Numerical Research on Convective Heat Transfer and Resistance Characteristics of Turbulent Duct Flow Containing Nanorod-Based Nanofluids

A coupled numerical model for nanorod-based suspension flow is constructed, and the convective heat transfer and resistance characteristics of the nanofluid duct flow are investigated. *e numerical results are verified by experimental results and theoretical models. Most of nanorods are located randomly in the bulk fluid, while particles near the wall aligned with the flow direction. Friction factor of nanofluids with nanorods increases with higher particle volume concentration or aspect ratio, but the increment reduces when the Reynolds number gets larger. *e relative Nusselt number is obtained to characterize the intensity of convective heat transfer.*e results show that the Nusselt number of nanofluids increases when the particle volume concentration or aspect ratio becomes larger. Compared to increasing the aspect ratio of nanorods, increasing the particle volume concentration would be more effective on enhancing the convective heat transfer intensity in industrial applications although it will cause a slight increase of resistance.


Introduction
Nanofluids, which are fluids with nanosized particles suspended, are considered to be next-generation mediums for advanced heat transfer or cooling technology [1].Since Choi put forward the concept firstly [2], the study of nanofluids has been intensively performed and reported in worldwide.Abnormal heat transfer characteristics were reported by experiments including the effective heat conductivity, natural and forced convection heat transfer, and boiling heat transfer of nanofluids [3][4][5].Factors that can influence thermophysical properties have been discussed in many literatures, such as nanoparticle size and shape, material of particle and base fluid, particle loading, temperature, and so on, but quantitative understanding is still a lack on the topic [1,3].Another hot topic is the flow resistance characteristic of nanofluids [6,7].
e pumping power needed for the bulk fluid did not increase dramatically as the particle loading grows in experiments [5], whereas the viscosity of nanofluids shows a marked increase after adding nanoparticles into base fluid [1,6].e rheology of nanofluids also leads to the complexity on characterizing physical properties of bulk flow [8,9].
Although many researchers have reviewed lots of measured experimental data and demonstrated that particle shape plays an important role on convective heat transfer characteristic of nanofluids [10,11], few models can be found in literatures proposed for nanofluid flow containing nonspherical nanoparticles.Nanorods, as well as nanotubes, nanofibers, and nanobelts are used more and more for nanofluid preparation because of larger aspect ratio and surface area compared with spherical nanoparticles.Such kind of one-dimensional nanomaterials are available for heat transfer application with the development of nanoscale synthesis technology [12].Xie et al. [13] first found the thermal conductivity of nanofluids with elongated nanoparticles is superior to that of spherical nanoparticles.Murshed et al. [14] measured enhanced thermal conductivity of water-based nanofluid containing TiO 2 nanorods with the aspect ratio of 4; they showed that the results give 12% higher than that predicted by classical Hamilton-Crosser model.
en, two models derived considering interfacial layer existence between particle and base fluid for the effective thermal conductivities of nanofluids with spherical and cylindrical nanoparticles [15].Yang and Han [16] investigated the thermal conductivity of nanofluids dispersed by Bi 2 Te 3 nanorod in perfluoro-n-hexane and oil.ey found the thermal conductivities of nanofluids decrease with increasing temperature due to nanorod aspect ratio.e results measured by Zhang et al. [17,18] employing carbon nanotubes (CNTs) agreed well with the predictions of the unit-cell model proposed by Yamada and Ota [19] by considering shape factor for cylindrical particles.By using the excluded volume concept, Koo et al. [20] constructed a model which can reproduce the nonlinear increase of the thermal conductivity for carbon nanotube and nanofiber suspensions.Timofeeva et al. [21] studied the particle shape effect on thermophysical properties of alumina-EG/H 2 O suspensions and concluded that elongated particles and agglomerates resulted in higher viscosity at the same volume fraction due to structural limitation of rotational enhancements in effective thermal conductivities diminished by interfacial effects.Molecular dynamics (MD) simulation based on the model developed by Ghosh et al. [22] have shown that cylindrical nanoparticles pick up thermal energy much faster than spherical nanoparticles during a collision with a block-shaped heat source [23].Comparative experimental results reported by Yu et al. [10] indicated that convective heat transfer characteristic of nanofluids containing nonspherical nanoparticles is affected significantly by the shear-induced alignment and orientational motion of particles.By considering the thickness and thermal conductivity of the interfacial nanolayer, Jiang et al. [24] proposed a model which provides good predictions for the effective thermal conductivity of CNTbased nanofluids.Other existing models are summarized by Yang et al. [11], and three similar theoretical models are proposed for nanofluids with finite cylindrical particles by anisotropy analysis [25][26][27].All of the three models give minimal relative errors to experimental results compared to other models.Lin et al. [7,28] took numerical simulation on the phaolefins-based nanofluids fluid flow containing cylindrical Al 2 O 3 nanoparticles in laminar and turbulent pipes, respectively.e results show that the friction factor of flow decreases when the Reynolds number and particle aspect ratio become larger.e Nusselt number of convective heat transfer increases with the increase of Reynolds number, particle aspect ratio, and volume concentration.By comparing the defined performance evaluation criterion (PEC), nanofluids containing rod-like nanoparticles with large aspect ratio and a suitable particle volume concentration are more effective for convective heat transfer process at higher Reynolds number.A coupled numerical model is constructed by Yuan et al. [29] to simulate convective heat transfer and resistance characteristics of TiO 2 /water nanofluids with cylindrical particles in laminar channel flow.e results show that the Nusselt number of nanofluid flow related to the Reynolds number, axial length, Prandtl number, and particle volume concentration and a fitted formula is proposed to predict the Nusselt number of nanofluid flow containing cylindrical nanoparticles.
As summarized above, high potential of rod-like nanoparticles in enhancing thermal conductivity of nanofluids has got more attention by researchers.Related reports focus mostly on viscosity and thermal conductivity properties, while progress made on convective heat transfer characteristics is still highly needed.In addition, lack of numerical modelling on nanofluid flow brings an obstacle to industrial applications.e objective of this work is to develop a coupled model for nanorod-based suspension flow and investigate the convective heat transfer and resistance characteristics of the nanofluid duct flow numerically.
e orientation and movement of nanorod are considered, and key factors that can influence the friction factor and Nusselt number of the fluid flow are discussed in sequence.

Governing Equations.
e duct flow for nanofluid is considered as impressible, and the governing equations include the continuity equation, modified Navier-Stokes equation with the additional term of nanorods, and the energy conservation equation [28,30,31]: where u i , p, ρ nf , and T are the nanofluid velocity, pressure, density, and temperature, respectively; μ is the dynamic viscosity of the pure fluid; μ a is the apparent viscosity; a kl and a ijkl are the mean second-and fourth-order tensors of particle orientation, respectively; is the rate-of-strain tensor; and C nf is the thermal diffusivity coefficient of the nanofluid, C T � C μ k 2 /εPr T (k is the turbulent kinetic energy, ε is the turbulent dissipation rate, C μ � 0.09, and turbulent Prandtl number, Pr T � 0.9) is the eddy thermal diffusivity coefficient.Substituting instantaneous velocity, pressure, temperature, rate-of-strain tensor, and tensors of particle orientation which consist of mean and fluctuation part in (1-3) and averaging, we have 2 Journal of Nanotechnology where U i , P, and T are the mean nanofluid velocity, pressure, and temperature, respectively, u i ′ is the fluctuation flow velocity, and ε kl is the mean rate-of-strain tensor.
In (5), the nanofluid density is where the subscripts "f " and "p" stand for pure fluid and nanoparticles, respectively, and ϕ e is the effective particle volume concentration, which is used for replacing the nominal particle volume fraction ϕ, in consideration of the particle aggregation: where s and s e are the effective size of aggregates and primary particle size, respectively, and f i is the fractal index and varies from 1.5 to 2.45 for those of rod-like nanoparticles.Yu et al. [10] gives the value as: s/s e � 1.48 and f i � 1.95 according to experimental data.Batchelor's theory was extended by Mackaplow and Shaqfeh [32] to account for two-body interactions and gave the apparent viscosity in (5): where n and l are the number density and half length of nanorods, respectively, μ is the dynamic viscosity of the pure fluid, and r is the particle aspect ratio, which is the ratio of length of diameter.e thermal diffusivity coefficient in ( 6) is [33,34] where the heat capacitance of the nanofluid is given by In (10), the classical Hamilton-Crosser model is employed for predicting thermal conductivity of nanorodbased nanofluid as follows: in which k f and k p are the thermal conductivity of pure fluid and nanoparticles, respectively, and K is the shape factor given by K � 3/c, where c is the particle sphericity, defined as the ratio of the surface area of a sphere with the same volume as that of the particle and the surface area of the particle.e model predicts well the thermal conductivity of nanofluid with ZnO nanoparticles with experimental data by Ferrouillat et al. [35] in rod-like shape.e Reynolds stress tensor −ρu i ′u j ′ in ( 5) is where eddy viscosity μ T � 0.09ρ nf k 2 /ε.For solving (5), the k-equation and ε-equation for turbulent flow are given by where C 1 �1.44,C 2 � 1.92, σ k � 1.0, and σ ε � 1.3.

Nanoparticle Orientation Distribution.
e mean second-and fourth-order tensors of particle orientation in (5) describe the orientation distribution of nanoparticle population in flow field; they can be calculated by Advani and Tucker [36] as where p i is a unit vector parallel to the particle's axis and ψ(p) is the probability density function for particle orientation at any position with p being the orientation vector.The governing equation of ψ(p) is the modified Fokker-Planck equation considering the rotary effect caused by Brownian movement [37,38]: where D rB is the Brownian rotary diffusion coefficient, z/zp j is the gradient operator projected onto the surface of the unit sphere, and _ p j is the particle angular velocity.Cintra and Tucker [39] expressed _ p i as where ω ij � (zu j /zx i − zu i /zx j )/2 is the vorticity tensor, λ � (r 2 − 1)/(r 2 + 1).e last term in ( 18) is introduced to model the behavior at higher concentrations, in which D rI is a rotary diffusion coefficient resulting from particle-particle interactions.Folgar and Tucker [40] suggested that D rI is isotropic and given by C I ����� 2ε ij ε ji

􏽱
, where the interaction coefficient C I is fixed to 0.01.
For cylindrical particles, the Brownian rotary diffusion coefficient in (17) is With aspect ratio r, the rotational friction coefficients around long axis and short axis are [41,42]. with Substituting instantaneous ψ and _ p j which consist of mean and fluctuation part in (17) and averaging, we have the mean equation of probability density functions for the particle orientation: where α ψx � 1.3(5k 2 ]/3ε) 1/2 and α ψp � 0.7(4ε/15]) 1/2 are the dispersion coefficients of linear and angular displacement in which ] is the kinetic viscosity of the pure fluid [43], which appear based on the dispersion mechanism: Integral on the angle after multiplying component of unit vector p i , ( 22) is transformed to the equation about a ij : e mean fourth-order particle orientation tensor needs to be closured by model.Considering the shearing-stretch feature of duct flow, the orthotropic closure model [44] is fit for (24): in which det(a ij ) is the determinant of second-order tensor a 2 .

General Dynamics Equation for
Rod-Like Nanoparticles.e particle volume concentration ϕ in ( 7), (11), and ( 12) is considered distribution inhomogeneity in the flow.In order to obtain the distribution of ϕ in the flow, it is necessary to solve the general dynamics equation for nanorods: Based on the Reynolds average, the above equation is transformed to where n(v) is the mean particle volume distribution function based on the volume and the last term on the left-hand side represents the change in n(v) resulting from turbulent diffusion and is usually assumed to be [45]: where ε i is the eddy diffusion coefficient which is a function of position and ε i � ] t /Sc t , where v t and Sc t are the turbulent viscosity and the Schmidt number, respectively.D tB in ( 26) is the Brownian translational diffusion coefficient for rod-like nanoparticles: and the translational friction coefficients parallel and perpendicular to the long axis are [41,42] with Substituting ( 29) in ( 27), we have averaged general dynamics equation for nanoparticles.With the moment transformation which involves multiplying the equation by v k and then integrating over the entire size regime, the equation for rod-like nanoparticles finally becomes where the zero moment and the first-order moment describe the total particle number density and volume of nanoparticles at a given point.

Numerical Method
Schematic of a turbulent duct flow containing nanorodbased nanofluid is shown in Figure 1. e length and diameter of the channel are L and D, respectively.e boundary conditions are given as follows: inlet: U x � U 0 , U y � 0, T � T in , m 1 � m 10 ; outlet: zU x /zx � 0, U y � 0, zT/zx � 0, zm 0 /zx � 0; and wall: U x � U y � 0, m 1 � 0, zm 0 /zx � 0. Notably, numerical simulations performed here are forced convection process with uniform wall temperature, other than those with uniform wall heat flux.
e ratio of channel length to height is L/D � 125.e nanofluid is a mixture of water and rigid ZnO nanorods.ermophysical properties of water and ZnO are presented in Table 1.e temperatures of inlet nanofluid and wall are T in � 293 K and T w � 323 K, respectively.Reynolds number of the fluid flow is defined as Re � u 0 D/] nf ranging from 2500 to 15,000.
e SIMPLEC algorithm is employed for solving coupled continuity equation and moment equation, where the convection item and diffusion item are discreted using QUICK scheme and second-order center difference, respectively.No slip condition is set for wall boundary condition.

Calculation Verification.
To validate the model and code for nanofluid flow, classical models and experimental results are used for comparing the numerical results of friction factor and the Nusselt number of duct fluid flow in Figures 2  and 3. e definition of f and Nu is as follows: where ∆p is the pressure drop, U av is the average velocity of the fluid in the flow direction, and h is the heat transfer coefficient.
Figure 2 shows the comparison of friction factor of nanofluids varying with the Reynolds number.e solid points in figure are measured by Ferrouillat et al. [35], and the solid curve is the Blasius equation which is f � 0.316Re −0.25 for turbulent duct flow.e numerical results are drawn by hollow points and are obtained by setting the nanorod aspect ratio as 8, because the aspect ratio of nanoparticles in reference experiments are about 6∼10.From the figure, the numerical results show agreement with the experimental results and the curve of Blasius equation.In turbulence regime, the friction factor of nanofluid flow drops slowly when the Reynolds number grows.
Comparison of calculated Nusselt number of nanofluids with Dittus-Boelter relation is shown in Figure 3. e classical Dittus-Boelter relation is commonly used for predicting the Nusselt number in fully developed turbulent duct flow, which is Nu � 0.023Re 0.8 Pr 0.4 .e difference of Nusselt number predicted by current model and D-B relation is within about 10%, and the difference declines in high Nu region.

Orientation Distribution of Nanorods.
e orientation of nanorods in fluid makes an effect on nanofluid fluid flow.
e orientation distribution of nanorods population can be described by the mean second-order orientation tensor of particle.e components of a ij in z direction are shown in Figure 4, in which the volume concentration and aspect ratio   are set as ϕ 1.4% and r 16, respectively.It can be seen from the gure that the distributions of components of a ii coincide with each other, and the median distribution of a ii which along the axial direction is 1/π.e value of a ii near the wall is larger that that of central area, which means that the nanorods align with the ow direction due to the existence of wall.e orientation state of nanorods can hardly keep in y-z component because a 23 is always zero in z direction.But a 13 is not zero near the wall, which illustrates the wall e ect on orientation state of nanorods.

Friction Factors.
e friction factor and pressure drop of the duct ow are of importance to the practical applications.Figure 5 shows the friction factors of nano uids at di erent volume concentrations as a function of the Reynolds number.e aspect ratio of nanorods in experiments and simulations are set as r 8.All the friction factors of nano uid ow are higher than those of base uid and decrease when the Reynolds number grows higher.e friction factor increases slightly with the increase of particle volume concentration, and the di erence diminishes when the Reynolds number becomes larger.
In Figure 6, the friction factor as a function of the Reynolds number by di erent aspect ratio is given, and the volume concentration ϕ is 0.93%.As can be seen in gure, the shape of nanoparticles takes e ect on friction factor of nano uid uid ow.Nanorods with larger aspect ratio bring more ow resistance when the Reynolds number is the same.When the Reynolds number gets larger, the friction factor of nano uid approaches to that of base uid, which means the additional pumping power for nano uid is not signi cant in high Reynolds number region.

Convective Heat Transfer.
e Nusselt number describes the ratio of convective heat transfer across the wall.e relative Nusselt number is de ned here as the ratio of the Nusselt number of nano uids to that of base uid, that is, Nu nf /Nu f .Figures 7 and 8 show the calculated and experimental results of Nu nf /Nu f as a function of the Reynolds number with di erent particle aspect ratios or di erent volume concentrations.e solid points give the measured Nu with particle volume concentration as 0.93% and reveal that the Nusselt number of nano uid is smaller than that of base uid when Re is less than 7000.than 1, which means the convective heat transfer of nanouid is better than base uid at the same Reynolds number.e Nu nf /Nu f is monotone increasing as a function of the Reynolds number, no matter the particle volume concentration or aspect ratio varies in gures.In Figure 7, the relative Nusselt number increases as the particle volume concentration gets larger.Similarly, in Figure 8, the relative Nusselt number increases as the particle aspect ratio gets larger.Both the particle loading and particle shape in uence the convective heat transfer characteristic obviously.Compared to increasing the aspect ratio of nanorods, increasing the particle volume concentration would be more e ective on enhancing the convective heat transfer intensity in industrial applications although it will cause a slight increase of resistance.

Conclusions
e convective heat transfer and resistance characteristics of the nano uid duct ow are investigated based on a coupled numerical model for nanorod-based suspension ow. e numerical results are veri ed by experimental results and theoretical models.Most of nanorods are located randomly in the bulk uid, while particles near the wall aligned with the ow direction.Friction factor of nano uids with nanorods increases with higher particle volume concentration or aspect ratio, but the increment reduces when the Reynolds number getting larger.
e relative Nusselt number is obtained to characterize the intensity of convective heat transfer.
e results show that the Nusselt number of nano uids increases when the particle volume concentration or aspect ratio becomes larger.Compared to increasing the aspect ratio of nanorods, increasing the particle volume concentration would be more e ective on enhancing the convective heat transfer intensity in industrial applications although it will cause a slight increase of resistance.

Figure 7 :Figure 8 :
Figure 7: Nu nf /Nu f as a function of the Reynolds number by di erent volume concentrations (r 8).

Figure 6 :
Figure 6: Friction factor as a function of the Reynolds number by di erent aspect ratios (ϕ 0.93%).
e overall experimental data indicate that Nu nf /Nu f grows as the Reynolds number gets larger except the point when Re is about 2100.e calculated Nu nf /Nu f by model in this paper is larger than that of experiments when the volume concentration is set as 0.93%.In addition, all the numerical results are greater