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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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The mechanical energy is not appreciably dissipated into heat during the breakup of the larger eddies into smaller ones, and is worthless for practical purposes. This mechanical energy is finally converted into heat as the micro eddies are dissipated. This energy transfer strongly influences the particle transport and the particle agglomerations (Komarneni 2003). Camp and Stein approached this problem in a stirred tank reactor by calculating the shear rate, from the dissipation rates as power input per unit mass of the fluid, .

5.10   2       Where,  is the kinematic viscosity of the fluid ( =   ).

5.11 N p n3 Da

–  –  –

The result is similar to the expression derived by (Saffman and Turner 1956) for particle collisions in isotropic turbulence, but with a slightly different numerical factor.

5.4.3 Diffusion- Controlled Agglomeration

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Here,  ij is the agglomeration rate constant or agglomeration kernel. (Smoluchowski 1916) Smoluchowski calculated the rates of diffusion of spherical particles of fraction i to a fixed sphere j.

Practically, the central sphere j is not fixed and hence the term mutual diffusion coefficient Dij is introduced. The rate of collisions is then,

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distance; Dij  Di  D j is the mutual diffusion coefficient such that both the particles move about each other. The value of Dij can be calculated from the Stokes-Einstein‟s equation. The agglomeration rate constant has a very important feature for monodispersed particles of nearly same size; the agglomeration kernel becomes almost independent of particle size. The term (ri  rj ) 2 has a constant value of about 4 when ri  rj are the same. In such a case the ri  rj

–  –  –

5.4.4 Relative Sedimentation The relative sedimentation is another important phenomenon when particles of different sizes or density are settling down from the suspension. Big particles settle faster and they capture the small particles on the course of their travel. The velocity can be easily calculated, assuming the spherical particles and using the Stoke‟s law for their sedimentation rate (Smoluchowski 1917). The rate equation can be written as,

–  –  –

Where, g is the acceleration due to gravity, and  s is the density of the particles and  L is the density of the fluid. This phenomenon is important in the final stages of the agglomeration where aggregate growth by sedimentation becomes dominant. But in our case, this phenomenon can be neglected as the particles are of submicron size (M.Elimelech, Gregory et al. 1995).

5.4.5 Effects of hydrodynamic interactions

The main assumption of the Smoluchowski theory is that the interparticle interactions are negligible until the point of contact such that the collision takes place with 100% efficiency.

But, in reality the hydrodynamic forces are not negligible and they have a significant role upon the colliding particles. When particles near each other to collide, the fluid in the space between the particles is squeezed out. Hence, the particles rotate relative to one another, such that they deviate from the linear path assumed in the Smoluchowski approach. This approach is called as rectilinear approach (Danijela Vorkapic and Themis Matsoukas 1998).

Another alternative approach to this is the curvilinear approach, in which the hydrodynamic forces cause the particle to rotate slightly around one another. The collision frequency functions are also modified to incorporate for the hydrodynamic forces. In the following Table 5-1, we summarized the different agglomeration kernels.

5.4.6 Comparison of Agglomeration Kernels:

–  –  –

5.4.7 Disintegration rate kernel Disintegration is usually first order with respect to particle concentration, This means, the larger the particle fraction concentration within the process chamber of a reactor the larger is the proability of stressing and as its results of disintegration or breakage. But it is dependent on the local hydrodynamic field acting on the particles. However, it is the balancing of opposing phenomena of agglomeration called redispersion that decides the agglomerate size.

The computer simulations of (Fair and Gemmell 1964) showed the importance of redispersion or breakage in the agglomeration modeling.

Different researchers have proposed different disintegration functions b(t, x, y) or probability density functions. The particle disintegration and breakage based on a series of different simplified kernels assumes that the initial particle size distribution is monodispersed or narrow distributed. The results show that the different assumptions can have effect on both the initial rates of reaction and the steady state concentrations (Spicer and Pratsinis 1996a).The disintegration rate is assumed to be a function of the particle volume (Pandya and Spielman 1982), S ( x)  S0 ( x) 5.20 where  =1/3 (Boadway 1978), consistent with the expectation that the breakage rate is proportional to the particle size x. The Selection rate for disintegration S has been used for flow shear rate as the essential stressing parameter. The break up rate coefficient can be expressed as (Kim and Kramer 2006), Si  A ( ) f (ri ) 5.21 The term Si is the selection rate for disintegration, A is the disintegration rate constant, ri is the particle radius,  and f are the fit parameter from experimental data and  is the shear rate. This shows that the selection rate for disintegration is a function of flow strain rate resulting from energy input and the geometric properties of the agglomerate (i.e. size, area, or volume).

Over the last many years several attempts have been made to model disintegration kernel. In the following sections different forms of disintegration kernels are discussed. Austin Kernel

In a disintegration process, particles are stressed and may break into two or many fragments.

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

The primary cumulative disintegration distribution function has the form first proposed by Austin as in Eq.5.22

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where S0 and  are positive constants has been used for simulation. Diemer Kernel There are several forms like binary disintegration (two fragments), ternary disintegration (three fragments) and normal disintegration where the fragments are distributed in lower size ranges. (Spicer and Pratsinis 1996) showed that binary disintegration function is easy to implement and can be comfortably applied to predict the average particle sizes, without the additional requirement of fitting coefficients.

Here we employed Diemer's generalized form of Hill and Ng's power-law breakage distribution (Diemer and Olson 2002) as in Eq.5.25 p x c ( y - x)c  ( c 1)( p -2)  c  (c  1)( p -1) ! 5.25 b ( x, y )  y pc  ( p -1) c ! c  (c  1)( p - 2) !

Here, the exponent p describes the number of fragments per disintegration event and c  0 determines the shape of the daughter particle distribution. In the following Table 5-3, we summarized the different disintegration kernels.

5.4.8 Comparison of Disintegration Kernels

–  –  –

5.5 Methods to Solve the Population Balance Equations Analytical solutions for agglomeration disintegration population balance equations are available only for a limited number of simplified problems and therefore numerical solutions are frequently needed to solve such equations. Sectional methods such as Cell Average Technique (CAT) and Fixed Pivot Technique (FVT) are used to solve population balances.

Several numerical techniques including the method of successive approximations (D.Ramkrishna 2000), the method of moments (Barrett and Jheeta 1996; Mahoney and Ramkrishna 2002), the finite element methods (Nicmanis and Hounslow 1996; Mahoney and Ramkrishna 2002), the finite volume schemes (Motz, Mitrovic et al. 2002; Verkoeijen, A.

Pouw et al. 2002; Filbet and Laurencot 2004) and Monte Carlo simulation methods (F. Einar Kruis, Arkadi Maisels et al. 2000; Lin, Lee et al. 2002; Maisels, Einar Kruis et al. 2004) can be found in the literature for solving PBEs.

Population balance equations related to agglomeration and disintegration can be expressed by continuous and discrete approaches. Some analytical solutions are available under certain conditions on kernels (Peterson 1986; Ziff 1991). Therefore, numerical solutions are required for these agglomeration disintegration models. The simultaneous agglomeration disintegration in the form of a continuous population balance equations can be shown by Eq. 5.26 cn (t, x) 1 x xmax    (t, x  y, y ) cn (t, x  y ) cn (t, y )dy  cn (t, x)   (t, x, y ) cn (t, y )dy t 2 xmin xmin xmax  b(t, x, y ) S (t, y ) cn (t, y )dy  S (t, x) cn (t, x) 5.26 x In all population balance equations mentioned above, the size variable may vary from 0 to ∞.

In order to apply a numerical scheme for the solution of the equation the first step is to fix a finite computational domain. Therefore, we consider truncated equations by replacing ∞ by a sufficiently large size xmax, with xmax   and we also define xmin  0. Furthermore, for the sake of simplicity we assume that the kernels are compact enough so that the total mass of the system remains conserved.

5.5.1 Numerical Methods

The different sectional methods are differ mainly in terms of freedom of discretization, grid choice and conserved properties (at least two) during the discretization. The division in a number of certain size fractions is well known in process engineering because they are simple to implement and produce exact numerical results of some selected properties.

In this paragraph, we discuss different numerical methods and learn more about the cell average technique.

Batterham approach Batterham utilized the concept of size domain in which masses were divided in a geometric series of 2 (Batterham, S.Hall et al. 1981). He considered that the particle size distribution is i 1 formed only by particles made of monomers of masses mi  2  mi 1 (if only particles made by 1, 2, 4, 8…monomers exist). He deduced equations that allowed the particle interactions at an appropriate rate and splitting of the particles so formed into permissible sizes in such a fashion that mass is conserved. Although each class i, is constituted only by the element formed by mi primary particles, it includes all the elements in the range [ 3 4 mi, 3 2mi ]. He extended the procedure to the break-up of the particles also.

Hounslow’s technique

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Litster elaborated this method for finer size geometric grids, whereas Hill and Ng developed similar equations for breakup and finer grids. The main disadvantage of these methods is their inflexibility in terms of grid and conservation of distribution properties, which is restricted to number and mass or volumes respectively.

Approach by Litster

Litster‟s approach is an extension of the Hounslow‟s method to consider the fine size ( i 1)/ q discretization, characterized by the elements: mi  2  mi 1, where q is a positive integer (J. D. Lister, D. J. Smit et al. 1995). The ability to calculate exactly the total particle number and the total particle mass is the same as in Hounslow‟s method. Each section i is formed by

–  –  –

Approach by Marchal Marchal considered the process of aggregation as a chemical reaction whose stoichiometric coefficients can be adjusted for mass conservation (Marchal, David et al. 1988). The particle size distribution is divided into h arbitrary fractions, whose boundary elements are formed by m i' monomers. Any section i includes all the particles made by a number of monomers in the range [ mi' 1, mi' ], i =1, 2…h. The element of mass m i' can be the representative element of section i, mi. Better accuracy can be obtained with the arithmetic mean: mi  (mi' 1  mi' ) / 2 Marchal claimed that the method can be applied to the breakup as well, provided that only two fragments are formed, but there is no explicit formulation available. When dealing with probability density function, this lacks adequacy.

Approach by Vanni

Vanni proposed a system of sectional representation for pure fragmentation systems, in which sections of arbitrary size is used (Vanni 2000). The section boundaries contain the elements mi', i =0, 1, 2…h, and the representative element of each section is mi. The only restriction on the choice of mi is, of course, that mi' 1  mi  mi'.

The fixed pivot technique (FPT) The main disadvantage of the above described discretization techniques is that they can be used for limited number of geometric grids. (Kumar and Ramkrishna 1996) proposed a new method called fixed pivot technique. PBE can be solved for any arbitrarily chosen grids by using the fixed pivot technique. In this method, they find the position of new born particles

–  –  –

Figure 5-2 Rearrangement of newly formed particles that do not coincide with an existing pivot: Fixed pivot Eq.5.28 consist of four terms. The first is the aggregation birth term and contains a factor responsible for the reallocation of the formed particles to the adjoining pivots if they do not coincide with a pivot. The second term describes the loss of particles due to aggregation (aggregation death) and does not require any reallocation since particles only disappear and are not formed. The third term (breakage birth) does require a factor for reallocation (  i,k ) based on the breakage distribution function (Eq.5.30). The fourth term describes the loss of particles due to breakup (breakage death) and since no particles are formed during this process this term also does not require any reallocation.

Note that, when a geometrical grid with factor 2 (volume-based) is used, Eqs.5.28 and

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