Modeling of the roller pressing of fibrous materials

The work is devoted to modeling the regularities of hydraulic pressure distribution and fluid removal from the fibrous material in the contact zone. It was found that the hydraulic pressure in the compression zone increases from zero at the initial point of contact to a maximum at the point of maximum compression of the fibrous material. The patterns of distribution of hydraulic pressure in the recovery zone depend on the length of the section, where the fluid moves from the layer of material to the elastic coatings. It was revealed that the amount of liquid removed at the beginning of the compression zone grows faster, then the growth rate is much lower, and at the end of the compression zone, the removed liquid is stabilized. The patterns of change in the removed fluid in the recovery zone depend on the angle, which determines the position of the point where the fluid changes direction.


Introduction
A special group in the technology of mechanical processing of materials in roller machines is the roller pressing of fibrous materials.
The roller pressing of fibrous material consists of two aspects -contact interaction and hydraulic interaction. In this case, a change in the indices of the first aspect affects the change in the indices of the second aspect and vice versa. Therefore, the study of one phenomenon without considering the second phenomenon does not allow obtaining reliable parameters of the technological process.
Following the two phenomena, the theory of roller pressing of fibrous materials presents a joint solution of two problems -contact and hydraulic problems.
*Corresponding author: shavkat-xurramov59@mail.ru Analysis of works devoted to the main hydraulic problems, such as modeling the regularities of the distribution of hydraulic pressure [9][10][11][12][13][14][15][16] and modeling the residual moisture content of fibrous material [17][18][19][20][21][22][23], showed that the existing models of the distribution of hydraulic pressure and changes in residual moisture were obtained with the introduction of two-roll models and materials that do not correspond to the real physical phenomena of the roller pressing of fibrous materials. In addition, the solutions of hydraulic problems in them are not related to the contact problems of two-roll squeezing modules.
The work is devoted to solving the problem of modeling the regularities of the distribution of hydraulic pressure and changes in the residual moisture content of the squeezed material, taking into account the results of solving the contact problems of tworoll squeezing modules.

Materials and Methods
Consider a two-roll squeezing module (Figure 1), in which the rollers have unequal diameters and elastic coatings made of materials of different stiffness and friction coefficients; both rollers are driven, the layer of material has a uniform thickness.
The lower roll contact curve (curve A 1 A 2 ) consists of two zones , and the recovery occurs in the zone 2 3 A A . First, consider the process of fluid filtration in the compression zone 3 1 A A . In the compression zone, the liquid moves from the material to the roller coatings along the polar radius [5].
To solve the problems of the theory of roller pressing, it is necessary to develop the models of the roll contact curves and filtration rates in the area of pressing.
The model of the contact curve of the lower roller in the compression zone for the considered two-roll module has the form [2]. 1 11 11 11 11 11 11 11 cos 1 1 cos In [16], models of the filtration rates of the liquid flowing through the contact curve of the lower roller were determined In [13], assuming the working hypothesis of the orthogonality of the maximum and minimum porosity, the applicability of the generalized Darcy's law to an anisotropic medium was established (5) and found the formula for the filtration coefficient depending on the direction min 2 max 2 sin cos are the hydraulic pressure and filtration rate in direction r ; is the filtration coefficient (minimum) along the axis Ox ;   is the angle that defines the direction r ;   is the fluid viscosity coefficient. According to formulas (3), (5) and (6), we obtain We substitute expression 11 11  d dr from equation (2) After integration, we obtain Expanding the logarithmic functions in a series and limiting ourselves to terms up to the fifth power relative 11  , we obtain where The integration constant 11 C is determined by the initial condition In the recovery zone, the liquid, to the left of the certain point 4 A , moves from the fibrous material to the roller covering, and to the right of this point, it moves to the fibrous material [13]. . Then it follows from equation (4), (5) and (6) 12 12 ) ( In the second section of the recovery zone 2 4 A A , the equation 12 12 12 cos  r y  holds. Using expressions similar to (1) and (2) and making assumptions that 11 11 sin    , we obtain . 1 sin cos 12 Integrating (14) Thus, the distribution of hydraulic pressure along the contact curve of the lower roller is described by formulas (10), (11) and (15).
The distribution of hydraulic pressure along the contact curve of the upper roller is determined similarly.   (19) where  B is the width of the fibrous material layer;   is the fluid density.
In the compression zone 3 1 A A , the strain of the layer of material is expressed by the following equation Integrating (20) and using the boundary condition 0 ) ( 11 11   Q , we obtain The amount of extracted liquid flowing through the surface of the compression zone is determined by its flow rate at point 3 A , that is, by moisture content ) 0 ( 11 Q : . 3 ) 0 ( 5 11 11 11    Q (22) In the first section of the recovery zone, the strain of the layer of material has the form . cos ) 0 ( after integrating and using the condition  14 12 The amount of pressed liquid flowed from the layer of material through the surfaces of the compression zone, and the first section of the recovery zone is determined by the moisture content ) ( where . 1 0 , The amount of liquid extracted from the material through the contact surfaces of the lower roller is determined by its flow rate at point 2 A , that is, by moisture content ) ( The amount of liquid that flowed through the contact surfaces of the upper roller is determined similarly: The amount of liquid extracted from the layer of material during the pressing process is equal to the sum of the number of liquids extracted through the contact surfaces of the lower and upper rollers: On the graphs of the distribution of hydraulic pressures in the contact zone of the rolls, shown in Figure 2, it follows that the hydraulic pressure in the compression zone increases from zero at the initial point of contact to a maximum at the point of maximum compression of the material. The distribution of hydraulic pressure in the contact zone depends on the length of its part, where the liquid passes from the material layer to the roll coating.