3.1. Thermal conductivity of nanofluid

Researchers have widely studied nanofluid thermal conductivity. Investigation the increase of analytical thermal conductivity a solid-liquid mixture by adding solid particles of micro-size balls into a liquid known as the Maxwell model.

$$k\_{\rm nf} = \frac{k\_p + 2k\_b + 2\rho \left(k\_p - k\_b\right)}{k\_p + 2k\_b - \rho \left(k\_p - k\_b\right)} k\_b \tag{1}$$

where φ is the volume fraction of the nanofluid, kb is the thermal conductivity of the base fluid, and kp is the thermal conductivity of the nanoparticles.

Hamilton and Crosser [13] proposed a model for nonspherical particles by introducing a shape factor n given by n = 3/φ. The thermal conductivity is expressed as follows:

$$k\_{\eta^{\circ}} = \frac{k\_{\mathcal{p}} + (n - 1)k\_{\mathcal{b}} - (n - 1)\varphi(k\_{\mathcal{b}} - k\_{\mathcal{p}})}{k\_{\mathcal{p}} + (n - 1)k\_{\mathcal{b}} - \varphi(k\_{\mathcal{b}} - k\_{\mathcal{p}})} k\_{\mathcal{b}} \tag{2}$$

A modified Maxwell's model was proposed by Xuan and Li [14] by considering the Brownian motion of the particles in the base fluid for the thermal conductivity enhancement given as

$$k\_{nf} = \frac{k\_p + 2k\_b + 2\left(k\_b - k\_p\right)\varrho}{k\_p + 2k\_b - \left(k\_b - k\_p\right)\varrho}k\_b + \frac{\rho\varrho\mathbb{C}\_p}{2}\sqrt{\frac{k\_b T}{3\pi r\_c \mu}}\tag{3}$$

where kB is the Boltzmann constant, rc is the apparent radius of the cluster, and μ is a dynamic viscosity.

Figure 3 shows the variation of knf =kb as a function of alumina volume fraction with and without CTAB along with its best fits with and without interfacial resistance. The knf =kb value

semi-concentrated nanofluids with 5–10 vol.% with aggregation of nanoparticles, and concen-

There are some existing theoretical formulas to estimate the viscosity of nanofluid. Among them, equation suggested by Einstein [22] is a pioneer in determining the viscosity equation. The assumptions based on the linear viscous fluid containing spherical particles and low

where μnf is the viscosity of nanofluid, μ<sup>b</sup> is the viscosity of the base fluid, and φ is the volume fraction. It is a linear increase of the viscosity with increasing volume concentration. This formula has a limitation that is very small particle concentration. Later on, many researchers

In 1952, Brinkman [23] extended Einstein's formula to be used with moderate particle concentrations, and this correlation has more acceptance among the researchers. For particle concen-

Considering the effect due to the Brownian motion of particles on the bulk stress of an approximately isotropic suspension of rigid and spherical particles, Batchelor [24] proposed

It is clear from the above two relations that, if the second or higher order of φ is ignored, then these formulas will be the same as Einstein's equation has been validated for a particle volume

Nguyen et al. [17] showed that both the Brinkman [23] and Batchelor [24] equations severely underestimate nanofluid viscosities, except at very low particle volume fractions (lower than 1%). They have proposed two correlations for nanofluids consisting 47 and 36 nm of Al2O<sup>3</sup>

μnf ¼ μ<sup>b</sup> � 0:904e

Both of these models determine the viscosity by only considering base fluid viscosity and the particle volume fraction. Furthermore, they proposed a correlation for computing CuO water

μnf ¼ μbð Þ 1 þ 2:5φ (4)

Performance Evaluation Criterion of Nanofluid http://dx.doi.org/10.5772/intechopen.74610 283

<sup>μ</sup>nf <sup>¼</sup> <sup>μ</sup>bð Þ <sup>1</sup> � <sup>φ</sup> <sup>2</sup>:<sup>5</sup> (5)

<sup>μ</sup>nf <sup>¼</sup> <sup>μ</sup><sup>b</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>:5<sup>φ</sup> <sup>þ</sup> <sup>6</sup>:5φ<sup>2</sup> (6)

<sup>μ</sup>nf <sup>¼</sup> <sup>μ</sup><sup>b</sup> <sup>1</sup> <sup>þ</sup> <sup>0</sup>:025<sup>φ</sup> <sup>þ</sup> <sup>0</sup>:015φ<sup>2</sup> (8)

<sup>0</sup>, <sup>1842</sup><sup>φ</sup> (7)

trated nanofluid with 10 vol.% concentration, is out of the usual nanofluids [19–21].

particle volume fractions (φ < 0.02). The suggested formula is as follows:

Theoretical investigations.

contributed to correct this formula [16].

the following formula in 1977:

fraction up to φ < 0:1 [16].

viscosity as shown in Eq. (9) [16]:

trations less than 4%, the expression is as follows:

nanoparticles with water, respectively, as follows:

Figure 3. Variation of knf =kb as a function of alumina nanoparticle volume fraction with and without CTAB and its best fit with an interfacial resistance.

of 70 CMC surfactant is also shown. The effective medium theory (EMT) fits on the experimental data indicated by the solid line, which shows a perfect agreement with empirical data. It can be seen that the value of knf/kb with pure surfactant was negative, while it was positive at all other concentrations of nanoparticles [15]. In general the higher the particle volume fraction, the higher the nanofluid thermal conductivity of nanofluids.
