Single particle suspension

The equilibrium condition experienced by a particle falling at velocity Uf in a stationary fluid is, of course, equivalent to that of a motionless particle suspended in an upwardly flowing fluid with velocity uy this latter situation represents the upper fluid velocity bound for homogeneous fluidization. 

The forces acting on the particle are the result of fluid-particle interaction f] and gravity fg. Under conditions of equilibrium, we have  

For the case of essentially spherical particles and Newtonian fluids, u, can be readily estimated from this relation over the entire flow regime of relevance to the fluidization process. 

A rigorous solution exists for fj for the limiting condition of very low fluid flow rates around a sphere - in which the fluid streamlines follow the contours of the sphere, with no separation at the upper surface (the so called creeping flow regime). This may be regarded to occur at particle Reynolds numbers Re below about 0.1  

Many industrial fluidized bed reactors operate within, or close to, this range. 

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Figure 1.3 Homogeneous fluidization - from packed bed to single particle suspension.

Any analysis of the homogeneously fluidized state must encompass the conditions of single particle suspension and fluid flow through fixed beds of particles these represent, respectively, the upper and lower bounds for fluidization as illustrated in Figure 1.3. 

For single particle suspension, it was found convenient to define the effective weight of a particle as the net effect of gravity and buoyancy eqn (2.9). On applying the same definition to a fluidized sphere in equilibrium, for which. 

A sudden increase in fluid flux from U to U2 gives rise to a net upward force on all the particles these therefore immediately start to accelerate upwards together, without change in void fraction ei, until the relative fluid-particle velocity and the interaction force drop back to their previous equilibrium levels from this point on it would appear that the bed should continue its upward motion, piston-like at constant velocity, as was suggested in Chapter 1 on the basis of the analogy with single-particle suspension. The fact that this does not happen in practice (fluidized beds would never form if it did) has to do with the instability of the interface separating the bottom of the particle piston from the clear fluid below. 

Another approximation, one of the most enduring empirical correlations in multiphase systems, is the Richardson-Zaki correlation for a single particle in a suspension (3) ... 

In selecting cloths made from synthetic materials, one must account for the fact that staple cloths provide a good retentivity of solid particles due to the short hairs on their surface. However, cake removal is often difficult from these cloths - more than from cloths of polyfilament and, especially, monofilament fibers. The type of fiber weave and pore size determine the degree of retentivity and permeability. The objective of the process, and the properties of particles, suspension and cake should be accounted for. The cloth selected in this maimer should be confirmed or corrected by laboratory tests. Such tests can be performed on a single filter. These tests, however, provide no information on progressive pore plugging and cloth wear. However, they do provide indications of expected filtrate pureness, capacity and final cake wetness. 

If a single particle is falling freely under gravity in an infinitely dilute suspension, it will accelerate until it reaches a steady-state velocity. This final velocity is known as the terminal settling velocity (t/t) and represents the maximum useful superficial velocity achievable in a fluidised bed. Thus, the contained particles will be elutriated from the column if the superficial velocity is above Ut, the value of which can be predicted using the Stokes equation... 

The foregoing expressions give the suspension velocity (Fs) relative to the single particle free settling velocity, V0, i.e., the Stokes velocity. However, it is not necessary that the particle settling conditions correspond to the Stokes regime to use these equations. As shown in Chapter 11, the Dallavalle equation can be used to calculate the single particle terminal velocity V0... 

Add the protein to be coupled to the particle suspension in an amount equal to 1-10 X molar excess over the calculated monolayer for the protein type to be coupled. Mix thoroughly to dissolve. Low concentrations of protein may result in particle aggregation, because a single protein molecule can react with more than one particle. 

This result follows from the Richardson-Zaki equation. In their original work, Richardson and Zaki (1954) studied batch sedimentation, in particular the settling of coarse solid particles through a liquid in a vertical cylinder with a closed bottom. Richardson and Zaki found that the settling speed uc of the equal-sized particles in the concentrated suspension was related to the terminal settling speed u, of a single particle in a large expanse of liquid by the equation... 

The conceptual difference between the flux and drift velocity may thus be illustrated further by considering a dilute colloidal suspension in thermal equilibrium in the presence of a gravitational field. In equilibrium, this system adopts a single-particle probability density /(R) oc with... 

An important limitation of this book is that we treat only phenomena in which particle-particle interactions are of negligible importance. Hence, direct application of the book is limited to single-particle systems or dilute suspensions. 

In this section, various issues concerning solid particles are presented. The analysis covers the most important particle properties (surface area, particle shape and size distribution, mechanical strength, and density) as well as the behavior of a single particle in suspension (terminal velocity) and of a number of particles in fluidization state. Finally, the diffusion of molecules in a porous particle (diffusion coefficients) is also discussed. 

Comparing this equation to that of a single particle (eq. (3.565)), it is evident that in applying the Archimedes principle to a particle in a fluidized suspension, it is an average suspension density, including the particle density, rather than that of the fluid alone, that determines the buoyancy force (Foscolo and Gibilaro, 1984). The gravity force is... 

The literature has been divided into two main headings (1) production and (2) size measurement of both single particles and sprays (including uniform clouds of droplets and solid suspensions). In many instances a single reference may give important information in each category. 

Inside the accelerating tube the particles are accelerated by the air flow from zero to a certain velocity, usually 60-70% of the gas flow velocity. During acceleration, the relative velocity between the particles and the gas flow is very high. On the other hand, the concentration of particles in the solid-gas suspension to be processed with impinging streams is generally very small, as mentioned above, so that the interaction between particles can be neglected. Therefore the movement of particles can be described approximately with Newton s motion equations for a single particle, as follows ... 

Charge transfer occurs when particles collide with each other or with a solid wall. For monodispersed dilute suspensions of gas-solid flows, Cheng and Soo (1970) presented a simple model for the charge transfer in a single scattering collision between two elastic particles. They developed an electrostatic theory based on this mechanism, to illustrate the interrelationship between the charging current on a ball probe and the particle mass flux in a dilute gas-solid suspension. This electrostatic ball probe theory was modified to account for the multiple scattering effect in a dense particle suspension [Zhu and Soo, 1992]. 

Thus, from Eqs. (4.16) and (4.17), Nup 10, indicating that the Nup of a single particle due to thermal conduction in a gas-solid suspension roughly varies from 2 to 10. A more detailed discussion on the variation of Nup with S/dp is given by Zabrodsky (1966). 

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