Generalized Collision Term Formulation

In this section we proceed deriving an explicit expression for the collision source term ( )coiiision In (2.78). 

In the subsequent model derivation it is convenient to divide the velocity of all particles in the gas into two groups, the small range of velocities that fall into the interval dc about c and all other velocities denoted by Ci. 

The number of particles that are removed from the phase element drdc in the time dt equals the total number of collisions that the c particles have with the other ci particles in the time dt. The term ( )collisionconsiders all collisions between pairs of molecules that eject one of them out of the interval dc about c. By definition, the number of particles which is thrown out of the phase element is given by  

The notation used henceforth for the pair distribution function is defined by / (r,c,ri,ci,t). 

The number of particles injected into dcdr due to collisions in the interval ( f )coiiision Considers all the binary collisions that can send one particle into the velocity interval dc about c. 

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This relation enable us to simplify the formulation of the general equation of change considerably. Fortunately, the fundamental fluid dynamic conservation equations of continuity, momentum, and energy are thus derived from the Boltzmann equation without actually determining the form of either the collision term or the distribution function /. 

The main challenge in formulating these equations is related to the definition of the collision operator. So far this approach has been restricted to the formulation of the population balance equation. That is, in most cases a general transport equation which is complemented with postulated source term formulations for the particle behavior is used. Randolph [80] and Randolph and Larson [81] used this approach deriving a microscopic population balance equation for the purpose of describing the behavior of particulate systems. Ramkrishna [79] provides further details on this approach considering also fluid particle systems. 

The dense gas collision operator is also formulated in accordance with the ideas presented in sect 2.4.3, but in this case the generalized form (2.182) is adopted. After expressing the operator in terms of k, the result is ... 

In the approach adopted in my first edition, the derivation and use of the general dynamic equation for the particle size distribution played a central role. This special form of a population balance equation incorporated the Smoluchowski theory of coagulation and gas-to-panicle conversion through a Liouville term with a set of special growth laws coagulation and gas-to-particle conversion are processes that take place within an elemental gas volume. Brownian diffusion and external force fields transport particles across the boundaries of the elemental volume. A major limitation on the formulation was the assumption that the panicles were liquid droplets that coalesced instantaneously after collision. 

Expression (67.Ill) can be considered as a "statistical formulation of the rate constant in that it represents a formal generalization of activated complex theory which is the usual form of the statistical theory of reaction rates. Actually, this expression is an exact collision theory rate equation, since it was derived from the basic equations (32.Ill) and (41. HI) without any approximations. Indeed, the notion of the activated complex has been introduced here only in a quite formal way, using equations (60.Ill) and (61.Ill) as a definition, which has permitted a change of variables only in order to make a pure mathematical transformation. Therefore, in all cases in which the activated complex could be defined as a virtual transition state in terms of a potential energy surface, the formula (67.HI) may be used as a rate equation equivalent to the collision theory expression (51.III). 

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