Since the growth rate is directly related to the supersaturation a, and the rotational speed to the power input, the power law can also be written as:

Bo = kiVsMY

Frequently measured values for b, k and j under steady-state conditions are 1 < b < 3, 0.6 < k < 0.7 and j"1 or 2. For nucleation dominated by crystal-impeller collisions j = 2, while for nucleation ruled by crystal-crystal collisions, j = 2.

and is thus related to the supersaturation via A^u, and to the interfacial tension y.

For the homogeneous nucleation rate, Jhomo (#m~3), the following equation can be derived after some simplifications:

In this Arrhenius type of expression, changes in the supersaturation ratio S in the pre-exponential factors A • S are of minor influence compared to changes in S in the exponential term. Various authors use dif-

Figure 4 Nucleation rate as a function of the supersaturation (A^). Jhomo and Jhetero, homogeneous and heterogeneous nucleation rate; B0, secondary nucleation rate.

Also the crystallizer geometry-like type of stirrers or pumps, and number of blades, as well as the scale of operation influence the attrition rate of the crystals. These effects were supposed to be included in kN of kN, that could only be established experimentally. The power law with its four parameters, used together with other kinetic parameters (e.g. those related to the growth of the crystals) seems to be perfectly adequate to describe the steady-state median crystal size in continuous crystallization processes.

The power law, however, fails to describe the dynamics of a crystallization process. If, for example, the median crystal size is plotted versus time from immediately after the start-up, often an oscillating behaviour is observed that dampens out until finally a steady-state value is reached (Figure 5). This oscillating behaviour can be explained by the observation that only crystals above a certain size breed secondary nuclei by attrition. The first nuclei are created by primary nucleation and grow out, which causes the supersaturation to decrease. When the mean size has reached its first peak, more of the larger crystals are withdrawn with the product than are grown into the larger crystal sizes by outgrowth of secondary nuclei. This happens because at the early stages no large crystals are available for breeding.

Some groups have tried to account for this phenomenon by adding a target efficiency to the power law that is a function of the crystal size. One author, Eek, allowed only crystals above a certain size to breed. This improved the simulations, although none of them was fully successful.

An attrition function for the crystals, was introduced by O'Meadhra based on the approach of Mersmann. He distributed the attrited volume over the small crystal sizes. In this way a birth function B(L) was calculated from the volumetric attrition rate. A disadvantage of this modelling, in common with the power law, is that it has no predictive value, since

Figure 5 Dynamic behaviour of the CSD. Continuous line, measured; dashed line, modelled with the power law.

the attrition function also must be determined experimentally for the particular crystallizer.

Gahn and Mersmann were the first to derive a secondary nucleation rate model based on physical attrition properties. Their approach comprises three consecutive steps and calculates the secondary nucleation rate of crystals that collide with the blades of an impeller.

1. A simplified flow pattern based on geometric considerations as presented by Ploss et al. is used to calculate the impact velocity and the chance of crystals from size class i to collide with an impeller segment j.

2. Subsequently a model was developed to calculate the volume of attrition fragments produced during a single collision of a crystal corner represented by a cone, and a hard flat surface of the impeller. The model relates the attrited volume of crystal i and segment j to the impact energy of the crystal collision via its relevant mechanical properties, such as the Vickers hardness, the fracture resistance of the substance and the shear modulus. The model assumes that the circulation time is sufficient to heal the damaged crystal corner before a subsequent collision of the same corner takes place. This assumption is often not valid for crystallizers up to 100 L. A minimum impact energy required to cause fracture can also be derived, and thus the minimum crystal size for a given velocity profile is prone to attrition. The model also provides a normalized number density function of the fragments with a minimum and maximum size of fragments formed at each collision of crystal i with impeller segment j. In general, the size distribution of fragments lies in the range from 2 to 100 |im.

3. In the third step the rate of secondary nucleation is linked to the rate of formation of attrition fragments. The amount of stress remaining in the fragments limits the number that grows into the population, because stress increases their chemical potential. Their real saturation concentration c* becomes:


Figure 5 Dynamic behaviour of the CSD. Continuous line, measured; dashed line, modelled with the power law.

The stress content of the fragments is directly related to their length, and the value of rK has to be determined experimentally, for example from experiments where the fines of a crystallizer are withdrawn, and allowed to grow out in a growth cell.

The formed fragments with size Lfragment can now be distributed with their respective length and stress content. A number of fragments will dissolve, and the rest will grow into the population with a size-dependent growth rate.

This model does not deliver a secondary nucleation rate, B0, with nuclei born at zero size or a birth term, B(L), for the distribution of secondary nuclei, but calculates from the number of collisions, and from the surviving fragments per collision, the number of new developing crystals. Since for this calculation an initial CSD is needed, iteration loops are always needed if this nucleation model is used as a predictive tool for secondary nucleation.

Both this model and that of O'Meadhra are able to describe the dynamic behaviour of crystallization processes.

Crystal Growth

Definitions of growth rate The growth rates of the crystallographically different faces (h kl) of a crystal can vary considerably. The growth rates of the crystal faces determine the shape of the crystal and, together with the growth mechanisms, also the crystal surface structure.

The growth rate of a particular crystal face (h k l) is mostly defined by its linear growth rate Rlin (m s"1), which refers to the growth rate of that face along the normal direction. An overall linear growth rate Rlin averaged over all different (h k l) faces can be defined in several ways. One definition which is often used relates Rlin to the increase of the crystal mass in time:

For spheres and cubes (or for crystals where L is based on the diameter of a sphere with the same volume), ka/kv = 6, and:

Figure 6 Diffusion of a growth unit towards and integration into the crystal surface layer.

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