Effective thermal conductivity of nanofluids – A new model taking into consideration Brownian motion

International Journal of Heat and Mass Transfer 99 (2016) 532–540

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Effective thermal conductivity of nanofluids – A new model taking into consideration Brownian motion Kedar N. Shukla a, Thomas M. Koller a, Michael H. Rausch a,b, Andreas P. Fröba a,b,⇑ a b

Erlangen Graduate School in Advanced Optical Technologies (SAOT), Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Paul-Gordan-Straße 6, D-91052 Erlangen, Germany Lehrstuhl für Technische Thermodynamik (LTT), Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Am Weichselgarten 8, D-91058 Erlangen, Germany

a r t i c l e

i n f o

Article history: Received 3 August 2015 Received in revised form 26 February 2016 Accepted 30 March 2016

Keywords: Effective thermal conductivity Modeling Nanofluids Nanoparticles

a b s t r a c t In this study, a new analytical model for the effective thermal conductivity of liquids containing dispersed spherical and non-spherical nanometer particles was developed. In addition to heat conduction in the base fluid and the nanoparticles, we also consider convective heat transfer caused by the Brownian motion of the particles. For nanoparticle suspensions, the latter mechanism has significant influence on the effective thermal conductivity, which is reduced compared to a system in which only conduction is considered. The simple model developed allows for the prediction of the effective thermal conductivity of nanofluids as a function of volume fraction, diameter, and shape of the nanoparticles as well as temperature. Due to the inconsistency of experimental data in the literature, the model has been compared with the established Hamilton–Crosser model and other empirical models for the systems aluminum oxide (Al2O3) and titanium dioxide (TiO2) suspended in water and ethylene glycol. The theoretical estimates show no anomalous enhancement of the effective thermal conductivity and agree very well with the Hamilton–Crosser model within relative deviations of less than 8% for volume fractions of spherical particles up to 0.25. In accordance with the Hamilton–Crosser model for non-spherical particles, our model reveals that a more distinct increase in the enhancement of the effective thermal conductivity can be achieved using non-spherical nanoparticles having a larger volume-specific surface area. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Over the past decade, the dispersions of nanometer-sized particles with typical diameters ranging from 1 to 100 nm in a liquid medium, usually called nanofluids, have been reported to possess substantially higher thermal conductivities than anticipated from Maxwell’s classical theory [1]. As shown in the review by Tertsinidou et al. [2], a large number of experimental results have reported an anomalous increase in the thermal conductivity of nanoparticle suspensions. This would make them very attractive as potential heat transfer fluids for many applications. However, results from other experiments have not shown any anomalous increase in thermal conductivity [2–5]. This has triggered controversy regarding the actual value of the thermal conductivity of nanofluids and the reliability of the experimental methods. To solve this problem, a fundamental understanding of the heat transfer mechanisms

⇑ Corresponding author at: Erlangen Graduate School in Advanced Optical Technologies (SAOT), Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Paul-Gordan-Straße 6, D-91052 Erlangen, Germany. Tel.: +49 9131 85 29789; fax: +49 9131 85 25851. E-mail address: [email protected] (A.P. Fröba). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.03.129 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

present in suspensions of nanoparticles is necessary, which may lead to a physical model for the description of their effective thermal conductivity. Debate continues regarding the potential causes of abnormal behavior ranging, e.g., from the effect of nanoparticle clustering [6] over the layering of the liquid at the liquid-particle interface [7] to the role of the Brownian motion of nanoparticles [8]. Regarding the impact of Brownian motion on the effective thermal conductivity, one opinion in the literature [8–10] is that this effect is the main reason for the high thermal conductivity of nanofluids. Yet, the molecular dynamics simulations performed by Evans et al. [11] reveal that the enhancement effect due to Brownian motion of the nanoparticles is rather insignificant. Therefore, disagreement also exists among researchers on the role of Brownian motion of nanoparticles. Maxwell was one of the first to use the effective medium theory to study the properties of a solid bulk material consisting of one material distributed as spherical inclusions within a continuous material [1]. His static model for the effective electrical conductivity of solid-based systems was used by Hamilton and Crosser [12] to determine the effective thermal conductivity of two-phase, two-component solids because of the similar mathematical

K.N. Shukla et al. / International Journal of Heat and Mass Transfer 99 (2016) 532–540

533

Nomenclature a abf A b dp dp,eq

empirical parameter in model of Lin and Lu [13] thermal diffusivity of base fluid (m2 s1) surface area of particle (m2) empirical parameter in model of Lin and Lu [13] diameter of spherical nanoparticle (m) volume-equivalent diameter of non-spherical nanoparticle (m) convective heat transfer coefficient (W m2 K1) thermal conductivity (W m1 K1) Boltzmann constant (1.3806488  1023 J K1) edge length of the cube (m) mass of nanoparticle (kg) empirical shape factor of particle number of nanoparticles Nußelt number Peclet number Prandtl number thermal resistance of bulk fluid due to heat conduction (K W1) effective thermal resistance of nanofluid (K W1) total thermal resistance of nanoparticle (K W1) thermal resistance of nanoparticle due to heat conduction (K W1)

h k kb L m n N Nu Pe Pr Rbf Reff Rp Rp,cond

formulations of the two transport phenomena. Their model is also applicable for systems of liquids containing solid particles with different shapes and predicts their effective thermal conductivity keff by

keff kp þ ðn  1Þkbf  ðn  1Þuðkbf  kp Þ ¼ : kp þ ðn  1Þkbf þ uðkbf  kp Þ kbf

ð1Þ

The terms kbf and kp denote the thermal conductivities of the base fluid (bf), that is the continuous phase, and the solid particle (p), that is the dispersed phase. u is the volume fraction of the dispersed particles, while n is an empirical shape factor. The latter is connected with the sphericity w via

n¼

3 : w

ð2Þ

The sphericity of a particle is defined as the ratio of the surface area of a sphere, Ap,sph, having the same volume as the particle, Vp, to the surface area of the particle, Ap, according to

w¼

p1=3 ð6V p Þ2=3 Ap

:

ð3Þ

Particles with strong deviations from spherical shape show smaller w and thus larger n values. For spherical particles with w = 1 and n = 3, Eq. (1) can be simplified to the original Maxwelllike equation

keff kp þ 2kbf  2uðkbf  kp Þ ¼ kp þ 2kbf þ uðkbf  kp Þ kbf

ð4Þ

for predicting the enhancement of the effective thermal conductivity of nanofluids containing spherical solid particles. The effective model of Hamilton and Crosser [12] can be applied to low particle volume fractions and shows no dependence on the particle diameter and a very weak dependence on the temperature. For kp  kbf and small u, Eq. (1) reduces to

keff ¼ 1 þ nu: kbf

ð5Þ

Rp,conv Re T up V

thermal resistance of nanoparticle due to convection (K W1) Reynolds number temperature (K) velocity of nanoparticle (m s1) volume (m3)

Greek symbols mbf kinematic viscosity of base fluid (m2 s1) u volume fraction of nanoparticles qp density of nanoparticle (kg m3) w sphericity of nanoparticle Subscripts bf base fluid cond pure conduction conv convection eff effective non-sph non-spherical p particle sph spherical

Eq. (5) presents a simple linear relation for the effective thermal conductivity of diluted suspensions. Based on the continuum model of Hamilton and Crosser [12], there have been some modifications considering additional effects. For example, Lu and Lin [13] considered near- and far-field pair interactions applicable to spherical and non-spherical inclusions and modified Eq. (5) to a second-order polynomial,

keff ¼ 1 þ au þ bu2 : kbf

ð6Þ

For spherical isotropic inclusions, the constants a and b are given as a = 2.25 and b = 2.27 for kp/kbf = 10 as well as a = 3.00 and b = 4.51 for kp/kbf ? 1. For non-spherical inclusions, anisotropic effects have to be considered which results in the formulation of an effective thermal conductivity tensor for an anisotropic medium. Besides the models described above, there are numerous other predictive methods – most of them empirical or complex – for the effective thermal conductivity that cannot all be mentioned here. A review about theoretical studies on the effective thermal conductivity of nanofluids is given by Kleinstreuer and Feng [14]. Recently, Tertsinidou et al. [2] evaluated extensive data on the effective thermal conductivity of nanoparticle suspensions containing aluminum oxide (Al2O3), copper (Cu), copper oxide (CuO), and titanium dioxide (TiO2) suspended in water (H2O) and ethylene glycol (EG). When results for the same thermodynamic system are obtained using proven experimental techniques, they concluded that the effective thermal conductivity of nanofluids exhibits no inconsistency with the continuum model of Hamilton and Crosser [12]. The broad span of values for the enhancement of effective thermal conductivity in the literature, with relative deviations of several tens of percents for a given system, is rather attributed to poor characterization of the thermodynamic system and/or the application of experimental techniques of unproven validity [2]. A systematic benchmark study [15] on the thermal conductivity of nanofluids, performed over 30 laboratories worldwide and using a variety of experimental techniques, has also

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u¼

pd3p 6L3

:

ð8Þ

Thus, the particle diameter can be related to the side length of the elementary cube by

dp ¼


1=3 6u

p

ð9Þ

L:

Based on Eq. (9), the model is restricted to dp/L < 1. This imposes a condition on the applicable particle volume fraction of u < p/6  0.52, i.e., the maximum packing of spherical particles in a nanofluid. However, such a dense packing is not realistic for a nanofluid in terms of its synthesis and the suspension stability. Thus, only reasonable values of u < 0.25 are considered within this study. Regarding non-spherical particles such as cylinders or plates of volume Vnon-sph, they can be regarded as theoretical spherical particles having the same volumes as the non-spherical particles. The volume-equivalent particle diameter of the non-spherical particles, dp,eq, is Fig. 1. Conceptual three-dimensional sketch of a cube with length L containing a spherical nanoparticle of diameter dp and bulk fluid molecules.

shown good agreement between the experimental data and the corresponding data predicted by the Hamilton–Crosser model [12]. Nevertheless, the continuum model does not explicitly account for effects that are induced by nanoparticles with respect to the effective thermal conductivity of nanofluids. Nanoparticles may cause additional energy transport due to Brownian motion because of their small size, large specific surface area, and morphology. The motion of nanoparticles and base fluid molecules is affected by the combined effect of hydrodynamic and Brownian forces that produce micro-convection in the nanofluids. The aim of the present work was to develop a new model for the effective thermal conductivity of macroscopically static nanofluids taking into account the heat transfer mechanisms caused by convection as well as thermal conduction of the particles and the base fluid. A comparison of the model with the established model of Hamilton and Crosser [12] and the predictions of Lu and Lin [13] is drawn for selected nanofluid systems as a function of the parameters volume fraction, particle diameter, particle shape, and temperature. 2. Description of the model To develop an analytical model for the thermal conductivity of liquids with suspended nanoparticles, the thermal resistances of the base fluid and the nanoparticles as well as of convection induced in the fluid due to Brownian motion of the nanoparticles are determined. Other possible heat transfer effects in form of thermal radiation or thermal diffusion of the nanoparticles due to a temperature gradient were found to be negligible. In the following, the model is derived on basis of a nanofluid system containing spherical particles. Furthermore, analogies or differences in connection with the modeling of nanofluids containing non-spherical particles such as cylinders, cubes, or plates are given. It is assumed that N nanoparticles of spherical shape with diameter dp are uniformly suspended in a volume V of the nanofluid. The volume fraction of the nanoparticles u is then defined by

Npdp : 6V

dp;eq ¼


1 6V non-sph 3

p

:

ð10Þ

This approach allows for the same formulation of the volume fraction of non-spherical particles in the bulk fluid as given in Eq. (8). Furthermore, it can be used to model the heat transfer in such systems analogously to that described in the following for systems with spherical particles. The basic idea of the present modeling approach is to treat the heat transfer problem in connection with nanofluids by the analysis of the corresponding thermal resistances present in such systems. Fig. 2 illustrates the corresponding circuit diagram for the total thermal resistance of the nanofluid Reff based on the cube shown in Fig. 1. The thermal resistance of the base fluid, Rbf, is considered to be parallel to the thermal resistance of the nanoparticle, Rp. Thus, Reff can be expressed by

1 1u u ¼ þ : Reff Rbf Rp

ð11Þ

Arranging the resistances of the bulk fluid and the particles in parallel is reasonable because a heat flux can be either conducted through the base fluid or through the particle along a onedimensional temperature gradient. The key for a realistic description of the thermal resistance of the nanofluid is that the base fluid as a continuum fluid phase is analogously treated as a continuum resistance. To account for the volumes in which the resistances of the two phases are present, the inverse values of the thermal resistances Rbf and Rp are weighted in Eq. (11) by the corresponding volume fractions (1u) and u. Here, the circumstance that the volume fractions in the elementary cube are the same as those in the total nanofluid system can be employed.

3

u¼

ð7Þ

The volume V is now divided into N equal parts such that each nanoparticle is located in a cube with a side of length L. Fig. 1 depicts a three-dimensional sketch of such a nanoparticle-cube system. From Eq. (7) it follows that

Fig. 2. Circuit diagram for the thermal resistances of the bulk fluid due to conduction and of the nanoparticle due to conduction and Brownian convection.

K.N. Shukla et al. / International Journal of Heat and Mass Transfer 99 (2016) 532–540

Reff and Rbf are defined by the corresponding thermal conductivities keff and kbf and the geometries of the cube in the form of

Reff ¼

L keff L2

¼

1 keff L

ð12Þ

Rbf ¼

L 2

kbf L

¼

1 kbf L

:

ð13Þ

The thermally induced Brownian motion creates a thermal boundary layer on the spherical nanoparticles. The thermal resistance caused by heat transfer between the nanoparticles and the boundary layer is modeled with the resistance of the sphere due to thermal conduction, Rp,cond, and with that due to convection at its surface, Rp,conv, in series,

Rp ¼ Rp;cond þ Rp;conv :

ð14Þ

A circuit in series in connection with the nanoparticle can be justified because any heat flux associated with the particle needs to cross the boundary layer acting as first thermal resistance and is conducted through the particle material being a second thermal resistance. Thus, the sum of these two resistances determines the achievable enhancement of the effective thermal conductivity of nanofluids compared to the thermal conductivity of the corresponding base fluid. In Eq. (14), the thermal resistance due to thermal conduction of the particle is modeled by d

Rp;cond ¼

1 þ 0:5 Lp : 2pkp dp

ð15Þ

This expression reflects the thermal resistance for an isothermal spherical nanoparticle with thermal conductivity kp buried in a semi-infinite medium with insulated surface [16]. Here, it is assumed that the nanoparticle is submerged in the base fluid under the influence of a temperature gradient. Using instead the expression for the thermal resistance of an isothermal sphere buried in an infinite medium without any temperature gradient (Rp,cond = 1/ (2pkpdp)) [16], no significant difference for the modeled effective thermal conductivity of nanofluids is found compared to the use of Eq. (15). For the modeling of Rp of non-spherical particles, it would be plausible to use corresponding expressions from Ref. [16]. Yet, the solutions regarding non-spherical particles are only valid for certain geometrical restrictions and the value of Rp depends on how the particles are aligned with respect to the direction of the heat flux. For example, a long cylindrical particle shows less resistance if perpendicularly oriented to the heat flux compared to a parallel configuration. Furthermore, it is impossible to predict the distribution of the spatial arrangement of the individual particles in the nanofluid. All this impedes an analytical modeling of the effective value of Rp for non-spherical anisotropic particles in comparison with that for isotropic spherical particles. Our calculations have shown that the thermal conduction resistance of non-spherical particles averaged over a broad range of spatial positions is comparable to that of a spherical particle with the same volume. Apparently, the volume of the particle itself, regardless of its shape, is responsible for the heat transfer due to thermal conduction through the particle in the nanofluid. Hence, the analogy of using the volume-equivalent diameter dp, eq in Eq. (15) is straightforward to model Rp for non-spherical particles. The second term in Eq. (14), the thermal resistance caused by convective heat transfer

Rp;conv ¼

1 ; Ap h

depends on the surface area of the particle Ap and the convective heat transfer coefficient h. Eq. (3) can be used to derive an expression for Ap for particles of arbitrary shape,

ð16Þ

Ap;sph pdp ¼ : w w 2

Ap ¼

and

535

ð17Þ

Decreasing sphericity of the particles (w < 1) results in an increasing surface area and, according to Eq. (16), in a decreasing value for Rp,conv. The heat transfer coefficient for a spherical particle is determined by

h¼

kbf Nu; dp

ð18Þ

where dp represents the characteristic length and Nu is the Nußelt number. For a flow around a sphere, the latter can be correlated with the Peclet number Pe by [17]

Nu ¼ 2 þ



 Pe 1 2 Pe 1 3 Pe þ Pe ln þ 0:2073Pe2 þ Pe ln : 2 4 2 16 2

ð19Þ

Eq. (19) holds for Pe < 1 and Reynolds numbers Re  1, which is fulfilled for macroscopically static nanofluids. The first term in Eq. (19) represents the minimum Nu number of 2 for stagnant spherical bodies suspended in a fluid, i.e., Re = 0. In this case, the convective heat transfer between particles and fluid is at its minimum and corresponds to pure conduction heat transfer in the fluid. The dimensionless numbers Pe, Re, and the Prandtl number Pr are given by

Pe ¼ Re  Pr

ð20Þ

with

Re ¼

up dp

mbf

ð21Þ

and

Pr ¼

mbf abf

:

ð22Þ

In Eqs. (21) and (22), up, mbf, and abf are the velocity of the nanoparticles as well as the kinematic viscosity and the thermal diffusivity of the bulk fluid. Also for the calculation of Re for a sphere, dp is the characteristic length. The velocity up required in Eq. (21) can be calculated from the root mean square velocity of a spherical nanoparticle of mass m due to thermal motion by

rffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3kb T 18kb T ¼ up ¼ : m pd3p qp

ð23Þ

Here, kb (=1.38064881023 JK1) is the Boltzmann constant, T the temperature, and qp the density of the nanoparticle. Eq. (23) is commonly used for the calculation of the Brownian velocity of nanoparticles in liquid fluids [9,10,18]. In our model, up is considered to be independent of the particle volume fraction. To our knowledge, no empirical correlations are available in literature to model the convective heat transfer coefficient in connection with finite non-spherical particles of cylindrical, cubic, or plate shape. Thus, we follow the analogy of treating the nonspherical particles as volume-equivalent spheres with diameter dp,eq. This diameter is used for the calculation of up, Re, Pe, Nu, and h according to Eqs. (23), (21), (20), (19), and (18), respectively. By substituting Eqs. (9), (12), (13), (15)–(18) into Eq. (11), we obtain the simple expression for the dimensionless effective thermal conductivity keff/kbf of nanofluids containing spherical or non-spherical particles

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2 31  1=3 6u
1=3
 1 þ 0:5 p keff 6 k w 7 bf þ 5 : ¼ ð1  uÞ þ p u4=3 6 4 Nu kbf p 2 kp ð24Þ Three different contributions can be seen in Eq. (24). The first term (1u) considers the influence of the bulk fluid in form of its volume fraction while the second term is related to the nanoparticles. In the latter term, the brackets include the contributions from conduction through the particles (associated with kp) and from convective heat transfer between the particles and the bulk fluid (given by w/Nu).

3. Results and discussion To analyze the quality of our model for the description of the effective thermal conductivity of nanofluid systems by Eq. (24), we selected the four simple systems Al2O3/H2O, TiO2/H2O, Al2O3/ EG, and TiO2/EG. These systems are common, inexpensive, and have been widely tested experimentally [2]. The origin of the thermophysical properties of the nanoparticles and base fluids at atmospheric pressure required for all subsequent calculations is summarized in the following. The thermal conductivity kp and density qp, which were both assumed to be independent of temperature, are taken taken from Tertsinidou [19] and Patel et al. [20] for Al2O3 (kp = 40 Wm1K1, qp = 3970 kgm3) and from Tertsinidou [19] for TiO2 (kp = 11.8 Wm1K1, qp = 4230 kgm3). The properties kbf, qbf, mbf, and abf for water (H2O) were taken from the Refprop database [21]. The corresponding data for ethylene glycol (EG) taken from Patel et al. [20] were correlated with appropriate fits as a function of temperature between 283.15 K and 363.15 K. For spherical particles which are commonly used in nanofluids and are of primary interest in this study, particle diameters were varied from 5 nm up to 50 nm. For comparison, also non-spherical nanoparticles such as cylindrical and plated particles with varying length-to-diameter ratio as well as cubic particles were investigated. In Fig. 3, the Peclet number Pe and the corresponding Nußelt number Nu are depicted as a function of the nanoparticle diameter dp of spheres for the four studied nanofluids at 300 K. Due to the use of a volume-equivalent diameter dp,eq for non-spherical particles, the same results are valid for corresponding systems. With increasing dp, the Pe number decreases almost asymptotically due to the decrease in the Re number where Re  d1/2 . For the conp sidered nanofluids, the type of liquid has a stronger influence on the Pe number than the type of nanoparticle. The trends for the Nu number are similar to those for the Pe number because for small Pe numbers, Eq. (19) is approximately Nu  2 + Pe/2 and hence also scales with d1/2 . No significant variation in both dimensionless p numbers is observed for varied temperatures. For example, Nu decreases from 2.0440 at 283.15 K to 2.0417 at 363.15 K for the Al2O3/H2O system with dp = 30 nm. For all studied systems, the decrease in the Pr values with increasing temperature compensates for the increase in the Re values at a given particle diameter. It should be noted that Jang and Choi [8] considered Nu = Re2Pr2 by assuming Reynolds and Prandtl numbers on the order of 1 and 10 for the typical nanofluids. While the Pr numbers calculated in this study for the systems investigated are comparable to those reported by Jang and Choi [8], their Re values are by almost two orders of magnitude larger than those calculated here. Their results imply enhanced Brownian convection and hence an anomalous increase in the effective thermal conductivity of the nanofluids. Contrary to their assumptions, our calculations show that the Reynolds number is much smaller than 1 and the corresponding Nu

Fig. 3. Peclet number Pe (a) and Nußelt number Nu (b) as a function of the diameter of the spherical nanoparticles dp for the studied nanofluids at a temperature of T = 300 K.

Fig. 4. Ratio of thermal resistance due to Brownian convection at the nanoparticles to that due to thermal conduction through the nanoparticles for the studied nanofluids containing spherical nanoparticles with a diameter of 30 nm at T = 300 K.

number is always about 2.1 and thus close to the minimum value of 2 for the nanoparticle suspensions, cf. Fig. 3b. In the following, the influence of the thermal resistance for the convective heat transfer at nanoparticles on the resulting effective thermal conductivity of the nanofluid is considered. The second term of Eq. (24) consists of the contributions from the thermal conduction through the nanoparticles as well as from the Brownian convection. The two corresponding resistances are modeled to be in series. In Fig. 4, the ratio of the thermal resistances attributed to convection and to conduction through the nanoparticles is shown as a function of the volume fraction u for all four systems containing spherical particles with a diameter of 30 nm at a temperature of 300 K.

K.N. Shukla et al. / International Journal of Heat and Mass Transfer 99 (2016) 532–540

The thermal resistance due to micro-convection in the boundary layer at the nanoparticle for all systems is by at least one order of magnitude larger than that due to conduction through the nanoparticle itself. Increasing particle volume fractions result in only a slight decrease in the thermal resistance ratio. The same trends are also found for varied particle size and temperature. This result can be related to the fact that Nu is close to 2 for all nanoparticle systems studied here. Thus, the convective resistance caused by Brownian motion and being proportional to 1/Nu  1/2 is more significant in Eq. (24) than the conduction resistance term proportional to kbf/kp. Depending on the liquid-nanoparticle combination, kbf/kp values between about 1/20 for TiO2/H2O and 1/160 for Al2O3/ EG can be found. In consequence, it can be deduced that the convective heat transfer resistance is the limiting factor for the enhancement of the effective thermal conductivity of nanofluids and may not be neglected in corresponding models. For non-spherical particles, the value for Rp,conv/Rp,cond is smaller than for spherical particles at a given particle volume, particle volume fraction, and nanofluid system. While Rp,cond is considered to be constant in our model, Rp,conv of non-spherical particles is smaller than that of spherical particles due to the larger surface area and thus lower sphericity w. From this, it can already be concluded here that the effective thermal conductivity of nanofluids with non-spherical particles should be larger than that of nanofluids with spherical particles with equal particle volume. More details will be given later on. In Fig. 5, the percentage enhancement factor (keff/kbf1) calculated according to Eq. (24) is shown as a function of the volume fraction u of spherical particles for the two systems Al2O3/EG and TiO2/H2O at a temperature of 300 K. As described above, these two systems differ most strongly regarding the ratio kbf/kp, which allows for a better visualization of the influence of the volume fraction, particle diameter, and convection contribution. Regarding the influence of convection, the enhancement found on basis of our model (Eq. (24)), which includes the effect of Brownian convection, is compared with a ‘‘theoretical” enhancement which does not account for this effect. The latter system would consist of static particles suspended in the liquid, neglecting any thermal resistance in the boundary layer where heat is transferred from the liquid to the solid particle. In this case, Rp,conv = 0 in Eq. (14) which is equivalent to neglecting the term w/Nu = 1/Nu in Eq. (24).

Fig. 5. Percentage enhancement factor 100  (keff/kbf  1) on the basis of Eq. (24) as a function of the volume fraction u for different diameters dp of spherical particles at T = 300 K for the nanofluids Al2O3/EG and TiO2/H2O.

537

It is obvious that neglecting the thermal resistance contribution from the convective heat transfer between particles and base fluid results in much larger enhancement factors because the thermal resistance of the nanoparticle is significantly reduced to only the resistance of conduction through the particle Rp,cond. (see Fig. 5). Consequently, the additionally considered thermal resistance between contacting particles and fluid molecules reduces the effective thermal conductivity of the nanofluid compared to a theoretical system featuring no convective resistance between particles and fluid. Of course, an enhancement of the convective heat transfer with, e.g., increasing Re, Pe, and thus Nu numbers decreases the corresponding resistance, but this effect is rather small with respect to keff. This is caused by the almost stagnant flow behavior of nanoparticles in the fluid. These findings valid for nanofluids with spherical particles are in contradiction to the widely spread opinion in the literature [8–10] that Brownian convection is mainly responsible for the enhancement of the effective thermal conductivity of nanofluids. Fig. 5 shows that the effective thermal conductivity increases with increasing u, with increasing slope for increasing u values. Mainly due to the about eight times larger kbf/kp ratio for Al2O3/ EG compared to TiO2/H2O, enhancement in the effective thermal conductivity is stronger for Al2O3/EG for a given particle size. For u = 0.25 and dp = 5 nm, a difference in the enhancement of about 11% is found for the two nanofluids. For all the systems studied, keff/kbf decreases with increasing particle size. This can be attributed to the decrease in the Re, Pe, and Nu numbers as shown in Fig. 3. Since Nu  d1/2 approximately holds, the effective thermal p conductivity of the nanofluids does not change significantly for dp > 25 nm. The influence of temperature on the effective thermal conductivity enhancement of nanofluids is even less pronounced for various u and dp values. The reason for this has been given in the discussion on the Pe and Nu numbers. For example, considering the system Al2O3/EG containing nanoparticles with dp = 5, 25, and 50 nm at u = 0.25, the differences in the enhancement factors in the temperature range between 283.15 K and 363.15 K are 1.3%, 0.43%, and 0.30%, respectively. To test the reliability of the present model, a comparison with experimental results seems straightforward. However, the current situation of experimental data in the literature shows an obscure picture of thermal conductivity enhancement for nanofluids in general [2]. For the system Al2O3/EG at 298 K, for instance, Oh et al. [22] measured the effective thermal conductivity of nanofluids containing spherical nanoparticles with a diameter of 45 nm by using a 3x method with an uncertainty of 2% and obtained an enhancement of about 7.5% with respect to the thermal conductivity at u = 0.03. In contrast, the measurement results of Xie et al. [23] from a transient short hot wire method specified with an uncertainty of less than 0.5% show an enhancement of about 27% for the same system and particle volume fraction, only using smaller spherical particles with a diameter of 26 nm. The enhancement of 9.0% predicted by the Hamilton–Crosser model [12] agrees with the result of Oh et al. [22] within their given uncertainty. Given this data discrepancy, we preferred to check our model by comparing with other common models in the literature, where the effective medium model of Hamilton and Crosser [12] is considered to be one of the most reliable ones [2]. At first, nanofluids containing spherical particles are investigated. In Fig. 6, the enhancement of the thermal conductivity (keff/kbf  1) predicted from Eq. (24) is given as a function of the volume fraction u for all four nanofluids at a particle diameter of dp = 25 nm and a temperature of T = 300 K. In addition to our calculations, we also applied the continuum model of Hamilton and Crosser [12] given by Eq. (4) and the empirical model of Lu and Lin [13] according to Eq. (6) with the reported values for a and b for the cases kp/kbf = 10 as well as kp/kbf ? 1.

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Fig. 6. Percentage enhancement factor 100  (keff/kbf  1) as a function of volume fraction u at a temperature of T = 300 K for the nanofluids (a) Al2O3/H2O, (b) Al2O3/EG, (c) TiO2/H2O, and (d) TiO2/EG containing spherical nanoparticles calculated from the present model (Eq. (24), dp = 25 nm), the Hamilton-Crosser model [12] (Eq. (4)), and the Lu-Lin model [13] (Eq. (6) for the ratios kp/kbf = 10 and kp/kbf ? 1).

For all the nanofluid systems tested, very good agreement between our model and the continuum model of Hamilton and Crosser [12] for various particle volume fractions is found. Regarding the enhancement factors, the relative deviations between the continuum model and ours are less than 8% for volume fractions between 0 and 0.25 for the studied conditions. Also for varied temperatures and particle diameters, this deviation is not exceeded due to the very weak influences of T and dp on keff/kbf in our model and the fact that the Hamilton–Crosser model [12] also shows only very weak temperature dependence and depends on the particle diameter. While our model provides lower values than the continuum model for low volume fractions, there is crossover for particle volume fractions that seems to depend on the kp/kbf value of the nanofluid. The lower this thermal conductivity ratio, the lower is the u value at which our model exceeds the Hamilton–Crosser model [12]. In this context, a comparison of our model with that of Lu and Lin [13] provides further information. Their correlations show that an increasing kp/kbf value results in a larger enhancement of the effective thermal conductivity due to the addition of particles with larger thermal conductivity. Our model shows the same trend when the four nanofluids with different kp/kbf ratios are compared, see Fig. 6. While for the TiO2/H2O system with kp/kbf = 19.3 our model fits better with the Lu-Lin model [13] for kp/kbf = 10, there is better agreement between our model in the case of the

Al2O3/EG system with kp/kbf = 159.1 and their model for kp/kbf ? 1. In conclusion, the very good agreement of the established model of Hamilton and Crosser [12] as well as other empirical models [13] with our model indicates that the convective heat transfer caused by the Brownian motion of the nanoparticles needs to be considered for the heat transfer in nanofluids. Further evidence for this can be given by the investigation of nanofluids with non-spherical particles. In Fig. 7, the enhancement of the thermal conductivity (keff/kbf  1) modeled according to our prediction from Eq. (24) is exemplarily shown as a function of the sphericity of the particle w for the nanofluid Al2O3/H2O containing particles with dp = dp,eq = 30 nm at a temperature of T = 300 K and volume fractions u of 0.05, 0.10, 0.15, and 0.20. Spherical particles are compared with cubic particles as well as seven cylindrical particles with varying aspect ratios, i.e., the ratio of length to diameter, which are specified in Fig. 7. While prolate cylindrical particles have aspect ratios larger than 1, they are smaller for oblate particles. A broad range of w values is covered ranging from about 0.22 for prolate cylinders with an aspect ratio of 200:1 to 1 for spheres. To check our modeled data, we use again the continuum model of Hamilton and Crosser [12] in its general form given by Eq. (1). Also for systems with non-spherical nanoparticles, our modeled data and those predicted by Hamilton and Crosser [12] agree well for the various particle geometries. Both models show that

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For static nanofluids considered in this study, this could be rather realized by increasing the volume-specific surface area than by increasing the Nu number. Thus, it would be preferable to use nanofluids containing particles with low sphericity such as elongated thin and/or porous nanoparticles, which would be represented ideally by, e.g., carbon nanotubes. Here, a much larger surface area per volume of the particles is provided for convective heat transfer compared to spherical nanoparticles. In case of moving or pumped nanofluids, also an increased Nu number may improve the heat transfer. Yet, this increase is limited by the increasing viscosity of nanofluids with increasing particle volume fraction, in particular using non-spherical particles [24]. 4. Conclusions

Fig. 7. Percentage enhancement factor 100  (keff/kbf  1) as a function of the sphericity of the particle w at a temperature of T = 300 K for the nanofluid Al2O3/ H2O containing volume-equivalent (dp = dp,eq = 30 nm) nanoparticles of different shape calculated from the present model (Eq. (24)) and the Hamilton-Crosser model [12] (Eq. (1)) for various volume fractions u. The ratio ‘‘length:diameter” is given for prolate (ratio > 1) and oblate (ratio < 1) particles.

decreasing sphericity of the nanoparticles goes along with a distinct increase in the effective thermal conductivity. For example, our model predicts that the enhancement for cylinders with an aspect ratio of 10:1 (w = 0.58) is already twice as large as that for spherical particles (w = 1). The reduced convective heat transfer resistance caused by the larger specific surface area of nonspherical particles compared to spherical particles seems to reasonably account for the increased enhancement factors in our developed model. For all volume fractions, the slopes for the increasing keff values with decreasing w values are larger for our model compared to that calculated from the Hamilton-Crosser model [12]. This deviation is especially pronounced for small w values below 0.5. The reason for this discrepancy might be related to the limited applicability range with respect to w in the Hamilton–Crosser model [12]. It was developed based on experimental results for mixtures of rubber and dispersed alumina particles with sphericities only between 0.58 and 1, resulting in the empirical correlation between w and the shape factor n given in Eq. (2). Yet, it is also possible that our approach of adopting volume-equivalent diameters in our model for non-spherical particles is not fully transferable to particle shapes which differ more and more from the spherical shape. Changing the particle–fluid combination leads to the same general dependencies as discussed above for the system Al2O3/H2O. The main difference is that for increasing kp/kbf for the nanofluid systems, better agreement between the present model and the Hamilton–Crosser model as a function of the sphericity of the particles is found, also for low w values. Furthermore, the trends regarding the distinct influence of the volume fraction and the negligible influences of temperature as well as particle diameter on the enhancement of the effective thermal conductivity found for systems containing spherical particles can also be observed for the corresponding systems containing non-spherical particles. Based on the above results, we can conclude that the heat transfer resistance due to convection between the particles and the base fluid seems to dominate the effective thermal conductivity. Hence, a strong increase in the effective thermal conductivity of nanofluids seems to be achievable only if the convective heat transfer resistance between particles and base fluid is strongly reduced.

A new analytical model to determine the effective thermal conductivity in fluids containing well-dispersed spherical and nonspherical nanoparticles was presented. The model takes thermal resistances in connection with the base fluid, the nanoparticles, and the micro-convection between the nanoparticles and the fluid due to Brownian motion of the particles into account. Furthermore, the model is based on well-defined properties and does not include any empirical constants. It has revealed the significant role of the considered convective heat transfer resistance in reducing the effective thermal conductivity of a nanofluid compared with a theoretical nanofluid showing no thermal contact resistance between particle and base fluid. The convective heat transfer resistance turned out to control the achievable thermal conductivity enhancement. Our model points out that the enhancement of the effective thermal conductivity of the nanofluid related to the thermal conductivity of the base fluid is strongly dependent on the particle volume fraction, but very weakly dependent on the diameter of the nanoparticle and the temperature. For four exemplary systems containing spherical Al2O3 or TiO2 nanoparticles suspended in water or ethylene glycol, very good agreement was found between our calculation results and the commonly recommended model of Hamilton and Crosser [12]. In accordance with this effective continuum model, our model does not show any anomalous enhancement of the effective thermal conductivity of nanofluids for low volume fractions of spherical nanoparticles as it is often reported in the literature. The presented model also suggests that a stronger enhancement in the effective thermal conductivity of nanofluids is found using cubic and especially prolate or oblate cylindrical particles due to their larger volume-specific surface areas. A further reduction of the convective heat transfer restrictions can be expected by the formation of rows of nanotubes having a large surface-to-volume ratio. Acknowledgements This work was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) by funding the Erlangen Graduate School in Advanced Optical Technologies (SAOT) within the German Excellence Initiative. K.N. Shukla acknowledges support from the Alexander von Humboldt Foundation in sponsoring a renewed research stay at the Friedrich-Alexander-University Erlangen-Nuremberg (FAU). References [1] J.C. Maxwell, A Treatise on Electricity and Magnetism, third ed., vol. I, Oxford, Clarendon, 1892. [2] 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 (7) (2015) 1367–1395. [3] P. Keblinski, J.A. Eastman, D.G. Cahill, Nanofluids for thermal transport, Mater. Today 8 (6) (2005) 36–44.

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