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. 

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... 

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... 

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... 

Figure 21.15 Droplet formation at the pore opening. Side view with vectors indicating the unit vectors, M and m at the pore perimeter and the advancing (0a) and receding (0r) contact angles. Vrj represents the crossflow velocity of continuous phase at height equal to droplet radius, and vm is the mean disperse-phase velocity.

Here mp and Up represent the mass and velocity vector of the particle, respectively. The right-hand side represents the total force acting on the dispersed phase particle. The sum of forces due to continuous phase pressure gradient, Fp, and due to gravity, Fq, can be written ... 

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... 

On approaching the transition point of a continuous-phase transition, a critical slowing down of the fluctuations occurs (t [a(T—Tc) + Dq2] x) and the amplitudes of the fluctuations increase (< q y [a(T— Tc) + Dq2] x) at wave vector q (Fig. 7a). It can be proved that apparent pseudodivergences of the relaxation rate (]/Tr)c (T Tc) 7 (with y = 1/2 in mean field theories) can be observed near a phase transition if Tqm <3 1 where cop is the probing frequency. In that case, the associated spectral density... 

Where a, p, pif, u- are the volume fraction, density, transfer coefficient (viscosity), and velocity vector of the phase respectively (indices 1 - continuous phase, and 2 - dispersed phase). 

Scattering is the dispersal of radiation at an object (particle) that differs in the relevant material properties from its environment (continuous phase). Static scattering experiments record the scattering signal as a function of the angle of observation 0 or—more generally— as function of the scattering vector q ... 

In general we may conclude that the choice of the particle state is determined by the variables needed to specify (i) the rate of change of those of direct interest to the application, and (ii) the birth and death processes. The particle state may generally be characterized by a finite dimensional vector, although in some cases it may not be sufficient. For example, in a diffusive mass transfer process of a solute from a population of liquid droplets to a surrounding continuous phase (e.g., liquid-liquid extraction) one would require a concentration profile in the droplet to calculate the transport rate. In this case, the concentration profile would be an infinite dimensional vector. Although mathematical machinery is conceivable for dealing with infinite dimensional state vectors, it is often possible to use finite dimensional approximations such as a truncated Fourier series expansion. Thus it is adequate for most practical applications to assume that the particle state can be described by a finite dimensional vector. ... 

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... 

The generalizations of (2.10.2) and (2.10.3) for the general vector case including continuous phase dependence are identified as follows. Let the rate of change of particle state be given by stochastic differential equations of the... 

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