eMoM: Exact method of moments—Nucleation and size dependent growth of nanoparticles

doi:10.1016/j.compchemeng.2020.106775

Computers & Chemical Engineering

Volume 136, 8 May 2020, 106775

https://doi.org/10.1016/j.compchemeng.2020.106775 Get rights and content

Abstract

In this study we present a reformulation for a broad class of population balance equations that model nucleation and size dependent growth. This formulation enables the definition of new numerical methods, which have two advantages compared to existing schemes in the literature (e.g. finite volume type methods and methods based on the evolution of moments): i) higher precision due to non-smoothing and ii) less run-time in comparison to finite volume algorithms of equivalent accuracy. The described formulation represents the solution of the considered balance equation in terms of the solution of a scalar integral equation, which can then be exploited to establish efficient numerical solvers. The computation of a solution of an initial boundary value problem in space-time is thus reduced to a scalar integral equation, which considerably reduces the computational effort required for its approximation. The integral equation describes the exact evolution of the third moment of the solution, justifying the name for this method: the exact method of moments (eMoM).

Section snippets

Introduction and problem definition

Crystallization from solution (Mullin, 2001, Dirksen, Ring, 1991, Mersmann, 2001), including rapid precipitation (Thanh, Maclean, Mahiddine, 2014, Dirksen, Ring, 1991, Thorat, Dalvi, 2012), is widely applied in industry. The two primary processes of crystallization involve the formation of new nuclei from solutions and their subsequent growth, both of which are driven by the thermodynamic supersaturation build-up. These processes are best described by population balance equations (PBE), where

Derivation of eMoM for a wide range of growth kinetics

In the following, we will present and validate a representation of the solution to Eqs. (1)–(3) in terms of the solution of a scalar fixed-point equation. Definition 2.1 provides the main ingredient for the explicit solution formula. Despite the fact that this definition is independent of the spatial coordinate, it contains all the necessary information to providing the explicit solution. This means that only this scalar fixed-point equation must be solved to obtain the desired solution, in

Numerical approximation in the case of diffusion-limited growth with zero initial datum

To numerically approximate the solution q, we have to approximate the solution of the fixed-point equation in Eq. (7). Therefore, we present a scheme for the case of diffusion limited growth, i.e. $G_{1} (x) = x^{- 1} \forall x \in R_{\geq x_{n}},$ as introduced in the first row of Table 1, in the case of vanishing initial datum (q₀ ≡ 0). Similar schemes can be derived for the growth rates presented in Table 1 and of course for seeded growth ( $q_{0} ≢ 0$ ). The underlying idea is to differentiate w with respect to time and then use an

Comparison with classical discretization schemes

To compare eMoM to the frequently used methods of moments (MoM, see e.g. Schwarzer et al., 2006), and finite-volume type schemes (FVM, see e.g. Qamar, Elsner, Angelov, Warnecke, Seidel-Morgenstern, 2006, Gunawan, Fusman, Braatz, 2004), we will use the number of degrees of freedom (DoF) to compare the simulations. In FVM-type schemes, the DoF are denoted by the product of the discretization w.r.t. the disperse property as well as the process time, i.e. ${DoF}_{eMoM} = N_{t} N_{x}, T_{FVM}^{sim} \in O (N_{t} N_{x}),$ where $N_{t}, N_{x} \in N$

Conclusion and outlook

Investigating the structure of the PBE, modelling nucleation and size-dependent growth of nanoparticles led to a reformulation which can be exploited numerically. The numerical algorithms obtained from the solution formula are superior to the well-known and widely used algorithms in the literature. Not only does the algorithm presented here perform faster with higher precision, it is also able to capture the principal dynamic effect even at low concentration. As demonstrated in Fig. 4 this is

CRediT authorship contribution statement

Lukas Pflug: Supervision, Conceptualization, Methodology, Formal analysis, Writing - original draft. Tobias Schikarski: Investigation, Validation, Writing - original draft. Alexander Keimer: Formal analysis, Writing - original draft. Wolfgang Peukert: Writing - original draft. Michael Stingl: Writing - original draft.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for their financial support within the priority programs SPP 1679 (PE427/25 and LE595/30) and for support within the Cluster of Excellence “Engineering of Advanced Materials” at the University of Erlangen-Nürnberg (FAU). We also thank the ”Bavaria California Technology Center” (BaCaTeC) for travel funding. The research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research

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eMoM: Exact method of moments—Nucleation and size dependent growth of nanoparticles

Abstract

Section snippets

Introduction and problem definition

Derivation of eMoM for a wide range of growth kinetics

Numerical approximation in the case of diffusion-limited growth with zero initial datum

Comparison with classical discretization schemes

Conclusion and outlook

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

Chem. Eng. Sci.

Chem. Eng. Sci.

J. Differ. Equ.

J. Math. Anal. Appl.

J. Aerosol Sci.

J. Cryst. Growth

Comput. Chem. Eng.

Chem. Eng. Sci.

Chem. Eng. Sci.

Chem. Eng. Sci.

Chem. Eng. J.

Particuology

High resolution algorithms for multidimensional population balance equations

AlChE J.