The Movement of a Mixture of Cotton with an Air Stream during Pneumatic Transport by Pipeline of Variable Cross Section

The article considers the movement of a mixture of air and raw cotton through a pipeline with a variable cross-section as a multi-speed heterogeneous medium. The regularities of the movement of components inside the pipeline, the equation of change in the porosity of cotton, air pressure and component velocities in time and along the transportation line are obtained. It was found that in the initial 20 25 m part of the pneumatic transport pipe there is a sharp decrease in pressure and air flow velocity, while the speed of cotton increases rapidly due to which there is a strong deformation of cotton stretching under the influence of aerodynamic force, which occurs due to the difference in the velocities of the components of the mixture, as a result of which the cotton loosens, and its porosity increases intensively.


Introduction
Depending on the number of components in the pipeline transport, one-component and multi-component media are distinguished. A single-component medium is a medium consisting of a single material or substance, which is called a homogeneous mixture, and a multi-component-of several materials or substances that differ in physical, mechanical, chemical and other properties, is called a heterogeneous mixture [1] [2]. The mathematical description of the motion of a DOI: 10.4236/eng.2019.118037 532 Engineering multicomponent medium is complicated by the difference in the reaction of each component to the movement of another. Therefore, to simplify the process of theoretical consideration, multicomponent media are taken as a one-component or two-component medium consisting of a continuous phase-air and a discrete phase-material, which ultimately gives more general and less accurate results.
Especially, when considering the processes of pneumatic transportation of materials, including raw cotton, the theory of motion of a discrete medium in a stationary or moving continuous medium has been adopted to date, which will give a more or less distorted idea of the process of moving air and material inside the pipeline [3]. More suitable, in our opinion, to study the process of pneumatic transportation theory, is the theory of multi-speed systems, according to which, all components are moved through the pipeline separately, with interdependent parameters, including speed, which is a new approach to the study of the processes of pneumatic transportation of materials. To ensure the universality of the laws, it should consider the motion of the mixture inside the pipeline with a variable cross section [4] [5].
The mixture of air with fiber particles is assumed to be a heterogeneous mixture, to describe the motion of which you can use the theory of multi-speed systems proposed in the work of Kh. A. Rakhmatullin [6].
Heterogeneous mixtures, as a rule, are described by a multi-speed model taking into account the dynamic effects arising due to the mismatch of the speeds of the individual phases. At the same time, we consider the air to be an ideal fluid and the internal force of interaction is determined through the normal pressure, which is common to the whole mixture, opposite to the direction of movement of the particles of the fibrous medium [7] [8].
The following designations are used in the research:  f-the coefficient of friction between the outside surface of the pipe; L-length of the pipe cross-section contour), m.

Study of the Motion of a Multicomponent Medium
Set the origin of the coordinates in the initial section of the pipeline. We direct the axis along the axis of the pipeline, the cross-sectional area of which varies according to the law, considering the process stationary, denoted by, respectively, the air velocity (index-) and cotton particles of raw mass (index-1) in an arbitrary section of the pipeline. We believe that in the cross section of the pipeline the air flow with speed acts on the moving mass of the mixture. The equations of one-dimensional motion of the components of the mixture and the laws of mass conservation, according to, are written in the form Equations (1) and (2) we bring to the form Excluding the derivative from the system ( ) Equations (3) and (4) imply that the densities and velocities of the components in an arbitrary section of the pipeline are expressed in terms of porosity  (7) with the help of relations (4), satisfies the equation: where: 0 0 0 0 0 00 00 10 10 The densities are expressed in terms of porosity by the Formulas (3) and (4).
In the process of transporting the mixture on the inner wall from the side of the particles of the solid component (cotton particles of raw) acts lateral pressure Equations (1) and (2) are written in the form: After excluding the derivative ( ) d d sp x from system (1) and (11), we obtain      there is no significant change in speed. At large values of the interaction coefficient, the nature of the change of parameters is preserved. The difference is only the intensity of the change. Figure 5 represents the change in the porosity of cotton in both cases at a distance of 50 m reaches the same value, 0.75 -0.80. Only with a large k, the growth of porosity in the initial 10 -20 meters is much stronger and when it reaches 30 m, the change is significantly stabilized. A, the change in pressure, on the contrary ( Figure 6), with smaller k decreases more strongly than relatively large k. This shows that k negatively affects the resistance force, i.e., with its increase, the resistance force decreases, and with decreasing, on the contrary, it increases.

Computer Processing and Analysis of the Obtained Regularities
In both cases ( Figure 7) and values, the air velocity shows a character of decline with almost the same intensity. Only if in the first graphs the stabilization of air velocity occurs at a value of 9 -10 m/s, then in the following graphs, i.e., at large values, the air velocity remains without changes after the value of 15 m/s. The speed of cotton ( Figure 8) also changes according to the previous pattern: at smaller values k n the intensity of the increase in speed is relatively low, but this increase is maintained for a long time, and at large values k n the growth rate is much higher, but after 30 meters the speed of cotton is almost stabilized. At the same time, the speed is about 9 -10 m/s. If you pay attention to the air speed, you can see its value-at a distance of 40 -50 meters it is about 14 -15 m/s, that is, the air is ahead of cotton with a relative speed equal to the difference of their speeds. The ratio of cotton velocity to air velocity will give the latency coefficient, which in our case is equal to the tabular values of k = 0.7 -0.75 [8]. This     ratio shows how late cotton is from air. The fact of the equality of theoretical values with the table values, which are established experimentally, proves the compliance of the established theoretical pattern with the natural pattern of movement of the cotton-air mixture in the transport pipelines of pneumatic installations and will provide an opportunity to offer the established formulas for the calculation of pneumatic transport and its design [11].
The resulting model is universal and can be used for calculations of pipelines with variable cross-section.

Findings
1) Research has established that during pneumatic transportation, in the initial 20 -25 meters of the pipeline, the porosity of transported cotton greatly increases, which indicates its exposure to tensile deformation and proves that cotton is loosened, fluffed, and trash is separated from it.
2) The entry of cotton into the pneumatic conveying pipeline leads to a sharp decrease in pressure and air velocity, and the speed of cotton in the initial 15 to 20 meters of the pipeline increases strongly, then with more moderate intensity.
3) The multi-speed model of the movement of a cotton-air mixture as a heterogeneous medium in pneumotransport pipelines more correctly describes the process of moving air and material during pneumatic transportation, which makes it possible to suggest its use for calculating pneumatic transport and in its design.