Convection-diffusion Model Research Articles

Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbation

<p style='text-indent:20px;'>A multiscale finite element method is proposed for addressing a singularly perturbed convection-diffusion model. In reference to the a priori estimate of the boundary layer location, we provide a graded recursion for the mesh adaption. On this mesh, the multiscale basis functions are able to capture the microscopic boundary layers effectively. Then with a global reduction, the multiscale functional space may efficiently reflect the macroscopic essence. The error estimate for the adaptive multiscale strategy is presented, and a high-order convergence is proved. Numerical results validate the robustness of this novel multiscale simulation for singular perturbation with small parameters, as a consequence, high accuracy and uniform superconvergence are obtained through computational reductions.</p>

Convection-Diffusion Model for Radon Migration in a Three-Dimensional Confined Space in Turbulent Conditions

Convection and diffusion are the main factors affecting radon migration. In this paper, a coupled diffusion-convection radon migration model is presented taking into account turbulence effects. In particular, the migration of radon is simulated in the framework of the k-ε turbulence model. The model equations are solved in a complex 3D domain by the finite element method (FEM). Special attention is paid to the case study about radon migration in an abandoned air defense shelter (AADS). The results show that air convection in a confined space has a great influence on the radon migration and the radon concentration is inversely proportional to the wind speed.

Numerical simulation study on diffusion of temperature drainage in wide and shallow water in a refinery integration project

The numerical simulation of tidal current field in wide and shallow waters (Naochao River) of Caofeidian industrial zone is carried out by using a two-dimensional hydrodynamic model, taking the discharge problem of temperature and drainage in a refinery integration project as an example. Taking the tidal current field as the hydrodynamic condition of the temperature drainage transport simulation, the concentration distribution of the temperature drainage in the receiving water area was predicted by using the two-dimensional convection-diffusion model of the temperature drainage, and the change of the temperature rise line and its influence were analyzed. The results showed that the water area affected by temperature rise in winter and summer were 0.28km2 and 0.35km2, respectively, when the project was discharged separately. The existing temperature displacement of China Resources Power Plant in the superposition area is: 19.1km2 in winter and 1.64km2 in summer under the influence of temperature rise. By analyzing the influence degree of wide and shallow water (Naochao River) drainage outlet on temperature rise, the conclusion that the layout of drainage outlet should be further optimized is given. conclusion that the layout of drainage outlet should be further optimized is given.

Design, mathematical modelling, and numerical simulation of a novel tubular photoelectrochemical reactor and experimental validation by residence time distribution and mass transfer coefficients

The present work introduces a new design of input/output distributors for tubular photoelectrochemical reactors (T-PECR) with concentric electrodes and external illumination. Expanded meshes of TiO2-coated titanium (outer electrode), and nickel-plated stainless steel (inner electrode) were used as photoanode and cathode, respectively. 3D-simulation studies of the hydrodynamic (laminar and turbulent regimes) in steady state and the mass conservation equation (convection-diffusion model) in the transitory regime were used to design the geometry of the input/output distributors, the T-PECR length, and the inter-electrode gap. Two restrictions were considered for an adequate T-PECR design: to minimize the inlet/outlet effects and thus achieve fully developed velocity profiles. Two experimental approaches were used to validate the theoretical results: (a) residence time distribution (RTD) obtained by an innovative methodology of “step-signal input” and (b) mass transfer coefficients obtained by the limiting current (IL) technique. Both regimes showed a concentric core-annular flow and stagnant zone in the reactor shell (between the photoanode and the reactor wall). In the laminar regime, the core flow (inside the cathode) is fully independent of the annular flow (between the cathode and anode). In the turbulent regime, the development of a crisscross flow through the diamond-shaped openings of the electrodes causes the mixing of both core and annular flows. The local mass transfer coefficients (km) on the photoanode surface and the global mass transfer coefficients (kaef) in the diamond-shaped openings of the expanded meshes, estimated with Comsol Multiphysics, are in good agreement with those determined by limiting current experiments and the Plug Dispersion Exchange Model (PDEM). Thus, the use of inlet and outlet distributors improves the homogeneous flow pattern within the T-PECR.

Influence of support layer resistance on oxygen fluxes through asymmetric membranes based on perovskite-type oxides SrTi1-xFexO3-δ

Asymmetric membranes of mixed ionic-electronic conducting perovskite-type oxides SrTi1-xFexO3-δ (STF, x = 0.3, 0.5 and 0.7) were prepared by inverse sequential tape-casting. Both porous support (~600 μm) and functional membrane layer (~20 μm) for a given membrane assembly were made from the same composition to ensure thermochemical compatibility between the layers. Oxygen fluxes were assessed in the range 650 -1020 °C, using either (non-pressurized) ambient air or pure oxygen as feed gas at the support side of the asymmetric membrane and argon as sweep gas. Notably, similar oxygen fluxes (~1.2 × 10−6 mol cm−2 s−1) are measured through the membranes of different compositions above 950 °C when using ambient air as feed gas. This observation is interpreted to reflect the major role of the support layer resistance in rate-limiting the oxygen fluxes through the STF asymmetric membranes, which conclusion is supported by comparison of the oxygen fluxes with those measured previously through asymmetric membranes of Ba0.5Sr0.5Co0.8Fe0.2O3-δ (BSCF) and La0.6Sr0.4Co0.2Fe0.8O3-δ (LSCF). A simple diffusion-convection model is used to account for the observed gas phase polarization in the porous support layers limiting the oxygen fluxes.

Interpolation of Boundary Condition at Time-Interval of Unknown Lenghth for the Problem of Convective Diffusion in a Three-Layered Water Filter

On the basis of mathematical model of convectivediffusion in a three-layered filter it is formulated a contactinitial-boundary value problem for description of mass transferof pollution accompanying the sorption processes. It is proposedthe algorithm for establishing the estimation of values of soughtfunction (concentration of pollution) at the lower boundary of thefilter on the basis of the interpolation of experimental data. It istaken into account that the right end of the interpolation segmentis unknown. It is determined the exact solutions of contact-initialboundaryvalue problems of mass transfer with provision forboth diffusive and convective mechanisms of transfer as well assorption processes, which is based on integral transformationsover space variables in the contacting regions. Is it designedsoftware and established regularities of convective diffusionprocess in the three-layered filter.

Mathematical Modelling of Vortex Dust Separator

The problem of the motion of a swirling flow in an axisymmetric channel is investigated numerically. The flow field calculation results have been obtained as the solutions of the Navier-Stokes equations. Various flow regimes with the formation of axial recirculation zones are presented. The convection-diffusion model is used for the determination of the flow particle concentration and the formation of typical sedimentation zones.

Natural and anthropogenic gamma-ray emitters and iron-bearing compounds distribution in peatlands of the southern Espinhaço mountain chain, Brazil

To help understand the dynamics of peatlands forming on the Southern Serra do Espinhaço, Brazil, the occurring natural (226Ra, 232Th, 40K) and anthropogenic (137Cs) activity of gamma-ray emitters located at three sites, namely the localities of São João da Chapada (CH), Pinheiro (PIN) and Pau de Fruta (PDF), was studied. The iron compounds were characterized by 57Fe Mössbauer spectroscopy. Differences on natural radionuclide levels and distributions were found among the peatland sites, indicating differences on mineral composition, hydric regimes and living. Significant correlation was found only for 232Th–226Ra activity values (Pearson correlation coefficient of − 0.5), which is different from other South American soils, where significant positive correlations among all radionuclides were determined. In PDF and CH, the dose rate results are lower than the worldwide mean value, but in PIN, they are higher because of the relatively high 226Ra concentration. The only anthropogenic radionuclide is 137Cs. The convection–diffusion model fits the CH and PDF activity profiles, although the PIN profile is likely affected by other soil mechanisms as well. Mössbauer spectra reveal Fe3+ in two different local environments and the occurrence of hematite in the upper samples of the CH site. The iron concentration decreases with depth. Contrary to soils of other regions, no significant correlation was found between iron species and the natural radioactivity. The studied peatlands exhibit clear differences between their natural and anthropogenic radionuclide distributions to those of mineral soils.

Biomathematical model for simulating abnormal orifice patterns in colonic crypts

Colonic polyps, which are abnormal growths in the colon, are a major concern in colon cancer diagnosis and prevention. Medical studies evidence that there is a correlation between histopathology and the shapes of the orifices in colonic crypts. We propose a biomathematical model for simulating the appearance of anomalous shapes for the orifices of colonic crypts, associated to an abnormal cell proliferation. It couples a mechanical model that is a mixed elastic/viscoelastic quasi-static model describing the deformation of the crypt orifice, with a convection-diffusion model that simulates the crypt cell dynamics in space and time. The coupling resides in the variation of pressure generated by abnormal proliferative cells that induce a mechanical force and originate the change in shape of the crypt orifice. Furthermore the model is formulated in a two-dimensional setting, for emulating the top view of the colonic mucosa, observed in vivo in colonoscopy images. The primary focus of this study is on the modeling of this complex biological phenomenon, by defining an appropriate reduced biomathematical model. Additionally, a numerical procedure to determine its solution is also addressed. The overall numerical simulations indicate that an excess of cell proliferation, in different crypt locations, creates some of the anomalous patterns of the colonic crypt orifices, observed in vivo in medical images.

New Approach to Modeling and Simulation of Chemical and Mass Transfer Processes in Column Apparatuses

Objective:The classical mass transfer theory is not applicable for modeling the mass transfer of chemical, absorption, adsorption and catalytic processes in column apparatuses, where the velocity distributions and interphase boundaries are unknown. The modeling of these processes is related with the creation of new type of convection-diffusion models (for qualitative analysis) and average-concentration models (for quantitative analysis), where the surface reactions are replaced by equivalent volume reaction, while the velocity and concentration distributions are replaced by average velocity and concentrations. The effect of the radial non-uniformity of the velocity in the average-concentration models is introduced by model parameters, which must be obtained experimentally.Methods:The new convection-diffusion and average-concentration models are obtained in the cases of different processes in column apparatuses: simple and complicated chemical reactions, physical and chemical absorption, physical and chemical adsorption, heterogeneous catalytic processes (physical and chemical adsorption mechanism). These models are presented in the monograph Chr. Boyadjiev, M. Doichinova, B. Boyadjiev, P. Popova-Krumova, “Modeling of Column Apparatus Processes” (Second edition), Springer- Verlag, Berlin Heidelberg, 2018.Results:Two hydrodynamic situations are considered, when the radial velocity component is equal to zero, in the cases of an axial modification of the radial non-uniformity of the axial velocity component and when the radial velocity component is not equal to zero.Conclusion:The use of experimental data, for the average concentrations at the column end, for a concrete process and column, permits to obtain the model parameters, related with the radial non-uniformity of the velocity. These parameter values permit to use the averageconcentration models for modeling of different processes.

PARAMETERS OF MATHEMATICAL MODELS FOR VERTICAL MIGRATION OF HEAVY METALS IN SOILS IN THE AREA OF LEAD AND ANTIMONY PLANT

The data on distribution of heavy metals in the soil profile (Yuan et al., 2017) of the province Guangxi in southern China in the zone influenced by lead and antimony plant were analyzed. The soil is sod-calcareous and rich in humus. The averaged over the years and depth (“apparent”) parameters of diffusion and convective-diffusion models of migration in the area of combined soil pollution (only soil aerogenic pollution and in combination with the sewage waters) are estimated. The largest values of the diffusion parameters were obtained for Zn and Cd (n∙10–7 сm2/sec), the smallest values were obtained for Pb and Sb (n∙10–8 сm2/sec). When soil was flooded by waste waters, a reliable convective component of migration for Zn and Sb was noted. At the same time the peak of Zn concentration moved to the depth of 40–60 cm. Under these conditions there was no clear inverse correlation between the migration mobility and the amount of sorption of elements by the soil.

Critical Conditions of Adhesion and Separation of Functionalized Nanoparticles on Polymer Grafted Substrates

Emerging technologies of production and processing of functionalized nanoparticles (NP) require advanced methods of NP characterization and separation. While various methods are available for NP separation by size, there are no efficient methods for NP separation by surface chemistry. Using extensive dissipative particle dynamics simulations, this work investigates the mechanisms of NP adhesion and flow in polymer brush (PB)-grafted pore channels searching for the conditions for size-independent separation of NPs that are similar to the critical conditions of liquid chromatography of polymers. We consider interactions of NPs functionalized by hydrophilic and hydrophobic ligands with PBs, in which conformation and adhesion properties are controlled by the solvent quality varied with the composition of thermodynamically good and poor solvent components. The NP-PB adhesion is characterized by the free energy landscape calculated by the ghost tweezers simulation method that mimics the experimental technique of optical tweezers. The NP Henry constant and the respective partition coefficient are calculated depending on the NP size and ligand composition at varying solvent quality. Our findings demonstrate that, with the decrease of solvent quality, the NP elution undergoes a transition from the size-exclusion mode with larger NPs having shorter retention time to the adsorption mode with the reverse order of elution. This transition, which occurs in a narrow range of solvent composition, signifies the so-called “critical” point of adsorption that strongly depends on the NP functionalization. The dynamics of NP axial dispersion in the isocratic and gradient elution modes is characterized employing a convective-diffusion model. We show that the NPs can be effectively separated by surface chemistry at the critical points of adsorption, using the gradient mode of interaction NP chromatography with controlled variation of the solvent composition.

Nonlinear dynamics of ion concentration polarization in capacitive deionization

Capacitive deionization (CDI) is an emerging desalination technique by utilizing the electrosorption of porous electrodes. Thus far, only linear concentration profiles in bulk electrolyte between the electrodes have been observed, during ion charging (ion depletion) or ion discharging (ion enrichment). This so-called linear ion concentration polarization (ICP) can be predicted by the simple convective-diffusion model. In this study, we demonstrate that capacitive charging can induce nonlinear concentration profiles in the electrolyte. To visualize nonlinear ICP dynamics, a miniaturized CDI platform is developed to observe in situ ion concentration profiles. Determined by the ion flux mismatching of electrosorption and bulk ion transport, we discovered four ICP regimes in CDI – steady, linear ICP, delayed-nonlinear ICP, and nonlinear ICP. We directly observed the signatures of nonlinear ICP: i) flat depletion zone with extremely low ion concentration, and ii) corresponding exponential current drop. Based on this observation, two scaling models are formulated to predict i) when the nonlinear ICP occurs, and ii) the propagation speed of the flat depletion zone. To the best of our knowledge, this is the first microscopic characterization of ICP in CDI systems, and it firmly establishes the existence of nonlinear ion transport dynamics in CDI's bulk electrolyte.

The nature of two transition times on chronopotentiograms of heterogeneous ion exchange membranes: 2D modelling

Chronopotentiometry is largely used for electrochemical characterization of ion-exchange membranes. Nevertheless, the theory is poorly developed in the range of overlimiting currents. This paper proposes a two dimensional convective-diffusion model of ion transport in electromembrane system in galvanostatic mode. We use direct numerical simulation taking into account electroconvection to develop a comprehensive analysis of mechanisms of reaching transition states, which depend on the parameters of electrically heterogeneous surface of ion exchange membrane. We find that the occurrence of two transition times is typical for the simulated electrodialysis systems with heterogeneous membranes. The value of the first transition time, τ1, is conditioned by the fact that the normal and tangential diffusion delivery of ions from the solution bulk to the conductive membrane areas reaches its limiting value. When τ1 is attained, electroconvective vortices appear at the conductive-nonconductive boundaries. They mix the near-surface solution in the vicinity of these boundaries and increase the delivery of the “fresh” solution to the conductive regions. The second transition time, τ2, is reached when electroconvection-diffusion ion delivery to the overall membrane surface attains its limiting value. At t = τ2, electroconvective vortices whose size is comparable with the diffusion layer thickness arise. Both τ1 and τ2 depend on the size and fraction of conductive regions. In the case of conventional commercial ion-exchange membranes τ1 < τSand and τ2 ≥ τSand.

Permeate flux increase by colloidal fouling control in a vibration enhanced reverse osmosis membrane desalination system

Experiments and Computational Fluid Dynamics (CFD) simulations were performed to investigate the colloidal fouling control of a vibration enhanced reverse osmosis (VERO) membrane system for up to 12 h of operation time. A porous cake mass transfer model for the reverse osmosis (RO) colloidal fouling analysis was derived by combining the cake filtration model, cake enhanced osmotic pressure (CEOP) effect, the critical flux theory, and the solute convection-diffusion model. The process of colloidal particle deposition and fouling formation on the membrane surface were visualized using this model. The permeate flux variation and cake layer distribution along the membrane were depicted. Systematic studies including different initial permeate fluxes, Reynolds numbers, particle concentrations, and vibration frequencies were carried out to investigate the system performance under different operation conditions. The results suggest that colloidal fouling induced permeate flux decline could be improved by the high-frequency vibration in the VERO module. Both simulations and experiments demonstrated that, with a fixed vibration amplitude, membrane module with higher vibration frequencies will have less NaCl accumulation and higher permeate flux under the influence of colloidal fouling.

Numerical modeling of the horizontal flow and concentration distribution of nitrogen within a stored-paddy bulk in a large warehouse: Poster

The insect population in grain stores can be kept under control by maintaining a high concentration of N2 gas throughout the grain bed. The development of controlled atmosphere storage technology for insect control requires an accurate prediction of the distribution of introduced gases in bulk grain. In this paper, based on the convective-diffusion model, the horizontal flow of N2, which was introduced into the paddy bulk in a large warehouse by means of the horizontal ventilation system, are modeled as fluid flow in a porous medium. The experimental data for N2 transfer and flow through ducts and bulk paddy were used to validate the model. The equations were solved using the finite difference method, and the predictions from the proposed model were in good agreement with the experimental results. The concentration distribution and flow uniformity of nitrogen in stored paddy were also analyzed during the nitrogen-filling procedure for CA storage. It was shown that it is feasible and practical to introduce nitrogen into stored bulk grain in a large warehouse by means of the horizontal ventilation system.

Mathematical Modeling of Solutes Migration under the Conditions of Groundwater Filtration by the Model with the k-Caputo Fractional Derivative

Within the framework of a new mathematical model of convective diffusion with the k-Caputo derivative, we simulate the dynamics of anomalous soluble substances migration under the conditions of two-dimensional steady-state plane-vertical filtration with a free surface. As a corresponding filtration scheme, we consider the scheme for the spread of pollution from rivers, canals, or storages of industrial wastes. On the base of a locally one-dimensional finite-difference scheme, we develop a numerical method for obtaining solutions of boundary value problem for fractional differential equation with k-Caputo derivative with respect to the time variable that describes the convective diffusion of salt solution. The results of numerical experiments on modeling the dynamics of the considered process are presented. The results that show an existence of a time lag in the process of diffusion field formation are presented.

Simulations of a Single-Phase Flow in a Compound Parabolic Concentrator Reactor

This paper deals with the analysis and interpretation of flow visualization and residence time distribution (RTD) in a compound parabolic concentrator (CPC) reactor using computational fluid dynamics (CFD). CFD was calculated under turbulent flow conditions solving the Reynolds averaged Navier–Stokes (RANS) equation expressed in terms of turbulent viscosity and the standard k−ε turbulent model in 3D. A 3D diffusion-convection model was implemented in the CPC reactor to determine the RTD. The fluid flow visualization and RTD were validated with experimental results. The CFD showed that the magnitude of the velocity field remains almost uniform in most of the bulk reactor, although near and inside the 90° connectors and the union segments, the velocity presented low- and high-speed zones. Comparisons of theoretical and experimental RTD curves showed that the k−ε model is appropriate to simulate the nonideal flow inside the CPC reactor under turbulent flow conditions.

A convection–diffusion model for gang territoriality

We present an agent-based model to simulate gang territorial development motivated by graffiti marking on a two-dimensional discrete lattice. For simplicity, we assume that there are two rival gangs present, and they compete for territory. In this model, agents represent gang members and move according to a biased random walk, adding graffiti with some probability as they move and preferentially avoiding the other gang’s graffiti. All agent interactions are indirect, with the interactions occurring through the graffiti field. We show numerically that as parameters vary, a phase transition occurs between a well-mixed state and a well-segregated state. The numerical results show that system mass, decay rate and graffiti rate influence the critical parameter. From the discrete model, we derive a continuum system of convection–diffusion equations for territorial development. Using the continuum equations, we perform a linear stability analysis to determine the stability of the equilibrium solutions and we find that we can determine the precise location of the phase transition in parameter space as a function of the system mass and the graffiti creation and decay rates.

Analysis of inlet and gap effect in hydrodynamics and mass transport performance of a multipurpose electrochemical reactor: CFD simulation and experimental validation

This work deals with the reaction environment analysis (hydrodynamic and mass transport characterization) inside the flow channel of a parallel plate electrolyzer. Different electrode gap and inlet configurations were tested using computational and experimental techniques. Computational methods are based on numerical solution of mathematical models that describe fluid dynamics inside reactor meanwhile, experimental characterization was carried out by means of residence time distribution (RTD) and limiting current tests.In order to evidence the phenomenological influence of hydrodynamic pattern generated by different design parameters, a 3-D diffusion-convection model was implemented taking into account the local velocity vectors obtained by solution of Navier-Stokes equations to obtain theoretical RTD and by diffusion-convection equation in order to describe hydrodynamics and overall mass transport performance. Average Re numbers considered in this work were between 102 ≤ Re ≤ 702, which are characteristic of a laminar flow regime.Velocity fields obtained by CFD analysis evidenced the existence of prominent jet flow zones, particularly when 45° inlet angle is used. These effects diminished with 60° inlet angle configuration. RTD analysis also showed important tailing, indicating presence of internal flow by-passing channeling effects. Mass flux distribution (associated with global mass transport coefficient and limiting current) obtained by CFD analysis evidenced the existence of non-homogeneous distribution provoked by the flow pattern inside of our reactor. From phenomenological point of view, the analysis carried out in this work, demonstrates that mass flux distribution is strongly influenced by geometric design parameters of a multipurpose parallel plate reactor. The higher mass transport coefficient was obtained with 7 mm of interelectrode gap and 60° inlet angle configuration.