A Monodisperse Population Balance Model for Nanoparticle Agglomeration in the Transition Regime

• Main Authors: Georgios A. Kelesidis, M. Reza Kholghy
• Format: Article
• Language: English
• Published: MDPI AG 2021-07-01
• Series: Materials
• Subjects: agglomeration; transition regime; population balance model; discrete element model; fractal-like structure
• Online Access: https://www.mdpi.com/1996-1944/14/14/3882

Nanoparticle agglomeration in the transition regime (e.g. at high pressures or low temperatures) is commonly simulated by population balance models for volume-equivalent spheres or agglomerates with a constant fractal-like structure. However, neglecting the fractal-like morphology of agglomerates or their evolving structure during coagulation results in an underestimation or overestimation of the mean mobility diameter, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>m</mi></msub></mrow></semantics></math></inline-formula>, by up to 93 or 49%, repectively. Here, a monodisperse population balance model (MPBM) is interfaced with robust relations derived by mesoscale discrete element modeling (DEM) that account for the realistic agglomerate structure and size distribution during coagulation in the transition regime. For example, the DEM-derived collision frequency, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>, for polydisperse agglomerates is 82 ± 35% larger than that of monodisperse ones and in excellent agreement with measurements of flame-made TiO₂ nanoparticles. Therefore, the number density, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mrow><mi>A</mi><mi>g</mi></mrow></msub></mrow></semantics></math></inline-formula>, mean, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>m</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula> and volume-equivalent diameter, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>v</mi></msub></mrow></semantics></math></inline-formula>, estimated here by coupling the MPBM with this <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> and power laws for the evolving agglomerate morphology are on par with those obtained by DEM during the coagulation of monodisperse and polydisperse primary particles at pressures between 1 and 5 bar. Most importantly, the MPBM-derived <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mrow><mi>A</mi><mi>g</mi></mrow></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>m</mi></msub></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>v</mi></msub></mrow></semantics></math></inline-formula> are in excellent agreement with the data for soot coagulation during low temperature sampling. As a result, the computationally affordable MPBM derived here accounting for the realistic nanoparticle agglomerate structure can be readily interfaced with computational fluid dynamics in order to accurately simulate nanoparticle agglomeration at high pressures or low temperatures that are present in engines or during sampling and atmospheric aging.