Motion particle

In this equation, we recognize the static and dynamic pressures—the latter is often called the velocity head —as the first and third terms on the left-hand side. They have been divided by the fluid density. 

This equation shows that static and dynamic pressures can be interchanged in the flow field. In areas where the velocity is high, the static pressure will be low and vice versa. This is the principle used in many flow meters, for instance pitot tubes and venturi meters. It is especially important to appreciate this interdependence between static and dynamic pressure when dealing with swirling flows. 

The left-hand side of Eq. (2.1.3) is sometimes called Bernoulli s trinomial . The second term is unimportant relative to the two others when discussing gas cyclones and swirl tubes, since the fluid density is relatively low, and height differences not very large. 

In an actual flow situation, the fluid is not frictionless. Frictional dissipation of mechanical energy will therefore cause Bernoulli s trinomial to decrease in the flow direction, i.e. the trinomial is no longer constant, but decreases 

Frictionless flow is, nevertheless, a reasonably good approximation in the outer part of the swirl in a cyclone, Bernoulli s trinomial does not change very much there. 

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The treatment of translational motion in three dimensions involves representation of particle motions in tenns... 

It is helpful to distinguish three different types of problem to which Newton s laws of motion may be applied. In the simplest case, no force acts on each particle between collisions. From one collision to the next, the position of the particle thus changes by v,5f, where v, is the (constant) velocity and 6t is the time between collisions. In the second situation, the particle experiences a constant force between collisions. An example of this type of motion would be that of a charged particle moving in tr uniform electric field. In the third case, the force on the particle depends on its position relative to the other particles. Here the motion is often very difficult, if not impossible, to describe analytically, due to the coupled nature of the particles motions. 

This concludes our discussion of the viscosity of polymer solutions per se, although various aspects of the viscous resistance to particle motion continue to appear in the remainder of the chapter. We began this chapter by discussing the intrinsic viscosity and the friction factor for rigid spheres. Now that we have developed the intrinsic viscosity well beyond that first introduction, we shall do the same (more or less) for the friction factor. We turn to this in the next section, considering the relationship between the friction factor and diffusion. 

Electrophoresis (qv), ie, the migration of small particles suspended in a polar Hquid in an electric field toward an electrode, is the best known effect. If a sample of the suspension is placed in a suitably designed ceU, with a d-c potential appHed across the ceU, and the particles are observed through a microscope, they can all be seen to move in one direction, toward one of the two electrodes. AH of the particles, regardless of their size, appear to move at the same velocity, as both the electrostatic force and resistance to particle motion depend on particle surface this velocity can be easily measured. 

The presence of a static magnetic field within a plasma affects microscopic particle motions and microscopic wave motions. The charged particles execute cyclotron motion and their trajectories are altered into heUces along the field lines. The radius of the helix, or the T,arm or radius, is given by the following ... 

Particle Motion. AH suspended micrometer-si2e particles are in motion due to the thermal energy they possess. At any given temperature, the average kinetic energy due to thermal motion of an individual particle is equal to kP where k is the Bolt2maim constant (k = the gas constant, R, divided by Avogadro s number) ... 

The motion oF Fine particles immersed in a moving tliiid is more greatly aFFected by tliiid drag Forces than that For similar large particles, For vei v small particles in a tliiid, particle motion approximates... 

Particle Motion and Scale-Up Veiy little fundamental information is published on centrifugal granulators. Qualitatively, good operation rehes on maintaining a smoothly rotating stable rope of tumbling... 

In the following development we consider a plane wave of infinite lateral extent traveling in the positive Xj direction (the wave front itself lies in the Xj, Xj plane). When discussing anisotropic materials we restrict discussion to those propagation directions which produce longitudinal particle motion only i.e., if u is the particle velocity, then Uj = Uj = 0. The <100>, <110>, and <111 > direction in cubic crystals have this property, for example. The derivations presented here are heuristic with emphasis on the essential qualitative features of plastic flow. References are provided for those interested in proper quantitative features of crystal anisotropy and nonlinear thermoelasticity. 

In the intervening years, molecular dynamics simulations of biomolecules have undergone an explosive development and been applied to a wide range of problems [3,4]. Two attributes of molecular dynamics simulations have played an essential role in their increasing use. The first is that simulations provide individual particle motions as a function of time so they can answer detailed questions about the properties of a system, often more easily than experiments. For many aspects of biomolecule function, it is these details... 

Two most appealing features of this model draw so much attention to it. First, although microscopically one has very little information about the parameters entering into (5.24), it is known [Caldeira and Leggett 1983] that when the bath responds linearly to the particle motion, the operators q and p satisfying (5.24) can always be constructed, and the only quantity entering into the various observables obtained from the model (5.24) is the spectral density... 

If force P is greater than zero, the particle will be in motion relative to the continuous phase at a certain velocity, w. At the beginning of the particle s motion, a resistance force develops in the continuous phase, R, directed at the opposite side of the particle motion. At low particle velocity (relative to the continuous phase), fluid layers running against the particle are moved apart smoothly in front of it and then come together smoothly behind the particle (Figure 14). The fluid layer does not intermix (a system analogous to laminar fluid flow in smoothly bent pipes). The particles of fluid nearest the solid surface will take the same time to pass the body as those at some distance away. 

Coefficient A and exponent a must be evaluated experimentally. Experiments have shown that A and a are themselves functions of the Reynolds number. Equation 47 shows that the resistance force increases with increasing velocity. If the force field (e.g., gravity) has the same potential at all points, a dynamic equilibrium between forces P and R develops shortly after the particle motion begins. As described earlier, at some distance from its start the particle falls at a constant velocity. If the acting force depends on the particle location in space, in a... 

Once these first estimates for the geometric dimensions of the cyclone have been obtained, a full theoretical analysis of the fluid and particle motions in the cyclone may be performed using the theoretical models given in Section 13.2.1.2. A substantial use of the expression (13.26) for the collection efficiency should be employed so that an updated design of the geometry of the cyclone can be obtained. 

The motion of a charged aerosol particle in a gas is governed by the electrostatic force and the aerodynamic forces. The theory dealing with the particle motion has been discussed in several books (see, e.g., Hinds- ). The electrostatic force F caused by the electric field E is given by... 

Particle trajectories can be calculated by utilizing the modern CFD (computational fluid dynamics) methods. In these calculations, the flow field is determined with numerical means, and particle motion is modeled by combining a deterministic component with a stochastic component caused by the air turbulence. This technique is probably an effective means for solving particle collection in complicated cleaning systems. Computers and computational techniques are being developed at a fast pace, and one can expect that practical computer programs for solving particle collection in electrostatic precipitators will become available in the future. 

Photophoresis Particle motion that takes place in the direction of radiation, due to the absorbed radiation warming one side of the particle more than the other. 

Crystals suspended in liquors emerging from crystallizers are normally passed to solid-liquid separation devices such as gravity settlers or thickeners that may subsequently feed filters to remove yet more liquid prior to drying. Here the transport processes of particle motion and the flow of fluids through porous media are important in determining equipment size, the operation of which may be intensified by application of a centrifugal force. 

A. Bohr (Copenhagen), B. Mottelson (Copenhagen) and J. Rainwater (New York) discovery of the connection between collective motion and particle motion in atomic nuclei and the development of the theory of the structure of the atomic nucleus based on this connection. 

Also in 1950 Sakliarov and Tamm proposed an idea for a controlled thermonuclear fusion reactor, the TOKAMAK (acronym for the Russian phrase for toroidal chamber with magnetic coiF ), which achieved the highest ratio of output power to input power of any fusion device of the twentieth centuiy. This reactor grew out of interest in a controlled nuclear fusion reaction, since 1950. Sakharov first considered electrostatic confinement, but soon came to the idea of magnetic confinement. Tamm joined the effort with his work on particle motion in a magnetic field, including cyclotron motion, drifts, and magnetic surfaces. Sakharov and Tamm realized that... 

The calculation and combination of the components of particle motion requires imposition of a coordinate system. Perhaps the most commoi) is the Cartesian system illustrated in Figure 2-8. Defining unit vectors i, j, and k along the coordinate axes X, y, and z, the position of some point in space, P, can be defined by a position vector, r ... 

It is easy to invent rules that conserve particle number, energy, momentum and so on, and to smooth out the apparent lack of structural symmetry (although we have cheated a little in our example of a random walk because the circular symmetry in this case is really a statistical phenomenon and not a reflection of the individual particle motion). The more interesting question is whether relativistically correct (i.e. Lorentz invariant) behavior can also be made to emerge on a Cartesian lattice. Toffoli ([toff89], [toffSOb]) showed that this is possible. 

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