# Operative relationship of a CHC without attrition

Irrespective of the crystal size, the rate of increase in mass is:

Therefore:

However (see section 4.2.7):

The crystallizerâ€™s productivity is:

The nucleation rate is equal to the productivity quotient (in kg.s^{-1}) by the mean crystal mass (see section 4.2.7).

However, we frequently write:

e : agitation power per cubic meter of slurry (W.m^{-3}).

By equating the two expressions of J, we obtain:

Let us replace AX with its value in the expression of P:

Or again:

Lifting the two members of the previous equation to power i = *â€”* comes

^{g}

to:

However, the M_{c} / P ratio is written:

By eliminating P between [4. Ibis] and [4.2], we obtain the desired relationship:

In general, j is close to i, and M_{c} scarcely intervenes in the equation [4.3]. Often, the value of i, greater than 1, is of the order of 2 and we have:

We observe that the crystal size is far from proportional to the residence time. This can be explained by the supersaturation AX decreasing with residence time.