Particle space continuum

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... 

The Reynolds transport theorem is a convenient device to derive conservation equations in continuum mechanics. Toward derivation of the general population balance equation, we envisage the application of this theorem to the deforming particle space continuum defined in the previous section. We assume that particles are embedded on this continuum at every point such that the distribution of particles is described by the continuous density function / (x, r, t). Let i//(x, r) be an extensive property associated with a single particle located at (x, r). 

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. 

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 fluid is regarded as a continuum, and its behavior is described in terms of macroscopic properties such as velocity, pressure, density and temperature, and their space and time derivatives. A fluid particle or point in a fluid is die smallest possible element of fluid whose macroscopic properties are not influenced by individual molecules. Figure 10-1 shows die center of a small element located at position (x, y, z) with die six faces labelled N, S, E, W, T, and B. Consider a small element of fluid with sides 6x, 6y, and 6z. A systematic account... 

Here the symbol q, indicates the 3-dimensional vector position of the f particle. To apply Hilbert space concepts to this theory we now postulate a one-to-one correspondence between the rays of and the points of the Schrodinger 3N hyperspace. Thus there exists a continuum of normalized eigenvectors in represented by the symbol 4i.4a.". > ... 

Typically, the arguments considered for a continuum depend on molecules being very small relative to the problem scale (i.e., the film thickness), as shown in Fig. 2(a), which implies a spatial averaging. One must choose a small region of space (the point), which contains many particles, but is still much smaller than the problem scale. If certain ratios remain constant as the region of space is reduced in size, i.e., if a limit exists, a smoothly varying continuum spatial averaged property (e.g., density) can be defined ... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

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). 

Suppose we now assign a physical meaning to the velocity v, representing it as the velocity of matter in the volume, V. Then if V always contains the same mass, it is a system volume. The properties defined for each point of the system represent those of a continuum in which the macroscopic character of the system is retained as we shrink to a point. Properties at a molecular or atomic level do not exist in this continuum context. Furthermore, since the system volume is fixed in mass, we can regard volume V to always enclose the same particles of matter as it moves in space. Each particle retains its continuum character and thermodynamic properties apply. 

Detonation, Steady and Nonsteady State in (Steady Flaw or Streamline Flow and Nonsteady State in). This is the case when every particle that flows past a fixed point in space will have the same q, p and P at that point independent of time. In this condition, every point of the fluid continuum has a corresponding fluid velocity vector q. The term streamlines signifies a family of curves which are everyr where tangent to q thus, the direction of each streamline is everywhere that of the motion of the fluid... 

In studying processes of accumulation of the Frenkel defects, one uses three different types of simple models the box, continuum, and discrete (lattice) models. In the simplest, box model, which was proposed first in [22], one studies the accumulation of complementary particles in boxes having a certain capacity, with walls impenetrable for diffusion of particles among the boxes. The continuum model treats respectively a continuous medium the intrinsic volume of similar defects at any point of the space is not bounded. In the model of a discrete medium a single cell (e.g., crystalline lattice site) cannot contain more than one defect (v or i). 

The three-dimensional particle in a box corresponds to the real life problem of gas molecules in a container, and is also sometimes used as a first approximation for the free conduction electrons in a metal. As we found for one dimension (Section 2.3), the allowed energy levels are extremely closely spaced in macroscopically sized boxes. For many purposes they can be regarded as a continuum, with no discernible energy gaps. Nevertheless, there are problems, for example in the theory of metals and in the calculation of thermodynamic properties of gases, where it is essential to take note of the existence of discrete quantized levels, rather than a true continuum. 

For multiphase flow that is normally encountered in fluidized bed reactors, there are two kinds of definitions of the micro-scale first, it is the scale with respect to the smaller one between Kolmogorov eddies and particles second, it is the scale with respect to the smallest space required for two-phase continuum. If the first definition is adopted, the... 

Let us briefly discuss the relationship between approaches which use basis sets and thus have a discrete single-particle spectrum and those which employ the Hartree-Fock hamiltonian, which has a continuous spectrum, directly. Consider an atom enclosed in a box of radius R, much greater than the atomic dimension. This replaces the continuous spectrum by a set of closely spaced discrete levels. The relationship between the matrix Hartree-Fock problem, which arises when basis sets of discrete functions are utilized, and the Hartree-Fock problem can be seen by letting the dimensions of the box increase to infinity. Calculations which use discrete basis sets are thus capable, in principle, of yielding exact expectation values of the hamiltonian and other operators. In using a discrete basis set, we replace integrals over the continuum which arise in the evaluation of expectation values by summations. The use of a discrete basis set may thus be regarded as a quadrature scheme. 

This is essentially the same effeet as the last, but occurs between layers within a single particle. There could well be a continuum of effects, ranging from movement of water molecules between or away from the surfaces of adjacent separate particles to similar movement involving layers within a particle. An intermediate situation in such a continuum might be movement of water into or out of interlayer spaces at re-entrants at whieh layers of a eontinuous structure are splayed apart, such as those shown in Fig. 8.4. 

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