Ironmaking
Optimization of Capillary Forces of Clutching Particles in Iron Ore Materials Granulation
2021 Volume 61 Issue 7 Pages 2066-2073
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2021 Volume 61 Issue 7 Pages 2066-2073
Optimization of the operating modes of the pelletizing drum requires the study of the physical processes occurring inside the granules. The purpose of this work is to develop a recommendation for improving the quality of the pelletizing of the charge based on the study of the forces of capillary interaction of particles in granules of granular agglomerate.
A deterministic model of capillary adhesion of two particles of arbitrary size and properties based on the calculation of the geometric parameters of the liquid meniscus has been developed. Taking into account the size of the system and to simplify the calculation of the interaction of particles of the sinter batch, the meniscus is represented by a body of revolution with a spherical concave surface.
The interaction of 3 mm and 0.1 mm fractions with an average wetting angle θ = 25° for equilibrium states according to the degree of water filling W0.1/3 = W0.1/0.1 and adhesion forces F0.1/3 = F0.1/0.1 was researched. A value of the degree of filling the meniscus Wopt is lower than the optimal values of W for particles with a particle size of R2/R1 = 0.1/0.1 and R2/R1 = 0.1/3, therefore, one should assert not about the optimal degree of filling the meniscus, but about the optimal range of degree filling the meniscus from Wopt = 0.086 to W0.1/3 = 0.21.
The granulating of bulk materials has a significant impact on the performance of the sintering process and the quality of the products produced. Despite the intense destruction of the granules during their drying on the belt, well-pelletized charge usually provides maximum gas permeability of the sintered layer.
The process of obtaining iron ore granules occurs in two stages.1,2) At the first stage, the formation of lumps of loose material of a rounded shape under the influence of local waterlogging occurs. The density of lumps increases during the rotation of the pelletizer, and excess moisture appears on the surface. At the second stage, an increase in the size of lumps and their transformation into granules by attaching small particles to the waterlogged surface occurs.
For each of the stages, such important interactions as capillary, which are manifested in three-phase (solid – liquid – gas) dispersed systems, must be distinguished. The predominance of capillary forces over other components of interparticle interaction has a positive effect on the formation of granules.
Controlling the pelletizing process is complicated by the peculiarities of the initial charge and processing equipment.3) The components of the sinter batch have a wide range of sizes and different properties. Repeated theoretical and practical studies have established that the sinter charge can be divided into three fractions - cored, clumped and intermediate. The size of each fraction has no clear boundaries. However, most often, the fraction with a particle size of 0–1 mm is clumped, 2–5 mm is cored and 1–2 mm is intermediate.
The quality of the pelletized charge is largely determined by the intensity of the impact on the granules. Mechanical influences caused by the operation of plant equipment are always present in real conditions when moving and storing the charge. If, under conditions of mechanical overloads and vibration, the acceleration is 3 g, then the particle size decreases by approximately 1.73 times, and at 5 g - by 2.23 times.4) The calculated limit parameters for unsatisfactory humidification quality will be even lower.
The rotational speed and length of the pelletizer drum determine the work required to form the pellet. The forces acting on the granules must be sufficient to compact it and not exceed the adhesion force of the particles. The optimal parameters of capillary interaction are difficult to determine and interdependent. The parameters corresponding to the acceptable pelletizing quality are usually established experimentally.
Many researchers are increasingly focusing on the analysis of capillary forces inside the pellets to optimize their fractional composition.5,6) However, a more complete picture of the ongoing processes will never lose its relevance.
Also in the applied sense, the solution of this problem is of considerable interest for conditions of low and close to zero field strengths of mass forces, when capillary effects are significant. The formation of an agglomerate is similar to the formation of condensation or coagulation structures; under the action of capillary adhesion forces, capillary structures arise, which then leave their mark on the structure and properties of materials obtained from such systems.
The study of the mechanism of action of capillary adhesion will optimize and predict the effect of moisture on the density, adhesion, rheological and molding properties of powder mixtures. This opens up the possibility of controlling the technological properties of raw materials in the production of concrete, silicate, ceramic and other building materials.
Most of the packing models for discrete media make it possible to judge only about the geometric configuration of chaotic or regular structures consisting of bodies of regular shape (as a rule, equivalent spheres are considered) without taking into account a number of factors characteristic of real granular materials.7,8) These factors include the presence in almost any material in its natural state of the third phase component-moisture.
Small spherical particles are placed in the gaps between large spheres, thereby simulating the formation of arched structures. The described procedure is performed in a loop until the system reaches a static state. Such a three-dimensional model does not reflect the principle of capillary particle interaction. The system of “contracting” (Δр+) and “tearing” forces (Δр−) of the surface tension of a liquid with a material must be applied to the obtained geometric structure of particles.
In work9) a mathematical model that allows to determine the force of interaction (adhesion) of particles participating in granule formation, depending on the amount of liquid at the point of contact and the wetting ability of the liquid (contact angle) is developed. The capillary cuff is divided into two arcs and the segment connecting them, without taking into account the size and sphericity of the system. Those, if the liquid is small, then it is distributed in discrete disconnected rings at the points of contact of the particles (capillary disconnected state or the state of a trapped liquid).
A certain angle θ, the so-called contact angle of wetting (by convention, is counted into the interior of the liquid phase), depending on the values of surface free energies at the point of contact of the three phases is established.9,10,11,12) The porosity of the surface of the particles of the starting material, which is understood as various microcracks, surface roughness, grooves and cracks, is represented by the replacement of microcracks with cones. This allows you to determine the volume of cavities on the surface of the particle, filled with water.
The conditions for touching a solid with a given contact angle θ and, for example, the total volume of the liquid as boundary conditions are usually specified.
Methods for determining the moisture content of the material W, depending on the granulometric composition of the feedstock and the conditions of its wetting, which make it possible to obtain their maximum density and the highest granule formation rate during the formation of granules at the maximum adhesion force Fadh, have been developed. Most often, such models are based on the processing of empirical data using additional coefficients or unacceptable simplifications.13,14)
As a result, a complete system of mathematical models should describe the process of granule formation of a bulk material, taking into account the kinetics of pelletizing, the interaction between the initial particles during the formation of granules, dynamic agglomeration, the ability of particles to form clusters among themselves, and surface porosity.
The purpose of the article is to develop scientific ideas about the capillary interaction of particles inside granules during their pelletizing and to develop recommendations for improving the process of their formation when preparing a charge for sintering.
A deterministic model of capillary adhesion of two particles of arbitrary size and properties based on the calculation of the geometric parameters of the liquid meniscus has been developed.
Real phase boundaries are thin layers of complex structure. In such boundary layers, substance molecules interact simultaneously with molecules of both phases. Therefore, the structure and properties of the transition layer differ significantly from the structure and properties of matter in the internal volumes of the phases. The effective thickness of the transition layer is very small and is of the order of the molecular interaction radius (about 10−9 m).15,16,17,18,19) A detailed examination of the structure of the transition layer is a very difficult task. Until now, despite significant efforts and the presence of serious studies, the molecular theory of transition layers is far from complete.
The very small thickness of the transition layer is an argument in favor of idealization, in which the phase boundary is treated simply as a geometric surface. This surface model is often referred to as a fracture surface; when passing through the surface, the properties of a substance (density, internal energy, velocity, etc.) change abruptly. The transition to the geometric interface of the phases greatly simplifies the analysis. However, it is obvious that with the introduction of the rupture surface model, information on the properties of the transition layer should not be lost. Therefore, the interphase boundaries should be associated with certain macroscopic (phenomenological) properties.
The admissibility of simplifications is confirmed by the indicator characterizing the size of the system at which capillary phenomena become significant.
| (1) |
If the size of the system L < b, then a spherical surface is formed. For water at a temperature of 20°C b = 3.8 mm. To a certain extent, in order to simplify the calculation of the interaction of particles of the agglomeration mixture, such a representation is permissible.
Otherwise, L > b the picture is fairly well known. The free surface is flat almost everywhere. Only near the walls, at distances approximately equal to b, is the curvature of the boundary observed or the surface has an elliptical shape.
Surface free energy (surface tension) at the boundary of two fluid phases: gas-liquid, liquid-liquid causes a surface pressure jump p [Pa] in phases proportional to the curvature of the boundary and is calculated by the well-known Laplace formula (1806)20)
| (2) |
The curvature of the surface H (z) is a second-order nonlinear differential operator. In particular, if the surface in the Cartesian coordinate system is defined as z = f (x, y) then
| (3) |
The average curvature of the surface at a given point, which (as is proved in differential geometry) is expressed in terms of the principal radiuses of curvature R1р [m] and R2р [m]
| (4) |
The radiuses R1р and R2р have the same sign if the corresponding centers of curvature lie on one side of the tangent plane at a point, and a different sign if on opposite sides.
In the system under consideration, capillary contracting forces are manifested when particles of radius R1 and R2 are pulled together due to the formation of a “meniscus”. The meniscus is a surface of revolution and also has two radiuses of curvature. Under the assumption of absolute wetting, the total force that must be overcome in order for the particles to begin to detach is
| (5) |
Formula (5) does not take into account the forces of mechanical adhesion of particles to each other.
The main difficulty for calculating the adhesion force F is to determine the values of the radii of curvature Rp1 and Rp2 (Fig. 1).
Cross-section of particles and water meniscus. (Online version in color.)
To calculate them, we perform a plane section through the centers of spherical particles.
The problem was solved in a two-dimensional Cartesian coordinate system (x, y). The first particle was placed at the origin of the coordinate system (0, 0). The second was displaced along the OX axis by a distance equal to the sum of the two radii of each particle (R1 + R2, 0). It is possible that the particles are not in contact and there is a gap D between them. It is not included in the current model.
To calculate the main radii of curvature of the meniscus, you need to know the following seven values:
• coordinates of points of intersection of particle circles О1 and О2 with meniscus circumference M - (х1, у1) and (х2, у2),
• radius of the circle of the meniscus Rp2 and coordinates of its center (хp, уp).
Knowing the coordinates of the center (хр, ур) of the meniscus circle, the second principal radius of curvature Rp1 is determined from the expression Rp1 = (yp – Rp2).
It should be borne in mind that the meniscus is in contact with the surfaces of particles with contact angles θ1 and θ2. Those, the circumference of the meniscus should intersect the circles of the particles O1 and O2 at angles θ1 and θ2, respectively (Fig. 1(a)).
The contact angle θ is a local characteristic of the surface at the point of contact with the liquid and gas and theoretically should not depend on the conditions in the bulk of the liquid phase. The given value of the angle θ acts as a boundary condition on the surface of the solid phase.
Real solid surfaces are always rough, due to which the experimentally determined values of the contact angle θe turn out to be less than the theoretical θt.
The circles of the particles O1, O2 and the meniscus M are described by the following functions, respectively
| (6) |
| (7) |
| (8) |
To calculate the main radii of curvature of the meniscus Rp1 and Rp2, the system of nonlinear equations of seven conditions with seven unknowns was
(10) – the difference between the functions of the circles of the O2 particle and the meniscus M ensures their intersection and the calculation of the coordinates of the points (х2, у2);
(11) - condition of equality of distances from the center of the meniscus circle (хр, ур) to intersection points (х1, у1) and (х2, у2);
(12) and (13) – the conditions for ensuring the specified angles of wetting of the meniscus fluid θ1 and θ2 with the surfaces of the particles O1 and O2, respectively;
(14) - the condition for observance of Newton’s third law;
(15) - the condition for providing a given amount of water in the meniscus (a given degree of filling the meniscus with liquid W).
In condition (11) distances from the center of the meniscus circle (хр, ур) to intersection points (х1, у1) and (х2, у2) calculated by
| (16) |
These distances are equal to the required meniscus radius Rр2.
The conditions (12) and (13) obtained from the geometric meaning of the derivative of the function as the tangent of the angle of inclination of the tangent to it at a given point. Function derivative y1(x) for circle О1 has the form
| (17) |
The functions of derivatives dy2(x)/dx for the particle circle О2 and dyр(x)/dx for the circumference of the meniscus М have a similar form
| (18) |
Equation (14) was introduced to comply with Newton’s third law, i.e. equality of the adhesion forces of two particles with each other. According to Pascal’s law, the pressure p inside the meniscus spreads evenly in all directions. The forces of capillary pressures of both particles must be the same, those
| (19) |
From (19) it follows that the areas S1 and S2 must be equal. In the longitudinal section, this means the equality of the corresponding arc lengths L from the intersection with the OX axis to the points (х1, у1) and (х2, у2). The length of the arc of circles O1 and O2 was calculated by the formula
| (20) |
Condition (15) for providing a given amount of water in the meniscus (a given degree of filling the meniscus with liquid W)
| (21) |
Cross-sectional area of the water meniscus SW(x1, x2, xp, yp, Rp) is equal to the difference between the areas under the function of the meniscus circumference and the corresponding areas of the parts of the circles O1 and O2 in the interval from x1 to x2, bounded by the OX axis,
| (22) |
Integral of the meniscus circumference function yp(x, xp, yp, Rp) calculated by the formula
| (23) |
Integrals of functions of particle circles y1(x) and y2(x) have a similar form.
The calculation of the proportion of water relative to the mass of particles is incorrect, because the number of contacts of particles of different diameters is an arbitrary value and depends on their size and packing density.
Therefore, an additional construction was performed - a line of the maximum theoretical volume (area) of liquid in the meniscus was drawn. This straight line corresponds to the border of the meniscus with a radius of curvature Rр2 → ∞, which contact angles θ1 and θ2 to particles О1 and О2 neglected.
We make additional geometric construction. With respect to the two considered particles, we construct a straight line tangent to the smaller circle, determine the length of the arc from the point of tangency to the OX axis and intersect this tangent with another circle at a point with the same arc length to the OX axis (Fig. 1(c)).
If the particle size is small, then as the fluid flows in, the cuffs gradually increase and merge with each other. For equal particles, such fusion occurs at an angle φ = 30°.
From the point of view of the correct geometry and adhesion forces, it follows that the meniscus, in order to maintain an equal wetting area and required contact angles, should tend to push the particles apart at a certain moisture content. For example, if the contact angles θ = 35°, then the maximum moisture capacity of such a material for equal particles is 6.65%. This means that further addition of moisture will tend to reduce the contact angle. However, this would contradict the molecular interaction of water and surface. Therefore, to restore the angle and increase the moisture content, the particles must be moved apart from each other or the shape of the meniscus surface must be changed to an elliptical one. Under the conditions under study, destruction of the granule is most likely.
Maximum possible theoretical water meniscus area Smax is equal to the difference of the areas under the function yS(x) for equal area lines S1 and S2 and corresponding areas of parts of circles О1 and О2 between х1 to х2, axle limited ОХ, (see Fig. 1(c))
| (24) |
We determine the coordinates of the points of intersection and tangency of the line AB with the circles O1 and O2, knowing which the area of the theoretical meniscus Smax is determined.
The equation for the derivative of the O2 particle circle has the form
| (25) |
Then the equation of the line tangent to point B on the circle
| (26) |
The intersection of the tangent and the circle O1 and the equality of the lengths of the arcs of the circles O1 and O2 will be ensured. Those, the system of equations was composed and solved with respect to хА and хB
| (27) |
The area under the tangent line in the interval from xA to xB bounded by the OX axis is equal to the area of the trapezoid ABCD and is calculated by the expression
| (28) |
The roots of the system of nonlinear Eqs. (9)-(15) were found by the steepest descent method.
The calculation results are applicable to all isotropic layers according to the Cavalieri - Acker principle.21)
Knowing the amount of water in the places of contact of particles, the relative humidity of the granules for the initial fractional composition of the charge is determined
| (29) |
A system of particles with radii of 1, 0.7, 0.5, 0.3, and 0.1 mm is considered below. Particle wetting angles θ and correspond to the range for sinter batch from 15° to 35°. At values of the particle radius
| (30) |
The volume of water in one meniscus per volume of water for all points of contact and recalculated for relative humidity for subsequent analyzes of granules consisting of particles of polyfraction composition must be recalculated. The maximum theoretical volume of water in the meniscus must be known for this purpose.
For the developed model, this will be the area of water in the meniscus section (Fig. 2).
Maximum theoretical cross-sectional area of the water meniscus. (Online version in color.)
Based on the simulation results, the value of the maximum theoretical cross-sectional area of the water meniscus can be calculated with sufficient accuracy by the formula
| (31) |
The results of modeling the main radii of curvature of the meniscus Rp1 and Rp2 are shown in Fig. 3.
Change of the main radii of curvature of the capillary meniscus of adhesion of two particles of different sizes and wettability (arrows indicate axis position): а - θ1 = θ2 = 15°; b - θ1 = θ2 = 25°; c - θ1 = θ2 = 35°; d - θ1 = 15°, θ2 = 35° ◆ - R2/R1 = 1/1; ◆ - R2/R1 = 0.7/1; ▲- R2/R1 = 0.5/1; × - R2/R1 = 0.3/1; ● - R2/R1 = 0.1/1. (Online version in color.)
It has been established that the main radii of curvature of the meniscus Rp1 and Rp2 largely depend on the ratio of the particle size, their contact angles and the amount of water in it.
The main radius Rp1 changes from 0 to 1 mm with an increase in the degree of filling of the meniscus W to 1. For identical particles of large diameter R1 = R2 = 1 mm, the change in Rp1 from the water fraction is most significant. This is due to the fact that the shape of the meniscus tends to a straight line parallel to the OX axis at a distance R1.
With a decrease in the size R2 to 0.1 mm, the value of the radius Rp1 decreases by more than 8 times. In this regard, its change occurs in a smaller range.
The value of the main radius Rp2 largely depends on the amount of water in the meniscus. Those, at the maximum filling of the meniscus with water, Rp2 → ∞ and means the destruction of the bond. Therefore, these moisture values do not correspond to optimal pelletizing. With increasing differences in particle sizes, the Rp2 radius is less susceptible to the amount of water in the meniscus and becomes commensurate with the Rp1 radius.
An increase in the contact angles θ1 and θ2 negatively affects the dependences of the main radii Rp1 and Rp2 on the addition of water to the meniscus. For equal particles R1 = R2 = 1 mm, with an increase in θ1 and θ2 from 15° to 35°, the permissible range of the degree of filling W, at which Rp1 is less than 1 mm, decreases from 61% to 32%, those approximately 2 times. The radius Rp1 cannot be greater than the radii R1 and R2. So when the wettability of the particles deteriorates, the optimum moisture content decreases.
The adhesion force of spherical particles of different diameters and contact angles was investigated (Fig. 4).
Adhesion forces of two particles of different diameters (the values of the markers correspond to Fig. 3). а - θ1 = θ2 = 15°; b - θ1 = θ2 = 25°; c - θ1 = θ2 = 35°; d - θ1 = 15°, θ2 = 35°. (Online version in color.)
The simulation results agree with the experimental data presented in.9)
From the simulation results, it follows that the value of the adhesion force of two particles depends on their sizes R1 and R2, contact angles θ1 and θ2 and the amount of water W in the meniscus and has an extreme dependence. The search for this maximum must be ensured in a production environment. According to the degree of influence, the control actions must first of all be carried out by the amount of water in the meniscus, then by the ratio of the particle sizes and wetting.
The maximum adhesion force for large identical particles R2/R1 = 1/1 is approximately 6 times higher than for particles with a larger size difference R2/R1 = 0.1/1. This is due to the large values of the main radii of curvature of the meniscus, i.e. larger area of contact of the water surface with particles. However, small particles adhere to large ones with less force, but their adhesion is more stable and durable at higher humidity. The adhesion force of the same particles decreases sharply with the addition of water. At humidity, which provides the maximum adhesion force of small fractions with large ones, the bonds of large particles with each other will break. The moisture content of the maximum adhesion force of large particles R2/R1 = 1/1 is approximately 3 times lower than for particles with a large difference in size R2/R1 = 0.1/1. It should also be borne in mind that, based on the concept of the structure of the granules, the same particles of small size (on average 0.1 mm) are located on its surface. The adhesion force F0.1/0.1 of such particles is ten times less than for particles R2/R1 = 1/1, and about 1.7 lower than for particles with a particle size of R2/R1 = 0.1/1. Therefore, granules with a core consisting of two or more medium-sized particles are often found.
Agglomeration charge is composed of components with different physical properties. In addition to particle size, wettability is also different. For example, for small coke, the wettability angle is θ = 25–35°, and for iron ore particles θ = 15–20°. Therefore, the applicability of the model for calculating the interaction of particles of different diameters with different contact angles is of research interest.
The results of modeling the interaction of two particles with the ratio of radii R2/R1 = 0.1/1 and contact angles θ1 = 15°, θ2 = 35° in Fig. 4(d), are presented. Comparing the diagrams in Figs. 4(d) and 4(c), it can be seen that such a system with an average contact angle θ1 = θ2 = 25° can be considered with a sufficient degree of accuracy. Those, to simplify further calculations, the capillary system can be simplified with the same average contact angle. It also follows that the capillary adhesion of two particles with low wettability is unlikely.
Of particular interest is the possibility of replacing two particles of different sizes with particles of equal size with the same adhesion forces, especially when changing the amount of water in the meniscus. Because the change in the adhesion force of two particles of the same and different size differ significantly when water is added, then it is impossible to obtain a complete equivalent replacement.
With imperfect wetting of materials, the adhesive force decreases slightly. However, since the equation includes the cosine of the contact angle, then at contact angles from 15° to 30° the decrease in force will not be as significant as from moisture. The decrease in the optimal humidity by 1.0–1.5%abs is much more significant.
Under the conditions under study, the internal state of a granule is characterized by two parameters - the distribution of forces and water along its depth. With a properly debugged pelletizing technology, these parameters should be in equilibrium. This means that the mass transfer of water inside the granule must be completed and all the forces of capillary interaction of the particles must be the same.
The first condition will occur when the degree of filling W with water of the menisci throughout the depth of the granule will be the same.
Let us consider in a simplified way the granules of the agglomeration mixture as the interaction of the lumping 3 mm and the lumped 0.1 mm fractions with an average wetting angle θ = 25°. Inside the granules, the interaction of the fine and coarse fractions R2/R1 = 0.1/3 will be. The outer layers will consist only of fine fractions R2/R1 = 0.1/0.1. Thus, the distribution of capillary adhesion forces will be characterized by two curves F0.1/0.1 for R2/R1 = 0.1/0.1 and F0.1/3 for R2/R1 = 0.1/0.3. The point of intersection of these curves corresponds to the equilibrium state in terms of the degree of filling with water W0.1/3 = W0.1/0.1 and adhesion forces F0.1/3 = F0.1/0.1.
It should be noted that the recalculation of the adhesion forces in accordance with formula (24) does not shift the curves along the ОХ axis, therefore, the degree of filling W corresponding to the maximum adhesion forces will not change its value with a proportional change in the particle size. However, the first curve decreases along ОY by a factor of 10, and the second - by a factor of 3.
Based on Fig. 4(b), it follows that for the conditions under study, the value of the optimal degree of filling the meniscus with water is Wopt = 0.086. The adhesion force of the particles will be F = 2.7·10−5 N.
This value of the degree of filling the meniscus Wopt is lower than the optimal values of W for particles with a particle size of R2/R1 = 0.1/0.1 and R2/R1 = 0.1/3. Therefore, a slight increase in moisture can improve the quality of pelletizing due to an increase in the adhesion forces of small particles among themselves and with large ones. However, in this case the system will go out from equilibrium. But this is possible for a pelletizing time of 1.5–2 minutes. Then, for such conditions, one should assert not about the optimal degree of filling the meniscus, but about the optimal range of the degree of filling the meniscus from Wopt = 0.086 to W0.1/3 = 0.21. But at the same time, the stratification of granules, established by researchers, is possible with impulse water supply to the pelletizer.22)
Further, modeling the placement of particles inside the granule, using the formula (23), the optimal relative humidity of the granule is determined.
Modeling the interaction of two particles by solving the developed system of equations (9.1)–(9.7) is a rather difficult task and requires special software tools. Therefore, in order to simplify the calculations, the following regression equations, allowing with sufficient accuracy to calculate the values of the principal radii of curvature in the range of the degree of filling of the meniscus from 0.1 to 0.6, are proposed
| (32) |
| (33) |
Substituting the values of Rp1 and Rp2, calculated by formulas (32) and (33), into expression (5), the adhesion force of the particles is determined.
(1) As a result of research using the developed deterministic mathematical model, the main radii of curvature of the water meniscus for particles of the agglomeration mixture of arbitrary size and contact angles are determined. For this purpose, a new indicator - the degree of filling the theoretical maximum volume of the meniscus is introduced.
(2) The force of capillary adhesion of two particles has a maximum. For particles with a large difference in size, the adhesion force is more stable when the moisture changes near the maximum adhesion force. As the wettability of the particles deteriorates, the optimum moisture content decreases.
(3) For the production conditions of pelletizing, reaching an equilibrium state of the capillary adhesion force and the concentration of water in the volume of the granule is unlikely. Therefore, it is more correct to assert about the range of optimal moisture content, starting from the equilibrium moisture content and ending with the moisture content of maximum adhesion of small and large fractions.
