Attrition modeling

For agitated suspensions, however, reeent work suggests that miero-attrition of parent erystals ean oeeur in agitated suspensions (Synowiee etai, 1993). Sueh breakage via miero-attrition will result in many fine erystals generated and a relatively unehanged parent partiele (ef. seeondary nueleation, see below) with a eorrespondingly more eomplex distribution funetion (Hill and Ng, 1995, 1996). These alternative attrition models are eonsidered in more detail later. 

Figure 11.5 A high-level attrition model must represent the success probabilities and time taken at each stage of work, the return loops where rework is necessary, and the criteria for abandoning a project entirely.

Attrition cannot normally be directly investigated in the large-scale process. It is, for example, impossible to analyze the whole bulk of material and it is nearly impossible or at least very expensive to vary parameters in a running industrial process. For that reason, attrition has to be investigated in small-scale experiments. The results of these experiments require a model or at least an idea of the governing attrition mechanisms to be applied to the large-scale process. In principle, there are two different philosophies of attrition modeling. 

In order to achieve such a comprehensive insight into the attrition phenomena of a given process, it is necessary to consider meticulously each individual effect of the various influencing factors related to the material properties, the system design, and the operating conditions. The present chapter attempts to summarize systematically the findings available in the open literature and thus to draw up a possible approach in the attrition modeling. 

However, besides the simple ranking there is quite often even a quantitative prediction of the proeess attrition requested. This requires both an attrition model with a precise description of the process stress and as an input parameter to the model precise information on the material s attritability under this specific type of stress. This calls for attrition/friability tests that duplicate the process stress entirely. As will be elucidated in Sec. 5, the stress in a given fluidized bed system will be generated from at least three sources, i.e., the grid jets, the bubbling bed, and the cyclones. For each there is a corresponding friability test procedure. 

The modeling of fluidized beds remains a difficult problem since the usual assumptions made for the heat and mass transfer processes in coal combustion in stagnant air are no longer vaUd. Furthermore, the prediction of bubble behavior, generation, growth, coalescence, stabiUty, and interaction with heat exchange tubes, as well as attrition and elutriation of particles, are not well understood and much more research needs to be done. Good reviews on various aspects of fluidized-bed combustion appear in References 121 and 122 (Table 2). 

The Andritz-Sprout-Bauer attrition miU (Fig. 20-51) is available in single- and double-runner models with 30- to 91-cm- (12- to 36-in.-) diameter disks and with power ranging up to 750 kW (1000 hp). By the use of a variety of plates and sheirconstruciions these units are represented in such applications as coarse granulating, piilveriz-ing, and shredding. 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

These models have been tested in a erystal attrition eell using alternately stainless steel and rubber eoated impellers to determine impaet and turbulent eontributions respeetively by Synowiee etal. (1993) as shown in Figure 5.13. 

Falope etal. (2001) extended the MSMPR model of agglomerative erystal preeipitation based on the Monte Carlo simulation teehnique to aeeount for partiele disruption by eonsidering two alternative partiele size reduetion meehanisms - one representative of partiele splitting into two parts of equal volume, the other representative of miero attrition. 

In 1914, F. W. Lanchester introduced a set of coupled ordinary differential equations-now commonly called the Lanchester Equationsl (LEs)-as models of attrition in modern warfare. Similar ideas were proposed around that time by [chaseSS] and [osip95]. These equations are formally equivalent to the Lotka-Volterra equations used for modeling the dynamics of interacting predator-prey populations [hof98]. The LEs have since served as the fundamental mathematical models upon which most modern theories of combat attrition are based, and are to this day embedded in many state-of-the-art military models of combat. [Taylor] provides a thorough mathematical discussion. 

Most traditional models focus on looking for equilibrium solutions among some set of (pre-defined) aggregate variables. The LEs are effectively mean-field equations, in which certain variables (i.e. attrition rate) are assumed to represent an entire force, the equilibrium state is explicitly solved for and declared the battle outcome. In contrast, ABMs focus on understanding the kinds of emergent patterns that might arise while the overall system is out of (or far from) equilibrium. 

A well-defined bed of particles does not exist in the fast-fluidization regime. Instead, the particles are distributed more or less uniformly throughout the reactor. The two-phase model does not apply. Typically, the cracking reactor is described with a pseudohomogeneous, axial dispersion model. The maximum contact time in such a reactor is quite limited because of the low catalyst densities and high gas velocities that prevail in a fast-fluidized or transport-line reactor. Thus, the reaction must be fast, or low conversions must be acceptable. Also, the catalyst must be quite robust to minimize particle attrition. 

The study of DMPK has changed from a descriptive to a much more predictive science [3]. This is driven by great progress in bioanalytics, development of in vitro assays and in silica modeling/simulation, and a much better basic understanding of the processes. Thus, and fortunately, ADME-related attrition has lowered from around 40% in 1990 to around 10% in 2005 [13]. 

Designing a model fluidized bed which simulates the hydrodynamics of a commercial bed requires accounting for all of the mechanical forces in the system. In some instances, convective heat transfer can also be scaled but, at present, proper scaling relationships for chemical reactions or hydromechanical effects, such as particle attrition or the rate of tube erosion, have not been established. 

Arena et al. (1983) investigated the coal attrition in a mixture with sand under hot but inert conditions. As they increased the sand particle size while keeping its mass in the bed constant, they observed an increase in the coal attrition rate. They interpreted their results by assuming that the abrasion energy is shared out on the entire material surface. On the same basis Ray et al. (1987a) developed their attrition rate distribution model for abrasion in a fluidized bed. 

Modeling of Jet-Induced Attrition. Werther and Xi (1993) compared the jet attrition of catalysts particles under steady state conditions with a comminution process. They suggested a model which considers the efficiency of such a process by relating the surface energy created by comminution to the kinetic energy that has been spent to produce this surface area. The attrition rate, RaJ, defined as the mass of attrited and elutriated fines per unit time produced by a single jet, is described by... 

Ghadiri et al. (1992b, 1994, 1995) developed a more fundamental approach. They consider the particles entrained into the jet and relate the production of attrited fines to the attrition rates obtained from single particle impact tests (cf. Sec. 4.3). According to their model, it should be possible to predict jet attrition rates in fluidized beds on the basis of single particle impact tests combined with a detailed description of the jet hydrodynamics. 

Modeling ofBubble-Induced Attrition. Merrick and Highley (1974) have modeled bubble-induced attrition as a comminution process. According to Rittinger s law of size reduction by abrasion (cfi, Perry, 1973), the rate of creation of new surface area AS Al is proportional to the rate of energy input Ah. At... 

Vaux (1978), Ulerich et al. (1980) and Vaux and Schruben (1983) proposed a mechanical model of bubble-induced attrition based on the kinetic energy of particles agitated by the bubble motion. Since the bubble velocity increases with bed height due to bubble coalescence, the collision force between particles increases with bed height as well. The authors conclude that the rate of bubble-induced attrition, Rbub, is then proportional to the product of excess gas velocity and bed mass or bed height, respectively,... 

Modeling of Cyclone Attrition. A very simple model of cyclone attrition may be formulated in analogy to the models discussed with respect to jet attrition and bubble-induced attrition (Reppenhagen and Werther, 1997). 

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