where ܹ (kg.s-1) is the water mass flow rate, ܮݏ (kg.s-1) soluble lactose flow rate, ܮܿ (kg.s-1) is the solid crystal component for total feed, overflow-recycled component and non-recycled feed component (subscripts F, Or and Fr respectively), and οݐ (s) is the simulation time step. Each individual fluid section has a soluble lactose (Equation 67) and water (Equation 77) component which changes with time depending on the crystal growth that occurs in a time step. Individual particles are matched to a section of fluid by searching the Particle Matrix, through the correlation of particle and fluid section height. This calculates the amount of lactose (Equation 68) and water (Equation 71) consumed by the particles and thus the amount to be removed from the fluid section. Fluid concentration (Equation 53), density (Equation 62) and viscosity (Equation 63) were recalculated after each time step. These changed the individual fluid sections volume (Equation 74) and vertical length within the crystallizer.
88
The CSC column individual fluid sections distribution and movement can be visualised in Figure 50. Starting from the bottom of the column at ݖ=0 and ܼ=0. With each time step the fluid section moves to a new height position until ݖ=m at ܼ=ܼ. The properties of each individual fluid section are recalculated with every time step.
Figure 50 Schematic of theoretical CSC model fluid sections
6.2.4.2 Particle Matrix
Individual particles are monitored and a specified number of individual particles enter the column with each newly inputted fluid section at height ܼ=0 m. From particle height, the Fluid Matrix was searched, and the new fluid section which the particle resides in was found for each time step. The selected fluid section provided information for fluid concentration (Equation 53 and Equation 54), velocity (Equation 60), density (Equation 62), and viscosity (Equation 63). This information was used in particle growth (Equation 51) and vertical movement calculations (Equation 59 and Equation 61). Particles will stop growing and moving in the column once height is less than ܼ=0 m (product) or greater than ܼ=ܼ (overflow). Modelling individual particles, with a search function matching a fluid section with particles, results in long simulation times. To reduce computer processing time but to obtain a certain particle or nuclei loading, model calculations for defined individual particle water and lactose
ݓǡ௧݈ݏǡ௧݈ܿǡ௧ݖǡ௧
ݓୀǡ௧ ݈ݏǡ௧ ݈ܿୀǡ௧ݖୀǡ௧
Overflow or Waste at Z=Z
Feed (with nuclei) at Z=0 m
୫ାଵǡ୲୫ାଵǡ୲ ୫ାଵǡ୲ݖାଵǡ௧ ୫ାଶǡ୲୫ାଶǡ୲ ୫ାଶǡ୲ݖାଶǡ௧ Crystallizer
Height, Z
89 consumption are multiplied by a multiplication factor. The multiplied amount of particles will therefore have the same properties, and not change the span calculations.
Individual particles entering the column at height ܼ=0 m movement can be visualised in Figure 51. Crystal movement occurs in the positive ܼ direction while crystal terminal settling velocity,
ݑ௧ǡ (m.s-1), is less than average fluid velocity ݑ (m.s-1), and growth occurs while particle column height position ݖ (m) is greater than ܼ=0 m and less than ܼ=ܼ. The properties of individual crystals are recalculated with each time step.
Figure 51 Schematic of theoretical CSC model individual particle movement
݀ǡ௧ ݖǡ௧ ݇௦ǡ Overflow or Waste at Z=Z Feed at Z=0 Crystallizer Height, Z Product Crystals at Z<0 ݑ௧ǡ ൏ ݑ ݑ௧ǡ ݑ
90
6.3 Theoretical Model Results
6.3.1 Batch Model Results
An appropriate growth rate distribution prediction model was needed for the theoretical CSC simulations to be carried out. The constant crystal growth (CCG) rate data used was from 6 hr batch grown control experiments (sampled from nucleated feed described in Section 5.4.4) for two types of lactose with statistically different inherent growth rates. For each type of lactose the experimental growth conditions were the same as well as the theoretical model simulation conditions. The growth rate distribution arises from converting Malvern MasterSizer volume distribution into a number weighted table of kg bins (assuming CCG), and within each kg bin a random kg is generated; the particle model randomly selects a kg value from this table. The results for the IGL-1 and 200M-1 batch data are shown in Table 10 and Table 11; an averaged result with the standard deviation is shown. The model was run three times, as each run was unique giving slightly different results due to the random allocation of growth rates to each crystal.
Condition d10 (μm) d50 (μm) d90 (μm) Span
Batch Growth Theoretical Model Predictions
4hr Growth 15.2±0.2 34.7±0.4 71.9±0.3 1.64±0.02
6hr Growth 22.5±0.5 53.9±1.7 116.4±11.2 1.74±0.15
16hr Growth 58.2±0.7 142.5±3.8 300.2±16.9 1.70±0.08
Batch Growth Experimental Results
4hr Growth 15.4±1.04 36.0±2.1 71.5±5.4 1.56±0.09
6hr Growth 21.2±1.5 53.7±3.7 114.9±11.2 1.74±0.14
16hr Growth 47.1±8.5 158.8±32.4 333.2±64.4 1.81±0.28
Table 10 Comparison of IGL-1 theoretical batch simulations with experimental batch values
Condition d10 (μm) d50 (μm) d90 (μm) Span
Batch Growth Theoretical Model Predictions
4hr Growth 7.3±0.3 24.5±1.2 48.1±5.8 1.66±0.15
6hr Growth 12.8±1.0 37.2±1.7 78.7±11.4 1.77±0.25
16hr Growth 42.5±1.4 95.1±4.6 189.7±18.1 1.54±0.10
Batch Growth Experimental Results
4hr Growth 10.3±0.3 20.7±0.5 36.6±1.6 1.27±0.06
6hr Growth 14.3±0.5 32.0±1.8 62.1±5.7 1.49±0.09
16hr Growth 43.7±2.3 113.0±7.0 208.7±10.0 1.46±0.04
91 The generated growth rate distributions allowed for the preliminary investigations into theoretical CSC model predictions to be carried out. The batch model code (Appendix A.9) was a simplified version of CSC theoretical model matrices code. Only one fluid section was specified with no movement, and the volume was specified to give a certain nuclei or crystal loading achieved in practice. Concentration change of the fluid section utilised Equation 53, Equation 67 and Equation 70. Individual nuclei were inputted at time zero (total number to give desired nuclei loading), with no movement function. The change in crystal size utilised Equation 51 to Equation 58, and correlating with the (single) fluid section. Fluid and particle properties were solved for each time step.
6.3.2 CSC Model Results
The CSC theoretical model used the starting assumption of a single fluid velocity throughout the column. When CCG kinetics are assumed, fast and slow growing crystals can be seen following the path in the constant radius column crystallizer shown in Figure 52. A slow growing crystal travels higher up the column than a fast growing crystal before growing to the terminal settling diameter, opposing the fluid flow, and settling out into the product stream. Using a single fluid velocity resulted in a narrow product PSD, approximately 61 μm crystals, when constant column geometry was assumed. Additional column geometries consisting of a cone section and constant radius section were also tested (Figure 53). The effect of the cone section changed the fluid velocities that the fast and slow growing crystals encountered with height, leading to a larger product span and these designs were dismissed.
Figure 52 Theoretical model particle height within the crystallizer column versus particle size at 16 hours of simulation time when single column fluid velocity is assumed
92
Figure 53 Model results for particle position within the crystallizer and particle size, crystallizer has a 10° cone angle starting at 40% of the total crystallizer height (0.35 m)
Column flow with a single velocity predicts a product span of ~0, which was a promising start for the CSC concept. The assumption of single fluid velocity, ݑ, however does not hold for low Reynolds systems. When operating at fluid Reynolds numbers less than 2000, a laminar flow profile occurs. Laminar fluid flow across the pipes radius, ݎ (m), can be approximated by the parabolic velocity profile, ݑ (m.s-1), Equation 80 and seen in Figure 54.
ݑ ൌ ʹݑሺͳ െ ݎଶΤ்ܴଶሻ