Heterogeneous nanofluids: natural convection heat transfer enhancement
© Oueslati and Bennacer; licensee Springer. 2011
Received: 3 December 2010
Accepted: 15 March 2011
Published: 15 March 2011
Convective heat transfer using different nanofluid types is investigated. The domain is differentially heated and nanofluids are treated as heterogeneous mixtures with weak solutal diffusivity and possible Soret separation. Owing to the pronounced Soret effect of these materials in combination with a considerable solutal expansion, the resulting solutal buoyancy forces could be significant and interact with the initial thermal convection. A modified formulation taking into account the thermal conductivity, viscosity versus nanofluids type and concentration and the spatial heterogeneous concentration induced by the Soret effect is presented. The obtained results, by solving numerically the full governing equations, are found to be in good agreement with the developed solution based on the scale analysis approach. The resulting convective flows are found to be dependent on the local particle concentration φ and the corresponding solutal to thermal buoyancy ratio N. The induced nanofluid heterogeneity showed a significant heat transfer modification. The heat transfer in natural convection increases with nanoparticle concentration but remains less than the enhancement previously underlined in forced convection case.
The existence of convection in double-diffusive systems, in which heat and salt diffuse at a different rate, was first recognised in the late 1950 s. Since then, this phenomenon has been studied extensively due to the fact that its importance has been recognised in many fields such as geophysics, astrophysics, ocean physics and industrial processes [1–4]. The first study concerning double diffusion in a binary fluid seems to be that of Nield . Relying on linear stability theory, the onset of motion in an initially motionless, stable concentration and stratified horizontal fluid layer heated from below was predicted by this author. This cross-effect regarding the Rayleigh-Bénard convection dealing with the bifurcation and the possible change in the critical thresholds (i.e. transitional Rayleigh number from conductive to convective motion) was also considered on the same period by Veronis . All the above studies are concerned with the effect of the regular diffusion of each component (heat and salt) on convection. However, in a wide variety of natural and industrial situations, besides the usual diffusion, cross-diffusion between the two agents may also be important. This phenomenon, known as the Soret effect, has been relatively less studied despite its importance for a fluid layer of a binary mixture (convection and stability). In recent studies, the problem of the double thermo-diffusion effects that occurs under natural convection in fluid or porous media was studied; see for example Bennacer et al. 
During the past ten years, a new class of fluids made up of metal nanoparticles in suspension in a liquid, called nanofluids, has appeared. Nanofluids are composed of nanoparticles that (size in general <100 nm) are suspended in a base fluid, as water or an organic solvent [8–10]. The formation of extremely stable colloidal systems with very tiny settling is a characteristic feature of some nanofluids, the stability of the suspension is naturally achieved by electrostatic stabilisation by adjusting the pH . The presence of nanoparticles causes a significant modification of thermal properties of the resulting mixture; in particular, nanofluid viscosity and thermal conductivity increase with particle volume fraction. Although the increase in thermal conductivity is a very important interest, there are also increases in the average temperature of nanofluids compared to that of base fluid and that because of the specific heat of nanofluids, which decreases compared to that of base fluid [12, 13]. The abnormal rise of the thermal conductivity in comparison with the pure fluid , especially for low particle concentrations, is not totally understood today. Some assumptions are based on particle deposition on the surface resulting in the formation of nano fins . There are a many recent studies that report experimental measurements of thermophysical properties of nanofluids, including specific heat, thermal conductivity and viscosity; some recent reports include [16–19]. There has been great attention in nanofluids generated by a variety of applications, ranging from laser-assisted drug delivery to electronic chip cooling.
Some previous research works were mainly concerned with heat transfer and properties of these fluids, see Choi , Eastman et al. , Maïga et al.  and Wang and Mujumdar . The natural convection of nanofluids deserves more attention in light compared to forced convection [24–26]. Recently, linear stability analysis, employed model incorporates the effects of Brownian motion and thermophoresis, for the onset of natural convection in a horizontal nanofluid layer . For vertical layer it was underlined the existence of an optimal particle volume concentration of 2% [28, 29], which maximises heat transfer.
The aim of this article is to study the increase of heat transfer taking into account both the variation of thermal conductivity and viscosity in the governing equations when using nanofluids for different types of metallic particles such as Al2O3, TiO2 and Cu. Indeed, for the modelling is as realistic as possible, we considered the Soret effect and the heterogeneity of concentration due to crossed effect.
the Soret effect is taken into account if α = 1, or ignored if a = 0.
The derivation of the coupled governing equations, under their dimensionless form, has been based on the reference quantities for length, velocity, temperature and concentration differences given by cavity height H*, υ/H*, and .
The heat transfer is characterised by the Nusselt number, which is based on the reference diffusive heat flux: qref = λf ΔT*/H*
It is noted that the thermal coefficient, βT, is usually a positive quantity. On the other hand, the solutal coefficient βS can be positive (N > 0) or negative (N < 0). For N > 0, the thermal and solutal boundary forces are both destabilizing, i.e. the two buoyancy components make aiding contributions, whilst for N < 0, they make opposing contributions. In the present nanofluide study we have weak concentration but strong buoyancy forces wich is similar to the classical binary mixtures .
The controlling thermo-physical properties are the nanofluid to base fluid ratio of thermal conductivity λr = λnf/λf, and viscosity ratio μr = μnf/μf. These characteristics are functions of the nanofluid mixture used and furthermore, space dependent due to the possible heterogeneity of nanoparticles concentration. The subscripts f, nf and r refer, respectively, to the base fluid, the nanofluid (effective properties) and relative nanofluid/base fluid ratio of the physical quantity under consideration
The average number along the active wall is given by (M = Nu or Sh).
In the above equations, Nu represents, as usual, the heat transfers across the walls of the cavity resulting from the combined action of convection and conduction. However, because the walls of the cavity are impermeable, Sh does not have its usual significance. Here, it is rather related to the concentration distribution within the cavity induced by the Soret effect (taken into account for α = 1, or ignored, i.e. a = 0) and by natural convection.
Numerical method and validation
In order to numerically solve the governing equations, a control volume approach is used. Central differences are used to approximate the advection-diffusion terms, i.e. the scheme is second-order accurate in space. By spatial integration over control volumes, the governing equations are converted into a system of algebraic equations. The latter are solved by a line-by-line iterative method, which is combined with a sweeping technique over the integration domain along x- and y-axes and a tri-diagonal matrix inversion algorithm. The SIMPLER algorithm is employed to solve the equations in a form of primitive variables. Non-uniform grids are used in the program, allowing fine grid spacing near the two horizontal walls. The convergence criteria are based on the conservation of mass, momentum, energy and species, and this is on both global and local basis.
Flow intensity in the centre of the cavity versus literature results (Pr = 0.7, A = 1, φ = 0, Sr = 0, N = 0)
To ensure that the results are mesh-size independent, different non-uniform n y × n x meshes (where n y and n x represent, respectively, the node numbers in the vertical and horizontal directions), namely 412 and 812, were thoroughly tested. The difference between results given by those grids was less than 1% for Nu, Sh and Ψc numbers. Hence, most of the calculations presented in this article were performed using an n y × n x = 612 grid. Such a grid system possesses very fine meshes near all boundaries. The solution was carried out for a validation test case with Pr = 0.71 in a narrow channel flow for a range of controlling parameters. The converged solution achieved with all absolute residues of the governing equations is less than 10-7. All numerical results presented hereafter are obtained with parameters A = 1, Pr = 6.2, Le = 3 and Sr = 2%.
Results and discussion
Figure 5 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.
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 . 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 9 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 10 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 11a: 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 effect of using different nanofluids on the thermal and dynamic fields of natural convection in a differentially heated square cavity was studied numerically. Indeed, the results revealed that one type of nanofluid is a key factor for improving heat transfer. The highest values are obtained when using Cu nanoparticles. However, increasing the value of the Rayleigh number is growing the heat transfer. Moreover, the results show the influence due to competing effects between nanoparticles and thermal dynamics, and we identified the flow control parameters for different currents. The results also confirmed that the character of the natural convection directly affects a significant increase in heat transfer with the concentration of particles. Nevertheless the percentage of particle nature greatly affects the heat transfer and fluid flow.
The crossover Soret effect, which is the origin of the spatial distribution of nanoparticles concentrations, and its influence on heat transfer and flow field were studied. The percentage of the optimal nanoparticles concentration that maximises heat transfer was found and it is related to the kind of particle used.
The estimated Soret coefficient was supposed in this study not depending on the nanoparticles but we underline that molecular size and the electrical charges could modify the value of such coefficient and experimental work is necessary to go through this question.
List of symbols
A: Aspect ratio of the enclosure, = L/H; C: Concentration; C p : Specific heat; D: Mass diffusivity; DT: Thermal mass diffusivity; g: Gravitational acceleration; H: Height of the enclosure; L: Width of the cavity; Le: Lewis number, = α/D; N: Buoyancy ratio, βSΔC*/βTΔT*; Nu: Nusselt number, Equation 7; p: Dimensionless pressure, = p*H/ρα; Pr: Prandtl number, υ/α; RT: Thermal Rayleigh number, = gβT ΔT*H3ρC p /υα; Rs: Solutal Rayleigh number, = gβS ΔS*H3ρC p /υλ; Sc: Schmidt nmber = υ/D; Sh: Sherwood number, Equation 7; S r : Soret coefficient = DT/D; ΔT*: Characteristic temperature difference; ΔC*: Characteristic concentration difference, ; (x, y): Dimensionless coordinate system, x*/H, y*/H; (u, v): Dimensionless velocity components, u*/(υ/H), v*/(υ/H);
α: Thermal diffusivity, λ/(ρC p ); βs: Solutal expansion coefficient; βT: Thermal expansion coefficient; θ: Dimensionless temperature, ; ϕ: Dimensionless concentration, ; φ: Particle volume fraction; λ: Fluid thermal conductivity; μ: Dynamic viscosity; υ: Kinematic viscosity; ρ: Fluid density; Ψ: Stream function;
r: Nanofluid to base fluid ratio; *: Dimensional variable;
C: Center; S: Solutal; nf: Nanofluid; f: Base fluid; p: Particle; T: Temperature; 0: Reference state
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