Search Results (1,150)

Classical eddy viscosity models add a viscosity term with a turbulent viscosity coefficient developed beginning with the Kolmogorov–Prandtl parameterization. Approximations of unknown accuracy of the unknown mixing lengths and turbulent kinetic energy are typically constructed by solving associated systems of nonlinear convection–diffusion-reaction equations with nonlinear boundary conditions. These often over-diffuse, so additional fixes are added such as wall laws, or different approximations are used in different regions (which must also be specified). Alternately, one can solve an ensemble of NSEs with perturbed data, compute the ensemble mean and fluctuation, and simply directly compute the turbulent viscosity parameterization. This idea is recent. From previous work it seems to be of a lower complexity and greater accuracy. It also produces parameterizations with the correct near-wall asymptotic behavior. The question then arises: Does this ensemble eddy viscosity approach over-diffuse solutions? This question is addressed herein. Full article

The Mittag-Leffler function holds significant importance in fractional calculus due to its extensive applications in addressing challenges across science, engineering, biology, hydrology, and earth sciences. Notably, the closed-form solution of a time-fractional model naturally emerges as the Mittag-Leffler function (MLF), necessitating precise and efficient computations. Consequently, numerical approximations are essential for accurately calculating the Mittag-Leffler function. In this study, we develop a straightforward yet precise real pole rational approximation for the Mittag-Leffler function. We demonstrate first-order convergence and L-acceptability, which aid in mitigating unwanted oscillations. Additionally, we create an effective and precise first-order generalized exponential time differencing scheme to solve the time-fractional reaction–diffusion equations. We obtain and prove the convergence result using Grönwall-type inequality. Several numerical experiments are conducted to confirm the efficiency and accuracy of the proposed numerical scheme compared with exact solutions. The computational efficiency of the proposed method is compared with another existing first-order numerical technique. Furthermore, our proposed scheme is crucial for developing higher-order predictor–corrector schemes for solving time-fractional models. Full article

This study employs molecular dynamics simulations to investigate the influence of functionalized UiO-66 materials (with -H, -NH₂, -NO₂, and -(OH)₂ groups) on the adsorption and diffusion behaviors of ethanol and waste oil before transesterification reactions. A multi-scale modeling approach, including a three-layer interfacial model, surface adsorption, and intra-framework adsorption, was utilized to systematically evaluate the effects of functionalization on structural properties, molecular diffusion, adsorption performance, and interfacial interactions. The simulation results reveal that functionalization enhances the intrinsic diffusivity of the metal–organic framework but generally suppresses the diffusion of ethanol and waste oil. The -(OH)₂ group exhibits the most significant diffusion hindrance due to steric effects and strong hydrogen bonding. Adsorption of waste oil is dominated by coordination and hydrophobic interactions, while ethanol adsorption relies on hydrogen bonding. Within the framework, functionalization does not improve ethanol adsorption capacity; instead, pristine UiO-66 shows the highest uptake due to its optimal pore size. Adsorption energy calculations on the (002) surface indicate that the -NO₂ group exhibits the strongest affinity for oleic acid, owing to its strong electronegativity and synergistic effects with metal sites. For polyunsaturated fatty acids, adsorption performance depends critically on the compatibility between the hydrophobic pore environment and molecular conformation. Ethanol adsorption is governed primarily by hydrogen bonding and metal coordination. This study provides molecular-level insights into the structure–function relationships governing pre-reaction adsorption and mass transport mechanisms of functionalized UiO-66 in transesterification reactions, providing a theoretical foundation for the rational design of efficient pre-reaction microenvironments in biodiesel catalysts. Full article

In liquid composite moulding processes, the curing behaviour of thermoset matrices plays a decisive role in determining manufacturing quality and cycle time. Premature demoulding may lead to insufficiently cured components, whereas excessively long curing times reduce production efficiency. Reliable monitoring and modelling of the curing process are therefore essential for process optimisation. In this study, the cure kinetics of a fast-curing epoxy resin system are modelled using the Grindling kinetic model, which accounts for diffusion-controlled reaction behaviour and vitrification effects. Model parameters are identified using both dynamic and isothermal differential scanning calorimetry (DSC) measurements. In addition, the glass transition temperature is described as a function of the degree of cure using the DiBenedetto relationship. To demonstrate the applicability of the model for process monitoring, an experimental mould equipped with temperature sensors was developed to simulate real-time estimation of the degree of cure during isothermal processing. The predicted degree of cure is validated by post-process DSC analysis of the manufactured samples. Initial comparisons reveal systematic deviations caused by temperature measurement uncertainties. After implementing a temperature correction based on experimentally determined sensor deviations, the predicted degree of cure shows significantly improved agreement with DSC measurements. The results demonstrate that combining kinetic modelling with temperature monitoring enables reliable real-time estimation of the curing state for fast-curing epoxy systems. The study also highlights the critical importance of accurate temperature measurement for curing monitoring and provides insights into the practical implementation of sensor-based monitoring strategies in liquid composite moulding processes. Full article

The feasibility of using brittle δ-Nb₃Al as the reinforcement phase in powder metallurgy nickel-based superalloys depends on both the preparation of near-spherical particles and their phase stability during hot isostatic pressing (HIP). In this study, irregular δ-Nb₃Al particles were converted into near-spherical reinforcement particles by controlled ball milling. The optimized milling condition for obtaining high-sphericity δ-Nb₃Al particles was 200 r/min for 20 h. The morphological evolution during ball milling clarifies a particle-rounding mechanism governed by edge elimination, fine-fragment adhesion, surface consolidation, and re-fragmentation. During subsequent HIP consolidation to introduce the particles into a nickel-based superalloy, extensive interdiffusion occurred between δ-Nb₃Al and the surrounding matrix, resulting in the formation of multilayer interfacial reaction zones and multiple Nb-rich secondary phases, including Laves-(Ni, Cr)₂Nb, Ni₆Nb₇, Nb solid solution, and Ni₃Nb. Quantitative analysis indicates that the retained volume fraction of δ-Nb3Al after HIP is only about 9.85%, much lower than the initial addition level. Combined with thermodynamic analysis based on the effective heat of formation model, the results show that the final phase constitution is governed by the coupled effects of diffusion kinetics and thermodynamic driving force. These findings clarify the intrinsic processing–microstructure–phase transition relationship in δ-Nb₃Al-reinforced powder metallurgy nickel-based superalloys, showing that ball milling controls the powder-state evolution of δ-Nb₃Al, whereas diffusion-driven interfacial reactions during HIP govern its retention and final phase constitution. Full article

CAR-T cell therapy has shown substantial efficacy in haematological malignancies, but its application to solid tumours remains limited by poor effector-cell infiltration, functional exhaustion, antigenic heterogeneity, and an immunosuppressive microenvironment. In this study, we develop a new spatiotemporal mathematical model of CAR-T therapy for solid tumours that integrates these resistance mechanisms within a single reaction–diffusion framework. The model is formulated as a system of partial differential equations describing functional and exhausted CAR-T cells, antigen-positive and antigen-low tumour subpopulations, and chemokine, immunosuppressive, and hypoxic fields. Steady-state analysis and finite-difference simulations showed that therapeutic outcome is governed by the interplay between CAR-T cell infiltration, exhaustion, and antigen escape. The model reproduces partial tumour regression followed by residual tumour persistence, therapy-driven enrichment of antigen-low cells, and reduced efficacy under stronger immunosuppressive and hypoxic conditions. In the combination therapy scenario considered here, repeated simulated CAR-T cell administration together with attenuation of the suppressive microenvironment improves tumour control. The proposed model provides a mechanistic basis for analysing resistance and for future optimisation studies of CAR-T therapy in solid tumours. Full article

This study focuses on the use of Physics-Informed Neural Networks (PINNs) to solve the 1D Advection–Diffusion–Reaction (ADR) equation. The performance of the PINN model is evaluated in comparison with the classical Crank–Nicolson Finite Difference Method (CNFDM) and validated against analytical solutions to assess improvements in accuracy, robustness, and flexibility. Quantitative analysis reveals that the PINN achieved a high level of accuracy with absolute errors ranging from approximately $2.13\times {10}^{-4}$ to $1.17\times {10}^{-3}$ across the spatial domain. The study utilizes a neural network architecture with two hidden layers of 80 neurons each, optimized through a two-stage training process involving Adam and L-BFGS optimizers. This work contributes to the growing field of physics-informed machine learning by demonstrating the strengths and quantitative reliability of the PINN technique for solving complex partial differential equations in transport phenomena. Full article

This study presents a transient mathematical model and its numerical solution for the moving-boundary problem related to substrate diffusion and reaction in enzymatic catalytic particles. The main focus is on bioreactor startup, where the initial substrate concentration inside the particles is zero, forming a dead core that shrinks over time and makes the catalytic effectiveness factor time-dependent. The substrate mass balance leads to a partial differential equation with a moving boundary, solved using the method of lines coupled with Newton’s method (MLN), implemented in Wolfram Mathematica (WM). The proposed approach was validated for zero- and first-order kinetics at steady state, whose analytical solutions are available. Compared to the method of orthogonal collocation on finite elements, the MLN offers advantages such as not requiring an initial concentration profile and simple implementation in WM. The results demonstrate that the proposed method provides accurate and physically consistent solutions, contributing to a better understanding of dead-core dynamics and supporting the design of heterogeneous bioreactors with immobilized enzymes. Full article

Mercury in jarosites is crucial for environmental management and metallurgy. These minerals can incorporate highly toxic heavy metals from mining waste into their structure. This study analyzes the decomposition of mercury jarosite in a Ca(OH)₂ medium, focusing on its topological, kinetic, and modeling characteristics. Topological analysis, XRD and SEM−EDS were performed. ICP−OES was used to analyze the mercury and sulfur ions diffusing from the mercury jarosite into the Ca(OH)₂ solution. The kinetic model that best fit the data was that of spherical particles of constant size with an unreacted core under chemical control. The XRD results did not show new crystallographic phases. SEM−EDS showed a partially decomposed particle indicating a halo and core. The experimental conditions included temperatures from 298.15 to 333.15 K, concentrations of 0.0071–0.23210 mol L⁻¹ Ca(OH)₂, particle diameters of 25–53 µm, and pH of 11.12–12.85. During the induction period, reaction orders of 1.04 and 0.44 were obtained, along with an activation energy of 77.580 kJ mol⁻¹. For the progressive conversion period, the reaction orders were 0.59 and 0.15, with an activation energy of 52.124 kJ mol⁻¹. The overall kinetic modeling showed favorable results, supporting the evolutionary process of the mercury jarosite decomposition reaction in an alkaline medium under different conditions. This allows prediction of when mercury could be released back into the environment in alkaline soils or lime barriers. Full article

We revisit a hyperbolic wildfire model based on reaction–diffusion dynamics with relaxation effects and extend it by incorporating an advection transport term that accounts for wind-driven fire spread. After a planar two-dimensional reformulation and non-dimensionalization of the model, the analysis is restricted to the minimal ignition regime characterized by the presence of a logistic reaction term governing the evolution of the fire-affected tree fraction. The focus of the study is to assess the influence of the effective wind velocity on the propagation dynamics of the fire-affected tree fraction. For this purpose, analytical solutions of the extended wildfire model are derived by applying the Simple Equations Method (SEsM) in its (1,1) variant using a Riccati-type ordinary differential equation as a simple equation. The obtained families of exact solutions describe physically relevant transition fronts connecting fire-unaffected and fully fire-affected states, or vice versa. Numerical simulations of the derived analytical solutions are performed to demonstrate how the internal front thickness and the profile morphology depend on the specific variant of the Riccati-type solution and on the magnitude of the effective wind velocity. A phase-plane stability and bifurcation analysis of the reduced traveling wave system is carried out. Hopf bifurcation thresholds with respect to the effective wind velocity parameter are identified, revealing transitions between monotone front propagation and oscillatory regimes. A regime map is constructed in the parameter plane spanned by the effective wind velocity and the traveling wave speed. This regime diagram delineates regions of qualitatively different propagation behavior, including monotone advancing fronts, possible oscillatory regimes, and regimes in which traveling wave fronts cease to exist. Full article

Hydrogen energy is pivotal to the global energy transition, and the development of high-efficiency, safe hydrogen storage technologies constitutes a prerequisite for its large-scale commercialization. Kinetic bottlenecks including slow reactions, delayed front propagation, and marked spatial heterogeneity driven by strong thermal–mass transfer coupling restrict the engineering application of solid-state metal hydrides. However, the current research mainly focusing on overall performance lacks a systematic understanding of the spatiotemporal evolution mechanisms and their intrinsic links to internal structural control. In this work, a 3D multiphysics model of a LaNi₅-based reactor is developed to systematically elucidate spatiotemporal evolution patterns, facilitating the proposal of a structural decoupling framework based on synergistic thermal–mass resistance reconfiguration. Both absorption and desorption show distinct three-stage evolution, shifting from kinetic dominance to transfer limitation: absorption causes core self-inhibition via heat-hydrogen supply mismatch, leading to much lower core than surface storage capacity; desorption results in significant inner-layer lag due to endothermic cooling-driven pressure drops. Thermal–mass coupling-induced inverted spatiotemporal evolution is identified as the root cause of spatial heterogeneity. Quantitative comparison of straight-pipe, spiral-tube, and honeycomb structures reveals that internal architectures achieve effective thermal–mass decoupling through expanded heat-exchange areas, reconstructed diffusion pathways, and optimized heat source distribution. Notably, the honeycomb structure with a parallel micro-unit network achieves 89.1% and 86.6% reductions in absorption and desorption times, respectively, showing superior dynamic performance and field uniformity. This study provides a theoretical basis for the mechanism-driven design and synergistic performance optimization of high-efficiency solid-state hydrogen storage reactors. Full article

In this paper, we extend some existing models of the Ebola virus disease through a hybrid impulsive conformable approach. The base of the introduced model is a class of partial differential equations that incorporate diffusion terms to describe the development of the Ebola virus disease in time and space. In the extended model, we have considered impulsive effects at fixed moments of time, which is of high significance in investigating opportunities for impulsive vaccination strategies and impulsive control drug treatment on disease evolution. In addition, conformable setting is proposed, which provides modeling flexibility without the complications inherent in classical fractional derivatives. Instead of studying the global stability of an equilibrium, the more general notion of stability of sets is introduced and analyzed. The main stability of sets results are obtained by using the impulsive conformable Lyapunov technique and comparison principle. The proposed framework, concepts and techniques may serve as effective tools for analyzing numerous phenomena in medicine and biology. Full article

In this work, iron–phosphorus-based composite biochar (FPBC) was prepared by modification with the leachate of spent LiFePO₄ batteries. The effects of solution pH, dosage, adsorption time, initial concentration, and temperature on the adsorption performance of FPBC were investigated by batch adsorption experiments with Pb(II) and Sb(III) as the target pollutants, and the adsorption mechanism was explored using SEM, BET, XPS, FTIR and XRD characterization. The results indicated that as the initial pH of the solution increased, the removal efficiency of FPBC for Pb(II) gradually increased, while the removal efficiency for Sb(III) remained largely unchanged. The removal of Pb(II) and Sb(III) by FPBC fitted the pseudo-second-order kinetic model and the three-step intraparticle diffusion model, indicating that their removal was primarily controlled by chemical adsorption. Isothermal adsorption studies revealed that FPBC adsorption of Pb(II) better fitted the Langmuir and D-R models, suggesting a monolayer-dominated adsorption process. In contrast, adsorption of Sb(III) fitted the Langmuir, Freundlich, and Temkin models, suggesting a combination of monolayer and multilayer adsorption characteristics. The maximum adsorption capacities of FPBC for Pb(II) and Sb(III) were 312.54 mg·g⁻¹ and 219.20 mg·g⁻¹ at 30 °C, which were approximately 12.85 and 3.37 times those of commercial corn stalk biochar (BC). Thermodynamic analysis confirmed that the removal of Pb(II) and Sb(III) by FPBC was a spontaneous and endothermic process. In addition, FPBC demonstrated strong selective adsorption of Pb(II) in the binary co-adsorption system of Pb(II) and Sb(III). Mechanism studies indicated that Pb(II) removal primarily occurred through co-precipitation, complexation, ion exchange, and electrostatic adsorption, while Sb(III) was mainly adsorbed by FPBC via redox reactions and complexation. Therefore, this work not only provides a low-cost, high-performance adsorbent for the remediation of water contaminated with Pb(II) and Sb(III), but also opens up new avenues for the resource recovery of the leachate of spent LiFePO₄ batteries. Full article

Decarburization during austenitization degrades the surface integrity and mechanical performance of tool steels, yet the quantitative influence of process pressure remains unclear. In this study, the effect of process pressure on the decarburization behavior of H13 tool steel was investigated. Specimens were austenitized at 920–1020 °C for 60 min under pressures ranging from 0.01 to 760 Torr. Carbon concentration profiles were measured by electron probe microanalysis, and hardness degradation and mass loss were evaluated. A one-dimensional diffusion model with a Robin boundary condition was applied to describe the coupled effects of carbon diffusion and surface reaction. High-vacuum conditions suppressed decarburization, whereas increasing pressure accelerated carbon loss, leading to deeper decarburized layers and pronounced hardness reduction. The model reproduced the experimental results and revealed a pressure-dependent transition in the dominant decarburization mechanism. Full article

Hydrogen plasma heating, a unique method for heating and reducing iron ore, is distinguished by its high heat, rapid reduction, and high efficiency, making it a promising technique in the metallurgy field. In this study, a non-transferred arc plasma heating system was used with Ar-H₂ as the working gas and acidic pellets as the raw material. The microstructures and elemental distributions of the slag and iron phases during the reduction process were examined using electron microscopy and energy-dispersive X-ray. The variation patterns of Fe-containing phases in the reduction products were found using X-ray diffraction and full-spectrum fitting refinement. The conversion rate of the oxidized pellets and the deoxidation conversion rate per area were estimated for various gas flow rates and reduction times. A reaction kinetics model was also used to study the reaction controlling step. The results showed that during the reduction process, with an H₂ flow rate of 4.5 L min⁻¹ and a 40 min reduction, the conversion(α) reached 99.89% and the purity of the reduced metallic iron reached 99.9%, achieving the industrial-grade 3N standard. Si and Al in the melt bath generated fayalite (Fe₂SiO₄) and hercynite (FeAl₂O₄) with Fe_xO. The deoxidation conversion rate per unit area was 1.11 g (cm² min)⁻¹. A three-dimensional diffusion-controlled model was used to describe the reduction process, and the mechanism function was 2/3(1 + α)^3/2[(1 + α)^1/3]⁻¹. The values of the reduction reaction rate constant (K) were 12.6 × 10⁻² s⁻¹ and 12.8 × 10⁻² s⁻¹ when the flow rates of H₂ gas were 3 and 4.5 L min⁻¹, respectively. The apparent activation energy was 21.9 kJ mol⁻¹. The empirical equation for the specific reduction rate was calculated as ln r = −2637.5/T − 0.407. Full article