# Heterogeneous nanofluids: natural convection heat transfer enhancement

- Fakhreddine Segni Oueslati
^{1}and - Rachid Bennacer
^{2}Email author

**6**:222

https://doi.org/10.1186/1556-276X-6-222

© Oueslati and Bennacer; licensee Springer. 2011

**Received: **3 December 2010

**Accepted: **15 March 2011

**Published: **15 March 2011

## Abstract

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.

## Introduction

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 [5]. 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 [6]. 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. [7]

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 [11]. 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 [14], 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 [15]. 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 [20], Eastman et al. [21], Maïga et al. [22] and Wang and Mujumdar [23]. 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 [27]. 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.

### Governing equations

*y*).

*D*, see [30]) is modelled as an incompressible fluid possessing an initial uniform particle concentration and constant physical properties except for the density, which varies with temperature and concentration according to the Boussinesq's approximation as follows:

*ρ*

_{0}is the reference fluid density at temperature and concentration ,

*β*

_{T}and

*β*

_{S}are, respectively, the thermal and solutal expansion coefficients, respectively. The microscopic mass flux, taking into account the Soret effect, is given by:

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: *q*_{ref} = *λ*_{f} Δ*T**/*H**

*R*

_{T}, the solutal to thermal buoyancy ratio,

*N*, the Prandtl number,

*Pr*, the Lewis number,

*Le*and

*a*(for Soret effect occurrence). These parameters are defined as:

It is noted that the thermal coefficient, *β*_{T}, is usually a positive quantity. On the other hand, the solutal coefficient *β*_{S} 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 [31].

The controlling thermo-physical properties are the nanofluid to base fluid ratio of thermal conductivity *λ*^{r} = *λ*_{nf}/*λ*_{f}, and viscosity ratio *μ*^{r} = *μ*_{nf}/*μ*_{f}. 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.

*R*

_{T}= 10

^{4}and 10

^{5}, the values of the stream function at the cavity centre are also given for comparison. Such a structure appears 'conventional' and similar to results reported in the literature [32, 33].

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 41^{2} and 81^{2}, were thoroughly tested. The difference between results given by those grids was less than 1% for *Nu*, *Sh* and Ψ_{c} numbers. Hence, most of the calculations presented in this article were performed using an *n*_{
y
} × *n*_{
x
} = 61^{2} 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%.

*p*and

*φ*, refer to the particles and particle volume fraction, respectively. The effective density and specific heat of the nanofluid can be estimated on the physical principle of the mixture rule as:

_{2}O

_{3}nanofluid):

## Results and discussion

*ρC*

_{ p })

^{r}, which is the most realistic in the physical sense that the relative density, which multiplies the relative specific heat (

*ρ*

^{r})(

*C*

_{ p })

^{r}which is used by several authors. Indeed, this differentiation is crucial since it greatly affects the results, which is illustrated in Figure 3a. Indeed, the comparison clearly shows that the relative specific heat capacity (

*ρ*

^{r})(

*C*

_{ p })

^{r}continues to grow with the particle fraction of nanofluid, in the case of classical formulation, when it decreases slightly for (

*ρC*

_{ p })

^{r}.

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.

_{2}O

_{3}. 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 6 obviously show that such heterogeneity of nanoparticle concentration induces extra buoyancy forces and would modify the momentum equilibrium. Also, Figure 6 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.

*Nu*for the base fluid is, i.e. fluid without particles, (φ = 0). As usual,

*Nu*increases with

*R*

_{T}. The variation of the relative Nusselt number

*Nu*

^{r}(nanofluid to base fluid) with respect to the particle volume fraction for different

*R*

_{T}is represented in Figure 7. The relative Nusselt number increases in the diffusive regime (low Rayleigh number,

*R*

_{T}< 10

^{3}) 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 7. These results show that the heat transfer is mainly conductive for low value of

*R*

_{T}. For intermediate to high values of

*R*

_{T},

*R*

_{T}= 10

^{4}, 10

^{5}and 10

^{6}, heat transfer first increases with particle volume fraction up to nearly (φ = 5% for

*R*

_{T}= 10

^{4}, φ = 6% for

*R*

_{T}= 10

^{5}, φ = 7% for

*R*

_{T}= 10

^{6}) 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

*R*

_{T}. 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.

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

_{2-}water, Al

_{2}O

_{3}-water and Cu-water for

*R*

_{T}= 10

^{4}. 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 TiO

_{2}-water and Al

_{2}O

_{3}-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 9 and 10). Indeed, Figure 11 presents the temperatures (a) and concentrations (b) in the middle horizontal plane of the square enclosure, for different nanofluids (

*R*

_{T}= 10

^{4},

*Pr*= 6.2,

*Le*= 3 and

*Sr*= 2%), illustrates the distinction of each type. From superposed streamlines and isotherms of both TiO

_{2}-water and Al

_{2}O

_{3}-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 TiO

_{2}-water and Al

_{2}O

_{3}-water are comparable. In opposition, this is not the case for the other two nanofluids Cu-water and Al

_{2}O

_{3}-water, which the isotherms and the streamlines show that the distributions are very distinct.

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 Al_{2}O_{3}-water and TiO_{2}-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 (*R*_{T} = 10^{4}, *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%), Al_{2}O_{3} (6%) and TiO_{2} (5%). The lowest heat transfer was obtained for TiO_{2}-water in view of the fact that TiO_{2} has the lowest value of thermal conductivity compared to Cu and Al_{2}O_{3}. However, the difference in the values of Al_{2}O_{3} and TiO_{2} is negligible compared to the value of Cu. The thermal conductivity of TiO_{2} is roughly one fifty of Cu. Yet, a unique property of Al_{2}O_{3} is its high specific heat compared to Cu and TiO_{2}. 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* = 10^{4}, *Pr* = 6.2, *Le* = 3 and Soret coefficient *Sr* = 2%.

*R*

_{T}= 10

^{4},

*Pr*= 6.2,

*Le*= 3 and

*Sr*= 2%) is shown on Figure 12. 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 (Al

_{2}O

_{3}and TiO

_{2}) 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.

## Conclusion

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.

## Abbreviations

### List of symbols

*A*: Aspect ratio of the enclosure, = *L*/*H*; *C*: Concentration; *C*_{
p
} : Specific heat; *D*: Mass diffusivity; *D*_{T}: Thermal mass diffusivity; *g*: Gravitational acceleration; *H*: Height of the enclosure; *L*: Width of the cavity; *Le*: Lewis number, = *α*/*D*; *N*: Buoyancy ratio, *β*_{S}ΔC*/*β*_{T}ΔT*; *Nu*: Nusselt number, Equation 7; *p*: Dimensionless pressure, = *p***H*/*ρα*; *Pr*: Prandtl number, υ/*α*; *R*_{T}: Thermal Rayleigh number, = *gβ*_{T} Δ*T***H*^{3}*ρC*_{
p
} /υ*α*; *R*_{s}: Solutal Rayleigh number, = *gβ*_{S} Δ*S***H*^{3}*ρC*_{
p
} /υ*λ*; *Sc*: Schmidt nmber = υ/*D*; *Sh*: Sherwood number, Equation 7; *S*_{
r
} : Soret coefficient = *D*_{T}/*D*; Δ*T**: Characteristic temperature difference; Δ*C**: Characteristic concentration difference,
; (*x*, *y*): Dimensionless coordinate system, *x**/*H*, *y**/*H*; (*u*, *v*): Dimensionless velocity components, *u**/(*υ*/*H*), *v**/(*υ*/*H*);

### Greek symbols

*α*: Thermal diffusivity, *λ*/(*ρC*_{
p
} ); *β*_{s}: Solutal expansion coefficient; *β*_{T}: 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;

### Superscripts

*r*: Nanofluid to base fluid ratio; *: Dimensional variable;

### Subscripts

C: Center; S: Solutal; nf: Nanofluid; f: Base fluid; p: Particle; T: Temperature; 0: Reference state

## Declarations

## Authors’ Affiliations

## References

- Hansen U, Yuen DA: Subcritical double-diffusive convection at infinite Prandtl number.
*Geophys Astrophys Fluid Dyn*1989, 47: 199–224. 10.1080/03091928908221821View ArticleGoogle Scholar - Ostrach S: Natural convection with combined driving forces.
*Physicochem Hydrodyn*1980, 1(4):233–247.Google Scholar - Schmith RW: Double diffusion in oceanography.
*Ann Rev Fluid Mech*1994, 26: 255–236. 10.1146/annurev.fl.26.010194.001351View ArticleGoogle Scholar - Turner JS: Double-diffusive phenomena.
*Ann Rev Fluid Mech*1974, 6: 37–56. 10.1146/annurev.fl.06.010174.000345View ArticleGoogle Scholar - Nield DA: The thermohaline Rayleigh-Jeffreys problem.
*J Fluid Mech*1967, 29: 545–558. 10.1017/S0022112067001028View ArticleGoogle Scholar - Veronis G: Effect of a stabilizing in thermohaline convection. J Fluid Mech 34: 315–368. 10.1017/S0022112068001916Google Scholar
- Bennacer R, Mahidjiba A, Vasseur P, Beji H, Duval R: The Soret effect on convection in a horizontal porous domain under cross temperature and concentration gradients.
*Int J Numer Methods Heat Fluid Flow*2003, 13(2):199–215. 10.1108/09615530310459342View ArticleGoogle Scholar - Ganguly S, Sikdar S, Basu S: Experimental investigation of the effective electrical conductivity of aluminium oxide nanofluids.
*Powder Technol*2009, 196: 326–330. 10.1016/j.powtec.2009.08.010View ArticleGoogle Scholar - Merabia S, Shenogin S, Joly L, Keblinski P, Barrat J-L: Heat transfer from nanoparticles: a corresponding state analysis.
*Proc Natl Acad Sci*2009, 106(36):15113–15118. 10.1073/pnas.0901372106View ArticleGoogle Scholar - Sefiane K, Bennacer R: Nanofluids droplets evaporation kinetics and wetting dynamics on rough heated substrates.
*Adv Colloid Interface*2009, 147–148: 263–271. 10.1016/j.cis.2008.09.011View ArticleGoogle Scholar - Anoop KB, Kabelac S, Sundararajan T, Das SK: Rheological and flow characteristics of nanofluids: Influence of electroviscous effects and particle agglomeration.
*J Appl Phys*2009, 106: 034909. 10.1063/1.3182807View ArticleGoogle Scholar - Bergman TL: Effect of reduced specific heats of nanofluids on single phase, laminar internal forced convection.
*Int J Heat Mass Transf*2009, 52: 1240–1244. 10.1016/j.ijheatmasstransfer.2008.08.019View ArticleGoogle Scholar - Lee J, Mudawar I: Assessment of the effectiveness of nanofluids for singlephase heat transfer in micro-channels.
*Int J Heat Mass Transf*2007, 50: 452–463. 10.1016/j.ijheatmasstransfer.2006.08.001View ArticleGoogle Scholar - Jain S, Patel HE, Das SK: Brownian dynamic simulation for the prediction of effective thermal conductivity of nanofluid.
*J Nanopart Res*2009, 11: 767–773. 10.1007/s11051-008-9454-4View ArticleGoogle Scholar - Nelson IC, Banerjee D, Ponnappan R: Flow loop experiments using polyalphaolefin.
*J Thermophys Heat Transf*2009, 23: 752–761. 10.2514/1.31033View ArticleGoogle Scholar - Namburu PK, Kulkarni DP, Dandekar A, Das DK: Experimental investigation of viscosity and specific heat and silicon dioxide nanofluids.
*Micro Nano Lett*2007, 2: 67–71. 10.1049/mnl:20070037View ArticleGoogle Scholar - Namburu PK, Kulkarni DP, Misra D, Das DK: Viscosity of copper oxide nanoparticles dispersed in ethylene glycol and water mixture.
*Exp Therm Fluid Sci*2007, 32: 397–402. 10.1016/j.expthermflusci.2007.05.001View ArticleGoogle Scholar - Vajjha RS, Das DK: Specific heat measurement of three nanofluids and development of new correlations.
*ASME J Heat Transf*2009, 131: 071601. 10.1115/1.3090813View ArticleGoogle Scholar - Zhou SQ, Ni R: Measurement of the specific heat capacity of water-based AL2O3 nanofluid.
*Appl Phys Lett*2009, 92: 093123. 10.1063/1.2890431View ArticleGoogle Scholar - Choi SUS: Enhancing thermal conductivity of fluids with nanoparticles. In
*Developments and Applications of Non-Newtonian Flows, FED. Volume 231/MD Volume 66*. Edited by: Siginer DA, Wang HP. New York: ASME; 1995:99–103.Google Scholar - Eastman SR, Phillpot S, Choi US, Keblinski P: Thermal Transport in Nanofluids.
*Annu Rev Mater Res*2004, 34: 219–46. 10.1146/annurev.matsci.34.052803.090621View ArticleGoogle Scholar - Maïga SE, Nguyen CT, Galanis N, Roy G: Heat transfer enhancement in forced convection laminar tube flow by using nanofluids.
*Proc Int Symp Adv Comput Heat Transf CHT04, April 19–24; Norway*2004.Google Scholar - Wang X, Mujumdar A: Thermal characteristics of tree-shaped microchannel nets for cooling of a rectangular heat sink.
*Int J Therm Sci*2006, 46(1):1–19. 10.1016/j.ijthermalsci.2006.06.010View ArticleGoogle Scholar - Abu-Nada E: Effects of variable viscosity and thermal conductivity of Al
_{2}O_{3}**-water nanofluid on heat transfer enhancement in natural convection.***Int J Heat Mass Transfer*2009, 30: 679–690.Google Scholar - Hwang KS, Jang SP, Choi SUS: Flow and convective heat transfer characteristics of water-based Al2O3 nanofluids in fully developed laminar flow regime.
*Int J Heat Mass Transf*2009, 52: 193–199. 10.1016/j.ijheatmasstransfer.2008.06.032View ArticleGoogle Scholar - Rea U, McKrell T, Hu L-W, Buongiorno J: Laminar convective heat transfer and viscous pressure loss of alumina-water and zirconia-water nanofluids.
*Int J Heat Mass Transf*2009, 52: 2042–2048. 10.1016/j.ijheatmasstransfer.2008.10.025View ArticleGoogle Scholar - Nield DA, Kuznetsov AV: The onset of convection in a horizontal nanofluid layer of finite depth European.
*J Mech B/Fluids*2010, 29: 217–223. 10.1016/j.euromechflu.2010.02.003View ArticleGoogle Scholar - Bennacer R, El Ganaoui M, Maré T, Nguyen CT: Natural convection of nanofluids in a cavity including Soret effect.
*Comput Therm Sci*2009, 1(4):425–440. 10.1615/ComputThermalScien.v1.i4.40View ArticleGoogle Scholar - Bennacer R, Mohamad AA, El Ganaoui M: Thermodiffusion in porous media: Multidomain constituent separation.
*Int J Heat Mass Transf*2009, 52: 1725–1733. 10.1016/j.ijheatmasstransfer.2008.09.026View ArticleGoogle Scholar - Pinheiro JP, Domingos R, Lopez R, Brayner R, Fiévet F, Wilkinson K: Determination of diffusion coefficients of nanoparticles and humic substances using scanning stripping chronopotentiometry (SSCP).
*Colloids Surf A: Physicochem Eng Aspects*2007, 295(1–3):200–208. 10.1016/j.colsurfa.2006.08.054View ArticleGoogle Scholar - Ryskin A, Muller HW, Pleiner H: Thermal convection in binary fluid mixtures with a weak concentration diffusivity, but strong solutal buoyancy forces.
*Phys Rev E*2003, 67: 046302. 10.1103/PhysRevE.67.046302View ArticleGoogle Scholar - Choukairy K, Bennacer R, Vasseur P: Natural convection in a vertical annulus boarded by an inner wall of finite thickness.
*Int Commun Heat Mass Transf*2004, 31: 501–512. 10.1016/S0735-1933(04)00031-4View ArticleGoogle Scholar - De Vahl Davis G: Natural convection in square cavity: a comparison exercise.
*Int J Numer Methods Fluids*1983, 3: 227–248. 10.1002/fld.1650030304View ArticleGoogle Scholar - Drew DA, Passman SL:
*Theory of Multicomponent Fluids*. Berlin: Springer; 1999.View ArticleGoogle Scholar - Brinkman HC: The viscosity of concentrated suspensions and solution.
*J Chem Phys*1952, 20: 571–581. 10.1063/1.1700493View ArticleGoogle Scholar - Keblinski P, Eastman JA, Cahill DG: Nanofluids for thermal transport.
*Mater Today*2005, 36–40. 10.1016/S1369-7021(05)70936-6Google Scholar - Nguyen CT, Desgranges F, Galanis N, Roy G, Maré T, Boucher S, Mintsa HA: Viscosity data for Al
_{2}O_{3}**-water nanofluid--hysteresis: is heat transfer enhancement using nanofluids reliable?***Int J Therm Sci*2008, 47: 103–111. 10.1016/j.ijthermalsci.2007.01.033View ArticleGoogle Scholar - Hamilton RL, Crosser OK: Thermal conductivity of heterogeneous two-component systems.
*I & EC Fundam*1962, 1(3):187–191.View ArticleGoogle Scholar - Akbarinia A, Behzadmehr A: Numerical study of laminar mixed convection of a nanofluid in horizontal curved tubes.
*Appl Therm Eng*2007, 27: 1327–1337. 10.1016/j.applthermaleng.2006.10.034View ArticleGoogle Scholar - Leal MA, Machado HA, Cotta RM: Integral transform solutions of transient natural convection in enclosures with variable fluid properties.
*Int J Heat Mass Transf*2007, 43: 3977–3990. 10.1016/S0017-9310(00)00023-5View ArticleGoogle Scholar - Sai BVKS, Seetharamu KN, Narayana PAA: Solution of transient laminar natural convection in a square cavity by an explicit finite element scheme.
*Numer Heat Transf A*1994, 25: 412–422. 10.1080/10407789408955968Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.