Discrete Random Walk Model Research Articles

On the accuracy of Eulerian-Lagrangian CFD simulations for spray evaporation in turbulent flow

CFD simulation of droplet evaporation in turbulent flows is challenging as the accuracy and reliability of the results strongly depend on the available sub-models and their modeling parameters. This study presents a systematic sensitivity analysis focused on the impact of the most widely used discrete random walk (DRW) model, the constant time scale coefficient (CL), the turbulence model, and the drag coefficient model. CFD simulations with the Eulerian-Lagrangian approach are employed. The analysis is based on grid-sensitivity analysis and validation with measurements of spray evaporation in a heated turbulent airflow. The results show that using the DRW model leads to a good agreement between the CFD results and the experimental data of droplet size and droplet mean velocity, attributed to the turbulent fluctuations inducing droplet dispersion. The best performance is observed for the standard k-ε turbulence model with CL = 0.30 and 0.45. This is mainly attributed to the reasonable interaction time between droplets and turbulent eddies at these CL values. The three drag coefficient models (i.e., Spherical, Ischii-Zuber, and Grace) lead to similar results due to the low droplet Reynolds number.

Numerical study on the gas-solid diffusion characteristics of airflow and dust particles in metal mine blasting excavation under three hybrid ventilation modes

Dust pollution has long been a challenge in the blasting excavation of underground metal mines, and uncontrolled diffusion of dust particles seriously endangers the health and safety of workers as well as the environment in underground space. Most existing research on dust diffusion focuses on fully mechanized coal mining face or borehole drilling, where the dust source emission characteristics differ significantly from metal mine blasting excavation. Furthermore, it is still unclear how the ventilation modes of Far-Pressing-Far-Absorption (FPFA), Near-Pressing-Far-Absorption (NPFA), and Far-Pressing-Near-Absorption (FPNA) affect the mechanism of dust diffusion in the breathing zone. In this work, gas-solid flow characteristics in a typical metal mine blasting tunnel are numerically studied based on Euler-Lagrange method. The interphase forces between airflow and dust particles are comprehensively modeled, and the particle diffusion effect caused by fluid turbulence is described by the discrete random walk model. Five typical regions for airflow disorder phenomena in the working tunnel are predicted, including the jet development region, the return airflow core region, the airflow reversal region, the airflow recirculation region, and the secondary flow region. The spatial distribution of dust exhibits nonuniform characteristics under turbulent airflow transport. The time results of the monitored dust concentration reveal that the junction of the working tunnel and the return airway is a key area of focus for the prevention and control of dust pollution. Among the three ventilation modes, the Far-Pressing-Near-Absorption (FPNA) ventilation mode has the best dust removal efficiency. The supply vent further from the blasting face allows the jet to fully develop, while the exhaust vent closer to the blasting face allows dust particles to be drawn in and discharged earlier. Under the condition of fixed total power consumption for the supply and exhaust fans, a larger exhaust airflow velocity helps remove dust from the breathing zone.

Analysis on the gas-solid variation characteristics and dust spatial distribution in a return micro-cyclone with a tangential inlet

The cyclone separator is widely used for the dedusting process in industry to extend its application scope, a novel return micro-cyclone with a tangential inlet was constructed considering its small and convenient feature. The cyclone was constructed by 3D printing technology, and the experimental system was established. A 3D fluid dynamics (CFD) model was developed with RNG k–ε model and discrete random walk (DRW) model. Discrete Phase Model (DPM) was applied for dust particles of different sizes in the cyclone to investigate the swirl and separation flow. The transient flow field, the tail swinging phenomenon, the outer and inner swirling airflows can be detected vividly. The numerical method predicts the quantity percentage of dust well with the deviations of 0.92 %, 1.15 %, 0.88 %, 0.69 % and 4 % of the experimental data for PM10–20, PM20–30, PM30–40, PM40–50 and PM50–60 when the initial air velocity is 7.3 m/s. There are few dust particles in the exhaust pipe, which are mainly fine particles of PM10 with few particles of PM10–20 and PM20–30 when the initial air velocity is low. The phenomenon of the dust ring at the top region is significant and varies with the air velocity. Dust concentration in different sections can provide further references to optimize the cyclone structure.

Gas-Solid Flow Characteristics of Airflow and Dust Particles in Blasting Excavation of Underground Metal Mine Tunnels.

Effective dust removal has long been a challenge in the blasting mining of underground metal mine tunnels, and uncontrolled dust diffusion seriously endangers workers' respiratory systems and the underground space safety environment. However, the vast majority of existing numerical studies on dust diffusion are focused on coal mine fully mechanized mining, which is different from metal mine blasting excavation in terms of stope structure and dust properties. Furthermore, the mechanism by which the forced and exhaust ventilation modes affect the diffusion characteristics of inhalable particles is unclear. In this work, gas-solid flow characteristics for dust diffusion in a typical metal mine blasting tunnel were numerically investigated based on the Euler-Lagrange method, where the blasting face instantly released 6.37 × 107 particles with 100 different sizes, ranging from 0.8 to 200 μm. The interphase forces between airflow and dust particles are comprehensively modeled, and the particle diffusion effect caused by fluid turbulence is described by a discrete random walk model. Detailed information for airflow turbulence and particle migration was revealed, and dust removal efficiencies for inhalable particulate matter (PM10) by forced, exhaust, and hybrid ventilation were analyzed. Numerical results predict a complex airflow pattern in the working roadway, including the jet-flow region, return airflow core region, airflow disorder region, and secondary flow region. Dust diffusion temporal characteristics can be divided into three stages, namely, the initial stage of dust generation, the efficient ventilation and dust removal stage, and the later stage of dust diffusion. Dust diffusion spatial characteristics indicate that under the Coanda wall attachment effect, the dust concentration exhibits nonuniform distribution in both vertical and horizontal directions of the return air roadway. The dust removal efficiency of hybrid ventilation on inhalable particles above respiratory height is better than that of forced ventilation, especially in the return air roadway. The additional exhaust air duct based on forced ventilation can discharge more inhalable particles from the tunnel.

A novel transparent cabin used in the classroom during the coronavirus pandemic: a CFD analysis

A coronavirus family is a diverse group of many viruses. Coronavirus disease 19 (COVID-19) spreads when an infected person breathes out droplets and very small particles that contain the virus. These droplets and particles can be breathed in by other people or land on their eyes, noses, or mouths. In this paper, the airflow distribution and the movement of coronavirus particles during normal breathing and sneezing in classrooms have been studied using a CFD model developed in ANSYS® 2022R2. The objective is to find ways to control the spread of the virus that enable us to practice academic activity and deal normally with the pandemic and the spread of the disease. Experiments were done with more than one turbulence model to know which was closest to the experiments as well as to determine the best number of meshes in the classroom. The effect of turbulent dispersion on particles is resolved using a discrete random walk model for the discrete phase and the RANS model for the continuous phase in a coupled Eulerian–Lagrangian method. Furthermore, that is done in two scenarios: the first is to find the best ventilation configuration by investigating the following parameters: the effect of air change per hour, the height of the air inlets and outlets, and the infected student's position. The second is to control the spread of the coronavirus in the classroom in the event of sneezing from an infected student by placing cabins and an air filter with optimal design installed at the top around each student. It was found that optimal ventilation is achieved when fresh air enters from the side walls of the classroom at a distance of 1 m from the floor and the air exits from the ceiling in the form of two rows, and the rate change of air per hour (ACH) is 4, which leads to energy savings. In addition, a novel transparent cabin is designed for the student to sit in while in the classroom, consisting of a high-efficiency particulate air filter (HEPA) that collects any contamination and recirculates it from the top of the cabin back into the classroom with different fan speeds. Through this study, this cabin with a filter was successfully able to prevent any sneeze particles inside from reaching the rest of the students in the classroom.

Numerical Simulation of Turbulent Structure and Particle Deposition in a Three-Dimensional Heat Transfer Pipe with Corrugation

When colloidal particles are deposited in a heat transfer channel, they increase the flow resistance in the channel, resulting in a substantial decrease in heat transfer efficiency. It is critical to have a comprehensive understanding of particle properties in heat transfer channels for practical engineering applications. This study employed the Reynolds stress model (RSM) and the discrete particle model (DPM) to simulate particle deposition in a 3D corrugated rough-walled channel. The turbulent diffusion of particles was modeled with the discrete random walk model (DRW). A user-defined function (UDF) was created for particle–wall contact, and an improved particle bounce deposition model was implemented. The research focused on investigating secondary flow near the corrugated wall, Q-value standards, turbulent kinetic energy distribution, and particle deposition through validation of velocity in the tube and particle deposition modeling. The study analyzed the impact of airflow velocity, particle size, corrugation height, and corrugation period on particle deposition efficiency. The findings suggest that the use of corrugated walls can significantly improve the efficiency of deposition for particles less than 20 μm in size. Specifically, particles with a diameter of 3 μm showed five times higher efficacy of deposition with a corrugation height of 24 mm compared to a smooth surface.

Numerical simulation of altering the raw meal inlet position in a novel swirl precalciner

Conducted a numerical simulation to model a novel swirl precalciner, investigating how altering in the position of the raw meal inlet affects the internal gas flow, temperature field, and component concentration field within the precalciner. Applied the Realizable k-? two-equation turbulent model to the continuous phase. For the particle phase (pulverized coal), employed the Discrete Particle Model and the discrete random walk model. Simulated the combustion of pulverized coal and the decomposition of calcium carbonate by using the Species Transport model combined with the Finite-Rate/Eddy-Dissipation model. Modeled the generation of NOx using a NOx model. The results show that, in comparison to the condition with four raw meal inlets, the six raw meal inlets condition has a better coupling of pulverized coal combustion and raw meal decomposition. The decomposition rate of raw meal has seen a slight improvement, and there is a significant improvement in the occurrence of localized high temperatures within the precalciner, resulting in a reduction of the outlet NOx concentration from 1251 ppm to 225 ppm.

A Computational Approach for the Characterization of Airborne Pathogen Transmission in Turbulent Molecular Communication Channels

Airborne pathogen transmission mechanisms play a key role in the spread of infectious diseases such as COVID-19. In this work, we propose a computational fluid dynamics (CFD) approach to model and statistically characterize airborne pathogen transmission via pathogen-laden particles in turbulent channels from a molecular communication viewpoint. To this end, turbulent flows induced by coughing and the turbulent dispersion of droplets and aerosols are modeled by using the Reynolds-averaged Navier-Stokes equations coupled with the realizable k-model and the discrete random walk model, respectively. Via simulations realized by a CFD simulator, statistical data for the number of received particles are obtained. These data are post-processed to obtain the statistical characterization of the turbulent effect in the reception and to derive the probability of infection. Our results reveal that the turbulence has an irregular effect on the probability of infection, which shows itself by the multi-modal distribution as a weighted sum of normal and Weibull distributions. Furthermore, it is shown that the turbulent MC channel is characterized via multi-modal, i.e., sum of weighted normal distributions, or stable distributions, depending on the air velocity.

Simulation of Turbulent Flow Structure and Particle Deposition in a Three-Dimensional Heat Transfer Duct with Convex Dimples

In engineering applications, dust deposition on the heat transfer channel greatly reduces the efficiency of heat transfer. Therefore, it is very significant to study the characteristics of particle deposition for thermal energy engineering applications. In this study, the Reynolds stress model (RSM) and the discrete phrase model (DPM) were used to simulate particle deposition in a 3D convex-dimpled rough channel. A discrete random walk model (DRW) was used for the turbulent diffusion of particles, and user-defined functions were developed for collisions between particles and walls. An improved deposition model of rebound between particles was developed. The flow structure, secondary flow, temperature distribution, Q criterion, and particle deposition distribution in the convex-dimpled rough channel were analyzed after a study of the grid independence and a numerical validation. The results showed that these mechanisms affected the flow structure in the flow field. For tiny particles (dp ≤ 10 μm), the presence of convex dimples promoted their deposition. The rates of particle deposition in the presence of convex dimples were 535, 768, 269, and 2 times higher than in smooth channels (particle sizes of 1, 3, 5, and 10 μm, respectively). However, for large particles (dp > 10 μm), although the presence of convex dimples had a certain effect on the location distribution of particle deposition, it had little effect on the deposition rates of large particles, which were 0.99, 0.98, 0.97 and 0.96 times those in the smooth channel, respectively.

Random walk model of mobility in glass formers.

A mechanism responsible for the generic features of the mean squared displacement and the decay of the orientational autocorrelator of a molecule in a glass forming liquid is poorly understood, where such a mechanism would be critical for creating the theory of glass transition. A discrete random walk model is proposed where, instead of a straight line, a walk is along a tortuous path consisting of blocks of switchback ramps. Subdiffusive regime, short-term dynamic heterogeneity, and existence of the α- and β-relaxation processes emerge naturally from the model. The model suggests that slowing of the rate of relaxation may be due to an increase in the number of the switchback ramps per block rather than due to growth of an energy barrier as typically assumed.

Effect of walls with large scale roughness in deposition efficiency for 90-degree square bend configurations

This paper presents a numerical study of particle deposition in turbulent flow across bends with large scale roughness. The k-ε RNG model with Enhanced Wall Functions (EWF) for wall treatment and a Lagrangian point particle tracking method with a discrete random walk (DRW) model for turbulence dispersion were chosen in order to study the effect of cavities and ribs on the particle deposition efficiency in 90-degree square bends. The study is a complementary approach to previous ones performed for duct and channel flows. Results show that while cavity-type (c-type) and rib-type (r-type) large scale roughness elements in the external wall of the bend have a low effect on particle deposition, there is significant enhancement for cases in which large scale elements are located also at the inner side of the bend inducing high increments of secondary slow and turbulent kinetic energy. This, in addition to the increment of deposition area, generates a global shift of the particle deposition curve, especially for low and intermediate Stokes numbers and cases with high curvature ratio. Our study shows that bends of square cross section, similar to those used in Heating, Ventilation, and Air Conditioning (HVAC) applications but modified with large scale roughness elements in their walls, can be a potential alternative for filtering applications to be considered and studied more in depth in the future.

Numerical Investigation of Covid-19 Infection Spread Expelled from Cough in an Isolation Ward Under Different Air Distribution Strategies

COVID-19 is a severe and rapidly spreading respiratory disease that can be transmitted through airborne particles, emitted from cough. This study investigated the influence of underfloor air distribution (UFAD) and overhead air distribution on the diffusion of the coughed particles emitted from two infected patients in an isolation ward. Additionally, the study examined the performance of mounting retractable covers around the exhausts on minimizing the dispersion of the particles. A coupled Eulerian-Lagrangian approach is adopted by using a discrete random walk model. The effect of Brownian force, drag force with Cunningham slip correction factor, turbulence dispersion, Rosin-Rammler, and the breakup is considered in the respiratory airborne coughed particles simulation. The model has a good agreement with the experimental data. The results show that overhead air distribution (case 1) disperses the particles faster to the occupied zone due to the strong mixing between downward inlet airflow and indoor air accompanied by infectious particles. The usage of retractable covers (case 2) significantly minimized the diffusion of the particles inside the ward and their residence time. The particles quantity drops by 53, 98.62, and 99.94 % at 2.0, 10.0, and 120.0 s, respectively, compared to case 1. Case 2 shows the best efficient protection and inhaled air quality for the HCW in all the walking areas inside the room. The upward inlet air in underfloor air distribution (case 3) keeps particles floating for a while at a higher level near the ceiling exhausts, enhancing removal efficiency. Also, it has a slower lateral dispersion of the particles than in case 1. Underfloor air distribution minimized the number of particles by 30.0 % at 120.0 s, compared to overhead air distribution. Inlet air location significantly affects the diffusion of infectious particles in the indoor environment.

Numerical study on particle-carrying ability of liquid nitrogen jet

ABSTRACT Liquid nitrogen (LN2) jet is expected to have the higher rock-breaking efficiency than water jet as its ultra-low temperature characteristic. It is considered that adding a few solid particles into LN2 jet can noticeably enhance its rock-breaking capability. The abrasive LN2 jet has a great application prospect in petroleum as well as geothermal exploitation. This study investigates the particle-carrying ability of LN2 jet by a 3D numerical model. The geometric model adopts a spindle shape that is more similar to the realistic cavity created by abrasive jet. The standard model and discrete phase model are used to calculate the LN2 flow and the particle trajectory, respectively. The Discrete Random Walk model is applied to describe the particle space movement in flow field. The results show that the particle in abrasive LN2 jet obtains the highest nozzle exit velocity in simulation conditions, 11.1% higher than in water jet and 5.1% higher than in SC-CO2 jet. The potential core of abrasive LN2 jet is longest in these three types of abrasive jets. That means LN2 jet has a well particle-carrying ability at high-speed state. However, the LN2 turbulent flow at out-of-nozzle region causes the violent fluctuation of particle speed. Parametric analysis indicates that the nozzle pressure drop and nozzle diameter in LN2 jet have the vital influences on particle velocity, while the ambient pressure and particle diameter affect particle velocity slightly. The simulation results reveal the particle-carrying ability of LN2 jet in different conditions. It would shed light on cutting capability of abrasive LN2 jet and promote its application in petroleum and geothermal exploitation.

Numerical analysis of the effect of roof types and porch on particle dispersion and deposition around a low-rise building

Outdoor air quality demonstrates one of the major challenges for preserving human health and the environment and also involves the indoor environment of buildings. Accordingly, besides urban morphology, building form should be considered as a passive strategy to reduce regional particle deposition and dispersion. In this framework, this study numerically pursues the influence of building shape on improving outdoor air quality around buildings. For this pursuit, eight cases of low-rise buildings with different roof types and a porch at different heights are compared with a simple building form with a flat roof to examine their impact on airflow and particle dispersion and deposition around low-rise buildings. The RNGk−ε turbulence model, in line with the discrete particle model (DPM) is implemented to model particulate flow and investigate particle dispersion and deposition. The discrete random walk (DRW) model is used to simulate the turbulent particle concentration. The results confirm that the combination of shed roof and porch can reduce the amount of particle deposition on the downstream wall of the building by about 20% less than the building with a flat roof. Moreover, using a porch at different heights integrated with a flat roof accumulates more particles on the downstream wall. Evidently, the vortex size influences particle concentration around buildings depending on the building shape. These findings will be helpful for designers regarding the impact of porch and roof types to improve outdoor air quality around buildings. Additionally, these promotions can be employed for decreasing particle deposition on buildings’ walls.

Assessment of stochastic models for predicting particle transport and deposition in turbulent pipe flows

Particle dispersion and deposition in turbulent pipe flows were studied numerically. The mean flow solution was calculated via a Reynolds Stress Model. Three different stochastic models were used to evaluate the effect of instantaneous fluctuations on entrained particles including the standard Discrete Random Walk (DRW) model available in Ansys-Fluent and two Continuous Random Walk (CRW) schemes that solve a normalized Langevin equation to better account for the effects of inhomogeneous turbulence on particle entrainment. Special attention is paid to the inconsistent directional vector resulting from solving the Langevin equation in non-rectangular geometries in location-specific coordinate systems. Concentration, axial distributions of deposition velocity, and deposition velocity versus particle relaxation time were compared with existing experiments and direct numerical simulation (DNS) using two different geometries. The validity of DNS data for near wall corrections were discussed. A third geometry representative of the hot section of a gas turbine was employed to study the capability of DRW and CRW on modeling spatially-developing flows. Comparisons were drawn between different stochastic models with an all-stick condition and experimental results. It was shown that the classical DRW model tends to overpredict the effect of entrance and does not yield good agreement in the deposition statistics. The CRW model shows a promising capability of modeling particle dispersion and deposition. However, the accuracy of CRW relies heavily upon the fidelity of the underlying mean flow solution and root-mean-square turbulence statistics.

CFD modelling of a powder spraying nozzle used for dry coating

The aim of these studies was to examine the suitability of the Euler–Lagrange approach for modelling the powder spraying process performed with a pneumatic nozzle used for dry coating in spout-fluid bed apparatus. Simulation results were validated using the digital PIV (Particle Image Velocimetry) method and measurement equipment constructed by the authors. The validation process led to the determination of the optimal Ansys Fluent solver settings, which ensure a satisfactory accuracy of the simulation results (the average relative error of all simulation points at the level of 10.5%). The experimental results are most accurately reflected by simulation results obtained with the use of the standard k − ε model that has the discrete random walk (DRW) model activated. The detailed analysis of the operation of the spout-fluid bed apparatus allowed for the examination of the influence of hydrodynamic parameters (the mass flow rate of the gas spraying the powder and plasticizer, as well as spouting gas) on the velocity and concentration of the dispersed phase in the device. The scientific investigation covered the analysis of the velocity profiles and the volume fraction of the coating powder in different zones of the column.

Extinction of Bistable Populations is Affected by the Shape of their Initial Spatial Distribution.

The question of whether biological populations survive or are eventually driven to extinction has long been examined using mathematical models. In this work, we study population survival or extinction using a stochastic, discrete lattice-based random walk model where individuals undergo movement, birth and death events. The discrete model is defined on a two-dimensional hexagonal lattice with periodic boundary conditions. A key feature of the discrete model is that crowding effects are introduced by specifying two different crowding functions that govern how local agent density influences movement events and birth/death events. The continuum limit description of the discrete model is a nonlinear reaction-diffusion equation, and we focus on crowding functions that lead to linear diffusion and a bistable source term that is often associated with the strong Allee effect. Using both the discrete and continuum modelling tools, we explore the complicated relationship between the long-term survival or extinction of the population and the initial spatial arrangement of the population. In particular, we study different spatial arrangements of initial distributions: (i) a well-mixed initial distribution where the initial density is independent of position in the domain; (ii) a vertical strip initial distribution where the initial density is independent of vertical position in the domain; and, (iii) several forms of two-dimensional initial distributions where the initial population is distributed in regions with different shapes. Our results indicate that the shape of the initial spatial distribution of the population affects extinction of bistable populations. All software required to solve the discrete and continuum models used in this work are available on GitHub .

CFD modeling of virtual mass force and pressure gradient force on deposition rate of asphaltene aggregates in oil wells

Asphaltene deposition in oil wells greatly causes production and associated monetary losses. In this work, the virtual mass force and pressure gradient force are considered in the momentum equation of asphaltene aggregates, since the rate of asphaltene particle density (1100 kg/m3) to crude oil density (886 kg/m3) is close to 1. Accordingly, the flow of oil-asphaltene particles was simulated within the Eulerian–Lagrangian framework in a production tubing. Subsequently, the shear stress turbulence model and discrete random walk model are applied to predict the deposition rate of asphaltene aggregates on production tubing. The simulation results show that the deposition rate considering these two forces agrees well with the experimental data. Moreover, the effect of the saffman lift force, virtual mass force, pressure gradient force, oil viscosity, particle density, wall roughness as well as crude oil density are investigated on the deposition rate of asphaltene aggregates on production tubing. The findings of this work can favor a better understand of the law of asphaltene deposition in oil wells.

A Validation Study for RANS Based Modelling of Swirling Pulverized Fuel Flames

A swirling pulverized coal flame is computationally investigated. A Eulerian–Lagrangian formulation is used to describe the two-phase flow. Turbulence is modelled within a RANS (Reynolds averaged numerical simulation) framework. Four turbulence viscosity- (TV) based models, namely the standard k-ε model, realizable k-ε model, renormalization group theory k-ε model, and the shear stress transport k-ω model are used. In addition, a Reynolds stress transport model (RSM) is employed. The models are assessed by comparing the predicted velocity fields with the measurements of other authors. In terms of overall average values, the agreement of the predictions to the measurements is observed to be within the range 20–40%. A better performance of the RSM compared to the TV models is observed, with a nearly twice as better overall agreement to the experiments, particularly for the swirl velocity. In the second part of the investigation, the resolution of the discrete particle phase in modelling the turbulent particle dispersion (TPD) and particle size distribution (SD) is investigated. Using the discrete random walk model for the TPD, it is shown that even five random walks are sufficient for an accuracy that is quite high, with a less than 1% mean deviation from the solution obtained by thirty random walks. The approximation of the measured SD is determined by a continuous Rosin–Rammler distribution function, and inaccuracies that can occur in its subsequent discretization are demonstrated and discussed. An investigation on the resolution of the SD by discrete particle size classes (SC) indicates that 12 SC are required for an accuracy with a less than 1% mean deviation from the solution with 18 SC. Although these numbers may not necessarily be claimed to be sufficiently universal, they may serve as guidance, at least for SD with similar characteristics.

Comment on the paper by R. Maniar and A. Bhattacharyay, [Random walk model for coordinate-dependent diffusion in a force field, Physica A 584 (2021) 126348

Paper (Maniar and Bhattacharyay, 2021) introduces a discrete random-walk (RW) model with a site dependent waiting time and an external force i.e. a site dependent drift. The RW is compared with a Smoluchowski equation in continuous space and time with space dependent diffusivity and space dependent damping. Using numerical simulations in Maniar and Bhattacharyay (2021) it is shown that the equilibrium distribution satisfies locally the Stokes–Einstein relation i.e. there is a constant temperature in the whole system. Here we show that the result can be recovered analytically with a continuous limit in the RW master equation. Moreover, we evidence that the RW model should be carefully defined in order to satisfy locally Stokes–Einstein relation, indeed a different definition of the discrete RW process describes a system that satisfies the Smoluchowski equation with constant damping, where the temperature is not uniform.