Particle state space

In this section the population balance modeling approach established by Randolph [95], Randolph and Larson [96], Himmelblau and Bischoff [35], and Ramkrishna [93, 94] is outlined. The population balance model is considered a concept for describing the evolution of populations of countable entities like bubble, drops and particles. In particular, in multiphase reactive flow the dispersed phase is treated as a population of particles distributed not only in physical space (i.e., in the ambient continuous phase) but also in an abstract property space [37, 95]. In the terminology of Hulburt and Katz [37], one refers to the spatial coordinates as external coordinates and the property coordinates as internal coordinates. The joint space of internal and external coordinates is referred to as the particle phase space. In this case the quantity of basic interest is a density function like the average number of particles per unit volume of the particle state space. The population balance may thus be considered an equation for the number density and regarded as a number balance for particles of a particular state. 

One of the most widely used approaches for the simulation of sprays is the stochastic discrete droplet model introduced by Williams [30]. In this approach, the droplets are described by a probability density fxmction (PDF),/(t,X), which represents the probable number of droplets per unit volume at time t and in state X. The state of a droplet is described by its parameters that are the coordinates in the particle state space. Typically, the state parameters include the location x, the velocity v, the radius r, the temperature Td, the deformation parameter y, and the rate of deformation y. As discussed in more detail in Chapter 16, this spray PDF is the solution of a spray transport equation, which in component form is given by... 

The particles of interest to us have both internal and external coordinates. The internal coordinates of the particle provide quantitative characterization of its distinguishing traits other than its location while the external coordinates merely denote the location of the particles in physical space. Thus, a particle is distinguished by its internal and external coordinates. We shall refer to the joint space of internal and external coordinates as the particle state space. One or more of either the internal and/or external coordinates may be discrete while the others may be continuous. Thus, the external coordinates may be discrete if particles can occupy only discrete sites in a lattice. There are several ways in which the internal coordinates may be discrete. A simple example is that of particle size in a population of particles, initially all of uniform size, undergoing pure aggregation, for in this case the particle size can only vary as integral multiples of the initial size. For a more exotic example, let the particle be an emulsion droplet (a liquid) in which a precipitation process is carried out producing a discrete number of precipitate particles. Then the number of precipitate particles may serve to describe the discrete internal coordinate of the droplet, which is the main entity of population balance. 

The particle population may be regarded as being randomly distributed in the particle state space, which includes both physical space and the space of internal coordinates. Our immediate concern, however, will be about large populations, which will display relatively deterministic behavior because the random behavior of individual particles will be averaged out. 

We postulate that there exists an average number density function defined on the particle state space,... 

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. 

We are now in a position to derive the population balance equation for the one-dimensional case. The reader interested in this may directly proceed to Section 2.7 since the next section prepares for derivation of the population balance equation for the general vectorial particle state space. 

We now consider the derivation of the population balance equation for the general particle state space. 

We recall the domain A t) in particle state space considered in Section 2.6, which is initially at and continuously deforming in time and space. For the present, the particles are regarded as firmly embedded in the deforming particle state continuum described in Section 2.5. The only way in which the number of particles in A t) can change is by birth and death processes. We assume that this occurs at the net birth rate of /i(x, r, Y, t) per unit volume of particle state space so that the number conservation may be written as... 

The arbitrariness of the domain of integration above and the continuity of the integrand together imply that the integrand must vanish everywhere in particle state space, leading to the population balance equation... 

The equation must be supplemented with initial and boundary conditions. The initial condition must clearly stipulate the distribution of particles in the particle state space, including internal and external coordinates. For the particular case in which the particles are all of the same internal state, say, x, it is most convenient to use the Dirac delta function (x — xj, which has the properties... 

Suppose Eq. (2.7.9) is integrated over the entire particle state space. Denote the boundary of the particle state space by for internal coordinates and for external coordinates. These boundaries may be completely bounded or may have all or parts of them stretching to infinity. In either case, we may use the divergence theorem to write... 

The population balance equations considered so far were for systems in which particles changed their states deterministically. Thus specification of the state of the particle and its environment was sufficient to determine the rate of change of state of that particle. Applications may, however, be encountered where the particle state may change randomly as determined, for example, by a set of stochastic differential equations. Since, however, the population balance equation is a deterministic equation, our desire is to seek the expected displacement of particles moving randomly in particle state space during an infinitesimal interval dt. 

We begin with revisiting the boundary condition (2.7.12), which represents the crucial boundary condition representing the birth of new particles at the boundary, which subsequently migrate to the interior of the particle state space. If the birth of new particles represented by the boundary condition (2.7.12) occurs at the expense of existing particles, then the right-hand side of the population balance equation (2.7.9) must include a corresponding sink term. 

Consider the problem in the general setting of the vector particle state space of Section 2.1 in an environment with a continuous phase vector as... 

Probability density function for particles from the breakup of a particle of state (x, r ) in an environment of state Y at time t that have state (x, r). This is a continuously distributed fraction over particle state space. 

The particle state space must include external coordinates because the exit of bubbles from the reactor can only be recognized by their location at the vertical end. In this regard, this consideration is identical to that in the example treated in Section 2.11.4. Also, bubble location is important in determining coalescence rate of bubbles. In addition, we must also entertain internal coordinates, including... 

We now show how the average number density can be calculated from the sample paths in prediscretized (time-invariant) domains of the particle state space. This calculation is considerably easier for the case where particle state does not vary with time during the quiescence period. Denote the... 

The equilibrium system A (with N particles) is possible to be identified with the (state) space it occupies, being defined by "one-particle" railing of cells ["one-particle (state) space], with the... 

Following the notation of Ramkrishna [186] the method is called the discrete fixed pivot method in which derivatives and integrals are represented by some finite difference approximations. Such discretizations were considered of coarse nature thus the population balance under consideration was essentially macroscopic in particle state space. The wide use of the finite difference schemes to solve PBEs is mainly due to the simple construction of these schemes. 

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