Crystallization mass balances

Disruption of an agglomerated particle L, S), produces a single primary crystal (Ld, 1) and the residual particle (Ldd, —1). Conservation of crystal mass balance requires the following equations to be satisfied ... 

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). 

Population balances and crystallization kinetics may be used to relate process variables to the crystal size distribution produced by the crystallizer. Such balances are coupled to the more familiar balances on mass and energy. It is assumed that the population distribution is a continuous function and that crystal size, surface area, and volume can be described by a characteristic dimension T. Area and volume shape factors are assumed to be constant, which is to say that the morphology of the crystal does not change with size. 

Mass Balance Constraints. Erom the schematic diagram of a continuous crystallizer shown ia Eigure 11, the foUowiag mass balance on solute can be constmcted ... 

The first, and simplest, step in predicting crystallizer performance is the calculation of crystal yield. This can easily be estimated from knowledge of solution concentration and equilibrium conditions permitting calculation of the overall mass balance... 

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. 

Figure 9. A schematic and ideal model showing how the residence time of the magma in a steady-state reservoir of constant mass M, replenished with an influx O of magma and thoroughly mixed, can be calculated from disequilibrium data, in the simplifying case where crystal fractionation is neglected (Pyle 1992). The mass balance equation describing the evolution through time of the concentration [N2] (number of atoms of the daughter nuclide per unit mass of magma) in the reservoir is ...

Example 10.7 A solution of sucrose in water is to be separated b. A mass balance on the solvent gives by crystallization in a continuous operation. The solubility of... 

The final product contains 30 per cent crystals and hence 70 per cent solution containing 40 per cent dissolved solids. The total percentage of dissolved and undissolved solids in the final product is 0.30 + (0.40 x 0.70) = 0.58 or 58 per cent, and the mass balance becomes ... 

When minerals fractionate from a crystallizing magma, mass balance requires that, in a linear plot of Cr vs Ni, the points representing the composition of the parental... 

The symbols used in Section 1.5 to describe the evolution of element i concentration in the solid and the liquid during fractional crystallization will be kept. Other parameters used in the present derivation are almost identical to those of DePaolo (1981) although reference to time, which is immaterial to the mass balance and equilibrium conditions, has been omitted. Let a be the subscript representing the assimilated material, and assume that country-rocks concentration Cj is constant. Mass balance requires... 

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. 

Diffusion modeling, on the other hand, also predicts that accnrate temperatnres can be obtained from refractory accessory minerals, if they occur in a rock that is modally dominated by a readily exchangeable mineral (Valley 2001). The basis of this approach is that the accessory mineral preserves the isotope composition from crystallization becanse of slow diffnsion while the dominant mineral preserves its isotope composition by mass balance because there are no other snfficiently abnn-dant exchangeable phases. 

An alternative scheme, proposed by Garside et al. (16,17), uses the dynamic desupersaturation data from a batch crystallization experiment. After formulating a solute mass balance, where mass deposition due to nucleation was negligible, expressions are derived to calculate g and kg in Equation 3 explicitly. Estimates of the first and second derivatives of the transient desupersaturation curve at time zero are required. The disadvantages of this scheme are that numerical differentiation of experimental data is quite inaccurate due to measurement noise, the nucleation parameters are not estimated, and the analysis is invalid if nucleation rates are significant. Other drawbacks of both methods are that they are limited to specific model formulations, i.e., growth and nucleation rate forms and crystallizer configurations. 

When the crystal purity is plotted against the total crystal mass in the slurry calculated from the mass balances, the purity decrease seems to start at some constant value of the crystal mass as shown in Figure 5. As mentioned earlier in the text, there are possibilities of existence of the D-enantiomer as small particles on the surface of the seed crystals. If we assume that the breeding of the D-enantiomer starts only when that enantiomer has grown to a certain size, the amount of the L-enantiomer crystals must have also increased to a certain value, the latter being proportional to the former. The crystallization kinetics of the both enantiomers are believed to be the same, the relative amounts of crystals of the both enantiomers must therefore be constant before nucleation of the D-enantiomer starts. 

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

After the zinc has been removed, the sulfuric acid rich solution is returned to dissolve the next batch of sludge. Over time, the sodium concentrations will read unacceptable levels in the electrolyte. A bleed stream from the zinc cells is constantly being neutralized and filtered. The saturated sodium sulfate solution thus created is crystallized out as sodium sulfate anhydrous for sale to the pulp and paper industry. Table one shows a complete mass balance for a typical batch. 

Nevertheless, the chemical potentials of SE s are frequently used instead of the chemical potentials of (independent) components of a crystalline system. Obviously, a crystal with its given crystal lattice structure is composed of SE s. They are characterized much more specifically than the crystal s chemical components, namely with regard to lattice site and electrical charge. The introduction of these two additional reference structures leads to additional balanced equations or constraints (beside the mass balances) and, therefore, SE s are not independent species in the sense of chemical thermodynamics, as are, for example, ( - 1) chemical components in an n-component system. 

Defect clustering is the result of defect interactions. Pair formation is the most common mode of clustering. Let us distinguish the following situations a) two point defects of the same sort form a defect pair (B + B = B2 = [B, B] V+V = V2 = [V, V]) and b) two different point defects form a defect pair (electronic defects can be included here). The main question concerns the (relative) concentration of pairs as a function of the independent thermodynamic variables (P, T, pk). Under isothermal, isobaric conditions and given a dilute solution of B impurities, the equilibrium condition for the pair formation reaction B + B = B2 is 2-pB = The mass balance reads NB + 2-NBi = NB, where NB denotes the overall B content in the matrix crystal. It follows, considering Eqns. (2.39) and (2.40), that... 

Let us also consider the pairing reaction B A -t-V A = [B, V] in an ionic crystal AX, where the dopant BA is a heterovalent cation and V A is the compensating cation vacancy. We define the degree of pairing to be NP = A[B>V T/VB. From the mass balance equation A B = AB+AjB Vj and the condition of electroneutrality jVv + A b = NyA, one finds for the case that the undoped AX crystal exhibits Schottky type disorder (which means that = Ks)... 

Groundwater-inflow rates as calculated by the solute and isotope mass-balance methods for several northern Wisconsin lakes are listed in Table I. Dissolved calcium was used as the solute tracer because it is the constituent whose concentration differs the most between groundwater and precipitation, the two input components to be separated by the method. In addition, calcium is nearly conservative in the soft-water, moderately acidic to cir-cum-neutral lakes in northern Wisconsin. Results from the two methods agree relatively well, except for Crystal Lake, where groundwater-flow reversals are frequent. 

There was always an induction period of 10 to 20 min before the benzene product reached its steady-state rate of production as detected by the mass spectrometer after the introduction of cyclohexane onto the crystal surface. This is shown in Fig. 22 for several catalyst temperatures. The catalyst was initially at 300 K. When steady-state reaction rates were obtained, the catalyst temperature was rapidly increased (in approximately 30 sec) to 423 K and the reaction rate monitored. This was repeated with heating to 573 and 723 K. The benzene desorbed during rapid heating of the catalyst surface is approximately 1 x 1013 molecules or less and represents only a small fraction of the carbon on the surface. The steady-state reaction rates at a given temperature are the same whether the catalyst was initially at that temperature or another. This induction period coincides with a higher than steady-state uptake of cyclohexane. A mass balance calculation on carbon, utilizing the known... 

Now suppose some of the solvent is evaporated in the crystallizer. Independent balances can be written on total and solute masses ... 

As shown by Eq. (54), growth rate G can be obtained from the slope of a plot of the log of population density against crystal size nucleation rate B° can be obtained from the same data by using the relationship given by Eq. (57), with n° being the intercept of the population density plot. Nucleation rates obtained by these procedures should be checked by comparison with values obtained from a mass balance (see the later discussion of Eq. (66)). 

A more complete procedure to compute the average thicknesses of the water and oil layers will be presented below. It is based on eq 3c, mass balances of components, and phase equilibrium equations. The calculations indicated that the interfacial tension of lamellar liquid crystals is very low, of the order of 10 5 N/m. It will be shown that y = 0 is always an excellent approximation of eq 3 c. 

II.2. Charged Lyotropic Lamellar Liquid Crystals. In the case of a charged system, two changes in the system of eqs 17 must be made.80 81 First y should be replaced in eq 17a, for the surfactant, by y + fy=mole fraction of the surfactant in water, Xsw, should be replaced in the same equation by the mole fraction of the surfactant in water, in the vicinity of the interface CX SW). Assuming that the surfactant in the aqueous phase is totally dissociated, the concentration of surfactant near the interface (Xsw) can be related to the average surfactant concentration in the water phase, Xsw, by using the equilibrium and mass balance relations... 

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