... 1 4 c:
�
1 2 •�
8 � 6 4 2 1 00 300 500 700 Crystal Size (�m) 900 1 1 00Figure 2-26 Particle size measurements of a-l actose crystals before growth and a fter growth of fou rs hours at the supersatu rations of 5.23 and 1 0. 1 8 (g Ca-Cas g/l OOg water); expressed on
linea r scale
In Figure 2-26 it is not clear if growth rate dispersion has occurred. There appears to be a widening of the crystal size distribution, but this may be a consequence of the non-linear grouping of the crystal sizes. When classifying the crystals into their respective size fractions the Malvern MasterSizer 2000 uses a scale that increases at the rate of 2 1/6. It can be seen in Figure 2-27, that using this classi fication, a wider range of crystal diameters is covered at larger size fractions.
14 12 C 10
�
Q) 8 Cl. 6 4 2 1100Figu re 2-27 The c rystal size distribution of a-lactose monohydrate c rystals before growth viewed as a h istogram
To better determine if growth rate dispersion was occurring, the growth of 1 00,000 crystals with a particle size distribution matching the initial size distribution measured by the Malvern was modelled using different growth conditions. The particle population was developed in MA TLAB and a random normal distribution function was used to assign size values to each particle within the range of each bin size. The model was used to examine the scenarios of a constant single growth rate, and distribution of crystal growth rates set around a mean growth rate that was the same as the constant growth rate.
To study the dispersion of growth rates within the crystal population two distribution functions were considered. Firstly a log normal distribution was used; this represents the most commonly considered distribution function and is used by ( Mitrovic, Zekic,
& Petrusevski, 1 999). Secondly a Weibull Distribution was used; this distribution is
commonly applied in lifetime analysis, and the reasons for its use here are expanded below.
The Weibull distribution is named after Swedish physicist Waloddi Weibul l, who derived it in 1 93 7 and presented it in the paper, A Statistical Distribution Function of
Wide Applicability. It has found wide use in the area of quality and production control
for lifetime analysis ( Dodson, 1 994). The distribution is described by Equation 2- 1 3 . Where fJ > 0 and A> 1 are the shape and scale factors respectively.
f(x) = J.jJxfJ-le-}.xP
Equation 2-1 3 Weibull d istribution function
One advantage of the Weibull distribution is that allows for censonng ( Lawless, 2003). Censoring is a situation where part of a population has been removed due to its failure time being outside the range of interest. For example, items that have defects observed immediately after production are not considered as part of the population as they never entered the consumer market. It is easy to relate this to the situation where slow growing crystals are not considered due to their failure to reach the critical mesh Size.
The use of the Weibull distribution has had considerable discussion in the field of forest science studying tree growth (Fleming, 200 1 ), (Knowe, Ahrens, & DeBell, 1 997) and ( Hynynen 1., Burkhart, & Lee Allen, 1 998). The popularity of the Weibull function is due its shape factor, which manipulates the shape of the distribution as is shown in Figure 2-28. 1 .5 B = 0.5 1 .0
f(x)
0.5 (l.O o 2 J(t)
Fig u re 2-28 Weibull distribution showing shape factors 0.5, 1 .5 and 3.0 ( Lawless, 2003)
Figure 2-29 shows the best-fit predictions for the crystal size distribution after growth had occurred using the growth rate distribution models and a constant growth rate for all crystals. The best growth rate d istributions were determined by running a loop in MA TLAB to minimise the sum of the differences between the predicted and measured particle size distributions.
.... c 25 -Initial -Single Rate -GRDNrnl - -GRDWbl -Measured
e
� ��
o 200 400 600Crystal Size (urn)
800 1000 1 200
Figure 2-29 Prediction of crystal size distribution using a single common growth rate and the two growth rate dispersion models
Table 2-5 shows the best-fit parameters and the sum of their squares for each of the growth rate distributions. For the Normal distribution, the standard deviation was the variable manipulated to obtain the best fit. The best fit for the Weibull distribution was obtained by varying the shape factor. 0 use of the scale factor was made, as this would have shifted the mean growth rate. The growth rate is reported as a constant value specific to the supersaturation at which the crystal s were grown.
Conditions G rowth Rate Variable Sum Difr
rate 0 . 3 8 8 None 3 64 . 2
0 . 3 8 8 St.dev -0.43 1 7 . 72
Weibul l 0 . 3 8 8 F - 1 . 1 7 3 . 05
Table 2-5 Pa rameters used in predicting final crystal size d istribution
It is observed that without growth rate dispersion the C S D prediction is less accurate. This observation shows that growth rate dispersion is a necessary component in a
lactose crystallisation model, validating the work of Butler, ( 1 998) and Shi et al
( 1 989). The flexible shape of the Weibull model provides a more accurate prediction than the more rigid log nomlal distribution. This is demonstrated by the sum of differences squared value shown in Table 2-5 being the smallest for the Weibull model.
The Weibull shape factor of 1 . 1 7 is also of interest. I n Figure 2-28 a shape factor of 1 .5 shows a distribution that lacks a fines tail, indicating some censoring has occurred, this compares to a shape factor of three where a normal distribution appears. The shape factor of 1 . 1 7 would indicate that some censoring of slow growers has occurred because of the sieving of the crystals. This assumes that the crystal growth rate under standard conditions is log-normally distributed. The work by Butler, ( 1 998) demonstrates this to be the case. The effect censoring of crystals of different sizes has on the shape of the growth rate distribution is an area where further work may be o f interest.
2.8 Conclusion
This section of work has provided a review of the literature on lactose crystal growth in a water solution. Carried out in conjunction with the review have been a number of experiments that have sought to validate the results reported by the literature. It has been found that when the work carried in the literature is viewed using common units the results are consistent. The results from this study are in agreement with the literature results.
One difference with that recently reported in the literature is the effect of temperature. In this work and in the maj ority of historical studies when they were viewed in terms of absolute a-lactose supersaturation, apart from its affect on solubility, temperature has no significantly observable effect on the rate at which lactose crystals grow. This differs from the most recently reported equation for lactose crystal growth, Butler, ( 1 998), where temperature was included as a variable. A comparison between the equation developed in this work and the equation previously reported found that at low supersaturations there was l ittle difference between the results predicted. However, at higher supersaturations and temperatures, beyond the limits of Butler's equation, the difference becomes significant. The equation developed in this work, Equation 2- 1 1 , will be used to model lactose crystal growth in the remainder of this work.
Growth rate dispersion was i nvestigated in this work and it was found to be a necessary factor in predicting the final crystal size distribution of a population of
c rystals. In exammmg the effect of growth rate dispersion two distributions were considered, the first was a log normal distribution and the second a Weibull distribution. The best predi ction of the measured fi nal distribution was obtained using the more flexible Weibull distribution. This distribution accounted for the censoring caused by sieving out a particular size fraction of crystals.
The results from this chapter will be used as a basis against which lactose crystal growth in permeate concentrates can be compared. This is the focus of the next chapter.