Steady state crystal growth

A continuous cooling crystallizer is required to produce potassium sulphate crystals (density pc = 2660kgm, volume shape factor a = 0.7) of 750pm median size Lm at the rate Pc = 1000 kg h. On the basis of pilot-plant trials, it is expected that the crystallizer will operate with steady-state nucleation/ growth kinetics expressed (equation 9.39 with j = 1 and i = 2) as B = 4 X IO MtG m s. Assuming MSMPR conditions and a magma density Mt = 250 kg m, estimate the crystallizer volume and other relevant operating conditions. 

If the crystallizer is now assumed to operate with a cleat feed (n = 0), at steady state (dn jdt = 0), and if the crystal growth rate G is invariant and a mean residence time T is defined as then the population balance can be written as... 

The population balance analysis of the idealized MSMPR crystallizer is a particularly elegant method for analysing crystal size distributions at steady state in order to determine crystal growth and nucleation kinetics. Unfortunately, the latter cannot currently be predicted a priori and must be measured, as considered in Chapter 5. Anomalies can occur in the data and their subsequent analysis, however, if the assumptions of the MSMPR crystallizer are not strictly met. 

In the MSMPR crystallizer at steady state, the increase of particle number density brought about by particle growth and agglomeration is compensated by withdrawal of the product from the crystallizer. 

If crystal agglomeration and breakage can be neglected and crystal growth is invariant then at steady state the familiar analytic form is... 

Finally, we were led to the last stage of research where we treated the crystallization from the melt in multiple chain systems [22-24]. In most cases, we considered relatively short chains made of 100 beads they were designed to be mobile and slightly stiff to accelerate crystallization. We could then observe the steady-state growth of chain-folded lamellae, and we discussed the growth rate vs. crystallization temperature. We also examined the molecular trajectories at the growth front. In addition, we also studied the spontaneous formation of fiber structures from an oriented amorphous state [25]. In this chapter of the book, we review our researches, which have been performed over the last seven years. We want to emphasize the potential power of the molecular simulation in the studies of polymer crystallization. 

Figure 9.16 Kinetic fractionation during crystal growth. Steady-state distribution of melt concentrations in the vicinity of a solid growing at the rate v for trace elements with different solid-liquid fractionation coefficients [equation (9.6.5), Tiller et al. (1953)]. The stippled area indicates the steady-state chemical boundary-layer with thickness <5 = <5>/v.

Growth and nucleation interact in a crystalliser in which both contribute to the final crystal size distribution (CSD) of the product. The importance of the population balance(37) is widely acknowledged. This is most easily appreciated by reference to the simple, idealised case of a mixed-suspension, mixed-product removal (MSMPR) crystalliser operated continuously in the steady state, where no crystals are present in the feed stream, all crystals are of the same shape, no crystals break down by attrition, and crystal growth rate is independent of crystal size. The crystal size distribution for steady state operation in terms of crystal size d and population density // (number of crystals per unit size per unit volume of the system), derived directly from the population balance over the system(37) is ... 

The development and refinement of population balance techniques for the description of the behavior of laboratory and industrial crystallizers led to the belief that with accurate values for the crystal growth and nucleation kinetics, a simple MSMPR type crystallizer could be accurately modelled in terms of its CSD. Unfortunately, accurate measurement of the CSD with laser light scattering particle size analyzers (especially of the small particles) has revealed that this is not true. In mar cases the CSD data obtained from steady state operation of a MSMPR crystallizer is not a straight line as expected but curves upward (1. 32. 33V This indicates more small particles than predicted... 

By means of our experimental method (twin+single crystal kinetics) steady state growth morphology can be predicted as a function of supersaturation and temperature. 

In this way two objectives ensue 1. We obtain the growth mechanism of the most important flat and stepped faces of the crystal. 2. We are able to foresee the global crystal morphology in a steady state for all temperature and supersaturation values. 

The position of A will depend on process conditions. For a well mixed crystallizer at steady state, point A will adjust until the rate of hemlhydrate dissolution equals the rate of gypsum growth (In mole units). In simplified terms. 

From the popuiation baiance for a MSMPR crystaiiizer operated under the steady-state condition, the population density n for size-independent crystal growth is given as... 

That is, the concentration in the crystal is the same as that in the initial melt at steady state. Therefore, the growth of the crystal does not affect the mass excess or deficiency in the melt anymore, meaning that the concentration profile (in interface-fixed reference frame) in the melt is at steady state. Steady state maybe reached only for elements whose concentration in a mineral can vary non-stoichiometrically. 

For crystal growth at constant rate, if the crystal composition can respond to interface melt composition through surface equilibrium, steady state may be reached (Smith et ah, 1956). At steady state, (dCldt) = 0 by definition. Hence,... 

When a steady state is reached, the boundary layer thickness is independent of time and so is the crystal growth rate (or melt consumption rate u). The concentration profile at the steady state is... 

Now consider the case of three-dimensional crystal dissolution. Let the radius of the crystal be a (which depends on time). In this case, the most often-used reference frame is fixed at the center of the crystal, i.e., lab-fixed reference frame (different from the case of one-dimensional crystal growth for which the reference frame is fixed at the interface) so that the problem has spherical symmetry. Ignore melt density variation. The crystal dissolution rate (u ) and melt growth rate at the interface (Ua) are related by the continuity equation with approximation of steady state ... 

With two further assumptions, (hi) the crystal growth rate is independent of crystal size L, and (iv) CSD reached steady state so that dnldt=0, the above equation becomes... 

For low selenosulphate concentrations, only the small crystals were formed, even in thicker films, and this was rationalized by the lower steady-state selenide concentration, which would favor cluster growth over ion-by-ion formation (the product of free lead and selenide ions needs to be larger than the solubility product of PbSe for ion-by-ion deposition to occur). An important difference between the citrate depositions and the NTA or hydroxide ones is that, even in the ion-by-ion citrate deposition, some low concentration of colloidal hydrated oxide was present, due to the relatively low complexing strength of citrate. The pH of the hydroxide baths (> 13) was much higher than that of the citrate or NT A baths (10.8). 

For a finite flux jA, there is a (steady state) shift of the AX crystal towards the side with the higher pXi. jx does not lead to such a shift. The shift velocity is / A-Vm(AX). Equation (4.104) can also be used to quantify the basic (one dimensional) metal oxidation experiment A+1/2X2 = AX shown in Figure4-4. In terms of thickness growth, one obtains from Eqn. (4.104) the expression... 

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