## 3 Results and Discussion
### Thermal and Hydraulic Characteristics on a Horizontal Microscale Backward-Facing Step  
Figure 15 shows the velocity profile across the horizontal microscale backwards-facing step at the streamwise location where the channel undergoes the sudden enlargement, that is, at the step edge. These numerical predictions are obtained with the single-phase approach and with the two-phase mixture model. Different combinations of phase two volume fraction and of phase two Al\({}_{2}\)O\({}_{3}\) nanoparticle volume fraction, as defined in Table 2, provide the same 1% Al\({}_{2}\)O\({}_{3}\) nanoparticle volume fraction in the mixture. The results are compared to corresponding numerical predictions by [PERSON] and [PERSON] [52] at the Reynolds number of 280. The length to height ratio \(L_{u}/h_{u}=250\) is much greater than the entry length to height ratio of 3.41 of a laminar channel flow [59]. This establishes a fully developed laminar channel flow, with its characteristic parabolic velocity profile in \(x_{2}\). The simulation is two-dimensional and therefore the flow is uniform spanwise. The predictions from all the combinations of phase two volume fraction in Table 2 as well as from the single-phase approach collapse on essentially a single, parabolic-shaped curve. The velocity profile is substantially symmetric about \(x_{2}=0\), suggesting the absence of any significant flow stratification effect, due to the relatively small difference in density between the first and the second phases in the two-phase mixture model. The numerical results from [PERSON] and [PERSON] [52] follow the same trend and their scatter about the parabolic-shaped data collapse from the current models is driven by the digitization process of the reference data from the article print. As shown by the flow visualizations in [PERSON] and [PERSON] [52], the flow separates at the backwards-facing step edge, creating a laminar shear layer that reattaches on the floor, about six step heights downstream. The streamline connecting the separation point and the reattachment point defines a clockwise flow recirculation region, which is relatively segregated from the inlet cold coolant feed running over the top of the step. This creates a region of elevated temperature, in which the more modest wall-normal thermal gradient as well as the more modest convective heat transport away from the wall produces low local values of Nusselt number. Figure 14: Variation of the minichannel second law efficiency \(\eta_{II}\) over the Reynolds number range 92 to 455. Predictions using the single-phase approach and the two-phase mixture model. This effect is shown in Figure 16, where the predictions of the local Nusselt number are plotted along the bottom heated wall. The results are compared to the experiments by [PERSON], et al. [51] and to numerical results by [PERSON] and [PERSON] [52] at the same Reynolds number of 280 as in Figure 15. Just downstream of the backwards-facing step face, which is located at \(x_{1}=0.1\) m, the local value of the Nusselt number is relatively low and it increases in \(x_{1}\) over the \(x_{1}\) range from 0.1 m to 0.104 m. Just downstream of the reattachment point, the cold flow from the top of the step meets the heated wall, creating both a strong wall-normal thermal gradient and the means for transporting heat away from the wall, by the increase in coolant specific internal energy. This produces the observed local heat transfer maximum. Further downstream, the temperature increment of the coolant due to the heat that it has picked up reduces its temperature difference with the wall and therefore its ability to extract further heat. This produces a monotonic decay in the local Nusselt number all the way to the exit of the duct. Figure 16 shows good agreement among the single-phase model and the mixture models with a second phase nanoparticle volume fraction less or equal to 50%. These predictions are consistent with the reference measurements reported in [PERSON], et al. [51].