Population densities

The hypersurface fomied from variations in the system s coordinates and momenta at//(p, q) = /Tis the microcanonical system s phase space, which, for a Hamiltonian with 3n coordinates, has a dimension of 6n -1. The assumption that the system s states are populated statistically means that the population density over the whole surface of the phase space is unifomi. Thus, the ratio of molecules at the dividing surface to the total molecules [dA(qi, p )/A]... 

This solution can be obtained explicitly either by matrix diagonalization or by other techniques (see chapter A3.4 and [42, 43]). In many cases the discrete quantum level labels in equation (A3.13.24) can be replaced by a continuous energy variable and the populations by a population density p(E), with replacement of the sum by appropriate integrals [Hj. This approach can be made the starting point of usefiil analytical solutions for certain simple model systems [H, 19, 44, 45 and 46]. 

For an ensemble of tire systems described above, we can represent tire time rate of change of tire population density as... 

In thermoluminescence dating, a sample of the material is heated, and the light emitted by the sample as a result of the de-excitations of the electrons or holes that are freed from the traps at luminescence centers is measured providing a measure of the trap population density. This signal is compared with one obtained from the same sample after a laboratory irradiation of known dose. The annual dose rate for the clay is calculated from determined concentrations of radioisotopes in the material and assumed or measured environmental radiation intensities. 

The influence of human activities in a stream drainage basin can be relatively simple and direct, as in the disposal of soluble organic and inorganic waste, or more subtie and complex, as in the conversion of prairie or forest land to agricultural use. Such effects can be expected to increase as population density and agricultural, industrial, and mining activities increase. 

Population density (n) has dimensions number/(volume)(length) it is a key quantity in the discussion of CSD, a function of the characteristic crystal dimension E, and it is defined so that it is independent of the magnitude of the system. When a total population density is used, the symbol is n and the units are number/length. Population density is defined by letting AiVbe the number of crystals per unit system volume in a size range from E to L + AL, so that... 

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. 

A mass (weight) density function, given the symbol m and having dimensions mass/(volume)(length), can be defined analogously to population density letting AM be the mass of crystals per unit system volume in the size range E to L + AL,... 

Recognizing that the mass of crystals in a sample is the product of the number of crystals and the mass of a single crystal, mass and population densities may be related by the expression... 

The function M used in the above equations is a cumulative mass disttibution function, representing the mass of crystals having a characteristic dimension less than U. The total mass of crystals per unit volume is related to population density by the equation... 

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. 

Analysis of equation 48 shows that a single sample taken either from inside the crystallizer or from the product stream will allow evaluation of nucleation and growth rates at the system conditions. Figure 12 shows a plot of typical population density data obtained from a crystallizer meeting the stated assumptions. The slope of the plot of such data maybe used to obtain the growth rate, and the product of the intercept and growth rate gives the nucleation rate. 

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... 

For systems following invariant growth the crystal population density in each size range decays exponentially with the inverse of the product of growth rate and residence time. For a continuous distribution, the population densities of the classified fines and the product crystals must be the same at size Accordingly, the population density for a crystallizer operating with classified-fines removal is given by... 

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.

Classified removal of course material also can be used, as shown in Figure 16. In a crystallizer equipped with idealized classified-product removal, crystals above some size ate removed at a rate Z times the removal rate expected for a perfecdy mixed crystallizer, and crystals smaller than are not removed at all. Larger crystals can be removed selectively through the use of an elutriation leg, hydrocyclones, or screens. Using the analysis of classified-fines removal systems as a guide, it can be shown that the crystal population density within the crystallizer magma is given by the equations... 

Fig. 17. Effect of classified-product removal on population densities within a crystallizer and in the crystallizer product.

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

Mote quantitative relationships of the CSD obtained from batch operations can be developed through formulation of a population balance. Using a population density defined in terms of the total crystallizer volume rather than on a specific basis (n = nU), the general population balance given by equation 42 can be modified in recognition of there being no feed or product streams ... 

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... 

Example 3 Population, Density, Growth and Nucleation Rate. 18-43... 

In industrial practice, the size-distribution cui ve usually is not actually construc ted. Instead, a mean value of the population density for any sieve fraction of interest (in essence, the population density of the particle of average dimension in that fraction) is determined directly as AN/AL, AN being the number of particles retained on the sieve and AL being the difference between the mesh sizes of the retaining sieve and its immediate predecessor. It is common to employ the units of (mm-L)" for n. 

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