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«Dissertation zur Erlangung des akademischen Grades Doktoringenieurin (Dr.-Ing.) von: Yashodhan Pramod Gokhale geboren am: 05. October 1981 in Pune, ...»

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Agglomeration process is most common in powder processing industries. Agglomeration in the fluidized bed takes place, if, after the drying of liquid bridges, solid bridges arise. There are a large number of theoretical models available in the literature for predicting whether or not two colliding particles stick together. These models involve a wide range of different assumptions about the mechanical properties of the particles and the system characteristics.

The particles form aggregates as a result of collisions and these aggregates have higher effective sizes than the primary particles, which build it. The increase in the effective size can be explained by a typical example. The increase in the size of the aggregates accounts for the removal of the particles in the layer of water above thermocline (a layer of water in an ocean or certain lakes, where the temperature gradient is greater than that of the warmer layer above and the colder layer below). Particles in lakes are in a continuous process of agglomeration and disintegration until the steady state is reached. The final size depends on the shear rate and the volume fraction. Also, agglomeration formation initiated in the atmosphere, induces premature fallout of fine-grained ash. Hence, the settling velocity is higher for the agglomeration than the dispersed ash particles.

In addition, agglomeration results in the modification of the effective surfaces of the particles.

This is important when they adhere pollutant particles with them. Aggregation reduces the particle surface area for condensation and/or chemical reaction. This can have important consequences for particle (e.g. aerosol or colloid) transport as larger particles settle more rapidly under gravity but diffuse more slowly.

5.3.2 Disintegration Sub-Process

Disintegration means particle size reduction or particles disperse into primary particles.

Disintegration has a significant effect on the number of particles. The total number of particles in a disintegration process increases while the total mass remains constant.

Disintegration process can be named as reversible agglomeration depending on the process used. The term breakage denotes the mechanical fracture of the coarse solid. It can be applied not only to systems in which solids undergo random breakage, but also to the mechanisms in which the solids form from existing particles by other mechanisms. Also, cell division by asexual means is an example of such process.

The breakage of a particle results from stressing at machine tools, or with other processes like comminution operations. The particles distributed according to the mass or volume is frequently used in process industries.

For these kinds of breakage processes, the size reduction of the solid material forms an example. The evolution of drop size distributions in a stirred liquid-liquid dispersion in which the dispersed fraction is small occurs mainly by drop breakage, since at the initial stages the coalescence effects are negligibly small. The subject of high shear flows of colloidal suspensions consisting of disintegrating the clusters of nanoparticles represents a vast field.

This is characterized by a wide spectrum of characteristic length and time scales. These shear flows occur in a broad range of technological applications such as processing of nanoparticles, and engineering disciplines. Ball milling, high shear mixing, or ultrasonication are commonly used to disintegrate the agglomerate nanoparticles.

During ball milling, breakage occurs due to impact and high shear fields. The shape of produced particles is irregular and many defects are introduced into the grain structure.

Efficiency of agglomeration and disintegration process depends on applied equipment and process conditions.

5.3.3 The Moment Form of the Population Balance

–  –  –

where qr (d ) is the frequency distribution of particle size d (J.Tomas 2007). If the internal coordinate d is taken as length, then the zeroth moment is equal unity, and first, second, and third moments are proportional to the averaged length, area and volume of particle collective, respectively. On the other hand if d denotes volume of the particles, then the zeroth and first moments are proportional to the total number and total mass of particles, respectively. The second moment is in this case proportional to the light scattered by particles in the Rayleigh limit(Kumar and Ramkrishna 1996). The moment forms of the population balance can be very powerful.

5.4 Kernels of the Agglomeration and Disintegration Kinetics

In our model, agglomeration and disintegration takes place simultaneously. The primary particles bind together to form agglomerates, while the agglomerates split into pieces, as shown in Figure 5-1. The model is based on the assumption that the porous agglomerate structure is formed with the nonporous primary particles.

5.4.1 Agglomeration rate kernel

Agglomeration, the growth of particles by collisions and subsequent bonding of smaller particles contained in the fluid, can be alternatively called as aggregation or flocculation according to the micro process used. Von Smoluchowski (Smoluchowski 1916) considered the process of aggregation as a series of chemical reactions, and developed equations describing the particle aggregation rates as well as expressions for the particle collisions in solution (Park and Rogak 2004).

The mathematical representation of agglomeration has been based on the consideration that the process consists of two micro processes: transport and adhesion. The transport step, which leads to the collision of two particles, is achieved by the virtue of the local variations in fluid/particle velocities arising throughImposed velocity gradients from mixing (Orthokinetic agglomeration) (2) The random thermal „Brownian‟ motion of the particles (Perikinetic agglomeration) (3) Differences in the settling velocities of the individual particles (driven by forced field of gravity or differential settling).

Adhesion is then, depending upon a number of short-range forces largely pertaining to the nature of the surface themselves (Park and Rogak 2004).

The fundamental assumption of the aggregation process is that it is a second-order rate constant process in which the rate of collision is the product of concentrations of the two colliding particles. Mathematically, the rate of successful collisions between particles of size di, d j is given by,

–  –  –

where  is the agglomeration efficiency,  (ri, rj ) is the collision frequency between particles of radius ri, rj and cni and cnj are the particle concentrations for particles of radius ri, rj, respectively.

The collision frequency  is a function of the micro process, i.e. Perikinetic, orthokinetic or differential sedimentation. The agglomeration efficiency gives  (values from 0 to 1) is a function of the probability of successful particle agglomeration events. In other words, larger the agglomeration probability then value of is larger. Thus, in effect,  is a measure of the transport efficiency leading to collisions, while  represents the percentage of those collisions, which results in successful agglomeration events. All the models are based on this one fundamental equation. The values of the parameters  and  are dependent upon the nature of particles to the micro processes of agglomeration and the kinetic flow regime. The research is devoted in finding the values for these two parameters and establishing equations.

Also the importance of cni and cnj are noted, for the overall rate always increases with particle concentration. The basic assumption is that the agglomeration rate is independent of the colloidal interactions and depends only on the particle transport mechanism.

The assumption is based on the short-range nature of interparticle forces, which is usually much less than the particle size, so that the particles come in contact before these forces play a role. The decoupling of transport and adhesion steps is a strong simplification in the agglomeration kinetics. At the moment, let us assume that every collision between the particles results in the formation of an aggregate (i.e. the agglomeration efficiency,  =1).

Hence, the agglomeration rate and the collision frequency are the same. Particle aggregation can be described as by the rate at which a certain size aggregate is being formed by smaller aggregates minus the rate at which the aggregate combines to form a larger aggregate from small aggregates.

The rate of change of concentration of k -fold aggregates, where k  i  j, can be given by the Smoluchowski equation as given in Eq.5.6 dcn 1 i k 1  5.6   i, j cni cnj  cnk  i, k cni, dt 2 i  j k i 1 i 1 Where i, j and k represent discrete fractions of particle sizes. The first term on the right hand side represents the rate of formation of k -fold aggregates such that the total volume is equal to the volume of the particle of size fraction k. The summation by this method means counting each collision twice and hence the factor ½ is included. The second term on the right hand side describes the loss of particles of size fraction k by virtue of their aggregation with other particle sizes. The important notable point is that the above equation is applicable only for irreversible aggregation since no term is included for the break-up of the aggregates, which is usually common in aggregating environments.

5.4.2 Convection-Controlled Agglomeration

Mostly in practice the natural movement of the particles due to the Brownian motion is insufficient to overcome the electrostatic repulsion barrier between the particles. This results in permanent agglomeration. Nearly all the aggregation processes contain some form of induced shear, due to laminar or turbulent fluid flow during stirring. The directional or random movements of the particles due to the laminar or turbulent fluid flow during stirring results in increase in the rate of interparticle collisions. Agglomeration resulting in this manner is called as convection-controlled agglomeration, or orthokinetic agglomeration.

The main difference between the orthokinetic agglomeration and the perikinetic agglomeration is the rate constants or kernels. In perikinetic agglomeration, as the agglomeration proceeds, the big agglomerates move slowly when compared with the small particles and hence there is reduction in the rate constant value. On the other hand, in the orthokinetic agglomeration, as the size of agglomerates grows bigger, it tries to catch more particles due to the shear and hence we observe increment in rate constant value. In short, perikinetic agglomeration is more predominant in the initial stages of agglomerate formation, while orthokinetic agglomeration wins the race in the later stages when the particle size grows bigger. Laminar Flow

–  –  –

of the fluid. The term n is defined as the number of revolution of the stirrer; D a is the impeller diameter and  is the kinematic viscosity of the fluid. At low velocities, the flow is laminar (Re 2100); there is no lateral mixing in this flow. At high velocities, the flow is turbulent (Re 4000) which is marked by a chaotic nature. In between 2100 and 4000, the transition regime exists (Patil, Andrews et al. 2001).

(Smoluchowski 1916) who considered only the case of uniform laminar shear did the first work on the rate of orthokinetic aggregation. These conditions are more theoretical in nature and seldom in practice but it is convenient to start with a simple case and then modify the result for other conditions. Laminar and turbulent phenomena share the same kinetics for agglomeration; the only difference is in the magnitude of the shear rates,, from the stirrer power input. Here also the assumption is the same like Brownian diffusion, i.e. the diffusion is due to the moving particle i to the fixed particle j (diffusion is due to the turbulent fluid motion instead of the Brownian motion). Smoluchowski assumed that the particles flow in straight streamlines and collide with other particles in different streamlines, according to their relative position. The moving particles on streamlines that bring their center within a distance ri  rj (the collision radius, Rij ) of the central particle will collide (Collet 2004).The total number of agglomeration events occurring between i and j particles in unit volume and unit time is given by,

–  –  – Turbulent Flow We have considered so far the unrealistic situation of the uniform laminar shear. But, in the actual process, the phenomenon of turbulence is the most dominated and needed one. In static media, the aggregation of the nanoparticles is due to the Brownian collisions whereas the larger particles settle due to gravity and have different settling velocities due to their sizes.

Therefore these collide and aggregate However, in many practical applications, it is necessary to keep the solid-liquid suspension in motion to homogenize it. In such cases, in spite of the flow pattern, the role of the local shear flow is dominant. The turbulent behavior of slurries in a stirred tank is a typical example (Komarneni 2003). Moreover, turbulence is a poorly understood phenomenon. Turbulence can be generated from contact of a flowing stream with solid boundaries, called wall turbulence or from contact between layers of fluid moving at different velocities called free turbulence (Patil, Andrews et al. 2001). Free turbulence is also called as sheared flow.

Turbulent flow consists of a mass of eddies of various sizes existing along with each other in the flowing stream. The continually forming large size eddies break into smaller eddies of micro turbulence, and were dissipated into heat. The more vigor the turbulence has, the more number of eddies are created. The eddies posses energy which is supplied by the potential energy of the bulk flow of the fluid. From the energy point of view, turbulence transfers energy from large eddies to smaller ones (micro turbulence).

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