Population density function

The arbitrary system volume on which n is based must be defined before the population density function has meaning. For example, the volume may be that of the slurry or the clear Hquor in the system. 

Moments of a distribution often provide information that can be used to characterize particulate matter. Theyth moment of the population density function n is defined as... 

It can be demonstrated that the total number of crystals, the total length, the total area, and the total volume of crystals, all in a unit of system volume, can be evaluated from the zero, first, second, and third moments of the population density function. 

The magma density Mj- (mass of crystals per unit volume of slurry or Hquor) may be obtained from the third moment of the population density function and is given by... 

The effects of each selective removal function on CSD can be described in terms of the population density function n. It is convenient to define flow rates in terms of clear Hquor, which requires the population s density function to be defined on a clear-Hquor basis. In the present discussion, only systems exhibiting invariant crystal growth are considered. 

The function of clear-Hquor advance can be illustrated by considering a simple operation, shown in Figure 13, in which Qcv < 0 volumetric flow rates of clear-Hquor fed to the crystallizer, in the clear-Hquor advance, and in the output slurry. In such systems the population density function is given by the expression... 

Clearly, the form of the population density function resulting from a clear-Hquor advance system is identical to that expected from perfectly mixed systems in which T., is identical to T,. Unless the increase in magma density associated with clear-Hquor advance results in significant increases in... 

Figure 15 shows how the population density function changes with the addition of classified-fines removal. It is apparent from the figure that fines removal increases the dominant crystal size, but it also increases the spread of the distribution. 

Fig. 15. Population density function for product from crystallizer with classified-fines removal. Cut size Lp = 150 /tm R = 3.7.

Fig. 18. Population density functions of crystals within a crystallizer, having both classified-fines and classified-product removal and of crystals in the...

Identification of an initial condition is difficult because of the problem of specifying the size distribution at the instant nucleation occurs. The difficulty is mitigated through the use of seeding which would mean that the initial population density function would correspond to that of the seed crystals ... 

Moments of the population density function, which are given by... 

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. 

Moments of the population density function given by Eqs. (74) through (76) can be evaluated in piecewise fashion ... 

Equation (77) is used to estimate the moments of the population density function within the crystallizer, not of the product distribution. (Recall that moments of the distribution within the crystallizer are often required for kinetic equations.) Assuming perfect classification, moments of the product distribution can be obtained from the expression ... 

The rate of cooling, or evaporation, or addition of diluent required to maintain specified conditions in a batch crystallizer often can be determined from a population-balance model. Moments of the population density function are used in the development of equations relating the control variable to time. As defined earlier, the moments are... 

Mathematically, the one-dimensional PSD function can be expressed as m(L), where n is the population density function and L is the characteristic crystal length. For cube-Uke (or spherical) crystals, the characteristic length is approximately the diameter of the crystal. For crystals of other shape there are various definitions for the characteristic length. Most commonly, the characteristic length of the particle with an irregular shape is defined as the equivalent diameter of a sphere which has the same behaviors under the measurement conditions, for example sieving, laser scattering, and sedimentation (Mullin 2(X)1, Chapter 2). 

Basically, the method of moments converts the set of PIDEs into a set of PDEs where each PDE represents a given moment of the population size density function. Defining the j-th moment of the population density function like... 

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 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 ... 

For size-dependent growth [i.e., kc = kc(L)], the mathematical treatment of Eq. (10.11) becomes more complicated. One solution method, which included the use of moments and an orthogonal polynomial to simulate the population density function, was discussed by Wey (1985). Tavare et al. (1980) developed a... 

One of the main challenges in batch crystallization is to control the supersaturation and nucleation during the initial stage of the batch run. During this period, very little crystal suspension is present on which solute can crystallize, so that high supersaturation and excessive nucleation often occur. Another difficulty associated with batch crystallization is the determination of the initial condition for the population density function. In an unseeded batch crystallizer, initial nucleation can occur by several mechanisms and usually occurs as an initial shower followed by a reduced nucleation rate. Thus, an initial size distribution exists and one... 

The above equation expresses the ratio of the mass of crystals in the y-th sieve class to the mass of crystals in the mass average size class, where, n(L) is the population density function approximated with a straight line at the steady state (see Fi e 3). 

A number of empirical size-dependent growth expressions have been developed. Of these, the ASL model given in Eq. (11.2-28) is the most commonly used. Substituting this equation into the differential population balance given by Eq. (11.2-31), the steady-state population density function can be derived as... 

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