## 4 Results and Discussion
We further analyzed our pressure drop data using the Ergun equation (Equation (4)), which is valid for velocities in the packed bed region when the bed is not fluidized. According to the Ergun equation, the pressure drop (\(\Delta P\)) depends on the bed void fraction (\(\varepsilon\)) and agglomerate diameter (\(D_{uv}\)). The bed void fraction (\(\varepsilon\)) was computed from the bed height using Equation (5). As seen in Table 2, these values are lower than the corresponding incipient values (\(\varepsilon_{mf}\)) reported in Table 1. This leaves diameter as the only remaining unknown parameter in the Ergun equation. As shown in Figure 6, we fitted our experimental data with the Ergun equation in order to determine the effective \begin{table}
\begin{tabular}{c c c} \hline \hline
**Incipient Fluidization** & \(\mathbf{U_{mf}}\) & \(\varepsilon_{mf}\) \\ \cline{2-3}  & **(mm/s)** & **(-/-)** \\ \hline Unassisted & 118.3 & 0.984 \\
4.5 vol \% particle & 47.0 & 0.973 \\
4.5 vol \% + 0.1 Hz pulse & 25.0 & 0.971 \\
8.6 vol \% particle & 49.4 & 0.970 \\
8.6 vol \% + 0.1 Hz pulse & 33.0 & 0.969 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Incipient fluidization parameters of the fluidized bed. Figure 5: Minimum fluidization calculations for nanoparticle bed fluidization at different assistance conditions. diameter of the agglomerates of nanoparticles present in the bed. The results obtained using regression analysis are reported in Table 2. The \(R^{2}\) values in most cases indicated an excellent fit with the experimental data (more than 90% in all cases). The effective diameter in the table is the average diameter for both nanoparticles as well as external particles, while the mean agglomerate diameter is the average diameter of agglomerates only, which was calculated using Equation (9). Clearly, particle mixing helped to decrease the size of nanoparticles agglomerates in the packed bed. The agglomerate diameter decreased by 33.2% and 48.4% with the addition of 4.5 vol % and 8.6 vol % group A particles, respectively. The agglomerate diameter further reduced to 39.6% with the combined effect of 4.5 vol % particle mixing and pulsed airflow as compared to the case of particle mixing alone. However, for the case of 8.6 vol % particle mixing and flow pulsation together, the agglomerate size was further reduced to 50.7 %. This reduction is not as significant as the one seen in the case of 4.5 vol % when flow pulsation was applied. These results indicate that the use of flow pulsation with particle mixing further enhanced the deagglomeration phenomenon. However, the effect of pulsed flow is more prominent in 4.5 vol % particle mixing than in 8.6 vol % particle mixing. The size reduction can be attributed to the presence of larger micron-sized Geldart group A particles in the bed of nanoparticles. The repeated collision of denser and larger external particles tends to fragment agglomerates of nanoparticles. Moreover, the interagglomerate force equilibrium is disturbed, owing to the presence of different external particles in their midst, which leads to deagglomeration. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Assisted Fluidization Techniques** & **Porosity (\(\epsilon\))** & \begin{tabular}{c} **Effective** \\ **Diameter** \\ \end{tabular} & \
\begin{tabular}{c} **Mean** \\ **Agglomerate** \\ **Diameter** \\ \end{tabular} &
\begin{tabular}{c} **Agglomerate** \\ **Size** \\ **Reduction** \\ \end{tabular} & \(R^{2}\) \\ \cline{2-6}  & **(-/-)** & **(\(\mu\)m)** & **(\(\mu\)m)** & **(\%)** & **(-/-)** \\ \hline Unassisted & 0.979 & 14.05 & 14.05 & - & 0.997 \\
4.5 vol \% particle & 0.971 & 11.62 & 9.50 & 32.4 & 0.910 \\
4.5 vol \% \% + 0.1 Hz pulse & 0.970 & 10.65 & 8.49 & 39.6 & 0.992 \\
8.6 vol \% particle & 0.967 & 11.50 & 7.25 & 48.4 & 0.924 \\
8.6 vol \% + 0.1 Hz pulse & 0.969 & 11.21 & 6.93 & 50.7 & 0.975 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Agglomerate size reduction under different fluidization conditions (external particle diameter: 56.5 \(\mu\)m).