**3. Heat Transfer Correlation Equations**

As seen in Tables 3 and 4, a limited number of numerically developed correlation equations have been proposed. It is necessary to stress the very limited range of applications of the presented correlations.



**Table 4.** Correlation equations for turbulent flow of nanofluids.

## **4. Conclusions**

As shown in Table 1, a single-phase approach was used in the majority of works. The conclusions of the researchers who used single-phase models and two-phase models regarding the comparison of both approaches are different. Some of them [22,23] observed that the accuracy of the single-phase model is similar to the two-phase models. Others [11,12] indicate that the two-phase models more faithfully reproduce the results of experimental research than the single-phase approach. It was stressed that for nanofluids, the number of particles in the computational domain, even for a very small NPC, is very large. Therefore, due to limitations of the software abilities, the Lagrangian–Eulerian approach is extremely difficult to implement.

Several investigators [19,30,31,34,39,47,51,57] implemented mechanisms influencing convective heat transfer of nanofluids previously considered by Buongiorno [65]. According to [19,30], Brownian motion and thermophoresis have a negligible impact on heat transfer during the forced convection of nanofluids. Moreover, Albojamal and Vafai [51] demonstrated that the Brownian motion, thermophoresis, virtual mass, pressure gradient, and Saffman lift force do not significantly influence mean HTC. It was shown in [47] that gravity, virtual mass, pressure gradient, Brownian, and lift forces have no impact on mean flow field. As stated in [57], gravity force has to be included in all numerical models.

There is also no consensus among researchers regarding the assumption of constant or temperature-dependent thermophysical properties of nanofluids. In [33,34,61], a greater accuracy of the model using temperature-dependent thermophysical properties was observed, compared to the model where the thermophysical properties were constant.

Generally, the standard k-ε turbulence model was used to model turbulent flows of nanofluids [14,16,20,26,29,33,37,40,55,60,62]. Only a few studies used other models of turbulence [4–8].

Virtually all the results of numerical studies indicate the intensification of the heat transfer of nanofluids compared to base liquids, regardless of the type of flow (laminar or turbulent), although the results of the experimental studies are not so clear, e.g., [66–68].

There is also full agreement that the addition of nanoparticles increases flow resistance (PD, FF, and WSS).

Many researchers point to the need to intensify research on the thermophysical properties of nanofluids, as they determine the accuracy of the single-phase model first of all. In this context, however, one must bear in mind the limitations of experimental research [69].

The heat transfer correlations proposed in the literature have a very limited range of applicability. No numerically developed correlation for determination of the friction factor of nanofluids flowing inside a horizontal smooth tube has been presented.

Future works should concern a benchmark solution for laminar and turbulent flow of nanofluids similar to that of Buongiorno et al. in relation to thermal conductivity [70]. It is necessary to conduct a comparative study of the influence of divergent thermophysical properties of base liquids on the final results concerning the influence of nanoparticles on heat transfer, as was carried out in [57].

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
