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Movement of a particle of a marl on a surface of a rotating cone

Designs of the overwhelming majority of dynamic separators have switching centres for uniform distribution of a material in a separation working area. Besides this appointment that these devices make essential impact on efficiency of separation is much important also, giving initial speeds to material particles at hit in the gas environment of a working area of separation.

Hence, designs of these devices and technological parametres such as: speed of rotation, a way and a place of feed of a material, can be used for management of separation process. Often meeting switching centre is the ordinary rotating cone. Thus, knowing speeds of a descent of particles from a cone and feature of its work, can have practical value for increase of an overall performance of dynamic separators.

Let's consider movement of a particle of a material in mass mпо surfaces rotating with frequency konusa.

Drawing 2.1 Representation of a trajectory of movement of a particle of a material on a rotating surface of a cone

The trajectory of movement of a particle of a material on an external surface of a rotating cone (Drawing 2.1), obviously, will represent a cone-shaped spiral line or a lot of a spiral line with variable distance r from a rotation axis.

According to the settlement scheme presented in drawing 2.1, the spiral trajectory in a plane of a perpendicular rotation axis can be described the following parities:

Where the turn corner otschityvaemyj from a positive direction of an axis OH with a cone rotating speed is connected by a following parity:

Here t - running time.

For the description of movement of a particle of a material along forming a cone, we will enter two-dimensional dekartovuju system koordinats the beginning in a point About і

According to the settlement scheme presented in drawing 2.2.

Drawing 2.2 Settlement scheme for the description of movement of a particle of a material in motionless system of co-ordinates ζ; 0 1 ∙, ψ

Let during some any moment of time the material particle on a surface of a rotating cone has following parametres r, zи ζ, which

According to the scheme in drawing 2.2 are connected among themselves and the geometrical sizes of the cone, following parities:

Let's accept, that on the particle of a material which is on an external surface of the rotating cone following forces operate: particle weight - td; force of reaction of bearer NR; centrifugal force F4и a frictional force about a cone surface.

The projection of these forces to an axis About 1ψпозволит to receive a following parity:

Where γ - value of a corner formed directing a cone with an axis About z;

m - Mass of a particle of a material, kg;

d - acceleration of free falling, m./s2

The corner γ is defined through parametres of a cone and expressed according to a parity:

On the basis of (2.6) we find, that the size of force of reaction of a bearer will be defined by a parity:

At the fixed value of a rotating speed of a conical surface ω according to (2.9) such distance gkr from a rotation axis is defined, at which size of force of reaction of bearer N rпринимает zero value:

Where (2.10) taking into account (2.7), (2.8) becomes:

Hence, prichastitsa a material in a plane of a perpendicular axis

Cone rotations are made by movement on a spiral trajectory (2.1), (2.2), and in sluchaechastitsa a material will roll down in steps on

Cone surfaces.




Using results of the mathematical description of process of movement of a particle on a rotating surface of a cone graphic dependences of speeds of a particle of a marl υr (at the left) and υφ (on the right), from parametre Lh, m and running time tдля is constructive-technological parametres of the industrial dynamic separator "polidor" 04000, resulted on drawings 2.3 and 2.4 have been received. Following is constructive-technological parametres of a separator were considered:

According to drawing 2.3 () it is visible, that peak values of radial making speed υrвыше a maximum value of tangential speed Dφна the same drawing 2.3 ().

Drawing of 2.3 Dependences of making speeds of movement of a particle of a marl on a rotating conical surface from tи LH: - υφ∙, - υr

Both dependences of drawing 2.3 reflect that fact, that with growth of parametre value Lн, speeds increase also. It is obvious, that dependences have strongly pronounced nonlinear character, thus final speeds depend on the running parametre values t which for the given dependences laid in a range from 0 to 1, and peak values of speeds are reached at peak values t.

Drawing 2.4 displays zavisimostikogda

Parametre Lhимеет concrete values, namely 0,1; 0,2 and 0,3 m. As have already been noted, peak values of speeds correspond to the greatest values L hпри identical t so from the schedule it is visible, that speed υrдля L h = 0,1 reaches values of 4,1 km/s at t = 0,5 when for L n = 0,3 at the same t value reaches 5,6 km/s. For speed υφ, at Lh = 0,1 and t = 0,5 value reaches 2,9 km/s, and at Lh = 0,3 value of speed makes already 4 km/s.

Drawing of 2.4 Dependences of speed of movement of a particle of a marl υrи υ φ, km/s at private parametre values L n, ().

The resulted mathematical description and graphic dependences can have practical application. So, for example, at designing of new cone-shaped switching centres and by optimisation of process of separation of existing devices, at the expense of change of designs submitting a material on a cone for the purpose of the legend of optimum value to initial parametre L n. It will allow to regulate speed of a descent of a particle from the switching centre and to make essential impact on separation process.

For an estimation of process of interaction of particles with a cone surface at loading through the hollow bulwark we will consider the problem on a parity of number of particles,

Being in contact to a surface of a cone and number of particles loadings being in volume.

According to the settlement scheme presented in drawing 2.5, cone volume

Drawing of 2.5 Particles loadings being in volume

Whence parametre L0может to be defined according to following expression:

56

Quantity of particles N κ, being in contact to a surface of cone Sκв the working area of loading of a material can find on the basis of expression:

On the basis of (2.44) it is found:

According to (2.43) and (2.45) it is found:

Owing to that, chtomozhno to conclude, that quantity of particles,

Contacting to a cone surface, there is less than quantity of the particles which are not coming into contact No »» N κ [81].

2.1

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A source: Trofimchenko Vladimir Nikolaevich. PERFECTION of PROCESS DEZAGREGATSII of PARTICLES of the MATERIAL In the DYNAMIC SEPARATOR With the DEVICE In the form of mnogozahodnyh TAPES. The dissertation on competition of a scientific degree of a Cand.Tech.Sci. Belgorod - 2017. 2017

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