Influence of particle concentration in feed streams

As mentioned before, the collisions between particles significantly affect the impinging stream process while the frequency of the collisions is related to the concentration of particles in the feeding stream. It would be of interest to make certain theoretical predictions for the relationship between the variables mentioned. The analysis presented by Elperin et al. [44] for this purpose is a valuable reference, which is briefly introduced below. 

Consider the impingement between two opposed particles-gas suspension streams from accelerating tubes of the same diameter. The assumptions made in the establishment of the model are (1) The streams are symmetrical with respect to both the jet axis and the impingement plane (2) The gas flow velocity and all the physical properties of gas and solid are kept constant and (3) The particles beyond collision penetrate into the opposing stream up to. rlllas, while any particle will be drawn out of the system immediately once it collides with another particle. 

Let Z0 denote the initial number of particles crossing the impingement plane and penetrating into the opposing stream at r = 0. Due to collision, the number of particles is reduced by dZ within the time interval dr. The reduction of particle number is assumed to obey the first-order kinetics, that is 

For a spherical particle, the effective cross section for collision, a, is 

If the mean relative velocity of the particle with respect to its neighboring particles is denoted by np r, then the collision number along a path of the length equal to upr, N, is given by 

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