**3. Thermal conductivity and rheological models**

#### **3.1. Thermal conductivity models**

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

#### *3.1.1. Maxwell model*

reduced convection of particles. This claim was in good agreement that clustering effect per-

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

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

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,

forms poorly on stability and thermal conductivity of nanofluid [26].

68 Drilling

a percolation pathway according to the percolation theory.

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:

are non-linearizing wun-eian ouuer anu is spueuca in stappe. \"maximum mouen is expresseu as: 
$$k\_{\ast \circ} = \left[ \frac{k\_{\flat} + 2 \, k\_{\flat} + 2 \, (k\_{\flat} - k\_{\flat} \, q)}{k\_{\flat} + 2 \, k\_{\flat} - 2 \, (k\_{\flat} - k\_{\flat} \, q)} \right] k\_{\flat \prime} \tag{1}$$

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 millimetre scales.

#### *3.1.2. Hamilton-Crosser model*

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 as particle to the surface area of the particle.

 *knf* <sup>=</sup> *kbf*[ 1 + *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)

where *n* can be represented with *n* = 3/ψ, ψ is the sphericity of the particle. Generally, n = 3 is taken for spherical particles while n = 6 is considered for cylindrical shape particles.

#### *3.1.3. Bruggeman model*

Unlike Maxwell model, Bruggeman model is applicable for two binary mixtures with no particle concentration limitations. However, Bruggeman model tends to deviate from Maxwell model at higher concentrations. The Bruggeman model is similar to that of Maxwell model as both models use the same assumption basis that the shape of particles are spherical. The Bruggeman model is written as follows:

$$\text{sq}\left(\frac{k\_p - k\_{\text{eff}}}{k\_p + 2\,k\_{\text{eff}}}\right) + (1 - \text{q})\left(\frac{k\_p - k\_{\text{eff}}}{k\_p + 2\,k\_{\text{eff}}}\right) = 0\tag{3}$$

*σ* = *μ*.*γ<sup>n</sup>* (5)

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

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

71

where μ is the fluid viscosity, σ is the shear stress, γ is the shear rate and n is the power law index of the material. Shear thinning behaviour exhibits itself at n < 1 while n > 1 converts the material into a shear thickening fluid. This model is only limited to a small shear rate range as

In this study, hydrogenated base oil (HBO) as base fluid and graphene oxide paste are supplied by a local company supplier. HBO is derived from vegetable oil through catalytic hydrotreating process and contains alkane chain branch between C15-C18. For the characterization of nanoparticles, graphene oxide paste was subjected to FTIR (Perkin Elmer) with wavenumber ranging from 500 to 4000 cm−1 and TEM (Zeiss Libra 200FE) analysis at magnification range at 20,000x to 800,000x values. The HBO and graphene oxide paste were homogenized through a hydrodynamic cavitation unit at a constant flow rate of 1.5 L/min for 3 hours duration with an average of 10 bars pressure. The orifice diameter and length are 1 mm and 30 mm respectively. The schematic diagram is as shown in **Figure 3**. The hydrogenated oil-based nanofluids were transferred to an ultrasonic bath (Bath Ultrasonic Branson 8510E – DTH) for further homogenization.

Thermal conductivity analysis of hydrogenated oil-based nanofluids are carried out with KD2 Pro Thermal Properties Analyser equipped with KS-1 sensor with dimension

**Figure 3.** Schematic diagram of hydrodynamic cavitation unit (HDV: hydrodynamic vessel, MV: mixer vessel, PG:

predictions from the model will deviate at a higher shear rate range.

**4. Experimental study**

**4.1. Homogenization process**

**4.2. Thermal conductivity properties analysis**

pressure gauge, RP: rotary pump, HDP: hydrodynamic pump).

where φ is the volume fraction of nanoparticles dispersed, kbf is the thermal conductivity of base fluid, kp as the thermal conductivity of nanoparticles and keff as the effective thermal conductivity of nanofluid.

#### **3.2. Rheological models**

Rheological models are used to determine the relationships between shear stress and shear rate as different applications possess different characteristics. Non-Newtonian models such as Bingham Plastic model [39] and Power Law model [40] are commonly used to predict rheological behaviours and are considered in this study.

#### *3.2.1. Bingham Plastic model*

Bingham Plastic fluids are unique as it has "infinite" viscosity until adequate stress is applied to initiate flow process. The Bingham Plastic model is as follows:

$$
\sigma = \sigma\_0 + \mu \gamma \tag{4}
$$

where σ is the shear stress, σ<sup>o</sup> is the limiting shear stress, μ is the viscosity and γ is the shear rate. The limiting shear stress is often referred to as Bingham Yield Stress of the material. This model is suitable for concentrated mixtures and colloidal systems possessing Bingham behaviours.

#### *3.2.2. Power Law model*

Generally known as Ostwald model, non-Newtonian materials behave with respect to shear rate to produce two effects, namely shear thinning and shear thickening. Shear thinning yield lower viscosity when subjected to higher shear rate while shear thickening contradicts. The thickening is normally associated with the increase in sample volume and is known as dilatancy. The Power Law model is as follows:

$$
\sigma = \mu \gamma^{\mu} \tag{5}
$$

where μ is the fluid viscosity, σ is the shear stress, γ is the shear rate and n is the power law index of the material. Shear thinning behaviour exhibits itself at n < 1 while n > 1 converts the material into a shear thickening fluid. This model is only limited to a small shear rate range as predictions from the model will deviate at a higher shear rate range.
