# Batch crystallization methods (presentation)

The aim is to create the initial distribution of crystals, which will then be grown.

If, at a given time, supersaturation happens to be too high, there can be a massive primary nucleation that spreads the product’s size distribution, decreasing its mean size.

Accordingly, supersaturation must be controlled. One method for this consists of using a supersaturation program, whereby the product’s granulometry should be close to a given granulometry. However, this method, proposed by Jones [JON 74a], is mathematically complex, and is incongruous in industrial practice.

On the other hand, predictive calculations were performed by Jones and Mullin [JON 74b]. These calculations, assisted by practical experience, showed that an operation performed at constant supersaturation provides larger crystal sizes than a “natural” cooling and narrower size distribution. Let us consider this in more detail:

1) “Natural” cooling or vaporization (not controlled).

The thermal power, initially strong, decreases exponentially. Supersaturation, initially high, leads to the formation of numerous seeds over a non-negligible time. The seeds are numerous and their size distribution is spread.

Thus, the product’s quality depends on the thermal power. The greater the power, the smaller the crystals and the greater the granulometry spread due to the high initial nucleation.

2) Constant supersaturation.

This method was recommended to narrow the granulometry and increase the crystal size. On first impression, if the supersaturation is constant, we can suppose that growth G and nucleation velocity J are also constant. According to this hypothesis, the thermal power increases considerably from the

beginning to the end of the operation. Indeed, we require 100 times more energy to increase the size of a crystal of 1 mm by 20 pm than to increase a crystal of 100 pm by the same amount. It would be difficult to apply such a method in industry.

In practice, for an effective (“controlled”) operation, we use neither of these methods.

**Figure 4.11. ***Ideal controlled crystallization*

3) Controlled process.

The purpose of batch crystallization is to obtain a narrow crystal centered on a single size. In this case, the population density presents a sharp spike for the desired crystal size.

To allow for a brief mathematical parenthesis, we will say that the P.D (population density) is represented by a Dirac impulse function 5( L_{P}). A better term would be Dirac “distribution”.

This function’s integral is the echelon function E(x). and the P.D. with an integral equal to 1 must be:

This objective can be approached in two ways: by seeding or by controlled supersaturation.