**2. Thermal conductivity and rheological properties**

Thermal conductivity enhancement is influenced by several factors, such as Brownian motion of nanoparticles, nanolayer, nanoparticle clustering and other external parameters such as volume fraction, nanoparticle size and temperature [23]. Jang and Choi [24] discovered the random motion of Brownian motion contributed to 6% of total thermal conductivity enhancement. Nanolayers shown in **Figure 2** act as a barrier for thermal conductance which lowers the overall thermal conductivity of nanofluid. There are instances where clustering of nanoparticles by Van der Waals forces induces local percolation structure that can enhance thermal conductance of nanofluids [23].

However, a fractal model developed [25] showed no changes in thermal conductivity properties of nanofluid from clustering effects as the enhancement effects are counterbalanced from

Wang et al. [32] dispersed graphene nanoparticles at low loadings into ionanofluid and was found to possess slightly lower viscosity at higher temperatures as compared to its counterpart base fluids due to the self-lubrication of graphene nanoparticles. Lu et al. [33] concluded rheological properties to be highly dependent on nanoparticle concentrations. At very low loadings, nanofluids with Newtonian behaviours can produce shear-thinning non-Newtonian behaviour when subjected to high nanoparticle concentrations due to strong particle-particle

Bio-Based Oil Drilling Fluid Improvements through Carbon-Based Nanoparticle Additives

http://dx.doi.org/10.5772/intechopen.74674

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Conventional thermal conductivity models are used for the prediction of thermal conductivity of nanofluids based on several main key parameters such as nanoparticle volume fraction (φ), thermal conductivity of nanoparticle (kp), thermal conductivity of base fluid (kbf) and shape factor (*n*) for nanoparticle types. Effective medium theory (EMT) models, such as, Maxwell model [34], Hamilton-Crosser model [35] and Bruggeman model [36], are static models that predict based on the assumptions that particles are motionless and heat transfer

Maxwell had developed the first EMT model to predict suspensions containing diluted particles (< 1 vol% concentration) [34]. The assumption basis of this model is that the particles are non-interacting with each other and is spherical in shape. Maxwell model is expressed as:

*kp* + 2 *kbf* + 2(*kp* − *kbf ϕ*)

where knf is thermal conductivity of nanofluid, kp is thermal conductivity of nanoparticle, kbf is thermal conductivity of base fluid and φ is the volume fraction of nanoparticle. The Maxwell model was reported to predict well for relatively large particle size at micro- and

The Hamilton-Crosser (HC) model is expressed when the thermal conductivity of particle is greater than the thermal conductivity of liquid by 100 times (kp/kbf > 100). The HC model is an extension of Maxwell's model which takes shape factor, *n*, of particles into account in calculation. The shape factor is defined as the ratio of surface area of the sphere with constant volume

*kp* + (*n* − 1) *kbf* + (*n* − 1)

(*kp* − *kbf*)*ϕ*

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ *kp* <sup>+</sup> (*<sup>n</sup>* <sup>−</sup> 1) *kbf* <sup>−</sup> (*kp* <sup>−</sup> *kbf*)*<sup>ϕ</sup>* ] (2)

1 +

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ *kp* <sup>+</sup> <sup>2</sup> *kbf* <sup>−</sup> <sup>2</sup>(*kp* <sup>−</sup> *kbf <sup>ϕ</sup>*)] *kbf* (1)

interactions interrupted by shear rates exceeding a specific critical value.

**3. Thermal conductivity and rheological models**

between both continuous and dispersed phases are diffusive [37].

**3.1. Thermal conductivity models**

*knf* <sup>=</sup> [

as particle to the surface area of the particle.

*knf* <sup>=</sup> *kbf*[

*3.1.1. Maxwell model*

millimetre scales.

*3.1.2. Hamilton-Crosser model*

**Figure 2.** Schematic cross diagram of nanolayers at solid/liquid interface of nanoparticles and liquid [38].

reduced convection of particles. This claim was in good agreement that clustering effect performs poorly on stability and thermal conductivity of nanofluid [26].

Hadadian et al. [22] prepared different masses of graphene oxide in 50 mL of distilled water and ethylene glycol and were subjected to 15 min of ultrasonication to produce a homogenous suspension. They yielded a maximum 30% thermal conductivity enhancement with 0.07 mass fraction graphene oxide, owing to the excellent geometry of graphene oxide such as high interfacial area and comprised of sheet-like arrangements favourable for formation of a percolation structure. Ijam et al. [27] added graphene oxide nanosheets ranging from 0.01 to 0.10 wt% into deionized water to be sonicated for 10 min before further diluted with ethylene glycol to obtain deionized water/ethylene glycol mixing ratio of 60:40. Their findings showed maximum thermal conductivity enhancement of 10.47% was obtained from maximum graphene oxide loading at 45°C in which they have highlighted the effects of sheet sizes to form a percolation pathway according to the percolation theory.

It is important to know the rheological behaviour of various types of fluids. The addition of nanoparticles into base fluids can alter the liquid's thermo-physical properties. Such enhancements are useful in heat transfer applications because of the high transfer enhancement in nanofluids. Therefore, the viscosity of fluid is greatly increased even at very low nanoparticle loadings [28]. Nevertheless, high viscosity properties enable solids such as drill cuttings to be suspended at stagnant conditions and prevents sagging process [29]. The trade-off for having high fluid viscosity incurs higher pumping costs of the fluid. Vajjha and Das [30] had proven nanoparticle concentrations greater than 3 vol% increases cost of pumping. Therefore, consideration for suitable nanoparticle selection should be taken into account for certain applications such as drilling purposes. Ijam et al. [27] compared shear stress and viscosity of graphene oxide-water nanofluids and concluded viscosity to function with respect to temperature. The increase in temperature weakens the intermolecular forces between particles to lower viscosity of nanofluids. Under high shear rate, viscosity of graphene oxide-water decreases exponentially until it reaches a point where it is independent of shear rate.

However, the rheological properties of nanofluids are still widely debatable among researchers. Fluctuating results were reported by various researchers stating addition of nanoparticles gives an increment or decrement of viscosity properties of nanofluids [31]. For example, Wang et al. [32] dispersed graphene nanoparticles at low loadings into ionanofluid and was found to possess slightly lower viscosity at higher temperatures as compared to its counterpart base fluids due to the self-lubrication of graphene nanoparticles. Lu et al. [33] concluded rheological properties to be highly dependent on nanoparticle concentrations. At very low loadings, nanofluids with Newtonian behaviours can produce shear-thinning non-Newtonian behaviour when subjected to high nanoparticle concentrations due to strong particle-particle interactions interrupted by shear rates exceeding a specific critical value.
