Collision between particles

Figure A3.1.3. The collision cylinder for collisions between particles with velocities v and v. The origin is placed at the centre of the particle with velocity v and the z-axis is in the direction of v - v. The spheres indicate tire range, a, of the intennolecular forces.

Consider a head-on collision between particles incoming along directions cp and Cfc+3. There are two possible outcomes such that both particle number and momentum are conserved the output must consist of two particles emerging either along directions c +i and 3 +4 (figure 9.10-b) or along Cfc i and (figure 9.10-c). We can either have the system always choose the same output channel, which... 

In its more advanced aspects, kinetic theory is based upon a description of the gas in terms of the probability of a particle having certain values of coordinates and velocity, at a given time. Particle interactions are developed by the ordinary laws of mechanics, and the results of these are averaged over the probability distribution. The probability distribution function that is used for a given macroscopic physical situation is determined by means of an equation, the Boltzmann transport equation, which describes the space, velocity, and time changes of the distribution function in terms of collisions between particles. This equation is usually solved to give the distribution function in terms of certain macroscopic functions thus, the macroscopic conditions imposed upon the gas are taken into account in the probability function description of the microscopic situation. 

Inverse Collisions.—The particle velocities resulting from a collision between particles of velocities vx and v2, having collision parameters 6 and e, have been denoted as v[ and v they may be found from Eqs. (1-21). Consider now the particle velocities resulting from a collision between particles of velocities v[ and v2, with collision parameters b and e let these final velocities be denoted by v[ and v . 

In deriving this relation it has been assumed that the distribution function does not change significantly in a distance b, so that the distribution function describing the number of v2 particles is evaluated at the same point in space as that for the vx particles. Since the number of vx particles is f(r,v1,t)drdv1, the number of collisions between particles of velocity vx and v2 in At is... 

Second Derivation of the Boltzmann Equation.—The derivation of the Boltzmann equation given in the first sections of this chapter suffers from the obvious defect that it is in no way connected with the fundamental law of statistical mechanics, i.e., LiouviUe s equation. As discussed in Section 12.6of The Mathematics of Physics and Chemistry, 2nd Ed.,22 the behavior of all systems of particles should be compatible with this equation, and, thus, one should be able to derive the Boltzmann equation from it. This has been avoided in the previous derivation by implicitly making statistical assumptions about the behavior of colliding particles that the number of collisions between particles of velocities v1 and v2 is taken proportional to /(v.i)/(v2) implies that there has been no previous relation between the particles (statistical independence) before collision. As noted previously, in a... 

The removal rate of particles or the rate of flotation from pulp is essentially governed by (i) collision between particles and bubbles (ii) adhesion of particles to bubbles and (iii) detachment of particles from bubbles. Keeping these factors in mind, one can arrive at the following relationship ... 

It will be assumed for the present considerations that sufficient binder is present in the granulator as determined by the binder/powder ratio and that the binder is appropriately spread on enough granular surfaces so as to ensure that most random collisions between particles will occur on binder-covered areas. It will also be assumed that the particles are more or less spherical having a characteristic dimension, a. The simplified system of two colliding particles is schematically shown in Fig. 21. The thickness of the liquid layer is taken to be h, while the liquid is characterized by its surface tension yand its viscosity /x. The relative velocity U0 is taken to be only the normal component between particles while the tangential component is neglected. 

Hindered settling results from collisions between particles and also between particles and the wall. In addition high particle concentrations reduce the flow area and increase the velocity of the fluid with a consequent decrease in settling rate. Furthermore particle concentrations increase the apparent density and dynamic viscosity of the fluid. 

Frequency of Collisions between Particles. Particles in suspension collide with each other as a consequence of at least three mechanisms of particle transport ... 

The rate of coagulation of particles in a liquid depends on the frequency of collisions between particles due to their relative motion. When this motion is due to Brownian movement coagulation is termed perikinetic when the relative motion is caused by velocity gradients coagulation is termed orthokinetic. 

The rate of coagulation depends on the height of the potential energy barrier. In the absence of a barrier, the rate of disappearance of the primary particles is controlled by the number of collisions between particles, i. e. 

One way around the difficulty with the collision term of eqn. (291) is to restrict attention to hard spheres, which only interact with one another when they axe in contact. In essence, the range of the interaction has been reduced almost to zero and the time of collision is then infinitesimally short. This has another very useful consequence. The probability of two collisions occurring simultaneously is negligible. Events just before and just after the collision can be described without reference to any other collision. A collision between particles a and b alters the velocity and position of both particles before and after the collision, but all other... 

Before introducing the chemical reaction into the picture of the system, it is necessary to explore the motion of one or two particles in a little more detail. Consider a collision between particles a and jS, and the time a little before and afterwards, such that several other collisions between the particle a which has co-ordinates r, v, and all other particles (jS ce) may occur. All the other particles which do not collide with a may be ignored. Averaging over all the positions of the N — 2) particles removes their influence (which does not affect the motion of particle a) and leads to a doublet distribution, f2. Then averaging over all positions of the particles (3 (which are colliding with particle a) gives an equation for the singlet distribution [541,542]... 

Only about 300 of the more than 3600 known isotopes occur naturally. The remainder have been made by nuclear transmutation, the change of one element into another. Such transmutation is often brought about by bombardment of an atom with a high-energy particle such as a proton, neutron, or a particle. In the ensuing collision between particle and atom, an unstable nucleus is momentarily... 

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. 

Equation (2.46) indicates that is positively proportional to lnP(r). Therefore > must be very small, if collisionless movement is wanted Vice versa, larger values of / , would lead to significant collisions. Elperin analyzed the case of the particles-air suspension impinging streams with clv= 2.25xI0 , m and / p= 1170 kg m and concluded the following (1) Collisionless flow can be expected when, < 10 4 and (2) When / o> (1-5) xl0 the collisions between particles become significant. 

Collisions between particles with smooth surfaces may be reasonably approximated as elastic impact of frictionless spheres. Assume that the deformation process during a collision is quasi-static so that the Hertzian contact theory can be applied to establish the relations among impact velocities, material properties, impact duration, elastic deformation, and impact force. 

Spiral mills create a high velocity helix of gas that rotates around the center of the jet mill. Solids are introduced via a venturi feed injector (Fig. 8.2) and become entrained in the turbulent helical flow. The resulting high energy collisions between particles as well as with the particles and the mill internals fracture particles to micron and submicron size. 

Fluidized bed jet mills work in similar fashion, with the grinding chamber oriented as a fluidized bed. Specially designed nozzles introduce the grinding gas at the bottom of the fluidized bed, creating high-intensity collisions between particles... 

The first factor in Eq. [87] is the free molecular limit for the rate of collisions between particles of mass mp, whereas the second factor can be interpreted as the collision efficiency (sticking probability). The collision efficiency becomes negligible for very small particles, since in such cases the overall interaction potential becomes vanishingly small. 

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