Article pubs.acs.org/IECR
Modified Prediction Model for Thermal Conductivity of Spherical Nanoparticle Suspensions (Nanofluids) By Introducing Static and Dynamic Mechanisms Wenzheng Cui,†,‡ Zhaojie Shen,*,† Jianguo Yang,† and Shaohua Wu‡ †
School of Automotive Engineering, Harbin Institute of Technology, Weihai 264209, China School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China
‡
ABSTRACT: Nanofluids possess significantly increased thermal conductivity that conventional heat conduction theories and models for nanoparticles−liquid suspensions cannot explain, which provides new scientific challenges. The aim of this study is to establish a modified prediction model for thermal conductivity of nanofluids that takes into account more comprehensive mechanisms. Based on effective medium theory, the role of dynamics and static mechanisms including the absorption liquid layer at the liquid/nanoparticle interface, the effect of aggregate structure of nanoparticles, and the impact of random motions of nanoparticles have been considered to establish the model. The parameters in the model have definite physical meaning and are more precise. For instance, the thermal conductivity of nanoparticles, kp, is replaced with thermal conductivity of nanoparticles aggregate, and the volume fraction of nanoparticles along with the absorption layer in nanofluids ϕ is used instead of the volume fraction of aggregate. It is found that by considering roundly the mechanisms for heat conduction enhancement, the predictions by the present model are closer to the real situation. For various types of nanofluids (with different materials including metal, metallic oxide and nonmetallic oxide, different volume fractions, or different nanoparticle diameters), the present model gives good predictions. And the prediction results of the present model also coincide with molecular dynamics simulation, which further proves the advantages of the present model. absorption layer at the liquid/particle interface.7,8 The prediction results of the models for Cu−ethylene glycol nanofluids agree well with the experimental results. Based on average polarization theory, Xue et al. established a prediction model for thermal conductivity of nanofluids primarily focused on the effect of the absorption layer.9 And the forecasting effect of the model for carbon nanotubes−oil nanofluids and Al2O3− H2O nanofluids are good. Researchers also investigated the absorption mechanism at the surface of nanoparticles and proposed prediction models for the thickness of the liquid layer.10−13 The nanoparticle clustering is supposed to be an important mechanism for the enhancement of thermal conductivity. Wang et al.14 presented a modified prediction model considering the nanoparticle clustering and absorption effect of nanoparticles. Based on the analysis for the prediction results, they proposed that if the effect of nanoparticle clustering could be considered in the modeling the prediction results would be better. Prasher et al. pointed out that the clustering of nanoparticles would form an effective passageway for heat conduction, which should be considered in the modeling of heat conduction in nanofluids.15 Xu et al. proved that fractal theory is suitable for describing the clustering structure of nanoparticles, which could be further used for modeling the thermal conductivity of nanofluids.16 According to the small scale effect, there are Brownian motion and thermal diffusion in nanofluids because nano-
1. INTRODUCTION The novel concept of nanofluids refers to the new class of nanotechnology-based heat transfer fluids that exhibit thermal properties superior to those of their host fluids or conventional particle fluid suspensions.1 Nanofluids possess dramatic improvements in the thermal properties with a very small amount of guest nanoparticles dispersed uniformly and suspended stably in host fluids, and therefore nanofluids have received much attention from researchers in recent years.2−4 In particular, the thermal conductivity has been demonstrated to be significantly improved by numerous experiments.5 However, the marvelous experimental discoveries clearly offer theoretical challenges because they show the fundamental limits of conventional heat conduction theories and models for nanoparticles−liquid suspensions. A number of theoretical studies have been carried out to elucidate the behavior of nanofluids, and several mathematical models have been developed for predicting the thermal conductivity of nanofluids. Keblinski et al. proposed possible reasons for explaining the heat transfer enhancement in nanofluids, including the molecular-level layering of the liquid at the liquid/particle interface, the effect of nanoparticle clustering, Brownian motion of nanoparticles in base fluid, and ballistic phonon transport in nanoparticles.6 Due to the nanoscale of suspending particles, the thickness of the liquid layer at the liquid/particle interface is comparable to the nanoparticle size. Therefore, the effect of the liquid layer should be taken into account, which is different from conventional large-scale particle suspensions. On the basis of classical Maxwell and Hamilton−Crosser models, Yu et al. proposed modified prediction models considering the mechanism of © 2014 American Chemical Society
Received: Revised: Accepted: Published: 18071
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mechanisms for modeling the thermal conductivity of nanofluids. The parameters used in the model have definite physical meaning and are meticulous. For several common types of nanofluids the prediction results of the present model are in agreement with experiments. We have also compared the prediction results of the present model with the results from molecular dynamics (MD) simulations, and the predictions are consistent with the simulation results.
particles run irregularly in the base liquid due to the Brownian force.17 And thermal conductivity of nanofluids will increase significantly due to the microconvection effect caused by the Brownian motion, as it will enhance the energy transmission between nanoparticles and the base liquids. Therefore, more and more researchers focused on the enhancement of thermal conductivity in nanofluids influenced by random Brownian motion of nanoparticles, and proposed some corrected prediction models for thermal conductivity of nanofluids.18−23 Xuan et al. have analyzed the energy transmission process in nanofluids based on Brownian motion and energy transmission of nanoparticles, and proposed a semiempirical theoretical model to calculate the thermal conductivity in nanofluids.24 The model takes into account almost all factors of nanoparticles for the thermal conductivity in nanofluids; however, thermal contact resistance between nanoparticles and the base liquid needs to be determined by experiments. Jang et al.17,18 proposed that it is the Brownian motion of nanoparticles that increased the thermal conductivity of nanofluids, established a dynamic model of thermal conductivity for nanofluids that took temperature into account, and defined Re number using random Brownian motion velocity of nanoparticles. Prasher et al.19,20 also considered the microconvection effect and established a thermal conductivity model for nanofluids. Koo et al.21,22 pointed out that the enhancement effect of thermal conductivity in nanofluids by Brownian motion of nanoparticles will be improved by smaller nanoparticles. Ren et al.23 and Kumar et al.25 proposed a new thermal conductivity prediction model based on Brownian motion mechanism of nanoparticles as well. Li et al. analyzed the mixing effect of base liquid and nanoparticles using simulation methods, and compared the results with experiments.26 It is indicated that the mixing state caused by Brownian motion will influence the thermal conductivity of nanofluids significantly. However, there are negative opinions, some researchers argue that the time scale of Brownian motion is too short to affect the thermal conductivity. Generally, the mechanism of Brownian motion of nanoparticle stills needs more in-depth exploration. Based on the above literature analysis, it could be found that the reason for explaining the improvement of thermal conductivity in nanofluids should include both the static and dynamic mechanisms. Therefore, the prediction models proposed recently have considered both the contributions of static and dynamic mechanisms. Murshed et al. modeled thermal conductivity of nanofluids on the basis of considering both the static and dynamic mechanisms.27 Their model could reflect many influence factors for the thermal conductivity, including the nanoparticle size, effect of absorption layer, Brownian motion, the interactions between nanoparticles, etc. Sitprasert et al.28 modified the Leong model,29 which is a dynamic model, and introduced the impact of the interface layer effect. And the prediction from the model is better. However, the existing research works on modeling the thermal conductivity of nanofluids are still insufficient. Most of the established models in these works only apply to some particular type of nanofluids. And some of the parameters used in the models lack definite physical meaning. Nevertheless, the idea of modeling thermal conductivity of nanofluids by combining static and dynamics mechanisms is correct. The present paper proposed a modified prediction model for the effective thermal conductivity of spherical nanoparticle suspensions (nanofluids). Based on effective medium theory, the present work considered both the static and dynamics
2. EFFECTIVE MEDIUM THEORY By applying effective medium theory for predicting thermal conductivity of solid particle suspensions, the Maxwell−Garnett (MG) model is the most widely used prediction model, which is written as k m + 2k f − 2ϕp(k f − k m) k = kf k m + 2k f + ϕp(k f − k m)
(1)
where k represents the effective thermal conductivity of solid particle suspension, kf and km are the thermal conductivity of base fluid and bulk material of particles, respectively, and ϕp is volume concentration of particles suspension. The MG model requires the suspension to have a relatively low concentration, besides the interaction between particles is not considered in the model, which means the distance between nanoparticles should be large enough. Based on the MG model, Jeffrey applied Green’s function method and relaxed the requirement of uniform configuration for particles. The formula, which is suitable for predicting suspensions with nonuniformly distributing nanoparticles and relatively large volume concentrations, is written as k = 1 + 3βϕp + 3β(β + Σ)ϕp2 kf
(2)
where Σ is a convergent series, which depends on the specific value of thermal conductivity of nanoparticle and base fluid km/ kf. β is a coefficient, which is determined by km/kf.
β=
α−1 α+2
(3)
where α is the specific value of thermal conductivity of nanoparticle and base fluid km/kf. α=
km kf
(4)
In addition, there are some other classical theoretical models for effective thermal conductivity of particle suspensions including the Hamilton−Crosser model,30 Davis model,31 Lu and Lin model,32 etc. The aims of classical models are for the prediction of large-scale particle suspensions; therefore, the factors that should be considered are less. The main parameters used in the calculation are macrothermal conductivities of solid particles and base liquid, and the particle volume concentration, etc. It has been confirmed that in the range of low volume concentrations of conventional large-scale particles, the classic models can give accurate predictions, and the difference between predicted results of several classical models is not significant. However, when the classical models are used for predicting thermal conductivity of nanofluids, the predictions differ much from experimental results, and the thermal conductivity of nanofluids are generally underestimated.7,8 The main reason for 18072
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nanoparticle diameter, A is the surface area of nanoparticle. Heat transfer coefficient H represents the total energy transfer process between nanoparticles and surrounding base fluid, which is calculated by
the invalidation of classical models is due to the application object difference. The classical models are established for largescale particle suspensions, and the microscopic factors are not considered in the models due to the microscale effect of largescale particles is not obvious. However, when it comes to nanofluids, some minor factors in the classical model (e.g., strong adsorption layer effect, particle movement due to the small-scale, etc.) become important and cannot be neglected. On the basis of the above analysis, scholars investigated the microscopic factors for explaining heat conducting enhancement in nanofluids, and proposed revised theoretical models based on classical theory for predicting thermal conductivity of nanofluids. Choi and Yu considered the absorption effect of liquid molecules on the solid surface of nanoparticles, and proposed R-MG model based on Maxwell equation. And the thermal conductivity of Cu-EG nanofluids was analyzed by using the modified model.7 The modified model is written as k m + 2k f + 2(k m − k f )(1 + κ )3 ϕp k nf = kf k m + 2k f − (k m − k f )(1 + κ )3 ϕp
td 1 1 1 = + + HA hf A f k mA m hK A p
where Af = 4π(rp + td) , rp represents the radius of nanoparticle, td is the thickness of dispersant layer, Ap = 4πrp2, Am = (Am + Af)/2, hK = (1/Ki) . Xuan’s model contains multiple factors that influence thermal conductivity of nanofluids, including stochastic dynamics and random heat transfer process of suspended nanoparticles, energy transfer process through nanoparticles and base fluid, volume concentration of nanoparticles, nanoparticle size, temperature of base fluid, and dispersant, etc. When the heat resistance between nanoparticle and base fluid Ki is determined by experiment, the predictions of the model agree well with experiment results. Furthermore, there are other revised theoretical models that give good prediction, such as Jiang’s model,35 Prasher’s model,36 Xue’s model,37 etc. Through comprehensive analysis of existing models, it could be found that most of the existing models are on the basis of classical models. By taking into account one or multiple strengthening mechanisms, the models are modified to be consistent with experimental results. Nevertheless, few of the existing models could be able to fully consider the static and dynamic mechanisms for the thermal conductivity enhancement of nanofluids. Through MD simulations, it has been revealed that the mechanisms for the strengthened conduction heat transfer in nanofluids include both static mechanisms such as microcosmic structure change and absorption layer effect, and dynamic mechanism due to the intense motion of nanoparticles in the base fluid. As a consequence, energy transfer process in a nanofluid could be written as the superposition of static and dynamic mechanisms:
(5)
where κ is the ratio of absorption layer thickness t and nanoparticle average radius, κ = (t/rp). The prediction of modified model is closer to experimental results. However, as the absorption layer is the only consideration in this revised model, further studies still need to be done. Wang applied fractal theory to model the aggregate structure of nanoparticles in base fluid. Meanwhile the absorption effect on the nanoparticle surface and size effect of nanoparticles have been considered. And their modified model is written as14 ∞ k (r )n(r )
(1 − ϕp) + 3ϕp ∫ k cl(r) + 2k dr k nf 0 cl f = ∞ k f n(r ) kf (1 − ϕp) + 3ϕp ∫ k (r) + 2k dr 0
cl
(6)
f
where kcl(r) is the thermal conductivity of aggregate according to the Bruggerman model,33 and n(r) is the particle size distribution function, which is calculated by ⎧ ⎡ ln(r / r ) ⎤2 ⎫ 1 ̅ n(r ) = exp⎨−⎢ ⎥⎬ r 2π ln σ ⎩ ⎣ 2π ln σ ⎦ ⎭ ⎪
⎪
⎪
⎪
qtotal = qS + qD
or − k nf ΔT = −k SΔT − kDΔT
(10)
That is k nf = k S + kD
(7)
(11)
where kS represents the strengthened conduction heat transfer contributed by static mechanisms, and kD represents the strengthened conduction heat transfer contributed by dynamic mechanisms.
where r ̅ is the geometric average of aggregate diameter; σ is the root-mean-square deviation of r, which can be set to classical value of 1.5. Wang’s model has considered both absorption effect and the aggregate phenomenon, which is ubiquitous in real nanofluids. Therefore, this model is consistent with the truth of nanofluids, and the predictions are closer to the experimental results. Nevertheless, the dynamic strengthening mechanism for thermal conductivity of nanofluids by adding nanoparticles is not considered. Xuan et al.34 proposed a semiempirical prediction model for thermal conductivity of nanofluids on the basis of Brownian Dynamics theory and take consideration of microconvection effect and absorption effect, which is written as k m + 2k f + 2ϕp(k m − k f ) 18ϕHAkBT k nf + = τ kf k m + 2k f − ϕp(k m − k f ) π 2ρd p6
(9)
2
3. RENOVATED MODEL FOR PREDICTING THERMAL CONDUCTIVITY OF NANOFLUID Based on the above analysis, most of the existing models take classical model as the starting point, and by introducing specific microcosmic mechanisms in nanofluids the classical model is renovated to be suitable for nanofluid thermal conductivity. By following this research approach, the Jeffrey model is chosen to be the foundation model in this work, as it has more extensive applicability. Through introducing static and dynamic mechanisms into the Jeffrey model, a renovated model for nanofluid thermal conductivity could be proposed. 3.1. Role of Static Mechanism. 3.1.1. Absorption Layer. The first thing to be considered is the effect of absorption layer absorbed at the surface of nanoparticle. According to the preparative technique, the existing forms of nanoparticles in base fluids include individuals and aggregates, as shown in
(8)
where kB is Boltzmann constant, T is temperature, τ is the total relaxation time constant, ρ is density of nanoparticle, dp is 18073
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On the condition that the initial volume concentration of nanoparticles in nanofluids is ϕp, when taking into consideration the volume of absorption layer, the volume fraction ϕ of nanoparticles along with the absorption layer is calculated as
Figure 1. Regardless of the single or aggregate form, nanoparticles always absorb their surrounding liquid molecules,
3 ⎛ 2tab ⎞ ⎟ϕ ϕ = ⎜⎜1 + d p ⎟⎠ p ⎝
where dp is the diameter of nanoparticle, and tab is obtained by eq 12. 3.1.2. Thermal Conductivity of Nanoparticles. Then the thermal conductivity of nanoparticle kp should be considered. Due to the nanometer scale of nanoparticles, the thermophysical properties that represent the energy transport law will show obviously scale dependence, which signifies the thermal conductivities of nanoparticles kp and bulk material km are not the same. Thus, for accuracy the thermal conductivity of nanoparticles kp should be determined. According to relaxation time approximation of Boltzmann thermal conduction equation, the effective thermal conductivity of nanoparticles kp is calculated as14
Figure 1. Existence forms of nanoparticles in base fluid.
and bring about more regular distributed liquid molecules. According to classical heat conduction theory, more regular distribution of molecules is more beneficial for heat transferring. In other words, the thermal conductivity of this absorption layer is larger than that of base fluid and is even close to that of solid materials. Based on the above consideration, the absorbed liquid molecules should be considered as an important part of nanoparticles and included for nanoparticle volume fraction calculation. It is widely recognized that single molecule absorption is the main form for low concentration suspensions, and the absorbed molecules at the solid surface are distributed as the hexagonal close packed.14 The thickness tab of absorption layer formed by absorbed liquid molecules can be estimated by the Langmuir monolayer equation, which is written as 1 tab = 3
⎛ ⎞1/3 ⎜ 4M ⎟ ⎜ρ N ⎟ ⎝ f A⎠
χ = rp/Λ
H2O ethylene glycol (EG) engine oil
18.016 62.068
998.2 1113.5
2.85 4.16
400−800
800−900
8.61
(15)
where rp is the radius of nanoparticle, and Λ is the phonon mean free path, which is calculated as
Λ=
10aTme γT
(16)
where a is the lattice constant, Tme is the melting temperature, T is room temperature in the Kelvin scale, and γ is the Gruneisen constant. Taking CuO nanoparticle of 50 nm in diameter for example, a is equal to 0.5 nm,14 specific value of thermodynamic temperatures Tme/T is 5.5, and γ is about 2, then the calculated result for Λ according to eq 16 is about 14 nm. Table 2 shows the phonon mean free path Λ of common nanoparticle materials that are calculated based on eq 16. Table 2. Phonon Mean Free Path of Common Nanoparticle Materials
Table 1. Thickness of Absorption Layer for Common Base Fluids density ρf (kg/m3)
(14)
where km is the thermal conductivity of nanoparticle bulk material, and χ is dimensionless radius, which is calculated as
(12)
molecular weight M
3χ /4 km 3χ /4 + 1
kp =
where M, ρf, and NA are molecular weight, densitym and Avogadro coefficient of the base liquid, respectively. According to eq 12, the thickness of absorption layer depends only on the properties of the base liquid rather than particle properties, but the compactness of absorption layer is correlated with surface compatibility of particle to the base liquid. It is worthy to note that average diameter, bulk material or surface property of nanoparticle have not been considered in eq 12, therefore the applicability of this equation for the nanometer surface should be further discussed. Thicknesses of absorption layer for common base fluids calculated by eq 12 are shown in Table 1.
base fluid
(13)
materials
melting point Tm (K)
phonon mean free path Λ (nm)
Ag Cu Al2O3 SiO2 CuO
1233.65 1356.18 2272.15 1983.15 1612.33
21 23 35 34 14
3.1.3. Aggregate Structure of Nanoparticles. Furthermore, aggregate structures of nanoparticles at different scales are always present in the base fluid. Thus, aggregate structure of nanoparticles should be considered as a whole and its thermal conductivity should be determined. According to Wang et al.,14 aggregates of nanoparticles can be described using the fractal theory, and the volume fraction of nanoparticles in an aggregate ϕp,cl is calculated as
thickness of absorption layer t (nm)
18074
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⎛ d ⎞D−3 = ⎜⎜ cl ⎟⎟ ⎝ dp ⎠
Article
aggregate as a whole to calculate its thermal conductivity has involved the thermal conductivity of base fluid, which includes the influence of liquid. Therefore, the present method is more accurate compared to the way that using thermal conductivity of nanoparticles kp. In eq 20, the volume fraction of nanoparticles along with the absorption layer ϕ is used instead of the volume fraction of nanoparticle aggregate. In the calculation, we used the volume of nanoparticles and its absorption layer to calculate the volume fraction. If we use the volume of nanoparticle cluster (V2) to calculate the volume fraction, then a lot of free base fluid molecules (V3) would be included into the volume fraction calculation because they are in the range of the cluster diameter, as shown in Figure 3. The
(17)
where dcl is the average diameter of an aggregate, dp is the diameter of a single nanoparticle, and D is the fractal dimension of nanoparticles aggregate. When there is only one nanoparticle in the aggregate (dcl = dp), the volume fraction of nanoparticles in an aggregate ϕp,cl is equal to 1 (ϕp,cl = 1). Then the thermal conductivity of an aggregate is calculated as D−3 D ⎛⎜ dcl ⎞⎟ kcl = ⎜ ⎟ k p + 3 ⎝ dp ⎠
where
D ⎛ dcl ⎞ ⎜ ⎟ 3 ⎝ dp ⎠
⎛ ⎛ ⎞ D − 3⎞ ⎜1 − D ⎜ dcl ⎟ ⎟k f ⎜ 3 ⎜⎝ d p ⎟⎠ ⎟ ⎝ ⎠
(18)
D−3
is the fraction of nanoparticles at the surface
of nanoparticles aggregate. Substituting eq 14 into eq 18, the thermal conductivity of nanoparticles aggregate is calculated as D−3 D − 3⎞ ⎛ 3χ /4 D ⎛⎜ dcl ⎞⎟ D ⎛⎜ dcl ⎞⎟ ⎟ ⎜ kcl = ⎜ ⎟ km + 1 − ⎜ ⎟ kf ⎜ 3 ⎝ dp ⎠ 3χ /4 + 1 3 ⎝ dp ⎠ ⎟ ⎝ ⎠
(19)
Finally, substituting kcl (thermal conductivity of the aggregate) and ϕ (volume fraction of nanoparticles along with the absorption layer in nanofluids) into the Jeffrey model obtains the following equation
Figure 3. Difference between the volume of nanoparticle with its absorption layer and the volume of cluster.
⎛ ⎞ kS 3β 3 9β 3 α + 2 3β 4 = 1 + 3βϕ + ϕ2⎜3β 2 + + + 6 + ...⎟ 4 16 2α + 3 kf ⎝ ⎠ 2 = 1 + 3βϕ + ϕ2(3β 2 + 3β Σ)
where β =
α−1 , α+2
and α =
kcl . kf
(20)
difference between the volume of nanoparticle with its absorption layer and the volume of cluster are large. Because the thermal conductivity of free base fluid is relatively lower than that of nanoparticle cluster or absorption layer, the wrongly counted volume of free base fluid (V3) would make the prediction results to be significantly higher. 3.2. Role of Dynamic Mechanism. The contribution of static mechanisms for thermal conductivity enhancement in nanofluids has been analyzed previously. Moreover, high-speed motions of nanoparticles in base fluid including random translation and rotation will further enhance energy transmission between solid and liquid and increase apparent thermal conductivity of nanofluids. Therefore, the dynamic mechanism of thermal conductivity enhancement owing to micromotions of nanoparticles should be investigated. The influences of nanoparticle motions on thermal conductivity of nanofluids have been discussed as followed. As the scale of a nanoparticle is close to a molecule, the influence of collisions between neighboring nanoparticles or between a nanoparticle and surrounding liquid molecules is obvious, and Brownian motion is the main form of nanoparticle motion. Generally, there are two kinds of forces that a nanoparticle receives, one kind is the viscous drag force, and the other is the irregular force applied by fluid molecules or the other nanoparticles. In this work, the Langevin equation is used to describe the force situation of a nanoparticle; for a particle with mass of m, the force situation in the x direction is given by39
c = 3β2 + 3βΣ is a coefficient of
ϕ2, the value of which is picked from Figure 2 according to the calculation result of lg α. And the maximum and minimum values of c are 4.51 and 0.588, respectively.
Figure 2. Determination for the coefficient of ϕ2.
Equation 20 is the static part of thermal conductivity model for nanofluids, which takes into account the static mechanisms of heat conduction enhancement. In the equation, the thermal conductivity of nanoparticles kp is replaced with thermal conductivity of nanoparticles aggregate kcl. Currently, the theoretical calculation for thermal conductivity of absorption layer kab has not been solved.14 Taking the nanoparticles
m 18075
d2x dx = −a + F (t ) + f dt dt 2
(21)
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Table 3. Comparison between the Prediction Results and Experimental Data author
base fluid
type of nanoparticle
nanoparticle size (nm)
volume fraction (%)
thermal conductivity ratio (knf/kf)
present model
maximum error (%)
Masuda42 Teng43 Chon44 Xie45 Vajjha46 Wang47 Eastman48 Lee49 Xuan24 Li50 Eastman48 Hwang51 Zhang52 Hwang51
water water water EG 60:40% EG/water EG EG EG water water EG water water water
Al2O3 Al2O3 Al2O3 Al2O3 Al2O3 Al2O3 CuO CuO Cu Cu Cu CuO SiO2 SiO2
13 20 13 25 53 28 35 24 100 20 10 33 7 12
1.3−4.3 0.001−0.005 1 1.7−5 1−4 5−8 1−4 1−4 1−5 1−3 0.33−0.55 1 0.5−3 1
1.107−1.318 1.018−1.065 1.081 1.097−1.294 1.069−1.159 1.246−1.404 1.050−1.227 1.060−1.242 1.078−1.434 1.120−1.289 1.041−1.101 1.05 1.016−1.084 1.03
1.097−1.322 1.010−1.043 1.072 1.103−1.303 1.048−1.195 1.276−1.445 1.061−1.245 1.047−1.212 1.090−1.459 1.096−1.293 1.061−1.122 1.08 1.014−1.088 1.03
0.9 2.1 0.9 0.8 3.1 2.9 3.7 3.2 3.1 2.1 2 2.9 0.4 0
where the first item on the right side of the equation represents the viscous drag force and a is the drag coefficient, the second item on the right side of the equation F(t) represents the irregular force and t is time, and the third item represents external field force. In a fluid with viscosity of μ, the drag force on a sphere with radius of r is av, and the drag coefficient a = 6πrμ. In eq 21, ignore the external field force and average the nanoparticle motions with a large number of spheres ⎛ dx ⎞ 2 1 d2(mx 2) a d( x 2) ⎜ ⎟ = − m − + xF(t ) ⎝ dt ⎠ 2 dt 2 2 dt
l=
1 ∂T q = − nmcl c pl 2 ∂x
2
(22)
ϕp(ρc p)p kD = kf 2k f
respectively,
⎤ 2kBT ⎡ a −a / mt − 1)⎥ ⎢ t + (e ⎦ a 2 ⎣m m m
( )
(24)
(25)
2kBT kT t= B t a 3πrμ
kBT 3πrclμf
is the displacement of the cluster per
k k k nf = S + D kf kf kf
For nanoparticles, the precondition of (a/m)t ≫ 1 is satisfied, then the above equation is reduced to Einstein’s relation x2 ≈
(29)
unit time, kB is Boltzmann constant and kB = 1.381 × 10−23 J/K, T is temperature, rcl is the radius of the cluster, and μf is the viscosity of base fluid. Equation 29 is the dynamic part of thermal conductivity enhancement for nanofluids. Combine eq 29 and eq 20 according to the superposition principle and the equation for predicting the thermal conductivity of nanofluids that takes into account both the static and dynamic mechanisms is obtained, which is written as
2
With boundary conditions of t = 0 and x = 0,v = 0, the solution of eq 24 is x2 =
kBT 3πrclμf
where ϕp is the volume fraction of nanoparticles, cp and ρ are specific heat and density of nanoparticle bulk material
(23)
2k T d (x ) a d( x ) + − B = 0. m dt m dt 2
(28)
where mcl represents the mass of nanoparticles-cluster and cp is specific heat capacity of bulk material of nanoparticles. Substituting nmcl = ρϕ to eq 28, the increasing ratio of thermal conductivity caused by high-speed motions of nanoparticles is given by40
And eq 22 is reduced to 2
(27)
For n clusters, in the unit time there are only half number of the clusters move along the direction of +x, then the energy transmitted is expressed as
where the second item on the right side of the equation xF(t ) represents the irregular force, which is equal to zero after the average. Through applying the equipartition theorem of energy the above equation can be written as 1 2 1 mv = kBT 2 2
kBT 3πrclμ
3 6 ⎛ ⎛ ϕp(ρc p)p 2tab ⎞ 2tab ⎞ 2 ⎟ ⎜ ⎟ ⎜ = 1 + 3β ⎜1 + ⎟ ϕp + c ⎜1 + d ⎟ ϕp + 2k d ⎝ ⎝ p ⎠ p ⎠ f
kBT 3πrclμf
(30)
(26)
Compared with existing prediction models for thermal conductivity of nanofluids, the present model takes into account the static and dynamic mechanisms of strengthened heat conduction in nanofluids simultaneously, and possesses more definite physics meaning. In addition, parameters used in the current model are more precise, which ensures the veracity of the prediction result. For instance, the thermal conductivity
According to eq 26, the mean square value for the displacement of sphere is in proportion to time tand temperature T, and is inversely proportional to μ. On the basis of eq 26, considering a nanoparticles-cluster with the turning radius of rcl, the displacement per unit time in the x direction of the cluster is 18076
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of nanoparticles kp is distinguished from that of bulk material of nanoparticles km and used in the prediction model as an independent parameter. And the thermal conductivity of nanoparticles-cluster kcl is introduced as an independent parameter to include the heat conduction of absorption layer, which further improves the prediction accuracy of the present model.
4. VERIFICATION OF THE MODEL WITH EXPERIMENTAL RESULTS Experimental verification of the model has been conducted in the present work, and the results are presented in this section. The comparison between the prediction results and experimental data is shown in Table 3. We have also compared the present prediction model with existing classical theoretical works, as shown in Figures 4−7. In these predictions, the Figure 6. Comparison of thermal conductivity of CuO−EG nanofluid between prediction and experiment.
Figure 4. Comparison of thermal conductivity of Al2O3−H2O nanofluid between prediction and experiment. Figure 7. Comparison of thermal conductivity of Cu−H2O nanofluid between prediction and experiment.
Through comparing the prediction results of the present model and existing experimental data, the present prediction model is proved to be quite effective for predicting thermal conductivity of common nanofluids, as shown in Table 3. For various types of nanofluids (with different materials including metal, metallic oxide and nonmetallic oxide, different volume fractions, or different nanoparticle diameters), the present model gives good predictions. The comparisons between the present prediction model and existing classical theoretical works are shown in Figures 4−7. Figure 4 shows the comparison between model prediction and experimental results42 for thermal conductivity of Al2O3−H2O nanofluids. It could be found in the figure that the present model successfully predicts the thermal conductivity of this type of nanofluids. The trend of prediction results versus nanoparticles volume fraction agrees well with that of experiments, and the error between prediction and experimental value is lower than 1%. Comparisons between prediction and experimental results45 for thermal conductivity of Al2O3−ethylene glycol nanofluids are shown in Figure 5, and the prediction results for the Al2O3−H2O nanofluids are very close to the experimental results as well. Comparisons between prediction and experimental results48 for thermal conductivity of CuO− ethylene glycol nanofluids are shown in Figure 6, and the
Figure 5. Comparison of thermal conductivity of Al2O3−EG nanofluid between prediction and experiment.
diameter of the cluster dcl is equal to 500 nm, the fractal dimension is 2.64,35 and the thickness of the absorption layer is estimated to be 1 nm.41 And the other parameters used in the predictions are shown in Table 4. The parameters listed in Table 4 are in accordance with the existing experimental works.24,42−52 18077
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Table 4. Data Used in Theoretical Prediction density ρ (kg/m3) specific heat cp (J/(kg·K)) thermal conductivity k (W/(m·K)) viscosity μ (kg/(m·s))
Cu
CuO
Al2O3
SiO2
8930 386 398
6310 525 32.9
4000 880 36
2650 733 1.38
H2O
EG
0.613 1.004 × 10−3
0.253 1.99 × 10−2
maximal error is 3.7%, which is acceptable. Comparisons between prediction and experimental results24 for the thermal conductivity of Cu−H2O nanofluids are shown in Figure 7, and the maximal error is 5.82%. From these figures, it is not difficult to find that the prediction results of the present model are more accurate than other existing models because more comprehensive mechanisms for thermal conductivity enhancement have been taken into account. However, it should be noted that the dispersion of the nanoparticles and aggregate structures, which have great influence on thermal conductivity of nanofluids, is heavily depended on the preparation process of nanofluids. Therefore, for specific nanofluids, if the average diameter scale and fractal dimension of nanoparticles-cluster could be measured by experiments, then more reliable prediction results would be obtained by using the prediction model. Figure 9. Comparing thermal conductivity of 4 nm Cu−Ar nanofluid between prediction and MD simulation.
5. VERIFICATION OF THE MODEL WITH MD SIMULATION The prediction results of the present model have also been compared to the MD simulation results. Considering Cu−Ar nanofluids containing spherical nanoparticles with diameters of 2, 4, and 6 nm, the predicted and MD simulated results are shown and compared in Figures 8−10, respectively. In the
Figure 10. Comparison of thermal conductivity of 6 nm Cu−Ar nanofluid between prediction and MD simulation.
Table 5. Data Used in MD Calculations Cu density ρ (kg/m ) specific heat cp (kJ/(kg·K)) thermal conductivity k (W/(m·K)) viscosity μ (kg/(m·s)) 3
Figure 8. Comparison of thermal conductivity of 2 nm Cu−Ar nanofluid between prediction and MD simulation.
predictions, the fractal dimension number is 2.64,35 and the thickness of absorption layer is determined to be 0.5 nm by MD simulation.53 The other parameters used in the predictions are listed in Table 5.54 It can be found from these figures that the predictions by the present model coincide with the MD simulated results well. Comparisons of the predictions and MD simulated results for the case that the nanofluid contains 2 nm diameter nanoparticle with four different nanoparticle volume fractions are shown in Figure 8. There is a error between the two cases of 1% and 3% volume fractions, and the biggest error, 6.58%, is obtained in
8930 0.386 398
liquid argon
0.128 2.64 × 10−4
the case of the 2% volume fraction. Comparisons of the predictions and MD simulated results for the case of 4 nm nanoparticle nanofluids with four different volume fractions are shown in Figure 9. It can be found that it has better agreement between the predictions and MD simulated results, and the biggest error, 4.8%, is obtained in the case of the 2% volume fraction. Comparisons of the predictions and MD simulated results for the case of 6 nm nanoparticle nanofluids with four different volume fractions are shown in Figure 10. The 18078
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M = molecular weight m = mass NA = Avogadro coefficient q = heat flux r = radius r ̅ = geometric average of aggregate T = temperature t = thickness
prediction results go closer to that of MD simulation, while the biggest error, 2.8%, is still in the case of the 2% volume fraction. Overall, the predictions of the present model agree well with MD simulations, which further proves the advantages of the present prediction model.
6. CONCLUSIONS In this paper, both the static and dynamic mechanisms for heat conduction enhancement in nanofluids have been considered to establish a prediction model for thermal conductivity. The parameters in the model have definite physical meaning and are more precise. The prediction results of the present model are in good agreement with the experiments and MD simulations. (1) On the basis of effective medium theory, we have proposed a model for thermal conductivity of nanofluids. In this model the roles of static and dynamics mechanisms are considered simultaneously. The parameters in the model have definite physical meaning and are more precise. For instance, the thermal conductivity of nanoparticles kp is replaced with thermal conductivity of nanoparticles aggregate; and the volume fraction of nanoparticles along with the absorption layer in nanofluids ϕ is used instead of the volume fraction of aggregate. (2) By considering roundly the mechanisms for heat conduction enhancement, the modeling for thermal conductivity of nanofluids is closer to the real situation. For various types of spherical nanoparticle suspensions (with different materials including metal, metallic oxide and nonmetallic oxide, different volume fractions, or different nanoparticle diameters), the present model gives good predictions. And the prediction results of the present model also coincide with MD simulation, which further proves the advantages of the present model.
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Greek Letters
γ Λ μ ρ σ τ ϕ χ
Gruneisen constant phonon mean free path viscosity density root-mean-square deviation of radius total relaxation time constant volume fraction dimensionless radius
Subscripts
ab cl D d f i m me nf p p,cl S total
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AUTHOR INFORMATION
absorption layer cluster dynamic dispersant fluid interface material melting nanofluids particle particle in cluster static total
REFERENCES
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Corresponding Author
*Z. Shen. Tel./Fax: +86-631-5687863. E-mail: shenzj@hitwh. edu.cn. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS China Postdoctoral Science Foundation funded project (Grant No. 2014T70330, 2013M540284). Project (HIT.NSRIF.2015116) supported by Natural Scientific Research Innovation Foundation in Harbin Institute of Technology. Project (HIT (WH) 201301) supported by Scientific Research Foundation of Harbin Institute of Technology at Weihai. We acknowledge the reviewers’ comments and suggestions very much, which were valuable in improving the quality of our paper.
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NOMENCLATURE A = heat exchange area a = lattice constant cp = specific heat capacity D = fractal dimension d = diameter H = heat transfer coefficient K = heat resistance k = thermal conductivity kB = Boltzmann constant l = displacement per unit time 18079
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