Numerical analysis of forced convection heat transfer in a rectangular micro-channel totally filled with Ag/ water nano fluid in slip flow regime using the lattice Boltzmann method

Numerical simulation reported on heat transfer and fluid flow in a two-dimensional rectangular micro channel totally filled with Ag/water. The first –order slip/jump boundary conditions were uniformly imposed to the up and bottom walls. The governing conservation equations are translated in dimensionless form using the thermal Single Relaxation Time (T-SRT) modified Lattice Boltzmann Method (LBM) with double distribution functions (DDFs). The viscous dissipations effects are adopted into the energy equation. Effects of nanoparticle volume fraction φ, slip coefficient, B, on the flow of Nano fluid and heat transfer were studied. The results were interpreter in terms of slip velocity; temperature jump and Nusselt number. Based on the results found, it can be concluded that decreasing the values of slip coefficient enhances the convective heat transfer coefficient and consequently the Nusselt number (Nu) but increases the slip velocity at the wall and temperature jump values. * Corresponding author: Kaouther.benltaifa@yahoo.com © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). E3S Web of Conferences 321, 04008 (2021) ICCHMT 2021 https://doi.org/10.1051/e3sconf/202132104008


Introduction
Heat transfer in fluids led to many practical and industrial applications, including transportation (combustion engines), energy supply, air conditioning, and electronics cooling…The recent considerable development of research treating with nanofluid for certain applications induce an improvement of heat transfers by introducing into a pure fluid a low concentration of nanoparticles. The intensity of heat transfer depends strongly on the conductivity and thermal capacity of the heat transfer fluids. Then, nanofluids are colloidal solutions obtained by dispersing solid particles of nanometric size in a base fluid. Therefore, some of these solutions have been found to contribute more in the enhancement of heat transfer under certain conditions. There are many studies concerning nanofluid in different geometries, some researchers have reported the flow and heat transfer of the nanofluid in microchannels [1][2][3][4]. For instance, Karimipour et al. [5]simulated the Cu-water nanofluid in a microchannel for slip condition. In addition, the lattice Boltzmann method are sued to simulate each problems related to heat transfer of nanofluids in microchannels [6,7]. According to the above literature, the present study deals with laminar forced convection heat transfer of Ag-Water nanofluids in a microchannel using Lattice Boltzmann method.
Our attention focused on the effects of emerging parameter's on the slip velocity, temperature jump and Nusselt number.

. Lattice Boltzmann model
The Lattice Boltzmann equation with a single relaxation time from the BGK model was adopted in this study and it can be expressed as: (2) f and g indicated the density momentum and internal energy distribution functions respectively.The discrete distribution function i f i f i and i g i g with velocity i c at position x and time t are given by [8]: i Z and i D represent the effect of heat dissipation and the material derivative along the direction i c , g W and g W are the hydrodynamic and thermal relaxation times, respectively. e i f and e i g are the equilibrium distribution function. In this study, the 2-D nine-bit model (D2Q9) is used (Fig.1). This model can be defined as : The hydrodynamic and thermal variables can be determined by [10]: The kinematic viscosity and the thermal diffusivity are given by :

Nanofluid
The properties of the nanofuid are obtained using the following relations [11] : Using the Brinkman model [11] the effettive dynamic viscosity is: The nanofluid thermal conductivity as a function of liquid and solid conductivities is expressed as following [ In wich e P is the Peclet number with the brownien motion velocity of particles B u and B k is the constant of Boltzmann.

Boundary condition
Non-equilibrium bounce back model, normal to the boundary, is used for inlet and outlet hydrodynamic boundary conditions. In this model, distribution functions are reflected in suitable ways to satisfy the equilibrium conditions and improve accuracy.
The unknown inlet and outlet thermal distribution functions are estimated using the known inlet temperature profile and nnon-equilibrium bounce back model as follow [12,13]: u u e dt f Z g g g g g g g u u u u Concerning the boundary conditions of the walls of the microchannels, the boundary condition of sliding is applied for the hydrodynamic field. Ngoma and Erchiqui [14] considered β for the slip length coefficient and defined the slip velocity s u for the liquid inside the microchannel on the fixed walls as follows : The dimensionless form is written as : To defined the slip velocity in LBM, the specular reflective bounce back model (combination of bounce back and specular boundary condition) is applied in this work. For example for the bottom wall, the unknown distribution functions are approximated by : numerical results for different grids are shown in Table 1, due to small difference between the results of the last two grid sizes, a uniform grid with 800 40 u was chosen to obtain the best agreement between accuracy and computation time.
To validate the developed code, the comparison of the values obtained by Santra et al. [18] for the average Nusselt number (for different Reynolds number:Re=50, Re=100and Re=200) of a forced convection of cold Cu-water nanofluid in a macro channel with hot walls. The figure 3 demonstrated good agreement with those of Santra et al. [18].
In the present computation, the Reynolds number and the Prandtl number are chosen to be Re /  It is worth recalling that the laminar forced convection heat transfer of a Ag/water in a microchannel is studied numerically by using Lattice Boltzmann method. values. Moreover, they increase with φ. However, this effect is more pronounced for x Nu . Figure 9 portrays that the temperature jump around the entrance region occurs a higher values, which has the most temperature gradient near the wall.

Conclusion
Laminar forced convection heat transfer of Ag-water nanofluid in a microchannel is simulated using a double population LBM-BGK method.The effects of different volume fractions of Argent nanoparticles and slip coefficient were investigated on the slip velocity, temperture jump and Nusselt number For Re=0,01.
The numerical obtained results confirmed higher φ corresponds to larger Nux. Moreover, it was stated that larger values of B induces a decreasing in Nux and larger values of s U and s T . At the entrance region, the temperature jump reaches a high values along the microchannel walls especially which has the most temperature gradient between the walls and nanofluid. As a result, to increase Nux in micro liquid flows, it is recommended to use nanofluid with φ = 4% and at low values of slip coefficient as like B = 0.005. However, the effect of φ is more pronounced compared to B.