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Archives of Mining Sciences 50, Issue 3 (2005) 289–305
289
MARIAN BROŻEK*, ANNA TURNO*
THE PHYSICAL MODEL OF PARTITION FUNCTION OF THE ENRICHMENT PROCESS
IN A HEAVY LIQUID
FIZYCZNY MODEL FUNKCJI ROZDZIAŁU PROCESU WZBOGACANIA W CIECZY CIĘŻKIEJ
The enrichment in a heavy suspension liquid is carried out in the conditions of turbulent motion.
In order to provide a constant liquid density in the entire volume of the separator working chamber the
medium should be kept in a continuous movement which prevents the medium solids from sedimentation.
Such a regime of separation leads to the phenomenon of particles scattering into inappropriate products.
The statistical character of the separation process is formulated numerically by the partition function, determining the probability when a particle of a given density finds its way to a given separation product.
The mathematical form of the partition function was derived basing upon the laws of statistical
physics and taking into account the external and internal forces acting upon the particle. It was assumed
that the distribution of potential energy along the vertical axis of the chamber separator is in agreement
with Boltzmann’s distribution.
The stable state of the system of particles of the lowest energy is constituted by the state in which
all the particles of densities higher than the liquid density are placed on the chamber bottom whereas the
particles of densities lower than the liquid density constitute a floating product. Any other state is the state
of higher energy. The particles are transported into this state as a result of the liquid turbulent motion.
The energy state higher than the basic state is connected with the phenomenon of particles dispersion.
The return of particles to the basic state is enhanced by the gravity force and liquid uplift force whereas
this return is hindered by inter-particle actions whose value is proportional to the volume concentration
of grains in the separator chamber.
Such assumptions result in obtaining the expressions for the partition function and ecart probable in
which the particle geometrical properties occur openly as well as the volume concentration of particles
in the separator chamber, depending upon process efficiency. The mathematical form of the separation
function was verified empirically in an industrial experiment.
Keywords: partition function, suspension liquids, physical model, ecart probable, particles energy
distribution
*
ZAKŁAD PRZERÓBKI KOPALIN, OCHRONY ŚRODOWISKA I UTYLIZACJI ODPADÓW, AKADEMIA GÓRNICZO-HUTNICZA, AL. MICKIEWICZA 30, 30-059 KRAKÓW, POLAND
290
Wzbogacanie w cieczy ciężkiej zawiesinowej przebiega w warunkach ruchu turbulentnego. Chcąc
zapewnić stałą gęstość cieczy w całej objętości komory roboczej wzbogacalnika należy utrzymywać
ośrodek w ciągłym ruchu przeciwdziałającym sedymentacji obciążnika. Taki reżim rozdziału prowadzi
do zjawiska rozproszenia ziaren do niewłaściwych sobie produktów. W związku z tym proces rozdziału
podlega prawom rachunku prawdopodobieństwa. Statystyczny charakter procesu rozdziału liczbowo
ujmuje funkcja rozdziału, określająca prawdopodobieństwo trafienia ziarna o określonej gęstości do
danego produktu rozdziału.
Matematyczną postać funkcji rozdziału wyprowadzono w oparciu o prawa fizyki statystycznej
z uwzględnieniem sił zewnętrznych i wewnętrznych działających na ziarno. Założono, że rozkład energii
potencjalnej wzdłuż osi pionowej wzbogacalnika komorowego jest zgodny z rozkładem Boltzmanna.
Stanem trwałym układu ziaren o najniższej energii jest stan w którym wszystkie ziarna o gęstości większej
od gęstości cieczy są na dnie komory natomiast ziarna o gęstości mniejszej od gęstości cieczy stanowią
produkt pływający. Każdy inny stan jest stanem o wyższej energii. Do tego stanu przenoszone są ziarna
na skutek turbulentnego ruchu cieczy. Stan energetyczny wyższy od stanu podstawowego związany jest
ze zjawiskiem rozproszenia ziarn. Powrotowi ziaren do stanu podstawowego sprzyjają siła ciężkości i siła
wyporu cieczy, natomiast utrudniają ten powrót oddziaływania wewnętrzne pomiędzy ziarnami, których
wartość jest proporcjonalna do objętościowej koncentracji ziaren w komorze wzbogacalnika.
Konsekwencją takich założeń jest uzyskanie wyrażeń na funkcję rozdziału i rozproszenie prawdopodobne w których w sposób jawny występują własności geometryczne ziarna oraz objętościowa koncentracja
ziaren w komorze wzbogacalnika, zależna od wydajności procesu. Matematyczna postać funkcji rozdziału
została zweryfikowana empirycznie w eksperymencie przemysłowym.
Słowa kluczowe: funkcja rozdziału, ciecze zawiesinowe, model fizyczny, rozproszenie prawdopodobne,
rozkład energii ziaren
1. Introduction
The enrichment process in dense medium occurs in turbulent conditions. To assure
the constant liquid density in the whole capacity of separator working chamber, it is
recommended to keep medium in a movement counteracting of solid phase sedimentation. Such separation rigor leads to particles scattering phenomenon and contamination
of separation products by the strange fractions. Because of this, the separation process
undergoes to the probability theory laws. Statistical character of the separation process
is shown numerically by the partition function and its graphical representation – the
partition curve (Tromp curve), which determines the probability of certain density grain
occurring in a given separation product. The partition function characterizes the separating properties of washers during enrichment process.
The partition function for tailings is determined by the following formula (Sztaba,
1956a, 1956b):
(1)
where: γo – yield of tailings, fo(ρ) – frequency distribution function of particles density
in tailings, f(ρ) – frequency distribution function of particles density in the feed.
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In theory of separation processes, the partition function is the kernel of integral
transform in the following integral equation (Tumidajski, 1997):
(2)
where: ρc – density of suspension liquid, where separation occurred.
By a such mathematical description of separation process, the product p(ρ,ρc)dρc will
indicate the probability of event of particle of density ρ occurring in separation product
obtained in the liquids of densities ρc and ρc + dρc. However g(ρc) is the frequency distribution function of separation products yield. So yield of the heavy product γo will
be equal to:
(3)
Knowing the dependence of particle density on valuable component or ash content, it
is possible to forecast separation results in base of equation (3). Therefore, the knowledge
of partition function, which is in fact the kernel of integral transform, is very important
in practice of separation.
The separation efficiency in industrial practice is measured by ecart probable or
imperfection. In turbulent motion conditions, the value of ecart probable depends on
factors related to geometric properties of feed particles, device characteristics and
operating parameters of the process. The variables, which characterize the geometric
properties of particle are its size and shape. The square average velocity of the turbulent liquid movement in the separator may be taken as variable characterizing device
work conditions. Its value depends on the type of separator. As an operating parameter
characterizing the process, feed rate is used, which influences on volume concentration
of particles in separator chamber.
Many empirical models exist, for which the partition curve is estimated by statistical
distribution functions of random variables as: normal, log-normal, exponential, Weibull,
gamma and beta distribution functions, as well as arc tangs, hyperbolic, parabolic and
power-exponential type functions (Mayer, 1967a,b; Tarjan, 1974; Trawiński, 1976;
Gottfried, 1978; Paszkowska, 1985; Jowett, 1986; Tamilmani & Kapur, 1986; Baguley
and Napier-Munn, 1996; Paul et al., 1998). All of these approaches have the phenomenological character, from which is not possible to determine the functional form of
the ecart probable. In next chapter, the physical model of separation function in dense
separator will be presented. This model allows to determine the influence of the factors
mentioned above on separation efficiency. To simplify of derivation of the mathematical form of partition function, the studies were conducted for dense media separator of
chamber type. This kinds of separators as for example DISA type separators, are being
applicable for coal cleaning in Poland.
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2. The forces acting on particle in separator’s working chamber
The separator’s working chamber is presented schematically on Fig. 1. It is filled with
dense liquid of density ρc to the height of 2h. Next to it, the vertical axis z direction is
marked. The working chamber was divided conventionally into two zones: the bottom
one is the heavy product zone (tailings – in case of coal, concentrate – in case of ores)
and the upper one is the light product zone (concentrate – in case of coal, tailings – in
case of ores).
Fig. 1. Scheme of the separator working chamber
Rys. 1. Schemat komory roboczej wzbogacalnika
In case of static conditions of separation process, the heavy product of density ρ > ρc
would concentrate at the bottom of working chamber and the light product of particle
density ρ < ρc on the liquid surface. These positions, for the particles mentioned above,
are the state of permanent equilibrium. Particles of density equal to the liquid density
(ρ = ρc) would occupy the neutral equilibrium position or in other words would be distributed evenly in whole capacity of working chamber.
However, because of the fact, that the separation process in the dense media separator
occurs in turbulent conditions of liquid movement, the particles suitable for the heavy
product get to the light product zone and vice versa. It results in contamination of separation products by unsuitable particles. This permeation of the particles to unsuitable
products is dependent on the relations between four forces operating on each particle.
These are force due to gravity, buoyancy force, force of hydrodynamic interactions of the
turbulent liquid movement on the particle and force of interactions between particles.
Gravity Fg reduced by buoyancy is equal to:
(4)
293
where: g – acceleration due gravity, ρ – particle density, ρc – dense medium density,
d – sieve particle diameter, k1 – volumetric particle shape factor equal to the ratio of the
particle volume to sphere volume of the sieve particle diameter d,
.
Force of turbulent liquid movement, according to the Newton-Rittinger equation is
expressed by the formula (Sztaba, 1976):
(5)
where: vc2 – square average velocity of turbulent liquid movement; K2 – dynamic (Newton’s) shape factor equal to the ratio of particle drag coefficient to sphere drag coefficient
(Ganser, 1993; Thompson & Clark, 1991).
The force of interactions between particles it is the internal friction force resulting
from collisions between particles. Particles in separator’s working chamber are moving
chaotically, so they change of their momentums in the moment of collision. The interaction force, according to the second Newton’s law is equal to the change of particle
momentum per unit time (Tichonov, 1973):
(6)
where: m – mass of particle, t – average time between collisions, v–p – average velocity
of particle between collisions.
Average time between collisions is equal to:
(7)
where:
– mean free path of particle between collisions,
– number
of all particles per unit volume, c – volume concentration of particles in the separator
working chamber.
Therefore the average time between collisions is expressed by the formula:
(8)
After substitution of equation (8) to equation (6), formula for the internal friction
force is given by:
(9)
Value of the force Fw is dependent on particle density. So the average value of this
force will be equal to:
294
(10)
Two first forces: gravity and buoyancy are separating of the feed particles on two
subsets – heavy (sinking) product and light (floating) product. The next two forces destroy this order working to equalize the concentration. So, these forces are responsible
for the effect of particle dispersion. The value of dispersion depends on mutual relation
between this two groups of forces.
3. The physical model of partition function
Partition function for sinking product will be the topic of consideration in this paragraph. It is possible to perform analogical analysis for floating product. According to
the definition of partition function, partition number for particles of sinking product is
expressed by the following formula:
(11)
where: ns – number of particles of narrow size fraction of density ρ occurring in sinking
product (heavy product zone), nf – number of particles of density ρ occurring in floating
product. Dividing numerator and denominator of equation (11) by ns it results that:
(12)
Farther consideration were based on energetic relations between particles of chosen
density ρ > ρc. As it was mentioned above, the position of particles of sinking product at
the bottom of working chamber it is the state of permanent equilibrium, so than the state
of lowest energy. Each other position is the state of instable equilibrium and it increases
the energy of particle. This increase leads in consequence to the scattering phenomenon
of the particles. The field of hydrodynamic force supplies this energy. The elevation of
particle on higher energy level is followed by the return of this particle on the ground
level. This return is supported by the effective force of gravity (force of gravity reduced
by buoyancy force), however is prevented by force of internal friction. As the effect of
the action of these forces, determines a certain state of equilibrium, from which results
the distribution of particle number of a given density along the z axis (along height of
working chamber). Mathematical form of this distribution function is formulated by
Boltzmann’s particles energy distribution law, which is for this case in the following form:
(13)
295
where: N(z) – number of particles of density ρ on the level z; No – number of particles
of density ρ on the level z = 0.
Boltzmann’s distribution function in the form (13) is analogous to the barometric
formula. Denoting
(14)
it is possible to note distribution function (13) in the following way:
(15)
The total particles number of density ρ occurring in the heavy product zone is equal to:
(16)
Likewise, total number of particles of density ρ occurring in zone of light product
is as follows:
(17)
Substituting equations (16) and (17) into equation (12) gives the following formula:
(18)
After substitution of G from equation (14) into equation (18) the analytical form of
the partition function for dense media enrichment process is given:
(19a)
or
(19b)
where:
296
In the preceding formulae, the density ρr denotes the partition density. The partition
density is the density of such particles, which statistically occurring in half in the heavy
product and in half in the light product. After the model derivation it was assumed that
liquid density is identical as partition density. It results from the fact that in energetic
meaning, the particles of density ρ = ρc occupy the neutral equilibrium position. They
are then distributed evenly in the whole liquid volume and therefore they might occur
with equal probability in both separation products. In the equations (19) model was generalized by substituting ρc by ρr, because in case of dense media separation, not only
liquid’s chaotic turbulent movements occurred, but for liquid constancy assurance, the
liquids rising currents are being applied, which cause the growth of the partition density
above dense media density. The additional hydrodynamic force related to rising motion
is being added to Archimedes buoyancy force, what in the macroscopic sense it is equivalent to apparent increase of the liquid density.
Function
is symmetric towards the straight line ρ = ρr. Fig. 2 presents pic-
torial form of the partition curve, plotted according to the equation (19), for two values
Fig. 2. Separation curve defined by equation (19): v1 > v2, v3 = 0
Rys. 2. Krzywa rozdziału określona wzorem (19) v1 > v2, v3 = 0
of velocity vc2. With decreasing of vc2, the slope of the curve increases in the point ρ = ρr.
If the mean quadratic velocity is equal to zero, then the separation conditions are static.
The mathematical form of the partition function in this case is received by calculating
the following limits of function (19a):
297
for
ρ > ρr
(20a)
for
ρ < ρr
(20b)
for
ρ = ρr
(20c)
So, as the result, the stepped Heaviside’s function is given, which is defined in the
following way (Byron & Fuller, 1975):
H(ρ – ρr) = 1
for
ρ > ρr
(21a)
H(ρ – ρr) = 0
for
ρ < ρr
(21b)
It is then the form of the partition function for ideal enrichment without particle
scattering (curve 3 on Fig. 2).
If mean quadratic velocity is increasing then the limit:
(22)
So, the partition curve is the horizontal line form (T = 1–). In this case no separation
2
occurs.
As presented calculations showed, the partition curve equation (19a) is universal
one. According to it is possible to obtain all particular events, related to dense media
enrichment.
4. Coefficient of separation efficiency
Ecart probable as the measure of separation efficiency is defined in the following
way (Stępiński, 1964):
(23)
where: ρ(T = 0.75) denotes particles density, for which the separation number is equal
0.75. In other words, particles of a such density would occur in the tailing with probability 75%. Analogical interpretation is also for density ρ(T = 0.25).
After calculation of densities mentioned above according to the equation (19), ecart
probable is equal to:
298
(24)
The equation (24) formulates the known experimental facts in analytical way. Ecart
probable is increasing as the medium (in which separation proceeds) turbulence increases
(it concerns vc2 but at the same time vp2), volume concentration of particles in separator chamber increases (increase of feed rate) and diameter of particles being enriched
decreases. Furthermore it is proportional to the partition density.
If the ecart probable will be expressed by the parameter of partition function, the
following equation will be received:
(25)
Substituting the M value from equation (23) to equation (19b), the partition function
is given, with ecart probable and separation density as the parameters:
(26)
The partition function expressed by the equation (26) is two-parametric function
of Ep and ρr parameters. By fitting this function to the empirical dependence the ecart
probable and partition density are calculated.
5. Experimental
Experimental investigations were carried out in Coal Preparation Plant of one of
hard coal mines, on two-product DISA separator at feed rate equal to 26.9 kg/s and
dense medium density 1.58 Mg/m3. The volumetric concentration of particles in working chamber is dependent on feed rate. The experiment scheme is presented on Fig. 3
(Turno, 2004).
After stabilization of the separation process, the samples of feed, concentrate and
tailing were taken. In base of samples of concentrate and tailing, the sieve analysis
was carried out as well as float and sink analyses of every size fraction in zinc chloride
solutions. The data from these analyses was the base to calculate the coordinates of the
partition curves for every size fraction.
The feed rate was calculated by the following equation:
(27)
299
Fig. 3. Scheme of experiment. W – feed rate, O – point of sampling
Rys. 3. Schemat doświadczenia
where: M - mass of the sample taken from the belt section of length L, vt – velocity of
belt movement, equal to 1.27 m/s.
Results of analyses of separation products (concentrate and tailing) are the basic
material to empirical verification of the theoretical model of the partition function.
6. Results and discussion
The results of float and sink analyses of separation products for each size fraction
are presented in Table 1.
As it was written in experiment description, the separation in DISA separator was
conducted at liquid density 1.58 Mg/m3. At precise separation, all particles of density
lower than 1.8 Mg/m3 should occur in the concentrate and particles of bigger density
to the tailing. In this case, the ecart probable is equal to zero. As it is shown in Table 1,
the ecart probable is equal to zero for particles bigger than 50 mm.
For the rest of size fractions, for which ecart probable is higher than zero, the partition curves were drawn in purpose to evaluate this scattering. To the partition numbers
calculated from experiment, the model equations of partition curve were being fitted
according to the equation (26), in which parameters are ecart probable and separation
density.
Figures 4-8 ilustrate the partition curves for each size fractions.
200-125
125-80
80-63
63-50
50-31.5
31.5-16
16-12
12-10
10-6
6-0
Size
[mm]
<1.3
K
O
3.70
0
0.90
0
0.50
0
0
0
2.12
0
3.15
0
1.31
0
0.24
0
0.19
0
0.03
0
1.3-1.4
K
O
15.60
0
4.00
0
4.50
0
3.35
0
8.24
0
17.90 0.04
4.27
0.03
0.73
0.01
0.68
0.01
0.22
0.01
1.4-1.5
K
O
0
0
1.40
0
0.29
0
0.28
0
2.15 0.01
2.90 0.07
0.88
0
0.10 0.01
0.10 0.01
0.02 0.01
Density fraction [Mg/m3]
1.5-1.6
1.6-1.7
K
O
K
O
0
0
0
0
0
0
0
0
0
0
0
1.20
0
0
0
0.33
0.25 0.12 0.14 1.80
0.64 0.14 0.16 1.10
0.16 0.02 0.06 0.08
0.03 0.01 0.01 0.02
0.02 0.02 0.01 0.02
0
0.01 0.01 0.01
1.7-1.8
K
O
0
0
0
0
0
0.33
0
0.36
0
3.70
0.03 2.00
0.01 0.16
0
0.02
0
0.02
0
0.01
Wychody masowe [kg] produktów rozdziału przy koncentracji c = 1,80%
Mass yields [kg] of the separation products at concentration c = 1.80%
1.8-2.0
K
O
0
0
0
0.90
0
1.20
0
0.60
0
4.00
0.01 4.30
0
0.38
0
0.07
0
0.04
0
0.01
K
0
0
0
0
0
0
0
0
0
0
>2.0
O
0
4.70
5.70
3.70
16.60
24.00
3.67
0.72
0.53
0.20
TABLICA 1
TABLE 1
300
301
Fig. 4. Partition curve for size fraction 50-31.5 mm (Ep = 28.9 kg/m3)
Rys. 4. Krzywa rozdziału dla klasy ziarnowej 50-31,5 mm (Ep = 28,9 kg/m3)
Fig. 5. Partition curve for size fraction 31.5-16 mm (Ep = 43.7 kg/m3)
Rys. 5. Krzywa rozdziału dla klasy ziarnowej 31,5-16 mm (Ep = 43,7 kg/m3)
302
Fig. 6. Partition curve for size fraction 16-12 mm (Ep = 57.3 kg/m3)
Rys. 6. Krzywa rozdziału dla klasy ziarnowej 16-12 mm (Ep = 57,3 kg/m3)
Fig. 7. Partition curve for size fraction 12-10 mm (Ep = 58.0 kg/m3)
Rys. 7. Krzywa rozdziału dla klasy ziarnowej 12-10 mm (Ep = 58,0 kg/m3)
303
Fig. 8. Partition curve for size fraction 10-6 mm (Ep = 66.0 kg/m3)
Rys. 8. Krzywa rozdziału dla klasy ziarnowej 10-6 mm (Ep = 66,0 kg/m3)
Model equations of partition curves for particular size fractions are as follows:
• size fraction 50-31,5 mm
(28)
• size fraction 31,5-16 mm
(29)
• size fraction 16-12 mm
(30)
• size fraction 12-10 mm
(31)
304
• size fraction 10-6 mm
(32)
In these equations, both separation densities and ecart probable were given in [Mg/m3].
The solid lines on figures 4-8 present the model dependencies and the circles give the
empirical values. These figures show that the goodness of heuristic model of partition
curve (derived from the statistical thermodynamics equations, with taking into consideration external forces acting on the particle and particle – particle interactions) with
empirical values obtained from the industrial experiment is very high.
The partition density higher than liquid density may occur because of snatching
these particles by bigger ones to the concentrate zone. This is done thanks to existence
of the whirls, which are being created behind big particles, as well as it is the result of
liquid rising currents operations. The additional convection force occurs then, adding
to the liquid buoyancy force, which is equivalent to increase of the liquid density. The
higher partition density for bigger particles is caused by the fact that fine particles are
something like contaminated solids, which rise the liquid density.
7. Final conclusions
1. The heuristic model of partition function, derived from the hypothesis of Boltzmann’s distribution function of particles energy in dense media separator working
chamber, by taking into consideration the character of internal an external forces acting
on particles was verified in industrial experiment. The parameters of this model are ecart
probable and separation density.
2. From the heuristic model of partition function, the equations of partition function
for ideal separation case are being derived when the ecart probable is equal to zero and
for the case of lack of separation. This is done by boundary transitions with quadratic
mean velocity respectively to zero and to infinity, respectively.
3. The ecart probable (calculated from the heuristic model of partition function) is
dependent explicite on the geometric properties of particles and volume concentration
of particles in separator working chamber (dependent on feed rate). Influence of the
geometric properties of particles and the interactions between themselves occurs because
of turbulent liquid motion in the separator working chamber, which is characterized by
square quadratic velocity. In this meaning, the turbulent liquid motion is the main cause
of the existence of particles scattering phenomenon.
4. The geometric properties of particles influence on ecart probable by particle size
and shape coefficients distribution functions. By enrichment of narrow size fraction it
305
is expected that the particles of the same density may occur in separation products with
different probability in dependence on the values of their shape coefficients. Although,
this hypothesis should be verified.
This work was done as part of University of Science and Technology Research Program
No.11.11.100.238
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REVIEW BY: PROF. DR HAB. INŻ. STANISŁAW CIERPISZ, GLIWICE
Received: 20 May 2005