Discrete Random Walk Model Research Articles

Numerical simulation of flow and heat transfer in connection of gasifier to the radiant syngas cooler

AbstractThe connection of gasifier to the radiant syngas cooler has been regarded as a key technology for heat recovery system. Multiphase flow and heat transfer processes presented in this work considers particle deposition and radiation model to mixture of non‐gray gas with particles. An axisymmetric simulation of the multiphase flow in an industrial scale connection is performed. The standard k‐ε model, Renormalization group (RNG) k‐ε model and Realizable k‐ε model turbulence model are proposed. The particle motion is modeled by discrete random walk model. The discrete ordinates model (DOM), P‐1 and discrete transfer model (DTRM) are used to model the radiative heat transfer. The effect of particles on the radiative heat transfer was taken into account when the DOM and P‐1 model were used. The absorption coefficient of the gas mixture is calculated by means of a weighted‐sum‐of‐gray‐gas (WSGG) model. The results with the DOM and P‐1 model are very similar and close to practical condition. A large number of particles are deposited on the cone of gasifier which is the top of connection. Maximum temperature difference is approximate 7 K when the cooling tube heights change from 0.5 m to 1.5 m. The temperature inside has a linear relationship with operating temperature. Copyright © 2009 Curtin University of Technology and John Wiley & Sons, Ltd.

Particle transport in turbulent flow using both Lagrangian and Eulerian formulations

Lagrangian and Eulerian models for particle transport by a turbulent fluid phase are presented. In both methods, particle distribution results from the action of applied forces (buoyancy, inertial, added mass and drag forces) and turbulent effects are shown. The carrier phase flow – which is solved by finite element method using a k–ε turbulence model – is assumed not to depend on the particles' motion. In the Lagrangian formulation the dynamic equation for the particles is solved. A discrete random walk model is used to account for the turbulent effects. In the Eulerian formulation, the particle concentration is calculated from a convection–diffusion equation using the terminal particles' velocity and turbulent diffusivity. Both models are compared to experimental measurements and analytical results; a good agreement is observed.

The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation

In this paper, we consider a Riesz fractional advection-dispersion equation (RFADE), which is derived from the kinetics of chaotic dynamics. The RFADE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivatives by the Riesz fractional derivatives of order a e (0, 1) and fi e (1, 2], respectively. We derive the fundamental solution for the Riesz fractional advection-dispersion equation with an initial condition (RFADE-IC). We investigate a discrete random walk model based on an explicit finite-difference approximation for the RFADE-IC and prove that the random walk model belongs to the domain of attraction of the corresponding stable distribution. We also present explicit and implicit difference approximations for the Riesz fractional advection-dispersion equation with initial and boundary conditions (RFADE-IBC) in a finite domain. Stability and convergence of these numerical methods for the RFADE-IBC are discussed. Some numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

The Practical Use of Lasers in General Practice

The use of lasers in general dentistry is now an accepted--and to some extent, expected--treatment modality. Laser use can be either an adjunct to other procedures or the main form of treatment itself. For many procedures, lasers are now becoming the treatment of choice by both clinicians and patients, and in some cases, the standard of care. Though there are as many uses for lasers in general practice as the imagination can provide, this manuscript will address those indications for use that have received marketing clearance by the U.S. Food and Drug Administration (FDA).

Stochastic Binding of Ca 2+ Ions in the Dyadic Cleft; Continuous versus Random Walk Description of Diffusion

Ca 2+ signaling in the dyadic cleft in ventricular myocytes is fundamentally discrete and stochastic. We study the stochastic binding of single Ca 2+ ions to receptors in the cleft using two different models of diffusion: a stochastic and discrete Random Walk (RW) model, and a deterministic continuous model. We investigate whether the latter model, together with a stochastic receptor model, can reproduce binding events registered in fully stochastic RW simulations. By evaluating the continuous model goodness-of-fit for a large range of parameters, we present evidence that it can. Further, we show that the large fluctuations in binding rate observed at the level of single time-steps are integrated and smoothed at the larger timescale of binding events, which explains the continuous model goodness-of-fit. With these results we demonstrate that the stochasticity and discreteness of the Ca 2+ signaling in the dyadic cleft, determined by single binding events, can be described using a deterministic model of Ca 2+ diffusion together with a stochastic model of the binding events, for a specific range of physiological relevant parameters. Time-consuming RW simulations can thus be avoided. We also present a new analytical model of bimolecular binding probabilities, which we use in the RW simulations and the statistical analysis.

Fundamental solution and discrete random walk model for a time-space fractional diffusion equation of distributed order

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β 1∈(0,1) and β 2∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.

Numerical Simulation of Gas-Solid Flow in a Precalciner of Cement Industry

Precalciner, in which endothermic raw meal calcination and exothermic fuel combustion proceed simultaneously, is a key equipment in dry process of cement production. To increase the precalcination degree and reduce the energy consumption, more and more attentions have been paid on modeling gas-solid flow field in precalciners. However, most of them aimed at just qualitative studies lacking in necessary further quantitative analysis of precalciner performance parameters. In this paper, combining qualitative studies and quantitative analysis, the gas-solid flow field was carried out aiming at an actual precalciner under operational-based boundary conditions. In Euler coordinate system the gas phase is expressed with k-ε model, in Lagrange coordinate system the solid phase is expressed with Discrete Phase Model(DPM), and the random effects of turbulence on the particle dispersion is accounted for with Discrete Random Walk (DRW) model. The predicted gas velocity field agrees well with the measured result, and the calculated raw meal concentration distribution is consistent with the actul condition. The results predicted that there is a simple spraying-liked flow field in the precalciner, with not only a non-uniform particle dispersion condition but also a low solid residence-time and residence-time ratio between solid and gas.

Effect of Height and Position of Dams on Inclusion Removal in a Six Strand Tundish

The Reynolds-averaged Navier–Stokes equations have been numerically solved to obtain a steady, three-dimensional velocity field inside the six-strand billet caster tundish using the standard k–ε model of turbulence. These steady flow fields so obtained are then used to predict the removal of inclusions from molten steel by solving the inclusion transport equation with the help of a commercial CFD software Fluent 6.1.22. The effects of height and position of dams in the tundish and size of the inclusion particles on percentage inclusion removal were studied. To simulate the chaotic effect of the turbulent eddies on the particle paths; a discrete random walk model was applied during inclusion trajectory calculations. The computational model was first validated with the results reported in the literature a good match was found between the two. It was found that with the increase in the height of the dams, the inclusion removal tendency increased whereas the shifting of the dams towards the outlets decreases the inclusion removal. An increase in the inclusion removal was found with an increase in the inclusion particle size.

Stochastic binding of Ca2+ ions in the dyadic cleft; continuous vs random walk description of diffusion

AbstractCa2+ signaling in the dyadic cleft in ventricular myocytes is fundamentally discrete and stochastic. In this paper we study the stochastic binding of single Ca2+ ions to receptors in the cleft using two different models of diffusion; a stochastic and discrete Random walk (RW) model, and a deterministic continuous model. We investigate if the latter model, together with a stochastic receptor model, can reproduce binding events registered in fully stochastic RW simulations. By evaluating the continuous model goodness‐of‐fit, we present evidences that it can. The large fluctuations in binding rate observed at the time level of single time steps are integrated and smoothed at the larger time scale of binding events, explaining the continuous model goodness‐of‐fit. With this we demonstrate that the stochasticity and discreteness of the Ca2+ signaling in the dyadic cleft, determined by single binding events, can be described with a deterministic model of Ca2+ diffusion together with a stochastic model of the binding events. Time consuming RW simulations can thus be avoided. We also present a new analytical model of bi‐molecular binding probabilities that is used in the RW simulations, and in the statistical analysis. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical Simulation of Electrolyte Two-Phase Flow Induced by Anode Bubbles in an Aluminum Reduction Cell

In the production process of aluminium reduction cells, the anode bubble laden layer has several important influences on the performance of the aluminium reduction cells. Especially for a "drained cathode cell", without the agitating of movement of the melted metal, the bath flow field could be more important. In this paper, the electrolyte two-phase flow fields were studied by using numerically simulation method based on a two-phase turbulence model combining the k - ? model and the Discrete Random Walk model. The results show that: the motion of the bubbles mostly exists within a thin layer under the anode, which results in inducing local electrolyte to flow around the anode in various circulation flows; the flow field in the anode-cathode space can be divided into three regions with different characters; the results also show the Driving action of bubbles is closely related to the current density, inclination of anode and the anode-cathode distance. In general, the increasing in the current density increases the electrolyte velocity and the turbulent kinetic energy. The decrease in ACD significantly enhances the uniformity of the electrolyte flow field in the anode-cathode space. The increase in anode inclination angle increases the velocity of the electrolyte in the anode-cathode space, which would be beneficial to improving the diffusion and dissolution of the alumina and reducing the resistance between the anode and cathode.

Particle dispersion and deposition in ventilated rooms: Testing and evaluation of different Eulerian and Lagrangian models

Indoor particle dispersion in a three-dimensional ventilated room is simulated by a Lagrangian discrete random walk (DRW) model and two Eulerian models: drift flux model and mixture model. The simulated results are compared with the published measured data to check the performance of the three models for indoor particle dispersion simulation. The deposition velocity of the particles is also computed and compared with published data. The turbulent airflow is modeled with the renormalization group (RNG) k− ε and a zero equation turbulence model. Comparison of the calculated air velocities with measurement shows that both the two turbulence models can simulate the airflow well for the presented case. For the Lagrangian DRW model, a post-process program is used to state the particle trajectories and transfer the results to particle concentration distribution. For Eulerian models, the effect of particle deposition towards wall surfaces is incorporated with a semi-empirical particle deposition model. The comparison shows that both the Lagrangian DRW model and drift flux model yield satisfactory predictions, while the predicted results by the mixture model are not satisfied. The deposition velocity obtained by the three models match the experimental data well.

Target Zone Life Expectancy: Discrete Models and Secular Attacks

This paper uses a discrete simple random walk model to analyse the predicted behaviour of an exchange rate between formal or informal barriers. Various phenomena are examined including secular attacks, where the exchange rate appears to be attracted to and bounce repeatedly against a barrier, producing a signal that could be read as the start of a traditional speculative attack.

Near-field characterization of effective optical interfaces

The properties of many heterogeneous media depend on both the surface roughness and the local variations of the optical properties. An effective optical interface is usually invoked to describe the characteristics of such media. Using approaches specific to near-field optics, the two influences can be decoupled and a quantitative assessment of their contributions can be performed. It is also shown that a discrete random-walk model can be used to determine the magnitude of the dielectric constant fluctuations at subwavelength scales which, in turn, describe the morphology of optically inhomogeneous media.

Discrete and Continuous Random Walk Models for Space-Time Fractional Diffusion

A mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. A more general approach is however provided by the integral equation for the so-called continuous time random walk (CTRW), which can be understood as a random walk subordinated to a renewal process. We show how this integral equation reduces to our fractional diffusion equations by a properly scaled passage to the limit of compressed waiting times and jumps. The essential assumption is that the probabilities for waiting times and jumps behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. Illustrating examples are given, numerical results and plots of simulations are displayed.

Photon diffusion in biological tissues

The use of laser-based optical techniques for medical imaging is an attractive alternative to other methods that utilize ionizing radiation. Beside being non-carcinogenic, it is non-invasive, the equipment is transportable, and the methodology can be used to examine properties of soft tissue. However, unlike x-ray photons, optical photons generated in the near-infrared suffer significant amounts of scattering by heterogeneous bodies (e.g., organelles) found in biological tissue. Thus, theory is required to interpret experimental data which appear in the form o spatially or temporally varying light patterns on the skin surface. There is a wide range of parameters over which either diffusion theory or the theory of lattice random walks can be called on to translate optical data into medically significant information embodied in optical parameters of the tissue. We discuss several problems in diffusion theory arising in the analysis of optical measurements, for tissues modeled by a semi-infinite or slab geometry, having either isotropic or anisotropic optical parameters. The measured quantities are related to the intensity of light re-emitted on the tissue surface. A brief discussion is given related to the telegrapher’s equation, which has been suggested as a simple way of incorporating the effects of forward scattering. Mention is made of calculations related to layered media which frequently occur in tissues such as skull and esophagus. Finally, we briefly discuss discrete random walk models for photon migration. These have recently been used to provide parameters conveying information related to the region interrogated by photons constrained to reappear on skin surface.

Computer Simulation of Pollutant Transport and Deposition Near Peace Bridge

ABSTRACT Using computer simulations, transport, dispersion, and deposition of particulate pollutants near the Peace Bridge in the city of Buffalo, New York, are studied. An unstructured computational grid of Peace Bridge and its vicinity is generated, and the wind flow in the area is simulated. The Reynolds stress transport (RST) and the k-ϵ models of FLUENT code are used for simulating the mean airflow condition. The instantaneous turbulence fluctuating velocity is simulated by a discrete random walk (DRW) model. A Lagrangian particle-tracking model is used, and dispersion and deposition of particulate emissions from the motor vehicle exhaust on the bridge and in the Peace Bridge Plaza (customs area) are analyzed. The pollutant transport model accounts for the drag and Brownian forces acting on the particles, in addition to the gravitational sedimentation effects. For the case of a northwesterly wind of 7.7 m/s and particulate emissions in the size range of 0.01 to 50 μm, the corresponding deposition rates on various surfaces are studied. The importance of wind turbulence and gravity on particle deposition are evaluated. It is found that the Peace Bridge Plaza provides for a large fraction of the pollutants.

Discrete and Continuous Random Walk Models for Space-Time Fractional Diffusion

A mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. A more general approach is however provided by the integral equation for the so-called continuous time random walk (CTRW), which can be understood as a random walk subordinated to a renewal process. We show how this integral equation reduces to our fractional diffusion equations by a properly scaled passage to the limit of compressed waiting times and jumps. The essential assumption is that the probabilities for waiting times and jumps behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. Illustrating examples are given, numerical results and plots of simulations are displayed.

Numerical 3D simulation of the erosion due to solid particle impact in the main stop valve of a steam turbine

Solid particle erosion in a steam turbine main stop valve bypass valve has been investigated by means of computational fluid dynamics. Previews attempts to couple fluid mechanics and erosion modeling and improvements in the hydrodynamics models together with improvements in the erosion models are reviewed. The solid particle bearing steam flow through the valve was investigated using a 3D numerical model and the finite volume code Fluent V6.0.12, looking for a reduction of the erosion process. The flow simulation was carried out for the valve original and modified designs with changes of the angle of particle impact on the valve surface. Numerical predictions have been carried out using the Renormalization Group (RNG) k– ε turbulence model. To account for the influence of turbulent fluid fluctuations on particle motion, the stochastic tracking Discrete Random Walk model is used, which includes the effect of instantaneous turbulent velocity fluctuations on the particle trajectories. The removal of wall material due to erosion is calculated using the Finnie model developed for ductile materials. The numerical predictions showed a 51% reduction of the erosion rate for the valve modified design due to changes of the particles trajectories and impingement angle (angle of particle impact). The results obtained show that numerical simulation can be used in a predictive manner to solve a real practical design problem.

Discrete random walk model to interpret the dispersal parameters of organisms

Abstract Most models for describing the dispersal of organisms have been developed by using diffusion equations with a diffusion coefficient. These equations, however, do not yield information that is readily interpretable in biological terms, because the biological meaning of the diffusion coefficient is not always clear. Discrete random walk models, in which organisms move into adjacent positions by a specific probability, seem to be superior in their biological tractability, although they have not been as widely used as diffusion equations because of their mathematical intractability. We reconstructed discrete random walk models of one dimension based on two assumptions: (1) moving organisms settle by a constant probability and (2) settled individuals are captured by traps by a constant probability. We also constructed a model that is applicable for a directional movement caused by environmental factors such as wind. We applied the model to a one-dimensional dispersal experiment on the ragweed beetle, Ophraella communa LeSage, an insect of the size of about 4 mm in adults. Both larvae and adults of this species preferably eat ragweed, Ambrosia artemisiifolia L. We planted ragweed plants at a place in a linear field of 100 m length and 20 m width. In mid-August, adult beetles dispersed actively along the linear field to find new food plants after they almost defoliated ragweed plants. Assuming a non-directional random walk, we performed the linear regression, which indicated that the movement of the adult O. communa in this season is approximately described by a discrete random movement in which an individual travels next 10 m with a probability of 0.906 during its life. The dispersion parameter estimated by the Poisson regression was much larger than 1, indicating that there is a considerable amount of fluctuation in the probability of capture or in other parameters. It is shown that a similar model is also applicable to a situation in which individuals are directly removed by traps from the moving population before settlement.

Discrete random walk models for space–time fractional diffusion

Abstract A physical–mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By space–time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz–Feller derivative of order α∈(0,2] and skewness θ (|θ|⩽min{α,2−α}), and the first-order time derivative with a Caputo derivative of order β∈(0,1]. Such evolution equation implies for the flux a fractional Fick’s law which accounts for spatial and temporal non-locality. The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation.