Exercise 17.1 Grain size sorting in a hydrocyclone

We study, in a simple geometrical configuration, the way a hydrocyclone can be used to sort fine and coarse particles contained in a fluid. We determine how the flow rates should be selected, based on given geometrical dimensions and particle characteristics, in order to perform this sorting in the most satisfactory maimer possible. 

It is assumed that the fluid contains large spherical particles of diameter d and fine spherical particles of diameter c 2 The density of the solid particles is assumed to be the same and these particles are assumed to be denser than the fluid 

It is assumed that the flow is uniform in the inlet section of the lower cavity z = H, R3 r R2) and in the outlet sections (z = 0, R3 r R2, z =H, a r Ra) of the apparams. Prove that, in the cylindrical coordinate system 0, 0 ) whose Oz axis coincides with the apparatus axis, the field of the axial and radial components of the velocity can be modeled as follows (z = 0 is on the bottom)  

Which of the two components Ur and is determined first How is the second component derived from the first  

We denote by rp t) and Zp(t) the coonlinates of a particle. Write the equations of motion for a solid particle in the domain a r R2 under the effect of centrifugation and advection by fluid flow. 

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