scholarly journals Стохастическая ячеистая модель очистки моторного масла от механических примесей объёмным фильтрованием

2020 ◽  
Author(s):  
G.P. Kicha ◽  
L.A. Semeniuk ◽  
M.I. Tarasov
Keyword(s):  
Pore Space ◽  
Planck Equation ◽  
Complex Interaction ◽  
Dispersed Phase ◽  
Generalized Coordinates ◽  
Stochastic Filtering ◽  
Physical Laws ◽  
Random Particle ◽  
Filtering Problem ◽  
Dimensionless Coordinate

Приведено описание ячеистой модели фильтрования, которая создавалась на сочетании вероятностно-статистических методов с точным описанием на основе физических законов поведения дисперсной фазы при сложном взаимодействии её с дисперсионной средой. Выделены основные силы, действующие на частицу в потоке при фильтровании. Показано, что наибольшее воздействие на отсев оказывают силы Лондона-Ван-дер-Ваальса и электрокинетические, обусловленные полярными молекулами продуктов старения масла и моюще-диспергирующими присадками, которыми оно легируется. Выяснено, что в наименьшей степени на отсев влияют силы тяжести и Архимеда. Объединение детерминированных и случайных воздействий на дисперсную фазу осуществлено на базе уравнения Колмогорова Фоккера Планка. Показаны методы его формирования так, чтобы рассматриваемому случайному переносу частиц в поровом пространстве, идентифицированному совокупностью обобщенных координат и скоростей, соответствовало уравнение для многомерной плотности вероятности, отождествляемой с концентрацией дисперсной фазы. Приведены расчётные формулы для компонентов скорости частиц, по которым можно рассчитать граничную траекторию, определить безразмерную координату и фракционный коэффициент отсева. Определены краевые условия стохастической задачи фильтрования. Записаны интегралы для расчета фракционного отсева через паток вероятности. Проанализированы возможности детерминированной и стохастической ячеистых моделей фильтрования, показана их адекватность.A description is given of a cellular filtering model that was created using a combination of probabilistic and statistical methods with an accurate description based on the physical laws of the behavior of the dispersed phase during its complex interaction with the dispersion medium. The main forces acting on a particle in a stream during filtration are identified. It has been shown that the London-Van der Waals forces and electro kinetic forces caused by polar molecules of oil aging products and detergent-dispersant additives with which it is doped have the greatest impact on screening. It has been found that gravity and Archimedes force have the leas effect on screening. The combination of deterministic and random effects on the dispersed phase was carried out based on the Kolmogorov Fokker Planck equation. The methods of its formation are shown so that the equation for the multidimensional probability density, identified with the concentration of the dispersed phase, corresponds to the random particle transport in the pore space, identified by the set of generalized coordinates and velocities. Calculation formulas are given for the particle velocity components by which one can calculate the boundary trajectory and determine the dimensionless coordinate and fractional dropout coefficient. The boundary conditions of the stochastic filtering problem are determined. The integrals for calculating the fractional dropout through the molasses of probability are recorded. The possibilities of deterministic and stochastic cellular filtration models are analyzed, and their adequacy is shown.

scholarly journals A model for discrete fracture-clay rock interaction incorporating electrostatic effects on transport

2020 ◽  
Author(s):  
Carl I. Steefel ◽  
Christophe Tournassat
Keyword(s):  
Pore Space ◽  
Planck Equation ◽  
Differential Algebraic Equations ◽  
Diffuse Layer ◽  
Clay Rock ◽  
Algebraic Equations ◽  
Advective Transport ◽  
Electrostatic Effects ◽  
Rock Interaction ◽  
Discrete Fracture

AbstractA model based on the code CrunchClay is presented for a fracture-clay matrix system that takes electrostatic effects on transport into account. The electrostatic effects on transport include those associated with the development of a diffusion potential as captured by the Nernst-Planck equation, and the formation of a diffuse layer bordering negatively charged clay particles within which partial anion exclusion occurs. The model is based on a dual continuum formulation that accounts for diffuse layer and bulk water pore space, providing a more flexible framework than is found in the classical mean electrostatic potential models. The diffuse layer model is obtained by volume averaging ion concentrations in the Poisson-Boltzmann equation, but also includes the treatment of longitudinal transport within this continuum. The calculation of transport within the bulk and diffuse layer porosity is based on a new formulation for the Nernst-Planck equation that considers averaging of diffusion coefficients and accumulation factors at grid cell interfaces. Equations for function residuals and the associated Jacobian matrix are presented such that the system of nonlinear differential-algebraic equations can be solved with Newton’s method. As an example, we consider a 2D system with a single discrete fracture within which flow and advective transport occurs that is coupled to diffusion in the clay-rich matrix. The simulation results demonstrate the lack of retardation for anions (e.g., 36Cl−) of the contaminant plume within the fracture flow system because they are largely excluded from the charged clay rock, while the migration of cations (e.g., 90Sr++) is more strongly attenuated. The diffusive loss of divalent cations in particular from the fracture is accentuated by their accumulation in the diffuse layer within the clay-rich matrix.

scholarly journals The stochastic filtering problem: a brief historical account

2014 ◽  
Vol 51 (A) ◽  
pp. 13-22
Author(s):  
Dan Crisan
Keyword(s):  
Risk Estimation ◽  
Historical Account ◽  
Stochastic Filtering ◽  
Research Areas ◽  
Computer Scientists ◽  
Classical Mathematics ◽  
The Subject ◽  
Mathematical Techniques ◽  
New Research ◽  
Filtering Problem

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

scholarly journals Heterogeneous population dynamical model: a filtering problem

2005 ◽  
Vol 42 (02) ◽  
pp. 346-361 ◽  
Author(s):  
A. Gerardi ◽  
P. Tardelli
Keyword(s):  
Finite Number ◽  
Conditional Distribution ◽  
Heterogeneous Population ◽  
Time Approximation ◽  
Stochastic Filtering ◽  
Discrete Time Approximation ◽  
Different Types ◽  
Number Of Classes ◽  
Filtering Problem ◽  
Over Time

We consider a heterogeneous population of identical particles divided into a finite number of classes according to their level of health. The partition can change over time, and a suitable exchangeability assumption is made to allow for having identical items of different types. The partition is not observed; we only observe the cardinality of a particular class. We discuss the problem of finding the conditional distribution of particle lifetimes, given such observations, using stochastic filtering techniques. In particular, a discrete-time approximation is given.

scholarly journals The stochastic filtering problem: a brief historical account

2014 ◽  
Vol 51 (A) ◽  
pp. 13-22 ◽  
Author(s):  
Dan Crisan
Keyword(s):  
Risk Estimation ◽  
Historical Account ◽  
Stochastic Filtering ◽  
Research Areas ◽  
Computer Scientists ◽  
Classical Mathematics ◽  
The Subject ◽  
Mathematical Techniques ◽  
New Research ◽  
Filtering Problem

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

Third-Order Continuous-Discrete Filtering for a Nonlinear Dynamical System

2014 ◽  
Vol 9 (3) ◽  
Author(s):  
Hiren G. Patel ◽  
Shambhu N. Sharma
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Dynamical System ◽  
Planck Equation ◽  
Extended Kalman Filtering ◽  
Third Order ◽  
Nonlinear Dynamical ◽  
Stochastic Filtering ◽  
Systems And Control ◽  
Backward Equation ◽  
And Control

Approximate higher-order filters are more attractive and popular in control and signal processing literature in contrast to the exact filter, since the analytical and numerical solutions of the nonlinear exact filter are not possible. The filtering model of this paper involves stochastic differential equation (SDE) formalism in combination with a nonlinear discrete observation equation. The theory of this paper is developed by adopting a unified systematic approach involving celebrated results of stochastic calculus. The Kolmogorov–Fokker–Planck equation in combination with the Kolmogorov backward equation plays the pivotal role to construct the theory of this paper “between the observations.” The conditional characteristic function is exploited to develop “filtering” at the observation instant. Subsequently, the efficacy of the filtering method of this paper is examined on the basis of its comparison with extended Kalman filtering and true state trajectories. This paper will be of interest to applied mathematicians and research communities in systems and control looking for stochastic filtering methods in theoretical studies as well as their application to real physical systems.

scholarly journals Heterogeneous population dynamical model: a filtering problem

2005 ◽  
Vol 42 (2) ◽  
pp. 346-361 ◽  
Author(s):  
A. Gerardi ◽  
P. Tardelli
Keyword(s):  
Finite Number ◽  
Conditional Distribution ◽  
Heterogeneous Population ◽  
Time Approximation ◽  
Stochastic Filtering ◽  
Discrete Time Approximation ◽  
Different Types ◽  
Number Of Classes ◽  
Filtering Problem ◽  
Over Time

We consider a heterogeneous population of identical particles divided into a finite number of classes according to their level of health. The partition can change over time, and a suitable exchangeability assumption is made to allow for having identical items of different types. The partition is not observed; we only observe the cardinality of a particular class. We discuss the problem of finding the conditional distribution of particle lifetimes, given such observations, using stochastic filtering techniques. In particular, a discrete-time approximation is given.

Correlation of Pore Structure and Permeability by SEM-Image Analysis

1990 ◽  
Vol 48 (1) ◽  
pp. 428-429
Author(s):  
C. A. Callender ◽  
Wm. C. Dawson ◽  
J. J. Funk
Keyword(s):  
Image Analysis ◽  
Pore Structure ◽  
Imaging System ◽  
Pore Space ◽  
Simulation Models ◽  
Image Analyzer ◽  
Imaging Data ◽  
Sem Images ◽  
Thin Sections ◽  
Rock Types

The geometric structure of pore space in some carbonate rocks can be correlated with petrophysical measurements by quantitatively analyzing binaries generated from SEM images. Reservoirs with similar porosities can have markedly different permeabilities. Image analysis identifies which characteristics of a rock are responsible for the permeability differences. Imaging data can explain unusual fluid flow patterns which, in turn, can improve production simulation models.Analytical SchemeOur sample suite consists of 30 Middle East carbonates having porosities ranging from 21 to 28% and permeabilities from 92 to 2153 md. Engineering tests reveal the lack of a consistent (predictable) relationship between porosity and permeability (Fig. 1). Finely polished thin sections were studied petrographically to determine rock texture. The studied thin sections represent four petrographically distinct carbonate rock types ranging from compacted, poorly-sorted, dolomitized, intraclastic grainstones to well-sorted, foraminiferal,ooid, peloidal grainstones. The samples were analyzed for pore structure by a Tracor Northern 5500 IPP 5B/80 image analyzer and a 80386 microprocessor-based imaging system. Between 30 and 50 SEM-generated backscattered electron images (frames) were collected per thin section. Binaries were created from the gray level that represents the pore space. Calculated values were averaged and the data analyzed to determine which geological pore structure characteristics actually affect permeability.

AN APPLICATION OF FOKKER–PLANCK EQUATION TO JOSEPHSON JUNCTION

1989 ◽  
Vol 9 (1) ◽  
pp. 109-120
Author(s):  
G. Liao ◽  
A.F. Lawrence ◽  
A.T. Abawi
Keyword(s):  
Josephson Junction ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Fokker Planck
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