. Therefore, three-parameter equations of state were introduced with substance dependent critical compressibility factor which led to a better representation of liquid density. We divided modifications to PT EOS into three categories: (a) Modify expression of parameters (a,b, and c) or alpha functions (b) Introduce new terms in the expression of the equation of state (c) Propose new mixing rules or modify them for the application to mixtures.The generalized Patel-Teja-Valderrama equation of state (PTV EOS) is one of the most successful and widely used modifications of PT EOS. Despite significant efforts to improve the accuracy and performance of PT EOS, an accurate equation of state that applies to all situations remains elusive, necessitating further study and development in the field.