In the following we introduce frequently used Notations for symbols and functions and commonly used Acronyms.
Acronyms
API Active Pharmaceutical Ingredient ASA Acetylsalicylic Acid (Aspirin) CSD Crystal Size Distribution EtOH Ethanol
FOM Full Order Model MW Molecular Weight
ODE Ordinary Differential Equation PBE Population Balance Equation PDE Partial Differential Equation
PIDE Partial Integro Differential Equation POD Proper Orthogonal Decomposition PSD Particle Size Distribution
RCPE Research Center Pharmaceutical Engineering ROM Reduced Order Model
SVD Singular Value Decomposition
Notations
Bagg(ξ, t) Term for birth due to aggregation.
c(t) Function for the APIs concentration in the solvent.
c∗(t) Function denoting the time dependent solubility.
Dagg(ξ, t) Term for death due to aggregation.
∆T The linear cooling rate.
∆t The step size for the temporal discretization.
∆x The step size for the spatial discretization.
G(t, µ) The growth function.
kv The volume shape factor for crystals.
µ The parameter tuple appearing inG(t, µ) withµ= (k1, k2, k3).
mdis Denotes the dissolved mass.
merr Denotes simulations mass error.
msol Denotes the solid mass.
mtotal Denotes total particle mass that was initially put into the system.
m Denotes generally mass.
ρ The density of particles in our model.
S(t) Function denoting the supersaturation.
T Temperature, if not appearing in a clearly other context.
t The time variable.
Vtank The tank volume.
Xc(t) Function for the solubility of ASA in EtOH.
ξ In our class model the particle length.
Population Balance Equations
In this chapter, population balance equations are introduced and the test reactor framework of a particulate model for crystallization processes is presented. Thereby, the dynamic model of this work based on the implementation from our collaborators at the Research Center Pharmaceutical Engineering (Graz, Austria) is pointed out in detail and its numerical realization is outlined.
Conclusive, numerical results for the full order population balance model are presented.
2.1 Introduction
Population balance equations (PBE) are defined as a class of PDEs, which are commonly used in chemistry, biology or engineering to describe the temporal behaviour of discrete physical entities – such as crystallization processes or cell population models, for example.
The model usually consists of partial differential equations, specifically of partial integro differential equations (PIDE). These PIDEs contain non-local operators, which is why numerical solutions are of high interest from a practical point of view.
A standard work dealing with PBEs is [34], where “various aspects of the methodology of pop-ulation balance necessary for its successful application”1 are presented. In that book, different multidimensional PBE frameworks are modelled and mathematical solution methods are explored – thus, [34] is highly recommended from a mathematical point of view. An extensive overview over the simulation and modelling of different types of PBEs can be found additionally in [29].
In praxis, there is a wide range of different models, depending on which process is modelled and how the framework is set up. To begin with, we have a look at a very general formulation of a one-dimensional PBE modelling crystallization processes and describe the contained terms as well as their physical meaning. The model presented in this work describes temporal evolution of the particle size distribution (PSD) for a seeded crystallization process in a well mixed reactor. This solution is cooled or heated over a certain period of time and therefore particles undergo growth, nucleation or aggregation processes.
A general formulation of that kind of population balance transport equation, as similarly formulated
1[34, p. 4]
for the first time in the publications of [15] or [36], can be stated as
∂f(ξ, t)
∂t +∂ G(ξ, t, µ)f(ξ, t)
∂ξ =Z(ξ, t) (2.1.1)
with the non-homogeneous part
Z(ξ, t) =Bnuc(ξ, t) +Bagg(ξ, t) +Bbreak(ξ, t)−Dagg(ξ, t)−Dbreak(ξ, t).
We call the functionf(ξ, t) “number density function”, it describes in our case the particle size distribution, or rather the number of particles with the internal coordinateξat the external time coordinatet. The variable ξrepresents the particle property, for instance, particle mass, length or volume. In praxis, multidimensional equations combining various properties are considered, see [12, 21]. Yet, this work will restrict to the one dimensional case withξrepresenting the characteristic particle size of lengthξ.
The second term on the left hand side of (2.1.1) contains particle growth, including the growth functionG(ξ, t, µ)>0, whereµ∈R3 is a parameter tuple determining the growth function. The parameter dependence of the growth functionG(ξ, t, µ) is emphasized as the parameterµis crucial for modelling the process and therefore, in focus of parameter estimations. Examples for parameter estimations in a similar context of particulate processes can be found, for example, in [13] or [32].
Remark 2.1.1. Throughout this work, we assume that Gis not dependent on the particle size, so it holds that
G(t, µ) =G(ξ, t, µ).
The right hand side of equation (2.1.1) describes birth and death processes the particles undergo.
The term Z(ξ, t) in (2.1) contains in the first to third place functions representing the birth of particles due to nucleation, aggregation and breakage. The last two terms stand for the death of particles due to breakage and aggregation. Figure 2.1 illustrates each particle formation process: the aggregation of two particles, the breakage of one particle, the growth of a particle or the nucleation of a particle through (usually mostly) non-particle matter.
This work is focused on simultaneous growth and aggregation processes, as similarly done in [2].
Thus, we have either a homogeneous equation withZ(ξ, t) = 0 taking only growth processes into account, or we have a non-homogeneous equation withZ(ξ, t) =Bagg(ξ, t)−Dagg(ξ, t) combining growth and aggregation.
Equation (2.1.1) requires initial and boundary conditions, which are generally stated as
f(ξ, t0) =f0(ξ) (2.1.2a)
f(0, t) = 0. (2.1.2b)
The latter condition, the boundary condition, means that there are no particles of zero size, which is physically reasonable. The initial conditionf0 could be, for instance, a log-normal distribution and is simply the initial particle distribution at the beginning of the experiment.
Figure 2.1: The four basic particle formation processes.