Studying the interaction process of a solid particles flow with the hydraulic classifier flowing part

The paper provides a theoretical description of the solid particles movement process along with the horizontal flow of slurry within the hydraulic classifier flow part. Three modes of movement of solid particles are described: deceleration relative to the slurry flow at a constant speed or its increase, acceleration relative to the slurry flow at a decrease in its speed, with the speed of the slurry flow. The complex equation of solid particles motion in a horizontal co-flow of liquid. A graphic description of the process is given. A laboratory installation was developed to study the interaction of the hydraulic classifier flow part with the slurry flow. Experimental studies were carried out and quantitative process parameters were determined during the movement of rolled quartz solid particles in the size range < 1.8 mm and < 0.2 mm. For the first time in crisis economic conditions, in order to confirm the reliability of previously obtained analytical dependencies to determine the rational parameters of hydraulic disintegrators and determine the operational characteristics of equipment, full-scale experimental studies of the JPD 300-120 suction dredger equipped with an innovative ejector soil suction head with a jet disintegrator system were carried out.


Introduction
The processing of fine-grained materials is traditionally carried out in gravitational apparatus with a horizontal movement of the carrier flow: horizontal classifiers, deslimators, screw classifiers, decanters, horizontal settling tanks, etc. [1] Despite the mass nature of the work aimed at studying the process of falling solid particles in a stationary liquid, the process of motion of solid particles relative to a moving horizontal flow of a carrier slurry remains practically unexplored.Unprovenly considered that in a carrier horizontal flow solid particles move with a flow velocity [2].
It is known that the rational size for gravitational processing, for example, quartz particles, is 0.15 < d < 5 mm.In this case, the separation range for horizontal classifiers can be considered the fineness 0.1 < d < 0.2 mm [3].
Thus, to justify the rational parameters of the apparatus for the gravitational processing of granular materials in a horizontal flow, it is sufficient to simulate the movement of solid particles which have a size withing the separation range expected values [4,5].

Methodology
The work used a complex research method.A theoretical study of the interaction process between the hydraulic classifier flow part and the flow of solid particles that move along with IOP Publishing doi:10.1088/1755-1315/1254/1/012047 2 the horizontal flow of the slurry is made in the form of physical and mathematical modeling.The process of co-movement of solid particles and slurry is described using the Lagrange method.
Experimental study of the process was carried out on the developed laboratory installation [6,7].Processing and analysis of experimental data were performed using the methods of mathematical statistics and planning of experiments, and the approximating dependences were obtained by the least squares method.Measurements of technological and design parameters were carried out using standard metric equipment.

A theoretical study
The characteristic modes of movement of solid particles in a horizontal slurry flow during hydraulic classification in a horizontal flow are: • movement of particles with deceleration relative to the carrier flow at a constant and increasing speed of its movement; • movement of particles with acceleration relative to the carrier flow with a decrease in the speed of its movement; • movement of particles with the speed of the carrier flow.
To describe the process of interaction of the horizontal hydraulic classifier flow part with a slurry flow that includes suspended solid particles moving along, we apply the Lagrange method.Let a solid particle move under the action of a horizontal fluid flow along it (figure 1).The horizontal projection of the velocity vector of a solid particle is directed along the motion of the liquid.The horizontal motion of a solid particle is characterized by its mass in a liquid m, force of the hydrodynamic pressure F av at the velocity of the carrier flow U x and resistance force P , at the speed of movement of a solid particle relative to the carrier flow v x (figure 1).
Figure 1.Scheme of the process of a solid particle motion in a horizontal liquid flow.
After transformations, the equation of motion of a solid particle in a horizontal liquid flow will take the form The integration constant C is defined as the initial velocity of the solid particle U p x0 .
Let us analyze the obtained dependence and consider three modes of motion of a solid particle relative to the carrier flow.
Let the solid particle be motionless, that is U p x = 0.In this case, the velocity of the liquid relative to the solid particle U x is equal to the velocity of the solid particle relative v x The connected movement of a solid particle and a carrier flow can occur in the range U x > v x ≥ 0, in this case, the equation of motion (2) takes the form The counter motion of a solid particle and a carrier flow is characterized by the mode of motion U x < v x , while the equation of motion will retain the form (2).
Graphical representation of this process, in the absence of relative motion of the carrier flow and solid particles in the original cut, that is Process analysis shows that within time t the function U p x is limited by the functions U x = const and U x .The speed of a solid particle movement, in the i-th section, relative to the carrier horizontal flow, moving at a constant velocity, is taken as v 1 xi , but with relation to the decelerating flow v 2 xi (figure 2).The graphical representation of this process is built on the basis of the following hypothesis.Carrier flow moves in laminar mode.At a time interval t, at a constant velocity of the carrier flow a solid particle can move with relative deceleration, that is When the carrier flow slows down, the solid particle can move with relative acceleration, U p xi ≥ U xi .That is, the particle velocity function is within The theoretical substantiation of the process of relative movement of solid particles and the carrier horizontal flow is very difficult.A rational and more informative method of modeling the process, in a narrow designated area, can be considered an experimental study, which will allow to determine the values of relative velocities v 1 xi , v 2 xi .One of the important factors influencing the process of relative motion of a solid particle in an carrier flow is the particle form.However, in the designated area of motion velocities and size particles, characterized by the number Re, there is a mode of motion of a solid particle close to laminar, at which, in accordance with the data of [6], the form of the particle has an insignificant effect on its motion.
The vertical projection of the particle velocity vector is characterized by the sediment process, and is determined by the absolute value of the particle weight in the liquid and the resistance force in liquid, directed in the direction opposite to the direction of movement (figure 1).
where F g -gravity, H; F A -strength of Archimedes, H; P -resistance force, H.
Due to the traditional nature of the use of raw materials with a high content of rounded quartz particles (construction and glass sands), the investigated area of fineness 0.15 < d < 5 mm turned out to be the most studied experimentally.The literature sources provide data from different authors who obtained experimental values of the velocity of free fall of quartz particles [8].Thus, in view of the presence of a sufficient amount of experimental material, in further studies, the final velocity of the fall of quartz particles will be taken according to the experimental data known in the literature.It should also be noted that at the indicated velocities of the carrier flow, the sedimentation process time is much longer than the time of unsteady motion of a solid particle, so this factor is not taken into account in the model.
When taking into account the relative displacement of a solid particle driven by a horizontal carrier flow in the horizontal and vertical directions, the formula for constructing the particle trajectory will take the form Let us replace the parameter that takes into account the relative vertical displacement by the experimental value of the falling velocity of a solid particle in water U y − v y = ω then where ω is experimental value of the speed of free fall of a solid particle, m/s.The integration constant C characterizes the initial conditions for the motion of a solid particle, and, when the origin of coordinates is located in the initial position of the solid particle, the constant C = 0.
Let us assume that the movement of a liquid and a solid particle proceeds without mutual displacement, that is v x = 0. Thus, we simplify the consideration of the process and equate the horizontal projection of the velocity of a solid particle and the velocity of a liquid U x p = U x .Then the equation of motion of a solid particle under the action of a horizontal flow takes the form x = U x ω dy + C.

and after integration
x = U x ω y (5)

Experimental studies
Laboratory studies of the interaction process of the horizontal hydraulic classifier flow part with a pulp flow that includes suspended solid particles are performed on the basis of specialized laboratory equipment.Laboratory equipment for studying the processes of gravitational processing of granular materials is based on the use of a horizontal multi-section classifier HMC.With the use of the HMC classifier, fundamental laboratory studies were carried out to study the movement of granular material particles in a horizontal carrier flow.Laboratory installation based on the HMC classifier, consists of a horizontal multi-section classifier 1, a jet feeder 2, a water supply system (figure 3).The horizontal multi-section classifier is an elongated vessel of rectangular section 1, in the lower part divided into 12 compartments 3, equipped with discharge nozzles 4. Power is supplied through the intake pipe 5, the sludge is drained through the drain pipe 6.
The jet feeder 2 is designed to prepare the slurry of specified parameters and feed it into the inlet pipe 5 of the horizontal multi-section classifier 1.
Preparation of the slurry in the jet feeder 2 is carried out by supplying water through a pipe branch with irrigation nozzles 7, the transport of the prepared slurry is carried out by means of a jet pump 8.For water supply of the pipe branch 7 and the jet pump 8, a water supply system is used.
The water supply system consists of a main valve 9, a control valve 10 and a pressure gauge 11 of the water supply system of the pipe branch 7, a control valve 12 and a pressure gauge 13 of the water supply system of the jet pump 8. Transportation of water and sludge is carried out through flexible pipes 14.The equipment is installed on support frames 15, 16.The source of water for the operation of laboratory equipment is the plumbing.
The complexity of modeling the process of gravitational processing of granular materials in a carrier horizontal flow lies in the need to take into account the horizontal and vertical movement of solid particles.At the same time, the following factors may have some influence on the solid particles motion process: co-flowing relative vertical and horizontal motion of solid particles; constraint of the solid particles movement; movement of solid particles relative to the carrier horizontal flow.
In order to take into account the above factors and quantify their influence on the process of granular materials gravitational processing, the obtained experimental data were processed.At the same time, the tasks of quantifying and assessing the need to take into account such factors were solved: • movement of solid particles relative to the carrier horizontal flow; • constraint of the solid particles movement; • movement of solid particles in various parts of the gravity processing process in a horizontal flow.

The movement of solid particles relative to the carrier flow
The study of the movement of solid particles from this size range was devoted to experimental studies carried out on a laboratory installation using a horizontal multi-section classifier HMC (figure 3).As a result of laboratory studies, the movement of rounded quartz particles in the size range < 1.8 mm and < 0.2 mm was studied.The calculation of the theoretical trajectory of particle motion was performed using the dependence (5) (figure 4).
The sedimentation process proceeded at an average horizontal flow velocity of 0.03 m/s.Processing of the experimental data showed that in the size range < 0.2 mm, more than 90% of solid particles sedimented in compartments No. 2-6, located in the theoretical area of sedimentation of rounded quartz particles with a size of 0.2 ... 0.1 mm (table 1, figure 5).Thus, the minimum value of the probability of sediment of a rounded quartz particles with a particle size of 0.2 ... 0.1 mm in the theoretical area of deposition is 90%.And when taking into account the presence of particles smaller than 0.1 mm in the used raw material, the theoretical area expands and the probability tends to 100%.
The purpose of laboratory studies was also to determine the effect of the concentration of solid particles in the feed on the deposition process.Processing of experimental data showed that the concentration of solids affects the constraint of particle movement, causing a decrease in the fall velocity (figure 6).In a detailed study of the mass characteristics of the material deposited in the area with a 90% probability of sediment of rounded quartz particles with a particles size of 0.1 ... 0.2 mm the average value of the mass difference in compartments No. 2-6 at a solid concentration in the slurry of 3% and 19% was 3%.An analysis of experimental data from laboratory studies of the movement of rounded quartz particles with a particle size of < 1.8 mm showed, that the influence of the solid concentration in the slurry within 0 ... 20% on the average size of the granular material deposited in the compartments does not go beyond 5%, and the average error is 1.6% (table 2, figure 7, 8).
This value indicates the fact of the insignificance of the influence of this factor on the process  of the solid particles sedimentation in the concentration range of 0...20%.Thus, the performed laboratory studies of the rounded quartz particles deposition process, under the influence of a horizontal carrier flow, confirm the relevance of the application of dependence (5) to calculate the parameters of the solid particles sedimantation with particle size < 1.8 mm and < 0.2 mm in the concentration range 0. . .20%.

Conclusions
The results of the theoretical studies of the interaction process between the flow part of a horizontal hydraulic classifier and the laminar slurry flow, which includes suspended solid particles, allow us to make the following conclusions: 1.The basis for the theoretical description of the gravitational sedimantation of solid particles during its passing and counter motion relative to the horizontal carrier flow was the Lagrange method.2. At the velosity U x of v x the liquid movement, and the speed U p x of movement of the solid particle relative to the liquid, the absolute speed of the solid particle in passing motion with the carrier flow can be in the ranges: 1 -U p x < U x ; 2 -U p x > U x ; 3 -U p x = U x , v x = 0. 3.An initial theoretical dependence was obtained to determine the path traveled by a solid particle under the conditions of gravitational sedimantation under the action of a horizontal carrier flow, which takes into account possible scenarios for the relative motion of a solid particle and liquid.

Figure 2 .
Figure 2. Scheme of the process of co-movement of a solid particle and a horizontal liquid flow.

Figure 3 .
Figure 3. Laboratory installation for the study of gravitational processing of granular materials based on HMC: (a) the general view of the installation; (b) a circuit scheme of the installation.

Figure 4 .Table 1 .
Figure 4. Theoretical trajectory of motion of rounded quartz particles: 1 -trajectory of movement of rounded quartz particles with size 0.2 mm; 2 -trajectory of movement of rounded quartz particles with size 0.1 mm

Figure 5 .
Figure 5. Sedimentation of rounded quartz particles with a particle size of < 0.2 mm when fed with a volumetric concentration (C o , %).

Figure 7 .
Figure 7. Sedimentation of rounded quartz particles with a particle size of < 1.8 mm when fed with a volumetric concentration (C o , %).

Table 2 .
Average size of deposited quartz particles with a size of < 1.8 mm, %.