External coordinates continuous

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)... 

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. 

The disperse phase is constituted by discrete elements. One of the main assumptions of our analysis is that the characteristic length scales of the elements are smaller than the characteristic length scale of the variation of properties of interest (i.e. chemical species concentration, temperature, continuous phase velocities). If this hypothesis holds, the particulate system can be described by a continuum or mean-field theory. Each element of the disperse phase is generally identified by a number of properties known as coordinates. Two elements are identical if they have identical values for their coordinates, otherwise elements are indistinguishable. Usually coordinates are classified as internal and external. External coordinates are spatial coordinates in fact, the position of the elements in physical space is not an internal property of the elements. Internal coordinates refer to more intimate properties of the elements such as their momenta (or velocities), their enthalpy... 

With pure advection processes, we refer to continuous phenomena that cause continuous changes in the external and internal coordinates. Continuous changes of the particle s position in real space are quantified by the real-space advection (or free-transport) term ... 

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. 

Continuous variables may be encountered more frequently in population balance analysis. They often arise as a natural solution to dealing with indefinite or variable discreteness. For example, a particle-splitting process where the products of splitting could conceivably have any size smaller than the parent particle is most naturally handled by assigning particle size as a continuous variable. The external coordinates denoting the position vector of (the centroid of) a particle describing continuous motion through space represent continuous variables. The temperature of a particle in a fluidized bed is another example of a continuous variable. 

The continuous phase variables, which affect the behavior of each particle, may be collated into a finite c-dimensional vector field. We thus define a continuous phase vector Y(r, t) = [7 (r, t), 2(1, t. .., l (r, t)], which is clearly a function only of the external coordinates r and time t. The evolution of this field in space and time is governed by the laws of transport and interaction with the particles. The actual governing equations must involve the number density of particles in the particulate phase, which must first be identified. 

In this example we consider a cell population in a batch stirred reactor where the cells are distributed according to their age, denoted t, ranging between 0 and 00. The main purpose of this example is to demonstrate the boundary condition that arises. Each cell beyond a certain age has a constant rate of division, say, k. The division of the cell of age t results in the loss of that cell, but also in the gain of two new cells of age zero each. The only particle state variable is the cell age t. No external coordinates are needed because the population is in a well-stirred reactor. The continuous phase is assumed to have no explicit influence on the cells, presumably because the necessary nutrients are present in saturating proportions. We again have the particle state domain = [0, 00). 

One of the most detailed discrete simulations in the literature is that of Zeitlin and Tavlarides (1972a,b,c), which addressed the evolution of drop size distributions in mechanically agitated liquid-liquid distributions produced in batch, semibatch, and continuous flow vessels. In these simulations, the flow vessel was divided into distinct spatial regions among which individual droplets commuted with specified velocities. Thus, both internal and external coordinates characterized the individual particle state. Drop... 

The population balance equation for continuous systems with one internal coordinate and one external coordinate is given by ... 

Weapons Convention is facilitated by the Organisation for the Prohibition of Chemical Weapons and the assistance received through the contributions of member states and the utilisation of experts within and external to the OPCW. It also involves the coordination and delivery of specialised services from national agencies and other international organisations involved in providing emergency humanitarian assistance. The OPCW will continue its work on the cooperative efforts with many member states to maintain the effort to development, implement and train for an effective delivery of assistance in accordance with the provisions of the Chemical Weapons Convention. 

The Schlosser-Marcus variational principle is derived for a single surface a that subdivides coordinate space 9i3 into two subvolumes rm and rout. This generalizes immediately to a model of space-filling atomic cells, enclosed for a molecule by an external cell extending to infinity. The continuity conditions for the orbital Hilbert space require i>out =a i>in This implies a vanishing Wronskian surface integral... 

When a component of a continuous fluid phase (a liquid solution or a gas mixture) is present in nonuniform concentration, at uniform and constant temperature and pressure and in the absence of external fields, that component diffuses in such a way as to tend to render its concentration uniform. For simplicity, let the concentration of a given substance be a function of only one coordinate x, which we shall take as the upward direction. The net flux Z] of the substance passing upward past a given fixed point Aq (i.e., amount per unit cross-sectional area per unit time) is under most conditions found to be proportional to the negative of the concentration gradient ... 

The steady drag is the component of the hydrod3mamic force acting on the particle surface in the continuous phase flow direction. One might, for example, imagine a uniform velocity in the z-direction as sketched in Fig 5.1, and describe the external flow using the Cartesian coordinate system, then the steady drag force is defined by [14, 30] ... 

Another possible scenario is that as a consequence of mass transfer only the number of primary particles changes, whereas their size remains more or less constant. This hypothesis seems to be realistic in the case of negative molar flux, J <0, or, in other words, in the case of shrinking particles. In fact, in this case it is more likely that the external particles will be consumed before the internal ones. The resulting expressions for the continuous rate of change of the two internal coordinates therefore read as... 

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