Particle multi-, systems

This section will first deal with the phases in particle-fluid two-phase flow by developing a mathematical model to quantify local hydrodynamic states. This analysis will reveal the insufficiency of the conditions for the conservation of mass and momentum alone in determining the hydrodynamic states of heterogeneous particle-fluid systems, and calls for a methodology different from what is used in analyzing dilute uniform flow. For this purpose the concept of multi-scale interaction between particles and fluid and the principle of energy minimization are proposed. 

The derived formulae are applied to the SiC-SisN4 multi-particle-matrix system, and calculated values of investigated parameters are in an excellent agreement with those from published experimental results. 

To derive the thermal stresses acting in the system of the homogeneously distributed spherical particles embedded in the infinite matrix, the multi-particle-matrix system is di-... 

Depending on the distribution of the spherical particles of the radius R (Fig. 2), the infinite matrix is divided into cubic cells of the dimension d, representing an inter-particle distance, and consequently the spherical particle volume Ixaction of the multi-particle-matrix system, v, as the ratio of the spherical particle volume to the cubic cell volume, is derived as [18]... 

Due to the isotropy of the multi-particle-matrix system, the shape S5unmetry of the spherical particle and the cubic cell, the symmetrical distribution of the cubic cells resulting Irom the matrix infinity, the thermal stresses are thus sufficient to be investigated within one twenty-fourth of the cubic cell (Fig. 3), then for r G (0,rc), (p G (0,tc/4), v (V34,jc/2), where Vc = OC, V34 = Z (x2, OC34), C is a point on the cubic cell surface C1C2C3C4, and then [18]... 

As presented in the section 3, concerning the v-dependences of thermal-stress related parameters of the SiC-Si3N4 particle-matrix system in the interval v e (0,k/6), the formulae related to the isotropic one-particle-matrix system represented by one spherical particle of the radius R embedded in the infinite matrix, as transformation of those related to the isotropic multi-particle-matrix system are for v = 0, are required to be derived. 

The paper is the continuation of the calculation published in [18] and [25], presenting the thermal stresses in the isotropie multi- and one-particle-matrix systems, respectively, originating during a cooling process as a consequence of the difference of thermal expansion coefficients between the particle and the matrix. The isotropic multi- and one-particle-matrix systems are represented by the homogeneously distributed spherical particles embedded in... 

Applying the derived formulae to the SiC-Si3N4 multi-and one particle-matrix systems, the main results included in the section 3 are as follows ... 

The effect of physical processes on reactor performance is more complex than for two-phase systems because both gas-liquid and liquid-solid interphase transport effects may be coupled with the intrinsic rate. The most common types of three-phase reactors are the slurry and trickle-bed reactors. These have found wide applications in the petroleum industry. A slurry reactor is a multi-phase flow reactor in which the reactant gas is bubbled through a solution containing solid catalyst particles. The reactor may operate continuously as a steady flow system with respect to both gas and liquid phases. Alternatively, a fixed charge of liquid is initially added to the stirred vessel, and the gas is continuously added such that the reactor is batch with respect to the liquid phase. This method is used in some hydrogenation reactions such as hydrogenation of oils in a slurry of nickel catalyst particles. Figure 4-15 shows a slurry-type reactor used for polymerization of ethylene in a sluiTy of solid catalyst particles in a solvent of cyclohexane. 

The remarkable stability of onion-like particles[15] suggests that single-shell graphitic molecules (giant fullerenes) containing thousands of atoms are unstable and would collapse to form multi-layer particles in this way the system is stabilized by the energy gain from the van der Waals interaction between shells [15,26,27],... 

We hope that macroscopic samples of quasi-spherical onion-like particles will soon become available, and then we will be able to characterize these systems in detail. Probably a new generation of carbon materi-aks can be generated by the three-dimensional packing of quasi-spherical multi-shell fullerenes. 

P. Franke, G. Inden. Diffusion controlled transformations in multi-particle systems. Z Metallkd 88 9 1, 1997. 

Thus the interacting multi-electron system can be simulated by the noninteracting electrons under the influence of the effective potential l eff(r)- Kohn and Sham [51] took advantage of the fact that the case of non-interacting electrons allows an exact computation of the particle density and kinetic energy as... 

Speed-up of mixing is known not only for mixing of miscible liquids, but also for multi-phase systems the mass-transfer efficiency can be improved. As an example, for a gas/liquid micro reactor, a mini packed-bed, values of the mass-transfer coefficient K a were determined to be 5-15 s [2]. This is two orders of magnitude larger than for typical conventional reactors having K a of 0.01-0.08 s . Using the same reactor filled with 50 pm catalyst particles for gas/Hquid/solid reactions, a 100-fold increase in the surface-to-volume ratio compared with the dimensions of laboratory trickle-bed catalyst particles (4-8 mm) is foimd. 

For multi-particle systems with cartesian coordinates ri, V2,, the classical... 

The postulates 1 to 6 of quantum meehanies as stated in Sections 3.7 and 7.2 apply to multi-particle systems provided that each of the particles is distinguishable from the others. For example, the nucleus and the electron in a hydrogen-like atom are readily distinguishable by their differing masses and charges. When a system contains two or more identical particles, however, postulates 1 to 6 are not sufficient to predict the properties of the system. These postulates must be augmented by an additional postulate. This chapter introduces this new postulate and discusses its consequences. 

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