Some aspects of extrapolation based on interpolation polynomials

The problem of extrapolation on the basis of interpolation polynomials is considered in the paper. A simple computational procedure is proposed to find the predicted value for a polynomial of any degree under conditions of a uniform grid. An algorithm for determining the best polynomial for extrapolation is proposed. To construction of integral transformation for operator of equation of convective diffusion under mixed boundary conditions.

Unidirectional consolidation of clay soils and flocculated suspensions

A model of the unidirectional consolidation of a clay soil or flocculated suspension between a series of parallel drains is developed. A convective-diffusion equation for the pore pressure is derived, and an equal-strain approximation leads to an expression for the average effective stress profile between the drains. The solution depends on a Peclet number quantifying the ratio of the bulk soil flow rate to the rate of consolidation. By adjusting the number, height and spacing of the drains, very high rates of dewatering can be achieved. A potential application of the method to the rapid dewatering of mine tailings is described.

3D MODELING OF BIOLOGICAL WASTEWATER TREATMENT IN AERATION TANK

Purpose. The main purpose of the article is to develop a 3D CFD model for modeling the process of biological wastewater treatment in an aeration tank. Methodology. For mathematical modeling of the process of biological wastewater treatment in the reactor, taking into account the flow hydrodynamics, geometric shape of the aeration tank, convective-diffusion transfer of the substrate and activated sludge, a 3D CFD model was built. The model is based on the three-dimensional equation of motion of an ideal liquid and the equation of mass conservation for the substrate, activated sludge. The field of sewage flow rate in the aeration tank is calculated based on the velocity potential equation. The process of biological transformation of the substrate is calculated on the basis of the Monod model. The splitting scheme was used for numerical integration of the equations of convective-diffusion transfer of activated sludge and substrate. The splitting is carried out in such a way to take into account the transfer of substrate (activated sludge) in only one direction at each step of splitting. The calculation of the unknown value of the substrate (activated sludge) concentration is carried out according to an explicit scheme. The Richardson method is used to numerically integrate the three-dimensional equation for the velocity potential, and the unknown value of the velocity potential is calculated by an explicit formula. Euler's method is used for numerical integration of equations describing the process of substrate transformation and change in activated sludge concentration (Monod model). Findings. The software implementation of the constructed 3D CFD model is carried out. A description of the structure of the developed software package is provided. The results of a computer experiment to study the process of wastewater treatment in an aeration tank with additional elements are presented. Originality. A new multifactor 3D CFD model has been developed, which allows quick assessing the efficiency of biological treatment in an aeration tank. Practical value. The constructed 3D CFD model can be used to analyze the efficiency of the aeration tank under different operating conditions at the stage of sketch design of wastewater treatment systems.

Particle Tracking Random Simulation of Pollutant Migration and Diffusion

A two-step Lagrangian particle tracing method was used to simulate the particle movement in convective diffusion. Firstly, the flow field distribution was given by finite difference method. Secondly convective diffusion was simulated by particle tracing method. Jet current through centre slit in the static flow and turbulent flow as well as the side discharge of pollutant were simulated as case study. In this paper, finite difference method is used to solve fluid velocity field distribution, and particle tracking method is used to solve convection-diffusion equation. The result shows that the data produced by particle tracing method is reasonable and the method is stable. This method can be applied to the calculation of the convective diffusion of pollutant and spread to other problems.

On the selection of fractional-differential model of convective diffusion with mass exchange

The paper considers two fractional-differential models of convective diffusion with mass exchange and proposes a decision-making algorithm for determining the optimal model at the time of concentration field observation. As for soils of fractal structure, direct experimental determination of model parameters’ values and type of mass exchange process is in many cases impossible, calibration and determination of the most adequate models is performed mainly solving inverse problems by, in particular, meta-heuristic algorithms that are computationally complex. In order to reduce the computational complexity, we study the qualitative differences between diffusion processes described by fractional-differential models with non-local mass exchange on the base of the Caputo derivative and local non-linear mass exchange based on the non-equilibrium sorption equation that corresponds to the description by the Caputo-Fabrizio derivative. We determine under which conditions both models within a given accuracy describe the same set of measurements at a certain moment of time. When the solutions are close at a certain initial stage of process development, the model with the Caputo derivative describes its faster approach to a steady state. Based on the obtained estimates of differences in solutions, a decision-making algorithm is proposed to determine the most accurate model and the values of its parameters concurrently with the acquisition of measurements. This algorithm’s usage reduces the time spent on solving inverse calibration problems.

Increasing the Accuracy of the Difference Scheme Using the Richardson Extrapolation Based on the Movable Node Method

A one-dimensional convective-diffusion problem is considered. To improve the quality of difference schemes, the method of moving nodes is used in combination with Richardson interpolation. Approximate analytical solutions and improved schemes are obtained. Numerical experiments carried out.