Particle inertia parameter

The preceding discussion shows that the sampling efficiency for thin L-shaped probes is a function of two parameters the deviation from the isokinetic conditions and the response of the particles to the deflection of the fluid streamlines upstream of the sampler. The deviation from the isokinetic conditions is a function of the velocity ratio (U/Uq), whereas the particle response is a function of the ratio of particle inertia to fluid drag. This ratio in a dimensionless form is known as the particle inertia parameter, the Stokes number, or the Barth number (K), defined as ... 

K particle inertia parameter based on probe radius (also called Stokes number or Barth number)... 

K particle inertia parameter, p,d502 Ubl 18p/ DSM n Richardson and Zaki s exponent (dimensionless)... 

Since there are three dimensions, two dimensionless groups, e.g., and defined in Chapter 5, suffice to describe the motion. If the motion is unsteady, it is necessary to introduce the particle density explicitly, since it determines the particle inertia as well as the net gravity force. Also, since L/r varies with time and position, a further parameter must be introduced. This may be the distance x moved since the start of the motion. Equation (11-1) is then replaced by... 

Nasr-El-Din and Shook (58) studied solids distribution in a vertical pipe downstream of a 90° elbow. They tested sand-water slurries of various solid concentrations and particle sizes. The slurry flows were turbulent, and the particle Stokes number (inertia parameter) based on the pipe diameter and bulk velocity varied from 0.5 to 3. The solids distribution downstream of the elbow was found to be a function of the radius of curvature of the elbow, solids concentration, and particle size. 

The Reynolds number based upon the angular velocity of the particle is not an independent parameter for a freely supported particle rather according to Eq. (89b) a homogeneous spherical particle will, in the absence of wall and particle inertia effects, rotate with the local angular velocity of the undisturbed flow at its center—at least at sufficiently small Reynolds numbers. This velocity is... 

The fact that the total number of particles must be conserved during the development of occasional disturbances in a uniform vertical flow or in a homogeneous fluidized bed in itself results in the formation of kinematic waves of constant amplitude, as was first demonstrated by Kynch [48]. Both particle inertia and the nonlinear dependence of the interphase interaction force on the suspension concentration cause an increase in this amplitude. This amounts to the appearance of a resultant flow instability with respect to infinitesimal concentration disturbances and with respect to other mean flow variable disturbances. Various dissipative effects can slow the rate at which instability develops, but cannot actually prevent its development. Therefore, investigating the linear stability of a flow without allowing for interparticle interaction leads inevitably to the conclusion that the flow always is unstable irrespective of its concentration and the physical parameters of its phases. This conclusion contradicts experimental evidence that proves suspension flows of sufficiently small particles in liquids to be hydrodynamically stable in wide concentration intervals [57-59]. Moreover, even flows of large particles in gases may be stable if the concentration is either very low or very high. 

Notice that the function exhibits many local minima besides the global minimum located at (0,0). The PSO parameters are as follows Ci = C2 = 1.5 and x = 0.7. The inertia parameter w was linearly adjusted from 1.5 to 0.5 along K = 25 steps. To do this use was made of iV = 10 particles. The right panel of the figure shows that the global minimum was successfully found by the algorithm without being trapped in local minima. 

The inertia parameter used is the ratio between the particle stopping distance and the diameter of the sampling nozzle D so that... 

The objective of the polarization model is to relate the material parameters, such as the dielectric properties of both the liquid and solid particles, the particle volume fraction, the electric field strength, etc., to the rheological properties of the whole suspension, in combination with other micro structure features such as fibrillatcd chains. A idealized physical model ER system—an uniform, hard dielectric sphere dispersed in a Newtonian continuous medium, is usually assumed for simplification reason, and this model is thus also called the idealized electrostatic polarization model. The hard sphere means that the particle is uncharged and there are no electrostatic and dispersion interactions between the particles and the dispersing medium before the application of an external electric field. For the idealized electrostatic polarization model, there are roughly two ways to deal with the suspensions One is to consider the Brownian motion of particle, and another is to ignore the Brownian motion and particle inertia. For both cases the anisotropic structure of such a hard sphere suspension is assumed to be represented by the pair correlation function g(r,0), derived by... 

Thus, the inertia of the tunneling particle leads to two opposite effects a decrease of the transition probability due to the reorganization along the coordinate of the center of mass and an increase of the transition probability due to the increase of the Franck-Condon factor of the tunneling particle. Unlike the result in Ref. 66, it is found in Ref. 67 that for ordinary relationships between the physical parameters, the inertia leads to an increase of the transition probability. 

Other than gas velocity and the physical properties of the feed, particle size is a parameter which has a significant effect. Smaller bed particles are more likely to form permanent bonds, and to quench, because of their smaller inertia. The force tending to pull two particles apart is equal to the product of the particle mass and the distance between the two centres of mass. For the case of two spherical particles joined together at their surfaces, this force is proportional to the particle diameter raised to the fourth power. Other cases approximate to this relationship. 

For low shear stresses in the dispersions, the characteristic velocity, of the relative particle motion is small enough for the Reynolds number, Re = pF L/ri, to be a small parameter, where L is a characteristic length scale. In this case, the inertia terms in Equations 5.247 and 5.249 can be neglected. Then, the system of equations becomes linear and the different types of hydrodynamic motion become additive e.g., the motion in the liquid flow can be presented as a superposition of elementary translation and rotational motions. 

Defects include (dust) particles on or within suspended interdigitated seismic masses, leading to electrical shorts and to deviations in moments of inertia. Another example relates to the appearance of crystallographic defects such as dislocations or stacking faults in epitaxial superstructures, causing leakage currents that might impair sensor reliability in the field. Defect densities may of course vary within a wafer, from wafer to wafer, and from lot to lot. Unfortunately, at present not many models are available that link tolerance bands of functional parameters to specific defects and their tolerable density distributions. 

Since the bubble surface is impermeable to liquid, the normal component of the liquid velocity on the surface is zero. As the distance from the bubble surface increases, the normal component of the liquid velocity also increases. The thickness of the liquid layer, in which the normal component of the liquid velocity decreases due to the effect of the bubble, is of the order of the bubble radius. The particle crosses this liquid layer due to the inertia path whereby deposition of a particle depends on the dimensionless Stokes parameter. 

Finally, the parameter R> represents the fraction of particles, once in contact, that stick to the surface. Generally, small particles do not possess sufficient inertia to bounce off a surface, and Rt is simply unity. We noted this earlier in arguing that the surface resistance rc for particles is usually taken to be zero. Whether a particle possesses sufficient inertia to rebound from a surface depends on its Stokes number. Zhang et al. (2001) use the following form for R suggested by Slinn (1982) ... 

The dimensionless parameter S is referred to as the Stokes number. It characterizes the measure of particle s inertia. At S 1, the particle s inertia is small, and it follows from (10.119) that the trajectory of the particle is dose to the streamline. At S 1, one finds from (10.119) that d rfdt x 0, i.e. trajectories are the straight lines. 

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