The Effect of Changing j

It is known that the nucleation is not only related to the supersaturation, but also the crystal in the batch crystallizer. In our previous work, we only considered the situation that the exponent on the third moment in Equation 13 was equal to one. In this section, we discuss the situation that this exponent may change. The equation will be like this:

3

The dimensionless third moment remains the same:

( )

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We want the dimensionless moments ODE set would be like this:

( )

Therefore, we define

(

)

The dimensionless crystal birth and crystal growth becomes

( )

The dimensionless initial seed crystal size distribution becomes:

( ) ( )

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with Equation 51, the equation becomes:

( )

Where the parameter of seed distribution function are defined as:

†

The dimensionless average size of the seed for j and j=1 are:

( )

Divide the above equations we have:

1

This is the new dimensionless average size of the seed for j value that is not one. Similarly we have the dimensionless seed distribution width:

( )

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Divide the above equations we have:

1 seed distribution average and width. However, the shifting is depends on µ₃, and it is depends on the chemicals. In this work, we consider the case where

µ3 is equal to one.

Figure 34: The growth rate for different j.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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We optimiza using the objective “minimizing µ'_3n”, From Figure 34,

optimized trajectory for different j value both tend to have high growth rate late in the batch. The major reason is that if the growth rate or supersaturation rises, the nucleation rate will rise larger than growth rate. And to achieve the goal of final production mass, the area under the growth rate trajectory has to be some fixed value. The question becomes that is it better to put more under line area at the beginning of the batch, or at the end of the batch.

It is also known that the third moment of the crystal will react slower than lower level of moments after the growth rate trajectory raises. Although using this late growth trajectory will causes large amount of the nuclei forms at the end of the batch, there is no time for them to growth up to enough mass that affects the objective function value. Therefore for objective function that minimizes the third moment of the nucleated crystal, the optimal growth trajectory is tending to be a late growth trajectory.

The major difference of the optimized growth trajectory is at the beginning of the batch. We can see that in Figure 34, optimal trajectory for j equal to zero tend to lower value at the beginning of the batch. For j equal to one the optimal growth trajectory is slightly higher than trajectory that for j

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equal to zero. For j equal to three, the beginning value of the optimal growth trajectory is the highest.

It is known that the nucleation rate is not related to the crystal amount in the batch crystallizer if the j equal to zero. This idea comes from Equation 49, when the j equal to zero, the nucleation rate becomes:

B=k Gb ^γ (78)

There is no third moment term in the equation, so the nucleation rate is no longer

related to the third moment of the crystals. Since the nucleation rate is only

depends on the supersaturation, then the optimal trajectory for minimizing the

mass of the nucleated crystal tend to have small supersaturation during the batch,

and have large supersaturation at the end of the batch such that even nucleation

happens, there’s no time for nucleated crystals to growth.

For j not equal to zero, this is the value widely used for crystallization

process. The nucleation rate is related to both the supersaturation and the mass of

the crystal in the batch crystallizer. If there’s more crystals in the batch

crystallizer, the nucleation increase. Then it comes to the problem that where to

put more area under the growth trajectory line. To minimize the mass of the

nucleated crystals, one strategy is to set higher supersaturation at both the

beginning and the end of the batch. The advantage that put more growth rate or

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higher supersaturation at the end of the batch is the same for the j equal to zero.

That is there is not time for newly formed nucleated crystals to accumulate

enough mass. And there is also an advantage to set the supersaturation to a higher

value at the beginning of the batch compared to the middle of the batch. The total

mass of crystals is always increasing during the batch process. Therefore for the

same value of supersaturation, at the beginning of the batch there will be less

nucleation than at the middle of the batch. So the optimal trajectory tends to have

a higher superstauration at both the beginning of the batch and the end of the

batch.

For j is not equal to one, different j values primarily affect the optimal

trajectory at the beginning of the batch. Higher powers of the moment term in

Equation 49 cause a lower total nucleation rate. Mathematically, the reason is that the numerical value of the third moment is less than one. Therefore for same level

of supersaturation and same value of the third moment, the nucleation rate for

higher j value is lower. Again, since the power of the moment is not zero, higher

third moment of the crystal will cause higher nucleation rate. During the batch,

the third moment is always increasing. Therefore for j equal to one, two and three,

having higher supersaturation at the beginning of the batch is better than at the

middle of the batch. Moreover, since for same level of supersaturation and same

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value of the third moment, the nucleation rate for higher j value is less, the

system with higher j values can tolerate higher supersaturation compared to

systems with smaller j. Therefore, for higher j value system, the optimal

trajectory tends to have a higher superaturation at the beginning of the batch

compared to lower j value systems.

Figure 35: The concentration trajectory for different j value.

Figure 35 shows the dimensionless concentration trajectory during the batch. Because of the definition, the dimensionless concentration decreases from one to zero during the batch. The dimensionless concentration trajectory

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t'

C'

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when j=3 decreases more evenly during the batch than the others. Since amount of the solute in the solution is consumed by crystal nucleation and crystal growth, and both of them is higher for higher j system, the solute in higher j system is consumed faster. Therefore, the dimensionless concentration trajectory for higher j value is decreasing faster.

Figure 36: Final crystal size distribution for different j value.

Figure 36 shows the weigh base final crystal size distribution. The area under each curve is proportional to mass of the crystal. There are two peaks in this figure for each j value. The peak closer to L =' 0 represents the nucleated

0 0.5 1 1.5 2 2.5 3

0 2 4 6 8 10 12 14 16

L' L'3 f'(L')

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crystals, and the other parabolic peaks represent the seed crystals. The objective function is to minimize the nucleated crystal mass, and the optimal result is better when j is larger. This is because the numerical value of the third moment is less than one, therefore for higher power on the moment term in Equation 49, the nucleation rate is actually less, which causes the suppression of the mass of nucleated crystals.

Figure 37: Different moment during the batch for different j value.

0 0.5 1

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