Empirical Expression Research Articles

Recovery of Low-concentration Waste Acid by Electrodialysis: Modelling and Validation

In light of the operational principles of Electrodialysis (ED), it is anticipated that this technology would significantly contribute to the recovery of waste acids through selective separation and subsequent proton concentration. However, the imperfect proton leakage characteristics of anion exchange membranes (AEMs) not only detrimentally affect the efficacy of ED-based acid recovery systems but also present considerable challenges for modeling endeavors. This study introduces a model based on the Nernst-Plank Equation at the cell pair scale, aimed at predicting ED performance. The model incorporates an empirical expression that links operational parameters, such as acid concentration and the concentration ratio between the concentrate and dilute compartments, to the permselectivities of AEMs in terms of anion transport numbers. Furthermore, both numerical and experimental analyses are performed to evaluate energy consumption across various operating conditions. The simulation outcomes derived from the proposed model exhibit a strong correlation with experimental data concerning acid transport (where the acid concentration increases from 0.25 M to 1.1 M through a two-stage concentration process), water migration (which demonstrates a nearly linear increase over time with applied currents, specifically a 15% increase under low-current conditions and a 100% increase under high-current conditions), and energy consumption. It is hoped that this model will aid in the design and optimization of ED-based acid reclamation processes, thereby enhancing their practical applications.

A new nanoindentation approach for determining both indentation size effect and continuous apparent activation volume during loading of nanocrystalline Ni and Ni-20 wt.% Fe alloy

A new nanoindentation approach is developed in this study to determine both indentation size effect (ISE) and continuous apparent activation volume () during loading of nanocrystalline nickel (NC Ni) and nanocrystalline nickel-20 wt.% iron alloy (NC Ni-Fe alloy). This approach involves conducting nanoindentation tests under loading modes controlled by both constant strain rate (CSR) and constant loading rate (CLR). Firstly, a new empirical expression for the parameter in the modified Nix-Gao model is established to study the strain rate dependence of the ISEs created in the CSR and CLR tests. Then, continuous data of each individual sample (indentation) during loading are obtained in the CLR test through the alternative use of continuous stiffness measurement technique. The stress dependence of is analyzed based on Duhamel et al.’s model, which considers the role of vacancies generated during inhomogeneous dislocation nucleation. Finally, based on the above experimental and theoretical work as well as transmission electron microscopy observation, the influences of loading mode, loading rate, and material properties (stacking fault energy) on resulting intrinsic hardness of NC Ni and NC Ni-Fe alloy are analyzed. The nanoindentation approach developed in this study provides profound insights into the mechanical behaviors during loading and underlying mechanisms of materials.

Cluster radioactivity of heavy elements within an improved Coulomb and Proximity potential model

The Coulomb and Proximity potential Model (CPPM) is improved by introducing an accurate empirical expression for the nuclear surface diffuseness parameter in the expression for the proximity potential. Within the improved CPPM, the decay half-lives of experimentally detected cluster emissions from heavy elements are calculated. The results of the calculations using the Universal Decay Law (UDL) [Qi et al., Phys. Rev. Lett. 103 (2009) 072501], the improved empirical formula of Sahu et al. (ImSahuB) [Wang et al., Chin. Phys. C 45 (2021) 044111], and the empirical formula of Horoi [J. Phys. G. Nucl. Part. Phys. 30 (2004) 945] are compared with the predictions of the improved CPPM. The lowest standard deviation is obtained for the predictions using the improved CPPM. The model’s predictive power increases significantly when the proximity potential is modified by including the diffuseness parameter. Furthermore, improved CPPM is employed to calculate the decay half-lives of undiscovered cluster emissions from heavy elements, a valuable information for upcoming experimental investigations. The calculated values are compared with the predictions of the UDL, Horoi formula and the improved empirical formula of Sahu et al. (ImSahuB). There is a high degree of agreement between the improved CPPM, UDL, and Horoi formula predictions. The ImSahuB formula predict decay half-life one or two order magnitudes lower than the predictions of the other models.

Empirical Stability Criteria for 3D Hierarchical Triple Systems. I. Circumbinary Planets

In this work we revisit the problem of the dynamical stability of hierarchical triple systems with applications to circumbinary planetary orbits. We derive critical semimajor axes based on simulating and analyzing the dynamical behavior of 3 × 108 binary star–planet configurations. For the first time, three-dimensional and eccentric planetary orbits are considered. We explore systems with a variety of binary and planetary mass ratios, binary and planetary eccentricities from 0 to 0.9, and orbital mutual inclinations ranging from 0° to 180°. Planetary masses range between the size of Mercury and the lower fusion boundary (approximately 13 Jupiter masses). The stability of each system is monitored over 106 planetary orbital periods. We provide empirical expressions in the form of multidimensional, parameterized fits for two borders that separate dynamically stable, unstable, and mixed zones. In addition, we offer a machine learning model trained on our data set as an alternative tool for predicting the stability of circumbinary planets. Both the empirical fits and the machine learning model are tested for their predictive capabilities against randomly generated circumbinary systems with very good results. The empirical formulae are also applied to the Kepler and TESS circumbinary systems, confirming that many planets orbit their host stars close to the stability limit of those systems. Finally, we present a REST application programming interface with a web-based application for convenient access to our simulation data set.

Comparative Analysis of Heat Transfer Coefficient Using Experimental Data and Empirical Expressions

This study aims to determine the heat transfer coefficient by comparing results obtained from both experimental methods and empirical expressions. A controlled experiment involving the flow of water through a polyethylene hose was conducted, and the heat transfer coefficient was calculated based on inlet and outlet temperatures. These findings were then compared with values derived from empirical correlations, specifically the Sieder-Tate equation, to assess accuracy and reliability. The experimental setup maintained consistent boundary conditions, including water inlet temperature, airflow rate, and hose orientation. The results indicated that the experimentally determined heat transfer coefficient was 48.30 W/(m²·K), while the empirical calculation yielded a value of 53.44 W/(m²·K). The slight discrepancy between these values highlights minor experimental errors and assumptions inherent in empirical models. Overall, the close agreement between the experimental and empirical values validates the use of empirical correlations for predicting heat transfer coefficients in similar configurations. This study provides valuable insights for optimizing heat transfer processes in various industrial and engineering applications, emphasizing the importance of experimental validation. Future work could explore extended parameter ranges, advanced measurement techniques, and numerical simulations to further enhance the accuracy and applicability of the findings.

A Simple Empirical Expression for Line-of-Sight Probability in Industrial Environment

A Simple Empirical Expression for Line-of-Sight Probability in Industrial Environment

Experimental Analysis of Weld Slag Separation in Pipelines

Abstract In order to effectively eliminate welding slag from pipelines, ensure equipment operates normally, and reduce production costs, experiments were conducted based on field construction operations, with a focus on the intensity and frequency of tapping. The study findings indicate that a tapping force of 341.8 N leads to a stabilization in the rate of welding slag removal. Following manual tapping of the pipelines, the welding slag removal rate is approximately 60 %, and it stabilizes gradually after four tapping instances. To provide guidance for on-site operations, an empirical expression capturing the pattern of welding slag removal was derived from experiments by establishing the correlation between tapping frequency, tapping force, and welding slag removal rate.

Dynamic dielectric properties of biomass fractionation system using ethylene glycol

The unclear interaction between microwaves and reaction systems limits the further application of microwave in biomass fractionation. This article explored the main factors and their influencing mechanisms that affect biomass fractionation system’s dielectric properties at the frequency of 915 MHz. The experimental results shown that as the temperature of the system increases from 30 to 140 °C, its dielectric constant varies between 23 and 36, while the dielectric loss ranges from 10.5 to 17. The dielectric properties of the system can be influenced by its moisture content, biomass particle content, and ion concentration, except for the intermediate products formed during the biomass fractionation process. Based on the experimental data, empirical expressions for the dielectric constant and dielectric loss factors of biomass fractionation systems as a function of temperature was constructed with the correlation coefficients of 0.9779 and 0.9902, respectively. This study introduces a novel approach to explore the changes in dielectric properties of non-equilibrium systems. The findings of this study are highly significant for the design and optimization of microwave heating processes.

A novel nonlinear pressurization method for counter-gravity casting of cross-sectional mutation structures

In order to ensure the filling integrity of complex counter-gravity casting and improve metallurgical quality, it is necessary to shorten the filling time while avoiding air entrainments. To address this contradiction, a novel nonlinear pressurization method was proposed in this study. Through systematically analyzing the relationship between critical gating velocity and stable filling height, a criterion for iterative calculation of nonlinear pressurization curve was established, and an empirical expression between nonlinear pressurizing speed and the filling height was obtained. Based on the empirical expression, a nonlinear pressurization curve can be designed according to the casting structures and initial pressurizing speeds. The above nonlinear pressure curve design method was validated through water filling experiments. It was proved that the nonlinear pressure curve can shorten the filling time while avoiding air entrainments. It provides important processing control method for improving the low-pressure casting performance of complex castings.

Modeling static friction behavior of elastic–plastic spherical adhesive microcontact in full-stick condition

The static friction behavior of an elastic–plastic spherical adhesive microcontact between a rigid flat and a deformable sphere under combined normal and tangential loading is studied by the finite element method (FEM). The contact between the sphere and the rigid flat is assumed to be full-stick, and the sliding inception is related to a loss of tangential stiffness. The intermolecular force between the rigid flat and the sphere is assessed by the Lennard–Jones (LJ) potential, which is applied to the sphere and the rigid flat by a user subroutine. The evolution of the adhesive force with tangential displacement in the full-stick condition is revealed. The results indicate that the increasing effect of adhesive energy on the static friction coefficient gradually diminishes with an increase in the adhesive energy and the external normal load. Finally, based on an extensive parametric study, an empirical dimensionless expression is obtained to predict the static friction coefficient of the spherical adhesive microcontact considering the intermolecular force.

Computational Study of Overtopping Phenomenon over Cylindrical Structures Including Mitigation Structures

Wave overtopping occurring in offshore wind renewable energy structures such as tension leg platforms (TLPs) or semi-submersible platforms is a phenomenon that is worth studying and preventing in order to extend the remaining useful life of the corresponding facilities. The behaviour of this phenomenon has been extensively reported for linear coastal defences like seawalls. However, no referenced study has treated the case of cylindrical structures typical of these applications to a similar extent. The aim of the present study is to define an empirical expression that portrays the relative overtopping rate over a vertical cylinder including a variety of bull-nose type mitigation structures to reduce the overtopping rate in the same fashion as for the linear structures characteristic of shoreline defences. Hydrodynamic interaction was studied by means of an experimentally validated numerical model applied to a non-impulsive regular wave regime and the results were compared with the case of a plain cylinder to evaluate the expected improvement in the overtopping performance. Four different types of parapets were added to the crest of the base cylinder, with different parapet height and horizontal extension, to see the influence of the geometry on the mitigation efficiency. Computational results confirmed the effectivity of the proposed solution in the overtopping reduction, though the singularity of each parapet geometry did not lead to an outstanding difference between the analysed options. Consequently, the resulting overtopping decrease in all the proposed geometries could be modelled by a unique specific Weibull-type function of the relative freeboard, which governed the phenomenon, showing a net reduction in comparison with the cylinder without the geometric modifications. In addition, the relationship between the reduced relative overtopping rate and the mean flow thickness over the vertical cylinder crest was studied as an alternative methodology to assess the potential damage caused by overtopping in real structures without complex volumetric measurements. The collection of computational results was fitted to a useful function, allowing for the definition of the overtopping discharge once the mean flow thickness was known.

Robust inference for causal mediation analysis of recurrent event data.

Recurrent events, including cardiovascular events, are commonly observed in biomedical studies. Understanding the effects of various treatments on recurrent events and investigating the underlying mediation mechanisms by which treatments may reduce the frequency of recurrent events are crucial tasks for researchers. Although causal inference methods for recurrent event data have been proposed, they cannot be used to assess mediation. This study proposed a novel methodology of causal mediation analysis that accommodates recurrent outcomes of interest in a given individual. A formal definition of causal estimands (direct and indirect effects) within a counterfactual framework is given, and empirical expressions for these effects are identified. To estimate these effects, a semiparametric estimator with triple robustness against model misspecification was developed. The proposed methodology was demonstrated in a real-world application. The method was applied to measure the effects of two diabetes drugs on the recurrence of cardiovascular disease and to examine the mediating role of kidney function in this process.

Wind buckling analysis of cylindrical shells with various geometric imperfections

This study addresses the complexity of buckling behavior in cylindrical shells subjected to non-uniform wind loading, emphasizing the significant impact of geometric parameters on buckling patterns. Cylinders with varying aspect ratios exhibit distinct linear and nonlinear buckling behaviors, complicating the determination of the most detrimental structural imperfections across different geometries. Notably, imperfections affecting stocky cylinders may be less impactful for slender ones. This paper introduced equations for calculating linear critical pressures under wind loading, followed by an extensive numerical analysis assessing the imperfection sensitivity in anchored cylindrical shells of uniform thickness with diverse aspect ratios. Two types of geometric imperfections were employed to assess their influence on the nonlinear buckling strength of cylinders with varying geometries. Results demonstrate that eigenmode imperfections predominantly compromise the buckling strength of stocky cylinders, whereas nonlinear incremental mode imperfections significantly influence the nonlinear critical pressures of intermediate-length and slender cylinders. Consequently, empirical expressions have been formulated to calculate the imperfection reduction factor, offering a precise evaluation of nonlinear buckling pressures in imperfect cylinders under wind loading.

Development and evaluation of MgO-grit iron aggregate heavy density concrete for high temperature radiation shielding: Experimental and machine learning approach

This research focuses on the development and evaluation of heavy density concrete (HDC) for radiation shielding, utilizing both experimental and machine learning techniques. Various HDC specimens with different proportions (25 %, 50 %, 75 %, and 100 %) of grit iron aggregate replacing normal weight aggregate, in addition to control specimens with no grit iron scale aggregate, were cast. These samples were subjected to testing at temperatures ranging from room temperature to 1200°C to assess properties such as compressive strength, rebound number, ultrasonic pulse velocity, density loss, mass loss, linear attenuation coefficient (LAC), mass attenuation coefficient (MAC), half-value layer (HVL), tenth-value layer (TVL), and mean free path (MFP). Ensemble learning algorithms were employed using experimental data to predict compressive strength and new empirical expressions were formulated for mechanical and radiation shielding properties, including LAC, HVL, and TVL. The addition of grit iron aggregate, combined with MgO, demonstrated a significant improvement in the mechanical and radiation shielding properties of HDC. This research holds promise for applications in nuclear reactors operating at high temperatures.

An activity coefficient model for mixed-solvent electrolyte mixtures based on Gibbs–Duhem’s equation: A case study of mixtures of water, KCl and ethanol

Thermodynamic properties of mixtures with several fluid components and electrolytes are challenging to model. Models are needed to develop and improve a wide range of electrochemical systems such as fuel cells, lithium-ion batteries, and processes with ion-exchange membranes. In the literature, activity coefficients are extracted using fundamentally different experimental methods that rely on electrochemical cells, vapour pressure measurements, solubility measurements, etc. The reference states of the activity coefficients obtained from these methods are likely to differ. This makes it difficult to combine or compare activity coefficient models from different sources. In this work, we present a method using Gibbs–Duhem’s equation for the development of activity coefficient models of mixtures that are based on experimental data from different methods. We use the ternary mixture of KCl, H2O and ethanol as example. First, empirical expressions are developed for the logarithm of the activity coefficients of KCl and ethanol. The expression for the activity coefficient of H2O is next derived using Gibbs–Duhem’s equation. The resulting three activity coefficient models are fitted to available experimentally data from several sources, generating linear and relatively low-complexity activity coefficient models. The empirical activity coefficient models are next compared to the electrolyte cubic plus association (e-CPA) equation of state. The models give saturation pressures in ternary mixtures that have average absolute relative deviations for water/ethanol of 8.3/3.4% for the empirical model and 11.8/3.3% for e-CPA. Experimental data reduction procedures for concentration cells, formation cells, and vapour pressure measurements are discussed, and the freedom to choose the reference state and the consequences are highlighted.

Estimating relative density from shallow depth CPTs in normally consolidated and overconsolidated siliceous sand

Current interpretation of relative density (<i>D<sub>r</sub></i>) based on CPT end resistance (<i>q<sub>c</sub></i>) data in sand relies on empirical expressions established from calibration chamber studies for which a deep failure penetration mechanism is attained. These expressions are mainly based on stress normalised <i>q<sub>c</sub></i> data. Previous studies have highlighted the importance of performing the normalisation procedure with respect to the mean effective stress (<i>p'</i>) in overconsolidated (OC) sand deposits instead of the vertical effective stress (<i>σ'<sub>v</sub></i>) that has been used in normally consolidated (NC) deposits. Due to a large variation in the coefficient of earth pressure at rest (<i>K<sub>0</sub></i>), on which <i>p'</i> depends, in the uppermost (3-5) m of OC sand, a recent study has demonstrated a rigorous unified approach for estimating a varying <i>K<sub>0</sub></i> with depth. However, the surficial shallow failure penetration effects are not accounted for, and consequently, interpretation of the upper (0.5-1.5) m of sand is still erroneous. This depth range is very important for low stress geotechnical applications such as offshore flowlines and mudmats. With a reinterpretation of previously published data, a new global model is presented that enables estimation of <i>D<sub>r</sub></i> from shallow CPTs in siliceous sand by taking into account both the shallow failure penetration effects (important in the uppermost 0.5-1.5 m) as well as the varying <i>K<sub>0</sub></i> with depth for OC sand (important in the uppermost 3-5 m).

Empirical relations for α -decay half-lives: The effect of deformation of daughter nuclei

The influence of the daughter nuclei deformations on the α decay's half-life is modeled by adding the term depending on the quadrupole deformation parameter value to the empirical relations for the α-decay half-lives. It is shown that the addition of the deformation-dependent term led to a better description of the experimental data by the empirical expressions for the half-lives of the α decay than the ones without it. A new empirical expression for a description of the α-decay half-lives is proposed. The empirical relation for the α decay half-lives for the transitions from the ground state to the ground state and the lowest 2+ state of the even-even nuclei is discussed. Published by the American Physical Society 2024

Effect of partition walls on the vibration serviceability of cross-laminated timber floors

The current standards need more specific serviceability criteria for Cross-Laminated Timber (CLT) floors. This paper presents a comprehensive analysis of a relevant case study in Norway, which presented unexpectedly high accelerations in the CLT floors after construction. The lack of a sufficiently detailed design process resulted in the poor vibration performance of the floors, which partially compromised their functionality. The paper thoroughly investigates and discusses the experimental dynamic characterization of seven CLT floors within the considered building. The authors estimated the modal parameters and serviceability metrics based on both Eurocode 5 (EC5) and the new draft version of it. Specifically, the authors compared the analytical predictions of the fundamental frequency and the root mean square acceleration against the corresponding experimental estimations. Using a sensor-roving approach, a dense sensor configuration was adopted to describe seven floors’ mode shapes and serviceability metrics. These metrics include mean square acceleration and Vibration Dose Value. The scenarios examined in this study shed light on the significance of partition walls, a factor not considered in the practical design for serviceability verifications. The presence of partition walls can enhance the vibration comfort of timber floors by providing a stiffening effect, leading to a significant increase in the first fundamental frequency. A parametric finite element model was developed using OpenSeesPy to investigate this phenomenon further. The model, firstly validated against selected experimental results, is employed to assess the influence of partition walls on floor dynamics. The model predicts the increase in the first frequency caused by randomly generated partition walls. The paper proposes an empirical expression calibrated on the simulated data to estimate the relative increment of the floor’s fundamental frequency based on the partition walls’ layout. This formulation can be included in the analytical expression enclosed in the EC5 draft to predict the fundamental frequency of timber floors.

Spreading dynamics of a microgel particle-laden thermosensitive polymer drop along smooth and nanofiber surfaces

The research focuses on the influence of 300-μm microgel particles in an aqueous solution of a thermosensitive biopolymer on the spreading and deformation of 3.7-mm drops. The drops impact a smooth hydrophilic and a rough hydrophobic surface. A mass fraction of microgel particles varies in a range of 0–0.2. A universal physical model of the spreading of thermosensitive polymer drops laden with microgel particles along surfaces with significantly different roughness is proposed. It explains the strong inhomogeneity of the contact line stretching due to the deceleration of the continuous phase flow by microgel particles and the increased flow vorticity because of the addition of the surface roughness factor. The validity of the proposed physical model is proven by qualitative and quantitative assessments of the contact line deformation when spreading. An empirical expression for the maximum spreading factor is derived, taking into account the properties of liquids, wall roughness, and microgel particle concentration; it reliably predicts when Re≈110−3100, the surface roughness is 0.5–125 nm, Ca=4.5×10−7, and the number of microgel particles in drops is up to 100. The expression was successfully tested during the modeling of arbitrary surface roughness and the increased concentration of microgel particles relative to those considered in experiments during the formation of a biopolymer layer. When developing the method of additive manufacturing of a functional layer, a practical correlation was established between the volume content of microgel particles, acting as potential containers for living cells, in a drop and the area of the biopolymer layer.

Fluid‐Driven Particle Migration and Its Impact on Hydraulic Transmissivity of Stressed Filled Fractures

AbstractThe accurate assessment of hydraulic transmissivity in rock fractures filled with particles is not only a scientific challenge but also a critical need for various industrial applications. However, the intricate dynamics of particle erosion and pore clogging that govern transmissivity evolution remain largely unexplored. In this study, we experimentally examine the fluid‐driven particle migration behavior in filled fractures and its consequent impact on fracture transmissivity under various hydraulic gradients, normal stresses, and fracture apertures. We find that escalating hydraulic gradients not only intensify particle erosion through amplified fluid drag forces and hydro‐mechanical coupling effects but also lead to an increase in the size of migrating particles, thereby augmenting pore clogging. The dynamics of erosion and clogging define four distinct migration phases within the filled fractures. Variations in normal stress and initial fracture aperture significantly alter the particle arrangement and the soil structure stability within the fractures, thereby modulating the progress of particle migration in response to hydraulic gradients. The pattern of particle migration in filled fractures dictates the development of the internal pore structure and normal deformation, ultimately affecting fracture transmissivity. We propose an empirical expression to encapsulate the comprehensive evolution of fracture transmissivity across different particle migration patterns. Our research advances the understanding of fluid‐driven particle migration within filled fractures and provides a practical tool for the precise determination of hydraulic properties of fractured rocks amidst complex geological settings.