High heat transfer performance is vital in many engineering applications, both at the micro and the macro level (i.e. chip cooling and building heating).

The conventional methods to increase the heat transfer rate are those of extending the exchange surface or using a better fluid. The first approach is usually not preferred, because it leads to an increase of thermal system dimensions. Therefore, the second option is more desirable, but it is constrained by the thermophysical properties of conventional heat transfer fluids (i.e. water, ethylene glycol, etc.).

Over the last several decades, scientists and engineers have tried to develop fluids, which provide better performances for a variety of thermal applications.

Applying nanotechnology to heat transfer, the new concept of 'nanofluid', introduced by Choi [1] in 1995, has been proposed to meet the new heat transfer challenges. This new kind of fluid is manufactured by dispersing an amount of solid nanoparticles in traditional heat transfer fluids.

Maxwell [2] was the first to show the possibility of increasing thermal properties, particularly conductivity, of a liquid by including a volume fraction of solid particles. However, the dimensions of the particles were in the order of millimetre or micrometre, hence problems such as mixture stability and a dramatic increase in mixture viscosity were detected.

Several investigations revealed that the dispersion of a small amount of different kinds of nanoparticles (i.e. Al_{2}O_{3}, CuO, TiO_{2}) in water or ethylene glycol exhibit enhanced thermal conductivity, as reviewed in [3–5].

Different concepts have been proposed to explain this enhancement in thermal performance, which results to be higher with respect to that of classical mixtures.

Li and Xuan [6] and Xuan and Roetzel [7] attributed the enhancement of heat transfer to the increased thermal dispersion resulting from the chaotic movements of nanoparticles, which accelerates the exchange of energy. Keblinski et al. [8] proposed different mechanisms that contribute to the increase of nanofluids heat transfer, among which are Brownian motion of the particles and molecular level layering at the liquid/particle interface. Also Wang et al. [9] explained the heat transfer enhancement with the interface layer between liquid and particles. Buongiorno [10] developed a very in-depth analysis of all the possible mechanisms of fluid particles slip during convection of nanofluids, concluding that the abnormal increase of heat transfer coefficient in turbulent regime is due to the variation of thermophysical properties within the boundary layer, because of the effect of the temperature gradient and thermophoresis.

Many authors [11–16] focused their analysis on the measurement of nanofluid thermal conductivity, showing a much larger value with respect to the classical theoretical predictions [17]. In a recent article, Buongiorno et al. [18] conducted an international benchmark exercise on nanofluid thermal conductivity measurements, which concluded that no anomalous enhancement of thermal conductivity was observed.

Other authors concentrated their research on the experimental analysis of nanofluids forced convection in laminar and turbulent regime. The works of Pak and Cho [19] and Xuan and Li [20] represent two outstanding contributions to the experimental study of turbulent convection of nanofluids. They developed two correlations for the calculations of Nusselt number, indicating a remarkable increase of heat transfer performance over the base fluid for the same Reynolds number.

Numerical investigations on nanofluids were carried out by two approaches. The first approach assumes that the continuum assumption is still valid for fluids with suspended nano-size particles. The other approach uses a two-phase model for better description of both the fluid and the solid phases.

The single phase model, with thermophysical properties all assumed to be constant with temperature, was employed in [21–24].

The two phase approach seems to be a better model to describe the nanofluid flow. In fact, the slip velocity between the fluid and particles may not be zero [10] due to several factors such as gravity, friction between the fluid and solid particles, Brownian forces and thermophoresis. The two phase approach provides a field description of the dynamics of each phase or, alternatively, the Lagrangian trajectories of individual particles coupled with the Eulerian description of the fluid flow field [25–30].

As can be seen, all the aforementioned literature is focused on the theoretical, experimental and numerical study of thermophysical properties and convection of nanofluids, but the modern design concept for a thermal system, pursues not only the enhancement of heat transfer performance, but also requests the minimal power requirements.

Enhancement of the heat transfer performance, usually, must be achieved at the expense of power input and this is also the case of nanofluids. In fact, in the study of nanofluid convection, there is the recurrent question of where is the position of the trade-off between the increase in heat transfer and pressure loss. Therefore, the optimal trade-off between heat transfer and power input requirement becomes a major issue in the design of a thermal system.

A modern approach for the optimization of a thermal system is based on the second law of thermodynamics. Particularly, the entropy generation is used as the parameter for evaluating the efficiency of the system. The system with minimum entropy generation is considered as the optimal design [31, 32].

In our opinion, an accurate way to handle this common problem is to analyze the entropy generation in order to ascertain the condition under which entropy generation is minimized.

In this paper, developing turbulent forced convection flow of a nanofluid in a channel with square transversal section is numerically investigated. Steady state of a two-dimensional symmetric flow is considered and the channel is heated at uniform heat flux. The study is carried out for water with alumina particles with a spherical size of 38 nm diameter. The main aim of the present work is to estimate the thermal and fluid flow fields and to find, by means of second law analysis, the channel optimal working condition under given boundary conditions and particles' concentration. An analytical procedure is also proposed to estimate the entropy generation and a comparison with the numerical results is accomplished.

To the authors' best knowledge, it seems that nanofluids forced convection in tubes, with square section in turbulent regime, has not been previously investigated. Moreover, it seems that the optimization by means of second law analysis is applied for the first time to nanofluids convection. The intention of this investigation is to try to bridge the information gap.