*5.2. Effective Viscosity*

The effective viscosity of nanofluids is part of the chain that determines the applicability of using such a category of suspensions in heat transfer applications. Since it is a transport property directly related to the dynamic performance of the heat transfer system, where an increase in colloidal viscosity would lead to an increase in the friction coefficient and thus a raise in the pressure losses in the system. The heat transfer system then compensates for this pressure difference by increasing the pumping power, and accordingly, more electrical power gets consumed. For such a reason, many research studies have been devoted to investigating the link between the nanofluids effective viscosity and the different parameters associated with the suspension, such as nanoparticles shape, size, concentration, dispersion stability, and mixture temperature [247–256]. Mena et al. [257] suggested that nanofluids fabricated with nanoparticles concentration of up to 13 vol. % behaves as Newtonian fluids (i.e., their viscosity is independent of shear strain). In addition, many researchers proved that the stability of nanofluids has an inverse relationship with their effective viscosity. Meaning that well-dispersed suspensions tend to have lower effective viscosity than those of poor stability [202,258–260]. If the viscosity of a shelved nanofluid was to be categorized according to its stability condition, then there would exist one to three different viscosity regions. To be more precise, a well-dispersed suspension would roughly have a uniform viscosity along its column, while three different viscosity regions would exist in the semi-stable case, and two different viscosity regions would form in the unstable separation scenario. Figure 17 demonstrates the three stability cases with their different viscosity regions. As for the nanofluid in its dynamic form, these viscosity regions would most likely still exist within the suspension while flowing in the hosting system. Knowing this, one can explain why the unstable suspension would require higher pumping power compared to the stable form of the same dispersion. To calculate the percentage of viscosity increase that the dispersed particles cause to the base fluids, the following equation can be used [154]:

$$\text{Viscosity increase } (\%) = \left(\frac{\mu\_{eff}}{\mu\_{bf}} - 1\right) \times 100\tag{10}$$

where *µe f f* and *µb f* are the effective viscosities of the nanofluid and the base fluid, respectively. Furthermore, the most common and widely used approach for determining nanofluids viscosity is via the rotational viscosity measurement method [261]. In this method, a shaft is inserted in the sample, after which the rotational speed and the torque per unit length of the shaft are used to determine the viscosity of the nanofluid. Other measuring techniques are also used, such as the capillary viscometer, concentric cylinder viscometer, rheonuclear magnetic resonance, and rheoscope [262–264]. To be noted that, according to Prasher et al. [265], in order for a nanofluid to improve the performance of its

*Nanomaterials* 

hosting thermal application, the increase in effective viscosity should not exceed four times the mixture enhanced effective thermal conductivity, otherwise the working fluid would not serve the hosting system in terms of overall performance. This can be theoretically calculated through the following equations [266]:

$$\frac{\left(\frac{\mu\_{eff}}{\mu\_{bf}}\right) - 1}{\left(\frac{k\_{tf}}{k\_{bf}}\right) - 1} < 4\tag{11}$$

**Figure 17.** Nanofluids viscosity classification, where (**a**) shows the stable, (**b**) illustrates semistable, and (**c**) demonstrates the unstable cases of the suspension. **Figure 17.** Nanofluids viscosity classification, where (**a**) shows the stable, (**b**) illustrates semistable, and (**c**) demonstrates the unstable cases of the suspension.

In another study, Akhavan-Zanjani et al. [267] investigated the thermal conductivity and viscosity of nanofluids made of graphene, water, and polyvinyl alcohol (PVA) surfactant. The wt % used were of 0.005–0.02% and the mass ratio of the PVA employed was 3:1. The authors found a significant increase in the as-prepared fluid thermal conductivity with a moderate raise in the viscosity. The highest recorded percentages for the thermal conductivity and viscosity were 10.30% and 4.95%, respectively. Iranmanesh et al. [268] analyzed the effect of two preparation parameters, namely; the concentration and temperature, on aqueous graphene nanosheets nanofluids thermal conductivity and viscosity. They used 0.075–0.1 wt % to fabricate their nanofluids using the two-step method at 20– 60 °C. The findings indicated that the wt % used had a clear effect on the viscosity and thermal conductivity on the prepared dispersion. Moreover, the temperature, as a parameter, was seen to have a larger influence on the level of the final product viscosity compared to the added solid concentration. Although the study avoided any employment of surfactants, the authors as-prepared suspension was stable for several days. Such an observation is not new and was also reported by other researchers, such as Mehrali et al. [159], where they successfully fabricated stabilized nanofluids made of the graphene–water mixture without the need for surfactants or graphene functionalization, but these are rare cases because of the attraction nature between the head groups of both particles and the base fluid molecules. Ghazatloo et al. [269] have developed a model that can predict the effective viscosity of CVD graphene–water and CVD graphene–EG nanofluids at ambient temperature. They used experimental measurements of the property and employed a commonly used model to form their correlation. Moreover, the researchers used 0.5–1.5 vol % of 60 nm graphene sheets with two separate base fluids (i.e., water and EG), after which the content was subjected to sonication for 45 min. For their water based nanoflu-In another study, Akhavan-Zanjani et al. [267] investigated the thermal conductivity and viscosity of nanofluids made of graphene, water, and polyvinyl alcohol (PVA) surfactant. The wt % used were of 0.005–0.02% and the mass ratio of the PVA employed was 3:1. The authors found a significant increase in the as-prepared fluid thermal conductivity with a moderate raise in the viscosity. The highest recorded percentages for the thermal conductivity and viscosity were 10.30% and 4.95%, respectively. Iranmanesh et al. [268] analyzed the effect of two preparation parameters, namely; the concentration and temperature, on aqueous graphene nanosheets nanofluids thermal conductivity and viscosity. They used 0.075–0.1 wt % to fabricate their nanofluids using the two-step method at 20–60 ◦C. The findings indicated that the wt % used had a clear effect on the viscosity and thermal conductivity on the prepared dispersion. Moreover, the temperature, as a parameter, was seen to have a larger influence on the level of the final product viscosity compared to the added solid concentration. Although the study avoided any employment of surfactants, the authors as-prepared suspension was stable for several days. Such an observation is not new and was also reported by other researchers, such as Mehrali et al. [159], where they successfully fabricated stabilized nanofluids made of the graphene–water mixture without the need for surfactants or graphene functionalization, but these are rare cases because of the attraction nature between the head groups of both particles and the base fluid molecules. Ghazatloo et al. [269] have developed a model that can predict the effective viscosity of CVD graphene–water and CVD graphene–EG nanofluids at ambient temperature. They used experimental measurements of the property and employed a commonly used model to form their correlation. Moreover, the researchers used 0.5–1.5 vol % of 60 nm graphene

ids, a volume ratio of 1.5:1 of SDBS surfactant to solid content was used to physical stabilize the dispersion. The outcome of their research indicated that the effective viscosity

had a significant effect on the property. Furthermore, the Batchelor model [270] showed some deviation from the experimental data of the as-prepared suspensions viscosity. This

sheets with two separate base fluids (i.e., water and EG), after which the content was subjected to sonication for 45 min. For their water based nanofluids, a volume ratio of 1.5:1 of SDBS surfactant to solid content was used to physical stabilize the dispersion. The outcome of their research indicated that the effective viscosity remarkably increased with the raise of vol. %, and hence the concentration as a parameter had a significant effect on the property. Furthermore, the Batchelor model [270] showed some deviation from the experimental data of the as-prepared suspensions viscosity. This variation in results were reduced by the newly developed model, which, unlike the previous model, considered the solid additive geometry. The comparison between the authors model, Batchelor model, and experimental results is demonstrated in Figure 18. variation in results were reduced by the newly developed model, which, unlike the previous model, considered the solid additive geometry. The comparison between the authors model, Batchelor model, and experimental results is demonstrated in Figure 18. On the other hand, in terms of the effective viscosity theoretical models developments, Table 5 lists these correlations with their year of development, dependent parameters, and limitations. From the formulas shown in Table 5, it can be concluded that most of the authors have used specific experimental conditions to come up with their correlations, and hence the majority of the models are limited to their operating conditions (i.e., they cannot be accounted as universal models).

*Nanomaterials* **2021**, *11*, x FOR PEER REVIEW 34 of 79

**Figure 18.** Comparison between the Ghazatloo et al. [269] model, Batchelor [270] model, and experimental effective viscosity, where (**a**) shows the results for graphene–water nanofluid of 0.5 vol. % (G/W-1), 1.0 vol. % (G/W-2), and 1.5 vol. % (G/W-3), and (**b**) illustrates the values for graphene–EG suspensions of 0.5 vol. % (G/EG-1), 1.0 vol. % (G/EG-2), and 1.5 vol. % (G/EG-3). **Figure 18.** Comparison between the Ghazatloo et al. [269] model, Batchelor [270] model, and experimental effective viscosity, where (**a**) shows the results for graphene–water nanofluid of 0.5 vol. % (G/W-1), 1.0 vol. % (G/W-2), and 1.5 vol. % (G/W-3), and (**b**) illustrates the values for graphene–EG suspensions of 0.5 vol. % (G/EG-1), 1.0 vol. % (G/EG-2), and 1.5 vol. % (G/EG-3).

On the other hand, in terms of the effective viscosity theoretical models developments, Table 5 lists these correlations with their year of development, dependent parameters, and limitations. From the formulas shown in Table 5, it can be concluded that most of the authors have used specific experimental conditions to come up with their correlations, and hence the majority of the models are limited to their operating conditions (i.e., they cannot be accounted as universal models).




Ansón-Casaos et al. [292]

Ilyas et al. [154] <sup>2020</sup>

2020

solids.

μ = μ ቀ1 − <sup>χ</sup>

μ = μ exp ൬ ܨܲ.ଵ

<sup>2</sup> ƒቁ ିଶ ; where χ is equal to 2.5 for spherical particles or can be replaced by a function, ݂൫ݎ൯, to determine the suspension property containing 1D and 2D dispersed

ଶܲ.ܨ − ܶ

− ܨܲ.ହ ƒ ଶ ; where ܨܲ.ଵ, ܨܲ.ଶ, and ܨܲ.ସ are the temperature fitting parameters in Kelvin, whereas ܨܲ.ଷ and ܨܲ.ହ are the dynamic viscosity fitting parameters in Pa.s. The values


**Table 5.** *Cont*.

ܶ <sup>൰</sup>

൰ + ܨܲ.ଷ ƒ exp ൬ܨܲ.ସ

 ƒ, and χ

 ܲܨ ,ƒ and ܶ

Suitable for SWCNTs and graphene oxide.

Suitable for ND dispersed in thermal oil and is valid for the range of 0 ≤ ƒ ≤ 1 and 298.65 ≤

T (K) ≤ 338.15.

