# Parameters to manage

Let us remove т and n_{o} between J, ф_{т} and L_{50}. We obtain:

The crystallizer design allows for 9_{T} to be controlled together with supersaturation, that is, J and G, in order to obtain the median dimension required. However, in reality, things are not this simple. If we increase the residence time, supersaturation decreases, and consequently, the value of G is such that the median size does not increase in proportion to т even when 9_{T} is proportional to т^{4}. In fact, L_{50} slowly increases with т and even begins to decrease if attrition becomes significant.

We observe that L_{50} varies similarly to 9_{T} at power 0.25. Accordingly, it is worthwhile for the crystal volume fraction 9_{T} to approach 0.20, that is, to approach its maximum value, for an acceptable agitation.

# CHC series without attrition

For a single device, we know that the population density expression is:

Let us now distinguish two cases:

1) Nucleation occurs only in the first crystallizer and the speed of growth is the same in all crystallizers.

In the second crystallizer, we can write out the following balance:
where V_{c} is the volume of a device so that:

We can verify that the solution to this equation is:

For k crystallizers, this can also be generalized like so:

2) Nucleation and growth are identical in m crystallizers, so:

This expression can be deduced from the previous case by noting that population is the total of m populations corresponding to k, varying by 1 to m.

Randolph and Larson [RAN 71] show that, in the two previous cases, the size of crystals is lower and distribution is tighter compared with the case of a single crystallizer. These authors give the surface distributions and the mass for two crystallizers.

Often, it is not for reasons of size distribution that crystallizers are placed in series, but to save energy:

- - evaporation for several effects;
- - cooling on two levels. The cooler levels operate with a refrigeration unit and the entry level is cooled by the mother liquor from the cooler level.

However, the abundance of slurry pumps can create attrition problems for certain products.