Two-dimensional Diffusion Model Research Articles

Spatial diffusion of potentially toxic elements in soils around non-ferrous metal mines

This study focuses on the diffusion patterns of principal ore-forming elements (Pb and Zn) and associated elements (Cd, Cu, Cr, and As) in lead-zinc ore. Sampling points in upwind and downwind directions of lead-zinc ore areas at various densities (1 N/km2 – 4 N/km2) were categorized. This study analyzed the statistical relationship between the content of PTEs in the soil around lead-zinc ore and the source strength and dominant wind direction, constructed one-dimensional and two-dimensional diffusion model, and simulated the EER scope caused by PTEs. The findings indicate that: (1) concerning source strength, the content of PTEs in soils of high-density ore aggregation areas is significantly higher than in low-density ore aggregation areas. However, the impact of source strength decreases with decreasing ore grade, with a difference in Pb content of 1.71 times among principal ore-forming elements and almost consistent Cd content among associated elements. (2) Regarding the transport pathways, for most PTEs, the inverse proportion coefficients downwind are higher than upwind, approximately 1.18–3.63 times, indicating greater migration distances of PTEs downwind due to atmospheric dispersion. (3) By establishing a two-dimensional risk diffusion model, the study simulates the maximum radius of risk diffusion (r = 5.7 km), the 50% probability radius (r = 3.1 km), and the minimum radius (r = 0.8 km) based on the maximum, median, and minimum values statistically obtained from the EER. This study provides a scientific basis for implementing preventive measures for PTEs accumulation in soil within different pollution ranges. Different risk prevention and control measures should be adopted for PTEs accumulation in soil within the three ranges after cutting off pollution sources. Subsequent research should further investigate the impact and contribution of atmospheric transmission and surface runoff on the diffusion of PTEs in areas with high risk near lead-zinc ore.

Spatiotemporal dynamics of prey-predator model incorporating Holling-type II functional response with fear and its carryover effects.

The recent focus in the fields of biology and ecology has centered on the significant attention given to the mathematical modeling and analyzing the spatiotemporal population distribution among species engaged in interactions. This paper explores the dynamics of the temporal and spatiotemporal delayed Bazykin-type prey-predator model, incorporating fear and its carryover effect. In our model, we incorporated a functional response of the Holling-type II. In the temporal model, a detailed dynamic analysis was carried out, investigating the positivity and boundedness of solutions, establishing the uniqueness and existence of positive interior equilibria, and examining both local and global stability. Additionally, we explored the presence of saddle-node, transcritical, and Hopf bifurcations varying attack rate parameter. The delayed system shows highly periodic behavior. Additionally, for the spatiotemporal model, we provide a complete analysis of local and global stability, and we derive the conditions for the existence of Turing instability for both self-diffusion and cross-diffusion, respectively. The two-dimensional diffusive model is further discussed, highlighting various Turing patterns, including holes, stripes, and hot and cold spots, along with their biological significance. Numerical simulations are executed to validate the analytical findings in both temporal and spatiotemporal models.

Numerical approximation of a two-dimensional fractional neutron diffusion model describing dynamics of neutron flux in a nuclear reactor

The goal of this work is to develop a high-order numerical technique to solve a two-dimensional fractional neutron diffusion (FND) model describing dynamical behavior of a lead-cooled fast reactor. This method is based on L1 technique for temporal discretization and a high-order compact Alternating Direction Implicit (ADI) finite difference scheme for the spatial discretization. Numerical simulations are conducted to demonstrate the applicability of the method. We calculate the neutronic flux in the core and examine how the order of the temporal-fractional derivative influences the neutronic flux. The results obtained with FND model for different values of anomalous diffusion coefficient are compared against those obtained with P1 model and classical neutron diffusion equation. In addition, the computational time (CPU time) is provided in order to justify the computational efficiency of the proposed method.

Complex pattern evolution of a two-dimensional space diffusion model of malware spread

In order to investigate the propagation mechanism of malware in cyber-physical systems (CPSs), the cross-diffusion in two-dimensional space is attempted to be introduced into a class of susceptible-infected (SI) malware propagation model depicted by partial differential equations (PDEs). Most of the traditional reaction-diffusion models of malware propagation only take into account the self-diffusion in one-dimensional space, but take less consideration of the cross-diffusion in two-dimensional space. This paper investigates the spatial diffusion behaviour of malware nodes spreading through physical devices. The formations of Turing patterns after homogeneous stationary instability triggered by Turing bifurcation are investigated by linear stability analysis and multiscale analysis methods. The conditions under the occurence of Hopf bifurcation and Turing bifurcation in the malware model are obtained. The amplitude equations are derived in the vicinity of the bifurcation point to explore the conditions for the formation of Turing patterns in two-dimensional space. And the corresponding patterns are obtained by varying the control parameters. It is shown that malicious virus nodes spread in different forms including hexagons, stripes and a mixture of the two. This paper will extend a new direction for the study of system security theory.

The Cauchy problem for an inviscid and non-diffusive Oldroyd-B model in two dimensions

A two-dimensional inviscid and diffusive Oldroyd-B model was investigated by Elgindi and Rousset (2015) where the global existence and uniqueness of the strong solution were established for arbitrarily large initial data. As pointed out by Bhave et al. (1991), since the diffusion coefficient is significantly smaller than other effects, it is interesting to study the non-diffusive model. In the present work, we obtain the global-in-time existence and uniqueness of the strong solution to the non-diffusive model with small initial data by deriving some uniform regularity estimates and taking vanishing diffusion limits. In addition, the large time behavior of the solution is studied and the optimal time-decay rates for each order of spatial derivatives are obtained. The main challenges focus on the lack of dissipation and regularity effects of the system and on the slower decay in the two-dimensional settings. A combination of the spectral analysis and the Fourier splitting method is adopted.

Shape optimization of the energy efficiency of building retrofitted facade

The current state of research indicates a necessity of further examination in both numerical and experimental studies related to optimizing shapes of building enclosures for the enhancement of their energy efficiency. The demand for research primarily arises due to the numerical complexities associated with optimizing shapes for this specific purpose. Consequently, the primary objective of this article is to address and bridge these gaps in the field. To achieve this, a two-dimensional steady-state heat diffusion model is assumed to represent the physical processes occurring within building facades of varying shapes. A third type boundary condition is applied to the exterior boundary, encompassing convective and incident short-wave solar radiation effects. The calculation of short-wave radiation accounts for factors such as sunlight exposure and shading, influenced by the surrounding urban environment. The internal boundary interfaces with the indoor ambient air, and thus, a Robin boundary condition is adopted. To tackle the computational demands while ensuring accuracy, the boundary element method (BEM) is employed by discretizing the domain boundary into discrete elements. Then, two heat transfer design objectives are define according to the period of investigations: ones related to enhancing heat transfer and ones focused on thermal insulation problem. Last, a real-world case study is conducted, considering a house wall under varying climate conditions throughout the year. Optimal shapes for the external wall boundary are determined with the constraint that the optimized facade utilizes the same amount of material as the reference flat one. The results demonstrate a substantial increase in energy efficiency compared to the reference flat wall case.

Dynamic behavior of sorption of europium onto biotite flakes under the condition of saline groundwater

This study discussed the interaction between Eu and biotite in flakes under the coexistence of Na ions, considering saline groundwater. The sorption experiments were conducted by mixing an Eu solution with Na or K ions and biotite flakes at pH 5. Resultantly, the sorption of Eu gradually increased because of the diffusion into the layered structure of biotite flakes. Thereafter, the sorption amount of Eu with Na ions was less than that without Na ions, and Eu was hardly sorbed with K ions. Additionally, the apparent diffusion coefficients were estimated to be in the order of 10−13 m2/s based on a two-dimensional diffusion model. These values were lower than the effective diffusion coefficients in plutonic rock, which are employed to assess the performance of geological disposal systems. This indicates that the sorption of Eu in plutonic rock is restricted by the diffusion process in biotite flakes.Graphical abstract

Thermal Decomposition Kinetics of Clinochlore at High Temperature and Its Implications

Abstract The dehydration kinetics of clinochlore were performed using a non-isothermal thermogravimetric analysis under a dynamic nitrogen atmosphere at ambient pressure. Clinochlore, with a grain size of 5–10 μm, was analyzed using heating rates of 2, 5, 10, 20, and 40 K/min at temperatures of up to 1260 K. Thermogravimetric data of clinochlore at different heating rates were modeled using the Kissinger method, Flynn-Wall-Ozawa method, and Coats-Redfern method. The dehydration of clinochlore is ascribed to the decomposition of brucite and talc layers. The dehydration temperature of the brucite layer is 730–920 K, which corresponds to a two-dimensional phase boundary model (R2) with an activation energy of 214.94 ± 7.79 kJ/mol and a pre-exponential factor of 12.52 ± 0.62 min−1. The dehydration temperature of the talc-like layer is 960–1140 K, which follows a two-dimensional diffusion model with an apparent activation energy of 419.49 ± 23.62 kJ/mol and a pre-exponential factor of 19.97 ± 1.76 min−1. Combined with the parameters obtained in this study, the fast dehydration of clinochlore can trigger earthquakes in the subduction zone.

Exploring the Characteristics and Path of Internationalization Development of Vocational Education Based on the Information Diffusion Model

Abstract Internationalization of higher education has become an inevitable trend under the impetus of world economic integration and informationization. This paper adopts the information diffusion technique in the fuzzy information optimization method to construct an assessment model for internationalization development of vocational education. Explicit sample data are turned into fuzzy sets, information diffusion points are built using a two-dimensional average information diffusion model, and the estimated loss and probability distribution of information diffusion are calculated to assess the internationalization development level of vocational education. The constructed indicators set people’s awareness of the internationalization development of vocational education and the dissemination effect, analyzed the internationalization development trend of vocational education in the past five years by combining the Google search data, and explored the impact of internationalization development of vocational education based on the indicators of output level and quality level. The results show that the communication effects of internationalization of vocational education are all rated between (3.3, 3.6), and the attention to the internationalization development of vocational education has increased from 10% to 70% in six years. This study provides suggestions and strategies for advancing the internationalization of higher vocational education in China, which is conducive to steadily increasing the internationalization of higher vocational education in China and improving the level of schooling.

Stepped pyrolysis: A novel approach for enhanced adsorbency and carbon in Pigeon pea stalk char

This article introduces a novel technique, stepped pyrolysis, for thermal degradation of biomass. The differential thermogram (DTG) of pigeon pea stalk was deconvoluted into their intrinsic bioconstituents and these deconvoluted peaks were used to decide the different stages or steps to finalize the thermal protocol to be executed for stepped pyrolysis. The stepped pyrolysis and traditional pyrolysis of pigeon pea stalk were conducted and the characterisation of outputs was done. The obtained result for the char produced by the stepped pyrolysis method showed a reduction of 5% in the volatile matter and an increment of 5% in fixed carbon content, over the product of traditional pyrolysis. The stepped pyrolysis increased the iodine number by 17% while reducing the methylene blue values by 31% as compared to the traditional pyrolysis. A significant reduction in activation energy by 21% for FWO, 18% for KAS, and 26% for Starink methods were recorded for stepped, over traditional pyrolysis. The pyrolysis reaction mechanism of stepped and traditional pyrolysis methods followed the second-order (F2) and two-dimensional diffusion model (D2). The stepped pyrolysis method produced a better quality product with more micropores and carbon-rich char than the traditional method.

Sensitivity Analysis of a 2D Stochastic Agent-Based and PDE Diffusion Model for Cancer-on-Chip Experiments

The present work extends a previous paper where an agent-based and two-dimensional partial differential diffusion model was introduced for describing immune cell dynamics (leukocytes) in cancer-on-chip experiments. In the present work, new features are introduced for the dynamics of leukocytes and for their interactions with tumor cells, improving the adherence of the model to what is observed in laboratory experiments. Each system’s solution realization is a family of biased random walk trajectories, affected by the chemotactic gradients and in turn affecting them. A sensitivity analysis with respect to the model parameters is performed in order to assess the effect of their variation on both tumor cells and on leukocyte dynamics.

Lithium Self-Diffusion in a Polymer Electrolyte for Solid-State Batteries: ToF-SIMS/ssNMR Correlative Characterization and Modeling Based on Lithium Isotopic Labeling.

Manufacturers aim to commercialize efficient and safe batteries by finding new strategies. Solid-state electrolytes can be seen as an opportunity to develop batteries with a high energy density. They allow the use of lithium foil as the anode, increasing the energy density. Also, they are composed of nonflammable materials making them safer than liquid electrolytes. However, to enhance the electrochemical performances of forthcoming solid-state lithium metal batteries, phenomena governing ionic conductivity have yet to be mastered in such devices. Lithium isotopic tracing was successfully used in previous works to further understand lithium ion transport mechanisms in batteries. Time-of-flight secondary ion mass spectrometry (ToF-SIMS) and 6/7Li high-resolution solid-state nuclear magnetic resonance (ssNMR) spectroscopy are two complementary techniques probing local and global scale, respectively. Both techniques can distinguish lithium isotopes. Here, four polymer membranes were elaborated with the same lithium concentration, but with various isotopic enrichments from 7.6 to 95.4% of 6Li. The selected material was a poly(ethylene oxide) (PEO) membrane containing lithium bis(trifluoromethylsulfonyl)imide (LiTFSI) as lithium salt. They are widely studied in the lithium battery field. First, reliable ToF-SIMS and ssNMR methodologies were validated in light of the converging results. They led to accurate determination of lithium isotopic abundance of polymer membranes with a 1 or a 2% uncertainty, respectively. Then, the developed methodologies were applied to characterize lithium self-diffusion in a polymer membrane. Furthermore, numerical simulations based on a two-dimensional diffusion model compared with ToF-SIMS analyses allowed us to extract a lithium self-diffusion coefficient of 1.6 × 10-12 m2·s-1 at 60 °C, which complements other published values. The robust methodologies described in this work can be extended to various applications and materials. They stand as powerful strategies to better understand lithium ionic transport, especially in multiphase materials, for example, in hybrid solid-state electrolytes.

Investigating the chloride ion resistance of cracked concrete through Reverse-seepage and Saturation based Action Anti-corrosion Tech (RS-AAT): Experimental and numerical analysis

This study investigates the chloride ions transport in cracked concrete with the impact of RS-AAT (Reverse-seepage and Saturation based Action Anti-corrosion Tech) in the environment of wet-dry cycling and soaking conditions. With the effect of reverse water pressure, the chloride ions accumulation surrounding the crack on the concrete surface and the chloride ions penetration within the concrete along the cracks are demonstrated by the indoor experiments. A two-dimensional convective diffusion model of chloride ions in cracked concrete is established to simulate the correlation between crack size and the change of reverse water pressure. Four types of cracks with widths of 0 mm, 0.2 mm, 0.4 mm and 1 mm, and a depth of 20 mm were selected. Three types of water pressures with values of 0.2 MPa, 0.3 MPa and 0.4 MPa were used for the technique. The experimental results show that the anti-convection effect induced by the reverse water pressure can inhibit the accumulation of chloride ions around the cracks and effectively limit the intrusion of chloride ions into the concrete protection layer. With the increase of reverse water pressure, the inhibition effect of seepage flow on chloride ions is more obvious. The experimental findings provide valuable references for investigating marine concrete durability and resistance to chloride ion erosion.

Numerical investigation on the migration of leaked pollutants after liquid pressurized pipeline leakage regarding oil and gas parallel pipelines situation

Buried parallel pipelines may cause environmental pollution and explosive accidents after leakage due to corrosion, third-party damage or other factors. Here, an actual oil pipeline in-situ is employed as a prototype to establish a two-dimensional incompressible leakage and diffusion model. The model was applied to analyze the effects of different parameters including pipe diameters, burial depths and flat spacing to other pipelines on the diffusion distance of spilled diesel fuel in porous media. The results show that the existence of natural gas pipeline can change both of the polluted shape and the distance range. The natural gas pipeline can effectively inhibit the diffusion range of diesel with the increase of pipe diameter and the decrease of buried depth. A smaller spacing between parallel transmission pipelines, also results in a reduced horizontal diffusion rang. Meanwhile, the dimensionless formulas are fitted based on the Π theorem regarding diesel diffusion range versus diameter of gas pipeline, buried depth, and spacing between parallel pipelines. This study can develop appropriate protocols for detecting and recovering contaminants from paralleling pipeline.

Two-dimensional interfacial diffusion model of inhibitory synaptic receptor dynamics

The diffusion-trapping of protein receptors in post-synaptic regions of a neuron’s plasma membrane plays a key role in determining the strength of synaptic connections and their regulation during learning and memory. In this paper, we construct and analyse a two-dimensional interfacial diffusion model of inhibitory synaptic receptor dynamics. The model involves three major components. First, the boundary of each synapse is treated as a semi-permeable interface due to the effects of cytoskeletal structures. Second, the effective diffusivity within a synapse is taken to be smaller than the extrasynaptic diffusivity due to the temporary binding to scaffold protein buffers within the synapse. Third, receptors from intracellular pools are inserted into the membrane extrasynaptically and internalized extrasynaptically and synaptically. We first solve the model equations for a single synapse in an unbounded domain and explore how the non-equilibrium steady-state number of synaptic receptors depends on model parameters including synaptic radius and the permeability of the synaptic interface. We then use matched asymptotic analysis to solve the corresponding problem of multiple synapses in a large, bounded domain. In particular, we show how diffusion mediates pairwise synaptic interactions that could provide a substrate for heterosynaptic plasticity. Finally, we indicate how to apply the model to the stochastic dynamics of single receptors.

Data Assimilation for the Two-Dimensional Ambipolar Diffusion Equation in Earth’s Ionosphere Model

The problem of variational data assimilation for the INM RAS two-dimensional diffusion model of the Earth’s ionosphere F region is considered. Total integral electron contents along given paths are used as observation data. The general statement of the problem in differential form is formulated, and its solvability is analyzed. Based on a regularized statement, an iterative algorithm for solving the assimilation problem is constructed, and its convergence is demonstrated. A finite-dimensional approximation is constructed, the numerical solution of the problem is implemented, and the stability and convergence of the difference scheme are proved. The quality of the reconstruction of electron concentration fields is examined in test numerical experiments. It is shown that a weakly perturbed solution is reconstructed with acceptable accuracy for both stationary and evolutionary statements in the case of vertical and slant integration paths.

Qualitative Numerical Analysis of a Free-Boundary Diffusive Logistic Model

A two-dimensional free-boundary diffusive logistic model with radial symmetry is considered. This model is used in various fields to describe the dynamics of spreading in different media: fire propagation, spreading of population or biological invasions. Due to the radial symmetry, the free boundary can be treated by a front-fixing approach resulting in a fixed-domain non-linear problem, which is solved by an explicit finite difference method. Qualitative numerical analysis establishes the stability, positivity and monotonicity conditions. Special attention is paid to the spreading–vanishing dichotomy and a numerical algorithm for the spreading–vanishing boundary is proposed. Theoretical statements are illustrated by numerical tests.

Fractionalizing, coupling and methods for the coupled system of two-dimensional heat diffusion models

<abstract><p>The present manuscript gives an overview of how two-dimensional heat diffusion models underwent a fractional transformation, system coupling as well as solution treatment. The governing diffusion models, which are endowed with Caputo's fractional-order derivatives in time $ t $, are suitably coupled using the (1) convection phenomenon, (2) interfacial coupling by considering the mechanism of a double-layered bar, and the (3) nonlinear coupling due to temperature-dependent thermal diffusivities. Semi-analytical and analytical methods are considered for the solution treatment. Moreover, we seek a computational environment to graphically illustrate the systems' response to different fractional orders in each case through the determined diffusional fields. Besides, we supply certain concluding notes at the end.</p></abstract>

Multilayer perceptron neural network applied to TG dynamic data of biopolymer chitosan – A robust tool to study the kinetics of solid thermal decomposition

• New artificial neural network to study kinetics of solid thermal decomposition. • MLP neural network applied in TG non-isothermal data to determine kinetic triplet. • Validation of new MLP is made using simulated data with kinetic model combination. • Accurate kinetic study of chitosan thermal decomposition process. • Multi-step reactions are described by combination of kinetic models. Thermogravimetric analysis (TG) was used to investigate the thermal degradation process of chitosan under dynamic condition. A methodology based on multilayer perceptron neural network (MLP) was proposed and allows quantifying the contribution of each kinetic models to better describe experimental data. Artificial neural networks (ANN) have been applied to chemical problems to reproduce experimental data with higher accuracy, acting in this way as a universal approximation method. In this work, the MLP architecture uses the approximation on the integral temperature by Coats and Redefern, isoconversional KAS methodology and integral kinetic models. The neurons in the hidden layer are activated by kinetic models, not only to reproduce the experimental data, but to bring physical information about the thermal process. The MLP validation was carried out using simulated data, the neural network retrieves the mechanism based on the residual error and consider the curve shape. For the chitosan experimental data, the MLP shows the lowest residual error to fit experimental data, which were performed at N 2 atmosphere under four heating rates: 2.5, 5.0, 7.5, and 10 ° C min - 1 . The activation energy was calculated as 98.1 to 183.3 kJ mol - 1 along the conversion degree. This increasing behavior of activation energy is due to initial polymer dehydration, which foments the enhancement of intramolecular interactions between the monomers and also the stronger intermolecular interactions between the interconnected layers of polymeric chains. Due to decomposition of hydroxyl and acetamido groups, the MLP determined contribution of two-dimensional diffusion model (D2) and for the breakdown of the glycosidic bond, MLP determined high contribution of the volume contraction model (R3) followed by surface area contraction model (R2). These results corroborate the expected behavior as solid interconnected material along the energy supply experiment.

Kinetic and Thermodynamic Analysis of Thermal Decomposition of Waste Virgin PE and Waste Recycled PE

Polyethylene is one of the key components of plastic wastes that can be utilized for resource recovery through pyrolysis method. Understanding of thermal decomposition properties and reaction mechanism of pyrolysis are necessary in designing an efficient reactor system. This study investigated the kinetics and thermodynamics parameters for individual waste virgin polyethylene (WVPE) and waste recycled polyethylene (WRPE) by using distributed activation energy model (DAEM). The calculated kinetic parameters (activation energy (Ea) and pre-exponential factor (A) were used to determine thermodynamic parameters (enthalpy (ΔH), Gibbs free energy (ΔG) and entropy (ΔS). The activation energy (Ea) values for the WVPE estimated at conversion interval of 5%-95% were in the range of 180.62 to 268.04 kJ/mol while for the WRPE, the values were between 125.58 to 243.08 kJ/mol. This indicates the influence of exposure to weathering and mechanical stress during recycling on the course of the WRPE degradation. It was also found that the pyrolysis reaction for both WVPE and WRPE were best fitted using the two-dimensional diffusion (D2) model. The WVPE exhibited higher enthalpy and lower ΔG compared to WRPE, suggesting that less energy is required to thermally degrade recycled waste PE into products of char, gases and pyro-oils. Both kinetics and thermodynamics analyses were useful for the development of the plastic waste management through pyrolysis process.