Heat transfer analysis of nanofluid flow through backward facing step

This paper presents the numerical predictions of hydrodynamic and thermal characteristics of nanofluid flow through backward facing step. The governing equations are solved through the finite volume method, as described by Patankar, by taking into account the associated boundary conditions. Empirical relations were used to give the effective dynamic viscosity and the thermal conductivity of the nanofluid. Effects of different key parameters such as Reynolds number, nanoparticle solid volume fraction and nanoparticle solid diameter on the heat transfer and fluid flow are investigated. The results are discussed in terms of the average Nusselt number and streamlines.


Introduction
Heat transfer enhancement using nanoparticles have gained the attention of several researchers. The mixture of these nanoparticles with a base fluid gives birth to nanofluid which responded the need to improve the thermal diffusivity of the fluid using metals and oxides with better thermal conductivity. Many experimental and numerical studies have considered nanofluid in various kinds of geometries Benzema et al. [1] reported a numerical investigation of mixed convection in an irregular ventilated cavity filled with Cu-Water nanofluid. They concluded that the heat transfer rate is enhanced by increasing the Ri number, Re number and nanoparticles volume fraction but decreases with the nanoparticles diameters increase. Considering a backward facing step, Abu-Nada [2] carried out a numerical analysis of forced convection of nanofluid using various types of nanoparticles. He noticed that the Re number has no effect on the heat transfer rate inside the recirculation zone and that nanoparticles with low thermal conductivity present higher value of the Nusselt number. The same study has been undertaking by Al-aswadi et al. [3], they examined the Re number and nanoparticles types effects on the static pressure and the skin friction coefficient. A mixed convection of nanofluid flow over a backward facing step with slotted baffle has been investigated by Heshmati et al. [4]. They noticed that the inclined slotted baffle is the most advantageous geometry with an appropriate Nusselt number and minimal pressure drop. An experimental study has been made by Abbasian [5] to examine the heat transfer and fluid flow characteristics of TiO2-Water nanofluid. They found that the best diameter of TiO2 nanoparticles is 20 nm that presents the highest heat transfer rate. Many other works have focused on the heat transfer characteristics of nanofluid [6]- [8]. The aim of this study is to examine the heat transfer characteristics of nanofluid flow over backward facing step when a portion of the bottom wall is exposed to uniform heat flux.

Analysis and modelling
The configuration of problem under investigation " Figure 1" includes suspension of TiO2 nanoparticles and Newtonian base fluid through a backward facing step. A portion of the lower wall is maintained at uniform heat flux whereas the remaining walls of the channel are kept insulated. Besides, the thermophysical properties of the nanofluid (given in Table1) are assumed to be constant except the thermal conductivity which will vary with the temperature using Corcion's formulation [9].  Under the assumptions cited above, the corresponding non-dimensional governing equations are written as follows: MATEC Web of Conferences 307, 01010 (2020) https://doi.org/10.1051/matecconf/202030701010 ICOME'17 and ICOME'18 The effective viscosity of nanofluid was be modeled by the experimental correlation given by Corcione [9].
µf is the apparent viscosity of the base fluid.
The same thing for the effective thermal conductivity of the nanofluid, an experimental correlation proposed by Corcione was be used.
where Reb is the nanoparticle Reynolds number defined as: f and µf are the mass density and the dynamic viscosity of the base fluid, respectively, while UB is the mean Brownian velocity.
The other parameters are given as follows: The local Nusselt number along the lower wall is derived as:

Numerical modelling
The finite volume method was used to discretize the governing system of equations (12)-(15) whereas the SIMPLE algorithm assisted to handle the pressurevelocity coupling. The discretized equations are solved by the line by line tri diagonal matrix algorithm (TDMA). A non-uniform grid size with finer grids at the walls was adopted. Convergence is declared when the maximum relative change between two consecutive iteration level fall below 10 -6 , for U, V, P and . In order to validate our computational code accuracy, average Nusselt number was compared with the study of Santra et al. [10] for different volume fraction and Reynold number. As may be seen on " Figure 2" the results are in excellent agreement with a maximum discrepancy of about 2.5%.

Results and discussion
The heat transfer and flow characteristics of nanofluid over backward facing step have been investigated in this study. All computations are carried out for Pr =10. Figure 3 shows  It is noticed that when the Re number increases, the presence of nanoparticles enhances more the heat transfer rate compared to the base fluid (without nanoparticles). The figure 5 presents the average Nusselt number for different nanoparticles volume fraction for Re = 400 and Re = 300. It is observed that increasing the nanoparticles volume fraction leads to an increase in the average Nusselt number, this can be explain by the fact that increase the nanoparticles volume fraction increases the thermal conductivity of the fluid which enhance the average Nusselt number.   It is shown that increasing the nanoparticles diameters decreases the average Nusselt number. It is clear that the less nanoparticle diameter d = 20 nm presents the highest heat transfer rate. It can be explained as for a fixed nanoparticle volume fraction, the nanoparticles with small diameter (d = 20 nm) are more numerous than nanoparticles with high diameter which offers a better mix of the fluid and consequently higher average Nusselt number.
In addition, for  = 0.06 the average Nusselt decrease strongly from diameter to other, for example from d = 20 nm to d = 40 nm, while this reduction is not so important for  = 0.01.
It is also observed that for a given nanoparticle diameter, the higher value of the average Nusselt number is observed for the higher nanoparticle volume fraction.

Conclusions
A numerical study of thermal characteristics of nanofluid through backward facing step has been considered. Wide ranges of Re number, nanoparticles volume fraction and nanoparticles diameters have been examined. The main conclusions are as follow: • Heat transfer is enhanced by increasing Re number, nanoparticles volume fraction and by decreasing the nanoparticles diameters.
• For the nanofluid ( = 0.06), the heat transfer enhancement is more important when the Re number augment comparing to the base fluid ( = 0.0).
• When the nanoparticles diameter decreases, the heat transfer decreases strongly for  = 0.06 comparing to  = 0.01.