Modeling and force analysis of drum devices based on the geometry of the material segment

The aim of the research is the numerical simulation of the drum mixer operation, including the identification of geometrical indicators of a material pile in a rotating drum for conducting the force analysis and determining the expended power. The influence of the required performance of the drum device on the parameters of its container with regard to the filling degree is determined analytically using the mixer as an example. The functional model of the central angle of the material segment on the filling degree of the container is established. The expressions for the parameters of a material segment in a rotating container are revealed. They allowed for determining the torque and power consumption of the drive based on the power analysis. The graphic material of changes in the calculated indicators in the modeling process is presented.


Introduction
A variety of drum devices with a similar principle of operation are used in economic activities of people. One of the most commonly used devices are drum mixers. They are used in construction, engineering, chemical and food industries, agriculture. When studying drum mixers, researchers are most interested in the justification of the quality indicators of the mixer and its performance [1][2][3][4]. For this purpose, both experimental methods [1][2][3][4] and computer numerical analysis are used [5,7]. In some cases, the drum is equipped with additional structural elements [1][2][3], or without them [4,5]. Another area of research is the force analysis of the mixer [6][7][8], which allows one to determine the mixer drive power. In recent years, the interest in drum devices has increased, since they have been used to produce graphene.
Drum devices are used to implement many of the processes that are based on mixing. The use of mixing paddles in the drum intensifies the process of mixing the components, reducing the mixing time. However, a side effect is the inevitability of falling portions of the material raised by the paddles on the main body of the product. This limits the use of paddle drum mixers for mixing particles with low strength. Such drum-type devices without paddles or with micro-paddles are used for the particular case of mixing, which is coating (for example, to cover the seeds with a shell, or for manufacture of pellets -liquid particles with adhered solid particles outside). Drum devices are also used in triers for segregation, removal of particles of a certain geometric size. Numerical analysis of the force impact allows for simulating a drum mixer for the given conditions of its design or performance. This makes it possible to optimize the parameters of the mixer and establish functional dependencies between the features of its design and performance.
At the moment, it is still a challenge to determine the power required to rotate the drum with a solid bulk material, depending on the degree of filling of the drum, the physicomechanical properties of the material of the mixture and the subcritical frequency of the drum rotation.
Use of computer programs based on the finite element method is an advanced method of research. However, it requires large investments in the purchase of expensive specialized computer programs and computers with great potential. Such investments are rarely economically justified for performing calculations in ordinary engineering problems solved under practical conditions on ordinary computers.
The aim of the research is the numerical simulation of the drum mixer operation, including the identification of geometrical indicators of a material pile in a rotating drum for conducting the force analysis and determining the expended power.

Experimental Part
In the course of the research, it was envisaged to use analytical methods based on the required performance or mixer parameters to establish the geometrical parameters of the material pile in the drum mixer container, to determine the dependence between the pile geometry and the current material forces in the rotating drum, torque and power consumption. To assess the numerical values of specific parameters of the mixer and its work on the basis of established formulas, the MathCAD mathematical pack was used.
The required duration T c of components mixing by drum mixers is 180-300 s [1,2,8]. Within this period, sufficiently rapid uniform distribution of components occurs throughout the volume of the mixture. After reaching certain quality of the mixture, further mixing of the components is impractical due to the stabilization process.
According to [7], the performance of a batch mixer can be calculated, kg/s: where M -the mass of the feed portion, kg; 0 T -the cycle time of the mixer, s; 0 V -the volume of the mixer, m 3 ; E -the filling degree of the mixer;  -the density of material pile, kg/m 3 ; 1 T -the duration of the component load, s; C T -the duration of mixing the components, s; 2 T -the duration of unloading of the components, s; 3 T -the duration of additional operations, s; D, L -the diameter and length of the container, m.
For continuous mixers, the mixer's volume must comply with the condition: where p Q -the productivity of the technological mixing line.
Let us consider a cross section of a rotating drum with material pile inside it (Figure 1). In a rotating drum, particles of the material are captured by a moving element (here -a container) and rise up along the walls of the cylindrical container 1. Reaching a certain angle of ascent, particles 3 fall down, forming the outer edge of material 2 (А 0 А 1 ) at an angle  -a dynamic angle of repose of the material relative to the horizontal, rad.
The value of the dynamic angle of material collapse  is determined for a specific material based on field observations. When the drum rotates, the material is constantly moving from the upper part of the material segment to its base. As a result, a new surface is constantly formed at the angle of dynamic collapse of the material. Therefore, when modeling, we accept the assumption of linearity of the collapse surface at the subcritical angular velocity, which forces the material to rotate with the drum. The external forces applied to the material particle M (as an analogue of the material elementary sector) are the pressure of the upstream material column (forming gravitation force G), the inertial effect of rotation (centrifugal force F c of the material column above the particle M along the radius of the container). Under the action of these forces, the material elementary sector is pressed against the inner wall of the container, creating the reaction force of the normal pressure N, as well as the friction force tr F of the material against the container wall. Overcoming these forces, there is a force applied to the friction of the rotating cylindrical wall of the container tr F onto the particle M. (4) In the presence of the container rotation, a moment of material friction against the walls is created. If the friction force tr F is small (with a small friction coefficient f of the material against the container's wall, or a large angle of the wall placement relative to the horizontal), then there is a possibility for the material segment to slide along the container's walls. This condition is described by the first equation in the system (4), where tr F N f   . Otherwise, the material is lifted to a certain height, corresponding to the location of the particles at a certain angle from the lowest point of the container. Then the material of the outer layers is poured out due to insufficient centrifugal forces, and the surface 2 is formed at an angle  of repose of material pile. This condition is described by the second equation in the system (4). The drum rotates by the drive, which overcomes the total moment M k (Hm) from the friction force of the particles of all j material elementary segments, corresponding to the angle interval Figure 2, i.e. from  0 to  1 : Drive power for the drum rotation P, W: For the force analysis of the movement of the material particles and the working body, it is necessary to know the magnitudes of the acting forces ( Figure 1) on a certain particle M, as an analogue of the material elementary sector: Therefore, the purpose of the research is to determine the functional dependences of the height of the vertical and radial material columns, allowing one to calculate the values and to establish the dependences of changes in their values to obtain a mathematical model of changes in the specified parameters, torque and power to the container's rotation drive.
Let us consider the location of the material in the container. The material is located along the side wall and forms an angle  relative to the horizontal. As the material in the container increases, the angle of repose practically does not change. However, the filled cross-sectional area changes (i.e. the area of the segment cut by chord А 0 А 1 changes) and, accordingly, the degree of the container filling in this crosssection also changes. The central angle for the segment is indicated as , rad. The angular coordinate of А 0 relative to the vertical (OX -axis) is indicated as  0 (rad.), А 1 (relative to the OX axis) indicated as  1 . The current angle, at the calculations in the interval ( 0 ;  1 ), is indicated as β. The angular coordinate of A`` (the vertical projection of A 1 on the lower part of the circle) relative to the vertical (OX axis) is indicated as  2 (rad.).
The circle area (m 2 ) is determined: The degree of filling of the circle's cross section: The segment area, m 2 : -the central angle of the material sector, rad. Consequently: The question is how to solve this equation and find the angle . There is no exact analytical solution. The solution to this problem can be provided by a computer program. The results of numerical simulation to determine this angle are presented below.
It is necessary to determine the interrelation of the geometric indicators of the sector, taking into account the dependences for . Let us consider the case (Figure 2) when the А 0 , А 1 points and the segment lie on one side of the vertical (X-axis). In this case, the angles  0 and  1 can be expressed through the known angles  and . In this case, Let us find the OB from OBN , where The angle from the vertical to OA is indicated as β.
For the arc A``A```, height of the material layer is: (12) The angles of the material segment are determined as: Coordinates of characteristic points: For the current i -values of the β angle in the range of angles ( 0 1 ;   ), heights of the i-layer of the material are, m: For the arc 0 At bigger angles -1 0 h  .

Results and discussion
Numerical studies were carried out to determine the angle .
The numerical values of the calculated sector area S (m 2 ) are shown in Figure 3, where the number of the circle diameter D (m) is indicated horizontally, and the number of the filling degree S z (fraction) -vertically. In the selection process of the numerical values of the central angle , its expression (10) was equated to zero. For this purpose, the point of transition of the sign from "minus" to "plus" was found, determining the difference between the right and the left parts of the expression (10). The interval of the specified differences G000 and G001 (as an indicator of non-conformity to zero of the difference between the values of the right and the left parts) is 0.1 relative to the numerical values of the angles    -4 -9.828·10 -4 -9.828·10 -4 -9.828·10 -4 -9.828·10 -4 -9.828·10 -4 -9.828·10 -4 -9.349·10 -4 -9.349·10 -4 -9.349·10 -4 -9.349·10 -4 -9.349·10 -4 -9.349·10 -4 -9.349·10 -4 8.591·10 -4 8.591·10 -4 8.591·10 -4 8.591·10 -4 8.591·10 -      Taking into account the values of heights h 1 and h o , the magnitudes of the acting forces change in a similar way (Figure 10). The nature of centrifugal force F c corresponds to the character of height h o , and the character of gravitation force G change -to the character of h 1 . The nature of the normal reaction N changes according to the interaction of these forces. Friction force F tr changes by analogy with the normal reaction N, according to the first equation of the system (4). In this case, when the numerical values of the friction force according to this formula are less than the values of the friction force according to the second equation of the system (4), reliable contact of the material and the surface of the container (β  60) is observed. Later, this condition is not satisfied, and there is a possibility of the material sliding along the container's wall. It is limited by the material particles located below.
These functions of friction force cause the occurrence of two variants of torques ( Figure 11). Comparing their total values, and translating into power consumption, we get two options of power. The calculated power options are 134 W and 140 W, the difference is 4.9 %. The filling degree of the container is 30 % with the diameter and length of the container of 0.6 m, the density of the material 650 kg/m 3 , and the friction coefficient f of the material along the wall is 0.5,  = 45.

Conclusion
Thus, the established expressions allow for the simulation of changes in the basic geometrical indicators of a material segment in cylindrical containers, providing calculation with an error not exceeding 5 %, which allows establishing the functions of friction force, torque and power of container rotation. The implementation of the identified dependences using computer programs (for example, MathCAD) allows both calculating the acting forces, moments and power consumption values for specific parameters, as well as carrying out numerical studies of this process, optimizing the mixer parameters.