Introduction to nanofluids

C H A P T E R

1 Introduction to nanofluids

1.1 History of nanofluids The idea of nanofluids (NFs) was first proposed by Choi [1] after performing experimental studies on various nanoparticle suspensions. NFs can be formed by dispersing metal or metallic oxide nanoparticles into a selected base fluid, such as water, oils, and ethylene glycol. The thermal conductivity, viscosity, specific heat, and density of the base fluid can be changed after adding nanoparticles. Many NFs have been investigated, with results showing a high degree of heat exchange with some pressure drop. The convective heat transfer rate and flow performance of Cu
water NFs in a straight tube were investigated by Xuan and Li [2], and they found that the Nusselt number could be increased by more than 39% for an NF with 2.0% of Cu nanoparticle volume fraction. Azmi et al. [3] considered the heat transfer coefficient of TiO2 NF in a circular tube under turbulent flow. They reported a maximum enhancement of 22.8% at 50 C for Nusselt number at 1.5% particle concentration. Experimental studies by Zhang et al. [4] showed that a maximum heat transfer enhancement of 10.6% could be obtained using Al2O3
water NFs through a circular microchannel. However, there are many problems that must be solved before an NF-based heat transfer system can be commercially available. This analysis can be obtained by numerical simulation, which can significantly reduce calculations, increase the range of experimental investigations, and provide theoretical guidance for system optimization. In this book a number of approaches or models are proposed to simulate NF heat transfer. The discrete phase model (DPM) is a two-phase model used to simulate the motion of particles through a base flow with a force
balance

Nanofluids DOI: https://doi.org/10.1016/B978-0-08-102933-6.00001-9

1

© 2020 Elsevier Ltd. All rights reserved.

2

1. Introduction to nanofluids

equation. In the DPM, the nanoparticles are tracked using the Lagrangian approach, while the governing equations for the base fluid are solved using the Eulerian approach. The biggest advantage of the Lagrangian
Eulerian approach (DPM) is that the thermophysical properties of the base fluid and the nanoparticles can be given separately, unlike the single-phase model in which the thermophysical properties need to be specified either through experimental data or various approximation model [5]. In fact, heat transfer with NFs in laminar flow using the DPM has proved the applicability of the DPM for engineering estimation of heat transfer performance [6,7]. Unfortunately, agglomeration of nanoparticles, one of the main problems NFs have in base fluids, occurs often due to van der Walls forces, electrical double-layer interaction forces, etc., which substantially decrease system performance as often reported. For instance, agglomeration of carbon nanotubes (CNTs) in an NF-based solar thermal collector will result in fouling, clogging, and a considerable reduction in the absorbance of incident solar rays [8]. NFs that are colloidal suspensions containing a kind of dispersed nanoparticles smaller than 100 nm (,100 nm) in a base fluid which show thermal properties superior to traditional fluid media. NFs have also been proposed as next-generation heat transfer fluids for numerous heat transfer applications [9]. In recent years, the thermal conductivity of NFs (the essential heat transfer property) has been studied extensively by researchers. Improvement of the thermal conductivity of various NFs has been widely reported and several proposed models have been developed based on various mechanisms as presented in the literature [10
14]. However, in many of the studies, the obtained thermal conductivities did not agree well with each other due to various measurement conditions and experimental deviations. Thus further theoretical investigation on the thermal conductivity of NFs is still required for a deeper understanding of the heat transfer mechanisms in suspensions of nanoparticles.

1.1.1 Preparing nanofluids To show the NF preparation processes, three kinds of aqueous NFs were prepared using TiO2, CNT, and SiO2 nanoparticles [15]. TiO2 (85 nm), SiO2 (12 nm), and CNTs (6.2 nm diameter and 1
5 μm length) nanoparticles were purchased from Sigma-Aldrich. Solutions with the chosen volume concentration of nanoparticles were obtained by mixing the appropriate amounts of distilled water and nanoparticles. The mixtures were stirred for 15 min for stable dispersion of the nanoparticles, and the solutions were sonicated for approximately 3 h. To obtain

Nanofluids

3

1.2 Structures and different types

TABLE 1.1

Thermophysical properties of some nanoparticles and base fluids.

Material/ properties

ρ (kg m23)

Cp (J kg21 K21)

k (W m21 K21)

μ (kg m21 s21)

β (K21) 3 1026

Water, H2O

997.1

4179

0.613

0.0010003

210

Ethylene glycol, (CH2OH)2

1076

2664

0.261

0.003036

CuO

6320

531.8

76.5




18.0

Al2O3

3970

765

40




8.5

TiO2

4250

686.2

8.9538




9.0

Fe3O4

5200

670

6




Cu

8933

385

401




16.7

Ag

10,500

235

429




18.9

characterization of the NFs, the morphology and microstructure of the samples were studied using scanning electron microscopy. As shown in Ref. [15] the average particle sizes were 85 and 12 nm (nanometer) for SiO2 and TiO2, respectively, and 6.2 nm diameter and 1
5 μm length for CNTs. Table 1.1 shows the physical properties of
these nanoparticles. The effective density ρnf , the effective heat capacity ρCp nf , and the thermal expansion ðρβ Þnf of the NFs are explained in the next chapter. Note that all experiments were performed for 0.001 g nanoparticles in 3 L water [15].

1.1.2 Synthesis of nanofluids A complete discussion on the different approaches to nanoparticle synthesis such as sol
gel and analysis methods like transmission electron microscopy (TEM) is not provided here but can be found in Refs. [16
20].

1.2 Structures and different types Comparison of single- and two-phase NF modeling can be found in the literature. For instance, a comparison of the results of single-phase and two-phase numerical methods for NFs in a circular tube was reported by Haghshenas Fard et al. [21]. They reported that the average relative error between the experimental data and computational fluid dynamics (CFD) results was 16% for a Cu
water single-phase model

Nanofluids

4

1. Introduction to nanofluids

while it was 8% for a two-phase model. In another numerical study, Go¨ktepe et al. [22] compared the two models for NF convection at the entrance of a uniformly heated tube and found the same results, further confirming that the accuracy of two-phase modeling is greater than single-phase modeling. In the following cases single-phase and twophase modelings are presented.

1.2.1 Case 1: Single phase: different shapes of nanoparticles in a wavy-wall square cavity filled with power-law non-Newtonian nanofluid Fig. 1.1 shows the geometry of the studied wavy-wall square cavity [23]. The left wavy wall is hold in cold temperature (TL) and the right flat wall is kept in a constant hot temperature (TH). Two insulated bottom and upper flat walls were fixed in their locations. The domain was filled with Fe3O4/non-Newtonian shear-thinning NF in the presence of a magnetic field. The thermophysical properties used are given in Table 1.1. A magnetic field with three different angles (γ 5 0, 30, and 60 degrees) was applied on the cavity. The flow was incompressible, steady, and laminar. The density variation was approximated by the standard Boussinesq model. A nondimensional cosine function with Am as the wave amplitude was used to simulate the wavy wall as shown in Fig. 1.1 [23]. In this single-phase case, the temperature and velocity fields in wavy square cavity of Fig. 1.1 were obtained by solving the continuity, Navier
Stokes, and energy equations. Therefore the governing equations are simplified by using the following assumptions:

FIGURE 1.1 Schematic of the wavy-wall square cavity and generated mesh [23].

Nanofluids

1.2 Structures and different types

5

• The flow is incompressible and NF is power-law non-Newtonian. • Relative movement between fluid and Fe3O4 particles is zero and thermal equilibrium exists between them. • The temperature and velocity fields are laminar, steady state, and 2D. • The effects of radiation and viscous dissipation are neglected. The governing equations (continuity, momentum, and energy equations) and definitions of dimensionless variables can be introduced as follows based on the above assumptions [23,24]: @v @u 1 50 (1.1) @y @x    2   
 @u @u 1 @P @ u @2 u 2 2 1u 1 μnf v 2 1 2 1σnf B v sin γ cos γ 2 u sin γ 5 @y @x ρnf @x @x2 @y "    2  @v @v 1 @P @ v @2 v 1 μnf 1 v 1u 5 1 gρnf β nf ðT 2 Tc Þ 2 @y @x ρnf @y @x2 @y2 #
 2 2 1 σnf B u sin γ cos γ 2 v cos γ


ρCp



 nf

  2  @T @T @ T @2 T 1u v 1 2 5 knf @y @x @x2 @y

(1.2)

(1.3)

(1.4)

To solve the governing equations for single-phase modeling, the NF effective properties are required, and can be calculated as a single phase by the equations introduced in the next section (Eqs. 1.27
1.32). (    2   )ðn21Þ 2 @u 2 @v @v @u 2 1 12 1 μf 5 N 2 @x @x @y @x

(1.5)

In this case, the Hamilton equation is applied to calculate the thermal conductivity of the NF:
 knf kp 1 ðm 1 1Þkf 2 ðm 1 1Þϕ kf 2 kp
 5 (1.6) kf kp 1 ðm 1 1Þkf 1 ϕ kf 2 kp It can be shown that for spherical nanoparticles m 5 3 and for other nanoparticle shapes, m can be calculated by using m 5 3=ψ and Table 1.2. In Eq. (1.5), N is the consistency coefficient and n is the power-law index. Therefore the deviation of n from 1.0 specifies the degree of deviation from Newtonian behavior. For n 6¼ 1, the constitute Eq. (1.5)

Nanofluids

6

1. Introduction to nanofluids

TABLE 1.2 Constants of Eq. (1.6). Nanoparticle shape

ψ

Spherical

1

Platelet

0.52

Cylindrical

0.62

Brick

0.81

represents a pseudoplastic fluid (n , 1) and for (n . 1) it represents a dilatant fluid. The boundary condition of the problem is defined as: 8 @T > > 50 Insulated walls: u 5 v 5 0; > < @y (1.7) > Wavy wall: u 5 v 5 0; T 5 TC > > : Right wall: u 5 v 5 0; T 5 TH The following dimensionless variables are considered in the solution: Τ5

T 2 TC uL vL y x pL2 ; U5 ; V5 ; Y 5 ; Χ 5 ; P 5
2 0:5 0:5 αf Ra αf Ra L L TH 2 TC ρf αf Ra (1.8)

Using Eq. (1.8), Eqs. (1.1)
(1.4) can be written in dimensionless form as: @V @U 1 50 (1.9) @Y @X 2 0 1 0 0 113 ρ μ μ @U @U @P Pr f 1 42 @ @ f @U A1 @ @ f @@U 1@V AA5 1U 52 1 pffiffiffiffiffiffi V @Y @X @X @Y N @Y @X Ra ρnf ð12ϕÞ2:5 @X N @X 2 3
 Ha2 Pr σnf =σf 5 1 1 pffiffiffiffiffiffi 4 V sin γ cos γ 2U sin2 γ 2:5 Ra ρnf =ρf ð12ϕÞ (1.10) 0 0 113 μf @V μf @U @V @V @V @P Pr ρf 1 @ @ 42 @ A1 @ @ 1 AA5 52 1 pffiffiffiffiffiffi V 1U @Y @X @Y @X N @Y @X Ra ρnf ð12ϕÞ2:5 @Y N @Y 2 3 ðρβ Þnf
 Ha2 Pr σnf =σf 5 1 1 pffiffiffiffiffiffi 4 U sin γ cos γ 2V cos2 γ 1Pr T 2:5 ρnf β f Ra ρnf =ρf ð12ϕÞ 2

0

1

(1.11)

Nanofluids

7

1.2 Structures and different types

" #  knf =kf @Τ @Τ 1 @2 Τ @2 Τ 
 1V 5 pffiffiffiffiffiffi
U 1 @X @Y Ra ρCp nf = ρCp f @X2 @Y2 With dimensionless parameters: ρf β f gL3 ðTH 2 TC Þ

Raf 5 ; αf μf

υf Pr 5 ; αf

sffiffiffiffiffi σf Ha 5 LB μf

8 > > Insulated walls: > <

U 5 V 5 0;

@T 50 @Y

> Wavy wall: > > : Right wall:

U 5 V 5 0; U 5 V 5 0;

T50 T51

(1.12)

(1.13)

(1.14)

To numerically solve these dimensionless governing equations, the finite element method was applied using FlexPDE commercial software. In addition, the summation of residual convergence criterion was considered to be less than 1024. To validate the independence of the grids, the Num of the wavy wall was considered as the criterion, and three different geometries of wavy walls with amplitude of Am 5 0.1, 0.2, and 0.3 were considered. In this case, the main goal is to model the NF to find its best heat transfer performance with the least possible amount of entropy generation. Table 1.1 lists the thermophysical properties of Fe3O4 nanoparticles. The following dimensionless parameters were considered in this study: • • • • •

Nanoparticles volume fraction (ϕ): changes from 2% to 6%. The wavy-wall amplitude (Am): varies from 0.1 to 0.3. Hartmann number (Ha): changes between 20 and 100. Magnetic field angle (γ): varies from 0 to 60. Nanoparticles shapes: platelet, cylindrical, spherical, and brick shapes.

Fig. 1.2 shows the streamline and isotherm contours in the three different geometries of the cavity. The contours were drawn for the Rayleigh number of 10,000, Ha 5 20, n 5 1, γ 5 0, and spherical nanoparticles with ϕ 5 2%. It can be observed that in all temperature contours the left wavy side is cooler than the right wall due to boundary conditions. By increasing the Am, the peak of the wavy shape will be more near to hot wall, which makes a separation in the natural flow pattern in the streamline. As seen in Fig. 1.2, for Am 5 0.3 the separation of streamlines completely occurred and two different areas were created. Also, in the temperature

Nanofluids

8

1. Introduction to nanofluids

Temp.

Streamline

A m = 0. 1

A m = 0. 2

A m = 0. 3

FIGURE 1.2 Effect of Am on temperature and streamlines [23].

contours there are lower temperatures in the cavity center due to the cold temperature of the wavy wall. Fig. 1.3 shows the effects of nanoparticle volume fraction on the contour results, and based on these results, Fig. 1.4 shows the effect of Am and ϕ on the Nusselt number. Increasing the ϕ caused an increase in the Nusselt number due to more heat transfer from greater nanoparticle volume fraction.

Nanofluids

1.2 Structures and different types

Temp.

9

Streamline

φ = 0.0 2

φ = 0.0 4

FIGURE 1.3 Effect of phi on temperature and streamlines [23].

The effects of the four described nanoparticle shapes (i.e., platelet, cylindrical, spherical, and brick) are depicted in Figs. 1.5 and 1.6. It can be seen that the maximum average Nusselt number occurred for the brick-shaped nanoparticles, while the spherical-shaped nanoparticles had the minimum Nusselt number.

1.2.2 Case 2: Two-phase nanofluid thermal analysis over a stretching infinite solar plate In this case, the aim is to show two-phase NF flow modeling [25]. An NF flow over a solar plate was considered by Khan et al. [26] as shown in Fig. 1.7. As in the last case, the flow was incompressible and induced due to the plate being stretched in two directions by nonlinear functions. The plate was kept at constant temperature and the nanoparticle mass flux at the wall was expected to be zero. The three-dimensional governing equations are [25,26]:

Nanofluids

10

1. Introduction to nanofluids

2

Average Nusselt

1.8

1.6

1.4

1.2 0.1

0.15

0.2

0.25

0.3

Wave amplitude (Am) 1.24

1.22

Average Nusselt

1.2

1.18

1.16

1.14

1.12

1.1

0.02

0.03 0.04 0.05 Nanoparticles volume fraction (ϕ)

FIGURE 1.4 Nusselt numbers in different Am and ϕ [23].

Nanofluids

0.06

1.2 Structures and different types

Temp.

11

Streamline

Platelet

Brick

Spherical

FIGURE 1.5 Effect of ψ on temperature and streamlines [23].

@u @v @w 1 1 50 @x @y @z

(1.15)

u

@u @u @u @2 u 1v 1w 5 υf 2 @x @y @z @z

(1.16)

u

@v @v @v @2 v 1 v 1 w 5 υf 2 @x @y @z @z

(1.17)

Nanofluids

12

1. Introduction to nanofluids

FIGURE 1.6 Nusselt numbers in different nanoparticle shapes [23].

FIGURE 1.7 Schematic of the nanofluid flow over a stretched solar plate [25].

"  # @T @T @T @2 T @C @T DT @T 2 1v 1w 5 α 2 1 τ DB 1 u @x @y @z @z @z @z TN @z u

  2 @C @C @C @2 C DT @ T 1v 1w 5 DB 2 1 @x @y @z @z TN @z2

(1.18)

(1.19)

where u and v are the velocities in the x and y directions, respectively; T is the temperature, C is the concentration; DB is the Brownian diffusion

Nanofluids

13

1.2 Structures and different types

coefficient of the diffusing species; and DT is the thermophoretic diffusion coefficient. By this assumption that the plate is infinite and stretched in two directions by nonlinear functions, the boundary conditions will be:
n
n u 5 uw 5 a x1y ; v 5 vw 5 b x1y ; w 5 0; T 5 Tw ;

DB

@C DT @T 1 50 @z TN @z

u-0; v-0; T-TN ; C-CN

at z 5 0

as z-0

By introducing these parameters:
n
n u 5 a x1y f 0 ðηÞ; v 5 a x1y g0 ðηÞ 0 1 n21


 2 pffiffiffiffiffiffiffi @ n 1 1 f 1 g 1 n 2 1 η f 0 1 g0 A w 5 2 aυf x1y 2 2 T 2 TN C 2 CN θðηÞ 5 ; φðηÞ 5 ;η5 Tw 2 TN CN

(1.20)

(1.21)

n21 sffiffiffiffiffi  2 a
x1y z υf

and substituting above variables into Eqs. (1.15)
(1.19), we have: 
 n11
f 1 g fv 2 n f 0 1 g0 f 0 5 0 2 
 n11
f 1 g gv 2 n f 0 1 g0 g0 5 0 gw 1 2  1 n11
θv 1 f 1 g θ0 1 Nb φ0 θ0 1 Nt θ02 5 0 Pr 2 n11 Nt Scðf 1 gÞφ0 1 θv 5 0 φv 1 2 Nb fw 1

(1.22) (1.23) (1.24) (1.25)

These systems of nonlinear equations can be solved by a powerful numerical or analytical method. In this study, the optimal collocation method (which will be introduced in Chapter 2: Mathematical analysis of nanofluids) was applied with these boundary conditions: f ð0Þ 5 0; f 0 ð0Þ 5 1; gð0Þ 5 0; g0 ð0Þ 5 λ; θð0Þ 5 1; Nb φ0 ð0Þ 1 Nt θ0 ð0Þ 5 0 f 0 ðNÞ-0; g0 ðNÞ-0; θðNÞ-0; φðNÞ-0 (1.26) where Pr (Prandtl number), Sc (Schmidt number), Nb (Brownian motion parameter), Nt (thermophoresis parameter), and λ 5 b/a (ratio of the stretching rate along y to x directions) are the defined parameters found in Ref. [26].

Nanofluids

14

1. Introduction to nanofluids

Here, the Keller
Box method was used to solve the problem in Maple 15.0 software. The Keller
Box scheme, as described in Chapter 3, Numerical analysis of nanofluids, is a face-based method for solving partial differential equations (PDEs) that has several great mathematical and physical properties. These properties are due to the fact that the scheme exactly discretizes partial derivatives and only makes estimations in the algebraic constitutive relations given in the PDE. The discrete calculus associated with the Keller
Box scheme is different from all other simulated numerical procedures. Essentially, Keller
Box is a variation of the finite volume method (FVM) in which unknown coefficients are stored at control volume faces rather than at the traditional cell centers. It is due to the fact that in space
time equations, the unknowns sit at the corners of the space
time control volume, which is a box in one space dimension on a fixed mesh. The original method [27] was distributed with parabolic initial value problems such as the unsteady heat transfer equation. First of all, the infinite range must be clear to solve the governing equations. This means that η can be considered as the infinite number. To discover this value three different numbers (4, 7, and 10) were studied, the results of which are presented in Fig. 1.8. As seen for all the depicted graphs (velocity, temperature, and nanoparticle concentrations) the two last values (i.e., 7 and 10) have the same profiles, so increasing this value has no significant effect on the results (7 was chosen in the solution procedure). The effect of the power-law index (n) on the x- and y-components of dimensionless velocities (f0 (η) and g0 (η) functions), temperature, and nanoparticle volume fraction boundary layer profiles (θ(η) and ϕ(η)) are depicted in Fig. 1.9. As can be seen, by increasing the power-law index, n, dimensionless velocity will increase in both the x- and y-directions. Also, it can be observed that the reductions of velocity in the x- and y-directions are roughly equal. The results show that both velocity profiles are decreasing functions of the power index (n). It is also obvious that the thermal boundary layer will be thinner for larger n values while the nanoparticle concentration profile will become thicker, which increases the rate of heat transfer from the sheet. Furthermore, it can also be understood that the greater the n value the faster the decline of θ. It has been suggested that increasing n can enhance the convective properties of the fluid since it will increase the deformation by the shear stress from the wall to the fluid. Fig. 1.10 demonstrates the effect of n and λ on the shear stress at the surface (i.e., fvð0Þ and gvð0Þ) and reduced Nusselt number (Nur). This means that increasing both the n and λ parameters reduces the shear stress, but increases the Nusselt number.

Nanofluids

1.2 Structures and different types

15

FIGURE 1.8 Effect of η on profiles when n 5 1; Nb 5 0:7; Nt 5 0:4; Sc 5 3; Pr 5 1; λ 5 0:5: [25].

Nanofluids

16

1. Introduction to nanofluids

FIGURE 1.9 Effect of power index (n) when ηN 5 7; Nb 5 0:7; Nt 5 0:4; Sc 5 3; Pr 5 1;

λ 5 0:5 [25].

Nanofluids

1.2 Structures and different types

17

FIGURE 1.10 Effect of n and λ on Nur, fvð0Þ, and gvð0Þ when ηN 5 7; Nb 5 0:7; Nt 5 0:4; Sc 5 3; Pr 5 1 [25].

Nanofluids

18

1. Introduction to nanofluids

1.3 Nanofluid properties The following are some of the most applicable formulas for NF properties. These demonstrations of NF properties are obtained based on the nanoparticle volume fraction as follows:


ρeff 5 ρs φ 1 ρf ð1 2 φÞ

 ρCp eff 5 ρCp s φ 1 ρCp f ð1 2 φÞ

(1.27)

β eff 5 β f ð1 2 φÞ 1 β s φ

(1.29)



μeff 5

μf

ð12φÞ2:5
 2 2φ kf 2 ks 1 2kf 1 ks
 keff 5 kf φ kf 2 ks 1 2kf 1 ks

3 σσsf 2 1 φ σnf



511 σs σs σf 1 2 2 2 1 φ σf σf

(1.28)

(1.30) (1.31)

(1.32)

As mentioned above, these equations are traditionally used to model NF properties. But many studies have been performed to present novel or more accurate formulas for these properties. In the following some of these equations are reviewed and an experimental study is presented.

1.3.1 Case 1: A modified multisphere Brownian model to predict the thermal conductivity of colloid suspension of wide volume fraction ranges As mentioned above, traditional thermal conductivity models based on classical effective medium theory, such as the Maxwell [28] model, fail to describe the improvement of thermal conductivity in NFs. This is due to the fact that these models only consider the effects of nanoparticle concentration. However, it is known that the enhancement of NF thermal conductivity is dependent on many factors, such as the nanoparticle material, volume fraction, particle size and shape, base fluid, temperature, chemical additives, etc. Improved theoretical models that consider the effects of various heat conduction mechanisms (i.e., Brownian motion of nanoparticle, nanolayer, clustering, and the nature of heat transport in nanoparticles) have been suggested [29]. Nanoconvection is one of the commonly accepted mechanisms for enhanced heat transfer of NFs, which is caused by the nanoparticle Brownian motion. A novel model considering the effect of this

Nanofluids

1.3 Nanofluid properties

19

micromixing due to the Brownian motion with the conventional static contribution as determined by the Maxwell model was established by Kumar et al. [30]. They derived a moving particle model from the Stokes
Einstein formula to explain the temperature effect. Prasher et al. [31] determined that the local convection caused by the Brownian motion of the nanoparticles is primarily responsible for the irregular enhancement of NF thermal conductivity through an order-ofmagnitude analysis of various possible mechanisms. They also noted that the thermal conductivity for colloidal suspensions with large particle sizes can be explained by traditional conduction-based theories such as the Maxwell
Garnett model. Consequently, they suggested a method combining the Maxwell
Garnett conduction model and the Brownian motion
induced convection from multiple nanoparticles called the multisphere Brownian model (MSBM). 1.3.1.1 Models for the thermal conductivity of colloidal suspensions By collecting the volume fraction data from the available theoretical models in the literature for the thermal conductivity of NFs, it can be seen that the validity of most existing models has to be limited to NFs with volume fractions larger than 0.1%. Summaries of some of the models for thermal conductivity of NFs and their volume fraction ranges are listed in Table 1.3 [29]. 1.3.1.2 Multisphere Brownian model The MSBM is one of the most reliable and commonly used models for predicting the thermal conductivity of NFs [34,35]. After investigating several available heat conduction mechanisms and related models, Prasher et al. [31] determined that the local convection heat transfer in the liquid due to the Brownian motion of the particles is the main contributor to the enhancement of the thermal conductivity in NFs. They suggested the effective thermal conductivity of the semiinfinite area based on the Brownian
Reynolds number as a part of the modified Maxwell
Garnett model. In addition, this model considers the effect of interfacial thermal resistance between nanoparticles and different base fluids. Considering all the abovementioned factors at the same time, the multisphere Brownian model can be written as:   !  kp ð1 1 2αÞ 1 2km 1 2ϕ kp ð1 2 αÞ 2 km kn
m 0:333  5 1 1 ARe Pr φ  (1.38) kf kp ð1 1 2αÞ 1 2km 2 ϕ kp ð1 2 αÞ 2 km where Re 5 1/ν f(18kbT/πρpdp)0.5, ν f is the kinematic viscosity of the liquid, α 5 2Rbkm/dp is the nanoparticle Biot number, Rb is the selective

Nanofluids

TABLE 1.3

Typical models for thermal conductivity of nanofluids proposed in the literature [29].

References Prasher et al. [31]

Models 0  1  kp ð1 1 2αÞ 1 2km 1 2ϕ kp ð1 2 αÞ 2 km kn
m 0:333 @ A   5 1 1 ARe Pr ϕ kf kp ð1 1 2αÞ 1 2km 2 ϕ kp ð1 2 αÞ 2 km vffiffiffiffiffiffiffiffiffiffiffiffi u 
 2Rb km 1 u18kb T ; km 5 kf 1 1 1=4 Re Pr ; Re 5 t α5 ν f πρp dp dp

Volume fraction

Eq.

ϕ . 1%

(1.33)

ϕ 5 1% and 4%

(1.34)

0:1% # ϕ # 3%

(1.35)

Al2 O3 : 1% # ϕ # 10% ZnO: 1% # ϕ # 7% CuO: 1% # ϕ # 6%

(1.36)

0:2% # ϕ # 9%

(1.37)



Chon et al. [32]

kn 5 1 1 64:7ϕ0:7460 kf Pr 5

Patel et al. [33] Vajjha and Das [34]

μf ρf αf

; Re 5

!0:3690 df dp

!0:7476 kp kf

Pr0:9955 Re1:2321

ρf kb T 3πμf 2 lf

 0:273  0:547  0:234 kp kn T 100 5 1 1 0:135 ϕ0:467 20 dp kf kf 2 # " vffiffiffiffiffiffiffiffiffi
3 uk T kp 1 2kf 2 2ϕ kf 2 kp u b 4 4 5
 f ðT; ϕÞ kf 1 5 3 10 βϕρf cpf t kn 5 ρp dp kp 1 2kf 1 ϕ kf 2 kp 1 T f ðT; ϕÞ 5 2:8217 3 10 ϕ 1 3:917 3 10 @ A T0
 1 23:0669 3 1022 ϕ 2 3:91123 3 1023


22

23



0

β 5 8:4407ð100ϕÞ21:07304 for Al2 O3 β 5 8:4407ð100ϕÞ21:07304 for ZnO Corcione [35]

β 5 9:881ð100ϕÞ20:9446 for CuO !10 !0:03 kn kp 0:4 0:66 T 5 1 1 4:4Re Pr ϕ0:66 kf Tfr kf Re 5

2ρf kb T πμ2f dp

1.3 Nanofluid properties

21

FIGURE 1.11 Schematic of the interaction among nanoparticles in colloid suspension of low volume fraction (A) and high volume fraction (B) [29].

interfacial thermal resistance, km is the matrix thermal conductivity, and A and m are empirical constants determined by experimental data. The coefficient A is independent of the fluid type and equals 40,000, while the value of the exponent m depends on the type of the base fluid. For water-based NFs and assuming Rb to be 0.77 3 1028 K m2 W21, m will have a value of 2.5% 6 15% by comparison with the experimental values. 1.3.1.3 Modification of the multisphere Brownian model In a stable suspension, Brownian nanoparticles move accidentally and thus will convey the large surrounding liquid to near places. As reported by Prasher et al. [31] the thickness of the thermal boundary layer will exceed the interparticle distance when the volume fraction is larger than 0.0055% at room temperature. This means that even at very small nanoparticle concentrations, convection currents due to multiple particles will interact with each other. The existence of φ in Eq. (1.38) indicates strong interaction between the convection currents from various spheres [31]. However, as shown in Fig. 1.11, the intensity of this interaction will be completely different when the volume fraction of the NFs changes. Obviously, the interaction among nanoparticles will be much weaker in colloid suspension of low volume fraction in which the average interparticle distance increases. Also, there are exponential coefficients for the volume fraction term in the mathematical expressions of some suggested models as shown in Table 1.3. As can be seen, nanoparticle volume fraction is considered to be a crucial factor in these models. Furthermore, it is tried to modify the exponential part of the item for volume fraction φ in MSBM. Assuming A has the same value of 40,000 as the original model, a modified multisphere Brownian model is proposed by:   !  kp ð1 1 2αÞ 1 2km 1 2ϕ kp ð1 2 αÞ 2 km kn
m 0:333 n   5 1 1 40; 000Re Pr φ kf kp ð1 1 2αÞ 1 2km 2 ϕ kp ð1 2 αÞ 2 km (1.39)

Nanofluids

22

1. Introduction to nanofluids

where the m and n parameters are modifiable empirical constants determined by experimental data. The next crucial step for utilization of the modified MSBM is the suitable determination of m and n parameters and their valid application ranges. Prasher et al. [31] concluded that m 5 2.5% 6 15% by comparing the MSBM with experimental data for water-based NFs. For the modified MSBM, it is also expected that the deviation in m is centered on m  2.5, so the major emphasis is on the determination of the n values. Validation of this assumption can be found in the next section. 1.3.1.4 Establishment of database with various experimental results To find a widely applicable model, typical data for water-based NFs containing alumina, titania, and copper oxide nanoparticles of volume fractions ranging from 0.01% to 10% were considered for our experiment: that is, Al2O3 (0.01%
0.03%) from Lee et al. [36], TiO2 (0.2%
1%) from Reddy and Rao [37], CuO (1%
3.5%) from Kazemibeydokhti et al. [38], and CuO (5%
7%) and Al2O3 (5%
10%) from Mintsa et al. [39]. Due to the lack of data available on the thermal conductivity of NFs at low particle volume fractions (,0.1%), we obtained the necessary data by experiment for two typical NFs [i.e., Al2O3 (0.001%
0.005%) and TiO2 (0.008%
0.1%)]. The thermal conductivity of NFs was measured using the laser flash
based thermal analyzer LFA467 HyperFlash from NETZSCH instruments. 1.3.1.5 Thermal conductivity measured in our experiments In Fig. 1.12, the measured results of the thermal conductivities of water-based Al2O3 (30 nm) and TiO2 (21 nm) NFs are presented with the low volume fractions. The effects of volume fraction and temperature on the thermal conductivity of NFs were also investigated simultaneously. Clearly, the thermal conductivity will increase with volume fraction increment. Fig. 1.12A shows a nearly linear increase of the thermal conductivity with an increasing volume fraction of nanoparticles. Likewise, the thermal conductivity also increases with the temperature. 1.3.1.6 Determining the parameters for the modified model Table 1.4 gives the best-fit values of m and n, from which m is also found to be 2.5% 6 15%, while n is dependent on the volume fraction. Fig. 1.13 represents a comparison between the modified MSBM and experimental outcomes from the literature. It indicates good agreement between the thermal conductivity ratio predicted by the modified model and the available experimental data. To find the validity of our proposed modified model, it was applied to the measured data obtained for kn under different temperatures. Fig. 1.14 (for TiO2
water NFs)

Nanofluids

23

1.3 Nanofluid properties

(A)

Thermal conductivity (W m–1K–1)

0.650 Al2O3 /H 2O nanofluid

0.648

0.646

0.644

0.642

0.640 1

2

3

4

5

3

Volume fraction ×10 (%)

Thermal conductivity (W m–1K–1)

(B) 0.67 TiO2 /H2O nanofluid

0.66 0.65 0.64 0.63 0.62 0.61 0.60 20

30

40

50

60

Temperature (ºC)

FIGURE 1.12 Thermal conductivities of (A) Al2O3/H2O nanofluid against volume fraction at temperature 50 C and (B) TiO2/H2O NF against temperature with volume fraction of 0.008% [29].

shows that the modified model matches very well with the data by assuming m 5 2.5 and n 5 0.725 when φ 5 0.008%, whereas the original MSBM significantly underpredicts the thermal conductivity of NFs when its particle volume fraction is quite low (i.e., φ 5 0.008%).

Nanofluids

24

1. Introduction to nanofluids

TABLE 1.4 Best-fit values of m and n for various reference data [29]. Type of particles

References

Volume fraction

Temperature 

m

n

Present work

Al2O3—30 nm

0.001%
0.005%

50 C

2.45

0.7

Present work

TiO2—21 nm

0.008%
0.1%

20 C
60 C

2.5

0.725
0.86

Lee et al. [36]

Al2O3—30 nm

0.01%
0.3%

21 C

2.5

0.75
0.89

Reddy and Rao [37]

TiO2—21 nm

0.2%
1%

30 C

2.5

0.89
1

Chon et al. [32]

Al2O3—47 nm

1%

20 C
60 C

2.415

1

Kazemibeydokhti et al. [38]

CuO—23 nm

1%
3.5%

27 C

2.43

1

Mintsa et al. [39]

CuO—29 nm

5%
7%

21 C
23 C

2.65

1

2.85

1

Mintsa et al. [39]

Al2O3—47 nm

5%
10%





21 C
23 C

Lee et al. [36] Al 2 O3 , 30 nm Reddy and Rao [37] TiO2 , 21 nm

Thermal conductivity ratio from database (kn/k f )

1.25

Chon et al. [32] Al 2 O3 , 47 nm Kazemibeydokhti et al. [38] CuO , 23 nm Mintsa et al. [39] Al 2 O3 , 47 nm Mintsa et al. CuO , 29 nm Present work , Al 2 O3 , 30 nm

1.20

Present work , TiO2 , 21 nm ——

Line of slope = 1

1.15

1.10

1.05

1.00 1.00

1.05

1.10

1.15

1.20

1.25

Thermal conductivity ratio predicted by our model (kn /k f ) FIGURE 1.13 Comparison of our modified model with the experimental results; the corresponding values of m and n are given in Table 1.4 [29].

Nanofluids

25

1.4 Benefits and applications

1.020

Thermal conductivity ratio (kn /k f )

φ = 0.008% TiO2 /water (data) φ = 0.008% TiO2 /water (modified MSBM) φ = 0.008% TiO2 /water (original MSBM)

1.015

1.010

1.005

1.000 20

30

40

50

60

Temperature (ºC)

FIGURE 1.14 Comparison of predicted data by original and modified MSBM with the experimental data over TiO2
water suspensions [29]. MSBM, Multisphere Brownian model.

1.3.1.7 Dependency of the n values on the volume fractions Fig. 1.15 shows the dependence of the n values on the volume fractions of NFs. The appropriate values of n were determined under different volume fractions by comparison of the modified MSBM with various experimental results. It was concluded that n has a value of unity when the volume fraction is larger than 1%. This means that the modified MSBM can be changed to the original model for high nanoparticle concentrations. Furthermore, n approximately equals 0.7 when the volume fractions of NFs are lower than 0.005%. Between these two volume fractions, n is observed to have a nearly linear relation with the logarithm of volume fraction. This dependence relationship can be described by a brief mathematical formula as follows: 8 for φ . 1% > < 1:0 n 5 0:13 3 logðφÞ 1 1:26 for φ 5 0:005%
1% (1.40) > : 0:7 for φ , 0:005%

1.4 Benefits and applications Most of the applications of NFs are covered in Chapter 6: Nanofluids analysis in different applications. In this section, we look at the

Nanofluids

26

1. Introduction to nanofluids

Lee et al. [36] Al 2 O3 , 30 nm Reddy and Rao [37] TiO2 , 21 nm

1.1

Chon et al. [32] Al 2 O3 , 47 nm Kazemibeydokhti et al. [38] CuO , 23 nm Mintsa et al. [39] Al 2 O3 , 47 nm Mintsa et al. CuO , 29 nm Present work , Al 2 O3 , 30 nm

1.0

Present work , TiO 2 , 21 nm

n

0.9

0.8

0.7

0.6 1E–3

0.01

0.1

1

10

Volume fraction (%) FIGURE 1.15

Dependence of the n values on the volume fractions of various nano-

fluids [29].

characteristic oscillation phenomenon after head-on collision of two NF droplets to show the benefits and applications of NFs [40]. These experiments aimed to study the oscillation characteristic of NF droplets by looking at one drop fall from height and impact with low velocity onto another drop on a fixed platform. Since drops undergo spreading and recoiling after impact, the images of the drops corresponding to different shape changes were recorded and then analyzed with digitization. The experimental setup used to study the drop impact consists of three subsystems, as shown in Fig. 1.16: 1. drops generation subsystem (syringe pump, injector, and needles with different sizes); 2. supporting syringe pump and platform subsystem (adjustable height for the drop falling); and 3. high-speed photographic optical subsystem. A single drop is shaped from a needle attached to a syringe pump. After growth, it separates from the needle due to its gravity. The impact velocity of the drop depends on the height of the platform. Then the drop falls vertically onto a sessile droplet on a horizontal platform.

Nanofluids

27

1.4 Benefits and applications

(A)

(B) D H

v Vertical direction

Horizontal direction

FIGURE 1.16 (A) Experimental setup for studying the oscillation characteristic when two droplet collide head-on. (B) Illustration of the oscillation process [40].

The platform is made of polytef in order to guarantee the high contact angle of the drop on the platform. By measuring with a TR200 roughness meter, the Ra value (absolute surface roughness) was determined to be 0.253 μm. To avoid affecting vibration, the whole apparatus was placed on a vibration isolation table. As mentioned above, in the third subsystem the high-speed video camera for continuously capturing images of the drop as it impacts, spreads, recoils, and rebounds on the substrate is essential equipment for this research. The camera system can record at a framing rate up to 20,000 full images per second, but we chose 250 images per second, which was enough to meet our research needs. Images of drops subsequent to oscillation were analyzed using the MATLAB software to obtain the shapes of the droplets quantitatively. After analyzing, we obtained the drop size change, impact velocity, and oscillation frequency. As presented in Fig. 1.16B, D is the spread radius, H is the height of the drop rebound, and t refers to time measured from the instant the drop impacts. TiO2
water NF with different nanoparticle mass fraction was used for the droplets, which was prepared with distilled water as the base fluid. In order to keep the NF suspension stable, sodium dodecyl benzene sulfonate surfactant was also added to the fluid. Also, to obtain homogeneous nanoparticle dispersion, ultrasonic irradiation was applied to it. The particle size and morphology of the samples were observed on TEM images from a FEI Tecnai G2 F30 transmission electron microscope at an accelerating voltage of 300 kV. Particle size distribution (PSD) evolution of TiO2 in the suspension was studied by dynamic light scattering using a Malvern laser particle analyzer (Spraytec 300, Malvern, United Kingdom). As mentioned above, three nanoparticle concentrations, 0.001%, 0.01%, and 0.1%, were used in our study and the images for the prepared NFs are shown in Fig. 1.17. As can be seen, the color of the fluid

Nanofluids

28

1. Introduction to nanofluids

0%

0.001% TiO2–water

0.01% TiO2–water

0.1% TiO2–water

FIGURE 1.17 Images for various TiO2
water nanofluids employed in this research [40].

Surface tensioin σ (mN m–1)

50

3.5

40 3.0 30

2.5

20

2.0

10

1.5

0%

0.001% 0.01% 0.1% Nanoparticle mass fraction

0.5%

Dynamic viscosity μ (mPa s–1)

4.0 Surface tension Dynamic viscosity

1.0

FIGURE 1.18 Surface tension and viscosity of TiO2
water nanofluid with various nanoparticle mass fraction [40].

turns from almost transparent to completely opaque or milky when the loading amount of TiO2 increases from 0% to 0.1%. Fig. 1.18 shows the experimentally measured surface tension σ and dynamic viscosity μ of NFs for different mass fractions. It can be observed that the very low amount of 0.001% TiO2 leads to a minor

Nanofluids

29

1.4 Benefits and applications

(A)

(B) 16 0.001% 0.01% 0.1%

14

Percent (%)

12 10 8 6 4 2 0 –2

0

200

400

600

800

Particle diameter (nm)

FIGURE 1.19 TEM image and nanoparticle diameter distribution of test NF. (A) TEM image of TiO2 nanoparticles; (B) nanoparticles diameter distribution in three different mass fraction NF [40]. NF, Nanofluid.

decrease of both surface tension and dynamic viscosity compared with pure water. Subsequently, with the increase in nanoparticle concentrations, the σ and μ will also increase. It must be noted that the increasing trend is not strictly linear. Abnormal fluctuation appears when the nanoparticle mass fraction is around 0.1% (see Fig. 1.18). However, it is interesting that the changing trends for both surface tension and dynamic viscosity are identical. As is known, the surface tension of a surfactant solution is determined by the physicochemical nature of the surfactant molecules, which both have hydrophilic and hydrophobic groups and consequently have a tendency to stay at the liquid/air interface. Actually, the surfactant molecules form the monomolecular film in the liquid surface and then decrease or increase the interfacial surface tension. It was found that the dynamic viscosity indicates the internal friction between the liquid molecules. In our case, the concentration of surfactant already guaranteed the formation of micelles and the layer. Therefore it is assumed that TiO2 addition could also change the interactions at the liquid surface and lead to fluctuation of the surface tension and viscosity. However, more experiments are required to confirm these assumptions. The initial kinetic energy (inertial force) and the surface energy (surface tension force) play the main roles in finding the dynamics of drop impact and spreading on a substrate in low-speed spreading experiments. The initial kinetic energy is mainly obtained by the impact velocity U and the size, or drop radius Dd of the falling drop, while the surface energy is related to the surface tension σ. A TEM image of a TiO2 nanoparticle which can be considered as an indication of nanoparticle aggregation when they are dispersed in water is shown in Fig. 1.19A. For additional validation of the occurrence of nanoparticle aggregation, the dynamic light scattering method was

Nanofluids

30

1. Introduction to nanofluids

t = 0 ms

t = 4 ms

t = 8 ms

t = 16 ms

t = 20 ms

t = 24 ms

t = 12 ms

t = 28 ms

FIGURE 1.20

The shape change of drop as a function of time (nanofluid mass fraction is 0.1%, U 5 0.6 m s21, initial droplet diameter is 2.4 mm) [40].

employed as depicted in Fig. 1.19B. As can be seen, the maximum nanoparticle diameter will reach 800 nm, with an initial size of 21 nm before being dispersed in water. Fig. 1.20 shows the evolution process of droplet shape after one drop impacts a sessile drop. As can be seen, after head-on collision and merging of the two drops, a new drop forms that then experiences spreading, recoiling, and rebounding. As the impacting velocity in our study was only 0.5
1.5 m s21, droplets did not break up during impact because their kinetic energy was too low to overcome surface tension. A droplet at maximum velocity U 5 1.5 m s21, a droplet diameter of 2.8 mm, and nanoparticle mass fraction of 0.001% were chosen to calculate the kinetic energy, the energy of surface tension, and the viscidity of the dissipated work, respectively. Fig. 1.21 shows the history of droplet shape change after one drop impacts another sessile drop. In this case, the diameter of the droplet is 2.4 mm and the dropping velocity for one of them is U 5 0.6 m s21, and for the other one is U 5 1.5 m s21. Fig. 1.21 shows the deformation, expansion, and contraction of the drop upon impact in dimensionless terms as normalized drop radius, D/Dd, and normalized height, H/Dd, as a function of time measured from the instant of impact. Right after two droplets coalesce, the initial kinetic energy of the falling drop will be transferred to the sessile drop, then a newly merged droplet will be formed and oscillating with time in both the horizontal and vertical directions simultaneously. The drop will undergo damped oscillations on the platform. By comparing Fig. 1.21A, C, E, and G, and B, D, F, and H, it can be observed that the vibration amplitude of the drop in the

Nanofluids

(B) 1.4

(A)

4.5

φ = 0%

φ = 0%

U = 0.6 m s–1 U = 1.5 m s–1

1.2

U = 1.5 m s–1 U = 0.6 m s–1

4.0

1.0

H/Dd

D/Dd

3.5 3.0 2.5

0.8 0.6 0.4

2.0

0.2 0

40

80

120

160

200

240

280

0

40

80

Time (ms)

120

160

200

240

280

Time (ms)

(D)

(C)

1.4

4.5

φ = 0.001%

φ = 0.001% U = 1.5 m s–1 U = 0.6 m s–1

D/Dd

3.5

U = 0.6 m s–1 U = 1.5 m s–1

1.2 1.0

H/Dd

4.0

0.8 0.6

3.0

0.4 0.2

2.5 0

40

80

120

160

200

240

280

0

40

80

Time (ms)

(E)4.5

120

160

200

240

280

Time (ms)

(F) φ = 0.01% U = 1.5 m s–1 U = 0.6 m s–1

4.0

φ = 0.01%

2.4

U = 0.6 m s–1 U = 1.5 m s–1

2.0

3.5

D/Dd

1.6

H/Dd

3.0

1.2

2.5 0.8

2.0 1.5

0.4

0

40

80

120

160

200

240

(G)4.5

0.0

280

Time (ms)

40

80

120

160

200

240

280

Time (ms) φ = 0.1%

0.8

φ = 0.1% U = 1.5 m s–1 U = 0.6 m s–1

4.0

0

(H)

U = 0.6 m s–1 U = 1.5 m s–1

0.7 0.6

H/Dd

D/Dd

3.5 3.0 2.5

0.5 0.4 0.3 0.2

2.0 0

40

80

120

160

Time (ms)

200

240

280

0.1

0

40

80

120

160

200

240

280

Time (ms)

FIGURE 1.21 Time courses for the shape evolution of TiO2
water nanofluid drops of various nanoparticle loadings impacting and spreading on a PTEF surface, Dd 5 2.4 mm, U 5 0.6, and 1.5 m s21, 0.05% SDBS. Dimensionless radius of spreading drops, D/Dd; dimensionless height of spreading drops, H/Dd [40]. PTFE, Polytetrafluoroethylene; SDBS, sodium dodecyl benzene sulfonate surfactant.

32

1. Introduction to nanofluids

FIGURE 1.22 The image of nanoparticle moving to the triple line obtained by a chatelier-type microscope [40].

vertical direction is stronger and clearer than that in the horizontal direction. Here, we assume that the locking effect will dominate at the interface between the substrate and drop. This effect is expected to restrict the motion of the drop on the radius direction [41]. In our experiments, it was also detected that after the droplet reached its maximum radius, the contact line remain fixed throughout the period of oscillation, while the contact angle changed in a range, as shown in Fig. 1.20. Fig. 1.21B, D, F, and H also confirms that the impact velocity has a substantial influence on droplet height vibration. Hong et al. [42] found that water-based Fe3O4 ferrofluids have non-Newtonian behavior. Clearly, the rheological behavior is also important in droplet oscillation. We examined the rheological properties of NFs and nanoparticle-free fluids in these cases. Basaran and DePaoli [43] noted that the damping frequency and rate of drop oscillation is a significant function of drop size, so that the frequency of oscillation rises as drop size falls. Fig. 1.22 shows the TiO2
water NFs droplet image with 0.001% mass fraction obtained by a chatelier-type microscope. The triple-phase contact line, which is formed on a solid surface by two mutually dissoluble fluids, is important for studying droplet dynamics. As can be seen from Fig. 1.22, the nanoparticle will move to the triple-phase contact line from the indicated circle after droplet contact with the solid surface. Moreover, it can be observed that nanoparticles cluster at the edge of the droplet, and from the rim to center, the nanoparticles become sparse [41]. With low-impact velocity, the surfactant has sufficient time to restructure on the surface, resulting in uniform spreading and with reduced uniform surface tension. Thus in accordance with that at lowimpact velocities, the lower surface tension causes the larger maximum spreading radius of the drop. With the impact velocity increase, we found that two main effects will influence the drop maximum spreading radius. First, based on the Gibbs
Marangoni effect, the surface tension

Nanofluids

1.5 Other forces on nanoparticles in base fluid

33

FIGURE 1.23 Schematic illustration of the characteristic oscillation phenomenon after head-on collision of two nanofluid droplets (A) before impacting, (B) process of spreading, and (C) process of recoiling [40].

of fluid with more surfactant molecules should be less than that with fewer surfactant molecules. In these cases, it is considered that there also exists a Marangoni effect. As shown before, after droplet impact, the surfactant molecules will move to the vicinity of the droplet contact line due to inertness, and consequently the surfactant concentration at the droplet edge will be higher than in the droplet center. Accordingly, the surface tension near the center of the drop will be larger than that near the contact line. Therefore this gradient of surface tension will inhibit the spreading of the drop. Second, the effect of particle inertia is enhanced with high velocity and particle concentration. This described effect will opposes the inhibition of surface tension gradient. Thus there may be a velocity value that gives rise to a stronger effect on particle inertia than the surface tension gradient. As a result, the maximum drop spreading radius (at high particle concentration) will be greater than low concentration under high velocity. Fig. 1.23 shows a schematic for this mechanism, illustrating nanoparticle drop shape evolution when there is a surfactant addition [41].

1.5 Other forces on nanoparticles in base fluid There are other forces applied on nanoparticles in base fluid. Drag force (Fd), gravity force (Fg), Brownian force (Fb), force due to

Nanofluids

34

1. Introduction to nanofluids

FIGURE 1.24 Physical model for the wavy-wavy cavity (A) and the quadrilateral mesh for the calculated domain when γ 5 1.77 rad and A 5 0.11 (B) [44].

thermophoresis effect (FT), force due to lift motion (Fv), force due to virtual mass (Fl), and force due to pressure gradient (Fp) per unit mass are the main forces on nanoparticles, which are discussed in the following two cases.

1.5.1 Case 1: Natural convection heat transfer in an NF-filled cavity with double sinusoidal wavy walls This case is based on the earlier study of Sheikholeslami et al. [45]. As schematically presented in Fig. 1.24, a 2D quarter circular cavity with two wavy walls was modeled to study the natural convection heat transfer process using four different kinds of NFs. The quadrilateral dominant meshes used in the FVM program are presented in Fig. 1.24. The inner and outer walls were maintained at constant temperatures Th and Tc, respectively. The other two boundaries were considered to be adiabatic. The wavy shape of the inner and outer walls were assumed to be the following cosine functions [44,45]: r 5 rin 1 A cos ðN ðζ ÞÞ

(1.41)

r 5 rout 1 A cos ðN ðζ ÞÞ

(1.42)

where rin and rout are the circle radii of the inner and outer walls; A and N are the amplitude and number of undulations, respectively; and ζ is the rotation angle. In this case, for simplification, it is assumed that A and N of the inner wall are 0.3 and 9, respectively, based on our previous optimization study [46]. The number of boundary layers is five and the maximum cell size is 0.01. There are nine meshes created by

Nanofluids

1.5 Other forces on nanoparticles in base fluid

35

automethod and it is assumed that the mesh quality is beyond 0.5, which means they meet the needs of our calculations. The wavy-wall cavity in this case is based on and transformed from a quarter of a cyclic annular cavity with a 9 μm inner radius and a 20 μm outer radius. The nanoparticle was tracked using the Lagrangian model and the governing equations are calculated under the Eulerian frame. The volume fraction parameters were calculated under an implicit scheme. Liquid water was used as the base fluid and various kinds of nanoparticles (Ag, CuO, Al2O3, and TiO2 with thermophysical properties in Table 1.1) with averaged diameter of 20 nm were added to it. The pressure
velocity coupling equation used the phase-coupled semiimplicit method for pressure-linked equations (SIMPLE) scheme. The solution procedure showed that all the cases converge within 1000 steps approximately. Thermophoretic and Brownian forces are not considered here, but the equations for laminar natural convection under the Eulerian model are: Continuity equation: @ρ 1 r ðρVÞ 5 0 @t



(1.43)

Momentum equation: @ ðρ vl Þ 1 r ðρl vl vl Þ 5 2 rp 1 r τ l 1 ρl g 2 Sp @t l





where Sp can be obtained by: Sp 5 Energy equation:

X

Fmp Δt

(1.45)

  @τ 1 vl rτ 5 r ðkrτ Þ ρl c @t



(1.44)



(1.46)

where τ 1 can be obtained by:  2 τ l 5 μl rvl 1 rvTl 2 μl r vl I 3



(1.47)

Eqs. (1.43)
(1.47) describe the movement of continuous water phase. ρl is the fluid density, vl is the velocity, t is the time, p is the pressure, g is the gravitational acceleration, c is the heat capacity, k is the thermal conductivity, and T is the fluid temperature, Here, τ is the stress tensor, F is the total force acting on a particle, I is the unit vector, and mp is the mass of the nanoparticle.

Nanofluids

36

1. Introduction to nanofluids

In the Lagrangian frame of reference, the equation of motion of a nanoparticle is the following equation based on the total force applied on the nanoparticles: dvp 5F dt

(1.48)

F 5 FD 1 FG 1 FL 1 FP 1 FV

(1.49)

where FD is the hydrodynamic drag force from the fluid, which can be calculated by applying the Stokes’ law [47]: FD 5 6πμl ri ðui 2 up Þ

(1.50)

FG is the force due to gravity: FG 5

gðρp 2 ρl Þ ρp

(1.51)

The Staffman’s lift force, FL, was derived by Staffman as [48]: FL 5

2KS v1=2 ρl dij ρp dp ðdlk dkl Þ1=4

ðvl 2 vp Þ

(1.52)

where KS 5 2.594 is a constant, dij is the deformation tensor defined as dij 5 1/2(vli,j 1 vlj,i), and FP is the force due to the gravity gradient: ! ρl (1.53) FP 5 vp rvl ρp



FV is the virtual mass force: FV 5

1 ρl d ðvl 2 vp Þ 2 ρp dt

(1.54)

where h is the surface heat transfer coefficient calculated by Ansys
Fluent 15.0. The following assumptions were used to simplify the problem. 1. The working fluid is considered as incompressible laminar flow across a cavity and maintains the condition of single phase. 2. The thermophysical properties of the nanoparticles are constant. 3. The two straight surfaces of the cavity are well insulated. 4. The diameter of each nanoparticle is uniform. After this simplification, commercial CFD Ansys
Fluent 15.0 software based on the FVM was used to obtain the numerical solution. The

Nanofluids

1.5 Other forces on nanoparticles in base fluid

37

computational domain and created mesh were produced using the commercial preprocessing code ICEM CFD 15.0. As mentioned before, a Lagrangian
Eulerian approach was used here. The nanoparticle was tracked using the Lagrangian approach while the governing equations for the base fluid were solved using the Eulerian approach. It is worth noting here that Bianco et al. [49] compared the accuracy of the single-phase model and Lagrangian
Eulerian approach in a laminar flow with water
Al2O3. They reported that the results from both models were similar. In another study, Kumar and Puranik [50] studied the convective heat transfer of NFs in turbulent flow using a Lagrangian
Eulerian approach. They compared the single-phase model with the Lagrangian
Eulerian model and determined that the Lagrangian
Eulerian approach was a more accurate model for simulating forced convection heat transfer when the NF has a particle volume fraction less than 0.5%. Since the nanoparticle concentrations in all of our examined cases are under 0.5%, the Lagrangian
Eulerian approach is used here to model the heat transfer process. The pressure terms in the governing equations were discretized using the second-order implicit term, whereas the momentum and energy equations were discretized using the second-order upwind scheme. The pressure
velocity coupling was implemented following the SIMPLE algorithm. Moreover, the absolute convergence criterion was set to be 1025 for the continuity equation and velocity equation while it was 1026 for the energy equation. The Boussinesq approximation was used for both water and nanoparticles to simulate natural convection. Pure water was added to the cavity first where it was under laminar natural convection and then the nanoparticles were added to the water and the Eulerian model was used to simulate the two-phase flow. The evolution of the streamline when the volume fraction of Ag rises from 0% to 0.9% is depicted in Fig. 1.25. This figure demonstrates that by increasing the volume fraction, the high mass flow rate region of the NF decreases as well as the low mass flow rate region. In the interior region, the low mass flow region (which is blue in Fig. 1.25) becomes smaller and the underpart of this blue region gradually disappears. The velocity tends to become uniform because when the Ag volume fraction rises from 0% to 0.9%, the viscosity of NFs increases, so the velocity wave conveys slower. The surface heat transfer coefficient (h) of Ag NF when the volume fraction increases from 0% to 0.9% is shown in Table 1.5. It is obvious that it increases quickly with increasing volume fraction and the increased speed is higher with increased volume fraction. Fig. 1.26 shows an obvious increase of the outer wall Nusselt number compared to a circular wavy enclosure, thus indicating that considering the circular outer wall to be sinusoidal is meaningful from a heat transfer viewpoint.

Nanofluids

38

1. Introduction to nanofluids

FIGURE 1.25 Streamline evolution of water phase with Ag nanofluid s at A 5 0.1, γ 5 0 for (A) ϕ 5 0%, (B) ϕ 5 0.1%, (C) ϕ 5 0.3%, (D) ϕ 5 0.5%, (E) ϕ 5 0.7%, and (F) ϕ 5 0.9% [44]. TABLE 1.5 Absolute surface heat transfer coefficient of outer wall of Ag nanofluids at A 5 0.1, γ 5 0 for ϕ 5 0%, ϕ 5 0.1%, ϕ 5 0.3%, ϕ 5 0.5%, ϕ 5 0.7%, and ϕ 5 0.9%. 22

have (W m

21

K )

ϕ50

ϕ 5 0.1%

ϕ 5 0.3%

ϕ 5 0.5%

ϕ 5 0.7%

ϕ 5 0.9%

38,811.5

41,101.11

44,588.64

50,133.55

55,852.2

64,291.73

1.5.2 Case 2: The effects of nanoparticle aggregation on convection heat transfer investigated using a combined NDDM and DPM method In this case the effect of nanoparticle heat transfer (including the forces applied on the nanoparticles) was investigated using a combined nanoparticle diameter distribution model (NDDM) and DPM method. First, we defined the NDDM and DPM methods [51]. 1.5.2.1 Nanoparticle diameter distribution model Based on our previous study [52], the process of nanoparticle aggregation includes two main steps, coagulation and fragmentation, and this process can be defined by the following population-balance equation:

Nanofluids

39

1.5 Other forces on nanoparticles in base fluid

22

CWE WWE

20

Local Nusselt number

18 16 14 12 10 8 6 4 0

10

20

30

40

50

60

70

80

90

ζ

FIGURE 1.26

Comparison of CWE with WWE at different angles [44]. CWE, Circular

wavy enclosure. i22 i21 X dNi X 1 2 5 2j2i11 β i21;j Ni21 Nj 1 β i21;i21 Ni21 2 Ni 2j2i β i;j Nj 2 dt j51 j51

2 Ni

imax 21 X

imax X β i;j Nj 2 Si Ni 1 Γ i;j Sj Nj

j51

(1.55)

j51

where Ni is the number concentration of flocs with volume Vi. The β i,j is the collision rate of particles. It is assumed that the collision is mainly produced by the shear deformation of the fluid as well as the Brownian motion, so β i,j can be considered as: rffiffiffiffiffiffiffiffiffiffiffiffiffi  1 1 3 8πkB T 2 Df Df σ β i;j 5 0:31Gvp xi 1xj 1 (1.56) m where G is the shear rate, vp is the volume of primary particle, Df is the fractal dimension, xi is the number of primary particles in an aggregate in section i, m is the effective mass of particles, kB is the Boltzmann constant, and σ is the equivalent diameter. The rate of particle breakage is called the “fragmentation kernel Si,” which is dependent on the shear rate and aggregate size as follows: 
 q  3=Df 1 η φtot G dc;i Si 5 kb v3p (1.57) τ dp

Nanofluids

40

1. Introduction to nanofluids

where η is a variety of correlations between the particle volume fraction and suspension viscosity and can be calculated by: η0 ηðΦtot Þ 5
(1.58) 2 12Φtot =φm where η0 is the viscosity of basic fluid of dilute solution and Φtot is the effective of total volume concentration of the aggregates: imax X

Φtot 5

Ni vc;i

(1.59)

i51

where vc,j is the collision volume of aggregate. Also, the fragment distribution function Гi,j in Eq. (1.55) has been assumed to be [51,52]:

2 j5i11 (1.60) Γi;j 5 0 else

1.5.2.1.1 Discrete phase model

As mentioned, in the DPM, the Lagrangian approach was used for nanoparticle tracking, while the Eulerian approach was applied on the governing equations for the base fluid solution. The coupling between the two approaches was reached by adding the interphase momentum and energy exchanges as source terms in the suitable governing equations. The governing equations are as follows: Continuity equation:
 @ρb 1 r ρb Vb 5 0 @t

(1.61)


 @ρb vb 1 r ρb vb vb 5 2 rp 1 r Tb 2 Sp 1 ρb g @t

(1.62)



Momentum equation:



Energy equation:



  @T 1 vb rT 5 r ½krT  ρb c @t





(1.63)

In Eqs. (1.61)
(1.63), ρb is the base fluid density, vb is the base fluid velocity, t is the time, p is the pressure, g is the acceleration of gravity, c is the heat capacity, k is the thermal conductivity, and T is the fluid temperature. αb in Eq. (1.64) is the stress tensor, which can be determined as:

Nanofluids

1.5 Other forces on nanoparticles in base fluid

41

 2 αb 5 μb rvb 1 rvTb 2 μb rvb I 3

(1.64)

where μb is the shear viscosity of the fluid phase and I is the unit vector. Sp in Eq. (1.65) is the source term in order to signify the momentum transfer between the base fluid and particle phases. Sp can be determined by computing the particle momentum variation as they pass through the control volume for the fluid phase, as shown by [51]: X Fmp rt (1.65) Sp 5 In Eq. (1.65), F is the total force acting on a particle in the base fluid and is determined to be the summation of drag force (Fd), gravity force (Fg), Brownian force (Fb), force due to thermophoresis effect (FT), force due to lift motion (Fv), force due to virtual mass (Fl), and force due to pressure gradient (Fp) per unit mass. Thus the relation can be written as [51]: F 5 Fd 1 Fg 1 Fb 1 FT 1 Fv 1 Fp 1 Fl

(1.66)

Using the Lagrangian frame of reference, the equation of motion for a nanoparticle is: dvp 5F dt

(1.67)

where vp is the particle velocity. The Stokes
Cunningham drag law is offered in Ansys
Fluent 14.5 for calculation of drag force per unit mass. It is given by: Fd 5

18μb d2p ρp Cc

(1.68)

where Cc is called the Cunningham correction factor and is defined as:

2λ 1:257 1 0:4e2ð1:1dp =2λÞ Cc 5 1 1 (1.69) dp where dp is the particle diameter and λ is the molecular mean-free path of fluid. The gravity force is also a significant force and can be obtained by:
 g ρd 2 ρb Fg 5 (1.70) ρp Because some researchers pointed out that the convective heat transfer coefficient in laminar flow is affected more by the thermal

Nanofluids

42

1. Introduction to nanofluids

conductivity than by the viscosity, the lift force plays a limited role in this process and can be neglected in the process as done in this case. The Brownian force Fb can be expressed as: rffiffiffiffiffiffiffiffi πS0 Fbi 5 ζ i (1.71) Δt where ζ i is the unit variance
independent Gaussian random number with zero mean and S0 is defined by: S0 5

216vkB T ρ 2 π2 ρb d5p ρp Cc

(1.72)

b

where kB is the Boltzmann constant and vb is the kinetic viscosity of fluid. The thermophoresis force, which acts on the small particles suspended in a fluid under temperature gradient, is called the thermophoretic force and can be given by: FT 5 2

6πdp μ2b Cs ðK 1 Ct KnÞ 1 r Τ ρb ð1 1 3Cm KnÞð1 1 2K 1 2Ct KnÞ mp T



(1.73)

where Cm 5 1.146, Cs 5 1.147, and Ct 5 2.18 are the momentum exchange coefficient, the thermal slip coefficient, and the temperature jump coefficient, respectively. Also, Kn is the Knudsen number. The pressure gradient produces another force acting on nanoparticles in the fluid and can be expressed as: ! ρb dvb (1.74) Fp 5 up ρp dx The heat exchange between the “discrete phase” and the “base fluid phase” can be calculated using the expression: mp cp


 dTp 5 hAp Tb 2 Tp dt

(1.75)

where Ap is the particle superficial area, Tb is the base fluid temperature, and Tp is the particle temperature. The last equation assumes that there is no heat transfer through the radiation process and the internal thermal resistance within the nanoparticle is insignificant. Finally, the heat transfer coefficient h can be considered from the Ranz and Marshall expression [51]: Nu 5

hdp ð1=3Þ 5 2 1 0:6Re0:5 p Pr Kb

Nanofluids

(1.76)

1.5 Other forces on nanoparticles in base fluid

43

Constant surface heat flux at wall

Pressure outlet

Velocity inlet

Nanofluids

Constant surface heat flux at wall

FIGURE 1.27 Three-dimensional physical model for the horizontal microchannel circular tube [51].

1.5.2.2 Problem description in second case and its solution A schematic of the computational setup is shown in Fig. 1.27. A tube with an inner diameter of 0.4 mm and length of 0.12 m was considered for the numerical simulation and 3D DPM were used for the solution. The defined boundary condition for this tube was as follows: the wall of the tube has a constant surface heat flux of 69.9 kw m22, the inlet condition is velocity inlet, and the outlet type is considered to be pressure outlet. All the nanoparticles were assumed to be spherical and their initial sizes were 60 nm. The inlet temperature of the fluid was set to 291 K and details for the DPM settings were as follows. For the nanoparticle injection “surface injection” was selected and nanoparticles were injected from the inlet surface. The mass flow rate of the particle was determined using the nanoparticle density, NF velocity, and volume fraction. The boundary condition for the inlet and outlet surfaces were all set as “Escape” for particles, while the tube wall was defined as “Reflect.” The nanoparticle temperature and velocity at the inlet were considered to be the same as those for the base fluid. Ansys
Fluent 15.0 based on the FVM was used to model and analyze this case. The preprocessing meshing software ICEM was applied for generating the grid. The SIMPLE algorithm was used to solve the governing equations. During the solution, the DPM was enabled when the base fluid medium achieved a converged solution. The option for interaction with the continuous phase was enabled along with the option for updating the continuous phase after each iteration in the DPM phase. The maximum number of steps was not constant for all runs, and was actually based on the particle velocity for particle tracking which causes that all the injected particles exit from the outlet, completely. The numerical solution was validated based on the available experimental outcomes of Zhang et al. [53] where Al2O3 NF was considered as the working fluid. The average values of the heat transfer coefficient number were used under the two different Re numbers as the evaluated index.

Nanofluids

44 (A)

1. Introduction to nanofluids (B)

0.35

φ = 0.5%

Re = 250 Re = 750

0.25 0.20 0.15 0.10 0.05

φ = 1%

0.35

Particle number fraction (Ni /Ntot )

Particle number fraction (Ni /Ntot )

0.30

Re = 500 Re = 1000

0.00

Re = 500 Re = 1000

0.30 0.25 0.20 0.15 0.10 0.05 0.00

1

10

100

1

10

Dimensionless f loc diameter (d i /dp )

100

Dimensionless f loc diameter (d i /dp )

(C) 0.35

(D) 0.30

φ = 2%

Re = 250 Re = 750

0.30

Re = 500 Re = 1000

Re = 500 0.25

Particle number fraction (Ni /Ntot )

Particle number fraction (Ni /Ntot )

Re = 250 Re = 750

0.25 0.20 0.15 0.10 0.05

ϕ = 0.5%

ϕ2

0.20

0.15

ϕ1 ϕ3

0.10

0.05

0.00

0.00 1

10

100

1

10

100

Dimensionless f loc diameter (d i /dp )

Dimensionless f loc diameter (d i /dp )

FIGURE 1.28 Particles diameter distribution for various Re numbers [51].

The particle size evolution and distribution were essentially the combined results of Brownian motion and shear deformation. As mentioned in the literature, Brownian motion plays a dominant role in particle coagulation procession in the early stage (i.e., when the particles are very small (below micrometer)). After the aggregated particle increases to a certain size as introduced by Eq. (1.56), shear deformation will dominate the evolution of the size distribution and the agglomeration rate will decrease. The PSD at the dynamic equilibrium state was demonstrated in Fig. 1.28. It can be seen that the particle distribution profile is like the normal distribution curve and thus it can be concluded that the decrease of shear rate will lead to particle size increase. In order to simplify the follow-up simulation of particle-laden flow, the particle number fraction was generally divided into three parts when dynamic balance was achieved. As the curve shape of the steady state particle diameter distribution was approximately a normal distribution, three sections (small, middle, and large particle diameter values) were chosen and the average particle diameter of each part was considered for calculation, as shown in Fig. 1.28D. A summary of the results is given in Table 1.6.

Nanofluids

45

1.5 Other forces on nanoparticles in base fluid

TABLE 1.6

Results of particle diameter distribution after agglomeration.

ϕ1

Average particle diameter (nm)

ϕ2

ϕ3

Average particle diameter (nm)

0.05%

179.54

0.43%

803.25

0.02%

2931.48

500

0.02%

100.41

0.14%

318.77

0.01%

2112.51

750

0.01%

76.95

0.48%

230.84

0.01%

1964.10

1000

0.03%

67.80

0.47%

159.39

0.01%

1714.50

250

0.01%

179.54

0.92%

923.36

0.06%

3212.99

500

0.06%

115.42

0.91%

401.62

0.03%

2280.04

750

0.03%

87.71

0.93%

253.01

0.04%

1964.10

1000

0.01%

67.80

0.98%

183.22

0.01%

1964.10

250

0.12%

133.22

1.82%

5065.01

0.06%

2469.80

500

0.14%

133.22

1.77%

506.01

0.09%

2469.80

750

0.09%

100.41

1.84%

318.77

0.07%

2112.51

1000

0.05%

79.65

1.87%

200.81

0.08%

1832.17

Initial particle (nm)

Volume fraction

Re number

60

0.50%

250

1%

2%

Average particle diameter (nm)

As is known, Reynolds numbers have a very significant effect on the heat transfer process. Fig. 1.29 shows comparisons of the heat transfer coefficients of Al2O3
water NFs with different nanoparticle volume fraction. As depicted in Fig. 1.29, it is clear that by adding Al2O3 nanoparticles to the water, its heat transfer coefficient can be improved and the enhancement is more significant with the increasing nanoparticle volume fractions. Furthermore, the heat transfer rate is improved as the Reynolds number increases. The main reason for the heat transfer improvements is due to the much higher thermal conductivity of the solid particles than the base liquid. As described before, after adding the nanoparticles, the properties of the base fluid will also change, resulting in enhanced heat transfer in the compound fluid interior. Furthermore, Brownian motion force or other forces acting on the nanoparticles in the fluid is another reason for heat transfer improvement, which could result in Brownian diffusion and heat diffusion. It is correct to note that in the current study, the h values of the base fluid (water) increased relatively slowly and had a tendency to level off at high Re numbers, but raise quickly for NFs. For example, from Re 250 to 1000, the heat transfer coefficient of 2% Al2O3
water NFs increases by 112.44%, which is larger than the increment of 96.23% for water. This may be due to the degree of microscale disturbance from the

Nanofluids

46

1. Introduction to nanofluids

11000 10000

Water 0.5% Al2 O3 1%

Al2 O3

2%

Al2 O3

h (w m–2 k–1)

9000 8000 7000 6000 5000 4000 3000 200

400

600

800

1000

Re FIGURE 1.29 Variations of the heat transfer coefficient with Reynolds for the Al2O3
water nanofluids with different nanoparticle volume fractions [51].

nanoparticles, which increases as Re increases and would be more significant when the Re number is higher [51]. Fig. 1.30 shows the nondimensional cross-sectional temperature distribution of the base fluid and NFs for various volume fractions at the outlet under Re 5 250. Comparisons of nondimensional temperature distribution before and after particle agglomeration are also presented in Fig. 1.30. In this case, with constant heat flux on the wall, nondimensional temperature can be considered as: T 5

T 2 Tin qw R=λ0



(1.77)

where R is the inner radius of the tube, qw is the heat flux, Tin is the inlet temperature of coolant, and λ0 is the NF thermal conductivity. A smooth parabolic profile of temperature distribution for base fluid can be found in Fig. 1.30, and temperature profiles after adding nanoparticles were similar to that of the base fluid. Also, it can be seen that by increasing the nanoparticle volume fractions, the nondimensional temperature numbers near the wall decreased. For instance, when the nanoparticle volume fraction increased from 1% to 2%, the maximum dimensionless temperature decreased from 0.85 to 0.65 before nanoparticle agglomeration. In the literature, Xuan and Li [54] also reported that the temperature gradient between the fluid and the wall increased and

Nanofluids

1.5 Other forces on nanoparticles in base fluid

47

FIGURE 1.30 Nondimensional temperature distribution of the base fluid and Al2O3 nanofluids at Re 5 250 (A) before agglomeration and (B) after agglomeration [51].

the heat transfer rate near the wall boundary can improved after adding nanoparticles to the base fluid. It is assumed that the nanoparticles with the same temperature as the liquid of the central region of the tube move to the area near the solid wall and become the heat sinking points by absorbing heat, and then quickly move back to the tube central region after transferring the heat to the solid wall. It is assumed that thermal equilibrium between the nanoparticle and the liquid near the center of the tube cross section is achieved rapidly due to the large specific surface area of the nanoparticles, which contributes to the heat exchange area between the two phases. Thus nanoparticles are an efficient medium of heat transfer resulting in high-temperature gradients

Nanofluids

48

1. Introduction to nanofluids

near the wall. It is also meaningful to note that by the volume fraction increasing, the heat transfer rate is also improved, and consequently the nondimensional temperature near the wall is reduced. Additionally, it can be observed from Fig. 1.30 that after nanoparticle clustering, the nondimensional temperature near the wall is greater. Therefore the temperature difference between the fluid and the wall is decreased, leading to a declined heat transfer rate. It is important to mention that the degree of Brownian motion will be weakened after nanoparticle clustering, which causes a significant reduction of the heat transfer performance of NFs.

References [1] S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME Publ. Fed. 231 (1995) 99
106. [2] Y. Xuan, Q. Li, Investigation on convective heat transfer and flow features of nanofluids, ASME J. Heat Transf., 125, 2003, pp. 151
155. [3] W.H. Azmi, K. Abdul Hamid, R. Mamat, K.V. Sharma, Effects of working temperature on thermo-physical properties and forced convection heat transfer of TiO2, nanofluids in water
ethylene glycol mixture, Appl. Therm. Eng. 106 (2016) 1190
1199. [4] H. Zhang, S. Shao, H. Xu, C. Tian, Heat transfer and flow features of Al2O3
water nanofluids flowing through a circular microchannel
experimental results and correlations, Appl. Therm. Eng. 61 (2013) 86
92. [5] N. Kumar, B.P. Puranik, Numerical study of convective heat transfer with nanofluids in turbulent flow using a Lagrangian-Eulerian approach, Appl. Therm. Eng. 111 (2016) 1674
1681. [6] M. Mahdavi, M. Sharifpur, J.P. Meyer, CFD modelling of heat transfer and pressure drops for nanofluids through vertical tubes in laminar flow by Lagrangian and Eulerian approaches, Int. J. Heat Fluid Flow 88 (2015) 803
813. [7] S. Sonawane, U. Bhandarkar, B. Puranik, Modeling forced convection nanofluid heat transfer using an Eulerian-Lagrangian approach, J. Therm. Sci. Eng. Appl. 8 (2016) 031001
031008. [8] J. Zhou, X. Wang, D. Song, D. Jing, The effects of nanoparticle aggregation on the convection heat transfer investigated by a combined NDDM and DPM method, Numer. Heat Transf. Part A Appl. 71 (7) (2017) 754
768. [9] S. Krishnamurthy, P. Bhattacharya, P.E. Phelan, R.S. Prasher, Enhanced mass transport in nanofluids, Nano Lett. 6 (2006) 419
423. [10] W. Yu, D.M. France, Review and comparison of nanofluid thermal conductivity and heat transfer enhancements, Heat Transf. Eng. 29 (2008) 432
460. [11] J.H. Lee, S.H. Lee, C. Choi, S. Jang, S. Choi, A review of thermal conductivity data, mechanisms and models for nanofluids, Int. J. Micronano Scale Transp. 1 (2015) 269
322. [12] C. Pang, J. Lee, H. Hong, Y. Kang, Heat conduction mechanism in nanofluids, J. Mech. Sci. Technol. 28 (2014) 2925
2936. [13] H. Aybar, M. Sharifpur, M.R. Azizian, M. Mehrabi, J.P. Meyer, A review of thermal conductivity models for nanofluids, Heat Transf. Eng. 36 (2015) 1085
1110. [14] G. Tertsinidou, M.J. Assael, W.A. Wakeham, The apparent thermal conductivity of liquids containing solid particles of nanometer dimensions: a critique, Int. J. Thermophys. 36 (2015) 1367
1395.

Nanofluids

References

49

[15] M. Hatami, G. Domairry, S.N. Mirzababaei, Experimental investigation of preparing and using the H2O based nanofluids in the heating process of HVAC system model, Int. J. Hydrog. Energy 42 (12) (2017) 7820
7825. [16] S.K. Das, S.U. Choi, W. Yu, T. Pradeep, Nanofluids Science and Technology, Wiley, 2008. [17] A.V. Rane, K. Kanny, V.K. Abitha, S. Thomas, Methods for synthesis of nanoparticles and fabrication of nanocomposites, in: Synthesis of Inorganic Nanomaterials, Elsevier, 2018, pp. 121
139. [18] S. Roy, C.K. Ghosh, C.K. Sarkar (Eds.), Nanotechnology: Synthesis to Applications., CRC Press, 2017. [19] M. Aliofkhazraei (Ed.), Handbook of Nanoparticles, Springer International Publishing, Cham, 2016. [20] M. Hosokawa, K. Nogi, M. Naito, T. Yokoyama (Eds.), Nanoparticle Technology Handbook., Elsevier, 2012. [21] M. Haghshenas Fard, M. Nasr Esfahany, M.R. Talaie, Numerical study of convective heat transfer of nanofluids in a circular tube two-phase model versus single-phase model, Int. Commun. Heat Mass Transf. 37 (2010) 91
97. [22] S. Go¨ktepe, K. Atalık, H. Ertu¨rk, Comparison of single and two-phase models for nanofluid convection at the entrance of a uniformly heated tube, Int. J. Therm. Sci. 80 (2014) 83
92. [23] M. Hatami, Different shapes of Fe3O4 nanoparticles on the free convection and entropy generation in a wavy-wall square cavity filled by power-law non-Newtonian nanofluid, Int. J. Heat Technol. 36 (2) (2018) 509
524. [24] G.H.R. Kefayati, Simulation of heat transfer and entropy generation of MHD natural convection of non-Newtonian nanofluid in an enclosure, Int. J. Heat Mass Transf. 92 (2016) 1066
1089. [25] S. Mosayebidorcheh, M. Vatani, M. Hatami, D.D. Ganji, Two-phase nanofluid thermal analysis over a stretching infinite solar plate using Keller Box Method (KBM), Iran. J. Chem. Chem. Eng. 37 (2018) 247
256. [26] J.A. Khan, M. Mustafa, T. Hayat, A. Alsaedi, Three-dimensional flow of nanofluid over a non-linearly stretching sheet: an application to solar energy, Int. J. Heat Mass Transf. 86 (2015) 158
164. [27] H.B. Keller, A new difference scheme for parabolic problems, in: B. Hubbard (Ed.), Numerical Solutions of Partial Differential Equations, Part II, Academic Press, New York, 1971, pp. 327
350. [28] J.C. Maxwell, A Treatise on Electricity and Magnetism, Clarendon, Oxford, 1892. [29] J. Jin, M. Hatami, D. Jing, Experimental investigation and prediction of the thermal conductivity of water-based oxide nanofluids with low volume fractions, J. Therm. Anal. Calorim. 135 (1) (2019) 257
269. [30] D.H. Kumar, H.E. Patel, V.R.R. Kumar, T. Sundararajan, T. Pradeep, S.K. Das, Model for heat conduction in nanofluids, Phys. Rev. Lett. 93 (2004) 144301. [31] R. Prasher, P. Bhattacharya, P.E. Phelan, R. Prasher, P. Bhattacharya, Brownianmotion-based convective-conductive model for the effective thermal conductivity of nanofluids, J. Heat Transf. 128 (2006) 588
595. [32] C.H. Chon, K.D. Kihm, S.P. Lee, S.U.S. Choi, Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3), thermal conductivity enhancement, Appl. Phys. Lett. 87 (2005) 395
403. [33] H.E. Patel, T. Sundararajan, S.K. Das, An experimental investigation into the thermal conductivity enhancement in oxide and metallic nanofluids, J. Nanopart. Res. 12 (2010) 1015
1031. [34] R.S. Vajjha, D.K. Das, Experimental determination of thermal conductivity of three nanofluids and development of new correlations, Int. J. Heat Mass Transf. 52 (2009) 4675
4682.

Nanofluids

50

1. Introduction to nanofluids

[35] M. Corcione, Empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids, Energy. Convers. Manag. 52 (2011) 789
793. [36] J.H. Lee, K.S. Hwang, S.P. Jang, B.H. Lee, J.H. Kim, S.U.S. Choi, et al., Effective viscosities and thermal conductivities of aqueous nanofluids containing low volume concentrations of Al2O3 nanoparticles, Int. J. Heat Mass Transf. 51 (2008) 2651
2656. [37] M.C.S. Reddy, V.V. Rao, Experimental studies on thermal conductivity of blends of ethylene glycol-water-based TiO2 nanofluids, Int. Commun. Heat Mass Transf. 46 (2013) 31
36. [38] A. Kazemibeydokhti, S.Z. Heris, N. Moghadam, M. Shariatiniasar, A.A. Hamidi, Experimental investigation of parameters affecting nanofluid effective thermal conductivity, Chem. Eng. Commun. 201 (2013) 1443
1444. [39] H.A. Mintsa, G. Roy, T.N. Cong, D. Doucet, New temperature dependent thermal conductivity data for water-based nanofluids, Int. J. Therm. Sci. 48 (2009) 363
371. [40] J. Zhou, Y. Wang, J. Geng, D. Jing, Characteristic oscillation phenomenon after headon collision of two nanofluid droplets, Phys. Fluids 30 (7) (2018) 072107. [41] M. Pasandideh-Fard, Y.M. Qiao, S. Chandra, J. Mostaghimi, Capillary effects during droplet impact on a solid surface, Phys. Fluids 8 (3) (1996) 650
659. [42] R.Y. Hong, Z.Q. Ren, Y.P. Han, H.Z. Li, Y. Zheng, J. Ding, Rheological properties of water-based Fe3O4 ferrofluids, Chem. Eng. Sci. 62 (21) (2007) 5912
5924. [43] O.A. Basaran, D.W. DePaoli, Nonlinear oscillations of pendant drops, Phys. Fluids 6 (9) (1994) 2923
2943. [44] W. Tang, M. Hatami, J. Zhou, D. Jing, Natural convection heat transfer in a nanofluid-filled cavity with double sinusoidal wavy walls of various phase deviations, Int. J. Heat Mass Transf. 115 (2017) 430
440. [45] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, S. Soleimani, MHD natural convection in a nanofluid filled inclined enclosure with sinusoidal wall using CVFVM, Neural Comput. Appl. 24 (3) (2014) 873
882. [46] M. Hatami, D. Song, D. Jing, Optimization of a circular-wavy cavity filled by nanofluid under the natural convection heat transfer condition, Int. J. Heat Mass Transf. 98 (2016) 758
767. [47] K. Apostolou, A.N. Hrymak, Discrete element simulation of liquid-particle flows, Comput. Chem. Eng. 32 (2008) 841
856. [48] P.G. Staffman, The lift on a small sphere in a slow shear flow, J. Fluid Mech. 22 (1965) 385. [49] V. Bianco, F. Chiacchio, O. Manca, S. Nardini, Numerical investigation of nanofluids forced convection in circular tubes, Appl. Therm. Eng. 29 (2009) 3632
3642. [50] N. Kumar, B.P. Puranik, Numerical study of convective heat transfer with nanofluid in turbulent flow using a Lagrangian-Eulerian approach, Appl. Therm. Eng. 111 (2017) 1674
1681. [51] J. Zhou, X. Wang, D. Song, D. Jing, The effects of nanoparticle aggregation on the convection heat transfer investigated by a combined NDDM and DPM method, Numer. Heat Transf. Part A Appl. 71 (7) (2017) 754
768. [52] D. Song, D. Jing, J. Geng, Y. Ren, A modified aggregation based model for the accurate prediction of particle distribution and viscosity in magnetic nanofluids, Powder Technol. 283 (2015) 561
569. [53] H. Zhang, S. Shao, H. Xu, C. Tian, Heat transfer and flow features of Al2O3
water nanofluids flowing through a circular microchannel
experimental results and correlations, Appl. Therm. Eng. 61 (2013) 86
92. [54] Y. Xuan, Q. Li, Investigation on convective heat transfer and flow features of nanofluids, ASME J. Heat Transf., 125, 2003, pp. 151
155.

Nanofluids