Structural Model of Mass Transfer in Critical Regimes of TwoPhase Flows
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Abstract
Physics of a twophase 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 Crosssection geometry Flow rate Velocity profile Reynolds criterion Froude criterion Archimedes criterion Level lines Flow profile Velocity gradient7.1 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.
where \( {r_i} \) is the initial contents of narrow sizeclass particles at a certain ith stage of the apparatus; \( r_i^*  \) quantity of the same particles passing from the ith stage to the overlying (i – 1)th stage; K or \( k  \) distribution coefficient.
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.
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
 1.
Particles are spherical.
 2.
Distribution of particles of any narrow size class over the crosssection of the apparatus is uniform due to their intense interaction with each other and with the apparatus walls and internal facilities.
 3.
Ascending twophase flow should be considered as a continuum with elevated density. As established, carrying capacity of dustladen 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 crosssection. It can be written in a general form as
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 monofraction in the crosssection 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 \( (v\, \geqslant \,0) \).
Thus, we can determine the limiting size of particles, above which the overflow of particles is certainly turbulent.
 1.
For any coordinate
 2.
Respectively, at \( f\left( {\frac{r}{R}} \right) < \sqrt {{\frac{{4B}}{{3\lambda }}}} \), for any coordinate r, \( K = 0 \) is valid
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 \( f\left( {\frac{r}{R}} \right) \).
A concrete expression of the distribution coefficient can be obtained using a concrete profile of the continuum over the apparatus crosssection. 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
 1.
Velocity gradient on the flow axis
 2.
Velocity gradient on the pipe wall
7.2.2 Laminar Overflow Regime
7.2.3 Intermediate Regime of Overflow
7.3 Analysis of Distribution Coefficient
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 suspensionbearing flows are the most widespread. The first one considers a twophase 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 twophase 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.
\( P \) is the mass of an isolated particle of a specified narrow size class.
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.
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%.
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).

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 cm^{2} 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 jth narrow size class).
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.
Obviously, the abovestated estimation should be considered as approximate, because it is based on a number of assumptions.
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.
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.
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

Character of the change in the continuous phase velocity field along the apparatus height

Presence of stagnant zones and the rate of the apparatus crosssection 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 nonuniform along its height. Operation of hollow (equilibrium) apparatuses of a constant crosssection 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 crosssection 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 expression (7.49) for square and rectangular crosssections 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 crosssection.
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.
Maximal deviation of the estimated curve from experimental points does not exceed 15%.
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
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.