where ݀ହ (m) is the corrected cut size, ݉ (-) is the classification index, οܲ (Pa) is the pressure drop, ܵ (-) is the volumetric flow split, ܥሺ݀ሻ (-) is the classification function, ܴ (-) is the recovery of water to the underflow, ܵሺ݀ሻ (-) is the selectivity function, ܦ (m) diameter of cyclone, ܦ (m) diameter of vortex finder, ܦ௨ (m) diameter of spigot, ܦ (m) diameter of inlet, ݄ (m) free vortex height, ܲ (Pa) is the cyclone inlet pressure, ܥ(%) is the percent solids in feed by volume, d (m) size of particle, ߩ (kg.m-3) density of feed pulp, ܨଵǡ ܨଶǡ ܨଷǡ ܨସ (-) are calibration parameters and ݇ଶ (-) is an exponent for the effect of solids density, and ܨሺ݀ሻ (-) is the feed PSD.
The result of a single hydrocyclone cut with a d50c=60 μm (Figure 27), achieved a span of 0.66, d50 of 56 μm and overall yield of 17%. The use of a hydrocyclone cut also satisfied the desired IGL production objectives for span and d50.
47 Figure 27 Hydrocyclone cumulative PSD with a d50c of 69μm
3.4 Potential IGL Production Methods Summary
The droplet method has great promise to produce a tailored narrow PSD, but was rejected due to scalability issues. Theoretical calculations were carried out for an inclined settler and hydrocyclone using a feed with an 80 μm log mean normal distribution. Two cuts were made with the inclined settlers, where performance was not affected by which way around cuts occurred (Figure 25 and Figure 26). Only one pass was made for the hydrocyclone feed (Figure 27), because attrition and yield problems are expected with more passes. A summary of the results can be seen in Table 5 below. Section 7.4.4.5 also shows that a hydrocyclone cut can still achieve a final span of less than 1 when the input span is higher (spans of 1.4-1.8).
Inclined Settler Hydrocyclone
Span = 0.57 Overall Yield = 20% Little to no attrition Dilution hassles
Relatively slow, but good separation results Best Option
Span = 0.66 Overall Yield = 17% Attrition problems expected Theoretically can handle concentrated
suspensions
Fast, but average separation results Table 5 Comparison of results for inclined settler and hydrocyclone
It was established in theory that inclined settlers and a hydrocyclone cut would work to obtain the desired IGL d50 and span, with the inclined settler chosen as the best option. However both these methods are external separation techniques with low yields. It is desired to obtain a narrow PSD directly from the crystallization process.
0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 350 Cummulativ e Par
ticle Size Distr
ibution Diameter (μm) Ff(D) Fu(D) Fo(D) Feed =(d90-d10)/d50 =(155.7-55.2)/92.9 =1.08 Overflow - Product =(d90-d10)/d50 =(74.6-38.1)/55.5 =0.66
48
3.5 Continuous Settling Crystallizer
Instead of externally processing a crystallization stream with an inclined settler, a crystallization step incorporating settling, termed the continuous settling crystallizer (CSC) is proposed for theoretical development and is discussed in detail in Chapter 6. The basis for the CSC involves feeding a pre-nucleated stream into the bottom of a vertical column which flows up to the top under laminar conditions. Growth of the nuclei and settling of large crystals occurs inside the column. Once the nuclei have grown to a large enough size, where the terminal particle settling velocity opposes the fluid velocity, they will settle out from the column. An inherently slow growing crystal will travel further up the column before growing to the terminal particle diameter compared to an inherently fast growing crystal. Different growth rate crystals settle out from the column at the same final particle size, counteracting growth rate dispersion.
3.6 Conclusions
Three potential methods for producing a narrow particle size distribution were studied in detail: droplet growth (crystallization), hydrocyclone (separation) and inclined settler (separation). Droplet growth counteracts growth rate dispersion by controlling the maximum size a crystal within the drop can reach. This process is ideal for tailoring a narrow particle size distribution; however issues occur with scalability and producing drops without contamination. A hydrocyclone separates fines from large particles using centrifugal force, while an inclined settler uses gravitational force. Both methods, in theory, are capable of processing a post- crystallized lactose stream to a span less than one, with the inclined settler being slightly more efficient. These methods were considered to be additional processing steps; therefore the novel method of combining a single nucleation event and incorporating laminar settling into a continuous crystallizer, termed the continuous settling crystallizer, was proposed for further development.
49
Chapter 4 – Lactose Stokes Shape Factor
Parts of this chapter have been published in a paper titled “Stokes Shape Factor for Lactose Crystals” (Shaffer, Paterson, Davies, & Hebbink, 2011).
4.1 Introduction
The previous chapter into potential new crystallizer designs for producing inhaler grade lactose (IGL) concluded that a crystallizer which incorporates settling would be a worthwhile development. Gravity settlers and classifiers used in the production of particles of a specific size range require knowledge of the particle settling parameters. Settling parameters can be readily predicted for spherical particles using established relationships for drag coefficients widely available in the literature and standard texts. Lactose crystals, however, are not spherical and while there are several approaches to estimating the settling velocity of a non- spherical particle, these all require information on particle shape. Generally, lactose crystals conform to a well-defined geometry, exhibiting a tomahawk shape (van Kreveld & Michaels, 1965; Walstra, Jenness, & Badings, 1984) as depicted in Figure 28. The two principal dimensions, as can be measured when the crystal is lying on a flat surface in its most stable position, are its height, B, and its width, A. But, despite their importance industrially, there is little definitive information on shape that can be used with confidence to predict their terminal settling velocity.
Figure 28 Diagram of tomahawk shaped alpha-lactose crystal; redrawn from van Kreveld and Michaels (1965)
Factors for characterizing particle shape are used in widely differing contexts, and are defined in many different ways. For example researchers (Bouwman, Bosma, Vonk, Wesselingh, &
50
Frijlink, 2004) were interested in quantifying the shape and roughness of granulated pharmaceutical powders and considered eight different shape factors. Their work concluded the combination of a roughness factor and a new projection shape factor gave a good indication of granule shape and roughness. Stokes shape factor has also been related to sphericity of non-spherical agricultural particles for flowability (Xie & Zhang, 2001).
In this work, interest resides in particle descriptors that enable the estimation of terminal settling velocity, ut (m.s-1), at very low particle Reynolds numbers, Rep (-), and attention has
been confined to the Stokes shape factor, kst (-).
For Rep <~1, Stokes law applies well, and the terminal velocity of a spherical particle is explicitly
related to particle density, ρs (kg.m-3), fluid density, ρf (kg.m-3), fluid viscosity, ߤ௩ (Pa.s), particle diameter, ݀ (m), and acceleration due to gravity, ݃ (m.s-2), by the expression given in Equation 45; see for example, Rhodes, (2008):
ݑ௧ ൌ݀ ଶ൫ߩ
௦െ ߩ൯݃ ͳͺߤ௩