## 4 Numerical Results and Discussion
### Discussion  
In this part, the results are discussed for various values of parameters containing the space ratio and particle diameter. The particle dispersions inside the duct for different particle diameters at \(Re\) = 100, \(\Phi\) = 0.01, and \(S/D\) = 3.5 are shown in Figure 5. This figure shows the locations of injected particles at a specific time, which are superimposed on vorticity contours. It is observed that the particles with diameters of 30 nm are dispersed at entire of the duct. Brownian force generated between the liquid molecules and the suspended particles causes the randomized dispersion of the particles. This force is larger in comparison to the inertial forces for smaller values of particle diameter. The particles with smaller diameter (\(dp\) = 30 nm) are not affected by the flow streamline and they diffuse through the streamlines. However, the particles with larger diameters (i.e., 0.1 and 0.25 um) accumulate in the vorticity regions around the perimeter of the vortices. It should be stated that the larger particles are under the influence of forces created by the vortices and vortical flow field. Finally, the inertia forces of particles dominate the centrifugal forces formed by the flow for the particles with diameter of 0.5 um and this causes the exit of particles from the vortices path line and tends them toward the duct surfaces. Figure 6 discloses the concentration contours for different values of space ratio at \(Re\) = 100, \(d_{p}\) = 30 nm, and \(\Phi\) = 0.01. It is observed that the concentrations of particles for all values of spaceratio are almost constant with a very low deposition near the duct surface for the regions before the obstacles. For these regions, the concentration for most parts of the duct keeps constant at the entrance concentration (\(\Phi\) = 0.01). Note that the particles impacting to the duct walls bounce from the wall due to the reflect assumption of boundary but the flow. Moreover, thermophoretic force for the current case where the liquid temperature is higher than the wall temperature pushes the particles back toward the wall [37]. This causes a deposition of particles near the duct surface. This figure also discloses that the mass diffusion boundary layer is very narrow but is growing along the duct length. Finally, the particles concentration enhances in the recirculating wake region at the downstream of the obstacles. This means that the particles with this size are affected by the flow in the near wake region and some particles are distributed inside the vortex pathways. Effects of particle diameters on the particle deposition at \(Re\) = 100, \(\Phi\) = 0.01, and \(S\)/\(D\) = 1.5 are disclosed in Figure 7. The deposition is defined as the ratio of the number of deposited particles on the all involved surfaces containing surfaces of obstacle and duct to the number of particles injected to the duct. It can be observed that the particle deposition enhances with an enhance in the particle diameter. This may be justified by the effect of gravity force on the particle deposition that becomes more significant for higher values of particle diameters. The particle deposition percentages are about 1.1%, 1.8%, 2.7%, and 4.6% for \(d_{p}\) = 30 nm, 0.1 \(\upmu\)m, 0.25 \(\upmu\)m, and 0.5 \(\upmu\)m, respectively. Note that the trap boundary condition with a restitution coefficient of 0 is considered as the particle boundary condition on the surfaces of the duct and obstacles for evaluating particle deposition because for the reflect boundary, the particle deposition is zero as the restitution coefficient is equal to 1. The effects of space ratio values on the particle deposition at \(Re=100\), \(\Phi=0.01\), and \(d_{p}=0.1\)\(\upmu\)m are disclosed in Figure 8. It is observed that the particle deposition enhances with an enhance in the value of space ratio.