Flow of particles

This flow will be related to the local concentration c of species i 

In a cylinder of section S and length /, let we denote by J, the number of particles crossing S by time unit. 

In this paragraph the basic relation between diffusional flow and concentration gradient (Pick s law), is presented. 

The same evaluation can be done for charged particles and will give the relation between electrical and diffusional transport known as Nernst Einstein relation. The expression of the charge flow will also give Ohm s law for ionic transport in an electrolyte. 

The relation between the diffusional and electrical mobility is then 

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The computation of F is relatively straightforward. We simply consider the free flow of particles into and out of the region in time t. An expression for this flow in the v-direction, for example, can be obtained by considering two thin layers of size v 6t5r Sr that contain particles that move into or out of a cell with its centre at (x,y,z) in time 51 (see figure A3.1.6. ... 

The right-hand side of this equation includes components with and without exponential Boltzmann factor but their sum equals the total flow of particles from the j th rotational level to the rest of the levels. After this correction both necessary demands, Eq. (4.65) and Eq. (4.66), are satisfied. This result is of great advantage since calculation of the impact operator with the rather simple semiclassical formula, Eq. (5.1), does not lead after correction to any principal difficulties. The set of equations (5.26) and (5.27) determine the operator r(0) consistently but not uniquely. Other recipes may be used as well (see Chapter 7 and [195]). 

The boundary conditions for the bed at the side wall with no net flow of particles across it are at... 

In fast-fluidized beds, which use extremely small catalyst particles, the flow of particles and fluid is cocurrent, and nearly PF, leading to high conversion and selectivity. However, severe erosion of equipment is common, and thus, the design of the solids recovery and recirculation system is important. 

Cold particles of glass ballotini are fluidised with heated air in a bed in which a constant flow of particles is maintained in a horizontal direction. When steady conditions have been reached, the temperatures recorded by a bare thermocouple immersed in the bed are ... 

Ballotini particles, 0.25 mm in diameter, are fluidised by hot air flowing at the rate of 0.2 kg/m2 cross-section of bed to give a bed of voidage 0.5 and a cross-flow of particles is maintained to remove the heat. Under steady state conditions, a small bare thermocouple immersed in the bed gives the following data ... 

Elgin, J. C. and Foust, H. C. Ind. Eng. Chem. 42 (1950) 1127. Countercurrent flow of particles through moving continuous liquid pressure drop and flooding in spray-type liquid towers. 

Fig. 7-8. Quantum d3mamic tunneling and statistically activated flow of particles across an energy barrier e = energy of a tunneling particle...

Measurable quantities of cotton dust were extracted from small cotton lint samples by means of high-velocity air jets. Dust removed in this manner was sized by a wire-mesh screen that restricted the flow of particles larger than the openings in the screen. Particles smaller than the openings in the screen were collected for gravimetric analysis. 

Mixed Flow of Particles of a Single Unchanging Size, Uniform Gas Composition... 

However, the onset of natural convection or a deliberate resort to stirring the solution may hold the interfacial concentrations at steady values. Under these steady-state conditions, one may use Fick s first law, and the equality of the flows of particles up to and across the interface for an electronation reaction can be written as... 

Fig. 3.38. Flow of particles through a fluidised bed showing a normalised C-curve which is identical with the exit age distribution function E...

Since the total rate of flow is one characteristic property of any gas inlet system, it becomes obvious from equ. (10.88) that the conductance L plays a key role in describing the flow of particles through a tube. L depends on the geometry of the tube and the properties of the particle flow inside the tube. Two limiting cases for the particle flow can be distinguished ... 

The flow conditions ranging from single scattering to boundary layer flow of particles are shown in Fig. 3.6. It is noted that the single scattering is bounded by the condition of db/A.b = 1. In terms of Eq. (3.156), the boundary of the single scattering is a function of the particle size, ball probe size, and particle concentration. 

Consider a mixture of two groups of particles with radii a and a2, and corresponding number concentrations of i and n2. Assume that the group of particle 1 is subjected to a simple shear flow of particle 2. The x-component of the collision force between particle 1 and particle 2 can be obtained, from Eq. (5.213), as... 

The particle convection is in general important in the overall bed-to-surface heat transfer. When particles or particle clusters contact the surface, relatively large local temperature gradients are developed. This rate of heat transfer can be enhanced with increased surface renewal rate or decreased cluster residence time in the convective flow of particles in contact with the surface. The particle-convective component hpc can be expressed by the following equation, which is an alternative form of Eq. (12.39) ... 

There are no longer things and persons but only the direct flow of particles. Consciousness will now leave your body and flow into the stream of wave rhythm. 

The rate of coagulation, of particles of mass mp is equal to the net rate of flow of particles of reduced mass mT across the sphere of influence of radius Rs, i.e.,... 

Ding, J., Lyczkowski, R. W., Sha, W. T., Altobelli, S. A., and Fukushima, E. Analysis of liquid-solids suspension velocities and concentrations obtained by NMR imaging, NSF/DIE Workshop on Flow of Particles and Fluids, Sep. 17-18, 1992, Gaithersburg, Maryland. 

As the second application of irreversible thermodynamics we consider the Soret effect (1893) for a two-component system a flow of particles under the influence of a temperature gradient produces a gradient in concentration. We are ultimately interested in the magnitude of this effect under steady state conditions. Let J0, Jlt J2 be the entropy and particle fluxes... 

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