The Particle Space Continuum

t) for external coordinates separately. These functions are assumed to be as smooth as necessary. Generally, explicit dependence of X on external coordinates r is unnecessary, although this is not an assumption forbidding analysis. 

Clearly, in the foregoing discussion, the change of particle state has been viewed as a deterministic process. It is conceivable, however, that in some situations the change could be occurring randomly in time. In other words the velocities just defined may be random processes in space and time. It will be of interest for us to address problems of this kind. For the present, however, we postpone discussion of this issue until later in this chapter. 

Since velocities through both internal and external coordinate spaces are defined, it is now possible to identify particle (number) fluxes, i.e., the number of particles flowing per unit time per unit area normal to the direction of the velocity. Thus / (x, r, t)R(x, r, Y, t) represents the particle flux through physical space and / (x, r, t)X x, r, Y, t) is the particle flux through internal coordinate space. Both fluxes are evaluated at time t and at the point (x, r) in particle state space. Indeed these fluxes are clearly important in the formulation of population balance equations. 

Following earlier work of the author (Ramkrishna, 1985), it is convenient to define a particle space continuum that pervades the space of internal and external coordinates. For reasons to be clarified subsequently, we shall deem the particles to be imbedded in this continuum. This continuum may be viewed as deforming in space and time in accordance with the field [X(x, r, Y, t), R(x, r, Y, t)] relative to fixed coordinates. Thus for any point on the continuum initially at (x, rj, its location at some subsequent time t may be described by coordinates [X(t x, rj, R(t x, rj] which must satisfy the differential equations 

This continuum should not be confused with the fluid phase in which the particles are physically dispersed. They are the same only when there is no relative motion between the particles and the fluid phase. 

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Consider an arbitrarily selected domain in the particle space continuum at some arbitrary reference time t = 0. Note that consists of a part A, in the space of internal coordinates part A in the space... 

Generally, the functions X and h inherit the spatial uniformity of so that X = X and h == h. The surface integral over the inlet domain is the volumetric flow rate of the particle space continuum (and not necessarily the continuous phase ) entering the domain whereas that over the outlet domain is the flow out. If it is assumed that there is no relative motion between the continuous phase and particles, then the volumetric flow rates above are also those of the particle-fluid mixture. 

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