Part 1 of this article focused on basic principles and criteria for process design and equipment selection. This final installment will examine the criteria for scaleup, examine use of simulation and highlight some recent innovations in crystallization technology.
An ideal scaleup for any crystallization process would lead to identical levels of supersaturation in all equivalent regions of the laboratory, pilot and plant units. Slurry densities should be the same. Contact times between the crystals and supersaturated liquor should be equal, the Reynolds numbers and Froude numbers of the fluid should be identical, and the contact frequency and energy between crystals, and between the agitator and crystals, should be the same. Ideally, shear forces imparted by the agitator to the slurry and mixing times would be equivalent.
Unfortunately, it's not a perfect world. These goals are often conflicting, even with geometric similarity, and achievement of the same degree of supersaturation and mixing parameters upon scaleup is impossible. In fact, geometric similarity assures that equivalency will never be attained.
Due to conflicting similarity goals, strict geometric scaleup of mixers is restrictive. For example, Reynolds number varies with ND2, the Froude number varies with N2D and the Weber number varies with N2D3. If one of these is used for scaleup, the other two will not scale proportionately. It is better to design a system with more degrees of freedom.
It's important to keep in mind that laboratory and pilot-scale vessels have shorter liquid flowpaths, in both the axial and radial directions, than commercial facilities, yielding reduced turnover times -- typically 1-3 sec. versus 20-60 sec. or more for common commercial equipment. Turnover time is defined as the active slurry volume divided by the axial flow rate generated by the impeller.
An effective way around this problem is to use different geometries for the smaller units. Because the laboratory or pilot facility is so "well mixed," one might consider using a narrower blade and smaller agitator-to-tank-diameter, or D/T ratio, in the small equipment. Narrowing the D/T limits the increase in shear rate variation that occurs if D/T is held constant during scaleup. The reduction in D/T for the pilot plant increases the distribution of shear rates in that vessel and makes the distribution closer to that of the production unit.
If a mixer is scaled up with geometric similarity and constant P/V, the maximum shear rate at the blade increases, being proportional to D1/3, while the average shear rate decreases, being proportional to D-2/3. This maximum shear rate may present a breakage and secondary nucleation problem for some crystallizations.
Table 1 presents the results of different agitator scaleup strategies when utilizing geometrically similar vessels. The first column describes a number of pertinent agitation parameters, while the second column normalizes the same data by assigning a value of 1.0 to represent the value that would be found for each in a typical 25-gal. pilot plant crystallizer.
Each of the four remaining columns shows the implications of a chosen strategy for geometric scaleup to a 3,125 gal. full-scale vessel. In carrying out a given strategy, one of the parameters is held constant (e.g., P/V in the case depicted by Column 3). The other values in the column show the effect of that strategy on the other parameters.
Column 3 demonstrates that, if P/V is to be held constant for a geometric scaleup, the agitator speed will drop and the agitator-generated slurry turnover (Q/V) within the vessel will fall off. This criterion yields the same local energy dissipation and, thus, specific power input for micromixing.
With larger vessels, the maximum macroscale shear rate in the impeller zone increases while the average decreases. The maximum increases due to the tip speed, while the average decreases in response to speed. Experiments have shown that scaling on the basis of constant P/V usually results in a decrease in secondary nucleation.
Column 4 assumes holding active volume turnover rate constant, which makes P/V proportional to the square of the impeller diameter, while maintaining constant speed. In this case, the residence-time distribution (RTD) is constant. This causes a large rise in pumping power and yields similar macromixing, constant circulation time and constant pumping rate per unit volume. This choice is impractical, suggesting that a pilot plant's attractive low blend time and circulating time cannot realistically be maintained during geometric scaleup. Severe breakage and secondary nucleation can result.
As indicated in Column 5, when specifying constant tip speed, ST, (therefore constant torque-per-unit-volume for turbulent flow conditions with Newtonian fluids) the maximum impeller zone shear rate remains constant, but the turnovers drop significantly, along with the P/V. It yields equal bulk fluid velocities and can be considered as an approximate scaleup for constant mesomixing, since blending of incoming reactant with bulk or a second reactant is closely linked to shear in the mixing zone. Some experimental evidence indicates that scaleup on the basis of tip speed will result in a decrease in secondary nucleation rate.
Column 6 shows that it is totally impractical or impossible to keep the impeller Reynolds number constant, because the P/V value and the turnover would become unrealistically low.
In conclusion, Table 1 shows that for any realistic scaleup:
The above analysis predicts that secondary nucleation will be reduced when scaling up on the basis of either constant P/V or constant ST, with the reduction being greater for the latter. However, turnovers and blending are further reduced when employing the tip speed criteria. If constant tip speed is selected, one must make sure that the velocities are adequate to maintain solids suspension and, especially for precipitations, that the micromixing is sufficient to reduce inhomogeneities that result in high levels of supersaturation. If not, primary nucleation becomes possible.