Crystallization crystal size distribution function

The first three columns of Table 10.5 show sieve data for a 100-cc slurry sample containing 21.0 g of solids taken from a 20,000-gal (75-m3) mixed suspension-mixed product removal crystallizer (MSMPR) producing cubic ammonium sulfate crystals. Solids density is 1.77 g/cm3, and the density of the clear liquor leaving the crystallizer is 1.18 g/cm3. The hot feed flows to the crystallizer at 374,000 lb/h (47 kg/s). Calculate the residence time r, the crystal size distribution function n, the growth rate G, the nucleation density n°, the nucleation birth rate B°, and the area-weighted average crystal size L3 2 for the product crystals. 

Calculate the crystal size distribution function n. The crystal size distribution for the ith sieve tray is n, = 1012M3AVTj/(L3ALj), where AIT, is the weight fraction retained on the ith screen, /., is the average screen size of material retained on the ith screen (see Example 10.7, step 2), and A/., is the difference in particle sizes on the ith screen (see Example 10.7, step 3). For instance, for the Tyler mesh 100 screen, nio = 1012(0.119)(0.040)/(1613 x 28) = 40.7 crystals per cubic centimeter per micron. Table 10.5 shows the results for each sieve screen. 

W Crystal weight W Weight of crystal product W Weight of crystal seeds W (l) Crystal size distribution function (weight basis) w Work of forming a new phase... 

Therefore the crystal size distribution function F(rp) (defined by (2.4.2d)) is ... 

It has a value of 0 at Pp = 0 and 1 at Pp = co (just like the crystal size distribution function fl(Pp) in Example 6.4.1). A density function m/Pp) of the crystal mass distrihution function vis- -vis the particle size Pp is obtained over a differential size range of Pp to Pp + dPpt... 

Sepa.ra.tlon, Sodium carbonate (soda ash) is recovered from a brine by first contacting the brine with carbon dioxide to form sodium bicarbonate. Sodium bicarbonate has a lower solubiUty than sodium carbonate, and it can be readily crystallized. The primary function of crystallization in this process is separation a high percentage of sodium bicarbonate is soHdified in a form that makes subsequent separation of the crystals from the mother hquor economical. With the available pressure drop across filters that separate Hquid and soHd, the capacity of the process is determined by the rate at which hquor flows through the filter cake. That rate is set by the crystal size distribution produced in the crystallizer. 

Population balances and crystallization kinetics may be used to relate process variables to the crystal size distribution produced by the crystallizer. Such balances are coupled to the more familiar balances on mass and energy. It is assumed that the population distribution is a continuous function and that crystal size, surface area, and volume can be described by a characteristic dimension T. Area and volume shape factors are assumed to be constant, which is to say that the morphology of the crystal does not change with size. 

Although magma density is a function of the kinetic parameters fP and G, it often can be measured iadependentiy. In such cases, it should be used as a constraint ia evaluating nucleation and growth rates from measured crystal size distributions (62), especially if the system of iaterest exhibits the characteristics of anomalous crystal growth. 

As an idealization of the classified-fines removal operation, assume that two streams are withdrawn from the crystallizer, one corresponding to the product stream and the other a fines removal stream. Such an arrangement is shown schematically in Figure 14. The flow rate of the clear solution in the product stream is designated and the flow rate of the clear solution in the fines removal stream is set as (R — 1) - Furthermore, assume that the device used to separate fines from larger crystals functions so that only crystals below an arbitrary size are in the fines removal stream and that all crystals below size have an equal probabiHty of being removed in the fines removal stream. Under these conditions, the crystal size distribution is characterized by two mean residence times, one for the fines and the other for crystals larger than These quantities are related by the equations... 

Aluminium on Silicon. Low Contact Resistance. Improved Corrosion Resistance c/f Evaporated A1. Grain Size and Crystal Size Distribution is Function of Acceleration Voltage. Crystal Orientation is strongly (111) under High Acceleration Voltage... 

The experimental results reported in this paper demonstrate the ability of a flat-bottom hydrocyclone to separate the coarse fraction of ammonium sulfate crystals from a slurry which contains crystals of a wide size range. It appears that the grade efficiency curve, which predicts the probability of a particle reporting to the underflow of the cyclone as a function of size, can be adjusted by a change in the underflow diameter of the hydrocyclone. These two observations lead to the suggestion to use hydrocyclone separation to reduce the crystal size distribution which is produced in crystallisers, whilst using a variable underflow diameter as an additional input for process control. 

Quantification of coarsening is complicated. During coarsening, all crystals are not the same size. There is a crystal size distribution (Figure 4-15). The distribution function may be log-normal ... 

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... 

Recently, crystallization has been widely operated in chemical industries. Although the crystal size distribution in the operation has been expressed by using various PSD functions, the most utilized PSD function is the log-normal PSD function. However, there is no physical background for applying the lognormal PSD function to the crystal size distribution. Additionally, when the crystal size distribution is expressed by various PSD functions, it becomes difficult to discuss the relationship between the parameters in the PSD and the operation condition. This is one of the obstacles in the development of particle technology. 

Crystal size distributions may be characterized usefully (though only partially) by a single crystal size and the spread of the distribution about that size. For example, the dominant crystal size represents the size about which the mass in the distribution is clustered. It is defined as the size, Ld, at which a unimodal mass density function is a maximum, as shown in Fig. 11 in other words, the dominant crystal size LD is found where dm /dL is zero. (The data used to construct Fig. 11 are from Table II.) As the mass density is related to the population density by ... 

Crystallizers are made more flexible by the introduction of selective removal devices that alter the residence time distributions of materials flowing from the crystallizer. Three removal functions—clear-liquor advance, classified-fines removal, and classified-product removal— and their idealized removal devices will be used here to illustrate how design and operating variables can be manipulated to alter crystal size distributions. Idealized representations of the three classification devices are illustrated in Fig. 17. 

Clear-liquor advance from what is called a double draw-off crystallizer is simply the removal of mother liquor without simultaneous removal of crystals. The primary action in classified-fines removal is preferential withdrawal from the crystallizer of crystals of a size below some specified value this may be coupled with the dissolution of the crystals removed as fines and the return of the resulting solution to the crystallizer. Classified-product removal is carried out to remove preferentially those crystals of a size larger than some specified value. In the following discussion, the effects of each of these selective removal functions on crystal size distributions will be described in terms of the population density function n. Only the ideal solid-liquid classification devices will be examined. It is convenient in the analyses to define flow rates in terms of clear liquor. Necessarily, then, the population density function is defined on a clear-liquor basis. 

Related Calculations. This procedure can be used to analyze either wet or dry solids particle size distributions. Particle size distributions from grinding or combustion and particles from crystallizers are described by the same mathematics. See Example 10.6 for an example of a situation in which the size distribution is based on a known size-distribution function rather than on an experimental sample. 

The effect of the crystal size distribution on these results was investigated using the observed distribution function. The prediction of the model for the average particle radius was compared with a prediction... 

When the value of s is known from measurements of crystal size distribution, this expression may be evaluated numerically to give W as a function of Dt/jx. The value of D//x may then be obtained by matching the experimental data to the theoretical curve. 

In the quadrature method of moments (QMOM) developed by McGraw [131], for the description of sulfuric acid-water aerosol dynamics (growth), a certain type of quadrature function approximations are introduced to approximate the evolution of the integrals determining the moments. Marchisio et al [122, 123] extended the QMOM for the application to aggregation-breakage processes. For the solution of crystallization and precipitation kernels the size distribution function is expressed using an expansion in delta functions [122, 123] ... 

Several patterns, similar to that shown in Figure 6.9, were produced for different thermal treatment temperatures. These patterns were modeled according to the method we have just described, and we extracted, in particular, the size distribution functions of the tetragonal zirconia crystals. The resulting curves ate shown in Figure 6.17. Clearly, increasing the thermal treatment temperature significantly widens the crystal size distribution. 

In an effort to probe catalytic sites and their stability, the zeolite catalyst, ZSM-5 was Investigated by impedance and fourier transform infrared spectroscopies as a function of aluminium substitution and cation exchange. Samples were provided by Chemistry Division, DSIR, with (Si + A1)/A1 ratios of ao, 1000, 500, 200, 136 and 40. Crystallite size and morphology varied somewhat with aluminium content but typically the samples had crystal size distributions in the range 0.2 pus to 2 (im. 

Characteristics of Crystal Size Distributions from MSMPR Crystallizers The preceding discussion centered on the development of expressions for (he population density function in terms of nacleation and growth kinetics. It is also possible to express (he properties of a crystal size distribution in terms of a mass density function m. The two density functions can be shown to be related by the expression... 

The constants in Eqs. (ll.2-46)-(l 1,2-51) are functions of b only, and they are given in Table 11.2-6. The table can be used to estimate the desired property of a crystal size distribution from the model para maters and Eqs. (1 1.2-46)-(]. 2-51), or it cna be used in evaluating the model parameters from experimental data... 

The effects of each of the selective removal functions on crystal size distributions can ha described in terms of the population density function n. If it is assumed that perfect classification of fines and product crystals is implemented, (hen the following expression for population density results ... 

Although many commercial crystallizers operate with some form of selective crystal removal, devices that implement these functions add to the complexity of the operation. In addition, Randolph et a .76 have established that classified-product removal can lead to cycling of the crystal size distribution. A review of the simulation and control of ciystal size distributions hes been provided by Randolph.77 Properties of the crystal size distribution have been given in terms of R and Z and the moments of the crystal size distribution.74... 

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