Supersaturation balance

Tavare and Garside ( ) developed a method to employ the time evolution of the CSD in a seeded isothermal batch crystallizer to estimate both growth and nucleation kinetics. In this method, a distinction is made between the seed (S) crystals and those which have nucleated (N crystals). The moment transformation of the population balance model is used to represent the N crystals. A supersaturation balance is written in terms of both the N and S crystals. Experimental size distribution data is used along with a parameter estimation technique to obtain the kinetic constants. The parameter estimation involves a Laplace transform of the experimentally determined size distribution data followed a linear least square analysis. Depending on the form of the nucleation equation employed four, six or eight parameters will be estimated. A nonlinear method of parameter estimation employing desupersaturation curve data has been developed by Witkowki et al (S5). 

An early attempt to establish an optimum cooling curve for batch cooling crystallization (Mullin and Nyvlt, 1971) considered a controlled operation at constant supersaturation, and used a calculation method based on a supersaturation balance in which nuclei were generated in a sequence of discrete time steps. 

A supersaturation balance on a batch cooling crystallizer gives... 

Since the population balance relationships have already been developed in some detail above (section 9.1.1) most of the following examples are based on the population balance approach and/or nomenclature, for reasons of continuity. It is important to understand, however, that other design methods are available, some based on mass or supersaturation balances rather than on population balances. For details of these other methods reference should be made to original literature sources, e.g., as summarized by Toyokura et al. (1984) and Nyvlt (1978, 1992). 

In practice, there will be some residual supersaturation (i.e. cy > cji) so the potential yield will be somewhat less, the precise amount can only be calculated via the coupled population and mass balances, as will be seen later (pp. 194, 195). 

In general, both nucieation and crystal growth depend on supersaturation and to lesser extent temperature and magma characteristics. Such data must therefore be collected to gain maximum benefit from the population balance approach (Jones and MuIIin, 1974 Jones, 1974). Further simplifications to the describing equations are also possible, however (as follows). 

It has been shown that an increase in crystallizer residence time, or decrease in feed concentration, reduces the working level of supersaturation. This decrease in supersaturation results in a decrease in both nucleation and crystal growth. This in turn leads to a decrease in crystal surface area. By mass balance, this then causes an increase in the working solute concentration and hence an increase in the working level of supersaturation and so on. There is thus a complex feedback loop within a continuous crystallizer, illustrated in Figure 7.11. 

Using the SFM, the influence of micromixing and mesomixing on the precipitation process and properties of the precipitate can be investigated. Mass and population balances can be applied to the individual compartments and to the overall reactor accounting for different levels of supersaturation in different zones of the reactor. 

The dissolution rate, according to the theory, does not depend on the mineral s saturation state. The precipitation rate, on the other hand, varies strongly with saturation, exceeding the dissolution rate only when the mineral is supersaturated. At the point of equilibrium, the dissolution rate matches the rate of precipitation so that the net rate of reaction is zero. There is, therefore, a strong conceptual link between the kinetic and thermodynamic interpretations equilibrium is the state in which the forward and reverse rates of a reaction balance. 

The scaling tendency of the lime or limestone processes for flue gas desulfurization is highly dependent upon the supersaturation ratios of calcium sulfate and calcium sulfite, particularly calcium sulfate. The supersaturation ratios cannot be measured directly. They are determined by measuring experimentally the molalities of dissolved sulfur dioxide, sulfate, carbon dioxide, chloride, sodium and potassium, calcium, magnesium, and pH. Then by calculation, the appropriate activities are determined, and the supersaturation ratio is determined. Using the method outlined in Section IV, the concentrations of all ions and ion-pairs can be readily determined. The search variables are the molalities of bisulfite, bicarbonate, calcium, magnesium, and sulfate ions. The objective function is defined from the mass balance expressions for dissolved sulfur dioxide, sulfate, carbon dioxide, calcium, and magnesium. This equation is... 

The measurement of the width of the metastable zone is discussed in Section 15.2.4, and typical data are shown in Table 15.2. Provided the actual solution concentration and the corresponding equilibrium saturation concentration at a given temperature are known, the supersaturation may be calculated from equations 15.1-15.3. Data on the solubility for two- and three-component systems have been presented by Seidell and Linkiv22 , Stephen et alS23, > and Broul et a/. 24. Supersaturation concentrations may be determined by measuring a concentration-dependent property of the system such as density or refractive index, preferably in situ on the plant. On industrial plant, both temperature and feedstock concentration can fluctuate, making the assessment of supersaturation difficult. Under these conditions, the use of a mass balance based on feedstock and exit-liquor concentrations and crystal production rates, averaged over a period of time, is usually an adequate approach. 

The balance between relative rates of aerobic respiration and water movement were considered in Section 4.3.4. We saw that a subsurfece concentration minimum, the oxygen minimum zone (OMZ), is a common characteristic of vertical profiles of dissolved oxygen and is produced by in situ respiration. Waters with O2 concentrations less than 2.0 ppm are termed hypoxic The term anoxic is applied to conditions when O2 is absent. (Some oceanographers use the term suboxic to refer to conditions where O2 concentrations fall below 0.2 ppm but are still detectable.) As illustrated by Figure 4.21b, this water column is hypoxic in the OMZ. The dissolved oxygen concentrations are presented as % saturations in Figure 4.21c. With the exception of the mixed layer, the water column is undersaturated with respect to dissolved oxygen with the most intense undersaturations present in mid-depths. Surface supersaturations are the result of O2 input from photosynthesis and bubble injection. 

The higher the relative supersaturation, the more likely nucleation becomes, and the faster crystal growth proceeds. Molecules or ions will remain dissolved provided that the conditions are energetically favorable. However, all molecules above a certain threshold solution activity will remain in, or become part of, the solid phase. Molecules are in equilibrium between the solid state and the dissolved state. The extent to which the equilibrium balance favors the dissolved phase indicates the degree of solubility. 

Figure 6 shows the size distributions for the samples taken from one of the runs, presented as the cumulative number oversize per ml of slurry. From the lateral shift of the size distributions, the growth rate can be determined. Figure 7 shows values of growth rate, G, supersaturation, s, and crystal content determined during the run. As a material balance check, the crystal contents were evaluated from direct measurements, from solution analyses and from the moments of the size distribution. The agreement was satisfactory. No evidence of size dependent growth or size dispersion was observed. 

Milk serum is supersaturated with calcium phosphate, the excess being present in the colloidal phase, as described above. The balance between the colloidal and soluble phases may be upset by various factors, including changes in temperature, dilution or concentration, addition of acid, alkali or salts. The solubility product for secondary calcium phosphate, [Ca2+][HPOr] is about 1.5 x 1(T5 or pKs = 4.85. 

In precipitation, particle formation is extremely fast due to high supersaturations which in turn lead to fast nucleation. At least in the beginning, size distributions are narrow with particle sizes around one 1 nm. Nanomilling in stirred media mills is characterized by relatively slow particle formation kinetics, particle sizes ranging from several microns down to 10 nm and high sohds volume concentrations of up to 40%. Large particles may scavenge the fine fractions. The evolution of the particle size distribution can be described for both cases by population balance equations (Eq. (7)),... 

The concentration of metals in atmospheric aerosols and rainwater (Table 7.1) is therefore a function of their sources. This includes both the occurrence of the metals in combustion processes and their volatility, as well as their occurrence in crustal dust and seawater. As a result of this, the size distribution of different metals is very different and depends on the balance of these sources. For a particular metal this distinction is similar in most global locations (Table 7.2), although some variability does occur as wind speed and distance from source exert an influence on the particle size distribution spectrum (Slinn, 1983). Once in the atmosphere particles can change size and composition to some extent by condensation of water vapour, by coagulation with other particles, by chemical reaction, or by activation (when supersaturated) to become cloud or fog droplets (Andreae et al., 1986 Arimoto et al., 1997 Seinfeld and Pandis, 1998). 

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