Fluid-particle flow definition

A particle drag coefficient Cd can now be defined as the drag force divided by the product of the dynamic pressure acting on the particle (i.e. the velocity head expressed as an absolute pressure) and the cross-sectional area of the particle. This definition is analogous to that of a friction factor in conventional fluid flow. Hence... 

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... 

Continuous Mixers In continuous mixers, exiting fluid particles experience both different shear rate histories and residence times therefore they have acquired different strains. Following the considerations outlined previously and parallel to the definition of residence-time distribution function, the SDF for a continuous mixer/(y) dy is defined as the fraction of exiting flow rate that experienced a strain between y and y I dy, or it is the probability of an entering fluid particle to acquire strain y. The cumulative SDF, F(y), defined by... 

The term incompressible flow is applied to any situation where changes in the density of a fluid particle are negligible [119]. A mathematical definition is... 

For each set of initial conditions, Eqs. (4.1)-(4.3) can be solved to And X ", U ", and The initial conditions are randomly selected from known distribution functions, and we can assume that there is an infinite number of possible combinations. Each combination is called a realization of the granular flow, and the set of all possible realizations forms an ensemble. Note that, because the particles have finite size, they cannot be located at the same point thus X " 4 X for n 4 m. Also, the collision operator will generate chaotic trajectories and thus the particle positions will become uncorrelated after a relatively small number of collisions. In contrast, for particles suspended in a fluid the collisions are suppressed and correlations can be long-lived and of long range. We will make these concepts more precise when we introduce fluid-particle systems later. While the exact nature of the particle correlations is not a factor in the definition of the multi-particle joint PDF introduced below, it is important to keep in mind that they will have... 

On the other hand, we can also calculate the Stokes drift velocity from the solution of Eq. 8. We can follow a certain fluid particle s path x(xq, t), where xq is a reference point independent of the time. From the definition of the flow velocity, we can write... 

For a laminar flow through a straight microchannel, the fluid particles move in definite paths called streamlines and there are no components of fluid velocity normal to the duct axis. The projection of Eq. 6 along the axial direction (z) gives... 

Figure 2.1. Definition of the intrinsic coordinate system associated with the path of a fluid particle for a steady plane flow...

Good heat transfer on the outside of the reactor tube is essential but not sufficient because the heat transfer is limited at low flow rates at the inside film coefficient in the reacting stream. The same holds between catalyst particles and the streaming fluid, as in the case between the fluid and inside tube wall. This is why these reactors frequently exhibit ignition-extinction phenomena and non-reproducibility of results. Laboratory research workers untrained in the field of reactor thermal stability usually observe that the rate is not a continuous function of the temperature, as the Arrhenius relationship predicts, but that a definite minimum temperature is required to start the reaction. This is not a property of the reaction but a characteristic of the given system consisting of a reaction and a particular reactor. 

In this definition, ps and pt are the solid and fluid densities, respectively. The characteristic diameter of the particles is ds (which is used in calculating the projected cross-sectional area of particle in the direction of the flow in the drag law). The kinematic viscosity of the fluid is vf and y is a characteristic strain rate for the flow. In a turbulent flow, y can be approximated by l/r when ds is smaller than the Kolmogorov length scale r. (Unless the turbulence is extremely intense, this will usually be the case for fine particles.) Based on the Stokes... 

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