In this article, despite the lack of experimental results, we use the relative specific heat capacity (ρCp)r, which is the most realistic in the physical sense that the relative density, which multiplies the relative specific heat (ρr)(Cp)r which is used by several authors. Indeed, this differentiation is crucial since it greatly affects the results, which is illustrated in Figure . Indeed, the comparison clearly shows that the relative specific heat capacity (ρr)(Cp)r continues to grow with the particle fraction of nanofluid, in the case of classical formulation, when it decreases slightly for (ρCp)r.
Specific heat capacity versus nanoparticle concentration (Al2O3).
As mentioned before, both viscosity and thermal conductivity increase and specific heat capacity decreases with particle concentration (Figure ).
Relative viscosity (a), thermal conductivity (b) and specific heat capacity (c) versus nanoparticle concentration for different kind of particles.
Based on the definition of the Nusselt number (Equation 7), the heat transfer in the case of homogeneous nanofluid is given by:
= 0 it recovers the pure fluid case
and the relative nanofluid heat transfer is given by:
This expression illustrates well the evolution of the relative natural convection heat transfer to the pure fluid case. The second group functions only of the particle volume fraction and the relative nanoparticles-to-base fluid viscosity and conductivity (see Equations 13 and 14), and the relative density and heat capacity are given below:
Let's define Δρr
The effect of the particle volume fraction on the heat transfer is shown on Figure . The same figure exposed a comparison between the classical homogeneous nanofluid model and the heterogeneous nanofluid model. We note, for both homogeneous and heterogeneous as well as the analytical solution, there is a maximum particles concentration above which the heat transfer begins to decrease. In fact the increase of nanofluid viscosity increases the friction, so the flow rate decreases which in turn induces a diminution of heat transfer. On the other hand, an increase of nanofluid thermal conductivity would necessarily enhance the heat transfer. So, it is important to discuss which of these two effects influences most the heat transfer?
Nanoparticle fraction effect on heat transfer (a = 0, A = 1, RT = 105).
Figure shows also, for the classical homogeneous nanofluid model case, that the numerical and analytical results are in good agreement and the maximum Nusselt is reached for particle volume fraction of 2%. Nevertheless, for the case of the heterogeneous fluid model, we can note that the Nusselt is more enhanced and reach a maximum for particle volume fraction of 5%. In fact, the considered thermodiffusion affects clearly the heat transfer and the flow.
When the Soret effect is considered, the nanoparticle concentration within the fluid is spatial dependant (heterogeneous fluid). Such heterogeneity induces a strong non-linear effect as the conductivity, viscosity and heat capacity and solutal buoyancy became spatial-dependant. This explains the strong coupling between the flow, the heat transfer (dependent on the flow and local thermal conductivity) and the concentration which, indeed, is also dependent on both the flow and thermal fields.
Figure shows a comparison between homogeneous (plotted by dashed lines) and heterogeneous (plotted by solid lines) cases on streamlines (on the left), isotherms (at the middle) and isoconcentrations (on the right) using the same nanoparticles Al2O3. The figure demonstrates that a single circulation cell is formed in the clockwise direction for all values of Rayleigh numbers. One can observe that the separation caused by the Soret effect clearly shows the importance of the heterogeneity of the nanoparticle concentration in the cavity. Such a spatial heterogeneity causes, in turn, a relatively important modification of the thermal field, which can modify the heat transfer rate by as much as 10%. It is worth noting that many previous results do not take into account the buoyancy forces effect caused by this heterogeneous distribution of particle concentration. Our results from Figure obviously show that such heterogeneity of nanoparticle concentration induces extra buoyancy forces and would modify the momentum equilibrium. Also, Figure illustrates an example of the resulting dynamic, thermal and species fields as well as the important changes related to the adding temperature and concentration effects.
Figure 6 Dynamic, thermal and concentration fields for homogeneous (plotted by dashed lines) and heterogeneous (plotted by solid lines) cases (RT = 104, Pr = 6.2, Le = 3, Sr = 0.5%, = 2%, N = 1.75).
The effect of the flow intensity on the optimum value of particle volume fraction observed previously is illustrated on Figure . For comparison and discussion purpose, the reference Nu
for the base fluid is, i.e. fluid without particles, (
= 0). As usual, Nu
increases with RT
. The variation of the relative Nusselt number Nur
(nanofluid to base fluid) with respect to the particle volume fraction for different RT
is represented in Figure . The relative Nusselt number increases in the diffusive regime (low Rayleigh number, RT
) as it is directly dependent on the apparent thermal conductivity. The relative heat transfer (i.e. nanofluid to base fluid) illustrates a decrease for higher Rayleigh number and is a direct consequence of the reference increase illustrated by Figure . These results show that the heat transfer is mainly conductive for low value of RT
. For intermediate to high values of RT
, heat transfer first increases with particle volume fraction up to nearly (
= 5% for RT
= 6% for RT
= 7% for RT
) and then decreases with increasing particle fraction. Such a result for a 'homogeneous' fluid is considered as the reference, based on which we present the relative increase with the concentration for different RT
. The heat transfer increases with increasing particle volume fraction in a monotonic manner for low Rayleigh numbers because of the increase of the fluid thermal conductivity-as the heat transfer mechanism is mainly conduction.
Effect of nanofluid concentration on relative heat transfer for different RT (a = 1, A = 1, Sr = 2%, Pr = 6.2 and Le = 3).
It should be noted that the increase of heat transfer does not exceed 5%. It is worth mentioning that there exists a major difference between the cases of natural convection and forced convection as analysed by others authors, see for example [39
]. Such a difference can be explained by the fact that in this study, the flow is not imposed, and hence appears to be more sensitive to a change of the fluid viscosity. The buoyancy strength is governed by the heating conditions imposed so that the intensity of the flow then decreases with increasing viscosity effect.
Nanoparticle type effect
Figure presents the comparison of streamlines and isotherms using different nanofluids: TiO2-water, Al2O3-water and Cu-water for RT = 104. However, we varied the Rayleigh number for different types of nanoparticles, from diffusive state to convection state. For all nanofluids, a single cell movement was observed in a clockwise direction. The values of the maximum stream function show that the intensity of flow is higher for Cu-water than that of TiO2-water and Al2O3-water. Hence, in the case of nanofluid heterogeneous solutal forces are in addition to heat one. The importance of solutal gradients, which differs from one type of nanofluid to another, directly affects the dynamic state and heat transfer (illustrated by figure and ). Indeed, Figure presents the temperatures (a) and concentrations (b) in the middle horizontal plane of the square enclosure, for different nanofluids (RT = 104, Pr = 6.2, Le = 3 and Sr = 2%), illustrates the distinction of each type. From superposed streamlines and isotherms of both TiO2-water and Al2O3-water nanofluids, we find that the dynamic and thermal fields are similar. This reproaches qualitative aspects explained by the fact that the values of thermophysical properties of TiO2-water and Al2O3-water are comparable. In opposition, this is not the case for the other two nanofluids Cu-water and Al2O3-water, which the isotherms and the streamlines show that the distributions are very distinct.
Figure 8 Dynamic, thermal and species fields for different nature of nanoparticle (RT = 104, = 2%, Pr = 6.2, Sr = 2% and Le = 3).
Nanoparticle fraction effect on heat transfer for different kind of particle: homogeneous case (a = 0, A = 1, RT = 105).
Effect of nature of nanoparticle on the nanofluid heat transfer: heterogeneous case (RT = 104, Pr = 6.2, Sr = 2% and Le = 3).
Figure 11 Temperature (a) and concentration (b) on the horizontal mid-plan (RT = 104, = 2%, Le = 3 and Sr = 2%).
Figure shows the variation of relative Nusselt number, according to analytic approach (Equation 16) with volume fraction using different nanoparticles. We can note that the heat transfer increases with increasing the volume fraction for all nanofluids. For the three nanoparticles one notices the existence of a maximum, which is achieved by increasing the concentration, beyond which the transfer begins to decrease. This finding is valid for Al2O3-water and TiO2-water but not for Cu-water. Indeed, the increase of thermophysical properties as a function of the nanoparticles, namely thermal conductivity, viscosity and specific heat capacity, affects the heat transfer and flow. So, increasing the viscosity with the nanoparticles is exacerbating the friction that causes a decrease in heat transfer. But in the case of Cu, which provides thermal conductivity and density that increases remarkably with the nanoparticles which outweighs the increase in the viscous effect and the specific heat capacity that decreases with the nanoparticles.
We present on Figure the variation of mean Nusselt number with volume fraction using different nanoparticles and different values of Rayleigh number. Results are presented for the case (RT = 104, Pr = 6.2, Le = 3 and Sr = 2%). The figure shows that the heat transfer increases about monotonically with increasing the volume fraction for all Rayleigh numbers and nanofluids. For the three nanoparticles one notices the existence of a maximum, which is achieved by increasing the concentration, beyond which the transfer begins to decrease, but this maximum differs for Cu (7%), Al2O3 (6%) and TiO2 (5%). The lowest heat transfer was obtained for TiO2-water in view of the fact that TiO2 has the lowest value of thermal conductivity compared to Cu and Al2O3. However, the difference in the values of Al2O3 and TiO2 is negligible compared to the value of Cu. The thermal conductivity of TiO2 is roughly one fifty of Cu. Yet, a unique property of Al2O3 is its high specific heat compared to Cu and TiO2. The Cu nanoparticles have high values of thermal diffusivity and, thus, this reduces temperature gradients which will affect the performance of Cu nanoparticles. As volume fraction of nanoparticles increases, difference for mean Nusselt number becomes larger especially at higher Rayleigh numbers due to increasing of domination of convection mode of heat transfer. In fact, the temperature gradients grow to be more pronounced, which is illustrate in Figure : the temperature along the middle plane of the square enclosure using different nanofluids for Ra = 104, Pr = 6.2, Le = 3 and Soret coefficient Sr = 2%.
The vertical velocity along the middle plane of the square enclosure using different nanofluids (for RT = 104, Pr = 6.2, Le = 3 and Sr = 2%) is shown on Figure . Due to the floating flow inside the enclosure, the velocity shows a parabolic variation near the isothermal walls. The vertical velocity is susceptible to the nature of nanoparticles where two types of nanoparticles (Al2O3 and TiO2) show similar vertical velocity but the third (Cu) is so different. This is explained in Equation 16 where the Brinkman formula shows that the viscosity of the nanofluid is only sensitive to the volume fraction of particles and not influenced by the type of nanoparticles and the expression of the buoyancy ration which is a function of the mass expansion coefficient that depends on the density of the nature of the particle. Indeed, the mass buoyancy force, in addition to the thermal buoyancy force, intensified the flow. Even then, the vertical velocity of nanofluid is higher than that of pure fluid. It means that particle suspension affects the flow field. The flow velocity is almost zero around the centre of the cavity. The profile also gives idea on flow rotation direction.
Figure 12 Vertical velocity on the horizontal mid-plan (RT = 104, = 2%, Le = 3 and Sr = 2%).