Crystal size distribution CSD

The mass of one crystal of a chosen characteristic size L (section 2.14.2) is given by ap L , where a is a volume shape factor (section 2.14.3) and pc is the crystal density. The number of crystals, dA, of size L in a mass dM is thus dA = dMjapcL . Assuming no nucleation, the number of seeds dN of size is equal to the number of product crystals dAp of size Lp, i.e. 

One use of the law is demonstrated by the following example. If a quantity of seed crystals of known CSD is added to a crystallizer, it is possible to estimate the CSD of the final product by the following procedure. 

Find the value of AL compatible with the product yield Mp calculated from solubility data and the mass of added seeds M,. This involves a trial and error evaluation of equation 9.1, which may be integrated graphically (plot Ms versus (1 + AL/L,) the area enclosed under M, = 0 and M, = mass of seeds added is equal to Mp). 

Plot the integral curve of equation 9.1 (Mp versus M,) using the correct value of AL. 

Plot the CSD curve, e.g. from sieve test data, of the product crystals (Mp versus Lp) remembering that Lp = L, + AL. For comparison purposes, this plot may be made on the same diagram as the CSD curve of the seed crystals (Ms versus L,). 

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General solution of the population balance is complex and normally requires numerical methods. Using the moment transformation of the population balance, however, it is possible to reduce the dimensionality of the population balance to that of the transport equations. It should also be noted, however, that although the mathematical effort to solve the population balance may therefore decrease considerably by use of a moment transformation, it always leads to a loss of information about the distribution of the variables with the particle size or any other internal co-ordinate. Full crystal size distribution (CSD) information can be recovered by numerical inversion of the leading moments (Pope, 1979 Randolph and Larson, 1988), but often just mean values suffice. 

Growth and nucleation interact in a crystalliser in which both contribute to the final crystal size distribution (CSD) of the product. The importance of the population balance(37) is widely acknowledged. This is most easily appreciated by reference to the simple, idealised case of a mixed-suspension, mixed-product removal (MSMPR) crystalliser operated continuously in the steady state, where no crystals are present in the feed stream, all crystals are of the same shape, no crystals break down by attrition, and crystal growth rate is independent of crystal size. The crystal size distribution for steady state operation in terms of crystal size d and population density // (number of crystals per unit size per unit volume of the system), derived directly from the population balance over the system(37) is ... 

The number of inputs which are available for controlling crystallisation processes is limited. Possible Inputs for a continuous evaporative crystallisation process are, crystalliser temperature, residence time and rate of evaporation. These Inputs affect the crystal size distribution (CSD) through overall changes in the nucleatlon rate, the number of new crystals per unit time, and the growth rate, the increase in linear size per unit time, and therefore do not discriminate directly with respect to size. Moreover, it has been observed that, for a 970 litre continuous crystalliser, the effect of the residence time and the production rate is limited. Size classification, on the other hand, does allow direct manipulation of the CSD. 

There is a clear need for other size classifiers which combine a high separation efficiency with flexibility and compactness. Hydrocyclones have a small volume, are simple in operation and are standard size classification equipment, for example in closed circuit grinding applications. The recent development of the flat-bottom hydrocyclone, which permits classification in the coarse size range, creates an additional motive to study the use of hydrocyclones for Crystal Size Distribution (CSD) control. Furthermore, throttling of a flat botom hydrocyclone does not necessarily provoke blockage but allows continuous control of its cut size when a controlled throttling valve is used. There is a clear incentive for its use in this application since it may provide an additional process input. 

The observed transients of the crystal size distribution (CSD) of industrial crystallizers are either caused by process disturbances or by instabilities in the crystallization process itself (1 ). Due to the introduction of an on-line CSD measurement technique (2), the control of CSD s in crystallization processes comes into sight. Another requirement to reach this goal is a dynamic model for the CSD in Industrial crystallizers. The dynamic model for a continuous crystallization process consists of a nonlinear partial difference equation coupled to one or two ordinary differential equations (2..iU and is completed by a set of algebraic relations for the growth and nucleatlon kinetics. The kinetic relations are empirical and contain a number of parameters which have to be estimated from the experimental data. Simulation of the experimental data in combination with a nonlinear parameter estimation is a powerful 1 technique to determine the kinetic parameters from the experimental... 

Marsh (1988), Cashman and Marsh (1988), and Cashman and Ferry (1988) investigated the application of crystal size distribution (CSD) theory (Randolph and Larson, 1971) to extract crystal growth rate and nucleation density. The following summary is based on the work of Marsh (1988). In the CSD method, the crystal population density, n(L), is defined as the number of crystals of a given size L per unit volume of rock. The cumulative distribution function N(L) is defined as... 

Cashman K.V. and Ferry J.M. (1988) Crystal size distribution (CSD) in rocks and the kinetics and dynamics of crystallization. III metamorphic crystallization. Contrib. Mineral. Petrol. 99, 401-415. 

Closure in crystal size distributions (CSD), verification of CSD calculations,... 

Figure 8.8. Crystal size distribution (CSD) plots of (a) plagioclase in igneous rock and (b) garnet porphyroblast in contact metamorphic rock [4].

Crystal size distribution (CSD) is measured with a series of standard screens. The openings of the various mesh sizes according to the U.S. Standard are listed in Example 6.6, and according to the British Standard in Figure 16.4. Table 12.1 is a complete listing. The size of a crystal is taken to be the average of the screen openings of successive sizes that just pass and just retain the crystal. 

Figure 16.4. Several ways of recording the same data of crystal size distribution (CSD) (MulUn, 1972). (a) The data, (b) Cumulative wt % retained or passed, against sieve aperture, (c) Log-log plot according to the RRS equation P = exp[(—d/dm)"J off this plot, = 850, dm = 1000, n = 1.8. (d) Differential polygon, (e) Differential histogram.

Most crystallization processes produce particles whose sizes cover a range of varying breadth. If the particles consist of single crystals, the resulting distribution is a crystal size distribution (CSD) on the other hand, if the particles consist of agglomerates or other combination of multiple crystals, the distribution is a particle size distribution. In either case, the distribution is expressed in terms of either population (number) or mass. The popu-... 

Crystal size distribution (CSD) is measured with a series of standard screens. The openings of the various mesh sizes according to the... 

This misconception is particularly common in crystallization. The hypothesis of a perfectly mixed system is, for crystallization and precipitation processes, labeled as mixed-suspension, mixed-product removal (MSMPR). With diis model the crystalUzer is modeled with a spatially homogeneous NDF, generally called the crystal-size distribution (CSD). However, the fact that the CSD is constant through the vessel does not mean that the rates of crystal nucleation, molecular growth, aggregation, and breakage are constant. 

As explained throughout the book, disperse multiphase systems are characterized by multiple phases, with one phase continuous and the others dispersed (i.e. in the form of distinct particles, droplets, or bubbles). The term polydisperse is used in this context to specify that the relevant properties characterizing the elements of the disperse phases, such as mass, momentum, or energy, change from element to element, generating what are commonly called distributions. Typical distributions, which are often used as characteristic signatures of multiphase systems, are, for example, a crystal-size distribution (CSD), a particle-size distribution (PSD), and a particle-velocity distribution. 

Seed the process because the nucleation sets the crystal habit and influences crystal size distribution (CSD). Importance of contact nucleation where crystals strike pump and mixer impellers. Crystal growth kinetics increase with temperature increase. Crystal growth rate = 0.1 to... 

Many important properties of the crystal size distribution (CSD) can be calculated utilizing the population density. The total number of particles per unit volume, Nj, is... 

Continuous MSMPR Precipitator. The population balance, which was put forward by Randolph and Larson (1962) and Hulbert and Katz (1964), provides the basis for modeling the crystal size distribution (CSD) in precipitation processes. For a continuous mixed-suspension, mixed-product-removal (CMSMPR) precipitator with no suspended solids in the feed streams, the population balance equation (PBE) can be written as (Randolph and Larson 1988)... 

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