Calculation Of Critical Points Research Articles

Deep-learning-based acceleration of critical point calculations

Computation of the critical point of complex fluid mixtures is an important part of understanding their thermodynamic phase behaviour. While algorithms for these calculations are well established, they are often slow when the number of constituting components is large. In this work, we propose a new procedure to significantly accelerate critical point calculations by leveraging deep neural network (DNN) models. A DNN model for critical point predictions of a given mixture is first trained based on the critical points of such a mixture with various compositions. The predictions of the DNN model are then used to initialize both of the commonly used algorithms for mixture critical point calculations: root finding and global minimization. We demonstrate that when using the DNN-based predictions to initialize the root-finding-based and optimization-based algorithms, we can achieve 50-90% and 80-90% reductions in the number of required iterations, respectively.

The use of guided tissue regeneration in endodontic Microsurgery: Setting a threshold

AimWe aimed to compare the radiographic outcomes of conventional and regenerative approaches in endodontic microsurgery (EMS) and set a critical defect size for healing in conventional and regenerative therapies. MethodologyThe study evaluated 53 root canal-treated teeth (33 patients) with periapical lesions. Among them, 19 teeth (35.8 %) were treated with regenerative treatment, whereas 34 teeth (64.1 %) were managed with the conventional approach. Conventional and regenerative approaches were performed by endodontic and periodontic residents under consultants’ supervision. Healing was evaluated after a minimum period of 6 months by comparing pre- and post-operative cone-beam computed tomography (CBCT) findings. The radiographic interpretation was conducted by a single examiner who was not participating in the surgeries and was blind on the type of treatment prior to CBCT evaluation. New healing criteria were proposed owing to the limitations on the present criteria in evaluating endodontic surgery after regenerative treatment. Critical measurements were calculated for each approach based on periapical lesion dimensions. ResultsThe regenerative approach presented significantly better healing than conventional treatment (mean, 1.21 and 1.59, respectively; p = 0.047). Based on the critical-point calculations, the conventional approach was effective in lesions of up to 3 mm depth and height, whereas the regenerative approach resulted in better healing rates in lesions with 3–9 mm depth and 3–6 mm height. ConclusionsPerforming the regenerative approach in EMS resulted in better healing rates than those of the conventional approach. The conventional approach is recommended for small periapical lesions, whereas the first had better results in larger lesions.

A Numerical Algorithm for Calculating Critical Points and Its Application to Predictive Mixture Models and Binary CO_2 Mixtures

The use of supercritical fluids in technical applications requires an accurate knowledge of their critical points. For mixtures, these can deviate significantly and without a linear dependency from the critical points of its individual pure components. Since even small amounts of admixture can have noticeable effects, this not only concerns blends of targeted compositions, but also unintentional mixtures for example caused by impurities. Within this work, a method for the calculation of critical points is presented which focuses on numerical robustness promoting a fast and reliable generation of results. Implemented into the thermodynamic property software TREND, with its mixture modeling capabilities, the method allows a flexible combination of different equations of state and mixture models which also includes predictive approaches. Against the background of an increasing relevance of mixtures based on supercritical CO_2 (sCO_2) for energy applications, critical lines are calculated and compared against experimental results for selected sCO_2-based mixtures recently considered for power plant applications. Herein, several combinations of equations of state (EoS) and mixture models are compared. Critical lines are calculated for the first time in this work with the combination of the multi-fluid mixture model with excess Gibbs energy (g^text {E}) models. It was found that the critical lines calculated with the combination of the multi-fluid mixture model with the g^text {E}-model COSMO-SAC yields good predictive results for the investigated CO_2 mixtures.

Stability and activity of PdCu clusters embedded on pyridinic N-doped graphene: a density functional theory investigation

The stability and reactivity of Pd6-nCun (n = 0−3) clusters supported on pyridinic N-doped graphene (PNG) were investigated using the auxiliary density functional theory method. First, the stability of Pd6-nCun (n = 0−3) clusters supported on PNG was analyzed using charge transfer, interaction energy, and electron density critical point calculations. Then, the reactivity of these clusters supported on PNG was computed using frontier molecular orbitals and oxygen adsorption energies. According to the calculated interaction energy, as the Cu content increased, the interaction energy between Pd6-nCun (n = 0−3) clusters and PNG increased. The charge transfer indicated that the clusters donate charge to the PNG. Based on the energy of the frontier orbitals, the Pd5Cu cluster supported on PNG exhibits the smallest HOMO–LUMO gap inferring that this system is the most reactive, whereas the less reactive system corresponds to the Pd4Cu2 cluster supported on PNG. Finally, the oxygen adsorption energies on Pd6 (−3.86 eV) and Pd5Cu (−3.87 eV) clusters supported on PNG were similar. However, the Pd5Cu cluster has a lower Pd content than the Pd6 cluster. Therefore, this work reveals that PdCu systems supported on PNG could be cheap and efficient catalysts for the oxygen reduction reaction.

Prediction of critical points for carbon dioxide-based binary mixtures by the Heidemann-Khalil approach

Prediction of critical point is of great interest for CO2-based binary mixture being the working fluid of power cycle. Compared with empirical correlation, the critical point of mixture can be calculated based on its rigorous thermodynamic criteria, with limited binary interaction parameters and consistency on vapor-liquid equilibrium and other thermodynamic properties. In this study, the critical points of CO2-based binary mixture, applicable for being working fluid of power cycle, including hydrocarbons, fluorocarbons, dimethyl ether, methanol, water and xenon, were studied based on the Peng-Robinson equation of state (PR EOS) with van der Waals (vdW) mixing rule, according to the Heidemann-Khalil approach. By comparing the predicted results with the experimental data in the literature, the applicability of this method to various mixtures is discussed. The characteristics of critical lines, type of phase diagram, and binary interaction parameters of each mixture were analyzed and discussed, and the results show that the Heidemann-Khalil approach is efficient for critical point calculation and prediction of CO2-based binary mixture.

Calculation of Critical Points

Miscibility gaps appear in many alloy systems and, consequently, spinodals and critical points. Three methods are presented for determining a critical point in a multicomponent system. The first method is Gibbs thermodynamic conditions for spinodal and critical point. The spinodal is where the determinant of Gibbs energy Hessian equals zero. The critical point is on the spinodal and where the determinant of another matrix, the Gibbs energy Hessian with any one row replaced by the gradient of the determinant of Gibbs energy Hessian, also equals zero. The second method for a critical point uses the fact that the third directional derivative of the Gibbs energy along the eigenvector direction of Gibbs energy Hessian must be zero at a critical point. The third method uses the geometric features of binodal, spinodal, and eigenvector of Gibbs energy Hessian to determine the position of a critical point on a spinodal curve. The conditions for a spinodal and a critical point in a multicomponent system can be expressed in terms of directional derivatives of Gibbs energy, which are the natural extension of the binary conditions. The ternary Al-Cu-Sn alloy system is used as an example to demonstrate how to apply these conditions to determine the critical points and critical lines. A proof is given in Appendix for the gauge invariance of the determinant of the Gibbs energy Hessian.

Deriving force fields with a multiscale approach: From ab initio calculations to molecular-based equations of state.

Using theoretical and computational tools for predicting thermophysical properties of fluid systems and the soft matter has always been of interest to the physical, chemical, and engineering sciences. Certainly, the ultimate goal is to be able to compute these macroscopic properties from first-principles calculations beginning with the very atomic constitution of matter. In this work, Mie potential parameters were obtained through dimer interaction energy curves derived from ab initio calculations to represent methane and substituted-methane molecules in a spherical one-site coarse-grained model. Bottom-up-based Mie potential parameters of this work were compared with top-down-based ones from the statistical associating fluid theory (SAFT) models for the calculation of thermodynamic properties and critical point by molecular dynamics simulations and SAFT-VR Mie equation of state. Results demonstrated that bottom-up-based Mie potential parameters when averaging the Mie potential parameters of a representative population of conformers provide values close to the top-down-based ones from SAFT models and predict well properties of tetrahedral molecules. This shows the level of consistency embedded in the SAFT-VR Mie family of models and confers the status of a purely predictive equation of state for SAFT-VR Mie when a reasonable model is considered to represent a molecule of interest.

Application of Generalized (Hyper-) Dual Numbers in Equation of State Modeling

The calculation of derivatives is ubiquitous in science and engineering. In thermodynamics, in particular, state properties can be expressed as derivatives of thermodynamic potentials. The manual differentiation of complex models can be tedious and error-prone. In this work, we revisit dual and hyper-dual numbers for the calculation of exact derivatives and show generalizations to higher order derivatives and derivatives with respect to vector quantities. The methods described in this paper are accompanied by an open source Rust implementation with Python bindings. Applications of the generalized (hyper-) dual numbers are given in the context of equation of state modeling and the calculation of critical points.

Critical point calculations by numerical inversion of functions

In this work, we propose a new approach to the problem of critical point calculation, based on the formulation of Heidemann and Khalil. This leads to a 2 × 2 system of nonlinear algebraic equations in temperature and molar volume, which makes possible the prediction of critical points of the mixture through an adaptation of the technique of inversion of functions from the plane to the plane, proposed by Malta, Saldanha, and Tomei. The results are compared to those obtained by three methodologies: (i) the classical method of Heidemann and Khalil, which uses a double-loop structure, also in terms of temperature and molar volume; (ii) the algorithm of Dimitrakopoulos, Jia, and Li, which employs a damped Newton algorithm and (iii) the methodology proposed by Nichita and Gomez, based on a stochastic algorithm. The proposed methodology proves to be robust and accurate in the prediction of critical points, as well as provides a global view of the nonlinear problem.

Behaviour of CFRP wrapped RC square columns under eccentric compressive loading

This paper presents the effects of carbon fibre reinforced polymer (CFRP) volumetric ratios on structural and failure behaviour of CFRP wrapped reinforced concrete (RC) square columns under concentric, eccentric and flexural loadings. The CFRP volumetric ratio is considered as the ratio of volume of CFRP to the volume of concrete. The applicability of existing confinement models to predict the load-moment interaction diagrams of such columns in eccentric loadings is also presented in this paper. Total 20 columns were tested. Dimensions of all columns were 175x175x800 mm with 20 mm rounded corners. Variable considered in this study were number of CFRP layers (e.g. 0, 1, 2 and 3 layers) which corresponds to CFRP volumetric ratio of 0, 0.3, 0.6 and 0.9%, respectively, increasing eccentricities (e.g. 0, 25, 35 and 50 mm) of the applied load and pure flexural loading. Results show that ultimate load capacity and axial deformation increase with increase in CFRP volumetric ratios irrespective of eccentricities. However, load capacity and deformation decrease with increasing eccentricities for any given CFRP volumetric ratio. Hoop strains also increase with increase in CFRP volumetric ratios and eccentricities. Stiffness of CFRP wrapped columns also increases with increase in CFRP volumetric ratio up to 0.6% and decreases with increase in eccentricities. Ductility of CFRP wrapped columns increases with increase in CFRP volumetric ratios and eccentricities. The algorithms for calculation of critical points for load-moment (P-M) interaction diagrams using analytical as well as numerical stress integrations are developed and compared to those of the column tests. P-M interaction diagram developed using existing confinement models agree well with the experimental P-M interaction diagrams of CFRP wrapped RC square columns under eccentric loading.

On the cation– $$\uppi $$ π interactions in 1,2-dihydro-1,2-azaborine

In recent years, it has become a strategy, and important area of research, to incorporate heteroatom into polycyclic aromatic compounds. Especially B/N containing aromatic compounds have been the topic of interest over the years, because of the isoelectronic nature of B–N bond with C=C bond. The substitution of B–N in place of C=C brings many important electronic and structural changes in the system. This work presents the investigation of the interaction between different cations and the $$\uppi $$ -cloud of the 1,2-dihydro-1,2-azaborine molecule using quantum chemical investigations at the MP2/aug-cc-pVDZ level of theory. The nature of interactions has been gauged by employing the energy decomposition analysis and molecular electron density critical point calculations. Further, the changes in N–H, B–H and C–H bond vibrational frequencies due to the cation binding were also investigated. Among the cations studied in this work, $$\hbox {Be}^{2+}$$ is found to provide the best compromise between size and charge as revealed by its high interaction energy and presence of electron density three critical points. SYNOPSIS. This work presents the investigation of the interaction between different cations and the $$\uppi $$ -cloud of the 1,2-dihydro-1,2-azaborine molecule at MP2/aug-cc-pVDZ level of theory. Among the cations studied in this work, $$\hbox {Be}^{2+}$$ is found to form the most stable complex and provides a best compromise between size and charge.

Basins of Attraction in the Calculation of Critical Points

Critical point calculations are extremely useful in Chemical Engineering context, since the correct prediction of the critical coordinates are important in various situations (for instance, in enhanced oil recovery). In this work, we proposed the use of the basins of attraction to show the convergence behavior of two strategies for critical point calculations in a system with two critical points, formed by methane + hydrogen sulfide. This kind of diagram – with complex convergence patterns – justifies the development of robust methodologies to solve systems of algebraic equations with multiple solutions.

Calculation of multicomponent vapor–liquid critical point using the Dividing Rectangles global optimization algorithm

ABSTRACTIn this paper, the Dividing Rectangles (D.R.) global optimization technique is used to predict the critical temperature and pressure for multicomponent systems. The D.R. optimization algorithm is a fast and reliable optimization method without any need for adjustable parameters, initial guesses or objective function derivation. For calculating the critical point the tangent plane distance (TPD) in terms of the Helmholtz energy is selected. The Peng Robinson (PR) equation of state is applied for determining thermodynamic relations. The procedure of the critical point calculation by means of global optimization is described stepwise in order to clarify the concept and method of programing. The D.R.-based computer program is tested for some multicomponent systems and the results are compared with the Simulated Annealing (SA) global optimization method and with experimental data. The results indicated that the number of the objective function calls (the main parameter that increases the speed and efficiency of an optimization algorithm) is reduced to one twentieth with almost the same accuracy.

Calculation of critical points for natural gas mixtures with the GERG-2008 equation of state

Critical points of natural gas mixtures containing linear alkane molecules (C1-n-C8), branch alkanes (i-C4-i-C5), nitrogen (N2), and carbon dioxide (CO2) are calculated using a criterion based upon the Helmholtz free energy. We use the GERG equation of state with its mixing rules together with needed derivatives evaluated numerically. A multivariable iterative method is used to solve the system of non-linear equations that appear in this procedure. Results are checked for stability using mechanical, diffusional and global stability tests. Our results are compared with the literature values. Maximum error for the temperature and pressure are 1.301 and 11.087% respectively. The predictive capability of the GERG-2008 equation of state is compared with PR, SRK, PSRK, SPHCT, and PC-SAFT equations of state. For binary mixtures, all equations of state are capable of representing global phase behavior types I to IV. Results show that the GERG-2008 equation of state with binary interaction parameters is adequate for the prediction of critical points.

Calculation of critical points from Helmholtz-energy-explicit mixture models

The calculation of the critical points for a mixture of fluids is of practical interest as the calculated critical points can be used to more reliably and efficiently construct phase envelopes. The number of stable critical points found can also provide insight into whether the mixture has an open or closed phase envelope.In this work we have developed a reliable method for determining all the critical points for a mixture that is modeled with Helmholtz-energy-explicit equations of state. This method extends the algorithms developed in the literature for simpler equations of state to these more complex mixture models. These Helmholtz-energy-explicit equations of state could be either multi-fluid models or transformations of simple cubic equations of state to Helmholtz-energy-explicit forms. This algorithm locks onto the first criticality contour (the spinodal) and traces it to high density, thereby locating all relevant critical points. The necessary analytic derivatives of the residual Helmholtz energy, numerically validated values of the derivatives for validation, sample code, and additional figures and information are provided in the supplemental material.

Local electric dipole moments: A generalized approach.

We present an approach for calculating local electric dipole moments for fragments of molecular or supramolecular systems. This is important for understanding chemical gating and solvent effects in nanoelectronics, atomic force microscopy, and intensities in infrared spectroscopy. Owing to the nonzero partial charge of most fragments, "naively" defined local dipole moments are origin-dependent. Inspired by previous work based on Bader's atoms-in-molecules (AIM) partitioning, we derive a definition of fragment dipole moments which achieves origin-independence by relying on internal reference points. Instead of bond critical points (BCPs) as in existing approaches, we use as few reference points as possible, which are located between the fragment and the remainder(s) of the system and may be chosen based on chemical intuition. This allows our approach to be used with AIM implementations that circumvent the calculation of critical points for reasons of computational efficiency, for cases where no BCPs are found due to large interfragment distances, and with local partitioning schemes other than AIM which do not provide BCPs. It is applicable to both covalently and noncovalently bound systems. © 2016 Wiley Periodicals, Inc.

Effect of a magnetic field on massive-star winds – I. Mass-loss and velocity for a dipole field

We generalize the Rigid-Field Hydrodynamic equations to accommodate arbitrary magnetic field topologies, resulting in a new Arbitrary Rigid-Field hydrodynamic (ARFHD) formalism. We undertake a critical point calculation of the steady-state ARFHD equations with a CAK-type radiative acceleration and determine the effects of a dipole magnetic field on the usual CAK mass-loss rate and velocity structure. Enforcing the proper optically-thin limit for the radiative line-acceleration is found to decrease both the mass-loss and wind acceleration, while rotation boosts both properties. We define optically-thin-correction and rotation parameters to quantify these effects on the global mass-loss rate and develop scaling laws for the surface mass-flux as a function of surface colatitude. These scaling laws are found to agree with previous laws derived from magnetohydrodynamic simulations of magnetospheres. The dipole magnetosphere velocity structure is found to differ from a global beta-velocity law, which contradicts a central assumption of the previously-developed XADM model of X-ray emission from magnetospheres.

Critical point calculation for binary mixtures of symmetric non-additive hard disks

We have calculated the values of critical packing fractions for the mixtures of symmetric non-additive hard disks. An interesting feature of the model is the fact that the internal energy is zero and the phase transitions are entropically driven. A cluster algorithm for Monte Carlo simulations in a semigrand ensemble was used. The finite size scaling analysis was employed to compute the critical packing fractions for infinite systems with high accuracy for a range of non-additivity parameters wider than in the previous studies.

Critical point calculations of multi-component reservoir fluids using nature-inspired metaheuristic algorithms

This study introduces the application of nature-inspired metaheuristic algorithms for performing critical point calculations in multicomponent reservoir fluids. These algorithms are Monkey – Krill Herd Hybrid (MAKHA), Intelligent Firefly Algorithm (IFA), Covariance Matrix Adaptation Evolution Strategy (CMAES), Artificial Bee Colony (ABC), Cuckoo Search (CS), Bare Bones Particle Swarm Optimization (BBPSO) and Flower Pollination Algorithm (FPA). Capabilities and limitations of these optimizers have been analyzed using black oil, volatile oil, and condensate reservoir fluids with fifty components. Results showed that BBPSO, IFA and FPA outperformed other nature-inspired methods for critical point calculations in tested fluids. In particular, BBPSO offered the best efficiency-reliability tradeoff for the accurate prediction of critical points in multicomponent mixtures.

Computation of Critical Points of Mixtures Using Particle Swarm Optimization with Low-Discrepancy Sequences

The prediction of critical points of thermodynamic systems is an important tool for modeling many high-pressure processes of theoretical and practical interest. In this article, the calculation of critical points of multicomponent mixtures is treated as a global minimization problem of a modified merit function associated with the criticality conditions obtained from the Gibbs tangent plane criterion, designed to discriminate the scale of the problem. The methodology used to solve the optimization problem is based on two versions of the particle swarm optimization (PSO), equipped with low-discrepancy sequences to prevent the sensitivity of the swarm with respect to the location of the initial population. To avoid a rapid decrease in the weight inertia, and to prevent stagnations near undesirable local minimizers, we present a modification of the PSO method, which uses different search cycles with the same inertia weight. This new version developed here is a fast and robust algorithm for solving the critical-point problem, via global optimization.