One-dimensional Diffusion Research Articles

Non-local eddy diffusivity model based on turbulent energy density in scale space

Recently, a non-local eddy diffusivity model for the turbulent scalar flux was proposed to improve the local model and was validated using direct numerical simulation (DNS) of homogeneous isotropic turbulence with an inhomogeneous mean scalar (Hamba, J. Fluid Mech., vol. 950, 2022, A38). The non-local eddy diffusivity was assumed to be proportional to the two-point velocity correlation that was expressed in terms of the energy spectrum. Because the Fourier transform of velocity in the homogeneous directions was used to define the energy spectrum, it is not yet understood whether the proposed model can be applied to inhomogeneous turbulence. Thus, this study aimed to improve the non-local model using the scale-space energy density instead of the energy spectrum. First, the scale-space energy density based on filtered velocities was examined using the DNS data of homogeneous isotropic turbulence to obtain its simple form corresponding to the Kolmogorov energy spectrum. Subsequently, the two-point velocity correlation was expressed in terms of the scale-space energy density. Using these expressions, a new non-local eddy diffusivity model was proposed and validated using the DNS data. The one-dimensional non-local eddy diffusivity obtained from the new model agrees with the DNS value. The temporal behaviour of the three-dimensional non-local eddy diffusivity was improved compared with the previous model. Because the scale-space energy density was already examined in turbulent channel flow, it is expected that the new non-local model can also be applied to inhomogeneous turbulence and is useful for gaining insight into turbulent scalar transport.

Study of the kinetics of aviation oils thermal conversion under non-isothermal conditions

The paper discusses the results of a kinetic study of the thermal decomposition of MS-8P, TN-98, and TN-600 aviation oils under conditions of continuous heating at a constant rate of 5 K/min to a temperature of 1 073 K. An integral method was used to describe the reaction mechanism and determine the macrokinetic parameters. It has been established that, from a phenomenological point of view, the average reaction of aviation oils conversion under the experimental conditions corresponds to the reaction model described by the surface-limited reaction equation (MS-8P), the power law (TN-98) and the model described by the three-dimensional diffusion-limited reaction equation (TN-600). When dividing the averaged reaction into two reactions (the first is completed at a temperature of 550–600 K, the second at a temperature of 638–655 K), it is determined that the first reaction is described by the reaction equation of the 2nd order (MS-8P), the first order (TN-98) and the reaction equation of one-dimensional diffusion (TN-600), and the second the reaction equation of the first order (three types of oil). The activation energy of the first reaction was 99 kJ/mol (MS-8P), 145.6 kJ/mol (TN-98) and 57.4 kJ/mol (TN-600), the value of the pre-exponential factor was – 144 241 567 min–1 (MS-8P), 62 161 395 942 min–1 (TN-98) and 236.16 min–1 (TN600). The activation energy of the second reaction is 160 kJ/mol (MS-8P), 91.6 kJ/mol (TN-98) and 127.1 kJ/mol (TN-600), the pre-exponential factor is 8.81 ‧ 1011 min–1 (MS-8P), 1.26 ‧ 104 min–1 (TN-98) and 2.04 ‧ 108 min–1 (TN-600). It is shown that the use of these values of the activation energy and the pre-exponential factor leads to agreement between the calculated values of the degree of decomposition of the studied oil samples and the experimental ones in the range of values of the degree of decomposition from 0 to 1.

A stochastic interpretation of the parametrix method

UDC 519.21 We revisit, in a didactic manner and by using stochastic analysis, the parametrix method and its application to unbiased simulation. We consider, in particular, the case of one-dimensional diffusions without drift.

Multidimensional Transport Experiment and Simulation of Chloride Ions in Concrete Subject to Simulated Dry and Wet Cycles in a Marine Environment.

Chloride ion erosion is an important factor affecting the durability of marine engineering concrete. In particular, concrete structures in wave splash and tidal zones are subjected to dry and wet cycles and multidimensional diffusion of chloride ions. To investigate the intricate diffusion of chloride ions within concrete under these dynamic conditions, we devised a comprehensive experiment. This experiment encompasses multiple dimensions, involving dry and wet cycles, as well as static immersion. The experiment intends to reveal how chloride ions are distributed in the concrete and clarify the changes that occur in its microstructure. Based on Fick's second law, the multidimensional diffusion model of chloride ions in concrete under the dry and wet cycles and static immersion was established by comprehensively considering the effects of chloride ion exposure time, environment temperature, relative humidity, and the action of dry and wet cycles. The results show that, under the same conditions, the chloride content in concrete decreases with the increase in penetration depth but increases with the increase in the chloride diffusion dimension and exposure time. Dry and wet cycles and multidimensional diffusion of chloride ions increase the development of cracks and pores in the concrete structure and generate large quantities of C3A·CaCl2·10H2O, which will exacerbate the chloride ion transport rate and penetration depth of concrete. Under the same exposure time and penetration depth, the chloride ion content in concrete under two-dimensional (2D) and three-dimensional (3D) diffusion under dry and wet cycles was 1.09~4.08 times higher than that under one-dimensional (1D) diffusion. The correlation coefficients between the simulation results of the multidimensional transport model of chloride ions in concrete under multi-factor coupling and the experimental results were all greater than 0.95, and the model can be utilized to predict the distribution of chloride ion concentration in concrete.

Computational analysis of an electrostatic separator design for removal of volatile organic compounds from indoor air

ABSTRACT Concentrations of volatile organic compounds (VOCs) in air can be reduced in electrostatic separators where VOCs are ionized using ion-molecule reactions, extracted using electric fields, and eliminated in a waste flow. Embodiments for such separator technology have been explored in only a few studies, despite the possible advantage of purification without adsorbent filters. In one design, based on ionization of VOCs in positive polarity with hydrated protons as reactant ions, efficiencies for removal were measured as 30–40% . The results were fitted to a one-dimensional convective diffusion model requiring an unexpectedly high production rate of reactant ions to match both the model and data. A realistic rate of reactant ion production was used in finite element method simulations (COMSOL) and demonstrated that low removal efficiency could be attributed to non-uniform patterns of sample flow and to incomplete mixing of VOCs with reactant ions. In analysis of complex systems, such as this model, even limited computational modeling can outperform a pure analytical approach and bring insights into limiting factors or system bottlenecks. Implications: In this work, we applied modern computational methods to understand the performance of an air purifier based on electrostatics and ionized volatile organic compounds (VOCs). These were described in the publication early 2000s. The model presented was one-dimensional and did not account for the effects of flow. In our multiphysics finite element models, the efficiency and operation of the filter is better explained by the patterns of flow and flow influences on ion distributions in electric fields. In general, this work helps using and applying computational modelling to understand and improve the performance bottlenecks in air purification system designs.

Stable limit theorems for additive functionals of one-dimensional diffusion processes

Stable limit theorems for additive functionals of one-dimensional diffusion processes

Inverse problem of determining the kernel of integro-differential fractional diffusion equation in bounded domain

In this paper, an inverse problem of determining a kernel in a one-dimensional integro-differential time-fractional diffusion equation with initial-boundary and overdetermination conditions is investigated. An auxiliary problem equivalent to the problem is introduced first. By Fourier method this auxilary problem is reduced to equivalent integral equations. Then, using estimates of the Mittag-Leffler function and successive aproximation method, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel which will be used in study of inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven.

Droplet absorption and spreading into thin layers of polymer hydrogels

From biological tissues to microactuators and absorption of solvents into layers of paint, macroscopically non-porous materials with the capacity to swell when in contact with a solvent are ubiquitous. In these systems, owing to strong solid–fluid interactions, chemically driven flows can yield large geometric changes. We study experimentally and theoretically the canonical problem of the swelling of a thin hydrogel layer by a single water drop. Using a bespoke experimental set-up, we observe fast absorption leading to a radially spreading axisymmetric blister. We use a fully three-dimensional linear poroelastic framework with nonlinear kinematic equations to obtain governing equations, which we then reduce with thin-layer scalings to a one-dimensional nonlinear diffusion equation for the evolution of the blister geometry. In the limits of large and small deformations, the evolution of the blister characteristic height and radius are self-similar, following power laws in time. Our experimental measurements show that the evolution of the blister is broadly within the theoretical predictions in the large and small deformation regimes. In the general intermediate deformation regime, the measurements are well captured by our reduced one-dimensional diffusion model, which does not require the sophisticated and computationally expensive numerical approaches necessary for the original two-dimensional nonlinear coupled transport problem. By adapting modelling techniques from the fluid dynamics of thin porous elastic layers to a polymer swelling problem, our modelling framework extends the range of polymer swelling problems that can be treated with semi-analytical methods. Moreover, our detailed experimental data can serve as a test case for future nonlinear poroelastic frameworks of swelling polymer materials.

New understanding of one-dimensional thermal glide of a nano-sized prismatic dislocation loop in bcc iron: an atomic scale study

Vacancy-type prismatic dislocation loops (PDLs) can be formed directly through cross-slip mechanisms of screw dislocations or via the accumulation and transformation of high-density vacancy-type defects. The concentration of vacancy-type defects in body-centered cubic (BCC) iron can be increased through plastic deformation and in the presence of hydrogen. Nano-sized PDLs can act as high-speed vacancy transport in BCC iron, causing embrittlement. Based on molecular dynamics simulations, this study comprehensively analyzes the one-dimensional diffusion of vacancy-type PDLs composed of 37–169 vacancies (V37-V169) at temperatures ranging from 100 to 1000 K. The activation energy of diffusion increased with decreasing PDL size for small loops (V37–V61) but remained almost constant for larger loops (>V91). This behavior was found to be associated with the tensile stress produced by the opposing edge dislocation segment in the loop where the distance is closer for a smaller PDL. That is, the tensile stress along the slip direction of edge dislocation causes the unstable stacking fault energy to increase, and the change of dislocation core structure further contributes to an increase in the activation energy for the diffusion of PDL. In contrast, an external compressive stress reduces the activation energy and pre-exponential factor of diffusion in small PDLs, as exemplified in V37 PDL.

Changes in Extracellular Microcystins (MCs) Accompanying Algae/Cyanobacteria Removal during Three Representative Algae/Cyanobacteria Inactivation Processes and an MC Diffusion Model in Still Water

This study addresses the lack of comparative research on algae/cyanobacteria elimination technologies in the existing literature. Our investigation focused on evaluating the performance of three commonly used algae/cyanobacteria removal processes: ultrasound (20.8 kHz), copper sulfate and biotic algicide (Bacillus subtilis). The evaluation considered both algae/cyanobacteria removal efficacy and the consequent changes in extracellular microcystins (MCs). To achieve this, we employed real eutrophic water as the test water. The ultrasound treatment demonstrated effective algae/cyanobacteria removal, with an average rate of algae/cyanobacteria decreasing (RAD) ranging from 0.50 to 0.99 µg chlorophyll a per liter per minute (µg chlorophyll a/L·min). On the other hand, the copper sulfate and biotic algicide treatments exhibited relatively lower rates of algae/cyanobacteria removal, with average RAD values of 0.21 to 0.38 µg chlorophyll a per liter per day (µg chlorophyll a/L·d) and 0.10 to 0.13 µg chlorophyll a per liter per day (µg chlorophyll a/L·d), respectively. Moreover, we observed significant increases in extracellular MCs in the ultrasound and copper sulfate treatments. The corresponding values of the increment of extracellular MCs accompanying removal per microgramme (µg) chlorophyll a (IEMARMC) were 0.34 to 2.43 µg MCs per µg chlorophyll a (µg MCs/µg chlorophyll a) and 18.13 to 185.08 µg MCs per µg chlorophyll a (µg MCs/µg chlorophyll a), respectively. However, in certain conditions where sufficient dosages (0.5 to 2 mg/L) and reaction time (≥8 days) are provided, the biotic algicide treatment could result in a decrease in MCs compared to an untreated control group (IEMARMC: −43.94 to −32.18 µg MCs per µg chlorophyll a). This suggests that the biotic algicide effectively degraded the MCs. In addition, we developed a one-dimensional MC diffusion model in still water based on Newton’s second law, which exhibits excellent simulation capabilities.

Inverse coefficient problem for a fractional-diffusion equation with a Bessel operator

The second initial-boundary value problem in a bounded domain for a fractional-diffusion equation with the Bessel operator and the Gerasimov-Caputo derivative is investigated. Theorems of existence and uniqueness of the solution of the inverse problem of determining the lowest coefficient in a one-dimensional fractional diffusion equation under the condition of integral observation are obtained. The Schauder principle was used to prove the existence of the solution.

A Consistent One-Dimensional Multigroup Diffusion Model for Molten Salt Reactor Neutronics Calculations

Molten Salt Reactors (MSRs) have recently gained resurged research and development interest in the advanced reactor community. Several computational tools are being developed to capture the strong neutronics/thermal-hydraulics coupling effect in this special reactor configuration. This paper presents a consistent one-dimensional (1D) multigroup neutron diffusion model for MSR analysis, with the primary aim for fast and accurate calculations for long transients, as well as sensitivity and uncertainty analysis of the reactor. A fictitious radial leakage cross section is introduced in the model to properly account for the radial leakage effects of the reactor. The leakage cross section and other consistent neutronics parameters are generated with the Monte Carlo code Serpent using high-fidelity three-dimensional (3D) models. The accuracy of the 1D consistent model is verified by the reference solution from the Monte Carlo model on the Molten Salt Reactor Experiment (MSRE) configuration. The 1D consistent model successfully reproduced the integrated flux from the 3D model and the reactor multiplication factor keff with the error in the range of 95 to 397 pcm (per cent mille), depending on discretized energy group structures. The developed model is also extended to estimate the reactivity loss due to fuel circulation in MSRE. The estimate of reactivity loss in dynamics analysis is in great agreement with the experimental data. This model functions as the first step in the development of a 1D fully neutronics/thermal-hydraulics coupled model for short- and long-term MSRE transient analysis.

Continuous-state branching processes with collisions: First passage times and duality

We introduce a class of one-dimensional positive Markov processes generalizing continuous-state branching processes (CBs), by taking into account a phenomenon of random collisions. Besides branching, characterized by a general mechanism Ψ, at a constant rate in time two particles are sampled uniformly in the population, collide and leave a mass of particles governed by a (sub)critical mechanism Σ. Such CB processes with collisions (CBCs) are shown to be the only Feller processes without negative jumps satisfying a Laplace duality relationship with one-dimensional diffusions on the half-line. This generalizes the duality observed for logistic CBs in Foucart (2019). Via time-change, CBCs are also related to an auxiliary class of Markov processes, called CB processes with spectrally positive migration (CBMs), recently introduced in Vidmar (2022). We find necessary and sufficient conditions for the boundaries 0 or ∞ to be attracting and for a limiting distribution to exist. The Laplace transform of the latter is provided. Under the assumption that the CBC process does not explode, the Laplace transforms of the first passage times below arbitrary levels are represented with the help of the solution of a second-order differential equation, whose coefficients express in terms of the Lévy–Khintchine functions Σ and Ψ. Sufficient conditions for non-explosion are given.

An inverse source time-fractional diffusion problem via an input-output mapping

In this paper, we investigate an inverse source problem involving a one-dimensional diffusion equation of a time-fractional RiemannLiouville derivative with 0 < α < 1. First, results on the existence and regularity of the weak solution of the direct problem are obtained. For the determination of the unknown time-dependent source term, we use a monotone and distinguishable input-output mapping defined by the additional over-determination integral data for the considered sub-diffusion problem. Finally, the uniqueness of the solution of the inverse problem is proved.

Numerical solution of one inverse problem for a diffusion model of a chemical-technological process

A chemical-technological process taking place in a chemical reactor with a chemical reaction of the second order is considered. A one-dimensional one-parameter diffusion model of the hydrodynamic flow in the reactor is proposed for the mathematical description of this process. Within the framework of the proposed model, the inverse problem is posed to determine the concentration of the selected reagent in the incoming flow, ensuring the implementation of a predetermined hydrodynamic regime at the reactor outlet. A discrete analogue of the inverse problem is constructed and the resulting difference problem is presented as a variational problem with local regularization. A special representation is proposed for the numerical solution of the variational problem. As a result, an explicit formula is obtained for determining the approximate concentration of the selected reagent in the incoming stream. The effectiveness of the proposed method is illustrated by numerical calculations for model problems

Complex Three-Dimensional Mathematical Model of the Ignition of a Coniferous Tree via a Cloud-to-Ground Lightning Discharge: Electrophysical, Thermophysical and Physico-Chemical Processes

Thunderstorms are the main natural source of forest fires. The ignition mechanism of trees begins with the impact of cloud-to-ground lightning discharge. A common drawback of all predicting systems is that they ignore the physical mechanism of forest fire as a result of thunderstorm activity. The purpose of this article is to develop a physically based mathematical model for the ignition of a coniferous tree via cloud-to-ground lightning discharge, taking into account thermophysical, electrophysical, and physicochemical processes. The novelty of the article is explained by the development of an improved mathematical model for the ignition of coniferous trees via cloud-to-ground lightning discharge, taking into account the processes of soot formation caused by the thermal decomposition phase of dry organic matter. Mathematically, the process of tree ignition is described by a system of non-stationary nonlinear differential equations of heat conduction and diffusion. In this research, a locally one-dimensional method is used to solve three-dimensional partial differential equations. The finite difference method is used to solve one-dimensional heat conduction and diffusion equations. Difference analogues of the equations are solved using the marching method. To resolve nonlinearity, a simple iteration method is used. Temperature distributions in a structurally inhomogeneous trunk of a coniferous tree, as well as distributions of volume fractions of phases and concentrations of gas mixture components, are obtained. The conditions for tree trunk ignition under conditions of thunderstorm activity are determined. As a result, a complex three-dimensional mathematical model is developed, which makes it possible to identify the conditions for the ignition of a coniferous tree trunk via cloud-to-ground lightning discharge.

On the Inverse Problem of Time-Dependent Coefficient in a Time Fractional Diffusion Problem by Newly Defined Monic Laquerre Wavelets

Abstract The primary aim of this research is to establish the time-dependent diffusion coefficient in a one-dimensional time fractional diffusion equation in Caputo sense by means of newly defined Monic Laquerre wavelets (MLW) and collocation points. We first give the definition of MLW by taking Monic Laquerre's polynomials into account. Later, time fractional diffusion problem is reduced into a system of ordinary fractional and algebraic equations by utilizing MLW. The residual power series method (RPSM) and the overdetermined data are applied to this system to determine the solution and the unknown time-dependent coefficient together in series form. In the end, illustrative examples are presented to show the stability and accuracy of the proposed wavelet method for the inverse problem of determining unknown time-dependent coefficient in fractional diffusion problems. The reliability of the proposed algorithm for the inverse problems is supported by high degree of accuracy in the given examples.

Thermal dehydration kinetics of SrB6O10·5H2O with different morphologies

The globular-like, sheet-like, coral-like SrB6O10·5H2O was prepared by hydrothermal method, and characterized by X-ray powder diffractometer(XRD), thermogravimetry(TG), differential thermal analysis(DTA), scanning electron microscopy(SEM). The activation energy of the thermal dehydration of SrB6O10·5H2O with different morphologies was calculated by Ozawa-Flynn-Wall (OFW) method. The most probable mechanism function of the dehydration reaction was determined by master-plots method, and the average pre-exponential factor of the thermal dehydration of the sample was calculated. The results show that the thermal dehydration of SrB6O10·5H2O is divided into two steps. The activation energy of thermal dehydration reaction is affected by the morphology and size. The activation energy of SrB6O10·5H2O in the first step of thermal dehydration is close, while there is a significant difference in the activation energy value of SrB6O10·5H2O in the second step of thermal dehydration. The order of activation energy, as well as the order of the average pre-exponential factor, is consistent with the size order of the constituent units of the sample microstructure. The first-step thermal dehydration mechanism functions of SrB6O10·5H2O (globular-like, sheet-like, coral-like) are all [-ln(1-α)]3, which belongs to random nucleation and growth, and the second-step thermal dehydration mechanism functions are α2(one-dimensional diffusion), α+(1-α)·ln(1-α) (two-dimensional diffusion), and [-ln(1-α)]3 (random nucleation and growth), the reasons for the difference of their two-step mechanistic functions are analyzed.

Quantum algorithm for smoothed particle hydrodynamics

We present a quantum computing algorithm for the smoothed particle hydrodynamics (SPH) method. We use a normalization procedure to encode the SPH operators and domain discretization in a quantum register. We then perform the SPH summation via an inner product of quantum registers. Using a one-dimensional function, we test the approach in a classical sense for the kernel sum and first and second derivatives of a one-dimensional function, using both the Gaussian and Wendland kernel functions, and compare various register sizes against analytical results. Error convergence is exponentially fast in the number of qubits. We extend the method to solve the one-dimensional advection and diffusion partial differential equations, which are commonly encountered in fluids simulations. This work provides a foundation for a more general SPH algorithm, eventually leading to highly efficient simulations of complex engineering problems on gate-based quantum computers.

Collocation-Based Approximation for a Time-Fractional Sub-Diffusion Model

We consider the numerical solution of a one-dimensional time-fractional diffusion problem, where the order of the Caputo time derivative belongs to (0, 1). Using the technique of the method of lines, we first develop from the original problem a system of fractional ordinary differential equations. Using an integral equation reformulation of this system, we study the regularity properties of the exact solution of the system of fractional differential equations and construct a piecewise polynomial collocation method to solve it numerically. We also investigate the convergence and the convergence order of the proposed method. To conclude, we present the results of some numerical experiments.