# Balance of population with attrition

The frequency of impacts between the impeller and the parent crystals is proportional to the number of these crystals coming into contact with the impeller:

Q: recirculation flow in the vat (m^{3}.s^{-1})

c: number of particles per cubic meter of the suspension (m^{-3})

By applying probability p, so that an impact is followed by attrition:

The apparition of fragments between L_{a} and L_{a} + AL_{a} in dimension is given by:

m(L_{a}): frequency of the number of fragments issued by a single impact (m^{-1})

The attrition parameters are ultimately:

- frequency of the number of fragments of size L:

- growth decreases with the size of crystals, as attrition is equivalent to a negative growth proportional to the square of the crystal dimension:

A crystallizerâ€™s general balance of attrition for continuous crystallization is written as:

In addition, for batch crystallization:

# Continuous homogenous crystallizer with attrition

By bringing together the fragments resulting from attrition and crystals coming from primary nucleation or seeding into a single equation, we obtain:

where Lmin is the minimum size (that of seeds) and, if L nears Lmin, the second term is dominant and strongly positive.

However, for low values of L, the derivative is cancelled where:

It then becomes negative.

Lmax is the maximum size allowed and, if L nears this value, the derivative of Ln(n) is cancelled for:

For example, if:

We should then have:

Finally, variations of Ln (n) will appear as in Figure 4.1.

**Figure 4.1. ***Ln (n) variations with the crystal size*