Crystallization mass rate

Mass and Energy Balances. The formulation of mass and energy balances follows procedures outlined ia many basic texts (2). The use of solubihties to calculate crystal production rates from a cooling crystallizer was demonstrated by the discussion of equations 1 and 2. Subsequent to determining the yield, the rate at which heat must be removed from such a crystallizer can be calculated from an energy balance ... 

The effects of a solvent on growth rates have been attributed to two sets of factors (28) one has to do with the effects of solvent on mass transfer of the solute through adjustments in viscosity, density, and diffusivity the second is concerned with the stmcture of the interface between crystal and solvent. The analysis (28) concludes that a solute-solvent system that has a high solubiUty is likely to produce a rough interface and, concomitandy, large crystal growth rates. 

Crystal growth rate may be expressed either as a rate of linear inerease of eharaeteristie dimension (i.e. veloeity) or as a mass deposition rate (i.e. mass flux). Expressed as a veloeity, the overall linear erystal growth rate, G (=dL/dt where L is the eharaeteristie dimension that is inereasing). The rate of ehange of... 

The data plotted in the figure clearly support the predicted positive dependence of crystal size on agitation rate. Precipitation in the crystal film both enhances mass transfer and depletes bulk solute concentration. Thus, in the clear film model plotted by broken lines, bulk crystal sizes are initially slightly smaller than those predicted by the crystal film model but quickly become much larger due to increased yield. Taken together, these data imply that while the initial mean crystal growth rate and mixing rate dependence of size are... 

Besides, without addictive AICI3 as a crystal conversion agent, phase composition of most neogenic Ti02 particles was anatase in our experiment. Conversions active energy finm anatase to rutile was 460 kJ/mol [5], with temperature arose, crystal conversion rate as well as mass fraction of rutile would increase [6,7]. Hence, after a lot of heat accumulated, phase composition of particle-sintered layer was rutile. 

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. 

Methods used for the measurement of crystal growth rates are either a) direct measurement of the linear growth rate of a chosen crystal face or b) indirect estimation of an overall linear growth rate from mass deposition rates measured on individual crystals or on groups of freely suspended crystals 35,41,47,48). 

In Figure 4 the corresponding changes in the crystal purity are shown. For the solutions with the initial supersaturation At=5 C, the purity was always 100% for three hours, while purity decrease started very soon for the solution having higher initial supersaturation such as At=10 C. Since with different initial supersaturations, the crystallization rate are very different, the purity decrease should be compared on the basis of either the supersaturation change or the crystal mass grown in the solution rather than the crystallization time. 

Because interface reaction and mass/heat transfer are sequential steps, crystal growth rate is controlled hy the slowest step of interface reactions and mass/heat transfer. For a crystal growing from its own melt, the growth rate may be controlled either by interface reaction or heat transfer because mass transfer is not necessary. For a crystal growing from a melt or an aqueous solution of different composition, the growth rate may be controlled either by interface reaction or mass transfer because heat transfer is much more rapid than mass transfer. Different controls lead to different consequences, including the following cases ... 

If the boundary motion is controlled by an independent process, then the boundary motion velocity is independent of diffusion. This can happen if the magma is gradually cooling and crystal growth rate is controlled both by temperature change and mass diffusion. This problem does not have a name. In this case, u depends on time or may be constant. If the dependence of u on time is known, the problem can also be solved. The Stefan problem and the constant-w problem are covered below. 

Below, the melt consumption rate u is distinguished from the crystal growth rate (they differ by the density ratio), and the concentration in terms of kg/m (denoted as C) is also distinguished from mass fraction (the same as weight percent, denoted as w). 

Mass balance at the boundary provides an equation relating crystal growth rate to other parameters ... 

For convective crystal dissolution, the dissolution rate is u = (p/p )bD/8. For diffusive crystal dissolution, the dissolution rate is u = diffusive boundary layer thickness as 5 = (Df), the diffusive crystal dissolution rate can be written as u = aD/5, where a is positively related to b through Equation 4-100. Therefore, mass-transfer-controlled crystal dissolution rates (and crystal growth rates, discussed below) are controlled by three parameters the diffusion coefficient D, the boundary layer thickness 5, and the compositional parameter b. The variation and magnitude of these parameters are summarized below. 

If zeolitic diffusion is sufficiently rapid so that the sorbate concentration through any particular crystal is essentially constant and in equilibrium with the macropore fluid just outside the crystal, the rate of mass transfer will be controlled by transport through the macropores of the pellet. Transport through the macropores may be assumed to occur by a diffusional process characterized by a constant pore diffusion coefficient Z)p. The relevant form of the diffusion equation, neglecting accumulation in the fluid phase within the macropores which is generally small in comparison with accumulation within the zeolite crystals, is... 

The nucleation rate must generate one nucleus for every crystal present in the product. In terms of M , the total mass rate of production of crystals,... 

No study has been made to discover which of the several resistances is important, but a simple rate equation can be written which states that the rate of the over-all process is some function of the extent of departure from equilibrium. The function is likely to be approximately linear in the departure, unless the intrinsic crystal growth rate or the nucleation rate is controlling, because the mass and heat transfer rates are usually linear over small ranges of temperature or pressure. The departure from equilibrium is the driving force and can be measured by either a temperature or a pressure difference. The temperature difference between that of the bulk slurry and the equilibrium vapor temperature is measured experimentally to 0.2° F. and lies in the range of 0.5° to 2° F. under normal operating conditions. 

One of the most controversial topics in the recent literature, with regard to partition coefficients in carbonates, has been the effect of precipitation rates on values of the partition coefficients. The fact that partition coefficients can be substantially influenced by crystal growth rates has been well established for years in the chemical literature, and interesting models have been produced to explain experimental observations (e.g., for a simple summary see Ohara and Reid, 1973). The two basic modes of control postulated involve mass transport properties and surface reaction kinetics. Without getting into detailed theory, it is perhaps sufficient to point out that kinetic influences can cause both increases and decreases in partition coefficients. At high rates of precipitation, there is even a chance for the physical process of occlusion of adsorbates to occur. In summary, there is no reason to expect that partition coefficients in calcite should not be precipitation rate dependent. Two major questions are (1) how sensitive to reaction rate are the partition coefficients of interest and (2) will this variation of partition coefficients with rate be of significance to important natural processes Unless the first question is acceptably answered, it will obviously be difficult to deal with the second question. 

The rate of change of a crystal mass dmciys/dt can be related to the rate of change in the crystal characteristic dimension (dL/dt = G) by the equation ... 

The rotatable reactor can also be used for reactions in fluids having suitably low (< 10"3 Torr) vapor pressure. In this mode, metal atoms are evaporated upwards into the cold liquid, which is spun as a thin band on the inner surface of the flask. Reactions with dissolved polymers can then be studied. Specially designed electron gun sources can be operated, without static discharge, under these potentially high organic vapor pressure conditions (6). Run-to-nin reproducibility is obtained by monitoring the metal atom deposition rate with a quartz crystal mass balance (thickness monitor). 

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