315321 A Model for the Size of Polar and Apolar Nanoparticle Agglomerates in a Fluidized Bed
Fluidization is emerging as a promising but challenging technique for processing of nanoparticles, for instance, to produce coated nanoparticle. Nanoparticles do not fluidize individually but form agglomerates due to strong interparticle forces, dominant at the nano-scale . The type of fluidization and transport phenomena inside the agglomerates strongly depend on the agglomerates structure and size, two variables in turn related to the forces present between particles and agglomerates.
The size of the fluidized agglomerates is commonly estimated with a balance between attractive interparticle forces, such as van der Waals, electrostatic and capillary force and separating forces, such as gravity and drag force. When dealing with hydrophobic (apolar) nanoparticles, capillary is neglected and van der Waals is the only interparticle attractive force usually considered. In the case of hydrophilic (polar) nanoparticles, it is common to add a capillary force to the interparticle forces if the fluidization is carried out with ambient air. If the fluidizing gas is dry, the approximation is similar to that with apolar nanoparticles and only van der Waals interactions are included. Tahmasebpoor et al.  showed that attractive forces between dry and polar particles is considerably larger than between dry and apolar nanoparticles. This was attributed to the formation of hydrogen bonds between the surfaces of the polar nanoparticles.
In this work, we propose a new model to estimate the average agglomerate size in dry fluidized beds by taking into account the type of surface, polar or apolar. We do this by including a term in the inter-particle forces that takes into consideration the possibility of formation of hydrogen bonds.
In our model, complex agglomerates are built by porous and spherical primary agglomerates . The attraction between primary agglomerates is described according to Rabinovichxs model , considering that the nanoparticles act as asperities in the primary agglomerates. The Hamaker coefficient has been corrected with the two-body summation approach to take into account the porosity of the primary agglomerates. The contribution of the hydrogen bond has been modeled taking into account the geometry of the contact point between agglomerates. The only separating force considered is gravity. The average agglomerate size is therefore the size that makes the Bond number of the agglomerates equal to unity.
The resulting average agglomerate size is a function of dp, ρp, Df1, Df2, d*, AH, N and α, where dp and ρp are the size and density of the nanoparticles, respectively. Df1, Df2 are the fractal dimensions of the primary and complex agglomerates, respectively. d* is the average size of the primary agglomerates. AH is the Hamaker coefficient of the nanoparticles material. N is an empirical constant related to the connectivity of the primary agglomerates, being 1.38 for all the nanopowders. α is an empirical constant that takes into account the formation of hydrogen bond between nanoparticles. α is 0 for apolar nanoparticles and ≈1.67e-3 N/m for polar nanoparticles.
The proposed model successfully approximates the size of most of the nanopowders reported in literature, both in conventional and in centrifuged beds. It has no systematic error as a function of the particle size, density or Hamaker coefficient. The average prediction error of the proposed model is ≈20 %, while for the model proposed by Valverde and Castellanos , it is ≈45 %.