Population Balance Modelling of Zirconia Nanoparticles in Supercritical Water Hydrothermal Synthesis

Document Type : Research Article


1 Department of Chemical Engineering, University of Kashan, Kashan, I.R. IRAN

2 Material and Nuclear Fuel Research School, Nuclear Science and Technology Research Institute, Tehran, I.R. IRAN


Like any other precipitation process, in supercritical water hydrothermal synthesis (SWHS), the need to improve product quality and minimize production cost requires understanding and optimization of Particle Size Distribution (PSD). In this work, using Population Balance Equation (PBE) containing nucleation and growth terms, the reactive precipitation of zirconia nanoparticles prepared by SWHS in the batch reactor was modeled. An optimization method using genetic algorithm function in MATLAB environment was developed to find simultaneously the kinetic parameters of nucleation and crystal growth rates, used for predicting PSD in PBE. The methodology developed evaluated kinetic parameters at comparable order of magnitudes to those presented in the literature, indicating a reasonable validation of the modeling method adopted. PSD results, however, showed a weak convergence of experimental and those predicted, suggesting that here, aggregation most likely played a considerable role in the PBE modeling of SWHS preparation of the zirconia nanoparticles.


Main Subjects

[1] Becker J., Hald P., Bremholm M., Pedersen J.S., Chevallier J., Iversen S.B., Iversen B.B, Critical Size of Crystalline ZrO2 Nanoparticles Synthesized in Near- and Supercritical Water and Supercritical Isopropyl Alcohol, ACS Nano, 2(5): 1058-1068 (2008).
[2] Hakuta Y., Ohashi T., Hayashi H., Arai K., Hydrothermal Synthesis of Zirconia Nanocrystals in Supercritical Water, J. Mater. Res. 19(8): 2230-2234 (2004).
[3] Ning G.-h., Zhao X.-p., Li J., Structure and optical Properties of MgxZn1−xO Nanoparticles Prepared  by Sol–Gel Method, Opt. Mater. 27: 1-5 (2004).
[4] Park S.B., Kang Y.C., Photocatalytic Activity of Nanometer Size ZnO Particles Prepared by Spray Pyrolysis, J. Aerosol Sci. 28: S473-S474 (1997).
[5] Yang Y., Chen H., Zhao B., Bao X., Size Control of ZnO Nanoparticles via Thermal Decomposition of Zinc Acetate Coated on Organic Additives, J. Cryst. Growth, 263: 447-453 (2004).
[6] Ismail A.A., El-Midany A., Abdel-Aal E.A., El-Shall H., Application of Statistical Design to Optimize the Preparation of ZnO Nanoparticles via Hydrothermal Technique, Mater. Lett. 59: 1924-1928 (2005).
[7] Sue K., Suzuki M., Arai K., Ohashi T., Ura H., Matsui K., Hakuta Y., Hayashi H., Watanabe M., Hiaki T., Size-Controlled Synthesis of Metal Oxide Nanoparticles with a Flow-Through Supercritical Water Method, Green Chem. 8:634-638 (2006).
[8] Zhou L., Wang Sh., Xu D., Guo Y, Impact of Mixing for the Production of CuO Nanoparticles in Supercritical Hydrothermal Synthesis, Ind. Eng. Chem. Res. 53: 481-493 (2014).
[9] Pirkhedri S., Anbia M., Rahimi R., Bandarchian F., Hydrothermal Synthesis of Flower-Like α-Quartz Nanostructures from Iran Kaolin, Iran. J. Chem. Chem. Eng. (IJCCE), 35 (3): 21-25 (2016).
[10] Akhoondi A., Ziarati, M., Khandan N., Hydrothermal Production of Highly Pure Nano Pyrite in a Stirred Reactor, Iran. J. Chem. Chem. Eng. (IJCCE), 33 (1): 15-19 (2014).
[12] Randolph A.D, Larson M.A. “Theory of Particulate Processes”, Academic Press, New York, (1988).
[13] Miller, S.M., Witkowski, W.R., Model Identification and Control of Solution Crystallization Processes, Ind Eng Chem Res., 32: 1275-1296 (1993).
[15] Masoodiyeh F., Karimi-Sabet J., Khanchi A.R, Mozdianfard M.R, Zirconia Nanoparticle Synthesis in Sub and Supercritical Water — Particle Morphology and Chemical Equilibria, Powder Technol., 269: 461-469 (2015).
[16] Palmer D.A., Fernandez-Prini R., Harvey A.H, “Aqueous systems at elevated temperatures and pressures, Physical Chemistry in Water, Steam and Hydrothermal Solutions”, A Project of The International Association for the Properties of Water and Steam, Elsevier, (2004).
[17] Il Lim Y., Le Lann J.M., Meyer X.M., Joulia X., Lee G., Yoon E.S., On the Solution of Population Balance Equations (PBE) with Accurate Front Tracking Methods in Practical Crystallization Processes, Chem. Eng. Sci. 57: 3715-3732 (2002).
[18] Testino A., Buscaglia V., Buscaglia M.T., Viviani M., Nanni P., Kinetic Modeling of Aqueous and Hydrothermal Synthesis of Barium Titanate (BaTiO3), Chem. Mater. 17: 5346-5356 (2005).
[19] Mullin J.W. “Crystallisation”, 4th ed., Butterworth Heinemann, Oxford, (2002).
[20] Nielsen A.E. “Kinetics of Precipitation”, Pergamon, Oxford, UK, (1964).
[21] Kumar S., Ramkrishna D, On the Solution of the Population balance Equations by Discretization - I. A Fixed Pivot Technique, Chem. Eng. Sci. 51: 1311-1332 (1996).
[24] Gunawan R., Ma D.L., Fujiwara M., Braatz R.D, Identification of Kinetic Parameters in a Multidimensional Crystallization Process, Int. J. Mod. Phys. B 16: 367 (2002).
[25] Hanhoun M., Montastruc L., Azzaro-Pantel C., Biscans B., Frèche M., Pibouleau L, Simultaneous Determination of Nucleation and Crystal Growth Kinetics of Struvite Using a Thermodynamic Modeling Approach, Chem. Eng. J. 215–216: 903-912 (2013).
[26] Zhu Z., Dorao C.A., Jakobsen H.A., A Least-Squares Method with Direct Minimization for the Solution of the Breakage-Coalescence Population Balance Equation, Math. Comput. Simulat. 79 (3): 716–727 (2008).
[27] Lin Y.L., Lee K., Matsoukas T, Solution of the Population Balance Equation Using Constant-Number Monte Carlo, Chem. Eng. Sci. 57(12): 2241–2252 (2002).
[28] Hounslow M.J., Ryall R.L., Marshall V.R., A Discretized Population Balance for Nucleation, Growth, and Aggregation, AIChE J. 34(11): 1821-1832 (1988).
[29] Diemer R.B., Olson J.H., A Moment Methodology for Coagulation and Breakage Problems: Part 2 – Moment Models and Distribution Reconstruction, Chem. Eng. Sci., 257(12): 2211–2228 (2002).
[30] Sommer M., Stenger F., Peukert W., Wagner N.J., Agglomeration and Breakage of Nanoparticles in Stirred Media Mills – A Comparison of Different Methods and Models, Chem. Eng. Sci., 61(1): 135–148 (2006).
[31] Marchisio D.L, Fox R.O., Solution of Population Balance Equations Using the Direct Quadrature Method of Moments, J. Aerosol Sci. 36(1): 43–73 (2005).
[32] Ramkrishna D., “Population Balances: Theory and Applications to Particulate Systems in Engineering”, San Diego, CA: Academic Press, (2000).
[33] Hulburt H.M., Katz S., Some Problems in Particle Technology – A Statistical Mechanical Formulation, Chem. Eng. Sci. 19(8): 555–574 (1964).
[35] Marchisio D., Quadrature Method of Moments for Aggregation–Breakage Processes, J. Colloid Interf. Sci., 258(2): 322–334 (2003).
[36] Qamar S., Warnecke G., Simulation of Multicomponent Flows Using High Order Central Schemes, Appl. Numer. Math, 52: 183-201 (2004).
[37] Qamar S., Warnecke G., A High Order Kinetic Flux-Spitting Method for the Special Relativistic Magnetohydrodynamics, J. Comput. Phys. 205: 182-204 (2005).
[38] Qamar S., Warnecke G., A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non Stiff Source Terms, Commun. Comput. Phys., 1: 451-480 (2006).
[39] Qamar S., Elsner M.P., Angelov I., Warnecke G., Seidel-Morgenstern A., Seidel-Morgenstern A.,
A Comparative Study of High Resolution Schemes for Solving Population Balances in Crystallization, Comput. Chem. Eng., 30: 1119-1131 (2006).
[40] Barbier E., Coste M., Genin A., Jung D., Lemoine C., Logette S., Muhr H, Simultaneous Determination of Nucleation and Crystal Growth Kinetics of Gypsum, Chem. Eng. Sci., 64: 363-369 (2009).
[41] Masoodiyeh, F., Mozdianfard, M.R., Karimi-Sabet, J., Modeling Zirconia Nanoparticles Prepared by Supercritical Water Hydrothermal Synthesis Using Population Balance Equation, Powder Technol., 317: 264–274 (2017).
[43] Smith B.R, Sweett F., The Crystallization of Calcium Sulfate Dehydrate, J. Colloid Interf. Sci. 37: 612-618 (1971).
[45] Karimi-Sabet, J., “Experimental Study and Modelling of Ultrafine Particles Formation by Using the Supercritical Fluids”. Ph.D. Thesis, Department of Chemical Engineering, Sharif University of Technology, Tehran, Iran. (2011).
[46] Chen M., Ma C.Y., Mahmud T., Darr J.A., Wang X.Z., Modelling and Simulation of Continuous Hydrothermal Flow Synthesis Process for Nano-Materials Manufacture, J. Supercrit. Fluids, 59: 131– 139 (2011).
[47] Szilágyi B., Muntean N., Barabás R., Ponta O., Lakatos B.G., Reaction Precipitation of Amorphous Calcium Phosphate: Population Balance Modelling and Kinetics, Chem. Eng. Res. Des., 93: 278–286 (2015).