Abstract
This work presents the fundamentals and exemplary applications of a generalized model for precipitation, aggregation and ripening processes including the formation of solid phases with two dimensions. The particle formation is governed by a widely applicable population balance approach. Solid formation processes are described via the numerically efficient Direct Quadrature Method of Moments (DQMOM), which can calculate the evolution of multiple solid phases simultaneously. The particle size distribution (PSD) is approximated by a summation of delta functions while the moment source term is approximated by a two-point quadrature. The moments to calculate the multivariate distributions are chosen carefully to represent the second order moments. Solid formation is based on the model of Haderlein et al. (2017) and is extended by a multidimensional aggregation model. Now, the influences of mixing, complex hydrochemistry and particle formation dynamics including nucleation, growth and aggregation on multiphase precipitation processes are modelled and simulated along independent dimensions with high efficiency.
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Abbreviations
- A:
-
Fraction of zone in mixing model [–]
- A′:
-
Fraction of zone in mixing model [–]
- B:
-
Fraction of zone in mixing model [–]
- B′:
-
Fraction of zone in mixing model [–]
- BHom:
-
Homogeneous nucleation rate [1/(m3*s)]
- BSec:
-
Secondary nucleation rate [1/(m3*s)]
- BHet:
-
Heterogeneous nucleation rate [1/(m3*s)]
- ci:
-
Concentration of species i [mol/l]
- c∞L:
-
Bulk concentration [mol/l]
- Dp:
-
Diffusion coefficient of phase p [m2/s]
- E:
-
Engulfment factor [1/s]
- f:
-
Particle size distribution density [1/m]
- fA,i:
-
Shape factor of phase i [–]
- g:
-
Residual describing the complex equilibria [–]
- Gi:
-
Diffusion limited growth rate in direction i [m/s]
- HA:
-
Hamaker constant [J]
- He:
-
Adsorption constant [–]
- hi,m:
-
Net particle formation rate of solid phase i in zone m [1/(m*s)]
- J:
-
Jacobian of the residual describing the complex equilibra [–]
- I:
-
Ion activity product of all solid phases [molx/lx]
- k:
-
Order of moment [–]
- kB:
-
Boltzmann constant [J/K]
- l:
-
Oder of mixed moment [–]
- ki:
-
Equilibrium constant of reaction i [molx/lx]
- KSP,P:
-
Solubility product of phase p [molx/lx]
- M:
-
Mixing matrix [–]
- Mi:
-
Molar mass of species i [kg/mol]
- N:
-
Total number of nodes for DQMOM [–]
- Ns:
-
Size of the property vector for node positions [–]
- ni,m:
-
Particle number density of phase i in zone m [1/(m*s)]
- n0:
-
Particle number at the beginning of the simulation [–]
- oj:
-
Logarithmic concentration of species j [–]
- R:
-
Ripening rate [m/s]
- Rs:
-
Reaction stoichiometry matrix [–]
- Ri:
-
Radius of interacting particle [m]
- Sp:
-
Supersaturation of solid phase p [–]
- Sξ:
-
Source term for moment transformation [1/(m*s)]
- Sh:
-
Sherwood number [–]
- T:
-
Temperature [K]
- U:
-
Molar balance stoichiometry matrix [–]
- ui:
-
Mean velocity of internal variable [m/s]
- VM:
-
Molecular volume [m3/mol]
- ΔVGrowth:
-
Volume growth during time step Δt [m3]
- wi:
-
Weighting factor for the Nelder-Mead optimization [–]
- Wij:
-
Fuchs stability ratio [–]
- Wtot:
-
Total interaction potential [J]
- x:
-
Particle diameter [m]
- xcrit:
-
Critical particle diameter [m]
- Y:
-
Normalized center-to-center distance of interacting particles [–]
- zi:
-
Ion charge of species i [–]
- β:
-
Sum of all sources Si [(m/s)k+l]
- βBrown:
-
Brownian aggregation kernel [m3/s]
- βTurb:
-
Turbulent aggregation kernel [m3/s]
- γ:
-
Interfacial energy [N/m]
- γi:
-
Activity coefficient of species i [–]
- ε:
-
Specific power input [W/kg]
- ε0:
-
Vacuum permittivity [A*s/(V*m)]
- εr:
-
Relative permittivity [–]
- ζ:
-
Target functional of the Nelder-Mead optimization [–]
- η:
-
Dynamic viscosity [Pa*s]
- θ:
-
Contact angle for heterogeneous nucleation [–]
- Θ:
-
Heaviside function [–]
- κ:
-
Debye length [1/m]
- µij:
-
Stoichiometric coefficient of component i in species j [–]
- ν:
-
Kinematic viscosity [m2/s]
- νij:
-
Stoichiometric coefficient of species j in reaction i [–]
- ξN,α:
-
Node position of property N of node α [m]
- π:
-
Archimedes constant [–]
- ρ:
-
Solid density [kg/m3]
- σ:
-
Standard deviation of particle size distribution [m]
- σi,p:
-
Stoichiometric coefficient of species i in solid phase p [–]
- Φ:
-
Solution of implicit equation [–]
- Ψi:
-
Surface potential of face i [C/m2]
- ωN,α:
-
Node weight of property N of node α [–]
References
Segets, D., Hartig, M.A.J., Gradl, J., Peukert, W.: A population balance model of quantum dot formation: oriented growth and ripening of ZnO. Chem. Eng. Sci. 70, 4–13 (2012)
Voigt, M., Klaumünzer, M., Thiem, H., Peukert, W.: Detailed analysis of the growth kinetics of ZnO nanorods in methanol. J. Phys. Chem. C 114, 6243–6249 (2010)
Encina, E.R., Distaso, M., Klupp Taylor, R.N., Peukert, W.: Synthesis of goethite α-FeOOH particles by air oxidation of ferrous hydroxide Fe(OH) 2 suspensions: insight on the formation mechanism. Crystal Growth Des. 15, 194–203 (2015)
Haderlein, M., Güldenpfennig, A., Segets, D., Peukert, W.: A widely applicable tool for modeling precipitation processes. Comput. Chem. Eng. 98, 197–208 (2017)
Hartig, M.A.J., Jacobsen, N., Peukert, W.: Multi-component and multi-phase population balance model: the case of georgeite formation as methanol catalyst precursor phase. Chem. Eng. Sci. 109, 158–170 (2014)
Gradl, J., Peukert, W.: Simultaneous 3D observation of different kinetic subprocesses for precipitation in a T-mixer. Chem. Eng. Sci. 64, 709–720 (2009)
Schikarski, T., Trzenschiok, H., Avila, M., Peukert, W.: Influence of mixing on the precipitation of organic nanoparticles: a lagrangian perspective on scale-up based on self-similar distributions. Chem. Eng. Technol. 23, 1635–1642 (2019)
Rollié, S., Briesen, H., Sundmacher, K.: Discrete bivariate population balance modelling of heteroaggregation processes. J. Colloid Interface Sci. 336, 551–564 (2009)
Haderlein, M., Segets, D., Gröschel, M., Pflug, L., Leugering, G., Peukert, W.: FIMOR: an efficient simulation for ZnO quantum dot ripening applied to the optimization of nanoparticle synthesis. Chem. Eng. J. 260, 706–715 (2015)
Skorych, V., Dosta, M., Hartge, E.-U., Heinrich, S.: Novel system for dynamic flowsheet simulation of solids processes. Powder Technol. 314, 665–679 (2017)
Bogacki, P., Shampine, L.F.: A 3(2) pair of Runge—Kutta formulas. Appl. Math. Lett. 2, 321–325 (1989)
Marchisio, D.L., Fox, R.O.: Solution of population balance equations using the direct quadrature method of moments. J. Aerosol Sci. 36, 43–73 (2005)
Bourne, J.R.: Mixing and the selectivity of chemical reactions. Org. Process Res. Dev. 7, 471–508 (2003)
Schwarzer, H.-C., Peukert, W.: Combined experimental/numerical study on the precipitation of nanoparticles. AIChE J. 50, 3234–3247 (2004)
Gradl, J., Schwarzer, H.-C., Schwertfirm, F., Manhart, M., Peukert, W.: Precipitation of nanoparticles in a T-mixer: coupling the particle population dynamics with hydrodynamics through direct numerical simulation. Chem. Eng. Process. 45, 908–916 (2006)
Baldyga, J., Bourne, J.R.: Simplification of micromixing calculations. I. Derivation and application of new model. Chem. Eng. J. 42, 83–92 (1989)
Bałdyga, J., Bourne, J.R.: Turbulent Mixing and Chemical Reactions. Wiley, Chichester (1999)
Schwarzer, H.-C.: Nanoparticle Precipitation: An Experimental and Numerical Investigation Including Mixing. Logos-Verl, Berlin (2005)
Guichardon, P., Falk, L., Villermaux, J.: Characterisation of micromixing efficiency by the iodide–iodate reaction system. Part II: kinetic study. Chem. Eng. Sci. 55, 4245–4253 (2000)
Commenge, J.-M., Falk, L.: Villermaux-Dushman protocol for experimental characterization of micromixers. Chem. Eng. Process. 50, 979–990 (2011)
Davies, C.W.: 397. The extent of dissociation of salts in water. Part VIII. An equation for the mean ionic activity coefficient of an electrolyte in water, and a revision of the dissociation constants of some sulphates. J. Chem. Soc. 2093–2098 (1938)
Lifshitz, I.M., Slyozov, V.V.: The kinetics of precipitation from supersaturated solid solutions. J. Phys. Chem. Solids 19, 35–50 (1961)
Zur, Fuchs N., der Koagulation, Theorie: Z. Phys. Chem. 171A, 199–208 (1934)
Monnin, C.: A thermodynamic model for the solubility of barite and celestite in electrolyte solutions and seawater to 200 °C and to 1 kbar. Chem. Geol. 153, 187–209 (1999)
Powell, K.J., Brown, P.L., Byrne, R.H., Gajda, T., Hefter, G., Sjöberg, S., Wanner, H.: Chemical speciation of environmentally significant metals with inorganic ligands Part 2: The Cu2+, OH−Cl−, CO32−, SO42−, and PO43− systems (IUPAC Technical Report). Pure Appl. Chem. 79, 95–950 (2007)
Reichle, R.A., McCurdy, K.G., Hepler, L.G.: Zinc hydroxide: solubility product and hydroxy-complex stability constants from 12.5–75 ℃. Can. J. Chem. 53, 841–3845 (1975)
Martell, E., Smith, R.M.: NIST Standard Reference Database 46. NIST Critically Selected Stability Constants of Metal Complexes: Version 8.0 (2016)
Alwan, A.K., Thomas, J.H., Williams, P.A.: Mineral formation from aqueous solution. Part III. The stability of aurichalcite, (Zn, Cu)5(CO3)2(OH)6, and rosasite (Cu, Zn)2(CO3)(OH)2. Transition Met. Chem. 5, 3–5 (1980)
Leussing, D.L., Kolthoff, I.M.: The solubility product of ferrous hydroxide and the ionization of the aquo-ferrous ion. J. Am. Chem. Soc. 75, 2476–2479 (1953)
Güldenpfennig, A., Distaso, M., Klupp Taylor, R.N., Peukert, W.: Modelling the two-dimensional growth and oriented attachment of goethite nanorods synthesized via oxidation of aqueous ferrous hydroxide slurries. Chem. Eng. J. 347, 798–807 (2018)
Russell, B., Payne, M., Ciacchi, L.C.: Density functional theory study of Fe(II) adsorption and oxidation on goethite surfaces. Phys. Rev. B 79, 165101–165114 (2009)
Rustad, J.R., Felmy, A.R., Hay, B.P.: Molecular statics calculations of proton binding to goethite surfaces: a new approach to estimation of stability constants for multisite surface complexation models. Geochim. Cosmochim. Acta 60, 1563–1576 (1996)
Israelachvili, J.N.: Intermolecular and surface forces. Academic Press, MA, Burlington (2011)
Güldenpfennig, A., Pflug, L., Peukert, W.: How to estimate material parameters for multiphase, multicomponent precipitation modeling. Cryst. Growth Des. 19, 2785–2793 (2019)
Akdas, T., Distaso, M., Kuhri, S., Winter, B., Birajdar, B., Spiecker, E., Guldi, D.M., Peukert, W.: The effects of post-processing on the surface and the optical properties of copper indium sulfide quantum dots. J. Colloid Interface Sci. 445, 337–347 (2015)
Süß, S., Michaud, V., Amsharov, K., Akhmetov, V., Kaspereit, M., Damm, C., Peukert, W.: Quantitative evaluation of fullerene separation by liquid chromatography. J. Phys. Chem. C 123, 16747–16756 (2019)
Süß, S., Metzger, C., Damm, C., Segets, D., Peukert, W.: Quantitative evaluation of nanoparticle classification by size-exclusion chromatography. Powder Technol. 339, 264–272 (2018)
Khomane, A.S., Hankare, P.P.: Structural, optical and electrical characterization of chemically deposited CdSe thin films. J. Alloys Compd. 489, 605–608 (2010)
Salaheldin, A.M., Walter, J., Herre, P., Levchuk, I., Jabbari, Y., Kolle, J.M., Brabec, C.J., Peukert, W., Segets, D.: Automated synthesis of quantum dot nanocrystals by hot injection: mixing induced self-focusing. Chem. Eng. J. 320, 232–243 (2017)
Acknowledgements
The authors gratefully acknowledge financial support of Deutsche Forschungsgemeinschaft (DFG) in the scope of SPP 1679 (Dynamic Simulation of Interconnected Solids Processes) coordinated by Prof. S. Heinrich.
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Michaud, M., Haderlein, M., Segets, D., Peukert, W. (2020). Flowsheet Simulation of Integrated Precipitation Processes. In: Heinrich, S. (eds) Dynamic Flowsheet Simulation of Solids Processes. Springer, Cham. https://doi.org/10.1007/978-3-030-45168-4_8
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