A Probabilistic‐Statistical Model of the Particle Classification Process in Small Hydrocyclone Classifiers

Krokhina, A. V.; Lvov, V.; Pavlikhin, G. P.

doi:10.1002/ceat.201600602

Cited by 7 publications

(19 citation statements)

References 6 publications

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“…In a previous study , it was proposed to describe the behavior of suspension particles with a size of d 1) ≤ 40 µm based on the particle density distribution function f ( x,t )d x ;

x = R^{n + 1} / R_{0}^{n + 1}

is the generalized dimensionless coordinate determined by the ratio of the current radius, R , to the radius of the cylindrical part of the apparatus, R 0 , and the power exponent, n , which is included in the generalized law of tangential velocity distributions in hydrocyclones (Eq. ):

true w'_{0} R_{0}^{normaln} = w' R^{normaln} = D = normalconst

…”

Section: Use Of the Properties Of Statistical Self‐similarity For Thementioning

confidence: 99%

“…In the presence of random components of the processes, the influence of which cannot be neglected , Eq. can be derived for the generalized coordinate S = R n +1 on the basis of the known Fokker‐Planck‐Kolmogorov kinetic equation :

true \frac{\partial f (S, t)}{\partial t} = - \frac{\partial}{\partial S} (W' (S) f (S, t)) + \frac{1}{2} \frac{\partial^{2}}{\partial S^{2}} (B (S) f (S, t))

…”

Section: Use Of the Properties Of Statistical Self‐similarity For Thementioning

confidence: 99%

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Practical Application of the Probabilistic‐Statistical Model of the Suspension Separation in Hydrocyclones

2019

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An effective practical approach that allows not only a significant reduction in the scope of practical experiments in the course of studying suspension separation processes in hydrocyclones, but also makes it possible to assess the intensity of random components of the processes and define the interrelation between such components and hydrodynamics of flows in a hydrocyclone is presented. Within the frames of the developed probabilistic‐statistical model of suspension separation in hydrocyclones on the basis of statistical self‐similarity properties, a relationship was found between determined and random components of the processes. This allowed transitioning from three‐parameter probability density functions for suspension particles in hydrocyclones to two‐parameter functions; thus significantly improving the efficiency of practical application of the developed model.

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“…In a previous study , it was proposed to describe the behavior of suspension particles with a size of d 1) ≤ 40 µm based on the particle density distribution function f ( x,t )d x ;

x = R^{n + 1} / R_{0}^{n + 1}

true w'_{0} R_{0}^{normaln} = w' R^{normaln} = D = normalconst

…”

Section: Use Of the Properties Of Statistical Self‐similarity For Thementioning

confidence: 99%

true \frac{\partial f (S, t)}{\partial t} = - \frac{\partial}{\partial S} (W' (S) f (S, t)) + \frac{1}{2} \frac{\partial^{2}}{\partial S^{2}} (B (S) f (S, t))

…”

Section: Use Of the Properties Of Statistical Self‐similarity For Thementioning

confidence: 99%

Practical Application of the Probabilistic‐Statistical Model of the Suspension Separation in Hydrocyclones

2019

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“…In it is proposed to describe the behavior of particles with a dimension of d ≤ 40 µm in suspension on the base of a density distribution function f(x,t)dx , where

x = R^{n + 1} / R_{0}^{n + 1}

stands for a dimensionless generalized coordinate, defined as a ratio of the current radius R to the radius of the cylindrical part R 0 and the power exponent n , that is included in the generalized law of circumferential velocity distributions in hydrocyclones:

\begin{matrix} w_{0}^{'} R_{0}^{n} = w' R^{n} = D = const \end{matrix}

, where

w_{0}^{'}

and w ′ are tangential components of a velocity at the hydrocyclone inlet and at the current radius R , respectively.…”

Section: Theoretical Analysis Of Asymptotic Properties Of a Probabilimentioning

confidence: 99%

“…In the presence of processes random components cannot be neglected . The following expression can be derived for the generalized coordinate

S = R^{n + 1}

on the base of the known Fokker‐Planck‐Kolmogorov kinetic equation :

true \frac{\partial f (S, t)}{\partial t} = - \frac{\partial}{\partial S} (W' (S) f (S, t)) + \frac{1}{2} \frac{\partial^{2}}{\partial S^{2}} (B (S) f (S, t))

…”

Section: Theoretical Analysis Of Asymptotic Properties Of a Probabilimentioning

confidence: 99%

“…As a result, the analytical solution of Eq. for the density distribution function f(x,t) in the limit case with n = 1 was obtained in in the following form:

true \begin{matrix} right1 & left f (x, t) = {x^{\frac{Θ}{2}} e^{- \frac{mx}{2}} {\sum_{i = 0}^{i = \infty} (\frac{Θ m}{(Θ + 2 i)} x)}^{\frac{Θ}{2}} e^{- \frac{Θ m}{2 (Θ + 2 i)}} C_{i} L_{i}^{(Θ - 1)} \\ right1 & left (\frac{Θ m}{(Θ + 2 i)} x) \exp (- \frac{k^{2} (Θ + i)}{b {(Θ + 2 i)}^{2}} it)} \end{matrix}

…”

Section: Theoretical Analysis Of Asymptotic Properties Of a Probabilimentioning

confidence: 99%

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Asymptotic Properties of a Probabilistic‐Statistical Model of Particle Classification in Hydrocyclones

2018

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The peculiarities of practical implementation of a probabilistic‐statistical model for a hydrodynamic stage of particle classification process of liquid‐solid polydisperse systems in cylinder‐conic hydrocyclones of small size have been investigated. Within reasonable assumptions, stationary solutions of the Fokker‐Planck‐Kolmogorov kinetic equation were obtained for the considered separation process. In order to describe changes in characteristics of suspension separation in hydrocyclones it was proposed to use stationary distributions, which parameters depend not only on hydraulic and dynamic features of flows inside an apparatus, but also are determined by relative magnitudes of the impact of particle classification and centrifugal forces in comparison with the intensity of random perturbations.

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Analysis of Hydrocyclone Geometry via Rapid Optimization Based on Computational Fluid Dynamics

et al. 2021

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Hydrocyclones exploit density gradients for the centrifugal separation of dispersions in a continuous liquid. Selection of the geometrics for optimal separation is case specific, like the media characteristics. The existing optimization method based on computational fluid dynamics (CFD) provides a powerful analytical tool but requires long computational times. The most common praxis for CFD optimization is via the single‐factor optimization method (SFOM). In this study, a novel approach is presented as an improved rapid optimization method that implements a dynamic‐mesh and user‐defined function optimization method (DUOM). The DUOM adapts the dynamic‐mesh approach from other applications to the optimization analysis of hydrocyclones. The DUOM reduced the computational time by 31.1 %, compared to the SFOM.

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A Probabilistic‐Statistical Model of the Particle Classification Process in Small Hydrocyclone Classifiers

Cited by 7 publications

References 6 publications

Practical Application of the Probabilistic‐Statistical Model of the Suspension Separation in Hydrocyclones

Practical Application of the Probabilistic‐Statistical Model of the Suspension Separation in Hydrocyclones

Asymptotic Properties of a Probabilistic‐Statistical Model of Particle Classification in Hydrocyclones

Analysis of Hydrocyclone Geometry via Rapid Optimization Based on Computational Fluid Dynamics

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