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 (m3.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 La and La + ALa in dimension is given by:
m(La): 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 crystallizer’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