Mass deposition rate supersaturation

As with nucleation, classical theories of crystal growth 3 20 2135 40-421 have not led to working relationships, and rates of crystallisation are usually expressed in terms of the supersaturation by empirical relationships. In essence, overall mass deposition rates, which can be measured in laboratory fluidised beds or agitated vessels, are needed for crystalliser design, and growth rates of individual crystal faces under different conditions are required for the specification of operating conditions. 

Because the rate of growth depends, in a complex way, on temperature, supersaturation, size, habit, system turbulence and so on, there is no simple was of expressing the rate of crystal growth, although, under carefully defined conditions, growth may be expressed as an overall mass deposition rate, RG (kg/m2 s), an overall linear growth rate, Gd(= Ad./At) (m/s) or as a mean linear velocity, // (= Ar/At) (m/s). Here d is some characteristic size of the crystal such as the equivalent aperture size, and r is the radius corresponding to the... 

There is no simple or generally accepted method of expressing the rate of growth of a crystal, since it has a complex dependence on temperature, supersaturation, size, habit, system turbulence, and so on. However, for carefully defined conditions crystal growth rates may be expressed as a mass deposition rate Rq (kgm s ), a mean linear velocity v(ms ) or an overall linear growth rate G (ms ). The relationships between these quantities are... 

For the purpose of simplification, the assumption has been made in Figure 11.3 that the time for a recirculation process is sufficient to reduce the supersaturation down to negligible levels. This reduction in supersaturation is, however, a function of the mean mass deposition rate and the growing crystal surface area present [2] ... 

The smaller the active crystal surface available, the slower the mass deposition rate d7w/df, and the larger the supersaturation remaining after each recirculation cycle. The point 0, in Figures 11.3 and 11.4 moves up, if crystal growth does not desupersaturate to Ac —> 0. As this residual supersaturation is added to the newly created supersaturation, it is certainly possible that the metastable zone width will be... 

The mass deposition rate dmidt (Eq. (9.2)) can be described as exponential function with supersaturation AC as exponent basis and the crystal surface A as linear factor. The proportionaUty factor /Cg considers the influence of the temperature. 

It was shown above that the total erystal number, surfaee area and the mass mean size are affeeted by the mean residenee time and the rates of nueleation and erystal growth respeetively. Sinee both these kinetie proeesses depend upon the working level of supersaturation whieh will itself depend on the amount of surfaee area available and erystal mass deposited, the question arises what will be the effeet of a ehange in residenee time on erystallizer performanee Consider the idealized MSMPR erystallizer depieted in Figure 7.8. 

Let us consider the situation that reaction (6.1) is relatively slow, or that the solublility of the reaction product P is moderate. The reaction is carried out in a continuous stirred reactor, filled with a slurry of F-particles. The relative supersaturation is so low, that there is no primary nucleation and all product is deposited on the existing F-surface. To maintain a steady state, small P-particles (nuclei) have to be added continuously, or they have to be formed by secondary nucleation, i.e. by rupture of formed particles. The number of small particles that are added or generated, per unit time, must be equal to the number of product particles that are removed from the reactor with the product flow. Let us further assume that the conversion in the reactor is practically complete. The reaction rate, which is given by the feed rate of the reactants, must equal the mass transfer rate to the particles ... 

The curve shown in Fig. 3 cannot proceed indefinitely in either direction. In the cathodic direction, the deposition of copper ions proceeds from solution until the rate at which the ions are supplied to the electrode becomes limited by mass-transfer processes. In the anodic direction, copper atoms are oxidized to form soluble copper ions. While the supply of copper atoms from the surface is essentially unlimited, the solubility of product salts is finite. Local mass-transport conditions control the supply rate so a current is reached at which the solution supersaturates, and an insulating salt-film barrier is created. At that point the current drops to a low level further increase in the potential does not significantly increase the current density. A plot of the current density as a function of the potential is shown in Fig. 5 for the zinc electrode in alkaline electrolyte. The sharp drop in potential is clearly observed at -0.9 V versus the standard hydrogen electrode (SHE). At more positive potentials the current density remains at a low level, and the electrode is said to be passivated. 

Chin et al. [16] developed the chemical composition-process relationships in CVD-SiC from H2-CH3SiCl3. The phase relationships given in Fig. 10 indicate that the two parameters of supersaturation and temperature are joined by the mass transport requirements in terms of the rate of deposition of silicon carbide. In this case, the rate is also dependent on the hydrogen concentration as it is required in the reaction to form SiC and is the carrier gas for the reactants. At low hydrogen levels, pyrolytic carbon is formed. 

The diffusion theory states that matter is deposited in a continuous way on the surface of a crystal at a rate proportional to the difference in concentration between the bulk and the surface of the crystal. The mathematical analysis is then the same as for other diffusion and mass transfer processes and makes use of the film concept. Sometimes, the film theory is considered to be an oversimplification for crystallization and is replaced by a random surface removal theory (20-23). For both theories the rate of crystal growth (dm/dt) is given by equation XVII, where m, is the mass of solid deposited in time t k, the mass transfer coefficient by diffusion. A, the surface area of the crystal, c, the concentration in the supersaturated solution and Cj, the concentration at the crystal-solution interface (3). For the stagnant film and random surface removal model, equations XVIII and XIX can be used, respectively (3,4) D is the diffusion coefficient, x, the film thickness and f, the fractionai rate of surface renewal. 

Deposition of amorphous silica (silica scale) onto working surfaces in geothermal power plants (Gunnarson and Arnorsson, 2003) and water desalinization plants (Weng, 1995) reduces rates of heat and mass transfer thereby reducing the efficiency of these enterprises. Silica seating occurs when solutions become supersaturated with respect to amorphotis silica because of cooling, evaporation of water, or a chemical reaction. The reaction between two monomers joins two Si atoms and eliminates a water molecule. 

The scaling rate, i.e. the mass of scale deposited per unit time was also influenced by supersaturation ratio [57]. The higher the supersaturation ratio the more scale forming components are readily available for the scale to form. Hence it was expected that at higher supersaturation ratios the scaling rates were faster. In this single pipe flow experiment, this relationship has been confirmed [1] as a linear equation of first order (R = 0.9932) between mass of scale deposited per unit time (kgs ) and supersaturation level. Figure 7 illustiates this relationship. 

Figure 7. Relationship between scaling rate and supersaturation (flow rate 0.5 ml per second, temp. 20 C, lun time 3 hours, coupon length 6 cm, coupon diameter 1.3 cm, 4>d mass of solid deposited in unit time or scaling rate, kg per second, (Cb - C ) supersaturation, mol L )...

Perhaps the two most important parameters that influence the final grain structure of a vapor-deposited film for a given material system are substrate temperature Tg and growth flux R. The growth flux is a quantity that represents the mass or volume flow rate onto the growth surface, and it is closely related to the degree of supersaturation of the vapor. The flux of film material onto the growth surface can be estimated by appeal to... 

Rates of precipitation. The rate of precipitation of iron from bismuth in a pure iron steel crucible is very rapid. Iron precipitated from bismuth, saturated at 615 C, as rapidly as the temperature could be lowered to 425°C. The addition of Zr plus Mg to liquid metal did not change the rapid precipitation of most of the iron from the bismuth under these same conditions, but produced a marked delay in the precipitation of the last amount of iron in excess of equilibrium solubility. An apparently stable supersaturation ratio of 2.0 was observed for more than 7 hr at 425°C in a pure iron crucible containing Bi - - 1000 ppm Mg - - 500 ppm Zr, and 1.7 for more than 48 hr at 450°C. In a 5% Cr steel crucible, a supersaturation ratio of iron in Bi -f- Mg-f Zr of 2.9 was observed after 24 hr at 425°C. This phenomenon may be due to the ability of the formed surface deposits to poison the effectiveness of the iron surface as a nucleation promotor or catalyst, the different supersaturations observed being due to the relative abilities of a Zr-Fe intermetallic compound or of ZrN to promote nucleation of iron. This observed. supersaturation suggests that mass transfer should be nearly eliminated in a circulating system in which the solubility ratio due to the temperature gradient docs not exceed the measured "stable supersaturation at the cold-leg temperature. 

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