# Structural Model of Mass Transfer in Critical Regimes of Two-Phase Flows

• Eugene Barsky
Chapter
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 93)

## Abstract

Physics of a two-phase flow motion is examined. The notion of distribution coefficient is substantiated. Balance and structural mathematical models of such flows are developed. Distribution coefficient formation is considered in laminar, transient and turbulent flow regimes. Analysis of this parameter is performed. It has allowed us to formulate a mathematical definition of distribution coefficient for the three flow regimes. Structural model adequacy to experimental data is demonstrated. It allows a prognostic estimation of the process of mass distribution of polyfractional mixture of particles in a flow.

## Keywords

Flow structure Distribution coefficient Cross-section geometry Flow rate Velocity profile Reynolds criterion Froude criterion Archimedes criterion Level lines Flow profile Velocity gradient

## 7.1 Validation of the Distribution Coefficient

Any apparatus, even a hollow one, can be conventionally represented as comprising a certain number of stages with a directional mass exchange between them. A cascade classifier comprising stages of the same or different construction gives the simplest idea of the staged character of the process.

The quantity characterizing redistribution of a narrow size class at a single stage can be presented in the form
$$k = \frac{{r_i^* }}{{{r_i}}},$$
(7.1)

where $${r_i}$$ is the initial contents of narrow size-class particles at a certain i-th stage of the apparatus; $$r_i^* -$$ quantity of the same particles passing from the i-th stage to the overlying (i – 1)th stage; K or $$k -$$ distribution coefficient.

A general schematic diagram of particles distribution over the apparatus height at their feeding to the $${i^* }$$-th stage is presented in Fig. 7.1a. In case of one stage, the process pattern is rather simple (Fig. 7.1b).

We take the initial content of particles of the same narrow class as a unity being clearly aware of the fact that it is fed to be classified in a mixture with other particles.

Fractional extraction degree for the entire apparatus is expressed by a function
$${F_f}(x) = \frac{{{r_f}}}{{{r_s}}}{\gamma_s},$$
(7.2)
where $${r_f}$$ is the narrow class quantity in the fine product output; $${r_s} -$$ quantity of the same class in the initial material to be classified; $${\gamma_s} -$$ fine product output.
It was proved that the value of fractional extraction into fine product j is described for any size class by the dependence
$${F_f}({x_j}) = \left| {\begin{array} {ll}{\frac{{1 - {\sigma^{z + 1 - {i^* }}}}}{{1 - {\sigma^{z + 1}}}};} & {k \ne 0.5,} \\{\frac{{z + 1 - {i^* }}}{{z + 1}};} & {k = 0.5,} \\{0;} & \!\!\!\!\!{k = 0,} \\\end{array} } \right.$$
(7.3)
where $$\sigma = \frac{{1 - k}}{k};$$ $${i^* }$$ is the number of the stage of material input into the apparatus.

The dependence (7.3) may serve the basis for designing equilibrium and cascade classifiers, if we manage to establish the dependence of the parameter k on its regime and structural properties of the process.

## 7.2 Physical Meaning of the Distribution Coefficient

When developing a structural model of the distribution coefficient, we make certain assumptions, namely:
1. 1.

Particles are spherical.

2. 2.

Distribution of particles of any narrow size class over the cross-section of the apparatus is uniform due to their intense interaction with each other and with the apparatus walls and internal facilities.

3. 3.

Ascending two-phase flow should be considered as a continuum with elevated density. As established, carrying capacity of dust-laden flow is higher than that of pure medium. It can be conventionally taken into account by increasing the effective flow density. The distribution of local velocities of the solid phase is a function of geometrical characteristics of the channel cross-section. It can be written in a general form as

$${u_r} = w \cdot f\left( {\frac{r}{R}} \right)$$
(7.4)
where $$f\left( {\frac{r}{R}} \right)$$ is a function connected with the cross-section geometry; $$r$$ is a characteristic coordinate of a certain point of the apparatus cross-section; $$R$$ is a characteristic limiting dimension of the apparatus cross-section; $${u_r}$$ is the local velocity of the continuous phase in a point with the coordinate r; $$w$$ is the mean flow velocity. Thus, the dependence (7.4) takes into account the shape of the channel cross-section.
According to the Newton–Rittinger law, the dynamic impact of a flow on an isolated particle is described by the relationship
$${F_r} = \lambda \frac{{\pi {d^2}}}{4}{\rho_n}\frac{{({u_r} - {v_r})}}{2},$$
where $$\lambda$$ is the resistance coefficient of a particle; $$\frac{{\pi {d^2}}}{4}$$ is the midlength section area of a particle; $${\rho_n}$$ is flow density; $${v_r}$$ is an absolute local velocity of a particle; $$({u_r} - {v_r})$$ is the particle velocity with respect to the flow.

The difference of absolute velocities is algebraic. The flow direction is chosen as a positive direction of $${u_r}$$ and $${v_r}$$ velocities.

If we take the total number of particles of a given mono-fraction in the cross-section under study as a unity, the distribution coefficient can be written as $$K = {n}$$, which is a relative number of particles of a specified narrow size class having the absolute velocity above or equal to zero $$(v\, \geqslant \,0)$$.

Considering the equilibrium of an isolated particle at the distance $${r_0}$$ from the axis, we obtain
$$\frac{{\pi {d^3}}}{6}(\rho - {\rho_0}) = \lambda \frac{{\pi {d^2}}}{4}{\rho_0}\frac{{{{({u_r} - {v_r})}^2}}}{2}.$$
(7.5)
Hence,
$${u_r} - {v_r} = w\sqrt {{\frac{4}{{3\lambda }} \cdot B}},$$
(7.6)
where $$B = \frac{{qd(\rho - {\rho_0})}}{{{w^2}{\rho_0}}}$$.
Let us examine the regime of turbulent overflow of a particle characterized by a constant resistance coefficient $$\lambda$$. In this case, the Reynolds criterion for a particle is
$${{\rm Re}_r} = \frac{{({u_r} - {v_r})d{\rho_o}}}{\mu } \,\geqslant 500,$$
where $$\mu$$ is the dynamic viscosity of the medium. Taking (7.6) into account, we obtain
$$\frac{{w\sqrt {{\frac{4}{{3\lambda }} \cdot B}} \cdot d{\rho_0}}}{\mu } \,\geqslant 500.$$
This condition corresponds (at $$\lambda = 0.5$$) to the expression
$$\sqrt {{\frac{8}{3}Ar}} \, \geqslant 500,$$
where Ar is the Archimedes criterion:
$$Ar = \frac{{q{d^2} \cdot \rho \cdot {\rho_0}}}{{{\mu^2}}}.$$

Thus, we can determine the limiting size of particles, above which the overflow of particles is certainly turbulent.

Now we examine some properties of laminar overflow of particles. In this case,
$${{\rm Re}_p} = \frac{{({u_r} - {v_r})d{\rho_0}}}{\mu } \,\leqslant 1.$$
It follows from equilibrium conditions that
$$\frac{{\pi {d^3}}}{6}q\rho = 3\pi \mu ({u_r} - {v_r})d.$$
Taking previous results into account, an expression for this case can be written as
$$Ar = 18{\rm Re}.$$
It is known that at a laminar overflow, the resistance coefficient of particles is
$$\lambda = \frac{{24}}{{{\rm Re} }} = \frac{{24\mu }}{{({u_r} - {v_r})d{\rho_0}}}.$$
Let us single out the following ratio from (7.5):
$$\frac{{{u_r} - {v_r}}}{w} = \sqrt {{\frac{4}{3}B\frac{{({u_r} - {v_r})d{\rho_0}}}{{24\mu }}}},$$
hence,
$$\frac{{{u_r} - {v_r}}}{w} = \frac{1}{{18}}{{\rm Re}_w}B,$$
where $${{\rm Re}_w}$$ is the Reynolds number estimated through the mean flow velocity.
Now we can revert to the relation (7.6). For particles of a narrow class with absolute velocity $${v_r}\; \geqslant\ 0,$$ we can write:
$${u_r} \,\geqslant\, w\sqrt {{\frac{{4B}}{{3\lambda }}}}.$$
Substituting this into (7.4), we obtain
$$f\left( {\frac{r}{R}} \right)\, \geqslant \, \sqrt {{\frac{4}{{3\lambda }} \cdot B}}.$$
(7.7)
Similarly, for particles with $${v_r} \;\leqslant\, 0$$,
$$f\left( {\frac{r}{R}} \right) \, \leqslant\, \sqrt {{\frac{{4B}}{{3\lambda }}.}}$$
(7.8)
Inequalities (7.7) and (7.8) enclose the following limiting cases:
1. 1.

For any coordinate

$$f\left( {\frac{r}{\, R}} \right) > \sqrt {{\frac{{4B}}{{3\lambda }}}}.$$
In this case, the distribution coefficient $$K = 1;$$
1. 2.

Respectively, at $$f\left( {\frac{r}{R}} \right) < \sqrt {{\frac{{4B}}{{3\lambda }}}}$$, for any coordinate r, $$K = 0$$ is valid

An intermediate case is characterized by level lines formed by certain coordinates according to the equality
$$f\left( {\frac{{{r_0}}}{R}} \right) = \sqrt {{\frac{{4B}}{{3\lambda }}}}.$$
(7.9)
We assume that this equation has one real root:
$$\frac{{{r_0}}}{R} = {f^{ - 1}}\left( {\frac{{4B}}{{3\lambda }}} \right).$$
(7.10)
Taking this into account, we find a corresponding area $${\omega_{{r_0}}}$$ for which the following is valid:
$$f\left( {\frac{{{r_i}}}{\,R}} \right) \,\geqslant \ f\left( {\frac{{{r_0}}}{R}} \right).$$

Then the distribution coefficient can be written as

$$K = \frac{{{\omega_{{r_0}}}}}{{{\omega_R}}} = c\left( {\frac{{{r_0}}}{R}} \right)$$ for a convex profile $$f\left( {\frac{r}{R}} \right)$$,

$$K = C\left[ {1 - {{\left( {\frac{{{r_0}}}{R}} \right)}^2}} \right]$$ for a concave profile $$f\left( {\frac{r}{R}} \right)$$.

The coefficient С characterizes the shape of level lines and cross-section. For example, for a circle, $$C = 1.$$ Substituting dependence (7.7) into the derived equations, we finally obtain
$$K = \phi \left( {\frac{B}{\lambda }} \right).$$
A similar dependence is valid for two or more roots of Eq. (7.8), which takes place in the case of complicated profiles $$f\left( {\frac{r}{R}} \right).$$ For example, in the case of a profile shown in Fig. 7.2, for a certain mono-fraction Eq. (7.8) has three real roots: $${r_{{0_1}}};{r_{{0_2}}};{r_{{0_3}}}.$$ The latter form isotaches with the account for the shape of the apparatus cross-section. Isotaches determine a corresponding total area $$\sum {{\omega_{{r_{oi}}}}}$$ for which the following is valid:
$$f\left( {\frac{r}{\,R}} \right) \, \geqslant \ f\left( {\frac{{{r_{oi}}}}{R}} \right).$$
Thus, for the present case,
$$\sum {{\omega_{{r_{0i}}}}} = {\omega_{{r_{01}}}} + {\omega_{{r_{02}}}} + {\omega_{{r_{03}}}}$$
and the distribution coefficient can be written as
$$K = \frac{{\sum {{\omega_{{r_{0i}}}}} }}{{{\omega_R}}}.$$
Since $$\sum {{\omega_{{r_{0i}}}}}$$ can be unambiguously expressed through $${r_{0i}}$$, which are roots of Eq. (7.8), the final expression of the distribution coefficient has the form
$$K = \phi \left( {\frac{B}{\lambda }} \right).$$
(7.11)

A concrete expression of the distribution coefficient can be obtained using a concrete profile of the continuum over the apparatus cross-section. Now we examine step by step possible cases of flow interaction with particles.

### 7.2.1 Turbulent Overflow of Particles and Turbulent Regime of the Medium Motion in the Apparatus

Velocity distribution of a continuum over the radius in equilibrium apparatuses of circular cross-section is usually expressed empirically as
$${u_r} = w\frac{{(n + 1)(n + 2)}}{2} \cdot {\left( {1 - \frac{r}{R}} \right)^n} = wf\left( {\frac{r}{R}} \right)$$
(7.12)
where n is a parameter depending on the regime of the medium motion and on the roughness of pipe walls $$(n \, < \,1).$$
Using Eq. (7.9), we can find the coordinate of isotach where the absolute velocity of a fixed monofraction is zero:
$$\frac{{(n + 1)(n + 2)}}{2} \cdot {\left( {1 - \frac{{{r_0}}}{R}} \right)^n} = \sqrt {{\frac{{4B}}{{3\lambda }}}},$$
and hence,
$$\frac{{{r_0}}}{R} = 1 - {\left[ {\frac{2}{{(n + 1)(n + 2)}}\sqrt {{\frac{{4B}}{{3\lambda }}}} } \right]^{\frac{1}{n}}}.$$
(7.13)
Here, the cross-section area for which
$$f\left( {\frac{r}{R}} \right) \geqslant f\left( {\frac{{{r_0}}}{R}} \right)$$
amounts to $${\omega_{{r_0}}} = \pi r_0^2.$$
Hence, the distribution coefficient is expressed by the ratio
$$K = {\left( {\frac{{{r_0}}}{R}} \right)^2}.$$
Then, taking Eq. (7.13) into account, we obtain
$$K = {\left\{ {1 - {{\left[ {\frac{{2 \cdot \sqrt {{4B/3\lambda }} }}{{(n + 1)(n + 2)}}} \right]}^{\frac{1}{n}}}} \right\}^2}.$$
Instead of dependence (7.12), we can examine a different profile of velocity distribution of a continuum over the radius:
$${u_r} = \frac{{n + 2}}{n} \cdot w\left[ {1 - {{\left( {\frac{r}{R}} \right)}^n}} \right]$$
(7.14)
where n is the flow turbulization degree $$(n = 2 \div \infty ).$$
Let us analyze the following cases:
1. 1.

Velocity gradient on the flow axis

The following relationship is valid for the dependence (7.14)
$$\frac{{du}}{{dr}} = - (n + 2) \cdot \frac{w}{R}{\left( {\frac{r}{R}} \right)^{n - 1}}$$
(7.15)
and for the dependence (7.13)
$$\frac{{du}}{{dr}} = - \frac{w}{R} \cdot \frac{{n(n + 1)(n + 2)}}{{2{{\left( {1 - r/R} \right)}^{1 - n}}}}.$$
(7.16)
Then according to Eq. (7.15), we obtain for Eq. (7.14):
$${\left( {\frac{{du}}{{dr}}} \right)_{r = 0}} = 0.$$
Respectively, we obtain for Eq. (7.12) from Eq. (7.16):
$${\left( {\frac{{du}}{{dr}}} \right)_{r = 0}} = - \frac{{f(0) \cdot n}}{R}.$$
Thus, Eq. (7.12), in contrast to Eq. (7.14), describes a function discontinuous on the flow axis.
1. 2.

Velocity gradient on the pipe wall

It follows from Eq. (7.15) for the function (7.14) that
$${\left( {\frac{{du}}{{dr}}} \right)_{r = R}} = - (n + 2) \cdot \frac{w}{R} = - \frac{{f(0) \cdot n}}{R},$$
that is, the higher flow turbulization degree and mean velocity, the greater the gradient.
For the dependence (7.12) we obtain, according to Eq. (7.16),
$${\left( {\frac{{du}}{{dr}}} \right)_{r = R}} = \infty,$$
which points to the transfer of infinite momentum and corresponds to infinite friction force.
1. 3.

The expression (7.14), in contrast to (7.12), combines all regimes of motion up to laminar. Based on Eq. (7.10), the radius forming the distribution coefficient is

$$\frac{{{r_0}}}{R} = {\left[ {1 - \frac{n}{{n + 2}} \cdot \sqrt {{\frac{{4B}}{{3\lambda }}}} } \right]^{\frac{1}{n}}}.$$
Hence, we obtain an expression for the distribution coefficient:
$$K = {\left[ {1 - \frac{n}{{n + 2}} \cdot \sqrt {{\frac{{4B}}{{3\lambda }}}} } \right]^{\frac{2}{n}}}.$$

### 7.2.2 Laminar Overflow Regime

In this case, the regime of medium motion in a channel is based on the dependence (7.14) for the flow structure. Thus, at n = 2 we deal with a parabolic velocity profile, at n > 8 – with a turbulent motion, at $$2 \,<\, n \,<\, 8 -$$with a transient regime. Then, according to Eq. (7.10),
$$\frac{{n + 2}}{n}\left[ {1 - {{\left( {\frac{{{r_0}}}{R}} \right)}^n}} \right] = \sqrt {{\frac{{4B}}{{3\lambda }}}}.$$
(7.17)
Using the expression for the resistance coefficient in this equation, we obtain
$$\frac{{n + 2}}{n}\left[ {1 - {{\left( {\frac{{{r_0}}}{R}} \right)}^n}} \right] = \sqrt {{\frac{4}{3} \cdot \frac{{{u_{{r_0}}}d{\rho_0}B}}{{24\mu }}}}$$
or
$$\frac{{n + 2}}{n}\left[ {1 - {{\left( {\frac{{{r_0}}}{R}} \right)}^n}} \right] = \sqrt {{\frac{{\frac{{n + 2}}{n} \cdot w\left[ {1 - {{\left( {\frac{{{r_0}}}{R}} \right)}^n}} \right]d{\rho_0}}}{{18\mu }} \cdot B}}.$$
Simplifying, we obtain
$$\frac{{n + 2}}{n}\left[ {1 - {{\left( {\frac{{{r_0}}}{R}} \right)}^n}} \right] = \frac{{{{{\rm Re} }_w}B}}{{18}}.$$
Taking into account the fact that the distribution coefficient $$K = {\left( {\frac{{{r_0}}}{R}} \right)^2}$$ and Re2B = Ar, we finally obtain the expression
$$K = {\left[ {1 - \frac{{\sqrt {{Ar \cdot B}} }}{{18}} \cdot \frac{n}{{n + 2}}} \right]^{\frac{2}{n}}}.$$
For a parabolic profile $$(n = 2)$$,
$$K = 1 - \frac{{\sqrt {{Ar \cdot B}} }}{{36}}.$$

### 7.2.3 Intermediate Regime of Overflow

Using a well-known dependence of the resistance coefficient on the criteria Re and Ar,
$$\lambda = \frac{4}{3} \cdot \frac{{Ar}}{{{{{\rm Re} }^2}}}$$
and an interpolation formula that is valid for all overflow regimes
$${\rm Re} = \frac{{Ar}}{{18 + 0.61\sqrt {{Ar}} }},$$
we obtain
$$\lambda = \frac{4}{3} \cdot \frac{{{{(18 + 0.61\sqrt {{Ar}} )}^2}}}{{Ar}}.$$
Substituting the latter into Eq. (7.17) and passing to the distribution coefficient, we can define
$$\frac{{n + 2}}{n}\left( {1 - {k^{\frac{n}{2}}}} \right) = \frac{{\sqrt {{Ar \cdot B}} }}{{18 + 0.61\sqrt {{Ar}} }}.$$
A generalized dependence of the distribution coefficient in an arbitrary regime of the medium motion and of an arbitrary regime of particles overflow acquires the form
$${K_{\sum {} }} = {\left[ {1 - \frac{n}{{n + 2}} \cdot \frac{{\sqrt {{Ar \cdot B}} }}{{(18 + 0.61\sqrt {{Ar}} )}}} \right]^{\frac{2}{n}}}.$$
(7.18)

## 7.3 Analysis of Distribution Coefficient

We analyze Eq. (7.18) for turbulent regimes $$(Ar \geqslant {10^5}).$$ Under such conditions, the summand 18 in the denominator can be neglected, and the formula is reduced to an expression
$$K = {\left( {1 - \frac{n}{{n + 2}} \cdot \sqrt {{\frac{8}{3} \cdot B}} } \right)^{\frac{2}{n}}}.$$

With the account for the mathematical model of the regular cascade (7.3), this expression well agrees with experimentally obtained dependences.

Two approaches to the study of suspension-bearing flows are the most widespread. The first one considers a two-phase flow as a continuum with averaged properties. Such a dispersoid is characterized by a certain mean velocity, density, etc.

As it is, this approach is unacceptable for a critical flow, since we have to divide the dispersoid into separate phases, because the process result is the separation of each monofraction, which constitute, in total, a discrete phase. Therefore, this approach can be successfully used, for example, for the description of such processes as pneumatic transport, and not for classification.

The second approach consists in a separate analysis of the behavior of each phase. Here, the classification process should take into account numerous random factors. This causes insuperable difficulties in the quantitative description of the results in an explicit form. Therefore, the applicability of this approach is limited. It allows solving the simplest problems of the behavior of two-phase flows and is absolutely inapplicable for describing the classification process on the whole.

As for the classification process, it seems expedient to apply a combined method. Its essence is a transition to a dispersoid with an effective carrying capacity on the basis of the evaluation of the continuum impact on a discrete phase and of the behavior and interaction of individual monofractions. Thus, both the continuum and each separate monofraction participate in the dispersoid formation, and the latter, in turn, affects the behavior of particles of each narrow size class. This implicitly reflects intraphase and interphase interactions on the basis of the continuum.

To substantiate the transition to a “divided” dispersoid, we evaluate the density of monofractions flow. The quantity of particles of a fixed monofraction passing through the apparatus cross-section per unit time can be expressed as
$${Q_{\sum {} }} = \frac{{G \cdot {r_s} \cdot {r_i}}}{P}$$
(7.19)
where G is the monofraction mass;

$$P$$ is the mass of an isolated particle of a specified narrow size class.

Here an ascending flow of monofraction particles is written as
$${Q_a} = \frac{{{G_f} \cdot {r_s} \cdot {r_i} \cdot K}}{P}$$
(7.20)
where $${r_i} \cdot K$$ is a relative flow of the particles under study from section i to the overlying section. The resulting ascending flow of the given monofraction is
$${Q_r} = \frac{{{G_f} \cdot {r_s} \cdot {F_f}}}{P}.$$
(7.21)

According to the regular cascade model, expressions (7.19), (7.20) depend on $$K,z,{i^* },i,$$ whereas Eq. (7.21) is independent of the section under study.

Assuming a uniform distribution of particles over the apparatus cross-section, we obtain expressions for the flow density of particles of a fixed narrow size class:
$${q_1} = \frac{{{G_f} \cdot {r_s}}}{{F \cdot P}}$$
for an isolated relative flow and
$$q = {q_1} \cdot {F_f}$$
(7.22)
for a resulting flow.
We transform the obtained expressions multiplying both the numerator and the denominator of the right-hand side by the volumetric flow of the continuous phase V
$$q = \frac{{{G_f} \cdot {r_s} \cdot V}}{{V \cdot F \cdot P}} = \frac{{\mu \cdot {r_s} \cdot V}}{{F \cdot P}} = \frac{{\mu \cdot {r_s} \cdot w}}{P}.$$
(7.23)

Let us examine the dependencies (7.22) and (7.23) on a concrete example. Thus, at the separation of periclase $$(\rho = 3,\!600\,{\hbox{kg/}}{{\hbox{m}}^3})$$ on an equilibrium apparatus in the regime of $$w = 2.83\,{\hbox{m/s}}$$ and at a consumed concentration $$\mu = 1.5\,{\hbox{kg/}}{{\hbox{m}}^3}$$, fine product output amounted to about 20%.

Granulometric composition of the initial material, fractional extraction degrees and particle density flows calculated using Eqs. (7.22) and (7.23) are given in Table 7.1.
Table 7.1

Flow densities of particles of different monofractions calculated by Eqs. (7.22) and (7.23)

 Narrow size class (mm) 0.14 0.2 + 0.14 0.3 + 0.2 0.5 + 0.3 Average size d (mm) 0.07 0.17 0.25 0.40 $${r_s}\%$$ 10.93 13.51 15.75 26.59 $${F_f}\%$$ 93 45.5 8.0 2.0 $$q{({\hbox{c}}{{\hbox{m}}^2} { \times {\rm s}})^{ - 1}}$$ $$71.8 \times {10^3}$$ $$6.2 \times {10^3}$$ $$2.3 \times {10^3}$$ $$0.94 \times {10^3}$$ $${q_{\sum {} }}{({\hbox{c}}{{\hbox{m}}^2}{\times {\rm s}})^{ - 1}}$$ $$66.8 \times {10^3}$$ $$2.8 \times {10^3}$$ 184 19

These data point to the fact that the densities of particle flows (especially of fine particles) in the apparatus are sufficiently high, although the fine product yield is low. It can be attributed to the fact that fine particles, catching up with coarse ones, exert additional impact on them in comparison with a continuum. Besides, the high density of particles averages and levels out this effect in time. This allows us to pass to the carrying capacity of the flow on the whole (to a dispersoid) and estimate its effect on particles of each narrow size class individually (a divided dispersoid).

It seems reasonable to compare the quantitative value of the particles flow density with experimental data. Thus, Razumov presents experimental data (under the conditions of vertical pneumatic transport) on the number of collisions between the suspension-carrying flow containing a mono-fraction of the size $$d = 2.3\,{\hbox{mm}}$$ and the motionless surface with the area $$1\,{\hbox{c}}{{\hbox{m}}^2}.$$ The characteristic parameters of the experimental conditions were as follows:
• Density of the medium $${\rho_0} = 1.29\,{\hbox{kg/}}{{\hbox{m}}^3}$$

• Density of the particles material $$\rho = 1,\!200\,{\hbox{kg/}}{{\hbox{m}}^3}$$

• Initial mass concentration $$3.5\,\frac{{{\text{kg/h}}}}{{{{\rm kg/h}}}}{,}$$ which corresponds to $$\mu = 4.515\,{\hbox{kg/}}{{\hbox{m}}^3}$$

• $${r_s} = 100\%,$$ since the experiment was performed on a monofraction

• Mean velocity of the medium flow varied within the limits of $$10 \div 17.5\,{\hbox{m/s}}$$

In the course of the experiments, 300–1,300 collisions were registered per second per 1 cm2 of surface placed into the ascending flow. Apparently, proceeding from experimental conditions, the number of collisions corresponds to the density of particles flow described by Eq. (7.22). Assuming, on the average, $$w = 14\,{\hbox{m/s}}$$, we obtain $$N = q = 827\,\frac{1}{{{\hbox{c}}{{\hbox{m}}^2} \cdot {\hbox{s}}}},$$ which is close to the average number of collisions registered in the experiment. For the velocities $$w = 10\,{\hbox{m/s}}$$ and $$w = 15\,{\hbox{m/s}}$$, the numbers of collisions determined by Eq. (7.22) are 590 and 1,033$$\frac{1}{{{\hbox{c}}{{\hbox{m}}^2} \cdot {\hbox{s}}}}$$, respectively. Apparently, these results give rather satisfactory estimations.

To pass to a divided dispersoid, which is different for particles of each narrow size class, we have to evaluate its important parameter – density $${\rho_{{n_j}}}$$ (dispersoid density for particles of the j-th narrow size class).

Let us examine two monofractions with particle masses m and M. We assume that N fine particles inelastically collide with one coarse particle imparting their momentum to it. To the first approximation, the quantity N can be evaluated through the ratio of flow densities of the monofractions under study.
$$N = \frac{{{q_m}}}{{{q_M}}} = \frac{{{r_m}}}{{{r_M}}} \cdot {\left( {\frac{{{d_m}}}{{{d_M}}}} \right)^3}.$$
(7.24)
As a result of inelastic collisions, the ensembles of fine and coarse particles acquire the same velocity $${v_{Km}} = {v_{KM}},$$ the velocity of fine particles having decreased from $${v_{Hm}}$$ to $${v_{KM}},$$ and that of coarse particles – increased from $${v_{HM}}$$ to $${v_{KM}}.$$ The change in the momentum of the fine particles ensemble amounts to
$$\Delta {L_m} = Nm({v_{Hm}} - {v_{Km}}),$$
and that of coarse particles – to
$$\Delta {L_M} = M({v_{KM}} - {v_{HM}}).$$
Obviously, $$\Delta {L_m} = \Delta {L_M}.$$ This stipulates the following:
$${v_{Km}} = {v_{KM}} = \frac{{Nm{v_{Hm}} + M{v_{HM}}}}{{Nm + M}}.$$
(7.25)
The conditions of a uniform motion of fine and coarse particles with initial velocities are as follows:
$${(u - {v_{Hm}})^2} = \frac{3}{{4\lambda }} \cdot \frac{\rho }{{{\rho_0}}} \cdot g{d_m},$$
(7.26)
$${(u - {v_{HM}})^2} = \frac{3}{{4\lambda }} \cdot \frac{\rho }{{{\rho_0}}} \cdot g{d_M}.$$
(7.27)
The conditions of a uniform motion of a coarse particle with a final velocity in a dispersoid is, respectively:
$${(u - {v_{KM}})^2} = \frac{3}{{4\lambda }} \cdot \frac{\rho }{{{\rho_0}}}g{d_M}.$$
(7.28)
Having divided term-by-term the expression (7.27) by (7.28), we obtain
$$\frac{\rho }{{{\rho_0}}} = {\left( {\frac{{u - {v_{KM}}}}{{u - {v_{Km}}}}} \right)^2}$$
(7.29)
From Eq. (7.25),
$$u - {v_{KM}} = u - \frac{{Nm{v_{Hm}} + m{v_{HM}}}}{{Nm + M}}.$$
(7.30)
After simple transformations, this expression acquires the form
$$u - {v_{KM}} = \frac{{Nm(u - {v_{Hm}}) + M(u - {v_{HM}})}}{{Nm + M}}.$$
(7.31)
Using Eq. (7.29)
$$\frac{{{\rho_n}}}{{{\rho_0}}} = \frac{{{{(Nm + M)}^2}}}{{{{\left( {Nm\frac{{u - {v_{Hm}}}}{{u - {v_{HM}}}} + M} \right)}^2}}}$$
(7.32)
or, taking into account both (7.26) and (7.27),
$$\frac{{{\rho_n}}}{{{\rho_0}}} = \frac{{{{(Nm + M)}^2}}}{{{{\left( {Nm\sqrt {{\frac{{{d_m}}}{{{d_M}}}}} + M} \right)}^2}}} = {\left( {\frac{{N\frac{m}{M} + 1}}{{N\frac{m}{M}\sqrt {{\frac{{{d_m}}}{{{d_M}}}}} + 1}}} \right)^2}.$$
(7.33)
Using Eq. (7.24) and taking into account the fact that $$\frac{m}{M} = {\left( {\frac{{{d_m}}}{{{d_M}}}} \right)^3},$$ we obtain
$$\frac{{{\rho_n}}}{{{\rho_0}}} = {\left( {\frac{{\frac{{{r_m}}}{{{r_M}}} + 1}}{{\frac{{{r_m}}}{{{r_M}}}\sqrt {{\frac{{{d_m}}}{{{d_M}}}}} + 1}}} \right)^2}.$$
(7.34)
Since $${\rho_n} = {\rho_0} + \Delta {\rho_n},$$ a relative increase in the dispersoid density is
$$\frac{{\Delta {\rho_n}}}{{{\rho_0}}} = \frac{{{\rho_n}}}{{{\rho_0}}} - 1.$$
With the account of n mono-fractions under study, which transfer the momentum to a coarser particle, relative increase in the dispersoid density is
$$\frac{{{\rho_n}}}{{{\rho_0}}} = {\sum\limits_{j = 1}^n {\left( {\frac{{{\rho_n}}}{{{\rho_0}}}} \right)}_j} - n + 1.$$
With the account for Eq. (7.34), the final expression for the dispersoid density is
$${\rho_n} = {\rho_0}\left[ {\sum\limits_{j = 1}^n {{{\left( {\frac{{\frac{{{r_m}}}{{{r_M}}} + 1}}{{\frac{{{r_m}}}{{{r_M}}}\sqrt {{\frac{{{d_m}}}{{{d_M}}}}} + 1}}} \right)}^2} + 1 - n} } \right].$$
(7.35)
To check the above-mentioned experiment on periclase classification using Eq. (7.35), we have calculated dispersoid flow densities for the mentioned mono-fractions
$$\begin{array}{lll} \;\begin{array}{ll} {{d_m} = 0.40\,{\hbox{mm}}} & {{\rho_n} = 2.29\,{\hbox{kg/}}{{\hbox{m}}^3}} \\\end{array} \hfill \\\begin{array} {ll}{{d_m} = 0.25\,{{\rm mm}}} & {{\rho_n} = 2.06\,{\hbox{kg/}}{{\hbox{m}}^3}} \\{{d_m} = 0.17\,{\hbox{mm}}} & {{\rho_n} = 1.70\,{\hbox{kg/}}{{\hbox{m}}^3}} \\\end{array} \hfill \\\end{array}$$

The obtained results point to an insignificant change in the flow density of the dispersoid affecting individual narrow classes of particles. On the average, in the present case we can assume $${\rho_n} = 2.0\,{\hbox{kg/}}{{\hbox{m}}^3} = const$$ as a first approximation for all monofractions. It is noteworthy that for materials without sharp granulometric differences, the difference in the mean flow density is insignificant. For more exact estimations (or for materials with extremely different compositions), it is recommended to use Eq. (7.35) for each monofraction.

Another important issue arising at a transition to the dispersoid is to evaluate its structure, the profile of its velocity distribution over the apparatus cross-section. Equations (7.26) and (7.27) implied a dispersoid with local effective velocities u. A transition to $${\rho_n}$$ was realized at the expense of the momentum transfer from fine particles to coarse ones. The flow density $${\rho_n}$$ was assumed to remain unchanged over the apparatus cross-section owing to the assumption of a uniform distribution of particles. Since the local carrying capacity per unit area is characterized by the product $${\rho_n}{u^2},$$ its profile should be affine or uniform with respect to the distribution of squared dispersoid velocities in the cross-section. Taking into account its affine transformation with the scale factor 0.5 and nonlinear square-root transformation, we obtain the profile of effective dispersoid velocities close to a parabolic one. Since the volume flow rate of a dispersoid should be equal to a continuum volume flow rate, the equation of dispersoid velocity distribution should be written as
$$f\left( {\frac{r}{R}} \right) = 2w\left[ {1 - {{\left( {\frac{r}{R}} \right)}^2}} \right],$$
(7.36)
which corresponds to the parameter $$n = 2.$$

Obviously, the above-stated estimation should be considered as approximate, because it is based on a number of assumptions.

Substituting $$n = 2$$ in Eq. (7.19) and replacing $${\rho_0}$$ with $${\rho_n},$$ we obtain
$$K = 1 - \sqrt {{0.4 \cdot B}}.$$
(7.37)
The obtained expression reflects adequately enough $${B_{\max }} = 2.5$$ and agrees sufficiently well with the description of experimental separation curves $${F_f}(d)$$ based on a cascade model (for turbulent regimes). In the case of an arbitrary regime of particles overflow, we derive from Eq. (7.19):
$$K = 1 - \frac{{\sqrt {{Ar \cdot B}} }}{{36 + 1.575\sqrt {{Ar}} }}.$$
(7.38)

Equations (7.37) and (7.38) are valid for $${\rho_0} = 1.2\,{\hbox{kg/}}{{\hbox{m}}^3}$$and $${\rho_n} = 2.0\,{\hbox{kg/}}{{\hbox{m}}^3}$$ appearing in the coefficients, and the criterion Ar and B are expressed, as before, through $${\rho_0}.$$

## 7.4 Analysis of Experimental Dependencies from the Standpoint of Structural Models

Principal regularities of the gravitational classification process were revealed experimentally on various cascade apparatuses. Now it has become possible to explain experimental facts from the standpoint of structural and cascade models.

First of all, it follows directly from Eqs. (7.37) and (7.19) that it is possible to plot a separation curve $${F_f}(x)$$ in any regime.

The same expressions allow us to take into account separation results depending on the number of stages in a cascade apparatus (classifier height) and on the material feeding place.

The effect of structural differences of various apparatuses on the fractioning process is taken into account by the application of different cascade models.

To check fractional separation curves for particles of different narrow size classes depending on the classification regime, as well as other regularities, respective estimations were performed for a shelf-type cascade apparatus comprising four stages $$(z = 4)$$ in the case of initial material feed from above $$({i^* } = 1).$$ Experimental data on quartzite separation in it $$(\rho = 2650\,{\hbox{kg/}}{{\hbox{m}}^3})$$ are presented in Fig. 7.3. The comparison of computation results $${F_f}({d_j},w)$$ obtained using Eqs. (7.37) and (7.19) with experimental data is presented in the same figure.
According to Eq. (7.37), the velocity $${w_0}$$ of the onset of fixed monofraction extraction (intersection of $${F_f}\left[ {{d_i},w} \right]$$ curve with the abscissa axis) is determined from the condition
$$K = 0 = 1 - \sqrt {{0.4 \cdot B}}.$$
Hence, neglecting $${\rho_0}$$ value in comparison with $${\rho_f},$$ we obtain
$${w_0} = \sqrt {{0.4 \cdot \frac{\rho }{{{\rho_0}}}gd}}.$$
This makes it possible to predict the ratio of the velocity $${w_0}$$ to the final settling velocity $${v_0}$$ of an individual particle of the specified size class in air. As known,
$${v_0} = \sqrt {{\frac{4}{{3\lambda }} \cdot \frac{\rho }{{{\rho_0}}} \cdot gd}}.$$
(7.39)
At the aerodynamic resistance factor $$\lambda = 0.5,$$ we obtain
$$\frac{{{v_0}}}{{{w_0}}} = \sqrt {{\frac{4}{{3 \cdot 0.5 \cdot 0.4}}}} = 2.59.$$
(7.40)
To check this ratio, $${w_0}$$ velocities were calculated using Eq. (7.39) for particles of all narrow size classes examined in the previous example, and compared with experimental $${w_0}$$ values. The aerodynamic resistance factor in Eq. (7.39) was determined using an adjusted dependence
$$\lambda = 0.5 + \frac{{29.2}}{{\sqrt {{Ar}} }} + \frac{{430}}{{Ar}}.$$
The comparison of the computed dependence (7.40) with experimental data is shown in Fig. 7.4.
A characteristic property of separation curves is their affinity. Its consequence is a unitary character of $${F_f}\left( {\frac{w}{{{w_{50}}}}} \right)$$ curve for all monofractions. A combined analysis of cascade and structural models confirms this fact. Thus, the distribution factor $${K_{50}}$$ for any 50%-extractable monofraction is determined from the equation
$$\frac{{1 - {{\left[ {\left( {1 - {K_{50}}} \right)/{K_{50}}} \right]}^{z + 1 - {i^* }}}}}{{{{\left[ {\left( {1 - {K_{50}}} \right)/{K_{50}}} \right]}^{z + 1}}}} = 0.5.$$
(7.41)
Substitute $${K_{50}}$$ value found from Eq. (7.41) into Eq. (7.37):
$${K_{50}} = 1 - \sqrt {{0.4\frac{\rho }{{{\rho_0}}} \cdot \frac{{gd}}{{w_{50}^2}}}}.$$
Hence,
$$0.4\frac{\rho }{{{\rho_0}}} \cdot gd = {(1 - {K_{50}})^2}w_{50}^2.$$
We substitute the obtained value $$0.4\frac{\rho }{{{\rho_0}}}gd$$ into Eq. (7.37) for an arbitrary distribution factor
$$K = 1 - \frac{{{w_{50}}}}{w}(1 - {K_{50}}).$$
Substitute the latter into Eq. (7.19):
$${F_f}\left( {\frac{w}{{{w_{50}}}}} \right) = \frac{{1 - {{\left\{ {1/\left[ {\left( {\frac{1}{{1 - {K_{50}}}}} \right)\frac{w}{{{w_{50}}}} - 1} \right]\,} \right\}}^{z + 1 - {i^* }}}}}{{1 - {{\left\{ {1/\left[ {\left( {\frac{1}{{1 - {K_{50}}}}} \right)\frac{w}{{{w_{50}}}} - 1} \right]} \right\}}^{z + 1}}}}.$$
(7.42)
The obtained expression (7.42) satisfies the unitary character of $${F_f}\left( {\frac{w}{{{w_{50}}}}} \right)$$ curve, whose plotting requires successive solution of Eq. (7.41), and then Eq. (7.42). In particular, for the case under study $$(z = 4,{i^* } = 1),$$ we derive from Eq. (7.41):
$${K_{50}} = 0.342;\;1 - \frac{1}{{{K_{50}}}} = 1.519.$$
In this case, the dependence (7.42) acquires the form
$${F_f}\left( {\frac{w}{{{w_{50}}}}} \right) = \frac{{1 - {{\left[ {1/\left( {1.519\frac{w}{{{w_{50}}}} - 1} \right)} \right]}^4}}}{{1 - {{\left( {\frac{1}{{1.519}}\frac{w}{{{w_{50}}}} - 1} \right)}^5}}}.$$
(7.43)
The comparison of results computed by Eq. (7.43) with experimental data is presented in Fig. 7.5.
As previously, the affinity of separation curves $${F_f}(x)$$ results in a unitary character of $${F_f}\left( {\frac{d}{{{d_{50}}}}} \right)$$ curve for all velocities. Using the solution of Eq. (7.41) with respect to $${K_{50}}$$ in (7.37), we obtain for an arbitrary regime:
$${K_{50}} = 1 - \sqrt {{0.4\frac{\rho }{{{\rho_0}}} \cdot \frac{{g{d_{50}}}}{{{w^2}}}}},$$
(7.44)
$$\frac{{{{(1 - {K_{50}})}^2}}}{{{d_{50}}}} = 0.4\frac{\rho }{{{\rho_0}}} \cdot \frac{g}{{{w^2}}}.$$
(7.45)
Passing to an arbitrary distribution factor, we obtain
$$K = 1 - (1 - {K_{50}}) \cdot \sqrt {{\frac{d}{{{d_{50}}}}}}.$$
(7.46)
Then the expression (7.19) acquires the form:
$${F_f}\left( {\frac{d}{{{d_{50}}}}} \right) = \frac{{1 - {{\left[ {\frac{{(1 - {K_{50}})\sqrt {{\frac{d}{{{d_{50}}}}}} }}{{1 - (1 - {K_{50}})\frac{d}{{{d_{50}}}}}}} \,\right]}^{z + 1 - {i^* }}}}}{{1 - {{\left[ {\frac{{(1 - {K_{50}})\sqrt {{\frac{d}{{{d_{50}}}}}} }}{{1 - (1 - {K_{50}})\sqrt {{\frac{d}{{{d_{50}}}}}} }}} \right]}^{z + 1}}}}.$$
(7.47)
Experimental check of the dependence (7.47) at $$z = 4;{i^* } = 1$$ is presented in Fig. 7.6.

It is noteworthy that we can prove in a similar way the unitary character of $${F_f}\left( {\frac{d}{{{d_{50}}}}} \right)$$ and $${F_f}\left( {\frac{w}{{{w_{s0}}}}} \right)$$ curves for an arbitrary fractional separation value.

Experimentally established universality of $${F_f}(Fr)$$ curve for various regimes and different monofractions can be directly revealed from the structural and cascade models. In fact, for a specific apparatus $$(z;{i^* })$$ and the density of material particles $$\rho,$$ fractional extraction is unambiguously determined by the parameter $$\frac{{gd}}{{{w^2}}}.$$ The comparison of the curve calculated by Eqs. (7.37) and (7.19) with experimental data in the case under study $$(z = 4;{i^* } = 1;$$ $$2,\!500\,{\hbox{kg/}}{{\hbox{m}}^3})$$ is presented in Fig. 7.7.
In a more general case, when the densities of materials to be classified differ, a universal dependence is a function of the generalized classification parameter $${F_f}(B).$$ This fact also follows directly from Eqs. (7.37) and (7.19). In particular, computed $${F_f}(B)$$ values and experimental data for the separation of different materials in an equilibrium apparatus of circular cross-section $$({D} = 100\,{\rm mm};$$ $$z = 9;$$ $${i^* } = 6;$$ $$\mu = 1.5 \, {\rm kg/m}^3)$$ are presented in Fig. 7.8.
To check the compliance of the structural model with the empirical dependence, we express the value of the parameter $$F{r_{50}}$$ using Eq. (7.19):
$${K_{50}} = 1 - \sqrt {{0.4\frac{{(\rho - {\rho_0})}}{{{\rho_0}}} \cdot F{r_{50}}}}.$$
Hence,
$$F{r_{50}} = \frac{{{{(1 - {K_{50}})}^2}}}{{0.4}} \cdot \frac{{{\rho_0}}}{{(\rho - {\rho_0})}}.$$
With the account for the fact that in this example $${K_{50}} = 0.342,$$ we obtain:
$$F{r_{50}} = 1.08\frac{{{\rho_0}}}{{(\rho - {\rho_0})}},$$
which is close to the experimental correlation.

On the whole, all the examples studied clearly point to the predominance of flow structure in the process of gravitational classification. The advantages of this approach are its simplicity and satisfactory compliance with basic experimental dependencies related to the gravitational classification process accumulated by today.

## 7.5 Check of the Structural Model Adequacy

The account for the flow structure allows a more objective approach to the prediction of fractionating results on apparatuses of different constructions. Without dwelling on extremely complicated patterns of flow formation in actual apparatuses, for their roughest examination we should single out three characteristic properties of a moving flow that can be connected with the apparatus construction:
• Character of the change in the continuous phase velocity field along the apparatus height

• Presence of stagnant zones and the rate of the apparatus cross-section filling with the moving flow

• Character of the material motion at the first stage of the process (at the feeding stage), intensity of its interaction with the flow and internal elements promoting concentration leveling and reduction of skips and depressions

Thus, for example, a velocity field of a continuum can be both uniform and non-uniform along its height. Operation of hollow (equilibrium) apparatuses of a constant cross-section is the closest to the former case. A uniform velocity field stipulates for an identical regime of flow interaction with particles at any level of the cross-section and predetermines the invariability of the distribution factor over the apparatus height. The model of regular cascade satisfies most completely such conditions of the process organization.

The intensity of the continuum interaction with particles depends not only on the nonuniform velocity field over the apparatus height, but also on its nonuniformity over the cross-section. For an apparatus of a circular cross-section, the structural model takes into account the influence of transverse nonuniformity of the flow. It is assumed obvious that the rate of filling the apparatus cross-section with a moving continuous phase amounts to 100%. The matter is different with apparatuses of square and rectangular cross-sections, where stagnant zones are formed in the corners, reducing to zero the effect of the continuum on the removal of particles. If we assume that the line of zero flow velocity in an apparatus of square cross-section can be approximated by a circumference inscribed into the square, the rate of filling such cross-section with an ascending flow amounts to
$${C_{sq}} = \frac{{{F_{cir}}}}{{{F_{sq}}}} = \frac{\pi }{4}.$$
Since the distribution factor (proceeding from the structural model) is determined through the ratio of areas, when estimating this factor for a square cross-section, a correction factor
$${C_{sq}} = \frac{\pi }{4}$$
should be introduced. Taking it into account,
$${k_{sq}} = \frac{\pi }{4} \cdot {K_0},$$
(7.48)
where $${K_0}$$ is the distribution factor for an apparatus of circular cross-section.
If we accept an inscribed ellipse as the zero velocity line for an apparatus of rectangular cross-section, the expression (7.48) will be also valid for determining $${k_{rec}}$$, since
$${C_{rec}} = \frac{{{F_{el}}}}{{{F_{rec}}}} = \frac{\pi }{4}.$$
(7.49)

The expression (7.49) for square and rectangular cross-sections is recommended only as a first approximation, since actual filling rates are somewhat higher.

Finally, at the first stage of the process, a significant fall of the majority of particles with respect to the level of their inlet is observed in the absence of intense interaction of particles with internal components of the apparatus and in the presence of stagnant zones. Apparently, the most favorable conditions for their skip are realized in a hollow apparatus of square or rectangular cross-section.

In the light of the statements above, we have made an attempt to predict quantitative results of fractionating process for several apparatuses of various designs. Computation results are presented in respective figures in comparison with experimental data.

Figure 7.9 shows an estimated curve and experimental data obtained at the classification of periclase with the density $$\rho = 3,\!600\,{\hbox{kg/}}{{\hbox{m}}^3}$$ in an equilibrium apparatus of circular cross-section $$({D} = 100\,{\hbox{mm}}).$$ The number of conventional sections $$z = 9,$$ feed sections $${i^* } = 6.$$ Consumed concentration of the material is $$\mu = 1.5\,{\hbox{kg/}}{{\hbox{m}}^3}.$$ Computations were carried out using the model of regular cascade (7.3) and structural model according to Eqs. (7.19) and (7.37). Average deviation of the estimated curve of $${F_f}(B)$$ dependence from experimental points in Fig. 7.9 amounts to $${ + 1.8}\% \div 2.1\%$$.
Figure 7.10 shows $${F_f}(B)$$ dependence at the classification of quartzite with the density $$\rho = 2,\!650\,{\hbox{kg/}}{{\hbox{m}}^3}$$ in an equilibrium apparatus of square cross-section with the dimensions $$100 \times 100\,{\hbox{m}}{{\hbox{m}}^2}.$$ The number of conventional sections $$z = 6,$$ feed sections $${i^* } = 3.$$ Consumed concentration of the material is $$\mu = 2\,{\hbox{kg/}}{{\hbox{m}}^3}.$$ Computations were carried out using the model of a regular cascade with a skip by 1.5 conventional sections.
$${F_f} = \frac{{1 - {\sigma^{2,5}}}}{{1 - {\sigma^7}}}.$$
The distribution factor was determined with a correction for square cross-section according to Eq. (7.48)
$${k_{sq}} = \frac{\pi }{4}\left[ {1 - \sqrt {{0.4 \cdot B}} } \right].$$

Maximal deviation of the estimated curve from experimental points does not exceed 15%.

Figure 7.11 shows $${F_f}(B)$$ dependence at the classification of quartzite with the density $$\rho = 2650\,{\hbox{kg/}}{{\hbox{m}}^3}$$ in an apparatus of rectangular cross-section of zigzag type. The number of sections $$z = 6;{i^* } = 3.$$ Consumed concentration is $$\mu = 2.0\,{\hbox{kg/}}{{\hbox{m}}^3}.$$ Computations were carried out using the model of a regular cascade:
$${F_f} = \frac{{1 - {\sigma^4}}}{{1 - {\sigma^7}}}.$$
The distribution factor was determined with a correction for square cross-section.
$${k_{sq}} = \frac{\pi }{4}\left[ {1 - \sqrt {{0.4 \cdot B}} } \right].$$

In all the cases, maximal deviation of the estimated curve from experimental points does not exceed 7%, which is within the limits of experimental accuracy.

## 7.6 Correlation Between the Structural and Cellular Models of the Process

Equation (7.6) can be written for a cell as follows:
$${u_r} - {v_r} = {u_r}\sqrt {{\frac{{4B}}{{3\lambda }}}},$$
since here we examine hydrodynamic conditions practically in one point of the flow. Hence,
$$1 - \frac{{{v_r}}}{{{u_r}}} = \sqrt {{\frac{{4B}}{{3\lambda }}}}$$
or
$$\frac{{{v_r}}}{{{u_r}}} = 1 - \sqrt {{\frac{{4B}}{{3\lambda }}}}.$$
According to (7.37), in this case we can write
$$\frac{{{v_r}}}{{{u_r}}} = \frac{{{v_r}}}{w} = k.$$
Then the limiting expression for coarse and fine particles extraction from a cell can be written as
$$f(E) = {e^{\frac{{\tau - E}}{\chi }}} = {e^{\frac{{\rho {v^2}}}{{{\rho_0}{w^2}}} - \frac{{gd\rho }}{{{w^2}{\rho_0}}}}}.$$
Then we obtain
$$f(E) = {e^{\frac{\rho }{{{\rho_0}}}{k^2}}} - B.$$
Since $$k = 1 - \sqrt {{\frac{{4B}}{{3\lambda }}}},$$ this dependence can be finally written as
$$f(E) = A \cdot {e^{ - \phi (B)}}$$
where $$A$$ is a constant value $$A = {e^{\frac{\rho }{{{\rho_0}}}}}$$; $$\phi (B)$$ is a function of the parameter $$B$$.

This dependence totally complies with empirical dependencies for actual separation curves, which were confirmed over and over again, but have not found as yet a clear theoretical justification.