The Continuous Phase Vector

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 some applications, a continuous phase balance may not be necessary because interaction between the population and the continuous phase may not bring about any (or a substantial enough) change in the continuous phase. In such cases, analysis of the population involves only the population balance equation. 

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The continuous phase vector Y(r, i) will include temperature (under nonisothermal situations), and concentrations of various chemical components that may be involved in transport between the continuous phase and the particles, and in chemical reactions in either phase. Liquid-liquid dispersed phase reactors are a common feature of the chemical process industry where the preceding processes are encountered. For the present we shall assume isothermal conditions and consider only concentration components in Y(r, t). Alternatively, this strategy would be appropriate even for nonisothermal situations if temperature were to be isolated as another variable to be dealt with through an energy transport equation. 

We now consider the well-stirred open system of Section 2.8 with the continuous phase vector represented by the spatially uniform Y t) in the domain Q, and at the entrance region Integrating Eq. (2.9.1) over the region and recognizing that the diffusive flux Jy must vanish everywhere, we obtain the equation... 

Apart from the drag force, there are three other important forces acting on a dispersed phase particle, namely lift force, virtual mass force and Basset history force. When the dispersed phase particle is rising through the non-uniform flow field of the continuous phase, it will experience a lift force due to vorticity or shear in the continuous phase flow field. Auton (1983) showed that the lift force is proportional to the vector product of the slip velocity and the curl of the liquid velocity. This suggests that lift force acts in a direction perpendicular to both, the direction of slip velocity... 

Recall that the continuous phase variables were described by the vector field Y(r, t). In general, the components of this vector field should encompass all continuous phase quantities that affect the behavior of single particles. These could include all dynamic quantities connected with the motion of the continuous phase, the local thermodynamic state variables such as pressure and temperature, concentrations of various chemical constituents, and so on. Clearly, this general setting is too enormously complex for fruitful applications so that it is necessary to suitably constrain our domain of interest. In this connection, the reader may recall our exclusion of the fluid mechanics of dispersions, so that we shall not be interested in the equation... 

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... 

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. 

First, the master density function is introduced in Section 7.1. The scalar particle state is discussed in detail in Section 7.1.1 with directions for generalization to the vector case in Section 7.1.2. Coupling with the continuous phase variables is ignored in the foregoing sections, but the necessary modifications for accommodating the environmental effect on the particles are discussed in Section 7.1.3. Thus, from Section 7.1, the basic implements of the stochastic theory of populations along with their probabilistic interpretations become available. These implements are the master density, moment densities that are called product densities, and the resulting mathematical machinery for the calculation of fluctuations. 

In a first step the velocity pattern was calculated for different inlet heights and different throughputs. These velocity vectors are shown in Figure 5. Now for each cell the droplet diameter was calculated when the droplet starts to settle (one droplet in each cell). This point is given when the upward velocity of the continuous phase is balanced by the sinking velocity of the droplets according to Stokes law ... 

In Eq. 9, E is the interfacial tension, p the pressure, Vy the undisturbed velocity gradient tensor and Vy its transpose, tjm is the viscosity of the continuous phase, V is the total volume of the system, n is the unit vector orthogonal to the interface between the two phases, u is the velocity at the interface, dA is the area of an interfacial element and the integrals are evaluated over the whole interfadal area of the system, A. Since the constituents are assumed to be Newtonian all nonlinear contributions to the stress a(t) are caused entirely by the deformation of the droplet interface. The unit vectors n and u describe this deformation and can be computed using the Maffettone-Minale (MM) model for different frequencies and amplitudes. The MM model uses a second rank, symmetric and positive definite... 

Vector representing the continuous phase variables Rate of change of particle mass in (4.187)... 

Components of the continued fraction vector in PD algorithm Instantaneous volume fraction of phase k —)... 

The settling velocity, is relative to the continuous Hquid phase where the particle or drop is suspended. If the Hquid medium exhibits a motion other than the rotational velocity, CO, the vector representing the Hquid-phase velocity should be combined with the settling velocity (eq. 2) to obtain a complete description of the motion of the particle (or drop). 

From the definition of a particle used in this book, it follows that the motion of the surrounding continuous phase is inherently three-dimensional. An important class of particle flows possesses axial symmetry. For axisymmetric flows of incompressible fluids, we define a stream function, ij/, called Stokes s stream function. The value of Imj/ at any point is the volumetric flow rate of fluid crossing any continuous surface whose outer boundary is a circle centered on the axis of symmetry and passing through the point in question. Clearly ij/ = 0 on the axis of symmetry. Stream surfaces are surfaces of constant ij/ and are parallel to the velocity vector, u, at every point. The intersection of a stream surface with a plane containing the axis of symmetry may be referred to as a streamline. The velocity components, and Uq, are related to ij/ in spherical-polar coordinates by... 

The United Nations Stockholm Treaty on persistent organic pollutants calls for the phase out of DDT but recognizes its efficacy as a deterrent to vector-borne diseases such as malaria and typhus. According to the treaty, the continued use of DDT is discouraged, but until effective economical alternatives are found, DDT use will be continued in countries with high rates of vector diseases. A number of developing countries still use DDT. It is applied primarily in the interior of homes to prevent malaria. Currently DDT is produced only in India and China, and current production volumes are unknown. 

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