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#### **Chapter 7 Provisional chapter**

#### **Heat Transfer of Ferrofluids Heat Transfer of Ferrofluids**

Seval Genc Seval Genc

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65912

#### **Abstract**

Magnetic field‐responsive materials are an important group of smart materials. They can adaptively change their physical properties due to external magnetic field. Magnetic liquids or ferrofluids are colloidal systems of ferro or ferrimagnetic single domain nanoparticles that are dispersed either in aqueous or in organic liquids. Currently, the research field is undergoing a transition taking into account the bulk forces in fluids, which are magnetically nonuniform. These researches enable scientists to develop promising new designs. Today, because of the advancement in technology and limited energy sources, engineering innovations are focused on development of alternative resources instead of current systems. In this chapter, it is aimed to give a brief review of the heat transfer of magnetic fluids based on different types of magnetic nanoparticles as well as some of the research and results of the heat transfer of magnetite‐based ferrofluids. The heat transfer of these materials was investigated under stationary conditions, and the heat transfer coefficient was calculated.

**Keywords:** ferrofluid, thermal conductivity, magnetic nanoparticles

## **1. Introduction**

Traditional heat transfer fluids such as water, oil, and ethylene glycol cause problems in the performance of engineering equipment such as heat exchangers and electronic devices due to their low thermal conductivity. To improve the performance of these devices, fluids with higher thermal conductivity have to substitute these fluids. An inventive way to increase the thermal conductivity of these fluids can be achieved by the use of the nanofluids [1]. A nanofluid is a new class of heat transfer fluids containing nanoparticles with the size range under 100 nm that are uniformly and stably suspended in a liquid. Compared to the thermal conductivity of the base fluids, nanofluids showed dramatic increase in the heat transfer due to the higher thermal

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

conductivity of these nanoparticles [2]. Intensive investigations on nanofluid containing metallic or nonmetallic nanoparticles such as TiO2, Al2O3, Cu, CuO, Ag, and carbon nanotubes are being conducted to enhance their potential applications in heat transfer [3, 4].

Among different kinds of researches on nanofluids containing metallic or nonmetallic nanoparticles, some of the studies have been focused on the nanofluids prepared by dispersing magnetic nanoparticles in a carrier liquid. These are called *ferrofluids*. They are colloidal suspensions of ultrafine single domain superparamagnetic nanoparticles of metallic materials (ferromagnetic materials) such as iron, cobalt, and nickel as well as their oxides (ferrimagnetic materials) such as magnetite (Fe3O4) and ferrites (MnZn, Co ferrites) in either polar or nonpolar liquid carriers [5–7]. These magnetic fluids are a specific subset of smart materials that can adaptively change their physical properties under an externally applied magnetic field [6].

The research and development on the preparation, characterization, and application of the ferrofluids have been studied since mid‐1960s which involve multidisciplinary sciences of chemistry, fluid mechanics, and magnetism. The most important advantage of these fluids is their ability to achieve a wide range of viscosity in a fraction of millisecond. The viscosity in the absence of magnetic field is called the "off‐state" viscosity. The off‐state viscosity of ferrofluids can go up to 2–500 mPa s depending on the concentration of the solid particles and the carrier liquid. Although they can respond to the action of external magnetic fields, stable ferrofluids show a relatively modest magneto‐rheological effect such as an increase in yield strength. Since the particle size of the magnetic phase is very small, under ordinary field strengths, thermal agitation gives rise to Brownian forces that can overcome the alignment of the dipoles. Therefore, ferrofluids exhibit field dependent viscosity, but they exhibit no yield stress (*τy* = 0) under magnetic field. Some properties of the ferrofluid are given in **Table 1**. The field dependent viscosity is given by Eq. 1 [8].

$$\frac{\Delta\eta}{\eta} = \frac{3}{2}\rho \frac{\frac{1}{2}aL(\alpha)}{1 + \frac{1}{2}aL(\alpha)}\sin^2\beta\tag{1}$$

where *α* = *µ*0*MdHV*/*kT.*

Magnetic control of the properties and behavior of these fluids are promising fields for advanced applications and a challenge for basic research and what makes these materials interesting. They are widely used in dynamic loudspeakers, computer hardware, dynamic sealing, electronic packaging, aerospace, and bioengineering [9]. Another important techno‐ logical application of magnetic fluids, which depends on the heat transfer, is its use as a voice coil coolant for modern loudspeakers, high power electric transformers, and in advanced energy conversion systems like solar collectors and magnetically controlled thermosyphons [10–12]. In some of the home appliances, such as refrigerators and ovens, heat transfer techniques are used in order to provide heating or cooling. Controlling the heat transfer in these appliances with these fluids may decrease the energy consumption.


**Table 1.** Some of the properties of ferrofluids [6].

conductivity of these nanoparticles [2]. Intensive investigations on nanofluid containing metallic or nonmetallic nanoparticles such as TiO2, Al2O3, Cu, CuO, Ag, and carbon nanotubes are being

Among different kinds of researches on nanofluids containing metallic or nonmetallic nanoparticles, some of the studies have been focused on the nanofluids prepared by dispersing magnetic nanoparticles in a carrier liquid. These are called *ferrofluids*. They are colloidal suspensions of ultrafine single domain superparamagnetic nanoparticles of metallic materials (ferromagnetic materials) such as iron, cobalt, and nickel as well as their oxides (ferrimagnetic materials) such as magnetite (Fe3O4) and ferrites (MnZn, Co ferrites) in either polar or nonpolar liquid carriers [5–7]. These magnetic fluids are a specific subset of smart materials that can adaptively change their physical properties under an externally applied magnetic field [6].

The research and development on the preparation, characterization, and application of the ferrofluids have been studied since mid‐1960s which involve multidisciplinary sciences of chemistry, fluid mechanics, and magnetism. The most important advantage of these fluids is their ability to achieve a wide range of viscosity in a fraction of millisecond. The viscosity in the absence of magnetic field is called the "off‐state" viscosity. The off‐state viscosity of ferrofluids can go up to 2–500 mPa s depending on the concentration of the solid particles and the carrier liquid. Although they can respond to the action of external magnetic fields, stable ferrofluids show a relatively modest magneto‐rheological effect such as an increase in yield strength. Since the particle size of the magnetic phase is very small, under ordinary field strengths, thermal agitation gives rise to Brownian forces that can overcome the alignment of the dipoles. Therefore, ferrofluids exhibit field dependent viscosity, but they exhibit no yield stress (*τy* = 0) under magnetic field. Some properties of the ferrofluid are given in **Table 1**. The

( )

*L*

Magnetic control of the properties and behavior of these fluids are promising fields for advanced applications and a challenge for basic research and what makes these materials interesting. They are widely used in dynamic loudspeakers, computer hardware, dynamic sealing, electronic packaging, aerospace, and bioengineering [9]. Another important techno‐ logical application of magnetic fluids, which depends on the heat transfer, is its use as a voice coil coolant for modern loudspeakers, high power electric transformers, and in advanced energy conversion systems like solar collectors and magnetically controlled thermosyphons [10–12]. In some of the home appliances, such as refrigerators and ovens, heat transfer techniques are used in order to provide heating or cooling. Controlling the heat transfer in

*L*

a a

a a

j

these appliances with these fluids may decrease the energy consumption.

h

<sup>D</sup> <sup>=</sup>

h

( )

2

 b

<sup>+</sup> (1)

*sin*

conducted to enhance their potential applications in heat transfer [3, 4].

142 Nanofluid Heat and Mass Transfer in Engineering Problems

field dependent viscosity is given by Eq. 1 [8].

where *α* = *µ*0*MdHV*/*kT.*

In the conventional nanofluids, the origin of the enhancement of thermal conductivity was thought to be due to the higher thermal conductivity of the nanoparticles (TiO2, Al2O3, Cu, etc.) than the carrier fluid. Since the thermal conductivity of common magnetic materials (Fe3O4) is relatively low, the investigations did not gain much attention until it was understood that the thermal conductivity of the solid material did not have much effect in the enhancement of the thermal conductivity of the dispersion [13]. By understanding the control of the thermal conductivity of the ferrofluid by magnetic field increased the intensity of the research. Using ferrofluids under an applied magnetic field for the heat transfer enhancement is more advan‐ tageous compared with the conventional nanofluids (nonmagnetic nanofluids). The advan‐ tages of the ferrofluids over conventional nanofluids can be summarized as thermomagnetic convection is more intense than the gravitational one, and the thermal conductivity and viscosity are tunable under magnetic field.

## **2. Preparation of ferrofluids**

Although pure metals (Fe, Co, Ni) possess the highest saturation magnetization, they are extremely sensitive to oxidation, hence the magnetic particles such as ferrites like magnetite (Fe3O4), maghemite (*γ*‐Fe2O3), or others (stoichiometric formula: MO·Fe2O3, where M is a divalent ion, M = Mn, Zn, Ni, Co, Fe) are commonly used in ferrofluids. And among these, nano‐sized iron oxide is the most widely used magnetic phase in ferrofluids. Various ap‐ proaches have been explored for synthesis and characterization of high quality magnetic iron oxide nanoparticles. For example, sol‐gel pyrolysis method was performed by Laokul et al. [14]. Synthesis of nanoparticles by thermal reductive decomposition method was performed by various scientists [15, 16]. Waje et al. performed mechanical alloying technique [17]. Hydrothermal technique was also used by various scientists to synthesize ferrite nanoparti‐ cles [18, 19]. However, the chemical method of coprecipitation of ferrous and ferric ions from solutions by addition of an alkali is a method which is very often used to prepare nanoparticles due to its low cost and simplicity [20]. Size reduction could be another method where magnetic powder of micron size is mixed with a solvent and a dispersant in a ball mill in order to grind for a period of several weeks [21].

The function of the carrier liquid is to provide a medium in which the magnetic powder is suspended. Ferrofluids used in different research and technology fields have been synthesized in carrier liquids such as water, silicone oil, synthetic or semi‐synthetic oil, mineral oil, lubricating oil, kerosene, and combinations of these and many other polar liquids [22–24]. Boiling temperature, vapor pressure at elevated temperature, and freezing point are important parameters to be considered when choosing the carrier liquid. The carrier liquid should be non‐reactive with the magnetic phase and also with the material used in the device. In terms of the heat transfer applications, the choice of the carrier fluid for the ferrofluid needs some additional requirements such as high conductivity, high heat capacity, and high thermal expansion coefficient. Water, oils, and ethylene glycol are considered as conventional heat transfer fluids and these can be good candidates for the carrier liquids. In recent years, studies on ferrofluids using ionic liquids have been reported, which seems to be a promising field of study [23, 24].

Colloidal stability of the ferrofluids is important in the technological applications. The stability is obtained by minimizing the agglomeration, which is maintained by the addition of the surfactants. The additives must be chosen to match the dielectric properties of the carrier liquid. Various surfactants such as silica, chitosan, polyvinyl alcohol (PVA), and ethylene glycol are usually used to coat the nanoparticles and to enhance dispersibility in aqueous medium [25– 28]. Oleic acid (OA) is a commonly used surfactant to stabilize magnetic nanoparticles synthesized by traditional coprecipitation method [22]. Antioxidation additives may also be added to prevent oxidation. In water‐based MR fluids, pH control additives are also used.

Magnetic nanoparticles tend to aggregate due to strong magnetic dipole‐dipole attraction between particles. Stability of the magnetic colloid depends on the thermal contribution and the balance between attractive (van der Waals and dipole‐dipole) and repulsive (steric and electrostatic) interactions. Under the magnetic field, the magnetic energy derives the particles to higher intensity regions; on the other hand, thermal energy forces the particles to wander around in the whole liquid. The stability against segregation is favored by the high ratio of the thermal energy to the magnetic energy. Stability against settling due to gravitational field is given by the ratio between gravitational energy and magnetic energy [8]. The two basic attractive interactions between the magnetic particles are dipole‐dipole and van der Waals‐ London interactions. The ratio of thermal energy (*kT*) to dipole‐dipole contact energy (*E*dipole = (*µ*0*M*<sup>2</sup> /12)*V*) must be greater than unity. The particle diameter is given by *D* ≤ (72 *kT*/*πµ*0*M*<sup>2</sup> ) 1/3, and the particle size is calculated as *D* ≤ 7.8 nm. The normal ferrofluids with the particle size of 10 nm are in the limits of agglomeration. Van der Waals forces arise due to the fluctuating electric dipole‐dipole forces. Preventing the contact of the particles is another necessity if a stable colloid is to be obtained [8].

The Brownian motion, electrostatic repulsion, and steric repulsion are the main mechanisms supporting the ferrofluid colloidal stability. Electrostatic interaction is the dominant mecha‐ nism in ionic ferrofluids, whereas steric repulsion is the dominant mechanism supporting the colloidal stability in organic‐based ferrofluids [23]. The agglomeration of particles suspended in a liquid can be prevented by creating mutually repelling charged double layers or by physically preventing the close approach of particles by steric hindrance which is provided by the surfactant molecules adsorbed onto the particle surface [8, 23]. As the thickness of the adsorbed polymer is increased, the stability of the dispersion increases [8, 23]. Wang and Huang showed that by retaining excess oleic acid in their ferrofluid, stable magnetic colloid was achieved by steric repulsion [8, 23].

## **3. Thermal conductivity of ferrofluids**

## **3.1. Experimental investigations**

The function of the carrier liquid is to provide a medium in which the magnetic powder is suspended. Ferrofluids used in different research and technology fields have been synthesized in carrier liquids such as water, silicone oil, synthetic or semi‐synthetic oil, mineral oil, lubricating oil, kerosene, and combinations of these and many other polar liquids [22–24]. Boiling temperature, vapor pressure at elevated temperature, and freezing point are important parameters to be considered when choosing the carrier liquid. The carrier liquid should be non‐reactive with the magnetic phase and also with the material used in the device. In terms of the heat transfer applications, the choice of the carrier fluid for the ferrofluid needs some additional requirements such as high conductivity, high heat capacity, and high thermal expansion coefficient. Water, oils, and ethylene glycol are considered as conventional heat transfer fluids and these can be good candidates for the carrier liquids. In recent years, studies on ferrofluids using ionic liquids have been reported, which seems to be a promising field of

Colloidal stability of the ferrofluids is important in the technological applications. The stability is obtained by minimizing the agglomeration, which is maintained by the addition of the surfactants. The additives must be chosen to match the dielectric properties of the carrier liquid. Various surfactants such as silica, chitosan, polyvinyl alcohol (PVA), and ethylene glycol are usually used to coat the nanoparticles and to enhance dispersibility in aqueous medium [25– 28]. Oleic acid (OA) is a commonly used surfactant to stabilize magnetic nanoparticles synthesized by traditional coprecipitation method [22]. Antioxidation additives may also be added to prevent oxidation. In water‐based MR fluids, pH control additives are also used.

Magnetic nanoparticles tend to aggregate due to strong magnetic dipole‐dipole attraction between particles. Stability of the magnetic colloid depends on the thermal contribution and the balance between attractive (van der Waals and dipole‐dipole) and repulsive (steric and electrostatic) interactions. Under the magnetic field, the magnetic energy derives the particles to higher intensity regions; on the other hand, thermal energy forces the particles to wander around in the whole liquid. The stability against segregation is favored by the high ratio of the thermal energy to the magnetic energy. Stability against settling due to gravitational field is given by the ratio between gravitational energy and magnetic energy [8]. The two basic attractive interactions between the magnetic particles are dipole‐dipole and van der Waals‐ London interactions. The ratio of thermal energy (*kT*) to dipole‐dipole contact energy (*E*dipole =

/12)*V*) must be greater than unity. The particle diameter is given by *D* ≤ (72 *kT*/*πµ*0*M*<sup>2</sup>

and the particle size is calculated as *D* ≤ 7.8 nm. The normal ferrofluids with the particle size of 10 nm are in the limits of agglomeration. Van der Waals forces arise due to the fluctuating electric dipole‐dipole forces. Preventing the contact of the particles is another necessity if a

The Brownian motion, electrostatic repulsion, and steric repulsion are the main mechanisms supporting the ferrofluid colloidal stability. Electrostatic interaction is the dominant mecha‐ nism in ionic ferrofluids, whereas steric repulsion is the dominant mechanism supporting the colloidal stability in organic‐based ferrofluids [23]. The agglomeration of particles suspended in a liquid can be prevented by creating mutually repelling charged double layers or by physically preventing the close approach of particles by steric hindrance which is provided by

) 1/3,

study [23, 24].

144 Nanofluid Heat and Mass Transfer in Engineering Problems

(*µ*0*M*<sup>2</sup>

stable colloid is to be obtained [8].

Thermal conductivity of ferrofluids has gained much attention in the last decade due to the significant enhancement compared to the nonmagnetic nanofluids. The increase in the thermal conductivity can occur both with and without the applied magnetic field. Experimental studies show that the change in the off‐state (when there is no magnetic field) thermal conductivity of ferrofluids could be due to volume fraction of magnetic phase, particle size distribution, temperature, surfactant, etc. On the other hand, when the magnetic field is applied, besides the factors mentioned above, the magnitude and direction of the applied magnetic field affect the thermal conductivity of the ferrofluids.

In the experimental studies of the thermal conductivity of ferrofluids, it has been observed that both the on‐state and off‐state thermal conductivities increase with the increase in the volume fraction of the magnetic phase. When the literature was reviewed, it was seen that most of the ferrofluids synthesized with magnetite (Fe3O4) which was produced by copre‐ ciptation method has been studied. Abareshi and coworkers synthesized ferrofluids by dispersing Fe3O4 nanoparticles in water [29]. They reported an increase of 11.5% in the off‐ state thermal conductivity as the particle loading increased to 3 vol% at 40°C. This increase was observed when the magnetic field was applied parallel to the heat flux. The study of Li et al. was also performed with water‐based ferrofluids and an increase in the on‐state (magnetic field, *H* = 19 kA/m) thermal conductivity of 11% for 1 vol% and 25% for 5 vol% magnetic nanoparticles was reported [30]. Philip et al. and Shima et al. [30, 31] investigated the thermal conductivity of kerosene‐based ferrofluids synthesized with Fe3O4 nanoparticles. When the magnetic field was applied parallel to the heat flux, they discovered a dramatic increase in the thermal conductivity. For a volume fraction of 6.3%, the increase was 300% at *H* = 7 kA/m field strength. No increase was observed when the magnetic field was applied perpendicular to the heat flux. The reason may be that the different directions of external magnetic field lead to quite different morphologies of the magnetic fluids that exerted quite different effects on the energy transport process inside the magnetic fluid [31]. They further explained that the chains formed by the particles provided more effective bridges for energy transport inside the ferrofluid along the direction of temperature gradient and as a result, the thermal process in the ferrofluid was enhanced. The anisotropic property of thermal conductivity was addressed in the theoretical study by Fu et al., and Blums et al [10, 32]. Blums et al. predict anisotropy of thermal conductivity in ferrofluids in the presence of a magnetic field [10]. In the research conducted by Gavili et al. the thermal conductivity of ferrofluids containing Fe3O4 nanoparticles suspended in deionized water under magnetic field was experimentally investigated [33]. According to their results, a ferrofluid with 5.0% volume fraction of nanoparticles with an average diameter of 10 nm enhanced the thermal conductivity more than 200% at 1000 Gauss magnetic field [33].

As it is seen from all the research and reports, it is evident the experimental results have been heterogeneous. The difference in the experimental outcomes may be due to magnetization, size distribution of the particles, the type of the carrier liquid, etc. The effect of the carrier liquid showed that the thermal conductivity ratio is higher for carrier liquid with a low thermal conductivity like common hydrocarbons. However, the absolute thermal conductivity of ferrofluid is higher for a carrier liquid with a high thermal conductivity.

In recent years, studies have been carried out to understand the thermal conductivity of ferrofluids synthesized by magnetic phase other than Fe3O4, especially with carbon nanotubes (CNT) has also been investigated by many scientists. Hong et al. [34] and Wensel et al. [35] experimentally measured the thermal conductivity of single wall carbon nanotubes coated by Fe2O3 nanoparticles suspended in water and they observed an approximately 10% increase in the thermal conductivity with 0.02% particle loading. Wright et al. reported thermal conductivity enhancement of single wall carbon nanotubes coated by Ni nanoparticles suspended in water [36]. Sundar et al. measured the thermal conductivity enhancement of the hybrid ferrofluid which was composed of carbon nanotube (CNT)—Fe3O4 and water [37]. They observed a thermal conductivity enhancement of 13.88–28.46% at 0.3% volume concentration in the temperature range of 25–60°C. Shahsavar et al. analyzed the thermal conductivity behavior of Fe3O4 and CNT hybrid ferrofluids and observed that the highest enhancement in the thermal conductivity was about 151% for 0.9% ferrofluid and 1.35% CNT [38].

The review of the literature on the experimental studies of the thermal conductivity of ferrofluids revealed that the thermal conductivity is enhanced by the volume fraction of the magnetic phase and the applied magnetic field. Next chapter will discuss the reasons for the abnormal enhancement in the thermal conductivity of the ferrofluid under the influence of applied magnetic field.

#### **3.2. Mechanisms of heat transfer enhancement**

In the thermal conductivity of conventional nanofluids and ferrofluids, the most discussed mechanisms have been Brownian motion and formation of particle chain/cluster structure [39]. The Brownian motion indicates the random movement of particles dispersed in liquid or gas, and the motion is due to collision with base fluid molecules, which makes particles undergo a random walk motion [40]. The Brownian motion could contribute to the thermal conductivity enhancement in two ways, namely, the direct contribution due to motion of nanoparticles that transports heat (diffusion of nanoparticles) and the indirect contribution due to the so called micro‐convection of fluid surrounding individual nanoparticles [40]. The diffusion of magnetic nanoparticles plays an important role at a low volume fraction (*ϕ* < 2%), which could be explained by the effective medium (Maxwell) theory rather than the effects associated with the Brownian motion‐induced hydrodynamics. The effective medium or mean‐field theory of Maxwell, which describes the effective macroscopic properties of the composite material as a function of the particle fraction and the material properties of the components, is most often used to analyze the thermal conductivity results of nanofluid experiments. For a nanofluid with non‐interacting spherical nanoparticles with low volume fraction, the theory predicts (Eq. 2)

field was experimentally investigated [33]. According to their results, a ferrofluid with 5.0% volume fraction of nanoparticles with an average diameter of 10 nm enhanced the thermal

As it is seen from all the research and reports, it is evident the experimental results have been heterogeneous. The difference in the experimental outcomes may be due to magnetization, size distribution of the particles, the type of the carrier liquid, etc. The effect of the carrier liquid showed that the thermal conductivity ratio is higher for carrier liquid with a low thermal conductivity like common hydrocarbons. However, the absolute thermal conductivity of

In recent years, studies have been carried out to understand the thermal conductivity of ferrofluids synthesized by magnetic phase other than Fe3O4, especially with carbon nanotubes (CNT) has also been investigated by many scientists. Hong et al. [34] and Wensel et al. [35] experimentally measured the thermal conductivity of single wall carbon nanotubes coated by Fe2O3 nanoparticles suspended in water and they observed an approximately 10% increase in the thermal conductivity with 0.02% particle loading. Wright et al. reported thermal conductivity enhancement of single wall carbon nanotubes coated by Ni nanoparticles suspended in water [36]. Sundar et al. measured the thermal conductivity enhancement of the hybrid ferrofluid which was composed of carbon nanotube (CNT)—Fe3O4 and water [37]. They observed a thermal conductivity enhancement of 13.88–28.46% at 0.3% volume concentration in the temperature range of 25–60°C. Shahsavar et al. analyzed the thermal conductivity behavior of Fe3O4 and CNT hybrid ferrofluids and observed that the highest enhancement in

the thermal conductivity was about 151% for 0.9% ferrofluid and 1.35% CNT [38].

applied magnetic field.

**3.2. Mechanisms of heat transfer enhancement**

The review of the literature on the experimental studies of the thermal conductivity of ferrofluids revealed that the thermal conductivity is enhanced by the volume fraction of the magnetic phase and the applied magnetic field. Next chapter will discuss the reasons for the abnormal enhancement in the thermal conductivity of the ferrofluid under the influence of

In the thermal conductivity of conventional nanofluids and ferrofluids, the most discussed mechanisms have been Brownian motion and formation of particle chain/cluster structure [39]. The Brownian motion indicates the random movement of particles dispersed in liquid or gas, and the motion is due to collision with base fluid molecules, which makes particles undergo a random walk motion [40]. The Brownian motion could contribute to the thermal conductivity enhancement in two ways, namely, the direct contribution due to motion of nanoparticles that transports heat (diffusion of nanoparticles) and the indirect contribution due to the so called micro‐convection of fluid surrounding individual nanoparticles [40]. The diffusion of magnetic nanoparticles plays an important role at a low volume fraction (*ϕ* < 2%), which could be explained by the effective medium (Maxwell) theory rather than the effects associated with the Brownian motion‐induced hydrodynamics. The effective medium or mean‐field theory of Maxwell, which describes the effective macroscopic properties of the composite material as a function of the particle fraction and the material properties of the components, is most often

conductivity more than 200% at 1000 Gauss magnetic field [33].

146 Nanofluid Heat and Mass Transfer in Engineering Problems

ferrofluid is higher for a carrier liquid with a high thermal conductivity.

$$\frac{\kappa}{\lambda\_f} = \frac{1 + 2\beta\Theta}{1 - \beta\Theta} \tag{2}$$

where *ϕ* is the nanoparticle volume fraction, *κ*p and *κ*<sup>f</sup> are thermal conductivity of the particle and the fluid, respectively. *β* = (*κ*p‐*κ*<sup>f</sup> )/( *κ*p + 2*κ*<sup>f</sup> ), and (*κ*p‐*κ*<sup>f</sup> ) is the difference between the thermal conductivities of the nanoparticle and the base fluid. However, in the study done by Vadasz et al. [41] and Keblinski et al. [42], the results of the thermal conductivity measurements showed divergence from the effective medium theory. One possible discrepancy between effective medium theory and the experimental results is the interparticle interactions, which can result in the formation of chain and cluster‐like formations. Philip and coworkers showed that the micro‐convection of the fluid medium around randomly moving nanoparticles did not affect the thermal conductivity of a nanofluid and the microconvection model overesti‐ mated the thermal conductivity values [43]. According to them, the conductivity enhancement in the ferrofluid at high volume fraction (*ϕ* < 2%) was due to the presence of dimmers or trimmers in the fluid. These results were in a reasonable agreement with the Maxwell‐Gannet model, especially at higher volume fractions. Clusters or chains of the particles may form heat bridges [31]. The form and magnitude of these structures vary and depend not only on the material of the carrier medium and the particles but also on the shape and size of the particles [31].

In ferrofluids, the interparticle interactions have even more important impact on the properties of the fluid, due to the chain‐like formation of the particles caused by the magnetic dipole interactions. The effect of this interaction can be seen, for example, in changes in the viscosity of ferrofluids, which depends on interparticle interaction [7, 44].

Magnetically induced structure formation only arises if the magnetic energy of the particles is larger than their thermal energy. The mechanism of thermal conductivity enhancement can be explained as follows: The magnetic particles in the ferrofluid are single domain and super‐ paramagnetic with magnetic moment *m* as mentioned above [28]. The interparticle dipole‐ dipole interaction, which is also called dipolar coupling, refers to the interaction between magnetic dipoles. The potential energy of the interaction *Ud* is given by Eq. (3),

$$U\_{d}\left(\dot{y}\right) = \begin{bmatrix} 3\left(m\_{l}\cdot r\_{\dot{y}}\right)\left(m\_{f}\cdot r\_{\dot{y}}\right) \\\\ r\_{\dot{y}}^{\mathcal{S}} \end{bmatrix} \Big/ \begin{array}{l} \left(m\_{l}\cdot m\_{j}\right) \Big/ \\ r\_{\dot{y}}^{\mathcal{S}} \end{array} \tag{3}$$

Suppose *mi* and *mj* are two magnetic moments in space and *rij* (= *ri* ‐ *rj* ) is the distance between the *i* th and the *j* th particles. The magnetic moments are oriented in random directions in the absence of magnetic field and the nanoparticles are influenced by the Brownian motion as the thermal energy exceeds the magnetic dipole attraction ( < ). In the presence of a magnetic field, the magnetic dipolar interaction becomes strong enough to dominate the thermal energy so that the magnetic particles start aligning in the direction of the magnetic field [31]. **Figure 1** gives the schematic drawing of the clustering/chain‐like formation under magnetic field.

**Figure 1.** Schematic drawing of the chain‐like formation of the magnetic particles in the fluid. (a) No magnetic field. (b) Applied magnetic field.

The lengths of the chains depend on the magnitude of the magnetic field. As the magnetic field increases, the particles start forming short chains along the direction of the magnetic field and the chains get longer as the magnetic field increases. Based on Philip's theory due to linear chain‐like structures of the magnetic nanoparticles, the percolation theory could support the abnormal enhancement of the ferrofluid. They stated that the maximum enhancement was observed when the chain‐like aggregates were well dispersed without clumping [31].

Although the thermal conductivity of ferrofluids enhances with increasing magnetic field, there are reports regarding a decrease in the thermal conductivity of these fluids. Shima et al. observed decrease in the thermal conductivity above 82 Gauss magnetic field [31]. They attributed this decrease to the "zippering" of the chains. The linear and thick aggregates with the aspect ratio due to zippering can collapse to the bottom of the cell, and hence the thermal conductivity cannot be measured. Gavili and coworkers observed that the thermal conduc‐ tivity dramatically decreased in the presence of magnetic field with increasing temperature. When the temperature of the ferrofluid increases, the chain‐like structure is broken due to the increase in the thermal velocity and consequently the thermal conductivity decreases [33].

Theoretical and experimental studies related with the thermal radiation and convection of the heat transfer of nanofluids have started to gain more attention in the recent years especially, in the field of engineering applications such as solar collectors and in space applications [45– 47].Thermal convection in magnetic fluids heated from below subjected to an external magnetic field causes a convection‐driving mechanism. The temperature difference causes a gradient in the magnetic field and as a result a magnetic force appears. Beyond a certain threshold a thermomagnetic convection is generated.

## **4. Experimental study**

thermal energy exceeds the magnetic dipole attraction ( < ). In the presence of a magnetic field, the magnetic dipolar interaction becomes strong enough to dominate the thermal energy so that the magnetic particles start aligning in the direction of the magnetic field [31]. **Figure 1** gives the schematic drawing of the clustering/chain‐like formation under

**Figure 1.** Schematic drawing of the chain‐like formation of the magnetic particles in the fluid. (a) No magnetic field. (b)

The lengths of the chains depend on the magnitude of the magnetic field. As the magnetic field increases, the particles start forming short chains along the direction of the magnetic field and the chains get longer as the magnetic field increases. Based on Philip's theory due to linear chain‐like structures of the magnetic nanoparticles, the percolation theory could support the abnormal enhancement of the ferrofluid. They stated that the maximum enhancement was

Although the thermal conductivity of ferrofluids enhances with increasing magnetic field, there are reports regarding a decrease in the thermal conductivity of these fluids. Shima et al. observed decrease in the thermal conductivity above 82 Gauss magnetic field [31]. They attributed this decrease to the "zippering" of the chains. The linear and thick aggregates with the aspect ratio due to zippering can collapse to the bottom of the cell, and hence the thermal conductivity cannot be measured. Gavili and coworkers observed that the thermal conduc‐ tivity dramatically decreased in the presence of magnetic field with increasing temperature. When the temperature of the ferrofluid increases, the chain‐like structure is broken due to the increase in the thermal velocity and consequently the thermal conductivity decreases [33].

Theoretical and experimental studies related with the thermal radiation and convection of the heat transfer of nanofluids have started to gain more attention in the recent years especially, in the field of engineering applications such as solar collectors and in space applications [45– 47].Thermal convection in magnetic fluids heated from below subjected to an external magnetic field causes a convection‐driving mechanism. The temperature difference causes a gradient in the magnetic field and as a result a magnetic force appears. Beyond a certain

threshold a thermomagnetic convection is generated.

observed when the chain‐like aggregates were well dispersed without clumping [31].

magnetic field.

148 Nanofluid Heat and Mass Transfer in Engineering Problems

Applied magnetic field.

## **4.1. Apparatus and data analysis**

An experimental setup is built according to the ISO 8301 numbered "Thermal insulation‐ determination of steady state thermal resistance and related properties—Heat flow meter apparatus" standard has been used in the heat transfer experiments. Experimental setup has been established as single‐specimen asymmetrical configuration according to the standard shown in **Figure 2**.

**Figure 2.** Single‐specimen asymmetrical configuration (U', U" are the cooling and heating units, respectively, and *H* = heat flux meter).

Schematic drawing containing the requirements of experimental setup such as testing unit, two water baths, data acquisition system, and computer is given in **Figure 3**. The testing unit was heated and cooled by water bath.

**Figure 3.** The illustration of experimental setup.

A more detailed representation of the testing unit given in **Figure 4**, which consisted of a space for sample, heat flux sensors, thermocouples, heating‐cooling sources, and polyurethane insulating material from outside to inside. The magnetic field was applied by neodymium permanent magnets parallel to the temperature gradient. The maximum magnetic field obtained was approximately 140 Gauss. One of the uncertainties of the setup could be the non‐ uniformity of magnetic field. The magnetic field was calculated as the average of the field from three different points on the radial direction. Another uncertainty could be heat loss. Although the experimental setup was isolated, there could still be some heat loss. For reproducibility of the data, each measurement was performed for five ferrofluid samples.

**Figure 4.** Testing unit.

In order to determine the temperature range at which the thermal conductivity of ferrofluids is more effective for different applications, the experiments were done in two different temperature intervals; from ‐20 to 0°C, and from 0 to 50°C and two different temperature differences. The temperatures difference between the hot and cold surfaces of the setup are given in **Table 2**.


**Table 2.** Temperature intervals and temperature differences at which the experiments were performed.

Besides the effect of temperature, following variables were considered during investigation of heat transfer of ferrofluids, different carrier liquid and concentration of the magnetic phase.

The thermal conductivity measurements have been carried out when the water baths came a steady state temperature. The heat transfer coefficient has been calculated by using Eq. 4,

$$
\kappa = \frac{A}{Q} \frac{\Delta T}{\Delta \mathbf{x}} \tag{4}
$$

where *A* is the cross‐sectional area, *Q* is heat flux, Δ*T* is the temperature difference, and Δ*x* is the height of the measuring cell.

#### **4.2. Ferrofluid preparation**

A more detailed representation of the testing unit given in **Figure 4**, which consisted of a space for sample, heat flux sensors, thermocouples, heating‐cooling sources, and polyurethane insulating material from outside to inside. The magnetic field was applied by neodymium permanent magnets parallel to the temperature gradient. The maximum magnetic field obtained was approximately 140 Gauss. One of the uncertainties of the setup could be the non‐ uniformity of magnetic field. The magnetic field was calculated as the average of the field from three different points on the radial direction. Another uncertainty could be heat loss. Although the experimental setup was isolated, there could still be some heat loss. For reproducibility of

In order to determine the temperature range at which the thermal conductivity of ferrofluids is more effective for different applications, the experiments were done in two different temperature intervals; from ‐20 to 0°C, and from 0 to 50°C and two different temperature differences. The temperatures difference between the hot and cold surfaces of the setup are

**Between 0 and -20°C interval Between 0 and 50°C interval** Temperature difference, 20 K Temperature difference, 35 K Temperature difference, 15 K Temperature difference, 20 K

**Table 2.** Temperature intervals and temperature differences at which the experiments were performed.

Besides the effect of temperature, following variables were considered during investigation of heat transfer of ferrofluids, different carrier liquid and concentration of the magnetic phase.

The thermal conductivity measurements have been carried out when the water baths came a steady state temperature. The heat transfer coefficient has been calculated by using Eq. 4,

the data, each measurement was performed for five ferrofluid samples.

150 Nanofluid Heat and Mass Transfer in Engineering Problems

**Figure 4.** Testing unit.

given in **Table 2**.

Synthesis of stable and well dispersible MR fluid is extremely important for mechanical and heat transfer applications. Synthesis of stable ferrofluid depends on concentration and viscosity of carrier fluid, concentration of magnetic phase, particle size, and surfactants. The ferrofluids in this study were synthesized in water and silicone oil (viscosity 350 cSt) with as received magnetite Fe3O4 nanopowder from Aldrich. The particle size was approximately 50 nm. Samples were synthesized as volumetric percentages of 5 and 20%. Surfactant was also added to prevent sedimentation. The names and description of the ferrofluids are given in **Table 3**.


**Table 3.** Description of the MR fluids used in this research.

In order to have a stable dispersion ball milling which could break up the agglomerated, was applied. Ball milling procedure was conducted with yttria stabilized zirconia grinding media with 0.5 mm diameter.

## **5. Results and discussion**

#### **5.1. Analysis of thermal conductivity in the temperature interval between 0 and -50°C.**

In this part of the study, base liquid, volume fraction (5 and 20 vol% Fe3O4), temperature (Δ*T* = 20 and 35 K), and dependence of the thermal conductivity of the ferrofluids were investigated.

## **5.2. Carrier liquid dependence of the 5 vol% Fe3O4-based ferrofluids**

Heat transfer coefficients of ferrofluids were analyzed for two different temperature intervals, and in each temperature interval, the thermal conductivity of the ferrofluids was investigated for two different temperature differences. The temperature intervals were chosen as 0 to 50°C and ‐20 to 0°C, and the temperature differences are Δ*T* = 35 K and Δ*T* = 20 K for the first interval and Δ*T* = 15 K and Δ*T* = 20 K for the second interval mentioned above.

In the first part of the study, the heat transfer was investigated for 5Fe3O4‐S and 5Fe3O4‐W at a temperature difference of 20 K. The increase in heat transfer coefficient in the presence of magnetic field of 134 Gauss was 7 and 18% for 5 Fe3O4‐S and 5 Fe3O4‐W‐type ferrofluids, respectively (**Figure 5**).

**Figure 5.** Thermal conductivity at 0–50°C interval and the temperature difference Δ*T* = 20 K.

As discussed above, the heat transfer increased as we increased the magnetic field. Although this increase could be attributed to the chain formation of the iron particles in the fluid, the effect of magnetic field is still not very clear. In **Figure 6**, the percent change in the thermal conductivity is given at temperature difference of Δ*T* = 35 K in temperature interval of 0–50°C

**Figure 6.** Thermal conductivity at 0–50°C interval and the temperature difference Δ*T* = 35 K.

As given in **Figure 6**, heat transfer coefficients for 5 Fe3O4‐S and 5 Fe3O4‐W at the highest magnetic field were measured as 0.47 and 0.56 W/m K, respectively. In addition, the increase in the thermal conductivity for 5 Fe3O4‐S was about 23% and water based was 5%. In either temperature differences (Δ*T* = 35 and 20 K), the thermal conductivity coefficient of silicone‐ based ferrofluid is less than that of the water‐based ferrofluid. Since the volume concentration of the magnetic phase is small, the thermal conductivity of the base liquid could be a factor in the increase of the thermal conductivity of ferrofluid. The thermal conductivity of water is more than that of the silicone oil.

## **5.3. Volume percent and temperature dependence of thermal conductivity**

and ‐20 to 0°C, and the temperature differences are Δ*T* = 35 K and Δ*T* = 20 K for the first interval

In the first part of the study, the heat transfer was investigated for 5Fe3O4‐S and 5Fe3O4‐W at a temperature difference of 20 K. The increase in heat transfer coefficient in the presence of magnetic field of 134 Gauss was 7 and 18% for 5 Fe3O4‐S and 5 Fe3O4‐W‐type ferrofluids,

and Δ*T* = 15 K and Δ*T* = 20 K for the second interval mentioned above.

**Figure 5.** Thermal conductivity at 0–50°C interval and the temperature difference Δ*T* = 20 K.

**Figure 6.** Thermal conductivity at 0–50°C interval and the temperature difference Δ*T* = 35 K.

As discussed above, the heat transfer increased as we increased the magnetic field. Although this increase could be attributed to the chain formation of the iron particles in the fluid, the effect of magnetic field is still not very clear. In **Figure 6**, the percent change in the thermal conductivity is given at temperature difference of Δ*T* = 35 K in temperature interval of 0–50°C

respectively (**Figure 5**).

152 Nanofluid Heat and Mass Transfer in Engineering Problems

In this part of the study, the volume dependence of the thermal conductivity was investigated. This investigation was performed at Δ*T* = 35 and 20 K as well. In **Figure 7**, it is observed that as the volume fraction is increased, the thermal conductivity coefficients for 5 Fe3O4‐S and 20 Fe3O4‐S ferrofluid also increased for Δ*T* = 35 K. At the highest magnetic field, the thermal conductivity coefficient, **k**, is measured as 0.51 and 0.47 W/mK for 20 and 5 vol%, respectively. The percent increase in both of the fluids was the same, which was approximately 24%.

When the temperature difference is kept smaller (Δ*T* = 20 K) in the same temperature interval, the thermal conductivity coefficient, **k**, is 0.51 and 0.47 W/m K (**Figure 10**) at 134 Gauss which were the same as the coefficients in Δ*T* = 35 K and the percent change of the thermal conduc‐ tivity for these two intervals was almost the same. When we compare **Figures 7** and **8** at a magnetic field of 134 Gauss we saw that the **k** values were the same for 20 and 35 K temperature differences. However, at zero magnetic field, they were different which made a difference in the percent increase. The percent increase in the 35 K difference is more than that of the 20 K difference.

**Figure 7.** Volume dependence of thermal conductivity between 0 and 50°C temperature interval and 35 K temperature difference.

**Figure 8.** Volume dependence of thermal conductivity between 0 and 50°C temperature interval and 20 K temperature difference.

**Figure 9.** Temperature difference dependence of 20 vol% Fe3O4 and silicone oil‐based ferrofluid.

**Figure 10.** Temperature difference dependence of 5 vol% Fe3O4 and silicone oil‐based ferrofluid.

**Figures 9** and **10** give a clearer view of the temperature difference dependence of the thermal conductivity for 20 Fe3O4‐S and 5 Fe3O4‐S‐type ferrofluid, respectively. In both graphs, the thermal conductivity enhancement is more for 35 K temperature difference. As the magnetic field increased, the k values reached the same value irrespective of the temperature difference.

## **5.4. Analysis of thermal conductivity in the temperature interval between -20 and 0°C.**

In this part of the study, the thermal conductivity of the for 5 Fe3O4‐S and 20 Fe3O4‐S ferrofluids were investigated in the temperature interval of ‐20 to 0°C in which the temperature differences were kept as 15 and 20 K.

#### *5.4.1. Volume percent and temperature dependence of thermal conductivity*

**Figure 8.** Volume dependence of thermal conductivity between 0 and 50°C temperature interval and 20 K temperature

**Figure 9.** Temperature difference dependence of 20 vol% Fe3O4 and silicone oil‐based ferrofluid.

**Figure 10.** Temperature difference dependence of 5 vol% Fe3O4 and silicone oil‐based ferrofluid.

difference.

154 Nanofluid Heat and Mass Transfer in Engineering Problems

The next set of measurements involved a lower temperature interval, such as from ‐20 to 0°C. In this interval, the temperature differences were taken as 15 and 20 K. **Figure 11** shows the change in the thermal conductivity with respect to the magnetic field when the temperature gradient was 20 K. The percentage decrease in the thermal conductivity of ferrofluids for 5 and 20 vol% ferrofluids at 134 Gauss was 33 and 34%, respectively. The volume dependence of the thermal conductivity was also observed in these measurements. The 20 vol% ferrofluid had a higher thermal conductivity. **Figure 12** shows the change in the thermal conductivity with respect to the magnetic field under a 15 K temperature difference. Unlike the results obtained in the higher temperature intervals, there was a very small increase in the conductivity, followed by a slight decrease as the magnetic field increased. All the fluids showed a similar trend. The same amount of decrease was observed in this range for two different magnetic phase concentrations. The percentage decrease in the thermal conductivity for 5 Fe3O4‐S and 20 Fe3O4‐S type ferrofluids at 134 Gauss was 2 and 3%, respectively. The volume dependence

**Figure 11.** Volume dependence of thermal conductivity between ‐20 and 0°C temperature interval and 20 K tempera‐ ture difference.

of the thermal conductivity was also observed in these measurements. The for 20 Fe3O4‐S type ferrofluid had a thermal conductivity of 42 W/K.m at 134 Gauss whereas for 5 Fe3O4‐S type ferrofluid had 37 W/K.m.

**Figure 12.** Volume dependence of thermal conductivity between ‐20 and 0°C temperature interval and temperature difference of 15 K.

The thermal conductivity depended on the temperature difference at very low magnetic fields. As the magnetic field increases the thermal conductivity became irrespective of the tempera‐ ture. This behavior is observed both for 5 Fe3O4‐S and 20 Fe3O4‐S (**Figures 13** and **14**, respec‐ tively).

**Figure 13.** Temperature difference dependence of 5 vol% Fe3O4 and silicone oil‐based ferrofluid (5Fe3O4‐S).

**Figure 14.** Temperature difference dependence of 20 vol% Fe3O4 and silicone oil‐based ferrofluid (20Fe3O4‐S).

## **6. Conclusion**

of the thermal conductivity was also observed in these measurements. The for 20 Fe3O4‐S type ferrofluid had a thermal conductivity of 42 W/K.m at 134 Gauss whereas for 5 Fe3O4‐S type

**Figure 12.** Volume dependence of thermal conductivity between ‐20 and 0°C temperature interval and temperature

The thermal conductivity depended on the temperature difference at very low magnetic fields. As the magnetic field increases the thermal conductivity became irrespective of the tempera‐ ture. This behavior is observed both for 5 Fe3O4‐S and 20 Fe3O4‐S (**Figures 13** and **14**, respec‐

**Figure 13.** Temperature difference dependence of 5 vol% Fe3O4 and silicone oil‐based ferrofluid (5Fe3O4‐S).

ferrofluid had 37 W/K.m.

156 Nanofluid Heat and Mass Transfer in Engineering Problems

difference of 15 K.

tively).

Thermal conductivity mechanism in liquid involves collision of the molecules and transfer of the energy and momentum with one another. Transfer of the kinetic energy occurs in the lower temperature part of the system when a molecule moves from a high temperature region to a region of lower temperature and this molecule gives up this energy via collision with lower energy molecules. In solids, on the other hand, thermal energy may be conducted by lattice vibrations. At low‐temperature ranges (‐20 to 0°C), the energy of the molecules in the liquid is not enough to cause collisions and lattice vibrations could be insufficient to conduct heat. The decrease of the thermal conductivity observed in the temperature range between 0 and ‐20°C could be due to the less energetic particles. We consider that the thermal conductivity of the ferrofluid is determined by the factors such as stability of the ferrofluid particle size and the viscosity of the base liquid. Another important point in the heat transfer of the ferrofluids could be the thermo‐convective instability of the magnetic fluids. The instability arises due to the stronger magnetization of the colder fluid which is drawn to the higher region and is displaced by the warmer fluid [5, 46, 47]

As the temperature decreases the density and viscosity of the base fluid, silicone oil, increase. The denser and more viscous fluid hinders the motion of the magnetic particles due to the temperature difference. Thus, lower temperatures inhibit the motion of particles in the ferrofluid which will prevent settling of the particles. The stability of the fluid can be affected in a negative way at higher temperatures and at lower magnetic fields because the density and viscosity of the silicone oil decrease at high temperature. The instability of the ferrofluid at low temperature could be the reason for the different thermal conductivities at low magnetic fields. At low temperature, chain formation between the magnetic particles is activated by the increase in magnetic field. Consequently, this increases the thermal conductivity of the magnetic fluid at low temperatures. As the magnetism increases, the effect of the temperature range diminishes and thermal conductivity reaches almost the same value.

The heat transfer characteristics of 5Fe3O4‐S and 20Fe3O4‐S ferrofluid were investigated in the presence of the magnetic field applied parallel to the temperature gradient. The thermal conductivity behavior of the ferrofluids in different temperature ranges was analyzed, and it was seen that the heat transfer was more effective at higher temperatures. The fluids showed an increase in the thermal conductivity in the temperature intervals from 0 to 50°C, and a decrease from ‐20 to 0°C.

## **Acknowledgements**

The work is supported by Scientific and Technological Council of Turkey, TUBITAK, under Grant 108M473.

## **Nomenclature**



## **Author details**

Seval Genc

increase in magnetic field. Consequently, this increases the thermal conductivity of the magnetic fluid at low temperatures. As the magnetism increases, the effect of the temperature

The heat transfer characteristics of 5Fe3O4‐S and 20Fe3O4‐S ferrofluid were investigated in the presence of the magnetic field applied parallel to the temperature gradient. The thermal conductivity behavior of the ferrofluids in different temperature ranges was analyzed, and it was seen that the heat transfer was more effective at higher temperatures. The fluids showed an increase in the thermal conductivity in the temperature intervals from 0 to 50°C, and a

The work is supported by Scientific and Technological Council of Turkey, TUBITAK, under

range diminishes and thermal conductivity reaches almost the same value.

decrease from ‐20 to 0°C.

158 Nanofluid Heat and Mass Transfer in Engineering Problems

**Acknowledgements**

Grant 108M473.

**Nomenclature**

Δ*η* (B): Change in viscosity under magnetic field (Pa s)

)

*κ*p: Thermal conductivity of the particle (W/m K)

) distance between the ith and the jth particle

: Thermal conductivity of the fluid

*U*d: Potential energy of the interaction

: Magnetic moments in space

*η* (0): Viscosity without magnetic field (Pa s)

*k*: Boltzmann constant (J/K)

*µ*0: Vacuum permeability (Vs/A m)

*T*: Temperature (K) *E*dipole: Dipole‐dipole energy

*M*: Magnetization (A/m) *M*d: Domain magnetization *H*: Magnetic field (Gauss)

*D*: Particle diameter (m)

*β*: Magnetic induction (T) *Φ*: Volume fraction

*V*: Volume (m3

*κ*f

*mi*, *mj*

*rij* : (= *ri* ‐ *rj* Address all correspondence to: sgenc@marmara.edu.tr

Engineering Faculty, Metallurgical and Materials Engineering, Marmara University, Istanbul, Turkey

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Lucian Pîslaru-Dănescu, Gabriela Telipan, Floriana D. Stoian, Sorin Holotescu and Oana Maria Marinică Lucian Pîslaru-Dănescu, Gabriela Telipan, Floriana D. Stoian, Sorin Holotescu and Oana Maria Marinică

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65556

## Abstract

In this study, we propose a new type of a cooling agent based on magnetic nanofluid for the purpose of replacing the classical cooling fluids in electrical power transformers. The magnetite (Fe3O4) nanoparticles were synthesized by the co-precipitation method from an aqueous medium of salts FeCl3x6H2O and FeSO4x7H2O in the molar ratio Fe3+/Fe2+ = 2:1, by alkalization with 10% aqueous solution of NaOH at 80°C, for 1 h. The size of the magnetite nanoparticles, as measured by X-ray diffraction method, was 14 nm and by scanning electron microscopy (SEM), they are between 10 and 30 nm. Magnetite powder was placed in oleic acid as a surfactant to prevent agglomeration of nanoparticles. The resulting mixture was dispersed in transformer oil UTR 40, with the role of carrier liquid. The magnetic, rheological, thermal and electrical characteristic properties of the obtained Fe3O4 transformer oil-based nanofluid were determined. A mathematical model and numerical simulation results are very useful for investigating the heat transfer performances of the magnetic nanofluid. Based on this study, it was tested the cooling performance of this magnetic nanofluid for two types of electrical power transformers as compared to classical methods. We also presented a microactuator based on the same magnetic nanofluid.

Keywords: colloidal magnetic Fe3O4 nanoparticles, X-ray diffraction, SEM, electrical transformer, magnetic nanofluid coolant, heat transfer, magnetic properties, microactuator, mass transfer, pulse width modulation, mathematical model, numerical simulation

## 1. Introduction

Magnetic nanofluids, known also as ferrofluids or magnetic liquids, are stable colloidal suspensions of superparamagnetic nanoparticles such as γ-Fe2O3, α-Fe2O3, Fe3O4, CoFe2O4,

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and eproduction in any medium, provided the original work is properly cited.

Mn1<sup>−</sup>xZnxFe2O4, in a carrier liquid (an organic solvent or water) [1–5]. In order to prevent the aggregation of magnetic nanoparticles and to attain a stable magnetic nanofluid, the nanoparticles are coated with a surfactant during the preparation process [1, 4–8]. The characterization of magnetic nanoparticles and of the obtained magnetic nanofluid is carried out by various techniques in order to determine their structural, magnetic, rheological and magnetorheological properties [1–11]. The development of the synthesis methods leads to the possibility of tailoring the magnetic nanofluids and consequently, their physical properties, such that it fulfills the requirements of a certain application [1, 5, 12–24].

Besides the well-known application of ferrofluids in sealing and lubrication, recent developments envisaged their potential in fields like actuation [22, 23], medicine [1, 16], biotechnology [14, 24], as cooling fluids [25–29] or as liquid core in power transformers [30, 31]. The applications in medicine and biotechnology require from the magnetic nanofluid to be biocompatible. Therefore, water-based magnetic nanofluids are the candidates for magnetic hyperthermia for cancer treatment and targeted drug delivery, as well magnetic separation for purification of cells, proteins or else. The applications envisaged in electrical engineering require from the proposed magnetic nanofluid to also have good thermal and insulating properties. These conditions can be fulfilled by the transformer oil-based magnetic nanofluids.

This chapter is addressing the application of a Fe3O4 magnetic nanofluid based on transformer oil, as cooling and insulating fluid of a power transformer. Thus, the preparation procedure and the characterization of structural, magnetic, rheological, thermal and electrical properties are presented. The mathematical model applied to the problem is introduced, and the numerical results for two power transformers are discussed. The use of this Fe3O4 magnetic nanofluid in a micro-actuation application is also presented.

## 2. The synthesis and complex characterization of nanofluid with colloidal magnetic Fe3O4 nanoparticles

## 2.1. Nanofluid synthesis

The materials used for the synthesis of nanofluid with colloidal magnetic Fe3O4 nanoparticles we can mention: hexahydrated ferric chloride (FeCl3x6H2O) of 99% purity obtained from Merck Germany; ferrous sulfate sheptahydrate (FeSO4x7H2O) of 98% purity purchased from Chimopar, Romania; sodium hydroxide (NaOH) of 99% purity and oleic acid (C18H34O2) of 99% purity provided by Riedel de Haen. Other chemicals were of analytic grade. The reagents were used without further purification. All solutions were prepared with deionized water.

The transformer oil-based ferrofluid with Fe3O4 nanoparticles was synthesized by chemical coprecipitation method [23], using FeCl3 +6H2O and FeSO4 +7H2O with a molar ratio of Fe2+/ Fe3+ = 1:2, dissolved in 300 ml of water and treated with NaOH 10%. The mixture was stirred at 80°C for 1 h. The resulting black color precipitate of Fe3O4 was washed with deionized water and magnetically decanted until a pH of 7 was reached. Then, 10 ml of HCl 0.1 N was added to the Fe3O4 precipitate for peptization. The mixture was washed and decanted again until pH 7, dried at 90°C and treated with acetone for water removal. A small part of the obtained powder was analyzed [X-ray diffraction, scanning electron microscopy (SEM) and elemental analysis EDX] for structural properties. The remaining part of the powder was treated with 2 ml of oleic acid as surfactant, and with 5 ml of toluene and heated at 90°C for toluene removal. Finally, the mixture was dispersed in 50 ml of UTR 40 transformer oil, under strong stirring for 20 h in order to obtain the magnetic nanofluid. Figure 1 schematically presents the transformer oilbased magnetic nanofluid of Fe3O4 synthesis.

## 2.2. Nanofluid characterization

Mn1<sup>−</sup>xZnxFe2O4, in a carrier liquid (an organic solvent or water) [1–5]. In order to prevent the aggregation of magnetic nanoparticles and to attain a stable magnetic nanofluid, the nanoparticles are coated with a surfactant during the preparation process [1, 4–8]. The characterization of magnetic nanoparticles and of the obtained magnetic nanofluid is carried out by various techniques in order to determine their structural, magnetic, rheological and magnetorheological properties [1–11]. The development of the synthesis methods leads to the possibility of tailoring the magnetic nanofluids and consequently, their physical properties, such that it

Besides the well-known application of ferrofluids in sealing and lubrication, recent developments envisaged their potential in fields like actuation [22, 23], medicine [1, 16], biotechnology [14, 24], as cooling fluids [25–29] or as liquid core in power transformers [30, 31]. The applications in medicine and biotechnology require from the magnetic nanofluid to be biocompatible. Therefore, water-based magnetic nanofluids are the candidates for magnetic hyperthermia for cancer treatment and targeted drug delivery, as well magnetic separation for purification of cells, proteins or else. The applications envisaged in electrical engineering require from the proposed magnetic nanofluid to also have good thermal and insulating properties. These

This chapter is addressing the application of a Fe3O4 magnetic nanofluid based on transformer oil, as cooling and insulating fluid of a power transformer. Thus, the preparation procedure and the characterization of structural, magnetic, rheological, thermal and electrical properties are presented. The mathematical model applied to the problem is introduced, and the numerical results for two power transformers are discussed. The use of this Fe3O4 magnetic nanofluid

2. The synthesis and complex characterization of nanofluid with colloidal

The materials used for the synthesis of nanofluid with colloidal magnetic Fe3O4 nanoparticles we can mention: hexahydrated ferric chloride (FeCl3x6H2O) of 99% purity obtained from Merck Germany; ferrous sulfate sheptahydrate (FeSO4x7H2O) of 98% purity purchased from Chimopar, Romania; sodium hydroxide (NaOH) of 99% purity and oleic acid (C18H34O2) of 99% purity provided by Riedel de Haen. Other chemicals were of analytic grade. The reagents were used without further purification. All solutions were prepared with deionized water.

The transformer oil-based ferrofluid with Fe3O4 nanoparticles was synthesized by chemical co-

Fe3+ = 1:2, dissolved in 300 ml of water and treated with NaOH 10%. The mixture was stirred at 80°C for 1 h. The resulting black color precipitate of Fe3O4 was washed with deionized water and magnetically decanted until a pH of 7 was reached. Then, 10 ml of HCl 0.1 N was added to the Fe3O4 precipitate for peptization. The mixture was washed and decanted again until pH 7, dried at 90°C and treated with acetone for water removal. A small part of the obtained powder

6H2O and FeSO4

+

7H2O with a molar ratio of Fe2+/

+

conditions can be fulfilled by the transformer oil-based magnetic nanofluids.

fulfills the requirements of a certain application [1, 5, 12–24].

164 Nanofluid Heat and Mass Transfer in Engineering Problems

in a micro-actuation application is also presented.

magnetic Fe3O4 nanoparticles

precipitation method [23], using FeCl3

2.1. Nanofluid synthesis

## 2.2.1. Powder characterization of Fe3O4 nanoparticles

The Fe3O4 powder was structurally characterized by X-ray diffraction using a diffractometer D8 ADVANCE type X Bruker-AXS in conditions: Cu-Kα radiation (γ = 1.5406 Å), 40 KV/40 mA,

Figure 1. Transformer oil-based magnetic nanofluid of Fe3O4 synthesis.

filter k<sup>β</sup> of Ni, in the 2θ range of 25–70°, using a step of 0.04° and measuring time on point of 1 s.

The XRD pattern of the powder is presented in Figure 2 and shows the peaks corresponding to the Fe3O4 highlighted by "hkl" Miller indices (220), (311), (400), (422), (511) and (440), [2, 4], which denote a spinel structure with lattice parameter a = 0.83778 nm, in accord with the literature data (JCPDS file no. 19-629). According to the (311) peak, the medium size of the crystallites determined by Scherrer formula (1) is 14 nm.

$$\mathbf{D} = \frac{0.9\lambda}{\mathbf{B}^\* \cos \theta},\tag{1}$$

where D is the crystallites medium size, γ is the wavelength of this X-ray (γ = 0.154059 nm), B<sup>∗</sup> is the full width at half maximum (FWHM) and θ is the half diffraction angle of crystal orientation peak.

The morphology of the sample was studied by SEM using a Carl Zeiss SMT FESEM-FIB Auriger type scanner. The elemental analysis (energy-dispersive X-ray spectroscopy EDX) was performed with an energy dispersive probe of Inca Energy 250 type Oxford Instruments LTD England coupled to SEM.

The topology of the Fe3O4 powder analyzed by scanning electron microscopy evidenced two types of surface: smooth surface and rough surface (fracture). A crystalline structure of the material was found for the first type of surface, which is composed of crystallites having

Figure 2. The Fe3O4 XRD pattern of the Fe3O4 nanoparticles.

average sizes between 10 and 30 nm (Figure 3) in good agreement with the diffraction analysis. For the second type of surface, a structure of acicular type agglomerates was found (Figure 4).

The elemental analysis confirms the presence of Fe3O4 (Figure 5) and shows that the resulting black powder contains 70.14%Fe, 24.96% O, 4.16% C and 0.74% Cl (Table 1). The presence of the Fe3O4 is exhibited by elemental Fe-peaks of about 6.45 and 0.75 keV. The high percentage of oxygen is related to its existence in the iron oxide.

The SEM image (Figure 5a) and the elemental energy dispersive X-ray analysis (Figure 5b) confirms the data determined by X-ray diffraction. The content of C and Cl represents the little impurities.

Figure 3. The SEM image for the first structure.

filter k<sup>β</sup> of Ni, in the 2θ range of 25–70°, using a step of 0.04° and measuring time on point

The XRD pattern of the powder is presented in Figure 2 and shows the peaks corresponding to the Fe3O4 highlighted by "hkl" Miller indices (220), (311), (400), (422), (511) and (440), [2, 4], which denote a spinel structure with lattice parameter a = 0.83778 nm, in accord with the literature data (JCPDS file no. 19-629). According to the (311) peak, the medium size of the

<sup>D</sup> <sup>¼</sup> <sup>0</sup>:9<sup>λ</sup>

where D is the crystallites medium size, γ is the wavelength of this X-ray (γ = 0.154059 nm), B<sup>∗</sup> is the full width at half maximum (FWHM) and θ is the half diffraction angle of crystal

The morphology of the sample was studied by SEM using a Carl Zeiss SMT FESEM-FIB Auriger type scanner. The elemental analysis (energy-dispersive X-ray spectroscopy EDX) was performed with an energy dispersive probe of Inca Energy 250 type Oxford Instruments

The topology of the Fe3O4 powder analyzed by scanning electron microscopy evidenced two types of surface: smooth surface and rough surface (fracture). A crystalline structure of the material was found for the first type of surface, which is composed of crystallites having

<sup>B</sup><sup>∗</sup>cos<sup>θ</sup> ; (1)

crystallites determined by Scherrer formula (1) is 14 nm.

166 Nanofluid Heat and Mass Transfer in Engineering Problems

of 1 s.

orientation peak.

LTD England coupled to SEM.

Figure 2. The Fe3O4 XRD pattern of the Fe3O4 nanoparticles.

Figure 4. The SEM image for the second structure.

Figure 5. The elemental analysis for Fe3O4 powder: (a) the SEM image and (b) the elemental energy dispersive X-ray analysis.


Table 1. Analysis data for Fe3O4 nanoparticles, energy dispersive.

2.2.2. Characteristic properties of the nanofluid with colloidal magnetic Fe3O4 nanoparticles used as a cooling fluid for power transformers

#### 2.2.2.1. Magnetic properties

The full magnetization curve and the hysteresis loop of the transformer oil-based magnetic nanofluid (MNF/UTR 40), with a solid volume fraction of the dispersed magnetite particles of 1.67%, were measured at room temperature (25°C), using a vibrating sample magnetometer— VSM 880—ADE Technologies USA, in the magnetic field range of 0–950 kA/m.

The magnetization M measured at the maximum value of the applied magnetic field, approx. 900 kA/m, is considered to be the nominal magnetization of the investigated sample. Also, the absence of hysteresis loop area indicates a specific behavior of soft magnetic material (Figure 6) with the magnetic characteristics shown in Table 2.

In Table 2, Mr represents the remnant magnetization and Hc is the coercive magnetic field; ρ24oC is the density of the magnetic fluid at 24°C and ϕFe3O4 is the solid volume fraction of the dispersed magnetite.

According to Shliomis [32], the magnetic behavior of a diluted magnetic nanofluid ðϕFe3O4 < 5Þ under the action of an external magnetic field is well described by the single-particle model,

Nanofluid with Colloidal Magnetic Fe3O4 Nanoparticles and Its Applications in Electrical Engineering http://dx.doi.org/10.5772/65556 169

Figure 6. Hysteresis loop and full magnetization curve of the transformer oil-based magnetic fluid sample shows specific behavior of soft magnetic materials: (a) hysteresis loop for UTR 40-based MNF sample and (b) full magnetization curve for UTR 40-based MNF sample.


Table 2. Physical properties of MNF/UTR 40 sample.

which states that the energy of dipolar interactions is lower than the thermal energy. In this case, the equilibrium static magnetization is a superposition of Langevin functions,

$$\mathbf{M} = \boldsymbol{\varphi}\_{\mathrm{m}} \mathbf{M}\_{\mathrm{d}} \left( \coth \xi - \frac{1}{\xi} \right) = \boldsymbol{\varphi}\_{\mathrm{m}} \mathbf{M}\_{\mathrm{d}} \mathrm{L}(\xi), \tag{2}$$

with

2.2.2. Characteristic properties of the nanofluid with colloidal magnetic Fe3O4 nanoparticles used as a

Figure 5. The elemental analysis for Fe3O4 powder: (a) the SEM image and (b) the elemental energy dispersive X-ray

Element Weight (%) Atomic (%) C K 4.16 10.89 O K 24.96 49.01 Cl K 0.74 0.66 Fe K 70.14 39.45 Totals 100.00 100.00

The full magnetization curve and the hysteresis loop of the transformer oil-based magnetic nanofluid (MNF/UTR 40), with a solid volume fraction of the dispersed magnetite particles of 1.67%, were measured at room temperature (25°C), using a vibrating sample magnetometer—

The magnetization M measured at the maximum value of the applied magnetic field, approx. 900 kA/m, is considered to be the nominal magnetization of the investigated sample. Also, the absence of hysteresis loop area indicates a specific behavior of soft magnetic material (Figure 6)

In Table 2, Mr represents the remnant magnetization and Hc is the coercive magnetic field; ρ24oC is the density of the magnetic fluid at 24°C and ϕFe3O4 is the solid volume fraction of the

According to Shliomis [32], the magnetic behavior of a diluted magnetic nanofluid ðϕFe3O4 < 5Þ under the action of an external magnetic field is well described by the single-particle model,

VSM 880—ADE Technologies USA, in the magnetic field range of 0–950 kA/m.

cooling fluid for power transformers

with the magnetic characteristics shown in Table 2.

Table 1. Analysis data for Fe3O4 nanoparticles, energy dispersive.

168 Nanofluid Heat and Mass Transfer in Engineering Problems

2.2.2.1. Magnetic properties

analysis.

dispersed magnetite.

$$
\xi = \frac{\pi \mu\_0 \mathbf{M\_d D\_m^3 H}}{6 \mathbf{k\_B T}},
\tag{3}
$$

representing the Langevin parameter. Herein Md ¼ 480 kA=m is the monodomenial magnetization of magnetite, <sup>μ</sup><sup>0</sup> <sup>¼</sup> <sup>4</sup><sup>π</sup> · <sup>10</sup><sup>−</sup><sup>7</sup> <sup>H</sup>=m is the magnetic permeability of vacuum, Dm is the magnetic diameter of the dispersed magnetite particles, H is the applied magnetic field, kB <sup>¼</sup> <sup>1</sup>:<sup>38</sup> · <sup>10</sup><sup>−</sup><sup>23</sup> <sup>J</sup>=K is the Boltzmann constant and T is the absolute temperature.

In low magnetic fields <sup>ð</sup><sup>&</sup>lt; 1 mTÞ, with <sup>ξ</sup> ! 0, the Langevin function becomes LðξÞ ! <sup>ξ</sup> <sup>3</sup>, that is a linear variation in sample magnetization with the applied field. Knowing that the initial magnetic susceptibility is χiL ¼ M=H, one can obtain:

$$\chi\_{\rm iL} = \frac{\pi \mu\_0 \rho\_{\rm m} \mathbf{M}\_{\rm d}^2 \mathbf{D}\_{\rm m}^3}{18 \mathbf{k}\_{\rm B} \mathbf{T}} \tag{4}$$

On the other hand, in the region of intense magnetic fields ðξ >> 1Þ, the Langevin function is given by LðξÞ ! <sup>1</sup><sup>−</sup> <sup>1</sup> ξ , and static magnetization of the magnetic nanofluid is approximated by the following relationship [33],

$$\mathbf{M} \models \boldsymbol{\varphi}\_{\mathrm{m}} \mathbf{M}\_{\mathrm{d}} \left( 1 - \frac{6 \mathbf{k}\_{\mathrm{B}} \mathbf{T}}{\pi \mu\_{0} \mathbf{M}\_{\mathrm{d}} \mathbf{D}\_{\mathrm{m}}^{3} \mathbf{H}} \right), \tag{5}$$

where ϕ<sup>m</sup> ¼ Ms=Md is the magnetic volume fraction, with Ms representing the saturation magnetization of the magnetic nanofluid sample. The above relationship shows that the magnetization reaches saturation for very high values of the magnetic field ðH ! ∞Þ.

In fact, in real ferrofluids, the dimensional polydispersity of the magnetic particles is a characteristic that cannot be neglected and, in the absence of inter-particle interactions, an accurate expression of magnetization is obtained [34],

$$\mathbf{M} = \mathbf{M}\_{\text{s}} \overset{\text{\textquotedblleft}}{\text{L}}(\xi)\mathbf{f}(\mathbf{x})\mathbf{dx},\tag{6}$$

where fðxÞ is the log-normal distribution function (Figure 7)

$$\mathbf{f}(\mathbf{x}) = \frac{1}{\mathbf{x}\mathbf{S}\sqrt{2\pi}} \exp\left(-\frac{\ln^2\frac{\mathbf{x}}{\mathbf{D}\_0}}{2\mathbf{S}^2}\right),\tag{7}$$

with x is the magnetic diameter of the magnetite particles; fðxÞdx representing the probability that the magnetic diameter of the magnetic particles to be in the range of ðx; x þ dxÞ; D0 is the dimensional distribution parameter, defined by the relationship lnðD0Þ ¼ 〈lnðxÞ〉; S is also a dimensional distribution parameter, representing the deviation of lnðxÞ value from lnðD0Þ [35].

Another important aspect regarding the magnetization evaluation that should be considered is the dependence of the dispersed nanoparticles magnetic moments with their magnetic diameters [36, 37]. Here, MsðxÞ ¼ nmðxÞ and the ferrofluid magnetization become

Figure 7. (a) TEM image of mono-layer covered magnetite nanoparticles with oleic acid and stably dispersed in hexane; (b) dimensional distribution of the magnetic particles physical diameters is well approximated by the log-normal distribution function.

Nanofluid with Colloidal Magnetic Fe3O4 Nanoparticles and Its Applications in Electrical Engineering http://dx.doi.org/10.5772/65556 171

$$\mathbf{M} = \underset{\mathbf{0}}{\text{m}} \underset{\mathbf{m}}{\text{m}}(\mathbf{x})\mathbf{L}(\boldsymbol{\xi})\mathbf{f}(\mathbf{x})\mathbf{dx},\tag{8}$$

with n representing the density of the dispersed magnetite particles in magnetic fluid and m is the dipolar magnetic moment.

The linear dependence of the initial magnetic susceptibility versus magnetic particle concentration in low fields, rel. (4), and the asymptotic variation in magnetization in intense magnetic fields, rel. (5), form the basic instruments of magnetogranulometric analysis, in order to determine the mean magnetic diameter of the dispersed magnetic particles,

$$
\langle \mathbf{D}\_{\rm m} \rangle = \mathbf{D}\_{0} \exp\left(\frac{\mathbf{S}^{2}}{2}\right),
\tag{9}
$$

and standard deviation

$$
\sigma = \sqrt{\left< \mathbf{D}\_{\rm m}^{2} \right> - \left< \mathbf{D}\_{\rm m} \right>^{2}},
$$

$$
\sigma = \mathbf{D}\_{0} \exp\left(\frac{\mathbf{S}^{2}}{2}\right) [\exp \mathbf{S}^{2} - 1]^{1/2},\tag{10}
$$

where the dimensional distribution parameters are evaluated first,

$$\mathbf{S} = \frac{1}{3} \sqrt{\ln \frac{\mathbf{3} \mathbf{\bar{}\_{i\rm iL}} \mathbf{H}\_0}{\mathbf{M}\_s}},\tag{11}$$

$$\mathbf{D}\_0^3 = \frac{6\mathbf{k\_B T}}{\pi \mu\_0 \mathbf{M\_d H\_0}} \sqrt{\frac{\mathbf{M\_s}}{3\chi\_{\text{iL}}\mathbf{H\_0}}},\tag{12}$$

and

<sup>M</sup>≅ϕmMd <sup>1</sup><sup>−</sup> 6kBT

magnetization reaches saturation for very high values of the magnetic field ðH ! ∞Þ.

M ¼ Ms ∫

xS ffiffiffiffiffiffi

<sup>2</sup><sup>π</sup> <sup>p</sup> exp � ln<sup>2</sup> <sup>x</sup>

with x is the magnetic diameter of the magnetite particles; fðxÞdx representing the probability that the magnetic diameter of the magnetic particles to be in the range of ðx; x þ dxÞ; D0 is the dimensional distribution parameter, defined by the relationship lnðD0Þ ¼ 〈lnðxÞ〉; S is also a dimensional distribution parameter, representing the deviation of lnðxÞ value from lnðD0Þ [35]. Another important aspect regarding the magnetization evaluation that should be considered is the dependence of the dispersed nanoparticles magnetic moments with their magnetic diame-

Figure 7. (a) TEM image of mono-layer covered magnetite nanoparticles with oleic acid and stably dispersed in hexane; (b) dimensional distribution of the magnetic particles physical diameters is well approximated by the log-normal distri-

D0 2S<sup>2</sup> !

<sup>f</sup>ðxÞ ¼ <sup>1</sup>

ters [36, 37]. Here, MsðxÞ ¼ nmðxÞ and the ferrofluid magnetization become

expression of magnetization is obtained [34],

170 Nanofluid Heat and Mass Transfer in Engineering Problems

bution function.

where fðxÞ is the log-normal distribution function (Figure 7)

where ϕ<sup>m</sup> ¼ Ms=Md is the magnetic volume fraction, with Ms representing the saturation magnetization of the magnetic nanofluid sample. The above relationship shows that the

In fact, in real ferrofluids, the dimensional polydispersity of the magnetic particles is a characteristic that cannot be neglected and, in the absence of inter-particle interactions, an accurate

> ∞ 0

πμ0MdD<sup>3</sup>

!

mH

; (5)

; (7)

LðξÞfðxÞdx; (6)

$$\mathbf{m} = \frac{\mu\_0 \mathbf{M}\_s \mathbf{H}\_0}{\mathbf{k}\_\mathbf{B} \mathbf{T}},\tag{13}$$

The expression of the dimensional distribution parameters was obtained from rel. (8), considering the above presented limit cases and evaluating the initial magnetic susceptibility and saturation magnetization, respectively.

In the linear region of small magnetic fields ðH < 1 kA=mÞ from static magnetization curve, measured for the MNF/UTR 40 sample, the initial magnetic susceptibility has been evaluated. Ms and H0 magnetic field were determined in the quasi-saturation region ðH > 700 kA=mÞ, where the contribution to the ferrofluid magnetization is given by the magnetic particle interactions with the applied magnetic field, inter-particle interactions being neglected. As a result, in the saturation region can be considered that the magnetic nanofluid magnetization varies linearly with 1=H according to Langevin's law, even for concentrated samples. In Figure 8, which represents the final part (quasi-saturation) of the magnetization curve in M ¼ fð1=HÞ representation, Ms is obtained as the intersection with the ordinate axis. Also, the ratio of the magnetization curve slope and the corresponding absolute value of Ms is the value of H0 field, where R<sup>2</sup> represents the measure of accuracy of the fit in linear regression (Table 3).

Magnetogranulometric analysis of the MNF/UTR 40 sample has revealed a mean magnetic diameter of the magnetite particles of 〈Dm〉 ¼ 6:46 nm and a standard deviation of σ ¼ 2:18 nm. The log-normal distribution parameters (rel. (11)–(13)), evaluated directly from the magnetization curve, were obtained through non-linear regression, using rel. (8). Considering the thickness of non-magnetic layer at the surface of magnetic nanoparticles of δ<sup>m</sup> ¼ 0:83 nm [33],

$$
\langle \mathbf{D}\_{\mathbf{p}} \rangle = \langle \mathbf{D}\_{\mathbf{m}} \rangle + 2 \cdot \delta\_{\mathbf{m}}, \tag{14}
$$

a value of 8.12 nm was determined for the mean physical diameter. Furthermore, a thickness of 1.9 nm of the oleic acid monolayer cover of magnetite particles δ<sup>s</sup> [38] leads to a hydrodynamic diameter of the particles of 〈Dh〉 ¼ 11:92 nm,

Figure 8. Obtaining Msand H0 field in the quasi-saturation region of the magnetization curve, in M ¼ fð1=HÞ representation, where the ferrofluid magnetization varies linearly with 1=H, according to Langevin's law.


Table 3. Initial magnetic susceptibility and saturation magnetization of the transformer oil-based magnetic fluid sample.

$$
\langle \mathbf{D}\_{\mathbf{h}} \rangle = \langle \mathbf{D}\_{\mathbf{p}} \rangle + 2 \cdot \delta\_{\mathbf{s}} \tag{15}
$$

Main results obtained through magnetogranulometric analysis of the MNF/UTR 40 sample are summarized in Table 4.

#### 2.2.2.2. Rheological properties

Figure 8, which represents the final part (quasi-saturation) of the magnetization curve in M ¼ fð1=HÞ representation, Ms is obtained as the intersection with the ordinate axis. Also, the ratio of the magnetization curve slope and the corresponding absolute value of Ms is the value of H0 field, where R<sup>2</sup> represents the measure of accuracy of the fit in linear regression (Table 3). Magnetogranulometric analysis of the MNF/UTR 40 sample has revealed a mean magnetic diameter of the magnetite particles of 〈Dm〉 ¼ 6:46 nm and a standard deviation of σ ¼ 2:18 nm. The log-normal distribution parameters (rel. (11)–(13)), evaluated directly from the magnetization curve, were obtained through non-linear regression, using rel. (8). Considering the thickness of non-magnetic layer at the surface of magnetic nanoparticles of

a value of 8.12 nm was determined for the mean physical diameter. Furthermore, a thickness of 1.9 nm of the oleic acid monolayer cover of magnetite particles δ<sup>s</sup> [38] leads to a hydrodynamic

Figure 8. Obtaining Msand H0 field in the quasi-saturation region of the magnetization curve, in M ¼ fð1=HÞ representa-

Sample <sup>χ</sup>iL (-) R<sup>2</sup> <sup>ð</sup>χiL<sup>Þ</sup> (-) Ms (kA/m) Ms (Gs) H0 (kA/m) R<sup>2</sup> <sup>ð</sup>Ms <sup>Þ</sup> (-) MNF/UTR 40 0.11 0.99467 4.24 53.23 34.72 0.98580

Table 3. Initial magnetic susceptibility and saturation magnetization of the transformer oil-based magnetic fluid sample.

tion, where the ferrofluid magnetization varies linearly with 1=H, according to Langevin's law.

〈Dp〉 ¼ 〈Dm〉 þ 2 � δm; (14)

δ<sup>m</sup> ¼ 0:83 nm [33],

diameter of the particles of 〈Dh〉 ¼ 11:92 nm,

172 Nanofluid Heat and Mass Transfer in Engineering Problems

Rheological investigations, carried out with an Anton Paar Physica MCR 300 rheometer using a double-gap concentric cylinder geometry, consisted of measuring the dynamic viscosity curves of the samples in the absence of the magnetic field. The shear rate, γ\_ , varied from 1 s−<sup>1</sup> to 1000 s−<sup>1</sup> , at different values of working temperature t ¼ ð20; 40; 60; 80Þ o C. The viscosity curves (Figure 9) measured for both the transformer oil-based magnetic nanofluid (MNF/ UTR 40) and the carrier liquid-transformer oil (UTR 40) showed that adding a small volume fraction of magnetic particles in the carrier ðφFe3O4 ≅ 1:67 %Þ leads to a very mild increase in the dynamic viscosity and the Newtonian behavior of the samples is preserved throughout the investigated temperature range.

This behavior indicates the absence of the magnetic particle interactions, that is, a very good stability of the sample, mainly due to the efficient steric stabilization. An Arrhenius-type relationship describes the sample viscosity behavior with temperature,


Table 4. Main properties of MNF/UTR 40 sample obtained through magnetogranulometric analysis, including the lognormal distribution parameters values.

Figure 9. Viscosity curves showed that the Newtonian behavior of the carrier (UTR 40) (a) is preserved throughout the temperature range, also for the magnetic nanofluid (MNF/UTR 40), (b) that contains a small amount of the dispersed magnetic particles, indicating an efficient steric stabilization.

$$
\eta = \eta\_{\rm ref} \exp\left[ -\frac{\rm Ea}{\rm R} \left( \frac{1}{T} - \frac{1}{T\_{\rm ref}} \right) \right], \tag{16}
$$

where Tref <sup>¼</sup> 20oC is considered the reference temperature; <sup>η</sup>refðPa � <sup>s</sup><sup>Þ</sup> is the dynamic viscosity corresponding to the reference absolute temperature; <sup>Ε</sup><sup>a</sup> <sup>ð</sup><sup>J</sup> � mol�<sup>1</sup> Þ means the activation energy and R <sup>¼</sup> <sup>8</sup>:31447 J � mol<sup>−</sup><sup>1</sup> K<sup>−</sup><sup>1</sup> is the ideal gas constant.

Considering a shear rate of 100 s−<sup>1</sup> , rel. (16) was used to fit the dependence η ¼ ηðTÞ, with activation energy as fit parameter (Figure 10).

In Table 5, it can be observed that the value of the viscous flow activation energy does not change after dispersing a small amount of surfacted magnetite particles in the carrier liquid. It can be concluded that the adding of small volume fractions of surfacted magnetite particles in a carrier liquidðϕFe3O4 < 5Þ, as transformer oil, has no significant influence on the rheological properties of the samples.

## 2.2.2.3. Thermal properties

The addition of metallic nanoparticles (magnetic or non-magnetic) or non-metallic (e.g., diamond nanoparticles) in transformer oils in order to improve their cooling performances is a solution that was demonstrated by several patents and associated research works (e.g. [39–45]). This paragraph analyzes the thermal properties of the magnetic nanofluid, which was tested for use as cooling and insulating medium in power transformers.

Determination of the effective thermal properties that characterize the cooling fluids in our study as well as their modeling using analytical formulae is representing a main problem of the topic in discussion. Irrespective of the considered approach, either theoretical or experimental, a representative element of the studied medium has to be chosen. It has to be

Figure 10. Arrhenius type dependence of the samples dynamic viscosity with temperature: (a) carrier liquid UTR 40 and (b) transformer oil-based magnetic nanofluid MNF/UTR 40.


Table 5. Viscous flow activation energy of the carrier liquid (UTR 40) and the magnetic nanofluid (MNF/UTR 40).

underlined that the determination of the properties of heterogeneous materials has to be made by complying with the request of representativeness of the studied volume, that is, the considered volume of the heterogeneous material must be sufficiently large in order to be statistically representative irrespective of the type of the carried out experiment [46, 47].

The main properties that are influencing the thermal behaviour of a material are the heat capacity, thermal conductivity, density, thermal expansion coefficient and thermal diffusivity (a property that depends on the first three). Various experimental studies determined that the effective properties of nanofluids (magnetic or non-magnetic) are dependent on the following characteristics of their components [25, 48–50]: the thermo-physical properties of the carrier fluid, nanoparticles and surfactant; nanoparticles volume fraction, size distribution, mean diameter and shape; temperature; magnetic field (in the case of magnetic nanoparticles).

A review of the reference literature regarding the main properties of the transformer oil-based fluids and their dependence with the temperature outlined the followings: specific heat is increasing as linear function with temperature; thermal conductivity is decreasing as a quasilinear function with temperature especially for transformer oils; dynamic viscosity is decreasing with the increasing temperature; the dielectric constant has relatively low values and decreases with the increasing temperature [51–53].

## 2.2.2.3.1. Thermal conductivity

<sup>η</sup> <sup>¼</sup> <sup>η</sup>refexp � Ea

corresponding to the reference absolute temperature; <sup>Ε</sup><sup>a</sup> <sup>ð</sup><sup>J</sup> � mol�<sup>1</sup>

tested for use as cooling and insulating medium in power transformers.

energy and R <sup>¼</sup> <sup>8</sup>:31447 J � mol<sup>−</sup><sup>1</sup>

properties of the samples.

2.2.2.3. Thermal properties

Considering a shear rate of 100 s−<sup>1</sup>

activation energy as fit parameter (Figure 10).

174 Nanofluid Heat and Mass Transfer in Engineering Problems

(b) transformer oil-based magnetic nanofluid MNF/UTR 40.

R

where Tref <sup>¼</sup> 20oC is considered the reference temperature; <sup>η</sup>refðPa � <sup>s</sup><sup>Þ</sup> is the dynamic viscosity

K<sup>−</sup><sup>1</sup> is the ideal gas constant.

In Table 5, it can be observed that the value of the viscous flow activation energy does not change after dispersing a small amount of surfacted magnetite particles in the carrier liquid. It can be concluded that the adding of small volume fractions of surfacted magnetite particles in a carrier liquidðϕFe3O4 < 5Þ, as transformer oil, has no significant influence on the rheological

The addition of metallic nanoparticles (magnetic or non-magnetic) or non-metallic (e.g., diamond nanoparticles) in transformer oils in order to improve their cooling performances is a solution that was demonstrated by several patents and associated research works (e.g. [39–45]). This paragraph analyzes the thermal properties of the magnetic nanofluid, which was

Determination of the effective thermal properties that characterize the cooling fluids in our study as well as their modeling using analytical formulae is representing a main problem of the topic in discussion. Irrespective of the considered approach, either theoretical or experimental, a representative element of the studied medium has to be chosen. It has to be

Figure 10. Arrhenius type dependence of the samples dynamic viscosity with temperature: (a) carrier liquid UTR 40 and

Sample <sup>φ</sup>Fe3O4 (%) <sup>Ε</sup>að<sup>J</sup> � mol�<sup>1</sup>

MNF/UTR 40 1.67 28.27

UTR 40 – 28.38

Table 5. Viscous flow activation energy of the carrier liquid (UTR 40) and the magnetic nanofluid (MNF/UTR 40).

1 <sup>T</sup> <sup>−</sup> <sup>1</sup> Tref

; (16)

, rel. (16) was used to fit the dependence η ¼ ηðTÞ, with

Þ means the activation

Þ

+103

+103 The thermal conductivity of magnetic nanofluids can be described as a function of several parameters, among the most important are the thermal conductivities of the carrier liquid and magnetic nanoparticles and their dependence on temperature and pressure, the volume fraction, the shape and the size distribution of the nanoparticles. The interfacial thermal resistance between the nanoparticles and the surrounding liquid is also considered. It has been proved numerically and experimentally that an applied magnetic field can affect the thermal conductivity of a magnetic nanofluid due to the consequent ordering of magnetic dipoles of the nanoparticles along the field lines [25, 54].

There are many Maxwell-type models developed for the thermal conductivity of mixtures (also named effective thermal conductivity—ETC), either solid matrix—solid filler or liquid carrier and dispersed nanoparticles that are based on the Maxwell model, which is recommended for low volume fraction of the filler/nanoparticles and considers that the nanoparticles are identical, spherical and non-interacting. The Holotescu-Stoian model, developed initially for the solid matrix—solid filler mixture and presented in Refs. [55, 56], introduced for the first time the filler particle size distribution in the Maxwell model. The expression for the effective thermal conductivity of the Holotescu-Stoian model, rel. (17), in the case of a magnetic nanofluid is [57],

$$\mathbf{k}\_{\rm e} = \mathbf{k}\_{\rm f} \, \frac{\mathbf{k}\_{\rm p} + 2 \; \mathbf{k}\_{\rm f} + 2 \; \boldsymbol{\varrho}\_{\rm e} \left(\mathbf{k}\_{\rm p} - \mathbf{k}\_{\rm f}\right)}{\mathbf{k}\_{\rm p} + 2 \; \mathbf{k}\_{\rm f} \cdot \boldsymbol{\varrho}\_{\rm e} \left(\mathbf{k}\_{\rm p} - \mathbf{k}\_{\rm f}\right)},\tag{17}$$

where ke is the effective thermal conductivity of the magnetic nanofluid, kp is the thermal conductivity of the magnetic nanoparticles, kf is the thermal conductivity of the carrier fluid, ϕ<sup>e</sup> is the equivalent volume fraction, defined by

$$
\varphi\_{\mathbf{e}} = \varphi\_{\text{Fe}\odot\mathbf{4}} \frac{\langle \left(\mathrm{D}\_{\text{m}} + \delta\right)^{3} \rangle^{2}}{\langle \left(\mathrm{D}\_{\text{m}} + \delta\right)^{2} \rangle^{3}},\tag{18}
$$

with ϕFe3O4 is the solid nanoparticles volume fraction (magnetite in this case),Dm is the magnetic diameter, δ ¼ 2δ<sup>m</sup> is the double thickness of the non-magnetic layer, and

$$
\langle \mathbf{u} \rangle = \underset{\mathbf{0}}{\overset{\circ}{\mathrm{uf}}} \mathbf{\hat{u}}(\mathbf{x}) \mathbf{dx} \tag{19}
$$

with u being a magnetic diameter dependent function and fðxÞ is the log-normal distribution function.

The relationship between physical (geometrical) diameter Dp, magnetic diameter Dm and δ is given by

$$\mathbf{D}\_{\mathbf{p}} = \mathbf{D}\_{\mathbf{m}} + \delta. \tag{20}$$

This model, confirmed by the experimental data [57], was applied to determine the thermal conductivity of the analyzed magnetic nanofluid sample. The results (at room temperature) are given in Table 6, and we observe that the addition of magnetite nanoparticles alone is increasing the thermal conductivity of the magnetic nanofluid. The transformer oil thermal conductivity is decreasing with the increasing temperature, thus the cooling performance can be diminished at normal operating conditions in power transformers (and other electrical equipments). We can conclude that the addition of the magnetite nanoparticles can counteract this disadvantage. Moreover, during the operation of a power transformer, for instance, the magnetic field is acting on the magnetic nanofluid, influencing its physical properties, and generating the magnetoconvection that can enhance the heat transfer, as shown in the next section.

#### 2.2.2.3.2. Specific heat

The specific heat of the magnetic nanofluid, at constant pressure, was determined using the following mixture formula [58]

$$\rho\_{\rm MNF}(\mathbf{T}) \cdot \mathbf{c\_{p,MNF}}(\mathbf{T}) = (1 - \boldsymbol{\varphi}) \cdot \rho\_{\rm UTR}(\mathbf{T}) \cdot \mathbf{c\_{p,UTR}}(\mathbf{T}) + \boldsymbol{\varphi}\_{\rm m} \cdot \rho\_{\rm NP}(\mathbf{T}) \cdot \mathbf{c\_{p,NP}}(\mathbf{T}),\tag{21}$$

where the transformer oil specific heat was determined by using

$$\mathbf{c}\_{\rm p,UTR}(\mathbf{T}) = 5.025 \cdot \mathbf{T} + 1789.50 \tag{22}$$

and the specific heat of the magnetite by using [59],


Table 6. Thermal expansion coefficients.

Nanofluid with Colloidal Magnetic Fe3O4 Nanoparticles and Its Applications in Electrical Engineering http://dx.doi.org/10.5772/65556 177

$$\mathbf{c}\_{\rm p,NP}(\mathbf{T}) = 0.6334 + 0.871 \cdot 10^{-3} \mathbf{T},\tag{23}$$

where T (K) is the absolute temperature of the solid and ρ is the mass density of the material.

The numerical results obtained for the magnetic nanofluid specific heat indicate a slightly decrease compared to that of the carrier liquid. In what it concerns the effect of an applied magnetic field on the specific heat capacity of a magnetic nanofluid, for a certain range of temperature, the reference literature indicates the influence of the nanoparticles volume fraction and nanofluid composition (carrier liquid and nanoparticles). Also, the magnitude and the applied field orientation relative to the gravitational field (as the experiments were conducted in gravitational field) should be considered. Korolev et al. [60] analyzed the influence of an applied magnetic field (oriented perpendicularly on gravity) on a transformer oil-based magnetic nanofluid with magnetite nanoparticles, having a solid volume fraction of 7.4%, in the temperature range from 15 to 80°C. The experiment showed that, for a certain temperature, the specific heat capacity has a maximum in the investigated range of the applied magnetic field, which, according to the authors, indicates the presence of a magneto-caloric effect.

#### 2.2.2.3.3. Thermal expansion coefficient

ϕ<sup>e</sup> ¼ ϕFe3O4

176 Nanofluid Heat and Mass Transfer in Engineering Problems

netic diameter, δ ¼ 2δ<sup>m</sup> is the double thickness of the non-magnetic layer, and

convection that can enhance the heat transfer, as shown in the next section.

where the transformer oil specific heat was determined by using

and the specific heat of the magnetite by using [59],

+

function.

given by

2.2.2.3.2. Specific heat

following mixture formula [58]

β (1/K) 7.15

Table 6. Thermal expansion coefficients.

〈u〉 ¼ ∫ ∞ 0

〈ðDm þ δÞ

〈ðDm þ δÞ

with ϕFe3O4 is the solid nanoparticles volume fraction (magnetite in this case),Dm is the mag-

with u being a magnetic diameter dependent function and fðxÞ is the log-normal distribution

The relationship between physical (geometrical) diameter Dp, magnetic diameter Dm and δ is

This model, confirmed by the experimental data [57], was applied to determine the thermal conductivity of the analyzed magnetic nanofluid sample. The results (at room temperature) are given in Table 6, and we observe that the addition of magnetite nanoparticles alone is increasing the thermal conductivity of the magnetic nanofluid. The transformer oil thermal conductivity is decreasing with the increasing temperature, thus the cooling performance can be diminished at normal operating conditions in power transformers (and other electrical equipments). We can conclude that the addition of the magnetite nanoparticles can counteract this disadvantage. Moreover, during the operation of a power transformer, for instance, the magnetic field is acting on the magnetic nanofluid, influencing its physical properties, and generating the magneto-

The specific heat of the magnetic nanofluid, at constant pressure, was determined using the

Property/sample UTR NP MNF\_UTR ΔX/XUTR ke (W/m K) 0.127 1.39 0.135 +6.29% cp (J/kg K) 1910.1 0.892 1867.8 −2.2%

+

10−<sup>4</sup> 1.2

ρMNFðTÞ � cp,MNFðTÞ¼ð1−ϕÞ � ρUTRðTÞ � cp,UTRðTÞ þ ϕ<sup>m</sup> � ρNPðTÞ � cp,NPðTÞ; (21)

cp,UTRðTÞ ¼ 5:025 � T þ 1789:50 (22)

10−<sup>4</sup> 6.44

+

10−<sup>4</sup> −9.94%

3 〉 2

2 〉

<sup>3</sup> ; (18)

ufðxÞdx (19)

Dp ¼ Dm þ δ: (20)

Similarly, to determine the thermal expansion coefficient of the magnetic nanofluid, a corresponding mixing formula was used [61],

$$\beta\_{\rm MNF} = \beta\_{\rm UTR} \left[ \frac{1}{1 + \frac{(1 - \wp\_{\rm Fo\_3O\_4})\rho\_{\rm UTR}}{\wp\_{\rm Fo\_3O\_4}\rho\_{\rm NP}} \beta\_{\rm UTR}} + \frac{1}{1 + \frac{\wp\_{\rm Fo\_3O\_4}}{1 - \wp\_{\rm Fo\_3O\_4}} \cdot \frac{\rho\_{\rm NP}}{\rho\_{\rm UTR}}} \right],\tag{24}$$

where βMNF [1/K] is the thermal expansion coefficient of MNF, βUTR is the thermal expansion coefficient of the transformer oil [62], βNP is the thermal expansion coefficient of the magnetite nanoparticles [63], <sup>ρ</sup>UTR <sup>¼</sup> <sup>0</sup>:867 g=cm<sup>3</sup> (at 20°C) and <sup>ρ</sup>MNF <sup>¼</sup> <sup>0</sup>:96 g=cm3 (at 24°C) are the measured densities of the transformer oil and the magnetic nanofluid, respectively.

The results of the calculations, summarized in Table 6, give the thermal properties at room temperature. We observe that the analyzed thermal properties have a diverging behavior. While the thermal expansion coefficient and specific heat are decreasing, the thermal conductivity is increasing. The last column is indicating the relative variation compared to the corresponding property of the carrier liquid (transformer oil).

#### 2.2.2.3.4. Evaluation of the heat transfer potential of the magnetic nanofluid

To determine the potential performance of the magnetic nanofluid for heat transfer, we considered the figure-of-merit (FOM), as defined for natural convection [64],

$$\text{FOM}\_{\text{NC}} = \left[ \beta \,\, \rho^2 \,\, \mathbf{c}\_{\text{P}} \,\, \mathbf{k}^{\frac{1}{w-1}} / \eta \right]^n,\tag{25}$$

where n ¼ 0:25 for laminar flow and n ¼ 0:33 for turbulent flow.


Table 7. FOM results, obtained for the carrier liquid and for the magnetic nanofluid.


Table 8. Effective electric permittivity of the Fe3O4 transformer oil-based magnetic nanofluid.

The results obtained for the FOM of the carrier liquid and the magnetic nanofluid, using the properties determined above, are presented in Table 7. The comparison of the relative increase of FOM indicates that the addition of nanoparticles is advantageous for both laminar and turbulent flow.

#### 2.2.2.4. Electric permittivity

As underlined above, the use of a magnetic nanofluid in electrical engineering applications imposes restrictions regarding its insulating properties. Transformer oils are known to be electrical insulators so they are an appropriate carrier liquid for a magnetic nanofluid used in such applications. If the magnetic nanoparticles volume fraction is kept in certain limits, the magnetic nanofluid preserves its insulating properties within the required limits, too [49, 50].

We estimated the effective electric permittivity of the magnetic nanofluid εMNF, using the Maxwell-Garnett equation for mixtures:

$$
\varepsilon\_{\rm MNF} = \varepsilon\_{\rm UTR} + 3\rho\_{\rm Fe\_3O\_4} \varepsilon\_{\rm UTR} \frac{\varepsilon\_{\rm NP} - \varepsilon\_{\rm UTR}}{\varepsilon\_{\rm NP} + 2\varepsilon\_{\rm UTR} - \rho\_{\rm Fe\_3O\_4} (\varepsilon\_{\rm NP} - \varepsilon\_{\rm UTR})},\tag{26}
$$

with εUTR is the electric permittivity of the transformer oil and εNP is the electric permittivity of the magnetite nanoparticles, ϕFe3O4 being the volume fraction of the magnetite nanoparticles.

The results are presented in Table 8, along with the relative difference between the values corresponding to the transformer oil and magnetic nanofluid, ε<sup>0</sup> being the free space permittivity, approximate equal to 8.85 +10−<sup>12</sup> F/m.

We observed that for the current volume fraction of magnetic nanoparticles, the insulating properties of the magnetic nanofluid remain very close to those of the carrier liquid (UTR 40). In what concerns the effect of working temperatures in the power transformer, experimental studies showed that electrical permittivity decreases with increasing temperature in the case of transformer oils [51].

## 3. Heat transfer and electromagnetic field by numerical simulation for the electrical transformer cooled by a specific nanofluid

A colloidal Fe3O4 specific nanofluid dispersed in oil transformer UTR 40, named MNF/UTR 40, is utilized in a number of technologically relevant applications where external magnetic fields are used to adjust their flow. A mathematical model and numerical simulation results are useful for investigating the heat transfer properties of the magnetic nanofluid. Based on this study, it was built and tested the experimental model: low power, medium voltage, single-phased transformer type TMOf-24-5" (Figure 18) and low power, medium voltage, single-phased transformer type TMOf2-36kV-40 kVA (Figure 19). First of all, it was used for the transformer the transformer oil UTR 40 as cooling and insulating liquid. After that, this oil was drained and the experimental model was filled with magnetic nanofluid based on transformer oil MNF/UTR 40. The MNF/UTR 40 specific nanofluid has been shown to provide both thermal and dielectric benefits to transformers, and can be utilized to improve cooling by enhancing fluid circulation within transformer windings, to increase transformer capacity to withstand lightning impulses, while also minimizing the effect of moisture on typical insulating fluids. Magnetic nanofluid flow may be influenced by external magnetic fields, and the retention force of a magnetic nanofluid can be adjusted by changing either the magnetization of the fluid or the magnetic field in the region. Opposite to usual magnetic fluids, the magnetizable nanofluids destined to heat transfer should have a low concentration of magnetic nanoparticles in order to make them competitive with the non-magnetic fluids.

## 3.1. Mathematical model

The results obtained for the FOM of the carrier liquid and the magnetic nanofluid, using the properties determined above, are presented in Table 7. The comparison of the relative increase of FOM indicates that the addition of nanoparticles is advantageous for both laminar and

Property/sample UTR NP MNF\_UTR Δε / εUTR

+

ε<sup>0</sup> 2.26

+

ε<sup>0</sup> 2.85%

ε<sup>0</sup> 81

Table 8. Effective electric permittivity of the Fe3O4 transformer oil-based magnetic nanofluid.

Property/sample UTR MNF\_UTR ΔX/XUTR FOM, n ¼ 0:25 18.61 19.57 +5.15% FOM, n ¼ 0:33 91.85 96.25 +4.8%

Table 7. FOM results, obtained for the carrier liquid and for the magnetic nanofluid.

+

As underlined above, the use of a magnetic nanofluid in electrical engineering applications imposes restrictions regarding its insulating properties. Transformer oils are known to be electrical insulators so they are an appropriate carrier liquid for a magnetic nanofluid used in such applications. If the magnetic nanoparticles volume fraction is kept in certain limits, the magnetic nanofluid preserves its insulating properties within the required limits, too [49, 50]. We estimated the effective electric permittivity of the magnetic nanofluid εMNF, using the

εUTR

10−<sup>12</sup> F/m.

with εUTR is the electric permittivity of the transformer oil and εNP is the electric permittivity of the magnetite nanoparticles, ϕFe3O4 being the volume fraction of the magnetite nano-

The results are presented in Table 8, along with the relative difference between the values corresponding to the transformer oil and magnetic nanofluid, ε<sup>0</sup> being the free space permit-

We observed that for the current volume fraction of magnetic nanoparticles, the insulating properties of the magnetic nanofluid remain very close to those of the carrier liquid (UTR 40). In what concerns the effect of working temperatures in the power transformer, experimental studies showed that electrical permittivity decreases with increasing temperature in the case of

εNP−εUTR

ðεNP−εUTRÞ

; (26)

εNP þ 2εUTR−ϕFe3O4

turbulent flow.

particles.

2.2.2.4. Electric permittivity

ε (F/m) 2.2

178 Nanofluid Heat and Mass Transfer in Engineering Problems

Maxwell-Garnett equation for mixtures:

tivity, approximate equal to 8.85

transformer oils [51].

εMNF ¼ εUTR þ 3ϕFe3O4

+

Several simplifying assumptions aimed and keeping the physical system within approachable software and hardware limits are requested and 2D models are best candidates, providing numerical simulation relevant results, of satisfactory accuracy. Following this path, we consider a 2D, Cartesian cross-sectional model, as shown in Figure 11.

The heat transfer and transport processes under the influence of the magnetic field for two prototypes electric transformer: low power mono-phased transformer (24 kVA), at medium voltage (20/√3//0,4/√3kV), TMOf-24-5 and low power mono-phased transformer (40 kVA), at medium voltage (30/√3//0,4/√3kV), prototype TMOf 2-36kV-40 kVA, are described by the following set of coupled partial differential equations [44]:

• electromagnetic field—quasi-steady, harmonic diffusion,

$$(\mathbf{j}\omega\sigma - \omega^2 \varepsilon\_0 \varepsilon\_r)\mathbf{A} + \nabla \times (\mu\_0^{-1} \mu\_r^{-1} \nabla \times \mathbf{A}) - \sigma \mathbf{u} \times (\nabla \times \mathbf{A}) = \mathbf{J}^{\mathbf{e}},\tag{27}$$

• momentum balance (Navier-Stokes),

$$
\rho \left[ \frac{\partial \mathbf{u}}{\partial \mathbf{t}} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right] = -\nabla \mathbf{p} + \underbrace{\mu\_0 (\mathbf{M} \cdot \nabla) \mathbf{H}}\_{\mathbf{f}\_{\text{reg}}} + \mu\_\mathbf{f} \nabla^2 \mathbf{u} + \mathbf{f}\_{\text{T}}, \tag{28}
$$

• mass conservation (incompressible flow),

$$
\nabla \mathbf{u} = \mathbf{0},
\tag{29}
$$

• heat transfer (energy equation),

$$
\rho \mathbf{c\_p} \left[ \frac{\partial \mathbf{T}}{\partial \mathbf{t}} + (\mathbf{u} \cdot \nabla) \mathbf{T} \right] = \mathbf{k} \nabla^2 \mathbf{T} + \mathbf{E} \cdot \mathbf{J}\_\varphi^e. \tag{30}
$$

Here: u is the velocity; p is the pressure; T is the absolute temperature; A is the magnetic vector field; M is the magnetization (in the magnetic nanofluid); H is the magnetic field strength; E is the electric field strength; J e <sup>ϕ</sup> is the angular component of the (external) current density (in the coil); f<sup>T</sup> is the buoyancy body force term; fmg is the magnetic body force term; σ is the electrical conductivity; k is the thermal conductivity; μ<sup>0</sup> is the magnetic permeability of vacuum (μair ¼ μr air � μ0≅μ0; μr air≅1); μ<sup>f</sup> is the kinematic viscosity; ρ is the mass density; cp is the specific heat. Heat transfer occurs by conduction in the solid regions of the system and by convection and diffusion in the fluid region. The temperature variation in the fluid region is responsible for a gravitational flow (Boussinesq approximation), whose structure depends on the cell geometric aspect ratio and thermal conditions. All subdomains have linear physical

Figure 11. The 2D simplified model and the FEM mesh made of triangular elements: (a) computational domain and (b) detailed view—windings, iron core, case.

properties, except for the magnetic nanofluid (based on transformer oil), whose magnetization characteristic is nonlinear.

• mass conservation (incompressible flow),

180 Nanofluid Heat and Mass Transfer in Engineering Problems

ρcp ∂T

e

<sup>∂</sup><sup>t</sup> þ ð<sup>u</sup> � <sup>∇</sup>Þ<sup>T</sup> 

<sup>¼</sup> <sup>k</sup>∇<sup>2</sup>

Here: u is the velocity; p is the pressure; T is the absolute temperature; A is the magnetic vector field; M is the magnetization (in the magnetic nanofluid); H is the magnetic field strength; E is

coil); f<sup>T</sup> is the buoyancy body force term; fmg is the magnetic body force term; σ is the electrical conductivity; k is the thermal conductivity; μ<sup>0</sup> is the magnetic permeability of vacuum (μair ¼ μr air � μ0≅μ0; μr air≅1); μ<sup>f</sup> is the kinematic viscosity; ρ is the mass density; cp is the specific heat. Heat transfer occurs by conduction in the solid regions of the system and by convection and diffusion in the fluid region. The temperature variation in the fluid region is responsible for a gravitational flow (Boussinesq approximation), whose structure depends on the cell geometric aspect ratio and thermal conditions. All subdomains have linear physical

Figure 11. The 2D simplified model and the FEM mesh made of triangular elements: (a) computational domain and (b)

T þ E � J e

<sup>ϕ</sup> is the angular component of the (external) current density (in the

• heat transfer (energy equation),

the electric field strength; J

detailed view—windings, iron core, case.

∇u ¼ 0; (29)

<sup>ϕ</sup>: (30)

The magnetic field magnetizes the magnetic nanofluid, and the corresponding body force term adds to the gravitational, thermal flow of the magnetic nanofluid coolant. The flow structure is the result of the two competing forces: thermal force and magnetic force. In this study, we deal with a super-paramagnetic magnetic nanofluid where the influence of the coercive magnetic field intensity, Hc or the remnant induction, Br is discarded. The constitutive law for the magnetic field of the magnetic fluid is then

$$\mathbf{B} = \mu\_0 (\mathbf{H} + \mathbf{M}).\tag{31}$$

The magnetic field produced by the electrical current in the coil magnetizes the fluid and is responsible for the magnetic body forces that influence the thermally induced flow. The magnetization of the magnetic fluid is approximated here by the analytic formula

$$\mathbf{M}\_{\mathbf{x},\mathbf{y}} = \mathbf{a} \cdot \arctan(\mathbf{b} \cdot \mathbf{H}\_{\mathbf{x},\mathbf{y}}),\tag{32}$$

with a <sup>¼</sup> <sup>10</sup><sup>4</sup> <sup>A</sup>=m and b <sup>¼</sup> <sup>3</sup> · <sup>10</sup><sup>−</sup><sup>5</sup> m=A are empiric constants. The magnetic body forces are then obtained out of the magnetic energy, by taking its derivatives with respect to the coordinates

$$\mathbf{f\_{mg}} = \mu\_0 (\mathbf{M} \cdot \nabla) \mathbf{H}.\tag{33}$$

The strategy that we used in the numerical simulation consists of solving for the magnetic field first, and then using the active power thus obtained as heat source in the heat transfer and flow parts of the problem. The obtained solutions are steady state [65] for heat transfer and flow and quasi-steady (harmonic) for the electromagnetic field.

The main dimensions (windings, iron core, case sizes) are those of the single-phased transformer considered in our study. The amperturns of the windings correspond to the nominal working point, when the iron core exhibits lower levels of magnetization—the amperturns are compensated.

## 3.2. Simulation results for mono-phased transformer of low power and medium voltage type TMOf-24-5 compared to the mono-phased transformer of low power and medium voltage type TMOf2-36kV-40 kVA

Numerical simulation of the mono-phased transformer of low power and medium voltage type TMOf-24-5 evidenced that the convective heat transfer in the channels between the windings and between the windings and core (3–5 mm) is less important in the overall process, therefore it was discarded, and only conduction heat transfer was accounted for in these areas. Figure 12 shows simulation results for a non-magnetic, regular cooling fluid—the temperature field (surface color map, Figure 12a–d), the thermal flow (streamlines and velocity vectors, Figure 12a–c), magnetic flux density (Figure 12d, surface color map, iso-lines of magnetic

Figure 12. Magnetic field, temperature and flow fields for mono-phased transformer of low power and medium voltage, type TMOf-24-5, (a) Heat transfer and thermal flow—upper part, (b) detail—temperature, buoyancy flow, (c) heat transfer and thermal flow—bottom part and (d) magnetic field.

vector field). Apparently, the thermal flow exhibits two minor recirculation areas—in the upper and lower part of the windings, by the top and bottom covers—trapped within a larger recirculation cell that develops by the lateral wall. For the heat transfer part of the problem, we assumed a convection (Robin) type boundary condition (the ambient temperature was assumed to be Tamb <sup>¼</sup> 300 K, with a heat transfer coefficient of h <sup>¼</sup> 2 W=m2K, that is, moderate natural convection.

The iron core, although less magnetized in this particular regime (compensated primary and secondary amperturns), plays a crucial role in the heat transfer problem.

When a colloidal Fe3O4 specific nanofluid MNF/UTR 40 is utilized as a coolant, magnetic body forces add to the thermal, gravitational body forces. Figure 13 displays the magnetic field and forces (magnetic and thermal). Apparently, the magnetic forces contribute differently to the overall convection flow: in the upper part of the cell they add to the buoyancy forces, whereas in the lower part they are opposite. However not unexpected, Eq. rel. (33), another important finding is the effect that the en-parts of the windings and the iron core have: these are regions of high gradient magnetic field strength, and it is here that the body magnetization forces are significant. The orientation of the magnetic forces versus the thermal forces is an important factor in providing an optimal design. We observe that the thermal gravitationally driven forces and the magnetic forces act concurrently in this plane, their combined effect being greater at the left and right end regions. Similarly, the heat transfer direction is from the hotter regions (core and windings) to the case, but with enhanced convection and increased heat removal efficiency. Comparing the two cooling options, that is, specific nanofluid MNF/UTR 40 (Figure 14b) versus regular coolant UTR 40 (Figure 14a) apparently the MNF/UTR 40 may do better in cooling the transformer. This essentially means a lower hot spot temperature by approximately 10° in this model and a more uniform temperature distribution. These results suggest that the vertical design of the low-power mono-phased transformer (40 kVA), at medium voltage (30/√3//0,4/√3kV) prototype TMOf 2-36kV-40kVA (Figure 15) may be advisable [66, 67]. This prototype TMOf 2-36kV-40kVA realized in compliance with the constructive

vector field). Apparently, the thermal flow exhibits two minor recirculation areas—in the upper and lower part of the windings, by the top and bottom covers—trapped within a larger recirculation cell that develops by the lateral wall. For the heat transfer part of the problem, we assumed a convection (Robin) type boundary condition (the ambient temperature was assumed to be Tamb <sup>¼</sup> 300 K, with a heat transfer coefficient of h <sup>¼</sup> 2 W=m2K, that is, moderate

Figure 12. Magnetic field, temperature and flow fields for mono-phased transformer of low power and medium voltage, type TMOf-24-5, (a) Heat transfer and thermal flow—upper part, (b) detail—temperature, buoyancy flow, (c) heat transfer

The iron core, although less magnetized in this particular regime (compensated primary and

secondary amperturns), plays a crucial role in the heat transfer problem.

natural convection.

and thermal flow—bottom part and (d) magnetic field.

182 Nanofluid Heat and Mass Transfer in Engineering Problems

Figure 13. Magnetic field and body forces when the coolant is a magnetic nanofluid, MNF/UTR 40: (a) detail (top) magnetic body forces, (b) detail (top)—buoyancy forces, (c) detail (bottom)—magnetic body forces, (d) detail (bottom) buoyancy forces.

Figure 14. 2D, axial model - magnetic flux density field, temperature distribution, and flow field for the horizontal design TMOf-24-5 type transformer: (a) the coolant is regular UTR 40 transformer oil and (b) the coolant is a magnetic nanofluid, MNF/UTR 40.

Figure 15. 2D, axial model - magnetic flux density field, temperature distribution, and flow field for the vertical design TMOf 2-36kV-40kVA type transformer: (a) the coolant is regular UTR 40 transformer oil and (b) the coolant is a magnetic nanofluid, MNF/UTR 40.

solutions with characteristics specific for the aimed purpose presents the following advantages: reduced weight and dimensions in comparison with the current power transformers that have the same rated voltage and rated power, following the intensification of the cooling effect in the presence of the specific nanofluid MNF/UTR 40 (Figure 15b). The number of convection zones is greater when the coolant is magnetic nanofluid MNF/UTR 40 (Figure 15b) as compared to the regular coolant, that is, the UTR 40 transformer oil (Figure 15a). Also, because of the execution form of the magnetic circuit and of the metallic construction (tank-bottom-lid), the construction of the power transformer in the aggregate is realized with a smaller consumption of the main materials: copper, magnetic steel sheet and the specific nanofluid MNF/UTR 40 are included.

## 4. Designing the electrical transformer cooled by nanofluid with colloidal magnetic Fe3O4 nanoparticles dispersed in UTR 40 transformer oil

The low-power mono-phased transformer (40 kVA), at medium voltage (30/√3//0,4/√3kV), prototype vertical design TMOf 2-36kV-40kVA type transformer has the active part (Figures 16 and 17), magnetic iron core with the high voltage (HV) and low voltage (LV) windings fixed in

Figure 16. The aggregate active parts, core and windings [44].

Figure 17. Magnetic cores and the core rolling device [44].

solutions with characteristics specific for the aimed purpose presents the following advantages: reduced weight and dimensions in comparison with the current power transformers that have the same rated voltage and rated power, following the intensification of the cooling effect in the presence of the specific nanofluid MNF/UTR 40 (Figure 15b). The number of convection zones is greater when the coolant is magnetic nanofluid MNF/UTR 40

Figure 15. 2D, axial model - magnetic flux density field, temperature distribution, and flow field for the vertical design TMOf 2-36kV-40kVA type transformer: (a) the coolant is regular UTR 40 transformer oil and (b) the coolant is a magnetic

Figure 14. 2D, axial model - magnetic flux density field, temperature distribution, and flow field for the horizontal design TMOf-24-5 type transformer: (a) the coolant is regular UTR 40 transformer oil and (b) the coolant is a magnetic nanofluid,

MNF/UTR 40.

184 Nanofluid Heat and Mass Transfer in Engineering Problems

nanofluid, MNF/UTR 40.

a finned metallic tank constituted from two parts air-proof assembled through a soldering that is soft and capable of elastic deformation for the taking-over of the variation with temperature of the cooling liquid volume (Figure 19). The magnetic circuit is coating type, constituted of two identical cores of rectangular shape (flat-core), back to back disposal (Figure 17). The aggregate active parts (Figure 16) are the magnetic core with the high voltage and low voltage windings, fixed on a metallic lid with their axes in a vertical position, the most convenient situation for the heat transfer enhancement by the nanoparticles in the presence of the electromagnetic field. With a view to performing comparative tests related to the use of magnetic nanofluid MNF/UTR 40 as cooling and insulating fluid transformers and regular UTR 40 transformer oil cooling, the mono-phased transformer of low power and medium voltage, the horizontal design TMOf-24-5 type transformer (Figure 18) and mono-phased transformer of low power and medium voltage the vertical design TMOf 2-36kV-40kVA type transformer (Figure 19) has been achieved [44]. The numerical simulation results show that the direction of the magnetizing force in comparison with the gravitational thermal force is an important

Figure 18. Mono-phased transformer of low power and medium voltage, the horizontal design TMOf-24-5 type transformer.

Nanofluid with Colloidal Magnetic Fe3O4 Nanoparticles and Its Applications in Electrical Engineering http://dx.doi.org/10.5772/65556 187

a finned metallic tank constituted from two parts air-proof assembled through a soldering that is soft and capable of elastic deformation for the taking-over of the variation with temperature of the cooling liquid volume (Figure 19). The magnetic circuit is coating type, constituted of two identical cores of rectangular shape (flat-core), back to back disposal (Figure 17). The aggregate active parts (Figure 16) are the magnetic core with the high voltage and low voltage windings, fixed on a metallic lid with their axes in a vertical position, the most convenient situation for the heat transfer enhancement by the nanoparticles in the presence of the electromagnetic field. With a view to performing comparative tests related to the use of magnetic nanofluid MNF/UTR 40 as cooling and insulating fluid transformers and regular UTR 40 transformer oil cooling, the mono-phased transformer of low power and medium voltage, the horizontal design TMOf-24-5 type transformer (Figure 18) and mono-phased transformer of low power and medium voltage the vertical design TMOf 2-36kV-40kVA type transformer (Figure 19) has been achieved [44]. The numerical simulation results show that the direction of the magnetizing force in comparison with the gravitational thermal force is an important

186 Nanofluid Heat and Mass Transfer in Engineering Problems

Figure 18. Mono-phased transformer of low power and medium voltage, the horizontal design TMOf-24-5 type trans-

former.

Figure 19. Mono-phased transformer of low power and medium voltage, the vertical design TMOf 2-36kV-40kVA type transformer [44].

element in assuring of an optimal heat transfer. Both numerical simulations as well as laboratory measurements [65–67] confirm the following aspects: about the usage of a magnetic nanofluid MNF/UTR 40 as cooling and insulating fluid for transformers, this provides for magnetization body forces that add to the thermal, gravitational forces. In the vertical layout of the transformer, these forces act concurrently with the thermal flow, and the overall effect is the enhancement of the heat transferred from the aggregate active parts (core and windings) to the ambient.

In both cases, first of all, the regular UTR 40 transformer oil as cooling and insulating fluid was used for the transformers. After that, this oil was drained and the transformers were filled with magnetic nanofluid MNF/UTR 40 as cooling and insulating fluid.

Figures 20 and 21 show the temperature on the surface of the ribbed tank when magnetic nanofluid MNF/UTR 40 is used for the vertical design TMOf 2-36kV-40kVA type transformer after 1 h of operation. Monitoring of the temperature was achieved with the thermographic camera, FLUKE Ti 20. The temperature does not exceed the value of 54°C. Magnetic nanofluid

Figure 20. Temperature distribution by thermographic imaging-the tank- for mono-phased transformer of low power and medium voltage, type TMOf2-36kV-40 kVA, after 1 h of operation.

Figure 21. 3D characteristic of the temperature depending on the X and Y coordinates, associated with the thermographic image in Figure 20.

MNF/UTR 40 provides also the increase of the transformer's capacity to sustain over-voltages and withstand better to degradation in time due to humidity, as compared to the regular UTR 40 transformer oil coolant. Thus, transformers with reduced dimensions and higher efficiency with loading capacity and extended life duration may be designed.

## 5. Nanofluid with colloidal magnetic Fe3O4 nanoparticles used in microactuation process

Based on the afore described magnetic nanofluid, we can make a microactuator whose operation complies with the principle of Pulse Width Modulation (PWM) [23, 68]. The output PWM rectangular pulse form for a pulse duty factor of 14% is presented in Figure 22. The PWM generator discharges on the microactuator magnetic nanofluid impedances, two windings L1 and L2 (Figure 23). The electromagnetic force developed by the microactuator and implicitly the movement of the magnetic nanofluid depends mainly on the windings excitation voltage pulse duty factor Ku %. Passing an electric current by the microactuator windings results a magnetic field. The net effect of this magnetic field is a mass transfer of the magnetic nanofluid.

Figure 22. The output PWM rectangular pulse form, for a pulse duty factor of 14%.

Figure 23. The microactuator with magnetic nanofluid during testing [68].

MNF/UTR 40 provides also the increase of the transformer's capacity to sustain over-voltages and withstand better to degradation in time due to humidity, as compared to the regular UTR 40 transformer oil coolant. Thus, transformers with reduced dimensions and higher efficiency

Figure 21. 3D characteristic of the temperature depending on the X and Y coordinates, associated with the thermographic

Figure 20. Temperature distribution by thermographic imaging-the tank- for mono-phased transformer of low power

and medium voltage, type TMOf2-36kV-40 kVA, after 1 h of operation.

188 Nanofluid Heat and Mass Transfer in Engineering Problems

with loading capacity and extended life duration may be designed.

image in Figure 20.

Also, the maximum amplitude of the excitation voltage is constant, Umax = 15 V. The RMS value of the current that goes through the coils of the actuator, for a fixed frequency of the PWM voltage, depends mainly on the pulse duty factor. Two windings, L1 and L2, are excited with a rectangular waveform, counter phase, in compliance with Figure 22.

## Acknowledgements

The authors express special thanks to Prof. Alexandru Mihail Morega, corresponding Member of the Romanian Academy, for valuable results concerning the numerical simulations. The research was performed with the support of UEFISCDI, PNCDI II Programme—Joint Applied Research Projects, Romania, Contract 63/2014, Environment energy harvesting hybrid system by photovoltaic and piezoelectric conversion, DC/DC transformation with MEMS integration and adaptive storage. Also, the numerical simulations were conducted in the Laboratory for Multiphysics Modeling, at UPB, with the support of the PNCDI-II "Parteneriate" Contract 21-043/2008.

## Nomenclature


Also, the maximum amplitude of the excitation voltage is constant, Umax = 15 V. The RMS value of the current that goes through the coils of the actuator, for a fixed frequency of the PWM voltage, depends mainly on the pulse duty factor. Two windings, L1 and L2, are excited

The authors express special thanks to Prof. Alexandru Mihail Morega, corresponding Member of the Romanian Academy, for valuable results concerning the numerical simulations. The research was performed with the support of UEFISCDI, PNCDI II Programme—Joint Applied Research Projects, Romania, Contract 63/2014, Environment energy harvesting hybrid system by photovoltaic and piezoelectric conversion, DC/DC transformation with MEMS integration and adaptive storage. Also, the numerical simulations were conducted in the Laboratory for Multiphysics Modeling, at UPB, with the support of the PNCDI-II "Parteneriate" Contract 21-043/2008.

with a rectangular waveform, counter phase, in compliance with Figure 22.

Acknowledgements

190 Nanofluid Heat and Mass Transfer in Engineering Problems

Nomenclature

ϑ ( o

ρ (kg/m<sup>3</sup>

D (nm) crystallites medium size

Mr (A/m) remnant magnetization Hc (A/m) coercive magnetic field

) mass density

ξ (-) Langevin parameter L(ξ)=cothξ-1/ξ (-) Langevin function

Dm (nm) magnetic diameter H (A/m) magnetic field strength kB (J/K) Boltzmann constant (1.38

T (K) absolute temperature

χiL (-) initial magnetic susceptibility MS (A/m) saturation magnetization ϕ<sup>m</sup> ¼ MS=Md (-) magnetic volume fraction fðxÞ (-) log-normal distribution function

M (A/m) magnetization

λ (nm) wavelength of the Cu-Kα radiation (0.154059 nm)

ϕFe3O4 (-) solid volume fraction of the dispersed magnetite Md (A/m) monodomenial magnetization of magnetite (480 kA/m)

) half diffraction angle of crystal orientation peak (Bragg angle)

+

10−<sup>23</sup> J/K)

+

10−<sup>7</sup> H/m)

B<sup>∗</sup> (rad) full width at half maximum (FWHM)

μ<sup>0</sup> (H/m) magnetic permeability of vacuum (4π

μ (H/m) magnetic permeability of the medium μ<sup>r</sup> (-) relative magnetic permeability (μ/μ0)



## Author details

Lucian Pîslaru-Dănescu1 \*, Gabriela Telipan<sup>1</sup> , Floriana D. Stoian<sup>2</sup> , Sorin Holotescu<sup>2</sup> and Oana Maria Marinică<sup>2</sup>

\*Address all correspondence to: lucian.pislaru@icpe-ca.ro


## References

[1] Vékás L, Bica D, Avdeev M. Magnetic nanoparticles and concentrated magnetic nanofluids: Synthesis, properties and some applications. China Particuology. 2007;5(1–2):43–49.

[2] Ervithayasuporn V, Kawakawi Y. Synthesis and characterization of core-shell type Fe3O4 nanoparticles in poly (organosilsesquixane). Journal of Colloid and Interface Science. 2009;332:389–393. DOI: 10.1016/j.jcis.2008.12.061

n (-) constant whose value depends on flow type; n = 0.25 for laminar flow and

+

10−<sup>12</sup> F/m)

, Sorin Holotescu<sup>2</sup> and Oana

n = 0.33 for turbulent flow

ε<sup>0</sup> (F/m) electric permittivity of free space (8.8541878176

εMNF (F/m) effective electric permittivity of the magnetic fluid

εNP (F/m) electric permittivity of the magnetite nanoparticles

) angular component of the current density

+

10−<sup>5</sup> m/A)

[1] Vékás L, Bica D, Avdeev M. Magnetic nanoparticles and concentrated magnetic nanofluids: Synthesis, properties and some applications. China Particuology. 2007;5(1–2):43–49.

, Floriana D. Stoian<sup>2</sup>

ε<sup>r</sup> (-) relative electric permittivity (ε<sup>r</sup> = ε/ε0)

εUTR (F/m) electric permittivity of the transformer oil

ε (F/m) electric permittivity

192 Nanofluid Heat and Mass Transfer in Engineering Problems

u (m/s) velocity ω (rad/s) angular speed

) pressure A (-) magnetic vector field E (V/m) electric field strength

fT (N) buoyancy body force term fmg (N) magnetic body force term σ (S/m) electrical conductivity

/s) kinematics viscosity B (T) magnetic field induction Br (T) remnant induction

a (A/m) empiric constant (10<sup>4</sup> A/m)

Tamb (K) ambient temperature (300 K) h (W/m<sup>2</sup> K) heat transfer coefficient

Umax (V) excitation voltage at maximum amplitude

\*Address all correspondence to: lucian.pislaru@icpe-ca.ro

2 Politehnica University of Timisoara, Timisoara, Romania

\*, Gabriela Telipan<sup>1</sup>

1 National Institute for Electrical Engineering ICPE-CA, Bucharest, Romania

b (m/A) empiric constant (3

p (N/m<sup>2</sup>

J e <sup>ϕ</sup> (A/m<sup>2</sup>

<sup>μ</sup><sup>f</sup> (m<sup>2</sup>

Author details

Maria Marinică<sup>2</sup>

References

Lucian Pîslaru-Dănescu1


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#### **Magnetic Nanofluids: Mechanism of Heat Generation and Transport and Their Biomedical Application Magnetic Nanofluids: Mechanism of Heat Generation and Transport and Their Biomedical Application**

Prem P. Vaishnava and Ronald J. Tackett Prem P. Vaishnava and Ronald J. Tackett

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/66389

#### **Abstract**

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198 Nanofluid Heat and Mass Transfer in Engineering Problems

6231963

Magnetic nanofluids, also known as ferrofluids, are a colloidal mixture of superpara‐ magnetic nanoparticles suspended in a carrier fluid. For most biomedical applications, such as magnetic fluid hyperthermia (MFH), 10‐ to 20‐nm mean diameter nanoparticles of Fe3O4 or γ‐Fe2O3 (or various ferrites such as CoFe2O4 and ZnFe2O4) are coated with an organic surfactant (such as dextran) and suspended in water. These ferrofluids exhibit both magnetic and fluidic properties and were first developed by NASA in the 1960s for space applications; however, in the twenty‐first century they have been the subject of intense investigation due to their technological and biomedical applications. In biomedicine, ferrofluids have a wide range of uses including by not limited to use as MRI contrast agents, for the targeted delivery of drugs, for DNA transfection, and in the MFH treatment of cancer. For technological applications, ferrofluids are investigated for cooling multiple electronic devices. This chapter describes common methods of ferrofluid synthesis, and their magnetic, structural, and morphological properties, as well as a discussion of the methods of heat generation and transport when exposed to RF alternating magnetic fields. The results of a number of investigations have been used to illustrate the application of MFH for the treatment of malignant tumors without the undesirable side‐effects of the more traditional radiation and chemotherapy treatment regimens.

**Keywords:** magnetic nanofluids, ferrofluids, hyperthermia, biomedical engineering

## **1. Introduction**

From the early times, all known magnetic materials have existed in the solid form either as permanent magnets or soft magnetic materials; however, scientists have dreamed about a

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

stable and durable liquid magnet for some time. The desire to create such a liquid became reality in the 1960s when Steven Papell of NASA initiated a project to control and direct liquid rocket fuel by converting nonmagnetic fuel into magnetically active fuel that could be controlled by the application of magnetic field in zero gravity conditions [1]. As a result of this work, Papell is credited as the first person to prepare a liquid magnetic material and patented this procedure in 1965.

The magnetic fluid developed by Papell was never used in the aerospace industry; however, it set the foundation for intense scientific research and development of magnetofluidic technology. Ron Rosensweig led the development of magnetic fluid mechanics which is a new branch of ferro‐hydrodynamics. In 1968 Rosensweig and Moskowitz founded a corporation (now known as FerroTech) which developed a product named Ferrofluid consisting of three components: nanosize (10–100 nm) iron oxide particle (usually magnetite, Fe3O4), a surfactant or a dispersant coating, and a carrier fluid. Surfactants are soap‐like materials that coat the nanoparticles to prevent the aggregation of nanoparticles which prevents suspension of the nanoparticles in the carrier fluid (**Figure 1**).

**Figure 1.** A magnet shapes a ferrofluid from underneath a glass sheet. In the 1960s, NASA developed the magnetic liquid technology as a way to move a spacecraft's fuel to its engines. *Image courtesy of Gregory Maxwell*.

In recent years, ferrofluids have been actively pursued as an adjuvant therapy together with chemotherapy and radiation to treat malignant tumors [2–8]. In addition, there has been a wealth of investigation into the flow and heat transfer properties of these nanofluids under various conditions and with various materials—studies that are key to the understanding of how these materials may be used for biomedical applications [9–19]. For hyperthermia therapy, magnetic nanoparticle (ferrofluids) solutions are directly injected into the tumor which is subsequently exposed to an alternating magnet of the frequency from 150 to 500 kHz. As energy from the magnetic field is absorbed by the nanoparticles (through processes to be described later), the ferrofluid and its surroundings (the tumor) are heated. Due to the increased temperature sensitivity of malignant neoplastic tissue (as compared to healthy cells), tumor necrosis can be initiated without damage to the surrounding tissue (**Figure 2**).

In addition, this procedure makes the malignant cells more sensitive to chemotherapy and radiation, allowing the application of these therapies in reduced doses to minimize their negative side effects. Successful management of magnetic fluid hyperthermia (MFH) treatment requires a clear understanding of the heat generation by the nanoparticles as well as the heat transfer from the ferrofluid to the malignant tissue. Heat generation in the nanoparticles occurs through three separate processes that can run in parallel. These mechanisms include hysteresis loss, Néel relaxation, and Brownian relaxation, all of which have been extensively studied [21]. The processes of heat transfer through particle‐particle interactions, tissue‐particle interac‐ tions, and by other mechanisms are much less understood and have not undergone the exhaustive investigation seen by the mechanisms involved in heat generation by magnetic nanoparticles. At the time of publication of this chapter, there are no known publications that report the effects of interparticle interaction or tissue‐particle interactions on heat transfer in malignant tissue.

**Figure 2.** Basic theme of hyperthermia treatment for killing cancer cells [20].

stable and durable liquid magnet for some time. The desire to create such a liquid became reality in the 1960s when Steven Papell of NASA initiated a project to control and direct liquid rocket fuel by converting nonmagnetic fuel into magnetically active fuel that could be controlled by the application of magnetic field in zero gravity conditions [1]. As a result of this work, Papell is credited as the first person to prepare a liquid magnetic material and patented

The magnetic fluid developed by Papell was never used in the aerospace industry; however, it set the foundation for intense scientific research and development of magnetofluidic technology. Ron Rosensweig led the development of magnetic fluid mechanics which is a new branch of ferro‐hydrodynamics. In 1968 Rosensweig and Moskowitz founded a corporation (now known as FerroTech) which developed a product named Ferrofluid consisting of three components: nanosize (10–100 nm) iron oxide particle (usually magnetite, Fe3O4), a surfactant or a dispersant coating, and a carrier fluid. Surfactants are soap‐like materials that coat the nanoparticles to prevent the aggregation of nanoparticles which prevents suspension of the

**Figure 1.** A magnet shapes a ferrofluid from underneath a glass sheet. In the 1960s, NASA developed the magnetic

In recent years, ferrofluids have been actively pursued as an adjuvant therapy together with chemotherapy and radiation to treat malignant tumors [2–8]. In addition, there has been a wealth of investigation into the flow and heat transfer properties of these nanofluids under various conditions and with various materials—studies that are key to the understanding of how these materials may be used for biomedical applications [9–19]. For hyperthermia therapy, magnetic nanoparticle (ferrofluids) solutions are directly injected into the tumor which is subsequently exposed to an alternating magnet of the frequency from 150 to 500 kHz. As energy from the magnetic field is absorbed by the nanoparticles (through processes to be described later), the ferrofluid and its surroundings (the tumor) are heated. Due to the increased temperature sensitivity of malignant neoplastic tissue (as compared to healthy cells),

liquid technology as a way to move a spacecraft's fuel to its engines. *Image courtesy of Gregory Maxwell*.

tumor necrosis can be initiated without damage to the surrounding tissue (**Figure 2**).

In addition, this procedure makes the malignant cells more sensitive to chemotherapy and radiation, allowing the application of these therapies in reduced doses to minimize their

this procedure in 1965.

nanoparticles in the carrier fluid (**Figure 1**).

200 Nanofluid Heat and Mass Transfer in Engineering Problems

Along with the investigations of heat generation in ferrofluids, much work has been done toward the understanding of heat conduction as exhibited by these materials. Conventional fluids used for heat transfer applications such as water, oil, and ethylene glycol have serious limitations in improving the performance of a host of electronic equipment; however, the suspension of magnetic nanoparticles in these fluids (creating a ferrofluid) provide an innovative method of improving the thermal conductivity of these materials. Many researchers have mixed various nanosized particles of different materials to study their heat transfer characteristics [22–25]. Ferrofluids exhibit both magnetic and fluidic properties thus, theoret‐ ically, allowing for the possibility to produce a material with a tunable viscosity, surface tension, temperature, vapor pressure, and stability in a hostile environment. Such nanofluids have potential applications in microelectronic devices operating at high speeds, high‐power engines, optical devices, and similar devices requiring advanced cooling systems. Ferrofluids have many exciting applications in biomedicine which include targeted drug delivery, MRI contrast agents, and magnetic fluid hyperthermia. This chapter will review not only the sources of heat generation in magnetic nanofluids, it will also discuss the means of energy transfer within the malignant tumors. A short discussion on the challenges in the clinical application of hyperthermia is also included.

## **2. Synthesis techniques for magnetic nanofluids**

Over the past several decades, different methods for the synthesis of superparamagnetic iron oxide nanoparticles (SPION) have been investigated [26]. This interest is predominantly driven by iron oxide's biocompatibility and high saturation magnetization which makes this material an ideal candidate for applications in biomedicine. While the synthesis of iron oxides can be quite complicated due to the many distinct species of iron oxides, iron hydroxides, and iron oxyhydroxides, synthesis can be achieved through various well‐defined processes discussed here.

#### **2.1. Coprecipitation of iron oxide nanoparticles**

The simplest and most efficient of all techniques for the production of SPION is that of the coprecipitation of iron salts using a strong base [26]. In this method, iron oxides, generally magnetite (Fe3O4) or maghemite (γFe2O3), are prepared through the aging of stoichiometric aqueous mixtures of iron(II) and iron(III) salts. Formation of Fe3O4 is achieved through the chemical reaction

$$\mathrm{Fe}^{2+} + 2\mathrm{Fe}^{3+} + 8\mathrm{OH}^- \rightarrow \mathrm{Fe}\_3\mathrm{O}\_4 + 4\mathrm{H}\_2\mathrm{O}.\tag{1}$$

Starting with a stoichiometric ratio of 2Fe3+:1Fe2+ in a nonoxidizing environment, the thermo‐ dynamics of this reaction dictate that full precipitation of Fe3O4 is expected at a pH between 8 and 14. While Fe3O4 has a high saturation magnetization (a desirable property for many biomedical applications), it is generally unstable and readily transforms into the less magnet‐ ically active γFe2O3 in the presence of oxygen through the reaction

$$\text{Fe}\_3\text{O}\_4 + 2\text{H}^+ \rightarrow \wp \text{Fe}\_3\text{O}\_4 + \text{Fe}^{2+} + \text{H}\_2\text{O}.\tag{2}$$

It is important to note that oxidation in air is not the only pathway through which Fe3O4 can transform into γFe2O3 as this process can occur through various electronic or ionic transfers depending on the pH of the suspension involved.

The coprecipitation process allows for rapid production of large amounts of nanoparticles; however, the technique only allows for limited control of particle size distributions as the growth of the crystals are controlled only by kinetic factors. The size of nanoparticles synthe‐ sized through coprecipitation can be tuned over the range from 2 to 17 nm with some success through adjustment of reaction pH, ionic strength, reaction temperature, the nature of the salts, or the Fe2+:Fe3+ concentration ratio. In addition, the addition of chelating organic anions or polymer surface complexion agents during nanoparticle formation can help to control the size of the resultant particles.

## **2.2. Reactions in constrained environments**

engines, optical devices, and similar devices requiring advanced cooling systems. Ferrofluids have many exciting applications in biomedicine which include targeted drug delivery, MRI contrast agents, and magnetic fluid hyperthermia. This chapter will review not only the sources of heat generation in magnetic nanofluids, it will also discuss the means of energy transfer within the malignant tumors. A short discussion on the challenges in the clinical application

Over the past several decades, different methods for the synthesis of superparamagnetic iron oxide nanoparticles (SPION) have been investigated [26]. This interest is predominantly driven by iron oxide's biocompatibility and high saturation magnetization which makes this material an ideal candidate for applications in biomedicine. While the synthesis of iron oxides can be quite complicated due to the many distinct species of iron oxides, iron hydroxides, and iron oxyhydroxides, synthesis can be achieved through various well‐defined processes discussed

The simplest and most efficient of all techniques for the production of SPION is that of the coprecipitation of iron salts using a strong base [26]. In this method, iron oxides, generally magnetite (Fe3O4) or maghemite (γFe2O3), are prepared through the aging of stoichiometric aqueous mixtures of iron(II) and iron(III) salts. Formation of Fe3O4 is achieved through the

Starting with a stoichiometric ratio of 2Fe3+:1Fe2+ in a nonoxidizing environment, the thermo‐ dynamics of this reaction dictate that full precipitation of Fe3O4 is expected at a pH between 8 and 14. While Fe3O4 has a high saturation magnetization (a desirable property for many biomedical applications), it is generally unstable and readily transforms into the less magnet‐

γ <sup>2</sup>

It is important to note that oxidation in air is not the only pathway through which Fe3O4 can transform into γFe2O3 as this process can occur through various electronic or ionic transfers

The coprecipitation process allows for rapid production of large amounts of nanoparticles; however, the technique only allows for limited control of particle size distributions as the

34 2 Fe Fe OH 2 8 4Fe O H O. (1)

3 4 3 4 <sup>2</sup> Fe O 2H Fe O Fe H O. + + +® ++ (2)

++ - ++ ® + 2 3

ically active γFe2O3 in the presence of oxygen through the reaction

depending on the pH of the suspension involved.

of hyperthermia is also included.

202 Nanofluid Heat and Mass Transfer in Engineering Problems

here.

chemical reaction

**2. Synthesis techniques for magnetic nanofluids**

**2.1. Coprecipitation of iron oxide nanoparticles**

Another method to produce SPION of well‐defined size uses synthetic and biological nano‐ reactors to provide size constrained environments for particle formation [27–33]. These processes include the use of amphoteric surfactants to create water‐swollen reversed micelle structures in monopolar solvents, apoferritin protein cages, dendrimers, cyclohexatrienes, and phospholipid membranes that form vesicles with iron oxide nanoparticles serving as solid supports.

## **2.3. Hydrothermal and high‐temperature reactions**

There are two main avenues through which the formation of ferrites can be achieved via hydrothermal conditions. These two reactions, hydrolysis and the oxidation or neutralization of mixed metal hydroxides, are very similar in nature; however, hydrolysis involves the use of ferrous salts.

In the hydrothermal synthesis of SPION, the reaction conditions (i.e., solvent, temperature, and time) generally have important effects on the outcome of the nanoparticles produced. It is generally accepted that the particle size increases with prolonged reaction time and that higher water content results in the creation of larger particles. By the nature of the hydrother‐ mal process, particle size is mainly controlled through the rate processes of nucleation and grain growth. Generally, with temperature held constant, these rates depend on reaction temperature with nucleation outpacing grain growth at higher temperatures, resulting in decreased particle size. Contrastingly, longer reaction times favor the grain growth process which results in larger particles.

The production of size‐monodisperse SPION with high levels of size control can be achieved [34] through the high‐temperature decomposition of iron‐based organic precursors, such as Fe(CO)5, Fe(acac)3, or Fe(Cup)3, using organic solvents and surfactants. While the nanoparticles produced through these methods are highly desirable due to their monodispersivity, there are barriers to scaled‐up production in terms of the hazards associated with the reactants and the high temperatures required.

## **2.4. Sol‐gel reactions**

An often used process for the wet‐chemical synthesis of metal‐oxide nanostructures is that of the sol‐gel method which is based on the hydroxylation and condensation of molecular precursors in solution, resulting in the production of a "sol" of nanometer‐sized particles [35– 37]. Heating results in further condensation and inorganic polymerization of the sol into a wet gel of a denominated three‐dimensional metal oxide network. Postreaction annealing is typically required to achieve the final crystallization of the particles. The main parameters which influence the kinetics, reaction growth, hydrolysis, and condensation (and consequently the structure and properties of the gel) are solvent, temperature, nature, precursor (salts) concentration, pH, and agitation.

## **2.5. Polyol methods**

The polyol process [38] is essentially a sol‐gel process in which a polyol, such as polyethylene glycol, is used as the solvent. In this technique, precursor compounds are dissolved in a liquid polyol which is subsequently heated (while stirring) to the boiling point of the polyol. During this process, the metal precursor becomes solubilized in the diol, forming an intermediate product which is then reduced to form metal nuclei that will serve nucleation centers for the formation of metal particles. Particle size is tunable through the variation of reaction temper‐ ature with smaller particles created at higher temperatures. In addition, size can be controlled through the induction of heterogeneous nucleation by adding foreign nuclei or forming foreign nuclei *in situ*.

## **2.6. Flow injection synthesis**

Confinement of the reaction space through the use of different matrices, such as emulsions, has been shown to produce particles with narrow size distributions and, in some cases, lead to desired particles morphologies. Salazar‐Alvarez et al. [39] developed a novel technique for the production of magnetite nanoparticles based on a flow injection synthesis (FIS) method. This technique consists of continuous or segmented mixing of reagents under laminar flow in a capillary reactor. The high reproducibility of this method that results from the plug flow and laminar flow conditions, high mixing homogeneity, and precise control, make this an appealing technique for producing Fe3O4 nanoparticles in the 2–7 nm range.

## **3. Surface modification of magnetic nanoparticles**

Once synthesized, magnetic nanoparticles are surfactant‐coated and dispersed in liquid carriers to create a ferrofluid. Surfactants used to coat magnetic nanoparticles are typically organic molecules such as oleic acid, tetramethyalammonium hydroxide (TMAH), lauric acid, and dextran. The purpose of these surfactants is to prevent clumping of the nanoparticles so as to prevent the formation of aggregates that become too heavy to remain dispersed in the liquid carrier. These molecules, which are typically composed of a polar head (or tail) and a nonpolar tail (or head), are typically absorbed to a nanoparticle on the one end while the other end sticks out into the carrier medium forming a micelle (either inverse or regular) around the particle. The electrostatic repulsion of the ends of these molecules prevents the agglomeration of the nanoparticles. **Figure 3** gives a brief description of various coating polymers (and types) used in this process.

Magnetic Nanofluids: Mechanism of Heat Generation and Transport and Their Biomedical Application http://dx.doi.org/10.5772/66389 205

**Figure 3.** Different polymers used for surface modification of magnetic nanoparticles [32].

## **4. Characterization of ferrofluids**

gel of a denominated three‐dimensional metal oxide network. Postreaction annealing is typically required to achieve the final crystallization of the particles. The main parameters which influence the kinetics, reaction growth, hydrolysis, and condensation (and consequently the structure and properties of the gel) are solvent, temperature, nature, precursor (salts)

The polyol process [38] is essentially a sol‐gel process in which a polyol, such as polyethylene glycol, is used as the solvent. In this technique, precursor compounds are dissolved in a liquid polyol which is subsequently heated (while stirring) to the boiling point of the polyol. During this process, the metal precursor becomes solubilized in the diol, forming an intermediate product which is then reduced to form metal nuclei that will serve nucleation centers for the formation of metal particles. Particle size is tunable through the variation of reaction temper‐ ature with smaller particles created at higher temperatures. In addition, size can be controlled through the induction of heterogeneous nucleation by adding foreign nuclei or forming

Confinement of the reaction space through the use of different matrices, such as emulsions, has been shown to produce particles with narrow size distributions and, in some cases, lead to desired particles morphologies. Salazar‐Alvarez et al. [39] developed a novel technique for the production of magnetite nanoparticles based on a flow injection synthesis (FIS) method. This technique consists of continuous or segmented mixing of reagents under laminar flow in a capillary reactor. The high reproducibility of this method that results from the plug flow and laminar flow conditions, high mixing homogeneity, and precise control, make this an appealing

Once synthesized, magnetic nanoparticles are surfactant‐coated and dispersed in liquid carriers to create a ferrofluid. Surfactants used to coat magnetic nanoparticles are typically organic molecules such as oleic acid, tetramethyalammonium hydroxide (TMAH), lauric acid, and dextran. The purpose of these surfactants is to prevent clumping of the nanoparticles so as to prevent the formation of aggregates that become too heavy to remain dispersed in the liquid carrier. These molecules, which are typically composed of a polar head (or tail) and a nonpolar tail (or head), are typically absorbed to a nanoparticle on the one end while the other end sticks out into the carrier medium forming a micelle (either inverse or regular) around the particle. The electrostatic repulsion of the ends of these molecules prevents the agglomeration of the nanoparticles. **Figure 3** gives a brief description of various coating polymers (and types)

technique for producing Fe3O4 nanoparticles in the 2–7 nm range.

**3. Surface modification of magnetic nanoparticles**

concentration, pH, and agitation.

204 Nanofluid Heat and Mass Transfer in Engineering Problems

**2.5. Polyol methods**

foreign nuclei *in situ*.

used in this process.

**2.6. Flow injection synthesis**

Powder X‐ray diffraction is the most abundantly used technique with which to study the crystal structure of the magnetic nanoparticles suspended in ferrofluid. In this technique, a monochromatic beam of X‐rays is incident on powder consisting of approximately 1010 nanoparticles packed at ∼50% density and the scattered X‐rays are collected as a function of the diffraction angle . A peak in X‐ray intensity is observed when the scattered beam from a set of lattice planes satisfies the Bragg condition, given by

$$2\triangle \sin \Theta = n\lambda\_\prime \tag{3}$$

where is the interplanar spacing characterized by the Miller indices (ℎ), is the diffraction angle, and is the wavelength of the X‐rays (typically 1.54 Å for machines using a copper‐ anode‐based X‐ray generator). For nanoparticles with a spherical morphology, the crystallite diameter, , can be determined using the Debye‐Scherer equation [40]

$$D = \frac{0.9\lambda}{\Gamma \cos \theta} \,\,\,\,\,\tag{4}$$

where Γ is the full‐width‐at‐half‐maximum (FWHM) of the X‐ray peak and is the diffraction angle about which the peak is centered. In most cases, the collected spectra is analyzed fit using a either Le Bail fit [41] (which matches the spectrum to a certain lattice type—where the peaks are located) or a Rietveld refinement [42] (which takes into account both the lattice type and form factors of the atoms contained on the lattice—where the peaks are located and their relative intensities) (**Figure 4**).

**Figure 4.** XRD spectrum taken for Fe3O4 nanoparticle ensemble. The open circles correspond to the observed X‐ray in‐ tensity and the line was generated using a Le Bail fitting routine (FullPROF) confirming the cubic () lattice with a lattice parameter of 8.36 Å. Using the full‐width‐at‐half‐maximum of the (311) peak in the Debye‐Scherrer equation, the average nanoparticle diameter was found to be 14 nm [43].

**Figure 5.** (a) A TEM image of Fe3O4 nanoparticles, (b) a high‐resolution TEM micrograph of an Fe3O4 nanoparticle, and (c) a histogram of the particle size distribution with an associated fit to a log‐normal distribution [44].

Using transmission electron microscopy (TEM), nanoparticles can be imaged and information about the morphology and size can be determined. TEM operates on the same principles as that of an optical microscope; however, it uses high energy electrons (typically 80–200 keV) as an illumination source instead of light. The small DeBroglie wavelengths of these electrons allow an electron microscope to overcome the optical resolution limits of light microscopes and can provide resolution down to subnanometer scales. **Figure 5** shows images collected using a JEOL JEM‐2010 TEM operating a 200 kV (200 keV electrons) to image Fe3O4 nanopar‐ ticles. The samples were prepared by sonicating the nanoparticles in ethanol at very low concentrations (∼250–500 μg/mL) and then pipetting a drop of the solution onto a carbon‐ coated (∼100 nm thickness) copper grid.

#### **4.1. Magnetic characterization**

where Γ is the full‐width‐at‐half‐maximum (FWHM) of the X‐ray peak and is the diffraction angle about which the peak is centered. In most cases, the collected spectra is analyzed fit using a either Le Bail fit [41] (which matches the spectrum to a certain lattice type—where the peaks are located) or a Rietveld refinement [42] (which takes into account both the lattice type and form factors of the atoms contained on the lattice—where the peaks are located and their

**Figure 4.** XRD spectrum taken for Fe3O4 nanoparticle ensemble. The open circles correspond to the observed X‐ray in‐ tensity and the line was generated using a Le Bail fitting routine (FullPROF) confirming the cubic () lattice with a lattice parameter of 8.36 Å. Using the full‐width‐at‐half‐maximum of the (311) peak in the Debye‐Scherrer equation,

**Figure 5.** (a) A TEM image of Fe3O4 nanoparticles, (b) a high‐resolution TEM micrograph of an Fe3O4 nanoparticle, and

(c) a histogram of the particle size distribution with an associated fit to a log‐normal distribution [44].

relative intensities) (**Figure 4**).

206 Nanofluid Heat and Mass Transfer in Engineering Problems

the average nanoparticle diameter was found to be 14 nm [43].

The magnetic properties (i.e., saturation magnetization, magnetic core radius, coercivity, etc.) of the ferrofluids can be determined through ac‐ and dc‐magnetometry using a physical properties measurement system (Quantum Design Model 6000 PPMS). Small cylinders made from 1266 stycast (Emerson & Cumming) epoxy can be constructed or polystyrene capsules can be purchased to confine the liquid for measurement. Ferrofluid of known concentration is sealed in the confinement vessel and left to cure for 24 h. The vessel is then stitched securely into a measurement straw, sealed with kapton tape, and loaded into the PPMS to measure magnetization curves as a function of field or temperature (**Figure 6**).

**Figure 6.** Low‐field (*H* = 100 Oe) ZFC‐FC moment of NiFe2O4 nanoparticles showing the blocking process at *T*<sup>1</sup> ∼ 45 K and a sudden increase of the magnetization in the FC branch at *T*<sup>2</sup> ∼ 6 K [45].

#### **4.2. Dynamic light scattering**

Dynamic light scattering is the most common method used for making reliable estimates of the hydrodynamic sizes of the colloidal particles. It is also known as quasi‐elastic light scattering or photon correlation spectroscopy. There are two distinct advantages associated with the use of DLS to measure sizes: the measurement time is short and the entire process is minimally labor intensive. During the measurement, the colloidal solution is exposed to a light beam and the scattered beam intensity is observed. The electric field of the light beam interacts with the particles in the medium and leads to a shift in the frequency and angular distribution of light.

A typical arrangement of the dynamic light scattering experiment is shown in **Figure 7**. The thermal motion of the nanoparticles in solution is random and the time‐dependent fluctuations in the scattered light are measured. The main goal of the experiment is to measure the diffusion coefficient of the particles in solution. To achieve this the scattered light intensity is measured over a range of scattering angles *θ* for a given time *t* in steps of Δ*t*. Typically in scattering experiments, the scattering angles *θ* is expressed as a magnitude of the scattering wave vector *q* given by [47]

$$q = \left(\frac{4\pi m}{\lambda}\right) \sin\left(\frac{\Theta}{2}\right) \tag{5}$$

**Figure 7.** Typical experimental setup for the DLS experiment [46].

where is the refractive index of the solution and *λ* is the wavelength of light in vacuum. Due to the Brownian motion of the particles, the time‐dependent fluctuation *I*(*q*,*t*) fluctuates around the average intensity (). A quantitative measure of the random fluctuations is the second‐ order correlation function given by

$$\log^2(\pi) = \left\langle I(t)I(t+\pi) \right\rangle / \left\langle I(t) \right\rangle^2 \tag{6}$$

The averaging is done over time. From the correlation function, the decay rate Γ is determined using the equation

$$\log^2(\tau) = A + \beta \mathrm{e} \times \mathrm{p} \left( -2\mathrm{"{r}"} \,\mathrm{r} \right) \tag{7}$$

In the above expression *A* is the baseline at infinite decay and *β* is the amplitude at zero decay. The diffusion constant *D* is related to the decay rate by,

$$\mathbf{D} = \mathfrak{N}^{\top} / \mathfrak{q}^{2} \tag{8}$$

Assuming spherical particles in Brownian motion, determination of the diffusion constant, , leads to the evaluation of the hydrodynamic radius, RH using the Stokes‐Einstein equation given by [48],

$$D = \frac{kT}{6\pi\eta R\_H} \tag{9}$$

where is the viscosity of the medium, is the absolute temperature, and is the Boltzmann constant. The DLS technique is very sensitive to the presence of small aggregates due to the scattering intensity being proportional to the sixth power of the particle radius.

#### **4.3. Magnetic hyperthermia system**

scattering or photon correlation spectroscopy. There are two distinct advantages associated with the use of DLS to measure sizes: the measurement time is short and the entire process is minimally labor intensive. During the measurement, the colloidal solution is exposed to a light beam and the scattered beam intensity is observed. The electric field of the light beam interacts with the particles in the medium and leads to a shift in the frequency and angular distribution

A typical arrangement of the dynamic light scattering experiment is shown in **Figure 7**. The thermal motion of the nanoparticles in solution is random and the time‐dependent fluctuations in the scattered light are measured. The main goal of the experiment is to measure the diffusion coefficient of the particles in solution. To achieve this the scattered light intensity is measured over a range of scattering angles *θ* for a given time *t* in steps of Δ*t*. Typically in scattering experiments, the scattering angles *θ* is expressed as a magnitude of the scattering wave vector

<sup>4</sup> sin

**Figure 7.** Typical experimental setup for the DLS experiment [46].

order correlation function given by

æ ö æö <sup>=</sup> ç ÷ ç÷ è ø èø *πn θ*

2

where is the refractive index of the solution and *λ* is the wavelength of light in vacuum. Due to the Brownian motion of the particles, the time‐dependent fluctuation *I*(*q*,*t*) fluctuates around the average intensity (). A quantitative measure of the random fluctuations is the second‐

*<sup>q</sup> <sup>λ</sup>* (5)

<sup>2</sup> <sup>2</sup> g() ()( ) / () t= t *I t I t+ I t* (6)

of light.

208 Nanofluid Heat and Mass Transfer in Engineering Problems

*q* given by [47]

A schematic representation of the magnetic hyperthermia system is shown in **Figure 8** (one of several commercially available hyperthermia systems available for purchase). The experimen‐ tal setup consists of a coil which can carry current provided by an amplifier. This coil essentially acts as an inductor (of inductance ) and is coupled in parallel to a capacitor (of capacitance ). This arrangement provides an oscillating magnetic field of a particular amplitude and frequency; amplitude being determined by the current through the coil and frequency determined by the values of *L* and *<sup>C</sup>* ( <sup>=</sup> <sup>1</sup> 2 ). The ferrofluid is placed within the coil and the temperature rise is measured using a fiber optic thermometer. The sample is enclosed in an insulating environment to minimize heat loss to the surrounding environment

The actual setup is shown in **Figure 8**. The magnetic hyperthermia measurements were carried out under ambient conditions and magnetic field strengths were measured using a simple pick up coil of known mutual inductance and measuring the induced emf with an oscilloscope. With the current set‐up magnetic fields of amplitude 140–240 Oe can be generated at frequen‐ cies ranging between 150 and 500 kHz (**Figure 9**).

The sample is placed in a small polystyrene vial insulated all around, including the top and the bottom, using a cotton sleeve. This was done to minimize the heat loss to the surroundings. The sample holder was then placed inside the solenoid such that the entire sample was well within the region of uniform magnetic field. This corresponds to the sample height being approximately 3 cm in the sample holder, in our set up. The temperature of the sample was monitored using an Optocon P/N FOTEMP1‐OEM fiber optic thermometer interfaced to the computer for automatic data collection every 2 or 5 s (**Figure 10**).

**Figure 8.** Schematic representation of a typical setup for measurement of the specific absorption rate (SAR) in a ferro‐ fluid [49].

**Figure 9.** Ambrell Easy Heat System with coil, sample holder, insulating cotton padding, and OPTOCON fiber optic thermometer.

Magnetic Nanofluids: Mechanism of Heat Generation and Transport and Their Biomedical Application http://dx.doi.org/10.5772/66389 211

**Figure 10.** The temperature vs. time plot for the AGFO (γ‐Fe2O3 in alginate), TGFO (γ‐Fe2O3 coated with tetramethy‐ lammonium hydroxide), and TFO (Fe3O4 coated with tetramethylammonium hydroxide) samples heated by a 250 Oe ac magnetic field oscillating at 125 kHz. The heating curve for a pure water (deionized, unfiltered) sample is also in‐ cluded as a reference. The solid lines drawn through the data are intended as guides to the eye [50].

#### **4.4. Heat generation in magnetic nanofluids**

The sample holder was then placed inside the solenoid such that the entire sample was well within the region of uniform magnetic field. This corresponds to the sample height being approximately 3 cm in the sample holder, in our set up. The temperature of the sample was monitored using an Optocon P/N FOTEMP1‐OEM fiber optic thermometer interfaced to the

**Figure 8.** Schematic representation of a typical setup for measurement of the specific absorption rate (SAR) in a ferro‐

**Figure 9.** Ambrell Easy Heat System with coil, sample holder, insulating cotton padding, and OPTOCON fiber optic

computer for automatic data collection every 2 or 5 s (**Figure 10**).

210 Nanofluid Heat and Mass Transfer in Engineering Problems

fluid [49].

thermometer.

Heat is generated in magnetic nanoparticles suspended in a carrier fluid when exposed to an oscillating magnetic field primarily by two mechanisms—Neel relaxation and Brownian relaxation. The process of heat generation is complicated because of the short thermal relax‐ ation time constants together with challenges of coupling sufficient power to obtain the de‐ sired temperature. In biomedical application it has been found that individual nanoparticles are not able to effectively heat cells and tissues in the presence of an oscillating magnetic field [51]. Nonetheless, various experiments have shown that sufficient nanoparticle heating can be achieved if the nanoparticles aggregate. Therefore, clustering of the nanoparticles is essential for useful heat generation [52, 53]. There are many investigations that point to heat generation by particle‐particle interactions. The precise process is not very clear whether it is the result of the combination effect or the individual process. More details of heat genera‐ tion by particle‐particle interaction is provided in the following section. Nonetheless, the heating effects are highly local in nature; the clustering and spatial distribution of the parti‐ cles play an important role in heating effects. Most of the literature is filled with the experi‐ mental results about the heat generation in ferrofluids. However, a study by Etheridge et al. [54] includes both experimental and numerical models of heat generation in a droplet con‐ taining magnetic nanoparticles. One of the goals of the study is to get an insight of the order of magnitude of volume power density (W/m3 ) required to achieve significant heating in evenly dispersed clusters of nanoparticles by using finite element method. The study also investigated heat transfer in multiple tumor geometry environments. Recently, Pearce et al. [55] developed and implemented a numerical model that included the bio distribution of the magnetic nanoparticles within the local boundary conditions.

As discussed above, magnetic heating in ferrofluids is produced primarily by two dissipation mechanisms, Neel and Brownian relaxation. In Neel relaxation [56], which dominates the magnetic relaxation for nanoparticles fixed in a solid matrix, the nanoparticle magnetic moment aligns with the external field by coherently flipping along the magnetocrystalline easy axes. Conversely, Brownian relaxation [57] requires the nanoparticle to undergo a physical rotation to align the moment with the applied field, which is suppressed for fixed nanoparti‐ cles. Magnetic dissipation in ferrofluids has been investigated in the framework of Neel and Brownian relaxation both theoretically and experimentally by considering the dependence on particle size and frequency, among other parameters [58].

The Neel relaxation time, which depends on the nanoparticle magnetic core volume *Vm*, the magnetocrystalline anisotropy constant *K*, and attempt frequency 1/*τ*o of the magnetic moment of the nanoparticle*,* is given by [56]

$$
\tau\_N = \frac{1}{2} \sqrt{\tau \tau} \tau\_0 \exp\left(\frac{K V\_m}{k\_B T}\right) \sqrt{\frac{k\_B T}{K V\_m}}.\tag{10}
$$

This expression shows that , and therefore the expected magnetic dissipation, has a strong explicit dependence on temperature. The Brownian relaxation time depends on the hydrody‐ namic volume of the nanoparticle/surfactant complex and is given by [57]

$$
\pi\_{\rm B} = \frac{\Im \eta V\_H}{k\_{\rm B} T}.\tag{11}
$$

Both of these processes act in parallel and the effective relaxation time given by

$$\frac{1}{\tau} = \frac{1}{\tau\_N} + \frac{1}{\tau\_B}.\tag{12}$$

is dominated by the shorter of the two processes (typically the Neel mechanism dominates). The energy absorbed by the nanoparticles in a solid matrix, where only Neel relaxation is relevant, is directly related to the dissipative component of the ferrofluid susceptibility given by [59]

$$\mathbf{P} = \pi \mu\_{\rm o} \mathbf{H}\_{\rm o}^{2} f \boldsymbol{\chi}\_{\rm o} \frac{2 \pi \mathbf{f} \mathbf{\tau}}{1 + \left(2 \pi \mathbf{f} \mathbf{\tau}\right)^{2}} \tag{13}$$

with *H*o and *f* the amplitude and frequency of the applied magnetic field and *χ*o the equilibrium susceptibility of the ferrofluids consisting of magnetic nanoparticles of volume fraction and domain magnetization , *Md*.

[55] developed and implemented a numerical model that included the bio distribution of

As discussed above, magnetic heating in ferrofluids is produced primarily by two dissipation mechanisms, Neel and Brownian relaxation. In Neel relaxation [56], which dominates the magnetic relaxation for nanoparticles fixed in a solid matrix, the nanoparticle magnetic moment aligns with the external field by coherently flipping along the magnetocrystalline easy axes. Conversely, Brownian relaxation [57] requires the nanoparticle to undergo a physical rotation to align the moment with the applied field, which is suppressed for fixed nanoparti‐ cles. Magnetic dissipation in ferrofluids has been investigated in the framework of Neel and Brownian relaxation both theoretically and experimentally by considering the dependence on

The Neel relaxation time, which depends on the nanoparticle magnetic core volume *Vm*, the magnetocrystalline anisotropy constant *K*, and attempt frequency 1/*τ*o of the magnetic moment

> <sup>1</sup> . <sup>2</sup> æ ö <sup>=</sup> ç ÷

è ø

This expression shows that , and therefore the expected magnetic dissipation, has a strong explicit dependence on temperature. The Brownian relaxation time depends on the hydrody‐

<sup>3</sup> <sup>=</sup> . *<sup>H</sup>*

*B ηV <sup>τ</sup>* *m B*

*B m KV k T*

*τ πτ exp k T KV* (10)

*k T* (11)

*N B ττ τ* (12)

*f* (13)

0

namic volume of the nanoparticle/surfactant complex and is given by [57]

*B*

Both of these processes act in parallel and the effective relaxation time given by

11 1 = + .

2

= pm c

2 f P H

is dominated by the shorter of the two processes (typically the Neel mechanism dominates). The energy absorbed by the nanoparticles in a solid matrix, where only Neel relaxation is relevant, is directly related to the dissipative component of the ferrofluid susceptibility given

Oo o 2

1 (2 f ) p t

+ pt

the magnetic nanoparticles within the local boundary conditions.

particle size and frequency, among other parameters [58].

*N*

of the nanoparticle*,* is given by [56]

212 Nanofluid Heat and Mass Transfer in Engineering Problems

by [59]

$$\chi^{\circ\_0} = \mathbf{3} \frac{\mathbb{X}\_{\cdot}}{\mathbb{X}\_{\cdot}} (\coth \mathbb{z} - \frac{1}{\mathbb{z}}), \quad \chi\_{\circ} = \frac{\mu\_{\mathrm{o}} \phi \mathbf{M}\_{\mathrm{d}} \mathbf{V}\_{\mathrm{m}}}{\mathbf{3} \mathbf{k}\_{\mathrm{B}} \mathbf{T}}, \text{ and } \nwarrow = \frac{\mathbf{M}\_{\mathrm{d}} \mathbf{H}\_{\mathrm{d}} \mathbf{V}\_{\mathrm{m}}}{\mathbf{k}\_{\mathrm{B}} \mathbf{T}}. \tag{14}$$

Since this power depends strongly on the mass of the magnetic material, it is represented by specific power, Π in units of W/g, and it is given by

$$
\Pi = \frac{\mathbf{M}\_{\text{sample}}}{\mathbf{m}\_{\text{Fe}\_3\text{O}\_4}} \mathbf{C} \frac{\Delta \mathbf{T}}{\Delta \mathbf{t}} \tag{15}
$$

where *M*sample and 34 are masses of sample and Fe3O4, respectively, *C* is the specific heat capacity of the carrier liquid (water), and Δ*T*/Δ*t* is the time rate of change of temperature due to magnetic heating.

Summarizing all the physical concepts, it becomes clear that the specific heating power depends on the magnetic hydrodynamic properties of the nanoparticles. Magnetic properties are a strong function of particle size distribution, nature of the carrier fluid, nature of surfactant materials, and amplitude and frequency of the applied field. It has been shown experimentally and through numerical simulation that there is a certain combination of particle size and field amplitude and frequency that produces the maximum amount of heat through the nanopar‐ ticles [60, 61]. Hydrodynamic properties are functions of viscosity and hydrodynamic size of the particles.

#### **4.5. Effects of particle‐particle interaction on heat generation in ferrofluids**

In a ferrofluid the coated superparamagnetic particles are dispersed in a carrier liquid and show a constant distance between each other. However, when injected in a tumor, agglomer‐ ation of the particles takes place due to many biological effects, distortion in the coating, magnetic attraction, or van der Waal forces. Depending on the concentration, various effects show up. For example, Castro et al. [62] found that 1% volume fraction is a small fraction agglomerated particles are dimers, but at 10% volume fraction about 10% of the particles are agglomerated at trimers. Chantrell [63] and Dutz and Hergt [64] studied the packing density of a system of identical uniaxial particles by numerical methods and found an exponential decrease of hysteresis losses down to 50% by changing packing density from 0.24 to 76.6%. The result could be interpreted that for ferromagnetic nanoparticles, an increase in packing density means decrease in the distance which in turn leads to decreasing coercivity, remnant magnetization, and hysteresis losses in the system of the nanoparticles. This decrease therefore is due to the increasing diploe‐dipole interaction that disturbs the energy configuration of the system. This hypothesis is supported by experimental studies where increasing coercivity and heating power were found for agglomerated nanoparticles in comparison to single particles [65, 66]. Multicore materials have interesting behavior. The coercivity and remnant magneti‐ zation are significantly lower than that for single core particles of comparable size, but the heating power is higher than that of the single core particles. Multicore particles are an interesting material for hyperthermia but their behavior is not completely understood [67].

## **4.6. Effects of particle‐tissue interaction on heat generation**

The amount of heat generated in magnetic nanoparticles depends on whether the particles are free to move in the fluid injected to a tumor or are fixed to the tumor tissue. Depending on the situation, either Neel relaxation or Brownian relaxation mechanism would determine the amount of heat generated in the nanoparticles. Many authors believe that smaller particles lose energy through Neel relaxation [68] and bigger particles through Brownian relaxation [59]. It is important for Brownian relaxation that the particle should be able to show Brownian rotation inside the tumor. In the absence of the Brownian rotation, only Neel relaxation would take place for superparamagnetic particles and hysteresis losses due to ferromagnetic particles. Immobilization of the magnetic nanoparticles was investigated by magnetoreflexometery [69] and vibrating sample magnetometry [70] and found that even for high magnetic fields up to 25 kA/m, which is an upper limit for the application of hyperthermia, the strong tissue‐particle interaction hampers the rotation of the particles. Only a limited number of investigations has been performed to study particle‐tissue interaction and its effects on heat generation and transfer. It will be useful to study immobilization in a number of tissues with nanoparticles of different coatings. Regmi et al. [71] performed an investigation by coating magnetic nanopar‐ ticles with fatty acids with different bond lengths. Such fatty acids, lauric acid, myrestic acid, and oleic acid, were suspended in water to tune Brownian relaxation. A strong dependence of bond length on SAR value was found. This investigation could be repeated with different tumor tissues to gain knowledge on Brownian relaxation on immobilization by varying bond lengths of the surface functionalization.

## **4.7. Mass and heat transfer modeling in malignant tumors**

Hyperthermia treatment has been known to induce cell death and eventually shrink the tumor [72, 73]. The main goal of hyperthermia treatment is to raise the temperature of the tumor high enough to kill cancerous cell while minimizing damage to the normal surrounding cells. Hyperthermia has been successfully used in combination with other modalities such as chemotherapy and radiation showing a considerable reduction in the size of the tumor [74– 78]. The efficacy of hyperthermia treatment depends on several factors including the moni‐ toring and determining the temperature of the tumor region, the total time of heating, and heating and mass conduction properties of the nanoparticles.

In recent years, considerable work has been performed to understand the physics of hyper‐ thermia through mathematical and computational modeling. Jain introduced a simple mathematical model to predict the temperature distribution during hyperthermia treatment in normal and neoplastic mammalian cells [79]. Volpe and Jain [80] studied at 45 term lumped mathematical model to examine the average temperature distribution and thermal responses of the body under different clinical whole body hyperthermia. In recent years, the efficiency of hyperthermia has been advanced due to the emergence of many nanoparticle systems. Gold‐ based nanoparticle, carbon‐based nanoparticles, and iron oxide nanoparticles have shown promising results in heating the tumors. Many studies have focused on maximizing SAR values in accordance with the tumors temperature [68, 81]. Three important studies are worth reporting here—von Maltzahn et al. [82] used a transient three‐dimensional finite elemental heat transfer model and experimental results to investigate the photo thermal tumor ablation by using gold nanorods. This result is important as it highlights the potential of numerical simulation coupled with the experiment to optimizing tumor therapy planning. Through axisymmetric three‐dimensional cell death and heat transfer model, Huang et al. [83] studied spatial temporal distribution of injured cancerous cell and temperature distribution in the human prostate. The results of the study successfully agreed with the experiments performed with gold nanorods solutions by the near‐infrared irradiation method.

The hyperthermia effects for the cancer treatment are governed by three processes: the nature of the blood profusion in the area of the tumor, the particle transport and interaction with the tissue, and lastly, the monitoring of the heat generation and transfer in the nanoparticles. From these three important effects, Nabil et al. [84] derived a model for couple heat and mass transport in the tumor environment and applied the results to hyperthermia cancer treatment. The study found that ferrofluid‐based hyperthermia treatment depends upon (1) tumor size and vascularity and (2) two mechanisms, perfusion and diffusion, regulated by the distribution of particle and temperature.

## **5. Limitations of hyperthermia**

system. This hypothesis is supported by experimental studies where increasing coercivity and heating power were found for agglomerated nanoparticles in comparison to single particles [65, 66]. Multicore materials have interesting behavior. The coercivity and remnant magneti‐ zation are significantly lower than that for single core particles of comparable size, but the heating power is higher than that of the single core particles. Multicore particles are an interesting material for hyperthermia but their behavior is not completely understood [67].

The amount of heat generated in magnetic nanoparticles depends on whether the particles are free to move in the fluid injected to a tumor or are fixed to the tumor tissue. Depending on the situation, either Neel relaxation or Brownian relaxation mechanism would determine the amount of heat generated in the nanoparticles. Many authors believe that smaller particles lose energy through Neel relaxation [68] and bigger particles through Brownian relaxation [59]. It is important for Brownian relaxation that the particle should be able to show Brownian rotation inside the tumor. In the absence of the Brownian rotation, only Neel relaxation would take place for superparamagnetic particles and hysteresis losses due to ferromagnetic particles. Immobilization of the magnetic nanoparticles was investigated by magnetoreflexometery [69] and vibrating sample magnetometry [70] and found that even for high magnetic fields up to 25 kA/m, which is an upper limit for the application of hyperthermia, the strong tissue‐particle interaction hampers the rotation of the particles. Only a limited number of investigations has been performed to study particle‐tissue interaction and its effects on heat generation and transfer. It will be useful to study immobilization in a number of tissues with nanoparticles of different coatings. Regmi et al. [71] performed an investigation by coating magnetic nanopar‐ ticles with fatty acids with different bond lengths. Such fatty acids, lauric acid, myrestic acid, and oleic acid, were suspended in water to tune Brownian relaxation. A strong dependence of bond length on SAR value was found. This investigation could be repeated with different tumor tissues to gain knowledge on Brownian relaxation on immobilization by varying bond

Hyperthermia treatment has been known to induce cell death and eventually shrink the tumor [72, 73]. The main goal of hyperthermia treatment is to raise the temperature of the tumor high enough to kill cancerous cell while minimizing damage to the normal surrounding cells. Hyperthermia has been successfully used in combination with other modalities such as chemotherapy and radiation showing a considerable reduction in the size of the tumor [74– 78]. The efficacy of hyperthermia treatment depends on several factors including the moni‐ toring and determining the temperature of the tumor region, the total time of heating, and

In recent years, considerable work has been performed to understand the physics of hyper‐ thermia through mathematical and computational modeling. Jain introduced a simple mathematical model to predict the temperature distribution during hyperthermia treatment in normal and neoplastic mammalian cells [79]. Volpe and Jain [80] studied at 45 term lumped

**4.6. Effects of particle‐tissue interaction on heat generation**

214 Nanofluid Heat and Mass Transfer in Engineering Problems

lengths of the surface functionalization.

**4.7. Mass and heat transfer modeling in malignant tumors**

heating and mass conduction properties of the nanoparticles.

In recent years, much progress has been made to bring hyperthermia from lab experiments to the clinic as an important modality to treat cancer; however, there are many challenges yet to be overcome. The first challenge is the alternating magnetic field applied to the body to kill cancer cells produces unwanted heating of the healthy tissues due to eddy currents. The absorbed power due to eddy current is given by

$$P\_{\rm cdd} = \sigma \text{ G(H.f.r)^2} \tag{16}$$

where is the electrical conductivity of the tissue, *G* is the geometric coefficient, and *r* the radius of the coil size. In an experimental study [85], a patient was exposed to a magnetic field where the product of magnetic field and frequency was kept at 4.85 × 108 A/ms with a coil of 30 cm diameter. The patient was able to tolerate the treatment for 1 h without major discomfort.

The second challenge is of uncontrolled movement of limbs happening especially when frequency below 100 kHz is used in hyperthermia treatment. This effect is caused by in‐ duced emf produced in neural cells when alternating magnetic field is applied [86]. Several investigations have been performed in the last decade to study the stimulation threshold in different tissues and found that the eddy current threshold is the limiting threshold for frequencies beyond several hundred kHz, the typical frequency range of ac field for hyper‐ thermia.

Another limitation of hyperthermia is the tumor size. With decreasing tumor size, the surface‐ to‐volume ratio would increase which leads to a stronger dissipation of the generated heat in the healthy tissue and would lower the temperature inside the tumor. Hergt et al. [87] found a relationship with size effect and SHP given by

$$SHP = \begin{array}{c} \frac{\Delta T 3 \lambda}{c R^2} \end{array} \tag{17}$$

where is the desired increase in temperature, is the heat conductivity of the tissue, assumed to be 0.64 W/K m, *c* is the magnetic nanoparticle concentration, and *R* the diameter of the tumor.

Yamada and coworkers [88] heated the cancer cell pellets of different sizes from 1 to 40 mm to 50°C for 10 min to achieve complete cancer cell killing. It was confirmed that large tumors require a smaller heat dose to achieve the target temperature. Another major challenge in hyperthermia treatment is the problem of appropriately administrating the ferrofluids to the tumor regions. It is rather impossible to achieve uniform distribution of the nanoparticles in the tumor region. The best method of administration appears to be multipoint injection into the tumor volume. Salloum et al. [89] reported the results of their investigation on optimum heating patterns induced by multiple nanoparticle injections in tumor models with irregular tumor shapes.

## **6. Thermal conductivity of ferrofluids**

The understanding of the process of heat generation in ferrofluid is important, but it is equally vital to understand mechanisms of heat transfer in ferrofluids, especially inside a tumor region from one point to another. Ferrofluids are industrially prepared magnetic fluids which consist of stable colloidal suspensions of small single‐domain ferromagnetic particles in suitable carrier liquids. Usually, these fluids do not conduct electric current and exhibit a nonlinear paramagnetic behavior. The variety of formulations available for ferrofluids permits a great number of applications, from medical to satellite and vacuum technologies [90]. The idea that thermal conductivity of a suspension can be increased by adding nanoparticles led to many early investigations where metallic or metal oxide nanoparticles such as TiO2, Al2O3, Cu, CuO, and Ag carbon nanotubes were used to show the effect [91]. The magnetic nanofluids have lower thermal conductivity as compared to the metals and oxides listed above; however, MNF shows considerable increase in the conductivity value in presence of an external magnetic field [92, 93]. At present there is no reliable theory that would explain or predict thermal conduc‐ tivity of ferrofluids. It is known that thermal conductivity depends on the thermal conductivity of the constituents of the ferrofluids such as magnetic nanoparticles, carrier fluids, and the surfactants. It also depends on volume fraction, surface area, shapes of the nanoparticles, and the temperature. Although there exists no theory of thermal conductivity of ferrofluids, a number of semiempirical relationships can predict and calculate thermal conductivity of two phase mixtures. These models are summarized in [91]. In brief, the theoretical model by Maxwell [94] gives thermal conductivity for solid‐liquid mixture of relatively large particles (micron size). The effective thermal conductivity is given by

$$k\_{c\circlearrowleft} = \frac{k\_p + \mathcal{D}k\_b + \mathcal{D}\left(k\_p - k\_b\right)\mathcal{D}}{k\_p + \mathcal{D}k\_b - \mathcal{D}\left(k\_p - k\_b\right)\mathcal{D}}k\_b\tag{18}$$

where *kp* is the thermal conductivity of the particle, *kb* is the thermal conductivity of the base fluid, and *ϕ* is the volume fraction of the suspension. The Maxwell model fails to provide a good match with the experimental data when the particle concentration is sufficiently high.

Thermal conductivity relation is modified when the interaction among the randomly distrib‐ uted particle in high concentration is considered. Bruggeman [95] proposed a model for binary mixture of homogenous spherical particles to calculate thermal conductivity by the following equation:

$$\mathcal{L}\mathcal{D}\left(\frac{k\_{\rho}-k\_{\epsilon\mathcal{Y}}}{k\_{\rho}+\mathcal{Z}k\_{\alpha\cdot}}\right)+\left(1-\mathcal{O}\right)\left(\frac{k\_{b-k\_{\sigma}}}{k\_{b}+\mathcal{Z}k\_{\alpha\cdot}}\right)=0\tag{19}$$

Bruggeman's model agrees well with the experimental results.

investigations have been performed in the last decade to study the stimulation threshold in different tissues and found that the eddy current threshold is the limiting threshold for frequencies beyond several hundred kHz, the typical frequency range of ac field for hyper‐

Another limitation of hyperthermia is the tumor size. With decreasing tumor size, the surface‐ to‐volume ratio would increase which leads to a stronger dissipation of the generated heat in the healthy tissue and would lower the temperature inside the tumor. Hergt et al. [87] found

where is the desired increase in temperature, is the heat conductivity of the tissue, assumed to be 0.64 W/K m, *c* is the magnetic nanoparticle concentration, and *R* the diameter of the tumor. Yamada and coworkers [88] heated the cancer cell pellets of different sizes from 1 to 40 mm to 50°C for 10 min to achieve complete cancer cell killing. It was confirmed that large tumors require a smaller heat dose to achieve the target temperature. Another major challenge in hyperthermia treatment is the problem of appropriately administrating the ferrofluids to the tumor regions. It is rather impossible to achieve uniform distribution of the nanoparticles in the tumor region. The best method of administration appears to be multipoint injection into the tumor volume. Salloum et al. [89] reported the results of their investigation on optimum heating patterns induced by multiple nanoparticle injections in tumor models with irregular

The understanding of the process of heat generation in ferrofluid is important, but it is equally vital to understand mechanisms of heat transfer in ferrofluids, especially inside a tumor region from one point to another. Ferrofluids are industrially prepared magnetic fluids which consist of stable colloidal suspensions of small single‐domain ferromagnetic particles in suitable carrier liquids. Usually, these fluids do not conduct electric current and exhibit a nonlinear paramagnetic behavior. The variety of formulations available for ferrofluids permits a great number of applications, from medical to satellite and vacuum technologies [90]. The idea that thermal conductivity of a suspension can be increased by adding nanoparticles led to many early investigations where metallic or metal oxide nanoparticles such as TiO2, Al2O3, Cu, CuO, and Ag carbon nanotubes were used to show the effect [91]. The magnetic nanofluids have lower thermal conductivity as compared to the metals and oxides listed above; however, MNF shows considerable increase in the conductivity value in presence of an external magnetic field [92, 93]. At present there is no reliable theory that would explain or predict thermal conduc‐ tivity of ferrofluids. It is known that thermal conductivity depends on the thermal conductivity of the constituents of the ferrofluids such as magnetic nanoparticles, carrier fluids, and the surfactants. It also depends on volume fraction, surface area, shapes of the nanoparticles, and

<sup>2</sup> , (17)

<sup>=</sup> 3

thermia.

tumor shapes.

a relationship with size effect and SHP given by

216 Nanofluid Heat and Mass Transfer in Engineering Problems

**6. Thermal conductivity of ferrofluids**

Hamilton and Crosser [96] studied a solid‐liquid mixture that contained nonspherical particles. They introduced a parameter, *n*, to account for the shape of the particles to give the thermal conductivity by the following equation:

$$k\_{cfl} = \frac{k\_p + (n-1)k\_b - (n-1)(k\_p - k\_b)\mathcal{Q}}{k\_p + (n-1)k\_b - (k\_p - k\_b)\mathcal{Q}} \, k\_b \tag{20}$$

The empirical shape factor was given by = <sup>3</sup> , where is the particle sphericity defined as

$$\psi = \frac{\text{surface area of a sphere with volume equal to that of the particle}}{\text{surface area of the particle}} \tag{21}$$

It is clear that the Maxwell model is the same as Hamilton‐Crosser model for sphericity, equal to 1.

Another modified Maxwell model was proposed by Yu and Choi [97] to account for the effect of the nanolayer by replacing the thermal conductivity of solid particle *kp* by modified thermal conductivity of the particle *kpe*. This model is based on effective medium theory given by Swartz et al. in 1995 [98]. The effective thermal conductivity of the particle is given by

$$k\_{\mu\epsilon} = \frac{\left[2\left(1-\gamma\right) + \left(1+\beta\right)^2 \left(1-2\gamma\right)\gamma\right]}{-\left(1-\gamma\right) + \left(1+\beta\right)^3 \left(1+2\gamma\right)} k\_{\mu} \tag{22}$$

where = is the ratio of the thermal conductivity of the nanolayer to the thermal conductivity of the particle and *β* = *h/r* is the ratio of the nanolayer thickness to the original particle radius.

In 2004, Yu and Choi [99] modified Hamilton‐Crosser model to include particle‐liquid interfacial layer for nonspherical particles given by

$$k\_{\epsilon\mathcal{Y}} = \left(1 + \frac{n\mathcal{Z}\_{\epsilon\mathcal{Y}}A}{1 - \mathcal{Z}\_{\epsilon\mathcal{Y}}A}\right)k\_b\tag{23}$$

where

$$A = \frac{1}{3} \sum\_{j=a,b,c} \frac{\left(k\_{p\parallel} - k\_b\right)}{k\_{p\parallel} + (n-1)k\_b} \tag{24}$$

and

$$\bigcirc\_{c \mid \mathcal{I}} = \bigcirc \sqrt{(a^2 + t)\big(b^2 + t\big)\big(c^2 + t\big)} / \sqrt{abc} \tag{25}$$

is the equivalent volume concentration of complex ellipsoids *a*( ) and = 3−, here *α* is an empirical parameter and *ψ* is the particle sphericity.

During the past 15 years, many more models have been proposed which are modifications of the existing models. Xue [100] proposed a model for effective thermal conductivity for nanofluid that is based on the average polarization theory that included the effects between interface between solid particles and carrier fluid. Xie et al. [101] proposed the interfacial nanolayer with linear thermal conductivity model to account for the effects of mono layer thickness, nanoparticle size, volume fraction, and thermal conductivities of fluid, nanoparti‐ cles. For metallic particles, Patel et al. [102] found a 9% increase in the thermal conductivity value at a very low concentration of 0.00026%. Philip et al. [22] found an unusually large enhancement of thermal conductivity observed in a nanofluid containing chain‐like aggregate. They showed that efficient transport of heat could take place through percolating paths. They used colloidal suspension of Fe3O4 nanoparticle of average size 6.7 nm diameter and coated with oleic acid suspended in kerosene oil.

The effect of magnetic field on a hybrid nanofluid containing tetramethylammonium hydrox‐ ide‐coated Fe3O4 nanoparticles and gum Arabic‐coated carbon nanotube was studied [103]. The effect of temperature on the time variation of thermal conductivity was also investigated. It was found that the viscosity of hybrid fluid increases with the strength of the field but it decreases with temperature. It was also noticed the nanofluid behaves as shear thinning fluid at low shear rate but exhibits Newtonian behavior at higher shear rates. Interestingly, the thermal conductivity exhibits a peak with the magnetic field.

## **6.1. Measurement of thermal conductivity of ferrofluids**

Another modified Maxwell model was proposed by Yu and Choi [97] to account for the effect of the nanolayer by replacing the thermal conductivity of solid particle *kp* by modified thermal conductivity of the particle *kpe*. This model is based on effective medium theory given by Swartz

> ( ) ( )( ) () ( )

1 (1 ) 1 2 é ù - ++ - ê ú ë û <sup>=</sup> -- ++ + *pe <sup>p</sup> γ β γγ*

21 1 1 2

*k k γβγ*

2

3

conductivity of the particle and *β* = *h/r* is the ratio of the nanolayer thickness to the original

In 2004, Yu and Choi [99] modified Hamilton‐Crosser model to include particle‐liquid

*n A k k*

1 ( ) 3 ( 1)

+ - å *pj b j=a,b,c*

is the equivalent volume concentration of complex ellipsoids *a*( ) and = 3−, here

During the past 15 years, many more models have been proposed which are modifications of the existing models. Xue [100] proposed a model for effective thermal conductivity for nanofluid that is based on the average polarization theory that included the effects between interface between solid particles and carrier fluid. Xie et al. [101] proposed the interfacial nanolayer with linear thermal conductivity model to account for the effects of mono layer


*k k*

*pj b*

( )( )( ) <sup>222</sup> Æ =Æ + + + / *eff a t b t c t abc* (25)

1 1 æ ö Æ = + ç ÷ - Æ è ø *eff eff b eff*

*A =*

*α* is an empirical parameter and *ψ* is the particle sphericity.

is the ratio of the thermal conductivity of the nanolayer to the thermal

*<sup>A</sup>* (23)

*k nk* (24)

(22)

et al. in 1995 [98]. The effective thermal conductivity of the particle is given by

where =

where

and

particle radius.

 

218 Nanofluid Heat and Mass Transfer in Engineering Problems

interfacial layer for nonspherical particles given by

There have been many methods developed over the years to measure thermal transport properties of ferrofluids; however, in recent years the transient hot wire (THW) method has been widely accepted. Although the principle involved in the measurement is very simple, it requires precise temperature sensing, automatic control data acquisition, and analysis technique to obtain accurate result. Because of the short measuring time and large number of parameters involved, it is essential to control measurement through computer. The measure‐ ments and control are achieved and integrated by using micrononverter quick start system using ADuC845 from analog devices. The data are transmitted to a computer using RS232 serial interface and then processed by Instrument Control Box in MATLAB or can be exported to Microsoft Excel and processed.

The principle of the measurement of thermal conductivity by the transient hot wire (THW) method involves detection of temporal temperature rise in a thin platinum wire that is immersed in the test material following the application of step‐wise electrical current [104]. The wire acts as a heat source and produces a time‐dependent temperature field within the test material. The theoretical model describing the THW technique is derived from an analyt‐ ical solution of the heat conduction equation for a linear heat source of radius *r* = 0 and length *l*–infinity of negligible thermal mass. The sink at temperature *T*0 is considered a homogeneous and isotropic material with constant thermal transport properties. When a constant electric current is applied, the wire instantly liberates heat energy per unit length, *q*, to the test sample, where it is conducted outward. The rise in temperature at a radial distance *r* from the heat source is given by [105]

$$
\Delta \mathbf{T}(\mathbf{r}, \mathbf{t}) = \frac{q}{4\pi k} \ln \left(\frac{4at}{r^2 \mathbf{C}}\right) \tag{26}
$$

where *t* is the time, *k* is the thermal conductivity, *a* is the thermal diffusivity, and *C* = exp(*γ*), *γ* = 0.5772157 is the Euler's constant. A plot of *T*(*r*, *t*) with ln(*t*) gives slope *s* = *q*/4*πk* from which *k* can be calculated (**Figure 11**).

**Figure 11.** Experimental set to measure thermal conductivity of a ferrofluid by transient hot wire method [106].

The experimental set up to determine thermal conductivity of a liquid consists of a testing cell that contains the liquid sample, a heating source, usually a platinum wire, and a sensor to measure the temperature at a known distance from the heating source. The testing cell is well isolated from the ambient conditions. A current is applied to the heating element for a very short amount of time, in the order of a few seconds, in a stepwise manner. A voltmeter connected to the sensor measures the temperature in a very short time interval, usually of the order of tens of milliseconds. The computer controls the measurement sequences, acquires data of temperature, and time for a given distance between the heating source and the sensor.

#### **6.2. Mechanisms of thermal conductivity enhancement**

Although many experiments have been performed that show the influence of external field on the enhancement of the thermal conductivity value for MNF, there are no viable explanations for the increased in thermal conductivity values. There are two important mechanisms that have been proposed thus far. These are Brownian motion and particle aggregation mecha‐ nisms.

#### *6.2.1. Brownian motion*

where *t* is the time, *k* is the thermal conductivity, *a* is the thermal diffusivity, and *C* = exp(*γ*), *γ* = 0.5772157 is the Euler's constant. A plot of *T*(*r*, *t*) with ln(*t*) gives slope *s* = *q*/4*πk* from

**Figure 11.** Experimental set to measure thermal conductivity of a ferrofluid by transient hot wire method [106].

**6.2. Mechanisms of thermal conductivity enhancement**

The experimental set up to determine thermal conductivity of a liquid consists of a testing cell that contains the liquid sample, a heating source, usually a platinum wire, and a sensor to measure the temperature at a known distance from the heating source. The testing cell is well isolated from the ambient conditions. A current is applied to the heating element for a very short amount of time, in the order of a few seconds, in a stepwise manner. A voltmeter connected to the sensor measures the temperature in a very short time interval, usually of the order of tens of milliseconds. The computer controls the measurement sequences, acquires data of temperature, and time for a given distance between the heating source and the sensor.

Although many experiments have been performed that show the influence of external field on the enhancement of the thermal conductivity value for MNF, there are no viable explanations for the increased in thermal conductivity values. There are two important mechanisms that

which *k* can be calculated (**Figure 11**).

220 Nanofluid Heat and Mass Transfer in Engineering Problems

In this mechanism, the suspended particles move in the liquid randomly and create collision with the liquid molecules which creates a random walk motion. Thermal conductivity increase in this mechanism can be considered in two ways, one due to the direct contribution through diffusion of nanoparticles that transport heat and the other indirectly due to the microcon‐ vection of fluid surrounding the nanoparticles. It has been suggested that the latter effect can set up a current of heat transfer between nanoparticles and the carrier fluid causing the increase in thermal conductivity. However, the recent studies have rejected the increase in thermal conductivity due to the formation of microconvection. Particle clustering and particle aggre‐ gation have been suggested as the reason for the increase. Phillips et al. [92] showed the microconvection of the fluid medium did not affect the thermal conductivity of a nanofluid. Through the analysis they showed that the model has overestimated the thermal conductivity value. Also, it was found that diffusion of the nanoparticles plays an important role at lower volume fraction which contributed to the presence of dimers and trimers in the suspension.

#### *6.2.2. Nanoparticle clustering*

Clustering and aggregation of nanoparticles, especially into linear chains, have attracted a lot of attention to understand and explain thermal conductivity of nanofluids. The theory behind this thinking is that clustering in nanofluids creates higher conducting paths for the heat flow thereby increasing the heat conduction. It was shown by Bishop et al. [107] that even in the absence of magnetic field, interparticle interaction can have self‐induced self‐assembly, which could influence heat conduction. In a self‐assembled group, the magnetic moment tends to align with the local field due to the neighboring particles or the external field. Such an effect leads to an anisotropy of the interaction which helps magnetic nanoparticles to form one‐ dimensional chains, wires, rings, two dimension aggregates, or three dimension super lattices. Thus, it is very important to understand the mechanism of aggregation, their distribution, morphology, and their interaction in the presence of a magnetic field for the determination of the thermal conductivity. A decrease in the thermal conductivity value with an increase in the mass of the nanoparticle is difficult to explain based on the existing models and theories [108].

## **7. Thermomagnetic convection**

An external magnetic field imposed on a ferrofluid with varying susceptibility produced by the presence of a temperature gradient, results in a nonuniform Kelvin body force, which leads to a form of heat transfer called thermomagnetic convection. This form of heat transfer can be useful when conventional convection heat transfer is inadequate, e.g., in miniature microscale devices or under reduced gravity conditions. A good understanding of the relationship between an imposed magnetic field, the resulting ferrofluid flow, and the temperature distribution is a prerequisite for the proper design and implementation of applications involving thermomagnetic convection. The Kelvin body force is given by

$$
\Delta f m = \mu \left( M \Delta \nabla B \right) \tag{27}
$$

where *M* is the magnetization and *B* is the magnetic induction. The Kelvin body force creates a static pressure field in the flow that is symmetric about the applied magnetic field producing a rotational force field. Such a symmetric field does not change the velocity profile and as a result convection inside the fluid cannot take place.

Ferrofluids can be used to transfer heat, since heat and mass transport in such magnetic fluids can be controlled using an external magnetic field. In 1970, Finlayson [109] first explained how an external magnetic field imposed on a ferrofluid with varying magnetic susceptibility, e.g., due to a temperature gradient, results in a nonuniform magnetic body force, which leads to thermomagnetic convection. This form of heat transfer can be useful for cases where conven‐ tional convection fails to provide adequate heat transfer, e.g., in miniature microscale devices or under reduced gravity conditions.

A comprehensive review [110] of thermomagnetic convection also shows that this form of convection can be correlated with a dimensionless magnetic Rayleigh number. Subsequently, this group explained that fluid motion occurs due to the presence of a Kelvin body force that has two terms. The first term can be treated as a magneto static pressure, while the second is important only if there is a spatial gradient of the fluid susceptibility, e.g., in a nonisothermal system. Colder fluid that has a larger magnetic susceptibility is attracted toward regions with larger field strength during thermomagnetic convection, which displaces warmer fluid of lower susceptibility. They showed that thermomagnetic convection can be correlated with a dimensionless magnetic Rayleigh number. Heat transfer due to this form of convection can be much more effective than buoyancy‐induced convection for systems with small dimensions, etc.

The heat transfer intensity is measured by Rayleigh number which is the sum of the thermo‐ magnetic and thermos gravitational part. The value of the Rayleigh number therefore depends on the magnetic field distribution, properties of the ferrofluid, and pyro magnetic coefficient, which is the degree of dependence of magnetization on temperature. Pyro magnetic coefficient is given by [111]:

Bb = b +b M ( ) T M (28)

where *βT* is the thermal expansion coefficient of the ferrofluid and *βM* denotes the relative pyro magnetic coefficient of dispersed material. For most heat transfer applications, ferrofluids are synthesized to have a greater pyro magnetic coefficient, that is, their curie temperature is closer to the operating temperature. In that respect Zn substituted ferrites are considered favorable for thermomagnetic convention applications. In particular, Mn0.5Zn0.5Fe2O4 are widely used due to their low Neel temperature and higher thermomagnetic coefficient.

## **7.1. Experimental studies on thermomagnetic convections**

Shuchi et al. [112] studied the effect of magnetic field on heat transfer for magnetic nanopar‐ ticles coated with polymers and suspended in an organic fluid and found that the heat transfer capability of the system is improved when the magnetic field is applied at the entrance region. Blums et al. [113] investigated the heat transfer from a nonmagnetic cylinder to a temperature‐ sensitive ferrofluid, MnxZn1‐xFe2O4 suspended in tertadecane. They applied uniform and nonuniform fields, directed perpendicular to the laminar‐free convection and found that thermal gravitation and thermomagnetic forces worked additively on heat transfer intensity in ferrofluid. The increase in the heat transfer was attributed to both the properties of the ferrofluid and the magnetic field gradient. They found that a higher heat transfer could be achieved under an increased magnetic field.

A few recent studies have found that the thermomagnetic convection is strongly influenced by external magnetic field and thermos physical properties of the ferrofluid. In particular, Stefan et al. [114] investigated the magneto viscous effects on thermomagnetic convection. They found an increase in the thermos‐viscous effect due to the morphology of the nanopar‐ ticles. Nanjundappa et al. [115] confirmed Bernard‐Mangaroni thermomagnetic convection in their theoretical investigation

## **7.2. Magnetoviscous effects**

distribution is a prerequisite for the proper design and implementation of applications

where *M* is the magnetization and *B* is the magnetic induction. The Kelvin body force creates a static pressure field in the flow that is symmetric about the applied magnetic field producing a rotational force field. Such a symmetric field does not change the velocity profile and as a

Ferrofluids can be used to transfer heat, since heat and mass transport in such magnetic fluids can be controlled using an external magnetic field. In 1970, Finlayson [109] first explained how an external magnetic field imposed on a ferrofluid with varying magnetic susceptibility, e.g., due to a temperature gradient, results in a nonuniform magnetic body force, which leads to thermomagnetic convection. This form of heat transfer can be useful for cases where conven‐ tional convection fails to provide adequate heat transfer, e.g., in miniature microscale devices

A comprehensive review [110] of thermomagnetic convection also shows that this form of convection can be correlated with a dimensionless magnetic Rayleigh number. Subsequently, this group explained that fluid motion occurs due to the presence of a Kelvin body force that has two terms. The first term can be treated as a magneto static pressure, while the second is important only if there is a spatial gradient of the fluid susceptibility, e.g., in a nonisothermal system. Colder fluid that has a larger magnetic susceptibility is attracted toward regions with larger field strength during thermomagnetic convection, which displaces warmer fluid of lower susceptibility. They showed that thermomagnetic convection can be correlated with a dimensionless magnetic Rayleigh number. Heat transfer due to this form of convection can be much more effective than buoyancy‐induced convection for systems with small dimensions,

The heat transfer intensity is measured by Rayleigh number which is the sum of the thermo‐ magnetic and thermos gravitational part. The value of the Rayleigh number therefore depends on the magnetic field distribution, properties of the ferrofluid, and pyro magnetic coefficient, which is the degree of dependence of magnetization on temperature. Pyro magnetic coefficient

where *βT* is the thermal expansion coefficient of the ferrofluid and *βM* denotes the relative pyro magnetic coefficient of dispersed material. For most heat transfer applications, ferrofluids are synthesized to have a greater pyro magnetic coefficient, that is, their curie temperature is closer to the operating temperature. In that respect Zn substituted ferrites are considered favorable

Bb = b +b M ( ) T M (28)

*fm µ M B* = Ñ ( ) . (27)

involving thermomagnetic convection. The Kelvin body force is given by

result convection inside the fluid cannot take place.

222 Nanofluid Heat and Mass Transfer in Engineering Problems

or under reduced gravity conditions.

etc.

is given by [111]:

When ferrofluids are subjected to a magnetic field, the suspended magnetic nanoparticles tend to align their dipoles in the direction of the magnetic field. This alignment impedes the movement of the particles and eventually increases the effective viscosity of the ferrofluids. The change in the viscosity with the application of the magnetic field is called magnetoviscous effects. This effect was first studied by McTague [116] who investigated the phenomenon by measuring the capillary flow of a highly diluted ferrofluid under the influence of parallel and perpendicular magnetic fields. The results show an increase in the viscosity of the ferrofluids under both orientations of the magnetic field. Interestingly, the increase in the viscosity value in the parallel direction was double that of the value in the perpendicular direction.

The experimental results of McTague were theoretically explained by Shilomis [117] who introduced a model consisting of internal rotation of a single magnetic nanoparticle to derive an expression for the viscosity for the two orientations of the magnetic field. Shilomis' model was based on two assumptions—all nanoparticles are magnetically hard and there is no interaction between the nanoparticles. In addition, the effect of shear rate was neglected. Shilomis's model was investigated experimentally by Ambacher et al. [118] and Odenbach and Gilly [119] using concentrated magnetic ferrofluid under high shear rates. The experimental results diverged from theoretical values. The difference in the values was explained by the presence of the strong dipole‐dipole interaction between magnetic nanoparticles and the formation of particle agglomeration. Since then, the magneto viscous effect has been exten‐ sively investigated by many researchers for many different ferrofluids.

The influence of nanoparticle size on the change in viscous behavior of the ferrofluid was studied by Odenbach and Raj [120] who found the effect is significant for bigger size nano‐ particle and negligible for particles of smaller diameter (about 10 nm). A similar effect was noticed [121–123] by increasing the strength of the magnetic field from 0 to 1 *T*. The authors also measured the Bingham's yield stress for different size particles. It was found that the maximum value of 246 Pa for bigger size particle was 15 times higher than that for the smaller particles. Recently, Patel et al. [124] dispersed a small fraction of micron‐sized nanoparticles in a dilute ferrofluid suspension. It was found that the small size magnetic nanoparticles tend to attach with the large particles. The result showed that the viscosity of the ferrofluid containing small magnetic nanoparticles increased significantly by adding a small fraction of large size nanoparticles. The viscosity was increased by 58% in zero field, while the field influenced viscosity was found to be 92% of the zero field value.

A number of studies were performed to investigate the effect of volume fraction on the magnetoviscous effects. Hezaveh et al. [125] investigated the rheological behavior of Fe3O4 magnetic nanoparticles with different volume fractions 5, 10, 15, 20, 25, and 30% in paraffin base. It was found that the ferrofluid behave like Newtonian in the low concentration regimen and like non‐Newtonian in the higher concentration. In a constant shear rate of 5 s‐1, fluid viscosities increased with increasing magnetic field up to the peak value around 0.3 kA/m and then declined constantly. The trends were clearly observed at high concentration (30 wt%). This phenomenon was explained by Hosseini et al. [126] with the fact that at high concentra‐ tion, the distance between the particles is less, therefore the interaction among the particles become stronger resulting in higher resistance to flow. Once they have exceeded the yield stress, the structures disintegrate, and thus viscosity decreases. The effect of viscosity was found less in low concentration due to fact that the particles are far apart. The influence of shear on the magnetoviscous effect was studied by Khosroshahi and Ghazanfari [127]. It was found that the ferrofluid containing Fe3O4/PVA exhibited non‐Newtonian shear thinning at a volume concentration of 7% under a constant magnetic field. The apparent viscosity decreased when the shear rate increased from 20 to 150 s‐1.

Rodriguez‐Arco et al. [128] studied the effect of coating on the magnetoviscous effect. After coating with citric acid, humic acid, and oleic acid, the mean diameters of the coated Fe3O4 nanoparticles were 7.7±1.3, 11.1±2.4, and 7.4±1.3 nm. As the thickness of the coating layer is increased, the magnetic energy of interaction was found to decrease. This phenomenon could hinder thermal motion, the growth of magnetic aggregates, and also weaken the magnetovis‐ cous effect.

Linke and Odenbach [129] studied the effect of the direction of the magnetic field on the magneto viscous behavior of the ferrofluids. The magneto viscous effect was found to be the highest when the magnetic field is perpendicular to the flow. The lowest magneto effect was observed for the parallel orientation to the flow because of the elongated microstructure in the direction of the flow. Gerth‐Noritzsch et al. [130] studied the dependence of the ratio of viscosity coefficient on shear rate for a ferrofluid in parallel and perpendicular configurations. The ratio between the relative changes of viscosity for parallel and perpendicular orientation on shear rate is given by:

$$R = \frac{\Delta \eta H\_{\parallel} \left( \gamma^{\bullet} \alpha \right)}{\Delta \eta H\_{\perp} \left( \gamma^{\bullet} \alpha \right)} \tag{29}$$

where Δ*η* is the relative change in viscosity, *H* is the strength of the magnetic field, and ˙ is shear rate at wall. The ratio *R* should be equal to 2 in order to avoid the hindrance of a single particle situation and for a complete structure disintegration [131, 132]. Thirupathi and Rajender [133] noticed the similar effect of shear rate on viscosity in Mn–Zn ferrite ferrofluid. The viscosity value was found to be maximum at 0.1 s‐1 shear rates for various applied magnetic fields and shear thinning can be detected at low shear rates. As shear rate increased, the fluid behavior changed from non‐Newtonian to Newtonian. For zero field to 1.30 *T*, viscosity slightly decreased until it approached the value of shear rate of 500 s‐1, whereby after this value, viscosity showed no more dependence on magnetic field strength. The authors emphasized that magnetoviscous effects in ferrofluid exist primarily due to the interactions between the particles in the agglomeration that were aligned in a straight chain and not because of the interaction between the chains [134]. Zubarev [135] also noticed that the effective coefficient of particle diffusion in the direction along the applied field is about one to two orders of magnitude greater compared to the direction perpendicular to the field.

## **8. Other applications**

formation of particle agglomeration. Since then, the magneto viscous effect has been exten‐

The influence of nanoparticle size on the change in viscous behavior of the ferrofluid was studied by Odenbach and Raj [120] who found the effect is significant for bigger size nano‐ particle and negligible for particles of smaller diameter (about 10 nm). A similar effect was noticed [121–123] by increasing the strength of the magnetic field from 0 to 1 *T*. The authors also measured the Bingham's yield stress for different size particles. It was found that the maximum value of 246 Pa for bigger size particle was 15 times higher than that for the smaller particles. Recently, Patel et al. [124] dispersed a small fraction of micron‐sized nanoparticles in a dilute ferrofluid suspension. It was found that the small size magnetic nanoparticles tend to attach with the large particles. The result showed that the viscosity of the ferrofluid containing small magnetic nanoparticles increased significantly by adding a small fraction of large size nanoparticles. The viscosity was increased by 58% in zero field, while the field

A number of studies were performed to investigate the effect of volume fraction on the magnetoviscous effects. Hezaveh et al. [125] investigated the rheological behavior of Fe3O4 magnetic nanoparticles with different volume fractions 5, 10, 15, 20, 25, and 30% in paraffin base. It was found that the ferrofluid behave like Newtonian in the low concentration regimen and like non‐Newtonian in the higher concentration. In a constant shear rate of 5 s‐1, fluid viscosities increased with increasing magnetic field up to the peak value around 0.3 kA/m and then declined constantly. The trends were clearly observed at high concentration (30 wt%). This phenomenon was explained by Hosseini et al. [126] with the fact that at high concentra‐ tion, the distance between the particles is less, therefore the interaction among the particles become stronger resulting in higher resistance to flow. Once they have exceeded the yield stress, the structures disintegrate, and thus viscosity decreases. The effect of viscosity was found less in low concentration due to fact that the particles are far apart. The influence of shear on the magnetoviscous effect was studied by Khosroshahi and Ghazanfari [127]. It was found that the ferrofluid containing Fe3O4/PVA exhibited non‐Newtonian shear thinning at a volume concentration of 7% under a constant magnetic field. The apparent viscosity decreased

Rodriguez‐Arco et al. [128] studied the effect of coating on the magnetoviscous effect. After coating with citric acid, humic acid, and oleic acid, the mean diameters of the coated Fe3O4 nanoparticles were 7.7±1.3, 11.1±2.4, and 7.4±1.3 nm. As the thickness of the coating layer is increased, the magnetic energy of interaction was found to decrease. This phenomenon could hinder thermal motion, the growth of magnetic aggregates, and also weaken the magnetovis‐

Linke and Odenbach [129] studied the effect of the direction of the magnetic field on the magneto viscous behavior of the ferrofluids. The magneto viscous effect was found to be the highest when the magnetic field is perpendicular to the flow. The lowest magneto effect was observed for the parallel orientation to the flow because of the elongated microstructure in the direction of the flow. Gerth‐Noritzsch et al. [130] studied the dependence of the ratio of viscosity coefficient on shear rate for a ferrofluid in parallel and perpendicular configurations. The ratio

sively investigated by many researchers for many different ferrofluids.

224 Nanofluid Heat and Mass Transfer in Engineering Problems

influenced viscosity was found to be 92% of the zero field value.

when the shear rate increased from 20 to 150 s‐1.

cous effect.

Magnetic nanoparticle ferrofluids have potential applications as heat transfer medium in energy conversion devices. They are commercially used such as loud speaker cooling. Some of the applications are reviewed here.

## **8.1. Energy convention devices**

Several theoretical and experimental studies have been performed to analyze heat transfer in parallel duct and loop shape channel‐type energy conversion devices based on thermomag‐ netic effects. Lian et al. [136] reported the performance of automatic energy transport in cooling devices based on the principles of a temperature‐sensitive ferrofluid and thermomagnetic effects. Their system consisted of a loop of permanent magnets, heat source, heat sink, and temperature‐sensitive magnetic fluid. All these parts were assembled in an automatic energy transfer device. By adjusting the magnetic field gradient and temperature gradient inside the ferrofluid, it was possible to control the energy transport in the device. Their result showed that the performance of their device depended on the structure of the loop.

## **8.2. Thermomagnetic‐based cooling**

One of the most important applications of the magnetic nanofluid is thermomagnetic convec‐ tion‐based cooling. As discussed earlier, thermomagnetic convection occurs in a ferrofluid that is placed under thermal gradient (producing varying susceptibility) and exposing to an external magnetic field. This action produces a nonuniform magnetic force called Kelvin body force that leads to thermomagnetic convection. One of the major challenges in such an application is to find a system that would retain its magnetic properties at high temperatures as most nanoparticles exhibit superparamagnetic behavior at room temperature and do not follow Curie Weiss law and therefore lose their magnetic properties at high temperatures. In spite of these challenges, many micro devices are being considered for thermomagnetic cooling. Several studies have been reported in the literature of heat transfer in such devices. Zablotsky et al. [137] studied the possibility of an application of surface cooling based on thermomagnetic convection. They found that the thermomagnetic convection could be enhanced if the heat source was located into the region of the magnetic field intensity. They concluded that it is important (1) to select a MNF with higher pyro magnetic coefficient: lower Curie temperature, higher saturation magnetization, and higher boiling point; and (2) adjust the magnetic field to its maximum value to maintain a balance between temperature gradient and magnetic field.

## **8.3. Thermal conduction‐based smart cooling**

Recently a smart system was developed where thermal conductivity was tuned with externally applied field. Philip et al. [138] developed a device that has the ability to tune thermal conductivity to viscosity ratio to remove heat and control vibration in the system. Such a device can have important applications in mircofludic devices, MEMS, and NEMS, and many other miniature devices. The system is still in a developmental stage to become fully utilized for such applications.

## **9. Future directions**

Thermal conduction in presence of magnetic field has many potential applications in micro‐ and nanodevices. Several studies have studied the connection between the anomalous thermal conductivity observed and the chai formation in the fluids. Magnetic nanoparticle ferrofluids have potential applications as heat transfer medium in energy conversion devices. They are commercially used for loud speaker cooling. Understanding the role of chain formation in the enhancement of thermal conductivity is the main challenge in this area. The future investiga‐ tion should focus on the following areas:


It is important that suitable models be developed and tested with experimental investigation. The use of magnetic nanofluid is very promising for exciting applications such as smart cooling, automatic energy devices, and automatic cooling devices. There are many challenges to understand thermal conduction in ferrofluid for biomedical applications such as hyper‐ thermia where heat energy is used for killing cancer cells. It is, however, not known how heat transfers in tumor regions from the magnetic nanoparticles fluid.

## **Acknowledgements**

**8.2. Thermomagnetic‐based cooling**

226 Nanofluid Heat and Mass Transfer in Engineering Problems

and magnetic field.

applications.

**9. Future directions**

tion should focus on the following areas:

and magnetic properties.

liquid on the thermomagnetic properties.

**8.3. Thermal conduction‐based smart cooling**

One of the most important applications of the magnetic nanofluid is thermomagnetic convec‐ tion‐based cooling. As discussed earlier, thermomagnetic convection occurs in a ferrofluid that is placed under thermal gradient (producing varying susceptibility) and exposing to an external magnetic field. This action produces a nonuniform magnetic force called Kelvin body force that leads to thermomagnetic convection. One of the major challenges in such an application is to find a system that would retain its magnetic properties at high temperatures as most nanoparticles exhibit superparamagnetic behavior at room temperature and do not follow Curie Weiss law and therefore lose their magnetic properties at high temperatures. In spite of these challenges, many micro devices are being considered for thermomagnetic cooling. Several studies have been reported in the literature of heat transfer in such devices. Zablotsky et al. [137] studied the possibility of an application of surface cooling based on thermomagnetic convection. They found that the thermomagnetic convection could be enhanced if the heat source was located into the region of the magnetic field intensity. They concluded that it is important (1) to select a MNF with higher pyro magnetic coefficient: lower Curie temperature, higher saturation magnetization, and higher boiling point; and (2) adjust the magnetic field to its maximum value to maintain a balance between temperature gradient

Recently a smart system was developed where thermal conductivity was tuned with externally applied field. Philip et al. [138] developed a device that has the ability to tune thermal conductivity to viscosity ratio to remove heat and control vibration in the system. Such a device can have important applications in mircofludic devices, MEMS, and NEMS, and many other miniature devices. The system is still in a developmental stage to become fully utilized for such

Thermal conduction in presence of magnetic field has many potential applications in micro‐ and nanodevices. Several studies have studied the connection between the anomalous thermal conductivity observed and the chai formation in the fluids. Magnetic nanoparticle ferrofluids have potential applications as heat transfer medium in energy conversion devices. They are commercially used for loud speaker cooling. Understanding the role of chain formation in the enhancement of thermal conductivity is the main challenge in this area. The future investiga‐

**1.** Understanding the role of particle size, shape, coating, and the properties of the carrier

**2.** Development of thermomagnetic models that would include particle size, morphology,

The authors would like to thank Provost James Zhang at Kettering University for the financial support through the Rodes Professorship Fellowship (PPV) and the Faculty Research Fellow‐ ship (PPV, RJT) awards. We specially thank Professor Julie Dean and Megan Allyn for their helpful discussions.

## **Author details**

Prem P. Vaishnava\* and Ronald J. Tackett

\*Address all correspondence to: pvaishna@kettering.edu

Department of Physics, Kettering University, Flint, MI, USA

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Provisional chapter

## **Energy Transfer in Mixed Convection MHD Flow of Nanofluid Containing Different Shapes of Nanoparticles in a Channel Filled with Saturated Porous Medium** Energy Transfer in Mixed Convection MHD Flow of Nanofluid Containing Different Shapes of Nanoparticles in a Channel Filled with Saturated Porous Medium

Aaiza Gul, Ilyas Khan and Sharidan Shafie Aaiza Gul, Ilyas Khan and Sharidan Shafie

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67367

## Abstract

Energy transfer in mixed convection unsteady magnetohydrodynamic (MHD) flow of an incompressible nanofluid inside a channel filled with a saturated porous medium is investigated. The walls of the channel are kept at constant temperature, and uniform magnetic field is applied perpendicular to the direction of the flow. Three different flow situations are discussed on the basis of physical boundary conditions. The problem is first written in terms of partial differential equations (PDEs), then reduces to ordinary differential equations (ODEs) by using a perturbation technique and solved for solutions of velocity and temperature. Four different shapes of nanoparticles inside ethylene glycol (C2H6O2) and water (H2O)-based nanofluids are used in equal volume fraction. The solutions of velocity and temperature are plotted graphically, and the physical behavior of the problem is discussed for different flow parameters. It is evaluated from this problem that viscosity and thermal conductivity are the dominant parameters responsible for different consequences of motion and temperature of nanofluids. Due to greater viscosity and thermal conductivity, C2H6O2-based nanofluid is regarded as better convectional base fluid assimilated to H2O.

Keywords: mixed convection, nanofluid, heat transfer, cylindrical-shaped nanoparticles, MHD flow, porous medium, analytical solutions

## 1. Introduction

#### 1.1. Nanofluids

Thermal conductivity is the important thermophysical property that plays an essential role in heat transfer enhancement. The internal property of the material is depended on the nature of

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

the fluids. Changing the nature of the fluids or materials can also be affected. In contrast, by applying external force, thermal convection can also be affected on the fluids. Thermal convections are depended on the nature of the fluids as well as on the fluid flow, fluid regime, geometry, etc. Heat transfer conventional fluids, such as ethylene glycol (EG), water, and oils, which are lubricant and kerosene oil have poor thermal conductivities compared to solids. On the other hand, solid particles exhibited rich thermal conductivities than those of conventional heat transfer fluids. The significant improvements in the thermal conductivity of the fluids, which are conventional base by adding up the nanosized particles, were considered by Choi [1] for the first time. The mixture of nanosized particles containing conventional-based fluids is usually known as nanofluids. More accurately, nanofluids are suspensions of nanosized particles in base fluids. The most ordinary nanoparticles used in nanofluids are oxides, carbides, and metals. Nanofluids are mainly used in electronic equipment, power generation, energy supply, production, and air conditioning. Vajjha and Das [2] investigated, for the first time, EG (60%) and water (40%) mixture that is a base fluid for the preparation of alumina (Al2O3), copper oxide ðCuOÞ, and zinc oxide ðZnOÞ nanofluids. At the equal temperature and concentration, CuO nanofluids have higher thermal conductivity than those of Al2O3 and ZnO nanofluids. Naik and Sundar [3] have prepared ðCuOÞ nanofluids with 70% propylene glycol and 30% water. As estimated, they found that CuO nanofluids have rich thermal conductivity and viscosity as compared to the other fluids that are base. Recently, Mansur et al. [4] have studied magnetohydrodynamic (MHD) stagnation-point flow of nanofluids over a permeable sheet for stretching and shrinking cases. They used bvp4c program in MATLAB to obtain the numerical solutions and computed results for different parameters.

The quality of nanofluids in terms of heat transfer performance depends on the volume fraction and types of nanoparticles as well as also depends on the shapes of nanoparticles. Many researchers have used nanoparticles in spherical shapes. However, the nanofluids containing nonspherical shapes of nanoparticles have higher thermal conductivities as compared to spherical ones. Hence, nonspherical shapes of nanoparticles are most suitable to be used. Nanoparticles other than spherical shapes have been used in this research due to their abovementioned properties. More closely, in this study we discuss four different types of nanoparticles, namely, cylinder, platelet, blade, and brick. Nanoparticles of nonspherical shapes with desirable properties are the main focus of present study. Recently, the development in the field of nanotechnology has shown that cylindrical-shaped nanoparticles are seven times more harmful than spherical-shaped nanoparticles in the deliverance of drugs to breast cancer. In the literature survey of nanofluids, we found that analytical studies for the suspension of a variety of shapes of nanoparticles containing EG and water-based fluids are not described yet. Although Timofeeva et al. [5] study the problem of Al2O3 nanofluids which are of different shaped nanoparticles, but they performed experimentally with theoretical modeling together in this study. More closely, they studied various shapes of Al2O3 nanoparticles in a base fluid mixture of EG and water of similar volumes. Improvement in the effective thermal conductivities due to particle shapes was mentioned by Timofeeva et al. Loganathan et al. [6] considered nanoparticles that are sphere shaped and analyzed radiation effects on an unsteady natural convection flow of nanofluids over an infinite vertical plate. They have found that the velocity of spherical silver ðAgÞ nanofluids is less than that of copper ðCuÞ; titanium dioxide ðTiO2Þ, and Al2O3 spherical nanofluids. Recently, Asma et al. [7] found exact solutions for free-convection flow of nanofluids at ramped wall temperature by using five various kinds of spherical-shaped nanoparticles.

the fluids. Changing the nature of the fluids or materials can also be affected. In contrast, by applying external force, thermal convection can also be affected on the fluids. Thermal convections are depended on the nature of the fluids as well as on the fluid flow, fluid regime, geometry, etc. Heat transfer conventional fluids, such as ethylene glycol (EG), water, and oils, which are lubricant and kerosene oil have poor thermal conductivities compared to solids. On the other hand, solid particles exhibited rich thermal conductivities than those of conventional heat transfer fluids. The significant improvements in the thermal conductivity of the fluids, which are conventional base by adding up the nanosized particles, were considered by Choi [1] for the first time. The mixture of nanosized particles containing conventional-based fluids is usually known as nanofluids. More accurately, nanofluids are suspensions of nanosized particles in base fluids. The most ordinary nanoparticles used in nanofluids are oxides, carbides, and metals. Nanofluids are mainly used in electronic equipment, power generation, energy supply, production, and air conditioning. Vajjha and Das [2] investigated, for the first time, EG (60%) and water (40%) mixture that is a base fluid for the preparation of alumina (Al2O3), copper oxide ðCuOÞ, and zinc oxide ðZnOÞ nanofluids. At the equal temperature and concentration, CuO nanofluids have higher thermal conductivity than those of Al2O3 and ZnO nanofluids. Naik and Sundar [3] have prepared ðCuOÞ nanofluids with 70% propylene glycol and 30% water. As estimated, they found that CuO nanofluids have rich thermal conductivity and viscosity as compared to the other fluids that are base. Recently, Mansur et al. [4] have studied magnetohydrodynamic (MHD) stagnation-point flow of nanofluids over a permeable sheet for stretching and shrinking cases. They used bvp4c program in MATLAB to obtain the

The quality of nanofluids in terms of heat transfer performance depends on the volume fraction and types of nanoparticles as well as also depends on the shapes of nanoparticles. Many researchers have used nanoparticles in spherical shapes. However, the nanofluids containing nonspherical shapes of nanoparticles have higher thermal conductivities as compared to spherical ones. Hence, nonspherical shapes of nanoparticles are most suitable to be used. Nanoparticles other than spherical shapes have been used in this research due to their abovementioned properties. More closely, in this study we discuss four different types of nanoparticles, namely, cylinder, platelet, blade, and brick. Nanoparticles of nonspherical shapes with desirable properties are the main focus of present study. Recently, the development in the field of nanotechnology has shown that cylindrical-shaped nanoparticles are seven times more harmful than spherical-shaped nanoparticles in the deliverance of drugs to breast cancer. In the literature survey of nanofluids, we found that analytical studies for the suspension of a variety of shapes of nanoparticles containing EG and water-based fluids are not described yet. Although Timofeeva et al. [5] study the problem of Al2O3 nanofluids which are of different shaped nanoparticles, but they performed experimentally with theoretical modeling together in this study. More closely, they studied various shapes of Al2O3 nanoparticles in a base fluid mixture of EG and water of similar volumes. Improvement in the effective thermal conductivities due to particle shapes was mentioned by Timofeeva et al. Loganathan et al. [6] considered nanoparticles that are sphere shaped and analyzed radiation effects on an unsteady natural convection flow of nanofluids over an infinite vertical plate. They have found that the velocity of spherical silver ðAgÞ nanofluids is less than that of copper ðCuÞ; titanium dioxide ðTiO2Þ, and Al2O3 spherical nanofluids. Recently, Asma et al.

numerical solutions and computed results for different parameters.

240 Nanofluid Heat and Mass Transfer in Engineering Problems

Sebdani et al. [8] studied heat transfer of Al2O3 water nanofluid in mixed convection flow inside a square cavity. Fan et al. [9] described mixed convection heat transfer in a horizontal channel filled with nanofluids. Tiwari and Das [10] and Sheikhzadeh et al. [11] investigated laminar mixed convection flow of a nanofluid in two-sided lid-driven enclosures. Furthermore, a magnetic field in nanofluids has diverse applications such as in the metallurgy and polymer industry where hydromagnetic techniques are used. Nadeem and Saleem [12] presented the unsteady flow of a rotating MHD nanofluid in a rotating cone in the presence of magnetic field. Al-Salem et al. [13] examined MHD mixed convection flow in a linearly heated cavity. The effects of variable viscosity and variable thermal conductivity on the MHD flow and heat transfer over a nonlinear stretching sheet was studied by Prasad et al. [14]. The problem of Darcy Forchheimer mixed convection heat and mass transfer in fluid-saturated porous media in the presence of thermophoresis was discussed by Rami et al. [15]. The effect of radiation and magnetic fields on the mixed convection stagnation-point flow over a vertical stretching sheet in a porous medium bounded by a stretching vertical plate was analyzed by Hayat et al. [16]. Some other studies on mixed convection flow of nanofluids are mentioned in Refs. [17–28]. Nanofluids due to the higher conduction or heat transfer rate of together with other several applications in the field of nanoscience have attracted the attention of researchers to perform future research. Altogether, several researchers are working experimentally, some of them are using numerical simulation. However, very limited research studies are obtainable on analytical side. Experimental research mainly focuses on the enhancement of heat transfer rate of nanofluids through thermal conduction. And this study also includes heat transfer through mixed convection. Several other efforts made on nanofluids are presented in Refs. [29–39].

In order to encounter the importance of MHD in nanofluids, Mansur et al. [4] explored the MHD stagnation point flow of nanofluids over a stretching/shrinking sheet with suction. Colla et al. [21] conducted water-based nanofluids characterization, thermal conductivity and viscosity measurements, and correlation. Abareshi et al. [40] investigated fabrication, characterization, and measurement of thermal conductivity of nanofluids. Borglin et al. [41] examined experimentally the flow of magnetic nanofluids in porous media. MHD effect on nanofluid with energy and hydrothermal behavior between two collateral plates was presented by Sheikholeslami et al. [42]. Sheikholeslami et al. [43] studied forced convection heat transfer in a semiannulus under the influence of a variable magnetic field. Some other research studies on electrically conducting nanofluids are mentioned in Refs. [44–50].

In view of the above literature, the existing research is concerned with the radiative heat transfer in mixed convection MHD flow of different shapes of Al2O3 in EG base nanofluid in a channel filled with saturated porous medium. The foremost focus of this study is the importance of cylindrical-shaped nanofluids on several flow parameters. The uniform constant magnetic field is applied at 90 to the flow and the nanofluids are supposed electrically conducting. No slip condition is taken at the boundary walls of the channel. Three different flow cases on the basis of appropriate boundary conditions are explored. Both of the boundary walls of the channel are at rest in the first case, while motion in the fluids is induced due to buoyancy force and external pressure gradient. In the second case, the right wall of the channel is oscillating in its own plane whereas in the third case, both of the bounding walls of the channel are set into oscillatory motions. The analytical solutions of velocity and temperature are obtained by using a perturbation technique. Graphs are plotted, and the physical behavior of the problem is discussed for different parameters of interest. Nusselt number and skin friction are also computed.

#### 2. Derivation and solutions of governing equations

Consider flow in oscillating form of an incompressible nanofluid inside vertical porous channel. Flow of the nanofluids is supposed electrically conducting under the influence of uniform constant magnetic field. The uniform magnetic field is applied perpendicular to the direction of the flow. Reynolds number is supposed small to ignore the effect of evoked magnetic field. The external electric and electric fields due to polarization of charges are taken zero in order to ignore the influence of electric force in Lorentz force. It is assumed that at time t ¼ 0, the flow is at constant temperature θ0: Right boundary of the channel ðy ¼ dÞ is maintained at constant temperature θwR, while the left boundary ðy ¼ 0Þ has a uniform temperature θwL. Coordinate axis is considered where flow of fluids is moving in the x-axis direction, while the y-axis is taken perpendicular to the velocity of the flow direction.

After derivation of governing equations for quantity of motion and energy under the assumption of Boussinesq approximation are as follow:

$$
\rho\_{\rm nf} \frac{\partial u}{\partial t} = -\frac{\partial p}{\partial x} + \mu\_{\rm nf} \frac{\partial^2 u}{\partial y^2} - \left(\sigma\_{\rm nf} B\_0^2 + \frac{\mu\_{\rm nf}}{\kappa\_1}\right) u + \left(\rho \beta\right)\_{\rm nf} g(\theta - \theta\_0), \tag{1}
$$

$$(\rho c\_p)\_{\rm nf} \frac{\partial \theta}{\partial t} = \kappa\_{\rm nf} \frac{\partial^2 \theta}{\partial y^2} - \frac{\partial q}{\partial y},\tag{2}$$

where u ¼ uðy, tÞ, θ ¼ θðy, tÞ, ρnf, μnf , σnf , κ<sup>1</sup> > 0, ðρβÞnf, g, ðρcpÞnf, κnf q specify the fluid velocity, temperature, density, dynamic viscosity, electrical conductivity, permeability of the porous medium, thermal expansion coefficient, gravitational acceleration, heat capacitance, thermal conductivity of nanofluids, and radiative heat flux in the x-direction. �∂p=∂x ¼ λ exp ðiωtÞ represents the pressure gradient of the flow in oscillatory form where ω is the oscillation parameter and λ is the amplitude of oscillation.

The Hamilton and Crosser [17] model for thermal conductivity and the Timofeeva et al. [5] model for calculating dynamic viscosity of nanofluids are used in order to encounter spherical and other than spherical shapes of nanoparticles inside nanofluids. From these models:

$$
\mu\_{\rm nf} = \quad \mu\_f(1 + a\phi + \phi^2 b), \tag{3}
$$

$$\frac{\kappa \mathbf{n} \mathbf{f}}{\kappa\_f} = \frac{\kappa\_s + (m - 1)\kappa\_f + (n - 1)(\kappa\_s - \kappa\_s)\phi}{\kappa\_s + (m - 1)\kappa\_f - (\kappa\_s - \kappa\_f)\phi},\tag{4}$$

where κ<sup>f</sup> and κ<sup>s</sup> are the thermal conductivities of the base fluid and solid nanoparticles,

respectively. The density ρnf, thermal expansion coefficient ðρβÞnf, heat capacitance ðρcpÞnf and thermal conductivity σnf , of nanofluids are derived by using the relations given by [6, 7]

$$(\rho c\_p)\_{\rm nf} = (1 - \phi)(\rho c\_p)\_f + \phi(\rho c\_p)\_s,\\ \rho\_{\rm nf} = (1 - \phi)\rho\_f + \phi \rho\_s,$$

$$(\rho \beta)\_{\rm nf} = (1 - \phi)(\rho \beta)\_f + \phi(\rho \beta)\_s,\\ \kappa\_{\rm nf} = \alpha\_{\rm nf}(\rho c\_p)\_{\rm nf},$$

$$\sigma\_{\rm nf} = \sigma\_f \left[ 1 + \frac{3(\sigma - 1)\phi}{(\sigma + 2) - (\sigma - 1)\phi} \right], \ \sigma = \frac{\sigma\_s}{\sigma\_f},\tag{5}$$

where φ is the volume fraction of the nanoparticles, ρ<sup>f</sup> and ρ<sup>s</sup> are the density of the base fluid and solid nanoparticles, respectively, μ<sup>f</sup> is the dynamic viscosity of the base fluid, ðcpÞ<sup>f</sup> and ðcpÞ<sup>s</sup> denote the specific heat at constant pressure corresponding to the base fluid and solid nanoparticles and σ<sup>f</sup> and σ<sup>s</sup> are electrical conductivities of base fluids and solid nanoparticles. The total term ðρcpÞ is known as heat capacitance.

Here a and b are shape constants and different for different shape of nanoparticles as presented in Table 1 [5], μ<sup>f</sup> , κ<sup>f</sup> , and κ<sup>s</sup> are the dynamic viscosity, thermal conductivity of the base fluid, and thermal conductivity of solid nanoparticles, respectively. The empirical shape factor n defined in Eq. (4) is equal to n ¼ 3=Ψ, where Ψ is the sphericity. Sphericity given in Hamilton and Crosser model is the ratio of surface area of the sphere to the surface area of real particle with equal volumes. The values of sphericity for different shapes of nanoparticles are given in Table 2 [5].


Table 1. Constants a and b empirical shape factors.

buoyancy force and external pressure gradient. In the second case, the right wall of the channel is oscillating in its own plane whereas in the third case, both of the bounding walls of the channel are set into oscillatory motions. The analytical solutions of velocity and temperature are obtained by using a perturbation technique. Graphs are plotted, and the physical behavior of the problem is discussed for different parameters of interest. Nusselt number and skin friction are also computed.

Consider flow in oscillating form of an incompressible nanofluid inside vertical porous channel. Flow of the nanofluids is supposed electrically conducting under the influence of uniform constant magnetic field. The uniform magnetic field is applied perpendicular to the direction of the flow. Reynolds number is supposed small to ignore the effect of evoked magnetic field. The external electric and electric fields due to polarization of charges are taken zero in order to ignore the influence of electric force in Lorentz force. It is assumed that at time t ¼ 0, the flow is at constant temperature θ0: Right boundary of the channel ðy ¼ dÞ is maintained at constant temperature θwR, while the left boundary ðy ¼ 0Þ has a uniform temperature θwL. Coordinate axis is considered where flow of fluids is moving in the x-axis direction, while the y-axis is

After derivation of governing equations for quantity of motion and energy under the assump-

where u ¼ uðy, tÞ, θ ¼ θðy, tÞ, ρnf, μnf , σnf , κ<sup>1</sup> > 0, ðρβÞnf, g, ðρcpÞnf, κnf q specify the fluid velocity, temperature, density, dynamic viscosity, electrical conductivity, permeability of the porous medium, thermal expansion coefficient, gravitational acceleration, heat capacitance, thermal conductivity of nanofluids, and radiative heat flux in the x-direction. �∂p=∂x ¼ λ exp ðiωtÞ represents the pressure gradient of the flow in oscillatory form where ω

The Hamilton and Crosser [17] model for thermal conductivity and the Timofeeva et al. [5] model for calculating dynamic viscosity of nanofluids are used in order to encounter spherical and other than spherical shapes of nanoparticles inside nanofluids. From these models:

<sup>μ</sup>nf <sup>¼</sup> <sup>μ</sup>fð<sup>1</sup> <sup>þ</sup> <sup>a</sup><sup>φ</sup> <sup>þ</sup> <sup>φ</sup><sup>2</sup>

<sup>¼</sup> <sup>κ</sup><sup>s</sup> þ ð<sup>m</sup> � <sup>1</sup>Þκ<sup>f</sup> þ ð<sup>n</sup> � <sup>1</sup>Þðκ<sup>s</sup> � <sup>κ</sup>sÞ<sup>φ</sup>

where κ<sup>f</sup> and κ<sup>s</sup> are the thermal conductivities of the base fluid and solid nanoparticles,

<sup>0</sup> <sup>þ</sup> <sup>μ</sup>nf κ1

> ∂<sup>2</sup>θ <sup>∂</sup>y<sup>2</sup> � <sup>∂</sup><sup>q</sup> ∂y

u þ ðρβÞ

nf gðθ � θ0Þ, (1)

, (2)

bÞ, (3)

<sup>κ</sup><sup>s</sup> þ ð<sup>m</sup> � <sup>1</sup>Þκ<sup>f</sup> � ðκ<sup>s</sup> � <sup>κ</sup>fÞ<sup>φ</sup> , (4)

∂<sup>2</sup>u

ðρcpÞnf

is the oscillation parameter and λ is the amplitude of oscillation.

κnf κf

<sup>∂</sup>y<sup>2</sup> � <sup>σ</sup>nfB<sup>2</sup>

∂θ <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>κ</sup>nf

2. Derivation and solutions of governing equations

242 Nanofluid Heat and Mass Transfer in Engineering Problems

taken perpendicular to the velocity of the flow direction.

tion of Boussinesq approximation are as follow:

ρnf ∂u <sup>∂</sup><sup>t</sup> ¼ � <sup>∂</sup><sup>p</sup> ∂x þ μnf


Table 2. Sphericity Ψ for different shapes nanoparticles.

In Eqs. (1) and (2), the ðρcpÞnf, ρnf, ðρβÞnf, κnf, and σnf of nanofluids are used from the relations given by Asma et al. [7], as

The temperature θ<sup>0</sup> and θ<sup>w</sup> of both walls of the channel is assumed high and both walls are emitting radiations [20]. Therefore, radiative heat flux as a function of temperature is given by

$$-\frac{\partial q}{\partial y} = 4\alpha\_0^2(\theta - \theta\_0),\tag{6}$$

where α<sup>0</sup> is the radiation absorption coefficient.

Using Eq. (6) into Eq. (2) gives

$$(\rho c\_p)\_{\rm nf} \frac{\partial \theta}{\partial t} = \kappa\_{\rm nf} \frac{\partial^2 \theta}{\partial y^2} - 4\alpha\_0^2(\theta - \theta\_0),\tag{7}$$

In order to convert dimensional partial differential equations (PDEs) into dimensionless PDEs, the following nondimensional variable are introduced

$$\begin{aligned} x^\* &= \frac{x}{d}, \ y^\* = \frac{y}{d}, \ u^\* = \frac{u}{\mathcal{U}\_0}, \ t^\* = \frac{t\mathcal{U}\_0}{d}, \ \theta^\* = \frac{\theta - \theta\_0}{\theta\_w - \theta\_0}, \\ p^\* &= \frac{d}{\mu\mathcal{U}\_0}p, \ \ \omega^\* = \frac{d\omega\_1}{\mathcal{U}\_0}, \end{aligned} \tag{8}$$

After dimensionalization (dropping for convenience) Eqs. (1) and (7) give

$$\left[ (1 - \phi) + \phi \frac{\rho\_s}{\rho\_f} \right] \text{Re}\frac{\partial u}{\partial t} = \lambda \exp\left( i\omega t \right) + \left( 1 + a\phi + b\phi^2 \right) \frac{\partial^2 u}{\partial y^2} - \tag{9}$$

$$\left[ 1 + \frac{3(\sigma - 1)\phi}{(\sigma + 2) - (\sigma - 1)\phi} \right] M^2 u - \frac{1}{K} (1 + a\phi + b\phi^2) + \left[ (1 - \phi) + \frac{\phi(\rho\theta)\_s}{(\rho\theta)\_f} \right] G\sigma\theta,$$

$$\frac{P e}{\lambda\_n} \left[ (1 - \phi) + \phi \frac{(\rho c\_p)\_s}{(\rho c\_p)\_f} \right] \frac{\partial \theta}{\partial t} = \frac{\partial^2 \theta}{\partial y^2} + \frac{N^2}{\lambda\_n} \theta,\tag{10}$$

where

$$\begin{split} Re &= \frac{\mathcal{U}\_{0}d}{\upsilon\_{f}}, \mathcal{M}^{2} = \frac{\sigma\_{f}B\_{0}^{2}d^{2}}{\mu\_{f}}, \mathcal{K} = \frac{\kappa\_{1}}{d^{2}}Gr = \frac{g\beta\_{f}d^{2}(\theta\_{w} - \theta\_{0})}{\upsilon\_{f}\mathcal{U}\_{0}}, \ P e = \frac{(\rho c\_{p})\_{f}d\mathcal{U}\_{0}}{\kappa\_{f}}, \\ \lambda\_{n} &= \frac{\kappa\_{n\ell}}{\kappa\_{f}} = \frac{\kappa\_{s} + (m-1)\kappa\_{f} + (n-1)(\kappa\_{s} - \kappa\_{s})\phi}{\kappa\_{s} + (m-1)\kappa\_{f} - (\kappa\_{s} - \kappa\_{f})\phi}, \ N^{2} = \frac{4d^{2}\alpha\_{0}^{2}}{\kappa\_{f}}. \end{split}$$

Here Re, M, Gr, Pe, and N denote the Reynolds number, magnetic parameter, the thermal Grashof number, the Peclet number, and the radiation parameter, respectively. Three different flow cases are considered in order to solve Eqs. (9) and (10). These are as follow:

#### 2.1. Case I: Stationary walls of the channel

In this case, the gap between the two plates of the channel is denoted by d and both plates are assumed stationary at y ¼ 0 and y ¼ d. The flow of nanofluids is unidirectional and moving with velocity in the x-axis. Both plates of the channel are maintained at constant and uniform temperature θ<sup>w</sup> and θ0. Thus, the appropriate boundary conditions are

$$
\mu(0, t) = 0, \quad \mu(d, t) = 0,\tag{11}
$$

$$
\theta(0, t) = \theta\_0, \ \theta(d, t) = \theta\_w. \tag{12}
$$

After reducing Eqs. (11) and (12) into dimensionless form, we get

Energy Transfer in Mixed Convection MHD Flow of Nanofluid Containing Different Shapes of Nanoparticles… http://dx.doi.org/10.5772/67367 245

$$
u(0, \ t) = 0; \quad \boldsymbol{u}(1, \ t) = 0, \ t > 0,\tag{13}$$

$$
\theta(0, \ t) = 0; \ \theta(1, \ t) = 1; \quad t > 0. \tag{14}
$$

Further simplification of Eqs. (9) and (10) after conversion of dimensional PDEs, we get

$$d\_0 \frac{\partial u}{\partial t} = \lambda \exp\left(i\omega t\right) + \phi\_2 \frac{\partial^2 u}{\partial y^2} - h\_0^2 u + d\_1 \Theta,\tag{15}$$

$$
\varepsilon\_0 \frac{\partial \theta}{\partial t} = \frac{\partial^2 \theta}{\partial y^2} + \varepsilon\_1 \theta,\tag{16}
$$

where

Using Eq. (6) into Eq. (2) gives

244 Nanofluid Heat and Mass Transfer in Engineering Problems

1 þ

Re <sup>¼</sup> <sup>U</sup>0<sup>d</sup> vf

> <sup>λ</sup><sup>n</sup> <sup>¼</sup> <sup>κ</sup>nf κf

2.1. Case I: Stationary walls of the channel

where

ðρcpÞnf

, <sup>y</sup><sup>∗</sup> <sup>¼</sup> <sup>y</sup> d

ρf

the following nondimensional variable are introduced

<sup>x</sup><sup>∗</sup> <sup>¼</sup> <sup>x</sup> d

<sup>p</sup><sup>∗</sup> <sup>¼</sup> <sup>d</sup> μU<sup>0</sup>

<sup>ð</sup><sup>1</sup> � <sup>φ</sup>Þ þ <sup>φ</sup> <sup>ρ</sup><sup>s</sup>

Pe λn

, <sup>M</sup><sup>2</sup> <sup>¼</sup> <sup>σ</sup><sup>f</sup> <sup>B</sup><sup>2</sup>

0d2 μf

3ðσ � 1Þφ ðσ þ 2Þ�ðσ � 1Þφ � �

" #

∂θ <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>κ</sup>nf

, <sup>u</sup><sup>∗</sup> <sup>¼</sup> <sup>u</sup> U<sup>0</sup> , t

> U<sup>0</sup> ,

<sup>p</sup>, <sup>ω</sup><sup>∗</sup> <sup>¼</sup> <sup>d</sup>ω<sup>1</sup>

After dimensionalization (dropping for convenience) Eqs. (1) and (7) give

Re <sup>∂</sup><sup>u</sup>

<sup>ð</sup><sup>1</sup> � <sup>φ</sup>Þ þ <sup>φ</sup> <sup>ð</sup>ρcpÞ<sup>s</sup>

, <sup>K</sup> <sup>¼</sup> <sup>κ</sup><sup>1</sup>

flow cases are considered in order to solve Eqs. (9) and (10). These are as follow:

temperature θ<sup>w</sup> and θ0. Thus, the appropriate boundary conditions are

After reducing Eqs. (11) and (12) into dimensionless form, we get

" #

M<sup>2</sup> <sup>u</sup> � <sup>1</sup> ∂<sup>2</sup>θ <sup>∂</sup>y<sup>2</sup> � <sup>4</sup>α<sup>2</sup>

In order to convert dimensional partial differential equations (PDEs) into dimensionless PDEs,

<sup>∗</sup> <sup>¼</sup> tU<sup>0</sup>

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>λ</sup> exp <sup>ð</sup>iωtÞþð<sup>1</sup> <sup>þ</sup> <sup>a</sup><sup>φ</sup> <sup>þ</sup> <sup>b</sup>φ<sup>2</sup>

<sup>K</sup> <sup>ð</sup><sup>1</sup> <sup>þ</sup> <sup>a</sup><sup>φ</sup> <sup>þ</sup> <sup>b</sup>φ<sup>2</sup>

∂θ <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup><sup>θ</sup> ∂y<sup>2</sup> þ

<sup>κ</sup><sup>s</sup> þ ð<sup>m</sup> � <sup>1</sup>Þκ<sup>f</sup> � ðκ<sup>s</sup> � <sup>κ</sup>fÞ<sup>φ</sup> , <sup>N</sup><sup>2</sup> <sup>¼</sup> <sup>4</sup>d<sup>2</sup>

ðρcpÞ<sup>f</sup>

<sup>d</sup><sup>2</sup> Gr <sup>¼</sup> <sup>g</sup>β<sup>f</sup> <sup>d</sup><sup>2</sup>

Here Re, M, Gr, Pe, and N denote the Reynolds number, magnetic parameter, the thermal Grashof number, the Peclet number, and the radiation parameter, respectively. Three different

In this case, the gap between the two plates of the channel is denoted by d and both plates are assumed stationary at y ¼ 0 and y ¼ d. The flow of nanofluids is unidirectional and moving with velocity in the x-axis. Both plates of the channel are maintained at constant and uniform

<sup>¼</sup> <sup>κ</sup><sup>s</sup> þ ð<sup>m</sup> � <sup>1</sup>Þκ<sup>f</sup> þ ð<sup>n</sup> � <sup>1</sup>Þðκ<sup>s</sup> � <sup>κ</sup>sÞ<sup>φ</sup>

<sup>d</sup> , <sup>θ</sup><sup>∗</sup> <sup>¼</sup> <sup>θ</sup> � <sup>θ</sup><sup>0</sup>

θ<sup>w</sup> � θ<sup>0</sup>

Þ ∂<sup>2</sup>u ∂y<sup>2</sup> �

" #

ðρβÞ<sup>f</sup>

, Pe <sup>¼</sup> <sup>ð</sup>ρcpÞ<sup>f</sup> dU<sup>0</sup> κf

> α2 0 κf :

Grθ,

,

θ, (10)

Þþ ð<sup>1</sup> � <sup>φ</sup>Þ þ <sup>φ</sup>ðρβÞ<sup>s</sup>

N2 λn

uð0, tÞ ¼ 0, uðd, tÞ ¼ 0, (11)

θð0, tÞ ¼ θ0, θðd, tÞ ¼ θw: (12)

ðθ<sup>w</sup> � θ0Þ vf U<sup>0</sup>

<sup>0</sup>ðθ � θ0Þ, (7)

,

(8)

(9)

$$d\_0 = \phi\_1 Re,\\ \phi\_1 = (1 - \phi) + \phi \frac{\rho\_s}{\rho\_f},\\ \phi\_2 = (1 + a\phi + b\phi^2),\\ h\_0^2 = \phi\_5 M^2 + 1/K,$$

$$\phi\_5 = \left[1 + \frac{3(\sigma - 1)\phi}{(\sigma + 2) - (\sigma - 1)\phi}\right],\\ d\_1 = \phi\_3 Gr,\\ \phi\_3 = (1 - \phi) + \phi \frac{(\rho\beta)\_s}{(\rho\beta)\_f},\\ e\_0^2 = \frac{Pe\phi\_4}{\lambda\_n},$$

$$\phi\_4 = \left[(1 - \phi) + \phi \frac{(\rho c\_p)\_s}{(\rho c\_p)\_f}\right],\\ e\_1^{-2} = \frac{N^2}{\lambda\_n}.$$

Perturb-type solutions are supposed to convert PDEs in Eqs. (15) and (16) under appropriate boundary conditions, and in Eqs. (11) and (12) into ordinary differential equations (ODEs) are

$$u(y,t) = [u\_0(y) + \varepsilon \exp\left(i\omega t\right)u\_1(y)],\tag{17}$$

$$\theta(y, t) = [\theta\_0(y) + \varepsilon \exp(i\omega t) \,\theta\_1(y)]. \tag{18}$$

Using Eqs. (17) and (18) into Eqs. (15) and (16), we obtain the following system of ordinary differential equations

$$\frac{d^2 u\_0(y)}{dy^2} - h\_1^2 u\_0(y) = d\_2 \theta\_0(y),\tag{19}$$

$$\frac{d^2 u\_1(y)}{dy^2} - h\_2^2 u\_1(y) = -\frac{\lambda}{\phi\_2},\tag{20}$$

$$\frac{d^2\theta\_0(y)}{dy^2} - \varepsilon\_1^2 \theta\_0(y) = 0,\tag{21}$$

$$\frac{d^2\theta\_1(y)}{dy^2} + h\_3^2 \theta\_1(y) = 0,\tag{22}$$

where

$$h\_1 = \sqrt{\frac{h\_0^2}{\phi\_2}}, \ d\_2 = \frac{d\_1}{\phi\_2}, \ h\_2 = \sqrt{\frac{h\_0^2 + i\omega d\_0}{\phi\_2}}, \ h\_3 = \sqrt{e\_1 - i\omega e\_0}.$$

The associated boundary conditions (Eqs. (13) and (14)) are reduce to

$$
u\_0(0) = 0; \ u\_0(1) = 0,\tag{23}$$

$$
\mu\_1(0) = 0; \ \mu\_1(1) = 0,\tag{24}
$$

$$
\theta\_0(0) = 0; \ \theta\_0(1) = 1,\tag{25}
$$

$$
\theta\_1(0) = 0; \ \theta\_1(1) = 0. \tag{26}
$$

Perturb solution for temperature in Eqs. (21) and (22) under appropriate boundary conditions (Eqs. (25) and (26)) give

$$\theta\_0(y) = \frac{\sin \left(e\_1 y\right)}{\sin \left(e\_1\right)},\tag{27}$$

$$
\theta\_1(y) = 0.\tag{28}
$$

Temperature of nanofluids, Eq. (18) using Eqs. (27) and (28) gives

$$
\theta(y, t) = \theta(y) = \frac{\sin\left(e\_1 y\right)}{\sin\left(e\_1\right)}.\tag{29}
$$

Perturb solution for temperature in Eqs. (19) and (20), using Eq. (27) under appropriate boundary conditions, Eqs. (23) and (24) yield

$$
\mu\_0(y) = c\_1 \sinh(h\_1 y) + c\_2 \cosh(h\_1 y) + \frac{d\_2}{(e\_1^2 + h\_1^2)} \frac{\sin(e\_1 y)}{\sin(e\_1)}.\tag{30}
$$

$$
\mu\_1(y) = c\_3 \sinh(h\_2 y) + c\_4 \cosh(h\_2 y) + \frac{\lambda}{h\_2^2 \phi\_2}.\tag{31}
$$

Here c1, c<sup>2</sup> , c<sup>3</sup> and c<sup>4</sup> are arbitrary constants given by

$$\mathcal{L}\_1 = -\frac{d\_2}{\sinh(h\_1)(e\_1^2 + h\_1^2)},\\\mathcal{c}\_2 = 0,\\\mathcal{c}\_3 = \frac{\lambda}{h\_2^2 \phi\_2} \frac{1}{\sinh(h\_2)} (\cosh(h\_2) - 1),\\\mathcal{c}\_4 = -\frac{\lambda}{h\_2^2 \phi\_2}.\tag{32}$$

Final velocity for nanofluids, substituting Eqs. (30)–(32) into Eq. (17), we obtain

$$u(y, t) = -\frac{d\_2 \sinh(h\_1 y)}{(e\_1^2 + h\_1^2) \sinh(h\_1)} + \frac{d\_2 \sin\left(e\_1 y\right)}{(e\_1^2 + h\_1^2) \sin\left(e\_1\right)}$$

$$= +\varepsilon \exp\left(i\omega t\right) \begin{bmatrix} \frac{\lambda\left(\cosh(h\_2) - 1\right) \sinh(h\_2 y)}{h\_2^2 \phi\_2 \sinh(h\_2)}\\ + \frac{h\_2^2 \phi\_2 \sinh(h\_2)}{h\_2^2 \phi\_2} \\ + \frac{\lambda}{h\_2^2 \phi\_2} \left(1 - \cosh(h\_2 y)\right) \end{bmatrix}.\tag{33}$$

#### 2.2. Case 2: right plate of the channel is oscillating in its own plane

In this case, the right wall of the channel (y ¼ d) is locate into oscillatory motion, while on other hand, the left wall ðy ¼ 0Þ is taken as stationary. The first boundary condition is the same, while the second boundary condition in dimensionless form modifies to

$$u(1,t) = H(t)\varepsilon \exp\left(i\omega t\right);\ t>0,\tag{34}$$

where HðtÞ is the Heaviside step function.

u0ð0Þ ¼ 0; u0ð1Þ ¼ 0, (23)

u1ð0Þ ¼ 0; u1ð1Þ ¼ 0, (24)

θ0ð0Þ ¼ 0; θ0ð1Þ ¼ 1, (25)

θ1ð0Þ ¼ 0; θ1ð1Þ ¼ 0: (26)

sin <sup>ð</sup>e1<sup>Þ</sup> , (27)

sin <sup>ð</sup>e1<sup>Þ</sup> : (29)

sin ðe1yÞ

<sup>ð</sup>coshðh2Þ � <sup>1</sup>Þ, <sup>c</sup><sup>4</sup> ¼ � <sup>λ</sup>

sin <sup>ð</sup>e1<sup>Þ</sup> : (30)

: (31)

h2 <sup>2</sup>φ<sup>2</sup> : (32)

(33)

θ1ðyÞ ¼ 0: (28)

Perturb solution for temperature in Eqs. (21) and (22) under appropriate boundary conditions

<sup>θ</sup>0ðyÞ ¼ sin <sup>ð</sup>e1y<sup>Þ</sup>

<sup>θ</sup>ðy, <sup>t</sup>Þ ¼ <sup>θ</sup>ðyÞ ¼ sin <sup>ð</sup>e1y<sup>Þ</sup>

Perturb solution for temperature in Eqs. (19) and (20), using Eq. (27) under appropriate

<sup>u</sup>1ðyÞ ¼ <sup>c</sup><sup>3</sup> sinhðh2yÞ þ <sup>c</sup><sup>4</sup> coshðh2yÞ þ <sup>λ</sup>

h2 <sup>2</sup>φ<sup>2</sup>

<sup>1</sup>Þsinhðh1Þ

h2

λðcoshðh2Þ � 1Þsinhðh2yÞ

<sup>2</sup>φ2sinhðh2Þ

1 � coshðh2yÞ

ðe2 <sup>1</sup> <sup>þ</sup> <sup>h</sup><sup>2</sup> 1Þ

1 sinhðh2Þ

þ

ðe2 <sup>1</sup> <sup>þ</sup> <sup>h</sup><sup>2</sup> h2 <sup>2</sup>φ<sup>2</sup>

d<sup>2</sup> sin ðe1yÞ

�

<sup>1</sup>Þ sin ðe1Þ

<sup>u</sup>0ðyÞ ¼ <sup>c</sup><sup>1</sup> sinhðh1yÞ þ <sup>c</sup><sup>2</sup> coshðh1yÞ þ <sup>d</sup><sup>2</sup>

, <sup>c</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>, <sup>c</sup><sup>3</sup> <sup>¼</sup> <sup>λ</sup>

Final velocity for nanofluids, substituting Eqs. (30)–(32) into Eq. (17), we obtain

<sup>u</sup>ðy, <sup>t</sup>޼� <sup>d</sup>2sinhðh1y<sup>Þ</sup> ðe2 <sup>1</sup> <sup>þ</sup> <sup>h</sup><sup>2</sup>

þ λ h2 <sup>2</sup>φ<sup>2</sup> �

Temperature of nanofluids, Eq. (18) using Eqs. (27) and (28) gives

boundary conditions, Eqs. (23) and (24) yield

Here c1, c<sup>2</sup> , c<sup>3</sup> and c<sup>4</sup> are arbitrary constants given by

<sup>1</sup> <sup>þ</sup> <sup>h</sup><sup>2</sup> 1Þ

þε exp ðiωtÞ

<sup>c</sup><sup>1</sup> ¼ � <sup>d</sup><sup>2</sup>

sinhðh1Þðe<sup>2</sup>

(Eqs. (25) and (26)) give

246 Nanofluid Heat and Mass Transfer in Engineering Problems

By using the same procedure as in Case-1, and the velocity solution is obtained as

$$u(y, t) = -\frac{d\_2 \sinh(h\_1 y)}{\sinh(h\_1)(e\_1^2 + h\_1^2)} + \frac{d\_2}{(e\_1^2 + h\_1^2)} \frac{\sin(e\_1 y)}{\sin(e\_1)}$$

$$+ \varepsilon \exp\left(i\omega t\right) \begin{bmatrix} \frac{\sinh(h\_2 y)}{\sinh(h\_2)} \left(H(t) + \frac{\lambda}{(h\_2^2 \phi\_2)} \left(\cosh(h\_2) - 1\right)\right) \\\\ - \frac{\lambda}{(h\_2^2 \phi\_2)} \cosh(h\_2 y) + \frac{\lambda}{(h\_2^2 \phi\_2)} \end{bmatrix} . \tag{35}$$

#### 2.3. Case 3: both plates of the channel are oscillating in its own plane

In this case, both plates of the channel are chosen into oscillatory motions. The dimensionless form of the boundary conditions is

$$
\mu(0, t) = \mu(1, t) = H(t)\varepsilon \exp\left(i\omega t\right); \ t > 0. \tag{36}
$$

The resulting expression for velocity is obtained as

$$u(y, t) = -\frac{d\_2 \sinh(h\_1 y)}{\sinh(h\_1)(e\_1^2 + h\_1^2)} + \frac{d\_2}{(e\_1^2 + h\_1^2)} \frac{\sin(\epsilon\_1 y)}{\sin(\epsilon\_1)}$$

$$+ \varepsilon \exp\left(i\omega t\right) \begin{bmatrix} \frac{\sinh(h\_2 y)}{\sinh(h\_2)} \left(\frac{H(t)\left(1 - \cosh(h\_2 y)\right)}{\lambda}\right) \\\\ \frac{\lambda}{(h\_2^2 \phi\_2)} \left(\cosh(h\_2) - 1\right) \\\\ + \left\{H(t) - \frac{\lambda}{(h\_2^2 \phi\_2)}\right\} \cosh(h\_2 y) + \frac{\lambda}{(h\_2^2 \phi\_2)} \end{bmatrix} . \tag{37}$$

#### 2.4. Evaluation of Nusselt number and skin-friction

The dimensionless derivations for Nusselt number and skin-frictions are evaluated from Eqs. (29), (33), (35) and (37) as follows:

$$Nu = \frac{e\_1}{\sin\left(e\_1\right)},\tag{38}$$

$$\tau\_1 = \tau\_1(\ t) = -\frac{d\_2 h\_1}{(e\_1^2 + h\_1^2)\sinh(h\_1)} + \frac{d\_2 b\_1}{(e\_1^2 + h\_1^2)\sin(e\_1)}\tag{39}$$

$$\tau\_2 = \tau\_1 + \tau\_2 + \dots + \tau\_n\tag{30}$$

$$+\varepsilon \exp\left(i\omega t\right) \left[\frac{\lambda\left(\cosh(h\_2) - 1\right)}{h\_2\phi\_2\sinh(h\_2)}\right].$$

$$\tau\_2 = \tau\_2(t) = -\frac{d\_2 h\_1}{(e\_1^2 + h\_1^2)\sinh(h\_1)} + \frac{d\_2 e\_1}{(e\_1^2 + h\_1^2)\sin(e\_1)}\tag{40}$$

$$\tau\_1 + \varepsilon \exp\left(i\omega t\right) \left[\frac{h\_2}{\sinh(h\_2)} \left\{H(t) + \frac{\lambda}{(h\_2\phi\_2)} \left(\cosh(h\_2) - 1\right)\right\} \right],$$

$$\tau\_3 = \tau\_3(t) = -\frac{d\_2h\_1}{(e\_1^2 + h\_1^2)\sinh(h\_1)} + \frac{d\_2b\_1}{(e\_1^2 + h\_1^2)\sin(e\_1)}$$

$$+\varepsilon \exp\left(i\omega t\right) \left[\frac{h\_2}{\sinh(h\_2)} \begin{Bmatrix} H(t) \left(1 - \cosh(h\_2)\right) \\\\ +\frac{\lambda}{(h\_2^2 \phi\_2)} \left(\cosh(h\_2) - 1\right) \end{Bmatrix}\right].\tag{41}$$

## 3. Graphical consequences and depiction

In this segment, graphical consequences are figured and debated. Similarly, influence of the radiation effect on heat transfer in mixed convection MHD flow of nanofluids inside a channel filled with a saturated porous medium is explored. Three different flow cases on the basis of appropriate boundary conditions are examined. Four dissimilar shapes of Al2O3 as solid nanoparticles, which are cylinder, platelet, brick, and blade, are dangling into conventional base fluids, ethylene glycol and water. The physical performance of the exhibited graphs is conferred for various embedded parameters. The numerical values of constants a and b (called shape factors) are chosen from Table 1, and sphericity Ψ is given in Table 2. It should be acclaimed that a and b coefficients vary highly with particle shape. The numerical values for the various shapes of nanoparticles (platelet, blade, cylinder and brick) at equal volumes are presented in Tables 1–3.

The geometry of the problem is shown in Figure 1. The impact of dissimilar shapes of alumina Al2O3 nanoparticles on the motion of ethylene glycol-based nanofluids is represented in Figure 2. It is viewed that the blade shape of alumina Al2O3 nanoparticles inside ethylene glycol-based nanofluids has the uppermost velocity chased by brick, platelet, and cylindricalshaped nanoparticles in ethylene glycol-based nanofluids. The impact of the shapes on the motion of nanofluids is because of the strong subjection of viscosity on particle shapes for φ < 0:1. It is exonerated from the present consequences that the elongated shaped nanoparticles like cylinder and platelet have larger viscosities as assimilated to nanofluids comprising square-shaped nanoparticles like brick and blade. The acquired consequences agree well with the experimental consequences predicted by Timofeeva et al.. A very small divergence is perceived in the present research, where the cylindrical-shaped nanoparticles have the larger viscosity, whereas from the experimental findings perceived by Timofeeva et al., the platelet has the larger viscosity. Timofeeva et al. had assimilated their consequences with the Hamilton and Crosser model and found that their consequences were equivalent with the Hamilton and Crosser model. In the current work, the model of Hamilton and Crosser is applied and found that analytical consequences in this research also identical to the experimental consequences of Timofeeva et al.


Table 3. Thermophysical properties of water and nanoparticles.

Nu <sup>¼</sup> <sup>e</sup><sup>1</sup>

<sup>τ</sup><sup>1</sup> <sup>¼</sup> <sup>τ</sup>1<sup>ð</sup> <sup>t</sup>޼� <sup>d</sup>2h<sup>1</sup> ðe2 <sup>1</sup> <sup>þ</sup> <sup>h</sup><sup>2</sup>

<sup>τ</sup><sup>2</sup> <sup>¼</sup> <sup>τ</sup>2ðt޼� <sup>d</sup>2h<sup>1</sup> ðe2 <sup>1</sup> <sup>þ</sup> <sup>h</sup><sup>2</sup>

<sup>τ</sup><sup>3</sup> <sup>¼</sup> <sup>τ</sup>3ðt޼� <sup>d</sup>2h<sup>1</sup> ðe2 <sup>1</sup> <sup>þ</sup> <sup>h</sup><sup>2</sup>

sinhðh2Þ

<sup>þ</sup><sup>ε</sup> exp <sup>ð</sup>iωt<sup>Þ</sup> <sup>h</sup><sup>2</sup>

248 Nanofluid Heat and Mass Transfer in Engineering Problems

<sup>þ</sup><sup>ε</sup> exp <sup>ð</sup>iωt<sup>Þ</sup> <sup>h</sup><sup>2</sup>

3. Graphical consequences and depiction

presented in Tables 1–3.

sin ðe1Þ

<sup>1</sup>Þsinhðh1Þ

<sup>1</sup>Þsinhðh1Þ

<sup>1</sup>Þsinhðh1Þ

HðtÞ �

8 >>><

>>>:

λ ðh2 <sup>2</sup>φ2Þ �

þ

In this segment, graphical consequences are figured and debated. Similarly, influence of the radiation effect on heat transfer in mixed convection MHD flow of nanofluids inside a channel filled with a saturated porous medium is explored. Three different flow cases on the basis of appropriate boundary conditions are examined. Four dissimilar shapes of Al2O3 as solid nanoparticles, which are cylinder, platelet, brick, and blade, are dangling into conventional base fluids, ethylene glycol and water. The physical performance of the exhibited graphs is conferred for various embedded parameters. The numerical values of constants a and b (called shape factors) are chosen from Table 1, and sphericity Ψ is given in Table 2. It should be acclaimed that a and b coefficients vary highly with particle shape. The numerical values for the various shapes of nanoparticles (platelet, blade, cylinder and brick) at equal volumes are

The geometry of the problem is shown in Figure 1. The impact of dissimilar shapes of alumina Al2O3 nanoparticles on the motion of ethylene glycol-based nanofluids is represented in Figure 2. It is viewed that the blade shape of alumina Al2O3 nanoparticles inside ethylene glycol-based nanofluids has the uppermost velocity chased by brick, platelet, and cylindricalshaped nanoparticles in ethylene glycol-based nanofluids. The impact of the shapes on the motion of nanofluids is because of the strong subjection of viscosity on particle shapes for φ < 0:1. It is exonerated from the present consequences that the elongated shaped nanoparticles like cylinder and platelet have larger viscosities as assimilated to nanofluids comprising square-shaped nanoparticles like brick and blade. The acquired consequences

<sup>þ</sup><sup>ε</sup> exp <sup>ð</sup>iωt<sup>Þ</sup> <sup>λ</sup>ðcoshðh2Þ � <sup>1</sup><sup>Þ</sup>

sinhðh2<sup>Þ</sup> <sup>H</sup>ðtÞ þ <sup>λ</sup>

þ

h2φ2sinhðh2Þ � �

ðh2φ2Þ

� �� � �

þ

þ

ðe2 <sup>1</sup> <sup>þ</sup> <sup>h</sup><sup>2</sup>

1 � coshðh2Þ

ðe2 <sup>1</sup> <sup>þ</sup> <sup>h</sup><sup>2</sup>

�

ðe2 <sup>1</sup> <sup>þ</sup> <sup>h</sup><sup>2</sup>

d2b<sup>1</sup>

:

d2e<sup>1</sup>

coshðh2Þ � 1

d2b<sup>1</sup>

�

coshðh2Þ � 1

<sup>1</sup>Þ sin ðe1Þ

<sup>1</sup>Þ sin ðe1Þ

<sup>1</sup>Þ sin ðe1Þ

�

9 >>>=

>>>;

, (38)

,

(39)

(40)

(41)

Figure 1. Physical model and coordinates system.

Figure 3 shows the result of dissimilar shapes of alumina Al2O3 nanoparticles on the motion of water H2O-based nanofluids. It is surely noticed that the cylindrical-shaped alumina Al2O3

Figure 2. Velocity profiles for different shapes of Al2O3 nanoparticles in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1, λ ¼ 1, K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

Figure 3. Velocity profiles for different shapes of Al2O3 nanoparticles in water-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1, λ ¼ 1, K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

nanoparticles in ethylene glycol-based nanofluids have the uppermost velocity chased by platelet, brick, and blade. Thus, in accordance with Hamilton and Crosser model, solution of elongated and thin shaped particles (high shape factor m) should have larger thermal conductivities, if the ratio of knf=kf is higher than 100. It is also predicted by Colla et al. that the thermal conductivity and viscosity increase with the increase of particle absorption due to which motion of nanofluids decreases. For that reason, the cylindrical-shaped alumina Al2O3 nanoparticles have the higher thermal conductivity chased by platelet, brick, and blade. Timofeeva et al. depicted the conclusion that, when the sphericity of nanoparticles is less than 0.6, the negative concession of heat flow resistance at the solid-liquid interface increases much faster than the particle shape concession. Thus, the inclusive thermal conductivity of solution starts decreasing less than sphericity of 0.6. However, it is increasing in the case of Hamilton and Crosser model because of the only concession of particle shape parameter m: Nevertheless, flow in this work is one-directional and one-dimensional; therefore, the negative concession of heat flow resistance is ignored. Timofeeva et al. reported the model knf=kf ¼ 1 þ ðc shape <sup>k</sup> <sup>þ</sup> <sup>c</sup>surface <sup>k</sup> Þ φ, for calculating the thermal conductivity of nanoparticles. In accordance with this model, c shape <sup>k</sup> and csurface <sup>k</sup> coefficients reflecting concessions to the effective thermal conductivity because of particle shape (positive influence) and because of surface resistance (negative effect), respectively. Particle shape coefficient c shape <sup>k</sup> was also derived from the Hamilton and Crosser's equation (1962).

A comparability of alumina Al2O3 in C2H6O2-based nanofluids with alumina Al2O3 in H2O-based nanofluids is shown in Figure 4. It is viewed that the motion of H2O-based nanofluids is larger than the motion of C2H6O2-based nanofluids. The thermal conductivity and viscosity of C2H6O2- and H2O-based nanofluids are also perceived by the Hamilton and Crosser model for equal value of φ. This consequence shows that C2H6O2-based nanofluids have larger thermal conductivity and viscosity than H2O-based nanofluids.

Figure 2. Velocity profiles for different shapes of Al2O3 nanoparticles in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1,

Figure 3. Velocity profiles for different shapes of Al2O3 nanoparticles in water-based nanofluids when Gr ¼ 0:1,

M ¼ 1, λ ¼ 1, K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

250 Nanofluid Heat and Mass Transfer in Engineering Problems

N ¼ 0:1, M ¼ 1, λ ¼ 1, K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

The influence of different solid nanoparticles on the motion of different nanofluids is shown in Figure 5. From this figure, it is notable that cylindrical-shaped Al2O3 in C2H6O2-based nanofluids has the uppermost motion chased by Fe3O4, TiO2, Cu, and Ag in C2H6O2-based nanofluids. This indicates that cylindrical-shaped silver Ag in C2H6O2-based nanofluids has the higher viscosity and thermal conductivity assimilated to Cu, TiO2, Fe3O4, and Al2O3 in C2H6O2-based nanofluids. One can view from this consequence that cylindrical-shaped Ag in C2H6O2-based nanofluids has better quality fluids assimilated to Fe3O4 cylindrical-shaped in C2H6O2-based nanofluids. This consequence is supported by the Hamilton and Crosser model that the viscosity and thermal conductivity of nanofluids are also influenced by nanoparticles φ, i.e., the viscosity and thermal conductivity increase with the increase in φ. Therefore, motion decreases with the increase in φ. This figure further shows that the viscosity of Al2O3 in C2H6O2-based nanofluids at φ is below 0.1, which increases nonlinearly with nanoparticles suspension. This consequence is found similar to the experimental consequence predicted by Colla et al.

Different φ of nonspherical cylindrical-shaped alumina Al2O3 nanoparticles on the motion of alumina Al2O3 in C2H6O2-based nanofluids is exhibited in Figure 6. It is view from this figure that with the increase of φ the motion of nanofluids is decreased. Due to this reason, the

Figure 4. Comparison of velocity profiles of Al2O3 in EG and water-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1, λ ¼ 1, K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

Figure 5. Velocity profiles of different nanoparticles in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1, λ ¼ 1, K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

Energy Transfer in Mixed Convection MHD Flow of Nanofluid Containing Different Shapes of Nanoparticles… http://dx.doi.org/10.5772/67367 253

Figure 6. Velocity profiles for different values of φ of Al2O3 in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1, λ ¼ 1, K ¼ 1, t ¼ 5, ω ¼ 0:2:

Figure 4. Comparison of velocity profiles of Al2O3 in EG and water-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1,

Figure 5. Velocity profiles of different nanoparticles in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1, λ ¼ 1,

λ ¼ 1, K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

252 Nanofluid Heat and Mass Transfer in Engineering Problems

K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

nanofluid becomes more viscous with the increase of φ, which governs to decrease the motion of nanofluids. The thermal conductivity of nanofluids also increases with the increase of φ. The experimental research by Colla et al. also supports these results.

Consequences for various values of radiation parameter N of Al2O3 in H2O-based nanofluids are displayed in Figure 7. It is observed that motion increases with the increase of N: This consequence agrees well with the consequence reported by Makinde and Mhone. Physically, this means that with the increase of N, the amount of heat energy transfers to the fluids also increases.

The graphical outcomes of the motion of nanofluids for several values of magnetic parameter M of alumina Al2O3 in H2O-based nanofluids are exhibited in Figure 8. Increasing magnetic parameter, M, results in the decrease of the motion of the alumina nanofluids. Increasing perpendicular magnetic field on the electrically conducting fluid imparts to a resistive force called Lorentz force, which is identical to drag force, and upon increasing the value of magnetic parameter, M, the drag force rises which has the tendency to reduce the motion of the nanofluid. The resistive force is maximum near the plates of the channel and minimum in the middle of the plates. Therefore, motion on the alumina nanofluids is maximum in the middle of the plates and minimum at the plates. The motion of alumina nanofluids for various numerical values of Grashof number, Gr of Al2O3 in H2O-based nanofluids is shown in Figure 9. It is concluded that an increase in Grashof number, Gr, governs to an increase in the motion of alumina nanofluids. An increase in Grashof number, Gr, rises temperature of alumina in ethylene glycol nanofluids, which govern to an increase in the upward buoyancy force. Therefore, motion in alumina nanofluids increases with Gr, because of the increment of buoyancy force. Figure 10 shows

Figure 7. Velocity profiles for different values of N of Al2O3 in EG-based nanofluids when Gr ¼ 0:1, M ¼ 1, λ ¼ 1, K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

Figure 8. Velocity profiles for different values of M of Al2O3 in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, λ ¼ 1, K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

Figure 9. Velocity profiles for different values of Gr of Al2O3 in EG-based nanofluids when N ¼ 0:1, M ¼ 1, λ ¼ 1, K ¼ 1, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

permeability parameter K: It is view that motion of Al2O3 in C2H6O2-based nanofluids increases with increasing values of permeability parameter K of Al2O3 in C2H6O2-based nanofluids because of the small friction force. Similarly, increasingK reduces the nanofluids friction within the plate walls and motion of alumina nanofluids enhances. In the second case, Figures 11–18 show the flow condition when the left wall is oscillating in its own plane and the right wall is stationary. Under the last conditions, when both plates of the channel are oscillating in their planes (Figures 19–26). From all these graphs, we found that they are qualitatively identical but different quantitatively to Figures 2–10. It can also viewed from Figures 11, 13, and 14 that the motion of Al2O3 in C2H6O2-based nanofluids at the right wall (y = 1) is not equal to zero. It is because of the reason of oscillating right plate of the channel. It should also be investigated from Figures 20, 24, and 26 that only the magnitude of motion of nanofluids is considered; therefore, the negative sign in motion is ignored in these figures and only shows that the motion of Al2O3 in C2H6O2-based nanofluids is reduced.

The influence of dissimilar particle shapes on the temperature of Al2O3 in H2O and C2H6O2 based nanofluids is presented in Figures 27 and 28. The temperature of both types of nanofluids is different for dissimilar shapes because of the various viscosity and thermal conductivity of these nanoparticles. It should be concluded that the influence of thermal conductivity increases with the increase of temperature. However, the viscosity decreases with the increase of temperature. It is view that an elongated shape of nanoparticles inside H2Obased nanofluids like cylinder and platelet has small temperature due to the larger viscosity and thermal conductivity while the blade shape of nanoparticles has the uppermost temperature because of least viscosity and thermal conductivity. The brick shape nanoparticles suspended fluids are lowest in the temperature range; however, it has low viscosity. This is

Figure 8. Velocity profiles for different values of M of Al2O3 in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, λ ¼ 1,

Figure 7. Velocity profiles for different values of N of Al2O3 in EG-based nanofluids when Gr ¼ 0:1, M ¼ 1, λ ¼ 1,

K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

254 Nanofluid Heat and Mass Transfer in Engineering Problems

Figure 10. Velocity profiles of different values of K of Al2O3 in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1, λ ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

Figure 11. Velocity profiles for different shapes of Al2O3 nanoparticles in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1, λ ¼ 1, K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

Energy Transfer in Mixed Convection MHD Flow of Nanofluid Containing Different Shapes of Nanoparticles… http://dx.doi.org/10.5772/67367 257

Figure 12. Velocity profiles for different shapes of Al2O3 nanoparticles in water-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1, λ ¼ 1, K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

Figure 10. Velocity profiles of different values of K of Al2O3 in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1,

Figure 11. Velocity profiles for different shapes of Al2O3 nanoparticles in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1,

λ ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

256 Nanofluid Heat and Mass Transfer in Engineering Problems

M ¼ 1, λ ¼ 1, K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

Figure 13. Comparison of velocity profiles of Al2O<sup>3</sup> in EG- and water-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1, λ ¼ 1, K ¼ 1, t ¼ 5, φ ¼ 0:04, ω ¼ 0:2:

Figure 14. Velocity profiles for different values of φ of Al2O3 in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1, λ ¼ 0:01, K ¼ 0:3, t ¼ 10, ω ¼ 0:2:

Figure 15. Velocity profiles for different values of N of Al2O3 in EG-based nanofluids when Gr ¼ 1, M ¼ 1, λ ¼ 0:01, K ¼ 0:2, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

Energy Transfer in Mixed Convection MHD Flow of Nanofluid Containing Different Shapes of Nanoparticles… http://dx.doi.org/10.5772/67367 259

Figure 16. Velocity profiles for different values of M of Al2O3 in EG-based nanofluids when Gr ¼ 1, N ¼ 0:1, λ ¼ 0:001, K ¼ 1, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

Figure 14. Velocity profiles for different values of φ of Al2O3 in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1,

Figure 15. Velocity profiles for different values of N of Al2O3 in EG-based nanofluids when Gr ¼ 1, M ¼ 1, λ ¼ 0:01,

λ ¼ 0:01, K ¼ 0:3, t ¼ 10, ω ¼ 0:2:

258 Nanofluid Heat and Mass Transfer in Engineering Problems

K ¼ 0:2, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

Figure 17. Velocity profiles for different values of Gr of Al2O3 in EG-based nanofluids when N ¼ 0:1, M ¼ 1, λ ¼ 0:01, K ¼ 0:2, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

because of the shear thinning behavior with temperature of nanofluids. Moreover, cylindricalshaped nanofluids also show shear thinning behavior. However, the effect is less dominant. All the other dissimilar shapes like platelet and blade show Newtonian behavior and independence of viscosity on shear rate. This shear thinning behavior is also analyzed experimentally by Timofeeva et al.

Figure 18. Velocity profiles for different values of K of Al2O3 in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1, λ ¼ 0:01, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

Figure 19. Velocity profiles for different shapes of Al2O3 nanoparticles in EG-based nanofluids when Gr ¼ 1, N ¼ 0:1, M ¼ 1, λ ¼ 0:01, K ¼ 1, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

Energy Transfer in Mixed Convection MHD Flow of Nanofluid Containing Different Shapes of Nanoparticles… http://dx.doi.org/10.5772/67367 261

the other dissimilar shapes like platelet and blade show Newtonian behavior and independence of viscosity on shear rate. This shear thinning behavior is also analyzed experimentally by

Figure 18. Velocity profiles for different values of K of Al2O3 in EG-based nanofluids when Gr ¼ 0:1, N ¼ 0:1, M ¼ 1,

Figure 19. Velocity profiles for different shapes of Al2O3 nanoparticles in EG-based nanofluids when Gr ¼ 1, N ¼ 0:1,

Timofeeva et al.

260 Nanofluid Heat and Mass Transfer in Engineering Problems

λ ¼ 0:01, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

M ¼ 1, λ ¼ 0:01, K ¼ 1, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

Figure 20. Velocity profiles for different shapes of Al2O3 nanoparticles in water-based nanofluids when Gr ¼ 1, N ¼ 0:1, M ¼ 1, λ ¼ 0:01, K ¼ 1, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

Figure 21. Comparison of velocity profiles of Al2O3 in EG- and water-based nanofluids when Gr ¼ 1, N ¼ 0:1, M ¼ 1, λ ¼ 0:01, K ¼ 1, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

Figure 22. Velocity profiles for different values of φ of Al2O3 in EG-based nanofluids when Gr ¼ 1, N ¼ 0:1, M ¼ 1, λ ¼ 0:01, K ¼ 1, t ¼ 10, ω ¼ 0:2:

Figure 23. Velocity profiles for different values of N of Al2O3 in EG-based nanofluids when Gr ¼ 1, M ¼ 1, λ ¼ 0:01, K ¼ 1, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

Energy Transfer in Mixed Convection MHD Flow of Nanofluid Containing Different Shapes of Nanoparticles… http://dx.doi.org/10.5772/67367 263

Figure 24. Velocity profiles for different values of M of Al2O3 in EG-based nanofluids when Gr ¼ 1, M ¼ 1, λ ¼ 0:01, K ¼ 1, t ¼ 10, φ ¼ 0:04, λ ¼ 0:01, ω ¼ 0:2:

Figure 22. Velocity profiles for different values of φ of Al2O3 in EG-based nanofluids when Gr ¼ 1, N ¼ 0:1, M ¼ 1,

Figure 23. Velocity profiles for different values of N of Al2O3 in EG-based nanofluids when Gr ¼ 1, M ¼ 1, λ ¼ 0:01,

λ ¼ 0:01, K ¼ 1, t ¼ 10, ω ¼ 0:2:

262 Nanofluid Heat and Mass Transfer in Engineering Problems

K ¼ 1, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

Figure 25. Velocity profiles for different values of Gr of Al2O3 in EG-based nanofluids when Gr ¼ 1, M ¼ 1, λ ¼ 0:01, K ¼ 1, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

Figure 26. Velocity profiles for different values of K of Al2O3 in EG-based nanofluids when Gr ¼ 1, M ¼ 1, λ ¼ 0:01, K ¼ 1, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

Figure 27. Temperature profiles for different shapes of Al2O3 nanoparticles in EG-based nanofluids when N ¼ 1:5, t ¼ 1:

A comparatively study of Al2O3 in H2O- and C2H6O2-based nanofluids is displayed in Figure 29. It is analyzed that both nanofluids are temperature dependent and the variation is found at the same rate for both types of nanofluids. This means that the influence of temperature of nanofluids on the thermal conductivity and viscosity of two types of base nanofluids may cause at the same rate. Figure 30 exhibits the influence of φ on the temperature of Al2O3 in C2H6O2 based nanofluids. It is evaluated that with the increase of φ temperature of the nanofluid increases because of the shear thinning nature. The viscosity of cylindrical-shaped nanoparticles inside water and C2H6O2-based nanofluids shows shear thinning nature at the highest suspension. This was also experimentally investigated by Timofeeva et al.

The graphical consequences of temperature of nanofluids for several values of N are displayed in Figure 31. It is indicated that the temperature of the cylindrical-shaped alumina Al2O3 nanoparticles in C2H6O2-based nanofluids shows larger oscillation with the increase of N: It is evaluated in the solution of the problem that temperature of the alumina nanofluids is oscillating and the influence of oscillation is increase with the increase of N: The increasing N means cooler or dense nanofluids or reduce the influence of energy transport to the nanofluids. The cylindrical-shape nanofluids have temperature dependent viscosity because of the shear thinning nature.

Figure 28. Temperature profiles for different shapes of Al2O3 nanoparticles in water-based nanofluids when N ¼ 1:5, t ¼ 1:

A comparatively study of Al2O3 in H2O- and C2H6O2-based nanofluids is displayed in Figure 29. It is analyzed that both nanofluids are temperature dependent and the variation is found at the same rate for both types of nanofluids. This means that the influence of temperature of nanofluids on the thermal conductivity and viscosity of two types of base nanofluids may cause at the same rate. Figure 30 exhibits the influence of φ on the temperature of Al2O3 in C2H6O2 based nanofluids. It is evaluated that with the increase of φ temperature of the nanofluid increases because of the shear thinning nature. The viscosity of cylindrical-shaped nanoparticles inside water and C2H6O2-based nanofluids shows shear thinning nature at the highest suspen-

Figure 27. Temperature profiles for different shapes of Al2O3 nanoparticles in EG-based nanofluids when N ¼ 1:5, t ¼ 1:

Figure 26. Velocity profiles for different values of K of Al2O3 in EG-based nanofluids when Gr ¼ 1, M ¼ 1, λ ¼ 0:01,

K ¼ 1, t ¼ 10, φ ¼ 0:04, ω ¼ 0:2:

264 Nanofluid Heat and Mass Transfer in Engineering Problems

sion. This was also experimentally investigated by Timofeeva et al.

Figure 29. Comparison of temperature profiles of Al2O3 in EG- and water-based nanofluids when N ¼ 1:5, t ¼ 1:

Figure 30. Temperature profiles for different values of φ of Al2O3 in EG-based nanofluids when N ¼ 1:5, t ¼ 1:

Figure 31. Temperature profiles for different values of N of Al2O3 in EG-based nanofluids when t ¼ 1:

#### 4. Conclusions

In this chapter, the influence of radiative heat transfer on mixed convection MHD flow of different shapes of Al2O3 in C2H6O2, and H2O base nanofluids in a channel filled with a saturated porous medium is analyzed. The two plates of the channel at finite distance with nonuniform wall temperature are chosen in a vertical direction under the influence of a perpendicular magnetic field. The governing PDEs are solved by the perturbation method for three different flow cases, and analytic solutions are evaluated. The influence of the dissimilar shapes of nanoparticles, namely, platelet, blade, cylinder, and brick of the same volume, on the motion of nanofluids and temperature of nanofluids is examined with different consequences. An elongated shape of nanoparticles inside base fluids like cylinder and platelet results in greater viscosity at the equal volume fraction due to structural limitation of rotational and transitional Brownian motion. The shear thinning nature of cylinder and blade shape of nanoparticles inside H2O and C2H6O2 is also investigated in this research. Viscosities and thermal conductivities of nanofluids are viewed depending on nanoparticle shapes, suspension of volume fraction, and base fluid of solid nanoparticles. The concluded results are as follow:


## Acknowledgements

The authors are grateful to the reviewers for their excellent comments to improve the quality of the present article. The authors would also like to acknowledge the Research Management Center-UTM for the financial support through vote numbers 4F109 and 03J62 for this research.

## Conflicts of interest

The authors declare that they have no conflicts of interest.

#### Nomenclature

4. Conclusions

In this chapter, the influence of radiative heat transfer on mixed convection MHD flow of different shapes of Al2O3 in C2H6O2, and H2O base nanofluids in a channel filled with a saturated porous medium is analyzed. The two plates of the channel at finite distance with nonuniform wall temperature are chosen in a vertical direction under the influence of a perpendicular magnetic field. The governing PDEs are solved by the perturbation method for three different flow cases, and analytic solutions are evaluated. The influence of the dissimilar shapes of nanoparticles, namely, platelet, blade, cylinder, and brick of the same volume, on the motion

Figure 31. Temperature profiles for different values of N of Al2O3 in EG-based nanofluids when t ¼ 1:

Figure 30. Temperature profiles for different values of φ of Al2O3 in EG-based nanofluids when N ¼ 1:5, t ¼ 1:

266 Nanofluid Heat and Mass Transfer in Engineering Problems



Roman letters

ðcpÞnf Heat capacity of nanofluids D Rate of strain tensor

dp Diameter of solid nanoparticles

e Internal energy per unit volume

f Function of temperature and volume fraction, etc.

E Total electric field

exp Exponential function

Gr Thermal Grashof number g Gravitational acceleration Hð:Þ Heaviside function

I Identity tensor

J Current density J ·B Lorentz force

kb Boltzmann constant

M Magnetic parameter m Mass of the flow of fluids N Radiation parameter Nu Nusselt number n Empirical shape factors

ph Hydrostatic pressure pd Dynamic pressure Pe Peclet number

Q Heat generation parameter

qr Magnitude of radiant heat flux q} Heat conduction per unit area

q} Magnitude of heat conduction per unit area

Radiant flux vector

p Pressure

H Total momentum of the system

i Cartesian unit vector in the x-direction

j Cartesian unit vector in the y-direction K Dimensionless permeability parameter ks Thermal conductivity of solid nanoparticles

kf Thermal conductivity of base fluids knf Thermal conductivity of nanofluids

k Cartesian unit vector in the z-direction

F Force

268 Nanofluid Heat and Mass Transfer in Engineering Problems


## Author details

Aaiza Gul1 , Ilyas Khan<sup>2</sup> \* and Sharidan Shafie<sup>1</sup>

\*Address all correspondence to: ilyaskhanqau@yahoo.com

1 Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Skudai, Malaysia

2 Basic Engineering Sciences Department, College of Engineering Majmaah University, Majmaah, Saudi Arabia

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## *Edited by Mohsen Sheikholeslami Kandelousi*

In the present book, nanofluid heat and mass transfer in engineering problems are investigated. The use of additives in the base fluid like water or ethylene glycol is one of the techniques applied to augment heat transfer. Newly, innovative nanometersized particles have been dispersed in the base fluid in heat transfer fluids. The fluids containing the solid nanometer-sized particle dispersion are called "nanofluids." At first, nanofluid heat and mass transfer over a stretching sheet are provided with various boundary conditions. Problems faced for simulating nanofluids are reported. Also, thermophysical properties of various nanofluids are presented. Nanofluid flow and heat transfer in the presence of magnetic field are investigated. Furthermore, applications for electrical and biomedical engineering are provided. Besides, applications of nanofluid in internal combustion engine are provided.

Nanofluid Heat and Mass Transfer in Engineering Problems

Nanofluid Heat

and Mass Transfer

in Engineering Problems

*Edited by Mohsen Sheikholeslami Kandelousi*

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