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Segregation modeling of particle rapid gravity flow
POWDER TECHNOLOGY ELSEVIER
Powder Technology 83 (1995) 95-103
Segregation modeling of particle rapid gravity flow V.N. Dolgunin, A.A. Ukolov Department of Machines and Apparatus, State Technical University, Tarnbov, Russia
Received 18 October 1991; revised 1 December 1994
Abstract
A model describing the size and density separation of granular materials undergoing gravity flow down a rough inclined chute is analyzed in the present paper. A dynamic model is developed on the basis of an equivalent mass transfer equation taking into account the mass transfer due to convection, quasi-diffusion and segregation. Segregation and quasi-diffusion kinetic parameters are developed from the gravity flow microstructural analysis based on Ackermann and Shen's [14] rheological model. An experimental and analytical method of investigation of the parameters of the granular material rapid gravity flow down a rough inclined chute is being developed. It is based on the analysis of the free fall phase of the particles that starts after the particles have been sent down the chute. The analogy between the patterns of rapid shear flow and molecular gas dynamics of the particles is used. The results of the particles of segregation modeling of granular material gravity flow down an inclined chute are compared with experimental data. Keyword$: Granular materials; Gravity shear flow; Mass transfer; Convection; Diffusion; Segregation
1. I n t r o d u c t i o n
The separation of solid particles in a mixture of granular materials flow has been well known for a long time. The essence of this phenomenon is that in the course of the flow, the nonuniformity of the mixture in some parts of the flow increases owing to separation and localization of particles which display similarity according to certain criterion. In its physical sense, this phenomenon is the inverse of mixing and is often called segregation. Segregation occurs in the course of many natural phenomena such as snow avalanches, rock falls, mudflows, subaqueous grain flows as well as during some technological processes connected with granular materials processing (mixing, granulation, screening, etc.). It also takes place during transportation of suspensions and loading and unloading operations. In the majority of these cases the flow of granular media can be referred to as rapid shear flow [1]. Rapid shear flow investigation is a field of granular media mechanics. Such flow is often called grain-inertia flow as, internally, momentum is carried by the inertia of the particles and exchanged during interparticle collisions. Segregation in granular shear flows has been scantily explored. Due to recent development in granulation techniques as well as a wider use of granular materials, 0032-5910/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSD1 0032-5910(94)02954-M
powders and composite materials, a great deal of attention has been paid to the segregation process study. However, so far no constitutive relationship between the process kinetics and the dispersed flow physical, mechanical and kinematic characteristics have been established. The existing models of segregation kinetics are either of speculative nature, for instance [1], or they use the body of mathematical statistics and probability theory [2], but do not reflect the physical nature of the process, accounting for their limited application. Besides, in the general case they fail to predict the direction in which the segregation proceeds. Among the existing physical models of the segregation process the percolation model seems to be the most widely used. It describes the segregation by a simple screening of smaller and denser particles in the direction of the gravity force. However, it is obvious that the percolation model cannot predict the direction of particle displacement in a number of cases, for instance, when they differ simultaneously both in density and in size and the bigger ones have a greater density. As a result of Bridgwater's experiments with an annular shear cell [3] it was found out that the segregation direction may be different even when particles vary only in size. A similar behaviour was noticed by the authors of the present paper for gravity flow down a rough inclined chute, the particles differing only in size or only in density (see Fig. 4 of the present paper).
V.N. Dolgunin, A.A. Ukolov / Powder Technology 83 (1955) 95-103
96
This paper presents a distribution dynamic modeling of granular materials for a particular but widespread case of a rapid shear motion. The modeling is based on a general equation of species transfer, taking into account convection transfer, quasi-diffusional mixing and particle segregation. The segregation flow is described using the physical model of particles hydromechanical separation [4]. The model developed earlier represented the shear flow as a number of elementary layers situated one above another (Fig. 1). To develop the physical segregation model we used the microstructural and continuum approaches described in [5]. These approaches are based on rapid granular flow theory. The microstructural approach allows us to determine the system geometrical parameters for a steady shear flow as well as to set up the equations for the determination of forces acting on the test particle when it moves towards the neighbouring layer (Fig. 2). The distance between the layers of diameter d particles is determined as follows l = bd
= [~-/6(1 - E)]°33d
(1)
and the mean distance between particles is described by the equation s = (b/bo -
1)d
(2)
where • is the specific free volume (SFV) of the layer, that is the ratio of the void volume to the layer volume; bo =b(eo), where Eo= 0.2595. 8d
Fig. 1. Determination of shear flow parameters.
Fig. 2. The formation of a support contact with an instantaneous axis of rotation.
A granular medium consisting of rough inelastic spheres of diameter d is considered. A test particle B of diameter D is placed in the medium. The analysis of non-uniform particle interaction was carried out with due regard for the impact momenta, frictional and gravitational forces, the force of rotation inertia being neglected. Equations for the moments of these forces are set up, relative to the particle instantaneous axis of rotation passing through check points O' O" (Fig.
2). The physical model is based on the assumption that the direction and velocity displacement of a spherical particle are determined according to the value and direction of the excess moment of forces acting on the test particle when it interacts with the other particles of the medium which is assumed to be of uniform composition [4]. The sum of the moments of impact momenta, frictional and gravity forces acting on the test particle relative to the instantaneous axis of rotation equals: M= Mc + Mv + MG
(3)
It is assumed [4] that the absolute value M conventionally determines the velocity of the test particle in the y-direction as a function of the most important physico-mechanical and geometrical characteristics of the flow and its components, that is particle size and density, friction and restitution coefficients, the layer shear rate and its specific free volume.
2. Segregation kinetics and dynamics of the distribution of solid particles in gravity flow down a rough chute
To develop a segregation model it is worthwhile to analyze a granular shear flow, as most flows induce some sort of shearing motion. The segregation dynamics are determined by the relative magnitude of at least two opposite flows, one of which impedes particles separation while the other one impedes their ideal mixing. The first flow is entropic in the broad sense of the term and may be modeled as a quasi-diffusional process. The other flow is formed due to hydromechanical interraction of nonuniform particles. During guided shear a conventional flow of the mixture components will also take place. It should also be taken into account during modeling of the distribution dynamics of particles. Using these parameters, determination of the functional dependence of one of the fractions of granular particles mixture concentration, c =f(t,xy,z) on time and flow coordinate, we may use the general equation of substance transfer:
V.N. Dolgunin, AM. Ukolov / Powder Technology 83 (1995) 95-103
ac Ot + div(u, c) = -
div(j~)
(4)
where Jc is the component flux at the point xy,z. Flux j¢ is the sum of two fluxes differing in nature and direction: j~=jdi~+js
(5)
Flux Js appears in the course of the segregation process, flux Jd~f, due to the medium quasi-diffusional mixing. In rapid shear flow [5] behaviour patterns which are determined basically by the presence of the fluctuating component V' of the velocity vector of a separate particle, quasi-diffusional mixing takes place owing to the accidental special distribution of the vector V'. In this connection one can trace an analogy between the process of mixing and diffusion in mixtures of dense gases, from which follows [6]: ~Jdltl= -Ddlf" grad c
(6)
For a quantitative estimation of flux is, it is necessary to formulate the segregation law. The segregation kinetics are determined on the basis of general kinetic behaviour patterns of the chemical technology: the speed of the process is directly proportional to the driving force and inversely proportional to the resistance: M ~=KsA
(7)
where M is the quantity of the mixture component transferred through surface F in time t, Ks is the segregation coefficient, h is the segregation driving force. The parameters of the segregation kinetics, Eq. (7) are expressed on the basis of the postulate that in similar conditions of shear flow, (the layer SFV, shear rate) the segregation flux is directly proportional to the degree of nonuniformity of the granular medium. In a granular medium there are two major aspects of nonuniformity: (1) nonuniformity of the properties of the particle (size, density, shape, roughness, elasticity, etc.); and (2) nonuniformity of the medium caused by adding one component to another. We have taken into account the fact that the degree of particle nonuniformity, with regard to the conditions of their interaction, determines the velocity of the particle displacement, that is the rate of the process. At the same rate, the segregation flux is proportional to the granular medium nonuniformity caused by adding one component to another. It is obvious that the segregation driving force is the continuous function of concentration c and A(0)=A(1)=0. Taking into account general kinetic behaviour patterns of technological processes, the driving force in the area of low concentration is considered
97
to be proportional to the concentration, that is A =c, and when the concentration approaches 1, A = l - c . Since the segregation driving force should also be a measure,of the nonuniformity of the medium, it can be reduced to the variance: A =C(1 - c )
(8)
Finally the flux of particle segregation in the granular medium is presented as follows:
lLl=Ks'c.(a-c)
(9)
In the present paper, the model of the segregation process during gravitational shear flow of granular materials is developed, since such a flow in the most widespread type of granular media movements, which is accompanied by an intensive particles separation. Let us consider a disperse medium which is a mixture of solid spherical particles of diameters d and D. The particles move along a rough inclined chute owing to gravitational force. The degree of roughness is half the diameter of the largest particles (Fig. 1). Let us also assume that the medium movement is two-dimensional, that is the flow parameters are invariable in the direction of the chute inclination as well in the direction perpendicular to the chute surface. Let us mark the slope of this surface relative to the horizon as a. In this case, to ensure the accepted condition concerning the steady flow, the value of a should be close to the value of the angle of repose [2]. Neglecting the particle quasi-diffusion in the direction of the chute and expressing its conventional flow in this direction through the average velocity of the medium translational motion U(y), from Eqs. (4), (6) and (9), we get the equation of component distribution dynamics c(t;cy) for a two-dimensional steady gravity flow:
ac_ at
a(Uc) + a-y ] [D ax
a~_Ks.c. ac
(l-c) (10)
where c is the test component concentration, t time, Dd~ the coefficient of particle quasi-diffusional mixing and Ks the segregation coefficient. The initial conditions are set by the equation: c(0, x, y) =f0')
(11)
Boundary conditions are formulated assuming that there is no flux of particles through the bed boundaries:
Dd~,"~,, -Ks'c-O-c)
=0
(12)
y~0, y~h
where h is the bed height. In accordance with the conditions adopted for the model the quasi-diffusional mixing during a rapid shear flow will be conditioned by an accidental character of special distribution of the fluctuating component V' of
V.N. Dolgunin, AM. Ukolov / Powder Technology 83 (1955) 95-103
98
the individual particle velocity vector. On the basis of this analogy and using the gas kinetic theory [6] we get: 1
O,m = ~ A. V'
(13)
where A is the average length of free path of a particle. Taking Eq. (2) into account, we formulate Eq. (13) in the following way: 1 Ddlf = ~ s. V'
(14)
where V' = 2f- s. Frequency f in Eq. (14) is calculated according to Shen and Ackermann [5], taking into account the changeable conditions of collision of particles considered to be uniform [4]. Eq. (10), for a general case of gravity flow, turns out to be greatly nonlinear, which can be indirectly derived from Bridgwater's investigation [3] and can be proved on the basis of the data given later (see Fig. 4). A complicated form of the function c(t,r~v) testifies to the existence of a reverse direction of the flux depending on the characteristics of the flow heterogeneity. It will be shown further that nonlinearity of Eq. (10) is connected not only with a great heterogeneity of the flow but also with its large influence on kinetic coefficients Ddif and Ks. In a general case the segregation rate depends on a great number of physico-mechanical and geometrical parameters of the particles (size, density, surface roughness, adhesion properties, shape, elasticity, etc.) as well as on the flow structural and kinematic characteristics. Obviously, it is not possible to determine definitely the process rate coefficient for such a case. Also some difficulties arise in the course of the experiment. These are the factors impeding the establishment of the theoretical basis for segregation with the use of general kinetic behaviour patterns of technological processes. To determine the values of the segregation coefficient we have used the model of process mechanism for spherical particles [4]. In accordance with this postulate, the process rate is proportional to the degree of nonuniformity of the granular components. Since the value of the resultant of the moment of forces (Eq. (3)) acting on the test particle determines the tendency of its displacement in accordance with the main properties of particles, inter-particle medium and flow parameters, we use this moment to express the degree of nonuniformity of components when they interact with each other. Taking into account the fact that in a bed of uniform particles there is an equal possibility for them to move in opposite directions, we can express the parameter of nonuniformity as the difference in the moments of forces AM. One of the moments, M is calculated in
accordance with Eq. (3) for nonuniform particles. The other one Mo, is calculated using the same dependence for the case of uniform particles, when the flow conditions being the same in both cases: AM=M-Mo
(15)
When the value of AM is negative, the test particle moves to the lower layer, when the value is positive, the particles moves to the upper layer. The absolute value of AM determines conventionally the rate of spherical particle segregation, depending on the most important characteristics of the medium and the flow, such as the size and density, the coefficients of friction and restitution at collision, the flow shear rate and SFV. The segregation coefficient Ks is determined as a supposedly linear function of parameter AM: Ks =KAM
(16)
where K is the rate coefficient which is equal to the average velocity of displacement of the mixture component per a unit of the excess moment AM, which is a measure of nonuniformity of spherical particles during interaction. Kinetic parameters Ddif and K~ used in Eqs. (10) and (12) are calculated in accordance with the function of velocity distribution U(y) and particles concentration P*0'). When solving the problem of segregation kinetics it is necessary to determine these functions. 3. The definition of the velocity profile and distribution of particles in gravity flow
Description of velocity profiles and solid particles distribution in rapid shear flows is given in [7-15]. Taking into account the main statements of the continuous medium mechanics, Goodman and Cowin [7] developed a model of a rapid shear flow of granular materials. Savage [8] adapted the model for particles rapid gravity flow down an incline. Kanatani [9] developed the continuous medium micropolar theory which describes the granular material flow and derived an equation determining the flow rate and medium density interrelationship. Later Ogawa et al. [10], Jenkins and Savage [11] and Savage [15] developed a model of rapid gravity flow based on the microstructural approach. These authors determined the stress tensor by analyzing momentum translation during particle collision and solving the system of equations on the conservation of momentum and fluctuation energy. But the assumptions made by the authors led to errors in the description of flows, these errors being quite apparent in numerical modeling of granular particles flow down a steep incline carried out by Hutter and Sheiwiller (12). Moreover, it was found out [12]
99
V.N. Dolgunin, AM. Ukolov / Powder Technology 83 (1995) 95-103
that the models require an exact statement of boundary conditions which are difficult to determine. The results of direct computer modeling of the spherical particles shear flow (Couette flow) microstructure obtained by Campbell and Brennen [13] are valuable for a more comprehensive understanding of granular materials rapid shear flow behaviour. One of the most important results of their computational experiment is the conclusion that the effective coefficient of friction determined as a shear stress to normal stress ratio depends on the solid phase concentration. Similar results were obtained experimentally by Savage [15]. Ackermann and Shen [14] used geometrical analysis of the shear flow microstructure and came to the conclusion that the gravity flow should be characterized by a considerable lateral mass transfer (quasi-diffusion) which must be taken into consideration for an adequate flow modeling. However, it seems that the authors [14] restrict the possibilities of the approach when they solve the problem of momentum transfer determination during particle collisions. They proceed from the assumption that the collision conditions do not depend on the solid phase concentration. A close consideration of all the papers [5,13,14] permits the conclusion that the assumptions made by the authors about the absence of momentum transfer due to particles quasi-diffusional displacement and the assumption that the effective coefficient of friction does not depend on concentration are very approximate. These models [5,7-11,14,15] predict the existence of a steady gravity flow only within a narrow range of bed inclination angle values, and allow the description of velocity profiles and solid particles distribution in full, especially in thin layers and the flow boundary areas. For all the reasons cited above it seems impossible to use the existing models for an adequate predicting of functions U(y) and P*0')Considerable experimental difficulties complicate the rapid gravity flow modeling of granular materials. It is rightly pointed out in [15] that rapid gravity flows of particles down a steep incline appear to be extremely complicated for an experimental study despite their apparent simplicity. The analysis of the results of such investigations of flows shows [12,15] that the main difficulties arise from a high flow sensitivity to the conditions at the bottom bed boundary as well as on the slightest disturbances. The present paper describes the method of determination of granular particle velocity and concentration profiles according to the bed thickness at the threshold of discharge from the incline by analyzing the free fall phase [16]. The method consists of sending a granular material down a stationary incline and collecting the particles in a tray containing a number of cells (Fig. 3).
%
Y
2 /---
IIIIIIIIIV Fig. 3. Schematic of experimental unit: 1, open channel; 2, tray; 3, cells; 4, slide-value; 5, dosage device.
In accordance with the method developed, the experimental data include the bed depth at the discharge threshold h in a steady flow, time t, the material distribution function G(Xl) along a certain horizontal coordinate axis xl located at a vertical distance H from the discharge threshold and the bed inclination angle a (Fig. 3). Assuming that particles scatter during their free fall, does not considerably influence the function G(xl) and taking into account the experimental data and the laws of a free failing body and mass conservation we obtain: x ~ - y sin a [lJ I = cos a [ ( H + y . c o s a - ( x 1 - y . s i n a).tg a~/2 2/g)] 1/2
Iol-p*(v)-- a(zl)
(17) (18)
where p*(y) is the solid particle distribution function. Eqs. (17) and (18) indirectly determine the unknown functions P*0') and U(y). Additional data is needed to solve equations which may be presented as an interrelationship function p*(y)=f(U). Many authors (for example, [11] and [13]) have pointed out that rapid shear flows are analogous to molecular gas dynamics. This allowed some of them [11] to apply a well-elaborated kinetic gas theory for solving the problem of granular materials flow dynamics. Let us determine the interrelationship function p*0')=f(U) using this analogy in the form of a wellknown law of dense gas state which establishes the ratio between molar volume, pressure and temperature. Adopting the following corresponding parameters of the granular medium: the shear flow dilatation ~ the hydrostatic pressure P analogue; and the granular material temperature O, [13] we can define their interrelationship in the following way: P~ = 0,9
(19)
100
V..N. Dolgunin, A.A. Ukolov / Powder Technology 83 (1955) 95-103
where ~b is a coefficient which depends on the physical and mechanical properties of the particles. The granular medium temperature is expressed by the following equation: 7rd 3
-]5-.o.(v") 2
(20)
where V' = 2fs. Taking [4] into consideration we obtain:
(V')Z=(2f.s)2=K(dU~ 2 \dy]
(21)
The granular medium pressure is expressed in the following integral form: h
P(Y) = I P*(Y)g cos a dy
(22)
h--y
We assume the moving bed specific free volume increment which takes place due to the bed dilatation to be the gas volume molar analogue: E - - Eo
= -1-E
(23)
where Eo is the SFV of the dense particles packing. Expressions (17), (18) and (19) with due regard to (20)-(23) form a closed system of equations, relative to the functions U(y), y(xl), e(y) and P(y), where y(xl) is the coordinate of the particles in bed, determined by the coordinate of the particles after falling, xa. The boundary condition of this equation system is formulated in accordance with the condition of adhesion at the flow bottom boundary. In this condition the particles at the bottom will have y = 0, U = 0 and xl is determined by 0 as a result of the vertical falling of the particles, that is why
U(O)=O
(24)
y(0)=0
(25)
The solution to this system was obtained by using successive approximations, the first approximation being 0*(3') = p* = constant. The proposed analytical and experimental method of granular materials rapid gravity flows creates favourable conditions for the segregation phenomenon investigation by analyzing the test component distribution in the flow of falling particles.
poly(propylene) grains, triple superphosphate and silica gel as well as their mixtures. The investigations were carried out in conditions similar to steady rapid gravity flow. These conditions were established by fixing the incline (its roughness being half the diameter of the biggest particle in the mixture) at an angle to the horizon equal to the angle of repose of the material. The desired roughness of the incline was achieved by fixing a perforated metal plate above the smooth base at a distance of half the diameter of the biggest particles. The dimension of the holes in the plate was equal to 2- to 3-times the diameter of the biggest particles in the mixture and the free cross-sectional area was 70% of area plate. During the experimental determination of velocity profiles at the threshold of discharge at various lengths of sending bed, it was established that the flow becomes steady at 0.15 m. Velocity profiles at a bed length of 0.15 and 0.6 m differ by about 5%. The particle bed depth at the threshold of discharge was estimated by a visual method. Fig. 4 shows (1) velocity profile; (2) the specific free volume; and (3) the test component distribution functions relative to the depth y of a mixture of rusty steel balls rolling down a rough incline, of length 0.45 m. The mixture consists of balls of diameter 6.6 mm and 7.0 mm, the latter diameter constituting 42% of the total mass. In accordance with the classification of gravity flow conditions proposed by Savage [15], the observed velocity profile is similar in its main features to a sliding flow which takes place when the rough incline slope approximates the angle of repose of the granular material. The function of free volume distribution according to the bed depth indicates that the maximum concentration of particles is observed in the middle of the bed.
•
o
x
0.0~
•
02
o
~'
I
o.'o
d
~.'5 v.~.~-,
0.0t5 001 •
t
0.005
4. Segregation modeling during particle gravity flow down an incline During the investigation of the segregation phenomenon, binary mixtures of granular materials differing in size were used, for instance: steel balls,
d3
o.'6
Fig. 4. VelocRy profile U(v) specific tree volume ~(y) and test component particlescO,) distributionprofilesduring steel ball gravity flow down a rough incline: I, U(y); 2, e(y); 3, c(y).
V.N. Dolgunin, AM. Ukolov / Powder Technology 83 (1995) 95-103
As the free surface is approached, the particle concentration falls almost to zero. In the bed base direction the concentration drops to a certain value and increases again in the vicinity of the incline surface. The observed pattern of particle distribution with maximum concentration in the middle of the layer was described in [12,15,16] and explained by particles slipping at the base and forming a rarefied cloud at the free surface. The distribution function obtained differs from that previously observed owing to the existence of a second extremum in the layer adjacent to the bed base where the minimum flow density is observed. The authors relate the availability of such an extremum to the specific boundary condition of sticking to the rough incline. Under such a condition, the particles of the bottom layer are detained on the incline surface and the area of the most considerable shear is situated at a certain distance from the incline, this distance approximating to the mean diameter of the particles. It is noteworthy that the free volume distribution function and the function of the test component concentration correlate and can be considered to be mirror images of one another. Thus, their analysis is of particular importance. Subjectivism in observing the correlation in the process of experiments is practically excluded as the parameters analyzed are determined independently. Randomness in the observed correlation is eliminated in the course of a great number of experimental investigations of various mixtures of granular materials. The fact that singular points coincide can be explained in the following way. The particles in contact with the surface of the incline possess a relatively small component of the oscillating displacement velocity which slows down their transfer to the upper layer. Thus, the test component concentration in the vicinity of the incline surface differs very little from the mean concentration of the mixture. Farther on from the incline surface, the shear velocity increases, thus leading to the formation of a rarefied interlayer. The particle concentration increases above this interlayer due to the decrease in the shear velocity, while in the middle the particle concentration increases and is maximum. In the rarefied interlayer and above it, particle agitation is much higher than at the incline surface. At large enough shear rates, large particles, being stress concentrators, move towards the layer-free surface. Thus one can observe a coincidence along the extreme values coordinate of the free volume distribution and test component function. The area in the vicinity of the layer-free surface is referred to as the particle cloud [15]. When there is nothing to restrict the bed volume, the particles of this area lose a certain arrangement which is typical of the central part of the flow. Owing to chaotic collisions in the rarefied medium, the particles move and jump above
101
the bed surface. In the course of the exchange of particle impact momenta the smallest and lightest ones acquire a higher speed. As a result they jump higher above the surface bed. In contrast, the bigger and heavier particles in the area, with a relatively low velocity gradient, move towards the bed bottom. As a result of such particle redistribution, the test component distribution function has the second extremum in the upper part of the flow nucleus. To verify the adequacy of the proposed segregation model, we compared the component distribution obtained experimentally from a mixture of superphosphate granules flowing down a rough incline with those calculated from Eqs. (10)-(12) and the kinetic parameters K~ and Ddi f from Eqs. (14) and (16). The initial mixture contained an equal mass parts of 3.0-3.25 mm and 3.5-3.75 mm particle size fractions. The experimental distribution functions U(y) and E(y) are obtained using the proposed experimental and analytical method for investigating the parameters of the rapid gravity flow at a 0.3 m incline length, On the basis of this distribution function, the kinetic parameters AM and Ddjf can be calculated. Their values, plotted against the bed depth coordinate AM(y) and Ddjf(y) are presented in Fig. 5. The modeling of the test component distribution function against the bed depth was carried out using the calculated dependence (Eq. (14)) for the quasidiffusional coefficient Dd~dy). An analogous segregation coefficient dependence Ks(y) was obtained by making an assumption that a linear correlation exists between Ks and AM. In the course of the segregation, kinetics modeling, it was determined that Ks = 1.1 × 104(AM). The comparison of the experimental and calculated functions of the test component distribution (large size fraction) at 0.3 m incline length is shown in Fig. 6. Fig. 7 shows the results of computational experiments on segregation kinetics in the gravity flow of mixtures of superphosphate granules moving down the incline. The computational experiment involve no adjustment
2 0,0t5
0.005
0
2
~
6
8
40
~M.IO-~N.~
Fig. 5. Determination of the segregation kinetic parameter AM and quasi-diffusion coefficient Dd~f against the bed depth y of superphosphate granules moving down a rough incline: 1, AM(y); 2, Ddir(y).
V.N. Dolgunin, A.A. Ukolov / Powder Technology 83 (1955) 95-103
102
5. Conclusions
o,0t
0.005 o Fig. 6. Experimental (1) and m o d e l (2) test c o m p o n e n t distribution in a bed of s u p e r p h o s p h a t e granules (a mixture of fractions of 3.0-3.25 m m size, 50%, a n d 3.5-3.75 m m , 50%) down a rough 0.3 m long incline.
dot5
OO . d~ O O .O5 0
2 i
I
I
i
1
Fig. 7. Experimental (1,3) and model (2,4) test c o m p o n e n t cO,) distribution in a layer of s u p e r p h o s p h a t e granules down a rough incline: 1,2, for a mixture of fractions having sizes: 3.0--3.25 m m , 50% and 3.5-3.75 ram, 50%, the incline being 0.15 m long; 3,4, for a mixture of fractions having sizes: 3.0--3.25 mm, 50% and 4.0-4.25 m m , 50%.
A model of nonuniform particle distribution dynamics for rapid gravity flow of a granular material down rough inclined surfaces has been developed. The model takes into account the test component transfer due to convection, quasi-diffusion and segregation. The segregation kinetic parameter is determined for spherical particles using the 'stress-strain' model developed by Ackermann and Shen as a function of the shear velocity and the solid phase concentration and is calculated against the main physical and mechanical properties of the particles and the medium between them. The quasi-diffusion coefficient is expressed by analogy with the gas molecular diffusion coefficient. An experimental and analytical investigation method of the parameters of granular material gravity flow down a rough incline has been developed. The method allows the determination of the velocity profile, as well as the free volume and the test component distribution at the discharge threshold by analyzing the experimental information received at the particle free-fall phase. The analysis is carded out using the intercalations between the shear velocity and free volume and assuming that a formal analogy exists between rapid shear granular material flow and gas molecular dynamics. Particle segregation dynamics in rapid gravity flow of a granular material down a rough incline has been modeled. The results of the modeling as well as the experimental and analytical information have been analyzed and show that these methods are satisfactory. It is possible to use them for the investigation of rapid shear flows of granular materials as well as predicting the course of segregation.
6. List of symbols
of the segregation coefficient Ks performed by selecting the coefficient of relative velocity K. Fig. 7 displays the model test component distribution for a mixture of fractions (3.0--3.25) mm and (3.5-3.75) mm at the incline length of 0.15 m (curve 2). This distribution was predicted, using kinetics parameters which have been obtained earlier from segregation modeling at the 0.3 m incline length, assuming the flow parameters at the incline threshold to be the same in the described cases. The figure also displays the experimental (curve 3) and calculated (curve 4) distribution of the test component at the discharge threshold when another mixture of superphosphate granules is flowing down. The mixture is composed of 3.0-3.25 mm and 4.0-4.25 mm fractions, of equal ratio. In this case, the calculated distribution function is obtained using the segregation coefficient, its value being equivalent to AM with due regard to the coefficient of relative velocity K obtained previously.
d D Ddif
f C(x) H J K
K~ l M P s
t
geometrical parameter concentration of test component of granular mixture diameter of particles diameter of test particle coefficient of diffusion frequency of collision material distribution function along a horizontal coordinate axis xl vertical distance flux of particles coefficient of relative rate of segregation segregation coefficient distance between the layers moment of forces acting on the particle analogue of hydrostatic pressure mean distance between particles time
I~.N. Dolgunin, A.A. Ukolov / Powder Technology 83 (1995) 95-103 u
V'
(x,y,xl)
mean velocity in x coordinate direction fluctuating component of the particle velocity Cartesian coordinate, Fig. 3
Greek letters ot
A E
~9 /(
p*
q,
bed inclination angle segregation driving force specific free volume bed dilatation granular medium temperature coefficient average length of free path of a particle solid particle distribution function coefficient
References
[1] R.A. Bagnold, proc. P~ Soc. London, Ser. A 225 (1954) 49--63. [2] V.F. Pershin, Teor. Osn. Khim. TekhnoL, 20 (1986) 508.
103
[3] D.J. Stephens and J. Bridgwater, Powder Technol., 21 (1978) 29. [4] V.N. Dolgunin, A.A. Ukolov and P.V. Klassen, Teor. Osn. Khira. Tekhnol., 26 (1992) 707. [5] H.H. Shen and N.L. Ackermann, Div. Eng. Mech., 108 (1982) 748. [6] J.H. Ferziger and H.G. Kaper, Mathematical Theory of Transport Processes in Gases, North-Holland, Amsterdam, 1972. [7] M.A. Goodman and S.C. Cowin, J. Fluid Mech., 45 (1971) 321. [8] S.B. Savage, ./. Fluid Mech., 92 (1979) 53. [9] K.I. Kanatani, Int. Z Eng. Sci., 17 (1979) 419. [10] S. Ogawa and A. Umemura, Z Appl. Math. Phys., 31 (1980) 482. [11] J.T. Jenkins and S.B. Savage, J. Fluid Mech., 130 (1983) 187. [12] K. Hutter and T. Sheiwiller, in J.T. Jenkins and M. Satake (eds.), Mechanics o f Granular Materials, Elsevier, Amsterdam, 1983, p. 283. [13] C.S. Campbell and C.E. Brermen, in J.T. Jenkins and M. Satake (eds.), Mechanics o f Granular Materials, Elsevier, Amsterdam, 1983, p. 313. [14] N.L. Aekermann and H.H. Shen, in J.T. Jenkins and M. Satake (eds.), Mechanics o f Granular Materials, Elsevier, Amsterdam, 1983, p. 295. [15] S.B. Savage, in J.T. Jenkins and M. Satake (eds.), Mechanics of Granular Materials, Elsevier, Amsterdam, 1983, p. 261. [16] D.A. Augenstein and R. Hogg, Powder TechnoL, 19 (1978) 205.
Powder Technology 83 (1995) 95-103
Segregation modeling of particle rapid gravity flow V.N. Dolgunin, A.A. Ukolov Department of Machines and Apparatus, State Technical University, Tarnbov, Russia
Received 18 October 1991; revised 1 December 1994
Abstract
A model describing the size and density separation of granular materials undergoing gravity flow down a rough inclined chute is analyzed in the present paper. A dynamic model is developed on the basis of an equivalent mass transfer equation taking into account the mass transfer due to convection, quasi-diffusion and segregation. Segregation and quasi-diffusion kinetic parameters are developed from the gravity flow microstructural analysis based on Ackermann and Shen's [14] rheological model. An experimental and analytical method of investigation of the parameters of the granular material rapid gravity flow down a rough inclined chute is being developed. It is based on the analysis of the free fall phase of the particles that starts after the particles have been sent down the chute. The analogy between the patterns of rapid shear flow and molecular gas dynamics of the particles is used. The results of the particles of segregation modeling of granular material gravity flow down an inclined chute are compared with experimental data. Keyword$: Granular materials; Gravity shear flow; Mass transfer; Convection; Diffusion; Segregation
1. I n t r o d u c t i o n
The separation of solid particles in a mixture of granular materials flow has been well known for a long time. The essence of this phenomenon is that in the course of the flow, the nonuniformity of the mixture in some parts of the flow increases owing to separation and localization of particles which display similarity according to certain criterion. In its physical sense, this phenomenon is the inverse of mixing and is often called segregation. Segregation occurs in the course of many natural phenomena such as snow avalanches, rock falls, mudflows, subaqueous grain flows as well as during some technological processes connected with granular materials processing (mixing, granulation, screening, etc.). It also takes place during transportation of suspensions and loading and unloading operations. In the majority of these cases the flow of granular media can be referred to as rapid shear flow [1]. Rapid shear flow investigation is a field of granular media mechanics. Such flow is often called grain-inertia flow as, internally, momentum is carried by the inertia of the particles and exchanged during interparticle collisions. Segregation in granular shear flows has been scantily explored. Due to recent development in granulation techniques as well as a wider use of granular materials, 0032-5910/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSD1 0032-5910(94)02954-M
powders and composite materials, a great deal of attention has been paid to the segregation process study. However, so far no constitutive relationship between the process kinetics and the dispersed flow physical, mechanical and kinematic characteristics have been established. The existing models of segregation kinetics are either of speculative nature, for instance [1], or they use the body of mathematical statistics and probability theory [2], but do not reflect the physical nature of the process, accounting for their limited application. Besides, in the general case they fail to predict the direction in which the segregation proceeds. Among the existing physical models of the segregation process the percolation model seems to be the most widely used. It describes the segregation by a simple screening of smaller and denser particles in the direction of the gravity force. However, it is obvious that the percolation model cannot predict the direction of particle displacement in a number of cases, for instance, when they differ simultaneously both in density and in size and the bigger ones have a greater density. As a result of Bridgwater's experiments with an annular shear cell [3] it was found out that the segregation direction may be different even when particles vary only in size. A similar behaviour was noticed by the authors of the present paper for gravity flow down a rough inclined chute, the particles differing only in size or only in density (see Fig. 4 of the present paper).
V.N. Dolgunin, A.A. Ukolov / Powder Technology 83 (1955) 95-103
96
This paper presents a distribution dynamic modeling of granular materials for a particular but widespread case of a rapid shear motion. The modeling is based on a general equation of species transfer, taking into account convection transfer, quasi-diffusional mixing and particle segregation. The segregation flow is described using the physical model of particles hydromechanical separation [4]. The model developed earlier represented the shear flow as a number of elementary layers situated one above another (Fig. 1). To develop the physical segregation model we used the microstructural and continuum approaches described in [5]. These approaches are based on rapid granular flow theory. The microstructural approach allows us to determine the system geometrical parameters for a steady shear flow as well as to set up the equations for the determination of forces acting on the test particle when it moves towards the neighbouring layer (Fig. 2). The distance between the layers of diameter d particles is determined as follows l = bd
= [~-/6(1 - E)]°33d
(1)
and the mean distance between particles is described by the equation s = (b/bo -
1)d
(2)
where • is the specific free volume (SFV) of the layer, that is the ratio of the void volume to the layer volume; bo =b(eo), where Eo= 0.2595. 8d
Fig. 1. Determination of shear flow parameters.
Fig. 2. The formation of a support contact with an instantaneous axis of rotation.
A granular medium consisting of rough inelastic spheres of diameter d is considered. A test particle B of diameter D is placed in the medium. The analysis of non-uniform particle interaction was carried out with due regard for the impact momenta, frictional and gravitational forces, the force of rotation inertia being neglected. Equations for the moments of these forces are set up, relative to the particle instantaneous axis of rotation passing through check points O' O" (Fig.
2). The physical model is based on the assumption that the direction and velocity displacement of a spherical particle are determined according to the value and direction of the excess moment of forces acting on the test particle when it interacts with the other particles of the medium which is assumed to be of uniform composition [4]. The sum of the moments of impact momenta, frictional and gravity forces acting on the test particle relative to the instantaneous axis of rotation equals: M= Mc + Mv + MG
(3)
It is assumed [4] that the absolute value M conventionally determines the velocity of the test particle in the y-direction as a function of the most important physico-mechanical and geometrical characteristics of the flow and its components, that is particle size and density, friction and restitution coefficients, the layer shear rate and its specific free volume.
2. Segregation kinetics and dynamics of the distribution of solid particles in gravity flow down a rough chute
To develop a segregation model it is worthwhile to analyze a granular shear flow, as most flows induce some sort of shearing motion. The segregation dynamics are determined by the relative magnitude of at least two opposite flows, one of which impedes particles separation while the other one impedes their ideal mixing. The first flow is entropic in the broad sense of the term and may be modeled as a quasi-diffusional process. The other flow is formed due to hydromechanical interraction of nonuniform particles. During guided shear a conventional flow of the mixture components will also take place. It should also be taken into account during modeling of the distribution dynamics of particles. Using these parameters, determination of the functional dependence of one of the fractions of granular particles mixture concentration, c =f(t,xy,z) on time and flow coordinate, we may use the general equation of substance transfer:
V.N. Dolgunin, AM. Ukolov / Powder Technology 83 (1995) 95-103
ac Ot + div(u, c) = -
div(j~)
(4)
where Jc is the component flux at the point xy,z. Flux j¢ is the sum of two fluxes differing in nature and direction: j~=jdi~+js
(5)
Flux Js appears in the course of the segregation process, flux Jd~f, due to the medium quasi-diffusional mixing. In rapid shear flow [5] behaviour patterns which are determined basically by the presence of the fluctuating component V' of the velocity vector of a separate particle, quasi-diffusional mixing takes place owing to the accidental special distribution of the vector V'. In this connection one can trace an analogy between the process of mixing and diffusion in mixtures of dense gases, from which follows [6]: ~Jdltl= -Ddlf" grad c
(6)
For a quantitative estimation of flux is, it is necessary to formulate the segregation law. The segregation kinetics are determined on the basis of general kinetic behaviour patterns of the chemical technology: the speed of the process is directly proportional to the driving force and inversely proportional to the resistance: M ~=KsA
(7)
where M is the quantity of the mixture component transferred through surface F in time t, Ks is the segregation coefficient, h is the segregation driving force. The parameters of the segregation kinetics, Eq. (7) are expressed on the basis of the postulate that in similar conditions of shear flow, (the layer SFV, shear rate) the segregation flux is directly proportional to the degree of nonuniformity of the granular medium. In a granular medium there are two major aspects of nonuniformity: (1) nonuniformity of the properties of the particle (size, density, shape, roughness, elasticity, etc.); and (2) nonuniformity of the medium caused by adding one component to another. We have taken into account the fact that the degree of particle nonuniformity, with regard to the conditions of their interaction, determines the velocity of the particle displacement, that is the rate of the process. At the same rate, the segregation flux is proportional to the granular medium nonuniformity caused by adding one component to another. It is obvious that the segregation driving force is the continuous function of concentration c and A(0)=A(1)=0. Taking into account general kinetic behaviour patterns of technological processes, the driving force in the area of low concentration is considered
97
to be proportional to the concentration, that is A =c, and when the concentration approaches 1, A = l - c . Since the segregation driving force should also be a measure,of the nonuniformity of the medium, it can be reduced to the variance: A =C(1 - c )
(8)
Finally the flux of particle segregation in the granular medium is presented as follows:
lLl=Ks'c.(a-c)
(9)
In the present paper, the model of the segregation process during gravitational shear flow of granular materials is developed, since such a flow in the most widespread type of granular media movements, which is accompanied by an intensive particles separation. Let us consider a disperse medium which is a mixture of solid spherical particles of diameters d and D. The particles move along a rough inclined chute owing to gravitational force. The degree of roughness is half the diameter of the largest particles (Fig. 1). Let us also assume that the medium movement is two-dimensional, that is the flow parameters are invariable in the direction of the chute inclination as well in the direction perpendicular to the chute surface. Let us mark the slope of this surface relative to the horizon as a. In this case, to ensure the accepted condition concerning the steady flow, the value of a should be close to the value of the angle of repose [2]. Neglecting the particle quasi-diffusion in the direction of the chute and expressing its conventional flow in this direction through the average velocity of the medium translational motion U(y), from Eqs. (4), (6) and (9), we get the equation of component distribution dynamics c(t;cy) for a two-dimensional steady gravity flow:
ac_ at
a(Uc) + a-y ] [D ax
a~_Ks.c. ac
(l-c) (10)
where c is the test component concentration, t time, Dd~ the coefficient of particle quasi-diffusional mixing and Ks the segregation coefficient. The initial conditions are set by the equation: c(0, x, y) =f0')
(11)
Boundary conditions are formulated assuming that there is no flux of particles through the bed boundaries:
Dd~,"~,, -Ks'c-O-c)
=0
(12)
y~0, y~h
where h is the bed height. In accordance with the conditions adopted for the model the quasi-diffusional mixing during a rapid shear flow will be conditioned by an accidental character of special distribution of the fluctuating component V' of
V.N. Dolgunin, AM. Ukolov / Powder Technology 83 (1955) 95-103
98
the individual particle velocity vector. On the basis of this analogy and using the gas kinetic theory [6] we get: 1
O,m = ~ A. V'
(13)
where A is the average length of free path of a particle. Taking Eq. (2) into account, we formulate Eq. (13) in the following way: 1 Ddlf = ~ s. V'
(14)
where V' = 2f- s. Frequency f in Eq. (14) is calculated according to Shen and Ackermann [5], taking into account the changeable conditions of collision of particles considered to be uniform [4]. Eq. (10), for a general case of gravity flow, turns out to be greatly nonlinear, which can be indirectly derived from Bridgwater's investigation [3] and can be proved on the basis of the data given later (see Fig. 4). A complicated form of the function c(t,r~v) testifies to the existence of a reverse direction of the flux depending on the characteristics of the flow heterogeneity. It will be shown further that nonlinearity of Eq. (10) is connected not only with a great heterogeneity of the flow but also with its large influence on kinetic coefficients Ddif and Ks. In a general case the segregation rate depends on a great number of physico-mechanical and geometrical parameters of the particles (size, density, surface roughness, adhesion properties, shape, elasticity, etc.) as well as on the flow structural and kinematic characteristics. Obviously, it is not possible to determine definitely the process rate coefficient for such a case. Also some difficulties arise in the course of the experiment. These are the factors impeding the establishment of the theoretical basis for segregation with the use of general kinetic behaviour patterns of technological processes. To determine the values of the segregation coefficient we have used the model of process mechanism for spherical particles [4]. In accordance with this postulate, the process rate is proportional to the degree of nonuniformity of the granular components. Since the value of the resultant of the moment of forces (Eq. (3)) acting on the test particle determines the tendency of its displacement in accordance with the main properties of particles, inter-particle medium and flow parameters, we use this moment to express the degree of nonuniformity of components when they interact with each other. Taking into account the fact that in a bed of uniform particles there is an equal possibility for them to move in opposite directions, we can express the parameter of nonuniformity as the difference in the moments of forces AM. One of the moments, M is calculated in
accordance with Eq. (3) for nonuniform particles. The other one Mo, is calculated using the same dependence for the case of uniform particles, when the flow conditions being the same in both cases: AM=M-Mo
(15)
When the value of AM is negative, the test particle moves to the lower layer, when the value is positive, the particles moves to the upper layer. The absolute value of AM determines conventionally the rate of spherical particle segregation, depending on the most important characteristics of the medium and the flow, such as the size and density, the coefficients of friction and restitution at collision, the flow shear rate and SFV. The segregation coefficient Ks is determined as a supposedly linear function of parameter AM: Ks =KAM
(16)
where K is the rate coefficient which is equal to the average velocity of displacement of the mixture component per a unit of the excess moment AM, which is a measure of nonuniformity of spherical particles during interaction. Kinetic parameters Ddif and K~ used in Eqs. (10) and (12) are calculated in accordance with the function of velocity distribution U(y) and particles concentration P*0'). When solving the problem of segregation kinetics it is necessary to determine these functions. 3. The definition of the velocity profile and distribution of particles in gravity flow
Description of velocity profiles and solid particles distribution in rapid shear flows is given in [7-15]. Taking into account the main statements of the continuous medium mechanics, Goodman and Cowin [7] developed a model of a rapid shear flow of granular materials. Savage [8] adapted the model for particles rapid gravity flow down an incline. Kanatani [9] developed the continuous medium micropolar theory which describes the granular material flow and derived an equation determining the flow rate and medium density interrelationship. Later Ogawa et al. [10], Jenkins and Savage [11] and Savage [15] developed a model of rapid gravity flow based on the microstructural approach. These authors determined the stress tensor by analyzing momentum translation during particle collision and solving the system of equations on the conservation of momentum and fluctuation energy. But the assumptions made by the authors led to errors in the description of flows, these errors being quite apparent in numerical modeling of granular particles flow down a steep incline carried out by Hutter and Sheiwiller (12). Moreover, it was found out [12]
99
V.N. Dolgunin, AM. Ukolov / Powder Technology 83 (1995) 95-103
that the models require an exact statement of boundary conditions which are difficult to determine. The results of direct computer modeling of the spherical particles shear flow (Couette flow) microstructure obtained by Campbell and Brennen [13] are valuable for a more comprehensive understanding of granular materials rapid shear flow behaviour. One of the most important results of their computational experiment is the conclusion that the effective coefficient of friction determined as a shear stress to normal stress ratio depends on the solid phase concentration. Similar results were obtained experimentally by Savage [15]. Ackermann and Shen [14] used geometrical analysis of the shear flow microstructure and came to the conclusion that the gravity flow should be characterized by a considerable lateral mass transfer (quasi-diffusion) which must be taken into consideration for an adequate flow modeling. However, it seems that the authors [14] restrict the possibilities of the approach when they solve the problem of momentum transfer determination during particle collisions. They proceed from the assumption that the collision conditions do not depend on the solid phase concentration. A close consideration of all the papers [5,13,14] permits the conclusion that the assumptions made by the authors about the absence of momentum transfer due to particles quasi-diffusional displacement and the assumption that the effective coefficient of friction does not depend on concentration are very approximate. These models [5,7-11,14,15] predict the existence of a steady gravity flow only within a narrow range of bed inclination angle values, and allow the description of velocity profiles and solid particles distribution in full, especially in thin layers and the flow boundary areas. For all the reasons cited above it seems impossible to use the existing models for an adequate predicting of functions U(y) and P*0')Considerable experimental difficulties complicate the rapid gravity flow modeling of granular materials. It is rightly pointed out in [15] that rapid gravity flows of particles down a steep incline appear to be extremely complicated for an experimental study despite their apparent simplicity. The analysis of the results of such investigations of flows shows [12,15] that the main difficulties arise from a high flow sensitivity to the conditions at the bottom bed boundary as well as on the slightest disturbances. The present paper describes the method of determination of granular particle velocity and concentration profiles according to the bed thickness at the threshold of discharge from the incline by analyzing the free fall phase [16]. The method consists of sending a granular material down a stationary incline and collecting the particles in a tray containing a number of cells (Fig. 3).
%
Y
2 /---
IIIIIIIIIV Fig. 3. Schematic of experimental unit: 1, open channel; 2, tray; 3, cells; 4, slide-value; 5, dosage device.
In accordance with the method developed, the experimental data include the bed depth at the discharge threshold h in a steady flow, time t, the material distribution function G(Xl) along a certain horizontal coordinate axis xl located at a vertical distance H from the discharge threshold and the bed inclination angle a (Fig. 3). Assuming that particles scatter during their free fall, does not considerably influence the function G(xl) and taking into account the experimental data and the laws of a free failing body and mass conservation we obtain: x ~ - y sin a [lJ I = cos a [ ( H + y . c o s a - ( x 1 - y . s i n a).tg a~/2 2/g)] 1/2
Iol-p*(v)-- a(zl)
(17) (18)
where p*(y) is the solid particle distribution function. Eqs. (17) and (18) indirectly determine the unknown functions P*0') and U(y). Additional data is needed to solve equations which may be presented as an interrelationship function p*(y)=f(U). Many authors (for example, [11] and [13]) have pointed out that rapid shear flows are analogous to molecular gas dynamics. This allowed some of them [11] to apply a well-elaborated kinetic gas theory for solving the problem of granular materials flow dynamics. Let us determine the interrelationship function p*0')=f(U) using this analogy in the form of a wellknown law of dense gas state which establishes the ratio between molar volume, pressure and temperature. Adopting the following corresponding parameters of the granular medium: the shear flow dilatation ~ the hydrostatic pressure P analogue; and the granular material temperature O, [13] we can define their interrelationship in the following way: P~ = 0,9
(19)
100
V..N. Dolgunin, A.A. Ukolov / Powder Technology 83 (1955) 95-103
where ~b is a coefficient which depends on the physical and mechanical properties of the particles. The granular medium temperature is expressed by the following equation: 7rd 3
-]5-.o.(v") 2
(20)
where V' = 2fs. Taking [4] into consideration we obtain:
(V')Z=(2f.s)2=K(dU~ 2 \dy]
(21)
The granular medium pressure is expressed in the following integral form: h
P(Y) = I P*(Y)g cos a dy
(22)
h--y
We assume the moving bed specific free volume increment which takes place due to the bed dilatation to be the gas volume molar analogue: E - - Eo
= -1-E
(23)
where Eo is the SFV of the dense particles packing. Expressions (17), (18) and (19) with due regard to (20)-(23) form a closed system of equations, relative to the functions U(y), y(xl), e(y) and P(y), where y(xl) is the coordinate of the particles in bed, determined by the coordinate of the particles after falling, xa. The boundary condition of this equation system is formulated in accordance with the condition of adhesion at the flow bottom boundary. In this condition the particles at the bottom will have y = 0, U = 0 and xl is determined by 0 as a result of the vertical falling of the particles, that is why
U(O)=O
(24)
y(0)=0
(25)
The solution to this system was obtained by using successive approximations, the first approximation being 0*(3') = p* = constant. The proposed analytical and experimental method of granular materials rapid gravity flows creates favourable conditions for the segregation phenomenon investigation by analyzing the test component distribution in the flow of falling particles.
poly(propylene) grains, triple superphosphate and silica gel as well as their mixtures. The investigations were carried out in conditions similar to steady rapid gravity flow. These conditions were established by fixing the incline (its roughness being half the diameter of the biggest particle in the mixture) at an angle to the horizon equal to the angle of repose of the material. The desired roughness of the incline was achieved by fixing a perforated metal plate above the smooth base at a distance of half the diameter of the biggest particles. The dimension of the holes in the plate was equal to 2- to 3-times the diameter of the biggest particles in the mixture and the free cross-sectional area was 70% of area plate. During the experimental determination of velocity profiles at the threshold of discharge at various lengths of sending bed, it was established that the flow becomes steady at 0.15 m. Velocity profiles at a bed length of 0.15 and 0.6 m differ by about 5%. The particle bed depth at the threshold of discharge was estimated by a visual method. Fig. 4 shows (1) velocity profile; (2) the specific free volume; and (3) the test component distribution functions relative to the depth y of a mixture of rusty steel balls rolling down a rough incline, of length 0.45 m. The mixture consists of balls of diameter 6.6 mm and 7.0 mm, the latter diameter constituting 42% of the total mass. In accordance with the classification of gravity flow conditions proposed by Savage [15], the observed velocity profile is similar in its main features to a sliding flow which takes place when the rough incline slope approximates the angle of repose of the granular material. The function of free volume distribution according to the bed depth indicates that the maximum concentration of particles is observed in the middle of the bed.
•
o
x
0.0~
•
02
o
~'
I
o.'o
d
~.'5 v.~.~-,
0.0t5 001 •
t
0.005
4. Segregation modeling during particle gravity flow down an incline During the investigation of the segregation phenomenon, binary mixtures of granular materials differing in size were used, for instance: steel balls,
d3
o.'6
Fig. 4. VelocRy profile U(v) specific tree volume ~(y) and test component particlescO,) distributionprofilesduring steel ball gravity flow down a rough incline: I, U(y); 2, e(y); 3, c(y).
V.N. Dolgunin, AM. Ukolov / Powder Technology 83 (1995) 95-103
As the free surface is approached, the particle concentration falls almost to zero. In the bed base direction the concentration drops to a certain value and increases again in the vicinity of the incline surface. The observed pattern of particle distribution with maximum concentration in the middle of the layer was described in [12,15,16] and explained by particles slipping at the base and forming a rarefied cloud at the free surface. The distribution function obtained differs from that previously observed owing to the existence of a second extremum in the layer adjacent to the bed base where the minimum flow density is observed. The authors relate the availability of such an extremum to the specific boundary condition of sticking to the rough incline. Under such a condition, the particles of the bottom layer are detained on the incline surface and the area of the most considerable shear is situated at a certain distance from the incline, this distance approximating to the mean diameter of the particles. It is noteworthy that the free volume distribution function and the function of the test component concentration correlate and can be considered to be mirror images of one another. Thus, their analysis is of particular importance. Subjectivism in observing the correlation in the process of experiments is practically excluded as the parameters analyzed are determined independently. Randomness in the observed correlation is eliminated in the course of a great number of experimental investigations of various mixtures of granular materials. The fact that singular points coincide can be explained in the following way. The particles in contact with the surface of the incline possess a relatively small component of the oscillating displacement velocity which slows down their transfer to the upper layer. Thus, the test component concentration in the vicinity of the incline surface differs very little from the mean concentration of the mixture. Farther on from the incline surface, the shear velocity increases, thus leading to the formation of a rarefied interlayer. The particle concentration increases above this interlayer due to the decrease in the shear velocity, while in the middle the particle concentration increases and is maximum. In the rarefied interlayer and above it, particle agitation is much higher than at the incline surface. At large enough shear rates, large particles, being stress concentrators, move towards the layer-free surface. Thus one can observe a coincidence along the extreme values coordinate of the free volume distribution and test component function. The area in the vicinity of the layer-free surface is referred to as the particle cloud [15]. When there is nothing to restrict the bed volume, the particles of this area lose a certain arrangement which is typical of the central part of the flow. Owing to chaotic collisions in the rarefied medium, the particles move and jump above
101
the bed surface. In the course of the exchange of particle impact momenta the smallest and lightest ones acquire a higher speed. As a result they jump higher above the surface bed. In contrast, the bigger and heavier particles in the area, with a relatively low velocity gradient, move towards the bed bottom. As a result of such particle redistribution, the test component distribution function has the second extremum in the upper part of the flow nucleus. To verify the adequacy of the proposed segregation model, we compared the component distribution obtained experimentally from a mixture of superphosphate granules flowing down a rough incline with those calculated from Eqs. (10)-(12) and the kinetic parameters K~ and Ddi f from Eqs. (14) and (16). The initial mixture contained an equal mass parts of 3.0-3.25 mm and 3.5-3.75 mm particle size fractions. The experimental distribution functions U(y) and E(y) are obtained using the proposed experimental and analytical method for investigating the parameters of the rapid gravity flow at a 0.3 m incline length, On the basis of this distribution function, the kinetic parameters AM and Ddjf can be calculated. Their values, plotted against the bed depth coordinate AM(y) and Ddjf(y) are presented in Fig. 5. The modeling of the test component distribution function against the bed depth was carried out using the calculated dependence (Eq. (14)) for the quasidiffusional coefficient Dd~dy). An analogous segregation coefficient dependence Ks(y) was obtained by making an assumption that a linear correlation exists between Ks and AM. In the course of the segregation, kinetics modeling, it was determined that Ks = 1.1 × 104(AM). The comparison of the experimental and calculated functions of the test component distribution (large size fraction) at 0.3 m incline length is shown in Fig. 6. Fig. 7 shows the results of computational experiments on segregation kinetics in the gravity flow of mixtures of superphosphate granules moving down the incline. The computational experiment involve no adjustment
2 0,0t5
0.005
0
2
~
6
8
40
~M.IO-~N.~
Fig. 5. Determination of the segregation kinetic parameter AM and quasi-diffusion coefficient Dd~f against the bed depth y of superphosphate granules moving down a rough incline: 1, AM(y); 2, Ddir(y).
V.N. Dolgunin, A.A. Ukolov / Powder Technology 83 (1955) 95-103
102
5. Conclusions
o,0t
0.005 o Fig. 6. Experimental (1) and m o d e l (2) test c o m p o n e n t distribution in a bed of s u p e r p h o s p h a t e granules (a mixture of fractions of 3.0-3.25 m m size, 50%, a n d 3.5-3.75 m m , 50%) down a rough 0.3 m long incline.
dot5
OO . d~ O O .O5 0
2 i
I
I
i
1
Fig. 7. Experimental (1,3) and model (2,4) test c o m p o n e n t cO,) distribution in a layer of s u p e r p h o s p h a t e granules down a rough incline: 1,2, for a mixture of fractions having sizes: 3.0--3.25 m m , 50% and 3.5-3.75 ram, 50%, the incline being 0.15 m long; 3,4, for a mixture of fractions having sizes: 3.0--3.25 mm, 50% and 4.0-4.25 m m , 50%.
A model of nonuniform particle distribution dynamics for rapid gravity flow of a granular material down rough inclined surfaces has been developed. The model takes into account the test component transfer due to convection, quasi-diffusion and segregation. The segregation kinetic parameter is determined for spherical particles using the 'stress-strain' model developed by Ackermann and Shen as a function of the shear velocity and the solid phase concentration and is calculated against the main physical and mechanical properties of the particles and the medium between them. The quasi-diffusion coefficient is expressed by analogy with the gas molecular diffusion coefficient. An experimental and analytical investigation method of the parameters of granular material gravity flow down a rough incline has been developed. The method allows the determination of the velocity profile, as well as the free volume and the test component distribution at the discharge threshold by analyzing the experimental information received at the particle free-fall phase. The analysis is carded out using the intercalations between the shear velocity and free volume and assuming that a formal analogy exists between rapid shear granular material flow and gas molecular dynamics. Particle segregation dynamics in rapid gravity flow of a granular material down a rough incline has been modeled. The results of the modeling as well as the experimental and analytical information have been analyzed and show that these methods are satisfactory. It is possible to use them for the investigation of rapid shear flows of granular materials as well as predicting the course of segregation.
6. List of symbols
of the segregation coefficient Ks performed by selecting the coefficient of relative velocity K. Fig. 7 displays the model test component distribution for a mixture of fractions (3.0--3.25) mm and (3.5-3.75) mm at the incline length of 0.15 m (curve 2). This distribution was predicted, using kinetics parameters which have been obtained earlier from segregation modeling at the 0.3 m incline length, assuming the flow parameters at the incline threshold to be the same in the described cases. The figure also displays the experimental (curve 3) and calculated (curve 4) distribution of the test component at the discharge threshold when another mixture of superphosphate granules is flowing down. The mixture is composed of 3.0-3.25 mm and 4.0-4.25 mm fractions, of equal ratio. In this case, the calculated distribution function is obtained using the segregation coefficient, its value being equivalent to AM with due regard to the coefficient of relative velocity K obtained previously.
d D Ddif
f C(x) H J K
K~ l M P s
t
geometrical parameter concentration of test component of granular mixture diameter of particles diameter of test particle coefficient of diffusion frequency of collision material distribution function along a horizontal coordinate axis xl vertical distance flux of particles coefficient of relative rate of segregation segregation coefficient distance between the layers moment of forces acting on the particle analogue of hydrostatic pressure mean distance between particles time
I~.N. Dolgunin, A.A. Ukolov / Powder Technology 83 (1995) 95-103 u
V'
(x,y,xl)
mean velocity in x coordinate direction fluctuating component of the particle velocity Cartesian coordinate, Fig. 3
Greek letters ot
A E
~9 /(
p*
q,
bed inclination angle segregation driving force specific free volume bed dilatation granular medium temperature coefficient average length of free path of a particle solid particle distribution function coefficient
References
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103
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