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Overall balance of the crystallizer population

Band (L, L + AL) sees its population varying during time 5t.

Crystals of length L grow from G(L)5t and enter the number section of n(L) G(L) 5t = nGSt.

n(L)AL is the number of crystals present in 1 m3 of slurry, between length L and L + AL.

Crystals of length L + AL also grow, leaving their section. Their growth is:

They leave their section at number:

Therefore, disregarding infinitely small terms of the second order:

The overall balance of the band is (entry minus exit):

Assuming a homogenous crystallizer, the population variation of the preceding band over time St is:

V is the crystallizer volume (m3).

If there is vaporization, the variation is:

Therefore, globally:

The balance for the device is written:

Hence:

However, this balance is incomplete.

We must account for:

  • 1) the disappearance of D crystals per unit of time and unit of volume of the crystallizer. This disappearance is due to dissolution;
  • 2) the apparition J of crystals per unit of time and unit of volume of the crystallizer. This apparition is due to primary and/or secondary nucleation (attrition);
  • 3) the extraction of production qs (L) ns(L). The extracted flow varies with the size of crystals as their residence times are not necessarily identical;
  • 4) the arrival of supply qana(L). However, most of the time, supply does not include crystals unless there is inflow of recycling returns;

The complete balance is written as:

The dimensions in this balance are as follows:

 
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