Particle state space number density

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. 

A mechanical property of a system is a function of the instantaneous state, Ft, of the system. For example, if A is a mechanical property, then A(t) = A(ri). Examples of mechanical properties are the kinetic energy of a single particle and the number density in the neighborhood of a point in the system. As time goes by a mechanical property will change unless it is a constant of the motion. The typical behavior of a mechanical property A corresponding to a given trajectory in phase space is illustrated in Fig. 2.3.1 b. 

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... 

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

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... 

A partial differential equation is then developed for the number density of particles in the phase space (analogous to the classical Liouville equation that expresses the conservation of probability in the phase space of a mechanical system) (32>. In other words, if the particle states (i.e. points in the particle phase space) are regarded at any moment as a continuum filling a suitable portion of the phase space, flowing with a velocity field specified by the function u , then one may ask for the density of this fluid streaming through the phase space, i.e. the number density function n(z,t) of particles in the phase space defined as the number of particles in the system at time t with phase coordinates in the range z (dz/2). 

Hulburt and Katz (HI7) developed a framework for the analysis of particulate systems with the population balance equation for a multivariate particle number density. This number density is defined over phase space which is characterized by a vector of the least number of independent coordinates attached to a particle distribution that allow complete description of the properties of the distribution. Phase space is composed of three external particle coordinates x and m internal particle coordinates Xj. The former (Xei, x 2, A es) refer to the spatial distribution of particles. The latter coordinate properties Ocu,Xa,. . , Xt ) give a quantitative description of the state of an individual particle, such as its mass, concentration, temperature, age, etc. In the case of a homogeneous dispersion such as in a well-mixed vessel the external coordinates are unnecessary whereas for a nonideal stirred vessel or tubular configuration they may be needed. Thus (x t)d represents the number of particles per unit volume of dispersion at time t in the incremental range x, x -I- d, where x represents both coordinate sets. The number density continuity equation in particle phase space is shown to be (HI 8, R6)... 

The mathematical description of a crystal size distribution and of its change in space and time makes use of the conservative character of the number of particles in space and state (i.e., particle size L). In the respective number balance, the particle size distribution is represented by the number density, see Hulburt and Katz (1964) and Randolph and Larson (1988),... 

We shall consider here a population of particles distinguished from one another by a finite dimensional vector x of internal coordinates and distributed uniformly in space. Further, we shall be concerned with the open system of Section 2.8 whose behavior is dictated by the population balance equation (2.8.3). Thus the number density in the feed,/i jn(x), may be assumed to be Nff x) where Nf is the total number density in the feed stream and /(x) is probability density of particle states in it. It will also be assumed that the continuous phase plays no role in the behavior of the system. Relaxing this assumption does not add to any conceptual difficulty, although it may increase the computational burden of the resulting simulation procedure. 

Limitation to ensembles that allow exchange of energy, but not of matter, with their environment is unnecessarily restrictive and unrealistic. What is required is an ensemble for which the particle numbers, Nj also appear as random variables. As pointed out before, the probability that a system has variable particle numbers N and occurs in a mechanical state (p, q) can not be interpreted as a classical phase density. In quantum statistics the situation is different. Because of second quantization the grand canonical ensemble, like the microcanonical and canonical ensembles, can be represented by means of a density operator in Hilbert space. 

We now introduce a set of preferred coordinates ), cf Refs. [Kiibler 1973 Zeh 1973 Zurek 1981 Zurek 1982 Zurek 2003], These are the relevant degrees of freedom coupled to the neutron probe. The density matrix needed in (13) is then the reduced one in the space spanned by these states, and it is obtained by tracing out the (huge number of the) remaining degrees of freedom belonging to the "environment" of the microscopic scattering system (e.g., a proton and its adjacent particles). To simplify notations, we denote this reduced density matrix by p too. 

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