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Original papers Manufacturing processes and systems
УДК 621.6.04.001.5 DOI: 10.17277/jamt.2022.03.pp.181-191
Evolution of separation effects under variable conditions of gravity flows of granular materials
© Oleg O. Ivanova, Viktor N. Dolgunin^, Alexander G. Tarakanova, Andrey A. Zhiloa, Vasiliy A. Pronina
Tambov State Technical University, 1, Leningradskaya St., Tambov, 392000, Russian Federation
И dolgunin-vn@yandex.ru
Abstract: The influence of the shear rate and the distribution of the void volume fraction in a rapid gravity flow of granular materials on the separation effects by size and density have been studied by means of mathematical modeling. The simulation was carried out on the basis of the process dynamics equation describing the distribution of nonuniform particles in the flow under the action of convection, quasi-diffusion and separation. Separation is considered as a result of conjugating the segregation and quasi-diffusion separation effects. Modeling is carried out by varying the shape of either the velocity profile or the void fraction profile. In order to fill the missing information in terms of structural or kinematic parameters of the gravity flow, the granular medium state equation is used. The state equation establishes the relationship between hydrostatic pressure, the fraction of void volume and the shear rate during rapid gravity shear. It has been found that the effect of shear flow segregation of particles on size has high intensity if high shear rates in the bed volume (75-85 s-1) takes place, the excess of which is accompanied by intensification of quasi-diffusion processes. The formation of a parabolic convex profile of the void volume fraction in the gravity flow at its maximum value in the central part of the flow contributes to an increase in the intensity of quasi-diffusion separation (migration).
Keywords: granular material; rapid gravity flow; separation by size and density; segregation; quasi-diffusion separation (migration); shear rate; void volume fraction.
For citation: Ivanov OO, Dolgunin VN, Tarakanov AG, Zhilo AA, Pronin VA. Evolution of separation effects under variable conditions of gravity flows of granular materials. Journal of Advanced Materials and Technologies. 2022;7(3):181-191. D0I:10.17277/jamt.2022.03.pp.181-191
Эволюция эффектов сепарации в переменных условиях гравитационных течений зернистых материалов
© О. О. Иванова, В. Н. ДолгунинаИ, А. Г. Тараканова, А. А. Жилоа, В. А. Пронин*
а Тамбовский государственный технический университет, ул. Ленинградская, 1, Тамбов, 392000, Российская Федерация
И dolgunin-vn@yandex.ru
Аннотация: Методом математического моделирования исследовано влияние скорости сдвига и распределения доли пустот в быстром гравитационном потоке зернистых материалов на эффекты сепарации по размеру и плотности. Моделирование проведено на базе уравнения динамики процесса, описывающего распределение неоднородных частиц в потоке под действием конвекции, квазидиффузии и сепарации. Сепарация рассматривается как результат сопряжения эффектов сегрегации и квазидиффузионной сепарации. Моделирование осуществляется при варьировании формой либо профиля скорости, либо профиля порозности при восполнении недостающей информации о структурно-кинематических параметрах гравитационного потока с использованием уравнения состояния зернистой среды. Уравнение состояния устанавливает взаимосвязь между гидростатическим давлением, объемной долей пустот и скоростью сдвига при быстром гравитационном сдвиге.
Установлено, что эффект сдвиговой поточной сегрегации частиц по размеру интенсивно протекает при поддержании в объеме слоя высоких значений скорости сдвига (75...85 с"1), превышение которых сопровождается интенсификацией квазидиффузионных процессов. Повышению интенсивности квазидиффузионной сепарации (миграции) способствует формирование в гравитационном потоке параболического выпуклого профиля порозности при максимальном ее значении в центральной части потока.
Ключевые слова: зернистый материал; быстрое гравитационное течение; сепарация по размеру и плотности; сегрегация; квазидиффузионная сепарация (миграция); скорость сдвига; объемная доля пустот.
Для цитирования: Ivanov OO, Dolgunin VN, Tarakanov AG, Zhilo AA, Pronin VA. Evolution of separation effects under variable conditions of gravity flows of granular materials. Journal of Advanced Materials and Technologies. 2022;7(3):181-191. D01:10.17277/jamt.2022.03.pp.181-191
1. Introduction
In most cases, processing technologies for granular materials deal with inhomogeneous granular media. If the process or its sub-operations, such as loading or unloading, are accompanied by a relative movement of particles, it leads to a spontaneous redistribution of particles in the process flow according to similar properties, which is called segregation. In cases where it is not possible to neglect the inhomogeneity of particles, there is a technological need to either suppress or intensify segregation, depending on how its effects can be assessed as negative or positive.
The choice of the method of influencing segregation to achieve the required technological effect depends on the dominant distinguishing feature of particle inhomogeneity and the conditions of their interaction in the process flow [1, 2]. Among various distinguishing features of particles, the presence of which can be the cause of segregation, the most significant are size and density [1], while typical conditions for interaction of particles in technological processes, accompanied by the most pronounced segregation, are created with a rapid gravity flow of granular materials [2].
Such flows are formed under the action of intense gravity shear [3-5]. At high rates of shear deformations, particles, along with the speed of their translational movement, acquire a spatially distributed component of the speed of chaotic displacements (fluctuations). In this case, shear stresses are generated mainly under the action of a flow of impact pulses, which the particles exchange with each other, interacting through the shear surface. In connection with the indicated properties of a rapid gravity flow, the state of granular media under appropriate conditions is often called "a gas of solid particles" [6-10].
The analysis shows [1, 11] that, on the one hand, the causes of intense segregation in rapid gravity flows are high shear strain rates, and structural-kinematic inhomogeneity, on the other hand. Due to
the indicated specifics of gravity flows, segregation processes of nonuniform particles with fundamentally different physical nature proceed intensively in them, one of which is due to the high concentration of stresses on nonuniform particles, and the other is due to the difference in their quasi-diffusion velocities with an inhomogeneous distribution of the solid phase in the flow [1]. In this regard, it makes sense to speak about the spontaneous redistribution of particles in a gravity flow as a separation process [1, 2], which is a consequence of conjugation of separation flows caused by particle segregation as a result of relaxation of local stress concentrators and quasi-diffusion separation (migration) under the influence of spatial structural inhomogeneity of the gravity flow.
The magnitude of the components of separation flows is determined in accordance with the general kinetic law of chemical technology processes as the product of the driving force and the velocity coefficient. The segregation flux is expressed as follows [2]:
js = KsCPbШ
(1)
where Am is the driving force of segregation, №m; c is the local value of concentration of control particles in the flow, kg-kg- ; pb is the bulk density
_3
under local flow conditions, kg-m ; Ks is the
segregation coefficient, (№s)-1, determined experimentally [2].
As the driving force of segregation, the excess moment of forces acting on the particle of the control component is used, which is calculated as
am = M - M0 = (Mg + Mf + Mc ) - (Mg + Mf + Mf),
_ (2) where M is the resulting moment of the combination of gravity forces M , friction Mf and impact
impulses Mc, acting on the control particle in the
flow of nonuniform particles; M0 is the resulting moment of similar forces acting on a particle of a conditionally uniform medium, the properties of which are defined as volume average [2].
The value of the flux of quasi-diffusion separation (migration) is calculated analytically in accordance with the expression [1]
_ 1
Jm = DmcPb~grad s ,
5
(3)
where 1/s (grad 5) is the driving force of migration in the direction of the gradient of the average distance between particles 5, which is calculated depending on
the average particle diameter and the local bulk
2 —1
density; Dm is the migration coefficient, m s . In the case of particle size and density differences, the migration coefficient is calculated as [1]
Dm =
m (c) s 2F f
d2
v m1d2
d 2
m2 d2
(4)
where F is the frequency of collisions for a particle of a conditionally uniform medium, s1, which is calculated from the energy balance condition in the elementary volume of the flow according to the dissipation energy [1]; d , m are the diameter (m) and mass (kg) of a particle of a conditionally uniform medium; d1, m1 and d2, m2 are the diameter and mass of particles of the first and second components of the granular medium.
The fundamentally different physical nature of the described separation fluxes implies the presence of a different correlation of their intensity from the physico-mechanical properties of the particles, kinematic and structural characteristics of the gravity flow. In [12], the flows of segregation and quasidiffusion separation were analyzed with the identification of areas of their dominance depending on the ratio of particle sizes and densities under ordinary conditions of a rapid gravity flow of granular material on a rough chute.
The aim of this work is to analyze the effect of possible variations in the structural and kinematic parameters of a rapid gravity flow of the granular material under some artificial conditions of its organization on the separation fluxes of particles that are nonuniform in size and density. In the analysis, special attention is paid to assessing the significance of the fluxes of quasi-diffusion separation and segregation in the overall effect of particle separation depending on the structural and kinematic parameters of the gravity flow.
2. Materials and Methods
The analysis of separation effects was carried out by the method of mathematical modeling of the separation dynamics of nonuniform particles in a gravity flow for various variants of its structural and kinematic characteristics. The simulation was performed on the basis of the general equation of process dynamics [1, 2], in which, along with separation fluxes (1) and (3), convection transfer flows in the shear direction x with velocity u and quasi-diffusion mixing of control particles in the transverse direction y are taken into account:
d(cpb ) d{iic pb )
d + —
dy
Pb
dt
dc
dx
Ddif--cDm
dy
d ln s
dy
Y
-- KcAM
(5)
The flow of quasi-diffusion mixing is determined by analytical calculation of the quasidiffusion coefficient Ddif [1]. Thus, to model the separation dynamics using (5), it is necessary to experimentally determine the only kinetic constant of segregation Ks according to the method described in [2]. Equation (5) is solved numerically under the following conditions at the bed boundaries
dc
dy
Ddif — = cDm
d ln s
dy
= KscAM|y=0,, = 0,
c(t, x = 0, y) = co
and initial condition
c(0, x, y) = c0 .
(6)
(7)
The analysis of the influence of structural and kinematic parameters of a rapid gravity flow of the granular material on the separation efficiency of its particles that are nonuniform in size and density was carried out by modeling their distributions in the flow depending on the shape of the velocity and void volume fraction profiles. According to the findings [12], presented in the form of a diagram in Fig. 1, segregation is the determining mechanism for separating the admixture of large particles from small ones. The analysis of the driving force of segregation shows [2] that one of the main parameters of the shear flow that determines the magnitude of the driving force is the shear rate. This conclusion substantiates the expediency of studying the effect of gravity flow velocity profiles with different shear rates on the efficiency of particle size separation.
2
PjPb 2.0
1.5 -
1.0 —
0.5
0
Fig. 1. Diagram of the areas of pronounced physical effects of separation of particle impurities which differ in size and density from particles of the basic component
of a rapid gravity flow with size db and density p b [12]
On the contrary, in accordance with the same diagram (Fig.1), the determining separation mechanism for particles of different densities is quasi-diffUsion separation (migration). The analysis of the kinetic equation (3) shows that the main parameter of the shear flow, which determines the magnitude of the driving force of migration, is the gradient of the average distance between particles, which directly correlates with the void volume fraction dynamics in the gravity flow. In accordance with this conclusion, it seems appropriate to study the influence of the shape of void volume fraction profiles on the efficiency of particles separation by density.
dux
The relationship between the shear rate -and
dy
void volume fraction s under conditions of a rapid gravity flow can be expressed using the state equation of the granular medium [1], in which flow dilatancy s is a void fraction function:
(dux ^2
ps = xE =
dy
(8)
where x is the coefficient of the state equation of the granular medium; E is temperature of the granular
medium defined as the kinetic energy of particles in
2 -2
their mutual displacements [1], kg-m -s ; 9 is the coefficient depending on the physico-mechanical properties of particles and conditions of their interaction in the flow [1].
Flow dilatancy, which is the relative increase in the bed volume under the action of shear deformation
per unit volume of the solid phase, is determined by the following relationship:
- s-s0 s =-
1 — s0
(9)
where s, so is the void volume fraction of the bed under conditions of shear deformation and at rest, respectively.
Hydrostatic pressure calculated as follows:
analog (N-m ) is
p(y)= jpb (y) & cos a dy, (10)
h-y
where h is the bed height, m; a is a slope angle.
Since the coefficient x of the state equation of the medium (8) can be taken as invariant with respect to the flow conditions of the granular material, and the coefficient 9 is uniquely determined depending on the conditions of particle interaction [1], it seems possible to carry out computational experiments on the separation dynamics based on equation (5) at virtual variation in velocity and void fraction profiles. In this case, the variation of one of these profiles is accompanied by an unambiguous change in the other profile in accordance with the state equation of the granular medium (8) at an invariant value of the gravity flow.
The study of the influence of gravity flow parameters on the effects of separation (segregation and migration) and mixing was carried out in accordance with the following methodology. In the process of particle separation by size, velocity profiles were set with different shear rates, for which, on the basis of the state equation (8), the corresponding void fraction profiles were determined. The velocity and void fraction profiles were used to simulate the separation dynamics of inhomogeneous particles with subsequent calculation of the coefficient of variation of their distribution in the flow in accordance with the following relationship
V = -
100
c
I
u (1—si )pi(c
Iu (1 - si )Pi(c)AHj
C - C)2, (11)
where c is the average concentration of impurity particles, kg-kg- ; ct is the concentration of impurity particles in the i-th sublayer of the flow, kg-kg-1; Uj is the velocity of particles of the i-th sublayer, m-s-1; pi is the average density of material particles, kg-m ; AHj is the thickness of the i-th sublayer, m.
In a similar way, the separation of particles by density was carried out with the only difference that at first they were set by the void fraction profiles of
a certain shape with its different gradients along the bed thickness, with further determination of the velocity profiles corresponding to them.
As the basic parameters of the gravity flow, the velocity u(y) and void fraction s(y) profiles were taken for the basic component of the model granular material - glass beads of the fraction (3.25-3.50)-10 m with an admixture of 12 % of the beads of the fraction (3.60-3.75)-10-3 m. This binary mixture was used in the study of the particle size separation process. In the study of density separation, a virtual mixture with a similar base component (beads with a
-3
density of 2500 kg-m ) was used, and virtual particles were used as an impurity with a concentration of 12 %, the only difference of which from the particles of the base component was their
-3
density equal to 4000 kg-m .
3. Results and Discussion
3.1. The influence of the velocity profile on the intensity of particle separation by size
The initial velocity profiles in the study of the separation dynamics of particles by size are shown in Fig. 2a. In this case, the profile presented at number 1 is experimental, and the remaining profiles are set from the condition of constancy and systematic increase in the shear rate, i.e. linear functions of velocity on the coordinate along the bed height. This provides conditions for a more definite reflection of the dependence of the intensity of particle separation by size on the shear rate. The void fraction profiles corresponding to the given velocity profiles are shown in Fig. 2b.
The flow parameters corresponding to the velocity and porosity profiles shown in Fig. 2 are used to model the particle size separation dynamics based on equation (5). To confirm the adequacy of the used mathematical model, modeling and experimental research results of the distribution profiles of the concentration of control large particles in the gravity flow, the structural and kinematic characteristics of which are determined by the velocity and porosity profiles presented in Fig. 2 under number 1, were compared. The adequacy of the calculated and experimental data (Fig. 3) is established by comparing the dispersions of adequacy and reproducibility according to the Fisher criterion at 95 % confidence level.
In order to explain the physical nature of the evolution of the in variation index of the distribution of control particles depending on the shear rate in the gravity flow, the simulation was performed for various conjugation options of separation effects: 1) the complex action of segregation and migration effects (Dm ^ 0, Ks ^ 0); 2) the action of only the migration effect (Dm ^ 0, Ks = 0); 3) the effect of only the segregation effect (Dm = 0, Ks ^ 0).
The assessment of the heterogeneity of the distribution of the control component in the mixture was carried out using the variation coefficient according to the formula (11). The dependence of the variation coefficient V on the shear rate is shown in Fig. 4. The given dependence V = f (dU/dy) has a pronounced maximum, which occurs at a shear rate in the particle flow equal to 80 s-1 at a fixed bed thickness.
7-10
2.0 1.6
1.2 0.8
0.4
0
1.0
(a)
U, m-s
-1
/v\ 3 -3
6(7), mi -m
(b)
Fig. 2. Particle velocity profiles (a) and corresponding void fraction profiles (b) in a rapid gravity flow of glass beads of fraction (3.25-3.50)-10-3 m with an admixture of 12 % of fraction beads (3.60-3.75)-10-3 m (profiles number 1 were obtained by the experimental-analytical method [1, 2]) at shear rates (s-1): 2 - 40; 3 - 50; 4 - 70; 5 - 80; 6 - 95; 7 - 110
y-102, m
Fig. 3. Assessment of the adequacy of modeling results of the distribution profiles of the concentration of large particles (+3.60-3.75 mm) in the mixture with small particles (+3.25-3.50 mm) in the gravity flow of beads
An explanation of the obtained dependence can be given on the basis of the analysis of the
contribution of segregation and migration effects in gravity flows to the separation process with a change in the shear rate in combination with void fraction profiles (Fig. 2b). Fig. 5 shows the distribution profiles of the concentration of control large particles in gravity flows, the shear rate in which is less (70 s-1) and more (95 s-1) than its critical value, corresponding to 80 s-1.
The analysis allows to conclude that an increase in the heterogeneity of the concentration distribution with an increase in the shear rate to its critical value occurs with a dominant and increasing role of segregation (Fig. 5a). The intensification of segregation occurs due to an increase in the driving
force of the process am (2) with an increase in the shear rate at a sufficiently high volume fraction of the solid phase. When the critical value of the shear rate is exceeded, quasi-diffusion effects begin to dominate in the gravity flow, and quasi-diffusion migration becomes the dominant separation effect. With an increase in the shear rate in this area, an intensive increase in void fraction occurs with a decrease in its gradient (Fig. 2b). Since this gradient directly determines the magnitude of the driving force of quasi-diffusion separation (3), the intensity of the latter decreases with increasing shear rate, and the effect of quasi-diffusion mixing increases in the gravity flow.
V , % 16 '
12 '
8 •
4 ■
0
40 80 duldy, s 1
Fig. 4. Variation coefficient of the concentration of large particles in the gravity flow of beads as a function of the shear rate (s-1)
3.2. The influence of the void fraction profile on the intensity of particle separation by density
In order to conduct a computational experiment to study the influence of flow parameters on the separation dynamics of particles differing in density, a hypothetical mixture of beads consisting of two fractions was used: 1) with a density of 2500 kg-m (88 %); 2) with a density of 4000 kg-m-3 (12 %) with a particle size of 3.25-3.50 mm. According to the diagram shown in Fig. 1, in the gravity flow of such particles, the dominant effect of separation is migration. Since the driving force of migration is the gradient of the average distance between particles (3), which is formed due to the spatial structural inhomogeneity of the gravity flow, it is advisable to set the void fraction profiles as the initial data for modeling in this case.
Among various possible options of void fraction profiles in the gravity flow on a rough slope, parabolic profiles are of the greatest interest from scientific and practical points of view. This is explained by the possibility of achieving the highest values of the gradient of the average distance between particles with such profiles. In this regard, two types of porosity profiles were used in this experiment: 1) parabolic convex (Fig. 6a), 2) parabolic concave (Fig. 7a), having different concentration gradients of the solid phase. The corresponding particle velocity profiles calculated using the state equation (8) are shown in Figs. 6b, 7b.
y, m
0.016
0.012
0.008
0.004
0.035
0.07
0.105
c, kg-kg
(a)
y, m
0.016
0.012
0.008
0.004
0
0.105 c, kg-kg1 —A— 3
0.035 0.07
—O— 1 —•— 2
(b)
Fig. 5. Distribution profiles of the impurity concentration of large particles for various modeling options:
1 - Dm * 0, Ks * 0; 2 - Dm * 0, Ks = 0; 3 - Dm = 0, Ks * 0 in the gravity flow of the granular medium at shear rates of 70 s-1 (a) and 95 s-1 (b)
y-102, m 2.0
1.(5
1.44
0.8
0
y K)2,m 5 4 3 2
0.4 0.5 0.(5 0.7 0.8 (a)
3 -3
6(y), m -m
2.0 \
1.6 . i i i
1.2 i i)
0.8 ¥A
0.44 ß
0
3 4 5 u(y), m- s 1 (b)
Fig. 6. Porosity profiles (a) of a parabolic convex type with void fraction in the central part of the bed s(y) (1 - 0.99; 2 - 0.9; 3 - 0.8; 4 - 0.7; 5 - 0.6) and the corresponding particle velocity profiles u(y) (b)
0
0.9 e(y), m3-m 3
2.9 3.44 u(y), m-s
-1
(b)
Fig. 7. Porosity profiles (a) of a parabolic concave type with void fraction in the central part of the bed e(y) (J - 0.42; 2 - 0.5; 3 - 0.6; 4 - 0.7; 5 - 0.8) and the corresponding particle velocity profiles u(y) (b)
In order to obtain information for comparative analysis and physical explanation of the separation
efficiency at various flow parameters for the specified mixture, the distribution profiles of the concentration of test dense particles were modeled. Modeling was carried out on the basis of equation (5) for three conjugation options of segregation and migration effects: 1) there is a cumulative manifestation of segregation and migration effects (Ks ^ 0, Dm ^ 0);
2) only the migration effect appears (Dm ^ 0, Ks = 0);
3) only the segregation effect appears: Dm = 0, Ks ^ 0.
The simulation results in the form of concentration distribution profiles were used to calculate the variation coefficients (11) for each variant of structural and kinematic parameters of the gravity flow. The dependence of the variation coefficient on the void fraction in the central part of the bed for variants of convex and concave parabolic profiles is shown in Fig. 8. The presented results indicate a systematic increase in the inhomogeneity of the distribution of control particles and, accordingly, the intensity of separation with an increase in the degree of dilatancy of the gravity
flow. This result indicates the dominance of the quasi-diffusion separation effect in rarefied flows, which is confirmed by the simulation results presented in Fig. 9. The results shown in the figure indicate the complete dominance of quasi-diffusion interaction of particles (migration and quasi-diffusion mixing mechanisms) at high values of void fraction in the gravity flow exceeding the value 0.7.
Particular attention is drawn to the fundamentally different shape of the velocity profiles for flows with convex and concave parabolic void fraction profiles. In the case of a convex void fraction profile, the velocity profile has an s-shape with a maximum shear rate in the central part of the bed with its minimum values at the flow boundaries (Fig. 6b). A flow with a concave parabolic void fraction profile is characterized by parabolic velocity profiles with a maximum shear rate at the lower boundary of the bed (Fig. 7b). The conditions of the most intense shear in the central part of the flow with a convex void fraction profile contribute to the formation of the gravity flow with high shear rate gradients and, as a result, void fraction (Fig. 6).
V, %
160" 120-8040--
00.4
0.5
0.6
0.7
0.8
09 e(y), m3-m-
Fig. 8. Variation coefficient of the impurity concentration of dense particles in the gravity flow of uniform-sized particles of beads of various densities with concave (J) and convex (2) parabolic profiles as a void fraction function
in the central part of the bed
3
0.020 0.016
0.012
0.008
0.004
.>", m
_i_i_i_i_i
0 0.035 0.070 0.105 0.140 s, kg-kg-1
Fig. 9. Concentration distribution profiles of dense particles in the gravity flow of uniform-sized particles of various densities with a parabolic concave void fraction profile at its value in the central part of the bed equal to 0.7 for various modeling options: 1 - Dm ^ 0, Ks ^ 0; 2 - Dm ^ 0, Ks = 0; 3 - Dm = 0, Ks ^ 0
In accordance with the kinetic dependence (3), the presence of large void fraction gradients leads to an intensive process of quasi-diffusion separation (migration) of particles with different density. This is reflected in the high values of the variation coefficient in the gravity flow with a convex parabolic void fraction profile (curve 1, Fig. 8).
The analysis of the profiles shown in Figs. 6 and 7, in combination with the concentration distribution profiles of dense particles, for example, Fig. 9, shows that in all cases the maximum values of the concentration of dense particles are observed in the flow regions with the smallest void fraction values. The most striking manifestation of the separation effect is achieved in the gravity flow with a parabolic convex void fraction profile at its maximum value in the central part of the bed. If in flows with parabolic concave void fraction profiles there are obvious signs of correspondence with the flow parameters and separation effects in gravity flows on a rough chute [1-3], then for flows with parabolic convex profiles, the Couette flow is suitable for identification only in limited cases [14-16].
4. Conclusions
The separation effects of particles that are nonuniform in size and density in a rapid gravity flow on a rough chute depending on the kinematic and structural parameters of the flow were studied. The study was carried out by the method of mathematical modeling of the distribution dynamics of nonuniform particles in a flow based on the general equation of the process dynamics, which takes into account the
transfer of particles under the influence of convection, quasi-diffusion and separation effects. The separation effect is modeled as a set of segregation and quasi-diffusion separation effects fundamentally different in physical nature, the first of which is due to the relaxation of stress concentrations on nonuniform particles, and the second is a consequence of the difference in the rates of their quasi-diffusion displacements under conditions of structural inhomogeneity of the gravity flow.
During simulation, either the velocity profile (for size separation) or the void fraction profile (for density separation) was varied. When varying the velocity profile, profiles with different shear rates were set, for which, using the state equation of the granular medium under rapid shear, the corresponding void fraction profiles were determined. On the contrary, when varying the void fraction profile, convex and concave parabolic profiles with different curvature were set, for which the corresponding velocity profiles were determined using the same state equation. The state equation establishes the relationship between hydrostatic pressure, void volume fraction, and shear rate in the rapid gravity flow of cohesionless granular materials and is used to determine the full range of structural and kinematic flow characteristics needed to model the dynamics of the separation process.
Based on the findings, it seems possible to conclude that in order to intensify the segregation effect in a rapid gravity flow, high shear rates (75-85 s-1) should be maintained in its volume, the excess of which is accompanied by intensification of
quasi-diffusion processes. An increase in the intensity of quasi-diffUsion separation (migration) is promoted by the formation of a parabolic convex void fraction profile in the gravity flow at its maximum value in the central part of the flow.
The results of the study allow expanding knowledge about the possibilities of intensifying the gravity separation of particles by size and density in a rapid gravity flow and can be used as a theoretical basis for developing technical solutions in technologies for processing granular materials.
5. Funding
The study was supported by the RFBR grant No. 09-08-97521.
6. Conflict of interests
The authors declare no conflict of interest.
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Information about the authors / Информация об авторах
Oleg O. Ivanov, Cand. Sc. (Eng.), Associate Professor, Vice-Rector for Innovation, Tambov State Technical University (TSTU), Tambov, Russian Federation; 0RCID 0000-0002-9868-5487; e-mail: iooc4@mail.ru
Viktor N. Dolgunin, D. Sc. (Eng.), Professor, TSTU, Tambov, Russian Federation; ORCID 0000-0002-62275224; e-mail: dolgunin-vn@yandex.ru
Иванов Олег Олегович, кандидат технических наук, доцент, проректор по инновационному развитию, Тамбовский государственный технический университет (ТГТУ), Тамбов, Российская Федерация; ORCID 0000-0002-9868-5487; e-mail: iooc4@mail.ru
Долгунин Виктор Николаевич, доктор технических наук, профессор, ТГТУ, Тамбов, Российская Федерация; ORCID 0000-0002-6227-5224; e-mail: dolgunin-vn@yandex. ru
Alexander G. Tarakanov, Postgraduate, TSTU, Tambov, Russian Federation; ORCID 0000-0002-4902-982X; e-mail: uazqaaz@gmail.com Andrey A. Zhilo, Postgraduate, TSTU, Tambov, Russian Federation; ORCID 0000-0001-5033-8315; e-mail: zhilo97@mail.ru
Vasiliy A. Pronin, Cand. Sc. (Eng.), Associate Professor, TSTU, Tambov, Russian Federation; ORCID 0000-0002-1507-2969; e-mail: ua3rbs65@mail.ru
Тараканов Александр Геннадьевич, аспирант, ТГТУ, Тамбов, Российская Федерация; ORCID 0000-0002-4902-982X; e-mail: uazqaaz@gmail.com Жило Андрей Андреевич, магистрант, ТГТУ, Тамбов, Российская Федерация; ORCID 0000-00015033-8315; e-mail: zhilo97@mail.ru Пронин Василий Александрович, кандидат технических наук, доцент, ТГТУ, Тамбов, Российская Федерация; ORCID 0000-0002-1507-2969; e-mail: ua3rbs65@mail.ru
Received 13 June 2022; Accepted 25 July 2022; Published 12 October 2022
Copyright: © Ivanov OO, Dolgunin VN, Tarakanov AG, Zhilo AA, Pronin VA, 2022. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
