British Association For Crystal Growth Annual Conference 2017
Morphological population balance equation modeling for milling
processes
1
1
Fabio Salvatori, Marco Mazzotti,
1
Separation Processes Laboratory, Institute of Process Engineering, ETH Zurich, CH 8092-Zurich,
Switzerland
[email protected]
In the pharmaceutical and food industry, many
products are commonly sold in powder form. The
properties of these powders, such as their biovailability
and processability, are strongly dependent on the size
and shape of the single crystals. The control of crystal
morphology during crystallization is a research topic of
great interest and several are the techniques
suggested to manipulate crystal habit. Lovette et al. [1]
advise the use of temperature cycles to this aim.
Another alternative involves the use of different solutesolvent combination or the use of specific additives
that can selectively adsorb on specific crystal facets
[2]. However, in the current industrial practice, shape
manipulation is intended as part of the downstream
processes. Here, comminution is the step deputed to
the production of crystals of more equant shape.
Despite the wide implementation of this milling step in
the formulation processes and the wide variety of Figure 1: Schematization of a milling process. On the
machineries developed to efficiently perform milling, left, the particle size and shape distribution of the
only a small amount of models, capable of predicting seeds at the beginning of the process. On the right,
the resulting properties of the products, is available. the simulated (red) and measured (blue) distribution
The number further reduces when considering of the products after comminution.
mathematical models taking into account the morphology of the particles.
In this work, we aim at developing a model, based on morphological population balance equations(MPBE),
capable of tracking the evolution of the size and shape during a continuous milling process of particles
characterized by a needle-like morphology. The model chosen for the single crystals is the generic particle
model as presented by Schorsch et al. [3] and shown in Figure 2. Under the assumption of constant
supersaturation, no temperature variations, perfectly segregated flow, and no growth and nucleation
phenomena, the MPBE reads as follows:
𝜕
𝑛(𝐿1 , 𝐿2 , 𝜏) = 𝐷 + 𝐵
𝜕𝜏
In this equation, which represents the basis of our model, 𝜏 is the residence time in the grinding chamber of
the suspension characterized by the particle size and shape distribution 𝑛(𝐿1 , 𝐿2 , 𝜏), while 𝐷 and 𝐵 represent
the death and birth terms due to breakage. Assuming rupture can only occur along planes perpendicular to
Figure 2: Model for the single crystal and the ensemble of particles. On the left, a
sketch of a real crystal with the different facets highlighted in different colors. In the
middle, the generic particle model approximation. On the right, an exemplary particle
size and shape distribution.
British Association For Crystal Growth Annual Conference 2017
the characteristic dimensions of the crystals, The equation for the death term can be written as follows:
𝐷 = −(𝐾1 (𝐿1 , 𝐿2 , 𝒛) + 𝐾2 (𝐿1 , 𝐿2 , 𝒛))𝑛(𝐿1 , 𝐿2 , 𝜏)
𝐾1 (𝐿1 , 𝐿2 , 𝒛)and 𝐾2 (𝐿1 , 𝐿2 , 𝒛) are the breakage frequencies, describing how fast a crystal is breaking. These
functions depend on properties characteristic of the particle itself, such as its sizes, as well as the operating
conditions 𝒛. The equation for the birth term can be written as follows:
∞
∞
𝐵 = ∫ 𝐾1 (𝑥, 𝐿2 , 𝒛)𝑛(𝑥, 𝐿2 , 𝜏)𝑔1 (𝐿1 , 𝑥)d𝑥 − ∫ 𝐾2 (𝐿1 , 𝑦, 𝒛)𝑛(𝐿1 , 𝑦, 𝜏)𝑔2 (𝐿2 , 𝑦)d𝑦
𝐿1
𝐿2
Here, 𝑔1 (𝐿1 , 𝑥) and 𝑔2 (𝐿2 , 𝑦) are the daughter distributions, functions used to model the breakage
mechanism (fracture, abrasion, …) and that accordingly model the distribution of the fragments formed.
Once the framework of the morphological population balance equation has been developed, equations for
the daughter distributions and the breakage frequencies are proposed, taking into account both the
mechanics of fracture and the physical properties of the ground crystals.
Figure 3: Scheme of the experimental rig used for the comminution experiments. The use of two
crystallizers and of a thermostat connected to the milling device ensures precise control of the suspension
temperature.
To verify the suitability for the model to accurately describe comminution processes and to fit the kinetic
parameters, a dedicated experimental campaign has been performed on the laboratory setup sketched in
Figure 3. Two different systems are analyzed, namely β L-Glutamic acid in water and γ D-Mannitol in
propan-2-ol. Suspensions are prepared at 25 °C in the first crystallizer and subsequently milled and collected
in the second vessel. Both the initial and final particle size and shape distributions are measured with the
flow-through cell, an in-house developed device [3]. Different experimental conditions, such as suspension
density and milling intensity, are investigated to get a wide pool of experimental data for parameter fitting. An
exemplary comparison between the experimental and the fitted outcome is illustrated in Figure 1.
References:
[1] M. A. Lovette, M. Muratore, M. F. Doherty, AIChE Journal, 2012, 58, 1465.
[2] M. Salvalaglio, T. Vetter, M. Mazzotti, M. Parrinello, Angewandte Chemie, 2013, 52, 13369.
[3] S. Schorsch, D. R. Ochsenbein, T. Vetter, M. Morari, M. Mazzotti, Chemical Engineering Science, 2014, 105, 155.